4. Solutions with finite mass are found by considering corrections to the YM Lagrangean consisting in higher therm of the Yang-Mills hierarchy. Such systems can occur in the low energy effective action of string theory. A discussion of the main properties of the solutions and the differences with respect to the four dimensional case is presented. The mass of these configurations is computed by using a counterterm method.\n\nIntroduction\n\nIf we allow for a negative cosmological constant, the solution of the matter free Einstein equations possessing the maximal number of symmetries is the anti-de Sitter (AdS) spacetime. Recently a considerable amount of interest has been focused on solutions of Einstein equations with this type of asymptotics. This interest is mainly motivated by the proposed correspondence between physical effects associated with gravitating fields propagating in AdS spacetime and those of a conformal field theory on the boundary of AdS spacetime [1,2].\n\nIn view of these developments, an examination of the classical solutions of gravitating fields in asymptotically AdS (AAdS) spacetimes seems appropriate. Recently, some authors have discussed the properties of gravitating SU(2) nonabelian fields with a negative cosmological constant Λ [3,4]. Considering the case of four spacetime dimensions, they obtained some surprising results, which are strikingly different from the results found in the asymptotically flat case. For example there are solutions for continuous intervals of the parameter space, rather then discrete points. The asymptotic values of the gauge potentials are arbitrary and there exist solutions supporting magnetic and electric fluxes without a Higgs field. Some of these solutions are stable against spherically symmetric linear perturbations. The literature on AAdS solutions with nonabelian fields is growing steadily, including stability analyses [5,6], the study of configurations with a NUT charge [7], topological black holes with nonabelian hair [8] as well as axially symmetric generalisations [9,10]. The existence of these solutions invalidates the AdS 4 version of the no hair conjecture, which states that the black holes are completely characterised by their mass, charge and angular momentum.\n\nHowever, all these studies approach the case of a four dimensional AAdS spacetime, and relatively little is known about higher dimensional AAdS solutions with non Abelian matter fields. Practically all that is known for d > 4 is the five dimensional nonabelian SU (2) solutions discussed in [11]. At the same time gauged supergravity theories playing an important role in AdS/CFT, generically contain non Abelian matter fields in the bulk, although in the literature only Abelian truncations are considered, to date. Thus, the examination of higher dimensional gravitating non Abelian solutions with Λ < 0 is a pertinent task.\n\nHigher dimensional asymptotically flat solutions of the Einstein-Yang-Mills (EYM) equations have recently been the subject of several studies. As found in [12], in five spacetime dimensions, the particle spectrum obtained by uplifting the d = 4 flat space YM instantons become completely destroyed by gravity, as a result of their scaling behaviour. However, by adding higher order 1 terms in the Yang-Mills (YM) hierarchy this obstacle due to scaling is removed and solutions in higher dimensions can be found. Such regular, static and spherically symmetric solutions in spacetime dimensions d ≤ 8 were presented in [13], and for d = 5, both globally regular and black hole solutions of the EYM system were found in [14]. The properties of these solutions are rather different from the familiar Bartnik-McKinnon (BK) solutions [16] to EYM in d = 4, and are somewhat more akin to the gravitating monopole solutions to EYM-Higgs (EYMH) [17]. This is because like in the latter case [17], where the VEV of the Higgs field features as an additional dimensional constant, here [13] also additional dimensional constants enter with each higher order YM curvature term 2 . They are however quite distinctive in their critical behaviour. The typical critical features discovered in [13,14] have recently been analysed and explained in [19].\n\nThese results can be systematically extended to all d ≥ 5 and one finds that no finite mass solutions can exist in EYM theory, unless one modifies the non Abelian action by adding higher order curvature terms 3 in the YM 2p-form curvature F (2p), the p = 1 case being the usual 2-form YM curvature. Without these higher order YM terms, only vortex-type finite energy solutions [12] exist, describing effective systems in 3 spacelike dimensions, and with a number of codimensions.\n\nIt is the purpose of this paper to examine the corresponding situation in higher dimensions, in the presence of a negative cosmological constant. It is interesting to inquire whether the introduction of a negative cosmological constant to these higher dimensional EYM models will lead to some new effects as it does in the d = 4 case, due to the different asymptotic structure of the spacetime. In the first place this would lead to our understanding of how the behaviour of EYM theory depends on the dimensionality of the spacetime. But such higher dimensional AAdS solutions might be relevant to superstring theory, namely in the context of solutions to various supergravities containing nonabelian matter fields. Here however we restrict our considerations to the simplest case of systems consisting only of gravitational and YM fields, namely to higher dimensional EYM model. In particular, we restrict to Einstein-Hilbert gravity and the first two members of the YM hierarchy, and hence to d ≤ 8, augmented with negative cosmological constant in d spacetime dimensions. As in [3,4], for the d = 4 case, we seek static spherically symmetric solutions in the d − 1 spacelike dimensions. We find both globally regular and black hole solutions with finite ADM mass. Unlike in the d = 4 case however, we find that for d > 4 and a negative cosmological constant, the properties of the AAdS solutions do not differ qualitatively from the asymptotically flat case.\n\nOur strategy is to first consider the usual YM model, namely the p = 1 member of the YM hierarchy, or the square of the 2-form curvature F (2). We present an argument for the absence of solutions with reasonable asymptotics for any spacetime dimension d ≥ 5. Although the EYM equations in this case present solutions approaching asymptotically the AdS background, the mass generically diverges. This can be seen as a simple version of the no hair theorem, holding for the EYM system in d > 4 dimensions. In other words, Schwarzschild-AdS black hole is the only static, spherically symmetric solution of the EYM system with finite mass. This is presented in Section 2, where in addition we have considered the special case d = 3, extending Deser's analysis [20] for Λ < 0.\n\nIn Section 3 we introduce the higher order YM hierarchy models featuring the terms F (2p), p ≥ 2. We derive the classical equations subject to our spherically symmetric ansatz, and present a detailed numerical study of of both regular and black hole solutions. As is the case with the usual EYM system, where the existence of finite mass regular solutions leads to the violation of the no hair conjecture, here too, these solutions are explicit counterexamples to this conjecture in AdS d spacetime.\n\nOne may ask about the possible relevance of these higher dimensional configurations within the AdS/CFT correspondence. In Section 4 we compute the boundary stress tensor and the mass and action of the solutions in a number of spacetime dimensions up to eight. In five and seven dimensions, the counterterm prescription of [21] gives an additional vacuum (Casimir) energy, which agrees with that found in the context of AdS/CFT correspondence. A counterterm based proposal to remove the divergences of a F (2) theory, such that the mass and action be finite, is also presented in Appendix B. We give our conclusions and remarks in the final section.\n\nEverewhere in this paper we employ the notations and conventions of [13]. 2 The F (2) model\n\n\nThe action principle\n\nWe start with the following action principle in d−spacetime dimensions\nI = M d d x √ −g 1 16πG (R − 2Λ) + L m − 1 8πG ∂M d 3 x √ −hK,(1)\nwhere R is the Ricci scalar associated with the spacetime metric g µν , Λ = −(d − 1)(d − 2)/(2ℓ 2 ) is the cosmological constant and G is the gravitational constant (following [14], we define also κ = 1/(8πG)). The matter term in the above relation\nL m = − 1 4 τ 1 tr F µν F µν(2)\nis the usual F 2 nonabelian action density,\nF µν = ∂ µ A ν − ∂ ν A µ − i[A µ , A ν ]\nbeing the gauge field strength tensor. Here τ 1 is the coupling constant of the model (in the usual notation τ 1 = 1/g 2 ).\n\nThe last term in (1) is the Hawking-Gibbons surface term [18], where K is the trace of the extrinsic curvature for the boundary ∂M and h is the induced metric of the boundary. Of course this term does not affect the equations of motion but it is relevant for the discussion of the mass and the action of the solutions, in Section 4.\n\nThe field equations are obtained by varying the action (1) with respect to the field variables g µν , A µ\nR µν − 1 2 g µν R + Λg µν = 8πGT µν ,(3)∇ µ F µν − i[A µ , F µν ] = 0,\nwhere the energy momentum tensor is defined by\nT µν = tr F µα F νβ g αβ − 1 4 g µν tr F αβ F αβ .(4)\n\nThe general Ansatz\n\nFor the case of a d-dimensional spacetime, we restrict to static fields that are spherically symmetric in the d − 1 spacelike dimensions, with a metric ansatz in terms of Schwarzschild coordinates\nds 2 = dr 2 N (r) + r 2 dΩ 2 d−2 − σ 2 (r)N (r)dt 2 ,(5)\nwith dΩ d−2 the d − 2 dimensional angular volume element and\nN = 1 − 2m(r) κr d−3 + r 2 ℓ 2 ,(6)\nthe function m(r) being related to the local mass-energy density up to some d−dependent factor. As discussed in [13], the choice of gauge group compatible with the symmetries of the line element (5) is somewhat flexible. In [13] the gauge group chosen was SO(d), in d dimensions. But the gauge field of the static solutions in question took their values in SO(d − 1). Thus in effect, it is possible to choose SO(d) in the first place. Now for even d, it is convenient to choose SO(d) since we can then avail of the chiral representations of the latter, although this is by no means obligatory. Adopting this criterion, namely to employ chiral representations, also for odd d, it is convenient to choose the gauge group to be SO(d − 1).\n\nWe shall therefore denote our representation matrices by SO ± (d), whered = d andd = d − 1 for even and odd d respectively.\n\nIn this unified notation (for odd and even d), the spherically symmetric Ansatz for the SO ± (d)-valued gauge fields then reads [13] \nA 0 = 0 , A i = 1 − w(r) r Σ (±) ijx j , Σ (±) ij = − 1 4 1 ± Γd +1 2 [Γ i , Γ j ] .(7)\nThe Γ's denote thed-dimensional gamma matrices and 1, j = 1, 2, ..., d − 1 for both cases. Inserting this ansatz into the action (1), the EYM field equations reduce to\n0 = r d−4 σN w ′ ′ − (d − 3)r d−6 σ(w 2 − 1)w,(8)m ′ = τ 1 2 r d−4 N w ′2 + (d − 3) (w 2 − 1) 2 2r 2 ,(9)σ ′ σ = τ 1 κ w ′2 r ,(10)\nwhich can also be derived from the reduced action\nS = dr σ m ′ − τ 1 2 r d−4 N w ′2 + (d − 3) w 2 − 1) 2 2r 2 .(11)\nFor a F 2 theory, the constants κ, τ 1 can always be absorbed by rescaling r → cr, Λ → Λ/c 2 and m → mκc d−3 , with c = τ 1 /κ. The above differential equations have two analytic solutions. One of them with w(r) = ±1, m(r) = M, σ(r) = 1 (12) corresponds to Schwarzschild-AdS spacetime. For w(r) = 0 we find a non Abelian generalisation of the magnetic-Reissner-Nordstrøm solution with σ(r) = 1 and\nm(r) = M 0 + τ 1 2 log r if d = 5, and m(r) = M 0 + τ 1 (d − 3) 4(d − 5) r d−5 for d = 5,(13)\nM 0 being an arbitrary constant. We can see that, although these solutions are asymptotically AdS, the mass defined in the usual way diverges.\n\n\nd = 3\n\nThe (2+1) dimensional case is rather special. Three dimensional gravity has provided many important clues about higher dimensional physics. This theory with Λ < 0 has non-trivial solutions, such as the BTZ blackhole spacetime [22], which provide an important testing ground for quantum gravity and the AdS/CFT correspondence. Many other types of 3d regular and black hole solutions with a negative cosmological constant have also been found by coupling matter fields to gravity in different ways. However, as proven in [20], there are no d = 3 asymptotically flat static solutions of the EYM equations. The arguments in [20] can easily be generalised for the AAdS case. We notice that for d = 3, the YM equation (8) implies the existence of a first integral w ′ = αr/(σN ) with α an arbitrary real constant. Therefore, assuming AdS 3 asymptotics, w ′ decays asymptotically as 1/r which from (9) implies a divergent value of m(r) as r → ∞. However, similar to the Λ = 0 case [23], this argument does not exclude the existence of nontrivial solutions of the field equations.\n\nHere we should remark that since for d = 3 we are dealing with SO(d − 1) = SO(2), the gauge group is Abelian and we recover Einstein-Maxwell theory with a negative cosmological constant, whose solutions are known in the literature. The corresponding solution with a vanishing electric field was found by Hirschmann and Welch [24] and has a line element 4\nds 2 = r 2 r 2 + c 2 log |r 2 /ℓ 2 − M | dr 2 (r 2 /ℓ 2 − M ) + (r 2 + c 2 log |r 2 /ℓ 2 − M |)dϕ 2 − (r 2 /ℓ 2 − M )dt 2 ,(14)\nthe magnetic potential being\nw(r) = w 0 + c √ 2 log |r 2 /ℓ 2 − M |,(15)\nwith w 0 , c and M arbitrary real constants, the BTZ metric being recovered for c = 0 (see also [25,26] for more details on this solution).\n\nOne can see that although the quasilocal mass defined in the usual way diverges as r → ∞, the metric still approaches the AdS 3 background. However, a similar problem appears for other d = 3 AdS solution e.g. for the electrically charged BTZ black hole [22], or for a self-interacting scalar field minimally coupled to gravity [27], in which cases it was possible to find a suitable mass definition. We expect the formalism developed in those cases to work also for the Hirschmann-Welch solution (14)- (15), but this lies outside the scope of the present work.\n\n\nd = 4\n\nFour dimensional black hole solutions of the equations (8)- (10) have been found in [3], the globally regular counterparts being discussed in [4]. Differing from the asymptotically flat case, for Λ < 0 there is a continuum of solutions in terms of the adjustable shooting parameter that specifies the initial conditions at the origin or at the event horizon. As a new feature, the asymptotic value of the gauge function w 0 is arbitrary. The spectrum has a finite number of continuous branches, depending on the value of Λ. When the parameter Λ approaches zero, an already-existing branch of solutions collapses to a single point in the moduli space. At the same time new branches of solutions emerge. A fractal structure in the moduli space has been noticed [28]. There are also nontrivial solutions stable against spherically symmetric linear perturbations, corresponding to stable configurations. The solutions are classified by non-Abelian magnetic charge and the ADM mass.\n\nNote also that the d = 4 EYM solutions with a negative cosmological constant Λ = −3/τ 1 have some relevance in AdS/CFT context. As proven in [29], for this value of the cosmological constant, an arbitrary solution (g µν , A (a) µ ) of the four dimensional EYM equations gives a solution of the equations of motion of the d = 11 supergravity. Based on this observation, an exact BPS-type EYM solution has been constructed in [10]. However, similar to some supersymmetric solutions in Einstein-Maxwell theory with Λ < 0, this Λ = −3/τ 1 configuration presents a naked central singularity.\n\n\nd ≥ 5\n\nAs discovered by Coleman [30] and Deser [31], there are no flat space static solutions of the YM equations, except for d = 5. However, the inclusion of gravity may change this picture, as seen from the famous d = 4 asymptotically flat Bartnik-McKinnon solutions [16]. In this case, the repulsive YM force is compensated by the attractive character of the gravity, and as a result we find both regular and black hole unstable configurations (see [32] for a fairly recent survey). As found in [12] the d = 5 particle-like solutions are destroyed by gravity, their mass diverging logarithmically, while w(r) presents an infinite number of nodes. The AAdS couterparts of the d = 5 asymptotically flat solutions are discussed in [11]. Although approaching asymptotically the AdS 5 background, the mass of these configurations also diverges logarithmically.\n\nAs conjectured by several authors, this result extends to higher dimensions. Following the approach in [11], we prove in Appendix A the nonexistence of asymptotically flat or AdS solutions with a finite mass in a F (2) EYM model given by (2) for any spacetime dimension d ≥ 5 (see also the discussion in Section 3.3). Therefore, the d−dimensional Schwarzschild-AdS configuration is the only finite mass solution of the equations (8)-(10) and a simple version of the no hair theorem seems to hold for the F (2) (usual) EYM system in d > 4.\n\nWe should remark that in deriving this result we assumed implicitely that m(r), σ(r), w(r) are smooth functions approaching finite values as r → ∞. A divergent asymptotic value of m(r) invalidates the proof presented in Appendix A and also the virial argumnets in Section 3.3. Therefore we cannot exclude the existence of spherically symmetric, nontrivial solutions of the field equations for any d > 4. However, the mass of these solutions generically diverges, although the spacetimes are still AAdS. The work of Ref. [11] presents an extensive discussion of such AAdS solutions for d = 5. Both regular and black hole solutions exist in d = 5 for compact intervals of the parameter that specifies the initial conditions at the origin or at the event horizon. Differing from the Λ = 0 case, the gauge field function w(r) does not oscillate between 1 and −1 but approaches asymptotically some finite value w 0 , the node number being finite. The masses of these solutions behave asymptotically as (w 2 0 − 1) log r, with all w 0 = ±1 solutions corresponding to pure gauge configurations.\n\nThe results we found by solving numerically the equations (8)-(10), for d = 6, 7, 8 and several negative values of Λ confirm that this is a generic behaviour for d > 4. The corresponding boundary conditions at the origin (or event horizon) are found by taking P = 1 in relations (26), (27) given in Section 3. Except for a divergent value of m(r) as r → ∞, according to\nm(r) = M 0 + τ 1 (d − 3) 4(d − 5) (w 2 0 − 1) 2 r d−5 ,(16)\nthe properties of these solutions are very similar to the the more familiar d = 4 case. For d > 5, the asymptotic value w 0 of the gauge field function w is also arbitrary, being fixed by the initial parameters w ′′ (0) or w(r h ) respectively, w 0 = ±1 corresponding to pure gauge configurations. Solutions for a compact interval of these parameters were found to exist, the general structure being Λ-dependent. Solutions with nodes in w(r) were also found. Typical d = 6 configurations with a regular origin are presented in Figure 1, for Λ = −0.01 and three different values of b = −w ′′ (0)/2. One can see that the mass function diverges linearly while σ(r) and w(r) asymptotically approach some finite values. In Figure 2 we plot the parameters M 0 (appearing in (16), which in Section 4.3 we argue that it can be taken as the renormalised mass of the solutions), w 0 , the value σ 0 of the metric function σ at the origin and the minimal value N m of the metric function N as a function of b for a family of d = 6 AAdS solutions with Λ = −1. This branch ends for some finite value of b, where σ(0) → 0. Black hole solutions have been found as well, presenting the same general features. Here also we find a continuum of solutions with arbitrary values of w 0 , the relevant parameter being the value of the gauge potential at the event horizon. Similar to d = 4, solutions appear to exist for any value of the event horizon radius. The drawback of the solutions in d > 5 described above is that their ADM masses are divergent, making their physical significance obscure (see, however, the discussion in Appendix B).\n\nOne may hope to find a different picture by including some other matter field in the action (1). Such fields should interact with the YM sector so as to compensate for the scaling behaviour of the non Abelian fields. This excludes the dilaton field, as it can be proven that the latter does not change this nonexistence result. Note that d = 5 finite mass spherically symmetric gravitating non Abelian solutions with a Liouville-type dilaton potential are known to exist [35]; however these solutions are asymptotically neither flat nor AdS.\n\nIn the next Section, we will remedy this problem of nonexistence by adding higher order YM curvature terms F (2p), (p > 1), to (2). The main role of these terms is to alter the scaling properties of the action density in (1). But logically, such a role can be played also by altering the gravitational part of (1), through the introduction of higher order (in the Riemann curvature) terms R(p), (p > 1). R(p) here denotes the generalised Ricci scalar constructed from the 2p-form antisymmetrised p-fold product of Riemann tensor 2-forms, the p = 1 member of which gives the Einstein-Hilbert action, the p = 2, the Gauss-Bonnet, etc. Here we have eschewed the possibility of employing additional R(p) terms instead of, or together with F (2p) YM terms, because in the present work we exclude the participation of fields other than gravitational and Yang-Mills. In particular, the exclusion of the dilaton renders the usefulness of higher order (Gauss-Bonnet) gravities trivial from a practical point of view.\n\nTo see this, consider the situation where a term scaling as L −2p is needed, i.e. that the term\nκ p √ −g R(p)\nmust be added to the density in (1). Using the metric Ansatz (5), and discarding purely boundary terms in the residual one dimensional (spherically symmetric) Lagrangean, the term L (p) to be added is\nL (p) = κ p (d − 2p)! (d − 1)! σ r d−2p−2 r d dr (1 − N ) p + (d − 2p − 1)(1 − N ) p(17)\nin spacetime dimension d, which vanishes identically in dimensions up to d = 2p. This is because in the given dimensions the p-th member of the gravitational hierarchy becomes a total divergence. But for given d it is necessary to have 2p ≥ d, whence (17) trivialises. The situation would be quite different if the dilaton field were included together with the higher order gravitational terms, preventing the p-gravity density becoming a total divergence. But this option is excluded here and we opt to the exclusive use of higher order p-YM densities.\n\nYet other fields occurring in supergravities might be considered, e.g. Kalb-Ramond, or totally antisymmetric tensor fields. But these being Abelian, their effect would be felt only in given dimensions, or, subject to much less stringent symmetries than the spherical. This also is not a flexible option, so we restrict our attention to p-YM densities only.\n\n\nNonabelian hierarchies 3.1 The Lagrangean and field equations\n\nSince finite mass spherically symmetric solutions play a central role in AdS/CFT, it is desirable that the nonexistence result presented in Section 2.5 be circumvented.\n\nA simple way to circumvent these arguments and to find nontrivial solutions is to to modify the matter Lagrangean by adding higher order terms in the YM hierarchy. As noted in footnote 1 , these are constructed exclusively from YM curvature 2p-forms. For Λ = 0, asymptotically flat, finite energy solutions of this modified EYM system are constructed in [13,14].\n\nSuch terms as we propose to add are predicted by string theory, and hence provide a link with the AdS/CFT correspondence too. But here we are guided predominantly by symmetry considerations and do not claim to be employing terms strictly following from superstring theory. The situation concerning higher order YM curvature terms in the string theory effective action is complex and as yet not fully resolved. While YM terms up to F 4 arise from (the non Abelian version of) the Born-Infeld action [36], it appears that this approach does not yield all the F 6 terms [37]. Terms of order F 6 and higher can also be obtained by employing the constraints of (maximal) supersymmetry [38]. The results of the various approaches are not identical.\n\nThe definition we use for superposed YM hierarchy is\nL m = − P p=1 1 2(2p)! τ p √ −g Tr F (2p) 2 ,(18)\nwhere F (2p) is the 2p-form p-fold totally antisymmetrised product of the SO(d) YM curvature 2-form F (2)\nF (2p) ≡ F µ1µ2...µ2p = F [µ1µ2 F µ3µ4 ...F µ2p−1µ2p] .(19)\nEven though the 2p-form (19) is dual to a total divergence, namely the divergence of the corresponding Chern-Simons form, the density (18) is never a total divergence since it is the square of one. But the 2p-form (19) vanishes by (anti)symmetry for d < 2p so that the upper limit in the summation in (18) is P = d 2 for even d and P = d−1 2 for odd d. We define the p-stress tensor pertaining to each term in (18) as\nT (p) µν = Tr F (2p) µλ1λ2...λ2p−1 F (2p) ν λ1λ2...λ2p−1 − 1 4p g µν Tr F (2p) λ1λ2...λ2p F (2p) λ1λ2...λ2p .(20)\nFor the particular spherically symmetric ansatz considered in Section 2, we express the reduced YM Lagrangean arising from (18) as\nL m = − P p=1 L (p) Y M (21) with L (p) Y M given by L (p) Y M = r d−2 σ τ p 2 · (2p)! (d − 2)! (d − [2p + 1])! W p−1 (2p)N 1 r dw dr 2 + (d − [2p + 1]) W(22)\nhaving used the shorthand notation\nW = w 2 − 1 r 2 2 .(23)\nFor this general ansatz, we find the field equations\nm ′ = P p=1 τ p r d−2 2(2p)! (d − 3)! (d − [2p + 1])! W p−1 (2p)N 1 r dw dr 2 + (d − [2p + 1]) W , σ ′ σ = (d − 3)! κr w ′2 P2 p=1 τ p W p−1 (d − [2p + 1])!(2p − 1)! ,(24)P p=1 d dr r d−4 στ p (d − [2p + 1])!(2p − 1)! W p−1 N w ′ = P p=1 r d−6 στ p 2w(w 2 − 1)W p−1 (d − [2p + 1])!(2p − 1)! N (p − 1)W w ′2 r 2 + d − 2p − 1 2 ,\nwhich can also be derived from the reduced action\nS = dr σ m ′ − P p=1 τ p r d−2 2(2p)! (d − 3)! (d − [2p + 1])! W p−1 (2p)N 1 r dw dr 2 + (d − [2p + 1]) W .(25)\nFor τ i = 0, i > 2 the equations (8)-(10) are recovered. Note also the σ equation decouples and can be treated separately.\n\n\nBoundary conditions\n\nThe asymptotic solutions to these equations can be systematically constructed in both regions, near the origin (or event horizon) and for r ≫ 1. The corresponding expansion as r → 0 is\nw(r) = 1 − br 2 + O(r 4 ), m(r) = P p=1 τ p (d − 3)!(4b 2 ) p 2(2p)!(d − [2p + 1])! r d−1 + O(r d+1 ),(26)σ(r) = σ 0 + (d − 3)! 2κ P p=1 τ p (4b 2 ) p (2p − 1)!(d − [2p + 1])! r 2 + O(r 4 ),\nand contains one essential parameter b (the value of σ(0) ≡ σ 0 can be fixed by rescaling the time coordinate). For black hole configurations with a regular, nonextremal event horizon at r = r h , the expression near the event horizon is\nm(r) = m h + m ′ (r h )(r − r h ) + O(r − r h ) 2 , w(r) = w h + w ′ (r h )(r − r h ) + O(r − r h ) 2 , (27) σ(r) = σ h + σ ′ h (r − r h ) + O(r − r h ) 2 , where m(r h ) = 1 2 κr d−3 h (1 + r 2 h ℓ 2 ), W h = (w 2 h − 1) 2 r 4 h , m ′ (r h ) = P p=1 τ p r d−2 h (d − 3)! 2(2p)!(d − 2p − 2)! W p h , N ′ h = d − 3 r h + (d − 1)r h ℓ 2 − 2m ′ (r h ) κr d−3 h ,(28)σ ′ h = σ h (d − 3)! κr h w ′2 h P p=1 τ p W p−1 h (2p − 1)!(d − [2p + 1])! , w ′ h = 1 N ′ h w h (w 2 h − 1) r 2 h P p=1 τpW p−1 h (2p−1)!(d−2p−2)! P p=1 τpW p−1 h (2p−1)!(d−2p−1)! ,\nthe value of the gauge field on the event horizon being the esential parameter. Here the obvious condition N ′ (r h ) > 0 imposes some limits on the event horizon radius as a function of w h for given (τ i , Λ).\n\nSince the field equations are invariant under w → −w, one can take w(0) = 1 and w(r h ) > 0 without any loss of generality.\n\nFor r ≫ 1 we find for both regular and black hole solutions\nw(r) = ±1 + w 1 r d−3 + . . . , m(r) = M − τ 1 (d − 3)w 2 1 8ℓ 2 1 r d−3 + . . . ,(29)σ(r) = 1 − w 2 1 (d − 3) 2 τ 1 2κ(d − 2) 1 r d−4 + . . . .\nThese boundary conditions are also shared by the asymptotically flat solutions (with a different decay of the mass function m(r), however), w = ±1 being again the only allowed values of the gauge function as r → ∞. Therefore, we expect to find a qualitatively similar picture in both cases. We will find in Section 5 that the constant M in the above relations is the ADM mass up to a d-dependent factor. However, in the discussion of numerical solutions we will refer to M to as the mass of the solutions.\n\n\nFurther relations\n\nThe form (25) of the reduced action allow to derive an useful virial relation. To this end, we use the scaling technique proposed in [33,34] for the case of spherically symmetric gravitating systems. Let us assume the existence of a globally regular solution m(r), σ(r), w(r) of the field equations (24), with suitable boundary conditions at the origin and at infinity. Then each member of the 1-parameter family\nm λ (r) ≡ m(λr), σ λ (r) ≡ σ(λr), w λ (r) ≡ w(λr)(30)\nassumes the same boundary values at r = 0 and r = ∞, and the action S λ ≡ S[m λ , σ λ , w λ ] must have a critical point at λ = 1, i.e. [dS/dλ] λ=1 = 0. Therefore we find the following virial relation satisfied by the finite energy solutions of the field equations (note that following [33], it is possible to write a similar relation for black hole configurations, also)\nP p=1 ∞ 0 dr σ τ p (d − 3)! 2(2p)!d − [2p + 1])! W p−1 (d − 4p − 1)(2pN w ′2 r 2 + (d − 2p − 1)W )(31)\n+2p\nw ′2 r 2 ( 2(d − 3)m κr d−3 + 2r 2 ℓ 2 ) = 0.\nFor p = 1, i.e. a F 2 -theory, the above relation reads\n∞ 0 dr σ r d−4 (d − 5)(N w ′2 + (w 2 − 1) 2 2r 2 ) + w ′2 ( 2m κr d−3 (d − 3) + 2r 2 ℓ 2 ) = 0,(32)\nwhich clearly shows that no nontrivial gravitating solution with finite mass exists for d > 4, since all terms in the integrant are strictly positive quantities. Therefore it becomes obvious that new terms in the YM-hierarchy should be introduced as the spacetime dimension increases. For a given d, the relation P > [(d + 1)/4] should be satisfied. As it happens, to go to 5 ≤ d < 9 it is necessary to include at least the second member of the YM hierarchy to provide the requisite scaling (similarly for 9 ≤ d < 13 it is necessary to include the thrid member of the YM hierarchy). In practice we add only the lowest order such term necessary.\n\nWe mention here also the Hawking temperature expression of the black hole solutions. For the line element (5), if we treat t as complex, then its imaginary part is a coordinate for a nonsingular Euclidean submanifold iff it is periodic with period\nβ = 4π N ′ (r h )σ(r h ) .(33)\nThen continuous Euclidean Green functions must have this period, so by standard arguments the Hawking temperature is (with k B =h = 1)\nT H = σ h 4πr h d − 3 − 2m ′ (r h ) κr d−3 h + (d − 1) r 2 h ℓ 2 ≤ σ h 4πr h d − 3 + (d − 1) r 2 h ℓ 2 .(34)\nThus the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum black hole of equal horizon area.\n\nIn the presence of higher oder terms in the YM action, dimensionless quantities are obtained by rescaling\nr → (τ 2 /τ 1 ) 1/4 , Λ → (τ 1 /τ 2 ) 1/2 , m(r) → m(r)κ(τ 1 /τ 2 ) (d−3)/4 ,(35)\nThis reveals the existence of one fundamental parameter which gives the strength of the gravitational interaction\nα 2 = τ 3/2 1 κτ 1/2 2 ,(36)\nand P − 2 independent coupling constants\nβ 2 k−2 = τ k τ 1 τ 1 τ 2 k−1 ,(37)\nwith k = 3, P − 2 (i.e. no such constants appear in a p = 2 system).\n\nFor the F (2) + F (4) systems in d = 6, 7, 8 considered in [13], there exist gravity decoupling solutions at α = 0, from which the gravitating solutions branch out to a maximum value of α max and then decrease. The second limit of α → 0 also exists. For the same system in d = 5 studied in [14] on the other hand, there exists no gravity decoupling limit but nonetheless the solution branches out from α = 0 by employing a scaling procedure [19], and again α increases to a value α max and decreases. But in this case it does not reach the α → 0 limit. Rather it stops at a new critical point α c around which it oscillates [14], which is a new type of critical point identified in [19] and named a 'conical' fixed point.\n\n\nNumerical solutions\n\nIn the present work, we restrict our attention to the simplest nontrivial cases with only the two terms p = 1 and 2 in the YM hierarchy. However, we have obtained some numerical results also for a P = 3 hierarchy with β 1 = 1, which will be briefly mentioned. For d = 5, the solutions we found have some special features which will be discussed separately.\n\nBoth regular and black hole solutions of the EYM-hierarchy equations appear to exist for any value of Λ. Given (α, d, Λ), AAdS solutions may exist for discrete set of shooting parameters b and w h respectively. We follow the usual approach and, by using a standard ordinary differential equation solver, we evaluate the initial conditions at r = 10 −5 (or r h + 10 −5 ) for global tolerance 10 −14 , adjusting for shooting parameters and integrating towards r → ∞, and looking for AAdS solutions with a finite mass.\n\nSimilar to the p = 1, d = 4 asymptotically flat case, it can be proven that for all AAdS (or Λ = 0) solutions, w(r) is confined within the strip |w(r)| < 1. This can proven as follows: we suppose the existence of solutions with w(r) > 1 for some interval of r. Therefore w must develop a maximum for some r 0 , w ′ (r 0 ) = 0 and w(r 0 ) > 1 with w ′′ (r 0 ) < 0. However, the equations (24) imply that in the region w > 1 the only extremum can be a minimum. Therefore the condition |w(r)| < 1 is always fullfiled. As a general feature, all solutions discussed in the rest of this section present only one node in the gauge function w(r). Similar to the Λ = 0 case, we could not find multinode solutions 5 .\n\nThe absence of multinode solutions in this F (2) + F (4) model with Λ = 0, in the relevant dimensions 5 ≤ d ≤ 8 is analytically explained in [19]. We expect that the relevant fixed point analysis yields qualitatively similar results also for |Λ| > 0. This is borne out by our numerical results.\n\nFor any regular solution, the metric functions m(r) and σ(r) always increase monotonically with growing r from m(0) = 0 and σ(0) = σ 0 at the origin to m(∞) = M and σ(∞) = 1, respectively. The gauge function always interpolates between w(0) = 1 and w(∞) = −1 without any local extrema. For black hole configurations, the behaviour of the functions m(r), σ(r) and w(r) is similar to that for regular solutions. The gauge potential w(r) starts from some finite value 0 < w(r h ) < 1 at the horizon and monotonically approaches −1 at infinity. The metric functions m(r) and σ(r) increase also monotonically with r. In the asymptotic region, the geometry corresponds to a Schwarzschild-AdS solution. However, although these solutions are static and have vanishing YM charges (w 2 (∞) = 1), they are different from the Schwarzschild-AdS black hole, and therefore are not fully characterised by the mass-parameter M .\n\n\nRegular solutions d = 5\n\nAs α 2 → 0, the YM equations present a nontrivial, finite energy solution in a fixed AdS background. This nongravitating configuration approaches in the Λ = 0 limit the YM instanton in four dimensional flat space [39].\n\nWhen α 2 increases, this solution gets deformed by gravity and the mass M decreases. At the same time, both the value σ(0) and the minimal value N m of the function N (r) decrease, as indicated in Figure 3. This branch of solutions exists up to a maximal value α max of the parameter α, which is smaller than the corresponding value in the asymptotically flat case [14]. For example, we find numerically α 2 max ≈ 0.3445 for Λ = −0.2 while the corresponding value for Λ = −0.01 is α 2 max ≈ 0.5322. (without a cosmological term, this branch extends up to α 2 max ≈ 0.5648.) Similar to the EYM theory with Λ = 0 [14], we found always another branch of solutions on the interval α 2 ∈ [α 2 cr(1) , α 2 max ] with α 2 cr(1) depending again on the value of Λ (e.g. α 2 cr(1) ≈ 0.2876 for Λ = −0.2). On this second branch of solutions, both σ(0) and N m continue to decrease but stay finite. However, a third branch of solutions exists for α 2 ∈ [α 2 cr(1) , α 2 cr(2) ] , on which the two quantities decrease further. A fourth branch of solutions has also been found, with a corresponding α 2 cr(3) close to α 2 cr (2) . Further branches of solutions, exhibiting more oscillations very likely exist but their study is a difficult numerical problem. Along this succession of branches, the main observation is that the value σ(0) decreases much faster than that of N m as illustrated in Figure 3. Also, the mass parameters do not increase significantly along these secondary branches. However, the shooting parameter b increases to very large values. The pattern strongly suggests that after a finite (or more likely infinite) number of oscilations of σ(0), the solution terminates into a singular solution with σ(0) = 0 and a finite value of N (0). This is the behaviour observed in [14] for the EYM theory with Λ = 0. The inclusion of a negative cosmological constant does not seem to qualitatively change the properties of the system, but leads to different values of the critical parameters. As in the Λ = 0 case, the dominant term at the gravity decoupling limit α → 0 is the F (2) term, the energy being given by the action of the usual instanton [39], while as α → α cr the dominant term is F (4). The mechanism for this effect is explained in [19] (for Λ = 0) and is supported by our numerical results here. The typical d = 5 globally regular solutions look very similar to the d = 6, 8 profiles presented in Figures 4, 5. \n\n\nRegular solutions d > 5\n\nThe solutions in this case resemble again the Λ = 0 situation. In the presence of suitable higher order term in the hierarchy, the YM equations admit finite energy solutions in a fixed AdS d A p = 1, AdS 4 exact solution was found in [40], its higher order generalisation (specifically for p = 2, AdS 8 ) for d > 4 being discussed in [41]. According to the standard arguments, this AdS soliton can be generalised in the presence of gravity, provided that the dimensionless coupling constant α is small enough. Therefore the gravitating solutions exist up to a maximal value α max of the gravitational coupling constant. This value α max for a given d depends on the value of Λ. For example in d = 6, α max (Λ = 0) = 0.12675; α max (Λ = −1) = 0.070422; in 8 dimensions we find α max (Λ = 0) = 0.002193 while α max (Λ = −5) = 6.69 × 10 −4 .\n\nWhen α increases, the mass of the gravitating solutions decreases while the function N (r) develops a local minimum N m which becomes deeper while gravity becomes stronger and the value σ(0) decreases from one. At the same time, the value of the shooting parameter b increases with α. Our numerical analysis for d ≤ 10 indicates that a second branch of regular solutions always exists, starting at α max . Along this second branch the values σ(0) and N m decrease monotonically with α, while b and M still increase. The mass of a second branch solution is always larger than the corresponding mass (for the same value of α) on the first branch. For d > 6, the numerical analysis suggests that this second branch persists up to α 2 ≃ 0 and that in this limit σ(0) approaches a very small value. As far as our numerical analysis indicates, the value N m tends to a finite value in this limit so that there occurs no horizon. Therefore two regular solutions seems always to exist for any α < α d .\n\nThe case d = 6 is special, since the numerical procedure fails to give reliable results for second branch solutions, starting with some α d , whose value is Λ-dependent (for example we found α d (Λ = −1) = 0.0435 while α d (Λ = 0) = 0.0573). The quantity σ(0) reaches a very small value as α → α d . The minimal value of N (r) remains finite so that no horizon is approached. We expect that a different parametrisation of the metric and variables would allow us to continue this second branch to α → 0 in this case, too.\n\nThe behaviours just described qualitatively duplicate those of the Λ = 0 case [13], and are analysed in [19]. Likewise, the solutions are dominated by the F (2) terms in the gravity decoupling limit α → 0, while at the other end (on the second mass-branch) it is the F (4) terms that dominate.\n\nTypical The results we found by including the p = 3 term in the YM-hierarchy for d = 9, 10 follows the same pattern. Although the picture gets more complicated by the existence of one more coupling parameter, two branches of solution are always found to exist. We noticed also the existence of a maximal value of α which is (Λ, β 1 ) dependent.\n\n\nBlack hole solutions d = 5\n\nAccording to the standard arguments, one can expect to find black hole generalisations for any regular configurations, at least for small values of the horizon radius r h . For completeness we discuss here the basic features of the AAdS black hole solutions.\n\nAgain, the case d = 5 is special. The properties of these AAdS solutions are rather similar to the five dimensional asymptotically flat black hole configurations discussed in [14]. First, black hole solutions seem to exist for all values of α for which regular solutions could be constructed. Also, solutions exist only for a limited region of the (r h , α) space.\n\nThe typical behaviour of solutions as a function of r h is presented in Figure 9, for a small value of α compared to the maximal value α max of the regular solutions, in which case we notice the existence of only one r h = 0 regular configuration. Starting from this regular solution and increasing the event horizon radius, we find a first branch of solutions which extends to a maximal value r h(max) . As seen in Figures 9, 10, the value of r h(max) depends on Λ, α. The Hawking temperature decreases on this branch, while the mass parameter increases; however, the variation of mass and σ(r h ) is relatively small. Extending backwards in r h , we find a second branch of solutions for r h < r h(max) . This second branch stops at some critical value r h(cr) , where the numerical iteration fails to converge. The value of σ(r h ) on this branch decreases drastically, as shown in Figure 9. Also, the Hawking temperature after initially increasing, strongly decreases for values near r h(cr) , approaching a very small value, while the increase of the total mass is still very small. Similar to the Λ = 0 case [14], higher branches of solutions on which the value σ(r h ) continues to decrease further to zero are likely to exist. However, the extension of these branches in r h will be very small, which makes their study difficult. An approach to this problem with different parametrisation appears to be necessary.\n\nHowever we find, that the global picture is changed by considering large enough values of α. In this case, more than one regular configuration exists for a given value of α. This situation is illustrated in Figure 10, for solutions with α 2 = 0.5. Two regular solutions exist for this particular value of α and we find two black hole branches connecting these r h = 0 configurations. Again, the mass of the second branch solutions is always larger than the corresponding mass on the first branch.\n\nPreliminary numerical results indicate an even more complicated picture for solutions with α near α max . In this case the configurations combine features of both types of solutions discussed above. Several branches of black hole solutions are found for the same α. These branches start from regular configurations and are possibly disconnected.\n\n\nBlack hole solutions d > 5\n\nAlthough predicted in [14], no discussion of the Λ = 0, d > 5 black holes is presented in literature.\n\nAgain, black hole counterparts appear to exist for any regular solution. However, solutions with the right asymptotics are found for a limited region of the (r h , α) space only. We plot in Figures 11-13 some results we found for d = 6, 7 and d = 8. Starting for a given α 0 < α max from a r h = 0 first branch regular solution, we found the existence of a branch of black hole solutions extending up to a maximal value of the event horizon radius r h = r max h . When r h increases, the mass and the Hawking temperature increase while the value σ(r h ) decreases from its value at the regular solution. A second branch of black hole solutions seems to appear always at r max h , extending backwards to a zero event horizon radius. This limiting solution corresponds to the second branch of the regular solution at this value of α = α 0 . Like in those d = 5 cases where there are two regular solutions for a given α, say α 0 , here too these two regular solutions are the r h → 0 limits of the corresponding black hole \"loop\", connecting the two regular solutions. Along this second branch the values σ(r h ) of the metric function σ on the event horizon decrease monotonically with r h , while the Hawking temperature strongly increases. The mass of the solution of the second branch is larger than the corresponding one on the first branch, as illustrated by Figures 10-12. It is interesting to note here from the point of view of numerics, that a black hole loop corresponding to the two r h → 0 limits of two regular solutions, say at α 0 , can be constructed numerically even when the numerical process for the higher mass branch of the regular solutions for this α 0 runs into difficulties.\n\nThe introduction of a negative cosmological constant does not appear to change this picture qualitatively. However, we notice a smaller value of r max h with increasing |Λ| and a larger value of M for the same α. In Figure 14, we present the profiles of the metric functions N and σ and gauge function w(r) for the same values of α 2 , Λ on the first and second branch for some d = 6 solutions. The dependence of these functions on the value of Λ is illustrated in Figure 15 for several d = 7 black hole solutions.\n\n\nA computation of mass and action\n\n\nThe counterterm method\n\nIt is well known that the generalisation of Komar's formula for AAdS spacetimes is not straightforward and requires the further subtraction of a background configuration in order to render a finite result for the mass.\n\nThis problem was addressed for the first time in the eighties, with different approaches (see for instance Ref. [42,43]). Another formalism to define conserved charges in AAdS spacetimes was proposed in [44] and uses conformal techniques to construct conserved quantities yielding the results obtained by Hamiltonian methods. Other more recent approaches to the same problem are presented in Ref. [45].\n\nAs expected, these different methods yield the same total mass for the spherically symmetric AAdS configurations considered in Section 3\nM ADM = (d − 2)Ω d−2 8πG M,(38)\nwhere Ω d−2 = 2π (d−1)/2 /Γ((d − 1)/2) is the area of a unit (d − 2)-dimensional sphere, and M is defined in the second member of (29). A procedure leading (for odd dimensions) to a different result has been proposed by Balasubramanian and Kraus [21]. This technique was inspired by AdS/CFT correspondence and consists in adding suitable counterterms I ct to the action of the theory in order to ensure the finiteness of the boundary stress tensor derived by the quasilocal energy definition [46]. These counterterms are built up with curvature invariants of a boundary ∂M (which is sent to infinity after the integration) and thus obviously they do not alter the bulk equations of motion. Unlike background subtraction, the counterterm approach does no require the identification of a reference spacetime. Given the potential relevance of the EYM solutions in an AdS/CFT context, we present here a computation of the boundary stress tensor and of the total mass, by using the counterterm prescription.\n\nThe following counterterms are sufficient to cancel divergences in a pure gravity theory for d ≤ 9, with several exceptions (see e. g. [47])\nI ct = − 1 8πG ∂M d d−1 x √ −h d − 2 ℓ + ℓΘ(d − 4) 2(d − 3) R + ℓ 3 Θ(d − 6) 2(d − 5)(d − 3) 2 R AB R AB − d − 1 4(d − 2) R 2(39)+ ℓ 5 Θ (d − 8) (d − 2) 3 (d − 4)(d − 6) 3d + 2 4(d − 1) RR AB R AB − d(d + 2) 16(d − 1) 2 R 3 −2R AB R CD R ABCD − d 4(d − 1) ∇ A R∇ A R + ∇ C R AB ∇ C R AB .\nHere R ABCD , R AB are the Riemann and Ricci tensors, R is the Ricci scalar for the boundary metric h and Θ(x) is the step function, which is equal to 1 for x ≥ 0 and zero otherwise; A, B, . . . indicate the intrinsic coordinates of the boundary. Using these counterterms one can construct a divergenceless boundary stress tensor, which is given by the variation of the total action at the boundary with respect to h AB . Its explicit expression, restricting for simplicity to d < 7, is\nT AB = 1 8πG (K AB − Kh AB − d − 2 ℓ h AB + ℓ d − 3 E AB ),(40)\nwhere\nK AB = − 1 2 (∇ A n B + ∇ B n A )\nis the extrinsic curvature defined in terms of the normal n A to the boundary, K is its trace, and E AB is the Einstein tensor of the intrinsic metric h AB . (The corresponding form of (40) for d = 7, 8 is given e.g. in [48]).\n\nThe result we find in this way for T AB is given by\nT B A = 1 8πGℓ M + p (−1) p ℓ d−3 Γ( 2p−1 2 ) 2 √ πΓ(p + 1) δ 2p,d−1 (d − 1)u A u B + δ B A 1 r d−1 + O( 1 r d ),(41)\nwhere u A = δ t A and p is an integer. We can use this approach to assign a mass-energy to an AAdS geometry by writing the boundary metric in an ADM form\nh AB dx A dx B = −N 2 Σ dt 2 + σ ab (dx a + N a σ dt)(dx b + N b σ dt)(42)\nand the definition of the energy in this context is\nE = ∂Σ d d−1 x √ σN Σ ǫ.(43)\nHere ǫ = u µ u ν T µν is the proper energy density while u µ is a timelike unit normal to Σ.\n\nIf there are matter fields on M, additional counterterms may be needed to regulate the action. This is the case of F (2) theory, discussed in Appendix B. The couinterterm action depends in this case not only on the boundary metric but also on the boundary value of the gauge field.\n\nHowever, we find that for P > 1 EYM solutions with Λ < 0 in d = 5, 6, 7, 8 dimensions, the prescription (39) removes all divergences. The use of higher order terms in the YM curvature, namely F (2p) forms with p > 1 introduced in Section 3, results in this regularising of the masses. The crucial point here is that these solutions approaches asymptotically a Schwarzschild-AdS background, and the YM asymptotic parameter w 1 appears only in the next to leading order of the T AB expression.\n\nThe mass-energy of solutions computed in this way is\nE = (d − 2)Ω d−2 8πG M + E 0(44)\nwhere, for 3 < d < 9\nE 0 = (d − 2)Ω d−2 16πG 3 4 ℓ 2 δ 5,d − 5 8 ℓ 4 δ d,7 .(45)\nThe additional term E 0 appearing in E for d = 5, (7) is the mass of pure global AdS 5, (7) and is usually interpreted as the energy dual to the Casimir energy of the CFT defined on a four (six) dimensional Einstein universe [21]. The metric restricted to the boundary h AB diverges due to an infinite conformal factor r 2 /ℓ 2 . The background metric upon which the dual field theory resides is\nγ AB = lim r→∞ ℓ 2 r 2 h AB .(46)\nFor the asymptotically AdS d solutions considered here, the (d − 1) dimensional boundary is the Einstein universe, with the line element\nγ AB dx A dx B = −dt 2 + ℓ 2 dΩ 2 d−2 .(47)\nIn light of the AdS/CFT correspondence, Balasubramanian and Kraus have interpreted Eq. (40) as < τ AB >= 2 √ −γ δS ef f δγAB , where < τ AB > is the expectation value of the CFT stress tensor. Then, the divergences which appear are simply the standard ultraviolet divergences of a quantum field theory and we can cancel them by adding local counterterms to the action. Corresponding to the boundary metric (47), the stress-energy tensor τ AB for the dual theory can be calculated using the following relation [49] √ −γγ AB < τ BC >= lim\nr→∞ √ −hh AB T BC .(48)\n\nAction and entropy\n\nThe above approach can be used to compute the Euclidean action and to prove in a rigorous way that the entropy of the EYM black hole solutions is one quarter of the event horizon area, as expected. Here we start by constructing the path integral [18] \nZ = D[g]D[Ψ]e −iI[g,Ψ](49)\nby integrating over all metrics and matter fields between some given initial and final hypersurfaces, Ψ corresponding here to the SU(2) potentials. By analytically continuing the time coordinate t → iτ , the path integral formally converges, and in the leading order one obtains Z ≃ e −I cl (50) where I cl is the classical action evaluated on the equations of motion of the gravity/matter system. The physical interpretation of this formalism is that the class of regular stationary metrics forms an ensemble of thermodynamic systems at equilibrium temperature T (see e.g. [50]). Z has the interpretation of partition function and we can define the free energy of the system F = −β −1 log Z. Therefore\nlog Z = −βF = S − βE,(51)\nor\nS = βE − I cl ,(52)\nstraightforwardly follows.\n\nTo compute I cl , we make use of the Einstein equations, replacing the R − 2Λ volume term with 2R t t − 16πGT t t . For our purely magnetic ansatz, the term T t t exactly cancels the matter field Lagrangean in the bulk action. The divergent contribution given by the surface integral term at infinity in R t t is also canceled by I surface + I ct and for 3 < d < 9 we arrive at the simple finite expression\nI cl = βΩ d−2 8πG M − r d−2 h ℓ 2 + 3 8 ℓ 2 δ d,5 − 5 16 ℓ 4 δ d,7 .(53)\nReplacing I cl now in (52) (where E is the mass-energy computed in Section 4.1), we find\nS = 1 4 Ω d−2 r d−2 h ,(54)\nwhich is one quarter of the event horizon area, as expected. ¿From the AdS/CFT correspondence, we expect the nonabelian hairy black holes to be described by some thermal states in a dual theory formulated in a Einstein universe background. The spherically symmetric solitons will correspond to zero-temperature states in the same theory. The existence of these hairy configurations suggest that there should be some observables in the dual CFT that encode the hair information.\n\n\nConclusions\n\nMotivated by recent results in EYM theory in four dimensional AAdS spacetime, we studied higher dimensional spherically symmetric solutions with non Abelian fields in the presence of a negative cosmological constant. Since the mass-energy of the AAdS configurations plays a central role in its application to the AdS/CFT correspondence, we emphasised this aspect of the solutions we found. The mass-energy of the usual EYM solutions (both asymptotically flat and AAdS), defined according to the standard prescription, always diverges in spacetime dimensions d > 4. One of the tasks performed in this work was a demonstration of this fact in the AAdS case. The main properties of these higher dimensional F (2) solutions resemble the d = 4 case, a continuum of solutions with arbitrary asymptotic values of the gauge function w(r) being found.\n\nThen we focused on two possible approaches of dealing with the divergences of the mass and action. One of these involved the regularisation of the mass-energy using a counterterm mechanisms. In this approach, it turns out that the counterterm action depends not only on the boundary metric, but also on the boundary values of the gauge fields. However, the masses of the solutions defined in this way may take negative values, leading us to the second approach.\n\nThe other, which formed the main thrust of the work, was to augment the action density of the system with higher order curvature terms, consisting of 2p-form curvatures F (2p). These terms were added to the usual YM system constructed from F (2). It resulted in EYM solutions supporting finite mass-energy in all spacetime dimensions 5 ≤ d ≤ 2p.\n\nConcerning the construction of regular finite energy classical AAdS solutions in higher dimensions, we restricted ourselves to systems consisting exclusively of gravitational and non Abelian gauge fields. The salient features of the resulting solutions are captured in this framework, the addition of other (string theory inspired) matter terms being deferred to later work. The asymptotically flat versions of the higher-p EYM systems having been studied in [13,14,19], our task here involved the introduction of a negative cosmological constant. The most succint way of listing our conclusions is:\n\n• The qualitative properties of the regular AAdS solutions in spacetime dimensions d = 5, 6, 7, 8 are the same as the corresponding asymptotically flat ones, namely -A one parameter family of solutions parametrised by the (dimensionless) gravitational coupling constant α start at α = 0 (the gravity decoupling limit) and exist up to a maximum α max , after which α decreases again and ends at critical value.\n\n-For d = 6, 7, 8 the value of α in the second, not that of gravity decoupling, endpoint becomes very small and stops. In the d = 8 case it actually vanishes. For d = 5 the value of α in the second endpoint reaches a finite critical value, where it does not stop, but oscillates around this critical value.\n\n-As long as the physically important F (2) term is present in the YM sector, there exist no multinode solutions.\n\n• Black hole counterparts appear to exist for any regular solution. The qualitative properties of the AAdS black hole solutions are similar to the asymptotically flat case:\n\n-Different from the four dimensional theory, the event horizon radius presents a maximal value. This maximal value is a function of the gravitational coupling constant α.\n\n-For d = 6, 7, 8 the black hole solutions form a loop connecting the two regular solutions with the same value of α. The solutions of the five dimensional theory are somehow special, presenting a complicated brach structure which depends on α.\n\nAxially symmetric generalizations of these solutions are likely to exist. We expect them to share the basic properties of the spherically symmetric configurations. The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ. In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ. \n\n\nd = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ.\n\nFigure 1 .Figure 2 .Figure 3 .\n123The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ.\n\nFigure 4 .\n4The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles.\n\nFigure 5 .\n5Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ.\n\n\nFigure 6a.\n\n\nFigure 6b.\n\nFigure 6\n6Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ.\n\n\nFigure 7a.\n\n\nFigure 7b.\n\nFigure 7\n7The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ.\n\n\nFigure 8a.\n\n\nFigure 8b.\n\nFigure 8\n8The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ.\n\nFigure 9b .\n9bFigure 9a.\n\nFigure 9\n9Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.\n\nFigure 10b .\n10bFigure 10a.\n\nFigure 10\n10The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ.\n\nFigure 11 .\n11The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ.\n\nFigure 12 .Figure 13 .\n1213The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ.\n\nFigure 14 .\n14The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01.\n\nFigure 15 .\n15Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ.\nThe only higher order curvature terms considered in this paper are those constructed from a 2p-form field such that the Lagrangean contains velocity square fields only.\nLike the gravitating monopoles, these have gravity decoupling limits except in d = 5 (and in d = 4p + 1 modulo 4), and in all odd spacetime dimensions, the flat space solutions are stabilised by a Pontryagin charge analogous to the magnetic charge of the monopole, provided that the representations of the gauge group are chosen suitably. In all even d however, they are like the BK solution in that they are not stabilised by a topological charge and are likewise sphalerons[15] 3 In principle higher order terms with the desired scaling can be chosen to consist both of the YM and the Riemann curvatures, but in practice we restrict to the YM hierarchy. The reason will be explained in section 2.5. Besides, it was found in[13] that the inclusion of Gauss-Bonnet terms does not result in any new qualitative features to the solutions.\nOne can also solve directly the field equations (8)-(10), but the solution takes a much more complicated form for the metric ansatz(5).\nMultinode solutions exist if the lowest order YM term is F (2p) with p ≥ 2, in the appropriate dimensions[19] d.\nz dz < 0,(62)which contradicts E(z 1 ) > 0. Therefore E(z) should vanish identically and one finds no d > 5 finite mass, spherically symmetric EYM configurations in a F (2) theory. Note that this argument does not exclude the existence of configuration with a diverging mass functions as r → ∞.\nAcknowledgementWe are inebted to Dieter Maison for advice and discussions. ER thanks Cristian Stelea for useful remarks on Section 2. This work was carried out in the framework of Enterprise-Ireland Basic Science Research Project SC/2003/390.Appendix A: A nonexistence proof for p = 1, d > 5 finite mass solutions with Λ ≤ 0Following the notation used in[11], we introduce a new variable z = 2 log r,(55)and rewrite the basic equations(8),(9)in the formwhere the metric function σ has been eliminated by using(10). The function N is given by(6), namely asTo devise a proof for the nonexistence of finite mass solutions of the above system, it is convenient to introduce the functionwhich, from(56)and(57)satisfies the equationThe relations in Appendix B of Ref.[11]are recovered for d = 5, where a nonexistence proof is presented for particle-like solutions with finite mass (the extension to black hole case is considered in[51]). Therefore in what follows we will take d > 5 only. The approximate form of the function E at the origin (or event horizon) and infinity is found by taking P = 1 in relations(26),(27)and(29)given in Section 3 and replacing in(58). It is obvious that E(r h ) < 0, since N (r h ) = 0; at the same time, the corresponding expression as r → 0 (i.e. z → −∞) isand we find E → −0 in this limit. Also, the relations(26),(27)give m(r = 0) > 0, m(r h ) > 0, which, together with the equation(56)implies that the mass function m(r) is positive definite.Besides, by replacing in (58) the asymptotic expressions (29) as r → ∞ it follows thati.e. E → +0 as z → ∞, for finite mass solutions. Therefore, if the solution is regular everywhere, E must vanish at some finite point z 0 , and dE/dz ≥ 0 there, with E > 0 for z > z 0 (when there are several positive roots of E, we take the largest one). However, another point should exist z 1 > z 0 such that dE/dz = 0 i.e. the function E should present a positive maximum for some value of z. Now we integrate the equation (59) between z 0 and z 1 and findAppendix B: A matter counterterm proposalIn this Section we comment on the issue of mass definition of AAdS solutions in a F (2) 2 (i.e. p = 1) theory, if we do not exercise the option of employing higher order YM survature terms. As proven in Section 2, although the spacetime is still AAdS, the mass function m(r) of these solutions generically diverges as r d−5 (or as log r for d = 5). AAdS solutions with a diverging ADM mass have been considered recently by some authors, mainly for a scalar field in the bulk (see e.g.[52]-[57]. In this case it might be possible to relax the standard asymptotic conditions without loosing the original symmetries, but modifying the charges in order to take into account the presence of matter fields. Similar to the case of scalar field, for d > 5 it is still possible to obtain a finite mass of EYM solutions in a F 2 theory by allowing I ct to depend not only on the boundary metric h AB , but also on the gauge field strength tensor. This means that the quasilocal stress-energy tensor (40) also acquires a contribution coming from the matter fields.We find that by adding a counterterm of the formto the expression (39), the divergence disappears. This yields a supplementary contribution to (40)The mass of the d > 5 solutions computed in this way iswhere M 0 is the constant appearing in the asymptotic expansion(16). It can also be proven that this prescription leads to a finite action and the entropy-area relation is satisfied. However, as seen inFigure 2, the parameter M 0 of the p = 1 solutions takes negative values, pointing to some pathological properties. It would be nice to have a rigorous derivation of the matter counterterm expression, possibly along the lines of Ref.[58]. Also, there remains the issue of the d = 5 solutions, whose logarithmic divergeces require a different approach.\n. J M Maldacena, arXiv:hep-th/9711200Adv. Theor. Math. Phys. 2231Int. J. Theor. Phys.J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].\n\n. E Witten, arXiv:hep-th/9802150Adv. Theor. Math. Phys. 2253E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 [arXiv:hep-th/9802150].\n\n. E Winstanley, arXiv:gr-qc/9812064Class. Quant. Grav. 161963E. Winstanley, Class. Quant. Grav. 16 (1999) 1963 [arXiv:gr-qc/9812064].\n\n. J Bjoraker, Y Hosotani, arXiv:hep-th/0002098Phys. Rev. D. 6243513J. Bjoraker and Y. Hosotani, Phys. Rev. D 62 (2000) 043513 [arXiv:hep-th/0002098].\n\n. O Sarbach, E Winstanley, arXiv:gr-qc/0102033Class. Quant. Grav. 182125O. Sarbach and E. Winstanley, Class. Quant. Grav. 18 (2001) 2125 [arXiv:gr-qc/0102033].\n\n. P Breitenlohner, D Maison, G Lavrelashvili, arXiv:gr-qc/0307029Class. Quant. Grav. 211667P. Breitenlohner, D. Maison and G. Lavrelashvili, Class. Quant. Grav. 21 (2004) 1667 [arXiv:gr-qc/0307029].\n\n. E Radu, arXiv:hep-th/0211120Phys. Rev. D. 6784030E. Radu, Phys. Rev. D 67 (2003) 084030 [arXiv:hep-th/0211120].\n\n. J J Van Der Bij, E Radu, arXiv:gr-qc/0107065Phys. Lett. B. 536107J. J. Van der Bij and E. Radu, Phys. Lett. B 536 (2002) 107 [arXiv:gr-qc/0107065].\n\n. E Radu, arXiv:gr-qc/0210074Phys. Lett. B. 548224E. Radu, Phys. Lett. B 548 (2002) 224 [arXiv:gr-qc/0210074];\n\n. E Radu, arXiv:gr-qc/0109015Phys. Rev. D. 6544005E. Radu, Phys. Rev. D 65 (2002) 044005 [arXiv:gr-qc/0109015].\n\n. E Radu, E Winstanley, arXiv:hep-th/0407248Phys. Rev. D. 7084023E. Radu and E. Winstanley, Phys. Rev. D 70 (2004) 084023 [arXiv:hep-th/0407248].\n\n. N Okuyama, K I Maeda, arXiv:gr-qc/0212022Phys. Rev. D. 67104012N. Okuyama and K. i. Maeda, Phys. Rev. D 67 (2003) 104012 [arXiv:gr-qc/0212022].\n\n. M S Volkov, arXiv:hep-th/0103038Phys. Lett. B. 524369M. S. Volkov, Phys. Lett. B 524 (2002) 369 [arXiv:hep-th/0103038].\n\n. Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0202141Class. Quant. Grav. 202765Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, Class. Quant. Grav. 20 (2003) 2765 [arXiv:hep-th/0202141].\n\n. Y Brihaye, A Chakrabarti, B Hartmann, D H Tchrakian, arXiv:hep-th/0212288Phys. Lett. B. 561161Y. Brihaye, A. Chakrabarti, B. Hartmann and D. H. Tchrakian, Phys. Lett. B 561 (2003) 161 [arXiv:hep-th/0212288].\n\n. Y Brihaye, D H Tchrakian, arXiv:hep-th/0410008J. Phys. A. 383679Y. Brihaye and D. H. Tchrakian, J. Phys. A 38 (2005) 3679 [arXiv:hep-th/0410008].\n\n. R Bartnik, J Mckinnon, Phys. Rev. Lett. 61141R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141.\n\n. P Breitenlohner, P Forgacs, D Maison, ibid. 442Nucl. Phys. B. 383126P. Breitenlohner, P. Forgacs and D. Maison, Nucl. Phys. B 383 (1992) 357; ibid. 442 (1995) 126\n\n. G W Gibbons, S W Hawking, Phys. Rev. D. 152752G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2752.\n\n. P Breitenlohner, D Maison, D H Tchrakian, arXiv:gr-qc/0508027P. Breitenlohner, D. Maison and D.H. Tchrakian, arXiv:gr-qc/0508027.\n\n. S Deser, Class. Quant. Grav. 11S. Deser, Class. Quant. Grav. 1 (1984) L1.\n\n. V Balasubramanian, P Kraus, arXiv:hep-th/9902121Commun. Math. Phys. 208413V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208 (1999) 413 [arXiv:hep-th/9902121].\n\n. M Banados, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849.\n\n. D Maison, private communicationD. Maison, private communication.\n\n. E W Hirschmann, D L Welch, arXiv:hep-th/9510181Phys. Rev. D. 535579E. W. Hirschmann and D. L. Welch, Phys. Rev. D 53 (1996) 5579 [arXiv:hep-th/9510181].\n\n. M Cataldo, P Salgado, Phys. Rev. D. 542971M. Cataldo and P. Salgado, Phys. Rev. D 54 (1996) 2971.\n\n. Y Kiem, D Park, arXiv:hep-th/9610008Phys. Rev. D. 556112Y. Kiem and D. Park, Phys. Rev. D 55 (1997) 6112 [arXiv:hep-th/9610008].\n\n. M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0201170Phys. Rev. D. 65104007M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 65 (2002) 104007 [arXiv:hep-th/0201170].\n\n. Y Hosotani, arXiv:gr-qc/0103069J. Math. Phys. 43597Y. Hosotani, J. Math. Phys. 43 (2002) 597 [arXiv:gr-qc/0103069].\n\n. C N Pope, Class. Quant. Grav. 277C. N. Pope, Class. Quant. Grav. 2 (1985) L77.\n\nClassical lumps and their quantum descendants. S Coleman, 297in Erice subnuclear physicsS. Coleman, Classical lumps and their quantum descendants, in Erice subnuclear physics, pp. 297, 1975.\n\n. S Deser, Phys. Lett. B. 64463S. Deser, Phys. Lett. B 64 (1976) 463.\n\n. M S Volkov, D V , arXiv:hep-th/9810070Phys. Rept. 3191M. S. Volkov and D. V. Gal'tsov, Phys. Rept. 319 (1999) 1 [arXiv:hep-th/9810070].\n\n. M Heusler, arXiv:gr-qc/9610019Helv. Phys. Acta. 69501M. Heusler, Helv. Phys. Acta 69 (1996) 501 [arXiv:gr-qc/9610019].\n\n. M Heusler, N Straumann, Class. Quantum. Grav. 92177M. Heusler, N. Straumann, Class. Quantum. Grav. 9, 2177, (1992).\n\n. A H Chamseddine, M S Volkov, arXiv:hep-th/0101202JHEP. 010423A. H. Chamseddine and M. S. Volkov, JHEP 0104 (2001) 023 [arXiv:hep-th/0101202];\n\n. G Bertoldi, arXiv:hep-th/0210048JHEP. 021042G. Bertoldi, JHEP 0210 (2002) 042 [arXiv:hep-th/0210048].\n\nBorn-Infeld action, supersymmetry and string theory. A A Tseytlin, Yuri Golfand memorial volume. M. ShifmanWorld ScientificA. A. Tseytlin, Born-Infeld action, supersymmetry and string theory, in Yuri Golfand memorial volume, ed. M. Shifman, World Scientific (2000).\n\n. E A Bergshoeff, M De Roo, A Sevrin, arXiv:hep-th/0011264Nucl. Phys. Proc. Suppl. 49433Fortsch. Phys.E. A. Bergshoeff, M. de Roo and A. Sevrin, Fortsch. Phys. 49 (2001) 433 [Nucl. Phys. Proc. Suppl. 102 (2001) 50] [arXiv:hep-th/0011264].\n\n. M Cederwall, B E W Nilsson, D Tsimpis, arXiv:hep-th/0102009JHEP. 010634M. Cederwall, B. E. W. Nilsson and D. Tsimpis, JHEP 0106 (2001) 034 [arXiv:hep-th/0102009].\n\n. A A Belavin, A M Polyakov, A S Shvarts, Y S Tyupkin, Phys. Lett. B. 5985A. A. Belavin, A. M. Polyakov, A. S. Shvarts and Y. S. Tyupkin, Phys. Lett. B 59 (1975) 85.\n\n. H Boutaleb-Joutei, A Chakrabarti, A Comtet, Phys. Rev. D. 201884H. Boutaleb-Joutei, A. Chakrabarti and A. Comtet, Phys. Rev. D 20 (1979) 1884.\n\n. Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0002144J. Math. Phys. 415490Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, J. Math. Phys. 41 (2001) 5490 [arXiv:hep-th/0002144].\n\n. L F Abbott, S Deser, Nucl. Phys. B. 19576L. F. Abbott and S. Deser, Nucl. Phys. B 195 76 (1982).\n\n. M Henneaux, C Teitelboim, Commun. Math. Phys. 98391M. Henneaux and C. Teitelboim, Commun. Math. Phys. 98 391 (1985).\n\n. A Ashtekar, A Magnon, Class. Quant. Grav. 139A. Ashtekar and A. Magnon, Class. Quant. Grav. 1 (1984) L39;\n\n. A Ashtekar, S Das, Class. Quant. Grav. 1717A. Ashtekar and S. Das, Class. Quant. Grav. 17 (2000) L17.\n\n. R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. Lett. 841647R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. Lett. 84 (2000) 1647;\n\n. R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. D. 6244002R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. D 62 (2000) 044002;\n\n. E W Mielke, Phys. Rev. D. 6344018E. W. Mielke, Phys. Rev. D 63 (2001) 044018.\n\n. J D Brown, J W York, Phys. Rev. D. 471407J. D. Brown and J. W. York, Phys. Rev. D 47 1407 (1993).\n\n. M M Taylor-Robinson, arXiv:hep-th/0002125.K.SkenderisInt. J. Mod. Phys. A. 16740M. M. Taylor-Robinson, arXiv:hep-th/0002125. K. Skenderis, Int. J. Mod. Phys. A 16 (2001) 740.\n\n. S Haro, S N Solodukhin, K Skenderis, Commun. Math. Phys. 217595S. de Haro, S. N. Solodukhin and K. Skenderis, Commun. Math. Phys. 217 (2001) 595.\n\n. S Das, R B Mann, arXiv:hep-th/0008028JHEP. 000833S. Das and R. B. Mann, JHEP 0008 (2000) 033 [arXiv:hep-th/0008028].\n\n. R C Myers, Phys. Rev. D. 6046002R. C. Myers, Phys. Rev. D 60 (1999) 046002.\n\n. R B Mann, arXiv:gr-qc/0211047Found. Phys. 3365R. B. Mann, Found. Phys. 33 (2003) 65 [arXiv:gr-qc/0211047].\n\n. Y Brihaye, B Hartmann, E Radu, arXiv:hep-th/0508028Y. Brihaye, B. Hartmann and E. Radu, arXiv:hep-th/0508028.\n\n. T Hertog, K Maeda, arXiv:hep-th/0404261JHEP. 040751T. Hertog and K. Maeda, JHEP 0407 (2004) 051 [arXiv:hep-th/0404261].\n\n. T Hertog, K Maeda, arXiv:hep-th/0409314Phys. Rev. D. 7124001T. Hertog and K. Maeda, Phys. Rev. D 71 (2005) 024001 [arXiv:hep-th/0409314].\n\n. M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0404236Phys. Rev. D. 7044034M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 70 (2004) 044034 [arXiv:hep-th/0404236].\n\n. E Radu, D H Tchrakian, arXiv:hep-th/0410154Class. Quant. Grav. 22879E. Radu and D. H. Tchrakian, Class. Quant. Grav. 22 (2005) 879 [arXiv:hep-th/0410154].\n\n. S Hollands, A Ishibashi, D Marolf, arXiv:hep-th/0503045Class. Quant. Grav. 222881S. Hollands, A. Ishibashi and D. Marolf, Class. Quant. Grav. 22 (2005) 2881 [arXiv:hep-th/0503045].\n\n. J T Liu, W A Sabra, arXiv:hep-th/0405171Phys. Rev. D. 7264021J. T. Liu and W. A. Sabra, Phys. Rev. D 72 (2005) 064021 [arXiv:hep-th/0405171].\n\n. A Batrachenko, J T Liu, R Mcnees, W A Sabra, W Y Wen, arXiv:hep-th/0408205JHEP. 050534A. Batrachenko, J. T. Liu, R. McNees, W. A. Sabra and W. Y. Wen, JHEP 0505 (2005) 034 [arXiv:hep-th/0408205].\n"},"corpus_id":{"kind":"number","value":14280244,"string":"14,280,244"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"gr-qc/0508033\",\n \"mag\": 2090903971,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevd.73.024006\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/gr-qc/0508033v2.pdf\"\n ],\n \"pdf_sha\": \"8a79b598ee83cc1be9e4275e2cea9708ebac76c4\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/gr-qc/0508033\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"arXiv:gr-qc/0508033v2 15 Dec 2005 No hair conjecture, nonabelian hierarchies and anti-de Sitter spacetime"},"authors":{"kind":"list like","value":["Eugen Radu \nDepartment of Mathematical Physics\nNational University of Ireland Maynooth\nMaynoothIreland","D H Tchrakian \nDepartment of Mathematical Physics\nNational University of Ireland Maynooth\nMaynoothIreland\n\nSchool of Theoretical Physics -DIAS\n10 Burlington RoadDublin 4Ireland","Eugen Radu \nDepartment of Mathematical Physics\nNational University of Ireland Maynooth\nMaynoothIreland","D H Tchrakian \nDepartment of Mathematical Physics\nNational University of Ireland Maynooth\nMaynoothIreland\n\nSchool of Theoretical Physics -DIAS\n10 Burlington RoadDublin 4Ireland"],"string":"[\n \"Eugen Radu \\nDepartment of Mathematical Physics\\nNational University of Ireland Maynooth\\nMaynoothIreland\",\n \"D H Tchrakian \\nDepartment of Mathematical Physics\\nNational University of Ireland Maynooth\\nMaynoothIreland\\n\\nSchool of Theoretical Physics -DIAS\\n10 Burlington RoadDublin 4Ireland\",\n \"Eugen Radu \\nDepartment of Mathematical Physics\\nNational University of Ireland Maynooth\\nMaynoothIreland\",\n \"D H Tchrakian \\nDepartment of Mathematical Physics\\nNational University of Ireland Maynooth\\nMaynoothIreland\\n\\nSchool of Theoretical Physics -DIAS\\n10 Burlington RoadDublin 4Ireland\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We consider globally regular and black holes solutions for the Einstein-Yang-Mills system with negative cosmological constant in d−spacetime dimensions. We find that the ADM mass of the spherically symmetric solutions generically diverges for d > 4. Solutions with finite mass are found by considering corrections to the YM Lagrangean consisting in higher therm of the Yang-Mills hierarchy. Such systems can occur in the low energy effective action of string theory. A discussion of the main properties of the solutions and the differences with respect to the four dimensional case is presented. The mass of these configurations is computed by using a counterterm method."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":""},{"title":"Introduction","content":"If we allow for a negative cosmological constant, the solution of the matter free Einstein equations possessing the maximal number of symmetries is the anti-de Sitter (AdS) spacetime. Recently a considerable amount of interest has been focused on solutions of Einstein equations with this type of asymptotics. This interest is mainly motivated by the proposed correspondence between physical effects associated with gravitating fields propagating in AdS spacetime and those of a conformal field theory on the boundary of AdS spacetime [1,2].\nIf we allow for a negative cosmological constant, the solution of the matter free Einstein equations possessing the maximal number of symmetries is the anti-de Sitter (AdS) spacetime. Recently a considerable amount of interest has been focused on solutions of Einstein equations with this type of asymptotics. This interest is mainly motivated by the proposed correspondence between physical effects associated with gravitating fields propagating in AdS spacetime and those of a conformal field theory on the boundary of AdS spacetime [1,2].\nIn view of these developments, an examination of the classical solutions of gravitating fields in asymptotically AdS (AAdS) spacetimes seems appropriate. Recently, some authors have discussed the properties of gravitating SU(2) nonabelian fields with a negative cosmological constant Λ [3,4]. Considering the case of four spacetime dimensions, they obtained some surprising results, which are strikingly different from the results found in the asymptotically flat case. For example there are solutions for continuous intervals of the parameter space, rather then discrete points. The asymptotic values of the gauge potentials are arbitrary and there exist solutions supporting magnetic and electric fluxes without a Higgs field. Some of these solutions are stable against spherically symmetric linear perturbations. The literature on AAdS solutions with nonabelian fields is growing steadily, including stability analyses [5,6], the study of configurations with a NUT charge [7], topological black holes with nonabelian hair [8] as well as axially symmetric generalisations [9,10]. The existence of these solutions invalidates the AdS 4 version of the no hair conjecture, which states that the black holes are completely characterised by their mass, charge and angular momentum.\nIn view of these developments, an examination of the classical solutions of gravitating fields in asymptotically AdS (AAdS) spacetimes seems appropriate. Recently, some authors have discussed the properties of gravitating SU(2) nonabelian fields with a negative cosmological constant Λ [3,4]. Considering the case of four spacetime dimensions, they obtained some surprising results, which are strikingly different from the results found in the asymptotically flat case. For example there are solutions for continuous intervals of the parameter space, rather then discrete points. The asymptotic values of the gauge potentials are arbitrary and there exist solutions supporting magnetic and electric fluxes without a Higgs field. Some of these solutions are stable against spherically symmetric linear perturbations. The literature on AAdS solutions with nonabelian fields is growing steadily, including stability analyses [5,6], the study of configurations with a NUT charge [7], topological black holes with nonabelian hair [8] as well as axially symmetric generalisations [9,10]. The existence of these solutions invalidates the AdS 4 version of the no hair conjecture, which states that the black holes are completely characterised by their mass, charge and angular momentum.\nHowever, all these studies approach the case of a four dimensional AAdS spacetime, and relatively little is known about higher dimensional AAdS solutions with non Abelian matter fields. Practically all that is known for d > 4 is the five dimensional nonabelian SU (2) solutions discussed in [11]. At the same time gauged supergravity theories playing an important role in AdS/CFT, generically contain non Abelian matter fields in the bulk, although in the literature only Abelian truncations are considered, to date. Thus, the examination of higher dimensional gravitating non Abelian solutions with Λ < 0 is a pertinent task.\nHowever, all these studies approach the case of a four dimensional AAdS spacetime, and relatively little is known about higher dimensional AAdS solutions with non Abelian matter fields. Practically all that is known for d > 4 is the five dimensional nonabelian SU (2) solutions discussed in [11]. At the same time gauged supergravity theories playing an important role in AdS/CFT, generically contain non Abelian matter fields in the bulk, although in the literature only Abelian truncations are considered, to date. Thus, the examination of higher dimensional gravitating non Abelian solutions with Λ < 0 is a pertinent task.\nHigher dimensional asymptotically flat solutions of the Einstein-Yang-Mills (EYM) equations have recently been the subject of several studies. As found in [12], in five spacetime dimensions, the particle spectrum obtained by uplifting the d = 4 flat space YM instantons become completely destroyed by gravity, as a result of their scaling behaviour. However, by adding higher order 1 terms in the Yang-Mills (YM) hierarchy this obstacle due to scaling is removed and solutions in higher dimensions can be found. Such regular, static and spherically symmetric solutions in spacetime dimensions d ≤ 8 were presented in [13], and for d = 5, both globally regular and black hole solutions of the EYM system were found in [14]. The properties of these solutions are rather different from the familiar Bartnik-McKinnon (BK) solutions [16] to EYM in d = 4, and are somewhat more akin to the gravitating monopole solutions to EYM-Higgs (EYMH) [17]. This is because like in the latter case [17], where the VEV of the Higgs field features as an additional dimensional constant, here [13] also additional dimensional constants enter with each higher order YM curvature term 2 . They are however quite distinctive in their critical behaviour. The typical critical features discovered in [13,14] have recently been analysed and explained in [19].\nHigher dimensional asymptotically flat solutions of the Einstein-Yang-Mills (EYM) equations have recently been the subject of several studies. As found in [12], in five spacetime dimensions, the particle spectrum obtained by uplifting the d = 4 flat space YM instantons become completely destroyed by gravity, as a result of their scaling behaviour. However, by adding higher order 1 terms in the Yang-Mills (YM) hierarchy this obstacle due to scaling is removed and solutions in higher dimensions can be found. Such regular, static and spherically symmetric solutions in spacetime dimensions d ≤ 8 were presented in [13], and for d = 5, both globally regular and black hole solutions of the EYM system were found in [14]. The properties of these solutions are rather different from the familiar Bartnik-McKinnon (BK) solutions [16] to EYM in d = 4, and are somewhat more akin to the gravitating monopole solutions to EYM-Higgs (EYMH) [17]. This is because like in the latter case [17], where the VEV of the Higgs field features as an additional dimensional constant, here [13] also additional dimensional constants enter with each higher order YM curvature term 2 . They are however quite distinctive in their critical behaviour. The typical critical features discovered in [13,14] have recently been analysed and explained in [19].\nThese results can be systematically extended to all d ≥ 5 and one finds that no finite mass solutions can exist in EYM theory, unless one modifies the non Abelian action by adding higher order curvature terms 3 in the YM 2p-form curvature F (2p), the p = 1 case being the usual 2-form YM curvature. Without these higher order YM terms, only vortex-type finite energy solutions [12] exist, describing effective systems in 3 spacelike dimensions, and with a number of codimensions.\nThese results can be systematically extended to all d ≥ 5 and one finds that no finite mass solutions can exist in EYM theory, unless one modifies the non Abelian action by adding higher order curvature terms 3 in the YM 2p-form curvature F (2p), the p = 1 case being the usual 2-form YM curvature. Without these higher order YM terms, only vortex-type finite energy solutions [12] exist, describing effective systems in 3 spacelike dimensions, and with a number of codimensions.\nIt is the purpose of this paper to examine the corresponding situation in higher dimensions, in the presence of a negative cosmological constant. It is interesting to inquire whether the introduction of a negative cosmological constant to these higher dimensional EYM models will lead to some new effects as it does in the d = 4 case, due to the different asymptotic structure of the spacetime. In the first place this would lead to our understanding of how the behaviour of EYM theory depends on the dimensionality of the spacetime. But such higher dimensional AAdS solutions might be relevant to superstring theory, namely in the context of solutions to various supergravities containing nonabelian matter fields. Here however we restrict our considerations to the simplest case of systems consisting only of gravitational and YM fields, namely to higher dimensional EYM model. In particular, we restrict to Einstein-Hilbert gravity and the first two members of the YM hierarchy, and hence to d ≤ 8, augmented with negative cosmological constant in d spacetime dimensions. As in [3,4], for the d = 4 case, we seek static spherically symmetric solutions in the d − 1 spacelike dimensions. We find both globally regular and black hole solutions with finite ADM mass. Unlike in the d = 4 case however, we find that for d > 4 and a negative cosmological constant, the properties of the AAdS solutions do not differ qualitatively from the asymptotically flat case.\nIt is the purpose of this paper to examine the corresponding situation in higher dimensions, in the presence of a negative cosmological constant. It is interesting to inquire whether the introduction of a negative cosmological constant to these higher dimensional EYM models will lead to some new effects as it does in the d = 4 case, due to the different asymptotic structure of the spacetime. In the first place this would lead to our understanding of how the behaviour of EYM theory depends on the dimensionality of the spacetime. But such higher dimensional AAdS solutions might be relevant to superstring theory, namely in the context of solutions to various supergravities containing nonabelian matter fields. Here however we restrict our considerations to the simplest case of systems consisting only of gravitational and YM fields, namely to higher dimensional EYM model. In particular, we restrict to Einstein-Hilbert gravity and the first two members of the YM hierarchy, and hence to d ≤ 8, augmented with negative cosmological constant in d spacetime dimensions. As in [3,4], for the d = 4 case, we seek static spherically symmetric solutions in the d − 1 spacelike dimensions. We find both globally regular and black hole solutions with finite ADM mass. Unlike in the d = 4 case however, we find that for d > 4 and a negative cosmological constant, the properties of the AAdS solutions do not differ qualitatively from the asymptotically flat case.\nOur strategy is to first consider the usual YM model, namely the p = 1 member of the YM hierarchy, or the square of the 2-form curvature F (2). We present an argument for the absence of solutions with reasonable asymptotics for any spacetime dimension d ≥ 5. Although the EYM equations in this case present solutions approaching asymptotically the AdS background, the mass generically diverges. This can be seen as a simple version of the no hair theorem, holding for the EYM system in d > 4 dimensions. In other words, Schwarzschild-AdS black hole is the only static, spherically symmetric solution of the EYM system with finite mass. This is presented in Section 2, where in addition we have considered the special case d = 3, extending Deser's analysis [20] for Λ < 0.\nOur strategy is to first consider the usual YM model, namely the p = 1 member of the YM hierarchy, or the square of the 2-form curvature F (2). We present an argument for the absence of solutions with reasonable asymptotics for any spacetime dimension d ≥ 5. Although the EYM equations in this case present solutions approaching asymptotically the AdS background, the mass generically diverges. This can be seen as a simple version of the no hair theorem, holding for the EYM system in d > 4 dimensions. In other words, Schwarzschild-AdS black hole is the only static, spherically symmetric solution of the EYM system with finite mass. This is presented in Section 2, where in addition we have considered the special case d = 3, extending Deser's analysis [20] for Λ < 0.\nIn Section 3 we introduce the higher order YM hierarchy models featuring the terms F (2p), p ≥ 2. We derive the classical equations subject to our spherically symmetric ansatz, and present a detailed numerical study of of both regular and black hole solutions. As is the case with the usual EYM system, where the existence of finite mass regular solutions leads to the violation of the no hair conjecture, here too, these solutions are explicit counterexamples to this conjecture in AdS d spacetime.\nIn Section 3 we introduce the higher order YM hierarchy models featuring the terms F (2p), p ≥ 2. We derive the classical equations subject to our spherically symmetric ansatz, and present a detailed numerical study of of both regular and black hole solutions. As is the case with the usual EYM system, where the existence of finite mass regular solutions leads to the violation of the no hair conjecture, here too, these solutions are explicit counterexamples to this conjecture in AdS d spacetime.\nOne may ask about the possible relevance of these higher dimensional configurations within the AdS/CFT correspondence. In Section 4 we compute the boundary stress tensor and the mass and action of the solutions in a number of spacetime dimensions up to eight. In five and seven dimensions, the counterterm prescription of [21] gives an additional vacuum (Casimir) energy, which agrees with that found in the context of AdS/CFT correspondence. A counterterm based proposal to remove the divergences of a F (2) theory, such that the mass and action be finite, is also presented in Appendix B. We give our conclusions and remarks in the final section.\nOne may ask about the possible relevance of these higher dimensional configurations within the AdS/CFT correspondence. In Section 4 we compute the boundary stress tensor and the mass and action of the solutions in a number of spacetime dimensions up to eight. In five and seven dimensions, the counterterm prescription of [21] gives an additional vacuum (Casimir) energy, which agrees with that found in the context of AdS/CFT correspondence. A counterterm based proposal to remove the divergences of a F (2) theory, such that the mass and action be finite, is also presented in Appendix B. We give our conclusions and remarks in the final section.\nEverewhere in this paper we employ the notations and conventions of [13]. 2 The F (2) model\nEverewhere in this paper we employ the notations and conventions of [13]. 2 The F (2) model"},{"title":"The action principle","content":""},{"title":"The action principle","content":"We start with the following action principle in d−spacetime dimensions\nWe start with the following action principle in d−spacetime dimensions\nwhere R is the Ricci scalar associated with the spacetime metric g µν , Λ = −(d − 1)(d − 2)/(2ℓ 2 ) is the cosmological constant and G is the gravitational constant (following [14], we define also κ = 1/(8πG)). The matter term in the above relation\nwhere R is the Ricci scalar associated with the spacetime metric g µν , Λ = −(d − 1)(d − 2)/(2ℓ 2 ) is the cosmological constant and G is the gravitational constant (following [14], we define also κ = 1/(8πG)). The matter term in the above relation\nis the usual F 2 nonabelian action density,\nis the usual F 2 nonabelian action density,\nbeing the gauge field strength tensor. Here τ 1 is the coupling constant of the model (in the usual notation τ 1 = 1/g 2 ).\nbeing the gauge field strength tensor. Here τ 1 is the coupling constant of the model (in the usual notation τ 1 = 1/g 2 ).\nThe last term in (1) is the Hawking-Gibbons surface term [18], where K is the trace of the extrinsic curvature for the boundary ∂M and h is the induced metric of the boundary. Of course this term does not affect the equations of motion but it is relevant for the discussion of the mass and the action of the solutions, in Section 4.\nThe last term in (1) is the Hawking-Gibbons surface term [18], where K is the trace of the extrinsic curvature for the boundary ∂M and h is the induced metric of the boundary. Of course this term does not affect the equations of motion but it is relevant for the discussion of the mass and the action of the solutions, in Section 4.\nThe field equations are obtained by varying the action (1) with respect to the field variables g µν , A µ\nThe field equations are obtained by varying the action (1) with respect to the field variables g µν , A µ\nwhere the energy momentum tensor is defined by\nwhere the energy momentum tensor is defined by"},{"title":"The general Ansatz","content":""},{"title":"The general Ansatz","content":"For the case of a d-dimensional spacetime, we restrict to static fields that are spherically symmetric in the d − 1 spacelike dimensions, with a metric ansatz in terms of Schwarzschild coordinates\nFor the case of a d-dimensional spacetime, we restrict to static fields that are spherically symmetric in the d − 1 spacelike dimensions, with a metric ansatz in terms of Schwarzschild coordinates\nwith dΩ d−2 the d − 2 dimensional angular volume element and\nwith dΩ d−2 the d − 2 dimensional angular volume element and\nthe function m(r) being related to the local mass-energy density up to some d−dependent factor. As discussed in [13], the choice of gauge group compatible with the symmetries of the line element (5) is somewhat flexible. In [13] the gauge group chosen was SO(d), in d dimensions. But the gauge field of the static solutions in question took their values in SO(d − 1). Thus in effect, it is possible to choose SO(d) in the first place. Now for even d, it is convenient to choose SO(d) since we can then avail of the chiral representations of the latter, although this is by no means obligatory. Adopting this criterion, namely to employ chiral representations, also for odd d, it is convenient to choose the gauge group to be SO(d − 1).\nthe function m(r) being related to the local mass-energy density up to some d−dependent factor. As discussed in [13], the choice of gauge group compatible with the symmetries of the line element (5) is somewhat flexible. In [13] the gauge group chosen was SO(d), in d dimensions. But the gauge field of the static solutions in question took their values in SO(d − 1). Thus in effect, it is possible to choose SO(d) in the first place. Now for even d, it is convenient to choose SO(d) since we can then avail of the chiral representations of the latter, although this is by no means obligatory. Adopting this criterion, namely to employ chiral representations, also for odd d, it is convenient to choose the gauge group to be SO(d − 1).\nWe shall therefore denote our representation matrices by SO ± (d), whered = d andd = d − 1 for even and odd d respectively.\nWe shall therefore denote our representation matrices by SO ± (d), whered = d andd = d − 1 for even and odd d respectively.\nIn this unified notation (for odd and even d), the spherically symmetric Ansatz for the SO ± (d)-valued gauge fields then reads [13]\nIn this unified notation (for odd and even d), the spherically symmetric Ansatz for the SO ± (d)-valued gauge fields then reads [13]\nThe Γ's denote thed-dimensional gamma matrices and 1, j = 1, 2, ..., d − 1 for both cases. Inserting this ansatz into the action (1), the EYM field equations reduce to\nThe Γ's denote thed-dimensional gamma matrices and 1, j = 1, 2, ..., d − 1 for both cases. Inserting this ansatz into the action (1), the EYM field equations reduce to\nwhich can also be derived from the reduced action\nwhich can also be derived from the reduced action\nFor a F 2 theory, the constants κ, τ 1 can always be absorbed by rescaling r → cr, Λ → Λ/c 2 and m → mκc d−3 , with c = τ 1 /κ. The above differential equations have two analytic solutions. One of them with w(r) = ±1, m(r) = M, σ(r) = 1 (12) corresponds to Schwarzschild-AdS spacetime. For w(r) = 0 we find a non Abelian generalisation of the magnetic-Reissner-Nordstrøm solution with σ(r) = 1 and\nFor a F 2 theory, the constants κ, τ 1 can always be absorbed by rescaling r → cr, Λ → Λ/c 2 and m → mκc d−3 , with c = τ 1 /κ. The above differential equations have two analytic solutions. One of them with w(r) = ±1, m(r) = M, σ(r) = 1 (12) corresponds to Schwarzschild-AdS spacetime. For w(r) = 0 we find a non Abelian generalisation of the magnetic-Reissner-Nordstrøm solution with σ(r) = 1 and\nM 0 being an arbitrary constant. We can see that, although these solutions are asymptotically AdS, the mass defined in the usual way diverges.\nM 0 being an arbitrary constant. We can see that, although these solutions are asymptotically AdS, the mass defined in the usual way diverges."},{"title":"d = 3","content":""},{"title":"d = 3","content":"The (2+1) dimensional case is rather special. Three dimensional gravity has provided many important clues about higher dimensional physics. This theory with Λ < 0 has non-trivial solutions, such as the BTZ blackhole spacetime [22], which provide an important testing ground for quantum gravity and the AdS/CFT correspondence. Many other types of 3d regular and black hole solutions with a negative cosmological constant have also been found by coupling matter fields to gravity in different ways. However, as proven in [20], there are no d = 3 asymptotically flat static solutions of the EYM equations. The arguments in [20] can easily be generalised for the AAdS case. We notice that for d = 3, the YM equation (8) implies the existence of a first integral w ′ = αr/(σN ) with α an arbitrary real constant. Therefore, assuming AdS 3 asymptotics, w ′ decays asymptotically as 1/r which from (9) implies a divergent value of m(r) as r → ∞. However, similar to the Λ = 0 case [23], this argument does not exclude the existence of nontrivial solutions of the field equations.\nThe (2+1) dimensional case is rather special. Three dimensional gravity has provided many important clues about higher dimensional physics. This theory with Λ < 0 has non-trivial solutions, such as the BTZ blackhole spacetime [22], which provide an important testing ground for quantum gravity and the AdS/CFT correspondence. Many other types of 3d regular and black hole solutions with a negative cosmological constant have also been found by coupling matter fields to gravity in different ways. However, as proven in [20], there are no d = 3 asymptotically flat static solutions of the EYM equations. The arguments in [20] can easily be generalised for the AAdS case. We notice that for d = 3, the YM equation (8) implies the existence of a first integral w ′ = αr/(σN ) with α an arbitrary real constant. Therefore, assuming AdS 3 asymptotics, w ′ decays asymptotically as 1/r which from (9) implies a divergent value of m(r) as r → ∞. However, similar to the Λ = 0 case [23], this argument does not exclude the existence of nontrivial solutions of the field equations.\nHere we should remark that since for d = 3 we are dealing with SO(d − 1) = SO(2), the gauge group is Abelian and we recover Einstein-Maxwell theory with a negative cosmological constant, whose solutions are known in the literature. The corresponding solution with a vanishing electric field was found by Hirschmann and Welch [24] and has a line element 4\nHere we should remark that since for d = 3 we are dealing with SO(d − 1) = SO(2), the gauge group is Abelian and we recover Einstein-Maxwell theory with a negative cosmological constant, whose solutions are known in the literature. The corresponding solution with a vanishing electric field was found by Hirschmann and Welch [24] and has a line element 4\nthe magnetic potential being\nthe magnetic potential being\nwith w 0 , c and M arbitrary real constants, the BTZ metric being recovered for c = 0 (see also [25,26] for more details on this solution).\nwith w 0 , c and M arbitrary real constants, the BTZ metric being recovered for c = 0 (see also [25,26] for more details on this solution).\nOne can see that although the quasilocal mass defined in the usual way diverges as r → ∞, the metric still approaches the AdS 3 background. However, a similar problem appears for other d = 3 AdS solution e.g. for the electrically charged BTZ black hole [22], or for a self-interacting scalar field minimally coupled to gravity [27], in which cases it was possible to find a suitable mass definition. We expect the formalism developed in those cases to work also for the Hirschmann-Welch solution (14)- (15), but this lies outside the scope of the present work.\nOne can see that although the quasilocal mass defined in the usual way diverges as r → ∞, the metric still approaches the AdS 3 background. However, a similar problem appears for other d = 3 AdS solution e.g. for the electrically charged BTZ black hole [22], or for a self-interacting scalar field minimally coupled to gravity [27], in which cases it was possible to find a suitable mass definition. We expect the formalism developed in those cases to work also for the Hirschmann-Welch solution (14)- (15), but this lies outside the scope of the present work."},{"title":"d = 4","content":""},{"title":"d = 4","content":"Four dimensional black hole solutions of the equations (8)- (10) have been found in [3], the globally regular counterparts being discussed in [4]. Differing from the asymptotically flat case, for Λ < 0 there is a continuum of solutions in terms of the adjustable shooting parameter that specifies the initial conditions at the origin or at the event horizon. As a new feature, the asymptotic value of the gauge function w 0 is arbitrary. The spectrum has a finite number of continuous branches, depending on the value of Λ. When the parameter Λ approaches zero, an already-existing branch of solutions collapses to a single point in the moduli space. At the same time new branches of solutions emerge. A fractal structure in the moduli space has been noticed [28]. There are also nontrivial solutions stable against spherically symmetric linear perturbations, corresponding to stable configurations. The solutions are classified by non-Abelian magnetic charge and the ADM mass.\nFour dimensional black hole solutions of the equations (8)- (10) have been found in [3], the globally regular counterparts being discussed in [4]. Differing from the asymptotically flat case, for Λ < 0 there is a continuum of solutions in terms of the adjustable shooting parameter that specifies the initial conditions at the origin or at the event horizon. As a new feature, the asymptotic value of the gauge function w 0 is arbitrary. The spectrum has a finite number of continuous branches, depending on the value of Λ. When the parameter Λ approaches zero, an already-existing branch of solutions collapses to a single point in the moduli space. At the same time new branches of solutions emerge. A fractal structure in the moduli space has been noticed [28]. There are also nontrivial solutions stable against spherically symmetric linear perturbations, corresponding to stable configurations. The solutions are classified by non-Abelian magnetic charge and the ADM mass.\nNote also that the d = 4 EYM solutions with a negative cosmological constant Λ = −3/τ 1 have some relevance in AdS/CFT context. As proven in [29], for this value of the cosmological constant, an arbitrary solution (g µν , A (a) µ ) of the four dimensional EYM equations gives a solution of the equations of motion of the d = 11 supergravity. Based on this observation, an exact BPS-type EYM solution has been constructed in [10]. However, similar to some supersymmetric solutions in Einstein-Maxwell theory with Λ < 0, this Λ = −3/τ 1 configuration presents a naked central singularity.\nNote also that the d = 4 EYM solutions with a negative cosmological constant Λ = −3/τ 1 have some relevance in AdS/CFT context. As proven in [29], for this value of the cosmological constant, an arbitrary solution (g µν , A (a) µ ) of the four dimensional EYM equations gives a solution of the equations of motion of the d = 11 supergravity. Based on this observation, an exact BPS-type EYM solution has been constructed in [10]. However, similar to some supersymmetric solutions in Einstein-Maxwell theory with Λ < 0, this Λ = −3/τ 1 configuration presents a naked central singularity."},{"title":"d ≥ 5","content":""},{"title":"d ≥ 5","content":"As discovered by Coleman [30] and Deser [31], there are no flat space static solutions of the YM equations, except for d = 5. However, the inclusion of gravity may change this picture, as seen from the famous d = 4 asymptotically flat Bartnik-McKinnon solutions [16]. In this case, the repulsive YM force is compensated by the attractive character of the gravity, and as a result we find both regular and black hole unstable configurations (see [32] for a fairly recent survey). As found in [12] the d = 5 particle-like solutions are destroyed by gravity, their mass diverging logarithmically, while w(r) presents an infinite number of nodes. The AAdS couterparts of the d = 5 asymptotically flat solutions are discussed in [11]. Although approaching asymptotically the AdS 5 background, the mass of these configurations also diverges logarithmically.\nAs discovered by Coleman [30] and Deser [31], there are no flat space static solutions of the YM equations, except for d = 5. However, the inclusion of gravity may change this picture, as seen from the famous d = 4 asymptotically flat Bartnik-McKinnon solutions [16]. In this case, the repulsive YM force is compensated by the attractive character of the gravity, and as a result we find both regular and black hole unstable configurations (see [32] for a fairly recent survey). As found in [12] the d = 5 particle-like solutions are destroyed by gravity, their mass diverging logarithmically, while w(r) presents an infinite number of nodes. The AAdS couterparts of the d = 5 asymptotically flat solutions are discussed in [11]. Although approaching asymptotically the AdS 5 background, the mass of these configurations also diverges logarithmically.\nAs conjectured by several authors, this result extends to higher dimensions. Following the approach in [11], we prove in Appendix A the nonexistence of asymptotically flat or AdS solutions with a finite mass in a F (2) EYM model given by (2) for any spacetime dimension d ≥ 5 (see also the discussion in Section 3.3). Therefore, the d−dimensional Schwarzschild-AdS configuration is the only finite mass solution of the equations (8)-(10) and a simple version of the no hair theorem seems to hold for the F (2) (usual) EYM system in d > 4.\nAs conjectured by several authors, this result extends to higher dimensions. Following the approach in [11], we prove in Appendix A the nonexistence of asymptotically flat or AdS solutions with a finite mass in a F (2) EYM model given by (2) for any spacetime dimension d ≥ 5 (see also the discussion in Section 3.3). Therefore, the d−dimensional Schwarzschild-AdS configuration is the only finite mass solution of the equations (8)-(10) and a simple version of the no hair theorem seems to hold for the F (2) (usual) EYM system in d > 4.\nWe should remark that in deriving this result we assumed implicitely that m(r), σ(r), w(r) are smooth functions approaching finite values as r → ∞. A divergent asymptotic value of m(r) invalidates the proof presented in Appendix A and also the virial argumnets in Section 3.3. Therefore we cannot exclude the existence of spherically symmetric, nontrivial solutions of the field equations for any d > 4. However, the mass of these solutions generically diverges, although the spacetimes are still AAdS. The work of Ref. [11] presents an extensive discussion of such AAdS solutions for d = 5. Both regular and black hole solutions exist in d = 5 for compact intervals of the parameter that specifies the initial conditions at the origin or at the event horizon. Differing from the Λ = 0 case, the gauge field function w(r) does not oscillate between 1 and −1 but approaches asymptotically some finite value w 0 , the node number being finite. The masses of these solutions behave asymptotically as (w 2 0 − 1) log r, with all w 0 = ±1 solutions corresponding to pure gauge configurations.\nWe should remark that in deriving this result we assumed implicitely that m(r), σ(r), w(r) are smooth functions approaching finite values as r → ∞. A divergent asymptotic value of m(r) invalidates the proof presented in Appendix A and also the virial argumnets in Section 3.3. Therefore we cannot exclude the existence of spherically symmetric, nontrivial solutions of the field equations for any d > 4. However, the mass of these solutions generically diverges, although the spacetimes are still AAdS. The work of Ref. [11] presents an extensive discussion of such AAdS solutions for d = 5. Both regular and black hole solutions exist in d = 5 for compact intervals of the parameter that specifies the initial conditions at the origin or at the event horizon. Differing from the Λ = 0 case, the gauge field function w(r) does not oscillate between 1 and −1 but approaches asymptotically some finite value w 0 , the node number being finite. The masses of these solutions behave asymptotically as (w 2 0 − 1) log r, with all w 0 = ±1 solutions corresponding to pure gauge configurations.\nThe results we found by solving numerically the equations (8)-(10), for d = 6, 7, 8 and several negative values of Λ confirm that this is a generic behaviour for d > 4. The corresponding boundary conditions at the origin (or event horizon) are found by taking P = 1 in relations (26), (27) given in Section 3. Except for a divergent value of m(r) as r → ∞, according to\nThe results we found by solving numerically the equations (8)-(10), for d = 6, 7, 8 and several negative values of Λ confirm that this is a generic behaviour for d > 4. The corresponding boundary conditions at the origin (or event horizon) are found by taking P = 1 in relations (26), (27) given in Section 3. Except for a divergent value of m(r) as r → ∞, according to\nthe properties of these solutions are very similar to the the more familiar d = 4 case. For d > 5, the asymptotic value w 0 of the gauge field function w is also arbitrary, being fixed by the initial parameters w ′′ (0) or w(r h ) respectively, w 0 = ±1 corresponding to pure gauge configurations. Solutions for a compact interval of these parameters were found to exist, the general structure being Λ-dependent. Solutions with nodes in w(r) were also found. Typical d = 6 configurations with a regular origin are presented in Figure 1, for Λ = −0.01 and three different values of b = −w ′′ (0)/2. One can see that the mass function diverges linearly while σ(r) and w(r) asymptotically approach some finite values. In Figure 2 we plot the parameters M 0 (appearing in (16), which in Section 4.3 we argue that it can be taken as the renormalised mass of the solutions), w 0 , the value σ 0 of the metric function σ at the origin and the minimal value N m of the metric function N as a function of b for a family of d = 6 AAdS solutions with Λ = −1. This branch ends for some finite value of b, where σ(0) → 0. Black hole solutions have been found as well, presenting the same general features. Here also we find a continuum of solutions with arbitrary values of w 0 , the relevant parameter being the value of the gauge potential at the event horizon. Similar to d = 4, solutions appear to exist for any value of the event horizon radius. The drawback of the solutions in d > 5 described above is that their ADM masses are divergent, making their physical significance obscure (see, however, the discussion in Appendix B).\nthe properties of these solutions are very similar to the the more familiar d = 4 case. For d > 5, the asymptotic value w 0 of the gauge field function w is also arbitrary, being fixed by the initial parameters w ′′ (0) or w(r h ) respectively, w 0 = ±1 corresponding to pure gauge configurations. Solutions for a compact interval of these parameters were found to exist, the general structure being Λ-dependent. Solutions with nodes in w(r) were also found. Typical d = 6 configurations with a regular origin are presented in Figure 1, for Λ = −0.01 and three different values of b = −w ′′ (0)/2. One can see that the mass function diverges linearly while σ(r) and w(r) asymptotically approach some finite values. In Figure 2 we plot the parameters M 0 (appearing in (16), which in Section 4.3 we argue that it can be taken as the renormalised mass of the solutions), w 0 , the value σ 0 of the metric function σ at the origin and the minimal value N m of the metric function N as a function of b for a family of d = 6 AAdS solutions with Λ = −1. This branch ends for some finite value of b, where σ(0) → 0. Black hole solutions have been found as well, presenting the same general features. Here also we find a continuum of solutions with arbitrary values of w 0 , the relevant parameter being the value of the gauge potential at the event horizon. Similar to d = 4, solutions appear to exist for any value of the event horizon radius. The drawback of the solutions in d > 5 described above is that their ADM masses are divergent, making their physical significance obscure (see, however, the discussion in Appendix B).\nOne may hope to find a different picture by including some other matter field in the action (1). Such fields should interact with the YM sector so as to compensate for the scaling behaviour of the non Abelian fields. This excludes the dilaton field, as it can be proven that the latter does not change this nonexistence result. Note that d = 5 finite mass spherically symmetric gravitating non Abelian solutions with a Liouville-type dilaton potential are known to exist [35]; however these solutions are asymptotically neither flat nor AdS.\nOne may hope to find a different picture by including some other matter field in the action (1). Such fields should interact with the YM sector so as to compensate for the scaling behaviour of the non Abelian fields. This excludes the dilaton field, as it can be proven that the latter does not change this nonexistence result. Note that d = 5 finite mass spherically symmetric gravitating non Abelian solutions with a Liouville-type dilaton potential are known to exist [35]; however these solutions are asymptotically neither flat nor AdS.\nIn the next Section, we will remedy this problem of nonexistence by adding higher order YM curvature terms F (2p), (p > 1), to (2). The main role of these terms is to alter the scaling properties of the action density in (1). But logically, such a role can be played also by altering the gravitational part of (1), through the introduction of higher order (in the Riemann curvature) terms R(p), (p > 1). R(p) here denotes the generalised Ricci scalar constructed from the 2p-form antisymmetrised p-fold product of Riemann tensor 2-forms, the p = 1 member of which gives the Einstein-Hilbert action, the p = 2, the Gauss-Bonnet, etc. Here we have eschewed the possibility of employing additional R(p) terms instead of, or together with F (2p) YM terms, because in the present work we exclude the participation of fields other than gravitational and Yang-Mills. In particular, the exclusion of the dilaton renders the usefulness of higher order (Gauss-Bonnet) gravities trivial from a practical point of view.\nIn the next Section, we will remedy this problem of nonexistence by adding higher order YM curvature terms F (2p), (p > 1), to (2). The main role of these terms is to alter the scaling properties of the action density in (1). But logically, such a role can be played also by altering the gravitational part of (1), through the introduction of higher order (in the Riemann curvature) terms R(p), (p > 1). R(p) here denotes the generalised Ricci scalar constructed from the 2p-form antisymmetrised p-fold product of Riemann tensor 2-forms, the p = 1 member of which gives the Einstein-Hilbert action, the p = 2, the Gauss-Bonnet, etc. Here we have eschewed the possibility of employing additional R(p) terms instead of, or together with F (2p) YM terms, because in the present work we exclude the participation of fields other than gravitational and Yang-Mills. In particular, the exclusion of the dilaton renders the usefulness of higher order (Gauss-Bonnet) gravities trivial from a practical point of view.\nTo see this, consider the situation where a term scaling as L −2p is needed, i.e. that the term\nTo see this, consider the situation where a term scaling as L −2p is needed, i.e. that the term\nmust be added to the density in (1). Using the metric Ansatz (5), and discarding purely boundary terms in the residual one dimensional (spherically symmetric) Lagrangean, the term L (p) to be added is\nmust be added to the density in (1). Using the metric Ansatz (5), and discarding purely boundary terms in the residual one dimensional (spherically symmetric) Lagrangean, the term L (p) to be added is\nin spacetime dimension d, which vanishes identically in dimensions up to d = 2p. This is because in the given dimensions the p-th member of the gravitational hierarchy becomes a total divergence. But for given d it is necessary to have 2p ≥ d, whence (17) trivialises. The situation would be quite different if the dilaton field were included together with the higher order gravitational terms, preventing the p-gravity density becoming a total divergence. But this option is excluded here and we opt to the exclusive use of higher order p-YM densities.\nin spacetime dimension d, which vanishes identically in dimensions up to d = 2p. This is because in the given dimensions the p-th member of the gravitational hierarchy becomes a total divergence. But for given d it is necessary to have 2p ≥ d, whence (17) trivialises. The situation would be quite different if the dilaton field were included together with the higher order gravitational terms, preventing the p-gravity density becoming a total divergence. But this option is excluded here and we opt to the exclusive use of higher order p-YM densities.\nYet other fields occurring in supergravities might be considered, e.g. Kalb-Ramond, or totally antisymmetric tensor fields. But these being Abelian, their effect would be felt only in given dimensions, or, subject to much less stringent symmetries than the spherical. This also is not a flexible option, so we restrict our attention to p-YM densities only.\nYet other fields occurring in supergravities might be considered, e.g. Kalb-Ramond, or totally antisymmetric tensor fields. But these being Abelian, their effect would be felt only in given dimensions, or, subject to much less stringent symmetries than the spherical. This also is not a flexible option, so we restrict our attention to p-YM densities only."},{"title":"Nonabelian hierarchies 3.1 The Lagrangean and field equations","content":""},{"title":"Nonabelian hierarchies 3.1 The Lagrangean and field equations","content":"Since finite mass spherically symmetric solutions play a central role in AdS/CFT, it is desirable that the nonexistence result presented in Section 2.5 be circumvented.\nSince finite mass spherically symmetric solutions play a central role in AdS/CFT, it is desirable that the nonexistence result presented in Section 2.5 be circumvented.\nA simple way to circumvent these arguments and to find nontrivial solutions is to to modify the matter Lagrangean by adding higher order terms in the YM hierarchy. As noted in footnote 1 , these are constructed exclusively from YM curvature 2p-forms. For Λ = 0, asymptotically flat, finite energy solutions of this modified EYM system are constructed in [13,14].\nA simple way to circumvent these arguments and to find nontrivial solutions is to to modify the matter Lagrangean by adding higher order terms in the YM hierarchy. As noted in footnote 1 , these are constructed exclusively from YM curvature 2p-forms. For Λ = 0, asymptotically flat, finite energy solutions of this modified EYM system are constructed in [13,14].\nSuch terms as we propose to add are predicted by string theory, and hence provide a link with the AdS/CFT correspondence too. But here we are guided predominantly by symmetry considerations and do not claim to be employing terms strictly following from superstring theory. The situation concerning higher order YM curvature terms in the string theory effective action is complex and as yet not fully resolved. While YM terms up to F 4 arise from (the non Abelian version of) the Born-Infeld action [36], it appears that this approach does not yield all the F 6 terms [37]. Terms of order F 6 and higher can also be obtained by employing the constraints of (maximal) supersymmetry [38]. The results of the various approaches are not identical.\nSuch terms as we propose to add are predicted by string theory, and hence provide a link with the AdS/CFT correspondence too. But here we are guided predominantly by symmetry considerations and do not claim to be employing terms strictly following from superstring theory. The situation concerning higher order YM curvature terms in the string theory effective action is complex and as yet not fully resolved. While YM terms up to F 4 arise from (the non Abelian version of) the Born-Infeld action [36], it appears that this approach does not yield all the F 6 terms [37]. Terms of order F 6 and higher can also be obtained by employing the constraints of (maximal) supersymmetry [38]. The results of the various approaches are not identical.\nThe definition we use for superposed YM hierarchy is\nThe definition we use for superposed YM hierarchy is\nwhere F (2p) is the 2p-form p-fold totally antisymmetrised product of the SO(d) YM curvature 2-form F (2)\nwhere F (2p) is the 2p-form p-fold totally antisymmetrised product of the SO(d) YM curvature 2-form F (2)\nEven though the 2p-form (19) is dual to a total divergence, namely the divergence of the corresponding Chern-Simons form, the density (18) is never a total divergence since it is the square of one. But the 2p-form (19) vanishes by (anti)symmetry for d < 2p so that the upper limit in the summation in (18) is P = d 2 for even d and P = d−1 2 for odd d. We define the p-stress tensor pertaining to each term in (18) as\nEven though the 2p-form (19) is dual to a total divergence, namely the divergence of the corresponding Chern-Simons form, the density (18) is never a total divergence since it is the square of one. But the 2p-form (19) vanishes by (anti)symmetry for d < 2p so that the upper limit in the summation in (18) is P = d 2 for even d and P = d−1 2 for odd d. We define the p-stress tensor pertaining to each term in (18) as\nFor the particular spherically symmetric ansatz considered in Section 2, we express the reduced YM Lagrangean arising from (18) as\nFor the particular spherically symmetric ansatz considered in Section 2, we express the reduced YM Lagrangean arising from (18) as\nhaving used the shorthand notation\nhaving used the shorthand notation\nFor this general ansatz, we find the field equations\nFor this general ansatz, we find the field equations\nwhich can also be derived from the reduced action\nwhich can also be derived from the reduced action\nFor τ i = 0, i > 2 the equations (8)-(10) are recovered. Note also the σ equation decouples and can be treated separately.\nFor τ i = 0, i > 2 the equations (8)-(10) are recovered. Note also the σ equation decouples and can be treated separately."},{"title":"Boundary conditions","content":""},{"title":"Boundary conditions","content":"The asymptotic solutions to these equations can be systematically constructed in both regions, near the origin (or event horizon) and for r ≫ 1. The corresponding expansion as r → 0 is\nThe asymptotic solutions to these equations can be systematically constructed in both regions, near the origin (or event horizon) and for r ≫ 1. The corresponding expansion as r → 0 is\nand contains one essential parameter b (the value of σ(0) ≡ σ 0 can be fixed by rescaling the time coordinate). For black hole configurations with a regular, nonextremal event horizon at r = r h , the expression near the event horizon is\nand contains one essential parameter b (the value of σ(0) ≡ σ 0 can be fixed by rescaling the time coordinate). For black hole configurations with a regular, nonextremal event horizon at r = r h , the expression near the event horizon is\nthe value of the gauge field on the event horizon being the esential parameter. Here the obvious condition N ′ (r h ) > 0 imposes some limits on the event horizon radius as a function of w h for given (τ i , Λ).\nthe value of the gauge field on the event horizon being the esential parameter. Here the obvious condition N ′ (r h ) > 0 imposes some limits on the event horizon radius as a function of w h for given (τ i , Λ).\nSince the field equations are invariant under w → −w, one can take w(0) = 1 and w(r h ) > 0 without any loss of generality.\nSince the field equations are invariant under w → −w, one can take w(0) = 1 and w(r h ) > 0 without any loss of generality.\nFor r ≫ 1 we find for both regular and black hole solutions\nFor r ≫ 1 we find for both regular and black hole solutions\nThese boundary conditions are also shared by the asymptotically flat solutions (with a different decay of the mass function m(r), however), w = ±1 being again the only allowed values of the gauge function as r → ∞. Therefore, we expect to find a qualitatively similar picture in both cases. We will find in Section 5 that the constant M in the above relations is the ADM mass up to a d-dependent factor. However, in the discussion of numerical solutions we will refer to M to as the mass of the solutions.\nThese boundary conditions are also shared by the asymptotically flat solutions (with a different decay of the mass function m(r), however), w = ±1 being again the only allowed values of the gauge function as r → ∞. Therefore, we expect to find a qualitatively similar picture in both cases. We will find in Section 5 that the constant M in the above relations is the ADM mass up to a d-dependent factor. However, in the discussion of numerical solutions we will refer to M to as the mass of the solutions."},{"title":"Further relations","content":""},{"title":"Further relations","content":"The form (25) of the reduced action allow to derive an useful virial relation. To this end, we use the scaling technique proposed in [33,34] for the case of spherically symmetric gravitating systems. Let us assume the existence of a globally regular solution m(r), σ(r), w(r) of the field equations (24), with suitable boundary conditions at the origin and at infinity. Then each member of the 1-parameter family\nThe form (25) of the reduced action allow to derive an useful virial relation. To this end, we use the scaling technique proposed in [33,34] for the case of spherically symmetric gravitating systems. Let us assume the existence of a globally regular solution m(r), σ(r), w(r) of the field equations (24), with suitable boundary conditions at the origin and at infinity. Then each member of the 1-parameter family\nassumes the same boundary values at r = 0 and r = ∞, and the action S λ ≡ S[m λ , σ λ , w λ ] must have a critical point at λ = 1, i.e. [dS/dλ] λ=1 = 0. Therefore we find the following virial relation satisfied by the finite energy solutions of the field equations (note that following [33], it is possible to write a similar relation for black hole configurations, also)\nassumes the same boundary values at r = 0 and r = ∞, and the action S λ ≡ S[m λ , σ λ , w λ ] must have a critical point at λ = 1, i.e. [dS/dλ] λ=1 = 0. Therefore we find the following virial relation satisfied by the finite energy solutions of the field equations (note that following [33], it is possible to write a similar relation for black hole configurations, also)\n+2p\n+2p\nFor p = 1, i.e. a F 2 -theory, the above relation reads\nFor p = 1, i.e. a F 2 -theory, the above relation reads\nwhich clearly shows that no nontrivial gravitating solution with finite mass exists for d > 4, since all terms in the integrant are strictly positive quantities. Therefore it becomes obvious that new terms in the YM-hierarchy should be introduced as the spacetime dimension increases. For a given d, the relation P > [(d + 1)/4] should be satisfied. As it happens, to go to 5 ≤ d < 9 it is necessary to include at least the second member of the YM hierarchy to provide the requisite scaling (similarly for 9 ≤ d < 13 it is necessary to include the thrid member of the YM hierarchy). In practice we add only the lowest order such term necessary.\nwhich clearly shows that no nontrivial gravitating solution with finite mass exists for d > 4, since all terms in the integrant are strictly positive quantities. Therefore it becomes obvious that new terms in the YM-hierarchy should be introduced as the spacetime dimension increases. For a given d, the relation P > [(d + 1)/4] should be satisfied. As it happens, to go to 5 ≤ d < 9 it is necessary to include at least the second member of the YM hierarchy to provide the requisite scaling (similarly for 9 ≤ d < 13 it is necessary to include the thrid member of the YM hierarchy). In practice we add only the lowest order such term necessary.\nWe mention here also the Hawking temperature expression of the black hole solutions. For the line element (5), if we treat t as complex, then its imaginary part is a coordinate for a nonsingular Euclidean submanifold iff it is periodic with period\nWe mention here also the Hawking temperature expression of the black hole solutions. For the line element (5), if we treat t as complex, then its imaginary part is a coordinate for a nonsingular Euclidean submanifold iff it is periodic with period\nThen continuous Euclidean Green functions must have this period, so by standard arguments the Hawking temperature is (with k B =h = 1)\nThen continuous Euclidean Green functions must have this period, so by standard arguments the Hawking temperature is (with k B =h = 1)\nThus the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum black hole of equal horizon area.\nThus the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum black hole of equal horizon area.\nIn the presence of higher oder terms in the YM action, dimensionless quantities are obtained by rescaling\nIn the presence of higher oder terms in the YM action, dimensionless quantities are obtained by rescaling\nThis reveals the existence of one fundamental parameter which gives the strength of the gravitational interaction\nThis reveals the existence of one fundamental parameter which gives the strength of the gravitational interaction\nand P − 2 independent coupling constants\nand P − 2 independent coupling constants\nwith k = 3, P − 2 (i.e. no such constants appear in a p = 2 system).\nwith k = 3, P − 2 (i.e. no such constants appear in a p = 2 system).\nFor the F (2) + F (4) systems in d = 6, 7, 8 considered in [13], there exist gravity decoupling solutions at α = 0, from which the gravitating solutions branch out to a maximum value of α max and then decrease. The second limit of α → 0 also exists. For the same system in d = 5 studied in [14] on the other hand, there exists no gravity decoupling limit but nonetheless the solution branches out from α = 0 by employing a scaling procedure [19], and again α increases to a value α max and decreases. But in this case it does not reach the α → 0 limit. Rather it stops at a new critical point α c around which it oscillates [14], which is a new type of critical point identified in [19] and named a 'conical' fixed point.\nFor the F (2) + F (4) systems in d = 6, 7, 8 considered in [13], there exist gravity decoupling solutions at α = 0, from which the gravitating solutions branch out to a maximum value of α max and then decrease. The second limit of α → 0 also exists. For the same system in d = 5 studied in [14] on the other hand, there exists no gravity decoupling limit but nonetheless the solution branches out from α = 0 by employing a scaling procedure [19], and again α increases to a value α max and decreases. But in this case it does not reach the α → 0 limit. Rather it stops at a new critical point α c around which it oscillates [14], which is a new type of critical point identified in [19] and named a 'conical' fixed point."},{"title":"Numerical solutions","content":""},{"title":"Numerical solutions","content":"In the present work, we restrict our attention to the simplest nontrivial cases with only the two terms p = 1 and 2 in the YM hierarchy. However, we have obtained some numerical results also for a P = 3 hierarchy with β 1 = 1, which will be briefly mentioned. For d = 5, the solutions we found have some special features which will be discussed separately.\nIn the present work, we restrict our attention to the simplest nontrivial cases with only the two terms p = 1 and 2 in the YM hierarchy. However, we have obtained some numerical results also for a P = 3 hierarchy with β 1 = 1, which will be briefly mentioned. For d = 5, the solutions we found have some special features which will be discussed separately.\nBoth regular and black hole solutions of the EYM-hierarchy equations appear to exist for any value of Λ. Given (α, d, Λ), AAdS solutions may exist for discrete set of shooting parameters b and w h respectively. We follow the usual approach and, by using a standard ordinary differential equation solver, we evaluate the initial conditions at r = 10 −5 (or r h + 10 −5 ) for global tolerance 10 −14 , adjusting for shooting parameters and integrating towards r → ∞, and looking for AAdS solutions with a finite mass.\nBoth regular and black hole solutions of the EYM-hierarchy equations appear to exist for any value of Λ. Given (α, d, Λ), AAdS solutions may exist for discrete set of shooting parameters b and w h respectively. We follow the usual approach and, by using a standard ordinary differential equation solver, we evaluate the initial conditions at r = 10 −5 (or r h + 10 −5 ) for global tolerance 10 −14 , adjusting for shooting parameters and integrating towards r → ∞, and looking for AAdS solutions with a finite mass.\nSimilar to the p = 1, d = 4 asymptotically flat case, it can be proven that for all AAdS (or Λ = 0) solutions, w(r) is confined within the strip |w(r)| < 1. This can proven as follows: we suppose the existence of solutions with w(r) > 1 for some interval of r. Therefore w must develop a maximum for some r 0 , w ′ (r 0 ) = 0 and w(r 0 ) > 1 with w ′′ (r 0 ) < 0. However, the equations (24) imply that in the region w > 1 the only extremum can be a minimum. Therefore the condition |w(r)| < 1 is always fullfiled. As a general feature, all solutions discussed in the rest of this section present only one node in the gauge function w(r). Similar to the Λ = 0 case, we could not find multinode solutions 5 .\nSimilar to the p = 1, d = 4 asymptotically flat case, it can be proven that for all AAdS (or Λ = 0) solutions, w(r) is confined within the strip |w(r)| < 1. This can proven as follows: we suppose the existence of solutions with w(r) > 1 for some interval of r. Therefore w must develop a maximum for some r 0 , w ′ (r 0 ) = 0 and w(r 0 ) > 1 with w ′′ (r 0 ) < 0. However, the equations (24) imply that in the region w > 1 the only extremum can be a minimum. Therefore the condition |w(r)| < 1 is always fullfiled. As a general feature, all solutions discussed in the rest of this section present only one node in the gauge function w(r). Similar to the Λ = 0 case, we could not find multinode solutions 5 .\nThe absence of multinode solutions in this F (2) + F (4) model with Λ = 0, in the relevant dimensions 5 ≤ d ≤ 8 is analytically explained in [19]. We expect that the relevant fixed point analysis yields qualitatively similar results also for |Λ| > 0. This is borne out by our numerical results.\nThe absence of multinode solutions in this F (2) + F (4) model with Λ = 0, in the relevant dimensions 5 ≤ d ≤ 8 is analytically explained in [19]. We expect that the relevant fixed point analysis yields qualitatively similar results also for |Λ| > 0. This is borne out by our numerical results.\nFor any regular solution, the metric functions m(r) and σ(r) always increase monotonically with growing r from m(0) = 0 and σ(0) = σ 0 at the origin to m(∞) = M and σ(∞) = 1, respectively. The gauge function always interpolates between w(0) = 1 and w(∞) = −1 without any local extrema. For black hole configurations, the behaviour of the functions m(r), σ(r) and w(r) is similar to that for regular solutions. The gauge potential w(r) starts from some finite value 0 < w(r h ) < 1 at the horizon and monotonically approaches −1 at infinity. The metric functions m(r) and σ(r) increase also monotonically with r. In the asymptotic region, the geometry corresponds to a Schwarzschild-AdS solution. However, although these solutions are static and have vanishing YM charges (w 2 (∞) = 1), they are different from the Schwarzschild-AdS black hole, and therefore are not fully characterised by the mass-parameter M .\nFor any regular solution, the metric functions m(r) and σ(r) always increase monotonically with growing r from m(0) = 0 and σ(0) = σ 0 at the origin to m(∞) = M and σ(∞) = 1, respectively. The gauge function always interpolates between w(0) = 1 and w(∞) = −1 without any local extrema. For black hole configurations, the behaviour of the functions m(r), σ(r) and w(r) is similar to that for regular solutions. The gauge potential w(r) starts from some finite value 0 < w(r h ) < 1 at the horizon and monotonically approaches −1 at infinity. The metric functions m(r) and σ(r) increase also monotonically with r. In the asymptotic region, the geometry corresponds to a Schwarzschild-AdS solution. However, although these solutions are static and have vanishing YM charges (w 2 (∞) = 1), they are different from the Schwarzschild-AdS black hole, and therefore are not fully characterised by the mass-parameter M ."},{"title":"Regular solutions d = 5","content":""},{"title":"Regular solutions d = 5","content":"As α 2 → 0, the YM equations present a nontrivial, finite energy solution in a fixed AdS background. This nongravitating configuration approaches in the Λ = 0 limit the YM instanton in four dimensional flat space [39].\nAs α 2 → 0, the YM equations present a nontrivial, finite energy solution in a fixed AdS background. This nongravitating configuration approaches in the Λ = 0 limit the YM instanton in four dimensional flat space [39].\nWhen α 2 increases, this solution gets deformed by gravity and the mass M decreases. At the same time, both the value σ(0) and the minimal value N m of the function N (r) decrease, as indicated in Figure 3. This branch of solutions exists up to a maximal value α max of the parameter α, which is smaller than the corresponding value in the asymptotically flat case [14]. For example, we find numerically α 2 max ≈ 0.3445 for Λ = −0.2 while the corresponding value for Λ = −0.01 is α 2 max ≈ 0.5322. (without a cosmological term, this branch extends up to α 2 max ≈ 0.5648.) Similar to the EYM theory with Λ = 0 [14], we found always another branch of solutions on the interval α 2 ∈ [α 2 cr(1) , α 2 max ] with α 2 cr(1) depending again on the value of Λ (e.g. α 2 cr(1) ≈ 0.2876 for Λ = −0.2). On this second branch of solutions, both σ(0) and N m continue to decrease but stay finite. However, a third branch of solutions exists for α 2 ∈ [α 2 cr(1) , α 2 cr(2) ] , on which the two quantities decrease further. A fourth branch of solutions has also been found, with a corresponding α 2 cr(3) close to α 2 cr (2) . Further branches of solutions, exhibiting more oscillations very likely exist but their study is a difficult numerical problem. Along this succession of branches, the main observation is that the value σ(0) decreases much faster than that of N m as illustrated in Figure 3. Also, the mass parameters do not increase significantly along these secondary branches. However, the shooting parameter b increases to very large values. The pattern strongly suggests that after a finite (or more likely infinite) number of oscilations of σ(0), the solution terminates into a singular solution with σ(0) = 0 and a finite value of N (0). This is the behaviour observed in [14] for the EYM theory with Λ = 0. The inclusion of a negative cosmological constant does not seem to qualitatively change the properties of the system, but leads to different values of the critical parameters. As in the Λ = 0 case, the dominant term at the gravity decoupling limit α → 0 is the F (2) term, the energy being given by the action of the usual instanton [39], while as α → α cr the dominant term is F (4). The mechanism for this effect is explained in [19] (for Λ = 0) and is supported by our numerical results here. The typical d = 5 globally regular solutions look very similar to the d = 6, 8 profiles presented in Figures 4, 5.\nWhen α 2 increases, this solution gets deformed by gravity and the mass M decreases. At the same time, both the value σ(0) and the minimal value N m of the function N (r) decrease, as indicated in Figure 3. This branch of solutions exists up to a maximal value α max of the parameter α, which is smaller than the corresponding value in the asymptotically flat case [14]. For example, we find numerically α 2 max ≈ 0.3445 for Λ = −0.2 while the corresponding value for Λ = −0.01 is α 2 max ≈ 0.5322. (without a cosmological term, this branch extends up to α 2 max ≈ 0.5648.) Similar to the EYM theory with Λ = 0 [14], we found always another branch of solutions on the interval α 2 ∈ [α 2 cr(1) , α 2 max ] with α 2 cr(1) depending again on the value of Λ (e.g. α 2 cr(1) ≈ 0.2876 for Λ = −0.2). On this second branch of solutions, both σ(0) and N m continue to decrease but stay finite. However, a third branch of solutions exists for α 2 ∈ [α 2 cr(1) , α 2 cr(2) ] , on which the two quantities decrease further. A fourth branch of solutions has also been found, with a corresponding α 2 cr(3) close to α 2 cr (2) . Further branches of solutions, exhibiting more oscillations very likely exist but their study is a difficult numerical problem. Along this succession of branches, the main observation is that the value σ(0) decreases much faster than that of N m as illustrated in Figure 3. Also, the mass parameters do not increase significantly along these secondary branches. However, the shooting parameter b increases to very large values. The pattern strongly suggests that after a finite (or more likely infinite) number of oscilations of σ(0), the solution terminates into a singular solution with σ(0) = 0 and a finite value of N (0). This is the behaviour observed in [14] for the EYM theory with Λ = 0. The inclusion of a negative cosmological constant does not seem to qualitatively change the properties of the system, but leads to different values of the critical parameters. As in the Λ = 0 case, the dominant term at the gravity decoupling limit α → 0 is the F (2) term, the energy being given by the action of the usual instanton [39], while as α → α cr the dominant term is F (4). The mechanism for this effect is explained in [19] (for Λ = 0) and is supported by our numerical results here. The typical d = 5 globally regular solutions look very similar to the d = 6, 8 profiles presented in Figures 4, 5."},{"title":"Regular solutions d > 5","content":""},{"title":"Regular solutions d > 5","content":"The solutions in this case resemble again the Λ = 0 situation. In the presence of suitable higher order term in the hierarchy, the YM equations admit finite energy solutions in a fixed AdS d A p = 1, AdS 4 exact solution was found in [40], its higher order generalisation (specifically for p = 2, AdS 8 ) for d > 4 being discussed in [41]. According to the standard arguments, this AdS soliton can be generalised in the presence of gravity, provided that the dimensionless coupling constant α is small enough. Therefore the gravitating solutions exist up to a maximal value α max of the gravitational coupling constant. This value α max for a given d depends on the value of Λ. For example in d = 6, α max (Λ = 0) = 0.12675; α max (Λ = −1) = 0.070422; in 8 dimensions we find α max (Λ = 0) = 0.002193 while α max (Λ = −5) = 6.69 × 10 −4 .\nThe solutions in this case resemble again the Λ = 0 situation. In the presence of suitable higher order term in the hierarchy, the YM equations admit finite energy solutions in a fixed AdS d A p = 1, AdS 4 exact solution was found in [40], its higher order generalisation (specifically for p = 2, AdS 8 ) for d > 4 being discussed in [41]. According to the standard arguments, this AdS soliton can be generalised in the presence of gravity, provided that the dimensionless coupling constant α is small enough. Therefore the gravitating solutions exist up to a maximal value α max of the gravitational coupling constant. This value α max for a given d depends on the value of Λ. For example in d = 6, α max (Λ = 0) = 0.12675; α max (Λ = −1) = 0.070422; in 8 dimensions we find α max (Λ = 0) = 0.002193 while α max (Λ = −5) = 6.69 × 10 −4 .\nWhen α increases, the mass of the gravitating solutions decreases while the function N (r) develops a local minimum N m which becomes deeper while gravity becomes stronger and the value σ(0) decreases from one. At the same time, the value of the shooting parameter b increases with α. Our numerical analysis for d ≤ 10 indicates that a second branch of regular solutions always exists, starting at α max . Along this second branch the values σ(0) and N m decrease monotonically with α, while b and M still increase. The mass of a second branch solution is always larger than the corresponding mass (for the same value of α) on the first branch. For d > 6, the numerical analysis suggests that this second branch persists up to α 2 ≃ 0 and that in this limit σ(0) approaches a very small value. As far as our numerical analysis indicates, the value N m tends to a finite value in this limit so that there occurs no horizon. Therefore two regular solutions seems always to exist for any α < α d .\nWhen α increases, the mass of the gravitating solutions decreases while the function N (r) develops a local minimum N m which becomes deeper while gravity becomes stronger and the value σ(0) decreases from one. At the same time, the value of the shooting parameter b increases with α. Our numerical analysis for d ≤ 10 indicates that a second branch of regular solutions always exists, starting at α max . Along this second branch the values σ(0) and N m decrease monotonically with α, while b and M still increase. The mass of a second branch solution is always larger than the corresponding mass (for the same value of α) on the first branch. For d > 6, the numerical analysis suggests that this second branch persists up to α 2 ≃ 0 and that in this limit σ(0) approaches a very small value. As far as our numerical analysis indicates, the value N m tends to a finite value in this limit so that there occurs no horizon. Therefore two regular solutions seems always to exist for any α < α d .\nThe case d = 6 is special, since the numerical procedure fails to give reliable results for second branch solutions, starting with some α d , whose value is Λ-dependent (for example we found α d (Λ = −1) = 0.0435 while α d (Λ = 0) = 0.0573). The quantity σ(0) reaches a very small value as α → α d . The minimal value of N (r) remains finite so that no horizon is approached. We expect that a different parametrisation of the metric and variables would allow us to continue this second branch to α → 0 in this case, too.\nThe case d = 6 is special, since the numerical procedure fails to give reliable results for second branch solutions, starting with some α d , whose value is Λ-dependent (for example we found α d (Λ = −1) = 0.0435 while α d (Λ = 0) = 0.0573). The quantity σ(0) reaches a very small value as α → α d . The minimal value of N (r) remains finite so that no horizon is approached. We expect that a different parametrisation of the metric and variables would allow us to continue this second branch to α → 0 in this case, too.\nThe behaviours just described qualitatively duplicate those of the Λ = 0 case [13], and are analysed in [19]. Likewise, the solutions are dominated by the F (2) terms in the gravity decoupling limit α → 0, while at the other end (on the second mass-branch) it is the F (4) terms that dominate.\nThe behaviours just described qualitatively duplicate those of the Λ = 0 case [13], and are analysed in [19]. Likewise, the solutions are dominated by the F (2) terms in the gravity decoupling limit α → 0, while at the other end (on the second mass-branch) it is the F (4) terms that dominate.\nTypical The results we found by including the p = 3 term in the YM-hierarchy for d = 9, 10 follows the same pattern. Although the picture gets more complicated by the existence of one more coupling parameter, two branches of solution are always found to exist. We noticed also the existence of a maximal value of α which is (Λ, β 1 ) dependent.\nTypical The results we found by including the p = 3 term in the YM-hierarchy for d = 9, 10 follows the same pattern. Although the picture gets more complicated by the existence of one more coupling parameter, two branches of solution are always found to exist. We noticed also the existence of a maximal value of α which is (Λ, β 1 ) dependent."},{"title":"Black hole solutions d = 5","content":""},{"title":"Black hole solutions d = 5","content":"According to the standard arguments, one can expect to find black hole generalisations for any regular configurations, at least for small values of the horizon radius r h . For completeness we discuss here the basic features of the AAdS black hole solutions.\nAccording to the standard arguments, one can expect to find black hole generalisations for any regular configurations, at least for small values of the horizon radius r h . For completeness we discuss here the basic features of the AAdS black hole solutions.\nAgain, the case d = 5 is special. The properties of these AAdS solutions are rather similar to the five dimensional asymptotically flat black hole configurations discussed in [14]. First, black hole solutions seem to exist for all values of α for which regular solutions could be constructed. Also, solutions exist only for a limited region of the (r h , α) space.\nAgain, the case d = 5 is special. The properties of these AAdS solutions are rather similar to the five dimensional asymptotically flat black hole configurations discussed in [14]. First, black hole solutions seem to exist for all values of α for which regular solutions could be constructed. Also, solutions exist only for a limited region of the (r h , α) space.\nThe typical behaviour of solutions as a function of r h is presented in Figure 9, for a small value of α compared to the maximal value α max of the regular solutions, in which case we notice the existence of only one r h = 0 regular configuration. Starting from this regular solution and increasing the event horizon radius, we find a first branch of solutions which extends to a maximal value r h(max) . As seen in Figures 9, 10, the value of r h(max) depends on Λ, α. The Hawking temperature decreases on this branch, while the mass parameter increases; however, the variation of mass and σ(r h ) is relatively small. Extending backwards in r h , we find a second branch of solutions for r h < r h(max) . This second branch stops at some critical value r h(cr) , where the numerical iteration fails to converge. The value of σ(r h ) on this branch decreases drastically, as shown in Figure 9. Also, the Hawking temperature after initially increasing, strongly decreases for values near r h(cr) , approaching a very small value, while the increase of the total mass is still very small. Similar to the Λ = 0 case [14], higher branches of solutions on which the value σ(r h ) continues to decrease further to zero are likely to exist. However, the extension of these branches in r h will be very small, which makes their study difficult. An approach to this problem with different parametrisation appears to be necessary.\nThe typical behaviour of solutions as a function of r h is presented in Figure 9, for a small value of α compared to the maximal value α max of the regular solutions, in which case we notice the existence of only one r h = 0 regular configuration. Starting from this regular solution and increasing the event horizon radius, we find a first branch of solutions which extends to a maximal value r h(max) . As seen in Figures 9, 10, the value of r h(max) depends on Λ, α. The Hawking temperature decreases on this branch, while the mass parameter increases; however, the variation of mass and σ(r h ) is relatively small. Extending backwards in r h , we find a second branch of solutions for r h < r h(max) . This second branch stops at some critical value r h(cr) , where the numerical iteration fails to converge. The value of σ(r h ) on this branch decreases drastically, as shown in Figure 9. Also, the Hawking temperature after initially increasing, strongly decreases for values near r h(cr) , approaching a very small value, while the increase of the total mass is still very small. Similar to the Λ = 0 case [14], higher branches of solutions on which the value σ(r h ) continues to decrease further to zero are likely to exist. However, the extension of these branches in r h will be very small, which makes their study difficult. An approach to this problem with different parametrisation appears to be necessary.\nHowever we find, that the global picture is changed by considering large enough values of α. In this case, more than one regular configuration exists for a given value of α. This situation is illustrated in Figure 10, for solutions with α 2 = 0.5. Two regular solutions exist for this particular value of α and we find two black hole branches connecting these r h = 0 configurations. Again, the mass of the second branch solutions is always larger than the corresponding mass on the first branch.\nHowever we find, that the global picture is changed by considering large enough values of α. In this case, more than one regular configuration exists for a given value of α. This situation is illustrated in Figure 10, for solutions with α 2 = 0.5. Two regular solutions exist for this particular value of α and we find two black hole branches connecting these r h = 0 configurations. Again, the mass of the second branch solutions is always larger than the corresponding mass on the first branch.\nPreliminary numerical results indicate an even more complicated picture for solutions with α near α max . In this case the configurations combine features of both types of solutions discussed above. Several branches of black hole solutions are found for the same α. These branches start from regular configurations and are possibly disconnected.\nPreliminary numerical results indicate an even more complicated picture for solutions with α near α max . In this case the configurations combine features of both types of solutions discussed above. Several branches of black hole solutions are found for the same α. These branches start from regular configurations and are possibly disconnected."},{"title":"Black hole solutions d > 5","content":""},{"title":"Black hole solutions d > 5","content":"Although predicted in [14], no discussion of the Λ = 0, d > 5 black holes is presented in literature.\nAlthough predicted in [14], no discussion of the Λ = 0, d > 5 black holes is presented in literature.\nAgain, black hole counterparts appear to exist for any regular solution. However, solutions with the right asymptotics are found for a limited region of the (r h , α) space only. We plot in Figures 11-13 some results we found for d = 6, 7 and d = 8. Starting for a given α 0 < α max from a r h = 0 first branch regular solution, we found the existence of a branch of black hole solutions extending up to a maximal value of the event horizon radius r h = r max h . When r h increases, the mass and the Hawking temperature increase while the value σ(r h ) decreases from its value at the regular solution. A second branch of black hole solutions seems to appear always at r max h , extending backwards to a zero event horizon radius. This limiting solution corresponds to the second branch of the regular solution at this value of α = α 0 . Like in those d = 5 cases where there are two regular solutions for a given α, say α 0 , here too these two regular solutions are the r h → 0 limits of the corresponding black hole \"loop\", connecting the two regular solutions. Along this second branch the values σ(r h ) of the metric function σ on the event horizon decrease monotonically with r h , while the Hawking temperature strongly increases. The mass of the solution of the second branch is larger than the corresponding one on the first branch, as illustrated by Figures 10-12. It is interesting to note here from the point of view of numerics, that a black hole loop corresponding to the two r h → 0 limits of two regular solutions, say at α 0 , can be constructed numerically even when the numerical process for the higher mass branch of the regular solutions for this α 0 runs into difficulties.\nAgain, black hole counterparts appear to exist for any regular solution. However, solutions with the right asymptotics are found for a limited region of the (r h , α) space only. We plot in Figures 11-13 some results we found for d = 6, 7 and d = 8. Starting for a given α 0 < α max from a r h = 0 first branch regular solution, we found the existence of a branch of black hole solutions extending up to a maximal value of the event horizon radius r h = r max h . When r h increases, the mass and the Hawking temperature increase while the value σ(r h ) decreases from its value at the regular solution. A second branch of black hole solutions seems to appear always at r max h , extending backwards to a zero event horizon radius. This limiting solution corresponds to the second branch of the regular solution at this value of α = α 0 . Like in those d = 5 cases where there are two regular solutions for a given α, say α 0 , here too these two regular solutions are the r h → 0 limits of the corresponding black hole \"loop\", connecting the two regular solutions. Along this second branch the values σ(r h ) of the metric function σ on the event horizon decrease monotonically with r h , while the Hawking temperature strongly increases. The mass of the solution of the second branch is larger than the corresponding one on the first branch, as illustrated by Figures 10-12. It is interesting to note here from the point of view of numerics, that a black hole loop corresponding to the two r h → 0 limits of two regular solutions, say at α 0 , can be constructed numerically even when the numerical process for the higher mass branch of the regular solutions for this α 0 runs into difficulties.\nThe introduction of a negative cosmological constant does not appear to change this picture qualitatively. However, we notice a smaller value of r max h with increasing |Λ| and a larger value of M for the same α. In Figure 14, we present the profiles of the metric functions N and σ and gauge function w(r) for the same values of α 2 , Λ on the first and second branch for some d = 6 solutions. The dependence of these functions on the value of Λ is illustrated in Figure 15 for several d = 7 black hole solutions.\nThe introduction of a negative cosmological constant does not appear to change this picture qualitatively. However, we notice a smaller value of r max h with increasing |Λ| and a larger value of M for the same α. In Figure 14, we present the profiles of the metric functions N and σ and gauge function w(r) for the same values of α 2 , Λ on the first and second branch for some d = 6 solutions. The dependence of these functions on the value of Λ is illustrated in Figure 15 for several d = 7 black hole solutions."},{"title":"A computation of mass and action","content":""},{"title":"A computation of mass and action","content":""},{"title":"The counterterm method","content":""},{"title":"The counterterm method","content":"It is well known that the generalisation of Komar's formula for AAdS spacetimes is not straightforward and requires the further subtraction of a background configuration in order to render a finite result for the mass.\nIt is well known that the generalisation of Komar's formula for AAdS spacetimes is not straightforward and requires the further subtraction of a background configuration in order to render a finite result for the mass.\nThis problem was addressed for the first time in the eighties, with different approaches (see for instance Ref. [42,43]). Another formalism to define conserved charges in AAdS spacetimes was proposed in [44] and uses conformal techniques to construct conserved quantities yielding the results obtained by Hamiltonian methods. Other more recent approaches to the same problem are presented in Ref. [45].\nThis problem was addressed for the first time in the eighties, with different approaches (see for instance Ref. [42,43]). Another formalism to define conserved charges in AAdS spacetimes was proposed in [44] and uses conformal techniques to construct conserved quantities yielding the results obtained by Hamiltonian methods. Other more recent approaches to the same problem are presented in Ref. [45].\nAs expected, these different methods yield the same total mass for the spherically symmetric AAdS configurations considered in Section 3\nAs expected, these different methods yield the same total mass for the spherically symmetric AAdS configurations considered in Section 3\nwhere Ω d−2 = 2π (d−1)/2 /Γ((d − 1)/2) is the area of a unit (d − 2)-dimensional sphere, and M is defined in the second member of (29). A procedure leading (for odd dimensions) to a different result has been proposed by Balasubramanian and Kraus [21]. This technique was inspired by AdS/CFT correspondence and consists in adding suitable counterterms I ct to the action of the theory in order to ensure the finiteness of the boundary stress tensor derived by the quasilocal energy definition [46]. These counterterms are built up with curvature invariants of a boundary ∂M (which is sent to infinity after the integration) and thus obviously they do not alter the bulk equations of motion. Unlike background subtraction, the counterterm approach does no require the identification of a reference spacetime. Given the potential relevance of the EYM solutions in an AdS/CFT context, we present here a computation of the boundary stress tensor and of the total mass, by using the counterterm prescription.\nwhere Ω d−2 = 2π (d−1)/2 /Γ((d − 1)/2) is the area of a unit (d − 2)-dimensional sphere, and M is defined in the second member of (29). A procedure leading (for odd dimensions) to a different result has been proposed by Balasubramanian and Kraus [21]. This technique was inspired by AdS/CFT correspondence and consists in adding suitable counterterms I ct to the action of the theory in order to ensure the finiteness of the boundary stress tensor derived by the quasilocal energy definition [46]. These counterterms are built up with curvature invariants of a boundary ∂M (which is sent to infinity after the integration) and thus obviously they do not alter the bulk equations of motion. Unlike background subtraction, the counterterm approach does no require the identification of a reference spacetime. Given the potential relevance of the EYM solutions in an AdS/CFT context, we present here a computation of the boundary stress tensor and of the total mass, by using the counterterm prescription.\nThe following counterterms are sufficient to cancel divergences in a pure gravity theory for d ≤ 9, with several exceptions (see e. g. [47])\nThe following counterterms are sufficient to cancel divergences in a pure gravity theory for d ≤ 9, with several exceptions (see e. g. [47])\nHere R ABCD , R AB are the Riemann and Ricci tensors, R is the Ricci scalar for the boundary metric h and Θ(x) is the step function, which is equal to 1 for x ≥ 0 and zero otherwise; A, B, . . . indicate the intrinsic coordinates of the boundary. Using these counterterms one can construct a divergenceless boundary stress tensor, which is given by the variation of the total action at the boundary with respect to h AB . Its explicit expression, restricting for simplicity to d < 7, is\nHere R ABCD , R AB are the Riemann and Ricci tensors, R is the Ricci scalar for the boundary metric h and Θ(x) is the step function, which is equal to 1 for x ≥ 0 and zero otherwise; A, B, . . . indicate the intrinsic coordinates of the boundary. Using these counterterms one can construct a divergenceless boundary stress tensor, which is given by the variation of the total action at the boundary with respect to h AB . Its explicit expression, restricting for simplicity to d < 7, is\nwhere\nwhere\nis the extrinsic curvature defined in terms of the normal n A to the boundary, K is its trace, and E AB is the Einstein tensor of the intrinsic metric h AB . (The corresponding form of (40) for d = 7, 8 is given e.g. in [48]).\nis the extrinsic curvature defined in terms of the normal n A to the boundary, K is its trace, and E AB is the Einstein tensor of the intrinsic metric h AB . (The corresponding form of (40) for d = 7, 8 is given e.g. in [48]).\nThe result we find in this way for T AB is given by\nThe result we find in this way for T AB is given by\nwhere u A = δ t A and p is an integer. We can use this approach to assign a mass-energy to an AAdS geometry by writing the boundary metric in an ADM form\nwhere u A = δ t A and p is an integer. We can use this approach to assign a mass-energy to an AAdS geometry by writing the boundary metric in an ADM form\nand the definition of the energy in this context is\nand the definition of the energy in this context is\nHere ǫ = u µ u ν T µν is the proper energy density while u µ is a timelike unit normal to Σ.\nHere ǫ = u µ u ν T µν is the proper energy density while u µ is a timelike unit normal to Σ.\nIf there are matter fields on M, additional counterterms may be needed to regulate the action. This is the case of F (2) theory, discussed in Appendix B. The couinterterm action depends in this case not only on the boundary metric but also on the boundary value of the gauge field.\nIf there are matter fields on M, additional counterterms may be needed to regulate the action. This is the case of F (2) theory, discussed in Appendix B. The couinterterm action depends in this case not only on the boundary metric but also on the boundary value of the gauge field.\nHowever, we find that for P > 1 EYM solutions with Λ < 0 in d = 5, 6, 7, 8 dimensions, the prescription (39) removes all divergences. The use of higher order terms in the YM curvature, namely F (2p) forms with p > 1 introduced in Section 3, results in this regularising of the masses. The crucial point here is that these solutions approaches asymptotically a Schwarzschild-AdS background, and the YM asymptotic parameter w 1 appears only in the next to leading order of the T AB expression.\nHowever, we find that for P > 1 EYM solutions with Λ < 0 in d = 5, 6, 7, 8 dimensions, the prescription (39) removes all divergences. The use of higher order terms in the YM curvature, namely F (2p) forms with p > 1 introduced in Section 3, results in this regularising of the masses. The crucial point here is that these solutions approaches asymptotically a Schwarzschild-AdS background, and the YM asymptotic parameter w 1 appears only in the next to leading order of the T AB expression.\nThe mass-energy of solutions computed in this way is\nThe mass-energy of solutions computed in this way is\nwhere, for 3 < d < 9\nwhere, for 3 < d < 9\nThe additional term E 0 appearing in E for d = 5, (7) is the mass of pure global AdS 5, (7) and is usually interpreted as the energy dual to the Casimir energy of the CFT defined on a four (six) dimensional Einstein universe [21]. The metric restricted to the boundary h AB diverges due to an infinite conformal factor r 2 /ℓ 2 . The background metric upon which the dual field theory resides is\nThe additional term E 0 appearing in E for d = 5, (7) is the mass of pure global AdS 5, (7) and is usually interpreted as the energy dual to the Casimir energy of the CFT defined on a four (six) dimensional Einstein universe [21]. The metric restricted to the boundary h AB diverges due to an infinite conformal factor r 2 /ℓ 2 . The background metric upon which the dual field theory resides is\nFor the asymptotically AdS d solutions considered here, the (d − 1) dimensional boundary is the Einstein universe, with the line element\nFor the asymptotically AdS d solutions considered here, the (d − 1) dimensional boundary is the Einstein universe, with the line element\nIn light of the AdS/CFT correspondence, Balasubramanian and Kraus have interpreted Eq. (40) as < τ AB >= 2 √ −γ δS ef f δγAB , where < τ AB > is the expectation value of the CFT stress tensor. Then, the divergences which appear are simply the standard ultraviolet divergences of a quantum field theory and we can cancel them by adding local counterterms to the action. Corresponding to the boundary metric (47), the stress-energy tensor τ AB for the dual theory can be calculated using the following relation [49] √ −γγ AB < τ BC >= lim\nIn light of the AdS/CFT correspondence, Balasubramanian and Kraus have interpreted Eq. (40) as < τ AB >= 2 √ −γ δS ef f δγAB , where < τ AB > is the expectation value of the CFT stress tensor. Then, the divergences which appear are simply the standard ultraviolet divergences of a quantum field theory and we can cancel them by adding local counterterms to the action. Corresponding to the boundary metric (47), the stress-energy tensor τ AB for the dual theory can be calculated using the following relation [49] √ −γγ AB < τ BC >= lim"},{"title":"Action and entropy","content":""},{"title":"Action and entropy","content":"The above approach can be used to compute the Euclidean action and to prove in a rigorous way that the entropy of the EYM black hole solutions is one quarter of the event horizon area, as expected. Here we start by constructing the path integral [18]\nThe above approach can be used to compute the Euclidean action and to prove in a rigorous way that the entropy of the EYM black hole solutions is one quarter of the event horizon area, as expected. Here we start by constructing the path integral [18]\nby integrating over all metrics and matter fields between some given initial and final hypersurfaces, Ψ corresponding here to the SU(2) potentials. By analytically continuing the time coordinate t → iτ , the path integral formally converges, and in the leading order one obtains Z ≃ e −I cl (50) where I cl is the classical action evaluated on the equations of motion of the gravity/matter system. The physical interpretation of this formalism is that the class of regular stationary metrics forms an ensemble of thermodynamic systems at equilibrium temperature T (see e.g. [50]). Z has the interpretation of partition function and we can define the free energy of the system F = −β −1 log Z. Therefore\nby integrating over all metrics and matter fields between some given initial and final hypersurfaces, Ψ corresponding here to the SU(2) potentials. By analytically continuing the time coordinate t → iτ , the path integral formally converges, and in the leading order one obtains Z ≃ e −I cl (50) where I cl is the classical action evaluated on the equations of motion of the gravity/matter system. The physical interpretation of this formalism is that the class of regular stationary metrics forms an ensemble of thermodynamic systems at equilibrium temperature T (see e.g. [50]). Z has the interpretation of partition function and we can define the free energy of the system F = −β −1 log Z. Therefore\nor\nor\nstraightforwardly follows.\nstraightforwardly follows.\nTo compute I cl , we make use of the Einstein equations, replacing the R − 2Λ volume term with 2R t t − 16πGT t t . For our purely magnetic ansatz, the term T t t exactly cancels the matter field Lagrangean in the bulk action. The divergent contribution given by the surface integral term at infinity in R t t is also canceled by I surface + I ct and for 3 < d < 9 we arrive at the simple finite expression\nTo compute I cl , we make use of the Einstein equations, replacing the R − 2Λ volume term with 2R t t − 16πGT t t . For our purely magnetic ansatz, the term T t t exactly cancels the matter field Lagrangean in the bulk action. The divergent contribution given by the surface integral term at infinity in R t t is also canceled by I surface + I ct and for 3 < d < 9 we arrive at the simple finite expression\nReplacing I cl now in (52) (where E is the mass-energy computed in Section 4.1), we find\nReplacing I cl now in (52) (where E is the mass-energy computed in Section 4.1), we find\nwhich is one quarter of the event horizon area, as expected. ¿From the AdS/CFT correspondence, we expect the nonabelian hairy black holes to be described by some thermal states in a dual theory formulated in a Einstein universe background. The spherically symmetric solitons will correspond to zero-temperature states in the same theory. The existence of these hairy configurations suggest that there should be some observables in the dual CFT that encode the hair information.\nwhich is one quarter of the event horizon area, as expected. ¿From the AdS/CFT correspondence, we expect the nonabelian hairy black holes to be described by some thermal states in a dual theory formulated in a Einstein universe background. The spherically symmetric solitons will correspond to zero-temperature states in the same theory. The existence of these hairy configurations suggest that there should be some observables in the dual CFT that encode the hair information."},{"title":"Conclusions","content":""},{"title":"Conclusions","content":"Motivated by recent results in EYM theory in four dimensional AAdS spacetime, we studied higher dimensional spherically symmetric solutions with non Abelian fields in the presence of a negative cosmological constant. Since the mass-energy of the AAdS configurations plays a central role in its application to the AdS/CFT correspondence, we emphasised this aspect of the solutions we found. The mass-energy of the usual EYM solutions (both asymptotically flat and AAdS), defined according to the standard prescription, always diverges in spacetime dimensions d > 4. One of the tasks performed in this work was a demonstration of this fact in the AAdS case. The main properties of these higher dimensional F (2) solutions resemble the d = 4 case, a continuum of solutions with arbitrary asymptotic values of the gauge function w(r) being found.\nMotivated by recent results in EYM theory in four dimensional AAdS spacetime, we studied higher dimensional spherically symmetric solutions with non Abelian fields in the presence of a negative cosmological constant. Since the mass-energy of the AAdS configurations plays a central role in its application to the AdS/CFT correspondence, we emphasised this aspect of the solutions we found. The mass-energy of the usual EYM solutions (both asymptotically flat and AAdS), defined according to the standard prescription, always diverges in spacetime dimensions d > 4. One of the tasks performed in this work was a demonstration of this fact in the AAdS case. The main properties of these higher dimensional F (2) solutions resemble the d = 4 case, a continuum of solutions with arbitrary asymptotic values of the gauge function w(r) being found.\nThen we focused on two possible approaches of dealing with the divergences of the mass and action. One of these involved the regularisation of the mass-energy using a counterterm mechanisms. In this approach, it turns out that the counterterm action depends not only on the boundary metric, but also on the boundary values of the gauge fields. However, the masses of the solutions defined in this way may take negative values, leading us to the second approach.\nThen we focused on two possible approaches of dealing with the divergences of the mass and action. One of these involved the regularisation of the mass-energy using a counterterm mechanisms. In this approach, it turns out that the counterterm action depends not only on the boundary metric, but also on the boundary values of the gauge fields. However, the masses of the solutions defined in this way may take negative values, leading us to the second approach.\nThe other, which formed the main thrust of the work, was to augment the action density of the system with higher order curvature terms, consisting of 2p-form curvatures F (2p). These terms were added to the usual YM system constructed from F (2). It resulted in EYM solutions supporting finite mass-energy in all spacetime dimensions 5 ≤ d ≤ 2p.\nThe other, which formed the main thrust of the work, was to augment the action density of the system with higher order curvature terms, consisting of 2p-form curvatures F (2p). These terms were added to the usual YM system constructed from F (2). It resulted in EYM solutions supporting finite mass-energy in all spacetime dimensions 5 ≤ d ≤ 2p.\nConcerning the construction of regular finite energy classical AAdS solutions in higher dimensions, we restricted ourselves to systems consisting exclusively of gravitational and non Abelian gauge fields. The salient features of the resulting solutions are captured in this framework, the addition of other (string theory inspired) matter terms being deferred to later work. The asymptotically flat versions of the higher-p EYM systems having been studied in [13,14,19], our task here involved the introduction of a negative cosmological constant. The most succint way of listing our conclusions is:\nConcerning the construction of regular finite energy classical AAdS solutions in higher dimensions, we restricted ourselves to systems consisting exclusively of gravitational and non Abelian gauge fields. The salient features of the resulting solutions are captured in this framework, the addition of other (string theory inspired) matter terms being deferred to later work. The asymptotically flat versions of the higher-p EYM systems having been studied in [13,14,19], our task here involved the introduction of a negative cosmological constant. The most succint way of listing our conclusions is:\n• The qualitative properties of the regular AAdS solutions in spacetime dimensions d = 5, 6, 7, 8 are the same as the corresponding asymptotically flat ones, namely -A one parameter family of solutions parametrised by the (dimensionless) gravitational coupling constant α start at α = 0 (the gravity decoupling limit) and exist up to a maximum α max , after which α decreases again and ends at critical value.\n• The qualitative properties of the regular AAdS solutions in spacetime dimensions d = 5, 6, 7, 8 are the same as the corresponding asymptotically flat ones, namely -A one parameter family of solutions parametrised by the (dimensionless) gravitational coupling constant α start at α = 0 (the gravity decoupling limit) and exist up to a maximum α max , after which α decreases again and ends at critical value.\n-For d = 6, 7, 8 the value of α in the second, not that of gravity decoupling, endpoint becomes very small and stops. In the d = 8 case it actually vanishes. For d = 5 the value of α in the second endpoint reaches a finite critical value, where it does not stop, but oscillates around this critical value.\n-For d = 6, 7, 8 the value of α in the second, not that of gravity decoupling, endpoint becomes very small and stops. In the d = 8 case it actually vanishes. For d = 5 the value of α in the second endpoint reaches a finite critical value, where it does not stop, but oscillates around this critical value.\n-As long as the physically important F (2) term is present in the YM sector, there exist no multinode solutions.\n-As long as the physically important F (2) term is present in the YM sector, there exist no multinode solutions.\n• Black hole counterparts appear to exist for any regular solution. The qualitative properties of the AAdS black hole solutions are similar to the asymptotically flat case:\n• Black hole counterparts appear to exist for any regular solution. The qualitative properties of the AAdS black hole solutions are similar to the asymptotically flat case:\n-Different from the four dimensional theory, the event horizon radius presents a maximal value. This maximal value is a function of the gravitational coupling constant α.\n-Different from the four dimensional theory, the event horizon radius presents a maximal value. This maximal value is a function of the gravitational coupling constant α.\n-For d = 6, 7, 8 the black hole solutions form a loop connecting the two regular solutions with the same value of α. The solutions of the five dimensional theory are somehow special, presenting a complicated brach structure which depends on α.\n-For d = 6, 7, 8 the black hole solutions form a loop connecting the two regular solutions with the same value of α. The solutions of the five dimensional theory are somehow special, presenting a complicated brach structure which depends on α.\nAxially symmetric generalizations of these solutions are likely to exist. We expect them to share the basic properties of the spherically symmetric configurations. The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ. In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.\nAxially symmetric generalizations of these solutions are likely to exist. We expect them to share the basic properties of the spherically symmetric configurations. The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ. In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ."},{"title":"Figure 1 .Figure 2 .Figure 3 .","content":""},{"title":"Figure 1 .Figure 2 .Figure 3 .","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 6","content":""},{"title":"Figure 6","content":""},{"title":"Figure 7","content":""},{"title":"Figure 7","content":""},{"title":"Figure 8","content":""},{"title":"Figure 8","content":""},{"title":"Figure 9b .","content":""},{"title":"Figure 9b .","content":""},{"title":"Figure 9","content":""},{"title":"Figure 9","content":""},{"title":"Figure 10b .","content":""},{"title":"Figure 10b .","content":""},{"title":"Figure 10","content":""},{"title":"Figure 10","content":""},{"title":"Figure 11 .","content":""},{"title":"Figure 11 .","content":""},{"title":"Figure 12 .Figure 13 .","content":""},{"title":"Figure 12 .Figure 13 .","content":""},{"title":"Figure 14 .","content":""},{"title":"Figure 14 .","content":""},{"title":"Figure 15 .","content":""},{"title":"Figure 15 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"\"\n },\n {\n \"title\": \"Introduction\",\n \"content\": \"If we allow for a negative cosmological constant, the solution of the matter free Einstein equations possessing the maximal number of symmetries is the anti-de Sitter (AdS) spacetime. Recently a considerable amount of interest has been focused on solutions of Einstein equations with this type of asymptotics. This interest is mainly motivated by the proposed correspondence between physical effects associated with gravitating fields propagating in AdS spacetime and those of a conformal field theory on the boundary of AdS spacetime [1,2].\\nIf we allow for a negative cosmological constant, the solution of the matter free Einstein equations possessing the maximal number of symmetries is the anti-de Sitter (AdS) spacetime. Recently a considerable amount of interest has been focused on solutions of Einstein equations with this type of asymptotics. This interest is mainly motivated by the proposed correspondence between physical effects associated with gravitating fields propagating in AdS spacetime and those of a conformal field theory on the boundary of AdS spacetime [1,2].\\nIn view of these developments, an examination of the classical solutions of gravitating fields in asymptotically AdS (AAdS) spacetimes seems appropriate. Recently, some authors have discussed the properties of gravitating SU(2) nonabelian fields with a negative cosmological constant Λ [3,4]. Considering the case of four spacetime dimensions, they obtained some surprising results, which are strikingly different from the results found in the asymptotically flat case. For example there are solutions for continuous intervals of the parameter space, rather then discrete points. The asymptotic values of the gauge potentials are arbitrary and there exist solutions supporting magnetic and electric fluxes without a Higgs field. Some of these solutions are stable against spherically symmetric linear perturbations. The literature on AAdS solutions with nonabelian fields is growing steadily, including stability analyses [5,6], the study of configurations with a NUT charge [7], topological black holes with nonabelian hair [8] as well as axially symmetric generalisations [9,10]. The existence of these solutions invalidates the AdS 4 version of the no hair conjecture, which states that the black holes are completely characterised by their mass, charge and angular momentum.\\nIn view of these developments, an examination of the classical solutions of gravitating fields in asymptotically AdS (AAdS) spacetimes seems appropriate. Recently, some authors have discussed the properties of gravitating SU(2) nonabelian fields with a negative cosmological constant Λ [3,4]. Considering the case of four spacetime dimensions, they obtained some surprising results, which are strikingly different from the results found in the asymptotically flat case. For example there are solutions for continuous intervals of the parameter space, rather then discrete points. The asymptotic values of the gauge potentials are arbitrary and there exist solutions supporting magnetic and electric fluxes without a Higgs field. Some of these solutions are stable against spherically symmetric linear perturbations. The literature on AAdS solutions with nonabelian fields is growing steadily, including stability analyses [5,6], the study of configurations with a NUT charge [7], topological black holes with nonabelian hair [8] as well as axially symmetric generalisations [9,10]. The existence of these solutions invalidates the AdS 4 version of the no hair conjecture, which states that the black holes are completely characterised by their mass, charge and angular momentum.\\nHowever, all these studies approach the case of a four dimensional AAdS spacetime, and relatively little is known about higher dimensional AAdS solutions with non Abelian matter fields. Practically all that is known for d > 4 is the five dimensional nonabelian SU (2) solutions discussed in [11]. At the same time gauged supergravity theories playing an important role in AdS/CFT, generically contain non Abelian matter fields in the bulk, although in the literature only Abelian truncations are considered, to date. Thus, the examination of higher dimensional gravitating non Abelian solutions with Λ < 0 is a pertinent task.\\nHowever, all these studies approach the case of a four dimensional AAdS spacetime, and relatively little is known about higher dimensional AAdS solutions with non Abelian matter fields. Practically all that is known for d > 4 is the five dimensional nonabelian SU (2) solutions discussed in [11]. At the same time gauged supergravity theories playing an important role in AdS/CFT, generically contain non Abelian matter fields in the bulk, although in the literature only Abelian truncations are considered, to date. Thus, the examination of higher dimensional gravitating non Abelian solutions with Λ < 0 is a pertinent task.\\nHigher dimensional asymptotically flat solutions of the Einstein-Yang-Mills (EYM) equations have recently been the subject of several studies. As found in [12], in five spacetime dimensions, the particle spectrum obtained by uplifting the d = 4 flat space YM instantons become completely destroyed by gravity, as a result of their scaling behaviour. However, by adding higher order 1 terms in the Yang-Mills (YM) hierarchy this obstacle due to scaling is removed and solutions in higher dimensions can be found. Such regular, static and spherically symmetric solutions in spacetime dimensions d ≤ 8 were presented in [13], and for d = 5, both globally regular and black hole solutions of the EYM system were found in [14]. The properties of these solutions are rather different from the familiar Bartnik-McKinnon (BK) solutions [16] to EYM in d = 4, and are somewhat more akin to the gravitating monopole solutions to EYM-Higgs (EYMH) [17]. This is because like in the latter case [17], where the VEV of the Higgs field features as an additional dimensional constant, here [13] also additional dimensional constants enter with each higher order YM curvature term 2 . They are however quite distinctive in their critical behaviour. The typical critical features discovered in [13,14] have recently been analysed and explained in [19].\\nHigher dimensional asymptotically flat solutions of the Einstein-Yang-Mills (EYM) equations have recently been the subject of several studies. As found in [12], in five spacetime dimensions, the particle spectrum obtained by uplifting the d = 4 flat space YM instantons become completely destroyed by gravity, as a result of their scaling behaviour. However, by adding higher order 1 terms in the Yang-Mills (YM) hierarchy this obstacle due to scaling is removed and solutions in higher dimensions can be found. Such regular, static and spherically symmetric solutions in spacetime dimensions d ≤ 8 were presented in [13], and for d = 5, both globally regular and black hole solutions of the EYM system were found in [14]. The properties of these solutions are rather different from the familiar Bartnik-McKinnon (BK) solutions [16] to EYM in d = 4, and are somewhat more akin to the gravitating monopole solutions to EYM-Higgs (EYMH) [17]. This is because like in the latter case [17], where the VEV of the Higgs field features as an additional dimensional constant, here [13] also additional dimensional constants enter with each higher order YM curvature term 2 . They are however quite distinctive in their critical behaviour. The typical critical features discovered in [13,14] have recently been analysed and explained in [19].\\nThese results can be systematically extended to all d ≥ 5 and one finds that no finite mass solutions can exist in EYM theory, unless one modifies the non Abelian action by adding higher order curvature terms 3 in the YM 2p-form curvature F (2p), the p = 1 case being the usual 2-form YM curvature. Without these higher order YM terms, only vortex-type finite energy solutions [12] exist, describing effective systems in 3 spacelike dimensions, and with a number of codimensions.\\nThese results can be systematically extended to all d ≥ 5 and one finds that no finite mass solutions can exist in EYM theory, unless one modifies the non Abelian action by adding higher order curvature terms 3 in the YM 2p-form curvature F (2p), the p = 1 case being the usual 2-form YM curvature. Without these higher order YM terms, only vortex-type finite energy solutions [12] exist, describing effective systems in 3 spacelike dimensions, and with a number of codimensions.\\nIt is the purpose of this paper to examine the corresponding situation in higher dimensions, in the presence of a negative cosmological constant. It is interesting to inquire whether the introduction of a negative cosmological constant to these higher dimensional EYM models will lead to some new effects as it does in the d = 4 case, due to the different asymptotic structure of the spacetime. In the first place this would lead to our understanding of how the behaviour of EYM theory depends on the dimensionality of the spacetime. But such higher dimensional AAdS solutions might be relevant to superstring theory, namely in the context of solutions to various supergravities containing nonabelian matter fields. Here however we restrict our considerations to the simplest case of systems consisting only of gravitational and YM fields, namely to higher dimensional EYM model. In particular, we restrict to Einstein-Hilbert gravity and the first two members of the YM hierarchy, and hence to d ≤ 8, augmented with negative cosmological constant in d spacetime dimensions. As in [3,4], for the d = 4 case, we seek static spherically symmetric solutions in the d − 1 spacelike dimensions. We find both globally regular and black hole solutions with finite ADM mass. Unlike in the d = 4 case however, we find that for d > 4 and a negative cosmological constant, the properties of the AAdS solutions do not differ qualitatively from the asymptotically flat case.\\nIt is the purpose of this paper to examine the corresponding situation in higher dimensions, in the presence of a negative cosmological constant. It is interesting to inquire whether the introduction of a negative cosmological constant to these higher dimensional EYM models will lead to some new effects as it does in the d = 4 case, due to the different asymptotic structure of the spacetime. In the first place this would lead to our understanding of how the behaviour of EYM theory depends on the dimensionality of the spacetime. But such higher dimensional AAdS solutions might be relevant to superstring theory, namely in the context of solutions to various supergravities containing nonabelian matter fields. Here however we restrict our considerations to the simplest case of systems consisting only of gravitational and YM fields, namely to higher dimensional EYM model. In particular, we restrict to Einstein-Hilbert gravity and the first two members of the YM hierarchy, and hence to d ≤ 8, augmented with negative cosmological constant in d spacetime dimensions. As in [3,4], for the d = 4 case, we seek static spherically symmetric solutions in the d − 1 spacelike dimensions. We find both globally regular and black hole solutions with finite ADM mass. Unlike in the d = 4 case however, we find that for d > 4 and a negative cosmological constant, the properties of the AAdS solutions do not differ qualitatively from the asymptotically flat case.\\nOur strategy is to first consider the usual YM model, namely the p = 1 member of the YM hierarchy, or the square of the 2-form curvature F (2). We present an argument for the absence of solutions with reasonable asymptotics for any spacetime dimension d ≥ 5. Although the EYM equations in this case present solutions approaching asymptotically the AdS background, the mass generically diverges. This can be seen as a simple version of the no hair theorem, holding for the EYM system in d > 4 dimensions. In other words, Schwarzschild-AdS black hole is the only static, spherically symmetric solution of the EYM system with finite mass. This is presented in Section 2, where in addition we have considered the special case d = 3, extending Deser's analysis [20] for Λ < 0.\\nOur strategy is to first consider the usual YM model, namely the p = 1 member of the YM hierarchy, or the square of the 2-form curvature F (2). We present an argument for the absence of solutions with reasonable asymptotics for any spacetime dimension d ≥ 5. Although the EYM equations in this case present solutions approaching asymptotically the AdS background, the mass generically diverges. This can be seen as a simple version of the no hair theorem, holding for the EYM system in d > 4 dimensions. In other words, Schwarzschild-AdS black hole is the only static, spherically symmetric solution of the EYM system with finite mass. This is presented in Section 2, where in addition we have considered the special case d = 3, extending Deser's analysis [20] for Λ < 0.\\nIn Section 3 we introduce the higher order YM hierarchy models featuring the terms F (2p), p ≥ 2. We derive the classical equations subject to our spherically symmetric ansatz, and present a detailed numerical study of of both regular and black hole solutions. As is the case with the usual EYM system, where the existence of finite mass regular solutions leads to the violation of the no hair conjecture, here too, these solutions are explicit counterexamples to this conjecture in AdS d spacetime.\\nIn Section 3 we introduce the higher order YM hierarchy models featuring the terms F (2p), p ≥ 2. We derive the classical equations subject to our spherically symmetric ansatz, and present a detailed numerical study of of both regular and black hole solutions. As is the case with the usual EYM system, where the existence of finite mass regular solutions leads to the violation of the no hair conjecture, here too, these solutions are explicit counterexamples to this conjecture in AdS d spacetime.\\nOne may ask about the possible relevance of these higher dimensional configurations within the AdS/CFT correspondence. In Section 4 we compute the boundary stress tensor and the mass and action of the solutions in a number of spacetime dimensions up to eight. In five and seven dimensions, the counterterm prescription of [21] gives an additional vacuum (Casimir) energy, which agrees with that found in the context of AdS/CFT correspondence. A counterterm based proposal to remove the divergences of a F (2) theory, such that the mass and action be finite, is also presented in Appendix B. We give our conclusions and remarks in the final section.\\nOne may ask about the possible relevance of these higher dimensional configurations within the AdS/CFT correspondence. In Section 4 we compute the boundary stress tensor and the mass and action of the solutions in a number of spacetime dimensions up to eight. In five and seven dimensions, the counterterm prescription of [21] gives an additional vacuum (Casimir) energy, which agrees with that found in the context of AdS/CFT correspondence. A counterterm based proposal to remove the divergences of a F (2) theory, such that the mass and action be finite, is also presented in Appendix B. We give our conclusions and remarks in the final section.\\nEverewhere in this paper we employ the notations and conventions of [13]. 2 The F (2) model\\nEverewhere in this paper we employ the notations and conventions of [13]. 2 The F (2) model\"\n },\n {\n \"title\": \"The action principle\",\n \"content\": \"\"\n },\n {\n \"title\": \"The action principle\",\n \"content\": \"We start with the following action principle in d−spacetime dimensions\\nWe start with the following action principle in d−spacetime dimensions\\nwhere R is the Ricci scalar associated with the spacetime metric g µν , Λ = −(d − 1)(d − 2)/(2ℓ 2 ) is the cosmological constant and G is the gravitational constant (following [14], we define also κ = 1/(8πG)). The matter term in the above relation\\nwhere R is the Ricci scalar associated with the spacetime metric g µν , Λ = −(d − 1)(d − 2)/(2ℓ 2 ) is the cosmological constant and G is the gravitational constant (following [14], we define also κ = 1/(8πG)). The matter term in the above relation\\nis the usual F 2 nonabelian action density,\\nis the usual F 2 nonabelian action density,\\nbeing the gauge field strength tensor. Here τ 1 is the coupling constant of the model (in the usual notation τ 1 = 1/g 2 ).\\nbeing the gauge field strength tensor. Here τ 1 is the coupling constant of the model (in the usual notation τ 1 = 1/g 2 ).\\nThe last term in (1) is the Hawking-Gibbons surface term [18], where K is the trace of the extrinsic curvature for the boundary ∂M and h is the induced metric of the boundary. Of course this term does not affect the equations of motion but it is relevant for the discussion of the mass and the action of the solutions, in Section 4.\\nThe last term in (1) is the Hawking-Gibbons surface term [18], where K is the trace of the extrinsic curvature for the boundary ∂M and h is the induced metric of the boundary. Of course this term does not affect the equations of motion but it is relevant for the discussion of the mass and the action of the solutions, in Section 4.\\nThe field equations are obtained by varying the action (1) with respect to the field variables g µν , A µ\\nThe field equations are obtained by varying the action (1) with respect to the field variables g µν , A µ\\nwhere the energy momentum tensor is defined by\\nwhere the energy momentum tensor is defined by\"\n },\n {\n \"title\": \"The general Ansatz\",\n \"content\": \"\"\n },\n {\n \"title\": \"The general Ansatz\",\n \"content\": \"For the case of a d-dimensional spacetime, we restrict to static fields that are spherically symmetric in the d − 1 spacelike dimensions, with a metric ansatz in terms of Schwarzschild coordinates\\nFor the case of a d-dimensional spacetime, we restrict to static fields that are spherically symmetric in the d − 1 spacelike dimensions, with a metric ansatz in terms of Schwarzschild coordinates\\nwith dΩ d−2 the d − 2 dimensional angular volume element and\\nwith dΩ d−2 the d − 2 dimensional angular volume element and\\nthe function m(r) being related to the local mass-energy density up to some d−dependent factor. As discussed in [13], the choice of gauge group compatible with the symmetries of the line element (5) is somewhat flexible. In [13] the gauge group chosen was SO(d), in d dimensions. But the gauge field of the static solutions in question took their values in SO(d − 1). Thus in effect, it is possible to choose SO(d) in the first place. Now for even d, it is convenient to choose SO(d) since we can then avail of the chiral representations of the latter, although this is by no means obligatory. Adopting this criterion, namely to employ chiral representations, also for odd d, it is convenient to choose the gauge group to be SO(d − 1).\\nthe function m(r) being related to the local mass-energy density up to some d−dependent factor. As discussed in [13], the choice of gauge group compatible with the symmetries of the line element (5) is somewhat flexible. In [13] the gauge group chosen was SO(d), in d dimensions. But the gauge field of the static solutions in question took their values in SO(d − 1). Thus in effect, it is possible to choose SO(d) in the first place. Now for even d, it is convenient to choose SO(d) since we can then avail of the chiral representations of the latter, although this is by no means obligatory. Adopting this criterion, namely to employ chiral representations, also for odd d, it is convenient to choose the gauge group to be SO(d − 1).\\nWe shall therefore denote our representation matrices by SO ± (d), whered = d andd = d − 1 for even and odd d respectively.\\nWe shall therefore denote our representation matrices by SO ± (d), whered = d andd = d − 1 for even and odd d respectively.\\nIn this unified notation (for odd and even d), the spherically symmetric Ansatz for the SO ± (d)-valued gauge fields then reads [13]\\nIn this unified notation (for odd and even d), the spherically symmetric Ansatz for the SO ± (d)-valued gauge fields then reads [13]\\nThe Γ's denote thed-dimensional gamma matrices and 1, j = 1, 2, ..., d − 1 for both cases. Inserting this ansatz into the action (1), the EYM field equations reduce to\\nThe Γ's denote thed-dimensional gamma matrices and 1, j = 1, 2, ..., d − 1 for both cases. Inserting this ansatz into the action (1), the EYM field equations reduce to\\nwhich can also be derived from the reduced action\\nwhich can also be derived from the reduced action\\nFor a F 2 theory, the constants κ, τ 1 can always be absorbed by rescaling r → cr, Λ → Λ/c 2 and m → mκc d−3 , with c = τ 1 /κ. The above differential equations have two analytic solutions. One of them with w(r) = ±1, m(r) = M, σ(r) = 1 (12) corresponds to Schwarzschild-AdS spacetime. For w(r) = 0 we find a non Abelian generalisation of the magnetic-Reissner-Nordstrøm solution with σ(r) = 1 and\\nFor a F 2 theory, the constants κ, τ 1 can always be absorbed by rescaling r → cr, Λ → Λ/c 2 and m → mκc d−3 , with c = τ 1 /κ. The above differential equations have two analytic solutions. One of them with w(r) = ±1, m(r) = M, σ(r) = 1 (12) corresponds to Schwarzschild-AdS spacetime. For w(r) = 0 we find a non Abelian generalisation of the magnetic-Reissner-Nordstrøm solution with σ(r) = 1 and\\nM 0 being an arbitrary constant. We can see that, although these solutions are asymptotically AdS, the mass defined in the usual way diverges.\\nM 0 being an arbitrary constant. We can see that, although these solutions are asymptotically AdS, the mass defined in the usual way diverges.\"\n },\n {\n \"title\": \"d = 3\",\n \"content\": \"\"\n },\n {\n \"title\": \"d = 3\",\n \"content\": \"The (2+1) dimensional case is rather special. Three dimensional gravity has provided many important clues about higher dimensional physics. This theory with Λ < 0 has non-trivial solutions, such as the BTZ blackhole spacetime [22], which provide an important testing ground for quantum gravity and the AdS/CFT correspondence. Many other types of 3d regular and black hole solutions with a negative cosmological constant have also been found by coupling matter fields to gravity in different ways. However, as proven in [20], there are no d = 3 asymptotically flat static solutions of the EYM equations. The arguments in [20] can easily be generalised for the AAdS case. We notice that for d = 3, the YM equation (8) implies the existence of a first integral w ′ = αr/(σN ) with α an arbitrary real constant. Therefore, assuming AdS 3 asymptotics, w ′ decays asymptotically as 1/r which from (9) implies a divergent value of m(r) as r → ∞. However, similar to the Λ = 0 case [23], this argument does not exclude the existence of nontrivial solutions of the field equations.\\nThe (2+1) dimensional case is rather special. Three dimensional gravity has provided many important clues about higher dimensional physics. This theory with Λ < 0 has non-trivial solutions, such as the BTZ blackhole spacetime [22], which provide an important testing ground for quantum gravity and the AdS/CFT correspondence. Many other types of 3d regular and black hole solutions with a negative cosmological constant have also been found by coupling matter fields to gravity in different ways. However, as proven in [20], there are no d = 3 asymptotically flat static solutions of the EYM equations. The arguments in [20] can easily be generalised for the AAdS case. We notice that for d = 3, the YM equation (8) implies the existence of a first integral w ′ = αr/(σN ) with α an arbitrary real constant. Therefore, assuming AdS 3 asymptotics, w ′ decays asymptotically as 1/r which from (9) implies a divergent value of m(r) as r → ∞. However, similar to the Λ = 0 case [23], this argument does not exclude the existence of nontrivial solutions of the field equations.\\nHere we should remark that since for d = 3 we are dealing with SO(d − 1) = SO(2), the gauge group is Abelian and we recover Einstein-Maxwell theory with a negative cosmological constant, whose solutions are known in the literature. The corresponding solution with a vanishing electric field was found by Hirschmann and Welch [24] and has a line element 4\\nHere we should remark that since for d = 3 we are dealing with SO(d − 1) = SO(2), the gauge group is Abelian and we recover Einstein-Maxwell theory with a negative cosmological constant, whose solutions are known in the literature. The corresponding solution with a vanishing electric field was found by Hirschmann and Welch [24] and has a line element 4\\nthe magnetic potential being\\nthe magnetic potential being\\nwith w 0 , c and M arbitrary real constants, the BTZ metric being recovered for c = 0 (see also [25,26] for more details on this solution).\\nwith w 0 , c and M arbitrary real constants, the BTZ metric being recovered for c = 0 (see also [25,26] for more details on this solution).\\nOne can see that although the quasilocal mass defined in the usual way diverges as r → ∞, the metric still approaches the AdS 3 background. However, a similar problem appears for other d = 3 AdS solution e.g. for the electrically charged BTZ black hole [22], or for a self-interacting scalar field minimally coupled to gravity [27], in which cases it was possible to find a suitable mass definition. We expect the formalism developed in those cases to work also for the Hirschmann-Welch solution (14)- (15), but this lies outside the scope of the present work.\\nOne can see that although the quasilocal mass defined in the usual way diverges as r → ∞, the metric still approaches the AdS 3 background. However, a similar problem appears for other d = 3 AdS solution e.g. for the electrically charged BTZ black hole [22], or for a self-interacting scalar field minimally coupled to gravity [27], in which cases it was possible to find a suitable mass definition. We expect the formalism developed in those cases to work also for the Hirschmann-Welch solution (14)- (15), but this lies outside the scope of the present work.\"\n },\n {\n \"title\": \"d = 4\",\n \"content\": \"\"\n },\n {\n \"title\": \"d = 4\",\n \"content\": \"Four dimensional black hole solutions of the equations (8)- (10) have been found in [3], the globally regular counterparts being discussed in [4]. Differing from the asymptotically flat case, for Λ < 0 there is a continuum of solutions in terms of the adjustable shooting parameter that specifies the initial conditions at the origin or at the event horizon. As a new feature, the asymptotic value of the gauge function w 0 is arbitrary. The spectrum has a finite number of continuous branches, depending on the value of Λ. When the parameter Λ approaches zero, an already-existing branch of solutions collapses to a single point in the moduli space. At the same time new branches of solutions emerge. A fractal structure in the moduli space has been noticed [28]. There are also nontrivial solutions stable against spherically symmetric linear perturbations, corresponding to stable configurations. The solutions are classified by non-Abelian magnetic charge and the ADM mass.\\nFour dimensional black hole solutions of the equations (8)- (10) have been found in [3], the globally regular counterparts being discussed in [4]. Differing from the asymptotically flat case, for Λ < 0 there is a continuum of solutions in terms of the adjustable shooting parameter that specifies the initial conditions at the origin or at the event horizon. As a new feature, the asymptotic value of the gauge function w 0 is arbitrary. The spectrum has a finite number of continuous branches, depending on the value of Λ. When the parameter Λ approaches zero, an already-existing branch of solutions collapses to a single point in the moduli space. At the same time new branches of solutions emerge. A fractal structure in the moduli space has been noticed [28]. There are also nontrivial solutions stable against spherically symmetric linear perturbations, corresponding to stable configurations. The solutions are classified by non-Abelian magnetic charge and the ADM mass.\\nNote also that the d = 4 EYM solutions with a negative cosmological constant Λ = −3/τ 1 have some relevance in AdS/CFT context. As proven in [29], for this value of the cosmological constant, an arbitrary solution (g µν , A (a) µ ) of the four dimensional EYM equations gives a solution of the equations of motion of the d = 11 supergravity. Based on this observation, an exact BPS-type EYM solution has been constructed in [10]. However, similar to some supersymmetric solutions in Einstein-Maxwell theory with Λ < 0, this Λ = −3/τ 1 configuration presents a naked central singularity.\\nNote also that the d = 4 EYM solutions with a negative cosmological constant Λ = −3/τ 1 have some relevance in AdS/CFT context. As proven in [29], for this value of the cosmological constant, an arbitrary solution (g µν , A (a) µ ) of the four dimensional EYM equations gives a solution of the equations of motion of the d = 11 supergravity. Based on this observation, an exact BPS-type EYM solution has been constructed in [10]. However, similar to some supersymmetric solutions in Einstein-Maxwell theory with Λ < 0, this Λ = −3/τ 1 configuration presents a naked central singularity.\"\n },\n {\n \"title\": \"d ≥ 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"d ≥ 5\",\n \"content\": \"As discovered by Coleman [30] and Deser [31], there are no flat space static solutions of the YM equations, except for d = 5. However, the inclusion of gravity may change this picture, as seen from the famous d = 4 asymptotically flat Bartnik-McKinnon solutions [16]. In this case, the repulsive YM force is compensated by the attractive character of the gravity, and as a result we find both regular and black hole unstable configurations (see [32] for a fairly recent survey). As found in [12] the d = 5 particle-like solutions are destroyed by gravity, their mass diverging logarithmically, while w(r) presents an infinite number of nodes. The AAdS couterparts of the d = 5 asymptotically flat solutions are discussed in [11]. Although approaching asymptotically the AdS 5 background, the mass of these configurations also diverges logarithmically.\\nAs discovered by Coleman [30] and Deser [31], there are no flat space static solutions of the YM equations, except for d = 5. However, the inclusion of gravity may change this picture, as seen from the famous d = 4 asymptotically flat Bartnik-McKinnon solutions [16]. In this case, the repulsive YM force is compensated by the attractive character of the gravity, and as a result we find both regular and black hole unstable configurations (see [32] for a fairly recent survey). As found in [12] the d = 5 particle-like solutions are destroyed by gravity, their mass diverging logarithmically, while w(r) presents an infinite number of nodes. The AAdS couterparts of the d = 5 asymptotically flat solutions are discussed in [11]. Although approaching asymptotically the AdS 5 background, the mass of these configurations also diverges logarithmically.\\nAs conjectured by several authors, this result extends to higher dimensions. Following the approach in [11], we prove in Appendix A the nonexistence of asymptotically flat or AdS solutions with a finite mass in a F (2) EYM model given by (2) for any spacetime dimension d ≥ 5 (see also the discussion in Section 3.3). Therefore, the d−dimensional Schwarzschild-AdS configuration is the only finite mass solution of the equations (8)-(10) and a simple version of the no hair theorem seems to hold for the F (2) (usual) EYM system in d > 4.\\nAs conjectured by several authors, this result extends to higher dimensions. Following the approach in [11], we prove in Appendix A the nonexistence of asymptotically flat or AdS solutions with a finite mass in a F (2) EYM model given by (2) for any spacetime dimension d ≥ 5 (see also the discussion in Section 3.3). Therefore, the d−dimensional Schwarzschild-AdS configuration is the only finite mass solution of the equations (8)-(10) and a simple version of the no hair theorem seems to hold for the F (2) (usual) EYM system in d > 4.\\nWe should remark that in deriving this result we assumed implicitely that m(r), σ(r), w(r) are smooth functions approaching finite values as r → ∞. A divergent asymptotic value of m(r) invalidates the proof presented in Appendix A and also the virial argumnets in Section 3.3. Therefore we cannot exclude the existence of spherically symmetric, nontrivial solutions of the field equations for any d > 4. However, the mass of these solutions generically diverges, although the spacetimes are still AAdS. The work of Ref. [11] presents an extensive discussion of such AAdS solutions for d = 5. Both regular and black hole solutions exist in d = 5 for compact intervals of the parameter that specifies the initial conditions at the origin or at the event horizon. Differing from the Λ = 0 case, the gauge field function w(r) does not oscillate between 1 and −1 but approaches asymptotically some finite value w 0 , the node number being finite. The masses of these solutions behave asymptotically as (w 2 0 − 1) log r, with all w 0 = ±1 solutions corresponding to pure gauge configurations.\\nWe should remark that in deriving this result we assumed implicitely that m(r), σ(r), w(r) are smooth functions approaching finite values as r → ∞. A divergent asymptotic value of m(r) invalidates the proof presented in Appendix A and also the virial argumnets in Section 3.3. Therefore we cannot exclude the existence of spherically symmetric, nontrivial solutions of the field equations for any d > 4. However, the mass of these solutions generically diverges, although the spacetimes are still AAdS. The work of Ref. [11] presents an extensive discussion of such AAdS solutions for d = 5. Both regular and black hole solutions exist in d = 5 for compact intervals of the parameter that specifies the initial conditions at the origin or at the event horizon. Differing from the Λ = 0 case, the gauge field function w(r) does not oscillate between 1 and −1 but approaches asymptotically some finite value w 0 , the node number being finite. The masses of these solutions behave asymptotically as (w 2 0 − 1) log r, with all w 0 = ±1 solutions corresponding to pure gauge configurations.\\nThe results we found by solving numerically the equations (8)-(10), for d = 6, 7, 8 and several negative values of Λ confirm that this is a generic behaviour for d > 4. The corresponding boundary conditions at the origin (or event horizon) are found by taking P = 1 in relations (26), (27) given in Section 3. Except for a divergent value of m(r) as r → ∞, according to\\nThe results we found by solving numerically the equations (8)-(10), for d = 6, 7, 8 and several negative values of Λ confirm that this is a generic behaviour for d > 4. The corresponding boundary conditions at the origin (or event horizon) are found by taking P = 1 in relations (26), (27) given in Section 3. Except for a divergent value of m(r) as r → ∞, according to\\nthe properties of these solutions are very similar to the the more familiar d = 4 case. For d > 5, the asymptotic value w 0 of the gauge field function w is also arbitrary, being fixed by the initial parameters w ′′ (0) or w(r h ) respectively, w 0 = ±1 corresponding to pure gauge configurations. Solutions for a compact interval of these parameters were found to exist, the general structure being Λ-dependent. Solutions with nodes in w(r) were also found. Typical d = 6 configurations with a regular origin are presented in Figure 1, for Λ = −0.01 and three different values of b = −w ′′ (0)/2. One can see that the mass function diverges linearly while σ(r) and w(r) asymptotically approach some finite values. In Figure 2 we plot the parameters M 0 (appearing in (16), which in Section 4.3 we argue that it can be taken as the renormalised mass of the solutions), w 0 , the value σ 0 of the metric function σ at the origin and the minimal value N m of the metric function N as a function of b for a family of d = 6 AAdS solutions with Λ = −1. This branch ends for some finite value of b, where σ(0) → 0. Black hole solutions have been found as well, presenting the same general features. Here also we find a continuum of solutions with arbitrary values of w 0 , the relevant parameter being the value of the gauge potential at the event horizon. Similar to d = 4, solutions appear to exist for any value of the event horizon radius. The drawback of the solutions in d > 5 described above is that their ADM masses are divergent, making their physical significance obscure (see, however, the discussion in Appendix B).\\nthe properties of these solutions are very similar to the the more familiar d = 4 case. For d > 5, the asymptotic value w 0 of the gauge field function w is also arbitrary, being fixed by the initial parameters w ′′ (0) or w(r h ) respectively, w 0 = ±1 corresponding to pure gauge configurations. Solutions for a compact interval of these parameters were found to exist, the general structure being Λ-dependent. Solutions with nodes in w(r) were also found. Typical d = 6 configurations with a regular origin are presented in Figure 1, for Λ = −0.01 and three different values of b = −w ′′ (0)/2. One can see that the mass function diverges linearly while σ(r) and w(r) asymptotically approach some finite values. In Figure 2 we plot the parameters M 0 (appearing in (16), which in Section 4.3 we argue that it can be taken as the renormalised mass of the solutions), w 0 , the value σ 0 of the metric function σ at the origin and the minimal value N m of the metric function N as a function of b for a family of d = 6 AAdS solutions with Λ = −1. This branch ends for some finite value of b, where σ(0) → 0. Black hole solutions have been found as well, presenting the same general features. Here also we find a continuum of solutions with arbitrary values of w 0 , the relevant parameter being the value of the gauge potential at the event horizon. Similar to d = 4, solutions appear to exist for any value of the event horizon radius. The drawback of the solutions in d > 5 described above is that their ADM masses are divergent, making their physical significance obscure (see, however, the discussion in Appendix B).\\nOne may hope to find a different picture by including some other matter field in the action (1). Such fields should interact with the YM sector so as to compensate for the scaling behaviour of the non Abelian fields. This excludes the dilaton field, as it can be proven that the latter does not change this nonexistence result. Note that d = 5 finite mass spherically symmetric gravitating non Abelian solutions with a Liouville-type dilaton potential are known to exist [35]; however these solutions are asymptotically neither flat nor AdS.\\nOne may hope to find a different picture by including some other matter field in the action (1). Such fields should interact with the YM sector so as to compensate for the scaling behaviour of the non Abelian fields. This excludes the dilaton field, as it can be proven that the latter does not change this nonexistence result. Note that d = 5 finite mass spherically symmetric gravitating non Abelian solutions with a Liouville-type dilaton potential are known to exist [35]; however these solutions are asymptotically neither flat nor AdS.\\nIn the next Section, we will remedy this problem of nonexistence by adding higher order YM curvature terms F (2p), (p > 1), to (2). The main role of these terms is to alter the scaling properties of the action density in (1). But logically, such a role can be played also by altering the gravitational part of (1), through the introduction of higher order (in the Riemann curvature) terms R(p), (p > 1). R(p) here denotes the generalised Ricci scalar constructed from the 2p-form antisymmetrised p-fold product of Riemann tensor 2-forms, the p = 1 member of which gives the Einstein-Hilbert action, the p = 2, the Gauss-Bonnet, etc. Here we have eschewed the possibility of employing additional R(p) terms instead of, or together with F (2p) YM terms, because in the present work we exclude the participation of fields other than gravitational and Yang-Mills. In particular, the exclusion of the dilaton renders the usefulness of higher order (Gauss-Bonnet) gravities trivial from a practical point of view.\\nIn the next Section, we will remedy this problem of nonexistence by adding higher order YM curvature terms F (2p), (p > 1), to (2). The main role of these terms is to alter the scaling properties of the action density in (1). But logically, such a role can be played also by altering the gravitational part of (1), through the introduction of higher order (in the Riemann curvature) terms R(p), (p > 1). R(p) here denotes the generalised Ricci scalar constructed from the 2p-form antisymmetrised p-fold product of Riemann tensor 2-forms, the p = 1 member of which gives the Einstein-Hilbert action, the p = 2, the Gauss-Bonnet, etc. Here we have eschewed the possibility of employing additional R(p) terms instead of, or together with F (2p) YM terms, because in the present work we exclude the participation of fields other than gravitational and Yang-Mills. In particular, the exclusion of the dilaton renders the usefulness of higher order (Gauss-Bonnet) gravities trivial from a practical point of view.\\nTo see this, consider the situation where a term scaling as L −2p is needed, i.e. that the term\\nTo see this, consider the situation where a term scaling as L −2p is needed, i.e. that the term\\nmust be added to the density in (1). Using the metric Ansatz (5), and discarding purely boundary terms in the residual one dimensional (spherically symmetric) Lagrangean, the term L (p) to be added is\\nmust be added to the density in (1). Using the metric Ansatz (5), and discarding purely boundary terms in the residual one dimensional (spherically symmetric) Lagrangean, the term L (p) to be added is\\nin spacetime dimension d, which vanishes identically in dimensions up to d = 2p. This is because in the given dimensions the p-th member of the gravitational hierarchy becomes a total divergence. But for given d it is necessary to have 2p ≥ d, whence (17) trivialises. The situation would be quite different if the dilaton field were included together with the higher order gravitational terms, preventing the p-gravity density becoming a total divergence. But this option is excluded here and we opt to the exclusive use of higher order p-YM densities.\\nin spacetime dimension d, which vanishes identically in dimensions up to d = 2p. This is because in the given dimensions the p-th member of the gravitational hierarchy becomes a total divergence. But for given d it is necessary to have 2p ≥ d, whence (17) trivialises. The situation would be quite different if the dilaton field were included together with the higher order gravitational terms, preventing the p-gravity density becoming a total divergence. But this option is excluded here and we opt to the exclusive use of higher order p-YM densities.\\nYet other fields occurring in supergravities might be considered, e.g. Kalb-Ramond, or totally antisymmetric tensor fields. But these being Abelian, their effect would be felt only in given dimensions, or, subject to much less stringent symmetries than the spherical. This also is not a flexible option, so we restrict our attention to p-YM densities only.\\nYet other fields occurring in supergravities might be considered, e.g. Kalb-Ramond, or totally antisymmetric tensor fields. But these being Abelian, their effect would be felt only in given dimensions, or, subject to much less stringent symmetries than the spherical. This also is not a flexible option, so we restrict our attention to p-YM densities only.\"\n },\n {\n \"title\": \"Nonabelian hierarchies 3.1 The Lagrangean and field equations\",\n \"content\": \"\"\n },\n {\n \"title\": \"Nonabelian hierarchies 3.1 The Lagrangean and field equations\",\n \"content\": \"Since finite mass spherically symmetric solutions play a central role in AdS/CFT, it is desirable that the nonexistence result presented in Section 2.5 be circumvented.\\nSince finite mass spherically symmetric solutions play a central role in AdS/CFT, it is desirable that the nonexistence result presented in Section 2.5 be circumvented.\\nA simple way to circumvent these arguments and to find nontrivial solutions is to to modify the matter Lagrangean by adding higher order terms in the YM hierarchy. As noted in footnote 1 , these are constructed exclusively from YM curvature 2p-forms. For Λ = 0, asymptotically flat, finite energy solutions of this modified EYM system are constructed in [13,14].\\nA simple way to circumvent these arguments and to find nontrivial solutions is to to modify the matter Lagrangean by adding higher order terms in the YM hierarchy. As noted in footnote 1 , these are constructed exclusively from YM curvature 2p-forms. For Λ = 0, asymptotically flat, finite energy solutions of this modified EYM system are constructed in [13,14].\\nSuch terms as we propose to add are predicted by string theory, and hence provide a link with the AdS/CFT correspondence too. But here we are guided predominantly by symmetry considerations and do not claim to be employing terms strictly following from superstring theory. The situation concerning higher order YM curvature terms in the string theory effective action is complex and as yet not fully resolved. While YM terms up to F 4 arise from (the non Abelian version of) the Born-Infeld action [36], it appears that this approach does not yield all the F 6 terms [37]. Terms of order F 6 and higher can also be obtained by employing the constraints of (maximal) supersymmetry [38]. The results of the various approaches are not identical.\\nSuch terms as we propose to add are predicted by string theory, and hence provide a link with the AdS/CFT correspondence too. But here we are guided predominantly by symmetry considerations and do not claim to be employing terms strictly following from superstring theory. The situation concerning higher order YM curvature terms in the string theory effective action is complex and as yet not fully resolved. While YM terms up to F 4 arise from (the non Abelian version of) the Born-Infeld action [36], it appears that this approach does not yield all the F 6 terms [37]. Terms of order F 6 and higher can also be obtained by employing the constraints of (maximal) supersymmetry [38]. The results of the various approaches are not identical.\\nThe definition we use for superposed YM hierarchy is\\nThe definition we use for superposed YM hierarchy is\\nwhere F (2p) is the 2p-form p-fold totally antisymmetrised product of the SO(d) YM curvature 2-form F (2)\\nwhere F (2p) is the 2p-form p-fold totally antisymmetrised product of the SO(d) YM curvature 2-form F (2)\\nEven though the 2p-form (19) is dual to a total divergence, namely the divergence of the corresponding Chern-Simons form, the density (18) is never a total divergence since it is the square of one. But the 2p-form (19) vanishes by (anti)symmetry for d < 2p so that the upper limit in the summation in (18) is P = d 2 for even d and P = d−1 2 for odd d. We define the p-stress tensor pertaining to each term in (18) as\\nEven though the 2p-form (19) is dual to a total divergence, namely the divergence of the corresponding Chern-Simons form, the density (18) is never a total divergence since it is the square of one. But the 2p-form (19) vanishes by (anti)symmetry for d < 2p so that the upper limit in the summation in (18) is P = d 2 for even d and P = d−1 2 for odd d. We define the p-stress tensor pertaining to each term in (18) as\\nFor the particular spherically symmetric ansatz considered in Section 2, we express the reduced YM Lagrangean arising from (18) as\\nFor the particular spherically symmetric ansatz considered in Section 2, we express the reduced YM Lagrangean arising from (18) as\\nhaving used the shorthand notation\\nhaving used the shorthand notation\\nFor this general ansatz, we find the field equations\\nFor this general ansatz, we find the field equations\\nwhich can also be derived from the reduced action\\nwhich can also be derived from the reduced action\\nFor τ i = 0, i > 2 the equations (8)-(10) are recovered. Note also the σ equation decouples and can be treated separately.\\nFor τ i = 0, i > 2 the equations (8)-(10) are recovered. Note also the σ equation decouples and can be treated separately.\"\n },\n {\n \"title\": \"Boundary conditions\",\n \"content\": \"\"\n },\n {\n \"title\": \"Boundary conditions\",\n \"content\": \"The asymptotic solutions to these equations can be systematically constructed in both regions, near the origin (or event horizon) and for r ≫ 1. The corresponding expansion as r → 0 is\\nThe asymptotic solutions to these equations can be systematically constructed in both regions, near the origin (or event horizon) and for r ≫ 1. The corresponding expansion as r → 0 is\\nand contains one essential parameter b (the value of σ(0) ≡ σ 0 can be fixed by rescaling the time coordinate). For black hole configurations with a regular, nonextremal event horizon at r = r h , the expression near the event horizon is\\nand contains one essential parameter b (the value of σ(0) ≡ σ 0 can be fixed by rescaling the time coordinate). For black hole configurations with a regular, nonextremal event horizon at r = r h , the expression near the event horizon is\\nthe value of the gauge field on the event horizon being the esential parameter. Here the obvious condition N ′ (r h ) > 0 imposes some limits on the event horizon radius as a function of w h for given (τ i , Λ).\\nthe value of the gauge field on the event horizon being the esential parameter. Here the obvious condition N ′ (r h ) > 0 imposes some limits on the event horizon radius as a function of w h for given (τ i , Λ).\\nSince the field equations are invariant under w → −w, one can take w(0) = 1 and w(r h ) > 0 without any loss of generality.\\nSince the field equations are invariant under w → −w, one can take w(0) = 1 and w(r h ) > 0 without any loss of generality.\\nFor r ≫ 1 we find for both regular and black hole solutions\\nFor r ≫ 1 we find for both regular and black hole solutions\\nThese boundary conditions are also shared by the asymptotically flat solutions (with a different decay of the mass function m(r), however), w = ±1 being again the only allowed values of the gauge function as r → ∞. Therefore, we expect to find a qualitatively similar picture in both cases. We will find in Section 5 that the constant M in the above relations is the ADM mass up to a d-dependent factor. However, in the discussion of numerical solutions we will refer to M to as the mass of the solutions.\\nThese boundary conditions are also shared by the asymptotically flat solutions (with a different decay of the mass function m(r), however), w = ±1 being again the only allowed values of the gauge function as r → ∞. Therefore, we expect to find a qualitatively similar picture in both cases. We will find in Section 5 that the constant M in the above relations is the ADM mass up to a d-dependent factor. However, in the discussion of numerical solutions we will refer to M to as the mass of the solutions.\"\n },\n {\n \"title\": \"Further relations\",\n \"content\": \"\"\n },\n {\n \"title\": \"Further relations\",\n \"content\": \"The form (25) of the reduced action allow to derive an useful virial relation. To this end, we use the scaling technique proposed in [33,34] for the case of spherically symmetric gravitating systems. Let us assume the existence of a globally regular solution m(r), σ(r), w(r) of the field equations (24), with suitable boundary conditions at the origin and at infinity. Then each member of the 1-parameter family\\nThe form (25) of the reduced action allow to derive an useful virial relation. To this end, we use the scaling technique proposed in [33,34] for the case of spherically symmetric gravitating systems. Let us assume the existence of a globally regular solution m(r), σ(r), w(r) of the field equations (24), with suitable boundary conditions at the origin and at infinity. Then each member of the 1-parameter family\\nassumes the same boundary values at r = 0 and r = ∞, and the action S λ ≡ S[m λ , σ λ , w λ ] must have a critical point at λ = 1, i.e. [dS/dλ] λ=1 = 0. Therefore we find the following virial relation satisfied by the finite energy solutions of the field equations (note that following [33], it is possible to write a similar relation for black hole configurations, also)\\nassumes the same boundary values at r = 0 and r = ∞, and the action S λ ≡ S[m λ , σ λ , w λ ] must have a critical point at λ = 1, i.e. [dS/dλ] λ=1 = 0. Therefore we find the following virial relation satisfied by the finite energy solutions of the field equations (note that following [33], it is possible to write a similar relation for black hole configurations, also)\\n+2p\\n+2p\\nFor p = 1, i.e. a F 2 -theory, the above relation reads\\nFor p = 1, i.e. a F 2 -theory, the above relation reads\\nwhich clearly shows that no nontrivial gravitating solution with finite mass exists for d > 4, since all terms in the integrant are strictly positive quantities. Therefore it becomes obvious that new terms in the YM-hierarchy should be introduced as the spacetime dimension increases. For a given d, the relation P > [(d + 1)/4] should be satisfied. As it happens, to go to 5 ≤ d < 9 it is necessary to include at least the second member of the YM hierarchy to provide the requisite scaling (similarly for 9 ≤ d < 13 it is necessary to include the thrid member of the YM hierarchy). In practice we add only the lowest order such term necessary.\\nwhich clearly shows that no nontrivial gravitating solution with finite mass exists for d > 4, since all terms in the integrant are strictly positive quantities. Therefore it becomes obvious that new terms in the YM-hierarchy should be introduced as the spacetime dimension increases. For a given d, the relation P > [(d + 1)/4] should be satisfied. As it happens, to go to 5 ≤ d < 9 it is necessary to include at least the second member of the YM hierarchy to provide the requisite scaling (similarly for 9 ≤ d < 13 it is necessary to include the thrid member of the YM hierarchy). In practice we add only the lowest order such term necessary.\\nWe mention here also the Hawking temperature expression of the black hole solutions. For the line element (5), if we treat t as complex, then its imaginary part is a coordinate for a nonsingular Euclidean submanifold iff it is periodic with period\\nWe mention here also the Hawking temperature expression of the black hole solutions. For the line element (5), if we treat t as complex, then its imaginary part is a coordinate for a nonsingular Euclidean submanifold iff it is periodic with period\\nThen continuous Euclidean Green functions must have this period, so by standard arguments the Hawking temperature is (with k B =h = 1)\\nThen continuous Euclidean Green functions must have this period, so by standard arguments the Hawking temperature is (with k B =h = 1)\\nThus the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum black hole of equal horizon area.\\nThus the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum black hole of equal horizon area.\\nIn the presence of higher oder terms in the YM action, dimensionless quantities are obtained by rescaling\\nIn the presence of higher oder terms in the YM action, dimensionless quantities are obtained by rescaling\\nThis reveals the existence of one fundamental parameter which gives the strength of the gravitational interaction\\nThis reveals the existence of one fundamental parameter which gives the strength of the gravitational interaction\\nand P − 2 independent coupling constants\\nand P − 2 independent coupling constants\\nwith k = 3, P − 2 (i.e. no such constants appear in a p = 2 system).\\nwith k = 3, P − 2 (i.e. no such constants appear in a p = 2 system).\\nFor the F (2) + F (4) systems in d = 6, 7, 8 considered in [13], there exist gravity decoupling solutions at α = 0, from which the gravitating solutions branch out to a maximum value of α max and then decrease. The second limit of α → 0 also exists. For the same system in d = 5 studied in [14] on the other hand, there exists no gravity decoupling limit but nonetheless the solution branches out from α = 0 by employing a scaling procedure [19], and again α increases to a value α max and decreases. But in this case it does not reach the α → 0 limit. Rather it stops at a new critical point α c around which it oscillates [14], which is a new type of critical point identified in [19] and named a 'conical' fixed point.\\nFor the F (2) + F (4) systems in d = 6, 7, 8 considered in [13], there exist gravity decoupling solutions at α = 0, from which the gravitating solutions branch out to a maximum value of α max and then decrease. The second limit of α → 0 also exists. For the same system in d = 5 studied in [14] on the other hand, there exists no gravity decoupling limit but nonetheless the solution branches out from α = 0 by employing a scaling procedure [19], and again α increases to a value α max and decreases. But in this case it does not reach the α → 0 limit. Rather it stops at a new critical point α c around which it oscillates [14], which is a new type of critical point identified in [19] and named a 'conical' fixed point.\"\n },\n {\n \"title\": \"Numerical solutions\",\n \"content\": \"\"\n },\n {\n \"title\": \"Numerical solutions\",\n \"content\": \"In the present work, we restrict our attention to the simplest nontrivial cases with only the two terms p = 1 and 2 in the YM hierarchy. However, we have obtained some numerical results also for a P = 3 hierarchy with β 1 = 1, which will be briefly mentioned. For d = 5, the solutions we found have some special features which will be discussed separately.\\nIn the present work, we restrict our attention to the simplest nontrivial cases with only the two terms p = 1 and 2 in the YM hierarchy. However, we have obtained some numerical results also for a P = 3 hierarchy with β 1 = 1, which will be briefly mentioned. For d = 5, the solutions we found have some special features which will be discussed separately.\\nBoth regular and black hole solutions of the EYM-hierarchy equations appear to exist for any value of Λ. Given (α, d, Λ), AAdS solutions may exist for discrete set of shooting parameters b and w h respectively. We follow the usual approach and, by using a standard ordinary differential equation solver, we evaluate the initial conditions at r = 10 −5 (or r h + 10 −5 ) for global tolerance 10 −14 , adjusting for shooting parameters and integrating towards r → ∞, and looking for AAdS solutions with a finite mass.\\nBoth regular and black hole solutions of the EYM-hierarchy equations appear to exist for any value of Λ. Given (α, d, Λ), AAdS solutions may exist for discrete set of shooting parameters b and w h respectively. We follow the usual approach and, by using a standard ordinary differential equation solver, we evaluate the initial conditions at r = 10 −5 (or r h + 10 −5 ) for global tolerance 10 −14 , adjusting for shooting parameters and integrating towards r → ∞, and looking for AAdS solutions with a finite mass.\\nSimilar to the p = 1, d = 4 asymptotically flat case, it can be proven that for all AAdS (or Λ = 0) solutions, w(r) is confined within the strip |w(r)| < 1. This can proven as follows: we suppose the existence of solutions with w(r) > 1 for some interval of r. Therefore w must develop a maximum for some r 0 , w ′ (r 0 ) = 0 and w(r 0 ) > 1 with w ′′ (r 0 ) < 0. However, the equations (24) imply that in the region w > 1 the only extremum can be a minimum. Therefore the condition |w(r)| < 1 is always fullfiled. As a general feature, all solutions discussed in the rest of this section present only one node in the gauge function w(r). Similar to the Λ = 0 case, we could not find multinode solutions 5 .\\nSimilar to the p = 1, d = 4 asymptotically flat case, it can be proven that for all AAdS (or Λ = 0) solutions, w(r) is confined within the strip |w(r)| < 1. This can proven as follows: we suppose the existence of solutions with w(r) > 1 for some interval of r. Therefore w must develop a maximum for some r 0 , w ′ (r 0 ) = 0 and w(r 0 ) > 1 with w ′′ (r 0 ) < 0. However, the equations (24) imply that in the region w > 1 the only extremum can be a minimum. Therefore the condition |w(r)| < 1 is always fullfiled. As a general feature, all solutions discussed in the rest of this section present only one node in the gauge function w(r). Similar to the Λ = 0 case, we could not find multinode solutions 5 .\\nThe absence of multinode solutions in this F (2) + F (4) model with Λ = 0, in the relevant dimensions 5 ≤ d ≤ 8 is analytically explained in [19]. We expect that the relevant fixed point analysis yields qualitatively similar results also for |Λ| > 0. This is borne out by our numerical results.\\nThe absence of multinode solutions in this F (2) + F (4) model with Λ = 0, in the relevant dimensions 5 ≤ d ≤ 8 is analytically explained in [19]. We expect that the relevant fixed point analysis yields qualitatively similar results also for |Λ| > 0. This is borne out by our numerical results.\\nFor any regular solution, the metric functions m(r) and σ(r) always increase monotonically with growing r from m(0) = 0 and σ(0) = σ 0 at the origin to m(∞) = M and σ(∞) = 1, respectively. The gauge function always interpolates between w(0) = 1 and w(∞) = −1 without any local extrema. For black hole configurations, the behaviour of the functions m(r), σ(r) and w(r) is similar to that for regular solutions. The gauge potential w(r) starts from some finite value 0 < w(r h ) < 1 at the horizon and monotonically approaches −1 at infinity. The metric functions m(r) and σ(r) increase also monotonically with r. In the asymptotic region, the geometry corresponds to a Schwarzschild-AdS solution. However, although these solutions are static and have vanishing YM charges (w 2 (∞) = 1), they are different from the Schwarzschild-AdS black hole, and therefore are not fully characterised by the mass-parameter M .\\nFor any regular solution, the metric functions m(r) and σ(r) always increase monotonically with growing r from m(0) = 0 and σ(0) = σ 0 at the origin to m(∞) = M and σ(∞) = 1, respectively. The gauge function always interpolates between w(0) = 1 and w(∞) = −1 without any local extrema. For black hole configurations, the behaviour of the functions m(r), σ(r) and w(r) is similar to that for regular solutions. The gauge potential w(r) starts from some finite value 0 < w(r h ) < 1 at the horizon and monotonically approaches −1 at infinity. The metric functions m(r) and σ(r) increase also monotonically with r. In the asymptotic region, the geometry corresponds to a Schwarzschild-AdS solution. However, although these solutions are static and have vanishing YM charges (w 2 (∞) = 1), they are different from the Schwarzschild-AdS black hole, and therefore are not fully characterised by the mass-parameter M .\"\n },\n {\n \"title\": \"Regular solutions d = 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Regular solutions d = 5\",\n \"content\": \"As α 2 → 0, the YM equations present a nontrivial, finite energy solution in a fixed AdS background. This nongravitating configuration approaches in the Λ = 0 limit the YM instanton in four dimensional flat space [39].\\nAs α 2 → 0, the YM equations present a nontrivial, finite energy solution in a fixed AdS background. This nongravitating configuration approaches in the Λ = 0 limit the YM instanton in four dimensional flat space [39].\\nWhen α 2 increases, this solution gets deformed by gravity and the mass M decreases. At the same time, both the value σ(0) and the minimal value N m of the function N (r) decrease, as indicated in Figure 3. This branch of solutions exists up to a maximal value α max of the parameter α, which is smaller than the corresponding value in the asymptotically flat case [14]. For example, we find numerically α 2 max ≈ 0.3445 for Λ = −0.2 while the corresponding value for Λ = −0.01 is α 2 max ≈ 0.5322. (without a cosmological term, this branch extends up to α 2 max ≈ 0.5648.) Similar to the EYM theory with Λ = 0 [14], we found always another branch of solutions on the interval α 2 ∈ [α 2 cr(1) , α 2 max ] with α 2 cr(1) depending again on the value of Λ (e.g. α 2 cr(1) ≈ 0.2876 for Λ = −0.2). On this second branch of solutions, both σ(0) and N m continue to decrease but stay finite. However, a third branch of solutions exists for α 2 ∈ [α 2 cr(1) , α 2 cr(2) ] , on which the two quantities decrease further. A fourth branch of solutions has also been found, with a corresponding α 2 cr(3) close to α 2 cr (2) . Further branches of solutions, exhibiting more oscillations very likely exist but their study is a difficult numerical problem. Along this succession of branches, the main observation is that the value σ(0) decreases much faster than that of N m as illustrated in Figure 3. Also, the mass parameters do not increase significantly along these secondary branches. However, the shooting parameter b increases to very large values. The pattern strongly suggests that after a finite (or more likely infinite) number of oscilations of σ(0), the solution terminates into a singular solution with σ(0) = 0 and a finite value of N (0). This is the behaviour observed in [14] for the EYM theory with Λ = 0. The inclusion of a negative cosmological constant does not seem to qualitatively change the properties of the system, but leads to different values of the critical parameters. As in the Λ = 0 case, the dominant term at the gravity decoupling limit α → 0 is the F (2) term, the energy being given by the action of the usual instanton [39], while as α → α cr the dominant term is F (4). The mechanism for this effect is explained in [19] (for Λ = 0) and is supported by our numerical results here. The typical d = 5 globally regular solutions look very similar to the d = 6, 8 profiles presented in Figures 4, 5.\\nWhen α 2 increases, this solution gets deformed by gravity and the mass M decreases. At the same time, both the value σ(0) and the minimal value N m of the function N (r) decrease, as indicated in Figure 3. This branch of solutions exists up to a maximal value α max of the parameter α, which is smaller than the corresponding value in the asymptotically flat case [14]. For example, we find numerically α 2 max ≈ 0.3445 for Λ = −0.2 while the corresponding value for Λ = −0.01 is α 2 max ≈ 0.5322. (without a cosmological term, this branch extends up to α 2 max ≈ 0.5648.) Similar to the EYM theory with Λ = 0 [14], we found always another branch of solutions on the interval α 2 ∈ [α 2 cr(1) , α 2 max ] with α 2 cr(1) depending again on the value of Λ (e.g. α 2 cr(1) ≈ 0.2876 for Λ = −0.2). On this second branch of solutions, both σ(0) and N m continue to decrease but stay finite. However, a third branch of solutions exists for α 2 ∈ [α 2 cr(1) , α 2 cr(2) ] , on which the two quantities decrease further. A fourth branch of solutions has also been found, with a corresponding α 2 cr(3) close to α 2 cr (2) . Further branches of solutions, exhibiting more oscillations very likely exist but their study is a difficult numerical problem. Along this succession of branches, the main observation is that the value σ(0) decreases much faster than that of N m as illustrated in Figure 3. Also, the mass parameters do not increase significantly along these secondary branches. However, the shooting parameter b increases to very large values. The pattern strongly suggests that after a finite (or more likely infinite) number of oscilations of σ(0), the solution terminates into a singular solution with σ(0) = 0 and a finite value of N (0). This is the behaviour observed in [14] for the EYM theory with Λ = 0. The inclusion of a negative cosmological constant does not seem to qualitatively change the properties of the system, but leads to different values of the critical parameters. As in the Λ = 0 case, the dominant term at the gravity decoupling limit α → 0 is the F (2) term, the energy being given by the action of the usual instanton [39], while as α → α cr the dominant term is F (4). The mechanism for this effect is explained in [19] (for Λ = 0) and is supported by our numerical results here. The typical d = 5 globally regular solutions look very similar to the d = 6, 8 profiles presented in Figures 4, 5.\"\n },\n {\n \"title\": \"Regular solutions d > 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Regular solutions d > 5\",\n \"content\": \"The solutions in this case resemble again the Λ = 0 situation. In the presence of suitable higher order term in the hierarchy, the YM equations admit finite energy solutions in a fixed AdS d A p = 1, AdS 4 exact solution was found in [40], its higher order generalisation (specifically for p = 2, AdS 8 ) for d > 4 being discussed in [41]. According to the standard arguments, this AdS soliton can be generalised in the presence of gravity, provided that the dimensionless coupling constant α is small enough. Therefore the gravitating solutions exist up to a maximal value α max of the gravitational coupling constant. This value α max for a given d depends on the value of Λ. For example in d = 6, α max (Λ = 0) = 0.12675; α max (Λ = −1) = 0.070422; in 8 dimensions we find α max (Λ = 0) = 0.002193 while α max (Λ = −5) = 6.69 × 10 −4 .\\nThe solutions in this case resemble again the Λ = 0 situation. In the presence of suitable higher order term in the hierarchy, the YM equations admit finite energy solutions in a fixed AdS d A p = 1, AdS 4 exact solution was found in [40], its higher order generalisation (specifically for p = 2, AdS 8 ) for d > 4 being discussed in [41]. According to the standard arguments, this AdS soliton can be generalised in the presence of gravity, provided that the dimensionless coupling constant α is small enough. Therefore the gravitating solutions exist up to a maximal value α max of the gravitational coupling constant. This value α max for a given d depends on the value of Λ. For example in d = 6, α max (Λ = 0) = 0.12675; α max (Λ = −1) = 0.070422; in 8 dimensions we find α max (Λ = 0) = 0.002193 while α max (Λ = −5) = 6.69 × 10 −4 .\\nWhen α increases, the mass of the gravitating solutions decreases while the function N (r) develops a local minimum N m which becomes deeper while gravity becomes stronger and the value σ(0) decreases from one. At the same time, the value of the shooting parameter b increases with α. Our numerical analysis for d ≤ 10 indicates that a second branch of regular solutions always exists, starting at α max . Along this second branch the values σ(0) and N m decrease monotonically with α, while b and M still increase. The mass of a second branch solution is always larger than the corresponding mass (for the same value of α) on the first branch. For d > 6, the numerical analysis suggests that this second branch persists up to α 2 ≃ 0 and that in this limit σ(0) approaches a very small value. As far as our numerical analysis indicates, the value N m tends to a finite value in this limit so that there occurs no horizon. Therefore two regular solutions seems always to exist for any α < α d .\\nWhen α increases, the mass of the gravitating solutions decreases while the function N (r) develops a local minimum N m which becomes deeper while gravity becomes stronger and the value σ(0) decreases from one. At the same time, the value of the shooting parameter b increases with α. Our numerical analysis for d ≤ 10 indicates that a second branch of regular solutions always exists, starting at α max . Along this second branch the values σ(0) and N m decrease monotonically with α, while b and M still increase. The mass of a second branch solution is always larger than the corresponding mass (for the same value of α) on the first branch. For d > 6, the numerical analysis suggests that this second branch persists up to α 2 ≃ 0 and that in this limit σ(0) approaches a very small value. As far as our numerical analysis indicates, the value N m tends to a finite value in this limit so that there occurs no horizon. Therefore two regular solutions seems always to exist for any α < α d .\\nThe case d = 6 is special, since the numerical procedure fails to give reliable results for second branch solutions, starting with some α d , whose value is Λ-dependent (for example we found α d (Λ = −1) = 0.0435 while α d (Λ = 0) = 0.0573). The quantity σ(0) reaches a very small value as α → α d . The minimal value of N (r) remains finite so that no horizon is approached. We expect that a different parametrisation of the metric and variables would allow us to continue this second branch to α → 0 in this case, too.\\nThe case d = 6 is special, since the numerical procedure fails to give reliable results for second branch solutions, starting with some α d , whose value is Λ-dependent (for example we found α d (Λ = −1) = 0.0435 while α d (Λ = 0) = 0.0573). The quantity σ(0) reaches a very small value as α → α d . The minimal value of N (r) remains finite so that no horizon is approached. We expect that a different parametrisation of the metric and variables would allow us to continue this second branch to α → 0 in this case, too.\\nThe behaviours just described qualitatively duplicate those of the Λ = 0 case [13], and are analysed in [19]. Likewise, the solutions are dominated by the F (2) terms in the gravity decoupling limit α → 0, while at the other end (on the second mass-branch) it is the F (4) terms that dominate.\\nThe behaviours just described qualitatively duplicate those of the Λ = 0 case [13], and are analysed in [19]. Likewise, the solutions are dominated by the F (2) terms in the gravity decoupling limit α → 0, while at the other end (on the second mass-branch) it is the F (4) terms that dominate.\\nTypical The results we found by including the p = 3 term in the YM-hierarchy for d = 9, 10 follows the same pattern. Although the picture gets more complicated by the existence of one more coupling parameter, two branches of solution are always found to exist. We noticed also the existence of a maximal value of α which is (Λ, β 1 ) dependent.\\nTypical The results we found by including the p = 3 term in the YM-hierarchy for d = 9, 10 follows the same pattern. Although the picture gets more complicated by the existence of one more coupling parameter, two branches of solution are always found to exist. We noticed also the existence of a maximal value of α which is (Λ, β 1 ) dependent.\"\n },\n {\n \"title\": \"Black hole solutions d = 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Black hole solutions d = 5\",\n \"content\": \"According to the standard arguments, one can expect to find black hole generalisations for any regular configurations, at least for small values of the horizon radius r h . For completeness we discuss here the basic features of the AAdS black hole solutions.\\nAccording to the standard arguments, one can expect to find black hole generalisations for any regular configurations, at least for small values of the horizon radius r h . For completeness we discuss here the basic features of the AAdS black hole solutions.\\nAgain, the case d = 5 is special. The properties of these AAdS solutions are rather similar to the five dimensional asymptotically flat black hole configurations discussed in [14]. First, black hole solutions seem to exist for all values of α for which regular solutions could be constructed. Also, solutions exist only for a limited region of the (r h , α) space.\\nAgain, the case d = 5 is special. The properties of these AAdS solutions are rather similar to the five dimensional asymptotically flat black hole configurations discussed in [14]. First, black hole solutions seem to exist for all values of α for which regular solutions could be constructed. Also, solutions exist only for a limited region of the (r h , α) space.\\nThe typical behaviour of solutions as a function of r h is presented in Figure 9, for a small value of α compared to the maximal value α max of the regular solutions, in which case we notice the existence of only one r h = 0 regular configuration. Starting from this regular solution and increasing the event horizon radius, we find a first branch of solutions which extends to a maximal value r h(max) . As seen in Figures 9, 10, the value of r h(max) depends on Λ, α. The Hawking temperature decreases on this branch, while the mass parameter increases; however, the variation of mass and σ(r h ) is relatively small. Extending backwards in r h , we find a second branch of solutions for r h < r h(max) . This second branch stops at some critical value r h(cr) , where the numerical iteration fails to converge. The value of σ(r h ) on this branch decreases drastically, as shown in Figure 9. Also, the Hawking temperature after initially increasing, strongly decreases for values near r h(cr) , approaching a very small value, while the increase of the total mass is still very small. Similar to the Λ = 0 case [14], higher branches of solutions on which the value σ(r h ) continues to decrease further to zero are likely to exist. However, the extension of these branches in r h will be very small, which makes their study difficult. An approach to this problem with different parametrisation appears to be necessary.\\nThe typical behaviour of solutions as a function of r h is presented in Figure 9, for a small value of α compared to the maximal value α max of the regular solutions, in which case we notice the existence of only one r h = 0 regular configuration. Starting from this regular solution and increasing the event horizon radius, we find a first branch of solutions which extends to a maximal value r h(max) . As seen in Figures 9, 10, the value of r h(max) depends on Λ, α. The Hawking temperature decreases on this branch, while the mass parameter increases; however, the variation of mass and σ(r h ) is relatively small. Extending backwards in r h , we find a second branch of solutions for r h < r h(max) . This second branch stops at some critical value r h(cr) , where the numerical iteration fails to converge. The value of σ(r h ) on this branch decreases drastically, as shown in Figure 9. Also, the Hawking temperature after initially increasing, strongly decreases for values near r h(cr) , approaching a very small value, while the increase of the total mass is still very small. Similar to the Λ = 0 case [14], higher branches of solutions on which the value σ(r h ) continues to decrease further to zero are likely to exist. However, the extension of these branches in r h will be very small, which makes their study difficult. An approach to this problem with different parametrisation appears to be necessary.\\nHowever we find, that the global picture is changed by considering large enough values of α. In this case, more than one regular configuration exists for a given value of α. This situation is illustrated in Figure 10, for solutions with α 2 = 0.5. Two regular solutions exist for this particular value of α and we find two black hole branches connecting these r h = 0 configurations. Again, the mass of the second branch solutions is always larger than the corresponding mass on the first branch.\\nHowever we find, that the global picture is changed by considering large enough values of α. In this case, more than one regular configuration exists for a given value of α. This situation is illustrated in Figure 10, for solutions with α 2 = 0.5. Two regular solutions exist for this particular value of α and we find two black hole branches connecting these r h = 0 configurations. Again, the mass of the second branch solutions is always larger than the corresponding mass on the first branch.\\nPreliminary numerical results indicate an even more complicated picture for solutions with α near α max . In this case the configurations combine features of both types of solutions discussed above. Several branches of black hole solutions are found for the same α. These branches start from regular configurations and are possibly disconnected.\\nPreliminary numerical results indicate an even more complicated picture for solutions with α near α max . In this case the configurations combine features of both types of solutions discussed above. Several branches of black hole solutions are found for the same α. These branches start from regular configurations and are possibly disconnected.\"\n },\n {\n \"title\": \"Black hole solutions d > 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Black hole solutions d > 5\",\n \"content\": \"Although predicted in [14], no discussion of the Λ = 0, d > 5 black holes is presented in literature.\\nAlthough predicted in [14], no discussion of the Λ = 0, d > 5 black holes is presented in literature.\\nAgain, black hole counterparts appear to exist for any regular solution. However, solutions with the right asymptotics are found for a limited region of the (r h , α) space only. We plot in Figures 11-13 some results we found for d = 6, 7 and d = 8. Starting for a given α 0 < α max from a r h = 0 first branch regular solution, we found the existence of a branch of black hole solutions extending up to a maximal value of the event horizon radius r h = r max h . When r h increases, the mass and the Hawking temperature increase while the value σ(r h ) decreases from its value at the regular solution. A second branch of black hole solutions seems to appear always at r max h , extending backwards to a zero event horizon radius. This limiting solution corresponds to the second branch of the regular solution at this value of α = α 0 . Like in those d = 5 cases where there are two regular solutions for a given α, say α 0 , here too these two regular solutions are the r h → 0 limits of the corresponding black hole \\\"loop\\\", connecting the two regular solutions. Along this second branch the values σ(r h ) of the metric function σ on the event horizon decrease monotonically with r h , while the Hawking temperature strongly increases. The mass of the solution of the second branch is larger than the corresponding one on the first branch, as illustrated by Figures 10-12. It is interesting to note here from the point of view of numerics, that a black hole loop corresponding to the two r h → 0 limits of two regular solutions, say at α 0 , can be constructed numerically even when the numerical process for the higher mass branch of the regular solutions for this α 0 runs into difficulties.\\nAgain, black hole counterparts appear to exist for any regular solution. However, solutions with the right asymptotics are found for a limited region of the (r h , α) space only. We plot in Figures 11-13 some results we found for d = 6, 7 and d = 8. Starting for a given α 0 < α max from a r h = 0 first branch regular solution, we found the existence of a branch of black hole solutions extending up to a maximal value of the event horizon radius r h = r max h . When r h increases, the mass and the Hawking temperature increase while the value σ(r h ) decreases from its value at the regular solution. A second branch of black hole solutions seems to appear always at r max h , extending backwards to a zero event horizon radius. This limiting solution corresponds to the second branch of the regular solution at this value of α = α 0 . Like in those d = 5 cases where there are two regular solutions for a given α, say α 0 , here too these two regular solutions are the r h → 0 limits of the corresponding black hole \\\"loop\\\", connecting the two regular solutions. Along this second branch the values σ(r h ) of the metric function σ on the event horizon decrease monotonically with r h , while the Hawking temperature strongly increases. The mass of the solution of the second branch is larger than the corresponding one on the first branch, as illustrated by Figures 10-12. It is interesting to note here from the point of view of numerics, that a black hole loop corresponding to the two r h → 0 limits of two regular solutions, say at α 0 , can be constructed numerically even when the numerical process for the higher mass branch of the regular solutions for this α 0 runs into difficulties.\\nThe introduction of a negative cosmological constant does not appear to change this picture qualitatively. However, we notice a smaller value of r max h with increasing |Λ| and a larger value of M for the same α. In Figure 14, we present the profiles of the metric functions N and σ and gauge function w(r) for the same values of α 2 , Λ on the first and second branch for some d = 6 solutions. The dependence of these functions on the value of Λ is illustrated in Figure 15 for several d = 7 black hole solutions.\\nThe introduction of a negative cosmological constant does not appear to change this picture qualitatively. However, we notice a smaller value of r max h with increasing |Λ| and a larger value of M for the same α. In Figure 14, we present the profiles of the metric functions N and σ and gauge function w(r) for the same values of α 2 , Λ on the first and second branch for some d = 6 solutions. The dependence of these functions on the value of Λ is illustrated in Figure 15 for several d = 7 black hole solutions.\"\n },\n {\n \"title\": \"A computation of mass and action\",\n \"content\": \"\"\n },\n {\n \"title\": \"A computation of mass and action\",\n \"content\": \"\"\n },\n {\n \"title\": \"The counterterm method\",\n \"content\": \"\"\n },\n {\n \"title\": \"The counterterm method\",\n \"content\": \"It is well known that the generalisation of Komar's formula for AAdS spacetimes is not straightforward and requires the further subtraction of a background configuration in order to render a finite result for the mass.\\nIt is well known that the generalisation of Komar's formula for AAdS spacetimes is not straightforward and requires the further subtraction of a background configuration in order to render a finite result for the mass.\\nThis problem was addressed for the first time in the eighties, with different approaches (see for instance Ref. [42,43]). Another formalism to define conserved charges in AAdS spacetimes was proposed in [44] and uses conformal techniques to construct conserved quantities yielding the results obtained by Hamiltonian methods. Other more recent approaches to the same problem are presented in Ref. [45].\\nThis problem was addressed for the first time in the eighties, with different approaches (see for instance Ref. [42,43]). Another formalism to define conserved charges in AAdS spacetimes was proposed in [44] and uses conformal techniques to construct conserved quantities yielding the results obtained by Hamiltonian methods. Other more recent approaches to the same problem are presented in Ref. [45].\\nAs expected, these different methods yield the same total mass for the spherically symmetric AAdS configurations considered in Section 3\\nAs expected, these different methods yield the same total mass for the spherically symmetric AAdS configurations considered in Section 3\\nwhere Ω d−2 = 2π (d−1)/2 /Γ((d − 1)/2) is the area of a unit (d − 2)-dimensional sphere, and M is defined in the second member of (29). A procedure leading (for odd dimensions) to a different result has been proposed by Balasubramanian and Kraus [21]. This technique was inspired by AdS/CFT correspondence and consists in adding suitable counterterms I ct to the action of the theory in order to ensure the finiteness of the boundary stress tensor derived by the quasilocal energy definition [46]. These counterterms are built up with curvature invariants of a boundary ∂M (which is sent to infinity after the integration) and thus obviously they do not alter the bulk equations of motion. Unlike background subtraction, the counterterm approach does no require the identification of a reference spacetime. Given the potential relevance of the EYM solutions in an AdS/CFT context, we present here a computation of the boundary stress tensor and of the total mass, by using the counterterm prescription.\\nwhere Ω d−2 = 2π (d−1)/2 /Γ((d − 1)/2) is the area of a unit (d − 2)-dimensional sphere, and M is defined in the second member of (29). A procedure leading (for odd dimensions) to a different result has been proposed by Balasubramanian and Kraus [21]. This technique was inspired by AdS/CFT correspondence and consists in adding suitable counterterms I ct to the action of the theory in order to ensure the finiteness of the boundary stress tensor derived by the quasilocal energy definition [46]. These counterterms are built up with curvature invariants of a boundary ∂M (which is sent to infinity after the integration) and thus obviously they do not alter the bulk equations of motion. Unlike background subtraction, the counterterm approach does no require the identification of a reference spacetime. Given the potential relevance of the EYM solutions in an AdS/CFT context, we present here a computation of the boundary stress tensor and of the total mass, by using the counterterm prescription.\\nThe following counterterms are sufficient to cancel divergences in a pure gravity theory for d ≤ 9, with several exceptions (see e. g. [47])\\nThe following counterterms are sufficient to cancel divergences in a pure gravity theory for d ≤ 9, with several exceptions (see e. g. [47])\\nHere R ABCD , R AB are the Riemann and Ricci tensors, R is the Ricci scalar for the boundary metric h and Θ(x) is the step function, which is equal to 1 for x ≥ 0 and zero otherwise; A, B, . . . indicate the intrinsic coordinates of the boundary. Using these counterterms one can construct a divergenceless boundary stress tensor, which is given by the variation of the total action at the boundary with respect to h AB . Its explicit expression, restricting for simplicity to d < 7, is\\nHere R ABCD , R AB are the Riemann and Ricci tensors, R is the Ricci scalar for the boundary metric h and Θ(x) is the step function, which is equal to 1 for x ≥ 0 and zero otherwise; A, B, . . . indicate the intrinsic coordinates of the boundary. Using these counterterms one can construct a divergenceless boundary stress tensor, which is given by the variation of the total action at the boundary with respect to h AB . Its explicit expression, restricting for simplicity to d < 7, is\\nwhere\\nwhere\\nis the extrinsic curvature defined in terms of the normal n A to the boundary, K is its trace, and E AB is the Einstein tensor of the intrinsic metric h AB . (The corresponding form of (40) for d = 7, 8 is given e.g. in [48]).\\nis the extrinsic curvature defined in terms of the normal n A to the boundary, K is its trace, and E AB is the Einstein tensor of the intrinsic metric h AB . (The corresponding form of (40) for d = 7, 8 is given e.g. in [48]).\\nThe result we find in this way for T AB is given by\\nThe result we find in this way for T AB is given by\\nwhere u A = δ t A and p is an integer. We can use this approach to assign a mass-energy to an AAdS geometry by writing the boundary metric in an ADM form\\nwhere u A = δ t A and p is an integer. We can use this approach to assign a mass-energy to an AAdS geometry by writing the boundary metric in an ADM form\\nand the definition of the energy in this context is\\nand the definition of the energy in this context is\\nHere ǫ = u µ u ν T µν is the proper energy density while u µ is a timelike unit normal to Σ.\\nHere ǫ = u µ u ν T µν is the proper energy density while u µ is a timelike unit normal to Σ.\\nIf there are matter fields on M, additional counterterms may be needed to regulate the action. This is the case of F (2) theory, discussed in Appendix B. The couinterterm action depends in this case not only on the boundary metric but also on the boundary value of the gauge field.\\nIf there are matter fields on M, additional counterterms may be needed to regulate the action. This is the case of F (2) theory, discussed in Appendix B. The couinterterm action depends in this case not only on the boundary metric but also on the boundary value of the gauge field.\\nHowever, we find that for P > 1 EYM solutions with Λ < 0 in d = 5, 6, 7, 8 dimensions, the prescription (39) removes all divergences. The use of higher order terms in the YM curvature, namely F (2p) forms with p > 1 introduced in Section 3, results in this regularising of the masses. The crucial point here is that these solutions approaches asymptotically a Schwarzschild-AdS background, and the YM asymptotic parameter w 1 appears only in the next to leading order of the T AB expression.\\nHowever, we find that for P > 1 EYM solutions with Λ < 0 in d = 5, 6, 7, 8 dimensions, the prescription (39) removes all divergences. The use of higher order terms in the YM curvature, namely F (2p) forms with p > 1 introduced in Section 3, results in this regularising of the masses. The crucial point here is that these solutions approaches asymptotically a Schwarzschild-AdS background, and the YM asymptotic parameter w 1 appears only in the next to leading order of the T AB expression.\\nThe mass-energy of solutions computed in this way is\\nThe mass-energy of solutions computed in this way is\\nwhere, for 3 < d < 9\\nwhere, for 3 < d < 9\\nThe additional term E 0 appearing in E for d = 5, (7) is the mass of pure global AdS 5, (7) and is usually interpreted as the energy dual to the Casimir energy of the CFT defined on a four (six) dimensional Einstein universe [21]. The metric restricted to the boundary h AB diverges due to an infinite conformal factor r 2 /ℓ 2 . The background metric upon which the dual field theory resides is\\nThe additional term E 0 appearing in E for d = 5, (7) is the mass of pure global AdS 5, (7) and is usually interpreted as the energy dual to the Casimir energy of the CFT defined on a four (six) dimensional Einstein universe [21]. The metric restricted to the boundary h AB diverges due to an infinite conformal factor r 2 /ℓ 2 . The background metric upon which the dual field theory resides is\\nFor the asymptotically AdS d solutions considered here, the (d − 1) dimensional boundary is the Einstein universe, with the line element\\nFor the asymptotically AdS d solutions considered here, the (d − 1) dimensional boundary is the Einstein universe, with the line element\\nIn light of the AdS/CFT correspondence, Balasubramanian and Kraus have interpreted Eq. (40) as < τ AB >= 2 √ −γ δS ef f δγAB , where < τ AB > is the expectation value of the CFT stress tensor. Then, the divergences which appear are simply the standard ultraviolet divergences of a quantum field theory and we can cancel them by adding local counterterms to the action. Corresponding to the boundary metric (47), the stress-energy tensor τ AB for the dual theory can be calculated using the following relation [49] √ −γγ AB < τ BC >= lim\\nIn light of the AdS/CFT correspondence, Balasubramanian and Kraus have interpreted Eq. (40) as < τ AB >= 2 √ −γ δS ef f δγAB , where < τ AB > is the expectation value of the CFT stress tensor. Then, the divergences which appear are simply the standard ultraviolet divergences of a quantum field theory and we can cancel them by adding local counterterms to the action. Corresponding to the boundary metric (47), the stress-energy tensor τ AB for the dual theory can be calculated using the following relation [49] √ −γγ AB < τ BC >= lim\"\n },\n {\n \"title\": \"Action and entropy\",\n \"content\": \"\"\n },\n {\n \"title\": \"Action and entropy\",\n \"content\": \"The above approach can be used to compute the Euclidean action and to prove in a rigorous way that the entropy of the EYM black hole solutions is one quarter of the event horizon area, as expected. Here we start by constructing the path integral [18]\\nThe above approach can be used to compute the Euclidean action and to prove in a rigorous way that the entropy of the EYM black hole solutions is one quarter of the event horizon area, as expected. Here we start by constructing the path integral [18]\\nby integrating over all metrics and matter fields between some given initial and final hypersurfaces, Ψ corresponding here to the SU(2) potentials. By analytically continuing the time coordinate t → iτ , the path integral formally converges, and in the leading order one obtains Z ≃ e −I cl (50) where I cl is the classical action evaluated on the equations of motion of the gravity/matter system. The physical interpretation of this formalism is that the class of regular stationary metrics forms an ensemble of thermodynamic systems at equilibrium temperature T (see e.g. [50]). Z has the interpretation of partition function and we can define the free energy of the system F = −β −1 log Z. Therefore\\nby integrating over all metrics and matter fields between some given initial and final hypersurfaces, Ψ corresponding here to the SU(2) potentials. By analytically continuing the time coordinate t → iτ , the path integral formally converges, and in the leading order one obtains Z ≃ e −I cl (50) where I cl is the classical action evaluated on the equations of motion of the gravity/matter system. The physical interpretation of this formalism is that the class of regular stationary metrics forms an ensemble of thermodynamic systems at equilibrium temperature T (see e.g. [50]). Z has the interpretation of partition function and we can define the free energy of the system F = −β −1 log Z. Therefore\\nor\\nor\\nstraightforwardly follows.\\nstraightforwardly follows.\\nTo compute I cl , we make use of the Einstein equations, replacing the R − 2Λ volume term with 2R t t − 16πGT t t . For our purely magnetic ansatz, the term T t t exactly cancels the matter field Lagrangean in the bulk action. The divergent contribution given by the surface integral term at infinity in R t t is also canceled by I surface + I ct and for 3 < d < 9 we arrive at the simple finite expression\\nTo compute I cl , we make use of the Einstein equations, replacing the R − 2Λ volume term with 2R t t − 16πGT t t . For our purely magnetic ansatz, the term T t t exactly cancels the matter field Lagrangean in the bulk action. The divergent contribution given by the surface integral term at infinity in R t t is also canceled by I surface + I ct and for 3 < d < 9 we arrive at the simple finite expression\\nReplacing I cl now in (52) (where E is the mass-energy computed in Section 4.1), we find\\nReplacing I cl now in (52) (where E is the mass-energy computed in Section 4.1), we find\\nwhich is one quarter of the event horizon area, as expected. ¿From the AdS/CFT correspondence, we expect the nonabelian hairy black holes to be described by some thermal states in a dual theory formulated in a Einstein universe background. The spherically symmetric solitons will correspond to zero-temperature states in the same theory. The existence of these hairy configurations suggest that there should be some observables in the dual CFT that encode the hair information.\\nwhich is one quarter of the event horizon area, as expected. ¿From the AdS/CFT correspondence, we expect the nonabelian hairy black holes to be described by some thermal states in a dual theory formulated in a Einstein universe background. The spherically symmetric solitons will correspond to zero-temperature states in the same theory. The existence of these hairy configurations suggest that there should be some observables in the dual CFT that encode the hair information.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"Motivated by recent results in EYM theory in four dimensional AAdS spacetime, we studied higher dimensional spherically symmetric solutions with non Abelian fields in the presence of a negative cosmological constant. Since the mass-energy of the AAdS configurations plays a central role in its application to the AdS/CFT correspondence, we emphasised this aspect of the solutions we found. The mass-energy of the usual EYM solutions (both asymptotically flat and AAdS), defined according to the standard prescription, always diverges in spacetime dimensions d > 4. One of the tasks performed in this work was a demonstration of this fact in the AAdS case. The main properties of these higher dimensional F (2) solutions resemble the d = 4 case, a continuum of solutions with arbitrary asymptotic values of the gauge function w(r) being found.\\nMotivated by recent results in EYM theory in four dimensional AAdS spacetime, we studied higher dimensional spherically symmetric solutions with non Abelian fields in the presence of a negative cosmological constant. Since the mass-energy of the AAdS configurations plays a central role in its application to the AdS/CFT correspondence, we emphasised this aspect of the solutions we found. The mass-energy of the usual EYM solutions (both asymptotically flat and AAdS), defined according to the standard prescription, always diverges in spacetime dimensions d > 4. One of the tasks performed in this work was a demonstration of this fact in the AAdS case. The main properties of these higher dimensional F (2) solutions resemble the d = 4 case, a continuum of solutions with arbitrary asymptotic values of the gauge function w(r) being found.\\nThen we focused on two possible approaches of dealing with the divergences of the mass and action. One of these involved the regularisation of the mass-energy using a counterterm mechanisms. In this approach, it turns out that the counterterm action depends not only on the boundary metric, but also on the boundary values of the gauge fields. However, the masses of the solutions defined in this way may take negative values, leading us to the second approach.\\nThen we focused on two possible approaches of dealing with the divergences of the mass and action. One of these involved the regularisation of the mass-energy using a counterterm mechanisms. In this approach, it turns out that the counterterm action depends not only on the boundary metric, but also on the boundary values of the gauge fields. However, the masses of the solutions defined in this way may take negative values, leading us to the second approach.\\nThe other, which formed the main thrust of the work, was to augment the action density of the system with higher order curvature terms, consisting of 2p-form curvatures F (2p). These terms were added to the usual YM system constructed from F (2). It resulted in EYM solutions supporting finite mass-energy in all spacetime dimensions 5 ≤ d ≤ 2p.\\nThe other, which formed the main thrust of the work, was to augment the action density of the system with higher order curvature terms, consisting of 2p-form curvatures F (2p). These terms were added to the usual YM system constructed from F (2). It resulted in EYM solutions supporting finite mass-energy in all spacetime dimensions 5 ≤ d ≤ 2p.\\nConcerning the construction of regular finite energy classical AAdS solutions in higher dimensions, we restricted ourselves to systems consisting exclusively of gravitational and non Abelian gauge fields. The salient features of the resulting solutions are captured in this framework, the addition of other (string theory inspired) matter terms being deferred to later work. The asymptotically flat versions of the higher-p EYM systems having been studied in [13,14,19], our task here involved the introduction of a negative cosmological constant. The most succint way of listing our conclusions is:\\nConcerning the construction of regular finite energy classical AAdS solutions in higher dimensions, we restricted ourselves to systems consisting exclusively of gravitational and non Abelian gauge fields. The salient features of the resulting solutions are captured in this framework, the addition of other (string theory inspired) matter terms being deferred to later work. The asymptotically flat versions of the higher-p EYM systems having been studied in [13,14,19], our task here involved the introduction of a negative cosmological constant. The most succint way of listing our conclusions is:\\n• The qualitative properties of the regular AAdS solutions in spacetime dimensions d = 5, 6, 7, 8 are the same as the corresponding asymptotically flat ones, namely -A one parameter family of solutions parametrised by the (dimensionless) gravitational coupling constant α start at α = 0 (the gravity decoupling limit) and exist up to a maximum α max , after which α decreases again and ends at critical value.\\n• The qualitative properties of the regular AAdS solutions in spacetime dimensions d = 5, 6, 7, 8 are the same as the corresponding asymptotically flat ones, namely -A one parameter family of solutions parametrised by the (dimensionless) gravitational coupling constant α start at α = 0 (the gravity decoupling limit) and exist up to a maximum α max , after which α decreases again and ends at critical value.\\n-For d = 6, 7, 8 the value of α in the second, not that of gravity decoupling, endpoint becomes very small and stops. In the d = 8 case it actually vanishes. For d = 5 the value of α in the second endpoint reaches a finite critical value, where it does not stop, but oscillates around this critical value.\\n-For d = 6, 7, 8 the value of α in the second, not that of gravity decoupling, endpoint becomes very small and stops. In the d = 8 case it actually vanishes. For d = 5 the value of α in the second endpoint reaches a finite critical value, where it does not stop, but oscillates around this critical value.\\n-As long as the physically important F (2) term is present in the YM sector, there exist no multinode solutions.\\n-As long as the physically important F (2) term is present in the YM sector, there exist no multinode solutions.\\n• Black hole counterparts appear to exist for any regular solution. The qualitative properties of the AAdS black hole solutions are similar to the asymptotically flat case:\\n• Black hole counterparts appear to exist for any regular solution. The qualitative properties of the AAdS black hole solutions are similar to the asymptotically flat case:\\n-Different from the four dimensional theory, the event horizon radius presents a maximal value. This maximal value is a function of the gravitational coupling constant α.\\n-Different from the four dimensional theory, the event horizon radius presents a maximal value. This maximal value is a function of the gravitational coupling constant α.\\n-For d = 6, 7, 8 the black hole solutions form a loop connecting the two regular solutions with the same value of α. The solutions of the five dimensional theory are somehow special, presenting a complicated brach structure which depends on α.\\n-For d = 6, 7, 8 the black hole solutions form a loop connecting the two regular solutions with the same value of α. The solutions of the five dimensional theory are somehow special, presenting a complicated brach structure which depends on α.\\nAxially symmetric generalizations of these solutions are likely to exist. We expect them to share the basic properties of the spherically symmetric configurations. The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ. In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.\\nAxially symmetric generalizations of these solutions are likely to exist. We expect them to share the basic properties of the spherically symmetric configurations. The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ. In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.\"\n },\n {\n \"title\": \"Figure 1 .Figure 2 .Figure 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 1 .Figure 2 .Figure 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 9b .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 9b .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 9\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 9\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 10b .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 10b .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 10\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 10\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 11 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 11 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 12 .Figure 13 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 12 .Figure 13 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 14 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 14 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 15 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 15 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"d = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ.","content":"d = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ."},{"id":"","caption":"d = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ.","content":"d = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ."},{"id":"","caption":"The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ.","content":"Figure 1 .Figure 2 .Figure 3 .\n123The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ."},{"id":"","caption":"The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ.","content":"Figure 1 .Figure 2 .Figure 3 .\n123The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ."},{"id":"","caption":"The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles.","content":"Figure 4 .\n4The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles."},{"id":"","caption":"The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles.","content":"Figure 4 .\n4The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles."},{"id":"","caption":"Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ.","content":"Figure 5 .\n5Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ."},{"id":"","caption":"Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ.","content":"Figure 5 .\n5Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ."},{"id":"","caption":"Figure 6a.","content":"Figure 6a."},{"id":"","caption":"Figure 6a.","content":"Figure 6a."},{"id":"","caption":"Figure 6b.","content":"Figure 6b."},{"id":"","caption":"Figure 6b.","content":"Figure 6b."},{"id":"","caption":"Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ.","content":"Figure 6\n6Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ."},{"id":"","caption":"Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ.","content":"Figure 6\n6Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ."},{"id":"","caption":"Figure 7a.","content":"Figure 7a."},{"id":"","caption":"Figure 7a.","content":"Figure 7a."},{"id":"","caption":"Figure 7b.","content":"Figure 7b."},{"id":"","caption":"Figure 7b.","content":"Figure 7b."},{"id":"","caption":"The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ.","content":"Figure 7\n7The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ."},{"id":"","caption":"The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ.","content":"Figure 7\n7The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ."},{"id":"","caption":"Figure 8a.","content":"Figure 8a."},{"id":"","caption":"Figure 8a.","content":"Figure 8a."},{"id":"","caption":"Figure 8b.","content":"Figure 8b."},{"id":"","caption":"Figure 8b.","content":"Figure 8b."},{"id":"","caption":"The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ.","content":"Figure 8\n8The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ."},{"id":"","caption":"The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ.","content":"Figure 8\n8The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ."},{"id":"","caption":"Figure 9a.","content":"Figure 9b .\n9bFigure 9a."},{"id":"","caption":"Figure 9a.","content":"Figure 9b .\n9bFigure 9a."},{"id":"","caption":"Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.","content":"Figure 9\n9Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ."},{"id":"","caption":"Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.","content":"Figure 9\n9Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ."},{"id":"","caption":"Figure 10a.","content":"Figure 10b .\n10bFigure 10a."},{"id":"","caption":"Figure 10a.","content":"Figure 10b .\n10bFigure 10a."},{"id":"","caption":"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ.","content":"Figure 10\n10The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ."},{"id":"","caption":"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ.","content":"Figure 10\n10The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ."},{"id":"","caption":"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ.","content":"Figure 11 .\n11The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ."},{"id":"","caption":"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ.","content":"Figure 11 .\n11The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ."},{"id":"","caption":"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ.","content":"Figure 12 .Figure 13 .\n1213The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ."},{"id":"","caption":"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ.","content":"Figure 12 .Figure 13 .\n1213The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ."},{"id":"","caption":"The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01.","content":"Figure 14 .\n14The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01."},{"id":"","caption":"The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01.","content":"Figure 14 .\n14The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01."},{"id":"","caption":"Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ.","content":"Figure 15 .\n15Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ."},{"id":"","caption":"Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ.","content":"Figure 15 .\n15Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"d = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ.\",\n \"content\": \"d = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"d = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ.\",\n \"content\": \"d = 6, 8 solutions are presented in Figures 4, 5 respectively. In Figures 5, 6 and 7 we plot some relevant quantities for d = 6, 7, 8 and several values of Λ. One can see that the qualitative behaviour of the functions m, σ, w does not change by changing the value of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ.\",\n \"content\": \"Figure 1 .Figure 2 .Figure 3 .\\n123The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ.\",\n \"content\": \"Figure 1 .Figure 2 .Figure 3 .\\n123The functions σ(r), w(r) and the ratio m(r)/r are plotted as functions of radius for typical d = 6 regular solutions in a F 2 EYM theory with Λ = −0.1 and several values of the parameter b = − 1 2 w ′′ (0). The value N m of the minimum of the metric function N (r), the parameter M 0 appearing in the asymptotic expansion of m(r), the asymptotic value w 0 of the gauge function w(r) as well as the value σ(0) of the metric function σ at the origin, are shown as functions of b for d = 6 solutions of F 2 theory with Λ = −1. The value N m of the minimum of the metric function N (r), the mass parameter M as well as the value of the metric function σ at the origin, σ(0), are shown for d = 5 solutions as functions of α 2 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles.\",\n \"content\": \"Figure 4 .\\n4The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles.\",\n \"content\": \"Figure 4 .\\n4The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of radius for typical d = 6 regular solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.1, Λ = −0.01. Here and inFigure 14the continuos line corresponds to an upper branch solution, the dotted line denoting lower branch profiles.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ.\",\n \"content\": \"Figure 5 .\\n5Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ.\",\n \"content\": \"Figure 5 .\\n5Typical globally regular solutions of d = 8 EYM theory with α 2 = 0.0005 and p = 1, 2 terms are plotted as functions of radius for several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 6a.\",\n \"content\": \"Figure 6a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 6a.\",\n \"content\": \"Figure 6a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 6b.\",\n \"content\": \"Figure 6b.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 6b.\",\n \"content\": \"Figure 6b.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ.\",\n \"content\": \"Figure 6\\n6Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ.\",\n \"content\": \"Figure 6\\n6Figure 6 The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0) (Figure 6a), and, the parameters M and b (Figure 6b), are shown for d = 6 solutions as functions of α 2 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 7a.\",\n \"content\": \"Figure 7a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 7a.\",\n \"content\": \"Figure 7a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 7b.\",\n \"content\": \"Figure 7b.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 7b.\",\n \"content\": \"Figure 7b.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ.\",\n \"content\": \"Figure 7\\n7The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ.\",\n \"content\": \"Figure 7\\n7The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 7a), and, the parameters M and b(Figure 7b), are shown for d = 7 solutions as functions of α 2 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 8a.\",\n \"content\": \"Figure 8a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 8a.\",\n \"content\": \"Figure 8a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 8b.\",\n \"content\": \"Figure 8b.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 8b.\",\n \"content\": \"Figure 8b.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ.\",\n \"content\": \"Figure 8\\n8The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ.\",\n \"content\": \"Figure 8\\n8The value N m of the minimum of the metric function N (r) and the value of the metric function σ at the origin σ(0)(Figure 8a), and the parameters M and b(Figure 8b), are shown for d = 8 solutions as functions of α 2 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 9a.\",\n \"content\": \"Figure 9b .\\n9bFigure 9a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 9a.\",\n \"content\": \"Figure 9b .\\n9bFigure 9a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.\",\n \"content\": \"Figure 9\\n9Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.\",\n \"content\": \"Figure 9\\n9Figure 9 In Figure 9a we plot the value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon (the magnified profiles of σ(r h ) are displayed in the window, to help distinguish these from the profiles of w(r h )). In Figure 9b, the mass parameter M and the Hawking temperature T H are presented. All profiles as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.02 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 10a.\",\n \"content\": \"Figure 10b .\\n10bFigure 10a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 10a.\",\n \"content\": \"Figure 10b .\\n10bFigure 10a.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ.\",\n \"content\": \"Figure 10\\n10The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ.\",\n \"content\": \"Figure 10\\n10The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon,(Figure 10a), as well as the mass parameter M and the Hawking temperature T H(Figure 10b) are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in five dimensions with α 2 = 0.5 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ.\",\n \"content\": \"Figure 11 .\\n11The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ.\",\n \"content\": \"Figure 11 .\\n11The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in six dimensions with α 2 = 0.066 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ.\",\n \"content\": \"Figure 12 .Figure 13 .\\n1213The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ.\",\n \"content\": \"Figure 12 .Figure 13 .\\n1213The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M and the Hawking temperature T H are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in seven dimensions with α 2 = 0.011 and several values of Λ. The value of the gauge field function at the horizon w(r h ) and σ(r h ), the value of the metric function σ at the horizon, as well as the mass parameter M are shown as functions of the event horizon radius r h for the p = 1, 2 black hole solutions in eight dimensions with α 2 = 0.002 and several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01.\",\n \"content\": \"Figure 14 .\\n14The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01.\",\n \"content\": \"Figure 14 .\\n14The profiles of the functions m(r), σ(r) and w(r) are plotted as functions of the radius for typical d = 6 black hole solutions in a EYM theory with p = 1, 2 terms and α 2 = 0.05, Λ = −0.01.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ.\",\n \"content\": \"Figure 15 .\\n15Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ.\",\n \"content\": \"Figure 15 .\\n15Typical black hole solutions of p = 1, 2 EYM theory in seven dimensions with α 2 = 0.001, r h = 0.28 are plotted as functions of the radius for several values of Λ.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". J M Maldacena, arXiv:hep-th/9711200Adv. Theor. Math. Phys. 2231Int. J. Theor. Phys.J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].","doi":"arXiv:hep-th/9711200"},{"text":". E Witten, arXiv:hep-th/9802150Adv. Theor. Math. Phys. 2253E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 [arXiv:hep-th/9802150].","doi":"arXiv:hep-th/9802150"},{"text":". E Winstanley, arXiv:gr-qc/9812064Class. Quant. Grav. 161963E. Winstanley, Class. Quant. Grav. 16 (1999) 1963 [arXiv:gr-qc/9812064].","doi":"arXiv:gr-qc/9812064"},{"text":". J Bjoraker, Y Hosotani, arXiv:hep-th/0002098Phys. Rev. D. 6243513J. Bjoraker and Y. Hosotani, Phys. Rev. D 62 (2000) 043513 [arXiv:hep-th/0002098].","doi":"arXiv:hep-th/0002098"},{"text":". O Sarbach, E Winstanley, arXiv:gr-qc/0102033Class. Quant. Grav. 182125O. Sarbach and E. Winstanley, Class. Quant. Grav. 18 (2001) 2125 [arXiv:gr-qc/0102033].","doi":"arXiv:gr-qc/0102033"},{"text":". P Breitenlohner, D Maison, G Lavrelashvili, arXiv:gr-qc/0307029Class. Quant. Grav. 211667P. Breitenlohner, D. Maison and G. Lavrelashvili, Class. Quant. Grav. 21 (2004) 1667 [arXiv:gr-qc/0307029].","doi":"arXiv:gr-qc/0307029"},{"text":". E Radu, arXiv:hep-th/0211120Phys. Rev. D. 6784030E. Radu, Phys. Rev. D 67 (2003) 084030 [arXiv:hep-th/0211120].","doi":"arXiv:hep-th/0211120"},{"text":". J J Van Der Bij, E Radu, arXiv:gr-qc/0107065Phys. Lett. B. 536107J. J. Van der Bij and E. Radu, Phys. Lett. B 536 (2002) 107 [arXiv:gr-qc/0107065].","doi":"arXiv:gr-qc/0107065"},{"text":". E Radu, arXiv:gr-qc/0210074Phys. Lett. B. 548224E. Radu, Phys. Lett. B 548 (2002) 224 [arXiv:gr-qc/0210074];","doi":"arXiv:gr-qc/0210074"},{"text":". E Radu, arXiv:gr-qc/0109015Phys. Rev. D. 6544005E. Radu, Phys. Rev. D 65 (2002) 044005 [arXiv:gr-qc/0109015].","doi":"arXiv:gr-qc/0109015"},{"text":". E Radu, E Winstanley, arXiv:hep-th/0407248Phys. Rev. D. 7084023E. Radu and E. Winstanley, Phys. Rev. D 70 (2004) 084023 [arXiv:hep-th/0407248].","doi":"arXiv:hep-th/0407248"},{"text":". N Okuyama, K I Maeda, arXiv:gr-qc/0212022Phys. Rev. D. 67104012N. Okuyama and K. i. Maeda, Phys. Rev. D 67 (2003) 104012 [arXiv:gr-qc/0212022].","doi":"arXiv:gr-qc/0212022"},{"text":". M S Volkov, arXiv:hep-th/0103038Phys. Lett. B. 524369M. S. Volkov, Phys. Lett. B 524 (2002) 369 [arXiv:hep-th/0103038].","doi":"arXiv:hep-th/0103038"},{"text":". Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0202141Class. Quant. Grav. 202765Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, Class. Quant. Grav. 20 (2003) 2765 [arXiv:hep-th/0202141].","doi":"arXiv:hep-th/0202141"},{"text":". Y Brihaye, A Chakrabarti, B Hartmann, D H Tchrakian, arXiv:hep-th/0212288Phys. Lett. B. 561161Y. Brihaye, A. Chakrabarti, B. Hartmann and D. H. Tchrakian, Phys. Lett. B 561 (2003) 161 [arXiv:hep-th/0212288].","doi":"arXiv:hep-th/0212288"},{"text":". Y Brihaye, D H Tchrakian, arXiv:hep-th/0410008J. Phys. A. 383679Y. Brihaye and D. H. Tchrakian, J. Phys. A 38 (2005) 3679 [arXiv:hep-th/0410008].","doi":"arXiv:hep-th/0410008"},{"text":". R Bartnik, J Mckinnon, Phys. Rev. Lett. 61141R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141.","doi":""},{"text":". P Breitenlohner, P Forgacs, D Maison, ibid. 442Nucl. Phys. B. 383126P. Breitenlohner, P. Forgacs and D. Maison, Nucl. Phys. B 383 (1992) 357; ibid. 442 (1995) 126","doi":"ibid. 442"},{"text":". G W Gibbons, S W Hawking, Phys. Rev. D. 152752G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2752.","doi":""},{"text":". P Breitenlohner, D Maison, D H Tchrakian, arXiv:gr-qc/0508027P. Breitenlohner, D. Maison and D.H. Tchrakian, arXiv:gr-qc/0508027.","doi":"arXiv:gr-qc/0508027"},{"text":". S Deser, Class. Quant. Grav. 11S. Deser, Class. Quant. Grav. 1 (1984) L1.","doi":""},{"text":". V Balasubramanian, P Kraus, arXiv:hep-th/9902121Commun. Math. Phys. 208413V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208 (1999) 413 [arXiv:hep-th/9902121].","doi":"arXiv:hep-th/9902121"},{"text":". M Banados, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849.","doi":""},{"text":". D Maison, private communicationD. Maison, private communication.","doi":""},{"text":". E W Hirschmann, D L Welch, arXiv:hep-th/9510181Phys. Rev. D. 535579E. W. Hirschmann and D. L. Welch, Phys. Rev. D 53 (1996) 5579 [arXiv:hep-th/9510181].","doi":"arXiv:hep-th/9510181"},{"text":". M Cataldo, P Salgado, Phys. Rev. D. 542971M. Cataldo and P. Salgado, Phys. Rev. D 54 (1996) 2971.","doi":""},{"text":". Y Kiem, D Park, arXiv:hep-th/9610008Phys. Rev. D. 556112Y. Kiem and D. Park, Phys. Rev. D 55 (1997) 6112 [arXiv:hep-th/9610008].","doi":"arXiv:hep-th/9610008"},{"text":". M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0201170Phys. Rev. D. 65104007M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 65 (2002) 104007 [arXiv:hep-th/0201170].","doi":"arXiv:hep-th/0201170"},{"text":". Y Hosotani, arXiv:gr-qc/0103069J. Math. Phys. 43597Y. Hosotani, J. Math. Phys. 43 (2002) 597 [arXiv:gr-qc/0103069].","doi":"arXiv:gr-qc/0103069"},{"text":". C N Pope, Class. Quant. Grav. 277C. N. Pope, Class. Quant. Grav. 2 (1985) L77.","doi":""},{"text":"Classical lumps and their quantum descendants. S Coleman, 297in Erice subnuclear physicsS. Coleman, Classical lumps and their quantum descendants, in Erice subnuclear physics, pp. 297, 1975.","doi":""},{"text":". S Deser, Phys. Lett. B. 64463S. Deser, Phys. Lett. B 64 (1976) 463.","doi":""},{"text":". M S Volkov, D V , arXiv:hep-th/9810070Phys. Rept. 3191M. S. Volkov and D. V. Gal'tsov, Phys. Rept. 319 (1999) 1 [arXiv:hep-th/9810070].","doi":"arXiv:hep-th/9810070"},{"text":". M Heusler, arXiv:gr-qc/9610019Helv. Phys. Acta. 69501M. Heusler, Helv. Phys. Acta 69 (1996) 501 [arXiv:gr-qc/9610019].","doi":"arXiv:gr-qc/9610019"},{"text":". M Heusler, N Straumann, Class. Quantum. Grav. 92177M. Heusler, N. Straumann, Class. Quantum. Grav. 9, 2177, (1992).","doi":""},{"text":". A H Chamseddine, M S Volkov, arXiv:hep-th/0101202JHEP. 010423A. H. Chamseddine and M. S. Volkov, JHEP 0104 (2001) 023 [arXiv:hep-th/0101202];","doi":"arXiv:hep-th/0101202"},{"text":". G Bertoldi, arXiv:hep-th/0210048JHEP. 021042G. Bertoldi, JHEP 0210 (2002) 042 [arXiv:hep-th/0210048].","doi":"arXiv:hep-th/0210048"},{"text":"Born-Infeld action, supersymmetry and string theory. A A Tseytlin, Yuri Golfand memorial volume. M. ShifmanWorld ScientificA. A. Tseytlin, Born-Infeld action, supersymmetry and string theory, in Yuri Golfand memorial volume, ed. M. Shifman, World Scientific (2000).","doi":""},{"text":". E A Bergshoeff, M De Roo, A Sevrin, arXiv:hep-th/0011264Nucl. Phys. Proc. Suppl. 49433Fortsch. Phys.E. A. Bergshoeff, M. de Roo and A. Sevrin, Fortsch. Phys. 49 (2001) 433 [Nucl. Phys. Proc. Suppl. 102 (2001) 50] [arXiv:hep-th/0011264].","doi":"arXiv:hep-th/0011264"},{"text":". M Cederwall, B E W Nilsson, D Tsimpis, arXiv:hep-th/0102009JHEP. 010634M. Cederwall, B. E. W. Nilsson and D. Tsimpis, JHEP 0106 (2001) 034 [arXiv:hep-th/0102009].","doi":"arXiv:hep-th/0102009"},{"text":". A A Belavin, A M Polyakov, A S Shvarts, Y S Tyupkin, Phys. Lett. B. 5985A. A. Belavin, A. M. Polyakov, A. S. Shvarts and Y. S. Tyupkin, Phys. Lett. B 59 (1975) 85.","doi":""},{"text":". H Boutaleb-Joutei, A Chakrabarti, A Comtet, Phys. Rev. D. 201884H. Boutaleb-Joutei, A. Chakrabarti and A. Comtet, Phys. Rev. D 20 (1979) 1884.","doi":""},{"text":". Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0002144J. Math. Phys. 415490Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, J. Math. Phys. 41 (2001) 5490 [arXiv:hep-th/0002144].","doi":"arXiv:hep-th/0002144"},{"text":". L F Abbott, S Deser, Nucl. Phys. B. 19576L. F. Abbott and S. Deser, Nucl. Phys. B 195 76 (1982).","doi":""},{"text":". M Henneaux, C Teitelboim, Commun. Math. Phys. 98391M. Henneaux and C. Teitelboim, Commun. Math. Phys. 98 391 (1985).","doi":""},{"text":". A Ashtekar, A Magnon, Class. Quant. Grav. 139A. Ashtekar and A. Magnon, Class. Quant. Grav. 1 (1984) L39;","doi":""},{"text":". A Ashtekar, S Das, Class. Quant. Grav. 1717A. Ashtekar and S. Das, Class. Quant. Grav. 17 (2000) L17.","doi":""},{"text":". R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. Lett. 841647R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. Lett. 84 (2000) 1647;","doi":""},{"text":". R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. D. 6244002R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. D 62 (2000) 044002;","doi":""},{"text":". E W Mielke, Phys. Rev. D. 6344018E. W. Mielke, Phys. Rev. D 63 (2001) 044018.","doi":""},{"text":". J D Brown, J W York, Phys. Rev. D. 471407J. D. Brown and J. W. York, Phys. Rev. D 47 1407 (1993).","doi":""},{"text":". M M Taylor-Robinson, arXiv:hep-th/0002125.K.SkenderisInt. J. Mod. Phys. A. 16740M. M. Taylor-Robinson, arXiv:hep-th/0002125. K. Skenderis, Int. J. Mod. Phys. A 16 (2001) 740.","doi":"arXiv:hep-th/0002125.K.Skenderis"},{"text":". S Haro, S N Solodukhin, K Skenderis, Commun. Math. Phys. 217595S. de Haro, S. N. Solodukhin and K. Skenderis, Commun. Math. Phys. 217 (2001) 595.","doi":""},{"text":". S Das, R B Mann, arXiv:hep-th/0008028JHEP. 000833S. Das and R. B. Mann, JHEP 0008 (2000) 033 [arXiv:hep-th/0008028].","doi":"arXiv:hep-th/0008028"},{"text":". R C Myers, Phys. Rev. D. 6046002R. C. Myers, Phys. Rev. D 60 (1999) 046002.","doi":""},{"text":". R B Mann, arXiv:gr-qc/0211047Found. Phys. 3365R. B. Mann, Found. Phys. 33 (2003) 65 [arXiv:gr-qc/0211047].","doi":"arXiv:gr-qc/0211047"},{"text":". Y Brihaye, B Hartmann, E Radu, arXiv:hep-th/0508028Y. Brihaye, B. Hartmann and E. Radu, arXiv:hep-th/0508028.","doi":"arXiv:hep-th/0508028"},{"text":". T Hertog, K Maeda, arXiv:hep-th/0404261JHEP. 040751T. Hertog and K. Maeda, JHEP 0407 (2004) 051 [arXiv:hep-th/0404261].","doi":"arXiv:hep-th/0404261"},{"text":". T Hertog, K Maeda, arXiv:hep-th/0409314Phys. Rev. D. 7124001T. Hertog and K. Maeda, Phys. Rev. D 71 (2005) 024001 [arXiv:hep-th/0409314].","doi":"arXiv:hep-th/0409314"},{"text":". M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0404236Phys. Rev. D. 7044034M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 70 (2004) 044034 [arXiv:hep-th/0404236].","doi":"arXiv:hep-th/0404236"},{"text":". E Radu, D H Tchrakian, arXiv:hep-th/0410154Class. Quant. Grav. 22879E. Radu and D. H. Tchrakian, Class. Quant. Grav. 22 (2005) 879 [arXiv:hep-th/0410154].","doi":"arXiv:hep-th/0410154"},{"text":". S Hollands, A Ishibashi, D Marolf, arXiv:hep-th/0503045Class. Quant. Grav. 222881S. Hollands, A. Ishibashi and D. Marolf, Class. Quant. Grav. 22 (2005) 2881 [arXiv:hep-th/0503045].","doi":"arXiv:hep-th/0503045"},{"text":". J T Liu, W A Sabra, arXiv:hep-th/0405171Phys. Rev. D. 7264021J. T. Liu and W. A. Sabra, Phys. Rev. D 72 (2005) 064021 [arXiv:hep-th/0405171].","doi":"arXiv:hep-th/0405171"},{"text":". A Batrachenko, J T Liu, R Mcnees, W A Sabra, W Y Wen, arXiv:hep-th/0408205JHEP. 050534A. Batrachenko, J. T. Liu, R. McNees, W. A. Sabra and W. Y. Wen, JHEP 0505 (2005) 034 [arXiv:hep-th/0408205].","doi":"arXiv:hep-th/0408205"},{"text":". J M Maldacena, arXiv:hep-th/9711200Adv. Theor. Math. Phys. 2231Int. J. Theor. Phys.J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].","doi":"arXiv:hep-th/9711200"},{"text":". E Witten, arXiv:hep-th/9802150Adv. Theor. Math. Phys. 2253E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 [arXiv:hep-th/9802150].","doi":"arXiv:hep-th/9802150"},{"text":". E Winstanley, arXiv:gr-qc/9812064Class. Quant. Grav. 161963E. Winstanley, Class. Quant. Grav. 16 (1999) 1963 [arXiv:gr-qc/9812064].","doi":"arXiv:gr-qc/9812064"},{"text":". J Bjoraker, Y Hosotani, arXiv:hep-th/0002098Phys. Rev. D. 6243513J. Bjoraker and Y. Hosotani, Phys. Rev. D 62 (2000) 043513 [arXiv:hep-th/0002098].","doi":"arXiv:hep-th/0002098"},{"text":". O Sarbach, E Winstanley, arXiv:gr-qc/0102033Class. Quant. Grav. 182125O. Sarbach and E. Winstanley, Class. Quant. Grav. 18 (2001) 2125 [arXiv:gr-qc/0102033].","doi":"arXiv:gr-qc/0102033"},{"text":". P Breitenlohner, D Maison, G Lavrelashvili, arXiv:gr-qc/0307029Class. Quant. Grav. 211667P. Breitenlohner, D. Maison and G. Lavrelashvili, Class. Quant. Grav. 21 (2004) 1667 [arXiv:gr-qc/0307029].","doi":"arXiv:gr-qc/0307029"},{"text":". E Radu, arXiv:hep-th/0211120Phys. Rev. D. 6784030E. Radu, Phys. Rev. D 67 (2003) 084030 [arXiv:hep-th/0211120].","doi":"arXiv:hep-th/0211120"},{"text":". J J Van Der Bij, E Radu, arXiv:gr-qc/0107065Phys. Lett. B. 536107J. J. Van der Bij and E. Radu, Phys. Lett. B 536 (2002) 107 [arXiv:gr-qc/0107065].","doi":"arXiv:gr-qc/0107065"},{"text":". E Radu, arXiv:gr-qc/0210074Phys. Lett. B. 548224E. Radu, Phys. Lett. B 548 (2002) 224 [arXiv:gr-qc/0210074];","doi":"arXiv:gr-qc/0210074"},{"text":". E Radu, arXiv:gr-qc/0109015Phys. Rev. D. 6544005E. Radu, Phys. Rev. D 65 (2002) 044005 [arXiv:gr-qc/0109015].","doi":"arXiv:gr-qc/0109015"},{"text":". E Radu, E Winstanley, arXiv:hep-th/0407248Phys. Rev. D. 7084023E. Radu and E. Winstanley, Phys. Rev. D 70 (2004) 084023 [arXiv:hep-th/0407248].","doi":"arXiv:hep-th/0407248"},{"text":". N Okuyama, K I Maeda, arXiv:gr-qc/0212022Phys. Rev. D. 67104012N. Okuyama and K. i. Maeda, Phys. Rev. D 67 (2003) 104012 [arXiv:gr-qc/0212022].","doi":"arXiv:gr-qc/0212022"},{"text":". M S Volkov, arXiv:hep-th/0103038Phys. Lett. B. 524369M. S. Volkov, Phys. Lett. B 524 (2002) 369 [arXiv:hep-th/0103038].","doi":"arXiv:hep-th/0103038"},{"text":". Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0202141Class. Quant. Grav. 202765Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, Class. Quant. Grav. 20 (2003) 2765 [arXiv:hep-th/0202141].","doi":"arXiv:hep-th/0202141"},{"text":". Y Brihaye, A Chakrabarti, B Hartmann, D H Tchrakian, arXiv:hep-th/0212288Phys. Lett. B. 561161Y. Brihaye, A. Chakrabarti, B. Hartmann and D. H. Tchrakian, Phys. Lett. B 561 (2003) 161 [arXiv:hep-th/0212288].","doi":"arXiv:hep-th/0212288"},{"text":". Y Brihaye, D H Tchrakian, arXiv:hep-th/0410008J. Phys. A. 383679Y. Brihaye and D. H. Tchrakian, J. Phys. A 38 (2005) 3679 [arXiv:hep-th/0410008].","doi":"arXiv:hep-th/0410008"},{"text":". R Bartnik, J Mckinnon, Phys. Rev. Lett. 61141R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141.","doi":""},{"text":". P Breitenlohner, P Forgacs, D Maison, ibid. 442Nucl. Phys. B. 383126P. Breitenlohner, P. Forgacs and D. Maison, Nucl. Phys. B 383 (1992) 357; ibid. 442 (1995) 126","doi":"ibid. 442"},{"text":". G W Gibbons, S W Hawking, Phys. Rev. D. 152752G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2752.","doi":""},{"text":". P Breitenlohner, D Maison, D H Tchrakian, arXiv:gr-qc/0508027P. Breitenlohner, D. Maison and D.H. Tchrakian, arXiv:gr-qc/0508027.","doi":"arXiv:gr-qc/0508027"},{"text":". S Deser, Class. Quant. Grav. 11S. Deser, Class. Quant. Grav. 1 (1984) L1.","doi":""},{"text":". V Balasubramanian, P Kraus, arXiv:hep-th/9902121Commun. Math. Phys. 208413V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208 (1999) 413 [arXiv:hep-th/9902121].","doi":"arXiv:hep-th/9902121"},{"text":". M Banados, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849.","doi":""},{"text":". D Maison, private communicationD. Maison, private communication.","doi":""},{"text":". E W Hirschmann, D L Welch, arXiv:hep-th/9510181Phys. Rev. D. 535579E. W. Hirschmann and D. L. Welch, Phys. Rev. D 53 (1996) 5579 [arXiv:hep-th/9510181].","doi":"arXiv:hep-th/9510181"},{"text":". M Cataldo, P Salgado, Phys. Rev. D. 542971M. Cataldo and P. Salgado, Phys. Rev. D 54 (1996) 2971.","doi":""},{"text":". Y Kiem, D Park, arXiv:hep-th/9610008Phys. Rev. D. 556112Y. Kiem and D. Park, Phys. Rev. D 55 (1997) 6112 [arXiv:hep-th/9610008].","doi":"arXiv:hep-th/9610008"},{"text":". M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0201170Phys. Rev. D. 65104007M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 65 (2002) 104007 [arXiv:hep-th/0201170].","doi":"arXiv:hep-th/0201170"},{"text":". Y Hosotani, arXiv:gr-qc/0103069J. Math. Phys. 43597Y. Hosotani, J. Math. Phys. 43 (2002) 597 [arXiv:gr-qc/0103069].","doi":"arXiv:gr-qc/0103069"},{"text":". C N Pope, Class. Quant. Grav. 277C. N. Pope, Class. Quant. Grav. 2 (1985) L77.","doi":""},{"text":"Classical lumps and their quantum descendants. S Coleman, 297in Erice subnuclear physicsS. Coleman, Classical lumps and their quantum descendants, in Erice subnuclear physics, pp. 297, 1975.","doi":""},{"text":". S Deser, Phys. Lett. B. 64463S. Deser, Phys. Lett. B 64 (1976) 463.","doi":""},{"text":". M S Volkov, D V , arXiv:hep-th/9810070Phys. Rept. 3191M. S. Volkov and D. V. Gal'tsov, Phys. Rept. 319 (1999) 1 [arXiv:hep-th/9810070].","doi":"arXiv:hep-th/9810070"},{"text":". M Heusler, arXiv:gr-qc/9610019Helv. Phys. Acta. 69501M. Heusler, Helv. Phys. Acta 69 (1996) 501 [arXiv:gr-qc/9610019].","doi":"arXiv:gr-qc/9610019"},{"text":". M Heusler, N Straumann, Class. Quantum. Grav. 92177M. Heusler, N. Straumann, Class. Quantum. Grav. 9, 2177, (1992).","doi":""},{"text":". A H Chamseddine, M S Volkov, arXiv:hep-th/0101202JHEP. 010423A. H. Chamseddine and M. S. Volkov, JHEP 0104 (2001) 023 [arXiv:hep-th/0101202];","doi":"arXiv:hep-th/0101202"},{"text":". G Bertoldi, arXiv:hep-th/0210048JHEP. 021042G. Bertoldi, JHEP 0210 (2002) 042 [arXiv:hep-th/0210048].","doi":"arXiv:hep-th/0210048"},{"text":"Born-Infeld action, supersymmetry and string theory. A A Tseytlin, Yuri Golfand memorial volume. M. ShifmanWorld ScientificA. A. Tseytlin, Born-Infeld action, supersymmetry and string theory, in Yuri Golfand memorial volume, ed. M. Shifman, World Scientific (2000).","doi":""},{"text":". E A Bergshoeff, M De Roo, A Sevrin, arXiv:hep-th/0011264Nucl. Phys. Proc. Suppl. 49433Fortsch. Phys.E. A. Bergshoeff, M. de Roo and A. Sevrin, Fortsch. Phys. 49 (2001) 433 [Nucl. Phys. Proc. Suppl. 102 (2001) 50] [arXiv:hep-th/0011264].","doi":"arXiv:hep-th/0011264"},{"text":". M Cederwall, B E W Nilsson, D Tsimpis, arXiv:hep-th/0102009JHEP. 010634M. Cederwall, B. E. W. Nilsson and D. Tsimpis, JHEP 0106 (2001) 034 [arXiv:hep-th/0102009].","doi":"arXiv:hep-th/0102009"},{"text":". A A Belavin, A M Polyakov, A S Shvarts, Y S Tyupkin, Phys. Lett. B. 5985A. A. Belavin, A. M. Polyakov, A. S. Shvarts and Y. S. Tyupkin, Phys. Lett. B 59 (1975) 85.","doi":""},{"text":". H Boutaleb-Joutei, A Chakrabarti, A Comtet, Phys. Rev. D. 201884H. Boutaleb-Joutei, A. Chakrabarti and A. Comtet, Phys. Rev. D 20 (1979) 1884.","doi":""},{"text":". Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0002144J. Math. Phys. 415490Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, J. Math. Phys. 41 (2001) 5490 [arXiv:hep-th/0002144].","doi":"arXiv:hep-th/0002144"},{"text":". L F Abbott, S Deser, Nucl. Phys. B. 19576L. F. Abbott and S. Deser, Nucl. Phys. B 195 76 (1982).","doi":""},{"text":". M Henneaux, C Teitelboim, Commun. Math. Phys. 98391M. Henneaux and C. Teitelboim, Commun. Math. Phys. 98 391 (1985).","doi":""},{"text":". A Ashtekar, A Magnon, Class. Quant. Grav. 139A. Ashtekar and A. Magnon, Class. Quant. Grav. 1 (1984) L39;","doi":""},{"text":". A Ashtekar, S Das, Class. Quant. Grav. 1717A. Ashtekar and S. Das, Class. Quant. Grav. 17 (2000) L17.","doi":""},{"text":". R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. Lett. 841647R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. Lett. 84 (2000) 1647;","doi":""},{"text":". R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. D. 6244002R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. D 62 (2000) 044002;","doi":""},{"text":". E W Mielke, Phys. Rev. D. 6344018E. W. Mielke, Phys. Rev. D 63 (2001) 044018.","doi":""},{"text":". J D Brown, J W York, Phys. Rev. D. 471407J. D. Brown and J. W. York, Phys. Rev. D 47 1407 (1993).","doi":""},{"text":". M M Taylor-Robinson, arXiv:hep-th/0002125.K.SkenderisInt. J. Mod. Phys. A. 16740M. M. Taylor-Robinson, arXiv:hep-th/0002125. K. Skenderis, Int. J. Mod. Phys. A 16 (2001) 740.","doi":"arXiv:hep-th/0002125.K.Skenderis"},{"text":". S Haro, S N Solodukhin, K Skenderis, Commun. Math. Phys. 217595S. de Haro, S. N. Solodukhin and K. Skenderis, Commun. Math. Phys. 217 (2001) 595.","doi":""},{"text":". S Das, R B Mann, arXiv:hep-th/0008028JHEP. 000833S. Das and R. B. Mann, JHEP 0008 (2000) 033 [arXiv:hep-th/0008028].","doi":"arXiv:hep-th/0008028"},{"text":". R C Myers, Phys. Rev. D. 6046002R. C. Myers, Phys. Rev. D 60 (1999) 046002.","doi":""},{"text":". R B Mann, arXiv:gr-qc/0211047Found. Phys. 3365R. B. Mann, Found. Phys. 33 (2003) 65 [arXiv:gr-qc/0211047].","doi":"arXiv:gr-qc/0211047"},{"text":". Y Brihaye, B Hartmann, E Radu, arXiv:hep-th/0508028Y. Brihaye, B. Hartmann and E. Radu, arXiv:hep-th/0508028.","doi":"arXiv:hep-th/0508028"},{"text":". T Hertog, K Maeda, arXiv:hep-th/0404261JHEP. 040751T. Hertog and K. Maeda, JHEP 0407 (2004) 051 [arXiv:hep-th/0404261].","doi":"arXiv:hep-th/0404261"},{"text":". T Hertog, K Maeda, arXiv:hep-th/0409314Phys. Rev. D. 7124001T. Hertog and K. Maeda, Phys. Rev. D 71 (2005) 024001 [arXiv:hep-th/0409314].","doi":"arXiv:hep-th/0409314"},{"text":". M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0404236Phys. Rev. D. 7044034M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 70 (2004) 044034 [arXiv:hep-th/0404236].","doi":"arXiv:hep-th/0404236"},{"text":". E Radu, D H Tchrakian, arXiv:hep-th/0410154Class. Quant. Grav. 22879E. Radu and D. H. Tchrakian, Class. Quant. Grav. 22 (2005) 879 [arXiv:hep-th/0410154].","doi":"arXiv:hep-th/0410154"},{"text":". S Hollands, A Ishibashi, D Marolf, arXiv:hep-th/0503045Class. Quant. Grav. 222881S. Hollands, A. Ishibashi and D. Marolf, Class. Quant. Grav. 22 (2005) 2881 [arXiv:hep-th/0503045].","doi":"arXiv:hep-th/0503045"},{"text":". J T Liu, W A Sabra, arXiv:hep-th/0405171Phys. Rev. D. 7264021J. T. Liu and W. A. Sabra, Phys. Rev. D 72 (2005) 064021 [arXiv:hep-th/0405171].","doi":"arXiv:hep-th/0405171"},{"text":". A Batrachenko, J T Liu, R Mcnees, W A Sabra, W Y Wen, arXiv:hep-th/0408205JHEP. 050534A. Batrachenko, J. T. Liu, R. McNees, W. A. Sabra and W. Y. Wen, JHEP 0505 (2005) 034 [arXiv:hep-th/0408205].","doi":"arXiv:hep-th/0408205"}],"string":"[\n {\n \"text\": \". J M Maldacena, arXiv:hep-th/9711200Adv. Theor. Math. Phys. 2231Int. J. Theor. Phys.J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].\",\n \"doi\": \"arXiv:hep-th/9711200\"\n },\n {\n \"text\": \". E Witten, arXiv:hep-th/9802150Adv. Theor. Math. Phys. 2253E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 [arXiv:hep-th/9802150].\",\n \"doi\": \"arXiv:hep-th/9802150\"\n },\n {\n \"text\": \". E Winstanley, arXiv:gr-qc/9812064Class. Quant. Grav. 161963E. Winstanley, Class. Quant. Grav. 16 (1999) 1963 [arXiv:gr-qc/9812064].\",\n \"doi\": \"arXiv:gr-qc/9812064\"\n },\n {\n \"text\": \". J Bjoraker, Y Hosotani, arXiv:hep-th/0002098Phys. Rev. D. 6243513J. Bjoraker and Y. Hosotani, Phys. Rev. D 62 (2000) 043513 [arXiv:hep-th/0002098].\",\n \"doi\": \"arXiv:hep-th/0002098\"\n },\n {\n \"text\": \". O Sarbach, E Winstanley, arXiv:gr-qc/0102033Class. Quant. Grav. 182125O. Sarbach and E. Winstanley, Class. Quant. Grav. 18 (2001) 2125 [arXiv:gr-qc/0102033].\",\n \"doi\": \"arXiv:gr-qc/0102033\"\n },\n {\n \"text\": \". P Breitenlohner, D Maison, G Lavrelashvili, arXiv:gr-qc/0307029Class. Quant. Grav. 211667P. Breitenlohner, D. Maison and G. Lavrelashvili, Class. Quant. Grav. 21 (2004) 1667 [arXiv:gr-qc/0307029].\",\n \"doi\": \"arXiv:gr-qc/0307029\"\n },\n {\n \"text\": \". E Radu, arXiv:hep-th/0211120Phys. Rev. D. 6784030E. Radu, Phys. Rev. D 67 (2003) 084030 [arXiv:hep-th/0211120].\",\n \"doi\": \"arXiv:hep-th/0211120\"\n },\n {\n \"text\": \". J J Van Der Bij, E Radu, arXiv:gr-qc/0107065Phys. Lett. B. 536107J. J. Van der Bij and E. Radu, Phys. Lett. B 536 (2002) 107 [arXiv:gr-qc/0107065].\",\n \"doi\": \"arXiv:gr-qc/0107065\"\n },\n {\n \"text\": \". E Radu, arXiv:gr-qc/0210074Phys. Lett. B. 548224E. Radu, Phys. Lett. B 548 (2002) 224 [arXiv:gr-qc/0210074];\",\n \"doi\": \"arXiv:gr-qc/0210074\"\n },\n {\n \"text\": \". E Radu, arXiv:gr-qc/0109015Phys. Rev. D. 6544005E. Radu, Phys. Rev. D 65 (2002) 044005 [arXiv:gr-qc/0109015].\",\n \"doi\": \"arXiv:gr-qc/0109015\"\n },\n {\n \"text\": \". E Radu, E Winstanley, arXiv:hep-th/0407248Phys. Rev. D. 7084023E. Radu and E. Winstanley, Phys. Rev. D 70 (2004) 084023 [arXiv:hep-th/0407248].\",\n \"doi\": \"arXiv:hep-th/0407248\"\n },\n {\n \"text\": \". N Okuyama, K I Maeda, arXiv:gr-qc/0212022Phys. Rev. D. 67104012N. Okuyama and K. i. Maeda, Phys. Rev. D 67 (2003) 104012 [arXiv:gr-qc/0212022].\",\n \"doi\": \"arXiv:gr-qc/0212022\"\n },\n {\n \"text\": \". M S Volkov, arXiv:hep-th/0103038Phys. Lett. B. 524369M. S. Volkov, Phys. Lett. B 524 (2002) 369 [arXiv:hep-th/0103038].\",\n \"doi\": \"arXiv:hep-th/0103038\"\n },\n {\n \"text\": \". Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0202141Class. Quant. Grav. 202765Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, Class. Quant. Grav. 20 (2003) 2765 [arXiv:hep-th/0202141].\",\n \"doi\": \"arXiv:hep-th/0202141\"\n },\n {\n \"text\": \". Y Brihaye, A Chakrabarti, B Hartmann, D H Tchrakian, arXiv:hep-th/0212288Phys. Lett. B. 561161Y. Brihaye, A. Chakrabarti, B. Hartmann and D. H. Tchrakian, Phys. Lett. B 561 (2003) 161 [arXiv:hep-th/0212288].\",\n \"doi\": \"arXiv:hep-th/0212288\"\n },\n {\n \"text\": \". Y Brihaye, D H Tchrakian, arXiv:hep-th/0410008J. Phys. A. 383679Y. Brihaye and D. H. Tchrakian, J. Phys. A 38 (2005) 3679 [arXiv:hep-th/0410008].\",\n \"doi\": \"arXiv:hep-th/0410008\"\n },\n {\n \"text\": \". R Bartnik, J Mckinnon, Phys. Rev. Lett. 61141R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Breitenlohner, P Forgacs, D Maison, ibid. 442Nucl. Phys. B. 383126P. Breitenlohner, P. Forgacs and D. Maison, Nucl. Phys. B 383 (1992) 357; ibid. 442 (1995) 126\",\n \"doi\": \"ibid. 442\"\n },\n {\n \"text\": \". G W Gibbons, S W Hawking, Phys. Rev. D. 152752G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2752.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Breitenlohner, D Maison, D H Tchrakian, arXiv:gr-qc/0508027P. Breitenlohner, D. Maison and D.H. Tchrakian, arXiv:gr-qc/0508027.\",\n \"doi\": \"arXiv:gr-qc/0508027\"\n },\n {\n \"text\": \". S Deser, Class. Quant. Grav. 11S. Deser, Class. Quant. Grav. 1 (1984) L1.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Balasubramanian, P Kraus, arXiv:hep-th/9902121Commun. Math. Phys. 208413V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208 (1999) 413 [arXiv:hep-th/9902121].\",\n \"doi\": \"arXiv:hep-th/9902121\"\n },\n {\n \"text\": \". M Banados, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Maison, private communicationD. Maison, private communication.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E W Hirschmann, D L Welch, arXiv:hep-th/9510181Phys. Rev. D. 535579E. W. Hirschmann and D. L. Welch, Phys. Rev. D 53 (1996) 5579 [arXiv:hep-th/9510181].\",\n \"doi\": \"arXiv:hep-th/9510181\"\n },\n {\n \"text\": \". M Cataldo, P Salgado, Phys. Rev. D. 542971M. Cataldo and P. Salgado, Phys. Rev. D 54 (1996) 2971.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Kiem, D Park, arXiv:hep-th/9610008Phys. Rev. D. 556112Y. Kiem and D. Park, Phys. Rev. D 55 (1997) 6112 [arXiv:hep-th/9610008].\",\n \"doi\": \"arXiv:hep-th/9610008\"\n },\n {\n \"text\": \". M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0201170Phys. Rev. D. 65104007M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 65 (2002) 104007 [arXiv:hep-th/0201170].\",\n \"doi\": \"arXiv:hep-th/0201170\"\n },\n {\n \"text\": \". Y Hosotani, arXiv:gr-qc/0103069J. Math. Phys. 43597Y. Hosotani, J. Math. Phys. 43 (2002) 597 [arXiv:gr-qc/0103069].\",\n \"doi\": \"arXiv:gr-qc/0103069\"\n },\n {\n \"text\": \". C N Pope, Class. Quant. Grav. 277C. N. Pope, Class. Quant. Grav. 2 (1985) L77.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Classical lumps and their quantum descendants. S Coleman, 297in Erice subnuclear physicsS. Coleman, Classical lumps and their quantum descendants, in Erice subnuclear physics, pp. 297, 1975.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Deser, Phys. Lett. B. 64463S. Deser, Phys. Lett. B 64 (1976) 463.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M S Volkov, D V , arXiv:hep-th/9810070Phys. Rept. 3191M. S. Volkov and D. V. Gal'tsov, Phys. Rept. 319 (1999) 1 [arXiv:hep-th/9810070].\",\n \"doi\": \"arXiv:hep-th/9810070\"\n },\n {\n \"text\": \". M Heusler, arXiv:gr-qc/9610019Helv. Phys. Acta. 69501M. Heusler, Helv. Phys. Acta 69 (1996) 501 [arXiv:gr-qc/9610019].\",\n \"doi\": \"arXiv:gr-qc/9610019\"\n },\n {\n \"text\": \". M Heusler, N Straumann, Class. Quantum. Grav. 92177M. Heusler, N. Straumann, Class. Quantum. Grav. 9, 2177, (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A H Chamseddine, M S Volkov, arXiv:hep-th/0101202JHEP. 010423A. H. Chamseddine and M. S. Volkov, JHEP 0104 (2001) 023 [arXiv:hep-th/0101202];\",\n \"doi\": \"arXiv:hep-th/0101202\"\n },\n {\n \"text\": \". G Bertoldi, arXiv:hep-th/0210048JHEP. 021042G. Bertoldi, JHEP 0210 (2002) 042 [arXiv:hep-th/0210048].\",\n \"doi\": \"arXiv:hep-th/0210048\"\n },\n {\n \"text\": \"Born-Infeld action, supersymmetry and string theory. A A Tseytlin, Yuri Golfand memorial volume. M. ShifmanWorld ScientificA. A. Tseytlin, Born-Infeld action, supersymmetry and string theory, in Yuri Golfand memorial volume, ed. M. Shifman, World Scientific (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E A Bergshoeff, M De Roo, A Sevrin, arXiv:hep-th/0011264Nucl. Phys. Proc. Suppl. 49433Fortsch. Phys.E. A. Bergshoeff, M. de Roo and A. Sevrin, Fortsch. Phys. 49 (2001) 433 [Nucl. Phys. Proc. Suppl. 102 (2001) 50] [arXiv:hep-th/0011264].\",\n \"doi\": \"arXiv:hep-th/0011264\"\n },\n {\n \"text\": \". M Cederwall, B E W Nilsson, D Tsimpis, arXiv:hep-th/0102009JHEP. 010634M. Cederwall, B. E. W. Nilsson and D. Tsimpis, JHEP 0106 (2001) 034 [arXiv:hep-th/0102009].\",\n \"doi\": \"arXiv:hep-th/0102009\"\n },\n {\n \"text\": \". A A Belavin, A M Polyakov, A S Shvarts, Y S Tyupkin, Phys. Lett. B. 5985A. A. Belavin, A. M. Polyakov, A. S. Shvarts and Y. S. Tyupkin, Phys. Lett. B 59 (1975) 85.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Boutaleb-Joutei, A Chakrabarti, A Comtet, Phys. Rev. D. 201884H. Boutaleb-Joutei, A. Chakrabarti and A. Comtet, Phys. Rev. D 20 (1979) 1884.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0002144J. Math. Phys. 415490Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, J. Math. Phys. 41 (2001) 5490 [arXiv:hep-th/0002144].\",\n \"doi\": \"arXiv:hep-th/0002144\"\n },\n {\n \"text\": \". L F Abbott, S Deser, Nucl. Phys. B. 19576L. F. Abbott and S. Deser, Nucl. Phys. B 195 76 (1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Henneaux, C Teitelboim, Commun. Math. Phys. 98391M. Henneaux and C. Teitelboim, Commun. Math. Phys. 98 391 (1985).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Ashtekar, A Magnon, Class. Quant. Grav. 139A. Ashtekar and A. Magnon, Class. Quant. Grav. 1 (1984) L39;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Ashtekar, S Das, Class. Quant. Grav. 1717A. Ashtekar and S. Das, Class. Quant. Grav. 17 (2000) L17.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. Lett. 841647R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. Lett. 84 (2000) 1647;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. D. 6244002R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. D 62 (2000) 044002;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E W Mielke, Phys. Rev. D. 6344018E. W. Mielke, Phys. Rev. D 63 (2001) 044018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Brown, J W York, Phys. Rev. D. 471407J. D. Brown and J. W. York, Phys. Rev. D 47 1407 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M M Taylor-Robinson, arXiv:hep-th/0002125.K.SkenderisInt. J. Mod. Phys. A. 16740M. M. Taylor-Robinson, arXiv:hep-th/0002125. K. Skenderis, Int. J. Mod. Phys. A 16 (2001) 740.\",\n \"doi\": \"arXiv:hep-th/0002125.K.Skenderis\"\n },\n {\n \"text\": \". S Haro, S N Solodukhin, K Skenderis, Commun. Math. Phys. 217595S. de Haro, S. N. Solodukhin and K. Skenderis, Commun. Math. Phys. 217 (2001) 595.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Das, R B Mann, arXiv:hep-th/0008028JHEP. 000833S. Das and R. B. Mann, JHEP 0008 (2000) 033 [arXiv:hep-th/0008028].\",\n \"doi\": \"arXiv:hep-th/0008028\"\n },\n {\n \"text\": \". R C Myers, Phys. Rev. D. 6046002R. C. Myers, Phys. Rev. D 60 (1999) 046002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R B Mann, arXiv:gr-qc/0211047Found. Phys. 3365R. B. Mann, Found. Phys. 33 (2003) 65 [arXiv:gr-qc/0211047].\",\n \"doi\": \"arXiv:gr-qc/0211047\"\n },\n {\n \"text\": \". Y Brihaye, B Hartmann, E Radu, arXiv:hep-th/0508028Y. Brihaye, B. Hartmann and E. Radu, arXiv:hep-th/0508028.\",\n \"doi\": \"arXiv:hep-th/0508028\"\n },\n {\n \"text\": \". T Hertog, K Maeda, arXiv:hep-th/0404261JHEP. 040751T. Hertog and K. Maeda, JHEP 0407 (2004) 051 [arXiv:hep-th/0404261].\",\n \"doi\": \"arXiv:hep-th/0404261\"\n },\n {\n \"text\": \". T Hertog, K Maeda, arXiv:hep-th/0409314Phys. Rev. D. 7124001T. Hertog and K. Maeda, Phys. Rev. D 71 (2005) 024001 [arXiv:hep-th/0409314].\",\n \"doi\": \"arXiv:hep-th/0409314\"\n },\n {\n \"text\": \". M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0404236Phys. Rev. D. 7044034M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 70 (2004) 044034 [arXiv:hep-th/0404236].\",\n \"doi\": \"arXiv:hep-th/0404236\"\n },\n {\n \"text\": \". E Radu, D H Tchrakian, arXiv:hep-th/0410154Class. Quant. Grav. 22879E. Radu and D. H. Tchrakian, Class. Quant. Grav. 22 (2005) 879 [arXiv:hep-th/0410154].\",\n \"doi\": \"arXiv:hep-th/0410154\"\n },\n {\n \"text\": \". S Hollands, A Ishibashi, D Marolf, arXiv:hep-th/0503045Class. Quant. Grav. 222881S. Hollands, A. Ishibashi and D. Marolf, Class. Quant. Grav. 22 (2005) 2881 [arXiv:hep-th/0503045].\",\n \"doi\": \"arXiv:hep-th/0503045\"\n },\n {\n \"text\": \". J T Liu, W A Sabra, arXiv:hep-th/0405171Phys. Rev. D. 7264021J. T. Liu and W. A. Sabra, Phys. Rev. D 72 (2005) 064021 [arXiv:hep-th/0405171].\",\n \"doi\": \"arXiv:hep-th/0405171\"\n },\n {\n \"text\": \". A Batrachenko, J T Liu, R Mcnees, W A Sabra, W Y Wen, arXiv:hep-th/0408205JHEP. 050534A. Batrachenko, J. T. Liu, R. McNees, W. A. Sabra and W. Y. Wen, JHEP 0505 (2005) 034 [arXiv:hep-th/0408205].\",\n \"doi\": \"arXiv:hep-th/0408205\"\n },\n {\n \"text\": \". J M Maldacena, arXiv:hep-th/9711200Adv. Theor. Math. Phys. 2231Int. J. Theor. Phys.J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].\",\n \"doi\": \"arXiv:hep-th/9711200\"\n },\n {\n \"text\": \". E Witten, arXiv:hep-th/9802150Adv. Theor. Math. Phys. 2253E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 [arXiv:hep-th/9802150].\",\n \"doi\": \"arXiv:hep-th/9802150\"\n },\n {\n \"text\": \". E Winstanley, arXiv:gr-qc/9812064Class. Quant. Grav. 161963E. Winstanley, Class. Quant. Grav. 16 (1999) 1963 [arXiv:gr-qc/9812064].\",\n \"doi\": \"arXiv:gr-qc/9812064\"\n },\n {\n \"text\": \". J Bjoraker, Y Hosotani, arXiv:hep-th/0002098Phys. Rev. D. 6243513J. Bjoraker and Y. Hosotani, Phys. Rev. D 62 (2000) 043513 [arXiv:hep-th/0002098].\",\n \"doi\": \"arXiv:hep-th/0002098\"\n },\n {\n \"text\": \". O Sarbach, E Winstanley, arXiv:gr-qc/0102033Class. Quant. Grav. 182125O. Sarbach and E. Winstanley, Class. Quant. Grav. 18 (2001) 2125 [arXiv:gr-qc/0102033].\",\n \"doi\": \"arXiv:gr-qc/0102033\"\n },\n {\n \"text\": \". P Breitenlohner, D Maison, G Lavrelashvili, arXiv:gr-qc/0307029Class. Quant. Grav. 211667P. Breitenlohner, D. Maison and G. Lavrelashvili, Class. Quant. Grav. 21 (2004) 1667 [arXiv:gr-qc/0307029].\",\n \"doi\": \"arXiv:gr-qc/0307029\"\n },\n {\n \"text\": \". E Radu, arXiv:hep-th/0211120Phys. Rev. D. 6784030E. Radu, Phys. Rev. D 67 (2003) 084030 [arXiv:hep-th/0211120].\",\n \"doi\": \"arXiv:hep-th/0211120\"\n },\n {\n \"text\": \". J J Van Der Bij, E Radu, arXiv:gr-qc/0107065Phys. Lett. B. 536107J. J. Van der Bij and E. Radu, Phys. Lett. B 536 (2002) 107 [arXiv:gr-qc/0107065].\",\n \"doi\": \"arXiv:gr-qc/0107065\"\n },\n {\n \"text\": \". E Radu, arXiv:gr-qc/0210074Phys. Lett. B. 548224E. Radu, Phys. Lett. B 548 (2002) 224 [arXiv:gr-qc/0210074];\",\n \"doi\": \"arXiv:gr-qc/0210074\"\n },\n {\n \"text\": \". E Radu, arXiv:gr-qc/0109015Phys. Rev. D. 6544005E. Radu, Phys. Rev. D 65 (2002) 044005 [arXiv:gr-qc/0109015].\",\n \"doi\": \"arXiv:gr-qc/0109015\"\n },\n {\n \"text\": \". E Radu, E Winstanley, arXiv:hep-th/0407248Phys. Rev. D. 7084023E. Radu and E. Winstanley, Phys. Rev. D 70 (2004) 084023 [arXiv:hep-th/0407248].\",\n \"doi\": \"arXiv:hep-th/0407248\"\n },\n {\n \"text\": \". N Okuyama, K I Maeda, arXiv:gr-qc/0212022Phys. Rev. D. 67104012N. Okuyama and K. i. Maeda, Phys. Rev. D 67 (2003) 104012 [arXiv:gr-qc/0212022].\",\n \"doi\": \"arXiv:gr-qc/0212022\"\n },\n {\n \"text\": \". M S Volkov, arXiv:hep-th/0103038Phys. Lett. B. 524369M. S. Volkov, Phys. Lett. B 524 (2002) 369 [arXiv:hep-th/0103038].\",\n \"doi\": \"arXiv:hep-th/0103038\"\n },\n {\n \"text\": \". Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0202141Class. Quant. Grav. 202765Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, Class. Quant. Grav. 20 (2003) 2765 [arXiv:hep-th/0202141].\",\n \"doi\": \"arXiv:hep-th/0202141\"\n },\n {\n \"text\": \". Y Brihaye, A Chakrabarti, B Hartmann, D H Tchrakian, arXiv:hep-th/0212288Phys. Lett. B. 561161Y. Brihaye, A. Chakrabarti, B. Hartmann and D. H. Tchrakian, Phys. Lett. B 561 (2003) 161 [arXiv:hep-th/0212288].\",\n \"doi\": \"arXiv:hep-th/0212288\"\n },\n {\n \"text\": \". Y Brihaye, D H Tchrakian, arXiv:hep-th/0410008J. Phys. A. 383679Y. Brihaye and D. H. Tchrakian, J. Phys. A 38 (2005) 3679 [arXiv:hep-th/0410008].\",\n \"doi\": \"arXiv:hep-th/0410008\"\n },\n {\n \"text\": \". R Bartnik, J Mckinnon, Phys. Rev. Lett. 61141R. Bartnik and J. McKinnon, Phys. Rev. Lett. 61 (1988) 141.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Breitenlohner, P Forgacs, D Maison, ibid. 442Nucl. Phys. B. 383126P. Breitenlohner, P. Forgacs and D. Maison, Nucl. Phys. B 383 (1992) 357; ibid. 442 (1995) 126\",\n \"doi\": \"ibid. 442\"\n },\n {\n \"text\": \". G W Gibbons, S W Hawking, Phys. Rev. D. 152752G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2752.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Breitenlohner, D Maison, D H Tchrakian, arXiv:gr-qc/0508027P. Breitenlohner, D. Maison and D.H. Tchrakian, arXiv:gr-qc/0508027.\",\n \"doi\": \"arXiv:gr-qc/0508027\"\n },\n {\n \"text\": \". S Deser, Class. Quant. Grav. 11S. Deser, Class. Quant. Grav. 1 (1984) L1.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Balasubramanian, P Kraus, arXiv:hep-th/9902121Commun. Math. Phys. 208413V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208 (1999) 413 [arXiv:hep-th/9902121].\",\n \"doi\": \"arXiv:hep-th/9902121\"\n },\n {\n \"text\": \". M Banados, C Teitelboim, J Zanelli, Phys. Rev. Lett. 691849M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Maison, private communicationD. Maison, private communication.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E W Hirschmann, D L Welch, arXiv:hep-th/9510181Phys. Rev. D. 535579E. W. Hirschmann and D. L. Welch, Phys. Rev. D 53 (1996) 5579 [arXiv:hep-th/9510181].\",\n \"doi\": \"arXiv:hep-th/9510181\"\n },\n {\n \"text\": \". M Cataldo, P Salgado, Phys. Rev. D. 542971M. Cataldo and P. Salgado, Phys. Rev. D 54 (1996) 2971.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Kiem, D Park, arXiv:hep-th/9610008Phys. Rev. D. 556112Y. Kiem and D. Park, Phys. Rev. D 55 (1997) 6112 [arXiv:hep-th/9610008].\",\n \"doi\": \"arXiv:hep-th/9610008\"\n },\n {\n \"text\": \". M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0201170Phys. Rev. D. 65104007M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 65 (2002) 104007 [arXiv:hep-th/0201170].\",\n \"doi\": \"arXiv:hep-th/0201170\"\n },\n {\n \"text\": \". Y Hosotani, arXiv:gr-qc/0103069J. Math. Phys. 43597Y. Hosotani, J. Math. Phys. 43 (2002) 597 [arXiv:gr-qc/0103069].\",\n \"doi\": \"arXiv:gr-qc/0103069\"\n },\n {\n \"text\": \". C N Pope, Class. Quant. Grav. 277C. N. Pope, Class. Quant. Grav. 2 (1985) L77.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Classical lumps and their quantum descendants. S Coleman, 297in Erice subnuclear physicsS. Coleman, Classical lumps and their quantum descendants, in Erice subnuclear physics, pp. 297, 1975.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Deser, Phys. Lett. B. 64463S. Deser, Phys. Lett. B 64 (1976) 463.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M S Volkov, D V , arXiv:hep-th/9810070Phys. Rept. 3191M. S. Volkov and D. V. Gal'tsov, Phys. Rept. 319 (1999) 1 [arXiv:hep-th/9810070].\",\n \"doi\": \"arXiv:hep-th/9810070\"\n },\n {\n \"text\": \". M Heusler, arXiv:gr-qc/9610019Helv. Phys. Acta. 69501M. Heusler, Helv. Phys. Acta 69 (1996) 501 [arXiv:gr-qc/9610019].\",\n \"doi\": \"arXiv:gr-qc/9610019\"\n },\n {\n \"text\": \". M Heusler, N Straumann, Class. Quantum. Grav. 92177M. Heusler, N. Straumann, Class. Quantum. Grav. 9, 2177, (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A H Chamseddine, M S Volkov, arXiv:hep-th/0101202JHEP. 010423A. H. Chamseddine and M. S. Volkov, JHEP 0104 (2001) 023 [arXiv:hep-th/0101202];\",\n \"doi\": \"arXiv:hep-th/0101202\"\n },\n {\n \"text\": \". G Bertoldi, arXiv:hep-th/0210048JHEP. 021042G. Bertoldi, JHEP 0210 (2002) 042 [arXiv:hep-th/0210048].\",\n \"doi\": \"arXiv:hep-th/0210048\"\n },\n {\n \"text\": \"Born-Infeld action, supersymmetry and string theory. A A Tseytlin, Yuri Golfand memorial volume. M. ShifmanWorld ScientificA. A. Tseytlin, Born-Infeld action, supersymmetry and string theory, in Yuri Golfand memorial volume, ed. M. Shifman, World Scientific (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E A Bergshoeff, M De Roo, A Sevrin, arXiv:hep-th/0011264Nucl. Phys. Proc. Suppl. 49433Fortsch. Phys.E. A. Bergshoeff, M. de Roo and A. Sevrin, Fortsch. Phys. 49 (2001) 433 [Nucl. Phys. Proc. Suppl. 102 (2001) 50] [arXiv:hep-th/0011264].\",\n \"doi\": \"arXiv:hep-th/0011264\"\n },\n {\n \"text\": \". M Cederwall, B E W Nilsson, D Tsimpis, arXiv:hep-th/0102009JHEP. 010634M. Cederwall, B. E. W. Nilsson and D. Tsimpis, JHEP 0106 (2001) 034 [arXiv:hep-th/0102009].\",\n \"doi\": \"arXiv:hep-th/0102009\"\n },\n {\n \"text\": \". A A Belavin, A M Polyakov, A S Shvarts, Y S Tyupkin, Phys. Lett. B. 5985A. A. Belavin, A. M. Polyakov, A. S. Shvarts and Y. S. Tyupkin, Phys. Lett. B 59 (1975) 85.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Boutaleb-Joutei, A Chakrabarti, A Comtet, Phys. Rev. D. 201884H. Boutaleb-Joutei, A. Chakrabarti and A. Comtet, Phys. Rev. D 20 (1979) 1884.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Brihaye, A Chakrabarti, D H Tchrakian, arXiv:hep-th/0002144J. Math. Phys. 415490Y. Brihaye, A. Chakrabarti and D. H. Tchrakian, J. Math. Phys. 41 (2001) 5490 [arXiv:hep-th/0002144].\",\n \"doi\": \"arXiv:hep-th/0002144\"\n },\n {\n \"text\": \". L F Abbott, S Deser, Nucl. Phys. B. 19576L. F. Abbott and S. Deser, Nucl. Phys. B 195 76 (1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Henneaux, C Teitelboim, Commun. Math. Phys. 98391M. Henneaux and C. Teitelboim, Commun. Math. Phys. 98 391 (1985).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Ashtekar, A Magnon, Class. Quant. Grav. 139A. Ashtekar and A. Magnon, Class. Quant. Grav. 1 (1984) L39;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Ashtekar, S Das, Class. Quant. Grav. 1717A. Ashtekar and S. Das, Class. Quant. Grav. 17 (2000) L17.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. Lett. 841647R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. Lett. 84 (2000) 1647;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Aros, M Contreras, R Olea, R Troncoso, J Zanelli, Phys. Rev. D. 6244002R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli, Phys. Rev. D 62 (2000) 044002;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E W Mielke, Phys. Rev. D. 6344018E. W. Mielke, Phys. Rev. D 63 (2001) 044018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Brown, J W York, Phys. Rev. D. 471407J. D. Brown and J. W. York, Phys. Rev. D 47 1407 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M M Taylor-Robinson, arXiv:hep-th/0002125.K.SkenderisInt. J. Mod. Phys. A. 16740M. M. Taylor-Robinson, arXiv:hep-th/0002125. K. Skenderis, Int. J. Mod. Phys. A 16 (2001) 740.\",\n \"doi\": \"arXiv:hep-th/0002125.K.Skenderis\"\n },\n {\n \"text\": \". S Haro, S N Solodukhin, K Skenderis, Commun. Math. Phys. 217595S. de Haro, S. N. Solodukhin and K. Skenderis, Commun. Math. Phys. 217 (2001) 595.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Das, R B Mann, arXiv:hep-th/0008028JHEP. 000833S. Das and R. B. Mann, JHEP 0008 (2000) 033 [arXiv:hep-th/0008028].\",\n \"doi\": \"arXiv:hep-th/0008028\"\n },\n {\n \"text\": \". R C Myers, Phys. Rev. D. 6046002R. C. Myers, Phys. Rev. D 60 (1999) 046002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R B Mann, arXiv:gr-qc/0211047Found. Phys. 3365R. B. Mann, Found. Phys. 33 (2003) 65 [arXiv:gr-qc/0211047].\",\n \"doi\": \"arXiv:gr-qc/0211047\"\n },\n {\n \"text\": \". Y Brihaye, B Hartmann, E Radu, arXiv:hep-th/0508028Y. Brihaye, B. Hartmann and E. Radu, arXiv:hep-th/0508028.\",\n \"doi\": \"arXiv:hep-th/0508028\"\n },\n {\n \"text\": \". T Hertog, K Maeda, arXiv:hep-th/0404261JHEP. 040751T. Hertog and K. Maeda, JHEP 0407 (2004) 051 [arXiv:hep-th/0404261].\",\n \"doi\": \"arXiv:hep-th/0404261\"\n },\n {\n \"text\": \". T Hertog, K Maeda, arXiv:hep-th/0409314Phys. Rev. D. 7124001T. Hertog and K. Maeda, Phys. Rev. D 71 (2005) 024001 [arXiv:hep-th/0409314].\",\n \"doi\": \"arXiv:hep-th/0409314\"\n },\n {\n \"text\": \". M Henneaux, C Martinez, R Troncoso, J Zanelli, arXiv:hep-th/0404236Phys. Rev. D. 7044034M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Phys. Rev. D 70 (2004) 044034 [arXiv:hep-th/0404236].\",\n \"doi\": \"arXiv:hep-th/0404236\"\n },\n {\n \"text\": \". E Radu, D H Tchrakian, arXiv:hep-th/0410154Class. Quant. Grav. 22879E. Radu and D. H. Tchrakian, Class. Quant. Grav. 22 (2005) 879 [arXiv:hep-th/0410154].\",\n \"doi\": \"arXiv:hep-th/0410154\"\n },\n {\n \"text\": \". S Hollands, A Ishibashi, D Marolf, arXiv:hep-th/0503045Class. Quant. Grav. 222881S. Hollands, A. Ishibashi and D. Marolf, Class. Quant. Grav. 22 (2005) 2881 [arXiv:hep-th/0503045].\",\n \"doi\": \"arXiv:hep-th/0503045\"\n },\n {\n \"text\": \". J T Liu, W A Sabra, arXiv:hep-th/0405171Phys. Rev. D. 7264021J. T. Liu and W. A. Sabra, Phys. Rev. D 72 (2005) 064021 [arXiv:hep-th/0405171].\",\n \"doi\": \"arXiv:hep-th/0405171\"\n },\n {\n \"text\": \". A Batrachenko, J T Liu, R Mcnees, W A Sabra, W Y Wen, arXiv:hep-th/0408205JHEP. 050534A. Batrachenko, J. T. Liu, R. McNees, W. A. Sabra and W. Y. Wen, JHEP 0505 (2005) 034 [arXiv:hep-th/0408205].\",\n \"doi\": \"arXiv:hep-th/0408205\"\n }\n]","isConversational":false}}},{"rowIdx":23,"cells":{"text":{"kind":"string","value":"\nThe dynamics of competitive learning: the role of updates and memory\n20 Jan 2012\n\nAjaz Ahmad Bhat \nTheoretical Science Department\nS N Bose National Centre\nBlock JD Sector III\nSalt Lake700098KolkataIndia\n\nAnita Mehta \nTheoretical Science Department\nS N Bose National Centre\nBlock JD Sector III\nSalt Lake700098KolkataIndia\n\nThe dynamics of competitive learning: the role of updates and memory\n20 Jan 2012(Dated: Revised version: January 20, 2013)numbers: 0570Jk8729lv8719Ge0250Le\nWe examine the effects of memory and different updating paradigms in a game-theoretic model of competitive learning, where agents are influenced in their choice of strategy by both the choices made by, and the consequent success rates of, their immediate neighbours. We apply parallel and sequential updates in all possible combinations to the two competing rules, and find, typically, that the phase diagram of the model consists of a disordered phase separating two ordered phases at coexistence. A major result is that the corresponding critical exponents belong to the generalised universality class of the voter model. When the two strategies are distinct but not too different, we find the expected linear response behaviour as a function of their difference. Finally, we look at the extreme situation when a superior strategy, accompanied by a short memory of earlier outcomes, is pitted against its inverse; interestingly, we find that a long memory of earlier outcomes can occasionally compensate for the choice of a globally inferior strategy.\n\nI. INTRODUCTION\n\nThe modelling of social behaviour is of increasing concern to statistical physicists [1]. Studies of social and biological systems often reveal that even when the interactions of a given individual are very localised in time and space, collective, regular behaviour can emerge: this is analogous to the cooperative behaviour manifested by emergent systems in the natural world. Such social regularities may well take the form of learning, when individuals adopt the behaviour of other individuals. From the perspective of game theory [2], this can be seen as an adoption of a particular strategy, whose result may or may not be associated with a favourable outcome. It is then quite reasonable to expect that the effectiveness of a strategy in yielding favourable outcomes should influence how likely it is to persist, and spread through the population; the resulting ideas of strategic learning [3] have found wide application, starting from economics [4] to cognitive science [5].\n\nAgainst the backdrop of the above ideas, a model of strategic learning was introduced in [6], with one of two possible strategies (denoted as + and − in the remainder of this paper) being available to each agent on a lattice: the agents were referred to as 'myopic' (aware only of their immediate neighbours) and 'memoryless' (unaware of their own and others' past outcomes) in the paper on technology diffusion [4] that inspired the above model [6,7]. The question on which this body of work has centred is: despite these handicaps, can agents overall learn to use the superior one of two available technologies? Briefly, each agent changes (or does not change) strategy based on two elementary rules at every time step: a * ajaz@bose.res.in † anita@bose.res.in majority-based rule, reflecting its tendency to align with its local neighbourhood, followed by a performance-based rule, where the agent adopts the strategy that 'wins' in its neighbourhood. This (relative) success is measured in terms of outcomes, where the probability of a successful outcome for strategy + (−) is p + (p − ). Also, the model of [6] added to the description of [4] by endowing the agents with memory: those agents who make their choices on the basis of the last payoff alone, are adjudged to be memoryless (with a corresponding parameter ε near 1), while those who allow for memories of earlier outcomes may make decisions that run counter to immediate evidence (ε small).\n\nSome related ideas have been examined in recent work. For example, the issue of consensus formation in a model of threshold learning [8] shows close analogies: in this model, the competition between the 'noisy' signals from the immediate neighbours of an agent (cf. the majority rule in [6]) and the acceptance threshold that agents require to change their state (cf. the memory threshold in the performance-based rule of [6]), determine the phase diagrams obtained. Recent studies of coevolving Glauber dynamics on networks [9] are also relevant, since the model of [6] can be viewed as a competition between the Glauber dynamics of two sets of Ising spins, corresponding to strategy and outcome respectively.\n\nIn the current paper, we take all these ideas further. First, we explore the effect of different updates. If new information propagates sequentially through the network, and the arrow of time is discernible in the decisions of individual agents, are the global phase diagrams any different from what they would be if information was transmitted and all decisions were taken simultaneously? Common sense tells us that sequential or parallel updates should make a difference to the nature of the phase diagram, and the results of the present paper confirm this. Also (unlike the work of [6,7] which examined the situa-tion at coexistence) we look in this paper at the effects of disparate strategies (p + = p − ). The final, and possibly most important issue, is that of memory, which acts as a threshold governing change [8]: what is the effect of the threshold ε, which tells the agent that longer-term inputs are significant, and need to be considered when making a decision? We will find that, indeed, a longer memory of earlier outcomes can sometimes make up for the choice of a globally inferior strategy.\n\nThe plan of this paper is as follows. In Section II, we review the model of [6]. In Section III, we discuss the behaviour of the model for a range of updating schemes, in the presence of memory. In Section IV, we examine the behaviour of the model away from coexistence, as a function of distinct parameter values for the two strategies; in particular we discuss here the role of memory. In the concluding section, we discuss our results and put them in the context of other recent work in the field.\n\n\nII. DEFINITION OF THE MODEL\n\nThe model of [6] involves two types of strategies, − and +, where the + strategy is globally superior [4] to the − strategy. As mentioned above, agents tend to follow the strategy adopted by the majority of their neighbours, modifying this choice in a second step (if necessary) according to which of these have proved to be the most successful.\n\nAssuming that the agents sit at the nodes of a ddimensional regular lattice with coordination number z = 2d, the efficiency of an agent at site i is represented by an Ising spin variable:\nη i (t) = +1 if i is + at time t, −1 if i is − at time t.(1)\nThe evolution dynamics of the lattice is governed by two rules. The first is a majority rule, which consists of the alignment of an agent with the local field (created by its nearest neighbours) acting upon it, according to:\nη i (t + τ 1 ) = +1 if h i (t) > 0, ±1 w.p 1 2 if h i (t) = 0, −1 if h i (t) < 0.(2)\nHere, the local field\nh i (t) = j(i) η j (t),(3)\nis the sum of the efficiencies of the z neighbouring agents j of site i and τ 1 is the associated time step. Next, a performance rule is applied. This starts with the assignment of an outcome σ i (another Ising-like variable, with values of ±1 corresponding to success and failure respectively) to each site i, according to the following rules:\n\nif η i (t) = +1,\nthen σ i (t + τ 2 ) = +1 w.p. p + −1 w.p. 1 − p + , if η i (t) = −1,then σ i (t + τ 2 ) = +1 w.p. p − −1 w.p. 1 − p − ,(4)\nwhere τ 2 is the associated time step and p ± are the probabilities of having a successful outcome for the corresponding strategy. With N + i and N − i denoting the total number of neighbours of a site i who have adopted strategies + and − respectively, and I + i (I − i ) denoting the number of successful outcomes within the set N + i (N − i ), the dynamical rules for site i are:\n\nif η i (t) = +1 and\nI + i (t) N + i (t) < I − i (t) N − i (t) , then η i (t + τ 3 ) = −1 w.p. ε + +1 w.p. 1 − ε + , if η i (t) = −1 and I − i (t) N − i (t) < I + i (t) N + i (t) , then η i (t + τ 3 ) = +1 w.p. ε − −1 w.p. 1 − ε − .(5)\nHere, the ratios Ii(t) Ni(t) are nothing but the average payoff assigned by an agent to each of the two strategies in its neighbourhood at time t (assuming that success yields a payoff of unity and failure, zero). Also, τ 3 is the associated time step and the parameters ε ± are indicators of the memory associated with each strategy. In their full generality, ε and p are independent variables: the choice of a particular strategy can be associated with either a short or a long memory. However, we would like in this paper to answer a question which was posed, but not answered in [6]: can the presence of a good memory compensate for the choice of an inferior strategy? We therefore examine the extreme situation when a globally superior strategy (p + ≫ p − ), combined with a shorter memory (ε + ≫ ε − ) is in competition with its inverse: this is the situation that will be studied in Section IV.\n\nSetting the timescales\nτ 2 → 0, τ 1 = τ 3 = 1,(6)\nthe above steps of the performance rule are recast as effective dynamical rules involving the efficiencies η i (t) and the associated local fields alone:\nif η i (t) = +1,then η i (t + 1) = +1 w.p. w + [h i (t)] −1 w.p. 1-w + [h i (t)], if η i (t) = −1,then η i (t + 1) = +1 w.p. w − [h i (t)] −1 w.p. 1-w − [h i (t)].(7)\nThe effective transition probabilities w ± (h) are evaluated by enumerating the 2 z possible realizations of the outcomes σ j of the sites neighbouring site i, and weighing them appropriately. For a 2-d square lattice, the possible local field values at the interfacial sites are 0 and ±2.\n\nThe corresponding transition probabilities for these field values are [6]:\nw + (+2) = 1 − ε + p − (1 − p 3 + ), w − (+2) = ε − (1 − p − )[1 − (1 − p + ) 3 ], w + (0) = 1 − ε + p − (1 − p + )(2 − p − − 2p + + 3p − p + ), w − (0) = ε − p + (1 − p − )(2 − p + − 2p − + 3p − p + ), w + (−2) = 1 − ε + (1 − p + )[1 − (1 − p − ) 3 ], w − (−2) = ε − p + (1 − p 3 − ).(8)\nIn [6], the model was explored at coexistence with an ordered sequential update applied to memoryless agents [4]:\np + = p − , ε + = ε − = 1.(9)\nIn the present paper, we go beyond this in two different ways. First, still at coexistence, we explore the effect of different updates on the p − ε phase diagram of the model: next, we examine the model away from coexistence, for distinct values of p ± and ε ± . The basic quantities considered hereafter are the magnetization M , staggered magnetization M stag and the energy E. These quantities are defined for a finite sample of N agents (or sites) and N z /2 bonds (or links), as\nM = 1 N i η i , E = 1 N z ij (1 − η i η j ) , M stag = 2 N i η i if i is odd or even.(10)\nIn the following we shall usually consider mean values M , E and M stag .\n\n\nIII. THE EFFECT OF FINITE MEMORY, AND OF DIFFERENT UPDATES\n\nWe begin this section with a review of the physical significance of updating schemes. Most generally, updates can be random or ordered as follows:\n\n• Random: Here, sites are chosen at random for the consecutive application of rules.\n\n• Ordered : Here, sites are chosen in an ordered fashion, i.e., after choosing every (i, j) th site, the (i, j + 1) th site is selected.\n\nSince the sociological basis for this work was the propagation of innovation through connected societies [4], we choose to deal only with ordered updates here. However, even ordered updates have two subclasses: parallel and sequential. Assume a condition A such that when an agent satisfies A, it changes strategy:\n\n• Sequential update: In this type of update, we check the condition A on the (i, j) th site, then update the efficiency of the site and proceed to the (i, j + 1) th site using the updated value of the (i, j) th site.\n\n• Parallel update: In this type of update, we check the condition A on the (i, j) th site, do not update the site but instead save the update-decision in memory, and proceed to the next site. Once the whole lattice is swept, all the saved updatedecisions are implemented 'simultaneously'.\n\nThe choice of different updates generally corresponds to different physical situations: it has been shown that it also leads to a disparity in the convergence time of the systems concerned [10,11]. We therefore examine all possible combinations for our two update rules:\n\nI parallel updates for both majority rule and performance rules (pp).\n\nII parallel update for majority rule and sequential update for performance rules (ps).\n\nIII sequential updates for both majority rule and performance rules (ss).\n\nIV sequential update for majority rule and parallel update for performance rules (sp).\n\nIn the following subsections, we explore the phase dynamics at coexistence for each of these update rules in turn, for both parameters p and ε. We state at the outset that all the updates (except for the sp update) which we consider, result in models which are in the general university class of the voter model [12]: the inverse energy 1/E(t) is thus always proportional to the logarithm of time, ln t. When, as in the case of the ss update, the value of the slope is exactly 2/π [6], the exact universality class of the voter model is retrieved.\n\n\nA. The ss update\n\nThis is the update that was used throughout [6]; however the phase behaviour of the model was there only explored for the parameter p, whereas here we extend it to the parameter ε. In Figure 1, we plot the inverse energy 1/E(t) in the p − ε plane at time t = 512 for a square lattice of size N = 64 2 . This phase diagram shows clearly the existence of a disordered paramagnetic phase embedded in a largely frozen phase elsewhere. The disordered phase exists for p c1 (= 0.56 ± 0.01) < p < p c2 (= 0.70 ± 0.01) when ε ≥ 0.980. Our results agree with those of [6] for ε = 1, and extend them all across the rest of the p − ε plane. We mention here that the average time required to reach consensus increases exponentially as p decreases in the frozen phase, leading to the presence of striped states [13] at limiting values of p. Figure 1 also makes it clear that the effect of increasing memory wipes out the disordered phase: this is as it should be, since the disordered phase is generated by the competition between the majority and performance-based rules, which is dulled by increasing memory. In Figure 3, we have plotted the inverse energy 1/E(t) against the natural logarithm of time ln t for values of p around the critical point p c2 = 0.70 ± 0.01. Each of the curves is obtained by averaging over 200 independent samples of size 256 2 . At the critical point, we obtain a straight line with a slope close to 2/π [6], a behaviour characteristic of the exact voter model [12] that corresponds to\nE(t) ≈ π/2 ln t .(11)\nSimilar behaviour is obtained at the other critical point p c1 , in agreement with [6]. The curve corresponding to p = pc2 = 0.70 (shown in red) has slope 2/π approximately [see Equation 11].\n\n\nB. The pp update\n\nIn this case, both environmental majority and performance-based rules are applied using parallel updates. As we will see, although the universality class of the model is qualitatively unchanged, this update results in the appearance of novel ordered phases compared to the ss update. As before, we first plot the phase diagram for all values of p and ε, then show snapshots of the dynamics, and finally get a more quantitative feel for the behaviour of key quantities as a function of p.\n\nAccordingly, Figure 4 (top and bottom), are plots of the absolute values of magnetization |M | and staggered magnetization |M stag |, at time t = 512 for a lattice size N = 100 2 , in the p-ε plane using pp updates. In these phase diagrams, we see clear evidence of the existence of two distinct frozen phases separated by a disordered phase. Looking along the line ε = 1, disorder prevails for p c1 < p < p c2 with p c1 = 0.43±0.01 and p c2 = 0.57±0.01. Notice the symmetry of the two critical points about p = 0.5: we shall have more to say about this later on.\n\nFor p below p c1 , there is a frozen phase characterised by overall alignment of spins: we call this the parallel frozen phase (PFP). For p above p c2 , the frozen phase that appears is characterised by an anti-parallel ordering of spins: we call this the anti-parallel frozen phase (AFP). We mention also that in the AFP, the lattice may have more than one anti-parallel domain, with thin frustrated chains running in between them. This frustration can be attributed to the inability of the different domains to align with each other under periodic boundary conditions. The disturbances caused by these chains (in quantities such as |M | or E) due to misalignment decrease as 1/ √ N and also appear to vanish for large times. Again, we notice that the phase transition disappears for low ε; in fact, at very low values of ε the evolving lattice may get trapped into striped states [13] at long times.\n\nThis can be understood as follows: the effect of a long memory (ε small) strongly reduces the relative impact of the performance-based rule. Depending on the value of ε, the performance rule may not be effective for several timesteps whereas the majority rule is implemented at every timestep. In the limit of vanishing ε, then, only the (zero-temperature) majority rule will be effective, leading to stripe formation as predicted by [13] for this situation. Figure 5 comprises snapshots of the dynamics of the model for a 2d square lattice of size N = 512 2 and at time t = 256, with random initial configurations. The plots show a portion of size 100 2 of the square lattice for three values of p: p = 0.41 (near the critical point p c1 between the PFP and the paramagnetic phase), p = 0.50 (within the paramagnetic phase) and p = 0.59 (near the critical point p c2 separating the paramagnetic phase from the AFP), with ε = 1. The snapshot at p = 0.41 shows the lattice evolving towards consensus (parallel alignment) with the formation of domains of one type only. The snapshot at p = 0.50 shows the lattice in its disordered phase, while the one at p = 0.59 shows that the nature of the lattice ordering is anti-parallel.\n\nTo investigate this more quantitatively, we plot the absolute value of magnetization |M |, the absolute value of staggered magnetization |M stag | and energy E(t) against p, with ε = 1.0, in Figure 6. These measurements were recorded using a square lattice of size N = 80 2 at time t = 10 6 . All the curves are averaged over 100 independent samples for each value of p. In the region p ≤ p c1 , the values of magnetization |M | and staggered magnetization |M stag | are both equal to unity at saturation, implying a parallel alignment of the sites; whereas for p above p c2 , the magnetization |M | is zero and the staggered magnetization |M stag | equals unity at saturation, indicating an anti-parallel alignment of the sites. The energy graph is consistent with this interpretation, given the definition of the energy in Equation 10: zero in the PFP, middling in the paramagnetic phase and unity in the AFP.\n\nIn order to confirm the voter-like nature of the critical points, we plot the inverse energy 1/E(t) against the natural logarithm of time ln t, choosing p values near both critical points (see Figure 7 and curve is an average over 200 independent samples. Exactly at the critical points p c1 = 0.43 and p c2 = 0.57, a linear behaviour of inverse energy with respect to ln t is found, with slopes of 1/2π and −1/5π respectively. While the critical exponents are those of the voter model [12], the values of the slope are different from 2/π: we find therefore that the pp update of the model belongs to the universality class of the generalised, rather than the exact, voter model [12]. To conclude this subsection: the main effect of the pp update is to change the nature of the ordering in one of the two frozen phases, so that anti-parallel ordering is found in the high-p frozen phase. As before, the effect of increasing memory (going to low ε) is to smear out the phase transitions to the disordered phase, by undermining the effect of the outcome-based rule whose com-petition with the majority rule causes the appearance of disorder. Such instances of mixed domains have been found in recent work on coevolving (parallel) dynamics [9]; some features of these results also appear in studies of threshold dynamics of societal systems [8]. For a real-life example of the AFP in the case of technology diffusion, we cite the results of [14] where the authors conclude that \"in technology clusters where direct competitors are right next door, leading firms generate innovations that are technologically very distant from their neighbours\" [14]. \n\n\nC. The ps update\n\nThe behaviour of the ps-updated model is qualitatively similar to that of the pp-updated model above. Again, there are two frozen phases PFP and AFP, separated by a disordered phase: the values of the critical points p c1 and p c2 are however shifted, such that the disordered region extends between p c1 = 0.31 ± 0.01 and p c2 = 0.69 ± 0.01 at ε = 1.0. We find once again that the two critical points are symmetrically placed with respect to p = 0.5, as in the pp update: we will give an argument for why this is so, in the following subsection.\n\nTo avoid repetition, we present only the phase diagram for the staggered magnetisation as a function of p and ε: Figure 9 shows the absolute value of the staggered magnetization |M stag | of the system at time t = 512 for a square lattice of size N = 100 2 . The paramagnetic region, with low values of |M stag | is coloured black in the figure, whereas the frozen regions (containing either parallel or anti-parallel ordering) with high values of |M stag |, are coloured yellow (light grey). These phases are investigated more quantitatively in Figure 10, where we plot the absolute value of magnetization |M |, energy E and the absolute value of staggered magnetization |M stag | against Finally, we present the variation of inverse energy with the natural logarithm of time, ln t, near the critical points p c1 and p c2 in Figure 11 and Figure 12 respectively. Each of the curves is an average over 200 independent runs. At criticality, both plots show a linear proportionality between 1/E(t) and ln t, with slopes of 4/3π and −4/15π at p c1 = 0.31 and p c2 = 0.69 respectively. Again, this indicates that the ps update of the model belongs to the generalised, rather than the exact, universality class of the voter model [12].\n\nTo conclude, the ps update yields qualitatively similar results to the pp update, with the appearance of two frozen phases PFP and AFP. Again, small values of ε indicating longer memories of outcomes, lead to a smearing out of the phase transition, because of the decreasing effectiveness of the outcome-based rule. The clue which explains all of the above, is the formation of 'active' or disparate bonds by the rules of the model under different updates: these are clearly the units of anti-parallel ordering. Consider thus configurations where a site is surrounded by a majority of its own kind: this would correspond to a local field of +2 for a +, and −2 for a −. Here the majority of the bonds are 'like' or 'inactive'. The transition probability for the increase of active bonds from such configurations is 1 − w + (+2) (or 1−w − (−2)) [see Equation 8]. The transition probabilities for the decrease of active bonds are given by an opposite scenario, yielding w − (+2) (or w + (−2)) [see Equation 8]. We plot two of these transition probabilities in Figure 13, corresponding respectively to an increase and a decrease of active bonds: the former peaks at p = 0.63 while the latter peaks at p = 0.37. The net probability of having active bonds is the difference between these two transition probabilities, and is plotted in Figure 14. We see from this that the probability of having active bonds is greatest at p = 0.79, and least at p = 0.21. The last ingredient that we need to explain the AFP phase in the pp and ps updates is the fact that once clusters with many active bonds, i.e. antiparallel ordering, are formed, the majority rule applied via the parallel update preserves such ordering. With all this in place we see that as expected, the AFP phase in both pp and ps updates shows up in qualitatively the same regions as predicted by Figure 14, with a peak, in both cases at around p = 0.79. Correspondingly, the PFP in both pp and ps updates shows up in the region predicted in this figure, with a peak in both cases at around p = 0.21. Notice (Figure 14) that the peak and the dip in the probability of active bonds are symmetric about p = 0.5, thus explaining the symmetry that we have observed in Figure 4 and Figure 9; p = 0.5 is thus the natural point for the appearance of the disordered phase in both pp and ps updates, as will be confirmed by an inspection of Figure 4, Figure 9 and Figure 14. The only remaining point to be explained is the appearance of the disordered phase in the ss update. In this case too, the analysis leading to Figure 14 for the probabilities of having active bonds remains valid. However, the sequential update of the majority rule always favours strictly parallel ordering, so that typically clusters of active bonds are destroyed once formed. When the probability of their formation is strongest, i.e. at p = 0.63 (see Figure 13), the competition between the majority and outcome-based rules is at its most intense, and a disordered phase may be expected to appear. Indeed, the mid-point of the disordered phase for the ss update is shown in Figure 1 to be in exact agreement with this predicted peak, given as it is by (p c1 + p c2 )/2 = 0.63.\n\n\nE. The sp update\n\nIn the case of this update, the phase diagram, Figure 15, shows nothing but a frozen phase. As is evident from the plot of inverse energy 1/E(t) versus ln t (Figure 16), there is a continuous increase in 1/E(t) for all values of p at ε = 1.0 (where the phase transition is expected to be the most visible). This suggests that the two rules, majority and performance-based, do not compete with each other at all (this is what had led to the appearance of the disordered phase in all the other updates). We suggest that this might be because the sequential update (with its more immediate conversions) in the case of the majority rule completely dominates the slower parallel update for the outcome-based rule: this in turn leads to an increasing tendency for consensus, independent of the value of p, with which our results are consistent. \n\n\nIV. AWAY FROM COEXISTENCE: WHEN THE STRATEGIES ARE DISTINCT\n\nEvidently, the real use of a competitive learning model such as this one is when the agents have a choice of distinct strategies. The full exploration of the behaviour of the model at coexistence as carried out in this paper as well as in earlier work [6,7] was aimed at an understanding of its phase diagram. However, in the exploration of the behaviour of the model away from coexistence, we hope to gain an understanding of the relative importance of parameters such as superiority of strategy (modelled by p) and memory (modelled by ε), when these are in competition. The behaviour in asymmetric conditions (using p + > p − and ε + > ε − ) is formulated in terms of the application of two biasing 'fields' [6] \nH = p + − p − , B = ε + − ε − ,(12)\nsuch that one strategy is favoured over the other. In the following subsection, we look at a linear response formulation of our question in terms of unequal p's, viewed as a biasing field, keeping ε the same for both strategies. In the final subsection, we look at unequal strategies as well as unequal memories, to find out whether inferior strategies applied with a good memory of past outcomes, can win overall.\n\n\nA. Linear response theory: strategies with unequal p\n\nLinear response theory is premised on the basis that an order parameter such as the magnetisation undergoes a sharp change in the neighbourhood of a critical point. In both the ss and pp updates of this model, there are two critical points p c1 and p c2 separating a paramagnetic phase from two frozen phases. In this subsection, we look at the linear response behaviour of the model in the vicinity of both critical points, starting from the disordered phase: clearly the response will depend both on the value of p as well as on the value of the biasing field H (defined in terms of the difference of the p's in Equation 12). In the following, we examine the response by choosing a given value of p, and writing p ± = p ± H/2, keeping ε fixed.\n\nWe first consider the ss-updated model. The quality of the fit to tanh(bH) is seen Figure 18: the black fitting curve almost completely coincides with a sample curve taken from Figure 17. These results also admit of an alternative representation, shown in Figure 19, where it is clear that the relative values of the bias correspond to different regions of domination of each strategy in phase space.\n\nWe next examine the linear response behaviour of the pp-updated model, again in the vicinity of the two critical points. Figure 20 is a plot showing the variation in magnetization M along the field H for different p values at ε = 1.0. For the lower values of p, in the vicinity of p c1 , we see very similar behaviour to that presented in Figure 17, corresponding to an expected tanh(bH) behaviour as shown in Figure 18: the PFP phase lying to the left of p c1 is, after all, identical to the frozen phases in the ss update. As we approach the vicinity of p c2 , the curves are markedly different: the nature of the ordered phase is one that corresponds to magnetisation values of 0 (see orange curve drawn using plus symbols in Figure 20), which is again consistent with the AFP phase that lies to the right of p c2 . To establish this more firmly we look at plots of the absolute value of the staggered magnetisation |M stag | as a function of bias H, in Figure 21. The green (triangle), blue (square) and black (plus) lines denote increasing values of p < p c2 , where the staggered magnetisation increasingly approaches zero, as expected in the disordered phase: however the red (star) line, corresponding to p > p c2 shows an abrupt jump in the value of |M stag | to unity. Combined with the analysis of the previous paragraph, this shows convincingly that the phase we refer to as AFP indeed corresponds to anti-parallel ordering.\n\nWe present below an alternative representation of the above results for ease of visualisation. In Figure 22, the magnetisation M is plotted in the p + -p − plane: as before, the regions of brown (black) (resp. yellow (light grey)) correspond to domination by − strategies (resp. + strategies). Notice, however, that the coexistence line has an island of very low magnetisation: in actual fact, this corresponds to the regions of both the paramagnetic and AFP phase. This is clearer in the plot of the absolute value of the staggered magnetisation |M stag |, shown in Figure 23, where the black portion of the island along the coexistence line corresponds to the disordered phase, while the faintly brown (grey) portion corresponds to the AFP.\n\nThese plots allow us to go beyond the previous analysis in defining the domain of stability of the AFP phase: we see clearly from Figure qualitative terms, this implies that at least in the absence of memory, when the two strategies have nearly equal success rates, neighbouring agents may adopt different strategies [14] in equilibrium.\n\nHaving thoroughly investigated the linear response regime for the ss-and pp-updated models, we will now examine the effect of the memory parameter ε in the next subsection.\n\nB. Role of memory parameters: the case of unequal ε\n\nThe principal competition in this model is that between two strategies with different global success rates p, which determines the relative dominance of each one in phase space. The memory parameter ε plays a more subtle role in this competition: although it cannot be a determinant of phase behaviour in the way that the success rates are (as a consequence of the rules elucidated in Equation 8), it can, as we will show, cause a surprising change in the dominance of an ostensibly superior strategy. In [6], it had been suggested that agents with inferior strategies and good memories might indeed win against agents who had better strategies but worse memories. Here, we will make this prediction more quantitative.\n\nThe phase diagram of the model away from coexistence involves four parameters, p ± , ε ± , so that its representation is a non-trivial problem. In the following, we choose to fix p + to 0.5, and to vary the other three parameters: a sample 3d plot is shown in Figure 24. We analyse the three visible faces in detail, before remarking on the phase behaviour within the cube: the colour coding is such that green (grey) represents dominance of + strategies, blue (black) represents dominance of − strategies, and other colours represent mixed states. • The leftmost face of the cube corresponds to the plane ε + = 0; this implies that the agents using + strategies will never convert, no matter what the outcome-based rule says. The minimum occupancy of + strategies for random configurations should thus be of the order of N/2, which can only increase depending on the conversions of agents using − strategies into the camp of the +'s. The bottom line corresponds to p − = 0, which is when such conversions are maximal (so that all N sites are +): the green (grey) colour is at its most pronounced here, changing gradually over to other colours only as ε − → 0 to the right of the line, when agents using − strategies too begin to refuse to convert, irrespective of the outcome rules. As the values of p − increase beyond 0.5 (the fixed value for p + ), we note that the dominance of the + strategy gradually gives way to states with a mixture of strategies; when ε − → 0, this tendency is at its most pronounced, while when ε − → 1, this is at its least pronounced, since local conversions can sometimes go against global success rates.\n\n• The front face of the cube corresponds to ε − = 0; this implies that agents using − strategies will never convert, no matter what the outcome-based rule says. This is a reflection of the previous case,\n\nwhere the minimum number of − sites is once again N/2, which can only be increased as the conversions from the +'s add to it.\n\n• The top face of the cube corresponds to p − = 1, where globally a predominance of the − strategy is expected. This is found over almost all the range of ε − except at low values of ε + , where agents using the + strategy refuse to convert, despite their globally poorer performance. The interior of the cube can show markedly different behaviour, which we illustrate via a sample slice shown in Figure 25. In this figure, red (lower black) and blue (upper black) regions correspond to the dominance of + and − strategies respectively. Here we set the value of p + to 0.7, and look at a slice of its phase space cube, as before: choosing ε + to be between 0.7 and 0.8, we look at the dominating strategy as a function of the variables ε − and p − . If the memory parameters had not existed, we would have expected the − strategy (blue (upper black) region in the figure) to predominate only for p − > 0.7; the red (lower black) region would have been covering the entire slice below this, corresponding to the dominance of the + strategy. However, the reality is rather different. The + strategy does indeed predominate for p − < 0.7, provided that agents using the − strategy have imperfect memory; but there is a striking predominance of the − strategy (even for very low values of p − ) provided that the memory of the agents employing this strategy, is much better than those of the other kind (ε − ≪ ε + ).\n\nA last feature to mention is the green (grey) region in Figure 25: here, there is a region of alternating + and − ordering (AFP), corresponding to ongoing competition between the two strategies. This phenomenon is most pronounced when the two strategies are equally successful, and are both accompanied by weak memories of earlier outcomes. In Figure 26, the structure of the full AFP is shown (by selecting phase points with low values of absolute magnetisation |M | and high absolute staggered magnetization |M stag |) as a function of p − and ε ± , fixing p + = 0.5.\n\n\nV. DISCUSSION\n\nThe work of this paper extends work done on a problem of strategic learning [6,7] which, although originally suggested by a problem on technology diffusion [4], has much wider ramifications (e.g., in relation to threshold learning dynamics [8]).\n\nIn any agent-based modelling scheme, it is important to know whether agents react sequentially or collectively to the spread of information. Our results show that these issues make a quantitative as well as a qualitative difference to the results, changing not just exponents but also the entire nature of the phase diagram in most cases. Given that, typically, the propagation of technologies through well-connected societies is of interest [4], we choose ordered rather than random updates, and examine the response of the model of [6] to all possible combinations of sequential and parallel updating. From the viewpoint of theoretical physics, a major result is that this model is robustly in the universality class of the voter model [12], for all but one of the updates. This strong relationship with the voter model results from the model of [6,7] being driven by interfacial noise alone, i.e. the absence of surface tension [12].\n\nAnother major result, still to do with updates, is the appearance of a phase of anti-parallel ordering (AFP) in the high-performing limits of p, for both the pp and the ps updates. While the technicalities behind this are explained in the text, we give here a more intuitive reason for this, from the perspective of strategic learning. The parallel scheme can be viewed as a more 'equilibrated' update than the sequential one, since it gives a chance for the entire lattice to be updated 'simultaneously'. It is then natural that in the regime that both agents are high-performing, they should be equally preferred: this lies behind the 'alternating' order inherent in the AFP regime. By contrast, since the sequential paradigm corresponds to a 'non-equilibrium' update, where every agent responds to the updated value of its neighbours, the above logic leads to a disordered phase where every prescription of the outcome-based rule is countermanded by the following majority rule. Using once again the illustration of propagating technologies [4]: when all the populace have equal and simultaneous access to information about two high-performing technologies, we will see the coexistence of both [14] (as predicted by the AFP phase), whereas when information about each one is passed on sequentially, the conflicting information so obtained can result in sheer disorder. Finally, we mention here that our investigation of different updates on this game-theoretic model has been applied to related game-theoretic models of cognitive learning and synaptic plasticity [15,16], where updates relate to the directionality of synapses in a network.\n\nMoving away from the domain of critical behaviour at coexistence, we have looked at the behaviour of the competitive learning model when the two strategies have distinct attributes (this, after all, is truer to the title of competitive learning!). To begin with, we have examined the response of the model to unequal strategies p ± , and have found in general that the smarter strategy wins (for equal values of the memory parameter ε), as might be expected. An interesting feature is that the region of anti-parallel ordering (AFP) found earlier still persists in the presence of bias, provided that the difference in p is below a well-defined bound: in other words, when two distinct strategies are almost equally successful, one will typically find that they can coexist in society. Finally, we have looked at the effect of memory: we have found that while memory has a secondary role in determining the phase behaviour of the model, it has a particularly striking effect in turning around the results of any bias in p. A major result of our paper is thus that decisions based on a good memory of earlier outcomes can, within limits, compensate for the choice of inferior strategies.\n\n\nACKNOWLEDGMENTS\n\nAAB would like to thank Dr. G. Mahajan and Mr. B. Chakraborty for helpful discussions. AM acknowledges the award of a grant from the DST (Govt. of India) through the project \"Generativity in Cognitive Networks\", through which AAB was supported.\n\nFIG. 1 .\n1(color online)[ss update] Phase diagram of the model with an ss update. Plot of the inverse energy 1/E(t) at time t = 512 for a square lattice of size N = 64 2 in the p-ε plane. The black region shows the disordered phase and the yellowish (light grey) region shows the frozen phase.\n\nFigure 2\n2shows snapshots of the dynamics of the model using a lattice of size N = 512 2 at times t = 8, t = 64, and t = 512 with random initial configurations and parameter values p = 0.72 (very close to the critical point p c2 ) and ε = 1.0. The plots reveal characteristically voter-like[12] coarsening behaviour.\n\nFIG. 2 .\n2(color online)[ss update] Snapshots of the dynamics of the ss-updated model. Each plot is a portion (of size 100 2 ) of a square lattice of N = 256 2 for p = 0.72 and ε = 1.0 at times t = 8 (top-left), t = 64 (top-right) and t = 512 (bottom).\n\nFIG. 3 .\n3(color online)[ss update] Plot of the inverse energy 1/E(t) versus the natural logarithm of time ln t for different values of p close to pc2 = 0.70, with ε set to 1.0. The lattice size N = 256 2 , and the p-values are indicated on the curves.\n\nFIG. 4 .\n4(color online)[pp update] Plot of the absolute value of the magnetization |M | (top) and the absolute value of the staggered magnetization |Mstag| (bottom) at time t = 512 for a lattice of N = 100 2 in the p-ε plane. In the top figure, the yellowish (light grey) region refers to the parallel frozen phase (PFP), while the black region includes both the antiparallel frozen phase (AFP), and the disordered region. In the bottom figure, the black region represents the disordered phase characterised by very low |Mstag|.\n\nFIG. 5 .\n5(color online)[pp update] Snapshots of the dynamics of the pp-updated model on a square lattice, for p = 0.41 (leftmost), p = 0.50 (centre) and p = 0.59 (rightmost) at time t = 512, with ε = 1.0. The yellow (light grey) and black colours represent the two strategies, while the greyish grid corresponds to anti-parallel arrangements of yellow (light grey) and black. The leftmost picture represents the PFP (see text), the centre one the disordered phase, and the rightmost one the AFP (see text).\n\nFigure 8 FIG. 6 .\n86(color online)[pp update] Plot of the absolute value of magnetization |M |, the absolute value of staggered magnetization |Mstag| and energy E against p, with ε = 1 and lattice size N = 80 2 . Each curve is drawn using symbols (and colour) as indicated in the legend.\n\nFIG. 7 .\n7(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.43 and ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.43 (shown in red) has slope 1/2π approximately.\n\nFIG. 8 .\n8(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.57, with ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.57 (shown in red) has slope −1/5π approximately.\n\nFIG. 9 .\n9(color online)[ps update] Phase diagram in the p-ε plane of the ps-updated model, with a plot of the absolute value of staggered magnetization |Mstag| at time t = 512, for a lattice size of N = 100 2 . The black region represents the disordered phase (very low |Mstag|), while the yellowish (light grey) region represents frozen phases with high |Mstag|. p, with ε equal to 1.0; each curve is an average over 100 independent runs. The region where both the magnetization |M | and staggered magnetization |M stag | curves saturate to 1, corresponds to parallel alignment, whereas |M | ≈ 0 with |M stag | ≈ 1 implies an anti-parallel alignment of the spin types. The energy graph is consistent with this interpretation, given the definition of the energy in Equation10: zero in the PFP, middling in the paramagnetic phase and unity in the AFP. 10. (color online)[ps update] Plot of the absolute value of magnetization |M |, energy E and the absolute value of staggered magnetization |Mstag| against p with ε set to 1.0 for a lattice size N = 100 2 . Each curve is drawn using symbols (and colour) as indicated in the legend.\n\nFIG\n. 11. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.31, with ε set to 1.0 for a lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.31 (shown in red) has a slope of approximately 4/3π.\n\nFIG\n. 12. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.69, with ε set to 1.0, for lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.69 (shown in red) has slope −4/15π approximately. D. Explanation for the nature of the phase diagrams for different updates In this subsection, we give arguments for the three most important features of the phase diagrams presented above: (i) The appearance of anti-parallel ordering in both pp and ps updates (ii) The symmetry of the PFP and the AFP phases in both pp and ps updates (iii) The positioning of the disordered phase in ss, pp and ps updates\n\n\nFIG. 13. (color online) Transition probabilities (1 − w+(+2)) (drawn as solid line (in green (grey))) and w−(+2) (drawn as dashed line (in black)) against p [from Equation 8].\n\n\nFIG. 14. (color online) The difference in the transition probabilities (1 − w+(+2)) and w−(+2) is plotted against p [seeEquation 8].\n\n\nFIG. 15. (color online)[sp update] Phase diagram of the spupdated model. Plot of the absolute value of magnetization |M | at time t = 512 for lattice size N = 64 2 in the p-ε plane. No phase transition is visible.\n\nFIG\n. 16. (color online)[sp update] Plot of the inverse energy 1/E(t) versus ln (t) for different values of p and ε = 1, for a lattice size N = 256 2 . The value of p for each curve is given by a different colour as indicated. No phase transition is visible.\n\n\nFigure 17shows a plot for magnetization M against the biasing field H at various values of p, that are within the paramagnetic phase at ε = 1.0. Each curve is obtained after averaging over 100 initial configurations using a square lattice of size N = 100 2 . For each p in the paramagnetic phase, we see a linear behaviour of M against H around H ∼ 0, with all subsequent increases in the field strength leading to saturation, as expected. For a given p value we observe a functional dependence of the formM = tanh(bH) where b ∝ (p central + p) 2 taking p central = (p c1 + p c2 )/2.\n\nFIG\n. 17. (color online)[ss update] Plot of magnetization M against biasing field H for different values of p. Each curve is drawn using different symbols (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0.\n\n\nFIG. 18. (color online)[ss update] Plot of magnetization M against field H for p = 0.58 and ε = 1.0 at time t = 2000 with N = 100 2 . The fit of a tanh(bH) curve almost completely overlaps with our numerical results.\n\nFIG. 19 .\n19(color online)[ss update] Plot of magnetization M in the p+-p− plane at time t = 7000, for a lattice of size N = 100 2 with ε = 1.0. The + strategy dominates in the yellow (light grey) region, while the − strategy dominates in the brown (black) region.\n\nFIGFIG\n22 and Figure 23 that the AFP phase exists for p > p c2 only if the biasing field is within the bounds defined by H = |p + − p − | ≤ 0.19 ± 0.02. In . 20. (color online)[pp update] Plot of magnetization M against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0. . 21. (color online)[pp update] Plot of the absolute value of staggered magnetization |Mstag| against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 64 2 and ε = 1.0.\n\nFIG. 22 .\n22(color online)[pp update] Plot of magnetization M in the p+-p− plane for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. As before, the regions of + and − strategy domination are coloured yellow (light grey) and brown (black); the orange (grey) region corresponds to both the paramagnetic and the AFP region (see text).FIG. 23. (color online)[pp update]Plot of staggered magnetization |Mstag| in the p+-p− plane, for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. Here, yellow (light grey) represents the region of parallel ordering, black represents the disordered phase, and the light brown (grey) represents AFP order.\n\n\nFIG. 24. (color online)[pp update] A 3D plot of Magnetization M with parameters ε+ (along x), ε− (along y) and p− (along z), setting p+ = 0.50 for a lattice of size N = 64 2 at time T = 2000. Green (grey) denotes the dominance of the +'s, while blue (black) denotes that of the −'s. The other colours represent cases of intermediate ordering.\n\n\n. 25. (color online)[pp update] A 3D plot of Magnetization M for the parameters ε+ (lying between 0.7 and 0.8), ε− and p−, with p+ = 0.70, for a lattice of size N = 64 2 at time T = 2000. The red (lower black) region represents the dominance of the + strategy and the blue (upper black) region represents the dominance of the − strategy. The green (grey) region represents AFP.\n\n\n26. (color online)[pp update] A 3D plot of absolute staggered magnetization |Mstag| with parameters ε+(along x), ε−(along y) and p−(along z) for p+ = 0.50 and lattice size N = 64 2 , at time T = 2000. The graph delineates the region of AFP for these parameter values.\n\nStatistical physics of social dynamics. C Castellano, S Fortunato, V Loreto, Rev. Mod. Phys. 81591646C Castellano, S Fortunato and V Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591646 (2009).\n\nD Fudenberg, D K Levine, The Theory of Learning in Games. CambridgeMIT PressD Fudenberg and D K Levine, The Theory of Learning in Games (MIT Press, Cambridge 1998).\n\nYoung H Payton, Strategic Learning and Its Limits. OxfordOxford University PressH Payton Young, Strategic Learning and Its Limits (Ox- ford University Press, Oxford 2004).\n\nTechnology diffusion by learning from neighbours. K Chatterjee, S H Xu, Adv. Appl. Prob. 36K Chatterjee and S H Xu, Technology diffusion by learn- ing from neighbours, Adv. Appl. Prob. 36, 355-376 (2004).\n\nBehavioural studies of strategic thinking in games. C F Camerer, Trends Cogn. Sci. 7C F Camerer, Behavioural studies of strategic thinking in games, Trends Cogn. Sci. 7, 225-231 (2003).\n\nModels of competitive learning: Complex dynamics, intermittent conversions, and oscillatory coarsening. A Mehta, J M Luck, Physical Review E. 60A Mehta and J M Luck, Models of competitive learning: Complex dynamics, intermittent conversions, and oscil- latory coarsening, Physical Review E 60, 5: 5218-5230 (1999).\n\nCompeting with oneself: introducing self-interaction in a model of competitive learning. G Mahajan, Mehta, Theory Biosci. 129G Mahajan and A Mehta, Competing with oneself: intro- ducing self-interaction in a model of competitive learning, Theory Biosci. 129, 271-282 (2010).\n\nJ C Gonzalez-Avella, V M Eguluz, M Marsili, M Vega-Redondo, Miguel San, Threshold Learning Dynamics in Social Networks. 620207J C Gonzalez-Avella, V M Eguluz, M Marsili, F Vega- Redondo and M San Miguel, Threshold Learning Dynam- ics in Social Networks, PLoS ONE 6, 5: e20207 (2011).\n\nCoevolution of Glauber-like Ising dynamics and topology. S Mandra, S Fortunato, C Castellano, Phys. Rev. E. 8056105S Mandra, S Fortunato and C Castellano, Coevolution of Glauber-like Ising dynamics and topology, Phys. Rev. E 80, 056105 (2009).\n\nSynchronous or asynchronous parallel dynamics. Which is more efficient?. Kanter, Physica D. 42I Kanter, Synchronous or asynchronous parallel dynam- ics. Which is more efficient?, Physica D 42, 273-280 (1990).\n\nParallel versus sequential updating for belief propagation decoding. H Kfir, Kanter, Physica A. 330H Kfir and I Kanter, Parallel versus sequential updating for belief propagation decoding, Physica A 330, 1-2: 259- 270 (2003).\n\nCritical Coarsening without Surface Tension: The Universality Class of the Voter Model. Dornic, Chate, H Chave, Hinrichsen, Phys. Rev. Lett. 8745701I Dornic, H Chate, J Chave, and H Hinrichsen, Critical Coarsening without Surface Tension: The Universality Class of the Voter Model, Phys. Rev. Lett. 87, 045701 (2001).\n\nFreezing into stripe states in two-dimensional ferromagnets and crossing probabilities in critical percolation. K Barros, S Krapivsky, Redner, Phys. Rev. E. 8040101K Barros, P L Krapivsky, and S Redner, Freezing into stripe states in two-dimensional ferromagnets and cross- ing probabilities in critical percolation, Phys. Rev. E 80, 040101 (2009).\n\nGlobal Competitors as Next-Door Neighbors: Competition and Geographic Concentration in the Semiconductor Industry. M Zhao, J Alcacer, Ross School of Business Paper. 1091M Zhao and J Alcacer, Global Competitors as Next-Door Neighbors: Competition and Geographic Concentration in the Semiconductor Industry, Ross School of Business Paper No. 1091 (2007).\n\nCompeting synapses with two timescales -a basis for learning and forgetting. G Mahajan, Mehta, Europhysics Letters. 9548008G Mahajan and A Mehta, Competing synapses with two timescales -a basis for learning and forgetting, Euro- physics Letters 95, 48008 (2011);\n\nLearning with a network of competing synapses. A A Bhat, G Mahajan, A Mehta, PLoS ONE. 625048A A Bhat, G Mahajan and A Mehta, Learning with a network of competing synapses, PLoS ONE 6, 9: e25048 (2011).\n"},"corpus_id":{"kind":"number","value":30280960,"string":"30,280,960"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1105.0523\",\n \"mag\": 2038555311,\n \"acl\": null,\n \"pubmed\": \"22400539\",\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physreve.85.011134\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1105.0523v3.pdf\"\n ],\n \"pdf_sha\": \"774622d2ddcd82d75f14f591d9c1558c095db812\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1105.0523\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"The dynamics of competitive learning: the role of updates and memory"},"authors":{"kind":"list like","value":["Ajaz Ahmad Bhat \nTheoretical Science Department\nS N Bose National Centre\nBlock JD Sector III\nSalt Lake700098KolkataIndia","Anita Mehta \nTheoretical Science Department\nS N Bose National Centre\nBlock JD Sector III\nSalt Lake700098KolkataIndia"],"string":"[\n \"Ajaz Ahmad Bhat \\nTheoretical Science Department\\nS N Bose National Centre\\nBlock JD Sector III\\nSalt Lake700098KolkataIndia\",\n \"Anita Mehta \\nTheoretical Science Department\\nS N Bose National Centre\\nBlock JD Sector III\\nSalt Lake700098KolkataIndia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We examine the effects of memory and different updating paradigms in a game-theoretic model of competitive learning, where agents are influenced in their choice of strategy by both the choices made by, and the consequent success rates of, their immediate neighbours. We apply parallel and sequential updates in all possible combinations to the two competing rules, and find, typically, that the phase diagram of the model consists of a disordered phase separating two ordered phases at coexistence. A major result is that the corresponding critical exponents belong to the generalised universality class of the voter model. When the two strategies are distinct but not too different, we find the expected linear response behaviour as a function of their difference. Finally, we look at the extreme situation when a superior strategy, accompanied by a short memory of earlier outcomes, is pitted against its inverse; interestingly, we find that a long memory of earlier outcomes can occasionally compensate for the choice of a globally inferior strategy."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"The modelling of social behaviour is of increasing concern to statistical physicists [1]. Studies of social and biological systems often reveal that even when the interactions of a given individual are very localised in time and space, collective, regular behaviour can emerge: this is analogous to the cooperative behaviour manifested by emergent systems in the natural world. Such social regularities may well take the form of learning, when individuals adopt the behaviour of other individuals. From the perspective of game theory [2], this can be seen as an adoption of a particular strategy, whose result may or may not be associated with a favourable outcome. It is then quite reasonable to expect that the effectiveness of a strategy in yielding favourable outcomes should influence how likely it is to persist, and spread through the population; the resulting ideas of strategic learning [3] have found wide application, starting from economics [4] to cognitive science [5].\nAgainst the backdrop of the above ideas, a model of strategic learning was introduced in [6], with one of two possible strategies (denoted as + and − in the remainder of this paper) being available to each agent on a lattice: the agents were referred to as 'myopic' (aware only of their immediate neighbours) and 'memoryless' (unaware of their own and others' past outcomes) in the paper on technology diffusion [4] that inspired the above model [6,7]. The question on which this body of work has centred is: despite these handicaps, can agents overall learn to use the superior one of two available technologies? Briefly, each agent changes (or does not change) strategy based on two elementary rules at every time step: a * ajaz@bose.res.in † anita@bose.res.in majority-based rule, reflecting its tendency to align with its local neighbourhood, followed by a performance-based rule, where the agent adopts the strategy that 'wins' in its neighbourhood. This (relative) success is measured in terms of outcomes, where the probability of a successful outcome for strategy + (−) is p + (p − ). Also, the model of [6] added to the description of [4] by endowing the agents with memory: those agents who make their choices on the basis of the last payoff alone, are adjudged to be memoryless (with a corresponding parameter ε near 1), while those who allow for memories of earlier outcomes may make decisions that run counter to immediate evidence (ε small).\nSome related ideas have been examined in recent work. For example, the issue of consensus formation in a model of threshold learning [8] shows close analogies: in this model, the competition between the 'noisy' signals from the immediate neighbours of an agent (cf. the majority rule in [6]) and the acceptance threshold that agents require to change their state (cf. the memory threshold in the performance-based rule of [6]), determine the phase diagrams obtained. Recent studies of coevolving Glauber dynamics on networks [9] are also relevant, since the model of [6] can be viewed as a competition between the Glauber dynamics of two sets of Ising spins, corresponding to strategy and outcome respectively.\nIn the current paper, we take all these ideas further. First, we explore the effect of different updates. If new information propagates sequentially through the network, and the arrow of time is discernible in the decisions of individual agents, are the global phase diagrams any different from what they would be if information was transmitted and all decisions were taken simultaneously? Common sense tells us that sequential or parallel updates should make a difference to the nature of the phase diagram, and the results of the present paper confirm this. Also (unlike the work of [6,7] which examined the situa-tion at coexistence) we look in this paper at the effects of disparate strategies (p + = p − ). The final, and possibly most important issue, is that of memory, which acts as a threshold governing change [8]: what is the effect of the threshold ε, which tells the agent that longer-term inputs are significant, and need to be considered when making a decision? We will find that, indeed, a longer memory of earlier outcomes can sometimes make up for the choice of a globally inferior strategy.\nThe plan of this paper is as follows. In Section II, we review the model of [6]. In Section III, we discuss the behaviour of the model for a range of updating schemes, in the presence of memory. In Section IV, we examine the behaviour of the model away from coexistence, as a function of distinct parameter values for the two strategies; in particular we discuss here the role of memory. In the concluding section, we discuss our results and put them in the context of other recent work in the field."},{"title":"II. DEFINITION OF THE MODEL","content":"The model of [6] involves two types of strategies, − and +, where the + strategy is globally superior [4] to the − strategy. As mentioned above, agents tend to follow the strategy adopted by the majority of their neighbours, modifying this choice in a second step (if necessary) according to which of these have proved to be the most successful.\nAssuming that the agents sit at the nodes of a ddimensional regular lattice with coordination number z = 2d, the efficiency of an agent at site i is represented by an Ising spin variable:\nThe evolution dynamics of the lattice is governed by two rules. The first is a majority rule, which consists of the alignment of an agent with the local field (created by its nearest neighbours) acting upon it, according to:\nHere, the local field\nis the sum of the efficiencies of the z neighbouring agents j of site i and τ 1 is the associated time step. Next, a performance rule is applied. This starts with the assignment of an outcome σ i (another Ising-like variable, with values of ±1 corresponding to success and failure respectively) to each site i, according to the following rules:\nif η i (t) = +1,\nwhere τ 2 is the associated time step and p ± are the probabilities of having a successful outcome for the corresponding strategy. With N + i and N − i denoting the total number of neighbours of a site i who have adopted strategies + and − respectively, and I + i (I − i ) denoting the number of successful outcomes within the set N + i (N − i ), the dynamical rules for site i are:\nif η i (t) = +1 and\nHere, the ratios Ii(t) Ni(t) are nothing but the average payoff assigned by an agent to each of the two strategies in its neighbourhood at time t (assuming that success yields a payoff of unity and failure, zero). Also, τ 3 is the associated time step and the parameters ε ± are indicators of the memory associated with each strategy. In their full generality, ε and p are independent variables: the choice of a particular strategy can be associated with either a short or a long memory. However, we would like in this paper to answer a question which was posed, but not answered in [6]: can the presence of a good memory compensate for the choice of an inferior strategy? We therefore examine the extreme situation when a globally superior strategy (p + ≫ p − ), combined with a shorter memory (ε + ≫ ε − ) is in competition with its inverse: this is the situation that will be studied in Section IV.\nSetting the timescales\nthe above steps of the performance rule are recast as effective dynamical rules involving the efficiencies η i (t) and the associated local fields alone:\nThe effective transition probabilities w ± (h) are evaluated by enumerating the 2 z possible realizations of the outcomes σ j of the sites neighbouring site i, and weighing them appropriately. For a 2-d square lattice, the possible local field values at the interfacial sites are 0 and ±2.\nThe corresponding transition probabilities for these field values are [6]:\nIn [6], the model was explored at coexistence with an ordered sequential update applied to memoryless agents [4]:\nIn the present paper, we go beyond this in two different ways. First, still at coexistence, we explore the effect of different updates on the p − ε phase diagram of the model: next, we examine the model away from coexistence, for distinct values of p ± and ε ± . The basic quantities considered hereafter are the magnetization M , staggered magnetization M stag and the energy E. These quantities are defined for a finite sample of N agents (or sites) and N z /2 bonds (or links), as\nIn the following we shall usually consider mean values M , E and M stag ."},{"title":"III. THE EFFECT OF FINITE MEMORY, AND OF DIFFERENT UPDATES","content":"We begin this section with a review of the physical significance of updating schemes. Most generally, updates can be random or ordered as follows:\n• Random: Here, sites are chosen at random for the consecutive application of rules.\n• Ordered : Here, sites are chosen in an ordered fashion, i.e., after choosing every (i, j) th site, the (i, j + 1) th site is selected.\nSince the sociological basis for this work was the propagation of innovation through connected societies [4], we choose to deal only with ordered updates here. However, even ordered updates have two subclasses: parallel and sequential. Assume a condition A such that when an agent satisfies A, it changes strategy:\n• Sequential update: In this type of update, we check the condition A on the (i, j) th site, then update the efficiency of the site and proceed to the (i, j + 1) th site using the updated value of the (i, j) th site.\n• Parallel update: In this type of update, we check the condition A on the (i, j) th site, do not update the site but instead save the update-decision in memory, and proceed to the next site. Once the whole lattice is swept, all the saved updatedecisions are implemented 'simultaneously'.\nThe choice of different updates generally corresponds to different physical situations: it has been shown that it also leads to a disparity in the convergence time of the systems concerned [10,11]. We therefore examine all possible combinations for our two update rules:\nI parallel updates for both majority rule and performance rules (pp).\nII parallel update for majority rule and sequential update for performance rules (ps).\nIII sequential updates for both majority rule and performance rules (ss).\nIV sequential update for majority rule and parallel update for performance rules (sp).\nIn the following subsections, we explore the phase dynamics at coexistence for each of these update rules in turn, for both parameters p and ε. We state at the outset that all the updates (except for the sp update) which we consider, result in models which are in the general university class of the voter model [12]: the inverse energy 1/E(t) is thus always proportional to the logarithm of time, ln t. When, as in the case of the ss update, the value of the slope is exactly 2/π [6], the exact universality class of the voter model is retrieved."},{"title":"A. The ss update","content":"This is the update that was used throughout [6]; however the phase behaviour of the model was there only explored for the parameter p, whereas here we extend it to the parameter ε. In Figure 1, we plot the inverse energy 1/E(t) in the p − ε plane at time t = 512 for a square lattice of size N = 64 2 . This phase diagram shows clearly the existence of a disordered paramagnetic phase embedded in a largely frozen phase elsewhere. The disordered phase exists for p c1 (= 0.56 ± 0.01) < p < p c2 (= 0.70 ± 0.01) when ε ≥ 0.980. Our results agree with those of [6] for ε = 1, and extend them all across the rest of the p − ε plane. We mention here that the average time required to reach consensus increases exponentially as p decreases in the frozen phase, leading to the presence of striped states [13] at limiting values of p. Figure 1 also makes it clear that the effect of increasing memory wipes out the disordered phase: this is as it should be, since the disordered phase is generated by the competition between the majority and performance-based rules, which is dulled by increasing memory. In Figure 3, we have plotted the inverse energy 1/E(t) against the natural logarithm of time ln t for values of p around the critical point p c2 = 0.70 ± 0.01. Each of the curves is obtained by averaging over 200 independent samples of size 256 2 . At the critical point, we obtain a straight line with a slope close to 2/π [6], a behaviour characteristic of the exact voter model [12] that corresponds to\nSimilar behaviour is obtained at the other critical point p c1 , in agreement with [6]. The curve corresponding to p = pc2 = 0.70 (shown in red) has slope 2/π approximately [see Equation 11]."},{"title":"B. The pp update","content":"In this case, both environmental majority and performance-based rules are applied using parallel updates. As we will see, although the universality class of the model is qualitatively unchanged, this update results in the appearance of novel ordered phases compared to the ss update. As before, we first plot the phase diagram for all values of p and ε, then show snapshots of the dynamics, and finally get a more quantitative feel for the behaviour of key quantities as a function of p.\nAccordingly, Figure 4 (top and bottom), are plots of the absolute values of magnetization |M | and staggered magnetization |M stag |, at time t = 512 for a lattice size N = 100 2 , in the p-ε plane using pp updates. In these phase diagrams, we see clear evidence of the existence of two distinct frozen phases separated by a disordered phase. Looking along the line ε = 1, disorder prevails for p c1 < p < p c2 with p c1 = 0.43±0.01 and p c2 = 0.57±0.01. Notice the symmetry of the two critical points about p = 0.5: we shall have more to say about this later on.\nFor p below p c1 , there is a frozen phase characterised by overall alignment of spins: we call this the parallel frozen phase (PFP). For p above p c2 , the frozen phase that appears is characterised by an anti-parallel ordering of spins: we call this the anti-parallel frozen phase (AFP). We mention also that in the AFP, the lattice may have more than one anti-parallel domain, with thin frustrated chains running in between them. This frustration can be attributed to the inability of the different domains to align with each other under periodic boundary conditions. The disturbances caused by these chains (in quantities such as |M | or E) due to misalignment decrease as 1/ √ N and also appear to vanish for large times. Again, we notice that the phase transition disappears for low ε; in fact, at very low values of ε the evolving lattice may get trapped into striped states [13] at long times.\nThis can be understood as follows: the effect of a long memory (ε small) strongly reduces the relative impact of the performance-based rule. Depending on the value of ε, the performance rule may not be effective for several timesteps whereas the majority rule is implemented at every timestep. In the limit of vanishing ε, then, only the (zero-temperature) majority rule will be effective, leading to stripe formation as predicted by [13] for this situation. Figure 5 comprises snapshots of the dynamics of the model for a 2d square lattice of size N = 512 2 and at time t = 256, with random initial configurations. The plots show a portion of size 100 2 of the square lattice for three values of p: p = 0.41 (near the critical point p c1 between the PFP and the paramagnetic phase), p = 0.50 (within the paramagnetic phase) and p = 0.59 (near the critical point p c2 separating the paramagnetic phase from the AFP), with ε = 1. The snapshot at p = 0.41 shows the lattice evolving towards consensus (parallel alignment) with the formation of domains of one type only. The snapshot at p = 0.50 shows the lattice in its disordered phase, while the one at p = 0.59 shows that the nature of the lattice ordering is anti-parallel.\nTo investigate this more quantitatively, we plot the absolute value of magnetization |M |, the absolute value of staggered magnetization |M stag | and energy E(t) against p, with ε = 1.0, in Figure 6. These measurements were recorded using a square lattice of size N = 80 2 at time t = 10 6 . All the curves are averaged over 100 independent samples for each value of p. In the region p ≤ p c1 , the values of magnetization |M | and staggered magnetization |M stag | are both equal to unity at saturation, implying a parallel alignment of the sites; whereas for p above p c2 , the magnetization |M | is zero and the staggered magnetization |M stag | equals unity at saturation, indicating an anti-parallel alignment of the sites. The energy graph is consistent with this interpretation, given the definition of the energy in Equation 10: zero in the PFP, middling in the paramagnetic phase and unity in the AFP.\nIn order to confirm the voter-like nature of the critical points, we plot the inverse energy 1/E(t) against the natural logarithm of time ln t, choosing p values near both critical points (see Figure 7 and curve is an average over 200 independent samples. Exactly at the critical points p c1 = 0.43 and p c2 = 0.57, a linear behaviour of inverse energy with respect to ln t is found, with slopes of 1/2π and −1/5π respectively. While the critical exponents are those of the voter model [12], the values of the slope are different from 2/π: we find therefore that the pp update of the model belongs to the universality class of the generalised, rather than the exact, voter model [12]. To conclude this subsection: the main effect of the pp update is to change the nature of the ordering in one of the two frozen phases, so that anti-parallel ordering is found in the high-p frozen phase. As before, the effect of increasing memory (going to low ε) is to smear out the phase transitions to the disordered phase, by undermining the effect of the outcome-based rule whose com-petition with the majority rule causes the appearance of disorder. Such instances of mixed domains have been found in recent work on coevolving (parallel) dynamics [9]; some features of these results also appear in studies of threshold dynamics of societal systems [8]. For a real-life example of the AFP in the case of technology diffusion, we cite the results of [14] where the authors conclude that \"in technology clusters where direct competitors are right next door, leading firms generate innovations that are technologically very distant from their neighbours\" [14]."},{"title":"C. The ps update","content":"The behaviour of the ps-updated model is qualitatively similar to that of the pp-updated model above. Again, there are two frozen phases PFP and AFP, separated by a disordered phase: the values of the critical points p c1 and p c2 are however shifted, such that the disordered region extends between p c1 = 0.31 ± 0.01 and p c2 = 0.69 ± 0.01 at ε = 1.0. We find once again that the two critical points are symmetrically placed with respect to p = 0.5, as in the pp update: we will give an argument for why this is so, in the following subsection.\nTo avoid repetition, we present only the phase diagram for the staggered magnetisation as a function of p and ε: Figure 9 shows the absolute value of the staggered magnetization |M stag | of the system at time t = 512 for a square lattice of size N = 100 2 . The paramagnetic region, with low values of |M stag | is coloured black in the figure, whereas the frozen regions (containing either parallel or anti-parallel ordering) with high values of |M stag |, are coloured yellow (light grey). These phases are investigated more quantitatively in Figure 10, where we plot the absolute value of magnetization |M |, energy E and the absolute value of staggered magnetization |M stag | against Finally, we present the variation of inverse energy with the natural logarithm of time, ln t, near the critical points p c1 and p c2 in Figure 11 and Figure 12 respectively. Each of the curves is an average over 200 independent runs. At criticality, both plots show a linear proportionality between 1/E(t) and ln t, with slopes of 4/3π and −4/15π at p c1 = 0.31 and p c2 = 0.69 respectively. Again, this indicates that the ps update of the model belongs to the generalised, rather than the exact, universality class of the voter model [12].\nTo conclude, the ps update yields qualitatively similar results to the pp update, with the appearance of two frozen phases PFP and AFP. Again, small values of ε indicating longer memories of outcomes, lead to a smearing out of the phase transition, because of the decreasing effectiveness of the outcome-based rule. The clue which explains all of the above, is the formation of 'active' or disparate bonds by the rules of the model under different updates: these are clearly the units of anti-parallel ordering. Consider thus configurations where a site is surrounded by a majority of its own kind: this would correspond to a local field of +2 for a +, and −2 for a −. Here the majority of the bonds are 'like' or 'inactive'. The transition probability for the increase of active bonds from such configurations is 1 − w + (+2) (or 1−w − (−2)) [see Equation 8]. The transition probabilities for the decrease of active bonds are given by an opposite scenario, yielding w − (+2) (or w + (−2)) [see Equation 8]. We plot two of these transition probabilities in Figure 13, corresponding respectively to an increase and a decrease of active bonds: the former peaks at p = 0.63 while the latter peaks at p = 0.37. The net probability of having active bonds is the difference between these two transition probabilities, and is plotted in Figure 14. We see from this that the probability of having active bonds is greatest at p = 0.79, and least at p = 0.21. The last ingredient that we need to explain the AFP phase in the pp and ps updates is the fact that once clusters with many active bonds, i.e. antiparallel ordering, are formed, the majority rule applied via the parallel update preserves such ordering. With all this in place we see that as expected, the AFP phase in both pp and ps updates shows up in qualitatively the same regions as predicted by Figure 14, with a peak, in both cases at around p = 0.79. Correspondingly, the PFP in both pp and ps updates shows up in the region predicted in this figure, with a peak in both cases at around p = 0.21. Notice (Figure 14) that the peak and the dip in the probability of active bonds are symmetric about p = 0.5, thus explaining the symmetry that we have observed in Figure 4 and Figure 9; p = 0.5 is thus the natural point for the appearance of the disordered phase in both pp and ps updates, as will be confirmed by an inspection of Figure 4, Figure 9 and Figure 14. The only remaining point to be explained is the appearance of the disordered phase in the ss update. In this case too, the analysis leading to Figure 14 for the probabilities of having active bonds remains valid. However, the sequential update of the majority rule always favours strictly parallel ordering, so that typically clusters of active bonds are destroyed once formed. When the probability of their formation is strongest, i.e. at p = 0.63 (see Figure 13), the competition between the majority and outcome-based rules is at its most intense, and a disordered phase may be expected to appear. Indeed, the mid-point of the disordered phase for the ss update is shown in Figure 1 to be in exact agreement with this predicted peak, given as it is by (p c1 + p c2 )/2 = 0.63."},{"title":"E. The sp update","content":"In the case of this update, the phase diagram, Figure 15, shows nothing but a frozen phase. As is evident from the plot of inverse energy 1/E(t) versus ln t (Figure 16), there is a continuous increase in 1/E(t) for all values of p at ε = 1.0 (where the phase transition is expected to be the most visible). This suggests that the two rules, majority and performance-based, do not compete with each other at all (this is what had led to the appearance of the disordered phase in all the other updates). We suggest that this might be because the sequential update (with its more immediate conversions) in the case of the majority rule completely dominates the slower parallel update for the outcome-based rule: this in turn leads to an increasing tendency for consensus, independent of the value of p, with which our results are consistent."},{"title":"IV. AWAY FROM COEXISTENCE: WHEN THE STRATEGIES ARE DISTINCT","content":"Evidently, the real use of a competitive learning model such as this one is when the agents have a choice of distinct strategies. The full exploration of the behaviour of the model at coexistence as carried out in this paper as well as in earlier work [6,7] was aimed at an understanding of its phase diagram. However, in the exploration of the behaviour of the model away from coexistence, we hope to gain an understanding of the relative importance of parameters such as superiority of strategy (modelled by p) and memory (modelled by ε), when these are in competition. The behaviour in asymmetric conditions (using p + > p − and ε + > ε − ) is formulated in terms of the application of two biasing 'fields' [6]\nsuch that one strategy is favoured over the other. In the following subsection, we look at a linear response formulation of our question in terms of unequal p's, viewed as a biasing field, keeping ε the same for both strategies. In the final subsection, we look at unequal strategies as well as unequal memories, to find out whether inferior strategies applied with a good memory of past outcomes, can win overall."},{"title":"A. Linear response theory: strategies with unequal p","content":"Linear response theory is premised on the basis that an order parameter such as the magnetisation undergoes a sharp change in the neighbourhood of a critical point. In both the ss and pp updates of this model, there are two critical points p c1 and p c2 separating a paramagnetic phase from two frozen phases. In this subsection, we look at the linear response behaviour of the model in the vicinity of both critical points, starting from the disordered phase: clearly the response will depend both on the value of p as well as on the value of the biasing field H (defined in terms of the difference of the p's in Equation 12). In the following, we examine the response by choosing a given value of p, and writing p ± = p ± H/2, keeping ε fixed.\nWe first consider the ss-updated model. The quality of the fit to tanh(bH) is seen Figure 18: the black fitting curve almost completely coincides with a sample curve taken from Figure 17. These results also admit of an alternative representation, shown in Figure 19, where it is clear that the relative values of the bias correspond to different regions of domination of each strategy in phase space.\nWe next examine the linear response behaviour of the pp-updated model, again in the vicinity of the two critical points. Figure 20 is a plot showing the variation in magnetization M along the field H for different p values at ε = 1.0. For the lower values of p, in the vicinity of p c1 , we see very similar behaviour to that presented in Figure 17, corresponding to an expected tanh(bH) behaviour as shown in Figure 18: the PFP phase lying to the left of p c1 is, after all, identical to the frozen phases in the ss update. As we approach the vicinity of p c2 , the curves are markedly different: the nature of the ordered phase is one that corresponds to magnetisation values of 0 (see orange curve drawn using plus symbols in Figure 20), which is again consistent with the AFP phase that lies to the right of p c2 . To establish this more firmly we look at plots of the absolute value of the staggered magnetisation |M stag | as a function of bias H, in Figure 21. The green (triangle), blue (square) and black (plus) lines denote increasing values of p < p c2 , where the staggered magnetisation increasingly approaches zero, as expected in the disordered phase: however the red (star) line, corresponding to p > p c2 shows an abrupt jump in the value of |M stag | to unity. Combined with the analysis of the previous paragraph, this shows convincingly that the phase we refer to as AFP indeed corresponds to anti-parallel ordering.\nWe present below an alternative representation of the above results for ease of visualisation. In Figure 22, the magnetisation M is plotted in the p + -p − plane: as before, the regions of brown (black) (resp. yellow (light grey)) correspond to domination by − strategies (resp. + strategies). Notice, however, that the coexistence line has an island of very low magnetisation: in actual fact, this corresponds to the regions of both the paramagnetic and AFP phase. This is clearer in the plot of the absolute value of the staggered magnetisation |M stag |, shown in Figure 23, where the black portion of the island along the coexistence line corresponds to the disordered phase, while the faintly brown (grey) portion corresponds to the AFP.\nThese plots allow us to go beyond the previous analysis in defining the domain of stability of the AFP phase: we see clearly from Figure qualitative terms, this implies that at least in the absence of memory, when the two strategies have nearly equal success rates, neighbouring agents may adopt different strategies [14] in equilibrium.\nHaving thoroughly investigated the linear response regime for the ss-and pp-updated models, we will now examine the effect of the memory parameter ε in the next subsection.\nB. Role of memory parameters: the case of unequal ε\nThe principal competition in this model is that between two strategies with different global success rates p, which determines the relative dominance of each one in phase space. The memory parameter ε plays a more subtle role in this competition: although it cannot be a determinant of phase behaviour in the way that the success rates are (as a consequence of the rules elucidated in Equation 8), it can, as we will show, cause a surprising change in the dominance of an ostensibly superior strategy. In [6], it had been suggested that agents with inferior strategies and good memories might indeed win against agents who had better strategies but worse memories. Here, we will make this prediction more quantitative.\nThe phase diagram of the model away from coexistence involves four parameters, p ± , ε ± , so that its representation is a non-trivial problem. In the following, we choose to fix p + to 0.5, and to vary the other three parameters: a sample 3d plot is shown in Figure 24. We analyse the three visible faces in detail, before remarking on the phase behaviour within the cube: the colour coding is such that green (grey) represents dominance of + strategies, blue (black) represents dominance of − strategies, and other colours represent mixed states. • The leftmost face of the cube corresponds to the plane ε + = 0; this implies that the agents using + strategies will never convert, no matter what the outcome-based rule says. The minimum occupancy of + strategies for random configurations should thus be of the order of N/2, which can only increase depending on the conversions of agents using − strategies into the camp of the +'s. The bottom line corresponds to p − = 0, which is when such conversions are maximal (so that all N sites are +): the green (grey) colour is at its most pronounced here, changing gradually over to other colours only as ε − → 0 to the right of the line, when agents using − strategies too begin to refuse to convert, irrespective of the outcome rules. As the values of p − increase beyond 0.5 (the fixed value for p + ), we note that the dominance of the + strategy gradually gives way to states with a mixture of strategies; when ε − → 0, this tendency is at its most pronounced, while when ε − → 1, this is at its least pronounced, since local conversions can sometimes go against global success rates.\n• The front face of the cube corresponds to ε − = 0; this implies that agents using − strategies will never convert, no matter what the outcome-based rule says. This is a reflection of the previous case,\nwhere the minimum number of − sites is once again N/2, which can only be increased as the conversions from the +'s add to it.\n• The top face of the cube corresponds to p − = 1, where globally a predominance of the − strategy is expected. This is found over almost all the range of ε − except at low values of ε + , where agents using the + strategy refuse to convert, despite their globally poorer performance. The interior of the cube can show markedly different behaviour, which we illustrate via a sample slice shown in Figure 25. In this figure, red (lower black) and blue (upper black) regions correspond to the dominance of + and − strategies respectively. Here we set the value of p + to 0.7, and look at a slice of its phase space cube, as before: choosing ε + to be between 0.7 and 0.8, we look at the dominating strategy as a function of the variables ε − and p − . If the memory parameters had not existed, we would have expected the − strategy (blue (upper black) region in the figure) to predominate only for p − > 0.7; the red (lower black) region would have been covering the entire slice below this, corresponding to the dominance of the + strategy. However, the reality is rather different. The + strategy does indeed predominate for p − < 0.7, provided that agents using the − strategy have imperfect memory; but there is a striking predominance of the − strategy (even for very low values of p − ) provided that the memory of the agents employing this strategy, is much better than those of the other kind (ε − ≪ ε + ).\nA last feature to mention is the green (grey) region in Figure 25: here, there is a region of alternating + and − ordering (AFP), corresponding to ongoing competition between the two strategies. This phenomenon is most pronounced when the two strategies are equally successful, and are both accompanied by weak memories of earlier outcomes. In Figure 26, the structure of the full AFP is shown (by selecting phase points with low values of absolute magnetisation |M | and high absolute staggered magnetization |M stag |) as a function of p − and ε ± , fixing p + = 0.5."},{"title":"V. DISCUSSION","content":"The work of this paper extends work done on a problem of strategic learning [6,7] which, although originally suggested by a problem on technology diffusion [4], has much wider ramifications (e.g., in relation to threshold learning dynamics [8]).\nIn any agent-based modelling scheme, it is important to know whether agents react sequentially or collectively to the spread of information. Our results show that these issues make a quantitative as well as a qualitative difference to the results, changing not just exponents but also the entire nature of the phase diagram in most cases. Given that, typically, the propagation of technologies through well-connected societies is of interest [4], we choose ordered rather than random updates, and examine the response of the model of [6] to all possible combinations of sequential and parallel updating. From the viewpoint of theoretical physics, a major result is that this model is robustly in the universality class of the voter model [12], for all but one of the updates. This strong relationship with the voter model results from the model of [6,7] being driven by interfacial noise alone, i.e. the absence of surface tension [12].\nAnother major result, still to do with updates, is the appearance of a phase of anti-parallel ordering (AFP) in the high-performing limits of p, for both the pp and the ps updates. While the technicalities behind this are explained in the text, we give here a more intuitive reason for this, from the perspective of strategic learning. The parallel scheme can be viewed as a more 'equilibrated' update than the sequential one, since it gives a chance for the entire lattice to be updated 'simultaneously'. It is then natural that in the regime that both agents are high-performing, they should be equally preferred: this lies behind the 'alternating' order inherent in the AFP regime. By contrast, since the sequential paradigm corresponds to a 'non-equilibrium' update, where every agent responds to the updated value of its neighbours, the above logic leads to a disordered phase where every prescription of the outcome-based rule is countermanded by the following majority rule. Using once again the illustration of propagating technologies [4]: when all the populace have equal and simultaneous access to information about two high-performing technologies, we will see the coexistence of both [14] (as predicted by the AFP phase), whereas when information about each one is passed on sequentially, the conflicting information so obtained can result in sheer disorder. Finally, we mention here that our investigation of different updates on this game-theoretic model has been applied to related game-theoretic models of cognitive learning and synaptic plasticity [15,16], where updates relate to the directionality of synapses in a network.\nMoving away from the domain of critical behaviour at coexistence, we have looked at the behaviour of the competitive learning model when the two strategies have distinct attributes (this, after all, is truer to the title of competitive learning!). To begin with, we have examined the response of the model to unequal strategies p ± , and have found in general that the smarter strategy wins (for equal values of the memory parameter ε), as might be expected. An interesting feature is that the region of anti-parallel ordering (AFP) found earlier still persists in the presence of bias, provided that the difference in p is below a well-defined bound: in other words, when two distinct strategies are almost equally successful, one will typically find that they can coexist in society. Finally, we have looked at the effect of memory: we have found that while memory has a secondary role in determining the phase behaviour of the model, it has a particularly striking effect in turning around the results of any bias in p. A major result of our paper is thus that decisions based on a good memory of earlier outcomes can, within limits, compensate for the choice of inferior strategies."},{"title":"ACKNOWLEDGMENTS","content":"AAB would like to thank Dr. G. Mahajan and Mr. B. Chakraborty for helpful discussions. AM acknowledges the award of a grant from the DST (Govt. of India) through the project \"Generativity in Cognitive Networks\", through which AAB was supported."},{"title":"FIG. 1 .","content":""},{"title":"Figure 2","content":""},{"title":"FIG. 2 .","content":""},{"title":"FIG. 3 .","content":""},{"title":"FIG. 4 .","content":""},{"title":"FIG. 5 .","content":""},{"title":"Figure 8 FIG. 6 .","content":""},{"title":"FIG. 7 .","content":""},{"title":"FIG. 8 .","content":""},{"title":"FIG. 9 .","content":""},{"title":"FIG","content":""},{"title":"FIG","content":""},{"title":"FIG","content":""},{"title":"FIG","content":""},{"title":"FIG. 19 .","content":""},{"title":"FIGFIG","content":""},{"title":"FIG. 22 .","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"The modelling of social behaviour is of increasing concern to statistical physicists [1]. Studies of social and biological systems often reveal that even when the interactions of a given individual are very localised in time and space, collective, regular behaviour can emerge: this is analogous to the cooperative behaviour manifested by emergent systems in the natural world. Such social regularities may well take the form of learning, when individuals adopt the behaviour of other individuals. From the perspective of game theory [2], this can be seen as an adoption of a particular strategy, whose result may or may not be associated with a favourable outcome. It is then quite reasonable to expect that the effectiveness of a strategy in yielding favourable outcomes should influence how likely it is to persist, and spread through the population; the resulting ideas of strategic learning [3] have found wide application, starting from economics [4] to cognitive science [5].\\nAgainst the backdrop of the above ideas, a model of strategic learning was introduced in [6], with one of two possible strategies (denoted as + and − in the remainder of this paper) being available to each agent on a lattice: the agents were referred to as 'myopic' (aware only of their immediate neighbours) and 'memoryless' (unaware of their own and others' past outcomes) in the paper on technology diffusion [4] that inspired the above model [6,7]. The question on which this body of work has centred is: despite these handicaps, can agents overall learn to use the superior one of two available technologies? Briefly, each agent changes (or does not change) strategy based on two elementary rules at every time step: a * ajaz@bose.res.in † anita@bose.res.in majority-based rule, reflecting its tendency to align with its local neighbourhood, followed by a performance-based rule, where the agent adopts the strategy that 'wins' in its neighbourhood. This (relative) success is measured in terms of outcomes, where the probability of a successful outcome for strategy + (−) is p + (p − ). Also, the model of [6] added to the description of [4] by endowing the agents with memory: those agents who make their choices on the basis of the last payoff alone, are adjudged to be memoryless (with a corresponding parameter ε near 1), while those who allow for memories of earlier outcomes may make decisions that run counter to immediate evidence (ε small).\\nSome related ideas have been examined in recent work. For example, the issue of consensus formation in a model of threshold learning [8] shows close analogies: in this model, the competition between the 'noisy' signals from the immediate neighbours of an agent (cf. the majority rule in [6]) and the acceptance threshold that agents require to change their state (cf. the memory threshold in the performance-based rule of [6]), determine the phase diagrams obtained. Recent studies of coevolving Glauber dynamics on networks [9] are also relevant, since the model of [6] can be viewed as a competition between the Glauber dynamics of two sets of Ising spins, corresponding to strategy and outcome respectively.\\nIn the current paper, we take all these ideas further. First, we explore the effect of different updates. If new information propagates sequentially through the network, and the arrow of time is discernible in the decisions of individual agents, are the global phase diagrams any different from what they would be if information was transmitted and all decisions were taken simultaneously? Common sense tells us that sequential or parallel updates should make a difference to the nature of the phase diagram, and the results of the present paper confirm this. Also (unlike the work of [6,7] which examined the situa-tion at coexistence) we look in this paper at the effects of disparate strategies (p + = p − ). The final, and possibly most important issue, is that of memory, which acts as a threshold governing change [8]: what is the effect of the threshold ε, which tells the agent that longer-term inputs are significant, and need to be considered when making a decision? We will find that, indeed, a longer memory of earlier outcomes can sometimes make up for the choice of a globally inferior strategy.\\nThe plan of this paper is as follows. In Section II, we review the model of [6]. In Section III, we discuss the behaviour of the model for a range of updating schemes, in the presence of memory. In Section IV, we examine the behaviour of the model away from coexistence, as a function of distinct parameter values for the two strategies; in particular we discuss here the role of memory. In the concluding section, we discuss our results and put them in the context of other recent work in the field.\"\n },\n {\n \"title\": \"II. DEFINITION OF THE MODEL\",\n \"content\": \"The model of [6] involves two types of strategies, − and +, where the + strategy is globally superior [4] to the − strategy. As mentioned above, agents tend to follow the strategy adopted by the majority of their neighbours, modifying this choice in a second step (if necessary) according to which of these have proved to be the most successful.\\nAssuming that the agents sit at the nodes of a ddimensional regular lattice with coordination number z = 2d, the efficiency of an agent at site i is represented by an Ising spin variable:\\nThe evolution dynamics of the lattice is governed by two rules. The first is a majority rule, which consists of the alignment of an agent with the local field (created by its nearest neighbours) acting upon it, according to:\\nHere, the local field\\nis the sum of the efficiencies of the z neighbouring agents j of site i and τ 1 is the associated time step. Next, a performance rule is applied. This starts with the assignment of an outcome σ i (another Ising-like variable, with values of ±1 corresponding to success and failure respectively) to each site i, according to the following rules:\\nif η i (t) = +1,\\nwhere τ 2 is the associated time step and p ± are the probabilities of having a successful outcome for the corresponding strategy. With N + i and N − i denoting the total number of neighbours of a site i who have adopted strategies + and − respectively, and I + i (I − i ) denoting the number of successful outcomes within the set N + i (N − i ), the dynamical rules for site i are:\\nif η i (t) = +1 and\\nHere, the ratios Ii(t) Ni(t) are nothing but the average payoff assigned by an agent to each of the two strategies in its neighbourhood at time t (assuming that success yields a payoff of unity and failure, zero). Also, τ 3 is the associated time step and the parameters ε ± are indicators of the memory associated with each strategy. In their full generality, ε and p are independent variables: the choice of a particular strategy can be associated with either a short or a long memory. However, we would like in this paper to answer a question which was posed, but not answered in [6]: can the presence of a good memory compensate for the choice of an inferior strategy? We therefore examine the extreme situation when a globally superior strategy (p + ≫ p − ), combined with a shorter memory (ε + ≫ ε − ) is in competition with its inverse: this is the situation that will be studied in Section IV.\\nSetting the timescales\\nthe above steps of the performance rule are recast as effective dynamical rules involving the efficiencies η i (t) and the associated local fields alone:\\nThe effective transition probabilities w ± (h) are evaluated by enumerating the 2 z possible realizations of the outcomes σ j of the sites neighbouring site i, and weighing them appropriately. For a 2-d square lattice, the possible local field values at the interfacial sites are 0 and ±2.\\nThe corresponding transition probabilities for these field values are [6]:\\nIn [6], the model was explored at coexistence with an ordered sequential update applied to memoryless agents [4]:\\nIn the present paper, we go beyond this in two different ways. First, still at coexistence, we explore the effect of different updates on the p − ε phase diagram of the model: next, we examine the model away from coexistence, for distinct values of p ± and ε ± . The basic quantities considered hereafter are the magnetization M , staggered magnetization M stag and the energy E. These quantities are defined for a finite sample of N agents (or sites) and N z /2 bonds (or links), as\\nIn the following we shall usually consider mean values M , E and M stag .\"\n },\n {\n \"title\": \"III. THE EFFECT OF FINITE MEMORY, AND OF DIFFERENT UPDATES\",\n \"content\": \"We begin this section with a review of the physical significance of updating schemes. Most generally, updates can be random or ordered as follows:\\n• Random: Here, sites are chosen at random for the consecutive application of rules.\\n• Ordered : Here, sites are chosen in an ordered fashion, i.e., after choosing every (i, j) th site, the (i, j + 1) th site is selected.\\nSince the sociological basis for this work was the propagation of innovation through connected societies [4], we choose to deal only with ordered updates here. However, even ordered updates have two subclasses: parallel and sequential. Assume a condition A such that when an agent satisfies A, it changes strategy:\\n• Sequential update: In this type of update, we check the condition A on the (i, j) th site, then update the efficiency of the site and proceed to the (i, j + 1) th site using the updated value of the (i, j) th site.\\n• Parallel update: In this type of update, we check the condition A on the (i, j) th site, do not update the site but instead save the update-decision in memory, and proceed to the next site. Once the whole lattice is swept, all the saved updatedecisions are implemented 'simultaneously'.\\nThe choice of different updates generally corresponds to different physical situations: it has been shown that it also leads to a disparity in the convergence time of the systems concerned [10,11]. We therefore examine all possible combinations for our two update rules:\\nI parallel updates for both majority rule and performance rules (pp).\\nII parallel update for majority rule and sequential update for performance rules (ps).\\nIII sequential updates for both majority rule and performance rules (ss).\\nIV sequential update for majority rule and parallel update for performance rules (sp).\\nIn the following subsections, we explore the phase dynamics at coexistence for each of these update rules in turn, for both parameters p and ε. We state at the outset that all the updates (except for the sp update) which we consider, result in models which are in the general university class of the voter model [12]: the inverse energy 1/E(t) is thus always proportional to the logarithm of time, ln t. When, as in the case of the ss update, the value of the slope is exactly 2/π [6], the exact universality class of the voter model is retrieved.\"\n },\n {\n \"title\": \"A. The ss update\",\n \"content\": \"This is the update that was used throughout [6]; however the phase behaviour of the model was there only explored for the parameter p, whereas here we extend it to the parameter ε. In Figure 1, we plot the inverse energy 1/E(t) in the p − ε plane at time t = 512 for a square lattice of size N = 64 2 . This phase diagram shows clearly the existence of a disordered paramagnetic phase embedded in a largely frozen phase elsewhere. The disordered phase exists for p c1 (= 0.56 ± 0.01) < p < p c2 (= 0.70 ± 0.01) when ε ≥ 0.980. Our results agree with those of [6] for ε = 1, and extend them all across the rest of the p − ε plane. We mention here that the average time required to reach consensus increases exponentially as p decreases in the frozen phase, leading to the presence of striped states [13] at limiting values of p. Figure 1 also makes it clear that the effect of increasing memory wipes out the disordered phase: this is as it should be, since the disordered phase is generated by the competition between the majority and performance-based rules, which is dulled by increasing memory. In Figure 3, we have plotted the inverse energy 1/E(t) against the natural logarithm of time ln t for values of p around the critical point p c2 = 0.70 ± 0.01. Each of the curves is obtained by averaging over 200 independent samples of size 256 2 . At the critical point, we obtain a straight line with a slope close to 2/π [6], a behaviour characteristic of the exact voter model [12] that corresponds to\\nSimilar behaviour is obtained at the other critical point p c1 , in agreement with [6]. The curve corresponding to p = pc2 = 0.70 (shown in red) has slope 2/π approximately [see Equation 11].\"\n },\n {\n \"title\": \"B. The pp update\",\n \"content\": \"In this case, both environmental majority and performance-based rules are applied using parallel updates. As we will see, although the universality class of the model is qualitatively unchanged, this update results in the appearance of novel ordered phases compared to the ss update. As before, we first plot the phase diagram for all values of p and ε, then show snapshots of the dynamics, and finally get a more quantitative feel for the behaviour of key quantities as a function of p.\\nAccordingly, Figure 4 (top and bottom), are plots of the absolute values of magnetization |M | and staggered magnetization |M stag |, at time t = 512 for a lattice size N = 100 2 , in the p-ε plane using pp updates. In these phase diagrams, we see clear evidence of the existence of two distinct frozen phases separated by a disordered phase. Looking along the line ε = 1, disorder prevails for p c1 < p < p c2 with p c1 = 0.43±0.01 and p c2 = 0.57±0.01. Notice the symmetry of the two critical points about p = 0.5: we shall have more to say about this later on.\\nFor p below p c1 , there is a frozen phase characterised by overall alignment of spins: we call this the parallel frozen phase (PFP). For p above p c2 , the frozen phase that appears is characterised by an anti-parallel ordering of spins: we call this the anti-parallel frozen phase (AFP). We mention also that in the AFP, the lattice may have more than one anti-parallel domain, with thin frustrated chains running in between them. This frustration can be attributed to the inability of the different domains to align with each other under periodic boundary conditions. The disturbances caused by these chains (in quantities such as |M | or E) due to misalignment decrease as 1/ √ N and also appear to vanish for large times. Again, we notice that the phase transition disappears for low ε; in fact, at very low values of ε the evolving lattice may get trapped into striped states [13] at long times.\\nThis can be understood as follows: the effect of a long memory (ε small) strongly reduces the relative impact of the performance-based rule. Depending on the value of ε, the performance rule may not be effective for several timesteps whereas the majority rule is implemented at every timestep. In the limit of vanishing ε, then, only the (zero-temperature) majority rule will be effective, leading to stripe formation as predicted by [13] for this situation. Figure 5 comprises snapshots of the dynamics of the model for a 2d square lattice of size N = 512 2 and at time t = 256, with random initial configurations. The plots show a portion of size 100 2 of the square lattice for three values of p: p = 0.41 (near the critical point p c1 between the PFP and the paramagnetic phase), p = 0.50 (within the paramagnetic phase) and p = 0.59 (near the critical point p c2 separating the paramagnetic phase from the AFP), with ε = 1. The snapshot at p = 0.41 shows the lattice evolving towards consensus (parallel alignment) with the formation of domains of one type only. The snapshot at p = 0.50 shows the lattice in its disordered phase, while the one at p = 0.59 shows that the nature of the lattice ordering is anti-parallel.\\nTo investigate this more quantitatively, we plot the absolute value of magnetization |M |, the absolute value of staggered magnetization |M stag | and energy E(t) against p, with ε = 1.0, in Figure 6. These measurements were recorded using a square lattice of size N = 80 2 at time t = 10 6 . All the curves are averaged over 100 independent samples for each value of p. In the region p ≤ p c1 , the values of magnetization |M | and staggered magnetization |M stag | are both equal to unity at saturation, implying a parallel alignment of the sites; whereas for p above p c2 , the magnetization |M | is zero and the staggered magnetization |M stag | equals unity at saturation, indicating an anti-parallel alignment of the sites. The energy graph is consistent with this interpretation, given the definition of the energy in Equation 10: zero in the PFP, middling in the paramagnetic phase and unity in the AFP.\\nIn order to confirm the voter-like nature of the critical points, we plot the inverse energy 1/E(t) against the natural logarithm of time ln t, choosing p values near both critical points (see Figure 7 and curve is an average over 200 independent samples. Exactly at the critical points p c1 = 0.43 and p c2 = 0.57, a linear behaviour of inverse energy with respect to ln t is found, with slopes of 1/2π and −1/5π respectively. While the critical exponents are those of the voter model [12], the values of the slope are different from 2/π: we find therefore that the pp update of the model belongs to the universality class of the generalised, rather than the exact, voter model [12]. To conclude this subsection: the main effect of the pp update is to change the nature of the ordering in one of the two frozen phases, so that anti-parallel ordering is found in the high-p frozen phase. As before, the effect of increasing memory (going to low ε) is to smear out the phase transitions to the disordered phase, by undermining the effect of the outcome-based rule whose com-petition with the majority rule causes the appearance of disorder. Such instances of mixed domains have been found in recent work on coevolving (parallel) dynamics [9]; some features of these results also appear in studies of threshold dynamics of societal systems [8]. For a real-life example of the AFP in the case of technology diffusion, we cite the results of [14] where the authors conclude that \\\"in technology clusters where direct competitors are right next door, leading firms generate innovations that are technologically very distant from their neighbours\\\" [14].\"\n },\n {\n \"title\": \"C. The ps update\",\n \"content\": \"The behaviour of the ps-updated model is qualitatively similar to that of the pp-updated model above. Again, there are two frozen phases PFP and AFP, separated by a disordered phase: the values of the critical points p c1 and p c2 are however shifted, such that the disordered region extends between p c1 = 0.31 ± 0.01 and p c2 = 0.69 ± 0.01 at ε = 1.0. We find once again that the two critical points are symmetrically placed with respect to p = 0.5, as in the pp update: we will give an argument for why this is so, in the following subsection.\\nTo avoid repetition, we present only the phase diagram for the staggered magnetisation as a function of p and ε: Figure 9 shows the absolute value of the staggered magnetization |M stag | of the system at time t = 512 for a square lattice of size N = 100 2 . The paramagnetic region, with low values of |M stag | is coloured black in the figure, whereas the frozen regions (containing either parallel or anti-parallel ordering) with high values of |M stag |, are coloured yellow (light grey). These phases are investigated more quantitatively in Figure 10, where we plot the absolute value of magnetization |M |, energy E and the absolute value of staggered magnetization |M stag | against Finally, we present the variation of inverse energy with the natural logarithm of time, ln t, near the critical points p c1 and p c2 in Figure 11 and Figure 12 respectively. Each of the curves is an average over 200 independent runs. At criticality, both plots show a linear proportionality between 1/E(t) and ln t, with slopes of 4/3π and −4/15π at p c1 = 0.31 and p c2 = 0.69 respectively. Again, this indicates that the ps update of the model belongs to the generalised, rather than the exact, universality class of the voter model [12].\\nTo conclude, the ps update yields qualitatively similar results to the pp update, with the appearance of two frozen phases PFP and AFP. Again, small values of ε indicating longer memories of outcomes, lead to a smearing out of the phase transition, because of the decreasing effectiveness of the outcome-based rule. The clue which explains all of the above, is the formation of 'active' or disparate bonds by the rules of the model under different updates: these are clearly the units of anti-parallel ordering. Consider thus configurations where a site is surrounded by a majority of its own kind: this would correspond to a local field of +2 for a +, and −2 for a −. Here the majority of the bonds are 'like' or 'inactive'. The transition probability for the increase of active bonds from such configurations is 1 − w + (+2) (or 1−w − (−2)) [see Equation 8]. The transition probabilities for the decrease of active bonds are given by an opposite scenario, yielding w − (+2) (or w + (−2)) [see Equation 8]. We plot two of these transition probabilities in Figure 13, corresponding respectively to an increase and a decrease of active bonds: the former peaks at p = 0.63 while the latter peaks at p = 0.37. The net probability of having active bonds is the difference between these two transition probabilities, and is plotted in Figure 14. We see from this that the probability of having active bonds is greatest at p = 0.79, and least at p = 0.21. The last ingredient that we need to explain the AFP phase in the pp and ps updates is the fact that once clusters with many active bonds, i.e. antiparallel ordering, are formed, the majority rule applied via the parallel update preserves such ordering. With all this in place we see that as expected, the AFP phase in both pp and ps updates shows up in qualitatively the same regions as predicted by Figure 14, with a peak, in both cases at around p = 0.79. Correspondingly, the PFP in both pp and ps updates shows up in the region predicted in this figure, with a peak in both cases at around p = 0.21. Notice (Figure 14) that the peak and the dip in the probability of active bonds are symmetric about p = 0.5, thus explaining the symmetry that we have observed in Figure 4 and Figure 9; p = 0.5 is thus the natural point for the appearance of the disordered phase in both pp and ps updates, as will be confirmed by an inspection of Figure 4, Figure 9 and Figure 14. The only remaining point to be explained is the appearance of the disordered phase in the ss update. In this case too, the analysis leading to Figure 14 for the probabilities of having active bonds remains valid. However, the sequential update of the majority rule always favours strictly parallel ordering, so that typically clusters of active bonds are destroyed once formed. When the probability of their formation is strongest, i.e. at p = 0.63 (see Figure 13), the competition between the majority and outcome-based rules is at its most intense, and a disordered phase may be expected to appear. Indeed, the mid-point of the disordered phase for the ss update is shown in Figure 1 to be in exact agreement with this predicted peak, given as it is by (p c1 + p c2 )/2 = 0.63.\"\n },\n {\n \"title\": \"E. The sp update\",\n \"content\": \"In the case of this update, the phase diagram, Figure 15, shows nothing but a frozen phase. As is evident from the plot of inverse energy 1/E(t) versus ln t (Figure 16), there is a continuous increase in 1/E(t) for all values of p at ε = 1.0 (where the phase transition is expected to be the most visible). This suggests that the two rules, majority and performance-based, do not compete with each other at all (this is what had led to the appearance of the disordered phase in all the other updates). We suggest that this might be because the sequential update (with its more immediate conversions) in the case of the majority rule completely dominates the slower parallel update for the outcome-based rule: this in turn leads to an increasing tendency for consensus, independent of the value of p, with which our results are consistent.\"\n },\n {\n \"title\": \"IV. AWAY FROM COEXISTENCE: WHEN THE STRATEGIES ARE DISTINCT\",\n \"content\": \"Evidently, the real use of a competitive learning model such as this one is when the agents have a choice of distinct strategies. The full exploration of the behaviour of the model at coexistence as carried out in this paper as well as in earlier work [6,7] was aimed at an understanding of its phase diagram. However, in the exploration of the behaviour of the model away from coexistence, we hope to gain an understanding of the relative importance of parameters such as superiority of strategy (modelled by p) and memory (modelled by ε), when these are in competition. The behaviour in asymmetric conditions (using p + > p − and ε + > ε − ) is formulated in terms of the application of two biasing 'fields' [6]\\nsuch that one strategy is favoured over the other. In the following subsection, we look at a linear response formulation of our question in terms of unequal p's, viewed as a biasing field, keeping ε the same for both strategies. In the final subsection, we look at unequal strategies as well as unequal memories, to find out whether inferior strategies applied with a good memory of past outcomes, can win overall.\"\n },\n {\n \"title\": \"A. Linear response theory: strategies with unequal p\",\n \"content\": \"Linear response theory is premised on the basis that an order parameter such as the magnetisation undergoes a sharp change in the neighbourhood of a critical point. In both the ss and pp updates of this model, there are two critical points p c1 and p c2 separating a paramagnetic phase from two frozen phases. In this subsection, we look at the linear response behaviour of the model in the vicinity of both critical points, starting from the disordered phase: clearly the response will depend both on the value of p as well as on the value of the biasing field H (defined in terms of the difference of the p's in Equation 12). In the following, we examine the response by choosing a given value of p, and writing p ± = p ± H/2, keeping ε fixed.\\nWe first consider the ss-updated model. The quality of the fit to tanh(bH) is seen Figure 18: the black fitting curve almost completely coincides with a sample curve taken from Figure 17. These results also admit of an alternative representation, shown in Figure 19, where it is clear that the relative values of the bias correspond to different regions of domination of each strategy in phase space.\\nWe next examine the linear response behaviour of the pp-updated model, again in the vicinity of the two critical points. Figure 20 is a plot showing the variation in magnetization M along the field H for different p values at ε = 1.0. For the lower values of p, in the vicinity of p c1 , we see very similar behaviour to that presented in Figure 17, corresponding to an expected tanh(bH) behaviour as shown in Figure 18: the PFP phase lying to the left of p c1 is, after all, identical to the frozen phases in the ss update. As we approach the vicinity of p c2 , the curves are markedly different: the nature of the ordered phase is one that corresponds to magnetisation values of 0 (see orange curve drawn using plus symbols in Figure 20), which is again consistent with the AFP phase that lies to the right of p c2 . To establish this more firmly we look at plots of the absolute value of the staggered magnetisation |M stag | as a function of bias H, in Figure 21. The green (triangle), blue (square) and black (plus) lines denote increasing values of p < p c2 , where the staggered magnetisation increasingly approaches zero, as expected in the disordered phase: however the red (star) line, corresponding to p > p c2 shows an abrupt jump in the value of |M stag | to unity. Combined with the analysis of the previous paragraph, this shows convincingly that the phase we refer to as AFP indeed corresponds to anti-parallel ordering.\\nWe present below an alternative representation of the above results for ease of visualisation. In Figure 22, the magnetisation M is plotted in the p + -p − plane: as before, the regions of brown (black) (resp. yellow (light grey)) correspond to domination by − strategies (resp. + strategies). Notice, however, that the coexistence line has an island of very low magnetisation: in actual fact, this corresponds to the regions of both the paramagnetic and AFP phase. This is clearer in the plot of the absolute value of the staggered magnetisation |M stag |, shown in Figure 23, where the black portion of the island along the coexistence line corresponds to the disordered phase, while the faintly brown (grey) portion corresponds to the AFP.\\nThese plots allow us to go beyond the previous analysis in defining the domain of stability of the AFP phase: we see clearly from Figure qualitative terms, this implies that at least in the absence of memory, when the two strategies have nearly equal success rates, neighbouring agents may adopt different strategies [14] in equilibrium.\\nHaving thoroughly investigated the linear response regime for the ss-and pp-updated models, we will now examine the effect of the memory parameter ε in the next subsection.\\nB. Role of memory parameters: the case of unequal ε\\nThe principal competition in this model is that between two strategies with different global success rates p, which determines the relative dominance of each one in phase space. The memory parameter ε plays a more subtle role in this competition: although it cannot be a determinant of phase behaviour in the way that the success rates are (as a consequence of the rules elucidated in Equation 8), it can, as we will show, cause a surprising change in the dominance of an ostensibly superior strategy. In [6], it had been suggested that agents with inferior strategies and good memories might indeed win against agents who had better strategies but worse memories. Here, we will make this prediction more quantitative.\\nThe phase diagram of the model away from coexistence involves four parameters, p ± , ε ± , so that its representation is a non-trivial problem. In the following, we choose to fix p + to 0.5, and to vary the other three parameters: a sample 3d plot is shown in Figure 24. We analyse the three visible faces in detail, before remarking on the phase behaviour within the cube: the colour coding is such that green (grey) represents dominance of + strategies, blue (black) represents dominance of − strategies, and other colours represent mixed states. • The leftmost face of the cube corresponds to the plane ε + = 0; this implies that the agents using + strategies will never convert, no matter what the outcome-based rule says. The minimum occupancy of + strategies for random configurations should thus be of the order of N/2, which can only increase depending on the conversions of agents using − strategies into the camp of the +'s. The bottom line corresponds to p − = 0, which is when such conversions are maximal (so that all N sites are +): the green (grey) colour is at its most pronounced here, changing gradually over to other colours only as ε − → 0 to the right of the line, when agents using − strategies too begin to refuse to convert, irrespective of the outcome rules. As the values of p − increase beyond 0.5 (the fixed value for p + ), we note that the dominance of the + strategy gradually gives way to states with a mixture of strategies; when ε − → 0, this tendency is at its most pronounced, while when ε − → 1, this is at its least pronounced, since local conversions can sometimes go against global success rates.\\n• The front face of the cube corresponds to ε − = 0; this implies that agents using − strategies will never convert, no matter what the outcome-based rule says. This is a reflection of the previous case,\\nwhere the minimum number of − sites is once again N/2, which can only be increased as the conversions from the +'s add to it.\\n• The top face of the cube corresponds to p − = 1, where globally a predominance of the − strategy is expected. This is found over almost all the range of ε − except at low values of ε + , where agents using the + strategy refuse to convert, despite their globally poorer performance. The interior of the cube can show markedly different behaviour, which we illustrate via a sample slice shown in Figure 25. In this figure, red (lower black) and blue (upper black) regions correspond to the dominance of + and − strategies respectively. Here we set the value of p + to 0.7, and look at a slice of its phase space cube, as before: choosing ε + to be between 0.7 and 0.8, we look at the dominating strategy as a function of the variables ε − and p − . If the memory parameters had not existed, we would have expected the − strategy (blue (upper black) region in the figure) to predominate only for p − > 0.7; the red (lower black) region would have been covering the entire slice below this, corresponding to the dominance of the + strategy. However, the reality is rather different. The + strategy does indeed predominate for p − < 0.7, provided that agents using the − strategy have imperfect memory; but there is a striking predominance of the − strategy (even for very low values of p − ) provided that the memory of the agents employing this strategy, is much better than those of the other kind (ε − ≪ ε + ).\\nA last feature to mention is the green (grey) region in Figure 25: here, there is a region of alternating + and − ordering (AFP), corresponding to ongoing competition between the two strategies. This phenomenon is most pronounced when the two strategies are equally successful, and are both accompanied by weak memories of earlier outcomes. In Figure 26, the structure of the full AFP is shown (by selecting phase points with low values of absolute magnetisation |M | and high absolute staggered magnetization |M stag |) as a function of p − and ε ± , fixing p + = 0.5.\"\n },\n {\n \"title\": \"V. DISCUSSION\",\n \"content\": \"The work of this paper extends work done on a problem of strategic learning [6,7] which, although originally suggested by a problem on technology diffusion [4], has much wider ramifications (e.g., in relation to threshold learning dynamics [8]).\\nIn any agent-based modelling scheme, it is important to know whether agents react sequentially or collectively to the spread of information. Our results show that these issues make a quantitative as well as a qualitative difference to the results, changing not just exponents but also the entire nature of the phase diagram in most cases. Given that, typically, the propagation of technologies through well-connected societies is of interest [4], we choose ordered rather than random updates, and examine the response of the model of [6] to all possible combinations of sequential and parallel updating. From the viewpoint of theoretical physics, a major result is that this model is robustly in the universality class of the voter model [12], for all but one of the updates. This strong relationship with the voter model results from the model of [6,7] being driven by interfacial noise alone, i.e. the absence of surface tension [12].\\nAnother major result, still to do with updates, is the appearance of a phase of anti-parallel ordering (AFP) in the high-performing limits of p, for both the pp and the ps updates. While the technicalities behind this are explained in the text, we give here a more intuitive reason for this, from the perspective of strategic learning. The parallel scheme can be viewed as a more 'equilibrated' update than the sequential one, since it gives a chance for the entire lattice to be updated 'simultaneously'. It is then natural that in the regime that both agents are high-performing, they should be equally preferred: this lies behind the 'alternating' order inherent in the AFP regime. By contrast, since the sequential paradigm corresponds to a 'non-equilibrium' update, where every agent responds to the updated value of its neighbours, the above logic leads to a disordered phase where every prescription of the outcome-based rule is countermanded by the following majority rule. Using once again the illustration of propagating technologies [4]: when all the populace have equal and simultaneous access to information about two high-performing technologies, we will see the coexistence of both [14] (as predicted by the AFP phase), whereas when information about each one is passed on sequentially, the conflicting information so obtained can result in sheer disorder. Finally, we mention here that our investigation of different updates on this game-theoretic model has been applied to related game-theoretic models of cognitive learning and synaptic plasticity [15,16], where updates relate to the directionality of synapses in a network.\\nMoving away from the domain of critical behaviour at coexistence, we have looked at the behaviour of the competitive learning model when the two strategies have distinct attributes (this, after all, is truer to the title of competitive learning!). To begin with, we have examined the response of the model to unequal strategies p ± , and have found in general that the smarter strategy wins (for equal values of the memory parameter ε), as might be expected. An interesting feature is that the region of anti-parallel ordering (AFP) found earlier still persists in the presence of bias, provided that the difference in p is below a well-defined bound: in other words, when two distinct strategies are almost equally successful, one will typically find that they can coexist in society. Finally, we have looked at the effect of memory: we have found that while memory has a secondary role in determining the phase behaviour of the model, it has a particularly striking effect in turning around the results of any bias in p. A major result of our paper is thus that decisions based on a good memory of earlier outcomes can, within limits, compensate for the choice of inferior strategies.\"\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"AAB would like to thank Dr. G. Mahajan and Mr. B. Chakraborty for helpful discussions. AM acknowledges the award of a grant from the DST (Govt. of India) through the project \\\"Generativity in Cognitive Networks\\\", through which AAB was supported.\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8 FIG. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 9 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 19 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIGFIG\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 22 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"(color online)[ss update] Phase diagram of the model with an ss update. Plot of the inverse energy 1/E(t) at time t = 512 for a square lattice of size N = 64 2 in the p-ε plane. The black region shows the disordered phase and the yellowish (light grey) region shows the frozen phase.","content":"FIG. 1 .\n1(color online)[ss update] Phase diagram of the model with an ss update. Plot of the inverse energy 1/E(t) at time t = 512 for a square lattice of size N = 64 2 in the p-ε plane. The black region shows the disordered phase and the yellowish (light grey) region shows the frozen phase."},{"id":"","caption":"shows snapshots of the dynamics of the model using a lattice of size N = 512 2 at times t = 8, t = 64, and t = 512 with random initial configurations and parameter values p = 0.72 (very close to the critical point p c2 ) and ε = 1.0. The plots reveal characteristically voter-like[12] coarsening behaviour.","content":"Figure 2\n2shows snapshots of the dynamics of the model using a lattice of size N = 512 2 at times t = 8, t = 64, and t = 512 with random initial configurations and parameter values p = 0.72 (very close to the critical point p c2 ) and ε = 1.0. The plots reveal characteristically voter-like[12] coarsening behaviour."},{"id":"","caption":"(color online)[ss update] Snapshots of the dynamics of the ss-updated model. Each plot is a portion (of size 100 2 ) of a square lattice of N = 256 2 for p = 0.72 and ε = 1.0 at times t = 8 (top-left), t = 64 (top-right) and t = 512 (bottom).","content":"FIG. 2 .\n2(color online)[ss update] Snapshots of the dynamics of the ss-updated model. Each plot is a portion (of size 100 2 ) of a square lattice of N = 256 2 for p = 0.72 and ε = 1.0 at times t = 8 (top-left), t = 64 (top-right) and t = 512 (bottom)."},{"id":"","caption":"(color online)[ss update] Plot of the inverse energy 1/E(t) versus the natural logarithm of time ln t for different values of p close to pc2 = 0.70, with ε set to 1.0. The lattice size N = 256 2 , and the p-values are indicated on the curves.","content":"FIG. 3 .\n3(color online)[ss update] Plot of the inverse energy 1/E(t) versus the natural logarithm of time ln t for different values of p close to pc2 = 0.70, with ε set to 1.0. The lattice size N = 256 2 , and the p-values are indicated on the curves."},{"id":"","caption":"(color online)[pp update] Plot of the absolute value of the magnetization |M | (top) and the absolute value of the staggered magnetization |Mstag| (bottom) at time t = 512 for a lattice of N = 100 2 in the p-ε plane. In the top figure, the yellowish (light grey) region refers to the parallel frozen phase (PFP), while the black region includes both the antiparallel frozen phase (AFP), and the disordered region. In the bottom figure, the black region represents the disordered phase characterised by very low |Mstag|.","content":"FIG. 4 .\n4(color online)[pp update] Plot of the absolute value of the magnetization |M | (top) and the absolute value of the staggered magnetization |Mstag| (bottom) at time t = 512 for a lattice of N = 100 2 in the p-ε plane. In the top figure, the yellowish (light grey) region refers to the parallel frozen phase (PFP), while the black region includes both the antiparallel frozen phase (AFP), and the disordered region. In the bottom figure, the black region represents the disordered phase characterised by very low |Mstag|."},{"id":"","caption":"(color online)[pp update] Snapshots of the dynamics of the pp-updated model on a square lattice, for p = 0.41 (leftmost), p = 0.50 (centre) and p = 0.59 (rightmost) at time t = 512, with ε = 1.0. The yellow (light grey) and black colours represent the two strategies, while the greyish grid corresponds to anti-parallel arrangements of yellow (light grey) and black. The leftmost picture represents the PFP (see text), the centre one the disordered phase, and the rightmost one the AFP (see text).","content":"FIG. 5 .\n5(color online)[pp update] Snapshots of the dynamics of the pp-updated model on a square lattice, for p = 0.41 (leftmost), p = 0.50 (centre) and p = 0.59 (rightmost) at time t = 512, with ε = 1.0. The yellow (light grey) and black colours represent the two strategies, while the greyish grid corresponds to anti-parallel arrangements of yellow (light grey) and black. The leftmost picture represents the PFP (see text), the centre one the disordered phase, and the rightmost one the AFP (see text)."},{"id":"","caption":"(color online)[pp update] Plot of the absolute value of magnetization |M |, the absolute value of staggered magnetization |Mstag| and energy E against p, with ε = 1 and lattice size N = 80 2 . Each curve is drawn using symbols (and colour) as indicated in the legend.","content":"Figure 8 FIG. 6 .\n86(color online)[pp update] Plot of the absolute value of magnetization |M |, the absolute value of staggered magnetization |Mstag| and energy E against p, with ε = 1 and lattice size N = 80 2 . Each curve is drawn using symbols (and colour) as indicated in the legend."},{"id":"","caption":"(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.43 and ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.43 (shown in red) has slope 1/2π approximately.","content":"FIG. 7 .\n7(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.43 and ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.43 (shown in red) has slope 1/2π approximately."},{"id":"","caption":"(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.57, with ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.57 (shown in red) has slope −1/5π approximately.","content":"FIG. 8 .\n8(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.57, with ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.57 (shown in red) has slope −1/5π approximately."},{"id":"","caption":"(color online)[ps update] Phase diagram in the p-ε plane of the ps-updated model, with a plot of the absolute value of staggered magnetization |Mstag| at time t = 512, for a lattice size of N = 100 2 . The black region represents the disordered phase (very low |Mstag|), while the yellowish (light grey) region represents frozen phases with high |Mstag|. p, with ε equal to 1.0; each curve is an average over 100 independent runs. The region where both the magnetization |M | and staggered magnetization |M stag | curves saturate to 1, corresponds to parallel alignment, whereas |M | ≈ 0 with |M stag | ≈ 1 implies an anti-parallel alignment of the spin types. The energy graph is consistent with this interpretation, given the definition of the energy in Equation10: zero in the PFP, middling in the paramagnetic phase and unity in the AFP. 10. (color online)[ps update] Plot of the absolute value of magnetization |M |, energy E and the absolute value of staggered magnetization |Mstag| against p with ε set to 1.0 for a lattice size N = 100 2 . Each curve is drawn using symbols (and colour) as indicated in the legend.","content":"FIG. 9 .\n9(color online)[ps update] Phase diagram in the p-ε plane of the ps-updated model, with a plot of the absolute value of staggered magnetization |Mstag| at time t = 512, for a lattice size of N = 100 2 . The black region represents the disordered phase (very low |Mstag|), while the yellowish (light grey) region represents frozen phases with high |Mstag|. p, with ε equal to 1.0; each curve is an average over 100 independent runs. The region where both the magnetization |M | and staggered magnetization |M stag | curves saturate to 1, corresponds to parallel alignment, whereas |M | ≈ 0 with |M stag | ≈ 1 implies an anti-parallel alignment of the spin types. The energy graph is consistent with this interpretation, given the definition of the energy in Equation10: zero in the PFP, middling in the paramagnetic phase and unity in the AFP. 10. (color online)[ps update] Plot of the absolute value of magnetization |M |, energy E and the absolute value of staggered magnetization |Mstag| against p with ε set to 1.0 for a lattice size N = 100 2 . Each curve is drawn using symbols (and colour) as indicated in the legend."},{"id":"","caption":". 11. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.31, with ε set to 1.0 for a lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.31 (shown in red) has a slope of approximately 4/3π.","content":"FIG\n. 11. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.31, with ε set to 1.0 for a lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.31 (shown in red) has a slope of approximately 4/3π."},{"id":"","caption":". 12. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.69, with ε set to 1.0, for lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.69 (shown in red) has slope −4/15π approximately. D. Explanation for the nature of the phase diagrams for different updates In this subsection, we give arguments for the three most important features of the phase diagrams presented above: (i) The appearance of anti-parallel ordering in both pp and ps updates (ii) The symmetry of the PFP and the AFP phases in both pp and ps updates (iii) The positioning of the disordered phase in ss, pp and ps updates","content":"FIG\n. 12. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.69, with ε set to 1.0, for lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.69 (shown in red) has slope −4/15π approximately. D. Explanation for the nature of the phase diagrams for different updates In this subsection, we give arguments for the three most important features of the phase diagrams presented above: (i) The appearance of anti-parallel ordering in both pp and ps updates (ii) The symmetry of the PFP and the AFP phases in both pp and ps updates (iii) The positioning of the disordered phase in ss, pp and ps updates"},{"id":"","caption":"FIG. 13. (color online) Transition probabilities (1 − w+(+2)) (drawn as solid line (in green (grey))) and w−(+2) (drawn as dashed line (in black)) against p [from Equation 8].","content":"FIG. 13. (color online) Transition probabilities (1 − w+(+2)) (drawn as solid line (in green (grey))) and w−(+2) (drawn as dashed line (in black)) against p [from Equation 8]."},{"id":"","caption":"FIG. 14. (color online) The difference in the transition probabilities (1 − w+(+2)) and w−(+2) is plotted against p [seeEquation 8].","content":"FIG. 14. (color online) The difference in the transition probabilities (1 − w+(+2)) and w−(+2) is plotted against p [seeEquation 8]."},{"id":"","caption":"FIG. 15. (color online)[sp update] Phase diagram of the spupdated model. Plot of the absolute value of magnetization |M | at time t = 512 for lattice size N = 64 2 in the p-ε plane. No phase transition is visible.","content":"FIG. 15. (color online)[sp update] Phase diagram of the spupdated model. Plot of the absolute value of magnetization |M | at time t = 512 for lattice size N = 64 2 in the p-ε plane. No phase transition is visible."},{"id":"","caption":". 16. (color online)[sp update] Plot of the inverse energy 1/E(t) versus ln (t) for different values of p and ε = 1, for a lattice size N = 256 2 . The value of p for each curve is given by a different colour as indicated. No phase transition is visible.","content":"FIG\n. 16. (color online)[sp update] Plot of the inverse energy 1/E(t) versus ln (t) for different values of p and ε = 1, for a lattice size N = 256 2 . The value of p for each curve is given by a different colour as indicated. No phase transition is visible."},{"id":"","caption":"Figure 17shows a plot for magnetization M against the biasing field H at various values of p, that are within the paramagnetic phase at ε = 1.0. Each curve is obtained after averaging over 100 initial configurations using a square lattice of size N = 100 2 . For each p in the paramagnetic phase, we see a linear behaviour of M against H around H ∼ 0, with all subsequent increases in the field strength leading to saturation, as expected. For a given p value we observe a functional dependence of the formM = tanh(bH) where b ∝ (p central + p) 2 taking p central = (p c1 + p c2 )/2.","content":"Figure 17shows a plot for magnetization M against the biasing field H at various values of p, that are within the paramagnetic phase at ε = 1.0. Each curve is obtained after averaging over 100 initial configurations using a square lattice of size N = 100 2 . For each p in the paramagnetic phase, we see a linear behaviour of M against H around H ∼ 0, with all subsequent increases in the field strength leading to saturation, as expected. For a given p value we observe a functional dependence of the formM = tanh(bH) where b ∝ (p central + p) 2 taking p central = (p c1 + p c2 )/2."},{"id":"","caption":". 17. (color online)[ss update] Plot of magnetization M against biasing field H for different values of p. Each curve is drawn using different symbols (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0.","content":"FIG\n. 17. (color online)[ss update] Plot of magnetization M against biasing field H for different values of p. Each curve is drawn using different symbols (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0."},{"id":"","caption":"FIG. 18. (color online)[ss update] Plot of magnetization M against field H for p = 0.58 and ε = 1.0 at time t = 2000 with N = 100 2 . The fit of a tanh(bH) curve almost completely overlaps with our numerical results.","content":"FIG. 18. (color online)[ss update] Plot of magnetization M against field H for p = 0.58 and ε = 1.0 at time t = 2000 with N = 100 2 . The fit of a tanh(bH) curve almost completely overlaps with our numerical results."},{"id":"","caption":"(color online)[ss update] Plot of magnetization M in the p+-p− plane at time t = 7000, for a lattice of size N = 100 2 with ε = 1.0. The + strategy dominates in the yellow (light grey) region, while the − strategy dominates in the brown (black) region.","content":"FIG. 19 .\n19(color online)[ss update] Plot of magnetization M in the p+-p− plane at time t = 7000, for a lattice of size N = 100 2 with ε = 1.0. The + strategy dominates in the yellow (light grey) region, while the − strategy dominates in the brown (black) region."},{"id":"","caption":"22 and Figure 23 that the AFP phase exists for p > p c2 only if the biasing field is within the bounds defined by H = |p + − p − | ≤ 0.19 ± 0.02. In . 20. (color online)[pp update] Plot of magnetization M against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0. . 21. (color online)[pp update] Plot of the absolute value of staggered magnetization |Mstag| against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 64 2 and ε = 1.0.","content":"FIGFIG\n22 and Figure 23 that the AFP phase exists for p > p c2 only if the biasing field is within the bounds defined by H = |p + − p − | ≤ 0.19 ± 0.02. In . 20. (color online)[pp update] Plot of magnetization M against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0. . 21. (color online)[pp update] Plot of the absolute value of staggered magnetization |Mstag| against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 64 2 and ε = 1.0."},{"id":"","caption":"(color online)[pp update] Plot of magnetization M in the p+-p− plane for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. As before, the regions of + and − strategy domination are coloured yellow (light grey) and brown (black); the orange (grey) region corresponds to both the paramagnetic and the AFP region (see text).FIG. 23. (color online)[pp update]Plot of staggered magnetization |Mstag| in the p+-p− plane, for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. Here, yellow (light grey) represents the region of parallel ordering, black represents the disordered phase, and the light brown (grey) represents AFP order.","content":"FIG. 22 .\n22(color online)[pp update] Plot of magnetization M in the p+-p− plane for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. As before, the regions of + and − strategy domination are coloured yellow (light grey) and brown (black); the orange (grey) region corresponds to both the paramagnetic and the AFP region (see text).FIG. 23. (color online)[pp update]Plot of staggered magnetization |Mstag| in the p+-p− plane, for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. Here, yellow (light grey) represents the region of parallel ordering, black represents the disordered phase, and the light brown (grey) represents AFP order."},{"id":"","caption":"FIG. 24. (color online)[pp update] A 3D plot of Magnetization M with parameters ε+ (along x), ε− (along y) and p− (along z), setting p+ = 0.50 for a lattice of size N = 64 2 at time T = 2000. Green (grey) denotes the dominance of the +'s, while blue (black) denotes that of the −'s. The other colours represent cases of intermediate ordering.","content":"FIG. 24. (color online)[pp update] A 3D plot of Magnetization M with parameters ε+ (along x), ε− (along y) and p− (along z), setting p+ = 0.50 for a lattice of size N = 64 2 at time T = 2000. Green (grey) denotes the dominance of the +'s, while blue (black) denotes that of the −'s. The other colours represent cases of intermediate ordering."},{"id":"","caption":". 25. (color online)[pp update] A 3D plot of Magnetization M for the parameters ε+ (lying between 0.7 and 0.8), ε− and p−, with p+ = 0.70, for a lattice of size N = 64 2 at time T = 2000. The red (lower black) region represents the dominance of the + strategy and the blue (upper black) region represents the dominance of the − strategy. The green (grey) region represents AFP.","content":". 25. (color online)[pp update] A 3D plot of Magnetization M for the parameters ε+ (lying between 0.7 and 0.8), ε− and p−, with p+ = 0.70, for a lattice of size N = 64 2 at time T = 2000. The red (lower black) region represents the dominance of the + strategy and the blue (upper black) region represents the dominance of the − strategy. The green (grey) region represents AFP."},{"id":"","caption":"26. (color online)[pp update] A 3D plot of absolute staggered magnetization |Mstag| with parameters ε+(along x), ε−(along y) and p−(along z) for p+ = 0.50 and lattice size N = 64 2 , at time T = 2000. The graph delineates the region of AFP for these parameter values.","content":"26. (color online)[pp update] A 3D plot of absolute staggered magnetization |Mstag| with parameters ε+(along x), ε−(along y) and p−(along z) for p+ = 0.50 and lattice size N = 64 2 , at time T = 2000. The graph delineates the region of AFP for these parameter values."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"(color online)[ss update] Phase diagram of the model with an ss update. Plot of the inverse energy 1/E(t) at time t = 512 for a square lattice of size N = 64 2 in the p-ε plane. The black region shows the disordered phase and the yellowish (light grey) region shows the frozen phase.\",\n \"content\": \"FIG. 1 .\\n1(color online)[ss update] Phase diagram of the model with an ss update. Plot of the inverse energy 1/E(t) at time t = 512 for a square lattice of size N = 64 2 in the p-ε plane. The black region shows the disordered phase and the yellowish (light grey) region shows the frozen phase.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"shows snapshots of the dynamics of the model using a lattice of size N = 512 2 at times t = 8, t = 64, and t = 512 with random initial configurations and parameter values p = 0.72 (very close to the critical point p c2 ) and ε = 1.0. The plots reveal characteristically voter-like[12] coarsening behaviour.\",\n \"content\": \"Figure 2\\n2shows snapshots of the dynamics of the model using a lattice of size N = 512 2 at times t = 8, t = 64, and t = 512 with random initial configurations and parameter values p = 0.72 (very close to the critical point p c2 ) and ε = 1.0. The plots reveal characteristically voter-like[12] coarsening behaviour.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[ss update] Snapshots of the dynamics of the ss-updated model. Each plot is a portion (of size 100 2 ) of a square lattice of N = 256 2 for p = 0.72 and ε = 1.0 at times t = 8 (top-left), t = 64 (top-right) and t = 512 (bottom).\",\n \"content\": \"FIG. 2 .\\n2(color online)[ss update] Snapshots of the dynamics of the ss-updated model. Each plot is a portion (of size 100 2 ) of a square lattice of N = 256 2 for p = 0.72 and ε = 1.0 at times t = 8 (top-left), t = 64 (top-right) and t = 512 (bottom).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[ss update] Plot of the inverse energy 1/E(t) versus the natural logarithm of time ln t for different values of p close to pc2 = 0.70, with ε set to 1.0. The lattice size N = 256 2 , and the p-values are indicated on the curves.\",\n \"content\": \"FIG. 3 .\\n3(color online)[ss update] Plot of the inverse energy 1/E(t) versus the natural logarithm of time ln t for different values of p close to pc2 = 0.70, with ε set to 1.0. The lattice size N = 256 2 , and the p-values are indicated on the curves.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[pp update] Plot of the absolute value of the magnetization |M | (top) and the absolute value of the staggered magnetization |Mstag| (bottom) at time t = 512 for a lattice of N = 100 2 in the p-ε plane. In the top figure, the yellowish (light grey) region refers to the parallel frozen phase (PFP), while the black region includes both the antiparallel frozen phase (AFP), and the disordered region. In the bottom figure, the black region represents the disordered phase characterised by very low |Mstag|.\",\n \"content\": \"FIG. 4 .\\n4(color online)[pp update] Plot of the absolute value of the magnetization |M | (top) and the absolute value of the staggered magnetization |Mstag| (bottom) at time t = 512 for a lattice of N = 100 2 in the p-ε plane. In the top figure, the yellowish (light grey) region refers to the parallel frozen phase (PFP), while the black region includes both the antiparallel frozen phase (AFP), and the disordered region. In the bottom figure, the black region represents the disordered phase characterised by very low |Mstag|.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[pp update] Snapshots of the dynamics of the pp-updated model on a square lattice, for p = 0.41 (leftmost), p = 0.50 (centre) and p = 0.59 (rightmost) at time t = 512, with ε = 1.0. The yellow (light grey) and black colours represent the two strategies, while the greyish grid corresponds to anti-parallel arrangements of yellow (light grey) and black. The leftmost picture represents the PFP (see text), the centre one the disordered phase, and the rightmost one the AFP (see text).\",\n \"content\": \"FIG. 5 .\\n5(color online)[pp update] Snapshots of the dynamics of the pp-updated model on a square lattice, for p = 0.41 (leftmost), p = 0.50 (centre) and p = 0.59 (rightmost) at time t = 512, with ε = 1.0. The yellow (light grey) and black colours represent the two strategies, while the greyish grid corresponds to anti-parallel arrangements of yellow (light grey) and black. The leftmost picture represents the PFP (see text), the centre one the disordered phase, and the rightmost one the AFP (see text).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[pp update] Plot of the absolute value of magnetization |M |, the absolute value of staggered magnetization |Mstag| and energy E against p, with ε = 1 and lattice size N = 80 2 . Each curve is drawn using symbols (and colour) as indicated in the legend.\",\n \"content\": \"Figure 8 FIG. 6 .\\n86(color online)[pp update] Plot of the absolute value of magnetization |M |, the absolute value of staggered magnetization |Mstag| and energy E against p, with ε = 1 and lattice size N = 80 2 . Each curve is drawn using symbols (and colour) as indicated in the legend.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.43 and ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.43 (shown in red) has slope 1/2π approximately.\",\n \"content\": \"FIG. 7 .\\n7(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.43 and ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.43 (shown in red) has slope 1/2π approximately.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.57, with ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.57 (shown in red) has slope −1/5π approximately.\",\n \"content\": \"FIG. 8 .\\n8(color online)[pp update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.57, with ε = 1, for a lattice of size N = 100 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.57 (shown in red) has slope −1/5π approximately.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[ps update] Phase diagram in the p-ε plane of the ps-updated model, with a plot of the absolute value of staggered magnetization |Mstag| at time t = 512, for a lattice size of N = 100 2 . The black region represents the disordered phase (very low |Mstag|), while the yellowish (light grey) region represents frozen phases with high |Mstag|. p, with ε equal to 1.0; each curve is an average over 100 independent runs. The region where both the magnetization |M | and staggered magnetization |M stag | curves saturate to 1, corresponds to parallel alignment, whereas |M | ≈ 0 with |M stag | ≈ 1 implies an anti-parallel alignment of the spin types. The energy graph is consistent with this interpretation, given the definition of the energy in Equation10: zero in the PFP, middling in the paramagnetic phase and unity in the AFP. 10. (color online)[ps update] Plot of the absolute value of magnetization |M |, energy E and the absolute value of staggered magnetization |Mstag| against p with ε set to 1.0 for a lattice size N = 100 2 . Each curve is drawn using symbols (and colour) as indicated in the legend.\",\n \"content\": \"FIG. 9 .\\n9(color online)[ps update] Phase diagram in the p-ε plane of the ps-updated model, with a plot of the absolute value of staggered magnetization |Mstag| at time t = 512, for a lattice size of N = 100 2 . The black region represents the disordered phase (very low |Mstag|), while the yellowish (light grey) region represents frozen phases with high |Mstag|. p, with ε equal to 1.0; each curve is an average over 100 independent runs. The region where both the magnetization |M | and staggered magnetization |M stag | curves saturate to 1, corresponds to parallel alignment, whereas |M | ≈ 0 with |M stag | ≈ 1 implies an anti-parallel alignment of the spin types. The energy graph is consistent with this interpretation, given the definition of the energy in Equation10: zero in the PFP, middling in the paramagnetic phase and unity in the AFP. 10. (color online)[ps update] Plot of the absolute value of magnetization |M |, energy E and the absolute value of staggered magnetization |Mstag| against p with ε set to 1.0 for a lattice size N = 100 2 . Each curve is drawn using symbols (and colour) as indicated in the legend.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 11. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.31, with ε set to 1.0 for a lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.31 (shown in red) has a slope of approximately 4/3π.\",\n \"content\": \"FIG\\n. 11. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc1 = 0.31, with ε set to 1.0 for a lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc1 = 0.31 (shown in red) has a slope of approximately 4/3π.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 12. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.69, with ε set to 1.0, for lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.69 (shown in red) has slope −4/15π approximately. D. Explanation for the nature of the phase diagrams for different updates In this subsection, we give arguments for the three most important features of the phase diagrams presented above: (i) The appearance of anti-parallel ordering in both pp and ps updates (ii) The symmetry of the PFP and the AFP phases in both pp and ps updates (iii) The positioning of the disordered phase in ss, pp and ps updates\",\n \"content\": \"FIG\\n. 12. (color online)[ps update] Plot of the inverse energy 1/E(t) versus ln t for different values of p close to pc2 = 0.69, with ε set to 1.0, for lattice size N = 256 2 . The p-values are indicated on the curves. The straight line corresponding to pc2 = 0.69 (shown in red) has slope −4/15π approximately. D. Explanation for the nature of the phase diagrams for different updates In this subsection, we give arguments for the three most important features of the phase diagrams presented above: (i) The appearance of anti-parallel ordering in both pp and ps updates (ii) The symmetry of the PFP and the AFP phases in both pp and ps updates (iii) The positioning of the disordered phase in ss, pp and ps updates\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 13. (color online) Transition probabilities (1 − w+(+2)) (drawn as solid line (in green (grey))) and w−(+2) (drawn as dashed line (in black)) against p [from Equation 8].\",\n \"content\": \"FIG. 13. (color online) Transition probabilities (1 − w+(+2)) (drawn as solid line (in green (grey))) and w−(+2) (drawn as dashed line (in black)) against p [from Equation 8].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 14. (color online) The difference in the transition probabilities (1 − w+(+2)) and w−(+2) is plotted against p [seeEquation 8].\",\n \"content\": \"FIG. 14. (color online) The difference in the transition probabilities (1 − w+(+2)) and w−(+2) is plotted against p [seeEquation 8].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 15. (color online)[sp update] Phase diagram of the spupdated model. Plot of the absolute value of magnetization |M | at time t = 512 for lattice size N = 64 2 in the p-ε plane. No phase transition is visible.\",\n \"content\": \"FIG. 15. (color online)[sp update] Phase diagram of the spupdated model. Plot of the absolute value of magnetization |M | at time t = 512 for lattice size N = 64 2 in the p-ε plane. No phase transition is visible.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 16. (color online)[sp update] Plot of the inverse energy 1/E(t) versus ln (t) for different values of p and ε = 1, for a lattice size N = 256 2 . The value of p for each curve is given by a different colour as indicated. No phase transition is visible.\",\n \"content\": \"FIG\\n. 16. (color online)[sp update] Plot of the inverse energy 1/E(t) versus ln (t) for different values of p and ε = 1, for a lattice size N = 256 2 . The value of p for each curve is given by a different colour as indicated. No phase transition is visible.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 17shows a plot for magnetization M against the biasing field H at various values of p, that are within the paramagnetic phase at ε = 1.0. Each curve is obtained after averaging over 100 initial configurations using a square lattice of size N = 100 2 . For each p in the paramagnetic phase, we see a linear behaviour of M against H around H ∼ 0, with all subsequent increases in the field strength leading to saturation, as expected. For a given p value we observe a functional dependence of the formM = tanh(bH) where b ∝ (p central + p) 2 taking p central = (p c1 + p c2 )/2.\",\n \"content\": \"Figure 17shows a plot for magnetization M against the biasing field H at various values of p, that are within the paramagnetic phase at ε = 1.0. Each curve is obtained after averaging over 100 initial configurations using a square lattice of size N = 100 2 . For each p in the paramagnetic phase, we see a linear behaviour of M against H around H ∼ 0, with all subsequent increases in the field strength leading to saturation, as expected. For a given p value we observe a functional dependence of the formM = tanh(bH) where b ∝ (p central + p) 2 taking p central = (p c1 + p c2 )/2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 17. (color online)[ss update] Plot of magnetization M against biasing field H for different values of p. Each curve is drawn using different symbols (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0.\",\n \"content\": \"FIG\\n. 17. (color online)[ss update] Plot of magnetization M against biasing field H for different values of p. Each curve is drawn using different symbols (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 18. (color online)[ss update] Plot of magnetization M against field H for p = 0.58 and ε = 1.0 at time t = 2000 with N = 100 2 . The fit of a tanh(bH) curve almost completely overlaps with our numerical results.\",\n \"content\": \"FIG. 18. (color online)[ss update] Plot of magnetization M against field H for p = 0.58 and ε = 1.0 at time t = 2000 with N = 100 2 . The fit of a tanh(bH) curve almost completely overlaps with our numerical results.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[ss update] Plot of magnetization M in the p+-p− plane at time t = 7000, for a lattice of size N = 100 2 with ε = 1.0. The + strategy dominates in the yellow (light grey) region, while the − strategy dominates in the brown (black) region.\",\n \"content\": \"FIG. 19 .\\n19(color online)[ss update] Plot of magnetization M in the p+-p− plane at time t = 7000, for a lattice of size N = 100 2 with ε = 1.0. The + strategy dominates in the yellow (light grey) region, while the − strategy dominates in the brown (black) region.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"22 and Figure 23 that the AFP phase exists for p > p c2 only if the biasing field is within the bounds defined by H = |p + − p − | ≤ 0.19 ± 0.02. In . 20. (color online)[pp update] Plot of magnetization M against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0. . 21. (color online)[pp update] Plot of the absolute value of staggered magnetization |Mstag| against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 64 2 and ε = 1.0.\",\n \"content\": \"FIGFIG\\n22 and Figure 23 that the AFP phase exists for p > p c2 only if the biasing field is within the bounds defined by H = |p + − p − | ≤ 0.19 ± 0.02. In . 20. (color online)[pp update] Plot of magnetization M against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 100 2 and ε = 1.0. . 21. (color online)[pp update] Plot of the absolute value of staggered magnetization |Mstag| against field H for different values of p, each curve indicated by a different symbol (and colour) as shown in the legend, at time t = 2000 with N = 64 2 and ε = 1.0.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(color online)[pp update] Plot of magnetization M in the p+-p− plane for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. As before, the regions of + and − strategy domination are coloured yellow (light grey) and brown (black); the orange (grey) region corresponds to both the paramagnetic and the AFP region (see text).FIG. 23. (color online)[pp update]Plot of staggered magnetization |Mstag| in the p+-p− plane, for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. Here, yellow (light grey) represents the region of parallel ordering, black represents the disordered phase, and the light brown (grey) represents AFP order.\",\n \"content\": \"FIG. 22 .\\n22(color online)[pp update] Plot of magnetization M in the p+-p− plane for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. As before, the regions of + and − strategy domination are coloured yellow (light grey) and brown (black); the orange (grey) region corresponds to both the paramagnetic and the AFP region (see text).FIG. 23. (color online)[pp update]Plot of staggered magnetization |Mstag| in the p+-p− plane, for a lattice of size N = 100 2 at time t = 7000, with ε = 1.0. Here, yellow (light grey) represents the region of parallel ordering, black represents the disordered phase, and the light brown (grey) represents AFP order.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 24. (color online)[pp update] A 3D plot of Magnetization M with parameters ε+ (along x), ε− (along y) and p− (along z), setting p+ = 0.50 for a lattice of size N = 64 2 at time T = 2000. Green (grey) denotes the dominance of the +'s, while blue (black) denotes that of the −'s. The other colours represent cases of intermediate ordering.\",\n \"content\": \"FIG. 24. (color online)[pp update] A 3D plot of Magnetization M with parameters ε+ (along x), ε− (along y) and p− (along z), setting p+ = 0.50 for a lattice of size N = 64 2 at time T = 2000. Green (grey) denotes the dominance of the +'s, while blue (black) denotes that of the −'s. The other colours represent cases of intermediate ordering.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 25. (color online)[pp update] A 3D plot of Magnetization M for the parameters ε+ (lying between 0.7 and 0.8), ε− and p−, with p+ = 0.70, for a lattice of size N = 64 2 at time T = 2000. The red (lower black) region represents the dominance of the + strategy and the blue (upper black) region represents the dominance of the − strategy. The green (grey) region represents AFP.\",\n \"content\": \". 25. (color online)[pp update] A 3D plot of Magnetization M for the parameters ε+ (lying between 0.7 and 0.8), ε− and p−, with p+ = 0.70, for a lattice of size N = 64 2 at time T = 2000. The red (lower black) region represents the dominance of the + strategy and the blue (upper black) region represents the dominance of the − strategy. The green (grey) region represents AFP.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"26. (color online)[pp update] A 3D plot of absolute staggered magnetization |Mstag| with parameters ε+(along x), ε−(along y) and p−(along z) for p+ = 0.50 and lattice size N = 64 2 , at time T = 2000. The graph delineates the region of AFP for these parameter values.\",\n \"content\": \"26. (color online)[pp update] A 3D plot of absolute staggered magnetization |Mstag| with parameters ε+(along x), ε−(along y) and p−(along z) for p+ = 0.50 and lattice size N = 64 2 , at time T = 2000. The graph delineates the region of AFP for these parameter values.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Statistical physics of social dynamics. C Castellano, S Fortunato, V Loreto, Rev. Mod. Phys. 81591646C Castellano, S Fortunato and V Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591646 (2009).","doi":""},{"text":"D Fudenberg, D K Levine, The Theory of Learning in Games. CambridgeMIT PressD Fudenberg and D K Levine, The Theory of Learning in Games (MIT Press, Cambridge 1998).","doi":""},{"text":"Young H Payton, Strategic Learning and Its Limits. OxfordOxford University PressH Payton Young, Strategic Learning and Its Limits (Ox- ford University Press, Oxford 2004).","doi":""},{"text":"Technology diffusion by learning from neighbours. K Chatterjee, S H Xu, Adv. Appl. Prob. 36K Chatterjee and S H Xu, Technology diffusion by learn- ing from neighbours, Adv. Appl. Prob. 36, 355-376 (2004).","doi":""},{"text":"Behavioural studies of strategic thinking in games. C F Camerer, Trends Cogn. Sci. 7C F Camerer, Behavioural studies of strategic thinking in games, Trends Cogn. Sci. 7, 225-231 (2003).","doi":""},{"text":"Models of competitive learning: Complex dynamics, intermittent conversions, and oscillatory coarsening. A Mehta, J M Luck, Physical Review E. 60A Mehta and J M Luck, Models of competitive learning: Complex dynamics, intermittent conversions, and oscil- latory coarsening, Physical Review E 60, 5: 5218-5230 (1999).","doi":""},{"text":"Competing with oneself: introducing self-interaction in a model of competitive learning. G Mahajan, Mehta, Theory Biosci. 129G Mahajan and A Mehta, Competing with oneself: intro- ducing self-interaction in a model of competitive learning, Theory Biosci. 129, 271-282 (2010).","doi":""},{"text":"J C Gonzalez-Avella, V M Eguluz, M Marsili, M Vega-Redondo, Miguel San, Threshold Learning Dynamics in Social Networks. 620207J C Gonzalez-Avella, V M Eguluz, M Marsili, F Vega- Redondo and M San Miguel, Threshold Learning Dynam- ics in Social Networks, PLoS ONE 6, 5: e20207 (2011).","doi":""},{"text":"Coevolution of Glauber-like Ising dynamics and topology. S Mandra, S Fortunato, C Castellano, Phys. Rev. E. 8056105S Mandra, S Fortunato and C Castellano, Coevolution of Glauber-like Ising dynamics and topology, Phys. Rev. E 80, 056105 (2009).","doi":""},{"text":"Synchronous or asynchronous parallel dynamics. Which is more efficient?. Kanter, Physica D. 42I Kanter, Synchronous or asynchronous parallel dynam- ics. Which is more efficient?, Physica D 42, 273-280 (1990).","doi":""},{"text":"Parallel versus sequential updating for belief propagation decoding. H Kfir, Kanter, Physica A. 330H Kfir and I Kanter, Parallel versus sequential updating for belief propagation decoding, Physica A 330, 1-2: 259- 270 (2003).","doi":""},{"text":"Critical Coarsening without Surface Tension: The Universality Class of the Voter Model. Dornic, Chate, H Chave, Hinrichsen, Phys. Rev. Lett. 8745701I Dornic, H Chate, J Chave, and H Hinrichsen, Critical Coarsening without Surface Tension: The Universality Class of the Voter Model, Phys. Rev. Lett. 87, 045701 (2001).","doi":""},{"text":"Freezing into stripe states in two-dimensional ferromagnets and crossing probabilities in critical percolation. K Barros, S Krapivsky, Redner, Phys. Rev. E. 8040101K Barros, P L Krapivsky, and S Redner, Freezing into stripe states in two-dimensional ferromagnets and cross- ing probabilities in critical percolation, Phys. Rev. E 80, 040101 (2009).","doi":""},{"text":"Global Competitors as Next-Door Neighbors: Competition and Geographic Concentration in the Semiconductor Industry. M Zhao, J Alcacer, Ross School of Business Paper. 1091M Zhao and J Alcacer, Global Competitors as Next-Door Neighbors: Competition and Geographic Concentration in the Semiconductor Industry, Ross School of Business Paper No. 1091 (2007).","doi":""},{"text":"Competing synapses with two timescales -a basis for learning and forgetting. G Mahajan, Mehta, Europhysics Letters. 9548008G Mahajan and A Mehta, Competing synapses with two timescales -a basis for learning and forgetting, Euro- physics Letters 95, 48008 (2011);","doi":""},{"text":"Learning with a network of competing synapses. A A Bhat, G Mahajan, A Mehta, PLoS ONE. 625048A A Bhat, G Mahajan and A Mehta, Learning with a network of competing synapses, PLoS ONE 6, 9: e25048 (2011).","doi":""}],"string":"[\n {\n \"text\": \"Statistical physics of social dynamics. C Castellano, S Fortunato, V Loreto, Rev. Mod. Phys. 81591646C Castellano, S Fortunato and V Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81, 591646 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Fudenberg, D K Levine, The Theory of Learning in Games. CambridgeMIT PressD Fudenberg and D K Levine, The Theory of Learning in Games (MIT Press, Cambridge 1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Young H Payton, Strategic Learning and Its Limits. OxfordOxford University PressH Payton Young, Strategic Learning and Its Limits (Ox- ford University Press, Oxford 2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Technology diffusion by learning from neighbours. K Chatterjee, S H Xu, Adv. Appl. Prob. 36K Chatterjee and S H Xu, Technology diffusion by learn- ing from neighbours, Adv. Appl. Prob. 36, 355-376 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Behavioural studies of strategic thinking in games. C F Camerer, Trends Cogn. Sci. 7C F Camerer, Behavioural studies of strategic thinking in games, Trends Cogn. Sci. 7, 225-231 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Models of competitive learning: Complex dynamics, intermittent conversions, and oscillatory coarsening. A Mehta, J M Luck, Physical Review E. 60A Mehta and J M Luck, Models of competitive learning: Complex dynamics, intermittent conversions, and oscil- latory coarsening, Physical Review E 60, 5: 5218-5230 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Competing with oneself: introducing self-interaction in a model of competitive learning. G Mahajan, Mehta, Theory Biosci. 129G Mahajan and A Mehta, Competing with oneself: intro- ducing self-interaction in a model of competitive learning, Theory Biosci. 129, 271-282 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J C Gonzalez-Avella, V M Eguluz, M Marsili, M Vega-Redondo, Miguel San, Threshold Learning Dynamics in Social Networks. 620207J C Gonzalez-Avella, V M Eguluz, M Marsili, F Vega- Redondo and M San Miguel, Threshold Learning Dynam- ics in Social Networks, PLoS ONE 6, 5: e20207 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Coevolution of Glauber-like Ising dynamics and topology. S Mandra, S Fortunato, C Castellano, Phys. Rev. E. 8056105S Mandra, S Fortunato and C Castellano, Coevolution of Glauber-like Ising dynamics and topology, Phys. Rev. E 80, 056105 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Synchronous or asynchronous parallel dynamics. Which is more efficient?. Kanter, Physica D. 42I Kanter, Synchronous or asynchronous parallel dynam- ics. Which is more efficient?, Physica D 42, 273-280 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Parallel versus sequential updating for belief propagation decoding. H Kfir, Kanter, Physica A. 330H Kfir and I Kanter, Parallel versus sequential updating for belief propagation decoding, Physica A 330, 1-2: 259- 270 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Critical Coarsening without Surface Tension: The Universality Class of the Voter Model. Dornic, Chate, H Chave, Hinrichsen, Phys. Rev. Lett. 8745701I Dornic, H Chate, J Chave, and H Hinrichsen, Critical Coarsening without Surface Tension: The Universality Class of the Voter Model, Phys. Rev. Lett. 87, 045701 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Freezing into stripe states in two-dimensional ferromagnets and crossing probabilities in critical percolation. K Barros, S Krapivsky, Redner, Phys. Rev. E. 8040101K Barros, P L Krapivsky, and S Redner, Freezing into stripe states in two-dimensional ferromagnets and cross- ing probabilities in critical percolation, Phys. Rev. E 80, 040101 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Global Competitors as Next-Door Neighbors: Competition and Geographic Concentration in the Semiconductor Industry. M Zhao, J Alcacer, Ross School of Business Paper. 1091M Zhao and J Alcacer, Global Competitors as Next-Door Neighbors: Competition and Geographic Concentration in the Semiconductor Industry, Ross School of Business Paper No. 1091 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Competing synapses with two timescales -a basis for learning and forgetting. G Mahajan, Mehta, Europhysics Letters. 9548008G Mahajan and A Mehta, Competing synapses with two timescales -a basis for learning and forgetting, Euro- physics Letters 95, 48008 (2011);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning with a network of competing synapses. A A Bhat, G Mahajan, A Mehta, PLoS ONE. 625048A A Bhat, G Mahajan and A Mehta, Learning with a network of competing synapses, PLoS ONE 6, 9: e25048 (2011).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":24,"cells":{"text":{"kind":"string","value":"\nAccepted for publication on ApJ RECOILING SUPERMASSIVE BLACK HOLES: A SEARCH IN THE NEARBY UNIVERSE\n\n\nD Lena \nA Robinson \nA Marconi \nD J Axon \nA Capetti \nD Merritt \n& D Batcheldor \nAccepted for publication on ApJ RECOILING SUPERMASSIVE BLACK HOLES: A SEARCH IN THE NEARBY UNIVERSE\nAccepted for publication on ApJPreprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: black hole physics -gravitational waves -galaxies: nuclei -galaxies: active - galaxies: interactions\nThe coalescence of a binary black hole can be accompanied by a large gravitational recoil due to anisotropic emission of gravitational waves. A recoiling supermassive black hole (SBH) can subsequently undergo long-lived oscillations in the potential well of its host galaxy, suggesting that offset SBHs may be common in the cores of massive ellipticals. We have analyzed HST archival images of 14 nearby core ellipticals, finding evidence for small ( 10 pc) displacements between the AGN (locating the SBH) and the center of the galaxy (the mean photocenter) in 10 of them. Excluding objects that may be affected by large-scale isophotal asymmetries, we consider six galaxies to have detected displacements, including M87, where a displacement was previously reported byBatcheldor et al. (2010). In individual objects, these displacements can be attributed to residual gravitational recoil oscillations following a major or minor merger within the last few Gyr. For plausible merger rates, however, there is a high probability of larger displacements than those observed, if SBH coalescence took place in these galaxies. Remarkably, the AGN-photocenter displacements are approximately aligned with the radio source axis in four of the six galaxies with displacements, including three of the four having relatively powerful kpc-scale jets. This suggests intrinsic asymmetries in radio jet power as a possible displacement mechanism, although approximate alignments are also expected for gravitational recoil. Orbital motion in SBH binaries and interactions with massive perturbers can produce the observed displacement amplitudes but do not offer a ready explanation for the alignments.\n\nINTRODUCTION\n\nSupermassive black holes (SBHs) have long been identified as the engines of active galactic nuclei (AGNs). More recently, compelling evidence has been amassed that they are present in the centers of almost all galaxies (Kormendy & Richstone 1995;Ferrarese & Ford 2005) and furthermore that the growth of the SBH is intimately linked to galaxy evolution (Ferrarese & Merritt 2000;Gebhardt et al. 2000; see Alexander & Hickox 2012 for a recent review).\n\nAs a consequence of energy losses due to dynamical friction, SBHs are generally expected to reside at the large-scale potential minimum of the host galaxy. Nevertheless, there are several mechanisms that seem capable of displacing the SBH from its equilibrium position (Batcheldor et al. 2010, hereafter B10). Most recently, interest has focused on gravitational recoil resulting from the coalescence of an SBH-SBH binary (e.g., Merritt et al. 2004;Campanelli et al. 2006;Baker et al. 2006). Other possibilities include orbital motion of SBH-binaries (Komossa 2006), asymmetric radio jets (Shklovski 1982) and interactions with massive perturbers such as globular clusters, or massive molecular clouds.\n\nThe first two of these mechanisms, in particular, might be expected to occur as an inevitable consequence of galaxy evolution. Cold dark matter hierarchical cosmologies predict that galaxies experience multiple mergers during their lifetime (e.g., Coles & Lucchin 2002;Springel et al. 2005), with minor mergers being more common than major mergers (e.g. Hilz et al. 2012). When two galaxies merge, their central SBHs are expected to form a binary system (Begelman, Blandford & Rees 1980) at the center of the merged system due to dynamical friction. Further tightening of the binary ensues through threebody interactions with stars, or gravitational drag from gas (Merritt & Milosavljević 2005;Mayer et al. 2007), until finally, energy loss due to gravitational wave emission irreversibly drives the two SBHs to coalescence 1 .\n\nIn the final stage of SBH-binary coalescence, anisotropic emission of gravitational waves will, in general, impart a recoil velocity to the coalesced object 1 Begelman, Blandford & Rees (1980) suggested that the evolution of a binary SBH might stall at separations of ∼ 1 pc once the supply of \"loss-cone stars\" (stars intersecting the orbit of the binary) was exhausted. However, gas dynamical torques can drive continued shrinking of the binary orbit (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007), and recent N -body simulations suggest that hardening of the binary due to interactions with stars can be efficient in non spherical potentials (Preto et al. 2011;Gualandris & Merritt 2012;Vasiliev et al. 2014). (Bekenstein 1973). Following a recent breakthrough allowing the orbits of spinning black holes to be computed to coalescence (Pretorius 2005;Campanelli et al. 2006;Baker et al. 2006), numerical relativity simulations have produced recoil velocities v ∼ 10 3 km s −1 , even reaching ∼ 5 × 10 3 km s −1 for certain spin configurations González et al. 2007;Tichy & Marronetti 2007;Lousto & Zlochower 2011;Lousto et al. 2012). Such velocities would cause large displacements of the coalesced SBH from the center of the galaxy or, in the most extreme cases, eject it entirely from the host (Merritt et al. 2004;Campanelli et al. 2007a;Volonteri, Gúltekin & Dotti 2010).\n\nRecoils exceeding the escape velocity of the host galaxy are expected to be relatively rare (e.g., Lousto et al. 2012, hereafter L12) and, more frequently, the SBH will undergo damped oscillations in the galaxy potential. Nbody simulations (Gualandris and Merritt 2008, hereafter GM08) have shown that moderately large kicks (∼ few 10 2 km s −1 , sufficient to eject the SBH from the core 2 but not from the galaxy) result in long-lived oscillations which damp on a timescale ∆t ∼ 1 Gyr. GM08 identified three dynamical phases for the particular case where the SBH is initially displaced beyond the core radius (r c ∼ 10 2 pc). In \"phase I\" the SBH undergoes damped harmonic motion at amplitudes greater than the core radius. These oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction. \"Phase II\" is characterized by oscillations of amplitude ∼ 10 − 100 pc between the SBH and the stellar core about the center of mass, which persist for up to ∼ 1 Gyr. Finally, in \"phase III\", the SBH has reached thermal equilibrium with the surrounding stars and experiences Brownian motion, signifying the end of the event.\n\nLarge gravitational recoil kicks would have a variety of observable consequences. In the case of SBHs associated with AGNs, most of the accretion disk and broad emission line gas will remain bound for recoils with velocities v kick 10 3 km s −1 and such systems might be observed as a displaced, or velocity-shifted AGNs (e.g., Merritt et al. 2004;Madau & Quataert 2004;Loeb 2007;Bonning, Shields & Salviander 2007;Zanotti et al. 2010). Stars moving with orbital velocities v < v kick would also remain bound to the SBH after the kick, so that a recoiling SBH would be associated with a \"hypercompact stellar system\" of high internal velocity dispersion (Merritt et al. 2009).\n\nPossible spectroscopic signatures of recoiling SBHs have been identified in a few individual active galaxies (Komossa et al. 2008;Shields et al. 2009;Robinson et al. 2010;Steinhardt et al. 2012), which would indicate large ( 10 3 km s −1 ) kick velocities. In addition, systematic searches of large SDSS AGN samples have revealed ∼ 100 objects that exhibit velocity shifts ∼ 10 3 km s −1 between the broad and narrow lines (Bonning, Shields & Salviander 2007;Eracleous et al. 2012;Tsalmantza et al. 2011). Some of these may be recoiling SBHs, or SBH binaries, but alternative explanations for the shifts involving extreme broad line region kinematics cannot be excluded. Perhaps the most interesting case is that of CIS-42, a galaxy which hosts two compact optical sources separated by ≈ 2.5 kpc and also exhibits a large velocity shift (∼ 1300 km s −1 ) between broad and narrow Hβ emission lines (Civano et al. 2010(Civano et al. , 2012Blecha et al. 2012).\n\nThe gravitational recoil candidates discovered to date have been identified on the basis of anomalously large velocity shifts ( 10 3 km s −1 ) or a large spatial offset ( 1 kpc). If they are indeed recoiling SBHs, we are observing them during the early stages in their dynamical evolution following a large kick; that is, during the large amplitude phase I oscillations of GM08, if the SBH remains bound to the galaxy. However, special configurations of the progenitor binary are required to produce large kicks, which are therefore expected to be relatively rare events (L12) and furthermore, the subsequent large amplitude phase I oscillations are relatively short-lived.\n\nOn the other hand, the N -body simulations of GM08 predict that, following a recoil kick sufficiently large to remove the SBH from the galaxy core, the decaying oscillations persist in phase II for ∼ 1 Gyr, comparable with the time between galaxy mergers (Hopkins et al. 2010, hereafter H10). This suggests that low amplitude SBH displacements resulting from post-merger SBH binary coalescence should be relatively common in the cores of bright elliptical galaxies. Thus, rather than searching for short-lived phase I oscillations following rare large kicks, an alternative approach to studying the incidence of SBH binary coalescence is to look for phase II oscillations in nearby ellipticals.\n\nSpectroscopic identification of phase II oscillations would be extremely difficult, if not impossible, as the associated velocity shifts would be comparable with gas motions due to rotation, or flows or turbulence driven by the AGN, or starburst activity. However, it is feasible to directly detect small-amplitude displacements between the AGN and the galaxy photocenter in high spatial resolution images of nearby galaxies. We recently analyzed archival Hubble Space Telescope Advanced Camera for Surveys images of the active giant elliptical M87 and found a projected offset of 6.8 ± 0.8 pc (≈ 0. 1) between the stellar photocenter and the AGN (B10; we note, however, that Gebhardt et al. 2011 did not find a significant offset in their near-infrared integral field spectroscopy data).\n\nHere we describe a photometric analysis of archival HST images of 14 nearby (d<100 Mpc), bright ellipticals hosting low-luminosity AGNs, with the aim of directly measuring spatial offsets comparable with the core radius of the galaxy (typically < 500 pc). Our method consists in measuring the relative positions of the AGN point source and the photocenter of the inner 2-3 kpc of the galaxy, assuming that the former marks the SBH position and the latter the minimum of the galactic potential. As long as the AGN is bright enough to be easily detected and faint enough that it does not overwhelm the host galaxy, it is possible to perform standard photometric analysis to locate the photocenter. The \"displacement\" is then computed as the relative distance between the AGN and photocenter.\n\nThe plan of this paper is as follows: in §2 we describe the sample; §3 describes the analysis methods used to determine the photocenter, SBH position and offset, and their uncertainties; in §4 the measured displacements are presented for each galaxy. In §5, we discuss the results in the context of gravitational recoil and other possible displacement mechanisms. Finally, our conclusions are summarized in §6. The results for selected galaxies are discussed in more detail in an Appendix.\n\n\nTHE SAMPLE\n\nThe sample selected for this study consists of 14 nearby, regular, core elliptical galaxies which host AGNs and for which HST images obtained with ACS, NIC-MOS2, WFPC2 or WFC3 are available in the Hubble Legacy Archive (HLA).\n\nHST observations have revealed that the central light distribution in nearly all nearby early type galaxies can be described by a singular surface brightness profile: Σ(r) ∼ r −γ as r → 0. 1, where γ ≥ 0. The slope of the innermost surface brightness distribution is bimodal in the sense that there is a paucity of galaxies which have 0.3 < γ < 0.5. Hence, early type galaxies have been classified in two families: \"core galaxies\", which have a shallower (γ ≤ 0.3) inner cusp within a \"break\" (or \"core\") radius r c ∼ 100 pc, and \"power law galaxies\", which have a steep cusp (γ ≥ 0.5) continuing into the HST resolution limit . These classifications correlate with several other galaxy properties (e.g., Faber et al. 1997), in particular, core galaxies are on average more luminous than power-law galaxies, with essentially all bright (M V < −22) ellipticals having cores.\n\nCore galaxies are promising systems in which to search for SBH binaries or SBHs displaced by gravitational wave-induced recoils. They are bright ellipticals which are often the dominant components of clusters or groups and are thus likely to have experienced a recent major merger leading to the formation of an SBH binary. The shallow inner surface brightness profile indicates a mass deficit relative to that implied by inward extrapolation of the steeper brightness profile prevailing at larger radii. This is predicted as a natural consequence of depopulation of the inner region of the galaxy by 3-body interactions between an SBH binary and stars crossing the orbit of the binary (\"loss-cone stars\" Merritt & Milosavljević 2005). Therefore, \"flat\" inner surface brightness profiles have been proposed as a clear footprint of the formation of a tightly bound SBH binary (e.g., Faber et al. 1997;Merritt 2006).\n\nIn addition, core galaxies tend to have regular photometric structures, and hence well-defined photocenters. In order to accurately locate the position of the SBH, we further require that each galaxy hosts a low luminosity AGN visible as a point source in the HST images.\n\nIn order to resolve the low amplitude offsets between photocenter and AGN expected in the phase II oscillations predicted by GM08, we select only galaxies within 100 Mpc.\n\nThe sample analyzed in this work was extracted from a sample of 29 core elliptical galaxies previously studied by Capetti and Balmaverde, in a series of three articles (Capetti & Balmaverde 2005;Capetti & Balmaverde 2006, hereafter CB05, BC06 and CB06 respectively). CB05 compiled a radio-flux limited sample of AGNs in early type hosts by selecting radio-detected sources from the VLA surveys of Wrobel & Heeschen (1991) and Sadler et al. (1989). The former is a northern sample of 216 early-type galaxies extracted from the CfA redshift survey (Huchra et al. 1983), with the following criteria: declination δ 1950 ≥ 0; photometric magnitude B ≤ 14; heliocentric velocity ≤ 3000 km s −1 ; morphological Hubble type T ≤ −1. The latter is a similar southern sample of 116 E and S0 galaxies with declination −45 ≤ δ ≤ −32. Both were observed at 5 GHz with a flux limit of ∼ 1 mJy.\n\nThe 65 objects with available archival HST images were classified as core or power-law galaxies on the basis of the slopes of their nuclear brightness profile as obtained by fitting a broken power law (the so-called \"Nuker law\"; Lauer et al. 1995;CB05;CB06). For the purpose of this study, we focus on the core galaxies listed in Table 1 of BC06 and impose two additional selection criteria, based on visual inspection of HST images: (i) the presence of an optically bright central point-like source (a low luminosity AGN); (ii) the absence of heavy nuclear obscuration and other photometric irregularities.\n\nThe 14 galaxies selected for this study are listed in Table 1 together with their distances, core radii and details of the archival data sets that were retrieved for photometric analysis. The sample includes two galaxies (NGC 4696 and 5419) that exhibit double nuclei, separated by ∼ 0. 25. In these cases, it is not known which of the two point sources is the AGN and therefore we assume that the brightest is the AGN (although we have measured displacements relative to both).\n\nThe radio source properties, including the total power at 5 GHz and the position angle of any extended structure, are summarized in Table 2.\n\n\nDATA ANALYSIS\n\nBroad-band images of the sample galaxies acquired with ACS, NICMOS2, WFPC2 and WFC3 were retrieved from the Hubble Legacy Archive (HLA), giving preference to images obtained with the highest spatial resolutions and red filters (i.e., F606W and longer wavelengths).\n\nWhenever available, several images taken with different instruments and filters, covering both optical and NIR bands, were obtained for each galaxy. A list of the images analyzed for each galaxy is given in Table 1. No additional processing steps were applied beyond the standard HLA reduction pipeline. As the goal of our analysis is simply to determine the relative positions of the isophotal center of the inner few kiloparsecs of the galaxy and the AGN point source within each individual image, this is sufficient for our purposes.\n\nEach image was analyzed according to the following main steps: (1) a mask was constructed to block image defects and distorting features such as dust lanes, jets or bright stars and globular clusters; (2) elliptical isophotes were fitted to the galaxy surface brightness distribution and the photocenter computed as the flux-weighted average of the isophote centers; (3) the position of the point source (assumed to locate the SBH) was determined by fitting a gaussian profile. See §3.1 and §3.2 for a more detailed description of these procedures. Sky subtraction was not performed as a uniform background will not affect the photometric analysis and in many cases it is not possible to measure the sky background as the galaxy covers the frame. (2) name as it appears in BC06 if different than the name given in the first column; (3) adopted distance in Mpc (from the HyperLeda extragalactic database); (4) method used to determine the distance: \"sbf\", surface brightness fluctuation; \"vvir\", from radial velocity, corrected for Local Group infall into Virgo; \"Dn − σ\", size-sigma relation (Dressler et al. 1987); (5) core radius as derived by CB05, in arcsec (pc); (6) SBH influence radius (GM•/σ 2 ) in arcsec (pc). Central velocity dispersions are given in Table 8. Black hole masses have been derived from the M• − σ relation given in Ferrarese & Ford (2005); (7) HST instrument/camera/filter used; (8) pixel scale in arcseconds/pixels as determined from direct measurement on the frames; (9) proposal identification number. B10 : this is a combined image, see Table 1 in B10 for further details. † A WFC3/IR image is available for NGC 4696, however the complex nuclear features and the pixel size of the camera do not allow us to infer the SBH position with the precision necessary for this work.\n\n\nThe inner photocenter\n\nEach image was analyzed with the IRAF task ellipse (Jedrzejewski 1987), which was used to fit elliptical isophotes to the surface brightness distribution. The first step was to construct an image mask in order to minimize distortions due to image defects and intrinsic photometric irregularities such as dust features, optical or NIR knots associated with jets, globular clusters, foreground stars, etc. If an exposure time map (the \"weight image\") was provided in the retrieved data set, this was used as an initial mask to eliminate cosmic rays, bad pixels, null areas and other image frame blemishes. Applying the initial mask, a first run of ellipse was performed to create a zeroth-order photometric model. To account for intrinsic irregularities in the surface brightness distribution, we then created a second mask by subtracting the zeroth-order photometric model from the original image and masking residuals exceeding ±nσ, where σ is the standard deviation of the residual image and n is a clipping parameter.\n\nTo allow ellipse to find the photometric peak, the central region was unmasked and a new run performed. This process was iterated, adjusting n in order to map distorting features in the maximum possible detail. The iteration was terminated when the mask converged and further runs did not add any significant detail.\n\nOnce the final mask was constructed, ellipse was rerun with the mask applied to determine the photocenter of the galaxy. Elliptical isophotes were fitted between minimum and maximum values of the semi-major axis (SMA) which were usually determined by the core radius (r c ) and the image size, respectively. In almost all cases, the minimum SMA was set equal to or greater than the value of r c as determined by CB05; Table 1. This was typically ∼ 2 , or ∼ 300 pc. In some cases, where a dusty disk is present in the center of the galaxy, the initial SMA was chosen such that the inner ellipse is larger than the disk. The only exception was NGC 4486 (M 87), which has a very large core. In this case, we adopted the SMA range 1-3 used by B10, who found that isophotes within 1 are influenced by the AGN point source. The maximum SMA was that of the largest ellipse that completely fits within the image area -that is, only complete isophotes were fitted. This limits our isophotal analysis to the inner few kiloparsecs of each galaxy; the SMA of the outermost ellipse fit varies in the range 1 r 4 kpc.\n\nBetween these limits, a series of ellipses was fitted to the 2-D surface brightness distribution, with the SMA being increased by two pixels at each step. Each fit returned values of the position angle, eccentricity and the pixel x − y coordinates of the center of the ellipse. Fi- Note. -(1) Optical identification;\n\n(2) approximate length of the radio jet/extension; (3) position angle (degrees E of N); (4) total power at 5 GHz [10 20 W]; (5) comments on the radio source morphology; (6) comments on the nature of the galaxy. † This value is computed for the inner 25 pc. References: a Slee et al. (1994), b Wrobel & Heeschen (1991, c Wrobel (1991), d Sadler et al. (1989, e Shurkin et al. (2008), f Cavagnolo et al. (2010), g Giroletti et al. (2005), h Baade & Minkowski (1954), i Nagar et al. (2002), l Stanger & Warwick (1986), m Taylor et al. (2006), n Killeen et al. (1986a), p Jones et al. (2000. [HG82]: Huchra & Geller (1982).\n\nnally, the photocenter was obtained as the flux-weighted average of the centers of the elliptical annuli generated by ellipse:\n(x pc , y pc ) = i x i j i i j i , i y i j i i j i(1)\nwhere x i and y i are the photocenter coordinates of the ith isophote and j i is a weight, given by the product of the area of the annulus defined by neighbouring isophotes, with the mean intensity within the annulus.\n\nTo assess the sensitivity of the mean photocenter to the weighting method, we computed an alternative weight function using the flux enclosed within each successive elliptical isophote. For comparison, the two weight functions are plotted as a function of SMA in Fig.1, for the NICMOS image of NGC 4261. The annular flux method produces an approximately uniform weight distribution beyond the innermost region (SMA < 0. 5), but typically decreases slowly at large SMA values. Not surprisingly, the enclosed flux method produces weights that increase monotonically with SMA, thus assigning much greater weight to the outer, fainter isophotes. This may be an undesirable characteristic in that isophotal distortions which may be present at fainter surface brightness levels (due to interactions with nearby companions, for example) might unduly influence the mean photocenter. Nevertheless, for the sources in our sample, the photocenters computed using the two methods are consistent within the errors in all cases. This result reflects the rather regular nature of the sample galaxies at the wavelengths under consideration.\n\nWe explored alternative methods for the measurement of the photocenter position, namely a moment based technique and a 2-D decomposition of the surface brightness distribution. In the first case, the photocenter position is obtained by weighting the coordinate of each pixel by its own intensity. This method seems to be particularly sensitive to the masking details and to the presence of asymmetries in the light distribution. The second ap-proach, performed with the GALFIT surface brightness fitting program (Peng et al. 2002), requires, at least in some cases, multiple components for a successful fit (e.g. two or more Sérsic profiles) casting doubt on the physical meaning of the recovered photocenter and the errors associated with it.\n\nThe IRAF task ellipse was specifically built to fit the light profile of elliptical galaxies with minimal assumptions; the results obtained seem to be weakly dependent on the masking details and errors associated with the fitted parameters have been thoroughly studied (Busko 1996). Therefore, we prefer to adopt the photocenter position determined with this approach.\n\n\nThe SBH position\n\nWe make the key assumption that the point-like source near the center of each galaxy is an AGN, which therefore locates the position of the SBH (see §5 for a detailed discussion of this assumption).\n\nThe AGN point sources are bright enough to be easily detected, but in our sample they are not so bright that the point spread function dominates the host galaxy. In order to determine the position of the SBH, we proceeded as follows. First, a median filtered version of the image was subtracted from the original producing a residual map in which the AGN is a prominent feature 3 . In the second step, we used the IRAF task center to fit gaussians to the marginal distributions of the residual point source along the x and y axes.\n\n\nThe photocenter-AGN displacement\n\nFor each image of each galaxy, the x and y components of the displacement were measured as the difference in pixels between the x and y co-ordinates of the photocenter and the AGN point source. The significance of the displacement is classified in two steps, considering the x and y components separately. Displacements smaller than 3σ, where σ is the error on the displacement (Section 3.4.1), are considered non-significant. If the displacement exceeds 3σ we make a further classification based on the distribution of the isophote center coordinates produced by ellipse. For each coordinate, x and y, the inter-quartile range (IQR, the difference between the upper and lower quartiles) of the isophote centers was computed. The corresponding displacement is then assigned significance levels of \"null\", \"low\", \"intermediate\" or \"high\" depending on its magnitude relative to the IQR:\n\n• null, ∆x, ∆y < 0.8 IQR • low, 0.8 IQR ≤ ∆x, ∆y < 1.6 IQR\n• intermediate, 1.6 IQR ≤ ∆x, ∆y < 2.4 IQR • high, ∆x, ∆y ≥ 2.4 IQR.\nThe threshold values are based on the fact that, for a normal distribution, 0.8 IQR is equivalent to 1σ. For an ideal, symmetric galaxy the isophotes would be concentric, giving IQR = 0. Any irregularities in the surface brightness distribution will cause a dispersion in the centers of the fitted elliptical isophotes, resulting in a nonzero IQR. Hence, by normalizing to the IQR, we allow for uncertainties in the recovered offsets due to asymmetric surface brightness distributions arising from, for example, dust lanes or tidal distortions.\n\n\nErrors and biases\n\n\nPosition uncertainties\n\nThe final error on the measured projected displacement is the combination of the uncertainties in the positions of the photocenter and the AGN point source. As the photocenter is determined by fitting elliptical isophotes, it is important to understand if the errors provided by ellipse are reasonable and robust. To investigate this, we used ellipse to analyze synthetic galaxy images constructed with GALFIT. We also used these simulated galaxies to estimate the uncertainties on both the AGN and photocenter positions.\n\nThe model images were constructed by fitting a Sérsic profile and a nuclear point source (a PSF generated with Tiny Tim, Krist 1993) to selected galaxies in our sample. We used NGC 4261 and 4278 to build models of NIC-MOS images, IC 1459 for WFPC2 images, NGC 1399 for WFC3 and IC 4296 for ACS images. The resulting model surface brightness distributions were populated with gaussian random noise so as to match the signalto-noise ratio (SN) of the original galaxy image 4 . Four realizations of each synthetic galaxy image were generated and each realization was analyzed using the methods described in Sections 3.1 and 3.2, superposing the masks obtained from the corresponding real galaxy images.\n\nEllipse errors: for each isophote fit, the \"true error\" on the isophote center is computed as the difference between the position returned by ellipse and the known photocenter of the synthetic galaxy (i.e., the center position of the Sérsic component fitted by Galfit). These \"true errors\" are compared to the errors on the positions of the isophote centers returned by ellipse in Fig.2. The \"true\" error distribution is slightly broader than that of the ellipse errors, but both distributions are highly peaked, with ∼ 50% of fits producing errors ≤ 0.1 pixels. A detailed study by Busko (1996) of the errors returned by ellipse shows that errors on ellipticity, position angle and center position are unbiased and accurate when the radial gradient relative error 5 is less than 50%. This condition is satisfied for our sample galaxies.\n\nPosition errors: the \"true errors\" on the (x,y) coordinates of the AGN and galaxy photocenter positions were also determined for each realization of the synthetic galaxy images, where the true error in each case is defined as the difference between the known positions of the AGN and photocenter (i.e., the positions returned by Galfit) and the values recovered using our analysis methods.\n\nFor signal-to-noise ratios characteristic of our data (S/N ∼ 100), the distributions of the \"true errors\" on the AGN and photocenter positions derived from both the ACS and WFPC2 synthetic galaxy images are narrow and confined within 0.1 pixel. We therefore adopt 0.1 pixel as a conservative estimate of the uncertainty on both the AGN and photocenter positions derived from ACS and WFPC2 data. The \"true error\" obtained from the WFC3 and NICMOS2 images exhibit broader distributions, with tails extending to 0.4 pixels. We therefore adopt the median values of the \"true errors\" as the un- Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \"true errors\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC. certainties for WFC3 and NICMOS2 data, yielding 0.2 pixels for the photocenter position in both cameras and respectively 0.1 and 0.2 pixels for the AGN.\n\nThese uncertainties represent the precision with which we can determine the AGN and photocenter positions for an ideal galaxy, in an image with noise characteristics representative of our data. The adopted uncertainties are summarized in Table 3, together with the equivalent angular distances and linear distances corresponding to the closest and furthest of our sample galaxies.\n\nMinimum detectable displacement: the uncertainties on the AGN and photocenter coordinates determine the minimum detectable displacement. We require a 3σ detection for a displacement to be considered significant. Combining the uncertainties on the photocenter and AGN in quadrature, the error on each component of the displacement is σ = 0.14 pxl, for ACS and WFPC2 data, σ = 0.22 pxl for WFC3, and σ = 0.28 pxl for NICMOS2 data. Therefore, the minimum displacements considered significant are ∆x = ∆y ≈ 0.4 pxl, for ACS and WFPC2, ∆x = ∆y ≈ 0.7 pxl for WFPC3, and ∆x = ∆y ≈ 0.9 pxl for NICMOS2.\n\n\nPossible biases\n\nAsymmetric surface brightness distributions: the galaxies in our sample were selected to be symmetric and regular, based on visual inspection of the images. However, in some cases, the galaxy extends beyond the edges of the frame and/or is not centered in the image frame. To eliminate the possibility of spurious offsets due to truncation of isophotes by the frame edge, the upper limit on the SMA range used in the surface brightness fits is taken to be that of the largest ellipse that fits completely within the frame (Sec. 3.1).\n\nIn addition, subtle intrinsic asymmetries might be present that were not revealed by visual inspection. As discussed by Jedrzejewski (1987), asymmetric surface brightness distributions (such as large-scale lopsidedness in the isophotes) will cause shifts in the isophote centers, which in turn will increase the IQR. Therefore, intrinsic surface brightness asymmetries (or other irregularities) will tend to reduce the level of significance assigned to any measured displacement (Sec. 3.3).\n\nA similar possibility is that the outer isophotes are distorted by tidal interactions with nearby galaxies. As our analysis is confined to the inner few kiloparsecs, such interactions are unlikely to significantly affect our results. Nevertheless, we have verified that the photocenter position does not change significantly when recomputed excluding isophote centers corresponding to ellipse SMAs greater than ∼ 85, 90 and 95% of that of the largest complete ellipse that fits within the image frame.\n\nIsophote twists and PSF effects: core galaxies are often characterized by strong isophote twisting at radii interior to the core radius, r c (Lauer et al. 2005). In addition, the inner isophotes are distorted by the point spread function; TINY TIM simulations show that for the HST instruments used, the PSF can affect the region within a radius r PSF ∼ 15 pixels. As typically r PSF r c , we mitigate both effects by setting r c as the minimum SMA for the ellipse fits.\n\nLopsided stellar nuclei (LSN): observations of the nuclear regions of nearby galaxies have revealed the presence of double nuclei with separations in the range 1-10 pc (Lauer et al. 1993(Lauer et al. , 1996(Lauer et al. , 2005Thatte et al. 2000;Debattista et al. 2006). One explanation for such configurations is that lopsided stellar orbit distributions tend to persist within the SBH sphere of gravitational influence since orbits do not precess in Keplerian potentials and are not, therefore, axisymmetrized by phase mixing (Peiris & Tremaine 2003). The best studied example is M31, in which the components of the double nucleus are separated by ≈ 0. 5 (≈ 2 pc). Peiris & Tremaine (2003) proposed a model for M31 in which the nucleus consists of an eccentric disk of stars orbiting the SBH, the latter being coincident with the fainter component.\n\nThe WFPC2 Planetary Camera (with which most of the images studied here were obtained) resolves spatial separations of ≈ 5 pc only for galaxies within 20 Mpc. It is possible, therefore, that some of the galaxies in our sample harbor unresolved double nuclei. This will not affect the determination of the galaxy photocenter, since the core is excluded from the isophote fitting. However, if the unresolved nuclei are of unequal brightness as in M31, this could result in a systemic error in the AGN position which may approach the PSF FWHM (∼ 1 − 2 pixels). This in turn would result in a spurious displacement.\n\n\nRESULTS\n\nThe results of the photometric analysis are presented in Table 4, where we list the (x, y) pixel co-ordinates of the mean photocenter and the point source, the magnitudes of the corresponding components of the displacement, the direction of each displacement component on the sky and the significance level of the displacement, determined as outlined in Sec. 3.3. Figures 3 and 4 illustrate examples of the results for galaxies in which, respectively, no significant displacement was found (NGC4373, Fig. 3) and in which the measured displacement is considered significant (NGC 4486,Fig. 4). In each figure, the top row of panels shows the original image and the residual images after subtraction of, respectively, the isophotal model generated by ellipse fits and a median filtered image. In the middle row are plotted the surface brightness profile and the x and y pixel coordinates of the isophotal centers, all as functions of the ellipse SMA. The positions of the photocenter and the AGN point source are also plotted. The leftmost panel of the bottom row is a scatter plot showing the distribution of the isophote centers, colour coded according to SMA. The locus of the cumulative mean photocenter position, computed outwards from the inner SMA limit, is also shown as a solid black line. The last two panels show the distributions of the (x, y) isophote center co-ordinates. The co-ordinates of the mean photocenter and the AGN point source are also plotted in all three panels. Similar figures for the remaining galaxies are presented in the Appendix (Sec. C), where the results for individual galaxies are also described in more detail (Sec. B).\n\nIn the case of NGC 4373, the x and y components of the displacement are < 1σ and thus consistent with zero. In the case of the ACS/HRC F814W image of NGC 4486, both the x and y components of the displacement are significant at the > 3σ level and their magnitudes are ≥ 2.4×IQR and accordingly classified as having a \"high\" significance level (Table 4). For four galaxies, two or more images obtained with different instruments and/or filters were analyzed, providing various combinations of wavelength and spatial resolution. The photocenter displacements in milliarcseconds, relative to the AGN point source, are plotted for these cases in Fig. 5a. There are two galaxies, NGC 4696 and 5419, in which a second point-like brightness peak is present. In these cases, the following discussion refers to the displacement relative to the primary point source (the brightest, which we take to be the AGN), unless otherwise noted.\n\nAGN-photocenter displacements exceeding the minimum detectable value (i.e., significant at 3σ level; Sec. 3.4.1) were measured in at least one direction (x or y), in at least one image, for ten out of the 14 galaxies in the sample: NGC 1399, 4168, 4278, 4486 (M87), 4636, 4696, 5419, 5846, IC 4296, and IC 4931. In the remaining galaxies, NGC 4261, 4373, 4552, and IC 1459, the AGN and photocenter are coincident within the position uncertainties (< 3σ).\n\nThe distribution of the isophote center co-ordinates provides an indication of the systematic uncertainty arising from photometric distortions due to effects such as those outlined in Sec. 3.4.2. As discussed in Sec. 3.3, we assign a significance level for each measured displacement based on its magnitude relative to the IQR, which characterizes the width of the isophote center distribution. Three galaxies exhibit displacements (in at least one direction, in one image) ∆x, ∆y ≥ 1.6×IQR, that we classify as having intermediate or high significance. These are NGC 4278, 4486 (M87) and 5846. Three more have displacements classified as having low significance: NGC 1399, 5419 and IC 4296 (∆x, ∆y ≥ 0.8×IQR).\n\nIn the two double nucleus galaxies, the photocenter position is much closer to the primary point source than the secondary. The offsets (relative to the primary) are ≈ 0.5 pxl, but given the large IQR in NGC 4696 the displacement is classified as not significant in this case. In neither galaxy does the photocenter lie between the two \"nuclei\" (Fig. 5b, 25 and 26).\n\nFor the galaxies where multiple images were analyzed, the displacements in mas derived from the different images are consistent to within 2σ for two of the four galaxies (NGC 1399 and 4278).\n\nIn the remaining two galaxies (NGC 4486 and IC 4296), the results from one or more images differ significantly from the others (Fig. 5a). In IC 4296, the offset measured from the red (F814W) WFPC2 image is significantly different from those measured from the ACS-F625W and NICMOS2 F160W images, which are consistent to within 2σ. This galaxy has a warped dusty disk oriented approximately E-W, extending ≈ 1 either side of the nucleus but it seems unlikely that this structure is the cause of the discrepancy, the origin of which remains unclear.\n\nSeveral ACS, NICMOS2 and WFPC2 images were analyzed for NGC 4486 (M87). For the ACS red optical image (F814W) we recover the B10 result, which indicates that the photocenter is displaced by ≈ 100 mas to the north-west of the nucleus. A smaller but less significant offset in approximately the same direction is measured from the WFPC2 F814W image. However, the photocenters derived from the NICMOS2 F110W, F160M and F222W image are consistent with the AGN position. As discussed in detail in Appendix B, the origin of these differences is unclear. Neither dust, nor the prominent optical jet (also visible in the NIR) appear to provide satisfactory explanations. The central region of the galaxy where the isophote fitting was performed (1 ≤ r ≤ 3 ) appears to be free of large scale dust features. The masking procedure should prevent the jet from distorting the isophote fits but in any case, even when the fits were repeated with different levels of masking and even with no mask at all, no significant differences in the mean photocenter position were found. As there is no compelling reason to favor or discard the results from any given image, we compute a weighted average displacement.\n\nWhether classified as significant, or not, the measured angular photocenter displacements are always small, typically a few×10 mas and almost without exception 100 mas. The only cases that exceed 100 mas are the displacements relative to the secondary point sources in the two double nucleus galaxies. Disregarding these, the projected linear displacements are all 10 pc. Therefore, for every galaxy analyzed, the measured displacement is a small fraction of the galaxy core radius; as a fraction of r c , the range in displacement magnitude is approximately 1-10%.\n\nIn our previous analysis of NGC 4486 (B10), we found that the galaxy photocenter (in ACS and WFPC images) is displaced in approximately the same direction as the jet, implying that the SBH is displaced in the counter-jet direction. It is therefore of interest to compare the direction of the measured photocenter displacements with the radio source axis for this sample. In Table 5 we give the position angle (PA) of the radio source, for those galaxies in which jets or jet-like extended structures have been observed, along with the PA of the measured displacement (as derived from the x, y components). Five galaxies are associated with relatively powerful, extended (∼ 10 kpc) radio sources: NGC 1399, 4261, 4486, 4696 and IC 4296 6 . Of these, NGC 4486 (M 87; 3C 274), NGC 4261 (3C 270) and IC 4296 (PKS 1333-33) are Fanaroff-Riley Type I (FRI) radio sources, characterized by prominent twin jets ending in kiloparsec-scale lobes (Fanaroff & Riley 1974). All three also have parsec-scale jets approximately aligned with the kpc-scale structure.\n\nIn NGC 4486, the displacement PA derived from our reanalysis of the ACS F814W image agrees with the B10 result. The direction of the weighted mean displacement, derived from displacements obtained from all the images, is also closely aligned with the jet direction. We find a similar result for IC 4296. As already noted, the results from the ACS and NICMOS2 images are consistent and indicate a displacement to the NW. If the discrepant WFPC2 result is disregarded, the weighted mean of the ACS and NICMOS2 displacements yields a PA only ∼ 4 • different from that of the large-scale radio axis, with the photocenter displaced on the side of the brighter north-western jet.\n\nThe displacement measured in NGC 4261 is not considered to be significant (< 3σ). Nevertheless, the derived PA is consistent (albeit within the large uncertainty) with that of the radio axis, and again, the offset is in the direction of the brighter western jet.\n\nNGC 1399 also has an FRI-like radio source morphology with a well-defined axis in PA≈ −10 • . The measured displacement (classified as low significance) has a consistent PA with a value of −17 ± 16 • . NGC 4696, is one of the galaxies with two optical point sources. Its large scale radio source does not exhibit well-defined jets, but is elongated approximately E-W over ∼ 10 kpc, with the ends of both \"arms\" bending south. At parsec scales there is a compact core with a one-sided jet emerging to the SW in PA∼ −150 • . However, it is not clear which of the two optical nuclei hosts the core radio source (Taylor et al. 2006), complicating the comparison with the photocenter displacement. If the secondary point source is identified as the AGN, then the photocenter, which is close to the primary point source, is displaced approximately in the counter-jet direction. On the other hand, if the primary point source is indeed the AGN, the displacement (which, in any case, is classified as having \"null\" significance) would be almost perpendicular to the jet. Three more galaxies have relatively weak, small radio sources that have jet-like features or elongations on subkiloparsec scales. The measured photocenter displacements are not aligned with the radio source axis in NGC 4552 or NGC 4636. NGC 4278 has a compact (∼ 3 pc) source consisting of a core with jet-like features emerging along a SE-NW axis on either side. These features gradually bend to the east and west, respectively, becoming fainter and more diffuse. The weighted mean photocenter position is displaced approximately in the initial direction of the SE jet. According to Giroletti et al. (2005)'s analysis of this source, the jet axis is closely aligned with our line of sight, with the SE component being the oppositely directed counter-jet.\n\n\nDISCUSSION\n\nIn this section we will examine our key assumption that the SBH position is marked by the nuclear point source. We will then discuss the possible origins of the observed offsets. Most of the displacement mechanisms considered below have been previously discussed in B10. Nevertheless, we summarize them here in order to provide context for our new results.\n\n5.1. The AGN nature of the nuclear point sources Our sample of 14 core elliptical galaxies was selected from the 26 (out of 29) studied by BC06 that were identified as containing AGN based on the presence of i) an unresolved optical or X-ray source; ii) an \"AGN-like\" optical spectrum, or iii) radio jets. Most of our sample of 14 exhibit two or more AGN signatures in addition to the presence of a radio source and optical point source nucleus. Optical or UV variability has been detected in HST observations of NGC 4486 (Perlman et al. 2003 . Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers. (NGC 4278,4636 and NGC 4168), or emission line ratios indicative of AGN photoionization (NGC 4261,4486,4552,5846) have also been detected in several galaxies (Ho et al. 1997). In addition, the radio sources in most objects feature compact ( 1 pc) cores (NGC 1399, 4168, 4373, 4552, 5419, IC 1459) and/or parsec or kpc-scale jets (NGC 4261,4278,4486,4696,IC 4296; see Appendix B for references). Only IC 4931, which by comparison has been relatively little studied, lacks supporting evidence indicating the presence of an AGN in the form of X-ray, radio or line emission.\n\nThere is, therefore, plenty of evidence that these galaxies (with the possible exception of IC 4931) host AGNs. However, it does not necessarily follow (except in cases where optical/UV variability has been observed) that the optical or NIR point sources are themselves manifestations of the AGN, rather than, for example, nuclear star clusters. Low luminosity radio loud AGNs are thought to be powered by radiatively inefficient accretion flows (see, for example, Ho 2008; Balmaverde et al. 2008, and references therein), with most of the accretion power channeled into the kinetic energy of the radio jets. Detailed studies of NGC4486 (Di Matteo et al. 2003), IC 1459 (Fabbiano et al. 2003) and IC 4296 (Pellegrini et al. 2003) show that this is the case for at least three galaxies of our sample.\n\nIn low luminosity radio galaxies, the optical, near infrared and X-ray luminosities of the nuclei correlate tightly with the core radio luminosity, implying a common origin in non-thermal emission from the jet (Chiaberge et al. 1999;Capetti & Balmaverde 2005;Baldi et al. 2010). This is supported by polarization measurements: Capetti et al. (2007) found that the optical nuclei of the nine nearest FR I radio galaxies in the 3C catalogue, including two galaxies from our sample, NGC 4486 (M87, 3C 274) and NGC 4261 (3C 270), have high polarizations (2 − 11%), which they attribute to synchrotron emission. The optical or UV variability of the point sources in NGC 4486 and three other galaxies, noted above, is also consistent with jet synchrotron emission. More generally, BC06 found that their sample of core elliptical galaxies, from which this sample is drawn, also exhibit optical-radio and X-ray-radio correlations and indeed form a continuous distribution with the radio galaxies, extending these correlations to lower luminosities. Therefore, there is direct evidence, in the form of optical polarization and/or optical/UV variability that the nuclear sources in five of our sample galaxies are produced by synchrotron emission associated with the radio core source, presumably the base of the jet. It is reasonable to assume, given the correlations found by BC06, that the same is true for the rest of the sample.\n\nThe base of the jet is, in turn, very close to the SBH. Arguably the best studied jet is that of M87. Using multifrequency observations made with the Very Long Baseline Array, Hada et al. (2011) were able to show that the radio core at 43 GHz is located within 14-23 Schwarzchild radii of the SBH and accretion flow. In the optical, the lower optical depth should move the peak of the emission even closer to the SBH.\n\nIn summary, we contend that there are good reasons to believe that the optical or near-infrared point sources in our galaxies reveal the position of the SBH.\n\n\nRecoiling SBHs\n\nHere we consider if the measured SBH displacements are consistent with residual oscillations due to gravitational recoils generated by coalescence of an SBH binary. GM08 used N -body simulations to study the post recoil dynamical evolution of an SBH in representative core elliptical galaxy potentials. They found that when the recoil velocity is sufficient to eject the SBH from the core (v kick in the range ∼ 40 − 90% of the escape velocity, v esc ), the subsequent motion is characterized by three phases. The initial large-amplitide (>r c ) oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction (phase I). When the amplitude becomes comparable with the core radius, r c , the SBH and the stellar core undergo long-lived oscillations about their center of mass, which persist for ∼ 1 Gyr (phase II). Finally, the SBH reaches thermal equilibrium with the stars, experiencing low-amplitude Brownian motion (phase III).\n\nAs our measured displacements are 0.1 r c , it is highly unlikely that they represent phase I oscillations. This would require either fortuitous timing (the SBH would have to be caught whilst passing through or close to the equilibrium position) and/or orientation (the oscillation direction would have to be closely aligned with the line of sight) for each galaxy. We also rule out phase III Brownian motion due to interactions with individual stars, which will produce negligibly small amplitudes in real galaxies (GM08). The possibility of Brownian oscillation due to interactions with massive perturbers is discussed in Section 5.3. Here, we focus on phase II which, as already noted, is characterized by long-lived damped oscillations at amplitudes < r c .\n\nIn this phase, the characteristic damping time is given by GM08 as:\nτ ≈ 15 σ 3 G 2 ρM • ≈ 2 × 10 9 σ −3.86 300 r 2 500 yr (2)\nwhere the approximation on the right-hand side makes use of the M • -σ relation (Ferrarese & Ford 2005), σ being the 1D stellar velocity dispersion, with r 500 ≡ r c /(500 pc) and σ 300 ≡ σ/300 km s −1 . For our galaxies, eq. 2 yields values of τ in the range 0.02 − 4.85 Gyr, with an average of 1.2 Gyr (Table 8). The rms amplitude of the SBH motion with respect to the galaxy center is expected to evolve as:\nr rms (t) ≈ r c e −(t−tc)/2τ , t > t c(3)\nwhere t c is the time at which the oscillation amplitude has decayed to a scale comparable to the core radius r c (B10). Simulations indicate that the galaxy merger rate is a strong function of redshift and galaxy mass (e.g., Fakhouri et al. 2010, H10). The merger rate for brightest cluster galaxies has recently been determined observationally by Lidman et al. (2013), who find ≈ 0.4 mergers per Gyr at z ∼ 1, implying a mean time between mergers of ≈ 2.5 Gyr 7 . If we take this as representative of our sample, equations 2 and 3 together suggest that a \"typical\" rms displacement (ignoring projection effects) is ∼ 0.2 r c . This suggests that larger displacements than were actually observed might be expected if due to post gravitational recoil oscillations.\n\nHowever, equation 3 describes the oscillation amplitude rather than the instantaneous displacement and moreover the above estimate does not account for projection effects, or the range in damping timescales characterizing our sample galaxies, or for variations in the merger rate. In order to investigate in more detail the likelihood of obtaining the observed displacements in our sample, if they result from post-recoil oscillations, we have constructed a simple Monte-Carlo simulation based on the GM08 N -body simulations. Details of the method are given in Appendix A. However, we start from the assumption that, after any kick large enough to move the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy is given by:\nR(t) = R 0 e −∆t/τ sin(ω c ∆t)(4)\nwhere ∆t is the elapsed time since the kick, i.e., the time since the last merger, τ is the damping time given by eq.2 and ω c is the SBH oscillation frequency calculated from eq. A2. We set R 0 = r c , since phase II begins when the oscillation amplitude is roughly equal to the core radius. We then suppose that each galaxy is observed at a random time, ∆t, since its last merger, where the probability that the merger occurred at time ∆t follows an exponential distribution characterized by a mean timebetween-mergers, t m . This distribution is sampled to generate corresponding values of R i ≡ R(t i ), which are then projected onto the sky plane assuming that the recoil kicks have random directions. The distribution of projected displacement for each galaxy is then used to compute the probability, p i , of observing a displacement larger than that actually measured, given a kick of sufficient magnitude. These probabilities were computed for two values of the mean time between mergers: t m = 5.0 Gyr and t m = 0.4 Gyr, which were derived from the galaxy merger rate models of H10 for redshifts z < 1, mass ratios q > 0.1 and galaxy masses log(M gal /M ) > 11 and log(M gal /M ) > 12, respectively (see Appendix A for details). The two values of t m are representative of the mass range covered by our sample, and illustrate the effect on the displacement probabilities of the strong galaxy mass dependence of the merger rate.\n\nThe probabilities, p i , are listed for each galaxy in Table 6 for both values of t m . The probabilities P (d > x) of observing a projected displacement d exceeding distance\n\nx are plotted as functions of x in Figures 7a and 7b, for each galaxy, along with the observed displacement.\n\nAs would be expected, the smaller value of t m results in larger values of p i ; clearly if t m τ , there is a greater chance of observing a large displacement than when t m >> τ . For t m = 0.4 Gyr, p i > 0.5 for all but two galaxies (NGC 4278 and NGC 4552) and p i ≥ 0.95 for all 6 galaxies that have masses ≥ 10 12 M (for which assumed fraction of mass accreted by a BCG during a merger. Therefore, this merger rate relies on model-dependent assumptions about the dark-matter haloes surrounding the galaxies.\n\nthis value of t m is presumably most relevant). On the other hand, for t m = 5.0 Gyr, all but four galaxies have p i < 0.5, including five of the six having displacements considered significant at the level of > 0.8 IQR (the exception is M87, for which p i = 0.75).\n\nConsidering the sample as a whole, the probability of not observing a displacement larger than those actually measured in any of the galaxies in the sample is simply P = i (1 − p i ). For t m = 0.4 Gyr, this is negligibly small, P ∼ 10 −17 . Even for t m = 5.0 Gyr, P ≈ 3 × 10 −4 , indicating that it is statistically unlikely that larger displacements were not observed, given the occurrence of a recoil kick sufficient to trigger phase II oscillations.\n\nHowever, it is unlikely that all the galaxies in the sample experienced a recoil kick big enough to move the SBH beyond the core radius. Assuming that the potential experienced by the SBH can be approximated as a harmonic potential (φ = 1/2 ω 2 x 2 ), we estimated the kick velocity required to displace the SBH out to the core radius for each galaxy, obtaining values in the range 159 v kick 325 km s −1 , with an average of v kick ≈ 240 km s −1 (the values for each galaxy are listed in Table 8).\n\nL12 have studied kick velocity distributions for \"hangup\" kicks, which arise in binary configurations where the SBH spins are partially aligned with the orbital angular momentum by pre-merger accretion. For spin magnitude and orientation distributions derived from accretion simulations and a mass ratio distribution based on galaxy merger studies (including both minor and major mergers), they generate probability distributions for the magnitude of the recoil velocity for two extreme cases of \"hot\" and \"cold\" gas disks (i.e., corresponding to adiabatic indices γ = 5/3 and 7/5, respectively). The former regime is likely to be most applicable to the ellipticals in our sample and in this case, probabilities range from p k ≈ 66% for kicks > 100 km s −1 to ≈ 33% for kicks > 300 km s −1 . Our Monte-Carlo simulation assumes that the coalescence event produces a recoil large enough to produce phase II oscillations (i.e., the values given in Table 8). As the probabilities p i are conditional on the occurrence of a kick of sufficient magnitude, the probability that both events will occur is p k × p i . There is thus a smaller probability of observing a displacement larger than actually measured for each galaxy. Using the kick probabilities given by L12, we find that the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 8 × 10 −4 for t m = 0.4 Gyr and P ≈ 5 × 10 −2 for t m = 5.0 Gyr.\n\nAs the merger rate increases with redshift, we may have underestimated the time-between-mergers by integrating over the redshift range z = 1 → 0. However, the higher value of t m used in the simulation implies a merger rate (∼ 0.2 mergers Gyr −1 ) consistent with that given by the models of H10 for log(M gal /M ) > 11 galaxies at z = 0 (see their Fig. 3). The rate for major mergers is a factor 2-3 lower than the rate for major and minor mergers combined, which we used to estimate t m . However, although the magnitude of the recoil velocity is a function of SBH mass ratio (with the largest recoils occurring for mass ratios characteristic of major mergers), even small (q > 0.1) mass ratios are capable of producing kicks ∼ 200 km s −1 , large enough to trigger phase II oscillations in our galaxies (Lousto et al. 2010). Mi- Note. -(1) Optical name;\n\n(2) instrument/camera/filter on HST;\n\n(3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) nuclear point source position in pixels; (7 -8) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (9) offset in pixel normalized by the corresponding IQR; (10) offset in pc; (11) direction of the offset; (12) type of displacement as defined in §3.3. : for further details see Table 1 in B10. nor mergers were, in any case, included in computing the kick velocity distribution presented by L12.\n\nThus, even allowing for the distribution of kick velocities, the Monte-Carlo simulation suggests that it is highly likely that displacements larger than those actually measured would have been observed in this sample if each galaxy has experienced at least one merger leading to an SBH-binary coalescence and gravitational recoil within the last few Gyr. In fact, our simulations suggest that t m 30 − 40 Gyr, corresponding to a merger rate ∼ 0.03 mergers Gyr −1 , is required for a > 50% chance of not observing larger displacements in the sample.\n\nIf, indeed, the formation of a binary SBH is an inevitable outcome of galaxy mergers, explanations must be sought in the evolution of the binary or that of the recoiling SBH. Possibilities include (1) the binary stalls before reaching the radius at which gravitational wave emission drives rapid coalescence; (2) if coalescence occurs, the mass ratio or spin-orbit properties of the binary are such as to typically preclude recoil velocities large enough to displace the SBH to the core radius ( 240 km s −1 ); or (3) the damping time for the recoil oscillations may be shorter than predicted by the pure N -body simulations.\n\nAlternatively, it is possible that galaxy mergers are much more infrequent than inferred from studies based on the current ΛCDM framework (see Kroupa 2014, and references therein for a detailed discussion).\n\nAlthough our results imply that SBH displacements due to gravitational recoil are much less common than might be inferred from theoretical considerations, this does not preclude the possibility that observed displacements in individual galaxies are due to recoil oscillations. Indeed, the p i values listed in Table 6 indicate that for t m = 5.0 Gyr, the chance of not observing a larger displacement, given a sufficiently large kick, is 30% for all objects having a displacement considered significant. Even for t m = 0.4 Gyr, there are still three of these galaxies for which 1 − p i ≥ 0.25. The likelihood that the recovered offset is due to SBH coalescence is discussed for individual galaxies in Appendix B.\n\nAssuming that the measured displacements are indeed due to phase II recoil oscillations, we have used eq. 3 to estimate the merger epoch for those galaxies with displacements rated at least as \"low significance\" in terms of the isophote centroid IQR. Allowing for the time taken for the SBH binary to form and subsequently coalesce in the center of the merged galaxy, and for phase I oscillations, the elapsed times since the merger are typically ∼ several Gyr, comparable with the mean time between galaxy mergers estimated here for galaxy masses log(M gal /M ) > 11, and consistent with the observationally determined merger rate for brightest cluster galaxies (Lidman et al. 2013).\n\nEq.2, used to estimate the SBH damping time, relies on the M • -σ relation (Ferrarese & Ford 2005). To test the dependence of our result on the value adopted for M • we use eq.6 in Lauer et al. (2007) to estimate SBH masses using the M • -M V relation (see Table 9 and Appendix A for more details). Using these alternative masses, and taking into account the kick velocity distribution computed by L12, we derive new values for P obtaining 1 × 10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5 Gyr. The likelihood of our result is somewhat increased for t m = 5 Gyr, however there is still a large probability of finding displacements larger than those measured, leaving our basic conclusion unchanged.\n\nNo useful constraints can be obtained on the merger epoch from consideration of the time necessary to replenish the stellar population of the core. This is roughly the relaxation time, T rel , at the influence radius of the central SBH, which can be estimated as:\n\nT r (r i ) ≈ 1.2 × 10 11 σ 7.47 100 yr (5) Merritt (2013, eq. 3.5). Given that the measured velocity dispersion is always > 100 km s −1 in the centers of our galaxies (see Table 8), it is clear that the replenishment time is much longer than the timescales involved in the SBH merger-recoil event (∼ 10 9 yr). In general, coalescence of an SBH-binary results in a re-orientation of the SBH spin axis, leading to a sudden flip in the direction of the associated radio jet (Merritt & Ekers 2002). Therefore, the radio source morphology may act as a \"signpost\" of SBH coalescence. However, powerful extended radio sources have lifetimes ∼ 10 6 −10 8 yr (e.g., Parma et al. 1999;O'Dea et al. 2009;Antognini, Bird & Martini 2012), much shorter than the time-between-mergers, even for massive galaxies. Thus even if a spin flip took place, morphological traces of this event, such as a change in jet direction, would not necessarily be evident in the current (post-coalescence) radio source. The radio source properties of individual galaxies are discussed in Appendix B.\n\nIt is notable that in four of the six galaxies in which significant displacements have been found, the displacements are also approximately aligned with the radio jet axis (Section 4 and Table 5). This number includes three of the four that have powerful kpc-scale jets (Section 4 and Table 5), suggesting jet power asymmetries as a possible displacement mechanism (as discussed below). However, such alignments do not necessarily argue against gravitational recoil. In their statistical study of the coalescence of spinning black hole binaries Lousto et al. (2010) find that the recoil velocity is preferentially aligned (or counter-aligned) with the orbital angular momentum of the progenitor binary and moreover, that the spin direction distribution of the recoiling SBH peaks at an angle of ≈ 25 • to the orbital angular momentum (for an equal mass progenitor binary; the probability distribution is broader and peaks at larger angles for smaller mass ratios). This suggests, assuming that the jet traces the spin axis of the recoiling SBH, a tendency for the displacement to be somewhat, but not greatly, misaligned with the jet axis, consistent with what is observed in these objects.\n\n\nOther displacement mechanisms\n\nWe now consider several other mechanisms that may plausibly produce SBH displacements.\n\n1. Asymmetric jets: if the AGN jets are intrinsically asymmetric in power output, the resulting net thrust can push the SBH away from the original equilibrium position (Shklovski 1982;Saslaw & Whittle 1988 Note. -(1) Optical name; (2) instrument/camera/filter on HST; (3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) and (7) nuclear point source position in pixels; (8 -9) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (10) offset in pixel normalized by the corresponding IQR; (11) offset in pc; (12) direction of the offset; (13) type of displacement as defined in §3.3. Table 4). a This value changes to −8 ± 16 • when only WFPC2 results are considered. b This value changes to 312 ± 8 • (132 ± 8 • ) when only ACS and NICMOS2 results are considered. See Table 2 for references on the radio jet PA. Note. -Probability to observe a projected displacement larger than the value actually observed in this work (in units of core radii) at a random time after the last kick. Probabilities are computed for a mean time between galactic mergers tm = 0.4 Gyr and 5 Gyr. The symbol \" * \" indicates 3σ offsets that were classified as \"null\" after the normalization for the co-ordinates IQR. The symbol \"<\" indicates offsets that do not reach the 3σ level.\n\nthis scenario:\na • ≈ 2.1 × 10 −6 f jetṁ cm s −2(6)\nwhereṁ ≡Ṁ a /Ṁ edd and f jet = L jet /L a 8 ,Ṁ a is the mass accretion rate,Ṁ edd ≈ 2.2 M • /10 8 M M yr −1 the Eddington accretion rate, L a the accretion luminosity and L jet is the jet luminosity. For asymmetrical, oppositely directed jets L jet is the difference in the luminosities of the two jets.\n\nUnder the assumption that the restoring force from the galaxy is negligible (a reasonable approximation for the low-density cores of \"core galaxies\"), Eq. 6 can be integrated to obtain an expression for the displacement:\n\n∆r ≈ 340 f jetṁ t 2 6 pc (7) where t 6 is the time over which the SBH is accelerated to produce an offset ∆r.\n\nThe best candidates for jet thrust displacements are the four galaxies (NGC 1399, 4261, 4486 and IC 4296) that have relatively powerful kpc-scale jets (Section 4 and Table 2). Of these, we did not detect a significant displacement in NGC 4261, which has an FR I radio source. NGC 4486 and IC 4296 also host FR I radio sources and NGC 1399 has an FR I-like morphology, despite its relatively low power. Interestingly, all three of these galaxies exhibit photocenter displacements that are at least approximately (within 20 • ) in the jet direction (Table 5). This implies that the SBH is displaced relative to its equilibrium position in the counter-jet direction, as might be expected if the displacements are related to intrinsic asymmetries in jet power. NGC 4486 (M87) has already been discussed in detail by B10, who concluded that a jet power asymmetry amounting to ∼ 3% of the accretion luminosity can explain the observed displacement, for a radio source lifetime ∼ 10 8 yr. We find similar results in the cases of NGC 1399 and IC 4296, where the observed displacement can be produced for a jet asymmetry < 1% of the Bondi accretion luminosity, again for a radio source lifetime ∼ 10 8 yr (See Appendix B for details).\n\nA close alignment between the displacement and the (initial) jet direction is also found for the low power pcscale radio source in NGC 4278. In this case, Giroletti et al. (2005)'s interpretation of the radio data implies that the photocenter is displaced in the counter-jet direction, with the jet axis being closely aligned with the line of sight. Assuming that the SBH is displaced along the jet axis, the de-projected magnitude of the displacement would be ∼ 100 − 200 pc. Nevertheless, it seems possible that this could result from jet thrust, if sustained for ∼ 10 8 yr (Appendix B).\n\nThe double nucleus galaxy NGC 4696 does not show well-defined jets on kpc scales, but has a one-sided pcscale jet in PA≈ −150 • . The displacement relative to the brighter nucleus is considered non-significant because of the large IQR. However, as already noted, it is not known which of the two optical nuclei hosts the AGN producing the radio jet. If it is the fainter nucleus, the photocenter displacement is approximately in the counter-jet direction, which is not consistent with jet acceleration of the SBH (which would cause the SBH to be displaced in the counter-jet direction).\n\nThe remaining two objects which exhibit displacements considered to be significant (NGC 5419 and 5846) do not have well-defined jets on either parsec or kiloparsec scales.\n\n2. Stalled SBH binaries: in the aftermath of a galaxy-galaxy merger, the SBH binary orbit shrinks at first due to dynamical friction and subsequently through slingshot ejection of stars intersecting the orbit. Investigations of quasi-steady spherical models suggested that the evolution of the binary stalls at separations ∼ 1 pc, due to a paucity of interacting stars, rather than hardening to the point at which gravitational wave emission drives the final inspiral to coalescence (the so-called \"final parsec problem\"; e.g., Merritt & Milosavljević 2005). Based on N -body simulations, Merritt (2006) estimated \"typical\" semi-major axes for stalled binaries, finding a stall 10 pc for an SBH mass ratio q = 0.5 and a stall 3 pc for q = 0.1, respectively. If the binary center of mass is located at the photocenter, the displacements of the primary and secondary components would be ∆r 1 ∼ a stall q/(1 + q) and ∆r 2 ∼ a stall /(1 + q), respectively, giving ∆r 1 ∼ 3 (0.3) pc and ∆r 2 ∼ 7 (3) pc for q = 0.5 (0.1). Thus stalled binaries could produce displacements comparable with our results, particularly if the secondary SBH is accreting and the mass ratio is near unity.\n\nIt has been argued that stalling can be avoided in galaxies containing significant amounts of nuclear gas (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007). Even in purely stellar nuclei, N -body simulations sometimes find that evolution of the binary can continue efficiently due to the presence of centrophilic orbits (Khan et al. 2011;Preto et al. 2011;Gualandris & Merritt 2012). However the existing gas-dynamical simulations probably do not yet have enough spatial resolution to follow the binary's evolution to sub-parsec scales, . Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position). and the N -body simulations appear to not yet have large enough N that their results can be robustly extrapolated to the much larger-N regime of real galaxies (Vasiliev et al. 2014). Another possibility, discussed by Antonini & Merritt (2012), is accretion of a less massive galaxy by a giant elliptical, such as NGC 4486, which has a pre-existing depleted core (presumably the result of the evolution of SBH binaries formed in previous mergers). In such situations, dynamical friction is very inefficient in the core due to the lack of stars moving slower than the sinking object. The orbital eccentricity can increase rapidly while the apoapsis hardly changes, resulting in a slowly evolving SBH binary in a highly eccentric orbit. The simulations presented by Antonini & Merritt (2012) indicate that in an M87-like core, a low mass ratio binary (q ≈ 10 −3 ) can persist over a Hubble time in an increasingly eccentric orbit with a semi major axis ∼ 10 pc. If the secondary SBH is accreting, it will be visible as an off-center AGN; if both components of the SBH binary are accreting, a double nucleus might be observed (perhaps as in NGC 4696 and NGC 5419).\n\nHelical distortion or \"wiggling\" of parsec-scale radio jets has been linked to putative SBH binaries in several AGNs, with the jet wiggles being variously attributed to orbital motion, precession of the accretion disk around the jet-emitting black hole or to geodetic precession (e.g., Begelman, Blandford & Rees 1980;Roos et al. 1993;Katz 1997;Romero et al. 2000;Lobanov & Roland 2005). However, the periods (∼ 3 years) and separations (∼ 10 −2 pc) typically inferred from analyses of jet wiggles are much smaller than would be the case for a stalling binary. If the ∼ pc-scale displacements measured in this work are interpreted as SBH binary orbits, periods ∼ 10 4−5 years are implied for total masses ∼ 10 8−9 M . Geodetic precession is insignificant at these separations. The wavelength of jet wiggles caused by orbital motion is given by λ jet ∼ v app t orbit , where v app = βc sin i/(1 − β cos i) is the apparent jet velocity (for a jet speed v jet = βc and inclination to the line of sight, i) and t orbit is the orbital period. Therefore, assuming β ∼ 1, a pc-scale binary will produce very long wavelength wiggles in the jet (∼ 10 kpc). In general, due to the combination of orbital and jet velocities, the jet will precess on the surface of a cone which, for a pc-scale binary of mass ∼ 10 9 M will have a half-opening angle 1 • (see Equation 7 in Roos et al. 1993). Thus, orbital motion would cause only small curvatures in the jet over ∼ kpc scales, which would be difficult to discern as the jet loses collimation, or to distinguish from jet bending due to environmental effects, such as ram pressure. This will also be the case for disk precession, since the precession period exceeds the orbital period. Therefore, although the jet morphology has been mapped in detail from pc to kpc scales for a number of our sample galaxies (see Table 2 for references), these observations are unlikely to provide unambiguous clues as to the presence, or not, of a stalled pc-scale binary.\n\n3. Massive perturbers: galaxy centers host a variety of potential perturbers with masses ranging from ∼ 1M (e.g., stellar mass black holes and neutron stars) to ∼ 10 4 − 10 7 M , such as giant molecular clouds and stellar clusters.\n\nGravitational interactions with these objects will cause the SBH to undergo a type of Brownian motion, with the amplitude of the root-mean-square displacement given by:\n∆r rms ≈ m M • 0.5 r c (8)\nwherem is proportional to the second moment of the mass distribution of the massive perturbers:\n\nm ≡ n(m)m 2 dm n(m)mdm with n(m) dm being the number of perturbers with masses in the range m, m + dm (Merritt 2013, eq. 7.63). The offsets measured in this work range from 0.01 to 0.1 r c , but are typically ∼ 0.01 r c (Table 6). For a typical SBH mass of order 10 9 M (Table 9), rms displacements ∼ 0.01r c can be generated by Brownian motion form ∼ 10 5 M .\n\nThe mass functions of globular clusters, gas clumps and giant molecular clouds in the inner 100 pc of the Galaxy have been estimated by Perets et al. (2007). For comparison, using the mass functions presented in their paper, we find thatm can be as high as 10 5 M for a population of giant molecular clouds. The Milky Way, of course, is very different from the galaxies studied here. However, as recently discussed by Antonini & Merritt (2012), the low efficiency of dynamical friction in low density cores favors the formation of a population of stalling massive objects in the cores of giant ellipticals.\n\nTherefore, it seems plausible that displacements of order those observed could result from Brownian motion due to a population of massive perturbers. In several galaxies, the observed displacements are approximately aligned with the kpc-scale radio jets. This suggests that the displacement is not random, at least in these cases, favoring other mechanisms (i.e, gravitational recoil or jet thrust) that are expected to offset the SBH in the jet direction. Nevertheless, we conclude that Brownian motion cannot be excluded as the cause of the displacement in any individual galaxy, particularly those where the offsets are not aligned with kpc-scale jets, i.e. NGC 4278, 5846 and 5419.\n\n\nSUMMARY AND CONCLUSIONS\n\nWe have analyzed HST archival images of 14 nearby core elliptical galaxies, each of which hosts a central point-like source associated with a low-luminosity AGN, in order to search for offsets between the AGN and the galaxy photocenter. Such AGN-photocenter displacements are possible signposts of gravitational recoils resulting from the coalescence of an SBH binary.\n\nWe find significant (> 3σ) differences between the positions of the nuclear optical (or NIR) point source and the mean photocenter of the galaxy, as determined from isophote fits, in ten of the 14 galaxies in the sample. Assuming that the mean photocenter locates the minimum of the galactic potential well and that the point source locates the position of the AGN and hence the SBH, these results imply that the SBH is displaced from its equilibrium position by angular distances ranging between 20 3.8 ± 0.7 338 ± 7\n\nGalaxies with two nuclear sources NGC 5419 7.5 ± 1.7 62 ± 2 252 ± 13 346 ± 2\n\nNote. -Projected offsets of the photocenter with respect to the AGN and their possible origin. When multiple images have been analyzed, values presented here are the error weighted average. Position angles (PA) are given in degrees East from North. NPS = nuclear point source. and 90 mas, or projected linear distances in the range 1 − 10 pc. As spurious offsets may occur as a result of large-scale isophotal asymmetries, only displacements of magnitude > 0.8 × the inter-quartile range of the distribution of isophote center coordinates (equivalent to 1σ for a gaussian distribution) are considered \"real\". There are six galaxies that exhibit displacements > 0.8 IQR with three of these (NGC 4278, NGC 4486 and NGC 5846) having displacements > 1.6 IQR (equivalent to 2σ). In every case, the measured displacement of the SBH relative to the galaxy photocenter is a small fraction (1-10%) of the galaxy core radius, which is typically r c ∼ 300 pc, for these galaxies.\n\nApproximate alignments between the SBHphotocenter displacements and the radio source axis were found in four of the six galaxies considered to have significant displacements, including three of the four that have FR I, or FR I-like radio sources with relatively powerful and well-defined kpc-scale jets. Indeed, in every case in which there is both a significant displacement and an unambiguous jet, the two are approximately aligned.\n\nLacking detailed knowledge of the merger history of the galaxies, or of the SBH binary parameters (such as mass ratio and spin configuration) that determine the recoil velocity, it is not possible to directly test the hypothesis that the displacements are caused by residual gravitational recoil oscillations. Instead, we used a simple Monte-Carlo model to investigate if the measured displacements are consistent with gravitational recoil. We find that the displacements in individual objects can plausibly be attributed to residual gravitational recoil oscillations following a major or minor merger within the last few Gyr. However, for plausible merger rates there is a high probability of larger displacements than actually observed, if SBH coalescence events took place in these galaxies.\n\nThat larger displacements were not observed suggests that the frequency of gravitational recoil kicks large enough to trigger long-lived oscillations is lower than predicted, perhaps because the evolution of the SBH binary typically results in a configuration that suppresses recoil kicks with velocities 200 km s −1 . Alternatively, the post-recoil oscillations may be damped more quickly than predicted by pure N -body simulations. In either case, gas may play an important role (e.g., Dotti et al. 2010;Sijacki, Springel & Haehnelt 2011). Otherwise, it is possible that galaxy mergers are much more infrequent than implied by the current ΛCDM paradigm (Kroupa 2014).\n\nSeveral other mechanisms are capable of producing the observed displacements, with the observed alignments between the SBH-photocenter displacements and the radio source axis favoring jet acceleration in some objects. An approximate displacement-radio axis alignment is also expected for gravitational recoil, but not for orbital motion in pre-coalescence SBH binaries or interactions with massive perturbers. However, both of the latter mechanisms are capable of producing displacement amplitudes comparable to those observed and cannot be ruled out in individual objects.\n\nIn general, it is not possible to unambiguously distinguish between different mechanisms (including recoil) on the basis of the displacement measurements alone for individual galaxies. However, with a larger sample it may be possible to distinguish mechanisms using statistical arguments. Thus, for jet acceleration the displacement direction should be strongly correlated with the radio jet, with the amplitude correlating with jet power. In the case of gravitational recoil, a weaker correlation with jet direction might be expected. However, no such correlations are expected for binary SBH, or massive perturbers.\n\n\nACKNOWLEDGMENTS\n\nWe wish to honor the memory of our great friend, colleague and mentor David Axon. We thank the anonymous referee for the valuable and timely comments that improved the paper. DL thanks E. Vasiliev, R. Mittal, F. Antonini and M. Richmond for helpful discussions. AM acknowledges support from the Italian National Institute for Astrophysics (INAF) through PRIN-INAF 2011 \"Black hole growth and AGN feedback through the cosmic time\" and from the Italian ministry for school, university and research (MIUR) through PRIN-MIUR 2010-2011 \"The dark Universe and the cosmic evolution of baryons: from current surveys to Euclid\". DM was supported by the National Science Foundation under grant no. AST 1211602 and by the National Aeronautics and Space Administration under grant no. NNX13AG92G. Support for program AR-11771 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr) and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Center (CADC/NRC/CSA). Figure 6. HST images of the galaxies in the sample. The number at the bottom represents the length of the whole axes.\n\n\nAPPENDIX\n\n\nA: DISTRIBUTION OF RECOIL DISPLACEMENTS\n\nHere we outline the method used to estimate the probability distributions for displacements resulting from postrecoil oscillations of a kicked SBH. For our purposes, of the three phases predicted by GM08, only phase II (when the oscillation amplitude becomes comparable with the galaxy core radius) is of interest. The initial, large amplitude phase I oscillations are short-lived, while in phase III the residual oscillations are indistinguishable from Brownian motion (which is considered separately in Section 5.3).\n\nIn the GM08 N -body simulations, phase II oscillations occur for kick velocities in the range v kick ∼ 0.4 − 0.9 v esc . The simulations also indicate that the subsequent evolution does not depend strongly on the magnitude of the initial kick. Therefore, we assume that, given any initial kick large enough to displace the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy evolves during phase II as a damped harmonic oscillator:\nR(t) = R 0 e −∆t/τ sin(ω c ∆t).(A1)\nThe initial amplitude, R 0 , is set by the condition that phase II begins when the oscillation amplitude is comparable to the core radius; ∆t is the time since the kick, i.e., the time since the last merger, and ω c is the SBH oscillation frequency:\nω c = 3 4 σ 2 c r 2 c . (A2) Equation A2\nis obtained by combining the radial oscillation frequency in a spherical core of uniform density, ω c = [(4π/3)Gρ c ] 1/2 , with equation 13 in GM08. The damping time, τ , is given by equation 2. The adopted parameter values are listed in Table 8.\n\nSuppose that we observe a galaxy at a random time since its last merger. For simplicity, we assume that galaxy mergers occur stochastically and can be represented as a Poisson process. The probability that the last merger occurred at time ∆t ago is then:\nP (∆t) = t −1 m e −∆t/tm , ∆t > 0 (A3)\nwhich is associated with the cumulative distribution:\nP (< ∆t) = 1 − e −∆t/tm (A4)\nwhere t m is the mean time between galaxy mergers. The galaxy merger rate is uncertain but simulations indicate that it is a function of redshift, galaxy mass and mass ratio (e.g., Bell et al. 2006;Conselice, Yang & Bluck 2009;Hopkins et al. 2010;Burke & Collins 2013). Here, since the phase II damping times are typically ∼ 1 Gyr (Table 8), we consider redshifts z ≤ 1 and estimate the mean time between galaxy mergers as the ratio between the lookback time corresponding to z = 1 and the number of mergers, N , during that time. The latter was determined using the analytical fits to semi-analytical models for galaxy merger rates given by H10. Integrating the merger rate as given by equation 5 in H10 over redshift, we find for the number of mergers,\nN = t H A(M min ) 1 0 (1 + z) β(Mmin)−5/2 dz(A5)\nwhere t H = 13.7 Gyr is the Hubble time, the normalization constant A(M min ) and the slope β(M min ) are given as functions of the galaxy mass threshold M min , for mergers with mass ratio q > 0.1, by equations 9 & 10 in H10. For log M gal ≥ log M min = 11, we find N ≈ 1.5 for the number of mergers per galaxy between z = 0 and z = 1. However, we note that N could vary between 0.75 and 3, allowing for systematic uncertainties of a factor two in A(M min ) and 0.2 in β(M min ). The look back time at z = 1 is t L = 7.731 Gyr, for H 0 =71 km s −1 Mpc −1 , Ω M = 0.27 and Ω vac = 0.73, yielding a mean time between galactic mergers t m ≈ 5 Gyr. The simple analytical functions provided by H10 do not accurately reproduce the numerical results for the most massive galaxies. Therefore, we use an average merger rate for z ≤ 1 estimated from their Figure 3, for log M gal > 12. The rate for q > 0.1 mergers is roughly constant for z ≤ 1 and it seems reasonable to adopt an average value of 2.5 mergers/galaxy/Gyr. This corresponds to t m ≈ 0.4 Gyr.\n\nThe stellar mass, M gal , is derived for each galaxy from the relation:\nΥ = 10 0.4(M K −M K ) M M (A6)\nwhere Υ is the mass-to-light ratio and M K is the K-band magnitude given by BC06. For the mass-to-light ratio, we use a representative value, Υ = 3.1, computed as the average value of the ratio between the virial mass and the K-band luminosity for the sample of nearby core ellipticals studied by Sani et al. (2011).\n\nAs the derived values of M gal span the range 11 log M gal 12 we compute displacement probability distributions for the limiting cases t m = 0.4 Gyr and t m = 5.0 Gyr. For comparison, Lidman et al. (2013) find an observationallyderived major merger rate of ≈ 0.38 mergers Gyr −1 for brightest cluster galaxies at z ∼ 1, implying t m ≈ 2.6 Gyr.\n\nFrom the cumulative distributions given by eq. A4, we extract random times in order to generate a set of R i ≡ R(t i ) from equation A1 for each galaxy, for both values of t m . Projected displacements are then calculated:\nR i,p = R i sin(θ i ) (A7)\nwhere θ i is the angle between the displacement direction, assumed to be randomly oriented, and the line of sight.\n\nFinally, the distribution of the projected offsets is computed. In Figs. 7a and 7b we plot the probability, P (d > x), of observing a displacement, d, larger than a given value, x, as a function of x. Values of the measured displacements are also plotted. The probability of generating a projected displacement larger than the value actually measured is listed for each galaxy in Table 6, for both values of t m .\n\nThe damping time computed from eq.2 depends on the mass of the SBH. Measurement uncertainties for SBH masses in galaxies beyond the Local Group are typically large and are probably dominated by poorly-understood systematics, given that none of these galaxies exhibits a clear central rise in the stellar velocities (Merritt 2013, section 2.2). In Table 9 we list SBH masses estimated from the M • -σ relation (Ferrarese & Ford 2005), from the M • -M V relation for bright galaxies (M V < −19, Lauer et al. 2007) and from direct measurements (stellar or gas dynamics modeling). For most of our galaxies the discrepancy in the masses recovered with different methods is small. We compute damping times using SBH masses from the M • -M V relation finding a smaller average damping timeτ = 0.66 Gyr, although the new damping times for individual galaxies are not systematically smaller than those computed using the M • -σ SBH masses. We then compute new values for the probabilities presented in Table 6 and the corresponding cumulative probabilities of not observing displacements larger than those actually measured in this work (P). We obtain P = 10 −16 (10 −3 ), instead of 10 −17 (10 −4 ), for an average time between galactic mergers t m = 5 (0.4) Gyr. After taking into account the kick probabilities given by L12, as specified in §5.2, the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 1×10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5.0 Gyr. The different masses adopted for the SBH caused, therefore, a difference in P of one order of magnitude. Nevertheless, even for t m = 5 Gyr, there remains a large probability of finding displacements larger than those actually observed, if these galaxies experienced gravitational recoil events within the last few Gyr.\n\n\nB: NOTES ON INDIVIDUAL SOURCES\n\nIn this section, the galaxies are ordered according to the significance assigned to the displacement.\n\n\nNGC 4278 (high/intermediate)\n\nNGC 4278 is a large elliptical near the center of the dense Coma I cloud which is characterized by the presence of an extended dust distribution extending over an arc ranging from N to NW relative to the photocenter (Carollo et al. 1997). Radio interferometry (VLBA) observations reveal a low-power, parsec-scale source (∼ 45 mas) with twin mildly-relativistic S-shaped jets emerging from a compact, flat spectrum core (Giroletti et al. 2005). Within the inner 7 mas the jets are oriented SE-NW but then bend to the east and west, respectively, forming an \"S\" shape. Giroletti et al. argue that the jets are closely aligned with the line of sight (with a viewing angle ∼ 2 − 4 • ) with the NW jet approaching and the bends being explained by amplification of intrinsically small deviations due to projection effects. These characteristics and the presence of atomic hydrogen suggest that the galaxy may have experienced a minor merger with another member of the cloud (Gerin & Casoli 1994).\n\nPhotometric analysis. Three images, obtained with ACS/WFC-F850LP, WFPC2/PC-F814W and NIC-MOS2/F160W, respectively, were analyzed for this galaxy (Figs 15, 16 and 17). Photocenter-AGN displacements of intermediate or high significance are detected in all three. The uncertainty-weighted average displacement has an amplitude ∆r ≥ 7.6 ± 0.4 pc and direction SE (PA = 152 ± 3 • ). The photocenter positions derived from each image are plotted relative to the AGN position in Fig.5a. The photometric analysis of the optical ACS and WFPC2 images shows that the y co-ordinate of the isophote centers migrates southward at intermediate SMA, then northward at larger SMA. These irregularities are not present in the NICMOS2 image and are likely due to the dust N-NW of the center.\n\nOffset origin. The extended dust distribution produces a large-scale asymmetry in surface brightness, which is particularly evident in optical and UV wavelengths. This may bias the photocenter, shifting it in the opposite direction (S-SE) to the dust. As the AGN is a point source, its position should not be significantly affected by the dust distribution and thus it would appear to be shifted N-NW relative to the photocenter. Therefore, the offsets recovered Note. -(1) Galaxy name;\n\n(2) core radius from CB05; (3) stellar velocity dispersion from Hyperleda; (4) stellar mass of the galaxy; (5) dynamical time; (6) damping time; (7) kick velocity required to displace the SBH up to the core radius. The dominant source of errors in the quantities presented above is the stellar velocity dispersion. Note. -(1) Galaxy name, (2) masses from the M• − σ (Ferrarese & Ford 2005) using the velocity dispersion specified in Table 8 from ACS and WFPC2 data may be partly due to asymmetric extinction. However, the effects of dust are not apparent in the NIR (NICMOS2) image, from which an offset of similar magnitude and direction was recovered. Therefore, we believe that the measured displacement is not simply an artifact of extinction, although this may contribute. Relative to the core radius, NGC 4278 has the largest displacement (≈ 0.1r c ; Table 6) in our sample. If it is due to phase II oscillations of the SBH following gravitational recoil (see §5.2), we estimate that the probability of observing a projected displacement larger than that actually measured is 4 (36)% for a mean time-between-mergers of t m = 5 (0.4) Gyr (Table 6) and given a kick large enough to displace the SBH at least as far as the core radius. The \"hot\" disk recoil velocity distribution of L12 indicates a 45% probability of a sufficiently large kick ( 200 km s −1 , Table 8) occurring in an SBH coalescence event. Therefore, we consider residual oscillations following gravitational recoil to be a plausible explanation for the displacement.\n\nIn this case, we can estimate the merger epoch. Supposing the SBH to be undergoing phase II oscillations, we combine eq.2 and 3. Using the measured (projected) offset as a lower limit for r rms , r rms 7.6 pc, and taking the values of r c (core radius) and σ (stellar velocity dispersion) from Table 8, we find that (t − t c ) 5τ ≈ 0.7 Gyr has elapsed since the amplitude of the oscillation damped down to r ∼ r c . This implies that the onset of phase II occurred no later than ∼ 0.7 Gyr ago. As phase I is shorter than phase II, this scenario is consistent with an SBH-SBH coalescence event that occurred within 2(t − t c ) ∼ 1.4 Gyr. The total evolution time from the beginning of the galactic merger to SBH binary coalescence is suggested to be of order 10 8 yr (Gualandris & Merritt 2012). Adding this time to the estimated kick epoch, we infer that a galaxy merger that took place up to ∼ 1.5 Gyr in the past could be responsible for a recoil kick consistent with the observed displacement.\n\nAlthough the displacement is approximately in the direction of elongation of the mas-scale radio core (155 • over the inner ∼ 7 mas), the low power (P 5GHz ≈ 70 × 10 20 W, Wrobel 1991) and compact nature (∼ 3 pc) of the radio source argue against jet thrust as the cause of the offset.\n\nWe find no compelling reasons to exclude either a stalled binary on an eccentric orbit, or massive perturbers (Section 5.3) as possible causes of the displacement.\n\n\nNGC 5846 (intermediate)\n\nNGC 5846 is the dominant component of a small, compact group of ≈ 50 galaxies (Huchra & Geller 1982). A companion galaxy, NGC5846A, is located 0.7 arcmin to the S, but there is no evidence of an interaction. The presence of dust lanes in an apparent spiral pattern approximately centered on the compact radio core suggests a past merger, or accretion of a gas rich galaxy (Moellenhoff et al. 1992;Forbes et al. 1996). The optical nucleus has a LINER spectrum (Ho et al. 1997) and HST imaging (Masegosa et al. 2011) reveals extended Hα emission in a wide structure extending up to 2 W of the nucleus. The presence of two radio emitting bubbles located roughly symmetrically at ∼ 0.6 kpc from the center and a \"ghost\" X-ray cavity suggest that the galaxy recently experienced stronger AGN activity, for which Machacek et al. (2011) derive a duty cycle of ∼ 10 Myr. The radio morphology is complex: Filho et al. (2004) report VLBA observations at 2.3, 5 and 15 GHz which resolve the core into multiple blobs roughly aligned along the N-S axis. The nature of this structure is unclear, possibilities include an AGN jet, or compact supernovae remnants (e.g. Tarchi et al. 2000).\n\nPhotometric Analysis. One image, obtained with WFPC2/PC -F814W was analyzed for this galaxy (Figure 24). We find a photocenter-AGN displacement ∆r 8.2 ± 2.5 pc in the direction W-SW (PA = 253 ± 8 • ) which is considered to be of intermediate significance. The isophote centers show systematic migrations of ≈ 2 pixels in both x and y coordinates, from W to E and S to N, respectively, as the SMA decreases from 300 to 200 pixels. These migrations are possibly the result of the dust lanes to the S and NE of the nucleus. The y-component of the displacement is not significant. The x-component is considered to be of intermediate significance (1.6 IQR) and is unlikely to be an artifact of extinction, which in this case, would bias the photocenter such as to increase the x-displacement. Of more concern is the determination of the AGN position. The central source is relatively weak and embedded in an asymmetric structure that affects the gaussian fits used to determine the AGN position. We therefore assign a larger uncertainty to the x-component of the AGN position.\n\nOffset origin. For t m = 5 (0.4) Gyr, we estimate a 25 (90)% probability of finding a displacement larger than that observed, if due to phase II recoil oscillations. The probability of a recoil kick large enough ( 240 km s −1 ; Table 8) to initiate these oscillations is ∼45% (L12).\n\nAs for NGC 4278, we use equations 2 and 3 and values from Table 8 to estimate the time since the onset of phase II, finding (t-t c ) 6.2 τ or 4 Gyr. This implies an upper limit to the time elapsed since the putative SBH-binary coalescence event of 2(t-t c ) or 8 Gyr.\n\nAlthough, as noted, there is \"fossil\" evidence for previous jet activity, the AGN does not currently produce a powerful radio jet, so it seems unlikely that the displacement is due to jet acceleration. However, interactions with massive perturbers or an SBH binary on a highly eccentric orbit remain possibilities. NGC 4486 (M87,intermediate) NGC 4486 (M87) is the dominant galaxy of the Virgo Cluster. It hosts the best studied extragalactic jet, which exhibits prominent knots at optical and IR wavelengths yet, apart from this, the brightness profile is regular and featureless. The peculiarly large population of globular clusters, its size and location, and the presence of stellar streams support the hypothesis that the galaxy has experienced a number of minor mergers during its lifetime (Janowiecki et al. 2010). A remarkable feature is the unusually large core, which has a radius r c ∼ 9. 41 (CB05). This characteristic makes M87 a special case in our sample in that for all the other galaxies we restricted the isophotal analysis to radii larger than the core radius, whereas for M87 we analyzed isophotes within the core radius in order to be consistent with B10.\n\nPhotometric analysis. A total of six images obtained with ACS/HRC, WFPC2/PC and NICMOS2 were analyzed (see Table 1), including the combined ACS images (ACS/HRC/F606W and ACS/HRC/F814W) previously analyzed by B10. The results of the photometric analysis are shown in Figs. 4 and 18-22 and the measured displacements relative to the AGN are plotted in Fig. 5a. In all images, we analyzed isophotes within the range 1 ≤ r ≤ 3 , for consistency with B10. The displacements measured in different images vary in amplitude and significance. Notably, none of the three NICMOS2 images (F110W, F160W and F222W) exhibit significant displacements at the ≥ 3σ level. Furthermore, the 1σ uncertainties, when converted to linear distances, do not encompass the significant displacements recovered from the optical images. The displacements measured from the ACS and WFPC2 images also differ in amplitude but there is a tendency for the photocenter to be displaced NW of the AGN, roughly in the direction of the radio jet. The isophote centers are reasonably stable -although there are deviations ∼ 1 − 2 pixels (≤ 0. 1), systematic trends with SMA are not generally evident . The most significant displacement is obtained from the ACS/HRC F814 image, for which the isophote centers cluster tightly around the mean photocenter (Fig. 4). The measured displacement of the photocenter relative to the AGN is ∆r = 7.7 ± 0.3 pc NW, implying that the SBH is displaced by ∆r in the counter-jet direction. This result is in agreement with the displacement obtained by B10 from a similar analysis. The main differences between the method used by B10 and that employed in this paper are the iterative technique used here to generate the mask and the weight function used to derive the isophotal center: while B10 used a conventional uncertainty weight, here we weight for the light content of each isophotal annulus (see §3.1). The weighted average displacement has a magnitude 4.3 ± 0.2 pc and direction NW (PA = 307 ± 1 • ). Offset origin. The presence of a large number of bright globular clusters visible in all the images analyzed allows us to define a common reference frame which we can use to compare results from images obtained with different instrument/filter combinations. After selecting several globular clusters as reference points and matching their positions, we plotted the AGN and photocenter positions determined from the different images in the reference frame defined by the ACS image (Fig. 8). The AGN position as measured from the different images is fairly stable whereas the photocenters exhibit differences in position much larger than the scatter (∼ 0.5 pixels) in the AGN position. This indicates that the differences in the displacements are due largely to differences in the positions of the photocenters. The origin of these discrepancies is unclear. Unlike other galaxies, there is no evidence of dust extinction or other asymmetries that may account for different results at different wavelengths.\n\nWe conducted several trials in order to determine the influence of (i) the degree of masking applied to the jet, (ii) the bright knot visible NW of the nucleus in the ACS image. The results were found to be independent of the degree of masking; no significant differences in the photocenter positions were found when the analysis was performed with no masking at all or with heavy jet/knot-masking.\n\nAs a fraction of the rather large core radius, the weighted mean displacement is only ≈ 0.01 r c (Table 6). We estimate a 85 (99)% probability of finding a displacement larger than that observed for t m = 5 (0.4) Gyr, assuming phase II recoil oscillations were triggered. The probability of obtaining the relatively large recoil kick ( 325 km s −1 ; Table 8) required for phase II oscillations is ≈ 33% (L12).\n\nThe apparent close alignment of the offset with the jet direction favors jet acceleration but is also generally consistent with gravitational recoil. However, as there is no reason to expect such an alignment if the displacement is due to gravitational interactions with massive perturbers, or orbital motion in a SBH-binary, these mechanisms are disfavored. A detailed discussion of the possible displacement mechanisms is given in §4 of B10.\n\n\nNGC 1399 (low)\n\nNGC 1399 is a regular giant E1 and is the central and dominant galaxy of the Fornax cluster (Ferguson 1989). It is associated with a low power radio source characterized by two nearly bi-symmetric jets extending ∼ 10 kpc, approximately N and S of the core (PA ≈ 9 • W of N, ). Shurkin et al. (2008) report the presence of X-ray cavities associated with the radio lobes. NGC 1399 has been the subject of a number of dynamical studies (e.g. Houghton et al. (2006) and references therein). In particular, Houghton et al. performed near-infrared adaptive optics assisted spectroscopic observations of the central 1. 5 of the galaxy and found evidence of a kinematically decoupled core and offset asymmetric isophotes that they considered to be consistent with the presence of a M31-like double nucleus. The asymmetric isophotes, elongated in the direction E-SE, are localized in a region much smaller than that used for our isophotal analysis (r > 2. 5).\n\nPhotometric analysis. Four images, obtained with WFPC2/PC -F606W, F814W and WFC3/IR -F110W, F160W, were analyzed for this galaxy. From the F606W image we find a low significance displacement of amplitude 1.7 ± 0.4 pc, in almost the same N-NW direction as the radio jet axis, PA = −8±16 • . A 2σ offset in the same direction was recovered from the WFPC2/PC -F814W image, which is consistent within the errors with the F606W result. Offsets recovered from the WFC3 images are classified as non-significant, but, given their large error bars, they are consistent with the WFPC2 results (Fig. 5a). The error weighted offset, computed including results from both WFPC2 and WFC3, is 1.5 ± 0.4 pc with PA = -17 ± 16 • . The isophote fitting analysis (Figs. 9 to 12) reveals a remarkably regular photometric structure well modeled by concentric elliptical isophotes. For both images, the isophote center co-ordinates vary little with SMA and are very tightly clustered, with narrow ( 0.5 pixel) distributions in x and y.\n\nOffset origin. The weighted mean displacement is small compared to the core radius, ≈ 0.01 r c (Table 6). A minimum kick velocity 300 km s −1 (Table 8), is required to trigger phase II recoil oscillations; the probability of generating a kick of at least this magnitude is ∼ 33% (L12). Assuming that phase II recoil oscillations were triggered, we estimate the probability of finding a displacement larger than that observed to be 11 (75)%, for t m = 5 (0.4) Gyr.\n\nIf the observed displacement is due to a recoiling SBH, we estimate that (t -t c ) 1.6 Gyr has elapsed since the oscillation damped below the core radius, using τ = 0.17 Gyr and σ = 346 km s −1 ( Table 8).\n\nThe close alignment of the displacement with the radio axis is consistent with both recoil and jet acceleration as the cause of the displacement. If the latter, eq. 7 implies that, for a Bondi accretion rateṀ a ∼ 0.04 M yr −1 (Loewenstein et al. 2001) and a 10 9 M black hole, an asymmetry in the radio jet power in the range 3 × 10 −4 f jet 3 × 10 −3 that persists for t ∼ 10 8 yr would be needed to produce an offset 1.8 ∆r 18 pc.\n\nOther mechanisms (interactions with massive perturbers and a stalled SBH binary), are disfavored by the alignment between the displacement and jet direction. The presence of an unresolved M31-like double nucleus as suggested by Houghton et al. (2006) might also explain the measured offset, although as in the case of a binary SBH, the alignment with the radio source axis would have to be regarded as fortuitous.\n\n\nIC 4296 (low)\n\nThe properties of this galaxy have been extensively studied in a series of papers by Killeen et al. (1986a,b) and . It is a fairly circular elliptical which is the brightest component in the small group A3565. It is associated with the extended low-to-intermediate luminosity radio source PKS 1333-33 (Mills et al. 1960), which has two quasi-symmetric slightly curved jets with an average position angle ∼ 130 • and terminating in lobes ∼ 200 kpc from the core (Killeen et al. 1986a). The curvature along the jets is consistent with the motion of the galaxy through the group inter-galactic medium ). On the parsec scale, a jet and counter-jet emerge from the unresolved core in PA ≈ 140 • , consistent with the large-scale radio source, with the brighter jet on the NW side. The brightness asymmetry is attributed to Doppler boosting (Pellegrini et al. 2003). Isophotal studies suggest tidal interactions with the smaller companion IC 4299 (Younis et al. 1985). Nevertheless, Efstathiou et al. (1980) show that the isophotal ellipticity and position angle are constant at all radii from 2.4 to 48.8 arcsec (a region much larger than that studied in this work, 1. 5 < ∆r < 6. 5), which suggests that any distortions due to interactions with companion galaxies are negligible over this scale.\n\nPhotometric analysis. Three images, obtained with ACS/HRC-F625W, WFPC2/PC-F814W and NICMOS2-F160W, were analyzed for this galaxy. In the ACS image a displacement of ∆r ≥ 5.2 ± 1 pc was found. On the basis of the IQR, that is considered to be of low significance. The displacement obtained from the NICMOS2 image is not considered significant, but is consistent with the ACS result, albeit within large errors. For the WFPC2 image, the y-component of the displacement is consistent with the ACS result but the x component exhibits a marginally significant difference and in fact indicates a displacement in the opposite direction (E). However, neither component of the WFPC2 displacement is classified as significant. These results are summarized in Fig. 5a.\n\nThe discrepancy in the photocenter positions appears to be linked to systematic migrations in the isophote center coordinates determined from the fits in the NICMOS2 and WFPC2 images: there are systematic migrations northwards in the NICMOS2 image (Fig. 31) and eastwards in the WFPC2 image (Fig. 30) resulting in distributions with large IQRs in the y and x co-ordinates, respectively. In contrast, the co-ordinates extracted from the ACS image (Fig. 29) show little variation with SMA and cluster quite symmetrically around the photocenter. This galaxy exhibits a warped dusty disk, which crosses the nucleus in a roughly E-W direction and is clearly visible in all three images (Figs. 29,30 and 31). This feature is too small (r 0. 9) to affect the photocenter position, which is calculated from isophote fits over radii 1. 5 ≤ r ≤ 6. 5. However, it does create large gradients in the background brightness around the nucleus which affect the gaussian fits to the central peak, especially when large fitting-boxes are used. The effects are particularly noticeable for the WFPC2 image, where the AGN position systematically shifts South-East as the size of the fitting-box is increased. This was mitigated by reducing the size of the fitting-box until stable values were found for the AGN co-ordinates.\n\nThe average error-weighted offset obtained for this galaxy from the ACS, NICMOS2 and WFPC2 images is ∆r ≥ 3.8 ± 0.7 pc in PA = -22 ± 7 • E of N.\n\nOffset origin. The status of IC 4296 as the brightest member of its group and the fact that it has a kinematically peculiar core (Killeen et al. 1986a) suggest that this galaxy has experienced minor mergers during the past few Gyr.\n\nThe weighted mean displacement is a small fraction of the core radius, ≈ 0.01 r c ( Table 6). The probability of generating a recoil kick large enough to trigger phase II recoil oscillations ( 200 km s −1 ; Table 8), is ∼ 45% (L12) and given that phase II oscillations were triggered, we estimate a 37 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr.\n\nUsing eqs. 2 and 3 with σ = 333 km s −1 , r rms ≥ 3.8 pc and r c = 347 pc, we find (t − t c ) 9 τ ≈ 5.8 Gyr. In this case, the low density of the large core is responsible for the long damping time.\n\nIt has been pointed out by Killeen et al. (1986a) that on kpc scales the radio jets exhibit transverse oscillations with wavelengths ∼ 10 kpc and an amplitude comparable with the jet width. Similar oscillation wavelengths are expected from precession due to orbital motion in a ∼ pc scale SBH binary (see Section 5.3). However, found that, in detail, the oscillation pattern is not consistent with precession and favor instead helical Kelvin-Helmholtz instabilities.\n\nIf we disregard the discrepant WFPC2 result, the direction of the weighted mean displacement derived from the NICMOS2 and ACS images is PA = 132±8 • . In projection, therefore, the displacement is remarkably well-aligned with the radio source axis, suggesting jet acceleration as a plausible candidate for the displacement mechanism. Pellegrini et al. (2003) infer a viewing angle θ = 63.5 ± 3.5 • for the parsec-scale jets. Assuming that the displacement is in the same direction in space as the jet, the de-projected amplitude is ∆r jet = 4.3 ± 3 pc.\n\nGiven that the galaxy is characterized by a low density core, taking the age of the radio source to be t ∼ 10 8 yr ,Ṁ a =ṁ Bondi ∼ 0.02 M yr −1 (Pellegrini et al. 2003), M • ∼ 10 9 M from the M − σ relation and 3 ∆r 30 pc, we find from eq. 7 that 10 −3 f jet 9.7 × 10 −3 .\n\nTherefore, an asymmetry in jet power amounting to 1% of the accretion luminosity could account for the observed offset if it persists for ∼ 10 8 yr.\n\nThe close alignment with the radio jet axis argues against the displacement being caused by gravitational interactions with massive perturbers.\n\n\nNGC 5419 (low, double nucleus)\n\nThis galaxy is the dominant member of the cluster Abell S753 (Abell et al. 1989). A low surface brightness X-ray halo is centered on the galaxy and extends over a radius r ≈ 16 (190 h −1 kpc). An unusual diffuse radio source, PKS B1400-33, is associated with the cluster, but is offset relative to the X-ray emission and NGC 5419 (Subrahmanyan et al. 2003). The galaxy is located just beyond the NW edge of the extended radio emission and itself is associated with a bright, compact radio source. The low surface brightness and steep spectral index of the diffuse source suggest that it is a relic, with a spectral age of 5 × 10 8 yr. However, it is unclear if the relic was created by an earlier episode of powerful activity in NGC 5419, or whether it is a relic lobe injected into the cluster by a previously active double radio source located outside the cluster (the scenario favored by Subrahmanyan et al., although an optical counterpart has not been firmly identified). If the relic originated from NGC 5419, the spatial offset may be due to proper motion of the galaxy or to buoyancy of the synchrotron plasma (e.g., McNamara et al. 2001).\n\nPhotometric analysis. The galaxy has a double nucleus (Lauer et al. 2005;, which is well-resolved in the WFPC2 planetary camera image (WFPC2/PC-F555W). The two nuclei are separated by 0. 27, with the fainter (secondary) nucleus located almost at the south of the brighter (primary) nucleus. The photocenter is offset by ∆r 1 7.5 ± 1.7 pc W-SW (PA = 252 ± 12 • ) from the primary and ∆r 2 62 ± 2 pc N-NW (PA = 346 ± 2 • ) from the secondary nucleus.\n\nA periodic pattern is visible in the residual image obtained after subtraction of the photometric model (middle panel in Fig. 26). This is likely to be due to the pronounced \"boxiness\" of the isophotes, which could not be perfectly reproduced even though higher harmonics were included in the model. The isophote center co-ordinates cluster fairly tightly around the photocenter, which is closest to the primary nucleus, and generally show little variation with SMA (Fig. 26), with the exception of a small migration (≈ 0. 05) eastward of the photocenter in the inner 170 pixels (8. 5).\n\nOffset origin. The projected separation between the two nuclei (∆r ≈ 75 pc) is slightly smaller than the estimated SBH influence radius (r h ≈ 90 pc) and much smaller than the core radius (r c ≈ 500 pc; Table 1). Hence, it is possible that the secondary nucleus is a second SBH, or even the nucleus of a satellite galaxy, on a slowly-evolving orbit in the low-density core (Antonini & Merritt 2012).\n\nThe displacement of the photocenter relative to the primary nucleus is a small fraction of the core radius, ≈ 0.02r c ( Table 6). The probability of generating a recoil kick velocity large enough to trigger phase II recoil oscillations (v 300 km s −1 , Table 8), is ∼ 33% (L12) and given that phase II oscillations occur, we estimate a 46 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr. Using eqs. 2 and 3 with σ = 351 km s −1 , r rms ≥ 7 pc and r c = 500 pc, we find (t − t c ) 9 τ ≈ 10 Gyr, a factor ten larger than the estimated spectral age of the diffuse radio source. Therefore, if an SBH binary coalescence event is responsible for the displacement, it seems likely that it pre-dates the AGN phase that created the radio relic (if indeed NGC 5419 is the origin of the relic).\n\nNotes on other galaxies In the following galaxies, photocenter -SBH displacements were not considered significant.\n\n\nNGC 4168\n\nNGC 4168 is an E2 in the Virgo cluster. Broad Hα has been detected in its nuclear optical spectrum, resulting in classification as a Seyfert 1.9 (Ho et al. 1997). VLBA observations at 8.4 GHz show an unresolved radio source with no evidence of jets (Anderson et al. 2004). Although the galaxy appears very regular in the WFPC2-WF/F606W image, the photometric analysis reveals systematic variations in the isophote centers with SMA that are much larger in amplitude than the displacement between the mean photocenter and the AGN position (Fig. 13). In particular, there is an abrupt shift to the E in the x co-ordinate and a systematic migration to the S in y, for SMA 260 pixels (13 ). The measured displacements in x and y are < 0.8 IQR and therefore considered \"null\" results.\n\n\nNGC 4261\n\nNGC 4261 is a massive elliptical (E2-3) in the outskirts of the Virgo cluster. Nolthenius (1993) identifies this galaxy as the dominant member of a group of 33 galaxies, the Virgo W cloud. It is associated with a bright FR I radio source, 3C 270, which has twin jets close to the plane of the sky, θ = 63 ± 3 • , and aligned on the axis W-E, PA = 87 ± 8 • , with the western radio component slightly brighter than the eastern (Jones et al. 2000;Worrall et al. 2010). Deep imaging reveals the presence of tidal tails to the NW and a tidal fan SE (Tal et al. 2009) while Giordano et al. (2005) find anisotropies in the globular cluster distribution. Cappellari et al. (2007) report the presence of a kpc-scale kinematically decoupled core hosting a 100 pc scale disk of dust, cool molecular and atomic gas (Jaffe et al. 1993;Jaffe & McNamara 1994). Jaffe et al. (1996 studied the central region of this galaxy in detail using HST/WFPC2/PC1 V (filter F547M), R (F675W) and I (F791W) images. After correcting the R images for line emission, they determined that the central dusty disk is characterized by a rotation axis which closely matches the axis of the jets but it is not coaxial with the semi-major axis of the galaxy. The disk is not centered on the AGN, nor on the photocenter, which they find is itself displaced relative to the AGN by ∆r = 3.3 ± 1.3 pc NE.\n\nIn this study, we chose to analyze a NICMOS2 F160W image (with a pixel size of 0. 05), since the nuclear source is bright and dust extinction and contamination due to emission from ionized gas are minimized. The galaxy exhibits boxy isophotes which are difficult to model accurately. Even though the third and forth harmonics were included in the isophote fits, the residual image still shows a spiral pattern suggesting that a more complex model perhaps involving multiple components would be necessary to successfully model the surface brightness distribution (Fig. 14). Nevertheless, the isophote centers are relatively stable, except for a slow migration to the N for SMA 120 pixels (6 ). The mean photocenter displacement relative to the AGN is ∆r = 3.3 ± 2.2 pc W-NW (PA = 297 ± 38 • ). The x component of the displacement is large compared to the IQR. However, because of the large uncertainty in positions derived from NICMOS2 images (0.23 pixels), we do not consider this offset significant.\n\n\nNGC 4373\n\nThis galaxy is located in the outskirts of the Hydra-Centaurus supercluster and is associated with a compact radio source (Sadler et al. 1989). It forms an isolated pair with IC 3290 (Sadler & Sharp 1984), which is located 2 W-SW of NGC 4373. Grützbauch et al. (2007) suggest that there is an ongoing strong interaction with the companion. Our photometric analysis of the WFPC2/PC-F814W image reveals boxy isophotes which produce cross-shaped residuals, even though higher harmonics were included in the isophote fits (Fig. 3). The y coordinate of the isophote center position has a narrow distribution centered on the mean photocenter and shows only a slight drift northward, for SMA > 170 pixel (8. 5). The x co-ordinate exhibits a systematic migration eastwards from the photocenter, resulting in a broader, skewed distribution. The mean photocenter is consistent with the AGN position within the uncertainties.\n\n\nNGC 4552\n\nNGC 4552 is a typical giant elliptical in the Virgo cluster which hosts a very low-luminosity AGN. HST imaging and spectroscopy reveal a variable UV point source that has optical emission lines characteristic of AGN (Cappellari et al. 1999;Maoz et al. 2005). Interestingly, a second, transient UV source of comparable luminosity, but offset from the first by 0. 14, was present in 1991, but was not detected in 1993 Renzini et al. (1995); Cappellari et al. (1999); Maoz et al. (2005). Radio observations at 8.4 (Filho et al. 2000) and 5 GHz (Nagar et al. 2002) show the presence of a flat spectrum radio core. Nagar et al. (2002) also report variability at the ∼ 20% level on a ∼ 1 year timescale and find evidence of pc-scale jets, or extensions, along the East-West axis. NGC 4552 exhibits several morphological features indicative of past mergers, including three concentric shells at 5 , 7 and 10 from the nucleus (Malin 1979). Traces of a recent minor merger with a gas rich satellite galaxy have been identified in the presence of dust patches in the inner region (van Dokkum & Franx 1995), and Hα emission extending up to ∼ 2.5 kpc from the center (Trinchieri & di Serego Alighieri 1991; Macchetto et al. 1996).\n\nThe photometric analysis of the WFPC2/PC-F814W image shows that the y-coordinate of the isophote center migrates systematically southward relative to the photocenter for SMA > 200 pixels (10 ) with a maximum shift 0. 2 (Fig. 23). There is a smaller westward shift in the x coordinate. The inner 0. 3 of the residual image shows evidence of an arc-like dust lane partly encircling the central source with an opening toward the S. This feature was previously noted by Carollo et al. (1997) and Cappellari et al. (1999) (see Fig. 12 in the latter) and is the cause of the increased uncertainty on the y co-ordinate of the AGN position. We did not find a significant (> 3σ) displacement in this object.\n\n\nNGC 4636\n\nNGC 4636 is a giant elliptical in the outskirts of the Virgo cluster. Its nucleus has a LINER emission line spectrum, in which a weak broad Hα line has been detected (Ho et al. 1997). It has a jet-like radio source extending ≈ 1.5 kpc either side of the nucleus in position angle θ ≈ 33 • (Birkinshaw & Davies 1985). NGC 4636 is also associated with a luminous, extended X-ray source with a peculiar morphology featuring two symmetric, arm-like extensions emerging from the bright central region and extending up to ≈ 8 kpc along the NE-SW axis. Other fainter arm-like features emerge from the nucleus and Jones et al. (2002) suggest that these features trace cyclic nuclear outburst events occurring on a timescale ∼ 10 7 yr. According to Temi et al. (2003), the observed FIR fluxes (at 60, 90 and 180 µm) suggest the accretion of dust in a recent (within ∼ 10 8 yr) merger with a gas-rich galaxy.\n\nOur isophotal modeling of the WFPC2/PC-F814W image indicates the presence of faint dust lanes inspiraling toward the nucleus (Fig. 24). There is a systematic shift in the isophote centers to the north and west relative to the photocenter for SMA > 170 pixels (8. 5), which results in broad distributions in the x and y co-ordinates (Fig. 24). The photocenter position is consistent with the AGN position in the y direction, but exhibits a > 3σ offset in x. Nevertheless, as the IQR of the isophote center coordinate distribution exceeds the displacement we consider this a null result.\n\n\nNGC 4696 (double nucleus)\n\nNGC 4696 is the dominant galaxy of the Centaurus cluster. There is an extensive literature on the multiwavewelength properties of this galaxy, which has a secondary nucleus (Laine et al. 2003) offset by ≈ 0. 3 to the SW of the primary (Laine et al. 2003 and Fig. 25). The double nucleus is partially surrounded by a complex of dust lanes and associated emission line filaments, the most prominent of which is in the form of a spiral leading towards the center. These features have been interpreted as evidence of a recent minor merger with a gas-rich galaxy (Sparks et al. 1989;Farage et al. 2010; alternatively the emission-line gas may have been \"drawn out\" by rising radio bubbles Canning et al. 2011). The galaxy is associated with the moderately powerful radio source PKS 1246-410, which extends ≈ 10 kpc along an approximately E-W axis, with the ends of both arms bending southwards. The radio source wraps around the region of brightest X-ray emission, suggesting interactions between the radio plasma and the hot thermal gas (Taylor et al. 2006). VLBA observations also obtained by Taylor et al. (2006) reveal a relatively weak core and a one-sided jet extending over ≈ 25 pc in PA = −150 • .\n\nSubtraction of our photometric model from the ACS/WFC-F814W image (Fig. 25) clearly shows the dust features around the nucleus. The positive residuals probably trace the emission-line filaments. The isophote centers exhibit large excursions, resulting in broad distributions in both x and (especially) y coordinates. Although we measure a 3σ offset in x between the brightest nucleus and the mean photocenter, we consider this a null result because of the large IQR.\n\n\nIC 1459\n\nIC 1459 is a giant elliptical galaxy (E3-4) and a member of a loose group (identified as \"group 15\" in Huchra & Geller 1982) of ten galaxies, which are mainly spirals. Despite its relatively unremarkable appearance to casual inspection, this galaxy turns out to be notably peculiar (see Forbes et al. 1995 for a detailed analysis): it has one of the faster counter rotating kinematically distinct cores (Franx & Illingworth 1988), faint stellar spiral arms (Malin 1985), outer shells (Forbes et al. 1995) and irregular dust lanes or patches near the nucleus (Forbes et al. 1994). It has a hard X-ray point source (González-Martín et al. 2009) and a strong, compact radio core at 5 GHz (Slee et al. 1994).\n\nWe analyzed a WFPC2/PC-F814W image (Fig. 28). After subtraction of the photometric model, the residual image shows a periodic pattern, indicating that the isophotes deviate from perfect ellipses. Systematic variations in the isophote center co-ordinates are seen in both x and y, with increasing scatter also present for SMA > 230 pixels (11. 5). We did not find a significant (> 3σ) displacement in this galaxy.\n\n\nIC 4931\n\nThis relatively little studied galaxy is the brightest member of the group A3656 (Postman & Lauer 1995). Its radio source is relatively weak, with a total radio flux at 5 GHz of 0.9 mJy (see Table 2 to compare with other galaxies). It is also detected at 1.4 GHz with a flux density of 22 mJy (Brown et al. 2011). Apart from the optical nucleus and the radio emission, there is no supporting evidence indicating the presence of an AGN such as an X-ray point source, or line emission.\n\nOur analysis of the WFPC2/PC-F814 image (Fig. 32) shows large systematic variations in the isophote center coordinates, with the x coordinate in particular migrating up to 0. 4 W relative the photocenter for SM A 100 pixels (5 ). The co-ordinate distributions are among the broadest of the whole sample, spanning a range of 0. 5 in x. We measured a ∼ 3σ offset between the photocenter and the AGN in the x co-ordinate (E), but this is considered a null result because of the large IQR.\n\nC: IMAGES Figure 9. As in Fig.3 for galaxy NGC 1399, WFPC2/PC -F606W, scale=0. 05/pxl. \n\nFigure 1 .\n1Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA.\n\nFigure 2 .\n2Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \"true errors\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC.\n\nFigure 3\n3Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers.\n\nFigure 4 .\n4Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3.\n\n\nFigure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position).\n\nFigure 5b .\n5bAs inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2.\n\nFigure 7a .\n7aProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\n\nFigure 7b .\n7bProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\n\n\n, (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000).\n\nFigure 8 .\n8M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear.\n\nFigure 18 .\n18As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl.\n\nFigure 21 .\n21As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl.\n\nFigure 22 .\n22As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl.\n\nTable 1 THE SAMPLE\n1Galaxy \nDistance \nrc \nr h \nInstrument \nPixel Scale \nID \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n\nNGC 1399 \n18 ± 1 \nsbf \n2.19 (189) \n0.99 (86) \nWFPC2/PC/F606W \n0.05 \n8214 \nWFPC2/PC/F814W \n0.05 \n5990 \nWFC3/IR/F110W \n0.09 \n11712 \nWFC3/IR/F160W \n0.09 \n11712 \nNGC 4168 \nUGC 7203 \n31 ± 6 \nsbf \n2.02 (303) \n0.09 (14) \nWFPC2/PC/F702W \n0.05 \n6357 \nNGC 4261 \nUGC 7360 \n30 ± 2 \nsbf \n1.62 (237) \n0.428 (62) \nNICMOS2/F160W \n0.05 \n7868 \nNGC 4278 \nUGC 7386 \n18 ± 1 \nsbf \n0.97 (83) \n0.334 (29) \nACS/WFC/F850LP \n0.05 \n10835 \nWFPC2/PC/F814W \n0.05 \n5454 \nNICMOS2/F160W \n0.05 \n7886 \nNGC 4373 \n40 ± 2 \nsbf \n1.19 (269) \n0.167 (32) \nWFPC2/PC/F814W \n0.05 \n5214 \nNGC 4486 \nUGC 7654 (M87) \n16 ± 1 \nsbf \n9.41 (733) \n1 (78) \nACS/HRC/F606W \n0.025 \nB10 \nACS/HRC/F814W \n0.025 \nB10 \nNICMOS2/F110W \n0.05 \n7171 \nNICMOS2/F160W \n0.05 \n7171 \nNICMOS2/F222M \n0.05 \n7171 \nWFPC2/PC/F814W \n0.05 \n8592 \nNGC 4552 \nUGC 7760 \n15 ± 1 \nsbf \n0.49 (36) \n0.48 (35) \nWFPC2/PC/F814W \n0.05 \n6099 \nNGC 4636 \nUGC 7878 \n13 ± 1 \nsbf \n3.44 (219) \n0.297 (19) \nWFPC2/PC/F814W \n0.05 \n8686 \nNGC 4696 \n37 ± 2 \nsbf \n1.4 (251) \n0.2 (36) \nACS/WFC/F814W \n0.05 \n9427 \n\n † \n\nNGC 5419 \n49 ± 9 \nDn − σ \n2.11 (499) \n0.38 (90) \nWFPC2/PC/F555W \n0.05 \n6587 \nNGC 5846 \nUGC 9706 \n25 ± 2 \nsbf \n1.52 (183) \n0.25 (30) \nWFPC2/PC/F814W \n0.05 \n5920 \nIC 1459 \n29 ± 4 \nsbf \n1.82 (258) \n0.43 (60) \nWFPC2/PC/F814W \n0.05 \n5454 \nIC 4296 \n50 ± 3 \nsbf \n1.44 (347) \n0.32 (77) \nACS/HRC/F625W \n0.025 \n9838 \nWFPC2/PC/F814W \n0.05 \n5910 \nNICMOS2/F160W \n0.05 \n7453 \nIC 4931 \n∼ 84 \nvvir \n0.99 (403) \n0.13 (51) \nWFPC2/PC/F814W \n0.05 \n8683 \n\nNote. -(1) Source name; \n\nTable 2\n2NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIESGalaxy \nRadio Source \nNature of the galaxy \nScale \nPA \nP 5GHz \nComments \n\nNGC 1399 \n10 kpc e \n−10 • \n137 a \nLinear, almost symmetric. \nDominant component of Fornax cluster. \nNGC 4168 \n-\n-\n7 ± 0.3 b \nUnresolved radio emission \nE2 in Virgo cluster. \nNGC 4261 \n30 kpc f \n87 ± 8 •p \n12657 ± 610 c Linear, symmetric lobes. W jet brighter than E. Massive E in the outskirts of Virgo cluster. \nNGC 4278 \n1.4 pc g \n100 • − 155 • \n69 ± 2 c \nS-shaped, inner jet brightest to the N. \nLarge E near the center of Coma I cloud. \nNGC 4373 \n-\n-\n40 d \nCompact radio source. \nOutskirts of Hydra-Centaurus supercluster. \nNGC 4486 1.5 kpc h \n260 • \n43552 ± 242 c Linear on kpc scale. \nVirgo Cluster BCG. \nNGC 4552 \n0.5 pc i \n∼ 50 • \n12 ± 0.4 c \nTwin pc-scale extension. \nMassive E in Virgo cluster. \nNGC 4636 \n1 kpc l \n∼ 35 • \n16 ± 3 c \nZ shaped. No radio lobes. \nMassive E in the outskirts of Virgo cluster. \nNGC 4696 1.6 kpc m \n−150 • † \n3345 a \nBroad, one sided on pc-scale. \nCentaurus Cluster BCG. \nNGC 5419 \n-\n-\n1719 a \nLow surface brightness radio relic. \nDominant component of group S753. \nNGC 5846 \n-\n-\n4 ± 0.2 b \nComplex morphology. \nDominant component of group [HG82]50. \nIC 1459 \n-\n-\n541 a \nCompact source. \nGiant E in group [HG82]15. \nIC 4296 \n240 kpc n \n130 •n \n5912 a \nLinear, NW component slightly brighter. \nGiant E, BCG of group A3565. \nIC 4931 \n-\n-\n7 d \n-\nBCG of A3656. \n\n\n\nTable 3\n3PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.Precision \nACS \nWFPC2 \nNICMOS2 WFC3 \nHRC \nWFC \nPC \nWF \n\n0.1 pxl \n2.5 \n5 \n5 \n10 \n9 \nmas \n0.2 \n0.3 \n0.3 \n0.6 \n0.6 \npc \n1 \n2 \n2 \n4.1 \n3.7 \npc \n0.2 pxl \n10 \n18 \nmas \n0.6 \n1.1 \npc \n4 \n7.3 \npc \n\n\n\n\n), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines\n\nTable 4\n4MEASURED PROJECTED OFFSETSGalaxy \nInstrument \nregion \nX \nY \nOffset \nDir Type \n\nHST \nIQR \nPC \nNPS \n[pxl] \n[mas] \n[IQR] \n[pc] \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n(10) \n(11) (12) \n\nNGC 1399 WFPC2/PC -F606W [2.5:15] 0.19 \n520 ± 0.1 \n519.9 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.5 ± 0.7 0.4 ± 0.6 \nW \nnon sig \n0.28 \n465.4 ± 0.1 \n465 ± 0.1 \n0.4 ± 0.14 \n20 ± 7 \n1.4 ± 0.5 1.7 ± 0.6 \nN \nlow \n\nWFPC2/PC -F814W \n0.29 \n502.9 ± 0.1 \n502.9 ± 0.1 \n0 ± 0.14 \n0 ± 7 \n0 ± 0.5 \n0 ± 0.6 \n-\nnon sig \n0.27 \n438.6 ± 0.1 \n438.3 ± 0.1 \n0.3 ± 0.14 \n15 ± 7 \n1.1 ± 0.5 1.3 ± 0.6 \nN \nnon sig \n\nWFC3/IR -F110W \n0.13 \n759.9 ± 0.2 \n759.7 ± 0.1 \n0.2 ± 0.22 \n18 ± 20 1.5 ± 1.7 1.6 ± 1.8 \nW \nnon sig \n0.14 \n725.4 ± 0.2 \n725.3 ± 0.1 \n0.1 ± 0.22 \n9 ± 20 \n0.7 ± 1.6 0.8 ± 1.8 \nN \nnon sig \n\nWFC3/IR -F160W \n0.18 \n759.9 ± 0.2 \n759.7 ± 0.1 \n0.2 ± 0.22 \n18 ± 20 1.1 ± 1.2 1.6 ± 1.8 \nW \nnon sig \n0.08 \n725.4 ± 0.2 \n725.4 ± 0.1 \n0 ± 0.22 \n0 ± 20 \n0 ± 2.8 \n0 ± 1.8 \n-\nnon sig \n\nNGC 4168 WFPC2/PC -F702W [2.1:8.5] 1.76 \n409.6 ± 0.1 \n408.9 ± 0.1 \n0.7 ± 0.14 \n35 ± 7 \n0.4 ± 0.1 \n5 ± 1 \nW \nnull \n1.47 \n425.1 ± 0.1 \n424.6 ± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.3 ± 0.1 \n4 ± 1 \nN \nnull \n\nNGC 4261 NICMOS2 -F160W \n[1.7:7.5] 0.08 \n200 ± 0.2 \n199.6 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 \n5 ± 3.7 \n2.9 ± 2.2 \nW \nnon sig \n0.26 \n182.3 ± 0.2 \n182.1 ± 0.2 \n0.2 ± 0.3 \n10 ± 15 0.8 ± 1.1 1.5 ± 2.2 \nN \nnon sig \n\nNGC 4278 ACS/WFC -F850LP \n[1.2:9.1] 0.31 1165.6 ± 0.1 1166.4 ± 0.1 0.8 ± 0.14 \n40 ± 7 \n2.6 ± 0.5 3.4 ± 0.6 \nE \nhigh \n0.83 \n3160 ± 0.1 \n3161.1 ± 0.1 1.1 ± 0.14 \n55 ± 7 \n1.3 ± 0.2 4.7 ± 0.6 \nS \nlow \n\nWFPC2/PC -F814W \n0.44 \n370.2 ± 0.1 \n370.6 ± 0.1 \n0.4 ± 0.14 \n22 ± 7 \n1 ± 0.3 \n1.9 ± 0.6 \nE \nlow \n0.87 \n357.3 ± 0.1 \n359 ± 0.1 \n1.7 ± 0.14 \n84 ± 7 \n1.9 ± 0.2 7.2 ± 0.6 \nS \nint \n\nNICMOS2 -F160W \n0.26 \n245.8 ± 0.2 \n246.4 ± 0.2 \n0.6 ± 0.3 \n30 ± 15 2.3 ± 1.2 2.6 ± 1.3 \nE \nnon sig \n0.62 \n224.8 ± 0.2 \n225.8 ± 0.2 \n1 ± 0.3 \n50 ± 15 1.6 ± 0.5 4.3 ± 1.3 \nS \nint \n\nNGC 4373 WFPC2/PC -F814W [1.2:10] 0.58 2328.7 ± 0.1 2328.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.2 ± 0.2 \n2 ± 2.7 \nE \nnon sig \n0.16 \n870.7 ± 0.1 \n870.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.6 ± 0.9 \n2 ± 2.7 \nS \nnon sig \n\nNGC 4486 ACS/HRC -F606W \n[1:3] \n2.78 \n419.1 ± 0.1 \n682.1 ± 0.1 \n1.4 ± 0.14 \n35 ± 4 \n0.5 ± 0.1 2.7 ± 0.3 \nW \nnull \n0.98 \n417.7 ± 0.1 \n679.5 ± 0.1 \n2.6 ± 0.14 \n65 ± 4 \n2.7 ± 0.1 5.1 ± 0.3 \nN \nhigh \n\nACS/HRC -F814W \n1.03 \n553.1 ± 0.1 \n549.6 ± 0.1 \n3.5 ± 0.14 \n87 ± 4 \n3.4 ± 0.1 6.8 ± 0.3 \nW \nhigh \n0.8 \n551.8 ± 0.1 \n549.9 ± 0.1 \n1.9 ± 0.14 \n47 ± 4 \n2.4 ± 0.1 3.6 ± 0.3 \nN \nhigh \n\nNICMOS2 -F110W \n0.42 \n207.2 ± 0.2 \n206.8 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 \n1 ± 0.7 \n1.6 ± 1.2 \nW \nnon sig \n0.96 \n277.9 ± 0.2 \n277.7 ± 0.2 \n0.2 ± 0.3 \n10 ± 15 0.2 ± 0.3 0.8 ± 1.2 \nN \nnon sig \n\nNICMOS2 -F160W \n0.65 \n207.5 ± 0.2 \n207.4 ± 0.2 \n0.1 ± 0.3 \n5 ± 15 \n0.2 ± 0.5 0.4 ± 1.2 \nW \nnon sig \n0.76 \n277.8 ± 0.2 \n277.7 ± 0.2 \n0.1 ± 0.3 \n5 ± 15 \n0.1 ± 0.4 0.4 ± 1.2 \nN \nnon sig \n\nNICMOS2 -F222M \n0.51 \n207.6 ± 0.2 \n207.2 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 0.8 ± 0.6 1.6 ± 1.2 \nW \nnon sig \n0.59 \n277.3 ± 0.2 \n277.9 ± 0.2 \n0.6 ± 0.3 \n30 ± 15 \n1 ± 0.5 \n2.3 ± 1.2 \nS \nnon sig \n\nWFPC2/PC -F814W \n0.42 \n448.3 ± 0.1 \n447.6 ± 0.1 \n0.7 ± 0.14 \n35 ± 7 \n1.7 ± 0.3 2.7 ± 0.6 \nW \nlow \n0.58 \n517.2 ± 0.1 \n517 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.3 ± 0.2 0.8 ± 0.6 \nN \nnon sig \n\n\n\n\n). Kornreich & Lovelace (2008) determine the SBH acceleration for\n\nTable 4\n4MEASURED PROJECTED OFFSETS (continued) Galaxy \nInstrument \nregion \nX \nY \nOffset \nDir \nType \n\nHST \nIQR \nPC \nNPS1 \nNPS2 \n[pxl] \n[mas] \n[IQR] \n[pc] \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n(10) \n(11) \n(12) (13) \n\nNGC 4552 WFPC2/PC -F814W \n[1:15] \n0.42 \n538.9 ± 0.1 \n538.7 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.5 ± 0.3 \n0.7 ± 0.5 W \nnon sig \n0.89 \n500.8 ± 0.1 \n500.4 ± 0.2 \n0.4 ± 0.23 \n20 ± 12 \n0.4 ± 0.3 \n1.5 ± 0.9 N \nnon sig \n\nNGC 4636 WFPC2/PC -F814W \n[4:14] \n0.94 \n698 ± 0.1 \n697.4 ± 0.1 \n0.6 ± 0.14 \n30 ± 7 \n0.6 ± 0.2 \n1.9 ± 0.5 W \nnull \n0.93 \n569.9 ± 0.1 \n570 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.1 ± 0.2 \n0.3 ± 0.5 S \nnon sig \n\nNGC 4696 \nACS/WFC -F814W \n[1.5:16] \n2.15 1582.9 ± 0.1 1583.4 ± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.2 ± 0.07 \n4.5 ± 1.3 E \nnull \n3.78 2977.6 ± 0.1 2977.8 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.05 ± 0.04 1.8 ± 1.3 S \nnon sig \n\n1586.1 ± 0.1 3.2 ± 0.14 160 ± 5 \n1.5 ± 0.05 \n29 ± 1.3 E \nlow \n2973.2 ± 0.1 4.4 ± 0.14 220 ± 5 \n1.2 ± 0.03 \n39 ± 1.3 N \nlow \n\nNGC 5419 WFPC2/PC -F555W \n[2:13.5] \n0.5 \n398.4 ± 0.1 \n397.8 ± 0.1 \n0.6 ± 0.14 \n30 ± 7 \n1.2 ± 0.3 \n7 ± 2 \nW \nlow \n0.22 \n447.6 ± 0.1 \n447.8 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.9 ± 0.6 \n2 ± 2 \nS \nnon sig \n\n397.1 ± 0.1 \n1.3 ± 0.14 \n65 ± 7 \n2.6 ± 0.2 \n15 ± 2 \nW \nhigh \n442.5 ± 0.1 \n5.1 ± 0.14 255 ± 7 23.2 ± 0.5 \n60 ± 2 \nN \nhigh \n\nNGC 5846 WFPC2/PC -F814W [1.6:15.8] 0.84 \n519.5 ± 0.1 \n518.2 ± 0.4 \n1.3 ± 0.4 \n65 ± 21 \n1.6 ± 0.5 \n7.8 ± 2.5 W \nint \n1.05 \n466 ± 0.1 \n466.4 ± 0.1 \n0.4 ± 0.14 \n20 ± 7 \n0.4 ± 0.1 \n2.4 ± 0.9 S \nnull \n\nIC 1459 \nWFPC2/PC -F814W \n[2.5:15] \n0.42 \n536.4 ± 0.1 \n536.7 ± 0.1 \n0.3 ± 0.14 \n15 ± 7 \n0.7 ± 0.3 \n2.1 ± 1 \nE \nnon sig \n0.50 \n510.5 ± 0.1 \n510.7 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.4 ± 0.3 \n1.4 ± 1 \nS \nnon sig \n\nIC 4296 \nACS/HRC -F625W \n[1.5:6.5] \n0.49 \n833.3 ± 0.1 \n832.6 ± 0.2 \n0.7 ± 0.22 \n18 ± 6 \n1.4 ± 0.4 \n4.2 ± 1.2 W \nlow \n0.38 \n785.3 ± 0.1 \n784.8 ± 0.1 \n0.5 ± 0.14 \n13 ± 4 \n1.3 ± 0.3 \n3 ± 0.6 \nN \nlow \n\nWFPC2/PC -F814W \n0.39 \n551.5 ± 0.1 \n552 ± 0.18 \n0.5 ± 0.2 \n25 ± 10 \n1.3 ± 0.5 \n6 ± 2 \nE \nnon sig \n0.13 \n515 ± 0.1 \n514.9 ± 0.18 \n0.1 ± 0.2 \n5 ± 10 \n0.8 ± 1.6 \n1 ± 2 \nN \nnon sig \n\nNICMOS2 -F160W \n0.13 \n420.8 ± 0.2 \n420.6 ± 0.2 \n0.2 ± 0.3 \n13 ± 14 \n1.9 ± 2.2 \n2.8 ± 3.2 W \nnon sig \n1.07 \n253.4 ± 0.2 \n252.7 ± 0.1 \n0.7 ± 0.2 \n37 ± 11 \n0.7 ± 0.2 \n8.3 ± 2.5 N \nnull \n\nIC 4931 \nWFPC2/PC -F814W \n[1:15] \n1.29 \n403.6 ± 0.1 \n404.1± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.4 ± 0.1 \n10 ± 3 \nE \nnull \n0.66 \n505.9 ± 0.1 \n505.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.2 ± 0.2 \n2 ± 3 \nN \nnon sig \n\n\n\nTable 5\n5RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (seeGalaxy \nPA (deg) \njet \ndisplacement \nNPS1 \nNPS2 \n\nkpc-scale jets: \nNGC 1399 \n−10 \n−17 ± 16 a \nlow \nNGC 4261 \n87 ± 8 \n297 ± 38 (117 ± 38) \nnon significant \nNGC 4486 \n290 ± 3 \n307 ± 1 (127 ± 1) \n\n ‡ \n\nNGC 4696 \n210 (+30) 112 ± 15 \n36 ± 1 \n\n ‡ \n\nIC 4296 \n130 \n338 ± 7 (158 ± 8) b \nlow \n\npc-scale jets: \nNGC 4278 \n100 − 155 152 ± 3 \n\n ‡ \n\nNGC 4552 \n50 \n333 ± 21 \nlow \nNGC 4636 \n35 \n261 ± 9 (81 ± 9) \nnull \n\n\n\nTable 6 PROBABILITIES\n6Galaxy \nd i [rc] p i (tm = 0.4 Gyr) p i (tm = 5 Gyr) \n\nNGC 1399 \n0.01 \n0.75 \n0.11 \nNGC 4168 * 0.02 \n0.98 \n0.89 \nNGC 4261 <0.01 \n0.94 \n0.23 \nNGC 4278 \n0.1 \n0.36 \n0.04 \nNGC 4373 <0.01 \n0.99 \n0.58 \nNGC 4486 \n0.01 \n0.99 \n0.85 \nNGC 4552 <0.05 \n0.10 \n0.01 \nNGC 4636 * 0.01 \n0.99 \n0.69 \nNGC 4696 * 0.02 \n0.97 \n0.42 \nNGC 5419 \n0.02 \n0.97 \n0.46 \nNGC 5846 \n0.04 \n0.90 \n0.25 \nIC 1459 \n<0.01 \n0.92 \n0.21 \nIC 4296 \n0.01 \n0.97 \n0.37 \nIC 4931 \n * 0.03 \n0.95 \n0.46 \n\n\n\nTable 7 SUMMARY\n7Galaxy \nOffset (pc) \nPA (deg) \nSuspected Origin \nNPS1 \nNPS2 \nNPS1 \nNPS2 \nGravitational \nJet \nStalling \nMassive \nRecoil \nThrust SBH-SBH \nPerturbers \n\nNGC 1399 1.5 ± 0.4 \n-17 ± 16 \nNGC 4278 7.6 ± 0.4 \n152 ± 3 \nNGC 4486 4.3 ± 0.2 \n307 ± 1 \nNGC 5846 8.2 ± 2.5 \n253 ± 8 \nIC 4296 \n\n\nTable 8\n8PARAMETERS OF RECOIL OSCILLATION MODELGalaxy \nrc \nσ \nlog(M gal ) τ \n2π/ωc v kick \n[pc] [km s −1 ] [M ] \n[Gyr] [Gyr] \n[km s −1 ] \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n\nNGC 1399 189 346 \n11.70 \n0.16 \n0.004 \n300 \nNGC 4168 303 184 \n11.44 \n4.85 \n0.012 \n159 \nNGC 4261 237 309 \n11.85 \n0.40 \n0.005 \n273 \nNGC 4278 83 \n237 \n10.99 \n0.14 \n0.002 \n205 \nNGC 4373 269 246 \n12.01 \n1.25 \n0.008 \n213 \nNGC 4486 733 334 \n12.00 \n2.84 \n0.014 \n325 \nNGC 4552 36 \n252 \n10.55 \n0.02 \n0.001 \n218 \nNGC 4636 219 203 \n11.58 \n1.73 \n0.008 \n176 \nNGC 4696 251 254 \n12.08 \n0.96 \n0.007 \n220 \nNGC 5419 499 351 \n12.26 \n1.09 \n0.01 \n304 \nNGC 5846 183 239 \n11.83 \n0.64 \n0.005 \n207 \nIC 1459 \n258 306 \n11.68 \n0.49 \n0.005 \n294 \nIC 4296 \n347 333 \n12.17 \n0.64 \n0.011 \n196 \nIC 4931 \n397 288 \n12.12 \n1.52 \n0.01 \n249 \n\naverage \n286 277 \n11.73 \n1.20 \n0.007 \n239 \n\n\n\nTable 9\n9SBH MASSESGalaxy \nlog(M•/M ) \nM•-σ M•-M V \nstars \ngas \n(1) \n(2) \n(3) \n(4) \n(5) \n\nNGC 1399 \n9.38 \n8.71 \n8.64 a , 9.01 b \nNGC 4168 \n8.04 \n8.56 \nNGC 4261 \n9.14 \n8.81 \n8.69 c \nNGC 4278 \n8.58 \n8.17 \nNGC 4373 \n8.66 \n8.72 \nNGC 4486 \n9.30 \n9.04 \n9.74 d \nNGC 4552 \n8.71 \n8.49 \nNGC 4636 \n8.25 \n8.60 \nNGC 4696 \n8.72 \n9.90 \nNGC 5419 \n9.41 \n9.39 \nNGC 5846 \n8.60 \n8.64 \nIC 1459 \n9.12 \n8.94 \n9.40 e \n8.53 -8.98 e, † , 8.15 -8.75 f \nIC 4296 \n9.30 \n8.89 \nIC 4931 \n8.99 \n9.45 \n\naverage \n8.87 \n8.88 \n\n\nThe core (or break) radius, rc, is defined as the radius where the projected surface brightness distribution of the galaxy becomes shallower than the inward extrapolation of the outer surface brightness profile.\nWe prefer to subtract a median filtered image instead of the photometric model because in the former method we can control the size of the structures that are removed. That is we can change the dimensions of the box over which the median is computed. This determines the degree of smoothing of the filter and hence the size of the structures removed after the subtraction.\nFor this purpose, the signal-to-noise ratio is defined as the ratio between the flux of the nuclear point source to the standard deviation of the residual measured in the outer regions of the galaxy after subtracting the photometric model from the image.5 The radial gradient relative error (σ /I ) is the relative error on the radial gradient of the intensity. This is a default output of ellipse.\nWe omit NGC 5419 from this list, since the origin of its unusual radio source is unclear, see Section B.\nNote that this quantity was derived by making use of merger time-scales calibrated against virtual galaxy catalogs from the Millennium Simulation(Springel et al. 2005) and it depends on the\nf jet ≤ 1, unless energy is extracted from the SBH rotation(Blandford & Znajek 1977).\nFigure 10. As inFig.3for galaxy NGC 1399, WFPC2/PC -F814W, scale=0. 05/pxl.\nFigure 11. As inFig.3for galaxy NGC 1399, WFC3/IR -F110W, scale=0. 09/pxl.\nFigure 12. As inFig.3for galaxy NGC 1399, WFC3/IR -F160W, scale=0. 09/pxl.\nFigure 13. As inFig.3for galaxy NGC 4168, WFPC2/PC -F702W, scale=0. 05/pxl.\nFigure 14. As inFig.3for galaxy NGC 4261, NICMOS2 -F160W, scale=0. 05/pxl.\nFigure 15. As inFig.3for galaxy NGC 4278, ACS/WFC -F850LP, scale=0. 05/pxl.\nFigure 16. As inFig.3for galaxy NGC 4278, WFPC2/PC -F814W, scale=0. 05/pxl.\nFigure 17. As inFig.3for galaxy NGC 4278, NICMOS2 -F160W, scale=0. 05/pxl.\nFigure 19. As inFig.3for galaxy NGC 4486 (M 87), NICMOS2 -F110W, scale=0. 05/pxl.\nFigure 20. As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F160W, scale=0. 05/pxl.\nFigure 23. As inFig.3for galaxy NGC 4552, WFPC2/PC -F814W, scale=0. 05/pxl.\nFigure 24. As inFig.3for galaxy NGC 4636, WFPC2/PC -F814W, scale=0. 05/pxl.\nFigure 25. As inFig.3for galaxy NGC 4696, ACS/WFC -F814W, scale=0. 05/pxl.\nFigure 26. As inFig.3for galaxy NGC 5419, WFPC2/PC -F555W, scale=0. 05/pxl.\nFigure 27. As inFig.3for galaxy NGC 5846, WFPC2/PC -F814W, scale=0. 05/pxl.\nFigure 28. As inFig.3for galaxy IC 1459, WFPC2/PC -F814W, scale=0. 05/pxl.\nFigure 29. As inFig.3for galaxy IC 4296, ACS/HRC -F625W, scale=0. 025/pxl.\nFigure 30. As inFig.3for galaxy IC 4296, WFPC2/PC -F814W, scale=0. 05/pxl.\nFigure 31. As inFig.3for galaxy IC 4296, NICMOS2 -F160W, scale=0. 05/pxl.\nFigure 32. As inFig.3for galaxy IC 4931, WFPC2/PC -F814W, scale=0. 05/pxl.\n\n. G O Abell, H G Corwin, Jr, R P Olowin, ApJS. 701Abell, G. O., Corwin, H. G., Jr., & Olowin, R. P. 1989, ApJS, 70, 1\n\n. D M Alexander, R C Hickox, NAR. 5693Alexander, D. M., & Hickox, R. C. 2012, NAR, 56, 93\n\n. J M Anderson, J S Ulvestad, L C Ho, ApJ. 60342Anderson, J. M., Ulvestad, J. S., & Ho, L. C. 2004, ApJ, 603, 42\n\n. J Antognini, J Bird, P Martini, ApJ. 756116Antognini, J., Bird, J. & Martini, P. 2012, ApJ, 756, 116\n\n. F Antonini, D Merritt, ApJ. 74583Antonini, F., & Merritt, D. 2012, ApJ, 745, 83\n\n. W Baade, R Minkowski, ApJ. 119215Baade, W., & Minkowski, R. 1954, ApJ, 119, 215\n\n. J N Bahcall, R A Wolf, ApJ. 209214Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214\n\n. R D Baldi, M Chiaberge, A Capetti, W Sparks, F D Macchetto, C P O'dea, D J Axon, S A Baum, A C Quillen, ApJ. 7252426Baldi, R. D., Chiaberge, M., Capetti, A., Sparks, W., Macchetto, F. D., O'Dea, C. P., Axon, D. J., Baum, S. A. & Quillen, A. C. 2010, ApJ, 725, 2426\n\n. B Balmaverde, R D Baldi, A Capetti, A&A. 486119Balmaverde, B., Baldi, R. D., & Capetti, A. 2008, A&A, 486, 119\n\n. B Balmaverde, A Capetti, A&A. 447BC0697Balmaverde, B., & Capetti, A. 2006, A&A, 447, 97 (BC06)\n\n. J G Baker, J Centrella, D.-I Choi, M Koppitz, J Van Meter, Phys. Rev. Lett. 96111102Baker, J. G. , Centrella, J. , Choi,D.-I., Koppitz, M., & van Meter, J. 2006, Phys. Rev. Lett. 96, 111102\n\n. D Batcheldor, A Robinson, D J Axon, E S Perlman, D Merritt, ApJ. 717B106Batcheldor, D., Robinson, A., Axon, D. J., Perlman, E. S., & Merritt, D. 2010, ApJ, 717, L6 (B10)\n\n. M C Begelman, R D Blandford, M J Rees, Nature. 287307Begelman, M.C., Blandford, R.D. & Rees, M.J., 1980, Nature, 287, 307\n\n. J D Bekenstein, ApJ. 183657Bekenstein, J. D. 1973, ApJ, 183, 657\n\n. E F Bell, S Phleps, R S Somerville, C Wolf, A Borch, K Meisenheimer, ApJ. 652270Bell, E. F., Phleps, S., Somerville, R. S., Wolf, C., Borch, A. & Meisenheimer, K. 2006, ApJ, 652, 270\n\n. M Birkinshaw, R L Davies, ApJ. 29132Birkinshaw, M., & Davies, R. L. 1985, ApJ, 291, 32\n\n. R D Blandford, R L Znajek, MNRAS. 179433Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433\n\n. L Blecha, F Civano, M Elvis, A Loeb, MNRAS. 4281341Blecha, L., Civano, F., Elvis, M. & Loeb, A. 2013, MNRAS, 428,1341\n\n. E W Bonning, G A Shields, S Salviander, ApJ. 66613Bonning, E. W., Shields, G. A. & Salviander, S. 2007, ApJ, 666, 13\n\n. M J I Brown, B T Jannuzi, D J E Floyd, J Mould, R , ApJ. 73141Brown, M. J. I., Jannuzi, B. T., Floyd, D. J. E., Mould, J., R. 2011, ApJ, 731, L41\n\n. C Burke, C A Collins, MNRAS. 4342856Burke, C. & Collins, C. A., 2013, MNRAS, 434, 2856\n\nI C Busko, Astronomical Data Analysis Software and Systems V. 101139Busko, I. C. 1996, Astronomical Data Analysis Software and Systems V, 101, 139\n\n. M Campanelli, C O Lousto, P Marronetti, Y Zlochower, Phys. Rev. Lett. 96111101Campanelli, M., Lousto, C. O., Marronetti, P., & Zlochower, Y. 2006, Phys. Rev. Lett. 96, 111101\n\n. M Campanelli, C O Lousto, Y Zlochower, D Merritt, Physical Review Letters. 98231102Campanelli, M., Lousto, C. O., Zlochower, Y., & Merritt, D. 2007, Physical Review Letters, 98, 231102\n\n. M Campanelli, C Lousto, Y Zlochower, D Merritt, ApJ. 6595Campanelli, M., Lousto, C., Zlochower, Y., & Merritt, D. 2007, ApJ, 659, L5\n\n. R E A Canning, A C Fabian, R M Johnstone, J S Sanders, C S Crawford, G J Ferland, N A Hatch, MNRAS. 4173080Canning, R.E.A., Fabian, A. C., Johnstone, R. M., Sanders, J. S., Crawford, C. S., Ferland, G. J., Hatch, N. A. 2011, MNRAS, 417, 3080\n\n. A Capetti, D J Axon, M Chiaberge, A&A. 471137Capetti, A., Axon, D. J., Chiaberge, M., et al. 2007, A&A, 471, 137\n\n. A Capetti, B Balmaverde, A&A. 440CB0573Capetti, A., & Balmaverde, B. 2005, A&A, 440, 73 (CB05)\n\n. A Capetti, B Balmaverde, A&A. 453CB0627Capetti, A., & Balmaverde, B. 2006, A&A, 453, 27 (CB06)\n\n. M Cappellari, A Renzini, L Greggio, ApJ. 519117Cappellari, M., Renzini, A., Greggio, L., et al. 1999, ApJ, 519, 117\n\n. M Cappellari, E K Verolme, R P Van Der Marel, ApJ. 578787Cappellari, M., Verolme, E. K., van der Marel, R. P., et al. 2002, ApJ, 578, 787\n\n. M Cappellari, E Emsellem, R Bacon, MNRAS. 379418Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418\n\n. A Cardullo, E M Corsini, A Beifiori, A&A. 508641Cardullo, A., Corsini, E. M., Beifiori, A., et al. 2009, A&A, 508, 641\n\n. C M Carollo, M Franx, G D Illingworth, D A Forbes, ApJ. 481710Carollo, C. M., Franx, M., Illingworth, G. D., & Forbes, D. A. 1997, ApJ, 481, 710\n\n. K W Cavagnolo, B R Mcnamara, P E J Nulsen, ApJ. 7201066Cavagnolo, K. W., McNamara, B. R., Nulsen, P. E. J., et al. 2010, ApJ, 720, 1066\n\n. M Chiaberge, A Capetti, A Celotti, A&A. 34977Chiaberge, M., Capetti, A., & Celotti, A. 1999, A&A, 349, 77\n\n. F Civano, M Elvis, G Lanzuisi, ApJ. 717209Civano, F., Elvis, M., Lanzuisi, G., et al. 2010, ApJ, 717, 209\n\n. F Civano, M Elvis, G Lanzuisi, ApJ. 75249Civano, F., Elvis, M., Lanzuisi, G., et al. 2012, ApJ, 752, 49\n\nP Coles, F Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure. New YorkWiley2nd ed.Coles, P., & Lucchin, F. 2002, Cosmology: The Origin and Evolution of Cosmic Structure (2nd ed.; New York: Wiley)\n\n. C J Conselice, C Yang, A F L Bluck, MNRAS. 3941956Conselice, C. J., Yang, C. & Bluck, A. F. L. 2009, MNRAS, 394, 1956\n\n. J Cuadra, P J Armitage, R D Alexander, M C Begelman, MNRAS. 3931423Cuadra, J., Armitage, P. J. Alexander, R. D. & Begelman, M. C. 2009, MNRAS, 393, 1423\n\n. V P Debattista, I Ferreras, A Pasquali, ApJ. 65197Debattista, V. P., Ferreras, I., Pasquali, A., et al. 2006, ApJ, 651, L97\n\nThird Reference Catalogue of Bright Galaxies Di Matteo. G De Vaucouleurs, A De Vaucouleurs, H G Corwin, Jr, ApJ. 582133de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Jr., et al. 1991, Third Reference Catalogue of Bright Galaxies Di Matteo, T., Allen, S. W., Fabian, A. C., Wilson, A. S., & Young, A. J. 2003, ApJ, 582, 133\n\n. M Dotti, M Volonteri, A Perego, M Colpi, M Ruszkowski, F Haardt, MNRAS. 402682Dotti, M., Volonteri, M., Perego, A., Colpi, M., Ruszkowski, M. & Haardt, F. 2010, MNRAS, 402, 682\n\n. A Dressler, D Lynden-Bell, D Burstein, ApJ. 31342Dressler, A., Lynden-Bell, D., Burstein, D., et al. 1987, ApJ, 313, 42\n\n. G Efstathiou, R S Ellis, D Carter, MNRAS. 193931Efstathiou, G., Ellis, R. S., & Carter, D. 1980, MNRAS, 193, 931\n\n. M Eracleous, T A Boroson, J P Halpern, J Liu, ApJS. 20123Eracleous, M., Boroson, T. A., Halpern, J. P., & Liu, J. 2012, ApJS, 201, 23\n\n. A Escala, R B Larson, P S Coppi, D Mardones, ApJ. 630152Escala, A., Larson, R. B., Coppi, P. S. & Mardones, D. 2005, ApJ, 630, 152\n\n. G Fabbiano, M Elvis, S Markoff, ApJ. 588175Fabbiano, G., Elvis, M., Markoff, S., et al. 2003, ApJ, 588, 175\n\n. S M Faber, S Tremaine, E A Ajhar, AJ. 1141771Faber, S. M., Tremaine, S., Ajhar, E. A., et al. 1997, AJ, 114, 1771\n\n. B L Fanaroff, J M Riley, MNRAS. 16731Fanaroff, B. L., & Riley, J. M. 1974, MNRAS, 167, 31P\n\n. C L Farage, P J Mcgregor, M A Dopita, G V Bicknell, ApJ. 724267Farage, C. L., McGregor, P. J., Dopita, M. A., & Bicknell, G. V. 2010, ApJ, 724, 267\n\n. M Favata, S A Hughes, D E Holz, ApJ. 6075Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ, 607, L5\n\n. H C Ferguson, AJ. 98367Ferguson, H. C. 1989, AJ, 98, 367\n\n. L Ferrarese, H C Ford, W Jaffe, ApJ. 470444Ferrarese, L., Ford, H. C., & Jaffe, W. 1996, ApJ, 470, 444\n\n. L Ferrarese, D Merritt, ApJ. 5399Ferrarese, L., & Merritt, D. 2000, ApJ, 539, L9\n\n. L Ferrarese, H Ford, Sci. Rev. 116523Ferrarese, L., Ford, H., 2005, Sci. Rev., 116, 523\n\n. M E Filho, P D Barthel, L C Ho, ApJS. 12993Filho, M. E., Barthel, P. D., & Ho, L. C. 2000, ApJS, 129, 93\n\n. M E Filho, F Fraternali, S Markoff, A&A. 418429Filho, M. E., Fraternali, F., Markoff, S., et al. 2004, A&A, 418, 429\n\n. D A Forbes, M Franx, G D Illingworth, ApJ. 42849Forbes, D. A., Franx, M., & Illingworth, G. D. 1994, ApJ, 428, L49\n\n. D A Forbes, J P Brodie, J Huchra, AJ. 1122448Forbes, D. A., Brodie, J. P., & Huchra, J. 1996, AJ, 112, 2448\n\n. D A Forbes, D B Reitzel, G M Williger, AJ. 1091576Forbes, D. A., Reitzel, D. B., & Williger, G. M. 1995, AJ, 109, 1576\n\n. M Franx, G D Illingworth, ApJ. 32755Franx, M., & Illingworth, G. D. 1988, ApJ, 327, L55\n\n. O Fakhouri, C.-P Ma, M Boylan-Kolchin, MNRAS. 4062267Fakhouri, O., Ma, C.-P., & Boylan-Kolchin, M. 2010, MNRAS, 406, 2267\n\n. K Gebhardt, R Bender, G Bower, ApJ. 53913Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13\n\n. K Gebhardt, T R Lauer, J Pinkney, ApJ. 6711321Gebhardt, K., Lauer, T. R., Pinkney, J., et al. 2007, ApJ, 671, 1321\n\n. K Gebhardt, J Adams, D Richstone, ApJ. 729119Gebhardt, K., Adams, J., Richstone, D., et al. 2011, ApJ, 729, 119\n\n. M Gerin, F Casoli, A&A. 29049Gerin, M., & Casoli, F. 1994, A&A, 290, 49\n\n. M Giroletti, G B Taylor, G Giovannini, ApJ. 622178Giroletti, M., Taylor, G. B., & Giovannini, G. 2005, ApJ, 622, 178\n\n. L Giordano, L Cortese, G Trinchieri, A Wolter, M Colpi, G Gavazzi, L Mayer, ApJ. 634272Giordano, L., Cortese, L., Trinchieri, G., Wolter, A., Colpi, M., Gavazzi, G., & Mayer, L. 2005, ApJ, 634, 272\n\n. J A González, M Hannam, U Sperhake, B Brügmann, S Husa, Phys. Rev. Lett. 98231101González, J. A., Hannam, M., Sperhake, U., Brügmann, B., & Husa, S. 2007, Phys. Rev. Lett., 98, 231101\n\n. O González-Martín, J Masegosa, I Márquez, M Guainazzi, E Jiménez-Bailón, A&A. 5061107González-Martín, O., Masegosa, J., Márquez, I., Guainazzi, M., & Jiménez-Bailón, E. 2009, A&A, 506, 1107\n\n. R Grützbauch, G Trinchieri, R Rampazzo, AJ. 133220Grützbauch, R., Trinchieri, G., Rampazzo, R., et al. 2007, AJ, 133, 220\n\n. A Gualandris, D Merritt, A Gualandris, D Merritt, ApJ. 678GM0874ApJGualandris, A., & Merritt, D. 2008, ApJ, 678, 780 (GM08) Gualandris, A., & Merritt, D. 2012, ApJ, 744, 74\n\n. K Hada, A Doi, M Kino, Nature. 477185Hada, K., Doi, A., Kino, M., et al. 2011, Nature, 477, 185\n\n. M Hilz, T Naab, J P Ostriker, arXiv:1206.5004Hilz, M., Naab, T., & Ostriker, J. P. 2012, arXiv:1206.5004\n\n. L C Ho, ARA&A. 46475Ho, L. C. 2008, ARA&A, 46, 475\n\n. L C Ho, A V Filippenko, W L W Sargent, C Y Peng, ApJS. 112391Ho, L. C., Filippenko, A. V., Sargent, W. L. W. & Peng, C. Y. 1997, ApJS, 112, 391\n\n. P F Hopkins, K Bundy, D Croton, L Hernquist, D Keres, S Khochfar, K Stewart, A Wetzel, J D Younger, ApJ. 715H10202Hopkins, P. F., Bundy, K., Croton, D., Hernquist, L., Keres, D., Khochfar, S., Stewart, K., Wetzel, A. & Younger, J. D. 2010, ApJ, 715, 202 (H10)\n\n. R C W Houghton, J Magorrian, M Sarzi, MNRAS. 3672Houghton, R. C. W., Magorrian, J., Sarzi, M., et al. 2006, MNRAS, 367, 2\n\n. J P Huchra, M J Geller, ApJ. 257423Huchra, J. P., & Geller, M. J. 1982, ApJ, 257, 423\n\n. J Huchra, M Davis, D Latham, J Tonry, ApJS. 5289Huchra, J., Davis, M., Latham, D., & Tonry, J. 1983, ApJS, 52, 89\n\n. W Jaffe, H C Ford, L Ferrarese, Van Den, F Bosch, R W Connell, Nature. 364213Jaffe, W., Ford, H. C., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1993, Nature, 364, 213\n\n. W Jaffe, B R Mcnamara, ApJ. 434110Jaffe, W., & McNamara, B. R. 1994, ApJ, 434, 110\n\n. W Jaffe, H Ford, L Ferrarese, Van Den, F Bosch, R W Connell, ApJ. 460214Jaffe, W., Ford, H., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1996, ApJ, 460, 214\n\n. S Janowiecki, J C Mihos, P Harding, ApJ. 715972Janowiecki, S., Mihos, J. C., Harding, P., et al. 2010, ApJ, 715, 972\n\n. R I Jedrzejewski, MNRAS. 226747Jedrzejewski, R. I. 1987, MNRAS, 226, 747\n\n. D L Jones, A E Wehrle, D L Meier, B G Piner, ApJ. 534165Jones, D. L., Wehrle, A. E., Meier, D. L., & Piner, B. G. 2000, ApJ, 534, 165\n\n. C Jones, W Forman, A Vikhlinin, ApJ. 567115Jones, C., Forman, W., Vikhlinin, A., et al. 2002, ApJ, 567, L115\n\n. J I Katz, ApJ. 478527Katz, J. I. 1997, ApJ, 478, 527\n\n. F M Khan, A Just, D Merritt, ApJ. 73289Khan, F. M., Just, A., & Merritt, D. 2011, ApJ, 732, 89\n\n. N E B Killeen, G V Bicknell, R D Ekers, ApJ. 302306Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1986, ApJ, 302, 306\n\n. N E B Killeen, G V Bicknell, D Carter, ApJ. 30945Killeen, N. E. B., Bicknell, G. V., & Carter, D. 1986, ApJ, 309, 45\n\n. N E B Killeen, G V Bicknell, ApJ. 324198Killeen, N. E. B., & Bicknell, G. V. 1988, ApJ, 324, 198\n\n. N E B Killeen, G V Bicknell, R D Ekers, ApJ. 325180Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1988, ApJ, 325, 180\n\n. S Komossa, H Zhou, H Lu, ApJ. 67881Komossa, S., Zhou, H., & Lu, H. 2008, ApJ, 678, L81\n\n. S Komossa, Mem. Soc. Astron. Italiana. 77733Komossa, S. 2006, Mem. Soc. Astron. Italiana, 77, 733\n\n. J Kormendy, D Richstone ; Ara&a Kornreich, D A Lovelace, R V E , 681..104KJ. Kormendy, D. Richstone, 1995, ARA&A Kornreich D. A. and Lovelace R. V. E., 2008ApJ...681..104K\n\nJ Krist, Astronomical Data Analysis Software and Systems II. 52536Krist, J. 1993, Astronomical Data Analysis Software and Systems II, 52, 536\n\n. P Kroupa, arXiv:1406.4860Kroupa, P. 2014, arXiv:1406.4860\n\n. S Laine, R P Van Der Marel, T R Lauer, M Postman, C P O'dea, F N Owen, AJ. 125478Laine, S., van der Marel, R. P., Lauer, T. R., Postman, M., O'Dea, C. P., & Owen, F. N. 2003, AJ, 125, 478\n\n. T R Lauer, S M Faber, E J Groth, AJ. 1061436Lauer, T. R., Faber, S. M., Groth, E. J., et al. 1993, AJ, 106, 1436\n\n. T R Lauer, E A Ajhar, Y.-I Byun, AJ. 1102622Lauer, T. R., Ajhar, E. A., Byun, Y.-I., et al. 1995, AJ, 110, 2622\n\n. T R Lauer, S Tremaine, E A Ajhar, ApJ. 47179Lauer, T. R., Tremaine, S., Ajhar, E. A., et al. 1996, ApJ, 471, L79\n\n. T R Lauer, S M Faber, K Gebhardt, AJ. 1292138Lauer, T. R., Faber, S. M., Gebhardt, K., et al. 2005, AJ, 129, 2138\n\n. T R Lauer, S M Faber, D Richstone, ApJ. 662808Lauer, T. R., Faber, S. M., Richstone, D., et al. 2007, ApJ, 662, 808\n\n. C Lidman, G Iacobuta, A E Bauer, MNRAS. 433825Lidman, C., Iacobuta, G., Bauer, A. E., et al. 2013, MNRAS, 433, 825\n\n. A P Lobanov, J Roland, A&A. 431831Lobanov, A. P., & Roland, J. 2005, A&A, 431, 831\n\n. A Loeb, Phrvl, 9941103Loeb, A., PhRvL, 99, 041103L\n\n. M Loewenstein, R F Mushotzky, L Angelini, K A Arnaud, E Quataert, ApJ. 55521Loewenstein, M., Mushotzky, R. F., Angelini, L., Arnaud, K. A., & Quataert, E. 2001, ApJ, 555, L21\n\n. C O Lousto, H Nakano, Y Zlochower, M Campanelli, Phys. Rev. D. 8184023Lousto, C. O., Nakano, H., Zlochower, Y., & Campanelli, M. 2010, Phys. Rev. D 81, 084023\n\n. C O Lousto, Y Zlochower, Phys. Rev. Lett. 107231102Lousto, C. O. & Zlochower, Y. 2011, Phys. Rev. Lett. 107, 231102\n\n. C O Lousto, Y Zlochower, M Dotti, M Volonteri, arXiv:1201.1923(L12Lousto, C. O., Zlochower, Y., Dotti, M., & Volonteri, M. 2012, arXiv:1201.1923 (L12)\n\n. F Macchetto, M Pastoriza, N Caon, A&AS. 120463Macchetto, F., Pastoriza, M., Caon, N., et al. 1996, A&AS, 120, 463\n\n. D Maoz, N M Nagar, H Falcke, A S Wilson, ApJ. 625699Maoz, D., Nagar, N. M., Falcke, H., & Wilson, A. S. 2005, ApJ, 625, 699\n\n. B R Mcnamara, ApJ. 562149McNamara, B. R., et al. 2001, ApJ, 562, L149\n\n. M E Machacek, D Jerius, R Kraft, ApJ. 74315Machacek, M. E., Jerius, D., Kraft, R., et al. 2011, ApJ, 743, 15\n\n. P Madau, E Quataert, ApJ. 60617Madau, P. and Quataert, E. 2004, ApJ, 606, 17\n\n. D F Malin, Nature. 277279Malin, D. F. 1979, Nature, 277, 279\n\nD F Malin, New Aspects of Galaxy Photometry. D. F. Malin, New Aspects of Galaxy Photometry, 1985\n\n. L Mayer, S Kazantzidis, P Madau, Science. 3161874Mayer, L., Kazantzidis, S., Madau, P., et al. 2007, Science, 316, 1874\n\n. J Masegosa, I Márquez, A Ramirez, O González-Martín, A&A. 52723Masegosa, J., Márquez, I., Ramirez, A., & González-Martín, O. 2011, A&A, 527, A23\n\nD Merritt, Dynamics and Evolution of Galactic Nuclei. Princeton, NJPrinceton Univ. PressMerritt, D. 2013, Dynamics and Evolution of Galactic Nuclei (Princeton, NJ: Princeton Univ. Press)\n\n. D Merritt, R D Ekers, Science. 2971310Merritt, D., & Ekers, R. D. 2002, Science, 297, 1310\n\n. D Merritt, M Milosavljević, M Favata, S A Hughes, D E Holz, ApJ. 6079Merritt, D., Milosavljević, M., Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ 607, L9.\n\n. D Merritt, M Milosavljević, Living Reviews in Relativity. 88Merritt, D., & Milosavljević, M. 2005, Living Reviews in Relativity, 8, 8\n\n. D Merritt, ApJ. 648976Merritt, D. 2006, ApJ, 648, 976\n\n. D Merritt, J D Schnittman, S Komossa, ApJ. 6991690Merritt, D., Schnittman, J. D., & Komossa, S. 2009, ApJ, 699, 1690\n\n. B Y Mills, O B Slee, E R Hill, Australian Journal of Physics. 13676Mills, B. Y., Slee, O. B., & Hill, E. R. 1960, Australian Journal of Physics, 13, 676\n\n. C Moellenhoff, E Hummel, R Bender, A&A. 25535Moellenhoff, C., Hummel, E., & Bender, R. 1992, A&A, 255, 35\n\n. N M Nagar, H Falcke, A S Wilson, J S Ulvestad, A&A. 39253Nagar, N. M., Falcke, H., Wilson, A. S., & Ulvestad, J. S. 2002, A&A, 392, 53\n\n. R Nolthenius, ApJS. 851Nolthenius, R. 1993, ApJS, 85, 1\n\n. R W O'connell, J R Martin, J D Crane, ApJ. 635305O'Connell, R. W., Martin, J. R., Crane, J. D., et al. 2005, ApJ, 635, 305\n\n. C P O'dea, R A Daly, P Kharb, K A Freeman, S A Baum, A&A. 494471O'Dea, C. P., Daly, R. A., Kharb, P., Freeman, K. A. & Baum, S. A. 2009, A&A, 494, 471\n\n. P Parma, M Murgia, R Morganti, A Capetti, H R De Ruiter, R Fanti, A&A. 3447Parma, P., Murgia, M., Morganti, R., Capetti, A., de Ruiter, H. R. & Fanti, R. 1999, A&A, 344, 7\n\n. S Pellegrini, T Venturi, A Comastri, ApJ. 585677Pellegrini, S., Venturi, T., Comastri, A., et al. 2003, ApJ, 585, 677\n\n. H V Peiris, S Tremaine, ApJ. Peiris, H. V. & Tremaine, S. 2003, ApJ, 237-257\n\n. C Y Peng, L C Ho, C D Impey, H.-W Rix, AJ. 124266Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266\n\n. H B Perets, C Hopman, T Alexander, ApJ. 656709Perets, H. B., Hopman, C., & Alexander, T. 2007, ApJ, 656, 709\n\n. E S Perlman, D E Harris, J A Biretta, W B Sparks, F D Macchetto, ApJ. 59965Perlman, E. S., Harris, D. E., Biretta, J. A., Sparks, W. B., & Macchetto, F. D. 2003, ApJ, 599, L65\n\n. M Postman, T R Lauer, ApJ. 44028Postman, M., & Lauer, T. R. 1995, ApJ, 440, 28\n\n. M Preto, I Berentzen, P Berczik, R Spurzem, ApJ. 73226Preto, M., Berentzen, I., Berczik, P., & Spurzem, R. 2011, ApJ, 732, L26\n\n. F Pretorius, Phys. Rev. Lett. 95121101Pretorius, F. 2005, Phys. Rev. Lett. 95, 121101\n\n. A Renzini, L Greggio, S Di Serego Alighieri, Nature. 37839Renzini, A., Greggio, L., di Serego Alighieri, S., et al. 1995, Nature, 378, 39\n\n. A Robinson, S Young, D J Axon, P Kharb, J E Smith, ApJ. 717122Robinson, A., Young, S., Axon, D. J., Kharb, P., & Smith, J. E. 2010, ApJ, 717, L122\n\n. G E Romero, L Chajet, Z Abraham, J H Fan, A&A. 36057Romero, G. E., Chajet, L., Abraham, Z., & Fan, J. H. 2000, A&A, 360, 57\n\n. N Roos, J S Kaastra, C A Hummel, ApJ. 409130Roos, N., Kaastra, J. S., & Hummel, C. A. 1993, ApJ, 409, 130\n\n. E M Sadler, N A Sharp, A&A. 133216Sadler, E. M., & Sharp, N. A. 1984, A&A, 133, 216\n\n. E M Sadler, C R Jenkins, C G Kotanyi, MNRAS. 240591Sadler, E. M., Jenkins, C. R., & Kotanyi, C. G. 1989, MNRAS, 240, 591\n\n. E Sani, A Marconi, L K Hunt, G Risaliti, MNRAS. 4131479Sani, E., Marconi, A., Hunt, L. K., & Risaliti, G. 2011, MNRAS, 413, 1479\n\n. W C Saslaw, M Whittle, ApJ. 325142Saslaw, W. C., & Whittle, M. 1988, ApJ, 325, 142\n\nExtragalactic Radio Sources. I S Shklovsky, 97475Shklovsky, I. S. 1982, Extragalactic Radio Sources, 97, 475\n\n. G A Shields, D J Rosario, K L Smith, ApJ. 707936Shields, G. A., Rosario, D. J., Smith, K. L., et al. 2009, ApJ, 707, 936\n\n. K Shurkin, R J H Dunn, G Gentile, G B Taylor, S W Allen, MNRAS. 383923Shurkin, K., Dunn, R. J. H., Gentile, G., Taylor, G. B., & Allen, S. W. 2008, MNRAS, 383, 923\n\n. D Sijacki, V Springel, M G Haehnelt, MNRAS. 4143656Sijacki, D., Springel, V. & Haehnelt, M. G. 2011, MNRAS, 414, 3656\n\n. O B Slee, E M Sadler, J E Reynolds, R D Ekers, MNRAS. 269928Slee, O. B., Sadler, E. M., Reynolds, J. E., & Ekers, R. D. 1994, MNRAS, 269, 928\n\n. W B Sparks, F Macchetto, D Golombek, ApJ. 345153Sparks, W. B., Macchetto, F., & Golombek, D. 1989, ApJ, 345, 153\n\n. V Springel, S D M White, A Jenkins, Nature. 435629Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629\n\n. V J Stanger, R S Warwick, MNRAS. 220363Stanger, V. J., & Warwick, R. S. 1986, MNRAS, 220, 363\n\n. C L Steinhardt, M Schramm, J D Silverman, ApJ. 75924Steinhardt, C. L., Schramm, M., Silverman, J. D., et al. 2012, ApJ, 759, 24\n\n. R Subrahmanyan, A J Beasley, W M Goss, K Golap, R W Hunstead, AJ. 1251095Subrahmanyan, R., Beasley, A. J., Goss, W. M., Golap, K., & Hunstead, R. W. 2003, AJ, 125, 1095\n\n. G B Taylor, J S Sanders, A C Fabian, S W Allen, MNRAS. 365705Taylor, G. B., Sanders, J. S., Fabian, A. C., & Allen, S. W. 2006, MNRAS, 365, 705\n\n. T Tal, P G Van Dokkum, J Nelan, R Bezanson, AJ. 1381417Tal, T., van Dokkum, P. G., Nelan, J., & Bezanson, R. 2009, AJ, 138, 1417\n\n. A Tarchi, N Neininger, A Greve, A&A. 35895Tarchi, A., Neininger, N., Greve, A., et al. 2000, A&A, 358, 95\n\n. N Thatte, M Tecza, R Genzel, A&A. 36447Thatte, N., Tecza, M., & Genzel, R. 2000, A&A, 364, L47\n\n. P Temi, W G Mathews, F Brighenti, J D Bregman, ApJ. 585121Temi, P., Mathews, W. G., Brighenti, F., & Bregman, J. D. 2003, ApJ, 585, L121\n\n. W Tichy, P Marronetti, Phys. Rev. D. 061502. Trinchieri, G., & di Serego Alighieri761647AJTichy, W. & Marronetti, P. 2007, Phys. Rev. D 76, 061502. Trinchieri, G., & di Serego Alighieri, S. 1991, AJ, 101, 1647\n\n. P Tsalmantza, R Decarli, M Dotti, D W Hogg, 73820Tsalmantza, P., Decarli, R., Dotti, M. & Hogg, D. W., 2011ApJ, 738, 20\n\n. E Vasiliev, F Antonini, D Merritt, P G Van Dokkum, M Franx, ApJ. 7852027AJVasiliev, E., Antonini, F., & Merritt, D. 2014, ApJ, 785, 163 van Dokkum, P. G., & Franx, M. 1995, AJ, 110, 2027\n\n. G A Verdoes Kleijn, R P Van Der Marel, C M Carollo, P T De Zeeuw, AJ. 1201221Verdoes Kleijn, G. A., van der Marel, R. P., Carollo, C. M., & de Zeeuw, P. T. 2000, AJ, 120, 1221\n\n. M Volonteri, K Gúltekin, M Dotti, 4042143Volonteri, M.; Gúltekin, K.; Dotti, M., 2010MNRAS, 404, 2143\n\n. D M Worrall, M Birkinshaw, E O'sullivan, A Zezas, A Wolter, G Trinchieri, G Fabbiano, MNRAS. 408701Worrall, D. M., Birkinshaw, M., O'Sullivan, E., Zezas, A., Wolter, A., Trinchieri, G., & Fabbiano, G. 2010, MNRAS, 408, 701\n\n. J M Wrobel, AJ. 101127Wrobel, J. M. 1991, AJ, 101, 127\n\n. J M Wrobel, D S Heeschen, AJ. 101148Wrobel, J. M., & Heeschen, D. S. 1991, AJ, 101, 148\n\n. S Younis, J Meaburn, P Stewart, A&A. 147178Younis, S., Meaburn, J., & Stewart, P. 1985, A&A, 147, 178\n\n. O Zanotti, L Rezzolla, L Del Zanna, C Palenzuela, A&A. 5238Zanotti, O., Rezzolla, L., Del Zanna, L., & Palenzuela, C. 2010, A&A, 523, A8\n"},"corpus_id":{"kind":"number","value":7331754,"string":"7,331,754"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1409.3976\",\n \"mag\": 3106526245,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/0004-637x/795/2/146\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1409.3976v1.pdf\"\n ],\n \"pdf_sha\": \"b59b6b3c83a88c14e53b6f939f1279937a6f2e6e\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1409.3976\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Accepted for publication on ApJ RECOILING SUPERMASSIVE BLACK HOLES: A SEARCH IN THE NEARBY UNIVERSE"},"authors":{"kind":"list like","value":["D Lena","A Robinson","A Marconi","D J Axon","A Capetti","D Merritt","& D Batcheldor","D Lena","A Robinson","A Marconi","D J Axon","A Capetti","D Merritt","& D Batcheldor"],"string":"[\n \"D Lena\",\n \"A Robinson\",\n \"A Marconi\",\n \"D J Axon\",\n \"A Capetti\",\n \"D Merritt\",\n \"& D Batcheldor\",\n \"D Lena\",\n \"A Robinson\",\n \"A Marconi\",\n \"D J Axon\",\n \"A Capetti\",\n \"D Merritt\",\n \"& D Batcheldor\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The coalescence of a binary black hole can be accompanied by a large gravitational recoil due to anisotropic emission of gravitational waves. A recoiling supermassive black hole (SBH) can subsequently undergo long-lived oscillations in the potential well of its host galaxy, suggesting that offset SBHs may be common in the cores of massive ellipticals. We have analyzed HST archival images of 14 nearby core ellipticals, finding evidence for small ( 10 pc) displacements between the AGN (locating the SBH) and the center of the galaxy (the mean photocenter) in 10 of them. Excluding objects that may be affected by large-scale isophotal asymmetries, we consider six galaxies to have detected displacements, including M87, where a displacement was previously reported byBatcheldor et al. (2010). In individual objects, these displacements can be attributed to residual gravitational recoil oscillations following a major or minor merger within the last few Gyr. For plausible merger rates, however, there is a high probability of larger displacements than those observed, if SBH coalescence took place in these galaxies. Remarkably, the AGN-photocenter displacements are approximately aligned with the radio source axis in four of the six galaxies with displacements, including three of the four having relatively powerful kpc-scale jets. This suggests intrinsic asymmetries in radio jet power as a possible displacement mechanism, although approximate alignments are also expected for gravitational recoil. Orbital motion in SBH binaries and interactions with massive perturbers can produce the observed displacement amplitudes but do not offer a ready explanation for the alignments."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":""},{"title":"INTRODUCTION","content":"Supermassive black holes (SBHs) have long been identified as the engines of active galactic nuclei (AGNs). More recently, compelling evidence has been amassed that they are present in the centers of almost all galaxies (Kormendy & Richstone 1995;Ferrarese & Ford 2005) and furthermore that the growth of the SBH is intimately linked to galaxy evolution (Ferrarese & Merritt 2000;Gebhardt et al. 2000; see Alexander & Hickox 2012 for a recent review).\nSupermassive black holes (SBHs) have long been identified as the engines of active galactic nuclei (AGNs). More recently, compelling evidence has been amassed that they are present in the centers of almost all galaxies (Kormendy & Richstone 1995;Ferrarese & Ford 2005) and furthermore that the growth of the SBH is intimately linked to galaxy evolution (Ferrarese & Merritt 2000;Gebhardt et al. 2000; see Alexander & Hickox 2012 for a recent review).\nAs a consequence of energy losses due to dynamical friction, SBHs are generally expected to reside at the large-scale potential minimum of the host galaxy. Nevertheless, there are several mechanisms that seem capable of displacing the SBH from its equilibrium position (Batcheldor et al. 2010, hereafter B10). Most recently, interest has focused on gravitational recoil resulting from the coalescence of an SBH-SBH binary (e.g., Merritt et al. 2004;Campanelli et al. 2006;Baker et al. 2006). Other possibilities include orbital motion of SBH-binaries (Komossa 2006), asymmetric radio jets (Shklovski 1982) and interactions with massive perturbers such as globular clusters, or massive molecular clouds.\nAs a consequence of energy losses due to dynamical friction, SBHs are generally expected to reside at the large-scale potential minimum of the host galaxy. Nevertheless, there are several mechanisms that seem capable of displacing the SBH from its equilibrium position (Batcheldor et al. 2010, hereafter B10). Most recently, interest has focused on gravitational recoil resulting from the coalescence of an SBH-SBH binary (e.g., Merritt et al. 2004;Campanelli et al. 2006;Baker et al. 2006). Other possibilities include orbital motion of SBH-binaries (Komossa 2006), asymmetric radio jets (Shklovski 1982) and interactions with massive perturbers such as globular clusters, or massive molecular clouds.\nThe first two of these mechanisms, in particular, might be expected to occur as an inevitable consequence of galaxy evolution. Cold dark matter hierarchical cosmologies predict that galaxies experience multiple mergers during their lifetime (e.g., Coles & Lucchin 2002;Springel et al. 2005), with minor mergers being more common than major mergers (e.g. Hilz et al. 2012). When two galaxies merge, their central SBHs are expected to form a binary system (Begelman, Blandford & Rees 1980) at the center of the merged system due to dynamical friction. Further tightening of the binary ensues through threebody interactions with stars, or gravitational drag from gas (Merritt & Milosavljević 2005;Mayer et al. 2007), until finally, energy loss due to gravitational wave emission irreversibly drives the two SBHs to coalescence 1 .\nThe first two of these mechanisms, in particular, might be expected to occur as an inevitable consequence of galaxy evolution. Cold dark matter hierarchical cosmologies predict that galaxies experience multiple mergers during their lifetime (e.g., Coles & Lucchin 2002;Springel et al. 2005), with minor mergers being more common than major mergers (e.g. Hilz et al. 2012). When two galaxies merge, their central SBHs are expected to form a binary system (Begelman, Blandford & Rees 1980) at the center of the merged system due to dynamical friction. Further tightening of the binary ensues through threebody interactions with stars, or gravitational drag from gas (Merritt & Milosavljević 2005;Mayer et al. 2007), until finally, energy loss due to gravitational wave emission irreversibly drives the two SBHs to coalescence 1 .\nIn the final stage of SBH-binary coalescence, anisotropic emission of gravitational waves will, in general, impart a recoil velocity to the coalesced object 1 Begelman, Blandford & Rees (1980) suggested that the evolution of a binary SBH might stall at separations of ∼ 1 pc once the supply of \"loss-cone stars\" (stars intersecting the orbit of the binary) was exhausted. However, gas dynamical torques can drive continued shrinking of the binary orbit (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007), and recent N -body simulations suggest that hardening of the binary due to interactions with stars can be efficient in non spherical potentials (Preto et al. 2011;Gualandris & Merritt 2012;Vasiliev et al. 2014). (Bekenstein 1973). Following a recent breakthrough allowing the orbits of spinning black holes to be computed to coalescence (Pretorius 2005;Campanelli et al. 2006;Baker et al. 2006), numerical relativity simulations have produced recoil velocities v ∼ 10 3 km s −1 , even reaching ∼ 5 × 10 3 km s −1 for certain spin configurations González et al. 2007;Tichy & Marronetti 2007;Lousto & Zlochower 2011;Lousto et al. 2012). Such velocities would cause large displacements of the coalesced SBH from the center of the galaxy or, in the most extreme cases, eject it entirely from the host (Merritt et al. 2004;Campanelli et al. 2007a;Volonteri, Gúltekin & Dotti 2010).\nIn the final stage of SBH-binary coalescence, anisotropic emission of gravitational waves will, in general, impart a recoil velocity to the coalesced object 1 Begelman, Blandford & Rees (1980) suggested that the evolution of a binary SBH might stall at separations of ∼ 1 pc once the supply of \"loss-cone stars\" (stars intersecting the orbit of the binary) was exhausted. However, gas dynamical torques can drive continued shrinking of the binary orbit (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007), and recent N -body simulations suggest that hardening of the binary due to interactions with stars can be efficient in non spherical potentials (Preto et al. 2011;Gualandris & Merritt 2012;Vasiliev et al. 2014). (Bekenstein 1973). Following a recent breakthrough allowing the orbits of spinning black holes to be computed to coalescence (Pretorius 2005;Campanelli et al. 2006;Baker et al. 2006), numerical relativity simulations have produced recoil velocities v ∼ 10 3 km s −1 , even reaching ∼ 5 × 10 3 km s −1 for certain spin configurations González et al. 2007;Tichy & Marronetti 2007;Lousto & Zlochower 2011;Lousto et al. 2012). Such velocities would cause large displacements of the coalesced SBH from the center of the galaxy or, in the most extreme cases, eject it entirely from the host (Merritt et al. 2004;Campanelli et al. 2007a;Volonteri, Gúltekin & Dotti 2010).\nRecoils exceeding the escape velocity of the host galaxy are expected to be relatively rare (e.g., Lousto et al. 2012, hereafter L12) and, more frequently, the SBH will undergo damped oscillations in the galaxy potential. Nbody simulations (Gualandris and Merritt 2008, hereafter GM08) have shown that moderately large kicks (∼ few 10 2 km s −1 , sufficient to eject the SBH from the core 2 but not from the galaxy) result in long-lived oscillations which damp on a timescale ∆t ∼ 1 Gyr. GM08 identified three dynamical phases for the particular case where the SBH is initially displaced beyond the core radius (r c ∼ 10 2 pc). In \"phase I\" the SBH undergoes damped harmonic motion at amplitudes greater than the core radius. These oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction. \"Phase II\" is characterized by oscillations of amplitude ∼ 10 − 100 pc between the SBH and the stellar core about the center of mass, which persist for up to ∼ 1 Gyr. Finally, in \"phase III\", the SBH has reached thermal equilibrium with the surrounding stars and experiences Brownian motion, signifying the end of the event.\nRecoils exceeding the escape velocity of the host galaxy are expected to be relatively rare (e.g., Lousto et al. 2012, hereafter L12) and, more frequently, the SBH will undergo damped oscillations in the galaxy potential. Nbody simulations (Gualandris and Merritt 2008, hereafter GM08) have shown that moderately large kicks (∼ few 10 2 km s −1 , sufficient to eject the SBH from the core 2 but not from the galaxy) result in long-lived oscillations which damp on a timescale ∆t ∼ 1 Gyr. GM08 identified three dynamical phases for the particular case where the SBH is initially displaced beyond the core radius (r c ∼ 10 2 pc). In \"phase I\" the SBH undergoes damped harmonic motion at amplitudes greater than the core radius. These oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction. \"Phase II\" is characterized by oscillations of amplitude ∼ 10 − 100 pc between the SBH and the stellar core about the center of mass, which persist for up to ∼ 1 Gyr. Finally, in \"phase III\", the SBH has reached thermal equilibrium with the surrounding stars and experiences Brownian motion, signifying the end of the event.\nLarge gravitational recoil kicks would have a variety of observable consequences. In the case of SBHs associated with AGNs, most of the accretion disk and broad emission line gas will remain bound for recoils with velocities v kick 10 3 km s −1 and such systems might be observed as a displaced, or velocity-shifted AGNs (e.g., Merritt et al. 2004;Madau & Quataert 2004;Loeb 2007;Bonning, Shields & Salviander 2007;Zanotti et al. 2010). Stars moving with orbital velocities v < v kick would also remain bound to the SBH after the kick, so that a recoiling SBH would be associated with a \"hypercompact stellar system\" of high internal velocity dispersion (Merritt et al. 2009).\nLarge gravitational recoil kicks would have a variety of observable consequences. In the case of SBHs associated with AGNs, most of the accretion disk and broad emission line gas will remain bound for recoils with velocities v kick 10 3 km s −1 and such systems might be observed as a displaced, or velocity-shifted AGNs (e.g., Merritt et al. 2004;Madau & Quataert 2004;Loeb 2007;Bonning, Shields & Salviander 2007;Zanotti et al. 2010). Stars moving with orbital velocities v < v kick would also remain bound to the SBH after the kick, so that a recoiling SBH would be associated with a \"hypercompact stellar system\" of high internal velocity dispersion (Merritt et al. 2009).\nPossible spectroscopic signatures of recoiling SBHs have been identified in a few individual active galaxies (Komossa et al. 2008;Shields et al. 2009;Robinson et al. 2010;Steinhardt et al. 2012), which would indicate large ( 10 3 km s −1 ) kick velocities. In addition, systematic searches of large SDSS AGN samples have revealed ∼ 100 objects that exhibit velocity shifts ∼ 10 3 km s −1 between the broad and narrow lines (Bonning, Shields & Salviander 2007;Eracleous et al. 2012;Tsalmantza et al. 2011). Some of these may be recoiling SBHs, or SBH binaries, but alternative explanations for the shifts involving extreme broad line region kinematics cannot be excluded. Perhaps the most interesting case is that of CIS-42, a galaxy which hosts two compact optical sources separated by ≈ 2.5 kpc and also exhibits a large velocity shift (∼ 1300 km s −1 ) between broad and narrow Hβ emission lines (Civano et al. 2010(Civano et al. , 2012Blecha et al. 2012).\nPossible spectroscopic signatures of recoiling SBHs have been identified in a few individual active galaxies (Komossa et al. 2008;Shields et al. 2009;Robinson et al. 2010;Steinhardt et al. 2012), which would indicate large ( 10 3 km s −1 ) kick velocities. In addition, systematic searches of large SDSS AGN samples have revealed ∼ 100 objects that exhibit velocity shifts ∼ 10 3 km s −1 between the broad and narrow lines (Bonning, Shields & Salviander 2007;Eracleous et al. 2012;Tsalmantza et al. 2011). Some of these may be recoiling SBHs, or SBH binaries, but alternative explanations for the shifts involving extreme broad line region kinematics cannot be excluded. Perhaps the most interesting case is that of CIS-42, a galaxy which hosts two compact optical sources separated by ≈ 2.5 kpc and also exhibits a large velocity shift (∼ 1300 km s −1 ) between broad and narrow Hβ emission lines (Civano et al. 2010(Civano et al. , 2012Blecha et al. 2012).\nThe gravitational recoil candidates discovered to date have been identified on the basis of anomalously large velocity shifts ( 10 3 km s −1 ) or a large spatial offset ( 1 kpc). If they are indeed recoiling SBHs, we are observing them during the early stages in their dynamical evolution following a large kick; that is, during the large amplitude phase I oscillations of GM08, if the SBH remains bound to the galaxy. However, special configurations of the progenitor binary are required to produce large kicks, which are therefore expected to be relatively rare events (L12) and furthermore, the subsequent large amplitude phase I oscillations are relatively short-lived.\nThe gravitational recoil candidates discovered to date have been identified on the basis of anomalously large velocity shifts ( 10 3 km s −1 ) or a large spatial offset ( 1 kpc). If they are indeed recoiling SBHs, we are observing them during the early stages in their dynamical evolution following a large kick; that is, during the large amplitude phase I oscillations of GM08, if the SBH remains bound to the galaxy. However, special configurations of the progenitor binary are required to produce large kicks, which are therefore expected to be relatively rare events (L12) and furthermore, the subsequent large amplitude phase I oscillations are relatively short-lived.\nOn the other hand, the N -body simulations of GM08 predict that, following a recoil kick sufficiently large to remove the SBH from the galaxy core, the decaying oscillations persist in phase II for ∼ 1 Gyr, comparable with the time between galaxy mergers (Hopkins et al. 2010, hereafter H10). This suggests that low amplitude SBH displacements resulting from post-merger SBH binary coalescence should be relatively common in the cores of bright elliptical galaxies. Thus, rather than searching for short-lived phase I oscillations following rare large kicks, an alternative approach to studying the incidence of SBH binary coalescence is to look for phase II oscillations in nearby ellipticals.\nOn the other hand, the N -body simulations of GM08 predict that, following a recoil kick sufficiently large to remove the SBH from the galaxy core, the decaying oscillations persist in phase II for ∼ 1 Gyr, comparable with the time between galaxy mergers (Hopkins et al. 2010, hereafter H10). This suggests that low amplitude SBH displacements resulting from post-merger SBH binary coalescence should be relatively common in the cores of bright elliptical galaxies. Thus, rather than searching for short-lived phase I oscillations following rare large kicks, an alternative approach to studying the incidence of SBH binary coalescence is to look for phase II oscillations in nearby ellipticals.\nSpectroscopic identification of phase II oscillations would be extremely difficult, if not impossible, as the associated velocity shifts would be comparable with gas motions due to rotation, or flows or turbulence driven by the AGN, or starburst activity. However, it is feasible to directly detect small-amplitude displacements between the AGN and the galaxy photocenter in high spatial resolution images of nearby galaxies. We recently analyzed archival Hubble Space Telescope Advanced Camera for Surveys images of the active giant elliptical M87 and found a projected offset of 6.8 ± 0.8 pc (≈ 0. 1) between the stellar photocenter and the AGN (B10; we note, however, that Gebhardt et al. 2011 did not find a significant offset in their near-infrared integral field spectroscopy data).\nSpectroscopic identification of phase II oscillations would be extremely difficult, if not impossible, as the associated velocity shifts would be comparable with gas motions due to rotation, or flows or turbulence driven by the AGN, or starburst activity. However, it is feasible to directly detect small-amplitude displacements between the AGN and the galaxy photocenter in high spatial resolution images of nearby galaxies. We recently analyzed archival Hubble Space Telescope Advanced Camera for Surveys images of the active giant elliptical M87 and found a projected offset of 6.8 ± 0.8 pc (≈ 0. 1) between the stellar photocenter and the AGN (B10; we note, however, that Gebhardt et al. 2011 did not find a significant offset in their near-infrared integral field spectroscopy data).\nHere we describe a photometric analysis of archival HST images of 14 nearby (d<100 Mpc), bright ellipticals hosting low-luminosity AGNs, with the aim of directly measuring spatial offsets comparable with the core radius of the galaxy (typically < 500 pc). Our method consists in measuring the relative positions of the AGN point source and the photocenter of the inner 2-3 kpc of the galaxy, assuming that the former marks the SBH position and the latter the minimum of the galactic potential. As long as the AGN is bright enough to be easily detected and faint enough that it does not overwhelm the host galaxy, it is possible to perform standard photometric analysis to locate the photocenter. The \"displacement\" is then computed as the relative distance between the AGN and photocenter.\nHere we describe a photometric analysis of archival HST images of 14 nearby (d<100 Mpc), bright ellipticals hosting low-luminosity AGNs, with the aim of directly measuring spatial offsets comparable with the core radius of the galaxy (typically < 500 pc). Our method consists in measuring the relative positions of the AGN point source and the photocenter of the inner 2-3 kpc of the galaxy, assuming that the former marks the SBH position and the latter the minimum of the galactic potential. As long as the AGN is bright enough to be easily detected and faint enough that it does not overwhelm the host galaxy, it is possible to perform standard photometric analysis to locate the photocenter. The \"displacement\" is then computed as the relative distance between the AGN and photocenter.\nThe plan of this paper is as follows: in §2 we describe the sample; §3 describes the analysis methods used to determine the photocenter, SBH position and offset, and their uncertainties; in §4 the measured displacements are presented for each galaxy. In §5, we discuss the results in the context of gravitational recoil and other possible displacement mechanisms. Finally, our conclusions are summarized in §6. The results for selected galaxies are discussed in more detail in an Appendix.\nThe plan of this paper is as follows: in §2 we describe the sample; §3 describes the analysis methods used to determine the photocenter, SBH position and offset, and their uncertainties; in §4 the measured displacements are presented for each galaxy. In §5, we discuss the results in the context of gravitational recoil and other possible displacement mechanisms. Finally, our conclusions are summarized in §6. The results for selected galaxies are discussed in more detail in an Appendix."},{"title":"THE SAMPLE","content":""},{"title":"THE SAMPLE","content":"The sample selected for this study consists of 14 nearby, regular, core elliptical galaxies which host AGNs and for which HST images obtained with ACS, NIC-MOS2, WFPC2 or WFC3 are available in the Hubble Legacy Archive (HLA).\nThe sample selected for this study consists of 14 nearby, regular, core elliptical galaxies which host AGNs and for which HST images obtained with ACS, NIC-MOS2, WFPC2 or WFC3 are available in the Hubble Legacy Archive (HLA).\nHST observations have revealed that the central light distribution in nearly all nearby early type galaxies can be described by a singular surface brightness profile: Σ(r) ∼ r −γ as r → 0. 1, where γ ≥ 0. The slope of the innermost surface brightness distribution is bimodal in the sense that there is a paucity of galaxies which have 0.3 < γ < 0.5. Hence, early type galaxies have been classified in two families: \"core galaxies\", which have a shallower (γ ≤ 0.3) inner cusp within a \"break\" (or \"core\") radius r c ∼ 100 pc, and \"power law galaxies\", which have a steep cusp (γ ≥ 0.5) continuing into the HST resolution limit . These classifications correlate with several other galaxy properties (e.g., Faber et al. 1997), in particular, core galaxies are on average more luminous than power-law galaxies, with essentially all bright (M V < −22) ellipticals having cores.\nHST observations have revealed that the central light distribution in nearly all nearby early type galaxies can be described by a singular surface brightness profile: Σ(r) ∼ r −γ as r → 0. 1, where γ ≥ 0. The slope of the innermost surface brightness distribution is bimodal in the sense that there is a paucity of galaxies which have 0.3 < γ < 0.5. Hence, early type galaxies have been classified in two families: \"core galaxies\", which have a shallower (γ ≤ 0.3) inner cusp within a \"break\" (or \"core\") radius r c ∼ 100 pc, and \"power law galaxies\", which have a steep cusp (γ ≥ 0.5) continuing into the HST resolution limit . These classifications correlate with several other galaxy properties (e.g., Faber et al. 1997), in particular, core galaxies are on average more luminous than power-law galaxies, with essentially all bright (M V < −22) ellipticals having cores.\nCore galaxies are promising systems in which to search for SBH binaries or SBHs displaced by gravitational wave-induced recoils. They are bright ellipticals which are often the dominant components of clusters or groups and are thus likely to have experienced a recent major merger leading to the formation of an SBH binary. The shallow inner surface brightness profile indicates a mass deficit relative to that implied by inward extrapolation of the steeper brightness profile prevailing at larger radii. This is predicted as a natural consequence of depopulation of the inner region of the galaxy by 3-body interactions between an SBH binary and stars crossing the orbit of the binary (\"loss-cone stars\" Merritt & Milosavljević 2005). Therefore, \"flat\" inner surface brightness profiles have been proposed as a clear footprint of the formation of a tightly bound SBH binary (e.g., Faber et al. 1997;Merritt 2006).\nCore galaxies are promising systems in which to search for SBH binaries or SBHs displaced by gravitational wave-induced recoils. They are bright ellipticals which are often the dominant components of clusters or groups and are thus likely to have experienced a recent major merger leading to the formation of an SBH binary. The shallow inner surface brightness profile indicates a mass deficit relative to that implied by inward extrapolation of the steeper brightness profile prevailing at larger radii. This is predicted as a natural consequence of depopulation of the inner region of the galaxy by 3-body interactions between an SBH binary and stars crossing the orbit of the binary (\"loss-cone stars\" Merritt & Milosavljević 2005). Therefore, \"flat\" inner surface brightness profiles have been proposed as a clear footprint of the formation of a tightly bound SBH binary (e.g., Faber et al. 1997;Merritt 2006).\nIn addition, core galaxies tend to have regular photometric structures, and hence well-defined photocenters. In order to accurately locate the position of the SBH, we further require that each galaxy hosts a low luminosity AGN visible as a point source in the HST images.\nIn addition, core galaxies tend to have regular photometric structures, and hence well-defined photocenters. In order to accurately locate the position of the SBH, we further require that each galaxy hosts a low luminosity AGN visible as a point source in the HST images.\nIn order to resolve the low amplitude offsets between photocenter and AGN expected in the phase II oscillations predicted by GM08, we select only galaxies within 100 Mpc.\nIn order to resolve the low amplitude offsets between photocenter and AGN expected in the phase II oscillations predicted by GM08, we select only galaxies within 100 Mpc.\nThe sample analyzed in this work was extracted from a sample of 29 core elliptical galaxies previously studied by Capetti and Balmaverde, in a series of three articles (Capetti & Balmaverde 2005;Capetti & Balmaverde 2006, hereafter CB05, BC06 and CB06 respectively). CB05 compiled a radio-flux limited sample of AGNs in early type hosts by selecting radio-detected sources from the VLA surveys of Wrobel & Heeschen (1991) and Sadler et al. (1989). The former is a northern sample of 216 early-type galaxies extracted from the CfA redshift survey (Huchra et al. 1983), with the following criteria: declination δ 1950 ≥ 0; photometric magnitude B ≤ 14; heliocentric velocity ≤ 3000 km s −1 ; morphological Hubble type T ≤ −1. The latter is a similar southern sample of 116 E and S0 galaxies with declination −45 ≤ δ ≤ −32. Both were observed at 5 GHz with a flux limit of ∼ 1 mJy.\nThe sample analyzed in this work was extracted from a sample of 29 core elliptical galaxies previously studied by Capetti and Balmaverde, in a series of three articles (Capetti & Balmaverde 2005;Capetti & Balmaverde 2006, hereafter CB05, BC06 and CB06 respectively). CB05 compiled a radio-flux limited sample of AGNs in early type hosts by selecting radio-detected sources from the VLA surveys of Wrobel & Heeschen (1991) and Sadler et al. (1989). The former is a northern sample of 216 early-type galaxies extracted from the CfA redshift survey (Huchra et al. 1983), with the following criteria: declination δ 1950 ≥ 0; photometric magnitude B ≤ 14; heliocentric velocity ≤ 3000 km s −1 ; morphological Hubble type T ≤ −1. The latter is a similar southern sample of 116 E and S0 galaxies with declination −45 ≤ δ ≤ −32. Both were observed at 5 GHz with a flux limit of ∼ 1 mJy.\nThe 65 objects with available archival HST images were classified as core or power-law galaxies on the basis of the slopes of their nuclear brightness profile as obtained by fitting a broken power law (the so-called \"Nuker law\"; Lauer et al. 1995;CB05;CB06). For the purpose of this study, we focus on the core galaxies listed in Table 1 of BC06 and impose two additional selection criteria, based on visual inspection of HST images: (i) the presence of an optically bright central point-like source (a low luminosity AGN); (ii) the absence of heavy nuclear obscuration and other photometric irregularities.\nThe 65 objects with available archival HST images were classified as core or power-law galaxies on the basis of the slopes of their nuclear brightness profile as obtained by fitting a broken power law (the so-called \"Nuker law\"; Lauer et al. 1995;CB05;CB06). For the purpose of this study, we focus on the core galaxies listed in Table 1 of BC06 and impose two additional selection criteria, based on visual inspection of HST images: (i) the presence of an optically bright central point-like source (a low luminosity AGN); (ii) the absence of heavy nuclear obscuration and other photometric irregularities.\nThe 14 galaxies selected for this study are listed in Table 1 together with their distances, core radii and details of the archival data sets that were retrieved for photometric analysis. The sample includes two galaxies (NGC 4696 and 5419) that exhibit double nuclei, separated by ∼ 0. 25. In these cases, it is not known which of the two point sources is the AGN and therefore we assume that the brightest is the AGN (although we have measured displacements relative to both).\nThe 14 galaxies selected for this study are listed in Table 1 together with their distances, core radii and details of the archival data sets that were retrieved for photometric analysis. The sample includes two galaxies (NGC 4696 and 5419) that exhibit double nuclei, separated by ∼ 0. 25. In these cases, it is not known which of the two point sources is the AGN and therefore we assume that the brightest is the AGN (although we have measured displacements relative to both).\nThe radio source properties, including the total power at 5 GHz and the position angle of any extended structure, are summarized in Table 2.\nThe radio source properties, including the total power at 5 GHz and the position angle of any extended structure, are summarized in Table 2."},{"title":"DATA ANALYSIS","content":""},{"title":"DATA ANALYSIS","content":"Broad-band images of the sample galaxies acquired with ACS, NICMOS2, WFPC2 and WFC3 were retrieved from the Hubble Legacy Archive (HLA), giving preference to images obtained with the highest spatial resolutions and red filters (i.e., F606W and longer wavelengths).\nBroad-band images of the sample galaxies acquired with ACS, NICMOS2, WFPC2 and WFC3 were retrieved from the Hubble Legacy Archive (HLA), giving preference to images obtained with the highest spatial resolutions and red filters (i.e., F606W and longer wavelengths).\nWhenever available, several images taken with different instruments and filters, covering both optical and NIR bands, were obtained for each galaxy. A list of the images analyzed for each galaxy is given in Table 1. No additional processing steps were applied beyond the standard HLA reduction pipeline. As the goal of our analysis is simply to determine the relative positions of the isophotal center of the inner few kiloparsecs of the galaxy and the AGN point source within each individual image, this is sufficient for our purposes.\nWhenever available, several images taken with different instruments and filters, covering both optical and NIR bands, were obtained for each galaxy. A list of the images analyzed for each galaxy is given in Table 1. No additional processing steps were applied beyond the standard HLA reduction pipeline. As the goal of our analysis is simply to determine the relative positions of the isophotal center of the inner few kiloparsecs of the galaxy and the AGN point source within each individual image, this is sufficient for our purposes.\nEach image was analyzed according to the following main steps: (1) a mask was constructed to block image defects and distorting features such as dust lanes, jets or bright stars and globular clusters; (2) elliptical isophotes were fitted to the galaxy surface brightness distribution and the photocenter computed as the flux-weighted average of the isophote centers; (3) the position of the point source (assumed to locate the SBH) was determined by fitting a gaussian profile. See §3.1 and §3.2 for a more detailed description of these procedures. Sky subtraction was not performed as a uniform background will not affect the photometric analysis and in many cases it is not possible to measure the sky background as the galaxy covers the frame. (2) name as it appears in BC06 if different than the name given in the first column; (3) adopted distance in Mpc (from the HyperLeda extragalactic database); (4) method used to determine the distance: \"sbf\", surface brightness fluctuation; \"vvir\", from radial velocity, corrected for Local Group infall into Virgo; \"Dn − σ\", size-sigma relation (Dressler et al. 1987); (5) core radius as derived by CB05, in arcsec (pc); (6) SBH influence radius (GM•/σ 2 ) in arcsec (pc). Central velocity dispersions are given in Table 8. Black hole masses have been derived from the M• − σ relation given in Ferrarese & Ford (2005); (7) HST instrument/camera/filter used; (8) pixel scale in arcseconds/pixels as determined from direct measurement on the frames; (9) proposal identification number. B10 : this is a combined image, see Table 1 in B10 for further details. † A WFC3/IR image is available for NGC 4696, however the complex nuclear features and the pixel size of the camera do not allow us to infer the SBH position with the precision necessary for this work.\nEach image was analyzed according to the following main steps: (1) a mask was constructed to block image defects and distorting features such as dust lanes, jets or bright stars and globular clusters; (2) elliptical isophotes were fitted to the galaxy surface brightness distribution and the photocenter computed as the flux-weighted average of the isophote centers; (3) the position of the point source (assumed to locate the SBH) was determined by fitting a gaussian profile. See §3.1 and §3.2 for a more detailed description of these procedures. Sky subtraction was not performed as a uniform background will not affect the photometric analysis and in many cases it is not possible to measure the sky background as the galaxy covers the frame. (2) name as it appears in BC06 if different than the name given in the first column; (3) adopted distance in Mpc (from the HyperLeda extragalactic database); (4) method used to determine the distance: \"sbf\", surface brightness fluctuation; \"vvir\", from radial velocity, corrected for Local Group infall into Virgo; \"Dn − σ\", size-sigma relation (Dressler et al. 1987); (5) core radius as derived by CB05, in arcsec (pc); (6) SBH influence radius (GM•/σ 2 ) in arcsec (pc). Central velocity dispersions are given in Table 8. Black hole masses have been derived from the M• − σ relation given in Ferrarese & Ford (2005); (7) HST instrument/camera/filter used; (8) pixel scale in arcseconds/pixels as determined from direct measurement on the frames; (9) proposal identification number. B10 : this is a combined image, see Table 1 in B10 for further details. † A WFC3/IR image is available for NGC 4696, however the complex nuclear features and the pixel size of the camera do not allow us to infer the SBH position with the precision necessary for this work."},{"title":"The inner photocenter","content":""},{"title":"The inner photocenter","content":"Each image was analyzed with the IRAF task ellipse (Jedrzejewski 1987), which was used to fit elliptical isophotes to the surface brightness distribution. The first step was to construct an image mask in order to minimize distortions due to image defects and intrinsic photometric irregularities such as dust features, optical or NIR knots associated with jets, globular clusters, foreground stars, etc. If an exposure time map (the \"weight image\") was provided in the retrieved data set, this was used as an initial mask to eliminate cosmic rays, bad pixels, null areas and other image frame blemishes. Applying the initial mask, a first run of ellipse was performed to create a zeroth-order photometric model. To account for intrinsic irregularities in the surface brightness distribution, we then created a second mask by subtracting the zeroth-order photometric model from the original image and masking residuals exceeding ±nσ, where σ is the standard deviation of the residual image and n is a clipping parameter.\nEach image was analyzed with the IRAF task ellipse (Jedrzejewski 1987), which was used to fit elliptical isophotes to the surface brightness distribution. The first step was to construct an image mask in order to minimize distortions due to image defects and intrinsic photometric irregularities such as dust features, optical or NIR knots associated with jets, globular clusters, foreground stars, etc. If an exposure time map (the \"weight image\") was provided in the retrieved data set, this was used as an initial mask to eliminate cosmic rays, bad pixels, null areas and other image frame blemishes. Applying the initial mask, a first run of ellipse was performed to create a zeroth-order photometric model. To account for intrinsic irregularities in the surface brightness distribution, we then created a second mask by subtracting the zeroth-order photometric model from the original image and masking residuals exceeding ±nσ, where σ is the standard deviation of the residual image and n is a clipping parameter.\nTo allow ellipse to find the photometric peak, the central region was unmasked and a new run performed. This process was iterated, adjusting n in order to map distorting features in the maximum possible detail. The iteration was terminated when the mask converged and further runs did not add any significant detail.\nTo allow ellipse to find the photometric peak, the central region was unmasked and a new run performed. This process was iterated, adjusting n in order to map distorting features in the maximum possible detail. The iteration was terminated when the mask converged and further runs did not add any significant detail.\nOnce the final mask was constructed, ellipse was rerun with the mask applied to determine the photocenter of the galaxy. Elliptical isophotes were fitted between minimum and maximum values of the semi-major axis (SMA) which were usually determined by the core radius (r c ) and the image size, respectively. In almost all cases, the minimum SMA was set equal to or greater than the value of r c as determined by CB05; Table 1. This was typically ∼ 2 , or ∼ 300 pc. In some cases, where a dusty disk is present in the center of the galaxy, the initial SMA was chosen such that the inner ellipse is larger than the disk. The only exception was NGC 4486 (M 87), which has a very large core. In this case, we adopted the SMA range 1-3 used by B10, who found that isophotes within 1 are influenced by the AGN point source. The maximum SMA was that of the largest ellipse that completely fits within the image area -that is, only complete isophotes were fitted. This limits our isophotal analysis to the inner few kiloparsecs of each galaxy; the SMA of the outermost ellipse fit varies in the range 1 r 4 kpc.\nOnce the final mask was constructed, ellipse was rerun with the mask applied to determine the photocenter of the galaxy. Elliptical isophotes were fitted between minimum and maximum values of the semi-major axis (SMA) which were usually determined by the core radius (r c ) and the image size, respectively. In almost all cases, the minimum SMA was set equal to or greater than the value of r c as determined by CB05; Table 1. This was typically ∼ 2 , or ∼ 300 pc. In some cases, where a dusty disk is present in the center of the galaxy, the initial SMA was chosen such that the inner ellipse is larger than the disk. The only exception was NGC 4486 (M 87), which has a very large core. In this case, we adopted the SMA range 1-3 used by B10, who found that isophotes within 1 are influenced by the AGN point source. The maximum SMA was that of the largest ellipse that completely fits within the image area -that is, only complete isophotes were fitted. This limits our isophotal analysis to the inner few kiloparsecs of each galaxy; the SMA of the outermost ellipse fit varies in the range 1 r 4 kpc.\nBetween these limits, a series of ellipses was fitted to the 2-D surface brightness distribution, with the SMA being increased by two pixels at each step. Each fit returned values of the position angle, eccentricity and the pixel x − y coordinates of the center of the ellipse. Fi- Note. -(1) Optical identification;\nBetween these limits, a series of ellipses was fitted to the 2-D surface brightness distribution, with the SMA being increased by two pixels at each step. Each fit returned values of the position angle, eccentricity and the pixel x − y coordinates of the center of the ellipse. Fi- Note. -(1) Optical identification;\n(2) approximate length of the radio jet/extension; (3) position angle (degrees E of N); (4) total power at 5 GHz [10 20 W]; (5) comments on the radio source morphology; (6) comments on the nature of the galaxy. † This value is computed for the inner 25 pc. References: a Slee et al. (1994), b Wrobel & Heeschen (1991, c Wrobel (1991), d Sadler et al. (1989, e Shurkin et al. (2008), f Cavagnolo et al. (2010), g Giroletti et al. (2005), h Baade & Minkowski (1954), i Nagar et al. (2002), l Stanger & Warwick (1986), m Taylor et al. (2006), n Killeen et al. (1986a), p Jones et al. (2000. [HG82]: Huchra & Geller (1982).\n(2) approximate length of the radio jet/extension; (3) position angle (degrees E of N); (4) total power at 5 GHz [10 20 W]; (5) comments on the radio source morphology; (6) comments on the nature of the galaxy. † This value is computed for the inner 25 pc. References: a Slee et al. (1994), b Wrobel & Heeschen (1991, c Wrobel (1991), d Sadler et al. (1989, e Shurkin et al. (2008), f Cavagnolo et al. (2010), g Giroletti et al. (2005), h Baade & Minkowski (1954), i Nagar et al. (2002), l Stanger & Warwick (1986), m Taylor et al. (2006), n Killeen et al. (1986a), p Jones et al. (2000. [HG82]: Huchra & Geller (1982).\nnally, the photocenter was obtained as the flux-weighted average of the centers of the elliptical annuli generated by ellipse:\nnally, the photocenter was obtained as the flux-weighted average of the centers of the elliptical annuli generated by ellipse:\nwhere x i and y i are the photocenter coordinates of the ith isophote and j i is a weight, given by the product of the area of the annulus defined by neighbouring isophotes, with the mean intensity within the annulus.\nwhere x i and y i are the photocenter coordinates of the ith isophote and j i is a weight, given by the product of the area of the annulus defined by neighbouring isophotes, with the mean intensity within the annulus.\nTo assess the sensitivity of the mean photocenter to the weighting method, we computed an alternative weight function using the flux enclosed within each successive elliptical isophote. For comparison, the two weight functions are plotted as a function of SMA in Fig.1, for the NICMOS image of NGC 4261. The annular flux method produces an approximately uniform weight distribution beyond the innermost region (SMA < 0. 5), but typically decreases slowly at large SMA values. Not surprisingly, the enclosed flux method produces weights that increase monotonically with SMA, thus assigning much greater weight to the outer, fainter isophotes. This may be an undesirable characteristic in that isophotal distortions which may be present at fainter surface brightness levels (due to interactions with nearby companions, for example) might unduly influence the mean photocenter. Nevertheless, for the sources in our sample, the photocenters computed using the two methods are consistent within the errors in all cases. This result reflects the rather regular nature of the sample galaxies at the wavelengths under consideration.\nTo assess the sensitivity of the mean photocenter to the weighting method, we computed an alternative weight function using the flux enclosed within each successive elliptical isophote. For comparison, the two weight functions are plotted as a function of SMA in Fig.1, for the NICMOS image of NGC 4261. The annular flux method produces an approximately uniform weight distribution beyond the innermost region (SMA < 0. 5), but typically decreases slowly at large SMA values. Not surprisingly, the enclosed flux method produces weights that increase monotonically with SMA, thus assigning much greater weight to the outer, fainter isophotes. This may be an undesirable characteristic in that isophotal distortions which may be present at fainter surface brightness levels (due to interactions with nearby companions, for example) might unduly influence the mean photocenter. Nevertheless, for the sources in our sample, the photocenters computed using the two methods are consistent within the errors in all cases. This result reflects the rather regular nature of the sample galaxies at the wavelengths under consideration.\nWe explored alternative methods for the measurement of the photocenter position, namely a moment based technique and a 2-D decomposition of the surface brightness distribution. In the first case, the photocenter position is obtained by weighting the coordinate of each pixel by its own intensity. This method seems to be particularly sensitive to the masking details and to the presence of asymmetries in the light distribution. The second ap-proach, performed with the GALFIT surface brightness fitting program (Peng et al. 2002), requires, at least in some cases, multiple components for a successful fit (e.g. two or more Sérsic profiles) casting doubt on the physical meaning of the recovered photocenter and the errors associated with it.\nWe explored alternative methods for the measurement of the photocenter position, namely a moment based technique and a 2-D decomposition of the surface brightness distribution. In the first case, the photocenter position is obtained by weighting the coordinate of each pixel by its own intensity. This method seems to be particularly sensitive to the masking details and to the presence of asymmetries in the light distribution. The second ap-proach, performed with the GALFIT surface brightness fitting program (Peng et al. 2002), requires, at least in some cases, multiple components for a successful fit (e.g. two or more Sérsic profiles) casting doubt on the physical meaning of the recovered photocenter and the errors associated with it.\nThe IRAF task ellipse was specifically built to fit the light profile of elliptical galaxies with minimal assumptions; the results obtained seem to be weakly dependent on the masking details and errors associated with the fitted parameters have been thoroughly studied (Busko 1996). Therefore, we prefer to adopt the photocenter position determined with this approach.\nThe IRAF task ellipse was specifically built to fit the light profile of elliptical galaxies with minimal assumptions; the results obtained seem to be weakly dependent on the masking details and errors associated with the fitted parameters have been thoroughly studied (Busko 1996). Therefore, we prefer to adopt the photocenter position determined with this approach."},{"title":"The SBH position","content":""},{"title":"The SBH position","content":"We make the key assumption that the point-like source near the center of each galaxy is an AGN, which therefore locates the position of the SBH (see §5 for a detailed discussion of this assumption).\nWe make the key assumption that the point-like source near the center of each galaxy is an AGN, which therefore locates the position of the SBH (see §5 for a detailed discussion of this assumption).\nThe AGN point sources are bright enough to be easily detected, but in our sample they are not so bright that the point spread function dominates the host galaxy. In order to determine the position of the SBH, we proceeded as follows. First, a median filtered version of the image was subtracted from the original producing a residual map in which the AGN is a prominent feature 3 . In the second step, we used the IRAF task center to fit gaussians to the marginal distributions of the residual point source along the x and y axes.\nThe AGN point sources are bright enough to be easily detected, but in our sample they are not so bright that the point spread function dominates the host galaxy. In order to determine the position of the SBH, we proceeded as follows. First, a median filtered version of the image was subtracted from the original producing a residual map in which the AGN is a prominent feature 3 . In the second step, we used the IRAF task center to fit gaussians to the marginal distributions of the residual point source along the x and y axes."},{"title":"The photocenter-AGN displacement","content":""},{"title":"The photocenter-AGN displacement","content":"For each image of each galaxy, the x and y components of the displacement were measured as the difference in pixels between the x and y co-ordinates of the photocenter and the AGN point source. The significance of the displacement is classified in two steps, considering the x and y components separately. Displacements smaller than 3σ, where σ is the error on the displacement (Section 3.4.1), are considered non-significant. If the displacement exceeds 3σ we make a further classification based on the distribution of the isophote center coordinates produced by ellipse. For each coordinate, x and y, the inter-quartile range (IQR, the difference between the upper and lower quartiles) of the isophote centers was computed. The corresponding displacement is then assigned significance levels of \"null\", \"low\", \"intermediate\" or \"high\" depending on its magnitude relative to the IQR:\nFor each image of each galaxy, the x and y components of the displacement were measured as the difference in pixels between the x and y co-ordinates of the photocenter and the AGN point source. The significance of the displacement is classified in two steps, considering the x and y components separately. Displacements smaller than 3σ, where σ is the error on the displacement (Section 3.4.1), are considered non-significant. If the displacement exceeds 3σ we make a further classification based on the distribution of the isophote center coordinates produced by ellipse. For each coordinate, x and y, the inter-quartile range (IQR, the difference between the upper and lower quartiles) of the isophote centers was computed. The corresponding displacement is then assigned significance levels of \"null\", \"low\", \"intermediate\" or \"high\" depending on its magnitude relative to the IQR:\n• null, ∆x, ∆y < 0.8 IQR • low, 0.8 IQR ≤ ∆x, ∆y < 1.6 IQR\n• null, ∆x, ∆y < 0.8 IQR • low, 0.8 IQR ≤ ∆x, ∆y < 1.6 IQR\nThe threshold values are based on the fact that, for a normal distribution, 0.8 IQR is equivalent to 1σ. For an ideal, symmetric galaxy the isophotes would be concentric, giving IQR = 0. Any irregularities in the surface brightness distribution will cause a dispersion in the centers of the fitted elliptical isophotes, resulting in a nonzero IQR. Hence, by normalizing to the IQR, we allow for uncertainties in the recovered offsets due to asymmetric surface brightness distributions arising from, for example, dust lanes or tidal distortions.\nThe threshold values are based on the fact that, for a normal distribution, 0.8 IQR is equivalent to 1σ. For an ideal, symmetric galaxy the isophotes would be concentric, giving IQR = 0. Any irregularities in the surface brightness distribution will cause a dispersion in the centers of the fitted elliptical isophotes, resulting in a nonzero IQR. Hence, by normalizing to the IQR, we allow for uncertainties in the recovered offsets due to asymmetric surface brightness distributions arising from, for example, dust lanes or tidal distortions."},{"title":"Errors and biases","content":""},{"title":"Errors and biases","content":""},{"title":"Position uncertainties","content":""},{"title":"Position uncertainties","content":"The final error on the measured projected displacement is the combination of the uncertainties in the positions of the photocenter and the AGN point source. As the photocenter is determined by fitting elliptical isophotes, it is important to understand if the errors provided by ellipse are reasonable and robust. To investigate this, we used ellipse to analyze synthetic galaxy images constructed with GALFIT. We also used these simulated galaxies to estimate the uncertainties on both the AGN and photocenter positions.\nThe final error on the measured projected displacement is the combination of the uncertainties in the positions of the photocenter and the AGN point source. As the photocenter is determined by fitting elliptical isophotes, it is important to understand if the errors provided by ellipse are reasonable and robust. To investigate this, we used ellipse to analyze synthetic galaxy images constructed with GALFIT. We also used these simulated galaxies to estimate the uncertainties on both the AGN and photocenter positions.\nThe model images were constructed by fitting a Sérsic profile and a nuclear point source (a PSF generated with Tiny Tim, Krist 1993) to selected galaxies in our sample. We used NGC 4261 and 4278 to build models of NIC-MOS images, IC 1459 for WFPC2 images, NGC 1399 for WFC3 and IC 4296 for ACS images. The resulting model surface brightness distributions were populated with gaussian random noise so as to match the signalto-noise ratio (SN) of the original galaxy image 4 . Four realizations of each synthetic galaxy image were generated and each realization was analyzed using the methods described in Sections 3.1 and 3.2, superposing the masks obtained from the corresponding real galaxy images.\nThe model images were constructed by fitting a Sérsic profile and a nuclear point source (a PSF generated with Tiny Tim, Krist 1993) to selected galaxies in our sample. We used NGC 4261 and 4278 to build models of NIC-MOS images, IC 1459 for WFPC2 images, NGC 1399 for WFC3 and IC 4296 for ACS images. The resulting model surface brightness distributions were populated with gaussian random noise so as to match the signalto-noise ratio (SN) of the original galaxy image 4 . Four realizations of each synthetic galaxy image were generated and each realization was analyzed using the methods described in Sections 3.1 and 3.2, superposing the masks obtained from the corresponding real galaxy images.\nEllipse errors: for each isophote fit, the \"true error\" on the isophote center is computed as the difference between the position returned by ellipse and the known photocenter of the synthetic galaxy (i.e., the center position of the Sérsic component fitted by Galfit). These \"true errors\" are compared to the errors on the positions of the isophote centers returned by ellipse in Fig.2. The \"true\" error distribution is slightly broader than that of the ellipse errors, but both distributions are highly peaked, with ∼ 50% of fits producing errors ≤ 0.1 pixels. A detailed study by Busko (1996) of the errors returned by ellipse shows that errors on ellipticity, position angle and center position are unbiased and accurate when the radial gradient relative error 5 is less than 50%. This condition is satisfied for our sample galaxies.\nEllipse errors: for each isophote fit, the \"true error\" on the isophote center is computed as the difference between the position returned by ellipse and the known photocenter of the synthetic galaxy (i.e., the center position of the Sérsic component fitted by Galfit). These \"true errors\" are compared to the errors on the positions of the isophote centers returned by ellipse in Fig.2. The \"true\" error distribution is slightly broader than that of the ellipse errors, but both distributions are highly peaked, with ∼ 50% of fits producing errors ≤ 0.1 pixels. A detailed study by Busko (1996) of the errors returned by ellipse shows that errors on ellipticity, position angle and center position are unbiased and accurate when the radial gradient relative error 5 is less than 50%. This condition is satisfied for our sample galaxies.\nPosition errors: the \"true errors\" on the (x,y) coordinates of the AGN and galaxy photocenter positions were also determined for each realization of the synthetic galaxy images, where the true error in each case is defined as the difference between the known positions of the AGN and photocenter (i.e., the positions returned by Galfit) and the values recovered using our analysis methods.\nPosition errors: the \"true errors\" on the (x,y) coordinates of the AGN and galaxy photocenter positions were also determined for each realization of the synthetic galaxy images, where the true error in each case is defined as the difference between the known positions of the AGN and photocenter (i.e., the positions returned by Galfit) and the values recovered using our analysis methods.\nFor signal-to-noise ratios characteristic of our data (S/N ∼ 100), the distributions of the \"true errors\" on the AGN and photocenter positions derived from both the ACS and WFPC2 synthetic galaxy images are narrow and confined within 0.1 pixel. We therefore adopt 0.1 pixel as a conservative estimate of the uncertainty on both the AGN and photocenter positions derived from ACS and WFPC2 data. The \"true error\" obtained from the WFC3 and NICMOS2 images exhibit broader distributions, with tails extending to 0.4 pixels. We therefore adopt the median values of the \"true errors\" as the un- Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \"true errors\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC. certainties for WFC3 and NICMOS2 data, yielding 0.2 pixels for the photocenter position in both cameras and respectively 0.1 and 0.2 pixels for the AGN.\nFor signal-to-noise ratios characteristic of our data (S/N ∼ 100), the distributions of the \"true errors\" on the AGN and photocenter positions derived from both the ACS and WFPC2 synthetic galaxy images are narrow and confined within 0.1 pixel. We therefore adopt 0.1 pixel as a conservative estimate of the uncertainty on both the AGN and photocenter positions derived from ACS and WFPC2 data. The \"true error\" obtained from the WFC3 and NICMOS2 images exhibit broader distributions, with tails extending to 0.4 pixels. We therefore adopt the median values of the \"true errors\" as the un- Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \"true errors\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC. certainties for WFC3 and NICMOS2 data, yielding 0.2 pixels for the photocenter position in both cameras and respectively 0.1 and 0.2 pixels for the AGN.\nThese uncertainties represent the precision with which we can determine the AGN and photocenter positions for an ideal galaxy, in an image with noise characteristics representative of our data. The adopted uncertainties are summarized in Table 3, together with the equivalent angular distances and linear distances corresponding to the closest and furthest of our sample galaxies.\nThese uncertainties represent the precision with which we can determine the AGN and photocenter positions for an ideal galaxy, in an image with noise characteristics representative of our data. The adopted uncertainties are summarized in Table 3, together with the equivalent angular distances and linear distances corresponding to the closest and furthest of our sample galaxies.\nMinimum detectable displacement: the uncertainties on the AGN and photocenter coordinates determine the minimum detectable displacement. We require a 3σ detection for a displacement to be considered significant. Combining the uncertainties on the photocenter and AGN in quadrature, the error on each component of the displacement is σ = 0.14 pxl, for ACS and WFPC2 data, σ = 0.22 pxl for WFC3, and σ = 0.28 pxl for NICMOS2 data. Therefore, the minimum displacements considered significant are ∆x = ∆y ≈ 0.4 pxl, for ACS and WFPC2, ∆x = ∆y ≈ 0.7 pxl for WFPC3, and ∆x = ∆y ≈ 0.9 pxl for NICMOS2.\nMinimum detectable displacement: the uncertainties on the AGN and photocenter coordinates determine the minimum detectable displacement. We require a 3σ detection for a displacement to be considered significant. Combining the uncertainties on the photocenter and AGN in quadrature, the error on each component of the displacement is σ = 0.14 pxl, for ACS and WFPC2 data, σ = 0.22 pxl for WFC3, and σ = 0.28 pxl for NICMOS2 data. Therefore, the minimum displacements considered significant are ∆x = ∆y ≈ 0.4 pxl, for ACS and WFPC2, ∆x = ∆y ≈ 0.7 pxl for WFPC3, and ∆x = ∆y ≈ 0.9 pxl for NICMOS2."},{"title":"Possible biases","content":""},{"title":"Possible biases","content":"Asymmetric surface brightness distributions: the galaxies in our sample were selected to be symmetric and regular, based on visual inspection of the images. However, in some cases, the galaxy extends beyond the edges of the frame and/or is not centered in the image frame. To eliminate the possibility of spurious offsets due to truncation of isophotes by the frame edge, the upper limit on the SMA range used in the surface brightness fits is taken to be that of the largest ellipse that fits completely within the frame (Sec. 3.1).\nAsymmetric surface brightness distributions: the galaxies in our sample were selected to be symmetric and regular, based on visual inspection of the images. However, in some cases, the galaxy extends beyond the edges of the frame and/or is not centered in the image frame. To eliminate the possibility of spurious offsets due to truncation of isophotes by the frame edge, the upper limit on the SMA range used in the surface brightness fits is taken to be that of the largest ellipse that fits completely within the frame (Sec. 3.1).\nIn addition, subtle intrinsic asymmetries might be present that were not revealed by visual inspection. As discussed by Jedrzejewski (1987), asymmetric surface brightness distributions (such as large-scale lopsidedness in the isophotes) will cause shifts in the isophote centers, which in turn will increase the IQR. Therefore, intrinsic surface brightness asymmetries (or other irregularities) will tend to reduce the level of significance assigned to any measured displacement (Sec. 3.3).\nIn addition, subtle intrinsic asymmetries might be present that were not revealed by visual inspection. As discussed by Jedrzejewski (1987), asymmetric surface brightness distributions (such as large-scale lopsidedness in the isophotes) will cause shifts in the isophote centers, which in turn will increase the IQR. Therefore, intrinsic surface brightness asymmetries (or other irregularities) will tend to reduce the level of significance assigned to any measured displacement (Sec. 3.3).\nA similar possibility is that the outer isophotes are distorted by tidal interactions with nearby galaxies. As our analysis is confined to the inner few kiloparsecs, such interactions are unlikely to significantly affect our results. Nevertheless, we have verified that the photocenter position does not change significantly when recomputed excluding isophote centers corresponding to ellipse SMAs greater than ∼ 85, 90 and 95% of that of the largest complete ellipse that fits within the image frame.\nA similar possibility is that the outer isophotes are distorted by tidal interactions with nearby galaxies. As our analysis is confined to the inner few kiloparsecs, such interactions are unlikely to significantly affect our results. Nevertheless, we have verified that the photocenter position does not change significantly when recomputed excluding isophote centers corresponding to ellipse SMAs greater than ∼ 85, 90 and 95% of that of the largest complete ellipse that fits within the image frame.\nIsophote twists and PSF effects: core galaxies are often characterized by strong isophote twisting at radii interior to the core radius, r c (Lauer et al. 2005). In addition, the inner isophotes are distorted by the point spread function; TINY TIM simulations show that for the HST instruments used, the PSF can affect the region within a radius r PSF ∼ 15 pixels. As typically r PSF r c , we mitigate both effects by setting r c as the minimum SMA for the ellipse fits.\nIsophote twists and PSF effects: core galaxies are often characterized by strong isophote twisting at radii interior to the core radius, r c (Lauer et al. 2005). In addition, the inner isophotes are distorted by the point spread function; TINY TIM simulations show that for the HST instruments used, the PSF can affect the region within a radius r PSF ∼ 15 pixels. As typically r PSF r c , we mitigate both effects by setting r c as the minimum SMA for the ellipse fits.\nLopsided stellar nuclei (LSN): observations of the nuclear regions of nearby galaxies have revealed the presence of double nuclei with separations in the range 1-10 pc (Lauer et al. 1993(Lauer et al. , 1996(Lauer et al. , 2005Thatte et al. 2000;Debattista et al. 2006). One explanation for such configurations is that lopsided stellar orbit distributions tend to persist within the SBH sphere of gravitational influence since orbits do not precess in Keplerian potentials and are not, therefore, axisymmetrized by phase mixing (Peiris & Tremaine 2003). The best studied example is M31, in which the components of the double nucleus are separated by ≈ 0. 5 (≈ 2 pc). Peiris & Tremaine (2003) proposed a model for M31 in which the nucleus consists of an eccentric disk of stars orbiting the SBH, the latter being coincident with the fainter component.\nLopsided stellar nuclei (LSN): observations of the nuclear regions of nearby galaxies have revealed the presence of double nuclei with separations in the range 1-10 pc (Lauer et al. 1993(Lauer et al. , 1996(Lauer et al. , 2005Thatte et al. 2000;Debattista et al. 2006). One explanation for such configurations is that lopsided stellar orbit distributions tend to persist within the SBH sphere of gravitational influence since orbits do not precess in Keplerian potentials and are not, therefore, axisymmetrized by phase mixing (Peiris & Tremaine 2003). The best studied example is M31, in which the components of the double nucleus are separated by ≈ 0. 5 (≈ 2 pc). Peiris & Tremaine (2003) proposed a model for M31 in which the nucleus consists of an eccentric disk of stars orbiting the SBH, the latter being coincident with the fainter component.\nThe WFPC2 Planetary Camera (with which most of the images studied here were obtained) resolves spatial separations of ≈ 5 pc only for galaxies within 20 Mpc. It is possible, therefore, that some of the galaxies in our sample harbor unresolved double nuclei. This will not affect the determination of the galaxy photocenter, since the core is excluded from the isophote fitting. However, if the unresolved nuclei are of unequal brightness as in M31, this could result in a systemic error in the AGN position which may approach the PSF FWHM (∼ 1 − 2 pixels). This in turn would result in a spurious displacement.\nThe WFPC2 Planetary Camera (with which most of the images studied here were obtained) resolves spatial separations of ≈ 5 pc only for galaxies within 20 Mpc. It is possible, therefore, that some of the galaxies in our sample harbor unresolved double nuclei. This will not affect the determination of the galaxy photocenter, since the core is excluded from the isophote fitting. However, if the unresolved nuclei are of unequal brightness as in M31, this could result in a systemic error in the AGN position which may approach the PSF FWHM (∼ 1 − 2 pixels). This in turn would result in a spurious displacement."},{"title":"RESULTS","content":""},{"title":"RESULTS","content":"The results of the photometric analysis are presented in Table 4, where we list the (x, y) pixel co-ordinates of the mean photocenter and the point source, the magnitudes of the corresponding components of the displacement, the direction of each displacement component on the sky and the significance level of the displacement, determined as outlined in Sec. 3.3. Figures 3 and 4 illustrate examples of the results for galaxies in which, respectively, no significant displacement was found (NGC4373, Fig. 3) and in which the measured displacement is considered significant (NGC 4486,Fig. 4). In each figure, the top row of panels shows the original image and the residual images after subtraction of, respectively, the isophotal model generated by ellipse fits and a median filtered image. In the middle row are plotted the surface brightness profile and the x and y pixel coordinates of the isophotal centers, all as functions of the ellipse SMA. The positions of the photocenter and the AGN point source are also plotted. The leftmost panel of the bottom row is a scatter plot showing the distribution of the isophote centers, colour coded according to SMA. The locus of the cumulative mean photocenter position, computed outwards from the inner SMA limit, is also shown as a solid black line. The last two panels show the distributions of the (x, y) isophote center co-ordinates. The co-ordinates of the mean photocenter and the AGN point source are also plotted in all three panels. Similar figures for the remaining galaxies are presented in the Appendix (Sec. C), where the results for individual galaxies are also described in more detail (Sec. B).\nThe results of the photometric analysis are presented in Table 4, where we list the (x, y) pixel co-ordinates of the mean photocenter and the point source, the magnitudes of the corresponding components of the displacement, the direction of each displacement component on the sky and the significance level of the displacement, determined as outlined in Sec. 3.3. Figures 3 and 4 illustrate examples of the results for galaxies in which, respectively, no significant displacement was found (NGC4373, Fig. 3) and in which the measured displacement is considered significant (NGC 4486,Fig. 4). In each figure, the top row of panels shows the original image and the residual images after subtraction of, respectively, the isophotal model generated by ellipse fits and a median filtered image. In the middle row are plotted the surface brightness profile and the x and y pixel coordinates of the isophotal centers, all as functions of the ellipse SMA. The positions of the photocenter and the AGN point source are also plotted. The leftmost panel of the bottom row is a scatter plot showing the distribution of the isophote centers, colour coded according to SMA. The locus of the cumulative mean photocenter position, computed outwards from the inner SMA limit, is also shown as a solid black line. The last two panels show the distributions of the (x, y) isophote center co-ordinates. The co-ordinates of the mean photocenter and the AGN point source are also plotted in all three panels. Similar figures for the remaining galaxies are presented in the Appendix (Sec. C), where the results for individual galaxies are also described in more detail (Sec. B).\nIn the case of NGC 4373, the x and y components of the displacement are < 1σ and thus consistent with zero. In the case of the ACS/HRC F814W image of NGC 4486, both the x and y components of the displacement are significant at the > 3σ level and their magnitudes are ≥ 2.4×IQR and accordingly classified as having a \"high\" significance level (Table 4). For four galaxies, two or more images obtained with different instruments and/or filters were analyzed, providing various combinations of wavelength and spatial resolution. The photocenter displacements in milliarcseconds, relative to the AGN point source, are plotted for these cases in Fig. 5a. There are two galaxies, NGC 4696 and 5419, in which a second point-like brightness peak is present. In these cases, the following discussion refers to the displacement relative to the primary point source (the brightest, which we take to be the AGN), unless otherwise noted.\nIn the case of NGC 4373, the x and y components of the displacement are < 1σ and thus consistent with zero. In the case of the ACS/HRC F814W image of NGC 4486, both the x and y components of the displacement are significant at the > 3σ level and their magnitudes are ≥ 2.4×IQR and accordingly classified as having a \"high\" significance level (Table 4). For four galaxies, two or more images obtained with different instruments and/or filters were analyzed, providing various combinations of wavelength and spatial resolution. The photocenter displacements in milliarcseconds, relative to the AGN point source, are plotted for these cases in Fig. 5a. There are two galaxies, NGC 4696 and 5419, in which a second point-like brightness peak is present. In these cases, the following discussion refers to the displacement relative to the primary point source (the brightest, which we take to be the AGN), unless otherwise noted.\nAGN-photocenter displacements exceeding the minimum detectable value (i.e., significant at 3σ level; Sec. 3.4.1) were measured in at least one direction (x or y), in at least one image, for ten out of the 14 galaxies in the sample: NGC 1399, 4168, 4278, 4486 (M87), 4636, 4696, 5419, 5846, IC 4296, and IC 4931. In the remaining galaxies, NGC 4261, 4373, 4552, and IC 1459, the AGN and photocenter are coincident within the position uncertainties (< 3σ).\nAGN-photocenter displacements exceeding the minimum detectable value (i.e., significant at 3σ level; Sec. 3.4.1) were measured in at least one direction (x or y), in at least one image, for ten out of the 14 galaxies in the sample: NGC 1399, 4168, 4278, 4486 (M87), 4636, 4696, 5419, 5846, IC 4296, and IC 4931. In the remaining galaxies, NGC 4261, 4373, 4552, and IC 1459, the AGN and photocenter are coincident within the position uncertainties (< 3σ).\nThe distribution of the isophote center co-ordinates provides an indication of the systematic uncertainty arising from photometric distortions due to effects such as those outlined in Sec. 3.4.2. As discussed in Sec. 3.3, we assign a significance level for each measured displacement based on its magnitude relative to the IQR, which characterizes the width of the isophote center distribution. Three galaxies exhibit displacements (in at least one direction, in one image) ∆x, ∆y ≥ 1.6×IQR, that we classify as having intermediate or high significance. These are NGC 4278, 4486 (M87) and 5846. Three more have displacements classified as having low significance: NGC 1399, 5419 and IC 4296 (∆x, ∆y ≥ 0.8×IQR).\nThe distribution of the isophote center co-ordinates provides an indication of the systematic uncertainty arising from photometric distortions due to effects such as those outlined in Sec. 3.4.2. As discussed in Sec. 3.3, we assign a significance level for each measured displacement based on its magnitude relative to the IQR, which characterizes the width of the isophote center distribution. Three galaxies exhibit displacements (in at least one direction, in one image) ∆x, ∆y ≥ 1.6×IQR, that we classify as having intermediate or high significance. These are NGC 4278, 4486 (M87) and 5846. Three more have displacements classified as having low significance: NGC 1399, 5419 and IC 4296 (∆x, ∆y ≥ 0.8×IQR).\nIn the two double nucleus galaxies, the photocenter position is much closer to the primary point source than the secondary. The offsets (relative to the primary) are ≈ 0.5 pxl, but given the large IQR in NGC 4696 the displacement is classified as not significant in this case. In neither galaxy does the photocenter lie between the two \"nuclei\" (Fig. 5b, 25 and 26).\nIn the two double nucleus galaxies, the photocenter position is much closer to the primary point source than the secondary. The offsets (relative to the primary) are ≈ 0.5 pxl, but given the large IQR in NGC 4696 the displacement is classified as not significant in this case. In neither galaxy does the photocenter lie between the two \"nuclei\" (Fig. 5b, 25 and 26).\nFor the galaxies where multiple images were analyzed, the displacements in mas derived from the different images are consistent to within 2σ for two of the four galaxies (NGC 1399 and 4278).\nFor the galaxies where multiple images were analyzed, the displacements in mas derived from the different images are consistent to within 2σ for two of the four galaxies (NGC 1399 and 4278).\nIn the remaining two galaxies (NGC 4486 and IC 4296), the results from one or more images differ significantly from the others (Fig. 5a). In IC 4296, the offset measured from the red (F814W) WFPC2 image is significantly different from those measured from the ACS-F625W and NICMOS2 F160W images, which are consistent to within 2σ. This galaxy has a warped dusty disk oriented approximately E-W, extending ≈ 1 either side of the nucleus but it seems unlikely that this structure is the cause of the discrepancy, the origin of which remains unclear.\nIn the remaining two galaxies (NGC 4486 and IC 4296), the results from one or more images differ significantly from the others (Fig. 5a). In IC 4296, the offset measured from the red (F814W) WFPC2 image is significantly different from those measured from the ACS-F625W and NICMOS2 F160W images, which are consistent to within 2σ. This galaxy has a warped dusty disk oriented approximately E-W, extending ≈ 1 either side of the nucleus but it seems unlikely that this structure is the cause of the discrepancy, the origin of which remains unclear.\nSeveral ACS, NICMOS2 and WFPC2 images were analyzed for NGC 4486 (M87). For the ACS red optical image (F814W) we recover the B10 result, which indicates that the photocenter is displaced by ≈ 100 mas to the north-west of the nucleus. A smaller but less significant offset in approximately the same direction is measured from the WFPC2 F814W image. However, the photocenters derived from the NICMOS2 F110W, F160M and F222W image are consistent with the AGN position. As discussed in detail in Appendix B, the origin of these differences is unclear. Neither dust, nor the prominent optical jet (also visible in the NIR) appear to provide satisfactory explanations. The central region of the galaxy where the isophote fitting was performed (1 ≤ r ≤ 3 ) appears to be free of large scale dust features. The masking procedure should prevent the jet from distorting the isophote fits but in any case, even when the fits were repeated with different levels of masking and even with no mask at all, no significant differences in the mean photocenter position were found. As there is no compelling reason to favor or discard the results from any given image, we compute a weighted average displacement.\nSeveral ACS, NICMOS2 and WFPC2 images were analyzed for NGC 4486 (M87). For the ACS red optical image (F814W) we recover the B10 result, which indicates that the photocenter is displaced by ≈ 100 mas to the north-west of the nucleus. A smaller but less significant offset in approximately the same direction is measured from the WFPC2 F814W image. However, the photocenters derived from the NICMOS2 F110W, F160M and F222W image are consistent with the AGN position. As discussed in detail in Appendix B, the origin of these differences is unclear. Neither dust, nor the prominent optical jet (also visible in the NIR) appear to provide satisfactory explanations. The central region of the galaxy where the isophote fitting was performed (1 ≤ r ≤ 3 ) appears to be free of large scale dust features. The masking procedure should prevent the jet from distorting the isophote fits but in any case, even when the fits were repeated with different levels of masking and even with no mask at all, no significant differences in the mean photocenter position were found. As there is no compelling reason to favor or discard the results from any given image, we compute a weighted average displacement.\nWhether classified as significant, or not, the measured angular photocenter displacements are always small, typically a few×10 mas and almost without exception 100 mas. The only cases that exceed 100 mas are the displacements relative to the secondary point sources in the two double nucleus galaxies. Disregarding these, the projected linear displacements are all 10 pc. Therefore, for every galaxy analyzed, the measured displacement is a small fraction of the galaxy core radius; as a fraction of r c , the range in displacement magnitude is approximately 1-10%.\nWhether classified as significant, or not, the measured angular photocenter displacements are always small, typically a few×10 mas and almost without exception 100 mas. The only cases that exceed 100 mas are the displacements relative to the secondary point sources in the two double nucleus galaxies. Disregarding these, the projected linear displacements are all 10 pc. Therefore, for every galaxy analyzed, the measured displacement is a small fraction of the galaxy core radius; as a fraction of r c , the range in displacement magnitude is approximately 1-10%.\nIn our previous analysis of NGC 4486 (B10), we found that the galaxy photocenter (in ACS and WFPC images) is displaced in approximately the same direction as the jet, implying that the SBH is displaced in the counter-jet direction. It is therefore of interest to compare the direction of the measured photocenter displacements with the radio source axis for this sample. In Table 5 we give the position angle (PA) of the radio source, for those galaxies in which jets or jet-like extended structures have been observed, along with the PA of the measured displacement (as derived from the x, y components). Five galaxies are associated with relatively powerful, extended (∼ 10 kpc) radio sources: NGC 1399, 4261, 4486, 4696 and IC 4296 6 . Of these, NGC 4486 (M 87; 3C 274), NGC 4261 (3C 270) and IC 4296 (PKS 1333-33) are Fanaroff-Riley Type I (FRI) radio sources, characterized by prominent twin jets ending in kiloparsec-scale lobes (Fanaroff & Riley 1974). All three also have parsec-scale jets approximately aligned with the kpc-scale structure.\nIn our previous analysis of NGC 4486 (B10), we found that the galaxy photocenter (in ACS and WFPC images) is displaced in approximately the same direction as the jet, implying that the SBH is displaced in the counter-jet direction. It is therefore of interest to compare the direction of the measured photocenter displacements with the radio source axis for this sample. In Table 5 we give the position angle (PA) of the radio source, for those galaxies in which jets or jet-like extended structures have been observed, along with the PA of the measured displacement (as derived from the x, y components). Five galaxies are associated with relatively powerful, extended (∼ 10 kpc) radio sources: NGC 1399, 4261, 4486, 4696 and IC 4296 6 . Of these, NGC 4486 (M 87; 3C 274), NGC 4261 (3C 270) and IC 4296 (PKS 1333-33) are Fanaroff-Riley Type I (FRI) radio sources, characterized by prominent twin jets ending in kiloparsec-scale lobes (Fanaroff & Riley 1974). All three also have parsec-scale jets approximately aligned with the kpc-scale structure.\nIn NGC 4486, the displacement PA derived from our reanalysis of the ACS F814W image agrees with the B10 result. The direction of the weighted mean displacement, derived from displacements obtained from all the images, is also closely aligned with the jet direction. We find a similar result for IC 4296. As already noted, the results from the ACS and NICMOS2 images are consistent and indicate a displacement to the NW. If the discrepant WFPC2 result is disregarded, the weighted mean of the ACS and NICMOS2 displacements yields a PA only ∼ 4 • different from that of the large-scale radio axis, with the photocenter displaced on the side of the brighter north-western jet.\nIn NGC 4486, the displacement PA derived from our reanalysis of the ACS F814W image agrees with the B10 result. The direction of the weighted mean displacement, derived from displacements obtained from all the images, is also closely aligned with the jet direction. We find a similar result for IC 4296. As already noted, the results from the ACS and NICMOS2 images are consistent and indicate a displacement to the NW. If the discrepant WFPC2 result is disregarded, the weighted mean of the ACS and NICMOS2 displacements yields a PA only ∼ 4 • different from that of the large-scale radio axis, with the photocenter displaced on the side of the brighter north-western jet.\nThe displacement measured in NGC 4261 is not considered to be significant (< 3σ). Nevertheless, the derived PA is consistent (albeit within the large uncertainty) with that of the radio axis, and again, the offset is in the direction of the brighter western jet.\nThe displacement measured in NGC 4261 is not considered to be significant (< 3σ). Nevertheless, the derived PA is consistent (albeit within the large uncertainty) with that of the radio axis, and again, the offset is in the direction of the brighter western jet.\nNGC 1399 also has an FRI-like radio source morphology with a well-defined axis in PA≈ −10 • . The measured displacement (classified as low significance) has a consistent PA with a value of −17 ± 16 • . NGC 4696, is one of the galaxies with two optical point sources. Its large scale radio source does not exhibit well-defined jets, but is elongated approximately E-W over ∼ 10 kpc, with the ends of both \"arms\" bending south. At parsec scales there is a compact core with a one-sided jet emerging to the SW in PA∼ −150 • . However, it is not clear which of the two optical nuclei hosts the core radio source (Taylor et al. 2006), complicating the comparison with the photocenter displacement. If the secondary point source is identified as the AGN, then the photocenter, which is close to the primary point source, is displaced approximately in the counter-jet direction. On the other hand, if the primary point source is indeed the AGN, the displacement (which, in any case, is classified as having \"null\" significance) would be almost perpendicular to the jet. Three more galaxies have relatively weak, small radio sources that have jet-like features or elongations on subkiloparsec scales. The measured photocenter displacements are not aligned with the radio source axis in NGC 4552 or NGC 4636. NGC 4278 has a compact (∼ 3 pc) source consisting of a core with jet-like features emerging along a SE-NW axis on either side. These features gradually bend to the east and west, respectively, becoming fainter and more diffuse. The weighted mean photocenter position is displaced approximately in the initial direction of the SE jet. According to Giroletti et al. (2005)'s analysis of this source, the jet axis is closely aligned with our line of sight, with the SE component being the oppositely directed counter-jet.\nNGC 1399 also has an FRI-like radio source morphology with a well-defined axis in PA≈ −10 • . The measured displacement (classified as low significance) has a consistent PA with a value of −17 ± 16 • . NGC 4696, is one of the galaxies with two optical point sources. Its large scale radio source does not exhibit well-defined jets, but is elongated approximately E-W over ∼ 10 kpc, with the ends of both \"arms\" bending south. At parsec scales there is a compact core with a one-sided jet emerging to the SW in PA∼ −150 • . However, it is not clear which of the two optical nuclei hosts the core radio source (Taylor et al. 2006), complicating the comparison with the photocenter displacement. If the secondary point source is identified as the AGN, then the photocenter, which is close to the primary point source, is displaced approximately in the counter-jet direction. On the other hand, if the primary point source is indeed the AGN, the displacement (which, in any case, is classified as having \"null\" significance) would be almost perpendicular to the jet. Three more galaxies have relatively weak, small radio sources that have jet-like features or elongations on subkiloparsec scales. The measured photocenter displacements are not aligned with the radio source axis in NGC 4552 or NGC 4636. NGC 4278 has a compact (∼ 3 pc) source consisting of a core with jet-like features emerging along a SE-NW axis on either side. These features gradually bend to the east and west, respectively, becoming fainter and more diffuse. The weighted mean photocenter position is displaced approximately in the initial direction of the SE jet. According to Giroletti et al. (2005)'s analysis of this source, the jet axis is closely aligned with our line of sight, with the SE component being the oppositely directed counter-jet."},{"title":"DISCUSSION","content":""},{"title":"DISCUSSION","content":"In this section we will examine our key assumption that the SBH position is marked by the nuclear point source. We will then discuss the possible origins of the observed offsets. Most of the displacement mechanisms considered below have been previously discussed in B10. Nevertheless, we summarize them here in order to provide context for our new results.\nIn this section we will examine our key assumption that the SBH position is marked by the nuclear point source. We will then discuss the possible origins of the observed offsets. Most of the displacement mechanisms considered below have been previously discussed in B10. Nevertheless, we summarize them here in order to provide context for our new results.\n5.1. The AGN nature of the nuclear point sources Our sample of 14 core elliptical galaxies was selected from the 26 (out of 29) studied by BC06 that were identified as containing AGN based on the presence of i) an unresolved optical or X-ray source; ii) an \"AGN-like\" optical spectrum, or iii) radio jets. Most of our sample of 14 exhibit two or more AGN signatures in addition to the presence of a radio source and optical point source nucleus. Optical or UV variability has been detected in HST observations of NGC 4486 (Perlman et al. 2003 . Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers. (NGC 4278,4636 and NGC 4168), or emission line ratios indicative of AGN photoionization (NGC 4261,4486,4552,5846) have also been detected in several galaxies (Ho et al. 1997). In addition, the radio sources in most objects feature compact ( 1 pc) cores (NGC 1399, 4168, 4373, 4552, 5419, IC 1459) and/or parsec or kpc-scale jets (NGC 4261,4278,4486,4696,IC 4296; see Appendix B for references). Only IC 4931, which by comparison has been relatively little studied, lacks supporting evidence indicating the presence of an AGN in the form of X-ray, radio or line emission.\n5.1. The AGN nature of the nuclear point sources Our sample of 14 core elliptical galaxies was selected from the 26 (out of 29) studied by BC06 that were identified as containing AGN based on the presence of i) an unresolved optical or X-ray source; ii) an \"AGN-like\" optical spectrum, or iii) radio jets. Most of our sample of 14 exhibit two or more AGN signatures in addition to the presence of a radio source and optical point source nucleus. Optical or UV variability has been detected in HST observations of NGC 4486 (Perlman et al. 2003 . Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers. (NGC 4278,4636 and NGC 4168), or emission line ratios indicative of AGN photoionization (NGC 4261,4486,4552,5846) have also been detected in several galaxies (Ho et al. 1997). In addition, the radio sources in most objects feature compact ( 1 pc) cores (NGC 1399, 4168, 4373, 4552, 5419, IC 1459) and/or parsec or kpc-scale jets (NGC 4261,4278,4486,4696,IC 4296; see Appendix B for references). Only IC 4931, which by comparison has been relatively little studied, lacks supporting evidence indicating the presence of an AGN in the form of X-ray, radio or line emission.\nThere is, therefore, plenty of evidence that these galaxies (with the possible exception of IC 4931) host AGNs. However, it does not necessarily follow (except in cases where optical/UV variability has been observed) that the optical or NIR point sources are themselves manifestations of the AGN, rather than, for example, nuclear star clusters. Low luminosity radio loud AGNs are thought to be powered by radiatively inefficient accretion flows (see, for example, Ho 2008; Balmaverde et al. 2008, and references therein), with most of the accretion power channeled into the kinetic energy of the radio jets. Detailed studies of NGC4486 (Di Matteo et al. 2003), IC 1459 (Fabbiano et al. 2003) and IC 4296 (Pellegrini et al. 2003) show that this is the case for at least three galaxies of our sample.\nThere is, therefore, plenty of evidence that these galaxies (with the possible exception of IC 4931) host AGNs. However, it does not necessarily follow (except in cases where optical/UV variability has been observed) that the optical or NIR point sources are themselves manifestations of the AGN, rather than, for example, nuclear star clusters. Low luminosity radio loud AGNs are thought to be powered by radiatively inefficient accretion flows (see, for example, Ho 2008; Balmaverde et al. 2008, and references therein), with most of the accretion power channeled into the kinetic energy of the radio jets. Detailed studies of NGC4486 (Di Matteo et al. 2003), IC 1459 (Fabbiano et al. 2003) and IC 4296 (Pellegrini et al. 2003) show that this is the case for at least three galaxies of our sample.\nIn low luminosity radio galaxies, the optical, near infrared and X-ray luminosities of the nuclei correlate tightly with the core radio luminosity, implying a common origin in non-thermal emission from the jet (Chiaberge et al. 1999;Capetti & Balmaverde 2005;Baldi et al. 2010). This is supported by polarization measurements: Capetti et al. (2007) found that the optical nuclei of the nine nearest FR I radio galaxies in the 3C catalogue, including two galaxies from our sample, NGC 4486 (M87, 3C 274) and NGC 4261 (3C 270), have high polarizations (2 − 11%), which they attribute to synchrotron emission. The optical or UV variability of the point sources in NGC 4486 and three other galaxies, noted above, is also consistent with jet synchrotron emission. More generally, BC06 found that their sample of core elliptical galaxies, from which this sample is drawn, also exhibit optical-radio and X-ray-radio correlations and indeed form a continuous distribution with the radio galaxies, extending these correlations to lower luminosities. Therefore, there is direct evidence, in the form of optical polarization and/or optical/UV variability that the nuclear sources in five of our sample galaxies are produced by synchrotron emission associated with the radio core source, presumably the base of the jet. It is reasonable to assume, given the correlations found by BC06, that the same is true for the rest of the sample.\nIn low luminosity radio galaxies, the optical, near infrared and X-ray luminosities of the nuclei correlate tightly with the core radio luminosity, implying a common origin in non-thermal emission from the jet (Chiaberge et al. 1999;Capetti & Balmaverde 2005;Baldi et al. 2010). This is supported by polarization measurements: Capetti et al. (2007) found that the optical nuclei of the nine nearest FR I radio galaxies in the 3C catalogue, including two galaxies from our sample, NGC 4486 (M87, 3C 274) and NGC 4261 (3C 270), have high polarizations (2 − 11%), which they attribute to synchrotron emission. The optical or UV variability of the point sources in NGC 4486 and three other galaxies, noted above, is also consistent with jet synchrotron emission. More generally, BC06 found that their sample of core elliptical galaxies, from which this sample is drawn, also exhibit optical-radio and X-ray-radio correlations and indeed form a continuous distribution with the radio galaxies, extending these correlations to lower luminosities. Therefore, there is direct evidence, in the form of optical polarization and/or optical/UV variability that the nuclear sources in five of our sample galaxies are produced by synchrotron emission associated with the radio core source, presumably the base of the jet. It is reasonable to assume, given the correlations found by BC06, that the same is true for the rest of the sample.\nThe base of the jet is, in turn, very close to the SBH. Arguably the best studied jet is that of M87. Using multifrequency observations made with the Very Long Baseline Array, Hada et al. (2011) were able to show that the radio core at 43 GHz is located within 14-23 Schwarzchild radii of the SBH and accretion flow. In the optical, the lower optical depth should move the peak of the emission even closer to the SBH.\nThe base of the jet is, in turn, very close to the SBH. Arguably the best studied jet is that of M87. Using multifrequency observations made with the Very Long Baseline Array, Hada et al. (2011) were able to show that the radio core at 43 GHz is located within 14-23 Schwarzchild radii of the SBH and accretion flow. In the optical, the lower optical depth should move the peak of the emission even closer to the SBH.\nIn summary, we contend that there are good reasons to believe that the optical or near-infrared point sources in our galaxies reveal the position of the SBH.\nIn summary, we contend that there are good reasons to believe that the optical or near-infrared point sources in our galaxies reveal the position of the SBH."},{"title":"Recoiling SBHs","content":""},{"title":"Recoiling SBHs","content":"Here we consider if the measured SBH displacements are consistent with residual oscillations due to gravitational recoils generated by coalescence of an SBH binary. GM08 used N -body simulations to study the post recoil dynamical evolution of an SBH in representative core elliptical galaxy potentials. They found that when the recoil velocity is sufficient to eject the SBH from the core (v kick in the range ∼ 40 − 90% of the escape velocity, v esc ), the subsequent motion is characterized by three phases. The initial large-amplitide (>r c ) oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction (phase I). When the amplitude becomes comparable with the core radius, r c , the SBH and the stellar core undergo long-lived oscillations about their center of mass, which persist for ∼ 1 Gyr (phase II). Finally, the SBH reaches thermal equilibrium with the stars, experiencing low-amplitude Brownian motion (phase III).\nHere we consider if the measured SBH displacements are consistent with residual oscillations due to gravitational recoils generated by coalescence of an SBH binary. GM08 used N -body simulations to study the post recoil dynamical evolution of an SBH in representative core elliptical galaxy potentials. They found that when the recoil velocity is sufficient to eject the SBH from the core (v kick in the range ∼ 40 − 90% of the escape velocity, v esc ), the subsequent motion is characterized by three phases. The initial large-amplitide (>r c ) oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction (phase I). When the amplitude becomes comparable with the core radius, r c , the SBH and the stellar core undergo long-lived oscillations about their center of mass, which persist for ∼ 1 Gyr (phase II). Finally, the SBH reaches thermal equilibrium with the stars, experiencing low-amplitude Brownian motion (phase III).\nAs our measured displacements are 0.1 r c , it is highly unlikely that they represent phase I oscillations. This would require either fortuitous timing (the SBH would have to be caught whilst passing through or close to the equilibrium position) and/or orientation (the oscillation direction would have to be closely aligned with the line of sight) for each galaxy. We also rule out phase III Brownian motion due to interactions with individual stars, which will produce negligibly small amplitudes in real galaxies (GM08). The possibility of Brownian oscillation due to interactions with massive perturbers is discussed in Section 5.3. Here, we focus on phase II which, as already noted, is characterized by long-lived damped oscillations at amplitudes < r c .\nAs our measured displacements are 0.1 r c , it is highly unlikely that they represent phase I oscillations. This would require either fortuitous timing (the SBH would have to be caught whilst passing through or close to the equilibrium position) and/or orientation (the oscillation direction would have to be closely aligned with the line of sight) for each galaxy. We also rule out phase III Brownian motion due to interactions with individual stars, which will produce negligibly small amplitudes in real galaxies (GM08). The possibility of Brownian oscillation due to interactions with massive perturbers is discussed in Section 5.3. Here, we focus on phase II which, as already noted, is characterized by long-lived damped oscillations at amplitudes < r c .\nIn this phase, the characteristic damping time is given by GM08 as:\nIn this phase, the characteristic damping time is given by GM08 as:\nwhere the approximation on the right-hand side makes use of the M • -σ relation (Ferrarese & Ford 2005), σ being the 1D stellar velocity dispersion, with r 500 ≡ r c /(500 pc) and σ 300 ≡ σ/300 km s −1 . For our galaxies, eq. 2 yields values of τ in the range 0.02 − 4.85 Gyr, with an average of 1.2 Gyr (Table 8). The rms amplitude of the SBH motion with respect to the galaxy center is expected to evolve as:\nwhere the approximation on the right-hand side makes use of the M • -σ relation (Ferrarese & Ford 2005), σ being the 1D stellar velocity dispersion, with r 500 ≡ r c /(500 pc) and σ 300 ≡ σ/300 km s −1 . For our galaxies, eq. 2 yields values of τ in the range 0.02 − 4.85 Gyr, with an average of 1.2 Gyr (Table 8). The rms amplitude of the SBH motion with respect to the galaxy center is expected to evolve as:\nwhere t c is the time at which the oscillation amplitude has decayed to a scale comparable to the core radius r c (B10). Simulations indicate that the galaxy merger rate is a strong function of redshift and galaxy mass (e.g., Fakhouri et al. 2010, H10). The merger rate for brightest cluster galaxies has recently been determined observationally by Lidman et al. (2013), who find ≈ 0.4 mergers per Gyr at z ∼ 1, implying a mean time between mergers of ≈ 2.5 Gyr 7 . If we take this as representative of our sample, equations 2 and 3 together suggest that a \"typical\" rms displacement (ignoring projection effects) is ∼ 0.2 r c . This suggests that larger displacements than were actually observed might be expected if due to post gravitational recoil oscillations.\nwhere t c is the time at which the oscillation amplitude has decayed to a scale comparable to the core radius r c (B10). Simulations indicate that the galaxy merger rate is a strong function of redshift and galaxy mass (e.g., Fakhouri et al. 2010, H10). The merger rate for brightest cluster galaxies has recently been determined observationally by Lidman et al. (2013), who find ≈ 0.4 mergers per Gyr at z ∼ 1, implying a mean time between mergers of ≈ 2.5 Gyr 7 . If we take this as representative of our sample, equations 2 and 3 together suggest that a \"typical\" rms displacement (ignoring projection effects) is ∼ 0.2 r c . This suggests that larger displacements than were actually observed might be expected if due to post gravitational recoil oscillations.\nHowever, equation 3 describes the oscillation amplitude rather than the instantaneous displacement and moreover the above estimate does not account for projection effects, or the range in damping timescales characterizing our sample galaxies, or for variations in the merger rate. In order to investigate in more detail the likelihood of obtaining the observed displacements in our sample, if they result from post-recoil oscillations, we have constructed a simple Monte-Carlo simulation based on the GM08 N -body simulations. Details of the method are given in Appendix A. However, we start from the assumption that, after any kick large enough to move the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy is given by:\nHowever, equation 3 describes the oscillation amplitude rather than the instantaneous displacement and moreover the above estimate does not account for projection effects, or the range in damping timescales characterizing our sample galaxies, or for variations in the merger rate. In order to investigate in more detail the likelihood of obtaining the observed displacements in our sample, if they result from post-recoil oscillations, we have constructed a simple Monte-Carlo simulation based on the GM08 N -body simulations. Details of the method are given in Appendix A. However, we start from the assumption that, after any kick large enough to move the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy is given by:\nwhere ∆t is the elapsed time since the kick, i.e., the time since the last merger, τ is the damping time given by eq.2 and ω c is the SBH oscillation frequency calculated from eq. A2. We set R 0 = r c , since phase II begins when the oscillation amplitude is roughly equal to the core radius. We then suppose that each galaxy is observed at a random time, ∆t, since its last merger, where the probability that the merger occurred at time ∆t follows an exponential distribution characterized by a mean timebetween-mergers, t m . This distribution is sampled to generate corresponding values of R i ≡ R(t i ), which are then projected onto the sky plane assuming that the recoil kicks have random directions. The distribution of projected displacement for each galaxy is then used to compute the probability, p i , of observing a displacement larger than that actually measured, given a kick of sufficient magnitude. These probabilities were computed for two values of the mean time between mergers: t m = 5.0 Gyr and t m = 0.4 Gyr, which were derived from the galaxy merger rate models of H10 for redshifts z < 1, mass ratios q > 0.1 and galaxy masses log(M gal /M ) > 11 and log(M gal /M ) > 12, respectively (see Appendix A for details). The two values of t m are representative of the mass range covered by our sample, and illustrate the effect on the displacement probabilities of the strong galaxy mass dependence of the merger rate.\nwhere ∆t is the elapsed time since the kick, i.e., the time since the last merger, τ is the damping time given by eq.2 and ω c is the SBH oscillation frequency calculated from eq. A2. We set R 0 = r c , since phase II begins when the oscillation amplitude is roughly equal to the core radius. We then suppose that each galaxy is observed at a random time, ∆t, since its last merger, where the probability that the merger occurred at time ∆t follows an exponential distribution characterized by a mean timebetween-mergers, t m . This distribution is sampled to generate corresponding values of R i ≡ R(t i ), which are then projected onto the sky plane assuming that the recoil kicks have random directions. The distribution of projected displacement for each galaxy is then used to compute the probability, p i , of observing a displacement larger than that actually measured, given a kick of sufficient magnitude. These probabilities were computed for two values of the mean time between mergers: t m = 5.0 Gyr and t m = 0.4 Gyr, which were derived from the galaxy merger rate models of H10 for redshifts z < 1, mass ratios q > 0.1 and galaxy masses log(M gal /M ) > 11 and log(M gal /M ) > 12, respectively (see Appendix A for details). The two values of t m are representative of the mass range covered by our sample, and illustrate the effect on the displacement probabilities of the strong galaxy mass dependence of the merger rate.\nThe probabilities, p i , are listed for each galaxy in Table 6 for both values of t m . The probabilities P (d > x) of observing a projected displacement d exceeding distance\nThe probabilities, p i , are listed for each galaxy in Table 6 for both values of t m . The probabilities P (d > x) of observing a projected displacement d exceeding distance\nx are plotted as functions of x in Figures 7a and 7b, for each galaxy, along with the observed displacement.\nx are plotted as functions of x in Figures 7a and 7b, for each galaxy, along with the observed displacement.\nAs would be expected, the smaller value of t m results in larger values of p i ; clearly if t m τ , there is a greater chance of observing a large displacement than when t m >> τ . For t m = 0.4 Gyr, p i > 0.5 for all but two galaxies (NGC 4278 and NGC 4552) and p i ≥ 0.95 for all 6 galaxies that have masses ≥ 10 12 M (for which assumed fraction of mass accreted by a BCG during a merger. Therefore, this merger rate relies on model-dependent assumptions about the dark-matter haloes surrounding the galaxies.\nAs would be expected, the smaller value of t m results in larger values of p i ; clearly if t m τ , there is a greater chance of observing a large displacement than when t m >> τ . For t m = 0.4 Gyr, p i > 0.5 for all but two galaxies (NGC 4278 and NGC 4552) and p i ≥ 0.95 for all 6 galaxies that have masses ≥ 10 12 M (for which assumed fraction of mass accreted by a BCG during a merger. Therefore, this merger rate relies on model-dependent assumptions about the dark-matter haloes surrounding the galaxies.\nthis value of t m is presumably most relevant). On the other hand, for t m = 5.0 Gyr, all but four galaxies have p i < 0.5, including five of the six having displacements considered significant at the level of > 0.8 IQR (the exception is M87, for which p i = 0.75).\nthis value of t m is presumably most relevant). On the other hand, for t m = 5.0 Gyr, all but four galaxies have p i < 0.5, including five of the six having displacements considered significant at the level of > 0.8 IQR (the exception is M87, for which p i = 0.75).\nConsidering the sample as a whole, the probability of not observing a displacement larger than those actually measured in any of the galaxies in the sample is simply P = i (1 − p i ). For t m = 0.4 Gyr, this is negligibly small, P ∼ 10 −17 . Even for t m = 5.0 Gyr, P ≈ 3 × 10 −4 , indicating that it is statistically unlikely that larger displacements were not observed, given the occurrence of a recoil kick sufficient to trigger phase II oscillations.\nConsidering the sample as a whole, the probability of not observing a displacement larger than those actually measured in any of the galaxies in the sample is simply P = i (1 − p i ). For t m = 0.4 Gyr, this is negligibly small, P ∼ 10 −17 . Even for t m = 5.0 Gyr, P ≈ 3 × 10 −4 , indicating that it is statistically unlikely that larger displacements were not observed, given the occurrence of a recoil kick sufficient to trigger phase II oscillations.\nHowever, it is unlikely that all the galaxies in the sample experienced a recoil kick big enough to move the SBH beyond the core radius. Assuming that the potential experienced by the SBH can be approximated as a harmonic potential (φ = 1/2 ω 2 x 2 ), we estimated the kick velocity required to displace the SBH out to the core radius for each galaxy, obtaining values in the range 159 v kick 325 km s −1 , with an average of v kick ≈ 240 km s −1 (the values for each galaxy are listed in Table 8).\nHowever, it is unlikely that all the galaxies in the sample experienced a recoil kick big enough to move the SBH beyond the core radius. Assuming that the potential experienced by the SBH can be approximated as a harmonic potential (φ = 1/2 ω 2 x 2 ), we estimated the kick velocity required to displace the SBH out to the core radius for each galaxy, obtaining values in the range 159 v kick 325 km s −1 , with an average of v kick ≈ 240 km s −1 (the values for each galaxy are listed in Table 8).\nL12 have studied kick velocity distributions for \"hangup\" kicks, which arise in binary configurations where the SBH spins are partially aligned with the orbital angular momentum by pre-merger accretion. For spin magnitude and orientation distributions derived from accretion simulations and a mass ratio distribution based on galaxy merger studies (including both minor and major mergers), they generate probability distributions for the magnitude of the recoil velocity for two extreme cases of \"hot\" and \"cold\" gas disks (i.e., corresponding to adiabatic indices γ = 5/3 and 7/5, respectively). The former regime is likely to be most applicable to the ellipticals in our sample and in this case, probabilities range from p k ≈ 66% for kicks > 100 km s −1 to ≈ 33% for kicks > 300 km s −1 . Our Monte-Carlo simulation assumes that the coalescence event produces a recoil large enough to produce phase II oscillations (i.e., the values given in Table 8). As the probabilities p i are conditional on the occurrence of a kick of sufficient magnitude, the probability that both events will occur is p k × p i . There is thus a smaller probability of observing a displacement larger than actually measured for each galaxy. Using the kick probabilities given by L12, we find that the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 8 × 10 −4 for t m = 0.4 Gyr and P ≈ 5 × 10 −2 for t m = 5.0 Gyr.\nL12 have studied kick velocity distributions for \"hangup\" kicks, which arise in binary configurations where the SBH spins are partially aligned with the orbital angular momentum by pre-merger accretion. For spin magnitude and orientation distributions derived from accretion simulations and a mass ratio distribution based on galaxy merger studies (including both minor and major mergers), they generate probability distributions for the magnitude of the recoil velocity for two extreme cases of \"hot\" and \"cold\" gas disks (i.e., corresponding to adiabatic indices γ = 5/3 and 7/5, respectively). The former regime is likely to be most applicable to the ellipticals in our sample and in this case, probabilities range from p k ≈ 66% for kicks > 100 km s −1 to ≈ 33% for kicks > 300 km s −1 . Our Monte-Carlo simulation assumes that the coalescence event produces a recoil large enough to produce phase II oscillations (i.e., the values given in Table 8). As the probabilities p i are conditional on the occurrence of a kick of sufficient magnitude, the probability that both events will occur is p k × p i . There is thus a smaller probability of observing a displacement larger than actually measured for each galaxy. Using the kick probabilities given by L12, we find that the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 8 × 10 −4 for t m = 0.4 Gyr and P ≈ 5 × 10 −2 for t m = 5.0 Gyr.\nAs the merger rate increases with redshift, we may have underestimated the time-between-mergers by integrating over the redshift range z = 1 → 0. However, the higher value of t m used in the simulation implies a merger rate (∼ 0.2 mergers Gyr −1 ) consistent with that given by the models of H10 for log(M gal /M ) > 11 galaxies at z = 0 (see their Fig. 3). The rate for major mergers is a factor 2-3 lower than the rate for major and minor mergers combined, which we used to estimate t m . However, although the magnitude of the recoil velocity is a function of SBH mass ratio (with the largest recoils occurring for mass ratios characteristic of major mergers), even small (q > 0.1) mass ratios are capable of producing kicks ∼ 200 km s −1 , large enough to trigger phase II oscillations in our galaxies (Lousto et al. 2010). Mi- Note. -(1) Optical name;\nAs the merger rate increases with redshift, we may have underestimated the time-between-mergers by integrating over the redshift range z = 1 → 0. However, the higher value of t m used in the simulation implies a merger rate (∼ 0.2 mergers Gyr −1 ) consistent with that given by the models of H10 for log(M gal /M ) > 11 galaxies at z = 0 (see their Fig. 3). The rate for major mergers is a factor 2-3 lower than the rate for major and minor mergers combined, which we used to estimate t m . However, although the magnitude of the recoil velocity is a function of SBH mass ratio (with the largest recoils occurring for mass ratios characteristic of major mergers), even small (q > 0.1) mass ratios are capable of producing kicks ∼ 200 km s −1 , large enough to trigger phase II oscillations in our galaxies (Lousto et al. 2010). Mi- Note. -(1) Optical name;\n(2) instrument/camera/filter on HST;\n(2) instrument/camera/filter on HST;\n(3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) nuclear point source position in pixels; (7 -8) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (9) offset in pixel normalized by the corresponding IQR; (10) offset in pc; (11) direction of the offset; (12) type of displacement as defined in §3.3. : for further details see Table 1 in B10. nor mergers were, in any case, included in computing the kick velocity distribution presented by L12.\n(3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) nuclear point source position in pixels; (7 -8) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (9) offset in pixel normalized by the corresponding IQR; (10) offset in pc; (11) direction of the offset; (12) type of displacement as defined in §3.3. : for further details see Table 1 in B10. nor mergers were, in any case, included in computing the kick velocity distribution presented by L12.\nThus, even allowing for the distribution of kick velocities, the Monte-Carlo simulation suggests that it is highly likely that displacements larger than those actually measured would have been observed in this sample if each galaxy has experienced at least one merger leading to an SBH-binary coalescence and gravitational recoil within the last few Gyr. In fact, our simulations suggest that t m 30 − 40 Gyr, corresponding to a merger rate ∼ 0.03 mergers Gyr −1 , is required for a > 50% chance of not observing larger displacements in the sample.\nThus, even allowing for the distribution of kick velocities, the Monte-Carlo simulation suggests that it is highly likely that displacements larger than those actually measured would have been observed in this sample if each galaxy has experienced at least one merger leading to an SBH-binary coalescence and gravitational recoil within the last few Gyr. In fact, our simulations suggest that t m 30 − 40 Gyr, corresponding to a merger rate ∼ 0.03 mergers Gyr −1 , is required for a > 50% chance of not observing larger displacements in the sample.\nIf, indeed, the formation of a binary SBH is an inevitable outcome of galaxy mergers, explanations must be sought in the evolution of the binary or that of the recoiling SBH. Possibilities include (1) the binary stalls before reaching the radius at which gravitational wave emission drives rapid coalescence; (2) if coalescence occurs, the mass ratio or spin-orbit properties of the binary are such as to typically preclude recoil velocities large enough to displace the SBH to the core radius ( 240 km s −1 ); or (3) the damping time for the recoil oscillations may be shorter than predicted by the pure N -body simulations.\nIf, indeed, the formation of a binary SBH is an inevitable outcome of galaxy mergers, explanations must be sought in the evolution of the binary or that of the recoiling SBH. Possibilities include (1) the binary stalls before reaching the radius at which gravitational wave emission drives rapid coalescence; (2) if coalescence occurs, the mass ratio or spin-orbit properties of the binary are such as to typically preclude recoil velocities large enough to displace the SBH to the core radius ( 240 km s −1 ); or (3) the damping time for the recoil oscillations may be shorter than predicted by the pure N -body simulations.\nAlternatively, it is possible that galaxy mergers are much more infrequent than inferred from studies based on the current ΛCDM framework (see Kroupa 2014, and references therein for a detailed discussion).\nAlternatively, it is possible that galaxy mergers are much more infrequent than inferred from studies based on the current ΛCDM framework (see Kroupa 2014, and references therein for a detailed discussion).\nAlthough our results imply that SBH displacements due to gravitational recoil are much less common than might be inferred from theoretical considerations, this does not preclude the possibility that observed displacements in individual galaxies are due to recoil oscillations. Indeed, the p i values listed in Table 6 indicate that for t m = 5.0 Gyr, the chance of not observing a larger displacement, given a sufficiently large kick, is 30% for all objects having a displacement considered significant. Even for t m = 0.4 Gyr, there are still three of these galaxies for which 1 − p i ≥ 0.25. The likelihood that the recovered offset is due to SBH coalescence is discussed for individual galaxies in Appendix B.\nAlthough our results imply that SBH displacements due to gravitational recoil are much less common than might be inferred from theoretical considerations, this does not preclude the possibility that observed displacements in individual galaxies are due to recoil oscillations. Indeed, the p i values listed in Table 6 indicate that for t m = 5.0 Gyr, the chance of not observing a larger displacement, given a sufficiently large kick, is 30% for all objects having a displacement considered significant. Even for t m = 0.4 Gyr, there are still three of these galaxies for which 1 − p i ≥ 0.25. The likelihood that the recovered offset is due to SBH coalescence is discussed for individual galaxies in Appendix B.\nAssuming that the measured displacements are indeed due to phase II recoil oscillations, we have used eq. 3 to estimate the merger epoch for those galaxies with displacements rated at least as \"low significance\" in terms of the isophote centroid IQR. Allowing for the time taken for the SBH binary to form and subsequently coalesce in the center of the merged galaxy, and for phase I oscillations, the elapsed times since the merger are typically ∼ several Gyr, comparable with the mean time between galaxy mergers estimated here for galaxy masses log(M gal /M ) > 11, and consistent with the observationally determined merger rate for brightest cluster galaxies (Lidman et al. 2013).\nAssuming that the measured displacements are indeed due to phase II recoil oscillations, we have used eq. 3 to estimate the merger epoch for those galaxies with displacements rated at least as \"low significance\" in terms of the isophote centroid IQR. Allowing for the time taken for the SBH binary to form and subsequently coalesce in the center of the merged galaxy, and for phase I oscillations, the elapsed times since the merger are typically ∼ several Gyr, comparable with the mean time between galaxy mergers estimated here for galaxy masses log(M gal /M ) > 11, and consistent with the observationally determined merger rate for brightest cluster galaxies (Lidman et al. 2013).\nEq.2, used to estimate the SBH damping time, relies on the M • -σ relation (Ferrarese & Ford 2005). To test the dependence of our result on the value adopted for M • we use eq.6 in Lauer et al. (2007) to estimate SBH masses using the M • -M V relation (see Table 9 and Appendix A for more details). Using these alternative masses, and taking into account the kick velocity distribution computed by L12, we derive new values for P obtaining 1 × 10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5 Gyr. The likelihood of our result is somewhat increased for t m = 5 Gyr, however there is still a large probability of finding displacements larger than those measured, leaving our basic conclusion unchanged.\nEq.2, used to estimate the SBH damping time, relies on the M • -σ relation (Ferrarese & Ford 2005). To test the dependence of our result on the value adopted for M • we use eq.6 in Lauer et al. (2007) to estimate SBH masses using the M • -M V relation (see Table 9 and Appendix A for more details). Using these alternative masses, and taking into account the kick velocity distribution computed by L12, we derive new values for P obtaining 1 × 10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5 Gyr. The likelihood of our result is somewhat increased for t m = 5 Gyr, however there is still a large probability of finding displacements larger than those measured, leaving our basic conclusion unchanged.\nNo useful constraints can be obtained on the merger epoch from consideration of the time necessary to replenish the stellar population of the core. This is roughly the relaxation time, T rel , at the influence radius of the central SBH, which can be estimated as:\nNo useful constraints can be obtained on the merger epoch from consideration of the time necessary to replenish the stellar population of the core. This is roughly the relaxation time, T rel , at the influence radius of the central SBH, which can be estimated as:\nT r (r i ) ≈ 1.2 × 10 11 σ 7.47 100 yr (5) Merritt (2013, eq. 3.5). Given that the measured velocity dispersion is always > 100 km s −1 in the centers of our galaxies (see Table 8), it is clear that the replenishment time is much longer than the timescales involved in the SBH merger-recoil event (∼ 10 9 yr). In general, coalescence of an SBH-binary results in a re-orientation of the SBH spin axis, leading to a sudden flip in the direction of the associated radio jet (Merritt & Ekers 2002). Therefore, the radio source morphology may act as a \"signpost\" of SBH coalescence. However, powerful extended radio sources have lifetimes ∼ 10 6 −10 8 yr (e.g., Parma et al. 1999;O'Dea et al. 2009;Antognini, Bird & Martini 2012), much shorter than the time-between-mergers, even for massive galaxies. Thus even if a spin flip took place, morphological traces of this event, such as a change in jet direction, would not necessarily be evident in the current (post-coalescence) radio source. The radio source properties of individual galaxies are discussed in Appendix B.\nT r (r i ) ≈ 1.2 × 10 11 σ 7.47 100 yr (5) Merritt (2013, eq. 3.5). Given that the measured velocity dispersion is always > 100 km s −1 in the centers of our galaxies (see Table 8), it is clear that the replenishment time is much longer than the timescales involved in the SBH merger-recoil event (∼ 10 9 yr). In general, coalescence of an SBH-binary results in a re-orientation of the SBH spin axis, leading to a sudden flip in the direction of the associated radio jet (Merritt & Ekers 2002). Therefore, the radio source morphology may act as a \"signpost\" of SBH coalescence. However, powerful extended radio sources have lifetimes ∼ 10 6 −10 8 yr (e.g., Parma et al. 1999;O'Dea et al. 2009;Antognini, Bird & Martini 2012), much shorter than the time-between-mergers, even for massive galaxies. Thus even if a spin flip took place, morphological traces of this event, such as a change in jet direction, would not necessarily be evident in the current (post-coalescence) radio source. The radio source properties of individual galaxies are discussed in Appendix B.\nIt is notable that in four of the six galaxies in which significant displacements have been found, the displacements are also approximately aligned with the radio jet axis (Section 4 and Table 5). This number includes three of the four that have powerful kpc-scale jets (Section 4 and Table 5), suggesting jet power asymmetries as a possible displacement mechanism (as discussed below). However, such alignments do not necessarily argue against gravitational recoil. In their statistical study of the coalescence of spinning black hole binaries Lousto et al. (2010) find that the recoil velocity is preferentially aligned (or counter-aligned) with the orbital angular momentum of the progenitor binary and moreover, that the spin direction distribution of the recoiling SBH peaks at an angle of ≈ 25 • to the orbital angular momentum (for an equal mass progenitor binary; the probability distribution is broader and peaks at larger angles for smaller mass ratios). This suggests, assuming that the jet traces the spin axis of the recoiling SBH, a tendency for the displacement to be somewhat, but not greatly, misaligned with the jet axis, consistent with what is observed in these objects.\nIt is notable that in four of the six galaxies in which significant displacements have been found, the displacements are also approximately aligned with the radio jet axis (Section 4 and Table 5). This number includes three of the four that have powerful kpc-scale jets (Section 4 and Table 5), suggesting jet power asymmetries as a possible displacement mechanism (as discussed below). However, such alignments do not necessarily argue against gravitational recoil. In their statistical study of the coalescence of spinning black hole binaries Lousto et al. (2010) find that the recoil velocity is preferentially aligned (or counter-aligned) with the orbital angular momentum of the progenitor binary and moreover, that the spin direction distribution of the recoiling SBH peaks at an angle of ≈ 25 • to the orbital angular momentum (for an equal mass progenitor binary; the probability distribution is broader and peaks at larger angles for smaller mass ratios). This suggests, assuming that the jet traces the spin axis of the recoiling SBH, a tendency for the displacement to be somewhat, but not greatly, misaligned with the jet axis, consistent with what is observed in these objects."},{"title":"Other displacement mechanisms","content":""},{"title":"Other displacement mechanisms","content":"We now consider several other mechanisms that may plausibly produce SBH displacements.\nWe now consider several other mechanisms that may plausibly produce SBH displacements.\n1. Asymmetric jets: if the AGN jets are intrinsically asymmetric in power output, the resulting net thrust can push the SBH away from the original equilibrium position (Shklovski 1982;Saslaw & Whittle 1988 Note. -(1) Optical name; (2) instrument/camera/filter on HST; (3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) and (7) nuclear point source position in pixels; (8 -9) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (10) offset in pixel normalized by the corresponding IQR; (11) offset in pc; (12) direction of the offset; (13) type of displacement as defined in §3.3. Table 4). a This value changes to −8 ± 16 • when only WFPC2 results are considered. b This value changes to 312 ± 8 • (132 ± 8 • ) when only ACS and NICMOS2 results are considered. See Table 2 for references on the radio jet PA. Note. -Probability to observe a projected displacement larger than the value actually observed in this work (in units of core radii) at a random time after the last kick. Probabilities are computed for a mean time between galactic mergers tm = 0.4 Gyr and 5 Gyr. The symbol \" * \" indicates 3σ offsets that were classified as \"null\" after the normalization for the co-ordinates IQR. The symbol \"<\" indicates offsets that do not reach the 3σ level.\n1. Asymmetric jets: if the AGN jets are intrinsically asymmetric in power output, the resulting net thrust can push the SBH away from the original equilibrium position (Shklovski 1982;Saslaw & Whittle 1988 Note. -(1) Optical name; (2) instrument/camera/filter on HST; (3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) and (7) nuclear point source position in pixels; (8 -9) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (10) offset in pixel normalized by the corresponding IQR; (11) offset in pc; (12) direction of the offset; (13) type of displacement as defined in §3.3. Table 4). a This value changes to −8 ± 16 • when only WFPC2 results are considered. b This value changes to 312 ± 8 • (132 ± 8 • ) when only ACS and NICMOS2 results are considered. See Table 2 for references on the radio jet PA. Note. -Probability to observe a projected displacement larger than the value actually observed in this work (in units of core radii) at a random time after the last kick. Probabilities are computed for a mean time between galactic mergers tm = 0.4 Gyr and 5 Gyr. The symbol \" * \" indicates 3σ offsets that were classified as \"null\" after the normalization for the co-ordinates IQR. The symbol \"<\" indicates offsets that do not reach the 3σ level.\nthis scenario:\nthis scenario:\nwhereṁ ≡Ṁ a /Ṁ edd and f jet = L jet /L a 8 ,Ṁ a is the mass accretion rate,Ṁ edd ≈ 2.2 M • /10 8 M M yr −1 the Eddington accretion rate, L a the accretion luminosity and L jet is the jet luminosity. For asymmetrical, oppositely directed jets L jet is the difference in the luminosities of the two jets.\nwhereṁ ≡Ṁ a /Ṁ edd and f jet = L jet /L a 8 ,Ṁ a is the mass accretion rate,Ṁ edd ≈ 2.2 M • /10 8 M M yr −1 the Eddington accretion rate, L a the accretion luminosity and L jet is the jet luminosity. For asymmetrical, oppositely directed jets L jet is the difference in the luminosities of the two jets.\nUnder the assumption that the restoring force from the galaxy is negligible (a reasonable approximation for the low-density cores of \"core galaxies\"), Eq. 6 can be integrated to obtain an expression for the displacement:\nUnder the assumption that the restoring force from the galaxy is negligible (a reasonable approximation for the low-density cores of \"core galaxies\"), Eq. 6 can be integrated to obtain an expression for the displacement:\n∆r ≈ 340 f jetṁ t 2 6 pc (7) where t 6 is the time over which the SBH is accelerated to produce an offset ∆r.\n∆r ≈ 340 f jetṁ t 2 6 pc (7) where t 6 is the time over which the SBH is accelerated to produce an offset ∆r.\nThe best candidates for jet thrust displacements are the four galaxies (NGC 1399, 4261, 4486 and IC 4296) that have relatively powerful kpc-scale jets (Section 4 and Table 2). Of these, we did not detect a significant displacement in NGC 4261, which has an FR I radio source. NGC 4486 and IC 4296 also host FR I radio sources and NGC 1399 has an FR I-like morphology, despite its relatively low power. Interestingly, all three of these galaxies exhibit photocenter displacements that are at least approximately (within 20 • ) in the jet direction (Table 5). This implies that the SBH is displaced relative to its equilibrium position in the counter-jet direction, as might be expected if the displacements are related to intrinsic asymmetries in jet power. NGC 4486 (M87) has already been discussed in detail by B10, who concluded that a jet power asymmetry amounting to ∼ 3% of the accretion luminosity can explain the observed displacement, for a radio source lifetime ∼ 10 8 yr. We find similar results in the cases of NGC 1399 and IC 4296, where the observed displacement can be produced for a jet asymmetry < 1% of the Bondi accretion luminosity, again for a radio source lifetime ∼ 10 8 yr (See Appendix B for details).\nThe best candidates for jet thrust displacements are the four galaxies (NGC 1399, 4261, 4486 and IC 4296) that have relatively powerful kpc-scale jets (Section 4 and Table 2). Of these, we did not detect a significant displacement in NGC 4261, which has an FR I radio source. NGC 4486 and IC 4296 also host FR I radio sources and NGC 1399 has an FR I-like morphology, despite its relatively low power. Interestingly, all three of these galaxies exhibit photocenter displacements that are at least approximately (within 20 • ) in the jet direction (Table 5). This implies that the SBH is displaced relative to its equilibrium position in the counter-jet direction, as might be expected if the displacements are related to intrinsic asymmetries in jet power. NGC 4486 (M87) has already been discussed in detail by B10, who concluded that a jet power asymmetry amounting to ∼ 3% of the accretion luminosity can explain the observed displacement, for a radio source lifetime ∼ 10 8 yr. We find similar results in the cases of NGC 1399 and IC 4296, where the observed displacement can be produced for a jet asymmetry < 1% of the Bondi accretion luminosity, again for a radio source lifetime ∼ 10 8 yr (See Appendix B for details).\nA close alignment between the displacement and the (initial) jet direction is also found for the low power pcscale radio source in NGC 4278. In this case, Giroletti et al. (2005)'s interpretation of the radio data implies that the photocenter is displaced in the counter-jet direction, with the jet axis being closely aligned with the line of sight. Assuming that the SBH is displaced along the jet axis, the de-projected magnitude of the displacement would be ∼ 100 − 200 pc. Nevertheless, it seems possible that this could result from jet thrust, if sustained for ∼ 10 8 yr (Appendix B).\nA close alignment between the displacement and the (initial) jet direction is also found for the low power pcscale radio source in NGC 4278. In this case, Giroletti et al. (2005)'s interpretation of the radio data implies that the photocenter is displaced in the counter-jet direction, with the jet axis being closely aligned with the line of sight. Assuming that the SBH is displaced along the jet axis, the de-projected magnitude of the displacement would be ∼ 100 − 200 pc. Nevertheless, it seems possible that this could result from jet thrust, if sustained for ∼ 10 8 yr (Appendix B).\nThe double nucleus galaxy NGC 4696 does not show well-defined jets on kpc scales, but has a one-sided pcscale jet in PA≈ −150 • . The displacement relative to the brighter nucleus is considered non-significant because of the large IQR. However, as already noted, it is not known which of the two optical nuclei hosts the AGN producing the radio jet. If it is the fainter nucleus, the photocenter displacement is approximately in the counter-jet direction, which is not consistent with jet acceleration of the SBH (which would cause the SBH to be displaced in the counter-jet direction).\nThe double nucleus galaxy NGC 4696 does not show well-defined jets on kpc scales, but has a one-sided pcscale jet in PA≈ −150 • . The displacement relative to the brighter nucleus is considered non-significant because of the large IQR. However, as already noted, it is not known which of the two optical nuclei hosts the AGN producing the radio jet. If it is the fainter nucleus, the photocenter displacement is approximately in the counter-jet direction, which is not consistent with jet acceleration of the SBH (which would cause the SBH to be displaced in the counter-jet direction).\nThe remaining two objects which exhibit displacements considered to be significant (NGC 5419 and 5846) do not have well-defined jets on either parsec or kiloparsec scales.\nThe remaining two objects which exhibit displacements considered to be significant (NGC 5419 and 5846) do not have well-defined jets on either parsec or kiloparsec scales.\n2. Stalled SBH binaries: in the aftermath of a galaxy-galaxy merger, the SBH binary orbit shrinks at first due to dynamical friction and subsequently through slingshot ejection of stars intersecting the orbit. Investigations of quasi-steady spherical models suggested that the evolution of the binary stalls at separations ∼ 1 pc, due to a paucity of interacting stars, rather than hardening to the point at which gravitational wave emission drives the final inspiral to coalescence (the so-called \"final parsec problem\"; e.g., Merritt & Milosavljević 2005). Based on N -body simulations, Merritt (2006) estimated \"typical\" semi-major axes for stalled binaries, finding a stall 10 pc for an SBH mass ratio q = 0.5 and a stall 3 pc for q = 0.1, respectively. If the binary center of mass is located at the photocenter, the displacements of the primary and secondary components would be ∆r 1 ∼ a stall q/(1 + q) and ∆r 2 ∼ a stall /(1 + q), respectively, giving ∆r 1 ∼ 3 (0.3) pc and ∆r 2 ∼ 7 (3) pc for q = 0.5 (0.1). Thus stalled binaries could produce displacements comparable with our results, particularly if the secondary SBH is accreting and the mass ratio is near unity.\n2. Stalled SBH binaries: in the aftermath of a galaxy-galaxy merger, the SBH binary orbit shrinks at first due to dynamical friction and subsequently through slingshot ejection of stars intersecting the orbit. Investigations of quasi-steady spherical models suggested that the evolution of the binary stalls at separations ∼ 1 pc, due to a paucity of interacting stars, rather than hardening to the point at which gravitational wave emission drives the final inspiral to coalescence (the so-called \"final parsec problem\"; e.g., Merritt & Milosavljević 2005). Based on N -body simulations, Merritt (2006) estimated \"typical\" semi-major axes for stalled binaries, finding a stall 10 pc for an SBH mass ratio q = 0.5 and a stall 3 pc for q = 0.1, respectively. If the binary center of mass is located at the photocenter, the displacements of the primary and secondary components would be ∆r 1 ∼ a stall q/(1 + q) and ∆r 2 ∼ a stall /(1 + q), respectively, giving ∆r 1 ∼ 3 (0.3) pc and ∆r 2 ∼ 7 (3) pc for q = 0.5 (0.1). Thus stalled binaries could produce displacements comparable with our results, particularly if the secondary SBH is accreting and the mass ratio is near unity.\nIt has been argued that stalling can be avoided in galaxies containing significant amounts of nuclear gas (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007). Even in purely stellar nuclei, N -body simulations sometimes find that evolution of the binary can continue efficiently due to the presence of centrophilic orbits (Khan et al. 2011;Preto et al. 2011;Gualandris & Merritt 2012). However the existing gas-dynamical simulations probably do not yet have enough spatial resolution to follow the binary's evolution to sub-parsec scales, . Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position). and the N -body simulations appear to not yet have large enough N that their results can be robustly extrapolated to the much larger-N regime of real galaxies (Vasiliev et al. 2014). Another possibility, discussed by Antonini & Merritt (2012), is accretion of a less massive galaxy by a giant elliptical, such as NGC 4486, which has a pre-existing depleted core (presumably the result of the evolution of SBH binaries formed in previous mergers). In such situations, dynamical friction is very inefficient in the core due to the lack of stars moving slower than the sinking object. The orbital eccentricity can increase rapidly while the apoapsis hardly changes, resulting in a slowly evolving SBH binary in a highly eccentric orbit. The simulations presented by Antonini & Merritt (2012) indicate that in an M87-like core, a low mass ratio binary (q ≈ 10 −3 ) can persist over a Hubble time in an increasingly eccentric orbit with a semi major axis ∼ 10 pc. If the secondary SBH is accreting, it will be visible as an off-center AGN; if both components of the SBH binary are accreting, a double nucleus might be observed (perhaps as in NGC 4696 and NGC 5419).\nIt has been argued that stalling can be avoided in galaxies containing significant amounts of nuclear gas (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007). Even in purely stellar nuclei, N -body simulations sometimes find that evolution of the binary can continue efficiently due to the presence of centrophilic orbits (Khan et al. 2011;Preto et al. 2011;Gualandris & Merritt 2012). However the existing gas-dynamical simulations probably do not yet have enough spatial resolution to follow the binary's evolution to sub-parsec scales, . Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position). and the N -body simulations appear to not yet have large enough N that their results can be robustly extrapolated to the much larger-N regime of real galaxies (Vasiliev et al. 2014). Another possibility, discussed by Antonini & Merritt (2012), is accretion of a less massive galaxy by a giant elliptical, such as NGC 4486, which has a pre-existing depleted core (presumably the result of the evolution of SBH binaries formed in previous mergers). In such situations, dynamical friction is very inefficient in the core due to the lack of stars moving slower than the sinking object. The orbital eccentricity can increase rapidly while the apoapsis hardly changes, resulting in a slowly evolving SBH binary in a highly eccentric orbit. The simulations presented by Antonini & Merritt (2012) indicate that in an M87-like core, a low mass ratio binary (q ≈ 10 −3 ) can persist over a Hubble time in an increasingly eccentric orbit with a semi major axis ∼ 10 pc. If the secondary SBH is accreting, it will be visible as an off-center AGN; if both components of the SBH binary are accreting, a double nucleus might be observed (perhaps as in NGC 4696 and NGC 5419).\nHelical distortion or \"wiggling\" of parsec-scale radio jets has been linked to putative SBH binaries in several AGNs, with the jet wiggles being variously attributed to orbital motion, precession of the accretion disk around the jet-emitting black hole or to geodetic precession (e.g., Begelman, Blandford & Rees 1980;Roos et al. 1993;Katz 1997;Romero et al. 2000;Lobanov & Roland 2005). However, the periods (∼ 3 years) and separations (∼ 10 −2 pc) typically inferred from analyses of jet wiggles are much smaller than would be the case for a stalling binary. If the ∼ pc-scale displacements measured in this work are interpreted as SBH binary orbits, periods ∼ 10 4−5 years are implied for total masses ∼ 10 8−9 M . Geodetic precession is insignificant at these separations. The wavelength of jet wiggles caused by orbital motion is given by λ jet ∼ v app t orbit , where v app = βc sin i/(1 − β cos i) is the apparent jet velocity (for a jet speed v jet = βc and inclination to the line of sight, i) and t orbit is the orbital period. Therefore, assuming β ∼ 1, a pc-scale binary will produce very long wavelength wiggles in the jet (∼ 10 kpc). In general, due to the combination of orbital and jet velocities, the jet will precess on the surface of a cone which, for a pc-scale binary of mass ∼ 10 9 M will have a half-opening angle 1 • (see Equation 7 in Roos et al. 1993). Thus, orbital motion would cause only small curvatures in the jet over ∼ kpc scales, which would be difficult to discern as the jet loses collimation, or to distinguish from jet bending due to environmental effects, such as ram pressure. This will also be the case for disk precession, since the precession period exceeds the orbital period. Therefore, although the jet morphology has been mapped in detail from pc to kpc scales for a number of our sample galaxies (see Table 2 for references), these observations are unlikely to provide unambiguous clues as to the presence, or not, of a stalled pc-scale binary.\nHelical distortion or \"wiggling\" of parsec-scale radio jets has been linked to putative SBH binaries in several AGNs, with the jet wiggles being variously attributed to orbital motion, precession of the accretion disk around the jet-emitting black hole or to geodetic precession (e.g., Begelman, Blandford & Rees 1980;Roos et al. 1993;Katz 1997;Romero et al. 2000;Lobanov & Roland 2005). However, the periods (∼ 3 years) and separations (∼ 10 −2 pc) typically inferred from analyses of jet wiggles are much smaller than would be the case for a stalling binary. If the ∼ pc-scale displacements measured in this work are interpreted as SBH binary orbits, periods ∼ 10 4−5 years are implied for total masses ∼ 10 8−9 M . Geodetic precession is insignificant at these separations. The wavelength of jet wiggles caused by orbital motion is given by λ jet ∼ v app t orbit , where v app = βc sin i/(1 − β cos i) is the apparent jet velocity (for a jet speed v jet = βc and inclination to the line of sight, i) and t orbit is the orbital period. Therefore, assuming β ∼ 1, a pc-scale binary will produce very long wavelength wiggles in the jet (∼ 10 kpc). In general, due to the combination of orbital and jet velocities, the jet will precess on the surface of a cone which, for a pc-scale binary of mass ∼ 10 9 M will have a half-opening angle 1 • (see Equation 7 in Roos et al. 1993). Thus, orbital motion would cause only small curvatures in the jet over ∼ kpc scales, which would be difficult to discern as the jet loses collimation, or to distinguish from jet bending due to environmental effects, such as ram pressure. This will also be the case for disk precession, since the precession period exceeds the orbital period. Therefore, although the jet morphology has been mapped in detail from pc to kpc scales for a number of our sample galaxies (see Table 2 for references), these observations are unlikely to provide unambiguous clues as to the presence, or not, of a stalled pc-scale binary.\n3. Massive perturbers: galaxy centers host a variety of potential perturbers with masses ranging from ∼ 1M (e.g., stellar mass black holes and neutron stars) to ∼ 10 4 − 10 7 M , such as giant molecular clouds and stellar clusters.\n3. Massive perturbers: galaxy centers host a variety of potential perturbers with masses ranging from ∼ 1M (e.g., stellar mass black holes and neutron stars) to ∼ 10 4 − 10 7 M , such as giant molecular clouds and stellar clusters.\nGravitational interactions with these objects will cause the SBH to undergo a type of Brownian motion, with the amplitude of the root-mean-square displacement given by:\nGravitational interactions with these objects will cause the SBH to undergo a type of Brownian motion, with the amplitude of the root-mean-square displacement given by:\nwherem is proportional to the second moment of the mass distribution of the massive perturbers:\nwherem is proportional to the second moment of the mass distribution of the massive perturbers:\nm ≡ n(m)m 2 dm n(m)mdm with n(m) dm being the number of perturbers with masses in the range m, m + dm (Merritt 2013, eq. 7.63). The offsets measured in this work range from 0.01 to 0.1 r c , but are typically ∼ 0.01 r c (Table 6). For a typical SBH mass of order 10 9 M (Table 9), rms displacements ∼ 0.01r c can be generated by Brownian motion form ∼ 10 5 M .\nm ≡ n(m)m 2 dm n(m)mdm with n(m) dm being the number of perturbers with masses in the range m, m + dm (Merritt 2013, eq. 7.63). The offsets measured in this work range from 0.01 to 0.1 r c , but are typically ∼ 0.01 r c (Table 6). For a typical SBH mass of order 10 9 M (Table 9), rms displacements ∼ 0.01r c can be generated by Brownian motion form ∼ 10 5 M .\nThe mass functions of globular clusters, gas clumps and giant molecular clouds in the inner 100 pc of the Galaxy have been estimated by Perets et al. (2007). For comparison, using the mass functions presented in their paper, we find thatm can be as high as 10 5 M for a population of giant molecular clouds. The Milky Way, of course, is very different from the galaxies studied here. However, as recently discussed by Antonini & Merritt (2012), the low efficiency of dynamical friction in low density cores favors the formation of a population of stalling massive objects in the cores of giant ellipticals.\nThe mass functions of globular clusters, gas clumps and giant molecular clouds in the inner 100 pc of the Galaxy have been estimated by Perets et al. (2007). For comparison, using the mass functions presented in their paper, we find thatm can be as high as 10 5 M for a population of giant molecular clouds. The Milky Way, of course, is very different from the galaxies studied here. However, as recently discussed by Antonini & Merritt (2012), the low efficiency of dynamical friction in low density cores favors the formation of a population of stalling massive objects in the cores of giant ellipticals.\nTherefore, it seems plausible that displacements of order those observed could result from Brownian motion due to a population of massive perturbers. In several galaxies, the observed displacements are approximately aligned with the kpc-scale radio jets. This suggests that the displacement is not random, at least in these cases, favoring other mechanisms (i.e, gravitational recoil or jet thrust) that are expected to offset the SBH in the jet direction. Nevertheless, we conclude that Brownian motion cannot be excluded as the cause of the displacement in any individual galaxy, particularly those where the offsets are not aligned with kpc-scale jets, i.e. NGC 4278, 5846 and 5419.\nTherefore, it seems plausible that displacements of order those observed could result from Brownian motion due to a population of massive perturbers. In several galaxies, the observed displacements are approximately aligned with the kpc-scale radio jets. This suggests that the displacement is not random, at least in these cases, favoring other mechanisms (i.e, gravitational recoil or jet thrust) that are expected to offset the SBH in the jet direction. Nevertheless, we conclude that Brownian motion cannot be excluded as the cause of the displacement in any individual galaxy, particularly those where the offsets are not aligned with kpc-scale jets, i.e. NGC 4278, 5846 and 5419."},{"title":"SUMMARY AND CONCLUSIONS","content":""},{"title":"SUMMARY AND CONCLUSIONS","content":"We have analyzed HST archival images of 14 nearby core elliptical galaxies, each of which hosts a central point-like source associated with a low-luminosity AGN, in order to search for offsets between the AGN and the galaxy photocenter. Such AGN-photocenter displacements are possible signposts of gravitational recoils resulting from the coalescence of an SBH binary.\nWe have analyzed HST archival images of 14 nearby core elliptical galaxies, each of which hosts a central point-like source associated with a low-luminosity AGN, in order to search for offsets between the AGN and the galaxy photocenter. Such AGN-photocenter displacements are possible signposts of gravitational recoils resulting from the coalescence of an SBH binary.\nWe find significant (> 3σ) differences between the positions of the nuclear optical (or NIR) point source and the mean photocenter of the galaxy, as determined from isophote fits, in ten of the 14 galaxies in the sample. Assuming that the mean photocenter locates the minimum of the galactic potential well and that the point source locates the position of the AGN and hence the SBH, these results imply that the SBH is displaced from its equilibrium position by angular distances ranging between 20 3.8 ± 0.7 338 ± 7\nWe find significant (> 3σ) differences between the positions of the nuclear optical (or NIR) point source and the mean photocenter of the galaxy, as determined from isophote fits, in ten of the 14 galaxies in the sample. Assuming that the mean photocenter locates the minimum of the galactic potential well and that the point source locates the position of the AGN and hence the SBH, these results imply that the SBH is displaced from its equilibrium position by angular distances ranging between 20 3.8 ± 0.7 338 ± 7\nGalaxies with two nuclear sources NGC 5419 7.5 ± 1.7 62 ± 2 252 ± 13 346 ± 2\nGalaxies with two nuclear sources NGC 5419 7.5 ± 1.7 62 ± 2 252 ± 13 346 ± 2\nNote. -Projected offsets of the photocenter with respect to the AGN and their possible origin. When multiple images have been analyzed, values presented here are the error weighted average. Position angles (PA) are given in degrees East from North. NPS = nuclear point source. and 90 mas, or projected linear distances in the range 1 − 10 pc. As spurious offsets may occur as a result of large-scale isophotal asymmetries, only displacements of magnitude > 0.8 × the inter-quartile range of the distribution of isophote center coordinates (equivalent to 1σ for a gaussian distribution) are considered \"real\". There are six galaxies that exhibit displacements > 0.8 IQR with three of these (NGC 4278, NGC 4486 and NGC 5846) having displacements > 1.6 IQR (equivalent to 2σ). In every case, the measured displacement of the SBH relative to the galaxy photocenter is a small fraction (1-10%) of the galaxy core radius, which is typically r c ∼ 300 pc, for these galaxies.\nNote. -Projected offsets of the photocenter with respect to the AGN and their possible origin. When multiple images have been analyzed, values presented here are the error weighted average. Position angles (PA) are given in degrees East from North. NPS = nuclear point source. and 90 mas, or projected linear distances in the range 1 − 10 pc. As spurious offsets may occur as a result of large-scale isophotal asymmetries, only displacements of magnitude > 0.8 × the inter-quartile range of the distribution of isophote center coordinates (equivalent to 1σ for a gaussian distribution) are considered \"real\". There are six galaxies that exhibit displacements > 0.8 IQR with three of these (NGC 4278, NGC 4486 and NGC 5846) having displacements > 1.6 IQR (equivalent to 2σ). In every case, the measured displacement of the SBH relative to the galaxy photocenter is a small fraction (1-10%) of the galaxy core radius, which is typically r c ∼ 300 pc, for these galaxies.\nApproximate alignments between the SBHphotocenter displacements and the radio source axis were found in four of the six galaxies considered to have significant displacements, including three of the four that have FR I, or FR I-like radio sources with relatively powerful and well-defined kpc-scale jets. Indeed, in every case in which there is both a significant displacement and an unambiguous jet, the two are approximately aligned.\nApproximate alignments between the SBHphotocenter displacements and the radio source axis were found in four of the six galaxies considered to have significant displacements, including three of the four that have FR I, or FR I-like radio sources with relatively powerful and well-defined kpc-scale jets. Indeed, in every case in which there is both a significant displacement and an unambiguous jet, the two are approximately aligned.\nLacking detailed knowledge of the merger history of the galaxies, or of the SBH binary parameters (such as mass ratio and spin configuration) that determine the recoil velocity, it is not possible to directly test the hypothesis that the displacements are caused by residual gravitational recoil oscillations. Instead, we used a simple Monte-Carlo model to investigate if the measured displacements are consistent with gravitational recoil. We find that the displacements in individual objects can plausibly be attributed to residual gravitational recoil oscillations following a major or minor merger within the last few Gyr. However, for plausible merger rates there is a high probability of larger displacements than actually observed, if SBH coalescence events took place in these galaxies.\nLacking detailed knowledge of the merger history of the galaxies, or of the SBH binary parameters (such as mass ratio and spin configuration) that determine the recoil velocity, it is not possible to directly test the hypothesis that the displacements are caused by residual gravitational recoil oscillations. Instead, we used a simple Monte-Carlo model to investigate if the measured displacements are consistent with gravitational recoil. We find that the displacements in individual objects can plausibly be attributed to residual gravitational recoil oscillations following a major or minor merger within the last few Gyr. However, for plausible merger rates there is a high probability of larger displacements than actually observed, if SBH coalescence events took place in these galaxies.\nThat larger displacements were not observed suggests that the frequency of gravitational recoil kicks large enough to trigger long-lived oscillations is lower than predicted, perhaps because the evolution of the SBH binary typically results in a configuration that suppresses recoil kicks with velocities 200 km s −1 . Alternatively, the post-recoil oscillations may be damped more quickly than predicted by pure N -body simulations. In either case, gas may play an important role (e.g., Dotti et al. 2010;Sijacki, Springel & Haehnelt 2011). Otherwise, it is possible that galaxy mergers are much more infrequent than implied by the current ΛCDM paradigm (Kroupa 2014).\nThat larger displacements were not observed suggests that the frequency of gravitational recoil kicks large enough to trigger long-lived oscillations is lower than predicted, perhaps because the evolution of the SBH binary typically results in a configuration that suppresses recoil kicks with velocities 200 km s −1 . Alternatively, the post-recoil oscillations may be damped more quickly than predicted by pure N -body simulations. In either case, gas may play an important role (e.g., Dotti et al. 2010;Sijacki, Springel & Haehnelt 2011). Otherwise, it is possible that galaxy mergers are much more infrequent than implied by the current ΛCDM paradigm (Kroupa 2014).\nSeveral other mechanisms are capable of producing the observed displacements, with the observed alignments between the SBH-photocenter displacements and the radio source axis favoring jet acceleration in some objects. An approximate displacement-radio axis alignment is also expected for gravitational recoil, but not for orbital motion in pre-coalescence SBH binaries or interactions with massive perturbers. However, both of the latter mechanisms are capable of producing displacement amplitudes comparable to those observed and cannot be ruled out in individual objects.\nSeveral other mechanisms are capable of producing the observed displacements, with the observed alignments between the SBH-photocenter displacements and the radio source axis favoring jet acceleration in some objects. An approximate displacement-radio axis alignment is also expected for gravitational recoil, but not for orbital motion in pre-coalescence SBH binaries or interactions with massive perturbers. However, both of the latter mechanisms are capable of producing displacement amplitudes comparable to those observed and cannot be ruled out in individual objects.\nIn general, it is not possible to unambiguously distinguish between different mechanisms (including recoil) on the basis of the displacement measurements alone for individual galaxies. However, with a larger sample it may be possible to distinguish mechanisms using statistical arguments. Thus, for jet acceleration the displacement direction should be strongly correlated with the radio jet, with the amplitude correlating with jet power. In the case of gravitational recoil, a weaker correlation with jet direction might be expected. However, no such correlations are expected for binary SBH, or massive perturbers.\nIn general, it is not possible to unambiguously distinguish between different mechanisms (including recoil) on the basis of the displacement measurements alone for individual galaxies. However, with a larger sample it may be possible to distinguish mechanisms using statistical arguments. Thus, for jet acceleration the displacement direction should be strongly correlated with the radio jet, with the amplitude correlating with jet power. In the case of gravitational recoil, a weaker correlation with jet direction might be expected. However, no such correlations are expected for binary SBH, or massive perturbers."},{"title":"ACKNOWLEDGMENTS","content":""},{"title":"ACKNOWLEDGMENTS","content":"We wish to honor the memory of our great friend, colleague and mentor David Axon. We thank the anonymous referee for the valuable and timely comments that improved the paper. DL thanks E. Vasiliev, R. Mittal, F. Antonini and M. Richmond for helpful discussions. AM acknowledges support from the Italian National Institute for Astrophysics (INAF) through PRIN-INAF 2011 \"Black hole growth and AGN feedback through the cosmic time\" and from the Italian ministry for school, university and research (MIUR) through PRIN-MIUR 2010-2011 \"The dark Universe and the cosmic evolution of baryons: from current surveys to Euclid\". DM was supported by the National Science Foundation under grant no. AST 1211602 and by the National Aeronautics and Space Administration under grant no. NNX13AG92G. Support for program AR-11771 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr) and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Center (CADC/NRC/CSA). Figure 6. HST images of the galaxies in the sample. The number at the bottom represents the length of the whole axes.\nWe wish to honor the memory of our great friend, colleague and mentor David Axon. We thank the anonymous referee for the valuable and timely comments that improved the paper. DL thanks E. Vasiliev, R. Mittal, F. Antonini and M. Richmond for helpful discussions. AM acknowledges support from the Italian National Institute for Astrophysics (INAF) through PRIN-INAF 2011 \"Black hole growth and AGN feedback through the cosmic time\" and from the Italian ministry for school, university and research (MIUR) through PRIN-MIUR 2010-2011 \"The dark Universe and the cosmic evolution of baryons: from current surveys to Euclid\". DM was supported by the National Science Foundation under grant no. AST 1211602 and by the National Aeronautics and Space Administration under grant no. NNX13AG92G. Support for program AR-11771 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr) and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Center (CADC/NRC/CSA). Figure 6. HST images of the galaxies in the sample. The number at the bottom represents the length of the whole axes."},{"title":"APPENDIX","content":""},{"title":"APPENDIX","content":""},{"title":"A: DISTRIBUTION OF RECOIL DISPLACEMENTS","content":""},{"title":"A: DISTRIBUTION OF RECOIL DISPLACEMENTS","content":"Here we outline the method used to estimate the probability distributions for displacements resulting from postrecoil oscillations of a kicked SBH. For our purposes, of the three phases predicted by GM08, only phase II (when the oscillation amplitude becomes comparable with the galaxy core radius) is of interest. The initial, large amplitude phase I oscillations are short-lived, while in phase III the residual oscillations are indistinguishable from Brownian motion (which is considered separately in Section 5.3).\nHere we outline the method used to estimate the probability distributions for displacements resulting from postrecoil oscillations of a kicked SBH. For our purposes, of the three phases predicted by GM08, only phase II (when the oscillation amplitude becomes comparable with the galaxy core radius) is of interest. The initial, large amplitude phase I oscillations are short-lived, while in phase III the residual oscillations are indistinguishable from Brownian motion (which is considered separately in Section 5.3).\nIn the GM08 N -body simulations, phase II oscillations occur for kick velocities in the range v kick ∼ 0.4 − 0.9 v esc . The simulations also indicate that the subsequent evolution does not depend strongly on the magnitude of the initial kick. Therefore, we assume that, given any initial kick large enough to displace the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy evolves during phase II as a damped harmonic oscillator:\nIn the GM08 N -body simulations, phase II oscillations occur for kick velocities in the range v kick ∼ 0.4 − 0.9 v esc . The simulations also indicate that the subsequent evolution does not depend strongly on the magnitude of the initial kick. Therefore, we assume that, given any initial kick large enough to displace the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy evolves during phase II as a damped harmonic oscillator:\nThe initial amplitude, R 0 , is set by the condition that phase II begins when the oscillation amplitude is comparable to the core radius; ∆t is the time since the kick, i.e., the time since the last merger, and ω c is the SBH oscillation frequency:\nThe initial amplitude, R 0 , is set by the condition that phase II begins when the oscillation amplitude is comparable to the core radius; ∆t is the time since the kick, i.e., the time since the last merger, and ω c is the SBH oscillation frequency:\nis obtained by combining the radial oscillation frequency in a spherical core of uniform density, ω c = [(4π/3)Gρ c ] 1/2 , with equation 13 in GM08. The damping time, τ , is given by equation 2. The adopted parameter values are listed in Table 8.\nis obtained by combining the radial oscillation frequency in a spherical core of uniform density, ω c = [(4π/3)Gρ c ] 1/2 , with equation 13 in GM08. The damping time, τ , is given by equation 2. The adopted parameter values are listed in Table 8.\nSuppose that we observe a galaxy at a random time since its last merger. For simplicity, we assume that galaxy mergers occur stochastically and can be represented as a Poisson process. The probability that the last merger occurred at time ∆t ago is then:\nSuppose that we observe a galaxy at a random time since its last merger. For simplicity, we assume that galaxy mergers occur stochastically and can be represented as a Poisson process. The probability that the last merger occurred at time ∆t ago is then:\nwhich is associated with the cumulative distribution:\nwhich is associated with the cumulative distribution:\nwhere t m is the mean time between galaxy mergers. The galaxy merger rate is uncertain but simulations indicate that it is a function of redshift, galaxy mass and mass ratio (e.g., Bell et al. 2006;Conselice, Yang & Bluck 2009;Hopkins et al. 2010;Burke & Collins 2013). Here, since the phase II damping times are typically ∼ 1 Gyr (Table 8), we consider redshifts z ≤ 1 and estimate the mean time between galaxy mergers as the ratio between the lookback time corresponding to z = 1 and the number of mergers, N , during that time. The latter was determined using the analytical fits to semi-analytical models for galaxy merger rates given by H10. Integrating the merger rate as given by equation 5 in H10 over redshift, we find for the number of mergers,\nwhere t m is the mean time between galaxy mergers. The galaxy merger rate is uncertain but simulations indicate that it is a function of redshift, galaxy mass and mass ratio (e.g., Bell et al. 2006;Conselice, Yang & Bluck 2009;Hopkins et al. 2010;Burke & Collins 2013). Here, since the phase II damping times are typically ∼ 1 Gyr (Table 8), we consider redshifts z ≤ 1 and estimate the mean time between galaxy mergers as the ratio between the lookback time corresponding to z = 1 and the number of mergers, N , during that time. The latter was determined using the analytical fits to semi-analytical models for galaxy merger rates given by H10. Integrating the merger rate as given by equation 5 in H10 over redshift, we find for the number of mergers,\nwhere t H = 13.7 Gyr is the Hubble time, the normalization constant A(M min ) and the slope β(M min ) are given as functions of the galaxy mass threshold M min , for mergers with mass ratio q > 0.1, by equations 9 & 10 in H10. For log M gal ≥ log M min = 11, we find N ≈ 1.5 for the number of mergers per galaxy between z = 0 and z = 1. However, we note that N could vary between 0.75 and 3, allowing for systematic uncertainties of a factor two in A(M min ) and 0.2 in β(M min ). The look back time at z = 1 is t L = 7.731 Gyr, for H 0 =71 km s −1 Mpc −1 , Ω M = 0.27 and Ω vac = 0.73, yielding a mean time between galactic mergers t m ≈ 5 Gyr. The simple analytical functions provided by H10 do not accurately reproduce the numerical results for the most massive galaxies. Therefore, we use an average merger rate for z ≤ 1 estimated from their Figure 3, for log M gal > 12. The rate for q > 0.1 mergers is roughly constant for z ≤ 1 and it seems reasonable to adopt an average value of 2.5 mergers/galaxy/Gyr. This corresponds to t m ≈ 0.4 Gyr.\nwhere t H = 13.7 Gyr is the Hubble time, the normalization constant A(M min ) and the slope β(M min ) are given as functions of the galaxy mass threshold M min , for mergers with mass ratio q > 0.1, by equations 9 & 10 in H10. For log M gal ≥ log M min = 11, we find N ≈ 1.5 for the number of mergers per galaxy between z = 0 and z = 1. However, we note that N could vary between 0.75 and 3, allowing for systematic uncertainties of a factor two in A(M min ) and 0.2 in β(M min ). The look back time at z = 1 is t L = 7.731 Gyr, for H 0 =71 km s −1 Mpc −1 , Ω M = 0.27 and Ω vac = 0.73, yielding a mean time between galactic mergers t m ≈ 5 Gyr. The simple analytical functions provided by H10 do not accurately reproduce the numerical results for the most massive galaxies. Therefore, we use an average merger rate for z ≤ 1 estimated from their Figure 3, for log M gal > 12. The rate for q > 0.1 mergers is roughly constant for z ≤ 1 and it seems reasonable to adopt an average value of 2.5 mergers/galaxy/Gyr. This corresponds to t m ≈ 0.4 Gyr.\nThe stellar mass, M gal , is derived for each galaxy from the relation:\nThe stellar mass, M gal , is derived for each galaxy from the relation:\nwhere Υ is the mass-to-light ratio and M K is the K-band magnitude given by BC06. For the mass-to-light ratio, we use a representative value, Υ = 3.1, computed as the average value of the ratio between the virial mass and the K-band luminosity for the sample of nearby core ellipticals studied by Sani et al. (2011).\nwhere Υ is the mass-to-light ratio and M K is the K-band magnitude given by BC06. For the mass-to-light ratio, we use a representative value, Υ = 3.1, computed as the average value of the ratio between the virial mass and the K-band luminosity for the sample of nearby core ellipticals studied by Sani et al. (2011).\nAs the derived values of M gal span the range 11 log M gal 12 we compute displacement probability distributions for the limiting cases t m = 0.4 Gyr and t m = 5.0 Gyr. For comparison, Lidman et al. (2013) find an observationallyderived major merger rate of ≈ 0.38 mergers Gyr −1 for brightest cluster galaxies at z ∼ 1, implying t m ≈ 2.6 Gyr.\nAs the derived values of M gal span the range 11 log M gal 12 we compute displacement probability distributions for the limiting cases t m = 0.4 Gyr and t m = 5.0 Gyr. For comparison, Lidman et al. (2013) find an observationallyderived major merger rate of ≈ 0.38 mergers Gyr −1 for brightest cluster galaxies at z ∼ 1, implying t m ≈ 2.6 Gyr.\nFrom the cumulative distributions given by eq. A4, we extract random times in order to generate a set of R i ≡ R(t i ) from equation A1 for each galaxy, for both values of t m . Projected displacements are then calculated:\nFrom the cumulative distributions given by eq. A4, we extract random times in order to generate a set of R i ≡ R(t i ) from equation A1 for each galaxy, for both values of t m . Projected displacements are then calculated:\nwhere θ i is the angle between the displacement direction, assumed to be randomly oriented, and the line of sight.\nwhere θ i is the angle between the displacement direction, assumed to be randomly oriented, and the line of sight.\nFinally, the distribution of the projected offsets is computed. In Figs. 7a and 7b we plot the probability, P (d > x), of observing a displacement, d, larger than a given value, x, as a function of x. Values of the measured displacements are also plotted. The probability of generating a projected displacement larger than the value actually measured is listed for each galaxy in Table 6, for both values of t m .\nFinally, the distribution of the projected offsets is computed. In Figs. 7a and 7b we plot the probability, P (d > x), of observing a displacement, d, larger than a given value, x, as a function of x. Values of the measured displacements are also plotted. The probability of generating a projected displacement larger than the value actually measured is listed for each galaxy in Table 6, for both values of t m .\nThe damping time computed from eq.2 depends on the mass of the SBH. Measurement uncertainties for SBH masses in galaxies beyond the Local Group are typically large and are probably dominated by poorly-understood systematics, given that none of these galaxies exhibits a clear central rise in the stellar velocities (Merritt 2013, section 2.2). In Table 9 we list SBH masses estimated from the M • -σ relation (Ferrarese & Ford 2005), from the M • -M V relation for bright galaxies (M V < −19, Lauer et al. 2007) and from direct measurements (stellar or gas dynamics modeling). For most of our galaxies the discrepancy in the masses recovered with different methods is small. We compute damping times using SBH masses from the M • -M V relation finding a smaller average damping timeτ = 0.66 Gyr, although the new damping times for individual galaxies are not systematically smaller than those computed using the M • -σ SBH masses. We then compute new values for the probabilities presented in Table 6 and the corresponding cumulative probabilities of not observing displacements larger than those actually measured in this work (P). We obtain P = 10 −16 (10 −3 ), instead of 10 −17 (10 −4 ), for an average time between galactic mergers t m = 5 (0.4) Gyr. After taking into account the kick probabilities given by L12, as specified in §5.2, the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 1×10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5.0 Gyr. The different masses adopted for the SBH caused, therefore, a difference in P of one order of magnitude. Nevertheless, even for t m = 5 Gyr, there remains a large probability of finding displacements larger than those actually observed, if these galaxies experienced gravitational recoil events within the last few Gyr.\nThe damping time computed from eq.2 depends on the mass of the SBH. Measurement uncertainties for SBH masses in galaxies beyond the Local Group are typically large and are probably dominated by poorly-understood systematics, given that none of these galaxies exhibits a clear central rise in the stellar velocities (Merritt 2013, section 2.2). In Table 9 we list SBH masses estimated from the M • -σ relation (Ferrarese & Ford 2005), from the M • -M V relation for bright galaxies (M V < −19, Lauer et al. 2007) and from direct measurements (stellar or gas dynamics modeling). For most of our galaxies the discrepancy in the masses recovered with different methods is small. We compute damping times using SBH masses from the M • -M V relation finding a smaller average damping timeτ = 0.66 Gyr, although the new damping times for individual galaxies are not systematically smaller than those computed using the M • -σ SBH masses. We then compute new values for the probabilities presented in Table 6 and the corresponding cumulative probabilities of not observing displacements larger than those actually measured in this work (P). We obtain P = 10 −16 (10 −3 ), instead of 10 −17 (10 −4 ), for an average time between galactic mergers t m = 5 (0.4) Gyr. After taking into account the kick probabilities given by L12, as specified in §5.2, the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 1×10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5.0 Gyr. The different masses adopted for the SBH caused, therefore, a difference in P of one order of magnitude. Nevertheless, even for t m = 5 Gyr, there remains a large probability of finding displacements larger than those actually observed, if these galaxies experienced gravitational recoil events within the last few Gyr."},{"title":"B: NOTES ON INDIVIDUAL SOURCES","content":""},{"title":"B: NOTES ON INDIVIDUAL SOURCES","content":"In this section, the galaxies are ordered according to the significance assigned to the displacement.\nIn this section, the galaxies are ordered according to the significance assigned to the displacement."},{"title":"NGC 4278 (high/intermediate)","content":""},{"title":"NGC 4278 (high/intermediate)","content":"NGC 4278 is a large elliptical near the center of the dense Coma I cloud which is characterized by the presence of an extended dust distribution extending over an arc ranging from N to NW relative to the photocenter (Carollo et al. 1997). Radio interferometry (VLBA) observations reveal a low-power, parsec-scale source (∼ 45 mas) with twin mildly-relativistic S-shaped jets emerging from a compact, flat spectrum core (Giroletti et al. 2005). Within the inner 7 mas the jets are oriented SE-NW but then bend to the east and west, respectively, forming an \"S\" shape. Giroletti et al. argue that the jets are closely aligned with the line of sight (with a viewing angle ∼ 2 − 4 • ) with the NW jet approaching and the bends being explained by amplification of intrinsically small deviations due to projection effects. These characteristics and the presence of atomic hydrogen suggest that the galaxy may have experienced a minor merger with another member of the cloud (Gerin & Casoli 1994).\nNGC 4278 is a large elliptical near the center of the dense Coma I cloud which is characterized by the presence of an extended dust distribution extending over an arc ranging from N to NW relative to the photocenter (Carollo et al. 1997). Radio interferometry (VLBA) observations reveal a low-power, parsec-scale source (∼ 45 mas) with twin mildly-relativistic S-shaped jets emerging from a compact, flat spectrum core (Giroletti et al. 2005). Within the inner 7 mas the jets are oriented SE-NW but then bend to the east and west, respectively, forming an \"S\" shape. Giroletti et al. argue that the jets are closely aligned with the line of sight (with a viewing angle ∼ 2 − 4 • ) with the NW jet approaching and the bends being explained by amplification of intrinsically small deviations due to projection effects. These characteristics and the presence of atomic hydrogen suggest that the galaxy may have experienced a minor merger with another member of the cloud (Gerin & Casoli 1994).\nPhotometric analysis. Three images, obtained with ACS/WFC-F850LP, WFPC2/PC-F814W and NIC-MOS2/F160W, respectively, were analyzed for this galaxy (Figs 15, 16 and 17). Photocenter-AGN displacements of intermediate or high significance are detected in all three. The uncertainty-weighted average displacement has an amplitude ∆r ≥ 7.6 ± 0.4 pc and direction SE (PA = 152 ± 3 • ). The photocenter positions derived from each image are plotted relative to the AGN position in Fig.5a. The photometric analysis of the optical ACS and WFPC2 images shows that the y co-ordinate of the isophote centers migrates southward at intermediate SMA, then northward at larger SMA. These irregularities are not present in the NICMOS2 image and are likely due to the dust N-NW of the center.\nPhotometric analysis. Three images, obtained with ACS/WFC-F850LP, WFPC2/PC-F814W and NIC-MOS2/F160W, respectively, were analyzed for this galaxy (Figs 15, 16 and 17). Photocenter-AGN displacements of intermediate or high significance are detected in all three. The uncertainty-weighted average displacement has an amplitude ∆r ≥ 7.6 ± 0.4 pc and direction SE (PA = 152 ± 3 • ). The photocenter positions derived from each image are plotted relative to the AGN position in Fig.5a. The photometric analysis of the optical ACS and WFPC2 images shows that the y co-ordinate of the isophote centers migrates southward at intermediate SMA, then northward at larger SMA. These irregularities are not present in the NICMOS2 image and are likely due to the dust N-NW of the center.\nOffset origin. The extended dust distribution produces a large-scale asymmetry in surface brightness, which is particularly evident in optical and UV wavelengths. This may bias the photocenter, shifting it in the opposite direction (S-SE) to the dust. As the AGN is a point source, its position should not be significantly affected by the dust distribution and thus it would appear to be shifted N-NW relative to the photocenter. Therefore, the offsets recovered Note. -(1) Galaxy name;\nOffset origin. The extended dust distribution produces a large-scale asymmetry in surface brightness, which is particularly evident in optical and UV wavelengths. This may bias the photocenter, shifting it in the opposite direction (S-SE) to the dust. As the AGN is a point source, its position should not be significantly affected by the dust distribution and thus it would appear to be shifted N-NW relative to the photocenter. Therefore, the offsets recovered Note. -(1) Galaxy name;\n(2) core radius from CB05; (3) stellar velocity dispersion from Hyperleda; (4) stellar mass of the galaxy; (5) dynamical time; (6) damping time; (7) kick velocity required to displace the SBH up to the core radius. The dominant source of errors in the quantities presented above is the stellar velocity dispersion. Note. -(1) Galaxy name, (2) masses from the M• − σ (Ferrarese & Ford 2005) using the velocity dispersion specified in Table 8 from ACS and WFPC2 data may be partly due to asymmetric extinction. However, the effects of dust are not apparent in the NIR (NICMOS2) image, from which an offset of similar magnitude and direction was recovered. Therefore, we believe that the measured displacement is not simply an artifact of extinction, although this may contribute. Relative to the core radius, NGC 4278 has the largest displacement (≈ 0.1r c ; Table 6) in our sample. If it is due to phase II oscillations of the SBH following gravitational recoil (see §5.2), we estimate that the probability of observing a projected displacement larger than that actually measured is 4 (36)% for a mean time-between-mergers of t m = 5 (0.4) Gyr (Table 6) and given a kick large enough to displace the SBH at least as far as the core radius. The \"hot\" disk recoil velocity distribution of L12 indicates a 45% probability of a sufficiently large kick ( 200 km s −1 , Table 8) occurring in an SBH coalescence event. Therefore, we consider residual oscillations following gravitational recoil to be a plausible explanation for the displacement.\n(2) core radius from CB05; (3) stellar velocity dispersion from Hyperleda; (4) stellar mass of the galaxy; (5) dynamical time; (6) damping time; (7) kick velocity required to displace the SBH up to the core radius. The dominant source of errors in the quantities presented above is the stellar velocity dispersion. Note. -(1) Galaxy name, (2) masses from the M• − σ (Ferrarese & Ford 2005) using the velocity dispersion specified in Table 8 from ACS and WFPC2 data may be partly due to asymmetric extinction. However, the effects of dust are not apparent in the NIR (NICMOS2) image, from which an offset of similar magnitude and direction was recovered. Therefore, we believe that the measured displacement is not simply an artifact of extinction, although this may contribute. Relative to the core radius, NGC 4278 has the largest displacement (≈ 0.1r c ; Table 6) in our sample. If it is due to phase II oscillations of the SBH following gravitational recoil (see §5.2), we estimate that the probability of observing a projected displacement larger than that actually measured is 4 (36)% for a mean time-between-mergers of t m = 5 (0.4) Gyr (Table 6) and given a kick large enough to displace the SBH at least as far as the core radius. The \"hot\" disk recoil velocity distribution of L12 indicates a 45% probability of a sufficiently large kick ( 200 km s −1 , Table 8) occurring in an SBH coalescence event. Therefore, we consider residual oscillations following gravitational recoil to be a plausible explanation for the displacement.\nIn this case, we can estimate the merger epoch. Supposing the SBH to be undergoing phase II oscillations, we combine eq.2 and 3. Using the measured (projected) offset as a lower limit for r rms , r rms 7.6 pc, and taking the values of r c (core radius) and σ (stellar velocity dispersion) from Table 8, we find that (t − t c ) 5τ ≈ 0.7 Gyr has elapsed since the amplitude of the oscillation damped down to r ∼ r c . This implies that the onset of phase II occurred no later than ∼ 0.7 Gyr ago. As phase I is shorter than phase II, this scenario is consistent with an SBH-SBH coalescence event that occurred within 2(t − t c ) ∼ 1.4 Gyr. The total evolution time from the beginning of the galactic merger to SBH binary coalescence is suggested to be of order 10 8 yr (Gualandris & Merritt 2012). Adding this time to the estimated kick epoch, we infer that a galaxy merger that took place up to ∼ 1.5 Gyr in the past could be responsible for a recoil kick consistent with the observed displacement.\nIn this case, we can estimate the merger epoch. Supposing the SBH to be undergoing phase II oscillations, we combine eq.2 and 3. Using the measured (projected) offset as a lower limit for r rms , r rms 7.6 pc, and taking the values of r c (core radius) and σ (stellar velocity dispersion) from Table 8, we find that (t − t c ) 5τ ≈ 0.7 Gyr has elapsed since the amplitude of the oscillation damped down to r ∼ r c . This implies that the onset of phase II occurred no later than ∼ 0.7 Gyr ago. As phase I is shorter than phase II, this scenario is consistent with an SBH-SBH coalescence event that occurred within 2(t − t c ) ∼ 1.4 Gyr. The total evolution time from the beginning of the galactic merger to SBH binary coalescence is suggested to be of order 10 8 yr (Gualandris & Merritt 2012). Adding this time to the estimated kick epoch, we infer that a galaxy merger that took place up to ∼ 1.5 Gyr in the past could be responsible for a recoil kick consistent with the observed displacement.\nAlthough the displacement is approximately in the direction of elongation of the mas-scale radio core (155 • over the inner ∼ 7 mas), the low power (P 5GHz ≈ 70 × 10 20 W, Wrobel 1991) and compact nature (∼ 3 pc) of the radio source argue against jet thrust as the cause of the offset.\nAlthough the displacement is approximately in the direction of elongation of the mas-scale radio core (155 • over the inner ∼ 7 mas), the low power (P 5GHz ≈ 70 × 10 20 W, Wrobel 1991) and compact nature (∼ 3 pc) of the radio source argue against jet thrust as the cause of the offset.\nWe find no compelling reasons to exclude either a stalled binary on an eccentric orbit, or massive perturbers (Section 5.3) as possible causes of the displacement.\nWe find no compelling reasons to exclude either a stalled binary on an eccentric orbit, or massive perturbers (Section 5.3) as possible causes of the displacement."},{"title":"NGC 5846 (intermediate)","content":""},{"title":"NGC 5846 (intermediate)","content":"NGC 5846 is the dominant component of a small, compact group of ≈ 50 galaxies (Huchra & Geller 1982). A companion galaxy, NGC5846A, is located 0.7 arcmin to the S, but there is no evidence of an interaction. The presence of dust lanes in an apparent spiral pattern approximately centered on the compact radio core suggests a past merger, or accretion of a gas rich galaxy (Moellenhoff et al. 1992;Forbes et al. 1996). The optical nucleus has a LINER spectrum (Ho et al. 1997) and HST imaging (Masegosa et al. 2011) reveals extended Hα emission in a wide structure extending up to 2 W of the nucleus. The presence of two radio emitting bubbles located roughly symmetrically at ∼ 0.6 kpc from the center and a \"ghost\" X-ray cavity suggest that the galaxy recently experienced stronger AGN activity, for which Machacek et al. (2011) derive a duty cycle of ∼ 10 Myr. The radio morphology is complex: Filho et al. (2004) report VLBA observations at 2.3, 5 and 15 GHz which resolve the core into multiple blobs roughly aligned along the N-S axis. The nature of this structure is unclear, possibilities include an AGN jet, or compact supernovae remnants (e.g. Tarchi et al. 2000).\nNGC 5846 is the dominant component of a small, compact group of ≈ 50 galaxies (Huchra & Geller 1982). A companion galaxy, NGC5846A, is located 0.7 arcmin to the S, but there is no evidence of an interaction. The presence of dust lanes in an apparent spiral pattern approximately centered on the compact radio core suggests a past merger, or accretion of a gas rich galaxy (Moellenhoff et al. 1992;Forbes et al. 1996). The optical nucleus has a LINER spectrum (Ho et al. 1997) and HST imaging (Masegosa et al. 2011) reveals extended Hα emission in a wide structure extending up to 2 W of the nucleus. The presence of two radio emitting bubbles located roughly symmetrically at ∼ 0.6 kpc from the center and a \"ghost\" X-ray cavity suggest that the galaxy recently experienced stronger AGN activity, for which Machacek et al. (2011) derive a duty cycle of ∼ 10 Myr. The radio morphology is complex: Filho et al. (2004) report VLBA observations at 2.3, 5 and 15 GHz which resolve the core into multiple blobs roughly aligned along the N-S axis. The nature of this structure is unclear, possibilities include an AGN jet, or compact supernovae remnants (e.g. Tarchi et al. 2000).\nPhotometric Analysis. One image, obtained with WFPC2/PC -F814W was analyzed for this galaxy (Figure 24). We find a photocenter-AGN displacement ∆r 8.2 ± 2.5 pc in the direction W-SW (PA = 253 ± 8 • ) which is considered to be of intermediate significance. The isophote centers show systematic migrations of ≈ 2 pixels in both x and y coordinates, from W to E and S to N, respectively, as the SMA decreases from 300 to 200 pixels. These migrations are possibly the result of the dust lanes to the S and NE of the nucleus. The y-component of the displacement is not significant. The x-component is considered to be of intermediate significance (1.6 IQR) and is unlikely to be an artifact of extinction, which in this case, would bias the photocenter such as to increase the x-displacement. Of more concern is the determination of the AGN position. The central source is relatively weak and embedded in an asymmetric structure that affects the gaussian fits used to determine the AGN position. We therefore assign a larger uncertainty to the x-component of the AGN position.\nPhotometric Analysis. One image, obtained with WFPC2/PC -F814W was analyzed for this galaxy (Figure 24). We find a photocenter-AGN displacement ∆r 8.2 ± 2.5 pc in the direction W-SW (PA = 253 ± 8 • ) which is considered to be of intermediate significance. The isophote centers show systematic migrations of ≈ 2 pixels in both x and y coordinates, from W to E and S to N, respectively, as the SMA decreases from 300 to 200 pixels. These migrations are possibly the result of the dust lanes to the S and NE of the nucleus. The y-component of the displacement is not significant. The x-component is considered to be of intermediate significance (1.6 IQR) and is unlikely to be an artifact of extinction, which in this case, would bias the photocenter such as to increase the x-displacement. Of more concern is the determination of the AGN position. The central source is relatively weak and embedded in an asymmetric structure that affects the gaussian fits used to determine the AGN position. We therefore assign a larger uncertainty to the x-component of the AGN position.\nOffset origin. For t m = 5 (0.4) Gyr, we estimate a 25 (90)% probability of finding a displacement larger than that observed, if due to phase II recoil oscillations. The probability of a recoil kick large enough ( 240 km s −1 ; Table 8) to initiate these oscillations is ∼45% (L12).\nOffset origin. For t m = 5 (0.4) Gyr, we estimate a 25 (90)% probability of finding a displacement larger than that observed, if due to phase II recoil oscillations. The probability of a recoil kick large enough ( 240 km s −1 ; Table 8) to initiate these oscillations is ∼45% (L12).\nAs for NGC 4278, we use equations 2 and 3 and values from Table 8 to estimate the time since the onset of phase II, finding (t-t c ) 6.2 τ or 4 Gyr. This implies an upper limit to the time elapsed since the putative SBH-binary coalescence event of 2(t-t c ) or 8 Gyr.\nAs for NGC 4278, we use equations 2 and 3 and values from Table 8 to estimate the time since the onset of phase II, finding (t-t c ) 6.2 τ or 4 Gyr. This implies an upper limit to the time elapsed since the putative SBH-binary coalescence event of 2(t-t c ) or 8 Gyr.\nAlthough, as noted, there is \"fossil\" evidence for previous jet activity, the AGN does not currently produce a powerful radio jet, so it seems unlikely that the displacement is due to jet acceleration. However, interactions with massive perturbers or an SBH binary on a highly eccentric orbit remain possibilities. NGC 4486 (M87,intermediate) NGC 4486 (M87) is the dominant galaxy of the Virgo Cluster. It hosts the best studied extragalactic jet, which exhibits prominent knots at optical and IR wavelengths yet, apart from this, the brightness profile is regular and featureless. The peculiarly large population of globular clusters, its size and location, and the presence of stellar streams support the hypothesis that the galaxy has experienced a number of minor mergers during its lifetime (Janowiecki et al. 2010). A remarkable feature is the unusually large core, which has a radius r c ∼ 9. 41 (CB05). This characteristic makes M87 a special case in our sample in that for all the other galaxies we restricted the isophotal analysis to radii larger than the core radius, whereas for M87 we analyzed isophotes within the core radius in order to be consistent with B10.\nAlthough, as noted, there is \"fossil\" evidence for previous jet activity, the AGN does not currently produce a powerful radio jet, so it seems unlikely that the displacement is due to jet acceleration. However, interactions with massive perturbers or an SBH binary on a highly eccentric orbit remain possibilities. NGC 4486 (M87,intermediate) NGC 4486 (M87) is the dominant galaxy of the Virgo Cluster. It hosts the best studied extragalactic jet, which exhibits prominent knots at optical and IR wavelengths yet, apart from this, the brightness profile is regular and featureless. The peculiarly large population of globular clusters, its size and location, and the presence of stellar streams support the hypothesis that the galaxy has experienced a number of minor mergers during its lifetime (Janowiecki et al. 2010). A remarkable feature is the unusually large core, which has a radius r c ∼ 9. 41 (CB05). This characteristic makes M87 a special case in our sample in that for all the other galaxies we restricted the isophotal analysis to radii larger than the core radius, whereas for M87 we analyzed isophotes within the core radius in order to be consistent with B10.\nPhotometric analysis. A total of six images obtained with ACS/HRC, WFPC2/PC and NICMOS2 were analyzed (see Table 1), including the combined ACS images (ACS/HRC/F606W and ACS/HRC/F814W) previously analyzed by B10. The results of the photometric analysis are shown in Figs. 4 and 18-22 and the measured displacements relative to the AGN are plotted in Fig. 5a. In all images, we analyzed isophotes within the range 1 ≤ r ≤ 3 , for consistency with B10. The displacements measured in different images vary in amplitude and significance. Notably, none of the three NICMOS2 images (F110W, F160W and F222W) exhibit significant displacements at the ≥ 3σ level. Furthermore, the 1σ uncertainties, when converted to linear distances, do not encompass the significant displacements recovered from the optical images. The displacements measured from the ACS and WFPC2 images also differ in amplitude but there is a tendency for the photocenter to be displaced NW of the AGN, roughly in the direction of the radio jet. The isophote centers are reasonably stable -although there are deviations ∼ 1 − 2 pixels (≤ 0. 1), systematic trends with SMA are not generally evident . The most significant displacement is obtained from the ACS/HRC F814 image, for which the isophote centers cluster tightly around the mean photocenter (Fig. 4). The measured displacement of the photocenter relative to the AGN is ∆r = 7.7 ± 0.3 pc NW, implying that the SBH is displaced by ∆r in the counter-jet direction. This result is in agreement with the displacement obtained by B10 from a similar analysis. The main differences between the method used by B10 and that employed in this paper are the iterative technique used here to generate the mask and the weight function used to derive the isophotal center: while B10 used a conventional uncertainty weight, here we weight for the light content of each isophotal annulus (see §3.1). The weighted average displacement has a magnitude 4.3 ± 0.2 pc and direction NW (PA = 307 ± 1 • ). Offset origin. The presence of a large number of bright globular clusters visible in all the images analyzed allows us to define a common reference frame which we can use to compare results from images obtained with different instrument/filter combinations. After selecting several globular clusters as reference points and matching their positions, we plotted the AGN and photocenter positions determined from the different images in the reference frame defined by the ACS image (Fig. 8). The AGN position as measured from the different images is fairly stable whereas the photocenters exhibit differences in position much larger than the scatter (∼ 0.5 pixels) in the AGN position. This indicates that the differences in the displacements are due largely to differences in the positions of the photocenters. The origin of these discrepancies is unclear. Unlike other galaxies, there is no evidence of dust extinction or other asymmetries that may account for different results at different wavelengths.\nPhotometric analysis. A total of six images obtained with ACS/HRC, WFPC2/PC and NICMOS2 were analyzed (see Table 1), including the combined ACS images (ACS/HRC/F606W and ACS/HRC/F814W) previously analyzed by B10. The results of the photometric analysis are shown in Figs. 4 and 18-22 and the measured displacements relative to the AGN are plotted in Fig. 5a. In all images, we analyzed isophotes within the range 1 ≤ r ≤ 3 , for consistency with B10. The displacements measured in different images vary in amplitude and significance. Notably, none of the three NICMOS2 images (F110W, F160W and F222W) exhibit significant displacements at the ≥ 3σ level. Furthermore, the 1σ uncertainties, when converted to linear distances, do not encompass the significant displacements recovered from the optical images. The displacements measured from the ACS and WFPC2 images also differ in amplitude but there is a tendency for the photocenter to be displaced NW of the AGN, roughly in the direction of the radio jet. The isophote centers are reasonably stable -although there are deviations ∼ 1 − 2 pixels (≤ 0. 1), systematic trends with SMA are not generally evident . The most significant displacement is obtained from the ACS/HRC F814 image, for which the isophote centers cluster tightly around the mean photocenter (Fig. 4). The measured displacement of the photocenter relative to the AGN is ∆r = 7.7 ± 0.3 pc NW, implying that the SBH is displaced by ∆r in the counter-jet direction. This result is in agreement with the displacement obtained by B10 from a similar analysis. The main differences between the method used by B10 and that employed in this paper are the iterative technique used here to generate the mask and the weight function used to derive the isophotal center: while B10 used a conventional uncertainty weight, here we weight for the light content of each isophotal annulus (see §3.1). The weighted average displacement has a magnitude 4.3 ± 0.2 pc and direction NW (PA = 307 ± 1 • ). Offset origin. The presence of a large number of bright globular clusters visible in all the images analyzed allows us to define a common reference frame which we can use to compare results from images obtained with different instrument/filter combinations. After selecting several globular clusters as reference points and matching their positions, we plotted the AGN and photocenter positions determined from the different images in the reference frame defined by the ACS image (Fig. 8). The AGN position as measured from the different images is fairly stable whereas the photocenters exhibit differences in position much larger than the scatter (∼ 0.5 pixels) in the AGN position. This indicates that the differences in the displacements are due largely to differences in the positions of the photocenters. The origin of these discrepancies is unclear. Unlike other galaxies, there is no evidence of dust extinction or other asymmetries that may account for different results at different wavelengths.\nWe conducted several trials in order to determine the influence of (i) the degree of masking applied to the jet, (ii) the bright knot visible NW of the nucleus in the ACS image. The results were found to be independent of the degree of masking; no significant differences in the photocenter positions were found when the analysis was performed with no masking at all or with heavy jet/knot-masking.\nWe conducted several trials in order to determine the influence of (i) the degree of masking applied to the jet, (ii) the bright knot visible NW of the nucleus in the ACS image. The results were found to be independent of the degree of masking; no significant differences in the photocenter positions were found when the analysis was performed with no masking at all or with heavy jet/knot-masking.\nAs a fraction of the rather large core radius, the weighted mean displacement is only ≈ 0.01 r c (Table 6). We estimate a 85 (99)% probability of finding a displacement larger than that observed for t m = 5 (0.4) Gyr, assuming phase II recoil oscillations were triggered. The probability of obtaining the relatively large recoil kick ( 325 km s −1 ; Table 8) required for phase II oscillations is ≈ 33% (L12).\nAs a fraction of the rather large core radius, the weighted mean displacement is only ≈ 0.01 r c (Table 6). We estimate a 85 (99)% probability of finding a displacement larger than that observed for t m = 5 (0.4) Gyr, assuming phase II recoil oscillations were triggered. The probability of obtaining the relatively large recoil kick ( 325 km s −1 ; Table 8) required for phase II oscillations is ≈ 33% (L12).\nThe apparent close alignment of the offset with the jet direction favors jet acceleration but is also generally consistent with gravitational recoil. However, as there is no reason to expect such an alignment if the displacement is due to gravitational interactions with massive perturbers, or orbital motion in a SBH-binary, these mechanisms are disfavored. A detailed discussion of the possible displacement mechanisms is given in §4 of B10.\nThe apparent close alignment of the offset with the jet direction favors jet acceleration but is also generally consistent with gravitational recoil. However, as there is no reason to expect such an alignment if the displacement is due to gravitational interactions with massive perturbers, or orbital motion in a SBH-binary, these mechanisms are disfavored. A detailed discussion of the possible displacement mechanisms is given in §4 of B10."},{"title":"NGC 1399 (low)","content":""},{"title":"NGC 1399 (low)","content":"NGC 1399 is a regular giant E1 and is the central and dominant galaxy of the Fornax cluster (Ferguson 1989). It is associated with a low power radio source characterized by two nearly bi-symmetric jets extending ∼ 10 kpc, approximately N and S of the core (PA ≈ 9 • W of N, ). Shurkin et al. (2008) report the presence of X-ray cavities associated with the radio lobes. NGC 1399 has been the subject of a number of dynamical studies (e.g. Houghton et al. (2006) and references therein). In particular, Houghton et al. performed near-infrared adaptive optics assisted spectroscopic observations of the central 1. 5 of the galaxy and found evidence of a kinematically decoupled core and offset asymmetric isophotes that they considered to be consistent with the presence of a M31-like double nucleus. The asymmetric isophotes, elongated in the direction E-SE, are localized in a region much smaller than that used for our isophotal analysis (r > 2. 5).\nNGC 1399 is a regular giant E1 and is the central and dominant galaxy of the Fornax cluster (Ferguson 1989). It is associated with a low power radio source characterized by two nearly bi-symmetric jets extending ∼ 10 kpc, approximately N and S of the core (PA ≈ 9 • W of N, ). Shurkin et al. (2008) report the presence of X-ray cavities associated with the radio lobes. NGC 1399 has been the subject of a number of dynamical studies (e.g. Houghton et al. (2006) and references therein). In particular, Houghton et al. performed near-infrared adaptive optics assisted spectroscopic observations of the central 1. 5 of the galaxy and found evidence of a kinematically decoupled core and offset asymmetric isophotes that they considered to be consistent with the presence of a M31-like double nucleus. The asymmetric isophotes, elongated in the direction E-SE, are localized in a region much smaller than that used for our isophotal analysis (r > 2. 5).\nPhotometric analysis. Four images, obtained with WFPC2/PC -F606W, F814W and WFC3/IR -F110W, F160W, were analyzed for this galaxy. From the F606W image we find a low significance displacement of amplitude 1.7 ± 0.4 pc, in almost the same N-NW direction as the radio jet axis, PA = −8±16 • . A 2σ offset in the same direction was recovered from the WFPC2/PC -F814W image, which is consistent within the errors with the F606W result. Offsets recovered from the WFC3 images are classified as non-significant, but, given their large error bars, they are consistent with the WFPC2 results (Fig. 5a). The error weighted offset, computed including results from both WFPC2 and WFC3, is 1.5 ± 0.4 pc with PA = -17 ± 16 • . The isophote fitting analysis (Figs. 9 to 12) reveals a remarkably regular photometric structure well modeled by concentric elliptical isophotes. For both images, the isophote center co-ordinates vary little with SMA and are very tightly clustered, with narrow ( 0.5 pixel) distributions in x and y.\nPhotometric analysis. Four images, obtained with WFPC2/PC -F606W, F814W and WFC3/IR -F110W, F160W, were analyzed for this galaxy. From the F606W image we find a low significance displacement of amplitude 1.7 ± 0.4 pc, in almost the same N-NW direction as the radio jet axis, PA = −8±16 • . A 2σ offset in the same direction was recovered from the WFPC2/PC -F814W image, which is consistent within the errors with the F606W result. Offsets recovered from the WFC3 images are classified as non-significant, but, given their large error bars, they are consistent with the WFPC2 results (Fig. 5a). The error weighted offset, computed including results from both WFPC2 and WFC3, is 1.5 ± 0.4 pc with PA = -17 ± 16 • . The isophote fitting analysis (Figs. 9 to 12) reveals a remarkably regular photometric structure well modeled by concentric elliptical isophotes. For both images, the isophote center co-ordinates vary little with SMA and are very tightly clustered, with narrow ( 0.5 pixel) distributions in x and y.\nOffset origin. The weighted mean displacement is small compared to the core radius, ≈ 0.01 r c (Table 6). A minimum kick velocity 300 km s −1 (Table 8), is required to trigger phase II recoil oscillations; the probability of generating a kick of at least this magnitude is ∼ 33% (L12). Assuming that phase II recoil oscillations were triggered, we estimate the probability of finding a displacement larger than that observed to be 11 (75)%, for t m = 5 (0.4) Gyr.\nOffset origin. The weighted mean displacement is small compared to the core radius, ≈ 0.01 r c (Table 6). A minimum kick velocity 300 km s −1 (Table 8), is required to trigger phase II recoil oscillations; the probability of generating a kick of at least this magnitude is ∼ 33% (L12). Assuming that phase II recoil oscillations were triggered, we estimate the probability of finding a displacement larger than that observed to be 11 (75)%, for t m = 5 (0.4) Gyr.\nIf the observed displacement is due to a recoiling SBH, we estimate that (t -t c ) 1.6 Gyr has elapsed since the oscillation damped below the core radius, using τ = 0.17 Gyr and σ = 346 km s −1 ( Table 8).\nIf the observed displacement is due to a recoiling SBH, we estimate that (t -t c ) 1.6 Gyr has elapsed since the oscillation damped below the core radius, using τ = 0.17 Gyr and σ = 346 km s −1 ( Table 8).\nThe close alignment of the displacement with the radio axis is consistent with both recoil and jet acceleration as the cause of the displacement. If the latter, eq. 7 implies that, for a Bondi accretion rateṀ a ∼ 0.04 M yr −1 (Loewenstein et al. 2001) and a 10 9 M black hole, an asymmetry in the radio jet power in the range 3 × 10 −4 f jet 3 × 10 −3 that persists for t ∼ 10 8 yr would be needed to produce an offset 1.8 ∆r 18 pc.\nThe close alignment of the displacement with the radio axis is consistent with both recoil and jet acceleration as the cause of the displacement. If the latter, eq. 7 implies that, for a Bondi accretion rateṀ a ∼ 0.04 M yr −1 (Loewenstein et al. 2001) and a 10 9 M black hole, an asymmetry in the radio jet power in the range 3 × 10 −4 f jet 3 × 10 −3 that persists for t ∼ 10 8 yr would be needed to produce an offset 1.8 ∆r 18 pc.\nOther mechanisms (interactions with massive perturbers and a stalled SBH binary), are disfavored by the alignment between the displacement and jet direction. The presence of an unresolved M31-like double nucleus as suggested by Houghton et al. (2006) might also explain the measured offset, although as in the case of a binary SBH, the alignment with the radio source axis would have to be regarded as fortuitous.\nOther mechanisms (interactions with massive perturbers and a stalled SBH binary), are disfavored by the alignment between the displacement and jet direction. The presence of an unresolved M31-like double nucleus as suggested by Houghton et al. (2006) might also explain the measured offset, although as in the case of a binary SBH, the alignment with the radio source axis would have to be regarded as fortuitous."},{"title":"IC 4296 (low)","content":""},{"title":"IC 4296 (low)","content":"The properties of this galaxy have been extensively studied in a series of papers by Killeen et al. (1986a,b) and . It is a fairly circular elliptical which is the brightest component in the small group A3565. It is associated with the extended low-to-intermediate luminosity radio source PKS 1333-33 (Mills et al. 1960), which has two quasi-symmetric slightly curved jets with an average position angle ∼ 130 • and terminating in lobes ∼ 200 kpc from the core (Killeen et al. 1986a). The curvature along the jets is consistent with the motion of the galaxy through the group inter-galactic medium ). On the parsec scale, a jet and counter-jet emerge from the unresolved core in PA ≈ 140 • , consistent with the large-scale radio source, with the brighter jet on the NW side. The brightness asymmetry is attributed to Doppler boosting (Pellegrini et al. 2003). Isophotal studies suggest tidal interactions with the smaller companion IC 4299 (Younis et al. 1985). Nevertheless, Efstathiou et al. (1980) show that the isophotal ellipticity and position angle are constant at all radii from 2.4 to 48.8 arcsec (a region much larger than that studied in this work, 1. 5 < ∆r < 6. 5), which suggests that any distortions due to interactions with companion galaxies are negligible over this scale.\nThe properties of this galaxy have been extensively studied in a series of papers by Killeen et al. (1986a,b) and . It is a fairly circular elliptical which is the brightest component in the small group A3565. It is associated with the extended low-to-intermediate luminosity radio source PKS 1333-33 (Mills et al. 1960), which has two quasi-symmetric slightly curved jets with an average position angle ∼ 130 • and terminating in lobes ∼ 200 kpc from the core (Killeen et al. 1986a). The curvature along the jets is consistent with the motion of the galaxy through the group inter-galactic medium ). On the parsec scale, a jet and counter-jet emerge from the unresolved core in PA ≈ 140 • , consistent with the large-scale radio source, with the brighter jet on the NW side. The brightness asymmetry is attributed to Doppler boosting (Pellegrini et al. 2003). Isophotal studies suggest tidal interactions with the smaller companion IC 4299 (Younis et al. 1985). Nevertheless, Efstathiou et al. (1980) show that the isophotal ellipticity and position angle are constant at all radii from 2.4 to 48.8 arcsec (a region much larger than that studied in this work, 1. 5 < ∆r < 6. 5), which suggests that any distortions due to interactions with companion galaxies are negligible over this scale.\nPhotometric analysis. Three images, obtained with ACS/HRC-F625W, WFPC2/PC-F814W and NICMOS2-F160W, were analyzed for this galaxy. In the ACS image a displacement of ∆r ≥ 5.2 ± 1 pc was found. On the basis of the IQR, that is considered to be of low significance. The displacement obtained from the NICMOS2 image is not considered significant, but is consistent with the ACS result, albeit within large errors. For the WFPC2 image, the y-component of the displacement is consistent with the ACS result but the x component exhibits a marginally significant difference and in fact indicates a displacement in the opposite direction (E). However, neither component of the WFPC2 displacement is classified as significant. These results are summarized in Fig. 5a.\nPhotometric analysis. Three images, obtained with ACS/HRC-F625W, WFPC2/PC-F814W and NICMOS2-F160W, were analyzed for this galaxy. In the ACS image a displacement of ∆r ≥ 5.2 ± 1 pc was found. On the basis of the IQR, that is considered to be of low significance. The displacement obtained from the NICMOS2 image is not considered significant, but is consistent with the ACS result, albeit within large errors. For the WFPC2 image, the y-component of the displacement is consistent with the ACS result but the x component exhibits a marginally significant difference and in fact indicates a displacement in the opposite direction (E). However, neither component of the WFPC2 displacement is classified as significant. These results are summarized in Fig. 5a.\nThe discrepancy in the photocenter positions appears to be linked to systematic migrations in the isophote center coordinates determined from the fits in the NICMOS2 and WFPC2 images: there are systematic migrations northwards in the NICMOS2 image (Fig. 31) and eastwards in the WFPC2 image (Fig. 30) resulting in distributions with large IQRs in the y and x co-ordinates, respectively. In contrast, the co-ordinates extracted from the ACS image (Fig. 29) show little variation with SMA and cluster quite symmetrically around the photocenter. This galaxy exhibits a warped dusty disk, which crosses the nucleus in a roughly E-W direction and is clearly visible in all three images (Figs. 29,30 and 31). This feature is too small (r 0. 9) to affect the photocenter position, which is calculated from isophote fits over radii 1. 5 ≤ r ≤ 6. 5. However, it does create large gradients in the background brightness around the nucleus which affect the gaussian fits to the central peak, especially when large fitting-boxes are used. The effects are particularly noticeable for the WFPC2 image, where the AGN position systematically shifts South-East as the size of the fitting-box is increased. This was mitigated by reducing the size of the fitting-box until stable values were found for the AGN co-ordinates.\nThe discrepancy in the photocenter positions appears to be linked to systematic migrations in the isophote center coordinates determined from the fits in the NICMOS2 and WFPC2 images: there are systematic migrations northwards in the NICMOS2 image (Fig. 31) and eastwards in the WFPC2 image (Fig. 30) resulting in distributions with large IQRs in the y and x co-ordinates, respectively. In contrast, the co-ordinates extracted from the ACS image (Fig. 29) show little variation with SMA and cluster quite symmetrically around the photocenter. This galaxy exhibits a warped dusty disk, which crosses the nucleus in a roughly E-W direction and is clearly visible in all three images (Figs. 29,30 and 31). This feature is too small (r 0. 9) to affect the photocenter position, which is calculated from isophote fits over radii 1. 5 ≤ r ≤ 6. 5. However, it does create large gradients in the background brightness around the nucleus which affect the gaussian fits to the central peak, especially when large fitting-boxes are used. The effects are particularly noticeable for the WFPC2 image, where the AGN position systematically shifts South-East as the size of the fitting-box is increased. This was mitigated by reducing the size of the fitting-box until stable values were found for the AGN co-ordinates.\nThe average error-weighted offset obtained for this galaxy from the ACS, NICMOS2 and WFPC2 images is ∆r ≥ 3.8 ± 0.7 pc in PA = -22 ± 7 • E of N.\nThe average error-weighted offset obtained for this galaxy from the ACS, NICMOS2 and WFPC2 images is ∆r ≥ 3.8 ± 0.7 pc in PA = -22 ± 7 • E of N.\nOffset origin. The status of IC 4296 as the brightest member of its group and the fact that it has a kinematically peculiar core (Killeen et al. 1986a) suggest that this galaxy has experienced minor mergers during the past few Gyr.\nOffset origin. The status of IC 4296 as the brightest member of its group and the fact that it has a kinematically peculiar core (Killeen et al. 1986a) suggest that this galaxy has experienced minor mergers during the past few Gyr.\nThe weighted mean displacement is a small fraction of the core radius, ≈ 0.01 r c ( Table 6). The probability of generating a recoil kick large enough to trigger phase II recoil oscillations ( 200 km s −1 ; Table 8), is ∼ 45% (L12) and given that phase II oscillations were triggered, we estimate a 37 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr.\nThe weighted mean displacement is a small fraction of the core radius, ≈ 0.01 r c ( Table 6). The probability of generating a recoil kick large enough to trigger phase II recoil oscillations ( 200 km s −1 ; Table 8), is ∼ 45% (L12) and given that phase II oscillations were triggered, we estimate a 37 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr.\nUsing eqs. 2 and 3 with σ = 333 km s −1 , r rms ≥ 3.8 pc and r c = 347 pc, we find (t − t c ) 9 τ ≈ 5.8 Gyr. In this case, the low density of the large core is responsible for the long damping time.\nUsing eqs. 2 and 3 with σ = 333 km s −1 , r rms ≥ 3.8 pc and r c = 347 pc, we find (t − t c ) 9 τ ≈ 5.8 Gyr. In this case, the low density of the large core is responsible for the long damping time.\nIt has been pointed out by Killeen et al. (1986a) that on kpc scales the radio jets exhibit transverse oscillations with wavelengths ∼ 10 kpc and an amplitude comparable with the jet width. Similar oscillation wavelengths are expected from precession due to orbital motion in a ∼ pc scale SBH binary (see Section 5.3). However, found that, in detail, the oscillation pattern is not consistent with precession and favor instead helical Kelvin-Helmholtz instabilities.\nIt has been pointed out by Killeen et al. (1986a) that on kpc scales the radio jets exhibit transverse oscillations with wavelengths ∼ 10 kpc and an amplitude comparable with the jet width. Similar oscillation wavelengths are expected from precession due to orbital motion in a ∼ pc scale SBH binary (see Section 5.3). However, found that, in detail, the oscillation pattern is not consistent with precession and favor instead helical Kelvin-Helmholtz instabilities.\nIf we disregard the discrepant WFPC2 result, the direction of the weighted mean displacement derived from the NICMOS2 and ACS images is PA = 132±8 • . In projection, therefore, the displacement is remarkably well-aligned with the radio source axis, suggesting jet acceleration as a plausible candidate for the displacement mechanism. Pellegrini et al. (2003) infer a viewing angle θ = 63.5 ± 3.5 • for the parsec-scale jets. Assuming that the displacement is in the same direction in space as the jet, the de-projected amplitude is ∆r jet = 4.3 ± 3 pc.\nIf we disregard the discrepant WFPC2 result, the direction of the weighted mean displacement derived from the NICMOS2 and ACS images is PA = 132±8 • . In projection, therefore, the displacement is remarkably well-aligned with the radio source axis, suggesting jet acceleration as a plausible candidate for the displacement mechanism. Pellegrini et al. (2003) infer a viewing angle θ = 63.5 ± 3.5 • for the parsec-scale jets. Assuming that the displacement is in the same direction in space as the jet, the de-projected amplitude is ∆r jet = 4.3 ± 3 pc.\nGiven that the galaxy is characterized by a low density core, taking the age of the radio source to be t ∼ 10 8 yr ,Ṁ a =ṁ Bondi ∼ 0.02 M yr −1 (Pellegrini et al. 2003), M • ∼ 10 9 M from the M − σ relation and 3 ∆r 30 pc, we find from eq. 7 that 10 −3 f jet 9.7 × 10 −3 .\nGiven that the galaxy is characterized by a low density core, taking the age of the radio source to be t ∼ 10 8 yr ,Ṁ a =ṁ Bondi ∼ 0.02 M yr −1 (Pellegrini et al. 2003), M • ∼ 10 9 M from the M − σ relation and 3 ∆r 30 pc, we find from eq. 7 that 10 −3 f jet 9.7 × 10 −3 .\nTherefore, an asymmetry in jet power amounting to 1% of the accretion luminosity could account for the observed offset if it persists for ∼ 10 8 yr.\nTherefore, an asymmetry in jet power amounting to 1% of the accretion luminosity could account for the observed offset if it persists for ∼ 10 8 yr.\nThe close alignment with the radio jet axis argues against the displacement being caused by gravitational interactions with massive perturbers.\nThe close alignment with the radio jet axis argues against the displacement being caused by gravitational interactions with massive perturbers."},{"title":"NGC 5419 (low, double nucleus)","content":""},{"title":"NGC 5419 (low, double nucleus)","content":"This galaxy is the dominant member of the cluster Abell S753 (Abell et al. 1989). A low surface brightness X-ray halo is centered on the galaxy and extends over a radius r ≈ 16 (190 h −1 kpc). An unusual diffuse radio source, PKS B1400-33, is associated with the cluster, but is offset relative to the X-ray emission and NGC 5419 (Subrahmanyan et al. 2003). The galaxy is located just beyond the NW edge of the extended radio emission and itself is associated with a bright, compact radio source. The low surface brightness and steep spectral index of the diffuse source suggest that it is a relic, with a spectral age of 5 × 10 8 yr. However, it is unclear if the relic was created by an earlier episode of powerful activity in NGC 5419, or whether it is a relic lobe injected into the cluster by a previously active double radio source located outside the cluster (the scenario favored by Subrahmanyan et al., although an optical counterpart has not been firmly identified). If the relic originated from NGC 5419, the spatial offset may be due to proper motion of the galaxy or to buoyancy of the synchrotron plasma (e.g., McNamara et al. 2001).\nThis galaxy is the dominant member of the cluster Abell S753 (Abell et al. 1989). A low surface brightness X-ray halo is centered on the galaxy and extends over a radius r ≈ 16 (190 h −1 kpc). An unusual diffuse radio source, PKS B1400-33, is associated with the cluster, but is offset relative to the X-ray emission and NGC 5419 (Subrahmanyan et al. 2003). The galaxy is located just beyond the NW edge of the extended radio emission and itself is associated with a bright, compact radio source. The low surface brightness and steep spectral index of the diffuse source suggest that it is a relic, with a spectral age of 5 × 10 8 yr. However, it is unclear if the relic was created by an earlier episode of powerful activity in NGC 5419, or whether it is a relic lobe injected into the cluster by a previously active double radio source located outside the cluster (the scenario favored by Subrahmanyan et al., although an optical counterpart has not been firmly identified). If the relic originated from NGC 5419, the spatial offset may be due to proper motion of the galaxy or to buoyancy of the synchrotron plasma (e.g., McNamara et al. 2001).\nPhotometric analysis. The galaxy has a double nucleus (Lauer et al. 2005;, which is well-resolved in the WFPC2 planetary camera image (WFPC2/PC-F555W). The two nuclei are separated by 0. 27, with the fainter (secondary) nucleus located almost at the south of the brighter (primary) nucleus. The photocenter is offset by ∆r 1 7.5 ± 1.7 pc W-SW (PA = 252 ± 12 • ) from the primary and ∆r 2 62 ± 2 pc N-NW (PA = 346 ± 2 • ) from the secondary nucleus.\nPhotometric analysis. The galaxy has a double nucleus (Lauer et al. 2005;, which is well-resolved in the WFPC2 planetary camera image (WFPC2/PC-F555W). The two nuclei are separated by 0. 27, with the fainter (secondary) nucleus located almost at the south of the brighter (primary) nucleus. The photocenter is offset by ∆r 1 7.5 ± 1.7 pc W-SW (PA = 252 ± 12 • ) from the primary and ∆r 2 62 ± 2 pc N-NW (PA = 346 ± 2 • ) from the secondary nucleus.\nA periodic pattern is visible in the residual image obtained after subtraction of the photometric model (middle panel in Fig. 26). This is likely to be due to the pronounced \"boxiness\" of the isophotes, which could not be perfectly reproduced even though higher harmonics were included in the model. The isophote center co-ordinates cluster fairly tightly around the photocenter, which is closest to the primary nucleus, and generally show little variation with SMA (Fig. 26), with the exception of a small migration (≈ 0. 05) eastward of the photocenter in the inner 170 pixels (8. 5).\nA periodic pattern is visible in the residual image obtained after subtraction of the photometric model (middle panel in Fig. 26). This is likely to be due to the pronounced \"boxiness\" of the isophotes, which could not be perfectly reproduced even though higher harmonics were included in the model. The isophote center co-ordinates cluster fairly tightly around the photocenter, which is closest to the primary nucleus, and generally show little variation with SMA (Fig. 26), with the exception of a small migration (≈ 0. 05) eastward of the photocenter in the inner 170 pixels (8. 5).\nOffset origin. The projected separation between the two nuclei (∆r ≈ 75 pc) is slightly smaller than the estimated SBH influence radius (r h ≈ 90 pc) and much smaller than the core radius (r c ≈ 500 pc; Table 1). Hence, it is possible that the secondary nucleus is a second SBH, or even the nucleus of a satellite galaxy, on a slowly-evolving orbit in the low-density core (Antonini & Merritt 2012).\nOffset origin. The projected separation between the two nuclei (∆r ≈ 75 pc) is slightly smaller than the estimated SBH influence radius (r h ≈ 90 pc) and much smaller than the core radius (r c ≈ 500 pc; Table 1). Hence, it is possible that the secondary nucleus is a second SBH, or even the nucleus of a satellite galaxy, on a slowly-evolving orbit in the low-density core (Antonini & Merritt 2012).\nThe displacement of the photocenter relative to the primary nucleus is a small fraction of the core radius, ≈ 0.02r c ( Table 6). The probability of generating a recoil kick velocity large enough to trigger phase II recoil oscillations (v 300 km s −1 , Table 8), is ∼ 33% (L12) and given that phase II oscillations occur, we estimate a 46 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr. Using eqs. 2 and 3 with σ = 351 km s −1 , r rms ≥ 7 pc and r c = 500 pc, we find (t − t c ) 9 τ ≈ 10 Gyr, a factor ten larger than the estimated spectral age of the diffuse radio source. Therefore, if an SBH binary coalescence event is responsible for the displacement, it seems likely that it pre-dates the AGN phase that created the radio relic (if indeed NGC 5419 is the origin of the relic).\nThe displacement of the photocenter relative to the primary nucleus is a small fraction of the core radius, ≈ 0.02r c ( Table 6). The probability of generating a recoil kick velocity large enough to trigger phase II recoil oscillations (v 300 km s −1 , Table 8), is ∼ 33% (L12) and given that phase II oscillations occur, we estimate a 46 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr. Using eqs. 2 and 3 with σ = 351 km s −1 , r rms ≥ 7 pc and r c = 500 pc, we find (t − t c ) 9 τ ≈ 10 Gyr, a factor ten larger than the estimated spectral age of the diffuse radio source. Therefore, if an SBH binary coalescence event is responsible for the displacement, it seems likely that it pre-dates the AGN phase that created the radio relic (if indeed NGC 5419 is the origin of the relic).\nNotes on other galaxies In the following galaxies, photocenter -SBH displacements were not considered significant.\nNotes on other galaxies In the following galaxies, photocenter -SBH displacements were not considered significant."},{"title":"NGC 4168","content":""},{"title":"NGC 4168","content":"NGC 4168 is an E2 in the Virgo cluster. Broad Hα has been detected in its nuclear optical spectrum, resulting in classification as a Seyfert 1.9 (Ho et al. 1997). VLBA observations at 8.4 GHz show an unresolved radio source with no evidence of jets (Anderson et al. 2004). Although the galaxy appears very regular in the WFPC2-WF/F606W image, the photometric analysis reveals systematic variations in the isophote centers with SMA that are much larger in amplitude than the displacement between the mean photocenter and the AGN position (Fig. 13). In particular, there is an abrupt shift to the E in the x co-ordinate and a systematic migration to the S in y, for SMA 260 pixels (13 ). The measured displacements in x and y are < 0.8 IQR and therefore considered \"null\" results.\nNGC 4168 is an E2 in the Virgo cluster. Broad Hα has been detected in its nuclear optical spectrum, resulting in classification as a Seyfert 1.9 (Ho et al. 1997). VLBA observations at 8.4 GHz show an unresolved radio source with no evidence of jets (Anderson et al. 2004). Although the galaxy appears very regular in the WFPC2-WF/F606W image, the photometric analysis reveals systematic variations in the isophote centers with SMA that are much larger in amplitude than the displacement between the mean photocenter and the AGN position (Fig. 13). In particular, there is an abrupt shift to the E in the x co-ordinate and a systematic migration to the S in y, for SMA 260 pixels (13 ). The measured displacements in x and y are < 0.8 IQR and therefore considered \"null\" results."},{"title":"NGC 4261","content":""},{"title":"NGC 4261","content":"NGC 4261 is a massive elliptical (E2-3) in the outskirts of the Virgo cluster. Nolthenius (1993) identifies this galaxy as the dominant member of a group of 33 galaxies, the Virgo W cloud. It is associated with a bright FR I radio source, 3C 270, which has twin jets close to the plane of the sky, θ = 63 ± 3 • , and aligned on the axis W-E, PA = 87 ± 8 • , with the western radio component slightly brighter than the eastern (Jones et al. 2000;Worrall et al. 2010). Deep imaging reveals the presence of tidal tails to the NW and a tidal fan SE (Tal et al. 2009) while Giordano et al. (2005) find anisotropies in the globular cluster distribution. Cappellari et al. (2007) report the presence of a kpc-scale kinematically decoupled core hosting a 100 pc scale disk of dust, cool molecular and atomic gas (Jaffe et al. 1993;Jaffe & McNamara 1994). Jaffe et al. (1996 studied the central region of this galaxy in detail using HST/WFPC2/PC1 V (filter F547M), R (F675W) and I (F791W) images. After correcting the R images for line emission, they determined that the central dusty disk is characterized by a rotation axis which closely matches the axis of the jets but it is not coaxial with the semi-major axis of the galaxy. The disk is not centered on the AGN, nor on the photocenter, which they find is itself displaced relative to the AGN by ∆r = 3.3 ± 1.3 pc NE.\nNGC 4261 is a massive elliptical (E2-3) in the outskirts of the Virgo cluster. Nolthenius (1993) identifies this galaxy as the dominant member of a group of 33 galaxies, the Virgo W cloud. It is associated with a bright FR I radio source, 3C 270, which has twin jets close to the plane of the sky, θ = 63 ± 3 • , and aligned on the axis W-E, PA = 87 ± 8 • , with the western radio component slightly brighter than the eastern (Jones et al. 2000;Worrall et al. 2010). Deep imaging reveals the presence of tidal tails to the NW and a tidal fan SE (Tal et al. 2009) while Giordano et al. (2005) find anisotropies in the globular cluster distribution. Cappellari et al. (2007) report the presence of a kpc-scale kinematically decoupled core hosting a 100 pc scale disk of dust, cool molecular and atomic gas (Jaffe et al. 1993;Jaffe & McNamara 1994). Jaffe et al. (1996 studied the central region of this galaxy in detail using HST/WFPC2/PC1 V (filter F547M), R (F675W) and I (F791W) images. After correcting the R images for line emission, they determined that the central dusty disk is characterized by a rotation axis which closely matches the axis of the jets but it is not coaxial with the semi-major axis of the galaxy. The disk is not centered on the AGN, nor on the photocenter, which they find is itself displaced relative to the AGN by ∆r = 3.3 ± 1.3 pc NE.\nIn this study, we chose to analyze a NICMOS2 F160W image (with a pixel size of 0. 05), since the nuclear source is bright and dust extinction and contamination due to emission from ionized gas are minimized. The galaxy exhibits boxy isophotes which are difficult to model accurately. Even though the third and forth harmonics were included in the isophote fits, the residual image still shows a spiral pattern suggesting that a more complex model perhaps involving multiple components would be necessary to successfully model the surface brightness distribution (Fig. 14). Nevertheless, the isophote centers are relatively stable, except for a slow migration to the N for SMA 120 pixels (6 ). The mean photocenter displacement relative to the AGN is ∆r = 3.3 ± 2.2 pc W-NW (PA = 297 ± 38 • ). The x component of the displacement is large compared to the IQR. However, because of the large uncertainty in positions derived from NICMOS2 images (0.23 pixels), we do not consider this offset significant.\nIn this study, we chose to analyze a NICMOS2 F160W image (with a pixel size of 0. 05), since the nuclear source is bright and dust extinction and contamination due to emission from ionized gas are minimized. The galaxy exhibits boxy isophotes which are difficult to model accurately. Even though the third and forth harmonics were included in the isophote fits, the residual image still shows a spiral pattern suggesting that a more complex model perhaps involving multiple components would be necessary to successfully model the surface brightness distribution (Fig. 14). Nevertheless, the isophote centers are relatively stable, except for a slow migration to the N for SMA 120 pixels (6 ). The mean photocenter displacement relative to the AGN is ∆r = 3.3 ± 2.2 pc W-NW (PA = 297 ± 38 • ). The x component of the displacement is large compared to the IQR. However, because of the large uncertainty in positions derived from NICMOS2 images (0.23 pixels), we do not consider this offset significant."},{"title":"NGC 4373","content":""},{"title":"NGC 4373","content":"This galaxy is located in the outskirts of the Hydra-Centaurus supercluster and is associated with a compact radio source (Sadler et al. 1989). It forms an isolated pair with IC 3290 (Sadler & Sharp 1984), which is located 2 W-SW of NGC 4373. Grützbauch et al. (2007) suggest that there is an ongoing strong interaction with the companion. Our photometric analysis of the WFPC2/PC-F814W image reveals boxy isophotes which produce cross-shaped residuals, even though higher harmonics were included in the isophote fits (Fig. 3). The y coordinate of the isophote center position has a narrow distribution centered on the mean photocenter and shows only a slight drift northward, for SMA > 170 pixel (8. 5). The x co-ordinate exhibits a systematic migration eastwards from the photocenter, resulting in a broader, skewed distribution. The mean photocenter is consistent with the AGN position within the uncertainties.\nThis galaxy is located in the outskirts of the Hydra-Centaurus supercluster and is associated with a compact radio source (Sadler et al. 1989). It forms an isolated pair with IC 3290 (Sadler & Sharp 1984), which is located 2 W-SW of NGC 4373. Grützbauch et al. (2007) suggest that there is an ongoing strong interaction with the companion. Our photometric analysis of the WFPC2/PC-F814W image reveals boxy isophotes which produce cross-shaped residuals, even though higher harmonics were included in the isophote fits (Fig. 3). The y coordinate of the isophote center position has a narrow distribution centered on the mean photocenter and shows only a slight drift northward, for SMA > 170 pixel (8. 5). The x co-ordinate exhibits a systematic migration eastwards from the photocenter, resulting in a broader, skewed distribution. The mean photocenter is consistent with the AGN position within the uncertainties."},{"title":"NGC 4552","content":""},{"title":"NGC 4552","content":"NGC 4552 is a typical giant elliptical in the Virgo cluster which hosts a very low-luminosity AGN. HST imaging and spectroscopy reveal a variable UV point source that has optical emission lines characteristic of AGN (Cappellari et al. 1999;Maoz et al. 2005). Interestingly, a second, transient UV source of comparable luminosity, but offset from the first by 0. 14, was present in 1991, but was not detected in 1993 Renzini et al. (1995); Cappellari et al. (1999); Maoz et al. (2005). Radio observations at 8.4 (Filho et al. 2000) and 5 GHz (Nagar et al. 2002) show the presence of a flat spectrum radio core. Nagar et al. (2002) also report variability at the ∼ 20% level on a ∼ 1 year timescale and find evidence of pc-scale jets, or extensions, along the East-West axis. NGC 4552 exhibits several morphological features indicative of past mergers, including three concentric shells at 5 , 7 and 10 from the nucleus (Malin 1979). Traces of a recent minor merger with a gas rich satellite galaxy have been identified in the presence of dust patches in the inner region (van Dokkum & Franx 1995), and Hα emission extending up to ∼ 2.5 kpc from the center (Trinchieri & di Serego Alighieri 1991; Macchetto et al. 1996).\nNGC 4552 is a typical giant elliptical in the Virgo cluster which hosts a very low-luminosity AGN. HST imaging and spectroscopy reveal a variable UV point source that has optical emission lines characteristic of AGN (Cappellari et al. 1999;Maoz et al. 2005). Interestingly, a second, transient UV source of comparable luminosity, but offset from the first by 0. 14, was present in 1991, but was not detected in 1993 Renzini et al. (1995); Cappellari et al. (1999); Maoz et al. (2005). Radio observations at 8.4 (Filho et al. 2000) and 5 GHz (Nagar et al. 2002) show the presence of a flat spectrum radio core. Nagar et al. (2002) also report variability at the ∼ 20% level on a ∼ 1 year timescale and find evidence of pc-scale jets, or extensions, along the East-West axis. NGC 4552 exhibits several morphological features indicative of past mergers, including three concentric shells at 5 , 7 and 10 from the nucleus (Malin 1979). Traces of a recent minor merger with a gas rich satellite galaxy have been identified in the presence of dust patches in the inner region (van Dokkum & Franx 1995), and Hα emission extending up to ∼ 2.5 kpc from the center (Trinchieri & di Serego Alighieri 1991; Macchetto et al. 1996).\nThe photometric analysis of the WFPC2/PC-F814W image shows that the y-coordinate of the isophote center migrates systematically southward relative to the photocenter for SMA > 200 pixels (10 ) with a maximum shift 0. 2 (Fig. 23). There is a smaller westward shift in the x coordinate. The inner 0. 3 of the residual image shows evidence of an arc-like dust lane partly encircling the central source with an opening toward the S. This feature was previously noted by Carollo et al. (1997) and Cappellari et al. (1999) (see Fig. 12 in the latter) and is the cause of the increased uncertainty on the y co-ordinate of the AGN position. We did not find a significant (> 3σ) displacement in this object.\nThe photometric analysis of the WFPC2/PC-F814W image shows that the y-coordinate of the isophote center migrates systematically southward relative to the photocenter for SMA > 200 pixels (10 ) with a maximum shift 0. 2 (Fig. 23). There is a smaller westward shift in the x coordinate. The inner 0. 3 of the residual image shows evidence of an arc-like dust lane partly encircling the central source with an opening toward the S. This feature was previously noted by Carollo et al. (1997) and Cappellari et al. (1999) (see Fig. 12 in the latter) and is the cause of the increased uncertainty on the y co-ordinate of the AGN position. We did not find a significant (> 3σ) displacement in this object."},{"title":"NGC 4636","content":""},{"title":"NGC 4636","content":"NGC 4636 is a giant elliptical in the outskirts of the Virgo cluster. Its nucleus has a LINER emission line spectrum, in which a weak broad Hα line has been detected (Ho et al. 1997). It has a jet-like radio source extending ≈ 1.5 kpc either side of the nucleus in position angle θ ≈ 33 • (Birkinshaw & Davies 1985). NGC 4636 is also associated with a luminous, extended X-ray source with a peculiar morphology featuring two symmetric, arm-like extensions emerging from the bright central region and extending up to ≈ 8 kpc along the NE-SW axis. Other fainter arm-like features emerge from the nucleus and Jones et al. (2002) suggest that these features trace cyclic nuclear outburst events occurring on a timescale ∼ 10 7 yr. According to Temi et al. (2003), the observed FIR fluxes (at 60, 90 and 180 µm) suggest the accretion of dust in a recent (within ∼ 10 8 yr) merger with a gas-rich galaxy.\nNGC 4636 is a giant elliptical in the outskirts of the Virgo cluster. Its nucleus has a LINER emission line spectrum, in which a weak broad Hα line has been detected (Ho et al. 1997). It has a jet-like radio source extending ≈ 1.5 kpc either side of the nucleus in position angle θ ≈ 33 • (Birkinshaw & Davies 1985). NGC 4636 is also associated with a luminous, extended X-ray source with a peculiar morphology featuring two symmetric, arm-like extensions emerging from the bright central region and extending up to ≈ 8 kpc along the NE-SW axis. Other fainter arm-like features emerge from the nucleus and Jones et al. (2002) suggest that these features trace cyclic nuclear outburst events occurring on a timescale ∼ 10 7 yr. According to Temi et al. (2003), the observed FIR fluxes (at 60, 90 and 180 µm) suggest the accretion of dust in a recent (within ∼ 10 8 yr) merger with a gas-rich galaxy.\nOur isophotal modeling of the WFPC2/PC-F814W image indicates the presence of faint dust lanes inspiraling toward the nucleus (Fig. 24). There is a systematic shift in the isophote centers to the north and west relative to the photocenter for SMA > 170 pixels (8. 5), which results in broad distributions in the x and y co-ordinates (Fig. 24). The photocenter position is consistent with the AGN position in the y direction, but exhibits a > 3σ offset in x. Nevertheless, as the IQR of the isophote center coordinate distribution exceeds the displacement we consider this a null result.\nOur isophotal modeling of the WFPC2/PC-F814W image indicates the presence of faint dust lanes inspiraling toward the nucleus (Fig. 24). There is a systematic shift in the isophote centers to the north and west relative to the photocenter for SMA > 170 pixels (8. 5), which results in broad distributions in the x and y co-ordinates (Fig. 24). The photocenter position is consistent with the AGN position in the y direction, but exhibits a > 3σ offset in x. Nevertheless, as the IQR of the isophote center coordinate distribution exceeds the displacement we consider this a null result."},{"title":"NGC 4696 (double nucleus)","content":""},{"title":"NGC 4696 (double nucleus)","content":"NGC 4696 is the dominant galaxy of the Centaurus cluster. There is an extensive literature on the multiwavewelength properties of this galaxy, which has a secondary nucleus (Laine et al. 2003) offset by ≈ 0. 3 to the SW of the primary (Laine et al. 2003 and Fig. 25). The double nucleus is partially surrounded by a complex of dust lanes and associated emission line filaments, the most prominent of which is in the form of a spiral leading towards the center. These features have been interpreted as evidence of a recent minor merger with a gas-rich galaxy (Sparks et al. 1989;Farage et al. 2010; alternatively the emission-line gas may have been \"drawn out\" by rising radio bubbles Canning et al. 2011). The galaxy is associated with the moderately powerful radio source PKS 1246-410, which extends ≈ 10 kpc along an approximately E-W axis, with the ends of both arms bending southwards. The radio source wraps around the region of brightest X-ray emission, suggesting interactions between the radio plasma and the hot thermal gas (Taylor et al. 2006). VLBA observations also obtained by Taylor et al. (2006) reveal a relatively weak core and a one-sided jet extending over ≈ 25 pc in PA = −150 • .\nNGC 4696 is the dominant galaxy of the Centaurus cluster. There is an extensive literature on the multiwavewelength properties of this galaxy, which has a secondary nucleus (Laine et al. 2003) offset by ≈ 0. 3 to the SW of the primary (Laine et al. 2003 and Fig. 25). The double nucleus is partially surrounded by a complex of dust lanes and associated emission line filaments, the most prominent of which is in the form of a spiral leading towards the center. These features have been interpreted as evidence of a recent minor merger with a gas-rich galaxy (Sparks et al. 1989;Farage et al. 2010; alternatively the emission-line gas may have been \"drawn out\" by rising radio bubbles Canning et al. 2011). The galaxy is associated with the moderately powerful radio source PKS 1246-410, which extends ≈ 10 kpc along an approximately E-W axis, with the ends of both arms bending southwards. The radio source wraps around the region of brightest X-ray emission, suggesting interactions between the radio plasma and the hot thermal gas (Taylor et al. 2006). VLBA observations also obtained by Taylor et al. (2006) reveal a relatively weak core and a one-sided jet extending over ≈ 25 pc in PA = −150 • .\nSubtraction of our photometric model from the ACS/WFC-F814W image (Fig. 25) clearly shows the dust features around the nucleus. The positive residuals probably trace the emission-line filaments. The isophote centers exhibit large excursions, resulting in broad distributions in both x and (especially) y coordinates. Although we measure a 3σ offset in x between the brightest nucleus and the mean photocenter, we consider this a null result because of the large IQR.\nSubtraction of our photometric model from the ACS/WFC-F814W image (Fig. 25) clearly shows the dust features around the nucleus. The positive residuals probably trace the emission-line filaments. The isophote centers exhibit large excursions, resulting in broad distributions in both x and (especially) y coordinates. Although we measure a 3σ offset in x between the brightest nucleus and the mean photocenter, we consider this a null result because of the large IQR."},{"title":"IC 1459","content":""},{"title":"IC 1459","content":"IC 1459 is a giant elliptical galaxy (E3-4) and a member of a loose group (identified as \"group 15\" in Huchra & Geller 1982) of ten galaxies, which are mainly spirals. Despite its relatively unremarkable appearance to casual inspection, this galaxy turns out to be notably peculiar (see Forbes et al. 1995 for a detailed analysis): it has one of the faster counter rotating kinematically distinct cores (Franx & Illingworth 1988), faint stellar spiral arms (Malin 1985), outer shells (Forbes et al. 1995) and irregular dust lanes or patches near the nucleus (Forbes et al. 1994). It has a hard X-ray point source (González-Martín et al. 2009) and a strong, compact radio core at 5 GHz (Slee et al. 1994).\nIC 1459 is a giant elliptical galaxy (E3-4) and a member of a loose group (identified as \"group 15\" in Huchra & Geller 1982) of ten galaxies, which are mainly spirals. Despite its relatively unremarkable appearance to casual inspection, this galaxy turns out to be notably peculiar (see Forbes et al. 1995 for a detailed analysis): it has one of the faster counter rotating kinematically distinct cores (Franx & Illingworth 1988), faint stellar spiral arms (Malin 1985), outer shells (Forbes et al. 1995) and irregular dust lanes or patches near the nucleus (Forbes et al. 1994). It has a hard X-ray point source (González-Martín et al. 2009) and a strong, compact radio core at 5 GHz (Slee et al. 1994).\nWe analyzed a WFPC2/PC-F814W image (Fig. 28). After subtraction of the photometric model, the residual image shows a periodic pattern, indicating that the isophotes deviate from perfect ellipses. Systematic variations in the isophote center co-ordinates are seen in both x and y, with increasing scatter also present for SMA > 230 pixels (11. 5). We did not find a significant (> 3σ) displacement in this galaxy.\nWe analyzed a WFPC2/PC-F814W image (Fig. 28). After subtraction of the photometric model, the residual image shows a periodic pattern, indicating that the isophotes deviate from perfect ellipses. Systematic variations in the isophote center co-ordinates are seen in both x and y, with increasing scatter also present for SMA > 230 pixels (11. 5). We did not find a significant (> 3σ) displacement in this galaxy."},{"title":"IC 4931","content":""},{"title":"IC 4931","content":"This relatively little studied galaxy is the brightest member of the group A3656 (Postman & Lauer 1995). Its radio source is relatively weak, with a total radio flux at 5 GHz of 0.9 mJy (see Table 2 to compare with other galaxies). It is also detected at 1.4 GHz with a flux density of 22 mJy (Brown et al. 2011). Apart from the optical nucleus and the radio emission, there is no supporting evidence indicating the presence of an AGN such as an X-ray point source, or line emission.\nThis relatively little studied galaxy is the brightest member of the group A3656 (Postman & Lauer 1995). Its radio source is relatively weak, with a total radio flux at 5 GHz of 0.9 mJy (see Table 2 to compare with other galaxies). It is also detected at 1.4 GHz with a flux density of 22 mJy (Brown et al. 2011). Apart from the optical nucleus and the radio emission, there is no supporting evidence indicating the presence of an AGN such as an X-ray point source, or line emission.\nOur analysis of the WFPC2/PC-F814 image (Fig. 32) shows large systematic variations in the isophote center coordinates, with the x coordinate in particular migrating up to 0. 4 W relative the photocenter for SM A 100 pixels (5 ). The co-ordinate distributions are among the broadest of the whole sample, spanning a range of 0. 5 in x. We measured a ∼ 3σ offset between the photocenter and the AGN in the x co-ordinate (E), but this is considered a null result because of the large IQR.\nOur analysis of the WFPC2/PC-F814 image (Fig. 32) shows large systematic variations in the isophote center coordinates, with the x coordinate in particular migrating up to 0. 4 W relative the photocenter for SM A 100 pixels (5 ). The co-ordinate distributions are among the broadest of the whole sample, spanning a range of 0. 5 in x. We measured a ∼ 3σ offset between the photocenter and the AGN in the x co-ordinate (E), but this is considered a null result because of the large IQR.\nC: IMAGES Figure 9. As in Fig.3 for galaxy NGC 1399, WFPC2/PC -F606W, scale=0. 05/pxl.\nC: IMAGES Figure 9. As in Fig.3 for galaxy NGC 1399, WFPC2/PC -F606W, scale=0. 05/pxl."},{"title":"Figure 1 .","content":""},{"title":"Figure 1 .","content":""},{"title":"Figure 2 .","content":""},{"title":"Figure 2 .","content":""},{"title":"Figure 3","content":""},{"title":"Figure 3","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 5b .","content":""},{"title":"Figure 5b .","content":""},{"title":"Figure 7a .","content":""},{"title":"Figure 7a .","content":""},{"title":"Figure 7b .","content":""},{"title":"Figure 7b .","content":""},{"title":"Figure 8 .","content":""},{"title":"Figure 8 .","content":""},{"title":"Figure 18 .","content":""},{"title":"Figure 18 .","content":""},{"title":"Figure 21 .","content":""},{"title":"Figure 21 .","content":""},{"title":"Figure 22 .","content":""},{"title":"Figure 22 .","content":""},{"title":"Table 1 THE SAMPLE","content":""},{"title":"Table 1 THE SAMPLE","content":""},{"title":"Table 2","content":""},{"title":"Table 2","content":""},{"title":"Table 3","content":""},{"title":"Table 3","content":""},{"title":"Table 4","content":""},{"title":"Table 4","content":""},{"title":"Table 4","content":""},{"title":"Table 4","content":""},{"title":"Table 5","content":""},{"title":"Table 5","content":""},{"title":"Table 6 PROBABILITIES","content":""},{"title":"Table 6 PROBABILITIES","content":""},{"title":"Table 7 SUMMARY","content":""},{"title":"Table 7 SUMMARY","content":""},{"title":"Table 8","content":""},{"title":"Table 8","content":""},{"title":"Table 9","content":""},{"title":"Table 9","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"\"\n },\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"Supermassive black holes (SBHs) have long been identified as the engines of active galactic nuclei (AGNs). More recently, compelling evidence has been amassed that they are present in the centers of almost all galaxies (Kormendy & Richstone 1995;Ferrarese & Ford 2005) and furthermore that the growth of the SBH is intimately linked to galaxy evolution (Ferrarese & Merritt 2000;Gebhardt et al. 2000; see Alexander & Hickox 2012 for a recent review).\\nSupermassive black holes (SBHs) have long been identified as the engines of active galactic nuclei (AGNs). More recently, compelling evidence has been amassed that they are present in the centers of almost all galaxies (Kormendy & Richstone 1995;Ferrarese & Ford 2005) and furthermore that the growth of the SBH is intimately linked to galaxy evolution (Ferrarese & Merritt 2000;Gebhardt et al. 2000; see Alexander & Hickox 2012 for a recent review).\\nAs a consequence of energy losses due to dynamical friction, SBHs are generally expected to reside at the large-scale potential minimum of the host galaxy. Nevertheless, there are several mechanisms that seem capable of displacing the SBH from its equilibrium position (Batcheldor et al. 2010, hereafter B10). Most recently, interest has focused on gravitational recoil resulting from the coalescence of an SBH-SBH binary (e.g., Merritt et al. 2004;Campanelli et al. 2006;Baker et al. 2006). Other possibilities include orbital motion of SBH-binaries (Komossa 2006), asymmetric radio jets (Shklovski 1982) and interactions with massive perturbers such as globular clusters, or massive molecular clouds.\\nAs a consequence of energy losses due to dynamical friction, SBHs are generally expected to reside at the large-scale potential minimum of the host galaxy. Nevertheless, there are several mechanisms that seem capable of displacing the SBH from its equilibrium position (Batcheldor et al. 2010, hereafter B10). Most recently, interest has focused on gravitational recoil resulting from the coalescence of an SBH-SBH binary (e.g., Merritt et al. 2004;Campanelli et al. 2006;Baker et al. 2006). Other possibilities include orbital motion of SBH-binaries (Komossa 2006), asymmetric radio jets (Shklovski 1982) and interactions with massive perturbers such as globular clusters, or massive molecular clouds.\\nThe first two of these mechanisms, in particular, might be expected to occur as an inevitable consequence of galaxy evolution. Cold dark matter hierarchical cosmologies predict that galaxies experience multiple mergers during their lifetime (e.g., Coles & Lucchin 2002;Springel et al. 2005), with minor mergers being more common than major mergers (e.g. Hilz et al. 2012). When two galaxies merge, their central SBHs are expected to form a binary system (Begelman, Blandford & Rees 1980) at the center of the merged system due to dynamical friction. Further tightening of the binary ensues through threebody interactions with stars, or gravitational drag from gas (Merritt & Milosavljević 2005;Mayer et al. 2007), until finally, energy loss due to gravitational wave emission irreversibly drives the two SBHs to coalescence 1 .\\nThe first two of these mechanisms, in particular, might be expected to occur as an inevitable consequence of galaxy evolution. Cold dark matter hierarchical cosmologies predict that galaxies experience multiple mergers during their lifetime (e.g., Coles & Lucchin 2002;Springel et al. 2005), with minor mergers being more common than major mergers (e.g. Hilz et al. 2012). When two galaxies merge, their central SBHs are expected to form a binary system (Begelman, Blandford & Rees 1980) at the center of the merged system due to dynamical friction. Further tightening of the binary ensues through threebody interactions with stars, or gravitational drag from gas (Merritt & Milosavljević 2005;Mayer et al. 2007), until finally, energy loss due to gravitational wave emission irreversibly drives the two SBHs to coalescence 1 .\\nIn the final stage of SBH-binary coalescence, anisotropic emission of gravitational waves will, in general, impart a recoil velocity to the coalesced object 1 Begelman, Blandford & Rees (1980) suggested that the evolution of a binary SBH might stall at separations of ∼ 1 pc once the supply of \\\"loss-cone stars\\\" (stars intersecting the orbit of the binary) was exhausted. However, gas dynamical torques can drive continued shrinking of the binary orbit (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007), and recent N -body simulations suggest that hardening of the binary due to interactions with stars can be efficient in non spherical potentials (Preto et al. 2011;Gualandris & Merritt 2012;Vasiliev et al. 2014). (Bekenstein 1973). Following a recent breakthrough allowing the orbits of spinning black holes to be computed to coalescence (Pretorius 2005;Campanelli et al. 2006;Baker et al. 2006), numerical relativity simulations have produced recoil velocities v ∼ 10 3 km s −1 , even reaching ∼ 5 × 10 3 km s −1 for certain spin configurations González et al. 2007;Tichy & Marronetti 2007;Lousto & Zlochower 2011;Lousto et al. 2012). Such velocities would cause large displacements of the coalesced SBH from the center of the galaxy or, in the most extreme cases, eject it entirely from the host (Merritt et al. 2004;Campanelli et al. 2007a;Volonteri, Gúltekin & Dotti 2010).\\nIn the final stage of SBH-binary coalescence, anisotropic emission of gravitational waves will, in general, impart a recoil velocity to the coalesced object 1 Begelman, Blandford & Rees (1980) suggested that the evolution of a binary SBH might stall at separations of ∼ 1 pc once the supply of \\\"loss-cone stars\\\" (stars intersecting the orbit of the binary) was exhausted. However, gas dynamical torques can drive continued shrinking of the binary orbit (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007), and recent N -body simulations suggest that hardening of the binary due to interactions with stars can be efficient in non spherical potentials (Preto et al. 2011;Gualandris & Merritt 2012;Vasiliev et al. 2014). (Bekenstein 1973). Following a recent breakthrough allowing the orbits of spinning black holes to be computed to coalescence (Pretorius 2005;Campanelli et al. 2006;Baker et al. 2006), numerical relativity simulations have produced recoil velocities v ∼ 10 3 km s −1 , even reaching ∼ 5 × 10 3 km s −1 for certain spin configurations González et al. 2007;Tichy & Marronetti 2007;Lousto & Zlochower 2011;Lousto et al. 2012). Such velocities would cause large displacements of the coalesced SBH from the center of the galaxy or, in the most extreme cases, eject it entirely from the host (Merritt et al. 2004;Campanelli et al. 2007a;Volonteri, Gúltekin & Dotti 2010).\\nRecoils exceeding the escape velocity of the host galaxy are expected to be relatively rare (e.g., Lousto et al. 2012, hereafter L12) and, more frequently, the SBH will undergo damped oscillations in the galaxy potential. Nbody simulations (Gualandris and Merritt 2008, hereafter GM08) have shown that moderately large kicks (∼ few 10 2 km s −1 , sufficient to eject the SBH from the core 2 but not from the galaxy) result in long-lived oscillations which damp on a timescale ∆t ∼ 1 Gyr. GM08 identified three dynamical phases for the particular case where the SBH is initially displaced beyond the core radius (r c ∼ 10 2 pc). In \\\"phase I\\\" the SBH undergoes damped harmonic motion at amplitudes greater than the core radius. These oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction. \\\"Phase II\\\" is characterized by oscillations of amplitude ∼ 10 − 100 pc between the SBH and the stellar core about the center of mass, which persist for up to ∼ 1 Gyr. Finally, in \\\"phase III\\\", the SBH has reached thermal equilibrium with the surrounding stars and experiences Brownian motion, signifying the end of the event.\\nRecoils exceeding the escape velocity of the host galaxy are expected to be relatively rare (e.g., Lousto et al. 2012, hereafter L12) and, more frequently, the SBH will undergo damped oscillations in the galaxy potential. Nbody simulations (Gualandris and Merritt 2008, hereafter GM08) have shown that moderately large kicks (∼ few 10 2 km s −1 , sufficient to eject the SBH from the core 2 but not from the galaxy) result in long-lived oscillations which damp on a timescale ∆t ∼ 1 Gyr. GM08 identified three dynamical phases for the particular case where the SBH is initially displaced beyond the core radius (r c ∼ 10 2 pc). In \\\"phase I\\\" the SBH undergoes damped harmonic motion at amplitudes greater than the core radius. These oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction. \\\"Phase II\\\" is characterized by oscillations of amplitude ∼ 10 − 100 pc between the SBH and the stellar core about the center of mass, which persist for up to ∼ 1 Gyr. Finally, in \\\"phase III\\\", the SBH has reached thermal equilibrium with the surrounding stars and experiences Brownian motion, signifying the end of the event.\\nLarge gravitational recoil kicks would have a variety of observable consequences. In the case of SBHs associated with AGNs, most of the accretion disk and broad emission line gas will remain bound for recoils with velocities v kick 10 3 km s −1 and such systems might be observed as a displaced, or velocity-shifted AGNs (e.g., Merritt et al. 2004;Madau & Quataert 2004;Loeb 2007;Bonning, Shields & Salviander 2007;Zanotti et al. 2010). Stars moving with orbital velocities v < v kick would also remain bound to the SBH after the kick, so that a recoiling SBH would be associated with a \\\"hypercompact stellar system\\\" of high internal velocity dispersion (Merritt et al. 2009).\\nLarge gravitational recoil kicks would have a variety of observable consequences. In the case of SBHs associated with AGNs, most of the accretion disk and broad emission line gas will remain bound for recoils with velocities v kick 10 3 km s −1 and such systems might be observed as a displaced, or velocity-shifted AGNs (e.g., Merritt et al. 2004;Madau & Quataert 2004;Loeb 2007;Bonning, Shields & Salviander 2007;Zanotti et al. 2010). Stars moving with orbital velocities v < v kick would also remain bound to the SBH after the kick, so that a recoiling SBH would be associated with a \\\"hypercompact stellar system\\\" of high internal velocity dispersion (Merritt et al. 2009).\\nPossible spectroscopic signatures of recoiling SBHs have been identified in a few individual active galaxies (Komossa et al. 2008;Shields et al. 2009;Robinson et al. 2010;Steinhardt et al. 2012), which would indicate large ( 10 3 km s −1 ) kick velocities. In addition, systematic searches of large SDSS AGN samples have revealed ∼ 100 objects that exhibit velocity shifts ∼ 10 3 km s −1 between the broad and narrow lines (Bonning, Shields & Salviander 2007;Eracleous et al. 2012;Tsalmantza et al. 2011). Some of these may be recoiling SBHs, or SBH binaries, but alternative explanations for the shifts involving extreme broad line region kinematics cannot be excluded. Perhaps the most interesting case is that of CIS-42, a galaxy which hosts two compact optical sources separated by ≈ 2.5 kpc and also exhibits a large velocity shift (∼ 1300 km s −1 ) between broad and narrow Hβ emission lines (Civano et al. 2010(Civano et al. , 2012Blecha et al. 2012).\\nPossible spectroscopic signatures of recoiling SBHs have been identified in a few individual active galaxies (Komossa et al. 2008;Shields et al. 2009;Robinson et al. 2010;Steinhardt et al. 2012), which would indicate large ( 10 3 km s −1 ) kick velocities. In addition, systematic searches of large SDSS AGN samples have revealed ∼ 100 objects that exhibit velocity shifts ∼ 10 3 km s −1 between the broad and narrow lines (Bonning, Shields & Salviander 2007;Eracleous et al. 2012;Tsalmantza et al. 2011). Some of these may be recoiling SBHs, or SBH binaries, but alternative explanations for the shifts involving extreme broad line region kinematics cannot be excluded. Perhaps the most interesting case is that of CIS-42, a galaxy which hosts two compact optical sources separated by ≈ 2.5 kpc and also exhibits a large velocity shift (∼ 1300 km s −1 ) between broad and narrow Hβ emission lines (Civano et al. 2010(Civano et al. , 2012Blecha et al. 2012).\\nThe gravitational recoil candidates discovered to date have been identified on the basis of anomalously large velocity shifts ( 10 3 km s −1 ) or a large spatial offset ( 1 kpc). If they are indeed recoiling SBHs, we are observing them during the early stages in their dynamical evolution following a large kick; that is, during the large amplitude phase I oscillations of GM08, if the SBH remains bound to the galaxy. However, special configurations of the progenitor binary are required to produce large kicks, which are therefore expected to be relatively rare events (L12) and furthermore, the subsequent large amplitude phase I oscillations are relatively short-lived.\\nThe gravitational recoil candidates discovered to date have been identified on the basis of anomalously large velocity shifts ( 10 3 km s −1 ) or a large spatial offset ( 1 kpc). If they are indeed recoiling SBHs, we are observing them during the early stages in their dynamical evolution following a large kick; that is, during the large amplitude phase I oscillations of GM08, if the SBH remains bound to the galaxy. However, special configurations of the progenitor binary are required to produce large kicks, which are therefore expected to be relatively rare events (L12) and furthermore, the subsequent large amplitude phase I oscillations are relatively short-lived.\\nOn the other hand, the N -body simulations of GM08 predict that, following a recoil kick sufficiently large to remove the SBH from the galaxy core, the decaying oscillations persist in phase II for ∼ 1 Gyr, comparable with the time between galaxy mergers (Hopkins et al. 2010, hereafter H10). This suggests that low amplitude SBH displacements resulting from post-merger SBH binary coalescence should be relatively common in the cores of bright elliptical galaxies. Thus, rather than searching for short-lived phase I oscillations following rare large kicks, an alternative approach to studying the incidence of SBH binary coalescence is to look for phase II oscillations in nearby ellipticals.\\nOn the other hand, the N -body simulations of GM08 predict that, following a recoil kick sufficiently large to remove the SBH from the galaxy core, the decaying oscillations persist in phase II for ∼ 1 Gyr, comparable with the time between galaxy mergers (Hopkins et al. 2010, hereafter H10). This suggests that low amplitude SBH displacements resulting from post-merger SBH binary coalescence should be relatively common in the cores of bright elliptical galaxies. Thus, rather than searching for short-lived phase I oscillations following rare large kicks, an alternative approach to studying the incidence of SBH binary coalescence is to look for phase II oscillations in nearby ellipticals.\\nSpectroscopic identification of phase II oscillations would be extremely difficult, if not impossible, as the associated velocity shifts would be comparable with gas motions due to rotation, or flows or turbulence driven by the AGN, or starburst activity. However, it is feasible to directly detect small-amplitude displacements between the AGN and the galaxy photocenter in high spatial resolution images of nearby galaxies. We recently analyzed archival Hubble Space Telescope Advanced Camera for Surveys images of the active giant elliptical M87 and found a projected offset of 6.8 ± 0.8 pc (≈ 0. 1) between the stellar photocenter and the AGN (B10; we note, however, that Gebhardt et al. 2011 did not find a significant offset in their near-infrared integral field spectroscopy data).\\nSpectroscopic identification of phase II oscillations would be extremely difficult, if not impossible, as the associated velocity shifts would be comparable with gas motions due to rotation, or flows or turbulence driven by the AGN, or starburst activity. However, it is feasible to directly detect small-amplitude displacements between the AGN and the galaxy photocenter in high spatial resolution images of nearby galaxies. We recently analyzed archival Hubble Space Telescope Advanced Camera for Surveys images of the active giant elliptical M87 and found a projected offset of 6.8 ± 0.8 pc (≈ 0. 1) between the stellar photocenter and the AGN (B10; we note, however, that Gebhardt et al. 2011 did not find a significant offset in their near-infrared integral field spectroscopy data).\\nHere we describe a photometric analysis of archival HST images of 14 nearby (d<100 Mpc), bright ellipticals hosting low-luminosity AGNs, with the aim of directly measuring spatial offsets comparable with the core radius of the galaxy (typically < 500 pc). Our method consists in measuring the relative positions of the AGN point source and the photocenter of the inner 2-3 kpc of the galaxy, assuming that the former marks the SBH position and the latter the minimum of the galactic potential. As long as the AGN is bright enough to be easily detected and faint enough that it does not overwhelm the host galaxy, it is possible to perform standard photometric analysis to locate the photocenter. The \\\"displacement\\\" is then computed as the relative distance between the AGN and photocenter.\\nHere we describe a photometric analysis of archival HST images of 14 nearby (d<100 Mpc), bright ellipticals hosting low-luminosity AGNs, with the aim of directly measuring spatial offsets comparable with the core radius of the galaxy (typically < 500 pc). Our method consists in measuring the relative positions of the AGN point source and the photocenter of the inner 2-3 kpc of the galaxy, assuming that the former marks the SBH position and the latter the minimum of the galactic potential. As long as the AGN is bright enough to be easily detected and faint enough that it does not overwhelm the host galaxy, it is possible to perform standard photometric analysis to locate the photocenter. The \\\"displacement\\\" is then computed as the relative distance between the AGN and photocenter.\\nThe plan of this paper is as follows: in §2 we describe the sample; §3 describes the analysis methods used to determine the photocenter, SBH position and offset, and their uncertainties; in §4 the measured displacements are presented for each galaxy. In §5, we discuss the results in the context of gravitational recoil and other possible displacement mechanisms. Finally, our conclusions are summarized in §6. The results for selected galaxies are discussed in more detail in an Appendix.\\nThe plan of this paper is as follows: in §2 we describe the sample; §3 describes the analysis methods used to determine the photocenter, SBH position and offset, and their uncertainties; in §4 the measured displacements are presented for each galaxy. In §5, we discuss the results in the context of gravitational recoil and other possible displacement mechanisms. Finally, our conclusions are summarized in §6. The results for selected galaxies are discussed in more detail in an Appendix.\"\n },\n {\n \"title\": \"THE SAMPLE\",\n \"content\": \"\"\n },\n {\n \"title\": \"THE SAMPLE\",\n \"content\": \"The sample selected for this study consists of 14 nearby, regular, core elliptical galaxies which host AGNs and for which HST images obtained with ACS, NIC-MOS2, WFPC2 or WFC3 are available in the Hubble Legacy Archive (HLA).\\nThe sample selected for this study consists of 14 nearby, regular, core elliptical galaxies which host AGNs and for which HST images obtained with ACS, NIC-MOS2, WFPC2 or WFC3 are available in the Hubble Legacy Archive (HLA).\\nHST observations have revealed that the central light distribution in nearly all nearby early type galaxies can be described by a singular surface brightness profile: Σ(r) ∼ r −γ as r → 0. 1, where γ ≥ 0. The slope of the innermost surface brightness distribution is bimodal in the sense that there is a paucity of galaxies which have 0.3 < γ < 0.5. Hence, early type galaxies have been classified in two families: \\\"core galaxies\\\", which have a shallower (γ ≤ 0.3) inner cusp within a \\\"break\\\" (or \\\"core\\\") radius r c ∼ 100 pc, and \\\"power law galaxies\\\", which have a steep cusp (γ ≥ 0.5) continuing into the HST resolution limit . These classifications correlate with several other galaxy properties (e.g., Faber et al. 1997), in particular, core galaxies are on average more luminous than power-law galaxies, with essentially all bright (M V < −22) ellipticals having cores.\\nHST observations have revealed that the central light distribution in nearly all nearby early type galaxies can be described by a singular surface brightness profile: Σ(r) ∼ r −γ as r → 0. 1, where γ ≥ 0. The slope of the innermost surface brightness distribution is bimodal in the sense that there is a paucity of galaxies which have 0.3 < γ < 0.5. Hence, early type galaxies have been classified in two families: \\\"core galaxies\\\", which have a shallower (γ ≤ 0.3) inner cusp within a \\\"break\\\" (or \\\"core\\\") radius r c ∼ 100 pc, and \\\"power law galaxies\\\", which have a steep cusp (γ ≥ 0.5) continuing into the HST resolution limit . These classifications correlate with several other galaxy properties (e.g., Faber et al. 1997), in particular, core galaxies are on average more luminous than power-law galaxies, with essentially all bright (M V < −22) ellipticals having cores.\\nCore galaxies are promising systems in which to search for SBH binaries or SBHs displaced by gravitational wave-induced recoils. They are bright ellipticals which are often the dominant components of clusters or groups and are thus likely to have experienced a recent major merger leading to the formation of an SBH binary. The shallow inner surface brightness profile indicates a mass deficit relative to that implied by inward extrapolation of the steeper brightness profile prevailing at larger radii. This is predicted as a natural consequence of depopulation of the inner region of the galaxy by 3-body interactions between an SBH binary and stars crossing the orbit of the binary (\\\"loss-cone stars\\\" Merritt & Milosavljević 2005). Therefore, \\\"flat\\\" inner surface brightness profiles have been proposed as a clear footprint of the formation of a tightly bound SBH binary (e.g., Faber et al. 1997;Merritt 2006).\\nCore galaxies are promising systems in which to search for SBH binaries or SBHs displaced by gravitational wave-induced recoils. They are bright ellipticals which are often the dominant components of clusters or groups and are thus likely to have experienced a recent major merger leading to the formation of an SBH binary. The shallow inner surface brightness profile indicates a mass deficit relative to that implied by inward extrapolation of the steeper brightness profile prevailing at larger radii. This is predicted as a natural consequence of depopulation of the inner region of the galaxy by 3-body interactions between an SBH binary and stars crossing the orbit of the binary (\\\"loss-cone stars\\\" Merritt & Milosavljević 2005). Therefore, \\\"flat\\\" inner surface brightness profiles have been proposed as a clear footprint of the formation of a tightly bound SBH binary (e.g., Faber et al. 1997;Merritt 2006).\\nIn addition, core galaxies tend to have regular photometric structures, and hence well-defined photocenters. In order to accurately locate the position of the SBH, we further require that each galaxy hosts a low luminosity AGN visible as a point source in the HST images.\\nIn addition, core galaxies tend to have regular photometric structures, and hence well-defined photocenters. In order to accurately locate the position of the SBH, we further require that each galaxy hosts a low luminosity AGN visible as a point source in the HST images.\\nIn order to resolve the low amplitude offsets between photocenter and AGN expected in the phase II oscillations predicted by GM08, we select only galaxies within 100 Mpc.\\nIn order to resolve the low amplitude offsets between photocenter and AGN expected in the phase II oscillations predicted by GM08, we select only galaxies within 100 Mpc.\\nThe sample analyzed in this work was extracted from a sample of 29 core elliptical galaxies previously studied by Capetti and Balmaverde, in a series of three articles (Capetti & Balmaverde 2005;Capetti & Balmaverde 2006, hereafter CB05, BC06 and CB06 respectively). CB05 compiled a radio-flux limited sample of AGNs in early type hosts by selecting radio-detected sources from the VLA surveys of Wrobel & Heeschen (1991) and Sadler et al. (1989). The former is a northern sample of 216 early-type galaxies extracted from the CfA redshift survey (Huchra et al. 1983), with the following criteria: declination δ 1950 ≥ 0; photometric magnitude B ≤ 14; heliocentric velocity ≤ 3000 km s −1 ; morphological Hubble type T ≤ −1. The latter is a similar southern sample of 116 E and S0 galaxies with declination −45 ≤ δ ≤ −32. Both were observed at 5 GHz with a flux limit of ∼ 1 mJy.\\nThe sample analyzed in this work was extracted from a sample of 29 core elliptical galaxies previously studied by Capetti and Balmaverde, in a series of three articles (Capetti & Balmaverde 2005;Capetti & Balmaverde 2006, hereafter CB05, BC06 and CB06 respectively). CB05 compiled a radio-flux limited sample of AGNs in early type hosts by selecting radio-detected sources from the VLA surveys of Wrobel & Heeschen (1991) and Sadler et al. (1989). The former is a northern sample of 216 early-type galaxies extracted from the CfA redshift survey (Huchra et al. 1983), with the following criteria: declination δ 1950 ≥ 0; photometric magnitude B ≤ 14; heliocentric velocity ≤ 3000 km s −1 ; morphological Hubble type T ≤ −1. The latter is a similar southern sample of 116 E and S0 galaxies with declination −45 ≤ δ ≤ −32. Both were observed at 5 GHz with a flux limit of ∼ 1 mJy.\\nThe 65 objects with available archival HST images were classified as core or power-law galaxies on the basis of the slopes of their nuclear brightness profile as obtained by fitting a broken power law (the so-called \\\"Nuker law\\\"; Lauer et al. 1995;CB05;CB06). For the purpose of this study, we focus on the core galaxies listed in Table 1 of BC06 and impose two additional selection criteria, based on visual inspection of HST images: (i) the presence of an optically bright central point-like source (a low luminosity AGN); (ii) the absence of heavy nuclear obscuration and other photometric irregularities.\\nThe 65 objects with available archival HST images were classified as core or power-law galaxies on the basis of the slopes of their nuclear brightness profile as obtained by fitting a broken power law (the so-called \\\"Nuker law\\\"; Lauer et al. 1995;CB05;CB06). For the purpose of this study, we focus on the core galaxies listed in Table 1 of BC06 and impose two additional selection criteria, based on visual inspection of HST images: (i) the presence of an optically bright central point-like source (a low luminosity AGN); (ii) the absence of heavy nuclear obscuration and other photometric irregularities.\\nThe 14 galaxies selected for this study are listed in Table 1 together with their distances, core radii and details of the archival data sets that were retrieved for photometric analysis. The sample includes two galaxies (NGC 4696 and 5419) that exhibit double nuclei, separated by ∼ 0. 25. In these cases, it is not known which of the two point sources is the AGN and therefore we assume that the brightest is the AGN (although we have measured displacements relative to both).\\nThe 14 galaxies selected for this study are listed in Table 1 together with their distances, core radii and details of the archival data sets that were retrieved for photometric analysis. The sample includes two galaxies (NGC 4696 and 5419) that exhibit double nuclei, separated by ∼ 0. 25. In these cases, it is not known which of the two point sources is the AGN and therefore we assume that the brightest is the AGN (although we have measured displacements relative to both).\\nThe radio source properties, including the total power at 5 GHz and the position angle of any extended structure, are summarized in Table 2.\\nThe radio source properties, including the total power at 5 GHz and the position angle of any extended structure, are summarized in Table 2.\"\n },\n {\n \"title\": \"DATA ANALYSIS\",\n \"content\": \"\"\n },\n {\n \"title\": \"DATA ANALYSIS\",\n \"content\": \"Broad-band images of the sample galaxies acquired with ACS, NICMOS2, WFPC2 and WFC3 were retrieved from the Hubble Legacy Archive (HLA), giving preference to images obtained with the highest spatial resolutions and red filters (i.e., F606W and longer wavelengths).\\nBroad-band images of the sample galaxies acquired with ACS, NICMOS2, WFPC2 and WFC3 were retrieved from the Hubble Legacy Archive (HLA), giving preference to images obtained with the highest spatial resolutions and red filters (i.e., F606W and longer wavelengths).\\nWhenever available, several images taken with different instruments and filters, covering both optical and NIR bands, were obtained for each galaxy. A list of the images analyzed for each galaxy is given in Table 1. No additional processing steps were applied beyond the standard HLA reduction pipeline. As the goal of our analysis is simply to determine the relative positions of the isophotal center of the inner few kiloparsecs of the galaxy and the AGN point source within each individual image, this is sufficient for our purposes.\\nWhenever available, several images taken with different instruments and filters, covering both optical and NIR bands, were obtained for each galaxy. A list of the images analyzed for each galaxy is given in Table 1. No additional processing steps were applied beyond the standard HLA reduction pipeline. As the goal of our analysis is simply to determine the relative positions of the isophotal center of the inner few kiloparsecs of the galaxy and the AGN point source within each individual image, this is sufficient for our purposes.\\nEach image was analyzed according to the following main steps: (1) a mask was constructed to block image defects and distorting features such as dust lanes, jets or bright stars and globular clusters; (2) elliptical isophotes were fitted to the galaxy surface brightness distribution and the photocenter computed as the flux-weighted average of the isophote centers; (3) the position of the point source (assumed to locate the SBH) was determined by fitting a gaussian profile. See §3.1 and §3.2 for a more detailed description of these procedures. Sky subtraction was not performed as a uniform background will not affect the photometric analysis and in many cases it is not possible to measure the sky background as the galaxy covers the frame. (2) name as it appears in BC06 if different than the name given in the first column; (3) adopted distance in Mpc (from the HyperLeda extragalactic database); (4) method used to determine the distance: \\\"sbf\\\", surface brightness fluctuation; \\\"vvir\\\", from radial velocity, corrected for Local Group infall into Virgo; \\\"Dn − σ\\\", size-sigma relation (Dressler et al. 1987); (5) core radius as derived by CB05, in arcsec (pc); (6) SBH influence radius (GM•/σ 2 ) in arcsec (pc). Central velocity dispersions are given in Table 8. Black hole masses have been derived from the M• − σ relation given in Ferrarese & Ford (2005); (7) HST instrument/camera/filter used; (8) pixel scale in arcseconds/pixels as determined from direct measurement on the frames; (9) proposal identification number. B10 : this is a combined image, see Table 1 in B10 for further details. † A WFC3/IR image is available for NGC 4696, however the complex nuclear features and the pixel size of the camera do not allow us to infer the SBH position with the precision necessary for this work.\\nEach image was analyzed according to the following main steps: (1) a mask was constructed to block image defects and distorting features such as dust lanes, jets or bright stars and globular clusters; (2) elliptical isophotes were fitted to the galaxy surface brightness distribution and the photocenter computed as the flux-weighted average of the isophote centers; (3) the position of the point source (assumed to locate the SBH) was determined by fitting a gaussian profile. See §3.1 and §3.2 for a more detailed description of these procedures. Sky subtraction was not performed as a uniform background will not affect the photometric analysis and in many cases it is not possible to measure the sky background as the galaxy covers the frame. (2) name as it appears in BC06 if different than the name given in the first column; (3) adopted distance in Mpc (from the HyperLeda extragalactic database); (4) method used to determine the distance: \\\"sbf\\\", surface brightness fluctuation; \\\"vvir\\\", from radial velocity, corrected for Local Group infall into Virgo; \\\"Dn − σ\\\", size-sigma relation (Dressler et al. 1987); (5) core radius as derived by CB05, in arcsec (pc); (6) SBH influence radius (GM•/σ 2 ) in arcsec (pc). Central velocity dispersions are given in Table 8. Black hole masses have been derived from the M• − σ relation given in Ferrarese & Ford (2005); (7) HST instrument/camera/filter used; (8) pixel scale in arcseconds/pixels as determined from direct measurement on the frames; (9) proposal identification number. B10 : this is a combined image, see Table 1 in B10 for further details. † A WFC3/IR image is available for NGC 4696, however the complex nuclear features and the pixel size of the camera do not allow us to infer the SBH position with the precision necessary for this work.\"\n },\n {\n \"title\": \"The inner photocenter\",\n \"content\": \"\"\n },\n {\n \"title\": \"The inner photocenter\",\n \"content\": \"Each image was analyzed with the IRAF task ellipse (Jedrzejewski 1987), which was used to fit elliptical isophotes to the surface brightness distribution. The first step was to construct an image mask in order to minimize distortions due to image defects and intrinsic photometric irregularities such as dust features, optical or NIR knots associated with jets, globular clusters, foreground stars, etc. If an exposure time map (the \\\"weight image\\\") was provided in the retrieved data set, this was used as an initial mask to eliminate cosmic rays, bad pixels, null areas and other image frame blemishes. Applying the initial mask, a first run of ellipse was performed to create a zeroth-order photometric model. To account for intrinsic irregularities in the surface brightness distribution, we then created a second mask by subtracting the zeroth-order photometric model from the original image and masking residuals exceeding ±nσ, where σ is the standard deviation of the residual image and n is a clipping parameter.\\nEach image was analyzed with the IRAF task ellipse (Jedrzejewski 1987), which was used to fit elliptical isophotes to the surface brightness distribution. The first step was to construct an image mask in order to minimize distortions due to image defects and intrinsic photometric irregularities such as dust features, optical or NIR knots associated with jets, globular clusters, foreground stars, etc. If an exposure time map (the \\\"weight image\\\") was provided in the retrieved data set, this was used as an initial mask to eliminate cosmic rays, bad pixels, null areas and other image frame blemishes. Applying the initial mask, a first run of ellipse was performed to create a zeroth-order photometric model. To account for intrinsic irregularities in the surface brightness distribution, we then created a second mask by subtracting the zeroth-order photometric model from the original image and masking residuals exceeding ±nσ, where σ is the standard deviation of the residual image and n is a clipping parameter.\\nTo allow ellipse to find the photometric peak, the central region was unmasked and a new run performed. This process was iterated, adjusting n in order to map distorting features in the maximum possible detail. The iteration was terminated when the mask converged and further runs did not add any significant detail.\\nTo allow ellipse to find the photometric peak, the central region was unmasked and a new run performed. This process was iterated, adjusting n in order to map distorting features in the maximum possible detail. The iteration was terminated when the mask converged and further runs did not add any significant detail.\\nOnce the final mask was constructed, ellipse was rerun with the mask applied to determine the photocenter of the galaxy. Elliptical isophotes were fitted between minimum and maximum values of the semi-major axis (SMA) which were usually determined by the core radius (r c ) and the image size, respectively. In almost all cases, the minimum SMA was set equal to or greater than the value of r c as determined by CB05; Table 1. This was typically ∼ 2 , or ∼ 300 pc. In some cases, where a dusty disk is present in the center of the galaxy, the initial SMA was chosen such that the inner ellipse is larger than the disk. The only exception was NGC 4486 (M 87), which has a very large core. In this case, we adopted the SMA range 1-3 used by B10, who found that isophotes within 1 are influenced by the AGN point source. The maximum SMA was that of the largest ellipse that completely fits within the image area -that is, only complete isophotes were fitted. This limits our isophotal analysis to the inner few kiloparsecs of each galaxy; the SMA of the outermost ellipse fit varies in the range 1 r 4 kpc.\\nOnce the final mask was constructed, ellipse was rerun with the mask applied to determine the photocenter of the galaxy. Elliptical isophotes were fitted between minimum and maximum values of the semi-major axis (SMA) which were usually determined by the core radius (r c ) and the image size, respectively. In almost all cases, the minimum SMA was set equal to or greater than the value of r c as determined by CB05; Table 1. This was typically ∼ 2 , or ∼ 300 pc. In some cases, where a dusty disk is present in the center of the galaxy, the initial SMA was chosen such that the inner ellipse is larger than the disk. The only exception was NGC 4486 (M 87), which has a very large core. In this case, we adopted the SMA range 1-3 used by B10, who found that isophotes within 1 are influenced by the AGN point source. The maximum SMA was that of the largest ellipse that completely fits within the image area -that is, only complete isophotes were fitted. This limits our isophotal analysis to the inner few kiloparsecs of each galaxy; the SMA of the outermost ellipse fit varies in the range 1 r 4 kpc.\\nBetween these limits, a series of ellipses was fitted to the 2-D surface brightness distribution, with the SMA being increased by two pixels at each step. Each fit returned values of the position angle, eccentricity and the pixel x − y coordinates of the center of the ellipse. Fi- Note. -(1) Optical identification;\\nBetween these limits, a series of ellipses was fitted to the 2-D surface brightness distribution, with the SMA being increased by two pixels at each step. Each fit returned values of the position angle, eccentricity and the pixel x − y coordinates of the center of the ellipse. Fi- Note. -(1) Optical identification;\\n(2) approximate length of the radio jet/extension; (3) position angle (degrees E of N); (4) total power at 5 GHz [10 20 W]; (5) comments on the radio source morphology; (6) comments on the nature of the galaxy. † This value is computed for the inner 25 pc. References: a Slee et al. (1994), b Wrobel & Heeschen (1991, c Wrobel (1991), d Sadler et al. (1989, e Shurkin et al. (2008), f Cavagnolo et al. (2010), g Giroletti et al. (2005), h Baade & Minkowski (1954), i Nagar et al. (2002), l Stanger & Warwick (1986), m Taylor et al. (2006), n Killeen et al. (1986a), p Jones et al. (2000. [HG82]: Huchra & Geller (1982).\\n(2) approximate length of the radio jet/extension; (3) position angle (degrees E of N); (4) total power at 5 GHz [10 20 W]; (5) comments on the radio source morphology; (6) comments on the nature of the galaxy. † This value is computed for the inner 25 pc. References: a Slee et al. (1994), b Wrobel & Heeschen (1991, c Wrobel (1991), d Sadler et al. (1989, e Shurkin et al. (2008), f Cavagnolo et al. (2010), g Giroletti et al. (2005), h Baade & Minkowski (1954), i Nagar et al. (2002), l Stanger & Warwick (1986), m Taylor et al. (2006), n Killeen et al. (1986a), p Jones et al. (2000. [HG82]: Huchra & Geller (1982).\\nnally, the photocenter was obtained as the flux-weighted average of the centers of the elliptical annuli generated by ellipse:\\nnally, the photocenter was obtained as the flux-weighted average of the centers of the elliptical annuli generated by ellipse:\\nwhere x i and y i are the photocenter coordinates of the ith isophote and j i is a weight, given by the product of the area of the annulus defined by neighbouring isophotes, with the mean intensity within the annulus.\\nwhere x i and y i are the photocenter coordinates of the ith isophote and j i is a weight, given by the product of the area of the annulus defined by neighbouring isophotes, with the mean intensity within the annulus.\\nTo assess the sensitivity of the mean photocenter to the weighting method, we computed an alternative weight function using the flux enclosed within each successive elliptical isophote. For comparison, the two weight functions are plotted as a function of SMA in Fig.1, for the NICMOS image of NGC 4261. The annular flux method produces an approximately uniform weight distribution beyond the innermost region (SMA < 0. 5), but typically decreases slowly at large SMA values. Not surprisingly, the enclosed flux method produces weights that increase monotonically with SMA, thus assigning much greater weight to the outer, fainter isophotes. This may be an undesirable characteristic in that isophotal distortions which may be present at fainter surface brightness levels (due to interactions with nearby companions, for example) might unduly influence the mean photocenter. Nevertheless, for the sources in our sample, the photocenters computed using the two methods are consistent within the errors in all cases. This result reflects the rather regular nature of the sample galaxies at the wavelengths under consideration.\\nTo assess the sensitivity of the mean photocenter to the weighting method, we computed an alternative weight function using the flux enclosed within each successive elliptical isophote. For comparison, the two weight functions are plotted as a function of SMA in Fig.1, for the NICMOS image of NGC 4261. The annular flux method produces an approximately uniform weight distribution beyond the innermost region (SMA < 0. 5), but typically decreases slowly at large SMA values. Not surprisingly, the enclosed flux method produces weights that increase monotonically with SMA, thus assigning much greater weight to the outer, fainter isophotes. This may be an undesirable characteristic in that isophotal distortions which may be present at fainter surface brightness levels (due to interactions with nearby companions, for example) might unduly influence the mean photocenter. Nevertheless, for the sources in our sample, the photocenters computed using the two methods are consistent within the errors in all cases. This result reflects the rather regular nature of the sample galaxies at the wavelengths under consideration.\\nWe explored alternative methods for the measurement of the photocenter position, namely a moment based technique and a 2-D decomposition of the surface brightness distribution. In the first case, the photocenter position is obtained by weighting the coordinate of each pixel by its own intensity. This method seems to be particularly sensitive to the masking details and to the presence of asymmetries in the light distribution. The second ap-proach, performed with the GALFIT surface brightness fitting program (Peng et al. 2002), requires, at least in some cases, multiple components for a successful fit (e.g. two or more Sérsic profiles) casting doubt on the physical meaning of the recovered photocenter and the errors associated with it.\\nWe explored alternative methods for the measurement of the photocenter position, namely a moment based technique and a 2-D decomposition of the surface brightness distribution. In the first case, the photocenter position is obtained by weighting the coordinate of each pixel by its own intensity. This method seems to be particularly sensitive to the masking details and to the presence of asymmetries in the light distribution. The second ap-proach, performed with the GALFIT surface brightness fitting program (Peng et al. 2002), requires, at least in some cases, multiple components for a successful fit (e.g. two or more Sérsic profiles) casting doubt on the physical meaning of the recovered photocenter and the errors associated with it.\\nThe IRAF task ellipse was specifically built to fit the light profile of elliptical galaxies with minimal assumptions; the results obtained seem to be weakly dependent on the masking details and errors associated with the fitted parameters have been thoroughly studied (Busko 1996). Therefore, we prefer to adopt the photocenter position determined with this approach.\\nThe IRAF task ellipse was specifically built to fit the light profile of elliptical galaxies with minimal assumptions; the results obtained seem to be weakly dependent on the masking details and errors associated with the fitted parameters have been thoroughly studied (Busko 1996). Therefore, we prefer to adopt the photocenter position determined with this approach.\"\n },\n {\n \"title\": \"The SBH position\",\n \"content\": \"\"\n },\n {\n \"title\": \"The SBH position\",\n \"content\": \"We make the key assumption that the point-like source near the center of each galaxy is an AGN, which therefore locates the position of the SBH (see §5 for a detailed discussion of this assumption).\\nWe make the key assumption that the point-like source near the center of each galaxy is an AGN, which therefore locates the position of the SBH (see §5 for a detailed discussion of this assumption).\\nThe AGN point sources are bright enough to be easily detected, but in our sample they are not so bright that the point spread function dominates the host galaxy. In order to determine the position of the SBH, we proceeded as follows. First, a median filtered version of the image was subtracted from the original producing a residual map in which the AGN is a prominent feature 3 . In the second step, we used the IRAF task center to fit gaussians to the marginal distributions of the residual point source along the x and y axes.\\nThe AGN point sources are bright enough to be easily detected, but in our sample they are not so bright that the point spread function dominates the host galaxy. In order to determine the position of the SBH, we proceeded as follows. First, a median filtered version of the image was subtracted from the original producing a residual map in which the AGN is a prominent feature 3 . In the second step, we used the IRAF task center to fit gaussians to the marginal distributions of the residual point source along the x and y axes.\"\n },\n {\n \"title\": \"The photocenter-AGN displacement\",\n \"content\": \"\"\n },\n {\n \"title\": \"The photocenter-AGN displacement\",\n \"content\": \"For each image of each galaxy, the x and y components of the displacement were measured as the difference in pixels between the x and y co-ordinates of the photocenter and the AGN point source. The significance of the displacement is classified in two steps, considering the x and y components separately. Displacements smaller than 3σ, where σ is the error on the displacement (Section 3.4.1), are considered non-significant. If the displacement exceeds 3σ we make a further classification based on the distribution of the isophote center coordinates produced by ellipse. For each coordinate, x and y, the inter-quartile range (IQR, the difference between the upper and lower quartiles) of the isophote centers was computed. The corresponding displacement is then assigned significance levels of \\\"null\\\", \\\"low\\\", \\\"intermediate\\\" or \\\"high\\\" depending on its magnitude relative to the IQR:\\nFor each image of each galaxy, the x and y components of the displacement were measured as the difference in pixels between the x and y co-ordinates of the photocenter and the AGN point source. The significance of the displacement is classified in two steps, considering the x and y components separately. Displacements smaller than 3σ, where σ is the error on the displacement (Section 3.4.1), are considered non-significant. If the displacement exceeds 3σ we make a further classification based on the distribution of the isophote center coordinates produced by ellipse. For each coordinate, x and y, the inter-quartile range (IQR, the difference between the upper and lower quartiles) of the isophote centers was computed. The corresponding displacement is then assigned significance levels of \\\"null\\\", \\\"low\\\", \\\"intermediate\\\" or \\\"high\\\" depending on its magnitude relative to the IQR:\\n• null, ∆x, ∆y < 0.8 IQR • low, 0.8 IQR ≤ ∆x, ∆y < 1.6 IQR\\n• null, ∆x, ∆y < 0.8 IQR • low, 0.8 IQR ≤ ∆x, ∆y < 1.6 IQR\\nThe threshold values are based on the fact that, for a normal distribution, 0.8 IQR is equivalent to 1σ. For an ideal, symmetric galaxy the isophotes would be concentric, giving IQR = 0. Any irregularities in the surface brightness distribution will cause a dispersion in the centers of the fitted elliptical isophotes, resulting in a nonzero IQR. Hence, by normalizing to the IQR, we allow for uncertainties in the recovered offsets due to asymmetric surface brightness distributions arising from, for example, dust lanes or tidal distortions.\\nThe threshold values are based on the fact that, for a normal distribution, 0.8 IQR is equivalent to 1σ. For an ideal, symmetric galaxy the isophotes would be concentric, giving IQR = 0. Any irregularities in the surface brightness distribution will cause a dispersion in the centers of the fitted elliptical isophotes, resulting in a nonzero IQR. Hence, by normalizing to the IQR, we allow for uncertainties in the recovered offsets due to asymmetric surface brightness distributions arising from, for example, dust lanes or tidal distortions.\"\n },\n {\n \"title\": \"Errors and biases\",\n \"content\": \"\"\n },\n {\n \"title\": \"Errors and biases\",\n \"content\": \"\"\n },\n {\n \"title\": \"Position uncertainties\",\n \"content\": \"\"\n },\n {\n \"title\": \"Position uncertainties\",\n \"content\": \"The final error on the measured projected displacement is the combination of the uncertainties in the positions of the photocenter and the AGN point source. As the photocenter is determined by fitting elliptical isophotes, it is important to understand if the errors provided by ellipse are reasonable and robust. To investigate this, we used ellipse to analyze synthetic galaxy images constructed with GALFIT. We also used these simulated galaxies to estimate the uncertainties on both the AGN and photocenter positions.\\nThe final error on the measured projected displacement is the combination of the uncertainties in the positions of the photocenter and the AGN point source. As the photocenter is determined by fitting elliptical isophotes, it is important to understand if the errors provided by ellipse are reasonable and robust. To investigate this, we used ellipse to analyze synthetic galaxy images constructed with GALFIT. We also used these simulated galaxies to estimate the uncertainties on both the AGN and photocenter positions.\\nThe model images were constructed by fitting a Sérsic profile and a nuclear point source (a PSF generated with Tiny Tim, Krist 1993) to selected galaxies in our sample. We used NGC 4261 and 4278 to build models of NIC-MOS images, IC 1459 for WFPC2 images, NGC 1399 for WFC3 and IC 4296 for ACS images. The resulting model surface brightness distributions were populated with gaussian random noise so as to match the signalto-noise ratio (SN) of the original galaxy image 4 . Four realizations of each synthetic galaxy image were generated and each realization was analyzed using the methods described in Sections 3.1 and 3.2, superposing the masks obtained from the corresponding real galaxy images.\\nThe model images were constructed by fitting a Sérsic profile and a nuclear point source (a PSF generated with Tiny Tim, Krist 1993) to selected galaxies in our sample. We used NGC 4261 and 4278 to build models of NIC-MOS images, IC 1459 for WFPC2 images, NGC 1399 for WFC3 and IC 4296 for ACS images. The resulting model surface brightness distributions were populated with gaussian random noise so as to match the signalto-noise ratio (SN) of the original galaxy image 4 . Four realizations of each synthetic galaxy image were generated and each realization was analyzed using the methods described in Sections 3.1 and 3.2, superposing the masks obtained from the corresponding real galaxy images.\\nEllipse errors: for each isophote fit, the \\\"true error\\\" on the isophote center is computed as the difference between the position returned by ellipse and the known photocenter of the synthetic galaxy (i.e., the center position of the Sérsic component fitted by Galfit). These \\\"true errors\\\" are compared to the errors on the positions of the isophote centers returned by ellipse in Fig.2. The \\\"true\\\" error distribution is slightly broader than that of the ellipse errors, but both distributions are highly peaked, with ∼ 50% of fits producing errors ≤ 0.1 pixels. A detailed study by Busko (1996) of the errors returned by ellipse shows that errors on ellipticity, position angle and center position are unbiased and accurate when the radial gradient relative error 5 is less than 50%. This condition is satisfied for our sample galaxies.\\nEllipse errors: for each isophote fit, the \\\"true error\\\" on the isophote center is computed as the difference between the position returned by ellipse and the known photocenter of the synthetic galaxy (i.e., the center position of the Sérsic component fitted by Galfit). These \\\"true errors\\\" are compared to the errors on the positions of the isophote centers returned by ellipse in Fig.2. The \\\"true\\\" error distribution is slightly broader than that of the ellipse errors, but both distributions are highly peaked, with ∼ 50% of fits producing errors ≤ 0.1 pixels. A detailed study by Busko (1996) of the errors returned by ellipse shows that errors on ellipticity, position angle and center position are unbiased and accurate when the radial gradient relative error 5 is less than 50%. This condition is satisfied for our sample galaxies.\\nPosition errors: the \\\"true errors\\\" on the (x,y) coordinates of the AGN and galaxy photocenter positions were also determined for each realization of the synthetic galaxy images, where the true error in each case is defined as the difference between the known positions of the AGN and photocenter (i.e., the positions returned by Galfit) and the values recovered using our analysis methods.\\nPosition errors: the \\\"true errors\\\" on the (x,y) coordinates of the AGN and galaxy photocenter positions were also determined for each realization of the synthetic galaxy images, where the true error in each case is defined as the difference between the known positions of the AGN and photocenter (i.e., the positions returned by Galfit) and the values recovered using our analysis methods.\\nFor signal-to-noise ratios characteristic of our data (S/N ∼ 100), the distributions of the \\\"true errors\\\" on the AGN and photocenter positions derived from both the ACS and WFPC2 synthetic galaxy images are narrow and confined within 0.1 pixel. We therefore adopt 0.1 pixel as a conservative estimate of the uncertainty on both the AGN and photocenter positions derived from ACS and WFPC2 data. The \\\"true error\\\" obtained from the WFC3 and NICMOS2 images exhibit broader distributions, with tails extending to 0.4 pixels. We therefore adopt the median values of the \\\"true errors\\\" as the un- Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \\\"true errors\\\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC. certainties for WFC3 and NICMOS2 data, yielding 0.2 pixels for the photocenter position in both cameras and respectively 0.1 and 0.2 pixels for the AGN.\\nFor signal-to-noise ratios characteristic of our data (S/N ∼ 100), the distributions of the \\\"true errors\\\" on the AGN and photocenter positions derived from both the ACS and WFPC2 synthetic galaxy images are narrow and confined within 0.1 pixel. We therefore adopt 0.1 pixel as a conservative estimate of the uncertainty on both the AGN and photocenter positions derived from ACS and WFPC2 data. The \\\"true error\\\" obtained from the WFC3 and NICMOS2 images exhibit broader distributions, with tails extending to 0.4 pixels. We therefore adopt the median values of the \\\"true errors\\\" as the un- Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \\\"true errors\\\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC. certainties for WFC3 and NICMOS2 data, yielding 0.2 pixels for the photocenter position in both cameras and respectively 0.1 and 0.2 pixels for the AGN.\\nThese uncertainties represent the precision with which we can determine the AGN and photocenter positions for an ideal galaxy, in an image with noise characteristics representative of our data. The adopted uncertainties are summarized in Table 3, together with the equivalent angular distances and linear distances corresponding to the closest and furthest of our sample galaxies.\\nThese uncertainties represent the precision with which we can determine the AGN and photocenter positions for an ideal galaxy, in an image with noise characteristics representative of our data. The adopted uncertainties are summarized in Table 3, together with the equivalent angular distances and linear distances corresponding to the closest and furthest of our sample galaxies.\\nMinimum detectable displacement: the uncertainties on the AGN and photocenter coordinates determine the minimum detectable displacement. We require a 3σ detection for a displacement to be considered significant. Combining the uncertainties on the photocenter and AGN in quadrature, the error on each component of the displacement is σ = 0.14 pxl, for ACS and WFPC2 data, σ = 0.22 pxl for WFC3, and σ = 0.28 pxl for NICMOS2 data. Therefore, the minimum displacements considered significant are ∆x = ∆y ≈ 0.4 pxl, for ACS and WFPC2, ∆x = ∆y ≈ 0.7 pxl for WFPC3, and ∆x = ∆y ≈ 0.9 pxl for NICMOS2.\\nMinimum detectable displacement: the uncertainties on the AGN and photocenter coordinates determine the minimum detectable displacement. We require a 3σ detection for a displacement to be considered significant. Combining the uncertainties on the photocenter and AGN in quadrature, the error on each component of the displacement is σ = 0.14 pxl, for ACS and WFPC2 data, σ = 0.22 pxl for WFC3, and σ = 0.28 pxl for NICMOS2 data. Therefore, the minimum displacements considered significant are ∆x = ∆y ≈ 0.4 pxl, for ACS and WFPC2, ∆x = ∆y ≈ 0.7 pxl for WFPC3, and ∆x = ∆y ≈ 0.9 pxl for NICMOS2.\"\n },\n {\n \"title\": \"Possible biases\",\n \"content\": \"\"\n },\n {\n \"title\": \"Possible biases\",\n \"content\": \"Asymmetric surface brightness distributions: the galaxies in our sample were selected to be symmetric and regular, based on visual inspection of the images. However, in some cases, the galaxy extends beyond the edges of the frame and/or is not centered in the image frame. To eliminate the possibility of spurious offsets due to truncation of isophotes by the frame edge, the upper limit on the SMA range used in the surface brightness fits is taken to be that of the largest ellipse that fits completely within the frame (Sec. 3.1).\\nAsymmetric surface brightness distributions: the galaxies in our sample were selected to be symmetric and regular, based on visual inspection of the images. However, in some cases, the galaxy extends beyond the edges of the frame and/or is not centered in the image frame. To eliminate the possibility of spurious offsets due to truncation of isophotes by the frame edge, the upper limit on the SMA range used in the surface brightness fits is taken to be that of the largest ellipse that fits completely within the frame (Sec. 3.1).\\nIn addition, subtle intrinsic asymmetries might be present that were not revealed by visual inspection. As discussed by Jedrzejewski (1987), asymmetric surface brightness distributions (such as large-scale lopsidedness in the isophotes) will cause shifts in the isophote centers, which in turn will increase the IQR. Therefore, intrinsic surface brightness asymmetries (or other irregularities) will tend to reduce the level of significance assigned to any measured displacement (Sec. 3.3).\\nIn addition, subtle intrinsic asymmetries might be present that were not revealed by visual inspection. As discussed by Jedrzejewski (1987), asymmetric surface brightness distributions (such as large-scale lopsidedness in the isophotes) will cause shifts in the isophote centers, which in turn will increase the IQR. Therefore, intrinsic surface brightness asymmetries (or other irregularities) will tend to reduce the level of significance assigned to any measured displacement (Sec. 3.3).\\nA similar possibility is that the outer isophotes are distorted by tidal interactions with nearby galaxies. As our analysis is confined to the inner few kiloparsecs, such interactions are unlikely to significantly affect our results. Nevertheless, we have verified that the photocenter position does not change significantly when recomputed excluding isophote centers corresponding to ellipse SMAs greater than ∼ 85, 90 and 95% of that of the largest complete ellipse that fits within the image frame.\\nA similar possibility is that the outer isophotes are distorted by tidal interactions with nearby galaxies. As our analysis is confined to the inner few kiloparsecs, such interactions are unlikely to significantly affect our results. Nevertheless, we have verified that the photocenter position does not change significantly when recomputed excluding isophote centers corresponding to ellipse SMAs greater than ∼ 85, 90 and 95% of that of the largest complete ellipse that fits within the image frame.\\nIsophote twists and PSF effects: core galaxies are often characterized by strong isophote twisting at radii interior to the core radius, r c (Lauer et al. 2005). In addition, the inner isophotes are distorted by the point spread function; TINY TIM simulations show that for the HST instruments used, the PSF can affect the region within a radius r PSF ∼ 15 pixels. As typically r PSF r c , we mitigate both effects by setting r c as the minimum SMA for the ellipse fits.\\nIsophote twists and PSF effects: core galaxies are often characterized by strong isophote twisting at radii interior to the core radius, r c (Lauer et al. 2005). In addition, the inner isophotes are distorted by the point spread function; TINY TIM simulations show that for the HST instruments used, the PSF can affect the region within a radius r PSF ∼ 15 pixels. As typically r PSF r c , we mitigate both effects by setting r c as the minimum SMA for the ellipse fits.\\nLopsided stellar nuclei (LSN): observations of the nuclear regions of nearby galaxies have revealed the presence of double nuclei with separations in the range 1-10 pc (Lauer et al. 1993(Lauer et al. , 1996(Lauer et al. , 2005Thatte et al. 2000;Debattista et al. 2006). One explanation for such configurations is that lopsided stellar orbit distributions tend to persist within the SBH sphere of gravitational influence since orbits do not precess in Keplerian potentials and are not, therefore, axisymmetrized by phase mixing (Peiris & Tremaine 2003). The best studied example is M31, in which the components of the double nucleus are separated by ≈ 0. 5 (≈ 2 pc). Peiris & Tremaine (2003) proposed a model for M31 in which the nucleus consists of an eccentric disk of stars orbiting the SBH, the latter being coincident with the fainter component.\\nLopsided stellar nuclei (LSN): observations of the nuclear regions of nearby galaxies have revealed the presence of double nuclei with separations in the range 1-10 pc (Lauer et al. 1993(Lauer et al. , 1996(Lauer et al. , 2005Thatte et al. 2000;Debattista et al. 2006). One explanation for such configurations is that lopsided stellar orbit distributions tend to persist within the SBH sphere of gravitational influence since orbits do not precess in Keplerian potentials and are not, therefore, axisymmetrized by phase mixing (Peiris & Tremaine 2003). The best studied example is M31, in which the components of the double nucleus are separated by ≈ 0. 5 (≈ 2 pc). Peiris & Tremaine (2003) proposed a model for M31 in which the nucleus consists of an eccentric disk of stars orbiting the SBH, the latter being coincident with the fainter component.\\nThe WFPC2 Planetary Camera (with which most of the images studied here were obtained) resolves spatial separations of ≈ 5 pc only for galaxies within 20 Mpc. It is possible, therefore, that some of the galaxies in our sample harbor unresolved double nuclei. This will not affect the determination of the galaxy photocenter, since the core is excluded from the isophote fitting. However, if the unresolved nuclei are of unequal brightness as in M31, this could result in a systemic error in the AGN position which may approach the PSF FWHM (∼ 1 − 2 pixels). This in turn would result in a spurious displacement.\\nThe WFPC2 Planetary Camera (with which most of the images studied here were obtained) resolves spatial separations of ≈ 5 pc only for galaxies within 20 Mpc. It is possible, therefore, that some of the galaxies in our sample harbor unresolved double nuclei. This will not affect the determination of the galaxy photocenter, since the core is excluded from the isophote fitting. However, if the unresolved nuclei are of unequal brightness as in M31, this could result in a systemic error in the AGN position which may approach the PSF FWHM (∼ 1 − 2 pixels). This in turn would result in a spurious displacement.\"\n },\n {\n \"title\": \"RESULTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"RESULTS\",\n \"content\": \"The results of the photometric analysis are presented in Table 4, where we list the (x, y) pixel co-ordinates of the mean photocenter and the point source, the magnitudes of the corresponding components of the displacement, the direction of each displacement component on the sky and the significance level of the displacement, determined as outlined in Sec. 3.3. Figures 3 and 4 illustrate examples of the results for galaxies in which, respectively, no significant displacement was found (NGC4373, Fig. 3) and in which the measured displacement is considered significant (NGC 4486,Fig. 4). In each figure, the top row of panels shows the original image and the residual images after subtraction of, respectively, the isophotal model generated by ellipse fits and a median filtered image. In the middle row are plotted the surface brightness profile and the x and y pixel coordinates of the isophotal centers, all as functions of the ellipse SMA. The positions of the photocenter and the AGN point source are also plotted. The leftmost panel of the bottom row is a scatter plot showing the distribution of the isophote centers, colour coded according to SMA. The locus of the cumulative mean photocenter position, computed outwards from the inner SMA limit, is also shown as a solid black line. The last two panels show the distributions of the (x, y) isophote center co-ordinates. The co-ordinates of the mean photocenter and the AGN point source are also plotted in all three panels. Similar figures for the remaining galaxies are presented in the Appendix (Sec. C), where the results for individual galaxies are also described in more detail (Sec. B).\\nThe results of the photometric analysis are presented in Table 4, where we list the (x, y) pixel co-ordinates of the mean photocenter and the point source, the magnitudes of the corresponding components of the displacement, the direction of each displacement component on the sky and the significance level of the displacement, determined as outlined in Sec. 3.3. Figures 3 and 4 illustrate examples of the results for galaxies in which, respectively, no significant displacement was found (NGC4373, Fig. 3) and in which the measured displacement is considered significant (NGC 4486,Fig. 4). In each figure, the top row of panels shows the original image and the residual images after subtraction of, respectively, the isophotal model generated by ellipse fits and a median filtered image. In the middle row are plotted the surface brightness profile and the x and y pixel coordinates of the isophotal centers, all as functions of the ellipse SMA. The positions of the photocenter and the AGN point source are also plotted. The leftmost panel of the bottom row is a scatter plot showing the distribution of the isophote centers, colour coded according to SMA. The locus of the cumulative mean photocenter position, computed outwards from the inner SMA limit, is also shown as a solid black line. The last two panels show the distributions of the (x, y) isophote center co-ordinates. The co-ordinates of the mean photocenter and the AGN point source are also plotted in all three panels. Similar figures for the remaining galaxies are presented in the Appendix (Sec. C), where the results for individual galaxies are also described in more detail (Sec. B).\\nIn the case of NGC 4373, the x and y components of the displacement are < 1σ and thus consistent with zero. In the case of the ACS/HRC F814W image of NGC 4486, both the x and y components of the displacement are significant at the > 3σ level and their magnitudes are ≥ 2.4×IQR and accordingly classified as having a \\\"high\\\" significance level (Table 4). For four galaxies, two or more images obtained with different instruments and/or filters were analyzed, providing various combinations of wavelength and spatial resolution. The photocenter displacements in milliarcseconds, relative to the AGN point source, are plotted for these cases in Fig. 5a. There are two galaxies, NGC 4696 and 5419, in which a second point-like brightness peak is present. In these cases, the following discussion refers to the displacement relative to the primary point source (the brightest, which we take to be the AGN), unless otherwise noted.\\nIn the case of NGC 4373, the x and y components of the displacement are < 1σ and thus consistent with zero. In the case of the ACS/HRC F814W image of NGC 4486, both the x and y components of the displacement are significant at the > 3σ level and their magnitudes are ≥ 2.4×IQR and accordingly classified as having a \\\"high\\\" significance level (Table 4). For four galaxies, two or more images obtained with different instruments and/or filters were analyzed, providing various combinations of wavelength and spatial resolution. The photocenter displacements in milliarcseconds, relative to the AGN point source, are plotted for these cases in Fig. 5a. There are two galaxies, NGC 4696 and 5419, in which a second point-like brightness peak is present. In these cases, the following discussion refers to the displacement relative to the primary point source (the brightest, which we take to be the AGN), unless otherwise noted.\\nAGN-photocenter displacements exceeding the minimum detectable value (i.e., significant at 3σ level; Sec. 3.4.1) were measured in at least one direction (x or y), in at least one image, for ten out of the 14 galaxies in the sample: NGC 1399, 4168, 4278, 4486 (M87), 4636, 4696, 5419, 5846, IC 4296, and IC 4931. In the remaining galaxies, NGC 4261, 4373, 4552, and IC 1459, the AGN and photocenter are coincident within the position uncertainties (< 3σ).\\nAGN-photocenter displacements exceeding the minimum detectable value (i.e., significant at 3σ level; Sec. 3.4.1) were measured in at least one direction (x or y), in at least one image, for ten out of the 14 galaxies in the sample: NGC 1399, 4168, 4278, 4486 (M87), 4636, 4696, 5419, 5846, IC 4296, and IC 4931. In the remaining galaxies, NGC 4261, 4373, 4552, and IC 1459, the AGN and photocenter are coincident within the position uncertainties (< 3σ).\\nThe distribution of the isophote center co-ordinates provides an indication of the systematic uncertainty arising from photometric distortions due to effects such as those outlined in Sec. 3.4.2. As discussed in Sec. 3.3, we assign a significance level for each measured displacement based on its magnitude relative to the IQR, which characterizes the width of the isophote center distribution. Three galaxies exhibit displacements (in at least one direction, in one image) ∆x, ∆y ≥ 1.6×IQR, that we classify as having intermediate or high significance. These are NGC 4278, 4486 (M87) and 5846. Three more have displacements classified as having low significance: NGC 1399, 5419 and IC 4296 (∆x, ∆y ≥ 0.8×IQR).\\nThe distribution of the isophote center co-ordinates provides an indication of the systematic uncertainty arising from photometric distortions due to effects such as those outlined in Sec. 3.4.2. As discussed in Sec. 3.3, we assign a significance level for each measured displacement based on its magnitude relative to the IQR, which characterizes the width of the isophote center distribution. Three galaxies exhibit displacements (in at least one direction, in one image) ∆x, ∆y ≥ 1.6×IQR, that we classify as having intermediate or high significance. These are NGC 4278, 4486 (M87) and 5846. Three more have displacements classified as having low significance: NGC 1399, 5419 and IC 4296 (∆x, ∆y ≥ 0.8×IQR).\\nIn the two double nucleus galaxies, the photocenter position is much closer to the primary point source than the secondary. The offsets (relative to the primary) are ≈ 0.5 pxl, but given the large IQR in NGC 4696 the displacement is classified as not significant in this case. In neither galaxy does the photocenter lie between the two \\\"nuclei\\\" (Fig. 5b, 25 and 26).\\nIn the two double nucleus galaxies, the photocenter position is much closer to the primary point source than the secondary. The offsets (relative to the primary) are ≈ 0.5 pxl, but given the large IQR in NGC 4696 the displacement is classified as not significant in this case. In neither galaxy does the photocenter lie between the two \\\"nuclei\\\" (Fig. 5b, 25 and 26).\\nFor the galaxies where multiple images were analyzed, the displacements in mas derived from the different images are consistent to within 2σ for two of the four galaxies (NGC 1399 and 4278).\\nFor the galaxies where multiple images were analyzed, the displacements in mas derived from the different images are consistent to within 2σ for two of the four galaxies (NGC 1399 and 4278).\\nIn the remaining two galaxies (NGC 4486 and IC 4296), the results from one or more images differ significantly from the others (Fig. 5a). In IC 4296, the offset measured from the red (F814W) WFPC2 image is significantly different from those measured from the ACS-F625W and NICMOS2 F160W images, which are consistent to within 2σ. This galaxy has a warped dusty disk oriented approximately E-W, extending ≈ 1 either side of the nucleus but it seems unlikely that this structure is the cause of the discrepancy, the origin of which remains unclear.\\nIn the remaining two galaxies (NGC 4486 and IC 4296), the results from one or more images differ significantly from the others (Fig. 5a). In IC 4296, the offset measured from the red (F814W) WFPC2 image is significantly different from those measured from the ACS-F625W and NICMOS2 F160W images, which are consistent to within 2σ. This galaxy has a warped dusty disk oriented approximately E-W, extending ≈ 1 either side of the nucleus but it seems unlikely that this structure is the cause of the discrepancy, the origin of which remains unclear.\\nSeveral ACS, NICMOS2 and WFPC2 images were analyzed for NGC 4486 (M87). For the ACS red optical image (F814W) we recover the B10 result, which indicates that the photocenter is displaced by ≈ 100 mas to the north-west of the nucleus. A smaller but less significant offset in approximately the same direction is measured from the WFPC2 F814W image. However, the photocenters derived from the NICMOS2 F110W, F160M and F222W image are consistent with the AGN position. As discussed in detail in Appendix B, the origin of these differences is unclear. Neither dust, nor the prominent optical jet (also visible in the NIR) appear to provide satisfactory explanations. The central region of the galaxy where the isophote fitting was performed (1 ≤ r ≤ 3 ) appears to be free of large scale dust features. The masking procedure should prevent the jet from distorting the isophote fits but in any case, even when the fits were repeated with different levels of masking and even with no mask at all, no significant differences in the mean photocenter position were found. As there is no compelling reason to favor or discard the results from any given image, we compute a weighted average displacement.\\nSeveral ACS, NICMOS2 and WFPC2 images were analyzed for NGC 4486 (M87). For the ACS red optical image (F814W) we recover the B10 result, which indicates that the photocenter is displaced by ≈ 100 mas to the north-west of the nucleus. A smaller but less significant offset in approximately the same direction is measured from the WFPC2 F814W image. However, the photocenters derived from the NICMOS2 F110W, F160M and F222W image are consistent with the AGN position. As discussed in detail in Appendix B, the origin of these differences is unclear. Neither dust, nor the prominent optical jet (also visible in the NIR) appear to provide satisfactory explanations. The central region of the galaxy where the isophote fitting was performed (1 ≤ r ≤ 3 ) appears to be free of large scale dust features. The masking procedure should prevent the jet from distorting the isophote fits but in any case, even when the fits were repeated with different levels of masking and even with no mask at all, no significant differences in the mean photocenter position were found. As there is no compelling reason to favor or discard the results from any given image, we compute a weighted average displacement.\\nWhether classified as significant, or not, the measured angular photocenter displacements are always small, typically a few×10 mas and almost without exception 100 mas. The only cases that exceed 100 mas are the displacements relative to the secondary point sources in the two double nucleus galaxies. Disregarding these, the projected linear displacements are all 10 pc. Therefore, for every galaxy analyzed, the measured displacement is a small fraction of the galaxy core radius; as a fraction of r c , the range in displacement magnitude is approximately 1-10%.\\nWhether classified as significant, or not, the measured angular photocenter displacements are always small, typically a few×10 mas and almost without exception 100 mas. The only cases that exceed 100 mas are the displacements relative to the secondary point sources in the two double nucleus galaxies. Disregarding these, the projected linear displacements are all 10 pc. Therefore, for every galaxy analyzed, the measured displacement is a small fraction of the galaxy core radius; as a fraction of r c , the range in displacement magnitude is approximately 1-10%.\\nIn our previous analysis of NGC 4486 (B10), we found that the galaxy photocenter (in ACS and WFPC images) is displaced in approximately the same direction as the jet, implying that the SBH is displaced in the counter-jet direction. It is therefore of interest to compare the direction of the measured photocenter displacements with the radio source axis for this sample. In Table 5 we give the position angle (PA) of the radio source, for those galaxies in which jets or jet-like extended structures have been observed, along with the PA of the measured displacement (as derived from the x, y components). Five galaxies are associated with relatively powerful, extended (∼ 10 kpc) radio sources: NGC 1399, 4261, 4486, 4696 and IC 4296 6 . Of these, NGC 4486 (M 87; 3C 274), NGC 4261 (3C 270) and IC 4296 (PKS 1333-33) are Fanaroff-Riley Type I (FRI) radio sources, characterized by prominent twin jets ending in kiloparsec-scale lobes (Fanaroff & Riley 1974). All three also have parsec-scale jets approximately aligned with the kpc-scale structure.\\nIn our previous analysis of NGC 4486 (B10), we found that the galaxy photocenter (in ACS and WFPC images) is displaced in approximately the same direction as the jet, implying that the SBH is displaced in the counter-jet direction. It is therefore of interest to compare the direction of the measured photocenter displacements with the radio source axis for this sample. In Table 5 we give the position angle (PA) of the radio source, for those galaxies in which jets or jet-like extended structures have been observed, along with the PA of the measured displacement (as derived from the x, y components). Five galaxies are associated with relatively powerful, extended (∼ 10 kpc) radio sources: NGC 1399, 4261, 4486, 4696 and IC 4296 6 . Of these, NGC 4486 (M 87; 3C 274), NGC 4261 (3C 270) and IC 4296 (PKS 1333-33) are Fanaroff-Riley Type I (FRI) radio sources, characterized by prominent twin jets ending in kiloparsec-scale lobes (Fanaroff & Riley 1974). All three also have parsec-scale jets approximately aligned with the kpc-scale structure.\\nIn NGC 4486, the displacement PA derived from our reanalysis of the ACS F814W image agrees with the B10 result. The direction of the weighted mean displacement, derived from displacements obtained from all the images, is also closely aligned with the jet direction. We find a similar result for IC 4296. As already noted, the results from the ACS and NICMOS2 images are consistent and indicate a displacement to the NW. If the discrepant WFPC2 result is disregarded, the weighted mean of the ACS and NICMOS2 displacements yields a PA only ∼ 4 • different from that of the large-scale radio axis, with the photocenter displaced on the side of the brighter north-western jet.\\nIn NGC 4486, the displacement PA derived from our reanalysis of the ACS F814W image agrees with the B10 result. The direction of the weighted mean displacement, derived from displacements obtained from all the images, is also closely aligned with the jet direction. We find a similar result for IC 4296. As already noted, the results from the ACS and NICMOS2 images are consistent and indicate a displacement to the NW. If the discrepant WFPC2 result is disregarded, the weighted mean of the ACS and NICMOS2 displacements yields a PA only ∼ 4 • different from that of the large-scale radio axis, with the photocenter displaced on the side of the brighter north-western jet.\\nThe displacement measured in NGC 4261 is not considered to be significant (< 3σ). Nevertheless, the derived PA is consistent (albeit within the large uncertainty) with that of the radio axis, and again, the offset is in the direction of the brighter western jet.\\nThe displacement measured in NGC 4261 is not considered to be significant (< 3σ). Nevertheless, the derived PA is consistent (albeit within the large uncertainty) with that of the radio axis, and again, the offset is in the direction of the brighter western jet.\\nNGC 1399 also has an FRI-like radio source morphology with a well-defined axis in PA≈ −10 • . The measured displacement (classified as low significance) has a consistent PA with a value of −17 ± 16 • . NGC 4696, is one of the galaxies with two optical point sources. Its large scale radio source does not exhibit well-defined jets, but is elongated approximately E-W over ∼ 10 kpc, with the ends of both \\\"arms\\\" bending south. At parsec scales there is a compact core with a one-sided jet emerging to the SW in PA∼ −150 • . However, it is not clear which of the two optical nuclei hosts the core radio source (Taylor et al. 2006), complicating the comparison with the photocenter displacement. If the secondary point source is identified as the AGN, then the photocenter, which is close to the primary point source, is displaced approximately in the counter-jet direction. On the other hand, if the primary point source is indeed the AGN, the displacement (which, in any case, is classified as having \\\"null\\\" significance) would be almost perpendicular to the jet. Three more galaxies have relatively weak, small radio sources that have jet-like features or elongations on subkiloparsec scales. The measured photocenter displacements are not aligned with the radio source axis in NGC 4552 or NGC 4636. NGC 4278 has a compact (∼ 3 pc) source consisting of a core with jet-like features emerging along a SE-NW axis on either side. These features gradually bend to the east and west, respectively, becoming fainter and more diffuse. The weighted mean photocenter position is displaced approximately in the initial direction of the SE jet. According to Giroletti et al. (2005)'s analysis of this source, the jet axis is closely aligned with our line of sight, with the SE component being the oppositely directed counter-jet.\\nNGC 1399 also has an FRI-like radio source morphology with a well-defined axis in PA≈ −10 • . The measured displacement (classified as low significance) has a consistent PA with a value of −17 ± 16 • . NGC 4696, is one of the galaxies with two optical point sources. Its large scale radio source does not exhibit well-defined jets, but is elongated approximately E-W over ∼ 10 kpc, with the ends of both \\\"arms\\\" bending south. At parsec scales there is a compact core with a one-sided jet emerging to the SW in PA∼ −150 • . However, it is not clear which of the two optical nuclei hosts the core radio source (Taylor et al. 2006), complicating the comparison with the photocenter displacement. If the secondary point source is identified as the AGN, then the photocenter, which is close to the primary point source, is displaced approximately in the counter-jet direction. On the other hand, if the primary point source is indeed the AGN, the displacement (which, in any case, is classified as having \\\"null\\\" significance) would be almost perpendicular to the jet. Three more galaxies have relatively weak, small radio sources that have jet-like features or elongations on subkiloparsec scales. The measured photocenter displacements are not aligned with the radio source axis in NGC 4552 or NGC 4636. NGC 4278 has a compact (∼ 3 pc) source consisting of a core with jet-like features emerging along a SE-NW axis on either side. These features gradually bend to the east and west, respectively, becoming fainter and more diffuse. The weighted mean photocenter position is displaced approximately in the initial direction of the SE jet. According to Giroletti et al. (2005)'s analysis of this source, the jet axis is closely aligned with our line of sight, with the SE component being the oppositely directed counter-jet.\"\n },\n {\n \"title\": \"DISCUSSION\",\n \"content\": \"\"\n },\n {\n \"title\": \"DISCUSSION\",\n \"content\": \"In this section we will examine our key assumption that the SBH position is marked by the nuclear point source. We will then discuss the possible origins of the observed offsets. Most of the displacement mechanisms considered below have been previously discussed in B10. Nevertheless, we summarize them here in order to provide context for our new results.\\nIn this section we will examine our key assumption that the SBH position is marked by the nuclear point source. We will then discuss the possible origins of the observed offsets. Most of the displacement mechanisms considered below have been previously discussed in B10. Nevertheless, we summarize them here in order to provide context for our new results.\\n5.1. The AGN nature of the nuclear point sources Our sample of 14 core elliptical galaxies was selected from the 26 (out of 29) studied by BC06 that were identified as containing AGN based on the presence of i) an unresolved optical or X-ray source; ii) an \\\"AGN-like\\\" optical spectrum, or iii) radio jets. Most of our sample of 14 exhibit two or more AGN signatures in addition to the presence of a radio source and optical point source nucleus. Optical or UV variability has been detected in HST observations of NGC 4486 (Perlman et al. 2003 . Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers. (NGC 4278,4636 and NGC 4168), or emission line ratios indicative of AGN photoionization (NGC 4261,4486,4552,5846) have also been detected in several galaxies (Ho et al. 1997). In addition, the radio sources in most objects feature compact ( 1 pc) cores (NGC 1399, 4168, 4373, 4552, 5419, IC 1459) and/or parsec or kpc-scale jets (NGC 4261,4278,4486,4696,IC 4296; see Appendix B for references). Only IC 4931, which by comparison has been relatively little studied, lacks supporting evidence indicating the presence of an AGN in the form of X-ray, radio or line emission.\\n5.1. The AGN nature of the nuclear point sources Our sample of 14 core elliptical galaxies was selected from the 26 (out of 29) studied by BC06 that were identified as containing AGN based on the presence of i) an unresolved optical or X-ray source; ii) an \\\"AGN-like\\\" optical spectrum, or iii) radio jets. Most of our sample of 14 exhibit two or more AGN signatures in addition to the presence of a radio source and optical point source nucleus. Optical or UV variability has been detected in HST observations of NGC 4486 (Perlman et al. 2003 . Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers. (NGC 4278,4636 and NGC 4168), or emission line ratios indicative of AGN photoionization (NGC 4261,4486,4552,5846) have also been detected in several galaxies (Ho et al. 1997). In addition, the radio sources in most objects feature compact ( 1 pc) cores (NGC 1399, 4168, 4373, 4552, 5419, IC 1459) and/or parsec or kpc-scale jets (NGC 4261,4278,4486,4696,IC 4296; see Appendix B for references). Only IC 4931, which by comparison has been relatively little studied, lacks supporting evidence indicating the presence of an AGN in the form of X-ray, radio or line emission.\\nThere is, therefore, plenty of evidence that these galaxies (with the possible exception of IC 4931) host AGNs. However, it does not necessarily follow (except in cases where optical/UV variability has been observed) that the optical or NIR point sources are themselves manifestations of the AGN, rather than, for example, nuclear star clusters. Low luminosity radio loud AGNs are thought to be powered by radiatively inefficient accretion flows (see, for example, Ho 2008; Balmaverde et al. 2008, and references therein), with most of the accretion power channeled into the kinetic energy of the radio jets. Detailed studies of NGC4486 (Di Matteo et al. 2003), IC 1459 (Fabbiano et al. 2003) and IC 4296 (Pellegrini et al. 2003) show that this is the case for at least three galaxies of our sample.\\nThere is, therefore, plenty of evidence that these galaxies (with the possible exception of IC 4931) host AGNs. However, it does not necessarily follow (except in cases where optical/UV variability has been observed) that the optical or NIR point sources are themselves manifestations of the AGN, rather than, for example, nuclear star clusters. Low luminosity radio loud AGNs are thought to be powered by radiatively inefficient accretion flows (see, for example, Ho 2008; Balmaverde et al. 2008, and references therein), with most of the accretion power channeled into the kinetic energy of the radio jets. Detailed studies of NGC4486 (Di Matteo et al. 2003), IC 1459 (Fabbiano et al. 2003) and IC 4296 (Pellegrini et al. 2003) show that this is the case for at least three galaxies of our sample.\\nIn low luminosity radio galaxies, the optical, near infrared and X-ray luminosities of the nuclei correlate tightly with the core radio luminosity, implying a common origin in non-thermal emission from the jet (Chiaberge et al. 1999;Capetti & Balmaverde 2005;Baldi et al. 2010). This is supported by polarization measurements: Capetti et al. (2007) found that the optical nuclei of the nine nearest FR I radio galaxies in the 3C catalogue, including two galaxies from our sample, NGC 4486 (M87, 3C 274) and NGC 4261 (3C 270), have high polarizations (2 − 11%), which they attribute to synchrotron emission. The optical or UV variability of the point sources in NGC 4486 and three other galaxies, noted above, is also consistent with jet synchrotron emission. More generally, BC06 found that their sample of core elliptical galaxies, from which this sample is drawn, also exhibit optical-radio and X-ray-radio correlations and indeed form a continuous distribution with the radio galaxies, extending these correlations to lower luminosities. Therefore, there is direct evidence, in the form of optical polarization and/or optical/UV variability that the nuclear sources in five of our sample galaxies are produced by synchrotron emission associated with the radio core source, presumably the base of the jet. It is reasonable to assume, given the correlations found by BC06, that the same is true for the rest of the sample.\\nIn low luminosity radio galaxies, the optical, near infrared and X-ray luminosities of the nuclei correlate tightly with the core radio luminosity, implying a common origin in non-thermal emission from the jet (Chiaberge et al. 1999;Capetti & Balmaverde 2005;Baldi et al. 2010). This is supported by polarization measurements: Capetti et al. (2007) found that the optical nuclei of the nine nearest FR I radio galaxies in the 3C catalogue, including two galaxies from our sample, NGC 4486 (M87, 3C 274) and NGC 4261 (3C 270), have high polarizations (2 − 11%), which they attribute to synchrotron emission. The optical or UV variability of the point sources in NGC 4486 and three other galaxies, noted above, is also consistent with jet synchrotron emission. More generally, BC06 found that their sample of core elliptical galaxies, from which this sample is drawn, also exhibit optical-radio and X-ray-radio correlations and indeed form a continuous distribution with the radio galaxies, extending these correlations to lower luminosities. Therefore, there is direct evidence, in the form of optical polarization and/or optical/UV variability that the nuclear sources in five of our sample galaxies are produced by synchrotron emission associated with the radio core source, presumably the base of the jet. It is reasonable to assume, given the correlations found by BC06, that the same is true for the rest of the sample.\\nThe base of the jet is, in turn, very close to the SBH. Arguably the best studied jet is that of M87. Using multifrequency observations made with the Very Long Baseline Array, Hada et al. (2011) were able to show that the radio core at 43 GHz is located within 14-23 Schwarzchild radii of the SBH and accretion flow. In the optical, the lower optical depth should move the peak of the emission even closer to the SBH.\\nThe base of the jet is, in turn, very close to the SBH. Arguably the best studied jet is that of M87. Using multifrequency observations made with the Very Long Baseline Array, Hada et al. (2011) were able to show that the radio core at 43 GHz is located within 14-23 Schwarzchild radii of the SBH and accretion flow. In the optical, the lower optical depth should move the peak of the emission even closer to the SBH.\\nIn summary, we contend that there are good reasons to believe that the optical or near-infrared point sources in our galaxies reveal the position of the SBH.\\nIn summary, we contend that there are good reasons to believe that the optical or near-infrared point sources in our galaxies reveal the position of the SBH.\"\n },\n {\n \"title\": \"Recoiling SBHs\",\n \"content\": \"\"\n },\n {\n \"title\": \"Recoiling SBHs\",\n \"content\": \"Here we consider if the measured SBH displacements are consistent with residual oscillations due to gravitational recoils generated by coalescence of an SBH binary. GM08 used N -body simulations to study the post recoil dynamical evolution of an SBH in representative core elliptical galaxy potentials. They found that when the recoil velocity is sufficient to eject the SBH from the core (v kick in the range ∼ 40 − 90% of the escape velocity, v esc ), the subsequent motion is characterized by three phases. The initial large-amplitide (>r c ) oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction (phase I). When the amplitude becomes comparable with the core radius, r c , the SBH and the stellar core undergo long-lived oscillations about their center of mass, which persist for ∼ 1 Gyr (phase II). Finally, the SBH reaches thermal equilibrium with the stars, experiencing low-amplitude Brownian motion (phase III).\\nHere we consider if the measured SBH displacements are consistent with residual oscillations due to gravitational recoils generated by coalescence of an SBH binary. GM08 used N -body simulations to study the post recoil dynamical evolution of an SBH in representative core elliptical galaxy potentials. They found that when the recoil velocity is sufficient to eject the SBH from the core (v kick in the range ∼ 40 − 90% of the escape velocity, v esc ), the subsequent motion is characterized by three phases. The initial large-amplitide (>r c ) oscillations are damped relatively quickly (∼ 10 7 yr) by dynamical friction (phase I). When the amplitude becomes comparable with the core radius, r c , the SBH and the stellar core undergo long-lived oscillations about their center of mass, which persist for ∼ 1 Gyr (phase II). Finally, the SBH reaches thermal equilibrium with the stars, experiencing low-amplitude Brownian motion (phase III).\\nAs our measured displacements are 0.1 r c , it is highly unlikely that they represent phase I oscillations. This would require either fortuitous timing (the SBH would have to be caught whilst passing through or close to the equilibrium position) and/or orientation (the oscillation direction would have to be closely aligned with the line of sight) for each galaxy. We also rule out phase III Brownian motion due to interactions with individual stars, which will produce negligibly small amplitudes in real galaxies (GM08). The possibility of Brownian oscillation due to interactions with massive perturbers is discussed in Section 5.3. Here, we focus on phase II which, as already noted, is characterized by long-lived damped oscillations at amplitudes < r c .\\nAs our measured displacements are 0.1 r c , it is highly unlikely that they represent phase I oscillations. This would require either fortuitous timing (the SBH would have to be caught whilst passing through or close to the equilibrium position) and/or orientation (the oscillation direction would have to be closely aligned with the line of sight) for each galaxy. We also rule out phase III Brownian motion due to interactions with individual stars, which will produce negligibly small amplitudes in real galaxies (GM08). The possibility of Brownian oscillation due to interactions with massive perturbers is discussed in Section 5.3. Here, we focus on phase II which, as already noted, is characterized by long-lived damped oscillations at amplitudes < r c .\\nIn this phase, the characteristic damping time is given by GM08 as:\\nIn this phase, the characteristic damping time is given by GM08 as:\\nwhere the approximation on the right-hand side makes use of the M • -σ relation (Ferrarese & Ford 2005), σ being the 1D stellar velocity dispersion, with r 500 ≡ r c /(500 pc) and σ 300 ≡ σ/300 km s −1 . For our galaxies, eq. 2 yields values of τ in the range 0.02 − 4.85 Gyr, with an average of 1.2 Gyr (Table 8). The rms amplitude of the SBH motion with respect to the galaxy center is expected to evolve as:\\nwhere the approximation on the right-hand side makes use of the M • -σ relation (Ferrarese & Ford 2005), σ being the 1D stellar velocity dispersion, with r 500 ≡ r c /(500 pc) and σ 300 ≡ σ/300 km s −1 . For our galaxies, eq. 2 yields values of τ in the range 0.02 − 4.85 Gyr, with an average of 1.2 Gyr (Table 8). The rms amplitude of the SBH motion with respect to the galaxy center is expected to evolve as:\\nwhere t c is the time at which the oscillation amplitude has decayed to a scale comparable to the core radius r c (B10). Simulations indicate that the galaxy merger rate is a strong function of redshift and galaxy mass (e.g., Fakhouri et al. 2010, H10). The merger rate for brightest cluster galaxies has recently been determined observationally by Lidman et al. (2013), who find ≈ 0.4 mergers per Gyr at z ∼ 1, implying a mean time between mergers of ≈ 2.5 Gyr 7 . If we take this as representative of our sample, equations 2 and 3 together suggest that a \\\"typical\\\" rms displacement (ignoring projection effects) is ∼ 0.2 r c . This suggests that larger displacements than were actually observed might be expected if due to post gravitational recoil oscillations.\\nwhere t c is the time at which the oscillation amplitude has decayed to a scale comparable to the core radius r c (B10). Simulations indicate that the galaxy merger rate is a strong function of redshift and galaxy mass (e.g., Fakhouri et al. 2010, H10). The merger rate for brightest cluster galaxies has recently been determined observationally by Lidman et al. (2013), who find ≈ 0.4 mergers per Gyr at z ∼ 1, implying a mean time between mergers of ≈ 2.5 Gyr 7 . If we take this as representative of our sample, equations 2 and 3 together suggest that a \\\"typical\\\" rms displacement (ignoring projection effects) is ∼ 0.2 r c . This suggests that larger displacements than were actually observed might be expected if due to post gravitational recoil oscillations.\\nHowever, equation 3 describes the oscillation amplitude rather than the instantaneous displacement and moreover the above estimate does not account for projection effects, or the range in damping timescales characterizing our sample galaxies, or for variations in the merger rate. In order to investigate in more detail the likelihood of obtaining the observed displacements in our sample, if they result from post-recoil oscillations, we have constructed a simple Monte-Carlo simulation based on the GM08 N -body simulations. Details of the method are given in Appendix A. However, we start from the assumption that, after any kick large enough to move the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy is given by:\\nHowever, equation 3 describes the oscillation amplitude rather than the instantaneous displacement and moreover the above estimate does not account for projection effects, or the range in damping timescales characterizing our sample galaxies, or for variations in the merger rate. In order to investigate in more detail the likelihood of obtaining the observed displacements in our sample, if they result from post-recoil oscillations, we have constructed a simple Monte-Carlo simulation based on the GM08 N -body simulations. Details of the method are given in Appendix A. However, we start from the assumption that, after any kick large enough to move the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy is given by:\\nwhere ∆t is the elapsed time since the kick, i.e., the time since the last merger, τ is the damping time given by eq.2 and ω c is the SBH oscillation frequency calculated from eq. A2. We set R 0 = r c , since phase II begins when the oscillation amplitude is roughly equal to the core radius. We then suppose that each galaxy is observed at a random time, ∆t, since its last merger, where the probability that the merger occurred at time ∆t follows an exponential distribution characterized by a mean timebetween-mergers, t m . This distribution is sampled to generate corresponding values of R i ≡ R(t i ), which are then projected onto the sky plane assuming that the recoil kicks have random directions. The distribution of projected displacement for each galaxy is then used to compute the probability, p i , of observing a displacement larger than that actually measured, given a kick of sufficient magnitude. These probabilities were computed for two values of the mean time between mergers: t m = 5.0 Gyr and t m = 0.4 Gyr, which were derived from the galaxy merger rate models of H10 for redshifts z < 1, mass ratios q > 0.1 and galaxy masses log(M gal /M ) > 11 and log(M gal /M ) > 12, respectively (see Appendix A for details). The two values of t m are representative of the mass range covered by our sample, and illustrate the effect on the displacement probabilities of the strong galaxy mass dependence of the merger rate.\\nwhere ∆t is the elapsed time since the kick, i.e., the time since the last merger, τ is the damping time given by eq.2 and ω c is the SBH oscillation frequency calculated from eq. A2. We set R 0 = r c , since phase II begins when the oscillation amplitude is roughly equal to the core radius. We then suppose that each galaxy is observed at a random time, ∆t, since its last merger, where the probability that the merger occurred at time ∆t follows an exponential distribution characterized by a mean timebetween-mergers, t m . This distribution is sampled to generate corresponding values of R i ≡ R(t i ), which are then projected onto the sky plane assuming that the recoil kicks have random directions. The distribution of projected displacement for each galaxy is then used to compute the probability, p i , of observing a displacement larger than that actually measured, given a kick of sufficient magnitude. These probabilities were computed for two values of the mean time between mergers: t m = 5.0 Gyr and t m = 0.4 Gyr, which were derived from the galaxy merger rate models of H10 for redshifts z < 1, mass ratios q > 0.1 and galaxy masses log(M gal /M ) > 11 and log(M gal /M ) > 12, respectively (see Appendix A for details). The two values of t m are representative of the mass range covered by our sample, and illustrate the effect on the displacement probabilities of the strong galaxy mass dependence of the merger rate.\\nThe probabilities, p i , are listed for each galaxy in Table 6 for both values of t m . The probabilities P (d > x) of observing a projected displacement d exceeding distance\\nThe probabilities, p i , are listed for each galaxy in Table 6 for both values of t m . The probabilities P (d > x) of observing a projected displacement d exceeding distance\\nx are plotted as functions of x in Figures 7a and 7b, for each galaxy, along with the observed displacement.\\nx are plotted as functions of x in Figures 7a and 7b, for each galaxy, along with the observed displacement.\\nAs would be expected, the smaller value of t m results in larger values of p i ; clearly if t m τ , there is a greater chance of observing a large displacement than when t m >> τ . For t m = 0.4 Gyr, p i > 0.5 for all but two galaxies (NGC 4278 and NGC 4552) and p i ≥ 0.95 for all 6 galaxies that have masses ≥ 10 12 M (for which assumed fraction of mass accreted by a BCG during a merger. Therefore, this merger rate relies on model-dependent assumptions about the dark-matter haloes surrounding the galaxies.\\nAs would be expected, the smaller value of t m results in larger values of p i ; clearly if t m τ , there is a greater chance of observing a large displacement than when t m >> τ . For t m = 0.4 Gyr, p i > 0.5 for all but two galaxies (NGC 4278 and NGC 4552) and p i ≥ 0.95 for all 6 galaxies that have masses ≥ 10 12 M (for which assumed fraction of mass accreted by a BCG during a merger. Therefore, this merger rate relies on model-dependent assumptions about the dark-matter haloes surrounding the galaxies.\\nthis value of t m is presumably most relevant). On the other hand, for t m = 5.0 Gyr, all but four galaxies have p i < 0.5, including five of the six having displacements considered significant at the level of > 0.8 IQR (the exception is M87, for which p i = 0.75).\\nthis value of t m is presumably most relevant). On the other hand, for t m = 5.0 Gyr, all but four galaxies have p i < 0.5, including five of the six having displacements considered significant at the level of > 0.8 IQR (the exception is M87, for which p i = 0.75).\\nConsidering the sample as a whole, the probability of not observing a displacement larger than those actually measured in any of the galaxies in the sample is simply P = i (1 − p i ). For t m = 0.4 Gyr, this is negligibly small, P ∼ 10 −17 . Even for t m = 5.0 Gyr, P ≈ 3 × 10 −4 , indicating that it is statistically unlikely that larger displacements were not observed, given the occurrence of a recoil kick sufficient to trigger phase II oscillations.\\nConsidering the sample as a whole, the probability of not observing a displacement larger than those actually measured in any of the galaxies in the sample is simply P = i (1 − p i ). For t m = 0.4 Gyr, this is negligibly small, P ∼ 10 −17 . Even for t m = 5.0 Gyr, P ≈ 3 × 10 −4 , indicating that it is statistically unlikely that larger displacements were not observed, given the occurrence of a recoil kick sufficient to trigger phase II oscillations.\\nHowever, it is unlikely that all the galaxies in the sample experienced a recoil kick big enough to move the SBH beyond the core radius. Assuming that the potential experienced by the SBH can be approximated as a harmonic potential (φ = 1/2 ω 2 x 2 ), we estimated the kick velocity required to displace the SBH out to the core radius for each galaxy, obtaining values in the range 159 v kick 325 km s −1 , with an average of v kick ≈ 240 km s −1 (the values for each galaxy are listed in Table 8).\\nHowever, it is unlikely that all the galaxies in the sample experienced a recoil kick big enough to move the SBH beyond the core radius. Assuming that the potential experienced by the SBH can be approximated as a harmonic potential (φ = 1/2 ω 2 x 2 ), we estimated the kick velocity required to displace the SBH out to the core radius for each galaxy, obtaining values in the range 159 v kick 325 km s −1 , with an average of v kick ≈ 240 km s −1 (the values for each galaxy are listed in Table 8).\\nL12 have studied kick velocity distributions for \\\"hangup\\\" kicks, which arise in binary configurations where the SBH spins are partially aligned with the orbital angular momentum by pre-merger accretion. For spin magnitude and orientation distributions derived from accretion simulations and a mass ratio distribution based on galaxy merger studies (including both minor and major mergers), they generate probability distributions for the magnitude of the recoil velocity for two extreme cases of \\\"hot\\\" and \\\"cold\\\" gas disks (i.e., corresponding to adiabatic indices γ = 5/3 and 7/5, respectively). The former regime is likely to be most applicable to the ellipticals in our sample and in this case, probabilities range from p k ≈ 66% for kicks > 100 km s −1 to ≈ 33% for kicks > 300 km s −1 . Our Monte-Carlo simulation assumes that the coalescence event produces a recoil large enough to produce phase II oscillations (i.e., the values given in Table 8). As the probabilities p i are conditional on the occurrence of a kick of sufficient magnitude, the probability that both events will occur is p k × p i . There is thus a smaller probability of observing a displacement larger than actually measured for each galaxy. Using the kick probabilities given by L12, we find that the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 8 × 10 −4 for t m = 0.4 Gyr and P ≈ 5 × 10 −2 for t m = 5.0 Gyr.\\nL12 have studied kick velocity distributions for \\\"hangup\\\" kicks, which arise in binary configurations where the SBH spins are partially aligned with the orbital angular momentum by pre-merger accretion. For spin magnitude and orientation distributions derived from accretion simulations and a mass ratio distribution based on galaxy merger studies (including both minor and major mergers), they generate probability distributions for the magnitude of the recoil velocity for two extreme cases of \\\"hot\\\" and \\\"cold\\\" gas disks (i.e., corresponding to adiabatic indices γ = 5/3 and 7/5, respectively). The former regime is likely to be most applicable to the ellipticals in our sample and in this case, probabilities range from p k ≈ 66% for kicks > 100 km s −1 to ≈ 33% for kicks > 300 km s −1 . Our Monte-Carlo simulation assumes that the coalescence event produces a recoil large enough to produce phase II oscillations (i.e., the values given in Table 8). As the probabilities p i are conditional on the occurrence of a kick of sufficient magnitude, the probability that both events will occur is p k × p i . There is thus a smaller probability of observing a displacement larger than actually measured for each galaxy. Using the kick probabilities given by L12, we find that the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 8 × 10 −4 for t m = 0.4 Gyr and P ≈ 5 × 10 −2 for t m = 5.0 Gyr.\\nAs the merger rate increases with redshift, we may have underestimated the time-between-mergers by integrating over the redshift range z = 1 → 0. However, the higher value of t m used in the simulation implies a merger rate (∼ 0.2 mergers Gyr −1 ) consistent with that given by the models of H10 for log(M gal /M ) > 11 galaxies at z = 0 (see their Fig. 3). The rate for major mergers is a factor 2-3 lower than the rate for major and minor mergers combined, which we used to estimate t m . However, although the magnitude of the recoil velocity is a function of SBH mass ratio (with the largest recoils occurring for mass ratios characteristic of major mergers), even small (q > 0.1) mass ratios are capable of producing kicks ∼ 200 km s −1 , large enough to trigger phase II oscillations in our galaxies (Lousto et al. 2010). Mi- Note. -(1) Optical name;\\nAs the merger rate increases with redshift, we may have underestimated the time-between-mergers by integrating over the redshift range z = 1 → 0. However, the higher value of t m used in the simulation implies a merger rate (∼ 0.2 mergers Gyr −1 ) consistent with that given by the models of H10 for log(M gal /M ) > 11 galaxies at z = 0 (see their Fig. 3). The rate for major mergers is a factor 2-3 lower than the rate for major and minor mergers combined, which we used to estimate t m . However, although the magnitude of the recoil velocity is a function of SBH mass ratio (with the largest recoils occurring for mass ratios characteristic of major mergers), even small (q > 0.1) mass ratios are capable of producing kicks ∼ 200 km s −1 , large enough to trigger phase II oscillations in our galaxies (Lousto et al. 2010). Mi- Note. -(1) Optical name;\\n(2) instrument/camera/filter on HST;\\n(2) instrument/camera/filter on HST;\\n(3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) nuclear point source position in pixels; (7 -8) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (9) offset in pixel normalized by the corresponding IQR; (10) offset in pc; (11) direction of the offset; (12) type of displacement as defined in §3.3. : for further details see Table 1 in B10. nor mergers were, in any case, included in computing the kick velocity distribution presented by L12.\\n(3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) nuclear point source position in pixels; (7 -8) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (9) offset in pixel normalized by the corresponding IQR; (10) offset in pc; (11) direction of the offset; (12) type of displacement as defined in §3.3. : for further details see Table 1 in B10. nor mergers were, in any case, included in computing the kick velocity distribution presented by L12.\\nThus, even allowing for the distribution of kick velocities, the Monte-Carlo simulation suggests that it is highly likely that displacements larger than those actually measured would have been observed in this sample if each galaxy has experienced at least one merger leading to an SBH-binary coalescence and gravitational recoil within the last few Gyr. In fact, our simulations suggest that t m 30 − 40 Gyr, corresponding to a merger rate ∼ 0.03 mergers Gyr −1 , is required for a > 50% chance of not observing larger displacements in the sample.\\nThus, even allowing for the distribution of kick velocities, the Monte-Carlo simulation suggests that it is highly likely that displacements larger than those actually measured would have been observed in this sample if each galaxy has experienced at least one merger leading to an SBH-binary coalescence and gravitational recoil within the last few Gyr. In fact, our simulations suggest that t m 30 − 40 Gyr, corresponding to a merger rate ∼ 0.03 mergers Gyr −1 , is required for a > 50% chance of not observing larger displacements in the sample.\\nIf, indeed, the formation of a binary SBH is an inevitable outcome of galaxy mergers, explanations must be sought in the evolution of the binary or that of the recoiling SBH. Possibilities include (1) the binary stalls before reaching the radius at which gravitational wave emission drives rapid coalescence; (2) if coalescence occurs, the mass ratio or spin-orbit properties of the binary are such as to typically preclude recoil velocities large enough to displace the SBH to the core radius ( 240 km s −1 ); or (3) the damping time for the recoil oscillations may be shorter than predicted by the pure N -body simulations.\\nIf, indeed, the formation of a binary SBH is an inevitable outcome of galaxy mergers, explanations must be sought in the evolution of the binary or that of the recoiling SBH. Possibilities include (1) the binary stalls before reaching the radius at which gravitational wave emission drives rapid coalescence; (2) if coalescence occurs, the mass ratio or spin-orbit properties of the binary are such as to typically preclude recoil velocities large enough to displace the SBH to the core radius ( 240 km s −1 ); or (3) the damping time for the recoil oscillations may be shorter than predicted by the pure N -body simulations.\\nAlternatively, it is possible that galaxy mergers are much more infrequent than inferred from studies based on the current ΛCDM framework (see Kroupa 2014, and references therein for a detailed discussion).\\nAlternatively, it is possible that galaxy mergers are much more infrequent than inferred from studies based on the current ΛCDM framework (see Kroupa 2014, and references therein for a detailed discussion).\\nAlthough our results imply that SBH displacements due to gravitational recoil are much less common than might be inferred from theoretical considerations, this does not preclude the possibility that observed displacements in individual galaxies are due to recoil oscillations. Indeed, the p i values listed in Table 6 indicate that for t m = 5.0 Gyr, the chance of not observing a larger displacement, given a sufficiently large kick, is 30% for all objects having a displacement considered significant. Even for t m = 0.4 Gyr, there are still three of these galaxies for which 1 − p i ≥ 0.25. The likelihood that the recovered offset is due to SBH coalescence is discussed for individual galaxies in Appendix B.\\nAlthough our results imply that SBH displacements due to gravitational recoil are much less common than might be inferred from theoretical considerations, this does not preclude the possibility that observed displacements in individual galaxies are due to recoil oscillations. Indeed, the p i values listed in Table 6 indicate that for t m = 5.0 Gyr, the chance of not observing a larger displacement, given a sufficiently large kick, is 30% for all objects having a displacement considered significant. Even for t m = 0.4 Gyr, there are still three of these galaxies for which 1 − p i ≥ 0.25. The likelihood that the recovered offset is due to SBH coalescence is discussed for individual galaxies in Appendix B.\\nAssuming that the measured displacements are indeed due to phase II recoil oscillations, we have used eq. 3 to estimate the merger epoch for those galaxies with displacements rated at least as \\\"low significance\\\" in terms of the isophote centroid IQR. Allowing for the time taken for the SBH binary to form and subsequently coalesce in the center of the merged galaxy, and for phase I oscillations, the elapsed times since the merger are typically ∼ several Gyr, comparable with the mean time between galaxy mergers estimated here for galaxy masses log(M gal /M ) > 11, and consistent with the observationally determined merger rate for brightest cluster galaxies (Lidman et al. 2013).\\nAssuming that the measured displacements are indeed due to phase II recoil oscillations, we have used eq. 3 to estimate the merger epoch for those galaxies with displacements rated at least as \\\"low significance\\\" in terms of the isophote centroid IQR. Allowing for the time taken for the SBH binary to form and subsequently coalesce in the center of the merged galaxy, and for phase I oscillations, the elapsed times since the merger are typically ∼ several Gyr, comparable with the mean time between galaxy mergers estimated here for galaxy masses log(M gal /M ) > 11, and consistent with the observationally determined merger rate for brightest cluster galaxies (Lidman et al. 2013).\\nEq.2, used to estimate the SBH damping time, relies on the M • -σ relation (Ferrarese & Ford 2005). To test the dependence of our result on the value adopted for M • we use eq.6 in Lauer et al. (2007) to estimate SBH masses using the M • -M V relation (see Table 9 and Appendix A for more details). Using these alternative masses, and taking into account the kick velocity distribution computed by L12, we derive new values for P obtaining 1 × 10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5 Gyr. The likelihood of our result is somewhat increased for t m = 5 Gyr, however there is still a large probability of finding displacements larger than those measured, leaving our basic conclusion unchanged.\\nEq.2, used to estimate the SBH damping time, relies on the M • -σ relation (Ferrarese & Ford 2005). To test the dependence of our result on the value adopted for M • we use eq.6 in Lauer et al. (2007) to estimate SBH masses using the M • -M V relation (see Table 9 and Appendix A for more details). Using these alternative masses, and taking into account the kick velocity distribution computed by L12, we derive new values for P obtaining 1 × 10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5 Gyr. The likelihood of our result is somewhat increased for t m = 5 Gyr, however there is still a large probability of finding displacements larger than those measured, leaving our basic conclusion unchanged.\\nNo useful constraints can be obtained on the merger epoch from consideration of the time necessary to replenish the stellar population of the core. This is roughly the relaxation time, T rel , at the influence radius of the central SBH, which can be estimated as:\\nNo useful constraints can be obtained on the merger epoch from consideration of the time necessary to replenish the stellar population of the core. This is roughly the relaxation time, T rel , at the influence radius of the central SBH, which can be estimated as:\\nT r (r i ) ≈ 1.2 × 10 11 σ 7.47 100 yr (5) Merritt (2013, eq. 3.5). Given that the measured velocity dispersion is always > 100 km s −1 in the centers of our galaxies (see Table 8), it is clear that the replenishment time is much longer than the timescales involved in the SBH merger-recoil event (∼ 10 9 yr). In general, coalescence of an SBH-binary results in a re-orientation of the SBH spin axis, leading to a sudden flip in the direction of the associated radio jet (Merritt & Ekers 2002). Therefore, the radio source morphology may act as a \\\"signpost\\\" of SBH coalescence. However, powerful extended radio sources have lifetimes ∼ 10 6 −10 8 yr (e.g., Parma et al. 1999;O'Dea et al. 2009;Antognini, Bird & Martini 2012), much shorter than the time-between-mergers, even for massive galaxies. Thus even if a spin flip took place, morphological traces of this event, such as a change in jet direction, would not necessarily be evident in the current (post-coalescence) radio source. The radio source properties of individual galaxies are discussed in Appendix B.\\nT r (r i ) ≈ 1.2 × 10 11 σ 7.47 100 yr (5) Merritt (2013, eq. 3.5). Given that the measured velocity dispersion is always > 100 km s −1 in the centers of our galaxies (see Table 8), it is clear that the replenishment time is much longer than the timescales involved in the SBH merger-recoil event (∼ 10 9 yr). In general, coalescence of an SBH-binary results in a re-orientation of the SBH spin axis, leading to a sudden flip in the direction of the associated radio jet (Merritt & Ekers 2002). Therefore, the radio source morphology may act as a \\\"signpost\\\" of SBH coalescence. However, powerful extended radio sources have lifetimes ∼ 10 6 −10 8 yr (e.g., Parma et al. 1999;O'Dea et al. 2009;Antognini, Bird & Martini 2012), much shorter than the time-between-mergers, even for massive galaxies. Thus even if a spin flip took place, morphological traces of this event, such as a change in jet direction, would not necessarily be evident in the current (post-coalescence) radio source. The radio source properties of individual galaxies are discussed in Appendix B.\\nIt is notable that in four of the six galaxies in which significant displacements have been found, the displacements are also approximately aligned with the radio jet axis (Section 4 and Table 5). This number includes three of the four that have powerful kpc-scale jets (Section 4 and Table 5), suggesting jet power asymmetries as a possible displacement mechanism (as discussed below). However, such alignments do not necessarily argue against gravitational recoil. In their statistical study of the coalescence of spinning black hole binaries Lousto et al. (2010) find that the recoil velocity is preferentially aligned (or counter-aligned) with the orbital angular momentum of the progenitor binary and moreover, that the spin direction distribution of the recoiling SBH peaks at an angle of ≈ 25 • to the orbital angular momentum (for an equal mass progenitor binary; the probability distribution is broader and peaks at larger angles for smaller mass ratios). This suggests, assuming that the jet traces the spin axis of the recoiling SBH, a tendency for the displacement to be somewhat, but not greatly, misaligned with the jet axis, consistent with what is observed in these objects.\\nIt is notable that in four of the six galaxies in which significant displacements have been found, the displacements are also approximately aligned with the radio jet axis (Section 4 and Table 5). This number includes three of the four that have powerful kpc-scale jets (Section 4 and Table 5), suggesting jet power asymmetries as a possible displacement mechanism (as discussed below). However, such alignments do not necessarily argue against gravitational recoil. In their statistical study of the coalescence of spinning black hole binaries Lousto et al. (2010) find that the recoil velocity is preferentially aligned (or counter-aligned) with the orbital angular momentum of the progenitor binary and moreover, that the spin direction distribution of the recoiling SBH peaks at an angle of ≈ 25 • to the orbital angular momentum (for an equal mass progenitor binary; the probability distribution is broader and peaks at larger angles for smaller mass ratios). This suggests, assuming that the jet traces the spin axis of the recoiling SBH, a tendency for the displacement to be somewhat, but not greatly, misaligned with the jet axis, consistent with what is observed in these objects.\"\n },\n {\n \"title\": \"Other displacement mechanisms\",\n \"content\": \"\"\n },\n {\n \"title\": \"Other displacement mechanisms\",\n \"content\": \"We now consider several other mechanisms that may plausibly produce SBH displacements.\\nWe now consider several other mechanisms that may plausibly produce SBH displacements.\\n1. Asymmetric jets: if the AGN jets are intrinsically asymmetric in power output, the resulting net thrust can push the SBH away from the original equilibrium position (Shklovski 1982;Saslaw & Whittle 1988 Note. -(1) Optical name; (2) instrument/camera/filter on HST; (3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) and (7) nuclear point source position in pixels; (8 -9) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (10) offset in pixel normalized by the corresponding IQR; (11) offset in pc; (12) direction of the offset; (13) type of displacement as defined in §3.3. Table 4). a This value changes to −8 ± 16 • when only WFPC2 results are considered. b This value changes to 312 ± 8 • (132 ± 8 • ) when only ACS and NICMOS2 results are considered. See Table 2 for references on the radio jet PA. Note. -Probability to observe a projected displacement larger than the value actually observed in this work (in units of core radii) at a random time after the last kick. Probabilities are computed for a mean time between galactic mergers tm = 0.4 Gyr and 5 Gyr. The symbol \\\" * \\\" indicates 3σ offsets that were classified as \\\"null\\\" after the normalization for the co-ordinates IQR. The symbol \\\"<\\\" indicates offsets that do not reach the 3σ level.\\n1. Asymmetric jets: if the AGN jets are intrinsically asymmetric in power output, the resulting net thrust can push the SBH away from the original equilibrium position (Shklovski 1982;Saslaw & Whittle 1988 Note. -(1) Optical name; (2) instrument/camera/filter on HST; (3) lower and upper limit, in arcseconds, of the region analyzed; (4) inter-quartile range in pixel, this is the difference between 75th and 25th percentile of the isophotal center dataset; (5) photocenter position in pixel on the frame; (6) and (7) nuclear point source position in pixels; (8 -9) offset of the isophotal center with respect to the nuclear point source in pixels and milliarcseconds; (10) offset in pixel normalized by the corresponding IQR; (11) offset in pc; (12) direction of the offset; (13) type of displacement as defined in §3.3. Table 4). a This value changes to −8 ± 16 • when only WFPC2 results are considered. b This value changes to 312 ± 8 • (132 ± 8 • ) when only ACS and NICMOS2 results are considered. See Table 2 for references on the radio jet PA. Note. -Probability to observe a projected displacement larger than the value actually observed in this work (in units of core radii) at a random time after the last kick. Probabilities are computed for a mean time between galactic mergers tm = 0.4 Gyr and 5 Gyr. The symbol \\\" * \\\" indicates 3σ offsets that were classified as \\\"null\\\" after the normalization for the co-ordinates IQR. The symbol \\\"<\\\" indicates offsets that do not reach the 3σ level.\\nthis scenario:\\nthis scenario:\\nwhereṁ ≡Ṁ a /Ṁ edd and f jet = L jet /L a 8 ,Ṁ a is the mass accretion rate,Ṁ edd ≈ 2.2 M • /10 8 M M yr −1 the Eddington accretion rate, L a the accretion luminosity and L jet is the jet luminosity. For asymmetrical, oppositely directed jets L jet is the difference in the luminosities of the two jets.\\nwhereṁ ≡Ṁ a /Ṁ edd and f jet = L jet /L a 8 ,Ṁ a is the mass accretion rate,Ṁ edd ≈ 2.2 M • /10 8 M M yr −1 the Eddington accretion rate, L a the accretion luminosity and L jet is the jet luminosity. For asymmetrical, oppositely directed jets L jet is the difference in the luminosities of the two jets.\\nUnder the assumption that the restoring force from the galaxy is negligible (a reasonable approximation for the low-density cores of \\\"core galaxies\\\"), Eq. 6 can be integrated to obtain an expression for the displacement:\\nUnder the assumption that the restoring force from the galaxy is negligible (a reasonable approximation for the low-density cores of \\\"core galaxies\\\"), Eq. 6 can be integrated to obtain an expression for the displacement:\\n∆r ≈ 340 f jetṁ t 2 6 pc (7) where t 6 is the time over which the SBH is accelerated to produce an offset ∆r.\\n∆r ≈ 340 f jetṁ t 2 6 pc (7) where t 6 is the time over which the SBH is accelerated to produce an offset ∆r.\\nThe best candidates for jet thrust displacements are the four galaxies (NGC 1399, 4261, 4486 and IC 4296) that have relatively powerful kpc-scale jets (Section 4 and Table 2). Of these, we did not detect a significant displacement in NGC 4261, which has an FR I radio source. NGC 4486 and IC 4296 also host FR I radio sources and NGC 1399 has an FR I-like morphology, despite its relatively low power. Interestingly, all three of these galaxies exhibit photocenter displacements that are at least approximately (within 20 • ) in the jet direction (Table 5). This implies that the SBH is displaced relative to its equilibrium position in the counter-jet direction, as might be expected if the displacements are related to intrinsic asymmetries in jet power. NGC 4486 (M87) has already been discussed in detail by B10, who concluded that a jet power asymmetry amounting to ∼ 3% of the accretion luminosity can explain the observed displacement, for a radio source lifetime ∼ 10 8 yr. We find similar results in the cases of NGC 1399 and IC 4296, where the observed displacement can be produced for a jet asymmetry < 1% of the Bondi accretion luminosity, again for a radio source lifetime ∼ 10 8 yr (See Appendix B for details).\\nThe best candidates for jet thrust displacements are the four galaxies (NGC 1399, 4261, 4486 and IC 4296) that have relatively powerful kpc-scale jets (Section 4 and Table 2). Of these, we did not detect a significant displacement in NGC 4261, which has an FR I radio source. NGC 4486 and IC 4296 also host FR I radio sources and NGC 1399 has an FR I-like morphology, despite its relatively low power. Interestingly, all three of these galaxies exhibit photocenter displacements that are at least approximately (within 20 • ) in the jet direction (Table 5). This implies that the SBH is displaced relative to its equilibrium position in the counter-jet direction, as might be expected if the displacements are related to intrinsic asymmetries in jet power. NGC 4486 (M87) has already been discussed in detail by B10, who concluded that a jet power asymmetry amounting to ∼ 3% of the accretion luminosity can explain the observed displacement, for a radio source lifetime ∼ 10 8 yr. We find similar results in the cases of NGC 1399 and IC 4296, where the observed displacement can be produced for a jet asymmetry < 1% of the Bondi accretion luminosity, again for a radio source lifetime ∼ 10 8 yr (See Appendix B for details).\\nA close alignment between the displacement and the (initial) jet direction is also found for the low power pcscale radio source in NGC 4278. In this case, Giroletti et al. (2005)'s interpretation of the radio data implies that the photocenter is displaced in the counter-jet direction, with the jet axis being closely aligned with the line of sight. Assuming that the SBH is displaced along the jet axis, the de-projected magnitude of the displacement would be ∼ 100 − 200 pc. Nevertheless, it seems possible that this could result from jet thrust, if sustained for ∼ 10 8 yr (Appendix B).\\nA close alignment between the displacement and the (initial) jet direction is also found for the low power pcscale radio source in NGC 4278. In this case, Giroletti et al. (2005)'s interpretation of the radio data implies that the photocenter is displaced in the counter-jet direction, with the jet axis being closely aligned with the line of sight. Assuming that the SBH is displaced along the jet axis, the de-projected magnitude of the displacement would be ∼ 100 − 200 pc. Nevertheless, it seems possible that this could result from jet thrust, if sustained for ∼ 10 8 yr (Appendix B).\\nThe double nucleus galaxy NGC 4696 does not show well-defined jets on kpc scales, but has a one-sided pcscale jet in PA≈ −150 • . The displacement relative to the brighter nucleus is considered non-significant because of the large IQR. However, as already noted, it is not known which of the two optical nuclei hosts the AGN producing the radio jet. If it is the fainter nucleus, the photocenter displacement is approximately in the counter-jet direction, which is not consistent with jet acceleration of the SBH (which would cause the SBH to be displaced in the counter-jet direction).\\nThe double nucleus galaxy NGC 4696 does not show well-defined jets on kpc scales, but has a one-sided pcscale jet in PA≈ −150 • . The displacement relative to the brighter nucleus is considered non-significant because of the large IQR. However, as already noted, it is not known which of the two optical nuclei hosts the AGN producing the radio jet. If it is the fainter nucleus, the photocenter displacement is approximately in the counter-jet direction, which is not consistent with jet acceleration of the SBH (which would cause the SBH to be displaced in the counter-jet direction).\\nThe remaining two objects which exhibit displacements considered to be significant (NGC 5419 and 5846) do not have well-defined jets on either parsec or kiloparsec scales.\\nThe remaining two objects which exhibit displacements considered to be significant (NGC 5419 and 5846) do not have well-defined jets on either parsec or kiloparsec scales.\\n2. Stalled SBH binaries: in the aftermath of a galaxy-galaxy merger, the SBH binary orbit shrinks at first due to dynamical friction and subsequently through slingshot ejection of stars intersecting the orbit. Investigations of quasi-steady spherical models suggested that the evolution of the binary stalls at separations ∼ 1 pc, due to a paucity of interacting stars, rather than hardening to the point at which gravitational wave emission drives the final inspiral to coalescence (the so-called \\\"final parsec problem\\\"; e.g., Merritt & Milosavljević 2005). Based on N -body simulations, Merritt (2006) estimated \\\"typical\\\" semi-major axes for stalled binaries, finding a stall 10 pc for an SBH mass ratio q = 0.5 and a stall 3 pc for q = 0.1, respectively. If the binary center of mass is located at the photocenter, the displacements of the primary and secondary components would be ∆r 1 ∼ a stall q/(1 + q) and ∆r 2 ∼ a stall /(1 + q), respectively, giving ∆r 1 ∼ 3 (0.3) pc and ∆r 2 ∼ 7 (3) pc for q = 0.5 (0.1). Thus stalled binaries could produce displacements comparable with our results, particularly if the secondary SBH is accreting and the mass ratio is near unity.\\n2. Stalled SBH binaries: in the aftermath of a galaxy-galaxy merger, the SBH binary orbit shrinks at first due to dynamical friction and subsequently through slingshot ejection of stars intersecting the orbit. Investigations of quasi-steady spherical models suggested that the evolution of the binary stalls at separations ∼ 1 pc, due to a paucity of interacting stars, rather than hardening to the point at which gravitational wave emission drives the final inspiral to coalescence (the so-called \\\"final parsec problem\\\"; e.g., Merritt & Milosavljević 2005). Based on N -body simulations, Merritt (2006) estimated \\\"typical\\\" semi-major axes for stalled binaries, finding a stall 10 pc for an SBH mass ratio q = 0.5 and a stall 3 pc for q = 0.1, respectively. If the binary center of mass is located at the photocenter, the displacements of the primary and secondary components would be ∆r 1 ∼ a stall q/(1 + q) and ∆r 2 ∼ a stall /(1 + q), respectively, giving ∆r 1 ∼ 3 (0.3) pc and ∆r 2 ∼ 7 (3) pc for q = 0.5 (0.1). Thus stalled binaries could produce displacements comparable with our results, particularly if the secondary SBH is accreting and the mass ratio is near unity.\\nIt has been argued that stalling can be avoided in galaxies containing significant amounts of nuclear gas (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007). Even in purely stellar nuclei, N -body simulations sometimes find that evolution of the binary can continue efficiently due to the presence of centrophilic orbits (Khan et al. 2011;Preto et al. 2011;Gualandris & Merritt 2012). However the existing gas-dynamical simulations probably do not yet have enough spatial resolution to follow the binary's evolution to sub-parsec scales, . Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position). and the N -body simulations appear to not yet have large enough N that their results can be robustly extrapolated to the much larger-N regime of real galaxies (Vasiliev et al. 2014). Another possibility, discussed by Antonini & Merritt (2012), is accretion of a less massive galaxy by a giant elliptical, such as NGC 4486, which has a pre-existing depleted core (presumably the result of the evolution of SBH binaries formed in previous mergers). In such situations, dynamical friction is very inefficient in the core due to the lack of stars moving slower than the sinking object. The orbital eccentricity can increase rapidly while the apoapsis hardly changes, resulting in a slowly evolving SBH binary in a highly eccentric orbit. The simulations presented by Antonini & Merritt (2012) indicate that in an M87-like core, a low mass ratio binary (q ≈ 10 −3 ) can persist over a Hubble time in an increasingly eccentric orbit with a semi major axis ∼ 10 pc. If the secondary SBH is accreting, it will be visible as an off-center AGN; if both components of the SBH binary are accreting, a double nucleus might be observed (perhaps as in NGC 4696 and NGC 5419).\\nIt has been argued that stalling can be avoided in galaxies containing significant amounts of nuclear gas (e.g., Escala et al. 2005;Cuadra et al. 2009;Mayer et al. 2007). Even in purely stellar nuclei, N -body simulations sometimes find that evolution of the binary can continue efficiently due to the presence of centrophilic orbits (Khan et al. 2011;Preto et al. 2011;Gualandris & Merritt 2012). However the existing gas-dynamical simulations probably do not yet have enough spatial resolution to follow the binary's evolution to sub-parsec scales, . Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position). and the N -body simulations appear to not yet have large enough N that their results can be robustly extrapolated to the much larger-N regime of real galaxies (Vasiliev et al. 2014). Another possibility, discussed by Antonini & Merritt (2012), is accretion of a less massive galaxy by a giant elliptical, such as NGC 4486, which has a pre-existing depleted core (presumably the result of the evolution of SBH binaries formed in previous mergers). In such situations, dynamical friction is very inefficient in the core due to the lack of stars moving slower than the sinking object. The orbital eccentricity can increase rapidly while the apoapsis hardly changes, resulting in a slowly evolving SBH binary in a highly eccentric orbit. The simulations presented by Antonini & Merritt (2012) indicate that in an M87-like core, a low mass ratio binary (q ≈ 10 −3 ) can persist over a Hubble time in an increasingly eccentric orbit with a semi major axis ∼ 10 pc. If the secondary SBH is accreting, it will be visible as an off-center AGN; if both components of the SBH binary are accreting, a double nucleus might be observed (perhaps as in NGC 4696 and NGC 5419).\\nHelical distortion or \\\"wiggling\\\" of parsec-scale radio jets has been linked to putative SBH binaries in several AGNs, with the jet wiggles being variously attributed to orbital motion, precession of the accretion disk around the jet-emitting black hole or to geodetic precession (e.g., Begelman, Blandford & Rees 1980;Roos et al. 1993;Katz 1997;Romero et al. 2000;Lobanov & Roland 2005). However, the periods (∼ 3 years) and separations (∼ 10 −2 pc) typically inferred from analyses of jet wiggles are much smaller than would be the case for a stalling binary. If the ∼ pc-scale displacements measured in this work are interpreted as SBH binary orbits, periods ∼ 10 4−5 years are implied for total masses ∼ 10 8−9 M . Geodetic precession is insignificant at these separations. The wavelength of jet wiggles caused by orbital motion is given by λ jet ∼ v app t orbit , where v app = βc sin i/(1 − β cos i) is the apparent jet velocity (for a jet speed v jet = βc and inclination to the line of sight, i) and t orbit is the orbital period. Therefore, assuming β ∼ 1, a pc-scale binary will produce very long wavelength wiggles in the jet (∼ 10 kpc). In general, due to the combination of orbital and jet velocities, the jet will precess on the surface of a cone which, for a pc-scale binary of mass ∼ 10 9 M will have a half-opening angle 1 • (see Equation 7 in Roos et al. 1993). Thus, orbital motion would cause only small curvatures in the jet over ∼ kpc scales, which would be difficult to discern as the jet loses collimation, or to distinguish from jet bending due to environmental effects, such as ram pressure. This will also be the case for disk precession, since the precession period exceeds the orbital period. Therefore, although the jet morphology has been mapped in detail from pc to kpc scales for a number of our sample galaxies (see Table 2 for references), these observations are unlikely to provide unambiguous clues as to the presence, or not, of a stalled pc-scale binary.\\nHelical distortion or \\\"wiggling\\\" of parsec-scale radio jets has been linked to putative SBH binaries in several AGNs, with the jet wiggles being variously attributed to orbital motion, precession of the accretion disk around the jet-emitting black hole or to geodetic precession (e.g., Begelman, Blandford & Rees 1980;Roos et al. 1993;Katz 1997;Romero et al. 2000;Lobanov & Roland 2005). However, the periods (∼ 3 years) and separations (∼ 10 −2 pc) typically inferred from analyses of jet wiggles are much smaller than would be the case for a stalling binary. If the ∼ pc-scale displacements measured in this work are interpreted as SBH binary orbits, periods ∼ 10 4−5 years are implied for total masses ∼ 10 8−9 M . Geodetic precession is insignificant at these separations. The wavelength of jet wiggles caused by orbital motion is given by λ jet ∼ v app t orbit , where v app = βc sin i/(1 − β cos i) is the apparent jet velocity (for a jet speed v jet = βc and inclination to the line of sight, i) and t orbit is the orbital period. Therefore, assuming β ∼ 1, a pc-scale binary will produce very long wavelength wiggles in the jet (∼ 10 kpc). In general, due to the combination of orbital and jet velocities, the jet will precess on the surface of a cone which, for a pc-scale binary of mass ∼ 10 9 M will have a half-opening angle 1 • (see Equation 7 in Roos et al. 1993). Thus, orbital motion would cause only small curvatures in the jet over ∼ kpc scales, which would be difficult to discern as the jet loses collimation, or to distinguish from jet bending due to environmental effects, such as ram pressure. This will also be the case for disk precession, since the precession period exceeds the orbital period. Therefore, although the jet morphology has been mapped in detail from pc to kpc scales for a number of our sample galaxies (see Table 2 for references), these observations are unlikely to provide unambiguous clues as to the presence, or not, of a stalled pc-scale binary.\\n3. Massive perturbers: galaxy centers host a variety of potential perturbers with masses ranging from ∼ 1M (e.g., stellar mass black holes and neutron stars) to ∼ 10 4 − 10 7 M , such as giant molecular clouds and stellar clusters.\\n3. Massive perturbers: galaxy centers host a variety of potential perturbers with masses ranging from ∼ 1M (e.g., stellar mass black holes and neutron stars) to ∼ 10 4 − 10 7 M , such as giant molecular clouds and stellar clusters.\\nGravitational interactions with these objects will cause the SBH to undergo a type of Brownian motion, with the amplitude of the root-mean-square displacement given by:\\nGravitational interactions with these objects will cause the SBH to undergo a type of Brownian motion, with the amplitude of the root-mean-square displacement given by:\\nwherem is proportional to the second moment of the mass distribution of the massive perturbers:\\nwherem is proportional to the second moment of the mass distribution of the massive perturbers:\\nm ≡ n(m)m 2 dm n(m)mdm with n(m) dm being the number of perturbers with masses in the range m, m + dm (Merritt 2013, eq. 7.63). The offsets measured in this work range from 0.01 to 0.1 r c , but are typically ∼ 0.01 r c (Table 6). For a typical SBH mass of order 10 9 M (Table 9), rms displacements ∼ 0.01r c can be generated by Brownian motion form ∼ 10 5 M .\\nm ≡ n(m)m 2 dm n(m)mdm with n(m) dm being the number of perturbers with masses in the range m, m + dm (Merritt 2013, eq. 7.63). The offsets measured in this work range from 0.01 to 0.1 r c , but are typically ∼ 0.01 r c (Table 6). For a typical SBH mass of order 10 9 M (Table 9), rms displacements ∼ 0.01r c can be generated by Brownian motion form ∼ 10 5 M .\\nThe mass functions of globular clusters, gas clumps and giant molecular clouds in the inner 100 pc of the Galaxy have been estimated by Perets et al. (2007). For comparison, using the mass functions presented in their paper, we find thatm can be as high as 10 5 M for a population of giant molecular clouds. The Milky Way, of course, is very different from the galaxies studied here. However, as recently discussed by Antonini & Merritt (2012), the low efficiency of dynamical friction in low density cores favors the formation of a population of stalling massive objects in the cores of giant ellipticals.\\nThe mass functions of globular clusters, gas clumps and giant molecular clouds in the inner 100 pc of the Galaxy have been estimated by Perets et al. (2007). For comparison, using the mass functions presented in their paper, we find thatm can be as high as 10 5 M for a population of giant molecular clouds. The Milky Way, of course, is very different from the galaxies studied here. However, as recently discussed by Antonini & Merritt (2012), the low efficiency of dynamical friction in low density cores favors the formation of a population of stalling massive objects in the cores of giant ellipticals.\\nTherefore, it seems plausible that displacements of order those observed could result from Brownian motion due to a population of massive perturbers. In several galaxies, the observed displacements are approximately aligned with the kpc-scale radio jets. This suggests that the displacement is not random, at least in these cases, favoring other mechanisms (i.e, gravitational recoil or jet thrust) that are expected to offset the SBH in the jet direction. Nevertheless, we conclude that Brownian motion cannot be excluded as the cause of the displacement in any individual galaxy, particularly those where the offsets are not aligned with kpc-scale jets, i.e. NGC 4278, 5846 and 5419.\\nTherefore, it seems plausible that displacements of order those observed could result from Brownian motion due to a population of massive perturbers. In several galaxies, the observed displacements are approximately aligned with the kpc-scale radio jets. This suggests that the displacement is not random, at least in these cases, favoring other mechanisms (i.e, gravitational recoil or jet thrust) that are expected to offset the SBH in the jet direction. Nevertheless, we conclude that Brownian motion cannot be excluded as the cause of the displacement in any individual galaxy, particularly those where the offsets are not aligned with kpc-scale jets, i.e. NGC 4278, 5846 and 5419.\"\n },\n {\n \"title\": \"SUMMARY AND CONCLUSIONS\",\n \"content\": \"\"\n },\n {\n \"title\": \"SUMMARY AND CONCLUSIONS\",\n \"content\": \"We have analyzed HST archival images of 14 nearby core elliptical galaxies, each of which hosts a central point-like source associated with a low-luminosity AGN, in order to search for offsets between the AGN and the galaxy photocenter. Such AGN-photocenter displacements are possible signposts of gravitational recoils resulting from the coalescence of an SBH binary.\\nWe have analyzed HST archival images of 14 nearby core elliptical galaxies, each of which hosts a central point-like source associated with a low-luminosity AGN, in order to search for offsets between the AGN and the galaxy photocenter. Such AGN-photocenter displacements are possible signposts of gravitational recoils resulting from the coalescence of an SBH binary.\\nWe find significant (> 3σ) differences between the positions of the nuclear optical (or NIR) point source and the mean photocenter of the galaxy, as determined from isophote fits, in ten of the 14 galaxies in the sample. Assuming that the mean photocenter locates the minimum of the galactic potential well and that the point source locates the position of the AGN and hence the SBH, these results imply that the SBH is displaced from its equilibrium position by angular distances ranging between 20 3.8 ± 0.7 338 ± 7\\nWe find significant (> 3σ) differences between the positions of the nuclear optical (or NIR) point source and the mean photocenter of the galaxy, as determined from isophote fits, in ten of the 14 galaxies in the sample. Assuming that the mean photocenter locates the minimum of the galactic potential well and that the point source locates the position of the AGN and hence the SBH, these results imply that the SBH is displaced from its equilibrium position by angular distances ranging between 20 3.8 ± 0.7 338 ± 7\\nGalaxies with two nuclear sources NGC 5419 7.5 ± 1.7 62 ± 2 252 ± 13 346 ± 2\\nGalaxies with two nuclear sources NGC 5419 7.5 ± 1.7 62 ± 2 252 ± 13 346 ± 2\\nNote. -Projected offsets of the photocenter with respect to the AGN and their possible origin. When multiple images have been analyzed, values presented here are the error weighted average. Position angles (PA) are given in degrees East from North. NPS = nuclear point source. and 90 mas, or projected linear distances in the range 1 − 10 pc. As spurious offsets may occur as a result of large-scale isophotal asymmetries, only displacements of magnitude > 0.8 × the inter-quartile range of the distribution of isophote center coordinates (equivalent to 1σ for a gaussian distribution) are considered \\\"real\\\". There are six galaxies that exhibit displacements > 0.8 IQR with three of these (NGC 4278, NGC 4486 and NGC 5846) having displacements > 1.6 IQR (equivalent to 2σ). In every case, the measured displacement of the SBH relative to the galaxy photocenter is a small fraction (1-10%) of the galaxy core radius, which is typically r c ∼ 300 pc, for these galaxies.\\nNote. -Projected offsets of the photocenter with respect to the AGN and their possible origin. When multiple images have been analyzed, values presented here are the error weighted average. Position angles (PA) are given in degrees East from North. NPS = nuclear point source. and 90 mas, or projected linear distances in the range 1 − 10 pc. As spurious offsets may occur as a result of large-scale isophotal asymmetries, only displacements of magnitude > 0.8 × the inter-quartile range of the distribution of isophote center coordinates (equivalent to 1σ for a gaussian distribution) are considered \\\"real\\\". There are six galaxies that exhibit displacements > 0.8 IQR with three of these (NGC 4278, NGC 4486 and NGC 5846) having displacements > 1.6 IQR (equivalent to 2σ). In every case, the measured displacement of the SBH relative to the galaxy photocenter is a small fraction (1-10%) of the galaxy core radius, which is typically r c ∼ 300 pc, for these galaxies.\\nApproximate alignments between the SBHphotocenter displacements and the radio source axis were found in four of the six galaxies considered to have significant displacements, including three of the four that have FR I, or FR I-like radio sources with relatively powerful and well-defined kpc-scale jets. Indeed, in every case in which there is both a significant displacement and an unambiguous jet, the two are approximately aligned.\\nApproximate alignments between the SBHphotocenter displacements and the radio source axis were found in four of the six galaxies considered to have significant displacements, including three of the four that have FR I, or FR I-like radio sources with relatively powerful and well-defined kpc-scale jets. Indeed, in every case in which there is both a significant displacement and an unambiguous jet, the two are approximately aligned.\\nLacking detailed knowledge of the merger history of the galaxies, or of the SBH binary parameters (such as mass ratio and spin configuration) that determine the recoil velocity, it is not possible to directly test the hypothesis that the displacements are caused by residual gravitational recoil oscillations. Instead, we used a simple Monte-Carlo model to investigate if the measured displacements are consistent with gravitational recoil. We find that the displacements in individual objects can plausibly be attributed to residual gravitational recoil oscillations following a major or minor merger within the last few Gyr. However, for plausible merger rates there is a high probability of larger displacements than actually observed, if SBH coalescence events took place in these galaxies.\\nLacking detailed knowledge of the merger history of the galaxies, or of the SBH binary parameters (such as mass ratio and spin configuration) that determine the recoil velocity, it is not possible to directly test the hypothesis that the displacements are caused by residual gravitational recoil oscillations. Instead, we used a simple Monte-Carlo model to investigate if the measured displacements are consistent with gravitational recoil. We find that the displacements in individual objects can plausibly be attributed to residual gravitational recoil oscillations following a major or minor merger within the last few Gyr. However, for plausible merger rates there is a high probability of larger displacements than actually observed, if SBH coalescence events took place in these galaxies.\\nThat larger displacements were not observed suggests that the frequency of gravitational recoil kicks large enough to trigger long-lived oscillations is lower than predicted, perhaps because the evolution of the SBH binary typically results in a configuration that suppresses recoil kicks with velocities 200 km s −1 . Alternatively, the post-recoil oscillations may be damped more quickly than predicted by pure N -body simulations. In either case, gas may play an important role (e.g., Dotti et al. 2010;Sijacki, Springel & Haehnelt 2011). Otherwise, it is possible that galaxy mergers are much more infrequent than implied by the current ΛCDM paradigm (Kroupa 2014).\\nThat larger displacements were not observed suggests that the frequency of gravitational recoil kicks large enough to trigger long-lived oscillations is lower than predicted, perhaps because the evolution of the SBH binary typically results in a configuration that suppresses recoil kicks with velocities 200 km s −1 . Alternatively, the post-recoil oscillations may be damped more quickly than predicted by pure N -body simulations. In either case, gas may play an important role (e.g., Dotti et al. 2010;Sijacki, Springel & Haehnelt 2011). Otherwise, it is possible that galaxy mergers are much more infrequent than implied by the current ΛCDM paradigm (Kroupa 2014).\\nSeveral other mechanisms are capable of producing the observed displacements, with the observed alignments between the SBH-photocenter displacements and the radio source axis favoring jet acceleration in some objects. An approximate displacement-radio axis alignment is also expected for gravitational recoil, but not for orbital motion in pre-coalescence SBH binaries or interactions with massive perturbers. However, both of the latter mechanisms are capable of producing displacement amplitudes comparable to those observed and cannot be ruled out in individual objects.\\nSeveral other mechanisms are capable of producing the observed displacements, with the observed alignments between the SBH-photocenter displacements and the radio source axis favoring jet acceleration in some objects. An approximate displacement-radio axis alignment is also expected for gravitational recoil, but not for orbital motion in pre-coalescence SBH binaries or interactions with massive perturbers. However, both of the latter mechanisms are capable of producing displacement amplitudes comparable to those observed and cannot be ruled out in individual objects.\\nIn general, it is not possible to unambiguously distinguish between different mechanisms (including recoil) on the basis of the displacement measurements alone for individual galaxies. However, with a larger sample it may be possible to distinguish mechanisms using statistical arguments. Thus, for jet acceleration the displacement direction should be strongly correlated with the radio jet, with the amplitude correlating with jet power. In the case of gravitational recoil, a weaker correlation with jet direction might be expected. However, no such correlations are expected for binary SBH, or massive perturbers.\\nIn general, it is not possible to unambiguously distinguish between different mechanisms (including recoil) on the basis of the displacement measurements alone for individual galaxies. However, with a larger sample it may be possible to distinguish mechanisms using statistical arguments. Thus, for jet acceleration the displacement direction should be strongly correlated with the radio jet, with the amplitude correlating with jet power. In the case of gravitational recoil, a weaker correlation with jet direction might be expected. However, no such correlations are expected for binary SBH, or massive perturbers.\"\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"We wish to honor the memory of our great friend, colleague and mentor David Axon. We thank the anonymous referee for the valuable and timely comments that improved the paper. DL thanks E. Vasiliev, R. Mittal, F. Antonini and M. Richmond for helpful discussions. AM acknowledges support from the Italian National Institute for Astrophysics (INAF) through PRIN-INAF 2011 \\\"Black hole growth and AGN feedback through the cosmic time\\\" and from the Italian ministry for school, university and research (MIUR) through PRIN-MIUR 2010-2011 \\\"The dark Universe and the cosmic evolution of baryons: from current surveys to Euclid\\\". DM was supported by the National Science Foundation under grant no. AST 1211602 and by the National Aeronautics and Space Administration under grant no. NNX13AG92G. Support for program AR-11771 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr) and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Center (CADC/NRC/CSA). Figure 6. HST images of the galaxies in the sample. The number at the bottom represents the length of the whole axes.\\nWe wish to honor the memory of our great friend, colleague and mentor David Axon. We thank the anonymous referee for the valuable and timely comments that improved the paper. DL thanks E. Vasiliev, R. Mittal, F. Antonini and M. Richmond for helpful discussions. AM acknowledges support from the Italian National Institute for Astrophysics (INAF) through PRIN-INAF 2011 \\\"Black hole growth and AGN feedback through the cosmic time\\\" and from the Italian ministry for school, university and research (MIUR) through PRIN-MIUR 2010-2011 \\\"The dark Universe and the cosmic evolution of baryons: from current surveys to Euclid\\\". DM was supported by the National Science Foundation under grant no. AST 1211602 and by the National Aeronautics and Space Administration under grant no. NNX13AG92G. Support for program AR-11771 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr) and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Center (CADC/NRC/CSA). Figure 6. HST images of the galaxies in the sample. The number at the bottom represents the length of the whole axes.\"\n },\n {\n \"title\": \"APPENDIX\",\n \"content\": \"\"\n },\n {\n \"title\": \"APPENDIX\",\n \"content\": \"\"\n },\n {\n \"title\": \"A: DISTRIBUTION OF RECOIL DISPLACEMENTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"A: DISTRIBUTION OF RECOIL DISPLACEMENTS\",\n \"content\": \"Here we outline the method used to estimate the probability distributions for displacements resulting from postrecoil oscillations of a kicked SBH. For our purposes, of the three phases predicted by GM08, only phase II (when the oscillation amplitude becomes comparable with the galaxy core radius) is of interest. The initial, large amplitude phase I oscillations are short-lived, while in phase III the residual oscillations are indistinguishable from Brownian motion (which is considered separately in Section 5.3).\\nHere we outline the method used to estimate the probability distributions for displacements resulting from postrecoil oscillations of a kicked SBH. For our purposes, of the three phases predicted by GM08, only phase II (when the oscillation amplitude becomes comparable with the galaxy core radius) is of interest. The initial, large amplitude phase I oscillations are short-lived, while in phase III the residual oscillations are indistinguishable from Brownian motion (which is considered separately in Section 5.3).\\nIn the GM08 N -body simulations, phase II oscillations occur for kick velocities in the range v kick ∼ 0.4 − 0.9 v esc . The simulations also indicate that the subsequent evolution does not depend strongly on the magnitude of the initial kick. Therefore, we assume that, given any initial kick large enough to displace the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy evolves during phase II as a damped harmonic oscillator:\\nIn the GM08 N -body simulations, phase II oscillations occur for kick velocities in the range v kick ∼ 0.4 − 0.9 v esc . The simulations also indicate that the subsequent evolution does not depend strongly on the magnitude of the initial kick. Therefore, we assume that, given any initial kick large enough to displace the SBH beyond the core radius, the distance, R, of the SBH from the center of the galaxy evolves during phase II as a damped harmonic oscillator:\\nThe initial amplitude, R 0 , is set by the condition that phase II begins when the oscillation amplitude is comparable to the core radius; ∆t is the time since the kick, i.e., the time since the last merger, and ω c is the SBH oscillation frequency:\\nThe initial amplitude, R 0 , is set by the condition that phase II begins when the oscillation amplitude is comparable to the core radius; ∆t is the time since the kick, i.e., the time since the last merger, and ω c is the SBH oscillation frequency:\\nis obtained by combining the radial oscillation frequency in a spherical core of uniform density, ω c = [(4π/3)Gρ c ] 1/2 , with equation 13 in GM08. The damping time, τ , is given by equation 2. The adopted parameter values are listed in Table 8.\\nis obtained by combining the radial oscillation frequency in a spherical core of uniform density, ω c = [(4π/3)Gρ c ] 1/2 , with equation 13 in GM08. The damping time, τ , is given by equation 2. The adopted parameter values are listed in Table 8.\\nSuppose that we observe a galaxy at a random time since its last merger. For simplicity, we assume that galaxy mergers occur stochastically and can be represented as a Poisson process. The probability that the last merger occurred at time ∆t ago is then:\\nSuppose that we observe a galaxy at a random time since its last merger. For simplicity, we assume that galaxy mergers occur stochastically and can be represented as a Poisson process. The probability that the last merger occurred at time ∆t ago is then:\\nwhich is associated with the cumulative distribution:\\nwhich is associated with the cumulative distribution:\\nwhere t m is the mean time between galaxy mergers. The galaxy merger rate is uncertain but simulations indicate that it is a function of redshift, galaxy mass and mass ratio (e.g., Bell et al. 2006;Conselice, Yang & Bluck 2009;Hopkins et al. 2010;Burke & Collins 2013). Here, since the phase II damping times are typically ∼ 1 Gyr (Table 8), we consider redshifts z ≤ 1 and estimate the mean time between galaxy mergers as the ratio between the lookback time corresponding to z = 1 and the number of mergers, N , during that time. The latter was determined using the analytical fits to semi-analytical models for galaxy merger rates given by H10. Integrating the merger rate as given by equation 5 in H10 over redshift, we find for the number of mergers,\\nwhere t m is the mean time between galaxy mergers. The galaxy merger rate is uncertain but simulations indicate that it is a function of redshift, galaxy mass and mass ratio (e.g., Bell et al. 2006;Conselice, Yang & Bluck 2009;Hopkins et al. 2010;Burke & Collins 2013). Here, since the phase II damping times are typically ∼ 1 Gyr (Table 8), we consider redshifts z ≤ 1 and estimate the mean time between galaxy mergers as the ratio between the lookback time corresponding to z = 1 and the number of mergers, N , during that time. The latter was determined using the analytical fits to semi-analytical models for galaxy merger rates given by H10. Integrating the merger rate as given by equation 5 in H10 over redshift, we find for the number of mergers,\\nwhere t H = 13.7 Gyr is the Hubble time, the normalization constant A(M min ) and the slope β(M min ) are given as functions of the galaxy mass threshold M min , for mergers with mass ratio q > 0.1, by equations 9 & 10 in H10. For log M gal ≥ log M min = 11, we find N ≈ 1.5 for the number of mergers per galaxy between z = 0 and z = 1. However, we note that N could vary between 0.75 and 3, allowing for systematic uncertainties of a factor two in A(M min ) and 0.2 in β(M min ). The look back time at z = 1 is t L = 7.731 Gyr, for H 0 =71 km s −1 Mpc −1 , Ω M = 0.27 and Ω vac = 0.73, yielding a mean time between galactic mergers t m ≈ 5 Gyr. The simple analytical functions provided by H10 do not accurately reproduce the numerical results for the most massive galaxies. Therefore, we use an average merger rate for z ≤ 1 estimated from their Figure 3, for log M gal > 12. The rate for q > 0.1 mergers is roughly constant for z ≤ 1 and it seems reasonable to adopt an average value of 2.5 mergers/galaxy/Gyr. This corresponds to t m ≈ 0.4 Gyr.\\nwhere t H = 13.7 Gyr is the Hubble time, the normalization constant A(M min ) and the slope β(M min ) are given as functions of the galaxy mass threshold M min , for mergers with mass ratio q > 0.1, by equations 9 & 10 in H10. For log M gal ≥ log M min = 11, we find N ≈ 1.5 for the number of mergers per galaxy between z = 0 and z = 1. However, we note that N could vary between 0.75 and 3, allowing for systematic uncertainties of a factor two in A(M min ) and 0.2 in β(M min ). The look back time at z = 1 is t L = 7.731 Gyr, for H 0 =71 km s −1 Mpc −1 , Ω M = 0.27 and Ω vac = 0.73, yielding a mean time between galactic mergers t m ≈ 5 Gyr. The simple analytical functions provided by H10 do not accurately reproduce the numerical results for the most massive galaxies. Therefore, we use an average merger rate for z ≤ 1 estimated from their Figure 3, for log M gal > 12. The rate for q > 0.1 mergers is roughly constant for z ≤ 1 and it seems reasonable to adopt an average value of 2.5 mergers/galaxy/Gyr. This corresponds to t m ≈ 0.4 Gyr.\\nThe stellar mass, M gal , is derived for each galaxy from the relation:\\nThe stellar mass, M gal , is derived for each galaxy from the relation:\\nwhere Υ is the mass-to-light ratio and M K is the K-band magnitude given by BC06. For the mass-to-light ratio, we use a representative value, Υ = 3.1, computed as the average value of the ratio between the virial mass and the K-band luminosity for the sample of nearby core ellipticals studied by Sani et al. (2011).\\nwhere Υ is the mass-to-light ratio and M K is the K-band magnitude given by BC06. For the mass-to-light ratio, we use a representative value, Υ = 3.1, computed as the average value of the ratio between the virial mass and the K-band luminosity for the sample of nearby core ellipticals studied by Sani et al. (2011).\\nAs the derived values of M gal span the range 11 log M gal 12 we compute displacement probability distributions for the limiting cases t m = 0.4 Gyr and t m = 5.0 Gyr. For comparison, Lidman et al. (2013) find an observationallyderived major merger rate of ≈ 0.38 mergers Gyr −1 for brightest cluster galaxies at z ∼ 1, implying t m ≈ 2.6 Gyr.\\nAs the derived values of M gal span the range 11 log M gal 12 we compute displacement probability distributions for the limiting cases t m = 0.4 Gyr and t m = 5.0 Gyr. For comparison, Lidman et al. (2013) find an observationallyderived major merger rate of ≈ 0.38 mergers Gyr −1 for brightest cluster galaxies at z ∼ 1, implying t m ≈ 2.6 Gyr.\\nFrom the cumulative distributions given by eq. A4, we extract random times in order to generate a set of R i ≡ R(t i ) from equation A1 for each galaxy, for both values of t m . Projected displacements are then calculated:\\nFrom the cumulative distributions given by eq. A4, we extract random times in order to generate a set of R i ≡ R(t i ) from equation A1 for each galaxy, for both values of t m . Projected displacements are then calculated:\\nwhere θ i is the angle between the displacement direction, assumed to be randomly oriented, and the line of sight.\\nwhere θ i is the angle between the displacement direction, assumed to be randomly oriented, and the line of sight.\\nFinally, the distribution of the projected offsets is computed. In Figs. 7a and 7b we plot the probability, P (d > x), of observing a displacement, d, larger than a given value, x, as a function of x. Values of the measured displacements are also plotted. The probability of generating a projected displacement larger than the value actually measured is listed for each galaxy in Table 6, for both values of t m .\\nFinally, the distribution of the projected offsets is computed. In Figs. 7a and 7b we plot the probability, P (d > x), of observing a displacement, d, larger than a given value, x, as a function of x. Values of the measured displacements are also plotted. The probability of generating a projected displacement larger than the value actually measured is listed for each galaxy in Table 6, for both values of t m .\\nThe damping time computed from eq.2 depends on the mass of the SBH. Measurement uncertainties for SBH masses in galaxies beyond the Local Group are typically large and are probably dominated by poorly-understood systematics, given that none of these galaxies exhibits a clear central rise in the stellar velocities (Merritt 2013, section 2.2). In Table 9 we list SBH masses estimated from the M • -σ relation (Ferrarese & Ford 2005), from the M • -M V relation for bright galaxies (M V < −19, Lauer et al. 2007) and from direct measurements (stellar or gas dynamics modeling). For most of our galaxies the discrepancy in the masses recovered with different methods is small. We compute damping times using SBH masses from the M • -M V relation finding a smaller average damping timeτ = 0.66 Gyr, although the new damping times for individual galaxies are not systematically smaller than those computed using the M • -σ SBH masses. We then compute new values for the probabilities presented in Table 6 and the corresponding cumulative probabilities of not observing displacements larger than those actually measured in this work (P). We obtain P = 10 −16 (10 −3 ), instead of 10 −17 (10 −4 ), for an average time between galactic mergers t m = 5 (0.4) Gyr. After taking into account the kick probabilities given by L12, as specified in §5.2, the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 1×10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5.0 Gyr. The different masses adopted for the SBH caused, therefore, a difference in P of one order of magnitude. Nevertheless, even for t m = 5 Gyr, there remains a large probability of finding displacements larger than those actually observed, if these galaxies experienced gravitational recoil events within the last few Gyr.\\nThe damping time computed from eq.2 depends on the mass of the SBH. Measurement uncertainties for SBH masses in galaxies beyond the Local Group are typically large and are probably dominated by poorly-understood systematics, given that none of these galaxies exhibits a clear central rise in the stellar velocities (Merritt 2013, section 2.2). In Table 9 we list SBH masses estimated from the M • -σ relation (Ferrarese & Ford 2005), from the M • -M V relation for bright galaxies (M V < −19, Lauer et al. 2007) and from direct measurements (stellar or gas dynamics modeling). For most of our galaxies the discrepancy in the masses recovered with different methods is small. We compute damping times using SBH masses from the M • -M V relation finding a smaller average damping timeτ = 0.66 Gyr, although the new damping times for individual galaxies are not systematically smaller than those computed using the M • -σ SBH masses. We then compute new values for the probabilities presented in Table 6 and the corresponding cumulative probabilities of not observing displacements larger than those actually measured in this work (P). We obtain P = 10 −16 (10 −3 ), instead of 10 −17 (10 −4 ), for an average time between galactic mergers t m = 5 (0.4) Gyr. After taking into account the kick probabilities given by L12, as specified in §5.2, the probability of not observing a displacement larger than those actually measured in the entire sample is P ≈ 1×10 −3 , instead of 8×10 −4 , for t m = 0.4 Gyr and P ≈ 0.14 instead of 0.05 for t m = 5.0 Gyr. The different masses adopted for the SBH caused, therefore, a difference in P of one order of magnitude. Nevertheless, even for t m = 5 Gyr, there remains a large probability of finding displacements larger than those actually observed, if these galaxies experienced gravitational recoil events within the last few Gyr.\"\n },\n {\n \"title\": \"B: NOTES ON INDIVIDUAL SOURCES\",\n \"content\": \"\"\n },\n {\n \"title\": \"B: NOTES ON INDIVIDUAL SOURCES\",\n \"content\": \"In this section, the galaxies are ordered according to the significance assigned to the displacement.\\nIn this section, the galaxies are ordered according to the significance assigned to the displacement.\"\n },\n {\n \"title\": \"NGC 4278 (high/intermediate)\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 4278 (high/intermediate)\",\n \"content\": \"NGC 4278 is a large elliptical near the center of the dense Coma I cloud which is characterized by the presence of an extended dust distribution extending over an arc ranging from N to NW relative to the photocenter (Carollo et al. 1997). Radio interferometry (VLBA) observations reveal a low-power, parsec-scale source (∼ 45 mas) with twin mildly-relativistic S-shaped jets emerging from a compact, flat spectrum core (Giroletti et al. 2005). Within the inner 7 mas the jets are oriented SE-NW but then bend to the east and west, respectively, forming an \\\"S\\\" shape. Giroletti et al. argue that the jets are closely aligned with the line of sight (with a viewing angle ∼ 2 − 4 • ) with the NW jet approaching and the bends being explained by amplification of intrinsically small deviations due to projection effects. These characteristics and the presence of atomic hydrogen suggest that the galaxy may have experienced a minor merger with another member of the cloud (Gerin & Casoli 1994).\\nNGC 4278 is a large elliptical near the center of the dense Coma I cloud which is characterized by the presence of an extended dust distribution extending over an arc ranging from N to NW relative to the photocenter (Carollo et al. 1997). Radio interferometry (VLBA) observations reveal a low-power, parsec-scale source (∼ 45 mas) with twin mildly-relativistic S-shaped jets emerging from a compact, flat spectrum core (Giroletti et al. 2005). Within the inner 7 mas the jets are oriented SE-NW but then bend to the east and west, respectively, forming an \\\"S\\\" shape. Giroletti et al. argue that the jets are closely aligned with the line of sight (with a viewing angle ∼ 2 − 4 • ) with the NW jet approaching and the bends being explained by amplification of intrinsically small deviations due to projection effects. These characteristics and the presence of atomic hydrogen suggest that the galaxy may have experienced a minor merger with another member of the cloud (Gerin & Casoli 1994).\\nPhotometric analysis. Three images, obtained with ACS/WFC-F850LP, WFPC2/PC-F814W and NIC-MOS2/F160W, respectively, were analyzed for this galaxy (Figs 15, 16 and 17). Photocenter-AGN displacements of intermediate or high significance are detected in all three. The uncertainty-weighted average displacement has an amplitude ∆r ≥ 7.6 ± 0.4 pc and direction SE (PA = 152 ± 3 • ). The photocenter positions derived from each image are plotted relative to the AGN position in Fig.5a. The photometric analysis of the optical ACS and WFPC2 images shows that the y co-ordinate of the isophote centers migrates southward at intermediate SMA, then northward at larger SMA. These irregularities are not present in the NICMOS2 image and are likely due to the dust N-NW of the center.\\nPhotometric analysis. Three images, obtained with ACS/WFC-F850LP, WFPC2/PC-F814W and NIC-MOS2/F160W, respectively, were analyzed for this galaxy (Figs 15, 16 and 17). Photocenter-AGN displacements of intermediate or high significance are detected in all three. The uncertainty-weighted average displacement has an amplitude ∆r ≥ 7.6 ± 0.4 pc and direction SE (PA = 152 ± 3 • ). The photocenter positions derived from each image are plotted relative to the AGN position in Fig.5a. The photometric analysis of the optical ACS and WFPC2 images shows that the y co-ordinate of the isophote centers migrates southward at intermediate SMA, then northward at larger SMA. These irregularities are not present in the NICMOS2 image and are likely due to the dust N-NW of the center.\\nOffset origin. The extended dust distribution produces a large-scale asymmetry in surface brightness, which is particularly evident in optical and UV wavelengths. This may bias the photocenter, shifting it in the opposite direction (S-SE) to the dust. As the AGN is a point source, its position should not be significantly affected by the dust distribution and thus it would appear to be shifted N-NW relative to the photocenter. Therefore, the offsets recovered Note. -(1) Galaxy name;\\nOffset origin. The extended dust distribution produces a large-scale asymmetry in surface brightness, which is particularly evident in optical and UV wavelengths. This may bias the photocenter, shifting it in the opposite direction (S-SE) to the dust. As the AGN is a point source, its position should not be significantly affected by the dust distribution and thus it would appear to be shifted N-NW relative to the photocenter. Therefore, the offsets recovered Note. -(1) Galaxy name;\\n(2) core radius from CB05; (3) stellar velocity dispersion from Hyperleda; (4) stellar mass of the galaxy; (5) dynamical time; (6) damping time; (7) kick velocity required to displace the SBH up to the core radius. The dominant source of errors in the quantities presented above is the stellar velocity dispersion. Note. -(1) Galaxy name, (2) masses from the M• − σ (Ferrarese & Ford 2005) using the velocity dispersion specified in Table 8 from ACS and WFPC2 data may be partly due to asymmetric extinction. However, the effects of dust are not apparent in the NIR (NICMOS2) image, from which an offset of similar magnitude and direction was recovered. Therefore, we believe that the measured displacement is not simply an artifact of extinction, although this may contribute. Relative to the core radius, NGC 4278 has the largest displacement (≈ 0.1r c ; Table 6) in our sample. If it is due to phase II oscillations of the SBH following gravitational recoil (see §5.2), we estimate that the probability of observing a projected displacement larger than that actually measured is 4 (36)% for a mean time-between-mergers of t m = 5 (0.4) Gyr (Table 6) and given a kick large enough to displace the SBH at least as far as the core radius. The \\\"hot\\\" disk recoil velocity distribution of L12 indicates a 45% probability of a sufficiently large kick ( 200 km s −1 , Table 8) occurring in an SBH coalescence event. Therefore, we consider residual oscillations following gravitational recoil to be a plausible explanation for the displacement.\\n(2) core radius from CB05; (3) stellar velocity dispersion from Hyperleda; (4) stellar mass of the galaxy; (5) dynamical time; (6) damping time; (7) kick velocity required to displace the SBH up to the core radius. The dominant source of errors in the quantities presented above is the stellar velocity dispersion. Note. -(1) Galaxy name, (2) masses from the M• − σ (Ferrarese & Ford 2005) using the velocity dispersion specified in Table 8 from ACS and WFPC2 data may be partly due to asymmetric extinction. However, the effects of dust are not apparent in the NIR (NICMOS2) image, from which an offset of similar magnitude and direction was recovered. Therefore, we believe that the measured displacement is not simply an artifact of extinction, although this may contribute. Relative to the core radius, NGC 4278 has the largest displacement (≈ 0.1r c ; Table 6) in our sample. If it is due to phase II oscillations of the SBH following gravitational recoil (see §5.2), we estimate that the probability of observing a projected displacement larger than that actually measured is 4 (36)% for a mean time-between-mergers of t m = 5 (0.4) Gyr (Table 6) and given a kick large enough to displace the SBH at least as far as the core radius. The \\\"hot\\\" disk recoil velocity distribution of L12 indicates a 45% probability of a sufficiently large kick ( 200 km s −1 , Table 8) occurring in an SBH coalescence event. Therefore, we consider residual oscillations following gravitational recoil to be a plausible explanation for the displacement.\\nIn this case, we can estimate the merger epoch. Supposing the SBH to be undergoing phase II oscillations, we combine eq.2 and 3. Using the measured (projected) offset as a lower limit for r rms , r rms 7.6 pc, and taking the values of r c (core radius) and σ (stellar velocity dispersion) from Table 8, we find that (t − t c ) 5τ ≈ 0.7 Gyr has elapsed since the amplitude of the oscillation damped down to r ∼ r c . This implies that the onset of phase II occurred no later than ∼ 0.7 Gyr ago. As phase I is shorter than phase II, this scenario is consistent with an SBH-SBH coalescence event that occurred within 2(t − t c ) ∼ 1.4 Gyr. The total evolution time from the beginning of the galactic merger to SBH binary coalescence is suggested to be of order 10 8 yr (Gualandris & Merritt 2012). Adding this time to the estimated kick epoch, we infer that a galaxy merger that took place up to ∼ 1.5 Gyr in the past could be responsible for a recoil kick consistent with the observed displacement.\\nIn this case, we can estimate the merger epoch. Supposing the SBH to be undergoing phase II oscillations, we combine eq.2 and 3. Using the measured (projected) offset as a lower limit for r rms , r rms 7.6 pc, and taking the values of r c (core radius) and σ (stellar velocity dispersion) from Table 8, we find that (t − t c ) 5τ ≈ 0.7 Gyr has elapsed since the amplitude of the oscillation damped down to r ∼ r c . This implies that the onset of phase II occurred no later than ∼ 0.7 Gyr ago. As phase I is shorter than phase II, this scenario is consistent with an SBH-SBH coalescence event that occurred within 2(t − t c ) ∼ 1.4 Gyr. The total evolution time from the beginning of the galactic merger to SBH binary coalescence is suggested to be of order 10 8 yr (Gualandris & Merritt 2012). Adding this time to the estimated kick epoch, we infer that a galaxy merger that took place up to ∼ 1.5 Gyr in the past could be responsible for a recoil kick consistent with the observed displacement.\\nAlthough the displacement is approximately in the direction of elongation of the mas-scale radio core (155 • over the inner ∼ 7 mas), the low power (P 5GHz ≈ 70 × 10 20 W, Wrobel 1991) and compact nature (∼ 3 pc) of the radio source argue against jet thrust as the cause of the offset.\\nAlthough the displacement is approximately in the direction of elongation of the mas-scale radio core (155 • over the inner ∼ 7 mas), the low power (P 5GHz ≈ 70 × 10 20 W, Wrobel 1991) and compact nature (∼ 3 pc) of the radio source argue against jet thrust as the cause of the offset.\\nWe find no compelling reasons to exclude either a stalled binary on an eccentric orbit, or massive perturbers (Section 5.3) as possible causes of the displacement.\\nWe find no compelling reasons to exclude either a stalled binary on an eccentric orbit, or massive perturbers (Section 5.3) as possible causes of the displacement.\"\n },\n {\n \"title\": \"NGC 5846 (intermediate)\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 5846 (intermediate)\",\n \"content\": \"NGC 5846 is the dominant component of a small, compact group of ≈ 50 galaxies (Huchra & Geller 1982). A companion galaxy, NGC5846A, is located 0.7 arcmin to the S, but there is no evidence of an interaction. The presence of dust lanes in an apparent spiral pattern approximately centered on the compact radio core suggests a past merger, or accretion of a gas rich galaxy (Moellenhoff et al. 1992;Forbes et al. 1996). The optical nucleus has a LINER spectrum (Ho et al. 1997) and HST imaging (Masegosa et al. 2011) reveals extended Hα emission in a wide structure extending up to 2 W of the nucleus. The presence of two radio emitting bubbles located roughly symmetrically at ∼ 0.6 kpc from the center and a \\\"ghost\\\" X-ray cavity suggest that the galaxy recently experienced stronger AGN activity, for which Machacek et al. (2011) derive a duty cycle of ∼ 10 Myr. The radio morphology is complex: Filho et al. (2004) report VLBA observations at 2.3, 5 and 15 GHz which resolve the core into multiple blobs roughly aligned along the N-S axis. The nature of this structure is unclear, possibilities include an AGN jet, or compact supernovae remnants (e.g. Tarchi et al. 2000).\\nNGC 5846 is the dominant component of a small, compact group of ≈ 50 galaxies (Huchra & Geller 1982). A companion galaxy, NGC5846A, is located 0.7 arcmin to the S, but there is no evidence of an interaction. The presence of dust lanes in an apparent spiral pattern approximately centered on the compact radio core suggests a past merger, or accretion of a gas rich galaxy (Moellenhoff et al. 1992;Forbes et al. 1996). The optical nucleus has a LINER spectrum (Ho et al. 1997) and HST imaging (Masegosa et al. 2011) reveals extended Hα emission in a wide structure extending up to 2 W of the nucleus. The presence of two radio emitting bubbles located roughly symmetrically at ∼ 0.6 kpc from the center and a \\\"ghost\\\" X-ray cavity suggest that the galaxy recently experienced stronger AGN activity, for which Machacek et al. (2011) derive a duty cycle of ∼ 10 Myr. The radio morphology is complex: Filho et al. (2004) report VLBA observations at 2.3, 5 and 15 GHz which resolve the core into multiple blobs roughly aligned along the N-S axis. The nature of this structure is unclear, possibilities include an AGN jet, or compact supernovae remnants (e.g. Tarchi et al. 2000).\\nPhotometric Analysis. One image, obtained with WFPC2/PC -F814W was analyzed for this galaxy (Figure 24). We find a photocenter-AGN displacement ∆r 8.2 ± 2.5 pc in the direction W-SW (PA = 253 ± 8 • ) which is considered to be of intermediate significance. The isophote centers show systematic migrations of ≈ 2 pixels in both x and y coordinates, from W to E and S to N, respectively, as the SMA decreases from 300 to 200 pixels. These migrations are possibly the result of the dust lanes to the S and NE of the nucleus. The y-component of the displacement is not significant. The x-component is considered to be of intermediate significance (1.6 IQR) and is unlikely to be an artifact of extinction, which in this case, would bias the photocenter such as to increase the x-displacement. Of more concern is the determination of the AGN position. The central source is relatively weak and embedded in an asymmetric structure that affects the gaussian fits used to determine the AGN position. We therefore assign a larger uncertainty to the x-component of the AGN position.\\nPhotometric Analysis. One image, obtained with WFPC2/PC -F814W was analyzed for this galaxy (Figure 24). We find a photocenter-AGN displacement ∆r 8.2 ± 2.5 pc in the direction W-SW (PA = 253 ± 8 • ) which is considered to be of intermediate significance. The isophote centers show systematic migrations of ≈ 2 pixels in both x and y coordinates, from W to E and S to N, respectively, as the SMA decreases from 300 to 200 pixels. These migrations are possibly the result of the dust lanes to the S and NE of the nucleus. The y-component of the displacement is not significant. The x-component is considered to be of intermediate significance (1.6 IQR) and is unlikely to be an artifact of extinction, which in this case, would bias the photocenter such as to increase the x-displacement. Of more concern is the determination of the AGN position. The central source is relatively weak and embedded in an asymmetric structure that affects the gaussian fits used to determine the AGN position. We therefore assign a larger uncertainty to the x-component of the AGN position.\\nOffset origin. For t m = 5 (0.4) Gyr, we estimate a 25 (90)% probability of finding a displacement larger than that observed, if due to phase II recoil oscillations. The probability of a recoil kick large enough ( 240 km s −1 ; Table 8) to initiate these oscillations is ∼45% (L12).\\nOffset origin. For t m = 5 (0.4) Gyr, we estimate a 25 (90)% probability of finding a displacement larger than that observed, if due to phase II recoil oscillations. The probability of a recoil kick large enough ( 240 km s −1 ; Table 8) to initiate these oscillations is ∼45% (L12).\\nAs for NGC 4278, we use equations 2 and 3 and values from Table 8 to estimate the time since the onset of phase II, finding (t-t c ) 6.2 τ or 4 Gyr. This implies an upper limit to the time elapsed since the putative SBH-binary coalescence event of 2(t-t c ) or 8 Gyr.\\nAs for NGC 4278, we use equations 2 and 3 and values from Table 8 to estimate the time since the onset of phase II, finding (t-t c ) 6.2 τ or 4 Gyr. This implies an upper limit to the time elapsed since the putative SBH-binary coalescence event of 2(t-t c ) or 8 Gyr.\\nAlthough, as noted, there is \\\"fossil\\\" evidence for previous jet activity, the AGN does not currently produce a powerful radio jet, so it seems unlikely that the displacement is due to jet acceleration. However, interactions with massive perturbers or an SBH binary on a highly eccentric orbit remain possibilities. NGC 4486 (M87,intermediate) NGC 4486 (M87) is the dominant galaxy of the Virgo Cluster. It hosts the best studied extragalactic jet, which exhibits prominent knots at optical and IR wavelengths yet, apart from this, the brightness profile is regular and featureless. The peculiarly large population of globular clusters, its size and location, and the presence of stellar streams support the hypothesis that the galaxy has experienced a number of minor mergers during its lifetime (Janowiecki et al. 2010). A remarkable feature is the unusually large core, which has a radius r c ∼ 9. 41 (CB05). This characteristic makes M87 a special case in our sample in that for all the other galaxies we restricted the isophotal analysis to radii larger than the core radius, whereas for M87 we analyzed isophotes within the core radius in order to be consistent with B10.\\nAlthough, as noted, there is \\\"fossil\\\" evidence for previous jet activity, the AGN does not currently produce a powerful radio jet, so it seems unlikely that the displacement is due to jet acceleration. However, interactions with massive perturbers or an SBH binary on a highly eccentric orbit remain possibilities. NGC 4486 (M87,intermediate) NGC 4486 (M87) is the dominant galaxy of the Virgo Cluster. It hosts the best studied extragalactic jet, which exhibits prominent knots at optical and IR wavelengths yet, apart from this, the brightness profile is regular and featureless. The peculiarly large population of globular clusters, its size and location, and the presence of stellar streams support the hypothesis that the galaxy has experienced a number of minor mergers during its lifetime (Janowiecki et al. 2010). A remarkable feature is the unusually large core, which has a radius r c ∼ 9. 41 (CB05). This characteristic makes M87 a special case in our sample in that for all the other galaxies we restricted the isophotal analysis to radii larger than the core radius, whereas for M87 we analyzed isophotes within the core radius in order to be consistent with B10.\\nPhotometric analysis. A total of six images obtained with ACS/HRC, WFPC2/PC and NICMOS2 were analyzed (see Table 1), including the combined ACS images (ACS/HRC/F606W and ACS/HRC/F814W) previously analyzed by B10. The results of the photometric analysis are shown in Figs. 4 and 18-22 and the measured displacements relative to the AGN are plotted in Fig. 5a. In all images, we analyzed isophotes within the range 1 ≤ r ≤ 3 , for consistency with B10. The displacements measured in different images vary in amplitude and significance. Notably, none of the three NICMOS2 images (F110W, F160W and F222W) exhibit significant displacements at the ≥ 3σ level. Furthermore, the 1σ uncertainties, when converted to linear distances, do not encompass the significant displacements recovered from the optical images. The displacements measured from the ACS and WFPC2 images also differ in amplitude but there is a tendency for the photocenter to be displaced NW of the AGN, roughly in the direction of the radio jet. The isophote centers are reasonably stable -although there are deviations ∼ 1 − 2 pixels (≤ 0. 1), systematic trends with SMA are not generally evident . The most significant displacement is obtained from the ACS/HRC F814 image, for which the isophote centers cluster tightly around the mean photocenter (Fig. 4). The measured displacement of the photocenter relative to the AGN is ∆r = 7.7 ± 0.3 pc NW, implying that the SBH is displaced by ∆r in the counter-jet direction. This result is in agreement with the displacement obtained by B10 from a similar analysis. The main differences between the method used by B10 and that employed in this paper are the iterative technique used here to generate the mask and the weight function used to derive the isophotal center: while B10 used a conventional uncertainty weight, here we weight for the light content of each isophotal annulus (see §3.1). The weighted average displacement has a magnitude 4.3 ± 0.2 pc and direction NW (PA = 307 ± 1 • ). Offset origin. The presence of a large number of bright globular clusters visible in all the images analyzed allows us to define a common reference frame which we can use to compare results from images obtained with different instrument/filter combinations. After selecting several globular clusters as reference points and matching their positions, we plotted the AGN and photocenter positions determined from the different images in the reference frame defined by the ACS image (Fig. 8). The AGN position as measured from the different images is fairly stable whereas the photocenters exhibit differences in position much larger than the scatter (∼ 0.5 pixels) in the AGN position. This indicates that the differences in the displacements are due largely to differences in the positions of the photocenters. The origin of these discrepancies is unclear. Unlike other galaxies, there is no evidence of dust extinction or other asymmetries that may account for different results at different wavelengths.\\nPhotometric analysis. A total of six images obtained with ACS/HRC, WFPC2/PC and NICMOS2 were analyzed (see Table 1), including the combined ACS images (ACS/HRC/F606W and ACS/HRC/F814W) previously analyzed by B10. The results of the photometric analysis are shown in Figs. 4 and 18-22 and the measured displacements relative to the AGN are plotted in Fig. 5a. In all images, we analyzed isophotes within the range 1 ≤ r ≤ 3 , for consistency with B10. The displacements measured in different images vary in amplitude and significance. Notably, none of the three NICMOS2 images (F110W, F160W and F222W) exhibit significant displacements at the ≥ 3σ level. Furthermore, the 1σ uncertainties, when converted to linear distances, do not encompass the significant displacements recovered from the optical images. The displacements measured from the ACS and WFPC2 images also differ in amplitude but there is a tendency for the photocenter to be displaced NW of the AGN, roughly in the direction of the radio jet. The isophote centers are reasonably stable -although there are deviations ∼ 1 − 2 pixels (≤ 0. 1), systematic trends with SMA are not generally evident . The most significant displacement is obtained from the ACS/HRC F814 image, for which the isophote centers cluster tightly around the mean photocenter (Fig. 4). The measured displacement of the photocenter relative to the AGN is ∆r = 7.7 ± 0.3 pc NW, implying that the SBH is displaced by ∆r in the counter-jet direction. This result is in agreement with the displacement obtained by B10 from a similar analysis. The main differences between the method used by B10 and that employed in this paper are the iterative technique used here to generate the mask and the weight function used to derive the isophotal center: while B10 used a conventional uncertainty weight, here we weight for the light content of each isophotal annulus (see §3.1). The weighted average displacement has a magnitude 4.3 ± 0.2 pc and direction NW (PA = 307 ± 1 • ). Offset origin. The presence of a large number of bright globular clusters visible in all the images analyzed allows us to define a common reference frame which we can use to compare results from images obtained with different instrument/filter combinations. After selecting several globular clusters as reference points and matching their positions, we plotted the AGN and photocenter positions determined from the different images in the reference frame defined by the ACS image (Fig. 8). The AGN position as measured from the different images is fairly stable whereas the photocenters exhibit differences in position much larger than the scatter (∼ 0.5 pixels) in the AGN position. This indicates that the differences in the displacements are due largely to differences in the positions of the photocenters. The origin of these discrepancies is unclear. Unlike other galaxies, there is no evidence of dust extinction or other asymmetries that may account for different results at different wavelengths.\\nWe conducted several trials in order to determine the influence of (i) the degree of masking applied to the jet, (ii) the bright knot visible NW of the nucleus in the ACS image. The results were found to be independent of the degree of masking; no significant differences in the photocenter positions were found when the analysis was performed with no masking at all or with heavy jet/knot-masking.\\nWe conducted several trials in order to determine the influence of (i) the degree of masking applied to the jet, (ii) the bright knot visible NW of the nucleus in the ACS image. The results were found to be independent of the degree of masking; no significant differences in the photocenter positions were found when the analysis was performed with no masking at all or with heavy jet/knot-masking.\\nAs a fraction of the rather large core radius, the weighted mean displacement is only ≈ 0.01 r c (Table 6). We estimate a 85 (99)% probability of finding a displacement larger than that observed for t m = 5 (0.4) Gyr, assuming phase II recoil oscillations were triggered. The probability of obtaining the relatively large recoil kick ( 325 km s −1 ; Table 8) required for phase II oscillations is ≈ 33% (L12).\\nAs a fraction of the rather large core radius, the weighted mean displacement is only ≈ 0.01 r c (Table 6). We estimate a 85 (99)% probability of finding a displacement larger than that observed for t m = 5 (0.4) Gyr, assuming phase II recoil oscillations were triggered. The probability of obtaining the relatively large recoil kick ( 325 km s −1 ; Table 8) required for phase II oscillations is ≈ 33% (L12).\\nThe apparent close alignment of the offset with the jet direction favors jet acceleration but is also generally consistent with gravitational recoil. However, as there is no reason to expect such an alignment if the displacement is due to gravitational interactions with massive perturbers, or orbital motion in a SBH-binary, these mechanisms are disfavored. A detailed discussion of the possible displacement mechanisms is given in §4 of B10.\\nThe apparent close alignment of the offset with the jet direction favors jet acceleration but is also generally consistent with gravitational recoil. However, as there is no reason to expect such an alignment if the displacement is due to gravitational interactions with massive perturbers, or orbital motion in a SBH-binary, these mechanisms are disfavored. A detailed discussion of the possible displacement mechanisms is given in §4 of B10.\"\n },\n {\n \"title\": \"NGC 1399 (low)\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 1399 (low)\",\n \"content\": \"NGC 1399 is a regular giant E1 and is the central and dominant galaxy of the Fornax cluster (Ferguson 1989). It is associated with a low power radio source characterized by two nearly bi-symmetric jets extending ∼ 10 kpc, approximately N and S of the core (PA ≈ 9 • W of N, ). Shurkin et al. (2008) report the presence of X-ray cavities associated with the radio lobes. NGC 1399 has been the subject of a number of dynamical studies (e.g. Houghton et al. (2006) and references therein). In particular, Houghton et al. performed near-infrared adaptive optics assisted spectroscopic observations of the central 1. 5 of the galaxy and found evidence of a kinematically decoupled core and offset asymmetric isophotes that they considered to be consistent with the presence of a M31-like double nucleus. The asymmetric isophotes, elongated in the direction E-SE, are localized in a region much smaller than that used for our isophotal analysis (r > 2. 5).\\nNGC 1399 is a regular giant E1 and is the central and dominant galaxy of the Fornax cluster (Ferguson 1989). It is associated with a low power radio source characterized by two nearly bi-symmetric jets extending ∼ 10 kpc, approximately N and S of the core (PA ≈ 9 • W of N, ). Shurkin et al. (2008) report the presence of X-ray cavities associated with the radio lobes. NGC 1399 has been the subject of a number of dynamical studies (e.g. Houghton et al. (2006) and references therein). In particular, Houghton et al. performed near-infrared adaptive optics assisted spectroscopic observations of the central 1. 5 of the galaxy and found evidence of a kinematically decoupled core and offset asymmetric isophotes that they considered to be consistent with the presence of a M31-like double nucleus. The asymmetric isophotes, elongated in the direction E-SE, are localized in a region much smaller than that used for our isophotal analysis (r > 2. 5).\\nPhotometric analysis. Four images, obtained with WFPC2/PC -F606W, F814W and WFC3/IR -F110W, F160W, were analyzed for this galaxy. From the F606W image we find a low significance displacement of amplitude 1.7 ± 0.4 pc, in almost the same N-NW direction as the radio jet axis, PA = −8±16 • . A 2σ offset in the same direction was recovered from the WFPC2/PC -F814W image, which is consistent within the errors with the F606W result. Offsets recovered from the WFC3 images are classified as non-significant, but, given their large error bars, they are consistent with the WFPC2 results (Fig. 5a). The error weighted offset, computed including results from both WFPC2 and WFC3, is 1.5 ± 0.4 pc with PA = -17 ± 16 • . The isophote fitting analysis (Figs. 9 to 12) reveals a remarkably regular photometric structure well modeled by concentric elliptical isophotes. For both images, the isophote center co-ordinates vary little with SMA and are very tightly clustered, with narrow ( 0.5 pixel) distributions in x and y.\\nPhotometric analysis. Four images, obtained with WFPC2/PC -F606W, F814W and WFC3/IR -F110W, F160W, were analyzed for this galaxy. From the F606W image we find a low significance displacement of amplitude 1.7 ± 0.4 pc, in almost the same N-NW direction as the radio jet axis, PA = −8±16 • . A 2σ offset in the same direction was recovered from the WFPC2/PC -F814W image, which is consistent within the errors with the F606W result. Offsets recovered from the WFC3 images are classified as non-significant, but, given their large error bars, they are consistent with the WFPC2 results (Fig. 5a). The error weighted offset, computed including results from both WFPC2 and WFC3, is 1.5 ± 0.4 pc with PA = -17 ± 16 • . The isophote fitting analysis (Figs. 9 to 12) reveals a remarkably regular photometric structure well modeled by concentric elliptical isophotes. For both images, the isophote center co-ordinates vary little with SMA and are very tightly clustered, with narrow ( 0.5 pixel) distributions in x and y.\\nOffset origin. The weighted mean displacement is small compared to the core radius, ≈ 0.01 r c (Table 6). A minimum kick velocity 300 km s −1 (Table 8), is required to trigger phase II recoil oscillations; the probability of generating a kick of at least this magnitude is ∼ 33% (L12). Assuming that phase II recoil oscillations were triggered, we estimate the probability of finding a displacement larger than that observed to be 11 (75)%, for t m = 5 (0.4) Gyr.\\nOffset origin. The weighted mean displacement is small compared to the core radius, ≈ 0.01 r c (Table 6). A minimum kick velocity 300 km s −1 (Table 8), is required to trigger phase II recoil oscillations; the probability of generating a kick of at least this magnitude is ∼ 33% (L12). Assuming that phase II recoil oscillations were triggered, we estimate the probability of finding a displacement larger than that observed to be 11 (75)%, for t m = 5 (0.4) Gyr.\\nIf the observed displacement is due to a recoiling SBH, we estimate that (t -t c ) 1.6 Gyr has elapsed since the oscillation damped below the core radius, using τ = 0.17 Gyr and σ = 346 km s −1 ( Table 8).\\nIf the observed displacement is due to a recoiling SBH, we estimate that (t -t c ) 1.6 Gyr has elapsed since the oscillation damped below the core radius, using τ = 0.17 Gyr and σ = 346 km s −1 ( Table 8).\\nThe close alignment of the displacement with the radio axis is consistent with both recoil and jet acceleration as the cause of the displacement. If the latter, eq. 7 implies that, for a Bondi accretion rateṀ a ∼ 0.04 M yr −1 (Loewenstein et al. 2001) and a 10 9 M black hole, an asymmetry in the radio jet power in the range 3 × 10 −4 f jet 3 × 10 −3 that persists for t ∼ 10 8 yr would be needed to produce an offset 1.8 ∆r 18 pc.\\nThe close alignment of the displacement with the radio axis is consistent with both recoil and jet acceleration as the cause of the displacement. If the latter, eq. 7 implies that, for a Bondi accretion rateṀ a ∼ 0.04 M yr −1 (Loewenstein et al. 2001) and a 10 9 M black hole, an asymmetry in the radio jet power in the range 3 × 10 −4 f jet 3 × 10 −3 that persists for t ∼ 10 8 yr would be needed to produce an offset 1.8 ∆r 18 pc.\\nOther mechanisms (interactions with massive perturbers and a stalled SBH binary), are disfavored by the alignment between the displacement and jet direction. The presence of an unresolved M31-like double nucleus as suggested by Houghton et al. (2006) might also explain the measured offset, although as in the case of a binary SBH, the alignment with the radio source axis would have to be regarded as fortuitous.\\nOther mechanisms (interactions with massive perturbers and a stalled SBH binary), are disfavored by the alignment between the displacement and jet direction. The presence of an unresolved M31-like double nucleus as suggested by Houghton et al. (2006) might also explain the measured offset, although as in the case of a binary SBH, the alignment with the radio source axis would have to be regarded as fortuitous.\"\n },\n {\n \"title\": \"IC 4296 (low)\",\n \"content\": \"\"\n },\n {\n \"title\": \"IC 4296 (low)\",\n \"content\": \"The properties of this galaxy have been extensively studied in a series of papers by Killeen et al. (1986a,b) and . It is a fairly circular elliptical which is the brightest component in the small group A3565. It is associated with the extended low-to-intermediate luminosity radio source PKS 1333-33 (Mills et al. 1960), which has two quasi-symmetric slightly curved jets with an average position angle ∼ 130 • and terminating in lobes ∼ 200 kpc from the core (Killeen et al. 1986a). The curvature along the jets is consistent with the motion of the galaxy through the group inter-galactic medium ). On the parsec scale, a jet and counter-jet emerge from the unresolved core in PA ≈ 140 • , consistent with the large-scale radio source, with the brighter jet on the NW side. The brightness asymmetry is attributed to Doppler boosting (Pellegrini et al. 2003). Isophotal studies suggest tidal interactions with the smaller companion IC 4299 (Younis et al. 1985). Nevertheless, Efstathiou et al. (1980) show that the isophotal ellipticity and position angle are constant at all radii from 2.4 to 48.8 arcsec (a region much larger than that studied in this work, 1. 5 < ∆r < 6. 5), which suggests that any distortions due to interactions with companion galaxies are negligible over this scale.\\nThe properties of this galaxy have been extensively studied in a series of papers by Killeen et al. (1986a,b) and . It is a fairly circular elliptical which is the brightest component in the small group A3565. It is associated with the extended low-to-intermediate luminosity radio source PKS 1333-33 (Mills et al. 1960), which has two quasi-symmetric slightly curved jets with an average position angle ∼ 130 • and terminating in lobes ∼ 200 kpc from the core (Killeen et al. 1986a). The curvature along the jets is consistent with the motion of the galaxy through the group inter-galactic medium ). On the parsec scale, a jet and counter-jet emerge from the unresolved core in PA ≈ 140 • , consistent with the large-scale radio source, with the brighter jet on the NW side. The brightness asymmetry is attributed to Doppler boosting (Pellegrini et al. 2003). Isophotal studies suggest tidal interactions with the smaller companion IC 4299 (Younis et al. 1985). Nevertheless, Efstathiou et al. (1980) show that the isophotal ellipticity and position angle are constant at all radii from 2.4 to 48.8 arcsec (a region much larger than that studied in this work, 1. 5 < ∆r < 6. 5), which suggests that any distortions due to interactions with companion galaxies are negligible over this scale.\\nPhotometric analysis. Three images, obtained with ACS/HRC-F625W, WFPC2/PC-F814W and NICMOS2-F160W, were analyzed for this galaxy. In the ACS image a displacement of ∆r ≥ 5.2 ± 1 pc was found. On the basis of the IQR, that is considered to be of low significance. The displacement obtained from the NICMOS2 image is not considered significant, but is consistent with the ACS result, albeit within large errors. For the WFPC2 image, the y-component of the displacement is consistent with the ACS result but the x component exhibits a marginally significant difference and in fact indicates a displacement in the opposite direction (E). However, neither component of the WFPC2 displacement is classified as significant. These results are summarized in Fig. 5a.\\nPhotometric analysis. Three images, obtained with ACS/HRC-F625W, WFPC2/PC-F814W and NICMOS2-F160W, were analyzed for this galaxy. In the ACS image a displacement of ∆r ≥ 5.2 ± 1 pc was found. On the basis of the IQR, that is considered to be of low significance. The displacement obtained from the NICMOS2 image is not considered significant, but is consistent with the ACS result, albeit within large errors. For the WFPC2 image, the y-component of the displacement is consistent with the ACS result but the x component exhibits a marginally significant difference and in fact indicates a displacement in the opposite direction (E). However, neither component of the WFPC2 displacement is classified as significant. These results are summarized in Fig. 5a.\\nThe discrepancy in the photocenter positions appears to be linked to systematic migrations in the isophote center coordinates determined from the fits in the NICMOS2 and WFPC2 images: there are systematic migrations northwards in the NICMOS2 image (Fig. 31) and eastwards in the WFPC2 image (Fig. 30) resulting in distributions with large IQRs in the y and x co-ordinates, respectively. In contrast, the co-ordinates extracted from the ACS image (Fig. 29) show little variation with SMA and cluster quite symmetrically around the photocenter. This galaxy exhibits a warped dusty disk, which crosses the nucleus in a roughly E-W direction and is clearly visible in all three images (Figs. 29,30 and 31). This feature is too small (r 0. 9) to affect the photocenter position, which is calculated from isophote fits over radii 1. 5 ≤ r ≤ 6. 5. However, it does create large gradients in the background brightness around the nucleus which affect the gaussian fits to the central peak, especially when large fitting-boxes are used. The effects are particularly noticeable for the WFPC2 image, where the AGN position systematically shifts South-East as the size of the fitting-box is increased. This was mitigated by reducing the size of the fitting-box until stable values were found for the AGN co-ordinates.\\nThe discrepancy in the photocenter positions appears to be linked to systematic migrations in the isophote center coordinates determined from the fits in the NICMOS2 and WFPC2 images: there are systematic migrations northwards in the NICMOS2 image (Fig. 31) and eastwards in the WFPC2 image (Fig. 30) resulting in distributions with large IQRs in the y and x co-ordinates, respectively. In contrast, the co-ordinates extracted from the ACS image (Fig. 29) show little variation with SMA and cluster quite symmetrically around the photocenter. This galaxy exhibits a warped dusty disk, which crosses the nucleus in a roughly E-W direction and is clearly visible in all three images (Figs. 29,30 and 31). This feature is too small (r 0. 9) to affect the photocenter position, which is calculated from isophote fits over radii 1. 5 ≤ r ≤ 6. 5. However, it does create large gradients in the background brightness around the nucleus which affect the gaussian fits to the central peak, especially when large fitting-boxes are used. The effects are particularly noticeable for the WFPC2 image, where the AGN position systematically shifts South-East as the size of the fitting-box is increased. This was mitigated by reducing the size of the fitting-box until stable values were found for the AGN co-ordinates.\\nThe average error-weighted offset obtained for this galaxy from the ACS, NICMOS2 and WFPC2 images is ∆r ≥ 3.8 ± 0.7 pc in PA = -22 ± 7 • E of N.\\nThe average error-weighted offset obtained for this galaxy from the ACS, NICMOS2 and WFPC2 images is ∆r ≥ 3.8 ± 0.7 pc in PA = -22 ± 7 • E of N.\\nOffset origin. The status of IC 4296 as the brightest member of its group and the fact that it has a kinematically peculiar core (Killeen et al. 1986a) suggest that this galaxy has experienced minor mergers during the past few Gyr.\\nOffset origin. The status of IC 4296 as the brightest member of its group and the fact that it has a kinematically peculiar core (Killeen et al. 1986a) suggest that this galaxy has experienced minor mergers during the past few Gyr.\\nThe weighted mean displacement is a small fraction of the core radius, ≈ 0.01 r c ( Table 6). The probability of generating a recoil kick large enough to trigger phase II recoil oscillations ( 200 km s −1 ; Table 8), is ∼ 45% (L12) and given that phase II oscillations were triggered, we estimate a 37 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr.\\nThe weighted mean displacement is a small fraction of the core radius, ≈ 0.01 r c ( Table 6). The probability of generating a recoil kick large enough to trigger phase II recoil oscillations ( 200 km s −1 ; Table 8), is ∼ 45% (L12) and given that phase II oscillations were triggered, we estimate a 37 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr.\\nUsing eqs. 2 and 3 with σ = 333 km s −1 , r rms ≥ 3.8 pc and r c = 347 pc, we find (t − t c ) 9 τ ≈ 5.8 Gyr. In this case, the low density of the large core is responsible for the long damping time.\\nUsing eqs. 2 and 3 with σ = 333 km s −1 , r rms ≥ 3.8 pc and r c = 347 pc, we find (t − t c ) 9 τ ≈ 5.8 Gyr. In this case, the low density of the large core is responsible for the long damping time.\\nIt has been pointed out by Killeen et al. (1986a) that on kpc scales the radio jets exhibit transverse oscillations with wavelengths ∼ 10 kpc and an amplitude comparable with the jet width. Similar oscillation wavelengths are expected from precession due to orbital motion in a ∼ pc scale SBH binary (see Section 5.3). However, found that, in detail, the oscillation pattern is not consistent with precession and favor instead helical Kelvin-Helmholtz instabilities.\\nIt has been pointed out by Killeen et al. (1986a) that on kpc scales the radio jets exhibit transverse oscillations with wavelengths ∼ 10 kpc and an amplitude comparable with the jet width. Similar oscillation wavelengths are expected from precession due to orbital motion in a ∼ pc scale SBH binary (see Section 5.3). However, found that, in detail, the oscillation pattern is not consistent with precession and favor instead helical Kelvin-Helmholtz instabilities.\\nIf we disregard the discrepant WFPC2 result, the direction of the weighted mean displacement derived from the NICMOS2 and ACS images is PA = 132±8 • . In projection, therefore, the displacement is remarkably well-aligned with the radio source axis, suggesting jet acceleration as a plausible candidate for the displacement mechanism. Pellegrini et al. (2003) infer a viewing angle θ = 63.5 ± 3.5 • for the parsec-scale jets. Assuming that the displacement is in the same direction in space as the jet, the de-projected amplitude is ∆r jet = 4.3 ± 3 pc.\\nIf we disregard the discrepant WFPC2 result, the direction of the weighted mean displacement derived from the NICMOS2 and ACS images is PA = 132±8 • . In projection, therefore, the displacement is remarkably well-aligned with the radio source axis, suggesting jet acceleration as a plausible candidate for the displacement mechanism. Pellegrini et al. (2003) infer a viewing angle θ = 63.5 ± 3.5 • for the parsec-scale jets. Assuming that the displacement is in the same direction in space as the jet, the de-projected amplitude is ∆r jet = 4.3 ± 3 pc.\\nGiven that the galaxy is characterized by a low density core, taking the age of the radio source to be t ∼ 10 8 yr ,Ṁ a =ṁ Bondi ∼ 0.02 M yr −1 (Pellegrini et al. 2003), M • ∼ 10 9 M from the M − σ relation and 3 ∆r 30 pc, we find from eq. 7 that 10 −3 f jet 9.7 × 10 −3 .\\nGiven that the galaxy is characterized by a low density core, taking the age of the radio source to be t ∼ 10 8 yr ,Ṁ a =ṁ Bondi ∼ 0.02 M yr −1 (Pellegrini et al. 2003), M • ∼ 10 9 M from the M − σ relation and 3 ∆r 30 pc, we find from eq. 7 that 10 −3 f jet 9.7 × 10 −3 .\\nTherefore, an asymmetry in jet power amounting to 1% of the accretion luminosity could account for the observed offset if it persists for ∼ 10 8 yr.\\nTherefore, an asymmetry in jet power amounting to 1% of the accretion luminosity could account for the observed offset if it persists for ∼ 10 8 yr.\\nThe close alignment with the radio jet axis argues against the displacement being caused by gravitational interactions with massive perturbers.\\nThe close alignment with the radio jet axis argues against the displacement being caused by gravitational interactions with massive perturbers.\"\n },\n {\n \"title\": \"NGC 5419 (low, double nucleus)\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 5419 (low, double nucleus)\",\n \"content\": \"This galaxy is the dominant member of the cluster Abell S753 (Abell et al. 1989). A low surface brightness X-ray halo is centered on the galaxy and extends over a radius r ≈ 16 (190 h −1 kpc). An unusual diffuse radio source, PKS B1400-33, is associated with the cluster, but is offset relative to the X-ray emission and NGC 5419 (Subrahmanyan et al. 2003). The galaxy is located just beyond the NW edge of the extended radio emission and itself is associated with a bright, compact radio source. The low surface brightness and steep spectral index of the diffuse source suggest that it is a relic, with a spectral age of 5 × 10 8 yr. However, it is unclear if the relic was created by an earlier episode of powerful activity in NGC 5419, or whether it is a relic lobe injected into the cluster by a previously active double radio source located outside the cluster (the scenario favored by Subrahmanyan et al., although an optical counterpart has not been firmly identified). If the relic originated from NGC 5419, the spatial offset may be due to proper motion of the galaxy or to buoyancy of the synchrotron plasma (e.g., McNamara et al. 2001).\\nThis galaxy is the dominant member of the cluster Abell S753 (Abell et al. 1989). A low surface brightness X-ray halo is centered on the galaxy and extends over a radius r ≈ 16 (190 h −1 kpc). An unusual diffuse radio source, PKS B1400-33, is associated with the cluster, but is offset relative to the X-ray emission and NGC 5419 (Subrahmanyan et al. 2003). The galaxy is located just beyond the NW edge of the extended radio emission and itself is associated with a bright, compact radio source. The low surface brightness and steep spectral index of the diffuse source suggest that it is a relic, with a spectral age of 5 × 10 8 yr. However, it is unclear if the relic was created by an earlier episode of powerful activity in NGC 5419, or whether it is a relic lobe injected into the cluster by a previously active double radio source located outside the cluster (the scenario favored by Subrahmanyan et al., although an optical counterpart has not been firmly identified). If the relic originated from NGC 5419, the spatial offset may be due to proper motion of the galaxy or to buoyancy of the synchrotron plasma (e.g., McNamara et al. 2001).\\nPhotometric analysis. The galaxy has a double nucleus (Lauer et al. 2005;, which is well-resolved in the WFPC2 planetary camera image (WFPC2/PC-F555W). The two nuclei are separated by 0. 27, with the fainter (secondary) nucleus located almost at the south of the brighter (primary) nucleus. The photocenter is offset by ∆r 1 7.5 ± 1.7 pc W-SW (PA = 252 ± 12 • ) from the primary and ∆r 2 62 ± 2 pc N-NW (PA = 346 ± 2 • ) from the secondary nucleus.\\nPhotometric analysis. The galaxy has a double nucleus (Lauer et al. 2005;, which is well-resolved in the WFPC2 planetary camera image (WFPC2/PC-F555W). The two nuclei are separated by 0. 27, with the fainter (secondary) nucleus located almost at the south of the brighter (primary) nucleus. The photocenter is offset by ∆r 1 7.5 ± 1.7 pc W-SW (PA = 252 ± 12 • ) from the primary and ∆r 2 62 ± 2 pc N-NW (PA = 346 ± 2 • ) from the secondary nucleus.\\nA periodic pattern is visible in the residual image obtained after subtraction of the photometric model (middle panel in Fig. 26). This is likely to be due to the pronounced \\\"boxiness\\\" of the isophotes, which could not be perfectly reproduced even though higher harmonics were included in the model. The isophote center co-ordinates cluster fairly tightly around the photocenter, which is closest to the primary nucleus, and generally show little variation with SMA (Fig. 26), with the exception of a small migration (≈ 0. 05) eastward of the photocenter in the inner 170 pixels (8. 5).\\nA periodic pattern is visible in the residual image obtained after subtraction of the photometric model (middle panel in Fig. 26). This is likely to be due to the pronounced \\\"boxiness\\\" of the isophotes, which could not be perfectly reproduced even though higher harmonics were included in the model. The isophote center co-ordinates cluster fairly tightly around the photocenter, which is closest to the primary nucleus, and generally show little variation with SMA (Fig. 26), with the exception of a small migration (≈ 0. 05) eastward of the photocenter in the inner 170 pixels (8. 5).\\nOffset origin. The projected separation between the two nuclei (∆r ≈ 75 pc) is slightly smaller than the estimated SBH influence radius (r h ≈ 90 pc) and much smaller than the core radius (r c ≈ 500 pc; Table 1). Hence, it is possible that the secondary nucleus is a second SBH, or even the nucleus of a satellite galaxy, on a slowly-evolving orbit in the low-density core (Antonini & Merritt 2012).\\nOffset origin. The projected separation between the two nuclei (∆r ≈ 75 pc) is slightly smaller than the estimated SBH influence radius (r h ≈ 90 pc) and much smaller than the core radius (r c ≈ 500 pc; Table 1). Hence, it is possible that the secondary nucleus is a second SBH, or even the nucleus of a satellite galaxy, on a slowly-evolving orbit in the low-density core (Antonini & Merritt 2012).\\nThe displacement of the photocenter relative to the primary nucleus is a small fraction of the core radius, ≈ 0.02r c ( Table 6). The probability of generating a recoil kick velocity large enough to trigger phase II recoil oscillations (v 300 km s −1 , Table 8), is ∼ 33% (L12) and given that phase II oscillations occur, we estimate a 46 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr. Using eqs. 2 and 3 with σ = 351 km s −1 , r rms ≥ 7 pc and r c = 500 pc, we find (t − t c ) 9 τ ≈ 10 Gyr, a factor ten larger than the estimated spectral age of the diffuse radio source. Therefore, if an SBH binary coalescence event is responsible for the displacement, it seems likely that it pre-dates the AGN phase that created the radio relic (if indeed NGC 5419 is the origin of the relic).\\nThe displacement of the photocenter relative to the primary nucleus is a small fraction of the core radius, ≈ 0.02r c ( Table 6). The probability of generating a recoil kick velocity large enough to trigger phase II recoil oscillations (v 300 km s −1 , Table 8), is ∼ 33% (L12) and given that phase II oscillations occur, we estimate a 46 (97)% probability of finding a displacement larger than that observed, for t m = 5 (0.4) Gyr. Using eqs. 2 and 3 with σ = 351 km s −1 , r rms ≥ 7 pc and r c = 500 pc, we find (t − t c ) 9 τ ≈ 10 Gyr, a factor ten larger than the estimated spectral age of the diffuse radio source. Therefore, if an SBH binary coalescence event is responsible for the displacement, it seems likely that it pre-dates the AGN phase that created the radio relic (if indeed NGC 5419 is the origin of the relic).\\nNotes on other galaxies In the following galaxies, photocenter -SBH displacements were not considered significant.\\nNotes on other galaxies In the following galaxies, photocenter -SBH displacements were not considered significant.\"\n },\n {\n \"title\": \"NGC 4168\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 4168\",\n \"content\": \"NGC 4168 is an E2 in the Virgo cluster. Broad Hα has been detected in its nuclear optical spectrum, resulting in classification as a Seyfert 1.9 (Ho et al. 1997). VLBA observations at 8.4 GHz show an unresolved radio source with no evidence of jets (Anderson et al. 2004). Although the galaxy appears very regular in the WFPC2-WF/F606W image, the photometric analysis reveals systematic variations in the isophote centers with SMA that are much larger in amplitude than the displacement between the mean photocenter and the AGN position (Fig. 13). In particular, there is an abrupt shift to the E in the x co-ordinate and a systematic migration to the S in y, for SMA 260 pixels (13 ). The measured displacements in x and y are < 0.8 IQR and therefore considered \\\"null\\\" results.\\nNGC 4168 is an E2 in the Virgo cluster. Broad Hα has been detected in its nuclear optical spectrum, resulting in classification as a Seyfert 1.9 (Ho et al. 1997). VLBA observations at 8.4 GHz show an unresolved radio source with no evidence of jets (Anderson et al. 2004). Although the galaxy appears very regular in the WFPC2-WF/F606W image, the photometric analysis reveals systematic variations in the isophote centers with SMA that are much larger in amplitude than the displacement between the mean photocenter and the AGN position (Fig. 13). In particular, there is an abrupt shift to the E in the x co-ordinate and a systematic migration to the S in y, for SMA 260 pixels (13 ). The measured displacements in x and y are < 0.8 IQR and therefore considered \\\"null\\\" results.\"\n },\n {\n \"title\": \"NGC 4261\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 4261\",\n \"content\": \"NGC 4261 is a massive elliptical (E2-3) in the outskirts of the Virgo cluster. Nolthenius (1993) identifies this galaxy as the dominant member of a group of 33 galaxies, the Virgo W cloud. It is associated with a bright FR I radio source, 3C 270, which has twin jets close to the plane of the sky, θ = 63 ± 3 • , and aligned on the axis W-E, PA = 87 ± 8 • , with the western radio component slightly brighter than the eastern (Jones et al. 2000;Worrall et al. 2010). Deep imaging reveals the presence of tidal tails to the NW and a tidal fan SE (Tal et al. 2009) while Giordano et al. (2005) find anisotropies in the globular cluster distribution. Cappellari et al. (2007) report the presence of a kpc-scale kinematically decoupled core hosting a 100 pc scale disk of dust, cool molecular and atomic gas (Jaffe et al. 1993;Jaffe & McNamara 1994). Jaffe et al. (1996 studied the central region of this galaxy in detail using HST/WFPC2/PC1 V (filter F547M), R (F675W) and I (F791W) images. After correcting the R images for line emission, they determined that the central dusty disk is characterized by a rotation axis which closely matches the axis of the jets but it is not coaxial with the semi-major axis of the galaxy. The disk is not centered on the AGN, nor on the photocenter, which they find is itself displaced relative to the AGN by ∆r = 3.3 ± 1.3 pc NE.\\nNGC 4261 is a massive elliptical (E2-3) in the outskirts of the Virgo cluster. Nolthenius (1993) identifies this galaxy as the dominant member of a group of 33 galaxies, the Virgo W cloud. It is associated with a bright FR I radio source, 3C 270, which has twin jets close to the plane of the sky, θ = 63 ± 3 • , and aligned on the axis W-E, PA = 87 ± 8 • , with the western radio component slightly brighter than the eastern (Jones et al. 2000;Worrall et al. 2010). Deep imaging reveals the presence of tidal tails to the NW and a tidal fan SE (Tal et al. 2009) while Giordano et al. (2005) find anisotropies in the globular cluster distribution. Cappellari et al. (2007) report the presence of a kpc-scale kinematically decoupled core hosting a 100 pc scale disk of dust, cool molecular and atomic gas (Jaffe et al. 1993;Jaffe & McNamara 1994). Jaffe et al. (1996 studied the central region of this galaxy in detail using HST/WFPC2/PC1 V (filter F547M), R (F675W) and I (F791W) images. After correcting the R images for line emission, they determined that the central dusty disk is characterized by a rotation axis which closely matches the axis of the jets but it is not coaxial with the semi-major axis of the galaxy. The disk is not centered on the AGN, nor on the photocenter, which they find is itself displaced relative to the AGN by ∆r = 3.3 ± 1.3 pc NE.\\nIn this study, we chose to analyze a NICMOS2 F160W image (with a pixel size of 0. 05), since the nuclear source is bright and dust extinction and contamination due to emission from ionized gas are minimized. The galaxy exhibits boxy isophotes which are difficult to model accurately. Even though the third and forth harmonics were included in the isophote fits, the residual image still shows a spiral pattern suggesting that a more complex model perhaps involving multiple components would be necessary to successfully model the surface brightness distribution (Fig. 14). Nevertheless, the isophote centers are relatively stable, except for a slow migration to the N for SMA 120 pixels (6 ). The mean photocenter displacement relative to the AGN is ∆r = 3.3 ± 2.2 pc W-NW (PA = 297 ± 38 • ). The x component of the displacement is large compared to the IQR. However, because of the large uncertainty in positions derived from NICMOS2 images (0.23 pixels), we do not consider this offset significant.\\nIn this study, we chose to analyze a NICMOS2 F160W image (with a pixel size of 0. 05), since the nuclear source is bright and dust extinction and contamination due to emission from ionized gas are minimized. The galaxy exhibits boxy isophotes which are difficult to model accurately. Even though the third and forth harmonics were included in the isophote fits, the residual image still shows a spiral pattern suggesting that a more complex model perhaps involving multiple components would be necessary to successfully model the surface brightness distribution (Fig. 14). Nevertheless, the isophote centers are relatively stable, except for a slow migration to the N for SMA 120 pixels (6 ). The mean photocenter displacement relative to the AGN is ∆r = 3.3 ± 2.2 pc W-NW (PA = 297 ± 38 • ). The x component of the displacement is large compared to the IQR. However, because of the large uncertainty in positions derived from NICMOS2 images (0.23 pixels), we do not consider this offset significant.\"\n },\n {\n \"title\": \"NGC 4373\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 4373\",\n \"content\": \"This galaxy is located in the outskirts of the Hydra-Centaurus supercluster and is associated with a compact radio source (Sadler et al. 1989). It forms an isolated pair with IC 3290 (Sadler & Sharp 1984), which is located 2 W-SW of NGC 4373. Grützbauch et al. (2007) suggest that there is an ongoing strong interaction with the companion. Our photometric analysis of the WFPC2/PC-F814W image reveals boxy isophotes which produce cross-shaped residuals, even though higher harmonics were included in the isophote fits (Fig. 3). The y coordinate of the isophote center position has a narrow distribution centered on the mean photocenter and shows only a slight drift northward, for SMA > 170 pixel (8. 5). The x co-ordinate exhibits a systematic migration eastwards from the photocenter, resulting in a broader, skewed distribution. The mean photocenter is consistent with the AGN position within the uncertainties.\\nThis galaxy is located in the outskirts of the Hydra-Centaurus supercluster and is associated with a compact radio source (Sadler et al. 1989). It forms an isolated pair with IC 3290 (Sadler & Sharp 1984), which is located 2 W-SW of NGC 4373. Grützbauch et al. (2007) suggest that there is an ongoing strong interaction with the companion. Our photometric analysis of the WFPC2/PC-F814W image reveals boxy isophotes which produce cross-shaped residuals, even though higher harmonics were included in the isophote fits (Fig. 3). The y coordinate of the isophote center position has a narrow distribution centered on the mean photocenter and shows only a slight drift northward, for SMA > 170 pixel (8. 5). The x co-ordinate exhibits a systematic migration eastwards from the photocenter, resulting in a broader, skewed distribution. The mean photocenter is consistent with the AGN position within the uncertainties.\"\n },\n {\n \"title\": \"NGC 4552\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 4552\",\n \"content\": \"NGC 4552 is a typical giant elliptical in the Virgo cluster which hosts a very low-luminosity AGN. HST imaging and spectroscopy reveal a variable UV point source that has optical emission lines characteristic of AGN (Cappellari et al. 1999;Maoz et al. 2005). Interestingly, a second, transient UV source of comparable luminosity, but offset from the first by 0. 14, was present in 1991, but was not detected in 1993 Renzini et al. (1995); Cappellari et al. (1999); Maoz et al. (2005). Radio observations at 8.4 (Filho et al. 2000) and 5 GHz (Nagar et al. 2002) show the presence of a flat spectrum radio core. Nagar et al. (2002) also report variability at the ∼ 20% level on a ∼ 1 year timescale and find evidence of pc-scale jets, or extensions, along the East-West axis. NGC 4552 exhibits several morphological features indicative of past mergers, including three concentric shells at 5 , 7 and 10 from the nucleus (Malin 1979). Traces of a recent minor merger with a gas rich satellite galaxy have been identified in the presence of dust patches in the inner region (van Dokkum & Franx 1995), and Hα emission extending up to ∼ 2.5 kpc from the center (Trinchieri & di Serego Alighieri 1991; Macchetto et al. 1996).\\nNGC 4552 is a typical giant elliptical in the Virgo cluster which hosts a very low-luminosity AGN. HST imaging and spectroscopy reveal a variable UV point source that has optical emission lines characteristic of AGN (Cappellari et al. 1999;Maoz et al. 2005). Interestingly, a second, transient UV source of comparable luminosity, but offset from the first by 0. 14, was present in 1991, but was not detected in 1993 Renzini et al. (1995); Cappellari et al. (1999); Maoz et al. (2005). Radio observations at 8.4 (Filho et al. 2000) and 5 GHz (Nagar et al. 2002) show the presence of a flat spectrum radio core. Nagar et al. (2002) also report variability at the ∼ 20% level on a ∼ 1 year timescale and find evidence of pc-scale jets, or extensions, along the East-West axis. NGC 4552 exhibits several morphological features indicative of past mergers, including three concentric shells at 5 , 7 and 10 from the nucleus (Malin 1979). Traces of a recent minor merger with a gas rich satellite galaxy have been identified in the presence of dust patches in the inner region (van Dokkum & Franx 1995), and Hα emission extending up to ∼ 2.5 kpc from the center (Trinchieri & di Serego Alighieri 1991; Macchetto et al. 1996).\\nThe photometric analysis of the WFPC2/PC-F814W image shows that the y-coordinate of the isophote center migrates systematically southward relative to the photocenter for SMA > 200 pixels (10 ) with a maximum shift 0. 2 (Fig. 23). There is a smaller westward shift in the x coordinate. The inner 0. 3 of the residual image shows evidence of an arc-like dust lane partly encircling the central source with an opening toward the S. This feature was previously noted by Carollo et al. (1997) and Cappellari et al. (1999) (see Fig. 12 in the latter) and is the cause of the increased uncertainty on the y co-ordinate of the AGN position. We did not find a significant (> 3σ) displacement in this object.\\nThe photometric analysis of the WFPC2/PC-F814W image shows that the y-coordinate of the isophote center migrates systematically southward relative to the photocenter for SMA > 200 pixels (10 ) with a maximum shift 0. 2 (Fig. 23). There is a smaller westward shift in the x coordinate. The inner 0. 3 of the residual image shows evidence of an arc-like dust lane partly encircling the central source with an opening toward the S. This feature was previously noted by Carollo et al. (1997) and Cappellari et al. (1999) (see Fig. 12 in the latter) and is the cause of the increased uncertainty on the y co-ordinate of the AGN position. We did not find a significant (> 3σ) displacement in this object.\"\n },\n {\n \"title\": \"NGC 4636\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 4636\",\n \"content\": \"NGC 4636 is a giant elliptical in the outskirts of the Virgo cluster. Its nucleus has a LINER emission line spectrum, in which a weak broad Hα line has been detected (Ho et al. 1997). It has a jet-like radio source extending ≈ 1.5 kpc either side of the nucleus in position angle θ ≈ 33 • (Birkinshaw & Davies 1985). NGC 4636 is also associated with a luminous, extended X-ray source with a peculiar morphology featuring two symmetric, arm-like extensions emerging from the bright central region and extending up to ≈ 8 kpc along the NE-SW axis. Other fainter arm-like features emerge from the nucleus and Jones et al. (2002) suggest that these features trace cyclic nuclear outburst events occurring on a timescale ∼ 10 7 yr. According to Temi et al. (2003), the observed FIR fluxes (at 60, 90 and 180 µm) suggest the accretion of dust in a recent (within ∼ 10 8 yr) merger with a gas-rich galaxy.\\nNGC 4636 is a giant elliptical in the outskirts of the Virgo cluster. Its nucleus has a LINER emission line spectrum, in which a weak broad Hα line has been detected (Ho et al. 1997). It has a jet-like radio source extending ≈ 1.5 kpc either side of the nucleus in position angle θ ≈ 33 • (Birkinshaw & Davies 1985). NGC 4636 is also associated with a luminous, extended X-ray source with a peculiar morphology featuring two symmetric, arm-like extensions emerging from the bright central region and extending up to ≈ 8 kpc along the NE-SW axis. Other fainter arm-like features emerge from the nucleus and Jones et al. (2002) suggest that these features trace cyclic nuclear outburst events occurring on a timescale ∼ 10 7 yr. According to Temi et al. (2003), the observed FIR fluxes (at 60, 90 and 180 µm) suggest the accretion of dust in a recent (within ∼ 10 8 yr) merger with a gas-rich galaxy.\\nOur isophotal modeling of the WFPC2/PC-F814W image indicates the presence of faint dust lanes inspiraling toward the nucleus (Fig. 24). There is a systematic shift in the isophote centers to the north and west relative to the photocenter for SMA > 170 pixels (8. 5), which results in broad distributions in the x and y co-ordinates (Fig. 24). The photocenter position is consistent with the AGN position in the y direction, but exhibits a > 3σ offset in x. Nevertheless, as the IQR of the isophote center coordinate distribution exceeds the displacement we consider this a null result.\\nOur isophotal modeling of the WFPC2/PC-F814W image indicates the presence of faint dust lanes inspiraling toward the nucleus (Fig. 24). There is a systematic shift in the isophote centers to the north and west relative to the photocenter for SMA > 170 pixels (8. 5), which results in broad distributions in the x and y co-ordinates (Fig. 24). The photocenter position is consistent with the AGN position in the y direction, but exhibits a > 3σ offset in x. Nevertheless, as the IQR of the isophote center coordinate distribution exceeds the displacement we consider this a null result.\"\n },\n {\n \"title\": \"NGC 4696 (double nucleus)\",\n \"content\": \"\"\n },\n {\n \"title\": \"NGC 4696 (double nucleus)\",\n \"content\": \"NGC 4696 is the dominant galaxy of the Centaurus cluster. There is an extensive literature on the multiwavewelength properties of this galaxy, which has a secondary nucleus (Laine et al. 2003) offset by ≈ 0. 3 to the SW of the primary (Laine et al. 2003 and Fig. 25). The double nucleus is partially surrounded by a complex of dust lanes and associated emission line filaments, the most prominent of which is in the form of a spiral leading towards the center. These features have been interpreted as evidence of a recent minor merger with a gas-rich galaxy (Sparks et al. 1989;Farage et al. 2010; alternatively the emission-line gas may have been \\\"drawn out\\\" by rising radio bubbles Canning et al. 2011). The galaxy is associated with the moderately powerful radio source PKS 1246-410, which extends ≈ 10 kpc along an approximately E-W axis, with the ends of both arms bending southwards. The radio source wraps around the region of brightest X-ray emission, suggesting interactions between the radio plasma and the hot thermal gas (Taylor et al. 2006). VLBA observations also obtained by Taylor et al. (2006) reveal a relatively weak core and a one-sided jet extending over ≈ 25 pc in PA = −150 • .\\nNGC 4696 is the dominant galaxy of the Centaurus cluster. There is an extensive literature on the multiwavewelength properties of this galaxy, which has a secondary nucleus (Laine et al. 2003) offset by ≈ 0. 3 to the SW of the primary (Laine et al. 2003 and Fig. 25). The double nucleus is partially surrounded by a complex of dust lanes and associated emission line filaments, the most prominent of which is in the form of a spiral leading towards the center. These features have been interpreted as evidence of a recent minor merger with a gas-rich galaxy (Sparks et al. 1989;Farage et al. 2010; alternatively the emission-line gas may have been \\\"drawn out\\\" by rising radio bubbles Canning et al. 2011). The galaxy is associated with the moderately powerful radio source PKS 1246-410, which extends ≈ 10 kpc along an approximately E-W axis, with the ends of both arms bending southwards. The radio source wraps around the region of brightest X-ray emission, suggesting interactions between the radio plasma and the hot thermal gas (Taylor et al. 2006). VLBA observations also obtained by Taylor et al. (2006) reveal a relatively weak core and a one-sided jet extending over ≈ 25 pc in PA = −150 • .\\nSubtraction of our photometric model from the ACS/WFC-F814W image (Fig. 25) clearly shows the dust features around the nucleus. The positive residuals probably trace the emission-line filaments. The isophote centers exhibit large excursions, resulting in broad distributions in both x and (especially) y coordinates. Although we measure a 3σ offset in x between the brightest nucleus and the mean photocenter, we consider this a null result because of the large IQR.\\nSubtraction of our photometric model from the ACS/WFC-F814W image (Fig. 25) clearly shows the dust features around the nucleus. The positive residuals probably trace the emission-line filaments. The isophote centers exhibit large excursions, resulting in broad distributions in both x and (especially) y coordinates. Although we measure a 3σ offset in x between the brightest nucleus and the mean photocenter, we consider this a null result because of the large IQR.\"\n },\n {\n \"title\": \"IC 1459\",\n \"content\": \"\"\n },\n {\n \"title\": \"IC 1459\",\n \"content\": \"IC 1459 is a giant elliptical galaxy (E3-4) and a member of a loose group (identified as \\\"group 15\\\" in Huchra & Geller 1982) of ten galaxies, which are mainly spirals. Despite its relatively unremarkable appearance to casual inspection, this galaxy turns out to be notably peculiar (see Forbes et al. 1995 for a detailed analysis): it has one of the faster counter rotating kinematically distinct cores (Franx & Illingworth 1988), faint stellar spiral arms (Malin 1985), outer shells (Forbes et al. 1995) and irregular dust lanes or patches near the nucleus (Forbes et al. 1994). It has a hard X-ray point source (González-Martín et al. 2009) and a strong, compact radio core at 5 GHz (Slee et al. 1994).\\nIC 1459 is a giant elliptical galaxy (E3-4) and a member of a loose group (identified as \\\"group 15\\\" in Huchra & Geller 1982) of ten galaxies, which are mainly spirals. Despite its relatively unremarkable appearance to casual inspection, this galaxy turns out to be notably peculiar (see Forbes et al. 1995 for a detailed analysis): it has one of the faster counter rotating kinematically distinct cores (Franx & Illingworth 1988), faint stellar spiral arms (Malin 1985), outer shells (Forbes et al. 1995) and irregular dust lanes or patches near the nucleus (Forbes et al. 1994). It has a hard X-ray point source (González-Martín et al. 2009) and a strong, compact radio core at 5 GHz (Slee et al. 1994).\\nWe analyzed a WFPC2/PC-F814W image (Fig. 28). After subtraction of the photometric model, the residual image shows a periodic pattern, indicating that the isophotes deviate from perfect ellipses. Systematic variations in the isophote center co-ordinates are seen in both x and y, with increasing scatter also present for SMA > 230 pixels (11. 5). We did not find a significant (> 3σ) displacement in this galaxy.\\nWe analyzed a WFPC2/PC-F814W image (Fig. 28). After subtraction of the photometric model, the residual image shows a periodic pattern, indicating that the isophotes deviate from perfect ellipses. Systematic variations in the isophote center co-ordinates are seen in both x and y, with increasing scatter also present for SMA > 230 pixels (11. 5). We did not find a significant (> 3σ) displacement in this galaxy.\"\n },\n {\n \"title\": \"IC 4931\",\n \"content\": \"\"\n },\n {\n \"title\": \"IC 4931\",\n \"content\": \"This relatively little studied galaxy is the brightest member of the group A3656 (Postman & Lauer 1995). Its radio source is relatively weak, with a total radio flux at 5 GHz of 0.9 mJy (see Table 2 to compare with other galaxies). It is also detected at 1.4 GHz with a flux density of 22 mJy (Brown et al. 2011). Apart from the optical nucleus and the radio emission, there is no supporting evidence indicating the presence of an AGN such as an X-ray point source, or line emission.\\nThis relatively little studied galaxy is the brightest member of the group A3656 (Postman & Lauer 1995). Its radio source is relatively weak, with a total radio flux at 5 GHz of 0.9 mJy (see Table 2 to compare with other galaxies). It is also detected at 1.4 GHz with a flux density of 22 mJy (Brown et al. 2011). Apart from the optical nucleus and the radio emission, there is no supporting evidence indicating the presence of an AGN such as an X-ray point source, or line emission.\\nOur analysis of the WFPC2/PC-F814 image (Fig. 32) shows large systematic variations in the isophote center coordinates, with the x coordinate in particular migrating up to 0. 4 W relative the photocenter for SM A 100 pixels (5 ). The co-ordinate distributions are among the broadest of the whole sample, spanning a range of 0. 5 in x. We measured a ∼ 3σ offset between the photocenter and the AGN in the x co-ordinate (E), but this is considered a null result because of the large IQR.\\nOur analysis of the WFPC2/PC-F814 image (Fig. 32) shows large systematic variations in the isophote center coordinates, with the x coordinate in particular migrating up to 0. 4 W relative the photocenter for SM A 100 pixels (5 ). The co-ordinate distributions are among the broadest of the whole sample, spanning a range of 0. 5 in x. We measured a ∼ 3σ offset between the photocenter and the AGN in the x co-ordinate (E), but this is considered a null result because of the large IQR.\\nC: IMAGES Figure 9. As in Fig.3 for galaxy NGC 1399, WFPC2/PC -F606W, scale=0. 05/pxl.\\nC: IMAGES Figure 9. As in Fig.3 for galaxy NGC 1399, WFPC2/PC -F606W, scale=0. 05/pxl.\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5b .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5b .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7a .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7a .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7b .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7b .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 18 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 18 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 21 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 21 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 22 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 22 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 THE SAMPLE\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 THE SAMPLE\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 6 PROBABILITIES\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 6 PROBABILITIES\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 7 SUMMARY\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 7 SUMMARY\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 8\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 8\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 9\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 9\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA.","content":"Figure 1 .\n1Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA."},{"id":"","caption":"Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA.","content":"Figure 1 .\n1Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA."},{"id":"","caption":"Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \"true errors\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC.","content":"Figure 2 .\n2Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \"true errors\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC."},{"id":"","caption":"Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \"true errors\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC.","content":"Figure 2 .\n2Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \"true errors\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC."},{"id":"","caption":"Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers.","content":"Figure 3\n3Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers."},{"id":"","caption":"Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers.","content":"Figure 3\n3Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers."},{"id":"","caption":"Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3.","content":"Figure 4 .\n4Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3."},{"id":"","caption":"Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3.","content":"Figure 4 .\n4Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3."},{"id":"","caption":"Figure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position).","content":"Figure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position)."},{"id":"","caption":"Figure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position).","content":"Figure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position)."},{"id":"","caption":"As inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2.","content":"Figure 5b .\n5bAs inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2."},{"id":"","caption":"As inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2.","content":"Figure 5b .\n5bAs inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2."},{"id":"","caption":"Probability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.","content":"Figure 7a .\n7aProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source."},{"id":"","caption":"Probability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.","content":"Figure 7a .\n7aProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source."},{"id":"","caption":"Probability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.","content":"Figure 7b .\n7bProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source."},{"id":"","caption":"Probability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.","content":"Figure 7b .\n7bProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source."},{"id":"","caption":", (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000).","content":", (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000)."},{"id":"","caption":", (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000).","content":", (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000)."},{"id":"","caption":"M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear.","content":"Figure 8 .\n8M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear."},{"id":"","caption":"M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear.","content":"Figure 8 .\n8M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear."},{"id":"","caption":"As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl.","content":"Figure 18 .\n18As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl."},{"id":"","caption":"As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl.","content":"Figure 18 .\n18As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl."},{"id":"","caption":"As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl.","content":"Figure 21 .\n21As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl."},{"id":"","caption":"As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl.","content":"Figure 21 .\n21As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl."},{"id":"","caption":"As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl.","content":"Figure 22 .\n22As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl."},{"id":"","caption":"As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl.","content":"Figure 22 .\n22As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl."},{"id":"","caption":"NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIES","content":"Table 1 THE SAMPLE\n1Galaxy \nDistance \nrc \nr h \nInstrument \nPixel Scale \nID \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n\nNGC 1399 \n18 ± 1 \nsbf \n2.19 (189) \n0.99 (86) \nWFPC2/PC/F606W \n0.05 \n8214 \nWFPC2/PC/F814W \n0.05 \n5990 \nWFC3/IR/F110W \n0.09 \n11712 \nWFC3/IR/F160W \n0.09 \n11712 \nNGC 4168 \nUGC 7203 \n31 ± 6 \nsbf \n2.02 (303) \n0.09 (14) \nWFPC2/PC/F702W \n0.05 \n6357 \nNGC 4261 \nUGC 7360 \n30 ± 2 \nsbf \n1.62 (237) \n0.428 (62) \nNICMOS2/F160W \n0.05 \n7868 \nNGC 4278 \nUGC 7386 \n18 ± 1 \nsbf \n0.97 (83) \n0.334 (29) \nACS/WFC/F850LP \n0.05 \n10835 \nWFPC2/PC/F814W \n0.05 \n5454 \nNICMOS2/F160W \n0.05 \n7886 \nNGC 4373 \n40 ± 2 \nsbf \n1.19 (269) \n0.167 (32) \nWFPC2/PC/F814W \n0.05 \n5214 \nNGC 4486 \nUGC 7654 (M87) \n16 ± 1 \nsbf \n9.41 (733) \n1 (78) \nACS/HRC/F606W \n0.025 \nB10 \nACS/HRC/F814W \n0.025 \nB10 \nNICMOS2/F110W \n0.05 \n7171 \nNICMOS2/F160W \n0.05 \n7171 \nNICMOS2/F222M \n0.05 \n7171 \nWFPC2/PC/F814W \n0.05 \n8592 \nNGC 4552 \nUGC 7760 \n15 ± 1 \nsbf \n0.49 (36) \n0.48 (35) \nWFPC2/PC/F814W \n0.05 \n6099 \nNGC 4636 \nUGC 7878 \n13 ± 1 \nsbf \n3.44 (219) \n0.297 (19) \nWFPC2/PC/F814W \n0.05 \n8686 \nNGC 4696 \n37 ± 2 \nsbf \n1.4 (251) \n0.2 (36) \nACS/WFC/F814W \n0.05 \n9427 \n\n † \n\nNGC 5419 \n49 ± 9 \nDn − σ \n2.11 (499) \n0.38 (90) \nWFPC2/PC/F555W \n0.05 \n6587 \nNGC 5846 \nUGC 9706 \n25 ± 2 \nsbf \n1.52 (183) \n0.25 (30) \nWFPC2/PC/F814W \n0.05 \n5920 \nIC 1459 \n29 ± 4 \nsbf \n1.82 (258) \n0.43 (60) \nWFPC2/PC/F814W \n0.05 \n5454 \nIC 4296 \n50 ± 3 \nsbf \n1.44 (347) \n0.32 (77) \nACS/HRC/F625W \n0.025 \n9838 \nWFPC2/PC/F814W \n0.05 \n5910 \nNICMOS2/F160W \n0.05 \n7453 \nIC 4931 \n∼ 84 \nvvir \n0.99 (403) \n0.13 (51) \nWFPC2/PC/F814W \n0.05 \n8683 \n\nNote. -(1) Source name;"},{"id":"","caption":"NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIES","content":"Table 1 THE SAMPLE\n1Galaxy \nDistance \nrc \nr h \nInstrument \nPixel Scale \nID \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n\nNGC 1399 \n18 ± 1 \nsbf \n2.19 (189) \n0.99 (86) \nWFPC2/PC/F606W \n0.05 \n8214 \nWFPC2/PC/F814W \n0.05 \n5990 \nWFC3/IR/F110W \n0.09 \n11712 \nWFC3/IR/F160W \n0.09 \n11712 \nNGC 4168 \nUGC 7203 \n31 ± 6 \nsbf \n2.02 (303) \n0.09 (14) \nWFPC2/PC/F702W \n0.05 \n6357 \nNGC 4261 \nUGC 7360 \n30 ± 2 \nsbf \n1.62 (237) \n0.428 (62) \nNICMOS2/F160W \n0.05 \n7868 \nNGC 4278 \nUGC 7386 \n18 ± 1 \nsbf \n0.97 (83) \n0.334 (29) \nACS/WFC/F850LP \n0.05 \n10835 \nWFPC2/PC/F814W \n0.05 \n5454 \nNICMOS2/F160W \n0.05 \n7886 \nNGC 4373 \n40 ± 2 \nsbf \n1.19 (269) \n0.167 (32) \nWFPC2/PC/F814W \n0.05 \n5214 \nNGC 4486 \nUGC 7654 (M87) \n16 ± 1 \nsbf \n9.41 (733) \n1 (78) \nACS/HRC/F606W \n0.025 \nB10 \nACS/HRC/F814W \n0.025 \nB10 \nNICMOS2/F110W \n0.05 \n7171 \nNICMOS2/F160W \n0.05 \n7171 \nNICMOS2/F222M \n0.05 \n7171 \nWFPC2/PC/F814W \n0.05 \n8592 \nNGC 4552 \nUGC 7760 \n15 ± 1 \nsbf \n0.49 (36) \n0.48 (35) \nWFPC2/PC/F814W \n0.05 \n6099 \nNGC 4636 \nUGC 7878 \n13 ± 1 \nsbf \n3.44 (219) \n0.297 (19) \nWFPC2/PC/F814W \n0.05 \n8686 \nNGC 4696 \n37 ± 2 \nsbf \n1.4 (251) \n0.2 (36) \nACS/WFC/F814W \n0.05 \n9427 \n\n † \n\nNGC 5419 \n49 ± 9 \nDn − σ \n2.11 (499) \n0.38 (90) \nWFPC2/PC/F555W \n0.05 \n6587 \nNGC 5846 \nUGC 9706 \n25 ± 2 \nsbf \n1.52 (183) \n0.25 (30) \nWFPC2/PC/F814W \n0.05 \n5920 \nIC 1459 \n29 ± 4 \nsbf \n1.82 (258) \n0.43 (60) \nWFPC2/PC/F814W \n0.05 \n5454 \nIC 4296 \n50 ± 3 \nsbf \n1.44 (347) \n0.32 (77) \nACS/HRC/F625W \n0.025 \n9838 \nWFPC2/PC/F814W \n0.05 \n5910 \nNICMOS2/F160W \n0.05 \n7453 \nIC 4931 \n∼ 84 \nvvir \n0.99 (403) \n0.13 (51) \nWFPC2/PC/F814W \n0.05 \n8683 \n\nNote. -(1) Source name;"},{"id":"","caption":"PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.","content":"Table 2\n2NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIESGalaxy \nRadio Source \nNature of the galaxy \nScale \nPA \nP 5GHz \nComments \n\nNGC 1399 \n10 kpc e \n−10 • \n137 a \nLinear, almost symmetric. \nDominant component of Fornax cluster. \nNGC 4168 \n-\n-\n7 ± 0.3 b \nUnresolved radio emission \nE2 in Virgo cluster. \nNGC 4261 \n30 kpc f \n87 ± 8 •p \n12657 ± 610 c Linear, symmetric lobes. W jet brighter than E. Massive E in the outskirts of Virgo cluster. \nNGC 4278 \n1.4 pc g \n100 • − 155 • \n69 ± 2 c \nS-shaped, inner jet brightest to the N. \nLarge E near the center of Coma I cloud. \nNGC 4373 \n-\n-\n40 d \nCompact radio source. \nOutskirts of Hydra-Centaurus supercluster. \nNGC 4486 1.5 kpc h \n260 • \n43552 ± 242 c Linear on kpc scale. \nVirgo Cluster BCG. \nNGC 4552 \n0.5 pc i \n∼ 50 • \n12 ± 0.4 c \nTwin pc-scale extension. \nMassive E in Virgo cluster. \nNGC 4636 \n1 kpc l \n∼ 35 • \n16 ± 3 c \nZ shaped. No radio lobes. \nMassive E in the outskirts of Virgo cluster. \nNGC 4696 1.6 kpc m \n−150 • † \n3345 a \nBroad, one sided on pc-scale. \nCentaurus Cluster BCG. \nNGC 5419 \n-\n-\n1719 a \nLow surface brightness radio relic. \nDominant component of group S753. \nNGC 5846 \n-\n-\n4 ± 0.2 b \nComplex morphology. \nDominant component of group [HG82]50. \nIC 1459 \n-\n-\n541 a \nCompact source. \nGiant E in group [HG82]15. \nIC 4296 \n240 kpc n \n130 •n \n5912 a \nLinear, NW component slightly brighter. \nGiant E, BCG of group A3565. \nIC 4931 \n-\n-\n7 d \n-\nBCG of A3656."},{"id":"","caption":"PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.","content":"Table 2\n2NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIESGalaxy \nRadio Source \nNature of the galaxy \nScale \nPA \nP 5GHz \nComments \n\nNGC 1399 \n10 kpc e \n−10 • \n137 a \nLinear, almost symmetric. \nDominant component of Fornax cluster. \nNGC 4168 \n-\n-\n7 ± 0.3 b \nUnresolved radio emission \nE2 in Virgo cluster. \nNGC 4261 \n30 kpc f \n87 ± 8 •p \n12657 ± 610 c Linear, symmetric lobes. W jet brighter than E. Massive E in the outskirts of Virgo cluster. \nNGC 4278 \n1.4 pc g \n100 • − 155 • \n69 ± 2 c \nS-shaped, inner jet brightest to the N. \nLarge E near the center of Coma I cloud. \nNGC 4373 \n-\n-\n40 d \nCompact radio source. \nOutskirts of Hydra-Centaurus supercluster. \nNGC 4486 1.5 kpc h \n260 • \n43552 ± 242 c Linear on kpc scale. \nVirgo Cluster BCG. \nNGC 4552 \n0.5 pc i \n∼ 50 • \n12 ± 0.4 c \nTwin pc-scale extension. \nMassive E in Virgo cluster. \nNGC 4636 \n1 kpc l \n∼ 35 • \n16 ± 3 c \nZ shaped. No radio lobes. \nMassive E in the outskirts of Virgo cluster. \nNGC 4696 1.6 kpc m \n−150 • † \n3345 a \nBroad, one sided on pc-scale. \nCentaurus Cluster BCG. \nNGC 5419 \n-\n-\n1719 a \nLow surface brightness radio relic. \nDominant component of group S753. \nNGC 5846 \n-\n-\n4 ± 0.2 b \nComplex morphology. \nDominant component of group [HG82]50. \nIC 1459 \n-\n-\n541 a \nCompact source. \nGiant E in group [HG82]15. \nIC 4296 \n240 kpc n \n130 •n \n5912 a \nLinear, NW component slightly brighter. \nGiant E, BCG of group A3565. \nIC 4931 \n-\n-\n7 d \n-\nBCG of A3656."},{"id":"","caption":"), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines","content":"Table 3\n3PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.Precision \nACS \nWFPC2 \nNICMOS2 WFC3 \nHRC \nWFC \nPC \nWF \n\n0.1 pxl \n2.5 \n5 \n5 \n10 \n9 \nmas \n0.2 \n0.3 \n0.3 \n0.6 \n0.6 \npc \n1 \n2 \n2 \n4.1 \n3.7 \npc \n0.2 pxl \n10 \n18 \nmas \n0.6 \n1.1 \npc \n4 \n7.3 \npc"},{"id":"","caption":"), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines","content":"Table 3\n3PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.Precision \nACS \nWFPC2 \nNICMOS2 WFC3 \nHRC \nWFC \nPC \nWF \n\n0.1 pxl \n2.5 \n5 \n5 \n10 \n9 \nmas \n0.2 \n0.3 \n0.3 \n0.6 \n0.6 \npc \n1 \n2 \n2 \n4.1 \n3.7 \npc \n0.2 pxl \n10 \n18 \nmas \n0.6 \n1.1 \npc \n4 \n7.3 \npc"},{"id":"","caption":"MEASURED PROJECTED OFFSETS","content":"), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines"},{"id":"","caption":"MEASURED PROJECTED OFFSETS","content":"), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines"},{"id":"","caption":"). Kornreich & Lovelace (2008) determine the SBH acceleration for","content":"Table 4\n4MEASURED PROJECTED OFFSETSGalaxy \nInstrument \nregion \nX \nY \nOffset \nDir Type \n\nHST \nIQR \nPC \nNPS \n[pxl] \n[mas] \n[IQR] \n[pc] \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n(10) \n(11) (12) \n\nNGC 1399 WFPC2/PC -F606W [2.5:15] 0.19 \n520 ± 0.1 \n519.9 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.5 ± 0.7 0.4 ± 0.6 \nW \nnon sig \n0.28 \n465.4 ± 0.1 \n465 ± 0.1 \n0.4 ± 0.14 \n20 ± 7 \n1.4 ± 0.5 1.7 ± 0.6 \nN \nlow \n\nWFPC2/PC -F814W \n0.29 \n502.9 ± 0.1 \n502.9 ± 0.1 \n0 ± 0.14 \n0 ± 7 \n0 ± 0.5 \n0 ± 0.6 \n-\nnon sig \n0.27 \n438.6 ± 0.1 \n438.3 ± 0.1 \n0.3 ± 0.14 \n15 ± 7 \n1.1 ± 0.5 1.3 ± 0.6 \nN \nnon sig \n\nWFC3/IR -F110W \n0.13 \n759.9 ± 0.2 \n759.7 ± 0.1 \n0.2 ± 0.22 \n18 ± 20 1.5 ± 1.7 1.6 ± 1.8 \nW \nnon sig \n0.14 \n725.4 ± 0.2 \n725.3 ± 0.1 \n0.1 ± 0.22 \n9 ± 20 \n0.7 ± 1.6 0.8 ± 1.8 \nN \nnon sig \n\nWFC3/IR -F160W \n0.18 \n759.9 ± 0.2 \n759.7 ± 0.1 \n0.2 ± 0.22 \n18 ± 20 1.1 ± 1.2 1.6 ± 1.8 \nW \nnon sig \n0.08 \n725.4 ± 0.2 \n725.4 ± 0.1 \n0 ± 0.22 \n0 ± 20 \n0 ± 2.8 \n0 ± 1.8 \n-\nnon sig \n\nNGC 4168 WFPC2/PC -F702W [2.1:8.5] 1.76 \n409.6 ± 0.1 \n408.9 ± 0.1 \n0.7 ± 0.14 \n35 ± 7 \n0.4 ± 0.1 \n5 ± 1 \nW \nnull \n1.47 \n425.1 ± 0.1 \n424.6 ± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.3 ± 0.1 \n4 ± 1 \nN \nnull \n\nNGC 4261 NICMOS2 -F160W \n[1.7:7.5] 0.08 \n200 ± 0.2 \n199.6 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 \n5 ± 3.7 \n2.9 ± 2.2 \nW \nnon sig \n0.26 \n182.3 ± 0.2 \n182.1 ± 0.2 \n0.2 ± 0.3 \n10 ± 15 0.8 ± 1.1 1.5 ± 2.2 \nN \nnon sig \n\nNGC 4278 ACS/WFC -F850LP \n[1.2:9.1] 0.31 1165.6 ± 0.1 1166.4 ± 0.1 0.8 ± 0.14 \n40 ± 7 \n2.6 ± 0.5 3.4 ± 0.6 \nE \nhigh \n0.83 \n3160 ± 0.1 \n3161.1 ± 0.1 1.1 ± 0.14 \n55 ± 7 \n1.3 ± 0.2 4.7 ± 0.6 \nS \nlow \n\nWFPC2/PC -F814W \n0.44 \n370.2 ± 0.1 \n370.6 ± 0.1 \n0.4 ± 0.14 \n22 ± 7 \n1 ± 0.3 \n1.9 ± 0.6 \nE \nlow \n0.87 \n357.3 ± 0.1 \n359 ± 0.1 \n1.7 ± 0.14 \n84 ± 7 \n1.9 ± 0.2 7.2 ± 0.6 \nS \nint \n\nNICMOS2 -F160W \n0.26 \n245.8 ± 0.2 \n246.4 ± 0.2 \n0.6 ± 0.3 \n30 ± 15 2.3 ± 1.2 2.6 ± 1.3 \nE \nnon sig \n0.62 \n224.8 ± 0.2 \n225.8 ± 0.2 \n1 ± 0.3 \n50 ± 15 1.6 ± 0.5 4.3 ± 1.3 \nS \nint \n\nNGC 4373 WFPC2/PC -F814W [1.2:10] 0.58 2328.7 ± 0.1 2328.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.2 ± 0.2 \n2 ± 2.7 \nE \nnon sig \n0.16 \n870.7 ± 0.1 \n870.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.6 ± 0.9 \n2 ± 2.7 \nS \nnon sig \n\nNGC 4486 ACS/HRC -F606W \n[1:3] \n2.78 \n419.1 ± 0.1 \n682.1 ± 0.1 \n1.4 ± 0.14 \n35 ± 4 \n0.5 ± 0.1 2.7 ± 0.3 \nW \nnull \n0.98 \n417.7 ± 0.1 \n679.5 ± 0.1 \n2.6 ± 0.14 \n65 ± 4 \n2.7 ± 0.1 5.1 ± 0.3 \nN \nhigh \n\nACS/HRC -F814W \n1.03 \n553.1 ± 0.1 \n549.6 ± 0.1 \n3.5 ± 0.14 \n87 ± 4 \n3.4 ± 0.1 6.8 ± 0.3 \nW \nhigh \n0.8 \n551.8 ± 0.1 \n549.9 ± 0.1 \n1.9 ± 0.14 \n47 ± 4 \n2.4 ± 0.1 3.6 ± 0.3 \nN \nhigh \n\nNICMOS2 -F110W \n0.42 \n207.2 ± 0.2 \n206.8 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 \n1 ± 0.7 \n1.6 ± 1.2 \nW \nnon sig \n0.96 \n277.9 ± 0.2 \n277.7 ± 0.2 \n0.2 ± 0.3 \n10 ± 15 0.2 ± 0.3 0.8 ± 1.2 \nN \nnon sig \n\nNICMOS2 -F160W \n0.65 \n207.5 ± 0.2 \n207.4 ± 0.2 \n0.1 ± 0.3 \n5 ± 15 \n0.2 ± 0.5 0.4 ± 1.2 \nW \nnon sig \n0.76 \n277.8 ± 0.2 \n277.7 ± 0.2 \n0.1 ± 0.3 \n5 ± 15 \n0.1 ± 0.4 0.4 ± 1.2 \nN \nnon sig \n\nNICMOS2 -F222M \n0.51 \n207.6 ± 0.2 \n207.2 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 0.8 ± 0.6 1.6 ± 1.2 \nW \nnon sig \n0.59 \n277.3 ± 0.2 \n277.9 ± 0.2 \n0.6 ± 0.3 \n30 ± 15 \n1 ± 0.5 \n2.3 ± 1.2 \nS \nnon sig \n\nWFPC2/PC -F814W \n0.42 \n448.3 ± 0.1 \n447.6 ± 0.1 \n0.7 ± 0.14 \n35 ± 7 \n1.7 ± 0.3 2.7 ± 0.6 \nW \nlow \n0.58 \n517.2 ± 0.1 \n517 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.3 ± 0.2 0.8 ± 0.6 \nN \nnon sig"},{"id":"","caption":"). Kornreich & Lovelace (2008) determine the SBH acceleration for","content":"Table 4\n4MEASURED PROJECTED OFFSETSGalaxy \nInstrument \nregion \nX \nY \nOffset \nDir Type \n\nHST \nIQR \nPC \nNPS \n[pxl] \n[mas] \n[IQR] \n[pc] \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n(10) \n(11) (12) \n\nNGC 1399 WFPC2/PC -F606W [2.5:15] 0.19 \n520 ± 0.1 \n519.9 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.5 ± 0.7 0.4 ± 0.6 \nW \nnon sig \n0.28 \n465.4 ± 0.1 \n465 ± 0.1 \n0.4 ± 0.14 \n20 ± 7 \n1.4 ± 0.5 1.7 ± 0.6 \nN \nlow \n\nWFPC2/PC -F814W \n0.29 \n502.9 ± 0.1 \n502.9 ± 0.1 \n0 ± 0.14 \n0 ± 7 \n0 ± 0.5 \n0 ± 0.6 \n-\nnon sig \n0.27 \n438.6 ± 0.1 \n438.3 ± 0.1 \n0.3 ± 0.14 \n15 ± 7 \n1.1 ± 0.5 1.3 ± 0.6 \nN \nnon sig \n\nWFC3/IR -F110W \n0.13 \n759.9 ± 0.2 \n759.7 ± 0.1 \n0.2 ± 0.22 \n18 ± 20 1.5 ± 1.7 1.6 ± 1.8 \nW \nnon sig \n0.14 \n725.4 ± 0.2 \n725.3 ± 0.1 \n0.1 ± 0.22 \n9 ± 20 \n0.7 ± 1.6 0.8 ± 1.8 \nN \nnon sig \n\nWFC3/IR -F160W \n0.18 \n759.9 ± 0.2 \n759.7 ± 0.1 \n0.2 ± 0.22 \n18 ± 20 1.1 ± 1.2 1.6 ± 1.8 \nW \nnon sig \n0.08 \n725.4 ± 0.2 \n725.4 ± 0.1 \n0 ± 0.22 \n0 ± 20 \n0 ± 2.8 \n0 ± 1.8 \n-\nnon sig \n\nNGC 4168 WFPC2/PC -F702W [2.1:8.5] 1.76 \n409.6 ± 0.1 \n408.9 ± 0.1 \n0.7 ± 0.14 \n35 ± 7 \n0.4 ± 0.1 \n5 ± 1 \nW \nnull \n1.47 \n425.1 ± 0.1 \n424.6 ± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.3 ± 0.1 \n4 ± 1 \nN \nnull \n\nNGC 4261 NICMOS2 -F160W \n[1.7:7.5] 0.08 \n200 ± 0.2 \n199.6 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 \n5 ± 3.7 \n2.9 ± 2.2 \nW \nnon sig \n0.26 \n182.3 ± 0.2 \n182.1 ± 0.2 \n0.2 ± 0.3 \n10 ± 15 0.8 ± 1.1 1.5 ± 2.2 \nN \nnon sig \n\nNGC 4278 ACS/WFC -F850LP \n[1.2:9.1] 0.31 1165.6 ± 0.1 1166.4 ± 0.1 0.8 ± 0.14 \n40 ± 7 \n2.6 ± 0.5 3.4 ± 0.6 \nE \nhigh \n0.83 \n3160 ± 0.1 \n3161.1 ± 0.1 1.1 ± 0.14 \n55 ± 7 \n1.3 ± 0.2 4.7 ± 0.6 \nS \nlow \n\nWFPC2/PC -F814W \n0.44 \n370.2 ± 0.1 \n370.6 ± 0.1 \n0.4 ± 0.14 \n22 ± 7 \n1 ± 0.3 \n1.9 ± 0.6 \nE \nlow \n0.87 \n357.3 ± 0.1 \n359 ± 0.1 \n1.7 ± 0.14 \n84 ± 7 \n1.9 ± 0.2 7.2 ± 0.6 \nS \nint \n\nNICMOS2 -F160W \n0.26 \n245.8 ± 0.2 \n246.4 ± 0.2 \n0.6 ± 0.3 \n30 ± 15 2.3 ± 1.2 2.6 ± 1.3 \nE \nnon sig \n0.62 \n224.8 ± 0.2 \n225.8 ± 0.2 \n1 ± 0.3 \n50 ± 15 1.6 ± 0.5 4.3 ± 1.3 \nS \nint \n\nNGC 4373 WFPC2/PC -F814W [1.2:10] 0.58 2328.7 ± 0.1 2328.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.2 ± 0.2 \n2 ± 2.7 \nE \nnon sig \n0.16 \n870.7 ± 0.1 \n870.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.6 ± 0.9 \n2 ± 2.7 \nS \nnon sig \n\nNGC 4486 ACS/HRC -F606W \n[1:3] \n2.78 \n419.1 ± 0.1 \n682.1 ± 0.1 \n1.4 ± 0.14 \n35 ± 4 \n0.5 ± 0.1 2.7 ± 0.3 \nW \nnull \n0.98 \n417.7 ± 0.1 \n679.5 ± 0.1 \n2.6 ± 0.14 \n65 ± 4 \n2.7 ± 0.1 5.1 ± 0.3 \nN \nhigh \n\nACS/HRC -F814W \n1.03 \n553.1 ± 0.1 \n549.6 ± 0.1 \n3.5 ± 0.14 \n87 ± 4 \n3.4 ± 0.1 6.8 ± 0.3 \nW \nhigh \n0.8 \n551.8 ± 0.1 \n549.9 ± 0.1 \n1.9 ± 0.14 \n47 ± 4 \n2.4 ± 0.1 3.6 ± 0.3 \nN \nhigh \n\nNICMOS2 -F110W \n0.42 \n207.2 ± 0.2 \n206.8 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 \n1 ± 0.7 \n1.6 ± 1.2 \nW \nnon sig \n0.96 \n277.9 ± 0.2 \n277.7 ± 0.2 \n0.2 ± 0.3 \n10 ± 15 0.2 ± 0.3 0.8 ± 1.2 \nN \nnon sig \n\nNICMOS2 -F160W \n0.65 \n207.5 ± 0.2 \n207.4 ± 0.2 \n0.1 ± 0.3 \n5 ± 15 \n0.2 ± 0.5 0.4 ± 1.2 \nW \nnon sig \n0.76 \n277.8 ± 0.2 \n277.7 ± 0.2 \n0.1 ± 0.3 \n5 ± 15 \n0.1 ± 0.4 0.4 ± 1.2 \nN \nnon sig \n\nNICMOS2 -F222M \n0.51 \n207.6 ± 0.2 \n207.2 ± 0.2 \n0.4 ± 0.3 \n20 ± 15 0.8 ± 0.6 1.6 ± 1.2 \nW \nnon sig \n0.59 \n277.3 ± 0.2 \n277.9 ± 0.2 \n0.6 ± 0.3 \n30 ± 15 \n1 ± 0.5 \n2.3 ± 1.2 \nS \nnon sig \n\nWFPC2/PC -F814W \n0.42 \n448.3 ± 0.1 \n447.6 ± 0.1 \n0.7 ± 0.14 \n35 ± 7 \n1.7 ± 0.3 2.7 ± 0.6 \nW \nlow \n0.58 \n517.2 ± 0.1 \n517 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.3 ± 0.2 0.8 ± 0.6 \nN \nnon sig"},{"id":"","caption":"MEASURED PROJECTED OFFSETS (continued)","content":"). Kornreich & Lovelace (2008) determine the SBH acceleration for"},{"id":"","caption":"MEASURED PROJECTED OFFSETS (continued)","content":"). Kornreich & Lovelace (2008) determine the SBH acceleration for"},{"id":"","caption":"RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (see","content":"Table 4\n4MEASURED PROJECTED OFFSETS (continued) Galaxy \nInstrument \nregion \nX \nY \nOffset \nDir \nType \n\nHST \nIQR \nPC \nNPS1 \nNPS2 \n[pxl] \n[mas] \n[IQR] \n[pc] \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n(10) \n(11) \n(12) (13) \n\nNGC 4552 WFPC2/PC -F814W \n[1:15] \n0.42 \n538.9 ± 0.1 \n538.7 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.5 ± 0.3 \n0.7 ± 0.5 W \nnon sig \n0.89 \n500.8 ± 0.1 \n500.4 ± 0.2 \n0.4 ± 0.23 \n20 ± 12 \n0.4 ± 0.3 \n1.5 ± 0.9 N \nnon sig \n\nNGC 4636 WFPC2/PC -F814W \n[4:14] \n0.94 \n698 ± 0.1 \n697.4 ± 0.1 \n0.6 ± 0.14 \n30 ± 7 \n0.6 ± 0.2 \n1.9 ± 0.5 W \nnull \n0.93 \n569.9 ± 0.1 \n570 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.1 ± 0.2 \n0.3 ± 0.5 S \nnon sig \n\nNGC 4696 \nACS/WFC -F814W \n[1.5:16] \n2.15 1582.9 ± 0.1 1583.4 ± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.2 ± 0.07 \n4.5 ± 1.3 E \nnull \n3.78 2977.6 ± 0.1 2977.8 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.05 ± 0.04 1.8 ± 1.3 S \nnon sig \n\n1586.1 ± 0.1 3.2 ± 0.14 160 ± 5 \n1.5 ± 0.05 \n29 ± 1.3 E \nlow \n2973.2 ± 0.1 4.4 ± 0.14 220 ± 5 \n1.2 ± 0.03 \n39 ± 1.3 N \nlow \n\nNGC 5419 WFPC2/PC -F555W \n[2:13.5] \n0.5 \n398.4 ± 0.1 \n397.8 ± 0.1 \n0.6 ± 0.14 \n30 ± 7 \n1.2 ± 0.3 \n7 ± 2 \nW \nlow \n0.22 \n447.6 ± 0.1 \n447.8 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.9 ± 0.6 \n2 ± 2 \nS \nnon sig \n\n397.1 ± 0.1 \n1.3 ± 0.14 \n65 ± 7 \n2.6 ± 0.2 \n15 ± 2 \nW \nhigh \n442.5 ± 0.1 \n5.1 ± 0.14 255 ± 7 23.2 ± 0.5 \n60 ± 2 \nN \nhigh \n\nNGC 5846 WFPC2/PC -F814W [1.6:15.8] 0.84 \n519.5 ± 0.1 \n518.2 ± 0.4 \n1.3 ± 0.4 \n65 ± 21 \n1.6 ± 0.5 \n7.8 ± 2.5 W \nint \n1.05 \n466 ± 0.1 \n466.4 ± 0.1 \n0.4 ± 0.14 \n20 ± 7 \n0.4 ± 0.1 \n2.4 ± 0.9 S \nnull \n\nIC 1459 \nWFPC2/PC -F814W \n[2.5:15] \n0.42 \n536.4 ± 0.1 \n536.7 ± 0.1 \n0.3 ± 0.14 \n15 ± 7 \n0.7 ± 0.3 \n2.1 ± 1 \nE \nnon sig \n0.50 \n510.5 ± 0.1 \n510.7 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.4 ± 0.3 \n1.4 ± 1 \nS \nnon sig \n\nIC 4296 \nACS/HRC -F625W \n[1.5:6.5] \n0.49 \n833.3 ± 0.1 \n832.6 ± 0.2 \n0.7 ± 0.22 \n18 ± 6 \n1.4 ± 0.4 \n4.2 ± 1.2 W \nlow \n0.38 \n785.3 ± 0.1 \n784.8 ± 0.1 \n0.5 ± 0.14 \n13 ± 4 \n1.3 ± 0.3 \n3 ± 0.6 \nN \nlow \n\nWFPC2/PC -F814W \n0.39 \n551.5 ± 0.1 \n552 ± 0.18 \n0.5 ± 0.2 \n25 ± 10 \n1.3 ± 0.5 \n6 ± 2 \nE \nnon sig \n0.13 \n515 ± 0.1 \n514.9 ± 0.18 \n0.1 ± 0.2 \n5 ± 10 \n0.8 ± 1.6 \n1 ± 2 \nN \nnon sig \n\nNICMOS2 -F160W \n0.13 \n420.8 ± 0.2 \n420.6 ± 0.2 \n0.2 ± 0.3 \n13 ± 14 \n1.9 ± 2.2 \n2.8 ± 3.2 W \nnon sig \n1.07 \n253.4 ± 0.2 \n252.7 ± 0.1 \n0.7 ± 0.2 \n37 ± 11 \n0.7 ± 0.2 \n8.3 ± 2.5 N \nnull \n\nIC 4931 \nWFPC2/PC -F814W \n[1:15] \n1.29 \n403.6 ± 0.1 \n404.1± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.4 ± 0.1 \n10 ± 3 \nE \nnull \n0.66 \n505.9 ± 0.1 \n505.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.2 ± 0.2 \n2 ± 3 \nN \nnon sig"},{"id":"","caption":"RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (see","content":"Table 4\n4MEASURED PROJECTED OFFSETS (continued) Galaxy \nInstrument \nregion \nX \nY \nOffset \nDir \nType \n\nHST \nIQR \nPC \nNPS1 \nNPS2 \n[pxl] \n[mas] \n[IQR] \n[pc] \n(1) \n(2) \n(3) \n(4) \n(5) \n(6) \n(7) \n(8) \n(9) \n(10) \n(11) \n(12) (13) \n\nNGC 4552 WFPC2/PC -F814W \n[1:15] \n0.42 \n538.9 ± 0.1 \n538.7 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.5 ± 0.3 \n0.7 ± 0.5 W \nnon sig \n0.89 \n500.8 ± 0.1 \n500.4 ± 0.2 \n0.4 ± 0.23 \n20 ± 12 \n0.4 ± 0.3 \n1.5 ± 0.9 N \nnon sig \n\nNGC 4636 WFPC2/PC -F814W \n[4:14] \n0.94 \n698 ± 0.1 \n697.4 ± 0.1 \n0.6 ± 0.14 \n30 ± 7 \n0.6 ± 0.2 \n1.9 ± 0.5 W \nnull \n0.93 \n569.9 ± 0.1 \n570 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.1 ± 0.2 \n0.3 ± 0.5 S \nnon sig \n\nNGC 4696 \nACS/WFC -F814W \n[1.5:16] \n2.15 1582.9 ± 0.1 1583.4 ± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.2 ± 0.07 \n4.5 ± 1.3 E \nnull \n3.78 2977.6 ± 0.1 2977.8 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.05 ± 0.04 1.8 ± 1.3 S \nnon sig \n\n1586.1 ± 0.1 3.2 ± 0.14 160 ± 5 \n1.5 ± 0.05 \n29 ± 1.3 E \nlow \n2973.2 ± 0.1 4.4 ± 0.14 220 ± 5 \n1.2 ± 0.03 \n39 ± 1.3 N \nlow \n\nNGC 5419 WFPC2/PC -F555W \n[2:13.5] \n0.5 \n398.4 ± 0.1 \n397.8 ± 0.1 \n0.6 ± 0.14 \n30 ± 7 \n1.2 ± 0.3 \n7 ± 2 \nW \nlow \n0.22 \n447.6 ± 0.1 \n447.8 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.9 ± 0.6 \n2 ± 2 \nS \nnon sig \n\n397.1 ± 0.1 \n1.3 ± 0.14 \n65 ± 7 \n2.6 ± 0.2 \n15 ± 2 \nW \nhigh \n442.5 ± 0.1 \n5.1 ± 0.14 255 ± 7 23.2 ± 0.5 \n60 ± 2 \nN \nhigh \n\nNGC 5846 WFPC2/PC -F814W [1.6:15.8] 0.84 \n519.5 ± 0.1 \n518.2 ± 0.4 \n1.3 ± 0.4 \n65 ± 21 \n1.6 ± 0.5 \n7.8 ± 2.5 W \nint \n1.05 \n466 ± 0.1 \n466.4 ± 0.1 \n0.4 ± 0.14 \n20 ± 7 \n0.4 ± 0.1 \n2.4 ± 0.9 S \nnull \n\nIC 1459 \nWFPC2/PC -F814W \n[2.5:15] \n0.42 \n536.4 ± 0.1 \n536.7 ± 0.1 \n0.3 ± 0.14 \n15 ± 7 \n0.7 ± 0.3 \n2.1 ± 1 \nE \nnon sig \n0.50 \n510.5 ± 0.1 \n510.7 ± 0.1 \n0.2 ± 0.14 \n10 ± 7 \n0.4 ± 0.3 \n1.4 ± 1 \nS \nnon sig \n\nIC 4296 \nACS/HRC -F625W \n[1.5:6.5] \n0.49 \n833.3 ± 0.1 \n832.6 ± 0.2 \n0.7 ± 0.22 \n18 ± 6 \n1.4 ± 0.4 \n4.2 ± 1.2 W \nlow \n0.38 \n785.3 ± 0.1 \n784.8 ± 0.1 \n0.5 ± 0.14 \n13 ± 4 \n1.3 ± 0.3 \n3 ± 0.6 \nN \nlow \n\nWFPC2/PC -F814W \n0.39 \n551.5 ± 0.1 \n552 ± 0.18 \n0.5 ± 0.2 \n25 ± 10 \n1.3 ± 0.5 \n6 ± 2 \nE \nnon sig \n0.13 \n515 ± 0.1 \n514.9 ± 0.18 \n0.1 ± 0.2 \n5 ± 10 \n0.8 ± 1.6 \n1 ± 2 \nN \nnon sig \n\nNICMOS2 -F160W \n0.13 \n420.8 ± 0.2 \n420.6 ± 0.2 \n0.2 ± 0.3 \n13 ± 14 \n1.9 ± 2.2 \n2.8 ± 3.2 W \nnon sig \n1.07 \n253.4 ± 0.2 \n252.7 ± 0.1 \n0.7 ± 0.2 \n37 ± 11 \n0.7 ± 0.2 \n8.3 ± 2.5 N \nnull \n\nIC 4931 \nWFPC2/PC -F814W \n[1:15] \n1.29 \n403.6 ± 0.1 \n404.1± 0.1 \n0.5 ± 0.14 \n25 ± 7 \n0.4 ± 0.1 \n10 ± 3 \nE \nnull \n0.66 \n505.9 ± 0.1 \n505.8 ± 0.1 \n0.1 ± 0.14 \n5 ± 7 \n0.2 ± 0.2 \n2 ± 3 \nN \nnon sig"},{"id":"","caption":"PARAMETERS OF RECOIL OSCILLATION MODEL","content":"Table 5\n5RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (seeGalaxy \nPA (deg) \njet \ndisplacement \nNPS1 \nNPS2 \n\nkpc-scale jets: \nNGC 1399 \n−10 \n−17 ± 16 a \nlow \nNGC 4261 \n87 ± 8 \n297 ± 38 (117 ± 38) \nnon significant \nNGC 4486 \n290 ± 3 \n307 ± 1 (127 ± 1) \n\n ‡ \n\nNGC 4696 \n210 (+30) 112 ± 15 \n36 ± 1 \n\n ‡ \n\nIC 4296 \n130 \n338 ± 7 (158 ± 8) b \nlow \n\npc-scale jets: \nNGC 4278 \n100 − 155 152 ± 3 \n\n ‡ \n\nNGC 4552 \n50 \n333 ± 21 \nlow \nNGC 4636 \n35 \n261 ± 9 (81 ± 9) \nnull"},{"id":"","caption":"PARAMETERS OF RECOIL OSCILLATION MODEL","content":"Table 5\n5RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (seeGalaxy \nPA (deg) \njet \ndisplacement \nNPS1 \nNPS2 \n\nkpc-scale jets: \nNGC 1399 \n−10 \n−17 ± 16 a \nlow \nNGC 4261 \n87 ± 8 \n297 ± 38 (117 ± 38) \nnon significant \nNGC 4486 \n290 ± 3 \n307 ± 1 (127 ± 1) \n\n ‡ \n\nNGC 4696 \n210 (+30) 112 ± 15 \n36 ± 1 \n\n ‡ \n\nIC 4296 \n130 \n338 ± 7 (158 ± 8) b \nlow \n\npc-scale jets: \nNGC 4278 \n100 − 155 152 ± 3 \n\n ‡ \n\nNGC 4552 \n50 \n333 ± 21 \nlow \nNGC 4636 \n35 \n261 ± 9 (81 ± 9) \nnull"},{"id":"","caption":"SBH MASSES","content":"Table 6 PROBABILITIES\n6Galaxy \nd i [rc] p i (tm = 0.4 Gyr) p i (tm = 5 Gyr) \n\nNGC 1399 \n0.01 \n0.75 \n0.11 \nNGC 4168 * 0.02 \n0.98 \n0.89 \nNGC 4261 <0.01 \n0.94 \n0.23 \nNGC 4278 \n0.1 \n0.36 \n0.04 \nNGC 4373 <0.01 \n0.99 \n0.58 \nNGC 4486 \n0.01 \n0.99 \n0.85 \nNGC 4552 <0.05 \n0.10 \n0.01 \nNGC 4636 * 0.01 \n0.99 \n0.69 \nNGC 4696 * 0.02 \n0.97 \n0.42 \nNGC 5419 \n0.02 \n0.97 \n0.46 \nNGC 5846 \n0.04 \n0.90 \n0.25 \nIC 1459 \n<0.01 \n0.92 \n0.21 \nIC 4296 \n0.01 \n0.97 \n0.37 \nIC 4931 \n * 0.03 \n0.95 \n0.46"},{"id":"","caption":"SBH MASSES","content":"Table 6 PROBABILITIES\n6Galaxy \nd i [rc] p i (tm = 0.4 Gyr) p i (tm = 5 Gyr) \n\nNGC 1399 \n0.01 \n0.75 \n0.11 \nNGC 4168 * 0.02 \n0.98 \n0.89 \nNGC 4261 <0.01 \n0.94 \n0.23 \nNGC 4278 \n0.1 \n0.36 \n0.04 \nNGC 4373 <0.01 \n0.99 \n0.58 \nNGC 4486 \n0.01 \n0.99 \n0.85 \nNGC 4552 <0.05 \n0.10 \n0.01 \nNGC 4636 * 0.01 \n0.99 \n0.69 \nNGC 4696 * 0.02 \n0.97 \n0.42 \nNGC 5419 \n0.02 \n0.97 \n0.46 \nNGC 5846 \n0.04 \n0.90 \n0.25 \nIC 1459 \n<0.01 \n0.92 \n0.21 \nIC 4296 \n0.01 \n0.97 \n0.37 \nIC 4931 \n * 0.03 \n0.95 \n0.46"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA.\",\n \"content\": \"Figure 1 .\\n1Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA.\",\n \"content\": \"Figure 1 .\\n1Alternative weighting functions for the mean photocenter, derived from the NICMOS2/F160W image of NGC 4261. Both functions are normalized to their maximum values. Solid line: each isophote center is weighted by the light content of the corresponding elliptical annulus. Dashed dotted: each isophote center is weighted by the light content enclosed by the ellipse of the corresponding SMA.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \\\"true errors\\\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC.\",\n \"content\": \"Figure 2 .\\n2Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \\\"true errors\\\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \\\"true errors\\\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC.\",\n \"content\": \"Figure 2 .\\n2Figure 2. Error distributions for the (x, y) co-ordinates of isophote center positions derived from ellipse fits to sixteen simulated galaxy images. For each galaxy, a series of ellipses was fitted to the surface brightness distribution, incrementing the SMA by two pixels at each step, as described in Sec. 3.1. The errors returned by the IRAF ellipse task are compared with the \\\"true errors\\\", defined as the difference between the known position of the center of the model galaxy and the center position returned by the ellipse fit. The errors on each isophote center returned by ellipse are broadly consistent with the true errors. The error distributions combine results from all sixteen simulated galaxy images. The model images were derived from Galfit fits to the following data: NGC 4261/NICMOS2, IC 1459/WFPC2-PC, NGC 1399/WFC3-IR, IC 4296/ACS-HRC.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers.\",\n \"content\": \"Figure 3\\n3Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers.\",\n \"content\": \"Figure 3\\n3Figure 3. Example of a galaxy where the displacement is not significant. NGC 4373, WFPC2/PC -F814W, scale=0. 05/pxl. First row -left to right: the galaxy Log(counts s −1 ), note that the color scale is adjusted to highlight the presence of the nuclear point source, emission is present over the whole field of view; galaxy after subtraction of the isophotal model; galaxy after subtraction of the median filter. Second row -surface brightness profile in counts s −1 ; second and third panels: x and y coordinates of the isophotal centers versus semi-major axis length. Blue diamond: AGN; red bullet: photocenter. The blue diamond and the red bullet are not positioned both at SMA=0 to make the figure clear and avoid superpositions. Third row -scatter plot of the isophote centers. The solid line (near the photocenter) is the cumulative photocenter computed including progressively all data from the core radius outward. Different colors for the data points are used to represent the centers of isophotes with SMA length in a given range. Defining w = 33%(SM Amax − rc): green: rc < SM A ≤ rc + w, orange: rc + w < SM A ≤ rc + 2w, brown: rc + 2w < SM A < SM Amax. Second and third panels: histograms of the distributions of the x and y coordinates of the isophotal centers.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3.\",\n \"content\": \"Figure 4 .\\n4Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3.\",\n \"content\": \"Figure 4 .\\n4Example of a galaxy where the displacement is significant. NGC 4486 (M 87), ACS/HRC/F814W, scale=0. 025/pxl. Caption as inFig.3.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position).\",\n \"content\": \"Figure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position).\",\n \"content\": \"Figure 5a. Offset of the isophotal center with respect to the SBH as measured on different instruments, at different frequencies. Each photocenter is centered on an ellipse whose axes represent the IQR of the isophote centers dataset used to derive the photocenter. Error bars represent the error on the offset. The diamond marks the origin of the reference frame (SBH position).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2.\",\n \"content\": \"Figure 5b .\\n5bAs inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2.\",\n \"content\": \"Figure 5b .\\n5bAs inFig.5afor the galaxies showing a double nuclear point source. The offset of the fainter nuclear point source with respect to the SBH is marked with a diamond and the label NPS2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Probability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\",\n \"content\": \"Figure 7a .\\n7aProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Probability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\",\n \"content\": \"Figure 7a .\\n7aProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 0.4 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Probability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\",\n \"content\": \"Figure 7b .\\n7bProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Probability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\",\n \"content\": \"Figure 7b .\\n7bProbability to observe a displacement larger than the value specified on the x-axis in units of the core radius for tm = 5 Gyr. The observed displacement is marked as a filled circle. The filled square (in NGC 4694 and 5419) marks the offset of the secondary point source.\"\n },\n {\n \"id\": \"\",\n \"caption\": \", (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000).\",\n \"content\": \", (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000).\"\n },\n {\n \"id\": \"\",\n \"caption\": \", (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000).\",\n \"content\": \", (3) masses from the M•-M V relation from eq.6 in Lauer et al. (2007); values for M V have been taken from Lauer et al. (2007), when not available (NGC 4373, 5846 and IC 4296) they have been taken from NED (V T values from the RC3 catalog, de Vaucouleurs et al. 1991), (4) masses from stellar orbit modeling, (5) masses derived from gas kinematics. Masses have been scaled to the distances assumed in this work. † The authors warn that these values might be affected by non-gravitational motions in the gas (i.e. inflows or outflows). References: a Gebhardt et al. (2007), b Houghton et al. (2006), c Ferrarese et al. (1996), d Gebhardt et al. (2011), e Cappellari et al. (2002), f Verdoes Kleijn et al. (2000).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear.\",\n \"content\": \"Figure 8 .\\n8M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear.\",\n \"content\": \"Figure 8 .\\n8M87 -Using a set of bright globular clusters visible in all the frames, we matched the AGN and isophotal center position with the coordinate system defined by the ACS frame. The result shows that while the AGN (diamonds) remains fairly stable, the isophotal center (crosses) has the larger scatter accounting for the discrepancies in the recovered offsets. The smaller error is associated with results from ACS, the larger with results from NICMOS. Results from only three images out of six are shown to make the image clear.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl.\",\n \"content\": \"Figure 18 .\\n18As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl.\",\n \"content\": \"Figure 18 .\\n18As in Fig.3 for galaxy NGC 4486 (M 87), ACS/HRC -F606W, scale=0. 025/pxl.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl.\",\n \"content\": \"Figure 21 .\\n21As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl.\",\n \"content\": \"Figure 21 .\\n21As in Fig.3 for galaxy NGC 4486 (M 87), NICMOS2 -F222M, scale=0. 05/pxl.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl.\",\n \"content\": \"Figure 22 .\\n22As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl.\",\n \"content\": \"Figure 22 .\\n22As in Fig.3 for galaxy NGC 4486 (M 87), WFPC2/PC -F814W, scale=0. 05/pxl.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIES\",\n \"content\": \"Table 1 THE SAMPLE\\n1Galaxy \\nDistance \\nrc \\nr h \\nInstrument \\nPixel Scale \\nID \\n(1) \\n(2) \\n(3) \\n(4) \\n(5) \\n(6) \\n(7) \\n(8) \\n(9) \\n\\nNGC 1399 \\n18 ± 1 \\nsbf \\n2.19 (189) \\n0.99 (86) \\nWFPC2/PC/F606W \\n0.05 \\n8214 \\nWFPC2/PC/F814W \\n0.05 \\n5990 \\nWFC3/IR/F110W \\n0.09 \\n11712 \\nWFC3/IR/F160W \\n0.09 \\n11712 \\nNGC 4168 \\nUGC 7203 \\n31 ± 6 \\nsbf \\n2.02 (303) \\n0.09 (14) \\nWFPC2/PC/F702W \\n0.05 \\n6357 \\nNGC 4261 \\nUGC 7360 \\n30 ± 2 \\nsbf \\n1.62 (237) \\n0.428 (62) \\nNICMOS2/F160W \\n0.05 \\n7868 \\nNGC 4278 \\nUGC 7386 \\n18 ± 1 \\nsbf \\n0.97 (83) \\n0.334 (29) \\nACS/WFC/F850LP \\n0.05 \\n10835 \\nWFPC2/PC/F814W \\n0.05 \\n5454 \\nNICMOS2/F160W \\n0.05 \\n7886 \\nNGC 4373 \\n40 ± 2 \\nsbf \\n1.19 (269) \\n0.167 (32) \\nWFPC2/PC/F814W \\n0.05 \\n5214 \\nNGC 4486 \\nUGC 7654 (M87) \\n16 ± 1 \\nsbf \\n9.41 (733) \\n1 (78) \\nACS/HRC/F606W \\n0.025 \\nB10 \\nACS/HRC/F814W \\n0.025 \\nB10 \\nNICMOS2/F110W \\n0.05 \\n7171 \\nNICMOS2/F160W \\n0.05 \\n7171 \\nNICMOS2/F222M \\n0.05 \\n7171 \\nWFPC2/PC/F814W \\n0.05 \\n8592 \\nNGC 4552 \\nUGC 7760 \\n15 ± 1 \\nsbf \\n0.49 (36) \\n0.48 (35) \\nWFPC2/PC/F814W \\n0.05 \\n6099 \\nNGC 4636 \\nUGC 7878 \\n13 ± 1 \\nsbf \\n3.44 (219) \\n0.297 (19) \\nWFPC2/PC/F814W \\n0.05 \\n8686 \\nNGC 4696 \\n37 ± 2 \\nsbf \\n1.4 (251) \\n0.2 (36) \\nACS/WFC/F814W \\n0.05 \\n9427 \\n\\n † \\n\\nNGC 5419 \\n49 ± 9 \\nDn − σ \\n2.11 (499) \\n0.38 (90) \\nWFPC2/PC/F555W \\n0.05 \\n6587 \\nNGC 5846 \\nUGC 9706 \\n25 ± 2 \\nsbf \\n1.52 (183) \\n0.25 (30) \\nWFPC2/PC/F814W \\n0.05 \\n5920 \\nIC 1459 \\n29 ± 4 \\nsbf \\n1.82 (258) \\n0.43 (60) \\nWFPC2/PC/F814W \\n0.05 \\n5454 \\nIC 4296 \\n50 ± 3 \\nsbf \\n1.44 (347) \\n0.32 (77) \\nACS/HRC/F625W \\n0.025 \\n9838 \\nWFPC2/PC/F814W \\n0.05 \\n5910 \\nNICMOS2/F160W \\n0.05 \\n7453 \\nIC 4931 \\n∼ 84 \\nvvir \\n0.99 (403) \\n0.13 (51) \\nWFPC2/PC/F814W \\n0.05 \\n8683 \\n\\nNote. -(1) Source name;\"\n },\n {\n \"id\": \"\",\n \"caption\": \"NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIES\",\n \"content\": \"Table 1 THE SAMPLE\\n1Galaxy \\nDistance \\nrc \\nr h \\nInstrument \\nPixel Scale \\nID \\n(1) \\n(2) \\n(3) \\n(4) \\n(5) \\n(6) \\n(7) \\n(8) \\n(9) \\n\\nNGC 1399 \\n18 ± 1 \\nsbf \\n2.19 (189) \\n0.99 (86) \\nWFPC2/PC/F606W \\n0.05 \\n8214 \\nWFPC2/PC/F814W \\n0.05 \\n5990 \\nWFC3/IR/F110W \\n0.09 \\n11712 \\nWFC3/IR/F160W \\n0.09 \\n11712 \\nNGC 4168 \\nUGC 7203 \\n31 ± 6 \\nsbf \\n2.02 (303) \\n0.09 (14) \\nWFPC2/PC/F702W \\n0.05 \\n6357 \\nNGC 4261 \\nUGC 7360 \\n30 ± 2 \\nsbf \\n1.62 (237) \\n0.428 (62) \\nNICMOS2/F160W \\n0.05 \\n7868 \\nNGC 4278 \\nUGC 7386 \\n18 ± 1 \\nsbf \\n0.97 (83) \\n0.334 (29) \\nACS/WFC/F850LP \\n0.05 \\n10835 \\nWFPC2/PC/F814W \\n0.05 \\n5454 \\nNICMOS2/F160W \\n0.05 \\n7886 \\nNGC 4373 \\n40 ± 2 \\nsbf \\n1.19 (269) \\n0.167 (32) \\nWFPC2/PC/F814W \\n0.05 \\n5214 \\nNGC 4486 \\nUGC 7654 (M87) \\n16 ± 1 \\nsbf \\n9.41 (733) \\n1 (78) \\nACS/HRC/F606W \\n0.025 \\nB10 \\nACS/HRC/F814W \\n0.025 \\nB10 \\nNICMOS2/F110W \\n0.05 \\n7171 \\nNICMOS2/F160W \\n0.05 \\n7171 \\nNICMOS2/F222M \\n0.05 \\n7171 \\nWFPC2/PC/F814W \\n0.05 \\n8592 \\nNGC 4552 \\nUGC 7760 \\n15 ± 1 \\nsbf \\n0.49 (36) \\n0.48 (35) \\nWFPC2/PC/F814W \\n0.05 \\n6099 \\nNGC 4636 \\nUGC 7878 \\n13 ± 1 \\nsbf \\n3.44 (219) \\n0.297 (19) \\nWFPC2/PC/F814W \\n0.05 \\n8686 \\nNGC 4696 \\n37 ± 2 \\nsbf \\n1.4 (251) \\n0.2 (36) \\nACS/WFC/F814W \\n0.05 \\n9427 \\n\\n † \\n\\nNGC 5419 \\n49 ± 9 \\nDn − σ \\n2.11 (499) \\n0.38 (90) \\nWFPC2/PC/F555W \\n0.05 \\n6587 \\nNGC 5846 \\nUGC 9706 \\n25 ± 2 \\nsbf \\n1.52 (183) \\n0.25 (30) \\nWFPC2/PC/F814W \\n0.05 \\n5920 \\nIC 1459 \\n29 ± 4 \\nsbf \\n1.82 (258) \\n0.43 (60) \\nWFPC2/PC/F814W \\n0.05 \\n5454 \\nIC 4296 \\n50 ± 3 \\nsbf \\n1.44 (347) \\n0.32 (77) \\nACS/HRC/F625W \\n0.025 \\n9838 \\nWFPC2/PC/F814W \\n0.05 \\n5910 \\nNICMOS2/F160W \\n0.05 \\n7453 \\nIC 4931 \\n∼ 84 \\nvvir \\n0.99 (403) \\n0.13 (51) \\nWFPC2/PC/F814W \\n0.05 \\n8683 \\n\\nNote. -(1) Source name;\"\n },\n {\n \"id\": \"\",\n \"caption\": \"PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.\",\n \"content\": \"Table 2\\n2NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIESGalaxy \\nRadio Source \\nNature of the galaxy \\nScale \\nPA \\nP 5GHz \\nComments \\n\\nNGC 1399 \\n10 kpc e \\n−10 • \\n137 a \\nLinear, almost symmetric. \\nDominant component of Fornax cluster. \\nNGC 4168 \\n-\\n-\\n7 ± 0.3 b \\nUnresolved radio emission \\nE2 in Virgo cluster. \\nNGC 4261 \\n30 kpc f \\n87 ± 8 •p \\n12657 ± 610 c Linear, symmetric lobes. W jet brighter than E. Massive E in the outskirts of Virgo cluster. \\nNGC 4278 \\n1.4 pc g \\n100 • − 155 • \\n69 ± 2 c \\nS-shaped, inner jet brightest to the N. \\nLarge E near the center of Coma I cloud. \\nNGC 4373 \\n-\\n-\\n40 d \\nCompact radio source. \\nOutskirts of Hydra-Centaurus supercluster. \\nNGC 4486 1.5 kpc h \\n260 • \\n43552 ± 242 c Linear on kpc scale. \\nVirgo Cluster BCG. \\nNGC 4552 \\n0.5 pc i \\n∼ 50 • \\n12 ± 0.4 c \\nTwin pc-scale extension. \\nMassive E in Virgo cluster. \\nNGC 4636 \\n1 kpc l \\n∼ 35 • \\n16 ± 3 c \\nZ shaped. No radio lobes. \\nMassive E in the outskirts of Virgo cluster. \\nNGC 4696 1.6 kpc m \\n−150 • † \\n3345 a \\nBroad, one sided on pc-scale. \\nCentaurus Cluster BCG. \\nNGC 5419 \\n-\\n-\\n1719 a \\nLow surface brightness radio relic. \\nDominant component of group S753. \\nNGC 5846 \\n-\\n-\\n4 ± 0.2 b \\nComplex morphology. \\nDominant component of group [HG82]50. \\nIC 1459 \\n-\\n-\\n541 a \\nCompact source. \\nGiant E in group [HG82]15. \\nIC 4296 \\n240 kpc n \\n130 •n \\n5912 a \\nLinear, NW component slightly brighter. \\nGiant E, BCG of group A3565. \\nIC 4931 \\n-\\n-\\n7 d \\n-\\nBCG of A3656.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.\",\n \"content\": \"Table 2\\n2NOTES ON THE RADIO SOURCES OF INDIVIDUAL GALAXIESGalaxy \\nRadio Source \\nNature of the galaxy \\nScale \\nPA \\nP 5GHz \\nComments \\n\\nNGC 1399 \\n10 kpc e \\n−10 • \\n137 a \\nLinear, almost symmetric. \\nDominant component of Fornax cluster. \\nNGC 4168 \\n-\\n-\\n7 ± 0.3 b \\nUnresolved radio emission \\nE2 in Virgo cluster. \\nNGC 4261 \\n30 kpc f \\n87 ± 8 •p \\n12657 ± 610 c Linear, symmetric lobes. W jet brighter than E. Massive E in the outskirts of Virgo cluster. \\nNGC 4278 \\n1.4 pc g \\n100 • − 155 • \\n69 ± 2 c \\nS-shaped, inner jet brightest to the N. \\nLarge E near the center of Coma I cloud. \\nNGC 4373 \\n-\\n-\\n40 d \\nCompact radio source. \\nOutskirts of Hydra-Centaurus supercluster. \\nNGC 4486 1.5 kpc h \\n260 • \\n43552 ± 242 c Linear on kpc scale. \\nVirgo Cluster BCG. \\nNGC 4552 \\n0.5 pc i \\n∼ 50 • \\n12 ± 0.4 c \\nTwin pc-scale extension. \\nMassive E in Virgo cluster. \\nNGC 4636 \\n1 kpc l \\n∼ 35 • \\n16 ± 3 c \\nZ shaped. No radio lobes. \\nMassive E in the outskirts of Virgo cluster. \\nNGC 4696 1.6 kpc m \\n−150 • † \\n3345 a \\nBroad, one sided on pc-scale. \\nCentaurus Cluster BCG. \\nNGC 5419 \\n-\\n-\\n1719 a \\nLow surface brightness radio relic. \\nDominant component of group S753. \\nNGC 5846 \\n-\\n-\\n4 ± 0.2 b \\nComplex morphology. \\nDominant component of group [HG82]50. \\nIC 1459 \\n-\\n-\\n541 a \\nCompact source. \\nGiant E in group [HG82]15. \\nIC 4296 \\n240 kpc n \\n130 •n \\n5912 a \\nLinear, NW component slightly brighter. \\nGiant E, BCG of group A3565. \\nIC 4931 \\n-\\n-\\n7 d \\n-\\nBCG of A3656.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines\",\n \"content\": \"Table 3\\n3PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.Precision \\nACS \\nWFPC2 \\nNICMOS2 WFC3 \\nHRC \\nWFC \\nPC \\nWF \\n\\n0.1 pxl \\n2.5 \\n5 \\n5 \\n10 \\n9 \\nmas \\n0.2 \\n0.3 \\n0.3 \\n0.6 \\n0.6 \\npc \\n1 \\n2 \\n2 \\n4.1 \\n3.7 \\npc \\n0.2 pxl \\n10 \\n18 \\nmas \\n0.6 \\n1.1 \\npc \\n4 \\n7.3 \\npc\"\n },\n {\n \"id\": \"\",\n \"caption\": \"), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines\",\n \"content\": \"Table 3\\n3PRECISION ON ISOPHOTAL CENTER AND AGN POSITION Note. -Values are given in mas (first row), in pc at the nearest galaxy (13 Mpc, second row) and at the farthest (84 Mpc, third row). Similarly for the other rows. Values for NICMOS2 are computed for a pixel scale of 0. 05/pixel.Precision \\nACS \\nWFPC2 \\nNICMOS2 WFC3 \\nHRC \\nWFC \\nPC \\nWF \\n\\n0.1 pxl \\n2.5 \\n5 \\n5 \\n10 \\n9 \\nmas \\n0.2 \\n0.3 \\n0.3 \\n0.6 \\n0.6 \\npc \\n1 \\n2 \\n2 \\n4.1 \\n3.7 \\npc \\n0.2 pxl \\n10 \\n18 \\nmas \\n0.6 \\n1.1 \\npc \\n4 \\n7.3 \\npc\"\n },\n {\n \"id\": \"\",\n \"caption\": \"MEASURED PROJECTED OFFSETS\",\n \"content\": \"), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines\"\n },\n {\n \"id\": \"\",\n \"caption\": \"MEASURED PROJECTED OFFSETS\",\n \"content\": \"), NGC1399(O'Connell et al. 2005), NGC4552(Cappellari et al. 1999;Maoz et al. 2005) and NGC 4278 (Cardullo et al. 2009). Hard X-ray point sources have been detected in NGC 4261, 4278, 4486, 4552, 5419 and IC 1459 (González-Martín et al. 2009). Weak broad Hα lines\"\n },\n {\n \"id\": \"\",\n \"caption\": \"). Kornreich & Lovelace (2008) determine the SBH acceleration for\",\n \"content\": \"Table 4\\n4MEASURED PROJECTED OFFSETSGalaxy \\nInstrument \\nregion \\nX \\nY \\nOffset \\nDir Type \\n\\nHST \\nIQR \\nPC \\nNPS \\n[pxl] \\n[mas] \\n[IQR] \\n[pc] \\n(1) \\n(2) \\n(3) \\n(4) \\n(5) \\n(6) \\n(7) \\n(8) \\n(9) \\n(10) \\n(11) (12) \\n\\nNGC 1399 WFPC2/PC -F606W [2.5:15] 0.19 \\n520 ± 0.1 \\n519.9 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.5 ± 0.7 0.4 ± 0.6 \\nW \\nnon sig \\n0.28 \\n465.4 ± 0.1 \\n465 ± 0.1 \\n0.4 ± 0.14 \\n20 ± 7 \\n1.4 ± 0.5 1.7 ± 0.6 \\nN \\nlow \\n\\nWFPC2/PC -F814W \\n0.29 \\n502.9 ± 0.1 \\n502.9 ± 0.1 \\n0 ± 0.14 \\n0 ± 7 \\n0 ± 0.5 \\n0 ± 0.6 \\n-\\nnon sig \\n0.27 \\n438.6 ± 0.1 \\n438.3 ± 0.1 \\n0.3 ± 0.14 \\n15 ± 7 \\n1.1 ± 0.5 1.3 ± 0.6 \\nN \\nnon sig \\n\\nWFC3/IR -F110W \\n0.13 \\n759.9 ± 0.2 \\n759.7 ± 0.1 \\n0.2 ± 0.22 \\n18 ± 20 1.5 ± 1.7 1.6 ± 1.8 \\nW \\nnon sig \\n0.14 \\n725.4 ± 0.2 \\n725.3 ± 0.1 \\n0.1 ± 0.22 \\n9 ± 20 \\n0.7 ± 1.6 0.8 ± 1.8 \\nN \\nnon sig \\n\\nWFC3/IR -F160W \\n0.18 \\n759.9 ± 0.2 \\n759.7 ± 0.1 \\n0.2 ± 0.22 \\n18 ± 20 1.1 ± 1.2 1.6 ± 1.8 \\nW \\nnon sig \\n0.08 \\n725.4 ± 0.2 \\n725.4 ± 0.1 \\n0 ± 0.22 \\n0 ± 20 \\n0 ± 2.8 \\n0 ± 1.8 \\n-\\nnon sig \\n\\nNGC 4168 WFPC2/PC -F702W [2.1:8.5] 1.76 \\n409.6 ± 0.1 \\n408.9 ± 0.1 \\n0.7 ± 0.14 \\n35 ± 7 \\n0.4 ± 0.1 \\n5 ± 1 \\nW \\nnull \\n1.47 \\n425.1 ± 0.1 \\n424.6 ± 0.1 \\n0.5 ± 0.14 \\n25 ± 7 \\n0.3 ± 0.1 \\n4 ± 1 \\nN \\nnull \\n\\nNGC 4261 NICMOS2 -F160W \\n[1.7:7.5] 0.08 \\n200 ± 0.2 \\n199.6 ± 0.2 \\n0.4 ± 0.3 \\n20 ± 15 \\n5 ± 3.7 \\n2.9 ± 2.2 \\nW \\nnon sig \\n0.26 \\n182.3 ± 0.2 \\n182.1 ± 0.2 \\n0.2 ± 0.3 \\n10 ± 15 0.8 ± 1.1 1.5 ± 2.2 \\nN \\nnon sig \\n\\nNGC 4278 ACS/WFC -F850LP \\n[1.2:9.1] 0.31 1165.6 ± 0.1 1166.4 ± 0.1 0.8 ± 0.14 \\n40 ± 7 \\n2.6 ± 0.5 3.4 ± 0.6 \\nE \\nhigh \\n0.83 \\n3160 ± 0.1 \\n3161.1 ± 0.1 1.1 ± 0.14 \\n55 ± 7 \\n1.3 ± 0.2 4.7 ± 0.6 \\nS \\nlow \\n\\nWFPC2/PC -F814W \\n0.44 \\n370.2 ± 0.1 \\n370.6 ± 0.1 \\n0.4 ± 0.14 \\n22 ± 7 \\n1 ± 0.3 \\n1.9 ± 0.6 \\nE \\nlow \\n0.87 \\n357.3 ± 0.1 \\n359 ± 0.1 \\n1.7 ± 0.14 \\n84 ± 7 \\n1.9 ± 0.2 7.2 ± 0.6 \\nS \\nint \\n\\nNICMOS2 -F160W \\n0.26 \\n245.8 ± 0.2 \\n246.4 ± 0.2 \\n0.6 ± 0.3 \\n30 ± 15 2.3 ± 1.2 2.6 ± 1.3 \\nE \\nnon sig \\n0.62 \\n224.8 ± 0.2 \\n225.8 ± 0.2 \\n1 ± 0.3 \\n50 ± 15 1.6 ± 0.5 4.3 ± 1.3 \\nS \\nint \\n\\nNGC 4373 WFPC2/PC -F814W [1.2:10] 0.58 2328.7 ± 0.1 2328.8 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.2 ± 0.2 \\n2 ± 2.7 \\nE \\nnon sig \\n0.16 \\n870.7 ± 0.1 \\n870.8 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.6 ± 0.9 \\n2 ± 2.7 \\nS \\nnon sig \\n\\nNGC 4486 ACS/HRC -F606W \\n[1:3] \\n2.78 \\n419.1 ± 0.1 \\n682.1 ± 0.1 \\n1.4 ± 0.14 \\n35 ± 4 \\n0.5 ± 0.1 2.7 ± 0.3 \\nW \\nnull \\n0.98 \\n417.7 ± 0.1 \\n679.5 ± 0.1 \\n2.6 ± 0.14 \\n65 ± 4 \\n2.7 ± 0.1 5.1 ± 0.3 \\nN \\nhigh \\n\\nACS/HRC -F814W \\n1.03 \\n553.1 ± 0.1 \\n549.6 ± 0.1 \\n3.5 ± 0.14 \\n87 ± 4 \\n3.4 ± 0.1 6.8 ± 0.3 \\nW \\nhigh \\n0.8 \\n551.8 ± 0.1 \\n549.9 ± 0.1 \\n1.9 ± 0.14 \\n47 ± 4 \\n2.4 ± 0.1 3.6 ± 0.3 \\nN \\nhigh \\n\\nNICMOS2 -F110W \\n0.42 \\n207.2 ± 0.2 \\n206.8 ± 0.2 \\n0.4 ± 0.3 \\n20 ± 15 \\n1 ± 0.7 \\n1.6 ± 1.2 \\nW \\nnon sig \\n0.96 \\n277.9 ± 0.2 \\n277.7 ± 0.2 \\n0.2 ± 0.3 \\n10 ± 15 0.2 ± 0.3 0.8 ± 1.2 \\nN \\nnon sig \\n\\nNICMOS2 -F160W \\n0.65 \\n207.5 ± 0.2 \\n207.4 ± 0.2 \\n0.1 ± 0.3 \\n5 ± 15 \\n0.2 ± 0.5 0.4 ± 1.2 \\nW \\nnon sig \\n0.76 \\n277.8 ± 0.2 \\n277.7 ± 0.2 \\n0.1 ± 0.3 \\n5 ± 15 \\n0.1 ± 0.4 0.4 ± 1.2 \\nN \\nnon sig \\n\\nNICMOS2 -F222M \\n0.51 \\n207.6 ± 0.2 \\n207.2 ± 0.2 \\n0.4 ± 0.3 \\n20 ± 15 0.8 ± 0.6 1.6 ± 1.2 \\nW \\nnon sig \\n0.59 \\n277.3 ± 0.2 \\n277.9 ± 0.2 \\n0.6 ± 0.3 \\n30 ± 15 \\n1 ± 0.5 \\n2.3 ± 1.2 \\nS \\nnon sig \\n\\nWFPC2/PC -F814W \\n0.42 \\n448.3 ± 0.1 \\n447.6 ± 0.1 \\n0.7 ± 0.14 \\n35 ± 7 \\n1.7 ± 0.3 2.7 ± 0.6 \\nW \\nlow \\n0.58 \\n517.2 ± 0.1 \\n517 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.3 ± 0.2 0.8 ± 0.6 \\nN \\nnon sig\"\n },\n {\n \"id\": \"\",\n \"caption\": \"). Kornreich & Lovelace (2008) determine the SBH acceleration for\",\n \"content\": \"Table 4\\n4MEASURED PROJECTED OFFSETSGalaxy \\nInstrument \\nregion \\nX \\nY \\nOffset \\nDir Type \\n\\nHST \\nIQR \\nPC \\nNPS \\n[pxl] \\n[mas] \\n[IQR] \\n[pc] \\n(1) \\n(2) \\n(3) \\n(4) \\n(5) \\n(6) \\n(7) \\n(8) \\n(9) \\n(10) \\n(11) (12) \\n\\nNGC 1399 WFPC2/PC -F606W [2.5:15] 0.19 \\n520 ± 0.1 \\n519.9 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.5 ± 0.7 0.4 ± 0.6 \\nW \\nnon sig \\n0.28 \\n465.4 ± 0.1 \\n465 ± 0.1 \\n0.4 ± 0.14 \\n20 ± 7 \\n1.4 ± 0.5 1.7 ± 0.6 \\nN \\nlow \\n\\nWFPC2/PC -F814W \\n0.29 \\n502.9 ± 0.1 \\n502.9 ± 0.1 \\n0 ± 0.14 \\n0 ± 7 \\n0 ± 0.5 \\n0 ± 0.6 \\n-\\nnon sig \\n0.27 \\n438.6 ± 0.1 \\n438.3 ± 0.1 \\n0.3 ± 0.14 \\n15 ± 7 \\n1.1 ± 0.5 1.3 ± 0.6 \\nN \\nnon sig \\n\\nWFC3/IR -F110W \\n0.13 \\n759.9 ± 0.2 \\n759.7 ± 0.1 \\n0.2 ± 0.22 \\n18 ± 20 1.5 ± 1.7 1.6 ± 1.8 \\nW \\nnon sig \\n0.14 \\n725.4 ± 0.2 \\n725.3 ± 0.1 \\n0.1 ± 0.22 \\n9 ± 20 \\n0.7 ± 1.6 0.8 ± 1.8 \\nN \\nnon sig \\n\\nWFC3/IR -F160W \\n0.18 \\n759.9 ± 0.2 \\n759.7 ± 0.1 \\n0.2 ± 0.22 \\n18 ± 20 1.1 ± 1.2 1.6 ± 1.8 \\nW \\nnon sig \\n0.08 \\n725.4 ± 0.2 \\n725.4 ± 0.1 \\n0 ± 0.22 \\n0 ± 20 \\n0 ± 2.8 \\n0 ± 1.8 \\n-\\nnon sig \\n\\nNGC 4168 WFPC2/PC -F702W [2.1:8.5] 1.76 \\n409.6 ± 0.1 \\n408.9 ± 0.1 \\n0.7 ± 0.14 \\n35 ± 7 \\n0.4 ± 0.1 \\n5 ± 1 \\nW \\nnull \\n1.47 \\n425.1 ± 0.1 \\n424.6 ± 0.1 \\n0.5 ± 0.14 \\n25 ± 7 \\n0.3 ± 0.1 \\n4 ± 1 \\nN \\nnull \\n\\nNGC 4261 NICMOS2 -F160W \\n[1.7:7.5] 0.08 \\n200 ± 0.2 \\n199.6 ± 0.2 \\n0.4 ± 0.3 \\n20 ± 15 \\n5 ± 3.7 \\n2.9 ± 2.2 \\nW \\nnon sig \\n0.26 \\n182.3 ± 0.2 \\n182.1 ± 0.2 \\n0.2 ± 0.3 \\n10 ± 15 0.8 ± 1.1 1.5 ± 2.2 \\nN \\nnon sig \\n\\nNGC 4278 ACS/WFC -F850LP \\n[1.2:9.1] 0.31 1165.6 ± 0.1 1166.4 ± 0.1 0.8 ± 0.14 \\n40 ± 7 \\n2.6 ± 0.5 3.4 ± 0.6 \\nE \\nhigh \\n0.83 \\n3160 ± 0.1 \\n3161.1 ± 0.1 1.1 ± 0.14 \\n55 ± 7 \\n1.3 ± 0.2 4.7 ± 0.6 \\nS \\nlow \\n\\nWFPC2/PC -F814W \\n0.44 \\n370.2 ± 0.1 \\n370.6 ± 0.1 \\n0.4 ± 0.14 \\n22 ± 7 \\n1 ± 0.3 \\n1.9 ± 0.6 \\nE \\nlow \\n0.87 \\n357.3 ± 0.1 \\n359 ± 0.1 \\n1.7 ± 0.14 \\n84 ± 7 \\n1.9 ± 0.2 7.2 ± 0.6 \\nS \\nint \\n\\nNICMOS2 -F160W \\n0.26 \\n245.8 ± 0.2 \\n246.4 ± 0.2 \\n0.6 ± 0.3 \\n30 ± 15 2.3 ± 1.2 2.6 ± 1.3 \\nE \\nnon sig \\n0.62 \\n224.8 ± 0.2 \\n225.8 ± 0.2 \\n1 ± 0.3 \\n50 ± 15 1.6 ± 0.5 4.3 ± 1.3 \\nS \\nint \\n\\nNGC 4373 WFPC2/PC -F814W [1.2:10] 0.58 2328.7 ± 0.1 2328.8 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.2 ± 0.2 \\n2 ± 2.7 \\nE \\nnon sig \\n0.16 \\n870.7 ± 0.1 \\n870.8 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.6 ± 0.9 \\n2 ± 2.7 \\nS \\nnon sig \\n\\nNGC 4486 ACS/HRC -F606W \\n[1:3] \\n2.78 \\n419.1 ± 0.1 \\n682.1 ± 0.1 \\n1.4 ± 0.14 \\n35 ± 4 \\n0.5 ± 0.1 2.7 ± 0.3 \\nW \\nnull \\n0.98 \\n417.7 ± 0.1 \\n679.5 ± 0.1 \\n2.6 ± 0.14 \\n65 ± 4 \\n2.7 ± 0.1 5.1 ± 0.3 \\nN \\nhigh \\n\\nACS/HRC -F814W \\n1.03 \\n553.1 ± 0.1 \\n549.6 ± 0.1 \\n3.5 ± 0.14 \\n87 ± 4 \\n3.4 ± 0.1 6.8 ± 0.3 \\nW \\nhigh \\n0.8 \\n551.8 ± 0.1 \\n549.9 ± 0.1 \\n1.9 ± 0.14 \\n47 ± 4 \\n2.4 ± 0.1 3.6 ± 0.3 \\nN \\nhigh \\n\\nNICMOS2 -F110W \\n0.42 \\n207.2 ± 0.2 \\n206.8 ± 0.2 \\n0.4 ± 0.3 \\n20 ± 15 \\n1 ± 0.7 \\n1.6 ± 1.2 \\nW \\nnon sig \\n0.96 \\n277.9 ± 0.2 \\n277.7 ± 0.2 \\n0.2 ± 0.3 \\n10 ± 15 0.2 ± 0.3 0.8 ± 1.2 \\nN \\nnon sig \\n\\nNICMOS2 -F160W \\n0.65 \\n207.5 ± 0.2 \\n207.4 ± 0.2 \\n0.1 ± 0.3 \\n5 ± 15 \\n0.2 ± 0.5 0.4 ± 1.2 \\nW \\nnon sig \\n0.76 \\n277.8 ± 0.2 \\n277.7 ± 0.2 \\n0.1 ± 0.3 \\n5 ± 15 \\n0.1 ± 0.4 0.4 ± 1.2 \\nN \\nnon sig \\n\\nNICMOS2 -F222M \\n0.51 \\n207.6 ± 0.2 \\n207.2 ± 0.2 \\n0.4 ± 0.3 \\n20 ± 15 0.8 ± 0.6 1.6 ± 1.2 \\nW \\nnon sig \\n0.59 \\n277.3 ± 0.2 \\n277.9 ± 0.2 \\n0.6 ± 0.3 \\n30 ± 15 \\n1 ± 0.5 \\n2.3 ± 1.2 \\nS \\nnon sig \\n\\nWFPC2/PC -F814W \\n0.42 \\n448.3 ± 0.1 \\n447.6 ± 0.1 \\n0.7 ± 0.14 \\n35 ± 7 \\n1.7 ± 0.3 2.7 ± 0.6 \\nW \\nlow \\n0.58 \\n517.2 ± 0.1 \\n517 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.3 ± 0.2 0.8 ± 0.6 \\nN \\nnon sig\"\n },\n {\n \"id\": \"\",\n \"caption\": \"MEASURED PROJECTED OFFSETS (continued)\",\n \"content\": \"). Kornreich & Lovelace (2008) determine the SBH acceleration for\"\n },\n {\n \"id\": \"\",\n \"caption\": \"MEASURED PROJECTED OFFSETS (continued)\",\n \"content\": \"). Kornreich & Lovelace (2008) determine the SBH acceleration for\"\n },\n {\n \"id\": \"\",\n \"caption\": \"RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (see\",\n \"content\": \"Table 4\\n4MEASURED PROJECTED OFFSETS (continued) Galaxy \\nInstrument \\nregion \\nX \\nY \\nOffset \\nDir \\nType \\n\\nHST \\nIQR \\nPC \\nNPS1 \\nNPS2 \\n[pxl] \\n[mas] \\n[IQR] \\n[pc] \\n(1) \\n(2) \\n(3) \\n(4) \\n(5) \\n(6) \\n(7) \\n(8) \\n(9) \\n(10) \\n(11) \\n(12) (13) \\n\\nNGC 4552 WFPC2/PC -F814W \\n[1:15] \\n0.42 \\n538.9 ± 0.1 \\n538.7 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.5 ± 0.3 \\n0.7 ± 0.5 W \\nnon sig \\n0.89 \\n500.8 ± 0.1 \\n500.4 ± 0.2 \\n0.4 ± 0.23 \\n20 ± 12 \\n0.4 ± 0.3 \\n1.5 ± 0.9 N \\nnon sig \\n\\nNGC 4636 WFPC2/PC -F814W \\n[4:14] \\n0.94 \\n698 ± 0.1 \\n697.4 ± 0.1 \\n0.6 ± 0.14 \\n30 ± 7 \\n0.6 ± 0.2 \\n1.9 ± 0.5 W \\nnull \\n0.93 \\n569.9 ± 0.1 \\n570 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.1 ± 0.2 \\n0.3 ± 0.5 S \\nnon sig \\n\\nNGC 4696 \\nACS/WFC -F814W \\n[1.5:16] \\n2.15 1582.9 ± 0.1 1583.4 ± 0.1 \\n0.5 ± 0.14 \\n25 ± 7 \\n0.2 ± 0.07 \\n4.5 ± 1.3 E \\nnull \\n3.78 2977.6 ± 0.1 2977.8 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.05 ± 0.04 1.8 ± 1.3 S \\nnon sig \\n\\n1586.1 ± 0.1 3.2 ± 0.14 160 ± 5 \\n1.5 ± 0.05 \\n29 ± 1.3 E \\nlow \\n2973.2 ± 0.1 4.4 ± 0.14 220 ± 5 \\n1.2 ± 0.03 \\n39 ± 1.3 N \\nlow \\n\\nNGC 5419 WFPC2/PC -F555W \\n[2:13.5] \\n0.5 \\n398.4 ± 0.1 \\n397.8 ± 0.1 \\n0.6 ± 0.14 \\n30 ± 7 \\n1.2 ± 0.3 \\n7 ± 2 \\nW \\nlow \\n0.22 \\n447.6 ± 0.1 \\n447.8 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.9 ± 0.6 \\n2 ± 2 \\nS \\nnon sig \\n\\n397.1 ± 0.1 \\n1.3 ± 0.14 \\n65 ± 7 \\n2.6 ± 0.2 \\n15 ± 2 \\nW \\nhigh \\n442.5 ± 0.1 \\n5.1 ± 0.14 255 ± 7 23.2 ± 0.5 \\n60 ± 2 \\nN \\nhigh \\n\\nNGC 5846 WFPC2/PC -F814W [1.6:15.8] 0.84 \\n519.5 ± 0.1 \\n518.2 ± 0.4 \\n1.3 ± 0.4 \\n65 ± 21 \\n1.6 ± 0.5 \\n7.8 ± 2.5 W \\nint \\n1.05 \\n466 ± 0.1 \\n466.4 ± 0.1 \\n0.4 ± 0.14 \\n20 ± 7 \\n0.4 ± 0.1 \\n2.4 ± 0.9 S \\nnull \\n\\nIC 1459 \\nWFPC2/PC -F814W \\n[2.5:15] \\n0.42 \\n536.4 ± 0.1 \\n536.7 ± 0.1 \\n0.3 ± 0.14 \\n15 ± 7 \\n0.7 ± 0.3 \\n2.1 ± 1 \\nE \\nnon sig \\n0.50 \\n510.5 ± 0.1 \\n510.7 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.4 ± 0.3 \\n1.4 ± 1 \\nS \\nnon sig \\n\\nIC 4296 \\nACS/HRC -F625W \\n[1.5:6.5] \\n0.49 \\n833.3 ± 0.1 \\n832.6 ± 0.2 \\n0.7 ± 0.22 \\n18 ± 6 \\n1.4 ± 0.4 \\n4.2 ± 1.2 W \\nlow \\n0.38 \\n785.3 ± 0.1 \\n784.8 ± 0.1 \\n0.5 ± 0.14 \\n13 ± 4 \\n1.3 ± 0.3 \\n3 ± 0.6 \\nN \\nlow \\n\\nWFPC2/PC -F814W \\n0.39 \\n551.5 ± 0.1 \\n552 ± 0.18 \\n0.5 ± 0.2 \\n25 ± 10 \\n1.3 ± 0.5 \\n6 ± 2 \\nE \\nnon sig \\n0.13 \\n515 ± 0.1 \\n514.9 ± 0.18 \\n0.1 ± 0.2 \\n5 ± 10 \\n0.8 ± 1.6 \\n1 ± 2 \\nN \\nnon sig \\n\\nNICMOS2 -F160W \\n0.13 \\n420.8 ± 0.2 \\n420.6 ± 0.2 \\n0.2 ± 0.3 \\n13 ± 14 \\n1.9 ± 2.2 \\n2.8 ± 3.2 W \\nnon sig \\n1.07 \\n253.4 ± 0.2 \\n252.7 ± 0.1 \\n0.7 ± 0.2 \\n37 ± 11 \\n0.7 ± 0.2 \\n8.3 ± 2.5 N \\nnull \\n\\nIC 4931 \\nWFPC2/PC -F814W \\n[1:15] \\n1.29 \\n403.6 ± 0.1 \\n404.1± 0.1 \\n0.5 ± 0.14 \\n25 ± 7 \\n0.4 ± 0.1 \\n10 ± 3 \\nE \\nnull \\n0.66 \\n505.9 ± 0.1 \\n505.8 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.2 ± 0.2 \\n2 ± 3 \\nN \\nnon sig\"\n },\n {\n \"id\": \"\",\n \"caption\": \"RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (see\",\n \"content\": \"Table 4\\n4MEASURED PROJECTED OFFSETS (continued) Galaxy \\nInstrument \\nregion \\nX \\nY \\nOffset \\nDir \\nType \\n\\nHST \\nIQR \\nPC \\nNPS1 \\nNPS2 \\n[pxl] \\n[mas] \\n[IQR] \\n[pc] \\n(1) \\n(2) \\n(3) \\n(4) \\n(5) \\n(6) \\n(7) \\n(8) \\n(9) \\n(10) \\n(11) \\n(12) (13) \\n\\nNGC 4552 WFPC2/PC -F814W \\n[1:15] \\n0.42 \\n538.9 ± 0.1 \\n538.7 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.5 ± 0.3 \\n0.7 ± 0.5 W \\nnon sig \\n0.89 \\n500.8 ± 0.1 \\n500.4 ± 0.2 \\n0.4 ± 0.23 \\n20 ± 12 \\n0.4 ± 0.3 \\n1.5 ± 0.9 N \\nnon sig \\n\\nNGC 4636 WFPC2/PC -F814W \\n[4:14] \\n0.94 \\n698 ± 0.1 \\n697.4 ± 0.1 \\n0.6 ± 0.14 \\n30 ± 7 \\n0.6 ± 0.2 \\n1.9 ± 0.5 W \\nnull \\n0.93 \\n569.9 ± 0.1 \\n570 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.1 ± 0.2 \\n0.3 ± 0.5 S \\nnon sig \\n\\nNGC 4696 \\nACS/WFC -F814W \\n[1.5:16] \\n2.15 1582.9 ± 0.1 1583.4 ± 0.1 \\n0.5 ± 0.14 \\n25 ± 7 \\n0.2 ± 0.07 \\n4.5 ± 1.3 E \\nnull \\n3.78 2977.6 ± 0.1 2977.8 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.05 ± 0.04 1.8 ± 1.3 S \\nnon sig \\n\\n1586.1 ± 0.1 3.2 ± 0.14 160 ± 5 \\n1.5 ± 0.05 \\n29 ± 1.3 E \\nlow \\n2973.2 ± 0.1 4.4 ± 0.14 220 ± 5 \\n1.2 ± 0.03 \\n39 ± 1.3 N \\nlow \\n\\nNGC 5419 WFPC2/PC -F555W \\n[2:13.5] \\n0.5 \\n398.4 ± 0.1 \\n397.8 ± 0.1 \\n0.6 ± 0.14 \\n30 ± 7 \\n1.2 ± 0.3 \\n7 ± 2 \\nW \\nlow \\n0.22 \\n447.6 ± 0.1 \\n447.8 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.9 ± 0.6 \\n2 ± 2 \\nS \\nnon sig \\n\\n397.1 ± 0.1 \\n1.3 ± 0.14 \\n65 ± 7 \\n2.6 ± 0.2 \\n15 ± 2 \\nW \\nhigh \\n442.5 ± 0.1 \\n5.1 ± 0.14 255 ± 7 23.2 ± 0.5 \\n60 ± 2 \\nN \\nhigh \\n\\nNGC 5846 WFPC2/PC -F814W [1.6:15.8] 0.84 \\n519.5 ± 0.1 \\n518.2 ± 0.4 \\n1.3 ± 0.4 \\n65 ± 21 \\n1.6 ± 0.5 \\n7.8 ± 2.5 W \\nint \\n1.05 \\n466 ± 0.1 \\n466.4 ± 0.1 \\n0.4 ± 0.14 \\n20 ± 7 \\n0.4 ± 0.1 \\n2.4 ± 0.9 S \\nnull \\n\\nIC 1459 \\nWFPC2/PC -F814W \\n[2.5:15] \\n0.42 \\n536.4 ± 0.1 \\n536.7 ± 0.1 \\n0.3 ± 0.14 \\n15 ± 7 \\n0.7 ± 0.3 \\n2.1 ± 1 \\nE \\nnon sig \\n0.50 \\n510.5 ± 0.1 \\n510.7 ± 0.1 \\n0.2 ± 0.14 \\n10 ± 7 \\n0.4 ± 0.3 \\n1.4 ± 1 \\nS \\nnon sig \\n\\nIC 4296 \\nACS/HRC -F625W \\n[1.5:6.5] \\n0.49 \\n833.3 ± 0.1 \\n832.6 ± 0.2 \\n0.7 ± 0.22 \\n18 ± 6 \\n1.4 ± 0.4 \\n4.2 ± 1.2 W \\nlow \\n0.38 \\n785.3 ± 0.1 \\n784.8 ± 0.1 \\n0.5 ± 0.14 \\n13 ± 4 \\n1.3 ± 0.3 \\n3 ± 0.6 \\nN \\nlow \\n\\nWFPC2/PC -F814W \\n0.39 \\n551.5 ± 0.1 \\n552 ± 0.18 \\n0.5 ± 0.2 \\n25 ± 10 \\n1.3 ± 0.5 \\n6 ± 2 \\nE \\nnon sig \\n0.13 \\n515 ± 0.1 \\n514.9 ± 0.18 \\n0.1 ± 0.2 \\n5 ± 10 \\n0.8 ± 1.6 \\n1 ± 2 \\nN \\nnon sig \\n\\nNICMOS2 -F160W \\n0.13 \\n420.8 ± 0.2 \\n420.6 ± 0.2 \\n0.2 ± 0.3 \\n13 ± 14 \\n1.9 ± 2.2 \\n2.8 ± 3.2 W \\nnon sig \\n1.07 \\n253.4 ± 0.2 \\n252.7 ± 0.1 \\n0.7 ± 0.2 \\n37 ± 11 \\n0.7 ± 0.2 \\n8.3 ± 2.5 N \\nnull \\n\\nIC 4931 \\nWFPC2/PC -F814W \\n[1:15] \\n1.29 \\n403.6 ± 0.1 \\n404.1± 0.1 \\n0.5 ± 0.14 \\n25 ± 7 \\n0.4 ± 0.1 \\n10 ± 3 \\nE \\nnull \\n0.66 \\n505.9 ± 0.1 \\n505.8 ± 0.1 \\n0.1 ± 0.14 \\n5 ± 7 \\n0.2 ± 0.2 \\n2 ± 3 \\nN \\nnon sig\"\n },\n {\n \"id\": \"\",\n \"caption\": \"PARAMETERS OF RECOIL OSCILLATION MODEL\",\n \"content\": \"Table 5\\n5RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (seeGalaxy \\nPA (deg) \\njet \\ndisplacement \\nNPS1 \\nNPS2 \\n\\nkpc-scale jets: \\nNGC 1399 \\n−10 \\n−17 ± 16 a \\nlow \\nNGC 4261 \\n87 ± 8 \\n297 ± 38 (117 ± 38) \\nnon significant \\nNGC 4486 \\n290 ± 3 \\n307 ± 1 (127 ± 1) \\n\\n ‡ \\n\\nNGC 4696 \\n210 (+30) 112 ± 15 \\n36 ± 1 \\n\\n ‡ \\n\\nIC 4296 \\n130 \\n338 ± 7 (158 ± 8) b \\nlow \\n\\npc-scale jets: \\nNGC 4278 \\n100 − 155 152 ± 3 \\n\\n ‡ \\n\\nNGC 4552 \\n50 \\n333 ± 21 \\nlow \\nNGC 4636 \\n35 \\n261 ± 9 (81 ± 9) \\nnull\"\n },\n {\n \"id\": \"\",\n \"caption\": \"PARAMETERS OF RECOIL OSCILLATION MODEL\",\n \"content\": \"Table 5\\n5RADIO JET AND PHOTOCENTER POSITION ANGLES Note. -Photocenter position angles are computed with respect to the nuclear point source. Angles increase east of north. When multiple values have been derived from different images an error weighted mean is computed. Values in parenthesis indicate the supplementary angles (i.e. θ + 180 • ). ‡ The significance changes for different instruments (seeGalaxy \\nPA (deg) \\njet \\ndisplacement \\nNPS1 \\nNPS2 \\n\\nkpc-scale jets: \\nNGC 1399 \\n−10 \\n−17 ± 16 a \\nlow \\nNGC 4261 \\n87 ± 8 \\n297 ± 38 (117 ± 38) \\nnon significant \\nNGC 4486 \\n290 ± 3 \\n307 ± 1 (127 ± 1) \\n\\n ‡ \\n\\nNGC 4696 \\n210 (+30) 112 ± 15 \\n36 ± 1 \\n\\n ‡ \\n\\nIC 4296 \\n130 \\n338 ± 7 (158 ± 8) b \\nlow \\n\\npc-scale jets: \\nNGC 4278 \\n100 − 155 152 ± 3 \\n\\n ‡ \\n\\nNGC 4552 \\n50 \\n333 ± 21 \\nlow \\nNGC 4636 \\n35 \\n261 ± 9 (81 ± 9) \\nnull\"\n },\n {\n \"id\": \"\",\n \"caption\": \"SBH MASSES\",\n \"content\": \"Table 6 PROBABILITIES\\n6Galaxy \\nd i [rc] p i (tm = 0.4 Gyr) p i (tm = 5 Gyr) \\n\\nNGC 1399 \\n0.01 \\n0.75 \\n0.11 \\nNGC 4168 * 0.02 \\n0.98 \\n0.89 \\nNGC 4261 <0.01 \\n0.94 \\n0.23 \\nNGC 4278 \\n0.1 \\n0.36 \\n0.04 \\nNGC 4373 <0.01 \\n0.99 \\n0.58 \\nNGC 4486 \\n0.01 \\n0.99 \\n0.85 \\nNGC 4552 <0.05 \\n0.10 \\n0.01 \\nNGC 4636 * 0.01 \\n0.99 \\n0.69 \\nNGC 4696 * 0.02 \\n0.97 \\n0.42 \\nNGC 5419 \\n0.02 \\n0.97 \\n0.46 \\nNGC 5846 \\n0.04 \\n0.90 \\n0.25 \\nIC 1459 \\n<0.01 \\n0.92 \\n0.21 \\nIC 4296 \\n0.01 \\n0.97 \\n0.37 \\nIC 4931 \\n * 0.03 \\n0.95 \\n0.46\"\n },\n {\n \"id\": \"\",\n \"caption\": \"SBH MASSES\",\n \"content\": \"Table 6 PROBABILITIES\\n6Galaxy \\nd i [rc] p i (tm = 0.4 Gyr) p i (tm = 5 Gyr) \\n\\nNGC 1399 \\n0.01 \\n0.75 \\n0.11 \\nNGC 4168 * 0.02 \\n0.98 \\n0.89 \\nNGC 4261 <0.01 \\n0.94 \\n0.23 \\nNGC 4278 \\n0.1 \\n0.36 \\n0.04 \\nNGC 4373 <0.01 \\n0.99 \\n0.58 \\nNGC 4486 \\n0.01 \\n0.99 \\n0.85 \\nNGC 4552 <0.05 \\n0.10 \\n0.01 \\nNGC 4636 * 0.01 \\n0.99 \\n0.69 \\nNGC 4696 * 0.02 \\n0.97 \\n0.42 \\nNGC 5419 \\n0.02 \\n0.97 \\n0.46 \\nNGC 5846 \\n0.04 \\n0.90 \\n0.25 \\nIC 1459 \\n<0.01 \\n0.92 \\n0.21 \\nIC 4296 \\n0.01 \\n0.97 \\n0.37 \\nIC 4931 \\n * 0.03 \\n0.95 \\n0.46\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". G O Abell, H G Corwin, Jr, R P Olowin, ApJS. 701Abell, G. O., Corwin, H. G., Jr., & Olowin, R. P. 1989, ApJS, 70, 1","doi":""},{"text":". D M Alexander, R C Hickox, NAR. 5693Alexander, D. M., & Hickox, R. C. 2012, NAR, 56, 93","doi":""},{"text":". J M Anderson, J S Ulvestad, L C Ho, ApJ. 60342Anderson, J. M., Ulvestad, J. S., & Ho, L. C. 2004, ApJ, 603, 42","doi":""},{"text":". J Antognini, J Bird, P Martini, ApJ. 756116Antognini, J., Bird, J. & Martini, P. 2012, ApJ, 756, 116","doi":""},{"text":". F Antonini, D Merritt, ApJ. 74583Antonini, F., & Merritt, D. 2012, ApJ, 745, 83","doi":""},{"text":". W Baade, R Minkowski, ApJ. 119215Baade, W., & Minkowski, R. 1954, ApJ, 119, 215","doi":""},{"text":". J N Bahcall, R A Wolf, ApJ. 209214Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214","doi":""},{"text":". R D Baldi, M Chiaberge, A Capetti, W Sparks, F D Macchetto, C P O'dea, D J Axon, S A Baum, A C Quillen, ApJ. 7252426Baldi, R. D., Chiaberge, M., Capetti, A., Sparks, W., Macchetto, F. D., O'Dea, C. P., Axon, D. J., Baum, S. A. & Quillen, A. C. 2010, ApJ, 725, 2426","doi":""},{"text":". B Balmaverde, R D Baldi, A Capetti, A&A. 486119Balmaverde, B., Baldi, R. D., & Capetti, A. 2008, A&A, 486, 119","doi":""},{"text":". B Balmaverde, A Capetti, A&A. 447BC0697Balmaverde, B., & Capetti, A. 2006, A&A, 447, 97 (BC06)","doi":""},{"text":". J G Baker, J Centrella, D.-I Choi, M Koppitz, J Van Meter, Phys. Rev. Lett. 96111102Baker, J. G. , Centrella, J. , Choi,D.-I., Koppitz, M., & van Meter, J. 2006, Phys. Rev. Lett. 96, 111102","doi":""},{"text":". D Batcheldor, A Robinson, D J Axon, E S Perlman, D Merritt, ApJ. 717B106Batcheldor, D., Robinson, A., Axon, D. J., Perlman, E. S., & Merritt, D. 2010, ApJ, 717, L6 (B10)","doi":""},{"text":". M C Begelman, R D Blandford, M J Rees, Nature. 287307Begelman, M.C., Blandford, R.D. & Rees, M.J., 1980, Nature, 287, 307","doi":""},{"text":". J D Bekenstein, ApJ. 183657Bekenstein, J. D. 1973, ApJ, 183, 657","doi":""},{"text":". E F Bell, S Phleps, R S Somerville, C Wolf, A Borch, K Meisenheimer, ApJ. 652270Bell, E. F., Phleps, S., Somerville, R. S., Wolf, C., Borch, A. & Meisenheimer, K. 2006, ApJ, 652, 270","doi":""},{"text":". M Birkinshaw, R L Davies, ApJ. 29132Birkinshaw, M., & Davies, R. L. 1985, ApJ, 291, 32","doi":""},{"text":". R D Blandford, R L Znajek, MNRAS. 179433Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433","doi":""},{"text":". L Blecha, F Civano, M Elvis, A Loeb, MNRAS. 4281341Blecha, L., Civano, F., Elvis, M. & Loeb, A. 2013, MNRAS, 428,1341","doi":""},{"text":". E W Bonning, G A Shields, S Salviander, ApJ. 66613Bonning, E. W., Shields, G. A. & Salviander, S. 2007, ApJ, 666, 13","doi":""},{"text":". M J I Brown, B T Jannuzi, D J E Floyd, J Mould, R , ApJ. 73141Brown, M. J. I., Jannuzi, B. T., Floyd, D. J. E., Mould, J., R. 2011, ApJ, 731, L41","doi":""},{"text":". C Burke, C A Collins, MNRAS. 4342856Burke, C. & Collins, C. A., 2013, MNRAS, 434, 2856","doi":""},{"text":"I C Busko, Astronomical Data Analysis Software and Systems V. 101139Busko, I. C. 1996, Astronomical Data Analysis Software and Systems V, 101, 139","doi":""},{"text":". M Campanelli, C O Lousto, P Marronetti, Y Zlochower, Phys. Rev. Lett. 96111101Campanelli, M., Lousto, C. O., Marronetti, P., & Zlochower, Y. 2006, Phys. Rev. Lett. 96, 111101","doi":""},{"text":". M Campanelli, C O Lousto, Y Zlochower, D Merritt, Physical Review Letters. 98231102Campanelli, M., Lousto, C. O., Zlochower, Y., & Merritt, D. 2007, Physical Review Letters, 98, 231102","doi":""},{"text":". M Campanelli, C Lousto, Y Zlochower, D Merritt, ApJ. 6595Campanelli, M., Lousto, C., Zlochower, Y., & Merritt, D. 2007, ApJ, 659, L5","doi":""},{"text":". R E A Canning, A C Fabian, R M Johnstone, J S Sanders, C S Crawford, G J Ferland, N A Hatch, MNRAS. 4173080Canning, R.E.A., Fabian, A. C., Johnstone, R. M., Sanders, J. S., Crawford, C. S., Ferland, G. J., Hatch, N. A. 2011, MNRAS, 417, 3080","doi":""},{"text":". A Capetti, D J Axon, M Chiaberge, A&A. 471137Capetti, A., Axon, D. J., Chiaberge, M., et al. 2007, A&A, 471, 137","doi":""},{"text":". A Capetti, B Balmaverde, A&A. 440CB0573Capetti, A., & Balmaverde, B. 2005, A&A, 440, 73 (CB05)","doi":""},{"text":". A Capetti, B Balmaverde, A&A. 453CB0627Capetti, A., & Balmaverde, B. 2006, A&A, 453, 27 (CB06)","doi":""},{"text":". M Cappellari, A Renzini, L Greggio, ApJ. 519117Cappellari, M., Renzini, A., Greggio, L., et al. 1999, ApJ, 519, 117","doi":""},{"text":". M Cappellari, E K Verolme, R P Van Der Marel, ApJ. 578787Cappellari, M., Verolme, E. K., van der Marel, R. P., et al. 2002, ApJ, 578, 787","doi":""},{"text":". M Cappellari, E Emsellem, R Bacon, MNRAS. 379418Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418","doi":""},{"text":". A Cardullo, E M Corsini, A Beifiori, A&A. 508641Cardullo, A., Corsini, E. M., Beifiori, A., et al. 2009, A&A, 508, 641","doi":""},{"text":". C M Carollo, M Franx, G D Illingworth, D A Forbes, ApJ. 481710Carollo, C. M., Franx, M., Illingworth, G. D., & Forbes, D. A. 1997, ApJ, 481, 710","doi":""},{"text":". K W Cavagnolo, B R Mcnamara, P E J Nulsen, ApJ. 7201066Cavagnolo, K. W., McNamara, B. R., Nulsen, P. E. J., et al. 2010, ApJ, 720, 1066","doi":""},{"text":". M Chiaberge, A Capetti, A Celotti, A&A. 34977Chiaberge, M., Capetti, A., & Celotti, A. 1999, A&A, 349, 77","doi":""},{"text":". F Civano, M Elvis, G Lanzuisi, ApJ. 717209Civano, F., Elvis, M., Lanzuisi, G., et al. 2010, ApJ, 717, 209","doi":""},{"text":". F Civano, M Elvis, G Lanzuisi, ApJ. 75249Civano, F., Elvis, M., Lanzuisi, G., et al. 2012, ApJ, 752, 49","doi":""},{"text":"P Coles, F Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure. New YorkWiley2nd ed.Coles, P., & Lucchin, F. 2002, Cosmology: The Origin and Evolution of Cosmic Structure (2nd ed.; New York: Wiley)","doi":""},{"text":". C J Conselice, C Yang, A F L Bluck, MNRAS. 3941956Conselice, C. J., Yang, C. & Bluck, A. F. L. 2009, MNRAS, 394, 1956","doi":""},{"text":". J Cuadra, P J Armitage, R D Alexander, M C Begelman, MNRAS. 3931423Cuadra, J., Armitage, P. J. Alexander, R. D. & Begelman, M. C. 2009, MNRAS, 393, 1423","doi":""},{"text":". V P Debattista, I Ferreras, A Pasquali, ApJ. 65197Debattista, V. P., Ferreras, I., Pasquali, A., et al. 2006, ApJ, 651, L97","doi":""},{"text":"Third Reference Catalogue of Bright Galaxies Di Matteo. G De Vaucouleurs, A De Vaucouleurs, H G Corwin, Jr, ApJ. 582133de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Jr., et al. 1991, Third Reference Catalogue of Bright Galaxies Di Matteo, T., Allen, S. W., Fabian, A. C., Wilson, A. S., & Young, A. J. 2003, ApJ, 582, 133","doi":""},{"text":". M Dotti, M Volonteri, A Perego, M Colpi, M Ruszkowski, F Haardt, MNRAS. 402682Dotti, M., Volonteri, M., Perego, A., Colpi, M., Ruszkowski, M. & Haardt, F. 2010, MNRAS, 402, 682","doi":""},{"text":". A Dressler, D Lynden-Bell, D Burstein, ApJ. 31342Dressler, A., Lynden-Bell, D., Burstein, D., et al. 1987, ApJ, 313, 42","doi":""},{"text":". G Efstathiou, R S Ellis, D Carter, MNRAS. 193931Efstathiou, G., Ellis, R. S., & Carter, D. 1980, MNRAS, 193, 931","doi":""},{"text":". M Eracleous, T A Boroson, J P Halpern, J Liu, ApJS. 20123Eracleous, M., Boroson, T. A., Halpern, J. P., & Liu, J. 2012, ApJS, 201, 23","doi":""},{"text":". A Escala, R B Larson, P S Coppi, D Mardones, ApJ. 630152Escala, A., Larson, R. B., Coppi, P. S. & Mardones, D. 2005, ApJ, 630, 152","doi":""},{"text":". G Fabbiano, M Elvis, S Markoff, ApJ. 588175Fabbiano, G., Elvis, M., Markoff, S., et al. 2003, ApJ, 588, 175","doi":""},{"text":". S M Faber, S Tremaine, E A Ajhar, AJ. 1141771Faber, S. M., Tremaine, S., Ajhar, E. A., et al. 1997, AJ, 114, 1771","doi":""},{"text":". B L Fanaroff, J M Riley, MNRAS. 16731Fanaroff, B. L., & Riley, J. M. 1974, MNRAS, 167, 31P","doi":""},{"text":". C L Farage, P J Mcgregor, M A Dopita, G V Bicknell, ApJ. 724267Farage, C. L., McGregor, P. J., Dopita, M. A., & Bicknell, G. V. 2010, ApJ, 724, 267","doi":""},{"text":". M Favata, S A Hughes, D E Holz, ApJ. 6075Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ, 607, L5","doi":""},{"text":". H C Ferguson, AJ. 98367Ferguson, H. C. 1989, AJ, 98, 367","doi":""},{"text":". L Ferrarese, H C Ford, W Jaffe, ApJ. 470444Ferrarese, L., Ford, H. C., & Jaffe, W. 1996, ApJ, 470, 444","doi":""},{"text":". L Ferrarese, D Merritt, ApJ. 5399Ferrarese, L., & Merritt, D. 2000, ApJ, 539, L9","doi":""},{"text":". L Ferrarese, H Ford, Sci. Rev. 116523Ferrarese, L., Ford, H., 2005, Sci. Rev., 116, 523","doi":""},{"text":". M E Filho, P D Barthel, L C Ho, ApJS. 12993Filho, M. E., Barthel, P. D., & Ho, L. C. 2000, ApJS, 129, 93","doi":""},{"text":". M E Filho, F Fraternali, S Markoff, A&A. 418429Filho, M. E., Fraternali, F., Markoff, S., et al. 2004, A&A, 418, 429","doi":""},{"text":". D A Forbes, M Franx, G D Illingworth, ApJ. 42849Forbes, D. A., Franx, M., & Illingworth, G. D. 1994, ApJ, 428, L49","doi":""},{"text":". D A Forbes, J P Brodie, J Huchra, AJ. 1122448Forbes, D. A., Brodie, J. P., & Huchra, J. 1996, AJ, 112, 2448","doi":""},{"text":". D A Forbes, D B Reitzel, G M Williger, AJ. 1091576Forbes, D. A., Reitzel, D. B., & Williger, G. M. 1995, AJ, 109, 1576","doi":""},{"text":". M Franx, G D Illingworth, ApJ. 32755Franx, M., & Illingworth, G. D. 1988, ApJ, 327, L55","doi":""},{"text":". O Fakhouri, C.-P Ma, M Boylan-Kolchin, MNRAS. 4062267Fakhouri, O., Ma, C.-P., & Boylan-Kolchin, M. 2010, MNRAS, 406, 2267","doi":""},{"text":". K Gebhardt, R Bender, G Bower, ApJ. 53913Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13","doi":""},{"text":". K Gebhardt, T R Lauer, J Pinkney, ApJ. 6711321Gebhardt, K., Lauer, T. R., Pinkney, J., et al. 2007, ApJ, 671, 1321","doi":""},{"text":". K Gebhardt, J Adams, D Richstone, ApJ. 729119Gebhardt, K., Adams, J., Richstone, D., et al. 2011, ApJ, 729, 119","doi":""},{"text":". M Gerin, F Casoli, A&A. 29049Gerin, M., & Casoli, F. 1994, A&A, 290, 49","doi":""},{"text":". M Giroletti, G B Taylor, G Giovannini, ApJ. 622178Giroletti, M., Taylor, G. B., & Giovannini, G. 2005, ApJ, 622, 178","doi":""},{"text":". L Giordano, L Cortese, G Trinchieri, A Wolter, M Colpi, G Gavazzi, L Mayer, ApJ. 634272Giordano, L., Cortese, L., Trinchieri, G., Wolter, A., Colpi, M., Gavazzi, G., & Mayer, L. 2005, ApJ, 634, 272","doi":""},{"text":". J A González, M Hannam, U Sperhake, B Brügmann, S Husa, Phys. Rev. Lett. 98231101González, J. A., Hannam, M., Sperhake, U., Brügmann, B., & Husa, S. 2007, Phys. Rev. Lett., 98, 231101","doi":""},{"text":". O González-Martín, J Masegosa, I Márquez, M Guainazzi, E Jiménez-Bailón, A&A. 5061107González-Martín, O., Masegosa, J., Márquez, I., Guainazzi, M., & Jiménez-Bailón, E. 2009, A&A, 506, 1107","doi":""},{"text":". R Grützbauch, G Trinchieri, R Rampazzo, AJ. 133220Grützbauch, R., Trinchieri, G., Rampazzo, R., et al. 2007, AJ, 133, 220","doi":""},{"text":". A Gualandris, D Merritt, A Gualandris, D Merritt, ApJ. 678GM0874ApJGualandris, A., & Merritt, D. 2008, ApJ, 678, 780 (GM08) Gualandris, A., & Merritt, D. 2012, ApJ, 744, 74","doi":""},{"text":". K Hada, A Doi, M Kino, Nature. 477185Hada, K., Doi, A., Kino, M., et al. 2011, Nature, 477, 185","doi":""},{"text":". M Hilz, T Naab, J P Ostriker, arXiv:1206.5004Hilz, M., Naab, T., & Ostriker, J. P. 2012, arXiv:1206.5004","doi":"arXiv:1206.5004"},{"text":". L C Ho, ARA&A. 46475Ho, L. C. 2008, ARA&A, 46, 475","doi":""},{"text":". L C Ho, A V Filippenko, W L W Sargent, C Y Peng, ApJS. 112391Ho, L. C., Filippenko, A. V., Sargent, W. L. W. & Peng, C. Y. 1997, ApJS, 112, 391","doi":""},{"text":". P F Hopkins, K Bundy, D Croton, L Hernquist, D Keres, S Khochfar, K Stewart, A Wetzel, J D Younger, ApJ. 715H10202Hopkins, P. F., Bundy, K., Croton, D., Hernquist, L., Keres, D., Khochfar, S., Stewart, K., Wetzel, A. & Younger, J. D. 2010, ApJ, 715, 202 (H10)","doi":""},{"text":". R C W Houghton, J Magorrian, M Sarzi, MNRAS. 3672Houghton, R. C. W., Magorrian, J., Sarzi, M., et al. 2006, MNRAS, 367, 2","doi":""},{"text":". J P Huchra, M J Geller, ApJ. 257423Huchra, J. P., & Geller, M. J. 1982, ApJ, 257, 423","doi":""},{"text":". J Huchra, M Davis, D Latham, J Tonry, ApJS. 5289Huchra, J., Davis, M., Latham, D., & Tonry, J. 1983, ApJS, 52, 89","doi":""},{"text":". W Jaffe, H C Ford, L Ferrarese, Van Den, F Bosch, R W Connell, Nature. 364213Jaffe, W., Ford, H. C., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1993, Nature, 364, 213","doi":""},{"text":". W Jaffe, B R Mcnamara, ApJ. 434110Jaffe, W., & McNamara, B. R. 1994, ApJ, 434, 110","doi":""},{"text":". W Jaffe, H Ford, L Ferrarese, Van Den, F Bosch, R W Connell, ApJ. 460214Jaffe, W., Ford, H., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1996, ApJ, 460, 214","doi":""},{"text":". S Janowiecki, J C Mihos, P Harding, ApJ. 715972Janowiecki, S., Mihos, J. C., Harding, P., et al. 2010, ApJ, 715, 972","doi":""},{"text":". R I Jedrzejewski, MNRAS. 226747Jedrzejewski, R. I. 1987, MNRAS, 226, 747","doi":""},{"text":". D L Jones, A E Wehrle, D L Meier, B G Piner, ApJ. 534165Jones, D. L., Wehrle, A. E., Meier, D. L., & Piner, B. G. 2000, ApJ, 534, 165","doi":""},{"text":". C Jones, W Forman, A Vikhlinin, ApJ. 567115Jones, C., Forman, W., Vikhlinin, A., et al. 2002, ApJ, 567, L115","doi":""},{"text":". J I Katz, ApJ. 478527Katz, J. I. 1997, ApJ, 478, 527","doi":""},{"text":". F M Khan, A Just, D Merritt, ApJ. 73289Khan, F. M., Just, A., & Merritt, D. 2011, ApJ, 732, 89","doi":""},{"text":". N E B Killeen, G V Bicknell, R D Ekers, ApJ. 302306Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1986, ApJ, 302, 306","doi":""},{"text":". N E B Killeen, G V Bicknell, D Carter, ApJ. 30945Killeen, N. E. B., Bicknell, G. V., & Carter, D. 1986, ApJ, 309, 45","doi":""},{"text":". N E B Killeen, G V Bicknell, ApJ. 324198Killeen, N. E. B., & Bicknell, G. V. 1988, ApJ, 324, 198","doi":""},{"text":". N E B Killeen, G V Bicknell, R D Ekers, ApJ. 325180Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1988, ApJ, 325, 180","doi":""},{"text":". S Komossa, H Zhou, H Lu, ApJ. 67881Komossa, S., Zhou, H., & Lu, H. 2008, ApJ, 678, L81","doi":""},{"text":". S Komossa, Mem. Soc. Astron. Italiana. 77733Komossa, S. 2006, Mem. Soc. Astron. Italiana, 77, 733","doi":""},{"text":". J Kormendy, D Richstone ; Ara&a Kornreich, D A Lovelace, R V E , 681..104KJ. Kormendy, D. Richstone, 1995, ARA&A Kornreich D. A. and Lovelace R. V. E., 2008ApJ...681..104K","doi":"681..104K"},{"text":"J Krist, Astronomical Data Analysis Software and Systems II. 52536Krist, J. 1993, Astronomical Data Analysis Software and Systems II, 52, 536","doi":""},{"text":". P Kroupa, arXiv:1406.4860Kroupa, P. 2014, arXiv:1406.4860","doi":"arXiv:1406.4860"},{"text":". S Laine, R P Van Der Marel, T R Lauer, M Postman, C P O'dea, F N Owen, AJ. 125478Laine, S., van der Marel, R. P., Lauer, T. R., Postman, M., O'Dea, C. P., & Owen, F. N. 2003, AJ, 125, 478","doi":""},{"text":". T R Lauer, S M Faber, E J Groth, AJ. 1061436Lauer, T. R., Faber, S. M., Groth, E. J., et al. 1993, AJ, 106, 1436","doi":""},{"text":". T R Lauer, E A Ajhar, Y.-I Byun, AJ. 1102622Lauer, T. R., Ajhar, E. A., Byun, Y.-I., et al. 1995, AJ, 110, 2622","doi":""},{"text":". T R Lauer, S Tremaine, E A Ajhar, ApJ. 47179Lauer, T. R., Tremaine, S., Ajhar, E. A., et al. 1996, ApJ, 471, L79","doi":""},{"text":". T R Lauer, S M Faber, K Gebhardt, AJ. 1292138Lauer, T. R., Faber, S. M., Gebhardt, K., et al. 2005, AJ, 129, 2138","doi":""},{"text":". T R Lauer, S M Faber, D Richstone, ApJ. 662808Lauer, T. R., Faber, S. M., Richstone, D., et al. 2007, ApJ, 662, 808","doi":""},{"text":". C Lidman, G Iacobuta, A E Bauer, MNRAS. 433825Lidman, C., Iacobuta, G., Bauer, A. E., et al. 2013, MNRAS, 433, 825","doi":""},{"text":". A P Lobanov, J Roland, A&A. 431831Lobanov, A. P., & Roland, J. 2005, A&A, 431, 831","doi":""},{"text":". A Loeb, Phrvl, 9941103Loeb, A., PhRvL, 99, 041103L","doi":""},{"text":". M Loewenstein, R F Mushotzky, L Angelini, K A Arnaud, E Quataert, ApJ. 55521Loewenstein, M., Mushotzky, R. F., Angelini, L., Arnaud, K. A., & Quataert, E. 2001, ApJ, 555, L21","doi":""},{"text":". C O Lousto, H Nakano, Y Zlochower, M Campanelli, Phys. Rev. D. 8184023Lousto, C. O., Nakano, H., Zlochower, Y., & Campanelli, M. 2010, Phys. Rev. D 81, 084023","doi":""},{"text":". C O Lousto, Y Zlochower, Phys. Rev. Lett. 107231102Lousto, C. O. & Zlochower, Y. 2011, Phys. Rev. Lett. 107, 231102","doi":""},{"text":". C O Lousto, Y Zlochower, M Dotti, M Volonteri, arXiv:1201.1923(L12Lousto, C. O., Zlochower, Y., Dotti, M., & Volonteri, M. 2012, arXiv:1201.1923 (L12)","doi":"arXiv:1201.1923(L12"},{"text":". F Macchetto, M Pastoriza, N Caon, A&AS. 120463Macchetto, F., Pastoriza, M., Caon, N., et al. 1996, A&AS, 120, 463","doi":""},{"text":". D Maoz, N M Nagar, H Falcke, A S Wilson, ApJ. 625699Maoz, D., Nagar, N. M., Falcke, H., & Wilson, A. S. 2005, ApJ, 625, 699","doi":""},{"text":". B R Mcnamara, ApJ. 562149McNamara, B. R., et al. 2001, ApJ, 562, L149","doi":""},{"text":". M E Machacek, D Jerius, R Kraft, ApJ. 74315Machacek, M. E., Jerius, D., Kraft, R., et al. 2011, ApJ, 743, 15","doi":""},{"text":". P Madau, E Quataert, ApJ. 60617Madau, P. and Quataert, E. 2004, ApJ, 606, 17","doi":""},{"text":". D F Malin, Nature. 277279Malin, D. F. 1979, Nature, 277, 279","doi":""},{"text":"D F Malin, New Aspects of Galaxy Photometry. D. F. Malin, New Aspects of Galaxy Photometry, 1985","doi":""},{"text":". L Mayer, S Kazantzidis, P Madau, Science. 3161874Mayer, L., Kazantzidis, S., Madau, P., et al. 2007, Science, 316, 1874","doi":""},{"text":". J Masegosa, I Márquez, A Ramirez, O González-Martín, A&A. 52723Masegosa, J., Márquez, I., Ramirez, A., & González-Martín, O. 2011, A&A, 527, A23","doi":""},{"text":"D Merritt, Dynamics and Evolution of Galactic Nuclei. Princeton, NJPrinceton Univ. PressMerritt, D. 2013, Dynamics and Evolution of Galactic Nuclei (Princeton, NJ: Princeton Univ. Press)","doi":""},{"text":". D Merritt, R D Ekers, Science. 2971310Merritt, D., & Ekers, R. D. 2002, Science, 297, 1310","doi":""},{"text":". D Merritt, M Milosavljević, M Favata, S A Hughes, D E Holz, ApJ. 6079Merritt, D., Milosavljević, M., Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ 607, L9.","doi":""},{"text":". D Merritt, M Milosavljević, Living Reviews in Relativity. 88Merritt, D., & Milosavljević, M. 2005, Living Reviews in Relativity, 8, 8","doi":""},{"text":". D Merritt, ApJ. 648976Merritt, D. 2006, ApJ, 648, 976","doi":""},{"text":". D Merritt, J D Schnittman, S Komossa, ApJ. 6991690Merritt, D., Schnittman, J. D., & Komossa, S. 2009, ApJ, 699, 1690","doi":""},{"text":". B Y Mills, O B Slee, E R Hill, Australian Journal of Physics. 13676Mills, B. Y., Slee, O. B., & Hill, E. R. 1960, Australian Journal of Physics, 13, 676","doi":""},{"text":". C Moellenhoff, E Hummel, R Bender, A&A. 25535Moellenhoff, C., Hummel, E., & Bender, R. 1992, A&A, 255, 35","doi":""},{"text":". N M Nagar, H Falcke, A S Wilson, J S Ulvestad, A&A. 39253Nagar, N. M., Falcke, H., Wilson, A. S., & Ulvestad, J. S. 2002, A&A, 392, 53","doi":""},{"text":". R Nolthenius, ApJS. 851Nolthenius, R. 1993, ApJS, 85, 1","doi":""},{"text":". R W O'connell, J R Martin, J D Crane, ApJ. 635305O'Connell, R. W., Martin, J. R., Crane, J. D., et al. 2005, ApJ, 635, 305","doi":""},{"text":". C P O'dea, R A Daly, P Kharb, K A Freeman, S A Baum, A&A. 494471O'Dea, C. P., Daly, R. A., Kharb, P., Freeman, K. A. & Baum, S. A. 2009, A&A, 494, 471","doi":""},{"text":". P Parma, M Murgia, R Morganti, A Capetti, H R De Ruiter, R Fanti, A&A. 3447Parma, P., Murgia, M., Morganti, R., Capetti, A., de Ruiter, H. R. & Fanti, R. 1999, A&A, 344, 7","doi":""},{"text":". S Pellegrini, T Venturi, A Comastri, ApJ. 585677Pellegrini, S., Venturi, T., Comastri, A., et al. 2003, ApJ, 585, 677","doi":""},{"text":". H V Peiris, S Tremaine, ApJ. Peiris, H. V. & Tremaine, S. 2003, ApJ, 237-257","doi":""},{"text":". C Y Peng, L C Ho, C D Impey, H.-W Rix, AJ. 124266Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266","doi":""},{"text":". H B Perets, C Hopman, T Alexander, ApJ. 656709Perets, H. B., Hopman, C., & Alexander, T. 2007, ApJ, 656, 709","doi":""},{"text":". E S Perlman, D E Harris, J A Biretta, W B Sparks, F D Macchetto, ApJ. 59965Perlman, E. S., Harris, D. E., Biretta, J. A., Sparks, W. B., & Macchetto, F. D. 2003, ApJ, 599, L65","doi":""},{"text":". M Postman, T R Lauer, ApJ. 44028Postman, M., & Lauer, T. R. 1995, ApJ, 440, 28","doi":""},{"text":". M Preto, I Berentzen, P Berczik, R Spurzem, ApJ. 73226Preto, M., Berentzen, I., Berczik, P., & Spurzem, R. 2011, ApJ, 732, L26","doi":""},{"text":". F Pretorius, Phys. Rev. Lett. 95121101Pretorius, F. 2005, Phys. Rev. Lett. 95, 121101","doi":""},{"text":". A Renzini, L Greggio, S Di Serego Alighieri, Nature. 37839Renzini, A., Greggio, L., di Serego Alighieri, S., et al. 1995, Nature, 378, 39","doi":""},{"text":". A Robinson, S Young, D J Axon, P Kharb, J E Smith, ApJ. 717122Robinson, A., Young, S., Axon, D. J., Kharb, P., & Smith, J. E. 2010, ApJ, 717, L122","doi":""},{"text":". G E Romero, L Chajet, Z Abraham, J H Fan, A&A. 36057Romero, G. E., Chajet, L., Abraham, Z., & Fan, J. H. 2000, A&A, 360, 57","doi":""},{"text":". N Roos, J S Kaastra, C A Hummel, ApJ. 409130Roos, N., Kaastra, J. S., & Hummel, C. A. 1993, ApJ, 409, 130","doi":""},{"text":". E M Sadler, N A Sharp, A&A. 133216Sadler, E. M., & Sharp, N. A. 1984, A&A, 133, 216","doi":""},{"text":". E M Sadler, C R Jenkins, C G Kotanyi, MNRAS. 240591Sadler, E. M., Jenkins, C. R., & Kotanyi, C. G. 1989, MNRAS, 240, 591","doi":""},{"text":". E Sani, A Marconi, L K Hunt, G Risaliti, MNRAS. 4131479Sani, E., Marconi, A., Hunt, L. K., & Risaliti, G. 2011, MNRAS, 413, 1479","doi":""},{"text":". W C Saslaw, M Whittle, ApJ. 325142Saslaw, W. C., & Whittle, M. 1988, ApJ, 325, 142","doi":""},{"text":"Extragalactic Radio Sources. I S Shklovsky, 97475Shklovsky, I. S. 1982, Extragalactic Radio Sources, 97, 475","doi":""},{"text":". G A Shields, D J Rosario, K L Smith, ApJ. 707936Shields, G. A., Rosario, D. J., Smith, K. L., et al. 2009, ApJ, 707, 936","doi":""},{"text":". K Shurkin, R J H Dunn, G Gentile, G B Taylor, S W Allen, MNRAS. 383923Shurkin, K., Dunn, R. J. H., Gentile, G., Taylor, G. B., & Allen, S. W. 2008, MNRAS, 383, 923","doi":""},{"text":". D Sijacki, V Springel, M G Haehnelt, MNRAS. 4143656Sijacki, D., Springel, V. & Haehnelt, M. G. 2011, MNRAS, 414, 3656","doi":""},{"text":". O B Slee, E M Sadler, J E Reynolds, R D Ekers, MNRAS. 269928Slee, O. B., Sadler, E. M., Reynolds, J. E., & Ekers, R. D. 1994, MNRAS, 269, 928","doi":""},{"text":". W B Sparks, F Macchetto, D Golombek, ApJ. 345153Sparks, W. B., Macchetto, F., & Golombek, D. 1989, ApJ, 345, 153","doi":""},{"text":". V Springel, S D M White, A Jenkins, Nature. 435629Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629","doi":""},{"text":". V J Stanger, R S Warwick, MNRAS. 220363Stanger, V. J., & Warwick, R. S. 1986, MNRAS, 220, 363","doi":""},{"text":". C L Steinhardt, M Schramm, J D Silverman, ApJ. 75924Steinhardt, C. L., Schramm, M., Silverman, J. D., et al. 2012, ApJ, 759, 24","doi":""},{"text":". R Subrahmanyan, A J Beasley, W M Goss, K Golap, R W Hunstead, AJ. 1251095Subrahmanyan, R., Beasley, A. J., Goss, W. M., Golap, K., & Hunstead, R. W. 2003, AJ, 125, 1095","doi":""},{"text":". G B Taylor, J S Sanders, A C Fabian, S W Allen, MNRAS. 365705Taylor, G. B., Sanders, J. S., Fabian, A. C., & Allen, S. W. 2006, MNRAS, 365, 705","doi":""},{"text":". T Tal, P G Van Dokkum, J Nelan, R Bezanson, AJ. 1381417Tal, T., van Dokkum, P. G., Nelan, J., & Bezanson, R. 2009, AJ, 138, 1417","doi":""},{"text":". A Tarchi, N Neininger, A Greve, A&A. 35895Tarchi, A., Neininger, N., Greve, A., et al. 2000, A&A, 358, 95","doi":""},{"text":". N Thatte, M Tecza, R Genzel, A&A. 36447Thatte, N., Tecza, M., & Genzel, R. 2000, A&A, 364, L47","doi":""},{"text":". P Temi, W G Mathews, F Brighenti, J D Bregman, ApJ. 585121Temi, P., Mathews, W. G., Brighenti, F., & Bregman, J. D. 2003, ApJ, 585, L121","doi":""},{"text":". W Tichy, P Marronetti, Phys. Rev. D. 061502. Trinchieri, G., & di Serego Alighieri761647AJTichy, W. & Marronetti, P. 2007, Phys. Rev. D 76, 061502. Trinchieri, G., & di Serego Alighieri, S. 1991, AJ, 101, 1647","doi":""},{"text":". P Tsalmantza, R Decarli, M Dotti, D W Hogg, 73820Tsalmantza, P., Decarli, R., Dotti, M. & Hogg, D. W., 2011ApJ, 738, 20","doi":""},{"text":". E Vasiliev, F Antonini, D Merritt, P G Van Dokkum, M Franx, ApJ. 7852027AJVasiliev, E., Antonini, F., & Merritt, D. 2014, ApJ, 785, 163 van Dokkum, P. G., & Franx, M. 1995, AJ, 110, 2027","doi":""},{"text":". G A Verdoes Kleijn, R P Van Der Marel, C M Carollo, P T De Zeeuw, AJ. 1201221Verdoes Kleijn, G. A., van der Marel, R. P., Carollo, C. M., & de Zeeuw, P. T. 2000, AJ, 120, 1221","doi":""},{"text":". M Volonteri, K Gúltekin, M Dotti, 4042143Volonteri, M.; Gúltekin, K.; Dotti, M., 2010MNRAS, 404, 2143","doi":""},{"text":". D M Worrall, M Birkinshaw, E O'sullivan, A Zezas, A Wolter, G Trinchieri, G Fabbiano, MNRAS. 408701Worrall, D. M., Birkinshaw, M., O'Sullivan, E., Zezas, A., Wolter, A., Trinchieri, G., & Fabbiano, G. 2010, MNRAS, 408, 701","doi":""},{"text":". J M Wrobel, AJ. 101127Wrobel, J. M. 1991, AJ, 101, 127","doi":""},{"text":". J M Wrobel, D S Heeschen, AJ. 101148Wrobel, J. M., & Heeschen, D. S. 1991, AJ, 101, 148","doi":""},{"text":". S Younis, J Meaburn, P Stewart, A&A. 147178Younis, S., Meaburn, J., & Stewart, P. 1985, A&A, 147, 178","doi":""},{"text":". O Zanotti, L Rezzolla, L Del Zanna, C Palenzuela, A&A. 5238Zanotti, O., Rezzolla, L., Del Zanna, L., & Palenzuela, C. 2010, A&A, 523, A8","doi":""},{"text":". G O Abell, H G Corwin, Jr, R P Olowin, ApJS. 701Abell, G. O., Corwin, H. G., Jr., & Olowin, R. P. 1989, ApJS, 70, 1","doi":""},{"text":". D M Alexander, R C Hickox, NAR. 5693Alexander, D. M., & Hickox, R. C. 2012, NAR, 56, 93","doi":""},{"text":". J M Anderson, J S Ulvestad, L C Ho, ApJ. 60342Anderson, J. M., Ulvestad, J. S., & Ho, L. C. 2004, ApJ, 603, 42","doi":""},{"text":". J Antognini, J Bird, P Martini, ApJ. 756116Antognini, J., Bird, J. & Martini, P. 2012, ApJ, 756, 116","doi":""},{"text":". F Antonini, D Merritt, ApJ. 74583Antonini, F., & Merritt, D. 2012, ApJ, 745, 83","doi":""},{"text":". W Baade, R Minkowski, ApJ. 119215Baade, W., & Minkowski, R. 1954, ApJ, 119, 215","doi":""},{"text":". J N Bahcall, R A Wolf, ApJ. 209214Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214","doi":""},{"text":". R D Baldi, M Chiaberge, A Capetti, W Sparks, F D Macchetto, C P O'dea, D J Axon, S A Baum, A C Quillen, ApJ. 7252426Baldi, R. D., Chiaberge, M., Capetti, A., Sparks, W., Macchetto, F. D., O'Dea, C. P., Axon, D. J., Baum, S. A. & Quillen, A. C. 2010, ApJ, 725, 2426","doi":""},{"text":". B Balmaverde, R D Baldi, A Capetti, A&A. 486119Balmaverde, B., Baldi, R. D., & Capetti, A. 2008, A&A, 486, 119","doi":""},{"text":". B Balmaverde, A Capetti, A&A. 447BC0697Balmaverde, B., & Capetti, A. 2006, A&A, 447, 97 (BC06)","doi":""},{"text":". J G Baker, J Centrella, D.-I Choi, M Koppitz, J Van Meter, Phys. Rev. Lett. 96111102Baker, J. G. , Centrella, J. , Choi,D.-I., Koppitz, M., & van Meter, J. 2006, Phys. Rev. Lett. 96, 111102","doi":""},{"text":". D Batcheldor, A Robinson, D J Axon, E S Perlman, D Merritt, ApJ. 717B106Batcheldor, D., Robinson, A., Axon, D. J., Perlman, E. S., & Merritt, D. 2010, ApJ, 717, L6 (B10)","doi":""},{"text":". M C Begelman, R D Blandford, M J Rees, Nature. 287307Begelman, M.C., Blandford, R.D. & Rees, M.J., 1980, Nature, 287, 307","doi":""},{"text":". J D Bekenstein, ApJ. 183657Bekenstein, J. D. 1973, ApJ, 183, 657","doi":""},{"text":". E F Bell, S Phleps, R S Somerville, C Wolf, A Borch, K Meisenheimer, ApJ. 652270Bell, E. F., Phleps, S., Somerville, R. S., Wolf, C., Borch, A. & Meisenheimer, K. 2006, ApJ, 652, 270","doi":""},{"text":". M Birkinshaw, R L Davies, ApJ. 29132Birkinshaw, M., & Davies, R. L. 1985, ApJ, 291, 32","doi":""},{"text":". R D Blandford, R L Znajek, MNRAS. 179433Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433","doi":""},{"text":". L Blecha, F Civano, M Elvis, A Loeb, MNRAS. 4281341Blecha, L., Civano, F., Elvis, M. & Loeb, A. 2013, MNRAS, 428,1341","doi":""},{"text":". E W Bonning, G A Shields, S Salviander, ApJ. 66613Bonning, E. W., Shields, G. A. & Salviander, S. 2007, ApJ, 666, 13","doi":""},{"text":". M J I Brown, B T Jannuzi, D J E Floyd, J Mould, R , ApJ. 73141Brown, M. J. I., Jannuzi, B. T., Floyd, D. J. E., Mould, J., R. 2011, ApJ, 731, L41","doi":""},{"text":". C Burke, C A Collins, MNRAS. 4342856Burke, C. & Collins, C. A., 2013, MNRAS, 434, 2856","doi":""},{"text":"I C Busko, Astronomical Data Analysis Software and Systems V. 101139Busko, I. C. 1996, Astronomical Data Analysis Software and Systems V, 101, 139","doi":""},{"text":". M Campanelli, C O Lousto, P Marronetti, Y Zlochower, Phys. Rev. Lett. 96111101Campanelli, M., Lousto, C. O., Marronetti, P., & Zlochower, Y. 2006, Phys. Rev. Lett. 96, 111101","doi":""},{"text":". M Campanelli, C O Lousto, Y Zlochower, D Merritt, Physical Review Letters. 98231102Campanelli, M., Lousto, C. O., Zlochower, Y., & Merritt, D. 2007, Physical Review Letters, 98, 231102","doi":""},{"text":". M Campanelli, C Lousto, Y Zlochower, D Merritt, ApJ. 6595Campanelli, M., Lousto, C., Zlochower, Y., & Merritt, D. 2007, ApJ, 659, L5","doi":""},{"text":". R E A Canning, A C Fabian, R M Johnstone, J S Sanders, C S Crawford, G J Ferland, N A Hatch, MNRAS. 4173080Canning, R.E.A., Fabian, A. C., Johnstone, R. M., Sanders, J. S., Crawford, C. S., Ferland, G. J., Hatch, N. A. 2011, MNRAS, 417, 3080","doi":""},{"text":". A Capetti, D J Axon, M Chiaberge, A&A. 471137Capetti, A., Axon, D. J., Chiaberge, M., et al. 2007, A&A, 471, 137","doi":""},{"text":". A Capetti, B Balmaverde, A&A. 440CB0573Capetti, A., & Balmaverde, B. 2005, A&A, 440, 73 (CB05)","doi":""},{"text":". A Capetti, B Balmaverde, A&A. 453CB0627Capetti, A., & Balmaverde, B. 2006, A&A, 453, 27 (CB06)","doi":""},{"text":". M Cappellari, A Renzini, L Greggio, ApJ. 519117Cappellari, M., Renzini, A., Greggio, L., et al. 1999, ApJ, 519, 117","doi":""},{"text":". M Cappellari, E K Verolme, R P Van Der Marel, ApJ. 578787Cappellari, M., Verolme, E. K., van der Marel, R. P., et al. 2002, ApJ, 578, 787","doi":""},{"text":". M Cappellari, E Emsellem, R Bacon, MNRAS. 379418Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418","doi":""},{"text":". A Cardullo, E M Corsini, A Beifiori, A&A. 508641Cardullo, A., Corsini, E. M., Beifiori, A., et al. 2009, A&A, 508, 641","doi":""},{"text":". C M Carollo, M Franx, G D Illingworth, D A Forbes, ApJ. 481710Carollo, C. M., Franx, M., Illingworth, G. D., & Forbes, D. A. 1997, ApJ, 481, 710","doi":""},{"text":". K W Cavagnolo, B R Mcnamara, P E J Nulsen, ApJ. 7201066Cavagnolo, K. W., McNamara, B. R., Nulsen, P. E. J., et al. 2010, ApJ, 720, 1066","doi":""},{"text":". M Chiaberge, A Capetti, A Celotti, A&A. 34977Chiaberge, M., Capetti, A., & Celotti, A. 1999, A&A, 349, 77","doi":""},{"text":". F Civano, M Elvis, G Lanzuisi, ApJ. 717209Civano, F., Elvis, M., Lanzuisi, G., et al. 2010, ApJ, 717, 209","doi":""},{"text":". F Civano, M Elvis, G Lanzuisi, ApJ. 75249Civano, F., Elvis, M., Lanzuisi, G., et al. 2012, ApJ, 752, 49","doi":""},{"text":"P Coles, F Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure. New YorkWiley2nd ed.Coles, P., & Lucchin, F. 2002, Cosmology: The Origin and Evolution of Cosmic Structure (2nd ed.; New York: Wiley)","doi":""},{"text":". C J Conselice, C Yang, A F L Bluck, MNRAS. 3941956Conselice, C. J., Yang, C. & Bluck, A. F. L. 2009, MNRAS, 394, 1956","doi":""},{"text":". J Cuadra, P J Armitage, R D Alexander, M C Begelman, MNRAS. 3931423Cuadra, J., Armitage, P. J. Alexander, R. D. & Begelman, M. C. 2009, MNRAS, 393, 1423","doi":""},{"text":". V P Debattista, I Ferreras, A Pasquali, ApJ. 65197Debattista, V. P., Ferreras, I., Pasquali, A., et al. 2006, ApJ, 651, L97","doi":""},{"text":"Third Reference Catalogue of Bright Galaxies Di Matteo. G De Vaucouleurs, A De Vaucouleurs, H G Corwin, Jr, ApJ. 582133de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Jr., et al. 1991, Third Reference Catalogue of Bright Galaxies Di Matteo, T., Allen, S. W., Fabian, A. C., Wilson, A. S., & Young, A. J. 2003, ApJ, 582, 133","doi":""},{"text":". M Dotti, M Volonteri, A Perego, M Colpi, M Ruszkowski, F Haardt, MNRAS. 402682Dotti, M., Volonteri, M., Perego, A., Colpi, M., Ruszkowski, M. & Haardt, F. 2010, MNRAS, 402, 682","doi":""},{"text":". A Dressler, D Lynden-Bell, D Burstein, ApJ. 31342Dressler, A., Lynden-Bell, D., Burstein, D., et al. 1987, ApJ, 313, 42","doi":""},{"text":". G Efstathiou, R S Ellis, D Carter, MNRAS. 193931Efstathiou, G., Ellis, R. S., & Carter, D. 1980, MNRAS, 193, 931","doi":""},{"text":". M Eracleous, T A Boroson, J P Halpern, J Liu, ApJS. 20123Eracleous, M., Boroson, T. A., Halpern, J. P., & Liu, J. 2012, ApJS, 201, 23","doi":""},{"text":". A Escala, R B Larson, P S Coppi, D Mardones, ApJ. 630152Escala, A., Larson, R. B., Coppi, P. S. & Mardones, D. 2005, ApJ, 630, 152","doi":""},{"text":". G Fabbiano, M Elvis, S Markoff, ApJ. 588175Fabbiano, G., Elvis, M., Markoff, S., et al. 2003, ApJ, 588, 175","doi":""},{"text":". S M Faber, S Tremaine, E A Ajhar, AJ. 1141771Faber, S. M., Tremaine, S., Ajhar, E. A., et al. 1997, AJ, 114, 1771","doi":""},{"text":". B L Fanaroff, J M Riley, MNRAS. 16731Fanaroff, B. L., & Riley, J. M. 1974, MNRAS, 167, 31P","doi":""},{"text":". C L Farage, P J Mcgregor, M A Dopita, G V Bicknell, ApJ. 724267Farage, C. L., McGregor, P. J., Dopita, M. A., & Bicknell, G. V. 2010, ApJ, 724, 267","doi":""},{"text":". M Favata, S A Hughes, D E Holz, ApJ. 6075Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ, 607, L5","doi":""},{"text":". H C Ferguson, AJ. 98367Ferguson, H. C. 1989, AJ, 98, 367","doi":""},{"text":". L Ferrarese, H C Ford, W Jaffe, ApJ. 470444Ferrarese, L., Ford, H. C., & Jaffe, W. 1996, ApJ, 470, 444","doi":""},{"text":". L Ferrarese, D Merritt, ApJ. 5399Ferrarese, L., & Merritt, D. 2000, ApJ, 539, L9","doi":""},{"text":". L Ferrarese, H Ford, Sci. Rev. 116523Ferrarese, L., Ford, H., 2005, Sci. Rev., 116, 523","doi":""},{"text":". M E Filho, P D Barthel, L C Ho, ApJS. 12993Filho, M. E., Barthel, P. D., & Ho, L. C. 2000, ApJS, 129, 93","doi":""},{"text":". M E Filho, F Fraternali, S Markoff, A&A. 418429Filho, M. E., Fraternali, F., Markoff, S., et al. 2004, A&A, 418, 429","doi":""},{"text":". D A Forbes, M Franx, G D Illingworth, ApJ. 42849Forbes, D. A., Franx, M., & Illingworth, G. D. 1994, ApJ, 428, L49","doi":""},{"text":". D A Forbes, J P Brodie, J Huchra, AJ. 1122448Forbes, D. A., Brodie, J. P., & Huchra, J. 1996, AJ, 112, 2448","doi":""},{"text":". D A Forbes, D B Reitzel, G M Williger, AJ. 1091576Forbes, D. A., Reitzel, D. B., & Williger, G. M. 1995, AJ, 109, 1576","doi":""},{"text":". M Franx, G D Illingworth, ApJ. 32755Franx, M., & Illingworth, G. D. 1988, ApJ, 327, L55","doi":""},{"text":". O Fakhouri, C.-P Ma, M Boylan-Kolchin, MNRAS. 4062267Fakhouri, O., Ma, C.-P., & Boylan-Kolchin, M. 2010, MNRAS, 406, 2267","doi":""},{"text":". K Gebhardt, R Bender, G Bower, ApJ. 53913Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13","doi":""},{"text":". K Gebhardt, T R Lauer, J Pinkney, ApJ. 6711321Gebhardt, K., Lauer, T. R., Pinkney, J., et al. 2007, ApJ, 671, 1321","doi":""},{"text":". K Gebhardt, J Adams, D Richstone, ApJ. 729119Gebhardt, K., Adams, J., Richstone, D., et al. 2011, ApJ, 729, 119","doi":""},{"text":". M Gerin, F Casoli, A&A. 29049Gerin, M., & Casoli, F. 1994, A&A, 290, 49","doi":""},{"text":". M Giroletti, G B Taylor, G Giovannini, ApJ. 622178Giroletti, M., Taylor, G. B., & Giovannini, G. 2005, ApJ, 622, 178","doi":""},{"text":". L Giordano, L Cortese, G Trinchieri, A Wolter, M Colpi, G Gavazzi, L Mayer, ApJ. 634272Giordano, L., Cortese, L., Trinchieri, G., Wolter, A., Colpi, M., Gavazzi, G., & Mayer, L. 2005, ApJ, 634, 272","doi":""},{"text":". J A González, M Hannam, U Sperhake, B Brügmann, S Husa, Phys. Rev. Lett. 98231101González, J. A., Hannam, M., Sperhake, U., Brügmann, B., & Husa, S. 2007, Phys. Rev. Lett., 98, 231101","doi":""},{"text":". O González-Martín, J Masegosa, I Márquez, M Guainazzi, E Jiménez-Bailón, A&A. 5061107González-Martín, O., Masegosa, J., Márquez, I., Guainazzi, M., & Jiménez-Bailón, E. 2009, A&A, 506, 1107","doi":""},{"text":". R Grützbauch, G Trinchieri, R Rampazzo, AJ. 133220Grützbauch, R., Trinchieri, G., Rampazzo, R., et al. 2007, AJ, 133, 220","doi":""},{"text":". A Gualandris, D Merritt, A Gualandris, D Merritt, ApJ. 678GM0874ApJGualandris, A., & Merritt, D. 2008, ApJ, 678, 780 (GM08) Gualandris, A., & Merritt, D. 2012, ApJ, 744, 74","doi":""},{"text":". K Hada, A Doi, M Kino, Nature. 477185Hada, K., Doi, A., Kino, M., et al. 2011, Nature, 477, 185","doi":""},{"text":". M Hilz, T Naab, J P Ostriker, arXiv:1206.5004Hilz, M., Naab, T., & Ostriker, J. P. 2012, arXiv:1206.5004","doi":"arXiv:1206.5004"},{"text":". L C Ho, ARA&A. 46475Ho, L. C. 2008, ARA&A, 46, 475","doi":""},{"text":". L C Ho, A V Filippenko, W L W Sargent, C Y Peng, ApJS. 112391Ho, L. C., Filippenko, A. V., Sargent, W. L. W. & Peng, C. Y. 1997, ApJS, 112, 391","doi":""},{"text":". P F Hopkins, K Bundy, D Croton, L Hernquist, D Keres, S Khochfar, K Stewart, A Wetzel, J D Younger, ApJ. 715H10202Hopkins, P. F., Bundy, K., Croton, D., Hernquist, L., Keres, D., Khochfar, S., Stewart, K., Wetzel, A. & Younger, J. D. 2010, ApJ, 715, 202 (H10)","doi":""},{"text":". R C W Houghton, J Magorrian, M Sarzi, MNRAS. 3672Houghton, R. C. W., Magorrian, J., Sarzi, M., et al. 2006, MNRAS, 367, 2","doi":""},{"text":". J P Huchra, M J Geller, ApJ. 257423Huchra, J. P., & Geller, M. J. 1982, ApJ, 257, 423","doi":""},{"text":". J Huchra, M Davis, D Latham, J Tonry, ApJS. 5289Huchra, J., Davis, M., Latham, D., & Tonry, J. 1983, ApJS, 52, 89","doi":""},{"text":". W Jaffe, H C Ford, L Ferrarese, Van Den, F Bosch, R W Connell, Nature. 364213Jaffe, W., Ford, H. C., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1993, Nature, 364, 213","doi":""},{"text":". W Jaffe, B R Mcnamara, ApJ. 434110Jaffe, W., & McNamara, B. R. 1994, ApJ, 434, 110","doi":""},{"text":". W Jaffe, H Ford, L Ferrarese, Van Den, F Bosch, R W Connell, ApJ. 460214Jaffe, W., Ford, H., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1996, ApJ, 460, 214","doi":""},{"text":". S Janowiecki, J C Mihos, P Harding, ApJ. 715972Janowiecki, S., Mihos, J. C., Harding, P., et al. 2010, ApJ, 715, 972","doi":""},{"text":". R I Jedrzejewski, MNRAS. 226747Jedrzejewski, R. I. 1987, MNRAS, 226, 747","doi":""},{"text":". D L Jones, A E Wehrle, D L Meier, B G Piner, ApJ. 534165Jones, D. L., Wehrle, A. E., Meier, D. L., & Piner, B. G. 2000, ApJ, 534, 165","doi":""},{"text":". C Jones, W Forman, A Vikhlinin, ApJ. 567115Jones, C., Forman, W., Vikhlinin, A., et al. 2002, ApJ, 567, L115","doi":""},{"text":". J I Katz, ApJ. 478527Katz, J. I. 1997, ApJ, 478, 527","doi":""},{"text":". F M Khan, A Just, D Merritt, ApJ. 73289Khan, F. M., Just, A., & Merritt, D. 2011, ApJ, 732, 89","doi":""},{"text":". N E B Killeen, G V Bicknell, R D Ekers, ApJ. 302306Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1986, ApJ, 302, 306","doi":""},{"text":". N E B Killeen, G V Bicknell, D Carter, ApJ. 30945Killeen, N. E. B., Bicknell, G. V., & Carter, D. 1986, ApJ, 309, 45","doi":""},{"text":". N E B Killeen, G V Bicknell, ApJ. 324198Killeen, N. E. B., & Bicknell, G. V. 1988, ApJ, 324, 198","doi":""},{"text":". N E B Killeen, G V Bicknell, R D Ekers, ApJ. 325180Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1988, ApJ, 325, 180","doi":""},{"text":". S Komossa, H Zhou, H Lu, ApJ. 67881Komossa, S., Zhou, H., & Lu, H. 2008, ApJ, 678, L81","doi":""},{"text":". S Komossa, Mem. Soc. Astron. Italiana. 77733Komossa, S. 2006, Mem. Soc. Astron. Italiana, 77, 733","doi":""},{"text":". J Kormendy, D Richstone ; Ara&a Kornreich, D A Lovelace, R V E , 681..104KJ. Kormendy, D. Richstone, 1995, ARA&A Kornreich D. A. and Lovelace R. V. E., 2008ApJ...681..104K","doi":"681..104K"},{"text":"J Krist, Astronomical Data Analysis Software and Systems II. 52536Krist, J. 1993, Astronomical Data Analysis Software and Systems II, 52, 536","doi":""},{"text":". P Kroupa, arXiv:1406.4860Kroupa, P. 2014, arXiv:1406.4860","doi":"arXiv:1406.4860"},{"text":". S Laine, R P Van Der Marel, T R Lauer, M Postman, C P O'dea, F N Owen, AJ. 125478Laine, S., van der Marel, R. P., Lauer, T. R., Postman, M., O'Dea, C. P., & Owen, F. N. 2003, AJ, 125, 478","doi":""},{"text":". T R Lauer, S M Faber, E J Groth, AJ. 1061436Lauer, T. R., Faber, S. M., Groth, E. J., et al. 1993, AJ, 106, 1436","doi":""},{"text":". T R Lauer, E A Ajhar, Y.-I Byun, AJ. 1102622Lauer, T. R., Ajhar, E. A., Byun, Y.-I., et al. 1995, AJ, 110, 2622","doi":""},{"text":". T R Lauer, S Tremaine, E A Ajhar, ApJ. 47179Lauer, T. R., Tremaine, S., Ajhar, E. A., et al. 1996, ApJ, 471, L79","doi":""},{"text":". T R Lauer, S M Faber, K Gebhardt, AJ. 1292138Lauer, T. R., Faber, S. M., Gebhardt, K., et al. 2005, AJ, 129, 2138","doi":""},{"text":". T R Lauer, S M Faber, D Richstone, ApJ. 662808Lauer, T. R., Faber, S. M., Richstone, D., et al. 2007, ApJ, 662, 808","doi":""},{"text":". C Lidman, G Iacobuta, A E Bauer, MNRAS. 433825Lidman, C., Iacobuta, G., Bauer, A. E., et al. 2013, MNRAS, 433, 825","doi":""},{"text":". A P Lobanov, J Roland, A&A. 431831Lobanov, A. P., & Roland, J. 2005, A&A, 431, 831","doi":""},{"text":". A Loeb, Phrvl, 9941103Loeb, A., PhRvL, 99, 041103L","doi":""},{"text":". M Loewenstein, R F Mushotzky, L Angelini, K A Arnaud, E Quataert, ApJ. 55521Loewenstein, M., Mushotzky, R. F., Angelini, L., Arnaud, K. A., & Quataert, E. 2001, ApJ, 555, L21","doi":""},{"text":". C O Lousto, H Nakano, Y Zlochower, M Campanelli, Phys. Rev. D. 8184023Lousto, C. O., Nakano, H., Zlochower, Y., & Campanelli, M. 2010, Phys. Rev. D 81, 084023","doi":""},{"text":". C O Lousto, Y Zlochower, Phys. Rev. Lett. 107231102Lousto, C. O. & Zlochower, Y. 2011, Phys. Rev. Lett. 107, 231102","doi":""},{"text":". C O Lousto, Y Zlochower, M Dotti, M Volonteri, arXiv:1201.1923(L12Lousto, C. O., Zlochower, Y., Dotti, M., & Volonteri, M. 2012, arXiv:1201.1923 (L12)","doi":"arXiv:1201.1923(L12"},{"text":". F Macchetto, M Pastoriza, N Caon, A&AS. 120463Macchetto, F., Pastoriza, M., Caon, N., et al. 1996, A&AS, 120, 463","doi":""},{"text":". D Maoz, N M Nagar, H Falcke, A S Wilson, ApJ. 625699Maoz, D., Nagar, N. M., Falcke, H., & Wilson, A. S. 2005, ApJ, 625, 699","doi":""},{"text":". B R Mcnamara, ApJ. 562149McNamara, B. R., et al. 2001, ApJ, 562, L149","doi":""},{"text":". M E Machacek, D Jerius, R Kraft, ApJ. 74315Machacek, M. E., Jerius, D., Kraft, R., et al. 2011, ApJ, 743, 15","doi":""},{"text":". P Madau, E Quataert, ApJ. 60617Madau, P. and Quataert, E. 2004, ApJ, 606, 17","doi":""},{"text":". D F Malin, Nature. 277279Malin, D. F. 1979, Nature, 277, 279","doi":""},{"text":"D F Malin, New Aspects of Galaxy Photometry. D. F. Malin, New Aspects of Galaxy Photometry, 1985","doi":""},{"text":". L Mayer, S Kazantzidis, P Madau, Science. 3161874Mayer, L., Kazantzidis, S., Madau, P., et al. 2007, Science, 316, 1874","doi":""},{"text":". J Masegosa, I Márquez, A Ramirez, O González-Martín, A&A. 52723Masegosa, J., Márquez, I., Ramirez, A., & González-Martín, O. 2011, A&A, 527, A23","doi":""},{"text":"D Merritt, Dynamics and Evolution of Galactic Nuclei. Princeton, NJPrinceton Univ. PressMerritt, D. 2013, Dynamics and Evolution of Galactic Nuclei (Princeton, NJ: Princeton Univ. Press)","doi":""},{"text":". D Merritt, R D Ekers, Science. 2971310Merritt, D., & Ekers, R. D. 2002, Science, 297, 1310","doi":""},{"text":". D Merritt, M Milosavljević, M Favata, S A Hughes, D E Holz, ApJ. 6079Merritt, D., Milosavljević, M., Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ 607, L9.","doi":""},{"text":". D Merritt, M Milosavljević, Living Reviews in Relativity. 88Merritt, D., & Milosavljević, M. 2005, Living Reviews in Relativity, 8, 8","doi":""},{"text":". D Merritt, ApJ. 648976Merritt, D. 2006, ApJ, 648, 976","doi":""},{"text":". D Merritt, J D Schnittman, S Komossa, ApJ. 6991690Merritt, D., Schnittman, J. D., & Komossa, S. 2009, ApJ, 699, 1690","doi":""},{"text":". B Y Mills, O B Slee, E R Hill, Australian Journal of Physics. 13676Mills, B. Y., Slee, O. B., & Hill, E. R. 1960, Australian Journal of Physics, 13, 676","doi":""},{"text":". C Moellenhoff, E Hummel, R Bender, A&A. 25535Moellenhoff, C., Hummel, E., & Bender, R. 1992, A&A, 255, 35","doi":""},{"text":". N M Nagar, H Falcke, A S Wilson, J S Ulvestad, A&A. 39253Nagar, N. M., Falcke, H., Wilson, A. S., & Ulvestad, J. S. 2002, A&A, 392, 53","doi":""},{"text":". R Nolthenius, ApJS. 851Nolthenius, R. 1993, ApJS, 85, 1","doi":""},{"text":". R W O'connell, J R Martin, J D Crane, ApJ. 635305O'Connell, R. W., Martin, J. R., Crane, J. D., et al. 2005, ApJ, 635, 305","doi":""},{"text":". C P O'dea, R A Daly, P Kharb, K A Freeman, S A Baum, A&A. 494471O'Dea, C. P., Daly, R. A., Kharb, P., Freeman, K. A. & Baum, S. A. 2009, A&A, 494, 471","doi":""},{"text":". P Parma, M Murgia, R Morganti, A Capetti, H R De Ruiter, R Fanti, A&A. 3447Parma, P., Murgia, M., Morganti, R., Capetti, A., de Ruiter, H. R. & Fanti, R. 1999, A&A, 344, 7","doi":""},{"text":". S Pellegrini, T Venturi, A Comastri, ApJ. 585677Pellegrini, S., Venturi, T., Comastri, A., et al. 2003, ApJ, 585, 677","doi":""},{"text":". H V Peiris, S Tremaine, ApJ. Peiris, H. V. & Tremaine, S. 2003, ApJ, 237-257","doi":""},{"text":". C Y Peng, L C Ho, C D Impey, H.-W Rix, AJ. 124266Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266","doi":""},{"text":". H B Perets, C Hopman, T Alexander, ApJ. 656709Perets, H. B., Hopman, C., & Alexander, T. 2007, ApJ, 656, 709","doi":""},{"text":". E S Perlman, D E Harris, J A Biretta, W B Sparks, F D Macchetto, ApJ. 59965Perlman, E. S., Harris, D. E., Biretta, J. A., Sparks, W. B., & Macchetto, F. D. 2003, ApJ, 599, L65","doi":""},{"text":". M Postman, T R Lauer, ApJ. 44028Postman, M., & Lauer, T. R. 1995, ApJ, 440, 28","doi":""},{"text":". M Preto, I Berentzen, P Berczik, R Spurzem, ApJ. 73226Preto, M., Berentzen, I., Berczik, P., & Spurzem, R. 2011, ApJ, 732, L26","doi":""},{"text":". F Pretorius, Phys. Rev. Lett. 95121101Pretorius, F. 2005, Phys. Rev. Lett. 95, 121101","doi":""},{"text":". A Renzini, L Greggio, S Di Serego Alighieri, Nature. 37839Renzini, A., Greggio, L., di Serego Alighieri, S., et al. 1995, Nature, 378, 39","doi":""},{"text":". A Robinson, S Young, D J Axon, P Kharb, J E Smith, ApJ. 717122Robinson, A., Young, S., Axon, D. J., Kharb, P., & Smith, J. E. 2010, ApJ, 717, L122","doi":""},{"text":". G E Romero, L Chajet, Z Abraham, J H Fan, A&A. 36057Romero, G. E., Chajet, L., Abraham, Z., & Fan, J. H. 2000, A&A, 360, 57","doi":""},{"text":". N Roos, J S Kaastra, C A Hummel, ApJ. 409130Roos, N., Kaastra, J. S., & Hummel, C. A. 1993, ApJ, 409, 130","doi":""},{"text":". E M Sadler, N A Sharp, A&A. 133216Sadler, E. M., & Sharp, N. A. 1984, A&A, 133, 216","doi":""},{"text":". E M Sadler, C R Jenkins, C G Kotanyi, MNRAS. 240591Sadler, E. M., Jenkins, C. R., & Kotanyi, C. G. 1989, MNRAS, 240, 591","doi":""},{"text":". E Sani, A Marconi, L K Hunt, G Risaliti, MNRAS. 4131479Sani, E., Marconi, A., Hunt, L. K., & Risaliti, G. 2011, MNRAS, 413, 1479","doi":""},{"text":". W C Saslaw, M Whittle, ApJ. 325142Saslaw, W. C., & Whittle, M. 1988, ApJ, 325, 142","doi":""},{"text":"Extragalactic Radio Sources. I S Shklovsky, 97475Shklovsky, I. S. 1982, Extragalactic Radio Sources, 97, 475","doi":""},{"text":". G A Shields, D J Rosario, K L Smith, ApJ. 707936Shields, G. A., Rosario, D. J., Smith, K. L., et al. 2009, ApJ, 707, 936","doi":""},{"text":". K Shurkin, R J H Dunn, G Gentile, G B Taylor, S W Allen, MNRAS. 383923Shurkin, K., Dunn, R. J. H., Gentile, G., Taylor, G. B., & Allen, S. W. 2008, MNRAS, 383, 923","doi":""},{"text":". D Sijacki, V Springel, M G Haehnelt, MNRAS. 4143656Sijacki, D., Springel, V. & Haehnelt, M. G. 2011, MNRAS, 414, 3656","doi":""},{"text":". O B Slee, E M Sadler, J E Reynolds, R D Ekers, MNRAS. 269928Slee, O. B., Sadler, E. M., Reynolds, J. E., & Ekers, R. D. 1994, MNRAS, 269, 928","doi":""},{"text":". W B Sparks, F Macchetto, D Golombek, ApJ. 345153Sparks, W. B., Macchetto, F., & Golombek, D. 1989, ApJ, 345, 153","doi":""},{"text":". V Springel, S D M White, A Jenkins, Nature. 435629Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629","doi":""},{"text":". V J Stanger, R S Warwick, MNRAS. 220363Stanger, V. J., & Warwick, R. S. 1986, MNRAS, 220, 363","doi":""},{"text":". C L Steinhardt, M Schramm, J D Silverman, ApJ. 75924Steinhardt, C. L., Schramm, M., Silverman, J. D., et al. 2012, ApJ, 759, 24","doi":""},{"text":". R Subrahmanyan, A J Beasley, W M Goss, K Golap, R W Hunstead, AJ. 1251095Subrahmanyan, R., Beasley, A. J., Goss, W. M., Golap, K., & Hunstead, R. W. 2003, AJ, 125, 1095","doi":""},{"text":". G B Taylor, J S Sanders, A C Fabian, S W Allen, MNRAS. 365705Taylor, G. B., Sanders, J. S., Fabian, A. C., & Allen, S. W. 2006, MNRAS, 365, 705","doi":""},{"text":". T Tal, P G Van Dokkum, J Nelan, R Bezanson, AJ. 1381417Tal, T., van Dokkum, P. G., Nelan, J., & Bezanson, R. 2009, AJ, 138, 1417","doi":""},{"text":". A Tarchi, N Neininger, A Greve, A&A. 35895Tarchi, A., Neininger, N., Greve, A., et al. 2000, A&A, 358, 95","doi":""},{"text":". N Thatte, M Tecza, R Genzel, A&A. 36447Thatte, N., Tecza, M., & Genzel, R. 2000, A&A, 364, L47","doi":""},{"text":". P Temi, W G Mathews, F Brighenti, J D Bregman, ApJ. 585121Temi, P., Mathews, W. G., Brighenti, F., & Bregman, J. D. 2003, ApJ, 585, L121","doi":""},{"text":". W Tichy, P Marronetti, Phys. Rev. D. 061502. Trinchieri, G., & di Serego Alighieri761647AJTichy, W. & Marronetti, P. 2007, Phys. Rev. D 76, 061502. Trinchieri, G., & di Serego Alighieri, S. 1991, AJ, 101, 1647","doi":""},{"text":". P Tsalmantza, R Decarli, M Dotti, D W Hogg, 73820Tsalmantza, P., Decarli, R., Dotti, M. & Hogg, D. W., 2011ApJ, 738, 20","doi":""},{"text":". E Vasiliev, F Antonini, D Merritt, P G Van Dokkum, M Franx, ApJ. 7852027AJVasiliev, E., Antonini, F., & Merritt, D. 2014, ApJ, 785, 163 van Dokkum, P. G., & Franx, M. 1995, AJ, 110, 2027","doi":""},{"text":". G A Verdoes Kleijn, R P Van Der Marel, C M Carollo, P T De Zeeuw, AJ. 1201221Verdoes Kleijn, G. A., van der Marel, R. P., Carollo, C. M., & de Zeeuw, P. T. 2000, AJ, 120, 1221","doi":""},{"text":". M Volonteri, K Gúltekin, M Dotti, 4042143Volonteri, M.; Gúltekin, K.; Dotti, M., 2010MNRAS, 404, 2143","doi":""},{"text":". D M Worrall, M Birkinshaw, E O'sullivan, A Zezas, A Wolter, G Trinchieri, G Fabbiano, MNRAS. 408701Worrall, D. M., Birkinshaw, M., O'Sullivan, E., Zezas, A., Wolter, A., Trinchieri, G., & Fabbiano, G. 2010, MNRAS, 408, 701","doi":""},{"text":". J M Wrobel, AJ. 101127Wrobel, J. M. 1991, AJ, 101, 127","doi":""},{"text":". J M Wrobel, D S Heeschen, AJ. 101148Wrobel, J. M., & Heeschen, D. S. 1991, AJ, 101, 148","doi":""},{"text":". S Younis, J Meaburn, P Stewart, A&A. 147178Younis, S., Meaburn, J., & Stewart, P. 1985, A&A, 147, 178","doi":""},{"text":". O Zanotti, L Rezzolla, L Del Zanna, C Palenzuela, A&A. 5238Zanotti, O., Rezzolla, L., Del Zanna, L., & Palenzuela, C. 2010, A&A, 523, A8","doi":""}],"string":"[\n {\n \"text\": \". G O Abell, H G Corwin, Jr, R P Olowin, ApJS. 701Abell, G. O., Corwin, H. G., Jr., & Olowin, R. P. 1989, ApJS, 70, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D M Alexander, R C Hickox, NAR. 5693Alexander, D. M., & Hickox, R. C. 2012, NAR, 56, 93\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J M Anderson, J S Ulvestad, L C Ho, ApJ. 60342Anderson, J. M., Ulvestad, J. S., & Ho, L. C. 2004, ApJ, 603, 42\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Antognini, J Bird, P Martini, ApJ. 756116Antognini, J., Bird, J. & Martini, P. 2012, ApJ, 756, 116\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Antonini, D Merritt, ApJ. 74583Antonini, F., & Merritt, D. 2012, ApJ, 745, 83\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Baade, R Minkowski, ApJ. 119215Baade, W., & Minkowski, R. 1954, ApJ, 119, 215\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J N Bahcall, R A Wolf, ApJ. 209214Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R D Baldi, M Chiaberge, A Capetti, W Sparks, F D Macchetto, C P O'dea, D J Axon, S A Baum, A C Quillen, ApJ. 7252426Baldi, R. D., Chiaberge, M., Capetti, A., Sparks, W., Macchetto, F. D., O'Dea, C. P., Axon, D. J., Baum, S. A. & Quillen, A. C. 2010, ApJ, 725, 2426\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Balmaverde, R D Baldi, A Capetti, A&A. 486119Balmaverde, B., Baldi, R. D., & Capetti, A. 2008, A&A, 486, 119\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Balmaverde, A Capetti, A&A. 447BC0697Balmaverde, B., & Capetti, A. 2006, A&A, 447, 97 (BC06)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J G Baker, J Centrella, D.-I Choi, M Koppitz, J Van Meter, Phys. Rev. Lett. 96111102Baker, J. G. , Centrella, J. , Choi,D.-I., Koppitz, M., & van Meter, J. 2006, Phys. Rev. Lett. 96, 111102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Batcheldor, A Robinson, D J Axon, E S Perlman, D Merritt, ApJ. 717B106Batcheldor, D., Robinson, A., Axon, D. J., Perlman, E. S., & Merritt, D. 2010, ApJ, 717, L6 (B10)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M C Begelman, R D Blandford, M J Rees, Nature. 287307Begelman, M.C., Blandford, R.D. & Rees, M.J., 1980, Nature, 287, 307\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Bekenstein, ApJ. 183657Bekenstein, J. D. 1973, ApJ, 183, 657\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E F Bell, S Phleps, R S Somerville, C Wolf, A Borch, K Meisenheimer, ApJ. 652270Bell, E. F., Phleps, S., Somerville, R. S., Wolf, C., Borch, A. & Meisenheimer, K. 2006, ApJ, 652, 270\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Birkinshaw, R L Davies, ApJ. 29132Birkinshaw, M., & Davies, R. L. 1985, ApJ, 291, 32\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R D Blandford, R L Znajek, MNRAS. 179433Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Blecha, F Civano, M Elvis, A Loeb, MNRAS. 4281341Blecha, L., Civano, F., Elvis, M. & Loeb, A. 2013, MNRAS, 428,1341\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E W Bonning, G A Shields, S Salviander, ApJ. 66613Bonning, E. W., Shields, G. A. & Salviander, S. 2007, ApJ, 666, 13\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M J I Brown, B T Jannuzi, D J E Floyd, J Mould, R , ApJ. 73141Brown, M. J. I., Jannuzi, B. T., Floyd, D. J. E., Mould, J., R. 2011, ApJ, 731, L41\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Burke, C A Collins, MNRAS. 4342856Burke, C. & Collins, C. A., 2013, MNRAS, 434, 2856\",\n \"doi\": \"\"\n },\n {\n \"text\": \"I C Busko, Astronomical Data Analysis Software and Systems V. 101139Busko, I. C. 1996, Astronomical Data Analysis Software and Systems V, 101, 139\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Campanelli, C O Lousto, P Marronetti, Y Zlochower, Phys. Rev. Lett. 96111101Campanelli, M., Lousto, C. O., Marronetti, P., & Zlochower, Y. 2006, Phys. Rev. Lett. 96, 111101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Campanelli, C O Lousto, Y Zlochower, D Merritt, Physical Review Letters. 98231102Campanelli, M., Lousto, C. O., Zlochower, Y., & Merritt, D. 2007, Physical Review Letters, 98, 231102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Campanelli, C Lousto, Y Zlochower, D Merritt, ApJ. 6595Campanelli, M., Lousto, C., Zlochower, Y., & Merritt, D. 2007, ApJ, 659, L5\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E A Canning, A C Fabian, R M Johnstone, J S Sanders, C S Crawford, G J Ferland, N A Hatch, MNRAS. 4173080Canning, R.E.A., Fabian, A. C., Johnstone, R. M., Sanders, J. S., Crawford, C. S., Ferland, G. J., Hatch, N. A. 2011, MNRAS, 417, 3080\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Capetti, D J Axon, M Chiaberge, A&A. 471137Capetti, A., Axon, D. J., Chiaberge, M., et al. 2007, A&A, 471, 137\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Capetti, B Balmaverde, A&A. 440CB0573Capetti, A., & Balmaverde, B. 2005, A&A, 440, 73 (CB05)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Capetti, B Balmaverde, A&A. 453CB0627Capetti, A., & Balmaverde, B. 2006, A&A, 453, 27 (CB06)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Cappellari, A Renzini, L Greggio, ApJ. 519117Cappellari, M., Renzini, A., Greggio, L., et al. 1999, ApJ, 519, 117\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Cappellari, E K Verolme, R P Van Der Marel, ApJ. 578787Cappellari, M., Verolme, E. K., van der Marel, R. P., et al. 2002, ApJ, 578, 787\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Cappellari, E Emsellem, R Bacon, MNRAS. 379418Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Cardullo, E M Corsini, A Beifiori, A&A. 508641Cardullo, A., Corsini, E. M., Beifiori, A., et al. 2009, A&A, 508, 641\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C M Carollo, M Franx, G D Illingworth, D A Forbes, ApJ. 481710Carollo, C. M., Franx, M., Illingworth, G. D., & Forbes, D. A. 1997, ApJ, 481, 710\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K W Cavagnolo, B R Mcnamara, P E J Nulsen, ApJ. 7201066Cavagnolo, K. W., McNamara, B. R., Nulsen, P. E. J., et al. 2010, ApJ, 720, 1066\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Chiaberge, A Capetti, A Celotti, A&A. 34977Chiaberge, M., Capetti, A., & Celotti, A. 1999, A&A, 349, 77\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Civano, M Elvis, G Lanzuisi, ApJ. 717209Civano, F., Elvis, M., Lanzuisi, G., et al. 2010, ApJ, 717, 209\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Civano, M Elvis, G Lanzuisi, ApJ. 75249Civano, F., Elvis, M., Lanzuisi, G., et al. 2012, ApJ, 752, 49\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P Coles, F Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure. New YorkWiley2nd ed.Coles, P., & Lucchin, F. 2002, Cosmology: The Origin and Evolution of Cosmic Structure (2nd ed.; New York: Wiley)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C J Conselice, C Yang, A F L Bluck, MNRAS. 3941956Conselice, C. J., Yang, C. & Bluck, A. F. L. 2009, MNRAS, 394, 1956\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Cuadra, P J Armitage, R D Alexander, M C Begelman, MNRAS. 3931423Cuadra, J., Armitage, P. J. Alexander, R. D. & Begelman, M. C. 2009, MNRAS, 393, 1423\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V P Debattista, I Ferreras, A Pasquali, ApJ. 65197Debattista, V. P., Ferreras, I., Pasquali, A., et al. 2006, ApJ, 651, L97\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Third Reference Catalogue of Bright Galaxies Di Matteo. G De Vaucouleurs, A De Vaucouleurs, H G Corwin, Jr, ApJ. 582133de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Jr., et al. 1991, Third Reference Catalogue of Bright Galaxies Di Matteo, T., Allen, S. W., Fabian, A. C., Wilson, A. S., & Young, A. J. 2003, ApJ, 582, 133\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Dotti, M Volonteri, A Perego, M Colpi, M Ruszkowski, F Haardt, MNRAS. 402682Dotti, M., Volonteri, M., Perego, A., Colpi, M., Ruszkowski, M. & Haardt, F. 2010, MNRAS, 402, 682\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Dressler, D Lynden-Bell, D Burstein, ApJ. 31342Dressler, A., Lynden-Bell, D., Burstein, D., et al. 1987, ApJ, 313, 42\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Efstathiou, R S Ellis, D Carter, MNRAS. 193931Efstathiou, G., Ellis, R. S., & Carter, D. 1980, MNRAS, 193, 931\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Eracleous, T A Boroson, J P Halpern, J Liu, ApJS. 20123Eracleous, M., Boroson, T. A., Halpern, J. P., & Liu, J. 2012, ApJS, 201, 23\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Escala, R B Larson, P S Coppi, D Mardones, ApJ. 630152Escala, A., Larson, R. B., Coppi, P. S. & Mardones, D. 2005, ApJ, 630, 152\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Fabbiano, M Elvis, S Markoff, ApJ. 588175Fabbiano, G., Elvis, M., Markoff, S., et al. 2003, ApJ, 588, 175\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S M Faber, S Tremaine, E A Ajhar, AJ. 1141771Faber, S. M., Tremaine, S., Ajhar, E. A., et al. 1997, AJ, 114, 1771\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B L Fanaroff, J M Riley, MNRAS. 16731Fanaroff, B. L., & Riley, J. M. 1974, MNRAS, 167, 31P\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C L Farage, P J Mcgregor, M A Dopita, G V Bicknell, ApJ. 724267Farage, C. L., McGregor, P. J., Dopita, M. A., & Bicknell, G. V. 2010, ApJ, 724, 267\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Favata, S A Hughes, D E Holz, ApJ. 6075Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ, 607, L5\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H C Ferguson, AJ. 98367Ferguson, H. C. 1989, AJ, 98, 367\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ferrarese, H C Ford, W Jaffe, ApJ. 470444Ferrarese, L., Ford, H. C., & Jaffe, W. 1996, ApJ, 470, 444\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ferrarese, D Merritt, ApJ. 5399Ferrarese, L., & Merritt, D. 2000, ApJ, 539, L9\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ferrarese, H Ford, Sci. Rev. 116523Ferrarese, L., Ford, H., 2005, Sci. Rev., 116, 523\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E Filho, P D Barthel, L C Ho, ApJS. 12993Filho, M. E., Barthel, P. D., & Ho, L. C. 2000, ApJS, 129, 93\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E Filho, F Fraternali, S Markoff, A&A. 418429Filho, M. E., Fraternali, F., Markoff, S., et al. 2004, A&A, 418, 429\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D A Forbes, M Franx, G D Illingworth, ApJ. 42849Forbes, D. A., Franx, M., & Illingworth, G. D. 1994, ApJ, 428, L49\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D A Forbes, J P Brodie, J Huchra, AJ. 1122448Forbes, D. A., Brodie, J. P., & Huchra, J. 1996, AJ, 112, 2448\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D A Forbes, D B Reitzel, G M Williger, AJ. 1091576Forbes, D. A., Reitzel, D. B., & Williger, G. M. 1995, AJ, 109, 1576\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Franx, G D Illingworth, ApJ. 32755Franx, M., & Illingworth, G. D. 1988, ApJ, 327, L55\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Fakhouri, C.-P Ma, M Boylan-Kolchin, MNRAS. 4062267Fakhouri, O., Ma, C.-P., & Boylan-Kolchin, M. 2010, MNRAS, 406, 2267\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Gebhardt, R Bender, G Bower, ApJ. 53913Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Gebhardt, T R Lauer, J Pinkney, ApJ. 6711321Gebhardt, K., Lauer, T. R., Pinkney, J., et al. 2007, ApJ, 671, 1321\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Gebhardt, J Adams, D Richstone, ApJ. 729119Gebhardt, K., Adams, J., Richstone, D., et al. 2011, ApJ, 729, 119\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Gerin, F Casoli, A&A. 29049Gerin, M., & Casoli, F. 1994, A&A, 290, 49\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Giroletti, G B Taylor, G Giovannini, ApJ. 622178Giroletti, M., Taylor, G. B., & Giovannini, G. 2005, ApJ, 622, 178\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Giordano, L Cortese, G Trinchieri, A Wolter, M Colpi, G Gavazzi, L Mayer, ApJ. 634272Giordano, L., Cortese, L., Trinchieri, G., Wolter, A., Colpi, M., Gavazzi, G., & Mayer, L. 2005, ApJ, 634, 272\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J A González, M Hannam, U Sperhake, B Brügmann, S Husa, Phys. Rev. Lett. 98231101González, J. A., Hannam, M., Sperhake, U., Brügmann, B., & Husa, S. 2007, Phys. Rev. Lett., 98, 231101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O González-Martín, J Masegosa, I Márquez, M Guainazzi, E Jiménez-Bailón, A&A. 5061107González-Martín, O., Masegosa, J., Márquez, I., Guainazzi, M., & Jiménez-Bailón, E. 2009, A&A, 506, 1107\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Grützbauch, G Trinchieri, R Rampazzo, AJ. 133220Grützbauch, R., Trinchieri, G., Rampazzo, R., et al. 2007, AJ, 133, 220\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Gualandris, D Merritt, A Gualandris, D Merritt, ApJ. 678GM0874ApJGualandris, A., & Merritt, D. 2008, ApJ, 678, 780 (GM08) Gualandris, A., & Merritt, D. 2012, ApJ, 744, 74\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Hada, A Doi, M Kino, Nature. 477185Hada, K., Doi, A., Kino, M., et al. 2011, Nature, 477, 185\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Hilz, T Naab, J P Ostriker, arXiv:1206.5004Hilz, M., Naab, T., & Ostriker, J. P. 2012, arXiv:1206.5004\",\n \"doi\": \"arXiv:1206.5004\"\n },\n {\n \"text\": \". L C Ho, ARA&A. 46475Ho, L. C. 2008, ARA&A, 46, 475\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L C Ho, A V Filippenko, W L W Sargent, C Y Peng, ApJS. 112391Ho, L. C., Filippenko, A. V., Sargent, W. L. W. & Peng, C. Y. 1997, ApJS, 112, 391\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P F Hopkins, K Bundy, D Croton, L Hernquist, D Keres, S Khochfar, K Stewart, A Wetzel, J D Younger, ApJ. 715H10202Hopkins, P. F., Bundy, K., Croton, D., Hernquist, L., Keres, D., Khochfar, S., Stewart, K., Wetzel, A. & Younger, J. D. 2010, ApJ, 715, 202 (H10)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R C W Houghton, J Magorrian, M Sarzi, MNRAS. 3672Houghton, R. C. W., Magorrian, J., Sarzi, M., et al. 2006, MNRAS, 367, 2\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J P Huchra, M J Geller, ApJ. 257423Huchra, J. P., & Geller, M. J. 1982, ApJ, 257, 423\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Huchra, M Davis, D Latham, J Tonry, ApJS. 5289Huchra, J., Davis, M., Latham, D., & Tonry, J. 1983, ApJS, 52, 89\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Jaffe, H C Ford, L Ferrarese, Van Den, F Bosch, R W Connell, Nature. 364213Jaffe, W., Ford, H. C., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1993, Nature, 364, 213\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Jaffe, B R Mcnamara, ApJ. 434110Jaffe, W., & McNamara, B. R. 1994, ApJ, 434, 110\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Jaffe, H Ford, L Ferrarese, Van Den, F Bosch, R W Connell, ApJ. 460214Jaffe, W., Ford, H., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1996, ApJ, 460, 214\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Janowiecki, J C Mihos, P Harding, ApJ. 715972Janowiecki, S., Mihos, J. C., Harding, P., et al. 2010, ApJ, 715, 972\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R I Jedrzejewski, MNRAS. 226747Jedrzejewski, R. I. 1987, MNRAS, 226, 747\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D L Jones, A E Wehrle, D L Meier, B G Piner, ApJ. 534165Jones, D. L., Wehrle, A. E., Meier, D. L., & Piner, B. G. 2000, ApJ, 534, 165\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Jones, W Forman, A Vikhlinin, ApJ. 567115Jones, C., Forman, W., Vikhlinin, A., et al. 2002, ApJ, 567, L115\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J I Katz, ApJ. 478527Katz, J. I. 1997, ApJ, 478, 527\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F M Khan, A Just, D Merritt, ApJ. 73289Khan, F. M., Just, A., & Merritt, D. 2011, ApJ, 732, 89\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E B Killeen, G V Bicknell, R D Ekers, ApJ. 302306Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1986, ApJ, 302, 306\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E B Killeen, G V Bicknell, D Carter, ApJ. 30945Killeen, N. E. B., Bicknell, G. V., & Carter, D. 1986, ApJ, 309, 45\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E B Killeen, G V Bicknell, ApJ. 324198Killeen, N. E. B., & Bicknell, G. V. 1988, ApJ, 324, 198\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E B Killeen, G V Bicknell, R D Ekers, ApJ. 325180Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1988, ApJ, 325, 180\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Komossa, H Zhou, H Lu, ApJ. 67881Komossa, S., Zhou, H., & Lu, H. 2008, ApJ, 678, L81\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Komossa, Mem. Soc. Astron. Italiana. 77733Komossa, S. 2006, Mem. Soc. Astron. Italiana, 77, 733\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Kormendy, D Richstone ; Ara&a Kornreich, D A Lovelace, R V E , 681..104KJ. Kormendy, D. Richstone, 1995, ARA&A Kornreich D. A. and Lovelace R. V. E., 2008ApJ...681..104K\",\n \"doi\": \"681..104K\"\n },\n {\n \"text\": \"J Krist, Astronomical Data Analysis Software and Systems II. 52536Krist, J. 1993, Astronomical Data Analysis Software and Systems II, 52, 536\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Kroupa, arXiv:1406.4860Kroupa, P. 2014, arXiv:1406.4860\",\n \"doi\": \"arXiv:1406.4860\"\n },\n {\n \"text\": \". S Laine, R P Van Der Marel, T R Lauer, M Postman, C P O'dea, F N Owen, AJ. 125478Laine, S., van der Marel, R. P., Lauer, T. R., Postman, M., O'Dea, C. P., & Owen, F. N. 2003, AJ, 125, 478\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, S M Faber, E J Groth, AJ. 1061436Lauer, T. R., Faber, S. M., Groth, E. J., et al. 1993, AJ, 106, 1436\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, E A Ajhar, Y.-I Byun, AJ. 1102622Lauer, T. R., Ajhar, E. A., Byun, Y.-I., et al. 1995, AJ, 110, 2622\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, S Tremaine, E A Ajhar, ApJ. 47179Lauer, T. R., Tremaine, S., Ajhar, E. A., et al. 1996, ApJ, 471, L79\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, S M Faber, K Gebhardt, AJ. 1292138Lauer, T. R., Faber, S. M., Gebhardt, K., et al. 2005, AJ, 129, 2138\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, S M Faber, D Richstone, ApJ. 662808Lauer, T. R., Faber, S. M., Richstone, D., et al. 2007, ApJ, 662, 808\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Lidman, G Iacobuta, A E Bauer, MNRAS. 433825Lidman, C., Iacobuta, G., Bauer, A. E., et al. 2013, MNRAS, 433, 825\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A P Lobanov, J Roland, A&A. 431831Lobanov, A. P., & Roland, J. 2005, A&A, 431, 831\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Loeb, Phrvl, 9941103Loeb, A., PhRvL, 99, 041103L\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Loewenstein, R F Mushotzky, L Angelini, K A Arnaud, E Quataert, ApJ. 55521Loewenstein, M., Mushotzky, R. F., Angelini, L., Arnaud, K. A., & Quataert, E. 2001, ApJ, 555, L21\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C O Lousto, H Nakano, Y Zlochower, M Campanelli, Phys. Rev. D. 8184023Lousto, C. O., Nakano, H., Zlochower, Y., & Campanelli, M. 2010, Phys. Rev. D 81, 084023\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C O Lousto, Y Zlochower, Phys. Rev. Lett. 107231102Lousto, C. O. & Zlochower, Y. 2011, Phys. Rev. Lett. 107, 231102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C O Lousto, Y Zlochower, M Dotti, M Volonteri, arXiv:1201.1923(L12Lousto, C. O., Zlochower, Y., Dotti, M., & Volonteri, M. 2012, arXiv:1201.1923 (L12)\",\n \"doi\": \"arXiv:1201.1923(L12\"\n },\n {\n \"text\": \". F Macchetto, M Pastoriza, N Caon, A&AS. 120463Macchetto, F., Pastoriza, M., Caon, N., et al. 1996, A&AS, 120, 463\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Maoz, N M Nagar, H Falcke, A S Wilson, ApJ. 625699Maoz, D., Nagar, N. M., Falcke, H., & Wilson, A. S. 2005, ApJ, 625, 699\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B R Mcnamara, ApJ. 562149McNamara, B. R., et al. 2001, ApJ, 562, L149\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E Machacek, D Jerius, R Kraft, ApJ. 74315Machacek, M. E., Jerius, D., Kraft, R., et al. 2011, ApJ, 743, 15\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Madau, E Quataert, ApJ. 60617Madau, P. and Quataert, E. 2004, ApJ, 606, 17\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D F Malin, Nature. 277279Malin, D. F. 1979, Nature, 277, 279\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D F Malin, New Aspects of Galaxy Photometry. D. F. Malin, New Aspects of Galaxy Photometry, 1985\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Mayer, S Kazantzidis, P Madau, Science. 3161874Mayer, L., Kazantzidis, S., Madau, P., et al. 2007, Science, 316, 1874\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Masegosa, I Márquez, A Ramirez, O González-Martín, A&A. 52723Masegosa, J., Márquez, I., Ramirez, A., & González-Martín, O. 2011, A&A, 527, A23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Merritt, Dynamics and Evolution of Galactic Nuclei. Princeton, NJPrinceton Univ. PressMerritt, D. 2013, Dynamics and Evolution of Galactic Nuclei (Princeton, NJ: Princeton Univ. Press)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, R D Ekers, Science. 2971310Merritt, D., & Ekers, R. D. 2002, Science, 297, 1310\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, M Milosavljević, M Favata, S A Hughes, D E Holz, ApJ. 6079Merritt, D., Milosavljević, M., Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ 607, L9.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, M Milosavljević, Living Reviews in Relativity. 88Merritt, D., & Milosavljević, M. 2005, Living Reviews in Relativity, 8, 8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, ApJ. 648976Merritt, D. 2006, ApJ, 648, 976\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, J D Schnittman, S Komossa, ApJ. 6991690Merritt, D., Schnittman, J. D., & Komossa, S. 2009, ApJ, 699, 1690\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Y Mills, O B Slee, E R Hill, Australian Journal of Physics. 13676Mills, B. Y., Slee, O. B., & Hill, E. R. 1960, Australian Journal of Physics, 13, 676\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Moellenhoff, E Hummel, R Bender, A&A. 25535Moellenhoff, C., Hummel, E., & Bender, R. 1992, A&A, 255, 35\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N M Nagar, H Falcke, A S Wilson, J S Ulvestad, A&A. 39253Nagar, N. M., Falcke, H., Wilson, A. S., & Ulvestad, J. S. 2002, A&A, 392, 53\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Nolthenius, ApJS. 851Nolthenius, R. 1993, ApJS, 85, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R W O'connell, J R Martin, J D Crane, ApJ. 635305O'Connell, R. W., Martin, J. R., Crane, J. D., et al. 2005, ApJ, 635, 305\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C P O'dea, R A Daly, P Kharb, K A Freeman, S A Baum, A&A. 494471O'Dea, C. P., Daly, R. A., Kharb, P., Freeman, K. A. & Baum, S. A. 2009, A&A, 494, 471\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Parma, M Murgia, R Morganti, A Capetti, H R De Ruiter, R Fanti, A&A. 3447Parma, P., Murgia, M., Morganti, R., Capetti, A., de Ruiter, H. R. & Fanti, R. 1999, A&A, 344, 7\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Pellegrini, T Venturi, A Comastri, ApJ. 585677Pellegrini, S., Venturi, T., Comastri, A., et al. 2003, ApJ, 585, 677\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H V Peiris, S Tremaine, ApJ. Peiris, H. V. & Tremaine, S. 2003, ApJ, 237-257\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Y Peng, L C Ho, C D Impey, H.-W Rix, AJ. 124266Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H B Perets, C Hopman, T Alexander, ApJ. 656709Perets, H. B., Hopman, C., & Alexander, T. 2007, ApJ, 656, 709\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E S Perlman, D E Harris, J A Biretta, W B Sparks, F D Macchetto, ApJ. 59965Perlman, E. S., Harris, D. E., Biretta, J. A., Sparks, W. B., & Macchetto, F. D. 2003, ApJ, 599, L65\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Postman, T R Lauer, ApJ. 44028Postman, M., & Lauer, T. R. 1995, ApJ, 440, 28\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Preto, I Berentzen, P Berczik, R Spurzem, ApJ. 73226Preto, M., Berentzen, I., Berczik, P., & Spurzem, R. 2011, ApJ, 732, L26\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Pretorius, Phys. Rev. Lett. 95121101Pretorius, F. 2005, Phys. Rev. Lett. 95, 121101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Renzini, L Greggio, S Di Serego Alighieri, Nature. 37839Renzini, A., Greggio, L., di Serego Alighieri, S., et al. 1995, Nature, 378, 39\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Robinson, S Young, D J Axon, P Kharb, J E Smith, ApJ. 717122Robinson, A., Young, S., Axon, D. J., Kharb, P., & Smith, J. E. 2010, ApJ, 717, L122\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G E Romero, L Chajet, Z Abraham, J H Fan, A&A. 36057Romero, G. E., Chajet, L., Abraham, Z., & Fan, J. H. 2000, A&A, 360, 57\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Roos, J S Kaastra, C A Hummel, ApJ. 409130Roos, N., Kaastra, J. S., & Hummel, C. A. 1993, ApJ, 409, 130\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E M Sadler, N A Sharp, A&A. 133216Sadler, E. M., & Sharp, N. A. 1984, A&A, 133, 216\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E M Sadler, C R Jenkins, C G Kotanyi, MNRAS. 240591Sadler, E. M., Jenkins, C. R., & Kotanyi, C. G. 1989, MNRAS, 240, 591\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Sani, A Marconi, L K Hunt, G Risaliti, MNRAS. 4131479Sani, E., Marconi, A., Hunt, L. K., & Risaliti, G. 2011, MNRAS, 413, 1479\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W C Saslaw, M Whittle, ApJ. 325142Saslaw, W. C., & Whittle, M. 1988, ApJ, 325, 142\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Extragalactic Radio Sources. I S Shklovsky, 97475Shklovsky, I. S. 1982, Extragalactic Radio Sources, 97, 475\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G A Shields, D J Rosario, K L Smith, ApJ. 707936Shields, G. A., Rosario, D. J., Smith, K. L., et al. 2009, ApJ, 707, 936\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Shurkin, R J H Dunn, G Gentile, G B Taylor, S W Allen, MNRAS. 383923Shurkin, K., Dunn, R. J. H., Gentile, G., Taylor, G. B., & Allen, S. W. 2008, MNRAS, 383, 923\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Sijacki, V Springel, M G Haehnelt, MNRAS. 4143656Sijacki, D., Springel, V. & Haehnelt, M. G. 2011, MNRAS, 414, 3656\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O B Slee, E M Sadler, J E Reynolds, R D Ekers, MNRAS. 269928Slee, O. B., Sadler, E. M., Reynolds, J. E., & Ekers, R. D. 1994, MNRAS, 269, 928\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W B Sparks, F Macchetto, D Golombek, ApJ. 345153Sparks, W. B., Macchetto, F., & Golombek, D. 1989, ApJ, 345, 153\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Springel, S D M White, A Jenkins, Nature. 435629Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V J Stanger, R S Warwick, MNRAS. 220363Stanger, V. J., & Warwick, R. S. 1986, MNRAS, 220, 363\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C L Steinhardt, M Schramm, J D Silverman, ApJ. 75924Steinhardt, C. L., Schramm, M., Silverman, J. D., et al. 2012, ApJ, 759, 24\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Subrahmanyan, A J Beasley, W M Goss, K Golap, R W Hunstead, AJ. 1251095Subrahmanyan, R., Beasley, A. J., Goss, W. M., Golap, K., & Hunstead, R. W. 2003, AJ, 125, 1095\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G B Taylor, J S Sanders, A C Fabian, S W Allen, MNRAS. 365705Taylor, G. B., Sanders, J. S., Fabian, A. C., & Allen, S. W. 2006, MNRAS, 365, 705\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Tal, P G Van Dokkum, J Nelan, R Bezanson, AJ. 1381417Tal, T., van Dokkum, P. G., Nelan, J., & Bezanson, R. 2009, AJ, 138, 1417\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Tarchi, N Neininger, A Greve, A&A. 35895Tarchi, A., Neininger, N., Greve, A., et al. 2000, A&A, 358, 95\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Thatte, M Tecza, R Genzel, A&A. 36447Thatte, N., Tecza, M., & Genzel, R. 2000, A&A, 364, L47\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Temi, W G Mathews, F Brighenti, J D Bregman, ApJ. 585121Temi, P., Mathews, W. G., Brighenti, F., & Bregman, J. D. 2003, ApJ, 585, L121\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Tichy, P Marronetti, Phys. Rev. D. 061502. Trinchieri, G., & di Serego Alighieri761647AJTichy, W. & Marronetti, P. 2007, Phys. Rev. D 76, 061502. Trinchieri, G., & di Serego Alighieri, S. 1991, AJ, 101, 1647\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Tsalmantza, R Decarli, M Dotti, D W Hogg, 73820Tsalmantza, P., Decarli, R., Dotti, M. & Hogg, D. W., 2011ApJ, 738, 20\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Vasiliev, F Antonini, D Merritt, P G Van Dokkum, M Franx, ApJ. 7852027AJVasiliev, E., Antonini, F., & Merritt, D. 2014, ApJ, 785, 163 van Dokkum, P. G., & Franx, M. 1995, AJ, 110, 2027\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G A Verdoes Kleijn, R P Van Der Marel, C M Carollo, P T De Zeeuw, AJ. 1201221Verdoes Kleijn, G. A., van der Marel, R. P., Carollo, C. M., & de Zeeuw, P. T. 2000, AJ, 120, 1221\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Volonteri, K Gúltekin, M Dotti, 4042143Volonteri, M.; Gúltekin, K.; Dotti, M., 2010MNRAS, 404, 2143\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D M Worrall, M Birkinshaw, E O'sullivan, A Zezas, A Wolter, G Trinchieri, G Fabbiano, MNRAS. 408701Worrall, D. M., Birkinshaw, M., O'Sullivan, E., Zezas, A., Wolter, A., Trinchieri, G., & Fabbiano, G. 2010, MNRAS, 408, 701\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J M Wrobel, AJ. 101127Wrobel, J. M. 1991, AJ, 101, 127\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J M Wrobel, D S Heeschen, AJ. 101148Wrobel, J. M., & Heeschen, D. S. 1991, AJ, 101, 148\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Younis, J Meaburn, P Stewart, A&A. 147178Younis, S., Meaburn, J., & Stewart, P. 1985, A&A, 147, 178\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Zanotti, L Rezzolla, L Del Zanna, C Palenzuela, A&A. 5238Zanotti, O., Rezzolla, L., Del Zanna, L., & Palenzuela, C. 2010, A&A, 523, A8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G O Abell, H G Corwin, Jr, R P Olowin, ApJS. 701Abell, G. O., Corwin, H. G., Jr., & Olowin, R. P. 1989, ApJS, 70, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D M Alexander, R C Hickox, NAR. 5693Alexander, D. M., & Hickox, R. C. 2012, NAR, 56, 93\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J M Anderson, J S Ulvestad, L C Ho, ApJ. 60342Anderson, J. M., Ulvestad, J. S., & Ho, L. C. 2004, ApJ, 603, 42\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Antognini, J Bird, P Martini, ApJ. 756116Antognini, J., Bird, J. & Martini, P. 2012, ApJ, 756, 116\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Antonini, D Merritt, ApJ. 74583Antonini, F., & Merritt, D. 2012, ApJ, 745, 83\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Baade, R Minkowski, ApJ. 119215Baade, W., & Minkowski, R. 1954, ApJ, 119, 215\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J N Bahcall, R A Wolf, ApJ. 209214Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R D Baldi, M Chiaberge, A Capetti, W Sparks, F D Macchetto, C P O'dea, D J Axon, S A Baum, A C Quillen, ApJ. 7252426Baldi, R. D., Chiaberge, M., Capetti, A., Sparks, W., Macchetto, F. D., O'Dea, C. P., Axon, D. J., Baum, S. A. & Quillen, A. C. 2010, ApJ, 725, 2426\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Balmaverde, R D Baldi, A Capetti, A&A. 486119Balmaverde, B., Baldi, R. D., & Capetti, A. 2008, A&A, 486, 119\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Balmaverde, A Capetti, A&A. 447BC0697Balmaverde, B., & Capetti, A. 2006, A&A, 447, 97 (BC06)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J G Baker, J Centrella, D.-I Choi, M Koppitz, J Van Meter, Phys. Rev. Lett. 96111102Baker, J. G. , Centrella, J. , Choi,D.-I., Koppitz, M., & van Meter, J. 2006, Phys. Rev. Lett. 96, 111102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Batcheldor, A Robinson, D J Axon, E S Perlman, D Merritt, ApJ. 717B106Batcheldor, D., Robinson, A., Axon, D. J., Perlman, E. S., & Merritt, D. 2010, ApJ, 717, L6 (B10)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M C Begelman, R D Blandford, M J Rees, Nature. 287307Begelman, M.C., Blandford, R.D. & Rees, M.J., 1980, Nature, 287, 307\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Bekenstein, ApJ. 183657Bekenstein, J. D. 1973, ApJ, 183, 657\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E F Bell, S Phleps, R S Somerville, C Wolf, A Borch, K Meisenheimer, ApJ. 652270Bell, E. F., Phleps, S., Somerville, R. S., Wolf, C., Borch, A. & Meisenheimer, K. 2006, ApJ, 652, 270\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Birkinshaw, R L Davies, ApJ. 29132Birkinshaw, M., & Davies, R. L. 1985, ApJ, 291, 32\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R D Blandford, R L Znajek, MNRAS. 179433Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Blecha, F Civano, M Elvis, A Loeb, MNRAS. 4281341Blecha, L., Civano, F., Elvis, M. & Loeb, A. 2013, MNRAS, 428,1341\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E W Bonning, G A Shields, S Salviander, ApJ. 66613Bonning, E. W., Shields, G. A. & Salviander, S. 2007, ApJ, 666, 13\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M J I Brown, B T Jannuzi, D J E Floyd, J Mould, R , ApJ. 73141Brown, M. J. I., Jannuzi, B. T., Floyd, D. J. E., Mould, J., R. 2011, ApJ, 731, L41\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Burke, C A Collins, MNRAS. 4342856Burke, C. & Collins, C. A., 2013, MNRAS, 434, 2856\",\n \"doi\": \"\"\n },\n {\n \"text\": \"I C Busko, Astronomical Data Analysis Software and Systems V. 101139Busko, I. C. 1996, Astronomical Data Analysis Software and Systems V, 101, 139\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Campanelli, C O Lousto, P Marronetti, Y Zlochower, Phys. Rev. Lett. 96111101Campanelli, M., Lousto, C. O., Marronetti, P., & Zlochower, Y. 2006, Phys. Rev. Lett. 96, 111101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Campanelli, C O Lousto, Y Zlochower, D Merritt, Physical Review Letters. 98231102Campanelli, M., Lousto, C. O., Zlochower, Y., & Merritt, D. 2007, Physical Review Letters, 98, 231102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Campanelli, C Lousto, Y Zlochower, D Merritt, ApJ. 6595Campanelli, M., Lousto, C., Zlochower, Y., & Merritt, D. 2007, ApJ, 659, L5\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E A Canning, A C Fabian, R M Johnstone, J S Sanders, C S Crawford, G J Ferland, N A Hatch, MNRAS. 4173080Canning, R.E.A., Fabian, A. C., Johnstone, R. M., Sanders, J. S., Crawford, C. S., Ferland, G. J., Hatch, N. A. 2011, MNRAS, 417, 3080\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Capetti, D J Axon, M Chiaberge, A&A. 471137Capetti, A., Axon, D. J., Chiaberge, M., et al. 2007, A&A, 471, 137\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Capetti, B Balmaverde, A&A. 440CB0573Capetti, A., & Balmaverde, B. 2005, A&A, 440, 73 (CB05)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Capetti, B Balmaverde, A&A. 453CB0627Capetti, A., & Balmaverde, B. 2006, A&A, 453, 27 (CB06)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Cappellari, A Renzini, L Greggio, ApJ. 519117Cappellari, M., Renzini, A., Greggio, L., et al. 1999, ApJ, 519, 117\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Cappellari, E K Verolme, R P Van Der Marel, ApJ. 578787Cappellari, M., Verolme, E. K., van der Marel, R. P., et al. 2002, ApJ, 578, 787\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Cappellari, E Emsellem, R Bacon, MNRAS. 379418Cappellari, M., Emsellem, E., Bacon, R., et al. 2007, MNRAS, 379, 418\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Cardullo, E M Corsini, A Beifiori, A&A. 508641Cardullo, A., Corsini, E. M., Beifiori, A., et al. 2009, A&A, 508, 641\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C M Carollo, M Franx, G D Illingworth, D A Forbes, ApJ. 481710Carollo, C. M., Franx, M., Illingworth, G. D., & Forbes, D. A. 1997, ApJ, 481, 710\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K W Cavagnolo, B R Mcnamara, P E J Nulsen, ApJ. 7201066Cavagnolo, K. W., McNamara, B. R., Nulsen, P. E. J., et al. 2010, ApJ, 720, 1066\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Chiaberge, A Capetti, A Celotti, A&A. 34977Chiaberge, M., Capetti, A., & Celotti, A. 1999, A&A, 349, 77\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Civano, M Elvis, G Lanzuisi, ApJ. 717209Civano, F., Elvis, M., Lanzuisi, G., et al. 2010, ApJ, 717, 209\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Civano, M Elvis, G Lanzuisi, ApJ. 75249Civano, F., Elvis, M., Lanzuisi, G., et al. 2012, ApJ, 752, 49\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P Coles, F Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure. New YorkWiley2nd ed.Coles, P., & Lucchin, F. 2002, Cosmology: The Origin and Evolution of Cosmic Structure (2nd ed.; New York: Wiley)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C J Conselice, C Yang, A F L Bluck, MNRAS. 3941956Conselice, C. J., Yang, C. & Bluck, A. F. L. 2009, MNRAS, 394, 1956\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Cuadra, P J Armitage, R D Alexander, M C Begelman, MNRAS. 3931423Cuadra, J., Armitage, P. J. Alexander, R. D. & Begelman, M. C. 2009, MNRAS, 393, 1423\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V P Debattista, I Ferreras, A Pasquali, ApJ. 65197Debattista, V. P., Ferreras, I., Pasquali, A., et al. 2006, ApJ, 651, L97\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Third Reference Catalogue of Bright Galaxies Di Matteo. G De Vaucouleurs, A De Vaucouleurs, H G Corwin, Jr, ApJ. 582133de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Jr., et al. 1991, Third Reference Catalogue of Bright Galaxies Di Matteo, T., Allen, S. W., Fabian, A. C., Wilson, A. S., & Young, A. J. 2003, ApJ, 582, 133\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Dotti, M Volonteri, A Perego, M Colpi, M Ruszkowski, F Haardt, MNRAS. 402682Dotti, M., Volonteri, M., Perego, A., Colpi, M., Ruszkowski, M. & Haardt, F. 2010, MNRAS, 402, 682\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Dressler, D Lynden-Bell, D Burstein, ApJ. 31342Dressler, A., Lynden-Bell, D., Burstein, D., et al. 1987, ApJ, 313, 42\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Efstathiou, R S Ellis, D Carter, MNRAS. 193931Efstathiou, G., Ellis, R. S., & Carter, D. 1980, MNRAS, 193, 931\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Eracleous, T A Boroson, J P Halpern, J Liu, ApJS. 20123Eracleous, M., Boroson, T. A., Halpern, J. P., & Liu, J. 2012, ApJS, 201, 23\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Escala, R B Larson, P S Coppi, D Mardones, ApJ. 630152Escala, A., Larson, R. B., Coppi, P. S. & Mardones, D. 2005, ApJ, 630, 152\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Fabbiano, M Elvis, S Markoff, ApJ. 588175Fabbiano, G., Elvis, M., Markoff, S., et al. 2003, ApJ, 588, 175\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S M Faber, S Tremaine, E A Ajhar, AJ. 1141771Faber, S. M., Tremaine, S., Ajhar, E. A., et al. 1997, AJ, 114, 1771\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B L Fanaroff, J M Riley, MNRAS. 16731Fanaroff, B. L., & Riley, J. M. 1974, MNRAS, 167, 31P\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C L Farage, P J Mcgregor, M A Dopita, G V Bicknell, ApJ. 724267Farage, C. L., McGregor, P. J., Dopita, M. A., & Bicknell, G. V. 2010, ApJ, 724, 267\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Favata, S A Hughes, D E Holz, ApJ. 6075Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ, 607, L5\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H C Ferguson, AJ. 98367Ferguson, H. C. 1989, AJ, 98, 367\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ferrarese, H C Ford, W Jaffe, ApJ. 470444Ferrarese, L., Ford, H. C., & Jaffe, W. 1996, ApJ, 470, 444\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ferrarese, D Merritt, ApJ. 5399Ferrarese, L., & Merritt, D. 2000, ApJ, 539, L9\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ferrarese, H Ford, Sci. Rev. 116523Ferrarese, L., Ford, H., 2005, Sci. Rev., 116, 523\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E Filho, P D Barthel, L C Ho, ApJS. 12993Filho, M. E., Barthel, P. D., & Ho, L. C. 2000, ApJS, 129, 93\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E Filho, F Fraternali, S Markoff, A&A. 418429Filho, M. E., Fraternali, F., Markoff, S., et al. 2004, A&A, 418, 429\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D A Forbes, M Franx, G D Illingworth, ApJ. 42849Forbes, D. A., Franx, M., & Illingworth, G. D. 1994, ApJ, 428, L49\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D A Forbes, J P Brodie, J Huchra, AJ. 1122448Forbes, D. A., Brodie, J. P., & Huchra, J. 1996, AJ, 112, 2448\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D A Forbes, D B Reitzel, G M Williger, AJ. 1091576Forbes, D. A., Reitzel, D. B., & Williger, G. M. 1995, AJ, 109, 1576\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Franx, G D Illingworth, ApJ. 32755Franx, M., & Illingworth, G. D. 1988, ApJ, 327, L55\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Fakhouri, C.-P Ma, M Boylan-Kolchin, MNRAS. 4062267Fakhouri, O., Ma, C.-P., & Boylan-Kolchin, M. 2010, MNRAS, 406, 2267\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Gebhardt, R Bender, G Bower, ApJ. 53913Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Gebhardt, T R Lauer, J Pinkney, ApJ. 6711321Gebhardt, K., Lauer, T. R., Pinkney, J., et al. 2007, ApJ, 671, 1321\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Gebhardt, J Adams, D Richstone, ApJ. 729119Gebhardt, K., Adams, J., Richstone, D., et al. 2011, ApJ, 729, 119\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Gerin, F Casoli, A&A. 29049Gerin, M., & Casoli, F. 1994, A&A, 290, 49\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Giroletti, G B Taylor, G Giovannini, ApJ. 622178Giroletti, M., Taylor, G. B., & Giovannini, G. 2005, ApJ, 622, 178\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Giordano, L Cortese, G Trinchieri, A Wolter, M Colpi, G Gavazzi, L Mayer, ApJ. 634272Giordano, L., Cortese, L., Trinchieri, G., Wolter, A., Colpi, M., Gavazzi, G., & Mayer, L. 2005, ApJ, 634, 272\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J A González, M Hannam, U Sperhake, B Brügmann, S Husa, Phys. Rev. Lett. 98231101González, J. A., Hannam, M., Sperhake, U., Brügmann, B., & Husa, S. 2007, Phys. Rev. Lett., 98, 231101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O González-Martín, J Masegosa, I Márquez, M Guainazzi, E Jiménez-Bailón, A&A. 5061107González-Martín, O., Masegosa, J., Márquez, I., Guainazzi, M., & Jiménez-Bailón, E. 2009, A&A, 506, 1107\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Grützbauch, G Trinchieri, R Rampazzo, AJ. 133220Grützbauch, R., Trinchieri, G., Rampazzo, R., et al. 2007, AJ, 133, 220\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Gualandris, D Merritt, A Gualandris, D Merritt, ApJ. 678GM0874ApJGualandris, A., & Merritt, D. 2008, ApJ, 678, 780 (GM08) Gualandris, A., & Merritt, D. 2012, ApJ, 744, 74\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Hada, A Doi, M Kino, Nature. 477185Hada, K., Doi, A., Kino, M., et al. 2011, Nature, 477, 185\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Hilz, T Naab, J P Ostriker, arXiv:1206.5004Hilz, M., Naab, T., & Ostriker, J. P. 2012, arXiv:1206.5004\",\n \"doi\": \"arXiv:1206.5004\"\n },\n {\n \"text\": \". L C Ho, ARA&A. 46475Ho, L. C. 2008, ARA&A, 46, 475\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L C Ho, A V Filippenko, W L W Sargent, C Y Peng, ApJS. 112391Ho, L. C., Filippenko, A. V., Sargent, W. L. W. & Peng, C. Y. 1997, ApJS, 112, 391\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P F Hopkins, K Bundy, D Croton, L Hernquist, D Keres, S Khochfar, K Stewart, A Wetzel, J D Younger, ApJ. 715H10202Hopkins, P. F., Bundy, K., Croton, D., Hernquist, L., Keres, D., Khochfar, S., Stewart, K., Wetzel, A. & Younger, J. D. 2010, ApJ, 715, 202 (H10)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R C W Houghton, J Magorrian, M Sarzi, MNRAS. 3672Houghton, R. C. W., Magorrian, J., Sarzi, M., et al. 2006, MNRAS, 367, 2\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J P Huchra, M J Geller, ApJ. 257423Huchra, J. P., & Geller, M. J. 1982, ApJ, 257, 423\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Huchra, M Davis, D Latham, J Tonry, ApJS. 5289Huchra, J., Davis, M., Latham, D., & Tonry, J. 1983, ApJS, 52, 89\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Jaffe, H C Ford, L Ferrarese, Van Den, F Bosch, R W Connell, Nature. 364213Jaffe, W., Ford, H. C., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1993, Nature, 364, 213\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Jaffe, B R Mcnamara, ApJ. 434110Jaffe, W., & McNamara, B. R. 1994, ApJ, 434, 110\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Jaffe, H Ford, L Ferrarese, Van Den, F Bosch, R W Connell, ApJ. 460214Jaffe, W., Ford, H., Ferrarese, L., van den Bosch, F., & O'Connell, R. W. 1996, ApJ, 460, 214\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Janowiecki, J C Mihos, P Harding, ApJ. 715972Janowiecki, S., Mihos, J. C., Harding, P., et al. 2010, ApJ, 715, 972\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R I Jedrzejewski, MNRAS. 226747Jedrzejewski, R. I. 1987, MNRAS, 226, 747\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D L Jones, A E Wehrle, D L Meier, B G Piner, ApJ. 534165Jones, D. L., Wehrle, A. E., Meier, D. L., & Piner, B. G. 2000, ApJ, 534, 165\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Jones, W Forman, A Vikhlinin, ApJ. 567115Jones, C., Forman, W., Vikhlinin, A., et al. 2002, ApJ, 567, L115\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J I Katz, ApJ. 478527Katz, J. I. 1997, ApJ, 478, 527\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F M Khan, A Just, D Merritt, ApJ. 73289Khan, F. M., Just, A., & Merritt, D. 2011, ApJ, 732, 89\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E B Killeen, G V Bicknell, R D Ekers, ApJ. 302306Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1986, ApJ, 302, 306\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E B Killeen, G V Bicknell, D Carter, ApJ. 30945Killeen, N. E. B., Bicknell, G. V., & Carter, D. 1986, ApJ, 309, 45\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E B Killeen, G V Bicknell, ApJ. 324198Killeen, N. E. B., & Bicknell, G. V. 1988, ApJ, 324, 198\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E B Killeen, G V Bicknell, R D Ekers, ApJ. 325180Killeen, N. E. B., Bicknell, G. V., & Ekers, R. D. 1988, ApJ, 325, 180\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Komossa, H Zhou, H Lu, ApJ. 67881Komossa, S., Zhou, H., & Lu, H. 2008, ApJ, 678, L81\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Komossa, Mem. Soc. Astron. Italiana. 77733Komossa, S. 2006, Mem. Soc. Astron. Italiana, 77, 733\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Kormendy, D Richstone ; Ara&a Kornreich, D A Lovelace, R V E , 681..104KJ. Kormendy, D. Richstone, 1995, ARA&A Kornreich D. A. and Lovelace R. V. E., 2008ApJ...681..104K\",\n \"doi\": \"681..104K\"\n },\n {\n \"text\": \"J Krist, Astronomical Data Analysis Software and Systems II. 52536Krist, J. 1993, Astronomical Data Analysis Software and Systems II, 52, 536\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Kroupa, arXiv:1406.4860Kroupa, P. 2014, arXiv:1406.4860\",\n \"doi\": \"arXiv:1406.4860\"\n },\n {\n \"text\": \". S Laine, R P Van Der Marel, T R Lauer, M Postman, C P O'dea, F N Owen, AJ. 125478Laine, S., van der Marel, R. P., Lauer, T. R., Postman, M., O'Dea, C. P., & Owen, F. N. 2003, AJ, 125, 478\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, S M Faber, E J Groth, AJ. 1061436Lauer, T. R., Faber, S. M., Groth, E. J., et al. 1993, AJ, 106, 1436\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, E A Ajhar, Y.-I Byun, AJ. 1102622Lauer, T. R., Ajhar, E. A., Byun, Y.-I., et al. 1995, AJ, 110, 2622\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, S Tremaine, E A Ajhar, ApJ. 47179Lauer, T. R., Tremaine, S., Ajhar, E. A., et al. 1996, ApJ, 471, L79\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, S M Faber, K Gebhardt, AJ. 1292138Lauer, T. R., Faber, S. M., Gebhardt, K., et al. 2005, AJ, 129, 2138\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R Lauer, S M Faber, D Richstone, ApJ. 662808Lauer, T. R., Faber, S. M., Richstone, D., et al. 2007, ApJ, 662, 808\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Lidman, G Iacobuta, A E Bauer, MNRAS. 433825Lidman, C., Iacobuta, G., Bauer, A. E., et al. 2013, MNRAS, 433, 825\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A P Lobanov, J Roland, A&A. 431831Lobanov, A. P., & Roland, J. 2005, A&A, 431, 831\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Loeb, Phrvl, 9941103Loeb, A., PhRvL, 99, 041103L\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Loewenstein, R F Mushotzky, L Angelini, K A Arnaud, E Quataert, ApJ. 55521Loewenstein, M., Mushotzky, R. F., Angelini, L., Arnaud, K. A., & Quataert, E. 2001, ApJ, 555, L21\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C O Lousto, H Nakano, Y Zlochower, M Campanelli, Phys. Rev. D. 8184023Lousto, C. O., Nakano, H., Zlochower, Y., & Campanelli, M. 2010, Phys. Rev. D 81, 084023\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C O Lousto, Y Zlochower, Phys. Rev. Lett. 107231102Lousto, C. O. & Zlochower, Y. 2011, Phys. Rev. Lett. 107, 231102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C O Lousto, Y Zlochower, M Dotti, M Volonteri, arXiv:1201.1923(L12Lousto, C. O., Zlochower, Y., Dotti, M., & Volonteri, M. 2012, arXiv:1201.1923 (L12)\",\n \"doi\": \"arXiv:1201.1923(L12\"\n },\n {\n \"text\": \". F Macchetto, M Pastoriza, N Caon, A&AS. 120463Macchetto, F., Pastoriza, M., Caon, N., et al. 1996, A&AS, 120, 463\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Maoz, N M Nagar, H Falcke, A S Wilson, ApJ. 625699Maoz, D., Nagar, N. M., Falcke, H., & Wilson, A. S. 2005, ApJ, 625, 699\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B R Mcnamara, ApJ. 562149McNamara, B. R., et al. 2001, ApJ, 562, L149\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E Machacek, D Jerius, R Kraft, ApJ. 74315Machacek, M. E., Jerius, D., Kraft, R., et al. 2011, ApJ, 743, 15\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Madau, E Quataert, ApJ. 60617Madau, P. and Quataert, E. 2004, ApJ, 606, 17\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D F Malin, Nature. 277279Malin, D. F. 1979, Nature, 277, 279\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D F Malin, New Aspects of Galaxy Photometry. D. F. Malin, New Aspects of Galaxy Photometry, 1985\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Mayer, S Kazantzidis, P Madau, Science. 3161874Mayer, L., Kazantzidis, S., Madau, P., et al. 2007, Science, 316, 1874\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Masegosa, I Márquez, A Ramirez, O González-Martín, A&A. 52723Masegosa, J., Márquez, I., Ramirez, A., & González-Martín, O. 2011, A&A, 527, A23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Merritt, Dynamics and Evolution of Galactic Nuclei. Princeton, NJPrinceton Univ. PressMerritt, D. 2013, Dynamics and Evolution of Galactic Nuclei (Princeton, NJ: Princeton Univ. Press)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, R D Ekers, Science. 2971310Merritt, D., & Ekers, R. D. 2002, Science, 297, 1310\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, M Milosavljević, M Favata, S A Hughes, D E Holz, ApJ. 6079Merritt, D., Milosavljević, M., Favata, M., Hughes, S. A. & Holz, D. E. 2004, ApJ 607, L9.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, M Milosavljević, Living Reviews in Relativity. 88Merritt, D., & Milosavljević, M. 2005, Living Reviews in Relativity, 8, 8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, ApJ. 648976Merritt, D. 2006, ApJ, 648, 976\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Merritt, J D Schnittman, S Komossa, ApJ. 6991690Merritt, D., Schnittman, J. D., & Komossa, S. 2009, ApJ, 699, 1690\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Y Mills, O B Slee, E R Hill, Australian Journal of Physics. 13676Mills, B. Y., Slee, O. B., & Hill, E. R. 1960, Australian Journal of Physics, 13, 676\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Moellenhoff, E Hummel, R Bender, A&A. 25535Moellenhoff, C., Hummel, E., & Bender, R. 1992, A&A, 255, 35\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N M Nagar, H Falcke, A S Wilson, J S Ulvestad, A&A. 39253Nagar, N. M., Falcke, H., Wilson, A. S., & Ulvestad, J. S. 2002, A&A, 392, 53\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Nolthenius, ApJS. 851Nolthenius, R. 1993, ApJS, 85, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R W O'connell, J R Martin, J D Crane, ApJ. 635305O'Connell, R. W., Martin, J. R., Crane, J. D., et al. 2005, ApJ, 635, 305\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C P O'dea, R A Daly, P Kharb, K A Freeman, S A Baum, A&A. 494471O'Dea, C. P., Daly, R. A., Kharb, P., Freeman, K. A. & Baum, S. A. 2009, A&A, 494, 471\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Parma, M Murgia, R Morganti, A Capetti, H R De Ruiter, R Fanti, A&A. 3447Parma, P., Murgia, M., Morganti, R., Capetti, A., de Ruiter, H. R. & Fanti, R. 1999, A&A, 344, 7\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Pellegrini, T Venturi, A Comastri, ApJ. 585677Pellegrini, S., Venturi, T., Comastri, A., et al. 2003, ApJ, 585, 677\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H V Peiris, S Tremaine, ApJ. Peiris, H. V. & Tremaine, S. 2003, ApJ, 237-257\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Y Peng, L C Ho, C D Impey, H.-W Rix, AJ. 124266Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H B Perets, C Hopman, T Alexander, ApJ. 656709Perets, H. B., Hopman, C., & Alexander, T. 2007, ApJ, 656, 709\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E S Perlman, D E Harris, J A Biretta, W B Sparks, F D Macchetto, ApJ. 59965Perlman, E. S., Harris, D. E., Biretta, J. A., Sparks, W. B., & Macchetto, F. D. 2003, ApJ, 599, L65\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Postman, T R Lauer, ApJ. 44028Postman, M., & Lauer, T. R. 1995, ApJ, 440, 28\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Preto, I Berentzen, P Berczik, R Spurzem, ApJ. 73226Preto, M., Berentzen, I., Berczik, P., & Spurzem, R. 2011, ApJ, 732, L26\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Pretorius, Phys. Rev. Lett. 95121101Pretorius, F. 2005, Phys. Rev. Lett. 95, 121101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Renzini, L Greggio, S Di Serego Alighieri, Nature. 37839Renzini, A., Greggio, L., di Serego Alighieri, S., et al. 1995, Nature, 378, 39\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Robinson, S Young, D J Axon, P Kharb, J E Smith, ApJ. 717122Robinson, A., Young, S., Axon, D. J., Kharb, P., & Smith, J. E. 2010, ApJ, 717, L122\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G E Romero, L Chajet, Z Abraham, J H Fan, A&A. 36057Romero, G. E., Chajet, L., Abraham, Z., & Fan, J. H. 2000, A&A, 360, 57\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Roos, J S Kaastra, C A Hummel, ApJ. 409130Roos, N., Kaastra, J. S., & Hummel, C. A. 1993, ApJ, 409, 130\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E M Sadler, N A Sharp, A&A. 133216Sadler, E. M., & Sharp, N. A. 1984, A&A, 133, 216\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E M Sadler, C R Jenkins, C G Kotanyi, MNRAS. 240591Sadler, E. M., Jenkins, C. R., & Kotanyi, C. G. 1989, MNRAS, 240, 591\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Sani, A Marconi, L K Hunt, G Risaliti, MNRAS. 4131479Sani, E., Marconi, A., Hunt, L. K., & Risaliti, G. 2011, MNRAS, 413, 1479\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W C Saslaw, M Whittle, ApJ. 325142Saslaw, W. C., & Whittle, M. 1988, ApJ, 325, 142\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Extragalactic Radio Sources. I S Shklovsky, 97475Shklovsky, I. S. 1982, Extragalactic Radio Sources, 97, 475\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G A Shields, D J Rosario, K L Smith, ApJ. 707936Shields, G. A., Rosario, D. J., Smith, K. L., et al. 2009, ApJ, 707, 936\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Shurkin, R J H Dunn, G Gentile, G B Taylor, S W Allen, MNRAS. 383923Shurkin, K., Dunn, R. J. H., Gentile, G., Taylor, G. B., & Allen, S. W. 2008, MNRAS, 383, 923\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Sijacki, V Springel, M G Haehnelt, MNRAS. 4143656Sijacki, D., Springel, V. & Haehnelt, M. G. 2011, MNRAS, 414, 3656\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O B Slee, E M Sadler, J E Reynolds, R D Ekers, MNRAS. 269928Slee, O. B., Sadler, E. M., Reynolds, J. E., & Ekers, R. D. 1994, MNRAS, 269, 928\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W B Sparks, F Macchetto, D Golombek, ApJ. 345153Sparks, W. B., Macchetto, F., & Golombek, D. 1989, ApJ, 345, 153\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Springel, S D M White, A Jenkins, Nature. 435629Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V J Stanger, R S Warwick, MNRAS. 220363Stanger, V. J., & Warwick, R. S. 1986, MNRAS, 220, 363\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C L Steinhardt, M Schramm, J D Silverman, ApJ. 75924Steinhardt, C. L., Schramm, M., Silverman, J. D., et al. 2012, ApJ, 759, 24\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Subrahmanyan, A J Beasley, W M Goss, K Golap, R W Hunstead, AJ. 1251095Subrahmanyan, R., Beasley, A. J., Goss, W. M., Golap, K., & Hunstead, R. W. 2003, AJ, 125, 1095\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G B Taylor, J S Sanders, A C Fabian, S W Allen, MNRAS. 365705Taylor, G. B., Sanders, J. S., Fabian, A. C., & Allen, S. W. 2006, MNRAS, 365, 705\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Tal, P G Van Dokkum, J Nelan, R Bezanson, AJ. 1381417Tal, T., van Dokkum, P. G., Nelan, J., & Bezanson, R. 2009, AJ, 138, 1417\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Tarchi, N Neininger, A Greve, A&A. 35895Tarchi, A., Neininger, N., Greve, A., et al. 2000, A&A, 358, 95\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Thatte, M Tecza, R Genzel, A&A. 36447Thatte, N., Tecza, M., & Genzel, R. 2000, A&A, 364, L47\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Temi, W G Mathews, F Brighenti, J D Bregman, ApJ. 585121Temi, P., Mathews, W. G., Brighenti, F., & Bregman, J. D. 2003, ApJ, 585, L121\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Tichy, P Marronetti, Phys. Rev. D. 061502. Trinchieri, G., & di Serego Alighieri761647AJTichy, W. & Marronetti, P. 2007, Phys. Rev. D 76, 061502. Trinchieri, G., & di Serego Alighieri, S. 1991, AJ, 101, 1647\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Tsalmantza, R Decarli, M Dotti, D W Hogg, 73820Tsalmantza, P., Decarli, R., Dotti, M. & Hogg, D. W., 2011ApJ, 738, 20\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Vasiliev, F Antonini, D Merritt, P G Van Dokkum, M Franx, ApJ. 7852027AJVasiliev, E., Antonini, F., & Merritt, D. 2014, ApJ, 785, 163 van Dokkum, P. G., & Franx, M. 1995, AJ, 110, 2027\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G A Verdoes Kleijn, R P Van Der Marel, C M Carollo, P T De Zeeuw, AJ. 1201221Verdoes Kleijn, G. A., van der Marel, R. P., Carollo, C. M., & de Zeeuw, P. T. 2000, AJ, 120, 1221\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Volonteri, K Gúltekin, M Dotti, 4042143Volonteri, M.; Gúltekin, K.; Dotti, M., 2010MNRAS, 404, 2143\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D M Worrall, M Birkinshaw, E O'sullivan, A Zezas, A Wolter, G Trinchieri, G Fabbiano, MNRAS. 408701Worrall, D. M., Birkinshaw, M., O'Sullivan, E., Zezas, A., Wolter, A., Trinchieri, G., & Fabbiano, G. 2010, MNRAS, 408, 701\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J M Wrobel, AJ. 101127Wrobel, J. M. 1991, AJ, 101, 127\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J M Wrobel, D S Heeschen, AJ. 101148Wrobel, J. M., & Heeschen, D. S. 1991, AJ, 101, 148\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Younis, J Meaburn, P Stewart, A&A. 147178Younis, S., Meaburn, J., & Stewart, P. 1985, A&A, 147, 178\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Zanotti, L Rezzolla, L Del Zanna, C Palenzuela, A&A. 5238Zanotti, O., Rezzolla, L., Del Zanna, L., & Palenzuela, C. 2010, A&A, 523, A8\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":25,"cells":{"text":{"kind":"string","value":"\nINVARIANT SUBVARIETIES WITH SMALL DYNAMICAL DEGREE\n15 Apr 2021\n\nYohsuke Matsuzawa \nSheng Meng \nTakahiro Shibata \nDe-Qi Zhang \nGuolei Zhong \nINVARIANT SUBVARIETIES WITH SMALL DYNAMICAL DEGREE\n15 Apr 2021arXiv:2005.13368v2 [math.AG]\nLet f : X → X be a dominant self-morphism of an algebraic variety.Consider the set Σ f ∞ of f -periodic subvarieties of small dynamical degree, the subsetWhen X is projective, we prove the finiteness of the set P f of f -invariant prime divisors with small dynamical degree, and give an optimal upper boundas n → ∞, where d 1 (f ) is the first dynamic degree. When X is an algebraic group (with f being a translation of an isogeny), or a (not necessarily complete) toric variety, wegive an optimal upper boundas n → ∞, which slightly generalizes a conjecture of S.-W. Zhang for polarized f . 1 2 YOHSUKE MATSUZAWA, SHENG MENG, TAKAHIRO SHIBATA, DE-QI ZHANG, GUOLEI ZHONGLet f : X → X be a dominant self-morphism of an algebraic variety X. Let V ⊆ X be an (irreducible) closed subvariety. We say thatAn f -periodic closed subvariety V is said to be of Small Dynamical Degree (SDD forDenote by Σ f ∞ the set of f -periodic (irreducible closed) subvarieties of small dynamicalDenote by S f ∞ the set of maximal elements in Σ f ∞ . Denote by S f (resp. P f )\n\nhold as n → ∞? Remark 1.4. (1) Question 1.3 is sort of a generalization of [Zha06,Conjecture 1.2.2] which claims that if X is projective and f is q-polarized, then ♯ Fix(f n ) = q n·dim(X) (1 + o(1)) as n → ∞. In this case, the set of fixed points Fix(f n ) is just S f n and Question 1.3\n\nhas a positive answer when X is further smooth and general fixed points have multiplicity one; see the argument after [Zha06, Conjecture 1.2.2] by using Lefschetz fixed point theorem and Serre's Theorem, i.e., [Zha06, Theorem 1.1.2] (which are available for smooth X, but there are no easily applicable results even for singular surfaces) and noting that Fix(f n ) is a finite set for each n ≥ 1.\n\n(2) Here is a concrete example showing that the upper bound of the inequality in Question 1.3 is optimal. Let A be an abelian variety and f := (−q) A the multiplication map by −q (with q > 1) on A. Then f is q 2 -polarized and d 1 (f ) = q 2 . Hence ♯S f 2n+1 = ♯Fix(f 2n+1 ) is the number of (q 2n+1 + 1)-torsion points. So we have\n(q 2 ) (2n+1) dim(A) < ♯S f 2n+1 = (q 2n+1 + 1) 2 dim(A) = (q 2 ) (2n+1) dim(A) (1 + o(1))\nas n → ∞. Theorems 1.5, 1.6 and 1.9 are our main results.\n\nThe key idea of our first result concerning isogenies (and their translations) of algebraic groups is to find a unique maximal periodic SDD subgroup; see Definition 4.5. We leave it to the readers to consider more general cases, e.g., spherical varieties, (almost) homogeneous varieties. We use the fact d 1 (f n ) = d 1 (f ) n for any n > 0 to get the following upper bound when applying theorems in Section 5. (1) (cf. Theorem 5.7) X is an algebraic group; and f = R a • g where g is an isogeny and R a is the right multiplication map by some a ∈ X (when X is semi-abelian, the self-morphism f is automatically of this form).\n\n(2) (cf. Theorem 5.8) X is a toric variety with the big torus T being f -invariant. Theorem 1.6 below is a simpler version of Theorem 8.1, and answers Question 1.1 (2) affirmatively: the finiteness of SDD prime divisors. The main idea is to apply the higher dimensional Hodge index theorem to construct sufficiently many pairwise disjoint prime divisors on a smooth model. In this way, we are able to construct an equivariant fibration to a curve which contracts those periodic SDD prime divisors; and then what left is simply to count the fixed points of a polarized endomorphism on a curve. The upper bound of the inequality in Theorem 1.6 above is optimal. Indeed, let X = Y × E, where Y is any projective variety and E is an elliptic curve and let f = id Y × f E where f E := (−q) E is the multiplication map by −q (with q > 1) on E. In this case, by the product formula (cf. Notation 2.1), d 1 (f ) = q 2 , and a prime divisor P ∈ P f n if and only if it is an f n -stable fibre of the natural projection X → E. Thus\n♯P f 2n+1 = ♯ Fix(f 2n+1 E ) = (q 2n+1 + 1) 2 = (d 1 (f )) 2n+1 (1 + o(1)) (cf. Remark 1.4).\nWhen X is projective, Question 1.1 (2) can be reformulated as follows.\n\nTheorem 1.7. Let f : X → X be a surjective endomorphism of a projective variety X. Then the following are equivalent.\n\n(1) The set P f ∞ := ∞ n=1 P f n is infinite. (2) Replacing X by its normalization, there is an (iteration of ) f -equivariant fibration π : X → C onto a smooth projective curve C such that g := f | C is d 1 (f )-polarized and the relative dynamical degree d 1 (f | π ) < d 1 (f ).\n\nIn particular, the set of f −1 -periodic prime divisors of small dynamical degree is finite. Corollary 1.8 answers Question 1.1 affirmatively for projective surfaces, which is a direct consequence of Theorem 1.6 and Proposition 2.6. However, we are currently unable to show the upper bound in Question 1.3 in full generality except cases in Theorem 1.5. Corollary 1.8. S f is finite for any surjective endomorphism f of a projective surface.\n\nThroughout this paper, we try to make our argument as algebraic as we can so that most of our results can be generalized to positive characteristic. We leave such generalization to the careful readers which will not be mentioned too much in this paper for the sake of a comfortable reading.\n\nWe end the introduction with Theorem 1.9, as an application of a technique used when considering algebraic groups (the commutative diagram in Proposition 4.3). This compound theorem has its own interest and extends the results in [ to the most general setting (without assumptions like smooth, characteristic 0, separable, int-amplified, etc.). Theorem 1.9 is a simpler version of Theorem 6.4. Theorem 1.9. (cf. Theorem 6.4) Let f : X → X be a surjective endomorphism of a normal projective variety over the field k of arbitrary characteristic. Assume d 1 (f ) > 1.\n\nThen we have:\n(1) f * D ∼ R d 1 (f )D for some nef R-Cartier divisor D.\n(2) Suppose f * B ≡ qB for some big divisor and q > 1. Then f is q-polarized.\n\n(3) Suppose f is q-polarized. Then any eigenvalue of f * | H 1 et (X,Q ℓ ) has modulus √ q.\n\n(4) Suppose f is q-polarized (resp. int-amplified). Then so is f | Alb(X) : Alb(X) → Alb(X). • A surjective endomorphism f : X → X of a projective variety is said to be int-amplified if f * H − H = L for some ample Cartier divisors H and L.\n\n• Let X be a normal projective variety. Denote by Cl(X) the Weil divisor class group on X, and by Cl • (X) ⊆ Cl(X) the subgroup of divisors algebraically equivalent to 0, which is a birational invariant (cf. [Kol18,Lemma 18]). The quotient\nCl ns (X) := Cl(X)/Cl • (X)\nis called the Néron-Severi class group of X (in the sense of Weil divisors).\n\nThere are natural inclusions Pic • (X) ⊆ Cl • (X) and NS(X) ⊆ Cl ns (X). Both of these are equalities when X is smooth.\n• Denote by ρ(X) := dim R NS(X) ⊗ Z R the usual Picard number of X. Denote by ρ ns (X) := dim R Cl ns (X) ⊗ Z R which is finite (cf. [Kol18,\nTheorem 17]). We refer readers to [Kol18,Section 3] and [Ful98, Chapter 10] for more information.\n\n• Let f : X X and g : Y Y be dominant self-maps of projective varieties.\n\nLet π : X Y be a dominant rational map such that g • π = π • f . We recall the following definitions and facts about the first dynamical degree.\n\n(1) The (first) dynamical degree of f is defined by\nd 1 (f ) := lim n→∞ ((f n ) * H · H dim(X)−1 ) 1/n\nwhere H is a nef and big Cartier divisor of X.\n\n(2) The relative (first) dynamical degree of f is defined by\nd 1 (f | π ) = lim n→∞ ((f n ) * H X · (π * H Y ) dim(Y ) · H dim(X)−dim(Y )−1 X ) 1/n\nwhere H X and H Y are respectively nef and big Cartier divisors of X and Y .\n\n(3) Product formula:\nd 1 (f ) = max{d 1 (f | π ), d 1 (g)}.\n(4) d 1 (f n ) = d 1 (f ) n for any n > 0.\n(5) d 1 (f ) = ρ(f * | N 1 (X) ), the spectral radius of f * | N 1 (X) , if f is a self-morphism.\n(Relative) dynamical degrees can also be defined, by the above formulas, for (not necessarily projective) varieties over the field k (allowed to be of arbitrary characteristic) by taking any birational projective models since they are birational invariants and independent of the choices of nef and big divisors above; indeed, d 1 (f ) is invariant under generically finite map; see [DS05] and [Dan19] for details.\n\n• Let f : X → X be a surjective endomorphism of a smooth projective variety.\n\nDenote by\nχ i (f ) := ρ(f * | H í et (X,Q ℓ ) ) where ρ is the spectral radius, H í et (X, Q ℓ )\nis the ℓ-adic cohomology, and ℓ is a prime different from the characteristic of the ground field k. When X is an abelian variety, a characteristic free proof by Hu (cf. [Hu19, Theorem 1.2 and Formula (3.6)]) showed that\nχ 2 (f ) = d 1 (f ) = χ 1 (f ) 2 .\n• A normal (not necessarily projective) variety X is a toric variety if X contains an algebraic torus T = (k * ) n as an (affine) open dense subset such that the natural multiplication action of T on itself extends to an action on the whole variety.\n\n• A commutative algebraic group S is semi-abelian if S is an extension of an abelian variety A by an algebraic torus T . • Let f : X → X be a dominant self-morphism of a variety, which is not necessarily\nprojective. A closed subvariety Y ⊆ X is called f -invariant (resp. f -periodic) if f (Y ) = Y (resp. f s (Y ) = Y for some s > 0). An f -periodic closed subvariety V is said to be of Small Dynamical Degree (SDD for short) if the first dynamical degree d 1 (f s | V ) < d 1 (f s ) for some s > 0 such that f s (V ) = V .\n• Denote by Σ f ∞ the set of f -periodic (irreducible closed) subvarieties of Small Dynamical Degree (SDD for short). We define the following subsets of Σ f ∞ :\nS f ∞ the subset of maximal f -periodic subvarieties in Σ f ∞ , S f the subset of S f ∞ , consisting of all f -invariant elements, P f the subset of S f ∞ , consisting of all f -invariant prime divisors, P f ∞ the subset of (maximal) f -periodic prime divisors in Σ f ∞ , P −1 f ∞ the subset of (maximal) f −1 -periodic prime divisors in Σ f ∞ .\n• An element D ∈ S f is called an f -invariant maximal f -periodic SDD subvariety.\n\n• Denote by Fix(f ) (resp. Per(f )) the set of fixed (resp. periodic) points of f .\n\nThe following lemma is basic, and is used in the proof of Proposition 8.5.\n\nLemma 2.2. Let σ : X → X be a birational morphism between normal projective varieties. Let L be a nef R-Cartier divisor on X and denote by L := σ * L. Let π : X → Y be a morphism to another projective variety Y such that L ≡ π * H with H an ample divisor on Y . Then π factors through a well-defined morphism π : X → Y with L ≡ π * H.\n\nProof. By Zariski's main theorem, σ * O X = O X . For every curve C lying in a fibre of σ, it follows from the projection formula that H · π * C = π * H · C = σ * L · C = 0. Since H is ample on Y , π(C) is a point and hence every fibre of σ will also be contracted by π. By the rigidity lemma (cf. [Deb01, Lemma 1.15]), π factors through a well-defined morphism\nπ : X → Y . So π * H = σ * σ * π * H ≡ σ * L = L.\nGiven an f -equivariant surjective morphism π : X → Y with g := f | Y , it is natural for us to compare those ♯S f , ♯P −1 f ∞ , ♯P f on f with the corresponding ones on g. When π is finite, we formulate the following lemma to bound ♯S g in terms of ♯S f d! with d = deg π. It is applied to prove the finiteness of ♯S g for the Q-abelian case in Corollary 5.2. Our upper bound for ♯S g in the proof of Lemma 2.3 is not optimal at all. Lemma 2.3. Let π : X → Y be a finite surjective morphism of degree d between projective varieties. Let f : X → X and g : Y → Y be two surjective endomorphisms such that\nπ • f = g • π. If ♯S f d! < ∞, then ♯S g < ∞. Conversely, if ♯S g < ∞, then ♯S f < ∞.\nProof. Since π : X → Y is finite and d 1 (f ) is invariant under finite map, π induces a surjective map π : S f ∞ → S g ∞ while π −1 induces a one-to-many surjective map π −1 :\nS g ∞ → S f ∞ .\nThen the lemma follows.\n\nWhen π is a birational morphism, we can only consider prime divisors rather than higher codimensional subvarieties due to the complexity of periodic subvarieties contained in the exceptional locus of π.\n\nLemma 2.4. Let π : X Y be a birational map between projective varieties. Let f : X → X and g : Y → Y be two surjective endomorphisms such that π • f = g • π.\n\nThen ♯P f n = ♯P g n + O(1) as n → ∞, and ♯P f ∞ is finite if and only if so is ♯P g ∞ .\n\nProof. By taking the graph Γ ⊆ X ×Y , our f lifts to a surjective endomorphism (f ×g)| Γ .\n\nSo it suffices for us to consider the case when π is a birational morphism. Let B be the closed subset of Y , over which π is not isomorphic. Let B 1 be the set of prime divisors in B and B 2 the set of prime divisors in π −1 (B). For each n > 0, we have a natural map\nϕ n : P f n \\B 2 → P g n \\B 1 ,\nsending D to π(D). Note that ϕ n admits an inverse, sending D ′ ∈ P g n \\B 1 to its proper transform π −1 * (D ′ ). In particular, setting K := ♯B 1 + ♯B 2 , we have:\n♯P g n − K ≤ ♯P f n ≤ ♯P g n + K.\nBy a similar proof, we have the following lemma.\n\nLemma 2.5. Let π : X Y be a birational map between projective varieties. Let f : X → X and g : Y → Y be two surjective endomorphisms such that π • f = g • π.\n\nThen\n♯P −1 f ∞ is finite if and only if so is ♯P −1 g ∞ .\nProposition 2.6. Let f : X → X be a surjective endomorphism of a projective variety.\n\nThen there are only finitely many points (as zero-dimensional subvarieties) in the set S f .\n\nProof. We may assume\nd 1 (f ) > 1. Let Y ⊆ S f be the subset of f -fixed closed points.\nSuppose the contrary that ♯Y is infinite. Let Z be an irreducible component of the closure\nof Y in X with dim(Z) > 0. Then f | Z = id Z and 1 = d 1 (f | Z ) < d 1 (f ), a contradiction\nto the maximality of the points in Y .\n\n\nConnection of two arithmetic dynamical conjectures\n\nIn this section, we work over a number field K. We will give the relation between two conjectures in Theorem 3.4, which also gives an arithmetic motivation of this paper. We refer to [MMSZ20, Section 2] for the notation involved in this section.\n\nThe following are the two main conjectures studied in [MMSZ20].\nConjecture 3.1. (Kawaguchi-Silverman Conjecture, [KS16]) Let f : X → X be a\nsurjective endomorphism of a projective variety X defined over a number field K. Let \nx ∈ X(K) such that its arithmetic degree α f (x) < d 1 (f ). Then the orbit O f (x) is not Zariski dense in X K := X × K K.Z f (d) := {x ∈ X(d) | α f (x) < d 1 (f )} is not Zariski dense in X K = X × K K for any positive constant d > 0.\nWe copy [MMSZ20, Question 8.4] in the following. Note that for the equivalence of the above two conjectures, we may only focus on Y ∈ S f ∞ in Question 3.3 (1) and (2); see the proof of Theorem 3.4.\nQuestion 3.3. ([MMSZ20, Question 8.4]) Let f : X → X be a surjective endomorphism on a projective variety with d 1 (f ) > 1. Fix d > 0. (1) Let Y be an f -invariant subvariety of X K . Then, is X(d) ∩ (f −s (Y ) \\ f −(s−1) (Y ))\nempty for s ≫ 1?\n\n(2) Is there a positive integer N such that Y (d) is empty for any f -periodic subvariety Y ⊆ X K of small dynamical degree whose period is larger than N? \n(3) (cf. Question 1.1 (1)) Is S f s | X K aProof. Let x ∈ Z f (d), i.e., α f (x) < d 1 (f ) and x ∈ X(d). By [MMSZ20, Lemma 2.8], O f (x) = {x} ∪ {f (x)} ∪ · · · ∪ {f t−1 (x)} ∪ s−1 i=0 f i (O f s (f t (x))),\nwhich is a union of irreducible closed subsets and Y 0 :\n= O f s (f t (x)) is f s -invariant. Con- jecture 3.1 implies that α f s (f t (x)) = d 1 (f s | Y 0 ). Note that α f s (f t (x)) = α f (x) s < d 1 (f ) s . Then Y 0 ∈ Σ f ∞ and Y 0 ⊆ Y for some Y ∈ S f s ′ with s ′ > 0. Note that f t (x) ∈ Y ∩ X(d). By Question 3.3 (2), there exists a constant N > 0 such that Y ∈ S f N for any Y satisfying Y ∩ X(d) = ∅. Question 3.3 (3) implies that S f N is finite. Together with Question 3.3 (1), there exists another constant M > 0 such that x ∈ Y ∈S f N , 0≤i≤M f −i (Y ). Therefore Z f (d) ⊆ Y ∈S f N , 0≤i≤M f −i (Y )\nwhich is not Zariski dense in X K .\n\n\nIsogenies of algebraic groups\n\nWe use the following notation throughout this section.\n\nNotation 4.1. Let G be a connected algebraic group with the identity element e. We focus on the isogeny (i.e., surjective group homomorphism with finite kernel)\ng : G → G with d 1 (g) > 1, and f := R a • g\nwhere R a is the right multiplication map by some a ∈ G. Note that there exists a G-\nequivariant embedding G ֒− → G to a (smooth) projective variety G (cf. [Bri10, Theorem 2]\n). Then the same proof for [MS18, Lemma 5.6] shows that\nd 1 (f ) = d 1 (g) > 1.\nWe recall that g and hence f are finite surjective endomorphisms. Proof.\nLet p := f (b)b −1 ∈ H. We have g(H) = g(Hb)g(b) −1 = f (Hb)a −1 (f (b)a −1 ) −1 = Hbf (b) −1 = Hp −1 = H.\nMoreover, We have the following commutative diagram:\nH (Rp•g)| H / / R b H R b Hb f | Hb / / Hb where R b : H → Hb is an isomorphism of varieties. So d 1 (f | Hb ) = d 1 ((R p •g)| H ) = d 1 (g| H ) (cf. [MS18, Lemma 5.6]).\nThe following result is a generalization of Lemma 4.2 to arbitrary closed subvariety of G, which plays a crucial role in many later results.\nProposition 4.3. Let X be an f -invariant (irreducible closed) subvariety of G. Fix some b ∈ X and denote by X e := Xb −1 . Let H be the smallest algebraic subgroup of G containing X e . Then g(H) = H, f (Hb) = Hb, and d 1 (g| H ) = d 1 (f | Hb ) ≤ d 1 (f | X ). Proof. Let p := f (b)b −1 ∈ X e ⊆ H. We have g(H) ⊇ g(X e ) = g(X)g(b) −1 = f (X)f (b) −1 = X e p −1 .\nThen X e ⊆ X e p −1 (X e p −1 ) −1 ⊆ g(H) and hence H ≤ g(H) by the construction of H.\n\nSince g is an isogeny, g(H) = H.\n\nConsider the alternate multiplication map s n : X ×(2n) → H defined by\ns n (x 1 b, . . . , x 2n b) = (x 1 b)(x 2 b) −1 · · · (x 2i+1 b)(x 2i+2 b) −1 · · · (x 2n−1 b)(x 2n b) −1 = x 1 x −1 2 · · · x 2i+1 x −1 2i+2 · · · x 2n−1 x −1 2n ∈ (X e X −1 e ) ·n\nThen the image of s n is (X e X −1 e ) ·n = H when n ≫ 1 by [Spr09, Proposition 2.2.6] and noting that e ∈ X e . Consider the following diagram with n ≫ 1:\n(*) X ×(2n) (f | X ) ×(2n) / / sn X ×(2n) sn H g| H / / H Note that f (x i b) = g(x i )pb.\nThen\ns n (f (x 1 b), . . . , f (x 2n b)) = s n (g(x 1 )pb, . . . , g(x 2n )pb) = g(x 1 )g(x 2 ) −1 · · · g(x 2n−1 )g(x 2n ) −1 = (g • s n )(x 1 b, . . . , x 2n b).\nSo the diagram commutes. By the product formula, we have\nd 1 (g| H ) ≤ d 1 ((f | X ) ×(2n) ) = d 1 (f | X ).\nOn the other hand, note that\nf (Hb) = g(Hb)a = Hg(b)a = Hf (b)b −1 b = (Hp)b = Hb. By Lemma 4.2, we have d 1 (f | Hb ) = d 1 (g| H ) ≤ d 1 (f | X ).\nLemma 4.4. There exists a unique X ∈ S g ∞ containing e ∈ G. Moreover, X is an algebraic subgroup and g-invariant.\n\nProof. Take X ∈ S g ∞ containing e. It exists because d 1 (g) > 1 and d 1 (g| {e} ) = 1. Take s > 0 such that g s (X) = X. Take the smallest (connected closed) algebraic subgroup H of G containing X. By Proposition 4.3, g s (H) = H and d 1 (\ng s | H ) ≤ d 1 (g s | X ) < d 1 (g s ).\nBy the maximality of X, we have X = H and it is an algebraic subgroup.\n\nTake another Y ∈ S g ∞ containing e and then take t > 0 such that g t (Y ) = Y .\n\nThe argument in the previous paragraph shows that Y is a (connected closed) algebraic subgroup. Let Z be the smallest algebraic group containing the g st -invariant irreducible closed subvariety XY . Apply the similar commutative diagram (*) and argument in the proof of Proposition 4.3,\n(X × Y ) ×(2n) (g st | X ×g st | Y ) ×(2n) / / sn (X × Y ) ×(2n) sn Z g st | Z / / Z where s n : (X × Y ) ×(2n) → Z = (XY )(XY ) −1 · · · (XY )(XY ) −1 defined by s n (x 1 , y 1 , . . . , x 2n , y 2n ) = (x 1 y 1 )(x 2 y 2 ) −1 · · · (x 2n−1 y 2n−1 )(x 2n y 2n ) −1\nis surjective with n ≫ 1. Then we have d 1 (g st | Z ) < d 1 (g st ) and hence Z = X = Y by the maximality of X and Y . This shows the uniqueness of X.\n\nNext we show that X is g-invariant. Note that g(X) is also g s -invariant and contains e. By the product formula, the following diagram shows d 1 (g s | g(X) ) ≤ d 1 (g s | X ) < d 1 (g s ):\nX g s | X / / g X g g(X) g s | g(X) / / g(X)\nTherefore, we have g(X) ⊆ X by the maximality and uniqueness of X. In particular, X is g-invariant since g is an isogeny. This finishes our proof.\n\nDefinition 4.5. With the notation in Lemma 4.4, we denote by\nG(f ) = G(g) := X the unique element of S g ∞ containing e ∈ G. Note that G(f ) is g-invariant and G(f m ) = G(g m ) = G(f ) for any m > 0.\nWe shall prove in the next section that G(f ) is actually a normal subgroup of G.\nProposition 4.6. A subvariety X ⊆ G is a member of S f if and only if X = G(f )b with f (b)b −1 ∈ G(f ) (= G(g)).\nProof. Assume that X ∈ S f . Fix some b ∈ X and denote by X e := Xb −1 which contains e ∈ G. Let H be the smallest algebraic subgroup of G containing X e . By Proposition\n4.3, g(H) = H, f (Hb) = Hb, and d 1 (g| H ) ≤ d 1 (f | X ) < d 1 (f ). Then f (b)b −1 ∈ X e ⊆ H ≤ G(g) by Lemma 4.4. Note that f (G(f )b) = g(G(g))g(b)a = (G(g)f (b)b −1 )b = G(f )b.\nBy Lemma 4.2,\nd 1 (f | G(f )b ) = d 1 (g| G(g) ) < d 1 (g) = d 1 (f ). Since X ⊆ Hb ⊆ G(f )b, we have X = G(f )b by the maximality of X. Conversely, assume that X = G(f )b with f (b)b −1 ∈ G(f ). Then X ∈ Σ f by the same argument above. Suppose that we have Y ∈ S f ∞ containing X. Take s > 0 such that Y ∈ S f s . Then Y = G(f s )c = G(f )c for some c ∈ G with f s (c)c −1 ∈ G(f ) by the\nprevious paragraph. But this implies that X = Y since both are translations of the same group G(f ). So X is maximal.\n\nThe following modified product formula will be frequently used in the next Theorem and also next section.\n\nLemma 4.7. Consider the commutative diagram of homomorphisms of connected algebraic groups:\n1 / / B g| B / / A π / / g C / / g| C 1 1 / / B / / A π / / C / / 1\nwhere g is an isogeny. Then π is f -equivariant and the following hold.\n(1) d 1 (g| π ) = d 1 (g| B ). In particular, d 1 (g) = max{d 1 (g| B ), d 1 (g| C )}. (2) Suppose d 1 (g| B ) < d 1 (g). Then A(f ) = π −1 (C(f | C )\n) and π induces a bijection between S f and S f | C .\nProof. First, note that π is f -equivariant with f | C = R π(a) • g| C .\nLet A act on A and C by the right multiplication. By Note that the graph of π is A-equivariant. Replacing A by the A-equivariant resolution of this graph, we may further assume that π is a well-defined algebraic fibration.\n\nBy the generic smoothness of π, there exists a general point c 1 = π(a 1 ) ∈ C such that the fibre A c 1 := π −1 (c 1 ) is an (irreducible) smooth projective variety. For any\nc 2 = π(a 2 ) ∈ C, we have A c 2 = A c 1 ·(c −1 1 c 2 ) = A c 1 · (a −1 1 a 2 )\n. So all the fibres of π over C are (irreducible) smooth projective varieties isomorphic to each other.\n\nWe refer to [DN11] for the definitions of critical value and critical set. Now we have that each point of C is not a critical value of π. Note that B is g-invariant and g isétale. Then\n\nA e (the closure of B in A) is not contained in the critical set of g. Therefore, we verified the assumptions for A e in [DN11, Remark 3.4] and hence d 1 (g| π ) = d 1 (g| A e ) = d 1 (g| B ).\n\nTogether with the product formula, (1) is proved.\nFor (2), d 1 (g) = d 1 (g| C ) by (1).\nWe apply the maximality and uniqueness of A(g) and C(g| C ) (cf. Lemma 4.4) and our new product formula (1) several times in this diagram. First, we have B ≤ A(g).\n\nNote that d 1 (g| π(A(g)) ) = d 1 (g| A(g) ) < d 1 (g). So we have π(A(g)) ≤ C(g| C ). Conversely,\nd 1 (g| π −1 (C(g| C )) ) = d 1 ((g| C )| C(g| C ) ) < d 1 (g| C ) = d 1 (g). So we have A(g) = π −1 (C(g| C )).\nFinally, we apply Proposition 4.6 for f and f | C , then we see that the induced map\nπ : S f → S f | C , via A(f )b → C(f | C )π(b)\nis a bijection. So (2) is proved.\n\nWe complete this section by considering the commutative algebraic groups first.\n\nTheorem 4.8. Let f : G → G be a dominant self-morphism of a connected commutative algebraic group G, such that f = R a • g where g is an isogeny and a ∈ G. Suppose d 1 (f ) > 1. Let G(f ) be as in Definition 4.5. Then \nπ : G → G := G/G(f ) is f, g-equivariant and 0 < ♯S f = ♯S g = ♯ Fix(f ) = ♯ Fix(g) < ∞ where f = f | G = R π(a) • g. Proof. We simply define f (x · G(f )) := f (x) · G(f ). It is well-defined and f • π = π • fd 1 (g| H ) = max{d 1 (g| π ), d 1 (g| H )} = d 1 (g| π ) = d 1 (g| G(g) ) < d 1 (g).\nThis contradicts the maximality of G(g).\n\n\nNow we have shown\n♯ Fix(g) < ∞. Then φ : G → G (x → g(x) · x −1 ) is an isogeny. Note that f (x) = x if and only if φ(x) = a −1 . So we have 0 < ♯ Fix(f ) = ♯ φ −1 (a −1 ) = ♯ φ −1 (e) = ♯ Fix(g) < ∞\nand the theorem is proved.\n\n\nProof of Theorem 1.5\n\nIn this section, we focus on the upper bound in Question 1.3. We first consider the case of abelian varieties.\n\nTheorem 5.1. Let f be a surjective endomorphism of an abelian variety A. Then\n♯S f ≤ ( d 1 (f ) + 1) 2(dim(A)−dim(A(f )))\n(cf. Definition 4.5 for A(f )).\n\nProof. We may assume d 1 (f ) > 1. Write f = R a • g where g is an isogeny and a ∈ A.\n\nSince d 1 (f ) = d 1 (g) and A(f ) = A(g), applying Theorem 4.8, we may assume that f is an isogeny, i.e., f = g; further we have an f -equivariant quotient morphism\nπ : A → A ′ := A/A(f ) such that 0 < ♯S f = ♯ Fix(f ) < ∞ where f ′ = f | A ′ . Denote by φ : A ′ → A ′ (x ′ → f ′ (x ′ ) · x ′−1 ) (\n\nsee the ending part of the proof of Theorem 4.8), which is an isogeny. Then\n♯ Fix(f ′ ) = deg(φ). Set m := dim(A ′ ) = dim(A) − dim(A(f )). We compute deg(φ) = χ 2m (φ) = ρ(φ * | H 2ḿ et (A ′ ,Q ℓ ) ) = ρ (∧ 2m φ * )| ∧ 2m H 1 et (A ′ ,Q ℓ ) = | det(φ * | H 1 et (A ′ ,Q ℓ ) )|. Note that φ * | H 1 et (A ′ ,Q ℓ ) = f ′ * | H 1 et (A ′ ,Q ℓ ) − id H 1 et (A ′ ,Q ℓ ) (cf. [Hu19, Theorem 2.2]). So, letting α 1 , α 2 , . . . , α 2m be the eigenvalues of f ′ * | H 1 et (A ′ ,Q ℓ ) , we have deg(φ) = 2m i=1 |α i − 1| ≤ (χ 1 (f ′ ) + 1) 2m . Note that χ 1 (f ′ ) 2 = d 1 (f ′ ) ≤ d 1 (f )\n\nby [Hu19, Formula (3.6)] and the product formula.\n\nThen the theorem follows.\n\nRecall that a normal projective variety X is said to be Q-abelian (cf. [NZ10]) if there is a finite surjective morphism A → X from an abelian variety A, which isétale in codimension one. When f is a group homomorphism, it induces the following commutative diagram where T is an algebraic torus and A = Alb(X) is an abelian variety:\n0 / / T f | T / / X / / f A / / f | A 0 0 / / T / / X / / A / / 0\nTheorem 5.3. Let f be a dominant self-morphism of a semi-abelian variety X. Then\n♯S f ≤ ( d 1 (f ) + 1) 2(dim(X)−dim(X(f ))) .\nProof. By the same argument as in the first paragraph of the proof of Theorem 5.1, we may replace f by g and X by X/X(f ) so that dim(X(f )) = 0 and both f and\nφ : X → X (x → f (x) · x −1 ) are isogenies.\nConsider the commutative diagram before Theorem 5. 3. Note that f | T is defined by a matrix M of size dim(T ) with integer coefficients and a non-zero determinant. Then We compute\n♯ Fix(f | T ) = deg(φ| T ) = det(M − id) = (λ 1 − 1) · · · (λ dim(T ) − 1) ≤ (d 1 (f | T ) + 1) dim(T ) .\nTogether with the computation in the proof of Theorem 5.1 for abelian varieties, we have\n♯ Fix(f ) = ♯ Fix(f | T ) · ♯ Fix(f | A ) ≤ (d 1 (f | T ) + 1) dim(T ) · ( d 1 (f | A ) + 1) 2 dim(A) ≤ ( d 1 (f ) + 1) 2 dim(X)\nwhere the last inequality follows from\nd 1 (f | T ) ≤ d 1 (f ) and d 1 (f | A ) ≤ d 1 (f ); see e.g.\n[MS18, Theorem 1.6] or Lemma 4.7. So the theorem is proved.\n\nRemark 5.4. In the proof of Theorem 5.3, we see that if X = T is an algebraic torus, then the computation indeed shows that ♯S f ≤ (d 1 (f ) + 1) dim(T )−dim(T (f )) .\n\nWe now prepare to deal with general algebraic groups. See, for instance, [MZ18a,§3] for several decomposition facts about algebraic groups. First, we need observations for unipotent groups and semisimple groups below.\n\nLemma 5.5. Let g : U → U be an isogeny of a (connected) unipotent group U. Then d 1 (g) = 1.\n\nProof. Since Ker(g) is a finite subgroup of the unipotent group U, it is trivial, so g is an automorphism. We show by induction on dim(U) and assume dim(U) > 0. Note that U is solvable. So its commutator subgroup U (1) is strictly smaller than U. Note that g(U (1) ) = U (1) . Consider the following commutative diagram:\n1 / / U (1) g| U (1) / / U / / g U/U (1) / / g| U/U (1) 1 1 / / U (1) / / U / / U/U (1) / / 1 Suppose U (1)\nis not trivial. By Lemma 4.7 and induction, we have d 1 (g) = max{d 1 (g| U (1) ), d 1 (g| U/U (1) )} = 1.\n\nSuppose U (1) is trivial. Then U is a commutative unipotent group which is isomorphic to k ×dim(U ) . One checks that g ∈ GL(U) since g is additive. By is finite. Replacing g by a power, we may assume g is in the connected algebraic group Aut • (S). We are done by the same argument at the ending part of Lemma 5.5.\n\nFinally, we are able to give the same upper bound for any algebraic group.\n\nTheorem 5.7. Let G be a connected algebraic group. Then G(f ) is normal in G and\n♯S f ≤ ( d 1 (f ) + 1) 2(dim(G)−dim(G(f )))\nfor any self-morphism f = R a • g where g is an isogeny and a ∈ G.\n\nProof. We may assume 1 < d 1 (f ) (= d 1 (g)). Let G a be the maximal connected affine normal subgroup of G, and U the unipotent radical of G a . Then g(G a ) = G a and g(U) = U. Consider the following commutative diagram:\n1 / / U g| U / / G π / / g G/U / / g| G/U 1 1 / / U / / G π / / G/U / / 1\nBy Lemma 5.5, we have d 1 (g| U ) = 1 < d 1 (g). By Lemma 4.7, π is f -equivariant and\nU ≤ G(f ) = π −1 (G(f | G/U )). Then G(f ) is normal in G if and only if (G/U)(f | G/U ) is\nnormal in G/U. Therefore, we may replace G by G/U and assume that U is trivial.\n\nLet S := G (1) be the commutator subgroup of G. Then g(S) = S, and S = (G a ) (1) which is a semisimple group since G a is a reductive group now. Consider the following commutative diagram:\n1 / / S g| S / / G π / / g G/S / / g| G/S 1 1 / / S / / G π / / G/S / / 1\nBy Lemma 5.6, we have d 1 (g| S ) = 1. By Lemma 4.7 and an argument similar to the previous paragraph, we may replace G by G/S and assume that G is commutative.\n\nNote that now G is a semi-abelian variety. In particular, G(f ) is normal in G, and the upper bound is as claimed, by Theorem 5.3 and Lemma 4.7.\n\nAt the end of this section, we treat the toric varieties. Let X be a toric variety and T the big torus. A subvariety of the form O = x · T is called a torus orbit on X. Note that X = O⊆X O is a finite disjoint union of all torus orbits on X.\n\nTheorem 5.8. Let X be a toric variety with the big torus T ⊆ X. Let f : X → X be a dominant self-morphism such that f (T ) ⊆ T . Then\n♯S f ≤ O⊆X (d 1 (f ) + 1) dim(O)\nwhere the sum runs over all the torus orbits on X.\n\nProof. By [Nak08, Lemma 3.4.3], f = R a • g such that g| T is an isogeny and g(x · t) = g(x) · g(t); see [Nak08, Section 3.4] for details. Here R a is the natural extension of the right multiplication map on T by a ∈ T . Since g| T is dominant, f (T ) = g(T ) · a = T .\nMoreover, f (x · T ) = g(x · T ) · a = g(x) · T .\nIn particular, f maps every torus orbit onto a torus orbit. Note that\nd 1 (f | T ) = d 1 (f ) since it is a birational invariant. Let V ∈ S f . Recall that f (V ) = V and d 1 (f | V ) < d 1 (f ). Note that V = O⊆X V ∩ O.\nThen there is a torus orbit O such that V ∩ O contains an open dense subset of V . So \nf (O) ∩ O ⊇ f (V ∩ O) ∩ (V ∩ O) = ∅ and hence f (O) = O. Note that d 1 (f | O ) = d 1 (g| O ) (cf. [MS18,f (x 0 ) = x 0 · b for some b ∈ T . Define f O : O → O via f O := R b −1 ·a • f | O .\nThen\nd 1 (f O ) = d 1 (f | O ) (cf. [MS18,\nLemma 5.6]). Define the natural surjective morphism\nπ : T → O via π(t) = x 0 · t. Note that f O • π(t) = f (x 0 · t) · b −1 · a = g(x 0 ) · g(t) · a · b −1 · a = x 0 · f (t) = π • f | T (t).\nBy the product formula,\nd 1 (f | O ) = d 1 (f O ) ≤ d 1 (f | T ) = d 1 (f ). If d 1 (f | O ) < d 1 (f ), then V = O by the maximality. If d 1 (f | O ) ≥ d 1 (f ), then V ∩ O ∈ S f | O . Applying Remark 5.4 to f | O , the number of such V is bounded by (d 1 (f ) + 1) dim(O) .\nThis finishes the proof.\n\nRemark 5.9. In Theorem 5.8, as we do not assume X is projective, f is not necessarily finite (nor surjective). Also during its proof, it is possible that O ⊆ Sing(X), and then 6. Reduction to polarized case: Proof of Theorem 1.9\nwe cannot deduce d 1 (f | O ) ≤ d 1 (f ) from [MS18,\nIn this section, the ground field k is allowed to have arbitrary characteristic. We will prove Theorem 6.4 which includes Theorem 1.9 as a special case. The connection with polarized endomorphism will give us finiteness for ♯P f .\n\nProposition 6.1. Let π : X Y be a dominant rational map between projective\n\nvarieties. Let f : X → X and g : Y → Y be surjective endomorphisms such that\ng • π = π • f , dim(Y ) > 0, and g is d 1 (f )-polarized.\nThen for n ≫ 1,\n♯P f n ≤ (d 1 (f )) n (1 + o(1)).\nProof. By Lemma 2.4, replacing π with its graph and taking the normalizations, we may assume that π is a surjective morphism and both X and Y are normal.\n\nTake the Stein factorization X → Y 1 → Y such that p 1 : X → Y 1 has connected fibres and p 2 : Y 1 → Y is finite. By [CMZ20, Lemma 5.2], f descends to a surjective\nendomorphism h : Y 1 → Y 1 such that p 1 • f = h • p 1 and p 2 • h = g • p 2 . Our h, like g, is also d 1 (f )-polarized.\nHence we may assume further π has connected fibres.\nFor any D ∈ P f ∞ = ∞ n=1 P f n , say D ∈ P f s , π(D) is a point; otherwise, d 1 (f s ) > d 1 (f s | D ) ≥ d 1 (g s | π(D) ) = d 1 (g s ) = d 1 (f s ) since both g s and g s | π(D) are (d 1 (f ) s =) d 1 (f s )-\npolarized, a contradiction. For the upper bound, we may assume P f ∞ is an infinite set whose elements are contained in fibres of π. Then dim Y = 1. Let B be the finite set of prime divisors in the (finitely many) reducible fibres of π. We have an injection\nϕ : P f n \\B → Fix(g n ),\nsending D to π(D). Then our argument follows from Proposition 6.2.\n\nThe following result is well known when the ground field k has char k = 0 and the proof is not that troublesome due to the Hodge decomposition. Our proof here works when char k is arbitrary, and is related to a generalized problem of Weil's Riemann Hypothesis.\n\nProposition 6.2. Let f : C → C be a non-isomorphic endomorphism on a smooth projective curve C. Then ♯ Fix(f ) ≤ deg f + 2c(deg f ) 1/2 + 1 where c is the genus of C.\n\nProof. Let d := d 1 (f ) = deg f ≥ 2. Then f is d-polarized, so is its induced endomorphism g : Alb(C) → Alb(C) by Theorem 6.4. Note that f has at most finitely many fixed points.\n\nApplying Lefschetz fixed-point formula, we have\n♯Fix(f ) ≤ 2 i=0 (−1) i Tr(f * | H í et (C,Q ℓ ) )\n. We are going to prove Theorem 6.4 (cf. Theorem 1.9). It has its own independent interest and various applications, and is also needed in proving Proposition 6. Remark 5.9] without the smooth and characteristic 0 assumptions.\nIdentifying H 1 et (C, Q ℓ ) with H 1 et (Alb(C), Q ℓ ), we have f * | H 1 et (C,Q ℓ ) = g * | H 1 et (Alb(C),Q ℓ ) . Since Alb(C) is an abelian variety, we have χ 1 (f ) 2 = χ 1 (g) 2 = d 1 (g) = d (cf. [Hu19, Formula\nWe prepare some notation. Let f be a surjective endomorphism of a normal projective variety X over the field k which is allowed to be of arbitrary characteristic. Let We need the following in proving Theorem 6.4. Proposition 6.3. Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Let λ be an eigenvalue of f * | Pic • (X) red , equivalently of (f ∨ ) * | H 1 (X ∨ ,Z ℓ ) . Then\nX ∨ := Pic • (X) red , f ∨ := f * | Pic • (X) red : X ∨ → X ∨ι f ≤ |λ| 2 ≤ d 1 (f )\nwhere ι f is the minimal modulus of eigenvalues of f * | N 1 (X) .\n\nProof. Let P ∈ Z[t] be the minimal polynomial of f ∨ : X ∨ → X ∨ , which is the minimal\npolynomial of (f ∨ ) * | H 1 (X ∨ ,Z ℓ ) .\nThen P (f ∨ ) = 0 (cf. [Hu19, Section 2]). Note also that \nP (f ∨∨ ) = P (f ∨ ) ∨ = 0.b 2 = d 1 (f ∨∨ ) and a 2 = ι f ∨∨ . Moreover, note that f | Alb(X) = f ∨∨ + t for some t ∈ X ∨∨ = Alb(X). Then we have ι f | Alb(X) = ι f ∨∨ = ι f ∨ and d 1 (f | Alb(X) ) = d 1 (f ∨∨ ) = d 1 (f ∨ ). Let λ\nbe an eigenvalue of f ∨ . Then λ is a root of P and hence a ≤ |λ| ≤ b.\n\nWe may assume 0 ∈ alb X (X). Note that the smallest abelian subvariety of A containing alb X (X) is A itself by the universal property. Applying the similar technique used in the proof of Proposition 4.3, we have the following commutative diagram\n(**) X ×(2n) f ×(2n) / / sn X ×(2n) sn Alb(X) f | Alb(X) / / Alb(X)\nwhere n ≫ 1 and s n : X ×(2n) → Alb(X) is defined by\ns n (x 1 , . . . , x 2n ) = n i=1 alb X (x i ) − 2n j=n+1 alb X (x j ).\nSince s * n : N 1 (Alb(X)) → N 1 (X ×(2n) ) is (equivariant) injective, we conclude:\nι f = ι f ×(2n) ≤ ι f | Alb(X) ≤ |λ| 2 ≤ d 1 (f | Alb(X) ) ≤ d 1 (f ×(2n) ) = d 1 (f ).\nSo the proposition is proved. (2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number.\n\nThen f is q-polarized.\n\n(3) Suppose f is q-polarized. Then any eigenvalue of f * | H 1 (X,Q ℓ ) has modulus √ q.\n\n(4) Suppose f is q-polarized (resp. int-amplified). Then so are the induced endomorphisms f | Alb(X) : Alb(X) → Alb(X) and f | Alb(X) : Alb(X) → Alb(X) via the Albanese morphism and Albanese map respectively.\n\nProof. We follow the notation in the proof of Proposition 6.3.\n(1) Denote by (f ∨ ) K := f ∨ ⊗ Z K and (X ∨ ) K := X ∨ ⊗ Z K. Note that d 1 (f ) ∈ K. If f * D ≡ d 1 (f )D, then f * D − d 1 (f )D =: ∆ ∈ (X ∨ ) K .\nNote that although (X ∨ ) K is in general an infinite dimensional K-vector space, the Jordan canonical form of (f ∨ ) K has all the Jordan blocks of size ≤ deg P (recall that P is the minimal polynomial of f ∨ ).\n\nBy Proposition 6.3, any diagonal entry λ further satisfies |λ| ≤ d 1 (f ) < d 1 (f ).\n\nIn\nparticular, (f ∨ ) K − d 1 (f ) id (X ∨ ) K is invertible. Thus, there is a K-Cartier divisor L ∈ X ∨ such that f * L − d 1 (f )L ∼ K ∆. Then f * D ′ ∼ K d 1 (f )D ′ for D ′ = D − L.\nFor the second half of (1), applying Perron-Frobenius theorem to the nef cone (cf. [Bir67]), there is a nef\nR-Cartier divisor D 1 such that f * D 1 ≡ d 1 (f )D 1 ;\nwe then apply the first half of (1).\n\n(2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number. (3) just follows from Proposition 6.3. \n\n\nHigher-dimensional Hodge Index Theorem\n\nIn this section, we recall the higher-dimensional Hodge index theorem and give an application (cf. Proposition 7.2) which will be crucially used in Section 8.\n\nThe following lemma is known in [DS04, Corollaire 3.5] for the case of compact Kähler manifolds. See [MZg20, Lemma 4.5] for its full algebraic version. The proof below is taken from [MZg20, Lemma 4.5] with slight simplification for the convenience of the readers.\n\nLemma 7.1. Let X be a normal projective variety. Let L ≡ 0 and D ≡ 0 be two pseudoeffective R-Cartier divisors such that L 2 and D 2 are pseudo-effective (cf. Notation 2.1).\n\nSuppose L · D ≡ w 0. Then L ≡ aD for some a > 0.\n\nProof. We may assume n := dim(X) ≥ 2. Fix very ample Cartier divisors A 1 , · · · , A n−1 on X. Denote by V := RD + RL, and\nW := {x ∈ N 1 (X) | x · A 1 · · · A n−1 = 0}\nsubspaces of N 1 (X). By [Zha16,Lemma 3.2], L ≡ w 0 and D ≡ w 0. Then a := L · A 1 · · · A n−1 /D · A 1 · · · A n−1 > 0, and V ∩ W is of dimension ≤ 1 and spanned by\nD := L − aD ∈ V ∩ W.\nFor each 1 ≤ i ≤ n − 1, consider the following bilinear form on N 1 (X):\nq i (x, y) := x · y · A 1 · · · A i−1 · A i+1 · · · A n−1 .\nChoose a general normal projective surface S on X such that S ≡ w A 1 · · · A i−1 ·A i+1 · · · A n−1 .\n\nThen q i (x, y) = x| S · y| S . By the Hodge index theorem on S, q i is semi-negative on W .\n\nNote that q i is semi-positive on V since L 2 , D 2 are pseudo-effective and q i (L, D) = 0.\n\nHence q i ( D, D) = 0. For any w ∈ W and λ ∈ R, we have q i (λ D − w, λ D − w) ≤ 0. Then the inequality q i (w, w) − 2λq i ( D, w) ≤ 0 holds for any λ ∈ R and w ∈ W . This happens only when q i ( D, w) = 0 for any w ∈ W .\n\nNote that W and A i span N 1 (X) because W is a hyperplane of N 1 (X) and A i ∈ W .\n\nNote also that q i ( D, A i ) = 0. Then\nq i ( D, x) = D · A 1 · · · A i−1 · x · A i+1 · · · A n−1 = 0\nfor any x ∈ N 1 (X). This implies that V ∩ W is independent of the choice of each A i . So D · x 1 · · · x n−1 = 0 for any divisors x 1 , · · · , x n−1 ∈ N 1 (X), which means D ≡ w 0. Note that D is R-Cartier. Therefore, D ≡ 0 by [Zha16, Lemma 3.2].\n\nProposition 7.2. Let X be a normal projective variety of dimension n, L a nef R-Cartier divisor, and D 1 , D 2 two non-zero effective R-Cartier divisors on X. Suppose D 1 and D 2 have no common components and D 1 ≡ D 2 . Suppose further that L · D 1 · H 1 · · · H n−2 = 0 for some nef and big divisors H 1 , · · · , H n−2 . Then L ≡ aD 1 ≡ aD 2 for some a > 0 and L 2 ≡ w D 1 · D 2 ≡ w 0. In particular, Supp D 1 and Supp D 2 are disjoint.\n\nProof. For each H i , we write H i ∼ A i + E i for some ample divisor A i and effective divisor E i . By the assumption, the intersection of (the numerically movable) D 1 with any prime divisor is pseudo-effective, hence so is D 1 · E i for each i. Then L · D 1 · H 1 · · · H n−2 = 0 implies that L · D 1 · A 1 · H 2 · · · H n−2 = 0.\n\nRepeating the same process, we have L · D 1 · A 1 · · · A n−2 = 0 and thus L · D 1 ≡ w 0. By Lemma 7.1 and noting D 2 1 ≡ w D 1 D 2 , we get L ≡ aD 1 (≡ aD 2 ) with a > 0. So we have\nL 2 ≡ w a 2 D 1 · D 2 ≡ w aL · D 1 ≡ w 0. If (Supp D 1 ) ∩ (Supp D 2 ) = ∅, then D 1 · D 2\nis a non-zero effective cycle and hence we get a contradiction with D 1 · D 2 ≡ w 0.\n\n8. Proofs of Theorems 1.6 and 1.7\n\nIn this section, we will prove Theorems 1.7 and 8.1 below which includes Theorem 1.6 as a simpler version and bounds the number ♯P f in terms of the Néron-Severi class group rank ρ ns (X), the Picard number ρ( X) of any projective resolution X of X, and the first dynamical degree d 1 (f ).\n\nTheorem 8.1. Let f : X → X be a surjective endomorphism of a normal projective variety. Let X → X be any projective resolution. Then\n♯P f ≤ max{(1 + d 1 (f )) 2 + a, (ρ ns (X) + 1) · (⌊ ρ( X) 2 ⌋ + 1)},\nwhere a is a positive constant depending only on X. In particular, for n ≫ 1, (1)).\n♯P f n ≤ (d 1 (f )) n · (1 + o\nWe now prepare to prove Theorem 8.1.\n\nNotation 8.2. Let f : X → X be a surjective endomorphism on a normal projective variety X of dimension n. We fix a projective resolution σ : X → X with E 1 , · · · , E k its exceptional prime divisors. For D ∈ P f , denote by D := σ −1 * (D), the proper transform of D. Denote by\nP f := { D = σ −1 * (D) | D ∈ P f }.\nBy Perron-Frobenius theorem (cf. [Bir67]), fix a nef R-Cartier divisor L f ≡ 0 on X with\nf * L f ≡ d 1 (f )L f .\nDenote by\nL f := σ * L f . Lemma 8.3. L f · D ≡ w 0 for any D ∈ P f . Proof. Note that (f | D ) * (L f | D ) ≡ d 1 (f )(L f | D )\nand d 1 (f | D ) < d 1 (f ). Then L f | D ≡ 0 and hence L f · D ≡ w 0.\n\nLemma 8.4. There exist at least 2⌊♯P f /(ρ ns (X) + 1)⌋ disjoint prime divisors D u in P f with L f · D u ≡ w 0.\n\nProof. Let m := ⌊♯P f /(ρ ns (X)+1)⌋. Then we have m pairwise disjoint subsets S 1 , · · · , S m of P f with ♯S t = ρ ns (X) + 1 for each 1 ≤ t ≤ m.\n\nNote that ρ( X) = ρ ns (X) + k where k is the number of σ-exceptional prime divisors (cf. [Kol18, Lemma 18]). By the assumption, we have the equality ♯(S t ∪ {E 1 , · · · , E k }) = ρ( X) + 1.\n\nFor each t, we then have a numerical equivalence of two non-zero effective divisors which are combinations of prime divisors in S t and σ-exceptional prime divisors such that they have no common component. We write them in the following form i∈It a i D i + E t ≡ j∈Jt b j D j + F t where I t and J t are two (possibly empty) disjoint subsets of S t , a i > 0, b j > 0, and E t , F t are effective σ-exceptional divisors. Here without ambiguity, we identify D i with the index i in I t .\n\nLet H be an ample divisor on X. By Lemma 8.3 and the projection formula,\nL f · ( i∈It a i D i + E t ) · (σ * H) n−2 = L f · ( i∈It a i D i ) · H n−2 = 0\nwhere D i = σ * ( D i ) ∈ P f . Applying Proposition 7.2, for each t, we may assume (after some scalar multiplication)\nL f ≡ i∈It a i D i + E t ≡ j∈Jt b j D j + F t\nand we also have\nL f 2 ≡ w L f · ( i∈It a i D i + E t ) ≡ w L f · ( j∈Jt b j D j + F t ) ≡ w 0.\nThis further implies that\n\n(1)\nL f · D i ≡ w L f · D j ≡ w L f · E ≡ w 0\nfor any i ∈ I t , j ∈ J t and any prime divisor E in E t and F t .\n\nWe claim that I t and J t are non-empty. Suppose the contrary, for example, that I t is empty. Then L f ≡ E t = 0. However, this contradicts with the negativity lemma (cf. [KM98, Lemma 3.39]).\n\nConsider the pair (I t , J t ′ ) for any t, t ′ . Note that\n(2) i∈It a i D i + E t ≡ j∈J t ′ b j D j + F t ′ (≡ L f )\nWe claim that E t and F t ′ have no common components. Readjust it a bit in the following way, where ∆ is a σ-exceptional effective R-divisor,\ni∈It a i D i + ( E t − ∆) ≡ j∈J t ′ b j D j + ( F t ′ − ∆),\nso that E t − ∆ and F t ′ − ∆ are effective but with no common components. Note that\nL f · ( i∈It a i D i + ( E t − ∆)) ≡ w 0.\nBy Proposition 7.2, for some e > 0, we have e L f ≡ i∈It a i D i + ( E t − ∆).\n\nThen (1 − e) L f ≡ ∆, and hence e ≤ 1 since both L f and ∆ are pseudo-effective and X is projective. By the negativity lemma again (cf. [KM98, Lemma 3.39]), e = 1 and ∆ ≡ 0, so ∆ = 0 and the claim is proved. By Proposition 7.2 and the display (2) above and since L f · L f ≡ w 0, we have D i ∩ D j = ∅ for any i ∈ I t and j ∈ J t ′ .\n\nThe same argument works for the pairs (I t , I t ′ ) and (J t , J t ′ ) with t = t ′ . In particular, we may pick one prime divisor in I t and one prime divisor in J t for each t, and all these 2m prime divisors are pairwise disjoint. This completes the proof.\n\nWith ♯P f large enough, we can construct an f -equivariant fibration over a curve.\n\nProposition 8.5. Suppose ♯P f ≥ (⌊ ρ( X) 2 ⌋ + 1) · (ρ ns (X) + 1). Then there exists a fibration π : X → C onto a smooth curve C such that the following hold.\n\n(1) L f ≡ π * H C for some ample R-Cartier divisor H C on C.\n\n(2) The fibration π factors through a fibration π : X → C such that L f ≡ π * H C .\n\n(3) There exists a d 1 (f )-polarized endomorphism g : C → C such that π • f = g • π.\n\nProof. (1) By Lemma 8.4, there are ρ( X) + 1 pairwise disjoint prime divisors D u in P f with L f · D u ≡ w 0. By [BPS16, Theorem 1.1], there is a fibration π : X → C onto a smooth projective curve such that π( D u ) is a point for each u and D u = π −1 ( π( D u )) for X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. Let B be the finite set of prime divisors in the reducible fibres of π. By the same argument as in the proof of Proposition 6.1 and [CMZ20, Lemma 7.5], we have an injection ϕ : P −1 f ∞ \\B → P −1 g ∞ , sending D to π(D). Note that there are at most finitely many g −1 -periodic points (cf. [MZ20, Corollary 3.8]). So the theorem is proved.\n\nCorollary 8.6. Let f : X → X be a surjective endomorphism of a normal projective variety X with ♯P f ∞ = ∞. Then D is either Q-Cartier or f −1 -periodic for any D ∈ P f ∞ .\n\nTheorem 1 . 5 .\n15Let f be a dominant self-morphism of a variety X. Then\n\n\n♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases.\n\n\nf n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞.\n\nConjecture 3. 2 .\n2(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set\n\n\nSo f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure.\n\nLemma 4. 2 .\n2Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H ).\n\n[ Bri10 ,\nBri10Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps.\n\n\nsince g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have\n\nCorollary 5. 2 .\n2S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2]).\n\nd 1\n1(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M.\n\n[ Bri10 ,\nBri10Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)\n\n\nany torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write\n\n\nLemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here.\n\n\n]). So the proposition is proved.\n\n\n2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,\n\n\nthe dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X).\n\n\nBy a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says\n\nTheorem 6. 4 .\n4Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 .\n\n( 4 )\n4By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 .\n\n\nCMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]\n\n\nFrobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.In Remark 1.4, the method of counting fixed points for polarized endomorphisms only \n\nholds in characteristic 0, because it applied Serre's theorem to compute the eigenvalues \n\nof f * | H i (X,Z) which heavily relied on a Kähler form. Serre's theorem is also a Kähler \n\nanalogue of Weil's conjecture on eigenvalues of the pullback of (the purely inseparable \n\nq-polarized) \n\n\nThen algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved.\nYOHSUKE MATSUZAWA, SHENG MENG, TAKAHIRO SHIBATA, DE-QI ZHANG, GUOLEI ZHONG\nYOHSUKE MATSUZAWA, SHENG MENG, TAKAHIRO SHIBATA, DE-QI ZHANG, GUOLEI ZHONG\nYOHSUKE MATSUZAWA, SHENG MENG, TAKAHIRO SHIBATA, DE-QI ZHANG, GUOLEI ZHONG\nYOHSUKE MATSUZAWA, SHENG MENG, TAKAHIRO SHIBATA, DE-QI ZHANG, GUOLEI ZHONG\nAcknowledgments. The authors are supported by a JSPS Overseas Research Fellow-Proof. We apply Theorem 1.7 and use the notation there. If D = π −1 (π(D)), then D is Q-Cartier. Otherwise, π −1 (π(D)) is reducible. Denote by Σ the finite subset of C over which π has reducible fibres. Then g −1 (Σ) = Σ by[CMZ20,Lemma 7.4]. Therefore D is an irreducible component of π −1 (Σ) and hence f −1 -periodic.\n)/b, (1) follows. (2) follows from (1) and Lemma 2. H C := T, H C := t π( D 1 )/b, (1) follows. (2) follows from (1) and Lemma 2.2.\n\nFor every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0. For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0\n\nSo every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. Deb01So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. [Deb01,\n\n. * L F ≡ F * Π * H C = Π * G * H C ≡ Qπ * H C ≡ Ql F, which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7f * L f ≡ f * π * H C = π * g * H C ≡ qπ * H C ≡ qL f , which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7.\n\nSuppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X ). Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X )\n\nBy Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4)⌋ + 1). By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4).\n\nBy taking normalization of X. we may assume that X is normal (cf. Lemmas 2.4 and 2.5)Proof of Theorem 1.7. By taking normalization of X, we may assume that X is normal (cf. Lemmas 2.4 and 2.5).\n\nBy [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c\"(2) ⇒ (1).\" By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c\n\nAfter a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. We may also assume that D = π −1 (π(D)) for somed 1 (f s | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. \"(1) ⇒ (2).\" After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. We may also assume that D = π −1 (π(D)) for some\n\nThen d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant. D , after iteration) fibration π : ReferencesD ∈ P f . Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant (after iteration) fibration π : References\n\nLinear transformations with invariant cones. G Birkhoff, Amer. Math. Monthly. 74G. Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274- 276.\n\nFamilies of disjoint divisors on varieties. F Bogomolov, A Pirutka, A Silberstein, Eur. J. Math. 24F. Bogomolov, A. Pirutka and A. Silberstein, Families of disjoint divisors on varieties, Eur. J. Math. 2 (2016), no. 4, 917-928.\n\nSome basic results on actions of nonaffine algebraic groups. M Brion, Symmetry and spaces. Boston, MA278BirkhäuserM. Brion, Some basic results on actions of nonaffine algebraic groups, in: Symmetry and spaces, 1-20, Progr. Math. 278, Birkhäuser, Boston, MA, 2010.\n\nPolarized endomorphisms of normal projective threefolds in arbitrary characteristic. P Cascini, S Meng, D.-Q Zhang, Math. Ann. 3781-2P. Cascini, S. Meng and D.-Q. Zhang, Polarized endomorphisms of normal projective threefolds in arbitrary characteristic, Math. Ann. 378 (2020), no. 1-2, 637-665.\n\nDegrees of iterates of rational maps on normal projective varieties. N.-B Dang, Proc. Lond. Math. Soc. 3N.-B. Dang, Degrees of iterates of rational maps on normal projective varieties, Proc. Lond. Math. Soc. (3) 121 (2020), no. 5, 1268-1310.\n\nHigher-dimensional algebraic geometry. O Debarre, Springer-VerlagNew YorkUniversitextO. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001.\n\n. P Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43P. Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43 (1974), 273-307.\n\nComparison of dynamical degrees for semi-conjugate meromorphic maps. T.-C Dinh, V.-A Nguyen, Comment. Math. Helv. 864T.-C. Dinh and V.-A. Nguyen, Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011), no. 4, 817-840.\n\nGroupes commutatifs d'automorphismes d'une variété kählérienne. T.-C Dinh, N Sibony, Duke Math. J. 1232T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne, Duke Math. J. 123 (2004), no. 2, 311-328.\n\nUne borne supérieure pour l'entropie topologique d'une application rationnelle. T.-C Dinh, N Sibony, Annals of Math. 2T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Annals of Math. (2) 161 (2005), no. 3, 1637-1644.\n\nQuestions on self-maps of algebraic varieties. N Fakhruddin, J. Ramanujan Math. Soc. 182N. Fakhruddin, Questions on self-maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109-122.\n\nFundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs. B Fantechi, L Göttsche, L Illusie, S L Kleiman, N Nitsure, A Vistoli, American Mathematical Society123Providence, RIB. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs, 123. American Mathematical Society, Providence, RI.\n\nW Fulton, Intersection theory. BerlinSpringer-VerlagSecond editionW. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998.\n\nRepresentation theory: a first course. W Fulton, J Harris, Graduate Texts in Mathematics. 129Springer-VerlagReadings in MathematicsW. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.\n\nAutomorphisms of affine algebraic groups. G Hochschild, G D Mostow, J. Algebra. 13G. Hochschild and G. D. Mostow, Automorphisms of affine algebraic groups, J. Algebra 13 (1969), 535-543.\n\nCohomological and numerical dynamical degrees on abelian varieties. F Hu, Algebra Number Theory. 138F. Hu, Cohomological and numerical dynamical degrees on abelian varieties, Algebra Number Theory 13 (2019), no. 8, 1941-1958.\n\nLogarithmic forms of algebraic varieties. S Iitaka, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 233S. Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 525-544.\n\nOn the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. S Kawaguchi, J Silverman, J. Reine Angew. Math. 713S. Kawaguchi and J. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. 713 (2016), 21-48.\n\nBirational geometry of algebraic varieties. J Kollár, S Mori, Cambridge Univ. PressCambridgeJ. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, Cam- bridge, 1998.\n\n. J Kollár, arXiv:1803.07596Mumford divisorsJ. Kollár, Mumford divisors, arXiv:1803.07596\n\nAbelian varieties. S Lang, Springer-VerlagNew York-BerlinS. Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983.\n\nCanonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture. J Lesieutre, M Satriano, 10.1093/imrn/rnz067Int. Math. Res. Not. J. Lesieutre and M. Satriano, Canonical heights on hyper-Kähler varieties and the Kawaguchi- Silverman conjecture, Int. Math. Res. Not., https://doi.org/10.1093/imrn/rnz067\n\nPulling back cohomology classes and dynamical degrees of monomial maps. J Lin, Bull. Soc. Math. France. 1404J. Lin, Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France 140 (2012), no. 4, 533-549 (2013).\n\nArithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Y Matsuzawa, K Sano, Systems. 406Y. Matsuzawa and K. Sano, Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Systems 40 (2020), no. 6, 1655-1672.\n\nY Matsuzawa, S Meng, T Shibata, D.-Q Zhang, arXiv:2002.10976Non-density of points of small arithmetic degrees. Y. Matsuzawa, S. Meng, T. Shibata and D.-Q. Zhang, Non-density of points of small arith- metic degrees, arXiv:2002.10976\n\nBuilding blocks of amplified endomorphisms of normal projective varieties. S Meng, Math. Z. 2943S. Meng, Building blocks of amplified endomorphisms of normal projective varieties, Math. Z. 294 (2020), no. 3, 1727-1747.\n\nJordan property for non-linear algebraic groups and projective varieties. S Meng, D.-Q Zhang, Amer. J. Math. 1404S. Meng and D.-Q. Zhang, Jordan property for non-linear algebraic groups and projective varieties, Amer. J. Math. 140 (2018), no. 4, 1133-1145.\n\nBuilding blocks of polarized endomorphisms of normal projective varieties. S Meng, D.-Q Zhang, Adv. Math. 325S. Meng and D.-Q. Zhang, Building blocks of polarized endomorphisms of normal projective varieties, Adv. Math. 325 (2018), 243-273.\n\nS Meng, D.-Q Zhang, Kawaguchi-Silverman, arXiv:1908.01605Conjecture for surjective endomorphisms. S. Meng and D.-Q. Zhang, Kawaguchi-Silverman Conjecture for surjective endomorphisms, arXiv:1908.01605\n\nSemi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism. S Meng, D.-Q Zhang, Math. Res. Lett. 272S. Meng and D.-Q. Zhang, Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism, Math. Res. Lett. 27 (2020), no. 2, 523-550.\n\nS Meng, G Zhong, arXiv:2005.03983Rigidity of rationally connected smooth projective varieties from dynamical viewpoints. S. Meng and G. Zhong, Rigidity of rationally connected smooth projective varieties from dy- namical viewpoints, arXiv:2005.03983\n\nOn complex normal projective surfaces admitting non-isomorphic surjective endomorphisms. N Nakayama, preprintN. Nakayama, On complex normal projective surfaces admitting non-isomorphic surjective en- domorphisms, preprint September 2008.\n\nPolarized endomorphisms of complex normal varieties. N Nakayama, D.-Q Zhang, Math. Ann. 3464N. Nakayama and D.-Q. Zhang, Polarized endomorphisms of complex normal varieties, Math. Ann. 346 (2010), no. 4, 991-1018.\n\nThe canonical heights for Jordan blocks of small eigenvalues. K Sano, arXiv:1712.07533preperiodic points, and the arithmetic degrees. K. Sano, The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees, arXiv:1712.07533\n\nLinear algebraic groups. T A Springer, Modern Birkhaüser Classics. Birkhaüser Boston IncT. A. Springer, Linear algebraic groups, Modern Birkhaüser Classics, Boston, MA: Birkhaüser Boston Inc., 2009.\n\nn-dimensional projective varieties with the action of an abelian group of rank n − 1. D.-Q Zhang, Trans. Amer. Math. Soc. 36812D.-Q. Zhang, n-dimensional projective varieties with the action of an abelian group of rank n − 1, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8849-8872.\n\nDistributions in algebraic dynamics. S.-W Zhang, Department of Mathematics. 10Brown UniversitySurv. Differ. Geom.S.-W. Zhang, Distributions in algebraic dynamics, Surv. Differ. Geom., 10 (2006), 381-430. Department of Mathematics, box 1917, Brown University, Rhode Island 02912, USA\n"},"corpus_id":{"kind":"number","value":218900910,"string":"218,900,910"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2005.13368\",\n \"mag\": 3154238011,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1093/imrn/rnab039\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2005.13368v2.pdf\"\n ],\n \"pdf_sha\": \"5c8b6f80d419f98ec6f3a5203a5ea0d84445a45a\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/2005.13368\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"INVARIANT SUBVARIETIES WITH SMALL DYNAMICAL DEGREE"},"authors":{"kind":"list like","value":["Yohsuke Matsuzawa","Sheng Meng","Takahiro Shibata","De-Qi Zhang","Guolei Zhong","Yohsuke Matsuzawa","Sheng Meng","Takahiro Shibata","De-Qi Zhang","Guolei Zhong"],"string":"[\n \"Yohsuke Matsuzawa\",\n \"Sheng Meng\",\n \"Takahiro Shibata\",\n \"De-Qi Zhang\",\n \"Guolei Zhong\",\n \"Yohsuke Matsuzawa\",\n \"Sheng Meng\",\n \"Takahiro Shibata\",\n \"De-Qi Zhang\",\n \"Guolei Zhong\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Let f : X → X be a dominant self-morphism of an algebraic variety.Consider the set Σ f ∞ of f -periodic subvarieties of small dynamical degree, the subsetWhen X is projective, we prove the finiteness of the set P f of f -invariant prime divisors with small dynamical degree, and give an optimal upper boundas n → ∞, where d 1 (f ) is the first dynamic degree. When X is an algebraic group (with f being a translation of an isogeny), or a (not necessarily complete) toric variety, wegive an optimal upper boundas n → ∞, which slightly generalizes a conjecture of S.-W. Zhang for polarized f . 1 2 YOHSUKE MATSUZAWA, SHENG MENG, TAKAHIRO SHIBATA, DE-QI ZHANG, GUOLEI ZHONGLet f : X → X be a dominant self-morphism of an algebraic variety X. Let V ⊆ X be an (irreducible) closed subvariety. We say thatAn f -periodic closed subvariety V is said to be of Small Dynamical Degree (SDD forDenote by Σ f ∞ the set of f -periodic (irreducible closed) subvarieties of small dynamicalDenote by S f ∞ the set of maximal elements in Σ f ∞ . Denote by S f (resp. P f )"},"sections":{"kind":"list like","value":[{"title":"Connection of two arithmetic dynamical conjectures","content":""},{"title":"Connection of two arithmetic dynamical conjectures","content":"In this section, we work over a number field K. We will give the relation between two conjectures in Theorem 3.4, which also gives an arithmetic motivation of this paper. We refer to [MMSZ20, Section 2] for the notation involved in this section.\nIn this section, we work over a number field K. We will give the relation between two conjectures in Theorem 3.4, which also gives an arithmetic motivation of this paper. We refer to [MMSZ20, Section 2] for the notation involved in this section.\nThe following are the two main conjectures studied in [MMSZ20].\nThe following are the two main conjectures studied in [MMSZ20].\nsurjective endomorphism of a projective variety X defined over a number field K. Let\nsurjective endomorphism of a projective variety X defined over a number field K. Let\nWe copy [MMSZ20, Question 8.4] in the following. Note that for the equivalence of the above two conjectures, we may only focus on Y ∈ S f ∞ in Question 3.3 (1) and (2); see the proof of Theorem 3.4.\nWe copy [MMSZ20, Question 8.4] in the following. Note that for the equivalence of the above two conjectures, we may only focus on Y ∈ S f ∞ in Question 3.3 (1) and (2); see the proof of Theorem 3.4.\nempty for s ≫ 1?\nempty for s ≫ 1?\n(2) Is there a positive integer N such that Y (d) is empty for any f -periodic subvariety Y ⊆ X K of small dynamical degree whose period is larger than N?\n(2) Is there a positive integer N such that Y (d) is empty for any f -periodic subvariety Y ⊆ X K of small dynamical degree whose period is larger than N?\nwhich is a union of irreducible closed subsets and Y 0 :\nwhich is a union of irreducible closed subsets and Y 0 :\nwhich is not Zariski dense in X K .\nwhich is not Zariski dense in X K ."},{"title":"Isogenies of algebraic groups","content":""},{"title":"Isogenies of algebraic groups","content":"We use the following notation throughout this section.\nWe use the following notation throughout this section.\nNotation 4.1. Let G be a connected algebraic group with the identity element e. We focus on the isogeny (i.e., surjective group homomorphism with finite kernel)\nNotation 4.1. Let G be a connected algebraic group with the identity element e. We focus on the isogeny (i.e., surjective group homomorphism with finite kernel)\nwhere R a is the right multiplication map by some a ∈ G. Note that there exists a G-\nwhere R a is the right multiplication map by some a ∈ G. Note that there exists a G-\n). Then the same proof for [MS18, Lemma 5.6] shows that\n). Then the same proof for [MS18, Lemma 5.6] shows that\nWe recall that g and hence f are finite surjective endomorphisms. Proof.\nWe recall that g and hence f are finite surjective endomorphisms. Proof.\nMoreover, We have the following commutative diagram:\nMoreover, We have the following commutative diagram:\nThe following result is a generalization of Lemma 4.2 to arbitrary closed subvariety of G, which plays a crucial role in many later results.\nThe following result is a generalization of Lemma 4.2 to arbitrary closed subvariety of G, which plays a crucial role in many later results.\nThen X e ⊆ X e p −1 (X e p −1 ) −1 ⊆ g(H) and hence H ≤ g(H) by the construction of H.\nThen X e ⊆ X e p −1 (X e p −1 ) −1 ⊆ g(H) and hence H ≤ g(H) by the construction of H.\nSince g is an isogeny, g(H) = H.\nSince g is an isogeny, g(H) = H.\nConsider the alternate multiplication map s n : X ×(2n) → H defined by\nConsider the alternate multiplication map s n : X ×(2n) → H defined by\nThen the image of s n is (X e X −1 e ) ·n = H when n ≫ 1 by [Spr09, Proposition 2.2.6] and noting that e ∈ X e . Consider the following diagram with n ≫ 1:\nThen the image of s n is (X e X −1 e ) ·n = H when n ≫ 1 by [Spr09, Proposition 2.2.6] and noting that e ∈ X e . Consider the following diagram with n ≫ 1:\nThen\nThen\nSo the diagram commutes. By the product formula, we have\nSo the diagram commutes. By the product formula, we have\nOn the other hand, note that\nOn the other hand, note that\nLemma 4.4. There exists a unique X ∈ S g ∞ containing e ∈ G. Moreover, X is an algebraic subgroup and g-invariant.\nLemma 4.4. There exists a unique X ∈ S g ∞ containing e ∈ G. Moreover, X is an algebraic subgroup and g-invariant.\nProof. Take X ∈ S g ∞ containing e. It exists because d 1 (g) > 1 and d 1 (g| {e} ) = 1. Take s > 0 such that g s (X) = X. Take the smallest (connected closed) algebraic subgroup H of G containing X. By Proposition 4.3, g s (H) = H and d 1 (\nProof. Take X ∈ S g ∞ containing e. It exists because d 1 (g) > 1 and d 1 (g| {e} ) = 1. Take s > 0 such that g s (X) = X. Take the smallest (connected closed) algebraic subgroup H of G containing X. By Proposition 4.3, g s (H) = H and d 1 (\nBy the maximality of X, we have X = H and it is an algebraic subgroup.\nBy the maximality of X, we have X = H and it is an algebraic subgroup.\nTake another Y ∈ S g ∞ containing e and then take t > 0 such that g t (Y ) = Y .\nTake another Y ∈ S g ∞ containing e and then take t > 0 such that g t (Y ) = Y .\nThe argument in the previous paragraph shows that Y is a (connected closed) algebraic subgroup. Let Z be the smallest algebraic group containing the g st -invariant irreducible closed subvariety XY . Apply the similar commutative diagram (*) and argument in the proof of Proposition 4.3,\nThe argument in the previous paragraph shows that Y is a (connected closed) algebraic subgroup. Let Z be the smallest algebraic group containing the g st -invariant irreducible closed subvariety XY . Apply the similar commutative diagram (*) and argument in the proof of Proposition 4.3,\nis surjective with n ≫ 1. Then we have d 1 (g st | Z ) < d 1 (g st ) and hence Z = X = Y by the maximality of X and Y . This shows the uniqueness of X.\nis surjective with n ≫ 1. Then we have d 1 (g st | Z ) < d 1 (g st ) and hence Z = X = Y by the maximality of X and Y . This shows the uniqueness of X.\nNext we show that X is g-invariant. Note that g(X) is also g s -invariant and contains e. By the product formula, the following diagram shows d 1 (g s | g(X) ) ≤ d 1 (g s | X ) < d 1 (g s ):\nNext we show that X is g-invariant. Note that g(X) is also g s -invariant and contains e. By the product formula, the following diagram shows d 1 (g s | g(X) ) ≤ d 1 (g s | X ) < d 1 (g s ):\nTherefore, we have g(X) ⊆ X by the maximality and uniqueness of X. In particular, X is g-invariant since g is an isogeny. This finishes our proof.\nTherefore, we have g(X) ⊆ X by the maximality and uniqueness of X. In particular, X is g-invariant since g is an isogeny. This finishes our proof.\nDefinition 4.5. With the notation in Lemma 4.4, we denote by\nDefinition 4.5. With the notation in Lemma 4.4, we denote by\nWe shall prove in the next section that G(f ) is actually a normal subgroup of G.\nWe shall prove in the next section that G(f ) is actually a normal subgroup of G.\nProof. Assume that X ∈ S f . Fix some b ∈ X and denote by X e := Xb −1 which contains e ∈ G. Let H be the smallest algebraic subgroup of G containing X e . By Proposition\nProof. Assume that X ∈ S f . Fix some b ∈ X and denote by X e := Xb −1 which contains e ∈ G. Let H be the smallest algebraic subgroup of G containing X e . By Proposition\nBy Lemma 4.2,\nBy Lemma 4.2,\nprevious paragraph. But this implies that X = Y since both are translations of the same group G(f ). So X is maximal.\nprevious paragraph. But this implies that X = Y since both are translations of the same group G(f ). So X is maximal.\nThe following modified product formula will be frequently used in the next Theorem and also next section.\nThe following modified product formula will be frequently used in the next Theorem and also next section.\nLemma 4.7. Consider the commutative diagram of homomorphisms of connected algebraic groups:\nLemma 4.7. Consider the commutative diagram of homomorphisms of connected algebraic groups:\nwhere g is an isogeny. Then π is f -equivariant and the following hold.\nwhere g is an isogeny. Then π is f -equivariant and the following hold.\n) and π induces a bijection between S f and S f | C .\n) and π induces a bijection between S f and S f | C .\nLet A act on A and C by the right multiplication. By Note that the graph of π is A-equivariant. Replacing A by the A-equivariant resolution of this graph, we may further assume that π is a well-defined algebraic fibration.\nLet A act on A and C by the right multiplication. By Note that the graph of π is A-equivariant. Replacing A by the A-equivariant resolution of this graph, we may further assume that π is a well-defined algebraic fibration.\nBy the generic smoothness of π, there exists a general point c 1 = π(a 1 ) ∈ C such that the fibre A c 1 := π −1 (c 1 ) is an (irreducible) smooth projective variety. For any\nBy the generic smoothness of π, there exists a general point c 1 = π(a 1 ) ∈ C such that the fibre A c 1 := π −1 (c 1 ) is an (irreducible) smooth projective variety. For any\n. So all the fibres of π over C are (irreducible) smooth projective varieties isomorphic to each other.\n. So all the fibres of π over C are (irreducible) smooth projective varieties isomorphic to each other.\nWe refer to [DN11] for the definitions of critical value and critical set. Now we have that each point of C is not a critical value of π. Note that B is g-invariant and g isétale. Then\nWe refer to [DN11] for the definitions of critical value and critical set. Now we have that each point of C is not a critical value of π. Note that B is g-invariant and g isétale. Then\nA e (the closure of B in A) is not contained in the critical set of g. Therefore, we verified the assumptions for A e in [DN11, Remark 3.4] and hence d 1 (g| π ) = d 1 (g| A e ) = d 1 (g| B ).\nA e (the closure of B in A) is not contained in the critical set of g. Therefore, we verified the assumptions for A e in [DN11, Remark 3.4] and hence d 1 (g| π ) = d 1 (g| A e ) = d 1 (g| B ).\nTogether with the product formula, (1) is proved.\nTogether with the product formula, (1) is proved.\nWe apply the maximality and uniqueness of A(g) and C(g| C ) (cf. Lemma 4.4) and our new product formula (1) several times in this diagram. First, we have B ≤ A(g).\nWe apply the maximality and uniqueness of A(g) and C(g| C ) (cf. Lemma 4.4) and our new product formula (1) several times in this diagram. First, we have B ≤ A(g).\nNote that d 1 (g| π(A(g)) ) = d 1 (g| A(g) ) < d 1 (g). So we have π(A(g)) ≤ C(g| C ). Conversely,\nNote that d 1 (g| π(A(g)) ) = d 1 (g| A(g) ) < d 1 (g). So we have π(A(g)) ≤ C(g| C ). Conversely,\nFinally, we apply Proposition 4.6 for f and f | C , then we see that the induced map\nFinally, we apply Proposition 4.6 for f and f | C , then we see that the induced map\nis a bijection. So (2) is proved.\nis a bijection. So (2) is proved.\nWe complete this section by considering the commutative algebraic groups first.\nWe complete this section by considering the commutative algebraic groups first.\nTheorem 4.8. Let f : G → G be a dominant self-morphism of a connected commutative algebraic group G, such that f = R a • g where g is an isogeny and a ∈ G. Suppose d 1 (f ) > 1. Let G(f ) be as in Definition 4.5. Then\nTheorem 4.8. Let f : G → G be a dominant self-morphism of a connected commutative algebraic group G, such that f = R a • g where g is an isogeny and a ∈ G. Suppose d 1 (f ) > 1. Let G(f ) be as in Definition 4.5. Then\nThis contradicts the maximality of G(g).\nThis contradicts the maximality of G(g)."},{"title":"Now we have shown","content":""},{"title":"Now we have shown","content":"and the theorem is proved.\nand the theorem is proved."},{"title":"Proof of Theorem 1.5","content":""},{"title":"Proof of Theorem 1.5","content":"In this section, we focus on the upper bound in Question 1.3. We first consider the case of abelian varieties.\nIn this section, we focus on the upper bound in Question 1.3. We first consider the case of abelian varieties.\nTheorem 5.1. Let f be a surjective endomorphism of an abelian variety A. Then\nTheorem 5.1. Let f be a surjective endomorphism of an abelian variety A. Then\n(cf. Definition 4.5 for A(f )).\n(cf. Definition 4.5 for A(f )).\nProof. We may assume d 1 (f ) > 1. Write f = R a • g where g is an isogeny and a ∈ A.\nProof. We may assume d 1 (f ) > 1. Write f = R a • g where g is an isogeny and a ∈ A.\nSince d 1 (f ) = d 1 (g) and A(f ) = A(g), applying Theorem 4.8, we may assume that f is an isogeny, i.e., f = g; further we have an f -equivariant quotient morphism\nSince d 1 (f ) = d 1 (g) and A(f ) = A(g), applying Theorem 4.8, we may assume that f is an isogeny, i.e., f = g; further we have an f -equivariant quotient morphism"},{"title":"see the ending part of the proof of Theorem 4.8), which is an isogeny. Then","content":""},{"title":"see the ending part of the proof of Theorem 4.8), which is an isogeny. Then","content":""},{"title":"by [Hu19, Formula (3.6)] and the product formula.","content":""},{"title":"by [Hu19, Formula (3.6)] and the product formula.","content":"Then the theorem follows.\nThen the theorem follows.\nRecall that a normal projective variety X is said to be Q-abelian (cf. [NZ10]) if there is a finite surjective morphism A → X from an abelian variety A, which isétale in codimension one. When f is a group homomorphism, it induces the following commutative diagram where T is an algebraic torus and A = Alb(X) is an abelian variety:\nRecall that a normal projective variety X is said to be Q-abelian (cf. [NZ10]) if there is a finite surjective morphism A → X from an abelian variety A, which isétale in codimension one. When f is a group homomorphism, it induces the following commutative diagram where T is an algebraic torus and A = Alb(X) is an abelian variety:\nTheorem 5.3. Let f be a dominant self-morphism of a semi-abelian variety X. Then\nTheorem 5.3. Let f be a dominant self-morphism of a semi-abelian variety X. Then\nProof. By the same argument as in the first paragraph of the proof of Theorem 5.1, we may replace f by g and X by X/X(f ) so that dim(X(f )) = 0 and both f and\nProof. By the same argument as in the first paragraph of the proof of Theorem 5.1, we may replace f by g and X by X/X(f ) so that dim(X(f )) = 0 and both f and\nConsider the commutative diagram before Theorem 5. 3. Note that f | T is defined by a matrix M of size dim(T ) with integer coefficients and a non-zero determinant. Then We compute\nConsider the commutative diagram before Theorem 5. 3. Note that f | T is defined by a matrix M of size dim(T ) with integer coefficients and a non-zero determinant. Then We compute\nTogether with the computation in the proof of Theorem 5.1 for abelian varieties, we have\nTogether with the computation in the proof of Theorem 5.1 for abelian varieties, we have\nwhere the last inequality follows from\nwhere the last inequality follows from\n[MS18, Theorem 1.6] or Lemma 4.7. So the theorem is proved.\n[MS18, Theorem 1.6] or Lemma 4.7. So the theorem is proved.\nRemark 5.4. In the proof of Theorem 5.3, we see that if X = T is an algebraic torus, then the computation indeed shows that ♯S f ≤ (d 1 (f ) + 1) dim(T )−dim(T (f )) .\nRemark 5.4. In the proof of Theorem 5.3, we see that if X = T is an algebraic torus, then the computation indeed shows that ♯S f ≤ (d 1 (f ) + 1) dim(T )−dim(T (f )) .\nWe now prepare to deal with general algebraic groups. See, for instance, [MZ18a,§3] for several decomposition facts about algebraic groups. First, we need observations for unipotent groups and semisimple groups below.\nWe now prepare to deal with general algebraic groups. See, for instance, [MZ18a,§3] for several decomposition facts about algebraic groups. First, we need observations for unipotent groups and semisimple groups below.\nLemma 5.5. Let g : U → U be an isogeny of a (connected) unipotent group U. Then d 1 (g) = 1.\nLemma 5.5. Let g : U → U be an isogeny of a (connected) unipotent group U. Then d 1 (g) = 1.\nProof. Since Ker(g) is a finite subgroup of the unipotent group U, it is trivial, so g is an automorphism. We show by induction on dim(U) and assume dim(U) > 0. Note that U is solvable. So its commutator subgroup U (1) is strictly smaller than U. Note that g(U (1) ) = U (1) . Consider the following commutative diagram:\nProof. Since Ker(g) is a finite subgroup of the unipotent group U, it is trivial, so g is an automorphism. We show by induction on dim(U) and assume dim(U) > 0. Note that U is solvable. So its commutator subgroup U (1) is strictly smaller than U. Note that g(U (1) ) = U (1) . Consider the following commutative diagram:\nis not trivial. By Lemma 4.7 and induction, we have d 1 (g) = max{d 1 (g| U (1) ), d 1 (g| U/U (1) )} = 1.\nis not trivial. By Lemma 4.7 and induction, we have d 1 (g) = max{d 1 (g| U (1) ), d 1 (g| U/U (1) )} = 1.\nSuppose U (1) is trivial. Then U is a commutative unipotent group which is isomorphic to k ×dim(U ) . One checks that g ∈ GL(U) since g is additive. By is finite. Replacing g by a power, we may assume g is in the connected algebraic group Aut • (S). We are done by the same argument at the ending part of Lemma 5.5.\nSuppose U (1) is trivial. Then U is a commutative unipotent group which is isomorphic to k ×dim(U ) . One checks that g ∈ GL(U) since g is additive. By is finite. Replacing g by a power, we may assume g is in the connected algebraic group Aut • (S). We are done by the same argument at the ending part of Lemma 5.5.\nFinally, we are able to give the same upper bound for any algebraic group.\nFinally, we are able to give the same upper bound for any algebraic group.\nTheorem 5.7. Let G be a connected algebraic group. Then G(f ) is normal in G and\nTheorem 5.7. Let G be a connected algebraic group. Then G(f ) is normal in G and\nfor any self-morphism f = R a • g where g is an isogeny and a ∈ G.\nfor any self-morphism f = R a • g where g is an isogeny and a ∈ G.\nProof. We may assume 1 < d 1 (f ) (= d 1 (g)). Let G a be the maximal connected affine normal subgroup of G, and U the unipotent radical of G a . Then g(G a ) = G a and g(U) = U. Consider the following commutative diagram:\nProof. We may assume 1 < d 1 (f ) (= d 1 (g)). Let G a be the maximal connected affine normal subgroup of G, and U the unipotent radical of G a . Then g(G a ) = G a and g(U) = U. Consider the following commutative diagram:\nBy Lemma 5.5, we have d 1 (g| U ) = 1 < d 1 (g). By Lemma 4.7, π is f -equivariant and\nBy Lemma 5.5, we have d 1 (g| U ) = 1 < d 1 (g). By Lemma 4.7, π is f -equivariant and\nnormal in G/U. Therefore, we may replace G by G/U and assume that U is trivial.\nnormal in G/U. Therefore, we may replace G by G/U and assume that U is trivial.\nLet S := G (1) be the commutator subgroup of G. Then g(S) = S, and S = (G a ) (1) which is a semisimple group since G a is a reductive group now. Consider the following commutative diagram:\nLet S := G (1) be the commutator subgroup of G. Then g(S) = S, and S = (G a ) (1) which is a semisimple group since G a is a reductive group now. Consider the following commutative diagram:\nBy Lemma 5.6, we have d 1 (g| S ) = 1. By Lemma 4.7 and an argument similar to the previous paragraph, we may replace G by G/S and assume that G is commutative.\nBy Lemma 5.6, we have d 1 (g| S ) = 1. By Lemma 4.7 and an argument similar to the previous paragraph, we may replace G by G/S and assume that G is commutative.\nNote that now G is a semi-abelian variety. In particular, G(f ) is normal in G, and the upper bound is as claimed, by Theorem 5.3 and Lemma 4.7.\nNote that now G is a semi-abelian variety. In particular, G(f ) is normal in G, and the upper bound is as claimed, by Theorem 5.3 and Lemma 4.7.\nAt the end of this section, we treat the toric varieties. Let X be a toric variety and T the big torus. A subvariety of the form O = x · T is called a torus orbit on X. Note that X = O⊆X O is a finite disjoint union of all torus orbits on X.\nAt the end of this section, we treat the toric varieties. Let X be a toric variety and T the big torus. A subvariety of the form O = x · T is called a torus orbit on X. Note that X = O⊆X O is a finite disjoint union of all torus orbits on X.\nTheorem 5.8. Let X be a toric variety with the big torus T ⊆ X. Let f : X → X be a dominant self-morphism such that f (T ) ⊆ T . Then\nTheorem 5.8. Let X be a toric variety with the big torus T ⊆ X. Let f : X → X be a dominant self-morphism such that f (T ) ⊆ T . Then\nwhere the sum runs over all the torus orbits on X.\nwhere the sum runs over all the torus orbits on X.\nProof. By [Nak08, Lemma 3.4.3], f = R a • g such that g| T is an isogeny and g(x · t) = g(x) · g(t); see [Nak08, Section 3.4] for details. Here R a is the natural extension of the right multiplication map on T by a ∈ T . Since g| T is dominant, f (T ) = g(T ) · a = T .\nProof. By [Nak08, Lemma 3.4.3], f = R a • g such that g| T is an isogeny and g(x · t) = g(x) · g(t); see [Nak08, Section 3.4] for details. Here R a is the natural extension of the right multiplication map on T by a ∈ T . Since g| T is dominant, f (T ) = g(T ) · a = T .\nIn particular, f maps every torus orbit onto a torus orbit. Note that\nIn particular, f maps every torus orbit onto a torus orbit. Note that\nThen there is a torus orbit O such that V ∩ O contains an open dense subset of V . So\nThen there is a torus orbit O such that V ∩ O contains an open dense subset of V . So\nThen\nThen\nLemma 5.6]). Define the natural surjective morphism\nLemma 5.6]). Define the natural surjective morphism\nBy the product formula,\nBy the product formula,\nThis finishes the proof.\nThis finishes the proof.\nRemark 5.9. In Theorem 5.8, as we do not assume X is projective, f is not necessarily finite (nor surjective). Also during its proof, it is possible that O ⊆ Sing(X), and then 6. Reduction to polarized case: Proof of Theorem 1.9\nRemark 5.9. In Theorem 5.8, as we do not assume X is projective, f is not necessarily finite (nor surjective). Also during its proof, it is possible that O ⊆ Sing(X), and then 6. Reduction to polarized case: Proof of Theorem 1.9\nIn this section, the ground field k is allowed to have arbitrary characteristic. We will prove Theorem 6.4 which includes Theorem 1.9 as a special case. The connection with polarized endomorphism will give us finiteness for ♯P f .\nIn this section, the ground field k is allowed to have arbitrary characteristic. We will prove Theorem 6.4 which includes Theorem 1.9 as a special case. The connection with polarized endomorphism will give us finiteness for ♯P f .\nProposition 6.1. Let π : X Y be a dominant rational map between projective\nProposition 6.1. Let π : X Y be a dominant rational map between projective\nvarieties. Let f : X → X and g : Y → Y be surjective endomorphisms such that\nvarieties. Let f : X → X and g : Y → Y be surjective endomorphisms such that\nThen for n ≫ 1,\nThen for n ≫ 1,\nProof. By Lemma 2.4, replacing π with its graph and taking the normalizations, we may assume that π is a surjective morphism and both X and Y are normal.\nProof. By Lemma 2.4, replacing π with its graph and taking the normalizations, we may assume that π is a surjective morphism and both X and Y are normal.\nTake the Stein factorization X → Y 1 → Y such that p 1 : X → Y 1 has connected fibres and p 2 : Y 1 → Y is finite. By [CMZ20, Lemma 5.2], f descends to a surjective\nTake the Stein factorization X → Y 1 → Y such that p 1 : X → Y 1 has connected fibres and p 2 : Y 1 → Y is finite. By [CMZ20, Lemma 5.2], f descends to a surjective\nHence we may assume further π has connected fibres.\nHence we may assume further π has connected fibres.\npolarized, a contradiction. For the upper bound, we may assume P f ∞ is an infinite set whose elements are contained in fibres of π. Then dim Y = 1. Let B be the finite set of prime divisors in the (finitely many) reducible fibres of π. We have an injection\npolarized, a contradiction. For the upper bound, we may assume P f ∞ is an infinite set whose elements are contained in fibres of π. Then dim Y = 1. Let B be the finite set of prime divisors in the (finitely many) reducible fibres of π. We have an injection\nsending D to π(D). Then our argument follows from Proposition 6.2.\nsending D to π(D). Then our argument follows from Proposition 6.2.\nThe following result is well known when the ground field k has char k = 0 and the proof is not that troublesome due to the Hodge decomposition. Our proof here works when char k is arbitrary, and is related to a generalized problem of Weil's Riemann Hypothesis.\nThe following result is well known when the ground field k has char k = 0 and the proof is not that troublesome due to the Hodge decomposition. Our proof here works when char k is arbitrary, and is related to a generalized problem of Weil's Riemann Hypothesis.\nProposition 6.2. Let f : C → C be a non-isomorphic endomorphism on a smooth projective curve C. Then ♯ Fix(f ) ≤ deg f + 2c(deg f ) 1/2 + 1 where c is the genus of C.\nProposition 6.2. Let f : C → C be a non-isomorphic endomorphism on a smooth projective curve C. Then ♯ Fix(f ) ≤ deg f + 2c(deg f ) 1/2 + 1 where c is the genus of C.\nProof. Let d := d 1 (f ) = deg f ≥ 2. Then f is d-polarized, so is its induced endomorphism g : Alb(C) → Alb(C) by Theorem 6.4. Note that f has at most finitely many fixed points.\nProof. Let d := d 1 (f ) = deg f ≥ 2. Then f is d-polarized, so is its induced endomorphism g : Alb(C) → Alb(C) by Theorem 6.4. Note that f has at most finitely many fixed points.\nApplying Lefschetz fixed-point formula, we have\nApplying Lefschetz fixed-point formula, we have\n. We are going to prove Theorem 6.4 (cf. Theorem 1.9). It has its own independent interest and various applications, and is also needed in proving Proposition 6. Remark 5.9] without the smooth and characteristic 0 assumptions.\n. We are going to prove Theorem 6.4 (cf. Theorem 1.9). It has its own independent interest and various applications, and is also needed in proving Proposition 6. Remark 5.9] without the smooth and characteristic 0 assumptions.\nWe prepare some notation. Let f be a surjective endomorphism of a normal projective variety X over the field k which is allowed to be of arbitrary characteristic. Let We need the following in proving Theorem 6.4. Proposition 6.3. Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Let λ be an eigenvalue of f * | Pic • (X) red , equivalently of (f ∨ ) * | H 1 (X ∨ ,Z ℓ ) . Then\nWe prepare some notation. Let f be a surjective endomorphism of a normal projective variety X over the field k which is allowed to be of arbitrary characteristic. Let We need the following in proving Theorem 6.4. Proposition 6.3. Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Let λ be an eigenvalue of f * | Pic • (X) red , equivalently of (f ∨ ) * | H 1 (X ∨ ,Z ℓ ) . Then\nwhere ι f is the minimal modulus of eigenvalues of f * | N 1 (X) .\nwhere ι f is the minimal modulus of eigenvalues of f * | N 1 (X) .\nProof. Let P ∈ Z[t] be the minimal polynomial of f ∨ : X ∨ → X ∨ , which is the minimal\nProof. Let P ∈ Z[t] be the minimal polynomial of f ∨ : X ∨ → X ∨ , which is the minimal\nThen P (f ∨ ) = 0 (cf. [Hu19, Section 2]). Note also that\nThen P (f ∨ ) = 0 (cf. [Hu19, Section 2]). Note also that\nbe an eigenvalue of f ∨ . Then λ is a root of P and hence a ≤ |λ| ≤ b.\nbe an eigenvalue of f ∨ . Then λ is a root of P and hence a ≤ |λ| ≤ b.\nWe may assume 0 ∈ alb X (X). Note that the smallest abelian subvariety of A containing alb X (X) is A itself by the universal property. Applying the similar technique used in the proof of Proposition 4.3, we have the following commutative diagram\nWe may assume 0 ∈ alb X (X). Note that the smallest abelian subvariety of A containing alb X (X) is A itself by the universal property. Applying the similar technique used in the proof of Proposition 4.3, we have the following commutative diagram\nwhere n ≫ 1 and s n : X ×(2n) → Alb(X) is defined by\nwhere n ≫ 1 and s n : X ×(2n) → Alb(X) is defined by\nSince s * n : N 1 (Alb(X)) → N 1 (X ×(2n) ) is (equivariant) injective, we conclude:\nSince s * n : N 1 (Alb(X)) → N 1 (X ×(2n) ) is (equivariant) injective, we conclude:\nSo the proposition is proved. (2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number.\nSo the proposition is proved. (2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number.\nThen f is q-polarized.\nThen f is q-polarized.\n(3) Suppose f is q-polarized. Then any eigenvalue of f * | H 1 (X,Q ℓ ) has modulus √ q.\n(3) Suppose f is q-polarized. Then any eigenvalue of f * | H 1 (X,Q ℓ ) has modulus √ q.\n(4) Suppose f is q-polarized (resp. int-amplified). Then so are the induced endomorphisms f | Alb(X) : Alb(X) → Alb(X) and f | Alb(X) : Alb(X) → Alb(X) via the Albanese morphism and Albanese map respectively.\n(4) Suppose f is q-polarized (resp. int-amplified). Then so are the induced endomorphisms f | Alb(X) : Alb(X) → Alb(X) and f | Alb(X) : Alb(X) → Alb(X) via the Albanese morphism and Albanese map respectively.\nProof. We follow the notation in the proof of Proposition 6.3.\nProof. We follow the notation in the proof of Proposition 6.3.\nNote that although (X ∨ ) K is in general an infinite dimensional K-vector space, the Jordan canonical form of (f ∨ ) K has all the Jordan blocks of size ≤ deg P (recall that P is the minimal polynomial of f ∨ ).\nNote that although (X ∨ ) K is in general an infinite dimensional K-vector space, the Jordan canonical form of (f ∨ ) K has all the Jordan blocks of size ≤ deg P (recall that P is the minimal polynomial of f ∨ ).\nBy Proposition 6.3, any diagonal entry λ further satisfies |λ| ≤ d 1 (f ) < d 1 (f ).\nBy Proposition 6.3, any diagonal entry λ further satisfies |λ| ≤ d 1 (f ) < d 1 (f ).\nIn\nIn\nFor the second half of (1), applying Perron-Frobenius theorem to the nef cone (cf. [Bir67]), there is a nef\nFor the second half of (1), applying Perron-Frobenius theorem to the nef cone (cf. [Bir67]), there is a nef\nwe then apply the first half of (1).\nwe then apply the first half of (1).\n(2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number. (3) just follows from Proposition 6.3.\n(2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number. (3) just follows from Proposition 6.3."},{"title":"Higher-dimensional Hodge Index Theorem","content":""},{"title":"Higher-dimensional Hodge Index Theorem","content":"In this section, we recall the higher-dimensional Hodge index theorem and give an application (cf. Proposition 7.2) which will be crucially used in Section 8.\nIn this section, we recall the higher-dimensional Hodge index theorem and give an application (cf. Proposition 7.2) which will be crucially used in Section 8.\nThe following lemma is known in [DS04, Corollaire 3.5] for the case of compact Kähler manifolds. See [MZg20, Lemma 4.5] for its full algebraic version. The proof below is taken from [MZg20, Lemma 4.5] with slight simplification for the convenience of the readers.\nThe following lemma is known in [DS04, Corollaire 3.5] for the case of compact Kähler manifolds. See [MZg20, Lemma 4.5] for its full algebraic version. The proof below is taken from [MZg20, Lemma 4.5] with slight simplification for the convenience of the readers.\nLemma 7.1. Let X be a normal projective variety. Let L ≡ 0 and D ≡ 0 be two pseudoeffective R-Cartier divisors such that L 2 and D 2 are pseudo-effective (cf. Notation 2.1).\nLemma 7.1. Let X be a normal projective variety. Let L ≡ 0 and D ≡ 0 be two pseudoeffective R-Cartier divisors such that L 2 and D 2 are pseudo-effective (cf. Notation 2.1).\nSuppose L · D ≡ w 0. Then L ≡ aD for some a > 0.\nSuppose L · D ≡ w 0. Then L ≡ aD for some a > 0.\nProof. We may assume n := dim(X) ≥ 2. Fix very ample Cartier divisors A 1 , · · · , A n−1 on X. Denote by V := RD + RL, and\nProof. We may assume n := dim(X) ≥ 2. Fix very ample Cartier divisors A 1 , · · · , A n−1 on X. Denote by V := RD + RL, and\nsubspaces of N 1 (X). By [Zha16,Lemma 3.2], L ≡ w 0 and D ≡ w 0. Then a := L · A 1 · · · A n−1 /D · A 1 · · · A n−1 > 0, and V ∩ W is of dimension ≤ 1 and spanned by\nsubspaces of N 1 (X). By [Zha16,Lemma 3.2], L ≡ w 0 and D ≡ w 0. Then a := L · A 1 · · · A n−1 /D · A 1 · · · A n−1 > 0, and V ∩ W is of dimension ≤ 1 and spanned by\nFor each 1 ≤ i ≤ n − 1, consider the following bilinear form on N 1 (X):\nFor each 1 ≤ i ≤ n − 1, consider the following bilinear form on N 1 (X):\nChoose a general normal projective surface S on X such that S ≡ w A 1 · · · A i−1 ·A i+1 · · · A n−1 .\nChoose a general normal projective surface S on X such that S ≡ w A 1 · · · A i−1 ·A i+1 · · · A n−1 .\nThen q i (x, y) = x| S · y| S . By the Hodge index theorem on S, q i is semi-negative on W .\nThen q i (x, y) = x| S · y| S . By the Hodge index theorem on S, q i is semi-negative on W .\nNote that q i is semi-positive on V since L 2 , D 2 are pseudo-effective and q i (L, D) = 0.\nNote that q i is semi-positive on V since L 2 , D 2 are pseudo-effective and q i (L, D) = 0.\nHence q i ( D, D) = 0. For any w ∈ W and λ ∈ R, we have q i (λ D − w, λ D − w) ≤ 0. Then the inequality q i (w, w) − 2λq i ( D, w) ≤ 0 holds for any λ ∈ R and w ∈ W . This happens only when q i ( D, w) = 0 for any w ∈ W .\nHence q i ( D, D) = 0. For any w ∈ W and λ ∈ R, we have q i (λ D − w, λ D − w) ≤ 0. Then the inequality q i (w, w) − 2λq i ( D, w) ≤ 0 holds for any λ ∈ R and w ∈ W . This happens only when q i ( D, w) = 0 for any w ∈ W .\nNote that W and A i span N 1 (X) because W is a hyperplane of N 1 (X) and A i ∈ W .\nNote that W and A i span N 1 (X) because W is a hyperplane of N 1 (X) and A i ∈ W .\nNote also that q i ( D, A i ) = 0. Then\nNote also that q i ( D, A i ) = 0. Then\nfor any x ∈ N 1 (X). This implies that V ∩ W is independent of the choice of each A i . So D · x 1 · · · x n−1 = 0 for any divisors x 1 , · · · , x n−1 ∈ N 1 (X), which means D ≡ w 0. Note that D is R-Cartier. Therefore, D ≡ 0 by [Zha16, Lemma 3.2].\nfor any x ∈ N 1 (X). This implies that V ∩ W is independent of the choice of each A i . So D · x 1 · · · x n−1 = 0 for any divisors x 1 , · · · , x n−1 ∈ N 1 (X), which means D ≡ w 0. Note that D is R-Cartier. Therefore, D ≡ 0 by [Zha16, Lemma 3.2].\nProposition 7.2. Let X be a normal projective variety of dimension n, L a nef R-Cartier divisor, and D 1 , D 2 two non-zero effective R-Cartier divisors on X. Suppose D 1 and D 2 have no common components and D 1 ≡ D 2 . Suppose further that L · D 1 · H 1 · · · H n−2 = 0 for some nef and big divisors H 1 , · · · , H n−2 . Then L ≡ aD 1 ≡ aD 2 for some a > 0 and L 2 ≡ w D 1 · D 2 ≡ w 0. In particular, Supp D 1 and Supp D 2 are disjoint.\nProposition 7.2. Let X be a normal projective variety of dimension n, L a nef R-Cartier divisor, and D 1 , D 2 two non-zero effective R-Cartier divisors on X. Suppose D 1 and D 2 have no common components and D 1 ≡ D 2 . Suppose further that L · D 1 · H 1 · · · H n−2 = 0 for some nef and big divisors H 1 , · · · , H n−2 . Then L ≡ aD 1 ≡ aD 2 for some a > 0 and L 2 ≡ w D 1 · D 2 ≡ w 0. In particular, Supp D 1 and Supp D 2 are disjoint.\nProof. For each H i , we write H i ∼ A i + E i for some ample divisor A i and effective divisor E i . By the assumption, the intersection of (the numerically movable) D 1 with any prime divisor is pseudo-effective, hence so is D 1 · E i for each i. Then L · D 1 · H 1 · · · H n−2 = 0 implies that L · D 1 · A 1 · H 2 · · · H n−2 = 0.\nProof. For each H i , we write H i ∼ A i + E i for some ample divisor A i and effective divisor E i . By the assumption, the intersection of (the numerically movable) D 1 with any prime divisor is pseudo-effective, hence so is D 1 · E i for each i. Then L · D 1 · H 1 · · · H n−2 = 0 implies that L · D 1 · A 1 · H 2 · · · H n−2 = 0.\nRepeating the same process, we have L · D 1 · A 1 · · · A n−2 = 0 and thus L · D 1 ≡ w 0. By Lemma 7.1 and noting D 2 1 ≡ w D 1 D 2 , we get L ≡ aD 1 (≡ aD 2 ) with a > 0. So we have\nRepeating the same process, we have L · D 1 · A 1 · · · A n−2 = 0 and thus L · D 1 ≡ w 0. By Lemma 7.1 and noting D 2 1 ≡ w D 1 D 2 , we get L ≡ aD 1 (≡ aD 2 ) with a > 0. So we have\nis a non-zero effective cycle and hence we get a contradiction with D 1 · D 2 ≡ w 0.\nis a non-zero effective cycle and hence we get a contradiction with D 1 · D 2 ≡ w 0.\n8. Proofs of Theorems 1.6 and 1.7\n8. Proofs of Theorems 1.6 and 1.7\nIn this section, we will prove Theorems 1.7 and 8.1 below which includes Theorem 1.6 as a simpler version and bounds the number ♯P f in terms of the Néron-Severi class group rank ρ ns (X), the Picard number ρ( X) of any projective resolution X of X, and the first dynamical degree d 1 (f ).\nIn this section, we will prove Theorems 1.7 and 8.1 below which includes Theorem 1.6 as a simpler version and bounds the number ♯P f in terms of the Néron-Severi class group rank ρ ns (X), the Picard number ρ( X) of any projective resolution X of X, and the first dynamical degree d 1 (f ).\nTheorem 8.1. Let f : X → X be a surjective endomorphism of a normal projective variety. Let X → X be any projective resolution. Then\nTheorem 8.1. Let f : X → X be a surjective endomorphism of a normal projective variety. Let X → X be any projective resolution. Then\nwhere a is a positive constant depending only on X. In particular, for n ≫ 1, (1)).\nwhere a is a positive constant depending only on X. In particular, for n ≫ 1, (1)).\nWe now prepare to prove Theorem 8.1.\nWe now prepare to prove Theorem 8.1.\nNotation 8.2. Let f : X → X be a surjective endomorphism on a normal projective variety X of dimension n. We fix a projective resolution σ : X → X with E 1 , · · · , E k its exceptional prime divisors. For D ∈ P f , denote by D := σ −1 * (D), the proper transform of D. Denote by\nNotation 8.2. Let f : X → X be a surjective endomorphism on a normal projective variety X of dimension n. We fix a projective resolution σ : X → X with E 1 , · · · , E k its exceptional prime divisors. For D ∈ P f , denote by D := σ −1 * (D), the proper transform of D. Denote by\nBy Perron-Frobenius theorem (cf. [Bir67]), fix a nef R-Cartier divisor L f ≡ 0 on X with\nBy Perron-Frobenius theorem (cf. [Bir67]), fix a nef R-Cartier divisor L f ≡ 0 on X with\nDenote by\nDenote by\nand d 1 (f | D ) < d 1 (f ). Then L f | D ≡ 0 and hence L f · D ≡ w 0.\nand d 1 (f | D ) < d 1 (f ). Then L f | D ≡ 0 and hence L f · D ≡ w 0.\nLemma 8.4. There exist at least 2⌊♯P f /(ρ ns (X) + 1)⌋ disjoint prime divisors D u in P f with L f · D u ≡ w 0.\nLemma 8.4. There exist at least 2⌊♯P f /(ρ ns (X) + 1)⌋ disjoint prime divisors D u in P f with L f · D u ≡ w 0.\nProof. Let m := ⌊♯P f /(ρ ns (X)+1)⌋. Then we have m pairwise disjoint subsets S 1 , · · · , S m of P f with ♯S t = ρ ns (X) + 1 for each 1 ≤ t ≤ m.\nProof. Let m := ⌊♯P f /(ρ ns (X)+1)⌋. Then we have m pairwise disjoint subsets S 1 , · · · , S m of P f with ♯S t = ρ ns (X) + 1 for each 1 ≤ t ≤ m.\nNote that ρ( X) = ρ ns (X) + k where k is the number of σ-exceptional prime divisors (cf. [Kol18, Lemma 18]). By the assumption, we have the equality ♯(S t ∪ {E 1 , · · · , E k }) = ρ( X) + 1.\nNote that ρ( X) = ρ ns (X) + k where k is the number of σ-exceptional prime divisors (cf. [Kol18, Lemma 18]). By the assumption, we have the equality ♯(S t ∪ {E 1 , · · · , E k }) = ρ( X) + 1.\nFor each t, we then have a numerical equivalence of two non-zero effective divisors which are combinations of prime divisors in S t and σ-exceptional prime divisors such that they have no common component. We write them in the following form i∈It a i D i + E t ≡ j∈Jt b j D j + F t where I t and J t are two (possibly empty) disjoint subsets of S t , a i > 0, b j > 0, and E t , F t are effective σ-exceptional divisors. Here without ambiguity, we identify D i with the index i in I t .\nFor each t, we then have a numerical equivalence of two non-zero effective divisors which are combinations of prime divisors in S t and σ-exceptional prime divisors such that they have no common component. We write them in the following form i∈It a i D i + E t ≡ j∈Jt b j D j + F t where I t and J t are two (possibly empty) disjoint subsets of S t , a i > 0, b j > 0, and E t , F t are effective σ-exceptional divisors. Here without ambiguity, we identify D i with the index i in I t .\nLet H be an ample divisor on X. By Lemma 8.3 and the projection formula,\nLet H be an ample divisor on X. By Lemma 8.3 and the projection formula,\nwhere D i = σ * ( D i ) ∈ P f . Applying Proposition 7.2, for each t, we may assume (after some scalar multiplication)\nwhere D i = σ * ( D i ) ∈ P f . Applying Proposition 7.2, for each t, we may assume (after some scalar multiplication)\nand we also have\nand we also have\nThis further implies that\nThis further implies that\n(1)\n(1)\nfor any i ∈ I t , j ∈ J t and any prime divisor E in E t and F t .\nfor any i ∈ I t , j ∈ J t and any prime divisor E in E t and F t .\nWe claim that I t and J t are non-empty. Suppose the contrary, for example, that I t is empty. Then L f ≡ E t = 0. However, this contradicts with the negativity lemma (cf. [KM98, Lemma 3.39]).\nWe claim that I t and J t are non-empty. Suppose the contrary, for example, that I t is empty. Then L f ≡ E t = 0. However, this contradicts with the negativity lemma (cf. [KM98, Lemma 3.39]).\nConsider the pair (I t , J t ′ ) for any t, t ′ . Note that\nConsider the pair (I t , J t ′ ) for any t, t ′ . Note that\nWe claim that E t and F t ′ have no common components. Readjust it a bit in the following way, where ∆ is a σ-exceptional effective R-divisor,\nWe claim that E t and F t ′ have no common components. Readjust it a bit in the following way, where ∆ is a σ-exceptional effective R-divisor,\nso that E t − ∆ and F t ′ − ∆ are effective but with no common components. Note that\nso that E t − ∆ and F t ′ − ∆ are effective but with no common components. Note that\nBy Proposition 7.2, for some e > 0, we have e L f ≡ i∈It a i D i + ( E t − ∆).\nBy Proposition 7.2, for some e > 0, we have e L f ≡ i∈It a i D i + ( E t − ∆).\nThen (1 − e) L f ≡ ∆, and hence e ≤ 1 since both L f and ∆ are pseudo-effective and X is projective. By the negativity lemma again (cf. [KM98, Lemma 3.39]), e = 1 and ∆ ≡ 0, so ∆ = 0 and the claim is proved. By Proposition 7.2 and the display (2) above and since L f · L f ≡ w 0, we have D i ∩ D j = ∅ for any i ∈ I t and j ∈ J t ′ .\nThen (1 − e) L f ≡ ∆, and hence e ≤ 1 since both L f and ∆ are pseudo-effective and X is projective. By the negativity lemma again (cf. [KM98, Lemma 3.39]), e = 1 and ∆ ≡ 0, so ∆ = 0 and the claim is proved. By Proposition 7.2 and the display (2) above and since L f · L f ≡ w 0, we have D i ∩ D j = ∅ for any i ∈ I t and j ∈ J t ′ .\nThe same argument works for the pairs (I t , I t ′ ) and (J t , J t ′ ) with t = t ′ . In particular, we may pick one prime divisor in I t and one prime divisor in J t for each t, and all these 2m prime divisors are pairwise disjoint. This completes the proof.\nThe same argument works for the pairs (I t , I t ′ ) and (J t , J t ′ ) with t = t ′ . In particular, we may pick one prime divisor in I t and one prime divisor in J t for each t, and all these 2m prime divisors are pairwise disjoint. This completes the proof.\nWith ♯P f large enough, we can construct an f -equivariant fibration over a curve.\nWith ♯P f large enough, we can construct an f -equivariant fibration over a curve.\nProposition 8.5. Suppose ♯P f ≥ (⌊ ρ( X) 2 ⌋ + 1) · (ρ ns (X) + 1). Then there exists a fibration π : X → C onto a smooth curve C such that the following hold.\nProposition 8.5. Suppose ♯P f ≥ (⌊ ρ( X) 2 ⌋ + 1) · (ρ ns (X) + 1). Then there exists a fibration π : X → C onto a smooth curve C such that the following hold.\n(1) L f ≡ π * H C for some ample R-Cartier divisor H C on C.\n(1) L f ≡ π * H C for some ample R-Cartier divisor H C on C.\n(2) The fibration π factors through a fibration π : X → C such that L f ≡ π * H C .\n(2) The fibration π factors through a fibration π : X → C such that L f ≡ π * H C .\n(3) There exists a d 1 (f )-polarized endomorphism g : C → C such that π • f = g • π.\n(3) There exists a d 1 (f )-polarized endomorphism g : C → C such that π • f = g • π.\nProof. (1) By Lemma 8.4, there are ρ( X) + 1 pairwise disjoint prime divisors D u in P f with L f · D u ≡ w 0. By [BPS16, Theorem 1.1], there is a fibration π : X → C onto a smooth projective curve such that π( D u ) is a point for each u and D u = π −1 ( π( D u )) for X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. Let B be the finite set of prime divisors in the reducible fibres of π. By the same argument as in the proof of Proposition 6.1 and [CMZ20, Lemma 7.5], we have an injection ϕ : P −1 f ∞ \\B → P −1 g ∞ , sending D to π(D). Note that there are at most finitely many g −1 -periodic points (cf. [MZ20, Corollary 3.8]). So the theorem is proved.\nProof. (1) By Lemma 8.4, there are ρ( X) + 1 pairwise disjoint prime divisors D u in P f with L f · D u ≡ w 0. By [BPS16, Theorem 1.1], there is a fibration π : X → C onto a smooth projective curve such that π( D u ) is a point for each u and D u = π −1 ( π( D u )) for X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. Let B be the finite set of prime divisors in the reducible fibres of π. By the same argument as in the proof of Proposition 6.1 and [CMZ20, Lemma 7.5], we have an injection ϕ : P −1 f ∞ \\B → P −1 g ∞ , sending D to π(D). Note that there are at most finitely many g −1 -periodic points (cf. [MZ20, Corollary 3.8]). So the theorem is proved.\nCorollary 8.6. Let f : X → X be a surjective endomorphism of a normal projective variety X with ♯P f ∞ = ∞. Then D is either Q-Cartier or f −1 -periodic for any D ∈ P f ∞ .\nCorollary 8.6. Let f : X → X be a surjective endomorphism of a normal projective variety X with ♯P f ∞ = ∞. Then D is either Q-Cartier or f −1 -periodic for any D ∈ P f ∞ ."},{"title":"Theorem 1 . 5 .","content":""},{"title":"Theorem 1 . 5 .","content":""},{"title":"Conjecture 3. 2 .","content":""},{"title":"Conjecture 3. 2 .","content":""},{"title":"Lemma 4. 2 .","content":""},{"title":"Lemma 4. 2 .","content":""},{"title":"[ Bri10 ,","content":""},{"title":"[ Bri10 ,","content":""},{"title":"Corollary 5. 2 .","content":""},{"title":"Corollary 5. 2 .","content":""},{"title":"d 1","content":""},{"title":"d 1","content":""},{"title":"[ Bri10 ,","content":""},{"title":"[ Bri10 ,","content":""},{"title":"Theorem 6. 4 .","content":""},{"title":"Theorem 6. 4 .","content":""},{"title":"( 4 )","content":""},{"title":"( 4 )","content":""}],"string":"[\n {\n \"title\": \"Connection of two arithmetic dynamical conjectures\",\n \"content\": \"\"\n },\n {\n \"title\": \"Connection of two arithmetic dynamical conjectures\",\n \"content\": \"In this section, we work over a number field K. We will give the relation between two conjectures in Theorem 3.4, which also gives an arithmetic motivation of this paper. We refer to [MMSZ20, Section 2] for the notation involved in this section.\\nIn this section, we work over a number field K. We will give the relation between two conjectures in Theorem 3.4, which also gives an arithmetic motivation of this paper. We refer to [MMSZ20, Section 2] for the notation involved in this section.\\nThe following are the two main conjectures studied in [MMSZ20].\\nThe following are the two main conjectures studied in [MMSZ20].\\nsurjective endomorphism of a projective variety X defined over a number field K. Let\\nsurjective endomorphism of a projective variety X defined over a number field K. Let\\nWe copy [MMSZ20, Question 8.4] in the following. Note that for the equivalence of the above two conjectures, we may only focus on Y ∈ S f ∞ in Question 3.3 (1) and (2); see the proof of Theorem 3.4.\\nWe copy [MMSZ20, Question 8.4] in the following. Note that for the equivalence of the above two conjectures, we may only focus on Y ∈ S f ∞ in Question 3.3 (1) and (2); see the proof of Theorem 3.4.\\nempty for s ≫ 1?\\nempty for s ≫ 1?\\n(2) Is there a positive integer N such that Y (d) is empty for any f -periodic subvariety Y ⊆ X K of small dynamical degree whose period is larger than N?\\n(2) Is there a positive integer N such that Y (d) is empty for any f -periodic subvariety Y ⊆ X K of small dynamical degree whose period is larger than N?\\nwhich is a union of irreducible closed subsets and Y 0 :\\nwhich is a union of irreducible closed subsets and Y 0 :\\nwhich is not Zariski dense in X K .\\nwhich is not Zariski dense in X K .\"\n },\n {\n \"title\": \"Isogenies of algebraic groups\",\n \"content\": \"\"\n },\n {\n \"title\": \"Isogenies of algebraic groups\",\n \"content\": \"We use the following notation throughout this section.\\nWe use the following notation throughout this section.\\nNotation 4.1. Let G be a connected algebraic group with the identity element e. We focus on the isogeny (i.e., surjective group homomorphism with finite kernel)\\nNotation 4.1. Let G be a connected algebraic group with the identity element e. We focus on the isogeny (i.e., surjective group homomorphism with finite kernel)\\nwhere R a is the right multiplication map by some a ∈ G. Note that there exists a G-\\nwhere R a is the right multiplication map by some a ∈ G. Note that there exists a G-\\n). Then the same proof for [MS18, Lemma 5.6] shows that\\n). Then the same proof for [MS18, Lemma 5.6] shows that\\nWe recall that g and hence f are finite surjective endomorphisms. Proof.\\nWe recall that g and hence f are finite surjective endomorphisms. Proof.\\nMoreover, We have the following commutative diagram:\\nMoreover, We have the following commutative diagram:\\nThe following result is a generalization of Lemma 4.2 to arbitrary closed subvariety of G, which plays a crucial role in many later results.\\nThe following result is a generalization of Lemma 4.2 to arbitrary closed subvariety of G, which plays a crucial role in many later results.\\nThen X e ⊆ X e p −1 (X e p −1 ) −1 ⊆ g(H) and hence H ≤ g(H) by the construction of H.\\nThen X e ⊆ X e p −1 (X e p −1 ) −1 ⊆ g(H) and hence H ≤ g(H) by the construction of H.\\nSince g is an isogeny, g(H) = H.\\nSince g is an isogeny, g(H) = H.\\nConsider the alternate multiplication map s n : X ×(2n) → H defined by\\nConsider the alternate multiplication map s n : X ×(2n) → H defined by\\nThen the image of s n is (X e X −1 e ) ·n = H when n ≫ 1 by [Spr09, Proposition 2.2.6] and noting that e ∈ X e . Consider the following diagram with n ≫ 1:\\nThen the image of s n is (X e X −1 e ) ·n = H when n ≫ 1 by [Spr09, Proposition 2.2.6] and noting that e ∈ X e . Consider the following diagram with n ≫ 1:\\nThen\\nThen\\nSo the diagram commutes. By the product formula, we have\\nSo the diagram commutes. By the product formula, we have\\nOn the other hand, note that\\nOn the other hand, note that\\nLemma 4.4. There exists a unique X ∈ S g ∞ containing e ∈ G. Moreover, X is an algebraic subgroup and g-invariant.\\nLemma 4.4. There exists a unique X ∈ S g ∞ containing e ∈ G. Moreover, X is an algebraic subgroup and g-invariant.\\nProof. Take X ∈ S g ∞ containing e. It exists because d 1 (g) > 1 and d 1 (g| {e} ) = 1. Take s > 0 such that g s (X) = X. Take the smallest (connected closed) algebraic subgroup H of G containing X. By Proposition 4.3, g s (H) = H and d 1 (\\nProof. Take X ∈ S g ∞ containing e. It exists because d 1 (g) > 1 and d 1 (g| {e} ) = 1. Take s > 0 such that g s (X) = X. Take the smallest (connected closed) algebraic subgroup H of G containing X. By Proposition 4.3, g s (H) = H and d 1 (\\nBy the maximality of X, we have X = H and it is an algebraic subgroup.\\nBy the maximality of X, we have X = H and it is an algebraic subgroup.\\nTake another Y ∈ S g ∞ containing e and then take t > 0 such that g t (Y ) = Y .\\nTake another Y ∈ S g ∞ containing e and then take t > 0 such that g t (Y ) = Y .\\nThe argument in the previous paragraph shows that Y is a (connected closed) algebraic subgroup. Let Z be the smallest algebraic group containing the g st -invariant irreducible closed subvariety XY . Apply the similar commutative diagram (*) and argument in the proof of Proposition 4.3,\\nThe argument in the previous paragraph shows that Y is a (connected closed) algebraic subgroup. Let Z be the smallest algebraic group containing the g st -invariant irreducible closed subvariety XY . Apply the similar commutative diagram (*) and argument in the proof of Proposition 4.3,\\nis surjective with n ≫ 1. Then we have d 1 (g st | Z ) < d 1 (g st ) and hence Z = X = Y by the maximality of X and Y . This shows the uniqueness of X.\\nis surjective with n ≫ 1. Then we have d 1 (g st | Z ) < d 1 (g st ) and hence Z = X = Y by the maximality of X and Y . This shows the uniqueness of X.\\nNext we show that X is g-invariant. Note that g(X) is also g s -invariant and contains e. By the product formula, the following diagram shows d 1 (g s | g(X) ) ≤ d 1 (g s | X ) < d 1 (g s ):\\nNext we show that X is g-invariant. Note that g(X) is also g s -invariant and contains e. By the product formula, the following diagram shows d 1 (g s | g(X) ) ≤ d 1 (g s | X ) < d 1 (g s ):\\nTherefore, we have g(X) ⊆ X by the maximality and uniqueness of X. In particular, X is g-invariant since g is an isogeny. This finishes our proof.\\nTherefore, we have g(X) ⊆ X by the maximality and uniqueness of X. In particular, X is g-invariant since g is an isogeny. This finishes our proof.\\nDefinition 4.5. With the notation in Lemma 4.4, we denote by\\nDefinition 4.5. With the notation in Lemma 4.4, we denote by\\nWe shall prove in the next section that G(f ) is actually a normal subgroup of G.\\nWe shall prove in the next section that G(f ) is actually a normal subgroup of G.\\nProof. Assume that X ∈ S f . Fix some b ∈ X and denote by X e := Xb −1 which contains e ∈ G. Let H be the smallest algebraic subgroup of G containing X e . By Proposition\\nProof. Assume that X ∈ S f . Fix some b ∈ X and denote by X e := Xb −1 which contains e ∈ G. Let H be the smallest algebraic subgroup of G containing X e . By Proposition\\nBy Lemma 4.2,\\nBy Lemma 4.2,\\nprevious paragraph. But this implies that X = Y since both are translations of the same group G(f ). So X is maximal.\\nprevious paragraph. But this implies that X = Y since both are translations of the same group G(f ). So X is maximal.\\nThe following modified product formula will be frequently used in the next Theorem and also next section.\\nThe following modified product formula will be frequently used in the next Theorem and also next section.\\nLemma 4.7. Consider the commutative diagram of homomorphisms of connected algebraic groups:\\nLemma 4.7. Consider the commutative diagram of homomorphisms of connected algebraic groups:\\nwhere g is an isogeny. Then π is f -equivariant and the following hold.\\nwhere g is an isogeny. Then π is f -equivariant and the following hold.\\n) and π induces a bijection between S f and S f | C .\\n) and π induces a bijection between S f and S f | C .\\nLet A act on A and C by the right multiplication. By Note that the graph of π is A-equivariant. Replacing A by the A-equivariant resolution of this graph, we may further assume that π is a well-defined algebraic fibration.\\nLet A act on A and C by the right multiplication. By Note that the graph of π is A-equivariant. Replacing A by the A-equivariant resolution of this graph, we may further assume that π is a well-defined algebraic fibration.\\nBy the generic smoothness of π, there exists a general point c 1 = π(a 1 ) ∈ C such that the fibre A c 1 := π −1 (c 1 ) is an (irreducible) smooth projective variety. For any\\nBy the generic smoothness of π, there exists a general point c 1 = π(a 1 ) ∈ C such that the fibre A c 1 := π −1 (c 1 ) is an (irreducible) smooth projective variety. For any\\n. So all the fibres of π over C are (irreducible) smooth projective varieties isomorphic to each other.\\n. So all the fibres of π over C are (irreducible) smooth projective varieties isomorphic to each other.\\nWe refer to [DN11] for the definitions of critical value and critical set. Now we have that each point of C is not a critical value of π. Note that B is g-invariant and g isétale. Then\\nWe refer to [DN11] for the definitions of critical value and critical set. Now we have that each point of C is not a critical value of π. Note that B is g-invariant and g isétale. Then\\nA e (the closure of B in A) is not contained in the critical set of g. Therefore, we verified the assumptions for A e in [DN11, Remark 3.4] and hence d 1 (g| π ) = d 1 (g| A e ) = d 1 (g| B ).\\nA e (the closure of B in A) is not contained in the critical set of g. Therefore, we verified the assumptions for A e in [DN11, Remark 3.4] and hence d 1 (g| π ) = d 1 (g| A e ) = d 1 (g| B ).\\nTogether with the product formula, (1) is proved.\\nTogether with the product formula, (1) is proved.\\nWe apply the maximality and uniqueness of A(g) and C(g| C ) (cf. Lemma 4.4) and our new product formula (1) several times in this diagram. First, we have B ≤ A(g).\\nWe apply the maximality and uniqueness of A(g) and C(g| C ) (cf. Lemma 4.4) and our new product formula (1) several times in this diagram. First, we have B ≤ A(g).\\nNote that d 1 (g| π(A(g)) ) = d 1 (g| A(g) ) < d 1 (g). So we have π(A(g)) ≤ C(g| C ). Conversely,\\nNote that d 1 (g| π(A(g)) ) = d 1 (g| A(g) ) < d 1 (g). So we have π(A(g)) ≤ C(g| C ). Conversely,\\nFinally, we apply Proposition 4.6 for f and f | C , then we see that the induced map\\nFinally, we apply Proposition 4.6 for f and f | C , then we see that the induced map\\nis a bijection. So (2) is proved.\\nis a bijection. So (2) is proved.\\nWe complete this section by considering the commutative algebraic groups first.\\nWe complete this section by considering the commutative algebraic groups first.\\nTheorem 4.8. Let f : G → G be a dominant self-morphism of a connected commutative algebraic group G, such that f = R a • g where g is an isogeny and a ∈ G. Suppose d 1 (f ) > 1. Let G(f ) be as in Definition 4.5. Then\\nTheorem 4.8. Let f : G → G be a dominant self-morphism of a connected commutative algebraic group G, such that f = R a • g where g is an isogeny and a ∈ G. Suppose d 1 (f ) > 1. Let G(f ) be as in Definition 4.5. Then\\nThis contradicts the maximality of G(g).\\nThis contradicts the maximality of G(g).\"\n },\n {\n \"title\": \"Now we have shown\",\n \"content\": \"\"\n },\n {\n \"title\": \"Now we have shown\",\n \"content\": \"and the theorem is proved.\\nand the theorem is proved.\"\n },\n {\n \"title\": \"Proof of Theorem 1.5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proof of Theorem 1.5\",\n \"content\": \"In this section, we focus on the upper bound in Question 1.3. We first consider the case of abelian varieties.\\nIn this section, we focus on the upper bound in Question 1.3. We first consider the case of abelian varieties.\\nTheorem 5.1. Let f be a surjective endomorphism of an abelian variety A. Then\\nTheorem 5.1. Let f be a surjective endomorphism of an abelian variety A. Then\\n(cf. Definition 4.5 for A(f )).\\n(cf. Definition 4.5 for A(f )).\\nProof. We may assume d 1 (f ) > 1. Write f = R a • g where g is an isogeny and a ∈ A.\\nProof. We may assume d 1 (f ) > 1. Write f = R a • g where g is an isogeny and a ∈ A.\\nSince d 1 (f ) = d 1 (g) and A(f ) = A(g), applying Theorem 4.8, we may assume that f is an isogeny, i.e., f = g; further we have an f -equivariant quotient morphism\\nSince d 1 (f ) = d 1 (g) and A(f ) = A(g), applying Theorem 4.8, we may assume that f is an isogeny, i.e., f = g; further we have an f -equivariant quotient morphism\"\n },\n {\n \"title\": \"see the ending part of the proof of Theorem 4.8), which is an isogeny. Then\",\n \"content\": \"\"\n },\n {\n \"title\": \"see the ending part of the proof of Theorem 4.8), which is an isogeny. Then\",\n \"content\": \"\"\n },\n {\n \"title\": \"by [Hu19, Formula (3.6)] and the product formula.\",\n \"content\": \"\"\n },\n {\n \"title\": \"by [Hu19, Formula (3.6)] and the product formula.\",\n \"content\": \"Then the theorem follows.\\nThen the theorem follows.\\nRecall that a normal projective variety X is said to be Q-abelian (cf. [NZ10]) if there is a finite surjective morphism A → X from an abelian variety A, which isétale in codimension one. When f is a group homomorphism, it induces the following commutative diagram where T is an algebraic torus and A = Alb(X) is an abelian variety:\\nRecall that a normal projective variety X is said to be Q-abelian (cf. [NZ10]) if there is a finite surjective morphism A → X from an abelian variety A, which isétale in codimension one. When f is a group homomorphism, it induces the following commutative diagram where T is an algebraic torus and A = Alb(X) is an abelian variety:\\nTheorem 5.3. Let f be a dominant self-morphism of a semi-abelian variety X. Then\\nTheorem 5.3. Let f be a dominant self-morphism of a semi-abelian variety X. Then\\nProof. By the same argument as in the first paragraph of the proof of Theorem 5.1, we may replace f by g and X by X/X(f ) so that dim(X(f )) = 0 and both f and\\nProof. By the same argument as in the first paragraph of the proof of Theorem 5.1, we may replace f by g and X by X/X(f ) so that dim(X(f )) = 0 and both f and\\nConsider the commutative diagram before Theorem 5. 3. Note that f | T is defined by a matrix M of size dim(T ) with integer coefficients and a non-zero determinant. Then We compute\\nConsider the commutative diagram before Theorem 5. 3. Note that f | T is defined by a matrix M of size dim(T ) with integer coefficients and a non-zero determinant. Then We compute\\nTogether with the computation in the proof of Theorem 5.1 for abelian varieties, we have\\nTogether with the computation in the proof of Theorem 5.1 for abelian varieties, we have\\nwhere the last inequality follows from\\nwhere the last inequality follows from\\n[MS18, Theorem 1.6] or Lemma 4.7. So the theorem is proved.\\n[MS18, Theorem 1.6] or Lemma 4.7. So the theorem is proved.\\nRemark 5.4. In the proof of Theorem 5.3, we see that if X = T is an algebraic torus, then the computation indeed shows that ♯S f ≤ (d 1 (f ) + 1) dim(T )−dim(T (f )) .\\nRemark 5.4. In the proof of Theorem 5.3, we see that if X = T is an algebraic torus, then the computation indeed shows that ♯S f ≤ (d 1 (f ) + 1) dim(T )−dim(T (f )) .\\nWe now prepare to deal with general algebraic groups. See, for instance, [MZ18a,§3] for several decomposition facts about algebraic groups. First, we need observations for unipotent groups and semisimple groups below.\\nWe now prepare to deal with general algebraic groups. See, for instance, [MZ18a,§3] for several decomposition facts about algebraic groups. First, we need observations for unipotent groups and semisimple groups below.\\nLemma 5.5. Let g : U → U be an isogeny of a (connected) unipotent group U. Then d 1 (g) = 1.\\nLemma 5.5. Let g : U → U be an isogeny of a (connected) unipotent group U. Then d 1 (g) = 1.\\nProof. Since Ker(g) is a finite subgroup of the unipotent group U, it is trivial, so g is an automorphism. We show by induction on dim(U) and assume dim(U) > 0. Note that U is solvable. So its commutator subgroup U (1) is strictly smaller than U. Note that g(U (1) ) = U (1) . Consider the following commutative diagram:\\nProof. Since Ker(g) is a finite subgroup of the unipotent group U, it is trivial, so g is an automorphism. We show by induction on dim(U) and assume dim(U) > 0. Note that U is solvable. So its commutator subgroup U (1) is strictly smaller than U. Note that g(U (1) ) = U (1) . Consider the following commutative diagram:\\nis not trivial. By Lemma 4.7 and induction, we have d 1 (g) = max{d 1 (g| U (1) ), d 1 (g| U/U (1) )} = 1.\\nis not trivial. By Lemma 4.7 and induction, we have d 1 (g) = max{d 1 (g| U (1) ), d 1 (g| U/U (1) )} = 1.\\nSuppose U (1) is trivial. Then U is a commutative unipotent group which is isomorphic to k ×dim(U ) . One checks that g ∈ GL(U) since g is additive. By is finite. Replacing g by a power, we may assume g is in the connected algebraic group Aut • (S). We are done by the same argument at the ending part of Lemma 5.5.\\nSuppose U (1) is trivial. Then U is a commutative unipotent group which is isomorphic to k ×dim(U ) . One checks that g ∈ GL(U) since g is additive. By is finite. Replacing g by a power, we may assume g is in the connected algebraic group Aut • (S). We are done by the same argument at the ending part of Lemma 5.5.\\nFinally, we are able to give the same upper bound for any algebraic group.\\nFinally, we are able to give the same upper bound for any algebraic group.\\nTheorem 5.7. Let G be a connected algebraic group. Then G(f ) is normal in G and\\nTheorem 5.7. Let G be a connected algebraic group. Then G(f ) is normal in G and\\nfor any self-morphism f = R a • g where g is an isogeny and a ∈ G.\\nfor any self-morphism f = R a • g where g is an isogeny and a ∈ G.\\nProof. We may assume 1 < d 1 (f ) (= d 1 (g)). Let G a be the maximal connected affine normal subgroup of G, and U the unipotent radical of G a . Then g(G a ) = G a and g(U) = U. Consider the following commutative diagram:\\nProof. We may assume 1 < d 1 (f ) (= d 1 (g)). Let G a be the maximal connected affine normal subgroup of G, and U the unipotent radical of G a . Then g(G a ) = G a and g(U) = U. Consider the following commutative diagram:\\nBy Lemma 5.5, we have d 1 (g| U ) = 1 < d 1 (g). By Lemma 4.7, π is f -equivariant and\\nBy Lemma 5.5, we have d 1 (g| U ) = 1 < d 1 (g). By Lemma 4.7, π is f -equivariant and\\nnormal in G/U. Therefore, we may replace G by G/U and assume that U is trivial.\\nnormal in G/U. Therefore, we may replace G by G/U and assume that U is trivial.\\nLet S := G (1) be the commutator subgroup of G. Then g(S) = S, and S = (G a ) (1) which is a semisimple group since G a is a reductive group now. Consider the following commutative diagram:\\nLet S := G (1) be the commutator subgroup of G. Then g(S) = S, and S = (G a ) (1) which is a semisimple group since G a is a reductive group now. Consider the following commutative diagram:\\nBy Lemma 5.6, we have d 1 (g| S ) = 1. By Lemma 4.7 and an argument similar to the previous paragraph, we may replace G by G/S and assume that G is commutative.\\nBy Lemma 5.6, we have d 1 (g| S ) = 1. By Lemma 4.7 and an argument similar to the previous paragraph, we may replace G by G/S and assume that G is commutative.\\nNote that now G is a semi-abelian variety. In particular, G(f ) is normal in G, and the upper bound is as claimed, by Theorem 5.3 and Lemma 4.7.\\nNote that now G is a semi-abelian variety. In particular, G(f ) is normal in G, and the upper bound is as claimed, by Theorem 5.3 and Lemma 4.7.\\nAt the end of this section, we treat the toric varieties. Let X be a toric variety and T the big torus. A subvariety of the form O = x · T is called a torus orbit on X. Note that X = O⊆X O is a finite disjoint union of all torus orbits on X.\\nAt the end of this section, we treat the toric varieties. Let X be a toric variety and T the big torus. A subvariety of the form O = x · T is called a torus orbit on X. Note that X = O⊆X O is a finite disjoint union of all torus orbits on X.\\nTheorem 5.8. Let X be a toric variety with the big torus T ⊆ X. Let f : X → X be a dominant self-morphism such that f (T ) ⊆ T . Then\\nTheorem 5.8. Let X be a toric variety with the big torus T ⊆ X. Let f : X → X be a dominant self-morphism such that f (T ) ⊆ T . Then\\nwhere the sum runs over all the torus orbits on X.\\nwhere the sum runs over all the torus orbits on X.\\nProof. By [Nak08, Lemma 3.4.3], f = R a • g such that g| T is an isogeny and g(x · t) = g(x) · g(t); see [Nak08, Section 3.4] for details. Here R a is the natural extension of the right multiplication map on T by a ∈ T . Since g| T is dominant, f (T ) = g(T ) · a = T .\\nProof. By [Nak08, Lemma 3.4.3], f = R a • g such that g| T is an isogeny and g(x · t) = g(x) · g(t); see [Nak08, Section 3.4] for details. Here R a is the natural extension of the right multiplication map on T by a ∈ T . Since g| T is dominant, f (T ) = g(T ) · a = T .\\nIn particular, f maps every torus orbit onto a torus orbit. Note that\\nIn particular, f maps every torus orbit onto a torus orbit. Note that\\nThen there is a torus orbit O such that V ∩ O contains an open dense subset of V . So\\nThen there is a torus orbit O such that V ∩ O contains an open dense subset of V . So\\nThen\\nThen\\nLemma 5.6]). Define the natural surjective morphism\\nLemma 5.6]). Define the natural surjective morphism\\nBy the product formula,\\nBy the product formula,\\nThis finishes the proof.\\nThis finishes the proof.\\nRemark 5.9. In Theorem 5.8, as we do not assume X is projective, f is not necessarily finite (nor surjective). Also during its proof, it is possible that O ⊆ Sing(X), and then 6. Reduction to polarized case: Proof of Theorem 1.9\\nRemark 5.9. In Theorem 5.8, as we do not assume X is projective, f is not necessarily finite (nor surjective). Also during its proof, it is possible that O ⊆ Sing(X), and then 6. Reduction to polarized case: Proof of Theorem 1.9\\nIn this section, the ground field k is allowed to have arbitrary characteristic. We will prove Theorem 6.4 which includes Theorem 1.9 as a special case. The connection with polarized endomorphism will give us finiteness for ♯P f .\\nIn this section, the ground field k is allowed to have arbitrary characteristic. We will prove Theorem 6.4 which includes Theorem 1.9 as a special case. The connection with polarized endomorphism will give us finiteness for ♯P f .\\nProposition 6.1. Let π : X Y be a dominant rational map between projective\\nProposition 6.1. Let π : X Y be a dominant rational map between projective\\nvarieties. Let f : X → X and g : Y → Y be surjective endomorphisms such that\\nvarieties. Let f : X → X and g : Y → Y be surjective endomorphisms such that\\nThen for n ≫ 1,\\nThen for n ≫ 1,\\nProof. By Lemma 2.4, replacing π with its graph and taking the normalizations, we may assume that π is a surjective morphism and both X and Y are normal.\\nProof. By Lemma 2.4, replacing π with its graph and taking the normalizations, we may assume that π is a surjective morphism and both X and Y are normal.\\nTake the Stein factorization X → Y 1 → Y such that p 1 : X → Y 1 has connected fibres and p 2 : Y 1 → Y is finite. By [CMZ20, Lemma 5.2], f descends to a surjective\\nTake the Stein factorization X → Y 1 → Y such that p 1 : X → Y 1 has connected fibres and p 2 : Y 1 → Y is finite. By [CMZ20, Lemma 5.2], f descends to a surjective\\nHence we may assume further π has connected fibres.\\nHence we may assume further π has connected fibres.\\npolarized, a contradiction. For the upper bound, we may assume P f ∞ is an infinite set whose elements are contained in fibres of π. Then dim Y = 1. Let B be the finite set of prime divisors in the (finitely many) reducible fibres of π. We have an injection\\npolarized, a contradiction. For the upper bound, we may assume P f ∞ is an infinite set whose elements are contained in fibres of π. Then dim Y = 1. Let B be the finite set of prime divisors in the (finitely many) reducible fibres of π. We have an injection\\nsending D to π(D). Then our argument follows from Proposition 6.2.\\nsending D to π(D). Then our argument follows from Proposition 6.2.\\nThe following result is well known when the ground field k has char k = 0 and the proof is not that troublesome due to the Hodge decomposition. Our proof here works when char k is arbitrary, and is related to a generalized problem of Weil's Riemann Hypothesis.\\nThe following result is well known when the ground field k has char k = 0 and the proof is not that troublesome due to the Hodge decomposition. Our proof here works when char k is arbitrary, and is related to a generalized problem of Weil's Riemann Hypothesis.\\nProposition 6.2. Let f : C → C be a non-isomorphic endomorphism on a smooth projective curve C. Then ♯ Fix(f ) ≤ deg f + 2c(deg f ) 1/2 + 1 where c is the genus of C.\\nProposition 6.2. Let f : C → C be a non-isomorphic endomorphism on a smooth projective curve C. Then ♯ Fix(f ) ≤ deg f + 2c(deg f ) 1/2 + 1 where c is the genus of C.\\nProof. Let d := d 1 (f ) = deg f ≥ 2. Then f is d-polarized, so is its induced endomorphism g : Alb(C) → Alb(C) by Theorem 6.4. Note that f has at most finitely many fixed points.\\nProof. Let d := d 1 (f ) = deg f ≥ 2. Then f is d-polarized, so is its induced endomorphism g : Alb(C) → Alb(C) by Theorem 6.4. Note that f has at most finitely many fixed points.\\nApplying Lefschetz fixed-point formula, we have\\nApplying Lefschetz fixed-point formula, we have\\n. We are going to prove Theorem 6.4 (cf. Theorem 1.9). It has its own independent interest and various applications, and is also needed in proving Proposition 6. Remark 5.9] without the smooth and characteristic 0 assumptions.\\n. We are going to prove Theorem 6.4 (cf. Theorem 1.9). It has its own independent interest and various applications, and is also needed in proving Proposition 6. Remark 5.9] without the smooth and characteristic 0 assumptions.\\nWe prepare some notation. Let f be a surjective endomorphism of a normal projective variety X over the field k which is allowed to be of arbitrary characteristic. Let We need the following in proving Theorem 6.4. Proposition 6.3. Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Let λ be an eigenvalue of f * | Pic • (X) red , equivalently of (f ∨ ) * | H 1 (X ∨ ,Z ℓ ) . Then\\nWe prepare some notation. Let f be a surjective endomorphism of a normal projective variety X over the field k which is allowed to be of arbitrary characteristic. Let We need the following in proving Theorem 6.4. Proposition 6.3. Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Let λ be an eigenvalue of f * | Pic • (X) red , equivalently of (f ∨ ) * | H 1 (X ∨ ,Z ℓ ) . Then\\nwhere ι f is the minimal modulus of eigenvalues of f * | N 1 (X) .\\nwhere ι f is the minimal modulus of eigenvalues of f * | N 1 (X) .\\nProof. Let P ∈ Z[t] be the minimal polynomial of f ∨ : X ∨ → X ∨ , which is the minimal\\nProof. Let P ∈ Z[t] be the minimal polynomial of f ∨ : X ∨ → X ∨ , which is the minimal\\nThen P (f ∨ ) = 0 (cf. [Hu19, Section 2]). Note also that\\nThen P (f ∨ ) = 0 (cf. [Hu19, Section 2]). Note also that\\nbe an eigenvalue of f ∨ . Then λ is a root of P and hence a ≤ |λ| ≤ b.\\nbe an eigenvalue of f ∨ . Then λ is a root of P and hence a ≤ |λ| ≤ b.\\nWe may assume 0 ∈ alb X (X). Note that the smallest abelian subvariety of A containing alb X (X) is A itself by the universal property. Applying the similar technique used in the proof of Proposition 4.3, we have the following commutative diagram\\nWe may assume 0 ∈ alb X (X). Note that the smallest abelian subvariety of A containing alb X (X) is A itself by the universal property. Applying the similar technique used in the proof of Proposition 4.3, we have the following commutative diagram\\nwhere n ≫ 1 and s n : X ×(2n) → Alb(X) is defined by\\nwhere n ≫ 1 and s n : X ×(2n) → Alb(X) is defined by\\nSince s * n : N 1 (Alb(X)) → N 1 (X ×(2n) ) is (equivariant) injective, we conclude:\\nSince s * n : N 1 (Alb(X)) → N 1 (X ×(2n) ) is (equivariant) injective, we conclude:\\nSo the proposition is proved. (2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number.\\nSo the proposition is proved. (2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number.\\nThen f is q-polarized.\\nThen f is q-polarized.\\n(3) Suppose f is q-polarized. Then any eigenvalue of f * | H 1 (X,Q ℓ ) has modulus √ q.\\n(3) Suppose f is q-polarized. Then any eigenvalue of f * | H 1 (X,Q ℓ ) has modulus √ q.\\n(4) Suppose f is q-polarized (resp. int-amplified). Then so are the induced endomorphisms f | Alb(X) : Alb(X) → Alb(X) and f | Alb(X) : Alb(X) → Alb(X) via the Albanese morphism and Albanese map respectively.\\n(4) Suppose f is q-polarized (resp. int-amplified). Then so are the induced endomorphisms f | Alb(X) : Alb(X) → Alb(X) and f | Alb(X) : Alb(X) → Alb(X) via the Albanese morphism and Albanese map respectively.\\nProof. We follow the notation in the proof of Proposition 6.3.\\nProof. We follow the notation in the proof of Proposition 6.3.\\nNote that although (X ∨ ) K is in general an infinite dimensional K-vector space, the Jordan canonical form of (f ∨ ) K has all the Jordan blocks of size ≤ deg P (recall that P is the minimal polynomial of f ∨ ).\\nNote that although (X ∨ ) K is in general an infinite dimensional K-vector space, the Jordan canonical form of (f ∨ ) K has all the Jordan blocks of size ≤ deg P (recall that P is the minimal polynomial of f ∨ ).\\nBy Proposition 6.3, any diagonal entry λ further satisfies |λ| ≤ d 1 (f ) < d 1 (f ).\\nBy Proposition 6.3, any diagonal entry λ further satisfies |λ| ≤ d 1 (f ) < d 1 (f ).\\nIn\\nIn\\nFor the second half of (1), applying Perron-Frobenius theorem to the nef cone (cf. [Bir67]), there is a nef\\nFor the second half of (1), applying Perron-Frobenius theorem to the nef cone (cf. [Bir67]), there is a nef\\nwe then apply the first half of (1).\\nwe then apply the first half of (1).\\n(2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number. (3) just follows from Proposition 6.3.\\n(2) Suppose f * B ≡ qB for some big R-Cartier divisor and q > 1 a rational number. (3) just follows from Proposition 6.3.\"\n },\n {\n \"title\": \"Higher-dimensional Hodge Index Theorem\",\n \"content\": \"\"\n },\n {\n \"title\": \"Higher-dimensional Hodge Index Theorem\",\n \"content\": \"In this section, we recall the higher-dimensional Hodge index theorem and give an application (cf. Proposition 7.2) which will be crucially used in Section 8.\\nIn this section, we recall the higher-dimensional Hodge index theorem and give an application (cf. Proposition 7.2) which will be crucially used in Section 8.\\nThe following lemma is known in [DS04, Corollaire 3.5] for the case of compact Kähler manifolds. See [MZg20, Lemma 4.5] for its full algebraic version. The proof below is taken from [MZg20, Lemma 4.5] with slight simplification for the convenience of the readers.\\nThe following lemma is known in [DS04, Corollaire 3.5] for the case of compact Kähler manifolds. See [MZg20, Lemma 4.5] for its full algebraic version. The proof below is taken from [MZg20, Lemma 4.5] with slight simplification for the convenience of the readers.\\nLemma 7.1. Let X be a normal projective variety. Let L ≡ 0 and D ≡ 0 be two pseudoeffective R-Cartier divisors such that L 2 and D 2 are pseudo-effective (cf. Notation 2.1).\\nLemma 7.1. Let X be a normal projective variety. Let L ≡ 0 and D ≡ 0 be two pseudoeffective R-Cartier divisors such that L 2 and D 2 are pseudo-effective (cf. Notation 2.1).\\nSuppose L · D ≡ w 0. Then L ≡ aD for some a > 0.\\nSuppose L · D ≡ w 0. Then L ≡ aD for some a > 0.\\nProof. We may assume n := dim(X) ≥ 2. Fix very ample Cartier divisors A 1 , · · · , A n−1 on X. Denote by V := RD + RL, and\\nProof. We may assume n := dim(X) ≥ 2. Fix very ample Cartier divisors A 1 , · · · , A n−1 on X. Denote by V := RD + RL, and\\nsubspaces of N 1 (X). By [Zha16,Lemma 3.2], L ≡ w 0 and D ≡ w 0. Then a := L · A 1 · · · A n−1 /D · A 1 · · · A n−1 > 0, and V ∩ W is of dimension ≤ 1 and spanned by\\nsubspaces of N 1 (X). By [Zha16,Lemma 3.2], L ≡ w 0 and D ≡ w 0. Then a := L · A 1 · · · A n−1 /D · A 1 · · · A n−1 > 0, and V ∩ W is of dimension ≤ 1 and spanned by\\nFor each 1 ≤ i ≤ n − 1, consider the following bilinear form on N 1 (X):\\nFor each 1 ≤ i ≤ n − 1, consider the following bilinear form on N 1 (X):\\nChoose a general normal projective surface S on X such that S ≡ w A 1 · · · A i−1 ·A i+1 · · · A n−1 .\\nChoose a general normal projective surface S on X such that S ≡ w A 1 · · · A i−1 ·A i+1 · · · A n−1 .\\nThen q i (x, y) = x| S · y| S . By the Hodge index theorem on S, q i is semi-negative on W .\\nThen q i (x, y) = x| S · y| S . By the Hodge index theorem on S, q i is semi-negative on W .\\nNote that q i is semi-positive on V since L 2 , D 2 are pseudo-effective and q i (L, D) = 0.\\nNote that q i is semi-positive on V since L 2 , D 2 are pseudo-effective and q i (L, D) = 0.\\nHence q i ( D, D) = 0. For any w ∈ W and λ ∈ R, we have q i (λ D − w, λ D − w) ≤ 0. Then the inequality q i (w, w) − 2λq i ( D, w) ≤ 0 holds for any λ ∈ R and w ∈ W . This happens only when q i ( D, w) = 0 for any w ∈ W .\\nHence q i ( D, D) = 0. For any w ∈ W and λ ∈ R, we have q i (λ D − w, λ D − w) ≤ 0. Then the inequality q i (w, w) − 2λq i ( D, w) ≤ 0 holds for any λ ∈ R and w ∈ W . This happens only when q i ( D, w) = 0 for any w ∈ W .\\nNote that W and A i span N 1 (X) because W is a hyperplane of N 1 (X) and A i ∈ W .\\nNote that W and A i span N 1 (X) because W is a hyperplane of N 1 (X) and A i ∈ W .\\nNote also that q i ( D, A i ) = 0. Then\\nNote also that q i ( D, A i ) = 0. Then\\nfor any x ∈ N 1 (X). This implies that V ∩ W is independent of the choice of each A i . So D · x 1 · · · x n−1 = 0 for any divisors x 1 , · · · , x n−1 ∈ N 1 (X), which means D ≡ w 0. Note that D is R-Cartier. Therefore, D ≡ 0 by [Zha16, Lemma 3.2].\\nfor any x ∈ N 1 (X). This implies that V ∩ W is independent of the choice of each A i . So D · x 1 · · · x n−1 = 0 for any divisors x 1 , · · · , x n−1 ∈ N 1 (X), which means D ≡ w 0. Note that D is R-Cartier. Therefore, D ≡ 0 by [Zha16, Lemma 3.2].\\nProposition 7.2. Let X be a normal projective variety of dimension n, L a nef R-Cartier divisor, and D 1 , D 2 two non-zero effective R-Cartier divisors on X. Suppose D 1 and D 2 have no common components and D 1 ≡ D 2 . Suppose further that L · D 1 · H 1 · · · H n−2 = 0 for some nef and big divisors H 1 , · · · , H n−2 . Then L ≡ aD 1 ≡ aD 2 for some a > 0 and L 2 ≡ w D 1 · D 2 ≡ w 0. In particular, Supp D 1 and Supp D 2 are disjoint.\\nProposition 7.2. Let X be a normal projective variety of dimension n, L a nef R-Cartier divisor, and D 1 , D 2 two non-zero effective R-Cartier divisors on X. Suppose D 1 and D 2 have no common components and D 1 ≡ D 2 . Suppose further that L · D 1 · H 1 · · · H n−2 = 0 for some nef and big divisors H 1 , · · · , H n−2 . Then L ≡ aD 1 ≡ aD 2 for some a > 0 and L 2 ≡ w D 1 · D 2 ≡ w 0. In particular, Supp D 1 and Supp D 2 are disjoint.\\nProof. For each H i , we write H i ∼ A i + E i for some ample divisor A i and effective divisor E i . By the assumption, the intersection of (the numerically movable) D 1 with any prime divisor is pseudo-effective, hence so is D 1 · E i for each i. Then L · D 1 · H 1 · · · H n−2 = 0 implies that L · D 1 · A 1 · H 2 · · · H n−2 = 0.\\nProof. For each H i , we write H i ∼ A i + E i for some ample divisor A i and effective divisor E i . By the assumption, the intersection of (the numerically movable) D 1 with any prime divisor is pseudo-effective, hence so is D 1 · E i for each i. Then L · D 1 · H 1 · · · H n−2 = 0 implies that L · D 1 · A 1 · H 2 · · · H n−2 = 0.\\nRepeating the same process, we have L · D 1 · A 1 · · · A n−2 = 0 and thus L · D 1 ≡ w 0. By Lemma 7.1 and noting D 2 1 ≡ w D 1 D 2 , we get L ≡ aD 1 (≡ aD 2 ) with a > 0. So we have\\nRepeating the same process, we have L · D 1 · A 1 · · · A n−2 = 0 and thus L · D 1 ≡ w 0. By Lemma 7.1 and noting D 2 1 ≡ w D 1 D 2 , we get L ≡ aD 1 (≡ aD 2 ) with a > 0. So we have\\nis a non-zero effective cycle and hence we get a contradiction with D 1 · D 2 ≡ w 0.\\nis a non-zero effective cycle and hence we get a contradiction with D 1 · D 2 ≡ w 0.\\n8. Proofs of Theorems 1.6 and 1.7\\n8. Proofs of Theorems 1.6 and 1.7\\nIn this section, we will prove Theorems 1.7 and 8.1 below which includes Theorem 1.6 as a simpler version and bounds the number ♯P f in terms of the Néron-Severi class group rank ρ ns (X), the Picard number ρ( X) of any projective resolution X of X, and the first dynamical degree d 1 (f ).\\nIn this section, we will prove Theorems 1.7 and 8.1 below which includes Theorem 1.6 as a simpler version and bounds the number ♯P f in terms of the Néron-Severi class group rank ρ ns (X), the Picard number ρ( X) of any projective resolution X of X, and the first dynamical degree d 1 (f ).\\nTheorem 8.1. Let f : X → X be a surjective endomorphism of a normal projective variety. Let X → X be any projective resolution. Then\\nTheorem 8.1. Let f : X → X be a surjective endomorphism of a normal projective variety. Let X → X be any projective resolution. Then\\nwhere a is a positive constant depending only on X. In particular, for n ≫ 1, (1)).\\nwhere a is a positive constant depending only on X. In particular, for n ≫ 1, (1)).\\nWe now prepare to prove Theorem 8.1.\\nWe now prepare to prove Theorem 8.1.\\nNotation 8.2. Let f : X → X be a surjective endomorphism on a normal projective variety X of dimension n. We fix a projective resolution σ : X → X with E 1 , · · · , E k its exceptional prime divisors. For D ∈ P f , denote by D := σ −1 * (D), the proper transform of D. Denote by\\nNotation 8.2. Let f : X → X be a surjective endomorphism on a normal projective variety X of dimension n. We fix a projective resolution σ : X → X with E 1 , · · · , E k its exceptional prime divisors. For D ∈ P f , denote by D := σ −1 * (D), the proper transform of D. Denote by\\nBy Perron-Frobenius theorem (cf. [Bir67]), fix a nef R-Cartier divisor L f ≡ 0 on X with\\nBy Perron-Frobenius theorem (cf. [Bir67]), fix a nef R-Cartier divisor L f ≡ 0 on X with\\nDenote by\\nDenote by\\nand d 1 (f | D ) < d 1 (f ). Then L f | D ≡ 0 and hence L f · D ≡ w 0.\\nand d 1 (f | D ) < d 1 (f ). Then L f | D ≡ 0 and hence L f · D ≡ w 0.\\nLemma 8.4. There exist at least 2⌊♯P f /(ρ ns (X) + 1)⌋ disjoint prime divisors D u in P f with L f · D u ≡ w 0.\\nLemma 8.4. There exist at least 2⌊♯P f /(ρ ns (X) + 1)⌋ disjoint prime divisors D u in P f with L f · D u ≡ w 0.\\nProof. Let m := ⌊♯P f /(ρ ns (X)+1)⌋. Then we have m pairwise disjoint subsets S 1 , · · · , S m of P f with ♯S t = ρ ns (X) + 1 for each 1 ≤ t ≤ m.\\nProof. Let m := ⌊♯P f /(ρ ns (X)+1)⌋. Then we have m pairwise disjoint subsets S 1 , · · · , S m of P f with ♯S t = ρ ns (X) + 1 for each 1 ≤ t ≤ m.\\nNote that ρ( X) = ρ ns (X) + k where k is the number of σ-exceptional prime divisors (cf. [Kol18, Lemma 18]). By the assumption, we have the equality ♯(S t ∪ {E 1 , · · · , E k }) = ρ( X) + 1.\\nNote that ρ( X) = ρ ns (X) + k where k is the number of σ-exceptional prime divisors (cf. [Kol18, Lemma 18]). By the assumption, we have the equality ♯(S t ∪ {E 1 , · · · , E k }) = ρ( X) + 1.\\nFor each t, we then have a numerical equivalence of two non-zero effective divisors which are combinations of prime divisors in S t and σ-exceptional prime divisors such that they have no common component. We write them in the following form i∈It a i D i + E t ≡ j∈Jt b j D j + F t where I t and J t are two (possibly empty) disjoint subsets of S t , a i > 0, b j > 0, and E t , F t are effective σ-exceptional divisors. Here without ambiguity, we identify D i with the index i in I t .\\nFor each t, we then have a numerical equivalence of two non-zero effective divisors which are combinations of prime divisors in S t and σ-exceptional prime divisors such that they have no common component. We write them in the following form i∈It a i D i + E t ≡ j∈Jt b j D j + F t where I t and J t are two (possibly empty) disjoint subsets of S t , a i > 0, b j > 0, and E t , F t are effective σ-exceptional divisors. Here without ambiguity, we identify D i with the index i in I t .\\nLet H be an ample divisor on X. By Lemma 8.3 and the projection formula,\\nLet H be an ample divisor on X. By Lemma 8.3 and the projection formula,\\nwhere D i = σ * ( D i ) ∈ P f . Applying Proposition 7.2, for each t, we may assume (after some scalar multiplication)\\nwhere D i = σ * ( D i ) ∈ P f . Applying Proposition 7.2, for each t, we may assume (after some scalar multiplication)\\nand we also have\\nand we also have\\nThis further implies that\\nThis further implies that\\n(1)\\n(1)\\nfor any i ∈ I t , j ∈ J t and any prime divisor E in E t and F t .\\nfor any i ∈ I t , j ∈ J t and any prime divisor E in E t and F t .\\nWe claim that I t and J t are non-empty. Suppose the contrary, for example, that I t is empty. Then L f ≡ E t = 0. However, this contradicts with the negativity lemma (cf. [KM98, Lemma 3.39]).\\nWe claim that I t and J t are non-empty. Suppose the contrary, for example, that I t is empty. Then L f ≡ E t = 0. However, this contradicts with the negativity lemma (cf. [KM98, Lemma 3.39]).\\nConsider the pair (I t , J t ′ ) for any t, t ′ . Note that\\nConsider the pair (I t , J t ′ ) for any t, t ′ . Note that\\nWe claim that E t and F t ′ have no common components. Readjust it a bit in the following way, where ∆ is a σ-exceptional effective R-divisor,\\nWe claim that E t and F t ′ have no common components. Readjust it a bit in the following way, where ∆ is a σ-exceptional effective R-divisor,\\nso that E t − ∆ and F t ′ − ∆ are effective but with no common components. Note that\\nso that E t − ∆ and F t ′ − ∆ are effective but with no common components. Note that\\nBy Proposition 7.2, for some e > 0, we have e L f ≡ i∈It a i D i + ( E t − ∆).\\nBy Proposition 7.2, for some e > 0, we have e L f ≡ i∈It a i D i + ( E t − ∆).\\nThen (1 − e) L f ≡ ∆, and hence e ≤ 1 since both L f and ∆ are pseudo-effective and X is projective. By the negativity lemma again (cf. [KM98, Lemma 3.39]), e = 1 and ∆ ≡ 0, so ∆ = 0 and the claim is proved. By Proposition 7.2 and the display (2) above and since L f · L f ≡ w 0, we have D i ∩ D j = ∅ for any i ∈ I t and j ∈ J t ′ .\\nThen (1 − e) L f ≡ ∆, and hence e ≤ 1 since both L f and ∆ are pseudo-effective and X is projective. By the negativity lemma again (cf. [KM98, Lemma 3.39]), e = 1 and ∆ ≡ 0, so ∆ = 0 and the claim is proved. By Proposition 7.2 and the display (2) above and since L f · L f ≡ w 0, we have D i ∩ D j = ∅ for any i ∈ I t and j ∈ J t ′ .\\nThe same argument works for the pairs (I t , I t ′ ) and (J t , J t ′ ) with t = t ′ . In particular, we may pick one prime divisor in I t and one prime divisor in J t for each t, and all these 2m prime divisors are pairwise disjoint. This completes the proof.\\nThe same argument works for the pairs (I t , I t ′ ) and (J t , J t ′ ) with t = t ′ . In particular, we may pick one prime divisor in I t and one prime divisor in J t for each t, and all these 2m prime divisors are pairwise disjoint. This completes the proof.\\nWith ♯P f large enough, we can construct an f -equivariant fibration over a curve.\\nWith ♯P f large enough, we can construct an f -equivariant fibration over a curve.\\nProposition 8.5. Suppose ♯P f ≥ (⌊ ρ( X) 2 ⌋ + 1) · (ρ ns (X) + 1). Then there exists a fibration π : X → C onto a smooth curve C such that the following hold.\\nProposition 8.5. Suppose ♯P f ≥ (⌊ ρ( X) 2 ⌋ + 1) · (ρ ns (X) + 1). Then there exists a fibration π : X → C onto a smooth curve C such that the following hold.\\n(1) L f ≡ π * H C for some ample R-Cartier divisor H C on C.\\n(1) L f ≡ π * H C for some ample R-Cartier divisor H C on C.\\n(2) The fibration π factors through a fibration π : X → C such that L f ≡ π * H C .\\n(2) The fibration π factors through a fibration π : X → C such that L f ≡ π * H C .\\n(3) There exists a d 1 (f )-polarized endomorphism g : C → C such that π • f = g • π.\\n(3) There exists a d 1 (f )-polarized endomorphism g : C → C such that π • f = g • π.\\nProof. (1) By Lemma 8.4, there are ρ( X) + 1 pairwise disjoint prime divisors D u in P f with L f · D u ≡ w 0. By [BPS16, Theorem 1.1], there is a fibration π : X → C onto a smooth projective curve such that π( D u ) is a point for each u and D u = π −1 ( π( D u )) for X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. Let B be the finite set of prime divisors in the reducible fibres of π. By the same argument as in the proof of Proposition 6.1 and [CMZ20, Lemma 7.5], we have an injection ϕ : P −1 f ∞ \\\\B → P −1 g ∞ , sending D to π(D). Note that there are at most finitely many g −1 -periodic points (cf. [MZ20, Corollary 3.8]). So the theorem is proved.\\nProof. (1) By Lemma 8.4, there are ρ( X) + 1 pairwise disjoint prime divisors D u in P f with L f · D u ≡ w 0. By [BPS16, Theorem 1.1], there is a fibration π : X → C onto a smooth projective curve such that π( D u ) is a point for each u and D u = π −1 ( π( D u )) for X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. Let B be the finite set of prime divisors in the reducible fibres of π. By the same argument as in the proof of Proposition 6.1 and [CMZ20, Lemma 7.5], we have an injection ϕ : P −1 f ∞ \\\\B → P −1 g ∞ , sending D to π(D). Note that there are at most finitely many g −1 -periodic points (cf. [MZ20, Corollary 3.8]). So the theorem is proved.\\nCorollary 8.6. Let f : X → X be a surjective endomorphism of a normal projective variety X with ♯P f ∞ = ∞. Then D is either Q-Cartier or f −1 -periodic for any D ∈ P f ∞ .\\nCorollary 8.6. Let f : X → X be a surjective endomorphism of a normal projective variety X with ♯P f ∞ = ∞. Then D is either Q-Cartier or f −1 -periodic for any D ∈ P f ∞ .\"\n },\n {\n \"title\": \"Theorem 1 . 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 1 . 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Conjecture 3. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Conjecture 3. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 4. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 4. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"[ Bri10 ,\",\n \"content\": \"\"\n },\n {\n \"title\": \"[ Bri10 ,\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 5. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 5. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"d 1\",\n \"content\": \"\"\n },\n {\n \"title\": \"d 1\",\n \"content\": \"\"\n },\n {\n \"title\": \"[ Bri10 ,\",\n \"content\": \"\"\n },\n {\n \"title\": \"[ Bri10 ,\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 6. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 6. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"( 4 )\",\n \"content\": \"\"\n },\n {\n \"title\": \"( 4 )\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Let f be a dominant self-morphism of a variety X. Then","content":"Theorem 1 . 5 .\n15Let f be a dominant self-morphism of a variety X. Then"},{"id":"","caption":"Let f be a dominant self-morphism of a variety X. Then","content":"Theorem 1 . 5 .\n15Let f be a dominant self-morphism of a variety X. Then"},{"id":"","caption":"♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases.","content":"♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases."},{"id":"","caption":"♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases.","content":"♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases."},{"id":"","caption":"f n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞.","content":"f n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞."},{"id":"","caption":"f n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞.","content":"f n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞."},{"id":"","caption":"(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set","content":"Conjecture 3. 2 .\n2(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set"},{"id":"","caption":"(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set","content":"Conjecture 3. 2 .\n2(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set"},{"id":"","caption":"So f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure.","content":"So f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure."},{"id":"","caption":"So f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure.","content":"So f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure."},{"id":"","caption":"Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H ).","content":"Lemma 4. 2 .\n2Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H )."},{"id":"","caption":"Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H ).","content":"Lemma 4. 2 .\n2Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H )."},{"id":"","caption":"Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps.","content":"[ Bri10 ,\nBri10Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps."},{"id":"","caption":"Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps.","content":"[ Bri10 ,\nBri10Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps."},{"id":"","caption":"since g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have","content":"since g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have"},{"id":"","caption":"since g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have","content":"since g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have"},{"id":"","caption":"S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2]).","content":"Corollary 5. 2 .\n2S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2])."},{"id":"","caption":"S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2]).","content":"Corollary 5. 2 .\n2S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2])."},{"id":"","caption":"(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M.","content":"d 1\n1(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M."},{"id":"","caption":"(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M.","content":"d 1\n1(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M."},{"id":"","caption":"Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)","content":"[ Bri10 ,\nBri10Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)"},{"id":"","caption":"Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)","content":"[ Bri10 ,\nBri10Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)"},{"id":"","caption":"any torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write","content":"any torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write"},{"id":"","caption":"any torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write","content":"any torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write"},{"id":"","caption":"Lemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here.","content":"Lemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here."},{"id":"","caption":"Lemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here.","content":"Lemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here."},{"id":"","caption":"]). So the proposition is proved.","content":"]). So the proposition is proved."},{"id":"","caption":"]). So the proposition is proved.","content":"]). So the proposition is proved."},{"id":"","caption":"2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,","content":"2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,"},{"id":"","caption":"2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,","content":"2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,"},{"id":"","caption":"the dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X).","content":"the dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X)."},{"id":"","caption":"the dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X).","content":"the dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X)."},{"id":"","caption":"By a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says","content":"By a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says"},{"id":"","caption":"By a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says","content":"By a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says"},{"id":"","caption":"Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 .","content":"Theorem 6. 4 .\n4Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 ."},{"id":"","caption":"Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 .","content":"Theorem 6. 4 .\n4Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 ."},{"id":"","caption":"By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 .","content":"( 4 )\n4By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 ."},{"id":"","caption":"By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 .","content":"( 4 )\n4By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 ."},{"id":"","caption":"CMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]","content":"CMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]"},{"id":"","caption":"CMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]","content":"CMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]"},{"id":"","caption":"Frobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.","content":"Frobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.In Remark 1.4, the method of counting fixed points for polarized endomorphisms only \n\nholds in characteristic 0, because it applied Serre's theorem to compute the eigenvalues \n\nof f * | H i (X,Z) which heavily relied on a Kähler form. Serre's theorem is also a Kähler \n\nanalogue of Weil's conjecture on eigenvalues of the pullback of (the purely inseparable \n\nq-polarized)"},{"id":"","caption":"Frobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.","content":"Frobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.In Remark 1.4, the method of counting fixed points for polarized endomorphisms only \n\nholds in characteristic 0, because it applied Serre's theorem to compute the eigenvalues \n\nof f * | H i (X,Z) which heavily relied on a Kähler form. Serre's theorem is also a Kähler \n\nanalogue of Weil's conjecture on eigenvalues of the pullback of (the purely inseparable \n\nq-polarized)"},{"id":"","caption":"Then algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved.","content":"Then algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved."},{"id":"","caption":"Then algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved.","content":"Then algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Let f be a dominant self-morphism of a variety X. Then\",\n \"content\": \"Theorem 1 . 5 .\\n15Let f be a dominant self-morphism of a variety X. Then\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let f be a dominant self-morphism of a variety X. Then\",\n \"content\": \"Theorem 1 . 5 .\\n15Let f be a dominant self-morphism of a variety X. Then\"\n },\n {\n \"id\": \"\",\n \"caption\": \"♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases.\",\n \"content\": \"♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases.\",\n \"content\": \"♯S f n ≤ (d 1 (f )) n·dim(X) (1 + o(1))holds as n → ∞, if (X, f ) fits one of the following cases.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"f n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞.\",\n \"content\": \"f n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"f n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞.\",\n \"content\": \"f n ≤ (d 1 (f )) n (1 + o(1))holds as n → ∞.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set\",\n \"content\": \"Conjecture 3. 2 .\\n2(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set\",\n \"content\": \"Conjecture 3. 2 .\\n2(small Arithmetic Non-Density Conjecture, [MMSZ20, Conjecture 1.4]) Let f : X → X be a surjective endomorphism of a projective variety X defined over a number field K. Then the set\"\n },\n {\n \"id\": \"\",\n \"caption\": \"So f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure.\",\n \"content\": \"So f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"So f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure.\",\n \"content\": \"So f -invariant (resp. f -periodic) closed subvarieties defined in Notation 2.1 are in the usual sense without taking closure.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H ).\",\n \"content\": \"Lemma 4. 2 .\\n2Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H ).\",\n \"content\": \"Lemma 4. 2 .\\n2Let H be a (connected closed) algebraic subgroup of G. Suppose f (Hb) = Hb for some b ∈ G. Then g(H) = H and d 1 (f | Hb ) = d 1 (g| H ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps.\",\n \"content\": \"[ Bri10 ,\\nBri10Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps.\",\n \"content\": \"[ Bri10 ,\\nBri10Theorem 2], we have Aequivariant open embeddings to smooth projective varieties A ֒→ A and C ֒→ C (taking A-equivariant resolution if necessary). Denote by π : A C the induced dominant rational map. Denote by g : A A and g| C : C C the induced dominant self-maps.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"since g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have\",\n \"content\": \"since g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"since g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have\",\n \"content\": \"since g(G(f )) = G(f ). By Lemma 4.7 and Proposition 4.6, S f = Fix(f ) and π induces a bijection between S f and Fix(f ). Consider the case ♯ Fix(g) = ∞. Then there is a positive dimensional (connected closed) algebraic subgroup H of G such that g| H = id H . Let H := π −1 (H) which is a g-invariant (connected closed) algebraic subgroup of G. By the product formula in Lemma 4.7, we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2]).\",\n \"content\": \"Corollary 5. 2 .\\n2S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2]).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2]).\",\n \"content\": \"Corollary 5. 2 .\\n2S f is finite for any surjective endomorphism f of a Q-abelian variety. Proof. By taking the Albanese closure π : A → X (cf. [CMZ20, Lemma 8.1]), f lifts to a surjective endomorphism f A : A → A of an abelian variety A. Then our corollary follows from Lemma 2.3 and Theorem 5.1. Next we treat the semi-abelian varieties. Let f : X → X be a (not necessarily dominant) self-morphism of a semi-abelian variety X. Then R f (e) −1 • f is a group homomorphism of X (cf. [Itk76, Theorem 2]).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M.\",\n \"content\": \"d 1\\n1(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M.\",\n \"content\": \"d 1\\n1(f | T ) is equal to the spectral radius of M (cf. [Lin12, Theorem 1.1]). Let λ 1 , · · · , λ dim(T ) be the diagonal entries of the Jordan canonical form of M.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)\",\n \"content\": \"[ Bri10 ,\\nBri10Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)\",\n \"content\": \"[ Bri10 ,\\nBri10Theorem 2], there is a GL(U)-equivariant open embedding U ֒− → X to some smooth projective variety X (taking equivariant resolution if necessary). Thus the connected group GL(U) and hence g act trivially on the lattice NS(X)/(torsion). So d 1 (g) = 1.Lemma 5.6. Let g : S → S be an isogeny of a (connected) semisimple group S. Then d 1 (g) = 1.Proof. First note that (deg g) n = deg g n = ♯ Ker(g n ) ≤ ♯Z(S) < ∞ for any n > 0, where Z(S) is the centre of S (a finite set). Thus g ∈ Aut(S), the group of algebraic group automorphisms. Let π : S → S be the universal cover of S which is finite. Then g lifts to g : S → S. By the product formula, we may replace S by S and assumeS is simply connected. By the homomorphisms theorem and [HM69, Proposition 2.2], Aut(S) ∼ = Aut(s) in the sense of algebraic groups via the differential map at e, where s is the Lie algebra of S. By [FH91, Proposition D.40], Aut(S)/ Aut • (S) ∼ = Aut(s)/ Aut • (s)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"any torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write\",\n \"content\": \"any torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write\"\n },\n {\n \"id\": \"\",\n \"caption\": \"any torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write\",\n \"content\": \"any torus orbit O with f (O) = O, we compute the number of V ∈ S f such that V ∩ O contains an open dense subset of V . Fix any point x 0 ∈ O. Write\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Lemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here.\",\n \"content\": \"Lemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Lemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here.\",\n \"content\": \"Lemma 5.3] where we need to take (projective) resolution of X. Of course, if X (and hence O) are projective, then there is no issue here.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"]). So the proposition is proved.\",\n \"content\": \"]). So the proposition is proved.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"]). So the proposition is proved.\",\n \"content\": \"]). So the proposition is proved.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,\",\n \"content\": \"2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,\",\n \"content\": \"2 in positive characteristic. It generalizes [CMZ20, Theorem 1.2 and Theorem 5.1] by removing the separable assumption, [MZ19, Proposition 3.3] by removing the surjective assumption, [NZ10, Lemma 2.3] without the polarized and characteristic 0 assumptions, and [San17,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X).\",\n \"content\": \"the dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X).\",\n \"content\": \"the dual of X and dual isogeny of f . Denote by Alb(X) := Pic • (Pic • (X) red ) = X ∨∨ the Albanese variety of X. There are two canonical maps alb X : X → Alb(X), alb X : X Alb(X) called Albanese morphism and Albanese map satisfying the universal properties in the categories of morphisms and rational maps, respectively (cf. [FGI+05, Remark 9.5.25], [Lan83, Chapter II.3] and [CMZ20, Section 5]). In particular, f descends to surjective endomorphisms on Alb(X) and Alb(X).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"By a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says\",\n \"content\": \"By a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says\"\n },\n {\n \"id\": \"\",\n \"caption\": \"By a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says\",\n \"content\": \"By a dual argument, P is also the minimal polynomial of f ∨∨ , i.e., the minimal polynomial of (f ∨∨ ) * | H 1 (X ∨∨ ,Z l ) . Let a and b be the minimal and maximal modulus of roots of P , respectively. By [Hu19, Formula (3.6)], b 2 = d 1 (f ∨ ). The same proof for [Hu19, Formula (3.6)] shows that a 2 = ι f ∨ . A similar argument for f ∨∨ says\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 .\",\n \"content\": \"Theorem 6. 4 .\\n4Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 .\",\n \"content\": \"Theorem 6. 4 .\\n4Let f : X → X be a (not necessarily separable) surjective endomorphism of a normal projective variety X over an algebraically closed field k of arbitrary characteristic. Assume d 1 (f ) > 1. Then we have: (1) Set K := Q or R. Suppose f * D ≡ d 1 (f )D for some K-Cartier divisor D. Then there exists another K-Cartier divisor D ′ ≡ D such that f * D ′ ∼ K d 1 (f )D ′ . In particular, f * D 1 ∼ R d 1 (f )D 1 for some nef R-Cartier divisor D 1 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 .\",\n \"content\": \"( 4 )\\n4By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 .\",\n \"content\": \"( 4 )\\n4By the commutative diagram (**) in the proof of Proposition 6.3, if f is q-polarized (resp. int-amplified), then so are f ×(2n) and f | Alb(X) (cf. [MZ18b, Lemma 3.10, Theorem 3.11]) for polarized endomorphism and [Men20, Lemma 3.4]). For the Albanese map alb X : XAlb(X), we may replace X by the normalization of the graph of alb X and argue like above (cf. [CMZ20, Proof of Theorem 1.2]). So (4) is proved.Remark 6.5. In Theorem 6.4 (1), we cannot replace d 1 (f ) by ι f because it is possible that some eigenvalue λ of f ∨ is ι f . By the intersection pairing and the projection formula, one can show in Theorem 6.4 (3) that any eigenvalue of f * | H 2 dim(X)−1 (X,Q ℓ ) has modulus q dim(X)−1/2 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"CMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]\",\n \"content\": \"CMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"CMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]\",\n \"content\": \"CMZ20, Theorem 1.2 and Theorem 5.1], [MZ19, Proposition 3.3], [NZ10, Lemma 2.3] and [San17, Remark 5.9]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Frobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.\",\n \"content\": \"Frobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.In Remark 1.4, the method of counting fixed points for polarized endomorphisms only \\n\\nholds in characteristic 0, because it applied Serre's theorem to compute the eigenvalues \\n\\nof f * | H i (X,Z) which heavily relied on a Kähler form. Serre's theorem is also a Kähler \\n\\nanalogue of Weil's conjecture on eigenvalues of the pullback of (the purely inseparable \\n\\nq-polarized)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Frobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.\",\n \"content\": \"Frobenius endomorphism F on cohomology. It was proved by Deligne's celebrated theorem: all eigenvalues of F * | H í et (X,Q ℓ ) have modulus q i/2 (cf. [Del74, Théorème 1.6]). By our Theorem 1.9 (3), this holds for every polarized endomorphism when i = 1.In Remark 1.4, the method of counting fixed points for polarized endomorphisms only \\n\\nholds in characteristic 0, because it applied Serre's theorem to compute the eigenvalues \\n\\nof f * | H i (X,Z) which heavily relied on a Kähler form. Serre's theorem is also a Kähler \\n\\nanalogue of Weil's conjecture on eigenvalues of the pullback of (the purely inseparable \\n\\nq-polarized)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Then algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved.\",\n \"content\": \"Then algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Then algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved.\",\n \"content\": \"Then algebraic integer q is an integer and f * H ≡ qH for some ample Cartier divisor H by the norm criterion; see [MZ18b, Propositions 1.1 and 2.9]. By (1), f * H ′ ∼ Q qH ′ for some ample Q-Cartier divisor H ′ ≡ H ′ . Replacing H ′ by a multiple, (2) is proved.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":")/b, (1) follows. (2) follows from (1) and Lemma 2. H C := T, H C := t π( D 1 )/b, (1) follows. (2) follows from (1) and Lemma 2.2.","doi":""},{"text":"For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0. For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0","doi":""},{"text":"So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. Deb01So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. [Deb01,","doi":""},{"text":". * L F ≡ F * Π * H C = Π * G * H C ≡ Qπ * H C ≡ Ql F, which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7f * L f ≡ f * π * H C = π * g * H C ≡ qπ * H C ≡ qL f , which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7.","doi":""},{"text":"Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X ). Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X )","doi":""},{"text":"By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4)⌋ + 1). By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4).","doi":""},{"text":"By taking normalization of X. we may assume that X is normal (cf. Lemmas 2.4 and 2.5)Proof of Theorem 1.7. By taking normalization of X, we may assume that X is normal (cf. Lemmas 2.4 and 2.5).","doi":""},{"text":"By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c\"(2) ⇒ (1).\" By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c","doi":""},{"text":"After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. We may also assume that D = π −1 (π(D)) for somed 1 (f s | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. \"(1) ⇒ (2).\" After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. We may also assume that D = π −1 (π(D)) for some","doi":""},{"text":"Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant. D , after iteration) fibration π : ReferencesD ∈ P f . Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant (after iteration) fibration π : References","doi":""},{"text":"Linear transformations with invariant cones. G Birkhoff, Amer. Math. Monthly. 74G. Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274- 276.","doi":""},{"text":"Families of disjoint divisors on varieties. F Bogomolov, A Pirutka, A Silberstein, Eur. J. Math. 24F. Bogomolov, A. Pirutka and A. Silberstein, Families of disjoint divisors on varieties, Eur. J. Math. 2 (2016), no. 4, 917-928.","doi":""},{"text":"Some basic results on actions of nonaffine algebraic groups. M Brion, Symmetry and spaces. Boston, MA278BirkhäuserM. Brion, Some basic results on actions of nonaffine algebraic groups, in: Symmetry and spaces, 1-20, Progr. Math. 278, Birkhäuser, Boston, MA, 2010.","doi":""},{"text":"Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. P Cascini, S Meng, D.-Q Zhang, Math. Ann. 3781-2P. Cascini, S. Meng and D.-Q. Zhang, Polarized endomorphisms of normal projective threefolds in arbitrary characteristic, Math. Ann. 378 (2020), no. 1-2, 637-665.","doi":""},{"text":"Degrees of iterates of rational maps on normal projective varieties. N.-B Dang, Proc. Lond. Math. Soc. 3N.-B. Dang, Degrees of iterates of rational maps on normal projective varieties, Proc. Lond. Math. Soc. (3) 121 (2020), no. 5, 1268-1310.","doi":""},{"text":"Higher-dimensional algebraic geometry. O Debarre, Springer-VerlagNew YorkUniversitextO. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001.","doi":""},{"text":". P Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43P. Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43 (1974), 273-307.","doi":""},{"text":"Comparison of dynamical degrees for semi-conjugate meromorphic maps. T.-C Dinh, V.-A Nguyen, Comment. Math. Helv. 864T.-C. Dinh and V.-A. Nguyen, Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011), no. 4, 817-840.","doi":""},{"text":"Groupes commutatifs d'automorphismes d'une variété kählérienne. T.-C Dinh, N Sibony, Duke Math. J. 1232T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne, Duke Math. J. 123 (2004), no. 2, 311-328.","doi":""},{"text":"Une borne supérieure pour l'entropie topologique d'une application rationnelle. T.-C Dinh, N Sibony, Annals of Math. 2T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Annals of Math. (2) 161 (2005), no. 3, 1637-1644.","doi":""},{"text":"Questions on self-maps of algebraic varieties. N Fakhruddin, J. Ramanujan Math. Soc. 182N. Fakhruddin, Questions on self-maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109-122.","doi":""},{"text":"Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs. B Fantechi, L Göttsche, L Illusie, S L Kleiman, N Nitsure, A Vistoli, American Mathematical Society123Providence, RIB. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs, 123. American Mathematical Society, Providence, RI.","doi":""},{"text":"W Fulton, Intersection theory. BerlinSpringer-VerlagSecond editionW. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998.","doi":""},{"text":"Representation theory: a first course. W Fulton, J Harris, Graduate Texts in Mathematics. 129Springer-VerlagReadings in MathematicsW. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.","doi":""},{"text":"Automorphisms of affine algebraic groups. G Hochschild, G D Mostow, J. Algebra. 13G. Hochschild and G. D. Mostow, Automorphisms of affine algebraic groups, J. Algebra 13 (1969), 535-543.","doi":""},{"text":"Cohomological and numerical dynamical degrees on abelian varieties. F Hu, Algebra Number Theory. 138F. Hu, Cohomological and numerical dynamical degrees on abelian varieties, Algebra Number Theory 13 (2019), no. 8, 1941-1958.","doi":""},{"text":"Logarithmic forms of algebraic varieties. S Iitaka, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 233S. Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 525-544.","doi":""},{"text":"On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. S Kawaguchi, J Silverman, J. Reine Angew. Math. 713S. Kawaguchi and J. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. 713 (2016), 21-48.","doi":""},{"text":"Birational geometry of algebraic varieties. J Kollár, S Mori, Cambridge Univ. PressCambridgeJ. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, Cam- bridge, 1998.","doi":""},{"text":". J Kollár, arXiv:1803.07596Mumford divisorsJ. Kollár, Mumford divisors, arXiv:1803.07596","doi":"arXiv:1803.07596"},{"text":"Abelian varieties. S Lang, Springer-VerlagNew York-BerlinS. Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983.","doi":""},{"text":"Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture. J Lesieutre, M Satriano, 10.1093/imrn/rnz067Int. Math. Res. Not. J. Lesieutre and M. Satriano, Canonical heights on hyper-Kähler varieties and the Kawaguchi- Silverman conjecture, Int. Math. Res. Not., https://doi.org/10.1093/imrn/rnz067","doi":"10.1093/imrn/rnz067"},{"text":"Pulling back cohomology classes and dynamical degrees of monomial maps. J Lin, Bull. Soc. Math. France. 1404J. Lin, Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France 140 (2012), no. 4, 533-549 (2013).","doi":""},{"text":"Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Y Matsuzawa, K Sano, Systems. 406Y. Matsuzawa and K. Sano, Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Systems 40 (2020), no. 6, 1655-1672.","doi":""},{"text":"Y Matsuzawa, S Meng, T Shibata, D.-Q Zhang, arXiv:2002.10976Non-density of points of small arithmetic degrees. Y. Matsuzawa, S. Meng, T. Shibata and D.-Q. Zhang, Non-density of points of small arith- metic degrees, arXiv:2002.10976","doi":"arXiv:2002.10976"},{"text":"Building blocks of amplified endomorphisms of normal projective varieties. S Meng, Math. Z. 2943S. Meng, Building blocks of amplified endomorphisms of normal projective varieties, Math. Z. 294 (2020), no. 3, 1727-1747.","doi":""},{"text":"Jordan property for non-linear algebraic groups and projective varieties. S Meng, D.-Q Zhang, Amer. J. Math. 1404S. Meng and D.-Q. Zhang, Jordan property for non-linear algebraic groups and projective varieties, Amer. J. Math. 140 (2018), no. 4, 1133-1145.","doi":""},{"text":"Building blocks of polarized endomorphisms of normal projective varieties. S Meng, D.-Q Zhang, Adv. Math. 325S. Meng and D.-Q. Zhang, Building blocks of polarized endomorphisms of normal projective varieties, Adv. Math. 325 (2018), 243-273.","doi":""},{"text":"S Meng, D.-Q Zhang, Kawaguchi-Silverman, arXiv:1908.01605Conjecture for surjective endomorphisms. S. Meng and D.-Q. Zhang, Kawaguchi-Silverman Conjecture for surjective endomorphisms, arXiv:1908.01605","doi":"arXiv:1908.01605"},{"text":"Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism. S Meng, D.-Q Zhang, Math. Res. Lett. 272S. Meng and D.-Q. Zhang, Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism, Math. Res. Lett. 27 (2020), no. 2, 523-550.","doi":""},{"text":"S Meng, G Zhong, arXiv:2005.03983Rigidity of rationally connected smooth projective varieties from dynamical viewpoints. S. Meng and G. Zhong, Rigidity of rationally connected smooth projective varieties from dy- namical viewpoints, arXiv:2005.03983","doi":"arXiv:2005.03983"},{"text":"On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms. N Nakayama, preprintN. Nakayama, On complex normal projective surfaces admitting non-isomorphic surjective en- domorphisms, preprint September 2008.","doi":""},{"text":"Polarized endomorphisms of complex normal varieties. N Nakayama, D.-Q Zhang, Math. Ann. 3464N. Nakayama and D.-Q. Zhang, Polarized endomorphisms of complex normal varieties, Math. Ann. 346 (2010), no. 4, 991-1018.","doi":""},{"text":"The canonical heights for Jordan blocks of small eigenvalues. K Sano, arXiv:1712.07533preperiodic points, and the arithmetic degrees. K. Sano, The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees, arXiv:1712.07533","doi":"arXiv:1712.07533"},{"text":"Linear algebraic groups. T A Springer, Modern Birkhaüser Classics. Birkhaüser Boston IncT. A. Springer, Linear algebraic groups, Modern Birkhaüser Classics, Boston, MA: Birkhaüser Boston Inc., 2009.","doi":""},{"text":"n-dimensional projective varieties with the action of an abelian group of rank n − 1. D.-Q Zhang, Trans. Amer. Math. Soc. 36812D.-Q. Zhang, n-dimensional projective varieties with the action of an abelian group of rank n − 1, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8849-8872.","doi":""},{"text":"Distributions in algebraic dynamics. S.-W Zhang, Department of Mathematics. 10Brown UniversitySurv. Differ. Geom.S.-W. Zhang, Distributions in algebraic dynamics, Surv. Differ. Geom., 10 (2006), 381-430. Department of Mathematics, box 1917, Brown University, Rhode Island 02912, USA","doi":""},{"text":")/b, (1) follows. (2) follows from (1) and Lemma 2. H C := T, H C := t π( D 1 )/b, (1) follows. (2) follows from (1) and Lemma 2.2.","doi":""},{"text":"For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0. For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0","doi":""},{"text":"So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. Deb01So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. [Deb01,","doi":""},{"text":". * L F ≡ F * Π * H C = Π * G * H C ≡ Qπ * H C ≡ Ql F, which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7f * L f ≡ f * π * H C = π * g * H C ≡ qπ * H C ≡ qL f , which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7.","doi":""},{"text":"Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X ). Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X )","doi":""},{"text":"By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4)⌋ + 1). By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4).","doi":""},{"text":"By taking normalization of X. we may assume that X is normal (cf. Lemmas 2.4 and 2.5)Proof of Theorem 1.7. By taking normalization of X, we may assume that X is normal (cf. Lemmas 2.4 and 2.5).","doi":""},{"text":"By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c\"(2) ⇒ (1).\" By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c","doi":""},{"text":"After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. We may also assume that D = π −1 (π(D)) for somed 1 (f s | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. \"(1) ⇒ (2).\" After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. We may also assume that D = π −1 (π(D)) for some","doi":""},{"text":"Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant. D , after iteration) fibration π : ReferencesD ∈ P f . Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant (after iteration) fibration π : References","doi":""},{"text":"Linear transformations with invariant cones. G Birkhoff, Amer. Math. Monthly. 74G. Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274- 276.","doi":""},{"text":"Families of disjoint divisors on varieties. F Bogomolov, A Pirutka, A Silberstein, Eur. J. Math. 24F. Bogomolov, A. Pirutka and A. Silberstein, Families of disjoint divisors on varieties, Eur. J. Math. 2 (2016), no. 4, 917-928.","doi":""},{"text":"Some basic results on actions of nonaffine algebraic groups. M Brion, Symmetry and spaces. Boston, MA278BirkhäuserM. Brion, Some basic results on actions of nonaffine algebraic groups, in: Symmetry and spaces, 1-20, Progr. Math. 278, Birkhäuser, Boston, MA, 2010.","doi":""},{"text":"Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. P Cascini, S Meng, D.-Q Zhang, Math. Ann. 3781-2P. Cascini, S. Meng and D.-Q. Zhang, Polarized endomorphisms of normal projective threefolds in arbitrary characteristic, Math. Ann. 378 (2020), no. 1-2, 637-665.","doi":""},{"text":"Degrees of iterates of rational maps on normal projective varieties. N.-B Dang, Proc. Lond. Math. Soc. 3N.-B. Dang, Degrees of iterates of rational maps on normal projective varieties, Proc. Lond. Math. Soc. (3) 121 (2020), no. 5, 1268-1310.","doi":""},{"text":"Higher-dimensional algebraic geometry. O Debarre, Springer-VerlagNew YorkUniversitextO. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001.","doi":""},{"text":". P Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43P. Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43 (1974), 273-307.","doi":""},{"text":"Comparison of dynamical degrees for semi-conjugate meromorphic maps. T.-C Dinh, V.-A Nguyen, Comment. Math. Helv. 864T.-C. Dinh and V.-A. Nguyen, Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011), no. 4, 817-840.","doi":""},{"text":"Groupes commutatifs d'automorphismes d'une variété kählérienne. T.-C Dinh, N Sibony, Duke Math. J. 1232T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne, Duke Math. J. 123 (2004), no. 2, 311-328.","doi":""},{"text":"Une borne supérieure pour l'entropie topologique d'une application rationnelle. T.-C Dinh, N Sibony, Annals of Math. 2T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Annals of Math. (2) 161 (2005), no. 3, 1637-1644.","doi":""},{"text":"Questions on self-maps of algebraic varieties. N Fakhruddin, J. Ramanujan Math. Soc. 182N. Fakhruddin, Questions on self-maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109-122.","doi":""},{"text":"Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs. B Fantechi, L Göttsche, L Illusie, S L Kleiman, N Nitsure, A Vistoli, American Mathematical Society123Providence, RIB. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs, 123. American Mathematical Society, Providence, RI.","doi":""},{"text":"W Fulton, Intersection theory. BerlinSpringer-VerlagSecond editionW. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998.","doi":""},{"text":"Representation theory: a first course. W Fulton, J Harris, Graduate Texts in Mathematics. 129Springer-VerlagReadings in MathematicsW. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.","doi":""},{"text":"Automorphisms of affine algebraic groups. G Hochschild, G D Mostow, J. Algebra. 13G. Hochschild and G. D. Mostow, Automorphisms of affine algebraic groups, J. Algebra 13 (1969), 535-543.","doi":""},{"text":"Cohomological and numerical dynamical degrees on abelian varieties. F Hu, Algebra Number Theory. 138F. Hu, Cohomological and numerical dynamical degrees on abelian varieties, Algebra Number Theory 13 (2019), no. 8, 1941-1958.","doi":""},{"text":"Logarithmic forms of algebraic varieties. S Iitaka, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 233S. Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 525-544.","doi":""},{"text":"On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. S Kawaguchi, J Silverman, J. Reine Angew. Math. 713S. Kawaguchi and J. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. 713 (2016), 21-48.","doi":""},{"text":"Birational geometry of algebraic varieties. J Kollár, S Mori, Cambridge Univ. PressCambridgeJ. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, Cam- bridge, 1998.","doi":""},{"text":". J Kollár, arXiv:1803.07596Mumford divisorsJ. Kollár, Mumford divisors, arXiv:1803.07596","doi":"arXiv:1803.07596"},{"text":"Abelian varieties. S Lang, Springer-VerlagNew York-BerlinS. Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983.","doi":""},{"text":"Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture. J Lesieutre, M Satriano, 10.1093/imrn/rnz067Int. Math. Res. Not. J. Lesieutre and M. Satriano, Canonical heights on hyper-Kähler varieties and the Kawaguchi- Silverman conjecture, Int. Math. Res. Not., https://doi.org/10.1093/imrn/rnz067","doi":"10.1093/imrn/rnz067"},{"text":"Pulling back cohomology classes and dynamical degrees of monomial maps. J Lin, Bull. Soc. Math. France. 1404J. Lin, Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France 140 (2012), no. 4, 533-549 (2013).","doi":""},{"text":"Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Y Matsuzawa, K Sano, Systems. 406Y. Matsuzawa and K. Sano, Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Systems 40 (2020), no. 6, 1655-1672.","doi":""},{"text":"Y Matsuzawa, S Meng, T Shibata, D.-Q Zhang, arXiv:2002.10976Non-density of points of small arithmetic degrees. Y. Matsuzawa, S. Meng, T. Shibata and D.-Q. Zhang, Non-density of points of small arith- metic degrees, arXiv:2002.10976","doi":"arXiv:2002.10976"},{"text":"Building blocks of amplified endomorphisms of normal projective varieties. S Meng, Math. Z. 2943S. Meng, Building blocks of amplified endomorphisms of normal projective varieties, Math. Z. 294 (2020), no. 3, 1727-1747.","doi":""},{"text":"Jordan property for non-linear algebraic groups and projective varieties. S Meng, D.-Q Zhang, Amer. J. Math. 1404S. Meng and D.-Q. Zhang, Jordan property for non-linear algebraic groups and projective varieties, Amer. J. Math. 140 (2018), no. 4, 1133-1145.","doi":""},{"text":"Building blocks of polarized endomorphisms of normal projective varieties. S Meng, D.-Q Zhang, Adv. Math. 325S. Meng and D.-Q. Zhang, Building blocks of polarized endomorphisms of normal projective varieties, Adv. Math. 325 (2018), 243-273.","doi":""},{"text":"S Meng, D.-Q Zhang, Kawaguchi-Silverman, arXiv:1908.01605Conjecture for surjective endomorphisms. S. Meng and D.-Q. Zhang, Kawaguchi-Silverman Conjecture for surjective endomorphisms, arXiv:1908.01605","doi":"arXiv:1908.01605"},{"text":"Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism. S Meng, D.-Q Zhang, Math. Res. Lett. 272S. Meng and D.-Q. Zhang, Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism, Math. Res. Lett. 27 (2020), no. 2, 523-550.","doi":""},{"text":"S Meng, G Zhong, arXiv:2005.03983Rigidity of rationally connected smooth projective varieties from dynamical viewpoints. S. Meng and G. Zhong, Rigidity of rationally connected smooth projective varieties from dy- namical viewpoints, arXiv:2005.03983","doi":"arXiv:2005.03983"},{"text":"On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms. N Nakayama, preprintN. Nakayama, On complex normal projective surfaces admitting non-isomorphic surjective en- domorphisms, preprint September 2008.","doi":""},{"text":"Polarized endomorphisms of complex normal varieties. N Nakayama, D.-Q Zhang, Math. Ann. 3464N. Nakayama and D.-Q. Zhang, Polarized endomorphisms of complex normal varieties, Math. Ann. 346 (2010), no. 4, 991-1018.","doi":""},{"text":"The canonical heights for Jordan blocks of small eigenvalues. K Sano, arXiv:1712.07533preperiodic points, and the arithmetic degrees. K. Sano, The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees, arXiv:1712.07533","doi":"arXiv:1712.07533"},{"text":"Linear algebraic groups. T A Springer, Modern Birkhaüser Classics. Birkhaüser Boston IncT. A. Springer, Linear algebraic groups, Modern Birkhaüser Classics, Boston, MA: Birkhaüser Boston Inc., 2009.","doi":""},{"text":"n-dimensional projective varieties with the action of an abelian group of rank n − 1. D.-Q Zhang, Trans. Amer. Math. Soc. 36812D.-Q. Zhang, n-dimensional projective varieties with the action of an abelian group of rank n − 1, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8849-8872.","doi":""},{"text":"Distributions in algebraic dynamics. S.-W Zhang, Department of Mathematics. 10Brown UniversitySurv. Differ. Geom.S.-W. Zhang, Distributions in algebraic dynamics, Surv. Differ. Geom., 10 (2006), 381-430. Department of Mathematics, box 1917, Brown University, Rhode Island 02912, USA","doi":""}],"string":"[\n {\n \"text\": \")/b, (1) follows. (2) follows from (1) and Lemma 2. H C := T, H C := t π( D 1 )/b, (1) follows. (2) follows from (1) and Lemma 2.2.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0. For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0\",\n \"doi\": \"\"\n },\n {\n \"text\": \"So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. Deb01So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. [Deb01,\",\n \"doi\": \"\"\n },\n {\n \"text\": \". * L F ≡ F * Π * H C = Π * G * H C ≡ Qπ * H C ≡ Ql F, which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7f * L f ≡ f * π * H C = π * g * H C ≡ qπ * H C ≡ qL f , which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X ). Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X )\",\n \"doi\": \"\"\n },\n {\n \"text\": \"By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\\\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4)⌋ + 1). By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\\\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"By taking normalization of X. we may assume that X is normal (cf. Lemmas 2.4 and 2.5)Proof of Theorem 1.7. By taking normalization of X, we may assume that X is normal (cf. Lemmas 2.4 and 2.5).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c\\\"(2) ⇒ (1).\\\" By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c\",\n \"doi\": \"\"\n },\n {\n \"text\": \"After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. We may also assume that D = π −1 (π(D)) for somed 1 (f s | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. \\\"(1) ⇒ (2).\\\" After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. We may also assume that D = π −1 (π(D)) for some\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant. D , after iteration) fibration π : ReferencesD ∈ P f . Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant (after iteration) fibration π : References\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Linear transformations with invariant cones. G Birkhoff, Amer. Math. Monthly. 74G. Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274- 276.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Families of disjoint divisors on varieties. F Bogomolov, A Pirutka, A Silberstein, Eur. J. Math. 24F. Bogomolov, A. Pirutka and A. Silberstein, Families of disjoint divisors on varieties, Eur. J. Math. 2 (2016), no. 4, 917-928.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Some basic results on actions of nonaffine algebraic groups. M Brion, Symmetry and spaces. Boston, MA278BirkhäuserM. Brion, Some basic results on actions of nonaffine algebraic groups, in: Symmetry and spaces, 1-20, Progr. Math. 278, Birkhäuser, Boston, MA, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. P Cascini, S Meng, D.-Q Zhang, Math. Ann. 3781-2P. Cascini, S. Meng and D.-Q. Zhang, Polarized endomorphisms of normal projective threefolds in arbitrary characteristic, Math. Ann. 378 (2020), no. 1-2, 637-665.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Degrees of iterates of rational maps on normal projective varieties. N.-B Dang, Proc. Lond. Math. Soc. 3N.-B. Dang, Degrees of iterates of rational maps on normal projective varieties, Proc. Lond. Math. Soc. (3) 121 (2020), no. 5, 1268-1310.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Higher-dimensional algebraic geometry. O Debarre, Springer-VerlagNew YorkUniversitextO. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43P. Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43 (1974), 273-307.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Comparison of dynamical degrees for semi-conjugate meromorphic maps. T.-C Dinh, V.-A Nguyen, Comment. Math. Helv. 864T.-C. Dinh and V.-A. Nguyen, Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011), no. 4, 817-840.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Groupes commutatifs d'automorphismes d'une variété kählérienne. T.-C Dinh, N Sibony, Duke Math. J. 1232T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne, Duke Math. J. 123 (2004), no. 2, 311-328.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Une borne supérieure pour l'entropie topologique d'une application rationnelle. T.-C Dinh, N Sibony, Annals of Math. 2T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Annals of Math. (2) 161 (2005), no. 3, 1637-1644.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Questions on self-maps of algebraic varieties. N Fakhruddin, J. Ramanujan Math. Soc. 182N. Fakhruddin, Questions on self-maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109-122.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs. B Fantechi, L Göttsche, L Illusie, S L Kleiman, N Nitsure, A Vistoli, American Mathematical Society123Providence, RIB. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs, 123. American Mathematical Society, Providence, RI.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W Fulton, Intersection theory. BerlinSpringer-VerlagSecond editionW. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Representation theory: a first course. W Fulton, J Harris, Graduate Texts in Mathematics. 129Springer-VerlagReadings in MathematicsW. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Automorphisms of affine algebraic groups. G Hochschild, G D Mostow, J. Algebra. 13G. Hochschild and G. D. Mostow, Automorphisms of affine algebraic groups, J. Algebra 13 (1969), 535-543.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cohomological and numerical dynamical degrees on abelian varieties. F Hu, Algebra Number Theory. 138F. Hu, Cohomological and numerical dynamical degrees on abelian varieties, Algebra Number Theory 13 (2019), no. 8, 1941-1958.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Logarithmic forms of algebraic varieties. S Iitaka, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 233S. Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 525-544.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. S Kawaguchi, J Silverman, J. Reine Angew. Math. 713S. Kawaguchi and J. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. 713 (2016), 21-48.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Birational geometry of algebraic varieties. J Kollár, S Mori, Cambridge Univ. PressCambridgeJ. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, Cam- bridge, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Kollár, arXiv:1803.07596Mumford divisorsJ. Kollár, Mumford divisors, arXiv:1803.07596\",\n \"doi\": \"arXiv:1803.07596\"\n },\n {\n \"text\": \"Abelian varieties. S Lang, Springer-VerlagNew York-BerlinS. Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture. J Lesieutre, M Satriano, 10.1093/imrn/rnz067Int. Math. Res. Not. J. Lesieutre and M. Satriano, Canonical heights on hyper-Kähler varieties and the Kawaguchi- Silverman conjecture, Int. Math. Res. Not., https://doi.org/10.1093/imrn/rnz067\",\n \"doi\": \"10.1093/imrn/rnz067\"\n },\n {\n \"text\": \"Pulling back cohomology classes and dynamical degrees of monomial maps. J Lin, Bull. Soc. Math. France. 1404J. Lin, Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France 140 (2012), no. 4, 533-549 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Y Matsuzawa, K Sano, Systems. 406Y. Matsuzawa and K. Sano, Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Systems 40 (2020), no. 6, 1655-1672.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Y Matsuzawa, S Meng, T Shibata, D.-Q Zhang, arXiv:2002.10976Non-density of points of small arithmetic degrees. Y. Matsuzawa, S. Meng, T. Shibata and D.-Q. Zhang, Non-density of points of small arith- metic degrees, arXiv:2002.10976\",\n \"doi\": \"arXiv:2002.10976\"\n },\n {\n \"text\": \"Building blocks of amplified endomorphisms of normal projective varieties. S Meng, Math. Z. 2943S. Meng, Building blocks of amplified endomorphisms of normal projective varieties, Math. Z. 294 (2020), no. 3, 1727-1747.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Jordan property for non-linear algebraic groups and projective varieties. S Meng, D.-Q Zhang, Amer. J. Math. 1404S. Meng and D.-Q. Zhang, Jordan property for non-linear algebraic groups and projective varieties, Amer. J. Math. 140 (2018), no. 4, 1133-1145.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Building blocks of polarized endomorphisms of normal projective varieties. S Meng, D.-Q Zhang, Adv. Math. 325S. Meng and D.-Q. Zhang, Building blocks of polarized endomorphisms of normal projective varieties, Adv. Math. 325 (2018), 243-273.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Meng, D.-Q Zhang, Kawaguchi-Silverman, arXiv:1908.01605Conjecture for surjective endomorphisms. S. Meng and D.-Q. Zhang, Kawaguchi-Silverman Conjecture for surjective endomorphisms, arXiv:1908.01605\",\n \"doi\": \"arXiv:1908.01605\"\n },\n {\n \"text\": \"Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism. S Meng, D.-Q Zhang, Math. Res. Lett. 272S. Meng and D.-Q. Zhang, Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism, Math. Res. Lett. 27 (2020), no. 2, 523-550.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Meng, G Zhong, arXiv:2005.03983Rigidity of rationally connected smooth projective varieties from dynamical viewpoints. S. Meng and G. Zhong, Rigidity of rationally connected smooth projective varieties from dy- namical viewpoints, arXiv:2005.03983\",\n \"doi\": \"arXiv:2005.03983\"\n },\n {\n \"text\": \"On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms. N Nakayama, preprintN. Nakayama, On complex normal projective surfaces admitting non-isomorphic surjective en- domorphisms, preprint September 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarized endomorphisms of complex normal varieties. N Nakayama, D.-Q Zhang, Math. Ann. 3464N. Nakayama and D.-Q. Zhang, Polarized endomorphisms of complex normal varieties, Math. Ann. 346 (2010), no. 4, 991-1018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The canonical heights for Jordan blocks of small eigenvalues. K Sano, arXiv:1712.07533preperiodic points, and the arithmetic degrees. K. Sano, The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees, arXiv:1712.07533\",\n \"doi\": \"arXiv:1712.07533\"\n },\n {\n \"text\": \"Linear algebraic groups. T A Springer, Modern Birkhaüser Classics. Birkhaüser Boston IncT. A. Springer, Linear algebraic groups, Modern Birkhaüser Classics, Boston, MA: Birkhaüser Boston Inc., 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"n-dimensional projective varieties with the action of an abelian group of rank n − 1. D.-Q Zhang, Trans. Amer. Math. Soc. 36812D.-Q. Zhang, n-dimensional projective varieties with the action of an abelian group of rank n − 1, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8849-8872.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Distributions in algebraic dynamics. S.-W Zhang, Department of Mathematics. 10Brown UniversitySurv. Differ. Geom.S.-W. Zhang, Distributions in algebraic dynamics, Surv. Differ. Geom., 10 (2006), 381-430. Department of Mathematics, box 1917, Brown University, Rhode Island 02912, USA\",\n \"doi\": \"\"\n },\n {\n \"text\": \")/b, (1) follows. (2) follows from (1) and Lemma 2. H C := T, H C := t π( D 1 )/b, (1) follows. (2) follows from (1) and Lemma 2.2.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0. For every curve B lying in a fibre of π, by (2) and the projection formula, we have π * (f * B) · H C = f * B · L f = B · f * L f = d 1 (f )B · L f = d 1 (f )π * B · H C = 0\",\n \"doi\": \"\"\n },\n {\n \"text\": \"So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. Deb01So every fibre of π will also be contracted by π • f . By the rigidity lemma (cf. [Deb01,\",\n \"doi\": \"\"\n },\n {\n \"text\": \". * L F ≡ F * Π * H C = Π * G * H C ≡ Qπ * H C ≡ Ql F, which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7f * L f ≡ f * π * H C = π * g * H C ≡ qπ * H C ≡ qL f , which implies q = d 1 (f ). Hence g is d 1 (f )-polarized. This proves (3). With all the preparations done, we are ready to prove Theorems 8.1, 1.6 and 1.7.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X ). Suppose ♯P f ≥ (ρ ns (X) + 1) · (⌊ ρ( X )\",\n \"doi\": \"\"\n },\n {\n \"text\": \"By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\\\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4)⌋ + 1). By Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := we have an injection ϕ : P f \\\\B → Fix(g), sending D to π(D). Let a := ♯B. By Lemma 6.2, we can conclude the theorem: ♯P f ≤ ♯ Fix(g) + a ≤ (1 + d 1 (f )) 2 + a. Proof of Theorem 1.6. Apply Theorem 8.1 to the normalization of X (cf. Lemma 2.4).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"By taking normalization of X. we may assume that X is normal (cf. Lemmas 2.4 and 2.5)Proof of Theorem 1.7. By taking normalization of X, we may assume that X is normal (cf. Lemmas 2.4 and 2.5).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c\\\"(2) ⇒ (1).\\\" By [Fak03, Theorem 5.1], Per(g) is Zariski dense in C. Since π is a fibration, it has irreducible and reduced general fibres. Therefore X c\",\n \"doi\": \"\"\n },\n {\n \"text\": \"After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. We may also assume that D = π −1 (π(D)) for somed 1 (f s | Xc ) = d 1 (f s | π ) < d 1 (f s ) by the assumption. So X c ∈ P f ∞ and ♯P f ∞ is infinite. \\\"(1) ⇒ (2).\\\" After a sufficient iteration of f , we may assume that ♯P f ≫ 1. Applying Proposition 8.5, there is an f -equivariant fibration π : X → C onto a smooth curve C such that g := f | C is d 1 (f )-polarized. We may also assume that D = π −1 (π(D)) for some\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant. D , after iteration) fibration π : ReferencesD ∈ P f . Then d 1 (f | π ) = d 1 (f | D ) < d 1 (f ), which completes the first part of our theorem. Now we prove the finiteness of P −1 f ∞ . Suppose the contrary that P −1 f ∞ is infinite. By the above equivalent condition, there is an f -equivariant (after iteration) fibration π : References\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Linear transformations with invariant cones. G Birkhoff, Amer. Math. Monthly. 74G. Birkhoff, Linear transformations with invariant cones, Amer. Math. Monthly 74 (1967), 274- 276.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Families of disjoint divisors on varieties. F Bogomolov, A Pirutka, A Silberstein, Eur. J. Math. 24F. Bogomolov, A. Pirutka and A. Silberstein, Families of disjoint divisors on varieties, Eur. J. Math. 2 (2016), no. 4, 917-928.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Some basic results on actions of nonaffine algebraic groups. M Brion, Symmetry and spaces. Boston, MA278BirkhäuserM. Brion, Some basic results on actions of nonaffine algebraic groups, in: Symmetry and spaces, 1-20, Progr. Math. 278, Birkhäuser, Boston, MA, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. P Cascini, S Meng, D.-Q Zhang, Math. Ann. 3781-2P. Cascini, S. Meng and D.-Q. Zhang, Polarized endomorphisms of normal projective threefolds in arbitrary characteristic, Math. Ann. 378 (2020), no. 1-2, 637-665.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Degrees of iterates of rational maps on normal projective varieties. N.-B Dang, Proc. Lond. Math. Soc. 3N.-B. Dang, Degrees of iterates of rational maps on normal projective varieties, Proc. Lond. Math. Soc. (3) 121 (2020), no. 5, 1268-1310.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Higher-dimensional algebraic geometry. O Debarre, Springer-VerlagNew YorkUniversitextO. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43P. Deligne, La conjecture de Weil. I, Inst. HautesÉtudes Sci. Publ. Math. 43 (1974), 273-307.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Comparison of dynamical degrees for semi-conjugate meromorphic maps. T.-C Dinh, V.-A Nguyen, Comment. Math. Helv. 864T.-C. Dinh and V.-A. Nguyen, Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011), no. 4, 817-840.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Groupes commutatifs d'automorphismes d'une variété kählérienne. T.-C Dinh, N Sibony, Duke Math. J. 1232T.-C. Dinh and N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne, Duke Math. J. 123 (2004), no. 2, 311-328.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Une borne supérieure pour l'entropie topologique d'une application rationnelle. T.-C Dinh, N Sibony, Annals of Math. 2T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Annals of Math. (2) 161 (2005), no. 3, 1637-1644.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Questions on self-maps of algebraic varieties. N Fakhruddin, J. Ramanujan Math. Soc. 182N. Fakhruddin, Questions on self-maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109-122.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs. B Fantechi, L Göttsche, L Illusie, S L Kleiman, N Nitsure, A Vistoli, American Mathematical Society123Providence, RIB. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs, 123. American Mathematical Society, Providence, RI.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W Fulton, Intersection theory. BerlinSpringer-VerlagSecond editionW. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Representation theory: a first course. W Fulton, J Harris, Graduate Texts in Mathematics. 129Springer-VerlagReadings in MathematicsW. Fulton and J. Harris, Representation theory: a first course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Automorphisms of affine algebraic groups. G Hochschild, G D Mostow, J. Algebra. 13G. Hochschild and G. D. Mostow, Automorphisms of affine algebraic groups, J. Algebra 13 (1969), 535-543.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cohomological and numerical dynamical degrees on abelian varieties. F Hu, Algebra Number Theory. 138F. Hu, Cohomological and numerical dynamical degrees on abelian varieties, Algebra Number Theory 13 (2019), no. 8, 1941-1958.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Logarithmic forms of algebraic varieties. S Iitaka, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 233S. Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 525-544.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties. S Kawaguchi, J Silverman, J. Reine Angew. Math. 713S. Kawaguchi and J. Silverman, On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math. 713 (2016), 21-48.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Birational geometry of algebraic varieties. J Kollár, S Mori, Cambridge Univ. PressCambridgeJ. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, Cam- bridge, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Kollár, arXiv:1803.07596Mumford divisorsJ. Kollár, Mumford divisors, arXiv:1803.07596\",\n \"doi\": \"arXiv:1803.07596\"\n },\n {\n \"text\": \"Abelian varieties. S Lang, Springer-VerlagNew York-BerlinS. Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Canonical heights on hyper-Kähler varieties and the Kawaguchi-Silverman conjecture. J Lesieutre, M Satriano, 10.1093/imrn/rnz067Int. Math. Res. Not. J. Lesieutre and M. Satriano, Canonical heights on hyper-Kähler varieties and the Kawaguchi- Silverman conjecture, Int. Math. Res. Not., https://doi.org/10.1093/imrn/rnz067\",\n \"doi\": \"10.1093/imrn/rnz067\"\n },\n {\n \"text\": \"Pulling back cohomology classes and dynamical degrees of monomial maps. J Lin, Bull. Soc. Math. France. 1404J. Lin, Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France 140 (2012), no. 4, 533-549 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Y Matsuzawa, K Sano, Systems. 406Y. Matsuzawa and K. Sano, Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties, Ergodic Theory Dynam. Systems 40 (2020), no. 6, 1655-1672.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Y Matsuzawa, S Meng, T Shibata, D.-Q Zhang, arXiv:2002.10976Non-density of points of small arithmetic degrees. Y. Matsuzawa, S. Meng, T. Shibata and D.-Q. Zhang, Non-density of points of small arith- metic degrees, arXiv:2002.10976\",\n \"doi\": \"arXiv:2002.10976\"\n },\n {\n \"text\": \"Building blocks of amplified endomorphisms of normal projective varieties. S Meng, Math. Z. 2943S. Meng, Building blocks of amplified endomorphisms of normal projective varieties, Math. Z. 294 (2020), no. 3, 1727-1747.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Jordan property for non-linear algebraic groups and projective varieties. S Meng, D.-Q Zhang, Amer. J. Math. 1404S. Meng and D.-Q. Zhang, Jordan property for non-linear algebraic groups and projective varieties, Amer. J. Math. 140 (2018), no. 4, 1133-1145.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Building blocks of polarized endomorphisms of normal projective varieties. S Meng, D.-Q Zhang, Adv. Math. 325S. Meng and D.-Q. Zhang, Building blocks of polarized endomorphisms of normal projective varieties, Adv. Math. 325 (2018), 243-273.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Meng, D.-Q Zhang, Kawaguchi-Silverman, arXiv:1908.01605Conjecture for surjective endomorphisms. S. Meng and D.-Q. Zhang, Kawaguchi-Silverman Conjecture for surjective endomorphisms, arXiv:1908.01605\",\n \"doi\": \"arXiv:1908.01605\"\n },\n {\n \"text\": \"Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism. S Meng, D.-Q Zhang, Math. Res. Lett. 272S. Meng and D.-Q. Zhang, Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism, Math. Res. Lett. 27 (2020), no. 2, 523-550.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Meng, G Zhong, arXiv:2005.03983Rigidity of rationally connected smooth projective varieties from dynamical viewpoints. S. Meng and G. Zhong, Rigidity of rationally connected smooth projective varieties from dy- namical viewpoints, arXiv:2005.03983\",\n \"doi\": \"arXiv:2005.03983\"\n },\n {\n \"text\": \"On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms. N Nakayama, preprintN. Nakayama, On complex normal projective surfaces admitting non-isomorphic surjective en- domorphisms, preprint September 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarized endomorphisms of complex normal varieties. N Nakayama, D.-Q Zhang, Math. Ann. 3464N. Nakayama and D.-Q. Zhang, Polarized endomorphisms of complex normal varieties, Math. Ann. 346 (2010), no. 4, 991-1018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The canonical heights for Jordan blocks of small eigenvalues. K Sano, arXiv:1712.07533preperiodic points, and the arithmetic degrees. K. Sano, The canonical heights for Jordan blocks of small eigenvalues, preperiodic points, and the arithmetic degrees, arXiv:1712.07533\",\n \"doi\": \"arXiv:1712.07533\"\n },\n {\n \"text\": \"Linear algebraic groups. T A Springer, Modern Birkhaüser Classics. Birkhaüser Boston IncT. A. Springer, Linear algebraic groups, Modern Birkhaüser Classics, Boston, MA: Birkhaüser Boston Inc., 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"n-dimensional projective varieties with the action of an abelian group of rank n − 1. D.-Q Zhang, Trans. Amer. Math. Soc. 36812D.-Q. Zhang, n-dimensional projective varieties with the action of an abelian group of rank n − 1, Trans. Amer. Math. Soc. 368 (2016), no. 12, 8849-8872.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Distributions in algebraic dynamics. S.-W Zhang, Department of Mathematics. 10Brown UniversitySurv. Differ. Geom.S.-W. Zhang, Distributions in algebraic dynamics, Surv. Differ. Geom., 10 (2006), 381-430. Department of Mathematics, box 1917, Brown University, Rhode Island 02912, USA\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":26,"cells":{"text":{"kind":"string","value":"\nRoom temperature infrared photodetectors with hybrid structure based on 2D materials\n\n\nTiande Liu \nSchool of Optical and Electronic Information\nHuazhong University of Science and Technology\n430074WuhanHubeiChina\n\nLei Tong \nSchool of Optical and Electronic Information\nHuazhong University of Science and Technology\n430074WuhanHubeiChina\n\nXinyu Huang \nSchool of Optical and Electronic Information\nHuazhong University of Science and Technology\n430074WuhanHubeiChina\n\nLei Ye \nSchool of Optical and Electronic Information\nHuazhong University of Science and Technology\n430074WuhanHubeiChina\n\nRoom temperature infrared photodetectors with hybrid structure based on 2D materials\n1 † These author contributed equally * Corresponding authors Lei Ye: leiye@hust.edu.cn 2two dimensional materialsheterostructureinfrared photodetectorsPACS: 6865-k, 7478Fk, 8560Gz\nTwo-dimensional (2D) materials, such as graphene, transition metal dichalcogenides (TMDs), black phosphorus (BP) and related derivatives, have attracted great attention due to their advantages of flexibility, strong light-matter interaction, broadband absorption and high carrier mobility, and have become a powerful contender for next-generation infrared photodetectors. However, since the thickness of twodimensional materials is on the order of nanometers, the absorption of two-dimensional materials is very weak, which limits the detection performance of 2D materials-based infrared photodetector. In order to solve this problem, scientific researchers have tried to use optimized device structures to combine with two-dimensional materials for improving the performance of infrared photodetector. In this review, we review the progress of room temperature infrared photodetectors with hybrid structure based on 2D materials in recent years, focusing mainly on 2D-nD (n = 0, 1, 2) heterostructures, the integration between 2D materials and on-chip or plasmonic structure. Finally, we summarize the current challenges and point out the future development direction.\n\nIntroduction\n\nInfrared (IR) photodetectors can convert infrared light signals into easily detectable electrical signals, which are widely used in imaging, communications, medical and night vision applications at present. [1,2] However, in order to obtain better detecting performance, the traditional infrared photodetectors based on the III-V or II-VI 3 semiconductors have to work at low temperatures to reduce dark currents. [3] Because of the necessity of large refrigeration equipment to realize low operating temperature, it leads to the greatly increase of the manufacturing cost and limitation of the application, making it difficult to achieve miniaturization, flexibility and portability.\n\nIn order to circumvent this disadvantage, new type semiconductors are proposed to use in the application of IR photodetectors, which can achieve low current at room temperature and high photocurrent. In recent years, two-dimensional materials with an atomic layer thickness, are verified to overcome the shortcomings induced by traditional semiconductors used in traditional IR detection due to their excellent electrical, optical and mechanical properties. [4][5][6][7][8][9][10][11][12][13] For example, graphene is the most studied two-dimensional materials with semi-metal, high mobility, broad spectrum response, flexibility, and tunable Fermi level properties, [14,15] suggesting that graphene can be fabricated into flexible infrared photodetectors with broad spectrum absorption, ultra-fast response. [16,17] In addition, transition metal dichalcogenides materials MX2 (M: transition metal atoms, X=S, Se, Te), which have the properties of strong light interactions and tunable bandgap varying with the number of layers, are also suitable as photosensitive materials for infrared photodetectors. Black phosphorus shows a direct bandgap property with the range of 0.3-1.3 eV, leading to its absorption spectrum covering the near-infrared, mid-infrared and far-infrared regions, even extending to the terahertz region. [18] Especially, wide spectral response range coupled with its unique anisotropic crystal structure, promises the potential of black phosphorus for infrared polarized photodetectors. [19] Although 2D materials have various excellent and unique 4 properties for the applications of photodetector, they cannot avoid such a problem that such thin thickness cannot absorb more light. For example, the thickness of monolayer graphene is 0.35 nm and absorbs only 2.3% of light. [20] Although the bandgap of monolayer TMDs is a direct bandgap, the whole absorption is also less than 10%. [21] In order to solve this problem, it is necessary to combine two-dimensional materials with other strong light absorbing materials, or to integrate with some structures that can enhance light absorption. For example, the band gap of quantum dots and nanoparticles with strong light absorption can be tuned by changing its size, [22,23] which is very suitable as a photosensitive layer for infrared photodetectors. Carbon nanotubes have excellent broadband absorption characteristics, [24] which is also very suitable for forming heterostructures with two-dimensional materials. Integrating two-dimensional materials with waveguides or cavities can enhance the photon absorption of 2D materials to improve detection performance. In addition, this integration also reveals the compatibility of two-dimensional materials with existing optoelectronic technologies. The surface plasmon resonance effect not only enhances light absorption but also selectively absorbs light by designing different shapes and sizes, [25] therefore the spectral response of these designed photodetectors can be extended to infrared or terahertz region that cannot be absorbed by intrinsic two-dimensional material, due to the surface plasmon resonance effect.\n\nIn this review, we review the development of infrared photodetectors with mixeddimensional structure between two-dimensional materials and other low-dimensional materials, focusing on quantum dots, single-wall carbon nanotubes, and 2D vertical or 5 lateral heterostructures composed of different two-dimensional materials. Moreover, we also discuss the integration of two-dimensional materials and on-chip structures, including waveguides and cavities. We also introduce the application of surface plasmon resonance in infrared photodetectors based on two-dimensional materials.\n\nFinally, we summarize the faced challenges currently in infrared photodetectors with hybrid structures based on two-dimensional materials and point out the future directions for further development.\n\n\nHeterostuctures\n\n\nQuantum dots on 2D materials\n\nIn order to meet the demand for high-speed response of infrared photodetectors, the materials used to fabricate the detectors is often required to have high carrier mobility to reduce the transit time of photogenerated carriers in conducting channel.\n\nThe carrier mobility of graphene is as high as 60,000 cm 2 V -1 s -1 , [26] which can fully meet this demand. However, the optical response of photodetectors based on intrinsic graphene is only a few tens of mA/W, [27,28] which is caused by weak light absorption due to its atomic-scale thickness. Compared with graphene, monolayer molybdenum disulfide (MoS2) has a direct band gap, so the absorption is stronger than that of graphene, but still not enough. Moreover, the band gap of molybdenum disulfide is 1.8 eV, and the corresponding absorption wavelength range is in the visible light range, leading to the limitation of directly fabrication for an infrared photodetector. [21,29] The problems of weak light absorption caused by the ultra-thin thickness and the 6 unsatisfactory photoresponse range due to broad band gap, have seriously hindered the development of infrared photodetectors based on two-dimensional materials. To solve this problem, scientific researchers have tried many methods. It is a good attempt to combine two-dimensional materials with quantum dots. [16,[30][31][32][33][34][35][36][37][38][39] Because quantum dots have stronger light absorption relative to 2D materials, and their absorption wavelength range can be tuned by changing its size, [22,40] and it can efficiently and selectively absorb photons and convert them into photogenerated electron-hole pairs. Therefore, the hybrid 2D-QDs structure is promising in the field of infrared detection technology.\n\n\nKonstantatos et al. demonstrated a hybrid graphene-PbS quantum dots\n\nphototransistor with ultrahigh gain of 10 8 and responsivity of 510 7 AW -1 . [32] The schematic diagram of a hybrid graphene-PbS quantum dots phototransistor is shown in Figure 1a. Monolayer or bilayer graphene was exfoliated with tape on a Si/SiO2 substrate, and then an 80-nm PbS quantum dots film was prepared by low-cost spin casting method. PbS QDs thin film acts as light-sensitive material due to the characteristics of strong and selective light absorption, while graphene acts as a conduction channel owing to its extremely high carrier mobility. The energy band diagram at the interface of graphene and PbS QDs is shown in Figure 1b. On account of the work function mismatch between graphene and PbS QDs, the band bending occurs at the interface. PbS QDs thin film absorbs light to create photogenerated electron-hole pairs, which are then separated by an built-in electric field at the interface that induced by band bending. The Dirac point (VD) will be drifted to higher backgate voltage due to illumination. For VBG<VD, holes are transferred from PbS QDs to 7 graphene channel, and electrons are still trapped in the PbS QDs. The resistance of graphene decreases and carrier transport in the channel is hole-dominated. For VBG>VD, holes are transferred to graphene and recombined with electrons induced by the backgate, so the resistance of graphene increases and carrier transport in the channel is electron-dominated.\n\nAfter the photogenerated holes are transferred to the graphene channel, they are driven by the source-drain voltage (VSD) to reach the drain. The linear dependence of the photocurrent on VSD is shown in Figure 1c. In order to achieve charge conservation in graphene, holes are continuously replenished from the source. Due to the long charge trapping time of the PbS QDs and the high mobility of the graphene, [26,41] holes in the channel can circulate multiple times during quantum dots trapped photogenerated electrons, resulting in ultrahigh photoconductive gain. The relationship between the responsivity and the applied back-gate voltage is shown in Figure 1d, which shows that the photoconductive gain of the photodetector can be adjusted by VBG. For VBG<VD, the measured responsivity can reach up to 510 7 A/W at VBG=-20 V; when VBG=4 V, the responsivity reduces to zero. This back-gate-tunable gain is significant so that the required gain can be adjusted based on the photosignal intensity that needs to be reproduced from Ref. [32] with permission from the Nature Publishing Group.\n\nAnother important two-dimensional material, transition metal dichalcogenides 9 (TMDs), has also been extensively studied in recent years. [43][44][45][46] However, due to the bandgap limitation and ultrathin thickness, the response spectral range and responsivity of TMDs-based infrared photodetectors are limited. Similarly, in order to enhance light absorption and broaden the response spectral range, a strategy for integrating TMDs materials with quantum dots has also been exploited. [31,35,36] Huo et al. demonstrated\n\nMoS2-HgTe QDs hybrid photodetectors beyond 2m. [31] The infrared detection capability of this detector is mainly caused by HgTe QDs thin film, which acts as a sensitizing layer, and the response spectrum range can be further extended to the midwave infrared and long-wave infrared ranges by adjusting the size of HgTe quantum dots. [47,48] A TiO2 buffer layer between the MoS2 transistor channel and the HgTe QDs sensitizing layer was used as a protective layer of the MoS2 channel and as an n-type electron acceptor medium to form an efficient p-n junction with the HgTe QDs at the interface facilitating the charge transfer to MoS2 channel. [31,49] Thus, the dark current can be adjusted by the gate voltage and the specific detectivity D* (~10 12 Jones) at a wavelength of 2 m can be obtained. The detector can operate at room temperature and exhibit sub-milliseconds response and high responsivity of ~10 6 A/W, which demonstrates that the hybrid 2D-QDs structure is an effective method to improve the performance of photodetector based on 2D materials.\n\nIn order to obtain high gain to make up for the weak light absorption of 2D materials, many infrared photodetectors based on hybrid 2D-QDs heterostructures are phototransistors that have an electronic passive sensitization layer composed of strong light absorbing material. [32,50,51] Thus, the external quantum efficiency (EQE), response 10 speed and linear dynamic range are limited by thin sensitizing layers thickness, long trapping times, and high density of the sensitizing centers trap-state, respectively. [32,41,52] In order to solve these problems, Nikitskiy et al. proposed a photodetector that integrates an electrically active colloidal QD photodiode with a graphene transistor. [34] The device is composed of a bottom graphene channel, a 300-nm-thick PbS QDs layer and a top ITO electrode. The photodiode operation and phototransistor operation of the hybrid 2D-QDs photodetector are shown in Figure 2a and b, respectively. The graphene-QDs photodiode responsivity and EQE are shown in Figure 2c. With increasing VTD, EQE increased from 10% at VTD=0 V to 75% at VTD=1.2 V (limited by reflection), indicating the importance of VTD for improving EQE. Figure 2d shows the EQE and responsivity in phototransistor operation, agreeing with the dependence of EQETD on VTD in photodiode operation, which demonstrates that the EQE of this integrated configuration is determined by the photodiode. Moreover, the responsivity is 510 6 V/W at VTD=1.2 V, indicating that the phototransistor still has a high gain of (a-f) are reproduced from Ref. [34] with permission from the Nature Publishing Group.\n\n\nNanowires combined with 2D materials\n\nOne-dimensional nanomaterials, such as single-wall carbon nanotubes (SWNTs), have unique and excellent optical and electrical properties due to 1D quantum confinement effect. [53] Because of the type of tube chiralities, metallic or semiconducting properties are exhibited, and the carrier mobility of semiconducting SWNTs can reach up to 10 5 cm 2 V -1 s -1 at room temperature. [54][55][56] SWNTs have broadband and tunable absorption, so it has great potential in the field of infrared detection. However, the photoresponsivity is limited by the small light-absorbing area, and the responsivity of single-tube photovoltaic devices is always <10 -3 A/W. [57] Paulus et al. reported that the junction between a metallic SWNT and graphene can perform effective charge transfer. [58] This is because that the SWNT is able to dope the graphene in a spatially controlled way without affecting its sp 2 lattice structure. [58] Liu et al.\n\ndemonstrated a high-performance photodetector based on a planar atomically thin SWNT-graphene hybrid film across visible to near-infrared range (400-1,550 nm). [50] Schematic of the photodetector and atomic force microscope (AFM) characterization of SWNT-graphene hybrid film are shown in Figure 3a. SWNTs have a diameter in the range of 1.0-1.6 nm, and the S11 and S22 bands are found to be located at ~1,800 and ~1,000 nm with higher SWNTs loadings. The photogenerated electron-hole pairs are generated at the SWNT/ graphene interface under optical illumination, and then 13 separated by the built-in electric field. Electrons are transferred from SWNTs to graphene, while holes are trapped in the SWNTs, forming a photogating effect. Figure 3b shows that the source-drain current (ISD) as a function of the applied back-gate voltage (VG). The Dirac point shifts toward the lower VG as the light increases,\n\nindicating that the graphene resistance can be effectively modulated by the photogating effect. Unlike other photodetectors that use the photogating effect to increase the responsivity, which generally has slow response speed, the photodetector based on hybrid SWNTs-graphene exhibits a fast response time of ~100 s ( Figure 3c). This is mainly due to the high mobility of graphene and SWNTs and fast charge transfer at the SWNTs-graphene interface, and the field-effect mobility of electrons and holes are measured as 3,920 and 3,663 cm 2 V -1 s -1 , respectively. The transit time of the hybrid SWNTs-graphene photodetector is estimated to be ~ 10 -9 s, so the photoconductive gain is ~ 10 5 and the gain-bandwidth product is ~10 9 Hz. Moreover, since SWNTs act as a sensitizing layer, the detector also exhibits the characteristics of a broadband photoresponse. The relationship between the responsivity and the incident optical power at different illumination wavelengths in the range of visible to near-infrared range (405, 532, 650, 980 and 1,550 nm) is shown in Figure 3d, which shows a high responsivity of >100 A/W. Further, Liu et al. fabricated a photodetector based on graphene/carbon nanotube hybrid films on a PET substrate and measured the performance under bending conditions. [59] It was found that this device has good robustness against repetitive bending, which fully demonstrated the huge application potential of this heterostructure in the field of flexible electronics. 14 Jariwala et al. demonstrated a gate-tunable carbon nanotube-MoS2 heterjunction pn diode. [60] Since both monolayer MoS2 and semiconducting SWNTs have direct bandgaps and tightly bound excitation states in absorption spectra, [61] the photodiode based on hybrid MoS2-SWNTs also exhibits excellent detection performance. The external quantum efficiency (EQE) and the response time are ~25% and <15 s, respectively. Figure 3e shows that photocurrent spectral response can be tuned by changing the gate voltage. The photocurrent in the near-infrared region continuously decreases as the gate voltage decreases. Furthermore, the performance of devices based on 1D-2D heterostructures can be further optimized by strain modulation. For example, the responsivity of the photodetector based on the graphene-ZnO nanowire heterostructure is improved by 26% under 0.44% tensile strain on the ZnO nanowire. [62] The above results show that it is significant to carry out the research of the hybrid 1D-2D heterostructure in the field of infrared detection.\n\n15 VD=-10 V. (a-d) are reproduced from Ref. [50] with permission from the Nature 16 Publishing Group. (e) are reproduced from Ref. [60] with permission from the National Academy of Sciences.\n\n\n2D vertical and lateral heterostructures 2D vertical heterostructures\n\nTwo-dimensional vertical heterostructure is fabricated by stacking different twodimensional materials in the vertical direction. This heterostructure is generally prepared by targeted transfer method or CVD growth technique. [63][64][65][66][67][68] 2D vertical heterostructures have many advantages, such as atomically sharp interface, no atomic diffusion between layers, and no strict lattice matching problems due to weak van der Waals interaction between 2D materials. Moreover, the strong interlayer coupling and ultrafast transfer of charge carriers make it have great potential in the field of optoelectronics and electronic devices. [68][69][70][71][72][73] In order to achieve infrared detecting, photodetectors must have a broad spectral response. However, the development of TMDs for infrared detection is hindered due to the lack of 2D materials with narrow bandgap. Monolayer -MoTe2 has a band gap of 1.1 eV and can absorb light in the near-infrared range. [74,75] The response spectral of the photodetectors based on the -MoTe2/MoS2 dichalcogenide heterojunction covers in the range from the visible to near-infrared (800 nm). [76] However, this is not enough for mid-infrared and far-infrared detection, and the response range of the photodetector should be further extended. Graphene can absorb light indiscriminately due to the gapless band structure and monolayer black phosphorene shows a direct optical band gap of 1.3 eV, [77,78] so they are effective absorption material in the infrared region.\n\n\nLong et al. demonstrated a broadband photodetector based on MoS2-graphene-\n\nWSe2 heterostructure covering visible to short-wavelength infrared range at room temperature. [79] The schematic diagram and optical image of the photodetector based on a MoS2-graphene-WSe2 heterostructure are shown in Figure 4a. The p-doped WSe2 is induced by large work-function metal Pd, whereas MoS2 flakes remain n-type due to\n\nFermi level pinning, [80] and thus forming a p-n junction with a strong internal electric field of ~210 8 V/m. Photogenerated electron-hole pairs can be effectively separated in the presence of the internal electric field. Therefore, the dark current of the device is effectively suppressed and the specific detectivity increases accordingly. Figure 4b shows the responsivity R and specific detectivity D* of the photodetector for wavelengths ranging from 400 to 2,400 nm with an interval of 100 nm. R and D* are as high as 10 4 A/W and 10 15 Jones in the visible region, while they decrease sharply to 100 mA/W and 10 9 Jones in the infrared wavelength of 2400 nm, respectively. Figure 4c explains the reason for the sharp changes in responsivity and specific detectivity.\n\nThe band gap of MoS2 (Eg1) and WSe2 (Eg2) are 1.88 eV and 1.65 eV, respectively, and the corresponding wavelengths are 660 nm and 750 nm. [21,29,81] When the incident photon energy h exceeds the bandgap of the material, the material can absorb photons to generate electron-hole pairs. Therefore, for h >Eg1>Eg2, MoS2, graphene and WSe2 all can generate photoelectron electron-hole pairs, resulting in a higher responsivity.\n\nFor h <Eg1<Eg2, only graphene can generate photoelectron electron-hole pairs, and thus the photocurrent is drastically reduced. Qiao et al. reported a broadband 18 photodetectors based on graphene-Bi2Te3 heterostructure (Figure 4d). [82] Bi2Te3 has only 2.7% lattice mismatch with graphene, so the graphene-Bi2Te3 heterostructure can be directly prepared on graphene by Van der Waals epitaxial growth method. [83] Figure 4e plots photocurrent of devices based on pure graphene and graphene-Bi2Te3 heterostructure as a function of source-drain voltage without applied gate bias, which shows that the latter is ten times higher than the former. Both graphene and Bi2Te3 absorb light to generate photoelectron-hole pairs, which are subsequently separated by a built-in electric field. Then, photogenerated electrons in graphene are transferred into Bi2Te3, while the photogenerated holes in Bi2Te3 are transferred into graphene, which suppresses the recombination of carriers and generates a larger photocurrent. Figure 4f shows are reproduced from Ref. [79] with permission from the American Chemical Society. 20 (d-f) are reproduced from Ref. [82] with permission from the American Chemical Society.\n\nSo far, highly polarization sensitive infrared photodetector based on twodimensional materials working at room temperature are still lacking. Ye et al. first demonstrated a broadband photodetector based on a vertical photogate heterostructure of BP-on-WSe2 with highly polarization sensitive for infrared detection (Figure 5a). [85] WSe2 acts as a conductive channel, while BP with a small direct bandgap and highly anisotropic crystal structure acts as a photogate. [18,19,77] Figure 5b shows the photoconductive gain G and detectivity D* of the photodetector for wavelengths ranging from 400 to 1,600 nm. The detectivity in the visible and infrared regions are 10 14 and 10 10 Jones, and the gain in the visible 21 and infrared regions are 10 6 and 10 2 , respectively, demonstrating excellent broadband detection performance. The optical image of the BP-on-WSe2 photodetector and the relationship between the photocurrent image of the photodetector at 1,550 nm and the polarization angle are shown in Figure 5c. The advantage of this device structure is that it can eliminate the disturbance of two-fold polarization-dependent photocurrent resulted from the geometric edge effect at the metal-BP edge and fully collect the photogenerated free carriers isotropically for highly polarization-sensitive infrared detection. [85] As can be seen from the photocurrent image, the polarization photoresponse originates from the overlap of BP and WSe2, and it changes obviously with the polarization angle changes. The polar plot in Figure 5d shows . [85] The ratio of the maximum responsivity to the minimum responsivity is about 6, showing high polarization sensitive infrared detection performance.\n\nSimilarly, the mid-wave infrared photodetector based on BP and MoS2 heterostructure designed by Bullock et al. also utilized the anisotropy of BP to achieve polarization detection, which is bias-selectable and operates without the need for external optics. [86] The device exhibits high EQE (35%) and D* (1.110 10 cm Hz 1/2 W -1 ) at room temperature.\n\nThe above results show that the 2D vertical heterostructure can effectively extend 22 the response range of the photodetector to achieve infrared detection. In addition, CVD epitaxial growth technology provides a guarantee for the scaled preparation of vertical heterostructures. By combining 2D materials with different unique and excellent physical properties, a multifunctional infrared detector with excellent detection performance can be prepared. heterostructures generally include chemical doping, [87,88] electrostatic doping, [89][90][91] and one-pot or two-step chemical vapor deposition (CVD) growth methods. [64,65,[92][93][94][95][96] Recently, Yu et al. demonstrated a lateral black phosphorene P-N junction formed via chemical doping for high performance near-infrared photodetector. [88] The n-type doping of few-layer black phosphorene is achieved by using benzyl viologen (BV) as an electron dopant, which can obtain a high electron concentration while still maintain the high carrier mobility of BP as the processes do not induce defects in the crystal lattices, and Al2O3 is used to selectively protect part of the area of a p-type BP at the 24 same time. [88] The electron doping concentration of BP can be adjusted by doping time, and the built-in electric field at the P-N junction can effectively separate photogenerated electron-hole pairs and reduce recombination efficiency. Photocurrent tests show that the off-current under 1.47 m illumination is 4 orders of magnitude higher than in dark condition, demonstrating that the photocurrent dominates over thermionic and tunneling currents in the photodiode operation. [88] The photodetector exhibits high responsivity of ~180 mA/W, fast response of tens of milliseconds and high detectivity in the range of 10 11 -10 13 Jones under infrared light illumination (=1.47 m).\n\nHowever, the current methods for preparing lateral heterojunctions have their own limitations. Chemical doping has the characteristic that it is difficult to find suitable dopants. The density of the carrier modulated by the electrostatic gating is relatively low, so the modulation effect is limited. [19,97] CVD technology is an effective method for large-scale preparation of lateral heterostructures, but the main focus is on epitaxial growth of TMDs with large band gap at present, and thus CVD technology has limitation in the field of infrared detection. The infrared light detection technology based on 2D lateral heterostructure needs further research and development.\n\n\nOn-chip structure\n\n\n2D materials on waveguide\n\n25 Two-dimensional materials show strong interaction with light, but the spectral response range and responsivity of the photodetector based on 2D materials are limited due to the reason of the atomically ultra-thin thickness and the limitation of the band gap. In order to obtain better infrared detection performance to achieve practical applications, it is necessary to enhance the infrared light absorption of two-dimensional materials. Integrating two-dimensional materials with waveguides is a good strategy to enhance light absorption. [91,[98][99][100][101][102][103][104] The propagation of light in the waveguide is accompanied by the evanescent field, which can be absorbed by the two-dimensional materials to generate photogenerated electron-hole pairs. [105][106][107][108] Different with the incident light perpendicularly incident on the two-dimensional materials, the in-plane absorption of two-dimensional materials makes its nanoscale thickness to be no longer a major limiting factor for light absorption. The length of the two-dimensional material in contact with the waveguide is on the order of micrometers, so the length of the interaction with the light is also on the order of micrometers and therefore can effectively enhance light absorption.\n\nRecently, a series of similar but independent waveguide integrated graphene photodetectors have been reported, [98,101,102] The absorption coefficient of metal electrode M increases with increasing electrode 26 width W. For W>100 nm (W>160 nm), the light absorption of the metal electrode will exceed the absorption of monolayer (bilayer) graphene. However, the contact resistance between metal and graphene also increases with decreasing width of electrode, so the metal electrode absorption coefficient and contact resistance should be considered comprehensively to obtain the best device performance. Figure 6b shows (c-d) are reproduced from Ref. [98] with permission from the Nature Publishing Group.\n\n(e-f) are reproduced from Ref. [102] with permission from the Nature Publishing Group.\n\n\nSimilarly, Youngblood et al. demonstrated a waveguide-integrated few-layer black\n\nphosphorus photodetector with high responsivity and low dark current. [103] Figure 7a shows the structure of the photodetector, having a graphene top-gate, a BP channel and the waveguide patterned on the silicon-on-insulator substrate. Since the incident light propagates along the waveguide, the optical interaction length is equal to the width of the black phosphorus covering on the waveguide (6.5 m). The device can absorb 78.7%\n\nof the optical power in the waveguide, of which only 17.5% is absorbed by the black phosphorus and the rest is absorbed by the graphene top-gate. [103] Figure 7c shows that the photocurrent versus the gate voltage VG and the bias voltage VDS. VG can electrostatically tune the hole doping concentration of black phosphorus and the BP is nearly intrinsic at VG=-8 V. When black phosphorus is in a low doping state (-10 V < VG < -1 V), the photocurrent is larger and its sign is positive relative to VDS, which suggests photocurrent generation mechanism is photovoltaic effect at this time. [109][110][111] 30\n\nFor VG > 0 V, BP is tuned to be more heavily n-doped, the photocurrent is getting smaller and its sign changes to negative relative to VDS, and the photocurrent generation mechanism is bolometric effect at this time. [109,112] The frequency response of the photocurrent shown in Figure 7b can also demonstrate the photocurrent generation mechanism at different gate voltages. The roll-off frequency is measured to be 2. light source and photodetector based on a p-n junction of bilayer MoTe2. [91] Figure 8a shows the device schematic, and Figure 8b shows LED mode and photodetector mode of device, respectively. The MoTe2 p-n junction is composed of p-type and n-type 32 doping that are electrostatically induced by left and right graphite top gates above the hBN dielectric layer. The bilayer MoTe2 not only has stronger excitation and longer emission wavelength than the monolayer MoTe2, but also has excitation caused by direct bandgap optical transitions similar to that of the monolayer. [74,114,115] There is a grating coupler for excitation and collection at the far end of the waveguide. Excitation Figure 8d shows that the PhC transmission peak is at 1,160 nm, but this peak disappears after MoTe2 transfer, which is mainly due to the interference of MoTe2 absorption. The normalized EL emission spectrum collected at the grating coupler represented by the blue line in Figure 8d, showing a narrow peak centred at 1,160 nm on top of a broader peak centred at 1,175 nm. [91] The former coincides with the PhC transmission peak, and the latter matches the free-space emission of the bilayer MoTe2.\n\nThe reason that the PhC transmission peak can be observed at the blue line is that the excited states of MoTe2 have been filled by the electrically excited carriers in the LED mode, so the absorption of MoTe2 will not cause interference. When the device is 33 operating in photodetector mode, the responsivity and external quantum efficiency of the photodetector at the wavelength of 1,160 nm are 4.8 mA/W and 0.5%, respectively. Figure 8e shows the temporal response in photodetector mode. The photocurrent response bandwidth is 200 MHz (limited by experimental set-up), which is much higher than other MoTe2 photodetectors based on photogating and photoconductive effect, [116,117] indicating the fast response of waveguide-integrated photodetector with bilayer MoTe2 p-n junction. As can be seen from Figure reproduced from Ref. [91] with permission from the Nature Publishing Group.\n\n\n2D materials on cavity\n\nThe integration of two-dimensional materials and cavities, including photonic crystal cavities, [118,119] optical microcavities [120,121] and optical ring resonators, [122] is also an effective method for enhancing the light absorption of 2D materials. The incident light at the resonant frequency of the cavity is absorbed many times by the atomically thin two-dimensional material in the cavity, thereby enhancing absorption. Schottky photodetector operating at communication band (1,550 nm). [124] Figure 9c shows the resonant cavity enhanced graphene/Si Schottky photodetector and the resonant structure consists of a λ/2 Si slab layer confined between SLG/Si top and Au bottom mirrors. [124] The response spectrum of the device with and without Au bottom mirror is shown in Figure 9d. A 3-fold external responsivity Rext enhancement as compared with no Au bottom mirror. The above result shows the integration of cavity with two-dimensional materials can achieve stronger and selective light absorption of 2D materials, so this method is possible to be applied to specific wavelength detection. reproduced from Ref. [123] with permission from the American Chemical Society. (cd) are reproduced from Ref. [124] with permission from the American Chemical Society. 38 \n\n\nPlasmonic structure\n\nIn order to further enhance the photoresponse of the photodetector and obtain a tunable response spectrum, plasmonic nanostructures can also be applied to the infrared photodetector based on two-dimensional materials. [125][126][127][128][129][130][131][132][133] response spectrum in the range of visible to near-infrared and explained the reasons for photocurrent enhancement. [131] The schematic of photodetector and optical image are shown in Figure 10a and b. The Fano resonance plasma clusters (dimers, heptamers, and nonamers) sandwiched between two layers of graphene are respectively located in the 1, 3 and 5 regions, while the 2 and 4 regions have no plasmonic structure ( Figure 10b). The Raman image in Figure 10b shows that the plasmonic antenna clusters of heptamers and nonamers significantly enhance Raman intensity due to resonance under 785 nm laser illumination. The photocurrent response in different regions in Figure 10c shows that the metal plasmonic antenna clusters can effectively enhance the photocurrent. As heptamers can provide larger near-field enhancement and higher 39 thermal electron yield, the enhancement effect of heptamers is significantly better than that of the dimer. In addition, Figure 10d shows that the photocurrent enhancement of the dimer plasmonic clusters is related to the polarization of the incident light,\n\nindicating that this structure has potential in the field of infrared polarization detection. Figure 10g shows the experimentally measured photocurrent as a function of different resonant wavelength (i.e. different heptamer diameters). It can be clearly seen that the photocurrent consists of plasmon-induced hot electrons and photogenerated carriers of plasmon-enhanced direct carrier excitation. The mechanism of photocurrent enhancement by plasmonic nanostructures is as follows. When the incident light resonates with the transmission window of the plasmonic clusters, a near-field enhancement effect occurs. [134] The light absorption of graphene is effectively enhanced and more electron-hole pairs are generated by direct excitation (DE). Moreover, hot 40 electrons (HE) generated in metal nanostructures can transfer into the conduction band of graphene over the Schottky barrier between metal and graphene. [135] Both plasmoninduced hot electrons and directly excited carriers can be driven by the source-drain bias into the circuit to from photocurrent. Because the contribution of absorption relative to scattering decreases as the size of the nanoparticles increases, the contribution of HE to photocurrent tends to saturate, while the contribution of DE plays a greater role with increasing the diameter of the heptamer. [131] Since hot electrons are generated by plasmon decay, the response spectra of photodetectors based on two-dimensional materials and metal plasmonic nanostructures are not limited by the bandgap of two-dimensional materials. [129,130] For example, bilayer MoS2 has an indirect bandgap of 1.65 eV, corresponding to a wavelength of 750 nm. [136] The hot electron-based bilayer MoS2 photodetector has a high responsivity of 5.2 A/W at 1,070 nm ( Figure 10h). [129] MoS2 field-effect phototransistors were also found to have negative infrared photoresponse, which is caused by bolometric effect, and exhibited responsivity of 2.3 A/W at 980 nm. [137] The above results show that plasma-induced hot electrons can achieve effective light detection with below bandgap photon illumination.\n\nNi et al. reported a photodetector based on plasmonic Si QD/graphene heterostructure that enables effective detection from UV to mid-infrared. [138] B-doped\n\nSi QDs not only enhance the light absorption of graphene in the infrared region due to surface plasmon resonance but also form photogating effect, which greatly improves the performance of the detector. Thus, the device exhibits ultrahigh gain of ~10 12 and 41 responsivity of ~10 9 A/W. is reproduced from Ref. [129] with permission from the American Chemical Society. Infrared photodetectors with hybrid structure based on two-dimensional materials still face many challenges and need us to explore further.\n\n\nFigure 10\n\n\n1) Low-cost, large-scale preparation\n\nMany of the currently reported hybrid structures based on two-dimensional materials are prepared just only through mechanical exfoliation and targeted transfer techniques, leading to poor reproducibility and relatively low yield of photodetectors.\n\nThis method can only meet the needs of scientific research and cannot be used for commercial applications. Although the 2D vertical or lateral heterostructure can be prepared by the CVD method and 2D-QDs structure can be prepared by the solutionprocessed method, which shows the potential of large-scale production, but they still have many limitations and need further improve.\n\n\n2) Device performance optimization\n\nThe reported overall performance parameters of infrared photodetectors with hybrid structure based on two-dimensional materials are excellent, but they are still far from the theoretical value. The excellent optical and electrical properties of twodimensional materials have not been fully exploited. In order to obtain better detection 49 performance, it is necessary to further study the interfacial charge transfer and photocurrent generation mechanism in hybrid structure based on two-dimensional materials for designing or optimizing various hybrid structures.\n\n\n3) Research on novel and suitable two-dimensional materials\n\nThere are various currently limitations for two-dimensional materials used in the field of infrared detection. For example, the band gap of graphene and many TMDs (such as MoS2, WSe2) are not conducive to the absorption of infrared light. The instability of MoTe2 and BP with an appropriately bandgap in the air also limits their use. In order to obtain better infrared detection performance, novel two-dimensional materials with suitable band gap and good stability have to be developed.\n\n\n4) Design of novel heterostructure with 2D materials\n\nMany existing heterostructures based on two-dimensional materials simply combine two materials that appear to be suitable. However, this combination may not be optimal, which may leads to the inability to maximize the advantages of heterostructure. It is a worthwhile research direction to further understand the properties of various two-dimensional materials through theoretical research and computational simulation methods, and to design a more rational and ingenious novel heterostructure.\n\nIt has the potential to dramatically improve the performance of the photodetector and even discover new phenomena.\n\n\n5) Interface research of heterostructures\n\nTwo-dimensional materials are particularly sensitive to interface problem due to their large specific surface area and atomic thickness. On the one hand, interface 50 pollution and the introduction of defects have a huge impact on the physical properties of two-dimensional materials. Therefore, how to obtain a clean and defect-free interface in the preparation process of heterostructure based on two-dimensional materials has always been a research hotspot. On the other hand, the study of charge transfer and interaction at the interface of two materials also helps to clarify the working mechanism of the photodetector and to improve performance of photodetector after the formation of the heterostructure.\n\n\nDeclaration of conflicts of Interest\n\nThe authors declare no conflict of interest.\n\n\nZheng et al. demonstrated an ambipolar broadband photodetector based on N, S codecorated graphene-PbS quantum dot heterostructure. [42] N, S codecorated graphene greatly reduces the p-type doping of graphene in air. The gate voltage required to modulate the Dirac point and the carrier type is significantly reduced, thereby 8 achieving gate tuneable mutual transformation of positive and negative photoresponse.The device exhibits excellent detection performance with responsivity of 10 4 A/W and specific detectivity of 10 12 Jones at 1550 nm.\n\nFigure 1\n1(a) The schematic diagram of a hybrid graphene-PbS quantum dots phototransistor. (b) The energy band diagram at the graphene/PbS QDs interface. (c) The linear dependence of the photocurrent on the applied drain-source voltage for different optical powers (46, 336, 516 pW). VBG=0. Inset: The total current (I=Idark + Iphoto) as a function of the applied drain-source voltage for different optical powers. (d) The responsivity as a function of the applied back-gate voltage VBG. VSD=5 V. (a-d) are\n\n\nPhotogenerated electron-hole pairs are created at the QDs layer, and then drifted by the built-in electric field at the interface and by the applied bias voltage to form photocurrent directly in photodiode operation. However, after the electron-hole pairs are separated, electrons remain in the QDs layer and holes are driven by depletion region on the graphene/PbS QD interface and transferred to graphene in phototransistor operation. For VTD>0, the width of the depletion region expands, increasing the efficient charge collection area, and thus the external quantum efficiency also increases.\n\n\n10 5 compared with the photodiode operation. With increasing VTD, the temporal11 response accelerates and the linear dynamic range extends. The electrical 3 dB bandwidth of the detector reaches 1.5 kHz for the optimum VTD, and the linear dynamic range extends from 75 dB at VTD=0 V to 110 dB at VTD=1.2 V. The above results prove that the design and optimization of infrared photodetectors based on hybrid 2D-QDs heterostructures can effectively enhance the detection performance.\n\nFigure 2\n2(a-b) Schematic of photodiode operation and phototransistor operation of the hybrid graphene-PbS QDs photodetector, respectively. (c) Responsivity and external quantum efficiency EQE of photodetector in photodiode operation as a function of VTD. Inset: energy band diagrams of device with and without VTD bias. (d) Responsivity and EQE of photodetector in phototransistor operation as a function of VTD. (e) Normalized photoresponse versus light modulation frequency. The specified VTD values are shown in the legend. Inset: The linear dependence of extracted 3 dB bandwidth values on applied VTD. (f) Responsivity of the detector as a function of incident irradiance in the range of 10 -5 to 10 Wm -2 , which shows a significant extension 12 of the linear dynamic range with increasing VTD.\n\nFigure 3\n3(a) Schematic of the photodetector and AFM characterization of SWNTgraphene hybrid film (middle: scale bar, 200 nm; right: scale bar, 20 nm). (b) The ISD-VG transfer curve of device with increasing 650 nm illumination powers. VSD=0.5 V. Inset: the optical image of the SWNT-graphene device (Scale bar, 10m). (c) Temporal response of the SWNT-graphene hybrid photodetector under 650 nm illumination. (d) The dependence of responsivities of photodetector on the optical power at different illumination wavelengths from visible to near-infrared range (405, 532, 650, 980 and 1,550 nm). (e) Photocurrent spectral response can be tuned by varying the gate voltage.\n\n\nthe dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. Due to the gapless nature of graphene and a small bandgap (0.15-0.3 eV) of Bi-2Te3, [84] both graphene and Bi2Te3 are effective absorbent materials for infrared light, and thus the detector exhibits broadband detection characteristics. The effective detection range can be extended to the near infrared (980 nm) and c-communication band (1,550nm). The responsivity is 35 A/W, 10 A/W and 0.22 A/W in the visible (532 nm), infrared (980 nm) and c-communication band (1,550 nm) regions, respectively.\n\nFigure 4\n4(a) Top: optical image of the photodetector based on a MoS2−graphene−WSe2 heterostructure. Scale bar, 5 m. Bottom: Schematic diagram of the device. (b) Responsivity and specific detectivity of the photodetector for wavelengths ranging from 400 to 2,400 nm with an interval of 100 nm. (c) Top: Spectra from ultraviolet to infrared. The band gap of MoS2 (Eg1) and WSe2 (Eg2) as well as corresponding wavelengths are plotted on the graph, respectively. Bottom left: For h >Eg1>Eg2, MoS2, graphene and WSe2 all can generate photogenerated electron-hole pairs. Bottom right: For h <Eg1<Eg2, only graphene can generate photogenerated electron-hole pairs. (d) The schematic diagram of the device based on graphene-Bi2Te3 heterostructure. (e) The dependence of photocurrent of devices based on pure graphene and graphene-Bi2Te3 heterostructure on source-drain voltage VSD. VG=0 V. (f) Dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. (a-c)\n\n\nthe relationship between the photocurrent and the polarization angle of the incident light. The maximum photocurrent, corresponding responsivity of ~40 mA/W, is taken along the horizontal axis (defined as 0 o polarization), while the minimum photocurrent, corresponding responsivity of ~6.8 mA/W, is taken along the vertical axis (defined as 90 o polarization)\n\nFigure 5\n5(a) Schematic diagram of the photodetector with BP-on-WSe2 photogate structure. (b) The photoconductive gain G and detectivity D* of the photodetector for wavelengths ranging from 400 to 1,600 nm at 1 mW/cm 2 incident light power density. 23 Vds=0.5 V. (c) Top: the photocurrent image of the photodetector under 1,550 nm illumination with different light polarizations (red arrows). Bottom left: the optical image of the photodetector with BP-on-WSe2 photogate structure. (d) The photocurrent as a function of light polarization at 100 mW/cm 2 incident light power density. Vds=0.5 V. (a-d) are reproduced from Ref.[85] with permission from Elsevier. 2D lateral heterostructures The heterostructures mentioned above are basically vertical, and the built-in electric field at the interface promotes the separation of photogenerated electron-hole pairs and charge transfer, thereby reducing the recombination and accelerating the response speed. In addition, two-dimensional lateral heterostructures, a heterostructure composed of materials with different work functions or homojunction composed of the same material but different doping types in the lateral direction, form a built-in electric field at the interface and play a similar role. At present, the methods for forming lateral\n\n\nand the device structures are shown in Figures 6a, c and e, respectively. Figure 6a shows a scanning electron microscope (SEM)image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration. It should be noted that in addition to the length L of graphene, the width W of central electrode S in this device also affects the light absorption of the graphene.\n\n\nthe absorption of graphene as a function of graphene length L at different widths, indicating that as long as the appropriate W and L values are obtained, the light absorption of graphene can be greatly enhanced. For example, when W = 100 nm, more than 50% light absorption can be obtained with only 22 m length bilayer graphene. Figure 6d shows an SEM image of a waveguide-integrated graphene photodetector with high responsivity and photocurrent image at zero bias. A 53 m bilayer graphene was overlaid on the waveguide, and two electrodes with waveguide distances of 100 nm and 3.5 m, respectively, were used to collect the photogenerated carriers. The lateral metaldoped junction overlapping the waveguide is formed by doping of metal electrode 100 nm away from the waveguide, and effectively promotes the separation of photogenerated electron-hole pairs. The photocurrent image shows that the built-in electric field between metal-doped graphene and graphene appears at the metal and graphene interface and leads to two narrow photocurrent regions. Moreover, the region of maximum photocurrent coincides with the position of waveguide. Wavelength transmission measurements show that the transmission loss are 0.1 dB and 4.8 dB before and after the graphene transfer, respectively, indicating that the optical absorption of the waveguide-integrated graphene is greatly increased. The 27 photodetectors demonstrated an ultrafast response rates exceeding 20 GHz at zero bias and a high response rate of 0.108 A/W at 1 V bias. Thanks to the gapless graphene, the spectral response of photodetector ranged from 1,450 nm to 1,590 nm, which can effectively detect the c-communication band of the near-infrared region. The response spectrum of the waveguide-integrated graphene photodetector can be further extended to the mid-infrared region. Xu and coworkers demonstrated a graphene/silicon heterostructure waveguide photodetector with a responsivity of up to 0.13 A/W at room temperature in the mid-IR region (2.75 m).[102] In addition, this photodetector exhibits a low dark current due to the presence of a graphene/silicon heterojunction potential barrier, and the on/off photocurrent ratio is 30 at a bias of -1.5 V for 2.75 m illumination at room temperature(Figure 6f). This work further explains the photocurrent generation mechanism of waveguide-integrated graphene in the visible light region, near infrared region and mid-infrared region. For the visible light region, since silicon has an indirect bandgap of 1.1 eV, photoresponse is mainly due to electronhole pairs generated after the silicon absorbs light and graphene acts as a photogenerated free carriers conduction channel in this process. For the near-infrared region (1.55 m), the photon energy is more than twice that of the graphene Fermi energy level and the infrared light cannot be absorbed by silicon. The photoexcitation of graphene takes place in a direct transition way that electrons are excited from the valence band to the conduction band. For the mid-infrared region (2.75 m), the photon energy is less than twice the energy difference between the Fermi level and the Dirac point, thus preventing direct transitions of graphene.[102] The excitation of graphene28 follows the indirect transition processes, and electrons are excited from Fermi level in the valence band to the conduction band with a large momentum mismatch under midinfrared light illumination.\n\nFigure 6\n6(a) SEM image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration (false colour). Inset: An enlarged view of the inside of the black dashed line. (b) The absorption of graphene as a function of graphene length L at different widths W. Solid line: bilayer graphene. Dashed line: monolayer graphene. (c) 29 Schematic of the chip-integrated graphene photodetector. (d) The spatially resolved photocurrent image of the waveguide-integrated graphene photodetector at zero bias. The black solid line shows the relative potential distribution across the graphene channel along the white dashed line. Inset: corresponding SEM image. (e) Schematic of a graphene/silicon heterostructure waveguide photodetector. (f) Photocurrent as a function of applied bias voltage with 2.75 m laser illumination at different light power.(a-b) are reproduced from Ref.[101] with permission from the Nature Publishing Group.\n\n\nis in a low doping state (VG=-8 V), showing a fast response due to photovoltaic effect. But the roll-off frequency is measured to be 0.2 MHz when BP is in a high doping state (VG=8 V), showing a slower response due to bolometric effect.\n\nFigure 7d 31 Figure 7\n7d317shows the intrinsic responsivity R and internal quantum efficiency IQE as a function of bias for different thicknesses. The intrinsic responsivity and internal quantum efficiency of 11.5-nm-thick BP device with top-gate are 135 mA/W and 10% at Vbias=-0.4 V, respectively. For 100-nm-thick BP device without top-gate, a larger bias voltage can be applied because there is no need to consider the breakdown of the gate insulation layer. Therefore, this photodetector shows better detection performance with high intrinsic responsivity (657 mA/W) and high internal quantum efficiency (50%) at Vbias=-0.4 V. Unlike gapless graphene, BP has a finite direct band gap,[113] which make the dark current of waveguide-integrated BP photodetector extreme low (220 nA) atVG=-8 V and Vbias=-0.4 V. (a) Schematic of the waveguide-integrated few-layer BP photodetector. (b) The frequency response of the waveguide-integrated BP photodetector at low doping state (VG=-8 V) and high doping state (VG=8 V). (c) Two-dimensional contour plot of photocurrent versus the gate voltage VG and the bias voltage VDS. (d) The intrinsic responsivity R and internal quantum efficiency IQE as a function of applied bias for different thicknesses. tBP: the thickness of BP. (a-d) are reproduced from Ref.[103] with permission from the Nature Publishing Group. Recently, Bie et al. reported a silicon photonic-crystal (PhC) waveguide-integrated\n\n\nlight is coupled into the waveguide and transmitted along the waveguide to the grating coupler in LED mode, while incident light is coupled into the waveguide at the grating coupler and detected by the MoTe2 p-n junction in the photodetector mode (Figure 8b).\n\nFigure 8c\n8cshows an EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. Two extra emission spots are observed on the grating coupler on both sides, which coincides with the aforementioned LED mode (Figure 8b). The orange and green lines in Figure 8d represent the transmission spectra of the PhC waveguide before and after the bilayer MoTe2 transfer, respectively. The orange curve in\n\n\n8f, the response spectrum of photodetector in the range from 1040 nm to 1240 nm and the largest responsivity is at 1,160 nm (the location of the PhC transmission peak), which shows the dependence of the responsivity on the wavelength. This dependence indicates that the photocurrent originates mainly from the light transmitted by the PhC waveguide, and thus the light beyond this spectral response range cannot be detected due to the bandgap of the bilayer MoTe2 and the transmission cut-off of the waveguide. This work shows that all active optical components for point-to-point interconnects are possible with 2D TMD devices transferred onto otherwise passive photonic integrated circuits.[91] \n\nFigure 8\n8(a) Schematic of the silicon photonic-crystal (PhC) waveguide-integrated light-emitting diode and photodetector based on bilayer MoTe2 p-n junction. (b) The schematic diagram of the device operating in LED mode and photodetector mode. The arrow indicates the direction of light propagation in the PhC waveguide. (c) EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. (d) The orange and green lines represent the transmission spectra of the waveguide before and after the bilayer MoTe2 transfer, respectively. The normalized EL emission spectrum collected at the grating coupler represented by the blue line. (e) The temporal response of device in photodetector mode at zero bias. The 35 response bandwidth of the device is 200 MHz. (f) The response spectrum of photodetector in the range from 1,040 nm to 1,240 nm, which shows the dependence of the responsivity on the wavelength due to photonic-crystal waveguide. (a-f) are\n\nFurchi\net al. achieved a 26-fold absorption enhancement of graphene (>60%) by monolithically integrating graphene with a Fabry-Pérot microcavity. [123] The schematic of the microcavity-integrated graphene photodetector and the corresponding electric field distribution are shown in Figure 9a. Two Bragg mirrors are alternately composed of quarter-wavelength thick materials with different refractive index, and graphene layer is sandwiched between these mirrors. The presence of the barrier layer ensures the position of resonance occur at the graphene. The incident light are reflected back and forth multiple times between two Bragg mirrors and absorbed by the graphene, getting a higher absorption, which agrees with the electric field distribution. The spectral response of the device is shown in Figure 9b, and it can be clearly seen that a significant photocurrent enhancement at the cavity resonance (855 nm). Atomic thin monolayer graphene (only 0.335 nm) can absorb more than 60% of light, indicating 36 that the integration of 2D materials with cavity has great potential in enhancing the light absorption of 2D materials. However, the enhancement of light absorption by the cavity occurs only at the resonant wavelength of the cavity. In other words, the enhancement of the cavity is at the expense of the broadband photodetection. The strategy of integrating cavity and two-dimensional materials cannot be used for broadband detectors, and thus it can only be used to enhance light absorption at specific wavelengths. For example, Casalino et al. demonstrated a vertically illuminated, resonant cavity enhanced, graphene−Si\n\nFigure 9\n9(a) The schematic of the resonant cavity enhanced graphene photodetector and the corresponding electric field distribution. Red arrow: the position of graphene. (b) The dashed lines indicates the calculation result: reflection R (red), transmission T (green) and absorption A (blue), while the solid lines indicates the measurement result: reflection (red) and photocurrent (blue). A 26-fold absorption enhancement at the cavity resonance wavelength (855 nm). Inset: calculated theoretical results. (c) Schematic diagram of the resonant cavity enhanced graphene/Si Schottky photodetector. (d) The response spectrum of the device with and without Au mirror. A 3-fold absorption enhancement can be observed in the presence of Au mirror. The blue shift of the resonance peak is caused by the difference between different devices. (a-b) are\n\n\nMetal plasmonic nanostructures mainly enhance the response of the detector in two ways. First, the nearfield enhancement capability of the plasmonic nanostructures can focus the light field in a small area, effectively enhancing the light absorption of the two-dimensional material. Moreover, the plasmonic structure can generate plasma-induced hot electrons due to plasmon decay under light illumination, and then hot electrons are injected into the device across the Schottky barrier between the metal and the two-dimensional material to form photocurrent. Fang et al. demonstrated a graphene-antenna sandwich photodetector with\n\nFigure\n10e and f show the measured photocurrent and the corresponding band diagram for different applied gate voltages, respectively. The gate voltage VG can effectively adjust the Fermi level of the graphene channel, and then obtain different energy band bending between the Fermi level of graphene channel and the Ti electrode-doped graphene. The steeper band bending results in a correspondingly larger built-in electric field, which can obtain higher the separation efficiency of the photogenerated electronhole pairs and thus produces a greater photocurrent. The resonance wavelength of the plasmonic clusters can be controlled by varying the diameter of disks. For example, when the diameter of heptamer plasmonic clusters changes from 80 nm to 180 nm, the corresponding resonance wavelength changes from 650 nm to 950 nm.\n\n\n(a) The schematic of graphene-antenna sandwich photodetector. (b) 42 The optical image of device before (left) and after (right) transfer of top layer graphene. Inset: Raman mapping of the device under 785 nm laser illumination. (c) Photocurrents measured in the photodetector with dimer array and heptamer array and without pattern exhibit antisymmetric properties. The results show that the metal plasmonic antenna clusters can effectively enhance the photocurrent. The line scan position is shown by the white arrows in the inset. (d) The polarization dependent properties of the photocurrent measured in dimer and heptamer devices. (e) Photocurrents measured at different gate voltages in a heptamer device. (f) Energy band diagram of devices with different gate voltages in a heptamer device. The purple dashed line indicates the Dirac point of graphene. (g) Photocurrent as a function of different resonant wavelength. Black triangles: the measured photocurrent (I EXP (λ)). Blue triangles: the calculated DE photocurrent (I DE (λ)). Solid red triangles: the estimated HE photocurrent contribution E DE (λ) = I EXP (λ) −I DE (λ). Hollow red triangles: the calculated I HE (λ). (h) Schematic of hot electron-based bilayer MoS2 with near-infrared laser illumination. (a-g) are reproduced from Ref.[131] with permission from the American Chemical Society. (h)\n\n\nhybrid structure based on two-dimensional materials. Hybrid 2D-QDs heterostructures possess two prominent advantages of strong light absorption contributed by quantum dots and high carrier mobility contributed by two-dimensional materials, and thus exhibiting extremely high gain and responsivity. The detection performance of photodetectors based on hybrid graphene-SWNTs structure is superior to that of photodetectors based on only graphene or SWNTs, showing high responsivity and fast response, in which the bandgap of SWNTs can be adjusted by the diameter of SWNTs to extend the spectral response range to infrared. The 2D-2D van der Waals heterostructure is also suitable for light detection in the infrared region because of ultrafast charge transfer at the interface, and the combination of two or more twodimensional materials with different optical and electrical properties not only compensates for the lack of both (eg. the combination of 2D materials with different bandgap structures can achieve broadband detection), but also exploits the characteristics of 2D materials to prepare a photodetector with specific function (such as high polarization sensitivity of a BP-on-WSe2 photogate structure). The integration of two-dimensional material on waveguide makes the light-matter interaction no longer limited by the ultrathin thickness of two-dimensional material, greatly increasing the light absorption of photodetectors to obtain a higher responsivity. In addition, this integration also shows that two-dimensional materials are compatible with the existing mature optical waveguide technology, providing a pathway for the commercialized practical application of two-dimensional materials. The integration of two-dimensional material with the cavity enhances the light absorption of the material at the resonant48 wavelength of the cavity. This selective absorption method is very suitable for precise detection of specific wavelengths. Metal plasmonic nanostructures can also enhance light absorption and photocurrent through near-field enhancement effects and hot electrons. In addition, plasma-induced hot electrons can extend the spectral response range, which is useful for infrared detection based on some two-dimensional materials with wide bandgap.\n\n\nBoth WSe2 and BP absorb light efficiently and generate photogenerated electron-hole pairs under visible illumination, which are then separated by the built-in electric field. The photo-generated electrons in BP are transferred to WSe2 and the photo-generated holes in WSe2 are transferred to BP, which effectively reduces the recombination efficiency. An increasing number of photo-generated electrons in WSe2 are driven by source-drain bias to form photocurrent, thereby enhancing the responsivity. However, due to the band gap limitation of WSe2, only BP with a small band gap is an effective absorption material to generate photogenerated free carriers under infrared illumination. So the photoresponse in the infrared region will be smaller than in the visible region. The responsivity is ~10 3 A/W under visible illumination (637 nm), while the responsivity is ~0.5 A/W under infrared illumination (1,550 nm).\n\nTable 1 .\n1Performance of room temperature infrared photodetectors with hybrid structure based on 2D materials. (UV: Ultraviolet. NIR: Near infrared. MIR: Midinfrared. BP: Black phosphorus. Gr: Graphene. QD: Quantum dot. FET: Field effect transistor.)In this review, we have reviewed various infrared photodetectors operating at roomDevice type \nDevice \nSpect Responsi Detectivi Respon EQE/I \ngain \nRef. \n\n\nAcknowledgementsThe authors acknowledge funding by National Natural Science Foundation of China (No. 61704061)\n. P R Norton, J B Campbell, S Horn, D A Reago, International Symposium on Optical Science and Technology. 11Norton P R, Campbell J B, Horn S B and Reago D A International Symposium on Optical Science and Technology, p. 11\n\n. A Rogalski, Opto-Electronics Review. 20279Rogalski A 2012 Opto-Electronics Review 20 279\n\n. A Rogalski, J Antoszewski, L Faraone, Journal of Applied Physics. 10591101Rogalski A, Antoszewski J and Faraone L 2009 Journal of Applied Physics 105 091101\n\n. M Xu, T Liang, M Shi, H Chen, Chemical Reviews. 1133766Xu M, Liang T, Shi M and Chen H 2013 Chemical Reviews 113 3766\n\n. S Z Butler, S M Hollen, L Cao, Y Cui, J A Gupta, Gutié Rrez H R, T Heinz, Butler S Z, Hollen S M, Cao L, Cui Y, Gupta J A, Gutié rrez H R, Heinz T F,\n\n. S S Hong, J Huang, A F Ismach, E Johnston-Halperin, M Kuno, V V Plashnitsa, R D Robinson, R S Ruoff, S Salahuddin, J Shan, L Shi, M G Spencer, M Terrones 51, Windl W Goldberger, J E , ACS Nano. 72898Hong S S, Huang J, Ismach A F, Johnston-Halperin E, Kuno M, Plashnitsa V V, Robinson R D, Ruoff R S, Salahuddin S, Shan J, Shi L, Spencer M G, Terrones 51 M, Windl W and Goldberger J E 2013 ACS Nano 7 2898\n\n. H X Wu Pei, Zhang Jian, Sun Lian, Acta Physica Sinica. 66218102Wu Pei H X, Zhang Jian, Sun Lian-Feng 2017 Acta Physica Sinica 66 218102\n\n. C Tan Wee, L Huang, J Ng Rui, L Wang, N Hasan Dihan Md, J Duffin Thorin, S Kumar Karuppannan, Nijhuis Christian, A , Lee C Ang, K W , Tan Wee C, Huang L, Ng Rui J, Wang L, Hasan Dihan Md N, Duffin Thorin J, Kumar Karuppannan S, Nijhuis Christian A, Lee C and Ang K W 2017\n\n. Advanced Materials. 301705039Advanced Materials 30 1705039\n\n. J Lai, X Liu, J Ma, Q Wang, K Zhang, X Ren, Y Liu, Q Gu, X Zhuo, W Lu, Y Wu, Y Li, J Feng, S Zhou, J-H Chen, D Sun, Advanced Materials. 301707152Lai J, Liu X, Ma J, Wang Q, Zhang K, Ren X, Liu Y, Gu Q, Zhuo X, Lu W, Wu Y, Li Y, Feng J, Zhou S, Chen J-H and Sun D 2018 Advanced Materials 30 1707152\n\n. Xuefei Li, X X , Yanqing Wu, Chin. Phys. B. 2637307Xuefei Li X X, Yanqing Wu 2017 Chin. Phys. B 26 37307\n\n. Jianlu Wang, W , Chin. Phys. B. 2637106Jianlu Wang W H 2017 Chin. Phys. B 26 37106\n\n. Yuanfang Yu, F M , Jun He, Zhenhua Ni, Chin. Phys. B. 2636801Yuanfang Yu F M, Jun He, Zhenhua Ni 2017 Chin. Phys. B 26 36801\n\n. Zi-Wei Li, Y-H H , Yu Li, Chin. Phys. B. 2636802Zi-Wei Li Y-H H, Yu Li, Zhe-Yu Fang 2017 Chin. Phys. B 26 36802\n\n. Xiaomu Wang, X , Chin. Phys. B. 2634203Xiaomu Wang X G 2017 Chin. Phys. B 26 34203\n\n. K S Novoselov, V I Fal′ko, L Colombo, P R Gellert, M Schwab, K Kim, Nature. 490192Novoselov K S, Fal′ko V I, Colombo L, Gellert P R, Schwab M G and Kim K 2012 Nature 490 192\n\n. M J Allen, V Tung, R B Kaner, Chemical Reviews. 110132Allen M J, Tung V C and Kaner R B 2010 Chemical Reviews 110 132\n\n. Z Sun, Z Liu, J Li, Tai G A, S Lau, Yan F , Advanced Materials. 245878Sun Z, Liu Z, Li J, Tai G a, Lau S P and Yan F 2012 Advanced Materials 24 5878\n\n. B Y Zhang, T Liu, B Meng, X Li, G Liang, X Hu, Q J Wang, Nature Communications. 41811Zhang B Y, Liu T, Meng B, Li X, Liang G, Hu X and Wang Q J 2013 Nature Communications 4 1811\n\n. A Castellanos-Gomez, The Journal of Physical Chemistry Letters. 64280Castellanos-Gomez A 2015 The Journal of Physical Chemistry Letters 6 4280\n\n. F Xia, Wang H Jia, Y , Nature Communications. 54458Xia F, Wang H and Jia Y 2014 Nature Communications 5 4458\n\n. R R Nair, P Blake, A N Grigorenko, K S Novoselov, T J Booth, T Stauber, N Peres, M R, A K Geim, Science. 3201308Nair R R, Blake P, Grigorenko A N, Novoselov K S, Booth T J, Stauber T, Peres N M R and Geim A K 2008 Science 320 1308\n\n. K F Mak, C Lee, J Hone, Shan J Heinz, T F , Physical Review Letters. 105136805Mak K F, Lee C, Hone J, Shan J and Heinz T F 2010 Physical Review Letters 105 136805\n\n. I Moreels, Y Justo, De Geyter, B Haustraete, K Martins, J Hens, Z , ACS Nano. 5Moreels I, Justo Y, De Geyter B, Haustraete K, Martins J C and Hens Z 2011 ACS Nano 5 2004\n\n. X Yu, Y Li, X Hu, D Zhang, Y Tao, Z Liu, Y He, M A Haque, Z Liu, T Wu, Q J Wang, Nature Communications. 94299Yu X, Li Y, Hu X, Zhang D, Tao Y, Liu Z, He Y, Haque M A, Liu Z, Wu T and Wang Q J 2018 Nature Communications 9 4299\n\n. A Eatemadi, H Daraee, H Karimkhanloo, M Kouhi, N Zarghami, A Akbarzadeh, M Abasi, Y Hanifehpour, S W Joo, Nanoscale Research Letters. 9393Eatemadi A, Daraee H, Karimkhanloo H, Kouhi M, Zarghami N, Akbarzadeh A, Abasi M, Hanifehpour Y and Joo S W 2014 Nanoscale Research Letters 9 393\n\n. M L Brongersma, N Halas, P Nordlander, Nature Nanotechnology. 1025Brongersma M L, Halas N J and Nordlander P 2015 Nature Nanotechnology 10 25\n\n. C R Dean, A F Young, I Meric, C Lee, L Wang, S Sorgenfrei, K Watanabe, T Taniguchi, P Kim, K L Shepard, J Hone, Nature Nanotechnology. 5722Dean C R, Young A F, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard K L and Hone J 2010 Nature Nanotechnology 5 722\n\n. T Mueller, F Xia, P Avouris, Nature Photonics. 4297Mueller T, Xia F and Avouris P 2010 Nature Photonics 4 297\n\n. F Xia, T Mueller, Y-M Lin, A Valdes-Garcia, P Avouris, Nature Nanotechnology. 4839Xia F, Mueller T, Lin Y-m, Valdes-Garcia A and Avouris P 2009 Nature Nanotechnology 4 839\n\n. A Splendiani, L Sun, Y Zhang, T Li, J Kim, C-Y Chim, Galli G Wang, F , Nano Letters. 101271Splendiani A, Sun L, Zhang Y, Li T, Kim J, Chim C-Y, Galli G and Wang F 2010 Nano Letters 10 1271\n\n. Y Che, Y Zhang, X Cao, H Zhang, X Song, M Cao, Y Yu, H Dai, J Yang, G Zhang, J Yao, ACS Applied Materials & Interfaces. 932001Che Y, Zhang Y, Cao X, Zhang H, Song X, Cao M, Yu Y, Dai H, Yang J, Zhang G and Yao J 2017 ACS Applied Materials & Interfaces 9 32001\n\n. N Huo, S Gupta, G Konstantatos, Advanced Materials. 1606576Huo N, Gupta S and Konstantatos G 2017 Advanced Materials 1606576\n\n. G Konstantatos, M Badioli, L Gaudreau, J Osmond, M Bernechea, F P G D Arquer, Gatti F Koppens, F H L , Nature Nanotechnology. 7363Konstantatos G, Badioli M, Gaudreau L, Osmond J, Bernechea M, Arquer F P G D, Gatti F and Koppens F H L 2012 Nature Nanotechnology 7 363\n\n. Q Nian, L Gao, Y Hu, B Deng, J Tang, G J Cheng, ACS Applied Materials & Interfaces. 944715Nian Q, Gao L, Hu Y, Deng B, Tang J and Cheng G J 2017 ACS Applied Materials & Interfaces 9 44715\n\n. I Nikitskiy, S Goossens, D Kufer, T Lasanta, G Navickaite, F Koppens, G Konstantatos, Nature Communications. 711954Nikitskiy I, Goossens S, Kufer D, Lasanta T, Navickaite G, Koppens F H and Konstantatos G 2016 Nature Communications 7 11954\n\n. D Kufer, I Nikitskiy, T Lasanta, G Navickaite, Koppens Frank, H L Konstantatos, G , Advanced Materials. 27176Kufer D, Nikitskiy I, Lasanta T, Navickaite G, Koppens Frank H L and Konstantatos G 2015 Advanced Materials 27 176\n\n. J Schornbaum, B Winter, Schießl Stefan, P Gannott, F Katsukis, G , Guldi Dirk, M , Spiecker E Zaumseil, J , Advanced Functional Materials. 245798Schornbaum J, Winter B, Schießl Stefan P, Gannott F, Katsukis G, Guldi Dirk M, Spiecker E and Zaumseil J 2014 Advanced Functional Materials 24 5798\n\n. Y Zhang, M Cao, X Song, J Wang, Y Che, H Dai, X Ding, G Zhang, J Yao, The Journal of Physical Chemistry C. 11921739Zhang Y, Cao M, Song X, Wang J, Che Y, Dai H, Ding X, Zhang G and Yao J 2015 The Journal of Physical Chemistry C 119 21739\n\n. D Zhang, L Gan, Y Cao, Q Wang, L Qi, X Guo, Advanced Materials. 242715Zhang D, Gan L, Cao Y, Wang Q, Qi L and Guo X 2012 Advanced Materials 24 2715\n\n. M Sun, Q Fang, D Xie, Y Sun, L Qian, J Xu, P Xiao, C Teng, W Li, T Ren, Y Zhang, Nano Research. 113233Sun M, Fang Q, Xie D, Sun Y, Qian L, Xu J, Xiao P, Teng C, Li W, Ren T and Zhang Y 2018 Nano Research 11 3233\n\n. C Giansante, I Infante, E Fabiano, R Grisorio, G Suranna, G Gigli, Journal of the American Chemical Society. 1371875Giansante C, Infante I, Fabiano E, Grisorio R, Suranna G P and Gigli G 2015 Journal of the American Chemical Society 137 1875\n\n. G Konstantatos, J Clifford, Levina L Sargent, E H , Nature Photonics. 1531Konstantatos G, Clifford J, Levina L and Sargent E H 2007 Nature Photonics 1 531\n\n. L Zheng, W Zhou, Z Ning, G Wang, X Cheng, W Hu, W Zhou, Z Liu, S Yang, K Xu, Luo M , Yu , Advanced Optical Materials. 01800985Zheng L, Zhou W, Ning Z, Wang G, Cheng X, Hu W, Zhou W, Liu Z, Yang S, Xu K, Luo M and Yu Y 2018 Advanced Optical Materials 0 1800985\n\n. B Radisavljevic, A Radenovic, J Brivio, V Giacometti, A Kis, Nature Nanotechnology. 6147Radisavljevic B, Radenovic A, Brivio J, Giacometti V and Kis A 2011 Nature Nanotechnology 6 147\n\n. W Jin, P-C Yeh, N Zaki, D Zhang, J T Sadowski, Al-Mahboob , A Van Der Zande, A M Chenet, D A Dadap, J I Herman, I P Sutter, P Hone, J Osgood, R M , Physical Review Letters. 111106801Jin W, Yeh P-C, Zaki N, Zhang D, Sadowski J T, Al-Mahboob A, van der Zande A M, Chenet D A, Dadap J I, Herman I P, Sutter P, Hone J and Osgood R M 2013 Physical Review Letters 111 106801\n\n. G Eda, H Yamaguchi, D Voiry, T Fujita, Chen M Chhowalla, M , Nano Letters. 115111Eda G, Yamaguchi H, Voiry D, Fujita T, Chen M and Chhowalla M 2011 Nano Letters 11 5111\n\n. D Wu, Y Ma, Y Niu, Q Liu, T Dong, S Zhang, J Niu, H Zhou, J Wei, Y Wang, Z Zhao, Wang , Science Advances. 4Wu D, Ma Y, Niu Y, Liu Q, Dong T, Zhang S, Niu J, Zhou H, Wei J, Wang Y, Zhao Z and Wang N 2018 Science Advances 4\n\n. S E Keuleyan, P Guyot-Sionnest, Delerue , C , Allan G , ACS Nano. 88676Keuleyan S E, Guyot-Sionnest P, Delerue C and Allan G 2014 ACS Nano 8 8676\n\n. E Lhuillier, M Scarafagio, P Hease, B Nadal, H Aubin, X Z Xu, N Lequeux, G Patriarche, S Ithurria, B Dubertret, Nano Letters. 161282Lhuillier E, Scarafagio M, Hease P, Nadal B, Aubin H, Xu X Z, Lequeux N, Patriarche G, Ithurria S and Dubertret B 2016 Nano Letters 16 1282\n\n. D Kufer, T Lasanta, M Bernechea, F H L Koppens, G Konstantatos, ACS Photonics. 31324Kufer D, Lasanta T, Bernechea M, Koppens F H L and Konstantatos G 2016 ACS Photonics 3 1324\n\n. Y Liu, F Wang, X Wang, X Wang, E Flahaut, X Liu, Y Li, X Wang, Y Xu, Y Shi, R Zhang, Nature Communications. 68589Liu Y, Wang F, Wang X, Wang X, Flahaut E, Liu X, Li Y, Wang X, Xu Y, Shi Y and Zhang R 2015 Nature Communications 6 8589\n\n. K Roy, M Padmanabhan, S Goswami, T P Sai, G Ramalingam, S Raghavan, A Ghosh, Nature Nanotechnology. 8826Roy K, Padmanabhan M, Goswami S, Sai T P, Ramalingam G, Raghavan S and Ghosh A 2013 Nature Nanotechnology 8 826\n\n. G Konstantatos, L Levina, A Fischer, E H Sargent, Nano Letters. 81446Konstantatos G, Levina L, Fischer A and Sargent E H 2008 Nano Letters 8 1446\n\n. P Avouris, M Freitag, V Perebeinos, Nature Photonics. 2341Avouris P, Freitag M and Perebeinos V 2008 Nature Photonics 2 341\n\n. R Saito, M Fujita, G Dresselhaus, M S Dresselhaus, Applied Physics Letters. 602204Saito R, Fujita M, Dresselhaus G and Dresselhaus M S 1992 Applied Physics Letters 60 2204\n\n. H Kataura, Y Kumazawa, Y Maniwa, I Umezu, S Suzuki, Y Ohtsuka, Y Achiba, Synthetic Metals. 1032555Kataura H, Kumazawa Y, Maniwa Y, Umezu I, Suzuki S, Ohtsuka Y and Achiba Y 1999 Synthetic Metals 103 2555\n\n. T Dürkop, S A Getty, Cobas E Fuhrer, M S , Nano Letters. 435Dürkop T, Getty S A, Cobas E and Fuhrer M S 2004 Nano Letters 4 35\n\n. N M Gabor, Z Zhong, K Bosnick, Park J Mceuen, P L , Science. 3251367Gabor N M, Zhong Z, Bosnick K, Park J and McEuen P L 2009 Science 325 1367\n\n. L C Paulus Geraldine, Wang Qing, H , Ulissi Zachary, W , Mcnicholas Thomas, P Vijayaraghavan, A Shih, C J , Jin Z , Strano Michael, S , Small. 91954Paulus Geraldine L C, Wang Qing H, Ulissi Zachary W, McNicholas Thomas P, Vijayaraghavan A, Shih C J, Jin Z and Strano Michael S 2012 Small 9 1954\n\n. Y Liu, Y Liu, S Qin, Y Xu, R Zhang, F Wang, Nano Research. 1054Liu Y, Liu Y, Qin S, Xu Y, Zhang R and Wang F 2017 Nano Research 10 1880 54\n\n. D Jariwala, V K Sangwan, C-C Wu, P L Prabhumirashi, M L Geier, T J Marks, L Lauhon, M C Hersam, Proceedings of the National Academy of Sciences. 11018076Jariwala D, Sangwan V K, Wu C-C, Prabhumirashi P L, Geier M L, Marks T J, Lauhon L J and Hersam M C 2013 Proceedings of the National Academy of Sciences 110 18076\n\n. K F Mak, K He, C Lee, G H Lee, J Hone, T Heinz, J Shan, Nature Materials. 12207Mak K F, He K, Lee C, Lee G H, Hone J, Heinz T F and Shan J 2012 Nature Materials 12 207\n\n. S Liu, Q Liao, Z Zhang, X Zhang, S Lu, L Zhou, M Hong, Z Kang, Y Zhang, Nano Research. 103476Liu S, Liao Q, Zhang Z, Zhang X, Lu S, Zhou L, Hong M, Kang Z and Zhang Y 2017 Nano Research 10 3476\n\n. F Pizzocchero, L Gammelgaard, B S Jessen, J M Caridad, L Wang, J Hone, P Bøggild, T J Booth, Nature Communications. 711894Pizzocchero F, Gammelgaard L, Jessen B S, Caridad J M, Wang L, Hone J, Bøggild P and Booth T J 2016 Nature Communications 7 11894\n\n. Y Gong, J Lin, X Wang, G Shi, S Lei, Z Lin, X Zou, G Ye, R Vajtai, B I Yakobson, H Terrones, M Terrones, Tay Beng, K Lou, J Pantelides, S T Liu, Z Zhou, W Ajayan, P M , Nature Materials. 131135Gong Y, Lin J, Wang X, Shi G, Lei S, Lin Z, Zou X, Ye G, Vajtai R, Yakobson B I, Terrones H, Terrones M, Tay Beng K, Lou J, Pantelides S T, Liu Z, Zhou W and Ajayan P M 2014 Nature Materials 13 1135\n\n. Y Gong, S Lei, G Ye, B Li, Y He, K Keyshar, X Zhang, Q Wang, J Lou, Z Liu, R Vajtai, W Zhou, P M Ajayan, Nano Letters. 156135Gong Y, Lei S, Ye G, Li B, He Y, Keyshar K, Zhang X, Wang Q, Lou J, Liu Z, Vajtai R, Zhou W and Ajayan P M 2015 Nano Letters 15 6135\n\n. Y He, Y Yang, Z Zhang, Y Gong, W Zhou, Z Hu, G Ye, X Zhang, E Bianco, S Lei, Jin Z Zou, X Yang, Y Zhang, Y Xie, E Lou, J Yakobson, B Vajtai, R , Li B Ajayan, P , Nano Letters. 163314He Y, Yang Y, Zhang Z, Gong Y, Zhou W, Hu Z, Ye G, Zhang X, Bianco E, Lei S, Jin Z, Zou X, Yang Y, Zhang Y, Xie E, Lou J, Yakobson B, Vajtai R, Li B and Ajayan P 2016 Nano Letters 16 3314\n\n. Y Yu, S Hu, L Su, L Huang, Y Liu, Jin Z Purezky, A A Geohegan, D B Kim, K W Zhang, Y Cao, L , Nano Letters. 15486Yu Y, Hu S, Su L, Huang L, Liu Y, Jin Z, Purezky A A, Geohegan D B, Kim K W, Zhang Y and Cao L 2015 Nano Letters 15 486\n\n. Q Wang, Y Wen, K Cai, R Cheng, L Yin, Y Zhang, J Li, Z Wang, F Wang, F Wang, T A Shifa, C Jiang, Yang H He, Science Advances. 4Wang Q, Wen Y, Cai K, Cheng R, Yin L, Zhang Y, Li J, Wang Z, Wang F, Wang F, Shifa T A, Jiang C, Yang H and He J 2018 Science Advances 4\n\n. H Fang, C Battaglia, C Carraro, S Nemsak, B Ozdol, J S Kang, H A Bechtel, S B Desai, F Kronast, A A Unal, G Conti, C Conlon, G K Palsson, M C Martin, A M Minor, C S Fadley, E Yablonovitch, R Maboudian, A Javey, Proceedings of the National Academy of Sciences. 1116198Fang H, Battaglia C, Carraro C, Nemsak S, Ozdol B, Kang J S, Bechtel H A, Desai S B, Kronast F, Unal A A, Conti G, Conlon C, Palsson G K, Martin M C, Minor A M, Fadley C S, Yablonovitch E, Maboudian R and Javey A 2014 Proceedings of the National Academy of Sciences 111 6198\n\n. X Hong, J Kim, S-F Shi, Y Zhang, Jin C Sun, Y Tongay, S Wu, J Zhang, Y Wang, F , Nature Nanotechnology. 9682Hong X, Kim J, Shi S-F, Zhang Y, Jin C, Sun Y, Tongay S, Wu J, Zhang Y and Wang F 2014 Nature Nanotechnology 9 682\n\n. F Ceballos, M Z Bellus, H-Y Chiu, H Zhao, ACS Nano. 812717Ceballos F, Bellus M Z, Chiu H-Y and Zhao H 2014 ACS Nano 8 12717\n\n. R Long, O V Prezhdo, Nano Letters. 161996Long R and Prezhdo O V 2016 Nano Letters 16 1996\n\n. Xia Wei, F-G Y , Chao Shen, Quan-Shan Lv, Chin. Phys. B. 2638504Xia Wei F-G Y, Chao Shen, Quan-Shan Lv, Kai-You Wang 2017 Chin. Phys. B 26 38504\n\n. I G Lezama, A Arora, A Ubaldini, C Barreteau, E Giannini, M Potemski, A F Morpurgo, Nano Letters. 152336Lezama I G, Arora A, Ubaldini A, Barreteau C, Giannini E, Potemski M and Morpurgo A F 2015 Nano Letters 15 2336\n\n. C Ruppert, O Aslan, T F Heinz, Nano Letters. 146231Ruppert C, Aslan O B and Heinz T F 2014 Nano Letters 14 6231\n\n. A Pezeshki, H Shokouh Seyed Hossein, T Nazari, K Oh, S Im, Advanced Materials. 283216Pezeshki A, Shokouh Seyed Hossein H, Nazari T, Oh K and Im S 2016 Advanced Materials 28 3216\n\n. H Liu, A T Neal, Z Zhu, Z Luo, X Xu, Tomá, P D Ye, ACS Nano. 84033Liu H, Neal A T, Zhu Z, Luo Z, Xu X, Tomá nek D and Ye P D 2014 ACS Nano 8 4033\n\n. V Tran, R Soklaski, Y Liang, Yang L , Physical Review B. 89235319Tran V, Soklaski R, Liang Y and Yang L 2014 Physical Review B 89 235319\n\n. M Long, E Liu, P Wang, A Gao, H Xia, W Luo, B Wang, J Zeng, Y Fu, K Xu, W Zhou, Y Lv, S Yao, M Lu, Y Chen, Z Ni, Y You, X Zhang, S Qin, Y Shi, W Hu, D Xing, F Miao, Nano Letters. 162254Long M, Liu E, Wang P, Gao A, Xia H, Luo W, Wang B, Zeng J, Fu Y, Xu K, Zhou W, Lv Y, Yao S, Lu M, Chen Y, Ni Z, You Y, Zhang X, Qin S, Shi Y, Hu W, Xing D and Miao F 2016 Nano Letters 16 2254\n\n. S Das, H-Y Chen, A Penumatcha, J Appenzeller, Nano Letters. 13100Das S, Chen H-Y, Penumatcha A V and Appenzeller J 2013 Nano Letters 13 100\n\n. J S Ross, S Wu, H Yu, N J Ghimire, A M Jones, G Aivazian, J Yan, D G Mandrus, Xiao D Yao, W Xu, X , Nature Communications. 41474Ross J S, Wu S, Yu H, Ghimire N J, Jones A M, Aivazian G, Yan J, Mandrus D G, Xiao D, Yao W and Xu X 2013 Nature Communications 4 1474\n\n. H Qiao, J Yuan, Z Xu, C Chen, S Lin, Y Wang, J Song, Y Liu, Q Khan, H Y Hoh, C-X Pan, Li S Bao, Q , ACS Nano. 91886Qiao H, Yuan J, Xu Z, Chen C, Lin S, Wang Y, Song J, Liu Y, Khan Q, Hoh H Y, Pan C-X, Li S and Bao Q 2015 ACS Nano 9 1886\n\n. W Dang, H Peng, H Li, Wang P Liu, Z , Nano Letters. 102870Dang W, Peng H, Li H, Wang P and Liu Z 2010 Nano Letters 10 2870\n\n. M Hasan, C L Kane, Reviews of Modern Physics. 823045Hasan M Z and Kane C L 2010 Reviews of Modern Physics 82 3045\n\n. L Ye, P Wang, W Luo, F Gong, L Liao, T Liu, L Tong, J Zang, J Xu, W Hu, Nano Energy. 3753Ye L, Wang P, Luo W, Gong F, Liao L, Liu T, Tong L, Zang J, Xu J and Hu W 2017 Nano Energy 37 53\n\n. J Bullock, M Amani, J Cho, Y-Z Chen, G H Ahn, V Adinolfi, V R Shrestha, Y Gao, K B Crozier, Y-L Chueh, A Javey, Nature Photonics. 12601Bullock J, Amani M, Cho J, Chen Y-Z, Ahn G H, Adinolfi V, Shrestha V R, Gao Y, Crozier K B, Chueh Y-L and Javey A 2018 Nature Photonics 12 601\n\n. M S Choi, D Qu, D Lee, X Liu, K Watanabe, T Taniguchi, W J Yoo, ACS Nano. 89332Choi M S, Qu D, Lee D, Liu X, Watanabe K, Taniguchi T and Yoo W J 2014 ACS Nano 8 9332\n\n. X Yu, S Zhang, H Zeng, Q J Wang, Nano Energy. 2534Yu X, Zhang S, Zeng H and Wang Q J 2016 Nano Energy 25 34\n\n. L Li, Y Yu, G J Ye, Q Ge, X Ou, H Wu, D Feng, X Chen, Y Zhang, Nature Nanotechnology. 9372Li L, Yu Y, Ye G J, Ge Q, Ou X, Wu H, Feng D, Chen X H and Zhang Y 2014 Nature Nanotechnology 9 372\n\n. N R Pradhan, D Rhodes, Y Xin, S Memaran, L Bhaskaran, M Siddiq, S Hill, P Ajayan, L Balicas, ACS Nano. 87923Pradhan N R, Rhodes D, Xin Y, Memaran S, Bhaskaran L, Siddiq M, Hill S, Ajayan P M and Balicas L 2014 ACS Nano 8 7923\n\n. Y-Q Bie, G Grosso, M Heuck, M M Furchi, Y Cao, J Zheng, D Bunandar, E Navarro-Moratalla, L Zhou, D K Efetov, T Taniguchi, K Watanabe, J Kong, D Englund, Jarillo-Herrero , P , Nature Nanotechnology. 121124Bie Y-Q, Grosso G, Heuck M, Furchi M M, Cao Y, Zheng J, Bunandar D, Navarro-Moratalla E, Zhou L, Efetov D K, Taniguchi T, Watanabe K, Kong J, Englund D and Jarillo-Herrero P 2017 Nature Nanotechnology 12 1124\n\n. X Duan, C Wang, J C Shaw, R Cheng, Y Chen, H Li, X Wu, Y Tang, Q Zhang, A Pan, J Jiang, R Yu, Y Huang, X Duan, Nature Nanotechnology. 9Duan X, Wang C, Shaw J C, Cheng R, Chen Y, Li H, Wu X, Tang Y, Zhang Q, Pan A, Jiang J, Yu R, Huang Y and Duan X 2014 Nature Nanotechnology 9 56 1024\n\n. C Huang, S Wu, A M Sanchez, J J P Peters, R Beanland, J S Ross, P Rivera, W Yao, D Cobden, X Xu, Nature Materials. 131096Huang C, Wu S, Sanchez A M, Peters J J P, Beanland R, Ross J S, Rivera P, Yao W, Cobden D H and Xu X 2014 Nature Materials 13 1096\n\n. M-Y Li, Y Shi, C-C Cheng, L-S Lu, Y-C Lin, H-L Tang, M-L Tsai, C-W Chu, K-H Wei, J-H He, Chang W-H , Suenaga K , Li L-J , Science. 349524Li M-Y, Shi Y, Cheng C-C, Lu L-S, Lin Y-C, Tang H-L, Tsai M-L, Chu C-W, Wei K-H, He J-H, Chang W-H, Suenaga K and Li L-J 2015 Science 349 524\n\n. P K Sahoo, S Memaran, Y Xin, L Balicas, H Gutié Rrez, Nature. 55363Sahoo P K, Memaran S, Xin Y, Balicas L and Gutié rrez H R 2018 Nature 553 63\n\n. Z Zhang, P Chen, X Duan, K Zang, J Luo, X Duan, Science. 357788Zhang Z, Chen P, Duan X, Zang K, Luo J and Duan X 2017 Science 357 788\n\n. Y Du, H Liu, Y Deng, P D Ye, ACS Nano. 810035Du Y, Liu H, Deng Y and Ye P D 2014 ACS Nano 8 10035\n\n. X Gan, R-J Shiue, Y Gao, I Meric, T F Heinz, K Shepard, J Hone, S Assefa, D Englund, Nature Photonics. 7883Gan X, Shiue R-J, Gao Y, Meric I, Heinz T F, Shepard K, Hone J, Assefa S and Englund D 2013 Nature Photonics 7 883\n\n. I Goykhman, U Sassi, B Desiatov, N Mazurski, S Milana, D De Fazio, A Eiden, J Khurgin, J Shappir, Levy U Ferrari, A C , Nano Letters. 163005Goykhman I, Sassi U, Desiatov B, Mazurski N, Milana S, de Fazio D, Eiden A, Khurgin J, Shappir J, Levy U and Ferrari A C 2016 Nano Letters 16 3005\n\n. P Ma, N Flöry, Y Salamin, B Baeuerle, A Emboras, A Josten, T Taniguchi, K Watanabe, L Novotny, Leuthold , ACS Photonics. Ma P, Flöry N, Salamin Y, Baeuerle B, Emboras A, Josten A, Taniguchi T, Watanabe K, Novotny L and Leuthold J 2018 ACS Photonics\n\n. A Pospischil, M Humer, M M Furchi, D Bachmann, R Guider, T Fromherz, T Mueller, Nature Photonics. 7892Pospischil A, Humer M, Furchi M M, Bachmann D, Guider R, Fromherz T and Mueller T 2013 Nature Photonics 7 892\n\n. X Wang, Z Cheng, K Xu, H Tsang, J-B Xu, Nature Photonics. 7888Wang X, Cheng Z, Xu K, Tsang H K and Xu J-B 2013 Nature Photonics 7 888\n\n. N Youngblood, C Chen, S Koester, M Li, Nature Photonics. 9247Youngblood N, Chen C, Koester S J and Li M 2015 Nature Photonics 9 247\n\nDongdong Yin. Yubing Wang, W Y , Qin Han, Xiaohong Yang, Han Ye, Qianqian Lv, Chin. Phys. B. 25118103Yubing Wang W Y, Qin Han, Xiaohong Yang, Han Ye, Qianqian Lv, Dongdong Yin 2016 Chin. Phys. B 25 118103\n\n. M Liu, X Yin, E Ulin-Avila, B Geng, T Zentgraf, L Ju, Wang F Zhang, X , Nature. 47464Liu M, Yin X, Ulin-Avila E, Geng B, Zentgraf T, Ju L, Wang F and Zhang X 2011 Nature 474 64\n\n. T J Echtermeyer, L Britnell, P K Jasnos, A Lombardo, R V Gorbachev, A N Grigorenko, A K Geim, A Ferrari, K S Novoselov, Nature Communications. 2458Echtermeyer T J, Britnell L, Jasnos P K, Lombardo A, Gorbachev R V, Grigorenko A N, Geim A K, Ferrari A C and Novoselov K S 2011 Nature Communications 2 458\n\n. T Gu, N Petrone, J F Mcmillan, A Van Der Zande, M Yu, G Q Lo, D L Kwong, J Hone, C W Wong, Nature Photonics. 6554Gu T, Petrone N, McMillan J F, van der Zande A, Yu M, Lo G Q, Kwong D L, Hone J and Wong C W 2012 Nature Photonics 6 554\n\n. Q Bao, H Zhang, B Wang, Z Ni, C H Y X Lim, Y Wang, D Tang, K P Loh, Nature Photonics. 5411Bao Q, Zhang H, Wang B, Ni Z, Lim C H Y X, Wang Y, Tang D Y and Loh K P 2011 Nature Photonics 5 411\n\n. M Freitag, T Low, F Xia, P Avouris, Nature Photonics. 753Freitag M, Low T, Xia F and Avouris P 2012 Nature Photonics 7 53\n\n. T Low, A S Rodin, A Carvalho, Y Jiang, H Wang, F Xia, Castro Neto, A , Physical Review B. 9075434Low T, Rodin A S, Carvalho A, Jiang Y, Wang H, Xia F and Castro Neto A H 2014 Physical Review B 90 075434\n\n. T Hong, B Chamlagain, W Z Lin, H J Chuang, M H Pan, Z Zhou, Y Q Xu, NANOSCALE. 68978Hong T, Chamlagain B, Lin W Z, Chuang H J, Pan M H, Zhou Z X and Xu Y Q 2014 NANOSCALE 6 8978\n\n. T Low, M Engel, M Steiner, P Avouris, Physical Review B. 9081408Low T, Engel M, Steiner M and Avouris P 2014 Physical Review B 90 081408\n\n. S Das, W Zhang, M Demarteau, A Hoffmann, M Dubey, A Roelofs, Nano Letters. 145733Das S, Zhang W, Demarteau M, Hoffmann A, Dubey M and Roelofs A 2014 Nano Letters 14 5733\n\n. C Robert, R Picard, D Lagarde, G Wang, J P Echeverry, F Cadiz, P Renucci, A Högele, T Amand, Marie X Gerber, I Urbaszek, B , Physical Review B. 94155425Robert C, Picard R, Lagarde D, Wang G, Echeverry J P, Cadiz F, Renucci P, Högele A, Amand T, Marie X, Gerber I C and Urbaszek B 2016 Physical Review B 94 155425\n\n. G Froehlicher, Lorchat E Berciaud, S , Physical Review B. 9485429Froehlicher G, Lorchat E and Berciaud S 2016 Physical Review B 94 085429\n\n. L Yin, X Zhan, K Xu, F Wang, Z Wang, Y Huang, Q Wang, C Jiang, J He, Applied Physics Letters. 10843503Yin L, Zhan X, Xu K, Wang F, Wang Z, Huang Y, Wang Q, Jiang C and He J 2016 Applied Physics Letters 108 043503\n\n. Octon Tobias, J Nagareddy, V K Russo, S , Craciun Monica, F Wright, C D , Advanced Optical Materials. 41750Octon Tobias J, Nagareddy V K, Russo S, Craciun Monica F and Wright C D 2016 Advanced Optical Materials 4 1750\n\n. X Gan, Y Gao, Fai Mak, K Yao, X Shiue, R-J Van Der Zande, A Trusheim, M E Hatami, F Heinz, T F Hone, J Englund, D , Applied Physics Letters. 103181119Gan X, Gao Y, Fai Mak K, Yao X, Shiue R-J, van der Zande A, Trusheim M E, Hatami F, Heinz T F, Hone J and Englund D 2013 Applied Physics Letters 103 181119\n\n. S Wu, S Buckley, J R Schaibley, L Feng, J Yan, D G Mandrus, F Hatami, W Yao, J Vučković, A Majumdar, X Xu, Nature. 52069Wu S, Buckley S, Schaibley J R, Feng L, Yan J, Mandrus D G, Hatami F, Yao W, Vučković J, Majumdar A and Xu X 2015 Nature 520 69\n\n. X Liu, T Galfsky, Z Sun, F Xia, E-C Lin, Y-H Lee, Ké Na-Cohen, S Menon, V M , Nature Photonics. 930Liu X, Galfsky T, Sun Z, Xia F, Lin E-c, Lee Y-H, Ké na-Cohen S and Menon V M 2014 Nature Photonics 9 30\n\n. S Dufferwiel, S Schwarz, F Withers, A A P Trichet, F Li, M Sich, Del Pozo-Zamudio, O Clark, C Nalitov, A Solnyshkov, D D Malpuech, G Novoselov, K S Smith, J M Skolnick, M S Krizhanovskii, D Tartakovskii, A I , Dufferwiel S, Schwarz S, Withers F, Trichet A A P, Li F, Sich M, Del Pozo- Zamudio O, Clark C, Nalitov A, Solnyshkov D D, Malpuech G, Novoselov K S, Smith J M, Skolnick M S, Krizhanovskii D N and Tartakovskii A I 2015\n\n. Nature Communications. 68579Nature Communications 6 8579\n\n. Y Ye, Z J Wong, X Lu, X Ni, H Zhu, X Chen, Wang Y Zhang, X , Nature Photonics. 9733Ye Y, Wong Z J, Lu X, Ni X, Zhu H, Chen X, Wang Y and Zhang X 2015 Nature Photonics 9 733\n\n. M Furchi, A Urich, A Pospischil, G Lilley, K Unterrainer, H Detz, P Klang, A M Andrews, W Schrenk, G Strasser, T Mueller, Nano Letters. 122773Furchi M, Urich A, Pospischil A, Lilley G, Unterrainer K, Detz H, Klang P, Andrews A M, Schrenk W, Strasser G and Mueller T 2012 Nano Letters 12 2773\n\n. M Casalino, U Sassi, I Goykhman, A Eiden, E Lidorikis, S Milana, De Fazio, D Tomarchio, F Iodice, M Coppola, G Ferrari, A C , ACS Nano. 1110955Casalino M, Sassi U, Goykhman I, Eiden A, Lidorikis E, Milana S, De Fazio D, Tomarchio F, Iodice M, Coppola G and Ferrari A C 2017 ACS Nano 11 10955\n\n. Q Guo, C Li, B Deng, S Yuan, F Guinea, F Xia, ACS Photonics. 42989Guo Q, Li C, Deng B, Yuan S, Guinea F and Xia F 2017 ACS Photonics 4 2989\n\n. R Kumar, A Sharma, M Kaur, S Husale, Advanced Optical Materials. 51700009Kumar R, Sharma A, Kaur M and Husale S 2017 Advanced Optical Materials 5 1700009\n\n. I J Luxmoore, P Q Liu, P Li, J Faist, G R Nash, ACS Photonics. 3936Luxmoore I J, Liu P Q, Li P, Faist J and Nash G R 2016 ACS Photonics 3 936\n\n. Z Xia, P Li, Y Wang, T Song, Q Zhang, B Sun, ACS Applied Materials & Interfaces. 724136Xia Z, Li P, Wang Y, Song T, Zhang Q and Sun B 2015 ACS Applied Materials & Interfaces 7 24136\n\n. W Wang, A Klots, D Prasai, Y Yang, K Bolotin, J Valentine, Nano Letters. 157440Wang W, Klots A, Prasai D, Yang Y, Bolotin K I and Valentine J 2015 Nano Letters 15 7440\n\n. T Hong, B Chamlagain, S Hu, S M Weiss, Z Zhou, Y-Q Xu, ACS Nano. 95357Hong T, Chamlagain B, Hu S, Weiss S M, Zhou Z and Xu Y-Q 2015 ACS Nano 9 5357\n\n. Z Fang, Z Liu, Y Wang, P M Ajayan, Nordlander P Halas, N J , Nano Letters. 123808Fang Z, Liu Z, Wang Y, Ajayan P M, Nordlander P and Halas N J 2012 Nano Letters 12 3808\n\n. Somayyeh Asgari, H R , Nosrat Granpayeh, Homayoon Oraizi, Chin. Phys. B. 2784212Somayyeh Asgari H R, Nosrat Granpayeh, Homayoon Oraizi 2018 Chin. Phys. B 27 84212\n\n. Z Chen, X Li, J Wang, L Tao, M Long, S-J Liang, L K Ang, C Shu, H Tsang, J-B Xu, ACS Nano. 11430Chen Z, Li X, Wang J, Tao L, Long M, Liang S-J, Ang L K, Shu C, Tsang H K and Xu J-B 2017 ACS Nano 11 430\n\n. M Stockman, Nature. 467541Stockman M I 2010 Nature 467 541\n\n. M W Knight, H Sobhani, P Nordlander, N J Halas, Science. 332702Knight M W, Sobhani H, Nordlander P and Halas N J 2011 Science 332 702\n\n. H S Lee, Min S-W Chang, Y-G Park, M K , Nam T Kim, H Kim, J H , Ryu S Im, S , Nano Letters. 123695Lee H S, Min S-W, Chang Y-G, Park M K, Nam T, Kim H, Kim J H, Ryu S and Im S 2012 Nano Letters 12 3695\n\n. J-Y Wu, Y T Chun, S Li, T Zhang, J Wang, P Shrestha, D Chu, Advanced Materials. 301705880Wu J-Y, Chun Y T, Li S, Zhang T, Wang J, Shrestha P K and Chu D 2018 Advanced Materials 30 1705880\n\n. Z Ni, L Ma, S Du, Y Xu, M Yuan, H Fang, Z Wang, M Xu, D Li, J Yang, W Hu, X Pi, Yang D , ACS Nano. 119854Ni Z, Ma L, Du S, Xu Y, Yuan M, Fang H, Wang Z, Xu M, Li D, Yang J, Hu W, Pi X and Yang D 2017 ACS Nano 11 9854\n\n. N Guo, F Gong, J Liu, Y Jia, S Zhao, L Liao, M Su, Z Fan, X Chen, W Lu, Xiao L Hu, W , ACS Applied Materials & Interfaces. 934489Guo N, Gong F, Liu J, Jia Y, Zhao S, Liao L, Su M, Fan Z, Chen X, Lu W, Xiao L and Hu W 2017 ACS Applied Materials & Interfaces 9 34489\n\n. M Massicotte, P Schmidt, F Vialla, K Watanabe, T Taniguchi, K Tielrooij, F H L Koppens, Nature Communications. 712174Massicotte M, Schmidt P, Vialla F, Watanabe K, Taniguchi T, Tielrooij K J and Koppens F H L 2016 Nature Communications 7 12174\n\n. N Huo, G Konstantatos, Nature Communications. 8572Huo N and Konstantatos G 2017 Nature Communications 8 572\n\n. J Kim, S Park, H Jang, N Koirala, J-B Lee, U J Kim, H-S Lee, Y-G Roh, H Lee, S Sim, S Cha, C In, J Park, J Lee, M Noh, J Moon, M Salehi, J Sung, S Chee, Kim J, Park S, Jang H, Koirala N, Lee J-B, Kim U J, Lee H-S, Roh Y-G, Lee H, Sim S, Cha S, In C, Park J, Lee J, Noh M, Moon J, Salehi M, Sung J, Chee S-\n\n. S , Ham M-H , Jo M-H Oh, S Ahn, J-H Hwang, S W , Kim D Choi, H , S, Ham M-H, Jo M-H, Oh S, Ahn J-H, Hwang S W, Kim D and Choi H 2017\n\n. ACS Photonics. 4482ACS Photonics 4 482\n"},"corpus_id":{"kind":"number","value":119252717,"string":"119,252,717"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1812.05286\",\n \"mag\": 2904579272,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/1674-1056/28/1/017302\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1812.05286v1.pdf\"\n ],\n \"pdf_sha\": \"1ec300759da99aa9d02c6e462849324302124207\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1812.05286\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Room temperature infrared photodetectors with hybrid structure based on 2D materials"},"authors":{"kind":"list like","value":["Tiande Liu \nSchool of Optical and Electronic Information\nHuazhong University of Science and Technology\n430074WuhanHubeiChina","Lei Tong \nSchool of Optical and Electronic Information\nHuazhong University of Science and Technology\n430074WuhanHubeiChina","Xinyu Huang \nSchool of Optical and Electronic Information\nHuazhong University of Science and Technology\n430074WuhanHubeiChina","Lei Ye \nSchool of Optical and Electronic Information\nHuazhong University of Science and Technology\n430074WuhanHubeiChina"],"string":"[\n \"Tiande Liu \\nSchool of Optical and Electronic Information\\nHuazhong University of Science and Technology\\n430074WuhanHubeiChina\",\n \"Lei Tong \\nSchool of Optical and Electronic Information\\nHuazhong University of Science and Technology\\n430074WuhanHubeiChina\",\n \"Xinyu Huang \\nSchool of Optical and Electronic Information\\nHuazhong University of Science and Technology\\n430074WuhanHubeiChina\",\n \"Lei Ye \\nSchool of Optical and Electronic Information\\nHuazhong University of Science and Technology\\n430074WuhanHubeiChina\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Two-dimensional (2D) materials, such as graphene, transition metal dichalcogenides (TMDs), black phosphorus (BP) and related derivatives, have attracted great attention due to their advantages of flexibility, strong light-matter interaction, broadband absorption and high carrier mobility, and have become a powerful contender for next-generation infrared photodetectors. However, since the thickness of twodimensional materials is on the order of nanometers, the absorption of two-dimensional materials is very weak, which limits the detection performance of 2D materials-based infrared photodetector. In order to solve this problem, scientific researchers have tried to use optimized device structures to combine with two-dimensional materials for improving the performance of infrared photodetector. In this review, we review the progress of room temperature infrared photodetectors with hybrid structure based on 2D materials in recent years, focusing mainly on 2D-nD (n = 0, 1, 2) heterostructures, the integration between 2D materials and on-chip or plasmonic structure. Finally, we summarize the current challenges and point out the future development direction."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Infrared (IR) photodetectors can convert infrared light signals into easily detectable electrical signals, which are widely used in imaging, communications, medical and night vision applications at present. [1,2] However, in order to obtain better detecting performance, the traditional infrared photodetectors based on the III-V or II-VI 3 semiconductors have to work at low temperatures to reduce dark currents. [3] Because of the necessity of large refrigeration equipment to realize low operating temperature, it leads to the greatly increase of the manufacturing cost and limitation of the application, making it difficult to achieve miniaturization, flexibility and portability.\nIn order to circumvent this disadvantage, new type semiconductors are proposed to use in the application of IR photodetectors, which can achieve low current at room temperature and high photocurrent. In recent years, two-dimensional materials with an atomic layer thickness, are verified to overcome the shortcomings induced by traditional semiconductors used in traditional IR detection due to their excellent electrical, optical and mechanical properties. [4][5][6][7][8][9][10][11][12][13] For example, graphene is the most studied two-dimensional materials with semi-metal, high mobility, broad spectrum response, flexibility, and tunable Fermi level properties, [14,15] suggesting that graphene can be fabricated into flexible infrared photodetectors with broad spectrum absorption, ultra-fast response. [16,17] In addition, transition metal dichalcogenides materials MX2 (M: transition metal atoms, X=S, Se, Te), which have the properties of strong light interactions and tunable bandgap varying with the number of layers, are also suitable as photosensitive materials for infrared photodetectors. Black phosphorus shows a direct bandgap property with the range of 0.3-1.3 eV, leading to its absorption spectrum covering the near-infrared, mid-infrared and far-infrared regions, even extending to the terahertz region. [18] Especially, wide spectral response range coupled with its unique anisotropic crystal structure, promises the potential of black phosphorus for infrared polarized photodetectors. [19] Although 2D materials have various excellent and unique 4 properties for the applications of photodetector, they cannot avoid such a problem that such thin thickness cannot absorb more light. For example, the thickness of monolayer graphene is 0.35 nm and absorbs only 2.3% of light. [20] Although the bandgap of monolayer TMDs is a direct bandgap, the whole absorption is also less than 10%. [21] In order to solve this problem, it is necessary to combine two-dimensional materials with other strong light absorbing materials, or to integrate with some structures that can enhance light absorption. For example, the band gap of quantum dots and nanoparticles with strong light absorption can be tuned by changing its size, [22,23] which is very suitable as a photosensitive layer for infrared photodetectors. Carbon nanotubes have excellent broadband absorption characteristics, [24] which is also very suitable for forming heterostructures with two-dimensional materials. Integrating two-dimensional materials with waveguides or cavities can enhance the photon absorption of 2D materials to improve detection performance. In addition, this integration also reveals the compatibility of two-dimensional materials with existing optoelectronic technologies. The surface plasmon resonance effect not only enhances light absorption but also selectively absorbs light by designing different shapes and sizes, [25] therefore the spectral response of these designed photodetectors can be extended to infrared or terahertz region that cannot be absorbed by intrinsic two-dimensional material, due to the surface plasmon resonance effect.\nIn this review, we review the development of infrared photodetectors with mixeddimensional structure between two-dimensional materials and other low-dimensional materials, focusing on quantum dots, single-wall carbon nanotubes, and 2D vertical or 5 lateral heterostructures composed of different two-dimensional materials. Moreover, we also discuss the integration of two-dimensional materials and on-chip structures, including waveguides and cavities. We also introduce the application of surface plasmon resonance in infrared photodetectors based on two-dimensional materials.\nFinally, we summarize the faced challenges currently in infrared photodetectors with hybrid structures based on two-dimensional materials and point out the future directions for further development."},{"title":"Heterostuctures","content":""},{"title":"Quantum dots on 2D materials","content":"In order to meet the demand for high-speed response of infrared photodetectors, the materials used to fabricate the detectors is often required to have high carrier mobility to reduce the transit time of photogenerated carriers in conducting channel.\nThe carrier mobility of graphene is as high as 60,000 cm 2 V -1 s -1 , [26] which can fully meet this demand. However, the optical response of photodetectors based on intrinsic graphene is only a few tens of mA/W, [27,28] which is caused by weak light absorption due to its atomic-scale thickness. Compared with graphene, monolayer molybdenum disulfide (MoS2) has a direct band gap, so the absorption is stronger than that of graphene, but still not enough. Moreover, the band gap of molybdenum disulfide is 1.8 eV, and the corresponding absorption wavelength range is in the visible light range, leading to the limitation of directly fabrication for an infrared photodetector. [21,29] The problems of weak light absorption caused by the ultra-thin thickness and the 6 unsatisfactory photoresponse range due to broad band gap, have seriously hindered the development of infrared photodetectors based on two-dimensional materials. To solve this problem, scientific researchers have tried many methods. It is a good attempt to combine two-dimensional materials with quantum dots. [16,[30][31][32][33][34][35][36][37][38][39] Because quantum dots have stronger light absorption relative to 2D materials, and their absorption wavelength range can be tuned by changing its size, [22,40] and it can efficiently and selectively absorb photons and convert them into photogenerated electron-hole pairs. Therefore, the hybrid 2D-QDs structure is promising in the field of infrared detection technology."},{"title":"Konstantatos et al. demonstrated a hybrid graphene-PbS quantum dots","content":"phototransistor with ultrahigh gain of 10 8 and responsivity of 510 7 AW -1 . [32] The schematic diagram of a hybrid graphene-PbS quantum dots phototransistor is shown in Figure 1a. Monolayer or bilayer graphene was exfoliated with tape on a Si/SiO2 substrate, and then an 80-nm PbS quantum dots film was prepared by low-cost spin casting method. PbS QDs thin film acts as light-sensitive material due to the characteristics of strong and selective light absorption, while graphene acts as a conduction channel owing to its extremely high carrier mobility. The energy band diagram at the interface of graphene and PbS QDs is shown in Figure 1b. On account of the work function mismatch between graphene and PbS QDs, the band bending occurs at the interface. PbS QDs thin film absorbs light to create photogenerated electron-hole pairs, which are then separated by an built-in electric field at the interface that induced by band bending. The Dirac point (VD) will be drifted to higher backgate voltage due to illumination. For VBG<VD, holes are transferred from PbS QDs to 7 graphene channel, and electrons are still trapped in the PbS QDs. The resistance of graphene decreases and carrier transport in the channel is hole-dominated. For VBG>VD, holes are transferred to graphene and recombined with electrons induced by the backgate, so the resistance of graphene increases and carrier transport in the channel is electron-dominated.\nAfter the photogenerated holes are transferred to the graphene channel, they are driven by the source-drain voltage (VSD) to reach the drain. The linear dependence of the photocurrent on VSD is shown in Figure 1c. In order to achieve charge conservation in graphene, holes are continuously replenished from the source. Due to the long charge trapping time of the PbS QDs and the high mobility of the graphene, [26,41] holes in the channel can circulate multiple times during quantum dots trapped photogenerated electrons, resulting in ultrahigh photoconductive gain. The relationship between the responsivity and the applied back-gate voltage is shown in Figure 1d, which shows that the photoconductive gain of the photodetector can be adjusted by VBG. For VBG<VD, the measured responsivity can reach up to 510 7 A/W at VBG=-20 V; when VBG=4 V, the responsivity reduces to zero. This back-gate-tunable gain is significant so that the required gain can be adjusted based on the photosignal intensity that needs to be reproduced from Ref. [32] with permission from the Nature Publishing Group.\nAnother important two-dimensional material, transition metal dichalcogenides 9 (TMDs), has also been extensively studied in recent years. [43][44][45][46] However, due to the bandgap limitation and ultrathin thickness, the response spectral range and responsivity of TMDs-based infrared photodetectors are limited. Similarly, in order to enhance light absorption and broaden the response spectral range, a strategy for integrating TMDs materials with quantum dots has also been exploited. [31,35,36] Huo et al. demonstrated\nMoS2-HgTe QDs hybrid photodetectors beyond 2m. [31] The infrared detection capability of this detector is mainly caused by HgTe QDs thin film, which acts as a sensitizing layer, and the response spectrum range can be further extended to the midwave infrared and long-wave infrared ranges by adjusting the size of HgTe quantum dots. [47,48] A TiO2 buffer layer between the MoS2 transistor channel and the HgTe QDs sensitizing layer was used as a protective layer of the MoS2 channel and as an n-type electron acceptor medium to form an efficient p-n junction with the HgTe QDs at the interface facilitating the charge transfer to MoS2 channel. [31,49] Thus, the dark current can be adjusted by the gate voltage and the specific detectivity D* (~10 12 Jones) at a wavelength of 2 m can be obtained. The detector can operate at room temperature and exhibit sub-milliseconds response and high responsivity of ~10 6 A/W, which demonstrates that the hybrid 2D-QDs structure is an effective method to improve the performance of photodetector based on 2D materials.\nIn order to obtain high gain to make up for the weak light absorption of 2D materials, many infrared photodetectors based on hybrid 2D-QDs heterostructures are phototransistors that have an electronic passive sensitization layer composed of strong light absorbing material. [32,50,51] Thus, the external quantum efficiency (EQE), response 10 speed and linear dynamic range are limited by thin sensitizing layers thickness, long trapping times, and high density of the sensitizing centers trap-state, respectively. [32,41,52] In order to solve these problems, Nikitskiy et al. proposed a photodetector that integrates an electrically active colloidal QD photodiode with a graphene transistor. [34] The device is composed of a bottom graphene channel, a 300-nm-thick PbS QDs layer and a top ITO electrode. The photodiode operation and phototransistor operation of the hybrid 2D-QDs photodetector are shown in Figure 2a and b, respectively. The graphene-QDs photodiode responsivity and EQE are shown in Figure 2c. With increasing VTD, EQE increased from 10% at VTD=0 V to 75% at VTD=1.2 V (limited by reflection), indicating the importance of VTD for improving EQE. Figure 2d shows the EQE and responsivity in phototransistor operation, agreeing with the dependence of EQETD on VTD in photodiode operation, which demonstrates that the EQE of this integrated configuration is determined by the photodiode. Moreover, the responsivity is 510 6 V/W at VTD=1.2 V, indicating that the phototransistor still has a high gain of (a-f) are reproduced from Ref. [34] with permission from the Nature Publishing Group."},{"title":"Nanowires combined with 2D materials","content":"One-dimensional nanomaterials, such as single-wall carbon nanotubes (SWNTs), have unique and excellent optical and electrical properties due to 1D quantum confinement effect. [53] Because of the type of tube chiralities, metallic or semiconducting properties are exhibited, and the carrier mobility of semiconducting SWNTs can reach up to 10 5 cm 2 V -1 s -1 at room temperature. [54][55][56] SWNTs have broadband and tunable absorption, so it has great potential in the field of infrared detection. However, the photoresponsivity is limited by the small light-absorbing area, and the responsivity of single-tube photovoltaic devices is always <10 -3 A/W. [57] Paulus et al. reported that the junction between a metallic SWNT and graphene can perform effective charge transfer. [58] This is because that the SWNT is able to dope the graphene in a spatially controlled way without affecting its sp 2 lattice structure. [58] Liu et al.\ndemonstrated a high-performance photodetector based on a planar atomically thin SWNT-graphene hybrid film across visible to near-infrared range (400-1,550 nm). [50] Schematic of the photodetector and atomic force microscope (AFM) characterization of SWNT-graphene hybrid film are shown in Figure 3a. SWNTs have a diameter in the range of 1.0-1.6 nm, and the S11 and S22 bands are found to be located at ~1,800 and ~1,000 nm with higher SWNTs loadings. The photogenerated electron-hole pairs are generated at the SWNT/ graphene interface under optical illumination, and then 13 separated by the built-in electric field. Electrons are transferred from SWNTs to graphene, while holes are trapped in the SWNTs, forming a photogating effect. Figure 3b shows that the source-drain current (ISD) as a function of the applied back-gate voltage (VG). The Dirac point shifts toward the lower VG as the light increases,\nindicating that the graphene resistance can be effectively modulated by the photogating effect. Unlike other photodetectors that use the photogating effect to increase the responsivity, which generally has slow response speed, the photodetector based on hybrid SWNTs-graphene exhibits a fast response time of ~100 s ( Figure 3c). This is mainly due to the high mobility of graphene and SWNTs and fast charge transfer at the SWNTs-graphene interface, and the field-effect mobility of electrons and holes are measured as 3,920 and 3,663 cm 2 V -1 s -1 , respectively. The transit time of the hybrid SWNTs-graphene photodetector is estimated to be ~ 10 -9 s, so the photoconductive gain is ~ 10 5 and the gain-bandwidth product is ~10 9 Hz. Moreover, since SWNTs act as a sensitizing layer, the detector also exhibits the characteristics of a broadband photoresponse. The relationship between the responsivity and the incident optical power at different illumination wavelengths in the range of visible to near-infrared range (405, 532, 650, 980 and 1,550 nm) is shown in Figure 3d, which shows a high responsivity of >100 A/W. Further, Liu et al. fabricated a photodetector based on graphene/carbon nanotube hybrid films on a PET substrate and measured the performance under bending conditions. [59] It was found that this device has good robustness against repetitive bending, which fully demonstrated the huge application potential of this heterostructure in the field of flexible electronics. 14 Jariwala et al. demonstrated a gate-tunable carbon nanotube-MoS2 heterjunction pn diode. [60] Since both monolayer MoS2 and semiconducting SWNTs have direct bandgaps and tightly bound excitation states in absorption spectra, [61] the photodiode based on hybrid MoS2-SWNTs also exhibits excellent detection performance. The external quantum efficiency (EQE) and the response time are ~25% and <15 s, respectively. Figure 3e shows that photocurrent spectral response can be tuned by changing the gate voltage. The photocurrent in the near-infrared region continuously decreases as the gate voltage decreases. Furthermore, the performance of devices based on 1D-2D heterostructures can be further optimized by strain modulation. For example, the responsivity of the photodetector based on the graphene-ZnO nanowire heterostructure is improved by 26% under 0.44% tensile strain on the ZnO nanowire. [62] The above results show that it is significant to carry out the research of the hybrid 1D-2D heterostructure in the field of infrared detection.\n15 VD=-10 V. (a-d) are reproduced from Ref. [50] with permission from the Nature 16 Publishing Group. (e) are reproduced from Ref. [60] with permission from the National Academy of Sciences."},{"title":"2D vertical and lateral heterostructures 2D vertical heterostructures","content":"Two-dimensional vertical heterostructure is fabricated by stacking different twodimensional materials in the vertical direction. This heterostructure is generally prepared by targeted transfer method or CVD growth technique. [63][64][65][66][67][68] 2D vertical heterostructures have many advantages, such as atomically sharp interface, no atomic diffusion between layers, and no strict lattice matching problems due to weak van der Waals interaction between 2D materials. Moreover, the strong interlayer coupling and ultrafast transfer of charge carriers make it have great potential in the field of optoelectronics and electronic devices. [68][69][70][71][72][73] In order to achieve infrared detecting, photodetectors must have a broad spectral response. However, the development of TMDs for infrared detection is hindered due to the lack of 2D materials with narrow bandgap. Monolayer -MoTe2 has a band gap of 1.1 eV and can absorb light in the near-infrared range. [74,75] The response spectral of the photodetectors based on the -MoTe2/MoS2 dichalcogenide heterojunction covers in the range from the visible to near-infrared (800 nm). [76] However, this is not enough for mid-infrared and far-infrared detection, and the response range of the photodetector should be further extended. Graphene can absorb light indiscriminately due to the gapless band structure and monolayer black phosphorene shows a direct optical band gap of 1.3 eV, [77,78] so they are effective absorption material in the infrared region."},{"title":"Long et al. demonstrated a broadband photodetector based on MoS2-graphene-","content":"WSe2 heterostructure covering visible to short-wavelength infrared range at room temperature. [79] The schematic diagram and optical image of the photodetector based on a MoS2-graphene-WSe2 heterostructure are shown in Figure 4a. The p-doped WSe2 is induced by large work-function metal Pd, whereas MoS2 flakes remain n-type due to\nFermi level pinning, [80] and thus forming a p-n junction with a strong internal electric field of ~210 8 V/m. Photogenerated electron-hole pairs can be effectively separated in the presence of the internal electric field. Therefore, the dark current of the device is effectively suppressed and the specific detectivity increases accordingly. Figure 4b shows the responsivity R and specific detectivity D* of the photodetector for wavelengths ranging from 400 to 2,400 nm with an interval of 100 nm. R and D* are as high as 10 4 A/W and 10 15 Jones in the visible region, while they decrease sharply to 100 mA/W and 10 9 Jones in the infrared wavelength of 2400 nm, respectively. Figure 4c explains the reason for the sharp changes in responsivity and specific detectivity.\nThe band gap of MoS2 (Eg1) and WSe2 (Eg2) are 1.88 eV and 1.65 eV, respectively, and the corresponding wavelengths are 660 nm and 750 nm. [21,29,81] When the incident photon energy h exceeds the bandgap of the material, the material can absorb photons to generate electron-hole pairs. Therefore, for h >Eg1>Eg2, MoS2, graphene and WSe2 all can generate photoelectron electron-hole pairs, resulting in a higher responsivity.\nFor h <Eg1<Eg2, only graphene can generate photoelectron electron-hole pairs, and thus the photocurrent is drastically reduced. Qiao et al. reported a broadband 18 photodetectors based on graphene-Bi2Te3 heterostructure (Figure 4d). [82] Bi2Te3 has only 2.7% lattice mismatch with graphene, so the graphene-Bi2Te3 heterostructure can be directly prepared on graphene by Van der Waals epitaxial growth method. [83] Figure 4e plots photocurrent of devices based on pure graphene and graphene-Bi2Te3 heterostructure as a function of source-drain voltage without applied gate bias, which shows that the latter is ten times higher than the former. Both graphene and Bi2Te3 absorb light to generate photoelectron-hole pairs, which are subsequently separated by a built-in electric field. Then, photogenerated electrons in graphene are transferred into Bi2Te3, while the photogenerated holes in Bi2Te3 are transferred into graphene, which suppresses the recombination of carriers and generates a larger photocurrent. Figure 4f shows are reproduced from Ref. [79] with permission from the American Chemical Society. 20 (d-f) are reproduced from Ref. [82] with permission from the American Chemical Society.\nSo far, highly polarization sensitive infrared photodetector based on twodimensional materials working at room temperature are still lacking. Ye et al. first demonstrated a broadband photodetector based on a vertical photogate heterostructure of BP-on-WSe2 with highly polarization sensitive for infrared detection (Figure 5a). [85] WSe2 acts as a conductive channel, while BP with a small direct bandgap and highly anisotropic crystal structure acts as a photogate. [18,19,77] Figure 5b shows the photoconductive gain G and detectivity D* of the photodetector for wavelengths ranging from 400 to 1,600 nm. The detectivity in the visible and infrared regions are 10 14 and 10 10 Jones, and the gain in the visible 21 and infrared regions are 10 6 and 10 2 , respectively, demonstrating excellent broadband detection performance. The optical image of the BP-on-WSe2 photodetector and the relationship between the photocurrent image of the photodetector at 1,550 nm and the polarization angle are shown in Figure 5c. The advantage of this device structure is that it can eliminate the disturbance of two-fold polarization-dependent photocurrent resulted from the geometric edge effect at the metal-BP edge and fully collect the photogenerated free carriers isotropically for highly polarization-sensitive infrared detection. [85] As can be seen from the photocurrent image, the polarization photoresponse originates from the overlap of BP and WSe2, and it changes obviously with the polarization angle changes. The polar plot in Figure 5d shows . [85] The ratio of the maximum responsivity to the minimum responsivity is about 6, showing high polarization sensitive infrared detection performance.\nSimilarly, the mid-wave infrared photodetector based on BP and MoS2 heterostructure designed by Bullock et al. also utilized the anisotropy of BP to achieve polarization detection, which is bias-selectable and operates without the need for external optics. [86] The device exhibits high EQE (35%) and D* (1.110 10 cm Hz 1/2 W -1 ) at room temperature.\nThe above results show that the 2D vertical heterostructure can effectively extend 22 the response range of the photodetector to achieve infrared detection. In addition, CVD epitaxial growth technology provides a guarantee for the scaled preparation of vertical heterostructures. By combining 2D materials with different unique and excellent physical properties, a multifunctional infrared detector with excellent detection performance can be prepared. heterostructures generally include chemical doping, [87,88] electrostatic doping, [89][90][91] and one-pot or two-step chemical vapor deposition (CVD) growth methods. [64,65,[92][93][94][95][96] Recently, Yu et al. demonstrated a lateral black phosphorene P-N junction formed via chemical doping for high performance near-infrared photodetector. [88] The n-type doping of few-layer black phosphorene is achieved by using benzyl viologen (BV) as an electron dopant, which can obtain a high electron concentration while still maintain the high carrier mobility of BP as the processes do not induce defects in the crystal lattices, and Al2O3 is used to selectively protect part of the area of a p-type BP at the 24 same time. [88] The electron doping concentration of BP can be adjusted by doping time, and the built-in electric field at the P-N junction can effectively separate photogenerated electron-hole pairs and reduce recombination efficiency. Photocurrent tests show that the off-current under 1.47 m illumination is 4 orders of magnitude higher than in dark condition, demonstrating that the photocurrent dominates over thermionic and tunneling currents in the photodiode operation. [88] The photodetector exhibits high responsivity of ~180 mA/W, fast response of tens of milliseconds and high detectivity in the range of 10 11 -10 13 Jones under infrared light illumination (=1.47 m).\nHowever, the current methods for preparing lateral heterojunctions have their own limitations. Chemical doping has the characteristic that it is difficult to find suitable dopants. The density of the carrier modulated by the electrostatic gating is relatively low, so the modulation effect is limited. [19,97] CVD technology is an effective method for large-scale preparation of lateral heterostructures, but the main focus is on epitaxial growth of TMDs with large band gap at present, and thus CVD technology has limitation in the field of infrared detection. The infrared light detection technology based on 2D lateral heterostructure needs further research and development."},{"title":"On-chip structure","content":""},{"title":"2D materials on waveguide","content":"25 Two-dimensional materials show strong interaction with light, but the spectral response range and responsivity of the photodetector based on 2D materials are limited due to the reason of the atomically ultra-thin thickness and the limitation of the band gap. In order to obtain better infrared detection performance to achieve practical applications, it is necessary to enhance the infrared light absorption of two-dimensional materials. Integrating two-dimensional materials with waveguides is a good strategy to enhance light absorption. [91,[98][99][100][101][102][103][104] The propagation of light in the waveguide is accompanied by the evanescent field, which can be absorbed by the two-dimensional materials to generate photogenerated electron-hole pairs. [105][106][107][108] Different with the incident light perpendicularly incident on the two-dimensional materials, the in-plane absorption of two-dimensional materials makes its nanoscale thickness to be no longer a major limiting factor for light absorption. The length of the two-dimensional material in contact with the waveguide is on the order of micrometers, so the length of the interaction with the light is also on the order of micrometers and therefore can effectively enhance light absorption.\nRecently, a series of similar but independent waveguide integrated graphene photodetectors have been reported, [98,101,102] The absorption coefficient of metal electrode M increases with increasing electrode 26 width W. For W>100 nm (W>160 nm), the light absorption of the metal electrode will exceed the absorption of monolayer (bilayer) graphene. However, the contact resistance between metal and graphene also increases with decreasing width of electrode, so the metal electrode absorption coefficient and contact resistance should be considered comprehensively to obtain the best device performance. Figure 6b shows (c-d) are reproduced from Ref. [98] with permission from the Nature Publishing Group.\n(e-f) are reproduced from Ref. [102] with permission from the Nature Publishing Group."},{"title":"Similarly, Youngblood et al. demonstrated a waveguide-integrated few-layer black","content":"phosphorus photodetector with high responsivity and low dark current. [103] Figure 7a shows the structure of the photodetector, having a graphene top-gate, a BP channel and the waveguide patterned on the silicon-on-insulator substrate. Since the incident light propagates along the waveguide, the optical interaction length is equal to the width of the black phosphorus covering on the waveguide (6.5 m). The device can absorb 78.7%\nof the optical power in the waveguide, of which only 17.5% is absorbed by the black phosphorus and the rest is absorbed by the graphene top-gate. [103] Figure 7c shows that the photocurrent versus the gate voltage VG and the bias voltage VDS. VG can electrostatically tune the hole doping concentration of black phosphorus and the BP is nearly intrinsic at VG=-8 V. When black phosphorus is in a low doping state (-10 V < VG < -1 V), the photocurrent is larger and its sign is positive relative to VDS, which suggests photocurrent generation mechanism is photovoltaic effect at this time. [109][110][111] 30\nFor VG > 0 V, BP is tuned to be more heavily n-doped, the photocurrent is getting smaller and its sign changes to negative relative to VDS, and the photocurrent generation mechanism is bolometric effect at this time. [109,112] The frequency response of the photocurrent shown in Figure 7b can also demonstrate the photocurrent generation mechanism at different gate voltages. The roll-off frequency is measured to be 2. light source and photodetector based on a p-n junction of bilayer MoTe2. [91] Figure 8a shows the device schematic, and Figure 8b shows LED mode and photodetector mode of device, respectively. The MoTe2 p-n junction is composed of p-type and n-type 32 doping that are electrostatically induced by left and right graphite top gates above the hBN dielectric layer. The bilayer MoTe2 not only has stronger excitation and longer emission wavelength than the monolayer MoTe2, but also has excitation caused by direct bandgap optical transitions similar to that of the monolayer. [74,114,115] There is a grating coupler for excitation and collection at the far end of the waveguide. Excitation Figure 8d shows that the PhC transmission peak is at 1,160 nm, but this peak disappears after MoTe2 transfer, which is mainly due to the interference of MoTe2 absorption. The normalized EL emission spectrum collected at the grating coupler represented by the blue line in Figure 8d, showing a narrow peak centred at 1,160 nm on top of a broader peak centred at 1,175 nm. [91] The former coincides with the PhC transmission peak, and the latter matches the free-space emission of the bilayer MoTe2.\nThe reason that the PhC transmission peak can be observed at the blue line is that the excited states of MoTe2 have been filled by the electrically excited carriers in the LED mode, so the absorption of MoTe2 will not cause interference. When the device is 33 operating in photodetector mode, the responsivity and external quantum efficiency of the photodetector at the wavelength of 1,160 nm are 4.8 mA/W and 0.5%, respectively. Figure 8e shows the temporal response in photodetector mode. The photocurrent response bandwidth is 200 MHz (limited by experimental set-up), which is much higher than other MoTe2 photodetectors based on photogating and photoconductive effect, [116,117] indicating the fast response of waveguide-integrated photodetector with bilayer MoTe2 p-n junction. As can be seen from Figure reproduced from Ref. [91] with permission from the Nature Publishing Group."},{"title":"2D materials on cavity","content":"The integration of two-dimensional materials and cavities, including photonic crystal cavities, [118,119] optical microcavities [120,121] and optical ring resonators, [122] is also an effective method for enhancing the light absorption of 2D materials. The incident light at the resonant frequency of the cavity is absorbed many times by the atomically thin two-dimensional material in the cavity, thereby enhancing absorption. Schottky photodetector operating at communication band (1,550 nm). [124] Figure 9c shows the resonant cavity enhanced graphene/Si Schottky photodetector and the resonant structure consists of a λ/2 Si slab layer confined between SLG/Si top and Au bottom mirrors. [124] The response spectrum of the device with and without Au bottom mirror is shown in Figure 9d. A 3-fold external responsivity Rext enhancement as compared with no Au bottom mirror. The above result shows the integration of cavity with two-dimensional materials can achieve stronger and selective light absorption of 2D materials, so this method is possible to be applied to specific wavelength detection. reproduced from Ref. [123] with permission from the American Chemical Society. (cd) are reproduced from Ref. [124] with permission from the American Chemical Society. 38"},{"title":"Plasmonic structure","content":"In order to further enhance the photoresponse of the photodetector and obtain a tunable response spectrum, plasmonic nanostructures can also be applied to the infrared photodetector based on two-dimensional materials. [125][126][127][128][129][130][131][132][133] response spectrum in the range of visible to near-infrared and explained the reasons for photocurrent enhancement. [131] The schematic of photodetector and optical image are shown in Figure 10a and b. The Fano resonance plasma clusters (dimers, heptamers, and nonamers) sandwiched between two layers of graphene are respectively located in the 1, 3 and 5 regions, while the 2 and 4 regions have no plasmonic structure ( Figure 10b). The Raman image in Figure 10b shows that the plasmonic antenna clusters of heptamers and nonamers significantly enhance Raman intensity due to resonance under 785 nm laser illumination. The photocurrent response in different regions in Figure 10c shows that the metal plasmonic antenna clusters can effectively enhance the photocurrent. As heptamers can provide larger near-field enhancement and higher 39 thermal electron yield, the enhancement effect of heptamers is significantly better than that of the dimer. In addition, Figure 10d shows that the photocurrent enhancement of the dimer plasmonic clusters is related to the polarization of the incident light,\nindicating that this structure has potential in the field of infrared polarization detection. Figure 10g shows the experimentally measured photocurrent as a function of different resonant wavelength (i.e. different heptamer diameters). It can be clearly seen that the photocurrent consists of plasmon-induced hot electrons and photogenerated carriers of plasmon-enhanced direct carrier excitation. The mechanism of photocurrent enhancement by plasmonic nanostructures is as follows. When the incident light resonates with the transmission window of the plasmonic clusters, a near-field enhancement effect occurs. [134] The light absorption of graphene is effectively enhanced and more electron-hole pairs are generated by direct excitation (DE). Moreover, hot 40 electrons (HE) generated in metal nanostructures can transfer into the conduction band of graphene over the Schottky barrier between metal and graphene. [135] Both plasmoninduced hot electrons and directly excited carriers can be driven by the source-drain bias into the circuit to from photocurrent. Because the contribution of absorption relative to scattering decreases as the size of the nanoparticles increases, the contribution of HE to photocurrent tends to saturate, while the contribution of DE plays a greater role with increasing the diameter of the heptamer. [131] Since hot electrons are generated by plasmon decay, the response spectra of photodetectors based on two-dimensional materials and metal plasmonic nanostructures are not limited by the bandgap of two-dimensional materials. [129,130] For example, bilayer MoS2 has an indirect bandgap of 1.65 eV, corresponding to a wavelength of 750 nm. [136] The hot electron-based bilayer MoS2 photodetector has a high responsivity of 5.2 A/W at 1,070 nm ( Figure 10h). [129] MoS2 field-effect phototransistors were also found to have negative infrared photoresponse, which is caused by bolometric effect, and exhibited responsivity of 2.3 A/W at 980 nm. [137] The above results show that plasma-induced hot electrons can achieve effective light detection with below bandgap photon illumination.\nNi et al. reported a photodetector based on plasmonic Si QD/graphene heterostructure that enables effective detection from UV to mid-infrared. [138] B-doped\nSi QDs not only enhance the light absorption of graphene in the infrared region due to surface plasmon resonance but also form photogating effect, which greatly improves the performance of the detector. Thus, the device exhibits ultrahigh gain of ~10 12 and 41 responsivity of ~10 9 A/W. is reproduced from Ref. [129] with permission from the American Chemical Society. Infrared photodetectors with hybrid structure based on two-dimensional materials still face many challenges and need us to explore further."},{"title":"Figure 10","content":""},{"title":"1) Low-cost, large-scale preparation","content":"Many of the currently reported hybrid structures based on two-dimensional materials are prepared just only through mechanical exfoliation and targeted transfer techniques, leading to poor reproducibility and relatively low yield of photodetectors.\nThis method can only meet the needs of scientific research and cannot be used for commercial applications. Although the 2D vertical or lateral heterostructure can be prepared by the CVD method and 2D-QDs structure can be prepared by the solutionprocessed method, which shows the potential of large-scale production, but they still have many limitations and need further improve."},{"title":"2) Device performance optimization","content":"The reported overall performance parameters of infrared photodetectors with hybrid structure based on two-dimensional materials are excellent, but they are still far from the theoretical value. The excellent optical and electrical properties of twodimensional materials have not been fully exploited. In order to obtain better detection 49 performance, it is necessary to further study the interfacial charge transfer and photocurrent generation mechanism in hybrid structure based on two-dimensional materials for designing or optimizing various hybrid structures."},{"title":"3) Research on novel and suitable two-dimensional materials","content":"There are various currently limitations for two-dimensional materials used in the field of infrared detection. For example, the band gap of graphene and many TMDs (such as MoS2, WSe2) are not conducive to the absorption of infrared light. The instability of MoTe2 and BP with an appropriately bandgap in the air also limits their use. In order to obtain better infrared detection performance, novel two-dimensional materials with suitable band gap and good stability have to be developed."},{"title":"4) Design of novel heterostructure with 2D materials","content":"Many existing heterostructures based on two-dimensional materials simply combine two materials that appear to be suitable. However, this combination may not be optimal, which may leads to the inability to maximize the advantages of heterostructure. It is a worthwhile research direction to further understand the properties of various two-dimensional materials through theoretical research and computational simulation methods, and to design a more rational and ingenious novel heterostructure.\nIt has the potential to dramatically improve the performance of the photodetector and even discover new phenomena."},{"title":"5) Interface research of heterostructures","content":"Two-dimensional materials are particularly sensitive to interface problem due to their large specific surface area and atomic thickness. On the one hand, interface 50 pollution and the introduction of defects have a huge impact on the physical properties of two-dimensional materials. Therefore, how to obtain a clean and defect-free interface in the preparation process of heterostructure based on two-dimensional materials has always been a research hotspot. On the other hand, the study of charge transfer and interaction at the interface of two materials also helps to clarify the working mechanism of the photodetector and to improve performance of photodetector after the formation of the heterostructure."},{"title":"Declaration of conflicts of Interest","content":"The authors declare no conflict of interest."},{"title":"Figure 1","content":""},{"title":"Figure 2","content":""},{"title":"Figure 3","content":""},{"title":"Figure 4","content":""},{"title":"Figure 5","content":""},{"title":"Figure 6","content":""},{"title":"Figure 7d 31 Figure 7","content":""},{"title":"Figure 8c","content":""},{"title":"Figure 8","content":""},{"title":"Furchi","content":""},{"title":"Figure 9","content":""},{"title":"Figure","content":""},{"title":"Table 1 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Infrared (IR) photodetectors can convert infrared light signals into easily detectable electrical signals, which are widely used in imaging, communications, medical and night vision applications at present. [1,2] However, in order to obtain better detecting performance, the traditional infrared photodetectors based on the III-V or II-VI 3 semiconductors have to work at low temperatures to reduce dark currents. [3] Because of the necessity of large refrigeration equipment to realize low operating temperature, it leads to the greatly increase of the manufacturing cost and limitation of the application, making it difficult to achieve miniaturization, flexibility and portability.\\nIn order to circumvent this disadvantage, new type semiconductors are proposed to use in the application of IR photodetectors, which can achieve low current at room temperature and high photocurrent. In recent years, two-dimensional materials with an atomic layer thickness, are verified to overcome the shortcomings induced by traditional semiconductors used in traditional IR detection due to their excellent electrical, optical and mechanical properties. [4][5][6][7][8][9][10][11][12][13] For example, graphene is the most studied two-dimensional materials with semi-metal, high mobility, broad spectrum response, flexibility, and tunable Fermi level properties, [14,15] suggesting that graphene can be fabricated into flexible infrared photodetectors with broad spectrum absorption, ultra-fast response. [16,17] In addition, transition metal dichalcogenides materials MX2 (M: transition metal atoms, X=S, Se, Te), which have the properties of strong light interactions and tunable bandgap varying with the number of layers, are also suitable as photosensitive materials for infrared photodetectors. Black phosphorus shows a direct bandgap property with the range of 0.3-1.3 eV, leading to its absorption spectrum covering the near-infrared, mid-infrared and far-infrared regions, even extending to the terahertz region. [18] Especially, wide spectral response range coupled with its unique anisotropic crystal structure, promises the potential of black phosphorus for infrared polarized photodetectors. [19] Although 2D materials have various excellent and unique 4 properties for the applications of photodetector, they cannot avoid such a problem that such thin thickness cannot absorb more light. For example, the thickness of monolayer graphene is 0.35 nm and absorbs only 2.3% of light. [20] Although the bandgap of monolayer TMDs is a direct bandgap, the whole absorption is also less than 10%. [21] In order to solve this problem, it is necessary to combine two-dimensional materials with other strong light absorbing materials, or to integrate with some structures that can enhance light absorption. For example, the band gap of quantum dots and nanoparticles with strong light absorption can be tuned by changing its size, [22,23] which is very suitable as a photosensitive layer for infrared photodetectors. Carbon nanotubes have excellent broadband absorption characteristics, [24] which is also very suitable for forming heterostructures with two-dimensional materials. Integrating two-dimensional materials with waveguides or cavities can enhance the photon absorption of 2D materials to improve detection performance. In addition, this integration also reveals the compatibility of two-dimensional materials with existing optoelectronic technologies. The surface plasmon resonance effect not only enhances light absorption but also selectively absorbs light by designing different shapes and sizes, [25] therefore the spectral response of these designed photodetectors can be extended to infrared or terahertz region that cannot be absorbed by intrinsic two-dimensional material, due to the surface plasmon resonance effect.\\nIn this review, we review the development of infrared photodetectors with mixeddimensional structure between two-dimensional materials and other low-dimensional materials, focusing on quantum dots, single-wall carbon nanotubes, and 2D vertical or 5 lateral heterostructures composed of different two-dimensional materials. Moreover, we also discuss the integration of two-dimensional materials and on-chip structures, including waveguides and cavities. We also introduce the application of surface plasmon resonance in infrared photodetectors based on two-dimensional materials.\\nFinally, we summarize the faced challenges currently in infrared photodetectors with hybrid structures based on two-dimensional materials and point out the future directions for further development.\"\n },\n {\n \"title\": \"Heterostuctures\",\n \"content\": \"\"\n },\n {\n \"title\": \"Quantum dots on 2D materials\",\n \"content\": \"In order to meet the demand for high-speed response of infrared photodetectors, the materials used to fabricate the detectors is often required to have high carrier mobility to reduce the transit time of photogenerated carriers in conducting channel.\\nThe carrier mobility of graphene is as high as 60,000 cm 2 V -1 s -1 , [26] which can fully meet this demand. However, the optical response of photodetectors based on intrinsic graphene is only a few tens of mA/W, [27,28] which is caused by weak light absorption due to its atomic-scale thickness. Compared with graphene, monolayer molybdenum disulfide (MoS2) has a direct band gap, so the absorption is stronger than that of graphene, but still not enough. Moreover, the band gap of molybdenum disulfide is 1.8 eV, and the corresponding absorption wavelength range is in the visible light range, leading to the limitation of directly fabrication for an infrared photodetector. [21,29] The problems of weak light absorption caused by the ultra-thin thickness and the 6 unsatisfactory photoresponse range due to broad band gap, have seriously hindered the development of infrared photodetectors based on two-dimensional materials. To solve this problem, scientific researchers have tried many methods. It is a good attempt to combine two-dimensional materials with quantum dots. [16,[30][31][32][33][34][35][36][37][38][39] Because quantum dots have stronger light absorption relative to 2D materials, and their absorption wavelength range can be tuned by changing its size, [22,40] and it can efficiently and selectively absorb photons and convert them into photogenerated electron-hole pairs. Therefore, the hybrid 2D-QDs structure is promising in the field of infrared detection technology.\"\n },\n {\n \"title\": \"Konstantatos et al. demonstrated a hybrid graphene-PbS quantum dots\",\n \"content\": \"phototransistor with ultrahigh gain of 10 8 and responsivity of 510 7 AW -1 . [32] The schematic diagram of a hybrid graphene-PbS quantum dots phototransistor is shown in Figure 1a. Monolayer or bilayer graphene was exfoliated with tape on a Si/SiO2 substrate, and then an 80-nm PbS quantum dots film was prepared by low-cost spin casting method. PbS QDs thin film acts as light-sensitive material due to the characteristics of strong and selective light absorption, while graphene acts as a conduction channel owing to its extremely high carrier mobility. The energy band diagram at the interface of graphene and PbS QDs is shown in Figure 1b. On account of the work function mismatch between graphene and PbS QDs, the band bending occurs at the interface. PbS QDs thin film absorbs light to create photogenerated electron-hole pairs, which are then separated by an built-in electric field at the interface that induced by band bending. The Dirac point (VD) will be drifted to higher backgate voltage due to illumination. For VBG<VD, holes are transferred from PbS QDs to 7 graphene channel, and electrons are still trapped in the PbS QDs. The resistance of graphene decreases and carrier transport in the channel is hole-dominated. For VBG>VD, holes are transferred to graphene and recombined with electrons induced by the backgate, so the resistance of graphene increases and carrier transport in the channel is electron-dominated.\\nAfter the photogenerated holes are transferred to the graphene channel, they are driven by the source-drain voltage (VSD) to reach the drain. The linear dependence of the photocurrent on VSD is shown in Figure 1c. In order to achieve charge conservation in graphene, holes are continuously replenished from the source. Due to the long charge trapping time of the PbS QDs and the high mobility of the graphene, [26,41] holes in the channel can circulate multiple times during quantum dots trapped photogenerated electrons, resulting in ultrahigh photoconductive gain. The relationship between the responsivity and the applied back-gate voltage is shown in Figure 1d, which shows that the photoconductive gain of the photodetector can be adjusted by VBG. For VBG<VD, the measured responsivity can reach up to 510 7 A/W at VBG=-20 V; when VBG=4 V, the responsivity reduces to zero. This back-gate-tunable gain is significant so that the required gain can be adjusted based on the photosignal intensity that needs to be reproduced from Ref. [32] with permission from the Nature Publishing Group.\\nAnother important two-dimensional material, transition metal dichalcogenides 9 (TMDs), has also been extensively studied in recent years. [43][44][45][46] However, due to the bandgap limitation and ultrathin thickness, the response spectral range and responsivity of TMDs-based infrared photodetectors are limited. Similarly, in order to enhance light absorption and broaden the response spectral range, a strategy for integrating TMDs materials with quantum dots has also been exploited. [31,35,36] Huo et al. demonstrated\\nMoS2-HgTe QDs hybrid photodetectors beyond 2m. [31] The infrared detection capability of this detector is mainly caused by HgTe QDs thin film, which acts as a sensitizing layer, and the response spectrum range can be further extended to the midwave infrared and long-wave infrared ranges by adjusting the size of HgTe quantum dots. [47,48] A TiO2 buffer layer between the MoS2 transistor channel and the HgTe QDs sensitizing layer was used as a protective layer of the MoS2 channel and as an n-type electron acceptor medium to form an efficient p-n junction with the HgTe QDs at the interface facilitating the charge transfer to MoS2 channel. [31,49] Thus, the dark current can be adjusted by the gate voltage and the specific detectivity D* (~10 12 Jones) at a wavelength of 2 m can be obtained. The detector can operate at room temperature and exhibit sub-milliseconds response and high responsivity of ~10 6 A/W, which demonstrates that the hybrid 2D-QDs structure is an effective method to improve the performance of photodetector based on 2D materials.\\nIn order to obtain high gain to make up for the weak light absorption of 2D materials, many infrared photodetectors based on hybrid 2D-QDs heterostructures are phototransistors that have an electronic passive sensitization layer composed of strong light absorbing material. [32,50,51] Thus, the external quantum efficiency (EQE), response 10 speed and linear dynamic range are limited by thin sensitizing layers thickness, long trapping times, and high density of the sensitizing centers trap-state, respectively. [32,41,52] In order to solve these problems, Nikitskiy et al. proposed a photodetector that integrates an electrically active colloidal QD photodiode with a graphene transistor. [34] The device is composed of a bottom graphene channel, a 300-nm-thick PbS QDs layer and a top ITO electrode. The photodiode operation and phototransistor operation of the hybrid 2D-QDs photodetector are shown in Figure 2a and b, respectively. The graphene-QDs photodiode responsivity and EQE are shown in Figure 2c. With increasing VTD, EQE increased from 10% at VTD=0 V to 75% at VTD=1.2 V (limited by reflection), indicating the importance of VTD for improving EQE. Figure 2d shows the EQE and responsivity in phototransistor operation, agreeing with the dependence of EQETD on VTD in photodiode operation, which demonstrates that the EQE of this integrated configuration is determined by the photodiode. Moreover, the responsivity is 510 6 V/W at VTD=1.2 V, indicating that the phototransistor still has a high gain of (a-f) are reproduced from Ref. [34] with permission from the Nature Publishing Group.\"\n },\n {\n \"title\": \"Nanowires combined with 2D materials\",\n \"content\": \"One-dimensional nanomaterials, such as single-wall carbon nanotubes (SWNTs), have unique and excellent optical and electrical properties due to 1D quantum confinement effect. [53] Because of the type of tube chiralities, metallic or semiconducting properties are exhibited, and the carrier mobility of semiconducting SWNTs can reach up to 10 5 cm 2 V -1 s -1 at room temperature. [54][55][56] SWNTs have broadband and tunable absorption, so it has great potential in the field of infrared detection. However, the photoresponsivity is limited by the small light-absorbing area, and the responsivity of single-tube photovoltaic devices is always <10 -3 A/W. [57] Paulus et al. reported that the junction between a metallic SWNT and graphene can perform effective charge transfer. [58] This is because that the SWNT is able to dope the graphene in a spatially controlled way without affecting its sp 2 lattice structure. [58] Liu et al.\\ndemonstrated a high-performance photodetector based on a planar atomically thin SWNT-graphene hybrid film across visible to near-infrared range (400-1,550 nm). [50] Schematic of the photodetector and atomic force microscope (AFM) characterization of SWNT-graphene hybrid film are shown in Figure 3a. SWNTs have a diameter in the range of 1.0-1.6 nm, and the S11 and S22 bands are found to be located at ~1,800 and ~1,000 nm with higher SWNTs loadings. The photogenerated electron-hole pairs are generated at the SWNT/ graphene interface under optical illumination, and then 13 separated by the built-in electric field. Electrons are transferred from SWNTs to graphene, while holes are trapped in the SWNTs, forming a photogating effect. Figure 3b shows that the source-drain current (ISD) as a function of the applied back-gate voltage (VG). The Dirac point shifts toward the lower VG as the light increases,\\nindicating that the graphene resistance can be effectively modulated by the photogating effect. Unlike other photodetectors that use the photogating effect to increase the responsivity, which generally has slow response speed, the photodetector based on hybrid SWNTs-graphene exhibits a fast response time of ~100 s ( Figure 3c). This is mainly due to the high mobility of graphene and SWNTs and fast charge transfer at the SWNTs-graphene interface, and the field-effect mobility of electrons and holes are measured as 3,920 and 3,663 cm 2 V -1 s -1 , respectively. The transit time of the hybrid SWNTs-graphene photodetector is estimated to be ~ 10 -9 s, so the photoconductive gain is ~ 10 5 and the gain-bandwidth product is ~10 9 Hz. Moreover, since SWNTs act as a sensitizing layer, the detector also exhibits the characteristics of a broadband photoresponse. The relationship between the responsivity and the incident optical power at different illumination wavelengths in the range of visible to near-infrared range (405, 532, 650, 980 and 1,550 nm) is shown in Figure 3d, which shows a high responsivity of >100 A/W. Further, Liu et al. fabricated a photodetector based on graphene/carbon nanotube hybrid films on a PET substrate and measured the performance under bending conditions. [59] It was found that this device has good robustness against repetitive bending, which fully demonstrated the huge application potential of this heterostructure in the field of flexible electronics. 14 Jariwala et al. demonstrated a gate-tunable carbon nanotube-MoS2 heterjunction pn diode. [60] Since both monolayer MoS2 and semiconducting SWNTs have direct bandgaps and tightly bound excitation states in absorption spectra, [61] the photodiode based on hybrid MoS2-SWNTs also exhibits excellent detection performance. The external quantum efficiency (EQE) and the response time are ~25% and <15 s, respectively. Figure 3e shows that photocurrent spectral response can be tuned by changing the gate voltage. The photocurrent in the near-infrared region continuously decreases as the gate voltage decreases. Furthermore, the performance of devices based on 1D-2D heterostructures can be further optimized by strain modulation. For example, the responsivity of the photodetector based on the graphene-ZnO nanowire heterostructure is improved by 26% under 0.44% tensile strain on the ZnO nanowire. [62] The above results show that it is significant to carry out the research of the hybrid 1D-2D heterostructure in the field of infrared detection.\\n15 VD=-10 V. (a-d) are reproduced from Ref. [50] with permission from the Nature 16 Publishing Group. (e) are reproduced from Ref. [60] with permission from the National Academy of Sciences.\"\n },\n {\n \"title\": \"2D vertical and lateral heterostructures 2D vertical heterostructures\",\n \"content\": \"Two-dimensional vertical heterostructure is fabricated by stacking different twodimensional materials in the vertical direction. This heterostructure is generally prepared by targeted transfer method or CVD growth technique. [63][64][65][66][67][68] 2D vertical heterostructures have many advantages, such as atomically sharp interface, no atomic diffusion between layers, and no strict lattice matching problems due to weak van der Waals interaction between 2D materials. Moreover, the strong interlayer coupling and ultrafast transfer of charge carriers make it have great potential in the field of optoelectronics and electronic devices. [68][69][70][71][72][73] In order to achieve infrared detecting, photodetectors must have a broad spectral response. However, the development of TMDs for infrared detection is hindered due to the lack of 2D materials with narrow bandgap. Monolayer -MoTe2 has a band gap of 1.1 eV and can absorb light in the near-infrared range. [74,75] The response spectral of the photodetectors based on the -MoTe2/MoS2 dichalcogenide heterojunction covers in the range from the visible to near-infrared (800 nm). [76] However, this is not enough for mid-infrared and far-infrared detection, and the response range of the photodetector should be further extended. Graphene can absorb light indiscriminately due to the gapless band structure and monolayer black phosphorene shows a direct optical band gap of 1.3 eV, [77,78] so they are effective absorption material in the infrared region.\"\n },\n {\n \"title\": \"Long et al. demonstrated a broadband photodetector based on MoS2-graphene-\",\n \"content\": \"WSe2 heterostructure covering visible to short-wavelength infrared range at room temperature. [79] The schematic diagram and optical image of the photodetector based on a MoS2-graphene-WSe2 heterostructure are shown in Figure 4a. The p-doped WSe2 is induced by large work-function metal Pd, whereas MoS2 flakes remain n-type due to\\nFermi level pinning, [80] and thus forming a p-n junction with a strong internal electric field of ~210 8 V/m. Photogenerated electron-hole pairs can be effectively separated in the presence of the internal electric field. Therefore, the dark current of the device is effectively suppressed and the specific detectivity increases accordingly. Figure 4b shows the responsivity R and specific detectivity D* of the photodetector for wavelengths ranging from 400 to 2,400 nm with an interval of 100 nm. R and D* are as high as 10 4 A/W and 10 15 Jones in the visible region, while they decrease sharply to 100 mA/W and 10 9 Jones in the infrared wavelength of 2400 nm, respectively. Figure 4c explains the reason for the sharp changes in responsivity and specific detectivity.\\nThe band gap of MoS2 (Eg1) and WSe2 (Eg2) are 1.88 eV and 1.65 eV, respectively, and the corresponding wavelengths are 660 nm and 750 nm. [21,29,81] When the incident photon energy h exceeds the bandgap of the material, the material can absorb photons to generate electron-hole pairs. Therefore, for h >Eg1>Eg2, MoS2, graphene and WSe2 all can generate photoelectron electron-hole pairs, resulting in a higher responsivity.\\nFor h <Eg1<Eg2, only graphene can generate photoelectron electron-hole pairs, and thus the photocurrent is drastically reduced. Qiao et al. reported a broadband 18 photodetectors based on graphene-Bi2Te3 heterostructure (Figure 4d). [82] Bi2Te3 has only 2.7% lattice mismatch with graphene, so the graphene-Bi2Te3 heterostructure can be directly prepared on graphene by Van der Waals epitaxial growth method. [83] Figure 4e plots photocurrent of devices based on pure graphene and graphene-Bi2Te3 heterostructure as a function of source-drain voltage without applied gate bias, which shows that the latter is ten times higher than the former. Both graphene and Bi2Te3 absorb light to generate photoelectron-hole pairs, which are subsequently separated by a built-in electric field. Then, photogenerated electrons in graphene are transferred into Bi2Te3, while the photogenerated holes in Bi2Te3 are transferred into graphene, which suppresses the recombination of carriers and generates a larger photocurrent. Figure 4f shows are reproduced from Ref. [79] with permission from the American Chemical Society. 20 (d-f) are reproduced from Ref. [82] with permission from the American Chemical Society.\\nSo far, highly polarization sensitive infrared photodetector based on twodimensional materials working at room temperature are still lacking. Ye et al. first demonstrated a broadband photodetector based on a vertical photogate heterostructure of BP-on-WSe2 with highly polarization sensitive for infrared detection (Figure 5a). [85] WSe2 acts as a conductive channel, while BP with a small direct bandgap and highly anisotropic crystal structure acts as a photogate. [18,19,77] Figure 5b shows the photoconductive gain G and detectivity D* of the photodetector for wavelengths ranging from 400 to 1,600 nm. The detectivity in the visible and infrared regions are 10 14 and 10 10 Jones, and the gain in the visible 21 and infrared regions are 10 6 and 10 2 , respectively, demonstrating excellent broadband detection performance. The optical image of the BP-on-WSe2 photodetector and the relationship between the photocurrent image of the photodetector at 1,550 nm and the polarization angle are shown in Figure 5c. The advantage of this device structure is that it can eliminate the disturbance of two-fold polarization-dependent photocurrent resulted from the geometric edge effect at the metal-BP edge and fully collect the photogenerated free carriers isotropically for highly polarization-sensitive infrared detection. [85] As can be seen from the photocurrent image, the polarization photoresponse originates from the overlap of BP and WSe2, and it changes obviously with the polarization angle changes. The polar plot in Figure 5d shows . [85] The ratio of the maximum responsivity to the minimum responsivity is about 6, showing high polarization sensitive infrared detection performance.\\nSimilarly, the mid-wave infrared photodetector based on BP and MoS2 heterostructure designed by Bullock et al. also utilized the anisotropy of BP to achieve polarization detection, which is bias-selectable and operates without the need for external optics. [86] The device exhibits high EQE (35%) and D* (1.110 10 cm Hz 1/2 W -1 ) at room temperature.\\nThe above results show that the 2D vertical heterostructure can effectively extend 22 the response range of the photodetector to achieve infrared detection. In addition, CVD epitaxial growth technology provides a guarantee for the scaled preparation of vertical heterostructures. By combining 2D materials with different unique and excellent physical properties, a multifunctional infrared detector with excellent detection performance can be prepared. heterostructures generally include chemical doping, [87,88] electrostatic doping, [89][90][91] and one-pot or two-step chemical vapor deposition (CVD) growth methods. [64,65,[92][93][94][95][96] Recently, Yu et al. demonstrated a lateral black phosphorene P-N junction formed via chemical doping for high performance near-infrared photodetector. [88] The n-type doping of few-layer black phosphorene is achieved by using benzyl viologen (BV) as an electron dopant, which can obtain a high electron concentration while still maintain the high carrier mobility of BP as the processes do not induce defects in the crystal lattices, and Al2O3 is used to selectively protect part of the area of a p-type BP at the 24 same time. [88] The electron doping concentration of BP can be adjusted by doping time, and the built-in electric field at the P-N junction can effectively separate photogenerated electron-hole pairs and reduce recombination efficiency. Photocurrent tests show that the off-current under 1.47 m illumination is 4 orders of magnitude higher than in dark condition, demonstrating that the photocurrent dominates over thermionic and tunneling currents in the photodiode operation. [88] The photodetector exhibits high responsivity of ~180 mA/W, fast response of tens of milliseconds and high detectivity in the range of 10 11 -10 13 Jones under infrared light illumination (=1.47 m).\\nHowever, the current methods for preparing lateral heterojunctions have their own limitations. Chemical doping has the characteristic that it is difficult to find suitable dopants. The density of the carrier modulated by the electrostatic gating is relatively low, so the modulation effect is limited. [19,97] CVD technology is an effective method for large-scale preparation of lateral heterostructures, but the main focus is on epitaxial growth of TMDs with large band gap at present, and thus CVD technology has limitation in the field of infrared detection. The infrared light detection technology based on 2D lateral heterostructure needs further research and development.\"\n },\n {\n \"title\": \"On-chip structure\",\n \"content\": \"\"\n },\n {\n \"title\": \"2D materials on waveguide\",\n \"content\": \"25 Two-dimensional materials show strong interaction with light, but the spectral response range and responsivity of the photodetector based on 2D materials are limited due to the reason of the atomically ultra-thin thickness and the limitation of the band gap. In order to obtain better infrared detection performance to achieve practical applications, it is necessary to enhance the infrared light absorption of two-dimensional materials. Integrating two-dimensional materials with waveguides is a good strategy to enhance light absorption. [91,[98][99][100][101][102][103][104] The propagation of light in the waveguide is accompanied by the evanescent field, which can be absorbed by the two-dimensional materials to generate photogenerated electron-hole pairs. [105][106][107][108] Different with the incident light perpendicularly incident on the two-dimensional materials, the in-plane absorption of two-dimensional materials makes its nanoscale thickness to be no longer a major limiting factor for light absorption. The length of the two-dimensional material in contact with the waveguide is on the order of micrometers, so the length of the interaction with the light is also on the order of micrometers and therefore can effectively enhance light absorption.\\nRecently, a series of similar but independent waveguide integrated graphene photodetectors have been reported, [98,101,102] The absorption coefficient of metal electrode M increases with increasing electrode 26 width W. For W>100 nm (W>160 nm), the light absorption of the metal electrode will exceed the absorption of monolayer (bilayer) graphene. However, the contact resistance between metal and graphene also increases with decreasing width of electrode, so the metal electrode absorption coefficient and contact resistance should be considered comprehensively to obtain the best device performance. Figure 6b shows (c-d) are reproduced from Ref. [98] with permission from the Nature Publishing Group.\\n(e-f) are reproduced from Ref. [102] with permission from the Nature Publishing Group.\"\n },\n {\n \"title\": \"Similarly, Youngblood et al. demonstrated a waveguide-integrated few-layer black\",\n \"content\": \"phosphorus photodetector with high responsivity and low dark current. [103] Figure 7a shows the structure of the photodetector, having a graphene top-gate, a BP channel and the waveguide patterned on the silicon-on-insulator substrate. Since the incident light propagates along the waveguide, the optical interaction length is equal to the width of the black phosphorus covering on the waveguide (6.5 m). The device can absorb 78.7%\\nof the optical power in the waveguide, of which only 17.5% is absorbed by the black phosphorus and the rest is absorbed by the graphene top-gate. [103] Figure 7c shows that the photocurrent versus the gate voltage VG and the bias voltage VDS. VG can electrostatically tune the hole doping concentration of black phosphorus and the BP is nearly intrinsic at VG=-8 V. When black phosphorus is in a low doping state (-10 V < VG < -1 V), the photocurrent is larger and its sign is positive relative to VDS, which suggests photocurrent generation mechanism is photovoltaic effect at this time. [109][110][111] 30\\nFor VG > 0 V, BP is tuned to be more heavily n-doped, the photocurrent is getting smaller and its sign changes to negative relative to VDS, and the photocurrent generation mechanism is bolometric effect at this time. [109,112] The frequency response of the photocurrent shown in Figure 7b can also demonstrate the photocurrent generation mechanism at different gate voltages. The roll-off frequency is measured to be 2. light source and photodetector based on a p-n junction of bilayer MoTe2. [91] Figure 8a shows the device schematic, and Figure 8b shows LED mode and photodetector mode of device, respectively. The MoTe2 p-n junction is composed of p-type and n-type 32 doping that are electrostatically induced by left and right graphite top gates above the hBN dielectric layer. The bilayer MoTe2 not only has stronger excitation and longer emission wavelength than the monolayer MoTe2, but also has excitation caused by direct bandgap optical transitions similar to that of the monolayer. [74,114,115] There is a grating coupler for excitation and collection at the far end of the waveguide. Excitation Figure 8d shows that the PhC transmission peak is at 1,160 nm, but this peak disappears after MoTe2 transfer, which is mainly due to the interference of MoTe2 absorption. The normalized EL emission spectrum collected at the grating coupler represented by the blue line in Figure 8d, showing a narrow peak centred at 1,160 nm on top of a broader peak centred at 1,175 nm. [91] The former coincides with the PhC transmission peak, and the latter matches the free-space emission of the bilayer MoTe2.\\nThe reason that the PhC transmission peak can be observed at the blue line is that the excited states of MoTe2 have been filled by the electrically excited carriers in the LED mode, so the absorption of MoTe2 will not cause interference. When the device is 33 operating in photodetector mode, the responsivity and external quantum efficiency of the photodetector at the wavelength of 1,160 nm are 4.8 mA/W and 0.5%, respectively. Figure 8e shows the temporal response in photodetector mode. The photocurrent response bandwidth is 200 MHz (limited by experimental set-up), which is much higher than other MoTe2 photodetectors based on photogating and photoconductive effect, [116,117] indicating the fast response of waveguide-integrated photodetector with bilayer MoTe2 p-n junction. As can be seen from Figure reproduced from Ref. [91] with permission from the Nature Publishing Group.\"\n },\n {\n \"title\": \"2D materials on cavity\",\n \"content\": \"The integration of two-dimensional materials and cavities, including photonic crystal cavities, [118,119] optical microcavities [120,121] and optical ring resonators, [122] is also an effective method for enhancing the light absorption of 2D materials. The incident light at the resonant frequency of the cavity is absorbed many times by the atomically thin two-dimensional material in the cavity, thereby enhancing absorption. Schottky photodetector operating at communication band (1,550 nm). [124] Figure 9c shows the resonant cavity enhanced graphene/Si Schottky photodetector and the resonant structure consists of a λ/2 Si slab layer confined between SLG/Si top and Au bottom mirrors. [124] The response spectrum of the device with and without Au bottom mirror is shown in Figure 9d. A 3-fold external responsivity Rext enhancement as compared with no Au bottom mirror. The above result shows the integration of cavity with two-dimensional materials can achieve stronger and selective light absorption of 2D materials, so this method is possible to be applied to specific wavelength detection. reproduced from Ref. [123] with permission from the American Chemical Society. (cd) are reproduced from Ref. [124] with permission from the American Chemical Society. 38\"\n },\n {\n \"title\": \"Plasmonic structure\",\n \"content\": \"In order to further enhance the photoresponse of the photodetector and obtain a tunable response spectrum, plasmonic nanostructures can also be applied to the infrared photodetector based on two-dimensional materials. [125][126][127][128][129][130][131][132][133] response spectrum in the range of visible to near-infrared and explained the reasons for photocurrent enhancement. [131] The schematic of photodetector and optical image are shown in Figure 10a and b. The Fano resonance plasma clusters (dimers, heptamers, and nonamers) sandwiched between two layers of graphene are respectively located in the 1, 3 and 5 regions, while the 2 and 4 regions have no plasmonic structure ( Figure 10b). The Raman image in Figure 10b shows that the plasmonic antenna clusters of heptamers and nonamers significantly enhance Raman intensity due to resonance under 785 nm laser illumination. The photocurrent response in different regions in Figure 10c shows that the metal plasmonic antenna clusters can effectively enhance the photocurrent. As heptamers can provide larger near-field enhancement and higher 39 thermal electron yield, the enhancement effect of heptamers is significantly better than that of the dimer. In addition, Figure 10d shows that the photocurrent enhancement of the dimer plasmonic clusters is related to the polarization of the incident light,\\nindicating that this structure has potential in the field of infrared polarization detection. Figure 10g shows the experimentally measured photocurrent as a function of different resonant wavelength (i.e. different heptamer diameters). It can be clearly seen that the photocurrent consists of plasmon-induced hot electrons and photogenerated carriers of plasmon-enhanced direct carrier excitation. The mechanism of photocurrent enhancement by plasmonic nanostructures is as follows. When the incident light resonates with the transmission window of the plasmonic clusters, a near-field enhancement effect occurs. [134] The light absorption of graphene is effectively enhanced and more electron-hole pairs are generated by direct excitation (DE). Moreover, hot 40 electrons (HE) generated in metal nanostructures can transfer into the conduction band of graphene over the Schottky barrier between metal and graphene. [135] Both plasmoninduced hot electrons and directly excited carriers can be driven by the source-drain bias into the circuit to from photocurrent. Because the contribution of absorption relative to scattering decreases as the size of the nanoparticles increases, the contribution of HE to photocurrent tends to saturate, while the contribution of DE plays a greater role with increasing the diameter of the heptamer. [131] Since hot electrons are generated by plasmon decay, the response spectra of photodetectors based on two-dimensional materials and metal plasmonic nanostructures are not limited by the bandgap of two-dimensional materials. [129,130] For example, bilayer MoS2 has an indirect bandgap of 1.65 eV, corresponding to a wavelength of 750 nm. [136] The hot electron-based bilayer MoS2 photodetector has a high responsivity of 5.2 A/W at 1,070 nm ( Figure 10h). [129] MoS2 field-effect phototransistors were also found to have negative infrared photoresponse, which is caused by bolometric effect, and exhibited responsivity of 2.3 A/W at 980 nm. [137] The above results show that plasma-induced hot electrons can achieve effective light detection with below bandgap photon illumination.\\nNi et al. reported a photodetector based on plasmonic Si QD/graphene heterostructure that enables effective detection from UV to mid-infrared. [138] B-doped\\nSi QDs not only enhance the light absorption of graphene in the infrared region due to surface plasmon resonance but also form photogating effect, which greatly improves the performance of the detector. Thus, the device exhibits ultrahigh gain of ~10 12 and 41 responsivity of ~10 9 A/W. is reproduced from Ref. [129] with permission from the American Chemical Society. Infrared photodetectors with hybrid structure based on two-dimensional materials still face many challenges and need us to explore further.\"\n },\n {\n \"title\": \"Figure 10\",\n \"content\": \"\"\n },\n {\n \"title\": \"1) Low-cost, large-scale preparation\",\n \"content\": \"Many of the currently reported hybrid structures based on two-dimensional materials are prepared just only through mechanical exfoliation and targeted transfer techniques, leading to poor reproducibility and relatively low yield of photodetectors.\\nThis method can only meet the needs of scientific research and cannot be used for commercial applications. Although the 2D vertical or lateral heterostructure can be prepared by the CVD method and 2D-QDs structure can be prepared by the solutionprocessed method, which shows the potential of large-scale production, but they still have many limitations and need further improve.\"\n },\n {\n \"title\": \"2) Device performance optimization\",\n \"content\": \"The reported overall performance parameters of infrared photodetectors with hybrid structure based on two-dimensional materials are excellent, but they are still far from the theoretical value. The excellent optical and electrical properties of twodimensional materials have not been fully exploited. In order to obtain better detection 49 performance, it is necessary to further study the interfacial charge transfer and photocurrent generation mechanism in hybrid structure based on two-dimensional materials for designing or optimizing various hybrid structures.\"\n },\n {\n \"title\": \"3) Research on novel and suitable two-dimensional materials\",\n \"content\": \"There are various currently limitations for two-dimensional materials used in the field of infrared detection. For example, the band gap of graphene and many TMDs (such as MoS2, WSe2) are not conducive to the absorption of infrared light. The instability of MoTe2 and BP with an appropriately bandgap in the air also limits their use. In order to obtain better infrared detection performance, novel two-dimensional materials with suitable band gap and good stability have to be developed.\"\n },\n {\n \"title\": \"4) Design of novel heterostructure with 2D materials\",\n \"content\": \"Many existing heterostructures based on two-dimensional materials simply combine two materials that appear to be suitable. However, this combination may not be optimal, which may leads to the inability to maximize the advantages of heterostructure. It is a worthwhile research direction to further understand the properties of various two-dimensional materials through theoretical research and computational simulation methods, and to design a more rational and ingenious novel heterostructure.\\nIt has the potential to dramatically improve the performance of the photodetector and even discover new phenomena.\"\n },\n {\n \"title\": \"5) Interface research of heterostructures\",\n \"content\": \"Two-dimensional materials are particularly sensitive to interface problem due to their large specific surface area and atomic thickness. On the one hand, interface 50 pollution and the introduction of defects have a huge impact on the physical properties of two-dimensional materials. Therefore, how to obtain a clean and defect-free interface in the preparation process of heterostructure based on two-dimensional materials has always been a research hotspot. On the other hand, the study of charge transfer and interaction at the interface of two materials also helps to clarify the working mechanism of the photodetector and to improve performance of photodetector after the formation of the heterostructure.\"\n },\n {\n \"title\": \"Declaration of conflicts of Interest\",\n \"content\": \"The authors declare no conflict of interest.\"\n },\n {\n \"title\": \"Figure 1\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 3\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7d 31 Figure 7\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8c\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8\",\n \"content\": \"\"\n },\n {\n \"title\": \"Furchi\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 9\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Zheng et al. demonstrated an ambipolar broadband photodetector based on N, S codecorated graphene-PbS quantum dot heterostructure. [42] N, S codecorated graphene greatly reduces the p-type doping of graphene in air. The gate voltage required to modulate the Dirac point and the carrier type is significantly reduced, thereby 8 achieving gate tuneable mutual transformation of positive and negative photoresponse.The device exhibits excellent detection performance with responsivity of 10 4 A/W and specific detectivity of 10 12 Jones at 1550 nm.","content":"Zheng et al. demonstrated an ambipolar broadband photodetector based on N, S codecorated graphene-PbS quantum dot heterostructure. [42] N, S codecorated graphene greatly reduces the p-type doping of graphene in air. The gate voltage required to modulate the Dirac point and the carrier type is significantly reduced, thereby 8 achieving gate tuneable mutual transformation of positive and negative photoresponse.The device exhibits excellent detection performance with responsivity of 10 4 A/W and specific detectivity of 10 12 Jones at 1550 nm."},{"id":"","caption":"(a) The schematic diagram of a hybrid graphene-PbS quantum dots phototransistor. (b) The energy band diagram at the graphene/PbS QDs interface. (c) The linear dependence of the photocurrent on the applied drain-source voltage for different optical powers (46, 336, 516 pW). VBG=0. Inset: The total current (I=Idark + Iphoto) as a function of the applied drain-source voltage for different optical powers. (d) The responsivity as a function of the applied back-gate voltage VBG. VSD=5 V. (a-d) are","content":"Figure 1\n1(a) The schematic diagram of a hybrid graphene-PbS quantum dots phototransistor. (b) The energy band diagram at the graphene/PbS QDs interface. (c) The linear dependence of the photocurrent on the applied drain-source voltage for different optical powers (46, 336, 516 pW). VBG=0. Inset: The total current (I=Idark + Iphoto) as a function of the applied drain-source voltage for different optical powers. (d) The responsivity as a function of the applied back-gate voltage VBG. VSD=5 V. (a-d) are"},{"id":"","caption":"Photogenerated electron-hole pairs are created at the QDs layer, and then drifted by the built-in electric field at the interface and by the applied bias voltage to form photocurrent directly in photodiode operation. However, after the electron-hole pairs are separated, electrons remain in the QDs layer and holes are driven by depletion region on the graphene/PbS QD interface and transferred to graphene in phototransistor operation. For VTD>0, the width of the depletion region expands, increasing the efficient charge collection area, and thus the external quantum efficiency also increases.","content":"Photogenerated electron-hole pairs are created at the QDs layer, and then drifted by the built-in electric field at the interface and by the applied bias voltage to form photocurrent directly in photodiode operation. However, after the electron-hole pairs are separated, electrons remain in the QDs layer and holes are driven by depletion region on the graphene/PbS QD interface and transferred to graphene in phototransistor operation. For VTD>0, the width of the depletion region expands, increasing the efficient charge collection area, and thus the external quantum efficiency also increases."},{"id":"","caption":"10 5 compared with the photodiode operation. With increasing VTD, the temporal11 response accelerates and the linear dynamic range extends. The electrical 3 dB bandwidth of the detector reaches 1.5 kHz for the optimum VTD, and the linear dynamic range extends from 75 dB at VTD=0 V to 110 dB at VTD=1.2 V. The above results prove that the design and optimization of infrared photodetectors based on hybrid 2D-QDs heterostructures can effectively enhance the detection performance.","content":"10 5 compared with the photodiode operation. With increasing VTD, the temporal11 response accelerates and the linear dynamic range extends. The electrical 3 dB bandwidth of the detector reaches 1.5 kHz for the optimum VTD, and the linear dynamic range extends from 75 dB at VTD=0 V to 110 dB at VTD=1.2 V. The above results prove that the design and optimization of infrared photodetectors based on hybrid 2D-QDs heterostructures can effectively enhance the detection performance."},{"id":"","caption":"(a-b) Schematic of photodiode operation and phototransistor operation of the hybrid graphene-PbS QDs photodetector, respectively. (c) Responsivity and external quantum efficiency EQE of photodetector in photodiode operation as a function of VTD. Inset: energy band diagrams of device with and without VTD bias. (d) Responsivity and EQE of photodetector in phototransistor operation as a function of VTD. (e) Normalized photoresponse versus light modulation frequency. The specified VTD values are shown in the legend. Inset: The linear dependence of extracted 3 dB bandwidth values on applied VTD. (f) Responsivity of the detector as a function of incident irradiance in the range of 10 -5 to 10 Wm -2 , which shows a significant extension 12 of the linear dynamic range with increasing VTD.","content":"Figure 2\n2(a-b) Schematic of photodiode operation and phototransistor operation of the hybrid graphene-PbS QDs photodetector, respectively. (c) Responsivity and external quantum efficiency EQE of photodetector in photodiode operation as a function of VTD. Inset: energy band diagrams of device with and without VTD bias. (d) Responsivity and EQE of photodetector in phototransistor operation as a function of VTD. (e) Normalized photoresponse versus light modulation frequency. The specified VTD values are shown in the legend. Inset: The linear dependence of extracted 3 dB bandwidth values on applied VTD. (f) Responsivity of the detector as a function of incident irradiance in the range of 10 -5 to 10 Wm -2 , which shows a significant extension 12 of the linear dynamic range with increasing VTD."},{"id":"","caption":"(a) Schematic of the photodetector and AFM characterization of SWNTgraphene hybrid film (middle: scale bar, 200 nm; right: scale bar, 20 nm). (b) The ISD-VG transfer curve of device with increasing 650 nm illumination powers. VSD=0.5 V. Inset: the optical image of the SWNT-graphene device (Scale bar, 10m). (c) Temporal response of the SWNT-graphene hybrid photodetector under 650 nm illumination. (d) The dependence of responsivities of photodetector on the optical power at different illumination wavelengths from visible to near-infrared range (405, 532, 650, 980 and 1,550 nm). (e) Photocurrent spectral response can be tuned by varying the gate voltage.","content":"Figure 3\n3(a) Schematic of the photodetector and AFM characterization of SWNTgraphene hybrid film (middle: scale bar, 200 nm; right: scale bar, 20 nm). (b) The ISD-VG transfer curve of device with increasing 650 nm illumination powers. VSD=0.5 V. Inset: the optical image of the SWNT-graphene device (Scale bar, 10m). (c) Temporal response of the SWNT-graphene hybrid photodetector under 650 nm illumination. (d) The dependence of responsivities of photodetector on the optical power at different illumination wavelengths from visible to near-infrared range (405, 532, 650, 980 and 1,550 nm). (e) Photocurrent spectral response can be tuned by varying the gate voltage."},{"id":"","caption":"the dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. Due to the gapless nature of graphene and a small bandgap (0.15-0.3 eV) of Bi-2Te3, [84] both graphene and Bi2Te3 are effective absorbent materials for infrared light, and thus the detector exhibits broadband detection characteristics. The effective detection range can be extended to the near infrared (980 nm) and c-communication band (1,550nm). The responsivity is 35 A/W, 10 A/W and 0.22 A/W in the visible (532 nm), infrared (980 nm) and c-communication band (1,550 nm) regions, respectively.","content":"the dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. Due to the gapless nature of graphene and a small bandgap (0.15-0.3 eV) of Bi-2Te3, [84] both graphene and Bi2Te3 are effective absorbent materials for infrared light, and thus the detector exhibits broadband detection characteristics. The effective detection range can be extended to the near infrared (980 nm) and c-communication band (1,550nm). The responsivity is 35 A/W, 10 A/W and 0.22 A/W in the visible (532 nm), infrared (980 nm) and c-communication band (1,550 nm) regions, respectively."},{"id":"","caption":"(a) Top: optical image of the photodetector based on a MoS2−graphene−WSe2 heterostructure. Scale bar, 5 m. Bottom: Schematic diagram of the device. (b) Responsivity and specific detectivity of the photodetector for wavelengths ranging from 400 to 2,400 nm with an interval of 100 nm. (c) Top: Spectra from ultraviolet to infrared. The band gap of MoS2 (Eg1) and WSe2 (Eg2) as well as corresponding wavelengths are plotted on the graph, respectively. Bottom left: For h >Eg1>Eg2, MoS2, graphene and WSe2 all can generate photogenerated electron-hole pairs. Bottom right: For h <Eg1<Eg2, only graphene can generate photogenerated electron-hole pairs. (d) The schematic diagram of the device based on graphene-Bi2Te3 heterostructure. (e) The dependence of photocurrent of devices based on pure graphene and graphene-Bi2Te3 heterostructure on source-drain voltage VSD. VG=0 V. (f) Dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. (a-c)","content":"Figure 4\n4(a) Top: optical image of the photodetector based on a MoS2−graphene−WSe2 heterostructure. Scale bar, 5 m. Bottom: Schematic diagram of the device. (b) Responsivity and specific detectivity of the photodetector for wavelengths ranging from 400 to 2,400 nm with an interval of 100 nm. (c) Top: Spectra from ultraviolet to infrared. The band gap of MoS2 (Eg1) and WSe2 (Eg2) as well as corresponding wavelengths are plotted on the graph, respectively. Bottom left: For h >Eg1>Eg2, MoS2, graphene and WSe2 all can generate photogenerated electron-hole pairs. Bottom right: For h <Eg1<Eg2, only graphene can generate photogenerated electron-hole pairs. (d) The schematic diagram of the device based on graphene-Bi2Te3 heterostructure. (e) The dependence of photocurrent of devices based on pure graphene and graphene-Bi2Te3 heterostructure on source-drain voltage VSD. VG=0 V. (f) Dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. (a-c)"},{"id":"","caption":"the relationship between the photocurrent and the polarization angle of the incident light. The maximum photocurrent, corresponding responsivity of ~40 mA/W, is taken along the horizontal axis (defined as 0 o polarization), while the minimum photocurrent, corresponding responsivity of ~6.8 mA/W, is taken along the vertical axis (defined as 90 o polarization)","content":"the relationship between the photocurrent and the polarization angle of the incident light. The maximum photocurrent, corresponding responsivity of ~40 mA/W, is taken along the horizontal axis (defined as 0 o polarization), while the minimum photocurrent, corresponding responsivity of ~6.8 mA/W, is taken along the vertical axis (defined as 90 o polarization)"},{"id":"","caption":"(a) Schematic diagram of the photodetector with BP-on-WSe2 photogate structure. (b) The photoconductive gain G and detectivity D* of the photodetector for wavelengths ranging from 400 to 1,600 nm at 1 mW/cm 2 incident light power density. 23 Vds=0.5 V. (c) Top: the photocurrent image of the photodetector under 1,550 nm illumination with different light polarizations (red arrows). Bottom left: the optical image of the photodetector with BP-on-WSe2 photogate structure. (d) The photocurrent as a function of light polarization at 100 mW/cm 2 incident light power density. Vds=0.5 V. (a-d) are reproduced from Ref.[85] with permission from Elsevier. 2D lateral heterostructures The heterostructures mentioned above are basically vertical, and the built-in electric field at the interface promotes the separation of photogenerated electron-hole pairs and charge transfer, thereby reducing the recombination and accelerating the response speed. In addition, two-dimensional lateral heterostructures, a heterostructure composed of materials with different work functions or homojunction composed of the same material but different doping types in the lateral direction, form a built-in electric field at the interface and play a similar role. At present, the methods for forming lateral","content":"Figure 5\n5(a) Schematic diagram of the photodetector with BP-on-WSe2 photogate structure. (b) The photoconductive gain G and detectivity D* of the photodetector for wavelengths ranging from 400 to 1,600 nm at 1 mW/cm 2 incident light power density. 23 Vds=0.5 V. (c) Top: the photocurrent image of the photodetector under 1,550 nm illumination with different light polarizations (red arrows). Bottom left: the optical image of the photodetector with BP-on-WSe2 photogate structure. (d) The photocurrent as a function of light polarization at 100 mW/cm 2 incident light power density. Vds=0.5 V. (a-d) are reproduced from Ref.[85] with permission from Elsevier. 2D lateral heterostructures The heterostructures mentioned above are basically vertical, and the built-in electric field at the interface promotes the separation of photogenerated electron-hole pairs and charge transfer, thereby reducing the recombination and accelerating the response speed. In addition, two-dimensional lateral heterostructures, a heterostructure composed of materials with different work functions or homojunction composed of the same material but different doping types in the lateral direction, form a built-in electric field at the interface and play a similar role. At present, the methods for forming lateral"},{"id":"","caption":"and the device structures are shown in Figures 6a, c and e, respectively. Figure 6a shows a scanning electron microscope (SEM)image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration. It should be noted that in addition to the length L of graphene, the width W of central electrode S in this device also affects the light absorption of the graphene.","content":"and the device structures are shown in Figures 6a, c and e, respectively. Figure 6a shows a scanning electron microscope (SEM)image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration. It should be noted that in addition to the length L of graphene, the width W of central electrode S in this device also affects the light absorption of the graphene."},{"id":"","caption":"the absorption of graphene as a function of graphene length L at different widths, indicating that as long as the appropriate W and L values are obtained, the light absorption of graphene can be greatly enhanced. For example, when W = 100 nm, more than 50% light absorption can be obtained with only 22 m length bilayer graphene. Figure 6d shows an SEM image of a waveguide-integrated graphene photodetector with high responsivity and photocurrent image at zero bias. A 53 m bilayer graphene was overlaid on the waveguide, and two electrodes with waveguide distances of 100 nm and 3.5 m, respectively, were used to collect the photogenerated carriers. The lateral metaldoped junction overlapping the waveguide is formed by doping of metal electrode 100 nm away from the waveguide, and effectively promotes the separation of photogenerated electron-hole pairs. The photocurrent image shows that the built-in electric field between metal-doped graphene and graphene appears at the metal and graphene interface and leads to two narrow photocurrent regions. Moreover, the region of maximum photocurrent coincides with the position of waveguide. Wavelength transmission measurements show that the transmission loss are 0.1 dB and 4.8 dB before and after the graphene transfer, respectively, indicating that the optical absorption of the waveguide-integrated graphene is greatly increased. The 27 photodetectors demonstrated an ultrafast response rates exceeding 20 GHz at zero bias and a high response rate of 0.108 A/W at 1 V bias. Thanks to the gapless graphene, the spectral response of photodetector ranged from 1,450 nm to 1,590 nm, which can effectively detect the c-communication band of the near-infrared region. The response spectrum of the waveguide-integrated graphene photodetector can be further extended to the mid-infrared region. Xu and coworkers demonstrated a graphene/silicon heterostructure waveguide photodetector with a responsivity of up to 0.13 A/W at room temperature in the mid-IR region (2.75 m).[102] In addition, this photodetector exhibits a low dark current due to the presence of a graphene/silicon heterojunction potential barrier, and the on/off photocurrent ratio is 30 at a bias of -1.5 V for 2.75 m illumination at room temperature(Figure 6f). This work further explains the photocurrent generation mechanism of waveguide-integrated graphene in the visible light region, near infrared region and mid-infrared region. For the visible light region, since silicon has an indirect bandgap of 1.1 eV, photoresponse is mainly due to electronhole pairs generated after the silicon absorbs light and graphene acts as a photogenerated free carriers conduction channel in this process. For the near-infrared region (1.55 m), the photon energy is more than twice that of the graphene Fermi energy level and the infrared light cannot be absorbed by silicon. The photoexcitation of graphene takes place in a direct transition way that electrons are excited from the valence band to the conduction band. For the mid-infrared region (2.75 m), the photon energy is less than twice the energy difference between the Fermi level and the Dirac point, thus preventing direct transitions of graphene.[102] The excitation of graphene28 follows the indirect transition processes, and electrons are excited from Fermi level in the valence band to the conduction band with a large momentum mismatch under midinfrared light illumination.","content":"the absorption of graphene as a function of graphene length L at different widths, indicating that as long as the appropriate W and L values are obtained, the light absorption of graphene can be greatly enhanced. For example, when W = 100 nm, more than 50% light absorption can be obtained with only 22 m length bilayer graphene. Figure 6d shows an SEM image of a waveguide-integrated graphene photodetector with high responsivity and photocurrent image at zero bias. A 53 m bilayer graphene was overlaid on the waveguide, and two electrodes with waveguide distances of 100 nm and 3.5 m, respectively, were used to collect the photogenerated carriers. The lateral metaldoped junction overlapping the waveguide is formed by doping of metal electrode 100 nm away from the waveguide, and effectively promotes the separation of photogenerated electron-hole pairs. The photocurrent image shows that the built-in electric field between metal-doped graphene and graphene appears at the metal and graphene interface and leads to two narrow photocurrent regions. Moreover, the region of maximum photocurrent coincides with the position of waveguide. Wavelength transmission measurements show that the transmission loss are 0.1 dB and 4.8 dB before and after the graphene transfer, respectively, indicating that the optical absorption of the waveguide-integrated graphene is greatly increased. The 27 photodetectors demonstrated an ultrafast response rates exceeding 20 GHz at zero bias and a high response rate of 0.108 A/W at 1 V bias. Thanks to the gapless graphene, the spectral response of photodetector ranged from 1,450 nm to 1,590 nm, which can effectively detect the c-communication band of the near-infrared region. The response spectrum of the waveguide-integrated graphene photodetector can be further extended to the mid-infrared region. Xu and coworkers demonstrated a graphene/silicon heterostructure waveguide photodetector with a responsivity of up to 0.13 A/W at room temperature in the mid-IR region (2.75 m).[102] In addition, this photodetector exhibits a low dark current due to the presence of a graphene/silicon heterojunction potential barrier, and the on/off photocurrent ratio is 30 at a bias of -1.5 V for 2.75 m illumination at room temperature(Figure 6f). This work further explains the photocurrent generation mechanism of waveguide-integrated graphene in the visible light region, near infrared region and mid-infrared region. For the visible light region, since silicon has an indirect bandgap of 1.1 eV, photoresponse is mainly due to electronhole pairs generated after the silicon absorbs light and graphene acts as a photogenerated free carriers conduction channel in this process. For the near-infrared region (1.55 m), the photon energy is more than twice that of the graphene Fermi energy level and the infrared light cannot be absorbed by silicon. The photoexcitation of graphene takes place in a direct transition way that electrons are excited from the valence band to the conduction band. For the mid-infrared region (2.75 m), the photon energy is less than twice the energy difference between the Fermi level and the Dirac point, thus preventing direct transitions of graphene.[102] The excitation of graphene28 follows the indirect transition processes, and electrons are excited from Fermi level in the valence band to the conduction band with a large momentum mismatch under midinfrared light illumination."},{"id":"","caption":"(a) SEM image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration (false colour). Inset: An enlarged view of the inside of the black dashed line. (b) The absorption of graphene as a function of graphene length L at different widths W. Solid line: bilayer graphene. Dashed line: monolayer graphene. (c) 29 Schematic of the chip-integrated graphene photodetector. (d) The spatially resolved photocurrent image of the waveguide-integrated graphene photodetector at zero bias. The black solid line shows the relative potential distribution across the graphene channel along the white dashed line. Inset: corresponding SEM image. (e) Schematic of a graphene/silicon heterostructure waveguide photodetector. (f) Photocurrent as a function of applied bias voltage with 2.75 m laser illumination at different light power.(a-b) are reproduced from Ref.[101] with permission from the Nature Publishing Group.","content":"Figure 6\n6(a) SEM image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration (false colour). Inset: An enlarged view of the inside of the black dashed line. (b) The absorption of graphene as a function of graphene length L at different widths W. Solid line: bilayer graphene. Dashed line: monolayer graphene. (c) 29 Schematic of the chip-integrated graphene photodetector. (d) The spatially resolved photocurrent image of the waveguide-integrated graphene photodetector at zero bias. The black solid line shows the relative potential distribution across the graphene channel along the white dashed line. Inset: corresponding SEM image. (e) Schematic of a graphene/silicon heterostructure waveguide photodetector. (f) Photocurrent as a function of applied bias voltage with 2.75 m laser illumination at different light power.(a-b) are reproduced from Ref.[101] with permission from the Nature Publishing Group."},{"id":"","caption":"is in a low doping state (VG=-8 V), showing a fast response due to photovoltaic effect. But the roll-off frequency is measured to be 0.2 MHz when BP is in a high doping state (VG=8 V), showing a slower response due to bolometric effect.","content":"is in a low doping state (VG=-8 V), showing a fast response due to photovoltaic effect. But the roll-off frequency is measured to be 0.2 MHz when BP is in a high doping state (VG=8 V), showing a slower response due to bolometric effect."},{"id":"","caption":"shows the intrinsic responsivity R and internal quantum efficiency IQE as a function of bias for different thicknesses. The intrinsic responsivity and internal quantum efficiency of 11.5-nm-thick BP device with top-gate are 135 mA/W and 10% at Vbias=-0.4 V, respectively. For 100-nm-thick BP device without top-gate, a larger bias voltage can be applied because there is no need to consider the breakdown of the gate insulation layer. Therefore, this photodetector shows better detection performance with high intrinsic responsivity (657 mA/W) and high internal quantum efficiency (50%) at Vbias=-0.4 V. Unlike gapless graphene, BP has a finite direct band gap,[113] which make the dark current of waveguide-integrated BP photodetector extreme low (220 nA) atVG=-8 V and Vbias=-0.4 V. (a) Schematic of the waveguide-integrated few-layer BP photodetector. (b) The frequency response of the waveguide-integrated BP photodetector at low doping state (VG=-8 V) and high doping state (VG=8 V). (c) Two-dimensional contour plot of photocurrent versus the gate voltage VG and the bias voltage VDS. (d) The intrinsic responsivity R and internal quantum efficiency IQE as a function of applied bias for different thicknesses. tBP: the thickness of BP. (a-d) are reproduced from Ref.[103] with permission from the Nature Publishing Group. Recently, Bie et al. reported a silicon photonic-crystal (PhC) waveguide-integrated","content":"Figure 7d 31 Figure 7\n7d317shows the intrinsic responsivity R and internal quantum efficiency IQE as a function of bias for different thicknesses. The intrinsic responsivity and internal quantum efficiency of 11.5-nm-thick BP device with top-gate are 135 mA/W and 10% at Vbias=-0.4 V, respectively. For 100-nm-thick BP device without top-gate, a larger bias voltage can be applied because there is no need to consider the breakdown of the gate insulation layer. Therefore, this photodetector shows better detection performance with high intrinsic responsivity (657 mA/W) and high internal quantum efficiency (50%) at Vbias=-0.4 V. Unlike gapless graphene, BP has a finite direct band gap,[113] which make the dark current of waveguide-integrated BP photodetector extreme low (220 nA) atVG=-8 V and Vbias=-0.4 V. (a) Schematic of the waveguide-integrated few-layer BP photodetector. (b) The frequency response of the waveguide-integrated BP photodetector at low doping state (VG=-8 V) and high doping state (VG=8 V). (c) Two-dimensional contour plot of photocurrent versus the gate voltage VG and the bias voltage VDS. (d) The intrinsic responsivity R and internal quantum efficiency IQE as a function of applied bias for different thicknesses. tBP: the thickness of BP. (a-d) are reproduced from Ref.[103] with permission from the Nature Publishing Group. Recently, Bie et al. reported a silicon photonic-crystal (PhC) waveguide-integrated"},{"id":"","caption":"light is coupled into the waveguide and transmitted along the waveguide to the grating coupler in LED mode, while incident light is coupled into the waveguide at the grating coupler and detected by the MoTe2 p-n junction in the photodetector mode (Figure 8b).","content":"light is coupled into the waveguide and transmitted along the waveguide to the grating coupler in LED mode, while incident light is coupled into the waveguide at the grating coupler and detected by the MoTe2 p-n junction in the photodetector mode (Figure 8b)."},{"id":"","caption":"shows an EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. Two extra emission spots are observed on the grating coupler on both sides, which coincides with the aforementioned LED mode (Figure 8b). The orange and green lines in Figure 8d represent the transmission spectra of the PhC waveguide before and after the bilayer MoTe2 transfer, respectively. The orange curve in","content":"Figure 8c\n8cshows an EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. Two extra emission spots are observed on the grating coupler on both sides, which coincides with the aforementioned LED mode (Figure 8b). The orange and green lines in Figure 8d represent the transmission spectra of the PhC waveguide before and after the bilayer MoTe2 transfer, respectively. The orange curve in"},{"id":"","caption":"8f, the response spectrum of photodetector in the range from 1040 nm to 1240 nm and the largest responsivity is at 1,160 nm (the location of the PhC transmission peak), which shows the dependence of the responsivity on the wavelength. This dependence indicates that the photocurrent originates mainly from the light transmitted by the PhC waveguide, and thus the light beyond this spectral response range cannot be detected due to the bandgap of the bilayer MoTe2 and the transmission cut-off of the waveguide. This work shows that all active optical components for point-to-point interconnects are possible with 2D TMD devices transferred onto otherwise passive photonic integrated circuits.[91]","content":"8f, the response spectrum of photodetector in the range from 1040 nm to 1240 nm and the largest responsivity is at 1,160 nm (the location of the PhC transmission peak), which shows the dependence of the responsivity on the wavelength. This dependence indicates that the photocurrent originates mainly from the light transmitted by the PhC waveguide, and thus the light beyond this spectral response range cannot be detected due to the bandgap of the bilayer MoTe2 and the transmission cut-off of the waveguide. This work shows that all active optical components for point-to-point interconnects are possible with 2D TMD devices transferred onto otherwise passive photonic integrated circuits.[91]"},{"id":"","caption":"(a) Schematic of the silicon photonic-crystal (PhC) waveguide-integrated light-emitting diode and photodetector based on bilayer MoTe2 p-n junction. (b) The schematic diagram of the device operating in LED mode and photodetector mode. The arrow indicates the direction of light propagation in the PhC waveguide. (c) EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. (d) The orange and green lines represent the transmission spectra of the waveguide before and after the bilayer MoTe2 transfer, respectively. The normalized EL emission spectrum collected at the grating coupler represented by the blue line. (e) The temporal response of device in photodetector mode at zero bias. The 35 response bandwidth of the device is 200 MHz. (f) The response spectrum of photodetector in the range from 1,040 nm to 1,240 nm, which shows the dependence of the responsivity on the wavelength due to photonic-crystal waveguide. (a-f) are","content":"Figure 8\n8(a) Schematic of the silicon photonic-crystal (PhC) waveguide-integrated light-emitting diode and photodetector based on bilayer MoTe2 p-n junction. (b) The schematic diagram of the device operating in LED mode and photodetector mode. The arrow indicates the direction of light propagation in the PhC waveguide. (c) EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. (d) The orange and green lines represent the transmission spectra of the waveguide before and after the bilayer MoTe2 transfer, respectively. The normalized EL emission spectrum collected at the grating coupler represented by the blue line. (e) The temporal response of device in photodetector mode at zero bias. The 35 response bandwidth of the device is 200 MHz. (f) The response spectrum of photodetector in the range from 1,040 nm to 1,240 nm, which shows the dependence of the responsivity on the wavelength due to photonic-crystal waveguide. (a-f) are"},{"id":"","caption":"et al. achieved a 26-fold absorption enhancement of graphene (>60%) by monolithically integrating graphene with a Fabry-Pérot microcavity. [123] The schematic of the microcavity-integrated graphene photodetector and the corresponding electric field distribution are shown in Figure 9a. Two Bragg mirrors are alternately composed of quarter-wavelength thick materials with different refractive index, and graphene layer is sandwiched between these mirrors. The presence of the barrier layer ensures the position of resonance occur at the graphene. The incident light are reflected back and forth multiple times between two Bragg mirrors and absorbed by the graphene, getting a higher absorption, which agrees with the electric field distribution. The spectral response of the device is shown in Figure 9b, and it can be clearly seen that a significant photocurrent enhancement at the cavity resonance (855 nm). Atomic thin monolayer graphene (only 0.335 nm) can absorb more than 60% of light, indicating 36 that the integration of 2D materials with cavity has great potential in enhancing the light absorption of 2D materials. However, the enhancement of light absorption by the cavity occurs only at the resonant wavelength of the cavity. In other words, the enhancement of the cavity is at the expense of the broadband photodetection. The strategy of integrating cavity and two-dimensional materials cannot be used for broadband detectors, and thus it can only be used to enhance light absorption at specific wavelengths. For example, Casalino et al. demonstrated a vertically illuminated, resonant cavity enhanced, graphene−Si","content":"Furchi\net al. achieved a 26-fold absorption enhancement of graphene (>60%) by monolithically integrating graphene with a Fabry-Pérot microcavity. [123] The schematic of the microcavity-integrated graphene photodetector and the corresponding electric field distribution are shown in Figure 9a. Two Bragg mirrors are alternately composed of quarter-wavelength thick materials with different refractive index, and graphene layer is sandwiched between these mirrors. The presence of the barrier layer ensures the position of resonance occur at the graphene. The incident light are reflected back and forth multiple times between two Bragg mirrors and absorbed by the graphene, getting a higher absorption, which agrees with the electric field distribution. The spectral response of the device is shown in Figure 9b, and it can be clearly seen that a significant photocurrent enhancement at the cavity resonance (855 nm). Atomic thin monolayer graphene (only 0.335 nm) can absorb more than 60% of light, indicating 36 that the integration of 2D materials with cavity has great potential in enhancing the light absorption of 2D materials. However, the enhancement of light absorption by the cavity occurs only at the resonant wavelength of the cavity. In other words, the enhancement of the cavity is at the expense of the broadband photodetection. The strategy of integrating cavity and two-dimensional materials cannot be used for broadband detectors, and thus it can only be used to enhance light absorption at specific wavelengths. For example, Casalino et al. demonstrated a vertically illuminated, resonant cavity enhanced, graphene−Si"},{"id":"","caption":"(a) The schematic of the resonant cavity enhanced graphene photodetector and the corresponding electric field distribution. Red arrow: the position of graphene. (b) The dashed lines indicates the calculation result: reflection R (red), transmission T (green) and absorption A (blue), while the solid lines indicates the measurement result: reflection (red) and photocurrent (blue). A 26-fold absorption enhancement at the cavity resonance wavelength (855 nm). Inset: calculated theoretical results. (c) Schematic diagram of the resonant cavity enhanced graphene/Si Schottky photodetector. (d) The response spectrum of the device with and without Au mirror. A 3-fold absorption enhancement can be observed in the presence of Au mirror. The blue shift of the resonance peak is caused by the difference between different devices. (a-b) are","content":"Figure 9\n9(a) The schematic of the resonant cavity enhanced graphene photodetector and the corresponding electric field distribution. Red arrow: the position of graphene. (b) The dashed lines indicates the calculation result: reflection R (red), transmission T (green) and absorption A (blue), while the solid lines indicates the measurement result: reflection (red) and photocurrent (blue). A 26-fold absorption enhancement at the cavity resonance wavelength (855 nm). Inset: calculated theoretical results. (c) Schematic diagram of the resonant cavity enhanced graphene/Si Schottky photodetector. (d) The response spectrum of the device with and without Au mirror. A 3-fold absorption enhancement can be observed in the presence of Au mirror. The blue shift of the resonance peak is caused by the difference between different devices. (a-b) are"},{"id":"","caption":"Metal plasmonic nanostructures mainly enhance the response of the detector in two ways. First, the nearfield enhancement capability of the plasmonic nanostructures can focus the light field in a small area, effectively enhancing the light absorption of the two-dimensional material. Moreover, the plasmonic structure can generate plasma-induced hot electrons due to plasmon decay under light illumination, and then hot electrons are injected into the device across the Schottky barrier between the metal and the two-dimensional material to form photocurrent. Fang et al. demonstrated a graphene-antenna sandwich photodetector with","content":"Metal plasmonic nanostructures mainly enhance the response of the detector in two ways. First, the nearfield enhancement capability of the plasmonic nanostructures can focus the light field in a small area, effectively enhancing the light absorption of the two-dimensional material. Moreover, the plasmonic structure can generate plasma-induced hot electrons due to plasmon decay under light illumination, and then hot electrons are injected into the device across the Schottky barrier between the metal and the two-dimensional material to form photocurrent. Fang et al. demonstrated a graphene-antenna sandwich photodetector with"},{"id":"","caption":"10e and f show the measured photocurrent and the corresponding band diagram for different applied gate voltages, respectively. The gate voltage VG can effectively adjust the Fermi level of the graphene channel, and then obtain different energy band bending between the Fermi level of graphene channel and the Ti electrode-doped graphene. The steeper band bending results in a correspondingly larger built-in electric field, which can obtain higher the separation efficiency of the photogenerated electronhole pairs and thus produces a greater photocurrent. The resonance wavelength of the plasmonic clusters can be controlled by varying the diameter of disks. For example, when the diameter of heptamer plasmonic clusters changes from 80 nm to 180 nm, the corresponding resonance wavelength changes from 650 nm to 950 nm.","content":"Figure\n10e and f show the measured photocurrent and the corresponding band diagram for different applied gate voltages, respectively. The gate voltage VG can effectively adjust the Fermi level of the graphene channel, and then obtain different energy band bending between the Fermi level of graphene channel and the Ti electrode-doped graphene. The steeper band bending results in a correspondingly larger built-in electric field, which can obtain higher the separation efficiency of the photogenerated electronhole pairs and thus produces a greater photocurrent. The resonance wavelength of the plasmonic clusters can be controlled by varying the diameter of disks. For example, when the diameter of heptamer plasmonic clusters changes from 80 nm to 180 nm, the corresponding resonance wavelength changes from 650 nm to 950 nm."},{"id":"","caption":"(a) The schematic of graphene-antenna sandwich photodetector. (b) 42 The optical image of device before (left) and after (right) transfer of top layer graphene. Inset: Raman mapping of the device under 785 nm laser illumination. (c) Photocurrents measured in the photodetector with dimer array and heptamer array and without pattern exhibit antisymmetric properties. The results show that the metal plasmonic antenna clusters can effectively enhance the photocurrent. The line scan position is shown by the white arrows in the inset. (d) The polarization dependent properties of the photocurrent measured in dimer and heptamer devices. (e) Photocurrents measured at different gate voltages in a heptamer device. (f) Energy band diagram of devices with different gate voltages in a heptamer device. The purple dashed line indicates the Dirac point of graphene. (g) Photocurrent as a function of different resonant wavelength. Black triangles: the measured photocurrent (I EXP (λ)). Blue triangles: the calculated DE photocurrent (I DE (λ)). Solid red triangles: the estimated HE photocurrent contribution E DE (λ) = I EXP (λ) −I DE (λ). Hollow red triangles: the calculated I HE (λ). (h) Schematic of hot electron-based bilayer MoS2 with near-infrared laser illumination. (a-g) are reproduced from Ref.[131] with permission from the American Chemical Society. (h)","content":"(a) The schematic of graphene-antenna sandwich photodetector. (b) 42 The optical image of device before (left) and after (right) transfer of top layer graphene. Inset: Raman mapping of the device under 785 nm laser illumination. (c) Photocurrents measured in the photodetector with dimer array and heptamer array and without pattern exhibit antisymmetric properties. The results show that the metal plasmonic antenna clusters can effectively enhance the photocurrent. The line scan position is shown by the white arrows in the inset. (d) The polarization dependent properties of the photocurrent measured in dimer and heptamer devices. (e) Photocurrents measured at different gate voltages in a heptamer device. (f) Energy band diagram of devices with different gate voltages in a heptamer device. The purple dashed line indicates the Dirac point of graphene. (g) Photocurrent as a function of different resonant wavelength. Black triangles: the measured photocurrent (I EXP (λ)). Blue triangles: the calculated DE photocurrent (I DE (λ)). Solid red triangles: the estimated HE photocurrent contribution E DE (λ) = I EXP (λ) −I DE (λ). Hollow red triangles: the calculated I HE (λ). (h) Schematic of hot electron-based bilayer MoS2 with near-infrared laser illumination. (a-g) are reproduced from Ref.[131] with permission from the American Chemical Society. (h)"},{"id":"","caption":"hybrid structure based on two-dimensional materials. Hybrid 2D-QDs heterostructures possess two prominent advantages of strong light absorption contributed by quantum dots and high carrier mobility contributed by two-dimensional materials, and thus exhibiting extremely high gain and responsivity. The detection performance of photodetectors based on hybrid graphene-SWNTs structure is superior to that of photodetectors based on only graphene or SWNTs, showing high responsivity and fast response, in which the bandgap of SWNTs can be adjusted by the diameter of SWNTs to extend the spectral response range to infrared. The 2D-2D van der Waals heterostructure is also suitable for light detection in the infrared region because of ultrafast charge transfer at the interface, and the combination of two or more twodimensional materials with different optical and electrical properties not only compensates for the lack of both (eg. the combination of 2D materials with different bandgap structures can achieve broadband detection), but also exploits the characteristics of 2D materials to prepare a photodetector with specific function (such as high polarization sensitivity of a BP-on-WSe2 photogate structure). The integration of two-dimensional material on waveguide makes the light-matter interaction no longer limited by the ultrathin thickness of two-dimensional material, greatly increasing the light absorption of photodetectors to obtain a higher responsivity. In addition, this integration also shows that two-dimensional materials are compatible with the existing mature optical waveguide technology, providing a pathway for the commercialized practical application of two-dimensional materials. The integration of two-dimensional material with the cavity enhances the light absorption of the material at the resonant48 wavelength of the cavity. This selective absorption method is very suitable for precise detection of specific wavelengths. Metal plasmonic nanostructures can also enhance light absorption and photocurrent through near-field enhancement effects and hot electrons. In addition, plasma-induced hot electrons can extend the spectral response range, which is useful for infrared detection based on some two-dimensional materials with wide bandgap.","content":"hybrid structure based on two-dimensional materials. Hybrid 2D-QDs heterostructures possess two prominent advantages of strong light absorption contributed by quantum dots and high carrier mobility contributed by two-dimensional materials, and thus exhibiting extremely high gain and responsivity. The detection performance of photodetectors based on hybrid graphene-SWNTs structure is superior to that of photodetectors based on only graphene or SWNTs, showing high responsivity and fast response, in which the bandgap of SWNTs can be adjusted by the diameter of SWNTs to extend the spectral response range to infrared. The 2D-2D van der Waals heterostructure is also suitable for light detection in the infrared region because of ultrafast charge transfer at the interface, and the combination of two or more twodimensional materials with different optical and electrical properties not only compensates for the lack of both (eg. the combination of 2D materials with different bandgap structures can achieve broadband detection), but also exploits the characteristics of 2D materials to prepare a photodetector with specific function (such as high polarization sensitivity of a BP-on-WSe2 photogate structure). The integration of two-dimensional material on waveguide makes the light-matter interaction no longer limited by the ultrathin thickness of two-dimensional material, greatly increasing the light absorption of photodetectors to obtain a higher responsivity. In addition, this integration also shows that two-dimensional materials are compatible with the existing mature optical waveguide technology, providing a pathway for the commercialized practical application of two-dimensional materials. The integration of two-dimensional material with the cavity enhances the light absorption of the material at the resonant48 wavelength of the cavity. This selective absorption method is very suitable for precise detection of specific wavelengths. Metal plasmonic nanostructures can also enhance light absorption and photocurrent through near-field enhancement effects and hot electrons. In addition, plasma-induced hot electrons can extend the spectral response range, which is useful for infrared detection based on some two-dimensional materials with wide bandgap."},{"id":"","caption":"Both WSe2 and BP absorb light efficiently and generate photogenerated electron-hole pairs under visible illumination, which are then separated by the built-in electric field. The photo-generated electrons in BP are transferred to WSe2 and the photo-generated holes in WSe2 are transferred to BP, which effectively reduces the recombination efficiency. An increasing number of photo-generated electrons in WSe2 are driven by source-drain bias to form photocurrent, thereby enhancing the responsivity. However, due to the band gap limitation of WSe2, only BP with a small band gap is an effective absorption material to generate photogenerated free carriers under infrared illumination. So the photoresponse in the infrared region will be smaller than in the visible region. The responsivity is ~10 3 A/W under visible illumination (637 nm), while the responsivity is ~0.5 A/W under infrared illumination (1,550 nm).","content":"Both WSe2 and BP absorb light efficiently and generate photogenerated electron-hole pairs under visible illumination, which are then separated by the built-in electric field. The photo-generated electrons in BP are transferred to WSe2 and the photo-generated holes in WSe2 are transferred to BP, which effectively reduces the recombination efficiency. An increasing number of photo-generated electrons in WSe2 are driven by source-drain bias to form photocurrent, thereby enhancing the responsivity. However, due to the band gap limitation of WSe2, only BP with a small band gap is an effective absorption material to generate photogenerated free carriers under infrared illumination. So the photoresponse in the infrared region will be smaller than in the visible region. The responsivity is ~10 3 A/W under visible illumination (637 nm), while the responsivity is ~0.5 A/W under infrared illumination (1,550 nm)."},{"id":"","caption":"Performance of room temperature infrared photodetectors with hybrid structure based on 2D materials. (UV: Ultraviolet. NIR: Near infrared. MIR: Midinfrared. BP: Black phosphorus. Gr: Graphene. QD: Quantum dot. FET: Field effect transistor.)In this review, we have reviewed various infrared photodetectors operating at room","content":"Table 1 .\n1Performance of room temperature infrared photodetectors with hybrid structure based on 2D materials. (UV: Ultraviolet. NIR: Near infrared. MIR: Midinfrared. BP: Black phosphorus. Gr: Graphene. QD: Quantum dot. FET: Field effect transistor.)In this review, we have reviewed various infrared photodetectors operating at roomDevice type \nDevice \nSpect Responsi Detectivi Respon EQE/I \ngain \nRef."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Zheng et al. demonstrated an ambipolar broadband photodetector based on N, S codecorated graphene-PbS quantum dot heterostructure. [42] N, S codecorated graphene greatly reduces the p-type doping of graphene in air. The gate voltage required to modulate the Dirac point and the carrier type is significantly reduced, thereby 8 achieving gate tuneable mutual transformation of positive and negative photoresponse.The device exhibits excellent detection performance with responsivity of 10 4 A/W and specific detectivity of 10 12 Jones at 1550 nm.\",\n \"content\": \"Zheng et al. demonstrated an ambipolar broadband photodetector based on N, S codecorated graphene-PbS quantum dot heterostructure. [42] N, S codecorated graphene greatly reduces the p-type doping of graphene in air. The gate voltage required to modulate the Dirac point and the carrier type is significantly reduced, thereby 8 achieving gate tuneable mutual transformation of positive and negative photoresponse.The device exhibits excellent detection performance with responsivity of 10 4 A/W and specific detectivity of 10 12 Jones at 1550 nm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) The schematic diagram of a hybrid graphene-PbS quantum dots phototransistor. (b) The energy band diagram at the graphene/PbS QDs interface. (c) The linear dependence of the photocurrent on the applied drain-source voltage for different optical powers (46, 336, 516 pW). VBG=0. Inset: The total current (I=Idark + Iphoto) as a function of the applied drain-source voltage for different optical powers. (d) The responsivity as a function of the applied back-gate voltage VBG. VSD=5 V. (a-d) are\",\n \"content\": \"Figure 1\\n1(a) The schematic diagram of a hybrid graphene-PbS quantum dots phototransistor. (b) The energy band diagram at the graphene/PbS QDs interface. (c) The linear dependence of the photocurrent on the applied drain-source voltage for different optical powers (46, 336, 516 pW). VBG=0. Inset: The total current (I=Idark + Iphoto) as a function of the applied drain-source voltage for different optical powers. (d) The responsivity as a function of the applied back-gate voltage VBG. VSD=5 V. (a-d) are\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Photogenerated electron-hole pairs are created at the QDs layer, and then drifted by the built-in electric field at the interface and by the applied bias voltage to form photocurrent directly in photodiode operation. However, after the electron-hole pairs are separated, electrons remain in the QDs layer and holes are driven by depletion region on the graphene/PbS QD interface and transferred to graphene in phototransistor operation. For VTD>0, the width of the depletion region expands, increasing the efficient charge collection area, and thus the external quantum efficiency also increases.\",\n \"content\": \"Photogenerated electron-hole pairs are created at the QDs layer, and then drifted by the built-in electric field at the interface and by the applied bias voltage to form photocurrent directly in photodiode operation. However, after the electron-hole pairs are separated, electrons remain in the QDs layer and holes are driven by depletion region on the graphene/PbS QD interface and transferred to graphene in phototransistor operation. For VTD>0, the width of the depletion region expands, increasing the efficient charge collection area, and thus the external quantum efficiency also increases.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"10 5 compared with the photodiode operation. With increasing VTD, the temporal11 response accelerates and the linear dynamic range extends. The electrical 3 dB bandwidth of the detector reaches 1.5 kHz for the optimum VTD, and the linear dynamic range extends from 75 dB at VTD=0 V to 110 dB at VTD=1.2 V. The above results prove that the design and optimization of infrared photodetectors based on hybrid 2D-QDs heterostructures can effectively enhance the detection performance.\",\n \"content\": \"10 5 compared with the photodiode operation. With increasing VTD, the temporal11 response accelerates and the linear dynamic range extends. The electrical 3 dB bandwidth of the detector reaches 1.5 kHz for the optimum VTD, and the linear dynamic range extends from 75 dB at VTD=0 V to 110 dB at VTD=1.2 V. The above results prove that the design and optimization of infrared photodetectors based on hybrid 2D-QDs heterostructures can effectively enhance the detection performance.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a-b) Schematic of photodiode operation and phototransistor operation of the hybrid graphene-PbS QDs photodetector, respectively. (c) Responsivity and external quantum efficiency EQE of photodetector in photodiode operation as a function of VTD. Inset: energy band diagrams of device with and without VTD bias. (d) Responsivity and EQE of photodetector in phototransistor operation as a function of VTD. (e) Normalized photoresponse versus light modulation frequency. The specified VTD values are shown in the legend. Inset: The linear dependence of extracted 3 dB bandwidth values on applied VTD. (f) Responsivity of the detector as a function of incident irradiance in the range of 10 -5 to 10 Wm -2 , which shows a significant extension 12 of the linear dynamic range with increasing VTD.\",\n \"content\": \"Figure 2\\n2(a-b) Schematic of photodiode operation and phototransistor operation of the hybrid graphene-PbS QDs photodetector, respectively. (c) Responsivity and external quantum efficiency EQE of photodetector in photodiode operation as a function of VTD. Inset: energy band diagrams of device with and without VTD bias. (d) Responsivity and EQE of photodetector in phototransistor operation as a function of VTD. (e) Normalized photoresponse versus light modulation frequency. The specified VTD values are shown in the legend. Inset: The linear dependence of extracted 3 dB bandwidth values on applied VTD. (f) Responsivity of the detector as a function of incident irradiance in the range of 10 -5 to 10 Wm -2 , which shows a significant extension 12 of the linear dynamic range with increasing VTD.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) Schematic of the photodetector and AFM characterization of SWNTgraphene hybrid film (middle: scale bar, 200 nm; right: scale bar, 20 nm). (b) The ISD-VG transfer curve of device with increasing 650 nm illumination powers. VSD=0.5 V. Inset: the optical image of the SWNT-graphene device (Scale bar, 10m). (c) Temporal response of the SWNT-graphene hybrid photodetector under 650 nm illumination. (d) The dependence of responsivities of photodetector on the optical power at different illumination wavelengths from visible to near-infrared range (405, 532, 650, 980 and 1,550 nm). (e) Photocurrent spectral response can be tuned by varying the gate voltage.\",\n \"content\": \"Figure 3\\n3(a) Schematic of the photodetector and AFM characterization of SWNTgraphene hybrid film (middle: scale bar, 200 nm; right: scale bar, 20 nm). (b) The ISD-VG transfer curve of device with increasing 650 nm illumination powers. VSD=0.5 V. Inset: the optical image of the SWNT-graphene device (Scale bar, 10m). (c) Temporal response of the SWNT-graphene hybrid photodetector under 650 nm illumination. (d) The dependence of responsivities of photodetector on the optical power at different illumination wavelengths from visible to near-infrared range (405, 532, 650, 980 and 1,550 nm). (e) Photocurrent spectral response can be tuned by varying the gate voltage.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. Due to the gapless nature of graphene and a small bandgap (0.15-0.3 eV) of Bi-2Te3, [84] both graphene and Bi2Te3 are effective absorbent materials for infrared light, and thus the detector exhibits broadband detection characteristics. The effective detection range can be extended to the near infrared (980 nm) and c-communication band (1,550nm). The responsivity is 35 A/W, 10 A/W and 0.22 A/W in the visible (532 nm), infrared (980 nm) and c-communication band (1,550 nm) regions, respectively.\",\n \"content\": \"the dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. Due to the gapless nature of graphene and a small bandgap (0.15-0.3 eV) of Bi-2Te3, [84] both graphene and Bi2Te3 are effective absorbent materials for infrared light, and thus the detector exhibits broadband detection characteristics. The effective detection range can be extended to the near infrared (980 nm) and c-communication band (1,550nm). The responsivity is 35 A/W, 10 A/W and 0.22 A/W in the visible (532 nm), infrared (980 nm) and c-communication band (1,550 nm) regions, respectively.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) Top: optical image of the photodetector based on a MoS2−graphene−WSe2 heterostructure. Scale bar, 5 m. Bottom: Schematic diagram of the device. (b) Responsivity and specific detectivity of the photodetector for wavelengths ranging from 400 to 2,400 nm with an interval of 100 nm. (c) Top: Spectra from ultraviolet to infrared. The band gap of MoS2 (Eg1) and WSe2 (Eg2) as well as corresponding wavelengths are plotted on the graph, respectively. Bottom left: For h >Eg1>Eg2, MoS2, graphene and WSe2 all can generate photogenerated electron-hole pairs. Bottom right: For h <Eg1<Eg2, only graphene can generate photogenerated electron-hole pairs. (d) The schematic diagram of the device based on graphene-Bi2Te3 heterostructure. (e) The dependence of photocurrent of devices based on pure graphene and graphene-Bi2Te3 heterostructure on source-drain voltage VSD. VG=0 V. (f) Dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. (a-c)\",\n \"content\": \"Figure 4\\n4(a) Top: optical image of the photodetector based on a MoS2−graphene−WSe2 heterostructure. Scale bar, 5 m. Bottom: Schematic diagram of the device. (b) Responsivity and specific detectivity of the photodetector for wavelengths ranging from 400 to 2,400 nm with an interval of 100 nm. (c) Top: Spectra from ultraviolet to infrared. The band gap of MoS2 (Eg1) and WSe2 (Eg2) as well as corresponding wavelengths are plotted on the graph, respectively. Bottom left: For h >Eg1>Eg2, MoS2, graphene and WSe2 all can generate photogenerated electron-hole pairs. Bottom right: For h <Eg1<Eg2, only graphene can generate photogenerated electron-hole pairs. (d) The schematic diagram of the device based on graphene-Bi2Te3 heterostructure. (e) The dependence of photocurrent of devices based on pure graphene and graphene-Bi2Te3 heterostructure on source-drain voltage VSD. VG=0 V. (f) Dependence of photocurrent on wavelengths ranging from 400 nm to 1,550 nm. (a-c)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the relationship between the photocurrent and the polarization angle of the incident light. The maximum photocurrent, corresponding responsivity of ~40 mA/W, is taken along the horizontal axis (defined as 0 o polarization), while the minimum photocurrent, corresponding responsivity of ~6.8 mA/W, is taken along the vertical axis (defined as 90 o polarization)\",\n \"content\": \"the relationship between the photocurrent and the polarization angle of the incident light. The maximum photocurrent, corresponding responsivity of ~40 mA/W, is taken along the horizontal axis (defined as 0 o polarization), while the minimum photocurrent, corresponding responsivity of ~6.8 mA/W, is taken along the vertical axis (defined as 90 o polarization)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) Schematic diagram of the photodetector with BP-on-WSe2 photogate structure. (b) The photoconductive gain G and detectivity D* of the photodetector for wavelengths ranging from 400 to 1,600 nm at 1 mW/cm 2 incident light power density. 23 Vds=0.5 V. (c) Top: the photocurrent image of the photodetector under 1,550 nm illumination with different light polarizations (red arrows). Bottom left: the optical image of the photodetector with BP-on-WSe2 photogate structure. (d) The photocurrent as a function of light polarization at 100 mW/cm 2 incident light power density. Vds=0.5 V. (a-d) are reproduced from Ref.[85] with permission from Elsevier. 2D lateral heterostructures The heterostructures mentioned above are basically vertical, and the built-in electric field at the interface promotes the separation of photogenerated electron-hole pairs and charge transfer, thereby reducing the recombination and accelerating the response speed. In addition, two-dimensional lateral heterostructures, a heterostructure composed of materials with different work functions or homojunction composed of the same material but different doping types in the lateral direction, form a built-in electric field at the interface and play a similar role. At present, the methods for forming lateral\",\n \"content\": \"Figure 5\\n5(a) Schematic diagram of the photodetector with BP-on-WSe2 photogate structure. (b) The photoconductive gain G and detectivity D* of the photodetector for wavelengths ranging from 400 to 1,600 nm at 1 mW/cm 2 incident light power density. 23 Vds=0.5 V. (c) Top: the photocurrent image of the photodetector under 1,550 nm illumination with different light polarizations (red arrows). Bottom left: the optical image of the photodetector with BP-on-WSe2 photogate structure. (d) The photocurrent as a function of light polarization at 100 mW/cm 2 incident light power density. Vds=0.5 V. (a-d) are reproduced from Ref.[85] with permission from Elsevier. 2D lateral heterostructures The heterostructures mentioned above are basically vertical, and the built-in electric field at the interface promotes the separation of photogenerated electron-hole pairs and charge transfer, thereby reducing the recombination and accelerating the response speed. In addition, two-dimensional lateral heterostructures, a heterostructure composed of materials with different work functions or homojunction composed of the same material but different doping types in the lateral direction, form a built-in electric field at the interface and play a similar role. At present, the methods for forming lateral\"\n },\n {\n \"id\": \"\",\n \"caption\": \"and the device structures are shown in Figures 6a, c and e, respectively. Figure 6a shows a scanning electron microscope (SEM)image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration. It should be noted that in addition to the length L of graphene, the width W of central electrode S in this device also affects the light absorption of the graphene.\",\n \"content\": \"and the device structures are shown in Figures 6a, c and e, respectively. Figure 6a shows a scanning electron microscope (SEM)image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration. It should be noted that in addition to the length L of graphene, the width W of central electrode S in this device also affects the light absorption of the graphene.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the absorption of graphene as a function of graphene length L at different widths, indicating that as long as the appropriate W and L values are obtained, the light absorption of graphene can be greatly enhanced. For example, when W = 100 nm, more than 50% light absorption can be obtained with only 22 m length bilayer graphene. Figure 6d shows an SEM image of a waveguide-integrated graphene photodetector with high responsivity and photocurrent image at zero bias. A 53 m bilayer graphene was overlaid on the waveguide, and two electrodes with waveguide distances of 100 nm and 3.5 m, respectively, were used to collect the photogenerated carriers. The lateral metaldoped junction overlapping the waveguide is formed by doping of metal electrode 100 nm away from the waveguide, and effectively promotes the separation of photogenerated electron-hole pairs. The photocurrent image shows that the built-in electric field between metal-doped graphene and graphene appears at the metal and graphene interface and leads to two narrow photocurrent regions. Moreover, the region of maximum photocurrent coincides with the position of waveguide. Wavelength transmission measurements show that the transmission loss are 0.1 dB and 4.8 dB before and after the graphene transfer, respectively, indicating that the optical absorption of the waveguide-integrated graphene is greatly increased. The 27 photodetectors demonstrated an ultrafast response rates exceeding 20 GHz at zero bias and a high response rate of 0.108 A/W at 1 V bias. Thanks to the gapless graphene, the spectral response of photodetector ranged from 1,450 nm to 1,590 nm, which can effectively detect the c-communication band of the near-infrared region. The response spectrum of the waveguide-integrated graphene photodetector can be further extended to the mid-infrared region. Xu and coworkers demonstrated a graphene/silicon heterostructure waveguide photodetector with a responsivity of up to 0.13 A/W at room temperature in the mid-IR region (2.75 m).[102] In addition, this photodetector exhibits a low dark current due to the presence of a graphene/silicon heterojunction potential barrier, and the on/off photocurrent ratio is 30 at a bias of -1.5 V for 2.75 m illumination at room temperature(Figure 6f). This work further explains the photocurrent generation mechanism of waveguide-integrated graphene in the visible light region, near infrared region and mid-infrared region. For the visible light region, since silicon has an indirect bandgap of 1.1 eV, photoresponse is mainly due to electronhole pairs generated after the silicon absorbs light and graphene acts as a photogenerated free carriers conduction channel in this process. For the near-infrared region (1.55 m), the photon energy is more than twice that of the graphene Fermi energy level and the infrared light cannot be absorbed by silicon. The photoexcitation of graphene takes place in a direct transition way that electrons are excited from the valence band to the conduction band. For the mid-infrared region (2.75 m), the photon energy is less than twice the energy difference between the Fermi level and the Dirac point, thus preventing direct transitions of graphene.[102] The excitation of graphene28 follows the indirect transition processes, and electrons are excited from Fermi level in the valence band to the conduction band with a large momentum mismatch under midinfrared light illumination.\",\n \"content\": \"the absorption of graphene as a function of graphene length L at different widths, indicating that as long as the appropriate W and L values are obtained, the light absorption of graphene can be greatly enhanced. For example, when W = 100 nm, more than 50% light absorption can be obtained with only 22 m length bilayer graphene. Figure 6d shows an SEM image of a waveguide-integrated graphene photodetector with high responsivity and photocurrent image at zero bias. A 53 m bilayer graphene was overlaid on the waveguide, and two electrodes with waveguide distances of 100 nm and 3.5 m, respectively, were used to collect the photogenerated carriers. The lateral metaldoped junction overlapping the waveguide is formed by doping of metal electrode 100 nm away from the waveguide, and effectively promotes the separation of photogenerated electron-hole pairs. The photocurrent image shows that the built-in electric field between metal-doped graphene and graphene appears at the metal and graphene interface and leads to two narrow photocurrent regions. Moreover, the region of maximum photocurrent coincides with the position of waveguide. Wavelength transmission measurements show that the transmission loss are 0.1 dB and 4.8 dB before and after the graphene transfer, respectively, indicating that the optical absorption of the waveguide-integrated graphene is greatly increased. The 27 photodetectors demonstrated an ultrafast response rates exceeding 20 GHz at zero bias and a high response rate of 0.108 A/W at 1 V bias. Thanks to the gapless graphene, the spectral response of photodetector ranged from 1,450 nm to 1,590 nm, which can effectively detect the c-communication band of the near-infrared region. The response spectrum of the waveguide-integrated graphene photodetector can be further extended to the mid-infrared region. Xu and coworkers demonstrated a graphene/silicon heterostructure waveguide photodetector with a responsivity of up to 0.13 A/W at room temperature in the mid-IR region (2.75 m).[102] In addition, this photodetector exhibits a low dark current due to the presence of a graphene/silicon heterojunction potential barrier, and the on/off photocurrent ratio is 30 at a bias of -1.5 V for 2.75 m illumination at room temperature(Figure 6f). This work further explains the photocurrent generation mechanism of waveguide-integrated graphene in the visible light region, near infrared region and mid-infrared region. For the visible light region, since silicon has an indirect bandgap of 1.1 eV, photoresponse is mainly due to electronhole pairs generated after the silicon absorbs light and graphene acts as a photogenerated free carriers conduction channel in this process. For the near-infrared region (1.55 m), the photon energy is more than twice that of the graphene Fermi energy level and the infrared light cannot be absorbed by silicon. The photoexcitation of graphene takes place in a direct transition way that electrons are excited from the valence band to the conduction band. For the mid-infrared region (2.75 m), the photon energy is less than twice the energy difference between the Fermi level and the Dirac point, thus preventing direct transitions of graphene.[102] The excitation of graphene28 follows the indirect transition processes, and electrons are excited from Fermi level in the valence band to the conduction band with a large momentum mismatch under midinfrared light illumination.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) SEM image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration (false colour). Inset: An enlarged view of the inside of the black dashed line. (b) The absorption of graphene as a function of graphene length L at different widths W. Solid line: bilayer graphene. Dashed line: monolayer graphene. (c) 29 Schematic of the chip-integrated graphene photodetector. (d) The spatially resolved photocurrent image of the waveguide-integrated graphene photodetector at zero bias. The black solid line shows the relative potential distribution across the graphene channel along the white dashed line. Inset: corresponding SEM image. (e) Schematic of a graphene/silicon heterostructure waveguide photodetector. (f) Photocurrent as a function of applied bias voltage with 2.75 m laser illumination at different light power.(a-b) are reproduced from Ref.[101] with permission from the Nature Publishing Group.\",\n \"content\": \"Figure 6\\n6(a) SEM image of a waveguide-integrated graphene photodetector using a GND-S-GND configuration (false colour). Inset: An enlarged view of the inside of the black dashed line. (b) The absorption of graphene as a function of graphene length L at different widths W. Solid line: bilayer graphene. Dashed line: monolayer graphene. (c) 29 Schematic of the chip-integrated graphene photodetector. (d) The spatially resolved photocurrent image of the waveguide-integrated graphene photodetector at zero bias. The black solid line shows the relative potential distribution across the graphene channel along the white dashed line. Inset: corresponding SEM image. (e) Schematic of a graphene/silicon heterostructure waveguide photodetector. (f) Photocurrent as a function of applied bias voltage with 2.75 m laser illumination at different light power.(a-b) are reproduced from Ref.[101] with permission from the Nature Publishing Group.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"is in a low doping state (VG=-8 V), showing a fast response due to photovoltaic effect. But the roll-off frequency is measured to be 0.2 MHz when BP is in a high doping state (VG=8 V), showing a slower response due to bolometric effect.\",\n \"content\": \"is in a low doping state (VG=-8 V), showing a fast response due to photovoltaic effect. But the roll-off frequency is measured to be 0.2 MHz when BP is in a high doping state (VG=8 V), showing a slower response due to bolometric effect.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"shows the intrinsic responsivity R and internal quantum efficiency IQE as a function of bias for different thicknesses. The intrinsic responsivity and internal quantum efficiency of 11.5-nm-thick BP device with top-gate are 135 mA/W and 10% at Vbias=-0.4 V, respectively. For 100-nm-thick BP device without top-gate, a larger bias voltage can be applied because there is no need to consider the breakdown of the gate insulation layer. Therefore, this photodetector shows better detection performance with high intrinsic responsivity (657 mA/W) and high internal quantum efficiency (50%) at Vbias=-0.4 V. Unlike gapless graphene, BP has a finite direct band gap,[113] which make the dark current of waveguide-integrated BP photodetector extreme low (220 nA) atVG=-8 V and Vbias=-0.4 V. (a) Schematic of the waveguide-integrated few-layer BP photodetector. (b) The frequency response of the waveguide-integrated BP photodetector at low doping state (VG=-8 V) and high doping state (VG=8 V). (c) Two-dimensional contour plot of photocurrent versus the gate voltage VG and the bias voltage VDS. (d) The intrinsic responsivity R and internal quantum efficiency IQE as a function of applied bias for different thicknesses. tBP: the thickness of BP. (a-d) are reproduced from Ref.[103] with permission from the Nature Publishing Group. Recently, Bie et al. reported a silicon photonic-crystal (PhC) waveguide-integrated\",\n \"content\": \"Figure 7d 31 Figure 7\\n7d317shows the intrinsic responsivity R and internal quantum efficiency IQE as a function of bias for different thicknesses. The intrinsic responsivity and internal quantum efficiency of 11.5-nm-thick BP device with top-gate are 135 mA/W and 10% at Vbias=-0.4 V, respectively. For 100-nm-thick BP device without top-gate, a larger bias voltage can be applied because there is no need to consider the breakdown of the gate insulation layer. Therefore, this photodetector shows better detection performance with high intrinsic responsivity (657 mA/W) and high internal quantum efficiency (50%) at Vbias=-0.4 V. Unlike gapless graphene, BP has a finite direct band gap,[113] which make the dark current of waveguide-integrated BP photodetector extreme low (220 nA) atVG=-8 V and Vbias=-0.4 V. (a) Schematic of the waveguide-integrated few-layer BP photodetector. (b) The frequency response of the waveguide-integrated BP photodetector at low doping state (VG=-8 V) and high doping state (VG=8 V). (c) Two-dimensional contour plot of photocurrent versus the gate voltage VG and the bias voltage VDS. (d) The intrinsic responsivity R and internal quantum efficiency IQE as a function of applied bias for different thicknesses. tBP: the thickness of BP. (a-d) are reproduced from Ref.[103] with permission from the Nature Publishing Group. Recently, Bie et al. reported a silicon photonic-crystal (PhC) waveguide-integrated\"\n },\n {\n \"id\": \"\",\n \"caption\": \"light is coupled into the waveguide and transmitted along the waveguide to the grating coupler in LED mode, while incident light is coupled into the waveguide at the grating coupler and detected by the MoTe2 p-n junction in the photodetector mode (Figure 8b).\",\n \"content\": \"light is coupled into the waveguide and transmitted along the waveguide to the grating coupler in LED mode, while incident light is coupled into the waveguide at the grating coupler and detected by the MoTe2 p-n junction in the photodetector mode (Figure 8b).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"shows an EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. Two extra emission spots are observed on the grating coupler on both sides, which coincides with the aforementioned LED mode (Figure 8b). The orange and green lines in Figure 8d represent the transmission spectra of the PhC waveguide before and after the bilayer MoTe2 transfer, respectively. The orange curve in\",\n \"content\": \"Figure 8c\\n8cshows an EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. Two extra emission spots are observed on the grating coupler on both sides, which coincides with the aforementioned LED mode (Figure 8b). The orange and green lines in Figure 8d represent the transmission spectra of the PhC waveguide before and after the bilayer MoTe2 transfer, respectively. The orange curve in\"\n },\n {\n \"id\": \"\",\n \"caption\": \"8f, the response spectrum of photodetector in the range from 1040 nm to 1240 nm and the largest responsivity is at 1,160 nm (the location of the PhC transmission peak), which shows the dependence of the responsivity on the wavelength. This dependence indicates that the photocurrent originates mainly from the light transmitted by the PhC waveguide, and thus the light beyond this spectral response range cannot be detected due to the bandgap of the bilayer MoTe2 and the transmission cut-off of the waveguide. This work shows that all active optical components for point-to-point interconnects are possible with 2D TMD devices transferred onto otherwise passive photonic integrated circuits.[91]\",\n \"content\": \"8f, the response spectrum of photodetector in the range from 1040 nm to 1240 nm and the largest responsivity is at 1,160 nm (the location of the PhC transmission peak), which shows the dependence of the responsivity on the wavelength. This dependence indicates that the photocurrent originates mainly from the light transmitted by the PhC waveguide, and thus the light beyond this spectral response range cannot be detected due to the bandgap of the bilayer MoTe2 and the transmission cut-off of the waveguide. This work shows that all active optical components for point-to-point interconnects are possible with 2D TMD devices transferred onto otherwise passive photonic integrated circuits.[91]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) Schematic of the silicon photonic-crystal (PhC) waveguide-integrated light-emitting diode and photodetector based on bilayer MoTe2 p-n junction. (b) The schematic diagram of the device operating in LED mode and photodetector mode. The arrow indicates the direction of light propagation in the PhC waveguide. (c) EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. (d) The orange and green lines represent the transmission spectra of the waveguide before and after the bilayer MoTe2 transfer, respectively. The normalized EL emission spectrum collected at the grating coupler represented by the blue line. (e) The temporal response of device in photodetector mode at zero bias. The 35 response bandwidth of the device is 200 MHz. (f) The response spectrum of photodetector in the range from 1,040 nm to 1,240 nm, which shows the dependence of the responsivity on the wavelength due to photonic-crystal waveguide. (a-f) are\",\n \"content\": \"Figure 8\\n8(a) Schematic of the silicon photonic-crystal (PhC) waveguide-integrated light-emitting diode and photodetector based on bilayer MoTe2 p-n junction. (b) The schematic diagram of the device operating in LED mode and photodetector mode. The arrow indicates the direction of light propagation in the PhC waveguide. (c) EL emission intensity image at room temperature cover on top of a false-colour optical image of the device. (d) The orange and green lines represent the transmission spectra of the waveguide before and after the bilayer MoTe2 transfer, respectively. The normalized EL emission spectrum collected at the grating coupler represented by the blue line. (e) The temporal response of device in photodetector mode at zero bias. The 35 response bandwidth of the device is 200 MHz. (f) The response spectrum of photodetector in the range from 1,040 nm to 1,240 nm, which shows the dependence of the responsivity on the wavelength due to photonic-crystal waveguide. (a-f) are\"\n },\n {\n \"id\": \"\",\n \"caption\": \"et al. achieved a 26-fold absorption enhancement of graphene (>60%) by monolithically integrating graphene with a Fabry-Pérot microcavity. [123] The schematic of the microcavity-integrated graphene photodetector and the corresponding electric field distribution are shown in Figure 9a. Two Bragg mirrors are alternately composed of quarter-wavelength thick materials with different refractive index, and graphene layer is sandwiched between these mirrors. The presence of the barrier layer ensures the position of resonance occur at the graphene. The incident light are reflected back and forth multiple times between two Bragg mirrors and absorbed by the graphene, getting a higher absorption, which agrees with the electric field distribution. The spectral response of the device is shown in Figure 9b, and it can be clearly seen that a significant photocurrent enhancement at the cavity resonance (855 nm). Atomic thin monolayer graphene (only 0.335 nm) can absorb more than 60% of light, indicating 36 that the integration of 2D materials with cavity has great potential in enhancing the light absorption of 2D materials. However, the enhancement of light absorption by the cavity occurs only at the resonant wavelength of the cavity. In other words, the enhancement of the cavity is at the expense of the broadband photodetection. The strategy of integrating cavity and two-dimensional materials cannot be used for broadband detectors, and thus it can only be used to enhance light absorption at specific wavelengths. For example, Casalino et al. demonstrated a vertically illuminated, resonant cavity enhanced, graphene−Si\",\n \"content\": \"Furchi\\net al. achieved a 26-fold absorption enhancement of graphene (>60%) by monolithically integrating graphene with a Fabry-Pérot microcavity. [123] The schematic of the microcavity-integrated graphene photodetector and the corresponding electric field distribution are shown in Figure 9a. Two Bragg mirrors are alternately composed of quarter-wavelength thick materials with different refractive index, and graphene layer is sandwiched between these mirrors. The presence of the barrier layer ensures the position of resonance occur at the graphene. The incident light are reflected back and forth multiple times between two Bragg mirrors and absorbed by the graphene, getting a higher absorption, which agrees with the electric field distribution. The spectral response of the device is shown in Figure 9b, and it can be clearly seen that a significant photocurrent enhancement at the cavity resonance (855 nm). Atomic thin monolayer graphene (only 0.335 nm) can absorb more than 60% of light, indicating 36 that the integration of 2D materials with cavity has great potential in enhancing the light absorption of 2D materials. However, the enhancement of light absorption by the cavity occurs only at the resonant wavelength of the cavity. In other words, the enhancement of the cavity is at the expense of the broadband photodetection. The strategy of integrating cavity and two-dimensional materials cannot be used for broadband detectors, and thus it can only be used to enhance light absorption at specific wavelengths. For example, Casalino et al. demonstrated a vertically illuminated, resonant cavity enhanced, graphene−Si\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) The schematic of the resonant cavity enhanced graphene photodetector and the corresponding electric field distribution. Red arrow: the position of graphene. (b) The dashed lines indicates the calculation result: reflection R (red), transmission T (green) and absorption A (blue), while the solid lines indicates the measurement result: reflection (red) and photocurrent (blue). A 26-fold absorption enhancement at the cavity resonance wavelength (855 nm). Inset: calculated theoretical results. (c) Schematic diagram of the resonant cavity enhanced graphene/Si Schottky photodetector. (d) The response spectrum of the device with and without Au mirror. A 3-fold absorption enhancement can be observed in the presence of Au mirror. The blue shift of the resonance peak is caused by the difference between different devices. (a-b) are\",\n \"content\": \"Figure 9\\n9(a) The schematic of the resonant cavity enhanced graphene photodetector and the corresponding electric field distribution. Red arrow: the position of graphene. (b) The dashed lines indicates the calculation result: reflection R (red), transmission T (green) and absorption A (blue), while the solid lines indicates the measurement result: reflection (red) and photocurrent (blue). A 26-fold absorption enhancement at the cavity resonance wavelength (855 nm). Inset: calculated theoretical results. (c) Schematic diagram of the resonant cavity enhanced graphene/Si Schottky photodetector. (d) The response spectrum of the device with and without Au mirror. A 3-fold absorption enhancement can be observed in the presence of Au mirror. The blue shift of the resonance peak is caused by the difference between different devices. (a-b) are\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Metal plasmonic nanostructures mainly enhance the response of the detector in two ways. First, the nearfield enhancement capability of the plasmonic nanostructures can focus the light field in a small area, effectively enhancing the light absorption of the two-dimensional material. Moreover, the plasmonic structure can generate plasma-induced hot electrons due to plasmon decay under light illumination, and then hot electrons are injected into the device across the Schottky barrier between the metal and the two-dimensional material to form photocurrent. Fang et al. demonstrated a graphene-antenna sandwich photodetector with\",\n \"content\": \"Metal plasmonic nanostructures mainly enhance the response of the detector in two ways. First, the nearfield enhancement capability of the plasmonic nanostructures can focus the light field in a small area, effectively enhancing the light absorption of the two-dimensional material. Moreover, the plasmonic structure can generate plasma-induced hot electrons due to plasmon decay under light illumination, and then hot electrons are injected into the device across the Schottky barrier between the metal and the two-dimensional material to form photocurrent. Fang et al. demonstrated a graphene-antenna sandwich photodetector with\"\n },\n {\n \"id\": \"\",\n \"caption\": \"10e and f show the measured photocurrent and the corresponding band diagram for different applied gate voltages, respectively. The gate voltage VG can effectively adjust the Fermi level of the graphene channel, and then obtain different energy band bending between the Fermi level of graphene channel and the Ti electrode-doped graphene. The steeper band bending results in a correspondingly larger built-in electric field, which can obtain higher the separation efficiency of the photogenerated electronhole pairs and thus produces a greater photocurrent. The resonance wavelength of the plasmonic clusters can be controlled by varying the diameter of disks. For example, when the diameter of heptamer plasmonic clusters changes from 80 nm to 180 nm, the corresponding resonance wavelength changes from 650 nm to 950 nm.\",\n \"content\": \"Figure\\n10e and f show the measured photocurrent and the corresponding band diagram for different applied gate voltages, respectively. The gate voltage VG can effectively adjust the Fermi level of the graphene channel, and then obtain different energy band bending between the Fermi level of graphene channel and the Ti electrode-doped graphene. The steeper band bending results in a correspondingly larger built-in electric field, which can obtain higher the separation efficiency of the photogenerated electronhole pairs and thus produces a greater photocurrent. The resonance wavelength of the plasmonic clusters can be controlled by varying the diameter of disks. For example, when the diameter of heptamer plasmonic clusters changes from 80 nm to 180 nm, the corresponding resonance wavelength changes from 650 nm to 950 nm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) The schematic of graphene-antenna sandwich photodetector. (b) 42 The optical image of device before (left) and after (right) transfer of top layer graphene. Inset: Raman mapping of the device under 785 nm laser illumination. (c) Photocurrents measured in the photodetector with dimer array and heptamer array and without pattern exhibit antisymmetric properties. The results show that the metal plasmonic antenna clusters can effectively enhance the photocurrent. The line scan position is shown by the white arrows in the inset. (d) The polarization dependent properties of the photocurrent measured in dimer and heptamer devices. (e) Photocurrents measured at different gate voltages in a heptamer device. (f) Energy band diagram of devices with different gate voltages in a heptamer device. The purple dashed line indicates the Dirac point of graphene. (g) Photocurrent as a function of different resonant wavelength. Black triangles: the measured photocurrent (I EXP (λ)). Blue triangles: the calculated DE photocurrent (I DE (λ)). Solid red triangles: the estimated HE photocurrent contribution E DE (λ) = I EXP (λ) −I DE (λ). Hollow red triangles: the calculated I HE (λ). (h) Schematic of hot electron-based bilayer MoS2 with near-infrared laser illumination. (a-g) are reproduced from Ref.[131] with permission from the American Chemical Society. (h)\",\n \"content\": \"(a) The schematic of graphene-antenna sandwich photodetector. (b) 42 The optical image of device before (left) and after (right) transfer of top layer graphene. Inset: Raman mapping of the device under 785 nm laser illumination. (c) Photocurrents measured in the photodetector with dimer array and heptamer array and without pattern exhibit antisymmetric properties. The results show that the metal plasmonic antenna clusters can effectively enhance the photocurrent. The line scan position is shown by the white arrows in the inset. (d) The polarization dependent properties of the photocurrent measured in dimer and heptamer devices. (e) Photocurrents measured at different gate voltages in a heptamer device. (f) Energy band diagram of devices with different gate voltages in a heptamer device. The purple dashed line indicates the Dirac point of graphene. (g) Photocurrent as a function of different resonant wavelength. Black triangles: the measured photocurrent (I EXP (λ)). Blue triangles: the calculated DE photocurrent (I DE (λ)). Solid red triangles: the estimated HE photocurrent contribution E DE (λ) = I EXP (λ) −I DE (λ). Hollow red triangles: the calculated I HE (λ). (h) Schematic of hot electron-based bilayer MoS2 with near-infrared laser illumination. (a-g) are reproduced from Ref.[131] with permission from the American Chemical Society. (h)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"hybrid structure based on two-dimensional materials. Hybrid 2D-QDs heterostructures possess two prominent advantages of strong light absorption contributed by quantum dots and high carrier mobility contributed by two-dimensional materials, and thus exhibiting extremely high gain and responsivity. The detection performance of photodetectors based on hybrid graphene-SWNTs structure is superior to that of photodetectors based on only graphene or SWNTs, showing high responsivity and fast response, in which the bandgap of SWNTs can be adjusted by the diameter of SWNTs to extend the spectral response range to infrared. The 2D-2D van der Waals heterostructure is also suitable for light detection in the infrared region because of ultrafast charge transfer at the interface, and the combination of two or more twodimensional materials with different optical and electrical properties not only compensates for the lack of both (eg. the combination of 2D materials with different bandgap structures can achieve broadband detection), but also exploits the characteristics of 2D materials to prepare a photodetector with specific function (such as high polarization sensitivity of a BP-on-WSe2 photogate structure). The integration of two-dimensional material on waveguide makes the light-matter interaction no longer limited by the ultrathin thickness of two-dimensional material, greatly increasing the light absorption of photodetectors to obtain a higher responsivity. In addition, this integration also shows that two-dimensional materials are compatible with the existing mature optical waveguide technology, providing a pathway for the commercialized practical application of two-dimensional materials. The integration of two-dimensional material with the cavity enhances the light absorption of the material at the resonant48 wavelength of the cavity. This selective absorption method is very suitable for precise detection of specific wavelengths. Metal plasmonic nanostructures can also enhance light absorption and photocurrent through near-field enhancement effects and hot electrons. In addition, plasma-induced hot electrons can extend the spectral response range, which is useful for infrared detection based on some two-dimensional materials with wide bandgap.\",\n \"content\": \"hybrid structure based on two-dimensional materials. Hybrid 2D-QDs heterostructures possess two prominent advantages of strong light absorption contributed by quantum dots and high carrier mobility contributed by two-dimensional materials, and thus exhibiting extremely high gain and responsivity. The detection performance of photodetectors based on hybrid graphene-SWNTs structure is superior to that of photodetectors based on only graphene or SWNTs, showing high responsivity and fast response, in which the bandgap of SWNTs can be adjusted by the diameter of SWNTs to extend the spectral response range to infrared. The 2D-2D van der Waals heterostructure is also suitable for light detection in the infrared region because of ultrafast charge transfer at the interface, and the combination of two or more twodimensional materials with different optical and electrical properties not only compensates for the lack of both (eg. the combination of 2D materials with different bandgap structures can achieve broadband detection), but also exploits the characteristics of 2D materials to prepare a photodetector with specific function (such as high polarization sensitivity of a BP-on-WSe2 photogate structure). The integration of two-dimensional material on waveguide makes the light-matter interaction no longer limited by the ultrathin thickness of two-dimensional material, greatly increasing the light absorption of photodetectors to obtain a higher responsivity. In addition, this integration also shows that two-dimensional materials are compatible with the existing mature optical waveguide technology, providing a pathway for the commercialized practical application of two-dimensional materials. The integration of two-dimensional material with the cavity enhances the light absorption of the material at the resonant48 wavelength of the cavity. This selective absorption method is very suitable for precise detection of specific wavelengths. Metal plasmonic nanostructures can also enhance light absorption and photocurrent through near-field enhancement effects and hot electrons. In addition, plasma-induced hot electrons can extend the spectral response range, which is useful for infrared detection based on some two-dimensional materials with wide bandgap.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Both WSe2 and BP absorb light efficiently and generate photogenerated electron-hole pairs under visible illumination, which are then separated by the built-in electric field. The photo-generated electrons in BP are transferred to WSe2 and the photo-generated holes in WSe2 are transferred to BP, which effectively reduces the recombination efficiency. An increasing number of photo-generated electrons in WSe2 are driven by source-drain bias to form photocurrent, thereby enhancing the responsivity. However, due to the band gap limitation of WSe2, only BP with a small band gap is an effective absorption material to generate photogenerated free carriers under infrared illumination. So the photoresponse in the infrared region will be smaller than in the visible region. The responsivity is ~10 3 A/W under visible illumination (637 nm), while the responsivity is ~0.5 A/W under infrared illumination (1,550 nm).\",\n \"content\": \"Both WSe2 and BP absorb light efficiently and generate photogenerated electron-hole pairs under visible illumination, which are then separated by the built-in electric field. The photo-generated electrons in BP are transferred to WSe2 and the photo-generated holes in WSe2 are transferred to BP, which effectively reduces the recombination efficiency. An increasing number of photo-generated electrons in WSe2 are driven by source-drain bias to form photocurrent, thereby enhancing the responsivity. However, due to the band gap limitation of WSe2, only BP with a small band gap is an effective absorption material to generate photogenerated free carriers under infrared illumination. So the photoresponse in the infrared region will be smaller than in the visible region. The responsivity is ~10 3 A/W under visible illumination (637 nm), while the responsivity is ~0.5 A/W under infrared illumination (1,550 nm).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Performance of room temperature infrared photodetectors with hybrid structure based on 2D materials. (UV: Ultraviolet. NIR: Near infrared. MIR: Midinfrared. BP: Black phosphorus. Gr: Graphene. QD: Quantum dot. FET: Field effect transistor.)In this review, we have reviewed various infrared photodetectors operating at room\",\n \"content\": \"Table 1 .\\n1Performance of room temperature infrared photodetectors with hybrid structure based on 2D materials. (UV: Ultraviolet. NIR: Near infrared. MIR: Midinfrared. BP: Black phosphorus. Gr: Graphene. QD: Quantum dot. FET: Field effect transistor.)In this review, we have reviewed various infrared photodetectors operating at roomDevice type \\nDevice \\nSpect Responsi Detectivi Respon EQE/I \\ngain \\nRef.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". P R Norton, J B Campbell, S Horn, D A Reago, International Symposium on Optical Science and Technology. 11Norton P R, Campbell J B, Horn S B and Reago D A International Symposium on Optical Science and Technology, p. 11","doi":""},{"text":". A Rogalski, Opto-Electronics Review. 20279Rogalski A 2012 Opto-Electronics Review 20 279","doi":""},{"text":". A Rogalski, J Antoszewski, L Faraone, Journal of Applied Physics. 10591101Rogalski A, Antoszewski J and Faraone L 2009 Journal of Applied Physics 105 091101","doi":""},{"text":". M Xu, T Liang, M Shi, H Chen, Chemical Reviews. 1133766Xu M, Liang T, Shi M and Chen H 2013 Chemical Reviews 113 3766","doi":""},{"text":". S Z Butler, S M Hollen, L Cao, Y Cui, J A Gupta, Gutié Rrez H R, T Heinz, Butler S Z, Hollen S M, Cao L, Cui Y, Gupta J A, Gutié rrez H R, Heinz T F,","doi":""},{"text":". S S Hong, J Huang, A F Ismach, E Johnston-Halperin, M Kuno, V V Plashnitsa, R D Robinson, R S Ruoff, S Salahuddin, J Shan, L Shi, M G Spencer, M Terrones 51, Windl W Goldberger, J E , ACS Nano. 72898Hong S S, Huang J, Ismach A F, Johnston-Halperin E, Kuno M, Plashnitsa V V, Robinson R D, Ruoff R S, Salahuddin S, Shan J, Shi L, Spencer M G, Terrones 51 M, Windl W and Goldberger J E 2013 ACS Nano 7 2898","doi":""},{"text":". H X Wu Pei, Zhang Jian, Sun Lian, Acta Physica Sinica. 66218102Wu Pei H X, Zhang Jian, Sun Lian-Feng 2017 Acta Physica Sinica 66 218102","doi":""},{"text":". C Tan Wee, L Huang, J Ng Rui, L Wang, N Hasan Dihan Md, J Duffin Thorin, S Kumar Karuppannan, Nijhuis Christian, A , Lee C Ang, K W , Tan Wee C, Huang L, Ng Rui J, Wang L, Hasan Dihan Md N, Duffin Thorin J, Kumar Karuppannan S, Nijhuis Christian A, Lee C and Ang K W 2017","doi":""},{"text":". Advanced Materials. 301705039Advanced Materials 30 1705039","doi":""},{"text":". J Lai, X Liu, J Ma, Q Wang, K Zhang, X Ren, Y Liu, Q Gu, X Zhuo, W Lu, Y Wu, Y Li, J Feng, S Zhou, J-H Chen, D Sun, Advanced Materials. 301707152Lai J, Liu X, Ma J, Wang Q, Zhang K, Ren X, Liu Y, Gu Q, Zhuo X, Lu W, Wu Y, Li Y, Feng J, Zhou S, Chen J-H and Sun D 2018 Advanced Materials 30 1707152","doi":""},{"text":". Xuefei Li, X X , Yanqing Wu, Chin. Phys. B. 2637307Xuefei Li X X, Yanqing Wu 2017 Chin. Phys. B 26 37307","doi":""},{"text":". Jianlu Wang, W , Chin. Phys. B. 2637106Jianlu Wang W H 2017 Chin. Phys. B 26 37106","doi":""},{"text":". Yuanfang Yu, F M , Jun He, Zhenhua Ni, Chin. Phys. B. 2636801Yuanfang Yu F M, Jun He, Zhenhua Ni 2017 Chin. Phys. B 26 36801","doi":""},{"text":". Zi-Wei Li, Y-H H , Yu Li, Chin. Phys. B. 2636802Zi-Wei Li Y-H H, Yu Li, Zhe-Yu Fang 2017 Chin. Phys. B 26 36802","doi":""},{"text":". Xiaomu Wang, X , Chin. Phys. B. 2634203Xiaomu Wang X G 2017 Chin. Phys. B 26 34203","doi":""},{"text":". K S Novoselov, V I Fal′ko, L Colombo, P R Gellert, M Schwab, K Kim, Nature. 490192Novoselov K S, Fal′ko V I, Colombo L, Gellert P R, Schwab M G and Kim K 2012 Nature 490 192","doi":""},{"text":". M J Allen, V Tung, R B Kaner, Chemical Reviews. 110132Allen M J, Tung V C and Kaner R B 2010 Chemical Reviews 110 132","doi":""},{"text":". Z Sun, Z Liu, J Li, Tai G A, S Lau, Yan F , Advanced Materials. 245878Sun Z, Liu Z, Li J, Tai G a, Lau S P and Yan F 2012 Advanced Materials 24 5878","doi":""},{"text":". B Y Zhang, T Liu, B Meng, X Li, G Liang, X Hu, Q J Wang, Nature Communications. 41811Zhang B Y, Liu T, Meng B, Li X, Liang G, Hu X and Wang Q J 2013 Nature Communications 4 1811","doi":""},{"text":". A Castellanos-Gomez, The Journal of Physical Chemistry Letters. 64280Castellanos-Gomez A 2015 The Journal of Physical Chemistry Letters 6 4280","doi":""},{"text":". F Xia, Wang H Jia, Y , Nature Communications. 54458Xia F, Wang H and Jia Y 2014 Nature Communications 5 4458","doi":""},{"text":". R R Nair, P Blake, A N Grigorenko, K S Novoselov, T J Booth, T Stauber, N Peres, M R, A K Geim, Science. 3201308Nair R R, Blake P, Grigorenko A N, Novoselov K S, Booth T J, Stauber T, Peres N M R and Geim A K 2008 Science 320 1308","doi":""},{"text":". K F Mak, C Lee, J Hone, Shan J Heinz, T F , Physical Review Letters. 105136805Mak K F, Lee C, Hone J, Shan J and Heinz T F 2010 Physical Review Letters 105 136805","doi":""},{"text":". I Moreels, Y Justo, De Geyter, B Haustraete, K Martins, J Hens, Z , ACS Nano. 5Moreels I, Justo Y, De Geyter B, Haustraete K, Martins J C and Hens Z 2011 ACS Nano 5 2004","doi":""},{"text":". X Yu, Y Li, X Hu, D Zhang, Y Tao, Z Liu, Y He, M A Haque, Z Liu, T Wu, Q J Wang, Nature Communications. 94299Yu X, Li Y, Hu X, Zhang D, Tao Y, Liu Z, He Y, Haque M A, Liu Z, Wu T and Wang Q J 2018 Nature Communications 9 4299","doi":""},{"text":". A Eatemadi, H Daraee, H Karimkhanloo, M Kouhi, N Zarghami, A Akbarzadeh, M Abasi, Y Hanifehpour, S W Joo, Nanoscale Research Letters. 9393Eatemadi A, Daraee H, Karimkhanloo H, Kouhi M, Zarghami N, Akbarzadeh A, Abasi M, Hanifehpour Y and Joo S W 2014 Nanoscale Research Letters 9 393","doi":""},{"text":". M L Brongersma, N Halas, P Nordlander, Nature Nanotechnology. 1025Brongersma M L, Halas N J and Nordlander P 2015 Nature Nanotechnology 10 25","doi":""},{"text":". C R Dean, A F Young, I Meric, C Lee, L Wang, S Sorgenfrei, K Watanabe, T Taniguchi, P Kim, K L Shepard, J Hone, Nature Nanotechnology. 5722Dean C R, Young A F, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard K L and Hone J 2010 Nature Nanotechnology 5 722","doi":""},{"text":". T Mueller, F Xia, P Avouris, Nature Photonics. 4297Mueller T, Xia F and Avouris P 2010 Nature Photonics 4 297","doi":""},{"text":". F Xia, T Mueller, Y-M Lin, A Valdes-Garcia, P Avouris, Nature Nanotechnology. 4839Xia F, Mueller T, Lin Y-m, Valdes-Garcia A and Avouris P 2009 Nature Nanotechnology 4 839","doi":""},{"text":". A Splendiani, L Sun, Y Zhang, T Li, J Kim, C-Y Chim, Galli G Wang, F , Nano Letters. 101271Splendiani A, Sun L, Zhang Y, Li T, Kim J, Chim C-Y, Galli G and Wang F 2010 Nano Letters 10 1271","doi":""},{"text":". Y Che, Y Zhang, X Cao, H Zhang, X Song, M Cao, Y Yu, H Dai, J Yang, G Zhang, J Yao, ACS Applied Materials & Interfaces. 932001Che Y, Zhang Y, Cao X, Zhang H, Song X, Cao M, Yu Y, Dai H, Yang J, Zhang G and Yao J 2017 ACS Applied Materials & Interfaces 9 32001","doi":""},{"text":". N Huo, S Gupta, G Konstantatos, Advanced Materials. 1606576Huo N, Gupta S and Konstantatos G 2017 Advanced Materials 1606576","doi":""},{"text":". G Konstantatos, M Badioli, L Gaudreau, J Osmond, M Bernechea, F P G D Arquer, Gatti F Koppens, F H L , Nature Nanotechnology. 7363Konstantatos G, Badioli M, Gaudreau L, Osmond J, Bernechea M, Arquer F P G D, Gatti F and Koppens F H L 2012 Nature Nanotechnology 7 363","doi":""},{"text":". Q Nian, L Gao, Y Hu, B Deng, J Tang, G J Cheng, ACS Applied Materials & Interfaces. 944715Nian Q, Gao L, Hu Y, Deng B, Tang J and Cheng G J 2017 ACS Applied Materials & Interfaces 9 44715","doi":""},{"text":". I Nikitskiy, S Goossens, D Kufer, T Lasanta, G Navickaite, F Koppens, G Konstantatos, Nature Communications. 711954Nikitskiy I, Goossens S, Kufer D, Lasanta T, Navickaite G, Koppens F H and Konstantatos G 2016 Nature Communications 7 11954","doi":""},{"text":". D Kufer, I Nikitskiy, T Lasanta, G Navickaite, Koppens Frank, H L Konstantatos, G , Advanced Materials. 27176Kufer D, Nikitskiy I, Lasanta T, Navickaite G, Koppens Frank H L and Konstantatos G 2015 Advanced Materials 27 176","doi":""},{"text":". J Schornbaum, B Winter, Schießl Stefan, P Gannott, F Katsukis, G , Guldi Dirk, M , Spiecker E Zaumseil, J , Advanced Functional Materials. 245798Schornbaum J, Winter B, Schießl Stefan P, Gannott F, Katsukis G, Guldi Dirk M, Spiecker E and Zaumseil J 2014 Advanced Functional Materials 24 5798","doi":""},{"text":". Y Zhang, M Cao, X Song, J Wang, Y Che, H Dai, X Ding, G Zhang, J Yao, The Journal of Physical Chemistry C. 11921739Zhang Y, Cao M, Song X, Wang J, Che Y, Dai H, Ding X, Zhang G and Yao J 2015 The Journal of Physical Chemistry C 119 21739","doi":""},{"text":". D Zhang, L Gan, Y Cao, Q Wang, L Qi, X Guo, Advanced Materials. 242715Zhang D, Gan L, Cao Y, Wang Q, Qi L and Guo X 2012 Advanced Materials 24 2715","doi":""},{"text":". M Sun, Q Fang, D Xie, Y Sun, L Qian, J Xu, P Xiao, C Teng, W Li, T Ren, Y Zhang, Nano Research. 113233Sun M, Fang Q, Xie D, Sun Y, Qian L, Xu J, Xiao P, Teng C, Li W, Ren T and Zhang Y 2018 Nano Research 11 3233","doi":""},{"text":". C Giansante, I Infante, E Fabiano, R Grisorio, G Suranna, G Gigli, Journal of the American Chemical Society. 1371875Giansante C, Infante I, Fabiano E, Grisorio R, Suranna G P and Gigli G 2015 Journal of the American Chemical Society 137 1875","doi":""},{"text":". G Konstantatos, J Clifford, Levina L Sargent, E H , Nature Photonics. 1531Konstantatos G, Clifford J, Levina L and Sargent E H 2007 Nature Photonics 1 531","doi":""},{"text":". L Zheng, W Zhou, Z Ning, G Wang, X Cheng, W Hu, W Zhou, Z Liu, S Yang, K Xu, Luo M , Yu , Advanced Optical Materials. 01800985Zheng L, Zhou W, Ning Z, Wang G, Cheng X, Hu W, Zhou W, Liu Z, Yang S, Xu K, Luo M and Yu Y 2018 Advanced Optical Materials 0 1800985","doi":""},{"text":". B Radisavljevic, A Radenovic, J Brivio, V Giacometti, A Kis, Nature Nanotechnology. 6147Radisavljevic B, Radenovic A, Brivio J, Giacometti V and Kis A 2011 Nature Nanotechnology 6 147","doi":""},{"text":". W Jin, P-C Yeh, N Zaki, D Zhang, J T Sadowski, Al-Mahboob , A Van Der Zande, A M Chenet, D A Dadap, J I Herman, I P Sutter, P Hone, J Osgood, R M , Physical Review Letters. 111106801Jin W, Yeh P-C, Zaki N, Zhang D, Sadowski J T, Al-Mahboob A, van der Zande A M, Chenet D A, Dadap J I, Herman I P, Sutter P, Hone J and Osgood R M 2013 Physical Review Letters 111 106801","doi":""},{"text":". G Eda, H Yamaguchi, D Voiry, T Fujita, Chen M Chhowalla, M , Nano Letters. 115111Eda G, Yamaguchi H, Voiry D, Fujita T, Chen M and Chhowalla M 2011 Nano Letters 11 5111","doi":""},{"text":". D Wu, Y Ma, Y Niu, Q Liu, T Dong, S Zhang, J Niu, H Zhou, J Wei, Y Wang, Z Zhao, Wang , Science Advances. 4Wu D, Ma Y, Niu Y, Liu Q, Dong T, Zhang S, Niu J, Zhou H, Wei J, Wang Y, Zhao Z and Wang N 2018 Science Advances 4","doi":""},{"text":". S E Keuleyan, P Guyot-Sionnest, Delerue , C , Allan G , ACS Nano. 88676Keuleyan S E, Guyot-Sionnest P, Delerue C and Allan G 2014 ACS Nano 8 8676","doi":""},{"text":". E Lhuillier, M Scarafagio, P Hease, B Nadal, H Aubin, X Z Xu, N Lequeux, G Patriarche, S Ithurria, B Dubertret, Nano Letters. 161282Lhuillier E, Scarafagio M, Hease P, Nadal B, Aubin H, Xu X Z, Lequeux N, Patriarche G, Ithurria S and Dubertret B 2016 Nano Letters 16 1282","doi":""},{"text":". D Kufer, T Lasanta, M Bernechea, F H L Koppens, G Konstantatos, ACS Photonics. 31324Kufer D, Lasanta T, Bernechea M, Koppens F H L and Konstantatos G 2016 ACS Photonics 3 1324","doi":""},{"text":". Y Liu, F Wang, X Wang, X Wang, E Flahaut, X Liu, Y Li, X Wang, Y Xu, Y Shi, R Zhang, Nature Communications. 68589Liu Y, Wang F, Wang X, Wang X, Flahaut E, Liu X, Li Y, Wang X, Xu Y, Shi Y and Zhang R 2015 Nature Communications 6 8589","doi":""},{"text":". K Roy, M Padmanabhan, S Goswami, T P Sai, G Ramalingam, S Raghavan, A Ghosh, Nature Nanotechnology. 8826Roy K, Padmanabhan M, Goswami S, Sai T P, Ramalingam G, Raghavan S and Ghosh A 2013 Nature Nanotechnology 8 826","doi":""},{"text":". G Konstantatos, L Levina, A Fischer, E H Sargent, Nano Letters. 81446Konstantatos G, Levina L, Fischer A and Sargent E H 2008 Nano Letters 8 1446","doi":""},{"text":". P Avouris, M Freitag, V Perebeinos, Nature Photonics. 2341Avouris P, Freitag M and Perebeinos V 2008 Nature Photonics 2 341","doi":""},{"text":". R Saito, M Fujita, G Dresselhaus, M S Dresselhaus, Applied Physics Letters. 602204Saito R, Fujita M, Dresselhaus G and Dresselhaus M S 1992 Applied Physics Letters 60 2204","doi":""},{"text":". H Kataura, Y Kumazawa, Y Maniwa, I Umezu, S Suzuki, Y Ohtsuka, Y Achiba, Synthetic Metals. 1032555Kataura H, Kumazawa Y, Maniwa Y, Umezu I, Suzuki S, Ohtsuka Y and Achiba Y 1999 Synthetic Metals 103 2555","doi":""},{"text":". T Dürkop, S A Getty, Cobas E Fuhrer, M S , Nano Letters. 435Dürkop T, Getty S A, Cobas E and Fuhrer M S 2004 Nano Letters 4 35","doi":""},{"text":". N M Gabor, Z Zhong, K Bosnick, Park J Mceuen, P L , Science. 3251367Gabor N M, Zhong Z, Bosnick K, Park J and McEuen P L 2009 Science 325 1367","doi":""},{"text":". L C Paulus Geraldine, Wang Qing, H , Ulissi Zachary, W , Mcnicholas Thomas, P Vijayaraghavan, A Shih, C J , Jin Z , Strano Michael, S , Small. 91954Paulus Geraldine L C, Wang Qing H, Ulissi Zachary W, McNicholas Thomas P, Vijayaraghavan A, Shih C J, Jin Z and Strano Michael S 2012 Small 9 1954","doi":""},{"text":". Y Liu, Y Liu, S Qin, Y Xu, R Zhang, F Wang, Nano Research. 1054Liu Y, Liu Y, Qin S, Xu Y, Zhang R and Wang F 2017 Nano Research 10 1880 54","doi":""},{"text":". D Jariwala, V K Sangwan, C-C Wu, P L Prabhumirashi, M L Geier, T J Marks, L Lauhon, M C Hersam, Proceedings of the National Academy of Sciences. 11018076Jariwala D, Sangwan V K, Wu C-C, Prabhumirashi P L, Geier M L, Marks T J, Lauhon L J and Hersam M C 2013 Proceedings of the National Academy of Sciences 110 18076","doi":""},{"text":". K F Mak, K He, C Lee, G H Lee, J Hone, T Heinz, J Shan, Nature Materials. 12207Mak K F, He K, Lee C, Lee G H, Hone J, Heinz T F and Shan J 2012 Nature Materials 12 207","doi":""},{"text":". S Liu, Q Liao, Z Zhang, X Zhang, S Lu, L Zhou, M Hong, Z Kang, Y Zhang, Nano Research. 103476Liu S, Liao Q, Zhang Z, Zhang X, Lu S, Zhou L, Hong M, Kang Z and Zhang Y 2017 Nano Research 10 3476","doi":""},{"text":". F Pizzocchero, L Gammelgaard, B S Jessen, J M Caridad, L Wang, J Hone, P Bøggild, T J Booth, Nature Communications. 711894Pizzocchero F, Gammelgaard L, Jessen B S, Caridad J M, Wang L, Hone J, Bøggild P and Booth T J 2016 Nature Communications 7 11894","doi":""},{"text":". Y Gong, J Lin, X Wang, G Shi, S Lei, Z Lin, X Zou, G Ye, R Vajtai, B I Yakobson, H Terrones, M Terrones, Tay Beng, K Lou, J Pantelides, S T Liu, Z Zhou, W Ajayan, P M , Nature Materials. 131135Gong Y, Lin J, Wang X, Shi G, Lei S, Lin Z, Zou X, Ye G, Vajtai R, Yakobson B I, Terrones H, Terrones M, Tay Beng K, Lou J, Pantelides S T, Liu Z, Zhou W and Ajayan P M 2014 Nature Materials 13 1135","doi":""},{"text":". Y Gong, S Lei, G Ye, B Li, Y He, K Keyshar, X Zhang, Q Wang, J Lou, Z Liu, R Vajtai, W Zhou, P M Ajayan, Nano Letters. 156135Gong Y, Lei S, Ye G, Li B, He Y, Keyshar K, Zhang X, Wang Q, Lou J, Liu Z, Vajtai R, Zhou W and Ajayan P M 2015 Nano Letters 15 6135","doi":""},{"text":". Y He, Y Yang, Z Zhang, Y Gong, W Zhou, Z Hu, G Ye, X Zhang, E Bianco, S Lei, Jin Z Zou, X Yang, Y Zhang, Y Xie, E Lou, J Yakobson, B Vajtai, R , Li B Ajayan, P , Nano Letters. 163314He Y, Yang Y, Zhang Z, Gong Y, Zhou W, Hu Z, Ye G, Zhang X, Bianco E, Lei S, Jin Z, Zou X, Yang Y, Zhang Y, Xie E, Lou J, Yakobson B, Vajtai R, Li B and Ajayan P 2016 Nano Letters 16 3314","doi":""},{"text":". Y Yu, S Hu, L Su, L Huang, Y Liu, Jin Z Purezky, A A Geohegan, D B Kim, K W Zhang, Y Cao, L , Nano Letters. 15486Yu Y, Hu S, Su L, Huang L, Liu Y, Jin Z, Purezky A A, Geohegan D B, Kim K W, Zhang Y and Cao L 2015 Nano Letters 15 486","doi":""},{"text":". Q Wang, Y Wen, K Cai, R Cheng, L Yin, Y Zhang, J Li, Z Wang, F Wang, F Wang, T A Shifa, C Jiang, Yang H He, Science Advances. 4Wang Q, Wen Y, Cai K, Cheng R, Yin L, Zhang Y, Li J, Wang Z, Wang F, Wang F, Shifa T A, Jiang C, Yang H and He J 2018 Science Advances 4","doi":""},{"text":". H Fang, C Battaglia, C Carraro, S Nemsak, B Ozdol, J S Kang, H A Bechtel, S B Desai, F Kronast, A A Unal, G Conti, C Conlon, G K Palsson, M C Martin, A M Minor, C S Fadley, E Yablonovitch, R Maboudian, A Javey, Proceedings of the National Academy of Sciences. 1116198Fang H, Battaglia C, Carraro C, Nemsak S, Ozdol B, Kang J S, Bechtel H A, Desai S B, Kronast F, Unal A A, Conti G, Conlon C, Palsson G K, Martin M C, Minor A M, Fadley C S, Yablonovitch E, Maboudian R and Javey A 2014 Proceedings of the National Academy of Sciences 111 6198","doi":""},{"text":". X Hong, J Kim, S-F Shi, Y Zhang, Jin C Sun, Y Tongay, S Wu, J Zhang, Y Wang, F , Nature Nanotechnology. 9682Hong X, Kim J, Shi S-F, Zhang Y, Jin C, Sun Y, Tongay S, Wu J, Zhang Y and Wang F 2014 Nature Nanotechnology 9 682","doi":""},{"text":". F Ceballos, M Z Bellus, H-Y Chiu, H Zhao, ACS Nano. 812717Ceballos F, Bellus M Z, Chiu H-Y and Zhao H 2014 ACS Nano 8 12717","doi":""},{"text":". R Long, O V Prezhdo, Nano Letters. 161996Long R and Prezhdo O V 2016 Nano Letters 16 1996","doi":""},{"text":". Xia Wei, F-G Y , Chao Shen, Quan-Shan Lv, Chin. Phys. B. 2638504Xia Wei F-G Y, Chao Shen, Quan-Shan Lv, Kai-You Wang 2017 Chin. Phys. B 26 38504","doi":""},{"text":". I G Lezama, A Arora, A Ubaldini, C Barreteau, E Giannini, M Potemski, A F Morpurgo, Nano Letters. 152336Lezama I G, Arora A, Ubaldini A, Barreteau C, Giannini E, Potemski M and Morpurgo A F 2015 Nano Letters 15 2336","doi":""},{"text":". C Ruppert, O Aslan, T F Heinz, Nano Letters. 146231Ruppert C, Aslan O B and Heinz T F 2014 Nano Letters 14 6231","doi":""},{"text":". A Pezeshki, H Shokouh Seyed Hossein, T Nazari, K Oh, S Im, Advanced Materials. 283216Pezeshki A, Shokouh Seyed Hossein H, Nazari T, Oh K and Im S 2016 Advanced Materials 28 3216","doi":""},{"text":". H Liu, A T Neal, Z Zhu, Z Luo, X Xu, Tomá, P D Ye, ACS Nano. 84033Liu H, Neal A T, Zhu Z, Luo Z, Xu X, Tomá nek D and Ye P D 2014 ACS Nano 8 4033","doi":""},{"text":". V Tran, R Soklaski, Y Liang, Yang L , Physical Review B. 89235319Tran V, Soklaski R, Liang Y and Yang L 2014 Physical Review B 89 235319","doi":""},{"text":". M Long, E Liu, P Wang, A Gao, H Xia, W Luo, B Wang, J Zeng, Y Fu, K Xu, W Zhou, Y Lv, S Yao, M Lu, Y Chen, Z Ni, Y You, X Zhang, S Qin, Y Shi, W Hu, D Xing, F Miao, Nano Letters. 162254Long M, Liu E, Wang P, Gao A, Xia H, Luo W, Wang B, Zeng J, Fu Y, Xu K, Zhou W, Lv Y, Yao S, Lu M, Chen Y, Ni Z, You Y, Zhang X, Qin S, Shi Y, Hu W, Xing D and Miao F 2016 Nano Letters 16 2254","doi":""},{"text":". S Das, H-Y Chen, A Penumatcha, J Appenzeller, Nano Letters. 13100Das S, Chen H-Y, Penumatcha A V and Appenzeller J 2013 Nano Letters 13 100","doi":""},{"text":". J S Ross, S Wu, H Yu, N J Ghimire, A M Jones, G Aivazian, J Yan, D G Mandrus, Xiao D Yao, W Xu, X , Nature Communications. 41474Ross J S, Wu S, Yu H, Ghimire N J, Jones A M, Aivazian G, Yan J, Mandrus D G, Xiao D, Yao W and Xu X 2013 Nature Communications 4 1474","doi":""},{"text":". H Qiao, J Yuan, Z Xu, C Chen, S Lin, Y Wang, J Song, Y Liu, Q Khan, H Y Hoh, C-X Pan, Li S Bao, Q , ACS Nano. 91886Qiao H, Yuan J, Xu Z, Chen C, Lin S, Wang Y, Song J, Liu Y, Khan Q, Hoh H Y, Pan C-X, Li S and Bao Q 2015 ACS Nano 9 1886","doi":""},{"text":". W Dang, H Peng, H Li, Wang P Liu, Z , Nano Letters. 102870Dang W, Peng H, Li H, Wang P and Liu Z 2010 Nano Letters 10 2870","doi":""},{"text":". M Hasan, C L Kane, Reviews of Modern Physics. 823045Hasan M Z and Kane C L 2010 Reviews of Modern Physics 82 3045","doi":""},{"text":". L Ye, P Wang, W Luo, F Gong, L Liao, T Liu, L Tong, J Zang, J Xu, W Hu, Nano Energy. 3753Ye L, Wang P, Luo W, Gong F, Liao L, Liu T, Tong L, Zang J, Xu J and Hu W 2017 Nano Energy 37 53","doi":""},{"text":". J Bullock, M Amani, J Cho, Y-Z Chen, G H Ahn, V Adinolfi, V R Shrestha, Y Gao, K B Crozier, Y-L Chueh, A Javey, Nature Photonics. 12601Bullock J, Amani M, Cho J, Chen Y-Z, Ahn G H, Adinolfi V, Shrestha V R, Gao Y, Crozier K B, Chueh Y-L and Javey A 2018 Nature Photonics 12 601","doi":""},{"text":". M S Choi, D Qu, D Lee, X Liu, K Watanabe, T Taniguchi, W J Yoo, ACS Nano. 89332Choi M S, Qu D, Lee D, Liu X, Watanabe K, Taniguchi T and Yoo W J 2014 ACS Nano 8 9332","doi":""},{"text":". X Yu, S Zhang, H Zeng, Q J Wang, Nano Energy. 2534Yu X, Zhang S, Zeng H and Wang Q J 2016 Nano Energy 25 34","doi":""},{"text":". L Li, Y Yu, G J Ye, Q Ge, X Ou, H Wu, D Feng, X Chen, Y Zhang, Nature Nanotechnology. 9372Li L, Yu Y, Ye G J, Ge Q, Ou X, Wu H, Feng D, Chen X H and Zhang Y 2014 Nature Nanotechnology 9 372","doi":""},{"text":". N R Pradhan, D Rhodes, Y Xin, S Memaran, L Bhaskaran, M Siddiq, S Hill, P Ajayan, L Balicas, ACS Nano. 87923Pradhan N R, Rhodes D, Xin Y, Memaran S, Bhaskaran L, Siddiq M, Hill S, Ajayan P M and Balicas L 2014 ACS Nano 8 7923","doi":""},{"text":". Y-Q Bie, G Grosso, M Heuck, M M Furchi, Y Cao, J Zheng, D Bunandar, E Navarro-Moratalla, L Zhou, D K Efetov, T Taniguchi, K Watanabe, J Kong, D Englund, Jarillo-Herrero , P , Nature Nanotechnology. 121124Bie Y-Q, Grosso G, Heuck M, Furchi M M, Cao Y, Zheng J, Bunandar D, Navarro-Moratalla E, Zhou L, Efetov D K, Taniguchi T, Watanabe K, Kong J, Englund D and Jarillo-Herrero P 2017 Nature Nanotechnology 12 1124","doi":""},{"text":". X Duan, C Wang, J C Shaw, R Cheng, Y Chen, H Li, X Wu, Y Tang, Q Zhang, A Pan, J Jiang, R Yu, Y Huang, X Duan, Nature Nanotechnology. 9Duan X, Wang C, Shaw J C, Cheng R, Chen Y, Li H, Wu X, Tang Y, Zhang Q, Pan A, Jiang J, Yu R, Huang Y and Duan X 2014 Nature Nanotechnology 9 56 1024","doi":""},{"text":". C Huang, S Wu, A M Sanchez, J J P Peters, R Beanland, J S Ross, P Rivera, W Yao, D Cobden, X Xu, Nature Materials. 131096Huang C, Wu S, Sanchez A M, Peters J J P, Beanland R, Ross J S, Rivera P, Yao W, Cobden D H and Xu X 2014 Nature Materials 13 1096","doi":""},{"text":". M-Y Li, Y Shi, C-C Cheng, L-S Lu, Y-C Lin, H-L Tang, M-L Tsai, C-W Chu, K-H Wei, J-H He, Chang W-H , Suenaga K , Li L-J , Science. 349524Li M-Y, Shi Y, Cheng C-C, Lu L-S, Lin Y-C, Tang H-L, Tsai M-L, Chu C-W, Wei K-H, He J-H, Chang W-H, Suenaga K and Li L-J 2015 Science 349 524","doi":""},{"text":". P K Sahoo, S Memaran, Y Xin, L Balicas, H Gutié Rrez, Nature. 55363Sahoo P K, Memaran S, Xin Y, Balicas L and Gutié rrez H R 2018 Nature 553 63","doi":""},{"text":". Z Zhang, P Chen, X Duan, K Zang, J Luo, X Duan, Science. 357788Zhang Z, Chen P, Duan X, Zang K, Luo J and Duan X 2017 Science 357 788","doi":""},{"text":". Y Du, H Liu, Y Deng, P D Ye, ACS Nano. 810035Du Y, Liu H, Deng Y and Ye P D 2014 ACS Nano 8 10035","doi":""},{"text":". X Gan, R-J Shiue, Y Gao, I Meric, T F Heinz, K Shepard, J Hone, S Assefa, D Englund, Nature Photonics. 7883Gan X, Shiue R-J, Gao Y, Meric I, Heinz T F, Shepard K, Hone J, Assefa S and Englund D 2013 Nature Photonics 7 883","doi":""},{"text":". I Goykhman, U Sassi, B Desiatov, N Mazurski, S Milana, D De Fazio, A Eiden, J Khurgin, J Shappir, Levy U Ferrari, A C , Nano Letters. 163005Goykhman I, Sassi U, Desiatov B, Mazurski N, Milana S, de Fazio D, Eiden A, Khurgin J, Shappir J, Levy U and Ferrari A C 2016 Nano Letters 16 3005","doi":""},{"text":". P Ma, N Flöry, Y Salamin, B Baeuerle, A Emboras, A Josten, T Taniguchi, K Watanabe, L Novotny, Leuthold , ACS Photonics. Ma P, Flöry N, Salamin Y, Baeuerle B, Emboras A, Josten A, Taniguchi T, Watanabe K, Novotny L and Leuthold J 2018 ACS Photonics","doi":""},{"text":". A Pospischil, M Humer, M M Furchi, D Bachmann, R Guider, T Fromherz, T Mueller, Nature Photonics. 7892Pospischil A, Humer M, Furchi M M, Bachmann D, Guider R, Fromherz T and Mueller T 2013 Nature Photonics 7 892","doi":""},{"text":". X Wang, Z Cheng, K Xu, H Tsang, J-B Xu, Nature Photonics. 7888Wang X, Cheng Z, Xu K, Tsang H K and Xu J-B 2013 Nature Photonics 7 888","doi":""},{"text":". N Youngblood, C Chen, S Koester, M Li, Nature Photonics. 9247Youngblood N, Chen C, Koester S J and Li M 2015 Nature Photonics 9 247","doi":""},{"text":"Dongdong Yin. Yubing Wang, W Y , Qin Han, Xiaohong Yang, Han Ye, Qianqian Lv, Chin. Phys. B. 25118103Yubing Wang W Y, Qin Han, Xiaohong Yang, Han Ye, Qianqian Lv, Dongdong Yin 2016 Chin. Phys. B 25 118103","doi":""},{"text":". M Liu, X Yin, E Ulin-Avila, B Geng, T Zentgraf, L Ju, Wang F Zhang, X , Nature. 47464Liu M, Yin X, Ulin-Avila E, Geng B, Zentgraf T, Ju L, Wang F and Zhang X 2011 Nature 474 64","doi":""},{"text":". T J Echtermeyer, L Britnell, P K Jasnos, A Lombardo, R V Gorbachev, A N Grigorenko, A K Geim, A Ferrari, K S Novoselov, Nature Communications. 2458Echtermeyer T J, Britnell L, Jasnos P K, Lombardo A, Gorbachev R V, Grigorenko A N, Geim A K, Ferrari A C and Novoselov K S 2011 Nature Communications 2 458","doi":""},{"text":". T Gu, N Petrone, J F Mcmillan, A Van Der Zande, M Yu, G Q Lo, D L Kwong, J Hone, C W Wong, Nature Photonics. 6554Gu T, Petrone N, McMillan J F, van der Zande A, Yu M, Lo G Q, Kwong D L, Hone J and Wong C W 2012 Nature Photonics 6 554","doi":""},{"text":". Q Bao, H Zhang, B Wang, Z Ni, C H Y X Lim, Y Wang, D Tang, K P Loh, Nature Photonics. 5411Bao Q, Zhang H, Wang B, Ni Z, Lim C H Y X, Wang Y, Tang D Y and Loh K P 2011 Nature Photonics 5 411","doi":""},{"text":". M Freitag, T Low, F Xia, P Avouris, Nature Photonics. 753Freitag M, Low T, Xia F and Avouris P 2012 Nature Photonics 7 53","doi":""},{"text":". T Low, A S Rodin, A Carvalho, Y Jiang, H Wang, F Xia, Castro Neto, A , Physical Review B. 9075434Low T, Rodin A S, Carvalho A, Jiang Y, Wang H, Xia F and Castro Neto A H 2014 Physical Review B 90 075434","doi":""},{"text":". T Hong, B Chamlagain, W Z Lin, H J Chuang, M H Pan, Z Zhou, Y Q Xu, NANOSCALE. 68978Hong T, Chamlagain B, Lin W Z, Chuang H J, Pan M H, Zhou Z X and Xu Y Q 2014 NANOSCALE 6 8978","doi":""},{"text":". T Low, M Engel, M Steiner, P Avouris, Physical Review B. 9081408Low T, Engel M, Steiner M and Avouris P 2014 Physical Review B 90 081408","doi":""},{"text":". S Das, W Zhang, M Demarteau, A Hoffmann, M Dubey, A Roelofs, Nano Letters. 145733Das S, Zhang W, Demarteau M, Hoffmann A, Dubey M and Roelofs A 2014 Nano Letters 14 5733","doi":""},{"text":". C Robert, R Picard, D Lagarde, G Wang, J P Echeverry, F Cadiz, P Renucci, A Högele, T Amand, Marie X Gerber, I Urbaszek, B , Physical Review B. 94155425Robert C, Picard R, Lagarde D, Wang G, Echeverry J P, Cadiz F, Renucci P, Högele A, Amand T, Marie X, Gerber I C and Urbaszek B 2016 Physical Review B 94 155425","doi":""},{"text":". G Froehlicher, Lorchat E Berciaud, S , Physical Review B. 9485429Froehlicher G, Lorchat E and Berciaud S 2016 Physical Review B 94 085429","doi":""},{"text":". L Yin, X Zhan, K Xu, F Wang, Z Wang, Y Huang, Q Wang, C Jiang, J He, Applied Physics Letters. 10843503Yin L, Zhan X, Xu K, Wang F, Wang Z, Huang Y, Wang Q, Jiang C and He J 2016 Applied Physics Letters 108 043503","doi":""},{"text":". Octon Tobias, J Nagareddy, V K Russo, S , Craciun Monica, F Wright, C D , Advanced Optical Materials. 41750Octon Tobias J, Nagareddy V K, Russo S, Craciun Monica F and Wright C D 2016 Advanced Optical Materials 4 1750","doi":""},{"text":". X Gan, Y Gao, Fai Mak, K Yao, X Shiue, R-J Van Der Zande, A Trusheim, M E Hatami, F Heinz, T F Hone, J Englund, D , Applied Physics Letters. 103181119Gan X, Gao Y, Fai Mak K, Yao X, Shiue R-J, van der Zande A, Trusheim M E, Hatami F, Heinz T F, Hone J and Englund D 2013 Applied Physics Letters 103 181119","doi":""},{"text":". S Wu, S Buckley, J R Schaibley, L Feng, J Yan, D G Mandrus, F Hatami, W Yao, J Vučković, A Majumdar, X Xu, Nature. 52069Wu S, Buckley S, Schaibley J R, Feng L, Yan J, Mandrus D G, Hatami F, Yao W, Vučković J, Majumdar A and Xu X 2015 Nature 520 69","doi":""},{"text":". X Liu, T Galfsky, Z Sun, F Xia, E-C Lin, Y-H Lee, Ké Na-Cohen, S Menon, V M , Nature Photonics. 930Liu X, Galfsky T, Sun Z, Xia F, Lin E-c, Lee Y-H, Ké na-Cohen S and Menon V M 2014 Nature Photonics 9 30","doi":""},{"text":". S Dufferwiel, S Schwarz, F Withers, A A P Trichet, F Li, M Sich, Del Pozo-Zamudio, O Clark, C Nalitov, A Solnyshkov, D D Malpuech, G Novoselov, K S Smith, J M Skolnick, M S Krizhanovskii, D Tartakovskii, A I , Dufferwiel S, Schwarz S, Withers F, Trichet A A P, Li F, Sich M, Del Pozo- Zamudio O, Clark C, Nalitov A, Solnyshkov D D, Malpuech G, Novoselov K S, Smith J M, Skolnick M S, Krizhanovskii D N and Tartakovskii A I 2015","doi":""},{"text":". Nature Communications. 68579Nature Communications 6 8579","doi":""},{"text":". Y Ye, Z J Wong, X Lu, X Ni, H Zhu, X Chen, Wang Y Zhang, X , Nature Photonics. 9733Ye Y, Wong Z J, Lu X, Ni X, Zhu H, Chen X, Wang Y and Zhang X 2015 Nature Photonics 9 733","doi":""},{"text":". M Furchi, A Urich, A Pospischil, G Lilley, K Unterrainer, H Detz, P Klang, A M Andrews, W Schrenk, G Strasser, T Mueller, Nano Letters. 122773Furchi M, Urich A, Pospischil A, Lilley G, Unterrainer K, Detz H, Klang P, Andrews A M, Schrenk W, Strasser G and Mueller T 2012 Nano Letters 12 2773","doi":""},{"text":". M Casalino, U Sassi, I Goykhman, A Eiden, E Lidorikis, S Milana, De Fazio, D Tomarchio, F Iodice, M Coppola, G Ferrari, A C , ACS Nano. 1110955Casalino M, Sassi U, Goykhman I, Eiden A, Lidorikis E, Milana S, De Fazio D, Tomarchio F, Iodice M, Coppola G and Ferrari A C 2017 ACS Nano 11 10955","doi":""},{"text":". Q Guo, C Li, B Deng, S Yuan, F Guinea, F Xia, ACS Photonics. 42989Guo Q, Li C, Deng B, Yuan S, Guinea F and Xia F 2017 ACS Photonics 4 2989","doi":""},{"text":". R Kumar, A Sharma, M Kaur, S Husale, Advanced Optical Materials. 51700009Kumar R, Sharma A, Kaur M and Husale S 2017 Advanced Optical Materials 5 1700009","doi":""},{"text":". I J Luxmoore, P Q Liu, P Li, J Faist, G R Nash, ACS Photonics. 3936Luxmoore I J, Liu P Q, Li P, Faist J and Nash G R 2016 ACS Photonics 3 936","doi":""},{"text":". Z Xia, P Li, Y Wang, T Song, Q Zhang, B Sun, ACS Applied Materials & Interfaces. 724136Xia Z, Li P, Wang Y, Song T, Zhang Q and Sun B 2015 ACS Applied Materials & Interfaces 7 24136","doi":""},{"text":". W Wang, A Klots, D Prasai, Y Yang, K Bolotin, J Valentine, Nano Letters. 157440Wang W, Klots A, Prasai D, Yang Y, Bolotin K I and Valentine J 2015 Nano Letters 15 7440","doi":""},{"text":". T Hong, B Chamlagain, S Hu, S M Weiss, Z Zhou, Y-Q Xu, ACS Nano. 95357Hong T, Chamlagain B, Hu S, Weiss S M, Zhou Z and Xu Y-Q 2015 ACS Nano 9 5357","doi":""},{"text":". Z Fang, Z Liu, Y Wang, P M Ajayan, Nordlander P Halas, N J , Nano Letters. 123808Fang Z, Liu Z, Wang Y, Ajayan P M, Nordlander P and Halas N J 2012 Nano Letters 12 3808","doi":""},{"text":". Somayyeh Asgari, H R , Nosrat Granpayeh, Homayoon Oraizi, Chin. Phys. B. 2784212Somayyeh Asgari H R, Nosrat Granpayeh, Homayoon Oraizi 2018 Chin. Phys. B 27 84212","doi":""},{"text":". Z Chen, X Li, J Wang, L Tao, M Long, S-J Liang, L K Ang, C Shu, H Tsang, J-B Xu, ACS Nano. 11430Chen Z, Li X, Wang J, Tao L, Long M, Liang S-J, Ang L K, Shu C, Tsang H K and Xu J-B 2017 ACS Nano 11 430","doi":""},{"text":". M Stockman, Nature. 467541Stockman M I 2010 Nature 467 541","doi":""},{"text":". M W Knight, H Sobhani, P Nordlander, N J Halas, Science. 332702Knight M W, Sobhani H, Nordlander P and Halas N J 2011 Science 332 702","doi":""},{"text":". H S Lee, Min S-W Chang, Y-G Park, M K , Nam T Kim, H Kim, J H , Ryu S Im, S , Nano Letters. 123695Lee H S, Min S-W, Chang Y-G, Park M K, Nam T, Kim H, Kim J H, Ryu S and Im S 2012 Nano Letters 12 3695","doi":""},{"text":". J-Y Wu, Y T Chun, S Li, T Zhang, J Wang, P Shrestha, D Chu, Advanced Materials. 301705880Wu J-Y, Chun Y T, Li S, Zhang T, Wang J, Shrestha P K and Chu D 2018 Advanced Materials 30 1705880","doi":""},{"text":". Z Ni, L Ma, S Du, Y Xu, M Yuan, H Fang, Z Wang, M Xu, D Li, J Yang, W Hu, X Pi, Yang D , ACS Nano. 119854Ni Z, Ma L, Du S, Xu Y, Yuan M, Fang H, Wang Z, Xu M, Li D, Yang J, Hu W, Pi X and Yang D 2017 ACS Nano 11 9854","doi":""},{"text":". N Guo, F Gong, J Liu, Y Jia, S Zhao, L Liao, M Su, Z Fan, X Chen, W Lu, Xiao L Hu, W , ACS Applied Materials & Interfaces. 934489Guo N, Gong F, Liu J, Jia Y, Zhao S, Liao L, Su M, Fan Z, Chen X, Lu W, Xiao L and Hu W 2017 ACS Applied Materials & Interfaces 9 34489","doi":""},{"text":". M Massicotte, P Schmidt, F Vialla, K Watanabe, T Taniguchi, K Tielrooij, F H L Koppens, Nature Communications. 712174Massicotte M, Schmidt P, Vialla F, Watanabe K, Taniguchi T, Tielrooij K J and Koppens F H L 2016 Nature Communications 7 12174","doi":""},{"text":". N Huo, G Konstantatos, Nature Communications. 8572Huo N and Konstantatos G 2017 Nature Communications 8 572","doi":""},{"text":". J Kim, S Park, H Jang, N Koirala, J-B Lee, U J Kim, H-S Lee, Y-G Roh, H Lee, S Sim, S Cha, C In, J Park, J Lee, M Noh, J Moon, M Salehi, J Sung, S Chee, Kim J, Park S, Jang H, Koirala N, Lee J-B, Kim U J, Lee H-S, Roh Y-G, Lee H, Sim S, Cha S, In C, Park J, Lee J, Noh M, Moon J, Salehi M, Sung J, Chee S-","doi":""},{"text":". S , Ham M-H , Jo M-H Oh, S Ahn, J-H Hwang, S W , Kim D Choi, H , S, Ham M-H, Jo M-H, Oh S, Ahn J-H, Hwang S W, Kim D and Choi H 2017","doi":""},{"text":". ACS Photonics. 4482ACS Photonics 4 482","doi":""}],"string":"[\n {\n \"text\": \". P R Norton, J B Campbell, S Horn, D A Reago, International Symposium on Optical Science and Technology. 11Norton P R, Campbell J B, Horn S B and Reago D A International Symposium on Optical Science and Technology, p. 11\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Rogalski, Opto-Electronics Review. 20279Rogalski A 2012 Opto-Electronics Review 20 279\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Rogalski, J Antoszewski, L Faraone, Journal of Applied Physics. 10591101Rogalski A, Antoszewski J and Faraone L 2009 Journal of Applied Physics 105 091101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Xu, T Liang, M Shi, H Chen, Chemical Reviews. 1133766Xu M, Liang T, Shi M and Chen H 2013 Chemical Reviews 113 3766\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Z Butler, S M Hollen, L Cao, Y Cui, J A Gupta, Gutié Rrez H R, T Heinz, Butler S Z, Hollen S M, Cao L, Cui Y, Gupta J A, Gutié rrez H R, Heinz T F,\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S S Hong, J Huang, A F Ismach, E Johnston-Halperin, M Kuno, V V Plashnitsa, R D Robinson, R S Ruoff, S Salahuddin, J Shan, L Shi, M G Spencer, M Terrones 51, Windl W Goldberger, J E , ACS Nano. 72898Hong S S, Huang J, Ismach A F, Johnston-Halperin E, Kuno M, Plashnitsa V V, Robinson R D, Ruoff R S, Salahuddin S, Shan J, Shi L, Spencer M G, Terrones 51 M, Windl W and Goldberger J E 2013 ACS Nano 7 2898\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H X Wu Pei, Zhang Jian, Sun Lian, Acta Physica Sinica. 66218102Wu Pei H X, Zhang Jian, Sun Lian-Feng 2017 Acta Physica Sinica 66 218102\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Tan Wee, L Huang, J Ng Rui, L Wang, N Hasan Dihan Md, J Duffin Thorin, S Kumar Karuppannan, Nijhuis Christian, A , Lee C Ang, K W , Tan Wee C, Huang L, Ng Rui J, Wang L, Hasan Dihan Md N, Duffin Thorin J, Kumar Karuppannan S, Nijhuis Christian A, Lee C and Ang K W 2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Advanced Materials. 301705039Advanced Materials 30 1705039\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Lai, X Liu, J Ma, Q Wang, K Zhang, X Ren, Y Liu, Q Gu, X Zhuo, W Lu, Y Wu, Y Li, J Feng, S Zhou, J-H Chen, D Sun, Advanced Materials. 301707152Lai J, Liu X, Ma J, Wang Q, Zhang K, Ren X, Liu Y, Gu Q, Zhuo X, Lu W, Wu Y, Li Y, Feng J, Zhou S, Chen J-H and Sun D 2018 Advanced Materials 30 1707152\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Xuefei Li, X X , Yanqing Wu, Chin. Phys. B. 2637307Xuefei Li X X, Yanqing Wu 2017 Chin. Phys. B 26 37307\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Jianlu Wang, W , Chin. Phys. B. 2637106Jianlu Wang W H 2017 Chin. Phys. B 26 37106\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Yuanfang Yu, F M , Jun He, Zhenhua Ni, Chin. Phys. B. 2636801Yuanfang Yu F M, Jun He, Zhenhua Ni 2017 Chin. Phys. B 26 36801\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Zi-Wei Li, Y-H H , Yu Li, Chin. Phys. B. 2636802Zi-Wei Li Y-H H, Yu Li, Zhe-Yu Fang 2017 Chin. Phys. B 26 36802\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Xiaomu Wang, X , Chin. Phys. B. 2634203Xiaomu Wang X G 2017 Chin. Phys. B 26 34203\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K S Novoselov, V I Fal′ko, L Colombo, P R Gellert, M Schwab, K Kim, Nature. 490192Novoselov K S, Fal′ko V I, Colombo L, Gellert P R, Schwab M G and Kim K 2012 Nature 490 192\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M J Allen, V Tung, R B Kaner, Chemical Reviews. 110132Allen M J, Tung V C and Kaner R B 2010 Chemical Reviews 110 132\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z Sun, Z Liu, J Li, Tai G A, S Lau, Yan F , Advanced Materials. 245878Sun Z, Liu Z, Li J, Tai G a, Lau S P and Yan F 2012 Advanced Materials 24 5878\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Y Zhang, T Liu, B Meng, X Li, G Liang, X Hu, Q J Wang, Nature Communications. 41811Zhang B Y, Liu T, Meng B, Li X, Liang G, Hu X and Wang Q J 2013 Nature Communications 4 1811\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Castellanos-Gomez, The Journal of Physical Chemistry Letters. 64280Castellanos-Gomez A 2015 The Journal of Physical Chemistry Letters 6 4280\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Xia, Wang H Jia, Y , Nature Communications. 54458Xia F, Wang H and Jia Y 2014 Nature Communications 5 4458\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R R Nair, P Blake, A N Grigorenko, K S Novoselov, T J Booth, T Stauber, N Peres, M R, A K Geim, Science. 3201308Nair R R, Blake P, Grigorenko A N, Novoselov K S, Booth T J, Stauber T, Peres N M R and Geim A K 2008 Science 320 1308\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K F Mak, C Lee, J Hone, Shan J Heinz, T F , Physical Review Letters. 105136805Mak K F, Lee C, Hone J, Shan J and Heinz T F 2010 Physical Review Letters 105 136805\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Moreels, Y Justo, De Geyter, B Haustraete, K Martins, J Hens, Z , ACS Nano. 5Moreels I, Justo Y, De Geyter B, Haustraete K, Martins J C and Hens Z 2011 ACS Nano 5 2004\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Yu, Y Li, X Hu, D Zhang, Y Tao, Z Liu, Y He, M A Haque, Z Liu, T Wu, Q J Wang, Nature Communications. 94299Yu X, Li Y, Hu X, Zhang D, Tao Y, Liu Z, He Y, Haque M A, Liu Z, Wu T and Wang Q J 2018 Nature Communications 9 4299\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Eatemadi, H Daraee, H Karimkhanloo, M Kouhi, N Zarghami, A Akbarzadeh, M Abasi, Y Hanifehpour, S W Joo, Nanoscale Research Letters. 9393Eatemadi A, Daraee H, Karimkhanloo H, Kouhi M, Zarghami N, Akbarzadeh A, Abasi M, Hanifehpour Y and Joo S W 2014 Nanoscale Research Letters 9 393\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M L Brongersma, N Halas, P Nordlander, Nature Nanotechnology. 1025Brongersma M L, Halas N J and Nordlander P 2015 Nature Nanotechnology 10 25\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C R Dean, A F Young, I Meric, C Lee, L Wang, S Sorgenfrei, K Watanabe, T Taniguchi, P Kim, K L Shepard, J Hone, Nature Nanotechnology. 5722Dean C R, Young A F, Meric I, Lee C, Wang L, Sorgenfrei S, Watanabe K, Taniguchi T, Kim P, Shepard K L and Hone J 2010 Nature Nanotechnology 5 722\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Mueller, F Xia, P Avouris, Nature Photonics. 4297Mueller T, Xia F and Avouris P 2010 Nature Photonics 4 297\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Xia, T Mueller, Y-M Lin, A Valdes-Garcia, P Avouris, Nature Nanotechnology. 4839Xia F, Mueller T, Lin Y-m, Valdes-Garcia A and Avouris P 2009 Nature Nanotechnology 4 839\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Splendiani, L Sun, Y Zhang, T Li, J Kim, C-Y Chim, Galli G Wang, F , Nano Letters. 101271Splendiani A, Sun L, Zhang Y, Li T, Kim J, Chim C-Y, Galli G and Wang F 2010 Nano Letters 10 1271\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Che, Y Zhang, X Cao, H Zhang, X Song, M Cao, Y Yu, H Dai, J Yang, G Zhang, J Yao, ACS Applied Materials & Interfaces. 932001Che Y, Zhang Y, Cao X, Zhang H, Song X, Cao M, Yu Y, Dai H, Yang J, Zhang G and Yao J 2017 ACS Applied Materials & Interfaces 9 32001\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Huo, S Gupta, G Konstantatos, Advanced Materials. 1606576Huo N, Gupta S and Konstantatos G 2017 Advanced Materials 1606576\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Konstantatos, M Badioli, L Gaudreau, J Osmond, M Bernechea, F P G D Arquer, Gatti F Koppens, F H L , Nature Nanotechnology. 7363Konstantatos G, Badioli M, Gaudreau L, Osmond J, Bernechea M, Arquer F P G D, Gatti F and Koppens F H L 2012 Nature Nanotechnology 7 363\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Q Nian, L Gao, Y Hu, B Deng, J Tang, G J Cheng, ACS Applied Materials & Interfaces. 944715Nian Q, Gao L, Hu Y, Deng B, Tang J and Cheng G J 2017 ACS Applied Materials & Interfaces 9 44715\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Nikitskiy, S Goossens, D Kufer, T Lasanta, G Navickaite, F Koppens, G Konstantatos, Nature Communications. 711954Nikitskiy I, Goossens S, Kufer D, Lasanta T, Navickaite G, Koppens F H and Konstantatos G 2016 Nature Communications 7 11954\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Kufer, I Nikitskiy, T Lasanta, G Navickaite, Koppens Frank, H L Konstantatos, G , Advanced Materials. 27176Kufer D, Nikitskiy I, Lasanta T, Navickaite G, Koppens Frank H L and Konstantatos G 2015 Advanced Materials 27 176\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Schornbaum, B Winter, Schießl Stefan, P Gannott, F Katsukis, G , Guldi Dirk, M , Spiecker E Zaumseil, J , Advanced Functional Materials. 245798Schornbaum J, Winter B, Schießl Stefan P, Gannott F, Katsukis G, Guldi Dirk M, Spiecker E and Zaumseil J 2014 Advanced Functional Materials 24 5798\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Zhang, M Cao, X Song, J Wang, Y Che, H Dai, X Ding, G Zhang, J Yao, The Journal of Physical Chemistry C. 11921739Zhang Y, Cao M, Song X, Wang J, Che Y, Dai H, Ding X, Zhang G and Yao J 2015 The Journal of Physical Chemistry C 119 21739\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Zhang, L Gan, Y Cao, Q Wang, L Qi, X Guo, Advanced Materials. 242715Zhang D, Gan L, Cao Y, Wang Q, Qi L and Guo X 2012 Advanced Materials 24 2715\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Sun, Q Fang, D Xie, Y Sun, L Qian, J Xu, P Xiao, C Teng, W Li, T Ren, Y Zhang, Nano Research. 113233Sun M, Fang Q, Xie D, Sun Y, Qian L, Xu J, Xiao P, Teng C, Li W, Ren T and Zhang Y 2018 Nano Research 11 3233\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Giansante, I Infante, E Fabiano, R Grisorio, G Suranna, G Gigli, Journal of the American Chemical Society. 1371875Giansante C, Infante I, Fabiano E, Grisorio R, Suranna G P and Gigli G 2015 Journal of the American Chemical Society 137 1875\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Konstantatos, J Clifford, Levina L Sargent, E H , Nature Photonics. 1531Konstantatos G, Clifford J, Levina L and Sargent E H 2007 Nature Photonics 1 531\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Zheng, W Zhou, Z Ning, G Wang, X Cheng, W Hu, W Zhou, Z Liu, S Yang, K Xu, Luo M , Yu , Advanced Optical Materials. 01800985Zheng L, Zhou W, Ning Z, Wang G, Cheng X, Hu W, Zhou W, Liu Z, Yang S, Xu K, Luo M and Yu Y 2018 Advanced Optical Materials 0 1800985\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Radisavljevic, A Radenovic, J Brivio, V Giacometti, A Kis, Nature Nanotechnology. 6147Radisavljevic B, Radenovic A, Brivio J, Giacometti V and Kis A 2011 Nature Nanotechnology 6 147\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Jin, P-C Yeh, N Zaki, D Zhang, J T Sadowski, Al-Mahboob , A Van Der Zande, A M Chenet, D A Dadap, J I Herman, I P Sutter, P Hone, J Osgood, R M , Physical Review Letters. 111106801Jin W, Yeh P-C, Zaki N, Zhang D, Sadowski J T, Al-Mahboob A, van der Zande A M, Chenet D A, Dadap J I, Herman I P, Sutter P, Hone J and Osgood R M 2013 Physical Review Letters 111 106801\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Eda, H Yamaguchi, D Voiry, T Fujita, Chen M Chhowalla, M , Nano Letters. 115111Eda G, Yamaguchi H, Voiry D, Fujita T, Chen M and Chhowalla M 2011 Nano Letters 11 5111\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Wu, Y Ma, Y Niu, Q Liu, T Dong, S Zhang, J Niu, H Zhou, J Wei, Y Wang, Z Zhao, Wang , Science Advances. 4Wu D, Ma Y, Niu Y, Liu Q, Dong T, Zhang S, Niu J, Zhou H, Wei J, Wang Y, Zhao Z and Wang N 2018 Science Advances 4\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S E Keuleyan, P Guyot-Sionnest, Delerue , C , Allan G , ACS Nano. 88676Keuleyan S E, Guyot-Sionnest P, Delerue C and Allan G 2014 ACS Nano 8 8676\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Lhuillier, M Scarafagio, P Hease, B Nadal, H Aubin, X Z Xu, N Lequeux, G Patriarche, S Ithurria, B Dubertret, Nano Letters. 161282Lhuillier E, Scarafagio M, Hease P, Nadal B, Aubin H, Xu X Z, Lequeux N, Patriarche G, Ithurria S and Dubertret B 2016 Nano Letters 16 1282\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Kufer, T Lasanta, M Bernechea, F H L Koppens, G Konstantatos, ACS Photonics. 31324Kufer D, Lasanta T, Bernechea M, Koppens F H L and Konstantatos G 2016 ACS Photonics 3 1324\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Liu, F Wang, X Wang, X Wang, E Flahaut, X Liu, Y Li, X Wang, Y Xu, Y Shi, R Zhang, Nature Communications. 68589Liu Y, Wang F, Wang X, Wang X, Flahaut E, Liu X, Li Y, Wang X, Xu Y, Shi Y and Zhang R 2015 Nature Communications 6 8589\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Roy, M Padmanabhan, S Goswami, T P Sai, G Ramalingam, S Raghavan, A Ghosh, Nature Nanotechnology. 8826Roy K, Padmanabhan M, Goswami S, Sai T P, Ramalingam G, Raghavan S and Ghosh A 2013 Nature Nanotechnology 8 826\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Konstantatos, L Levina, A Fischer, E H Sargent, Nano Letters. 81446Konstantatos G, Levina L, Fischer A and Sargent E H 2008 Nano Letters 8 1446\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Avouris, M Freitag, V Perebeinos, Nature Photonics. 2341Avouris P, Freitag M and Perebeinos V 2008 Nature Photonics 2 341\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Saito, M Fujita, G Dresselhaus, M S Dresselhaus, Applied Physics Letters. 602204Saito R, Fujita M, Dresselhaus G and Dresselhaus M S 1992 Applied Physics Letters 60 2204\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Kataura, Y Kumazawa, Y Maniwa, I Umezu, S Suzuki, Y Ohtsuka, Y Achiba, Synthetic Metals. 1032555Kataura H, Kumazawa Y, Maniwa Y, Umezu I, Suzuki S, Ohtsuka Y and Achiba Y 1999 Synthetic Metals 103 2555\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Dürkop, S A Getty, Cobas E Fuhrer, M S , Nano Letters. 435Dürkop T, Getty S A, Cobas E and Fuhrer M S 2004 Nano Letters 4 35\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N M Gabor, Z Zhong, K Bosnick, Park J Mceuen, P L , Science. 3251367Gabor N M, Zhong Z, Bosnick K, Park J and McEuen P L 2009 Science 325 1367\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L C Paulus Geraldine, Wang Qing, H , Ulissi Zachary, W , Mcnicholas Thomas, P Vijayaraghavan, A Shih, C J , Jin Z , Strano Michael, S , Small. 91954Paulus Geraldine L C, Wang Qing H, Ulissi Zachary W, McNicholas Thomas P, Vijayaraghavan A, Shih C J, Jin Z and Strano Michael S 2012 Small 9 1954\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Liu, Y Liu, S Qin, Y Xu, R Zhang, F Wang, Nano Research. 1054Liu Y, Liu Y, Qin S, Xu Y, Zhang R and Wang F 2017 Nano Research 10 1880 54\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Jariwala, V K Sangwan, C-C Wu, P L Prabhumirashi, M L Geier, T J Marks, L Lauhon, M C Hersam, Proceedings of the National Academy of Sciences. 11018076Jariwala D, Sangwan V K, Wu C-C, Prabhumirashi P L, Geier M L, Marks T J, Lauhon L J and Hersam M C 2013 Proceedings of the National Academy of Sciences 110 18076\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K F Mak, K He, C Lee, G H Lee, J Hone, T Heinz, J Shan, Nature Materials. 12207Mak K F, He K, Lee C, Lee G H, Hone J, Heinz T F and Shan J 2012 Nature Materials 12 207\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Liu, Q Liao, Z Zhang, X Zhang, S Lu, L Zhou, M Hong, Z Kang, Y Zhang, Nano Research. 103476Liu S, Liao Q, Zhang Z, Zhang X, Lu S, Zhou L, Hong M, Kang Z and Zhang Y 2017 Nano Research 10 3476\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Pizzocchero, L Gammelgaard, B S Jessen, J M Caridad, L Wang, J Hone, P Bøggild, T J Booth, Nature Communications. 711894Pizzocchero F, Gammelgaard L, Jessen B S, Caridad J M, Wang L, Hone J, Bøggild P and Booth T J 2016 Nature Communications 7 11894\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Gong, J Lin, X Wang, G Shi, S Lei, Z Lin, X Zou, G Ye, R Vajtai, B I Yakobson, H Terrones, M Terrones, Tay Beng, K Lou, J Pantelides, S T Liu, Z Zhou, W Ajayan, P M , Nature Materials. 131135Gong Y, Lin J, Wang X, Shi G, Lei S, Lin Z, Zou X, Ye G, Vajtai R, Yakobson B I, Terrones H, Terrones M, Tay Beng K, Lou J, Pantelides S T, Liu Z, Zhou W and Ajayan P M 2014 Nature Materials 13 1135\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Gong, S Lei, G Ye, B Li, Y He, K Keyshar, X Zhang, Q Wang, J Lou, Z Liu, R Vajtai, W Zhou, P M Ajayan, Nano Letters. 156135Gong Y, Lei S, Ye G, Li B, He Y, Keyshar K, Zhang X, Wang Q, Lou J, Liu Z, Vajtai R, Zhou W and Ajayan P M 2015 Nano Letters 15 6135\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y He, Y Yang, Z Zhang, Y Gong, W Zhou, Z Hu, G Ye, X Zhang, E Bianco, S Lei, Jin Z Zou, X Yang, Y Zhang, Y Xie, E Lou, J Yakobson, B Vajtai, R , Li B Ajayan, P , Nano Letters. 163314He Y, Yang Y, Zhang Z, Gong Y, Zhou W, Hu Z, Ye G, Zhang X, Bianco E, Lei S, Jin Z, Zou X, Yang Y, Zhang Y, Xie E, Lou J, Yakobson B, Vajtai R, Li B and Ajayan P 2016 Nano Letters 16 3314\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Yu, S Hu, L Su, L Huang, Y Liu, Jin Z Purezky, A A Geohegan, D B Kim, K W Zhang, Y Cao, L , Nano Letters. 15486Yu Y, Hu S, Su L, Huang L, Liu Y, Jin Z, Purezky A A, Geohegan D B, Kim K W, Zhang Y and Cao L 2015 Nano Letters 15 486\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Q Wang, Y Wen, K Cai, R Cheng, L Yin, Y Zhang, J Li, Z Wang, F Wang, F Wang, T A Shifa, C Jiang, Yang H He, Science Advances. 4Wang Q, Wen Y, Cai K, Cheng R, Yin L, Zhang Y, Li J, Wang Z, Wang F, Wang F, Shifa T A, Jiang C, Yang H and He J 2018 Science Advances 4\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Fang, C Battaglia, C Carraro, S Nemsak, B Ozdol, J S Kang, H A Bechtel, S B Desai, F Kronast, A A Unal, G Conti, C Conlon, G K Palsson, M C Martin, A M Minor, C S Fadley, E Yablonovitch, R Maboudian, A Javey, Proceedings of the National Academy of Sciences. 1116198Fang H, Battaglia C, Carraro C, Nemsak S, Ozdol B, Kang J S, Bechtel H A, Desai S B, Kronast F, Unal A A, Conti G, Conlon C, Palsson G K, Martin M C, Minor A M, Fadley C S, Yablonovitch E, Maboudian R and Javey A 2014 Proceedings of the National Academy of Sciences 111 6198\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Hong, J Kim, S-F Shi, Y Zhang, Jin C Sun, Y Tongay, S Wu, J Zhang, Y Wang, F , Nature Nanotechnology. 9682Hong X, Kim J, Shi S-F, Zhang Y, Jin C, Sun Y, Tongay S, Wu J, Zhang Y and Wang F 2014 Nature Nanotechnology 9 682\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Ceballos, M Z Bellus, H-Y Chiu, H Zhao, ACS Nano. 812717Ceballos F, Bellus M Z, Chiu H-Y and Zhao H 2014 ACS Nano 8 12717\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Long, O V Prezhdo, Nano Letters. 161996Long R and Prezhdo O V 2016 Nano Letters 16 1996\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Xia Wei, F-G Y , Chao Shen, Quan-Shan Lv, Chin. Phys. B. 2638504Xia Wei F-G Y, Chao Shen, Quan-Shan Lv, Kai-You Wang 2017 Chin. Phys. B 26 38504\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I G Lezama, A Arora, A Ubaldini, C Barreteau, E Giannini, M Potemski, A F Morpurgo, Nano Letters. 152336Lezama I G, Arora A, Ubaldini A, Barreteau C, Giannini E, Potemski M and Morpurgo A F 2015 Nano Letters 15 2336\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Ruppert, O Aslan, T F Heinz, Nano Letters. 146231Ruppert C, Aslan O B and Heinz T F 2014 Nano Letters 14 6231\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Pezeshki, H Shokouh Seyed Hossein, T Nazari, K Oh, S Im, Advanced Materials. 283216Pezeshki A, Shokouh Seyed Hossein H, Nazari T, Oh K and Im S 2016 Advanced Materials 28 3216\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Liu, A T Neal, Z Zhu, Z Luo, X Xu, Tomá, P D Ye, ACS Nano. 84033Liu H, Neal A T, Zhu Z, Luo Z, Xu X, Tomá nek D and Ye P D 2014 ACS Nano 8 4033\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Tran, R Soklaski, Y Liang, Yang L , Physical Review B. 89235319Tran V, Soklaski R, Liang Y and Yang L 2014 Physical Review B 89 235319\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Long, E Liu, P Wang, A Gao, H Xia, W Luo, B Wang, J Zeng, Y Fu, K Xu, W Zhou, Y Lv, S Yao, M Lu, Y Chen, Z Ni, Y You, X Zhang, S Qin, Y Shi, W Hu, D Xing, F Miao, Nano Letters. 162254Long M, Liu E, Wang P, Gao A, Xia H, Luo W, Wang B, Zeng J, Fu Y, Xu K, Zhou W, Lv Y, Yao S, Lu M, Chen Y, Ni Z, You Y, Zhang X, Qin S, Shi Y, Hu W, Xing D and Miao F 2016 Nano Letters 16 2254\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Das, H-Y Chen, A Penumatcha, J Appenzeller, Nano Letters. 13100Das S, Chen H-Y, Penumatcha A V and Appenzeller J 2013 Nano Letters 13 100\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Ross, S Wu, H Yu, N J Ghimire, A M Jones, G Aivazian, J Yan, D G Mandrus, Xiao D Yao, W Xu, X , Nature Communications. 41474Ross J S, Wu S, Yu H, Ghimire N J, Jones A M, Aivazian G, Yan J, Mandrus D G, Xiao D, Yao W and Xu X 2013 Nature Communications 4 1474\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Qiao, J Yuan, Z Xu, C Chen, S Lin, Y Wang, J Song, Y Liu, Q Khan, H Y Hoh, C-X Pan, Li S Bao, Q , ACS Nano. 91886Qiao H, Yuan J, Xu Z, Chen C, Lin S, Wang Y, Song J, Liu Y, Khan Q, Hoh H Y, Pan C-X, Li S and Bao Q 2015 ACS Nano 9 1886\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Dang, H Peng, H Li, Wang P Liu, Z , Nano Letters. 102870Dang W, Peng H, Li H, Wang P and Liu Z 2010 Nano Letters 10 2870\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Hasan, C L Kane, Reviews of Modern Physics. 823045Hasan M Z and Kane C L 2010 Reviews of Modern Physics 82 3045\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Ye, P Wang, W Luo, F Gong, L Liao, T Liu, L Tong, J Zang, J Xu, W Hu, Nano Energy. 3753Ye L, Wang P, Luo W, Gong F, Liao L, Liu T, Tong L, Zang J, Xu J and Hu W 2017 Nano Energy 37 53\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Bullock, M Amani, J Cho, Y-Z Chen, G H Ahn, V Adinolfi, V R Shrestha, Y Gao, K B Crozier, Y-L Chueh, A Javey, Nature Photonics. 12601Bullock J, Amani M, Cho J, Chen Y-Z, Ahn G H, Adinolfi V, Shrestha V R, Gao Y, Crozier K B, Chueh Y-L and Javey A 2018 Nature Photonics 12 601\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M S Choi, D Qu, D Lee, X Liu, K Watanabe, T Taniguchi, W J Yoo, ACS Nano. 89332Choi M S, Qu D, Lee D, Liu X, Watanabe K, Taniguchi T and Yoo W J 2014 ACS Nano 8 9332\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Yu, S Zhang, H Zeng, Q J Wang, Nano Energy. 2534Yu X, Zhang S, Zeng H and Wang Q J 2016 Nano Energy 25 34\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Li, Y Yu, G J Ye, Q Ge, X Ou, H Wu, D Feng, X Chen, Y Zhang, Nature Nanotechnology. 9372Li L, Yu Y, Ye G J, Ge Q, Ou X, Wu H, Feng D, Chen X H and Zhang Y 2014 Nature Nanotechnology 9 372\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N R Pradhan, D Rhodes, Y Xin, S Memaran, L Bhaskaran, M Siddiq, S Hill, P Ajayan, L Balicas, ACS Nano. 87923Pradhan N R, Rhodes D, Xin Y, Memaran S, Bhaskaran L, Siddiq M, Hill S, Ajayan P M and Balicas L 2014 ACS Nano 8 7923\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y-Q Bie, G Grosso, M Heuck, M M Furchi, Y Cao, J Zheng, D Bunandar, E Navarro-Moratalla, L Zhou, D K Efetov, T Taniguchi, K Watanabe, J Kong, D Englund, Jarillo-Herrero , P , Nature Nanotechnology. 121124Bie Y-Q, Grosso G, Heuck M, Furchi M M, Cao Y, Zheng J, Bunandar D, Navarro-Moratalla E, Zhou L, Efetov D K, Taniguchi T, Watanabe K, Kong J, Englund D and Jarillo-Herrero P 2017 Nature Nanotechnology 12 1124\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Duan, C Wang, J C Shaw, R Cheng, Y Chen, H Li, X Wu, Y Tang, Q Zhang, A Pan, J Jiang, R Yu, Y Huang, X Duan, Nature Nanotechnology. 9Duan X, Wang C, Shaw J C, Cheng R, Chen Y, Li H, Wu X, Tang Y, Zhang Q, Pan A, Jiang J, Yu R, Huang Y and Duan X 2014 Nature Nanotechnology 9 56 1024\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Huang, S Wu, A M Sanchez, J J P Peters, R Beanland, J S Ross, P Rivera, W Yao, D Cobden, X Xu, Nature Materials. 131096Huang C, Wu S, Sanchez A M, Peters J J P, Beanland R, Ross J S, Rivera P, Yao W, Cobden D H and Xu X 2014 Nature Materials 13 1096\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M-Y Li, Y Shi, C-C Cheng, L-S Lu, Y-C Lin, H-L Tang, M-L Tsai, C-W Chu, K-H Wei, J-H He, Chang W-H , Suenaga K , Li L-J , Science. 349524Li M-Y, Shi Y, Cheng C-C, Lu L-S, Lin Y-C, Tang H-L, Tsai M-L, Chu C-W, Wei K-H, He J-H, Chang W-H, Suenaga K and Li L-J 2015 Science 349 524\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P K Sahoo, S Memaran, Y Xin, L Balicas, H Gutié Rrez, Nature. 55363Sahoo P K, Memaran S, Xin Y, Balicas L and Gutié rrez H R 2018 Nature 553 63\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z Zhang, P Chen, X Duan, K Zang, J Luo, X Duan, Science. 357788Zhang Z, Chen P, Duan X, Zang K, Luo J and Duan X 2017 Science 357 788\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Du, H Liu, Y Deng, P D Ye, ACS Nano. 810035Du Y, Liu H, Deng Y and Ye P D 2014 ACS Nano 8 10035\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Gan, R-J Shiue, Y Gao, I Meric, T F Heinz, K Shepard, J Hone, S Assefa, D Englund, Nature Photonics. 7883Gan X, Shiue R-J, Gao Y, Meric I, Heinz T F, Shepard K, Hone J, Assefa S and Englund D 2013 Nature Photonics 7 883\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Goykhman, U Sassi, B Desiatov, N Mazurski, S Milana, D De Fazio, A Eiden, J Khurgin, J Shappir, Levy U Ferrari, A C , Nano Letters. 163005Goykhman I, Sassi U, Desiatov B, Mazurski N, Milana S, de Fazio D, Eiden A, Khurgin J, Shappir J, Levy U and Ferrari A C 2016 Nano Letters 16 3005\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Ma, N Flöry, Y Salamin, B Baeuerle, A Emboras, A Josten, T Taniguchi, K Watanabe, L Novotny, Leuthold , ACS Photonics. Ma P, Flöry N, Salamin Y, Baeuerle B, Emboras A, Josten A, Taniguchi T, Watanabe K, Novotny L and Leuthold J 2018 ACS Photonics\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Pospischil, M Humer, M M Furchi, D Bachmann, R Guider, T Fromherz, T Mueller, Nature Photonics. 7892Pospischil A, Humer M, Furchi M M, Bachmann D, Guider R, Fromherz T and Mueller T 2013 Nature Photonics 7 892\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Wang, Z Cheng, K Xu, H Tsang, J-B Xu, Nature Photonics. 7888Wang X, Cheng Z, Xu K, Tsang H K and Xu J-B 2013 Nature Photonics 7 888\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Youngblood, C Chen, S Koester, M Li, Nature Photonics. 9247Youngblood N, Chen C, Koester S J and Li M 2015 Nature Photonics 9 247\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dongdong Yin. Yubing Wang, W Y , Qin Han, Xiaohong Yang, Han Ye, Qianqian Lv, Chin. Phys. B. 25118103Yubing Wang W Y, Qin Han, Xiaohong Yang, Han Ye, Qianqian Lv, Dongdong Yin 2016 Chin. Phys. B 25 118103\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Liu, X Yin, E Ulin-Avila, B Geng, T Zentgraf, L Ju, Wang F Zhang, X , Nature. 47464Liu M, Yin X, Ulin-Avila E, Geng B, Zentgraf T, Ju L, Wang F and Zhang X 2011 Nature 474 64\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T J Echtermeyer, L Britnell, P K Jasnos, A Lombardo, R V Gorbachev, A N Grigorenko, A K Geim, A Ferrari, K S Novoselov, Nature Communications. 2458Echtermeyer T J, Britnell L, Jasnos P K, Lombardo A, Gorbachev R V, Grigorenko A N, Geim A K, Ferrari A C and Novoselov K S 2011 Nature Communications 2 458\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Gu, N Petrone, J F Mcmillan, A Van Der Zande, M Yu, G Q Lo, D L Kwong, J Hone, C W Wong, Nature Photonics. 6554Gu T, Petrone N, McMillan J F, van der Zande A, Yu M, Lo G Q, Kwong D L, Hone J and Wong C W 2012 Nature Photonics 6 554\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Q Bao, H Zhang, B Wang, Z Ni, C H Y X Lim, Y Wang, D Tang, K P Loh, Nature Photonics. 5411Bao Q, Zhang H, Wang B, Ni Z, Lim C H Y X, Wang Y, Tang D Y and Loh K P 2011 Nature Photonics 5 411\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Freitag, T Low, F Xia, P Avouris, Nature Photonics. 753Freitag M, Low T, Xia F and Avouris P 2012 Nature Photonics 7 53\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Low, A S Rodin, A Carvalho, Y Jiang, H Wang, F Xia, Castro Neto, A , Physical Review B. 9075434Low T, Rodin A S, Carvalho A, Jiang Y, Wang H, Xia F and Castro Neto A H 2014 Physical Review B 90 075434\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Hong, B Chamlagain, W Z Lin, H J Chuang, M H Pan, Z Zhou, Y Q Xu, NANOSCALE. 68978Hong T, Chamlagain B, Lin W Z, Chuang H J, Pan M H, Zhou Z X and Xu Y Q 2014 NANOSCALE 6 8978\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Low, M Engel, M Steiner, P Avouris, Physical Review B. 9081408Low T, Engel M, Steiner M and Avouris P 2014 Physical Review B 90 081408\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Das, W Zhang, M Demarteau, A Hoffmann, M Dubey, A Roelofs, Nano Letters. 145733Das S, Zhang W, Demarteau M, Hoffmann A, Dubey M and Roelofs A 2014 Nano Letters 14 5733\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Robert, R Picard, D Lagarde, G Wang, J P Echeverry, F Cadiz, P Renucci, A Högele, T Amand, Marie X Gerber, I Urbaszek, B , Physical Review B. 94155425Robert C, Picard R, Lagarde D, Wang G, Echeverry J P, Cadiz F, Renucci P, Högele A, Amand T, Marie X, Gerber I C and Urbaszek B 2016 Physical Review B 94 155425\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Froehlicher, Lorchat E Berciaud, S , Physical Review B. 9485429Froehlicher G, Lorchat E and Berciaud S 2016 Physical Review B 94 085429\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Yin, X Zhan, K Xu, F Wang, Z Wang, Y Huang, Q Wang, C Jiang, J He, Applied Physics Letters. 10843503Yin L, Zhan X, Xu K, Wang F, Wang Z, Huang Y, Wang Q, Jiang C and He J 2016 Applied Physics Letters 108 043503\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Octon Tobias, J Nagareddy, V K Russo, S , Craciun Monica, F Wright, C D , Advanced Optical Materials. 41750Octon Tobias J, Nagareddy V K, Russo S, Craciun Monica F and Wright C D 2016 Advanced Optical Materials 4 1750\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Gan, Y Gao, Fai Mak, K Yao, X Shiue, R-J Van Der Zande, A Trusheim, M E Hatami, F Heinz, T F Hone, J Englund, D , Applied Physics Letters. 103181119Gan X, Gao Y, Fai Mak K, Yao X, Shiue R-J, van der Zande A, Trusheim M E, Hatami F, Heinz T F, Hone J and Englund D 2013 Applied Physics Letters 103 181119\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Wu, S Buckley, J R Schaibley, L Feng, J Yan, D G Mandrus, F Hatami, W Yao, J Vučković, A Majumdar, X Xu, Nature. 52069Wu S, Buckley S, Schaibley J R, Feng L, Yan J, Mandrus D G, Hatami F, Yao W, Vučković J, Majumdar A and Xu X 2015 Nature 520 69\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Liu, T Galfsky, Z Sun, F Xia, E-C Lin, Y-H Lee, Ké Na-Cohen, S Menon, V M , Nature Photonics. 930Liu X, Galfsky T, Sun Z, Xia F, Lin E-c, Lee Y-H, Ké na-Cohen S and Menon V M 2014 Nature Photonics 9 30\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Dufferwiel, S Schwarz, F Withers, A A P Trichet, F Li, M Sich, Del Pozo-Zamudio, O Clark, C Nalitov, A Solnyshkov, D D Malpuech, G Novoselov, K S Smith, J M Skolnick, M S Krizhanovskii, D Tartakovskii, A I , Dufferwiel S, Schwarz S, Withers F, Trichet A A P, Li F, Sich M, Del Pozo- Zamudio O, Clark C, Nalitov A, Solnyshkov D D, Malpuech G, Novoselov K S, Smith J M, Skolnick M S, Krizhanovskii D N and Tartakovskii A I 2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Nature Communications. 68579Nature Communications 6 8579\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Ye, Z J Wong, X Lu, X Ni, H Zhu, X Chen, Wang Y Zhang, X , Nature Photonics. 9733Ye Y, Wong Z J, Lu X, Ni X, Zhu H, Chen X, Wang Y and Zhang X 2015 Nature Photonics 9 733\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Furchi, A Urich, A Pospischil, G Lilley, K Unterrainer, H Detz, P Klang, A M Andrews, W Schrenk, G Strasser, T Mueller, Nano Letters. 122773Furchi M, Urich A, Pospischil A, Lilley G, Unterrainer K, Detz H, Klang P, Andrews A M, Schrenk W, Strasser G and Mueller T 2012 Nano Letters 12 2773\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Casalino, U Sassi, I Goykhman, A Eiden, E Lidorikis, S Milana, De Fazio, D Tomarchio, F Iodice, M Coppola, G Ferrari, A C , ACS Nano. 1110955Casalino M, Sassi U, Goykhman I, Eiden A, Lidorikis E, Milana S, De Fazio D, Tomarchio F, Iodice M, Coppola G and Ferrari A C 2017 ACS Nano 11 10955\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Q Guo, C Li, B Deng, S Yuan, F Guinea, F Xia, ACS Photonics. 42989Guo Q, Li C, Deng B, Yuan S, Guinea F and Xia F 2017 ACS Photonics 4 2989\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Kumar, A Sharma, M Kaur, S Husale, Advanced Optical Materials. 51700009Kumar R, Sharma A, Kaur M and Husale S 2017 Advanced Optical Materials 5 1700009\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I J Luxmoore, P Q Liu, P Li, J Faist, G R Nash, ACS Photonics. 3936Luxmoore I J, Liu P Q, Li P, Faist J and Nash G R 2016 ACS Photonics 3 936\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z Xia, P Li, Y Wang, T Song, Q Zhang, B Sun, ACS Applied Materials & Interfaces. 724136Xia Z, Li P, Wang Y, Song T, Zhang Q and Sun B 2015 ACS Applied Materials & Interfaces 7 24136\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Wang, A Klots, D Prasai, Y Yang, K Bolotin, J Valentine, Nano Letters. 157440Wang W, Klots A, Prasai D, Yang Y, Bolotin K I and Valentine J 2015 Nano Letters 15 7440\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Hong, B Chamlagain, S Hu, S M Weiss, Z Zhou, Y-Q Xu, ACS Nano. 95357Hong T, Chamlagain B, Hu S, Weiss S M, Zhou Z and Xu Y-Q 2015 ACS Nano 9 5357\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z Fang, Z Liu, Y Wang, P M Ajayan, Nordlander P Halas, N J , Nano Letters. 123808Fang Z, Liu Z, Wang Y, Ajayan P M, Nordlander P and Halas N J 2012 Nano Letters 12 3808\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Somayyeh Asgari, H R , Nosrat Granpayeh, Homayoon Oraizi, Chin. Phys. B. 2784212Somayyeh Asgari H R, Nosrat Granpayeh, Homayoon Oraizi 2018 Chin. Phys. B 27 84212\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z Chen, X Li, J Wang, L Tao, M Long, S-J Liang, L K Ang, C Shu, H Tsang, J-B Xu, ACS Nano. 11430Chen Z, Li X, Wang J, Tao L, Long M, Liang S-J, Ang L K, Shu C, Tsang H K and Xu J-B 2017 ACS Nano 11 430\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Stockman, Nature. 467541Stockman M I 2010 Nature 467 541\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M W Knight, H Sobhani, P Nordlander, N J Halas, Science. 332702Knight M W, Sobhani H, Nordlander P and Halas N J 2011 Science 332 702\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H S Lee, Min S-W Chang, Y-G Park, M K , Nam T Kim, H Kim, J H , Ryu S Im, S , Nano Letters. 123695Lee H S, Min S-W, Chang Y-G, Park M K, Nam T, Kim H, Kim J H, Ryu S and Im S 2012 Nano Letters 12 3695\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J-Y Wu, Y T Chun, S Li, T Zhang, J Wang, P Shrestha, D Chu, Advanced Materials. 301705880Wu J-Y, Chun Y T, Li S, Zhang T, Wang J, Shrestha P K and Chu D 2018 Advanced Materials 30 1705880\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z Ni, L Ma, S Du, Y Xu, M Yuan, H Fang, Z Wang, M Xu, D Li, J Yang, W Hu, X Pi, Yang D , ACS Nano. 119854Ni Z, Ma L, Du S, Xu Y, Yuan M, Fang H, Wang Z, Xu M, Li D, Yang J, Hu W, Pi X and Yang D 2017 ACS Nano 11 9854\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Guo, F Gong, J Liu, Y Jia, S Zhao, L Liao, M Su, Z Fan, X Chen, W Lu, Xiao L Hu, W , ACS Applied Materials & Interfaces. 934489Guo N, Gong F, Liu J, Jia Y, Zhao S, Liao L, Su M, Fan Z, Chen X, Lu W, Xiao L and Hu W 2017 ACS Applied Materials & Interfaces 9 34489\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Massicotte, P Schmidt, F Vialla, K Watanabe, T Taniguchi, K Tielrooij, F H L Koppens, Nature Communications. 712174Massicotte M, Schmidt P, Vialla F, Watanabe K, Taniguchi T, Tielrooij K J and Koppens F H L 2016 Nature Communications 7 12174\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Huo, G Konstantatos, Nature Communications. 8572Huo N and Konstantatos G 2017 Nature Communications 8 572\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Kim, S Park, H Jang, N Koirala, J-B Lee, U J Kim, H-S Lee, Y-G Roh, H Lee, S Sim, S Cha, C In, J Park, J Lee, M Noh, J Moon, M Salehi, J Sung, S Chee, Kim J, Park S, Jang H, Koirala N, Lee J-B, Kim U J, Lee H-S, Roh Y-G, Lee H, Sim S, Cha S, In C, Park J, Lee J, Noh M, Moon J, Salehi M, Sung J, Chee S-\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S , Ham M-H , Jo M-H Oh, S Ahn, J-H Hwang, S W , Kim D Choi, H , S, Ham M-H, Jo M-H, Oh S, Ahn J-H, Hwang S W, Kim D and Choi H 2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \". ACS Photonics. 4482ACS Photonics 4 482\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":27,"cells":{"text":{"kind":"string","value":"\nSTRICHARTZ ESTIMATES FOR SCHRÖDINGER EQUATIONS ON IRRATIONAL TORI\n10 Mar 2014\n\nZihua Guo \nTadahiro Oh \nYuzhao Wang \nSTRICHARTZ ESTIMATES FOR SCHRÖDINGER EQUATIONS ON IRRATIONAL TORI\n10 Mar 2014arXiv:1306.4973v3 [math.AP] Dedicated to the memory of Professor Harold N. Shapiro (1922-2013)\nIn this paper, we prove new Strichartz estimates for linear Schrödinger equations posed on d-dimensional irrational tori. Then, we use these estimates to prove subcritical and critical local well-posedness results for nonlinear Schrödinger equations (NLS) on irrational tori.2010 Mathematics Subject Classification. 35Q55, 42B37. We use A B to denote an estimate of the form A ≤ CB for some C > 0. Similarly, we use A ∼ B to denote A B and B A.where I is a compact interval. Note that, in the compact setting, an estimate of the form (1.9) does not hold with I = R, unless p = ∞. By drawing an analogy to the Euclidean case M = R d , one may hope to have (1.9) for p ≥ 2(d+2) d . By combining the tools from number theory, such as a divisor counting argument and the Hardy-Littlewood circle method, and the Tomas-Stein restriction method from harmonic analysis, Bourgain [3] proved (1.9) for certain ranges of p: (i) p > 2(d+2) d when d = 1, 2, (ii) p > 4 when d = 3, and (iii) p > 2(d+4) d for higher dimensions d ≥ 4. It is 2 Obviously, the scaling associated to the linear Schrödinger equation discussed above for R d does not quite make sense on T d . We nonetheless call the estimate (1.9) scaling-invariant.\ni∂ t u − ∆u = ±|u| p−1 u u t=0 = u 0 ∈ H s (M ), (x, t) ∈ M × R (1.1)\nhas been studied extensively in different settings (for example, M = R d , T d , and certain classes of manifolds) over recent years [16,39,28,13,3,4,9,10,30,21,22,20,42]. See also the following monographs [36,12,38] for references therein. In the study of NLS (1.1), Strichartz estimates of the following type have played a fundamental role\ne −it∆ f L q t L r x (R×M ) f H s x (M ) ,(1.2)\nwhere f L q t L r 1 In particular, when M = R d , (1.2) is known to hold with s = 0, namely\nx = f (x, t) L r x L q t .e −it∆ f L q t L r x (R×R d ) f L 2 x (R d ) ,(1.3)\nif and only if (q, r) satisfies 2 q + d r = d 2 with 2 ≤ q, r ≤ ∞ and (q, r, d) = (2, ∞, 2). See [34,44,17,29]. This was first obtained for the case q = r by Strichartz [34] via the Fourier restriction method. It was then generalized by a combination of the duality argument and the following dispersive estimate:\ne −it∆ f L ∞ x (R d ) |t| − d 2 f L 1 x (R d ) . (1.4)\nThe endpoint case (q, r) = (2, 2d d−2 ), d = 2, was then proven in [29].\n\nNow, consider f ∈ L 2 (R d ) with supp f ⊂ [−N, N ] d . Then, as an immediate corollary to (1.3), we have the following Strichartz estimate on R d : \ne −it∆ f L p t,x (R×R d ) N d 2 − d+2 p f L 2 (R d ) ,(1.q = r = 2(d+2) d gives e −it∆ f L 2(d+2) d t,x (R×R d ) f L 2 (R d ) .\n(1.6)\n\nOn the other hand, by Bernstein's inequality [43,Chapter 5], we have The solution u to (1.8) is given by u(x, t) := e −it∆ f (x). Then, the rescaled function u λ (x, t) := u(λx, λ 2 t), λ > 0, is also a solution to (1.8) but with the rescaled initial condition f λ (x) := f (λx). Noting that supp f λ = λ · supp f , it is easy to see that the power of N in (1.5) is the only power that is consistent with this scaling. We point out that the inequalities (1.3), (1.6), and (1.7) are also scaling-invariant with respect to this scaling associated to the linear Schrödinger equation and that the scaling-invariance shows sharpness of these estimates. When M is a compact manifold, the Strichartz estimate (1.2) becomes much more difficult and much less is known. This is partially due to the fact that we do not have the dispersive estimate (1.4) on a compact manifold. Moreover, (1.2) requires deep understanding of the eigenvalues and the eigenfunctions of the Laplacian. In the following, we focus on the case when M is a standard flat torus T d = (R/Z) d , corresponding to the usual periodic boundary condition. Moreover, we restrict our attention to the diagonal case, i.e. q = r. Then, one would like to establish the following scaling-invariant 2 Strichartz estimate:\ne −it∆ f L ∞ t,x (R×R d ) N d 2 f L 2 (R d ) ,(1.\n. The general conjecture (1.11), however, remains open up to date. In [3,4,21,42], these Strichartz estimates were then applied to prove well-posedness results of NLS (1.1) on T d . See Subsection 1.3 for more on the well-posedness issue of (1.1).\n\nLet us conclude this subsection by stating the result by Herr [20]. He considered the quintic NLS on a three-dimensional Zoll manifold M , i.e. a compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period. One simplest example is the three dimensional sphere S 3 . By establishing the Strichartz estimate (1.9) on M (instead of T d ) with p > 4, he proved local well-posedness of the quintic NLS on a three-dimensional Zoll manifold M in the energy space H 1 (M ). As mentioned above, all geodesics on a Zoll manifold have a common minimal period. Hence, it is natural to ask if a Strichartz estimate of the form (1.9) holds on a manifold, where there is no common minimal period for geodesics. This leads us to the study of Strichartz estimates on an irrational torus T d α α α , since it is one of the simplest examples of manifolds with no common minimal period for geodesics.\n\n1.2. Strichartz estimate on irrational tori. In the remaining part of this paper, we focus on the case when M is an irrational torus T d α α α :\nM = T d α α α := d j=1\nR/(α j Z), α j > 0, j = 1, . . . , d.\n\n(1.12)\n\nAs the name suggests, we are mainly interested in the case when at least one α j is irrational. More generally, we are interested in the case when at least one α j is \"rationally independent\" of the remaining ones, i.e. there exists α j that can not be written as a linear combination of the other α k 's with rational coefficients. First consider the case when all α j 's are rational. Namely, M = T d α α α is a \"rational\" torus. In this case, the problem can be reduced to that on the standard torus T d by a simple geometric consideration. By writing α j = k j m j for some k j , m j ∈ N, let k be the least common multiple of k j 's. The basic idea is to view the scaled standard torus \nf (n) = 1 |T d α α α |ˆTd α α α f (x)e −2πi d j=1 n j x j α j dx = 1 |kT d |ˆk T d f (x)e −2πi d j=1 kn j α j x j k dx = F[ f ] k α 1 n 1 , . . . , k α d n d ,\nwhere n = (n 1 , . . . , n d ) ∈ Z d and | · | denotes the Lebesgue measure of a set. Namely, we have supp\nF[ f ] = k α α α · supp f := k α 1 n 1 , . . . , k α d n d ∈ Z d :\nn ∈ supp f . Therefore, we see that the Strichartz estimates of the form (1.10) on the standard torus T d also hold on our rational torus M = T d α α α , where the implicit constants further depends on α α α = (α 1 , . . . , α d ). When there is no α j that is rationally independent of the remaining ones, we can use spatial and temporal dilations to reduce the situation to the case of a rational torus above. Therefore, in the following, we assume that at least one α j is rationally independent of the remaining α k 's.\n\nBefore proceeding further, let us change the spatial domain M = T d α α α to the standard torus T d at the expense of modifying the Laplacian. By a change of spatial variables (x j → α j x j ), we see that (1.1) is equivalent to the following NLS on the usual torus T d = (R/Z) d :\ni∂ t u − ∆u = ±|u| p−1 u u t=0 = u 0 ∈ H s (T d ), (x, t) ∈ T d × R,(1.13)\nwhere the Laplace operator ∆ is now defined by\n∆f (n) = −4π 2 Q(n) f (n),(1.14)\nwith n = (n 1 , . . . , n d ) ∈ Z d and\nQ(n) = θ 1 n 2 1 + · · · + θ d n 2 d , 1 C ≤ θ j := 1 α 2 j ≤ C, j = 1, · · · , d. (1.15)\nWe point out that some estimates in the following depend on C in (1.15) but not on the specific arithmetic nature of θ j 's. Our main interest is to discuss well-posedness of the Cauchy problem (1.13) by first studying relevant Strichartz estimates in this setting. As compared to the problem on the standard torus T d , i.e. with Q(n) = |n| 2 = d j=1 n 2 j , it is a lot harder to study Strichartz estimates on irrational tori. The main reason for this difficulty is that the number theoretic tools such as a divisor counting argument and the Hardy-Littlewood circle method do not work well in this setting.\n\nPreviously, Bourgain [6] and Catoire-Wang [11] studied the Cauchy problem (1.13) on irrational tori and proved some local well-posedness results in subcritical Sobolev spaces. See Theorem 1.4 below. In the following, we investigate new Strichartz estimates on irrational tori and use them to prove well-posedness results of the Cauchy problem (1.13) in both subcritical and critical Sobolev spaces. In the rest of the paper, we assume that the Laplacian ∆ is defined by (1.14), unless stated otherwise, and define the linear Schrödinger evolution by 3\ne −it∆ f (x) = n∈Z d f (n)e 2πi(n·x+Q(n)t) ,(1.16)\nwhere Q(n) is as in (1.15). We first summarize the known Strichartz estimates. In the following, I denotes a compact interval in R. \ne −it∆ f L p t,x (I×T d ) K p,N f L 2 (T d ) , (1.17) for all f ∈ L 2 (T d ) with supp f ⊂ [−N, N ] d in(i) d = 2 [11] 4 : K 4,N = N 1 6 , (ii) d = 3 [6]: K 4,N = N 1 3 +ε , (iii) d ≥ 3 [11]: K 4,N = N d 4 − d 2(d+1) +ε when d is odd, and K 4,N = N d 4 − 1 2 +ε when d is even, (iv) d ≥ 2 [7]: K p,N = N ε for p = 2(d+1) d ,\nfor any small ε > 0.\n\nNote that the implicit constants in (1.17) depend on C in (1.15) and the length of the local-in-time interval I. The same comment applies to all the estimates in the remaining of the paper and we do not mention this dependence explicitly in the following. In [6], Bourgain also proved\ne −it∆ f L p t L 4 x (I×T 3 ) N 3 4 − 2 p f L 2 (T 3 ) (1.18)\nfor p > 16 3 . In this paper, we partially improve the known results in Theorem 1.1, and obtain some critical Strichartz estimates when p is large. We state our main result on the Strichartz estimates on irrational tori. 3 Strictly speaking, there is an extra factor of 2π in front of Q(n) in (1.15). However, such a factor can be eliminated by time dilation and thus, for simplicity of notations, we drop it in the following. 4 After the completion of this manuscript, we learned that this result in [11] was recently improved to K4,N = N 131 832 + by Demirbas [15]. While the proof in [11] is based on Jarník's argument [27], the proof in [15] is based on Huxley's counting estimate [24]. More recently, this result in [15] was improved to K4,N = N 1 8 + by Demeter [14]. See the footnote in Theorem 1.2 (ii). \ne −it∆ f L p t,x (I×T d ) N d 2 − d+2 p f L 2 (T d ) , (1.19) for all f ∈ L 2 (T d ) with supp f ⊂ [−N, N ] d ,\nprovided that d and p are in the following ranges:\n(i.a) d = 2: p > 20 3 , (i.b) d = 3 : p > 16 3 , (i.c) d = 4: p > 4, (i.d) d ≥ 5: p ≥ 4.\n(ii) Let ε > 0. Then, the Strichartz estimate with an ε-loss of regularity holds on an irrational torus: 5\ne −it∆ f L p t,x (I×T d ) N d 2 − d+2 p +ε f L 2 (T d ) , (1.20) for all f ∈ L 2 (T d ) with supp f ⊂ [−N, N ] d ,\nprovided that d and p are in the following ranges:\n(ii.a) d = 2: p ≥ 20 3 , (ii.b) d = 3: p = 16 3 , (ii.c) d = 4: p = 4.\nWhen d ≥ 3, we follow a relatively simple argument after Bourgain [6] and prove Theorem 1.2 in Subsection 2.1. When d = 2, this argument proves (1.20) only for p ≥ 8. In Subsection 2.2, we present a duality argument to prove (1.19) (when d = 2) for p > 12. In Section 3, we establish certain level set estimates and provide a full proof of Theorem 1.2 when d = 2.\n\nBy interpolating with Theorems 1.1 and 1.2, we can obtain Strichartz estimates for other values of p. The following corollary shows a summary of known Strichartz estimates on irrational tori at this point. 5 In a very recent preprint, Demeter [14] proved the Strichartz estimates (1.20) with an ε-loss on the standard torus T d for p ≥ 2(d+3) d . His argument is based on incidence geometry, without any number theory. As a result, the same result holds for irrational tori and hence improves our result in Theorem 1.2 (ii) in a significant manner. This also improves the values of s0 in some subcritical local well-posedness results below (Theorem 1.4 (i.a), (i.b), (ii.a), and (ii.b)). Note that the result in [14] comes with an ε-loss and thus it does not improve the scaling-invariant Strichartz estimate (1.19) in Theorem 1.2 (i) and critical local well-posedness results in Theorems 1.5 and 1.6.\nCorollary 1.3. Let ε > 0. Then, the following Strichartz estimates hold for all f ∈ L 2 (T d ) with supp f ⊂ [−N, N ] d :(i) d = 2: e −it∆ f L p t,x (I×T 2 ) N ε f L 2 (T 2 ) , for 2 < p ≤ 3, N 2 3 − 2 p +ε f L 2 (T 2 ) , for 3 < p < 4, N 1 6 f L 2 (T 2 ) , for p = 4, N 3 4 − 7 3p +ε f L 2 (T 2 ) , for 4 < p ≤ 20 3 , N 1− 4 p f L 2 (T 2 )\n, for p > 20 3 .\n(1.21) (ii) d = 3\n:\ne −it∆ f L p t,x (I×T 3 ) N ε f L 2 (T 3 ) , for 2 < p ≤ 8 3 , N 1− 8 3p +ε f L 2 (T 3 ) , for 8 3 < p ≤ 4, N 5 4 − 11 3p +ε f L 2 (T 3 ) , for 4 < p ≤ 16 3 , N 3 2 − 5 p f L 2 (T 3 ) ,\nfor p > 16 3 .\n(1.22) (iii) d = 4: e −it∆ f L p t,x (I×T 4 ) N ε f L 2 (T 4 ) , for 2 < p ≤ 5 2 , N 4 3 − 10 3p +ε f L 2 (T 4 ) , for 5 2 < p ≤ 4, N 2− 6 p f L 2 (T 4 ) , for p > 4.\n(1.23) (iv) d ≥ 5:\ne −it∆ f L p t,x (I×T d ) N ε f L 2 (T d ) , for 2 < p ≤ 2(d+1) d , N ( d 4 − 1 2 )( 2d d−1 − 4(d+1) p(d−1) )+ε f L 2 (T d ) , for 2(d+1) d < p < 4, N d 2 − d+2 p f L 2 (T d ) , for p ≥ 4.\n(1.24)\n\n1.3. Local well-posedness results of NLS on irrational tori. In the following, we apply these Strichartz estimates in Corollary 1.3 to the Cauchy problem of NLS on an irrational torus:\ni∂ t u − ∆u = ±|u| 2k u u t=0 = u 0 ∈ H s (T d ), (x, t) ∈ T d × R,(1.25)\nwhere k ∈ N is a positive integer and the Laplacian ∆ is defined by (1.14). First, recall the following notion. When M = R d , the Cauchy problem (1.1) enjoys the dilation symmetry. Namely, if u is a solution to (1.1) with respect to an initial condition u 0 , then the rescaled function u λ (x, t) := λ 2 p−1 u(λx, λ 2 t) is also a solution to (1.1) with the rescaled initial condition u 0,λ (x) := λ 2 p−1 u 0 (λx). We say that the Sobolev index s c is critical if the homogeneous Sobolev norm · Ḣsc (R d ) is invariant under this dilation symmetry. In particular, the critical Sobolev index is given by s c = d 2 − 2 p−1 . When M = R d , we may not have this natural dilation symmetry. Nonetheless, the notion of the critical Sobolev index provides us important heuristics. In terms of the Cauchy problem (1.25), the critical Sobolev index s c is given by\ns c = d 2 − 1 k . (1.26)\nFirst, we state local well-posedness in subcritical Sobolev spaces H s (T d ) with s > s c . \n(i) d = 2: (i.a) k = 1, s 0 = 1 3 [11], (i.b) k = 2, 3, 4, 5, s 0 = 7k−3 7k+5 , (i.c) k ≥ 5, s 0 = s c = 1 − 1 k , Note that the values of s 0 in (i.b) and (i.c) coincide when k = 5. (ii) d = 3: (ii.a) k = 1, s 0 = 2 3 [6], (ii.b) k = 2, s 0 = 53 52 , (ii.c) k ≥ 3, s 0 = s c = 3 2 − 1 k , (iii) d ≥ 4: k ≥ 1, s 0 = s c = d 2 − 1 k .\nAfter Bourgain's seminal paper [3], the Fourier restriction norm method, involving the X s,b -space, has been applied to study well-posedness of a wide class of equations. In our proof, we also employ the X s,b -spaces and by the standard argument, the proof is reduced to establishing certain multilinear Strichartz estimates.\n\nFurthermore, by applying the well-posedness theory involving the U p -and V p -spaces developed by Tataru, Koch, and their collaborators [30,18,21,22], we prove some critical local well-posedness. 6 Theorem 1.5 (Local well-posedness in critical spaces). Given d ≥ 2 and k ∈ N, let s c be the critical Sobolev index given by (1.26). Then, the Cauchy problem (1.25) on a ddimensional irrational torus T d is locally well-posed in the critical Sobolev space H sc (T d ) in the following cases:\n(i) d = 2: k ≥ 6, (ii) d = 3: k ≥ 3, (iii) d ≥ 4: k ≥ 2.\nOnce again, the proof is reduced to establishing certain multilinear Strichartz estimates. See Propositions 5.6 and 5.7.\n\nLastly, we briefly discuss the case of a partially irrational torus. Namely, we consider Strichartz estimates on an irrational torus T d α α α , when some of α j 's in (1.12) are rationally 6 In a very recent paper, Strunk [35] extended Theorem 1.5 to (i) k ≥ 3 when d = 2 and (ii) k = 2 when d = 3. The main idea in [35] is based on considering Strichartz estimates in mixed Lebesgue spaces L q t L r x to improve the multilinear Strichartz estimate (Proposition 5.7 below). This clever argument avoids the need of improving the scaling-invariant Strichartz estimate (1.19). dependent. In this case, we may obtain improvements over Theorem 1.2, yielding better local well-posedness results than those presented in Theorems 1.4 and 1.5. For simplicity of presentation, we only consider an example of the three-dimensional torus of the form T 2 × T α 3 , where two periods are the same. By a change of spatial variables as before, we consider the Cauchy problem (1.25), where the multiplier Q(n) in (1.15) is given by Q(n) = n 2 1 + n 2 2 + θ 3 n 2 3 , θ 3 > 0, (1.27) i.e. we set θ 1 = θ 2 = 1. Then, we have the following local well-posedness result for the energy-critical quintic NLS on a three-dimensional partially irrational torus.\n\nTheorem 1.6. Suppose that Q(n) is given by (1.27). Then, the energy-critical quintic NLS, (1.25) with k = 2, on T 3 is locally well-posed in the critical Sobolev space H 1 (T 3 ).\n\nPreviously, Herr-Tataru-Tzvetkov [21] proved local well-posedness in the energy space H 1 (T 3 ) of the energy-critical quintic NLS on the three-dimensional standard torus T 3 . By combining the results in [7] and [22], we also see that the energy-critical cubic NLS on the four-dimensional standard torus T 4 is local well-posedness in the energy space H 1 (T 4 ). See also the work by the third author [42] for some other critical local well-posedness results. The result in [42], however, does not cover an energy-critical setting. As mentioned earlier, Herr [20] proved local well-posedness in the energy space of the energy-critical quintic NLS on three-dimensional Zoll manifolds. We point out that Theorem 1.6 seems to be the first local well-posedness result of the energy-critical NLS in its energy space H 1 (T 3 ), where there is no common minimal period for geodesics. We present a sketch of the proof in Appendix B. More precisely, we revisit the argument in Section 3 and prove the sharp Strichartz estimate (1.19) on T 3 for p > 14 3 under the assumption (1.27). The rest follows from a slight modification of the argument in Section 5. Lastly, note that Theorem 1.6 combined with the conservation of mass and Hamiltonian yields small data global well-posedness of the quintic NLS, (1.25) with k = 2, in H 1 (T 3 ), just as in [21,20]. Recently, global well-posedness (for large data) of the energy-critical quintic NLS on the three-dimensional standard torus T 3 and on the three-dimensional sphere S 3 was obtained by Ionescu-Pausader [26] and Pausader-Tzvetkov-Wang [32], respectively. It would be of interest to investigate if global well-posedness of the energy-critical quintic NLS holds in the setting of Theorem 1.6. 7\n\nThis paper is organized as follows. In Section 2, we prove Theorem 1.2 when d ≥ 3 and partially when d = 2, via multilinear estimates and a duality argument. In Section 3, we establish certain level set estimates and prove Theorem 1.2 when d = 2. In Section 4, we prove local well-posedness results in subcritical Sobolev spaces (Theorem 1.4). In Section 5, we prove local well-posedness results in critical Sobolev spaces (Theorem 1.5).\n\nIn Appendix A, we present a proof of (2.1) below, using the Hardy-Littlewood circle method. In Appendix B, we sketch a proof of Theorem 1.6. 7 After Strunk's result [35], it is now of interest to study global well-posedness of the energy-critical quintic NLS on a general three-dimensional irrational torus in its energy space H 1 (T 3 ).\n\nAcknowledgments: Z. Guo is supported in part by NNSF of China (No.11371037, No.11271023) and Beijing Higher Education Young Elite Teacher Project. Y. Wang is supported by NNSF of China (No.11126247, No.11201143) and AARMS Postdoctoral Fellowship. The authors would like to thank the anonymous referee for thoughtful comments that have significantly improved the introduction of this paper. T. Oh would like to express his sincere gratitude to Professor Harold N. Shapiro for his support and teaching in mathematics, including the Hardy-Littlewood circle method used in this paper, as well as in life.\n\n\nStrichartz estimates: Part 1\n\nIn this section, we prove our main result (Theorem 1.2) for d ≥ 3 and present a partial proof for d = 2. In [6], Bourgain treated the three-dimensional case. His argument is based on the following estimate:ˆT \n0≤n≤N e 2πin 2 t r dt ∼ N r−2 ,(2.} n∈Z d , define F a (t) by F a (t) = n∈Z d c n c a−n e 2πi[Q(n)+Q(a−n)]t , (2.2) where Q(n) is as in (1.15). Then, for p ≥ 2, we have F a (t) L p t ([−1,1]) k∈Z |Q(n)+Q(a−n)−k|≤ 1 2 |c n c a−n | p ′ 1 p ′ , (2.3) where 1 p + 1 p ′ = 1.\nLemma 2.1 was used in [6] for the three-dimensional case. See also Lemma 2 in [11]. A proof for general dimensions in [11] relies on Schur's lemma. In the following, we give a direct proof for reader's convenience. \nF a (t) L 2 t ([−1,1]) ≤ F a (t)η(t) L 2 t (R) = n c n c a−n η τ − [Q(n) + Q(a − n)] L 2 τ = k∈Z n∈I k,a c n c a−n η τ − [Q(n) + Q(a − n)] L 2 τ = k∈Z B k (τ ) L 2 τ , where I k,a = n ∈ Z d : Q(n) + Q(a − n) − k ∈ (− 1 2 , 1 2 ] and B k (τ ) = n∈I k,a c n c a−n η τ − [Q(n) + Q(a − n)] . Noting that |B k (τ )| n∈I k,a |c n c a−n | τ − k −2 , we have k∈Z B k (τ ) 2 L 2 τ = k∈Z B k (τ ) 2 L 1 τ k,k ′ B k (τ )B ′ k (τ ) L 1 τ k,k ′ n∈I k,a n ′ ∈I k ′ ,a |c n c a−n ||c n ′ c a−n ′ |ˆR τ − k −2 τ − k ′ −2 dτ k,k ′ 1 k − k ′ 2\nn∈I k,a n ′ ∈I k ′ ,a |c n c a−n ||c n ′ c a−n ′ | By Cauchy-Schwarz inequality (in k) followed by Young inequality, we have\n≤ k n∈I k,a |c n c a−n | 2 1 2 k k ′ n ′ ∈I k ′ ,a |c n ′ c a−n ′ | k − k ′ 2 2 1 2 k n∈I k,a |c n c a−n | 2 .\nThis completes the proof of Lemma 2.1.\n\nNext, we state the main proposition. Theorem 1.2 then follows this proposition and Bernstein's inequality when d ≥ 3. (4,4). Then, we have\nProposition 2.2. Let f be a function on T d with supp f ⊂ [−N, N ] d . (i) Let d ≥ 3. Then, for p ≥ max 16 d +, 4 , we have e −it∆ f L p t,loc L 4 x N d 4 − 2 p f L 2 , (2.4) (ii) Suppose that d and p satisfy (ii.a) d = 2, p ≥ 8, (ii.b) (d, p) = (3, 16 3 ), or (ii.c) (d, p) =e −it∆ f L p t,loc L 4 x N d 4 − 2 p (log N ) 2 q f L 2 , (2.5)\nwhere q = p when d = 3, 4 and q = 8 when d = 2.\n\nBourgain proved (2.4) for d = 3. See Proposition 1.1 in [6]. Our proof follows the ideas developed in [6]. By setting p = 4 when d ≥ 5 and p = 16 d + when d = 3, 4, Proposition 2.2 yields the L 4 -Strichartz estimate, which improves the result in [11] for d ≥ 3. Note that the Strichartz estimate (2.4) is essentially sharp in higher dimensions (d ≥ 4). Indeed, on R d with d ≥ 2, by Sobolev inequality and interpolation of (1.6) and (1.7)\ne −it∆ f L p t L 4 x (R×R d ) N 2( 1 4 − 1 p ) e −it∆ f L 4 t,x (R×R d ) N d 4 − 2 p f L 2 (R d ) ,\nfor p ≥ 4. We first use Proposition 2.2 to prove Theorem 1.2 when d ≥ 3. \ne −it∆ f L p t,x N d 4 − d p e −it∆ f L p t L 4 x N d 2 − d+2 p f L 2 .\nBy repeating the same argument with Proposition 2.2 (ii), we obtain Theorem 1.2 (ii) when d = 3, 4. When d = 2, this yields Theorem 1.2 (ii) only for p ≥ 8.\n\nWe now present the proof of Proposition 2.2.\n\nProof of Proposition 2.2. (i) Let Q(n) be as in (1.15). Then, we have\n(e −it∆ f )(x) = n∈Z d f (n)e 2πi(n·x+Q(n)t) .\n(2.6)\n\nWith c n = f (n), let F a (t) be as in (2.2). Then, by Minkowski's integral inequality with p ≥ 4, we have \ne −it∆ f 2 L p t L 4 x = (e −it∆ f ) 2 L p 2 t L 2 x = a∈Z d |F a (t)| 2 1 2 L p 2 t ≤ a∈Z d F a (t) 2 L p 2 t 1/2 . (2.7) For ℓ ∈ Z, let A ℓ = {n ∈ Z d : |Q(n) − ℓ| ≤ 1} ∩ [−N, N ] d . Noting that Q(n) + Q(a − n) = 1 2 Q(2n − a) + Q(a) , the condition |Q(n) + Q(a − n) − k| ≤ 1 2 is equivalent to 2n ∈ a + A ℓ with ℓ = 2k − Q(a). Note that, {ℓ ∈ Z : 2n ∈ a + A ℓ } 1 for all n ∈ Z d .F a L p 2 t ≤ ℓ∈Z 2n∈a+A ℓ |c n c a−n | p p−2 p−2 p ℓ |A ℓ | p 2(p−2) 2n∈a+A ℓ |c n c a−n | 2 p 2(p−2) p−2 p ≤ ℓ |A ℓ | p p−4 p−4 2p ℓ 2n∈a+A ℓ |c n c a−n | 2 1 2 ∼ ℓ |A ℓ | p p−4 p−4 2p n∈Z d |c n c a−n | 2 1 2 .\n(2.8)\n\nFrom (2.7) and (2.8), we have\ne −it∆ f L p t L 4 x ≤ C ℓ |A ℓ | p p−4 p−4 4p f L 2 .\n(2.9)\n\nNow, let η(t) be a smooth function with a compact support I ⊂ R such that η ≥ 0 and\nη ≥ 1 on [−1, 1]. Now we estimate ℓ |A ℓ | p p−4 p−4 4p , using |A ℓ | ≤ˆ n∈Z d |n j |≤N e 2πiQ(n)t η(t)e −2πiℓt dt.\nIf p ≤ 8, then we have p p−4 ≥ 2. Then, by Hausdorff-Young's inequality, we have\nℓ |A ℓ | p p−4 p−4 4p ˆI d j=1 |n j |≤N e 2πiθ j n 2 j t p 4 dt 1 p d j=1 ˆI |n j |≤N e 2πiθ j n 2 j t dp 4 dt 1 dp ˆI 0≤n≤N e 2πin 2 t dp 4 dt 1 p .\n(2.10)\n\nNote that r = dp 4 > 4, since p > 16 d . Then, by an application of the Hardy-Littlewood circle method (see Appendix A), we havê \nI 0≤n≤N e 2πin 2 t r dt ∼ N r−2 ,(2.e −it∆ f L p t L 4 x ≤ CN 1 4 − 2 p e −it∆ f L 8 t L 4\nx .\n\n(2.12)\n\nThen, (2.4) follows from (2.12) and (2.4) for p = 8.\n\n(ii) When (d, p) = (2,8), (3, 16 3 ), or (4, 4), we have r = dp 4 = 4. In this case, (2.11) does not hold. Nonetheless, by Hua's inequality [40], we havê\nI 0≤n≤N e 2πin 2 t 4 dt N 2 (log N ) 2 . (2.13)\nSee also [25, (8.13)]. Then, (2.5) follows from (2.13) and repeating the computation in (i \nQ(n) = n 2 1 + θn 2 2 , 1 C ≤ θ ≤ C.\n(2.14)\n\nThen, the local-in-time Strichartz estimate can be expressed as n∈S N a n e 2πi(n·x+Q(n)t)\nL p t,x (I×T 2 ) ≤ K p,N n∈S N |a n | 2 1/2 , (2.15)\nwhere S N denotes the following set:\nS N := (n 1 , n 2 ) ∈ Z 2 : |n j | ≤ N, j = 1, 2 . (2.16)\nOur task is to seek for an optimal constant K p,N . By duality, (2.15) is equivalent to\nn∈S N f (n, Q(n)) 2 1 2 ≤ K p,N f L p ′ t,x (I×T 2 ) , (2.17) for any f ∈ L p ′ (I × T 2 ), where 1 p + 1 p ′ = 1.\nHere, the Fourier transform f is defined by\nf (n, τ ) =ˆRˆT 2 e −2πin·x e −2πiτ t 1 I (t)f (x, t) dx dt.\nThen, (1.19) for p > 12 follows once we prove the next proposition.\nProposition 2.3. For p > 12, we have K p,N N 1− 4 p . Namely, we have n∈S N f (n, Q(n)) 2 1 2 N 1− 4 p f 2 L p ′ t,x (I×T 2 )\n.\n(2.18) Remark 2.4. Recall that, on the standard torus T 2 , i.e. with Q(n) = n 2 1 + n 2 2 , Bourgain [3] proved K p,N N 1− 4 p for p > 4.\nHence, Proposition 2.3 states that, on an irrational torus, the same estimate for K p,N holds, but only for p > 12.\n\nProof. Without loss of generality, assume that I is centered at 0. Let R be a kernel defined by\nR(x, t) = n∈S N e 2πi(n·x+Q(n)t) . (2.19) Then, defining R θ by R θ (x, t) = |n|≤N e 2πi(nx+θn 2 t) , (2.20) we have R(x, t) = R 1 (x 1 , t)R θ (x 2 , t). From Proposition 3.114 in [3], we have R 1 (x, t) L p t,x (I×T) ≤ C p,I N 1− 3 p , (2.21)\nfor p > 6. Bourgain's argument is based on an application of the Hardy-Littlewood circle method. See also Lemma 2.4 in [23] for a simpler proof based on the Poisson summation formula. Note that (2.21) does not hold for p = 6. See Rogovskaya [33] and [3]. By Hölder's inequality, (2.21), and Sobolev inequality, we have \nR L p t,x (I×T 2 ) = ˆI R 1 (x 1 , t) p L p x 1 R θ (x 2 , t) p L p x 2 dt 1 p ≤ R 1 (x 1 , t) L p t,x 1 R θ (x 2 , t) L ∞ t (I;L p x 2 ) N 1− 3 p R θ (x 2 , t) L ∞ t (I;H 1 2 − 1 p x 2 ) N 2− 4 p .(2.n∈S N f (n , Q(n)) 2 = R * 1 I f, 1 I f ≤ R L p 2 t,x (2I×T 2 ) f 2 L p ′ t,x (I×T 2 ) N 2− 8 p f 2 L p ′ t,x (I×T 2 )\nas long as p > 12.\n\n\nStrichartz estimates: Part 2\n\n3.1. Level set estimates. In this section, we prove Theorem 1.2 when d = 2. The main ingredient is the level set estimates on irrational tori in Proposition 3.1 below. For level sets estimates on the usual torus T d , see [3,23]. It turns out that these level set estimates are useful only when d = 2, 3 (see Remark 3.3), but we state and prove the results for a general dimension. In the following, we assume that θ 1 = 1 in (1.15) for simplicity. Namely, we consider Q(n) = n 2 1 + θ 2 n 2 2 + · · · + θ d n 2 d .\n(3.1) Also, let S N = {n ∈ Z d : |n j | ≤ N, j = 1, . . . , d}. Proposition 3.1. Let I be a compact interval in R. Given f (x) = n∈S N c n e 2πin·x (3.2) such that c n ℓ 2 n = 1, define the distribution function A λ by A λ = (x, t) ∈ T 2 × I : e −it∆ f (x, t) > λ . (3.3) (i) For any ε > 0, we have |A λ | N 2(d−1) 1 1+6ε λ −6+ 24 1+6ε ε (3.4) for λ N d 2 − 1 4 +ε .\n(ii) Let q > 6. Then, there exists small ε > 0 such that (3.4) and (3.5), the implicit constants depend on ε > 0, q > 6, and |I|, but are independent of f .\n|A λ | N d 2 q−(d+2) λ −q (3.5) for λ N d 2 −ε . In\nWe present the proof of Proposition 3.1 in Subsections 3.2 and 3.3. In the following, we use Proposition 3.1 to prove Theorem 1.2 when d = 2. First, we present the proof of Theorem 1.2 (ii.a), i.e. we prove (1.20) for p ≥ 20 3 when d = 2. Recall that Catoire-Wang [11] proved\ne −it∆ f L 4 t,x (I×T d ) N 1 6 f L 2 (T d ) (3.6) for f ∈ T 2 with supp f ⊂ [−N, N ] 2 . Given f as in (3.2), let F (x, t) = e −it∆ f (x, t). By Cauchy-Schwarz inequality, we have F L ∞ t,x\nN. Then, with Proposition 3.1 (i) and (3.6), we havê\nI×T 2 |F (x, t)| p dxdt ≤ˆN 3 4 +ε |F | N |F (x, t)| p dxdt + N ( 3 4 +ε)(p−4)ˆ| F (x, t)| 4 dxdt N 2− 12 1+6ε εˆN N 3 4 + λ p−7+ 24 1+6ε ε dλ + N ( 3 4 +ε)(p−4)+ 2 3 N p−4+ ,\nwhere the last inequality holds as long as p ≥ 20 3 . This proves Theorem 1.2 (ii.a). By Proposition 3.1 (i) and (ii), Theorem 1.2 (i.a) follows in a similar manner. We omit details. \ne −it∆ f L p t,x (I×T d ) N ε f L 2 (T d ) (3.7)\nfor p = 2(d+1) , where x = min n∈Z |x − n| denotes the distance of x to the closest integer. Note that Φ is periodic with period 1. By taking a Fourier transform, we have\nΦ(k) = M ≤q<2M 1 q 2 c q (k) η(q −2 k),(3.9)\nwhere c q (k) denotes Ramanujan's sum: c q (k) := a∈Jq e −2πi a q k . Let φ(q) be the Euler's totient function defined by φ(q) = a∈Jq 1. Then, by Theorem 330 in [19], we have\nΦ(0) ∼ 1 M 2 M ≤q<2M φ(q) ∼ 1.\nNamely, Φ(0) is independent of M . Without loss of generality, assume that I is centered at 0. With Q(n) in (3.1), define R as in (2.19) (2.20). Now, letting χ be a smooth cutoff function support on 3I such that χ(t) ≡ 1 on 2I, define R 1 and R 2 by\n, where S N = {n ∈ Z d : |n j | ≤ N, j = 1, . . . , d}. Then, we have R(x, t) = R 1 (x 1 , t) d j=2 R θ j (x j , t), where R θ is defined inR 1 (x, t) = Φ(t) Φ(0) R(x, t)χ(t) and R 2 (x, t) = R(x, t)χ(t) − R 1 (x, t). (3.10)\nNoting that the intervals I ℓ,q,a := ℓ + a q + 1 200q 2 , ℓ + a q + 1 100q 2 are disjoint for distinct values of ℓ, a, and q ∼ M ≫ 1, it follows from Weyl's inequality [31, Theorem 1 on p. 41] that\n|R 1 (x 1 , t)| N q 1 2 + N 1 2 (log q) 1 2 + q 1 2 (log q) 1 2 M 1 2 (log M ) 1 2 (3.11)\nfor t ∈ I ℓ,q,a since q ∼ M ≥ N . Then, along with a trivial bound |R θ j (x j , t)| N , we obtain\nR 1 L ∞ t,x min N d−1 M 1 2 (log M ) 1 2 , N d .(3.12)\nNext, we consider R 2 . By expanding Φ(t) in the Fourier series, we have\nR 2 (x, t) = − 1 Φ(0) R(x, t)χ(t) k =0 Φ(k)e 2πikt .(3.13)\nFirst, recall the following lemma (Lemma 3.33 in [3]). Given M ∈ N and k ∈ Z, we have M ≤q<2M\n|c q (k)| d(k, M )M 1+ ,(3.14)\nwhere d(k, M ) denotes the number of divisors of k less than M . Then, by taking a Fourier transform of (3.13) with (3.9), (3.14), and d(k, M ) k 0+ , we have\n| R 2 (n, τ )| = 1 S N (n) k =0 Φ(k) Φ(0) χ(τ − Q(n) − k) 1 M 2 k =0 M ≤q<2M |c q (k)| η(q −2 k) χ(τ − Q(n) − k) 1 M 2 k =0 k 0+ M 1+ M 2 k 0+ 1 τ − Q(n) − k 10 M −1+ . (3.15) Define Θ λ (x, t) by Θ λ (x, t) = exp i arg(e −it∆ f (x)) · 1 A λ (x, t). (3.16)\nNote that supp Θ λ (x, ·) ⊂ I for each x ∈ T d . Then, by Cauchy-Schwarz inequality with (3.2), we have\nλ 2 |A λ | 2 ≤ ˆI ×T 2 e −it∆ f (x, t)Θ λ (x, t)dxdt 2 = n∈S N c n Θ λ (n, Q(n)) 2 ≤ n∈S N Θ λ (n, Q(n)) 2 = R * Θ λ , Θ λ = (Rχ) * Θ λ , Θ λ (3.17)\nBy (3.10), (3.12), and (3.15), we have\n= R 1 * Θ λ , Θ λ + R 2 * Θ λ , Θ λ ≤ R 1 L ∞ t,x Θ λ 2 L 1 t,x + R 2 L ∞ τ ℓ ∞ n Θ λ 2 L 2 t,x ≤ C 1 N d−1 M 1 2 +ε 1 |A λ | 2 + M −1+ε 2 |A λ | (3.18) for small ε 1 , ε 2 > 0. Now, choose M ≥ N such that N d−1 M 1 2 +ε 1 ∼ λ 2 . (3.19) The condition (3.19) with M ≥ N implies that λ N d 2 − 1 4 + ε 1 2 . Then, (3.18) yields |A λ | N 2(d−1) λ 4 1−ε 2 1+2ε 1 λ −2 N 2(d−1) 1 1+3ε 1 λ −6+ 12 1+3ε 1 ε 1\nby setting ε 2 = ε 2 (ε 1 ) such that 1−ε 2 1+2ε 1 = 1 1+3ε 1 . This proves (3.4) with ε = ε 1 2 . 3.3. Proof of Proposition 3.1 (ii). In this subsection, we prove the level set estimate (3.5), which is sharp for q > 6. The following argument is inspired by Bourgain's paper [3]. We first go over some basic setups, restricting our attention to t ∈ T.\n\nLet (3.20)\n\nThen, we have the following lemma. Here, x = min n∈Z |x − n| denotes the distance of x to the closest integer as before.\nLemma 3.4 (Lemma 3.18 in [3]). Let 1 ≤ a ≤ q ≤ N and (a, q) = 1 such that t − a q ≤ 1 qN . (3.21)\nThen, we have\n|K(x, t)| N q 1 2 1 + N t − a q 1 2 . (3.22)\nNote that the multiplier σ n in (3.20) avoids the logarithmic loss (when q ∼ N ) in Weyl's inequality (3.11) on the Weyl sum W N (x, t) = |n|≤N e 2πi(nx+n 2 t) . Indeed, by writing\nK = 1 N 2N −1\nk=N W k , we see that this regularizing effect in (3.22) is analogous to that of the Féjér kernel over the Dirichlet kernel.\n\nIn the following, we fix N ≫ 1, dyadic. For dyadic M ≤ N , let R M by\nR M = a q : a ∈ J q , M ≤ q < 2M ,\nwhere J q is as in (3.8). Let ψ(t) be a smooth cutoff function supported on\n1 2 + 1 10 ≤ |t| ≤ 2 − 1 10 such that j∈Z ψ(2 −j t) = 1 for t = 0. For s ∈ N with M ≤ 2 s < N let ω N,2 s (t) = ψ(2 s N t) and define ω N,N by ω N,N (t) = j≥log 2 N ψ(2 j N t), t = 0, 1, t = 0.\nNote that we have supp(ω N,2 s ) ⊂ |t| \nR M 1 + − 2 M 1 N , 2 M 1 N ∩ R M 2 + − 2 M 2 N , 2 M 2 N = ∅.\nRecall that by Dirichlet's theorem [ Now, we focus on our problem. Namely, we do not assume t ∈ T any longer. Given an interval I ⊂ R, assume that I is centered at 0 and let χ be a smooth cutoff function support on 3I such that χ(t) ≡ 1 on 2I as before. We define \n| ρ(k)| ≤ M ≤N 1 M, dyadic M ≤2 s ≤N F T ∈R M δ T (k) · ω N,2 s (k) M ≤N 1 M, dyadic M ≤2 s ≤N d(k, M )M 1+ 2 s N k 2 s N −100 N −1+K(x, t) = χ(t) n∈ S N σ n 1 e 2πi(n·x+Q(n)t) = χ(t)K(x 1 , t) d j=2 |n j |≤N e 2πi(n j x j +θ j n 2 j t) ,(3.|Λ M,s (x, t)| N d−1 N M 1 2 1 + (2 −s N ) 1 2 + M 2 2 s N N d− 1 2 N d−1 2 s N M 1 2 . (3.32)\nHence, from (3.32), we have Hence, from (3.37), we have\nf * Λ M,s L ∞ (I×T d ) N d−1 2 s N M 1 2 f L 1 (I×T d ) .f * Λ M,s L 2 (I×T d ) M 2 s N N 0+ f L 2 (I×T d ) . (3.38) Also, with the trivial bound d(k, M ) ≤ M in (3.36), we have f * Λ M,s L 2 (I×T d ) M 2+ 2 s N f L 2 (I×T d ) . (3.39)\nThe second estimate (3.39) is useful when M ≪ N ε .\n\nIn the following, we establish another estimate on f * Λ M,s L 2 (I×T d ) , using the following lemma from Bourgain [3]. \nf * Λ M,s L 2 (I×T d ) M 1+ 2 s N ˆ S N σ 2 n 1 | f (n, τ )| 2 k =0 d(k, M ) k 2 s N 100 τ − Q(n) − k 100 2 dτ 1 2 + M 2 2 s N 2− f L 2 (I×T d ) ,(3.41)\nwhere S N is as in (3.30). Given D > 0 (to be chosen later), separate the first term, depending on d(k, M ) ≤ D or > D. The contribution from d(k, M ) ≤ D can be estimated by\nDM 1+ 2 s N f L 2 (I×T d ) .(3.42)\nNext, we estimate the contribution from d(k, M ) > D. By Cauchy-Schwarz inequality, we have\nk =0 d(k, M ) k 2 s N 100 τ − Q(n) − k 100 2 ≤ k =0 d(k, M ) 2 k 2 s N 200 τ − Q(n) − k 100 k =0 1 τ − Q(n) − k 100 k =0 d(k, M ) 2 k 2 s N 200 τ − Q(n) − k 100 . (3.43)\nNow, we estimate the first term on the right-hand side of (3.41) after applying (3.43). By first integrating in τ , then summing over |n j | N for j = 1, . . . , d, applying Lemma 3.5 (with 2B and 2β instead of B and β), the trivial bound d(k, M ) ≤ M , and Hausdorff-Young's inequality, we have \nM 1+ N d 2 2 s N |k| 2 s N d(k, M ) 2 + ∞ j=1 |k|∼2 s+j N d(k, M ) 2 k 2 s N −200 1 2 f L ∞ τ ℓ ∞ n M 2+ N d 2 2 s N ∞ j=0 2 −200j (D −2B M 2β 2 s+j N + M 2B ) 1 2 f L 1 (I×T d ) N d 2 D −B M 2+β+ 2 s 2 N 1 2 + M 2+B+ 2 s N f L 1 (I×T d ) .f * Λ M,s L 2 (I×T d ) DM 1+ 2 s N f L 2 (I×T d ) + N d 2 D −B M 2+β+ 2 s 2 N 1 2 + M 2+B+ 2 s N f L 1 (I×T d ) . (3.45) Define Λ by Λ(x, t) = M ≤N 1 M, dyadic M ≤2 s ≤N Λ M,s (x, t),(3.46)\nwhere Λ M,s is as in (3.31). By (3.25), we have \nK − Λ (x, t) = 1 + M ≤N 1 M, dyadic M ≤2 s ≤N α M,s K(x, t)ρ(t).K − Λ L ∞ (I×T d ) N d− 1 2 . (3.48)\nHence, we have\n| f, f * (K − Λ) | N d− 1 2 f 2 L 1 . (3.49) Let p ∈ (1, 2) such that 1 p = 1 − θ 1 + θ 2 (3.50)\nfor some θ ∈ (0, 1). Note that p ′ θ = 2.\n\nCase (i): θ < 1 5 . By interpolating (3.33) and (3.39), we have\nf * Λ M,s L p ′ (I×T d ) N (d−1)(1−θ) (2 s N ) 1 2 − 3 2 θ M − 1 2 +( 5 2 +)θ f L p (I×T d ) .\n(3.51)\n\nThen, summing over dyadic M ≥ 1 and s with 2 s ≤ N , we have\nf * Λ L p ′ (I×T d ) N d−(d+2)θ f L p (I×T d ) ,(3.52)\nas long as θ > 0 satisfies − 1 2 + ( 5 2 +)θ ≤ 0, i.e.\n\nθ < 1 5 .\n\n(3.53)\n\nIn the following, we prove the level set estimate (3.5) for q > 10. Given f as in (3.2),\nlet F (x, t) = e −it∆ f (x, t). Let Θ λ (x, t) be as in (3.16), where A λ = {(x, t) ∈ T d × I : |F (x, t)| > λ}.\nThen, proceeding as in (3.17) with (3.49) and (3.52), we have\nλ 2 |A λ | 2 ≤ n∈S N Θ λ (n, Q(n)) 2 ≤ K * Θ λ , Θ λ ≤ | (K − Λ) * Θ λ , Θ λ | + | Λ * Θ λ , Θ λ | N d− 1 2 |A λ | 2 + N d−(d+2)θ |A λ | 2 p . (3.54) For λ ≫ N d 2 − 1 4 , (3.54) reduces to λ 2 |A λ | 2 N d−(d+2)θ |A λ | 2 p . (3.55)\nNoting that p ′ θ = 2 by (3.50), it follows from (3.55) that\n|A λ | N d 2 q−(d+2) λ −q ,\nwhere q := p ′ = 2 θ > 10. Note that we only needed to assume λ ≫ N Case (ii): 1 5 ≤ θ < 1 3 . Let M j , j = 1, 2 be dyadic numbers such that\nM 1 ∼ N d 2 λ δ 1 N εδ 1 and M 2 ∼ N δ 2 . (3.56)\nHere, we choose δ 1 , δ 2 > 0 such that M 1 ≪ M 2 . We divide Λ into three pieces: Λ = Λ 1 + Λ 2 + Λ 3 by setting Similarly, by interpolating (3.33) and (3.45), we have \nΛ j = M ∈I j M, dyadic M ≤2 s ≤N Λ M,s ,(3.f * Λ 1 L p ′ (I×T d ) M − 1 2 +( 5 2 +)θ 1 N d−(d+2)θ f L p (I×T d ) , (3.58) since θ ∈ [ 1 5 , 1 3 ).f * Λ M,s L p ′ (I×T d ) N (d−1)(1−θ) (2 s N ) 1 2 − 3 2 θ M − 1 2 +( 3 2 +)θ D θ f L p (I×T d ) + N (d−1)(1−θ) 2 s N M 1 2 (1−θ) N d 2 θ D −B M 2+β+ 2 s 2 N 1 2 + M 2+B+ 2 s N θ f L 1 (I×T d ) .f * Λ 2 L p ′ (I×T d ) N d−(d+2)θ f L p (I×T d ) + M − 1 2 −σθ 1 N d−( d 2 +1)θ + M − 1 2 +( 5 2 +B+)θ 2 N d−( d 2 +2)θ f L 1 (I×T d ) , (3.61) as long as − 1 2 + ( 3 2 + α+)θ ≤ 0, i.e. θ < 1 3 + 2α+ . (3.62)\nNote that (3.62) can be satisfied as long as θ < 1 3 by choosing α sufficiently small. Lastly, from (3.33) and (3.38), we have\nf * Λ M,s L p ′ (I×T d ) N (d−1)(1−θ) 2 s N M 1 2 − 3 2 θ N (0+)θ f L p (I×T d ) . (3.63)\nThen, summing over M ≥ M 2 ∼ N δ 2 and s with 2 s ≤ N , we have\nf * Λ 3 L p ′ (I×T d ) N (d−1)(1−θ) N 2 M 2 1 2 − 3 2 θ N (0+)θ f L p (I×T d ) N d−(d+2)θ f L p (I×T d ) , (3.64)\nas long as θ < 1 3 . Putting (3.58), (3.61), and (3.64) together with (3.53), we obtain\nf * Λ L p ′ (I×T d ) M − 1 2 +( 5 2 +)θ 1 N d−(d+2)θ f L p (I×T d ) + M − 1 2 −σθ 1 N d−( d 2 +1)θ + M − 1 2 +( 5 2 +B+)θ 2 N d−( d 2 +2)θ f L 1 (I×T d ) . (3.65)\nNow, we are ready to prove the level set estimate (3.5) for q > 6. Then, proceeding as in Case (i) with (3.49) and (3.65), we have\nλ 2 |A λ | 2 ≤ | (K − Λ) * Θ λ , Θ λ | + | Λ * Θ λ , Θ λ | N d− 1 2 |A λ | 2 + M − 1 2 +( 5 2 +)θ 1 N d−(d+2)θ |A λ | 2 p + M − 1 2 −σθ 1 N d−( d 2 +1)θ |A λ | 1+ 1 p + M − 1 2 +( 5 2 +B+)θ 2 N d−( d 2 +2)θ |A λ | 1+ 1 p . (3.66) Since λ ≫ N d 2 − 1 4 , (3.66) reduces to λ 2 |A λ | 2 M − 1 2 +( 5 2 +)θ 1 N d−(d+2)θ |A λ | 2 p + M − 1 2 −σθ 1 N d−( d 2 +1)θ |A λ | 1+ 1 p + M − 1 2 +( 5 2 +B+)θ 2 N d−( d 2 +2)θ |A λ | 1+ 1 p =: I + II + III. (3.67)\nFirst, suppose that λ 2 |A λ | 2 I holds. Recall from (3.50) that p ′ θ = 2. Then, with (3.56), we have\n|A λ | N d 2 λ (− p ′ 4 + 5 2 +)δ 1 N d 2 p ′ −(d+2) λ −p ′ N d 2 q−(d+2) λ −q (3.68)\nfor q > p ′ by choosing δ 1 = δ 1 (q, p ′ ) sufficiently small. Next, suppose that λ 2 |A λ | 2 II holds. Then, from (3.56), we have \n|A λ | N d 2 p ′ −(d+2) λ −p ′ N d 2 p ′ λ −p ′ M − p ′ 2 −2σ 1 N d 2 p ′ −(d+2) λ −p ′ ,(3.|A λ | N d 2 p ′ −(d+2) λ −p ′ N −2+εp ′ +(− p ′ 2 +5+2B+)δ 2 N d 2 p ′ −(d+2) λ −p ′ (3.70)\nas long as we have εp ′ ≤ 1 and δ 2 = δ 2 (p ′ , B) is sufficiently small such that (− p ′ 2 + 5 + 2B+)δ 2 ≤ 1.\n\nFinally, given q > 6, we choose θ < 1 3 such that q > p ′ = 2 θ > 6. Then, from (3.67), (3.68), (3.69), and (3.70) with λ ≤ N d 2 , we obtain\n|A λ | N d 2 q−(d+2) λ −q .\nThis completes the proof of Proposition 3.1 (ii).\n\n\nWell-posedness in subcritical spaces\n\nIn this section, we prove local well-posedness of NLS (1.25) on irrational tori in subcritical Sobolev spaces (Theorem 1.4). It turns out that the well-posed theory of (1.25) is very similar to that on the standard tours [3,4,5,6].\n\nIn the seminal paper [3], Bourgain introduced the X s,b -space whose norm is given by\nu X s,b (R×T d ) = n s τ − |n| 2 b u(n, τ ) L 2 τ ℓ 2 n (R×Z d ) , (4.1)\nwhere · = (1 + | · | 2 ) 1 2 . After establishing Strichartz estimates, he proved several wellposedness results of NLS on the standard torus T d . In our case, i.e. on an irrational torus, we need to replace the weight τ − |n| 2 b in (4.1) by τ − Q(n) b , where Q(n) is defined in (1.15). Then, by the standard X s,b -theory, it is known that certain multilinear Strichartz estimates imply well-posedness. More precisely, we have the following lemma. The proof of Lemma 4.1 is standard and we refer the readers to [3,4,6,11] for details.\nk+1 j=1 e −it∆ φ j L 2 t,x (I×T d ) N −s max k+1 j=1 N s j φ i L 2 x (T d ) , (4.2) for all φ j ∈ L 2 (T d ) with supp φ j ⊂ [−N j , N j ] d ,\nProof of Theorem 1.4. In view of Lemma 4.1, it suffices to prove the (k + 1)-linear estimate (4.2) for s > s 0 . Without loss of generality, assume that N 1 ≥ N 2 ≥ · · · ≥ N k+1 .\n\nCase (i): d = 2. When k = 1, the well-posedness result was already obtain in [11]. In the following, we first consider the case k = 2, 3, 4, 5. First, assume that supp\nφ 1 ⊂ [−N 2 , N 2 ] 2 .\nThen, by Hölder's inequality and (1.21) with 4(7k+5) 3(k+3) ∈ (4, 20 3 ], we have\nk+1 j=1 e −it∆ φ j L 2 t,x ≤ e −it∆ φ 1 L 4(7k+5) 3(k+3) e −it∆ φ 2 L 4(7k+5) 3(k+3) k+1 j=3 e −it∆ φ j L 7k+5 2 N 7k−3 7k+5 +2ε 2 φ 1 L 2 φ 2 L 2 k+1 j=3 N 7k−3 7k+5 +ε j φ j L 2 ≤ N −s max k+1 j=1 N s j φ i L 2 x , (4.3) for s > s 0 = 7k−3 7k+5 . In general, if supp φ 1 ⊂ [−N 1 , N 1 ] 2 , then we can write φ 1 = |j| N 1 N 2 φ 1j , where supp φ 1j ⊂ N 2 j + [−N 2 , N 2 ] 2 . Letting ψ 1j (x) = e −2πiN 2 j·x φ 1j (x), we have supp ψ 1j ⊂ [−N 2 , N 2 ] 2 .\nThen, by a change of variables and (1.21) (see [5]), we obtain\ne −it∆ φ 1j L 4(7k+5) 3(k+3) t,x = e −it∆ ψ 1j L 4(7k+5) 3(k+3) N 7k−3 2(7k+5) +ε 2 φ 1j L 2 x . (4.4)\nThen, by almost orthogonality with (4.3) and (4.4), we have\nk+1 j=1 e −it∆ φ j 2 L 2 t,x |j| N 1 N 2 e −it∆ φ 1j k+1 j=2 e it∆ φ j 2 L 2 t,x N −2s max k+1 j=1 N 2s j φ j 2 L 2 x .\nRemark 4.2. In view of the reduction above, we only prove (4.2), assuming supp φ 1 ⊂ [−N 2 , N 2 ] d in the following.\n\nNext, we consider the case k ≥ 5. By Hölder's inequality and (1.21) with 8k k+1 ∈ [ 20 3 , 10), we have\nk+1 j=1 e −it∆ φ j L 2 t,x ≤ e −it∆ φ 1 L 8k k+1 e −it∆ φ 2 L 8k k+1 k+1 j=3 e −it∆ φ j L 4 k N 1− 1 k +2ε 2 φ 1 L 2 φ 2 L 2 k+1 j=3 N 1− 1 k j φ j L 2 x .\nHence, (4.2) holds for s > s 0 = 1 − 1 k . Case (ii): d = 3. When k = 1, the well-posedness result was obtained in [6]. When k = 2, by Hölder's inequality and (1.22), we have\n3 j=1 e −it∆ φ j L 2 t,x ≤ e −it∆ φ 1 L 104 21 e −it∆ φ 2 L 104 21 e −it∆ φ 3 L 52 5 N 53 52 +2ε 2 N 53 52 3 3 j=1 φ j L 2 x .\nHence, (4.2) holds for s > s 0 = 53 52 . When k ≥ 3, by Hölder's inequality and (1.22), we have\nk+1 j=1 e −it∆ φ j L 2 t,x ≤ e −it∆ φ 1 L 20k 3k+2 e −it∆ φ 2 L 20k 3k+2 k+1 j=3 e −it∆ φ j L 5k N 3 2 − 1 k 2 φ 1 L 2 φ 2 L 2 k+1 j=3 N 3 2 − 1 k j φ j L 2 x .\nHence, (4.2) holds s ≥ s 0 = 3 2 − 1 k . Case (iii): d ≥ 4 and k ≥ 1. By Hölder's inequality with (1.23) or (1.24), we have\nk+1 j=1 e −it∆ φ j L 2 t,x ≤ e −it∆ φ 1 L 4(d+2)k dk+2 e −it∆ φ 2 L 4(d+2)k dk+2 k+1 j=3 e −it∆ φ j L (d+2)k N d 2 − 1 k +ε 2 φ 1 L 2 φ 2 L 2 k+1 j=3 N d 2 − 1 k j φ j L 2 x .\nHence, (4.2) holds for s > s 0 = d 2 − 1 k .\n\n\n5.\n\nWell-posedness in critical spaces 5.1. Function spaces. In this section, we prove local well-posedness of NLS (1.25) on irrational tori in critical Sobolev spaces H sc (T d ) (Theorem 1.5). In the following, we use the U p -and V p -spaces, developed by Tataru, Koch, and their collaborators [30,18,21,22]. These spaces have been very effective in establishing well-posedness of various dispersive PDEs in critical spaces. We briefly go over the basic definitions of function spaces and their properties. See Hadac-Herr-Koch [18] and Herr-Tataru-Tzvetkov [21] for detailed proofs.\n\nLet H be a separable Hilbert space over C. In particular, it will be either H s (T d ) or C. Let Z be the collection of finite partitions {t k } K k=0 of R: −∞ < t 0 < · · · < t K ≤ ∞. If t K = ∞, we use the convention u(t K ) := 0 for all functions u : R → H. We use 1 I to denote the sharp characteristic function of a set I ⊂ R.\nDefinition 5.1. Let 1 ≤ p < ∞.\n(i) A U p -atom is defined by a step function a : R → H of the form\na = K k=1 1 [t k−1 ,t k ) φ k−1 , where {t k } K k=0 ∈ Z and {φ k } K−1 k=0 ⊂ H with K−1 k=0 φ k p H = 1.\nThen, we define the atomic space U p (R; H) to be the collection of functions u : R → H of the form\nu = ∞ j=1 λ j a j ,\nwhere a j 's are U p -atoms and {λ j } j∈N ∈ ℓ 1 (N; C), (ii) We define V p (R; H) by the collection of functions u :\nR → H with u V p (R;H) < ∞, where the V p -norm is defined by u V p (R;H) := sup {t k } K k=0 ∈Z K k=1 u(t k ) − u(t k−1 ) p H 1 p .\nWe also define V p rc (R; H) to be the closed subspace of all right-continuous functions in V p (R; H) such that lim t→−∞ u(t) = 0.\n\n(iii) Let s ∈ R. We define U p ∆ H s (and V p ∆ H s , respectively) to be the spaces of all functions u : R → H s (T d ) such that the following U p ∆ H s -norm (and V p ∆ H s -norm, respectively) is finite:\nu U p ∆ H s := e it∆ u U p (R;H s ) and u V p ∆ H s := e it∆ u V p (R;H s ) .\nHere, the Laplacian ∆ is defined in terms of Q(n) as in (1.14).\n\nRemark 5.2. Note that the spaces U p (R; H), V p (R; H), and V p rc (R; H) are Banach spaces. Moreover, we have the following embeddings:\nU p (R; H) ֒→ V p rc (R; H) ֒→ U q (R; H) ֒→ L ∞ (R; H) for 1 ≤ p < q < ∞. Similar embeddings hold for U p ∆ H s and V p ∆ H s .\nNext, we state a transference principle and an interpolation result.\n\n\nLemma 5.3. (i) (transference principle) Suppose that we have\nT (e −it∆ φ 1 , . . . , e −it∆ φ k ) L p t L q x (R×T d ) k j=1 φ j L 2\nx for some 1 ≤ p, q ≤ ∞. Then, we have\nT (u 1 , . . . , u k ) L p t L q x (R×T d ) k j=1 u j U p ∆ L 2 x .\n(ii) (interpolation) Let E be a Banach space. Suppose that T : U p 1 × · · · × U p k → E is a bounded k-linear operator such that\nT (u 1 , . . . , u k ) E ≤ C 1 k j=1 u j U p j\nfor some p 1 , . . . , p k > 2. Moreover, assume that there exists C 2 ∈ (0, C 1 ] such that\nT (u 1 , . . . , u k ) E ≤ C 2 k j=1 u j U 2 .\nThen, we have\nT (u 1 , . . . , u k ) E ≤ C 2 ln C 1 C 2 + 1 k k j=1 u j V 2 for u j ∈ V 2 rc , j = 1, . . . , k.\nA transference principle as above has been commonly used in the Fourier restriction norm method. See [18,Proposition 2.19] for the proof of Lemma 5.3 (i). The proof of the interpolation result follows from extending the trilinear result in [21] to a general k-linear case. See also [18,Proposition 2.20].\n\nLet η : R → [0, 1] be an even smooth cutoff function supported on [− 8 5 , 8 5 ] such that η ≡ 1 on [− 5 4 , 5 4 ]. Given a dyadic number N ≥ 1, we set η 1 (ξ) = η(|ξ|) and\nη N (ξ) = η |ξ| N − η 2|ξ| N for N ≥ 2.\nThen, we define the Littlewood-Paley projection operator P N as the Fourier multiplier operator with symbol η N . Moreover, we define P ≤N by P ≤N = 1≤M ≤N P M . More generally, given a set R ⊂ Z d , we define P R to be the Fourier multiplier operator with symbol 1 R .\nDefinition 5.4. (i) Let s ∈ R.\nWe define X s to be the space of all functions u :\nR → H s (T d ) such that u X s < ∞, where the X s -norm is defined by u X s := n∈Z d n 2s e −itQ(n) u(n, t) 2 U 2 (Rt;C) 1 2 .\n(ii) Let s ∈ R. We define Y s to be the space of all functions u : R → H s (T d ) such that for every n ∈ Z d , the map t → e −itQ(n) u(n, t) is in V 2 rc (R t ; C) and u Y s < ∞, where the Y s -norm is defined by\nu Y s := n∈Z d n 2s e −itQ(n) u(n, t) 2 V 2 (Rt;C) 1 2 .\nRecall the following embeddings:\nU 2 ∆ H s ֒→ X s ֒→ Y s ֒→ V 2 ∆ H s . (5.2)\nGiven a time interval I ⊂ R, we define the restrictions X s (I) and Y s (I) of these spaces in the usual manner. We now state the linear estimates. Given f ∈ L 1 loc ([0, ∞); L 2 (T d )), define I(f ) by\nI(f )(t) :=ˆt 0 e −i(t−t ′ )∆ f (t ′ )dt ′ .\nLemma 5.5 (Linear estimates). Let s ≥ 0 and T > 0. Then, the following linear estimates hold:\ne −it∆ φ X s ([0,T )) ≤ φ H s , I(f ) X s ([0,T )) ≤ sup v∈Y −s ([0,T )) v Y −s =1 ˆT 0ˆT d f (x, t)v(x, t)dxdt , for all φ ∈ H s (T d ) and f ∈ L 1 ([0, T ); H s (T d )).\nNext, we present the crucial multilinear estimate.\n\nProposition 5.6. Let d and k satisfy\n(i) d = 2, k ≥ 6, (ii) d = 3, k ≥ 3, or (iii) d ≥ 4, k ≥ 2. (5.3)\nThen, the following multilinear estimate holds for all T ∈ (0, 1]:\nI 2k+1 j=1 u * j X sc ([0,T )) 2k+1 j=1 u j X s ([0,T )) 2k+1 ℓ=1 ℓ =j u ℓ X sc ([0,T )) , (5.4) for s ≥ s c = d 2 − 1 k > 0, where u * j denotes either u j or u j .\nOnce we prove Proposition 5.6, one can prove Theorem 1.5 by the fixed point argument as in [21,42]. We omit details. The remainder of this section is devoted to the proof of Proposition 5.6. Indeed, the multilinear estimate (5.4) follows once we prove the following multilinear Strichartz estimate. \nP N j e −it∆ φ j L 2 t,x (I×T d ) N k+1 N 1 + 1 N 2 δ P N 1 φ 1 L 2 x (T d ) k+1 j=2 N sc j P N j φ j L 2 x (T d ) , (5.5)\nfor any interval I ⊂ [0, 1] and for all φ j ∈ L 2 (T d ), j = 1, . . . , k + 1, and N 1 ≥ N 2 ≥ · · · ≥ N k+1 ≥ 1.\n\nIn Subsection 5.2, we first present the proof of Proposition 5.7. In Subsection 5.3, we then use the multilinear Strichartz estimate (5.5) to prove Proposition 5.6, thus yielding Theorem 1.5.\n\n\nMultilinear Strichartz estimate.\n\nIn this subsection, we use the sharp L p -Strichartz estimates (1.19) in Theorem 1.2 to prove the multilinear Strichartz estimate (5.5). The main idea is to refine the Strichartz estimate by considering frequency scales finer than the standard dyadic Littlewood-Paley localizations as in [21]. See Lemma 5.10.\n\nDefinition 5.8. We say that (d, p) ∈ N × R is an admissible pair if\n(i) d = 2, p > 20 3 , (ii) d = 3, p > 16 3 , or (iii) d ≥ 4, p > 4. (5.6)\nNote that, by Theorem 1.2, the Strichartz estimates with (d, p) in this range are sharp.\n\nGiven dyadic N ≥ 1, let C N denote the collection of cubes C ⊂ Z d of side length ∼ N with arbitrary center and orientation. Then, we can rewrite Theorem 1.2 in the following form.\n\nLemma 5.9. Let (d, p) be admissible and I ⊂ R be a bounded interval. Then, for all dyadic N ≥ 1, we have\nP N e −it∆ φ L p t,x (I×T d ) N d 2 − d+2 p P N φ L 2 x (T d ) . (5.7)\nMore generally, for all C ∈ C N , we have\nP C e −it∆ φ L p t,x (I×T d ) N d 2 − d+2 p P C φ L 2 x (T d ) . (5.8)\nThe Strichartz estimate (5.8) shows that the loss in (5.8) depends only on the size of the frequency support, not the position. See [5,21]. In order to exploit further orthogonality between different frequency pieces under the linear Schrödinger evolution, we need to decompose the frequency cubes C N . Let R M (N ) be the collection of all sets in Z d which are given as the intersection of a cube of side length 2N with a strip of \"width\" 2M , i.e. the collection of all sets of the form\nn 0 + [−N, N ] d ∩ n ∈ Z d : |a · θ θ θ n − A| ≤ M (5.9)\nwith some n 0 ∈ Z d , a ∈ R d , |a| = 1, A ∈ R. Here, the dot product a · θ θ θ n is given by\na · θ θ θ n = d j=1 θ j a j n j ,(5.10)\nwhere θ θ θ = (θ 1 , . . . , θ d ) is as in (1.15). Then, we have the following refinement of the Strichartz estimate. \nP R e −it∆ φ L p t,x (I×T d ) N d 2 − d+2 p M N δ P R φ L 2 x (T d ) ,(5.11)\nwhere 0 < δ < 1 2 − 10 3p when d = 2, 0 < δ < 1 2 − 8 3p when d = 3, and 0 < δ < 1 2 − 2 p when d ≥ 4.\n\n\nProof. By Bernstein's inequality, we have\nP R e −it∆ φ L ∞ t,x (I×T d ) M 1 2 N d−1 2 P R φ L 2 x (T d ) . (5.12)\nGiven admissible (d, p), write 1 p = θ q + 1−θ ∞ for some θ ∈ [0, 1), where q < p is given by q = 20 3 + when d = 2, q = 16 3 + when d = 3, and q = 4+ when d ≥ 4. Then, (5.11) follows from interpolating (5.8) (with p = q) and (5.12). Now, we are ready to prove the main multilinear Strichartz estimates (5.5).\n\nProof of Proposition 5.7. Let u j = e −it∆ φ j . Then, by almost orthogonality in spatial frequencies, it suffices to prove that there exists δ > 0 such that\nP C P N 1 u 1 k+1 j=2 P N j u j L 2 t,x N k+1 N 1 + 1 N 2 δ P C P N 1 φ 1 L 2 x k+1 j=2 N sc j P N j φ j L 2 x , (5.13)\nfor all cubes C ∈ C N 2 . Fix a cube C ∈ C N 2 and let n 0 be the center of C. Partition C = R ℓ into disjoint strips R ℓ with width M = max(N 2 2 /N 1 , 1), which are all orthogonal to n 0 with respect to the dot product · θ θ θ in (5.10), i.e. R ℓ is given by\nR ℓ = n ∈ C : n · θ θ θ n 0 ∈ |n 0 |M ℓ, |n 0 |M (ℓ + 1) , |ℓ| ∼ N 1 M .\nNote that we have R ℓ ∈ R M (N 2 ). By writing\nP C P N 1 u 1 k+1 j=2 P N j u j = ℓ P R ℓ P N 1 u 1 k+1 j=2 P N j u j ,(5.14)\nwe show that the sum is almost orthogonal in L 2 t,x . Since N 2 2 M 2 ℓ, we have\nQ(n 1 ) = 1 Q(n 0 ) |n 1 · θ θ θ n 0 | 2 + Q(n 1 − n 0 ) − 1 Q(n 0 ) |(n 1 − n 0 ) · θ θ θ n 0 | 2 = M 2 ℓ 2 + O(M 2 ℓ),\nfor n 1 ∈ R ℓ . Note that the multiplication by the factor k+1 j=2 P N j u j in (5.14) changes the time frequency at most by O(N 2 2 ). Hence, P R ℓ P N 1 u 1 k+1 j=2 P N j u j in (5.14) is localized at time frequency M 2 ℓ 2 + O(M 2 ℓ) = O(M 2 ℓ 2 ) for each ℓ. Therefore, the sum in (5.14) is almost orthogonal and we have\nP C P N 1 u 1 k+1 j=2 P N j u j 2 L 2 t,x ∼ ℓ P R ℓ P N 1 u 1 k+1 j=2 P N j u j 2 L 2 t,x .\n(5.15)\n\nWith d and k as in (5.3), let p d,k = (d + 2)k. Then, by Lemma 5.9, we have The existence of such p is implied by (5.3). Moreover, the lower bound on p guarantees that each (d, p) is admissible, while the upper bound on p guarantees that\nP N e −it∆ φ L p d,k t,x N sc P N φ L 2 x .(5.d − 2(d + 2) p − s c < 0. (5.18) Let q such that 2 p + k − 2 p d,k + 1 q = 1 2 . (5.19)\nThen, it follows from (5.17) that (d, q) is also admissible. By Hölder's inequality and Lemmata 5.9 and 5.10, we have\nP R ℓ P N 1 u 1 k j=2 P N j u j L 2 t,x P R ℓ P N 1 u 1 L p P N 2 u 2 L p k j=3 P N j u j L p d,k P N k+1 u k+1 L q N d− 2(d+2) p −sc 2 N d 2 − d+2 q −sc k+1 M N 2 δ P R ℓ P N 1 φ 1 L 2 x k+1 j=2 N sc j P N j φ j L 2 x , (5.20)\nfor some δ > 0. In view of (5.17) and (5.18), choose p such that\n−d + 2(d + 2) p + s c = δ.\nMoreover, from (5.18) and (5.19), we have\nd 2 − d + 2 q − s c = −d + 2(d + 2) p + s c = δ > 0. (5.21)\nThen, noting M N 2 ∼ N 2 N 1 + 1 N 2 , it follows from (5.20) that\nP R ℓ P N 1 u 1 k+1 j=2 P N j u j L 2 t,x N k+1 N 1 + 1 N 2 δ P R ℓ P N 1 φ 1 L 2 x k+1 j=2 N sc j P N j φ j L 2 x .\nFinally, by summing up the squares in (5.15) with respect to ℓ, we obtain (5.13) and hence (5.5). This completes the proof of Proposition 5.7.\n\n\n5.3.\n\nProof of Proposition 5.6. First, we state and prove an auxiliary lemma (Lemma 5.11), using Proposition 5.7. Let C N , N ≥ 1, be the collection of cubes C ⊂ Z d of side length ∼ N as before. Let (d, p) be admissible in the sense of Definition 5.8. Then, it follows from Lemma 5.9 with Lemma 5.3 (i) that\nP C e it∆ φ L p t,x (I×T d ) N d 2 − d+2 p P C φ U p ∆ L 2 x (5.22)\nfor all C ∈ C N .\n\nLemma 5.11. Let d and k satisfy (5.3). Then, there exists δ ′ > 0 such that\nk+1 j=1 P N j u j L 2 t,x (I×T d ) N k+1 N 1 + 1 N 2 δ ′ P N 1 u 1 Y 0 k+1 j=2 P N j u j Y sc ,(5.23)\nfor any interval I ⊂ [0, 1] and for all N 1 ≥ N 2 ≥ · · · ≥ N k+1 ≥ 1.\n\nProof. By almost orthogonality in spatial frequencies, it suffices to prove that there exists δ ′ > 0 such that\nP C P N 1 u 1 k+1 j=2 P N j u j L 2 t,x N k+1 N 1 + 1 N 2 δ ′ P C P N 1 u 1 Y 0 k+1 j=2 N sc j P N j u j Y 0 , (5.24)\nfor all cubes C ∈ C N 2 . Moreover, by the embedding (5.2), it suffices to prove (5.24), where the Y 0 -norm is replaced by the V 2 ∆ L 2 -norm. Furthermore, it suffices to prove that there exists λ > 0 such that the following two estimates hold:\nLHS of (5.24) N k+1 N 1 + 1 N 2 δ P C P N 1 u 1 U 2 ∆ L 2 k+1 j=2 N sc j P N j u j U 2 ∆ L 2 ,(5.25)\nand LHS of (5.24) \nN k+1 N 2 δ P C P N 1 u 1 U p ∆ L 2 k+1 j=2 N sc j P N j u j U p ∆ L 2 ,(5.P C P N 1 u 1 k j=2 P N j u j L 2 t,x P C P N 1 u 1 L p P N 2 u 2 L p k j=3 P N j u j L p d,k P N k+1 u k+1 L q N d− 2(d+2) p −sc 2 N d 2 − d+2 q −sc k+1 P C P N 1 u 1 U p ∆ L 2 k+1 j=2 N sc j P N j u j U q j ∆ L 2 ,(5.27)\nwhere q 2 = p, q j = p d,k for j = 3, . . . , k, and q k+1 = q. We conclude this section by presenting the proof of Proposition 5.6.\n\nProof of Proposition 5.6. Let I = [0, T ). In the following, we prove\nI P ≤N 2k+1 j=1 u * j X sc (I) 2k+1 j=1 u j X s (I) 2k+1 ℓ=1 ℓ =j u ℓ X sc (I)\nfor all N ≥ 1, where the implicit constant is independent of N . By Lemma 5.5, we have\nI P ≤N 2k+1 j=1 u * j X sc (I) ≤ sup v∈Y −s ([0,T )) v Y −s =1 ˆI ×T d 2k+1 j=1 u * j (x, t)P ≤N v(x, t)dxdt . (5.28)\nHence, with u 0 = P ≤N v, it suffices to show that\nˆI ×T d 2k+1 j=0 u * j (x, t)dxdt u 0 Y −s (I) 2k+1 j=1 u j X s 2k+1 ℓ=1 ℓ =j u ℓ X sc . (5.29)\nNow, dyadically decompose u * j = N j ≥1 P N j u * j . Without loss of generality, assume N 1 ≥ N 2 ≥ · · · ≥ N 2k+1 . Then, in order to have a non-trivial contribution on the left-hand side of (5.29), we must have N 1 ∼ max(N 0 , N 2 ).\nCase (i): N 0 ∼ N 1 .\nBy Lemma 5.11, we have\nˆI ×T d 2k+1 j=0 P N j u * j (x, t)dxdt ≤ k j=0 P N 2j u * 2j L 2 t,x k j=0 P N 2j+1 u * 2j+1 L 2 t,x N 2k N 0 + 1 N 2 δ ′ N 2k+1 N 1 + 1 N 3 δ ′ P N 0 u 0 Y −s P N 1 u 1 Y s 2k+1 j=2 P N j u j Y sc . (5.30)\nSumming (5.30) over dyadic blocks N 0 ∼ N 1 ≥ N 2 ≥ · · · ≥ N 2k+1 and by Cauchy-Schwarz inequality, we have LHS of (5.29)\nN 0 ∼N 1 P N 0 u 0 Y −s P N 1 u 1 Y s 2k+1 j=2 u j Y sc u 0 Y −s u 1 Y s 2k+1 j=2 u j Y sc ,\nyielding (5.29) in view of (5.2).\nCase (ii): N 2 ∼ N 1 ≫ N 0 . By Lemma 5.11 with N 1 ∼ N 2 , we have ˆI ×T d 2k+1 j=0 P N j u * j (x, t)dxdt N 0 N 1 s+sc N 2k+1 N 1 + 1 N 3 δ ′ P N 0 u 0 Y 0 P N 1 u 1 Y s 2k+1 j=2 P N j u j Y sc (5.31)\nSumming (5.31) over dyadic blocks N 0 (≪ N 1 ) and N 1 ∼ N 2 ≥ N 3 ≥ · · · ≥ N 2k+1 and by Cauchy-Schwarz inequality, we have LHS of (5.29)\nu 0 Y −s N 1 ∼N 2 P N 1 u 1 Y s P N 2 u 2 Y sc 2k+1 j=3 u j Y sc u 0 Y −s u 1 Y s 2k+1 j=2 u j Y sc .\nThis completes the proof of Proposition 5.6.\n\nAppendix A. On the Weyl sum estimate (2.11) In this appendix, we present a proof of (2.11). We decided to include the proof for the convenience of readers, in particular for those in PDEs. Since I = I(η) is a compact interval and the integrand is periodic with period 1, it suffices to shoŵ\n1 0 |F (t)| r dt ∼ N r−2 , (A.1)\nfor r > 4, where F (t) is the Weyl sum defined by\nF (t) = 0≤n≤N e 2πin 2 t .\nFor a, q ∈ N with 1 ≤ a ≤ q ≤ N and (a, q) = 1, define a major arc M(q, a) by\nM(q, a) = t ∈ [0, 1] : t − a q ≤ 1 100N 2 , (A.2)\nwhere x = min n∈Z |x − n| denotes the distance of x to the closest integer as before. Let M = a,q M(q, a). Note that we have t − a q ≤ 1 q 2 for t ∈ M. Then, by Weyl's inequality we have\n|F (t)| N q 1 2 + N 1 2 (log q) 1 2 + q 1 2 (log q) 1 2 . (A.3)\nHence, the contribution from the major arc M is estimated bŷ\nM |F (t)| r dt N q=1 q a=1 (a,q)=1 N r q r 2 (log q) r 2 1 N 2 ≤ N r−2 N q=1 q 1− r 2 + N r−2 ,\nsince r > 4.\n\nRemark A.1. Indeed, the contribution from the major arc M provides the lower bound in (A.1). We only need to consider the contribution from M(q, a) for odd q ≤ N 1 2 . Let S(q, a) be the Gauss sum given by S(q, a) = q n=1 e 2πin 2 a q . We have S(q, a) = √ q for odd q. Now, suppose that q is odd such that 1 ≤ q ≤ N 1 2 . Then, by Van der Corput's method [41] with S(q, a) = √ q, one can show that |F (t)| N q 1 2 for t ∈ M(q, a). Noting that M(q, a)\n\nare disjoint, we havê\nM |F (t)| r dt N 1 2 q=1 q,odd q a=1 (a,q)=1 N r q r 2 1 N 2 = N r−2 N 1 2 q=1 q,odd φ(q)q − r 2 ∼ N r−2 ,\nwhere φ(q) denotes Euler's function. This shows the lower bound in (A.1).\n\nNext, we estimate the contribution from the minor arc m := [0, 1] \\ M. Fix small ε > 0. Then, Dirichlet's theorem [40, Lemma 2.1] states that given t ∈ [0, 1], there exist integers a, q with 1 ≤ a ≤ q ≤ N 2−ε and (a, q) = 1 such that t − a q ≤ 1 qN 2−ε . Define Let t ∈ m 1 , i.e. we have t − a q ≤ 1 qN 2−ε for some q ≪ N ε . Then, by Lemmata 2.7 and 2.8 in [40] with |S(q, a)| q 1 2 , we have\n|F (t)| = q −1 S(q, a)v t − a q + O(N 2 200 ) N 1− ε 2 ,\nwhere v is defined in [40, (2.9)]. Applying Hua's inequality as before, we havê\nm 1 |F (t)| r dt ≤ sup t∈m 1 |F (t)| r−4ˆ1 0 |F (t)| 4 dt N 1− ε 2 r−4 N 2+ ≤ N r−2 .\nThis completes the proof of (A.1).\n\nAppendix B. On a partially irrational torus T 2 × T α 3\n\nIn this appendix, we present a sketch of the proof of Theorem 1.6. Thus, we set d = 3 and assume that (1.27) holds in the following. The main ingredient is an improvement of the Strichartz estimate on a partially irrational torus. 1) and (B.2), the implicit constants depend on ε > 0, q > 4, and |I|, but are independent of f . We first prove Proposition B.1 assuming Lemma B.2. Then, we sketch the proof of Theorem 1.6. We present the proof of Lemma B.2 at the end of this appendix.\n\nGiven f as in (3.2), let F (x, t) = e −it∆ f (x, t). By Cauchy-Schwarz inequality, we have F L ∞ t,x N 3 2 . Given p > 14 3 , let q ∈ (4, p). Then, by Lemma B.2 and Theorem 1.1 (ii), we havê for sufficiently small ε 1 , ε 2 > 0. Here, the condition p > 14 3 is needed in the last inequality. This proves Proposition B.1.\nI×T 3 |F (x, t)| p dxdt ≤ ˆN 1+ε 1 |F | N 3 2 −ε 2 +ˆN 3 2 −ε 2 |F | N 3 2 |F (x, t)| p dxdt + N (1+ε 1 )(p−4)ˆ| F (x, t)| 4 dxdt N 1− 4 1+4ε 1 ε 1ˆN 3 2 −ε 2 N 1+ε 1 λ p−4+ 8 1+4ε 1 ε 1 dλ + N 3 2 q−5ˆN\nNext, we briefly discuss how Theorem 1.6 follows from Proposition B.1. As in Section 5, the goal is to prove the multilinear estimate (5.4) in Proposition 5.6 for d = 3 and k = 2 (with s c = 1) under the assumption (1.27). In view of the argument in Subsection 5.3 (which holds without any change even in this case), it suffices to prove the multilinear Strichartz estimate (5.5) in Proposition 5.7 for d = 3 and k = 2 (with s c = 1). By repeating the proof of Lemma 5.10 with Proposition B.1, we see that (5.11) holds for p > 14 3 with δ ∈ (0, 1 2 − 7 3p ). In the proof of Proposition 5.7, the only change appears in (5.17) and we can choose p such that 14 3 < p < 20k 3k + 2 in this case. In particular, we can set k = 2 by choosing p ∈ ( 14 3 , 5). The rest of the argument follows exactly as in Section 5.\n\nIn the remaining part of the paper, we present the proof of Lemma B.2.\n\nProof of Lemma B.2. The proof follows the proof of Proposition 3.1 with small modifications. In the following, we only point out these modifications. Given an interval I ⊂ R, assume that I is centered at 0.\n\n(i) With Q(n) in (1.27), define R as in (2.19). Then, we have R(x, t) = 2 j=1 R 1 (x j , t) · R θ 3 (x 3 , t), where R θ is defined in (2.20). We also define R 1 and R 2 as in (3.10). Then, by (3.11), we have by setting ε 2 = ε 2 (ε 1 ) such that 1−ε 2 1+ε 1 = 1 1+2ε 1 . This proves (B.1) with ε = ε 1 2 . (ii) Define K by K(x, t) = χ(t)K(x 1 , t)K(x 2 , t) Also, with Λ as in (3.46) and K as in (B.5), it follows from (3.26) and (3.29) \nλ 2 |A λ | 2 ≤ R 1 L ∞ t,x Θ λ 2 L 1 t,x + R 2 L ∞ τ ℓ ∞ n Θ λ 2 L 2 t,x ≤ C 1 N M 1+ε 1 |A λ | 2 + M −1+ε 2 |A λ |(with N 1 = 1 100 N that K − Λ L ∞ (I×T d ) N 2 . (B.8)\nLet p be as in (3.50). In the following, we consider the case θ ≥ as long as θ < 1 2 .\n\n.\nBackground. The Cauchy problem of the nonlinear Schrödinger equation (NLS):\n\n\nf ∈ L 2 (R d ) with supp f ⊂ [−N, N ] d . By interpolating (1.6) and (1.7), we see that the estimate (1.5) holds for 2(d+2) d ≤ p ≤ ∞ and is sharp in view of sharpness of (1.6) and (1.7). Note that the estimate (1.5) is scaling-invariant in the following sense. Consider the linear Schrödinger equation: i∂ t u − ∆u = 0 u t=0 = f. (1.8)\n\nM\n:= kT d = R/(kZ) d as a disjoint union of parallel translates of the original rational torus M = T d α α α with α −1 j k copies in the x j -direction. Now, consider the linear Schrödinger equation (1.8) on M = T d α α α . By periodic extension, we can view this problem on the scaled standard torus M = kT d . Given an initial condition f and the solution u(t) = e −it∆ f on M , let f and u denote their periodic extensions on M , respectively. By uniqueness of solutions to the linear Schrödinger equation, we see that u(t) = e −it∆ f on M . Clearly, the Strichartz estimates on the standard torus T d also hold on the scaled standard torus M = kT d , where the implicit constants further depend on k.With α α α = (α 1 , . . . , α d ), we havef L 2 x ( M ) = C(α α α) f L 2 x (M ) and u(t) L p x ( M ) = C(α α α, p) u(t) L p x (M ). Moreover, letting f and F [ f ] denote the Fourier coefficients of f on T d α α α and f on kT d , respectively, we have\n\nTheorem 1. 4 (\n4Local well-posedness in subcritical spaces). Let d ≥ 2 and k ∈ N. Then, there exists s 0 = s 0 (k, d) such that the Cauchy problem (1.25) on a d-dimensional irrational torus T d is locally well-posed in H s (T d ) for s > s 0 in the following cases:\n\nRemark 3. 2 .\n2When d = 2, Proposition 3.1 (i) and (ii) basically says that the level set estimates (3.4) and (3.5) are sufficient in proving the Strichartz estimates (1.19) and (1.20) for p > 6 as long as λ is large: λ ≥ N 1 4 + . Hence, an improvement on Theorem 1.2 when d = 2 may be obtained if we can improve the lower bound on λ in Proposition 3.1 (i) or the L 4 -Strichartz estimate (3.6).\n\n.\nSee Proposition 8 and the comment afterward in [7]. Combining Proposition 3.1 (i) and (3.7), we obtain (1.20) only for p ≥ 2(3d+1) d . When d = 2, the combination of Proposition 3.1 (i) and (3.6) yields a better result. When d ≥ 3, Proposition 2.2 yields better results. We point out that, when d = 3, combining Proposition 3.1 (i) with Theorem 1.1 (ii) yields another proof of Theorem 1.2 (ii.b). 3.2. Proof of Proposition 3.1 (i). Let η be a smooth cutoff function supported on [ 1 200 , 1 100 ]. Given q ∈ N, define J q by J q = {a ∈ N : 1 ≤ a ≤ q, (a, q) = 1}. (3.8) Then, for given M ∈ N with M ≥ N , we define Φ(t) = M ≤q<2M a∈Jq η q 2 t − a q\n\n\n{σ n } n∈Z be the multiplier defined by σ n = 1 on [−N, N ], σ n = N −j N for n = N + j and n = −N − j, j = 1, . . . , N , and σ n = 0 for |n| ≥ 2N . Consider K(x, t) := n∈Z σ n e 2πi(nx+n 2 t) .\n\nM\n≤2 s ≤N ω N,2 s . (3.24) Then, it follows that Ω M,N ≡ 1 on [− 1 M N , 1 M N ] and supp Ω M,N ⊂ [− 2 M N , 2 M N ]. Let N 1 = 1 100 N . Note that, for M 1 < M 2 ≤ N 1 , we have\n\n= 0 .\n0Here, we used the fact that d(k, M ) k 0+ and M ≤ 2 s ≤ N . Now, for each M and s, we choose a coefficient α M,s such that F T ∈R M δ T * ω N,2 s (0) = α M,s ρ(0).\n\n\n30) where S N = {n ∈ Z d : |n j | ≤ N, j = 2, . . . , N } and K(x 1 , t) is as in(3.20).Define Λ M,s byΛ M,s (x, t) = K(x, t) T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t) .(3.31) Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have\n\n\nwe estimate | Λ M,s |. Denote the second factor in (3.31) by Φ M,s , i.e. let Φ M,s (t) = T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t).\n\n= 0 .\n0Φ M,s is periodic. Moreover, by (3.28), we have Φ M,s (0) = 0. Hence, we have Λ M,s (n, τ ) = σ n 1 d j=2 1 |n j |≤N k =0 Φ M,s (k) χ(τ − Q(n) − k). By summing | χ(τ − Q(n) − k)| τ − Q(n) − k −100 over k = 0, it follows from (3.35) and (3.36) that | Λ M,s (n, τ )| M 2 s N 1− . (3.37)\n\nLemma 3 . 5 (\n35Lemma 3.47 in [3]). Let d(k, M ) denote the number of divisors of k less than M . Then, for any β, B, D > 0, we have #{0 ≤ k ≤ N : d(k, M ) > D} < c β,B (D −B M β N + M B ). (3.40) Note that the constant in (3.40) is independent of D > 0, M, N ∈ N. From (3.35) and (3.36), we have\n\n\nfrom (3.41), (3.42), and (3.44) with M ≤ N , we have\n\n\n57) where I 1 = [1, M 1 ], I 2 = (M 1 , M 2 ], and I 3 = (M 2 , N 1 ] with N 1 = 1 100 N as before. Then, summing (3.51) over dyadic M ≤ M 1 and 2 s ≤ N , we have\n\n1\n1choose D ∼ M α for some small α > 0. Then, set β ≪ 1 and B ≫ summing (3.59) over dyadic M ∈ (M 1 , M 2 ] and s with 2 s ≤ N , we have\n\nLemma 4 . 1 .\n41Let s 0 > max(0, s c ) and I ⊂ R be a bounded interval. Suppose that the following multilinear Strichartz estimate holds for s > s 0 :\n\nj = 1 ,\n1. . . , k + 1, and N max := max(N 1 , · · · , N k+1 ). Then, the Cauchy problem (1.25) is locally well-posed in H s (T d ) for s > s 0 .\n\nu\nU p (R;H) := inf λ ℓ 1 : (5.1) holds with λ = {λ j } j∈N and some U p -atoms a j .\n\nLemma 5 . 10 .\n510Let (d, p) be admissible and I ⊂ R be a bounded interval. Then, for all 1 ≤ M ≤ N and R ∈ R M (N ), we have\n\nf\n(x, t) is as in(3.20). Let Λ M,s be as in(3.31). Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have|Λ M,s (x, * Λ M,s L ∞ (I×T d ) 2 s N 2 M f L 1 (I×T d ) . (B.7)\n\n1 3 .f\n3With M 1 and M 2 as in (3.56), write Λ = Λ 1 + Λ 2 + Λ 3 as before. See(3.57).By interpolating (B.7) and (3.39) and summing over dyadic M ≤ M 1 and 2 s ≤ N , we havef * Λ 1 L p ′ (I×T d ) M −1+(3+)θ 1 N 3−5θ f L p (I×T d ) , (B.9) since θ ∈ [ 1 3 , 1 2 ). Next, choose D ∼ M α for some small α = α(θ) > 0,and set β ≪ 1 and B ≫ 1 such that σ := −3 − β + αB− > 0. Then, by interpolating (B.7) and (3.45) and summing over dyadic M ∈ (M 1 , M 2 ] and 2 s ≤ N , we havef * Λ 2 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) L 1 (I×T d ) ,(B.10) as long as θ < 1 2+α+ . This can be guaranteed by choosing α = α(θ) > 0 sufficiently small for given θ <1 2 . Lastly, by interpolating (B.7) and (3.38) and summing over M ≥ M 2 ∼ N δ 2 and s with 2 s ≤ N , we have f * Λ 3 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) , (B.11)\n\n\nTheorem 1.1. The Strichartz estimate on a irrational torus is known to hold\n\n\nthe following cases:\n\n\nTheorem 1.2. (i) The following scaling-invariant Strichartz estimate holds on an irrational torus:\n\n\nProof of Theorem 1.2 for d ≥ 3. Suppose that p ≥ max( 16 d +, 4), satisfying the hypothesis of Theorem 1.2 (i). By Bernstein's inequality and Proposition 2.2 (i), we have\n\n\nThen, by Lemma 2.1 and Cauchy-Schwarz and Hölder's inequalities, we have\n\n\nIf p > 8, then by Bernstein's inequality (in t), we have11) \n\nyielding (2.10) N \n\nd \n\n4 − 2 \n\np . Hence, (2.4) follows from (2.9) in this case. \n\n\n\n). This completes the proof of Proposition 2.2. 2.2. Two dimensional case. For d = 2, the sharp estimate (1.19) is not covered by Proposition 2.2. In the following, we use a simple duality argument and prove the sharp Strichartz estimate (1.19) for p > 12. Without loss of generality, we assume that\n\n69 )\n69by making σ = σ(p ′ , δ 1 ) in (3.60) (and hence B = B(p ′ , δ 1 )) sufficiently large.Lastly, suppose that λ 2 |A λ | 2 III holds. By λ Nd \n\n2 −ε and (3.56), we have \n\n\n\n26 )\n26for some p > 2. Indeed, if (5.25) and (5.26) hold, then it follows from Lemma 5.3 (ii) that (5.24) holds with δ ′ < δ.The first estimate (5.25) directly follows from Proposition 5.7 and Lemma 5.3 (i). Hence, it remains to prove the second estimate (5.26). Let p, p d,k , and q be as in the proof of Proposition 5.7. Then, by Hölder's inequality with (5.22), we have\n\n\nFrom (5.17) and (5.19) with p d,k = (d + 2)k, we have q > p d,k > p > 2. Therefore, (5.26) follows from (5.27) with Remark 5.2 and (5.21).\n\n\nProposition B.1. Let d = 3 and I be a compact interval in R. Suppose that (1.27) holds. Then, the scaling-invariant Strichartz estimate (1.19) holds for p > 14 3 . The proof of Proposition B.1 is based on the following level set estimates under the assumption (1.27). Lemma B.2. Suppose that (1.27) holds. Given a compact interval I ⊂ R and f as in (3.2), let A λ = A λ (f ) be the distribution function defined by (3.3). (i) For any ε > 0, we have |A λ | N for λ N 1+ε . (ii) Let q > 4. Then, there exists small ε > 0 such that |A λ | N1 \n\n1+4ε λ −4+ 8 \n\n1+4ε ε \n\n(B.1) \n\n3 \n\n2 q−5 λ −q \n(B.2) \n\nfor λ N \n\n3 \n\n2 −ε . \nIn (B.\n\n\nB.4) for small ε 1 , ε 2 > 0. Choose M ≥ N such that N M 1+ε 1 ∼ λ 2 . This condition with M ≥ N implies that λ N 1+ ε 1 2 .Then, (B.4) yields|A λ | \nN \nλ 2 \n\n1−ε 2 \n\n1+ε 1 λ −2 N \n\n1 \n\n1+2ε 1 λ \n\n−4+ 4 \n\n1+2ε 1 \n\nε 1 \n\n\n\nOn the multilinear restriction and Kakeya conjectures. J Bennett, A Carbery, T Tao, Acta Math. 1962J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261-302.\n\nOn Λ(p)-subsets of squares. J Bourgain, Israel J. Math. 673J. Bourgain, On Λ(p)-subsets of squares, Israel J. Math. 67 (1989), no. 3, 291-311.\n\nFourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. J Bourgain, Geom. Funct. Anal. 3J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107-156.\n\nExponential sums and nonlinear Schrödinger equations. J Bourgain, Geom. Funct. Anal. 3J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal. 3 (1993), 157-178.\n\nNonlinear Schrödinger equations. J Bourgain, Hyperbolic equations and frequency interactions. Park City, UT; Providence, RIAmer. Math. Soc5J. Bourgain, Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions (Park City, UT, 1995), 3-157, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999.\n\nOn Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations. J Bourgain, Ann. of Math. Stud. 163Princeton Univ. PressJ. Bourgain, On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations, 1-20, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.\n\nMoment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. J Bourgain, Israel J. Math. 1931J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel J. Math. 193 (2013), no. 1, 441-458.\n\nBounds on oscillatory integral operators based on multilinear estimates. J Bourgain, L Guth, Geom. Funct. Anal. 216J. Bourgain, L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239-1295.\n\nStrichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. N Burq, P Gérard, N Tzvetkov, Amer. J. Math. 1263N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569-605.\n\nBilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. N Burq, P Gérard, N Tzvetkov, Invent. Math. 1591N. Burq, P. Gérard, N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), no. 1, 187-223.\n\nBounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. F Catoire, W.-M Wang, Commun. Pure Appl. Anal. 9F. Catoire, W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal. 9 (2010), 483-491.\n\nSemilinear Schrödinger equations. T Cazenave, Courant Lecture Notes in Mathematics. 10Courant Institute of Mathematical SciencesT. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York;\n\nThe Cauchy problem for the critical nonlinear Schrödinger equation in H s. T Cazenave, F Weissler, Nonlinear Anal. 1410T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 (1990), no. 10, 807-836.\n\nC Demeter, arXiv:1401.1873Incidence theory and restriction estimates. math.CAC. Demeter, Incidence theory and restriction estimates, arXiv:1401.1873 [math.CA].\n\nLocal well-posedness for 2-d Schrödinger equation on irrational tori and bounds on Sobolev norms. S Demirbas, arXiv:1307.0051math.APS. Demirbas, Local well-posedness for 2-d Schrödinger equation on irrational tori and bounds on Sobolev norms, arXiv:1307.0051 [math.AP].\n\nOn the global Cauchy problem for some nonlinear Schrödinger equations. J Ginibre, G Velo, Ann. Inst. H. Poincaré Anal. Non Linéaire. 14J. Ginibre, G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 309-323.\n\nSmoothing properties and retarded estimates for some dispersive evolution equations. J Ginibre, G Velo, Comm. Math. Phys. 1441J. Ginibre, G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations. Comm. Math. Phys. 144 (1992), no. 1, 163-188.\n\nErratum to \"Well-posedness and scattering for the KP-II equation in a critical space. M Hadac, S Herr, H Koch, Ann. Inst. H. Poincaré Anal. Non Linéaire. 263Ann. Inst. H. Poincaré Anal. Non LinéaireM. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, 917-941. Erratum to \"Well-posedness and scattering for the KP-II equation in a critical space\", Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 3, 971-972.\n\nAn introduction to the theory of numbers. G Hardy, E Wright, Oxford University PressNew YorkFifth edition. xvi+426 ppG. Hardy, E. Wright, An introduction to the theory of numbers, Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp.\n\nThe quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds. S Herr, Amer. J. Math. 1355S. Herr. The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds, Amer. J. Math. 135 (2013), no. 5, 1271-1290.\n\nGlobal well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in H 1 (T 3 ). S Herr, D Tataru, N Tzvetkov, Duke Math. J. 159S. Herr, D. Tataru, N. Tzvetkov, Global well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in H 1 (T 3 ), Duke Math. J., 159 (2011) 329-349.\n\nStrichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications. S Herr, D Tataru, N Tzvetkov, doi10.1515/crelle-2012-0013J. Reine Angew. Math. S. Herr, D. Tataru, N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications, J. Reine Angew. Math. doi 10.1515/crelle-2012-0013.\n\nY Hu, X Li, arXiv:1108.5164v1Discrete Fourier restriction associated with Schrödinger Equations. Y. Hu, X. Li, Discrete Fourier restriction associated with Schrödinger Equations, arXiv:1108.5164v1.\n\nExponential sums and lattice points. M Huxley, III, Proc. London Math. Soc. 3M. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591-609.\n\nAnalytic number theory. H Iwaniec, E Kowalski, American Mathematical Society53Providence, RIxii+615 ppH. Iwaniec, E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publi- cations, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp.\n\nThe energy-critical defocusing NLS on T 3 , Duke Math. A Ionescu, B Pausader, J. 1618A. Ionescu, B. Pausader, The energy-critical defocusing NLS on T 3 , Duke Math. J. 161 (2012), no. 8, 1581-1612.\n\nV Jarník, Über die Gitterpunkte auf konvexen Kurven. 24V. Jarník,Über die Gitterpunkte auf konvexen Kurven, (German) Math. Z. 24 (1926), no. 1, 500-518.\n\nOn nonlinear Schrödinger equations. T Kato, Ann. Inst. H. Poincaré Phys. Théor. 461T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113-129.\n\nEndpoint Strichartz estimates. M Keel, T Tao, Amer. J. Math. 1205M. Keel, T. Tao, Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), no. 5, 955-980.\n\nA priori bounds for the 1D cubic NLS in negative Sobolev spaces. H Koch, D Tataru, Art. ID rnm053. 36ppH. Koch, D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. (2007), no. 16, Art. ID rnm053, 36 pp.\n\nH Montgomery, CBMS Regional Conference Series in Mathematics, 84. Published for the Conference Board of the Mathematical Sciences. Washington, DC; Providence, RIAmerican Mathematical SocietyTen lectures on the interface between analytic number theory and harmonic analysis. xiv+220 ppH. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. xiv+220 pp.\n\n. B Pausader, N Tzvetkov, X Wang, arXiv:1210.3842Global regularity for the energy-critical NLS on S. 3to appear in Ann. Inst. H. Poincaré Anal. Non LinéaireB. Pausader, N. Tzvetkov, X. Wang, Global regularity for the energy-critical NLS on S 3 , arXiv:1210.3842 [mathAP], to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.\n\nAn asymptotic formula for the number of solutions of a system of equations. N Rogovskaya, Moskov. Gos. Univ. Russian) Diophantine approximations, Part II (RussianN. Rogovskaya, An asymptotic formula for the number of solutions of a system of equations, (Russian) Diophantine approximations, Part II (Russian), 78-84, Moskov. Gos. Univ., Moscow, 1986.\n\nRestrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. R S Strichartz, Duke Math. J. 443R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), no. 3, 705-714.\n\nN Strunk, arXiv:1401.6080Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions. math.APN. Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions arXiv:1401.6080 [math.AP].\n\nThe nonlinear Schrödinger equation. Self-focusing and wave collapse. C Sulem, P L Sulem, Applied Mathematical Sciences. 139Springer-VerlagC. Sulem, P.L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. xvi+350 pp.\n\nOn 2D nonlinear Schrödinger equations with data on R × T. H Takaoka, N Tzvetkov, J. Funct. Anal. 1822H. Takaoka, N. Tzvetkov, On 2D nonlinear Schrödinger equations with data on R × T, J. Funct. Anal. 182 (2001), no. 2, 427-442.\n\nT Tao, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences. Washington, DC; Providence, RIAmerican Mathematical SocietyNonlinear dispersive equations. Local and global analysis. xvi+373 ppT. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp.\n\nL 2 -solutions for nonlinear Schrödinger equations and nonlinear groups. Y Tsutsumi, Funkcial. Ekvac. 301Y. Tsutsumi, L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115-125.\n\nThe Hardy-Littlewood method. R Vaughan, Cambridge Tracts in Mathematics. 125Cambridge University PressSecond editionR. Vaughan, The Hardy-Littlewood method. Second edition, Cambridge Tracts in Mathematics, 125. Cambridge University Press, Cambridge, 1997. xiv+232 pp.\n\nThe method of trigonometrical sums in the theory of numbers, Translated from the Russian. I M Vinogradov, K. F. Roth and Anne DavenportDover Publications, IncMineola, NYReprint of the 1954 translation. x+180 ppI.M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport. Reprint of the 1954 translation. Dover Publications, Inc., Mineola, NY, 2004. x+180 pp.\n\nPeriodic nonlinear Schrödinger equation in critical H s (T n ) spaces. Y Wang, SIAM J. Math. Anal. 453Y. Wang, Periodic nonlinear Schrödinger equation in critical H s (T n ) spaces, SIAM J. Math. Anal. 45 (2013), no. 3, 1691-1703.\n\nT Wolff, Lectures on harmonic analysis. Providence, RIAmerican Mathematical Society29T. Wolff, Lectures on harmonic analysis, University Lecture Series, 29. American Mathematical Society, Providence, RI, 2003. x+137 pp.\n\nExistence of solutions for Schrödinger evolution equations. K Yajima, Comm. Math. Phys. 1103K. Yajima, Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys. 110 (1987), no. 3, 415-426.\n\n. Zihua Guo, BeijingSchool of Mathematical Sciences, Peking UniversityChina E-mail address: zihuaguo@math.pku.edu.cnZihua Guo, School of Mathematical Sciences, Peking University, Beijing 100871, China E-mail address: zihuaguo@math.pku.edu.cn\n\nand The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings. Mayfield Road, Edinburgh, EH9 3JZ, Scotland; Fine Hall, Washington Rd., Princeton, NJ 08544-1000, USA E-mailTadahiro Oh, School of Mathematics, The University of Edinburgh ; and Department of Mathematics, Princeton UniversityTadahiro Oh, School of Mathematics, The University of Edinburgh, and The Maxwell In- stitute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, and Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ 08544-1000, USA E-mail address: hiro.oh@ed.ac.uk\n"},"corpus_id":{"kind":"number","value":119646823,"string":"119,646,823"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1306.4973\",\n \"mag\": 2950866968,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1112/plms/pdu025\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1306.4973v3.pdf\"\n ],\n \"pdf_sha\": \"28aca2f3834a923e5feb62f5a65c21349143d756\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://arxiv.org/pdf/1306.4973\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"STRICHARTZ ESTIMATES FOR SCHRÖDINGER EQUATIONS ON IRRATIONAL TORI"},"authors":{"kind":"list like","value":["Zihua Guo","Tadahiro Oh","Yuzhao Wang"],"string":"[\n \"Zihua Guo\",\n \"Tadahiro Oh\",\n \"Yuzhao Wang\"\n]","isConversational":false},"abstract":{"kind":"string","value":"In this paper, we prove new Strichartz estimates for linear Schrödinger equations posed on d-dimensional irrational tori. Then, we use these estimates to prove subcritical and critical local well-posedness results for nonlinear Schrödinger equations (NLS) on irrational tori.2010 Mathematics Subject Classification. 35Q55, 42B37. We use A B to denote an estimate of the form A ≤ CB for some C > 0. Similarly, we use A ∼ B to denote A B and B A.where I is a compact interval. Note that, in the compact setting, an estimate of the form (1.9) does not hold with I = R, unless p = ∞. By drawing an analogy to the Euclidean case M = R d , one may hope to have (1.9) for p ≥ 2(d+2) d . By combining the tools from number theory, such as a divisor counting argument and the Hardy-Littlewood circle method, and the Tomas-Stein restriction method from harmonic analysis, Bourgain [3] proved (1.9) for certain ranges of p: (i) p > 2(d+2) d when d = 1, 2, (ii) p > 4 when d = 3, and (iii) p > 2(d+4) d for higher dimensions d ≥ 4. It is 2 Obviously, the scaling associated to the linear Schrödinger equation discussed above for R d does not quite make sense on T d . We nonetheless call the estimate (1.9) scaling-invariant."},"sections":{"kind":"list like","value":[{"title":"Strichartz estimates: Part 1","content":"In this section, we prove our main result (Theorem 1.2) for d ≥ 3 and present a partial proof for d = 2. In [6], Bourgain treated the three-dimensional case. His argument is based on the following estimate:ˆT\nLemma 2.1 was used in [6] for the three-dimensional case. See also Lemma 2 in [11]. A proof for general dimensions in [11] relies on Schur's lemma. In the following, we give a direct proof for reader's convenience.\nn∈I k,a n ′ ∈I k ′ ,a |c n c a−n ||c n ′ c a−n ′ | By Cauchy-Schwarz inequality (in k) followed by Young inequality, we have\nThis completes the proof of Lemma 2.1.\nNext, we state the main proposition. Theorem 1.2 then follows this proposition and Bernstein's inequality when d ≥ 3. (4,4). Then, we have\nwhere q = p when d = 3, 4 and q = 8 when d = 2.\nBourgain proved (2.4) for d = 3. See Proposition 1.1 in [6]. Our proof follows the ideas developed in [6]. By setting p = 4 when d ≥ 5 and p = 16 d + when d = 3, 4, Proposition 2.2 yields the L 4 -Strichartz estimate, which improves the result in [11] for d ≥ 3. Note that the Strichartz estimate (2.4) is essentially sharp in higher dimensions (d ≥ 4). Indeed, on R d with d ≥ 2, by Sobolev inequality and interpolation of (1.6) and (1.7)\nfor p ≥ 4. We first use Proposition 2.2 to prove Theorem 1.2 when d ≥ 3.\nBy repeating the same argument with Proposition 2.2 (ii), we obtain Theorem 1.2 (ii) when d = 3, 4. When d = 2, this yields Theorem 1.2 (ii) only for p ≥ 8.\nWe now present the proof of Proposition 2.2.\nProof of Proposition 2.2. (i) Let Q(n) be as in (1.15). Then, we have\n(2.6)\nWith c n = f (n), let F a (t) be as in (2.2). Then, by Minkowski's integral inequality with p ≥ 4, we have\n(2.8)\nFrom (2.7) and (2.8), we have\n(2.9)\nNow, let η(t) be a smooth function with a compact support I ⊂ R such that η ≥ 0 and\nIf p ≤ 8, then we have p p−4 ≥ 2. Then, by Hausdorff-Young's inequality, we have\n(2.10)\nNote that r = dp 4 > 4, since p > 16 d . Then, by an application of the Hardy-Littlewood circle method (see Appendix A), we havê\nx .\n(2.12)\nThen, (2.4) follows from (2.12) and (2.4) for p = 8.\n(ii) When (d, p) = (2,8), (3, 16 3 ), or (4, 4), we have r = dp 4 = 4. In this case, (2.11) does not hold. Nonetheless, by Hua's inequality [40], we havê\nSee also [25, (8.13)]. Then, (2.5) follows from (2.13) and repeating the computation in (i\n(2.14)\nThen, the local-in-time Strichartz estimate can be expressed as n∈S N a n e 2πi(n·x+Q(n)t)\nwhere S N denotes the following set:\nOur task is to seek for an optimal constant K p,N . By duality, (2.15) is equivalent to\nHere, the Fourier transform f is defined by\nThen, (1.19) for p > 12 follows once we prove the next proposition.\n.\nHence, Proposition 2.3 states that, on an irrational torus, the same estimate for K p,N holds, but only for p > 12.\nProof. Without loss of generality, assume that I is centered at 0. Let R be a kernel defined by\nfor p > 6. Bourgain's argument is based on an application of the Hardy-Littlewood circle method. See also Lemma 2.4 in [23] for a simpler proof based on the Poisson summation formula. Note that (2.21) does not hold for p = 6. See Rogovskaya [33] and [3]. By Hölder's inequality, (2.21), and Sobolev inequality, we have\nas long as p > 12."},{"title":"Strichartz estimates: Part 2","content":"3.1. Level set estimates. In this section, we prove Theorem 1.2 when d = 2. The main ingredient is the level set estimates on irrational tori in Proposition 3.1 below. For level sets estimates on the usual torus T d , see [3,23]. It turns out that these level set estimates are useful only when d = 2, 3 (see Remark 3.3), but we state and prove the results for a general dimension. In the following, we assume that θ 1 = 1 in (1.15) for simplicity. Namely, we consider Q(n) = n 2 1 + θ 2 n 2 2 + · · · + θ d n 2 d .\n(ii) Let q > 6. Then, there exists small ε > 0 such that (3.4) and (3.5), the implicit constants depend on ε > 0, q > 6, and |I|, but are independent of f .\nWe present the proof of Proposition 3.1 in Subsections 3.2 and 3.3. In the following, we use Proposition 3.1 to prove Theorem 1.2 when d = 2. First, we present the proof of Theorem 1.2 (ii.a), i.e. we prove (1.20) for p ≥ 20 3 when d = 2. Recall that Catoire-Wang [11] proved\nN. Then, with Proposition 3.1 (i) and (3.6), we havê\nwhere the last inequality holds as long as p ≥ 20 3 . This proves Theorem 1.2 (ii.a). By Proposition 3.1 (i) and (ii), Theorem 1.2 (i.a) follows in a similar manner. We omit details.\nfor p = 2(d+1) , where x = min n∈Z |x − n| denotes the distance of x to the closest integer. Note that Φ is periodic with period 1. By taking a Fourier transform, we have\nwhere c q (k) denotes Ramanujan's sum: c q (k) := a∈Jq e −2πi a q k . Let φ(q) be the Euler's totient function defined by φ(q) = a∈Jq 1. Then, by Theorem 330 in [19], we have\nNamely, Φ(0) is independent of M . Without loss of generality, assume that I is centered at 0. With Q(n) in (3.1), define R as in (2.19) (2.20). Now, letting χ be a smooth cutoff function support on 3I such that χ(t) ≡ 1 on 2I, define R 1 and R 2 by\nNoting that the intervals I ℓ,q,a := ℓ + a q + 1 200q 2 , ℓ + a q + 1 100q 2 are disjoint for distinct values of ℓ, a, and q ∼ M ≫ 1, it follows from Weyl's inequality [31, Theorem 1 on p. 41] that\nfor t ∈ I ℓ,q,a since q ∼ M ≥ N . Then, along with a trivial bound |R θ j (x j , t)| N , we obtain\nNext, we consider R 2 . By expanding Φ(t) in the Fourier series, we have\nFirst, recall the following lemma (Lemma 3.33 in [3]). Given M ∈ N and k ∈ Z, we have M ≤q<2M\nwhere d(k, M ) denotes the number of divisors of k less than M . Then, by taking a Fourier transform of (3.13) with (3.9), (3.14), and d(k, M ) k 0+ , we have\nNote that supp Θ λ (x, ·) ⊂ I for each x ∈ T d . Then, by Cauchy-Schwarz inequality with (3.2), we have\nBy (3.10), (3.12), and (3.15), we have\nby setting ε 2 = ε 2 (ε 1 ) such that 1−ε 2 1+2ε 1 = 1 1+3ε 1 . This proves (3.4) with ε = ε 1 2 . 3.3. Proof of Proposition 3.1 (ii). In this subsection, we prove the level set estimate (3.5), which is sharp for q > 6. The following argument is inspired by Bourgain's paper [3]. We first go over some basic setups, restricting our attention to t ∈ T.\nLet (3.20)\nThen, we have the following lemma. Here, x = min n∈Z |x − n| denotes the distance of x to the closest integer as before.\nThen, we have\nNote that the multiplier σ n in (3.20) avoids the logarithmic loss (when q ∼ N ) in Weyl's inequality (3.11) on the Weyl sum W N (x, t) = |n|≤N e 2πi(nx+n 2 t) . Indeed, by writing\nk=N W k , we see that this regularizing effect in (3.22) is analogous to that of the Féjér kernel over the Dirichlet kernel.\nIn the following, we fix N ≫ 1, dyadic. For dyadic M ≤ N , let R M by\nwhere J q is as in (3.8). Let ψ(t) be a smooth cutoff function supported on\nNote that we have supp(ω N,2 s ) ⊂ |t|\nRecall that by Dirichlet's theorem [ Now, we focus on our problem. Namely, we do not assume t ∈ T any longer. Given an interval I ⊂ R, assume that I is centered at 0 and let χ be a smooth cutoff function support on 3I such that χ(t) ≡ 1 on 2I as before. We define\nHence, from (3.32), we have Hence, from (3.37), we have\nThe second estimate (3.39) is useful when M ≪ N ε .\nIn the following, we establish another estimate on f * Λ M,s L 2 (I×T d ) , using the following lemma from Bourgain [3].\nwhere S N is as in (3.30). Given D > 0 (to be chosen later), separate the first term, depending on d(k, M ) ≤ D or > D. The contribution from d(k, M ) ≤ D can be estimated by\nNext, we estimate the contribution from d(k, M ) > D. By Cauchy-Schwarz inequality, we have\nNow, we estimate the first term on the right-hand side of (3.41) after applying (3.43). By first integrating in τ , then summing over |n j | N for j = 1, . . . , d, applying Lemma 3.5 (with 2B and 2β instead of B and β), the trivial bound d(k, M ) ≤ M , and Hausdorff-Young's inequality, we have\nwhere Λ M,s is as in (3.31). By (3.25), we have\nHence, we have\nfor some θ ∈ (0, 1). Note that p ′ θ = 2.\nCase (i): θ < 1 5 . By interpolating (3.33) and (3.39), we have\n(3.51)\nThen, summing over dyadic M ≥ 1 and s with 2 s ≤ N , we have\nas long as θ > 0 satisfies − 1 2 + ( 5 2 +)θ ≤ 0, i.e.\nθ < 1 5 .\n(3.53)\nIn the following, we prove the level set estimate (3.5) for q > 10. Given f as in (3.2),\nThen, proceeding as in (3.17) with (3.49) and (3.52), we have\nNoting that p ′ θ = 2 by (3.50), it follows from (3.55) that\nwhere q := p ′ = 2 θ > 10. Note that we only needed to assume λ ≫ N Case (ii): 1 5 ≤ θ < 1 3 . Let M j , j = 1, 2 be dyadic numbers such that\nHere, we choose δ 1 , δ 2 > 0 such that M 1 ≪ M 2 . We divide Λ into three pieces: Λ = Λ 1 + Λ 2 + Λ 3 by setting Similarly, by interpolating (3.33) and (3.45), we have\nNote that (3.62) can be satisfied as long as θ < 1 3 by choosing α sufficiently small. Lastly, from (3.33) and (3.38), we have\nThen, summing over M ≥ M 2 ∼ N δ 2 and s with 2 s ≤ N , we have\nas long as θ < 1 3 . Putting (3.58), (3.61), and (3.64) together with (3.53), we obtain\nNow, we are ready to prove the level set estimate (3.5) for q > 6. Then, proceeding as in Case (i) with (3.49) and (3.65), we have\nFirst, suppose that λ 2 |A λ | 2 I holds. Recall from (3.50) that p ′ θ = 2. Then, with (3.56), we have\nfor q > p ′ by choosing δ 1 = δ 1 (q, p ′ ) sufficiently small. Next, suppose that λ 2 |A λ | 2 II holds. Then, from (3.56), we have\nas long as we have εp ′ ≤ 1 and δ 2 = δ 2 (p ′ , B) is sufficiently small such that (− p ′ 2 + 5 + 2B+)δ 2 ≤ 1.\nFinally, given q > 6, we choose θ < 1 3 such that q > p ′ = 2 θ > 6. Then, from (3.67), (3.68), (3.69), and (3.70) with λ ≤ N d 2 , we obtain\nThis completes the proof of Proposition 3.1 (ii)."},{"title":"Well-posedness in subcritical spaces","content":"In this section, we prove local well-posedness of NLS (1.25) on irrational tori in subcritical Sobolev spaces (Theorem 1.4). It turns out that the well-posed theory of (1.25) is very similar to that on the standard tours [3,4,5,6].\nIn the seminal paper [3], Bourgain introduced the X s,b -space whose norm is given by\nwhere · = (1 + | · | 2 ) 1 2 . After establishing Strichartz estimates, he proved several wellposedness results of NLS on the standard torus T d . In our case, i.e. on an irrational torus, we need to replace the weight τ − |n| 2 b in (4.1) by τ − Q(n) b , where Q(n) is defined in (1.15). Then, by the standard X s,b -theory, it is known that certain multilinear Strichartz estimates imply well-posedness. More precisely, we have the following lemma. The proof of Lemma 4.1 is standard and we refer the readers to [3,4,6,11] for details.\nProof of Theorem 1.4. In view of Lemma 4.1, it suffices to prove the (k + 1)-linear estimate (4.2) for s > s 0 . Without loss of generality, assume that N 1 ≥ N 2 ≥ · · · ≥ N k+1 .\nCase (i): d = 2. When k = 1, the well-posedness result was already obtain in [11]. In the following, we first consider the case k = 2, 3, 4, 5. First, assume that supp\nThen, by Hölder's inequality and (1.21) with 4(7k+5) 3(k+3) ∈ (4, 20 3 ], we have\nThen, by a change of variables and (1.21) (see [5]), we obtain\nThen, by almost orthogonality with (4.3) and (4.4), we have\nRemark 4.2. In view of the reduction above, we only prove (4.2), assuming supp φ 1 ⊂ [−N 2 , N 2 ] d in the following.\nNext, we consider the case k ≥ 5. By Hölder's inequality and (1.21) with 8k k+1 ∈ [ 20 3 , 10), we have\nHence, (4.2) holds for s > s 0 = 1 − 1 k . Case (ii): d = 3. When k = 1, the well-posedness result was obtained in [6]. When k = 2, by Hölder's inequality and (1.22), we have\nHence, (4.2) holds for s > s 0 = 53 52 . When k ≥ 3, by Hölder's inequality and (1.22), we have\nHence, (4.2) holds s ≥ s 0 = 3 2 − 1 k . Case (iii): d ≥ 4 and k ≥ 1. By Hölder's inequality with (1.23) or (1.24), we have\nHence, (4.2) holds for s > s 0 = d 2 − 1 k ."},{"title":"5.","content":"Well-posedness in critical spaces 5.1. Function spaces. In this section, we prove local well-posedness of NLS (1.25) on irrational tori in critical Sobolev spaces H sc (T d ) (Theorem 1.5). In the following, we use the U p -and V p -spaces, developed by Tataru, Koch, and their collaborators [30,18,21,22]. These spaces have been very effective in establishing well-posedness of various dispersive PDEs in critical spaces. We briefly go over the basic definitions of function spaces and their properties. See Hadac-Herr-Koch [18] and Herr-Tataru-Tzvetkov [21] for detailed proofs.\nLet H be a separable Hilbert space over C. In particular, it will be either H s (T d ) or C. Let Z be the collection of finite partitions {t k } K k=0 of R: −∞ < t 0 < · · · < t K ≤ ∞. If t K = ∞, we use the convention u(t K ) := 0 for all functions u : R → H. We use 1 I to denote the sharp characteristic function of a set I ⊂ R.\n(i) A U p -atom is defined by a step function a : R → H of the form\nThen, we define the atomic space U p (R; H) to be the collection of functions u : R → H of the form\nwhere a j 's are U p -atoms and {λ j } j∈N ∈ ℓ 1 (N; C), (ii) We define V p (R; H) by the collection of functions u :\nWe also define V p rc (R; H) to be the closed subspace of all right-continuous functions in V p (R; H) such that lim t→−∞ u(t) = 0.\n(iii) Let s ∈ R. We define U p ∆ H s (and V p ∆ H s , respectively) to be the spaces of all functions u : R → H s (T d ) such that the following U p ∆ H s -norm (and V p ∆ H s -norm, respectively) is finite:\nHere, the Laplacian ∆ is defined in terms of Q(n) as in (1.14).\nRemark 5.2. Note that the spaces U p (R; H), V p (R; H), and V p rc (R; H) are Banach spaces. Moreover, we have the following embeddings:\nNext, we state a transference principle and an interpolation result."},{"title":"Lemma 5.3. (i) (transference principle) Suppose that we have","content":"x for some 1 ≤ p, q ≤ ∞. Then, we have\n(ii) (interpolation) Let E be a Banach space. Suppose that T : U p 1 × · · · × U p k → E is a bounded k-linear operator such that\nfor some p 1 , . . . , p k > 2. Moreover, assume that there exists C 2 ∈ (0, C 1 ] such that\nThen, we have\nA transference principle as above has been commonly used in the Fourier restriction norm method. See [18,Proposition 2.19] for the proof of Lemma 5.3 (i). The proof of the interpolation result follows from extending the trilinear result in [21] to a general k-linear case. See also [18,Proposition 2.20].\nLet η : R → [0, 1] be an even smooth cutoff function supported on [− 8 5 , 8 5 ] such that η ≡ 1 on [− 5 4 , 5 4 ]. Given a dyadic number N ≥ 1, we set η 1 (ξ) = η(|ξ|) and\nThen, we define the Littlewood-Paley projection operator P N as the Fourier multiplier operator with symbol η N . Moreover, we define P ≤N by P ≤N = 1≤M ≤N P M . More generally, given a set R ⊂ Z d , we define P R to be the Fourier multiplier operator with symbol 1 R .\nWe define X s to be the space of all functions u :\n(ii) Let s ∈ R. We define Y s to be the space of all functions u : R → H s (T d ) such that for every n ∈ Z d , the map t → e −itQ(n) u(n, t) is in V 2 rc (R t ; C) and u Y s < ∞, where the Y s -norm is defined by\nRecall the following embeddings:\nGiven a time interval I ⊂ R, we define the restrictions X s (I) and Y s (I) of these spaces in the usual manner. We now state the linear estimates. Given f ∈ L 1 loc ([0, ∞); L 2 (T d )), define I(f ) by\nLemma 5.5 (Linear estimates). Let s ≥ 0 and T > 0. Then, the following linear estimates hold:\nNext, we present the crucial multilinear estimate.\nProposition 5.6. Let d and k satisfy\nThen, the following multilinear estimate holds for all T ∈ (0, 1]:\nOnce we prove Proposition 5.6, one can prove Theorem 1.5 by the fixed point argument as in [21,42]. We omit details. The remainder of this section is devoted to the proof of Proposition 5.6. Indeed, the multilinear estimate (5.4) follows once we prove the following multilinear Strichartz estimate.\nfor any interval I ⊂ [0, 1] and for all φ j ∈ L 2 (T d ), j = 1, . . . , k + 1, and N 1 ≥ N 2 ≥ · · · ≥ N k+1 ≥ 1.\nIn Subsection 5.2, we first present the proof of Proposition 5.7. In Subsection 5.3, we then use the multilinear Strichartz estimate (5.5) to prove Proposition 5.6, thus yielding Theorem 1.5."},{"title":"Multilinear Strichartz estimate.","content":"In this subsection, we use the sharp L p -Strichartz estimates (1.19) in Theorem 1.2 to prove the multilinear Strichartz estimate (5.5). The main idea is to refine the Strichartz estimate by considering frequency scales finer than the standard dyadic Littlewood-Paley localizations as in [21]. See Lemma 5.10.\nDefinition 5.8. We say that (d, p) ∈ N × R is an admissible pair if\nNote that, by Theorem 1.2, the Strichartz estimates with (d, p) in this range are sharp.\nGiven dyadic N ≥ 1, let C N denote the collection of cubes C ⊂ Z d of side length ∼ N with arbitrary center and orientation. Then, we can rewrite Theorem 1.2 in the following form.\nLemma 5.9. Let (d, p) be admissible and I ⊂ R be a bounded interval. Then, for all dyadic N ≥ 1, we have\nMore generally, for all C ∈ C N , we have\nThe Strichartz estimate (5.8) shows that the loss in (5.8) depends only on the size of the frequency support, not the position. See [5,21]. In order to exploit further orthogonality between different frequency pieces under the linear Schrödinger evolution, we need to decompose the frequency cubes C N . Let R M (N ) be the collection of all sets in Z d which are given as the intersection of a cube of side length 2N with a strip of \"width\" 2M , i.e. the collection of all sets of the form\nwith some n 0 ∈ Z d , a ∈ R d , |a| = 1, A ∈ R. Here, the dot product a · θ θ θ n is given by\nwhere θ θ θ = (θ 1 , . . . , θ d ) is as in (1.15). Then, we have the following refinement of the Strichartz estimate.\nwhere 0 < δ < 1 2 − 10 3p when d = 2, 0 < δ < 1 2 − 8 3p when d = 3, and 0 < δ < 1 2 − 2 p when d ≥ 4."},{"title":"Proof. By Bernstein's inequality, we have","content":"Given admissible (d, p), write 1 p = θ q + 1−θ ∞ for some θ ∈ [0, 1), where q < p is given by q = 20 3 + when d = 2, q = 16 3 + when d = 3, and q = 4+ when d ≥ 4. Then, (5.11) follows from interpolating (5.8) (with p = q) and (5.12). Now, we are ready to prove the main multilinear Strichartz estimates (5.5).\nProof of Proposition 5.7. Let u j = e −it∆ φ j . Then, by almost orthogonality in spatial frequencies, it suffices to prove that there exists δ > 0 such that\nfor all cubes C ∈ C N 2 . Fix a cube C ∈ C N 2 and let n 0 be the center of C. Partition C = R ℓ into disjoint strips R ℓ with width M = max(N 2 2 /N 1 , 1), which are all orthogonal to n 0 with respect to the dot product · θ θ θ in (5.10), i.e. R ℓ is given by\nNote that we have R ℓ ∈ R M (N 2 ). By writing\nwe show that the sum is almost orthogonal in L 2 t,x . Since N 2 2 M 2 ℓ, we have\nfor n 1 ∈ R ℓ . Note that the multiplication by the factor k+1 j=2 P N j u j in (5.14) changes the time frequency at most by O(N 2 2 ). Hence, P R ℓ P N 1 u 1 k+1 j=2 P N j u j in (5.14) is localized at time frequency M 2 ℓ 2 + O(M 2 ℓ) = O(M 2 ℓ 2 ) for each ℓ. Therefore, the sum in (5.14) is almost orthogonal and we have\n(5.15)\nWith d and k as in (5.3), let p d,k = (d + 2)k. Then, by Lemma 5.9, we have The existence of such p is implied by (5.3). Moreover, the lower bound on p guarantees that each (d, p) is admissible, while the upper bound on p guarantees that\nThen, it follows from (5.17) that (d, q) is also admissible. By Hölder's inequality and Lemmata 5.9 and 5.10, we have\nfor some δ > 0. In view of (5.17) and (5.18), choose p such that\nMoreover, from (5.18) and (5.19), we have\nThen, noting M N 2 ∼ N 2 N 1 + 1 N 2 , it follows from (5.20) that\nFinally, by summing up the squares in (5.15) with respect to ℓ, we obtain (5.13) and hence (5.5). This completes the proof of Proposition 5.7."},{"title":"5.3.","content":"Proof of Proposition 5.6. First, we state and prove an auxiliary lemma (Lemma 5.11), using Proposition 5.7. Let C N , N ≥ 1, be the collection of cubes C ⊂ Z d of side length ∼ N as before. Let (d, p) be admissible in the sense of Definition 5.8. Then, it follows from Lemma 5.9 with Lemma 5.3 (i) that\nfor all C ∈ C N .\nLemma 5.11. Let d and k satisfy (5.3). Then, there exists δ ′ > 0 such that\nfor any interval I ⊂ [0, 1] and for all N 1 ≥ N 2 ≥ · · · ≥ N k+1 ≥ 1.\nProof. By almost orthogonality in spatial frequencies, it suffices to prove that there exists δ ′ > 0 such that\nfor all cubes C ∈ C N 2 . Moreover, by the embedding (5.2), it suffices to prove (5.24), where the Y 0 -norm is replaced by the V 2 ∆ L 2 -norm. Furthermore, it suffices to prove that there exists λ > 0 such that the following two estimates hold:\nand LHS of (5.24)\nwhere q 2 = p, q j = p d,k for j = 3, . . . , k, and q k+1 = q. We conclude this section by presenting the proof of Proposition 5.6.\nProof of Proposition 5.6. Let I = [0, T ). In the following, we prove\nfor all N ≥ 1, where the implicit constant is independent of N . By Lemma 5.5, we have\nHence, with u 0 = P ≤N v, it suffices to show that\nNow, dyadically decompose u * j = N j ≥1 P N j u * j . Without loss of generality, assume N 1 ≥ N 2 ≥ · · · ≥ N 2k+1 . Then, in order to have a non-trivial contribution on the left-hand side of (5.29), we must have N 1 ∼ max(N 0 , N 2 ).\nBy Lemma 5.11, we have\nSumming (5.30) over dyadic blocks N 0 ∼ N 1 ≥ N 2 ≥ · · · ≥ N 2k+1 and by Cauchy-Schwarz inequality, we have LHS of (5.29)\nyielding (5.29) in view of (5.2).\nSumming (5.31) over dyadic blocks N 0 (≪ N 1 ) and N 1 ∼ N 2 ≥ N 3 ≥ · · · ≥ N 2k+1 and by Cauchy-Schwarz inequality, we have LHS of (5.29)\nThis completes the proof of Proposition 5.6.\nAppendix A. On the Weyl sum estimate (2.11) In this appendix, we present a proof of (2.11). We decided to include the proof for the convenience of readers, in particular for those in PDEs. Since I = I(η) is a compact interval and the integrand is periodic with period 1, it suffices to shoŵ\nfor r > 4, where F (t) is the Weyl sum defined by\nFor a, q ∈ N with 1 ≤ a ≤ q ≤ N and (a, q) = 1, define a major arc M(q, a) by\nwhere x = min n∈Z |x − n| denotes the distance of x to the closest integer as before. Let M = a,q M(q, a). Note that we have t − a q ≤ 1 q 2 for t ∈ M. Then, by Weyl's inequality we have\nHence, the contribution from the major arc M is estimated bŷ\nsince r > 4.\nRemark A.1. Indeed, the contribution from the major arc M provides the lower bound in (A.1). We only need to consider the contribution from M(q, a) for odd q ≤ N 1 2 . Let S(q, a) be the Gauss sum given by S(q, a) = q n=1 e 2πin 2 a q . We have S(q, a) = √ q for odd q. Now, suppose that q is odd such that 1 ≤ q ≤ N 1 2 . Then, by Van der Corput's method [41] with S(q, a) = √ q, one can show that |F (t)| N q 1 2 for t ∈ M(q, a). Noting that M(q, a)\nare disjoint, we havê\nwhere φ(q) denotes Euler's function. This shows the lower bound in (A.1).\nNext, we estimate the contribution from the minor arc m := [0, 1] \\ M. Fix small ε > 0. Then, Dirichlet's theorem [40, Lemma 2.1] states that given t ∈ [0, 1], there exist integers a, q with 1 ≤ a ≤ q ≤ N 2−ε and (a, q) = 1 such that t − a q ≤ 1 qN 2−ε . Define Let t ∈ m 1 , i.e. we have t − a q ≤ 1 qN 2−ε for some q ≪ N ε . Then, by Lemmata 2.7 and 2.8 in [40] with |S(q, a)| q 1 2 , we have\nwhere v is defined in [40, (2.9)]. Applying Hua's inequality as before, we havê\nThis completes the proof of (A.1).\nAppendix B. On a partially irrational torus T 2 × T α 3\nIn this appendix, we present a sketch of the proof of Theorem 1.6. Thus, we set d = 3 and assume that (1.27) holds in the following. The main ingredient is an improvement of the Strichartz estimate on a partially irrational torus. 1) and (B.2), the implicit constants depend on ε > 0, q > 4, and |I|, but are independent of f . We first prove Proposition B.1 assuming Lemma B.2. Then, we sketch the proof of Theorem 1.6. We present the proof of Lemma B.2 at the end of this appendix.\nGiven f as in (3.2), let F (x, t) = e −it∆ f (x, t). By Cauchy-Schwarz inequality, we have F L ∞ t,x N 3 2 . Given p > 14 3 , let q ∈ (4, p). Then, by Lemma B.2 and Theorem 1.1 (ii), we havê for sufficiently small ε 1 , ε 2 > 0. Here, the condition p > 14 3 is needed in the last inequality. This proves Proposition B.1.\nNext, we briefly discuss how Theorem 1.6 follows from Proposition B.1. As in Section 5, the goal is to prove the multilinear estimate (5.4) in Proposition 5.6 for d = 3 and k = 2 (with s c = 1) under the assumption (1.27). In view of the argument in Subsection 5.3 (which holds without any change even in this case), it suffices to prove the multilinear Strichartz estimate (5.5) in Proposition 5.7 for d = 3 and k = 2 (with s c = 1). By repeating the proof of Lemma 5.10 with Proposition B.1, we see that (5.11) holds for p > 14 3 with δ ∈ (0, 1 2 − 7 3p ). In the proof of Proposition 5.7, the only change appears in (5.17) and we can choose p such that 14 3 < p < 20k 3k + 2 in this case. In particular, we can set k = 2 by choosing p ∈ ( 14 3 , 5). The rest of the argument follows exactly as in Section 5.\nIn the remaining part of the paper, we present the proof of Lemma B.2.\nProof of Lemma B.2. The proof follows the proof of Proposition 3.1 with small modifications. In the following, we only point out these modifications. Given an interval I ⊂ R, assume that I is centered at 0.\n(i) With Q(n) in (1.27), define R as in (2.19). Then, we have R(x, t) = 2 j=1 R 1 (x j , t) · R θ 3 (x 3 , t), where R θ is defined in (2.20). We also define R 1 and R 2 as in (3.10). Then, by (3.11), we have by setting ε 2 = ε 2 (ε 1 ) such that 1−ε 2 1+ε 1 = 1 1+2ε 1 . This proves (B.1) with ε = ε 1 2 . (ii) Define K by K(x, t) = χ(t)K(x 1 , t)K(x 2 , t) Also, with Λ as in (3.46) and K as in (B.5), it follows from (3.26) and (3.29)\nLet p be as in (3.50). In the following, we consider the case θ ≥ as long as θ < 1 2 ."},{"title":".","content":""},{"title":"M","content":""},{"title":"Theorem 1. 4 (","content":""},{"title":"Remark 3. 2 .","content":""},{"title":".","content":""},{"title":"M","content":""},{"title":"= 0 .","content":""},{"title":"= 0 .","content":""},{"title":"Lemma 3 . 5 (","content":""},{"title":"1","content":""},{"title":"Lemma 4 . 1 .","content":""},{"title":"j = 1 ,","content":""},{"title":"u","content":""},{"title":"Lemma 5 . 10 .","content":""},{"title":"f","content":""},{"title":"1 3 .f","content":""},{"title":"69 )","content":""},{"title":"26 )","content":""}],"string":"[\n {\n \"title\": \"Strichartz estimates: Part 1\",\n \"content\": \"In this section, we prove our main result (Theorem 1.2) for d ≥ 3 and present a partial proof for d = 2. In [6], Bourgain treated the three-dimensional case. His argument is based on the following estimate:ˆT\\nLemma 2.1 was used in [6] for the three-dimensional case. See also Lemma 2 in [11]. A proof for general dimensions in [11] relies on Schur's lemma. In the following, we give a direct proof for reader's convenience.\\nn∈I k,a n ′ ∈I k ′ ,a |c n c a−n ||c n ′ c a−n ′ | By Cauchy-Schwarz inequality (in k) followed by Young inequality, we have\\nThis completes the proof of Lemma 2.1.\\nNext, we state the main proposition. Theorem 1.2 then follows this proposition and Bernstein's inequality when d ≥ 3. (4,4). Then, we have\\nwhere q = p when d = 3, 4 and q = 8 when d = 2.\\nBourgain proved (2.4) for d = 3. See Proposition 1.1 in [6]. Our proof follows the ideas developed in [6]. By setting p = 4 when d ≥ 5 and p = 16 d + when d = 3, 4, Proposition 2.2 yields the L 4 -Strichartz estimate, which improves the result in [11] for d ≥ 3. Note that the Strichartz estimate (2.4) is essentially sharp in higher dimensions (d ≥ 4). Indeed, on R d with d ≥ 2, by Sobolev inequality and interpolation of (1.6) and (1.7)\\nfor p ≥ 4. We first use Proposition 2.2 to prove Theorem 1.2 when d ≥ 3.\\nBy repeating the same argument with Proposition 2.2 (ii), we obtain Theorem 1.2 (ii) when d = 3, 4. When d = 2, this yields Theorem 1.2 (ii) only for p ≥ 8.\\nWe now present the proof of Proposition 2.2.\\nProof of Proposition 2.2. (i) Let Q(n) be as in (1.15). Then, we have\\n(2.6)\\nWith c n = f (n), let F a (t) be as in (2.2). Then, by Minkowski's integral inequality with p ≥ 4, we have\\n(2.8)\\nFrom (2.7) and (2.8), we have\\n(2.9)\\nNow, let η(t) be a smooth function with a compact support I ⊂ R such that η ≥ 0 and\\nIf p ≤ 8, then we have p p−4 ≥ 2. Then, by Hausdorff-Young's inequality, we have\\n(2.10)\\nNote that r = dp 4 > 4, since p > 16 d . Then, by an application of the Hardy-Littlewood circle method (see Appendix A), we havê\\nx .\\n(2.12)\\nThen, (2.4) follows from (2.12) and (2.4) for p = 8.\\n(ii) When (d, p) = (2,8), (3, 16 3 ), or (4, 4), we have r = dp 4 = 4. In this case, (2.11) does not hold. Nonetheless, by Hua's inequality [40], we havê\\nSee also [25, (8.13)]. Then, (2.5) follows from (2.13) and repeating the computation in (i\\n(2.14)\\nThen, the local-in-time Strichartz estimate can be expressed as n∈S N a n e 2πi(n·x+Q(n)t)\\nwhere S N denotes the following set:\\nOur task is to seek for an optimal constant K p,N . By duality, (2.15) is equivalent to\\nHere, the Fourier transform f is defined by\\nThen, (1.19) for p > 12 follows once we prove the next proposition.\\n.\\nHence, Proposition 2.3 states that, on an irrational torus, the same estimate for K p,N holds, but only for p > 12.\\nProof. Without loss of generality, assume that I is centered at 0. Let R be a kernel defined by\\nfor p > 6. Bourgain's argument is based on an application of the Hardy-Littlewood circle method. See also Lemma 2.4 in [23] for a simpler proof based on the Poisson summation formula. Note that (2.21) does not hold for p = 6. See Rogovskaya [33] and [3]. By Hölder's inequality, (2.21), and Sobolev inequality, we have\\nas long as p > 12.\"\n },\n {\n \"title\": \"Strichartz estimates: Part 2\",\n \"content\": \"3.1. Level set estimates. In this section, we prove Theorem 1.2 when d = 2. The main ingredient is the level set estimates on irrational tori in Proposition 3.1 below. For level sets estimates on the usual torus T d , see [3,23]. It turns out that these level set estimates are useful only when d = 2, 3 (see Remark 3.3), but we state and prove the results for a general dimension. In the following, we assume that θ 1 = 1 in (1.15) for simplicity. Namely, we consider Q(n) = n 2 1 + θ 2 n 2 2 + · · · + θ d n 2 d .\\n(ii) Let q > 6. Then, there exists small ε > 0 such that (3.4) and (3.5), the implicit constants depend on ε > 0, q > 6, and |I|, but are independent of f .\\nWe present the proof of Proposition 3.1 in Subsections 3.2 and 3.3. In the following, we use Proposition 3.1 to prove Theorem 1.2 when d = 2. First, we present the proof of Theorem 1.2 (ii.a), i.e. we prove (1.20) for p ≥ 20 3 when d = 2. Recall that Catoire-Wang [11] proved\\nN. Then, with Proposition 3.1 (i) and (3.6), we havê\\nwhere the last inequality holds as long as p ≥ 20 3 . This proves Theorem 1.2 (ii.a). By Proposition 3.1 (i) and (ii), Theorem 1.2 (i.a) follows in a similar manner. We omit details.\\nfor p = 2(d+1) , where x = min n∈Z |x − n| denotes the distance of x to the closest integer. Note that Φ is periodic with period 1. By taking a Fourier transform, we have\\nwhere c q (k) denotes Ramanujan's sum: c q (k) := a∈Jq e −2πi a q k . Let φ(q) be the Euler's totient function defined by φ(q) = a∈Jq 1. Then, by Theorem 330 in [19], we have\\nNamely, Φ(0) is independent of M . Without loss of generality, assume that I is centered at 0. With Q(n) in (3.1), define R as in (2.19) (2.20). Now, letting χ be a smooth cutoff function support on 3I such that χ(t) ≡ 1 on 2I, define R 1 and R 2 by\\nNoting that the intervals I ℓ,q,a := ℓ + a q + 1 200q 2 , ℓ + a q + 1 100q 2 are disjoint for distinct values of ℓ, a, and q ∼ M ≫ 1, it follows from Weyl's inequality [31, Theorem 1 on p. 41] that\\nfor t ∈ I ℓ,q,a since q ∼ M ≥ N . Then, along with a trivial bound |R θ j (x j , t)| N , we obtain\\nNext, we consider R 2 . By expanding Φ(t) in the Fourier series, we have\\nFirst, recall the following lemma (Lemma 3.33 in [3]). Given M ∈ N and k ∈ Z, we have M ≤q<2M\\nwhere d(k, M ) denotes the number of divisors of k less than M . Then, by taking a Fourier transform of (3.13) with (3.9), (3.14), and d(k, M ) k 0+ , we have\\nNote that supp Θ λ (x, ·) ⊂ I for each x ∈ T d . Then, by Cauchy-Schwarz inequality with (3.2), we have\\nBy (3.10), (3.12), and (3.15), we have\\nby setting ε 2 = ε 2 (ε 1 ) such that 1−ε 2 1+2ε 1 = 1 1+3ε 1 . This proves (3.4) with ε = ε 1 2 . 3.3. Proof of Proposition 3.1 (ii). In this subsection, we prove the level set estimate (3.5), which is sharp for q > 6. The following argument is inspired by Bourgain's paper [3]. We first go over some basic setups, restricting our attention to t ∈ T.\\nLet (3.20)\\nThen, we have the following lemma. Here, x = min n∈Z |x − n| denotes the distance of x to the closest integer as before.\\nThen, we have\\nNote that the multiplier σ n in (3.20) avoids the logarithmic loss (when q ∼ N ) in Weyl's inequality (3.11) on the Weyl sum W N (x, t) = |n|≤N e 2πi(nx+n 2 t) . Indeed, by writing\\nk=N W k , we see that this regularizing effect in (3.22) is analogous to that of the Féjér kernel over the Dirichlet kernel.\\nIn the following, we fix N ≫ 1, dyadic. For dyadic M ≤ N , let R M by\\nwhere J q is as in (3.8). Let ψ(t) be a smooth cutoff function supported on\\nNote that we have supp(ω N,2 s ) ⊂ |t|\\nRecall that by Dirichlet's theorem [ Now, we focus on our problem. Namely, we do not assume t ∈ T any longer. Given an interval I ⊂ R, assume that I is centered at 0 and let χ be a smooth cutoff function support on 3I such that χ(t) ≡ 1 on 2I as before. We define\\nHence, from (3.32), we have Hence, from (3.37), we have\\nThe second estimate (3.39) is useful when M ≪ N ε .\\nIn the following, we establish another estimate on f * Λ M,s L 2 (I×T d ) , using the following lemma from Bourgain [3].\\nwhere S N is as in (3.30). Given D > 0 (to be chosen later), separate the first term, depending on d(k, M ) ≤ D or > D. The contribution from d(k, M ) ≤ D can be estimated by\\nNext, we estimate the contribution from d(k, M ) > D. By Cauchy-Schwarz inequality, we have\\nNow, we estimate the first term on the right-hand side of (3.41) after applying (3.43). By first integrating in τ , then summing over |n j | N for j = 1, . . . , d, applying Lemma 3.5 (with 2B and 2β instead of B and β), the trivial bound d(k, M ) ≤ M , and Hausdorff-Young's inequality, we have\\nwhere Λ M,s is as in (3.31). By (3.25), we have\\nHence, we have\\nfor some θ ∈ (0, 1). Note that p ′ θ = 2.\\nCase (i): θ < 1 5 . By interpolating (3.33) and (3.39), we have\\n(3.51)\\nThen, summing over dyadic M ≥ 1 and s with 2 s ≤ N , we have\\nas long as θ > 0 satisfies − 1 2 + ( 5 2 +)θ ≤ 0, i.e.\\nθ < 1 5 .\\n(3.53)\\nIn the following, we prove the level set estimate (3.5) for q > 10. Given f as in (3.2),\\nThen, proceeding as in (3.17) with (3.49) and (3.52), we have\\nNoting that p ′ θ = 2 by (3.50), it follows from (3.55) that\\nwhere q := p ′ = 2 θ > 10. Note that we only needed to assume λ ≫ N Case (ii): 1 5 ≤ θ < 1 3 . Let M j , j = 1, 2 be dyadic numbers such that\\nHere, we choose δ 1 , δ 2 > 0 such that M 1 ≪ M 2 . We divide Λ into three pieces: Λ = Λ 1 + Λ 2 + Λ 3 by setting Similarly, by interpolating (3.33) and (3.45), we have\\nNote that (3.62) can be satisfied as long as θ < 1 3 by choosing α sufficiently small. Lastly, from (3.33) and (3.38), we have\\nThen, summing over M ≥ M 2 ∼ N δ 2 and s with 2 s ≤ N , we have\\nas long as θ < 1 3 . Putting (3.58), (3.61), and (3.64) together with (3.53), we obtain\\nNow, we are ready to prove the level set estimate (3.5) for q > 6. Then, proceeding as in Case (i) with (3.49) and (3.65), we have\\nFirst, suppose that λ 2 |A λ | 2 I holds. Recall from (3.50) that p ′ θ = 2. Then, with (3.56), we have\\nfor q > p ′ by choosing δ 1 = δ 1 (q, p ′ ) sufficiently small. Next, suppose that λ 2 |A λ | 2 II holds. Then, from (3.56), we have\\nas long as we have εp ′ ≤ 1 and δ 2 = δ 2 (p ′ , B) is sufficiently small such that (− p ′ 2 + 5 + 2B+)δ 2 ≤ 1.\\nFinally, given q > 6, we choose θ < 1 3 such that q > p ′ = 2 θ > 6. Then, from (3.67), (3.68), (3.69), and (3.70) with λ ≤ N d 2 , we obtain\\nThis completes the proof of Proposition 3.1 (ii).\"\n },\n {\n \"title\": \"Well-posedness in subcritical spaces\",\n \"content\": \"In this section, we prove local well-posedness of NLS (1.25) on irrational tori in subcritical Sobolev spaces (Theorem 1.4). It turns out that the well-posed theory of (1.25) is very similar to that on the standard tours [3,4,5,6].\\nIn the seminal paper [3], Bourgain introduced the X s,b -space whose norm is given by\\nwhere · = (1 + | · | 2 ) 1 2 . After establishing Strichartz estimates, he proved several wellposedness results of NLS on the standard torus T d . In our case, i.e. on an irrational torus, we need to replace the weight τ − |n| 2 b in (4.1) by τ − Q(n) b , where Q(n) is defined in (1.15). Then, by the standard X s,b -theory, it is known that certain multilinear Strichartz estimates imply well-posedness. More precisely, we have the following lemma. The proof of Lemma 4.1 is standard and we refer the readers to [3,4,6,11] for details.\\nProof of Theorem 1.4. In view of Lemma 4.1, it suffices to prove the (k + 1)-linear estimate (4.2) for s > s 0 . Without loss of generality, assume that N 1 ≥ N 2 ≥ · · · ≥ N k+1 .\\nCase (i): d = 2. When k = 1, the well-posedness result was already obtain in [11]. In the following, we first consider the case k = 2, 3, 4, 5. First, assume that supp\\nThen, by Hölder's inequality and (1.21) with 4(7k+5) 3(k+3) ∈ (4, 20 3 ], we have\\nThen, by a change of variables and (1.21) (see [5]), we obtain\\nThen, by almost orthogonality with (4.3) and (4.4), we have\\nRemark 4.2. In view of the reduction above, we only prove (4.2), assuming supp φ 1 ⊂ [−N 2 , N 2 ] d in the following.\\nNext, we consider the case k ≥ 5. By Hölder's inequality and (1.21) with 8k k+1 ∈ [ 20 3 , 10), we have\\nHence, (4.2) holds for s > s 0 = 1 − 1 k . Case (ii): d = 3. When k = 1, the well-posedness result was obtained in [6]. When k = 2, by Hölder's inequality and (1.22), we have\\nHence, (4.2) holds for s > s 0 = 53 52 . When k ≥ 3, by Hölder's inequality and (1.22), we have\\nHence, (4.2) holds s ≥ s 0 = 3 2 − 1 k . Case (iii): d ≥ 4 and k ≥ 1. By Hölder's inequality with (1.23) or (1.24), we have\\nHence, (4.2) holds for s > s 0 = d 2 − 1 k .\"\n },\n {\n \"title\": \"5.\",\n \"content\": \"Well-posedness in critical spaces 5.1. Function spaces. In this section, we prove local well-posedness of NLS (1.25) on irrational tori in critical Sobolev spaces H sc (T d ) (Theorem 1.5). In the following, we use the U p -and V p -spaces, developed by Tataru, Koch, and their collaborators [30,18,21,22]. These spaces have been very effective in establishing well-posedness of various dispersive PDEs in critical spaces. We briefly go over the basic definitions of function spaces and their properties. See Hadac-Herr-Koch [18] and Herr-Tataru-Tzvetkov [21] for detailed proofs.\\nLet H be a separable Hilbert space over C. In particular, it will be either H s (T d ) or C. Let Z be the collection of finite partitions {t k } K k=0 of R: −∞ < t 0 < · · · < t K ≤ ∞. If t K = ∞, we use the convention u(t K ) := 0 for all functions u : R → H. We use 1 I to denote the sharp characteristic function of a set I ⊂ R.\\n(i) A U p -atom is defined by a step function a : R → H of the form\\nThen, we define the atomic space U p (R; H) to be the collection of functions u : R → H of the form\\nwhere a j 's are U p -atoms and {λ j } j∈N ∈ ℓ 1 (N; C), (ii) We define V p (R; H) by the collection of functions u :\\nWe also define V p rc (R; H) to be the closed subspace of all right-continuous functions in V p (R; H) such that lim t→−∞ u(t) = 0.\\n(iii) Let s ∈ R. We define U p ∆ H s (and V p ∆ H s , respectively) to be the spaces of all functions u : R → H s (T d ) such that the following U p ∆ H s -norm (and V p ∆ H s -norm, respectively) is finite:\\nHere, the Laplacian ∆ is defined in terms of Q(n) as in (1.14).\\nRemark 5.2. Note that the spaces U p (R; H), V p (R; H), and V p rc (R; H) are Banach spaces. Moreover, we have the following embeddings:\\nNext, we state a transference principle and an interpolation result.\"\n },\n {\n \"title\": \"Lemma 5.3. (i) (transference principle) Suppose that we have\",\n \"content\": \"x for some 1 ≤ p, q ≤ ∞. Then, we have\\n(ii) (interpolation) Let E be a Banach space. Suppose that T : U p 1 × · · · × U p k → E is a bounded k-linear operator such that\\nfor some p 1 , . . . , p k > 2. Moreover, assume that there exists C 2 ∈ (0, C 1 ] such that\\nThen, we have\\nA transference principle as above has been commonly used in the Fourier restriction norm method. See [18,Proposition 2.19] for the proof of Lemma 5.3 (i). The proof of the interpolation result follows from extending the trilinear result in [21] to a general k-linear case. See also [18,Proposition 2.20].\\nLet η : R → [0, 1] be an even smooth cutoff function supported on [− 8 5 , 8 5 ] such that η ≡ 1 on [− 5 4 , 5 4 ]. Given a dyadic number N ≥ 1, we set η 1 (ξ) = η(|ξ|) and\\nThen, we define the Littlewood-Paley projection operator P N as the Fourier multiplier operator with symbol η N . Moreover, we define P ≤N by P ≤N = 1≤M ≤N P M . More generally, given a set R ⊂ Z d , we define P R to be the Fourier multiplier operator with symbol 1 R .\\nWe define X s to be the space of all functions u :\\n(ii) Let s ∈ R. We define Y s to be the space of all functions u : R → H s (T d ) such that for every n ∈ Z d , the map t → e −itQ(n) u(n, t) is in V 2 rc (R t ; C) and u Y s < ∞, where the Y s -norm is defined by\\nRecall the following embeddings:\\nGiven a time interval I ⊂ R, we define the restrictions X s (I) and Y s (I) of these spaces in the usual manner. We now state the linear estimates. Given f ∈ L 1 loc ([0, ∞); L 2 (T d )), define I(f ) by\\nLemma 5.5 (Linear estimates). Let s ≥ 0 and T > 0. Then, the following linear estimates hold:\\nNext, we present the crucial multilinear estimate.\\nProposition 5.6. Let d and k satisfy\\nThen, the following multilinear estimate holds for all T ∈ (0, 1]:\\nOnce we prove Proposition 5.6, one can prove Theorem 1.5 by the fixed point argument as in [21,42]. We omit details. The remainder of this section is devoted to the proof of Proposition 5.6. Indeed, the multilinear estimate (5.4) follows once we prove the following multilinear Strichartz estimate.\\nfor any interval I ⊂ [0, 1] and for all φ j ∈ L 2 (T d ), j = 1, . . . , k + 1, and N 1 ≥ N 2 ≥ · · · ≥ N k+1 ≥ 1.\\nIn Subsection 5.2, we first present the proof of Proposition 5.7. In Subsection 5.3, we then use the multilinear Strichartz estimate (5.5) to prove Proposition 5.6, thus yielding Theorem 1.5.\"\n },\n {\n \"title\": \"Multilinear Strichartz estimate.\",\n \"content\": \"In this subsection, we use the sharp L p -Strichartz estimates (1.19) in Theorem 1.2 to prove the multilinear Strichartz estimate (5.5). The main idea is to refine the Strichartz estimate by considering frequency scales finer than the standard dyadic Littlewood-Paley localizations as in [21]. See Lemma 5.10.\\nDefinition 5.8. We say that (d, p) ∈ N × R is an admissible pair if\\nNote that, by Theorem 1.2, the Strichartz estimates with (d, p) in this range are sharp.\\nGiven dyadic N ≥ 1, let C N denote the collection of cubes C ⊂ Z d of side length ∼ N with arbitrary center and orientation. Then, we can rewrite Theorem 1.2 in the following form.\\nLemma 5.9. Let (d, p) be admissible and I ⊂ R be a bounded interval. Then, for all dyadic N ≥ 1, we have\\nMore generally, for all C ∈ C N , we have\\nThe Strichartz estimate (5.8) shows that the loss in (5.8) depends only on the size of the frequency support, not the position. See [5,21]. In order to exploit further orthogonality between different frequency pieces under the linear Schrödinger evolution, we need to decompose the frequency cubes C N . Let R M (N ) be the collection of all sets in Z d which are given as the intersection of a cube of side length 2N with a strip of \\\"width\\\" 2M , i.e. the collection of all sets of the form\\nwith some n 0 ∈ Z d , a ∈ R d , |a| = 1, A ∈ R. Here, the dot product a · θ θ θ n is given by\\nwhere θ θ θ = (θ 1 , . . . , θ d ) is as in (1.15). Then, we have the following refinement of the Strichartz estimate.\\nwhere 0 < δ < 1 2 − 10 3p when d = 2, 0 < δ < 1 2 − 8 3p when d = 3, and 0 < δ < 1 2 − 2 p when d ≥ 4.\"\n },\n {\n \"title\": \"Proof. By Bernstein's inequality, we have\",\n \"content\": \"Given admissible (d, p), write 1 p = θ q + 1−θ ∞ for some θ ∈ [0, 1), where q < p is given by q = 20 3 + when d = 2, q = 16 3 + when d = 3, and q = 4+ when d ≥ 4. Then, (5.11) follows from interpolating (5.8) (with p = q) and (5.12). Now, we are ready to prove the main multilinear Strichartz estimates (5.5).\\nProof of Proposition 5.7. Let u j = e −it∆ φ j . Then, by almost orthogonality in spatial frequencies, it suffices to prove that there exists δ > 0 such that\\nfor all cubes C ∈ C N 2 . Fix a cube C ∈ C N 2 and let n 0 be the center of C. Partition C = R ℓ into disjoint strips R ℓ with width M = max(N 2 2 /N 1 , 1), which are all orthogonal to n 0 with respect to the dot product · θ θ θ in (5.10), i.e. R ℓ is given by\\nNote that we have R ℓ ∈ R M (N 2 ). By writing\\nwe show that the sum is almost orthogonal in L 2 t,x . Since N 2 2 M 2 ℓ, we have\\nfor n 1 ∈ R ℓ . Note that the multiplication by the factor k+1 j=2 P N j u j in (5.14) changes the time frequency at most by O(N 2 2 ). Hence, P R ℓ P N 1 u 1 k+1 j=2 P N j u j in (5.14) is localized at time frequency M 2 ℓ 2 + O(M 2 ℓ) = O(M 2 ℓ 2 ) for each ℓ. Therefore, the sum in (5.14) is almost orthogonal and we have\\n(5.15)\\nWith d and k as in (5.3), let p d,k = (d + 2)k. Then, by Lemma 5.9, we have The existence of such p is implied by (5.3). Moreover, the lower bound on p guarantees that each (d, p) is admissible, while the upper bound on p guarantees that\\nThen, it follows from (5.17) that (d, q) is also admissible. By Hölder's inequality and Lemmata 5.9 and 5.10, we have\\nfor some δ > 0. In view of (5.17) and (5.18), choose p such that\\nMoreover, from (5.18) and (5.19), we have\\nThen, noting M N 2 ∼ N 2 N 1 + 1 N 2 , it follows from (5.20) that\\nFinally, by summing up the squares in (5.15) with respect to ℓ, we obtain (5.13) and hence (5.5). This completes the proof of Proposition 5.7.\"\n },\n {\n \"title\": \"5.3.\",\n \"content\": \"Proof of Proposition 5.6. First, we state and prove an auxiliary lemma (Lemma 5.11), using Proposition 5.7. Let C N , N ≥ 1, be the collection of cubes C ⊂ Z d of side length ∼ N as before. Let (d, p) be admissible in the sense of Definition 5.8. Then, it follows from Lemma 5.9 with Lemma 5.3 (i) that\\nfor all C ∈ C N .\\nLemma 5.11. Let d and k satisfy (5.3). Then, there exists δ ′ > 0 such that\\nfor any interval I ⊂ [0, 1] and for all N 1 ≥ N 2 ≥ · · · ≥ N k+1 ≥ 1.\\nProof. By almost orthogonality in spatial frequencies, it suffices to prove that there exists δ ′ > 0 such that\\nfor all cubes C ∈ C N 2 . Moreover, by the embedding (5.2), it suffices to prove (5.24), where the Y 0 -norm is replaced by the V 2 ∆ L 2 -norm. Furthermore, it suffices to prove that there exists λ > 0 such that the following two estimates hold:\\nand LHS of (5.24)\\nwhere q 2 = p, q j = p d,k for j = 3, . . . , k, and q k+1 = q. We conclude this section by presenting the proof of Proposition 5.6.\\nProof of Proposition 5.6. Let I = [0, T ). In the following, we prove\\nfor all N ≥ 1, where the implicit constant is independent of N . By Lemma 5.5, we have\\nHence, with u 0 = P ≤N v, it suffices to show that\\nNow, dyadically decompose u * j = N j ≥1 P N j u * j . Without loss of generality, assume N 1 ≥ N 2 ≥ · · · ≥ N 2k+1 . Then, in order to have a non-trivial contribution on the left-hand side of (5.29), we must have N 1 ∼ max(N 0 , N 2 ).\\nBy Lemma 5.11, we have\\nSumming (5.30) over dyadic blocks N 0 ∼ N 1 ≥ N 2 ≥ · · · ≥ N 2k+1 and by Cauchy-Schwarz inequality, we have LHS of (5.29)\\nyielding (5.29) in view of (5.2).\\nSumming (5.31) over dyadic blocks N 0 (≪ N 1 ) and N 1 ∼ N 2 ≥ N 3 ≥ · · · ≥ N 2k+1 and by Cauchy-Schwarz inequality, we have LHS of (5.29)\\nThis completes the proof of Proposition 5.6.\\nAppendix A. On the Weyl sum estimate (2.11) In this appendix, we present a proof of (2.11). We decided to include the proof for the convenience of readers, in particular for those in PDEs. Since I = I(η) is a compact interval and the integrand is periodic with period 1, it suffices to shoŵ\\nfor r > 4, where F (t) is the Weyl sum defined by\\nFor a, q ∈ N with 1 ≤ a ≤ q ≤ N and (a, q) = 1, define a major arc M(q, a) by\\nwhere x = min n∈Z |x − n| denotes the distance of x to the closest integer as before. Let M = a,q M(q, a). Note that we have t − a q ≤ 1 q 2 for t ∈ M. Then, by Weyl's inequality we have\\nHence, the contribution from the major arc M is estimated bŷ\\nsince r > 4.\\nRemark A.1. Indeed, the contribution from the major arc M provides the lower bound in (A.1). We only need to consider the contribution from M(q, a) for odd q ≤ N 1 2 . Let S(q, a) be the Gauss sum given by S(q, a) = q n=1 e 2πin 2 a q . We have S(q, a) = √ q for odd q. Now, suppose that q is odd such that 1 ≤ q ≤ N 1 2 . Then, by Van der Corput's method [41] with S(q, a) = √ q, one can show that |F (t)| N q 1 2 for t ∈ M(q, a). Noting that M(q, a)\\nare disjoint, we havê\\nwhere φ(q) denotes Euler's function. This shows the lower bound in (A.1).\\nNext, we estimate the contribution from the minor arc m := [0, 1] \\\\ M. Fix small ε > 0. Then, Dirichlet's theorem [40, Lemma 2.1] states that given t ∈ [0, 1], there exist integers a, q with 1 ≤ a ≤ q ≤ N 2−ε and (a, q) = 1 such that t − a q ≤ 1 qN 2−ε . Define Let t ∈ m 1 , i.e. we have t − a q ≤ 1 qN 2−ε for some q ≪ N ε . Then, by Lemmata 2.7 and 2.8 in [40] with |S(q, a)| q 1 2 , we have\\nwhere v is defined in [40, (2.9)]. Applying Hua's inequality as before, we havê\\nThis completes the proof of (A.1).\\nAppendix B. On a partially irrational torus T 2 × T α 3\\nIn this appendix, we present a sketch of the proof of Theorem 1.6. Thus, we set d = 3 and assume that (1.27) holds in the following. The main ingredient is an improvement of the Strichartz estimate on a partially irrational torus. 1) and (B.2), the implicit constants depend on ε > 0, q > 4, and |I|, but are independent of f . We first prove Proposition B.1 assuming Lemma B.2. Then, we sketch the proof of Theorem 1.6. We present the proof of Lemma B.2 at the end of this appendix.\\nGiven f as in (3.2), let F (x, t) = e −it∆ f (x, t). By Cauchy-Schwarz inequality, we have F L ∞ t,x N 3 2 . Given p > 14 3 , let q ∈ (4, p). Then, by Lemma B.2 and Theorem 1.1 (ii), we havê for sufficiently small ε 1 , ε 2 > 0. Here, the condition p > 14 3 is needed in the last inequality. This proves Proposition B.1.\\nNext, we briefly discuss how Theorem 1.6 follows from Proposition B.1. As in Section 5, the goal is to prove the multilinear estimate (5.4) in Proposition 5.6 for d = 3 and k = 2 (with s c = 1) under the assumption (1.27). In view of the argument in Subsection 5.3 (which holds without any change even in this case), it suffices to prove the multilinear Strichartz estimate (5.5) in Proposition 5.7 for d = 3 and k = 2 (with s c = 1). By repeating the proof of Lemma 5.10 with Proposition B.1, we see that (5.11) holds for p > 14 3 with δ ∈ (0, 1 2 − 7 3p ). In the proof of Proposition 5.7, the only change appears in (5.17) and we can choose p such that 14 3 < p < 20k 3k + 2 in this case. In particular, we can set k = 2 by choosing p ∈ ( 14 3 , 5). The rest of the argument follows exactly as in Section 5.\\nIn the remaining part of the paper, we present the proof of Lemma B.2.\\nProof of Lemma B.2. The proof follows the proof of Proposition 3.1 with small modifications. In the following, we only point out these modifications. Given an interval I ⊂ R, assume that I is centered at 0.\\n(i) With Q(n) in (1.27), define R as in (2.19). Then, we have R(x, t) = 2 j=1 R 1 (x j , t) · R θ 3 (x 3 , t), where R θ is defined in (2.20). We also define R 1 and R 2 as in (3.10). Then, by (3.11), we have by setting ε 2 = ε 2 (ε 1 ) such that 1−ε 2 1+ε 1 = 1 1+2ε 1 . This proves (B.1) with ε = ε 1 2 . (ii) Define K by K(x, t) = χ(t)K(x 1 , t)K(x 2 , t) Also, with Λ as in (3.46) and K as in (B.5), it follows from (3.26) and (3.29)\\nLet p be as in (3.50). In the following, we consider the case θ ≥ as long as θ < 1 2 .\"\n },\n {\n \"title\": \".\",\n \"content\": \"\"\n },\n {\n \"title\": \"M\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 1. 4 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Remark 3. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \".\",\n \"content\": \"\"\n },\n {\n \"title\": \"M\",\n \"content\": \"\"\n },\n {\n \"title\": \"= 0 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"= 0 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 3 . 5 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"1\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 4 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"j = 1 ,\",\n \"content\": \"\"\n },\n {\n \"title\": \"u\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5 . 10 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"f\",\n \"content\": \"\"\n },\n {\n \"title\": \"1 3 .f\",\n \"content\": \"\"\n },\n {\n \"title\": \"69 )\",\n \"content\": \"\"\n },\n {\n \"title\": \"26 )\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Background. The Cauchy problem of the nonlinear Schrödinger equation (NLS):","content":".\nBackground. The Cauchy problem of the nonlinear Schrödinger equation (NLS):"},{"id":"","caption":"f ∈ L 2 (R d ) with supp f ⊂ [−N, N ] d . By interpolating (1.6) and (1.7), we see that the estimate (1.5) holds for 2(d+2) d ≤ p ≤ ∞ and is sharp in view of sharpness of (1.6) and (1.7). Note that the estimate (1.5) is scaling-invariant in the following sense. Consider the linear Schrödinger equation: i∂ t u − ∆u = 0 u t=0 = f. (1.8)","content":"f ∈ L 2 (R d ) with supp f ⊂ [−N, N ] d . By interpolating (1.6) and (1.7), we see that the estimate (1.5) holds for 2(d+2) d ≤ p ≤ ∞ and is sharp in view of sharpness of (1.6) and (1.7). Note that the estimate (1.5) is scaling-invariant in the following sense. Consider the linear Schrödinger equation: i∂ t u − ∆u = 0 u t=0 = f. (1.8)"},{"id":"","caption":":= kT d = R/(kZ) d as a disjoint union of parallel translates of the original rational torus M = T d α α α with α −1 j k copies in the x j -direction. Now, consider the linear Schrödinger equation (1.8) on M = T d α α α . By periodic extension, we can view this problem on the scaled standard torus M = kT d . Given an initial condition f and the solution u(t) = e −it∆ f on M , let f and u denote their periodic extensions on M , respectively. By uniqueness of solutions to the linear Schrödinger equation, we see that u(t) = e −it∆ f on M . Clearly, the Strichartz estimates on the standard torus T d also hold on the scaled standard torus M = kT d , where the implicit constants further depend on k.With α α α = (α 1 , . . . , α d ), we havef L 2 x ( M ) = C(α α α) f L 2 x (M ) and u(t) L p x ( M ) = C(α α α, p) u(t) L p x (M ). Moreover, letting f and F [ f ] denote the Fourier coefficients of f on T d α α α and f on kT d , respectively, we have","content":"M\n:= kT d = R/(kZ) d as a disjoint union of parallel translates of the original rational torus M = T d α α α with α −1 j k copies in the x j -direction. Now, consider the linear Schrödinger equation (1.8) on M = T d α α α . By periodic extension, we can view this problem on the scaled standard torus M = kT d . Given an initial condition f and the solution u(t) = e −it∆ f on M , let f and u denote their periodic extensions on M , respectively. By uniqueness of solutions to the linear Schrödinger equation, we see that u(t) = e −it∆ f on M . Clearly, the Strichartz estimates on the standard torus T d also hold on the scaled standard torus M = kT d , where the implicit constants further depend on k.With α α α = (α 1 , . . . , α d ), we havef L 2 x ( M ) = C(α α α) f L 2 x (M ) and u(t) L p x ( M ) = C(α α α, p) u(t) L p x (M ). Moreover, letting f and F [ f ] denote the Fourier coefficients of f on T d α α α and f on kT d , respectively, we have"},{"id":"","caption":"Local well-posedness in subcritical spaces). Let d ≥ 2 and k ∈ N. Then, there exists s 0 = s 0 (k, d) such that the Cauchy problem (1.25) on a d-dimensional irrational torus T d is locally well-posed in H s (T d ) for s > s 0 in the following cases:","content":"Theorem 1. 4 (\n4Local well-posedness in subcritical spaces). Let d ≥ 2 and k ∈ N. Then, there exists s 0 = s 0 (k, d) such that the Cauchy problem (1.25) on a d-dimensional irrational torus T d is locally well-posed in H s (T d ) for s > s 0 in the following cases:"},{"id":"","caption":"When d = 2, Proposition 3.1 (i) and (ii) basically says that the level set estimates (3.4) and (3.5) are sufficient in proving the Strichartz estimates (1.19) and (1.20) for p > 6 as long as λ is large: λ ≥ N 1 4 + . Hence, an improvement on Theorem 1.2 when d = 2 may be obtained if we can improve the lower bound on λ in Proposition 3.1 (i) or the L 4 -Strichartz estimate (3.6).","content":"Remark 3. 2 .\n2When d = 2, Proposition 3.1 (i) and (ii) basically says that the level set estimates (3.4) and (3.5) are sufficient in proving the Strichartz estimates (1.19) and (1.20) for p > 6 as long as λ is large: λ ≥ N 1 4 + . Hence, an improvement on Theorem 1.2 when d = 2 may be obtained if we can improve the lower bound on λ in Proposition 3.1 (i) or the L 4 -Strichartz estimate (3.6)."},{"id":"","caption":"See Proposition 8 and the comment afterward in [7]. Combining Proposition 3.1 (i) and (3.7), we obtain (1.20) only for p ≥ 2(3d+1) d . When d = 2, the combination of Proposition 3.1 (i) and (3.6) yields a better result. When d ≥ 3, Proposition 2.2 yields better results. We point out that, when d = 3, combining Proposition 3.1 (i) with Theorem 1.1 (ii) yields another proof of Theorem 1.2 (ii.b). 3.2. Proof of Proposition 3.1 (i). Let η be a smooth cutoff function supported on [ 1 200 , 1 100 ]. Given q ∈ N, define J q by J q = {a ∈ N : 1 ≤ a ≤ q, (a, q) = 1}. (3.8) Then, for given M ∈ N with M ≥ N , we define Φ(t) = M ≤q<2M a∈Jq η q 2 t − a q","content":".\nSee Proposition 8 and the comment afterward in [7]. Combining Proposition 3.1 (i) and (3.7), we obtain (1.20) only for p ≥ 2(3d+1) d . When d = 2, the combination of Proposition 3.1 (i) and (3.6) yields a better result. When d ≥ 3, Proposition 2.2 yields better results. We point out that, when d = 3, combining Proposition 3.1 (i) with Theorem 1.1 (ii) yields another proof of Theorem 1.2 (ii.b). 3.2. Proof of Proposition 3.1 (i). Let η be a smooth cutoff function supported on [ 1 200 , 1 100 ]. Given q ∈ N, define J q by J q = {a ∈ N : 1 ≤ a ≤ q, (a, q) = 1}. (3.8) Then, for given M ∈ N with M ≥ N , we define Φ(t) = M ≤q<2M a∈Jq η q 2 t − a q"},{"id":"","caption":"{σ n } n∈Z be the multiplier defined by σ n = 1 on [−N, N ], σ n = N −j N for n = N + j and n = −N − j, j = 1, . . . , N , and σ n = 0 for |n| ≥ 2N . Consider K(x, t) := n∈Z σ n e 2πi(nx+n 2 t) .","content":"{σ n } n∈Z be the multiplier defined by σ n = 1 on [−N, N ], σ n = N −j N for n = N + j and n = −N − j, j = 1, . . . , N , and σ n = 0 for |n| ≥ 2N . Consider K(x, t) := n∈Z σ n e 2πi(nx+n 2 t) ."},{"id":"","caption":"≤2 s ≤N ω N,2 s . (3.24) Then, it follows that Ω M,N ≡ 1 on [− 1 M N , 1 M N ] and supp Ω M,N ⊂ [− 2 M N , 2 M N ]. Let N 1 = 1 100 N . Note that, for M 1 < M 2 ≤ N 1 , we have","content":"M\n≤2 s ≤N ω N,2 s . (3.24) Then, it follows that Ω M,N ≡ 1 on [− 1 M N , 1 M N ] and supp Ω M,N ⊂ [− 2 M N , 2 M N ]. Let N 1 = 1 100 N . Note that, for M 1 < M 2 ≤ N 1 , we have"},{"id":"","caption":"Here, we used the fact that d(k, M ) k 0+ and M ≤ 2 s ≤ N . Now, for each M and s, we choose a coefficient α M,s such that F T ∈R M δ T * ω N,2 s (0) = α M,s ρ(0).","content":"= 0 .\n0Here, we used the fact that d(k, M ) k 0+ and M ≤ 2 s ≤ N . Now, for each M and s, we choose a coefficient α M,s such that F T ∈R M δ T * ω N,2 s (0) = α M,s ρ(0)."},{"id":"","caption":"30) where S N = {n ∈ Z d : |n j | ≤ N, j = 2, . . . , N } and K(x 1 , t) is as in(3.20).Define Λ M,s byΛ M,s (x, t) = K(x, t) T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t) .(3.31) Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have","content":"30) where S N = {n ∈ Z d : |n j | ≤ N, j = 2, . . . , N } and K(x 1 , t) is as in(3.20).Define Λ M,s byΛ M,s (x, t) = K(x, t) T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t) .(3.31) Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have"},{"id":"","caption":"we estimate | Λ M,s |. Denote the second factor in (3.31) by Φ M,s , i.e. let Φ M,s (t) = T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t).","content":"we estimate | Λ M,s |. Denote the second factor in (3.31) by Φ M,s , i.e. let Φ M,s (t) = T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t)."},{"id":"","caption":"Φ M,s is periodic. Moreover, by (3.28), we have Φ M,s (0) = 0. Hence, we have Λ M,s (n, τ ) = σ n 1 d j=2 1 |n j |≤N k =0 Φ M,s (k) χ(τ − Q(n) − k). By summing | χ(τ − Q(n) − k)| τ − Q(n) − k −100 over k = 0, it follows from (3.35) and (3.36) that | Λ M,s (n, τ )| M 2 s N 1− . (3.37)","content":"= 0 .\n0Φ M,s is periodic. Moreover, by (3.28), we have Φ M,s (0) = 0. Hence, we have Λ M,s (n, τ ) = σ n 1 d j=2 1 |n j |≤N k =0 Φ M,s (k) χ(τ − Q(n) − k). By summing | χ(τ − Q(n) − k)| τ − Q(n) − k −100 over k = 0, it follows from (3.35) and (3.36) that | Λ M,s (n, τ )| M 2 s N 1− . (3.37)"},{"id":"","caption":"Lemma 3.47 in [3]). Let d(k, M ) denote the number of divisors of k less than M . Then, for any β, B, D > 0, we have #{0 ≤ k ≤ N : d(k, M ) > D} < c β,B (D −B M β N + M B ). (3.40) Note that the constant in (3.40) is independent of D > 0, M, N ∈ N. From (3.35) and (3.36), we have","content":"Lemma 3 . 5 (\n35Lemma 3.47 in [3]). Let d(k, M ) denote the number of divisors of k less than M . Then, for any β, B, D > 0, we have #{0 ≤ k ≤ N : d(k, M ) > D} < c β,B (D −B M β N + M B ). (3.40) Note that the constant in (3.40) is independent of D > 0, M, N ∈ N. From (3.35) and (3.36), we have"},{"id":"","caption":"from (3.41), (3.42), and (3.44) with M ≤ N , we have","content":"from (3.41), (3.42), and (3.44) with M ≤ N , we have"},{"id":"","caption":"57) where I 1 = [1, M 1 ], I 2 = (M 1 , M 2 ], and I 3 = (M 2 , N 1 ] with N 1 = 1 100 N as before. Then, summing (3.51) over dyadic M ≤ M 1 and 2 s ≤ N , we have","content":"57) where I 1 = [1, M 1 ], I 2 = (M 1 , M 2 ], and I 3 = (M 2 , N 1 ] with N 1 = 1 100 N as before. Then, summing (3.51) over dyadic M ≤ M 1 and 2 s ≤ N , we have"},{"id":"","caption":"choose D ∼ M α for some small α > 0. Then, set β ≪ 1 and B ≫ summing (3.59) over dyadic M ∈ (M 1 , M 2 ] and s with 2 s ≤ N , we have","content":"1\n1choose D ∼ M α for some small α > 0. Then, set β ≪ 1 and B ≫ summing (3.59) over dyadic M ∈ (M 1 , M 2 ] and s with 2 s ≤ N , we have"},{"id":"","caption":"Let s 0 > max(0, s c ) and I ⊂ R be a bounded interval. Suppose that the following multilinear Strichartz estimate holds for s > s 0 :","content":"Lemma 4 . 1 .\n41Let s 0 > max(0, s c ) and I ⊂ R be a bounded interval. Suppose that the following multilinear Strichartz estimate holds for s > s 0 :"},{"id":"","caption":". . . , k + 1, and N max := max(N 1 , · · · , N k+1 ). Then, the Cauchy problem (1.25) is locally well-posed in H s (T d ) for s > s 0 .","content":"j = 1 ,\n1. . . , k + 1, and N max := max(N 1 , · · · , N k+1 ). Then, the Cauchy problem (1.25) is locally well-posed in H s (T d ) for s > s 0 ."},{"id":"","caption":"U p (R;H) := inf λ ℓ 1 : (5.1) holds with λ = {λ j } j∈N and some U p -atoms a j .","content":"u\nU p (R;H) := inf λ ℓ 1 : (5.1) holds with λ = {λ j } j∈N and some U p -atoms a j ."},{"id":"","caption":"Let (d, p) be admissible and I ⊂ R be a bounded interval. Then, for all 1 ≤ M ≤ N and R ∈ R M (N ), we have","content":"Lemma 5 . 10 .\n510Let (d, p) be admissible and I ⊂ R be a bounded interval. Then, for all 1 ≤ M ≤ N and R ∈ R M (N ), we have"},{"id":"","caption":"(x, t) is as in(3.20). Let Λ M,s be as in(3.31). Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have|Λ M,s (x, * Λ M,s L ∞ (I×T d ) 2 s N 2 M f L 1 (I×T d ) . (B.7)","content":"f\n(x, t) is as in(3.20). Let Λ M,s be as in(3.31). Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have|Λ M,s (x, * Λ M,s L ∞ (I×T d ) 2 s N 2 M f L 1 (I×T d ) . (B.7)"},{"id":"","caption":"With M 1 and M 2 as in (3.56), write Λ = Λ 1 + Λ 2 + Λ 3 as before. See(3.57).By interpolating (B.7) and (3.39) and summing over dyadic M ≤ M 1 and 2 s ≤ N , we havef * Λ 1 L p ′ (I×T d ) M −1+(3+)θ 1 N 3−5θ f L p (I×T d ) , (B.9) since θ ∈ [ 1 3 , 1 2 ). Next, choose D ∼ M α for some small α = α(θ) > 0,and set β ≪ 1 and B ≫ 1 such that σ := −3 − β + αB− > 0. Then, by interpolating (B.7) and (3.45) and summing over dyadic M ∈ (M 1 , M 2 ] and 2 s ≤ N , we havef * Λ 2 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) L 1 (I×T d ) ,(B.10) as long as θ < 1 2+α+ . This can be guaranteed by choosing α = α(θ) > 0 sufficiently small for given θ <1 2 . Lastly, by interpolating (B.7) and (3.38) and summing over M ≥ M 2 ∼ N δ 2 and s with 2 s ≤ N , we have f * Λ 3 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) , (B.11)","content":"1 3 .f\n3With M 1 and M 2 as in (3.56), write Λ = Λ 1 + Λ 2 + Λ 3 as before. See(3.57).By interpolating (B.7) and (3.39) and summing over dyadic M ≤ M 1 and 2 s ≤ N , we havef * Λ 1 L p ′ (I×T d ) M −1+(3+)θ 1 N 3−5θ f L p (I×T d ) , (B.9) since θ ∈ [ 1 3 , 1 2 ). Next, choose D ∼ M α for some small α = α(θ) > 0,and set β ≪ 1 and B ≫ 1 such that σ := −3 − β + αB− > 0. Then, by interpolating (B.7) and (3.45) and summing over dyadic M ∈ (M 1 , M 2 ] and 2 s ≤ N , we havef * Λ 2 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) L 1 (I×T d ) ,(B.10) as long as θ < 1 2+α+ . This can be guaranteed by choosing α = α(θ) > 0 sufficiently small for given θ <1 2 . Lastly, by interpolating (B.7) and (3.38) and summing over M ≥ M 2 ∼ N δ 2 and s with 2 s ≤ N , we have f * Λ 3 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) , (B.11)"},{"id":"","caption":"Theorem 1.1. The Strichartz estimate on a irrational torus is known to hold","content":"Theorem 1.1. The Strichartz estimate on a irrational torus is known to hold"},{"id":"","caption":"the following cases:","content":"the following cases:"},{"id":"","caption":"Theorem 1.2. (i) The following scaling-invariant Strichartz estimate holds on an irrational torus:","content":"Theorem 1.2. (i) The following scaling-invariant Strichartz estimate holds on an irrational torus:"},{"id":"","caption":"Proof of Theorem 1.2 for d ≥ 3. Suppose that p ≥ max( 16 d +, 4), satisfying the hypothesis of Theorem 1.2 (i). By Bernstein's inequality and Proposition 2.2 (i), we have","content":"Proof of Theorem 1.2 for d ≥ 3. Suppose that p ≥ max( 16 d +, 4), satisfying the hypothesis of Theorem 1.2 (i). By Bernstein's inequality and Proposition 2.2 (i), we have"},{"id":"","caption":"Then, by Lemma 2.1 and Cauchy-Schwarz and Hölder's inequalities, we have","content":"Then, by Lemma 2.1 and Cauchy-Schwarz and Hölder's inequalities, we have"},{"id":"","caption":"If p > 8, then by Bernstein's inequality (in t), we have","content":"If p > 8, then by Bernstein's inequality (in t), we have11) \n\nyielding (2.10) N \n\nd \n\n4 − 2 \n\np . Hence, (2.4) follows from (2.9) in this case."},{"id":"","caption":"). This completes the proof of Proposition 2.2. 2.2. Two dimensional case. For d = 2, the sharp estimate (1.19) is not covered by Proposition 2.2. In the following, we use a simple duality argument and prove the sharp Strichartz estimate (1.19) for p > 12. Without loss of generality, we assume that","content":"). This completes the proof of Proposition 2.2. 2.2. Two dimensional case. For d = 2, the sharp estimate (1.19) is not covered by Proposition 2.2. In the following, we use a simple duality argument and prove the sharp Strichartz estimate (1.19) for p > 12. Without loss of generality, we assume that"},{"id":"","caption":"by making σ = σ(p ′ , δ 1 ) in (3.60) (and hence B = B(p ′ , δ 1 )) sufficiently large.Lastly, suppose that λ 2 |A λ | 2 III holds. By λ N","content":"69 )\n69by making σ = σ(p ′ , δ 1 ) in (3.60) (and hence B = B(p ′ , δ 1 )) sufficiently large.Lastly, suppose that λ 2 |A λ | 2 III holds. By λ Nd \n\n2 −ε and (3.56), we have"},{"id":"","caption":"for some p > 2. Indeed, if (5.25) and (5.26) hold, then it follows from Lemma 5.3 (ii) that (5.24) holds with δ ′ < δ.The first estimate (5.25) directly follows from Proposition 5.7 and Lemma 5.3 (i). Hence, it remains to prove the second estimate (5.26). Let p, p d,k , and q be as in the proof of Proposition 5.7. Then, by Hölder's inequality with (5.22), we have","content":"26 )\n26for some p > 2. Indeed, if (5.25) and (5.26) hold, then it follows from Lemma 5.3 (ii) that (5.24) holds with δ ′ < δ.The first estimate (5.25) directly follows from Proposition 5.7 and Lemma 5.3 (i). Hence, it remains to prove the second estimate (5.26). Let p, p d,k , and q be as in the proof of Proposition 5.7. Then, by Hölder's inequality with (5.22), we have"},{"id":"","caption":"From (5.17) and (5.19) with p d,k = (d + 2)k, we have q > p d,k > p > 2. Therefore, (5.26) follows from (5.27) with Remark 5.2 and (5.21).","content":"From (5.17) and (5.19) with p d,k = (d + 2)k, we have q > p d,k > p > 2. Therefore, (5.26) follows from (5.27) with Remark 5.2 and (5.21)."},{"id":"","caption":"Proposition B.1. Let d = 3 and I be a compact interval in R. Suppose that (1.27) holds. Then, the scaling-invariant Strichartz estimate (1.19) holds for p > 14 3 . The proof of Proposition B.1 is based on the following level set estimates under the assumption (1.27). Lemma B.2. Suppose that (1.27) holds. Given a compact interval I ⊂ R and f as in (3.2), let A λ = A λ (f ) be the distribution function defined by (3.3). (i) For any ε > 0, we have |A λ | N for λ N 1+ε . (ii) Let q > 4. Then, there exists small ε > 0 such that |A λ | N","content":"Proposition B.1. Let d = 3 and I be a compact interval in R. Suppose that (1.27) holds. Then, the scaling-invariant Strichartz estimate (1.19) holds for p > 14 3 . The proof of Proposition B.1 is based on the following level set estimates under the assumption (1.27). Lemma B.2. Suppose that (1.27) holds. Given a compact interval I ⊂ R and f as in (3.2), let A λ = A λ (f ) be the distribution function defined by (3.3). (i) For any ε > 0, we have |A λ | N for λ N 1+ε . (ii) Let q > 4. Then, there exists small ε > 0 such that |A λ | N1 \n\n1+4ε λ −4+ 8 \n\n1+4ε ε \n\n(B.1) \n\n3 \n\n2 q−5 λ −q \n(B.2) \n\nfor λ N \n\n3 \n\n2 −ε . \nIn (B."},{"id":"","caption":"B.4) for small ε 1 , ε 2 > 0. Choose M ≥ N such that N M 1+ε 1 ∼ λ 2 . This condition with M ≥ N implies that λ N 1+ ε 1 2 .Then, (B.4) yields","content":"B.4) for small ε 1 , ε 2 > 0. Choose M ≥ N such that N M 1+ε 1 ∼ λ 2 . This condition with M ≥ N implies that λ N 1+ ε 1 2 .Then, (B.4) yields|A λ | \nN \nλ 2 \n\n1−ε 2 \n\n1+ε 1 λ −2 N \n\n1 \n\n1+2ε 1 λ \n\n−4+ 4 \n\n1+2ε 1 \n\nε 1"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Background. The Cauchy problem of the nonlinear Schrödinger equation (NLS):\",\n \"content\": \".\\nBackground. The Cauchy problem of the nonlinear Schrödinger equation (NLS):\"\n },\n {\n \"id\": \"\",\n \"caption\": \"f ∈ L 2 (R d ) with supp f ⊂ [−N, N ] d . By interpolating (1.6) and (1.7), we see that the estimate (1.5) holds for 2(d+2) d ≤ p ≤ ∞ and is sharp in view of sharpness of (1.6) and (1.7). Note that the estimate (1.5) is scaling-invariant in the following sense. Consider the linear Schrödinger equation: i∂ t u − ∆u = 0 u t=0 = f. (1.8)\",\n \"content\": \"f ∈ L 2 (R d ) with supp f ⊂ [−N, N ] d . By interpolating (1.6) and (1.7), we see that the estimate (1.5) holds for 2(d+2) d ≤ p ≤ ∞ and is sharp in view of sharpness of (1.6) and (1.7). Note that the estimate (1.5) is scaling-invariant in the following sense. Consider the linear Schrödinger equation: i∂ t u − ∆u = 0 u t=0 = f. (1.8)\"\n },\n {\n \"id\": \"\",\n \"caption\": \":= kT d = R/(kZ) d as a disjoint union of parallel translates of the original rational torus M = T d α α α with α −1 j k copies in the x j -direction. Now, consider the linear Schrödinger equation (1.8) on M = T d α α α . By periodic extension, we can view this problem on the scaled standard torus M = kT d . Given an initial condition f and the solution u(t) = e −it∆ f on M , let f and u denote their periodic extensions on M , respectively. By uniqueness of solutions to the linear Schrödinger equation, we see that u(t) = e −it∆ f on M . Clearly, the Strichartz estimates on the standard torus T d also hold on the scaled standard torus M = kT d , where the implicit constants further depend on k.With α α α = (α 1 , . . . , α d ), we havef L 2 x ( M ) = C(α α α) f L 2 x (M ) and u(t) L p x ( M ) = C(α α α, p) u(t) L p x (M ). Moreover, letting f and F [ f ] denote the Fourier coefficients of f on T d α α α and f on kT d , respectively, we have\",\n \"content\": \"M\\n:= kT d = R/(kZ) d as a disjoint union of parallel translates of the original rational torus M = T d α α α with α −1 j k copies in the x j -direction. Now, consider the linear Schrödinger equation (1.8) on M = T d α α α . By periodic extension, we can view this problem on the scaled standard torus M = kT d . Given an initial condition f and the solution u(t) = e −it∆ f on M , let f and u denote their periodic extensions on M , respectively. By uniqueness of solutions to the linear Schrödinger equation, we see that u(t) = e −it∆ f on M . Clearly, the Strichartz estimates on the standard torus T d also hold on the scaled standard torus M = kT d , where the implicit constants further depend on k.With α α α = (α 1 , . . . , α d ), we havef L 2 x ( M ) = C(α α α) f L 2 x (M ) and u(t) L p x ( M ) = C(α α α, p) u(t) L p x (M ). Moreover, letting f and F [ f ] denote the Fourier coefficients of f on T d α α α and f on kT d , respectively, we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Local well-posedness in subcritical spaces). Let d ≥ 2 and k ∈ N. Then, there exists s 0 = s 0 (k, d) such that the Cauchy problem (1.25) on a d-dimensional irrational torus T d is locally well-posed in H s (T d ) for s > s 0 in the following cases:\",\n \"content\": \"Theorem 1. 4 (\\n4Local well-posedness in subcritical spaces). Let d ≥ 2 and k ∈ N. Then, there exists s 0 = s 0 (k, d) such that the Cauchy problem (1.25) on a d-dimensional irrational torus T d is locally well-posed in H s (T d ) for s > s 0 in the following cases:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"When d = 2, Proposition 3.1 (i) and (ii) basically says that the level set estimates (3.4) and (3.5) are sufficient in proving the Strichartz estimates (1.19) and (1.20) for p > 6 as long as λ is large: λ ≥ N 1 4 + . Hence, an improvement on Theorem 1.2 when d = 2 may be obtained if we can improve the lower bound on λ in Proposition 3.1 (i) or the L 4 -Strichartz estimate (3.6).\",\n \"content\": \"Remark 3. 2 .\\n2When d = 2, Proposition 3.1 (i) and (ii) basically says that the level set estimates (3.4) and (3.5) are sufficient in proving the Strichartz estimates (1.19) and (1.20) for p > 6 as long as λ is large: λ ≥ N 1 4 + . Hence, an improvement on Theorem 1.2 when d = 2 may be obtained if we can improve the lower bound on λ in Proposition 3.1 (i) or the L 4 -Strichartz estimate (3.6).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"See Proposition 8 and the comment afterward in [7]. Combining Proposition 3.1 (i) and (3.7), we obtain (1.20) only for p ≥ 2(3d+1) d . When d = 2, the combination of Proposition 3.1 (i) and (3.6) yields a better result. When d ≥ 3, Proposition 2.2 yields better results. We point out that, when d = 3, combining Proposition 3.1 (i) with Theorem 1.1 (ii) yields another proof of Theorem 1.2 (ii.b). 3.2. Proof of Proposition 3.1 (i). Let η be a smooth cutoff function supported on [ 1 200 , 1 100 ]. Given q ∈ N, define J q by J q = {a ∈ N : 1 ≤ a ≤ q, (a, q) = 1}. (3.8) Then, for given M ∈ N with M ≥ N , we define Φ(t) = M ≤q<2M a∈Jq η q 2 t − a q\",\n \"content\": \".\\nSee Proposition 8 and the comment afterward in [7]. Combining Proposition 3.1 (i) and (3.7), we obtain (1.20) only for p ≥ 2(3d+1) d . When d = 2, the combination of Proposition 3.1 (i) and (3.6) yields a better result. When d ≥ 3, Proposition 2.2 yields better results. We point out that, when d = 3, combining Proposition 3.1 (i) with Theorem 1.1 (ii) yields another proof of Theorem 1.2 (ii.b). 3.2. Proof of Proposition 3.1 (i). Let η be a smooth cutoff function supported on [ 1 200 , 1 100 ]. Given q ∈ N, define J q by J q = {a ∈ N : 1 ≤ a ≤ q, (a, q) = 1}. (3.8) Then, for given M ∈ N with M ≥ N , we define Φ(t) = M ≤q<2M a∈Jq η q 2 t − a q\"\n },\n {\n \"id\": \"\",\n \"caption\": \"{σ n } n∈Z be the multiplier defined by σ n = 1 on [−N, N ], σ n = N −j N for n = N + j and n = −N − j, j = 1, . . . , N , and σ n = 0 for |n| ≥ 2N . Consider K(x, t) := n∈Z σ n e 2πi(nx+n 2 t) .\",\n \"content\": \"{σ n } n∈Z be the multiplier defined by σ n = 1 on [−N, N ], σ n = N −j N for n = N + j and n = −N − j, j = 1, . . . , N , and σ n = 0 for |n| ≥ 2N . Consider K(x, t) := n∈Z σ n e 2πi(nx+n 2 t) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"≤2 s ≤N ω N,2 s . (3.24) Then, it follows that Ω M,N ≡ 1 on [− 1 M N , 1 M N ] and supp Ω M,N ⊂ [− 2 M N , 2 M N ]. Let N 1 = 1 100 N . Note that, for M 1 < M 2 ≤ N 1 , we have\",\n \"content\": \"M\\n≤2 s ≤N ω N,2 s . (3.24) Then, it follows that Ω M,N ≡ 1 on [− 1 M N , 1 M N ] and supp Ω M,N ⊂ [− 2 M N , 2 M N ]. Let N 1 = 1 100 N . Note that, for M 1 < M 2 ≤ N 1 , we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Here, we used the fact that d(k, M ) k 0+ and M ≤ 2 s ≤ N . Now, for each M and s, we choose a coefficient α M,s such that F T ∈R M δ T * ω N,2 s (0) = α M,s ρ(0).\",\n \"content\": \"= 0 .\\n0Here, we used the fact that d(k, M ) k 0+ and M ≤ 2 s ≤ N . Now, for each M and s, we choose a coefficient α M,s such that F T ∈R M δ T * ω N,2 s (0) = α M,s ρ(0).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"30) where S N = {n ∈ Z d : |n j | ≤ N, j = 2, . . . , N } and K(x 1 , t) is as in(3.20).Define Λ M,s byΛ M,s (x, t) = K(x, t) T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t) .(3.31) Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have\",\n \"content\": \"30) where S N = {n ∈ Z d : |n j | ≤ N, j = 2, . . . , N } and K(x 1 , t) is as in(3.20).Define Λ M,s byΛ M,s (x, t) = K(x, t) T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t) .(3.31) Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"we estimate | Λ M,s |. Denote the second factor in (3.31) by Φ M,s , i.e. let Φ M,s (t) = T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t).\",\n \"content\": \"we estimate | Λ M,s |. Denote the second factor in (3.31) by Φ M,s , i.e. let Φ M,s (t) = T ∈R M δ T * ω N,2 s (t) − α M,s ρ(t).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Φ M,s is periodic. Moreover, by (3.28), we have Φ M,s (0) = 0. Hence, we have Λ M,s (n, τ ) = σ n 1 d j=2 1 |n j |≤N k =0 Φ M,s (k) χ(τ − Q(n) − k). By summing | χ(τ − Q(n) − k)| τ − Q(n) − k −100 over k = 0, it follows from (3.35) and (3.36) that | Λ M,s (n, τ )| M 2 s N 1− . (3.37)\",\n \"content\": \"= 0 .\\n0Φ M,s is periodic. Moreover, by (3.28), we have Φ M,s (0) = 0. Hence, we have Λ M,s (n, τ ) = σ n 1 d j=2 1 |n j |≤N k =0 Φ M,s (k) χ(τ − Q(n) − k). By summing | χ(τ − Q(n) − k)| τ − Q(n) − k −100 over k = 0, it follows from (3.35) and (3.36) that | Λ M,s (n, τ )| M 2 s N 1− . (3.37)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Lemma 3.47 in [3]). Let d(k, M ) denote the number of divisors of k less than M . Then, for any β, B, D > 0, we have #{0 ≤ k ≤ N : d(k, M ) > D} < c β,B (D −B M β N + M B ). (3.40) Note that the constant in (3.40) is independent of D > 0, M, N ∈ N. From (3.35) and (3.36), we have\",\n \"content\": \"Lemma 3 . 5 (\\n35Lemma 3.47 in [3]). Let d(k, M ) denote the number of divisors of k less than M . Then, for any β, B, D > 0, we have #{0 ≤ k ≤ N : d(k, M ) > D} < c β,B (D −B M β N + M B ). (3.40) Note that the constant in (3.40) is independent of D > 0, M, N ∈ N. From (3.35) and (3.36), we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"from (3.41), (3.42), and (3.44) with M ≤ N , we have\",\n \"content\": \"from (3.41), (3.42), and (3.44) with M ≤ N , we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"57) where I 1 = [1, M 1 ], I 2 = (M 1 , M 2 ], and I 3 = (M 2 , N 1 ] with N 1 = 1 100 N as before. Then, summing (3.51) over dyadic M ≤ M 1 and 2 s ≤ N , we have\",\n \"content\": \"57) where I 1 = [1, M 1 ], I 2 = (M 1 , M 2 ], and I 3 = (M 2 , N 1 ] with N 1 = 1 100 N as before. Then, summing (3.51) over dyadic M ≤ M 1 and 2 s ≤ N , we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"choose D ∼ M α for some small α > 0. Then, set β ≪ 1 and B ≫ summing (3.59) over dyadic M ∈ (M 1 , M 2 ] and s with 2 s ≤ N , we have\",\n \"content\": \"1\\n1choose D ∼ M α for some small α > 0. Then, set β ≪ 1 and B ≫ summing (3.59) over dyadic M ∈ (M 1 , M 2 ] and s with 2 s ≤ N , we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let s 0 > max(0, s c ) and I ⊂ R be a bounded interval. Suppose that the following multilinear Strichartz estimate holds for s > s 0 :\",\n \"content\": \"Lemma 4 . 1 .\\n41Let s 0 > max(0, s c ) and I ⊂ R be a bounded interval. Suppose that the following multilinear Strichartz estimate holds for s > s 0 :\"\n },\n {\n \"id\": \"\",\n \"caption\": \". . . , k + 1, and N max := max(N 1 , · · · , N k+1 ). Then, the Cauchy problem (1.25) is locally well-posed in H s (T d ) for s > s 0 .\",\n \"content\": \"j = 1 ,\\n1. . . , k + 1, and N max := max(N 1 , · · · , N k+1 ). Then, the Cauchy problem (1.25) is locally well-posed in H s (T d ) for s > s 0 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"U p (R;H) := inf λ ℓ 1 : (5.1) holds with λ = {λ j } j∈N and some U p -atoms a j .\",\n \"content\": \"u\\nU p (R;H) := inf λ ℓ 1 : (5.1) holds with λ = {λ j } j∈N and some U p -atoms a j .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let (d, p) be admissible and I ⊂ R be a bounded interval. Then, for all 1 ≤ M ≤ N and R ∈ R M (N ), we have\",\n \"content\": \"Lemma 5 . 10 .\\n510Let (d, p) be admissible and I ⊂ R be a bounded interval. Then, for all 1 ≤ M ≤ N and R ∈ R M (N ), we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(x, t) is as in(3.20). Let Λ M,s be as in(3.31). Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have|Λ M,s (x, * Λ M,s L ∞ (I×T d ) 2 s N 2 M f L 1 (I×T d ) . (B.7)\",\n \"content\": \"f\\n(x, t) is as in(3.20). Let Λ M,s be as in(3.31). Then, from Lemma 3.4, (3.26), and (3.29) with M ≤ 2 s ≤ N , we have|Λ M,s (x, * Λ M,s L ∞ (I×T d ) 2 s N 2 M f L 1 (I×T d ) . (B.7)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"With M 1 and M 2 as in (3.56), write Λ = Λ 1 + Λ 2 + Λ 3 as before. See(3.57).By interpolating (B.7) and (3.39) and summing over dyadic M ≤ M 1 and 2 s ≤ N , we havef * Λ 1 L p ′ (I×T d ) M −1+(3+)θ 1 N 3−5θ f L p (I×T d ) , (B.9) since θ ∈ [ 1 3 , 1 2 ). Next, choose D ∼ M α for some small α = α(θ) > 0,and set β ≪ 1 and B ≫ 1 such that σ := −3 − β + αB− > 0. Then, by interpolating (B.7) and (3.45) and summing over dyadic M ∈ (M 1 , M 2 ] and 2 s ≤ N , we havef * Λ 2 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) L 1 (I×T d ) ,(B.10) as long as θ < 1 2+α+ . This can be guaranteed by choosing α = α(θ) > 0 sufficiently small for given θ <1 2 . Lastly, by interpolating (B.7) and (3.38) and summing over M ≥ M 2 ∼ N δ 2 and s with 2 s ≤ N , we have f * Λ 3 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) , (B.11)\",\n \"content\": \"1 3 .f\\n3With M 1 and M 2 as in (3.56), write Λ = Λ 1 + Λ 2 + Λ 3 as before. See(3.57).By interpolating (B.7) and (3.39) and summing over dyadic M ≤ M 1 and 2 s ≤ N , we havef * Λ 1 L p ′ (I×T d ) M −1+(3+)θ 1 N 3−5θ f L p (I×T d ) , (B.9) since θ ∈ [ 1 3 , 1 2 ). Next, choose D ∼ M α for some small α = α(θ) > 0,and set β ≪ 1 and B ≫ 1 such that σ := −3 − β + αB− > 0. Then, by interpolating (B.7) and (3.45) and summing over dyadic M ∈ (M 1 , M 2 ] and 2 s ≤ N , we havef * Λ 2 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) L 1 (I×T d ) ,(B.10) as long as θ < 1 2+α+ . This can be guaranteed by choosing α = α(θ) > 0 sufficiently small for given θ <1 2 . Lastly, by interpolating (B.7) and (3.38) and summing over M ≥ M 2 ∼ N δ 2 and s with 2 s ≤ N , we have f * Λ 3 L p ′ (I×T d ) N 3−5θ f L p (I×T d ) , (B.11)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 1.1. The Strichartz estimate on a irrational torus is known to hold\",\n \"content\": \"Theorem 1.1. The Strichartz estimate on a irrational torus is known to hold\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the following cases:\",\n \"content\": \"the following cases:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 1.2. (i) The following scaling-invariant Strichartz estimate holds on an irrational torus:\",\n \"content\": \"Theorem 1.2. (i) The following scaling-invariant Strichartz estimate holds on an irrational torus:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Proof of Theorem 1.2 for d ≥ 3. Suppose that p ≥ max( 16 d +, 4), satisfying the hypothesis of Theorem 1.2 (i). By Bernstein's inequality and Proposition 2.2 (i), we have\",\n \"content\": \"Proof of Theorem 1.2 for d ≥ 3. Suppose that p ≥ max( 16 d +, 4), satisfying the hypothesis of Theorem 1.2 (i). By Bernstein's inequality and Proposition 2.2 (i), we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Then, by Lemma 2.1 and Cauchy-Schwarz and Hölder's inequalities, we have\",\n \"content\": \"Then, by Lemma 2.1 and Cauchy-Schwarz and Hölder's inequalities, we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"If p > 8, then by Bernstein's inequality (in t), we have\",\n \"content\": \"If p > 8, then by Bernstein's inequality (in t), we have11) \\n\\nyielding (2.10) N \\n\\nd \\n\\n4 − 2 \\n\\np . Hence, (2.4) follows from (2.9) in this case.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"). This completes the proof of Proposition 2.2. 2.2. Two dimensional case. For d = 2, the sharp estimate (1.19) is not covered by Proposition 2.2. In the following, we use a simple duality argument and prove the sharp Strichartz estimate (1.19) for p > 12. Without loss of generality, we assume that\",\n \"content\": \"). This completes the proof of Proposition 2.2. 2.2. Two dimensional case. For d = 2, the sharp estimate (1.19) is not covered by Proposition 2.2. In the following, we use a simple duality argument and prove the sharp Strichartz estimate (1.19) for p > 12. Without loss of generality, we assume that\"\n },\n {\n \"id\": \"\",\n \"caption\": \"by making σ = σ(p ′ , δ 1 ) in (3.60) (and hence B = B(p ′ , δ 1 )) sufficiently large.Lastly, suppose that λ 2 |A λ | 2 III holds. By λ N\",\n \"content\": \"69 )\\n69by making σ = σ(p ′ , δ 1 ) in (3.60) (and hence B = B(p ′ , δ 1 )) sufficiently large.Lastly, suppose that λ 2 |A λ | 2 III holds. By λ Nd \\n\\n2 −ε and (3.56), we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"for some p > 2. Indeed, if (5.25) and (5.26) hold, then it follows from Lemma 5.3 (ii) that (5.24) holds with δ ′ < δ.The first estimate (5.25) directly follows from Proposition 5.7 and Lemma 5.3 (i). Hence, it remains to prove the second estimate (5.26). Let p, p d,k , and q be as in the proof of Proposition 5.7. Then, by Hölder's inequality with (5.22), we have\",\n \"content\": \"26 )\\n26for some p > 2. Indeed, if (5.25) and (5.26) hold, then it follows from Lemma 5.3 (ii) that (5.24) holds with δ ′ < δ.The first estimate (5.25) directly follows from Proposition 5.7 and Lemma 5.3 (i). Hence, it remains to prove the second estimate (5.26). Let p, p d,k , and q be as in the proof of Proposition 5.7. Then, by Hölder's inequality with (5.22), we have\"\n },\n {\n \"id\": \"\",\n \"caption\": \"From (5.17) and (5.19) with p d,k = (d + 2)k, we have q > p d,k > p > 2. Therefore, (5.26) follows from (5.27) with Remark 5.2 and (5.21).\",\n \"content\": \"From (5.17) and (5.19) with p d,k = (d + 2)k, we have q > p d,k > p > 2. Therefore, (5.26) follows from (5.27) with Remark 5.2 and (5.21).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Proposition B.1. Let d = 3 and I be a compact interval in R. Suppose that (1.27) holds. Then, the scaling-invariant Strichartz estimate (1.19) holds for p > 14 3 . The proof of Proposition B.1 is based on the following level set estimates under the assumption (1.27). Lemma B.2. Suppose that (1.27) holds. Given a compact interval I ⊂ R and f as in (3.2), let A λ = A λ (f ) be the distribution function defined by (3.3). (i) For any ε > 0, we have |A λ | N for λ N 1+ε . (ii) Let q > 4. Then, there exists small ε > 0 such that |A λ | N\",\n \"content\": \"Proposition B.1. Let d = 3 and I be a compact interval in R. Suppose that (1.27) holds. Then, the scaling-invariant Strichartz estimate (1.19) holds for p > 14 3 . The proof of Proposition B.1 is based on the following level set estimates under the assumption (1.27). Lemma B.2. Suppose that (1.27) holds. Given a compact interval I ⊂ R and f as in (3.2), let A λ = A λ (f ) be the distribution function defined by (3.3). (i) For any ε > 0, we have |A λ | N for λ N 1+ε . (ii) Let q > 4. Then, there exists small ε > 0 such that |A λ | N1 \\n\\n1+4ε λ −4+ 8 \\n\\n1+4ε ε \\n\\n(B.1) \\n\\n3 \\n\\n2 q−5 λ −q \\n(B.2) \\n\\nfor λ N \\n\\n3 \\n\\n2 −ε . \\nIn (B.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"B.4) for small ε 1 , ε 2 > 0. Choose M ≥ N such that N M 1+ε 1 ∼ λ 2 . This condition with M ≥ N implies that λ N 1+ ε 1 2 .Then, (B.4) yields\",\n \"content\": \"B.4) for small ε 1 , ε 2 > 0. Choose M ≥ N such that N M 1+ε 1 ∼ λ 2 . This condition with M ≥ N implies that λ N 1+ ε 1 2 .Then, (B.4) yields|A λ | \\nN \\nλ 2 \\n\\n1−ε 2 \\n\\n1+ε 1 λ −2 N \\n\\n1 \\n\\n1+2ε 1 λ \\n\\n−4+ 4 \\n\\n1+2ε 1 \\n\\nε 1\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"On the multilinear restriction and Kakeya conjectures. J Bennett, A Carbery, T Tao, Acta Math. 1962J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261-302.","doi":""},{"text":"On Λ(p)-subsets of squares. J Bourgain, Israel J. Math. 673J. Bourgain, On Λ(p)-subsets of squares, Israel J. Math. 67 (1989), no. 3, 291-311.","doi":""},{"text":"Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. J Bourgain, Geom. Funct. Anal. 3J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107-156.","doi":""},{"text":"Exponential sums and nonlinear Schrödinger equations. J Bourgain, Geom. Funct. Anal. 3J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal. 3 (1993), 157-178.","doi":""},{"text":"Nonlinear Schrödinger equations. J Bourgain, Hyperbolic equations and frequency interactions. Park City, UT; Providence, RIAmer. Math. Soc5J. Bourgain, Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions (Park City, UT, 1995), 3-157, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999.","doi":""},{"text":"On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations. J Bourgain, Ann. of Math. Stud. 163Princeton Univ. PressJ. Bourgain, On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations, 1-20, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.","doi":""},{"text":"Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. J Bourgain, Israel J. Math. 1931J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel J. Math. 193 (2013), no. 1, 441-458.","doi":""},{"text":"Bounds on oscillatory integral operators based on multilinear estimates. J Bourgain, L Guth, Geom. Funct. Anal. 216J. Bourgain, L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239-1295.","doi":""},{"text":"Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. N Burq, P Gérard, N Tzvetkov, Amer. J. Math. 1263N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569-605.","doi":""},{"text":"Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. N Burq, P Gérard, N Tzvetkov, Invent. Math. 1591N. Burq, P. Gérard, N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), no. 1, 187-223.","doi":""},{"text":"Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. F Catoire, W.-M Wang, Commun. Pure Appl. Anal. 9F. Catoire, W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal. 9 (2010), 483-491.","doi":""},{"text":"Semilinear Schrödinger equations. T Cazenave, Courant Lecture Notes in Mathematics. 10Courant Institute of Mathematical SciencesT. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York;","doi":""},{"text":"The Cauchy problem for the critical nonlinear Schrödinger equation in H s. T Cazenave, F Weissler, Nonlinear Anal. 1410T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 (1990), no. 10, 807-836.","doi":""},{"text":"C Demeter, arXiv:1401.1873Incidence theory and restriction estimates. math.CAC. Demeter, Incidence theory and restriction estimates, arXiv:1401.1873 [math.CA].","doi":""},{"text":"Local well-posedness for 2-d Schrödinger equation on irrational tori and bounds on Sobolev norms. S Demirbas, arXiv:1307.0051math.APS. Demirbas, Local well-posedness for 2-d Schrödinger equation on irrational tori and bounds on Sobolev norms, arXiv:1307.0051 [math.AP].","doi":""},{"text":"On the global Cauchy problem for some nonlinear Schrödinger equations. J Ginibre, G Velo, Ann. Inst. H. Poincaré Anal. Non Linéaire. 14J. Ginibre, G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 309-323.","doi":""},{"text":"Smoothing properties and retarded estimates for some dispersive evolution equations. J Ginibre, G Velo, Comm. Math. Phys. 1441J. Ginibre, G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations. Comm. Math. Phys. 144 (1992), no. 1, 163-188.","doi":""},{"text":"Erratum to \"Well-posedness and scattering for the KP-II equation in a critical space. M Hadac, S Herr, H Koch, Ann. Inst. H. Poincaré Anal. Non Linéaire. 263Ann. Inst. H. Poincaré Anal. Non LinéaireM. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, 917-941. Erratum to \"Well-posedness and scattering for the KP-II equation in a critical space\", Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 3, 971-972.","doi":""},{"text":"An introduction to the theory of numbers. G Hardy, E Wright, Oxford University PressNew YorkFifth edition. xvi+426 ppG. Hardy, E. Wright, An introduction to the theory of numbers, Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp.","doi":""},{"text":"The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds. S Herr, Amer. J. Math. 1355S. Herr. The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds, Amer. J. Math. 135 (2013), no. 5, 1271-1290.","doi":""},{"text":"Global well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in H 1 (T 3 ). S Herr, D Tataru, N Tzvetkov, Duke Math. J. 159S. Herr, D. Tataru, N. Tzvetkov, Global well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in H 1 (T 3 ), Duke Math. J., 159 (2011) 329-349.","doi":""},{"text":"Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications. S Herr, D Tataru, N Tzvetkov, doi10.1515/crelle-2012-0013J. Reine Angew. Math. S. Herr, D. Tataru, N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications, J. Reine Angew. Math. doi 10.1515/crelle-2012-0013.","doi":""},{"text":"Y Hu, X Li, arXiv:1108.5164v1Discrete Fourier restriction associated with Schrödinger Equations. Y. Hu, X. Li, Discrete Fourier restriction associated with Schrödinger Equations, arXiv:1108.5164v1.","doi":""},{"text":"Exponential sums and lattice points. M Huxley, III, Proc. London Math. Soc. 3M. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591-609.","doi":""},{"text":"Analytic number theory. H Iwaniec, E Kowalski, American Mathematical Society53Providence, RIxii+615 ppH. Iwaniec, E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publi- cations, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp.","doi":""},{"text":"The energy-critical defocusing NLS on T 3 , Duke Math. A Ionescu, B Pausader, J. 1618A. Ionescu, B. Pausader, The energy-critical defocusing NLS on T 3 , Duke Math. J. 161 (2012), no. 8, 1581-1612.","doi":""},{"text":"V Jarník, Über die Gitterpunkte auf konvexen Kurven. 24V. Jarník,Über die Gitterpunkte auf konvexen Kurven, (German) Math. Z. 24 (1926), no. 1, 500-518.","doi":""},{"text":"On nonlinear Schrödinger equations. T Kato, Ann. Inst. H. Poincaré Phys. Théor. 461T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113-129.","doi":""},{"text":"Endpoint Strichartz estimates. M Keel, T Tao, Amer. J. Math. 1205M. Keel, T. Tao, Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), no. 5, 955-980.","doi":""},{"text":"A priori bounds for the 1D cubic NLS in negative Sobolev spaces. H Koch, D Tataru, Art. ID rnm053. 36ppH. Koch, D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. (2007), no. 16, Art. ID rnm053, 36 pp.","doi":""},{"text":"H Montgomery, CBMS Regional Conference Series in Mathematics, 84. Published for the Conference Board of the Mathematical Sciences. Washington, DC; Providence, RIAmerican Mathematical SocietyTen lectures on the interface between analytic number theory and harmonic analysis. xiv+220 ppH. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. xiv+220 pp.","doi":""},{"text":". B Pausader, N Tzvetkov, X Wang, arXiv:1210.3842Global regularity for the energy-critical NLS on S. 3to appear in Ann. Inst. H. Poincaré Anal. Non LinéaireB. Pausader, N. Tzvetkov, X. Wang, Global regularity for the energy-critical NLS on S 3 , arXiv:1210.3842 [mathAP], to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.","doi":""},{"text":"An asymptotic formula for the number of solutions of a system of equations. N Rogovskaya, Moskov. Gos. Univ. Russian) Diophantine approximations, Part II (RussianN. Rogovskaya, An asymptotic formula for the number of solutions of a system of equations, (Russian) Diophantine approximations, Part II (Russian), 78-84, Moskov. Gos. Univ., Moscow, 1986.","doi":""},{"text":"Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. R S Strichartz, Duke Math. J. 443R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), no. 3, 705-714.","doi":""},{"text":"N Strunk, arXiv:1401.6080Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions. math.APN. Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions arXiv:1401.6080 [math.AP].","doi":""},{"text":"The nonlinear Schrödinger equation. Self-focusing and wave collapse. C Sulem, P L Sulem, Applied Mathematical Sciences. 139Springer-VerlagC. Sulem, P.L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. xvi+350 pp.","doi":""},{"text":"On 2D nonlinear Schrödinger equations with data on R × T. H Takaoka, N Tzvetkov, J. Funct. Anal. 1822H. Takaoka, N. Tzvetkov, On 2D nonlinear Schrödinger equations with data on R × T, J. Funct. Anal. 182 (2001), no. 2, 427-442.","doi":""},{"text":"T Tao, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences. Washington, DC; Providence, RIAmerican Mathematical SocietyNonlinear dispersive equations. Local and global analysis. xvi+373 ppT. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp.","doi":""},{"text":"L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups. Y Tsutsumi, Funkcial. Ekvac. 301Y. Tsutsumi, L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115-125.","doi":""},{"text":"The Hardy-Littlewood method. R Vaughan, Cambridge Tracts in Mathematics. 125Cambridge University PressSecond editionR. Vaughan, The Hardy-Littlewood method. Second edition, Cambridge Tracts in Mathematics, 125. Cambridge University Press, Cambridge, 1997. xiv+232 pp.","doi":""},{"text":"The method of trigonometrical sums in the theory of numbers, Translated from the Russian. I M Vinogradov, K. F. Roth and Anne DavenportDover Publications, IncMineola, NYReprint of the 1954 translation. x+180 ppI.M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport. Reprint of the 1954 translation. Dover Publications, Inc., Mineola, NY, 2004. x+180 pp.","doi":""},{"text":"Periodic nonlinear Schrödinger equation in critical H s (T n ) spaces. Y Wang, SIAM J. Math. Anal. 453Y. Wang, Periodic nonlinear Schrödinger equation in critical H s (T n ) spaces, SIAM J. Math. Anal. 45 (2013), no. 3, 1691-1703.","doi":""},{"text":"T Wolff, Lectures on harmonic analysis. Providence, RIAmerican Mathematical Society29T. Wolff, Lectures on harmonic analysis, University Lecture Series, 29. American Mathematical Society, Providence, RI, 2003. x+137 pp.","doi":""},{"text":"Existence of solutions for Schrödinger evolution equations. K Yajima, Comm. Math. Phys. 1103K. Yajima, Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys. 110 (1987), no. 3, 415-426.","doi":""},{"text":". Zihua Guo, BeijingSchool of Mathematical Sciences, Peking UniversityChina E-mail address: zihuaguo@math.pku.edu.cnZihua Guo, School of Mathematical Sciences, Peking University, Beijing 100871, China E-mail address: zihuaguo@math.pku.edu.cn","doi":""},{"text":"and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings. Mayfield Road, Edinburgh, EH9 3JZ, Scotland; Fine Hall, Washington Rd., Princeton, NJ 08544-1000, USA E-mailTadahiro Oh, School of Mathematics, The University of Edinburgh ; and Department of Mathematics, Princeton UniversityTadahiro Oh, School of Mathematics, The University of Edinburgh, and The Maxwell In- stitute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, and Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ 08544-1000, USA E-mail address: hiro.oh@ed.ac.uk","doi":""}],"string":"[\n {\n \"text\": \"On the multilinear restriction and Kakeya conjectures. J Bennett, A Carbery, T Tao, Acta Math. 1962J. Bennett, A. Carbery, T. Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261-302.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On Λ(p)-subsets of squares. J Bourgain, Israel J. Math. 673J. Bourgain, On Λ(p)-subsets of squares, Israel J. Math. 67 (1989), no. 3, 291-311.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. J Bourgain, Geom. Funct. Anal. 3J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107-156.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Exponential sums and nonlinear Schrödinger equations. J Bourgain, Geom. Funct. Anal. 3J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal. 3 (1993), 157-178.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonlinear Schrödinger equations. J Bourgain, Hyperbolic equations and frequency interactions. Park City, UT; Providence, RIAmer. Math. Soc5J. Bourgain, Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions (Park City, UT, 1995), 3-157, IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations. J Bourgain, Ann. of Math. Stud. 163Princeton Univ. PressJ. Bourgain, On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations, 1-20, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. J Bourgain, Israel J. Math. 1931J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel J. Math. 193 (2013), no. 1, 441-458.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bounds on oscillatory integral operators based on multilinear estimates. J Bourgain, L Guth, Geom. Funct. Anal. 216J. Bourgain, L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239-1295.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. N Burq, P Gérard, N Tzvetkov, Amer. J. Math. 1263N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569-605.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. N Burq, P Gérard, N Tzvetkov, Invent. Math. 1591N. Burq, P. Gérard, N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 (2005), no. 1, 187-223.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. F Catoire, W.-M Wang, Commun. Pure Appl. Anal. 9F. Catoire, W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal. 9 (2010), 483-491.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Semilinear Schrödinger equations. T Cazenave, Courant Lecture Notes in Mathematics. 10Courant Institute of Mathematical SciencesT. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Cauchy problem for the critical nonlinear Schrödinger equation in H s. T Cazenave, F Weissler, Nonlinear Anal. 1410T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 (1990), no. 10, 807-836.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Demeter, arXiv:1401.1873Incidence theory and restriction estimates. math.CAC. Demeter, Incidence theory and restriction estimates, arXiv:1401.1873 [math.CA].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Local well-posedness for 2-d Schrödinger equation on irrational tori and bounds on Sobolev norms. S Demirbas, arXiv:1307.0051math.APS. Demirbas, Local well-posedness for 2-d Schrödinger equation on irrational tori and bounds on Sobolev norms, arXiv:1307.0051 [math.AP].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the global Cauchy problem for some nonlinear Schrödinger equations. J Ginibre, G Velo, Ann. Inst. H. Poincaré Anal. Non Linéaire. 14J. Ginibre, G. Velo, On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 309-323.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Smoothing properties and retarded estimates for some dispersive evolution equations. J Ginibre, G Velo, Comm. Math. Phys. 1441J. Ginibre, G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations. Comm. Math. Phys. 144 (1992), no. 1, 163-188.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Erratum to \\\"Well-posedness and scattering for the KP-II equation in a critical space. M Hadac, S Herr, H Koch, Ann. Inst. H. Poincaré Anal. Non Linéaire. 263Ann. Inst. H. Poincaré Anal. Non LinéaireM. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 3, 917-941. Erratum to \\\"Well-posedness and scattering for the KP-II equation in a critical space\\\", Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 3, 971-972.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An introduction to the theory of numbers. G Hardy, E Wright, Oxford University PressNew YorkFifth edition. xvi+426 ppG. Hardy, E. Wright, An introduction to the theory of numbers, Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979. xvi+426 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds. S Herr, Amer. J. Math. 1355S. Herr. The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds, Amer. J. Math. 135 (2013), no. 5, 1271-1290.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Global well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in H 1 (T 3 ). S Herr, D Tataru, N Tzvetkov, Duke Math. J. 159S. Herr, D. Tataru, N. Tzvetkov, Global well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in H 1 (T 3 ), Duke Math. J., 159 (2011) 329-349.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications. S Herr, D Tataru, N Tzvetkov, doi10.1515/crelle-2012-0013J. Reine Angew. Math. S. Herr, D. Tataru, N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications, J. Reine Angew. Math. doi 10.1515/crelle-2012-0013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Y Hu, X Li, arXiv:1108.5164v1Discrete Fourier restriction associated with Schrödinger Equations. Y. Hu, X. Li, Discrete Fourier restriction associated with Schrödinger Equations, arXiv:1108.5164v1.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Exponential sums and lattice points. M Huxley, III, Proc. London Math. Soc. 3M. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591-609.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Analytic number theory. H Iwaniec, E Kowalski, American Mathematical Society53Providence, RIxii+615 ppH. Iwaniec, E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publi- cations, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The energy-critical defocusing NLS on T 3 , Duke Math. A Ionescu, B Pausader, J. 1618A. Ionescu, B. Pausader, The energy-critical defocusing NLS on T 3 , Duke Math. J. 161 (2012), no. 8, 1581-1612.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V Jarník, Über die Gitterpunkte auf konvexen Kurven. 24V. Jarník,Über die Gitterpunkte auf konvexen Kurven, (German) Math. Z. 24 (1926), no. 1, 500-518.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On nonlinear Schrödinger equations. T Kato, Ann. Inst. H. Poincaré Phys. Théor. 461T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113-129.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Endpoint Strichartz estimates. M Keel, T Tao, Amer. J. Math. 1205M. Keel, T. Tao, Endpoint Strichartz estimates. Amer. J. Math. 120 (1998), no. 5, 955-980.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A priori bounds for the 1D cubic NLS in negative Sobolev spaces. H Koch, D Tataru, Art. ID rnm053. 36ppH. Koch, D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. (2007), no. 16, Art. ID rnm053, 36 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"H Montgomery, CBMS Regional Conference Series in Mathematics, 84. Published for the Conference Board of the Mathematical Sciences. Washington, DC; Providence, RIAmerican Mathematical SocietyTen lectures on the interface between analytic number theory and harmonic analysis. xiv+220 ppH. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. xiv+220 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Pausader, N Tzvetkov, X Wang, arXiv:1210.3842Global regularity for the energy-critical NLS on S. 3to appear in Ann. Inst. H. Poincaré Anal. Non LinéaireB. Pausader, N. Tzvetkov, X. Wang, Global regularity for the energy-critical NLS on S 3 , arXiv:1210.3842 [mathAP], to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An asymptotic formula for the number of solutions of a system of equations. N Rogovskaya, Moskov. Gos. Univ. Russian) Diophantine approximations, Part II (RussianN. Rogovskaya, An asymptotic formula for the number of solutions of a system of equations, (Russian) Diophantine approximations, Part II (Russian), 78-84, Moskov. Gos. Univ., Moscow, 1986.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. R S Strichartz, Duke Math. J. 443R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44 (1977), no. 3, 705-714.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N Strunk, arXiv:1401.6080Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions. math.APN. Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions arXiv:1401.6080 [math.AP].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The nonlinear Schrödinger equation. Self-focusing and wave collapse. C Sulem, P L Sulem, Applied Mathematical Sciences. 139Springer-VerlagC. Sulem, P.L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. xvi+350 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On 2D nonlinear Schrödinger equations with data on R × T. H Takaoka, N Tzvetkov, J. Funct. Anal. 1822H. Takaoka, N. Tzvetkov, On 2D nonlinear Schrödinger equations with data on R × T, J. Funct. Anal. 182 (2001), no. 2, 427-442.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"T Tao, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences. Washington, DC; Providence, RIAmerican Mathematical SocietyNonlinear dispersive equations. Local and global analysis. xvi+373 ppT. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups. Y Tsutsumi, Funkcial. Ekvac. 301Y. Tsutsumi, L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115-125.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Hardy-Littlewood method. R Vaughan, Cambridge Tracts in Mathematics. 125Cambridge University PressSecond editionR. Vaughan, The Hardy-Littlewood method. Second edition, Cambridge Tracts in Mathematics, 125. Cambridge University Press, Cambridge, 1997. xiv+232 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The method of trigonometrical sums in the theory of numbers, Translated from the Russian. I M Vinogradov, K. F. Roth and Anne DavenportDover Publications, IncMineola, NYReprint of the 1954 translation. x+180 ppI.M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport. Reprint of the 1954 translation. Dover Publications, Inc., Mineola, NY, 2004. x+180 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Periodic nonlinear Schrödinger equation in critical H s (T n ) spaces. Y Wang, SIAM J. Math. Anal. 453Y. Wang, Periodic nonlinear Schrödinger equation in critical H s (T n ) spaces, SIAM J. Math. Anal. 45 (2013), no. 3, 1691-1703.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"T Wolff, Lectures on harmonic analysis. Providence, RIAmerican Mathematical Society29T. Wolff, Lectures on harmonic analysis, University Lecture Series, 29. American Mathematical Society, Providence, RI, 2003. x+137 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Existence of solutions for Schrödinger evolution equations. K Yajima, Comm. Math. Phys. 1103K. Yajima, Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys. 110 (1987), no. 3, 415-426.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Zihua Guo, BeijingSchool of Mathematical Sciences, Peking UniversityChina E-mail address: zihuaguo@math.pku.edu.cnZihua Guo, School of Mathematical Sciences, Peking University, Beijing 100871, China E-mail address: zihuaguo@math.pku.edu.cn\",\n \"doi\": \"\"\n },\n {\n \"text\": \"and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings. Mayfield Road, Edinburgh, EH9 3JZ, Scotland; Fine Hall, Washington Rd., Princeton, NJ 08544-1000, USA E-mailTadahiro Oh, School of Mathematics, The University of Edinburgh ; and Department of Mathematics, Princeton UniversityTadahiro Oh, School of Mathematics, The University of Edinburgh, and The Maxwell In- stitute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland, and Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ 08544-1000, USA E-mail address: hiro.oh@ed.ac.uk\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":28,"cells":{"text":{"kind":"string","value":"\nToric Symplectic Geometry and Full Spark Frames\nJul 2022\n\nTom Needham \nClayton Shonkwiler \nDepartment of Mathematics\nColorado State University\nFort CollinsCO\n\n\nDepartment of Mathematics\nFlorida State University\nTallahasseeFL\n\nToric Symplectic Geometry and Full Spark Frames\nJul 2022\nThe collection of d × N complex matrices with prescribed column norms and prescribed (nonzero) singular values forms a compact algebraic variety, which we refer to as a frame space. Elements of frame spaces-i.e., frames-are used to give robust representations of complex-valued signals, so that geometrical and measure-theoretic properties of frame spaces are of interest to the signal processing community. This paper is concerned with the following question: what is the probability that a frame drawn uniformly at random from a given frame space has the property that any subset of d of its columns gives a basis for C d ? We show that the probability is one, generalizing recent work of Cahill, Mixon, and Strawn. To prove this, we first show that frame spaces are related to highly structured objects called toric symplectic manifolds. This relationship elucidates the geometric meaning of eigensteps-certain spectral invariants of a frame-and should be a more broadly applicable tool for studying probabilistic questions about the structure of frame spaces. As another application of our symplectic perspective, we completely characterize the norm and spectral data for which the corresponding frame space has singularities, answering some open questions in the frame theory literature.\n\nIntroduction\n\nA frame in a Hilbert space (H,⟨⋅,⋅⟩) is traditionally defined as a collection { f i } i∈I of vectors in H so that for all v ∈ H we have\na v 2 ≤ i∈I ⟨v, f i ⟩ 2 ≤ b v 2(1)\nfor some numbers 0 < a ≤ b called frame bounds. In this paper, we focus on finite frames in complex Hilbert spaces, in which case H = C d for some integer d > 0 with its standard Hermitian inner product and norm, I = {1,...,N} is finite, and the above condition is equivalent to the collection { f 1 ,... , f N } being a spanning set for C d . Interest in finite frames is largely due to their application to robust signal representation. Modeling a signal as an element of a Hilbert space H, a frame allows one to take a sequence of \"measurements\" by recording the inner product of the signal with each of the frame vectors. This signal representation is more robust to noise in the signal or random erasures of measurements than a similar measurement scheme associated to an orthonormal basis, at least when the frame has certain properties [11,23,29]. These desirable properties for a frame are typically expressed as prescriptions for the norms of the frame vectors and for the spectrum of an operator associated to the frame, which we describe in more detail below. The collection of frames with prescribed norm and spectral data is easily seen to define an algebraic variety, and there is interest in the geometric structure of these frame varieties [8,18,49]. This paper explores the geometry of frame varieties through the lens of symplectic geometry. Symplectic geometry is a subfield of differential geometry that studies manifolds endowed with a certain geometric structure-to keep the paper accessible to the broader frame theory community, we have aimed to give self-contained expositions of the relevant ideas from symplectic geometry, but we will not go into details here in the introduction.\n\nThe main contributions of this paper are as follows:\n\n• Our main theorem, Theorem 1.1, says that with probability 1 every size-d subset of a random frame in C d with prescribed norms and spectral data is a basis. Frames satisfying this non-degeneracy condition are called full spark frames. This generalizes the complex case of Theorem 1.6 from Cahill, Mixon, and Strawn's paper [8], which established the genericity of the full spark condition for frames whose frame bounds are equal (this can be stated as a spectral constraint) and whose frame vectors are all unit-this is referred to as the space of finite unit-norm tight frames or FUNTFs.\n\n• We show in Theorem 2.25 that the space of frames with prescribed norm and spectral data is a smooth manifold, for generic choices of this data. In fact, we give necessary and sufficient conditions on the norms and eigenvalues which guarantee that the corresponding frame variety is smooth, and we describe the local geometry of singular varieties near their singular points. This theorem (together with surrounding results) generalizes work of Dykema and Strawn [18], which once again specializes to the space of FUNTFs, and answers generalizations of two open questions posed in [8].\n\n• Both of the previous two results are proved by novel applications of ideas from symplectic geometry to frame theory. Throughout the course of the paper, we show that many spaces of complex frames have natural interpretations from the symplectic point of view. In particular, we show in Theorem 3.20 that each space of frames with prescribed spectral and norm data has a dense open subset which projects onto a highly structured geometric object called a toric symplectic manifold. This geometric structure has measure-theoretic implications-we use it to prove our full spark theorem, but expect that it will be a useful tool for the future study of probability theory on frame spaces. Theorem 3.20 generalizes work of Flaschka and Millson [19], which is written in the context of pure symplectic geometry and makes no references to frames. The theorem also gives a symplectic interpretation of the frame-theoretic concept of eigensteps, introduced by Cahill et al. [7], and an auxiliary result used in its proof generalizes a theorem of Haga and Pegel [27].\n\nTo describe our results in detail, we now introduce more precise terminology and notation.\n\nNotation and statement of the main theorem. Let F d,N be the space of frames of N vectors in C d . Identifying a frame { f i } N i=1 ∈ F d,N with the d × N matrix whose columns are the f i represented in the standard basis, the space F d,N can be viewed as an open, dense subset of the space C d×N of d × N complex matrices.\n\nWhen a = b in (1), the frame satisfies a scaled Parseval identity\nN i=1 ⟨v, f i ⟩ 2 = a v 2 ,\nand such frames are called a-tight (or just tight); 1-tight frames are called Parseval frames.\n\nEach frame { f i } N i=1 ∈ F d,N has three related operators, which can be interpreted in terms of the d × N matrix F = f 1 f 2 ... f N :\n\n1. The analysis operator C d → C N is defined by v ↦ (⟨v, f 1 ⟩,... ,⟨v, f N ⟩) , or equivalently v ↦ F * v, where F * is the Hermitian adjoint (i.e., conjugate transpose) of F; 2. The synthesis operator C N → C d is defined by\n(w 1 ,... ,w N ) ↦ N i=1 w i f i , or equivalently w ↦ Fw;\n3. The frame operator C d → C d is the composition of the analysis and synthesis operators; i.e., v ↦ FF * v.\n\nA simple calculation shows that a frame is a-tight if and only if its frame operator is aI d , where I d is the identity map on C d .\n\nFrame operators are always Hermitian and positive-definite, so they have spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, which we will call the frame spectrum. If λ λ λ = (λ 1 ,...,λ d ), we will use F d,N λ λ λ to indicate the frames with frame spectrum λ λ λ . Notice that the a-tight frames are uniquely specified by their frame spectra: F d,N (a,...,a) is the space of all a-tight frames. In addition to specifying a frame operator, we also often want to fix the (squared) norms of the individual frame vectors. We can always permute the labels on the frame vectors, so it will be convenient in what follows to assume the norms are sorted in decreasing order. If r r r = (r 1 ,... ,r N ) is a non-increasing list of positive numbers r 1 ≥ ⋅⋅⋅ ≥ r N > 0, 1 1 We could also allow some of the r i to be zero, but this would complicate some statements below to no apparent benefit. we will use F d,N (r r r) to indicate the space of frames with f i 2 = r i . It is especially common to require that all the frame vectors have the same norm: f i 2 = r > 0 for all i, in which case the frame is an equal-norm frame; if r = 1, this is a unit-norm frame.\n\nIn general, the frame norms determine the trace of the frame operator:\nN i=1 r i = N i=1 f i 2 = tr F * F = tr FF * = d i=1 λ i ,(2)\nby the cyclic invariance of trace. Hence, for tight frames the frame operator must be r r r 2 d I d and a unit-norm tight frame must have λ i = N d for all i. If r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) with r 1 ≥ ⋅⋅⋅ ≥ r N > 0 and λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, we will use F d,N λ λ λ (r r r) ∶= F d,N λ λ λ ∩ F d,N (r r r) to denote the space of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ . This space has a natural probability measure: the (normalized) Hausdorff measure it inherits as a compact subset of C d×N . Equation (2) is not the only restriction imposed on the frame vector norms by the frame operator: the partial sums of the squared frame vector norms must be bounded above by the partial sums of the eigenvalues of the frame operator [12,19]. More precisely, there exists a frame { f i } N i=1 with f i 2 = r i > 0 and frame spectrum λ λ λ if and only if (2) holds and, for all k = 1,... ,d,\nk i=1 r i ≤ k i=1 λ i .(3)\nGiven λ λ λ , we will call a non-increasing list r r r of positive numbers satisfying (2) and (3) λ λ λ -admissible, and if all inequalities in (3) are strict we call the list strongly λ λ λ -admissible. This terminology comes from Casazza and Leon [12], who showed that F d,N λ λ λ (r r r) is non-empty if and only if r r r is λ λ λ -admissible. Although the terminology does not appear in the work of Casazza and Leon, in general one says that the vector λ λ λ majorizes the vector r r r in the case that the relations (3) hold [39].\nFor { f i } N i=1 ∈ F d,N , the spark of { f i } N i=1\nis the size of the smallest linearly dependent subset [16]. The spark is bounded above by d + 1, and a frame with spark equal to d + 1 is called a full-spark frame. Equivalently, a frame in F d,N is full spark if and only if all of its size-d subsets are bases. Full spark frames are often desirable, for example because they provide unique reconstructions of the largest possible class of sparse signals [2,16].\n\nWe are now ready to state our main theorem:\nTheorem 1.1. Suppose N > d ≥ 1.\nLet r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) be nonincreasing lists of positive real numbers. There are three mutually exclusive possibilities for the space F d,N λ λ λ (r r r) of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ :\n\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\n\n2. If r r r is λ λ λ -admissible but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\n\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\n\nIn particular, since r r r = (1,. Structure of the paper. Section 2 begins with an exposition of the relevant ideas and tools from symplectic geometry. These tools are then applied to show that various spaces of frames with prescribed data have symplectic structures. Then, parameters λ λ λ and r r r for which the frame variety F d,N λ λ λ (r r r) is a smooth manifold are characterized. Section 3 is devoted to showing that the space F d,N λ λ λ (r r r) has a dense open subset which projects onto a toric symplectic manifold (we define and describe the basic properties of such an object in Section 3.1). The main theorem is proved in Section 4, as an application of the toric symplectic structure from the previous section. An alternative proof using related tools from algebraic geometry is also sketched. The paper concludes with a brief discussion of future directions in Section 5.\n\n\nSymplectic Structure of Frame Varieties\n\nIn this section, we describe the symplectic structures on the frame varieties of interest; i.e., spaces of frames with prescribed frame spectra and/or norms. It has been previously observed in the literature that certain parameter choices may lead to frame spaces with singularities [8,18,49]. We give a precise characterization of this phenomenon and describe the local structure of singular points later in this section.\n\n\nSymplectic Geometry\n\nWe begin with a review of some concepts of symplectic geometry, with a focus on various notions of reducing a symplectic manifold by a Lie group action. We use [40] as our main reference for the basics of symplectic manifolds. The ideas we present are standard in the field of symplectic geometry; this subsection is mainly intended as a quick reference for non-experts and to standardize our notation.\n\nA symplectic form on a (smooth, real) manifold M is a closed, nondegenerate 2-form ω. For a point p ∈ M and tangent vectors X,Y ∈ T p M, we write ω p (X,Y ) ∈ R for the evaluation of ω on the vectors. The closedness condition means that the exterior derivative dω is identically zero and the nondegeneracy condition means that for every nonzero X ∈ T p M there exists Y ∈ T p M such that ω p (X,Y ) ≠ 0. A manifold endowed with a symplectic form is called a symplectic manifold, denoted (M,ω) or simply as M when the symplectic form is understood to be fixed. A simple argument shows that if a manifold M admits a symplectic form, then it must be even dimensional (over the reals).\n\nExample 2.1 (Complex n-Space). The prototypical example of a symplectic manifold is n-dimensional complex space C n , which is considered as a 2n-dimensional real manifold via the natural identification C n ≈ R 2n . For any n-tuple p of complex numbers, there is a natural isomorphism T p C n ≈ C n . Coordinates (x 1 + √ −1y 1 ,... ,x n + √ −1y n ) for C n correspond to real coordinates (x 1 ,... ,x n ,y 1 ,... ,y n ) in which a symplectic form is given by\nω = dx 1 ∧ dy 1 + ⋯ + dx n ∧ dy n .(4)\nThis is referred to as the standard symplectic form on C n . In complex coordinates, (4) is expressed concretely for p ∈ C n and Z = (z 1 ,... ,z n ),\nW = (w 1 ,...,w n ) ∈ T p C n ≈ C n by ω p (Z,W ) = −Im (w 1 z 1 + ⋯ + w n z n ) = −Im (W * Z) = −Im⟨Z,W ⟩ ,\nwhere Im denotes the imaginary part of a complex number and ⟨⋅,⋅⟩ is the standard Hermitian inner product on C n . The most important example of a symplectic manifold for our purposes is complex matrix space C d×N , which is just a reshaped version of the space C d⋅N described above. For a matrix F ∈ C d×N , we have a natural identification\nT F C d×N ≈ C d×N . The canonical symplectic form on C d×N is defined as ω F (X,Y ) = −Im tr(Y * X).\nThis is just a transformation of the canonical symplectic form on C d⋅N under the reshaping map.\n\nSince the space of frames F d,N ⊂ C d×N is an open submanifold, the standard symplectic structure on C d×N restricts to make F d,N a symplectic manifold.\n\nIn fact, every 2n-dimensional symplectic manifold is locally equivalent to C n with the standard symplectic form. Let us now make this statement precise. If Ψ ∶ N → M is a smooth map from a manifold N to a symplectic manifold (M,ω), the pullback form Ψ * ω on N is defined by\n(Ψ * ω) p (X,Y ) = ω Ψ(p) (DΨ(p)(X),DΨ(p)(Y)),\nfor X,Y ∈ T p N and where DΨ(p) ∶ T p N → T Ψ(p) M denotes the derivative of Ψ at p. If (N,η) is also a symplectic manifold and Ψ is a diffeomorphism with the property that Ψ * ω = η, then we say Ψ is a symplectomorphism and that (N,η) and (M,ω) are symplectomorphic. A fundamental result of symplectic geometry is Darboux's Theorem [40,Theorem 3.2.2]: every point in a 2n-dimensional symplectic manifold has an open neighborhood U such that the symplectic manifold (U,ω U ) is symplectomorphic to C n with the standard symplectic form.\n\nAn important aspect of symplectic geometry is the study of interactions between symplectic structures and certain group actions on their manifolds. Let G be a Lie group with Lie algebra g and suppose that G acts on a manifold M endowed with a symplectic form ω. For p ∈ M and g ∈ G, let g ⋅ p ∈ M denote the action of g on p. To each ξ ∈ g, one associates an infinitesimal vector field Y ξ on M via the formula\nY ξ p ∶= d dε ε=0 exp(εξ ) ⋅ p, where exp ∶ g → G is the exponential map of G. A map Φ ∶ M → g * ,\nwhere g * denotes the dual to g, is called a momentum map for the G-action if its derivative interacts with the symplectic form as follows. Let DΦ(p) ∶ T p M → T Φ(p) g * denote the derivative of Φ at p ∈ M. Then for each X ∈ T p M, DΦ(p)(X) ∈ T Φ(p) g * ≈ g * , where we use the natural isomorphism coming from the fact that g * is a vector space. Then DΦ(p)(X) ∶ g → R, and for each ξ ∈ g we require\nDΦ(p)(X)(ξ ) = ω p (Y ξ p ,X).\nWe also require that the momentum map Φ is equivariant, in the following sense. Recall that the adjoint action of G on g is defined, for each g ∈ G, by the map Ad g ∶ g → g which is the derivative at the identity of the conjugation map h ↦ ghg −1 . The corresponding coadjoint action of G on the dual Lie algebra g * is defined, for each g ∈ G, by the map Ad * g ∶ g * → g * given by Ad * g (χ)(ξ ) ∶= χ(Ad g −1 (ξ )). When G is a matrix group, both the adjoint and coadjoint actions can be interpreted as conjugation actions. The momentum map is required to be equivariant with respect to the given G-action on M and the coadjoint action on g * . Explicitly, this means that, for each g ∈ G and each p ∈ M,\nAd * g (Φ(p)) = Φ(g ⋅ p).\nIf a G-action admits a momentum map, then we say the action is Hamiltonian. Hamiltonian actions give the appropriate setting for performing a quotient operation in the symplectic category. [38,41]). Let (M,ω) be a symplectic manifold with a Hamiltonian G-action, let Φ ∶ M → g * be a momentum map for the action and let χ ∈ g * be a regular value such that G acts freely on the level set Φ −1 (χ). Then the manifold\nM χ G ∶= Φ −1 (χ) G,\ncalled the symplectic reduction or symplectic quotient over χ, admits a symplectic structure ω red which is uniquely characterized by the equation\nq * ω red = ι * ω, where q ∶ Φ −1 (χ) → M χ G is the quotient map and ι ∶ Φ −1 (χ) → M is the inclusion map.\nSee [5, Example III.2.18] for a construction of the complex projective space CP n−1 as a symplectic reduction C n ξ U(1), where U(1) acts on C n by scalar multiplication. Notice that, in the statement of Theorem 2.2, we required that G acts freely on the level set Φ −1 (χ). In particular, G acts on this level set, meaning that for any p ∈ M with Φ(p) = χ, it must be the case that Φ(g ⋅ p) = χ for all g ∈ G.\n\nBy the equivariance required in the definition of the momentum map Φ, this implies that Ad * g (χ) = χ for all g ∈ G; in other words, χ must be a fixed point of the coadjoint action of G. Conversely, G will act on the level set over any fixed point of the coadjoint action.\n\nMore generally, when χ ∈ g * is not a fixed point of the coadjoint action, we can still take a symplectic reduction over the coadjoint orbit of χ, defined to be the set\nO χ ∶= Ad * g (χ) g ∈ G .\nIn this case the equivariance of the momentum map ensures that G acts on Φ −1 (O χ ). [38,41]). Let M, ω, G and Φ be as above and let χ ∈ g * be a regular value. If the action of G on Φ −1 (O χ ) is free, then the manifold\n\n\nTheorem 2.3 (Marsden-Weinstein-Meyer Theorem for Coadjoint Orbits\nM O χ G ∶= Φ −1 (O χ ) G\nadmits a symplectic structure ω red which is uniquely characterized by q * ω red = ι * ω, where q denotes the projection\nmap Φ −1 (O χ ) → Φ −1 (O χ ) G and ι denotes the inclusion map Φ −1 (O χ ) → M.\nIt is straightforward to show that when χ ∈ g * is a regular value, the action of G on Φ −1 (O χ ) is at worst locally free. Theorem 2.3 easily generalizes to this setting, with the only modification being that M O χ G is a symplectic orbifold rather than a symplectic manifold.\n\nPerhaps surprisingly, if we are more concerned with the \"symplectic\" part of this statement than the \"manifold\" (or \"orbifold\") part, Sjamaar and Lerman [48] have shown that χ being a regular value is not an essential hypothesis in Theorems 2.2 and 2.3. When χ is a singular value of Φ then Φ −1 (O χ ) is not a manifold, but it nonetheless admits a G action and the quotient still admits a natural symplectic structure.\n\nTheorem 2.4 (Sjamaar-Lerman [48]). Let M, ω, G and Φ be as above and let χ ∈ g * . The symplectic reduction\nM O χ G ∶= Φ −1 (O χ ) G is a symplectic stratified space with symplectic structure ω red characterized by q * ω red = ι * ω, where ι again denotes the inclusion map Φ −1 (O χ ) → M.\nRoughly speaking, M O χ G being a symplectic stratified space means that it is the disjoint union of symplectic manifolds and that these manifolds fit together nicely. Somewhat remarkably, each connected component contains a unique manifold piece which is open and dense: Therefore, up to sets of lower dimension, the symplectic reduction even over a singular value of a proper momentum map is a connected symplectic manifold.\n\n\nFrame Spaces as Symplectic Quotients\n\nVarious spaces of frames can be endowed with natural symplectic structures via Theorems 2.3 and 2.4. To be precise, the symplectic manifolds of interest are actually quotients of frame spaces by certain symmetry groups. Throughout this subsection, fix dimensions d and N and a frame spectrum λ λ λ = (λ 1 ,...,λ d ) with λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0.\n\n\nFrames with Prescribed Frame Spectrum\n\nThe unitary group acts on F d,N λ λ λ by left matrix multiplication. We then consider the quotient space F d,N λ λ λ U(d), which has a natural symplectic structure, as we will show below.\n\nBefore stating the result, we observe that the space H(d) of d × d Hermitian matrices may be identified with the dual Lie algebra u(d) * via the isomorphism α ∶ H(d) → u(d) * taking ξ to the linear functional α ξ ∶ u(d) → R, defined by\nα ξ (η) = √ −1 2 tr(η * ξ ) = ⟨ √ −1 2 ξ ,η⟩.(5)\nWe begin with a preliminary result about this isomorphism. We define a U(d)-action on H(d) which we (with a slight abuse of terminology) refer to as the adjoint action:\nAd ∶ U(d) × H(d) → H(d) (A,ξ ) ↦ Ad A (ξ ) ∶= Aξ A * .\nLemma 2.7. The map α defined in (5) is equivariant with respect to the adjoint action on H(d) and the coadjoint action on u(d) * . That is, α Ad A (ξ ) = Ad * A α ξ . It follows that, under the map α, the coadjoint orbit of any element of u(d) * is identified with the collection of Hermitian matrices with a fixed spectrum.\n\nProof. The first part is a straightforward computation: for η ∈ u(d),\nα Ad A (ξ ) (η) = √ −1 2 tr(η * Aξ A * ) = √ −1 2 tr((A * ηA) * ξ ) = α ξ (A * ηA) = Ad * A α ξ (η).\nAny Hermitian matrix ξ with spectrum λ λ λ can be expressed via its eigendecomposition as ξ = Ad A (diag(λ λ λ )) for some A ∈ U(d), so that the second part of the claim follows by equivariance.\n\nBased on the second part of the lemma, we use O λ λ λ ⊂ H(d) to denote the set of matrices with fixed spectrum λ λ λ . We now describe the symplectic structure on the space of frames with prescribed frame spectrum.\n\n\nProposition 2.8 ( [43]). The action of U(d) on C d×N by left multiplication is Hamiltonian with momentum map\nΦ U(d) ∶ C d×N → H(d) ≈ u(d) * F ↦ −FF * . It follows that the space F d,N λ λ λ U(d) is the symplectic quotient C d×N O −λ λ λ U(d).\nIn particular, it has a natural symplectic structure.\n\nProof Sketch. We sketch the proof, since the constructions involved will be useful later. For details, see our previous paper [43].\n\nThe fact that Φ U(d) ∶ F ↦ −FF * is a momentum map (where we are specifically fixing the isomorphism (5)) follows by a computation, as shown in [43,Proposition 2]. We remark that our momentum map differs by a sign in the present paper, due to slightly different conventions. The fact that\nF d,N λ λ λ = Φ −1 U(d) (O −λ λ λ )\nfollows by Lemma 2.7. Moreover, the negative-definite Hermitian matrices are the regular values of Φ U(d) in H(d); we proved this by slightly tedious but essentially straightforward calculation in [43], but it also follows easily from Proposition 2.20 below, since the only unitary matrix which acts trivially on a spanning set of C d is the identity matrix. Therefore, the entries in λ λ λ being positive means that Theorem 2.3 applies and\nC d×N O −λ λ λ U(d) = Φ −1 U(d) (O −λ λ λ ) U(d) = F d,N λ λ λ U(d)\nis a symplectic manifold.\n\n\nIdentification with a Coadjoint Orbit\n\nThe symplectic structure of F d,N λ λ λ U(d) can alternatively be realized by identifying the frame space with a certain coadjoint orbit in the larger space of Hermitian matrices H(N). This perspective will be useful in the following subsection. In order to state the result precisely, we introduce some more terminology.\n\nIt is well known that any coadjoint orbit has a natural symplectic structure, called the Kirillov-Kostant-Souriau (KKS) symplectic form [5,II.3.c]), which we denote generically as ω KKS . Indeed, for an arbitrary Lie group G with Lie algebra g, the tangent space to a coadjoint orbit O β at β ∈ g * consists of vectors of the form ad * ξ β , where ξ ∈ g, and where ad * ξ is the coadjoint representation of g on g * , obtained as the derivative of the map Ad * ∶ G → Aut(g * ) at the identity. Then the KKS form is defined by\nω KKS β (ad * ξ β ,ad * ξ ′ β ) ∶= β ([ξ ,ξ ′ ]),(6)\nwhere [⋅,⋅] is the Lie bracket on g.\n\nBy another slight abuse of terminology, we define the adjoint representation of u(d) on H(d) by\nad ∶ u(d) × H(d) → H(d) (η,ξ ) ↦ ad η (ξ ) ∶= ηξ − ξ η.\nAn argument similar to that of Lemma 2.7 proves the following.\n\nLemma 2.9. The map α defined in (5) is equivariant with respect to the adjoint representation of u(d) on H(d) and its coadjoint representation on u(d) * . That is,\nα adη (ξ ) = ad * η α ξ .\nThe adjoint representation of u(d) on H(d) is obtained by differentiating the adjoint action of U(d) on H(d). It follows that the tangent space to O −λ λ λ ⊂ H(d) at ξ consists of vectors of the form ad η (ξ ) for η ∈ u(d).\nLemma 2.10. The pullback of ω KKS to O −λ λ λ ⊂ H(d) via α is given by α * ω KKS ξ (ad η (ξ ),ad η ′ (ξ )) = Im tr(ξ ηη ′ ). Proof. We have α * ω KKS ξ (ad η (ξ ),ad η ′ (ξ )) = ω KKS α ξ (Dα(ξ )(ad η (ξ )),Dα(ξ )(ad η ′ (ξ ))) = ω KKS α ξ (α ad η (ξ ) ,α ad η ′ (ξ ) ) = ω KKS α ξ (ad * η α ξ ,ad * η ′ α ξ ) = α ξ ([η,η ′ ]) = √ −1 2 tr((ηη ′ − η ′ η) * ξ ) = √ −1 2 ⋅ −2 √ −1 ⋅ Im tr(ξ ηη ′ ) = Im tr(ξ ηη ′ ),\nwhere the second-to-last line uses cyclic invariance and linearity of trace.\n\nWe can now describe precisely how F d,N λ λ λ U(d) is equivalent to a coadjoint orbit in H(N).\n\n\nProposition 2.11 ( [43]\n\n). Let λ λ λ be a frame spectrum and defineλ λ λ ∶= (λ 1 ,... ,λ d ,0,...,0) ∈ R N to be λ λ λ padded with\nN − d zeros. The space F d,N λ λ λ U(d) is diffeomorphic to Oλ λ λ ⊂ H(N), via the map [F] ↦ F * F (7)\ntaking a unitary equivalence class to its Gram matrix. Moreover, the map (7) is a symplectomorphism with respect to the reduced symplectic structure on F d,N λ λ λ U(d) and the pullback of the Kirillov-Kostant-Souriau form α * ω KKS described in Lemma 2.10.\n\nThe proof of the proposition makes use of the following lemma.\nLemma 2.12. Let [F] ∈ F d,N λ λ λ U(d). The tangent space T [F] F d,N λ λ λ U(d) is naturally isomorphic to the vector space {Fζ ζ ∈ u(N)} {ξ F ξ ∈ u(d)}. Proof. Any F ∈ F d,N λ λ λ has singular value decomposition of the form F = UΣ λ λ λ V * , where Σ λ λ λ ∶= diag(λ λ λ ) 1 2 0 . Therefore F d,N λ λ λ = {UΣ λ λ λ V * U ∈ U(d) and V ∈ U(N)}. We claim that T F F d,N λ λ λ = {ξ F − Fζ ξ ∈ u(d) and ζ ∈ u(N)}. Indeed, a smooth path F t in F d,N λ λ λ with F 0 = F = UΣ λ λ λ V * can be expressed as F t = U t Σ λ λ λ V * t for some smooth paths U t in U(d) with U 0 = U and V t ∈ U(N) with V 0 = V . Then the derivative of F t at t = 0 satisfieṡ F 0 =U 0 Σ λ λ λ V * 0 +U 0 Σ λ λ λV * 0 = ξUΣ λ λ λ V * −UΣ λ λ λ V * ζ = ξ F − Fζ ,\nfor ξ ∈ u(d) and ζ ∈ u(N) satisfyingU 0 = ξU andV 0 = ζV , respectively. By general principles of quotient manifolds, there is a natural isomorphism\nT [F] F d,N λ λ λ U(d) ≈ T F F d,N λ λ λ T F (U(d) ⋅ F). Indeed, the quotient map F d,N λ λ λ → F d,N λ λ λ U(d)\nis a submersion, so the rank theorem [37,Theorem 4.12] implies that, locally near F, it is an orthogonal projection with the fiber U(d) ⋅ F being sent to the origin. This gives\nT F F d,N λ λ λ ≈ T F (U(d) ⋅ F) ⊕ T [F ] F d,N λ λ λ U(d) , and the claim follows. Since T F (U(d) ⋅ F) = {ξ F ξ ∈ u(d)}, the lemma follows.\nProof of Proposition 2.11. The diffeomorphism result is Proposition 3 in our previous paper [43], so we only sketch the details here. Unitary equivalence classes of frames are uniquely determined by their Gram matrices (see, for example, [50, §3.4]), so we can identify F d,N λ λ λ U(d) with the elements of H(N) which arise as Gram matrices of frames in F d,N λ λ λ . Of course, the frame operator FF * and the Gram matrix F * F have the same nonzero eigenvalues, so specifying the spectrum of the frame operator also determines the spectrum of the Gram matrix by padding with N − d zeros, and it follows that this subset consists of those N × N Hermitian matrices with spectrumλ λ λ . In other words, (7) is a diffeomorphism between F d,N λ λ λ U(d) and Oλ λ λ . It remains to prove the symplectomorphism claim. For the rest of this proof, let γ denote the map (7). By Lemma 2.12, the tangent space to F d,N\nλ λ λ U(d) at [F] consists of equivalence classes (under the quotient operation in the lemma) [Fζ ], where ζ ∈ u(N). The derivative of γ at [F] in the direction [Fζ ] is given by Dγ([F])([Fζ ]) = F * Fζ + (Fζ ) * F = F * Fζ − ζ F * F = ad ζ (F * F).\nThen, using Lemma 2.10,\nγ * α * ω KKS [F ] (Fζ ,Fζ ′ ) = α * ω KKS F * F (ad ζ (F * F),ad ζ ′ (F * F)) = Im tr(F * Fζ ζ ′ ).\nOn the other hand, the reduced symplectic form on F d,N\nλ λ λ U(d) evaluates as −Im tr((Fζ ′ ) * (Fζ )) = −Im tr(F * Fζ (ζ ′ ) * ) = Im tr(F * Fζ ζ ′ ),\nwhere we have used the characterization of the reduced form, cyclic invariance of trace and the fact that ζ ′ is skew-Hermitian.\n\nRemark 2.13. Since Oλ λ λ ⊂ H(N) is identified with a coadjoint orbit, it is diffeomorphic, like all coadjoint orbits, to a flag manifold [5, §II.1.d]. Specifically, if λ λ λ consists of ℓ distinct eigenvalues with multiplicities k 1 ,... ,k ℓ and\nd i = k 1 + ⋅⋅⋅ + k i for i = 1,...,ℓ, then Oλ λ λ ≈ Fℓ(d 1 ,...,d ℓ ,N),\nthe flag manifold whose elements are nested sequences\nV 1 ⊂ ⋅⋅⋅ ⊂ V ℓ ⊂ V ℓ+1 = C N of subspaces with dimV i = d i .\nIn particular, when λ λ λ is constant then ℓ = 1 and k 1 = d, so the flag manifold is Fℓ(d,N), whose elements are all subspaces\nV 1 ⊂ C N with dimV 1 = d; that is, the Grassmannian Gr d (C N ) of d-dimensional subspaces of C N .\n\nFrames with Prescribed Frame Spectrum and Norms\n\nThe torus U(1) N can be realized as the subgroup of diagonal elements of U(N); that is, as the standard maximal torus T ≤ U(N). Thought of in this way, U(1) N ≈ T acts on C d×N by right matrix multiplication and, as we will see in a moment, this action preserves the frame spectrum, and hence restricts to an action on F d,N λ λ λ .\n\nFirst, though, it is worthwhile to pause and think about the right action of the full unitary group U(N) on C d×N . If we try to define the action of A ∈ U(N) on F ∈ C d×N by A ⋅ F = FA we quickly run into problems: after all, if\nA 1 ,A 2 ∈ U(N), then A 1 ⋅ (A 2 ⋅ F) = A 1 ⋅ (FA 2 ) = FA 2 A 1 ≠ FA 1 A 2 = (A 1 A 2 ) ⋅ F,\nunless A 1 and A 2 happen to commute. This is not an issue at the level of T, which is abelian, but it is preferable to define the T action to be consistent with an honest U(N) action. We get such an action by defining\nA ⋅ F ∶= FA * for A ∈ U(N) and F ∈ C d×N . In particular, if D ∈ T, define D ⋅ F ∶= FD * = FD.\nThis action preserves the frame spectrum: if F ∈ F d,N λ λ λ and D ∈ T, then the frame operator of D ⋅ F = FD * is (FD * )(FD * ) * = FD * DF * = FF * , which is the same as the frame operator of F, and hence D ⋅ F ∈ F d,N λ λ λ as well. At the frame level, the action of an element of the torus on a frame performs an independent phase rotation on each frame vector. The right action of T commutes with the left action of U(d), so the torus action descends to the quotient F d,N λ λ λ U(d), but it is not an effective action. After all, both T and U(d) contain a 1-parameter subgroup of scalar matrices, and the actions of these two subgroups cancel each other: if I k denotes the k × k identity matrix and e iθ ∈ U(1), then e iθ I d ∈ U(d), e iθ I N ∈ T and for any\nF ∈ F d,N λ λ λ we have (e iθ I N ) ⋅ ((e iθ I d )F) = (e iθ I d )F(e −iθ I N ) = F.\nWe can put this in a more standard context using our identification of F d,N λ λ λ U(d) with the coadjoint orbit Oλ λ λ ⊂ H(N) ≈ u(N) * , described in Proposition 2.11. The standard maximal torus of U(N) certainly acts on any coadjoint orbit of U(N), and in this case the action is just the conjugation action. The scalar matrices are the center Z(U(N)) of U(N), whose coadjoint action is trivial. In general, to get an effective action of the maximal torus of a Lie group on a coadjoint orbit of the group, one needs to take the quotient of the torus by the center of the group; in this case, this is simply\nG ∶= T Z(U(N)) ≈ U(1) N U(1) ≈ U(1) N−1 ,\nwhich can be identified with the subgroup of diagonal elements of U(N) whose last entry is 1. Now we see why it was worth defining the action of U(1) N (and hence G) on F d,N λ λ λ by D ⋅ F = FD * , since this corresponds exactly to the standard coadjoint action on H(N) ≈ u(N) * .\n\nIn this setting, it is easy to see that the action of G on Oλ λ λ ≈ F d,N λ λ λ U(d) is Hamiltonian:\nProposition 2.14. The action of G on Oλ λ λ is Hamiltonian, with momentum map Φ G ∶ Oλ λ λ → g * ≈ R N−1 recording the first N − 1 diagonal entries.\nProof. First of all, it is standard (see, e.g., [40,Example 5.3.11]) that the coadjoint action of U(N) on the coadjoint orbit Oλ λ λ is Hamiltonian with momentum map given by the inclusion Oλ λ λ ↪ u(N) * ≈ H(N); in fact, the analogous statement holds for arbitrary compact Lie groups acting on coadjoint orbits. Similarly, it is standard (see, e.g., [5, Proposition III.1.10]) that the action of the maximal torus T ≤ U(N) on Oλ λ λ is Hamiltonian with momentum map given by the composition\nOλ λ λ ↪ H(N) ≈ u(N) * → t * ,\nwhere t * is the Lie algebra of T and the projection u(N) * → t * is the one induced by the inclusion T ↪ U(N). After identifying t * with R N , this projection is easily seen to be the map which records the diagonal entries of a Hermitian matrix. 2 More generally, the same applies for any subgroup (see, e.g., [9, p. 213]), and so the action of G on Oλ λ λ is Hamiltonian with momentum map given by the composition\nOλ λ λ ↪ H(N) ≈ u(N) * → g * ,\nwhere the projection u(N) * → g * is the one induced by the inclusion G ↪ U(N). Under the identification g * ≈ R N−1 , this map just records all the diagonal entries but the last one (which in any case is determined by the others, since the trace of any element of Oλ λ λ is ∑ i λ i ). \n(N) on F d,N λ λ λ U(d) is Hamiltonian, with momentum map [F] → F * F. If F ∈ F d,N λ λ λ , then the (i,i) entry of F * F ∈ Oλ λ λ is simply ⟨ f i , f i ⟩ = f i 2 , so, reinterpreting in terms of frame data, the momentum map of the Hamiltonian G-action on F d,N λ λ λ U(d) is simply [F] ↦ ( f 1 2 ,... , f N−1 2 ). Hence, as a subset of F d,N λ λ λ U(d), the collection F d,N λ λ λ (r r r) U(d) of unitary equivalence classes of frames in F d,N λ λ λ (r r r) is precisely the level set Φ −1 G (r 1 ,...,r N−1 )\n. By the Schur-Horn theorem [30,46]\n, Φ −1 G (r 1 ,... ,r N−1 )\nis non-empty only if the vector (r 1 ,...,r N ) lies in the convex hull of the orbit ofλ λ λ = (λ 1 ,... ,λ d ,0,... ,0) under the action of the symmetric group S N which permutes entries. When r r r and λ λ λ are non-increasing lists, this condition is easily seen to be equivalent to r r r being λ λ λ -admissible, so F d,N λ λ λ (r r r) is non-empty precisely when r r r is λ λ λ -admissible. When r r r is λ λ λ -admissible, there is no guarantee that (r 1 ,...,r N−1 ) ∈ R N−1 ≈ g * is a regular value, but we can still apply Theorem 2.4 to see that the quotient of Φ −1 G (r 1 ,... ,r N−1 ) by G is at worst a symplectic stratified space. We have thus proved the first part of the following proposition:\nProposition 2.16. When r r r is λ λ λ -admissible, the (non-empty) space F d,N λ λ λ (r r r) (U(d) × G) is the symplectic quotient F d,N λ λ λ U(d) r r r G,\nwhich is a symplectic stratified space. 3 Alternatively,\nF d,N λ λ λ (r r r) (U(d) × G) can be viewed as the symplectic quotient C d×N O −λ λ λ ×{r r r} (U(d) × G).\nProof. We proved the first sentence above. For the second sentence, Sjamaar and Lerman [48,Theorem 4.1] showed that even when reducing over singular values, we can perform a reduction by a product group in stages. Hence, with\nΦ ∶ C d×N → H(d) × R N−1 being the (product) momentum map of the Hamiltonian action of U(d) × G on C d×N , C d×N O −λ λ λ ×{r r r} (U(d) × G) ≈ C d×N O −λ λ λ U(d) r r r G = F d,N λ λ λ U(d) r r r G. Remark 2.17. Proposition 2.16 says that F d,N λ λ λ (r r r) (U(d) × G)\nis, in general, a symplectic stratified space. Depending on the parameters λ λ λ and r r r, it may actually be a smooth manifold. We characterize the parameters such that F d,N λ λ λ (r r r) is a smooth manifold below in Theorem 2.25.\nNotice that the momentum map Φ ∶ C d×N → H(d) × R N−1 is given by F ↦ (−FF * ,( f 1 2 ,... , f N−1 2 )).\nIn particular, the space F d,N λ λ λ (r r r) is exactly the level set Φ −1 (O −λ λ λ × {r r r}). To simplify slightly, assume r r r is strongly λ λ λ -admissible. When F d,N λ λ λ (r r r) is a manifold (conditions for which we determine in Corollary 2.22 in the next section), then we know its codimension inside C d×N is equal to the codimension of O −λ λ λ × {r r r} inside H(d) × R N−1 . Even when F d,N λ λ λ (r r r) is not a manifold, we know from Theorem 2.5 that its quotient by U(d) × G contains an open dense subset which is a manifold. This subset is a stratum, and hence its (open, dense) preimage is also a manifold by Theorem 3.5 from Sjamaar and Lerman [48].\n\nIn other words, F d,N λ λ λ (r r r) contains an open, dense subset which is a manifold and which consists precisely of regular points of the momentum map Φ, so for the purposes of dimension-counting we may as well assume F d,N λ λ λ (r r r) is a manifold.\n\nTo that end,\ndim(O −λ λ λ × {r r r}) = dimO −λ λ λ = d 2 − k 2 1 − ⋅⋅⋅ − k 2 ℓ ,\nwhere, as in Remark 2.13, the k 1 ,... ,k ℓ are the multiplicities of the entries in the frame spectrum λ λ λ . Subtracting this from\nd 2 + N − 1 = dim(H(d) × R N−1 ) gives the codimension N − 1 + k 2 1 + ⋅⋅⋅ + k 2 ℓ of O −λ λ λ × {r r r} (inside H(d) × R N−1 )\nand hence also of F d,N λ λ λ (r r r) (inside C d×N ). This proves: Corollary 2.18. For r r r which is strongly λ λ λ -admissible,\ndim F d,N λ λ λ (r r r) = 2dN − N + 1 − ℓ j=1 k 2 j .\nHence,\ndim F d,N λ λ λ (r r r) (U(d) × G) = 2N(d − 1) + 2 − d 2 − ℓ j=1 k 2 j ,\nwhere the k j are the multiplicities of the entries in the frame spectrum λ λ λ .\n\n\nManifold Structure of Frame Spaces\n\nWhile Corollary 2.6 will allow us to apply the symplectic machinery even when F d,N λ λ λ (r r r) is singular, it is still interesting to understand when this space is a smooth manifold. We completely characterize parameters for which F d,N λ λ λ (r r r) is a manifold in the following subsection. We then give a detailed description of the local structure of singularities in singular frame spaces.\n\n\nCharacterizing Smooth Frame Spaces\n\nThe model result on manifold structures of frame spaces is due to Dykema and Strawn [18], who showed that the singular points in the space F d,N N d ,..., N d (1,...,1) of unit-norm tight frames must be orthodecomposable (we recall the definition of this property below). An immediate corollary is that the space of unit-norm tight frames is a manifold when N and d are relatively prime. More generally, Strawn [49] showed that the singular points in any space of frames with fixed frame vector norms and fixed frame operator (rather than frame spectrum) must be orthodecomposable.\n\nWe will prove the converse of these results, showing that the singular points (if any) of every F d,N λ λ λ (r r r) space are exactly the orthodecomposable frames. First, recall what it means for a frame to be orthodecomposable: \nDefinition 2.19. A frame { f i } N i=1 ⊂ C d is orthodecomposable ifV 1 ,...,V m ⊂ C d with C d = ⊕ m i=1 V i so that, for all k = 1,...,m, { f i } i∈P k is a frame for V k .\nIn our setting, we are realizing the space F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) as the inverse image of a coadjoint orbit under the momentum map Φ. This frame space will certainly be a manifold if it contains no critical points of Φ, so we now characterize the critical points of Φ.\n\nA standard part of the discussion around Theorems 2. Recall that a group action is locally free at p if the stabilizer of p is discrete. Therefore, Proposition 2.20 gives a characterization of critical points of a momentum map: p ∈ M is a critical point if and only if the stabilizer of p is continuous, and in particular contains a nontrivial one-parameter subgroup. This is the key to characterizing the critical points of Φ ∶ C d×N → H(d) × R N−1 .\n\n\nProposition 2.21. A frame F is a critical point of the momentum map Φ ∶ C d×N → H(d) × R N−1 of the Hamiltonian action of U(d) × G if and only if F is orthodecomposable.\n\nProof. Let F = f 1 ⋯ f N ∈ C d×N be a frame and let ζ ∶ R → G be a one-parameter subgroup such that ζ (t) ⋅ F = F for all t ∈ R. Then ζ uniquely corresponds to an element of the Lie algebra u(d) × g of U(d) × G, and in particular there exist matrices ξ ∈ u(d) and θ ∈ g such that ζ (t) = exp(t(ξ ,θ )) = (exp(tξ ),exp(tθ )) (see, e.g., [37,Propositions 20.1 and 20.5]). Under our concrete realization of G = T Z (U(N)) as the subgroup of diagonal unitary matrices with last entry equal to 1, θ will be of the form\nθ = √ −1 ⋅ diag(θ 1 ,... ,θ N−1 ,0)\nfor some θ 1 ,...,θ N−1 ∈ R. Then we have\n0 = d dt t=0 ζ (t) ⋅ F = d dt t=0 exp(tξ )F exp(−tθ ) = ξ F − Fθ .\nAt the frame vector level, this yields a system of equations\n⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ξ f j = √ −1θ j f j , j = 1,...,N − 1, ξ f N = 0.(8)\nEach √ −1θ j is therefore an eigenvalue of the skew-Hermitian matrix ξ with corresponding eigenvector f j . Moreover, any solution of (8) yields a one-parameter subgroup in the stabilizer of F. By the Spectral Theorem for normal matrices, the eigenspaces for distinct eigenvalues of ξ are orthogonal.\n\nWith the above setup, we are prepared to prove the claim. First, suppose that F is not a regular point of the momentum map. By Proposition 2.20, it is possible to find a one-parameter subgroup ζ ∶ R → G of the stabilizer of F which is nontrivial. In this case, ξ has at least two distinct eigenvalues and it follows that F is orthodecomposable, with frame vectors partitioned into the orthogonal eigenspaces of ξ .\n\nConversely, suppose that F is orthodecomposable. Let P 1 ,... ,P m denote the partition and V 1 ,... ,V m the subspaces appearing in Definition 2.19. Permuting columns of F as necessary, we can assume without loss of generality that P 1 = {1,...,k} for some k < N. For convenience, we can apply a unitary transformation to F so that f 1 ,... , f k span a coordinate plane V 1 ; say, the span of the first ℓ < d standard basis vectors. Since V 2 ,... ,V m are all orthogonal to V 1 , ⊕ m i=2 V i is the span of the remaining d − ℓ standard basis vectors. Then F has the form\nF = F 1 0 0 F 2 ,\nwhere the nonzero blocks F 1 and F 2 have sizes ℓ × k and (d − ℓ) × (N − k), respectively. Let\nξ ∶= diag( √ −1,⋯, √ −1 ℓ ,0,...,0 d−ℓ ) ∈ u(d) and θ ∶= diag( √ −1,..., √ −1 k ,0,... ,0 N−k ) ∈ u(1) N .\nIt is easy to check that ξ ,θ , and F satisfy the system (8) and therefore yield a nontrivial one-parameter subgroup in the stabilizer of F. By Proposition 2.20, F is not a regular point of the momentum map.\n\nThe characterization of singular points in F d,N λ λ λ (r r r) is now an easy corollary: Proof. From Proposition 2.21 we know that the critical points of the momentum map Φ are exactly the orthodecomposable frames. It is a nontrivial fact (see, e.g., [3,Theorem 5] or [48,Proposition 2.5]) that in a neighborhood of a critical point of a momentum map, the level set looks like the product of a quadratic cone and a manifold. Therefore, F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) has a quadratic singularity at each orthodecomposable frame. Since the structure of such singularities is itself of interest [8], we describe these cone singularities in detail in Section 2.3.2 below.\nCorollary\nOrthodecomposable frames are the only possible singularities: if F ∈ F d,N λ λ λ (r r r) is not orthodecomposable, then the differential DΦ F ∶ T F C d×N → T Φ(F) (H(d) × R N−1 ) is surjective. In particular, this implies Φ is transverse to a neighborhood of Φ(F) in O −λ λ λ × {r r r}, so it is standard that its inverse image-which is a neighborhood of F in F d,N λ λ λ (r r r)-is a submanifold of C d×N (see, e.g., [37, Theorem 6.30]), and in particular is smooth at F. Remark 2.23. Since every orthodecomposable frame is a singular point of the F d,N λ λ λ (r r r) space containing it, Corollary 2.22 solves the complex case of Cahill, Mixon, and Strawn's Problem 5.8 [8].\n\nTo make Corollary 2.22 more practical, it would be helpful to know, in terms of some computable conditions on the parameters d, N, λ λ λ , and r r r, exactly when F d,N λ λ λ (r r r) contains an orthodecomposable frame. To that end, suppose F ∈ F d,N λ λ λ (r r r) is orthodecomposable. Then, as in the proof of Proposition 2.20, after possibly permuting columns and applying a unitary transformation, we can realize F in block-diagonal form:\nF = F 1 0 0 F 2 ,\nwhere F 1 ∈ C ℓ×k and F 2 ∈ C (d−ℓ)×(N−k) are frames. But then both the Gram matrix and the frame operator associated to F are also block-diagonal:\nF * F = F * 1 F 1 0 0 F * 2 F 2 and FF * = F 1 F * 1 0 0 F 2 F * 2 .\nSince tr(F * 1 F 1 ) = tr(F 1 F * 1 ), we see that r 1 + ⋅⋅⋅ + r k = λ 1 + ⋅⋅⋅ + λ ℓ . More generally, since we permuted columns to get F into block-diagonal form, there exist proper subsets {r i 1 ,... ,r i k } ⊊ r r r and {λ i 1 ,... ,λ i ℓ } ⊊ λ λ λ so that\nr i 1 + ⋅⋅⋅ + r i k = λ i 1 + ⋅⋅⋅ + λ i ℓ .\nMoreover, let r r r ′ = (r ′ 1 ,... ,r ′ k ) ∶= (r i 1 ,... ,r i k ) and λ λ λ ′ = (λ ′ 1 ,... ,λ ′ ℓ ) ∶= (λ i 1 ,...,λ i ℓ ) and let r r r ′′ and λ λ λ ′′ be their respective complement vectors. Then we see that the columns { f i 1 ,... , f i k } give a frame for an ℓ-dimensional subspace of C d with spectrum λ i 1 ≥ ⋅⋅⋅ ≥ λ i ℓ > 0, which means that the frame space F ℓ,k λ λ λ ′ (r r r ′ ) is non-empty, and in particular that r r r ′ is λ λ λ ′ -admissible. Similarly, the complementary columns give an element of F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ), and hence r r r ′′ is λ λ λ ′′ -admissible. Summarizing, we have shown the following consequences of orthodecomposability:\nLemma 2.24. If F ∈ F d,N\nλ λ λ (r r r) is orthodecomposable, then there exist proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum (which implies that r r r ′′ and λ λ λ ′′ have the same sum as well).\n\nCombined with Corollary 2.22, this gives a sufficient condition for F d,N λ λ λ (r r r) to be a manifold. In fact, it is also a necessary condition:\nTheorem 2.25. F d,N\nλ λ λ (r r r) is a manifold if and only if there are no proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum.\n\nProof. If F d,N λ λ λ (r r r) is not a manifold, then Corollary 2.22 tells us that it contains some orthodecomposable frame and hence, by Lemma 2.24, there are such partitions of r r r and λ λ λ .\n\nConversely, suppose there are such partitions. Then, since r r r ′ is λ λ λ ′ -admissible, there exists a frame { f i 1 ,... , f i k } ∈ F ℓ,k λ λ λ ′ (r r r ′ ), and similarly there exists a frame {g j 1 ,... ,g j N−k } ∈ F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ). Embedding the former into the ℓ-dimensional subspace of C d spanned by the first ℓ standard basis vectors and the latter into the (d −ℓ)-dimensional orthogonal complement and combining them into a single list sorted by decreasing norm gives an orthodecomposable element-and hence a singular point-of F d,N λ λ λ (r r r), which is therefore not a manifold. Notice that Theorem 2.25 recovers Dykema and Strawn's result that the space of unit-norm tight frames is a manifold when N and d are relatively prime, and also implies that F d,N λ λ λ (r r r) is a manifold for generic r r r and λ λ λ .\n\n\nDescribing Singularities\n\nAs was mentioned above, Corollary 2.22 solves Problem 5.8 of [8] for complex FUNTFs. Problem 1.5 of the same paper asks for a description of the local geometry of the space of FUNTFs near orthodecomposable frames, and we now expand on the proof of Corollary 2.22 to solve this problem. We describe singularities for progressively more general classes of frames, with full details given in the simplest case and sketches given for more general cases.\n\n\nMinimally Orthodecomposable FUNTFs. A result of Arms, Marsden, and Moncrief gives a description of the local geometry near any point on the level set of a momentum map of an arbitrary symplectic manifold [3, Theorem 5]. This specializes to our setting to show that near any FUNTF\nF ∈ F d,N ( N d ,..., N d ) (1,...,1) = Φ −1 − N d ,... ,− N d ,(1,...,1) ,\nthere is a local diffeomorphism of the ambient space C d×N which takes a neighborhood of F in the level set onto the product of a quadratic cone C F and a smooth manifold. The cone is described explicitly as\nC F = X ∈ ker(DΦ(F)) ∩ ker(DΦ(F) ○ J) D 2 Φ(F)(X,X) ∈ image(DΦ(F)) ,\nwhere J denotes multiplication by √ −1, considered as a linear map on the real vector space T F C d×N , and D 2 Φ(F) is the Hessian of Φ at F, considered as a H(d)-valued bilinear form. Observe that if F is a regular point of Φ then C F is just a linear subspace of C d×N . We will show that C F is a singular cone when F is not a regular point.\n\nWriting\nF = f 1 ⋯ f N and X = x 1 ⋯ x N , we have DΦ(F)(X) = −FX * − XF * ,(2Re⟨ f j ,x j ⟩) N−1 j=1 and DΦ(F) ○ J(X) = √ −1FX * − √ −1XF * ,(2Im⟨ f j ,x j ⟩) N−1 j=1 , so that ker(DΦ(F)) ∩ ker(DΦ(F) ○ J) = {X ∈ C d×N FX * = 0 and ⟨ f j ,x j ⟩ = 0 ∀ j = 1,...,N − 1}.\nWe can also show that\nD 2 Φ(F)(X,X) = −2XX * ,(2 x j 2 ) N−1 j=1\n. Now suppose that F is orthodecomposable. As in the proof of Proposition 2.21, we can assume without loss of generality that\nF = F 1 0 0 F 2 ,\nfor some submatrices F 1 ∈ C d 1 ×N 1 and F 2 ∈ C d 2 ×N 2 (we use a different indexing convention for dimensions than in the proof of Proposition 2.21 for ease of generalization later on). Furthermore, suppose that the blocks F 1 and F 2 are not themselves orthodecomposable. We then claim that image(DΦ(F)) ⊥ = span I d 1 0 0 0 ,(1,...,1 N 1 ,0,...,0\nN 2 −1 ) ⊂ H(d) × R N−1 ,(9)\nwhere the orthogonal complement is taken with respect to the inner product ⟪⋅,⋅⟫ on H d×d × R N−1 defined by\n⟪(ζ 1 ,ξ 1 ),(ζ 2 ,ξ 2 )⟫ ∶= Re trace(ζ * 2 ζ 1 ) + ξ * 2 ξ 1 .\nIndeed, a straightforward computation shows that the vector given in (9) belongs to image(DΦ(F)) ⊥ , and we see that this space is one-dimensional as follows. By standard arguments, it suffices to show that the nullity of the dual map DΦ(F) * is equal to one and Equation 5.2.7 of [40] implies that it suffices to show that the map taking an element of H(d) × R N−1 to its infinitesimal vector field evaluated at F has one dimensional kernel. By arguments similar to those used in the proof of Proposition 2.21, this is equivalent to showing that the space of solutions to the system of equations (8) is one-dimensional, which holds under our assumption that F 1 and F 2 are not themselves orthodecomposable (using the fact that the eigenspaces of a skew-Hermitian matrix are orthogonal). Based on the work in the previous two paragraphs, the cone C F is described more explicitly as the set of X ∈ C d×N satisfying \nFX * = 0, ⟨ f j ,x j ⟩ = 0 ∀ j = 1,...,N − 1,(10)\nwhere we express X as a block matrix with blocks X km , k,m ∈ {1,2}, of size d k ×N m whose column vectors are denoted x km j . We can verify that the intersection of the subspace defined by (10) and the solution set of the indefinite quadratic equation (11) yields a singular cone by showing that it is not a linear subspace. To do so, choose\nX = 0 X 2 X 1 0 such that F 1 X * 1 = 0 and F 2 X * 2 = 0, so that FX * = 0.\nThe condition ⟨ f j ,x j ⟩ holds automatically due to the block structures of F and X, hence (10) is satisfied. The matrices X j can be chosen to be nonzero, since the dimension of the row space of F j is at most d j < N j . We can therefore scale the X j 's to have equal Frobenius norm, which means that (11) is also satisfied, so that X ∈ C F . Next, we observe that\nX ′ = 0 X 2 −X 1 0\nalso lies in C F . However, the matrix X − X ′ does not satisfy (11), and this proves our claim that C F is not a linear subspace.\n\nExample 2.26. Here we provide a concrete example of the structure described above. Let F denote the orthodecomposable unit norm tight frame\nF = 1 1 0 0 0 0 1 1 ∈ F 2,4 (2,2) (1,1,1,1) =∶ F .\nThe cone C F can be described explicitly as the set of 2 × 4 matrices X satisfying equations (10) and (11). Using the x km j notation from above, we write and, letting f j and x j denote the columns of F and X, respectively,\n0 = ⟨ f j ,x j ⟩ ∀ j ∈ {1,2,3,4} ⇔ x 11 1 = x 11 2 = x 22 1 = x 22 2 = 0.\nThe cone equation (11) Putting all of this together, the cone C F is\nC F = {A(λ ,θ ,φ ) λ ,θ ,φ ∈ R}, where A(λ ,θ ,φ ) ∶= λ ⎛ ⎝ 0 0 e √ −1θ −e √ −1θ e √ −1φ −e √ −1φ 0 0 ⎞ ⎠ .\nInterpreted as a subset of\nC 2 = {(λ 1 e √ −1θ ,λ 2 e √ −1φ\n)}, this is precisely the cone over the Clifford torus. Now consider the map defined by\nA(λ ,θ ,φ ) ↦ √ 1 − λ 2 F + A(λ ,θ ,φ ) = ⎛ ⎝ √ 1 − λ 2 √ 1 − λ 2 λ e √ −1θ −λ e √ −1θ λ e √ −1φ −λ e √ −1φ √ 1 − λ 2 √ 1 − λ 2 ⎞ ⎠ ,(12)\nwith domain {A(λ ,θ ,φ ) λ ∈ [0,1)}, a neighborhood of the origin in C F . We claim that the image of this map is an open subset of F intersected with a slice of the U(2) × U(1) 4 ) U(1)-action on C 2×4 through F-that is, a subset S ⊂ C 2×4 containing F such that the orbit of any point in C 2×4 sufficiently close to F has a unique representative in S. 4 Specifically, the slice S is the set of matrices of the form\n⎛ ⎝ α 1 α 2 β 1 e √ −1θ β 2 e √ −1θ γ 1 e √ −1φ γ 2 e √ −1φ δ 1 δ 2 ⎞ ⎠ ,\nwhere α j and δ j are positive real numbers, and β j , γ j , θ , φ are arbitrary real numbers. The proof that S is a slice is somewhat technical, so we omit it. It is then straightforward to show that F ∩ S is equal to the image of our map (i.e., points of the form (12)), which implies that a neighborhood of F in F is identified with the product of a neighborhood of the origin in C F and the group.\n\nGeneral Orthodecomposable FUNTFs. Now suppose that F is a general orthodecomposable FUNTF. After applying isometries as necessary, we can assume without loss of generality that F is a block diagonal matrix F = diag(F 1 ,... ,F ℓ ), where F j a non-orthodecomposable matrix of size d j × N j . Similar computations show that the cone C F consists of matrices X ∈ C d×N satisfying the linear conditions (10) and the system of quadratic equations\nm≠k ⎛ ⎝ N m j=1 x km j 2 − N k j=1 x mk j 2 ⎞ ⎠ = 0 ∀ k = 1,...,ℓ,\nwhere we express X as a block matrix with blocks X km of size d k ×N m and with column vectors denoted x km j . Arguments similar to the above show that C F is a singular cone.\n\nOrthodecomposable Frames in Arbitrary Frame Spaces. These descriptions of the local structure of FUNTF space near orthodecomposable frames do not intrinsically use the unit norm condition-they only rely on the assumption that the columns have some fixed collection of norms. The arguments therefore apply to describe singularities of spaces of tight frames with fixed norms.\n\nIn fact, similar local characterizations can be derived near orthodecomposable frames in any frame space F d,N λ λ λ (r r r) with an added technical step. The result of Arms, Marsden, and Moncrief specifically treats zero level sets of momentum maps, and extends trivially to handle level sets of fixed points of the Ad * -action on the dual of the Lie algebra. The general frame space F d,N λ λ λ (r r r) is a level set of a coadjoint orbit: F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}). The standard shifting trick of symplectic geometry (see, e.g., [48, p. 376]) can be used to realize F d,N λ λ λ (r r r) as the 0-level set of an associated momentum map on the symplectic manifold C d×N ×O −λ λ λ ; more specifically, this space is endowed with a product form, where the symplectic form on the second factor is −ω KKS with ω KKS being the canonical Kirillov-Kostant-Souriau symplectic form on a coadjoint orbit defined in (6). After applying the shifting trick, similar computations can be done to describe the local geometry of singular points.\n\n\nToric Geometry of Frame Space\n\n\nToric Symplectic Manifolds\n\nRecall that our goal is to show that the collection of full-spark frames in F d,N λ λ λ (r r r) has full measure. As mentioned in the introduction, the natural measure on F d,N λ λ λ (r r r) is the Hausdorff measure it inherits as a compact subset of C d×N . This measure is not so easy to get a handle on directly, but symplectic geometry provides a more tractable approach.\n\nAll symplectic manifolds are equipped with natural measures. Specifically, suppose (M,ω) is a symplectic manifold of dimension 2n. By the non-degeneracy of the symplectic form ω, the maximal wedge power ω ∧n = ω ∧ ⋅⋅⋅ ∧ ω n is nowhere-vanishing, and hence defines a volume form and associated symplectic or Liouville measure m ω : for a Borel set U ⊂ M, m ω (U) ∶= ∫ U ω ∧n . While F d,N λ λ λ (r r r) is not symplectic, by Proposition 2.16 its quotient\nF d,N λ λ λ (r r r) (U(d) × G) ≃ C d×N O −λ λ λ ×{r r r} (U(d) × G)\nis. Moreover, since C d×N is Kähler-i.e., it is a complex manifold with a Hermitian metric whose negative imaginary part defines the standard symplectic form and whose real part defines the standard Riemannian metric-so is the symplectic reduction, and the symplectic measure agrees with the pushforward of Hausdorff measure on F d,N λ λ λ (r r r) by the quotient map F d,N λ λ λ (r r r) → F d,N λ λ λ (r r r) (U(d) × G) [28,Theorem 3.1]. Therefore, to prove our main theorem it suffices to prove that the equivalence classes of full-spark frames have full measure in F d,N λ λ λ (r r r) (U(d) × G), which we can do symplectically. The key is that F d,N λ λ λ (r r r) (U(d) × G) admits a Hamiltonian action of a high-dimensional torus, which considerably simplifies the task of understanding the symplectic measure.\n\nIf a 2n-dimensional symplectic manifold (M,ω) admits a Hamiltonian action of a torus U(1) k , then the associated momentum map Φ ∶ M → u(1) k * ≃ R k has convex image and connected level sets:\n\nTheorem 3.1 (Atiyah [4] and Guillemin-Sternberg [24]). With notation as above:\n\n• For any v ∈ R k , Φ −1 (v) is either empty or connected.\n\n• The image Φ(M) is the convex hull of the images of the fixed points of the torus action.\n\nIn particular, when M is (relatively) compact, (the closure of) Φ(M) is a bounded convex polytope P called the moment polytope associated to the Hamiltonian torus action.\n\nIn general, the Duistermaat-Heckman theorem [17] precisely describes the relationship between the pushforward measure Φ * (m ω ) and Lebesgue measure on the moment polytope P. Since we will not need the full statement, we restrict to the case when k = n, that is when M admits a Hamiltonian action of a half-dimensional torus. In this case M is called a toric symplectic manifold. Toric symplectic manifolds are closely related to toric varieties [13] and are completely classified by the combinatorics of the moment polytope [15].\n\nTheorem 3.2 (Duistermaat-Heckman [17], see also [9,Chapter 30]). Let M be a 2n-dimensional toric symplectic manifold with moment polytope P. The pushforward measure Φ * (m ω ) is a constant multiple of Lebesgue measure on P.\n\n\nToric Structure of Frame Space\n\n\nCircle Actions on Frame Space\n\nWith Theorem 3.2 in mind, our strategy is to show that the top stratum of F d,N λ λ λ (r r r) (U(d) × G) is toric, and then to see that the image of the spark-deficient frames in the moment polytope has measure zero with respect to Lebesgue measure.\n\nWe begin by defining a torus action on the full space of frames F d,N and then show that our construction descends to the symplectic quotient. Let F ∈ F d,N with column vectors f i ∈ C d×1 , i = 1,... ,N. For each k = 1,...,N, let\nµ k1 ≥ µ k2 ≥ ⋯ ≥ µ kd ≥ 0 denote the d eigenvalues of the d × d Hermitian matrix f 1 f * 1 + f 2 f * 2 + ⋯ + f k f * k ,\narranged in decreasing order-each such sum is rank at most k, so we only need consider the first k eigenvalues if k < d. The quantities µ k j were dubbed eigensteps by Cahill et al. [7]. To get well-defined circle actions, we make the assumption that the eigenvalues µ k j , j = 1,...,k, are all distinctthe necessity of this assumption is explained below in Remark 3.4. Let u k1 ,... ,u k min(k,d) be the eigenvectors corresponding to the nonzero eigenvalues. For each k = 1,...,N and each j = 1,... ,min(k,d), define a U(1)-action on F, denoted\nφ k j ∶ U(1) × F d,N → F d,N ,\nby defining it at the frame vector level as\nφ k j (t, f i ) = exp(t √ −1u k j u * k j ) f i if 1 ≤ i ≤ k f i if k + 1 ≤ i ≤ N,(13)\nwhere t ∈ [0,2π).\nRemark 3.3. To be precise, we are identifying [0,2π) ≈ U(1) via t ↦ exp(t √ −1). This identification induces an isomorphism R ≈ u(1) via s ↦ √ −1s.(14)\nKeeping track of the exact isomorphism used in this identification will be useful later on.\n\n\nRemark 3.4.\n\nThe assumption that the eigenvalues µ k j are distinct means that the eigenspaces are all one-dimensional, which, in turn, gives a well-defined ordering of the eigenvectors u k j . Without the isolated eigenvalues assumption, some of these actions would degenerate to U(ℓ)-actions, with ℓ the multiplicity of a repeated eigenvalue.\n\nRemark 3.5. The action defined by (13) really defines a circle action; i.e., it is 2π-periodic. Indeed, this follows from Lemma 3.11 below, and is also shown in [19,Corollary 5.1.4].\n\nNext we show that this action on F d,N induces a well-defined action on F d,N λ λ λ (r r r) (U(d) × G). We do so in stages. Proposition 3.6. The circle action φ k j commutes with the T action.\n\nProof. We need to show that if two frames F 1 and F 2 lie in the same T orbit, then so do φ k j (t,F 1 ) and φ k j (t,F 2 ). Indeed, this holds since the formula for the action (13) makes it clear that\nφ k j (t, f i e − √ −1θ ) = φ k j (t, f i )e − √ −1θ\nfor all θ . Proof. The claim is clear when k + 1 ≤ i ≤ N, since the action is trivial in that case. It remains to check the claim when 1 ≤ i ≤ k. In this case, we have φ k j (t,A f i ) = exp(t √ −1Au k j u * k j A * )A f i , since (µ k j ,Au k j ) are the corresponding eigenpair for the kth partial frame operator of AF. In turn, we have\nexp(t √ −1Au k j u * k j A * ) f i = Aexp(t √ −1u k j u * k j )A * A f i = Aexp(t √ −1u k j u * k j ) f i = Aφ k j (t, f i ).\nSince the circle action φ k j is defined on F = f 1 ⋯ f N ∈ F d,N through the action on the individual frame vectors, we have the following corollary. The next proposition is obvious from the formula for φ k j .\n\nProposition 3.9. The circle action φ k j preserves norms of frame vectors.\n\nThe following proposition is similar, but is less obvious. The proof uses a lemma.\nLemma 3.11. If P is a d × d projection matrix, then exp(t √ −1P) = I d + (e t √ −1 − 1)P.\nProof. This follows from the Taylor series representation of the matrix exponential and the property that P 2 = P.\n\nProof of Proposition 3. 10.\nSuppose F = f 1 f 2 ⋯ f N ∈ C d×N\nis a frame and let S = FF * = f 1 f * 1 + ⋅⋅⋅ + f N f * N be its frame operator with spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0. Let k ∈ {1,...,N} and let S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k be the partial frame operator with spectrum µ k1 ≥ ⋅⋅⋅ ≥ µ kd ≥ 0. Let j ∈ {1,...,min(k,d)}, and consider the torus action associated with the eigenvalue µ k j . Then, by definition,\nt ⋅ F = exp(t √ −1u k j u * k j ) f 1 ⋯ exp(t √ −1u k j u * k j ) f k f k+1 ⋯ f N\nand the corresponding frame operator is\nS t ∶= (t ⋅ F)(t ⋅ F) * = exp(t √ −1u k j u * k j )( f 1 f * 1 + ⋅⋅⋅ + f k f * k )exp(−t √ −1u k j u * k j ) + f k+1 f * k+1 + ⋅⋅⋅ + f N f * N = (I d + (e t √ −1 − 1)u k j u * k j )S k (I d + (e −t √ −1 − 1)u k j u * k j ) + f k+1 f * k+1 + ⋅⋅⋅ + f N f * N = S + (e t √ −1 − 1)u k j u * k j S k + S k (e −t √ −1 − 1)u k j u * k j + (2 − e t √ −1 − e −t √ −1 )u k j u * k j S k u k j u * k j ,\nusing Lemma 3.11. For i 1 ≠ i 2 the vectors u ki 1 and u ki 2 are Hermitian orthogonal and hence the product of projections u ki 1 u * ki 1 u ki 2 u * ki 2 = 0. Combining this with the spectral decomposition S k = µ k1 u k1 u * k1 + ⋅⋅⋅ + µ kd u kd u * kd , and again using the fact that projections are idempotent, we see that\nS t − S = (e t √ −1 − 1) + (e −t √ −1 − 1) + (2 − e t √ −1 − e −t √ −1 ) µ k j u k j u * k j = 0,\nso the frame operator is invariant under the circle action.\n\nCombining Propositions 3.6, 3.9, and 3.10 with Corollary 3.8, we obtain:\n\nProposition 3.12. The circle action φ k j on F d,N descends to a well-defined action on F d,N λ λ λ (r r r) (U(d) × G).\n\n\nThe Momentum Map of the Circle Action\n\nFor each k = 1,...,N and j = 1,...,min(k,d), define a map\nΦ k j ∶ F d,N → R F ↦ µ k j ,\nwhere µ k j = µ k j (F) is (as above) the jth eigenvalue (in descending order) of the kth partial frame operator of F. Recall that we identify u(1) ≈ R via (14). We likewise identify u(1) * ≈ R, where R * ≈ R is identified via the pairing\n⟨s,t⟩ = s ⋅t 2(15)\non R × R. This allows us to state the following result.\n\nProposition 3.13. The map Φ k j ∶ F d,N → R is a momentum map for the circle action φ k j on the dense open subset of matrices whose kth partial frame operators have isolated jth eigenvalues.\n\nTo prove the proposition, we introduce some notation and technical lemmas. Let F = f 1 ⋯ f N ∈ F d,N and let S k = S k (F) = f 1 f * 1 + ⋯ + f k f * k be its partial frame operator. Then µ k j (F) is the jth eigenvalue (in decreasing order) of S k .\n\nWe first consider the mapμ j which takes a Hermitian k × k matrix to its jth eigenvalue. In what follows, let ⟨⋅,⋅⟩ denote the Frobenius inner product on F d,N . Lemma 3.14. Suppose that S is a k × k Hermitian matrix with isolated jth eigenvalueμ j (S). Thenμ j is a smooth map in a neighborhood of S. Moreover, its gradient at S is given by\n∇μ j (S) =ū jū * j ,\nwhereū j is the unit eigenvector associated toμ j (S).\n\nProof. Let S have isolated jth eigenvalue, which we denote simply as µ. We denote the associated unit eigenvector as u. Let S ′ be a variation of S. Denote the jth eigenvalue and unit eigenvector of S + εS ′ as µ(ε) and u(ε), respectively; we denote the ε-derivatives of these functions asμ(ε) andu(ε). Then\nd dε ε µ(ε) = d dε ε ⟨(S + εS ′ )u(ε),u(ε)⟩ = ⟨S ′ u,u⟩ + ⟨Su(0),u⟩ + ⟨Su,u(0)⟩ = ⟨S ′ u,u⟩ + ⟨u(0),Su⟩ + ⟨Su,u(0)⟩(16)\n= ⟨S ′ u,u⟩ + µ (⟨u(0),u⟩ + ⟨u,u(0))⟩ (17)\n= ⟨S ′ u,u⟩ = ⟨S ′ ,uu * ⟩,\nwhere (16) follows because S is Hermitian and (17) follows by the condition that u(ε) is constant in ε. Lemma 3.15. Let F = f 1 ⋯ f N ∈ F d,N and s ∈ R ≈ u(1) (using the identification (14)). The infinitesimal vector field associated to s which is induced by the action φ k j is given at F by\nF s ∶= √ −1s ⋅ u k j u * k j f 1 ⋯ f k 0 ⋯ 0 .\nProof. The infinitesimal vector field is given at the frame level by\nd dε ε=0 φ k j (εs, f i ) = d dε ε=0 exp(εs √ −1u k j u * k j ) f i if 1 ≤ i ≤ k f i if k + 1 ≤ i ≤ N, = √ −1s ⋅ u k j u * k j f i if 1 ≤ i ≤ k 0 if k + 1 ≤ i ≤ N.\nThe result follows.\n\nWe are now prepared to prove the proposition.\n\nProof of Proposition 3.13. We need to show that, for (14)), the following equation holds:\nF = f 1 ⋯ f N ∈ F d,N , X = x 1 ⋯ x N ∈ T F F d,N ≈ C d×N , and s ∈ R ≈ u(1) (viaDΦ k j (F)(X)(s) = ω F (F s ,X).(18)\nThis is accomplished by direct computation. The right hand side of (18) simplifies to\nω F (F s ,X) = Imtr(F s X * ) = Imtr √ −1s ⋅ u k j u * k j ⋅ f 1 ⋯ f k 0 ⋯ 0 ⋅ x 1 ⋯ x N * (19) = Im √ −1s ⋅ tr u k j u * k j ( f 1 x * 1 + ⋯ + f k x * k ) = s ⋅ Re tr u k j u * k j ( f 1 x * 1 + ⋯ + f k x * k ) ,(20)\nwhere (19) uses the expression for F s from Lemma 3.15. The left hand side of (18) becomes\nDΦ k j (F)(X)(s) = 1 2 s ⋅ D(μ j ○ S k )(F)(X) = 1 2 s ⋅ Dμ j (S k (F)) ⋅ DS k (F)(X),\nwhere the first line follows by our choice of identification R ≈ u(1) * (15) and where S k (F) is the kth partial frame operator of F andμ j is the jth eigenvalue function on d × d Hermitian matrices. A straightforward computation shows that\nDS k (F)(X) = ( f 1 x * 1 + x 1 f * 1 ) + ⋯ + ( f k x * k + x k f * k )\n. Putting this together with Lemma 3.14, our simplification continues as\n1 2 s ⋅ Dμ j (S k (F)) ⋅ DS k (F)(X) = 1 2 s ⋅ Re ⟨u k j u * k j ,( f 1 x * 1 + x 1 f * 1 ) + ⋯ + ( f k x * k + x k f * k )⟩ = 1 2 s ⋅ Re tr u k j u * k j (( f 1 x * 1 + x 1 f * 1 ) + ⋯ + ( f k x * k + x k f * k )) = 1 2 s ⋅ Re tr u k j u * k j ( f 1 x * 1 + f 2 x * 2 + ⋯ + f k x * k ) + Retr u k j u * k j (x 1 f * 1 + x 2 f * 2 + ⋯ + x k f * k ) (21) = s ⋅ Re tr u k j u * k j ( f 1 x * 1 + ⋯ + f k x * k ) ,(22)\nwhere the last line follows by the observation that the trace terms in (21) are conjugates. Since (22) and (20) agree, (18) has been established.\n\n\nDimension Counting\n\nWe now pause briefly to count the possible number of independent eigensteps µ k j . The µ k j associated to a frame define a map F d,N λ λ λ (r r r) → R m for some m depending on N and d; as has been previously observed [7,8,27], the image of this map is a convex polytope, whose dimension we now determine. To slightly simplify some calculations, we will assume in this section that r r r is strongly λ λ λ -admissible.\n\nSince each partial frame operator S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k is a rank-1 perturbation S k = S k−1 + f k f * k of the previous partial frame operator, Weyl's perturbation inequalities (see, e.g., [6,Chapter III]) imply that the eigenvalues of S k and S k−1 satisfy the interlacing inequalities ⋅⋅⋅ ≥ µ k, j ≥ µ k−1, j ≥ µ k, j+1 ≥ µ k−1, j+1 ≥ ...\n\nfor each k = 2,... ,N and j = 1,...,min{d,k}, where we use the notation µ k, j in place of µ k j when we need to disambiguate the index k from the index j. When N = 4 and d = 3, this produces the array\nλ 1 λ 2 λ 3 µ 31 µ 32 µ 33 µ 21 µ 22 µ 11 ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥ ≥\nwhere we recall that S N = S, and hence µ N, j = λ j for each j = 1,...,d.\n\nIn other words, entries in the array are greater than entries to the right, regardless of vertical position. With this convention in place, we can omit explicit inequalities without losing any information:\nλ 1 λ 2 λ 3 µ 31 µ 32 µ 33 µ 21 µ 22 µ 11\nIn general, then, the µ k, j satisfy the system of interlacing inequalities implied by the following diagram:\nλ 1 λ 2 λ 3 ... λ d µ N−1,1 µ N−1,2 µ N−1,3 ... µ N−1,d . . . . . . . . . . . . . . . µ d+1,1 µ d+1,2 µ d+1,3 ... µ d+1,d µ d,1 µ d,2 µ d,3 ... µ d,d . . . . . . . . . . . . µ 3,1 µ 3,2 µ 3,3 µ 2,1 µ 2,2 µ 1,1(23)\nIn this diagram, all entries are nonnegative; equivalently, we think of implicit zeros to the right of every row. The pattern of inequalities described by the above diagram is often called a Gelfand-Tsetlin pattern [14,21]; it corresponds to the Gelfand-Tsetlin integrable system on the flag manifold Oλ λ λ ≃ F d,N λ λ λ U(d) [26]. In fact, there is one further constraint: cyclic invariance of trace implies that the trace of each partial frame operator S k must equal the trace of the corresponding partial Gram matrix. That is, the sum of each row in (23) must equal the corresponding partial sum of squared frame vector norms:\nmin{k,d} j=1 µ k j = k j=1 r j(24)\nfor each k = 1,...,N. With this constraint in place, we can count the number of free parameters in (23) and, hence, the number of independent φ k j . Notice that there are d(2N−d−1) 2 total entries in (23), excluding the top row (which we already know is fixed):\nd(d+1) 2\nfor the triangle in the bottom d rows, and d(N − d − 1) for the parallelogram in the upper N − d rows. Since each row sum is fixed, we lose one free parameter for each of the rows but the top one, meaning we subtract N − 1 parameters.\n\nIf all the eigenvalues λ 1 ,... ,λ d of the frame operator are distinct, we are done. However, if λ j = λ j+1 , then this implies that λ j = µ N−1, j = λ j+1 . More generally, if λ j has multiplicity k, meaning that λ j = λ j+1 = ⋅⋅⋅ = λ j+k−1 , then this fixes an entire upside-down triangle in (23) with vertices at λ j , λ j+k−1 , and µ N−k, j , comprising k(k−1) 2 of the µ i, j . Therefore, if the eigenvalues λ 1 ,... ,λ d have multiplicities k 1 ,... ,k ℓ with k 1 + ⋅⋅⋅ + k ℓ = d, then the total number of free parameters in (23) is\nd(2N − d − 1) 2 − (N − 1) − ℓ j=1 k j (k j − 1) 2 = N(d − 1) + 1 − d 2 2 − 1 2 ℓ j=1 k 2 j .\nThe interlacing inequalities implicit in the diagram (23) together with the row sums (24) determine a convex polytope P d,N λ λ λ (r r r), which we call the eigenstep polytope (at least when λ λ λ and r r r are rational, these are sometimes called (weight-restricted) Gelfand-Tsetlin polytopes in the combinatorics literature [1,14]). In the course of the discussion above, we have proved: Proposition 3.16. When r r r is strongly λ λ λ -admissible, the eigenstep polytope has dimension\nd(λ λ λ ,r r r) ∶= dim P d,N λ λ λ (r r r) = N(d − 1) + 1 − d 2 2 − 1 2 ℓ j=1 k 2 j ,\nwhere k 1 ,... ,k ℓ are the multiplicities of the spectrum λ λ λ = (λ 1 ,...,λ d ).\n\nComparing to Corollary 2.18,\ndim P d,N λ λ λ (r r r) = 1 2 dim F d,N λ λ λ (r r r) (U(d) × G) .\nIn the case of unit-norm tight frames, \n\n\nHamiltonian Torus Action\n\nWe now return to the task of showing that there is a Hamiltonian torus action on our frame spaces; the dimension d(λ λ λ ,r r r) just computed will be the dimension of the torus.\n\nOf course, the torus action will simply be the product of the individual circle actions, which we will show commute using Proposition 3.7.\n\nProposition 3.17. The circle actions φ k j and φ mℓ commute for all k, j,m,ℓ. N . It suffices to prove the claim at the frame level, i.e.,\nProof. Let F = f 1 ⋯ f N ∈ F d,φ k j (t,φ mℓ (s, f i )) = φ mℓ (s,φ k j (t, f i ))(25)\nfor all f i . Without loss of generality, assume k ≤ m. If k + 1 ≤ i ≤ N, then φ k j is the trivial action, so (25) is obvious.\nOtherwise we have φ k j (t,φ ℓm (s, f i )) = φ k j (t,exp(s √ −1u mℓ u * mℓ ) f i ) = exp(s √ −1u mℓ u * mℓ )φ k j (t, f i ) (26) = φ mℓ (s,φ k j (t, f i )),\nwhere (26) 19. In conjunction with Theorem 3.1, this result can be used to show that the space of frames with prescribed eigensteps is connected. While this fact essentially follows from [7,Theorem 7], which was a key tool in the original proof of the Frame Homotopy Conjecture [8], the symplectic viewpoint puts this fact in a more general context. By Proposition 3.16, the torus acting on the open dense subset from Corollary 3.18 is half-dimensional. We have therefore proved the following theorem on the geometric structure of frame spaces. \n\n\nProof of Main Theorem\n\nWe are now prepared to prove the main theorem using the toric symplectic framework developed in the previous section. We also sketch a more algebraically-flavored proof using related tools from Geometric Invariant Theory [42].\n\n\nA Toric Symplectic Approach\n\nIt is clear that the three options (repeated below) are mutually exclusive and exhaust all possibilities, so it remains only to show that the conclusion in each part follows from the assumption.\n\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\n\nProof. As pointed out in Section 2.2.3, this is a consequence of the Schur-Horn theorem [30,46]; it is also a result of Casazza and Leon [12].\n\n2. If r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\n\nProof. Suppose r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible. Then F d,N λ λ λ (r r r) is nonempty by Casazza and Leon's result [12].\n\nSince r r r is not strongly λ λ λ -admissible, there exists k ∈ {1,...,d} so that\nk i=1 r i = k i=1 λ i .\n\nThe admissibility criterion requires that\n∑ N i=1 r i = ∑ d i=1 λ i , so it follows that N i=k+1 r i = d i=k+1 λ i .(27)\nIf k = d, then we see that r k+1 = ⋅⋅⋅ = r N = 0, which does not satisfy the hypothesis that the r i are all positive.\n\nOtherwise, suppose F = f 1 ⋯ f N ∈ F d,N λ λ λ (r r r), so that λ λ λ is the spectrum of FF * and f i 2 = r i for all i = 1,... ,N. \nr i = d i=k+1 λ i(28)\nusing (27). Now, consider the portion of the eigenstep inequality diagram (23) forF starting from the (N − k)th row: 1μN−1,2μN−1 \nλ 1 λ 2 λ 3 ... λ d µ N−1,f N f * N + ⋅⋅⋅ + f N−d+1 f * N−d+1 ,\nthe length-d collection of frame vectors f N−d+1 ,... , f N is linearly dependent, and hence F is spark-deficient. Since the choice of F was arbitrary, we see that F d,N λ λ λ (r r r) consists entirely of spark-deficient frames.\n\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\n\nProof. It suffices to show that the set of full spark frames is full measure in the dense open subset of F d,N λ λ λ (r r r) from Theorem 3.20, which we denoteŨ. We will prove a subclaim: the set D d λ λ λ (r r r) ⊂Ũ of frames whose first d columns are linearly dependent is measure zero. The full claim then follows easily. Indeed, consider the action of the symmetric group on N letters on the matrix space C d×N given by permuting columns. This representation of the symmetric group embeds it as a subgroup of U(N), so that the action on C d×N is by isometries. The action of a permutation σ restricts to an isometry of each frame space F d,N λ λ λ (r r r) onto its image F d,N λ λ λ (σ ⋅ r r r), where σ ⋅ r r r is the corresponding permutation of the norm vector. It follows that the set of spark-deficient frames is realized as ∪ σ D d λ λ λ (σ ⋅ r r r), a finite union of measure zero sets. We proceed by establishing the subclaim. \nLet Φ = ⨉ 1≤k≤N−1 1≤ j≤min{d,k} Φ k j ∶ U → P d,N λ λ λ (r r r)\ndenote the moment map for the torus action on U; that is, Φ is the eigenstep map. By the remarks in the previous paragraph, the condition that the first d columns of a frame F are linearly independent is equivalent to the statement that Φ dd ([F]) ≠ 0.\n\nAccording to Section 3.2.3, the moment polytope P d,N λ λ λ (r r r) is a convex subset of an affine subspace A ⊂ R d(2N−d−1) 2 . Thinking of the Φ k j as coordinates on the ambient space, the condition Φ dd = 0 defines a hyperplane. The intersection of this hyperplane with P d,N λ λ λ (r r r) is positive codimension (hence measure zero), unless A is contained in the hyperplane. It is easy to see that this is not the case, as it suffices to show the existence of a pattern of the form (23) satisfying the defining equalities of A such that µ dd ≠ 0-such examples are trivial to construct without the polytope inequality constraints.\n\nWe have so far shown that image of the set of equivalence classes of frames whose first d columns are linearly independent has full measure image in P d,N λ λ λ (r r r) under Φ. By the Duistermaat-Heckman Theorem (Theorem 3.2), this implies that the set has full measure in U. This completes the proof of the subclaim, and therefore completes the proof of the theorem.\n\n\nAn algebraic approach\n\nThe above proof is based on symplectic geometry, but, under additional rationality assumptions, there is also an argument using algebraic geometry that shows that the collection of full-spark frames is dense in F d,N λ λ λ (r r r). While we expect a similar argument to apply to arbitrary rational λ λ λ and r r r, in which case the objects of interest are more general weight varieties [35], for simplicity we limit ourselves to the case when λ λ λ and r r r are constant, so we are talking about (scaled) unit-norm tight frames. It will be convenient to rescale so that r r r = (d,... ,d), and hence λ λ λ = (N,... ,N); note that both are integer vectors.\n\nAs above, it suffices to show that the (equivalence classes of) full spark frames are open and dense in the symplectic reduction F d,N λ λ λ (r r r) (U(d) × G) ≈ C d×N O −λ λ λ ×{r r r} (U(d) × G).\n\nSince λ λ λ is constant, the coadjoint orbit O −λ λ λ consists of the single point −λ λ λ I d . Taking the reduction in stages, the above quotient is isomorphic to C d×N −λ λ λ I d U(d) r r r G = Gr d (C N ) r r r G using Proposition 2.11 and Remark 2.13.\n\nWork of Sjamaar [47], which is a culmination of a series of results by Kempf and Ness [32], Guillemin and Sternberg [25], Kirwan [33], and Ness [45] relating symplectic reductions and Geometric Invariant Theory (GIT) quotients [42], has the following consequence in our case: where the line bundle L r r r on the Grassmannian is linearized to correspond to the action of the algebraic torus H = {(t 1 ,... ,t N ) ∈ (C * ) N ∶ ∏ti = 1} on C N given by identifying t t t = (t 1 ,...,t N ) ∈ H with the diagonal unitary matrix diag(t t t r r r t 1 ,...,t t t r r r t N ), where t t t r r r = t r 1 1 ⋯t r N N is the character of H corresponding to the vector r r r (cf. [20,31]).\nIf F ∈ F d,N λ λ λ\nis a frame representing a point [F] ∈ Gr d (C N ), then the determinants of the d × d minors of F are precisely the Plücker coordinates of [F], and in general the Plücker coordinates generate the homogeneous coordinate ring of Gr d (C N ). By definition, the full-spark frames are precisely those for which none of the Plücker coordinates vanish.\n\nIn turn, the homogeneous coordinate ring R of the GIT quotient Gr d (C N ) L r r r H consists of the H-invariant homogeneous coordinates on Gr d (C N ). It is known [31,Lemma 4.5] that R is spanned by monomials in the Plücker coordinates so that the total number of Plücker coordinates involving the ith column is kr i for some integer k independent of i.\n\nIn our case, all r i = d, so taking the product of all the Plücker coordinates and raising it to the dth power gives a homogeneous coordinate on Gr d (C N ) L r r r H whose vanishing set is exactly the collection of (equivalence classes of) spark-deficient frames in the space of (scaled) unit-norm tight frames. Since this is the vanishing set of a homogeneous coordinate, it is a subvariety, and hence its complement-the collection of full-spark frames-is open in the Zariski topology, and in particular either empty or dense. Since there are full-spark frames in each space of unit-norm tight frames (for example, the first d rows of a scaled N × N discrete Fourier transform matrix [2]), the collection of fullspark frames cannot be empty, so it must be dense.\n\n\nDiscussion\n\nGiven a 2n-dimensional toric symplectic manifold M with moment polytope P, one can often find action-angle coordinates on M which take the form of a map α ∶ int(P) ×U(1) n → M which inverts the momentum map Φ ∶ M → P in the sense that Φ(α(p,t)) = p. In this case, Theorem 3.2 can be extended slightly to show that the image of α is a full-measure subset of M and that the map α is measure-preserving. Sampling P ×U(1) n with respect to the product of Lebesgue measure on P and the standard product measure on U(1) n and pushing forward by α gives a uniformly random sample from the symplectic measure on M (see, for example, the discussion in [10]).\n\nIn our setting, this means that coupling explicit action-angle coordinates on F d,N λ λ λ (r r r) (U(d)×G) with an algorithm for sampling P d,N λ λ λ (r r r) would give an algorithm for sampling random frames in F d,N λ λ λ (r r r). In particular, by Corollary 1.2, such an algorithm would provide endless quantities of full spark FUNTFs.\n\nIt is natural to ask whether the analog of Theorem 1.1 holds for real and for quaternionic frames. Symplectic geometry is not the right tool in either case, but it is very plausible that the algebraic proof sketched in Section 4.2 could be adapted to the real case to show that full spark frames are dense in real frame spaces. In a different direction, the perspective based on isotropy orbits and isoparametric submanifolds introduced in [44] seems like the most promising way to understand the measures on real and quaternionic frame spaces.\n\nFinally, the spaces F d,N λ λ λ (r r r) (U(d) × G) are examples of weight varieties [35], and Goldin [22] has determined the rational cohomology ring of certain weight varieties, including the quotient F d,N (λ ,...,λ ) (r r r) (U(d) × G) of the space of λ -tight frames with fixed frame vector norms whenever it is a manifold. In particular, this determines the rational cohomology ring of the (U(d) × G)-quotient of FUNTF space when N and d are relatively prime. What about in the non-manifold case or for more general frame spectra?\n\nTheorem 2 . 2 (\n22Marsden-Weinstein-Meyer Theorem for Regular Values\n\nTheorem 2 . 5 (\n25Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary.\n\nCorollary 2 . 6 .\n26Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.\n\n\nto H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U\n\n\nthere exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces\n\n\n2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p.\n\nand 0 =\n0⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2\n\nProposition 3. 7 .\n7Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N.\n\nCorollary 3. 8 .\n8The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N .\n\nProposition 3 . 10 .\n310The circle action φ k j preserves the frame operator.\n\nTheorem 3 . 20 .\n320The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold.\n\n\na new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields\n\n\nA frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U.\n\nTheorem 4. 1 .\n1For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,\n\n\nconsists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N..,1) is strongly λ λ λ = ( N \nd ,... , N \nd )-admissible whenever N > d and since F d,d \n(1,...,1) (1,...,1) \nN \n\nd ,..., N \n\nd \n\n(1,...,1) of \n\nunit-norm tight frames (FUNTFs) in C d . \n\nN \n\nd ,..., N \n\nd \n\n(1,...,1). \n\n\n\nr i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16.\n\n\nfollows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.Remark 3.\n\n\nGoing diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator,3 \n...μ N−1,d \n\nµ N−2,1μN−2,2μN−2,3 \n...μ N−2,d \n\n. . . \n. . . \n. . . \n. . . \n. . . \n\nµ N−k,1μN−k,2μN−k,3 \n...μ N−k,d \n\n\nIt is worth pointing out that this is precisely the setup for the symplectic proof of the Schur-Horn theorem; see Knutson's excellent paper[36] for more.\nHere we are slightly abusing notation. We are reducing over the point (r 1 ,... ,r N−1 ) ∈ R N−1 , so we should, strictly speaking, use the notation F d,N λ λ λ U(d) (r 1 ,...,r N−1 ) G. This is notationally cumbersome and, since for elements of F d,N λ λ λ the quantity r N is determined by (r 1 ,... ,r N−1 ) anyway, it seems overly pedantic to invent a new shorthand for the truncated vector. Hence, we will use r r r to indicate both (r 1 ,... ,r N ) and (r 1 ,... ,r N−1 ) and trust both ourselves and the reader to keep track of which we mean from context.\nStrictly speaking, we should consider the group modulo the isotropy group of F when constructing the slice, but we are suppressing this technical detail for the sake of brevity.\nAcknowledgmentsWe are very grateful for ongoing conversations about frames with various friends and colleagues, especially Jason Cantarella, Khazhgali Kozhasov, Emily King, Chris Manon, Augustin-Liviu Mare, Dustin Mixon, Louis Scharf, and Soledad Villar, and we thank the anonymous Referee #1 for our earlier paper[43], who pushed us to explore what the symplectic machinery could say about genericity of full spark frames. This work was supported by grants from the National Science Foundation (DMS-2107808, Tom Needham; DMS-2107700, Clayton Shonkwiler) and the Simons Foundation (#709150, Clayton Shonkwiler).\nGelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics. 5423Per AlexanderssonPer Alexandersson. Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics, 54:1-20, 2016. ↑23\n\nFull spark frames. Boris Alexeev, Jameson Cahill, Dustin G Mixon, Journal of Fourier Analysis and Applications. 18627Boris Alexeev, Jameson Cahill, and Dustin G. Mixon. Full spark frames. Journal of Fourier Analysis and Applications, 18(6):1167-1194, 2012. ↑3, ↑27\n\nSymmetry and bifurcations of momentum mappings. Judith M Arms, Jerrold E Marsden, Vincent Moncrief, Communications in Mathematical Physics. 78415Judith M. Arms, Jerrold E. Marsden, and Vincent Moncrief. Symmetry and bifurcations of momentum mappings. Communications in Mathematical Physics, 78(4):455-478, 1981. ↑13, ↑15\n\nConvexity and commuting Hamiltonians. Michael Francis, Atiyah , Bulletin of the London Mathematical Society. 14118Michael Francis Atiyah. Convexity and commuting Hamiltonians. Bulletin of the London Mathematical Society, 14(1):1-15, 1982. ↑18\n\nMichèle Audin, Torus Actions on Symplectic Manifolds. Basel9310second revised editionMichèle Audin. Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Birkhäuser Verlag, Basel, second revised edition, 2004. ↑5, ↑7, ↑9, ↑10\n\nMatrix Analysis. Rajendra Bhatia, Graduate Texts in Mathematics. 16922SpringerRajendra Bhatia. Matrix Analysis, volume 169 of Graduate Texts in Mathematics. Springer, New York, NY, 1997. ↑22\n\nConstructing finite frames of a given spectrum and set of lengths. Jameson Cahill, Matthew Fickus, Dustin G Mixon, Miriam J Poteet, Nate Strawn, Applied and Computational Harmonic Analysis. 35124Jameson Cahill, Matthew Fickus, Dustin G. Mixon, Miriam J. Poteet, and Nate Strawn. Constructing finite frames of a given spectrum and set of lengths. Applied and Computational Harmonic Analysis, 35(1):52-73, 2013. ↑2, ↑18, ↑22, ↑24\n\nConnectivity and irreducibility of algebraic varieties of finite unit norm tight frames. Jameson Cahill, Dustin G Mixon, Nate Strawn, SIAM Journal on Applied Algebra and Geometry. 1124↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22Jameson Cahill, Dustin G. Mixon, and Nate Strawn. Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. SIAM Journal on Applied Algebra and Geometry, 1(1):38-72, 2017. ↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22, ↑24\n\nLectures on Symplectic Geometry. Ana Cannas Da Silva, Lecture Notes in Mathematics. 176418Springer-VerlagAna Cannas da Silva. Lectures on Symplectic Geometry, volume 1764 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, Heidelberg, 2001. ↑10, ↑12, ↑18\n\nThe symplectic geometry of closed equilateral random walks in 3-space. Jason Cantarella, Clayton Shonkwiler, The Annals of Applied Probability. 26127Jason Cantarella and Clayton Shonkwiler. The symplectic geometry of closed equilateral random walks in 3- space. The Annals of Applied Probability, 26(1):549-596, 2016. ↑27\n\nEqual-norm tight frames with erasures. G Peter, Jelena Casazza, Kovačević, Advances in Computational Mathematics. 182-41Peter G. Casazza and Jelena Kovačević. Equal-norm tight frames with erasures. Advances in Computational Mathematics, 18(2-4):387-430, 2003. ↑1\n\nExistence and construction of finite frames with a given frame operator. G Peter, Manuel T Casazza, Leon, International Journal of Pure and Applied Mathematics. 63225Peter G. Casazza and Manuel T. Leon. Existence and construction of finite frames with a given frame operator. International Journal of Pure and Applied Mathematics, 63(2):149-157, 2010. ↑3, ↑25\n\nToric Varieties. David A Cox, John B Little, Henry K Schenck, Graduate Studies in Mathematics. 12418American Mathematical SocietyDavid A. Cox, John B. Little, and Henry K. Schenck. Toric Varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011. ↑18\n\nVertices of Gelfand-Tsetlin polytopes. A De Jesús, Tyrrell B Loera, Mcallister, Discrete & Computational Geometry. 32423Jesús A. De Loera and Tyrrell B. McAllister. Vertices of Gelfand-Tsetlin polytopes. Discrete & Computational Geometry, 32(4):459-470, 2004. ↑23\n\nHamiltoniens périodiques et images convexes de l'application moment. Thomas Delzant, 11618Bulletin de la Société Mathématique de FranceThomas Delzant. Hamiltoniens périodiques et images convexes de l'application moment. Bulletin de la Société Mathématique de France, 116(3):315-339, 1988. ↑18\n\nOptimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. L David, Michael Donoho, Elad, Proceedings of the National Academy of Sciences of the United States of America. the National Academy of Sciences of the United States of America1003David L. Donoho and Michael Elad. Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. Proceedings of the National Academy of Sciences of the United States of America, 100(5):2197- 2202, 2003. ↑3\n\nOn the variation in the cohomology of the symplectic form of the reduced phase space. J Johannes, Gerrit J Duistermaat, Heckman, Inventiones Mathematicae. 69218Johannes J. Duistermaat and Gerrit J. Heckman. On the variation in the cohomology of the symplectic form of the reduced phase space. Inventiones Mathematicae, 69(2):259-268, 1982. ↑18\n\nManifold structure of spaces of spherical tight frames. Ken Dykema, Nate Strawn, International Journal of Pure and Applied Mathematics. 28212Ken Dykema and Nate Strawn. Manifold structure of spaces of spherical tight frames. International Journal of Pure and Applied Mathematics, 28(2):217-256, 2006. ↑1, ↑2, ↑4, ↑12\n\nBending flows for sums of rank one matrices. Hermann Flaschka, John J Millson, Canadian Journal of Mathematics. 57124Hermann Flaschka and John J. Millson. Bending flows for sums of rank one matrices. Canadian Journal of Mathematics, 57(1):114-158, 2005. ↑2, ↑3, ↑19, ↑24\n\nToric degenerations of weight varieties and applications. Philip Foth, Yi Hu, Travaux mathématiques. 1627Philip Foth and Yi Hu. Toric degenerations of weight varieties and applications. Travaux mathématiques, 16:87- 105, 2005. ↑27\n\nFinite-dimensional representations of the group of unimodular matrices. M Israïl, Michael L Tsetlin, Doklady Akademii Nauk SSSR (N.S.). 7123Israïl M. Gel'fand and Michael L. Tsetlin. Finite-dimensional representations of the group of unimodular matri- ces. Doklady Akademii Nauk SSSR (N.S.), 71:825-828, 1950. ↑23\n\nThe cohomology ring of weight varieties and polygon spaces. Rebecca F Goldin, Advances in Mathematics. 160227Rebecca F. Goldin. The cohomology ring of weight varieties and polygon spaces. Advances in Mathematics, 160(2):175-204, 2001. ↑27\n\nQuantized frame expansions with erasures. K Vivek, Jelena Goyal, Jonathan A Kovačević, Kelner, Applied and Computational Harmonic Analysis. 1031Vivek K. Goyal, Jelena Kovačević, and Jonathan A. Kelner. Quantized frame expansions with erasures. Applied and Computational Harmonic Analysis, 10(3):203-233, 2001. ↑1\n\nConvexity properties of the moment mapping. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67318Victor Guillemin and Shlomo Sternberg. Convexity properties of the moment mapping. Inventiones Mathemati- cae, 67(3):491-513, 1982. ↑18\n\nGeometric quantization and multiplicities of group representations. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67327Victor Guillemin and Shlomo Sternberg. Geometric quantization and multiplicities of group representations. Inventiones Mathematicae, 67(3):515-538, 1982. ↑27\n\nThe Gelfand-Cetlin system and quantization of the complex flag manifolds. Victor Guillemin, Shlomo Sternberg, Journal of Functional Analysis. 52123Victor Guillemin and Shlomo Sternberg. The Gelfand-Cetlin system and quantization of the complex flag man- ifolds. Journal of Functional Analysis, 52(1):106-128, 1983. ↑23\n\nPolytopes of eigensteps of finite equal norm tight frames. Discrete & Computational Geometry. Tim Haga, Christoph Pegel, 5624Tim Haga and Christoph Pegel. Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computa- tional Geometry, 56(3):727-742, 2016. ↑2, ↑22, ↑24\n\nHyperkähler metrics and supersymmetry. Nigel James Hitchin, Anders Karlhede, Ulf Lindström, Martin Roček, Communications in Mathematical Physics. 108418Nigel James Hitchin, Anders Karlhede, Ulf Lindström, and Martin Roček. Hyperkähler metrics and supersym- metry. Communications in Mathematical Physics, 108(4):535-589, 1987. ↑18\n\nOptimal frames for erasures. B Roderick, Vern I Holmes, Paulsen, Linear Algebra and its Applications. 3771Roderick B. Holmes and Vern I. Paulsen. Optimal frames for erasures. Linear Algebra and its Applications, 377:31-51, 2004. ↑1\n\nDoubly stochastic matrices and the diagonal of a rotation matrix. Alfred Horn, American Journal of Mathematics. 76325Alfred Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathematics, 76(3):620-630, 1954. ↑11, ↑25\n\nThe Chevalley involution and a duality of weight varieties. J Benjamin, John J Howard, Millson, The Asian Journal of Mathematics. 8427Benjamin J. Howard and John J. Millson. The Chevalley involution and a duality of weight varieties. The Asian Journal of Mathematics, 8(4):685-732, 2004. ↑27\n\nThe length of vectors in representation spaces. George Kempf, Linda Ness, Lecture Notes in Mathematics. Knud Lønsted73227Springer-VerlagAlgebraic GeometryGeorge Kempf and Linda Ness. The length of vectors in representation spaces. In Knud Lønsted, editor, Alge- braic Geometry, volume 732 of Lecture Notes in Mathematics, pages 233-243. Springer-Verlag, Berlin, 1979. ↑27\n\nCohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Frances Kirwan, Princeton University Press627Princeton, NJFrances Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1984. ↑6, ↑27\n\nConvexity properties of the moment mapping. Frances Kirwan, III. Inventiones Mathematicae. 7736Frances Kirwan. Convexity properties of the moment mapping. III. Inventiones Mathematicae, 77(3):547-552, 1984. ↑6\n\nWeight Varieties. Allen Knutson, 2627Massachusetts Institute of TechnologyPhD thesisAllen Knutson. Weight Varieties. PhD thesis, Massachusetts Institute of Technology, 1996. ↑26, ↑27\n\nThe symplectic and algebraic geometry of Horn's problem. Allen Knutson, Linear Algebra and its Applications. 3191-310Allen Knutson. The symplectic and algebraic geometry of Horn's problem. Linear Algebra and its Applications, 319(1-3):61-81, 2000. ↑10\n\nIntroduction to Smooth Manifolds. John M Lee, Graduate Texts in Mathematics. 21813Springersecond editionJohn M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, NY, second edition, 2013. ↑9, ↑13\n\nReduction of symplectic manifolds with symmetry. Jerrold E Marsden, Alan Weinstein, Reports on Mathematical Physics. 516Jerrold E. Marsden and Alan Weinstein. Reduction of symplectic manifolds with symmetry. Reports on Mathe- matical Physics, 5(1):121-130, 1974. ↑5, ↑6\n\nInequalities: Theory of Majorization and Its Applications. Albert W Marshall, Ingram Olkin, Barry C Arnold, Springer Series in Statistics. 3Springersecond editionAlbert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: Theory of Majorization and Its Applica- tions. Springer Series in Statistics. Springer, New York, NY, second edition, 2011. ↑3\n\nIntroduction to Symplectic Topology. Dusa Mcduff, Dietmar Salamon, Oxford University Press1215third edition, 2017. ↑4, ↑5, ↑10Dusa McDuff and Dietmar Salamon. Introduction to Symplectic Topology. Oxford University Press, third edition, 2017. ↑4, ↑5, ↑10, ↑12, ↑15\n\nSymmetries and integrals in mechanics. Kenneth R Meyer, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia. Mauricio M. PeixotoSalvador, Brasil; New York, NYAcademic Press56Kenneth R. Meyer. Symmetries and integrals in mechanics. In Mauricio M. Peixoto, editor, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26-August 14, 1971, pages 259-272. Academic Press, New York, NY, 1973. ↑5, ↑6\n\nGeometric Invariant Theory. David Mumford, John Fogarty, Frances Kirwan, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3427Springer-VerlagDavid Mumford, John Fogarty, and Frances Kirwan. Geometric Invariant Theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1994. ↑24, ↑27\n\nSymplectic geometry and connectivity of spaces of frames. Tom Needham, Clayton Shonkwiler, Advances in Computational Mathematics. 47128Tom Needham and Clayton Shonkwiler. Symplectic geometry and connectivity of spaces of frames. Advances in Computational Mathematics, 47(1):5, 2021. ↑7, ↑8, ↑9, ↑28\n\nAdmissibility and frame homotopy for quaternionic frames. Tom Needham, Clayton Shonkwiler, Linear Algebra and its Applications. 64527Tom Needham and Clayton Shonkwiler. Admissibility and frame homotopy for quaternionic frames. Linear Algebra and its Applications, 645:237-255, 2022. ↑27\n\nA stratification of the null cone via the moment map. Linda Ness, American Journal of Mathematics. 106627Linda Ness. A stratification of the null cone via the moment map. American Journal of Mathematics, 106(6):1281-1329, 1984. ↑27\n\nUber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Issai Schur, Sitzungsberichte der Berliner Mathematischen Gesellschaft. 2225Issai Schur. Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungs- berichte der Berliner Mathematischen Gesellschaft, 22:9-20, 1923. ↑11, ↑25\n\nHolomorphic slices, symplectic reduction and multiplicities of representations. Reyer Sjamaar, The Annals of Mathematics, Second Series. 141127Reyer Sjamaar. Holomorphic slices, symplectic reduction and multiplicities of representations. The Annals of Mathematics, Second Series, 141(1):87-129, 1995. ↑27\n\nStratified symplectic spaces and reduction. Reyer Sjamaar, Eugene Lerman, The Annals of Mathematics, Second Series. 134217Reyer Sjamaar and Eugene Lerman. Stratified symplectic spaces and reduction. The Annals of Mathematics, Second Series, 134(2):375-422, 1991. ↑6, ↑11, ↑13, ↑17\n\nFinite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Nate Strawn, Journal of Fourier Analysis and Applications. 17512Nate Strawn. Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Journal of Fourier Analysis and Applications, 17(5):821-853, 2010. ↑1, ↑4, ↑12\n\nAn Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. F D Shayne, Waldron, Birkhäuser9New York, NYShayne F. D. Waldron. An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY, 2018. ↑9\n"},"corpus_id":{"kind":"number","value":239049635,"string":"239,049,635"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2110.11295\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1016/j.acha.2022.07.004\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2110.11295v2.pdf\"\n ],\n \"pdf_sha\": \"bd26916ac408f5d3bf3e637b42445163ab8c3601\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/2110.11295\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Toric Symplectic Geometry and Full Spark Frames"},"authors":{"kind":"list like","value":["Tom Needham","Clayton Shonkwiler \nDepartment of Mathematics\nColorado State University\nFort CollinsCO","Department of Mathematics\nFlorida State University\nTallahasseeFL","Tom Needham","Clayton Shonkwiler \nDepartment of Mathematics\nColorado State University\nFort CollinsCO","Department of Mathematics\nFlorida State University\nTallahasseeFL"],"string":"[\n \"Tom Needham\",\n \"Clayton Shonkwiler \\nDepartment of Mathematics\\nColorado State University\\nFort CollinsCO\",\n \"Department of Mathematics\\nFlorida State University\\nTallahasseeFL\",\n \"Tom Needham\",\n \"Clayton Shonkwiler \\nDepartment of Mathematics\\nColorado State University\\nFort CollinsCO\",\n \"Department of Mathematics\\nFlorida State University\\nTallahasseeFL\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The collection of d × N complex matrices with prescribed column norms and prescribed (nonzero) singular values forms a compact algebraic variety, which we refer to as a frame space. Elements of frame spaces-i.e., frames-are used to give robust representations of complex-valued signals, so that geometrical and measure-theoretic properties of frame spaces are of interest to the signal processing community. This paper is concerned with the following question: what is the probability that a frame drawn uniformly at random from a given frame space has the property that any subset of d of its columns gives a basis for C d ? We show that the probability is one, generalizing recent work of Cahill, Mixon, and Strawn. To prove this, we first show that frame spaces are related to highly structured objects called toric symplectic manifolds. This relationship elucidates the geometric meaning of eigensteps-certain spectral invariants of a frame-and should be a more broadly applicable tool for studying probabilistic questions about the structure of frame spaces. As another application of our symplectic perspective, we completely characterize the norm and spectral data for which the corresponding frame space has singularities, answering some open questions in the frame theory literature."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":""},{"title":"Introduction","content":"A frame in a Hilbert space (H,⟨⋅,⋅⟩) is traditionally defined as a collection { f i } i∈I of vectors in H so that for all v ∈ H we have\nA frame in a Hilbert space (H,⟨⋅,⋅⟩) is traditionally defined as a collection { f i } i∈I of vectors in H so that for all v ∈ H we have\nfor some numbers 0 < a ≤ b called frame bounds. In this paper, we focus on finite frames in complex Hilbert spaces, in which case H = C d for some integer d > 0 with its standard Hermitian inner product and norm, I = {1,...,N} is finite, and the above condition is equivalent to the collection { f 1 ,... , f N } being a spanning set for C d . Interest in finite frames is largely due to their application to robust signal representation. Modeling a signal as an element of a Hilbert space H, a frame allows one to take a sequence of \"measurements\" by recording the inner product of the signal with each of the frame vectors. This signal representation is more robust to noise in the signal or random erasures of measurements than a similar measurement scheme associated to an orthonormal basis, at least when the frame has certain properties [11,23,29]. These desirable properties for a frame are typically expressed as prescriptions for the norms of the frame vectors and for the spectrum of an operator associated to the frame, which we describe in more detail below. The collection of frames with prescribed norm and spectral data is easily seen to define an algebraic variety, and there is interest in the geometric structure of these frame varieties [8,18,49]. This paper explores the geometry of frame varieties through the lens of symplectic geometry. Symplectic geometry is a subfield of differential geometry that studies manifolds endowed with a certain geometric structure-to keep the paper accessible to the broader frame theory community, we have aimed to give self-contained expositions of the relevant ideas from symplectic geometry, but we will not go into details here in the introduction.\nfor some numbers 0 < a ≤ b called frame bounds. In this paper, we focus on finite frames in complex Hilbert spaces, in which case H = C d for some integer d > 0 with its standard Hermitian inner product and norm, I = {1,...,N} is finite, and the above condition is equivalent to the collection { f 1 ,... , f N } being a spanning set for C d . Interest in finite frames is largely due to their application to robust signal representation. Modeling a signal as an element of a Hilbert space H, a frame allows one to take a sequence of \"measurements\" by recording the inner product of the signal with each of the frame vectors. This signal representation is more robust to noise in the signal or random erasures of measurements than a similar measurement scheme associated to an orthonormal basis, at least when the frame has certain properties [11,23,29]. These desirable properties for a frame are typically expressed as prescriptions for the norms of the frame vectors and for the spectrum of an operator associated to the frame, which we describe in more detail below. The collection of frames with prescribed norm and spectral data is easily seen to define an algebraic variety, and there is interest in the geometric structure of these frame varieties [8,18,49]. This paper explores the geometry of frame varieties through the lens of symplectic geometry. Symplectic geometry is a subfield of differential geometry that studies manifolds endowed with a certain geometric structure-to keep the paper accessible to the broader frame theory community, we have aimed to give self-contained expositions of the relevant ideas from symplectic geometry, but we will not go into details here in the introduction.\nThe main contributions of this paper are as follows:\nThe main contributions of this paper are as follows:\n• Our main theorem, Theorem 1.1, says that with probability 1 every size-d subset of a random frame in C d with prescribed norms and spectral data is a basis. Frames satisfying this non-degeneracy condition are called full spark frames. This generalizes the complex case of Theorem 1.6 from Cahill, Mixon, and Strawn's paper [8], which established the genericity of the full spark condition for frames whose frame bounds are equal (this can be stated as a spectral constraint) and whose frame vectors are all unit-this is referred to as the space of finite unit-norm tight frames or FUNTFs.\n• Our main theorem, Theorem 1.1, says that with probability 1 every size-d subset of a random frame in C d with prescribed norms and spectral data is a basis. Frames satisfying this non-degeneracy condition are called full spark frames. This generalizes the complex case of Theorem 1.6 from Cahill, Mixon, and Strawn's paper [8], which established the genericity of the full spark condition for frames whose frame bounds are equal (this can be stated as a spectral constraint) and whose frame vectors are all unit-this is referred to as the space of finite unit-norm tight frames or FUNTFs.\n• We show in Theorem 2.25 that the space of frames with prescribed norm and spectral data is a smooth manifold, for generic choices of this data. In fact, we give necessary and sufficient conditions on the norms and eigenvalues which guarantee that the corresponding frame variety is smooth, and we describe the local geometry of singular varieties near their singular points. This theorem (together with surrounding results) generalizes work of Dykema and Strawn [18], which once again specializes to the space of FUNTFs, and answers generalizations of two open questions posed in [8].\n• We show in Theorem 2.25 that the space of frames with prescribed norm and spectral data is a smooth manifold, for generic choices of this data. In fact, we give necessary and sufficient conditions on the norms and eigenvalues which guarantee that the corresponding frame variety is smooth, and we describe the local geometry of singular varieties near their singular points. This theorem (together with surrounding results) generalizes work of Dykema and Strawn [18], which once again specializes to the space of FUNTFs, and answers generalizations of two open questions posed in [8].\n• Both of the previous two results are proved by novel applications of ideas from symplectic geometry to frame theory. Throughout the course of the paper, we show that many spaces of complex frames have natural interpretations from the symplectic point of view. In particular, we show in Theorem 3.20 that each space of frames with prescribed spectral and norm data has a dense open subset which projects onto a highly structured geometric object called a toric symplectic manifold. This geometric structure has measure-theoretic implications-we use it to prove our full spark theorem, but expect that it will be a useful tool for the future study of probability theory on frame spaces. Theorem 3.20 generalizes work of Flaschka and Millson [19], which is written in the context of pure symplectic geometry and makes no references to frames. The theorem also gives a symplectic interpretation of the frame-theoretic concept of eigensteps, introduced by Cahill et al. [7], and an auxiliary result used in its proof generalizes a theorem of Haga and Pegel [27].\n• Both of the previous two results are proved by novel applications of ideas from symplectic geometry to frame theory. Throughout the course of the paper, we show that many spaces of complex frames have natural interpretations from the symplectic point of view. In particular, we show in Theorem 3.20 that each space of frames with prescribed spectral and norm data has a dense open subset which projects onto a highly structured geometric object called a toric symplectic manifold. This geometric structure has measure-theoretic implications-we use it to prove our full spark theorem, but expect that it will be a useful tool for the future study of probability theory on frame spaces. Theorem 3.20 generalizes work of Flaschka and Millson [19], which is written in the context of pure symplectic geometry and makes no references to frames. The theorem also gives a symplectic interpretation of the frame-theoretic concept of eigensteps, introduced by Cahill et al. [7], and an auxiliary result used in its proof generalizes a theorem of Haga and Pegel [27].\nTo describe our results in detail, we now introduce more precise terminology and notation.\nTo describe our results in detail, we now introduce more precise terminology and notation.\nNotation and statement of the main theorem. Let F d,N be the space of frames of N vectors in C d . Identifying a frame { f i } N i=1 ∈ F d,N with the d × N matrix whose columns are the f i represented in the standard basis, the space F d,N can be viewed as an open, dense subset of the space C d×N of d × N complex matrices.\nNotation and statement of the main theorem. Let F d,N be the space of frames of N vectors in C d . Identifying a frame { f i } N i=1 ∈ F d,N with the d × N matrix whose columns are the f i represented in the standard basis, the space F d,N can be viewed as an open, dense subset of the space C d×N of d × N complex matrices.\nWhen a = b in (1), the frame satisfies a scaled Parseval identity\nWhen a = b in (1), the frame satisfies a scaled Parseval identity\nand such frames are called a-tight (or just tight); 1-tight frames are called Parseval frames.\nand such frames are called a-tight (or just tight); 1-tight frames are called Parseval frames.\nEach frame { f i } N i=1 ∈ F d,N has three related operators, which can be interpreted in terms of the d × N matrix F = f 1 f 2 ... f N :\nEach frame { f i } N i=1 ∈ F d,N has three related operators, which can be interpreted in terms of the d × N matrix F = f 1 f 2 ... f N :\n1. The analysis operator C d → C N is defined by v ↦ (⟨v, f 1 ⟩,... ,⟨v, f N ⟩) , or equivalently v ↦ F * v, where F * is the Hermitian adjoint (i.e., conjugate transpose) of F; 2. The synthesis operator C N → C d is defined by\n1. The analysis operator C d → C N is defined by v ↦ (⟨v, f 1 ⟩,... ,⟨v, f N ⟩) , or equivalently v ↦ F * v, where F * is the Hermitian adjoint (i.e., conjugate transpose) of F; 2. The synthesis operator C N → C d is defined by\n3. The frame operator C d → C d is the composition of the analysis and synthesis operators; i.e., v ↦ FF * v.\n3. The frame operator C d → C d is the composition of the analysis and synthesis operators; i.e., v ↦ FF * v.\nA simple calculation shows that a frame is a-tight if and only if its frame operator is aI d , where I d is the identity map on C d .\nA simple calculation shows that a frame is a-tight if and only if its frame operator is aI d , where I d is the identity map on C d .\nFrame operators are always Hermitian and positive-definite, so they have spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, which we will call the frame spectrum. If λ λ λ = (λ 1 ,...,λ d ), we will use F d,N λ λ λ to indicate the frames with frame spectrum λ λ λ . Notice that the a-tight frames are uniquely specified by their frame spectra: F d,N (a,...,a) is the space of all a-tight frames. In addition to specifying a frame operator, we also often want to fix the (squared) norms of the individual frame vectors. We can always permute the labels on the frame vectors, so it will be convenient in what follows to assume the norms are sorted in decreasing order. If r r r = (r 1 ,... ,r N ) is a non-increasing list of positive numbers r 1 ≥ ⋅⋅⋅ ≥ r N > 0, 1 1 We could also allow some of the r i to be zero, but this would complicate some statements below to no apparent benefit. we will use F d,N (r r r) to indicate the space of frames with f i 2 = r i . It is especially common to require that all the frame vectors have the same norm: f i 2 = r > 0 for all i, in which case the frame is an equal-norm frame; if r = 1, this is a unit-norm frame.\nFrame operators are always Hermitian and positive-definite, so they have spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, which we will call the frame spectrum. If λ λ λ = (λ 1 ,...,λ d ), we will use F d,N λ λ λ to indicate the frames with frame spectrum λ λ λ . Notice that the a-tight frames are uniquely specified by their frame spectra: F d,N (a,...,a) is the space of all a-tight frames. In addition to specifying a frame operator, we also often want to fix the (squared) norms of the individual frame vectors. We can always permute the labels on the frame vectors, so it will be convenient in what follows to assume the norms are sorted in decreasing order. If r r r = (r 1 ,... ,r N ) is a non-increasing list of positive numbers r 1 ≥ ⋅⋅⋅ ≥ r N > 0, 1 1 We could also allow some of the r i to be zero, but this would complicate some statements below to no apparent benefit. we will use F d,N (r r r) to indicate the space of frames with f i 2 = r i . It is especially common to require that all the frame vectors have the same norm: f i 2 = r > 0 for all i, in which case the frame is an equal-norm frame; if r = 1, this is a unit-norm frame.\nIn general, the frame norms determine the trace of the frame operator:\nIn general, the frame norms determine the trace of the frame operator:\nby the cyclic invariance of trace. Hence, for tight frames the frame operator must be r r r 2 d I d and a unit-norm tight frame must have λ i = N d for all i. If r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) with r 1 ≥ ⋅⋅⋅ ≥ r N > 0 and λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, we will use F d,N λ λ λ (r r r) ∶= F d,N λ λ λ ∩ F d,N (r r r) to denote the space of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ . This space has a natural probability measure: the (normalized) Hausdorff measure it inherits as a compact subset of C d×N . Equation (2) is not the only restriction imposed on the frame vector norms by the frame operator: the partial sums of the squared frame vector norms must be bounded above by the partial sums of the eigenvalues of the frame operator [12,19]. More precisely, there exists a frame { f i } N i=1 with f i 2 = r i > 0 and frame spectrum λ λ λ if and only if (2) holds and, for all k = 1,... ,d,\nby the cyclic invariance of trace. Hence, for tight frames the frame operator must be r r r 2 d I d and a unit-norm tight frame must have λ i = N d for all i. If r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) with r 1 ≥ ⋅⋅⋅ ≥ r N > 0 and λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, we will use F d,N λ λ λ (r r r) ∶= F d,N λ λ λ ∩ F d,N (r r r) to denote the space of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ . This space has a natural probability measure: the (normalized) Hausdorff measure it inherits as a compact subset of C d×N . Equation (2) is not the only restriction imposed on the frame vector norms by the frame operator: the partial sums of the squared frame vector norms must be bounded above by the partial sums of the eigenvalues of the frame operator [12,19]. More precisely, there exists a frame { f i } N i=1 with f i 2 = r i > 0 and frame spectrum λ λ λ if and only if (2) holds and, for all k = 1,... ,d,\nGiven λ λ λ , we will call a non-increasing list r r r of positive numbers satisfying (2) and (3) λ λ λ -admissible, and if all inequalities in (3) are strict we call the list strongly λ λ λ -admissible. This terminology comes from Casazza and Leon [12], who showed that F d,N λ λ λ (r r r) is non-empty if and only if r r r is λ λ λ -admissible. Although the terminology does not appear in the work of Casazza and Leon, in general one says that the vector λ λ λ majorizes the vector r r r in the case that the relations (3) hold [39].\nGiven λ λ λ , we will call a non-increasing list r r r of positive numbers satisfying (2) and (3) λ λ λ -admissible, and if all inequalities in (3) are strict we call the list strongly λ λ λ -admissible. This terminology comes from Casazza and Leon [12], who showed that F d,N λ λ λ (r r r) is non-empty if and only if r r r is λ λ λ -admissible. Although the terminology does not appear in the work of Casazza and Leon, in general one says that the vector λ λ λ majorizes the vector r r r in the case that the relations (3) hold [39].\nis the size of the smallest linearly dependent subset [16]. The spark is bounded above by d + 1, and a frame with spark equal to d + 1 is called a full-spark frame. Equivalently, a frame in F d,N is full spark if and only if all of its size-d subsets are bases. Full spark frames are often desirable, for example because they provide unique reconstructions of the largest possible class of sparse signals [2,16].\nis the size of the smallest linearly dependent subset [16]. The spark is bounded above by d + 1, and a frame with spark equal to d + 1 is called a full-spark frame. Equivalently, a frame in F d,N is full spark if and only if all of its size-d subsets are bases. Full spark frames are often desirable, for example because they provide unique reconstructions of the largest possible class of sparse signals [2,16].\nWe are now ready to state our main theorem:\nWe are now ready to state our main theorem:\nLet r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) be nonincreasing lists of positive real numbers. There are three mutually exclusive possibilities for the space F d,N λ λ λ (r r r) of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ :\nLet r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) be nonincreasing lists of positive real numbers. There are three mutually exclusive possibilities for the space F d,N λ λ λ (r r r) of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ :\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\n2. If r r r is λ λ λ -admissible but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\n2. If r r r is λ λ λ -admissible but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\nIn particular, since r r r = (1,. Structure of the paper. Section 2 begins with an exposition of the relevant ideas and tools from symplectic geometry. These tools are then applied to show that various spaces of frames with prescribed data have symplectic structures. Then, parameters λ λ λ and r r r for which the frame variety F d,N λ λ λ (r r r) is a smooth manifold are characterized. Section 3 is devoted to showing that the space F d,N λ λ λ (r r r) has a dense open subset which projects onto a toric symplectic manifold (we define and describe the basic properties of such an object in Section 3.1). The main theorem is proved in Section 4, as an application of the toric symplectic structure from the previous section. An alternative proof using related tools from algebraic geometry is also sketched. The paper concludes with a brief discussion of future directions in Section 5.\nIn particular, since r r r = (1,. Structure of the paper. Section 2 begins with an exposition of the relevant ideas and tools from symplectic geometry. These tools are then applied to show that various spaces of frames with prescribed data have symplectic structures. Then, parameters λ λ λ and r r r for which the frame variety F d,N λ λ λ (r r r) is a smooth manifold are characterized. Section 3 is devoted to showing that the space F d,N λ λ λ (r r r) has a dense open subset which projects onto a toric symplectic manifold (we define and describe the basic properties of such an object in Section 3.1). The main theorem is proved in Section 4, as an application of the toric symplectic structure from the previous section. An alternative proof using related tools from algebraic geometry is also sketched. The paper concludes with a brief discussion of future directions in Section 5."},{"title":"Symplectic Structure of Frame Varieties","content":""},{"title":"Symplectic Structure of Frame Varieties","content":"In this section, we describe the symplectic structures on the frame varieties of interest; i.e., spaces of frames with prescribed frame spectra and/or norms. It has been previously observed in the literature that certain parameter choices may lead to frame spaces with singularities [8,18,49]. We give a precise characterization of this phenomenon and describe the local structure of singular points later in this section.\nIn this section, we describe the symplectic structures on the frame varieties of interest; i.e., spaces of frames with prescribed frame spectra and/or norms. It has been previously observed in the literature that certain parameter choices may lead to frame spaces with singularities [8,18,49]. We give a precise characterization of this phenomenon and describe the local structure of singular points later in this section."},{"title":"Symplectic Geometry","content":""},{"title":"Symplectic Geometry","content":"We begin with a review of some concepts of symplectic geometry, with a focus on various notions of reducing a symplectic manifold by a Lie group action. We use [40] as our main reference for the basics of symplectic manifolds. The ideas we present are standard in the field of symplectic geometry; this subsection is mainly intended as a quick reference for non-experts and to standardize our notation.\nWe begin with a review of some concepts of symplectic geometry, with a focus on various notions of reducing a symplectic manifold by a Lie group action. We use [40] as our main reference for the basics of symplectic manifolds. The ideas we present are standard in the field of symplectic geometry; this subsection is mainly intended as a quick reference for non-experts and to standardize our notation.\nA symplectic form on a (smooth, real) manifold M is a closed, nondegenerate 2-form ω. For a point p ∈ M and tangent vectors X,Y ∈ T p M, we write ω p (X,Y ) ∈ R for the evaluation of ω on the vectors. The closedness condition means that the exterior derivative dω is identically zero and the nondegeneracy condition means that for every nonzero X ∈ T p M there exists Y ∈ T p M such that ω p (X,Y ) ≠ 0. A manifold endowed with a symplectic form is called a symplectic manifold, denoted (M,ω) or simply as M when the symplectic form is understood to be fixed. A simple argument shows that if a manifold M admits a symplectic form, then it must be even dimensional (over the reals).\nA symplectic form on a (smooth, real) manifold M is a closed, nondegenerate 2-form ω. For a point p ∈ M and tangent vectors X,Y ∈ T p M, we write ω p (X,Y ) ∈ R for the evaluation of ω on the vectors. The closedness condition means that the exterior derivative dω is identically zero and the nondegeneracy condition means that for every nonzero X ∈ T p M there exists Y ∈ T p M such that ω p (X,Y ) ≠ 0. A manifold endowed with a symplectic form is called a symplectic manifold, denoted (M,ω) or simply as M when the symplectic form is understood to be fixed. A simple argument shows that if a manifold M admits a symplectic form, then it must be even dimensional (over the reals).\nExample 2.1 (Complex n-Space). The prototypical example of a symplectic manifold is n-dimensional complex space C n , which is considered as a 2n-dimensional real manifold via the natural identification C n ≈ R 2n . For any n-tuple p of complex numbers, there is a natural isomorphism T p C n ≈ C n . Coordinates (x 1 + √ −1y 1 ,... ,x n + √ −1y n ) for C n correspond to real coordinates (x 1 ,... ,x n ,y 1 ,... ,y n ) in which a symplectic form is given by\nExample 2.1 (Complex n-Space). The prototypical example of a symplectic manifold is n-dimensional complex space C n , which is considered as a 2n-dimensional real manifold via the natural identification C n ≈ R 2n . For any n-tuple p of complex numbers, there is a natural isomorphism T p C n ≈ C n . Coordinates (x 1 + √ −1y 1 ,... ,x n + √ −1y n ) for C n correspond to real coordinates (x 1 ,... ,x n ,y 1 ,... ,y n ) in which a symplectic form is given by\nThis is referred to as the standard symplectic form on C n . In complex coordinates, (4) is expressed concretely for p ∈ C n and Z = (z 1 ,... ,z n ),\nThis is referred to as the standard symplectic form on C n . In complex coordinates, (4) is expressed concretely for p ∈ C n and Z = (z 1 ,... ,z n ),\nwhere Im denotes the imaginary part of a complex number and ⟨⋅,⋅⟩ is the standard Hermitian inner product on C n . The most important example of a symplectic manifold for our purposes is complex matrix space C d×N , which is just a reshaped version of the space C d⋅N described above. For a matrix F ∈ C d×N , we have a natural identification\nwhere Im denotes the imaginary part of a complex number and ⟨⋅,⋅⟩ is the standard Hermitian inner product on C n . The most important example of a symplectic manifold for our purposes is complex matrix space C d×N , which is just a reshaped version of the space C d⋅N described above. For a matrix F ∈ C d×N , we have a natural identification\nThis is just a transformation of the canonical symplectic form on C d⋅N under the reshaping map.\nThis is just a transformation of the canonical symplectic form on C d⋅N under the reshaping map.\nSince the space of frames F d,N ⊂ C d×N is an open submanifold, the standard symplectic structure on C d×N restricts to make F d,N a symplectic manifold.\nSince the space of frames F d,N ⊂ C d×N is an open submanifold, the standard symplectic structure on C d×N restricts to make F d,N a symplectic manifold.\nIn fact, every 2n-dimensional symplectic manifold is locally equivalent to C n with the standard symplectic form. Let us now make this statement precise. If Ψ ∶ N → M is a smooth map from a manifold N to a symplectic manifold (M,ω), the pullback form Ψ * ω on N is defined by\nIn fact, every 2n-dimensional symplectic manifold is locally equivalent to C n with the standard symplectic form. Let us now make this statement precise. If Ψ ∶ N → M is a smooth map from a manifold N to a symplectic manifold (M,ω), the pullback form Ψ * ω on N is defined by\nfor X,Y ∈ T p N and where DΨ(p) ∶ T p N → T Ψ(p) M denotes the derivative of Ψ at p. If (N,η) is also a symplectic manifold and Ψ is a diffeomorphism with the property that Ψ * ω = η, then we say Ψ is a symplectomorphism and that (N,η) and (M,ω) are symplectomorphic. A fundamental result of symplectic geometry is Darboux's Theorem [40,Theorem 3.2.2]: every point in a 2n-dimensional symplectic manifold has an open neighborhood U such that the symplectic manifold (U,ω U ) is symplectomorphic to C n with the standard symplectic form.\nfor X,Y ∈ T p N and where DΨ(p) ∶ T p N → T Ψ(p) M denotes the derivative of Ψ at p. If (N,η) is also a symplectic manifold and Ψ is a diffeomorphism with the property that Ψ * ω = η, then we say Ψ is a symplectomorphism and that (N,η) and (M,ω) are symplectomorphic. A fundamental result of symplectic geometry is Darboux's Theorem [40,Theorem 3.2.2]: every point in a 2n-dimensional symplectic manifold has an open neighborhood U such that the symplectic manifold (U,ω U ) is symplectomorphic to C n with the standard symplectic form.\nAn important aspect of symplectic geometry is the study of interactions between symplectic structures and certain group actions on their manifolds. Let G be a Lie group with Lie algebra g and suppose that G acts on a manifold M endowed with a symplectic form ω. For p ∈ M and g ∈ G, let g ⋅ p ∈ M denote the action of g on p. To each ξ ∈ g, one associates an infinitesimal vector field Y ξ on M via the formula\nAn important aspect of symplectic geometry is the study of interactions between symplectic structures and certain group actions on their manifolds. Let G be a Lie group with Lie algebra g and suppose that G acts on a manifold M endowed with a symplectic form ω. For p ∈ M and g ∈ G, let g ⋅ p ∈ M denote the action of g on p. To each ξ ∈ g, one associates an infinitesimal vector field Y ξ on M via the formula\nwhere g * denotes the dual to g, is called a momentum map for the G-action if its derivative interacts with the symplectic form as follows. Let DΦ(p) ∶ T p M → T Φ(p) g * denote the derivative of Φ at p ∈ M. Then for each X ∈ T p M, DΦ(p)(X) ∈ T Φ(p) g * ≈ g * , where we use the natural isomorphism coming from the fact that g * is a vector space. Then DΦ(p)(X) ∶ g → R, and for each ξ ∈ g we require\nwhere g * denotes the dual to g, is called a momentum map for the G-action if its derivative interacts with the symplectic form as follows. Let DΦ(p) ∶ T p M → T Φ(p) g * denote the derivative of Φ at p ∈ M. Then for each X ∈ T p M, DΦ(p)(X) ∈ T Φ(p) g * ≈ g * , where we use the natural isomorphism coming from the fact that g * is a vector space. Then DΦ(p)(X) ∶ g → R, and for each ξ ∈ g we require\nWe also require that the momentum map Φ is equivariant, in the following sense. Recall that the adjoint action of G on g is defined, for each g ∈ G, by the map Ad g ∶ g → g which is the derivative at the identity of the conjugation map h ↦ ghg −1 . The corresponding coadjoint action of G on the dual Lie algebra g * is defined, for each g ∈ G, by the map Ad * g ∶ g * → g * given by Ad * g (χ)(ξ ) ∶= χ(Ad g −1 (ξ )). When G is a matrix group, both the adjoint and coadjoint actions can be interpreted as conjugation actions. The momentum map is required to be equivariant with respect to the given G-action on M and the coadjoint action on g * . Explicitly, this means that, for each g ∈ G and each p ∈ M,\nWe also require that the momentum map Φ is equivariant, in the following sense. Recall that the adjoint action of G on g is defined, for each g ∈ G, by the map Ad g ∶ g → g which is the derivative at the identity of the conjugation map h ↦ ghg −1 . The corresponding coadjoint action of G on the dual Lie algebra g * is defined, for each g ∈ G, by the map Ad * g ∶ g * → g * given by Ad * g (χ)(ξ ) ∶= χ(Ad g −1 (ξ )). When G is a matrix group, both the adjoint and coadjoint actions can be interpreted as conjugation actions. The momentum map is required to be equivariant with respect to the given G-action on M and the coadjoint action on g * . Explicitly, this means that, for each g ∈ G and each p ∈ M,\nIf a G-action admits a momentum map, then we say the action is Hamiltonian. Hamiltonian actions give the appropriate setting for performing a quotient operation in the symplectic category. [38,41]). Let (M,ω) be a symplectic manifold with a Hamiltonian G-action, let Φ ∶ M → g * be a momentum map for the action and let χ ∈ g * be a regular value such that G acts freely on the level set Φ −1 (χ). Then the manifold\nIf a G-action admits a momentum map, then we say the action is Hamiltonian. Hamiltonian actions give the appropriate setting for performing a quotient operation in the symplectic category. [38,41]). Let (M,ω) be a symplectic manifold with a Hamiltonian G-action, let Φ ∶ M → g * be a momentum map for the action and let χ ∈ g * be a regular value such that G acts freely on the level set Φ −1 (χ). Then the manifold\ncalled the symplectic reduction or symplectic quotient over χ, admits a symplectic structure ω red which is uniquely characterized by the equation\ncalled the symplectic reduction or symplectic quotient over χ, admits a symplectic structure ω red which is uniquely characterized by the equation\nSee [5, Example III.2.18] for a construction of the complex projective space CP n−1 as a symplectic reduction C n ξ U(1), where U(1) acts on C n by scalar multiplication. Notice that, in the statement of Theorem 2.2, we required that G acts freely on the level set Φ −1 (χ). In particular, G acts on this level set, meaning that for any p ∈ M with Φ(p) = χ, it must be the case that Φ(g ⋅ p) = χ for all g ∈ G.\nSee [5, Example III.2.18] for a construction of the complex projective space CP n−1 as a symplectic reduction C n ξ U(1), where U(1) acts on C n by scalar multiplication. Notice that, in the statement of Theorem 2.2, we required that G acts freely on the level set Φ −1 (χ). In particular, G acts on this level set, meaning that for any p ∈ M with Φ(p) = χ, it must be the case that Φ(g ⋅ p) = χ for all g ∈ G.\nBy the equivariance required in the definition of the momentum map Φ, this implies that Ad * g (χ) = χ for all g ∈ G; in other words, χ must be a fixed point of the coadjoint action of G. Conversely, G will act on the level set over any fixed point of the coadjoint action.\nBy the equivariance required in the definition of the momentum map Φ, this implies that Ad * g (χ) = χ for all g ∈ G; in other words, χ must be a fixed point of the coadjoint action of G. Conversely, G will act on the level set over any fixed point of the coadjoint action.\nMore generally, when χ ∈ g * is not a fixed point of the coadjoint action, we can still take a symplectic reduction over the coadjoint orbit of χ, defined to be the set\nMore generally, when χ ∈ g * is not a fixed point of the coadjoint action, we can still take a symplectic reduction over the coadjoint orbit of χ, defined to be the set\nIn this case the equivariance of the momentum map ensures that G acts on Φ −1 (O χ ). [38,41]). Let M, ω, G and Φ be as above and let χ ∈ g * be a regular value. If the action of G on Φ −1 (O χ ) is free, then the manifold\nIn this case the equivariance of the momentum map ensures that G acts on Φ −1 (O χ ). [38,41]). Let M, ω, G and Φ be as above and let χ ∈ g * be a regular value. If the action of G on Φ −1 (O χ ) is free, then the manifold"},{"title":"Theorem 2.3 (Marsden-Weinstein-Meyer Theorem for Coadjoint Orbits","content":""},{"title":"Theorem 2.3 (Marsden-Weinstein-Meyer Theorem for Coadjoint Orbits","content":"admits a symplectic structure ω red which is uniquely characterized by q * ω red = ι * ω, where q denotes the projection\nadmits a symplectic structure ω red which is uniquely characterized by q * ω red = ι * ω, where q denotes the projection\nIt is straightforward to show that when χ ∈ g * is a regular value, the action of G on Φ −1 (O χ ) is at worst locally free. Theorem 2.3 easily generalizes to this setting, with the only modification being that M O χ G is a symplectic orbifold rather than a symplectic manifold.\nIt is straightforward to show that when χ ∈ g * is a regular value, the action of G on Φ −1 (O χ ) is at worst locally free. Theorem 2.3 easily generalizes to this setting, with the only modification being that M O χ G is a symplectic orbifold rather than a symplectic manifold.\nPerhaps surprisingly, if we are more concerned with the \"symplectic\" part of this statement than the \"manifold\" (or \"orbifold\") part, Sjamaar and Lerman [48] have shown that χ being a regular value is not an essential hypothesis in Theorems 2.2 and 2.3. When χ is a singular value of Φ then Φ −1 (O χ ) is not a manifold, but it nonetheless admits a G action and the quotient still admits a natural symplectic structure.\nPerhaps surprisingly, if we are more concerned with the \"symplectic\" part of this statement than the \"manifold\" (or \"orbifold\") part, Sjamaar and Lerman [48] have shown that χ being a regular value is not an essential hypothesis in Theorems 2.2 and 2.3. When χ is a singular value of Φ then Φ −1 (O χ ) is not a manifold, but it nonetheless admits a G action and the quotient still admits a natural symplectic structure.\nTheorem 2.4 (Sjamaar-Lerman [48]). Let M, ω, G and Φ be as above and let χ ∈ g * . The symplectic reduction\nTheorem 2.4 (Sjamaar-Lerman [48]). Let M, ω, G and Φ be as above and let χ ∈ g * . The symplectic reduction\nRoughly speaking, M O χ G being a symplectic stratified space means that it is the disjoint union of symplectic manifolds and that these manifolds fit together nicely. Somewhat remarkably, each connected component contains a unique manifold piece which is open and dense: Therefore, up to sets of lower dimension, the symplectic reduction even over a singular value of a proper momentum map is a connected symplectic manifold.\nRoughly speaking, M O χ G being a symplectic stratified space means that it is the disjoint union of symplectic manifolds and that these manifolds fit together nicely. Somewhat remarkably, each connected component contains a unique manifold piece which is open and dense: Therefore, up to sets of lower dimension, the symplectic reduction even over a singular value of a proper momentum map is a connected symplectic manifold."},{"title":"Frame Spaces as Symplectic Quotients","content":""},{"title":"Frame Spaces as Symplectic Quotients","content":"Various spaces of frames can be endowed with natural symplectic structures via Theorems 2.3 and 2.4. To be precise, the symplectic manifolds of interest are actually quotients of frame spaces by certain symmetry groups. Throughout this subsection, fix dimensions d and N and a frame spectrum λ λ λ = (λ 1 ,...,λ d ) with λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0.\nVarious spaces of frames can be endowed with natural symplectic structures via Theorems 2.3 and 2.4. To be precise, the symplectic manifolds of interest are actually quotients of frame spaces by certain symmetry groups. Throughout this subsection, fix dimensions d and N and a frame spectrum λ λ λ = (λ 1 ,...,λ d ) with λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0."},{"title":"Frames with Prescribed Frame Spectrum","content":""},{"title":"Frames with Prescribed Frame Spectrum","content":"The unitary group acts on F d,N λ λ λ by left matrix multiplication. We then consider the quotient space F d,N λ λ λ U(d), which has a natural symplectic structure, as we will show below.\nThe unitary group acts on F d,N λ λ λ by left matrix multiplication. We then consider the quotient space F d,N λ λ λ U(d), which has a natural symplectic structure, as we will show below.\nBefore stating the result, we observe that the space H(d) of d × d Hermitian matrices may be identified with the dual Lie algebra u(d) * via the isomorphism α ∶ H(d) → u(d) * taking ξ to the linear functional α ξ ∶ u(d) → R, defined by\nBefore stating the result, we observe that the space H(d) of d × d Hermitian matrices may be identified with the dual Lie algebra u(d) * via the isomorphism α ∶ H(d) → u(d) * taking ξ to the linear functional α ξ ∶ u(d) → R, defined by\nWe begin with a preliminary result about this isomorphism. We define a U(d)-action on H(d) which we (with a slight abuse of terminology) refer to as the adjoint action:\nWe begin with a preliminary result about this isomorphism. We define a U(d)-action on H(d) which we (with a slight abuse of terminology) refer to as the adjoint action:\nLemma 2.7. The map α defined in (5) is equivariant with respect to the adjoint action on H(d) and the coadjoint action on u(d) * . That is, α Ad A (ξ ) = Ad * A α ξ . It follows that, under the map α, the coadjoint orbit of any element of u(d) * is identified with the collection of Hermitian matrices with a fixed spectrum.\nLemma 2.7. The map α defined in (5) is equivariant with respect to the adjoint action on H(d) and the coadjoint action on u(d) * . That is, α Ad A (ξ ) = Ad * A α ξ . It follows that, under the map α, the coadjoint orbit of any element of u(d) * is identified with the collection of Hermitian matrices with a fixed spectrum.\nProof. The first part is a straightforward computation: for η ∈ u(d),\nProof. The first part is a straightforward computation: for η ∈ u(d),\nAny Hermitian matrix ξ with spectrum λ λ λ can be expressed via its eigendecomposition as ξ = Ad A (diag(λ λ λ )) for some A ∈ U(d), so that the second part of the claim follows by equivariance.\nAny Hermitian matrix ξ with spectrum λ λ λ can be expressed via its eigendecomposition as ξ = Ad A (diag(λ λ λ )) for some A ∈ U(d), so that the second part of the claim follows by equivariance.\nBased on the second part of the lemma, we use O λ λ λ ⊂ H(d) to denote the set of matrices with fixed spectrum λ λ λ . We now describe the symplectic structure on the space of frames with prescribed frame spectrum.\nBased on the second part of the lemma, we use O λ λ λ ⊂ H(d) to denote the set of matrices with fixed spectrum λ λ λ . We now describe the symplectic structure on the space of frames with prescribed frame spectrum."},{"title":"Proposition 2.8 ( [43]). The action of U(d) on C d×N by left multiplication is Hamiltonian with momentum map","content":""},{"title":"Proposition 2.8 ( [43]). The action of U(d) on C d×N by left multiplication is Hamiltonian with momentum map","content":"In particular, it has a natural symplectic structure.\nIn particular, it has a natural symplectic structure.\nProof Sketch. We sketch the proof, since the constructions involved will be useful later. For details, see our previous paper [43].\nProof Sketch. We sketch the proof, since the constructions involved will be useful later. For details, see our previous paper [43].\nThe fact that Φ U(d) ∶ F ↦ −FF * is a momentum map (where we are specifically fixing the isomorphism (5)) follows by a computation, as shown in [43,Proposition 2]. We remark that our momentum map differs by a sign in the present paper, due to slightly different conventions. The fact that\nThe fact that Φ U(d) ∶ F ↦ −FF * is a momentum map (where we are specifically fixing the isomorphism (5)) follows by a computation, as shown in [43,Proposition 2]. We remark that our momentum map differs by a sign in the present paper, due to slightly different conventions. The fact that\nfollows by Lemma 2.7. Moreover, the negative-definite Hermitian matrices are the regular values of Φ U(d) in H(d); we proved this by slightly tedious but essentially straightforward calculation in [43], but it also follows easily from Proposition 2.20 below, since the only unitary matrix which acts trivially on a spanning set of C d is the identity matrix. Therefore, the entries in λ λ λ being positive means that Theorem 2.3 applies and\nfollows by Lemma 2.7. Moreover, the negative-definite Hermitian matrices are the regular values of Φ U(d) in H(d); we proved this by slightly tedious but essentially straightforward calculation in [43], but it also follows easily from Proposition 2.20 below, since the only unitary matrix which acts trivially on a spanning set of C d is the identity matrix. Therefore, the entries in λ λ λ being positive means that Theorem 2.3 applies and\nis a symplectic manifold.\nis a symplectic manifold."},{"title":"Identification with a Coadjoint Orbit","content":""},{"title":"Identification with a Coadjoint Orbit","content":"The symplectic structure of F d,N λ λ λ U(d) can alternatively be realized by identifying the frame space with a certain coadjoint orbit in the larger space of Hermitian matrices H(N). This perspective will be useful in the following subsection. In order to state the result precisely, we introduce some more terminology.\nThe symplectic structure of F d,N λ λ λ U(d) can alternatively be realized by identifying the frame space with a certain coadjoint orbit in the larger space of Hermitian matrices H(N). This perspective will be useful in the following subsection. In order to state the result precisely, we introduce some more terminology.\nIt is well known that any coadjoint orbit has a natural symplectic structure, called the Kirillov-Kostant-Souriau (KKS) symplectic form [5,II.3.c]), which we denote generically as ω KKS . Indeed, for an arbitrary Lie group G with Lie algebra g, the tangent space to a coadjoint orbit O β at β ∈ g * consists of vectors of the form ad * ξ β , where ξ ∈ g, and where ad * ξ is the coadjoint representation of g on g * , obtained as the derivative of the map Ad * ∶ G → Aut(g * ) at the identity. Then the KKS form is defined by\nIt is well known that any coadjoint orbit has a natural symplectic structure, called the Kirillov-Kostant-Souriau (KKS) symplectic form [5,II.3.c]), which we denote generically as ω KKS . Indeed, for an arbitrary Lie group G with Lie algebra g, the tangent space to a coadjoint orbit O β at β ∈ g * consists of vectors of the form ad * ξ β , where ξ ∈ g, and where ad * ξ is the coadjoint representation of g on g * , obtained as the derivative of the map Ad * ∶ G → Aut(g * ) at the identity. Then the KKS form is defined by\nwhere [⋅,⋅] is the Lie bracket on g.\nwhere [⋅,⋅] is the Lie bracket on g.\nBy another slight abuse of terminology, we define the adjoint representation of u(d) on H(d) by\nBy another slight abuse of terminology, we define the adjoint representation of u(d) on H(d) by\nAn argument similar to that of Lemma 2.7 proves the following.\nAn argument similar to that of Lemma 2.7 proves the following.\nLemma 2.9. The map α defined in (5) is equivariant with respect to the adjoint representation of u(d) on H(d) and its coadjoint representation on u(d) * . That is,\nLemma 2.9. The map α defined in (5) is equivariant with respect to the adjoint representation of u(d) on H(d) and its coadjoint representation on u(d) * . That is,\nThe adjoint representation of u(d) on H(d) is obtained by differentiating the adjoint action of U(d) on H(d). It follows that the tangent space to O −λ λ λ ⊂ H(d) at ξ consists of vectors of the form ad η (ξ ) for η ∈ u(d).\nThe adjoint representation of u(d) on H(d) is obtained by differentiating the adjoint action of U(d) on H(d). It follows that the tangent space to O −λ λ λ ⊂ H(d) at ξ consists of vectors of the form ad η (ξ ) for η ∈ u(d).\nwhere the second-to-last line uses cyclic invariance and linearity of trace.\nwhere the second-to-last line uses cyclic invariance and linearity of trace.\nWe can now describe precisely how F d,N λ λ λ U(d) is equivalent to a coadjoint orbit in H(N).\nWe can now describe precisely how F d,N λ λ λ U(d) is equivalent to a coadjoint orbit in H(N)."},{"title":"Proposition 2.11 ( [43]","content":""},{"title":"Proposition 2.11 ( [43]","content":"). Let λ λ λ be a frame spectrum and defineλ λ λ ∶= (λ 1 ,... ,λ d ,0,...,0) ∈ R N to be λ λ λ padded with\n). Let λ λ λ be a frame spectrum and defineλ λ λ ∶= (λ 1 ,... ,λ d ,0,...,0) ∈ R N to be λ λ λ padded with\ntaking a unitary equivalence class to its Gram matrix. Moreover, the map (7) is a symplectomorphism with respect to the reduced symplectic structure on F d,N λ λ λ U(d) and the pullback of the Kirillov-Kostant-Souriau form α * ω KKS described in Lemma 2.10.\ntaking a unitary equivalence class to its Gram matrix. Moreover, the map (7) is a symplectomorphism with respect to the reduced symplectic structure on F d,N λ λ λ U(d) and the pullback of the Kirillov-Kostant-Souriau form α * ω KKS described in Lemma 2.10.\nThe proof of the proposition makes use of the following lemma.\nThe proof of the proposition makes use of the following lemma.\nfor ξ ∈ u(d) and ζ ∈ u(N) satisfyingU 0 = ξU andV 0 = ζV , respectively. By general principles of quotient manifolds, there is a natural isomorphism\nfor ξ ∈ u(d) and ζ ∈ u(N) satisfyingU 0 = ξU andV 0 = ζV , respectively. By general principles of quotient manifolds, there is a natural isomorphism\nis a submersion, so the rank theorem [37,Theorem 4.12] implies that, locally near F, it is an orthogonal projection with the fiber U(d) ⋅ F being sent to the origin. This gives\nis a submersion, so the rank theorem [37,Theorem 4.12] implies that, locally near F, it is an orthogonal projection with the fiber U(d) ⋅ F being sent to the origin. This gives\nProof of Proposition 2.11. The diffeomorphism result is Proposition 3 in our previous paper [43], so we only sketch the details here. Unitary equivalence classes of frames are uniquely determined by their Gram matrices (see, for example, [50, §3.4]), so we can identify F d,N λ λ λ U(d) with the elements of H(N) which arise as Gram matrices of frames in F d,N λ λ λ . Of course, the frame operator FF * and the Gram matrix F * F have the same nonzero eigenvalues, so specifying the spectrum of the frame operator also determines the spectrum of the Gram matrix by padding with N − d zeros, and it follows that this subset consists of those N × N Hermitian matrices with spectrumλ λ λ . In other words, (7) is a diffeomorphism between F d,N λ λ λ U(d) and Oλ λ λ . It remains to prove the symplectomorphism claim. For the rest of this proof, let γ denote the map (7). By Lemma 2.12, the tangent space to F d,N\nProof of Proposition 2.11. The diffeomorphism result is Proposition 3 in our previous paper [43], so we only sketch the details here. Unitary equivalence classes of frames are uniquely determined by their Gram matrices (see, for example, [50, §3.4]), so we can identify F d,N λ λ λ U(d) with the elements of H(N) which arise as Gram matrices of frames in F d,N λ λ λ . Of course, the frame operator FF * and the Gram matrix F * F have the same nonzero eigenvalues, so specifying the spectrum of the frame operator also determines the spectrum of the Gram matrix by padding with N − d zeros, and it follows that this subset consists of those N × N Hermitian matrices with spectrumλ λ λ . In other words, (7) is a diffeomorphism between F d,N λ λ λ U(d) and Oλ λ λ . It remains to prove the symplectomorphism claim. For the rest of this proof, let γ denote the map (7). By Lemma 2.12, the tangent space to F d,N\nThen, using Lemma 2.10,\nThen, using Lemma 2.10,\nOn the other hand, the reduced symplectic form on F d,N\nOn the other hand, the reduced symplectic form on F d,N\nwhere we have used the characterization of the reduced form, cyclic invariance of trace and the fact that ζ ′ is skew-Hermitian.\nwhere we have used the characterization of the reduced form, cyclic invariance of trace and the fact that ζ ′ is skew-Hermitian.\nRemark 2.13. Since Oλ λ λ ⊂ H(N) is identified with a coadjoint orbit, it is diffeomorphic, like all coadjoint orbits, to a flag manifold [5, §II.1.d]. Specifically, if λ λ λ consists of ℓ distinct eigenvalues with multiplicities k 1 ,... ,k ℓ and\nRemark 2.13. Since Oλ λ λ ⊂ H(N) is identified with a coadjoint orbit, it is diffeomorphic, like all coadjoint orbits, to a flag manifold [5, §II.1.d]. Specifically, if λ λ λ consists of ℓ distinct eigenvalues with multiplicities k 1 ,... ,k ℓ and\nthe flag manifold whose elements are nested sequences\nthe flag manifold whose elements are nested sequences\nIn particular, when λ λ λ is constant then ℓ = 1 and k 1 = d, so the flag manifold is Fℓ(d,N), whose elements are all subspaces\nIn particular, when λ λ λ is constant then ℓ = 1 and k 1 = d, so the flag manifold is Fℓ(d,N), whose elements are all subspaces"},{"title":"Frames with Prescribed Frame Spectrum and Norms","content":""},{"title":"Frames with Prescribed Frame Spectrum and Norms","content":"The torus U(1) N can be realized as the subgroup of diagonal elements of U(N); that is, as the standard maximal torus T ≤ U(N). Thought of in this way, U(1) N ≈ T acts on C d×N by right matrix multiplication and, as we will see in a moment, this action preserves the frame spectrum, and hence restricts to an action on F d,N λ λ λ .\nThe torus U(1) N can be realized as the subgroup of diagonal elements of U(N); that is, as the standard maximal torus T ≤ U(N). Thought of in this way, U(1) N ≈ T acts on C d×N by right matrix multiplication and, as we will see in a moment, this action preserves the frame spectrum, and hence restricts to an action on F d,N λ λ λ .\nFirst, though, it is worthwhile to pause and think about the right action of the full unitary group U(N) on C d×N . If we try to define the action of A ∈ U(N) on F ∈ C d×N by A ⋅ F = FA we quickly run into problems: after all, if\nFirst, though, it is worthwhile to pause and think about the right action of the full unitary group U(N) on C d×N . If we try to define the action of A ∈ U(N) on F ∈ C d×N by A ⋅ F = FA we quickly run into problems: after all, if\nunless A 1 and A 2 happen to commute. This is not an issue at the level of T, which is abelian, but it is preferable to define the T action to be consistent with an honest U(N) action. We get such an action by defining\nunless A 1 and A 2 happen to commute. This is not an issue at the level of T, which is abelian, but it is preferable to define the T action to be consistent with an honest U(N) action. We get such an action by defining\nThis action preserves the frame spectrum: if F ∈ F d,N λ λ λ and D ∈ T, then the frame operator of D ⋅ F = FD * is (FD * )(FD * ) * = FD * DF * = FF * , which is the same as the frame operator of F, and hence D ⋅ F ∈ F d,N λ λ λ as well. At the frame level, the action of an element of the torus on a frame performs an independent phase rotation on each frame vector. The right action of T commutes with the left action of U(d), so the torus action descends to the quotient F d,N λ λ λ U(d), but it is not an effective action. After all, both T and U(d) contain a 1-parameter subgroup of scalar matrices, and the actions of these two subgroups cancel each other: if I k denotes the k × k identity matrix and e iθ ∈ U(1), then e iθ I d ∈ U(d), e iθ I N ∈ T and for any\nThis action preserves the frame spectrum: if F ∈ F d,N λ λ λ and D ∈ T, then the frame operator of D ⋅ F = FD * is (FD * )(FD * ) * = FD * DF * = FF * , which is the same as the frame operator of F, and hence D ⋅ F ∈ F d,N λ λ λ as well. At the frame level, the action of an element of the torus on a frame performs an independent phase rotation on each frame vector. The right action of T commutes with the left action of U(d), so the torus action descends to the quotient F d,N λ λ λ U(d), but it is not an effective action. After all, both T and U(d) contain a 1-parameter subgroup of scalar matrices, and the actions of these two subgroups cancel each other: if I k denotes the k × k identity matrix and e iθ ∈ U(1), then e iθ I d ∈ U(d), e iθ I N ∈ T and for any\nWe can put this in a more standard context using our identification of F d,N λ λ λ U(d) with the coadjoint orbit Oλ λ λ ⊂ H(N) ≈ u(N) * , described in Proposition 2.11. The standard maximal torus of U(N) certainly acts on any coadjoint orbit of U(N), and in this case the action is just the conjugation action. The scalar matrices are the center Z(U(N)) of U(N), whose coadjoint action is trivial. In general, to get an effective action of the maximal torus of a Lie group on a coadjoint orbit of the group, one needs to take the quotient of the torus by the center of the group; in this case, this is simply\nWe can put this in a more standard context using our identification of F d,N λ λ λ U(d) with the coadjoint orbit Oλ λ λ ⊂ H(N) ≈ u(N) * , described in Proposition 2.11. The standard maximal torus of U(N) certainly acts on any coadjoint orbit of U(N), and in this case the action is just the conjugation action. The scalar matrices are the center Z(U(N)) of U(N), whose coadjoint action is trivial. In general, to get an effective action of the maximal torus of a Lie group on a coadjoint orbit of the group, one needs to take the quotient of the torus by the center of the group; in this case, this is simply\nwhich can be identified with the subgroup of diagonal elements of U(N) whose last entry is 1. Now we see why it was worth defining the action of U(1) N (and hence G) on F d,N λ λ λ by D ⋅ F = FD * , since this corresponds exactly to the standard coadjoint action on H(N) ≈ u(N) * .\nwhich can be identified with the subgroup of diagonal elements of U(N) whose last entry is 1. Now we see why it was worth defining the action of U(1) N (and hence G) on F d,N λ λ λ by D ⋅ F = FD * , since this corresponds exactly to the standard coadjoint action on H(N) ≈ u(N) * .\nIn this setting, it is easy to see that the action of G on Oλ λ λ ≈ F d,N λ λ λ U(d) is Hamiltonian:\nIn this setting, it is easy to see that the action of G on Oλ λ λ ≈ F d,N λ λ λ U(d) is Hamiltonian:\nProof. First of all, it is standard (see, e.g., [40,Example 5.3.11]) that the coadjoint action of U(N) on the coadjoint orbit Oλ λ λ is Hamiltonian with momentum map given by the inclusion Oλ λ λ ↪ u(N) * ≈ H(N); in fact, the analogous statement holds for arbitrary compact Lie groups acting on coadjoint orbits. Similarly, it is standard (see, e.g., [5, Proposition III.1.10]) that the action of the maximal torus T ≤ U(N) on Oλ λ λ is Hamiltonian with momentum map given by the composition\nProof. First of all, it is standard (see, e.g., [40,Example 5.3.11]) that the coadjoint action of U(N) on the coadjoint orbit Oλ λ λ is Hamiltonian with momentum map given by the inclusion Oλ λ λ ↪ u(N) * ≈ H(N); in fact, the analogous statement holds for arbitrary compact Lie groups acting on coadjoint orbits. Similarly, it is standard (see, e.g., [5, Proposition III.1.10]) that the action of the maximal torus T ≤ U(N) on Oλ λ λ is Hamiltonian with momentum map given by the composition\nwhere t * is the Lie algebra of T and the projection u(N) * → t * is the one induced by the inclusion T ↪ U(N). After identifying t * with R N , this projection is easily seen to be the map which records the diagonal entries of a Hermitian matrix. 2 More generally, the same applies for any subgroup (see, e.g., [9, p. 213]), and so the action of G on Oλ λ λ is Hamiltonian with momentum map given by the composition\nwhere t * is the Lie algebra of T and the projection u(N) * → t * is the one induced by the inclusion T ↪ U(N). After identifying t * with R N , this projection is easily seen to be the map which records the diagonal entries of a Hermitian matrix. 2 More generally, the same applies for any subgroup (see, e.g., [9, p. 213]), and so the action of G on Oλ λ λ is Hamiltonian with momentum map given by the composition\nwhere the projection u(N) * → g * is the one induced by the inclusion G ↪ U(N). Under the identification g * ≈ R N−1 , this map just records all the diagonal entries but the last one (which in any case is determined by the others, since the trace of any element of Oλ λ λ is ∑ i λ i ).\nwhere the projection u(N) * → g * is the one induced by the inclusion G ↪ U(N). Under the identification g * ≈ R N−1 , this map just records all the diagonal entries but the last one (which in any case is determined by the others, since the trace of any element of Oλ λ λ is ∑ i λ i ).\n. By the Schur-Horn theorem [30,46]\n. By the Schur-Horn theorem [30,46]\nis non-empty only if the vector (r 1 ,...,r N ) lies in the convex hull of the orbit ofλ λ λ = (λ 1 ,... ,λ d ,0,... ,0) under the action of the symmetric group S N which permutes entries. When r r r and λ λ λ are non-increasing lists, this condition is easily seen to be equivalent to r r r being λ λ λ -admissible, so F d,N λ λ λ (r r r) is non-empty precisely when r r r is λ λ λ -admissible. When r r r is λ λ λ -admissible, there is no guarantee that (r 1 ,...,r N−1 ) ∈ R N−1 ≈ g * is a regular value, but we can still apply Theorem 2.4 to see that the quotient of Φ −1 G (r 1 ,... ,r N−1 ) by G is at worst a symplectic stratified space. We have thus proved the first part of the following proposition:\nis non-empty only if the vector (r 1 ,...,r N ) lies in the convex hull of the orbit ofλ λ λ = (λ 1 ,... ,λ d ,0,... ,0) under the action of the symmetric group S N which permutes entries. When r r r and λ λ λ are non-increasing lists, this condition is easily seen to be equivalent to r r r being λ λ λ -admissible, so F d,N λ λ λ (r r r) is non-empty precisely when r r r is λ λ λ -admissible. When r r r is λ λ λ -admissible, there is no guarantee that (r 1 ,...,r N−1 ) ∈ R N−1 ≈ g * is a regular value, but we can still apply Theorem 2.4 to see that the quotient of Φ −1 G (r 1 ,... ,r N−1 ) by G is at worst a symplectic stratified space. We have thus proved the first part of the following proposition:\nwhich is a symplectic stratified space. 3 Alternatively,\nwhich is a symplectic stratified space. 3 Alternatively,\nProof. We proved the first sentence above. For the second sentence, Sjamaar and Lerman [48,Theorem 4.1] showed that even when reducing over singular values, we can perform a reduction by a product group in stages. Hence, with\nProof. We proved the first sentence above. For the second sentence, Sjamaar and Lerman [48,Theorem 4.1] showed that even when reducing over singular values, we can perform a reduction by a product group in stages. Hence, with\nis, in general, a symplectic stratified space. Depending on the parameters λ λ λ and r r r, it may actually be a smooth manifold. We characterize the parameters such that F d,N λ λ λ (r r r) is a smooth manifold below in Theorem 2.25.\nis, in general, a symplectic stratified space. Depending on the parameters λ λ λ and r r r, it may actually be a smooth manifold. We characterize the parameters such that F d,N λ λ λ (r r r) is a smooth manifold below in Theorem 2.25.\nIn particular, the space F d,N λ λ λ (r r r) is exactly the level set Φ −1 (O −λ λ λ × {r r r}). To simplify slightly, assume r r r is strongly λ λ λ -admissible. When F d,N λ λ λ (r r r) is a manifold (conditions for which we determine in Corollary 2.22 in the next section), then we know its codimension inside C d×N is equal to the codimension of O −λ λ λ × {r r r} inside H(d) × R N−1 . Even when F d,N λ λ λ (r r r) is not a manifold, we know from Theorem 2.5 that its quotient by U(d) × G contains an open dense subset which is a manifold. This subset is a stratum, and hence its (open, dense) preimage is also a manifold by Theorem 3.5 from Sjamaar and Lerman [48].\nIn particular, the space F d,N λ λ λ (r r r) is exactly the level set Φ −1 (O −λ λ λ × {r r r}). To simplify slightly, assume r r r is strongly λ λ λ -admissible. When F d,N λ λ λ (r r r) is a manifold (conditions for which we determine in Corollary 2.22 in the next section), then we know its codimension inside C d×N is equal to the codimension of O −λ λ λ × {r r r} inside H(d) × R N−1 . Even when F d,N λ λ λ (r r r) is not a manifold, we know from Theorem 2.5 that its quotient by U(d) × G contains an open dense subset which is a manifold. This subset is a stratum, and hence its (open, dense) preimage is also a manifold by Theorem 3.5 from Sjamaar and Lerman [48].\nIn other words, F d,N λ λ λ (r r r) contains an open, dense subset which is a manifold and which consists precisely of regular points of the momentum map Φ, so for the purposes of dimension-counting we may as well assume F d,N λ λ λ (r r r) is a manifold.\nIn other words, F d,N λ λ λ (r r r) contains an open, dense subset which is a manifold and which consists precisely of regular points of the momentum map Φ, so for the purposes of dimension-counting we may as well assume F d,N λ λ λ (r r r) is a manifold.\nTo that end,\nTo that end,\nwhere, as in Remark 2.13, the k 1 ,... ,k ℓ are the multiplicities of the entries in the frame spectrum λ λ λ . Subtracting this from\nwhere, as in Remark 2.13, the k 1 ,... ,k ℓ are the multiplicities of the entries in the frame spectrum λ λ λ . Subtracting this from\nand hence also of F d,N λ λ λ (r r r) (inside C d×N ). This proves: Corollary 2.18. For r r r which is strongly λ λ λ -admissible,\nand hence also of F d,N λ λ λ (r r r) (inside C d×N ). This proves: Corollary 2.18. For r r r which is strongly λ λ λ -admissible,\nHence,\nHence,\nwhere the k j are the multiplicities of the entries in the frame spectrum λ λ λ .\nwhere the k j are the multiplicities of the entries in the frame spectrum λ λ λ ."},{"title":"Manifold Structure of Frame Spaces","content":""},{"title":"Manifold Structure of Frame Spaces","content":"While Corollary 2.6 will allow us to apply the symplectic machinery even when F d,N λ λ λ (r r r) is singular, it is still interesting to understand when this space is a smooth manifold. We completely characterize parameters for which F d,N λ λ λ (r r r) is a manifold in the following subsection. We then give a detailed description of the local structure of singularities in singular frame spaces.\nWhile Corollary 2.6 will allow us to apply the symplectic machinery even when F d,N λ λ λ (r r r) is singular, it is still interesting to understand when this space is a smooth manifold. We completely characterize parameters for which F d,N λ λ λ (r r r) is a manifold in the following subsection. We then give a detailed description of the local structure of singularities in singular frame spaces."},{"title":"Characterizing Smooth Frame Spaces","content":""},{"title":"Characterizing Smooth Frame Spaces","content":"The model result on manifold structures of frame spaces is due to Dykema and Strawn [18], who showed that the singular points in the space F d,N N d ,..., N d (1,...,1) of unit-norm tight frames must be orthodecomposable (we recall the definition of this property below). An immediate corollary is that the space of unit-norm tight frames is a manifold when N and d are relatively prime. More generally, Strawn [49] showed that the singular points in any space of frames with fixed frame vector norms and fixed frame operator (rather than frame spectrum) must be orthodecomposable.\nThe model result on manifold structures of frame spaces is due to Dykema and Strawn [18], who showed that the singular points in the space F d,N N d ,..., N d (1,...,1) of unit-norm tight frames must be orthodecomposable (we recall the definition of this property below). An immediate corollary is that the space of unit-norm tight frames is a manifold when N and d are relatively prime. More generally, Strawn [49] showed that the singular points in any space of frames with fixed frame vector norms and fixed frame operator (rather than frame spectrum) must be orthodecomposable.\nWe will prove the converse of these results, showing that the singular points (if any) of every F d,N λ λ λ (r r r) space are exactly the orthodecomposable frames. First, recall what it means for a frame to be orthodecomposable:\nWe will prove the converse of these results, showing that the singular points (if any) of every F d,N λ λ λ (r r r) space are exactly the orthodecomposable frames. First, recall what it means for a frame to be orthodecomposable:\nIn our setting, we are realizing the space F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) as the inverse image of a coadjoint orbit under the momentum map Φ. This frame space will certainly be a manifold if it contains no critical points of Φ, so we now characterize the critical points of Φ.\nIn our setting, we are realizing the space F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) as the inverse image of a coadjoint orbit under the momentum map Φ. This frame space will certainly be a manifold if it contains no critical points of Φ, so we now characterize the critical points of Φ.\nA standard part of the discussion around Theorems 2. Recall that a group action is locally free at p if the stabilizer of p is discrete. Therefore, Proposition 2.20 gives a characterization of critical points of a momentum map: p ∈ M is a critical point if and only if the stabilizer of p is continuous, and in particular contains a nontrivial one-parameter subgroup. This is the key to characterizing the critical points of Φ ∶ C d×N → H(d) × R N−1 .\nA standard part of the discussion around Theorems 2. Recall that a group action is locally free at p if the stabilizer of p is discrete. Therefore, Proposition 2.20 gives a characterization of critical points of a momentum map: p ∈ M is a critical point if and only if the stabilizer of p is continuous, and in particular contains a nontrivial one-parameter subgroup. This is the key to characterizing the critical points of Φ ∶ C d×N → H(d) × R N−1 ."},{"title":"Proposition 2.21. A frame F is a critical point of the momentum map Φ ∶ C d×N → H(d) × R N−1 of the Hamiltonian action of U(d) × G if and only if F is orthodecomposable.","content":""},{"title":"Proposition 2.21. A frame F is a critical point of the momentum map Φ ∶ C d×N → H(d) × R N−1 of the Hamiltonian action of U(d) × G if and only if F is orthodecomposable.","content":"Proof. Let F = f 1 ⋯ f N ∈ C d×N be a frame and let ζ ∶ R → G be a one-parameter subgroup such that ζ (t) ⋅ F = F for all t ∈ R. Then ζ uniquely corresponds to an element of the Lie algebra u(d) × g of U(d) × G, and in particular there exist matrices ξ ∈ u(d) and θ ∈ g such that ζ (t) = exp(t(ξ ,θ )) = (exp(tξ ),exp(tθ )) (see, e.g., [37,Propositions 20.1 and 20.5]). Under our concrete realization of G = T Z (U(N)) as the subgroup of diagonal unitary matrices with last entry equal to 1, θ will be of the form\nProof. Let F = f 1 ⋯ f N ∈ C d×N be a frame and let ζ ∶ R → G be a one-parameter subgroup such that ζ (t) ⋅ F = F for all t ∈ R. Then ζ uniquely corresponds to an element of the Lie algebra u(d) × g of U(d) × G, and in particular there exist matrices ξ ∈ u(d) and θ ∈ g such that ζ (t) = exp(t(ξ ,θ )) = (exp(tξ ),exp(tθ )) (see, e.g., [37,Propositions 20.1 and 20.5]). Under our concrete realization of G = T Z (U(N)) as the subgroup of diagonal unitary matrices with last entry equal to 1, θ will be of the form\nfor some θ 1 ,...,θ N−1 ∈ R. Then we have\nfor some θ 1 ,...,θ N−1 ∈ R. Then we have\nAt the frame vector level, this yields a system of equations\nAt the frame vector level, this yields a system of equations\nEach √ −1θ j is therefore an eigenvalue of the skew-Hermitian matrix ξ with corresponding eigenvector f j . Moreover, any solution of (8) yields a one-parameter subgroup in the stabilizer of F. By the Spectral Theorem for normal matrices, the eigenspaces for distinct eigenvalues of ξ are orthogonal.\nEach √ −1θ j is therefore an eigenvalue of the skew-Hermitian matrix ξ with corresponding eigenvector f j . Moreover, any solution of (8) yields a one-parameter subgroup in the stabilizer of F. By the Spectral Theorem for normal matrices, the eigenspaces for distinct eigenvalues of ξ are orthogonal.\nWith the above setup, we are prepared to prove the claim. First, suppose that F is not a regular point of the momentum map. By Proposition 2.20, it is possible to find a one-parameter subgroup ζ ∶ R → G of the stabilizer of F which is nontrivial. In this case, ξ has at least two distinct eigenvalues and it follows that F is orthodecomposable, with frame vectors partitioned into the orthogonal eigenspaces of ξ .\nWith the above setup, we are prepared to prove the claim. First, suppose that F is not a regular point of the momentum map. By Proposition 2.20, it is possible to find a one-parameter subgroup ζ ∶ R → G of the stabilizer of F which is nontrivial. In this case, ξ has at least two distinct eigenvalues and it follows that F is orthodecomposable, with frame vectors partitioned into the orthogonal eigenspaces of ξ .\nConversely, suppose that F is orthodecomposable. Let P 1 ,... ,P m denote the partition and V 1 ,... ,V m the subspaces appearing in Definition 2.19. Permuting columns of F as necessary, we can assume without loss of generality that P 1 = {1,...,k} for some k < N. For convenience, we can apply a unitary transformation to F so that f 1 ,... , f k span a coordinate plane V 1 ; say, the span of the first ℓ < d standard basis vectors. Since V 2 ,... ,V m are all orthogonal to V 1 , ⊕ m i=2 V i is the span of the remaining d − ℓ standard basis vectors. Then F has the form\nConversely, suppose that F is orthodecomposable. Let P 1 ,... ,P m denote the partition and V 1 ,... ,V m the subspaces appearing in Definition 2.19. Permuting columns of F as necessary, we can assume without loss of generality that P 1 = {1,...,k} for some k < N. For convenience, we can apply a unitary transformation to F so that f 1 ,... , f k span a coordinate plane V 1 ; say, the span of the first ℓ < d standard basis vectors. Since V 2 ,... ,V m are all orthogonal to V 1 , ⊕ m i=2 V i is the span of the remaining d − ℓ standard basis vectors. Then F has the form\nwhere the nonzero blocks F 1 and F 2 have sizes ℓ × k and (d − ℓ) × (N − k), respectively. Let\nwhere the nonzero blocks F 1 and F 2 have sizes ℓ × k and (d − ℓ) × (N − k), respectively. Let\nIt is easy to check that ξ ,θ , and F satisfy the system (8) and therefore yield a nontrivial one-parameter subgroup in the stabilizer of F. By Proposition 2.20, F is not a regular point of the momentum map.\nIt is easy to check that ξ ,θ , and F satisfy the system (8) and therefore yield a nontrivial one-parameter subgroup in the stabilizer of F. By Proposition 2.20, F is not a regular point of the momentum map.\nThe characterization of singular points in F d,N λ λ λ (r r r) is now an easy corollary: Proof. From Proposition 2.21 we know that the critical points of the momentum map Φ are exactly the orthodecomposable frames. It is a nontrivial fact (see, e.g., [3,Theorem 5] or [48,Proposition 2.5]) that in a neighborhood of a critical point of a momentum map, the level set looks like the product of a quadratic cone and a manifold. Therefore, F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) has a quadratic singularity at each orthodecomposable frame. Since the structure of such singularities is itself of interest [8], we describe these cone singularities in detail in Section 2.3.2 below.\nThe characterization of singular points in F d,N λ λ λ (r r r) is now an easy corollary: Proof. From Proposition 2.21 we know that the critical points of the momentum map Φ are exactly the orthodecomposable frames. It is a nontrivial fact (see, e.g., [3,Theorem 5] or [48,Proposition 2.5]) that in a neighborhood of a critical point of a momentum map, the level set looks like the product of a quadratic cone and a manifold. Therefore, F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) has a quadratic singularity at each orthodecomposable frame. Since the structure of such singularities is itself of interest [8], we describe these cone singularities in detail in Section 2.3.2 below.\nOrthodecomposable frames are the only possible singularities: if F ∈ F d,N λ λ λ (r r r) is not orthodecomposable, then the differential DΦ F ∶ T F C d×N → T Φ(F) (H(d) × R N−1 ) is surjective. In particular, this implies Φ is transverse to a neighborhood of Φ(F) in O −λ λ λ × {r r r}, so it is standard that its inverse image-which is a neighborhood of F in F d,N λ λ λ (r r r)-is a submanifold of C d×N (see, e.g., [37, Theorem 6.30]), and in particular is smooth at F. Remark 2.23. Since every orthodecomposable frame is a singular point of the F d,N λ λ λ (r r r) space containing it, Corollary 2.22 solves the complex case of Cahill, Mixon, and Strawn's Problem 5.8 [8].\nOrthodecomposable frames are the only possible singularities: if F ∈ F d,N λ λ λ (r r r) is not orthodecomposable, then the differential DΦ F ∶ T F C d×N → T Φ(F) (H(d) × R N−1 ) is surjective. In particular, this implies Φ is transverse to a neighborhood of Φ(F) in O −λ λ λ × {r r r}, so it is standard that its inverse image-which is a neighborhood of F in F d,N λ λ λ (r r r)-is a submanifold of C d×N (see, e.g., [37, Theorem 6.30]), and in particular is smooth at F. Remark 2.23. Since every orthodecomposable frame is a singular point of the F d,N λ λ λ (r r r) space containing it, Corollary 2.22 solves the complex case of Cahill, Mixon, and Strawn's Problem 5.8 [8].\nTo make Corollary 2.22 more practical, it would be helpful to know, in terms of some computable conditions on the parameters d, N, λ λ λ , and r r r, exactly when F d,N λ λ λ (r r r) contains an orthodecomposable frame. To that end, suppose F ∈ F d,N λ λ λ (r r r) is orthodecomposable. Then, as in the proof of Proposition 2.20, after possibly permuting columns and applying a unitary transformation, we can realize F in block-diagonal form:\nTo make Corollary 2.22 more practical, it would be helpful to know, in terms of some computable conditions on the parameters d, N, λ λ λ , and r r r, exactly when F d,N λ λ λ (r r r) contains an orthodecomposable frame. To that end, suppose F ∈ F d,N λ λ λ (r r r) is orthodecomposable. Then, as in the proof of Proposition 2.20, after possibly permuting columns and applying a unitary transformation, we can realize F in block-diagonal form:\nwhere F 1 ∈ C ℓ×k and F 2 ∈ C (d−ℓ)×(N−k) are frames. But then both the Gram matrix and the frame operator associated to F are also block-diagonal:\nwhere F 1 ∈ C ℓ×k and F 2 ∈ C (d−ℓ)×(N−k) are frames. But then both the Gram matrix and the frame operator associated to F are also block-diagonal:\nSince tr(F * 1 F 1 ) = tr(F 1 F * 1 ), we see that r 1 + ⋅⋅⋅ + r k = λ 1 + ⋅⋅⋅ + λ ℓ . More generally, since we permuted columns to get F into block-diagonal form, there exist proper subsets {r i 1 ,... ,r i k } ⊊ r r r and {λ i 1 ,... ,λ i ℓ } ⊊ λ λ λ so that\nSince tr(F * 1 F 1 ) = tr(F 1 F * 1 ), we see that r 1 + ⋅⋅⋅ + r k = λ 1 + ⋅⋅⋅ + λ ℓ . More generally, since we permuted columns to get F into block-diagonal form, there exist proper subsets {r i 1 ,... ,r i k } ⊊ r r r and {λ i 1 ,... ,λ i ℓ } ⊊ λ λ λ so that\nMoreover, let r r r ′ = (r ′ 1 ,... ,r ′ k ) ∶= (r i 1 ,... ,r i k ) and λ λ λ ′ = (λ ′ 1 ,... ,λ ′ ℓ ) ∶= (λ i 1 ,...,λ i ℓ ) and let r r r ′′ and λ λ λ ′′ be their respective complement vectors. Then we see that the columns { f i 1 ,... , f i k } give a frame for an ℓ-dimensional subspace of C d with spectrum λ i 1 ≥ ⋅⋅⋅ ≥ λ i ℓ > 0, which means that the frame space F ℓ,k λ λ λ ′ (r r r ′ ) is non-empty, and in particular that r r r ′ is λ λ λ ′ -admissible. Similarly, the complementary columns give an element of F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ), and hence r r r ′′ is λ λ λ ′′ -admissible. Summarizing, we have shown the following consequences of orthodecomposability:\nMoreover, let r r r ′ = (r ′ 1 ,... ,r ′ k ) ∶= (r i 1 ,... ,r i k ) and λ λ λ ′ = (λ ′ 1 ,... ,λ ′ ℓ ) ∶= (λ i 1 ,...,λ i ℓ ) and let r r r ′′ and λ λ λ ′′ be their respective complement vectors. Then we see that the columns { f i 1 ,... , f i k } give a frame for an ℓ-dimensional subspace of C d with spectrum λ i 1 ≥ ⋅⋅⋅ ≥ λ i ℓ > 0, which means that the frame space F ℓ,k λ λ λ ′ (r r r ′ ) is non-empty, and in particular that r r r ′ is λ λ λ ′ -admissible. Similarly, the complementary columns give an element of F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ), and hence r r r ′′ is λ λ λ ′′ -admissible. Summarizing, we have shown the following consequences of orthodecomposability:\nλ λ λ (r r r) is orthodecomposable, then there exist proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum (which implies that r r r ′′ and λ λ λ ′′ have the same sum as well).\nλ λ λ (r r r) is orthodecomposable, then there exist proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum (which implies that r r r ′′ and λ λ λ ′′ have the same sum as well).\nCombined with Corollary 2.22, this gives a sufficient condition for F d,N λ λ λ (r r r) to be a manifold. In fact, it is also a necessary condition:\nCombined with Corollary 2.22, this gives a sufficient condition for F d,N λ λ λ (r r r) to be a manifold. In fact, it is also a necessary condition:\nλ λ λ (r r r) is a manifold if and only if there are no proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum.\nλ λ λ (r r r) is a manifold if and only if there are no proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum.\nProof. If F d,N λ λ λ (r r r) is not a manifold, then Corollary 2.22 tells us that it contains some orthodecomposable frame and hence, by Lemma 2.24, there are such partitions of r r r and λ λ λ .\nProof. If F d,N λ λ λ (r r r) is not a manifold, then Corollary 2.22 tells us that it contains some orthodecomposable frame and hence, by Lemma 2.24, there are such partitions of r r r and λ λ λ .\nConversely, suppose there are such partitions. Then, since r r r ′ is λ λ λ ′ -admissible, there exists a frame { f i 1 ,... , f i k } ∈ F ℓ,k λ λ λ ′ (r r r ′ ), and similarly there exists a frame {g j 1 ,... ,g j N−k } ∈ F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ). Embedding the former into the ℓ-dimensional subspace of C d spanned by the first ℓ standard basis vectors and the latter into the (d −ℓ)-dimensional orthogonal complement and combining them into a single list sorted by decreasing norm gives an orthodecomposable element-and hence a singular point-of F d,N λ λ λ (r r r), which is therefore not a manifold. Notice that Theorem 2.25 recovers Dykema and Strawn's result that the space of unit-norm tight frames is a manifold when N and d are relatively prime, and also implies that F d,N λ λ λ (r r r) is a manifold for generic r r r and λ λ λ .\nConversely, suppose there are such partitions. Then, since r r r ′ is λ λ λ ′ -admissible, there exists a frame { f i 1 ,... , f i k } ∈ F ℓ,k λ λ λ ′ (r r r ′ ), and similarly there exists a frame {g j 1 ,... ,g j N−k } ∈ F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ). Embedding the former into the ℓ-dimensional subspace of C d spanned by the first ℓ standard basis vectors and the latter into the (d −ℓ)-dimensional orthogonal complement and combining them into a single list sorted by decreasing norm gives an orthodecomposable element-and hence a singular point-of F d,N λ λ λ (r r r), which is therefore not a manifold. Notice that Theorem 2.25 recovers Dykema and Strawn's result that the space of unit-norm tight frames is a manifold when N and d are relatively prime, and also implies that F d,N λ λ λ (r r r) is a manifold for generic r r r and λ λ λ ."},{"title":"Describing Singularities","content":""},{"title":"Describing Singularities","content":"As was mentioned above, Corollary 2.22 solves Problem 5.8 of [8] for complex FUNTFs. Problem 1.5 of the same paper asks for a description of the local geometry of the space of FUNTFs near orthodecomposable frames, and we now expand on the proof of Corollary 2.22 to solve this problem. We describe singularities for progressively more general classes of frames, with full details given in the simplest case and sketches given for more general cases.\nAs was mentioned above, Corollary 2.22 solves Problem 5.8 of [8] for complex FUNTFs. Problem 1.5 of the same paper asks for a description of the local geometry of the space of FUNTFs near orthodecomposable frames, and we now expand on the proof of Corollary 2.22 to solve this problem. We describe singularities for progressively more general classes of frames, with full details given in the simplest case and sketches given for more general cases."},{"title":"Minimally Orthodecomposable FUNTFs. A result of Arms, Marsden, and Moncrief gives a description of the local geometry near any point on the level set of a momentum map of an arbitrary symplectic manifold [3, Theorem 5]. This specializes to our setting to show that near any FUNTF","content":""},{"title":"Minimally Orthodecomposable FUNTFs. A result of Arms, Marsden, and Moncrief gives a description of the local geometry near any point on the level set of a momentum map of an arbitrary symplectic manifold [3, Theorem 5]. This specializes to our setting to show that near any FUNTF","content":"there is a local diffeomorphism of the ambient space C d×N which takes a neighborhood of F in the level set onto the product of a quadratic cone C F and a smooth manifold. The cone is described explicitly as\nthere is a local diffeomorphism of the ambient space C d×N which takes a neighborhood of F in the level set onto the product of a quadratic cone C F and a smooth manifold. The cone is described explicitly as\nwhere J denotes multiplication by √ −1, considered as a linear map on the real vector space T F C d×N , and D 2 Φ(F) is the Hessian of Φ at F, considered as a H(d)-valued bilinear form. Observe that if F is a regular point of Φ then C F is just a linear subspace of C d×N . We will show that C F is a singular cone when F is not a regular point.\nwhere J denotes multiplication by √ −1, considered as a linear map on the real vector space T F C d×N , and D 2 Φ(F) is the Hessian of Φ at F, considered as a H(d)-valued bilinear form. Observe that if F is a regular point of Φ then C F is just a linear subspace of C d×N . We will show that C F is a singular cone when F is not a regular point.\nWriting\nWriting\nWe can also show that\nWe can also show that\n. Now suppose that F is orthodecomposable. As in the proof of Proposition 2.21, we can assume without loss of generality that\n. Now suppose that F is orthodecomposable. As in the proof of Proposition 2.21, we can assume without loss of generality that\nfor some submatrices F 1 ∈ C d 1 ×N 1 and F 2 ∈ C d 2 ×N 2 (we use a different indexing convention for dimensions than in the proof of Proposition 2.21 for ease of generalization later on). Furthermore, suppose that the blocks F 1 and F 2 are not themselves orthodecomposable. We then claim that image(DΦ(F)) ⊥ = span I d 1 0 0 0 ,(1,...,1 N 1 ,0,...,0\nfor some submatrices F 1 ∈ C d 1 ×N 1 and F 2 ∈ C d 2 ×N 2 (we use a different indexing convention for dimensions than in the proof of Proposition 2.21 for ease of generalization later on). Furthermore, suppose that the blocks F 1 and F 2 are not themselves orthodecomposable. We then claim that image(DΦ(F)) ⊥ = span I d 1 0 0 0 ,(1,...,1 N 1 ,0,...,0\nwhere the orthogonal complement is taken with respect to the inner product ⟪⋅,⋅⟫ on H d×d × R N−1 defined by\nwhere the orthogonal complement is taken with respect to the inner product ⟪⋅,⋅⟫ on H d×d × R N−1 defined by\nIndeed, a straightforward computation shows that the vector given in (9) belongs to image(DΦ(F)) ⊥ , and we see that this space is one-dimensional as follows. By standard arguments, it suffices to show that the nullity of the dual map DΦ(F) * is equal to one and Equation 5.2.7 of [40] implies that it suffices to show that the map taking an element of H(d) × R N−1 to its infinitesimal vector field evaluated at F has one dimensional kernel. By arguments similar to those used in the proof of Proposition 2.21, this is equivalent to showing that the space of solutions to the system of equations (8) is one-dimensional, which holds under our assumption that F 1 and F 2 are not themselves orthodecomposable (using the fact that the eigenspaces of a skew-Hermitian matrix are orthogonal). Based on the work in the previous two paragraphs, the cone C F is described more explicitly as the set of X ∈ C d×N satisfying\nIndeed, a straightforward computation shows that the vector given in (9) belongs to image(DΦ(F)) ⊥ , and we see that this space is one-dimensional as follows. By standard arguments, it suffices to show that the nullity of the dual map DΦ(F) * is equal to one and Equation 5.2.7 of [40] implies that it suffices to show that the map taking an element of H(d) × R N−1 to its infinitesimal vector field evaluated at F has one dimensional kernel. By arguments similar to those used in the proof of Proposition 2.21, this is equivalent to showing that the space of solutions to the system of equations (8) is one-dimensional, which holds under our assumption that F 1 and F 2 are not themselves orthodecomposable (using the fact that the eigenspaces of a skew-Hermitian matrix are orthogonal). Based on the work in the previous two paragraphs, the cone C F is described more explicitly as the set of X ∈ C d×N satisfying\nwhere we express X as a block matrix with blocks X km , k,m ∈ {1,2}, of size d k ×N m whose column vectors are denoted x km j . We can verify that the intersection of the subspace defined by (10) and the solution set of the indefinite quadratic equation (11) yields a singular cone by showing that it is not a linear subspace. To do so, choose\nwhere we express X as a block matrix with blocks X km , k,m ∈ {1,2}, of size d k ×N m whose column vectors are denoted x km j . We can verify that the intersection of the subspace defined by (10) and the solution set of the indefinite quadratic equation (11) yields a singular cone by showing that it is not a linear subspace. To do so, choose\nThe condition ⟨ f j ,x j ⟩ holds automatically due to the block structures of F and X, hence (10) is satisfied. The matrices X j can be chosen to be nonzero, since the dimension of the row space of F j is at most d j < N j . We can therefore scale the X j 's to have equal Frobenius norm, which means that (11) is also satisfied, so that X ∈ C F . Next, we observe that\nThe condition ⟨ f j ,x j ⟩ holds automatically due to the block structures of F and X, hence (10) is satisfied. The matrices X j can be chosen to be nonzero, since the dimension of the row space of F j is at most d j < N j . We can therefore scale the X j 's to have equal Frobenius norm, which means that (11) is also satisfied, so that X ∈ C F . Next, we observe that\nalso lies in C F . However, the matrix X − X ′ does not satisfy (11), and this proves our claim that C F is not a linear subspace.\nalso lies in C F . However, the matrix X − X ′ does not satisfy (11), and this proves our claim that C F is not a linear subspace.\nExample 2.26. Here we provide a concrete example of the structure described above. Let F denote the orthodecomposable unit norm tight frame\nExample 2.26. Here we provide a concrete example of the structure described above. Let F denote the orthodecomposable unit norm tight frame\nThe cone C F can be described explicitly as the set of 2 × 4 matrices X satisfying equations (10) and (11). Using the x km j notation from above, we write and, letting f j and x j denote the columns of F and X, respectively,\nThe cone C F can be described explicitly as the set of 2 × 4 matrices X satisfying equations (10) and (11). Using the x km j notation from above, we write and, letting f j and x j denote the columns of F and X, respectively,\nThe cone equation (11) Putting all of this together, the cone C F is\nThe cone equation (11) Putting all of this together, the cone C F is\nInterpreted as a subset of\nInterpreted as a subset of\n)}, this is precisely the cone over the Clifford torus. Now consider the map defined by\n)}, this is precisely the cone over the Clifford torus. Now consider the map defined by\nwith domain {A(λ ,θ ,φ ) λ ∈ [0,1)}, a neighborhood of the origin in C F . We claim that the image of this map is an open subset of F intersected with a slice of the U(2) × U(1) 4 ) U(1)-action on C 2×4 through F-that is, a subset S ⊂ C 2×4 containing F such that the orbit of any point in C 2×4 sufficiently close to F has a unique representative in S. 4 Specifically, the slice S is the set of matrices of the form\nwith domain {A(λ ,θ ,φ ) λ ∈ [0,1)}, a neighborhood of the origin in C F . We claim that the image of this map is an open subset of F intersected with a slice of the U(2) × U(1) 4 ) U(1)-action on C 2×4 through F-that is, a subset S ⊂ C 2×4 containing F such that the orbit of any point in C 2×4 sufficiently close to F has a unique representative in S. 4 Specifically, the slice S is the set of matrices of the form\nwhere α j and δ j are positive real numbers, and β j , γ j , θ , φ are arbitrary real numbers. The proof that S is a slice is somewhat technical, so we omit it. It is then straightforward to show that F ∩ S is equal to the image of our map (i.e., points of the form (12)), which implies that a neighborhood of F in F is identified with the product of a neighborhood of the origin in C F and the group.\nwhere α j and δ j are positive real numbers, and β j , γ j , θ , φ are arbitrary real numbers. The proof that S is a slice is somewhat technical, so we omit it. It is then straightforward to show that F ∩ S is equal to the image of our map (i.e., points of the form (12)), which implies that a neighborhood of F in F is identified with the product of a neighborhood of the origin in C F and the group.\nGeneral Orthodecomposable FUNTFs. Now suppose that F is a general orthodecomposable FUNTF. After applying isometries as necessary, we can assume without loss of generality that F is a block diagonal matrix F = diag(F 1 ,... ,F ℓ ), where F j a non-orthodecomposable matrix of size d j × N j . Similar computations show that the cone C F consists of matrices X ∈ C d×N satisfying the linear conditions (10) and the system of quadratic equations\nGeneral Orthodecomposable FUNTFs. Now suppose that F is a general orthodecomposable FUNTF. After applying isometries as necessary, we can assume without loss of generality that F is a block diagonal matrix F = diag(F 1 ,... ,F ℓ ), where F j a non-orthodecomposable matrix of size d j × N j . Similar computations show that the cone C F consists of matrices X ∈ C d×N satisfying the linear conditions (10) and the system of quadratic equations\nwhere we express X as a block matrix with blocks X km of size d k ×N m and with column vectors denoted x km j . Arguments similar to the above show that C F is a singular cone.\nwhere we express X as a block matrix with blocks X km of size d k ×N m and with column vectors denoted x km j . Arguments similar to the above show that C F is a singular cone.\nOrthodecomposable Frames in Arbitrary Frame Spaces. These descriptions of the local structure of FUNTF space near orthodecomposable frames do not intrinsically use the unit norm condition-they only rely on the assumption that the columns have some fixed collection of norms. The arguments therefore apply to describe singularities of spaces of tight frames with fixed norms.\nOrthodecomposable Frames in Arbitrary Frame Spaces. These descriptions of the local structure of FUNTF space near orthodecomposable frames do not intrinsically use the unit norm condition-they only rely on the assumption that the columns have some fixed collection of norms. The arguments therefore apply to describe singularities of spaces of tight frames with fixed norms.\nIn fact, similar local characterizations can be derived near orthodecomposable frames in any frame space F d,N λ λ λ (r r r) with an added technical step. The result of Arms, Marsden, and Moncrief specifically treats zero level sets of momentum maps, and extends trivially to handle level sets of fixed points of the Ad * -action on the dual of the Lie algebra. The general frame space F d,N λ λ λ (r r r) is a level set of a coadjoint orbit: F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}). The standard shifting trick of symplectic geometry (see, e.g., [48, p. 376]) can be used to realize F d,N λ λ λ (r r r) as the 0-level set of an associated momentum map on the symplectic manifold C d×N ×O −λ λ λ ; more specifically, this space is endowed with a product form, where the symplectic form on the second factor is −ω KKS with ω KKS being the canonical Kirillov-Kostant-Souriau symplectic form on a coadjoint orbit defined in (6). After applying the shifting trick, similar computations can be done to describe the local geometry of singular points.\nIn fact, similar local characterizations can be derived near orthodecomposable frames in any frame space F d,N λ λ λ (r r r) with an added technical step. The result of Arms, Marsden, and Moncrief specifically treats zero level sets of momentum maps, and extends trivially to handle level sets of fixed points of the Ad * -action on the dual of the Lie algebra. The general frame space F d,N λ λ λ (r r r) is a level set of a coadjoint orbit: F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}). The standard shifting trick of symplectic geometry (see, e.g., [48, p. 376]) can be used to realize F d,N λ λ λ (r r r) as the 0-level set of an associated momentum map on the symplectic manifold C d×N ×O −λ λ λ ; more specifically, this space is endowed with a product form, where the symplectic form on the second factor is −ω KKS with ω KKS being the canonical Kirillov-Kostant-Souriau symplectic form on a coadjoint orbit defined in (6). After applying the shifting trick, similar computations can be done to describe the local geometry of singular points."},{"title":"Toric Geometry of Frame Space","content":""},{"title":"Toric Geometry of Frame Space","content":""},{"title":"Toric Symplectic Manifolds","content":""},{"title":"Toric Symplectic Manifolds","content":"Recall that our goal is to show that the collection of full-spark frames in F d,N λ λ λ (r r r) has full measure. As mentioned in the introduction, the natural measure on F d,N λ λ λ (r r r) is the Hausdorff measure it inherits as a compact subset of C d×N . This measure is not so easy to get a handle on directly, but symplectic geometry provides a more tractable approach.\nRecall that our goal is to show that the collection of full-spark frames in F d,N λ λ λ (r r r) has full measure. As mentioned in the introduction, the natural measure on F d,N λ λ λ (r r r) is the Hausdorff measure it inherits as a compact subset of C d×N . This measure is not so easy to get a handle on directly, but symplectic geometry provides a more tractable approach.\nAll symplectic manifolds are equipped with natural measures. Specifically, suppose (M,ω) is a symplectic manifold of dimension 2n. By the non-degeneracy of the symplectic form ω, the maximal wedge power ω ∧n = ω ∧ ⋅⋅⋅ ∧ ω n is nowhere-vanishing, and hence defines a volume form and associated symplectic or Liouville measure m ω : for a Borel set U ⊂ M, m ω (U) ∶= ∫ U ω ∧n . While F d,N λ λ λ (r r r) is not symplectic, by Proposition 2.16 its quotient\nAll symplectic manifolds are equipped with natural measures. Specifically, suppose (M,ω) is a symplectic manifold of dimension 2n. By the non-degeneracy of the symplectic form ω, the maximal wedge power ω ∧n = ω ∧ ⋅⋅⋅ ∧ ω n is nowhere-vanishing, and hence defines a volume form and associated symplectic or Liouville measure m ω : for a Borel set U ⊂ M, m ω (U) ∶= ∫ U ω ∧n . While F d,N λ λ λ (r r r) is not symplectic, by Proposition 2.16 its quotient\nis. Moreover, since C d×N is Kähler-i.e., it is a complex manifold with a Hermitian metric whose negative imaginary part defines the standard symplectic form and whose real part defines the standard Riemannian metric-so is the symplectic reduction, and the symplectic measure agrees with the pushforward of Hausdorff measure on F d,N λ λ λ (r r r) by the quotient map F d,N λ λ λ (r r r) → F d,N λ λ λ (r r r) (U(d) × G) [28,Theorem 3.1]. Therefore, to prove our main theorem it suffices to prove that the equivalence classes of full-spark frames have full measure in F d,N λ λ λ (r r r) (U(d) × G), which we can do symplectically. The key is that F d,N λ λ λ (r r r) (U(d) × G) admits a Hamiltonian action of a high-dimensional torus, which considerably simplifies the task of understanding the symplectic measure.\nis. Moreover, since C d×N is Kähler-i.e., it is a complex manifold with a Hermitian metric whose negative imaginary part defines the standard symplectic form and whose real part defines the standard Riemannian metric-so is the symplectic reduction, and the symplectic measure agrees with the pushforward of Hausdorff measure on F d,N λ λ λ (r r r) by the quotient map F d,N λ λ λ (r r r) → F d,N λ λ λ (r r r) (U(d) × G) [28,Theorem 3.1]. Therefore, to prove our main theorem it suffices to prove that the equivalence classes of full-spark frames have full measure in F d,N λ λ λ (r r r) (U(d) × G), which we can do symplectically. The key is that F d,N λ λ λ (r r r) (U(d) × G) admits a Hamiltonian action of a high-dimensional torus, which considerably simplifies the task of understanding the symplectic measure.\nIf a 2n-dimensional symplectic manifold (M,ω) admits a Hamiltonian action of a torus U(1) k , then the associated momentum map Φ ∶ M → u(1) k * ≃ R k has convex image and connected level sets:\nIf a 2n-dimensional symplectic manifold (M,ω) admits a Hamiltonian action of a torus U(1) k , then the associated momentum map Φ ∶ M → u(1) k * ≃ R k has convex image and connected level sets:\nTheorem 3.1 (Atiyah [4] and Guillemin-Sternberg [24]). With notation as above:\nTheorem 3.1 (Atiyah [4] and Guillemin-Sternberg [24]). With notation as above:\n• For any v ∈ R k , Φ −1 (v) is either empty or connected.\n• For any v ∈ R k , Φ −1 (v) is either empty or connected.\n• The image Φ(M) is the convex hull of the images of the fixed points of the torus action.\n• The image Φ(M) is the convex hull of the images of the fixed points of the torus action.\nIn particular, when M is (relatively) compact, (the closure of) Φ(M) is a bounded convex polytope P called the moment polytope associated to the Hamiltonian torus action.\nIn particular, when M is (relatively) compact, (the closure of) Φ(M) is a bounded convex polytope P called the moment polytope associated to the Hamiltonian torus action.\nIn general, the Duistermaat-Heckman theorem [17] precisely describes the relationship between the pushforward measure Φ * (m ω ) and Lebesgue measure on the moment polytope P. Since we will not need the full statement, we restrict to the case when k = n, that is when M admits a Hamiltonian action of a half-dimensional torus. In this case M is called a toric symplectic manifold. Toric symplectic manifolds are closely related to toric varieties [13] and are completely classified by the combinatorics of the moment polytope [15].\nIn general, the Duistermaat-Heckman theorem [17] precisely describes the relationship between the pushforward measure Φ * (m ω ) and Lebesgue measure on the moment polytope P. Since we will not need the full statement, we restrict to the case when k = n, that is when M admits a Hamiltonian action of a half-dimensional torus. In this case M is called a toric symplectic manifold. Toric symplectic manifolds are closely related to toric varieties [13] and are completely classified by the combinatorics of the moment polytope [15].\nTheorem 3.2 (Duistermaat-Heckman [17], see also [9,Chapter 30]). Let M be a 2n-dimensional toric symplectic manifold with moment polytope P. The pushforward measure Φ * (m ω ) is a constant multiple of Lebesgue measure on P.\nTheorem 3.2 (Duistermaat-Heckman [17], see also [9,Chapter 30]). Let M be a 2n-dimensional toric symplectic manifold with moment polytope P. The pushforward measure Φ * (m ω ) is a constant multiple of Lebesgue measure on P."},{"title":"Toric Structure of Frame Space","content":""},{"title":"Toric Structure of Frame Space","content":""},{"title":"Circle Actions on Frame Space","content":""},{"title":"Circle Actions on Frame Space","content":"With Theorem 3.2 in mind, our strategy is to show that the top stratum of F d,N λ λ λ (r r r) (U(d) × G) is toric, and then to see that the image of the spark-deficient frames in the moment polytope has measure zero with respect to Lebesgue measure.\nWith Theorem 3.2 in mind, our strategy is to show that the top stratum of F d,N λ λ λ (r r r) (U(d) × G) is toric, and then to see that the image of the spark-deficient frames in the moment polytope has measure zero with respect to Lebesgue measure.\nWe begin by defining a torus action on the full space of frames F d,N and then show that our construction descends to the symplectic quotient. Let F ∈ F d,N with column vectors f i ∈ C d×1 , i = 1,... ,N. For each k = 1,...,N, let\nWe begin by defining a torus action on the full space of frames F d,N and then show that our construction descends to the symplectic quotient. Let F ∈ F d,N with column vectors f i ∈ C d×1 , i = 1,... ,N. For each k = 1,...,N, let\narranged in decreasing order-each such sum is rank at most k, so we only need consider the first k eigenvalues if k < d. The quantities µ k j were dubbed eigensteps by Cahill et al. [7]. To get well-defined circle actions, we make the assumption that the eigenvalues µ k j , j = 1,...,k, are all distinctthe necessity of this assumption is explained below in Remark 3.4. Let u k1 ,... ,u k min(k,d) be the eigenvectors corresponding to the nonzero eigenvalues. For each k = 1,...,N and each j = 1,... ,min(k,d), define a U(1)-action on F, denoted\narranged in decreasing order-each such sum is rank at most k, so we only need consider the first k eigenvalues if k < d. The quantities µ k j were dubbed eigensteps by Cahill et al. [7]. To get well-defined circle actions, we make the assumption that the eigenvalues µ k j , j = 1,...,k, are all distinctthe necessity of this assumption is explained below in Remark 3.4. Let u k1 ,... ,u k min(k,d) be the eigenvectors corresponding to the nonzero eigenvalues. For each k = 1,...,N and each j = 1,... ,min(k,d), define a U(1)-action on F, denoted\nby defining it at the frame vector level as\nby defining it at the frame vector level as\nwhere t ∈ [0,2π).\nwhere t ∈ [0,2π).\nKeeping track of the exact isomorphism used in this identification will be useful later on.\nKeeping track of the exact isomorphism used in this identification will be useful later on."},{"title":"Remark 3.4.","content":""},{"title":"Remark 3.4.","content":"The assumption that the eigenvalues µ k j are distinct means that the eigenspaces are all one-dimensional, which, in turn, gives a well-defined ordering of the eigenvectors u k j . Without the isolated eigenvalues assumption, some of these actions would degenerate to U(ℓ)-actions, with ℓ the multiplicity of a repeated eigenvalue.\nThe assumption that the eigenvalues µ k j are distinct means that the eigenspaces are all one-dimensional, which, in turn, gives a well-defined ordering of the eigenvectors u k j . Without the isolated eigenvalues assumption, some of these actions would degenerate to U(ℓ)-actions, with ℓ the multiplicity of a repeated eigenvalue.\nRemark 3.5. The action defined by (13) really defines a circle action; i.e., it is 2π-periodic. Indeed, this follows from Lemma 3.11 below, and is also shown in [19,Corollary 5.1.4].\nRemark 3.5. The action defined by (13) really defines a circle action; i.e., it is 2π-periodic. Indeed, this follows from Lemma 3.11 below, and is also shown in [19,Corollary 5.1.4].\nNext we show that this action on F d,N induces a well-defined action on F d,N λ λ λ (r r r) (U(d) × G). We do so in stages. Proposition 3.6. The circle action φ k j commutes with the T action.\nNext we show that this action on F d,N induces a well-defined action on F d,N λ λ λ (r r r) (U(d) × G). We do so in stages. Proposition 3.6. The circle action φ k j commutes with the T action.\nProof. We need to show that if two frames F 1 and F 2 lie in the same T orbit, then so do φ k j (t,F 1 ) and φ k j (t,F 2 ). Indeed, this holds since the formula for the action (13) makes it clear that\nProof. We need to show that if two frames F 1 and F 2 lie in the same T orbit, then so do φ k j (t,F 1 ) and φ k j (t,F 2 ). Indeed, this holds since the formula for the action (13) makes it clear that\nfor all θ . Proof. The claim is clear when k + 1 ≤ i ≤ N, since the action is trivial in that case. It remains to check the claim when 1 ≤ i ≤ k. In this case, we have φ k j (t,A f i ) = exp(t √ −1Au k j u * k j A * )A f i , since (µ k j ,Au k j ) are the corresponding eigenpair for the kth partial frame operator of AF. In turn, we have\nfor all θ . Proof. The claim is clear when k + 1 ≤ i ≤ N, since the action is trivial in that case. It remains to check the claim when 1 ≤ i ≤ k. In this case, we have φ k j (t,A f i ) = exp(t √ −1Au k j u * k j A * )A f i , since (µ k j ,Au k j ) are the corresponding eigenpair for the kth partial frame operator of AF. In turn, we have\nSince the circle action φ k j is defined on F = f 1 ⋯ f N ∈ F d,N through the action on the individual frame vectors, we have the following corollary. The next proposition is obvious from the formula for φ k j .\nSince the circle action φ k j is defined on F = f 1 ⋯ f N ∈ F d,N through the action on the individual frame vectors, we have the following corollary. The next proposition is obvious from the formula for φ k j .\nProposition 3.9. The circle action φ k j preserves norms of frame vectors.\nProposition 3.9. The circle action φ k j preserves norms of frame vectors.\nThe following proposition is similar, but is less obvious. The proof uses a lemma.\nThe following proposition is similar, but is less obvious. The proof uses a lemma.\nProof. This follows from the Taylor series representation of the matrix exponential and the property that P 2 = P.\nProof. This follows from the Taylor series representation of the matrix exponential and the property that P 2 = P.\nProof of Proposition 3. 10.\nProof of Proposition 3. 10.\nis a frame and let S = FF * = f 1 f * 1 + ⋅⋅⋅ + f N f * N be its frame operator with spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0. Let k ∈ {1,...,N} and let S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k be the partial frame operator with spectrum µ k1 ≥ ⋅⋅⋅ ≥ µ kd ≥ 0. Let j ∈ {1,...,min(k,d)}, and consider the torus action associated with the eigenvalue µ k j . Then, by definition,\nis a frame and let S = FF * = f 1 f * 1 + ⋅⋅⋅ + f N f * N be its frame operator with spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0. Let k ∈ {1,...,N} and let S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k be the partial frame operator with spectrum µ k1 ≥ ⋅⋅⋅ ≥ µ kd ≥ 0. Let j ∈ {1,...,min(k,d)}, and consider the torus action associated with the eigenvalue µ k j . Then, by definition,\nand the corresponding frame operator is\nand the corresponding frame operator is\nusing Lemma 3.11. For i 1 ≠ i 2 the vectors u ki 1 and u ki 2 are Hermitian orthogonal and hence the product of projections u ki 1 u * ki 1 u ki 2 u * ki 2 = 0. Combining this with the spectral decomposition S k = µ k1 u k1 u * k1 + ⋅⋅⋅ + µ kd u kd u * kd , and again using the fact that projections are idempotent, we see that\nusing Lemma 3.11. For i 1 ≠ i 2 the vectors u ki 1 and u ki 2 are Hermitian orthogonal and hence the product of projections u ki 1 u * ki 1 u ki 2 u * ki 2 = 0. Combining this with the spectral decomposition S k = µ k1 u k1 u * k1 + ⋅⋅⋅ + µ kd u kd u * kd , and again using the fact that projections are idempotent, we see that\nso the frame operator is invariant under the circle action.\nso the frame operator is invariant under the circle action.\nCombining Propositions 3.6, 3.9, and 3.10 with Corollary 3.8, we obtain:\nCombining Propositions 3.6, 3.9, and 3.10 with Corollary 3.8, we obtain:\nProposition 3.12. The circle action φ k j on F d,N descends to a well-defined action on F d,N λ λ λ (r r r) (U(d) × G).\nProposition 3.12. The circle action φ k j on F d,N descends to a well-defined action on F d,N λ λ λ (r r r) (U(d) × G)."},{"title":"The Momentum Map of the Circle Action","content":""},{"title":"The Momentum Map of the Circle Action","content":"For each k = 1,...,N and j = 1,...,min(k,d), define a map\nFor each k = 1,...,N and j = 1,...,min(k,d), define a map\nwhere µ k j = µ k j (F) is (as above) the jth eigenvalue (in descending order) of the kth partial frame operator of F. Recall that we identify u(1) ≈ R via (14). We likewise identify u(1) * ≈ R, where R * ≈ R is identified via the pairing\nwhere µ k j = µ k j (F) is (as above) the jth eigenvalue (in descending order) of the kth partial frame operator of F. Recall that we identify u(1) ≈ R via (14). We likewise identify u(1) * ≈ R, where R * ≈ R is identified via the pairing\non R × R. This allows us to state the following result.\non R × R. This allows us to state the following result.\nProposition 3.13. The map Φ k j ∶ F d,N → R is a momentum map for the circle action φ k j on the dense open subset of matrices whose kth partial frame operators have isolated jth eigenvalues.\nProposition 3.13. The map Φ k j ∶ F d,N → R is a momentum map for the circle action φ k j on the dense open subset of matrices whose kth partial frame operators have isolated jth eigenvalues.\nTo prove the proposition, we introduce some notation and technical lemmas. Let F = f 1 ⋯ f N ∈ F d,N and let S k = S k (F) = f 1 f * 1 + ⋯ + f k f * k be its partial frame operator. Then µ k j (F) is the jth eigenvalue (in decreasing order) of S k .\nTo prove the proposition, we introduce some notation and technical lemmas. Let F = f 1 ⋯ f N ∈ F d,N and let S k = S k (F) = f 1 f * 1 + ⋯ + f k f * k be its partial frame operator. Then µ k j (F) is the jth eigenvalue (in decreasing order) of S k .\nWe first consider the mapμ j which takes a Hermitian k × k matrix to its jth eigenvalue. In what follows, let ⟨⋅,⋅⟩ denote the Frobenius inner product on F d,N . Lemma 3.14. Suppose that S is a k × k Hermitian matrix with isolated jth eigenvalueμ j (S). Thenμ j is a smooth map in a neighborhood of S. Moreover, its gradient at S is given by\nWe first consider the mapμ j which takes a Hermitian k × k matrix to its jth eigenvalue. In what follows, let ⟨⋅,⋅⟩ denote the Frobenius inner product on F d,N . Lemma 3.14. Suppose that S is a k × k Hermitian matrix with isolated jth eigenvalueμ j (S). Thenμ j is a smooth map in a neighborhood of S. Moreover, its gradient at S is given by\nwhereū j is the unit eigenvector associated toμ j (S).\nwhereū j is the unit eigenvector associated toμ j (S).\nProof. Let S have isolated jth eigenvalue, which we denote simply as µ. We denote the associated unit eigenvector as u. Let S ′ be a variation of S. Denote the jth eigenvalue and unit eigenvector of S + εS ′ as µ(ε) and u(ε), respectively; we denote the ε-derivatives of these functions asμ(ε) andu(ε). Then\nProof. Let S have isolated jth eigenvalue, which we denote simply as µ. We denote the associated unit eigenvector as u. Let S ′ be a variation of S. Denote the jth eigenvalue and unit eigenvector of S + εS ′ as µ(ε) and u(ε), respectively; we denote the ε-derivatives of these functions asμ(ε) andu(ε). Then\n= ⟨S ′ u,u⟩ + µ (⟨u(0),u⟩ + ⟨u,u(0))⟩ (17)\n= ⟨S ′ u,u⟩ + µ (⟨u(0),u⟩ + ⟨u,u(0))⟩ (17)\nwhere (16) follows because S is Hermitian and (17) follows by the condition that u(ε) is constant in ε. Lemma 3.15. Let F = f 1 ⋯ f N ∈ F d,N and s ∈ R ≈ u(1) (using the identification (14)). The infinitesimal vector field associated to s which is induced by the action φ k j is given at F by\nwhere (16) follows because S is Hermitian and (17) follows by the condition that u(ε) is constant in ε. Lemma 3.15. Let F = f 1 ⋯ f N ∈ F d,N and s ∈ R ≈ u(1) (using the identification (14)). The infinitesimal vector field associated to s which is induced by the action φ k j is given at F by\nProof. The infinitesimal vector field is given at the frame level by\nProof. The infinitesimal vector field is given at the frame level by\nThe result follows.\nThe result follows.\nWe are now prepared to prove the proposition.\nWe are now prepared to prove the proposition.\nProof of Proposition 3.13. We need to show that, for (14)), the following equation holds:\nProof of Proposition 3.13. We need to show that, for (14)), the following equation holds:\nThis is accomplished by direct computation. The right hand side of (18) simplifies to\nThis is accomplished by direct computation. The right hand side of (18) simplifies to\nwhere (19) uses the expression for F s from Lemma 3.15. The left hand side of (18) becomes\nwhere (19) uses the expression for F s from Lemma 3.15. The left hand side of (18) becomes\nwhere the first line follows by our choice of identification R ≈ u(1) * (15) and where S k (F) is the kth partial frame operator of F andμ j is the jth eigenvalue function on d × d Hermitian matrices. A straightforward computation shows that\nwhere the first line follows by our choice of identification R ≈ u(1) * (15) and where S k (F) is the kth partial frame operator of F andμ j is the jth eigenvalue function on d × d Hermitian matrices. A straightforward computation shows that\n. Putting this together with Lemma 3.14, our simplification continues as\n. Putting this together with Lemma 3.14, our simplification continues as\nwhere the last line follows by the observation that the trace terms in (21) are conjugates. Since (22) and (20) agree, (18) has been established.\nwhere the last line follows by the observation that the trace terms in (21) are conjugates. Since (22) and (20) agree, (18) has been established."},{"title":"Dimension Counting","content":""},{"title":"Dimension Counting","content":"We now pause briefly to count the possible number of independent eigensteps µ k j . The µ k j associated to a frame define a map F d,N λ λ λ (r r r) → R m for some m depending on N and d; as has been previously observed [7,8,27], the image of this map is a convex polytope, whose dimension we now determine. To slightly simplify some calculations, we will assume in this section that r r r is strongly λ λ λ -admissible.\nWe now pause briefly to count the possible number of independent eigensteps µ k j . The µ k j associated to a frame define a map F d,N λ λ λ (r r r) → R m for some m depending on N and d; as has been previously observed [7,8,27], the image of this map is a convex polytope, whose dimension we now determine. To slightly simplify some calculations, we will assume in this section that r r r is strongly λ λ λ -admissible.\nSince each partial frame operator S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k is a rank-1 perturbation S k = S k−1 + f k f * k of the previous partial frame operator, Weyl's perturbation inequalities (see, e.g., [6,Chapter III]) imply that the eigenvalues of S k and S k−1 satisfy the interlacing inequalities ⋅⋅⋅ ≥ µ k, j ≥ µ k−1, j ≥ µ k, j+1 ≥ µ k−1, j+1 ≥ ...\nSince each partial frame operator S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k is a rank-1 perturbation S k = S k−1 + f k f * k of the previous partial frame operator, Weyl's perturbation inequalities (see, e.g., [6,Chapter III]) imply that the eigenvalues of S k and S k−1 satisfy the interlacing inequalities ⋅⋅⋅ ≥ µ k, j ≥ µ k−1, j ≥ µ k, j+1 ≥ µ k−1, j+1 ≥ ...\nfor each k = 2,... ,N and j = 1,...,min{d,k}, where we use the notation µ k, j in place of µ k j when we need to disambiguate the index k from the index j. When N = 4 and d = 3, this produces the array\nfor each k = 2,... ,N and j = 1,...,min{d,k}, where we use the notation µ k, j in place of µ k j when we need to disambiguate the index k from the index j. When N = 4 and d = 3, this produces the array\nwhere we recall that S N = S, and hence µ N, j = λ j for each j = 1,...,d.\nwhere we recall that S N = S, and hence µ N, j = λ j for each j = 1,...,d.\nIn other words, entries in the array are greater than entries to the right, regardless of vertical position. With this convention in place, we can omit explicit inequalities without losing any information:\nIn other words, entries in the array are greater than entries to the right, regardless of vertical position. With this convention in place, we can omit explicit inequalities without losing any information:\nIn general, then, the µ k, j satisfy the system of interlacing inequalities implied by the following diagram:\nIn general, then, the µ k, j satisfy the system of interlacing inequalities implied by the following diagram:\nIn this diagram, all entries are nonnegative; equivalently, we think of implicit zeros to the right of every row. The pattern of inequalities described by the above diagram is often called a Gelfand-Tsetlin pattern [14,21]; it corresponds to the Gelfand-Tsetlin integrable system on the flag manifold Oλ λ λ ≃ F d,N λ λ λ U(d) [26]. In fact, there is one further constraint: cyclic invariance of trace implies that the trace of each partial frame operator S k must equal the trace of the corresponding partial Gram matrix. That is, the sum of each row in (23) must equal the corresponding partial sum of squared frame vector norms:\nIn this diagram, all entries are nonnegative; equivalently, we think of implicit zeros to the right of every row. The pattern of inequalities described by the above diagram is often called a Gelfand-Tsetlin pattern [14,21]; it corresponds to the Gelfand-Tsetlin integrable system on the flag manifold Oλ λ λ ≃ F d,N λ λ λ U(d) [26]. In fact, there is one further constraint: cyclic invariance of trace implies that the trace of each partial frame operator S k must equal the trace of the corresponding partial Gram matrix. That is, the sum of each row in (23) must equal the corresponding partial sum of squared frame vector norms:\nfor each k = 1,...,N. With this constraint in place, we can count the number of free parameters in (23) and, hence, the number of independent φ k j . Notice that there are d(2N−d−1) 2 total entries in (23), excluding the top row (which we already know is fixed):\nfor each k = 1,...,N. With this constraint in place, we can count the number of free parameters in (23) and, hence, the number of independent φ k j . Notice that there are d(2N−d−1) 2 total entries in (23), excluding the top row (which we already know is fixed):\nfor the triangle in the bottom d rows, and d(N − d − 1) for the parallelogram in the upper N − d rows. Since each row sum is fixed, we lose one free parameter for each of the rows but the top one, meaning we subtract N − 1 parameters.\nfor the triangle in the bottom d rows, and d(N − d − 1) for the parallelogram in the upper N − d rows. Since each row sum is fixed, we lose one free parameter for each of the rows but the top one, meaning we subtract N − 1 parameters.\nIf all the eigenvalues λ 1 ,... ,λ d of the frame operator are distinct, we are done. However, if λ j = λ j+1 , then this implies that λ j = µ N−1, j = λ j+1 . More generally, if λ j has multiplicity k, meaning that λ j = λ j+1 = ⋅⋅⋅ = λ j+k−1 , then this fixes an entire upside-down triangle in (23) with vertices at λ j , λ j+k−1 , and µ N−k, j , comprising k(k−1) 2 of the µ i, j . Therefore, if the eigenvalues λ 1 ,... ,λ d have multiplicities k 1 ,... ,k ℓ with k 1 + ⋅⋅⋅ + k ℓ = d, then the total number of free parameters in (23) is\nIf all the eigenvalues λ 1 ,... ,λ d of the frame operator are distinct, we are done. However, if λ j = λ j+1 , then this implies that λ j = µ N−1, j = λ j+1 . More generally, if λ j has multiplicity k, meaning that λ j = λ j+1 = ⋅⋅⋅ = λ j+k−1 , then this fixes an entire upside-down triangle in (23) with vertices at λ j , λ j+k−1 , and µ N−k, j , comprising k(k−1) 2 of the µ i, j . Therefore, if the eigenvalues λ 1 ,... ,λ d have multiplicities k 1 ,... ,k ℓ with k 1 + ⋅⋅⋅ + k ℓ = d, then the total number of free parameters in (23) is\nThe interlacing inequalities implicit in the diagram (23) together with the row sums (24) determine a convex polytope P d,N λ λ λ (r r r), which we call the eigenstep polytope (at least when λ λ λ and r r r are rational, these are sometimes called (weight-restricted) Gelfand-Tsetlin polytopes in the combinatorics literature [1,14]). In the course of the discussion above, we have proved: Proposition 3.16. When r r r is strongly λ λ λ -admissible, the eigenstep polytope has dimension\nThe interlacing inequalities implicit in the diagram (23) together with the row sums (24) determine a convex polytope P d,N λ λ λ (r r r), which we call the eigenstep polytope (at least when λ λ λ and r r r are rational, these are sometimes called (weight-restricted) Gelfand-Tsetlin polytopes in the combinatorics literature [1,14]). In the course of the discussion above, we have proved: Proposition 3.16. When r r r is strongly λ λ λ -admissible, the eigenstep polytope has dimension\nwhere k 1 ,... ,k ℓ are the multiplicities of the spectrum λ λ λ = (λ 1 ,...,λ d ).\nwhere k 1 ,... ,k ℓ are the multiplicities of the spectrum λ λ λ = (λ 1 ,...,λ d ).\nComparing to Corollary 2.18,\nComparing to Corollary 2.18,\nIn the case of unit-norm tight frames,\nIn the case of unit-norm tight frames,"},{"title":"Hamiltonian Torus Action","content":""},{"title":"Hamiltonian Torus Action","content":"We now return to the task of showing that there is a Hamiltonian torus action on our frame spaces; the dimension d(λ λ λ ,r r r) just computed will be the dimension of the torus.\nWe now return to the task of showing that there is a Hamiltonian torus action on our frame spaces; the dimension d(λ λ λ ,r r r) just computed will be the dimension of the torus.\nOf course, the torus action will simply be the product of the individual circle actions, which we will show commute using Proposition 3.7.\nOf course, the torus action will simply be the product of the individual circle actions, which we will show commute using Proposition 3.7.\nProposition 3.17. The circle actions φ k j and φ mℓ commute for all k, j,m,ℓ. N . It suffices to prove the claim at the frame level, i.e.,\nProposition 3.17. The circle actions φ k j and φ mℓ commute for all k, j,m,ℓ. N . It suffices to prove the claim at the frame level, i.e.,\nfor all f i . Without loss of generality, assume k ≤ m. If k + 1 ≤ i ≤ N, then φ k j is the trivial action, so (25) is obvious.\nfor all f i . Without loss of generality, assume k ≤ m. If k + 1 ≤ i ≤ N, then φ k j is the trivial action, so (25) is obvious.\nwhere (26) 19. In conjunction with Theorem 3.1, this result can be used to show that the space of frames with prescribed eigensteps is connected. While this fact essentially follows from [7,Theorem 7], which was a key tool in the original proof of the Frame Homotopy Conjecture [8], the symplectic viewpoint puts this fact in a more general context. By Proposition 3.16, the torus acting on the open dense subset from Corollary 3.18 is half-dimensional. We have therefore proved the following theorem on the geometric structure of frame spaces.\nwhere (26) 19. In conjunction with Theorem 3.1, this result can be used to show that the space of frames with prescribed eigensteps is connected. While this fact essentially follows from [7,Theorem 7], which was a key tool in the original proof of the Frame Homotopy Conjecture [8], the symplectic viewpoint puts this fact in a more general context. By Proposition 3.16, the torus acting on the open dense subset from Corollary 3.18 is half-dimensional. We have therefore proved the following theorem on the geometric structure of frame spaces."},{"title":"Proof of Main Theorem","content":""},{"title":"Proof of Main Theorem","content":"We are now prepared to prove the main theorem using the toric symplectic framework developed in the previous section. We also sketch a more algebraically-flavored proof using related tools from Geometric Invariant Theory [42].\nWe are now prepared to prove the main theorem using the toric symplectic framework developed in the previous section. We also sketch a more algebraically-flavored proof using related tools from Geometric Invariant Theory [42]."},{"title":"A Toric Symplectic Approach","content":""},{"title":"A Toric Symplectic Approach","content":"It is clear that the three options (repeated below) are mutually exclusive and exhaust all possibilities, so it remains only to show that the conclusion in each part follows from the assumption.\nIt is clear that the three options (repeated below) are mutually exclusive and exhaust all possibilities, so it remains only to show that the conclusion in each part follows from the assumption.\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\nProof. As pointed out in Section 2.2.3, this is a consequence of the Schur-Horn theorem [30,46]; it is also a result of Casazza and Leon [12].\nProof. As pointed out in Section 2.2.3, this is a consequence of the Schur-Horn theorem [30,46]; it is also a result of Casazza and Leon [12].\n2. If r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\n2. If r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\nProof. Suppose r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible. Then F d,N λ λ λ (r r r) is nonempty by Casazza and Leon's result [12].\nProof. Suppose r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible. Then F d,N λ λ λ (r r r) is nonempty by Casazza and Leon's result [12].\nSince r r r is not strongly λ λ λ -admissible, there exists k ∈ {1,...,d} so that\nSince r r r is not strongly λ λ λ -admissible, there exists k ∈ {1,...,d} so that"},{"title":"The admissibility criterion requires that","content":""},{"title":"The admissibility criterion requires that","content":"If k = d, then we see that r k+1 = ⋅⋅⋅ = r N = 0, which does not satisfy the hypothesis that the r i are all positive.\nIf k = d, then we see that r k+1 = ⋅⋅⋅ = r N = 0, which does not satisfy the hypothesis that the r i are all positive.\nOtherwise, suppose F = f 1 ⋯ f N ∈ F d,N λ λ λ (r r r), so that λ λ λ is the spectrum of FF * and f i 2 = r i for all i = 1,... ,N.\nOtherwise, suppose F = f 1 ⋯ f N ∈ F d,N λ λ λ (r r r), so that λ λ λ is the spectrum of FF * and f i 2 = r i for all i = 1,... ,N.\nusing (27). Now, consider the portion of the eigenstep inequality diagram (23) forF starting from the (N − k)th row: 1μN−1,2μN−1\nusing (27). Now, consider the portion of the eigenstep inequality diagram (23) forF starting from the (N − k)th row: 1μN−1,2μN−1\nthe length-d collection of frame vectors f N−d+1 ,... , f N is linearly dependent, and hence F is spark-deficient. Since the choice of F was arbitrary, we see that F d,N λ λ λ (r r r) consists entirely of spark-deficient frames.\nthe length-d collection of frame vectors f N−d+1 ,... , f N is linearly dependent, and hence F is spark-deficient. Since the choice of F was arbitrary, we see that F d,N λ λ λ (r r r) consists entirely of spark-deficient frames.\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\nProof. It suffices to show that the set of full spark frames is full measure in the dense open subset of F d,N λ λ λ (r r r) from Theorem 3.20, which we denoteŨ. We will prove a subclaim: the set D d λ λ λ (r r r) ⊂Ũ of frames whose first d columns are linearly dependent is measure zero. The full claim then follows easily. Indeed, consider the action of the symmetric group on N letters on the matrix space C d×N given by permuting columns. This representation of the symmetric group embeds it as a subgroup of U(N), so that the action on C d×N is by isometries. The action of a permutation σ restricts to an isometry of each frame space F d,N λ λ λ (r r r) onto its image F d,N λ λ λ (σ ⋅ r r r), where σ ⋅ r r r is the corresponding permutation of the norm vector. It follows that the set of spark-deficient frames is realized as ∪ σ D d λ λ λ (σ ⋅ r r r), a finite union of measure zero sets. We proceed by establishing the subclaim.\nProof. It suffices to show that the set of full spark frames is full measure in the dense open subset of F d,N λ λ λ (r r r) from Theorem 3.20, which we denoteŨ. We will prove a subclaim: the set D d λ λ λ (r r r) ⊂Ũ of frames whose first d columns are linearly dependent is measure zero. The full claim then follows easily. Indeed, consider the action of the symmetric group on N letters on the matrix space C d×N given by permuting columns. This representation of the symmetric group embeds it as a subgroup of U(N), so that the action on C d×N is by isometries. The action of a permutation σ restricts to an isometry of each frame space F d,N λ λ λ (r r r) onto its image F d,N λ λ λ (σ ⋅ r r r), where σ ⋅ r r r is the corresponding permutation of the norm vector. It follows that the set of spark-deficient frames is realized as ∪ σ D d λ λ λ (σ ⋅ r r r), a finite union of measure zero sets. We proceed by establishing the subclaim.\ndenote the moment map for the torus action on U; that is, Φ is the eigenstep map. By the remarks in the previous paragraph, the condition that the first d columns of a frame F are linearly independent is equivalent to the statement that Φ dd ([F]) ≠ 0.\ndenote the moment map for the torus action on U; that is, Φ is the eigenstep map. By the remarks in the previous paragraph, the condition that the first d columns of a frame F are linearly independent is equivalent to the statement that Φ dd ([F]) ≠ 0.\nAccording to Section 3.2.3, the moment polytope P d,N λ λ λ (r r r) is a convex subset of an affine subspace A ⊂ R d(2N−d−1) 2 . Thinking of the Φ k j as coordinates on the ambient space, the condition Φ dd = 0 defines a hyperplane. The intersection of this hyperplane with P d,N λ λ λ (r r r) is positive codimension (hence measure zero), unless A is contained in the hyperplane. It is easy to see that this is not the case, as it suffices to show the existence of a pattern of the form (23) satisfying the defining equalities of A such that µ dd ≠ 0-such examples are trivial to construct without the polytope inequality constraints.\nAccording to Section 3.2.3, the moment polytope P d,N λ λ λ (r r r) is a convex subset of an affine subspace A ⊂ R d(2N−d−1) 2 . Thinking of the Φ k j as coordinates on the ambient space, the condition Φ dd = 0 defines a hyperplane. The intersection of this hyperplane with P d,N λ λ λ (r r r) is positive codimension (hence measure zero), unless A is contained in the hyperplane. It is easy to see that this is not the case, as it suffices to show the existence of a pattern of the form (23) satisfying the defining equalities of A such that µ dd ≠ 0-such examples are trivial to construct without the polytope inequality constraints.\nWe have so far shown that image of the set of equivalence classes of frames whose first d columns are linearly independent has full measure image in P d,N λ λ λ (r r r) under Φ. By the Duistermaat-Heckman Theorem (Theorem 3.2), this implies that the set has full measure in U. This completes the proof of the subclaim, and therefore completes the proof of the theorem.\nWe have so far shown that image of the set of equivalence classes of frames whose first d columns are linearly independent has full measure image in P d,N λ λ λ (r r r) under Φ. By the Duistermaat-Heckman Theorem (Theorem 3.2), this implies that the set has full measure in U. This completes the proof of the subclaim, and therefore completes the proof of the theorem."},{"title":"An algebraic approach","content":""},{"title":"An algebraic approach","content":"The above proof is based on symplectic geometry, but, under additional rationality assumptions, there is also an argument using algebraic geometry that shows that the collection of full-spark frames is dense in F d,N λ λ λ (r r r). While we expect a similar argument to apply to arbitrary rational λ λ λ and r r r, in which case the objects of interest are more general weight varieties [35], for simplicity we limit ourselves to the case when λ λ λ and r r r are constant, so we are talking about (scaled) unit-norm tight frames. It will be convenient to rescale so that r r r = (d,... ,d), and hence λ λ λ = (N,... ,N); note that both are integer vectors.\nThe above proof is based on symplectic geometry, but, under additional rationality assumptions, there is also an argument using algebraic geometry that shows that the collection of full-spark frames is dense in F d,N λ λ λ (r r r). While we expect a similar argument to apply to arbitrary rational λ λ λ and r r r, in which case the objects of interest are more general weight varieties [35], for simplicity we limit ourselves to the case when λ λ λ and r r r are constant, so we are talking about (scaled) unit-norm tight frames. It will be convenient to rescale so that r r r = (d,... ,d), and hence λ λ λ = (N,... ,N); note that both are integer vectors.\nAs above, it suffices to show that the (equivalence classes of) full spark frames are open and dense in the symplectic reduction F d,N λ λ λ (r r r) (U(d) × G) ≈ C d×N O −λ λ λ ×{r r r} (U(d) × G).\nAs above, it suffices to show that the (equivalence classes of) full spark frames are open and dense in the symplectic reduction F d,N λ λ λ (r r r) (U(d) × G) ≈ C d×N O −λ λ λ ×{r r r} (U(d) × G).\nSince λ λ λ is constant, the coadjoint orbit O −λ λ λ consists of the single point −λ λ λ I d . Taking the reduction in stages, the above quotient is isomorphic to C d×N −λ λ λ I d U(d) r r r G = Gr d (C N ) r r r G using Proposition 2.11 and Remark 2.13.\nSince λ λ λ is constant, the coadjoint orbit O −λ λ λ consists of the single point −λ λ λ I d . Taking the reduction in stages, the above quotient is isomorphic to C d×N −λ λ λ I d U(d) r r r G = Gr d (C N ) r r r G using Proposition 2.11 and Remark 2.13.\nWork of Sjamaar [47], which is a culmination of a series of results by Kempf and Ness [32], Guillemin and Sternberg [25], Kirwan [33], and Ness [45] relating symplectic reductions and Geometric Invariant Theory (GIT) quotients [42], has the following consequence in our case: where the line bundle L r r r on the Grassmannian is linearized to correspond to the action of the algebraic torus H = {(t 1 ,... ,t N ) ∈ (C * ) N ∶ ∏ti = 1} on C N given by identifying t t t = (t 1 ,...,t N ) ∈ H with the diagonal unitary matrix diag(t t t r r r t 1 ,...,t t t r r r t N ), where t t t r r r = t r 1 1 ⋯t r N N is the character of H corresponding to the vector r r r (cf. [20,31]).\nWork of Sjamaar [47], which is a culmination of a series of results by Kempf and Ness [32], Guillemin and Sternberg [25], Kirwan [33], and Ness [45] relating symplectic reductions and Geometric Invariant Theory (GIT) quotients [42], has the following consequence in our case: where the line bundle L r r r on the Grassmannian is linearized to correspond to the action of the algebraic torus H = {(t 1 ,... ,t N ) ∈ (C * ) N ∶ ∏ti = 1} on C N given by identifying t t t = (t 1 ,...,t N ) ∈ H with the diagonal unitary matrix diag(t t t r r r t 1 ,...,t t t r r r t N ), where t t t r r r = t r 1 1 ⋯t r N N is the character of H corresponding to the vector r r r (cf. [20,31]).\nis a frame representing a point [F] ∈ Gr d (C N ), then the determinants of the d × d minors of F are precisely the Plücker coordinates of [F], and in general the Plücker coordinates generate the homogeneous coordinate ring of Gr d (C N ). By definition, the full-spark frames are precisely those for which none of the Plücker coordinates vanish.\nis a frame representing a point [F] ∈ Gr d (C N ), then the determinants of the d × d minors of F are precisely the Plücker coordinates of [F], and in general the Plücker coordinates generate the homogeneous coordinate ring of Gr d (C N ). By definition, the full-spark frames are precisely those for which none of the Plücker coordinates vanish.\nIn turn, the homogeneous coordinate ring R of the GIT quotient Gr d (C N ) L r r r H consists of the H-invariant homogeneous coordinates on Gr d (C N ). It is known [31,Lemma 4.5] that R is spanned by monomials in the Plücker coordinates so that the total number of Plücker coordinates involving the ith column is kr i for some integer k independent of i.\nIn turn, the homogeneous coordinate ring R of the GIT quotient Gr d (C N ) L r r r H consists of the H-invariant homogeneous coordinates on Gr d (C N ). It is known [31,Lemma 4.5] that R is spanned by monomials in the Plücker coordinates so that the total number of Plücker coordinates involving the ith column is kr i for some integer k independent of i.\nIn our case, all r i = d, so taking the product of all the Plücker coordinates and raising it to the dth power gives a homogeneous coordinate on Gr d (C N ) L r r r H whose vanishing set is exactly the collection of (equivalence classes of) spark-deficient frames in the space of (scaled) unit-norm tight frames. Since this is the vanishing set of a homogeneous coordinate, it is a subvariety, and hence its complement-the collection of full-spark frames-is open in the Zariski topology, and in particular either empty or dense. Since there are full-spark frames in each space of unit-norm tight frames (for example, the first d rows of a scaled N × N discrete Fourier transform matrix [2]), the collection of fullspark frames cannot be empty, so it must be dense.\nIn our case, all r i = d, so taking the product of all the Plücker coordinates and raising it to the dth power gives a homogeneous coordinate on Gr d (C N ) L r r r H whose vanishing set is exactly the collection of (equivalence classes of) spark-deficient frames in the space of (scaled) unit-norm tight frames. Since this is the vanishing set of a homogeneous coordinate, it is a subvariety, and hence its complement-the collection of full-spark frames-is open in the Zariski topology, and in particular either empty or dense. Since there are full-spark frames in each space of unit-norm tight frames (for example, the first d rows of a scaled N × N discrete Fourier transform matrix [2]), the collection of fullspark frames cannot be empty, so it must be dense."},{"title":"Discussion","content":""},{"title":"Discussion","content":"Given a 2n-dimensional toric symplectic manifold M with moment polytope P, one can often find action-angle coordinates on M which take the form of a map α ∶ int(P) ×U(1) n → M which inverts the momentum map Φ ∶ M → P in the sense that Φ(α(p,t)) = p. In this case, Theorem 3.2 can be extended slightly to show that the image of α is a full-measure subset of M and that the map α is measure-preserving. Sampling P ×U(1) n with respect to the product of Lebesgue measure on P and the standard product measure on U(1) n and pushing forward by α gives a uniformly random sample from the symplectic measure on M (see, for example, the discussion in [10]).\nGiven a 2n-dimensional toric symplectic manifold M with moment polytope P, one can often find action-angle coordinates on M which take the form of a map α ∶ int(P) ×U(1) n → M which inverts the momentum map Φ ∶ M → P in the sense that Φ(α(p,t)) = p. In this case, Theorem 3.2 can be extended slightly to show that the image of α is a full-measure subset of M and that the map α is measure-preserving. Sampling P ×U(1) n with respect to the product of Lebesgue measure on P and the standard product measure on U(1) n and pushing forward by α gives a uniformly random sample from the symplectic measure on M (see, for example, the discussion in [10]).\nIn our setting, this means that coupling explicit action-angle coordinates on F d,N λ λ λ (r r r) (U(d)×G) with an algorithm for sampling P d,N λ λ λ (r r r) would give an algorithm for sampling random frames in F d,N λ λ λ (r r r). In particular, by Corollary 1.2, such an algorithm would provide endless quantities of full spark FUNTFs.\nIn our setting, this means that coupling explicit action-angle coordinates on F d,N λ λ λ (r r r) (U(d)×G) with an algorithm for sampling P d,N λ λ λ (r r r) would give an algorithm for sampling random frames in F d,N λ λ λ (r r r). In particular, by Corollary 1.2, such an algorithm would provide endless quantities of full spark FUNTFs.\nIt is natural to ask whether the analog of Theorem 1.1 holds for real and for quaternionic frames. Symplectic geometry is not the right tool in either case, but it is very plausible that the algebraic proof sketched in Section 4.2 could be adapted to the real case to show that full spark frames are dense in real frame spaces. In a different direction, the perspective based on isotropy orbits and isoparametric submanifolds introduced in [44] seems like the most promising way to understand the measures on real and quaternionic frame spaces.\nIt is natural to ask whether the analog of Theorem 1.1 holds for real and for quaternionic frames. Symplectic geometry is not the right tool in either case, but it is very plausible that the algebraic proof sketched in Section 4.2 could be adapted to the real case to show that full spark frames are dense in real frame spaces. In a different direction, the perspective based on isotropy orbits and isoparametric submanifolds introduced in [44] seems like the most promising way to understand the measures on real and quaternionic frame spaces.\nFinally, the spaces F d,N λ λ λ (r r r) (U(d) × G) are examples of weight varieties [35], and Goldin [22] has determined the rational cohomology ring of certain weight varieties, including the quotient F d,N (λ ,...,λ ) (r r r) (U(d) × G) of the space of λ -tight frames with fixed frame vector norms whenever it is a manifold. In particular, this determines the rational cohomology ring of the (U(d) × G)-quotient of FUNTF space when N and d are relatively prime. What about in the non-manifold case or for more general frame spectra?\nFinally, the spaces F d,N λ λ λ (r r r) (U(d) × G) are examples of weight varieties [35], and Goldin [22] has determined the rational cohomology ring of certain weight varieties, including the quotient F d,N (λ ,...,λ ) (r r r) (U(d) × G) of the space of λ -tight frames with fixed frame vector norms whenever it is a manifold. In particular, this determines the rational cohomology ring of the (U(d) × G)-quotient of FUNTF space when N and d are relatively prime. What about in the non-manifold case or for more general frame spectra?"},{"title":"Theorem 2 . 2 (","content":""},{"title":"Theorem 2 . 2 (","content":""},{"title":"Theorem 2 . 5 (","content":""},{"title":"Theorem 2 . 5 (","content":""},{"title":"Corollary 2 . 6 .","content":""},{"title":"Corollary 2 . 6 .","content":""},{"title":"and 0 =","content":""},{"title":"and 0 =","content":""},{"title":"Proposition 3. 7 .","content":""},{"title":"Proposition 3. 7 .","content":""},{"title":"Corollary 3. 8 .","content":""},{"title":"Corollary 3. 8 .","content":""},{"title":"Proposition 3 . 10 .","content":""},{"title":"Proposition 3 . 10 .","content":""},{"title":"Theorem 3 . 20 .","content":""},{"title":"Theorem 3 . 20 .","content":""},{"title":"Theorem 4. 1 .","content":""},{"title":"Theorem 4. 1 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"\"\n },\n {\n \"title\": \"Introduction\",\n \"content\": \"A frame in a Hilbert space (H,⟨⋅,⋅⟩) is traditionally defined as a collection { f i } i∈I of vectors in H so that for all v ∈ H we have\\nA frame in a Hilbert space (H,⟨⋅,⋅⟩) is traditionally defined as a collection { f i } i∈I of vectors in H so that for all v ∈ H we have\\nfor some numbers 0 < a ≤ b called frame bounds. In this paper, we focus on finite frames in complex Hilbert spaces, in which case H = C d for some integer d > 0 with its standard Hermitian inner product and norm, I = {1,...,N} is finite, and the above condition is equivalent to the collection { f 1 ,... , f N } being a spanning set for C d . Interest in finite frames is largely due to their application to robust signal representation. Modeling a signal as an element of a Hilbert space H, a frame allows one to take a sequence of \\\"measurements\\\" by recording the inner product of the signal with each of the frame vectors. This signal representation is more robust to noise in the signal or random erasures of measurements than a similar measurement scheme associated to an orthonormal basis, at least when the frame has certain properties [11,23,29]. These desirable properties for a frame are typically expressed as prescriptions for the norms of the frame vectors and for the spectrum of an operator associated to the frame, which we describe in more detail below. The collection of frames with prescribed norm and spectral data is easily seen to define an algebraic variety, and there is interest in the geometric structure of these frame varieties [8,18,49]. This paper explores the geometry of frame varieties through the lens of symplectic geometry. Symplectic geometry is a subfield of differential geometry that studies manifolds endowed with a certain geometric structure-to keep the paper accessible to the broader frame theory community, we have aimed to give self-contained expositions of the relevant ideas from symplectic geometry, but we will not go into details here in the introduction.\\nfor some numbers 0 < a ≤ b called frame bounds. In this paper, we focus on finite frames in complex Hilbert spaces, in which case H = C d for some integer d > 0 with its standard Hermitian inner product and norm, I = {1,...,N} is finite, and the above condition is equivalent to the collection { f 1 ,... , f N } being a spanning set for C d . Interest in finite frames is largely due to their application to robust signal representation. Modeling a signal as an element of a Hilbert space H, a frame allows one to take a sequence of \\\"measurements\\\" by recording the inner product of the signal with each of the frame vectors. This signal representation is more robust to noise in the signal or random erasures of measurements than a similar measurement scheme associated to an orthonormal basis, at least when the frame has certain properties [11,23,29]. These desirable properties for a frame are typically expressed as prescriptions for the norms of the frame vectors and for the spectrum of an operator associated to the frame, which we describe in more detail below. The collection of frames with prescribed norm and spectral data is easily seen to define an algebraic variety, and there is interest in the geometric structure of these frame varieties [8,18,49]. This paper explores the geometry of frame varieties through the lens of symplectic geometry. Symplectic geometry is a subfield of differential geometry that studies manifolds endowed with a certain geometric structure-to keep the paper accessible to the broader frame theory community, we have aimed to give self-contained expositions of the relevant ideas from symplectic geometry, but we will not go into details here in the introduction.\\nThe main contributions of this paper are as follows:\\nThe main contributions of this paper are as follows:\\n• Our main theorem, Theorem 1.1, says that with probability 1 every size-d subset of a random frame in C d with prescribed norms and spectral data is a basis. Frames satisfying this non-degeneracy condition are called full spark frames. This generalizes the complex case of Theorem 1.6 from Cahill, Mixon, and Strawn's paper [8], which established the genericity of the full spark condition for frames whose frame bounds are equal (this can be stated as a spectral constraint) and whose frame vectors are all unit-this is referred to as the space of finite unit-norm tight frames or FUNTFs.\\n• Our main theorem, Theorem 1.1, says that with probability 1 every size-d subset of a random frame in C d with prescribed norms and spectral data is a basis. Frames satisfying this non-degeneracy condition are called full spark frames. This generalizes the complex case of Theorem 1.6 from Cahill, Mixon, and Strawn's paper [8], which established the genericity of the full spark condition for frames whose frame bounds are equal (this can be stated as a spectral constraint) and whose frame vectors are all unit-this is referred to as the space of finite unit-norm tight frames or FUNTFs.\\n• We show in Theorem 2.25 that the space of frames with prescribed norm and spectral data is a smooth manifold, for generic choices of this data. In fact, we give necessary and sufficient conditions on the norms and eigenvalues which guarantee that the corresponding frame variety is smooth, and we describe the local geometry of singular varieties near their singular points. This theorem (together with surrounding results) generalizes work of Dykema and Strawn [18], which once again specializes to the space of FUNTFs, and answers generalizations of two open questions posed in [8].\\n• We show in Theorem 2.25 that the space of frames with prescribed norm and spectral data is a smooth manifold, for generic choices of this data. In fact, we give necessary and sufficient conditions on the norms and eigenvalues which guarantee that the corresponding frame variety is smooth, and we describe the local geometry of singular varieties near their singular points. This theorem (together with surrounding results) generalizes work of Dykema and Strawn [18], which once again specializes to the space of FUNTFs, and answers generalizations of two open questions posed in [8].\\n• Both of the previous two results are proved by novel applications of ideas from symplectic geometry to frame theory. Throughout the course of the paper, we show that many spaces of complex frames have natural interpretations from the symplectic point of view. In particular, we show in Theorem 3.20 that each space of frames with prescribed spectral and norm data has a dense open subset which projects onto a highly structured geometric object called a toric symplectic manifold. This geometric structure has measure-theoretic implications-we use it to prove our full spark theorem, but expect that it will be a useful tool for the future study of probability theory on frame spaces. Theorem 3.20 generalizes work of Flaschka and Millson [19], which is written in the context of pure symplectic geometry and makes no references to frames. The theorem also gives a symplectic interpretation of the frame-theoretic concept of eigensteps, introduced by Cahill et al. [7], and an auxiliary result used in its proof generalizes a theorem of Haga and Pegel [27].\\n• Both of the previous two results are proved by novel applications of ideas from symplectic geometry to frame theory. Throughout the course of the paper, we show that many spaces of complex frames have natural interpretations from the symplectic point of view. In particular, we show in Theorem 3.20 that each space of frames with prescribed spectral and norm data has a dense open subset which projects onto a highly structured geometric object called a toric symplectic manifold. This geometric structure has measure-theoretic implications-we use it to prove our full spark theorem, but expect that it will be a useful tool for the future study of probability theory on frame spaces. Theorem 3.20 generalizes work of Flaschka and Millson [19], which is written in the context of pure symplectic geometry and makes no references to frames. The theorem also gives a symplectic interpretation of the frame-theoretic concept of eigensteps, introduced by Cahill et al. [7], and an auxiliary result used in its proof generalizes a theorem of Haga and Pegel [27].\\nTo describe our results in detail, we now introduce more precise terminology and notation.\\nTo describe our results in detail, we now introduce more precise terminology and notation.\\nNotation and statement of the main theorem. Let F d,N be the space of frames of N vectors in C d . Identifying a frame { f i } N i=1 ∈ F d,N with the d × N matrix whose columns are the f i represented in the standard basis, the space F d,N can be viewed as an open, dense subset of the space C d×N of d × N complex matrices.\\nNotation and statement of the main theorem. Let F d,N be the space of frames of N vectors in C d . Identifying a frame { f i } N i=1 ∈ F d,N with the d × N matrix whose columns are the f i represented in the standard basis, the space F d,N can be viewed as an open, dense subset of the space C d×N of d × N complex matrices.\\nWhen a = b in (1), the frame satisfies a scaled Parseval identity\\nWhen a = b in (1), the frame satisfies a scaled Parseval identity\\nand such frames are called a-tight (or just tight); 1-tight frames are called Parseval frames.\\nand such frames are called a-tight (or just tight); 1-tight frames are called Parseval frames.\\nEach frame { f i } N i=1 ∈ F d,N has three related operators, which can be interpreted in terms of the d × N matrix F = f 1 f 2 ... f N :\\nEach frame { f i } N i=1 ∈ F d,N has three related operators, which can be interpreted in terms of the d × N matrix F = f 1 f 2 ... f N :\\n1. The analysis operator C d → C N is defined by v ↦ (⟨v, f 1 ⟩,... ,⟨v, f N ⟩) , or equivalently v ↦ F * v, where F * is the Hermitian adjoint (i.e., conjugate transpose) of F; 2. The synthesis operator C N → C d is defined by\\n1. The analysis operator C d → C N is defined by v ↦ (⟨v, f 1 ⟩,... ,⟨v, f N ⟩) , or equivalently v ↦ F * v, where F * is the Hermitian adjoint (i.e., conjugate transpose) of F; 2. The synthesis operator C N → C d is defined by\\n3. The frame operator C d → C d is the composition of the analysis and synthesis operators; i.e., v ↦ FF * v.\\n3. The frame operator C d → C d is the composition of the analysis and synthesis operators; i.e., v ↦ FF * v.\\nA simple calculation shows that a frame is a-tight if and only if its frame operator is aI d , where I d is the identity map on C d .\\nA simple calculation shows that a frame is a-tight if and only if its frame operator is aI d , where I d is the identity map on C d .\\nFrame operators are always Hermitian and positive-definite, so they have spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, which we will call the frame spectrum. If λ λ λ = (λ 1 ,...,λ d ), we will use F d,N λ λ λ to indicate the frames with frame spectrum λ λ λ . Notice that the a-tight frames are uniquely specified by their frame spectra: F d,N (a,...,a) is the space of all a-tight frames. In addition to specifying a frame operator, we also often want to fix the (squared) norms of the individual frame vectors. We can always permute the labels on the frame vectors, so it will be convenient in what follows to assume the norms are sorted in decreasing order. If r r r = (r 1 ,... ,r N ) is a non-increasing list of positive numbers r 1 ≥ ⋅⋅⋅ ≥ r N > 0, 1 1 We could also allow some of the r i to be zero, but this would complicate some statements below to no apparent benefit. we will use F d,N (r r r) to indicate the space of frames with f i 2 = r i . It is especially common to require that all the frame vectors have the same norm: f i 2 = r > 0 for all i, in which case the frame is an equal-norm frame; if r = 1, this is a unit-norm frame.\\nFrame operators are always Hermitian and positive-definite, so they have spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, which we will call the frame spectrum. If λ λ λ = (λ 1 ,...,λ d ), we will use F d,N λ λ λ to indicate the frames with frame spectrum λ λ λ . Notice that the a-tight frames are uniquely specified by their frame spectra: F d,N (a,...,a) is the space of all a-tight frames. In addition to specifying a frame operator, we also often want to fix the (squared) norms of the individual frame vectors. We can always permute the labels on the frame vectors, so it will be convenient in what follows to assume the norms are sorted in decreasing order. If r r r = (r 1 ,... ,r N ) is a non-increasing list of positive numbers r 1 ≥ ⋅⋅⋅ ≥ r N > 0, 1 1 We could also allow some of the r i to be zero, but this would complicate some statements below to no apparent benefit. we will use F d,N (r r r) to indicate the space of frames with f i 2 = r i . It is especially common to require that all the frame vectors have the same norm: f i 2 = r > 0 for all i, in which case the frame is an equal-norm frame; if r = 1, this is a unit-norm frame.\\nIn general, the frame norms determine the trace of the frame operator:\\nIn general, the frame norms determine the trace of the frame operator:\\nby the cyclic invariance of trace. Hence, for tight frames the frame operator must be r r r 2 d I d and a unit-norm tight frame must have λ i = N d for all i. If r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) with r 1 ≥ ⋅⋅⋅ ≥ r N > 0 and λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, we will use F d,N λ λ λ (r r r) ∶= F d,N λ λ λ ∩ F d,N (r r r) to denote the space of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ . This space has a natural probability measure: the (normalized) Hausdorff measure it inherits as a compact subset of C d×N . Equation (2) is not the only restriction imposed on the frame vector norms by the frame operator: the partial sums of the squared frame vector norms must be bounded above by the partial sums of the eigenvalues of the frame operator [12,19]. More precisely, there exists a frame { f i } N i=1 with f i 2 = r i > 0 and frame spectrum λ λ λ if and only if (2) holds and, for all k = 1,... ,d,\\nby the cyclic invariance of trace. Hence, for tight frames the frame operator must be r r r 2 d I d and a unit-norm tight frame must have λ i = N d for all i. If r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) with r 1 ≥ ⋅⋅⋅ ≥ r N > 0 and λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0, we will use F d,N λ λ λ (r r r) ∶= F d,N λ λ λ ∩ F d,N (r r r) to denote the space of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ . This space has a natural probability measure: the (normalized) Hausdorff measure it inherits as a compact subset of C d×N . Equation (2) is not the only restriction imposed on the frame vector norms by the frame operator: the partial sums of the squared frame vector norms must be bounded above by the partial sums of the eigenvalues of the frame operator [12,19]. More precisely, there exists a frame { f i } N i=1 with f i 2 = r i > 0 and frame spectrum λ λ λ if and only if (2) holds and, for all k = 1,... ,d,\\nGiven λ λ λ , we will call a non-increasing list r r r of positive numbers satisfying (2) and (3) λ λ λ -admissible, and if all inequalities in (3) are strict we call the list strongly λ λ λ -admissible. This terminology comes from Casazza and Leon [12], who showed that F d,N λ λ λ (r r r) is non-empty if and only if r r r is λ λ λ -admissible. Although the terminology does not appear in the work of Casazza and Leon, in general one says that the vector λ λ λ majorizes the vector r r r in the case that the relations (3) hold [39].\\nGiven λ λ λ , we will call a non-increasing list r r r of positive numbers satisfying (2) and (3) λ λ λ -admissible, and if all inequalities in (3) are strict we call the list strongly λ λ λ -admissible. This terminology comes from Casazza and Leon [12], who showed that F d,N λ λ λ (r r r) is non-empty if and only if r r r is λ λ λ -admissible. Although the terminology does not appear in the work of Casazza and Leon, in general one says that the vector λ λ λ majorizes the vector r r r in the case that the relations (3) hold [39].\\nis the size of the smallest linearly dependent subset [16]. The spark is bounded above by d + 1, and a frame with spark equal to d + 1 is called a full-spark frame. Equivalently, a frame in F d,N is full spark if and only if all of its size-d subsets are bases. Full spark frames are often desirable, for example because they provide unique reconstructions of the largest possible class of sparse signals [2,16].\\nis the size of the smallest linearly dependent subset [16]. The spark is bounded above by d + 1, and a frame with spark equal to d + 1 is called a full-spark frame. Equivalently, a frame in F d,N is full spark if and only if all of its size-d subsets are bases. Full spark frames are often desirable, for example because they provide unique reconstructions of the largest possible class of sparse signals [2,16].\\nWe are now ready to state our main theorem:\\nWe are now ready to state our main theorem:\\nLet r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) be nonincreasing lists of positive real numbers. There are three mutually exclusive possibilities for the space F d,N λ λ λ (r r r) of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ :\\nLet r r r = (r 1 ,... ,r N ) and λ λ λ = (λ 1 ,... ,λ d ) be nonincreasing lists of positive real numbers. There are three mutually exclusive possibilities for the space F d,N λ λ λ (r r r) of frames { f i } N i=1 with f i 2 = r i and frame spectrum λ λ λ :\\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\\n2. If r r r is λ λ λ -admissible but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\\n2. If r r r is λ λ λ -admissible but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\\nIn particular, since r r r = (1,. Structure of the paper. Section 2 begins with an exposition of the relevant ideas and tools from symplectic geometry. These tools are then applied to show that various spaces of frames with prescribed data have symplectic structures. Then, parameters λ λ λ and r r r for which the frame variety F d,N λ λ λ (r r r) is a smooth manifold are characterized. Section 3 is devoted to showing that the space F d,N λ λ λ (r r r) has a dense open subset which projects onto a toric symplectic manifold (we define and describe the basic properties of such an object in Section 3.1). The main theorem is proved in Section 4, as an application of the toric symplectic structure from the previous section. An alternative proof using related tools from algebraic geometry is also sketched. The paper concludes with a brief discussion of future directions in Section 5.\\nIn particular, since r r r = (1,. Structure of the paper. Section 2 begins with an exposition of the relevant ideas and tools from symplectic geometry. These tools are then applied to show that various spaces of frames with prescribed data have symplectic structures. Then, parameters λ λ λ and r r r for which the frame variety F d,N λ λ λ (r r r) is a smooth manifold are characterized. Section 3 is devoted to showing that the space F d,N λ λ λ (r r r) has a dense open subset which projects onto a toric symplectic manifold (we define and describe the basic properties of such an object in Section 3.1). The main theorem is proved in Section 4, as an application of the toric symplectic structure from the previous section. An alternative proof using related tools from algebraic geometry is also sketched. The paper concludes with a brief discussion of future directions in Section 5.\"\n },\n {\n \"title\": \"Symplectic Structure of Frame Varieties\",\n \"content\": \"\"\n },\n {\n \"title\": \"Symplectic Structure of Frame Varieties\",\n \"content\": \"In this section, we describe the symplectic structures on the frame varieties of interest; i.e., spaces of frames with prescribed frame spectra and/or norms. It has been previously observed in the literature that certain parameter choices may lead to frame spaces with singularities [8,18,49]. We give a precise characterization of this phenomenon and describe the local structure of singular points later in this section.\\nIn this section, we describe the symplectic structures on the frame varieties of interest; i.e., spaces of frames with prescribed frame spectra and/or norms. It has been previously observed in the literature that certain parameter choices may lead to frame spaces with singularities [8,18,49]. We give a precise characterization of this phenomenon and describe the local structure of singular points later in this section.\"\n },\n {\n \"title\": \"Symplectic Geometry\",\n \"content\": \"\"\n },\n {\n \"title\": \"Symplectic Geometry\",\n \"content\": \"We begin with a review of some concepts of symplectic geometry, with a focus on various notions of reducing a symplectic manifold by a Lie group action. We use [40] as our main reference for the basics of symplectic manifolds. The ideas we present are standard in the field of symplectic geometry; this subsection is mainly intended as a quick reference for non-experts and to standardize our notation.\\nWe begin with a review of some concepts of symplectic geometry, with a focus on various notions of reducing a symplectic manifold by a Lie group action. We use [40] as our main reference for the basics of symplectic manifolds. The ideas we present are standard in the field of symplectic geometry; this subsection is mainly intended as a quick reference for non-experts and to standardize our notation.\\nA symplectic form on a (smooth, real) manifold M is a closed, nondegenerate 2-form ω. For a point p ∈ M and tangent vectors X,Y ∈ T p M, we write ω p (X,Y ) ∈ R for the evaluation of ω on the vectors. The closedness condition means that the exterior derivative dω is identically zero and the nondegeneracy condition means that for every nonzero X ∈ T p M there exists Y ∈ T p M such that ω p (X,Y ) ≠ 0. A manifold endowed with a symplectic form is called a symplectic manifold, denoted (M,ω) or simply as M when the symplectic form is understood to be fixed. A simple argument shows that if a manifold M admits a symplectic form, then it must be even dimensional (over the reals).\\nA symplectic form on a (smooth, real) manifold M is a closed, nondegenerate 2-form ω. For a point p ∈ M and tangent vectors X,Y ∈ T p M, we write ω p (X,Y ) ∈ R for the evaluation of ω on the vectors. The closedness condition means that the exterior derivative dω is identically zero and the nondegeneracy condition means that for every nonzero X ∈ T p M there exists Y ∈ T p M such that ω p (X,Y ) ≠ 0. A manifold endowed with a symplectic form is called a symplectic manifold, denoted (M,ω) or simply as M when the symplectic form is understood to be fixed. A simple argument shows that if a manifold M admits a symplectic form, then it must be even dimensional (over the reals).\\nExample 2.1 (Complex n-Space). The prototypical example of a symplectic manifold is n-dimensional complex space C n , which is considered as a 2n-dimensional real manifold via the natural identification C n ≈ R 2n . For any n-tuple p of complex numbers, there is a natural isomorphism T p C n ≈ C n . Coordinates (x 1 + √ −1y 1 ,... ,x n + √ −1y n ) for C n correspond to real coordinates (x 1 ,... ,x n ,y 1 ,... ,y n ) in which a symplectic form is given by\\nExample 2.1 (Complex n-Space). The prototypical example of a symplectic manifold is n-dimensional complex space C n , which is considered as a 2n-dimensional real manifold via the natural identification C n ≈ R 2n . For any n-tuple p of complex numbers, there is a natural isomorphism T p C n ≈ C n . Coordinates (x 1 + √ −1y 1 ,... ,x n + √ −1y n ) for C n correspond to real coordinates (x 1 ,... ,x n ,y 1 ,... ,y n ) in which a symplectic form is given by\\nThis is referred to as the standard symplectic form on C n . In complex coordinates, (4) is expressed concretely for p ∈ C n and Z = (z 1 ,... ,z n ),\\nThis is referred to as the standard symplectic form on C n . In complex coordinates, (4) is expressed concretely for p ∈ C n and Z = (z 1 ,... ,z n ),\\nwhere Im denotes the imaginary part of a complex number and ⟨⋅,⋅⟩ is the standard Hermitian inner product on C n . The most important example of a symplectic manifold for our purposes is complex matrix space C d×N , which is just a reshaped version of the space C d⋅N described above. For a matrix F ∈ C d×N , we have a natural identification\\nwhere Im denotes the imaginary part of a complex number and ⟨⋅,⋅⟩ is the standard Hermitian inner product on C n . The most important example of a symplectic manifold for our purposes is complex matrix space C d×N , which is just a reshaped version of the space C d⋅N described above. For a matrix F ∈ C d×N , we have a natural identification\\nThis is just a transformation of the canonical symplectic form on C d⋅N under the reshaping map.\\nThis is just a transformation of the canonical symplectic form on C d⋅N under the reshaping map.\\nSince the space of frames F d,N ⊂ C d×N is an open submanifold, the standard symplectic structure on C d×N restricts to make F d,N a symplectic manifold.\\nSince the space of frames F d,N ⊂ C d×N is an open submanifold, the standard symplectic structure on C d×N restricts to make F d,N a symplectic manifold.\\nIn fact, every 2n-dimensional symplectic manifold is locally equivalent to C n with the standard symplectic form. Let us now make this statement precise. If Ψ ∶ N → M is a smooth map from a manifold N to a symplectic manifold (M,ω), the pullback form Ψ * ω on N is defined by\\nIn fact, every 2n-dimensional symplectic manifold is locally equivalent to C n with the standard symplectic form. Let us now make this statement precise. If Ψ ∶ N → M is a smooth map from a manifold N to a symplectic manifold (M,ω), the pullback form Ψ * ω on N is defined by\\nfor X,Y ∈ T p N and where DΨ(p) ∶ T p N → T Ψ(p) M denotes the derivative of Ψ at p. If (N,η) is also a symplectic manifold and Ψ is a diffeomorphism with the property that Ψ * ω = η, then we say Ψ is a symplectomorphism and that (N,η) and (M,ω) are symplectomorphic. A fundamental result of symplectic geometry is Darboux's Theorem [40,Theorem 3.2.2]: every point in a 2n-dimensional symplectic manifold has an open neighborhood U such that the symplectic manifold (U,ω U ) is symplectomorphic to C n with the standard symplectic form.\\nfor X,Y ∈ T p N and where DΨ(p) ∶ T p N → T Ψ(p) M denotes the derivative of Ψ at p. If (N,η) is also a symplectic manifold and Ψ is a diffeomorphism with the property that Ψ * ω = η, then we say Ψ is a symplectomorphism and that (N,η) and (M,ω) are symplectomorphic. A fundamental result of symplectic geometry is Darboux's Theorem [40,Theorem 3.2.2]: every point in a 2n-dimensional symplectic manifold has an open neighborhood U such that the symplectic manifold (U,ω U ) is symplectomorphic to C n with the standard symplectic form.\\nAn important aspect of symplectic geometry is the study of interactions between symplectic structures and certain group actions on their manifolds. Let G be a Lie group with Lie algebra g and suppose that G acts on a manifold M endowed with a symplectic form ω. For p ∈ M and g ∈ G, let g ⋅ p ∈ M denote the action of g on p. To each ξ ∈ g, one associates an infinitesimal vector field Y ξ on M via the formula\\nAn important aspect of symplectic geometry is the study of interactions between symplectic structures and certain group actions on their manifolds. Let G be a Lie group with Lie algebra g and suppose that G acts on a manifold M endowed with a symplectic form ω. For p ∈ M and g ∈ G, let g ⋅ p ∈ M denote the action of g on p. To each ξ ∈ g, one associates an infinitesimal vector field Y ξ on M via the formula\\nwhere g * denotes the dual to g, is called a momentum map for the G-action if its derivative interacts with the symplectic form as follows. Let DΦ(p) ∶ T p M → T Φ(p) g * denote the derivative of Φ at p ∈ M. Then for each X ∈ T p M, DΦ(p)(X) ∈ T Φ(p) g * ≈ g * , where we use the natural isomorphism coming from the fact that g * is a vector space. Then DΦ(p)(X) ∶ g → R, and for each ξ ∈ g we require\\nwhere g * denotes the dual to g, is called a momentum map for the G-action if its derivative interacts with the symplectic form as follows. Let DΦ(p) ∶ T p M → T Φ(p) g * denote the derivative of Φ at p ∈ M. Then for each X ∈ T p M, DΦ(p)(X) ∈ T Φ(p) g * ≈ g * , where we use the natural isomorphism coming from the fact that g * is a vector space. Then DΦ(p)(X) ∶ g → R, and for each ξ ∈ g we require\\nWe also require that the momentum map Φ is equivariant, in the following sense. Recall that the adjoint action of G on g is defined, for each g ∈ G, by the map Ad g ∶ g → g which is the derivative at the identity of the conjugation map h ↦ ghg −1 . The corresponding coadjoint action of G on the dual Lie algebra g * is defined, for each g ∈ G, by the map Ad * g ∶ g * → g * given by Ad * g (χ)(ξ ) ∶= χ(Ad g −1 (ξ )). When G is a matrix group, both the adjoint and coadjoint actions can be interpreted as conjugation actions. The momentum map is required to be equivariant with respect to the given G-action on M and the coadjoint action on g * . Explicitly, this means that, for each g ∈ G and each p ∈ M,\\nWe also require that the momentum map Φ is equivariant, in the following sense. Recall that the adjoint action of G on g is defined, for each g ∈ G, by the map Ad g ∶ g → g which is the derivative at the identity of the conjugation map h ↦ ghg −1 . The corresponding coadjoint action of G on the dual Lie algebra g * is defined, for each g ∈ G, by the map Ad * g ∶ g * → g * given by Ad * g (χ)(ξ ) ∶= χ(Ad g −1 (ξ )). When G is a matrix group, both the adjoint and coadjoint actions can be interpreted as conjugation actions. The momentum map is required to be equivariant with respect to the given G-action on M and the coadjoint action on g * . Explicitly, this means that, for each g ∈ G and each p ∈ M,\\nIf a G-action admits a momentum map, then we say the action is Hamiltonian. Hamiltonian actions give the appropriate setting for performing a quotient operation in the symplectic category. [38,41]). Let (M,ω) be a symplectic manifold with a Hamiltonian G-action, let Φ ∶ M → g * be a momentum map for the action and let χ ∈ g * be a regular value such that G acts freely on the level set Φ −1 (χ). Then the manifold\\nIf a G-action admits a momentum map, then we say the action is Hamiltonian. Hamiltonian actions give the appropriate setting for performing a quotient operation in the symplectic category. [38,41]). Let (M,ω) be a symplectic manifold with a Hamiltonian G-action, let Φ ∶ M → g * be a momentum map for the action and let χ ∈ g * be a regular value such that G acts freely on the level set Φ −1 (χ). Then the manifold\\ncalled the symplectic reduction or symplectic quotient over χ, admits a symplectic structure ω red which is uniquely characterized by the equation\\ncalled the symplectic reduction or symplectic quotient over χ, admits a symplectic structure ω red which is uniquely characterized by the equation\\nSee [5, Example III.2.18] for a construction of the complex projective space CP n−1 as a symplectic reduction C n ξ U(1), where U(1) acts on C n by scalar multiplication. Notice that, in the statement of Theorem 2.2, we required that G acts freely on the level set Φ −1 (χ). In particular, G acts on this level set, meaning that for any p ∈ M with Φ(p) = χ, it must be the case that Φ(g ⋅ p) = χ for all g ∈ G.\\nSee [5, Example III.2.18] for a construction of the complex projective space CP n−1 as a symplectic reduction C n ξ U(1), where U(1) acts on C n by scalar multiplication. Notice that, in the statement of Theorem 2.2, we required that G acts freely on the level set Φ −1 (χ). In particular, G acts on this level set, meaning that for any p ∈ M with Φ(p) = χ, it must be the case that Φ(g ⋅ p) = χ for all g ∈ G.\\nBy the equivariance required in the definition of the momentum map Φ, this implies that Ad * g (χ) = χ for all g ∈ G; in other words, χ must be a fixed point of the coadjoint action of G. Conversely, G will act on the level set over any fixed point of the coadjoint action.\\nBy the equivariance required in the definition of the momentum map Φ, this implies that Ad * g (χ) = χ for all g ∈ G; in other words, χ must be a fixed point of the coadjoint action of G. Conversely, G will act on the level set over any fixed point of the coadjoint action.\\nMore generally, when χ ∈ g * is not a fixed point of the coadjoint action, we can still take a symplectic reduction over the coadjoint orbit of χ, defined to be the set\\nMore generally, when χ ∈ g * is not a fixed point of the coadjoint action, we can still take a symplectic reduction over the coadjoint orbit of χ, defined to be the set\\nIn this case the equivariance of the momentum map ensures that G acts on Φ −1 (O χ ). [38,41]). Let M, ω, G and Φ be as above and let χ ∈ g * be a regular value. If the action of G on Φ −1 (O χ ) is free, then the manifold\\nIn this case the equivariance of the momentum map ensures that G acts on Φ −1 (O χ ). [38,41]). Let M, ω, G and Φ be as above and let χ ∈ g * be a regular value. If the action of G on Φ −1 (O χ ) is free, then the manifold\"\n },\n {\n \"title\": \"Theorem 2.3 (Marsden-Weinstein-Meyer Theorem for Coadjoint Orbits\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2.3 (Marsden-Weinstein-Meyer Theorem for Coadjoint Orbits\",\n \"content\": \"admits a symplectic structure ω red which is uniquely characterized by q * ω red = ι * ω, where q denotes the projection\\nadmits a symplectic structure ω red which is uniquely characterized by q * ω red = ι * ω, where q denotes the projection\\nIt is straightforward to show that when χ ∈ g * is a regular value, the action of G on Φ −1 (O χ ) is at worst locally free. Theorem 2.3 easily generalizes to this setting, with the only modification being that M O χ G is a symplectic orbifold rather than a symplectic manifold.\\nIt is straightforward to show that when χ ∈ g * is a regular value, the action of G on Φ −1 (O χ ) is at worst locally free. Theorem 2.3 easily generalizes to this setting, with the only modification being that M O χ G is a symplectic orbifold rather than a symplectic manifold.\\nPerhaps surprisingly, if we are more concerned with the \\\"symplectic\\\" part of this statement than the \\\"manifold\\\" (or \\\"orbifold\\\") part, Sjamaar and Lerman [48] have shown that χ being a regular value is not an essential hypothesis in Theorems 2.2 and 2.3. When χ is a singular value of Φ then Φ −1 (O χ ) is not a manifold, but it nonetheless admits a G action and the quotient still admits a natural symplectic structure.\\nPerhaps surprisingly, if we are more concerned with the \\\"symplectic\\\" part of this statement than the \\\"manifold\\\" (or \\\"orbifold\\\") part, Sjamaar and Lerman [48] have shown that χ being a regular value is not an essential hypothesis in Theorems 2.2 and 2.3. When χ is a singular value of Φ then Φ −1 (O χ ) is not a manifold, but it nonetheless admits a G action and the quotient still admits a natural symplectic structure.\\nTheorem 2.4 (Sjamaar-Lerman [48]). Let M, ω, G and Φ be as above and let χ ∈ g * . The symplectic reduction\\nTheorem 2.4 (Sjamaar-Lerman [48]). Let M, ω, G and Φ be as above and let χ ∈ g * . The symplectic reduction\\nRoughly speaking, M O χ G being a symplectic stratified space means that it is the disjoint union of symplectic manifolds and that these manifolds fit together nicely. Somewhat remarkably, each connected component contains a unique manifold piece which is open and dense: Therefore, up to sets of lower dimension, the symplectic reduction even over a singular value of a proper momentum map is a connected symplectic manifold.\\nRoughly speaking, M O χ G being a symplectic stratified space means that it is the disjoint union of symplectic manifolds and that these manifolds fit together nicely. Somewhat remarkably, each connected component contains a unique manifold piece which is open and dense: Therefore, up to sets of lower dimension, the symplectic reduction even over a singular value of a proper momentum map is a connected symplectic manifold.\"\n },\n {\n \"title\": \"Frame Spaces as Symplectic Quotients\",\n \"content\": \"\"\n },\n {\n \"title\": \"Frame Spaces as Symplectic Quotients\",\n \"content\": \"Various spaces of frames can be endowed with natural symplectic structures via Theorems 2.3 and 2.4. To be precise, the symplectic manifolds of interest are actually quotients of frame spaces by certain symmetry groups. Throughout this subsection, fix dimensions d and N and a frame spectrum λ λ λ = (λ 1 ,...,λ d ) with λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0.\\nVarious spaces of frames can be endowed with natural symplectic structures via Theorems 2.3 and 2.4. To be precise, the symplectic manifolds of interest are actually quotients of frame spaces by certain symmetry groups. Throughout this subsection, fix dimensions d and N and a frame spectrum λ λ λ = (λ 1 ,...,λ d ) with λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0.\"\n },\n {\n \"title\": \"Frames with Prescribed Frame Spectrum\",\n \"content\": \"\"\n },\n {\n \"title\": \"Frames with Prescribed Frame Spectrum\",\n \"content\": \"The unitary group acts on F d,N λ λ λ by left matrix multiplication. We then consider the quotient space F d,N λ λ λ U(d), which has a natural symplectic structure, as we will show below.\\nThe unitary group acts on F d,N λ λ λ by left matrix multiplication. We then consider the quotient space F d,N λ λ λ U(d), which has a natural symplectic structure, as we will show below.\\nBefore stating the result, we observe that the space H(d) of d × d Hermitian matrices may be identified with the dual Lie algebra u(d) * via the isomorphism α ∶ H(d) → u(d) * taking ξ to the linear functional α ξ ∶ u(d) → R, defined by\\nBefore stating the result, we observe that the space H(d) of d × d Hermitian matrices may be identified with the dual Lie algebra u(d) * via the isomorphism α ∶ H(d) → u(d) * taking ξ to the linear functional α ξ ∶ u(d) → R, defined by\\nWe begin with a preliminary result about this isomorphism. We define a U(d)-action on H(d) which we (with a slight abuse of terminology) refer to as the adjoint action:\\nWe begin with a preliminary result about this isomorphism. We define a U(d)-action on H(d) which we (with a slight abuse of terminology) refer to as the adjoint action:\\nLemma 2.7. The map α defined in (5) is equivariant with respect to the adjoint action on H(d) and the coadjoint action on u(d) * . That is, α Ad A (ξ ) = Ad * A α ξ . It follows that, under the map α, the coadjoint orbit of any element of u(d) * is identified with the collection of Hermitian matrices with a fixed spectrum.\\nLemma 2.7. The map α defined in (5) is equivariant with respect to the adjoint action on H(d) and the coadjoint action on u(d) * . That is, α Ad A (ξ ) = Ad * A α ξ . It follows that, under the map α, the coadjoint orbit of any element of u(d) * is identified with the collection of Hermitian matrices with a fixed spectrum.\\nProof. The first part is a straightforward computation: for η ∈ u(d),\\nProof. The first part is a straightforward computation: for η ∈ u(d),\\nAny Hermitian matrix ξ with spectrum λ λ λ can be expressed via its eigendecomposition as ξ = Ad A (diag(λ λ λ )) for some A ∈ U(d), so that the second part of the claim follows by equivariance.\\nAny Hermitian matrix ξ with spectrum λ λ λ can be expressed via its eigendecomposition as ξ = Ad A (diag(λ λ λ )) for some A ∈ U(d), so that the second part of the claim follows by equivariance.\\nBased on the second part of the lemma, we use O λ λ λ ⊂ H(d) to denote the set of matrices with fixed spectrum λ λ λ . We now describe the symplectic structure on the space of frames with prescribed frame spectrum.\\nBased on the second part of the lemma, we use O λ λ λ ⊂ H(d) to denote the set of matrices with fixed spectrum λ λ λ . We now describe the symplectic structure on the space of frames with prescribed frame spectrum.\"\n },\n {\n \"title\": \"Proposition 2.8 ( [43]). The action of U(d) on C d×N by left multiplication is Hamiltonian with momentum map\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 2.8 ( [43]). The action of U(d) on C d×N by left multiplication is Hamiltonian with momentum map\",\n \"content\": \"In particular, it has a natural symplectic structure.\\nIn particular, it has a natural symplectic structure.\\nProof Sketch. We sketch the proof, since the constructions involved will be useful later. For details, see our previous paper [43].\\nProof Sketch. We sketch the proof, since the constructions involved will be useful later. For details, see our previous paper [43].\\nThe fact that Φ U(d) ∶ F ↦ −FF * is a momentum map (where we are specifically fixing the isomorphism (5)) follows by a computation, as shown in [43,Proposition 2]. We remark that our momentum map differs by a sign in the present paper, due to slightly different conventions. The fact that\\nThe fact that Φ U(d) ∶ F ↦ −FF * is a momentum map (where we are specifically fixing the isomorphism (5)) follows by a computation, as shown in [43,Proposition 2]. We remark that our momentum map differs by a sign in the present paper, due to slightly different conventions. The fact that\\nfollows by Lemma 2.7. Moreover, the negative-definite Hermitian matrices are the regular values of Φ U(d) in H(d); we proved this by slightly tedious but essentially straightforward calculation in [43], but it also follows easily from Proposition 2.20 below, since the only unitary matrix which acts trivially on a spanning set of C d is the identity matrix. Therefore, the entries in λ λ λ being positive means that Theorem 2.3 applies and\\nfollows by Lemma 2.7. Moreover, the negative-definite Hermitian matrices are the regular values of Φ U(d) in H(d); we proved this by slightly tedious but essentially straightforward calculation in [43], but it also follows easily from Proposition 2.20 below, since the only unitary matrix which acts trivially on a spanning set of C d is the identity matrix. Therefore, the entries in λ λ λ being positive means that Theorem 2.3 applies and\\nis a symplectic manifold.\\nis a symplectic manifold.\"\n },\n {\n \"title\": \"Identification with a Coadjoint Orbit\",\n \"content\": \"\"\n },\n {\n \"title\": \"Identification with a Coadjoint Orbit\",\n \"content\": \"The symplectic structure of F d,N λ λ λ U(d) can alternatively be realized by identifying the frame space with a certain coadjoint orbit in the larger space of Hermitian matrices H(N). This perspective will be useful in the following subsection. In order to state the result precisely, we introduce some more terminology.\\nThe symplectic structure of F d,N λ λ λ U(d) can alternatively be realized by identifying the frame space with a certain coadjoint orbit in the larger space of Hermitian matrices H(N). This perspective will be useful in the following subsection. In order to state the result precisely, we introduce some more terminology.\\nIt is well known that any coadjoint orbit has a natural symplectic structure, called the Kirillov-Kostant-Souriau (KKS) symplectic form [5,II.3.c]), which we denote generically as ω KKS . Indeed, for an arbitrary Lie group G with Lie algebra g, the tangent space to a coadjoint orbit O β at β ∈ g * consists of vectors of the form ad * ξ β , where ξ ∈ g, and where ad * ξ is the coadjoint representation of g on g * , obtained as the derivative of the map Ad * ∶ G → Aut(g * ) at the identity. Then the KKS form is defined by\\nIt is well known that any coadjoint orbit has a natural symplectic structure, called the Kirillov-Kostant-Souriau (KKS) symplectic form [5,II.3.c]), which we denote generically as ω KKS . Indeed, for an arbitrary Lie group G with Lie algebra g, the tangent space to a coadjoint orbit O β at β ∈ g * consists of vectors of the form ad * ξ β , where ξ ∈ g, and where ad * ξ is the coadjoint representation of g on g * , obtained as the derivative of the map Ad * ∶ G → Aut(g * ) at the identity. Then the KKS form is defined by\\nwhere [⋅,⋅] is the Lie bracket on g.\\nwhere [⋅,⋅] is the Lie bracket on g.\\nBy another slight abuse of terminology, we define the adjoint representation of u(d) on H(d) by\\nBy another slight abuse of terminology, we define the adjoint representation of u(d) on H(d) by\\nAn argument similar to that of Lemma 2.7 proves the following.\\nAn argument similar to that of Lemma 2.7 proves the following.\\nLemma 2.9. The map α defined in (5) is equivariant with respect to the adjoint representation of u(d) on H(d) and its coadjoint representation on u(d) * . That is,\\nLemma 2.9. The map α defined in (5) is equivariant with respect to the adjoint representation of u(d) on H(d) and its coadjoint representation on u(d) * . That is,\\nThe adjoint representation of u(d) on H(d) is obtained by differentiating the adjoint action of U(d) on H(d). It follows that the tangent space to O −λ λ λ ⊂ H(d) at ξ consists of vectors of the form ad η (ξ ) for η ∈ u(d).\\nThe adjoint representation of u(d) on H(d) is obtained by differentiating the adjoint action of U(d) on H(d). It follows that the tangent space to O −λ λ λ ⊂ H(d) at ξ consists of vectors of the form ad η (ξ ) for η ∈ u(d).\\nwhere the second-to-last line uses cyclic invariance and linearity of trace.\\nwhere the second-to-last line uses cyclic invariance and linearity of trace.\\nWe can now describe precisely how F d,N λ λ λ U(d) is equivalent to a coadjoint orbit in H(N).\\nWe can now describe precisely how F d,N λ λ λ U(d) is equivalent to a coadjoint orbit in H(N).\"\n },\n {\n \"title\": \"Proposition 2.11 ( [43]\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 2.11 ( [43]\",\n \"content\": \"). Let λ λ λ be a frame spectrum and defineλ λ λ ∶= (λ 1 ,... ,λ d ,0,...,0) ∈ R N to be λ λ λ padded with\\n). Let λ λ λ be a frame spectrum and defineλ λ λ ∶= (λ 1 ,... ,λ d ,0,...,0) ∈ R N to be λ λ λ padded with\\ntaking a unitary equivalence class to its Gram matrix. Moreover, the map (7) is a symplectomorphism with respect to the reduced symplectic structure on F d,N λ λ λ U(d) and the pullback of the Kirillov-Kostant-Souriau form α * ω KKS described in Lemma 2.10.\\ntaking a unitary equivalence class to its Gram matrix. Moreover, the map (7) is a symplectomorphism with respect to the reduced symplectic structure on F d,N λ λ λ U(d) and the pullback of the Kirillov-Kostant-Souriau form α * ω KKS described in Lemma 2.10.\\nThe proof of the proposition makes use of the following lemma.\\nThe proof of the proposition makes use of the following lemma.\\nfor ξ ∈ u(d) and ζ ∈ u(N) satisfyingU 0 = ξU andV 0 = ζV , respectively. By general principles of quotient manifolds, there is a natural isomorphism\\nfor ξ ∈ u(d) and ζ ∈ u(N) satisfyingU 0 = ξU andV 0 = ζV , respectively. By general principles of quotient manifolds, there is a natural isomorphism\\nis a submersion, so the rank theorem [37,Theorem 4.12] implies that, locally near F, it is an orthogonal projection with the fiber U(d) ⋅ F being sent to the origin. This gives\\nis a submersion, so the rank theorem [37,Theorem 4.12] implies that, locally near F, it is an orthogonal projection with the fiber U(d) ⋅ F being sent to the origin. This gives\\nProof of Proposition 2.11. The diffeomorphism result is Proposition 3 in our previous paper [43], so we only sketch the details here. Unitary equivalence classes of frames are uniquely determined by their Gram matrices (see, for example, [50, §3.4]), so we can identify F d,N λ λ λ U(d) with the elements of H(N) which arise as Gram matrices of frames in F d,N λ λ λ . Of course, the frame operator FF * and the Gram matrix F * F have the same nonzero eigenvalues, so specifying the spectrum of the frame operator also determines the spectrum of the Gram matrix by padding with N − d zeros, and it follows that this subset consists of those N × N Hermitian matrices with spectrumλ λ λ . In other words, (7) is a diffeomorphism between F d,N λ λ λ U(d) and Oλ λ λ . It remains to prove the symplectomorphism claim. For the rest of this proof, let γ denote the map (7). By Lemma 2.12, the tangent space to F d,N\\nProof of Proposition 2.11. The diffeomorphism result is Proposition 3 in our previous paper [43], so we only sketch the details here. Unitary equivalence classes of frames are uniquely determined by their Gram matrices (see, for example, [50, §3.4]), so we can identify F d,N λ λ λ U(d) with the elements of H(N) which arise as Gram matrices of frames in F d,N λ λ λ . Of course, the frame operator FF * and the Gram matrix F * F have the same nonzero eigenvalues, so specifying the spectrum of the frame operator also determines the spectrum of the Gram matrix by padding with N − d zeros, and it follows that this subset consists of those N × N Hermitian matrices with spectrumλ λ λ . In other words, (7) is a diffeomorphism between F d,N λ λ λ U(d) and Oλ λ λ . It remains to prove the symplectomorphism claim. For the rest of this proof, let γ denote the map (7). By Lemma 2.12, the tangent space to F d,N\\nThen, using Lemma 2.10,\\nThen, using Lemma 2.10,\\nOn the other hand, the reduced symplectic form on F d,N\\nOn the other hand, the reduced symplectic form on F d,N\\nwhere we have used the characterization of the reduced form, cyclic invariance of trace and the fact that ζ ′ is skew-Hermitian.\\nwhere we have used the characterization of the reduced form, cyclic invariance of trace and the fact that ζ ′ is skew-Hermitian.\\nRemark 2.13. Since Oλ λ λ ⊂ H(N) is identified with a coadjoint orbit, it is diffeomorphic, like all coadjoint orbits, to a flag manifold [5, §II.1.d]. Specifically, if λ λ λ consists of ℓ distinct eigenvalues with multiplicities k 1 ,... ,k ℓ and\\nRemark 2.13. Since Oλ λ λ ⊂ H(N) is identified with a coadjoint orbit, it is diffeomorphic, like all coadjoint orbits, to a flag manifold [5, §II.1.d]. Specifically, if λ λ λ consists of ℓ distinct eigenvalues with multiplicities k 1 ,... ,k ℓ and\\nthe flag manifold whose elements are nested sequences\\nthe flag manifold whose elements are nested sequences\\nIn particular, when λ λ λ is constant then ℓ = 1 and k 1 = d, so the flag manifold is Fℓ(d,N), whose elements are all subspaces\\nIn particular, when λ λ λ is constant then ℓ = 1 and k 1 = d, so the flag manifold is Fℓ(d,N), whose elements are all subspaces\"\n },\n {\n \"title\": \"Frames with Prescribed Frame Spectrum and Norms\",\n \"content\": \"\"\n },\n {\n \"title\": \"Frames with Prescribed Frame Spectrum and Norms\",\n \"content\": \"The torus U(1) N can be realized as the subgroup of diagonal elements of U(N); that is, as the standard maximal torus T ≤ U(N). Thought of in this way, U(1) N ≈ T acts on C d×N by right matrix multiplication and, as we will see in a moment, this action preserves the frame spectrum, and hence restricts to an action on F d,N λ λ λ .\\nThe torus U(1) N can be realized as the subgroup of diagonal elements of U(N); that is, as the standard maximal torus T ≤ U(N). Thought of in this way, U(1) N ≈ T acts on C d×N by right matrix multiplication and, as we will see in a moment, this action preserves the frame spectrum, and hence restricts to an action on F d,N λ λ λ .\\nFirst, though, it is worthwhile to pause and think about the right action of the full unitary group U(N) on C d×N . If we try to define the action of A ∈ U(N) on F ∈ C d×N by A ⋅ F = FA we quickly run into problems: after all, if\\nFirst, though, it is worthwhile to pause and think about the right action of the full unitary group U(N) on C d×N . If we try to define the action of A ∈ U(N) on F ∈ C d×N by A ⋅ F = FA we quickly run into problems: after all, if\\nunless A 1 and A 2 happen to commute. This is not an issue at the level of T, which is abelian, but it is preferable to define the T action to be consistent with an honest U(N) action. We get such an action by defining\\nunless A 1 and A 2 happen to commute. This is not an issue at the level of T, which is abelian, but it is preferable to define the T action to be consistent with an honest U(N) action. We get such an action by defining\\nThis action preserves the frame spectrum: if F ∈ F d,N λ λ λ and D ∈ T, then the frame operator of D ⋅ F = FD * is (FD * )(FD * ) * = FD * DF * = FF * , which is the same as the frame operator of F, and hence D ⋅ F ∈ F d,N λ λ λ as well. At the frame level, the action of an element of the torus on a frame performs an independent phase rotation on each frame vector. The right action of T commutes with the left action of U(d), so the torus action descends to the quotient F d,N λ λ λ U(d), but it is not an effective action. After all, both T and U(d) contain a 1-parameter subgroup of scalar matrices, and the actions of these two subgroups cancel each other: if I k denotes the k × k identity matrix and e iθ ∈ U(1), then e iθ I d ∈ U(d), e iθ I N ∈ T and for any\\nThis action preserves the frame spectrum: if F ∈ F d,N λ λ λ and D ∈ T, then the frame operator of D ⋅ F = FD * is (FD * )(FD * ) * = FD * DF * = FF * , which is the same as the frame operator of F, and hence D ⋅ F ∈ F d,N λ λ λ as well. At the frame level, the action of an element of the torus on a frame performs an independent phase rotation on each frame vector. The right action of T commutes with the left action of U(d), so the torus action descends to the quotient F d,N λ λ λ U(d), but it is not an effective action. After all, both T and U(d) contain a 1-parameter subgroup of scalar matrices, and the actions of these two subgroups cancel each other: if I k denotes the k × k identity matrix and e iθ ∈ U(1), then e iθ I d ∈ U(d), e iθ I N ∈ T and for any\\nWe can put this in a more standard context using our identification of F d,N λ λ λ U(d) with the coadjoint orbit Oλ λ λ ⊂ H(N) ≈ u(N) * , described in Proposition 2.11. The standard maximal torus of U(N) certainly acts on any coadjoint orbit of U(N), and in this case the action is just the conjugation action. The scalar matrices are the center Z(U(N)) of U(N), whose coadjoint action is trivial. In general, to get an effective action of the maximal torus of a Lie group on a coadjoint orbit of the group, one needs to take the quotient of the torus by the center of the group; in this case, this is simply\\nWe can put this in a more standard context using our identification of F d,N λ λ λ U(d) with the coadjoint orbit Oλ λ λ ⊂ H(N) ≈ u(N) * , described in Proposition 2.11. The standard maximal torus of U(N) certainly acts on any coadjoint orbit of U(N), and in this case the action is just the conjugation action. The scalar matrices are the center Z(U(N)) of U(N), whose coadjoint action is trivial. In general, to get an effective action of the maximal torus of a Lie group on a coadjoint orbit of the group, one needs to take the quotient of the torus by the center of the group; in this case, this is simply\\nwhich can be identified with the subgroup of diagonal elements of U(N) whose last entry is 1. Now we see why it was worth defining the action of U(1) N (and hence G) on F d,N λ λ λ by D ⋅ F = FD * , since this corresponds exactly to the standard coadjoint action on H(N) ≈ u(N) * .\\nwhich can be identified with the subgroup of diagonal elements of U(N) whose last entry is 1. Now we see why it was worth defining the action of U(1) N (and hence G) on F d,N λ λ λ by D ⋅ F = FD * , since this corresponds exactly to the standard coadjoint action on H(N) ≈ u(N) * .\\nIn this setting, it is easy to see that the action of G on Oλ λ λ ≈ F d,N λ λ λ U(d) is Hamiltonian:\\nIn this setting, it is easy to see that the action of G on Oλ λ λ ≈ F d,N λ λ λ U(d) is Hamiltonian:\\nProof. First of all, it is standard (see, e.g., [40,Example 5.3.11]) that the coadjoint action of U(N) on the coadjoint orbit Oλ λ λ is Hamiltonian with momentum map given by the inclusion Oλ λ λ ↪ u(N) * ≈ H(N); in fact, the analogous statement holds for arbitrary compact Lie groups acting on coadjoint orbits. Similarly, it is standard (see, e.g., [5, Proposition III.1.10]) that the action of the maximal torus T ≤ U(N) on Oλ λ λ is Hamiltonian with momentum map given by the composition\\nProof. First of all, it is standard (see, e.g., [40,Example 5.3.11]) that the coadjoint action of U(N) on the coadjoint orbit Oλ λ λ is Hamiltonian with momentum map given by the inclusion Oλ λ λ ↪ u(N) * ≈ H(N); in fact, the analogous statement holds for arbitrary compact Lie groups acting on coadjoint orbits. Similarly, it is standard (see, e.g., [5, Proposition III.1.10]) that the action of the maximal torus T ≤ U(N) on Oλ λ λ is Hamiltonian with momentum map given by the composition\\nwhere t * is the Lie algebra of T and the projection u(N) * → t * is the one induced by the inclusion T ↪ U(N). After identifying t * with R N , this projection is easily seen to be the map which records the diagonal entries of a Hermitian matrix. 2 More generally, the same applies for any subgroup (see, e.g., [9, p. 213]), and so the action of G on Oλ λ λ is Hamiltonian with momentum map given by the composition\\nwhere t * is the Lie algebra of T and the projection u(N) * → t * is the one induced by the inclusion T ↪ U(N). After identifying t * with R N , this projection is easily seen to be the map which records the diagonal entries of a Hermitian matrix. 2 More generally, the same applies for any subgroup (see, e.g., [9, p. 213]), and so the action of G on Oλ λ λ is Hamiltonian with momentum map given by the composition\\nwhere the projection u(N) * → g * is the one induced by the inclusion G ↪ U(N). Under the identification g * ≈ R N−1 , this map just records all the diagonal entries but the last one (which in any case is determined by the others, since the trace of any element of Oλ λ λ is ∑ i λ i ).\\nwhere the projection u(N) * → g * is the one induced by the inclusion G ↪ U(N). Under the identification g * ≈ R N−1 , this map just records all the diagonal entries but the last one (which in any case is determined by the others, since the trace of any element of Oλ λ λ is ∑ i λ i ).\\n. By the Schur-Horn theorem [30,46]\\n. By the Schur-Horn theorem [30,46]\\nis non-empty only if the vector (r 1 ,...,r N ) lies in the convex hull of the orbit ofλ λ λ = (λ 1 ,... ,λ d ,0,... ,0) under the action of the symmetric group S N which permutes entries. When r r r and λ λ λ are non-increasing lists, this condition is easily seen to be equivalent to r r r being λ λ λ -admissible, so F d,N λ λ λ (r r r) is non-empty precisely when r r r is λ λ λ -admissible. When r r r is λ λ λ -admissible, there is no guarantee that (r 1 ,...,r N−1 ) ∈ R N−1 ≈ g * is a regular value, but we can still apply Theorem 2.4 to see that the quotient of Φ −1 G (r 1 ,... ,r N−1 ) by G is at worst a symplectic stratified space. We have thus proved the first part of the following proposition:\\nis non-empty only if the vector (r 1 ,...,r N ) lies in the convex hull of the orbit ofλ λ λ = (λ 1 ,... ,λ d ,0,... ,0) under the action of the symmetric group S N which permutes entries. When r r r and λ λ λ are non-increasing lists, this condition is easily seen to be equivalent to r r r being λ λ λ -admissible, so F d,N λ λ λ (r r r) is non-empty precisely when r r r is λ λ λ -admissible. When r r r is λ λ λ -admissible, there is no guarantee that (r 1 ,...,r N−1 ) ∈ R N−1 ≈ g * is a regular value, but we can still apply Theorem 2.4 to see that the quotient of Φ −1 G (r 1 ,... ,r N−1 ) by G is at worst a symplectic stratified space. We have thus proved the first part of the following proposition:\\nwhich is a symplectic stratified space. 3 Alternatively,\\nwhich is a symplectic stratified space. 3 Alternatively,\\nProof. We proved the first sentence above. For the second sentence, Sjamaar and Lerman [48,Theorem 4.1] showed that even when reducing over singular values, we can perform a reduction by a product group in stages. Hence, with\\nProof. We proved the first sentence above. For the second sentence, Sjamaar and Lerman [48,Theorem 4.1] showed that even when reducing over singular values, we can perform a reduction by a product group in stages. Hence, with\\nis, in general, a symplectic stratified space. Depending on the parameters λ λ λ and r r r, it may actually be a smooth manifold. We characterize the parameters such that F d,N λ λ λ (r r r) is a smooth manifold below in Theorem 2.25.\\nis, in general, a symplectic stratified space. Depending on the parameters λ λ λ and r r r, it may actually be a smooth manifold. We characterize the parameters such that F d,N λ λ λ (r r r) is a smooth manifold below in Theorem 2.25.\\nIn particular, the space F d,N λ λ λ (r r r) is exactly the level set Φ −1 (O −λ λ λ × {r r r}). To simplify slightly, assume r r r is strongly λ λ λ -admissible. When F d,N λ λ λ (r r r) is a manifold (conditions for which we determine in Corollary 2.22 in the next section), then we know its codimension inside C d×N is equal to the codimension of O −λ λ λ × {r r r} inside H(d) × R N−1 . Even when F d,N λ λ λ (r r r) is not a manifold, we know from Theorem 2.5 that its quotient by U(d) × G contains an open dense subset which is a manifold. This subset is a stratum, and hence its (open, dense) preimage is also a manifold by Theorem 3.5 from Sjamaar and Lerman [48].\\nIn particular, the space F d,N λ λ λ (r r r) is exactly the level set Φ −1 (O −λ λ λ × {r r r}). To simplify slightly, assume r r r is strongly λ λ λ -admissible. When F d,N λ λ λ (r r r) is a manifold (conditions for which we determine in Corollary 2.22 in the next section), then we know its codimension inside C d×N is equal to the codimension of O −λ λ λ × {r r r} inside H(d) × R N−1 . Even when F d,N λ λ λ (r r r) is not a manifold, we know from Theorem 2.5 that its quotient by U(d) × G contains an open dense subset which is a manifold. This subset is a stratum, and hence its (open, dense) preimage is also a manifold by Theorem 3.5 from Sjamaar and Lerman [48].\\nIn other words, F d,N λ λ λ (r r r) contains an open, dense subset which is a manifold and which consists precisely of regular points of the momentum map Φ, so for the purposes of dimension-counting we may as well assume F d,N λ λ λ (r r r) is a manifold.\\nIn other words, F d,N λ λ λ (r r r) contains an open, dense subset which is a manifold and which consists precisely of regular points of the momentum map Φ, so for the purposes of dimension-counting we may as well assume F d,N λ λ λ (r r r) is a manifold.\\nTo that end,\\nTo that end,\\nwhere, as in Remark 2.13, the k 1 ,... ,k ℓ are the multiplicities of the entries in the frame spectrum λ λ λ . Subtracting this from\\nwhere, as in Remark 2.13, the k 1 ,... ,k ℓ are the multiplicities of the entries in the frame spectrum λ λ λ . Subtracting this from\\nand hence also of F d,N λ λ λ (r r r) (inside C d×N ). This proves: Corollary 2.18. For r r r which is strongly λ λ λ -admissible,\\nand hence also of F d,N λ λ λ (r r r) (inside C d×N ). This proves: Corollary 2.18. For r r r which is strongly λ λ λ -admissible,\\nHence,\\nHence,\\nwhere the k j are the multiplicities of the entries in the frame spectrum λ λ λ .\\nwhere the k j are the multiplicities of the entries in the frame spectrum λ λ λ .\"\n },\n {\n \"title\": \"Manifold Structure of Frame Spaces\",\n \"content\": \"\"\n },\n {\n \"title\": \"Manifold Structure of Frame Spaces\",\n \"content\": \"While Corollary 2.6 will allow us to apply the symplectic machinery even when F d,N λ λ λ (r r r) is singular, it is still interesting to understand when this space is a smooth manifold. We completely characterize parameters for which F d,N λ λ λ (r r r) is a manifold in the following subsection. We then give a detailed description of the local structure of singularities in singular frame spaces.\\nWhile Corollary 2.6 will allow us to apply the symplectic machinery even when F d,N λ λ λ (r r r) is singular, it is still interesting to understand when this space is a smooth manifold. We completely characterize parameters for which F d,N λ λ λ (r r r) is a manifold in the following subsection. We then give a detailed description of the local structure of singularities in singular frame spaces.\"\n },\n {\n \"title\": \"Characterizing Smooth Frame Spaces\",\n \"content\": \"\"\n },\n {\n \"title\": \"Characterizing Smooth Frame Spaces\",\n \"content\": \"The model result on manifold structures of frame spaces is due to Dykema and Strawn [18], who showed that the singular points in the space F d,N N d ,..., N d (1,...,1) of unit-norm tight frames must be orthodecomposable (we recall the definition of this property below). An immediate corollary is that the space of unit-norm tight frames is a manifold when N and d are relatively prime. More generally, Strawn [49] showed that the singular points in any space of frames with fixed frame vector norms and fixed frame operator (rather than frame spectrum) must be orthodecomposable.\\nThe model result on manifold structures of frame spaces is due to Dykema and Strawn [18], who showed that the singular points in the space F d,N N d ,..., N d (1,...,1) of unit-norm tight frames must be orthodecomposable (we recall the definition of this property below). An immediate corollary is that the space of unit-norm tight frames is a manifold when N and d are relatively prime. More generally, Strawn [49] showed that the singular points in any space of frames with fixed frame vector norms and fixed frame operator (rather than frame spectrum) must be orthodecomposable.\\nWe will prove the converse of these results, showing that the singular points (if any) of every F d,N λ λ λ (r r r) space are exactly the orthodecomposable frames. First, recall what it means for a frame to be orthodecomposable:\\nWe will prove the converse of these results, showing that the singular points (if any) of every F d,N λ λ λ (r r r) space are exactly the orthodecomposable frames. First, recall what it means for a frame to be orthodecomposable:\\nIn our setting, we are realizing the space F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) as the inverse image of a coadjoint orbit under the momentum map Φ. This frame space will certainly be a manifold if it contains no critical points of Φ, so we now characterize the critical points of Φ.\\nIn our setting, we are realizing the space F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) as the inverse image of a coadjoint orbit under the momentum map Φ. This frame space will certainly be a manifold if it contains no critical points of Φ, so we now characterize the critical points of Φ.\\nA standard part of the discussion around Theorems 2. Recall that a group action is locally free at p if the stabilizer of p is discrete. Therefore, Proposition 2.20 gives a characterization of critical points of a momentum map: p ∈ M is a critical point if and only if the stabilizer of p is continuous, and in particular contains a nontrivial one-parameter subgroup. This is the key to characterizing the critical points of Φ ∶ C d×N → H(d) × R N−1 .\\nA standard part of the discussion around Theorems 2. Recall that a group action is locally free at p if the stabilizer of p is discrete. Therefore, Proposition 2.20 gives a characterization of critical points of a momentum map: p ∈ M is a critical point if and only if the stabilizer of p is continuous, and in particular contains a nontrivial one-parameter subgroup. This is the key to characterizing the critical points of Φ ∶ C d×N → H(d) × R N−1 .\"\n },\n {\n \"title\": \"Proposition 2.21. A frame F is a critical point of the momentum map Φ ∶ C d×N → H(d) × R N−1 of the Hamiltonian action of U(d) × G if and only if F is orthodecomposable.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 2.21. A frame F is a critical point of the momentum map Φ ∶ C d×N → H(d) × R N−1 of the Hamiltonian action of U(d) × G if and only if F is orthodecomposable.\",\n \"content\": \"Proof. Let F = f 1 ⋯ f N ∈ C d×N be a frame and let ζ ∶ R → G be a one-parameter subgroup such that ζ (t) ⋅ F = F for all t ∈ R. Then ζ uniquely corresponds to an element of the Lie algebra u(d) × g of U(d) × G, and in particular there exist matrices ξ ∈ u(d) and θ ∈ g such that ζ (t) = exp(t(ξ ,θ )) = (exp(tξ ),exp(tθ )) (see, e.g., [37,Propositions 20.1 and 20.5]). Under our concrete realization of G = T Z (U(N)) as the subgroup of diagonal unitary matrices with last entry equal to 1, θ will be of the form\\nProof. Let F = f 1 ⋯ f N ∈ C d×N be a frame and let ζ ∶ R → G be a one-parameter subgroup such that ζ (t) ⋅ F = F for all t ∈ R. Then ζ uniquely corresponds to an element of the Lie algebra u(d) × g of U(d) × G, and in particular there exist matrices ξ ∈ u(d) and θ ∈ g such that ζ (t) = exp(t(ξ ,θ )) = (exp(tξ ),exp(tθ )) (see, e.g., [37,Propositions 20.1 and 20.5]). Under our concrete realization of G = T Z (U(N)) as the subgroup of diagonal unitary matrices with last entry equal to 1, θ will be of the form\\nfor some θ 1 ,...,θ N−1 ∈ R. Then we have\\nfor some θ 1 ,...,θ N−1 ∈ R. Then we have\\nAt the frame vector level, this yields a system of equations\\nAt the frame vector level, this yields a system of equations\\nEach √ −1θ j is therefore an eigenvalue of the skew-Hermitian matrix ξ with corresponding eigenvector f j . Moreover, any solution of (8) yields a one-parameter subgroup in the stabilizer of F. By the Spectral Theorem for normal matrices, the eigenspaces for distinct eigenvalues of ξ are orthogonal.\\nEach √ −1θ j is therefore an eigenvalue of the skew-Hermitian matrix ξ with corresponding eigenvector f j . Moreover, any solution of (8) yields a one-parameter subgroup in the stabilizer of F. By the Spectral Theorem for normal matrices, the eigenspaces for distinct eigenvalues of ξ are orthogonal.\\nWith the above setup, we are prepared to prove the claim. First, suppose that F is not a regular point of the momentum map. By Proposition 2.20, it is possible to find a one-parameter subgroup ζ ∶ R → G of the stabilizer of F which is nontrivial. In this case, ξ has at least two distinct eigenvalues and it follows that F is orthodecomposable, with frame vectors partitioned into the orthogonal eigenspaces of ξ .\\nWith the above setup, we are prepared to prove the claim. First, suppose that F is not a regular point of the momentum map. By Proposition 2.20, it is possible to find a one-parameter subgroup ζ ∶ R → G of the stabilizer of F which is nontrivial. In this case, ξ has at least two distinct eigenvalues and it follows that F is orthodecomposable, with frame vectors partitioned into the orthogonal eigenspaces of ξ .\\nConversely, suppose that F is orthodecomposable. Let P 1 ,... ,P m denote the partition and V 1 ,... ,V m the subspaces appearing in Definition 2.19. Permuting columns of F as necessary, we can assume without loss of generality that P 1 = {1,...,k} for some k < N. For convenience, we can apply a unitary transformation to F so that f 1 ,... , f k span a coordinate plane V 1 ; say, the span of the first ℓ < d standard basis vectors. Since V 2 ,... ,V m are all orthogonal to V 1 , ⊕ m i=2 V i is the span of the remaining d − ℓ standard basis vectors. Then F has the form\\nConversely, suppose that F is orthodecomposable. Let P 1 ,... ,P m denote the partition and V 1 ,... ,V m the subspaces appearing in Definition 2.19. Permuting columns of F as necessary, we can assume without loss of generality that P 1 = {1,...,k} for some k < N. For convenience, we can apply a unitary transformation to F so that f 1 ,... , f k span a coordinate plane V 1 ; say, the span of the first ℓ < d standard basis vectors. Since V 2 ,... ,V m are all orthogonal to V 1 , ⊕ m i=2 V i is the span of the remaining d − ℓ standard basis vectors. Then F has the form\\nwhere the nonzero blocks F 1 and F 2 have sizes ℓ × k and (d − ℓ) × (N − k), respectively. Let\\nwhere the nonzero blocks F 1 and F 2 have sizes ℓ × k and (d − ℓ) × (N − k), respectively. Let\\nIt is easy to check that ξ ,θ , and F satisfy the system (8) and therefore yield a nontrivial one-parameter subgroup in the stabilizer of F. By Proposition 2.20, F is not a regular point of the momentum map.\\nIt is easy to check that ξ ,θ , and F satisfy the system (8) and therefore yield a nontrivial one-parameter subgroup in the stabilizer of F. By Proposition 2.20, F is not a regular point of the momentum map.\\nThe characterization of singular points in F d,N λ λ λ (r r r) is now an easy corollary: Proof. From Proposition 2.21 we know that the critical points of the momentum map Φ are exactly the orthodecomposable frames. It is a nontrivial fact (see, e.g., [3,Theorem 5] or [48,Proposition 2.5]) that in a neighborhood of a critical point of a momentum map, the level set looks like the product of a quadratic cone and a manifold. Therefore, F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) has a quadratic singularity at each orthodecomposable frame. Since the structure of such singularities is itself of interest [8], we describe these cone singularities in detail in Section 2.3.2 below.\\nThe characterization of singular points in F d,N λ λ λ (r r r) is now an easy corollary: Proof. From Proposition 2.21 we know that the critical points of the momentum map Φ are exactly the orthodecomposable frames. It is a nontrivial fact (see, e.g., [3,Theorem 5] or [48,Proposition 2.5]) that in a neighborhood of a critical point of a momentum map, the level set looks like the product of a quadratic cone and a manifold. Therefore, F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}) has a quadratic singularity at each orthodecomposable frame. Since the structure of such singularities is itself of interest [8], we describe these cone singularities in detail in Section 2.3.2 below.\\nOrthodecomposable frames are the only possible singularities: if F ∈ F d,N λ λ λ (r r r) is not orthodecomposable, then the differential DΦ F ∶ T F C d×N → T Φ(F) (H(d) × R N−1 ) is surjective. In particular, this implies Φ is transverse to a neighborhood of Φ(F) in O −λ λ λ × {r r r}, so it is standard that its inverse image-which is a neighborhood of F in F d,N λ λ λ (r r r)-is a submanifold of C d×N (see, e.g., [37, Theorem 6.30]), and in particular is smooth at F. Remark 2.23. Since every orthodecomposable frame is a singular point of the F d,N λ λ λ (r r r) space containing it, Corollary 2.22 solves the complex case of Cahill, Mixon, and Strawn's Problem 5.8 [8].\\nOrthodecomposable frames are the only possible singularities: if F ∈ F d,N λ λ λ (r r r) is not orthodecomposable, then the differential DΦ F ∶ T F C d×N → T Φ(F) (H(d) × R N−1 ) is surjective. In particular, this implies Φ is transverse to a neighborhood of Φ(F) in O −λ λ λ × {r r r}, so it is standard that its inverse image-which is a neighborhood of F in F d,N λ λ λ (r r r)-is a submanifold of C d×N (see, e.g., [37, Theorem 6.30]), and in particular is smooth at F. Remark 2.23. Since every orthodecomposable frame is a singular point of the F d,N λ λ λ (r r r) space containing it, Corollary 2.22 solves the complex case of Cahill, Mixon, and Strawn's Problem 5.8 [8].\\nTo make Corollary 2.22 more practical, it would be helpful to know, in terms of some computable conditions on the parameters d, N, λ λ λ , and r r r, exactly when F d,N λ λ λ (r r r) contains an orthodecomposable frame. To that end, suppose F ∈ F d,N λ λ λ (r r r) is orthodecomposable. Then, as in the proof of Proposition 2.20, after possibly permuting columns and applying a unitary transformation, we can realize F in block-diagonal form:\\nTo make Corollary 2.22 more practical, it would be helpful to know, in terms of some computable conditions on the parameters d, N, λ λ λ , and r r r, exactly when F d,N λ λ λ (r r r) contains an orthodecomposable frame. To that end, suppose F ∈ F d,N λ λ λ (r r r) is orthodecomposable. Then, as in the proof of Proposition 2.20, after possibly permuting columns and applying a unitary transformation, we can realize F in block-diagonal form:\\nwhere F 1 ∈ C ℓ×k and F 2 ∈ C (d−ℓ)×(N−k) are frames. But then both the Gram matrix and the frame operator associated to F are also block-diagonal:\\nwhere F 1 ∈ C ℓ×k and F 2 ∈ C (d−ℓ)×(N−k) are frames. But then both the Gram matrix and the frame operator associated to F are also block-diagonal:\\nSince tr(F * 1 F 1 ) = tr(F 1 F * 1 ), we see that r 1 + ⋅⋅⋅ + r k = λ 1 + ⋅⋅⋅ + λ ℓ . More generally, since we permuted columns to get F into block-diagonal form, there exist proper subsets {r i 1 ,... ,r i k } ⊊ r r r and {λ i 1 ,... ,λ i ℓ } ⊊ λ λ λ so that\\nSince tr(F * 1 F 1 ) = tr(F 1 F * 1 ), we see that r 1 + ⋅⋅⋅ + r k = λ 1 + ⋅⋅⋅ + λ ℓ . More generally, since we permuted columns to get F into block-diagonal form, there exist proper subsets {r i 1 ,... ,r i k } ⊊ r r r and {λ i 1 ,... ,λ i ℓ } ⊊ λ λ λ so that\\nMoreover, let r r r ′ = (r ′ 1 ,... ,r ′ k ) ∶= (r i 1 ,... ,r i k ) and λ λ λ ′ = (λ ′ 1 ,... ,λ ′ ℓ ) ∶= (λ i 1 ,...,λ i ℓ ) and let r r r ′′ and λ λ λ ′′ be their respective complement vectors. Then we see that the columns { f i 1 ,... , f i k } give a frame for an ℓ-dimensional subspace of C d with spectrum λ i 1 ≥ ⋅⋅⋅ ≥ λ i ℓ > 0, which means that the frame space F ℓ,k λ λ λ ′ (r r r ′ ) is non-empty, and in particular that r r r ′ is λ λ λ ′ -admissible. Similarly, the complementary columns give an element of F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ), and hence r r r ′′ is λ λ λ ′′ -admissible. Summarizing, we have shown the following consequences of orthodecomposability:\\nMoreover, let r r r ′ = (r ′ 1 ,... ,r ′ k ) ∶= (r i 1 ,... ,r i k ) and λ λ λ ′ = (λ ′ 1 ,... ,λ ′ ℓ ) ∶= (λ i 1 ,...,λ i ℓ ) and let r r r ′′ and λ λ λ ′′ be their respective complement vectors. Then we see that the columns { f i 1 ,... , f i k } give a frame for an ℓ-dimensional subspace of C d with spectrum λ i 1 ≥ ⋅⋅⋅ ≥ λ i ℓ > 0, which means that the frame space F ℓ,k λ λ λ ′ (r r r ′ ) is non-empty, and in particular that r r r ′ is λ λ λ ′ -admissible. Similarly, the complementary columns give an element of F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ), and hence r r r ′′ is λ λ λ ′′ -admissible. Summarizing, we have shown the following consequences of orthodecomposability:\\nλ λ λ (r r r) is orthodecomposable, then there exist proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum (which implies that r r r ′′ and λ λ λ ′′ have the same sum as well).\\nλ λ λ (r r r) is orthodecomposable, then there exist proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum (which implies that r r r ′′ and λ λ λ ′′ have the same sum as well).\\nCombined with Corollary 2.22, this gives a sufficient condition for F d,N λ λ λ (r r r) to be a manifold. In fact, it is also a necessary condition:\\nCombined with Corollary 2.22, this gives a sufficient condition for F d,N λ λ λ (r r r) to be a manifold. In fact, it is also a necessary condition:\\nλ λ λ (r r r) is a manifold if and only if there are no proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum.\\nλ λ λ (r r r) is a manifold if and only if there are no proper partitions r r r = r r r ′ ⊔ r r r ′′ and λ λ λ = λ λ λ ′ ⊔ λ λ λ ′′ so that r r r ′ is λ λ λ ′ -admissible, r r r ′′ is λ λ λ ′′ -admissible, and r r r ′ and λ λ λ ′ have the same sum.\\nProof. If F d,N λ λ λ (r r r) is not a manifold, then Corollary 2.22 tells us that it contains some orthodecomposable frame and hence, by Lemma 2.24, there are such partitions of r r r and λ λ λ .\\nProof. If F d,N λ λ λ (r r r) is not a manifold, then Corollary 2.22 tells us that it contains some orthodecomposable frame and hence, by Lemma 2.24, there are such partitions of r r r and λ λ λ .\\nConversely, suppose there are such partitions. Then, since r r r ′ is λ λ λ ′ -admissible, there exists a frame { f i 1 ,... , f i k } ∈ F ℓ,k λ λ λ ′ (r r r ′ ), and similarly there exists a frame {g j 1 ,... ,g j N−k } ∈ F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ). Embedding the former into the ℓ-dimensional subspace of C d spanned by the first ℓ standard basis vectors and the latter into the (d −ℓ)-dimensional orthogonal complement and combining them into a single list sorted by decreasing norm gives an orthodecomposable element-and hence a singular point-of F d,N λ λ λ (r r r), which is therefore not a manifold. Notice that Theorem 2.25 recovers Dykema and Strawn's result that the space of unit-norm tight frames is a manifold when N and d are relatively prime, and also implies that F d,N λ λ λ (r r r) is a manifold for generic r r r and λ λ λ .\\nConversely, suppose there are such partitions. Then, since r r r ′ is λ λ λ ′ -admissible, there exists a frame { f i 1 ,... , f i k } ∈ F ℓ,k λ λ λ ′ (r r r ′ ), and similarly there exists a frame {g j 1 ,... ,g j N−k } ∈ F d−ℓ,N−k λ λ λ ′′ (r r r ′′ ). Embedding the former into the ℓ-dimensional subspace of C d spanned by the first ℓ standard basis vectors and the latter into the (d −ℓ)-dimensional orthogonal complement and combining them into a single list sorted by decreasing norm gives an orthodecomposable element-and hence a singular point-of F d,N λ λ λ (r r r), which is therefore not a manifold. Notice that Theorem 2.25 recovers Dykema and Strawn's result that the space of unit-norm tight frames is a manifold when N and d are relatively prime, and also implies that F d,N λ λ λ (r r r) is a manifold for generic r r r and λ λ λ .\"\n },\n {\n \"title\": \"Describing Singularities\",\n \"content\": \"\"\n },\n {\n \"title\": \"Describing Singularities\",\n \"content\": \"As was mentioned above, Corollary 2.22 solves Problem 5.8 of [8] for complex FUNTFs. Problem 1.5 of the same paper asks for a description of the local geometry of the space of FUNTFs near orthodecomposable frames, and we now expand on the proof of Corollary 2.22 to solve this problem. We describe singularities for progressively more general classes of frames, with full details given in the simplest case and sketches given for more general cases.\\nAs was mentioned above, Corollary 2.22 solves Problem 5.8 of [8] for complex FUNTFs. Problem 1.5 of the same paper asks for a description of the local geometry of the space of FUNTFs near orthodecomposable frames, and we now expand on the proof of Corollary 2.22 to solve this problem. We describe singularities for progressively more general classes of frames, with full details given in the simplest case and sketches given for more general cases.\"\n },\n {\n \"title\": \"Minimally Orthodecomposable FUNTFs. A result of Arms, Marsden, and Moncrief gives a description of the local geometry near any point on the level set of a momentum map of an arbitrary symplectic manifold [3, Theorem 5]. This specializes to our setting to show that near any FUNTF\",\n \"content\": \"\"\n },\n {\n \"title\": \"Minimally Orthodecomposable FUNTFs. A result of Arms, Marsden, and Moncrief gives a description of the local geometry near any point on the level set of a momentum map of an arbitrary symplectic manifold [3, Theorem 5]. This specializes to our setting to show that near any FUNTF\",\n \"content\": \"there is a local diffeomorphism of the ambient space C d×N which takes a neighborhood of F in the level set onto the product of a quadratic cone C F and a smooth manifold. The cone is described explicitly as\\nthere is a local diffeomorphism of the ambient space C d×N which takes a neighborhood of F in the level set onto the product of a quadratic cone C F and a smooth manifold. The cone is described explicitly as\\nwhere J denotes multiplication by √ −1, considered as a linear map on the real vector space T F C d×N , and D 2 Φ(F) is the Hessian of Φ at F, considered as a H(d)-valued bilinear form. Observe that if F is a regular point of Φ then C F is just a linear subspace of C d×N . We will show that C F is a singular cone when F is not a regular point.\\nwhere J denotes multiplication by √ −1, considered as a linear map on the real vector space T F C d×N , and D 2 Φ(F) is the Hessian of Φ at F, considered as a H(d)-valued bilinear form. Observe that if F is a regular point of Φ then C F is just a linear subspace of C d×N . We will show that C F is a singular cone when F is not a regular point.\\nWriting\\nWriting\\nWe can also show that\\nWe can also show that\\n. Now suppose that F is orthodecomposable. As in the proof of Proposition 2.21, we can assume without loss of generality that\\n. Now suppose that F is orthodecomposable. As in the proof of Proposition 2.21, we can assume without loss of generality that\\nfor some submatrices F 1 ∈ C d 1 ×N 1 and F 2 ∈ C d 2 ×N 2 (we use a different indexing convention for dimensions than in the proof of Proposition 2.21 for ease of generalization later on). Furthermore, suppose that the blocks F 1 and F 2 are not themselves orthodecomposable. We then claim that image(DΦ(F)) ⊥ = span I d 1 0 0 0 ,(1,...,1 N 1 ,0,...,0\\nfor some submatrices F 1 ∈ C d 1 ×N 1 and F 2 ∈ C d 2 ×N 2 (we use a different indexing convention for dimensions than in the proof of Proposition 2.21 for ease of generalization later on). Furthermore, suppose that the blocks F 1 and F 2 are not themselves orthodecomposable. We then claim that image(DΦ(F)) ⊥ = span I d 1 0 0 0 ,(1,...,1 N 1 ,0,...,0\\nwhere the orthogonal complement is taken with respect to the inner product ⟪⋅,⋅⟫ on H d×d × R N−1 defined by\\nwhere the orthogonal complement is taken with respect to the inner product ⟪⋅,⋅⟫ on H d×d × R N−1 defined by\\nIndeed, a straightforward computation shows that the vector given in (9) belongs to image(DΦ(F)) ⊥ , and we see that this space is one-dimensional as follows. By standard arguments, it suffices to show that the nullity of the dual map DΦ(F) * is equal to one and Equation 5.2.7 of [40] implies that it suffices to show that the map taking an element of H(d) × R N−1 to its infinitesimal vector field evaluated at F has one dimensional kernel. By arguments similar to those used in the proof of Proposition 2.21, this is equivalent to showing that the space of solutions to the system of equations (8) is one-dimensional, which holds under our assumption that F 1 and F 2 are not themselves orthodecomposable (using the fact that the eigenspaces of a skew-Hermitian matrix are orthogonal). Based on the work in the previous two paragraphs, the cone C F is described more explicitly as the set of X ∈ C d×N satisfying\\nIndeed, a straightforward computation shows that the vector given in (9) belongs to image(DΦ(F)) ⊥ , and we see that this space is one-dimensional as follows. By standard arguments, it suffices to show that the nullity of the dual map DΦ(F) * is equal to one and Equation 5.2.7 of [40] implies that it suffices to show that the map taking an element of H(d) × R N−1 to its infinitesimal vector field evaluated at F has one dimensional kernel. By arguments similar to those used in the proof of Proposition 2.21, this is equivalent to showing that the space of solutions to the system of equations (8) is one-dimensional, which holds under our assumption that F 1 and F 2 are not themselves orthodecomposable (using the fact that the eigenspaces of a skew-Hermitian matrix are orthogonal). Based on the work in the previous two paragraphs, the cone C F is described more explicitly as the set of X ∈ C d×N satisfying\\nwhere we express X as a block matrix with blocks X km , k,m ∈ {1,2}, of size d k ×N m whose column vectors are denoted x km j . We can verify that the intersection of the subspace defined by (10) and the solution set of the indefinite quadratic equation (11) yields a singular cone by showing that it is not a linear subspace. To do so, choose\\nwhere we express X as a block matrix with blocks X km , k,m ∈ {1,2}, of size d k ×N m whose column vectors are denoted x km j . We can verify that the intersection of the subspace defined by (10) and the solution set of the indefinite quadratic equation (11) yields a singular cone by showing that it is not a linear subspace. To do so, choose\\nThe condition ⟨ f j ,x j ⟩ holds automatically due to the block structures of F and X, hence (10) is satisfied. The matrices X j can be chosen to be nonzero, since the dimension of the row space of F j is at most d j < N j . We can therefore scale the X j 's to have equal Frobenius norm, which means that (11) is also satisfied, so that X ∈ C F . Next, we observe that\\nThe condition ⟨ f j ,x j ⟩ holds automatically due to the block structures of F and X, hence (10) is satisfied. The matrices X j can be chosen to be nonzero, since the dimension of the row space of F j is at most d j < N j . We can therefore scale the X j 's to have equal Frobenius norm, which means that (11) is also satisfied, so that X ∈ C F . Next, we observe that\\nalso lies in C F . However, the matrix X − X ′ does not satisfy (11), and this proves our claim that C F is not a linear subspace.\\nalso lies in C F . However, the matrix X − X ′ does not satisfy (11), and this proves our claim that C F is not a linear subspace.\\nExample 2.26. Here we provide a concrete example of the structure described above. Let F denote the orthodecomposable unit norm tight frame\\nExample 2.26. Here we provide a concrete example of the structure described above. Let F denote the orthodecomposable unit norm tight frame\\nThe cone C F can be described explicitly as the set of 2 × 4 matrices X satisfying equations (10) and (11). Using the x km j notation from above, we write and, letting f j and x j denote the columns of F and X, respectively,\\nThe cone C F can be described explicitly as the set of 2 × 4 matrices X satisfying equations (10) and (11). Using the x km j notation from above, we write and, letting f j and x j denote the columns of F and X, respectively,\\nThe cone equation (11) Putting all of this together, the cone C F is\\nThe cone equation (11) Putting all of this together, the cone C F is\\nInterpreted as a subset of\\nInterpreted as a subset of\\n)}, this is precisely the cone over the Clifford torus. Now consider the map defined by\\n)}, this is precisely the cone over the Clifford torus. Now consider the map defined by\\nwith domain {A(λ ,θ ,φ ) λ ∈ [0,1)}, a neighborhood of the origin in C F . We claim that the image of this map is an open subset of F intersected with a slice of the U(2) × U(1) 4 ) U(1)-action on C 2×4 through F-that is, a subset S ⊂ C 2×4 containing F such that the orbit of any point in C 2×4 sufficiently close to F has a unique representative in S. 4 Specifically, the slice S is the set of matrices of the form\\nwith domain {A(λ ,θ ,φ ) λ ∈ [0,1)}, a neighborhood of the origin in C F . We claim that the image of this map is an open subset of F intersected with a slice of the U(2) × U(1) 4 ) U(1)-action on C 2×4 through F-that is, a subset S ⊂ C 2×4 containing F such that the orbit of any point in C 2×4 sufficiently close to F has a unique representative in S. 4 Specifically, the slice S is the set of matrices of the form\\nwhere α j and δ j are positive real numbers, and β j , γ j , θ , φ are arbitrary real numbers. The proof that S is a slice is somewhat technical, so we omit it. It is then straightforward to show that F ∩ S is equal to the image of our map (i.e., points of the form (12)), which implies that a neighborhood of F in F is identified with the product of a neighborhood of the origin in C F and the group.\\nwhere α j and δ j are positive real numbers, and β j , γ j , θ , φ are arbitrary real numbers. The proof that S is a slice is somewhat technical, so we omit it. It is then straightforward to show that F ∩ S is equal to the image of our map (i.e., points of the form (12)), which implies that a neighborhood of F in F is identified with the product of a neighborhood of the origin in C F and the group.\\nGeneral Orthodecomposable FUNTFs. Now suppose that F is a general orthodecomposable FUNTF. After applying isometries as necessary, we can assume without loss of generality that F is a block diagonal matrix F = diag(F 1 ,... ,F ℓ ), where F j a non-orthodecomposable matrix of size d j × N j . Similar computations show that the cone C F consists of matrices X ∈ C d×N satisfying the linear conditions (10) and the system of quadratic equations\\nGeneral Orthodecomposable FUNTFs. Now suppose that F is a general orthodecomposable FUNTF. After applying isometries as necessary, we can assume without loss of generality that F is a block diagonal matrix F = diag(F 1 ,... ,F ℓ ), where F j a non-orthodecomposable matrix of size d j × N j . Similar computations show that the cone C F consists of matrices X ∈ C d×N satisfying the linear conditions (10) and the system of quadratic equations\\nwhere we express X as a block matrix with blocks X km of size d k ×N m and with column vectors denoted x km j . Arguments similar to the above show that C F is a singular cone.\\nwhere we express X as a block matrix with blocks X km of size d k ×N m and with column vectors denoted x km j . Arguments similar to the above show that C F is a singular cone.\\nOrthodecomposable Frames in Arbitrary Frame Spaces. These descriptions of the local structure of FUNTF space near orthodecomposable frames do not intrinsically use the unit norm condition-they only rely on the assumption that the columns have some fixed collection of norms. The arguments therefore apply to describe singularities of spaces of tight frames with fixed norms.\\nOrthodecomposable Frames in Arbitrary Frame Spaces. These descriptions of the local structure of FUNTF space near orthodecomposable frames do not intrinsically use the unit norm condition-they only rely on the assumption that the columns have some fixed collection of norms. The arguments therefore apply to describe singularities of spaces of tight frames with fixed norms.\\nIn fact, similar local characterizations can be derived near orthodecomposable frames in any frame space F d,N λ λ λ (r r r) with an added technical step. The result of Arms, Marsden, and Moncrief specifically treats zero level sets of momentum maps, and extends trivially to handle level sets of fixed points of the Ad * -action on the dual of the Lie algebra. The general frame space F d,N λ λ λ (r r r) is a level set of a coadjoint orbit: F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}). The standard shifting trick of symplectic geometry (see, e.g., [48, p. 376]) can be used to realize F d,N λ λ λ (r r r) as the 0-level set of an associated momentum map on the symplectic manifold C d×N ×O −λ λ λ ; more specifically, this space is endowed with a product form, where the symplectic form on the second factor is −ω KKS with ω KKS being the canonical Kirillov-Kostant-Souriau symplectic form on a coadjoint orbit defined in (6). After applying the shifting trick, similar computations can be done to describe the local geometry of singular points.\\nIn fact, similar local characterizations can be derived near orthodecomposable frames in any frame space F d,N λ λ λ (r r r) with an added technical step. The result of Arms, Marsden, and Moncrief specifically treats zero level sets of momentum maps, and extends trivially to handle level sets of fixed points of the Ad * -action on the dual of the Lie algebra. The general frame space F d,N λ λ λ (r r r) is a level set of a coadjoint orbit: F d,N λ λ λ (r r r) = Φ −1 (O −λ λ λ × {r r r}). The standard shifting trick of symplectic geometry (see, e.g., [48, p. 376]) can be used to realize F d,N λ λ λ (r r r) as the 0-level set of an associated momentum map on the symplectic manifold C d×N ×O −λ λ λ ; more specifically, this space is endowed with a product form, where the symplectic form on the second factor is −ω KKS with ω KKS being the canonical Kirillov-Kostant-Souriau symplectic form on a coadjoint orbit defined in (6). After applying the shifting trick, similar computations can be done to describe the local geometry of singular points.\"\n },\n {\n \"title\": \"Toric Geometry of Frame Space\",\n \"content\": \"\"\n },\n {\n \"title\": \"Toric Geometry of Frame Space\",\n \"content\": \"\"\n },\n {\n \"title\": \"Toric Symplectic Manifolds\",\n \"content\": \"\"\n },\n {\n \"title\": \"Toric Symplectic Manifolds\",\n \"content\": \"Recall that our goal is to show that the collection of full-spark frames in F d,N λ λ λ (r r r) has full measure. As mentioned in the introduction, the natural measure on F d,N λ λ λ (r r r) is the Hausdorff measure it inherits as a compact subset of C d×N . This measure is not so easy to get a handle on directly, but symplectic geometry provides a more tractable approach.\\nRecall that our goal is to show that the collection of full-spark frames in F d,N λ λ λ (r r r) has full measure. As mentioned in the introduction, the natural measure on F d,N λ λ λ (r r r) is the Hausdorff measure it inherits as a compact subset of C d×N . This measure is not so easy to get a handle on directly, but symplectic geometry provides a more tractable approach.\\nAll symplectic manifolds are equipped with natural measures. Specifically, suppose (M,ω) is a symplectic manifold of dimension 2n. By the non-degeneracy of the symplectic form ω, the maximal wedge power ω ∧n = ω ∧ ⋅⋅⋅ ∧ ω n is nowhere-vanishing, and hence defines a volume form and associated symplectic or Liouville measure m ω : for a Borel set U ⊂ M, m ω (U) ∶= ∫ U ω ∧n . While F d,N λ λ λ (r r r) is not symplectic, by Proposition 2.16 its quotient\\nAll symplectic manifolds are equipped with natural measures. Specifically, suppose (M,ω) is a symplectic manifold of dimension 2n. By the non-degeneracy of the symplectic form ω, the maximal wedge power ω ∧n = ω ∧ ⋅⋅⋅ ∧ ω n is nowhere-vanishing, and hence defines a volume form and associated symplectic or Liouville measure m ω : for a Borel set U ⊂ M, m ω (U) ∶= ∫ U ω ∧n . While F d,N λ λ λ (r r r) is not symplectic, by Proposition 2.16 its quotient\\nis. Moreover, since C d×N is Kähler-i.e., it is a complex manifold with a Hermitian metric whose negative imaginary part defines the standard symplectic form and whose real part defines the standard Riemannian metric-so is the symplectic reduction, and the symplectic measure agrees with the pushforward of Hausdorff measure on F d,N λ λ λ (r r r) by the quotient map F d,N λ λ λ (r r r) → F d,N λ λ λ (r r r) (U(d) × G) [28,Theorem 3.1]. Therefore, to prove our main theorem it suffices to prove that the equivalence classes of full-spark frames have full measure in F d,N λ λ λ (r r r) (U(d) × G), which we can do symplectically. The key is that F d,N λ λ λ (r r r) (U(d) × G) admits a Hamiltonian action of a high-dimensional torus, which considerably simplifies the task of understanding the symplectic measure.\\nis. Moreover, since C d×N is Kähler-i.e., it is a complex manifold with a Hermitian metric whose negative imaginary part defines the standard symplectic form and whose real part defines the standard Riemannian metric-so is the symplectic reduction, and the symplectic measure agrees with the pushforward of Hausdorff measure on F d,N λ λ λ (r r r) by the quotient map F d,N λ λ λ (r r r) → F d,N λ λ λ (r r r) (U(d) × G) [28,Theorem 3.1]. Therefore, to prove our main theorem it suffices to prove that the equivalence classes of full-spark frames have full measure in F d,N λ λ λ (r r r) (U(d) × G), which we can do symplectically. The key is that F d,N λ λ λ (r r r) (U(d) × G) admits a Hamiltonian action of a high-dimensional torus, which considerably simplifies the task of understanding the symplectic measure.\\nIf a 2n-dimensional symplectic manifold (M,ω) admits a Hamiltonian action of a torus U(1) k , then the associated momentum map Φ ∶ M → u(1) k * ≃ R k has convex image and connected level sets:\\nIf a 2n-dimensional symplectic manifold (M,ω) admits a Hamiltonian action of a torus U(1) k , then the associated momentum map Φ ∶ M → u(1) k * ≃ R k has convex image and connected level sets:\\nTheorem 3.1 (Atiyah [4] and Guillemin-Sternberg [24]). With notation as above:\\nTheorem 3.1 (Atiyah [4] and Guillemin-Sternberg [24]). With notation as above:\\n• For any v ∈ R k , Φ −1 (v) is either empty or connected.\\n• For any v ∈ R k , Φ −1 (v) is either empty or connected.\\n• The image Φ(M) is the convex hull of the images of the fixed points of the torus action.\\n• The image Φ(M) is the convex hull of the images of the fixed points of the torus action.\\nIn particular, when M is (relatively) compact, (the closure of) Φ(M) is a bounded convex polytope P called the moment polytope associated to the Hamiltonian torus action.\\nIn particular, when M is (relatively) compact, (the closure of) Φ(M) is a bounded convex polytope P called the moment polytope associated to the Hamiltonian torus action.\\nIn general, the Duistermaat-Heckman theorem [17] precisely describes the relationship between the pushforward measure Φ * (m ω ) and Lebesgue measure on the moment polytope P. Since we will not need the full statement, we restrict to the case when k = n, that is when M admits a Hamiltonian action of a half-dimensional torus. In this case M is called a toric symplectic manifold. Toric symplectic manifolds are closely related to toric varieties [13] and are completely classified by the combinatorics of the moment polytope [15].\\nIn general, the Duistermaat-Heckman theorem [17] precisely describes the relationship between the pushforward measure Φ * (m ω ) and Lebesgue measure on the moment polytope P. Since we will not need the full statement, we restrict to the case when k = n, that is when M admits a Hamiltonian action of a half-dimensional torus. In this case M is called a toric symplectic manifold. Toric symplectic manifolds are closely related to toric varieties [13] and are completely classified by the combinatorics of the moment polytope [15].\\nTheorem 3.2 (Duistermaat-Heckman [17], see also [9,Chapter 30]). Let M be a 2n-dimensional toric symplectic manifold with moment polytope P. The pushforward measure Φ * (m ω ) is a constant multiple of Lebesgue measure on P.\\nTheorem 3.2 (Duistermaat-Heckman [17], see also [9,Chapter 30]). Let M be a 2n-dimensional toric symplectic manifold with moment polytope P. The pushforward measure Φ * (m ω ) is a constant multiple of Lebesgue measure on P.\"\n },\n {\n \"title\": \"Toric Structure of Frame Space\",\n \"content\": \"\"\n },\n {\n \"title\": \"Toric Structure of Frame Space\",\n \"content\": \"\"\n },\n {\n \"title\": \"Circle Actions on Frame Space\",\n \"content\": \"\"\n },\n {\n \"title\": \"Circle Actions on Frame Space\",\n \"content\": \"With Theorem 3.2 in mind, our strategy is to show that the top stratum of F d,N λ λ λ (r r r) (U(d) × G) is toric, and then to see that the image of the spark-deficient frames in the moment polytope has measure zero with respect to Lebesgue measure.\\nWith Theorem 3.2 in mind, our strategy is to show that the top stratum of F d,N λ λ λ (r r r) (U(d) × G) is toric, and then to see that the image of the spark-deficient frames in the moment polytope has measure zero with respect to Lebesgue measure.\\nWe begin by defining a torus action on the full space of frames F d,N and then show that our construction descends to the symplectic quotient. Let F ∈ F d,N with column vectors f i ∈ C d×1 , i = 1,... ,N. For each k = 1,...,N, let\\nWe begin by defining a torus action on the full space of frames F d,N and then show that our construction descends to the symplectic quotient. Let F ∈ F d,N with column vectors f i ∈ C d×1 , i = 1,... ,N. For each k = 1,...,N, let\\narranged in decreasing order-each such sum is rank at most k, so we only need consider the first k eigenvalues if k < d. The quantities µ k j were dubbed eigensteps by Cahill et al. [7]. To get well-defined circle actions, we make the assumption that the eigenvalues µ k j , j = 1,...,k, are all distinctthe necessity of this assumption is explained below in Remark 3.4. Let u k1 ,... ,u k min(k,d) be the eigenvectors corresponding to the nonzero eigenvalues. For each k = 1,...,N and each j = 1,... ,min(k,d), define a U(1)-action on F, denoted\\narranged in decreasing order-each such sum is rank at most k, so we only need consider the first k eigenvalues if k < d. The quantities µ k j were dubbed eigensteps by Cahill et al. [7]. To get well-defined circle actions, we make the assumption that the eigenvalues µ k j , j = 1,...,k, are all distinctthe necessity of this assumption is explained below in Remark 3.4. Let u k1 ,... ,u k min(k,d) be the eigenvectors corresponding to the nonzero eigenvalues. For each k = 1,...,N and each j = 1,... ,min(k,d), define a U(1)-action on F, denoted\\nby defining it at the frame vector level as\\nby defining it at the frame vector level as\\nwhere t ∈ [0,2π).\\nwhere t ∈ [0,2π).\\nKeeping track of the exact isomorphism used in this identification will be useful later on.\\nKeeping track of the exact isomorphism used in this identification will be useful later on.\"\n },\n {\n \"title\": \"Remark 3.4.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Remark 3.4.\",\n \"content\": \"The assumption that the eigenvalues µ k j are distinct means that the eigenspaces are all one-dimensional, which, in turn, gives a well-defined ordering of the eigenvectors u k j . Without the isolated eigenvalues assumption, some of these actions would degenerate to U(ℓ)-actions, with ℓ the multiplicity of a repeated eigenvalue.\\nThe assumption that the eigenvalues µ k j are distinct means that the eigenspaces are all one-dimensional, which, in turn, gives a well-defined ordering of the eigenvectors u k j . Without the isolated eigenvalues assumption, some of these actions would degenerate to U(ℓ)-actions, with ℓ the multiplicity of a repeated eigenvalue.\\nRemark 3.5. The action defined by (13) really defines a circle action; i.e., it is 2π-periodic. Indeed, this follows from Lemma 3.11 below, and is also shown in [19,Corollary 5.1.4].\\nRemark 3.5. The action defined by (13) really defines a circle action; i.e., it is 2π-periodic. Indeed, this follows from Lemma 3.11 below, and is also shown in [19,Corollary 5.1.4].\\nNext we show that this action on F d,N induces a well-defined action on F d,N λ λ λ (r r r) (U(d) × G). We do so in stages. Proposition 3.6. The circle action φ k j commutes with the T action.\\nNext we show that this action on F d,N induces a well-defined action on F d,N λ λ λ (r r r) (U(d) × G). We do so in stages. Proposition 3.6. The circle action φ k j commutes with the T action.\\nProof. We need to show that if two frames F 1 and F 2 lie in the same T orbit, then so do φ k j (t,F 1 ) and φ k j (t,F 2 ). Indeed, this holds since the formula for the action (13) makes it clear that\\nProof. We need to show that if two frames F 1 and F 2 lie in the same T orbit, then so do φ k j (t,F 1 ) and φ k j (t,F 2 ). Indeed, this holds since the formula for the action (13) makes it clear that\\nfor all θ . Proof. The claim is clear when k + 1 ≤ i ≤ N, since the action is trivial in that case. It remains to check the claim when 1 ≤ i ≤ k. In this case, we have φ k j (t,A f i ) = exp(t √ −1Au k j u * k j A * )A f i , since (µ k j ,Au k j ) are the corresponding eigenpair for the kth partial frame operator of AF. In turn, we have\\nfor all θ . Proof. The claim is clear when k + 1 ≤ i ≤ N, since the action is trivial in that case. It remains to check the claim when 1 ≤ i ≤ k. In this case, we have φ k j (t,A f i ) = exp(t √ −1Au k j u * k j A * )A f i , since (µ k j ,Au k j ) are the corresponding eigenpair for the kth partial frame operator of AF. In turn, we have\\nSince the circle action φ k j is defined on F = f 1 ⋯ f N ∈ F d,N through the action on the individual frame vectors, we have the following corollary. The next proposition is obvious from the formula for φ k j .\\nSince the circle action φ k j is defined on F = f 1 ⋯ f N ∈ F d,N through the action on the individual frame vectors, we have the following corollary. The next proposition is obvious from the formula for φ k j .\\nProposition 3.9. The circle action φ k j preserves norms of frame vectors.\\nProposition 3.9. The circle action φ k j preserves norms of frame vectors.\\nThe following proposition is similar, but is less obvious. The proof uses a lemma.\\nThe following proposition is similar, but is less obvious. The proof uses a lemma.\\nProof. This follows from the Taylor series representation of the matrix exponential and the property that P 2 = P.\\nProof. This follows from the Taylor series representation of the matrix exponential and the property that P 2 = P.\\nProof of Proposition 3. 10.\\nProof of Proposition 3. 10.\\nis a frame and let S = FF * = f 1 f * 1 + ⋅⋅⋅ + f N f * N be its frame operator with spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0. Let k ∈ {1,...,N} and let S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k be the partial frame operator with spectrum µ k1 ≥ ⋅⋅⋅ ≥ µ kd ≥ 0. Let j ∈ {1,...,min(k,d)}, and consider the torus action associated with the eigenvalue µ k j . Then, by definition,\\nis a frame and let S = FF * = f 1 f * 1 + ⋅⋅⋅ + f N f * N be its frame operator with spectrum λ 1 ≥ ⋅⋅⋅ ≥ λ d > 0. Let k ∈ {1,...,N} and let S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k be the partial frame operator with spectrum µ k1 ≥ ⋅⋅⋅ ≥ µ kd ≥ 0. Let j ∈ {1,...,min(k,d)}, and consider the torus action associated with the eigenvalue µ k j . Then, by definition,\\nand the corresponding frame operator is\\nand the corresponding frame operator is\\nusing Lemma 3.11. For i 1 ≠ i 2 the vectors u ki 1 and u ki 2 are Hermitian orthogonal and hence the product of projections u ki 1 u * ki 1 u ki 2 u * ki 2 = 0. Combining this with the spectral decomposition S k = µ k1 u k1 u * k1 + ⋅⋅⋅ + µ kd u kd u * kd , and again using the fact that projections are idempotent, we see that\\nusing Lemma 3.11. For i 1 ≠ i 2 the vectors u ki 1 and u ki 2 are Hermitian orthogonal and hence the product of projections u ki 1 u * ki 1 u ki 2 u * ki 2 = 0. Combining this with the spectral decomposition S k = µ k1 u k1 u * k1 + ⋅⋅⋅ + µ kd u kd u * kd , and again using the fact that projections are idempotent, we see that\\nso the frame operator is invariant under the circle action.\\nso the frame operator is invariant under the circle action.\\nCombining Propositions 3.6, 3.9, and 3.10 with Corollary 3.8, we obtain:\\nCombining Propositions 3.6, 3.9, and 3.10 with Corollary 3.8, we obtain:\\nProposition 3.12. The circle action φ k j on F d,N descends to a well-defined action on F d,N λ λ λ (r r r) (U(d) × G).\\nProposition 3.12. The circle action φ k j on F d,N descends to a well-defined action on F d,N λ λ λ (r r r) (U(d) × G).\"\n },\n {\n \"title\": \"The Momentum Map of the Circle Action\",\n \"content\": \"\"\n },\n {\n \"title\": \"The Momentum Map of the Circle Action\",\n \"content\": \"For each k = 1,...,N and j = 1,...,min(k,d), define a map\\nFor each k = 1,...,N and j = 1,...,min(k,d), define a map\\nwhere µ k j = µ k j (F) is (as above) the jth eigenvalue (in descending order) of the kth partial frame operator of F. Recall that we identify u(1) ≈ R via (14). We likewise identify u(1) * ≈ R, where R * ≈ R is identified via the pairing\\nwhere µ k j = µ k j (F) is (as above) the jth eigenvalue (in descending order) of the kth partial frame operator of F. Recall that we identify u(1) ≈ R via (14). We likewise identify u(1) * ≈ R, where R * ≈ R is identified via the pairing\\non R × R. This allows us to state the following result.\\non R × R. This allows us to state the following result.\\nProposition 3.13. The map Φ k j ∶ F d,N → R is a momentum map for the circle action φ k j on the dense open subset of matrices whose kth partial frame operators have isolated jth eigenvalues.\\nProposition 3.13. The map Φ k j ∶ F d,N → R is a momentum map for the circle action φ k j on the dense open subset of matrices whose kth partial frame operators have isolated jth eigenvalues.\\nTo prove the proposition, we introduce some notation and technical lemmas. Let F = f 1 ⋯ f N ∈ F d,N and let S k = S k (F) = f 1 f * 1 + ⋯ + f k f * k be its partial frame operator. Then µ k j (F) is the jth eigenvalue (in decreasing order) of S k .\\nTo prove the proposition, we introduce some notation and technical lemmas. Let F = f 1 ⋯ f N ∈ F d,N and let S k = S k (F) = f 1 f * 1 + ⋯ + f k f * k be its partial frame operator. Then µ k j (F) is the jth eigenvalue (in decreasing order) of S k .\\nWe first consider the mapμ j which takes a Hermitian k × k matrix to its jth eigenvalue. In what follows, let ⟨⋅,⋅⟩ denote the Frobenius inner product on F d,N . Lemma 3.14. Suppose that S is a k × k Hermitian matrix with isolated jth eigenvalueμ j (S). Thenμ j is a smooth map in a neighborhood of S. Moreover, its gradient at S is given by\\nWe first consider the mapμ j which takes a Hermitian k × k matrix to its jth eigenvalue. In what follows, let ⟨⋅,⋅⟩ denote the Frobenius inner product on F d,N . Lemma 3.14. Suppose that S is a k × k Hermitian matrix with isolated jth eigenvalueμ j (S). Thenμ j is a smooth map in a neighborhood of S. Moreover, its gradient at S is given by\\nwhereū j is the unit eigenvector associated toμ j (S).\\nwhereū j is the unit eigenvector associated toμ j (S).\\nProof. Let S have isolated jth eigenvalue, which we denote simply as µ. We denote the associated unit eigenvector as u. Let S ′ be a variation of S. Denote the jth eigenvalue and unit eigenvector of S + εS ′ as µ(ε) and u(ε), respectively; we denote the ε-derivatives of these functions asμ(ε) andu(ε). Then\\nProof. Let S have isolated jth eigenvalue, which we denote simply as µ. We denote the associated unit eigenvector as u. Let S ′ be a variation of S. Denote the jth eigenvalue and unit eigenvector of S + εS ′ as µ(ε) and u(ε), respectively; we denote the ε-derivatives of these functions asμ(ε) andu(ε). Then\\n= ⟨S ′ u,u⟩ + µ (⟨u(0),u⟩ + ⟨u,u(0))⟩ (17)\\n= ⟨S ′ u,u⟩ + µ (⟨u(0),u⟩ + ⟨u,u(0))⟩ (17)\\nwhere (16) follows because S is Hermitian and (17) follows by the condition that u(ε) is constant in ε. Lemma 3.15. Let F = f 1 ⋯ f N ∈ F d,N and s ∈ R ≈ u(1) (using the identification (14)). The infinitesimal vector field associated to s which is induced by the action φ k j is given at F by\\nwhere (16) follows because S is Hermitian and (17) follows by the condition that u(ε) is constant in ε. Lemma 3.15. Let F = f 1 ⋯ f N ∈ F d,N and s ∈ R ≈ u(1) (using the identification (14)). The infinitesimal vector field associated to s which is induced by the action φ k j is given at F by\\nProof. The infinitesimal vector field is given at the frame level by\\nProof. The infinitesimal vector field is given at the frame level by\\nThe result follows.\\nThe result follows.\\nWe are now prepared to prove the proposition.\\nWe are now prepared to prove the proposition.\\nProof of Proposition 3.13. We need to show that, for (14)), the following equation holds:\\nProof of Proposition 3.13. We need to show that, for (14)), the following equation holds:\\nThis is accomplished by direct computation. The right hand side of (18) simplifies to\\nThis is accomplished by direct computation. The right hand side of (18) simplifies to\\nwhere (19) uses the expression for F s from Lemma 3.15. The left hand side of (18) becomes\\nwhere (19) uses the expression for F s from Lemma 3.15. The left hand side of (18) becomes\\nwhere the first line follows by our choice of identification R ≈ u(1) * (15) and where S k (F) is the kth partial frame operator of F andμ j is the jth eigenvalue function on d × d Hermitian matrices. A straightforward computation shows that\\nwhere the first line follows by our choice of identification R ≈ u(1) * (15) and where S k (F) is the kth partial frame operator of F andμ j is the jth eigenvalue function on d × d Hermitian matrices. A straightforward computation shows that\\n. Putting this together with Lemma 3.14, our simplification continues as\\n. Putting this together with Lemma 3.14, our simplification continues as\\nwhere the last line follows by the observation that the trace terms in (21) are conjugates. Since (22) and (20) agree, (18) has been established.\\nwhere the last line follows by the observation that the trace terms in (21) are conjugates. Since (22) and (20) agree, (18) has been established.\"\n },\n {\n \"title\": \"Dimension Counting\",\n \"content\": \"\"\n },\n {\n \"title\": \"Dimension Counting\",\n \"content\": \"We now pause briefly to count the possible number of independent eigensteps µ k j . The µ k j associated to a frame define a map F d,N λ λ λ (r r r) → R m for some m depending on N and d; as has been previously observed [7,8,27], the image of this map is a convex polytope, whose dimension we now determine. To slightly simplify some calculations, we will assume in this section that r r r is strongly λ λ λ -admissible.\\nWe now pause briefly to count the possible number of independent eigensteps µ k j . The µ k j associated to a frame define a map F d,N λ λ λ (r r r) → R m for some m depending on N and d; as has been previously observed [7,8,27], the image of this map is a convex polytope, whose dimension we now determine. To slightly simplify some calculations, we will assume in this section that r r r is strongly λ λ λ -admissible.\\nSince each partial frame operator S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k is a rank-1 perturbation S k = S k−1 + f k f * k of the previous partial frame operator, Weyl's perturbation inequalities (see, e.g., [6,Chapter III]) imply that the eigenvalues of S k and S k−1 satisfy the interlacing inequalities ⋅⋅⋅ ≥ µ k, j ≥ µ k−1, j ≥ µ k, j+1 ≥ µ k−1, j+1 ≥ ...\\nSince each partial frame operator S k = f 1 f * 1 + ⋅⋅⋅ + f k f * k is a rank-1 perturbation S k = S k−1 + f k f * k of the previous partial frame operator, Weyl's perturbation inequalities (see, e.g., [6,Chapter III]) imply that the eigenvalues of S k and S k−1 satisfy the interlacing inequalities ⋅⋅⋅ ≥ µ k, j ≥ µ k−1, j ≥ µ k, j+1 ≥ µ k−1, j+1 ≥ ...\\nfor each k = 2,... ,N and j = 1,...,min{d,k}, where we use the notation µ k, j in place of µ k j when we need to disambiguate the index k from the index j. When N = 4 and d = 3, this produces the array\\nfor each k = 2,... ,N and j = 1,...,min{d,k}, where we use the notation µ k, j in place of µ k j when we need to disambiguate the index k from the index j. When N = 4 and d = 3, this produces the array\\nwhere we recall that S N = S, and hence µ N, j = λ j for each j = 1,...,d.\\nwhere we recall that S N = S, and hence µ N, j = λ j for each j = 1,...,d.\\nIn other words, entries in the array are greater than entries to the right, regardless of vertical position. With this convention in place, we can omit explicit inequalities without losing any information:\\nIn other words, entries in the array are greater than entries to the right, regardless of vertical position. With this convention in place, we can omit explicit inequalities without losing any information:\\nIn general, then, the µ k, j satisfy the system of interlacing inequalities implied by the following diagram:\\nIn general, then, the µ k, j satisfy the system of interlacing inequalities implied by the following diagram:\\nIn this diagram, all entries are nonnegative; equivalently, we think of implicit zeros to the right of every row. The pattern of inequalities described by the above diagram is often called a Gelfand-Tsetlin pattern [14,21]; it corresponds to the Gelfand-Tsetlin integrable system on the flag manifold Oλ λ λ ≃ F d,N λ λ λ U(d) [26]. In fact, there is one further constraint: cyclic invariance of trace implies that the trace of each partial frame operator S k must equal the trace of the corresponding partial Gram matrix. That is, the sum of each row in (23) must equal the corresponding partial sum of squared frame vector norms:\\nIn this diagram, all entries are nonnegative; equivalently, we think of implicit zeros to the right of every row. The pattern of inequalities described by the above diagram is often called a Gelfand-Tsetlin pattern [14,21]; it corresponds to the Gelfand-Tsetlin integrable system on the flag manifold Oλ λ λ ≃ F d,N λ λ λ U(d) [26]. In fact, there is one further constraint: cyclic invariance of trace implies that the trace of each partial frame operator S k must equal the trace of the corresponding partial Gram matrix. That is, the sum of each row in (23) must equal the corresponding partial sum of squared frame vector norms:\\nfor each k = 1,...,N. With this constraint in place, we can count the number of free parameters in (23) and, hence, the number of independent φ k j . Notice that there are d(2N−d−1) 2 total entries in (23), excluding the top row (which we already know is fixed):\\nfor each k = 1,...,N. With this constraint in place, we can count the number of free parameters in (23) and, hence, the number of independent φ k j . Notice that there are d(2N−d−1) 2 total entries in (23), excluding the top row (which we already know is fixed):\\nfor the triangle in the bottom d rows, and d(N − d − 1) for the parallelogram in the upper N − d rows. Since each row sum is fixed, we lose one free parameter for each of the rows but the top one, meaning we subtract N − 1 parameters.\\nfor the triangle in the bottom d rows, and d(N − d − 1) for the parallelogram in the upper N − d rows. Since each row sum is fixed, we lose one free parameter for each of the rows but the top one, meaning we subtract N − 1 parameters.\\nIf all the eigenvalues λ 1 ,... ,λ d of the frame operator are distinct, we are done. However, if λ j = λ j+1 , then this implies that λ j = µ N−1, j = λ j+1 . More generally, if λ j has multiplicity k, meaning that λ j = λ j+1 = ⋅⋅⋅ = λ j+k−1 , then this fixes an entire upside-down triangle in (23) with vertices at λ j , λ j+k−1 , and µ N−k, j , comprising k(k−1) 2 of the µ i, j . Therefore, if the eigenvalues λ 1 ,... ,λ d have multiplicities k 1 ,... ,k ℓ with k 1 + ⋅⋅⋅ + k ℓ = d, then the total number of free parameters in (23) is\\nIf all the eigenvalues λ 1 ,... ,λ d of the frame operator are distinct, we are done. However, if λ j = λ j+1 , then this implies that λ j = µ N−1, j = λ j+1 . More generally, if λ j has multiplicity k, meaning that λ j = λ j+1 = ⋅⋅⋅ = λ j+k−1 , then this fixes an entire upside-down triangle in (23) with vertices at λ j , λ j+k−1 , and µ N−k, j , comprising k(k−1) 2 of the µ i, j . Therefore, if the eigenvalues λ 1 ,... ,λ d have multiplicities k 1 ,... ,k ℓ with k 1 + ⋅⋅⋅ + k ℓ = d, then the total number of free parameters in (23) is\\nThe interlacing inequalities implicit in the diagram (23) together with the row sums (24) determine a convex polytope P d,N λ λ λ (r r r), which we call the eigenstep polytope (at least when λ λ λ and r r r are rational, these are sometimes called (weight-restricted) Gelfand-Tsetlin polytopes in the combinatorics literature [1,14]). In the course of the discussion above, we have proved: Proposition 3.16. When r r r is strongly λ λ λ -admissible, the eigenstep polytope has dimension\\nThe interlacing inequalities implicit in the diagram (23) together with the row sums (24) determine a convex polytope P d,N λ λ λ (r r r), which we call the eigenstep polytope (at least when λ λ λ and r r r are rational, these are sometimes called (weight-restricted) Gelfand-Tsetlin polytopes in the combinatorics literature [1,14]). In the course of the discussion above, we have proved: Proposition 3.16. When r r r is strongly λ λ λ -admissible, the eigenstep polytope has dimension\\nwhere k 1 ,... ,k ℓ are the multiplicities of the spectrum λ λ λ = (λ 1 ,...,λ d ).\\nwhere k 1 ,... ,k ℓ are the multiplicities of the spectrum λ λ λ = (λ 1 ,...,λ d ).\\nComparing to Corollary 2.18,\\nComparing to Corollary 2.18,\\nIn the case of unit-norm tight frames,\\nIn the case of unit-norm tight frames,\"\n },\n {\n \"title\": \"Hamiltonian Torus Action\",\n \"content\": \"\"\n },\n {\n \"title\": \"Hamiltonian Torus Action\",\n \"content\": \"We now return to the task of showing that there is a Hamiltonian torus action on our frame spaces; the dimension d(λ λ λ ,r r r) just computed will be the dimension of the torus.\\nWe now return to the task of showing that there is a Hamiltonian torus action on our frame spaces; the dimension d(λ λ λ ,r r r) just computed will be the dimension of the torus.\\nOf course, the torus action will simply be the product of the individual circle actions, which we will show commute using Proposition 3.7.\\nOf course, the torus action will simply be the product of the individual circle actions, which we will show commute using Proposition 3.7.\\nProposition 3.17. The circle actions φ k j and φ mℓ commute for all k, j,m,ℓ. N . It suffices to prove the claim at the frame level, i.e.,\\nProposition 3.17. The circle actions φ k j and φ mℓ commute for all k, j,m,ℓ. N . It suffices to prove the claim at the frame level, i.e.,\\nfor all f i . Without loss of generality, assume k ≤ m. If k + 1 ≤ i ≤ N, then φ k j is the trivial action, so (25) is obvious.\\nfor all f i . Without loss of generality, assume k ≤ m. If k + 1 ≤ i ≤ N, then φ k j is the trivial action, so (25) is obvious.\\nwhere (26) 19. In conjunction with Theorem 3.1, this result can be used to show that the space of frames with prescribed eigensteps is connected. While this fact essentially follows from [7,Theorem 7], which was a key tool in the original proof of the Frame Homotopy Conjecture [8], the symplectic viewpoint puts this fact in a more general context. By Proposition 3.16, the torus acting on the open dense subset from Corollary 3.18 is half-dimensional. We have therefore proved the following theorem on the geometric structure of frame spaces.\\nwhere (26) 19. In conjunction with Theorem 3.1, this result can be used to show that the space of frames with prescribed eigensteps is connected. While this fact essentially follows from [7,Theorem 7], which was a key tool in the original proof of the Frame Homotopy Conjecture [8], the symplectic viewpoint puts this fact in a more general context. By Proposition 3.16, the torus acting on the open dense subset from Corollary 3.18 is half-dimensional. We have therefore proved the following theorem on the geometric structure of frame spaces.\"\n },\n {\n \"title\": \"Proof of Main Theorem\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proof of Main Theorem\",\n \"content\": \"We are now prepared to prove the main theorem using the toric symplectic framework developed in the previous section. We also sketch a more algebraically-flavored proof using related tools from Geometric Invariant Theory [42].\\nWe are now prepared to prove the main theorem using the toric symplectic framework developed in the previous section. We also sketch a more algebraically-flavored proof using related tools from Geometric Invariant Theory [42].\"\n },\n {\n \"title\": \"A Toric Symplectic Approach\",\n \"content\": \"\"\n },\n {\n \"title\": \"A Toric Symplectic Approach\",\n \"content\": \"It is clear that the three options (repeated below) are mutually exclusive and exhaust all possibilities, so it remains only to show that the conclusion in each part follows from the assumption.\\nIt is clear that the three options (repeated below) are mutually exclusive and exhaust all possibilities, so it remains only to show that the conclusion in each part follows from the assumption.\\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\\n1. If r r r is not λ λ λ -admissible, then F d,N λ λ λ (r r r) = ∅.\\nProof. As pointed out in Section 2.2.3, this is a consequence of the Schur-Horn theorem [30,46]; it is also a result of Casazza and Leon [12].\\nProof. As pointed out in Section 2.2.3, this is a consequence of the Schur-Horn theorem [30,46]; it is also a result of Casazza and Leon [12].\\n2. If r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\\n2. If r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible, then F d,N λ λ λ (r r r) is nonempty but consists entirely of frames which are not full spark.\\nProof. Suppose r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible. Then F d,N λ λ λ (r r r) is nonempty by Casazza and Leon's result [12].\\nProof. Suppose r r r is λ λ λ -admissible, but not strongly λ λ λ -admissible. Then F d,N λ λ λ (r r r) is nonempty by Casazza and Leon's result [12].\\nSince r r r is not strongly λ λ λ -admissible, there exists k ∈ {1,...,d} so that\\nSince r r r is not strongly λ λ λ -admissible, there exists k ∈ {1,...,d} so that\"\n },\n {\n \"title\": \"The admissibility criterion requires that\",\n \"content\": \"\"\n },\n {\n \"title\": \"The admissibility criterion requires that\",\n \"content\": \"If k = d, then we see that r k+1 = ⋅⋅⋅ = r N = 0, which does not satisfy the hypothesis that the r i are all positive.\\nIf k = d, then we see that r k+1 = ⋅⋅⋅ = r N = 0, which does not satisfy the hypothesis that the r i are all positive.\\nOtherwise, suppose F = f 1 ⋯ f N ∈ F d,N λ λ λ (r r r), so that λ λ λ is the spectrum of FF * and f i 2 = r i for all i = 1,... ,N.\\nOtherwise, suppose F = f 1 ⋯ f N ∈ F d,N λ λ λ (r r r), so that λ λ λ is the spectrum of FF * and f i 2 = r i for all i = 1,... ,N.\\nusing (27). Now, consider the portion of the eigenstep inequality diagram (23) forF starting from the (N − k)th row: 1μN−1,2μN−1\\nusing (27). Now, consider the portion of the eigenstep inequality diagram (23) forF starting from the (N − k)th row: 1μN−1,2μN−1\\nthe length-d collection of frame vectors f N−d+1 ,... , f N is linearly dependent, and hence F is spark-deficient. Since the choice of F was arbitrary, we see that F d,N λ λ λ (r r r) consists entirely of spark-deficient frames.\\nthe length-d collection of frame vectors f N−d+1 ,... , f N is linearly dependent, and hence F is spark-deficient. Since the choice of F was arbitrary, we see that F d,N λ λ λ (r r r) consists entirely of spark-deficient frames.\\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\\n3. If r r r is strongly λ λ λ -admissible, then full spark frames have full measure in F d,N λ λ λ (r r r).\\nProof. It suffices to show that the set of full spark frames is full measure in the dense open subset of F d,N λ λ λ (r r r) from Theorem 3.20, which we denoteŨ. We will prove a subclaim: the set D d λ λ λ (r r r) ⊂Ũ of frames whose first d columns are linearly dependent is measure zero. The full claim then follows easily. Indeed, consider the action of the symmetric group on N letters on the matrix space C d×N given by permuting columns. This representation of the symmetric group embeds it as a subgroup of U(N), so that the action on C d×N is by isometries. The action of a permutation σ restricts to an isometry of each frame space F d,N λ λ λ (r r r) onto its image F d,N λ λ λ (σ ⋅ r r r), where σ ⋅ r r r is the corresponding permutation of the norm vector. It follows that the set of spark-deficient frames is realized as ∪ σ D d λ λ λ (σ ⋅ r r r), a finite union of measure zero sets. We proceed by establishing the subclaim.\\nProof. It suffices to show that the set of full spark frames is full measure in the dense open subset of F d,N λ λ λ (r r r) from Theorem 3.20, which we denoteŨ. We will prove a subclaim: the set D d λ λ λ (r r r) ⊂Ũ of frames whose first d columns are linearly dependent is measure zero. The full claim then follows easily. Indeed, consider the action of the symmetric group on N letters on the matrix space C d×N given by permuting columns. This representation of the symmetric group embeds it as a subgroup of U(N), so that the action on C d×N is by isometries. The action of a permutation σ restricts to an isometry of each frame space F d,N λ λ λ (r r r) onto its image F d,N λ λ λ (σ ⋅ r r r), where σ ⋅ r r r is the corresponding permutation of the norm vector. It follows that the set of spark-deficient frames is realized as ∪ σ D d λ λ λ (σ ⋅ r r r), a finite union of measure zero sets. We proceed by establishing the subclaim.\\ndenote the moment map for the torus action on U; that is, Φ is the eigenstep map. By the remarks in the previous paragraph, the condition that the first d columns of a frame F are linearly independent is equivalent to the statement that Φ dd ([F]) ≠ 0.\\ndenote the moment map for the torus action on U; that is, Φ is the eigenstep map. By the remarks in the previous paragraph, the condition that the first d columns of a frame F are linearly independent is equivalent to the statement that Φ dd ([F]) ≠ 0.\\nAccording to Section 3.2.3, the moment polytope P d,N λ λ λ (r r r) is a convex subset of an affine subspace A ⊂ R d(2N−d−1) 2 . Thinking of the Φ k j as coordinates on the ambient space, the condition Φ dd = 0 defines a hyperplane. The intersection of this hyperplane with P d,N λ λ λ (r r r) is positive codimension (hence measure zero), unless A is contained in the hyperplane. It is easy to see that this is not the case, as it suffices to show the existence of a pattern of the form (23) satisfying the defining equalities of A such that µ dd ≠ 0-such examples are trivial to construct without the polytope inequality constraints.\\nAccording to Section 3.2.3, the moment polytope P d,N λ λ λ (r r r) is a convex subset of an affine subspace A ⊂ R d(2N−d−1) 2 . Thinking of the Φ k j as coordinates on the ambient space, the condition Φ dd = 0 defines a hyperplane. The intersection of this hyperplane with P d,N λ λ λ (r r r) is positive codimension (hence measure zero), unless A is contained in the hyperplane. It is easy to see that this is not the case, as it suffices to show the existence of a pattern of the form (23) satisfying the defining equalities of A such that µ dd ≠ 0-such examples are trivial to construct without the polytope inequality constraints.\\nWe have so far shown that image of the set of equivalence classes of frames whose first d columns are linearly independent has full measure image in P d,N λ λ λ (r r r) under Φ. By the Duistermaat-Heckman Theorem (Theorem 3.2), this implies that the set has full measure in U. This completes the proof of the subclaim, and therefore completes the proof of the theorem.\\nWe have so far shown that image of the set of equivalence classes of frames whose first d columns are linearly independent has full measure image in P d,N λ λ λ (r r r) under Φ. By the Duistermaat-Heckman Theorem (Theorem 3.2), this implies that the set has full measure in U. This completes the proof of the subclaim, and therefore completes the proof of the theorem.\"\n },\n {\n \"title\": \"An algebraic approach\",\n \"content\": \"\"\n },\n {\n \"title\": \"An algebraic approach\",\n \"content\": \"The above proof is based on symplectic geometry, but, under additional rationality assumptions, there is also an argument using algebraic geometry that shows that the collection of full-spark frames is dense in F d,N λ λ λ (r r r). While we expect a similar argument to apply to arbitrary rational λ λ λ and r r r, in which case the objects of interest are more general weight varieties [35], for simplicity we limit ourselves to the case when λ λ λ and r r r are constant, so we are talking about (scaled) unit-norm tight frames. It will be convenient to rescale so that r r r = (d,... ,d), and hence λ λ λ = (N,... ,N); note that both are integer vectors.\\nThe above proof is based on symplectic geometry, but, under additional rationality assumptions, there is also an argument using algebraic geometry that shows that the collection of full-spark frames is dense in F d,N λ λ λ (r r r). While we expect a similar argument to apply to arbitrary rational λ λ λ and r r r, in which case the objects of interest are more general weight varieties [35], for simplicity we limit ourselves to the case when λ λ λ and r r r are constant, so we are talking about (scaled) unit-norm tight frames. It will be convenient to rescale so that r r r = (d,... ,d), and hence λ λ λ = (N,... ,N); note that both are integer vectors.\\nAs above, it suffices to show that the (equivalence classes of) full spark frames are open and dense in the symplectic reduction F d,N λ λ λ (r r r) (U(d) × G) ≈ C d×N O −λ λ λ ×{r r r} (U(d) × G).\\nAs above, it suffices to show that the (equivalence classes of) full spark frames are open and dense in the symplectic reduction F d,N λ λ λ (r r r) (U(d) × G) ≈ C d×N O −λ λ λ ×{r r r} (U(d) × G).\\nSince λ λ λ is constant, the coadjoint orbit O −λ λ λ consists of the single point −λ λ λ I d . Taking the reduction in stages, the above quotient is isomorphic to C d×N −λ λ λ I d U(d) r r r G = Gr d (C N ) r r r G using Proposition 2.11 and Remark 2.13.\\nSince λ λ λ is constant, the coadjoint orbit O −λ λ λ consists of the single point −λ λ λ I d . Taking the reduction in stages, the above quotient is isomorphic to C d×N −λ λ λ I d U(d) r r r G = Gr d (C N ) r r r G using Proposition 2.11 and Remark 2.13.\\nWork of Sjamaar [47], which is a culmination of a series of results by Kempf and Ness [32], Guillemin and Sternberg [25], Kirwan [33], and Ness [45] relating symplectic reductions and Geometric Invariant Theory (GIT) quotients [42], has the following consequence in our case: where the line bundle L r r r on the Grassmannian is linearized to correspond to the action of the algebraic torus H = {(t 1 ,... ,t N ) ∈ (C * ) N ∶ ∏ti = 1} on C N given by identifying t t t = (t 1 ,...,t N ) ∈ H with the diagonal unitary matrix diag(t t t r r r t 1 ,...,t t t r r r t N ), where t t t r r r = t r 1 1 ⋯t r N N is the character of H corresponding to the vector r r r (cf. [20,31]).\\nWork of Sjamaar [47], which is a culmination of a series of results by Kempf and Ness [32], Guillemin and Sternberg [25], Kirwan [33], and Ness [45] relating symplectic reductions and Geometric Invariant Theory (GIT) quotients [42], has the following consequence in our case: where the line bundle L r r r on the Grassmannian is linearized to correspond to the action of the algebraic torus H = {(t 1 ,... ,t N ) ∈ (C * ) N ∶ ∏ti = 1} on C N given by identifying t t t = (t 1 ,...,t N ) ∈ H with the diagonal unitary matrix diag(t t t r r r t 1 ,...,t t t r r r t N ), where t t t r r r = t r 1 1 ⋯t r N N is the character of H corresponding to the vector r r r (cf. [20,31]).\\nis a frame representing a point [F] ∈ Gr d (C N ), then the determinants of the d × d minors of F are precisely the Plücker coordinates of [F], and in general the Plücker coordinates generate the homogeneous coordinate ring of Gr d (C N ). By definition, the full-spark frames are precisely those for which none of the Plücker coordinates vanish.\\nis a frame representing a point [F] ∈ Gr d (C N ), then the determinants of the d × d minors of F are precisely the Plücker coordinates of [F], and in general the Plücker coordinates generate the homogeneous coordinate ring of Gr d (C N ). By definition, the full-spark frames are precisely those for which none of the Plücker coordinates vanish.\\nIn turn, the homogeneous coordinate ring R of the GIT quotient Gr d (C N ) L r r r H consists of the H-invariant homogeneous coordinates on Gr d (C N ). It is known [31,Lemma 4.5] that R is spanned by monomials in the Plücker coordinates so that the total number of Plücker coordinates involving the ith column is kr i for some integer k independent of i.\\nIn turn, the homogeneous coordinate ring R of the GIT quotient Gr d (C N ) L r r r H consists of the H-invariant homogeneous coordinates on Gr d (C N ). It is known [31,Lemma 4.5] that R is spanned by monomials in the Plücker coordinates so that the total number of Plücker coordinates involving the ith column is kr i for some integer k independent of i.\\nIn our case, all r i = d, so taking the product of all the Plücker coordinates and raising it to the dth power gives a homogeneous coordinate on Gr d (C N ) L r r r H whose vanishing set is exactly the collection of (equivalence classes of) spark-deficient frames in the space of (scaled) unit-norm tight frames. Since this is the vanishing set of a homogeneous coordinate, it is a subvariety, and hence its complement-the collection of full-spark frames-is open in the Zariski topology, and in particular either empty or dense. Since there are full-spark frames in each space of unit-norm tight frames (for example, the first d rows of a scaled N × N discrete Fourier transform matrix [2]), the collection of fullspark frames cannot be empty, so it must be dense.\\nIn our case, all r i = d, so taking the product of all the Plücker coordinates and raising it to the dth power gives a homogeneous coordinate on Gr d (C N ) L r r r H whose vanishing set is exactly the collection of (equivalence classes of) spark-deficient frames in the space of (scaled) unit-norm tight frames. Since this is the vanishing set of a homogeneous coordinate, it is a subvariety, and hence its complement-the collection of full-spark frames-is open in the Zariski topology, and in particular either empty or dense. Since there are full-spark frames in each space of unit-norm tight frames (for example, the first d rows of a scaled N × N discrete Fourier transform matrix [2]), the collection of fullspark frames cannot be empty, so it must be dense.\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"Given a 2n-dimensional toric symplectic manifold M with moment polytope P, one can often find action-angle coordinates on M which take the form of a map α ∶ int(P) ×U(1) n → M which inverts the momentum map Φ ∶ M → P in the sense that Φ(α(p,t)) = p. In this case, Theorem 3.2 can be extended slightly to show that the image of α is a full-measure subset of M and that the map α is measure-preserving. Sampling P ×U(1) n with respect to the product of Lebesgue measure on P and the standard product measure on U(1) n and pushing forward by α gives a uniformly random sample from the symplectic measure on M (see, for example, the discussion in [10]).\\nGiven a 2n-dimensional toric symplectic manifold M with moment polytope P, one can often find action-angle coordinates on M which take the form of a map α ∶ int(P) ×U(1) n → M which inverts the momentum map Φ ∶ M → P in the sense that Φ(α(p,t)) = p. In this case, Theorem 3.2 can be extended slightly to show that the image of α is a full-measure subset of M and that the map α is measure-preserving. Sampling P ×U(1) n with respect to the product of Lebesgue measure on P and the standard product measure on U(1) n and pushing forward by α gives a uniformly random sample from the symplectic measure on M (see, for example, the discussion in [10]).\\nIn our setting, this means that coupling explicit action-angle coordinates on F d,N λ λ λ (r r r) (U(d)×G) with an algorithm for sampling P d,N λ λ λ (r r r) would give an algorithm for sampling random frames in F d,N λ λ λ (r r r). In particular, by Corollary 1.2, such an algorithm would provide endless quantities of full spark FUNTFs.\\nIn our setting, this means that coupling explicit action-angle coordinates on F d,N λ λ λ (r r r) (U(d)×G) with an algorithm for sampling P d,N λ λ λ (r r r) would give an algorithm for sampling random frames in F d,N λ λ λ (r r r). In particular, by Corollary 1.2, such an algorithm would provide endless quantities of full spark FUNTFs.\\nIt is natural to ask whether the analog of Theorem 1.1 holds for real and for quaternionic frames. Symplectic geometry is not the right tool in either case, but it is very plausible that the algebraic proof sketched in Section 4.2 could be adapted to the real case to show that full spark frames are dense in real frame spaces. In a different direction, the perspective based on isotropy orbits and isoparametric submanifolds introduced in [44] seems like the most promising way to understand the measures on real and quaternionic frame spaces.\\nIt is natural to ask whether the analog of Theorem 1.1 holds for real and for quaternionic frames. Symplectic geometry is not the right tool in either case, but it is very plausible that the algebraic proof sketched in Section 4.2 could be adapted to the real case to show that full spark frames are dense in real frame spaces. In a different direction, the perspective based on isotropy orbits and isoparametric submanifolds introduced in [44] seems like the most promising way to understand the measures on real and quaternionic frame spaces.\\nFinally, the spaces F d,N λ λ λ (r r r) (U(d) × G) are examples of weight varieties [35], and Goldin [22] has determined the rational cohomology ring of certain weight varieties, including the quotient F d,N (λ ,...,λ ) (r r r) (U(d) × G) of the space of λ -tight frames with fixed frame vector norms whenever it is a manifold. In particular, this determines the rational cohomology ring of the (U(d) × G)-quotient of FUNTF space when N and d are relatively prime. What about in the non-manifold case or for more general frame spectra?\\nFinally, the spaces F d,N λ λ λ (r r r) (U(d) × G) are examples of weight varieties [35], and Goldin [22] has determined the rational cohomology ring of certain weight varieties, including the quotient F d,N (λ ,...,λ ) (r r r) (U(d) × G) of the space of λ -tight frames with fixed frame vector norms whenever it is a manifold. In particular, this determines the rational cohomology ring of the (U(d) × G)-quotient of FUNTF space when N and d are relatively prime. What about in the non-manifold case or for more general frame spectra?\"\n },\n {\n \"title\": \"Theorem 2 . 2 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2 . 2 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2 . 5 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2 . 5 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 2 . 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 2 . 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"and 0 =\",\n \"content\": \"\"\n },\n {\n \"title\": \"and 0 =\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 3. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 3. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3 . 10 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3 . 10 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 3 . 20 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 3 . 20 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4. 1 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Marsden-Weinstein-Meyer Theorem for Regular Values","content":"Theorem 2 . 2 (\n22Marsden-Weinstein-Meyer Theorem for Regular Values"},{"id":"","caption":"Marsden-Weinstein-Meyer Theorem for Regular Values","content":"Theorem 2 . 2 (\n22Marsden-Weinstein-Meyer Theorem for Regular Values"},{"id":"","caption":"Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary.","content":"Theorem 2 . 5 (\n25Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary."},{"id":"","caption":"Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary.","content":"Theorem 2 . 5 (\n25Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary."},{"id":"","caption":"Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.","content":"Corollary 2 . 6 .\n26Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense."},{"id":"","caption":"Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.","content":"Corollary 2 . 6 .\n26Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense."},{"id":"","caption":"to H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U","content":"to H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U"},{"id":"","caption":"to H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U","content":"to H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U"},{"id":"","caption":"there exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces","content":"there exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces"},{"id":"","caption":"there exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces","content":"there exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces"},{"id":"","caption":"2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p.","content":"2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p."},{"id":"","caption":"2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p.","content":"2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p."},{"id":"","caption":"⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2","content":"and 0 =\n0⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2"},{"id":"","caption":"⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2","content":"and 0 =\n0⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2"},{"id":"","caption":"Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N.","content":"Proposition 3. 7 .\n7Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N."},{"id":"","caption":"Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N.","content":"Proposition 3. 7 .\n7Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N."},{"id":"","caption":"The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N .","content":"Corollary 3. 8 .\n8The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N ."},{"id":"","caption":"The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N .","content":"Corollary 3. 8 .\n8The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N ."},{"id":"","caption":"The circle action φ k j preserves the frame operator.","content":"Proposition 3 . 10 .\n310The circle action φ k j preserves the frame operator."},{"id":"","caption":"The circle action φ k j preserves the frame operator.","content":"Proposition 3 . 10 .\n310The circle action φ k j preserves the frame operator."},{"id":"","caption":"The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold.","content":"Theorem 3 . 20 .\n320The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold."},{"id":"","caption":"The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold.","content":"Theorem 3 . 20 .\n320The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold."},{"id":"","caption":"a new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields","content":"a new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields"},{"id":"","caption":"a new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields","content":"a new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields"},{"id":"","caption":"A frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U.","content":"A frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U."},{"id":"","caption":"A frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U.","content":"A frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U."},{"id":"","caption":"For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,","content":"Theorem 4. 1 .\n1For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,"},{"id":"","caption":"For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,","content":"Theorem 4. 1 .\n1For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,"},{"id":"","caption":"consists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N","content":"consists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N..,1) is strongly λ λ λ = ( N \nd ,... , N \nd )-admissible whenever N > d and since F d,d \n(1,...,1) (1,...,1) \nN \n\nd ,..., N \n\nd \n\n(1,...,1) of \n\nunit-norm tight frames (FUNTFs) in C d . \n\nN \n\nd ,..., N \n\nd \n\n(1,...,1)."},{"id":"","caption":"consists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N","content":"consists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N..,1) is strongly λ λ λ = ( N \nd ,... , N \nd )-admissible whenever N > d and since F d,d \n(1,...,1) (1,...,1) \nN \n\nd ,..., N \n\nd \n\n(1,...,1) of \n\nunit-norm tight frames (FUNTFs) in C d . \n\nN \n\nd ,..., N \n\nd \n\n(1,...,1)."},{"id":"","caption":"r i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16.","content":"r i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16."},{"id":"","caption":"r i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16.","content":"r i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16."},{"id":"","caption":"follows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.","content":"follows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.Remark 3."},{"id":"","caption":"follows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.","content":"follows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.Remark 3."},{"id":"","caption":"Going diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator","content":"Going diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator,3 \n...μ N−1,d \n\nµ N−2,1μN−2,2μN−2,3 \n...μ N−2,d \n\n. . . \n. . . \n. . . \n. . . \n. . . \n\nµ N−k,1μN−k,2μN−k,3 \n...μ N−k,d"},{"id":"","caption":"Going diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator","content":"Going diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator,3 \n...μ N−1,d \n\nµ N−2,1μN−2,2μN−2,3 \n...μ N−2,d \n\n. . . \n. . . \n. . . \n. . . \n. . . \n\nµ N−k,1μN−k,2μN−k,3 \n...μ N−k,d"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Marsden-Weinstein-Meyer Theorem for Regular Values\",\n \"content\": \"Theorem 2 . 2 (\\n22Marsden-Weinstein-Meyer Theorem for Regular Values\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Marsden-Weinstein-Meyer Theorem for Regular Values\",\n \"content\": \"Theorem 2 . 2 (\\n22Marsden-Weinstein-Meyer Theorem for Regular Values\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary.\",\n \"content\": \"Theorem 2 . 5 (\\n25Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary.\",\n \"content\": \"Theorem 2 . 5 (\\n25Sjamaar-Lerman[48]). Let M, ω, G, Φ, and χ be as above.Then each connected component of the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.Kirwan's proof that level sets of proper momentum maps are connected-this follows from [34, Remark 3.1], which handles the case where the domain is compact, together with the discussion surrounding [33, Remark 9.1], which applies to proper moment maps-yields the following immediate corollary.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.\",\n \"content\": \"Corollary 2 . 6 .\\n26Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.\",\n \"content\": \"Corollary 2 . 6 .\\n26Let M, ω, G, Φ, and χ be as above. If Φ is proper (for example, if G is compact), then the symplectic reduction M O χ G has a unique open stratum which is a manifold and is connected and dense.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"to H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U\",\n \"content\": \"to H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U\"\n },\n {\n \"id\": \"\",\n \"caption\": \"to H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U\",\n \"content\": \"to H(N) is equivariant with respect to the right multiplication action of U(N), A ⋅ [F] = [FA * ], and the Ad-action of U(N) onH(N). This observation, together with Proposition 2.11 and the first paragraph of the proof of Proposition 2.14, shows that the action of U\"\n },\n {\n \"id\": \"\",\n \"caption\": \"there exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces\",\n \"content\": \"there exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces\"\n },\n {\n \"id\": \"\",\n \"caption\": \"there exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces\",\n \"content\": \"there exists a partition P 1 ,... ,P m of {1,...,N} and pairwise orthogonal subspaces\"\n },\n {\n \"id\": \"\",\n \"caption\": \"2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p.\",\n \"content\": \"2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p.\",\n \"content\": \"2 and 2.3 (see, e.g., [40, Proof of Proposition 5.4.13] or [9, §23.2.1]) is the following characterization of regular points of momentum maps: Proposition 2.20. Let H be a Lie group and let (M,ω) be a symplectic manifold admitting a Hamiltonian H-action with momentum map Ψ ∶ M → h * . Then p ∈ M is a regular point of Ψ if and only if the action of H is locally free at p.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2\",\n \"content\": \"and 0 =\\n0⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2\"\n },\n {\n \"id\": \"\",\n \"caption\": \"⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2\",\n \"content\": \"and 0 =\\n0⟪ −2XX * ,(2 x j 2 ) N−1 j=1 , I d 1 0 0 0 ,(1,...,1,0,...,0) ⟫ = 2\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N.\",\n \"content\": \"Proposition 3. 7 .\\n7Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N.\",\n \"content\": \"Proposition 3. 7 .\\n7Let F = f 1 ⋯ f N ∈ F d,N . The circle action φ k j on the vectors of F is U(d)-equivariant. That is, for all A ∈ U(d), φ k j (t,A f i ) = Aφ k j (t, f i )for all t ∈ [0,2π) and all i = 1,...,N.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N .\",\n \"content\": \"Corollary 3. 8 .\\n8The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N .\",\n \"content\": \"Corollary 3. 8 .\\n8The circle action φ k j on F d,N commutes with the U(d) action. That is, for all A ∈ U(d), φ k j (t,AF) = Aφ k j (t,F) for all t ∈ [0,2π) and all F ∈ F d,N .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The circle action φ k j preserves the frame operator.\",\n \"content\": \"Proposition 3 . 10 .\\n310The circle action φ k j preserves the frame operator.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The circle action φ k j preserves the frame operator.\",\n \"content\": \"Proposition 3 . 10 .\\n310The circle action φ k j preserves the frame operator.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold.\",\n \"content\": \"Theorem 3 . 20 .\\n320The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold.\",\n \"content\": \"Theorem 3 . 20 .\\n320The space of frames F d,N λ λ λ (r r r) with prescribed spectrum and norms contains a dense open subset which is a (U(d) × G)-bundle over a toric symplectic manifold.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields\",\n \"content\": \"a new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields\",\n \"content\": \"a new frameF = f N ⋯ f 1 by reversing the order of the columns of F. Ifμ ℓ, j are the eigensteps ofF, ℓ = 1,...,N − 1. In particular, letting ℓ = N − k yields\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U.\",\n \"content\": \"A frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U.\",\n \"content\": \"A frame F = f 1 ⋯ f N ∈Ũ has linearly independent columns f 1 ,... , f d if and only if the partial frame operatorS d = f 1 f * 1 + ⋯ + f d f * d isfull rank, which holds if and only if the smallest eigenvalue of S d is positive. Observe that these conditions are well-defined on the (U(d) × G)-equivalence class of F, denoted [F]. Moreover, the quotient mapŨ → U ⊂ F d,N λ λ λ (r r r) (U(d) × G) is a Riemannian submersion (with respect to the real parts of the respective Kähler structures) onto its (dense, open) image. It is therefore sufficient to establish the subclaim for the set U.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,\",\n \"content\": \"Theorem 4. 1 .\\n1For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,\",\n \"content\": \"Theorem 4. 1 .\\n1For arbitrary r r r consisting of positive integers, Gr d (C N ) r r r G is isomorphic, as a complex projective variety, to the GIT quotient Gr d (C N ) L r r r H,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"consists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N\",\n \"content\": \"consists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N..,1) is strongly λ λ λ = ( N \\nd ,... , N \\nd )-admissible whenever N > d and since F d,d \\n(1,...,1) (1,...,1) \\nN \\n\\nd ,..., N \\n\\nd \\n\\n(1,...,1) of \\n\\nunit-norm tight frames (FUNTFs) in C d . \\n\\nN \\n\\nd ,..., N \\n\\nd \\n\\n(1,...,1).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"consists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N\",\n \"content\": \"consists of orthonormal bases, which are certainly full spark, we have: Corollary 1.2. For any N ≥ d ≥ 1, the full-spark frames have full measure inside the space F d,N This result essentially recovers the complex case of [8, Theorem 1.6], where the authors showed that full spark frames form an open dense subset of F d,N N d ,..., N d (1,...,1). Our corollary slightly sharpens this result, in that we are able to refer precisely to the canonical probability measure on F d,N..,1) is strongly λ λ λ = ( N \\nd ,... , N \\nd )-admissible whenever N > d and since F d,d \\n(1,...,1) (1,...,1) \\nN \\n\\nd ,..., N \\n\\nd \\n\\n(1,...,1) of \\n\\nunit-norm tight frames (FUNTFs) in C d . \\n\\nN \\n\\nd ,..., N \\n\\nd \\n\\n(1,...,1).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"r i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16.\",\n \"content\": \"r i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"r i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16.\",\n \"content\": \"r i = 1 for all i and λ j = N d for all j, so k 1 = d and the above dimension simplifies to dim P d,N N d ,..., N d (1,...,1) = (d − 1)(N − d − 1), so we see that a result of Flaschka-Millson [19, Proposition 6.2.2] and Haga-Pegel [27, Theorem 3.2] is a special case of Proposition 3.16.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"follows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.\",\n \"content\": \"follows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.Remark 3.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"follows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.\",\n \"content\": \"follows by Proposition 3.7, since the exponential is unitary.Corollary 3.18. If r r r is strongly λ λ λ -admissible, there is a Hamiltonian action of the torus U(1) d(λ λ λ ,r r r) on an open, dense subset of F d,N λ λ λ (r r r) (U(d) × G). The associated moment polytope is the eigenstep polytope P d,N λ λ λ (r r r).Proof. We know from Corollary 2.6 that F d,N λ λ λ (r r r) (U(d) × G) contains an open, dense symplectic manifold which, by Corollary 2.22, consists of orbits of non-orthodecomposable frames. This symplectic manifold contains an open, dense subset of frames whose partial frame operators each have as many distinct eigenvalues as possible (given the eigenstep constraints). Choosing d(λ λ λ ,r r r) free variables according to the dimension counting argument of Section 3.2.3 yields a Hamiltonian torus action on this open set.Remark 3.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Going diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator\",\n \"content\": \"Going diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator,3 \\n...μ N−1,d \\n\\nµ N−2,1μN−2,2μN−2,3 \\n...μ N−2,d \\n\\n. . . \\n. . . \\n. . . \\n. . . \\n. . . \\n\\nµ N−k,1μN−k,2μN−k,3 \\n...μ N−k,d\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Going diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator\",\n \"content\": \"Going diagonally up and to the right, we see thatμ N−k,i ≥ λ k+i for all i = 1,...,d − k. The only way this can be reconciled with(28) is ifμN−k,1 = λ k+1 , ... ,μ N−k,d−k = λ d , and henceμ N−k,d−k+1 = ⋅⋅⋅ =μ N−k,d = 0.But thenμ N−k,d ≥μ d,d , so it follows thatμ d,d = 0. Since this is an eigenvalue of the partial frame operator,3 \\n...μ N−1,d \\n\\nµ N−2,1μN−2,2μN−2,3 \\n...μ N−2,d \\n\\n. . . \\n. . . \\n. . . \\n. . . \\n. . . \\n\\nµ N−k,1μN−k,2μN−k,3 \\n...μ N−k,d\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics. 5423Per AlexanderssonPer Alexandersson. Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics, 54:1-20, 2016. ↑23","doi":""},{"text":"Full spark frames. Boris Alexeev, Jameson Cahill, Dustin G Mixon, Journal of Fourier Analysis and Applications. 18627Boris Alexeev, Jameson Cahill, and Dustin G. Mixon. Full spark frames. Journal of Fourier Analysis and Applications, 18(6):1167-1194, 2012. ↑3, ↑27","doi":""},{"text":"Symmetry and bifurcations of momentum mappings. Judith M Arms, Jerrold E Marsden, Vincent Moncrief, Communications in Mathematical Physics. 78415Judith M. Arms, Jerrold E. Marsden, and Vincent Moncrief. Symmetry and bifurcations of momentum mappings. Communications in Mathematical Physics, 78(4):455-478, 1981. ↑13, ↑15","doi":""},{"text":"Convexity and commuting Hamiltonians. Michael Francis, Atiyah , Bulletin of the London Mathematical Society. 14118Michael Francis Atiyah. Convexity and commuting Hamiltonians. Bulletin of the London Mathematical Society, 14(1):1-15, 1982. ↑18","doi":""},{"text":"Michèle Audin, Torus Actions on Symplectic Manifolds. Basel9310second revised editionMichèle Audin. Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Birkhäuser Verlag, Basel, second revised edition, 2004. ↑5, ↑7, ↑9, ↑10","doi":""},{"text":"Matrix Analysis. Rajendra Bhatia, Graduate Texts in Mathematics. 16922SpringerRajendra Bhatia. Matrix Analysis, volume 169 of Graduate Texts in Mathematics. Springer, New York, NY, 1997. ↑22","doi":""},{"text":"Constructing finite frames of a given spectrum and set of lengths. Jameson Cahill, Matthew Fickus, Dustin G Mixon, Miriam J Poteet, Nate Strawn, Applied and Computational Harmonic Analysis. 35124Jameson Cahill, Matthew Fickus, Dustin G. Mixon, Miriam J. Poteet, and Nate Strawn. Constructing finite frames of a given spectrum and set of lengths. Applied and Computational Harmonic Analysis, 35(1):52-73, 2013. ↑2, ↑18, ↑22, ↑24","doi":""},{"text":"Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. Jameson Cahill, Dustin G Mixon, Nate Strawn, SIAM Journal on Applied Algebra and Geometry. 1124↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22Jameson Cahill, Dustin G. Mixon, and Nate Strawn. Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. SIAM Journal on Applied Algebra and Geometry, 1(1):38-72, 2017. ↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22, ↑24","doi":""},{"text":"Lectures on Symplectic Geometry. Ana Cannas Da Silva, Lecture Notes in Mathematics. 176418Springer-VerlagAna Cannas da Silva. Lectures on Symplectic Geometry, volume 1764 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, Heidelberg, 2001. ↑10, ↑12, ↑18","doi":""},{"text":"The symplectic geometry of closed equilateral random walks in 3-space. Jason Cantarella, Clayton Shonkwiler, The Annals of Applied Probability. 26127Jason Cantarella and Clayton Shonkwiler. The symplectic geometry of closed equilateral random walks in 3- space. The Annals of Applied Probability, 26(1):549-596, 2016. ↑27","doi":""},{"text":"Equal-norm tight frames with erasures. G Peter, Jelena Casazza, Kovačević, Advances in Computational Mathematics. 182-41Peter G. Casazza and Jelena Kovačević. Equal-norm tight frames with erasures. Advances in Computational Mathematics, 18(2-4):387-430, 2003. ↑1","doi":""},{"text":"Existence and construction of finite frames with a given frame operator. G Peter, Manuel T Casazza, Leon, International Journal of Pure and Applied Mathematics. 63225Peter G. Casazza and Manuel T. Leon. Existence and construction of finite frames with a given frame operator. International Journal of Pure and Applied Mathematics, 63(2):149-157, 2010. ↑3, ↑25","doi":""},{"text":"Toric Varieties. David A Cox, John B Little, Henry K Schenck, Graduate Studies in Mathematics. 12418American Mathematical SocietyDavid A. Cox, John B. Little, and Henry K. Schenck. Toric Varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011. ↑18","doi":""},{"text":"Vertices of Gelfand-Tsetlin polytopes. A De Jesús, Tyrrell B Loera, Mcallister, Discrete & Computational Geometry. 32423Jesús A. De Loera and Tyrrell B. McAllister. Vertices of Gelfand-Tsetlin polytopes. Discrete & Computational Geometry, 32(4):459-470, 2004. ↑23","doi":""},{"text":"Hamiltoniens périodiques et images convexes de l'application moment. Thomas Delzant, 11618Bulletin de la Société Mathématique de FranceThomas Delzant. Hamiltoniens périodiques et images convexes de l'application moment. Bulletin de la Société Mathématique de France, 116(3):315-339, 1988. ↑18","doi":""},{"text":"Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. L David, Michael Donoho, Elad, Proceedings of the National Academy of Sciences of the United States of America. the National Academy of Sciences of the United States of America1003David L. Donoho and Michael Elad. Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. Proceedings of the National Academy of Sciences of the United States of America, 100(5):2197- 2202, 2003. ↑3","doi":""},{"text":"On the variation in the cohomology of the symplectic form of the reduced phase space. J Johannes, Gerrit J Duistermaat, Heckman, Inventiones Mathematicae. 69218Johannes J. Duistermaat and Gerrit J. Heckman. On the variation in the cohomology of the symplectic form of the reduced phase space. Inventiones Mathematicae, 69(2):259-268, 1982. ↑18","doi":""},{"text":"Manifold structure of spaces of spherical tight frames. Ken Dykema, Nate Strawn, International Journal of Pure and Applied Mathematics. 28212Ken Dykema and Nate Strawn. Manifold structure of spaces of spherical tight frames. International Journal of Pure and Applied Mathematics, 28(2):217-256, 2006. ↑1, ↑2, ↑4, ↑12","doi":""},{"text":"Bending flows for sums of rank one matrices. Hermann Flaschka, John J Millson, Canadian Journal of Mathematics. 57124Hermann Flaschka and John J. Millson. Bending flows for sums of rank one matrices. Canadian Journal of Mathematics, 57(1):114-158, 2005. ↑2, ↑3, ↑19, ↑24","doi":""},{"text":"Toric degenerations of weight varieties and applications. Philip Foth, Yi Hu, Travaux mathématiques. 1627Philip Foth and Yi Hu. Toric degenerations of weight varieties and applications. Travaux mathématiques, 16:87- 105, 2005. ↑27","doi":""},{"text":"Finite-dimensional representations of the group of unimodular matrices. M Israïl, Michael L Tsetlin, Doklady Akademii Nauk SSSR (N.S.). 7123Israïl M. Gel'fand and Michael L. Tsetlin. Finite-dimensional representations of the group of unimodular matri- ces. Doklady Akademii Nauk SSSR (N.S.), 71:825-828, 1950. ↑23","doi":""},{"text":"The cohomology ring of weight varieties and polygon spaces. Rebecca F Goldin, Advances in Mathematics. 160227Rebecca F. Goldin. The cohomology ring of weight varieties and polygon spaces. Advances in Mathematics, 160(2):175-204, 2001. ↑27","doi":""},{"text":"Quantized frame expansions with erasures. K Vivek, Jelena Goyal, Jonathan A Kovačević, Kelner, Applied and Computational Harmonic Analysis. 1031Vivek K. Goyal, Jelena Kovačević, and Jonathan A. Kelner. Quantized frame expansions with erasures. Applied and Computational Harmonic Analysis, 10(3):203-233, 2001. ↑1","doi":""},{"text":"Convexity properties of the moment mapping. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67318Victor Guillemin and Shlomo Sternberg. Convexity properties of the moment mapping. Inventiones Mathemati- cae, 67(3):491-513, 1982. ↑18","doi":""},{"text":"Geometric quantization and multiplicities of group representations. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67327Victor Guillemin and Shlomo Sternberg. Geometric quantization and multiplicities of group representations. Inventiones Mathematicae, 67(3):515-538, 1982. ↑27","doi":""},{"text":"The Gelfand-Cetlin system and quantization of the complex flag manifolds. Victor Guillemin, Shlomo Sternberg, Journal of Functional Analysis. 52123Victor Guillemin and Shlomo Sternberg. The Gelfand-Cetlin system and quantization of the complex flag man- ifolds. Journal of Functional Analysis, 52(1):106-128, 1983. ↑23","doi":""},{"text":"Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computational Geometry. Tim Haga, Christoph Pegel, 5624Tim Haga and Christoph Pegel. Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computa- tional Geometry, 56(3):727-742, 2016. ↑2, ↑22, ↑24","doi":""},{"text":"Hyperkähler metrics and supersymmetry. Nigel James Hitchin, Anders Karlhede, Ulf Lindström, Martin Roček, Communications in Mathematical Physics. 108418Nigel James Hitchin, Anders Karlhede, Ulf Lindström, and Martin Roček. Hyperkähler metrics and supersym- metry. Communications in Mathematical Physics, 108(4):535-589, 1987. ↑18","doi":""},{"text":"Optimal frames for erasures. B Roderick, Vern I Holmes, Paulsen, Linear Algebra and its Applications. 3771Roderick B. Holmes and Vern I. Paulsen. Optimal frames for erasures. Linear Algebra and its Applications, 377:31-51, 2004. ↑1","doi":""},{"text":"Doubly stochastic matrices and the diagonal of a rotation matrix. Alfred Horn, American Journal of Mathematics. 76325Alfred Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathematics, 76(3):620-630, 1954. ↑11, ↑25","doi":""},{"text":"The Chevalley involution and a duality of weight varieties. J Benjamin, John J Howard, Millson, The Asian Journal of Mathematics. 8427Benjamin J. Howard and John J. Millson. The Chevalley involution and a duality of weight varieties. The Asian Journal of Mathematics, 8(4):685-732, 2004. ↑27","doi":""},{"text":"The length of vectors in representation spaces. George Kempf, Linda Ness, Lecture Notes in Mathematics. Knud Lønsted73227Springer-VerlagAlgebraic GeometryGeorge Kempf and Linda Ness. The length of vectors in representation spaces. In Knud Lønsted, editor, Alge- braic Geometry, volume 732 of Lecture Notes in Mathematics, pages 233-243. Springer-Verlag, Berlin, 1979. ↑27","doi":""},{"text":"Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Frances Kirwan, Princeton University Press627Princeton, NJFrances Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1984. ↑6, ↑27","doi":""},{"text":"Convexity properties of the moment mapping. Frances Kirwan, III. Inventiones Mathematicae. 7736Frances Kirwan. Convexity properties of the moment mapping. III. Inventiones Mathematicae, 77(3):547-552, 1984. ↑6","doi":""},{"text":"Weight Varieties. Allen Knutson, 2627Massachusetts Institute of TechnologyPhD thesisAllen Knutson. Weight Varieties. PhD thesis, Massachusetts Institute of Technology, 1996. ↑26, ↑27","doi":""},{"text":"The symplectic and algebraic geometry of Horn's problem. Allen Knutson, Linear Algebra and its Applications. 3191-310Allen Knutson. The symplectic and algebraic geometry of Horn's problem. Linear Algebra and its Applications, 319(1-3):61-81, 2000. ↑10","doi":""},{"text":"Introduction to Smooth Manifolds. John M Lee, Graduate Texts in Mathematics. 21813Springersecond editionJohn M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, NY, second edition, 2013. ↑9, ↑13","doi":""},{"text":"Reduction of symplectic manifolds with symmetry. Jerrold E Marsden, Alan Weinstein, Reports on Mathematical Physics. 516Jerrold E. Marsden and Alan Weinstein. Reduction of symplectic manifolds with symmetry. Reports on Mathe- matical Physics, 5(1):121-130, 1974. ↑5, ↑6","doi":""},{"text":"Inequalities: Theory of Majorization and Its Applications. Albert W Marshall, Ingram Olkin, Barry C Arnold, Springer Series in Statistics. 3Springersecond editionAlbert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: Theory of Majorization and Its Applica- tions. Springer Series in Statistics. Springer, New York, NY, second edition, 2011. ↑3","doi":""},{"text":"Introduction to Symplectic Topology. Dusa Mcduff, Dietmar Salamon, Oxford University Press1215third edition, 2017. ↑4, ↑5, ↑10Dusa McDuff and Dietmar Salamon. Introduction to Symplectic Topology. Oxford University Press, third edition, 2017. ↑4, ↑5, ↑10, ↑12, ↑15","doi":""},{"text":"Symmetries and integrals in mechanics. Kenneth R Meyer, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia. Mauricio M. PeixotoSalvador, Brasil; New York, NYAcademic Press56Kenneth R. Meyer. Symmetries and integrals in mechanics. In Mauricio M. Peixoto, editor, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26-August 14, 1971, pages 259-272. Academic Press, New York, NY, 1973. ↑5, ↑6","doi":""},{"text":"Geometric Invariant Theory. David Mumford, John Fogarty, Frances Kirwan, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3427Springer-VerlagDavid Mumford, John Fogarty, and Frances Kirwan. Geometric Invariant Theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1994. ↑24, ↑27","doi":""},{"text":"Symplectic geometry and connectivity of spaces of frames. Tom Needham, Clayton Shonkwiler, Advances in Computational Mathematics. 47128Tom Needham and Clayton Shonkwiler. Symplectic geometry and connectivity of spaces of frames. Advances in Computational Mathematics, 47(1):5, 2021. ↑7, ↑8, ↑9, ↑28","doi":""},{"text":"Admissibility and frame homotopy for quaternionic frames. Tom Needham, Clayton Shonkwiler, Linear Algebra and its Applications. 64527Tom Needham and Clayton Shonkwiler. Admissibility and frame homotopy for quaternionic frames. Linear Algebra and its Applications, 645:237-255, 2022. ↑27","doi":""},{"text":"A stratification of the null cone via the moment map. Linda Ness, American Journal of Mathematics. 106627Linda Ness. A stratification of the null cone via the moment map. American Journal of Mathematics, 106(6):1281-1329, 1984. ↑27","doi":""},{"text":"Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Issai Schur, Sitzungsberichte der Berliner Mathematischen Gesellschaft. 2225Issai Schur. Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungs- berichte der Berliner Mathematischen Gesellschaft, 22:9-20, 1923. ↑11, ↑25","doi":""},{"text":"Holomorphic slices, symplectic reduction and multiplicities of representations. Reyer Sjamaar, The Annals of Mathematics, Second Series. 141127Reyer Sjamaar. Holomorphic slices, symplectic reduction and multiplicities of representations. The Annals of Mathematics, Second Series, 141(1):87-129, 1995. ↑27","doi":""},{"text":"Stratified symplectic spaces and reduction. Reyer Sjamaar, Eugene Lerman, The Annals of Mathematics, Second Series. 134217Reyer Sjamaar and Eugene Lerman. Stratified symplectic spaces and reduction. The Annals of Mathematics, Second Series, 134(2):375-422, 1991. ↑6, ↑11, ↑13, ↑17","doi":""},{"text":"Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Nate Strawn, Journal of Fourier Analysis and Applications. 17512Nate Strawn. Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Journal of Fourier Analysis and Applications, 17(5):821-853, 2010. ↑1, ↑4, ↑12","doi":""},{"text":"An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. F D Shayne, Waldron, Birkhäuser9New York, NYShayne F. D. Waldron. An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY, 2018. ↑9","doi":""},{"text":"Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics. 5423Per AlexanderssonPer Alexandersson. Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics, 54:1-20, 2016. ↑23","doi":""},{"text":"Full spark frames. Boris Alexeev, Jameson Cahill, Dustin G Mixon, Journal of Fourier Analysis and Applications. 18627Boris Alexeev, Jameson Cahill, and Dustin G. Mixon. Full spark frames. Journal of Fourier Analysis and Applications, 18(6):1167-1194, 2012. ↑3, ↑27","doi":""},{"text":"Symmetry and bifurcations of momentum mappings. Judith M Arms, Jerrold E Marsden, Vincent Moncrief, Communications in Mathematical Physics. 78415Judith M. Arms, Jerrold E. Marsden, and Vincent Moncrief. Symmetry and bifurcations of momentum mappings. Communications in Mathematical Physics, 78(4):455-478, 1981. ↑13, ↑15","doi":""},{"text":"Convexity and commuting Hamiltonians. Michael Francis, Atiyah , Bulletin of the London Mathematical Society. 14118Michael Francis Atiyah. Convexity and commuting Hamiltonians. Bulletin of the London Mathematical Society, 14(1):1-15, 1982. ↑18","doi":""},{"text":"Michèle Audin, Torus Actions on Symplectic Manifolds. Basel9310second revised editionMichèle Audin. Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Birkhäuser Verlag, Basel, second revised edition, 2004. ↑5, ↑7, ↑9, ↑10","doi":""},{"text":"Matrix Analysis. Rajendra Bhatia, Graduate Texts in Mathematics. 16922SpringerRajendra Bhatia. Matrix Analysis, volume 169 of Graduate Texts in Mathematics. Springer, New York, NY, 1997. ↑22","doi":""},{"text":"Constructing finite frames of a given spectrum and set of lengths. Jameson Cahill, Matthew Fickus, Dustin G Mixon, Miriam J Poteet, Nate Strawn, Applied and Computational Harmonic Analysis. 35124Jameson Cahill, Matthew Fickus, Dustin G. Mixon, Miriam J. Poteet, and Nate Strawn. Constructing finite frames of a given spectrum and set of lengths. Applied and Computational Harmonic Analysis, 35(1):52-73, 2013. ↑2, ↑18, ↑22, ↑24","doi":""},{"text":"Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. Jameson Cahill, Dustin G Mixon, Nate Strawn, SIAM Journal on Applied Algebra and Geometry. 1124↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22Jameson Cahill, Dustin G. Mixon, and Nate Strawn. Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. SIAM Journal on Applied Algebra and Geometry, 1(1):38-72, 2017. ↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22, ↑24","doi":""},{"text":"Lectures on Symplectic Geometry. Ana Cannas Da Silva, Lecture Notes in Mathematics. 176418Springer-VerlagAna Cannas da Silva. Lectures on Symplectic Geometry, volume 1764 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, Heidelberg, 2001. ↑10, ↑12, ↑18","doi":""},{"text":"The symplectic geometry of closed equilateral random walks in 3-space. Jason Cantarella, Clayton Shonkwiler, The Annals of Applied Probability. 26127Jason Cantarella and Clayton Shonkwiler. The symplectic geometry of closed equilateral random walks in 3- space. The Annals of Applied Probability, 26(1):549-596, 2016. ↑27","doi":""},{"text":"Equal-norm tight frames with erasures. G Peter, Jelena Casazza, Kovačević, Advances in Computational Mathematics. 182-41Peter G. Casazza and Jelena Kovačević. Equal-norm tight frames with erasures. Advances in Computational Mathematics, 18(2-4):387-430, 2003. ↑1","doi":""},{"text":"Existence and construction of finite frames with a given frame operator. G Peter, Manuel T Casazza, Leon, International Journal of Pure and Applied Mathematics. 63225Peter G. Casazza and Manuel T. Leon. Existence and construction of finite frames with a given frame operator. International Journal of Pure and Applied Mathematics, 63(2):149-157, 2010. ↑3, ↑25","doi":""},{"text":"Toric Varieties. David A Cox, John B Little, Henry K Schenck, Graduate Studies in Mathematics. 12418American Mathematical SocietyDavid A. Cox, John B. Little, and Henry K. Schenck. Toric Varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011. ↑18","doi":""},{"text":"Vertices of Gelfand-Tsetlin polytopes. A De Jesús, Tyrrell B Loera, Mcallister, Discrete & Computational Geometry. 32423Jesús A. De Loera and Tyrrell B. McAllister. Vertices of Gelfand-Tsetlin polytopes. Discrete & Computational Geometry, 32(4):459-470, 2004. ↑23","doi":""},{"text":"Hamiltoniens périodiques et images convexes de l'application moment. Thomas Delzant, 11618Bulletin de la Société Mathématique de FranceThomas Delzant. Hamiltoniens périodiques et images convexes de l'application moment. Bulletin de la Société Mathématique de France, 116(3):315-339, 1988. ↑18","doi":""},{"text":"Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. L David, Michael Donoho, Elad, Proceedings of the National Academy of Sciences of the United States of America. the National Academy of Sciences of the United States of America1003David L. Donoho and Michael Elad. Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. Proceedings of the National Academy of Sciences of the United States of America, 100(5):2197- 2202, 2003. ↑3","doi":""},{"text":"On the variation in the cohomology of the symplectic form of the reduced phase space. J Johannes, Gerrit J Duistermaat, Heckman, Inventiones Mathematicae. 69218Johannes J. Duistermaat and Gerrit J. Heckman. On the variation in the cohomology of the symplectic form of the reduced phase space. Inventiones Mathematicae, 69(2):259-268, 1982. ↑18","doi":""},{"text":"Manifold structure of spaces of spherical tight frames. Ken Dykema, Nate Strawn, International Journal of Pure and Applied Mathematics. 28212Ken Dykema and Nate Strawn. Manifold structure of spaces of spherical tight frames. International Journal of Pure and Applied Mathematics, 28(2):217-256, 2006. ↑1, ↑2, ↑4, ↑12","doi":""},{"text":"Bending flows for sums of rank one matrices. Hermann Flaschka, John J Millson, Canadian Journal of Mathematics. 57124Hermann Flaschka and John J. Millson. Bending flows for sums of rank one matrices. Canadian Journal of Mathematics, 57(1):114-158, 2005. ↑2, ↑3, ↑19, ↑24","doi":""},{"text":"Toric degenerations of weight varieties and applications. Philip Foth, Yi Hu, Travaux mathématiques. 1627Philip Foth and Yi Hu. Toric degenerations of weight varieties and applications. Travaux mathématiques, 16:87- 105, 2005. ↑27","doi":""},{"text":"Finite-dimensional representations of the group of unimodular matrices. M Israïl, Michael L Tsetlin, Doklady Akademii Nauk SSSR (N.S.). 7123Israïl M. Gel'fand and Michael L. Tsetlin. Finite-dimensional representations of the group of unimodular matri- ces. Doklady Akademii Nauk SSSR (N.S.), 71:825-828, 1950. ↑23","doi":""},{"text":"The cohomology ring of weight varieties and polygon spaces. Rebecca F Goldin, Advances in Mathematics. 160227Rebecca F. Goldin. The cohomology ring of weight varieties and polygon spaces. Advances in Mathematics, 160(2):175-204, 2001. ↑27","doi":""},{"text":"Quantized frame expansions with erasures. K Vivek, Jelena Goyal, Jonathan A Kovačević, Kelner, Applied and Computational Harmonic Analysis. 1031Vivek K. Goyal, Jelena Kovačević, and Jonathan A. Kelner. Quantized frame expansions with erasures. Applied and Computational Harmonic Analysis, 10(3):203-233, 2001. ↑1","doi":""},{"text":"Convexity properties of the moment mapping. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67318Victor Guillemin and Shlomo Sternberg. Convexity properties of the moment mapping. Inventiones Mathemati- cae, 67(3):491-513, 1982. ↑18","doi":""},{"text":"Geometric quantization and multiplicities of group representations. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67327Victor Guillemin and Shlomo Sternberg. Geometric quantization and multiplicities of group representations. Inventiones Mathematicae, 67(3):515-538, 1982. ↑27","doi":""},{"text":"The Gelfand-Cetlin system and quantization of the complex flag manifolds. Victor Guillemin, Shlomo Sternberg, Journal of Functional Analysis. 52123Victor Guillemin and Shlomo Sternberg. The Gelfand-Cetlin system and quantization of the complex flag man- ifolds. Journal of Functional Analysis, 52(1):106-128, 1983. ↑23","doi":""},{"text":"Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computational Geometry. Tim Haga, Christoph Pegel, 5624Tim Haga and Christoph Pegel. Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computa- tional Geometry, 56(3):727-742, 2016. ↑2, ↑22, ↑24","doi":""},{"text":"Hyperkähler metrics and supersymmetry. Nigel James Hitchin, Anders Karlhede, Ulf Lindström, Martin Roček, Communications in Mathematical Physics. 108418Nigel James Hitchin, Anders Karlhede, Ulf Lindström, and Martin Roček. Hyperkähler metrics and supersym- metry. Communications in Mathematical Physics, 108(4):535-589, 1987. ↑18","doi":""},{"text":"Optimal frames for erasures. B Roderick, Vern I Holmes, Paulsen, Linear Algebra and its Applications. 3771Roderick B. Holmes and Vern I. Paulsen. Optimal frames for erasures. Linear Algebra and its Applications, 377:31-51, 2004. ↑1","doi":""},{"text":"Doubly stochastic matrices and the diagonal of a rotation matrix. Alfred Horn, American Journal of Mathematics. 76325Alfred Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathematics, 76(3):620-630, 1954. ↑11, ↑25","doi":""},{"text":"The Chevalley involution and a duality of weight varieties. J Benjamin, John J Howard, Millson, The Asian Journal of Mathematics. 8427Benjamin J. Howard and John J. Millson. The Chevalley involution and a duality of weight varieties. The Asian Journal of Mathematics, 8(4):685-732, 2004. ↑27","doi":""},{"text":"The length of vectors in representation spaces. George Kempf, Linda Ness, Lecture Notes in Mathematics. Knud Lønsted73227Springer-VerlagAlgebraic GeometryGeorge Kempf and Linda Ness. The length of vectors in representation spaces. In Knud Lønsted, editor, Alge- braic Geometry, volume 732 of Lecture Notes in Mathematics, pages 233-243. Springer-Verlag, Berlin, 1979. ↑27","doi":""},{"text":"Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Frances Kirwan, Princeton University Press627Princeton, NJFrances Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1984. ↑6, ↑27","doi":""},{"text":"Convexity properties of the moment mapping. Frances Kirwan, III. Inventiones Mathematicae. 7736Frances Kirwan. Convexity properties of the moment mapping. III. Inventiones Mathematicae, 77(3):547-552, 1984. ↑6","doi":""},{"text":"Weight Varieties. Allen Knutson, 2627Massachusetts Institute of TechnologyPhD thesisAllen Knutson. Weight Varieties. PhD thesis, Massachusetts Institute of Technology, 1996. ↑26, ↑27","doi":""},{"text":"The symplectic and algebraic geometry of Horn's problem. Allen Knutson, Linear Algebra and its Applications. 3191-310Allen Knutson. The symplectic and algebraic geometry of Horn's problem. Linear Algebra and its Applications, 319(1-3):61-81, 2000. ↑10","doi":""},{"text":"Introduction to Smooth Manifolds. John M Lee, Graduate Texts in Mathematics. 21813Springersecond editionJohn M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, NY, second edition, 2013. ↑9, ↑13","doi":""},{"text":"Reduction of symplectic manifolds with symmetry. Jerrold E Marsden, Alan Weinstein, Reports on Mathematical Physics. 516Jerrold E. Marsden and Alan Weinstein. Reduction of symplectic manifolds with symmetry. Reports on Mathe- matical Physics, 5(1):121-130, 1974. ↑5, ↑6","doi":""},{"text":"Inequalities: Theory of Majorization and Its Applications. Albert W Marshall, Ingram Olkin, Barry C Arnold, Springer Series in Statistics. 3Springersecond editionAlbert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: Theory of Majorization and Its Applica- tions. Springer Series in Statistics. Springer, New York, NY, second edition, 2011. ↑3","doi":""},{"text":"Introduction to Symplectic Topology. Dusa Mcduff, Dietmar Salamon, Oxford University Press1215third edition, 2017. ↑4, ↑5, ↑10Dusa McDuff and Dietmar Salamon. Introduction to Symplectic Topology. Oxford University Press, third edition, 2017. ↑4, ↑5, ↑10, ↑12, ↑15","doi":""},{"text":"Symmetries and integrals in mechanics. Kenneth R Meyer, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia. Mauricio M. PeixotoSalvador, Brasil; New York, NYAcademic Press56Kenneth R. Meyer. Symmetries and integrals in mechanics. In Mauricio M. Peixoto, editor, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26-August 14, 1971, pages 259-272. Academic Press, New York, NY, 1973. ↑5, ↑6","doi":""},{"text":"Geometric Invariant Theory. David Mumford, John Fogarty, Frances Kirwan, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3427Springer-VerlagDavid Mumford, John Fogarty, and Frances Kirwan. Geometric Invariant Theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1994. ↑24, ↑27","doi":""},{"text":"Symplectic geometry and connectivity of spaces of frames. Tom Needham, Clayton Shonkwiler, Advances in Computational Mathematics. 47128Tom Needham and Clayton Shonkwiler. Symplectic geometry and connectivity of spaces of frames. Advances in Computational Mathematics, 47(1):5, 2021. ↑7, ↑8, ↑9, ↑28","doi":""},{"text":"Admissibility and frame homotopy for quaternionic frames. Tom Needham, Clayton Shonkwiler, Linear Algebra and its Applications. 64527Tom Needham and Clayton Shonkwiler. Admissibility and frame homotopy for quaternionic frames. Linear Algebra and its Applications, 645:237-255, 2022. ↑27","doi":""},{"text":"A stratification of the null cone via the moment map. Linda Ness, American Journal of Mathematics. 106627Linda Ness. A stratification of the null cone via the moment map. American Journal of Mathematics, 106(6):1281-1329, 1984. ↑27","doi":""},{"text":"Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Issai Schur, Sitzungsberichte der Berliner Mathematischen Gesellschaft. 2225Issai Schur. Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungs- berichte der Berliner Mathematischen Gesellschaft, 22:9-20, 1923. ↑11, ↑25","doi":""},{"text":"Holomorphic slices, symplectic reduction and multiplicities of representations. Reyer Sjamaar, The Annals of Mathematics, Second Series. 141127Reyer Sjamaar. Holomorphic slices, symplectic reduction and multiplicities of representations. The Annals of Mathematics, Second Series, 141(1):87-129, 1995. ↑27","doi":""},{"text":"Stratified symplectic spaces and reduction. Reyer Sjamaar, Eugene Lerman, The Annals of Mathematics, Second Series. 134217Reyer Sjamaar and Eugene Lerman. Stratified symplectic spaces and reduction. The Annals of Mathematics, Second Series, 134(2):375-422, 1991. ↑6, ↑11, ↑13, ↑17","doi":""},{"text":"Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Nate Strawn, Journal of Fourier Analysis and Applications. 17512Nate Strawn. Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Journal of Fourier Analysis and Applications, 17(5):821-853, 2010. ↑1, ↑4, ↑12","doi":""},{"text":"An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. F D Shayne, Waldron, Birkhäuser9New York, NYShayne F. D. Waldron. An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY, 2018. ↑9","doi":""}],"string":"[\n {\n \"text\": \"Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics. 5423Per AlexanderssonPer Alexandersson. Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics, 54:1-20, 2016. ↑23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Full spark frames. Boris Alexeev, Jameson Cahill, Dustin G Mixon, Journal of Fourier Analysis and Applications. 18627Boris Alexeev, Jameson Cahill, and Dustin G. Mixon. Full spark frames. Journal of Fourier Analysis and Applications, 18(6):1167-1194, 2012. ↑3, ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Symmetry and bifurcations of momentum mappings. Judith M Arms, Jerrold E Marsden, Vincent Moncrief, Communications in Mathematical Physics. 78415Judith M. Arms, Jerrold E. Marsden, and Vincent Moncrief. Symmetry and bifurcations of momentum mappings. Communications in Mathematical Physics, 78(4):455-478, 1981. ↑13, ↑15\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convexity and commuting Hamiltonians. Michael Francis, Atiyah , Bulletin of the London Mathematical Society. 14118Michael Francis Atiyah. Convexity and commuting Hamiltonians. Bulletin of the London Mathematical Society, 14(1):1-15, 1982. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Michèle Audin, Torus Actions on Symplectic Manifolds. Basel9310second revised editionMichèle Audin. Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Birkhäuser Verlag, Basel, second revised edition, 2004. ↑5, ↑7, ↑9, ↑10\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Matrix Analysis. Rajendra Bhatia, Graduate Texts in Mathematics. 16922SpringerRajendra Bhatia. Matrix Analysis, volume 169 of Graduate Texts in Mathematics. Springer, New York, NY, 1997. ↑22\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Constructing finite frames of a given spectrum and set of lengths. Jameson Cahill, Matthew Fickus, Dustin G Mixon, Miriam J Poteet, Nate Strawn, Applied and Computational Harmonic Analysis. 35124Jameson Cahill, Matthew Fickus, Dustin G. Mixon, Miriam J. Poteet, and Nate Strawn. Constructing finite frames of a given spectrum and set of lengths. Applied and Computational Harmonic Analysis, 35(1):52-73, 2013. ↑2, ↑18, ↑22, ↑24\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. Jameson Cahill, Dustin G Mixon, Nate Strawn, SIAM Journal on Applied Algebra and Geometry. 1124↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22Jameson Cahill, Dustin G. Mixon, and Nate Strawn. Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. SIAM Journal on Applied Algebra and Geometry, 1(1):38-72, 2017. ↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22, ↑24\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lectures on Symplectic Geometry. Ana Cannas Da Silva, Lecture Notes in Mathematics. 176418Springer-VerlagAna Cannas da Silva. Lectures on Symplectic Geometry, volume 1764 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, Heidelberg, 2001. ↑10, ↑12, ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The symplectic geometry of closed equilateral random walks in 3-space. Jason Cantarella, Clayton Shonkwiler, The Annals of Applied Probability. 26127Jason Cantarella and Clayton Shonkwiler. The symplectic geometry of closed equilateral random walks in 3- space. The Annals of Applied Probability, 26(1):549-596, 2016. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Equal-norm tight frames with erasures. G Peter, Jelena Casazza, Kovačević, Advances in Computational Mathematics. 182-41Peter G. Casazza and Jelena Kovačević. Equal-norm tight frames with erasures. Advances in Computational Mathematics, 18(2-4):387-430, 2003. ↑1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Existence and construction of finite frames with a given frame operator. G Peter, Manuel T Casazza, Leon, International Journal of Pure and Applied Mathematics. 63225Peter G. Casazza and Manuel T. Leon. Existence and construction of finite frames with a given frame operator. International Journal of Pure and Applied Mathematics, 63(2):149-157, 2010. ↑3, ↑25\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Toric Varieties. David A Cox, John B Little, Henry K Schenck, Graduate Studies in Mathematics. 12418American Mathematical SocietyDavid A. Cox, John B. Little, and Henry K. Schenck. Toric Varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vertices of Gelfand-Tsetlin polytopes. A De Jesús, Tyrrell B Loera, Mcallister, Discrete & Computational Geometry. 32423Jesús A. De Loera and Tyrrell B. McAllister. Vertices of Gelfand-Tsetlin polytopes. Discrete & Computational Geometry, 32(4):459-470, 2004. ↑23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hamiltoniens périodiques et images convexes de l'application moment. Thomas Delzant, 11618Bulletin de la Société Mathématique de FranceThomas Delzant. Hamiltoniens périodiques et images convexes de l'application moment. Bulletin de la Société Mathématique de France, 116(3):315-339, 1988. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. L David, Michael Donoho, Elad, Proceedings of the National Academy of Sciences of the United States of America. the National Academy of Sciences of the United States of America1003David L. Donoho and Michael Elad. Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. Proceedings of the National Academy of Sciences of the United States of America, 100(5):2197- 2202, 2003. ↑3\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the variation in the cohomology of the symplectic form of the reduced phase space. J Johannes, Gerrit J Duistermaat, Heckman, Inventiones Mathematicae. 69218Johannes J. Duistermaat and Gerrit J. Heckman. On the variation in the cohomology of the symplectic form of the reduced phase space. Inventiones Mathematicae, 69(2):259-268, 1982. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Manifold structure of spaces of spherical tight frames. Ken Dykema, Nate Strawn, International Journal of Pure and Applied Mathematics. 28212Ken Dykema and Nate Strawn. Manifold structure of spaces of spherical tight frames. International Journal of Pure and Applied Mathematics, 28(2):217-256, 2006. ↑1, ↑2, ↑4, ↑12\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bending flows for sums of rank one matrices. Hermann Flaschka, John J Millson, Canadian Journal of Mathematics. 57124Hermann Flaschka and John J. Millson. Bending flows for sums of rank one matrices. Canadian Journal of Mathematics, 57(1):114-158, 2005. ↑2, ↑3, ↑19, ↑24\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Toric degenerations of weight varieties and applications. Philip Foth, Yi Hu, Travaux mathématiques. 1627Philip Foth and Yi Hu. Toric degenerations of weight varieties and applications. Travaux mathématiques, 16:87- 105, 2005. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Finite-dimensional representations of the group of unimodular matrices. M Israïl, Michael L Tsetlin, Doklady Akademii Nauk SSSR (N.S.). 7123Israïl M. Gel'fand and Michael L. Tsetlin. Finite-dimensional representations of the group of unimodular matri- ces. Doklady Akademii Nauk SSSR (N.S.), 71:825-828, 1950. ↑23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The cohomology ring of weight varieties and polygon spaces. Rebecca F Goldin, Advances in Mathematics. 160227Rebecca F. Goldin. The cohomology ring of weight varieties and polygon spaces. Advances in Mathematics, 160(2):175-204, 2001. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantized frame expansions with erasures. K Vivek, Jelena Goyal, Jonathan A Kovačević, Kelner, Applied and Computational Harmonic Analysis. 1031Vivek K. Goyal, Jelena Kovačević, and Jonathan A. Kelner. Quantized frame expansions with erasures. Applied and Computational Harmonic Analysis, 10(3):203-233, 2001. ↑1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convexity properties of the moment mapping. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67318Victor Guillemin and Shlomo Sternberg. Convexity properties of the moment mapping. Inventiones Mathemati- cae, 67(3):491-513, 1982. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geometric quantization and multiplicities of group representations. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67327Victor Guillemin and Shlomo Sternberg. Geometric quantization and multiplicities of group representations. Inventiones Mathematicae, 67(3):515-538, 1982. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Gelfand-Cetlin system and quantization of the complex flag manifolds. Victor Guillemin, Shlomo Sternberg, Journal of Functional Analysis. 52123Victor Guillemin and Shlomo Sternberg. The Gelfand-Cetlin system and quantization of the complex flag man- ifolds. Journal of Functional Analysis, 52(1):106-128, 1983. ↑23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computational Geometry. Tim Haga, Christoph Pegel, 5624Tim Haga and Christoph Pegel. Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computa- tional Geometry, 56(3):727-742, 2016. ↑2, ↑22, ↑24\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hyperkähler metrics and supersymmetry. Nigel James Hitchin, Anders Karlhede, Ulf Lindström, Martin Roček, Communications in Mathematical Physics. 108418Nigel James Hitchin, Anders Karlhede, Ulf Lindström, and Martin Roček. Hyperkähler metrics and supersym- metry. Communications in Mathematical Physics, 108(4):535-589, 1987. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal frames for erasures. B Roderick, Vern I Holmes, Paulsen, Linear Algebra and its Applications. 3771Roderick B. Holmes and Vern I. Paulsen. Optimal frames for erasures. Linear Algebra and its Applications, 377:31-51, 2004. ↑1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Doubly stochastic matrices and the diagonal of a rotation matrix. Alfred Horn, American Journal of Mathematics. 76325Alfred Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathematics, 76(3):620-630, 1954. ↑11, ↑25\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Chevalley involution and a duality of weight varieties. J Benjamin, John J Howard, Millson, The Asian Journal of Mathematics. 8427Benjamin J. Howard and John J. Millson. The Chevalley involution and a duality of weight varieties. The Asian Journal of Mathematics, 8(4):685-732, 2004. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The length of vectors in representation spaces. George Kempf, Linda Ness, Lecture Notes in Mathematics. Knud Lønsted73227Springer-VerlagAlgebraic GeometryGeorge Kempf and Linda Ness. The length of vectors in representation spaces. In Knud Lønsted, editor, Alge- braic Geometry, volume 732 of Lecture Notes in Mathematics, pages 233-243. Springer-Verlag, Berlin, 1979. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Frances Kirwan, Princeton University Press627Princeton, NJFrances Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1984. ↑6, ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convexity properties of the moment mapping. Frances Kirwan, III. Inventiones Mathematicae. 7736Frances Kirwan. Convexity properties of the moment mapping. III. Inventiones Mathematicae, 77(3):547-552, 1984. ↑6\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Weight Varieties. Allen Knutson, 2627Massachusetts Institute of TechnologyPhD thesisAllen Knutson. Weight Varieties. PhD thesis, Massachusetts Institute of Technology, 1996. ↑26, ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The symplectic and algebraic geometry of Horn's problem. Allen Knutson, Linear Algebra and its Applications. 3191-310Allen Knutson. The symplectic and algebraic geometry of Horn's problem. Linear Algebra and its Applications, 319(1-3):61-81, 2000. ↑10\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to Smooth Manifolds. John M Lee, Graduate Texts in Mathematics. 21813Springersecond editionJohn M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, NY, second edition, 2013. ↑9, ↑13\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reduction of symplectic manifolds with symmetry. Jerrold E Marsden, Alan Weinstein, Reports on Mathematical Physics. 516Jerrold E. Marsden and Alan Weinstein. Reduction of symplectic manifolds with symmetry. Reports on Mathe- matical Physics, 5(1):121-130, 1974. ↑5, ↑6\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Inequalities: Theory of Majorization and Its Applications. Albert W Marshall, Ingram Olkin, Barry C Arnold, Springer Series in Statistics. 3Springersecond editionAlbert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: Theory of Majorization and Its Applica- tions. Springer Series in Statistics. Springer, New York, NY, second edition, 2011. ↑3\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to Symplectic Topology. Dusa Mcduff, Dietmar Salamon, Oxford University Press1215third edition, 2017. ↑4, ↑5, ↑10Dusa McDuff and Dietmar Salamon. Introduction to Symplectic Topology. Oxford University Press, third edition, 2017. ↑4, ↑5, ↑10, ↑12, ↑15\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Symmetries and integrals in mechanics. Kenneth R Meyer, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia. Mauricio M. PeixotoSalvador, Brasil; New York, NYAcademic Press56Kenneth R. Meyer. Symmetries and integrals in mechanics. In Mauricio M. Peixoto, editor, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26-August 14, 1971, pages 259-272. Academic Press, New York, NY, 1973. ↑5, ↑6\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geometric Invariant Theory. David Mumford, John Fogarty, Frances Kirwan, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3427Springer-VerlagDavid Mumford, John Fogarty, and Frances Kirwan. Geometric Invariant Theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1994. ↑24, ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Symplectic geometry and connectivity of spaces of frames. Tom Needham, Clayton Shonkwiler, Advances in Computational Mathematics. 47128Tom Needham and Clayton Shonkwiler. Symplectic geometry and connectivity of spaces of frames. Advances in Computational Mathematics, 47(1):5, 2021. ↑7, ↑8, ↑9, ↑28\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Admissibility and frame homotopy for quaternionic frames. Tom Needham, Clayton Shonkwiler, Linear Algebra and its Applications. 64527Tom Needham and Clayton Shonkwiler. Admissibility and frame homotopy for quaternionic frames. Linear Algebra and its Applications, 645:237-255, 2022. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A stratification of the null cone via the moment map. Linda Ness, American Journal of Mathematics. 106627Linda Ness. A stratification of the null cone via the moment map. American Journal of Mathematics, 106(6):1281-1329, 1984. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Issai Schur, Sitzungsberichte der Berliner Mathematischen Gesellschaft. 2225Issai Schur. Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungs- berichte der Berliner Mathematischen Gesellschaft, 22:9-20, 1923. ↑11, ↑25\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Holomorphic slices, symplectic reduction and multiplicities of representations. Reyer Sjamaar, The Annals of Mathematics, Second Series. 141127Reyer Sjamaar. Holomorphic slices, symplectic reduction and multiplicities of representations. The Annals of Mathematics, Second Series, 141(1):87-129, 1995. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stratified symplectic spaces and reduction. Reyer Sjamaar, Eugene Lerman, The Annals of Mathematics, Second Series. 134217Reyer Sjamaar and Eugene Lerman. Stratified symplectic spaces and reduction. The Annals of Mathematics, Second Series, 134(2):375-422, 1991. ↑6, ↑11, ↑13, ↑17\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Nate Strawn, Journal of Fourier Analysis and Applications. 17512Nate Strawn. Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Journal of Fourier Analysis and Applications, 17(5):821-853, 2010. ↑1, ↑4, ↑12\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. F D Shayne, Waldron, Birkhäuser9New York, NYShayne F. D. Waldron. An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY, 2018. ↑9\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics. 5423Per AlexanderssonPer Alexandersson. Gelfand-Tsetlin polytopes and the integer decomposition property. European Journal of Combinatorics, 54:1-20, 2016. ↑23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Full spark frames. Boris Alexeev, Jameson Cahill, Dustin G Mixon, Journal of Fourier Analysis and Applications. 18627Boris Alexeev, Jameson Cahill, and Dustin G. Mixon. Full spark frames. Journal of Fourier Analysis and Applications, 18(6):1167-1194, 2012. ↑3, ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Symmetry and bifurcations of momentum mappings. Judith M Arms, Jerrold E Marsden, Vincent Moncrief, Communications in Mathematical Physics. 78415Judith M. Arms, Jerrold E. Marsden, and Vincent Moncrief. Symmetry and bifurcations of momentum mappings. Communications in Mathematical Physics, 78(4):455-478, 1981. ↑13, ↑15\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convexity and commuting Hamiltonians. Michael Francis, Atiyah , Bulletin of the London Mathematical Society. 14118Michael Francis Atiyah. Convexity and commuting Hamiltonians. Bulletin of the London Mathematical Society, 14(1):1-15, 1982. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Michèle Audin, Torus Actions on Symplectic Manifolds. Basel9310second revised editionMichèle Audin. Torus Actions on Symplectic Manifolds, volume 93 of Progress in Mathematics. Birkhäuser Verlag, Basel, second revised edition, 2004. ↑5, ↑7, ↑9, ↑10\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Matrix Analysis. Rajendra Bhatia, Graduate Texts in Mathematics. 16922SpringerRajendra Bhatia. Matrix Analysis, volume 169 of Graduate Texts in Mathematics. Springer, New York, NY, 1997. ↑22\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Constructing finite frames of a given spectrum and set of lengths. Jameson Cahill, Matthew Fickus, Dustin G Mixon, Miriam J Poteet, Nate Strawn, Applied and Computational Harmonic Analysis. 35124Jameson Cahill, Matthew Fickus, Dustin G. Mixon, Miriam J. Poteet, and Nate Strawn. Constructing finite frames of a given spectrum and set of lengths. Applied and Computational Harmonic Analysis, 35(1):52-73, 2013. ↑2, ↑18, ↑22, ↑24\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. Jameson Cahill, Dustin G Mixon, Nate Strawn, SIAM Journal on Applied Algebra and Geometry. 1124↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22Jameson Cahill, Dustin G. Mixon, and Nate Strawn. Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames. SIAM Journal on Applied Algebra and Geometry, 1(1):38-72, 2017. ↑1, ↑2, ↑3, ↑4, ↑13, ↑14, ↑22, ↑24\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lectures on Symplectic Geometry. Ana Cannas Da Silva, Lecture Notes in Mathematics. 176418Springer-VerlagAna Cannas da Silva. Lectures on Symplectic Geometry, volume 1764 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, Heidelberg, 2001. ↑10, ↑12, ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The symplectic geometry of closed equilateral random walks in 3-space. Jason Cantarella, Clayton Shonkwiler, The Annals of Applied Probability. 26127Jason Cantarella and Clayton Shonkwiler. The symplectic geometry of closed equilateral random walks in 3- space. The Annals of Applied Probability, 26(1):549-596, 2016. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Equal-norm tight frames with erasures. G Peter, Jelena Casazza, Kovačević, Advances in Computational Mathematics. 182-41Peter G. Casazza and Jelena Kovačević. Equal-norm tight frames with erasures. Advances in Computational Mathematics, 18(2-4):387-430, 2003. ↑1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Existence and construction of finite frames with a given frame operator. G Peter, Manuel T Casazza, Leon, International Journal of Pure and Applied Mathematics. 63225Peter G. Casazza and Manuel T. Leon. Existence and construction of finite frames with a given frame operator. International Journal of Pure and Applied Mathematics, 63(2):149-157, 2010. ↑3, ↑25\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Toric Varieties. David A Cox, John B Little, Henry K Schenck, Graduate Studies in Mathematics. 12418American Mathematical SocietyDavid A. Cox, John B. Little, and Henry K. Schenck. Toric Varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vertices of Gelfand-Tsetlin polytopes. A De Jesús, Tyrrell B Loera, Mcallister, Discrete & Computational Geometry. 32423Jesús A. De Loera and Tyrrell B. McAllister. Vertices of Gelfand-Tsetlin polytopes. Discrete & Computational Geometry, 32(4):459-470, 2004. ↑23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hamiltoniens périodiques et images convexes de l'application moment. Thomas Delzant, 11618Bulletin de la Société Mathématique de FranceThomas Delzant. Hamiltoniens périodiques et images convexes de l'application moment. Bulletin de la Société Mathématique de France, 116(3):315-339, 1988. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. L David, Michael Donoho, Elad, Proceedings of the National Academy of Sciences of the United States of America. the National Academy of Sciences of the United States of America1003David L. Donoho and Michael Elad. Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ 1 minimization. Proceedings of the National Academy of Sciences of the United States of America, 100(5):2197- 2202, 2003. ↑3\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the variation in the cohomology of the symplectic form of the reduced phase space. J Johannes, Gerrit J Duistermaat, Heckman, Inventiones Mathematicae. 69218Johannes J. Duistermaat and Gerrit J. Heckman. On the variation in the cohomology of the symplectic form of the reduced phase space. Inventiones Mathematicae, 69(2):259-268, 1982. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Manifold structure of spaces of spherical tight frames. Ken Dykema, Nate Strawn, International Journal of Pure and Applied Mathematics. 28212Ken Dykema and Nate Strawn. Manifold structure of spaces of spherical tight frames. International Journal of Pure and Applied Mathematics, 28(2):217-256, 2006. ↑1, ↑2, ↑4, ↑12\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bending flows for sums of rank one matrices. Hermann Flaschka, John J Millson, Canadian Journal of Mathematics. 57124Hermann Flaschka and John J. Millson. Bending flows for sums of rank one matrices. Canadian Journal of Mathematics, 57(1):114-158, 2005. ↑2, ↑3, ↑19, ↑24\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Toric degenerations of weight varieties and applications. Philip Foth, Yi Hu, Travaux mathématiques. 1627Philip Foth and Yi Hu. Toric degenerations of weight varieties and applications. Travaux mathématiques, 16:87- 105, 2005. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Finite-dimensional representations of the group of unimodular matrices. M Israïl, Michael L Tsetlin, Doklady Akademii Nauk SSSR (N.S.). 7123Israïl M. Gel'fand and Michael L. Tsetlin. Finite-dimensional representations of the group of unimodular matri- ces. Doklady Akademii Nauk SSSR (N.S.), 71:825-828, 1950. ↑23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The cohomology ring of weight varieties and polygon spaces. Rebecca F Goldin, Advances in Mathematics. 160227Rebecca F. Goldin. The cohomology ring of weight varieties and polygon spaces. Advances in Mathematics, 160(2):175-204, 2001. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantized frame expansions with erasures. K Vivek, Jelena Goyal, Jonathan A Kovačević, Kelner, Applied and Computational Harmonic Analysis. 1031Vivek K. Goyal, Jelena Kovačević, and Jonathan A. Kelner. Quantized frame expansions with erasures. Applied and Computational Harmonic Analysis, 10(3):203-233, 2001. ↑1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convexity properties of the moment mapping. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67318Victor Guillemin and Shlomo Sternberg. Convexity properties of the moment mapping. Inventiones Mathemati- cae, 67(3):491-513, 1982. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geometric quantization and multiplicities of group representations. Victor Guillemin, Shlomo Sternberg, Inventiones Mathematicae. 67327Victor Guillemin and Shlomo Sternberg. Geometric quantization and multiplicities of group representations. Inventiones Mathematicae, 67(3):515-538, 1982. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Gelfand-Cetlin system and quantization of the complex flag manifolds. Victor Guillemin, Shlomo Sternberg, Journal of Functional Analysis. 52123Victor Guillemin and Shlomo Sternberg. The Gelfand-Cetlin system and quantization of the complex flag man- ifolds. Journal of Functional Analysis, 52(1):106-128, 1983. ↑23\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computational Geometry. Tim Haga, Christoph Pegel, 5624Tim Haga and Christoph Pegel. Polytopes of eigensteps of finite equal norm tight frames. Discrete & Computa- tional Geometry, 56(3):727-742, 2016. ↑2, ↑22, ↑24\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hyperkähler metrics and supersymmetry. Nigel James Hitchin, Anders Karlhede, Ulf Lindström, Martin Roček, Communications in Mathematical Physics. 108418Nigel James Hitchin, Anders Karlhede, Ulf Lindström, and Martin Roček. Hyperkähler metrics and supersym- metry. Communications in Mathematical Physics, 108(4):535-589, 1987. ↑18\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal frames for erasures. B Roderick, Vern I Holmes, Paulsen, Linear Algebra and its Applications. 3771Roderick B. Holmes and Vern I. Paulsen. Optimal frames for erasures. Linear Algebra and its Applications, 377:31-51, 2004. ↑1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Doubly stochastic matrices and the diagonal of a rotation matrix. Alfred Horn, American Journal of Mathematics. 76325Alfred Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathematics, 76(3):620-630, 1954. ↑11, ↑25\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Chevalley involution and a duality of weight varieties. J Benjamin, John J Howard, Millson, The Asian Journal of Mathematics. 8427Benjamin J. Howard and John J. Millson. The Chevalley involution and a duality of weight varieties. The Asian Journal of Mathematics, 8(4):685-732, 2004. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The length of vectors in representation spaces. George Kempf, Linda Ness, Lecture Notes in Mathematics. Knud Lønsted73227Springer-VerlagAlgebraic GeometryGeorge Kempf and Linda Ness. The length of vectors in representation spaces. In Knud Lønsted, editor, Alge- braic Geometry, volume 732 of Lecture Notes in Mathematics, pages 233-243. Springer-Verlag, Berlin, 1979. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Frances Kirwan, Princeton University Press627Princeton, NJFrances Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1984. ↑6, ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convexity properties of the moment mapping. Frances Kirwan, III. Inventiones Mathematicae. 7736Frances Kirwan. Convexity properties of the moment mapping. III. Inventiones Mathematicae, 77(3):547-552, 1984. ↑6\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Weight Varieties. Allen Knutson, 2627Massachusetts Institute of TechnologyPhD thesisAllen Knutson. Weight Varieties. PhD thesis, Massachusetts Institute of Technology, 1996. ↑26, ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The symplectic and algebraic geometry of Horn's problem. Allen Knutson, Linear Algebra and its Applications. 3191-310Allen Knutson. The symplectic and algebraic geometry of Horn's problem. Linear Algebra and its Applications, 319(1-3):61-81, 2000. ↑10\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to Smooth Manifolds. John M Lee, Graduate Texts in Mathematics. 21813Springersecond editionJohn M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, NY, second edition, 2013. ↑9, ↑13\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reduction of symplectic manifolds with symmetry. Jerrold E Marsden, Alan Weinstein, Reports on Mathematical Physics. 516Jerrold E. Marsden and Alan Weinstein. Reduction of symplectic manifolds with symmetry. Reports on Mathe- matical Physics, 5(1):121-130, 1974. ↑5, ↑6\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Inequalities: Theory of Majorization and Its Applications. Albert W Marshall, Ingram Olkin, Barry C Arnold, Springer Series in Statistics. 3Springersecond editionAlbert W. Marshall, Ingram Olkin, and Barry C. Arnold. Inequalities: Theory of Majorization and Its Applica- tions. Springer Series in Statistics. Springer, New York, NY, second edition, 2011. ↑3\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to Symplectic Topology. Dusa Mcduff, Dietmar Salamon, Oxford University Press1215third edition, 2017. ↑4, ↑5, ↑10Dusa McDuff and Dietmar Salamon. Introduction to Symplectic Topology. Oxford University Press, third edition, 2017. ↑4, ↑5, ↑10, ↑12, ↑15\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Symmetries and integrals in mechanics. Kenneth R Meyer, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia. Mauricio M. PeixotoSalvador, Brasil; New York, NYAcademic Press56Kenneth R. Meyer. Symmetries and integrals in mechanics. In Mauricio M. Peixoto, editor, Dynamical Systems: Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26-August 14, 1971, pages 259-272. Academic Press, New York, NY, 1973. ↑5, ↑6\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geometric Invariant Theory. David Mumford, John Fogarty, Frances Kirwan, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3427Springer-VerlagDavid Mumford, John Fogarty, and Frances Kirwan. Geometric Invariant Theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1994. ↑24, ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Symplectic geometry and connectivity of spaces of frames. Tom Needham, Clayton Shonkwiler, Advances in Computational Mathematics. 47128Tom Needham and Clayton Shonkwiler. Symplectic geometry and connectivity of spaces of frames. Advances in Computational Mathematics, 47(1):5, 2021. ↑7, ↑8, ↑9, ↑28\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Admissibility and frame homotopy for quaternionic frames. Tom Needham, Clayton Shonkwiler, Linear Algebra and its Applications. 64527Tom Needham and Clayton Shonkwiler. Admissibility and frame homotopy for quaternionic frames. Linear Algebra and its Applications, 645:237-255, 2022. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A stratification of the null cone via the moment map. Linda Ness, American Journal of Mathematics. 106627Linda Ness. A stratification of the null cone via the moment map. American Journal of Mathematics, 106(6):1281-1329, 1984. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Issai Schur, Sitzungsberichte der Berliner Mathematischen Gesellschaft. 2225Issai Schur. Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungs- berichte der Berliner Mathematischen Gesellschaft, 22:9-20, 1923. ↑11, ↑25\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Holomorphic slices, symplectic reduction and multiplicities of representations. Reyer Sjamaar, The Annals of Mathematics, Second Series. 141127Reyer Sjamaar. Holomorphic slices, symplectic reduction and multiplicities of representations. The Annals of Mathematics, Second Series, 141(1):87-129, 1995. ↑27\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stratified symplectic spaces and reduction. Reyer Sjamaar, Eugene Lerman, The Annals of Mathematics, Second Series. 134217Reyer Sjamaar and Eugene Lerman. Stratified symplectic spaces and reduction. The Annals of Mathematics, Second Series, 134(2):375-422, 1991. ↑6, ↑11, ↑13, ↑17\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Nate Strawn, Journal of Fourier Analysis and Applications. 17512Nate Strawn. Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. Journal of Fourier Analysis and Applications, 17(5):821-853, 2010. ↑1, ↑4, ↑12\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. F D Shayne, Waldron, Birkhäuser9New York, NYShayne F. D. Waldron. An Introduction to Finite Tight Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, New York, NY, 2018. ↑9\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":29,"cells":{"text":{"kind":"string","value":"\nOptical Tweezers: A Comprehensive Tutorial from Calibration to Applications\n\n\nJan Gieseler \nICFO -Institut de Ciències Fotòniques\nThe Barcelona Institute of Science and Technology\n08860CastelldefelsBarcelonaSpain\n\nDepartment of Physics\nHarvard University\n17 Oxford StCambridgeMAUSA\n\nJuan Ruben Gomez-Solano \nInstituto de Física\nUniversidad Nacional Autónoma de México\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\n\nAlessandro Magazzù \nDepartment of Physics\nUniversity of Gothenburg\n41296GothenburgSweden\n\nIsaac Pérez Castillo \nInstituto de Física\nUniversidad Nacional Autónoma de México\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\n\nLondon Mathematical Laboratory\n18 Margravine GardensW6 8RHLondonUnited Kingdom\n\nLaura Pérez García \nDepartment of Physics\nUniversity of Gothenburg\n41296GothenburgSweden\n\nMarta Gironella-Torrent \nDepartament de Fsica de la Matèria Condensada\nFacultat de Física\nSmall Biosystems Lab\nUniversitat de Barcelona\n08028BarcelonaSpain\n\nXavier Viader-Godoy \nDepartament de Fsica de la Matèria Condensada\nFacultat de Física\nSmall Biosystems Lab\nUniversitat de Barcelona\n08028BarcelonaSpain\n\nFelix Ritort \nDepartament de Fsica de la Matèria Condensada\nFacultat de Física\nSmall Biosystems Lab\nUniversitat de Barcelona\n08028BarcelonaSpain\n\nGiuseppe Pesce \nDepartment of Physics\nUniverisità degli di Studi di Napoli \"Federico II\" Complesso universitario monte S. Angelo\nVia Cintia80126NapoliItaly\n\nAlejandro V Arzola \nInstituto de Física\nUniversidad Nacional Autónoma de México\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\n\nKaren Volke-Sepulveda \nInstituto de Física\nUniversidad Nacional Autónoma de México\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\n\nGiovanni Volpe *giovanni.volpe@physics.gu.se \nDepartment of Physics\nUniversity of Gothenburg\n41296GothenburgSweden\n\nOptical Tweezers: A Comprehensive Tutorial from Calibration to Applications\n\nSince their invention in 1986 by Arthur Ashkin and colleagues, optical tweezers have become an essential tool in several fields of physics, spectroscopy, biology, nanotechnology, and thermodynamics. In this Tutorial, we provide a primer on how to calibrate optical tweezers and how to use them for advanced applications. After a brief general introduction on optical tweezers, we focus on describing and comparing the various available calibration techniques. Then, we discuss some cutting-edge applications of optical tweezers in a liquid medium, namely to study single-molecule and single-cell mechanics, microrheology, colloidal interactions, statistical physics, and transport phenomena. Finally, we consider optical tweezers in vacuum, where the absence of a viscous medium offers vastly different dynamics and presents new challenges. We conclude with some perspectives for the field and the future application of optical tweezers. This Tutorial provides both a step-by-step guide ideal for non-specialists entering the field and a comprehensive manual of advanced techniques useful for expert practitioners. All the examples are complemented by the sample data and software necessary to reproduce them.\n\nIntroduction\n\nIn 1970, Arthur Ashkin [1] demonstrated that a focused laser beam can accelerate, decelerate, and even stably trap micrometer-sized neutral particles. Later, in 1986, Ashkin and colleagues reported the first realization of an optical tweezers [2,3]: a tightly focused beam of light capable of holding microscopic particles in three dimensions. Thanks to this work, Ashkin was awarded a share of the 2018 Nobel Prize in Physics [4]. Furthermore, one of Ashkin's co-authors, Steven Chu, would go on to use optical tweezing in his work on cooling and trapping atoms; this research earned Chu a share of the 1997 Nobel Prize in Physics [5].\n\nIn the late 1980s, Ashkin and colleagues used optical tweezers to manipulate biological samples, starting from an individual tobacco mosaic virus and Escherichia coli bacterium [6]. In 1993, Lucien Ghislain and Watt Webb invented photonic force microscopy [?], where an optically trapped particle is used as an extremely sensitive cantilever to probe microscopic force fields ranging from femtonewtons (10 −15 N) to piconewtons (10 −12 N). Since the early 1990s, optical force spectroscopy has been utilized to characterize the mechanical properties of biomolecules and biological motors [7][8][9]. Optical tweezers have also been employed in many fields of physics [10][11][12][13][14][15], nanotechnology [16], spectroscopy [17], nanothermodynamics [18,19], soft matter [20], and biology [21].\n\nAn optical tweezers consist of a laser beam focused by a high numerical aperture (NA) microscope objective. A microscopic particle (whose refractive index is higher than that of its embedding medium) can then be trapped near the focal spot because of the emergence of scattering and gradient optical forces: the scattering forces are due to the radiation pressure of the light beam and act along the direction of propagation of the beam; and the gradient forces pull the particle towards the high-intensity focal spot. Using optical tweezers, a very broad range of particles has been trapped, as shown in Fig. 1.\n\nThere are already several resources to build and operate optical tweezers available in the literature. In the 2015 book Optical Tweezers: Principles and Applications [22], Phillip. H. Jones, Onofrio M. Maragó and Giovanni Volpe have discussed in detail the theory behind optical tweezers as well as how to build and operate optical tweezers setups. There are also several other tutorials on specific aspects of optical trapping. In particular, there are computational toolboxes to calculate optical forces using T-matrix methods by Timo Nieminen et al. [23] and geometrical optics by Agnese Callegari et al. [24]. There is a tutorial on how to simulate the Brownian motion of an optically trapped particle [25]. There are several tutorials on how to build optical tweezers setups by Stephen Smith et al. [26], John Bechhoefer and Scott Wilson [27], Angela Mellish and Andrew Wilson [28], David Appleyard et al. [29], Woei Lee et al. [30], Manoj Mathew et al. [31], and Giuseppe Pesce et al. [32].\n\nIn this Tutorial, we complete and complement these already available resources providing a primer on how to calibrate optical tweezers and how to use them for advanced applications. In particular, after a brief general introduction on optical tweezers (section 2), we focus on describing and comparing the various calibration techniques that are available (section 3). Then, we discuss some of the most exciting cutting-edge applications of optical tweezers in a liquid medium (section 4), namely to study single-molecule and single-cell mechanics, microrheology, colloidal interactions, statistical physics, and transport phenomena. Finally, we consider optical tweezers in vacuum, which features different dynamics than a viscous medium, so that vacuum operation comes with its own experimental challenges and calibration methods (section 5). This Tutorial provides both a step-by-step guide ideal for non-specialists entering the field and a comprehensive manual of advanced techniques useful for expert practitioners. All the examples are complemented by the sample data and the software necessary to reproduce them [33].\n\n\nGetting started\n\nIn this section we provide some essential background information on optical tweezers. First, we will describe a basic optical tweezers setup (section 2.1). Then, we will briefly discuss the Brownian motion of an optically trapped particle and the most basic optical tweezers calibration techniques (section 2.2). Finally, we will give a brief overview of more advanced optical tweezers and of alternative approaches for the trapping and manipulation of microscopic matter (section 2.3).\n\n\nA simple setup\n\nAn optical tweezers is a highly-focused laser beam that can trap microscopic particles. In practice, an optical tweezers can be built by focusing a laser beam through a microscope, which can be either commercial or homemade. As illustrated in Fig. 2, a basic optical tweezers setup is comprised of three parts: trapping, imaging, and position detection.\n\nTrapping. In most applications, the laser is continuous wave (CW), whose wavelength is in the visible or near-infrared region of the spectrum, and whose power is between 10 mW and 1 W.\n\nThe laser beam is expanded by a telescope to overfill the back-aperture of a high-NA microscope objective, is directed by a dichroic mirror towards the objective, and finally is focused by the objective to a diffraction-limited spot within the sample holder. This focal spot is the optical tweezers, which can trap a microscopic particle. Usually, the object to be trapped is suspended in an aqueous solution, which is held between a microscope slide and a cover slip; sometimes the particles to be trapped are suspended in other liquids, or in gases, or even in vacuum (section 5).\n\nImaging. Typically, the trapping and imaging objectives are the same. The light for the imaging is provided by a condenser, which is an integral part of any microscope. It is convenient to image through a dichroic beam splitter, which reflects the trapping light to the microscope, but also transmits some of the imaging light from the condenser to a camera. In both cases, some color filters are typically required in front of the camera to reduce the intensity of the trapping beam to an acceptable level such that the camera is not overexposed.\n\n\nPosition detection.\n\nTo use the optically trapped particle as a microscopic force transducer, it is necessary to track its position. This can be done by analyzing either the videos of the particle acquired by the camera (digital video microscopy), or the time series acquired through a position detector. In this latter case, the condenser collects the forward scattered light and projects it onto a position detector (typically either a quadrant photodetector or a position sensing detector). This detects the changes in the forward scattered light pattern due to the changes in the particle's position.\n\n\nOptical tweezers calibration\n\nAn optically trapped particle can be used as an extremely sensitive microscopic force transducer that can both exert and measure forces ranging from hundreds of piconewtons down to a few femtonewtons.\n\nAn optically trapped particle experiences a harmonic force that tends to keep it near a stable equilibrium position. Considering only one dimension, for small displacements from the equilibrium position, the force acting on a colloidal particle positioned at x within a trap centered at x eq is\nF ot = −κ x (x − x eq ),(1)\nwhere κ x is the trap stiffness along the x-direction. The resulting trapping potential is harmonic:\nU(x) = 1 2 κ x (x − x eq ) 2 .(2)\nA typical trajectory of an optically trapped particle can be seen in Fig. 3(a). The particle keeps on moving due to the presence of Brownian fluctuations. Therefore, an optically trapped particle is in a dynamic equilibrium between the thermal noise continuously pushing it out of the trap and the optical forces driving it towards the equilibrium position. For the particle to remain within the optical trap, the optical potential well must be sufficiently deep. The depth of the optical potential is typically characterized in units of the thermal energy k B T, where k B is the Boltzmann constant and T is the absolute temperature. This quantity gives a characteristic energy scale for mesoscopic phenomena. The potential well of an optical tweezers should be at least a few k B T deep to be able to confine a particle, as shown in Fig. 3(b).\n\nThe analysis of the thermal motion of the optically trapped particle in the optical potential provides information about the local forces acting on the particle. For example, since the particle is held in a harmonic trapping potential, its position distribution is Gaussian, as shown in Fig. 3(c), By sampling this probability distribution, it is possible to measure κ x using the equipartition theorem. The equipartition theorem states that for a system at thermodynamic equilibrium at absolute temperature T, the energy associated with each harmonic degree of freedom is equal to 1 2 k B T. Therefore, U(x) = 1 2 κ x (x − x eq ) 2 = 1 2 κ x σ 2 x = 1 2 k B T, where σ 2 x = (x − x eq ) 2 is the variance of the trapped particle position, and\nρ(x) = ρ 0 exp − κ x (x − x eq ) 2 2k B T .(3)κ x = k B T σ 2 x .(4)\nAs we will see in section 3, there are more sophisticated methods to use the thermal fluctuations of an optically trapped particle to calibrate its stiffness. In particular, it is possible to also exploit the temporal evolution of the particle in the optical trap, for example by calculating its correlation and power spectral density.\n\nOnce an optical tweezers is calibrated, a constant and homogeneous external force F ext, x shifts the equilibrium position of the trap. The value of the force can be obtained as:\nF ext,x = κ x ∆x eq ,(5)\nwhere ∆x eq is the average particle displacement from the original equilibrium position without the external force.\n\n\nAdvanced and alternative approaches\n\nGoing beyond the basic setup introduced in the previous section, more advanced setups become necessary to extend the optical manipulation toolkit, introducing, e.g., optical spanners [34], optical rheometers [35], optical stretchers [36], and optical sorters [37][38][39]. These tools allow one to manipulate multiple particles at once [40][41][42][43], rotate particles [44][45][46][47], measure the mechanical properties of biological samples [48], sort particles [49], trap particles with refractive index lower than their medium [50][51][52][53], induce optical binding and collective dynamics [11,[54][55][56][57][58][59]. Complex optical potentials, such as periodic and random patterns, provide experimental models to study microscopic and non-linear transport mechanisms [60][61][62][63][64][65][66][67]. Advanced optical manipulation has been employed also to build and study micromachines [68][69][70] and optofluidic devices [37,[71][72][73][74][75][76]. Furthermore, optical manipulation setups have also been successfully employed to investigate the properties of light itself, employing structured light beams with peculiar intensity, phase and polarization distributions [34,44,[77][78][79][80][81].\n\nIn this section, we will provide a brief overview of the main advanced optical manipulation techniques. We will consider only the overdamped regime (i.e., particles immerse in a viscous medium like water), since the case of underdamped systems involves many different experimental aspects that will be treated in (section 5). Also, unless otherwise stated, we will assume that the refractive index of the particles under consideration is higher than that of the surrounding medium.\n\n\nBeam steering\n\nMost advanced optical micromanipulation setups require to steer the trapping beam to control the position of the optical tweezers. The beam steering device can be as simple as a mirror on a gimbal mount (preferred over conventional kinematic mounts to avoid mirror shifting). For example, this simple approach can be used to move the optical trap within the sample instead of moving the sample with a translation stage while keeping the optical trap fixed. More sophisticated beam-steering devices are also often employed, such as acousto-optic deflectors and spatial light modulators (SLMs).\n\nWhen implementing beam steering, it is essential to ensure that the available light reaches the back aperture of the microscope objective, irrespective of the beam direction. To this end, a telescope should be added (4f-system) to create conjugate planes between the beam steering device and the back aperture of the objective, as shown in Fig. 4. This telescope can also be used to magnify the beam to slightly overfill the back aperture of the objective (to improve trapping), and a spatial filter can be added at Fourier plane 1 (to remove unwanted higher-order diffraction patterns), which is conjugated with Fourier plane 2. Fig. 4. Beam steering mechanism. The trapping beam is steered by the steering device (e.g., a galvo-mirror, an acousto-optic deflector, or a spatial light modulator). To ensure that the trapping beam reaches the back aperture of the microscope objective (O) while being steered, a telescope is added along the beam path (lenses L 1 and L 2 ) in a 4f-configuration ( f 1 and f 2 are the focal lengths of L 1 and L 2 respectively).\n\n\nPolarization splitting\n\nThe simplest alternative to have two neighboring traps while avoiding the interference between them is to split the original beam into two orthogonally polarized beams [40,82], as shown in Fig. 5. If the main control parameter of a given experiment is the distance between the traps, a single steering device can be enough, keeping one trap fixed while the other one is shifted. This works fine for a small distance between the traps. If the traps need to be moved over a large area, it might be useful to use two steering mirrors to minimize the distortion of each trap. Dynamical control of each trap can be achieved if the gimbal mounts are operated by actuators along the two axes, giving rise to a continuous motion along arbitrary trajectories in two dimensions in the sample plane. Finally, this twin-trap approach can also be used in combination with other advanced techniques, if the aim is to superimpose two complex patterns without interference between them. 1 and GM 2 should be mounted on gimbal mounts, and the distance between each of them and the lens L 1 should be equal to the focal length of L 1 (Fig. 4). For alignment purposes, both traps should be initially superimposed, arriving exactly at the same point in the sample plane. The intensity ratio between the traps can also be controlled, by placing a polarizer followed by a λ/2-retarder at the entrance of the Mach-Zehnder setup. As the retarder is rotated, the proportion of light reflected and transmitted through PBS 1 will change. DM: Dichroic mirror. M: mirror. O: microscope objective. C: Condenser.\n\n\nTime-sharing optical traps\n\nA single beam can be used to generate multiple optical tweezers if it is scanned sufficiently fast, i.e., with a frequency high enough to ensure that an optically trapped particle cannot diffuse away from the optical trap region between consecutive visits of the laser beam. This technique is known as time-sharing or laser-scanning optical trap. It can be performed with different deflecting devices, such as acousto-optic deflectors (AODs), mirror galvanometers (galvo-mirrors), electro-optic deflectors (EODs), and piezo-mirrors [40,[83][84][85][86]. Often, the beam deflecting devices have only one degree of freedom so that, to achieve beam motion over the whole sample plane, it is necessary to use two beam deflecting devices, one for each transverse direction, as illustrated in Fig. 6.\n\nThe angular range and the scanning frequency of the beam deflecting device determines the number of traps that can be created simultaneously and the extent of the trapping area. There is usually a compromise between the scanning speed and the deflection angular range. The first setup of time-sharing optical tweezers was built with galvo-mirrors [83], which could reach scanning frequencies of the order of 10 Hz. Nowadays, there are similar commercial systems that can reach up to 100 Hz, with a angular range of up to 25 • ; however, the angular range in the optical trapping setup is limited by the maximum acceptance angle of the microscope objective, which is just a few degrees at most. AODs can attain frequencies from tens to hundreds of kilohertz with angular ranges of up to 3 • , which makes them very suitable for trapping applications. For example, these devices have been used in studies of DNA stretching and molecular motors [40]. However, as the AOD working principle is based on the generation of diffraction gratings by means of the propagation of acoustic waves in a glass or quartz cell, resulting in a periodic refractive index modulation, it is worth to keep in mind that the diffraction efficiency can be at Fig. 6. Time-sharing optical tweezers. The trapping beam is time-shared between several optical traps using two beam deflector devices along the xand y-directions (X-BDD and Y-BDD, respectively). Two telescopes are needed (L 1 -L 2 and L 3 -L 4 ) to create conjugate planes between the two deflector devices as well as with the back aperture of the microscope objective O. most around 70%. EODs are even faster than AODs, but with smaller angular ranges of fractions of a degree. Piezo-mirrors are more limited, reaching frequencies of hundreds of Hertz with angular ranges of only a few hundredths of a degree.\n\nIf the position of a given trap is slightly shifted from one visit to the next one, the bead will follow it, making it possible to describe continuous trajectories within the plane. The average intensity in each trap can also be controlled by means of the number of visits of the scanning beam in a full cycle. For example, if a trapping site is visited twice as much as another one, it will experience twice the average intensity.\n\n\nInterferometric optical traps\n\nInterference patterns can trap multiple particles simultaneously and generate ordered structures [54,55,87]. For example, consider the simplest interference pattern consisting of alternating bright and dark fringes. The dynamics of the particles depend on their size with respect to the characteristic size of the fringes. Particles smaller than the fringes are attracted towards the bright fringes, while remaining free to move along the fringes. However, much more interesting dynamics emerge when the particle size is comparable or even larger than the fringes [39,49,88]. This has been called the size effect and will be explained in more detail in section 4.6.\n\nTo generate alternating bright and dark fringes, we need two collimated beams travelling with a relative angle: the larger this angle, the smaller the fringe period (the smallest fringe pattern is half wavelength). One possibility is to use a Mach-Zehnder interferometer using non-polarizing beam splitters ( Fig. 7(a)). A 4 f -system is formed by the lens L 1 and the microscope objective that conjugates the mirror on the gimbal mount and the focal plane of the objective (the objective can be replaced by an aspherical lens, since the high-NA requirement may be relaxed in this case). While an extended interference pattern generates strong intensity gradients in the transverse plane, the intensity gradient along the propagation direction is usually not enough to trap in 3D so that the particles are pushed by the scattering force against either the bottom or the top surface of the sample cell.1 Fig. 7(b) shows an alternative configuration for the interferometer module. In both configurations (Figs. 7(a) and 7(b)) a single mirror mounted on a gimbal mount is enough to control the fringe period for small interference angles.\n\nAlternative configurations are possible, an example of which is illustrated in Fig. 7 two beams are redirected by mirrors M 1 and M 2 towards a right-angle prism, which reflects them towards the objective, which in turn focuses them on the sample. The right-angle prism is mounted on a single-axis translation stage to control the distance between the reflected beams and, therefore, the interference angle. A low NA objective is advisable for this configuration to minimize the changes in the intensity distribution along the z-axis within the sample depth.\n\nAn extreme case of interference pattern is a speckle light field, which arises from the interference of a multitude of coherent waves with random phases, generated when laser light is, e.g., reflected from a rough surface or transmitted through a turbid medium. Speckle patterns have also been used for optical manipulation [65,89,90].\n\n\nHolographic optical tweezers\n\nThe most versatile optical tweezers setups are arguably based on holographic optical tweezers (HOTs) [41][42][43][91][92][93]. The basic working principle of HOTs is shown in Fig. 8. The wavefront of an incident beam is reshaped by a hologram projected on a spatial light modulator (SLM) and then relayed by a series of lenses to an objective that then generates the desired light pattern at the sample. In practice, there are many possible different HOT configurations. Discussing their details is out of the scope of this Tutorial, but a comprehensive review of the topic (including hologram computing algorithms, a numerical toolbox, and a comparison in terms of the main parameters such as efficiency, average intensity, uniformity, and percentage standard error) can a b c d\n\nFourier-plane phase mask Image-plane phase mask In the cases (a)-(c), the light pattern is obtained at the Fourier plane ( Fig. 8(a)), while in case (d) it is at the image plane ( Fig. 8(b)).\n\nbe found in Ref. [22]. Here, we will focus our attention on only two approaches.\n\nIn the first approach, the desired light pattern at the sample is the Fourier transform of the modulated laser beam right after the SLM. It is necessary to include a 4 f -system between the SLM and the back aperture of the objective to conjugate the two optical planes and to perform spatial filtering (Fourier Plane 1 in Fig. 8(a)). This approach works particularly well to generate multiple focal spots distributed in predetermined locations in space, which serve as individual traps. The simplest case is a single steerable trap ( Fig. 9(a)), and the corresponding hologram is a superposition of a blazed diffraction grating (which determines the lateral position of the focal spot) and a Fresnel lens (which shifts the focal spot along the propagation axis). For the generation of multiple traps, there are different algorithms to calculate the hologram, such as the superposition of gratings and lenses, the random mask encoding, the Gerchberg-Saxton algorithm, and the adaptive-additive algorithm [22]. An example of a phase mask that produces a set of individual traps can be seen in Fig. 9(b). Structured beams (e.g., Laguerre-Gaussian beams, Fig. 9(c)) can also be straightforwardly created with this approach. Furthermore, continuous optical potentials with arbitrary shapes can be produced as well by calculating the corresponding hologram with the aforementioned algorithms [22].\n\nIn the second approach, a phase profile related to the desired light pattern is directly projected on the SLM and relayed on the objective image plane (this requires an even number of lenses between the two planes, Fig. 8(b)). This configuration is specially suitable when the field of interest is not meant to be tightly focused (low-NA optics), e.g., using phase gratings to split the beam and generate interference fringes at the objective image plane [94]. Also, Bessel beams ( Fig. 9(d)), Mathieu beams, and nondiffracting beams are typically generated at the image plane instead of at the Fourier plane [45,52,79,[95][96][97][98][99][100][101][102].\n\nTypical switching speeds for nematic liquid crystal SLMs can reach 60 Hz with a large number of discrete phase levels (typically 256), whereas ferroelectric liquid crystal SLMs can switch at a rate of up to 1 kHz, but with only two phase levels (0 and π).\n\n\nCalibration\n\nAs we have seen in section 2.2, an optical trap acts as a microscopic Hookeian spring with a fixed stiffness so that the restoring force is proportional to the displacement of the optically trapped particle from the center of the optical trap. The calibration of an optical tweezers entails the determination of the value of the optical trap stiffness, which in general depends on the properties of both the light beam and the particle. The calibration of an optical tweezers is essential to use it as a microscopic force transducer, i.e., to measure and exert forces in the femtonewton and piconewton range.\n\nIn principle, it should be possible to determine the value of the stiffness for a given experimental configuration where the properties of the focused laser beam and the optically trapped particle are known. In fact, this has been done in certain cases [67,[103][104][105]. However, this is in general a cumbersome task because an exact description of the focused beam and its interaction with particles depends on several parameters that are very difficult to know accurately and may vary drastically even within the same experiment. For example, just the theoretical description of the focused beam requires precise information about the focusing lens, the thickness of the coverslip, the refractive index of the media, and the position of the focus inside the sample. Moreover, even with all this information at hand, the interaction between a focused beam and a trapped object is very complex, requiring to solve a computationally intensive electromagnetic scattering problem [22]. Therefore, the development and improvement of in-situ calibration methods have been very important to extend the practical applications of optical tweezers.\n\nThe calibration methods can be classified as either passive or active. Passive methods (sections 3.2 to 3.10) use information about the Brownian trajectory of a particle trapped in a static optical tweezers. Active methods (section 3.11) analyze the response of the particle to a known time-dependent perturbation to the optical trap exerted by an external force, e.g., generated by a periodic movement of the sample cell or of the position of the optical tweezers. The force exerted by the optical tweezers can also be directly determined by measuring how the trapping light is scattered using a digital camera or a photodetector (section 3.12). Most of the mathematical details regarding the non-linear fitting are in section 3.13.\n\nTo illustrate these calibration methods, we have carried out a series of experiments ( Table 1). The experimental data and the MATLAB implementation of the calibration techniques explained in this section are provided as supplementary materials [33]. An overview and comparison of the calibration methods is in Table 2 and the results obtained on the test datasets is in Table 3.\n\nWe remark that all methods we are going to describe rely on the knowledge of the particle position in units of length (i.e., meters). However, the detection devices used in optical tweezers setups (e.g., CCD cameras, quadrant photodiodes, and position sensitive detectors) provide signals in pixels or volts. Therefore, it is necessary to determine the conversion factor from pixels or volts to meters. For CCD camera, the pixel-to-meter conversion factor can be determined by standard microscopy techniques (e.g., by using a target with know dimensions) [22,32]. For quadrant photodiodes or position sensitive detectors, the volt-to-meter conversion factor can be determine by scanning an optically trapped particle across a (much weaker) detection beam, in which case one need to have at least two beams. It is also possible to perform this calibration self-consistently using only the position signal of an optically trapped particle acquired by a photodetector, when the diffusion coefficient D of the particle is know (e.g., when the particle radius, the medium viscosity and the temperature are known). The basic idea is that the knowledge of D determines the scaling factor of the functions of the particle position used in the calibration process. The details go beyond this Tutorial and can be found in Ref. [22].\n\n\nParameter\n\n\nValue\n\n\nParticle\n\nPolystyrene particle, radius a = Table 1. Experimental parameters. A microscopic particle immersed in water is trapped with a tightly-focused laser beam in a standard optical tweezers configuration and tracked using a quadrant photodiode at three different laser powers.\n\n\nTime series acquisition and notation\n\nWe consider a spherical particle of radius a immersed in a fluid with viscosity η at an absolute temperature T. According to Stokes' law, the friction coefficient is given by:\nγ = 6πηa,(6)\nwhile the diffusion constant is given by the Einstein-Smoluchowski relation\nD = k B T γ ,(7)\nwhere k B = 1.3806 · 10 −5 pN µm K −1 is Boltzmann's constant. The trajectory x(t) of an overdamped Brownian particle is described by the overdamped Langevin equation\ndx(t) dt = 1 γ F(x) + √ 2DW x (t),(8)\nwhere F(x) is a deterministic force exerted on the particle (in our case the optical force) and W x (t) is a white noise with zero mean and Dirac-delta-correlated variance, i.e., W x (t)W x (t ) = δ(t −t ). Experimentally, the particle's position is recorded at discrete times, yielding a time series {x } N =1 , where x ≡ x(t ) at times t = ∆t with = 1, . . . , N. To provide estimates with error bars, we can either repeat the experiment M times or divide the time series of a single experiment of size S into M blocks of N = S/M data points each. Either way, we end up with a set of times series {x (m) } N =1 for m = 1, . . . , M. A summary of the notation is in Table 4.\nPotential analysis (section 3.2) ρ(x) = ρ 0 e − U (x) k B T Result. It infers only κ x .\nAdvantages. Works for any potential (not only harmonic potentials); it does not require fast acquisition but long acquisition times (in a harmonic optical trap, T τ ot, x , where τ ot, x is the optical trap characteristic time). Disadvantages. It does not provide the dynamical properties of the optically trapped particle (e.g., γ or D).\nEquipartition method (section 3.3) κ x = k B T (x−xeq) 2\nResult. It infers only κ x .\n\nAdvantages. The simplest and easiest method to implement; it does not require fast acquisition. Disadvantages. It is restricted to harmonic potentials; it requires large datasets taken for long times T τ ot, x ; the inferred values are very sensitive to errors in the position detection.\n\n\nMean squared displacement (MSD) analysis (section 3.4)\nMSD x (τ) = 2k B T κx 1 − e − |τ | τ ot, x\nResult. It infers both κ x and γ. Advantages. it tends to provide accurate values. Disadvantages. It only applies to harmonic potentials; it is very slow due to the large number of operations in estimating MSD; it requires fitting a non-linear function; experimental errors are correlated making estimation from least squares more involved.\n\n\nAutocorrelation function (ACF) analysis (section 3.5)\nC x (τ) = k B T κx e − |τ | τ ot, x\nResult. It infers both κ x and γ. Advantages. Algorithmically faster than MSD. The fitting can be transformed into a linear one. Disadvantages. It only applies to harmonic potentials; it requires fitting a function; experimental errors are correlated making estimation from least squares more involved.\n\nPower spectral density (PSD) analysis (section 3.6)\nP x ( f ) = 1 2π 2 D f 2 c, x + f 2\nResult. It infers both κ x and γ. Advantages. Algorithmically faster than MSD and ACF; it provides accurate results; it can filter unwanted noise. Disadvantages. It only applies to harmonic potentials; one must fit a function, but this can be transformed into a linear problem. Key reference: [106] Force reconstruction via maximum-likelihood-estimator (FORMA) analysis (section 3.8)\nL {x n } N n=1 | {x 0n } N n=1\n, κ x /γ, D\n\n\nResult.\n\nIt infers both κ x and γ. Advantages. It is computationally fast to implement; it does not require a large dataset; the estimates are given by simple analytical expressions; it can be extended to any dimension and non-conservative forces. Disadvantages. It works for ∆t τ ot, x , although this can be generalized; estimates for the friction coefficient are sensitive to the exposure time. Key reference: [107] Bayesian inference analysis (section 3.9)\nP(κ x /γ, D) = L {xn } N n=1 |{x 0n } N n=1 , κx /γ, D P 0 (κx /γ, D) Z\nResult. It infers both κ x and γ. Advantages. It is computationally fast to implement; it can handle small datasets and include information a priori; the estimates are given by simple analytical expressions; it gives the whole probability of the estimates; it can be extended to any dimension and non-conservative forces. Disadvantages. it works for ∆t τ ot, x , although this can be generalized; estimates for the friction coefficient are sensitive to the exposure time. Key reference: [108] Table 2. Overview and comparison of calibration methods in one dimension in the passive configuration. In this table we summarize the practical details of each method, together with advantages, disadvantages and a key reference, whenever possible.\n\n\nPotential analysis\n\nThe potential method is fairly easy to implement and works for any conservative force with a potential that enables equilibrium of the particle in an accessible time. For a conservative force\nF(x) = − dU(x) dx , where U(x)\nis the equilibrium potential, and the probability density of finding the particle at a given position x at thermal equilibrium follows the Maxwell-Boltzmann distribution: where ρ 0 is a normalization factor ensuring that ∫ ρ(x)dx = 1. The potential U(x) can then be reconstructed as\nρ(x) = ρ 0 e − U (x) k B T ,(9)U(x) = −k B T ln ρ(x) ρ 0 .(10)\nThe expression in equation (9) is an equilibrium distribution that theoretically corresponds to the probability of finding a particle at a given position x from an statistical ensemble of independent particles. In practice, this distribution is usually computed from data sets coming from the trajectory of a particle, consisting of data points not necessarily uncorrelated. For example, in a harmonic potential, two points along the trajectory x(t) and x(t ) are uncorrelated if |t − t | τ ot, x , where τ ot, x = γ/κ x is the characteristic relaxation time of the trap. Thus, to ensure that we have enough independent samples T s ∆t τ ot, x (the second condition ensures that the samples are uncorrelated). An important limitation of the potential methods is that, to sample rare events (e.g., the anharmonicity of the potential), one requires a number of measurements N ∼ 1 ρ(x) , and depending on the point x, N can be extraordinarily large. On the other hand, being an equilibrium method, the potential method does not rely on the knowledge of the viscosity and the radius of the particle to obtain an estimate for the trap stiffness.\n\n\nNotation\n\n\nExplanation\n\nx ≡ x(t ) position of particle at time t = ∆t Table 3. 1. Determine the common maximum and minimum values of the position variable, denoted x max and x min , respectively.\n{x (m) } N\nAs an example, Fig. 10 illustrates the experimental distributions and potentials for a particle optically trapped at three different powers (corresponding to experiments I, II, and III, Table 1). By means of either a linear (black solid lines) or non-linear fit (dashed red lines), we are able to estimate the value of κ (ex)\n\nx for the experiments I, II and III (using P = 50 bins and the M = 5 experiments to estimate error bars for each bin). These estimates of κ (ex)\n\nx are given in Table 3.\n\n\nEquipartition method\n\nThe equipartition method is a very simple and easy to implement method, which does not even require fitting a function. The equipartition method is a particular case of the potential method where we assume a harmonic potential:\nU(x) = κ x 2 (x − x eq ) 2 ,(14)\nso that the particle position distribution corresponds to the Gaussian distribution\nρ(x) = κ x 2πk B T exp − κ x 2k B T (x − x eq ) 2 .(15)\nThus, the thermal average of the potential is\nU(x) = 1 2 κ x (x − x eq ) 2 = κ x 2 ∫ ∞ −∞ ρ(x)(x − x eq ) 2 dx = 1 2 k B T,(16)\nwhich is a restatement of the equipartition theorem, imposing that the energy associated with the harmonic potential degree of freedom is equal to 1 2 k B T. This provides a direct method to estimate κ x as\nκ x = k B T (x − x eq ) 2 ,(17)\nwhere we can estimate the equilibrium average (x − x eq ) 2 by a time average. As for the potential method, the trajectory needs to be long enough to probe the equilibrium distribution (T s ∆t τ ot, x ) and work with the uncorrelated subsample {x (m) } L =1 .2 For each trajectory m = 1, . . . , M, an estimate of the ensemble average (x − x eq ) 2 is given by its sample variance\nσ (ex)2 x,m = 1 L L =1 x (m) − x (ex) eq 2 ,(18)\nwhere\nx (ex) eq,m = 1 L L =1x\n(m) corresponds to the sample mean of the m-th experiment. Thus, for each experiment we obtain\nκ (ex) x,m = k B T σ (ex)2 x,m .(19)\nUsing the results of the M experiments, we calculate its mean value and variance, viz.\nκ (ex) x = 1 M M m=1 κ (ex) x,m , σ 2 (κ x ) = 1 M − 1 M m=1 κ (ex) x,m − κ (ex) x 2 .(20)\nUsing this approach on the experiments I, II and III (Table 1), yields the estimates of κ x reported in Table 3. These estimates are consistent with those obtained using the potential method albeit with larger errors. The harmonic potential corresponding to these estimates of the stiffness is plotted with black circles in Fig. 10 and is also in good agreement with the directly measured potential (blue circles).\n\nSince the estimation of κ x depends on the inverse of the variance of the position, any random error in the position detection will always lead to an underestimation of the stiffness. On the other hand, if the data position is retrieved after a band-pass filter operation, the variance will diminish and hence the stiffness will be overestimated.\n\n\nMean squared displacement analysis\n\nDifferently from the previous two methods, the mean squared displacement (MSD) analysis takes advantage of the time-correlated properties of the motion of the Brownian particle in the harmonic potential. From the Langevin equation (8) with linear restoring force F(x) = −κ x x, the MSD is:\nMSD x (τ) = [x(t + τ) − x(t)] 2 = 2k B T κ x 1 − e − |τ | τ ot, x ,(21)\nwhere (· · · ) denotes time average and τ ot, x = γ κ x is the characteristic relaxation time.3 MSD x (τ) comprises two different time regimes: for τ τ ot,x , MSD x (τ) ≈ 2k B T γ |τ|, the particle exhibits 2It is possible also to use a correlated time series as long as the sampling is done at regular time intervals. 3Following Ref. [109], we first see that the formal solution of the Langevin equation (8) is given by\nx(t) = x(0)e −t /τot, x + √ 2D ∫ t 0 dsW x (s)e −(t −s)/τot, x .(22)\nFrom here one can find, using the properties of white noise of W x (s), that\nx(t 1 )x(t 2 ) = x 2 (0)e −(t 1 +t 2 )/τot, x + Dτ ot, x e −|t 1 −t 2 |/τot, x − e −(t 1 +t 2 )/τot, x .(23)\nThis result can be used to derive the expression for the MSD x (τ) given by equation (21). free diffusion, while for τ τ ot, x , MSD x (τ) ≈ 2k B T κ x , the particle feels the harmonic potential and we recover the equipartition result given by equation (17).\n\nIf we measure MSD x (τ) experimentally (blue markers in Figs. 11(a-c)), we can estimate the trap stiffness κ x from a nonlinear fit to equation (21). This requires that ∆t < τ ot, x , such that the time series {x (m) n } N n=1 is correlated for each m = 1, . . . , M. Denoting t j = j∆t for j = 1, . . . , N and τ = ∆t, the experimental estimate for the MSD is:\nMSD (ex) x,m (τ ) = 1 N − N − j=1 [x (m) j+ − x (m) j ] 2 .(24)\nFrom the estimates {MSD (ex)\n\nx,m (τ )} M m=1 , we obtain their experimental mean and covariance matrix:\nMSD (ex) x (τ ) = 1 M M m=1 MSD (ex) x,m (τ ), ∆ [MSD (ex) ] , = 1 M − 1 M m=1 MSD (ex) x,m (τ ) − MSD (ex) x (τ ) MSD (ex) x,m (τ ) − MSD (ex) x (τ ) . Using the experimental dataset {MSD (ex) x (τ ), ∆ [MSD (ex) ] ,\n} and the theoretical model of the MSD in equation (21), we construct χ 2 according to equation (73) and minimize it numerically to find the estimates κ x and τ ot,x (red shaded lines in Figs. 11(a-c)). There are several aspects to consider when proceeding with the non-linear fitting: First, since the experimental data is time-correlated, we need the covariance matrix when inferring the parameters from the non-linear fitting. Second, as the MSD shows two distinct time-regimes, we must be sure to have a good balance of points in both regimes to avoid overweighting one respect to the other. Third, the experimental error of the MSD increases as the lag time grows since the number of data points used to estimate MSD diminishes. Fourth, errors in position detection have a strong impact particularly in the short time regime, because the MSD is proportional to the position squared. It is good practice to exclude the short times from the MSD fit, up to a fraction of the expected τ ot,x , as we did in Fig. 11.\n\nFor the experimental data recorded in experiments I, II and III (Table 1), we perform the nonlinear fitting on the interval [0.5τ ot,x , 6τ ot,x ] and estimate the errors as explained in section 3.13. This provides us with the estimates for κ (ex)\n\nx and τ (ex)\n\not, x , from which γ = κ x τ ot, x and D is given by the Einstein-Smoluchowski relation (7). All these estimates are reported in Table 3.\n\n\nExperimental MSD\n\nNon-linear fitting Fig. 11. Mean square displacement (MSD) analysis. Comparison between the theoretical expression (21) of the MSD and the experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The red dashed lines are the non-linear fitting and the blue markers with error bars are the experimental estimates. The data are fitted in the interval [0.5τ ot,x , 6τ ot,x ] (the upper limit is shown by the dashed vertical lines). The estimates are given in Table 3.\n\nThe MSD analysis has a few clear advantages over the equilibrium methods described in the previous two sections. First, the MSD uses more information so that the trap stiffness is estimated more precisely. Second, the MSD analysis can infer also dynamic properties, namely the friction coefficient, which is not directly accessible with the equilibrium methods. However, care must be taken when fitting a function with correlated errors and to have a good balance of data points between the short-time and long-time regimes.\n\n\nAutocorrelation function (ACF)\n\nStarting from the Langevin equation (8) with linear restoring force F(x) = −κ x x, the position autocorrelation function (ACF) is [109] \nC x (τ) = x(t + τ)x(t) = k B T κ x e − |τ | τ ot, x .(25)\nFrom a correlated time series sampled at regular time steps ∆t, we can estimate the ACF by using\nC (ex) x,m (τ ) = 1 N − N − j=1 x (m) j+ x (m) j ,(26)\nwhich gives a set of estimates {C (ex)\n\nx,m (τ )} M m=1 for each experiment m = 1, . . . , M (blue markers with error bars in Figs. 12(a-c)). From these estimates, we obtain their experimental mean and covariance matrix:\nC (ex) x (τ ) = 1 M M m=1 C (ex) x,m (τ ),(27)∆ [C (ex) x ] , = 1 M − 1 M m=1 C (ex) x,m (τ ) − C (ex) x (τ ) C (ex) x,m (τ ) − C (ex) x (τ ) .(28)\nUsing the the autocorrelation equation (25), we can build χ 2 (κ x , τ ot, x ) according to equation (73) and minimize with respect to the parameters κ x and τ ot, x , solving the resulting set of equations by means of non-linear procedures using iterative solutions (red dashed lines in Figs. 12(a-c)). Another possibility is to use the logarithm of the autocorrelation instead, i.e., Experimental ACF Linear fitting Non-linear fitting (25) of the ACF and the experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The black solid lines and red dashed lines are the linear and non-linear fitting, respectively, while the blue markers with error bars are the experimental estimates. The data are fitted on the interval [0, 2.5τ ot,x ] (the upper limit is shown by the dashed vertical lines). The estimates are given in Table 3. ln C x (τ) = ln(k B T/κ x ) − τ/τ ot, x and perform a linear regression, in which case the experimental dataset is {ln C (ex)\n\nx (τ )} together with their covariance matrix for the M experiments (black solid lines in Figs. 12(a-c)). As in the MSD analysis, from the estimates for κ x and τ ot, x we can determine estimates for γ = κ x τ ot, x and D = k B T/γ.\n\nAs in the MSD analysis, the ACF analysis requires ∆t τ ot,x and T s τ ot,x . Furthermore, since the ACF function decays exponentially with time, only the values corresponding to short times, around the relaxation time, are relevant to get the information of the confining potential.\n\nThe estimates for κ x , τ ot, x , γ and D performing the fittings on the interval [0, 2.5τ ot,x ] are provided in Table 1. The ACF analysis results are comparable to those obtained with the MSD analysis, both using non-linear and analytical linear regression. However, the errors for κ (ex)\n\nx are bigger than those obtained from the MSD analysis, which can be expected from the expression of the ACF (equation (25)), whose initial value is solely determined by the trap stiffness and not the decaying behavior so that we only have one data point to estimate κ (ex)\n\nx . An important difference is also the computation time: while the ACF is computed with very efficient algorithms based on the Fast Fourier Transform, the MSD requires a rather inefficient numerical computation.\n\n\nPower spectral density (PSD)\n\nCalibration of an optical tweezer by means of the power spectral density (PSD) is considered to be one of the most reliable methods due to its ability to remove common sources of noise with relative ease since one works in frequency space. In contrast to other methods, the PSD analysis is very robust to unwanted periodic noise in the extracted trajectory of the particle, due to, for example, illumination, electronic noise, fan noise of different electronic components inside the laboratory, or even some resonant vibration that is difficult to isolate with optical tables. The method explained in this section is based on Ref. [106].\n\nStarting from the Langevin equation (8) with F(x) = −κ x x and performing a Fourier transform, one obtains the PSD, which has a Lorentzian form and is the expected value of the average energy in the frequency domain, Fig. 13. Power spectral density analysis. Comparison between the theoretical expression (29) and experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The blue dots are the experimental estimates |x k | 2 T s given by equation (37), the solid red lines correspond to its expected value P (ex) α = |x k | 2 /T s n w over the n w frequencies belonging to the subdomain α, and the black dashed lines are the result of the linear fitting. The vertical dotted lines indicate the maximum frequency used to perform the fitting; in this example, f max ≡ max{ f α } = f Nyq /4 = f s /8. The estimates are given in Table 3.\nP( f ) = | x( f )| 2 = 1 2π 2 D f 2 c, x + f 2 ,(29)\nwhere x( f ) is\nx( f ) = 1 √ T ∫ T 0 x(t)e −i2π f t dt,\nand f c, x = κ x /(2πγ) = 1/(2πτ ot, x ) is the corner frequency. To derive equation (29), we have used the fact that the spectrum of the zero-mean white noise, | W x ( f )| 2 , has an exponential distribution with an expected value | W x ( f )| 2 = 1. Experimentally, we have a trajectory sampled with frequency f s = 1/∆t up to time T s , i.e., a time series x j = x(t j ) with t j = j∆t and j = 1, . . . , N, which is used to perform the discrete Fourier transform (DFT):x\nk = ∆t N j=1 e i2π f k t j x j = ∆t N j=1 e i2π jk/N x j .(30)\nIt is expected that |x k | 2 /T s ≈ |x( f k )| 2 when f s f c, x , so it is important to sample the trajectory at frequencies some orders of magnitude higher than the corner frequency. Furthermore, the discrete PSD will suffer from aliasing. A simple formula for the expected value of the aliased PSD can be derived from the discretized Einstein-Ornstein-Uhlenbeck process [106], yielding\nP (aliased) k = |x k | T s = (∆x) 2 ∆t 1 + c 2 − 2c cos(2π f k ∆t/N) ,(31)where ∆x = ((1 − c 2 )D/2π f c, x ) 1/2 and c = e −2π f c, x / f s .\nAs long as we sample at frequencies some orders of magnitude higher than the corner frequency, we can mitigate the aliasing by simply cropping the spectrum at a frequency below the Nyquist frequency f Nyq = f s /2. In principle, the expected value in the PSD can be estimated by averaging the experimental values |x k | 2 /T s over a large set of M experiments. In practice, one would need to acquire hundreds of experiments to have a good estimation. Therefore, it is common to apply a compression process to obtain a good estimation of the expected value with only one or a few experiments. Windowing is the most common compression process: it divides the frequency domain in N w narrow and equidistant subdomains with n w values, whose average represents the expected value at the corresponding mean frequency of the subdomain. This gives rise to the reduced experimental dataset for the PSD P (ex) α = |x k | 2 /T s n w with average frequency f α , with α = 0, 1, 2, . . . , N w − 1. With this procedure, the error in the expected value is σ (ex) α = P (ex) α / √ n w . If multiple realizations of the experiment are available, it is also possible to first compress the experiments and then average the results, giving an error σ (ex) α = P (ex) α / √ Mn w . Once the experimental PSD is estimated, the parameters f c, x and D are estimated by least-square fitting. Using directly the model (29) and the data set\n{ f α , P (ex) α } N w −1 α=0 with their associated weights, W α = 1/(σ (ex) α ) 2 , we can build χ 2 ( f c, x , D)\naccording to equation (73) and then apply the non-linear fitting procedure obtaining a numerical solution for the estimates. Alternatively, an analytical linear fitting can be performed using the experimental dataset { f 2 α , 1/P (ex) α } N w −1 α=0 with weights given by W α = 1/(P (ex) −1 α / √ n w ) 2 and the linear model 1/P( f ) = 2π 2 f 2 /D + 2π 2 f 2 c, x /D. Finally, an analytical solution can also be found [106] which minimizes\nχ 2 ( f 2 c, x , D) = 1 N w N w −1 α=0 P (ex) α − P( f α ; f 2 c, x , D) P( f α ; f 2 c, x , D)/ √ n w 2 .(32)\nThe analytical solutions for the estimates using χ 2 defined by equation (32) are\nf (ex) c, x = S 0,1 S 2,2 − S 1,1 S 1,2 S 1,1 S 0,2 − S 0,1 S 1,2 ,(33)D (ex) = 2π 2 S 0,2 S 2,2 − S 2 1,1 S 1,1 S 0,2 − S 0,1 S 1,2 ,(34)\nwhere\nS p,q = N w k f 2p k P (ex) q k .\nThe errors for these estimates are\nσ( f (ex) c,x ) f (ex) c, x = 1 (u − v) f (ex) c, x T s ,(35)σ(D (ex) ) D (ex) = u (x max − x min )π f (ex) c, x T s ,(36)\nwhere\nx min = f min / f (ex) c, x , x max = f max / f (ex) c,x , u = 2x max 1+x 2 max − 2x min 1+x 2 min + 2 arctan x max −x min 1+x max x min , v = 4\nx max −x min arctan 2 x max −x min 1+x max x min , and f min and f max correspond to the minimum and maximum on the list of frequencies { f α }.\n\nThe blue dots in Figs. 13(a-c) show the values |x k | 2 /T s computed by a fast Fourier transform for the trajectories acquired in experiments I, II, and III (Table 1):\n|x k | 2 T s = ∆t 2 T s |FFT{{x j } N j=1 } k | 2 with f k = k T s for k = 0, 1, 2, . . . , N/2.(37)\nTheir expected values are obtained by compression using N w = 500 (solid red lines in Figs. 13(ac)) and used to fit the Lorentzian function given by equation (29) using equations (33) and (34) (dashed black lines in Figs. 13(a-c)). To avoid aliasing (which gives rise to the plateau at high frequencies), the fitting was performed only with data corresponding to frequencies smaller than f max = f Nyq /4 = f s /8. From this fitting, we obtain the estimates for f c, x and D and, from these, the estimates for κ x , τ ot,x , and γ. Repeating this procedure for M = 5 experiments, we obtain the mean values of the estimates and their errors. All these estimates are reported in Table 3.\n\n\nDrift method\n\nThe drift method [110][111][112][113] relies on the fact that the drag force must be equal to the optical force. From the Langevin equation (8), this implies that\nF(x 0 ) = 6πηa ∆x x 0 ∆t ,(38)\nwhere ∆x x 0 is the local drift, which is the average displacement of a particle when located at initial position x 0 and under the external force F x 0 during a time interval ∆t. From an experimental trajectory the local drift can be estimated as the conditional average\n∆x x 0 = 1 S S i=1 [x i (t + ∆t) − x i (t) | |x i (t) ≈ x 0 ] ,(39)\nwhere the S corresponds to the number of times the initial position of the particle is close to x 0 . When applying this method there are several caveats. First, the size of the volume element near x 0 must be small enough to probe correctly the spatial variations of the force field. Second, the position of the particle must be measured with a precision much smaller than that of the volume element. Finally, equation (38) is valid when the diffusion coefficient is constant, while the presence of a space-dependent diffusion coefficient (e.g., due to hydrodynamics) leads to the emergence of spurious drift [112,113].\n\nIn the presence of a spurious drift, a diffusion gradient term need to be added to equation (38) leading to\nF(x 0 ) = γ(x 0 ) ∆x x 0 ∆t − αγ(x 0 ) dD(x 0 ) dx 0 ,(40)\nwhere α may take values in the interval [0, 1], depending on the specific stochastic process being modelled. In particular, α = 1 corresponds to a colloidal particle in a thermal bath [114][115][116].\n\n\nForce reconstruction via maximum-likelihood-estimator analysis (FORMA)\n\nThe force reconstruction via maximum-likelihood-estimator analysis (FORMA) was introduced recently to retrieve the force field acting on a Brownian particle from the analysis of its displacements [107]. This method is able to estimate accurately the conservative and nonconservative components of the force field with important advantages over the previously discussed techniques, being parameter-free, requiring ten-fold less data and executing orders-ofmagnitude faster [107]. Let us consider a particle in a harmonic potential and suppose that after a time interval ∆t τ ot, x the Brownian particle is at a position x, having started at position x 0 . Repeating this measurement N times, the likelihood of observing this set of measurements is\nL({x n } N n=1 |{x 0n } N n=1 , κ x /γ, D) = 1 (4πD∆t) N /2 exp − 1 4D N n=1 ∆t x n − x 0n ∆t + x 0n κ x γ 2 ,(41)\nwhich depends on the parameters κ x /γ and D. By maximizing the logarithm of the likelihood given by equation (41) with respect to the parameters κ x /γ and D, one obtains the so-called maximum likelihood estimators (MLEs) for a given time series {x (m) } N =1 :\nκ (ex) x,m γ = − N −1 n=1 x (m) n x (m) n+1 −x (m) n ∆t N −1 n=1 [x (m) n ] 2 , D (ex) m = ∆t 2N N −1 n=1 x (m) n+1 − x (m) n ∆t + κ (ex) x,m γ x (m) n 2 .(42)\nFrom these equations and the Einstein-Smoluchowski relation (7), it is straightforward to estimate κ (ex)\n\nx,m , γ, and D (ex) m . The standard errors for these estimates can be obtained from the mean and standard deviations from repeated experiments: using multiple time series m = 1, . . . , M to obtain the estimates {κ (ex) x,m } M m=1 and {D (ex) m } M m=1 yields the experimental mean and variances:\nκ (ex) x = 1 M M m=1 κ (ex) x,m , ∆ (κ x ) = 1 M − 1 N exp m=1 κ (ex) x,m − κ (ex) x 2 , D (ex) = 1 M N exp m=1 D (ex) m , ∆ (D) = 1 M − 1 M m=1 D (ex) m − D (ex) 2 .(43)\nFORMA provides also an alternative way to get error estimates for the MLEs given by equation (42) for a single experiment, without having to repeat the experiment. Indeed, if we perform a Taylor expansion of the log-likelihood around the MLEs given by equation (42) up to second order, we obtain a Gaussian approximation for L({x n } N n=1 |{x 0n } N n=1 ). This yields the following standard deviations for κ (ex)\n\nx and D (ex)\nσ D (ex) = D (ex) 2 N , σ κ (ex) x = 2D (ex) ∆t N n=1 x 2 0n .(44)\nWhen applying FORMA, ∆t τ ot, x and the exposure time should be much smaller than ∆t (i.e., as close as possible to an instantaneous recording of the particle position).\n\nWe report the estimates for the experimental data in Table 3, obtained using formulas (42) and (43). Comparing these results with the ones obtained by the other methods, we can see some discrepancies in the diffusion coefficient and hence in the friction coefficient. This is due to the fact that the condition that ∆t τ ot, x and that the exposure time be much smaller than ∆t does not hold true as the laser power is increased, causing the relaxation time to decrease.\n\nSince FORMA only depends on the local variables of the trajectory, that is the positions and their corresponding displacements, it allows to retrieve the forces of arbitrary optical landscapes, including non-conservative forces, and it does not need a trajectory with regular sampling, commonly required in the other methods. This is particularly useful when the problem is to retrieve the forces of extended potentials, where the particle neither reaches the equilibrium nor is able to explore the whole space by itself, such as for a dual optical tweezer or a speckle pattern [107]. The main drawback in using FORMA is the need for high spatial resolution and fast sampling. The latter can be addressed by generalizing the solution for any arbitrary time interval ∆t, as recently introduced in Ref. [117].\n\n\nBayesian inference\n\nWhile FORMA uses a non-Bayesian linear regression to infer the properties of the trap and the Brownian particle [107], other methods have used Bayesian inference to infer optical forces [117][118][119][120][121]. Recently, we have derived a Bayesian approach using conjugate priors, which renders very simple formulas for the posterior distribution, with known moments at all orders and marginal distributions [108]. Here, we provide an overview of this method.\n\nLet us suppose that we want to infer D and κ x /γ. Let P 0 (κ x /γ, D) be their a priori probability distribution. Then, given a time series, the posterior probability is given by Bayes' rule:\nP(κ x /γ, D) = L({x n } N n=1 |{x 0n } N n=1 , κ x /γ, D)P 0 (κ x /γ, D) Z(45)\nwith Z being a normalization factor. Choosing the a-priori probability P 0 (κ x /γ, D) to be a so-called conjugate prior for the model's likelihood,4 we obtain the following marginal posterior distribution for the trap stiffness and the diffusion constant\nP D (x) = InvGamm (x|α N , β N ) , P κ x /γ (x) = Γ 2α N +1 2 Γ(α N ) 1 √ 2πα N α Ñ γ N β N 1 + 1 2α N α N β NγN (x − K N ) 2 −(2α N +1)/2 ,(46)\nwhere InvGamm(x) is the inverse Gamma distribution, and the parameters α N , β N , γ N , and K N 4A conjugate prior is a particular type of a priori distribution, chosen so that the posterior distribution belongs to the same family of distributions of such prior. One may intuitively think of conjugate priors as eigenfunctions of equation (45). are determined by the measurement record:\nβ N = β 0 + 1 2 N n=1 [x n+1 − x n (1 − K N ∆t)] 2 ∆t + 1 2γ 0 (K N − K 0 ) 2 , α N = α 0 + N 2 , K N = − 1 ∆t −1 + ∆t N n=1 x n+1 x n + 1−K 0 ∆t γ 0 1 γ 0 + ∆t N n=1 x 2 n , 1 γ N = 1 γ 0 + ∆t N n=1 x 2 n ,(47)\nThe experimental estimates D (ex) and κ (ex)\n\nx are the mean values and variances of the posterior distributions P D and P κ x /γ :\nD (ex) = ∫ dx xP D (x) = β N 2(α N − 1) , σ 2 D (ex) = β 2 N 4(α N − 1) 2 (α N − 2) , κ (ex) x γ = ∫ dx xP κ x (x) = K N , σ 2 κ x /γ = γ N β N α N − 1 .(48)\nFormulas for higher cumulants can also be calculated analytically from equations (46). The hyper-parameters α 0 , β 0 , γ 0 , and K 0 , determine the a priori information about the model's parameters. For example, α 0 , β 0 , γ 0 , and K 0 can be chosen such that for N = 0 (i.e., without experimental data), we obtain their theoretical values and systematic uncertainties (i.e., D (th) ± σ D (th) and κ (th)\n\nx /γ ± σ κ (th) x /γ ):\nα 0 = 2 + (D (th) ) 2 σ 2 D (th) , β 0 = 2D (th) [(D (th) ) 2 + σ 2 D (th) ] σ 2 D (th) , K 0 = κ (th) x γ , γ 0 = σ 2 κ (th) x /γ 2D (th) .(49)\nFrom Table 1, D (th) = 0.21 µm 2 s −1 and, adding an error of 10 %, allows us to fix the values α 0 and β 0 . For the theoretical value of the inverse relaxation time, we are more in the dark, as we do not know the trap stiffness before the calibration procedure; thus, we can use an estimate obtained by another method to determine the values of K 0 and γ 0 . Using Bayesian inference on one of the time series {x (m) n }, we obtain the results reported in Table 3. For the estimates regarding the friction and diffusion coefficients, we notice that the present method has the same issue concerning the exposure time as FORMA.\n\nThe Bayesian inference method extracts all statistical information from the measured data, so that it provides an unbiased estimator of the experimental parameters. Moreover, it can provide reasonable estimates of the parameters also when little data is available, thanks to the use of prior information. The Bayesian inference method can also be generalized to more than one dimension for any time interval ∆t.\n\n\nMeasurement of non-conservative force fields\n\nWhile the previous methods rely on the forces being conservative, non-conservative forces often play also a role in experiments. The drift method can be straightforwardly applied to measure any force field [111]. Also, the ACF method has been adapted to measure non-conservative forces using cross-correlations [122][123][124]. Here, we briefly show how FORMA can be adapted to measure non-conservative forces [107], while the details for the Bayesian method can be found in [108].\n\nLet us consider the two-dimensional overdamped Langevin equation\nr = 1 γ F(r) + √ 2Dw,(50)\nwhere F(r) is the force field and w is a vector of independent white noise. The Taylor expansion of F(r) around r = 0 yields\nF(r) = F 0 + J 0 · r + O(r 2 ),(51)\nwhere F 0 = F(0) and J 0 = J(0) are the force and the Jacobian at the point r = 0, respectively. We further assume r = 0 to be an equilibrium point so that F 0 = 0. Discretizing the Langevin equation (50), we obtain\nf n = γ ∆r n ∆t = J 0 · r n + σw n ,(52)\nwhere σ = 2Dγ 2 ∆t and w n is a vector of independent Gaussian numbers with zero mean and unit variance. Equation (52) can be understood as a multivariate linear regression model: for a given set of pairs of data points {(r n , f n )} N n=1 , we construct the likelihood for (52), similarly as the expression (41) for the one-dimensional case, and maximize it with respect to J 0 to obtain the corresponding MLE J 0 , which is given by the Moore-Penrose pseudo-inverse:\nJ 0 = R T R −1 R T F,(53)\nwhere R and F are N × 2 matrices given by\nR = r T 1 . . . r T N , F = f T 1 . . . f T N .(54)\nFrom the MLE J 0 , we can estimate the force as F = J 0 · r. This result can be naturally split into conservative and non-conservative parts, corresponding to the symmetric and antisymmetric decompositions of J 0 = J c + J r , respectively. The symmetric part can be rewritten as\nJ c = 1 2 J 0 + J T 0 = R(θ ) −κ 1 0 0 −κ 2 R −1 (θ ),(55)\nwith R a rotation matrix and κ 1 and κ 2 the trap stiffness along the principal axes; θ represent the orientation of these axes with respect to the Cartesian system of reference. The rotational part is in turn given by\nJ r = 1 2 J 0 − J T 0 = 0 −γΩ γΩ 0 ,(56)\nwith Ω being the angular frequency at which the particle would orbit around the center of the trap in the absence of Brownian noise.\n\n\nActive calibration techniques\n\nAll calibration methods discussed so far are passive, i.e., they are based on the determination of the trap stiffness from the stochastic motion of the particle in thermal equilibrium with the surrounding fluid within a harmonic potential. Alternatively, the optical tweezers calibration can be performed by actively applying an external force on the particle and measuring its response. A simple active method consists of exposing an optically trapped particle to an external uniform flow in the x-direction with a known velocity v fluid so that the particle experiences a mean drag force F drag = −γv fluid . Consequently, the particle undergoes a mean displacement ∆x = x − x eq from the trap center x eq so that the restoring optical force balances the viscous drag. The stiffness can then be measured as\nκ = γv fluid ∆x .(57)\nIn practice, a uniform flow can be applied by moving the sample at constant speed −v fluid , e.g., by means of a piezoelectric translation stage, while keeping the particle far away from the cell walls to avoid hydrodynamics interactions which change the value of friction coefficient γ with respect to the value given by Stokes' law [125]. Another active method requires moving sinusoidally the position of the sample stage x stage , i.e.,\nx stage (t) = A sin(2π f stage t),(58)\nwhile keeping the optical tweezers at fixed x eq = 0 [126]. This method not only provides the value of the trap stiffness, but also the drag coefficient and the spatial conversion factor of the measured particle position x (typically in volts or pixels) into actual distances. The Langevin equation for the motion of the particle in the case of the oscillating stage is\ndx(t) dt − v stage (t) = −2π f c, x x(t) + √ 2DW x (t),(59)\nwhere v stage (t) = 2π f stage A cos(2π f stage t) is the velocity of the stage and W x is a white Gaussian noise. The corresponding PSD of the particle position is\nP( f ) = D π 2 1 f 2 + f 2 c, x thermal contribution + A 2 2 1 + f 2 c, x f 2 stage δ( f − f stage ) mechanical response ,(60)\nwhere the thermal contribution is equal to that given by equation (29), while the mechanical response is that of the particle to the excitation frequency f = f stage . Because the measurement time in an experiment is finite, t msr < ∞ , the Dirac-delta peak in equation (60) becomes a spike with finite height P stage = P( f stage ) and width ∆ f . If the total measurement time is chosen to be an integer multiple of the driving frequency of the stage, then the spike reduces to a single datum P J at frequency f J = f stage for some integer J, whereas ∆ f = 2δ f , where δ f corresponds to the frequency resolution of the PSD. In this case, the discrete version of equation (60) for the j-th data point is\nP j = D π 2 1 f 2 j + f 2 c, x + A 2 t msr 2 1 + f 2 c, x f 2 j δ j,J ,(61)\nwhere δ j.J represents the Kronecker delta. Therefore, disregarding the single point ( f J , P J ), a fit P fit ( f ) to the one-sided Lorentzian function in equation (61) allows to compute the diffusion coefficient D (1) and the corner frequency f c, x (section 3.6), from which the trap stiffness can be determined as\nκ (1) = 2π f c,x k B T D (1) .(62)\nIf x has not been yet converted into units of length, the values and units of D (1) and κ (1) are not correct. A further step using the active component of the spectrum in equation (61) allows to get the calibration factor α of x into units of length: x → αx, where α can have units of m/V or m/pixel depending on whether the particle position was recorded with a quadrant photodiode, a position sensitive detector or a camera. The power contained in the triangular spike centered at\nj = J, W exp = 1 2 [P J − P fit ( f J )]2δ f , where P fit ( f J ) = D (1) /π 2 f 2 J + f 2 c, x\n, and the theoretical value given in\nequation (60), W th = A 2 2 1+ f 2 c, x / f 2 stage\n, must be equal. Then, α 2 W exp = W th , from which\nα = A 2 1 + f 2 c, x f 2 J [P J − P fit ( f J )]δ f .(63)\nHence, the corrected values with the right units of the trap stiffness and the friction coefficient are\nκ = 2π f c, x k B T α 2 D (1)(64)\nand\nγ = k B T α 2 D (1) .(65)\nAn alternative to the aforementioned method is to move the position of the center of the trap, while keeping the sample cell at rest, i.e. x stage = 0 and x eq (t) = A sin(2π f eq t). This can be achieved for example with an acousto-optic deflector or a galvomirror. In this case, unlike the case of an external flow, an effective oscillatory force k x eq (t) = k A sin(2π f eq t) is exerted on the trapped particle and the corresponding Langevin equation is\ndx(t) dt = −2π f c, x x(t) − x eq (t) + √ 2DW x (t).(66)\nThe corresponding power spectral of the particle position is given in this case by\nP( f ) = D π 2 1 f 2 + f 2 c,x + A 2 2 1 + f 2 eq f 2 c, x δ f − f eq .(67)\nwhile its discretized form is\nP j = D π 2 1 f 2 j + f 2 c, x + A 2 t msr 2 1 + f 2 j f 2 c, x δ j,J ,(68)\nThus, once the PSD of the particle position is computed, the calibration of α, κ and γ proceeds similar to the case of the oscillatory flow created by moving the stage (equations (62)-(65)). Active calibration methods can also be applied to less trivial situations, for instance for particles trapped in viscoleastic fluids. In such a case, the instantaneous drag force on the particle at time t is not simply given by γ dx(t) dt as in equations (59) and (66), but it is related to the frequency-dependent rheological properties of the medium, which are a priori unknown. This will be discussed in detail in section 4.3.\n\n\nDirect optical force measurement\n\nAll the calibration methods discussed until now provide an indirect measurement of the optical forces, relying on the properties of the motion of the particle in the optical potential, and therefore depend on the assumption of a spherical particle and a harmonic optical potential. It is also possible to measure directly the optical force [127][128][129].\n\nDirect measurement of optical forces is based on recording the change of momentum between the ingoing and outgoing light, since the exerted optical force is equal to the change in the momentum flux of the trapping light: let I(θ, ϕ) be the intensity scattered into the direction (θ, ϕ) given in spherical coordinates, then the momentum flux density p of the outgoing light is\np = 1 c I(θ, ϕ) sin θ cos ϕ sin θ sin ϕ cos θ .(69)\nThe optical force acting on the particle corresponds to the difference between the total momentum flux of ingoing and outgoing light, where the incoming momentum flux is given by precisely the same formula but with opposite sign. To correctly apply this method, one needs to detect all of the scattered light. Further details of how this can be achieved experimentally can be found in Refs. [127][128][129]. Strickly speaking this is a rather unfeasible task. However, for most practical applications in which the refractive index of the observed particle is similar to that of the solvent, light scatters only slightly, making its experimental collection possible.\n\n\nMathematical details on error analysis and fitting\n\nMost of the calibration techniques described above use a physical observable Ψ that depends on a single trajectory. Let us denote its experimental estimate for trajectory m as\nΨ (ex) m (α) ≡ Ψ (ex) m (α, {x (m) } N =1 )\n, where α may denote either a time dependence or a spatial one. For example, in the potential method α represents the position associated to the histogram bins, while for the MSD α corresponds to the discrete time. We can then use the M experiments to calculate the experimental mean and the experimental covariance of Ψ. These are:\nΨ (ex) (α) = 1 M M m=1 Ψ (ex) m (α) ,(70)∆ (Ψ) α,β = 1 M − 1 M m=1 Ψ (ex) m (α) − Ψ (ex) (α) Ψ (ex) m (β) − Ψ (ex) (β) + 2 η 1 ,η 2 ,... (α). (71)\nImportantly, the experimental covariance matrix ∆ (Ψ) α,β in equation (71) has two terms. The first term corresponds to the experiment-to-experiment fluctuations that arise naturally due to the intrinsic randomness of the physical problem as well as due to the correlation between different points α and β. The second term ( 2 η 1 ,η 2 ,... ) captures the variances due to some known error in some variables (η 1 , η 2 , . . .) that are needed to compute Ψ (ex) q (α) as, for instance, uncertainties δα in the value of α, which can be estimated by standard error propagation techniques.5 5Using propagation of errors, the contribution to the total variance by the errors in the uncorrelated variables\nη 1 , η 2 , . . . is 2 η 1 , η 2 , ... (α) ≈ 1 M M m=1 ∂Ψ (ex) m (α) ∂η 1 2 δ 2 η 1 + ∂Ψ (ex) m (α) ∂η 2 2 δ 2 η 2 + . . . ,(72)\nwhere δ η 1 , δ η 2 , . . . are the errors associated to the corresponding variables [130].\n\nOnce Ψ (ex) (α) and ∆ (Ψ) α,β have been calculated, they are fitted to the corresponding physical observable Ψ(α; θ) to infer the unknown parameters θ = (θ 1 , . . . , θ q ) of the model. There are several approaches to perform this task. The most popular are based on least-square regressions (LSR), which defines the best estimates of the parameters as those that minimize the mean squared error.6 A fantastic book dealing with these matters is that by Peter Young [130].\n\nThe general least-square method used in this Tutorial relies on the weighted χ 2 function\nχ 2 (θ) = P α,β=1 Ψ (ex) (α) − Ψ(α; θ) R α,β Ψ (ex) (β) − Ψ(β; θ) ,(73)\nwhere R αβ is a symmetric semi-positive definite matrix that captures the correlation in the experimental dataset. The goal is to find the set of parameters θ ( ) that minimizes χ 2 (θ).\n\nWhen R is taken to be [∆ (Ψ) ] −1 , this is the so-called generalized least-square method or the correlated χ-squared method. The standard weighted least-square method is recovered when the off-diagonal terms of the experimental covariance are assumed to be zero so that\n[∆ (Ψ) ] −1 αβ = diag 1/∆ (Ψ) 11 , . . . , 1/∆ (Ψ)\nPP . If we do not have access to experimental errors, we can perform the standard least-square method by assuming the matrix ∆ (Ψ) to be equal to the identity matrix. In some cases, the inverse of the sample covariance matrix is an ill-posed problem and it is highly recommended to use the standard method of weighted least squares but including the experimental covariance matrix in the error estimation of the parameters [132]. This is the approach we have used to estimate errors for the MSD and ACF methods.\n\nIf the expression χ 2 (θ) is a simple function of the parameters, the minimization can be carried out analytically. For example, this is the case when Ψ(α; θ) is linear with respect to the model's parameters, that is\nΨ(α; θ) = q j=1 θ j Φ j (α),(74)\nwhere {Φ j (α)} is a set of functions, giving rise to the well established linear least-square fitting.\n\nIn some cases, a simple redefinition of variables can transform a non-linear fitting into a linear one, as, for instance, in the potential method. When it is not possible to obtain explicit forms for θ ( ) , one must rely on numerical minimizers, being the standard ones the Levenberg-Marquardt-Girard-Wynne-Morrison and the Trust-Region, implemented already in most of the standard computational packages for mathematical analysis, such as Python, MATLAB, Mathematica, and R. However, the computational methods given in these packages do not normally allow to use the experimental covariance matrix to estimate the covariance matrix of the estimated parameters. In this case, it is recommended to use the packages provided in [132], which we also have implemented in this Tutorial.\n\n\nApplications\n\nWe will now describe how optical tweezers have been used in several advanced applications. For each field of application, we provide a brief introduction and a detailed description of some paradigmatic experiments. In this section, we focus on experiments with particles optically trapped in a liquid environment, while we discuss experiments with particles optically trapped in air or vacuum in section 5.\n\n\nSingle-molecule mechanics\n\nThe study of single-molecule mechanics using force spectroscopy techniques has rapidly expanded over the past three decades [133][134][135][136][137][138][139][140]. Many labs worldwide use single-molecule methods 6Less common, but very powerful methods, are the robust least-squares, including the least absolute regression (LAR) and the Bisquare weights [131], both very useful to minimize the effect of outliers.\n\nto manipulate and track individual molecules. Broadly speaking, single-molecule methods can be classified into two large families depending on whether they manipulate molecules by exerting forces or they visualize molecules in action using appropriate markers. The latter include single-molecule fluorescence used to monitor the time evolution of fluorescently labeled molecular complexes in vitro and in vivo. Manipulation methods are capable of exerting forces from femtonewton to nanonewton. For example, they have been used to study the elastic properties of biopolymers [141], the thermodynamics and kinetics of intramolecular (e.g., molecular folding) [142,143] and intermolecular (e.g., ligand binding) interactions [144], molecular motors [145], and fundamental problems in nonequilibrium physics [146,147]. The field of force spectroscopy includes techniques such as laser optical tweezers [22], magnetic tweezers [148], atomic force microscopy [149], acoustic force spectroscopy [150], electrostatic trapping [151], microfluidics [152], microneedles [153], biomembrane force probe [154], and centrifugal force [155]. These techniques tend to compromise between measurement accuracy and high-throughput capabilities. While magnetic tweezers and acoustic force spectroscopy achieve high throughput, optical tweezers and atomic force microscopy remain the most accurate techniques in the low force (piconewton) and high force (nanonewton) regimes, respectively. Piconewton forces are the most relevant at the molecular level, making optical tweezers an excellent technique to accurately characterize molecular reactions.\n\nIn this section, we describe in detail how optical tweezers can be used to investigate molecular folding using short DNA hairpins (a double-helix stem of a few tens of base pairs terminated by a loop) as model systems. We will describe how to synthesize a short DNA hairpin, how to unzip it by stretching it while holding it between a micropipette and an optical tweezers, and how to analyze the obtained data. This experiment provides an ideal example of the use of single-molecule mechanics to determine elastic behaviour of biopolymers, to measure the intramolecular weak interactions that stabilize native molecular structures, and to extract the free energy of biomolecule formation from out-of-equilibrium pulling experiments. The experimental setup we use is based on the miniTweezers setup developed by Smith, Cui and Bustamante [127].\n\n\nMicrofluidics chamber\n\nThe schematic of the experimental setup is presented in Fig. 14(a). The experiments are performed in a microfluidics chamber placed vertically within the setup. As schematically shown in Figs. 14(b) and 14(c), this microfluidics chamber has three channels: we will refer to the three channels as the upper, central and lower channel. The propagation of both laser beams is perpendicular to the chamber surface (z-axis). The experimental area is restricted to the central channel, where the object of study (biomolecule or cell) is held by two beads, one held by the optical trap and the other held by a glass micropipette, as shown in Fig. 14(d). The upper and lower channels are used to supply the two types of coated beads used in the experiments.\n\nThe microfluidics chamber is realized by sandwiching two layers of parafilm (Parafilm M, Bemis) between two coverslips (No. 2, dimensions 24 mm × 60 mm × 200 µm), as shown in Fig. 14(c). In detail, the steps to prepare the chamber are:\n\n1. The entrance and exit holes are drilled in one of the glass coverslips using a laser cutter.\n\nThe coverslips are cleaned with a solution of 70% ethanol.7 tube (King precision glass, Inc., inside diameter=0.04 mm, outside diameter=0.08 mm, length=6.00 mm, glass type KG-33), as described in section 4.1.3.\n\n\n5.\n\nThis glass micropipette is placed on top of the parafilm layer perpendicular to the channels with the tip positioned in the central channel, as shown in Figs. 14(b) and 14(c).\n\n6. The upper and lower channels are connected to the central one via glass dispenser tubes (King precision glass, Inc., inside diameter=0.04 mm, outside diameter=0.10 mm, length=6.00 mm, glass type KG-33) cut using a diamond-tip cutter to obtain a clean cut.\n\n7. The second parafilm layer is placed on top and the chamber is closed using the second glass coverslip (previously cleaned with 70% ethanol), as shown in Fig. 14(c).\n\n8. The chamber is sealed by heating it on a hot plate at 120 • C while exerting a pressure of about 1 Kg in all the chambers' surface (either by placing a weight on top of the microfluidics chamber or, more simply, by exerting pressure by hand). To prevent the glass from breaking, it is recommended to sandwich the chamber between two thicker glass slides to homogenize the pressure applied to their surfaces.\n\n9. The chamber is placed in the metallic mount, as shown in the inset in Fig. 14(a). The correct alignment between the holes drilled in the glass coverslip and the holes in the mount is critical to ensure the flow in the chamber. It is also very important to tighten the screws to avoid buffer losses between the plastic tubes and the glass chamber, since this would cause flows inside the chamber. Do this carefully, because tightening the screws too much could break the glass coverslip.\n\n10. The side of the micropipette coming out of the chamber is cut to get rid of the excess tube, leaving only about ∼ 3 cm, and is introduced into a polyethylene tube (polyethylene tubing, Warner Instruments, PE-10/100; outside diameter = 0.61 mm, inside diameter = 0.28 mm). This tube is then fixed to the mount with tape. The pipette is easily breakable, so the tube needs to be placed as straight as possible. The connection is then sealed by using a special glue (Norland, NOA-61; UV Curing Optical Adhesives). The glue is placed between the the glass tube of the pipette and the polyethylene tube to fill the void outer space between tube and pipette via capillarity. After observing that the glue has entered into the tube, the UV-curation is performed by leaving the whole chamber for ∼ 25 min under a UV lamp.\n\nSince the glue could reach the end of the glass tube of the micropipette and block it, the chamber has to be placed under the UV-radiation as fast as possible. After the curation, a syringe (1 ml Luer Lock, HSW SOFT-JECT U100 Insulin Henke Sass Wolf) is inserted using a needle (BD Microlance 30G × 1/2\" -0.30 mm × 13 mm) at the end of the tube to create suction at the tip of the micropipette.\n\n11. The holes of the mount are connected with silicone-rubber tubes (Tygon 3350, Saint-Gobain, .031 ID X .093 OD X 50 FT TYGON 335) via nylon socket screws (Nylon set 8 − 32 × 3/8\", Product-Components, previously drilling a hole with a number 45 drill bit, of 0.82\"), as shown in Fig. 14(c). A segment of ∼ 20 cm polyethylene tube (Polyethylene tubing, Warner Instruments, PE-50/100: outside diameter = 0.97 mm, inside diameter = 0.58 mm) is inserted into the silicone-rubber tubes. The three exit tubes are connected to a trash (any small plastic container with a capacity volume ∼ 100 ml), while the three entry channels are connected to a syringe using a polyethylene tube (Polyethylene tubing, Warner Instruments, PE-50/100: outside diameter = 0.97 mm, inside diameter = 0.58 mm) which is inserted into the silicone-rubber tubes on one end, and a needle (HSW FINE-JECT 23G × 1\" -0.6 mm × 25 mm).\n\n\nThe miniTweezers setup\n\nThe miniTweezers [127] (Fig. 14(a)) consists of two focused counter-propagating laser beams (P = 200 mW, λ = 845 nm) that create a single optical trap [156]. It employs high-NA objectives (NA = 1.2), but underfills them to be able to collect almost all scattered light to measure the change of light momentum using position sensitive detectors (PSDs), as described in section 3.12.\n\nBefore the objectives, a pellicle diverts ∼8% of each laser beam to a secondary PSD to determine the optical trap position. The remaining ∼ 92% is collimated by using a lens and it is introduced into the optical axis by using a polarizing beam-splitter that selects the horizontally polarized light. Then the linearly polarized light of each laser is circularly polarized by a quarter waveplate8. The laser beams are focalized by their corresponding objectives and form the optical trap, which can hold dielectric objects with a refraction index higher than the surrounding aqueous medium (e.g., polystyrene beads) and exert forces to them. The exiting light is collected by the opposite objective and is converted to vertically polarized light by another quarter-wave plate. The vertically polarized light can be extracted from the optical path using two polarizing beam-splitters and relay lenses that redirect the light to the PSDs that measure the intensity of the beam (i.e., its exerted force). The miniTweezers has a resolution of 0.1 pN and 1 nm at a 1 kHz acquisition rate. The chamber where the experiments are performed (section 4.1.1) is placed between the two objectives and held with a metallic mount with three stages to permit movement along the x-, yand z-directions. There are two possible ways to manipulate the position of the optical trap with respect to the chamber depending on the precision that is needed. For large displacements (up to hundreds of micrometers), the whole chamber is displaced along the x-, y-and z-directions using stepmotors. For fine displacements (less than a few micrometers, but with a resolution of ∼ 1 nm and rates of displacement up to ∼ 50 nm s −1 ), the optical trap is displaced along the xand y-directions by a 2D piezoelectric motor attached to the tip of the optical fiber (wiggler) used to inject the laser into the optical setup.\n\nBefore any experiment, the setup needs to be aligned going through the following steps:\n\n1. Remove any trapped object.\n\n2. Move the chamber with the stepmotors so that the micropipette is a few micrometers away from the optical trap, in the so-called \"working zone\".\n\n3. Remove any voltage applied to the piezoelectric motor.\n\n4. Turn off the LED and remove the light filter, to be able to see the optical trap on the camera.\n\n\n5.\n\nDecrease the laser power until it is possible to distinguish both lasers.\n\n6. Using the screws of the kinematic mount, move the B laser (the one that has the real image shown on the screen) until it is superposed on the A laser.\n\n7. Put back on the light filter, turn on the LED, and increase the laser power to its working value.\n\n8. Set the current values recorded by the PSD as the zero force baseline along the xand y-directions (center of the light spot) and along the z-direction (size of light spot).\n\n9. Trap a bead with the optical trap by moving the chamber with the stepmotors close to the appropriate dispenser tube. 10. Bring the trapped bead to the working zone.\n\n11. Fine-tune the alignment of the B laser using the screws of the kinematic mount controlling the wiggler of its fiber so that the xy-force signals from both force-PSDs are as close as zero as possible, i.e., both lasers exert an xy-force as close to zero as possible.\n\n12. Fine-tune the position of one of the objectives along the z-directions so that the z-force from both force-PSDs is as close to zero as possible.\n\n8The use of quarter-wave plates ensures that the light coming from the two laser beams do not interact with each other and guarantees that the light reflected from the particle is not returned to the laser but is reflected to the opposite PSD [157]. 13. For both lasers, move the kinematic mount of the mirror that deflects the light that gets towards the position-PSD (between the aspherical lenses and the PSD) until its signal is zeroed (i.e., the light hits the center of the position-PSD).\n\n\nPipette making\n\nA micropipette is used to hold a microparticle by air suction, as shown in Fig. 14(b). The tip of this micropipette needs to have an inside diameter ∼ 1 µm, large enough to exert sufficient suction and small enough not to let the microparticles flow inside it. While one might use a commercial pipette puller to produce the pipettes, here, we will explain how a glass tube can be pulled to produce such micropipette using the homemade device shown in Fig. 15 [127,158]. This homemade pipette puller consists of a plastic platform with two parallel metallic bars, which hold a stage and along which a plastic weight can slide up and down. The two ends of the glass tube are fixed with screws to the stage and the weight, respectively. Thus, the glass tube is held in tension due to the pull exerted by gravity on the weight. The central part of the glass tube passes through a platinum wire; the two ends of the wire are connected to an electric supply, which provides a current ramp from 0 to ∼ 6 A in ∼ 8 s. The exact maximum intensity and time of the ramp have to be tuned to get the appropriate diameter and shape of the pipette (in general, longer times and higher intensities correspond to smaller tips). Specifically, the steps required to produce a micropipette are:\n\n1. Unplug the wire from the electric supply and place the pipette puller horizontally.\n\n2. Insert the glass tube through the stage, the platinum coil, and the plastic weight. Fig. 15. Pipette puller. First, the glass tube is carefully centered within a coiled platinum wire, as shown in the inset. Then, one end of the tube is attached to the puller, while the other is attached to a weight that pulls the tube down. Finally, an electric current intensity ramp is applied through the platinum wire, heating the adjacent glass tube, which during the melting is pulled down, creating a micropipette. 3. Carefully center the glass tube within the platinum wire. 4. Screw the glass tube at the stage and plastic weight. 5. Place the platform vertically (as shown in Fig. 15), carefully to avoid any sudden hit that may break the glass tube.\n\n6. Apply the current ramp. The weight will drop into the platform. The geometry of the platinum wire is of a critical importance to obtain the proper shape and size of the micropipette. For that reason, when the weight drops into the platform, we have to ensure that the remaining glass tube of the stage is not touching the platinum wire, otherwise it would get stuck to it.\n\n7. Unscrew the plastic weight and (with extreme caution and preferably with ethanol-cleaned tweezers) take the micropipette and place it into the chamber being built.9\n\n\nPreparation of the beads\n\nBiomolecules such as DNA, RNA and proteins are chains whose units are either nucleic bases or aminoacids [159]. These molecules can be conveniently manipulated through beads attached at their ends. Antigen-antibody connections can be used to attach a specific bead at each end, taking advantage of the fact that these connections are extremely specific [160]. In particular, avidin/streptavidin-biotin10 and digoxigenin-anti-digoxigenin connections are often employed in single-molecule force-spectroscopy experiments, because they can hold forces up to ∼ 100 pN at typical optical-tweezers loading rates of ∼ 1 pN s −1 [162]. Each bead is coated with a different molecule, which specifically binds to the cognate tails. Streptavidin-coated (SA) beads can be directly purchased (SPHERO streptavidin -polystyrene particles, 0.5% w/y, 2.17 µm, 5 ml). Anti-digoxigenin-coated (AD) beads are purchased as G-protein-coated polystyrene beads (Kisker Biotechnologies -G-coated polystyrene particles, 0.5% w/y, 3.18 µm, 5 ml), which must then be activated with anti-digoxigenin. The difference in size between these beads permits one to easily distinguish them by microscopy.\n\nBuffers. The following buffers are required for the preparation of the beads: PBS (pH 7.0) NaCl 0.14 M, KCl 2.7 mM, K 2 HPO 4 · 3H 2 O 61 mM, KH 2 PO 4 39 mM, NaN 3 (sodium azide) 0.02%. To prepare 50 ml: fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water; add 0.406 g of NaCl, 0.01 g of KCl, 0.696 g of K 2 HPO 4 · 3H 2 O, 0.265 g of KH 2 PO 4 , and 1 g of NaN 3 ; dissolve using a magnetic mixer; add Milli-Q water until reaching 50 ml; check the pH and add NaOH until the solution reaches a pH 7.0.\n\nPBS (pH 7.4) NaCl 0.14 M, KCl 2.7 mM, K 2 HPO 4 · 3H 2 O 80.2 mM, KH 2 PO 4 20 mM, NaN 3 (Sodium azide) 0.02%. For preparing 50 ml: follow the same procedure as for the previous buffer, adjusting to pH 7.4.\n\nAntibody crosslinker buffer (pH 7.4) Na 2 HPO 4 100 mM, NaCl 100 mM. To prepare 10 ml: fill ∼ 7 ml of a 10 ml Falcon tube with Milli-Q water; add 0.142 g of Na 2 HPO 4 and 0.058 g of NaCl; shake well until the salts have dissolved; add Milli-Q water until the 10 ml of total volume is reached; check the pH and add NaOH until the solution reaches pH 7.4.\n\n9The micropipette easily breaks or gets blocked by dust particles at the minimum contact of the tip of the glass tube with anything, so it needs to be handled with extreme care.\n\n10The interaction between streptavidin and biotin is one of the strongest non-covalent interactions in Nature [161].\n\nAll salts can be acquired from any chemical distributor (e.g., Sigma Aldrich). All products have to be of a biomolecular grade of purity. The water for preparing all buffers has to be Milli-Q water.11 For the RNA experiments, use RNAse-free water for all preparations (and make sure the pHmeter is also cleaned with RNAse-free water).\n\n\nSA beads.\n\nFor the already coated SA beads, the protocol consists in exchanging the buffer where they are preserved with PBS (pH 7.4). The procedure (∼ 30 min) for the preparation of 1 ml of SA beads (which should be sufficient to perform experiments for several months) is:\n\n1. Homogeneously resuspend the SA beads. To do so, place the container with the purchased particles into a vortex mixer for several seconds. An additional step of several seconds of sonication further improves the resuspension.\n\n2. Take 1 ml of the resuspended SA beads and place it in a new Eppendorf tube.\n\n3. Centrifuge the Eppendorf tube at 10000 rpm for 5 min. The dense SA beads will precipitate, while the buffer will float above them.\n\n4. Extract the overnatant buffer and add 1 ml PBS (pH 7.4). Mix the SA beads and buffer. Sonicate a few seconds and centrifuge at 10000 rpm for 5 min, again. Extract the overnatant, resuspend in 1 ml PBS (pH 7.4) and sonicate during several seconds. This step is performed to exchange the buffer.\n\n5. Aliquote in 20 tubes (50 µl each). This allows for an optimal sterile preservation of the beads.\n\n6. Store at 4 • C for up to ∼ 6 months. 3. Homogeneously resuspend the G-coated beads. To do so, place the container with the purchased particles into a vortex mixer for several seconds. An additional step of several seconds of sonication improves the resuspension.\n\n4. Centrifuge at 5000 rpm for 2 min. The dense beads will precipitate, while the buffer float above them.\n\n5. Extract the overnatant buffer and add 1 ml antibody crosslinker buffer (pH 7.4). Mix the beads and the buffer, sonicate for a few seconds, and centrifuge at 5000 rpm for 2 min.\n\n6. Repeat the previous step (washing) and resuspend within 1 ml antibody crosslinker buffer (pH 7.4).\n\n7. Add 60 µl of anti-DIG antibody and 30 µl of freshly dissolved DMP crosslinker buffer.12\n\n8. Tumble at room temperature for 60 min.\n\n9. Centrifuge at 5000 rpm for 2 min.\n\n10. Wash twice with 1 ml PBS (pH 7.0), resuspend with the same buffer, and sonicate during several seconds.\n\n11. Aliquote in 20 tubes (50 µl each).\n\n12. Store at 4 • C for up to ∼ 6 months. For the SA beads, no incubation is required, dilute 1 µl of SA beads in 1 ml of the molecular buffer in an Eppendorf tube.\n\n12The volume of added anti-DIG antibody will depend on the coating of the beads: it may need to be higher if the vendor supplies the beads with a higher density of coating. Nevertheless, the proportions of anti-DIG and DMP crosslinker buffer have to be preserved (2:1).\n\n13TRIS is used in the formulation of buffer solutions in the pH range between 7.5 and 8.5.\n\n14EDTA is widely used for scavenging metallic ions, including divalent ones, which most enzymes need to be active. For this reason, it is widely used as a food preservative or stabilizer. In our case, it inactivates DNAses and RNAses, preventing nucleic acid degradation.\n\n15Typically, the molecule is concentrated and needs to be diluted 1:10 to 1:100.\n\n\nSynthesis of a short DNA hairpin\n\nAs shown in Fig. 14(d), a typical single-molecule experiment is performed using a short DNA hairpin of a few tens of basepairs (bp). A DNA hairpin is a single-stranded DNA molecule that forms a double helix stem ended by a loop. The hairpin is flanked by two double stranded helices that act as handles. Handles are molecular spacers used to manipulate the DNA hairpin. These spacers also avoid non-specific interactions between the DNA hairpin and the beads when performing the experiments. The handles are tagged with biotin and digoxigenin at one end to specifically attach them to the coated beads. The 29-bp handles are chosen because they provide higher rigidity than longer handles and higher signal-to-noise ratio measurements [163].\n\nTo synthesize short DNA hairpins with 29-bp dsDNA handles, the desired DNA sequence (that we will denote as CD4) is purchased in a series of oligonucleotides that self-assemble into the hairpin and handles [163]. Briefly, a 29-bp sequence is used to make the handles, with the hairpin sequence flanked by the handles at both sides (Table 5). For very short hairpins (≤ 13 bp), the hairpin sequence can be ordered as a single oligonucleotide; however, for longer hairpins, it is useful to split the molecular construct into two oligonucleotides that are annealed and ligated together. Finally, the dsDNA handles are created by annealing a third oligonucleotide that is complementary to the handle region (\"splint\"). The sequences of the oligonucleotides used for the CD4 DNA hairpin used in this section are shown in Table 5. One oligonucleotide ( Table 5, CD4-handleA) is purchased with a biotin at its beginning (5'-biotinylated), whereas the oligonucleotide containing the opposite handle (Table 5, CD4-handleB) is purchased 5'-phosphorylated and then tailed at its 3'-end with multiple digoxigenins. All three oligonucleotides can be purchased from companies such as Eurofins, Fisher Scientific or Sigma Aldrich. Table 5. Oligonucleotides. Oligonucleotides used to synthesize the hairpin by a tailing, annealing and ligation reaction. The splint is used to the create the 29-bp double-stranded DNA handles. The hairpin loop is highlighted in bold and the sequences corresponding to the handles are highlighted in blue.\n\nTo complete the DNA hairpin synthesis, three main reactions have to be performed:\n\n1. Tailing reaction: As we have already noted, one oligonucleotide is already purchased with the biotin at its 5' end. However, to be able to stretch the hairpin, digoxigenins need to be added to the other end of the hairpin by performing the tailing reaction described in Table 6.\n\nAfter the tailing steps, the oligos are purified using the QIAquick Nucleotide Removal Kit (Qiagen) and eluted in 50 µl Tris-Cl 10 mM, giving rise to a final concentration, assuming a 100% efficiency, of ∼ 2 µM.\n\n\nAnnealing reaction:\n\nThe final construct is then assembled in an equimolar reaction. Since the sequences are complementary, at equal concentration for all the oligonucleotides, the equilibrium structure is the formed hairpin with dsDNA handles (Fig. 14(c)). The annealing is performed in two steps : (a) CD4-handleA + splint and CD4-handleB + splint; (b) mix of the two previous reactions. The annealing protocol is described in Table 7. Table 7. Annealing reaction. Mix the items for each reaction in a separate Eppendorf tube. Keep both reactions at 95 • C for 1 min, 80 • C for 10 min, decrease by 0.5 • C every 10 min down to 40 • C. Hold the temperature, mix both reactions and decrease by 0.5 • C every 20 min down to 10 • C. The decrease in temperature can also be achieved by letting the tubes inside a thermal bath cooling down with the heater off; however, the usage of a thermocycler is recommended, since it allows for a better temperature control.\n\n3. Ligation: The molecular assembly is almost completed. All the Watson-Crick bonds of the structure have been formed. However, the backbone of the two oligos of the stem of the hairpin are still not covalently bonded. The final step is to connect the phosphorylated 5' end of CD4-handleB oligo with the C3' end of CD4-handleB. To do so, the assembly is then ligated in an overnight reaction using T4 DNA ligase (New England Biolabs) at 16 • C. The ligase is heat inactivated at 65 • C for 10 min, as shown in Table 8.\n\n\nPulling experiments\n\nParadigmatic manipulation experiments one can perform with a DNA hairpin are pulling experiments. The DNA hairpin is tethered between the two beads and the force it experiences is cyclically varied between a minimum and a maximum force value. Starting at a low force (typically 0 to 5 pN) with the hairpin folded, the trap is moved away from the pipette at a constant pulling speed and the exerted force is steadily increased. At a given point, the hairpin will reach a force high enough to unravel the double-stranded DNA helix structure reaching the unfolded state. This mechanically induced denaturation process is called unzipping. The stretching of the unfolded hairpin keeps on until a maximum force is reached. Afterwards, the same reverse protocol is applied: the trap is moved backwards at the same pulling speed and the applied force decreased until reaching the minimum force value. In this reverse process (called rezipping) the DNA will refold to the hairpin native state when the applied force is low enough. Once the applied force reaches the initial minimum value a new pulling cycle starts again. The CD4 DNA hairpin is a good molecular marker that can be used for force calibration. It reversibly unfolds and folds at 14.7 ± 0.3 pN at standard conditions (T = 298 K, 1 M NaCl). The detailed procedure to perform the pulling experiment is:\n\n1. The three channels are filled with the molecular buffer described in section 4.1.4 through the syringes connected to the channels. The buffers have to be kept in the fridge and filtered using a 10 ml syringe and a filter (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International) to prevent microorganism growth.\n\n2. The chamber (already held in the metallic mount as described in section 4.1.1) is placed between the objectives.\n\n3. The buffer containing SA beads is flowed in the upper channel and AD beads in the lower channel (the channels can be interchanged).\n\n4. The microfluidics chamber is moved using the stepper motors to bring the optical trap close to the dispenser tube of the upper supplier channel to capture a SA bead.\n\n5. After a SA bead is captured by the optical trap, the microfluidics chamber is moved back to position the trapped SA bead on the tip of the micropipette. The syringe is then pulled and the SA bead is held by air suction at the tip of the pipette.\n\n6. The microfluidics chamber is then moved close to the dispenser tube of the lower channel supplying AD beads.\n\n7. Once an AD bead is captured (paying special attention not to trap more than a single bead), the AD bead is brought close to the SA bead, i.e., the chamber is moved so that the optical trap is in the working zone.\n\n8. The tether is formed by poking the SA bead held by the micropipette with the optically trapped AD bead, until a non-zero force is recorded upon retracting the AD from the SA bead. 9. Check that only one molecule has been attached to the bead. To do so, the easiest way is to move the optical trap away from the bead in the pipette and verify that a sudden decrease of the force by ∼ 1.5 pN occurs at ∼ 15pN (red lines in Fig. 16(a)), which indicates the unfolding of the DNA molecule. Then, by moving inwards the optical trap, a similar increase of the force should occur at ∼ 14pN (blue lines in Fig. 16(a)), which indicates that the hairpin is folding back. This equilibrium unzipping-rezipping force has been found to be 14.7 ± 0.3pN at standard conditions (see above).\n\n10. Now, the pulling experiment can start by cyclically varying the distance L between the pipette-held particle and the center of the trap (as shown in Fig. 14(d)). In this way, repeated unfolding and refolding pulling cycles (∼ 100) are performed over the same molecule while recording the force and the trap position. Some representative data are shown in Fig. 16(a), with a typical pulling speed of 100 nm s −1 . To extract the free energy of formation of the hairpin, as we will see in the next section, it is key that the hairpin is folded at L = L 0 and unfolded at L = L 1 .\n\n\nExtracting the free-energy of hairpin formation\n\nOne of the main applications of single molecule experiments is the possibility to extract free energy differences from irreversible work measurements using fluctuation theorems [164]. From the force-distance curves ( Fig. 16(a)) the work W exerted on the molecule between the starting trap position, L 0 , and the final one, L 1 can be directly measured as the area below the curve:\nW = ∫ L 1 L 0 f dL.(75)\nFor quasi-static processes (i.e., sufficiently slow pulls) the work equals the free energy difference ∆G FU between the initial L 0 (folded) and final L 1 (unfolded) trap positions. However, this is not true for irreversible pulls which exhibit hysteresis between the stretching (forward) and releasing (reverse) processes. The difference in the area between consecutive unfolding and folding cycles is the work dissipated by the system, which is positive in average according to the second law of thermodynamics. Noteworthy, since we are not in the thermodynamic limit (as we are dealing with small systems), there may be trajectories that violate the second law of thermodynamics. Fluctuation theorems state that the work distributions measured along the forward and reverse processes fulfil a symmetry relation [164]. One of the main consequences of such a relation is that forward and reverse work distributions, despite being different due to the hysteresis, cross each other at the value of ∆G FU , independently of how irreversible the pulling process is. To extract the work distributions we use equation (75) to compute the work for all the forward (stretching) and reverse (releasing) cycles. The probability distributions of the work for the forward and reverse processes are shown in Fig. 16(b) by the red and blue histograms, respectively. The crossing point of the forward and reverse work probability distributions corresponds to the difference in free energy ∆G FU between the folded (L 0 ) and unfolded (L 1 ) states. The work histogram in Fig. 16 The free energy difference between the folded and unfolded states can be written as:\n∆G FU = ∆G 0 FU + ∆W ssDNA FU − ∆W dipole FU + ∆W handles FU + ∆W bead FU ,(76)\nwhere the term ∆G 0 FU is the free energy of formation of the molecule, which is the quantity we are interested in obtaining, and the other contributions are the reversible work differences related to the different parts of the experimental setup (ssDNA, handles, bead, hairpin dipole, as shown in Fig. 14(d)). These contributions can be computed as follows:\n1. ∆W ssDNA FU\nis the reversible work needed to stretch the unfolded molecule, which is modeled with an inextensible Worm-like chain [165]:\nf ssDNA (x) = k B T p 1 4 1 − x nl 0 2 − 1 4 + x nl 0 ,(77)\nwhere f ssDNA is the force, x is the molecular extension, n is the number of monomers of the molecule (n = 44 bases for our case), l 0 = 0.59 nm is the contour length per monomer, and p = 1.35 nm is the persistence length (parameters from Ref. [166]). The stretching contribution of this released ssDNA is then computed as\n∆W ssDNA FU = ∫ x( f max ) 0 f ssDNA (x)dx.(78)\nThe value we obtain is ∆W ssDNA\nFU = 19.0 k B T. 2. ∆W dipole FU\nis the reversible work needed to stretch the folded molecule, which is modelled as a single dipole of length d = 2 nm [166], corresponding to the double-helix diameter [167]:\nx dipole ( f ) = d tanh f d k B T − k B T f d(79)\nand\n∆W dipole FU = ∆ f ∆x − ∫ f max f min x dipole ( f )df ,(80)\nwhere\n∆ f = f max − f min and ∆x = x( f max ) − x( f min ). The value we obtain is ∆W dipole FU = 4.6 k B T.\n3. Finally, the last two contributions to equation (76) can be computed by considering the slope of the force-distance curves just before the rip, where the distance corresponds to the optical trap position (L), as shown in Fig. 16(a). The slope κ eff can be easily obtained by linearly fitting the data points right before the unfolding event. This yields the effective stiffness of two serially connected springs: the trapped bead of stiffness κ B , and the handles stiffness, κ handles :\nκ −1 eff = κ −1 B + κ −1 handles .\nAssuming that the hairpin orientation has a much higher stiffness than the handles and the optical trap, the combined work can be written as [166]:\n∆W handles FU + ∆W bead FU ≈ f 2 max − f 2 min 2 κ eff ,(81)\nwhere we have considered that the stiffness of the folded molecule is very large and consequently does not contribute to κ eff . The value we obtain is ∆W handles\nFU + ∆W bead FU = 282 k B T.\nFrom equations (78), (80) and (81), and from the crossing point of the forward and reverse work probability distributions (∆G FU ), we obtain all the terms needed to calculate the free energy of formation of the hairpin, ∆G 0 FU , from equation (76). The free energy of formation at zero force we obtain for this molecule is ∆G 0 FU = 51.9 k B T, which is in agreement with that of the sequence of the CD4 hairpin, ∆G 0 FU = 50.6 k B T, using the values provided in Ref. [168]. Typical errors of ∆G 0 FU are on the order of several k B T, corresponding, in this case, to a 5% to 10% relative error.\n\n\nOther applications to single-molecule mechanics and outlook\n\n\"Take a single DNA molecule and pull from its extremities while recording the force-extension curve until it gets fully straightened.\" This thought experiment, which was just a dream a few decades ago, has now become standard in many research labs worldwide. Since the seminal works on single-molecule manipulation almost 30 years ago [7][8][9], the number of applications of single-molecule mechanics has kept on growing. Applications might be classified in two large sets: those aimed to address specific biological questions (e.g., the mechanics of maintenance and regulatory enzymes in the genome, the energetics and kinetics of DNA hybridization, RNA and protein folding, molecular footprinting of proteins and ligands binding DNA), and those aimed to enlighten fundamental physico-chemical questions (e.g., polymer physics, chemical reactions, nonequilibrium physics) using biomolecular matter as model systems. The reader can have a look at published reviews specifically addressing these fields to discover that the future of the single-molecule field has a long way ahead to run [133][134][135][136][137][138][139][140]. Most optical-tweezers and single-molecule technology is still instrumental craftwork not widely available in research institutes and labs worldwide, yet it is becoming steadily more popular among biologists aiming to enter the field. Future developments in the field will be the parallelization of measurements in optical tweezers making it high-throughput either using holographic techniques [43] (recently extended also for acoustic tweezing [169]), combining single-molecule fluorescence with forcemeasurement techniques [170][171][172], controlled force resistive pulse sensing [173,174], and plasmonic trapping [13]. Finally, a major step would be to apply force spectroscopy techniques to ex-vivo and in-vivo conditions.\n\n\nSingle-cell mechanics\n\nBiological microscopic objects such as organelles and living cells have been optically manipulated since the late 1980s [175,176]. In Ref. [177], the swimming motion of optically trapped bacteria was recorded while the trapping forces were measured providing the first example of single-cell optical manipulation. Since then, biomechanics at the cellular scale has kept on growing in terms of techniques and objects of study [178,179]. Techniques such as atomic force microscopy [180,181], optical tweezers [182][183][184], particle tracking microscopy [185,186], traction microscopy [187][188][189], magnetic twisting cytometry [190], acoustic force spectroscopy [191], and micropipette aspiration [192] have become commonly employed to investigate the response of cellular objects to mechanical stresses. Essential cellular processes (e.g., cell division) and collective effects (e.g., cell signaling and migration in tissues) are currently being investigated with such techniques.\n\nIn this section, we will demonstrate one of the simplest cell manipulation experiments with optical tweezers: the mechanical stretching of a single red blood cell (RBC) to measure its rigidity to an externally applied deformation or strain.\n\n\nRBC buffer preparation\n\nThe RBC buffer should mimic the physiological conditions found in human blood. It can be prepared as a solution of 130 mM NaCl, 20 mM K/Na phosphate buffer at 7.4 pH, 10 mM glucose, and 1 mg ml −1 BSA (Bovine Serum Albumin) [193]. It is convenient to prepare higher concentration stocks (1 M) of every solution and store them in the fridge (higher concentration of salts together with low temperature prevent microbial contamination), keeping them in 50 ml Falcon tubes for repeated use. To prepare a 50 ml Falcon tube containing the RBC buffer, follow these steps:\n\n1. Add 6.5 ml of 1 M NaCl, 1 ml of 1 M K/Na phosphate buffer at 7.4 pH,16 and 0.5 ml of 1 M glucose into an empty 50 ml Falcon tube.\n\n2. Add 50 g of BSA.\n\n3. Add 42 ml of milli-Q water.\n\n4. Mix the RBC buffer and filter it (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International).\n\n\nPlace the filtered solution into a new 50 ml Falcon tube.\n\nAll remaining unused RBC buffer must be re-filtered on a daily basis to prevent molecular aggregation and bacterial growth. After three days, it is recommended to dispose of the remaining RBC buffer because the re-filtering will not be enough to ensure proper conservation.\n\n\nBlood extraction\n\nTo obtain human blood, the easiest way is to prick the finger of a healthy donor. Use of diabetic testing lancets will ensure sterilization of the pricking device. The extraction is made following these steps:\n\n1. Disinfect the finger of the healthy donor with ethanol and dry it with handkerchief paper to remove any remaining liquid.\n\n2. Prick the donor's finger with the lancet and apply some pressure to extract a drop of blood. A single drop of blood will be enough to perform experiments during an entire day.\n\n3. With a 200 µl pipette take 10 µl of blood from the finger.\n\n4. Dilute as fast as possible the blood drop in 1 ml of RBC buffer inside a 1.5 ml Eppendorf tube to prevent degradation of the RBCs. \n\n\nBead preparation\n\nThe simplest option to attach the beads to the RBCs is to take advantage of non-specific interactions between polystyrene beads and RBCs.17 To do so:\n\n1. Add 1 ml RBC buffer in a 1.5 ml Eppendorf tube.\n\n2. Homogeneously resuspend the polystyrene beads by placing the purchased beads (Microbead NIST Traceable Particle Size Standard, 3.00 µm, Polysciences, Inc.) into a vortex mixer for several seconds. An additional step of several seconds of sonication improves the resuspension.\n\n3. Add 3 µl of the bead stock solution to the Eppendorf tube.\n\nIt is recommended to have two Eppendorf tubes with bead solution available to ensure continued supply of the chamber with beads during the experiment.\n\n\nSetup preparation\n\nThe experiment can be performed using the same microfluidics chamber and miniTweezers setup presented in section 4.1. In particular, the procedure to build the chamber and to place it into the mount is the same as described in section 4.1.1, the only difference being a longer plastic tube (40 cm) to connect one of the syringes to the inlet of the central channel. Once the chamber is fixed into the mount with the syringes connected through the plastic tubes, the next steps are:\n\n1. Flow 3 ml of RBC buffer in each of the three channels to produce an homogeneous environment in the chamber.\n\n2. Identify which dispenser tube (upper/lower) has the most clear exit (i.e., the one displaying the most sharp perpendicular cut in the video image) to the central channel. This will simplify trapping the beads. 5. Introduce the mount with the chamber between the two objectives of the optical tweezers.\n\n6. Connect the syringe with the larger plastic tube (the one of the central channel) to a pump to apply a controlled flow to the central channel when it is needed. We will use this flow to stretch the RBC-bead dumbbell captured in the optical trap, thereby preventing thermaland photo-damage of the cell.\n\n\nConstruction of the cell-bead configuration\n\nTo perform cellular pulling experiments, the RBC need to be held between two beads attached to the cell membrane on opposites sides, as shown in Fig. 17(d). One bead is held by the micropipette by air suction, while the other bead is held by the optical trap. The use of beads as handles to manipulate the RBC minimizes the direct exposure of the RBC to the laser light. The steps of the procedure to obtain this cellular configuration are:\n\n1. Align the lasers without a bead (steps 1-8 described section 4.1.2).\n\n17There are also several options to treat the beads and RBC to specifically attach them, e.g., using the lectin protein Concanavalin A [191] or biotinylating the RBC [194] 2. Identify the position of the pipette and of the two dispenser tubes that connect the upper and the lower channels with the central one.\n\n3. Move the chamber using the stepmotors and place the optical trap close to the exit of the dispenser tube that contains the bead solution.\n\n4. Flow some bead solution pushing the syringe either by hand or using a pump, and trap a bead with the optical trap.\n\n5. Move the trapped bead next to the pipette tip in the working zone.\n\n6. Align the lasers with the trapped bead (steps 9-13 described section 4.1.2).\n\n7. With the bead already trapped, try different flow values with the pump connected to the syringe of the central channel until the force reaches ∼ 10 pN due to the RBC buffer flow. This will be the approximate flow that we will need to keep the optically trapped bead separated from the RBC (Figs. 17a and 17b) after we obtain an attachment.\n\n8. Turn off the pump.\n\n9. Move the trapped bead to the area of the blood dispenser tube exit. It is important not to put the bead exactly at the exit of the tube to prevent losing the bead due to the RBC solution flow.\n\n10. Flow some RBC solution applying a small pressure to the syringe. RBCs will flow into the central channel.\n\n11. Move the trapped bead close to a RBC and try to make a connection by poking the surface of the RBC with the bead. The connection is formed when the RBC follows the trapped bead when moved.\n\n12. Move the chamber using the stepmotors to place the bead and the attached RBC next to the micropipette.\n\n13. Turn on the pump. The configuration should look like Fig. 17(a). The RBC is attached to the bead and partially trapped by the lasers. 14. If the flow is not enough to separate the RBC from the trap and, therefore, to obtain the configuration of Fig. 17(b), it is necessary to move the chamber using the stepmotors along the horizontal direction to exert an extra force and separate the RBC from the optically trapped bead.\n\n15. Move the bead close to the tip of the pipette. 16. Carefully apply air suction to immobilize the bead. This step is critical because a large enough air suction force could absorb the RBC inside the pipette.\n\n17. Move the optical trap away from the bead. If the bead is properly fixed to the tip of the pipette, the bead will stay there. If this is not the case, when the optical trap is moved the bead will follow it. 23. Try to make a connection between the trapped bead and the RBC as in Fig. 17(c). Notice that the two beads are on opposite sides of the RBC. 24. Check that the new connection between the bead and the RBC is done by moving the optical trap to the same direction as the pump flow. If the RBC is deformed while moving the trap, the attachment is done. 25. Turn off the pump to remove the flow of the central channel. 26. Nip very carefully the plastic tube that is connected to the entrance of the central channel with a clamp to prevent flows due to the difference of pressure at both ends of the plastic tubes.\n\n27. Move the stepmotors to obtain the configuration shown in Fig. 17(d), aiming at having a zero reading for the force along the y-direction.\n\n\nExperiments\n\nBy cyclically stretching and releasing the RBC, it is possible to obtain the force-extension curves from which the stiffness and deformability of the RBC can be extracted. Starting with a RBC in the configuration described in the previous section ( Fig. 17(d)), it is possible to stretch the cell by moving the optically trapped bead outwards (i.e., away from the micropipette) and release it by moving it inwards (i.e., towards the micropipette).18 Follow these steps to define the stretching-realising protocol:\n\n18A range for the optical trap displacement of at least ∼ 8 µm is necessary to perform the RBC stretching. Thus, it is recommended to move the optical trap towards the lower limit of its y-displacement, before starting the pulling protocol.\n\nStarting from the minimum force, the RBC is pulled at the constant velocity until the maximum force. After reaching this maximum force, the RBC is pushed at the same constant velocity to the minimum force. During this process, the displacements and forces along the RBC stretching direction (y-direction) are recorded for both the A-and B-trap in the miniTweezers setup ( Fig. 14(a)); then, the two displacements are averaged to measure the trap displacement, while the two forces are summed to obtain the total force acting on the RBC. The whole process is called pulling cycle. Recording 5 pulling cycles for each RBC is typically enough for the subsequent analysis.\n\n\nData analysis\n\nThe data analysis aims at computing the RBC stiffness knowing the stiffness of the optical trap κ trap . The experiment measures the trap displacement ∆x trap , not the cellular extension ∆x cell . Thus, ∆x cell can be computed from ∆x trap , κ trap , and the measured optical force f trap . For each intermediate trap position from L 0 to L f , we are able to transform L into x cell substracting the 19Depending on the laser power, it could be very hard to reach the 0 pN force as the RBC will be attracted to the optical trap.\n\n20Above 20 pN, as we have not applied any specific coating to the beads, a very thin membrane tube (theter) forms and alters the force-extension curve.\n\n\nRBC\n\nΔL Δx trap Fig. 18. Sketch of RBC extension under applied force. The red ellipse represents the RBC, the light blue circles represent the beads, and the small green circle represents the optical trap. The cell extension is ∆x cell , while ∆x trap is the bead displacement respect to the trap center, and ∆L is the displacement of the optical trap. Note that ∆x cell and ∆x trap are distances along the y-axis. trap contribution (Fig. 18), as we show in the following expression:21\n∆x cell = ∆L − ∆x trap = ∆L − f trap κ trap ,(82)\nwhere ∆L is the variation of the trap position between that at minimum force and that at maximum force. The resulting force-cellular-extension curve is shown in Fig. 19(a) for the second pulling cycle for two different RBCs. 22 The RBC stiffness can be measured from the slope of the force-cellular-extension curves. As can be observed in Fig. 19(a), the force-cellular-extension curve is not a straight line. This indicates that the stiffness of the RBC depends on the force applied to the cell. For this reason, the slope is computed for five different force windows whose stiffness values are represented in Fig. 19(b). The RBC stiffness increases as we increase the applied force, the stiffness curves of the stretching and releasing cross each other between 6 and 10 pN, and the releasing stiffness is higher than the stretching stiffness at high forces while it is lower at low forces. The expected values for RBC stiffness in this specific geometry are in the range between 3 pN µm −1 and 10 pN µm −1 . The dispersion between different RBCs, in terms of force-cellular-extension curves, can be very large. This can be due to the fact that the RBCs are not separated by density so that the RBCs can have very different ages and, thus, very different physiological properties [196].\n\n\nOther applications to single-cell mechanics and outlook\n\nUntil now, the manipulation of single cells with optical tweezers has not been extensively investigated as single-molecule manipulation. The main reason is the difficulty inherent to the geometry of the pulling assays when working with cells. Indeed, while the specific labeling of molecules makes it possible to design the most diverse pulling geometries, this is more difficult to achieve when manipulating single cells which ultimately require the physical contact between the optically trapped bead and the lipid cell membrane. Additionally, single cell manipulation with optical tweezers is feasible on cells in suspension (such as RBC, lymphocytes, senescent, stem cells), whereas adherent cells are harder to manipulate, often requiring confocal microscopy 21This is the same analysis done to obtain the molecular extension in long nucleic acids (≈ 10 kbp double-stranded nucleic acids or ≈ 1 kbp single-stranded nucleic acid [195]).\n\n22It is best to analyze the second pulling curve because the first one could exhibit a special transient behavior.\n\nsetups with optical traps very close to the glass substrate. For the latter class of cells, other techniques (e.g., atomic force microscopy, traction microscopy, magnetic twisting cytometry) turn out to be more informative. Finally, let us mention that while most single cell manipulation studies are carried out in in-vitro conditions (e.g., on the Petri dish), it will be necessary to expand the use of these manipulation techniques to more relevant environments such as ex-vivo tissues.\n\nIn this case, the mechanical response of a single cell in a tissue-like environment is expected to differ a lot with respect to the isolated cell. Overall, this makes the future of single cell mechanics very promising beyond biology, with implications in physiology and medicine.\n\n\nMicrorheology\n\nRheology investigates the mechanical response of soft materials such as simple liquids, polymer solutions, micellar fluids, colloidal suspensions, liquid crystals, gels, and foams [20,197]. Specifically, it is concerned with flow and deformation of such materials under applied stress or strain. Depending on the specific details of their microstructure, different kinds of rheological behavior can be observed. For instance, for most simple liquids, the rheological response is Newtonian: upon imposing a given strain rate, the resulting stress is proportional to it, thereby dissipating energy instantaneously by viscous resistance. Instead, complex fluids and gels, whose components are usually long macromolecules suspended in a viscous solvent, exhibit viscoelasticity, i.e., both viscous (liquid-like) and elastic (solid-like) responses are possible [198]; therefore, they are able to store and dissipate energy depending on the time-scale of observation.\n\nThe goal of rheology is to provide quantitative parameters, e.g., viscosities, relaxation moduli and creep compliances, that relate the stress and the strain (or the strain rate) of the material under deformation. To this end, the most straightforward techniques consist in imposing stresses and strains in a controlled manner by means of a rheometer to the material sample of interest and then measuring the corresponding mechanical response. Despite their relative simplicity, these methods require typically milliliter-sized sample volumes, thus being unsuitable for soft matter systems that are expensive or difficult to find in abundance, such as biological fluids and newly synthesized materials. Microrheology overcomes this major drawback by measuring mechanical properties of a soft material sample using colloidal probes directly embedded in it. The rheological information of the material is fully inferred from the motion of the probe, which requires only the detection and tracking of its position over time. Therefore, very small sample amounts, typically ranging from pico-to microliters, are needed. This in turn allows to carry out in-situ rheological measurements which are inaccessible to macroscopic methods, e.g., biofluids within living tissues and cells, soft interfaces and membranes. Furthermore, microrheology offers several advantages over conventional macroscopic rheometry. For instance, unlike bulk rheology, which provides quantities averaged over the entire macroscopic sample, microrheology permits to explore local mechanical properties at the length-scale of the micron-sized probe. This is particularly useful for investigating heterogeneous materials and fluids close to interfaces, where submillimetric spatial variations of the rheological parameters occur. In addition, undesirable artifacts that can arise due to inertial effects when using a macroscopic rheometer are automatically ruled out in microrheology due to the extremely low Reynolds numbers (Re 10 −3 ) of the flows induced by the motion of a micron-sized probe. This expands significantly the available frequency range over which the linear viscoelastic spectrum of the system can be determined without inertial corrections, typically from the range [10 −1 Hz, 10 2 Hz] up to the range [10 −1 Hz, 10 6 Hz]. Besides, conventional rheometers are usually designed for either highly elastic or highly viscous materials, but they are not sensitive enough to measure low viscosities and elastic moduli; then, another advantage of microrheology is its capability to detect very weak viscoelastic parameters. Moreover, the typical forces and energies involved in microrheology, either from the thermal motion of the material molecules or those externally exerted by optical tweezers, are of the order of ∼pN and ∼k B T, respectively. This guarantees that the microstructure of the sample is not irreversibly destroyed by the motion of the probe. 20. Microrheology with an optically trapped particle. (a) Sketch of the general response of the shear stress σ(t) of a viscoelastic material to a step-like shear strain (t).\n\n\nBasic setup and concepts\nϵ t) ( a b ϵ 0 t σ t) ( t t=0 x ot y x z Fig.\nFor t ≥ 0, the ratio σ(t)/ 0 defines the stress relaxation modulus G(t). (b) Sketch of the typical experimental system used in microrheology: a spherical colloidal bead (radius a), embedded in a soft material, is trapped by optical tweezers. If the position of the optical trap is fixed (x ot (t) ≡ 0, passive microrheology), the particle is only subject to the thermal collisions of the molecules of the surrounding fluid microstructure. If x ot (t) is varied in time, an additional time-dependent force is exerted on the probe particle (active microrheology).\n\nOptical tweezers provide the ideal tool to manipulate probe particles in microrheology [20,[199][200][201][202], because they permit one to tailor the forces that can be exerted to locally strain the sample.\n\nTo extract meaningful information by means of standard microrheological methods, some experimental conditions must be fulfilled. First of all, the material must be homogeneous and isotropic at the scale of the probe, which is usually valid when the relevant length-scale of the material, e.g., the mesh size of gels and semidilute polymer solutions, is much smaller than the probe size. In this case, the environment behaves as a continuum and its mechanical response can be fully characterized by a single scalar function, e.g., the stress relaxation modulus G(t). For example, upon applying a step-like shear strain at t = 0 to a material, (t) = 0 Θ(t), where Θ is the Heaviside function, the stress relaxation modulus is given by\nG(t) = σ(t) 0 for t ≥ 0,(83)\nwhere σ(t) is the resulting shear stress ( Fig. 20(a)). For a viscoelastic liquid, G(t) decays to zero over a finite time, while for a viscoelastic solid, it decays to a constant non-zero value. In both cases, the drag force experienced by an embedded particle at time t can be written as\nF drag (t) = − ∫ t −∞ dt Γ(t − t ) r(t ),(84)\nwhere r(t ) is the instantaneous particle velocity at time t < t and the minus sign indicates that the drag force is opposed to the instantaneous velocity, whereas Γ is a memory function related to G(t), which weights the role of the previous history of the particle motion on its current drag force due to the temporal correlations induced by the surrounding medium. For example, for a spherical particle of radius a, the memory function is Γ(t) = 6πaG(t), which can be regarded as a generalization of the Stokes' law (equation (6)). For a Newtonian fluid with shear viscosity η, the relaxation modulus is G(t) = 2ηδ(t),23 which corresponds to an instantaneous response of the drag force F drag (t) = −6πηa r(t) to the particle velocity r(t). Moreover, for some techniques, it is required that the sample is in thermal equilibrium and that the forces externally applied to the probe are small enough to keep the material in its linear response regime. Under such conditions, the time evolution of a single particle coordinate, x, is described by the generalized Langevin equation [203] \n0 = − ∫ t −∞ dt Γ(t − t ) x(t ) − κ[x(t) − x ot (t)] + ζ(t),(85)\nwhere x ot represents the x-component of the central position of the optical trap, while ζ is a Gaussian noise of zero-mean, ζ(t) = 0, and autocorrelation\nζ(t)ζ(t ) = k B T Γ(|t − t |).\nDepending on how optical tweezers are employed to manipulate a colloidal probe embedded in a soft material ( Fig. 20(b)), two kinds of microrheological techniques exist: passive and active [197,201]. In the following sections, we describe how to prepare a prototypical viscoelastic fluid (section 4.3.2) as well as how to determine rheological parameters of simple liquids using passive and active microrheology (sections 4.3.3 and 4.3.4).\n\n\nPreparation of a viscoelastic fluid and setup\n\nAn equimolar wormlike micellar solution made of the surfactant cetylpyridinium chloride (CPyCl) and the salt sodium salicylate (NaSal), both dissolved in water, represents a prototypical viscoelastic fluid. To prepare it, overnight mixing of the two components in deionized water is required at approximately 50 • C, after which a homogenous and isotropic viscoelastic fluid results. 24 To get reproducible rheological parameters of the fluid, the conductivity of water should be kept at most at 10 µS m −1 (milli-Q water is a good option for this purpose). Depending on the surfactant/salt concentration, different kinds of microstructures in the aqueous solvent are formed [204]. For instance, above the first critical micelle concentration but below 4.5 mM, the surfactant molecules aggregate into spherical and cylindrical micelles of radius 2 to 3 nm, which result in a weakly viscoelastic behavior of the dilute solution. Increasing further the concentration up to 10 mM, the surfactant molecules self-organize in long flexible cylindrical micelles, which form, overlap, and deform dynamically in the solvent. Such wormlike micelles have a radius from 2 to 3 nm and a contour length ranging from 100 nm to 1 µm, while their persistence length and mesh size are of order 10 nm for concentrations between 5 and 10 mM. To perform optical-tweezers-based microrheology, a very small amount of micron-sized beads must be suspended in a microliter volume of solution. Special care is needed to avoid air bubbles inside the fluid, which can be easily formed due to the presence of the surfactant. Sonication during a few minutes of the fluid with the dispersed colloidal particles is recommended before being loaded into the sample cell. Such a sample cell consist of a standard chamber (cross-sectional area about 1 cm 2 ) made of a glass slide and a coverslip, separated by a distance of approximately 100 µm by means of, e.g., spacer tape or parafilm strips. The chamber must be laterally sealed with UV-curable optical adhesive after inserting a small volume (∼ 10 µl) of the suspension, in order to avoid evaporation of the fluid and drift. During the measurements, the fluid sample must be kept thermally coupled to a thermostat at constant temperature T within an accuracy range of ±1 K, as the rheological properties of the fluid strongly depend on temperature. It is 23Here, we use the following convention for the Dirac delta ∫ ∞ 0 delt a(t )dt = 1/2. Thus, the factor 2 is needed in the relaxation modulus G(t) to recover the correct friction coefficient.\n\n24It is recommended to wear gloves and safety googles when handling CPyCl, as direct contact with power can cause skin and eye irritation. crucial to maintain T above the so-called Krafft temperature [205], which is 18 • C for the specific micellar solution described here. Below the Krafft temperature, the surfactant crystallizes, thus suppressing the viscoelasticity of the fluid.\n\nTo illustrate the microrheological techniques described in the following, two Newtonian fluids with distinct viscosities at room temperature are used. The first one is deoinized water, whose conductivity must be less than 10 µS m −1 in order to facilitate the sample cell preparation. The second Newtonian fluid is propylene glycol n-propyl ether (PNP), also known as 1-propoxy-2propanol, which can be directly purchased with a purity ≈ 99% so that no further purification is needed. On the other hand, two viscoelastic fluids are employed: an equimolar CPyCl/NaSal micellar solution at 4 mM and one at 5 mM. The different fluids and their main properties are summarized in Table 9. Table 9. Microrheology fluids. Fluids used to illustrate microrheology with an optically trapped particle.\n\nAll the experiments described in this section are performed in a standard optical tweezers setup implemented with an oil-immersion microscope objective (100×, NA = 1.4) which tightly focuses a Gaussian laser beam (λ = 1070 nm) onto the sample. A single spherical particle is trapped and kept at least 10 µ away from any wall of the sample cell in order to avoid hydrodynamic interactions, which can significantly affect the values of the fluid's rheological parameters with respect to those obtained from bulk measurements. Imaging and position detection can be realized by conventional video microscopy. In particular, for the determination of the linear viscoelastic spectrum by passive microrheology, it is required that the acquisition frequency of the camera must be above about 1000 frames per second. Therefore, the use of a position sensitive detector is an alternative for this purpose. Furthermore, beam steering, which is crucial for active microrheology, is achieved here by means of galvomirrors (section 2.3.1).\n\n\nPassive microrheology\n\nPassive microrhelogy is the simplest approach to study the local mechanical response of soft matter. As its name suggests, the embedded colloidal bead is used as a passive element, whose thermal motion provides all the necessary information to extract, via fluctuation-dissipation relations, the frequency-dependent rheological parameters of the material under investigation. To prevent sedimentation as well as significantly large diffusive displacements of the probe from its initial position, which would otherwise restrict the measurement time of the particle position x(t), a static optical trap can be used to confine the particle motion around a mean fixed position, i.e., x ot (t) = 0, as depicted in Fig. 20(b). Here, we present two direct applications of passive microrheology with optical tweezers for viscous and viscoelastic liquids.\n\nDetermination of zero-shear viscosities. For a Newtonian liquid, the autocorrelation function of the particle position provides a straightforward way to determine its viscosity η. From equation (85), using the instantaneous memory function Γ(t) = 2γδ(t) for a purely viscous liquid, the expression for its autocorrelation function can be readily derived (section 3.5):\nx(t)x(0) = k B T κ e −t/τ ot for t ≥ 0,(86)\nwhere . . . represents an ensemble average over different realizations of the thermal noise, while τ ot = γ/κ is a decay rate related to the viscous friction coefficient γ of the probe and the trap stiffness κ. In practice, the ensemble average can be replaced by a time average along the stochastic trajectory x(t), provided that the total measurement time of x(t) is much larger than τ ot . For a spherical particle of radius a, the friction coefficient is γ = 6πaη, where η is the fluid viscosity. Then, once the trap stiffness κ is known (e.g., by the equipartition method: κ = k B T x(t) 2 , section 3.3), the experimental autocorrelation function of x(t) can be fitted to the right-hand side of equation (86). Therefore, η can be determined from the fitting parameter τ ot by means of η = κτ ot 6πa . This is illustrated in Fig. 21, where the viscosities at T = 25 • C of two Newtonian fluids, water and propylene glycol n-propyl ether (PNP), are computed by means of this passive method. In Fig. 21(a), we plot the typical trajectories x(t) of particles embedded in these liquids, trapped by optical tweezers at approximately 40 µm away from the cell walls. The corresponding autocorrelation functions of x(t) are shown in Fig. 21(b), where we show that the experimental data (symbols) can be very well fitted to equation (86) (dashed lines). From the equipartition relation, we first determine the values of the trap stiffness for each case: κ = 1.24 pN µm −1 for the particle in water (diameter 2a = 2.73 µm) and κ = 0.90 pN µm −1 for the particle in PNP (diameter 2a = 3.25 µm). Next, from the fitting parameters τ ot , we find the corresponding values of the viscosities of both fluids: 0.9 mPa s for water and 4.5 mPa s for PNP, which agree very well with their bulk values.\n\nThis technique can be extended to determine steady-state flow properties of viscoelastic fluids, which exhibit both liquid-and solid-like responses at times shorter than a typical time τ. Such a time-scale reflects the stress relaxation of the elastic material microstructure suspended in the solvent. For a viscoelastic micellar solution, τ originates from the continuous formation and breaking of the micelles as well as the reptation modes of the wormlike structures, and can range from a few milliseconds to several seconds in the semidilute regime [206]. Unlike viscoelastic solids (e.g., gels), viscoelastic fluids are able to flow with constant shear rate at sufficiently long times, t τ, after applying constant shear stress at t = 0. Therefore, they are characterized by a zero-shear viscosity η 0 , which in absence of a confining potential, leads to a long-time diffusion of an embedded spherical particle, where the diffusion coefficient is given by k B T 6πaη 0 . In the presence of an optical trap, from equation (85) it can be shown that the position autocorrelation function has the form\nx(t)x(0) = k B T κ Ae −t/τ ot for τ t < τ ot ,(87)\nwhere A < 1 is a prefactor that depends mainly on κ and τ. Equation (87) is valid provided that τ ot τ [207], which must be verified a posteriori. To fulfill this condition of large time-scale separation, sufficiently small values of the trap stiffness are needed. In Figs. 21(a) and 21(b), we illustrate this method for a 2a = 3.25 µm silica particle suspended in a dilute micellar solution (concentration 4 mM, T = 295 K), trapped by optical tweezers with stiffness κ = 1.34 pN µm −1 . At this concentration, the viscoelastic fluid is composed of non-overlapping micellar structures, thus resulting in typical relaxation times, τ, of order of milliseconds. We check that equation (87) is a very good approximation to the experimental position autocorrelation function, where only the first experimental point slightly deviates from the fit. The fitting parameter τ ot = 0.16 s yields the value η 0 = 6.9 mPa s, which is also in agreement with reported bulk measurements [204].\n\n\nDetermination of storage and loss moduli. The complex shear modulus\n, G * ( f ) = G ( f ) + iG ( f )\n, is a quantity used in rheology to characterize linear viscoelasticity of soft matter in the frequency domain [208]. G ( f ) is the storage modulus and G ( f ) is the loss modulus, which account for the elastic energy in phase with the applied strain and the out-of-phase viscous dissipation of the material, respectively. For instance, for a purely viscous liquid (viscosity η), G ( f ) = 0 and G ( f ) = 2π f η, whereas for an elastic solid (elastic modulus G 0 ), G ( f ) = G 0 , and G ( f ) = 0. In general, for viscoelastic materials, both storage and loss moduli are non-zero. G * ( f ) is closely related to the Fourier transform,G( f ), of the stress relaxation modulus, G(t), by \nG * ( f ) = 2πi fG( f ).(88)\nTherefore, for an applied oscillatory shear strain ( f ) = 0 e 2πi f t with amplitude 0 and frequency f , the resulting shear stress is also oscillatory and can be expressed as G * ( f ) ( f ).\n\nA very helpful expression for G * ( f ) in terms of the equilibrium fluctuations of the position x(t) of a trapped particle can be derived from the fluctuation-dissipation theorem. The latter relates the linear response function χ(t) of x(t) to a perturbative force F(t), defined as\nx(t) F = x(0) F + ∫ t 0 dt χ(t − t )F(t ),(89)\nwith the equilibrium (F(t) = 0) autocorrelation function of x(t), i.e., x(t)x(0) . In equation (89), the nonequilibrium ensemble average . . . F is defined in presence of the time-dependent perturbation. From equations (85) and (89), the Fourier transform of\nχ(t),χ( f ) = ∫ ∞ −∞ dte −2πi f t χ(t) = χ ( f ) + i χ ( f ), can be directly related in frequency domain to the complex shear modulus G * ( f ) by means of G * ( f ) = 1 6πa 1 χ( f ) − κ .(90)\nNote that equation (90) shows that the elastic contribution of the optical tweezers stiffness to the mechanical response of the system must be subtracted for a correct determination of the complex shear modulus of the surrounding medium. Moreover, equation (90) involves directly the inverse ofχ( f ), which can be determined by either passive or active microrheology, as will be shown. Therefore, this expression is applicable for both types of microrheological techniques. In particular, in the case of passive microrheology, by making use of the fluctuation-dissipation theorem in frequency domain and the Kramers-Kronig relations, one can find both the real, χ ( f ), and the imaginary, χ ( f ), parts of the response functionχ( f ):\nχ ( f ) = 2π k B T P ∫ ∞ 0 dν ν 2 S(ν) ν 2 − f 2 ,(91)χ ( f ) = π f 2k B T S( f ),(92)\nwhere S( f ) = |x( f )| 2 is the one-sided PSD of x(t) in thermal equilibrium, i.e., the Fourier transform of x(t)x(0) , whereas P denotes the Cauchy principal value of the integral. Thus, once χ ( f ) and χ ( f ) are determined from the experimentally recorded trajectory of the trapped particle, the storage modulus G ( f ) and loss modulus G ( f ) can be computed over the available frequency range [0, f max ] from equation (90). For a given sampling frequency f s of x(t), f max is fixed by the Nyquist frequency: f max = 1 2 f s . This technique needs a rather high sampling frequency of x(t) to avoid a large underestimation of the integral in equation (91).\n\nIn Fig. 22, we apply the previous method to characterize the linear viscoelasticity of the wormlike micellar solution in the semidilute regime at 5 mM and T = 20 • C. First, we compute the equilibrium PSD, S( f ), of a long trajectory x(t) measured over 25 min at a sampling frequency f s = 2000 Hz. Since any discrete Fourier transform involved in the calculation of S( f ) leads inexorably to a noisy curve, the spectral profile must be smoothed before numerically computing the integral of equation (91). In Fig. 22(a), we show the original noisy profile of S( f ) (blue line) obtained by means of a discrete Fourier transform computed with 2 17 points. A polynomial fit (dashed black line) is performed to smooth the PSD profile. Then, a direct numerical calculation using equations (90)- (92) leads to the values of the storage and loss moduli shown in Fig. 22(b) over the available frequency interval 0 Hz ≤ f ≤ 1000 Hz. At high frequencies approaching 1 2 f s , an abrupt decrease of the numerical values of both G ( f ) and G ( f ) is observed. This is an artifact due to the finite frequency range, which imposes a cut-off in the required Kramers-Kronig integral transformation [209]. Therefore, the physically meaningful values of the components of the complex modulus G * ( f ) are only those at frequencies one decade below the Nyquist frequency [210].\n\n\nActive microrheology\n\nIn active microrheology, the trapped colloidal probe is externally forced through the material to locally strain it, and its mechanical response is directly measured [211]. From this response, G * ( f ) can be determined by means of equation (90). Different kinds of perturbations are possible. For instance, the probe particle can be trapped by static optical tweezers while the whole sample cell is moved by a piezo stage (section 3.11). Here, we will focus on a perturbation created by the oscillatory motion of the position x ot (t) of the optical trap, which exerts a time-dependent force F(t) = κx ot (t) on the particle according to equation (85) (Fig. 20(b)).\n\nAs active microrheology provides a direct measurement of the linear response function χ(t) defined by equation (89), it does not necessarily require that the material under investigation is in thermal equilibrium. Consequently, this technique is suitable for the investigation of rheological properties of general viscoelastic fluids and solids, including out-of-equilibrium soft matter, such as actin networks [212,213], physical gels [214], and glassy colloidal suspensions [215,216]. It is particularly useful for the determination of the storage G ( f ) and loss G ( f ) modulus at a specific driving frequency f d , for which a single-frequency oscillatory perturbation force\nF(t) = κX 0 cos(2π f d t + φ)(93)\ncan be directly applied to the colloidal probe, where X 0 is the maximum displacement of the optical trap and φ is the initial phase. Therefore, a previous passive calibration of κ is needed to know the applied force (section 3). In order for both the optical trapping and the material microstructure to remain in the linear response regime, X 0 must be sufficiently small. This is generally fulfilled if the typical energy injected by F(t) is at most of the order of ∼ k B T, i.e., X 0 k B T/κ. In Figure 23, we show some results to illustrate the implementation of active microrheology to characterize the rheological properties of a wormlike micellar solution at 5 mM and T = 20 • C (Table 9). In the inset of Figure 23(a), the black line shows the time evolution of F(t)/κ, which results from the sinusoidal motion of the optical trap (equation (93) with X 0 = 200 nm, κ = 1.0 pN µm −1 , f d = 1 Hz). The blue line plots the trajectory x(t) of an actively driven particle (2a = 2 µm), which clearly exhibits an oscillatory behavior with a time delay relative to F(t) due to the resistance of the surrounding viscoelastic fluid. Here, the crucial step is to compute the Fourier transform of the linear response function from the measurement of x(t) and x ot (t). From equation (89), this is given byχ\n( f ) = x( f ) F κx ot ( f ) ,(94)\nwhere the numerator is the value at frequency f of the non-equilibrium ensemble average of the Fourier transform of x(t), in presence of F(t) with the same initial phase φ, whereas the denominator involves the Fourier transform of the trap position. Due to the sinusoidal form of x ot (t), the denominator is zero for all frequencies f different from f d . Therefore, in practice it is customary to compute the inverse of equation (94) and to approximate it at f d by\n1 χ( f d ) = κ x ot ( f d )x * ( f d ) φ |x( f d )| 2 φ ,(95)\nwhere the numerator is the Fourier transform of the cross-correlation function between the trap position and the particle position, while the denominator is the value of the PSD of x(t), both quantities at frequency f d , first computed for a given F(t) and then averaged over different realizations of φ. In Fig. 23(a), we plot the PSD of the particle position x(t) in presence of the force F(t) at different driving frequencies f d . The pronounced peaks at f = f d result from the sinusoidal form of F(t)\n\n, while for f f d , the contribution to S( f ) is only due to thermal noise in the fluid. In Fig. 23(b), we plot as symbols the values of the storage and loss moduli computed by means of equations (90) at the distinct driving frequencies. For comparison, we also represent as dotted and dashed lines the corresponding values determined by passive microrheology, demonstrating the rather good agreement between both methods.\n\nAlthough more elaborate to implement in experiments, active microrheology is less prone to numerical artifacts than passive microrheology. This is because it is based on a measurement of the mechanical response of the particle position at a single frequency, which does not require any indirect numerical integration over a discrete finite frequency interval as those involved in the Kramers-Kronig relations.\n\n\nOther microrheological applications and outlook\n\nIn the previous sections, we have described some basic applications of microrheology with optical tweezers. More advanced methods, which are beyond the scope of this Tutorial, focus on investigating flow and deformation properties of complex materials which do not trivially fulfill the general conditions described above. Some important examples are the following:\n\nInterfacial microrheology. Near a liquid-solid, a liquid-liquid or a liquid-vapor interface, the viscosity of liquids becomes anisotropic and exhibits a dependence on the closest distance to the interface due to the specific hydrodynamic boundary conditions. Passive microrheology can be applied to determine such a spatial dependence, provided that the three spatial coordinates of the probe particle, embedded in the liquid phase of interest, can be tracked [217][218][219].\n\nTwo-point microrheology. The motion of pairs of colloidal particles not so far from each other is strongly correlated when suspended in a fluid. Since the flow and strain fields around one of them entrain the second particle, the motion of the latter encodes information of the rheological properties of the fluid at the location of the former. Therefore, a dual-beam optical tweezers can be used to fix their mean separation and perform two-point microrheology. Here, one particle plays the role of a control element, either thermally or externally driven, while the second one is used as a passive probe. This technique is particularly useful to investigate the mechanical response of heterogeneous and anisotropic soft materials, such as gels and polymer solutions [220,221].\n\nTransient microrheology. Using a combination of both active and passive microrheology, the transient behavior of viscoelastic materials either upon flow startup or cessation can be investigated. For example, during an active period, an optically trapped particle can be driven though the viscoelastic fluid, followed by a sudden release from the tweezers by shutting off the laser, during which its position is recorded. During this non-equilibrium passive period, the particle motion is subject to the recovery of the initially strained fluid microstructure. Consequently, this technique allows to measure in a straightforward manner relaxation times and relaxation moduli of viscoelastic materials [222,223].\n\nMicrorheology has evolved during the last two decades into a powerful experimental tool to investigate complex soft materials. Most of its current applications rely on a well-established theoretical framework based on equilibrium statistical mechanics, low-Reynolds-number hydrodynamics and linear response theory. Nevertheless, more recent approaches aim to investigate non-linear rheological properties, e.g., shear thinning of polymer solutions, by means of colloidal probes driven at very high velocities or under large-amplitude oscillations [20]. Although the implementation of such microrheological techniques does not represent a big experimental challenge, the interpretation of the data in terms of meaningful parameters and its connection with bulk quantities is not trivial. A complete understanding of such a wealth of information will certainly rely on current theoretical advancements in non-equilibrium soft matter systems [224], which are able to find a direct link between the motion of the probe and the microstructural deformation of the surrounding medium.\n\n\nColloidal interactions\n\nA colloidal system comprises a continuum medium (the solvent, such as water) and a disperse phase (the solute, such as small colloidal particles from 10 nm to 10 µm). Due to their dimensions, colloidal particles experience thermal fluctuations and undergo Brownian motion [225]. Since their properties can be described using statistical physics ensembles, colloidal suspensions represent an ideal model system for statistical physics and they can be considered as \"big atoms\" [226] mimicking atomic systems at different time and length scales, which make their dynamics accessible by optical experiments. Therefore, colloidal suspensions have been used to investigate phase transitions between gas, liquid, solid, and crystalline phases [227,228], crystal nucleation [229][230][231], glassy states [232], and vapor-liquid interfaces [233]. Colloidal suspensions are also employed in industry to stabilize emulsions (oil in water and water in oil) and foams [234,235]. Colloidal interactions play a key role in all these applications, from food industry [236] to nanotechnology [237]. The main colloidal interactions are:\n\nVan der Waals forces. Van der Waals forces arise between neutral particles such as atoms, molecules or colloids. They have an electromagnetic origin due to the interaction between the dipoles of the singles particles. They are repulsive at short distance range, due to repulsion between the atoms' electron clouds, and attractive at larger distance. Their intensity quickly decreases with the inter-particle distance following a power law [238,239].\n\nDouble-layer forces. Double-layer forces are very important for the stability of many biological systems and for the formation of colloidal crystals [240,241]. They result from the spontaneous charging of the particle surfaces when immersed in a liquid, due to ionic adsorption or dissociation. They can be both attractive or repulsive.\n\nDLVO theory forces. Double-layer forces and van der Waals forces are combined in the DLVO theory (named after Derjaguin, Landau, Verwey und Overbeek) [242] to calculate the interaction between spherical colloidal particles [243,244].\n\nSteric forces. Steric forces are repulsive nonbonding interactions influencing the spatial conformation and the reactivity of ions and molecules. They can be used to promote or prevent flocculation (steric stabilization) of colloidal solutions. They play an important role in chemistry, biochemistry, pharmacology and food industry, where they are often employed to design the physical properties of food and pharmaceutical products by steric stabilization [236,[245][246][247], as well as to treat waste water and to purify drinkable water by flocculation of metals and airborne particles [248,249].\n\nDepletion forces. Depletion forces are attractive forces arising in a suspension of large and small particles [250]. Usually, the large particles are colloids (typically, at least a fraction of a micrometer), while the small particles can be smaller colloids, micelles, solvent macromolecules, or dissolved ions [250][251][252][253][254].\n\nCritical Casimir forces. Critical Casimir forces emerge between objects (e.g., colloidal particles) immersed in a critical mixture close to its critical point [255][256][257]. Althougth these forces were first predicted theoretically in 1978 by M. E. Fisher and P. G. de Gennes [255] in analogy to quantum-electrodynamical (QED) Casimir forces [258], they have been measured directly only recently [259,260] proving their relevance for soft matter and nanotechnological applications thanks to their piconewton strength and nanometric action range [261][262][263].\n\nHere, we will focus our attention on how to measure critical Casimir forces. In particular, we will describe a typical experiment to quantify the dynamic effects of critical Casimir forces between two colloidal particles.\n\n\nCritical Casimir forces\n\nCritical Casimir forces take place between objects immersed in a critical mixture when they confine the mixture composition fluctuations arising near the critical temperature. These forces are typically in the piconewton and nanometer ranges, they are tunable as a function of temperature, and they feature an exquisite dependence on the surface properties of the involved objects [256,257,264,265]. Attractive critical Casimir forces arise whenever the density fluctuations are confined between objects with the same surface properties (e.g., between two hydrophilic or hydrophobic particles). Repulsive critical Casimir forces take place between objects with opposite surface properties (e.g., between a hydrophilic and a hydrophobic particle).\n\n\nExperiment outline\n\nIn the following sections, we will present in detail how to perform an experiment where the effects of critical Casimir forces on the free dynamic of a pair of colloidal particles are measured using blinking optical tweezers [263,[266][267][268].\n\nWe employ hydrophilic silica microspheres (diameter d = 2.06 ± 0.05 µm, Microparticles GmbH) dispersed in a water-2.6-lutidine mixture at the critical lutidine mass fraction c c L = 0.286, corresponding to a lower critical point at the temperature T c 34 • C [269,270]. Due to their extremely sensitive dependence on temperature, we have to control and stabilize the sample temperature to within ±2 mK via a feedback controller [260,263].\n\nUsing two holographic optical traps, we trap two microparticles in the bulk of the critical mixture at a surface-to-surface distance of ≈ 300 nm. This distance is larger than the range of the electrostatic repulsion between the particle surfaces, but still comparable with largest action range of critical Casimir forces. Whenever the two optical potentials are switched off (by chopping the laser beam), the colloids diffuse in the bulk of the critical mixture.\n\nWhen the temperature T of the solution is sufficiently lower than T c (i.e., for ∆T = T c − T 500 mK) no critical Casimir forces are observed and the particle diffuses freely in the medium. Approaching the critical temperature T c (i.e., for ∆T → 0), density fluctuations start to take place in the critical mixture leading to attractive critical Casimir forces between the two colloids, affecting their diffusion and reducing the average inter-particle distance. To obtain sufficient statistics of the dynamics of the colloids, we need to repeat the entire blinking process for several times (i.e, about 400 times) for each value of ∆T.\n\n\nExperimental setup and feedback temperature controller\n\nThe experimental setup requires holographic optical tweezers, digital video microscopy, and feedback temperature control [260,263]. The holographic optical tweezers (section 2.3.5) is realized using a phase-only spatial light modulator, a laser beam with a wavelength of 532 nm, and a oil-immersion objective (100×, NA = 1.30). The resulting traps can be periodically switched on and off by a chopper.\n\nThe most crucial part of the setup is the temperature controller to stabilize the sample temperature to within ±2 mK. This is realized using a two-stage stabilization protocol. The first temperature stabilization is achieved by keeping constant the temperature of the sample holder to within ±50 mK by a circulating water chiller. The second and finer temperature stabilization to within ±2 mK is achieved by a temperature feedback controller applied through the objective. The controller reads in real time the temperature of the objective and stabilizes it by a Peltier element thermically connected to the objective, which is the closest element to the investigated volume. An important aspect to take into account is the thermal isolation of the sample area as well as of the entire setup to prevent thermal fluctuations due to air flow and to room temperature fluctuations.\n\n\nPotential analysis\n\nThe motion of the two colloids is recorded by a camera with acquisition frequency 300 fps and the acquired videos can be analyzed by digital video microscopy [22,271] to obtain the trajectories r 1 (t) and r 2 (t) of the centers of the two particles projected onto the x y-plane, where r l (t) = (x l (t), y l (t)) with l = 1, 2 labelling the particles. Once these trajectories are obtained, it is possible to calculate their relative distance r(t) = |r 2 (t) − r 1 (t)| for different values of T (Fig. 24). The time evolution of the probability density distribution, for all the inter-particle distances r(t) at specific values of ∆T, are reported in Figs. 24(a-d). For ∆T = 456 ± 2 mK, far away from the critical point ( Fig. 24(a)), the particles diffuse freely and there is no critical Casimir forces affecting their behaviour. Increasing the temperature, ∆T = 200 ± 2 mK ( Fig. 24(b)) and ∆T = 163 ± 2 mK (Fig. 24(c)), density fluctuations start to take place in the mixture and critical Casimir forces arise affecting the dynamics of the colloids. We can observe that occasionally they cause adhesion between the colloids as can be inferred from the emergence of a peak in the inter-particle distance probability density at r ≈ 2.16 µm. This peak is due to the gradual emergence of attractive critical Casimir forces between the particles, approaching T c , and indicate that critical Casimir forces are strong enough to produce particle adhesion also in the presence of the optical potentials. The peak position indicates the region where the repulsive electrostatic forces and the attractive critical Casimir forces are balanced [263]. When the temperature T is very close to T c (∆T = 108 ± 2 mK, Fig. 24(d)), strong attractive critical Casimir forces inhibit the free diffusion of the particles, which often adhere to each other so that the values of r lie Sometimes r(t) is smaller than d because a displacement of the colloids along the vertical z-axis causes their projections onto the xy-plane to overlap. This occurs more frequently in the presence of critical Casimir forces and particularly if the particles are temporarily stuck together. (e-h) Equilibrium distribution P eq (r) of the inter-particle distance r(0) (i.e., when the optical tweezers are switched off) for two optically trapped colloids at temperatures (f) ∆T = 456 ± 2 mK, (g) 200 ± 2 mK, (h) 163 ± 2 mK, and (i) 108 ± 2 mK. Each histogram is obtained from 400 different experimental values. The solid black lines are the theoretical distribution of r(0), obtained via Monte Carlo integration (10 6 samples) of two optically trapped particles subjected to the theoretical total potential V(R 1 , R 2 ) (equation (96)). within a small region resulting from the equilibrium between the repulsive electrostatic forces and the attractive critical Casimir forces [263].\n\nFor each value of ∆T, all the initial inter-particle distances r 1,2 (0) represent the equilibrium distribution of initial positions R l = (x l , y l , z l ) with l = 1, 2 of the two colloids subjected to a total potential including the optical potentials V ot,1 (R 1 ) + V ot,2 (R 2 ), the repulsive electrostatic potential V es (ρ), and eventually the potential of the critical Casimir forces V C (ρ):\nV(R 1 , R 2 ) = V ot,1 (R 1 ) + V ot,2 (R 2 ) + V es (ρ) + V C (ρ),(96)\nwhere ρ = |R 2 − R 1 | − d is the actual surface-to-surface distance between the two colloids. Note that the projected distance r introduced above is generically smaller than the actual center-to-center distance |R 2 − R 1 |, due to possible displacements of the colloids along the vertical z-direction.\n\nAssuming that the optical potentials V ot,1 and V ot,2 reported in equation (96) are harmonic, we have:\nV ot,l (R l ) = 1 2 k l (R l − R 0,l ) 2 ,(97)\nwhere the centers R 0,l and the stiffnesses k l of traps l = 1, 2 can be determined experimentally by the separate calibration of the two optical traps. These initial values are fixed in all the experiment and in the following analysis.\n\nFor the electrostatic repulsion potential V es , we can consider the simple expression [260,270] \nV es (ρ) = k B T c e −(ρ−ρ es )/ D ,(98)\nwhere ρ is the surface-to-surface distance between the colloids, ρ es is an effective parameter, which depends on the surface charges, while D is the Debye screening length [260,270]. For the potential V C of the critical Casimir forces, we can consider the theoretical prediction based on the Derjaguin approximation [270]:\nV C (ρ) = k B T c d 4ρ Θ(ρ/ξ),(99)\nwhere Θ is a universal scaling function [272][273][274] and ξ is the bulk correlation length of the critical fluctuation of the density fluctuations of the critical mixture. Whenever the optical potential is switched on, the particles trajectories evolve under the action of the total potential V(R 1 , R 2 ), and after a sufficiently long time, they reach the equilibrium distribution:\nP eq (R 1 , R 2 ) ∝ exp[−V(R 1 , R 2 )/(k B T)].(100)\nThe histograms in Figs. 24(e-h) represent the experimental values of the equilibrium distribution P eq and the solid black lines are the corresponding theoretical results obtained from the Monte Carlo integration of equation (100). When the mixure temperature is far away from T c (∆T = 456 ± 2 mK, Fig. 24(e)), the probability distribution P eq (r) is very well approximated by a Gaussian distribution centered at the value r 2.40 µm, corresponding to experimental distance r 0 between the centres of the two optical traps [263]. Reducing ∆T (Figs. 24(e-h)), a peak arises at r 2.16 µm on the left flank of the Gaussian distribution, becoming more dominant at the expense of the Gaussian distribution. Starting from probability distribution P eq (r), we fit its experimental values by Monte Carlo integration of equation (100) based on the theoretical potential V(R 1 , R 2 ). In particular, we use the experimental values of V ot,l (R l ) (equation (97)) and Θ(ρ/ξ) (equation (99)) as input functions, and ρ es , D , and ξ as fitting parameters. In doing this, we assume that the fitting parameters ρ es and D have the same values ( D 13 nm and ρ es 95 nm) for all T, while the correlation length ξ is specific to each T. Once the values of ξ have been obtained, we calculate the real temperature of the solution by fitting the experimental temperature values measured at the objective with the theoretically expected temperature dependence of ξ, according to the method reported in Refs. [260,263]. (105)) fitted to the experimental data. The colors refer to data taken at ∆T = 456 ± 2 mK (blue), 273 ± 2 mK (light blue), 200 ± 2 mK (green), 163 ± 2 mK (yellow), 127 ± 2 mK (orange), and 108 ± 2 mK (red). Errors bars represent the standard deviation of the experimental values. The solid lines represent the theoretical predictions accounting for the effect of the hydrodynamic interaction between the colloids [275]. The nature of the deviations observed in D at short and long distances is discussed in the main text.\n\n\nDrift analysis\n\nThe relative position and the distance between the two particles can be used to determine the values of their diffusion coefficient D(r) and their drift velocity v(r) as function of their inter-particle distance r. Due to the hydrodynamic interaction between the two colloids, the particle diffusion coefficient depends on the distance between the particles and is different along the directions parallel and perpendicular to the line connecting the centers of the two particles (see equation (5.5) in Ref. [275]). For this analysis, we consider the particle trajectories as a sequence of values r i , numbered by i and acquired at times it s , where t s is the time between sampling. We can decompose the i-th displacement ∆r (n) i = r i+n − r i into its parallel and perpendicular components:\n∆r (n) i = ∆r (n) i ·r i(101)\nand\n∆r (n) i⊥ = ∆r (n) i · (ẑ ×r i ),(102)\nwherer i = r i /r i andẑ is the unit vector along the z-direction, which is perpendicular to the x y-plane of observation where the position vectors r i lie. Then, the parallel and perpendicular diffusion coefficients are [263]:\nD (r) = 1 2 |∆r (n=3) i | 2 3t s r i ∈ [r − δr, r + δr] ,(103)\nand D ⊥ (r) = 1 2 where δr represent the amplitude of a small interval around r. The measured normalized values of D ⊥ and D (symbols) are reported in Fig. 25 as functions of the ratio r/d, together with the theoretical prediction obtained in Ref. [275] (solid line) for no-slip boundary conditions. In particular, D ⊥ and D are normalized by the bulk diffusion constant\n|∆r (n=3) i⊥ | 2 3t s r i ∈ [r − δr, r + δr] ,(104)D 0 = k B T 3πηd 0.22 µm 2 s −1 ,(105)\nwhere η 2 · 10 −3 Ns m −2 is the viscosity of the mixture close to T c [276]. The values of D ⊥ and D in Fig. 25 shows satisfactory agreement, with a systematic discrepancy emerging only in D for r/d 1.05 due to the limited experimental acquisition rate, which does not allow us to resolve times shorter than 3 ms. 25 The parallel and perpendicular drift velocities can be measured using:\nv (r) = ∆r (n=10) i 10t s r i ∈ [r − δr, r + δr] ,(106)\nand\nv ⊥ (r) = ∆r (n=10) i⊥ 10t s r i ∈ [r − δr, r + δr] .(107)\nFor large ∆T (Fig. 25(a) (107)). Here, notice that v ⊥ vanishes in all investigated cases and shows no temperature dependence because all the forces at play in this experiment act along the direction which connects the centers of the particles [263].\n\n\nOther colloidal interactions and outlook\n\nCritical Casimir forces can play an important role in nanoscience and nanotechnology thanks to their piconewton strength, nanometric action range, fine tunability as a function of temperature, and particle surface dependence. They can be employed to manipulate micro and nano-objects (e.g., by controllable periodic deformations of chains), to assemble micro and nano-devices (e.g., via the self-assembly of colloidal molecules [277,278]), and to drive nanomachines (e.g., by powering rotators [279]) at the nano and micro-meter scale.\n\nBeyond critical Casimir forces, other colloidal interactions have also been studied using optical tweezers. Recently, Van der Waals forces have been directly measured between two isolated optically trapped rubidium atoms [280] opening the path to the atomic modeling of micro-and nano-structured devices controlled by these forces [281]. Double layer forces can also play an important role in nanotechnology due to their ability to tune the transport and deposition rate of micro and nano-sized particles. Steric forces play an important role in tuning the rates and the activation energies of most chemical reactions, crucial in chemistry, biochemistry, and pharmacology. Depletion forces are extensively used to stabilize colloidal solutions by flocculation, to produce the programmable self-assembly of colloidal nanostructures [282], and to drive cellular organization [252].\n\n\nStatistical physics\n\nThe advent of minituarization methods and microscopic manipulation techniques, as those discussed in this Tutorial, has opened the door to understand fundamental concepts and ideas at the foundations of statistical physics, allowing us to explore how macroscopic laws emerge from their microscopic dynamics. In this section, we only discuss experiments performed in a liquid. For related work in vacuum, see section 5 and Ref. [19]. We will start discussing the theory of Kramers transitions and stochastic resonance (section 4.5.1). We will continue exploring non-equilibrium fluctuation-dissipation theorems (section 4.5.2) and introducing stochastic thermodynamics (section 4.5.3). Finally, we will explain how the Carnot's cycle and efficiency of macroscopic engines can be scaled down to the microscopic realm, and conclude with a discussion of the concept of Maxwell's demons and Szilard engines (section 4.5.4).\n\n\nKramers' transitions and stochastic resonance\n\nUsing optical trapping techniques, Simon et al. [283] and later McCann et al. [284] studied the escape of a Brownian particle from a potential well, visualizing for the first time thermally activated processes and giving in this way a neat experimental proof of Kramers' theory [285].\n\nKramers' theory in the overdamped limit predicts the mean rate for the activation of processes over an energy barrier ∆U to be\n1 τ ≈ 1 τ 0 e −∆U/k B T ,(108)\nwhere τ is the mean residence time, also called Kramers' time and τ 0 is the overall relaxation time of the well, which can be expressed in terms of the angular frequencies near the stable\n(U A (x) = κ A x 2 /2 = mω A x 2 /2) and unstable (U S (x) = κ S x 2 /2 = mω S x 2 /2) critical points as follows, τ 0 = πγ m|ω S |ω A = πγ |κ S |κ A ,(109)\nwith γ the friction coefficient and m the mass of the particle. For the two-dimensional case, the expression for 1/τ 0 takes the following form [286]:\n1 τ 0 = m|w (x) S |w (x) A w (y) A πγw (y) S = |κ (x) S |κ (x) A κ (y) A πγ κ (y) S ,(110)\nwhere the superscripts (x) and (y) indicate the parallel and perpendicular directions to the axis joining the stable and unstable critical points, respectively. Fig. 27(a) shows a schematic of the configuration of the optical tweezers to generate a double-well potential.27 A silica bead28 of diameter d p = 0.96 µm is confined in a bistable potential generated by two parallel optical tweezers with orthogonal polarizations and separated by a distance d = 0.8 µm, which are generated by focusing two Gaussian beams with a water immersion objective with NA = 1.2. These two beams are generated with two polarizing beam splitters according to the scheme shown in Fig. 5 (section 2.3.2). The mirrors in the beam splitter are controlled with piezo actuators with 30 nm step resolution leading to a resolution of the position of the optical traps in the sample cell below 20 nm. The trapping region is set 27Alternative experimental configurations have also been used to realize a bistable optical potential. For example, using two nanoholes in a metallic film [287], sculpting the bistable potential with a spatial light modulator [288], or positioning time-shared optical traps in two closely-separated positions using acousto-optical deflectors [289].\n\n28Interestingly, also 1 µm polystyrene particles can be trapped with this setup, but with these particles we have not been able to observe any bistable behavior. We believe this is because the higher weight and lower refractive index of the silica beads make them more stably trapped along the axial direction of the beams allowing us to explore relatively larger distances between optical tweezers in a range where the bistable potential is neatly observed. A typical trajectory of the Brownian particle in this double-well potential is shown in Fig. 27(b). Along the x-direction joining the equilibrium points, the particle spends most of the time near the equilibrium points A and B, occasionally crossing the saddle point S. Along the perpendicular y-direction, the motion does not appear to be affected by the presence of two neighboring traps. Knowing the friction coefficient of the particle, this trajectory permits us to determine the Kramers' time using equations (108) and (110), where the characteristic frequencies (or equivalently their stiffnesses) at the critical points can be straightforwardly obtained using FORMA (section 3.8). Fig. 27(c) shows the force field in the xy-plane reconstructed using FORMA, from which the Table 10. Characterization of equilibrium points in a bistable potential. The theoretical mean crossing rates (1/τ (th) ) obtained using equations (108) and (110) Table 10 provides the stiffnesses of the equilibrium points (κ x and κ y ), the energy barriers (∆U), and the overall relaxation times (1/τ 0 ), from which we estimate the mean crossing rates (1/τ (th) ) using equations (108) and (110). The average crossing rate can be straightforwardly obtained from the trajectory of the bead by detecting the times at which the particle crosses the saddle point, shown in Fig. 27(f). For the validity of the approximations within Kramers' theory, these crossing events have to be rare, i.e., the time spent by the particle near the stable positions must be always longer than the relaxation time τ 0 defined by the stiffnesses of the critical points, limiting the ideal situations to the cases where the energy barrier is very high in comparison to k B T. This is a homogeneous Poisson process, where each escape event occurs at random with small probability [283]. Therefore, we expect the escape times to be exponentially distributed (red curves in Fig. 27(f)):\nPoint κ x (pN µm −1 ) κ y (pN µm −1 ) ∆U (k B T) 1 τ 0 (s −1 ) 1 τ (th) (s −1 ) 1 τ (ex) (s −1 ) A 0.p(τ) = p 0 e −τ/τ ,(111)\nwhere p 0 = 1/τ is the mean crossing rate and τ is the mean residence time. The resulting experimental mean rates are reported in Table 10 (1/τ (ex) ), showing very good agreement with data obtained by means of Kramers' assumptions (1/τ (th) ). Bistable potentials are often employed to study non-linear dynamics. For example, a bistable potential such as that described in this section has been used to demonstrate the emergence of stochastic resonance when the optical tweezers are modulated in time harmonically, but asymmetrically (i.e., one potential well gets deeper while the other one gets shallower) [283]: as the modulation time approaches the Kramers escape time, the escape probability and the activation rate increases. A different process of synchronization between two bistable systems was also demonstrated when two optically generated bistable systems with one bead each in water are set very close to let the particles to interact [288]: when the two bistable systems have slightly different escape rates, the particles synchronize their jumps and lock their rates to an intermediate value between those featured by the two non-interacting systems.\n\n\nNon-equilibrium fluctuation-dissipation relations\n\nIn its simplest form the fluctuation-dissipation theorem (FDT) [290] can be expressed as k B T µ = D (Einstein-Smoluchowski equation (7)): it provides a direct relationship between the mobility µ = 1/γ, which measures the response of its velocity to an external force, and its diffusion coefficient D.\n\nMore broadly, the FDT states that for a system in contact with a heat bath at temperature T, upon applying a small external perturbation h which changes the energy U of the system as U → U − hV, where V is the variable conjugate to h with respect to the energy, the following relation holds for any observable Q [291]:\nk B T R(t − s) = ∂ s C(t − s) for t ≥ s,(112)\nwhere\nR(t − s) = δ Q(t) h δh(s) h=0\nis the linear response function of Q at time t to the perturbation h at time s, δ δh denotes a functional derivative, i.e.\nQ(t) h = Q(t) 0 + ∫ t −∞ ds R(t − s)h(s),(113)\nand C(t − s) = Q(t)V(s) 0 is the two-time correlation function between the observable Q and V. The brackets . . . h denote a non-equilibrium ensemble average in presence of the perturbation h, whereas . . . 0 represent an equilibrium average for h = 0. The expression (112) of the FDT can be re-written in the integral form as\nk B T χ(t) = C(0) − C(t) for t ≥ 0,(114)\nwhere χ(t) = ∫ t 0 R(t − s)ds is the integrated response function, which is usually much easier to implement and measure than R(t).\n\nThe importance of equations (112) and (114) for experiments lies on the fact that linear response functions, which characterize states slightly away from equilibrium, and equilibrium correlation functions can be interchangeably determined by an appropriate choice of the varibles h, Q and V, depending on the specific conditions. For example, for a colloidal particle suspended in water at constant temperature T and trapped by optical tweezers with stiffness κ, the integrated response function of the particle position x with respect to the trap center x OT to an external time-dependent force can be determined by applying a Heaviside-like perturbative displacement of the trap position x OT (s) = x 0 Θ(s) and then by measuring the resulting time evolution of the ratio x(t) h κ x 0 at t ≥ 0. Then, it follows that R(t) = 1\nκ x 0 d x(t) h dt\n. Equivalently, it can be indirectly computed by means of the equilibrium quantity 1 k B T ∂ s C(t − s), where t ≥ s, Q = x is the observable of interest, V = x is the variable conjugate to the perturbative force h = κx OT , and C(t − s) = x(t)x(s) 0 , for which no external force is needed. Using the explicit expression for the positional correlation function in thermal equilibrium (section 3.5), x(t)x(s) 0 = k B T κ exp − κ |t−s | γ , one finds R(t) = 1 γ exp − κt γ , t ≥ 0, from which the time evolution of x(t) h in response to any external force h can be determined by means of equation (113).\n\n\nFluctuation-dissipation relation for non-equilibrium steady states (NESS).\n\nOptical tweezers can be used to explore some generalizations of equation (112) around non-equilibrium steady states (NESS) for mesoscopic systems such as colloidal particles, molecular motors, and biomolecules [147]. In particular, a 1D model system that has been extensively studied within the context of non-equilibrium statistical physics consists of an overdamped particle with friction coefficient γ, moving on a circle of radius R circle and polar coordinate θ, across a periodic potential U(θ) = U(θ + 2π) under the action of a constant force f 0 and a thermally fluctuating force ξ [164]. The dynamics of the polar coordinate θ is described by the Langevin equation\ndθ dt = −∂ θ [Aφ(θ)] + F + ζ,(115)\nwhere φ(θ) = U(θ) max{U(θ)} is the normalized potential profile, A = max{U(θ)} γa 2 is its amplitude, F = 1 γa f 0 is a non-conservative force term,29 and ζ = 1 γa ξ is a Gaussian white noise of zero 29On a circle, any constant force f 0 0 is non-conservative, because it cannot be expressed as the derivative of a potential due to the periodic boundary conditions. mean, i.e., ζ = 0, and autocorrelation ζ(t)ζ(s) = 2D θ δ(t − s), where the angular diffusion coefficient along the circle is given by D θ = D a 2 = k B T γa 2 . Due to the non-conservative force, the system lacks detailed balance: for constant values of F > 0, A > 0 and D θ > 0, and in absence of time-dependent perturbations, θ reaches a NESS, characterized by a distribution ρ NESS (θ), which is different from the equilibrium Boltzmann distribution ρ eq (θ) ∝ exp − Aφ(θ) D θ . Under such conditions, the particle is able to go beyond the potential barrier and explore the whole circle due to the combined effects of thermal fluctuations and the non-conservative force, thereby developing a constant non-vanishing probability current given by\nj = {F − ∂ θ [Aφ(θ)]}ρ NESS (θ) − D θ ∂ θ ρ NESS (θ) > 0.(116)\nAround this NESS, the expression (112) of the FDT can be generalized by the following relation [292] \nk B T R(t − s) = ∂ s C(t − s) − b(t − s), t ≥ s,(117)\nwhere . . . 0 and . . . h represent now a non-equilibrium ensemble average in the NESS (h = 0) and around the perturbed NESS (h 0), respectively, while the additional term\nb(t − s) = Q(t)v 0 (t)∂ θ V(s) 0(118)\ntakes into account the extent of the \"violation\" of the conventional FDT due to the non-vanishing current j. Here, the observables Q and V as well as v 0 are functions of θ, while v 0 is called the mean local velocity, and is given by v 0 (θ) = j/ρ NESS (θ). Note that the generalized expression (117) around a NESS reduces to the conventional FDT (112) around thermal equilibrium, when j = 0. The expression (117) of the generalized FDT can also be written in the integral form\nk B T χ(t) = C(0) − C(t) − B(t) for t ≥ 0,(119)\nwhere B(t) = ∫ t 0 b(t − s)ds, and represents the generalization of equation (114) for NESS. Similar experimental approaches based on scanning optical tweezers have been followed to investigate fluctuation-response relations for other kinds of non-equilibrium states [293][294][295][296][297].\n\n\nExperimental implementation of a NESS.\n\nWe illustrate the generalization of the FDT around a NESS with optical tweezers. To experimentally realize an overdamped system whose stochastic motion is well-modelled by equation (115), we consider a spherical silica particle (radius a = 1 µm) suspended in water at constant temperature (T = 20 • C, γ = 1.89 · 10 −8 kg s −1 ). The particle is subject to a toroidal optical trap, which can be created by a conventional single trap, whose position is deflected on the plane transverse to the beam axis either by two perpendicular galvanometric mirrors [293,298], or by a pair of perpendicular acousto-optic deflectors (AOD) [299]. For the latter approach, one applies simultaneously sinusoidal frequency modulations to the acoustic waves created inside each AOD X and Y, ∆F X,Y (t) = ∆F sin(2π f R t + α X,Y ), with amplitude ∆F ∼ MHz, scanning frequency f R , and phase difference |α X − α Y | = π 2 . When f R is kept fixed, the incident laser beam is deflected in a periodic manner. In this way the beam focus describes a circular trajectory inside the sample cell, θ OT (t) = 2π f R t + α, of constant radius R circle at frequency f R on the plane perpendicular to the beam propagation. To achieve a periodic force landscape as described by the Langevin model (115), the scanning frequency of the laser beam must be on the order of f R ∼ 10 2 Hz. For a particle with radius a = 1 µm, the optical setup must be built in such away that the final radius of the circular path described by the tweezers is R circle ≈ 2 − 5 µm. Smaller values of R circle lead to a Mexican-hat potential U(θ) in which the particle can spontaneously jump from θ to θ + π, similar to a Kramer process in a double-well potential (section 4.5.1), while larger values of R circle yield very slow dynamics for which it becomes experimentally difficult to sample θ with good statistics. For R circle = 4.12 µm, Fig. 28. Experimental realization of a NESS (a) Typical unwrapped (i.e., defined on the interval [0, ∞)) trajectories θ(t) of a colloidal particle in a NESS. The dashed line represents the mean drift due to the non-zero probability current j induced by the non-conservative term F. Inset: example of wrapped (i.e., defined over [0, 2π)) trajectory θ(t). (b) NESS probability density function of θ, defined over [0, 2π) (bars) and reconstructed potential energy U(θ) (solid line). The arrow indicates the direction of the non-conservative force f 0 , which shifts the maximum of ρ NESS (θ) to the right relative to the minimum of U(θ). Inset: comparison between the NESS distribution ρ NESS (θ) and the equilibrium one ρ eq (θ) (F = 0, sharp peak). (c) Correlation function between the observable Q(θ) = sin θ and the variable V(θ) = sin θ. f R = 200 Hz, and an infrared laser (λ = 1064 nm) with incident power of 30 mW, the resulting scanning speed of the focused beam is v R = 2πR circle f R ≈ 5 mm s −1 , which is so high that it is not able to trap and drag continuously the particle through the fluid because the viscous frictional force due to the surrounding water quickly exceeds the optical trapping force. Consequently, the beam only kicks the particle a small distance δs γv R κ along the polar direction θ at each rotation. During the absence of the beam (≈ 1/ f R = 5 ms) the particle undergoes free diffusion over a length scale l D = D f R 40 nm R circle in the directions radial and perpendicular to the circle. The beam kicks the particle again during the next arrival, thus preventing its escape by diffusion away from the region scanned by the beam focus. Therefore, the particle motion is effectively confined to a torus of major radius R circle and minor radius ∼ l D /2, where the angular position θ is the only relevant dynamical degree of freedom. In addition, a periodic intensity profile can be created along the main circle by sinusoidally modulating in time the laser power P(t) with an amplitude ∆P = 2.1 mW around the mean value of P 0 = 30 mW, at the same frequency as the scanning frequency f R , i.e., P(t) = P 0 + ∆P sin(2π f R t + α X ). These optical conditions result in the desired force landscape: a constant non-conservative force f 0 associated to the mean kick and a static periodic potential U(θ) due to the conservative force exerted by the sinusoidal light intensity profile [299,300].\n\nThe instantaneous angular coordinate θ of the optically driven particle can be obtained from its 2D trajectories (x, y), by θ = arctan(y/x) ( Fig. 28(a)). The effective parameters F, A and the profile φ(θ) must be determined from an ensemble of independent NESS trajectories θ(t): at least 100 trajectories, each with at least a full rotation on the circle are needed to find the probability current j (dashed line in Fig. 28(a)) and the corresponding nonequilibrium distristriution ρ NESS (θ) (histogram in Fig. 28(b)).\n\nDepending on the quantity of interest, θ must be defined either on the interval [0, 2π) or [0, ∞). For example, the probability current must be computed as [299] \nj = 1 2π d θ(t) 0 dt ,(120)\nwhere θ(t) 0 is the average of an ensemble of independent NESS trajectories θ(t), defined on [0, ∞) ( Fig. 28(a)). This permits us to verify that θ(t) 0 is linear in t, thus leading to a constant j > 0. The quantity j −1 represents the typical time the particle needs to perform a full cycle around the circle, which corresponds to the longest relaxation time of the NESS. For example, for the experiment presented here j −1 = 26 s, which is much larger than the typical equilibrium relaxation time of a particle in water trapped by optical tweezers (∼ ms). Therefore, it is expected that the particle motion in this NESS has very long-lived correlations. In Fig. 28(b), we plot the experimental profile of the NESS distribution ρ NESS (θ), computed from all the data points of the trajectories on [0, 2π). From the experimental profile of ρ NESS (θ) and the value of j, the full force landscape of the particle can be reconstructed [301]. First, taking into account the periodic boundary conditions of the NESS distribution, i.e. ρ NESS (0) = ρ NESS (2π), from equation (116) we get the non-conservative term\nF = j 2π ∫ 2π 0 dθ 1 ρ NESS (θ) ,(121)\nwhereas the potential is given by\nAφ(θ) = −D θ log ρ NESS (θ) + ∫ θ 0 dθ F − j ρ NESS (θ ) .(122)\nThe experimental values of the parameters considered here yield F = 0.85 rad s −1 and A = 0.87 rad 2 s −1 , which correspond to a non-conservative force f 0 = γaF = 66 fN and an energy potential amplitude max{U(θ)} = γa 2 A = 68.8 k B T, while the normalized potential profile is sinusoidal, φ(θ) = sin θ (red solid line in Fig. 28(b)). In thermal equilibrium (F = 0), the particle motion would be tightly confined around the minimum of such a deep non-linear potential at φ m = 3π/2, with a width ∆θ ≈ k B T γa 2 A = 0.12 rad and a vanishingly small probability to cross the potential barrier of height 2A = 137.6 k B T (inset in Fig. 28(b)). However, under the action of the non-conservative force, the distribution of θ is much broader, with a maximum shifted in the direction of the resulting current j > 0. This reflects the highly non-equilibrium conditions of the system. Indeed, they can be better assessed by means of the generalized FDT relation (119). We focus on the observable Q(θ) = sin θ, which is periodic on the circle and corresponds to the dimensionless potential energy of the particle φ(θ) = U(θ)/max{U(θ)}.30 A variation in the potential amplitude, A → A + δ A translates into the following change in the potential energy: γa 2 Aφ(θ) → γa 2 Aφ(θ) − (−γa 2 δ A)φ(θ). Therefore, by noting that a variation h in a control parameter leads to a change in the potential energy, i.e., U → U − hV, we conclude that in this specific example the integrated response function of Q(θ) to the perturbation h = −γa 2 δ A is given by equation (119), where the variable conjugate to the perturbation is V(θ) = sin θ, whereas ∂ θ V(θ) = cos θ in equation (118). In Fig. 28(c), we plot the NESS correlation function C(t) = sin θ(t) sin θ(0) 0 , revealing an oscillatory behavior with a typical time-scale ≈ 25 s, which corresponds to j −1 . Here, to improve the statistics from the available data, the NESS ensemble average sin θ(t) sin θ(0) 0 is computed first by taking a time average\nC t max (t) = 1 t max − t ∫ t max −t 0 du sin θ(u) sin θ(u + t),(123)\nalong a given realization of θ(t), where t max > j −1 to span a full cycle of the probability current, and then by performing an additional ensemble average C t max (t) over different independent NESS trajectories θ(t). The same procedure must be carried out for the correlation B(t) defined through equation (118). In Fig. 28(d), we plot both the NESS terms C(0) − C(t) and B(t) involved in the integral generalized FDT (119). We observe that both terms are of the same order of magnitude, thus confirming that the experimental conditions of the system are far from thermal equilibrium. Note that, without the corrective term b defined in equation (118), a determination of the integrated response function χ(t) by means of the equilibrium FDT (114), and consequently R(t), would be wrong. This is shown in the inset of Fig. 28(d) , where we compare the numerical time derivative of 1\nk B T [C(0) − C(t)] with that of 1 k B T [C(0) − C(t) − B(t)]\n, which shows that without taking into account the corrective term B, the response function R(t) would be highly overestimated around a NESS.\n\n\nStochastic Thermodynamics\n\nStochastic thermodynamics is an emerging branch of statistical physics that extends concepts of classical thermodynamics such as heat, work and entropy production to the level of single stochastic trajectories for systems in contact with a heat bath and driven arbitrarily far from thermal equilibrium by means of a well-specified experimental protocol [164]. It provides a number of statistical relationships involving these fluctuating quantities, which generalize the macroscopic laws of thermodynamics and describe the non-equilibrium energetics of many mesoscopic systems of interest in soft matter and biophysics, such as colloidal particles, polymers, biomolecules, molecular motors, biochemical networks, and microelectromechanical systems [147,302]. Thus, stochastic thermodynamics represents nowadays a solid theoretical framework for applications ranging from single-molecule biomechanics (section 4.1) to optimization protocols and the efficiency of microscopic heat engines (section 4.5.4) .\n\nHere, we present some general notions of stochastic thermodynamics using a paradigmatic system: a colloidal particle embedded in a viscous liquid (friction coefficient γ), in contact with a thermostat at constant temperature T and subject to a possibly time-dependent force. The 1D motion of a single coordinate x of the particle can be described by the Langevin equation\nx = µ − ∂U(x, λ) ∂ x + f (x, λ) + ξ,(124)\nwhere U(x, λ) is the potential energy which leads to a conservative part −∂U(x, λ)/∂ x in the total force, whereas f (x, λ) represents a non-conservative force, as that described in section 4.5.2. Both forces can be time-dependent through the control paramenter λ, which can be externally varied according to a prescribed protocol from λ(0) to λ(t) during the time interval [0, t]. In addition, in equation (124) µ = 1/γ is the particle mobility, whereas ξ represents a Gaussian white noise with zero-mean and autocorrelation function ξ(t)ξ(s) = 2Dδ(t − s), which accounts for the thermal fluctuations of the particle velocity. Stochastic thermodynamics requires that the properties of the heat bath are not affected by the presence of external driving forces, thus implying that D = k B T µ regardless of the strength of −∂U(x, λ)/∂ x and f (x, λ). Note that a statistically equivalent description of the system modelled by equation (124) can be given by means of the Fokker-Planck equation for the probability density function of x at time t\n∂ ρ(x, t) ∂t = − ∂ j(x, t) ∂ x ,(125)\nwhere the probability current is defined as\nj(x, t) = µ − ∂U(x, λ(t)) ∂ x + f (x, λ(t)) ρ(x, t) − D ∂ ρ(x, t) ∂ x ,(126)\nsupplemented with the initial condition ρ(x, t = 0) = ρ 0 (x). Both ρ(x, t) and j(x, t) allow to directly assess how far the system is from thermal equilibrium conditions, for which j = 0 and ρ ∝ exp − U(x) k B T .\n\n\nFirst Law.\n\nTo generalize classical thermodynamic concepts to the level of an individual stochastic realization of the process x(t), we first note that either a temporal variation of the control parameter λ during the infinitesimal time duration dt or a non-zero non-conservative force, f 0, can be identified as an infinitesimal work increment done on the system:\nδW = ∂U ∂λ λdt + f dx.(127)\nThen, from equation (127) an inifinitesimal variation dU in the potential energy during dt can be written as\ndU = ∂U ∂λ λdt + ∂U ∂ x dx, = δW − f − ∂U ∂ x dx.\nThus, by defining the infinitesimal increment of the heat dissipated into the medium as the product of the total force acting on the particle times the infinitesimal displacement dx,\nδQ = f − ∂U ∂ x dx,(128)\nthe expression (128) can be regarded as a generalization of the First Law of thermodynamics [303] dU = δW − δQ on the level of the stochastic process x(t). From the Langevin equation (124), the work done on the system and the heat dissipated into the medium over the finite time interval [0, τ], are\nW τ [x(t)] = ∫ τ 0 dt ∂U(x(t), λ(t)) ∂λ λ(t) + f x(t) ,(130)Q τ [x(t)] = ∫ τ 0 dt f − ∂U(x(t), λ(t)) ∂ x x(t), = 1 µ ∫ τ 0 dt x(t) 2 − 1 µ ∫ τ 0 dtξ(t) x(t),(131)\nrespectively, where the notation [x(t)] highlights the fact that both quantities fluctuate over time and depend on the specific stochastic realization of x(t) up to time τ. The terms which involve dx(t) = dt x(t) in the integrals of equations (130) and (131) must be interpreted in the Stratonovich sense [304]. Note that the definition of the heat given by equations (128) and (131) is physically meaningful, since the first term on the right-hand side of equation (131) is positive definite and represents the energy delivered to the thermostat, while the second term can be either positive or negative and corresponds to the thermal fluctuating energy injected by the medium. Moreover, for systems in thermal equilibrium (i.e., f = 0 and λ = 0), equation (129) consistently reduces to dU = −δQ, where any increment (decrease) in the potential energy of the particle is only due to a fluctuation of the heat taken up from (delivered to) the thermostat, being both on average equal to zero. We analyze the previous concepts in the specific case of a colloidal particle (diameter 2a = 2.73 µm) suspended in water at constant T = 295 K, for which the friction coefficient is γ = 2.44 · 10 −8 kg s −1 . The particle is subject to the harmonic potential of an optical trap, where the position of its minimum plays the role of the control parameter λ in equation (124). In the case where the optical trap is uniformly translated, λ varies linearly over time at constant speed v and, under the initial condition λ(t = 0) = 0, U(x, λ(t)) = 1 2 k (x − vt) 2 , λ(t) = vt, and λ(t) = v. Without loss of generality, we consider the simplest situation where all the forces acting on the system are conservative (i.e., f = 0), thus being the temporal variation of λ the source of non-equilibrium work and dissipated heat. This well-defined protocol can be implemented by means of the deflection of the trapping beam using galvanometric mirrors or acousto-optics deflectors (section 2.3.1). Although with a different thermodyamic interpretation,31 a closely related protocol can be carried out by moving the sample cell at constant speed v by means of a piezoelectric stage (section 3.11) [305]. Fig. 29(a) shows a typical experimental trajectory x(t) of such a colloidal particle, as well as the time evolution of the position of the minimum of the optical trap, λ(t). For times much longer than the trap relaxation time (≈ 21 ms), the dragged particle reaches a steady state, whose statistical properties do not depend on time. In this case, from the definition (130), one can find the work done by the optical tweezers on the trapped particle over a time interval of duration τ along a given stochastic realization of x(t):\nW τ = −kv ∫ τ 0 dt [x(t) − vt] , = 1 2 kv 2 τ 2 − kv ∫ τ 0 dt x(t),(132)\nwhere we have dropped the argument [x(t)] for the sake of simplicity. Since for this control parameter and a harmonic potential, equation (124) is linear and ξ is Gaussian, x must also be 31When moving the sample stage at speed v, an external flow is applied to the trapped particle, which results in an extra term in the viscous drag force acting on the particle, i.e. γ( x − v). On the other hand, the potential energy remains unaffected by v, U(x) = κ x 2 /2. This leads to expressions for the work and heat which are different from those obtained when performing a motion of the optical trap.\n\nGaussian and, thus, W τ is Gaussian as well, with mean\nW τ = γv 2 τ(133)\nand variance\n[W τ − W τ ] 2 = 2k B T γv 2 τ + γ k exp − kτ γ − γ k .(134)\nIndeed, using the experimental trajectory x(t) (Fig. 29(a)), we compute numerically W τ by means of equation (132) and check that its mean and standard deviation satisfy equations (133) and (134), as shown by the blue circles in Figs. 29(b) and 29(c), respectively. The distribution of W τ for different values of τ is plotted in Fig. 29(d), where we verify that P(W τ ) is Gaussian. Note that, while W τ is always positive and increases linearly with the observation time τ as the trap drags the particle through the fluid, there can be negative work fluctuations, where the system loses mechanical energy [305]. These rare events, which are at first glance in contradiction with macroscopic intuition, are due to thermal fluctuations, which are not taken into account by classical thermodynamics. The role of such fluctuations, quantified by the ratio [W τ − W τ ] 2 / W τ , is very prominent at short time-scales and become negligible as τ approaches the limit of a macroscopic observation time, i.e., τ γ k . The heat dissipated into the medium due to the motion at constant speed of the trap can be derived from equation (131) as\nQ τ = −k ∫ τ 0 dt [x(t) − vt] x(t) = −k ∫ τ 0 [x(t) − vt] dx(t).(135)\nIn this case, some care is needed to compute the stochastic integral (132) from the experimental values of x(t). As previously mentioned, Stranotovich integration must be used to get physically meaningful results.32 Specifically, from the collection of N experimental data points x(t j ), defined at each time t j = j−1 N −1 τ over [0, τ], where j = 1, . . . , N, a mid-point discretization of the force −k[x(t) − vt] with respect to the stochastic increment dx(t) must be performed\nQ τ = − k 2 N −1 j=1 x(t j+1 ) + x(t j ) − (t j+1 + t j )v [x(t j+1 ) − x(t j )].(136)\nIn Fig. 29(e) we show the distribution of Q τ for different values of τ, computed by means of equation (136). In this case, all discrete time steps have a constant value fixed by the inverse of the experimental sampling frequency f s = 145 Hz, i.e., t j+1 − t j = f −1 s = 0.007 s. It should be noted that, although the mean value of Q τ coincides with that of W τ for all τ ≥ 0 in a steady state, as shown in Fig 29(b), their distributions are quite different, see Fig.29(e). In this case, the distribution of the heat, P(Q τ ) has non-Gaussian tails, with standard deviation larger than the variance of the work fluctuations (Fig. 29(c)). In addition, similar to the behavior of the stochastic work, rare events where Q τ is negative are also possible for small values of τ at which thermal fluctuations are significant, i.e., the heat spontaneously flows from the surrounding medium to the particle even if mechanical work is done on it.\n\nFinally, in Fig. 29(f) we plot the distribution P(∆U τ ) of the variation of the potential energy over a time interval of duration τ:\n∆U τ = U(x(τ), λ(0)) − U(x(0), λ(0)) = k 2 [x(τ) − vτ] 2 − x(0) 2 .(137)\n32While equation (132) for the work simply requires the product of the deterministic time increment dt with the stochastic particle position relative to the position of the trap, x(t) − vt, the expression for the heat (equation (135)) involves the stochastic increment of the particle position, dx times x(t) − vt.\n\nIn this case, while the mean value of ∆U τ is equal to 0 for all values of τ ( Fig. 29(b)), consistent with the fact that in a steady state the potential energy of the system remains constant on average, its distribution is strongly non-Gaussian, due to the non-linear dependence of ∆U τ on x(t), with standard deviation which quickly saturates to a constant value, as observed in Fig. 29(c).\n\nThe statistical properties of W τ , Q τ and ∆U τ for this simple experiment illustrate the main essence of stochastic thermodynamics. Even though the average values of such quantities satisfy the expected thermodynamics behavior, the probability of observing rare events, which are not described by the classical thermodynamics, is not negligible for mesoscopic systems at sufficiently short time scales.\n\nSecond Law and entropy. Stochastic thermodynamics also offers the possibility to extend the concept of entropy to stochastic trajectories of systems in non-equilibrium states, and provides quantitative relationships that generalize the macroscopic Second Law of thermodynamics. In the case of a system described by equation (124), the variation of the total entropy is defined over a given time interval [0, τ] as\n∆S τ = Q τ [x(t)] T − k B ln ρ(x(τ), t) ρ(x(0), 0) .(138)\nThe first term on the right-hand side of equation (138) can be identified as a change in entropy due to the heat dissipated into the medium, defined in equation (131), while the second term is called stochastic entropy, and must be determined by solving first the corresponding Fokker-Planck equation (125) and then evaluating the solution ρ(x, t) along the specific stochastic trajectory x(t) with initial condition x(0) at t = 0. Therefore, unlike work, heat and potential energy, the stochastic entropy does not only depend on the particular realization of x(t) but on the ensemble through the quantity ρ(x, t). For example, for the linear motion (λ(t) = vt) of the position of the optical trap considered in the experiment above, the solution of the Fokker-Planck equation (125) is\nρ(x, t) ∝ exp − k(x−vt+ γ v k ) 2 2k B T\nunder the condition that the system has already achieved a steady state in the comoving frame of the trap. In this case, the variation of the stochastic entropy over the time interval [0, τ] is given by k 2T\n\nx(τ) − vτ + γv k 2 − x(0) + γv k 2 , from which the corresponding change of the total entropy can be computed using equations (131) and (138).\n\nRemarkably, the change in the total entropy defined by equation (138) satisfies the integral fluctuation theorem (IFT) [306] exp − 1 k B ∆S τ = 1 (139) for all measurement times τ and any kind of protocol involving a non-equilibrium temporal variation of the control parameter λ from λ(0) to λ(τ), and possibly including non-conservative forces f 0 that break detailed balance. Using the Jensen's inequality, e a ≥ e a , equation (139) implies that ∆S τ ≥ 0, which corresponds to one of the expressions of the Second Law of thermodynamics. Therefore, the IFT represents a refinement of the Second Law of classical thermodynamics. For the experiment above, we verify the IFT for different values of τ in the inset of Figure 29(e), where the left-hand side of equation (139) is numerically determined using the computed values of ∆S τ . Depending on the specific protocol for the variation of the control parameter λ and on the state of the system (e.g., transient or steady), alternative versions of the Second Law can be derived within the context of stochastic thermodynamics. For example, when driving the system by changing the control parameter from λ(0) to λ(τ), the non-equilibrium work W τ done on the system, defined by equation (130), and the corresponding difference in free energy ∆F = F(λ(τ)) − F(λ(0)), satisfies the relation [307] \nexp − W τ k B T = exp − ∆F k B T(140)\nfor any arbitrary non-equilibrium protocol λ(t), 0 ≤ t ≤ τ , provided that the system is initally in thermal equilibrium at t = 0, and all the forces acting on the system are conservative (i.e., f = 0 in equation (124)). Equation (140) is called the Jarzynski relation [307], and derives directly from the IFT (139), in which case the total entropy takes the particular form ∆S τ = (W τ −∆F)/T, where W τ − ∆F is known as dissipated work. A closely related expression, from which the Jarzynski relation (140) can alternatively be derived is the so-called Crooks fluctuation theorem [308]. It quantifies the ratio between the probability density of the work spent in the forward process, ρ(W τ ), when driving the system from thermal equilibrium at λ(0) through an arbitrary nonequilibrium protocol up to λ(τ), and the distribution of the work done in the reversed process, ρ(W τ ), where the control parameter is varied asλ(t) = λ(τ − t), starting in thermal equilibrium atλ(0) = λ(τ). The expression of the Crooks fluctuations theorem is\nρ(W τ = −W) ρ(W τ = W) = exp − (W − ∆F) k B T .(141)\nThe practical importance of the relations (140) and (141) is that, unlike quasistatic experimental protocols based on classical thermodynamics, they allow to determine free energy differences between two equilibrium states from non-equilibrium measurements of the work performed at an arbitrarily fast rate, for example when studying biomolecules (section 4.1).\n\n\nCarnot cycles and Maxwell's demons\n\nIn this section, we will explore how optical tweezers can be used to get a deeper understanding of Carnot cycles, work efficiency and Maxwell engines. Although the results presented in this section have not been implemented by us in the laboratory directly, we provide sufficient instructions for the readers to be able to readily realize these experiments.\n\n\nCarnot cycles and microscopic engines\n\nIn 1824 Sadi Carnot tried to answer basic questions about the efficiency, construction and operation of macroscopic heat engines [309]. This seminal work gave birth to thermodynamics, for which Carnot is considered one of its fathers. It also introduced concepts that later were used by Clausius and Kelvin to define the concept of entropy and introduce the Second Law of thermodynamics. Almost two centuries later, similar questions posed by Carnot on big, chunky, greasy, macroscopic heat engines, are finally being explored in the microscopic realm [18]. Colloquially Carnot's cycle refers to a process in which one extracts work from a system coupled to two reservoirs at different temperatures T c < T h . More precisely, given a system interacting with two thermal baths, Carnot's cycle consists of four steps:\n\n1. An isothermal expansion at temperature T h , during which an amount of heat Q h is absorbed by the system 2. An adiabatic expansion from T h to T c .\n\n3. An isothermal contraction at temperature T c , during which an amount of heat Q c is released from the system.\n\n4. An adiabatic contraction from T c to T h , returning the system back to its original state. The four steps of this cycle are depicted in Fig. 30(a) in the (S, T)-plane, where S refers to the entropy. For a generic fluid with volume V and pressure P, Carnot's cycle in the (V, P)-plane is shown in Fig. 30(b). During one cycle, the extracted work is given by the First Law of thermodynamics (142) and the entropy production is\n− W = Q h + Q c∆S tot = − Q h T h − Q c T c .(143)\nThis implies that for a reversible operation (i.e., for a process for which ∆S tot = 0) we obtain the following relationship between the heat exchange and the temperature of both baths:\nQ c Q h = − T c T h .(144)\nCarnot showed that, for a reversible process, the efficiency of Carnot's cycle is\nη C ≡ −W Q h = Q h + Q c Q h = 1 − T c T h ,(145)\nwhere the efficiency η = −W/Q h is defined as the ratio between the extracted work and the absorbed heat. Ideally, for the process to be reversible, it must be performed infinitely slowly or, more precisely in a time τ much slower than any characteristic relaxation time of the system. This automatically means that the output power, defined as P ≡ −W/τ is zero for an ideal reversible heat engine.\n\nIn reality, Carnot's cycle must be performed in a finite time τ, which implies a trade-off between output power and efficiency. Due to the Second Law of thermodynamics, , which imposes ∆S tot > 0, we have that η < η C .\n\nMoreover, practical heat engines have additioanl limitations that further reduce their efficiency [310]. For example, accounting for heat transfer yields a maximum power output given by the Novikov-Curzon-Ahlborn formula [311,312] \nη NCA = 1 − T c T h ,(146)\nwhich has been confirmed experimentally with different protocols [313]. Another closed thermodynamic cycle, the Stirling cycle, is often used to describe engines that work by compressing and expanding a working fluid at different temperatures.33 This is also composed of four steps, namely:\n\n1. An isothermal compression during which heat is extracted towards the cold bath T c .\n\n2. An isochoric heat-addition.\n\n3. An isothermal expansion during which the system absorbs heat from the hot bath T h .\n\n\nAn isochoric heat-removal.\n\nA sketch of Stirling's cycle can be seen in Fig. 30(c).\n\nIn the last decade, various research groups have probed the concept of Carnot efficiency in microscopic heat engines, particularly focusing on single-particle or colloidal heat engines. The first of these colloidal heat engines implemented a Stirling engine by using an optically trapped particle [314], as shown in Fig. 30(d). In this design, a colloidal particle plays the role of the working fluid, while the optical trap can be understood as the engine cylinder where the compression and expansion occur. The cycle includes the following phases:\n\n1. The isothermal compression by increasing the trap stiffness (i.e., increasing the laser output power).\n\n2. The isochoric head-addition by heating up the colloidal suspension from T c to T h .\n\n3. The isothermal expansion by decreasing the trap stiffness (i.e., decreasing the laser power).\n\n\nThe isochoric heat-removal by cooling down the colloidal suspension from T h to T c .\n\nBeing a microscopic engine, the associated extracted work is a stochastic variable that depends strongly on the cycle duration τ. The first experimental realization of this microscopic Stirling cycle [314] operated between two baths at T c = 22 • C and T h = 90 • C with the maximum mean power −W/τ obtained for τ = 7 s. In these conditions, the average work extracted per cycle was about 4.5 · 10 −22 J, In 2014, a micrometer-sized equivalent of a piston steam engine was realized in an optical tweezer, where a single colloidal microparticle plays the role of the piston. The particle absorbs a small amount of light enabling to locally raise the temperature when the particle is exposed to light. At the beginning of the cycle, the gradient force of an optical tweezers pulls the particle towards the laser focus, giving rise to a dramatic increase in temperature and, consequently, 33The Stirling's cycle has the same efficiency as Carnot's cycle. the explosive vaporization of the surrounding liquid. This causes a bubble, which pushes the particle out of the focal region. As a consequence, the particle cools down and is pulled back again by the optical tweezers [315]. To achieve the sudden bubble formation, this experiment exploited a superheated liquid, which becomes unstable and relaxes back towards equilibrium via the nucleation of a vapor cavitation bubble. The mean extracted work per cycle exceeded that found in Ref. [314] by a striking six to eight orders of magnitude, because both the cycle frequency and maximum displacements were around two orders of magnitude larger.\n\nThese two approaches to microscopic heat engines rely on laser heating of the solution where the particle is immersed. The heat exchange rates of the heating and cooling processes effectively limit the range of accessible temperatures and the timescales of the experiments. This technical difficulty can be overcome by adding a synthetic noise to the Brownian motion [316]. This permits one to heat without having to raise or lower the physical temperature of the colloidal suspension, allowing fast heating and cooling as well as removing temperature limits, at least in principle. This method is called white-noise technique. Mathematically, the idea works as follows. Consider the underdamped one-dimensional Brownian particle:\nm d 2 x(t) dt 2 = −γ dx(t) dt − κ(t)x + ξ(t) + η(t) ,(147)\nwhere ξ(t) corresponds to the usual thermal noise with\nξ(t) = 0 , ξ(t)ξ(t ) = 2k B T δ(t − t ) ,(148)\nwhile η(t) is, for example, an electrostatic noise such that\nη(t) = 0 , η(t)η(t ) = σ 2 noise δ(t − t ) .(149)\nConsider, in addittion, that these two noise sources are independent. This model is statistically equivalent to an effective model for stochastic thermodynamics,\nm d 2 x(t) dt 2 = −γ dx(t) dt − κ(t)x + ξ eff (t) ,(150)\nwhere ξ eff (t) can be understood as thermal noise with effective temperature34\nT eff = T + σ 2 noise 2k B γ .(151)\nThe white-noise techniques allow to reach effective temperatures as high as T = 3300 K. The synthetic noise can be generated by placing the colloidal suspension in an electrophoretic fluid chamber with two electrodes connected to a computer-controlled electric generator that produces a sequence of noisy random electric fields, whose statistical properties closely resemble that of white Gaussian noise [316]. A similar effect of adding a random force applied to the particle can be achieved by modulating the position of the trap by using an acousto-optic deflector [317,318]. A remarkable step forward was achieved in a series of works where the concept of microadiabaticity was introduced [319], and later used for the creation of the first Brownian Carnot engine by concatenating isothermal and microadiabatic protocols [320]. The basic idea is as follows. Recall that, macroscopically, an adiabatic process occurs without exchange of heat (or mass) between the system and its surroundings (i.e., Q(t) = 0). For microscopic systems, the amount of heat is a random variable, so the simplest generalization is to consider those processes for which the average amount of exchanged heat is zero (i.e., Q(t) = 0). Moreover, if we further 34See also section 5.7, where we discuss feedback cooling of trapped particles in vacuum, which can also be discussed in terms of an effective temperature. assume that the process is performed quasi-statically, we also have that the average value of the variation of the total entropy is zero, ∆S tot (t) = 0, and, consequently, the process is isentropic, ∆S sys (t) = 0. In this scenario, a trick to implement a microadiabatic process consists in using protocols which keep the system's entropy constant. Importantly, this process can be quantified fairly precisely. Indeed, since the system's entropy is given by\nS sys (t) = −k B log P t (x(t), v(t))(152)\nand assuming that the system is close to equilibrium, i.e.,\nP t (x(t), v(t)) = 1 Z e − H (t ) k B T (t ) ,(153)\nwhere H(t) is the Hermitian of the system, we obtain that\nS sys (t) ∝ x 2 (t) v 2 (t) ∝ T 2 (t) κ(t) .(154)\nThus, a pseudo-adiabatic process corresponds to keeping T 2 (t)/κ(t) constant. Notice that the microscopic Carnot engine was also explored in the work in Ref. [321], but only considering the configurational space and not the phase space. [322] addressed the issue that the Second Law of thermodynamics might be violated by the inclusion of \"intelligent beings\" capable of processing information. In its original version, an ideal gas at temperature T is enclosed in an isolated container divided into two equal parts by a fixed wall. In this wall, there is a trap door operated by a \"demon\" who will open it if an incoming particle from the left (right) is faster (slower) than the average. After a while, the right compartment will have the fastest particles and, consequently, a higher temperature. Therefore, the effect of the demon is to generate a temperature gradient, which can be used to obtain work from the system, without the corresponding energy expenditure, thus violating the Second Law.\n\n\nMaxwell's demons and Szilard engines. The thought experiment known as Maxwell's demon\n\nTo better understand the origin of such violation, Leo Szilard proposed his eponymous simplified model. The Szilard engine consists of a single particle within a container divided into two equal partitions separated by a movable, frictionless wall [323]. When the container is put into contact with a single thermal reservoir at temperature T, the movable wall may then be used to extract work (e.g., by lifting a weight), as the particle transfers kinetic energy in successive elastic collisions. The isothermal volume expansion results in a work extraction W ext = k B T ln(V f /V 0 ), where V 0 and V f are the initial and final volumes of the partition that contains the particle. Since the final volume V f is that of the total box, then V 0 /V f = 1/2 is the fraction of the volume that contains the particle. This implies, in turn, that the expected work extracted from the engine is simply k B T ln 2, which is proportional to the Shannon entropy measuring the uncertainty on the initial position of the particle in a symmetric partition.\n\nLeon Brillouin suggested that the work extraction of the Szilard engine is precisely compensated by the work of measurement and, therefore, no net extraction of work is possible [324]. After a decade, Rolf Landauer introduced a concept of logical irreversibility [325] and Charles H. Bennett explained that the work extraction is precisely compensated by the erasing process of the memory apparatus [326]. Recently, a modified version of the Second Law was derived [327,328], the Jarzynski-Sagawa-Ueda relation, which has also been confirmed experimentally [329]. This latter result claims that the work extraction is compensated by the mutual information between the system and the measurement apparatus which is involved in both the measurement and erasing processes [330]. In a relative recent experiment [331], the authors used a colloidal suspension to implement a Szilard engine to explore how the entropy decreases due to the spontaneous symmetry breaking that is the cornerstone of the Szilard engine and Landauer's principle, and to measure directly this entropic change for a Brownian particle in a bistable potential implemented by two optical traps. Very recently, a continuous versions of the Maxwell's demon, which is capable of extracting arbitrarily large amounts of work per cycle by repeated measurements of the state of a system, was proposed and was experimentally implemented in a single DNA hairpin pulling experiment [332].\n\n\nOther applications to statistical physics and outlook\n\nIn this section, we have presented some well-known applications of optical tweezers to investigate both equilibrium and non-equilibrium properties of model systems in statistical physics. Here, we briefly mention some emerging areas where optical tweezers also represent a powerful tool to study statistical quantities of microscopic systems.\n\n\nSynchronization. Coordinated motion of self-sustained oscillators mediated by means of\n\nhydrodynamic interactions is a fundamental behavior in many biological systems, such as cilia and flagella. Using optically trapped microparticles, it has been possible to elucidate and directly visualize different aspects of the hydrodynamic synchronization of oscillators and rotors. There are different approaches to define these oscillators, for example with light-propelled chiral particles [333], using angular momentum in optical tweezers to propel birefringent or asymmetric particles [46], by means of rotating energy landscapes [334][335][336] or, as we already mentioned in section 4.5.1, by means of bistable potentials sculpted with light [288]. In all these cases, when two of these self-sustained oscillators are close enough, the hydrodynamic interactions perturb and tend to synchronize their phases and their frequencies. Since all of these experiments have been realized at microscopic scale, thermal fluctuations are not negligible and escape transitions play an important role in their dynamics. Active matter. Active particle systems, such as motile microorganisms, are non-equilibrium systems capable of converting autonomously the energy from their surroundings into directed motion. In recent years, also the possibility of developing artificial active matter systems has been extensively explored [337,338]. Optical tweezers can be used to probe such phenomena and their connection to non-equilibrium relations [339].\n\n\nPhase transitions.\n\nOptical tweezers can be used in a variety of situations to induce phase transitions and to probe critical phenomena [260]. One can, for example, use optical lattices to control colloidal phase transitions via tunable interactions [340][341][342].\n\nFurther advances in optical trapping could also enable the possibility to address more fundamental questions on statistical aspects of small systems, such as memory effects in non-equilibrium systems. Moreover, most of the studies discussed in this section deal with the behavior of individual particles, but it is natural to wonder how the properties of these systems change in many-body systems. Equally interesting would be to explore how microengines or active baths can be used to perform practical tasks, such as energy harvesting at the microscale.\n\n\nTransport in extended potential landscapes\n\nMicroscopic particles in a fluid are constantly diffusing: the smaller the particle, the faster it diffuses, following a power law dependence D ∝ 1/a, where a is the particle radius. As D is a tiny quantity in typical conditions (of the order of µm 2 s −1 ), diffusive transport is very slow. For example, on average a 1 µm particle in water takes approximately 26 days to move by about 1 mm away from its original position. Transport can be greatly enhanced by external forces. These forces generate directed motion, where diffusion generally plays a pernicious role, since the particle will be gradually dispersed away from the path defined otherwise by the force. In Nature, several microorganisms have developed different strategies for exploring their environment in a more efficient and precise way than that of simple diffusion, for example using flagella or cilia.\n\nMicroscopic objects are often subject to the influence of structured force landscapes, which may either boost or inhibit their motion. For example, some molecular motors inside eukaryotic cells move along the paths defined by microtubules that exhibit a periodic structure of two proteins, acting as a periodic potential. A paradigmatic model that serves as a starting point for studying microscopic dynamics is the motion of a Brownian particle in a corrugated potential landscape under the action of an external force. Being able to study and understand these transport mechanisms is not only important from the fundamental science perspective, but it also opens up the possibility to reproduce them and develop new techniques to perform tasks, such as particle sorting and delivery in lab-on-a-chip platforms.\n\nAs we will see in detail in this section, colloids are ideal prototypes for the study of microscopic transport phenomena, and extended light patterns provide a powerful and versatile way to create reconfigurable potential energy landscapes in optical micromanipulation systems. We will be primarily concerned with conservative optical forces in this section, but there are also interesting light-induced transport phenomena that emerge in the presence of non-conservative forces (e.g., scattering forces), as we will briefly comment on. Table 11 presents some examples of extended light patterns used in optical manipulation: the images on the left illustrate the optical field of interest generated at the sample (i.e., at the focal plane of the microscope objective or the last lens of the optical trapping setup); the images on the right provide information about how these patterns can be experimentally generated. As explained in section 2.3, there are different ways for creating extended light patterns, but here we will focus our attention on the two most common approaches using interference or diffractive optical elements.\n\n\nOptical forces and potentials in extended light patterns\n\nA subwavelength particle is pulled by the conservative optical gradient forces towards the brightest regions of an extended light pattern. In this case, the particle moves in an extended potential energy landscape mimicking the shape of the intensity pattern, with valleys at maxima and hills at minima of intensity. In this regime, the magnitude of the forces, or the depth of the valleys, increases in proportion to the cube of the particle radius and linearly with optical power. However, for particles whose size is comparable or larger than the wavelength, the shape of the energy landscape might be remarkably different from the intensity distribution.\n\nImportantly, when working with extended patterns of light, one should keep in mind that the larger the area covered by the pattern, the lower the intensity for a given optical power and, therefore, the magnitude of the optical force decreases. Hence, it is important to estimate the appropriate area of the pattern so that the average intensity is at least of the order of 0.1 mW µm −2 .\n\nOne-dimensional potentials Let us analyze the simplest case: an extended pattern of fringes obtained from the interference of two collimated beams, as that in the first row of Table 11. At the back aperture of the microscope objective, the Fourier transform of the two beams corresponds to a pair of focused spots, whose separation (2s) determines, together with the focal length of the microscope objective ( f ), the period of the interference pattern at the sample plane: Λ = λ f 2s , where λ is the wavelength of the laser in the background medium. The corresponding intensity distribution is given by\nI(x, y) = 2I 0 1 + cos 2πx Λ = 4I 0 cos 2 πx Λ ,(155)\n\nSinusoidal pattern (periodic in 1D) produced by the interference of two beams (left) and its Fourier spectrum (right).\n\nAny interferometer is suitable to produce this pattern, such as Fresnel biprisms [343] or a Mach-Zehnder interferometer [39,344]. A diffraction grating generated with a SLM can also be used to split the beam [94].\n\nSawtooth pattern (periodic in 1D) (left) generated by the superposition of two orthogonally polarized sinusoidal patterns (right). A three-armed Mach-Zehnder interferometer can produce two patterns of fringes, one with twice the spatial period of the other, by controlling the polarization of the input beams. The phase difference and intensity ratio determines the asymmetry [63,345].\n\n\nOblique lattice (periodic in 2D): Interferometric pattern of four beams (left) and its Fourier spectrum (right).\n\nThese patterns can be generated using a tandem of two interferometers or using diffraction gratings designed to split the input beam into multiple beams travelling along the desired directions [344].\n\n\nRectangular lattice (periodic in 2D): Diffractive pattern (left) generated with a phase hologram (right).\n\nHolographic phase masks can be designed to produce the target distributions of individual light spots using iterative optimization algorithms [22]. These can be displayed on a SLM and dynamically reconfigured.\n\n\n> >\n\n\nQuasi-periodic lattice: Interferometric pattern of five equidistant beams (left) and its Fourier spectrum (right).\n\nA pentagon-shaped diffraction grating can be displayed on a phase-only SLM to split the input beam into five beams travelling at the desired angle.\n\n\nZeroth-order Bessel beam (left) and its annular Fourier spectrum (right).\n\nBessel beams can be obtained efficiently with a conical lens or axicon. Any nondiffracting beam can also be generated by means of a properly encoded phase mask displayed on a SLM [100,101].\n\n\nLattice of asymmetric traps in 2D (left) generated by the superposition of orthogonally polarized identical patterns.\n\nA light lattice generated with a phase hologram can be introduced into a polarization splitting Mach-Zehnder device to produce two identical patterns, whose relative positions and intensity ratio determine the asymmetry of the resulting pattern [66].\n\nSpeckle pattern (left) generated with a random phase mask (right). Speckle can be generated by passing a laser beam through a diffuser plate [89], by coupling the laser into a multimode optical fiber [65,346], or with a random phase mask displayed on a SLM [107]. where we have assumed that both interfering beams are plane waves with equal intensity, I 0 .\n\nIn reality, there is always an envelope function modulating the intensity, often with a Gaussian shape I 0 (x, y) = A exp[−2(x 2 + y 2 ) 2 /w 2 0 ], but as long as w 0 Λ, the plane-wave assumption works reasonably well in the central region of the envelope.\n\nThere are several approaches with different levels of complexity for calculating the optical force exerted by this pattern on a dielectric particle of diameter D p > λ. In all cases, the force is also periodic with the same periodicity as the intensity profile, so it can be written as\nF(x) = A sin(2πx/Λ).(156)\nNotice that there is a π/2-phase difference between the expressions for the intensity and that for the force, since both maxima and minima of the intensity function correspond to zeroes of the force, representing either stable or unstable equilibrium positions. The amplitude factor A is proportional to the intensity of light and, more importantly, it depends on the size of the particle and its refractive index, as well as on the period Λ of the pattern [38,39,347], i.e. A = A(D p , n p , Λ).\n\nTo illustrate this fact, Fig. 31(a) shows the dependence of the normalized amplitude of the force (Q = A/2I 0 ) as a function of the period, for two silica spheres (refractive index n p = 1.46) with diameters D p = 2 µm and 4 µm, respectively, immersed in water (refractive index n m = 1.33). If Λ is larger than the bead size, the particle feels a potential with equilibrium points at the intensity maxima, corresponding to negative values of Q, similar to what happens with subwavelength particles. There is a global minimum in each curve of Fig. 31(a), which represents the optimum period, slightly larger than the diameter of the sphere, that maximizes the magnitude of the force. Now, as the period is progressively reduced, the magnitude of the force decreases until it vanishes, meaning that a light pattern with such a spatial period will have no effect on the sphere. If Λ becomes even shorter, the force changes its sign (Q > 0) and the optical potential reverses, having stable equilibrium points in places where the intensity is minimum. This is schematically illustrated in Figs. 31(b-c), showing each particle at one of its respective equilibrium positions in the same intensity pattern (background), which has the period Λ = 2 µm (marked by the vertical dashed line in Fig. 31(a)). The resulting energy potentials for these particles, U(x), are also plotted, exhibiting a sign reverse with respect to one another.\n\nIn general, the most important aspect for determining the trapping positions in an extended pattern is the relative size of the particle with respect to the characteristic size of the pattern. In that sense, it is convenient to plot Q in terms of the ratio Λ/D p . Accordingly, Fig. 31(d) depicts the normalized force amplitude as a function of Λ/D p for spheres made of silica n p = 1.46 (solid curve) and polystyrene n p = 1.6 (dashed curve). The red and blue vertical dashed lines indicate the values of the parameter Λ/D p corresponding to the examples in Figs. 31(b) and 31(c), respectively. This kind of pattern has been widely used for optical manipulation of multiple particles and optical sorting [38,39,87,94,344,348,349].\n\nThe particle-dependent response to the structure of a light pattern is sometimes referred to as the size effect of the optical forces and takes place regardless of the complexity of the pattern. As the optical landscape becomes more intricate, nevertheless, not only the magnitude and the sign of the optical force may change like in our simple example, but also the shape of the potential may be drastically altered depending on the size of the particle. For example, Fig. 32 shows the case of a sawtooth pattern of fringes, also shown in the second row of Table 11. This pattern can be generated by superimposing three beams with linear polarization oriented along the horizontal, vertical and diagonal directions, so that the latter partially interferes with the former two, creating two patterns of fringes orthogonally polarized, one with twice the period of the other [63]. Given the asymmetric intensity in the fringes, a consequent asymmetry could be expected in the potential, and this is true for the smaller particle (D a = 2 µm) shown in Fig. 32. However, a larger particle (D b = 4 µm) experiences a completely different potential, with a slight . Particle-dependent potential or size effect in asymmetric patterns. Asymmetric pattern of light with period Λ = 4 µm (background) and the resulting optical potentials, U a (x) and U b (x), for two silica beads with diameters D a = 2 µm and D b = 4 µm, respectively. The shape of the optical potential for a given pattern of light can be very different depending on the size of the particle. The potential was computed using the ray optics model. asymmetry in the opposite direction. The modification of the potential can be even more radical for other particle sizes. The application of this effect for the transport of particles was analyzed in detail in Refs. [63,345].\n\n\nTwo-dimensional potentials\n\nWhen the periodicity of the pattern is in 2D the size effect might give rise to different behaviors along each symmetry axis, and thus the equilibrium positions for a given particle might be unpredictable without performing an explicit calculation of the optical potential. Different kinds of light lattices in 2D have been used to study optical binding and colloidal crystallization [54,55], to explore structural phase transitions in colloidal monolayers [340,350,351], to tailor bandgap materials [352], and to optically sort particles in combination with microfluidics [344]. Periodic light lattices in 2D can be produced by the interference of at least three beams. For example, the coherent superposition of three waves, depending on the directions of their wave vectors, can lead to centered-rectangular or hexagonal lattices, whereas the interference of four waves can produce square, oblique or rectangular lattices. The example in the third row of Table 11 corresponds to an oblique or rhombic lattice. An alternative approach to generate 2D-periodic patterns is by means of phase holograms [41], such as the rectangular array of individual spots illustrated in the fourth row of Table 11. In contrast with the multiple interference fields, diffractive patterns can act as a collection of individual traps for 3D confinement, overcoming the size effect and hence facilitating the control of the equilibrium positions of individual particles. A SLM can be used as well to display diffractive gratings and produce multiple interfering beams [94], whilst the 2D-trapping capabilities can be extended to manipulate ordered structures in 3D by taking advantage of the self-imaging Talbot effect [353]. Another interesting case is that of a quasi-periodic pattern arising from the superposition of five waves, which is shown in the fifth row of Table 11. This has been used to study the formation of quasi crystalline structures in colloidal monolayers [354,355].\n\nA type of extended patterns with circular, elliptical or parabolic symmetry in the transverse plane correspond to Bessel, Mathieu and parabolic nondiffracting beams, respectively [95,96,98]. Their Fourier spectrum at the back aperture of the microscope objective lies on a thin annulus, where the particular beam geometry is determined only by the intensity and phase modulation around the annulus. The optical forces in a Bessel beam, like the pattern depicted in the sixth row of Table 11, exhibit very interesting features due to the size effect [356][357][358], and similar attributes can be expected in the other types of beams. Namely, the multi-ringed structure of the intensity pattern has an envelope that decreases radially outwards. Depending on its size with respect to the pattern, a particle might respond predominantly to the fine structure or to the envelope. This can be used as a passive mechanism to sort particles [359] or to build what is known as a washboard potential [360].\n\nAnother version of an asymmetric periodic pattern, but now in 2D, is illustrated in the seventh row of Table 11. This is generated by superimposing two identical light lattices, orthogonally polarized, produced with a single phase hologram [66]. By controlling the relative position and the intensity ratio between the two patterns, the asymmetry of the light lattice can be tailored. In general, periodic and asymmetric patterns are very important in the study of the ratchet effect [60,62,63,66], which is the emergence of directed transport in the presence of unbiased external forces due to a spatiotemporal symmetry breaking and, furthermore, is thought to be at the core of the operation mechanism of some biological engines [361].\n\nRandom light patterns like speckle (bottom row of Table 11) are also very important in the context of particle diffusion and transport. In practice, speckle can be generated by transmitting a laser beam through a glass diffuser, by coupling the laser into a multimode optical fiber, or by encoding a random phase mask on a SLM. A very interesting aspect of the interaction of Brownian particles with a random potential landscape is its major role in diffusion, and how it depends on the several parameters of the potential [65,89,346,362].\n\n\nTransport mechanisms\n\nSo far, we have examined optical forces and potentials in a static situation. Now we will examine the conditions for inducing directed transport in the system, i.e., a net current of particles in a given direction through their interaction with light and, possibly, other external forces. To address this subject, we will start by looking at the general equation of motion for a single particle interacting with an optical potential in the overdamped regime, which is a good description of the micromanipulation systems under consideration. For the sake of simplicity, only one dimension will be examined, this is:\nγ x = − d dx U(x, t) + F(t) + ξ(t),(157)\nwhere γ represents an effective drag coefficient, U(x, t) is the optical potential, F(t) denotes an external force, and ξ(t) is the thermal noise, having a correlation function ξ(t)ξ(t ) = 2Dδ(t −t ), with D = k B T/γ being the diffusion coefficient. Although noise usually plays an important role in microscopic-scale dynamics and some transport mechanisms even rely on it, this can be neglected under certain conditions, giving rise to a deterministic dynamics, ξ(t) ≈ 0. For example, Brownian motion of silica or latex spheres immersed in water at room temperature is very subtle if the diameter is D 5 µm, and it becomes practically negligible for D 10 µm.\n\nIn general terms, we can distinguish between two main transport mechanisms: a dynamic potential and a static potential in the presence of an external driving force. A very simple realization of the former case corresponds to a light pattern of fringes that moves with a constant speed v with respect to the static sample. This corresponds to a travelling periodic potential: U(x, t) = U(x − vt). The dynamics of the particle can be analyzed in a reference frame that moves along with the potential by making the change of variable u = x − vt in equation (157), leading to\nγ u = − d du U(u) − γv + ξ(t).(158)\nNotice that this is equivalent to adding a constant drag force to a static pattern and thus the following discussion will be valid for the case of a constant external driving force. Whether the particle will follow the pattern or not depends on the relative magnitude between the optical force, F op (u) = −dU/du, and the drag force, F d = −γv, in the moving reference frame. The constant force has the effect of tilting the potential energy landscape, giving rise to what is sometimes referred to as a washboard potential. If |F op | > |F d |, the particle will not be able to escape from the well and, therefore, it will be locked-in to the pattern, moving with the same velocity as the pattern with respect to a fixed reference frame. A particle with this behavior has been called a Brownian surfer [363]. If |F op | < |F d |, the particle will be pulled out from the potential well by the drag force in the moving reference frame or, in the fixed frame, it will not follow the pattern but it will diffuse around its initial position. This is known as a Brownian swimmer [363]. Depending on the thermal noise level, there might be intermediate situations, in which the particle moves in the same direction as the pattern but with a reduced speed v p < v. This happens when |F op | |F d | or, in terms of the potential energy barrier between neighboring wells, when ∆U k B T, so that thermal energy may sometimes help the particle to surmount the potential barrier between consecutive wells. Due to the size effect discussed above, in a polydisperse mixture of particles, some of them may follow the pattern while others may not, depending on their size and refractive index, or even if two types of particles move along with the pattern, their average speeds may be different. Therefore, a moving potential can be used as a mechanism to separate particles of different characteristics, i.e. as an optical sorting device.\n\nIn practice, there are several ways to move an interference pattern of fringes. Let us consider, for example, an optical micromanipulation setup with a Mach-Zehnder interferometer. We can express the resulting interference pattern at the sample plane as I(x, y) = 4I 0 cos 2 (πx/Λ + ϕ(t)/2), where ϕ(t) represents a time-dependent phase difference introduced in one arm of the interferometer. To generate a travelling potential, ϕ(t) should be a linear function of time. An elegant way to achieve this is by using the angular Doppler effect [364,365], arising when a circularly polarized beam passes through a rotating λ/2-plate, which gives rise to a frequency shift ∆ω = ±2Ω, where Ω is the rotation speed of the λ/2-plate. This can be easily implemented in the Mach-Zehnder interferometer [348], as illustrated in Fig. 33(a). A polarizing beam splitter (PBS) is followed by a λ/4-plate in each arm of the interferometer, producing orthogonal circular polarization states. The rotating λ/2-plate is inserted in one arm, such that it inverts the handedness of the circular polarization at the same time that it introduces an angular Doppler frequency shift. A non-polarizing beam splitter is placed at the output of the interferometer to superimpose the beams. The resulting phase difference between the two beams is ϕ(t) = (∆ω)t = ±2Ωt, where the sign depends on the rotation direction of the λ/2-plate with respect to the polarization handedness [365]. At the sample plane, this will define the direction of motion of the fringes (±x), whereas the speed of motion is v = Ω/k.\n\nThere are other alternatives for producing motion of the fringes. For example, a very simple option, which can be easily implemented in different types of interferometers, is to introduce a thin glass plate in the path of one of the interfering beams [343]. As illustrated in Fig. 33(b) for the particular case of a Mach-Zehnder interferometer, by gradually tilting the glass plate a small angle θ(t), the optical path length of the corresponding beam is modified, resulting in a relative phase shift between the two beams and the consequent motion of the pattern. A straightforward analysis using geometrical optics yields that, if θ 1, then the change in the optical path length is given by ∆(θ) ≈ T g /2(1 − 1/n) 2 θ 2 , where T g is the thickness of the glass plate and n its refractive index. Therefore, as ∆(θ) is not linear, the phase shift is not linear either: ϕ(t) = k∆(θ(t)). Although this complicates the theoretical description of the system, it is a cheap and accessible choice if the aim is just to make a demonstration of particle transport.\n\nAnother configuration that provides full control of the phase shift ϕ(t) makes use of a displacement of one of the mirrors in a Mach-Zehnder interferometer. For example, in the first interferometric optical tweezers setup [87], depicted in Fig. 33(c), an extra beam splitter is placed in one arm of the interferometer (BS1), so that mirror M1 retroreflects the beam and, mounted on a motorized translation stage, it is moved with a constant speed: D(t) = vt. This produces a phase shift between the interfering beams of the form ϕ(t) = 2k D(t), leading to a continuous motion of the fringes at the same speed as the mirror (v), while keeping the spatial period of the fringes unaltered. Indeed, the period of the fringes is a function of the lateral distance between the output beams and the focal length of lens L1 (see Fig. 33(c)). Notice, however, that the presence of the beam splitter BS1 yields an additional intensity loss. To equalize the intensity in both arms of the interferometer, two λ/2-plates should be used, one at the entrance of the setup in combination with a polarizing beam splitter (PBS) to adjust the amount of light in each arm, and the other one to set the same polarization in the two beams.\n\nA variant of this idea, which avoids the extra beam splitter, is shown in Fig. 33(d). A periodic and asymmetric displacement of the mirror M1 might be equivalent to a continuous travelling pattern under certain conditions. This can be done by mounting the mirror on a piezo-stage35 driven with a sawtooth function [39]:\nD(t) = v 1 t if 0 < t ≤ τ −v 2 t if τ < t ≤ τ + δt ,(159)\nwhere τ δt, v 1 = D max /τ and v 2 = D max /δt, with D max denoting the maximum displacement of the mirror, which should be small to prevent distortion in the period of the fringes. The frequency of D(t) is f = 1/T, with T = τ + δt the period of the function: D(t + T) = D(t). The motion of the mirror is related to the phase shift of the pattern as ϕ(t) = 2 √ 2k D(t), implying that the fringes at the sample mimic the sawtooth-like displacement, with speeds given by\nu i = 2 √ 2Λv i /λ, where i = 1, 2.\nThe main idea is that the particle is able to follow the pattern in the direction of the slower motion (u 1 ), but not during the fast motion (u 2 ) in the opposite direction, resulting in a unidirectional transport. Of course, the selection of the parameters36 D max , f , and τ, along with the spatial period of the fringes Λ, is crucial to optimize the current of particles. For example, if D max = mλ/(2\n\n\n√\n\n2), with m a positive integer, the particle will advance a distance mΛ in one direction and will end up at another minimum of the potential after an entire cycle, provided that Λ is such that optimizes the optical force for the given particle size. Figs. 34(a-b) illustrate the resulting particle current for two sizes of particles as a function of Λ and Ω = 2π f , respectively. An optimum can be clearly identified in each case. This device was used as an optical sieve to separate particles according to their size and it can also be used to sort particles according to their refractive indices [39]. In the latter case, the strategy is to start with a low intensity and gradually increase the optical power up to a value for which only the particles with the highest refractive index contrast with respect to the medium will respond.\n\nA related approach consists in generating phase gratings with a SLM aimed at splitting an incident beam to produce interference. The advantage is that the angular separation between the beams can be varied by changing the grating, so that the spatial period of the fringes can be tuned and its orientation can be modified as well. Furthermore, the motion of the pattern can be also controlled with the SLM, by adding a continuous phase shift to one of the beams. Such a device has been used to move and sort living objects and particles of different shapes [94].\n\nIt is worth mentioning that the techniques discussed above can be extended to more complex interference patterns, such as rotating beams, allowing circular motion of beads and structures [?,366]. Similar ideas have been applied as well to build optical conveyors that allow bidirectional transport in 3D. These kinds of devices use coaxial beams, either counter-propagating to generate axial standing waves or co-propagating, and not just Gaussian beams but also other structured fields, like Laguerre-Gaussian and Bessel beams [38,357,[367][368][369], including evanescent fields [349]. The operation principle is basically the same as before: to manage the motion of an interference field by adding a controlled phase shift between the beams.\n\n35Any programmable motorized translation stage would be useful and, for example, stepper motors are cheaper, at the cost of loosing precision and speed in the motion control.\n\n36There are alternative choices, but in any case, the dynamic phase ϕ(t) is defined by a set of three independent parameters, set through the motion function of the mirror. frequency Ω of the sawtooth function for the optimum spatial period in each case and fixed power P = 130 mW. The two particle kinds feature different optimal currents as a function of Λ and Ω, enabling to classify them with high throughput. Data from Ref. [39].\n\nThere have also been different approaches to yield moving patterns based on digital holography. Among the first schemes, we can mention a mechanism called optical peristalsis, in which a lattice of optical traps is created and controlled with a set of phase holograms displayed on a SLM [61]. Following a periodic sequence of three states, the pattern is consecutively displaced by one-third of the lattice constant, reaching the initial state again after an entire cycle. A particle trapped in a given potential well is released when that pattern is turned off and it is immediately attracted towards the nearest shifted well once the next-state pattern is turned on. By repeating this process, the particle is transferred from one well to the neighboring trap after each cycle, experiencing a constant motion in one direction, resembling the operation of a peristaltic pump. The geometry of the lattice can be varied to achieve, for example, a radial current in circular patterns. If the particle's diffusion during the time between consecutive states is negligible, the process is deterministic and the current is given by v = L/T, where L and T are the lattice constant and the temporal period of the sequence, respectively.\n\n\nOther transport mechanisms and applications\n\nAs we have already mentioned, a travelling potential is dynamically equivalent to having a particle in a fixed light pattern subject to a constant drag or, more generally, a driving force. The drag force can be provided, for example, by tilting the sample [347], by moving the sample with respect to the pattern [63,370], or by means of a controlled microfluidic flow. The latter option has the important advantage of being suitable to integrate into lab-on-chip platforms [76,371]. In addition, sorting capabilities can be boosted by using 2D light lattices instead of 1D-periodic patterns, offering new control parameters and thus more versatility. Indeed, biological or colloidal assays flowing through two-dimensional patterns may be deflected from the body of the flowing fluid depending on the orientation of the lattice with respect to the flow and on the strength of the optical force, giving rise to discrete angle deviations or fractionation [344,372,373]. Using this approach, a high throughput of sorted samples can be practically and efficiently achieved. Also, microfluidic techniques can be combined with other schemes of optomechanical control, like dynamic or blinking potentials [374].\n\nA fundamentally different and intriguing mechanism to selectively move objects at the microscopic scale is based on the ratchet effect. In contrast to previous examples where there exist a biased external field to induce a net current of particles, the ratchet effect makes use of non-biased time-varying forces to induce transport. Since these forces are of zero-mean in time and in space, a non-trivial symmetry has to be broken to observe a net current [361]. The light patterns represented in the second and seventh rows of Table 11 are examples of ratchet potentials with a space asymmetry, yet the symmetry breaking may be much more subtle, including variations in time. Actually, the spatial structure and time-modulation of the potential along with the time-dependence of the external force define many different types of ratchets [60,62,63,66,[375][376][377][378]. Moreover, although most of these systems are noise-assisted when particle diffusion is non-negligible, there are also deterministic ratchets [63]. The rich dynamics arising in these systems is due to the delicate interplay among the whole set of encompassed parameters and manifests itself in the diverse phenomena that can be observed, including bidirectional transport depending on size, currents in different directions with respect to the driving, current reversals and chaotic behavior. Interestingly, the ratchet mechanism is ubiquitous, as it appears in many biological, physical and chemical systems, and its study has paved the way to broaden our understanding of transport processes at the micro-and nanoscales [361].\n\nSo far we have looked mainly at ordered patterns or light lattices, but random patterns have also shown an excellent selectivity for sorting particles [65,89], with the additional advantage of making feasible to sort and select a continuous throughput of chosen constituents in turbid media, common in many biological assays.\n\nAlthough we did not deal with scattering forces in our previous discussion, it is important to remind that the use of scattering rather than gradient forces was really the first means to transport and separate particles. Indeed, since his earliest experiments in 1970, Ashkin noted that radiation pressure acting on particles of different sizes along the propagation axis of a moderately focused beam was proportional to the size of the particles, so that the beads were guided along the beam axis with different velocities [1]. This idea is the basis of the so-called optical chromatography, in which a Gaussian light beam opposes a fluid flow, causing that particles of various sizes and/or refractive indices find different equilibrium positions along the beam axis where the scattering force is balanced by the force due to the fluid flow [379,380]. Other schemes combine the action of scattering and gradient forces. Consider, for example, the interference pattern of fringes discussed above. In addition to the transverse gradient force, there is usually a scattering force along the propagation direction, which may push the particles selectively according to their relative size with respect to the fringes, so that small particles, attracted to the bright fringes, will be pushed much stronger than larger particles attracted to the dark fringes. Using this interplay between gradient forces and scattering forces, it is possible to activate transport with an all-optical static configuration [381]. These kinds of ideas have been extended further to include the use of evanescent waves generated on the surface of prisms (Kretschmann configuration) or with total internal reflection (TIR) microscope objectives [49,382,383]. An advantage is that the use of gradient and scattering forces arising from evanescent waves in near-field photonic configurations allows one to scale down to tens of nanometers the size of the manipulated objects [49,384].\n\nAnother puzzling transport mechanism appears in the so-called tractor beam. This is a counterintuitive effect, in which particles move opposite to the direction of the beam propagation due to a pulling scattering force [385]. In the case of a highly nonparaxial wave field, whose angular spectrum components subtend large angles with respect to the main propagation axis, part of the diffracted radiation may indeed have larger wave vector components along the propagation axis than those of the incident field, giving rise to strong forward scattering. The net effect might be a negative radiation pressure (pulling force), depending on the shape of the object, its size, and its interaction with the wave field. It is important to stress though, that this effect is not based on intensity variations and it is not derived from a potential, in contrast with the conservative gradient force acting in conventional optical tweezers. Remarkably, this phenomenon can happen also in rotational dynamics, where the spin angular momentum of light can exert transverse non-conservative forces in the opposite direction than expected in a chiral object [386,387].\n\nThe simplest description of the dynamical mechanisms previously discussed considers diluted samples, and this is usually a good approximation even if the sample is not well diluted and some interactions exist. When the interactions are strong, a more complete description becomes necessary to fully understand the phenomenology. The origin of the interactions can be manifold, like hydrodynamic, electrical or optical (optical binding), and they may promote collective behaviors such as dynamical ordering, phase transitions or cluster formation [94,342,385,[388][389][390][391].\n\nUnderstanding and controlling transport from the mesoscale down to the nanoscale is still a key topic in biophysics, nanomedicine and nanotechnology. There is still plenty of room for basic and applied research on the many aspects involved in transport of microscopic objects, including studies on self-propelled particles [337]. Especially in biological environments, the role of light-induced forces has been barely studied so far. On the other hand, many of the ideas and techniques discussed here are being extended to other realms where force fields can be used for contactless manipulation, either light, like in plasmonic [13,16,[392][393][394][395], optoelectronic [396] and some recent thermophoretic devices [397,398], or without light, like in acoustic manipulation [169,[399][400][401]. This range of techniques has broaden the trapping conditions and the accessible range of particle sizes and force magnitudes, going down to femtonewtons for nanoparticle maneuvering and up to micronewtons to trap millimiter-sized beads, and possibly beyond in both directions in the near future. Also, there is a great effort invested in turning the basic concepts into real applications, specially in the development of lab-on-chip platforms, where many modern areas can converge to perform practical tasks, including optomechanics, microfluidics, acoustofluidics, nanoplasmonics, thermoplasmonics, optoelectronics, near-field photonics, spectroscopy, to name but a few. A more extended discussion on light-induced mechanisms of controlled transport can be found in a very recent and comprehensive review [402].\n\n\nOptical tweezers in vacuum\n\nEven though Ashkin conducted his pioneering experiments on optical forces and radiation pressure in a gas or vacuum environment [1,[403][404][405], for many decades optical forces have been primarily harnessed to conduct optical tweezers experiments in liquids or for trapping and cooling of atoms [5]. Only recently there has been a renewed interest to use optical tweezers in vacuum. The fist single-beam gradient-force optical trap in air was demonstrated in Ref. [406] 10 years after its realization in water [2] and it took another 15 years to demonstrate single beam optical levitation in high vacuum [407].\n\nThis section will introduce the basic concepts for realizing experiments with optically levitated particles in high vacuum. We will discuss the implementation with a single beam optical trap, even though other geometries for optical trapping such as parabolic mirrors [408,409] or counter-propagating beams [410,411] have also been realized. Specifically, we will focus on the practical aspects such as particle loading, sources of noise, feedback stabilization, calibration, and provide the literature references that will allow the reader to independently implement optical levitation experiments in high vacuum. We begin in section 5.1 with a concise review over the applications of levitated particles in high vacuum.\n\n\nApplications of optical levitation\n\nIn vacuum the motion of an optically levitated dielectric nanosphere is well isolated from the environment. This makes it a promising candidate to demonstrate quantum behavior of a macroscopic object at room temperature [412][413][414]. To observe quantum effects, the motional energy of the levitated particle has to be reduced by more than seven orders of magnitude to the level of a single quantum excitation. Ongoing efforts into this direction have already reached impressive cooling rates using feedback cooling [407,[415][416][417][418] and cavity cooling schemes [419][420][421]. While the ultimate cooling limit, i.e., the quantum ground state, has already been reached in a range of nano and micromechanical systems [422,423], ground state cooling of a levitated37 nanoparticle has only be claimed recently [424]. The quantum mechanical ground state is an important milestone towards preparation of these massive objects in non-classical quantum states that are otherwise only known from elementary particles [425], atoms, ions [426], and molecules [427,428].\n\nInvestigating quantum mechanical states of motion of massive objects allows for experimental tests of fundamental physics in unexplored parameter regimes [429][430][431][432][433][434]. Preparation of nontrivial quantum mechanical states requires either a strong optomechanical interaction [435][436][437] to map non-classical states of light onto mechanical motion [438], a projective quantum measurement such as photon counting [439][440][441], or a quantum nonlinear interaction, which can be provided for example by a defect inside the levitated particle such as a nitrogen-vacancy (NV) center in diamond [442][443][444]. However, the necessary coupling strengths have yet to be demonstrated experimentally and optical levitation experiments with nanodiamonds have been limited by optical absorption [445][446][447][448][449].\n\nUltimately, levitated particles in the quantum regime might benefit from their simple mechanical mode structure. Conventional mechanical oscillators have a series of normal vibrational modes. Typically, only one mode is considered and the remaining modes are neglected. However, in measurement-based approaches to quantum state preparation [450,451], the displacement of the resonator is measured with a light pulse that is short compared to the oscillation frequency. The light pulse measures over a large bandwidth and thus measures not only the mode of interest but it also measures the displacement due to all the other modes that fall within the bandwidth. As a consequence, these so-called spectator modes add noise, which ultimately limits the efficiency at which the mode of interest can be measured and controlled [452]. Inside nanoparticles, the frequencies of bulk modes are several orders of magnitude higher than those of the center-ofmass motion. As a consequence, one usually only observes three modes corresponding to the center-of-mass motion in each dimension when working with spherical particles. With less symmetrical particles additional torsional [453] and precession [454] modes have been observed.\n\nThe extreme isolation from the environment also means that the mechanical quality factor (Q-factor) is very large and, hence, the thermo-mechanical noise of these nanomechanical resonators is very low. For this reason, levitated particles can be used as nanomechanical sensors for force [455][456][457], torque [453], and acceleration [458]. Applications of these sensors are envisioned to detect exotic forces [459,460], such as non-Newtonian gravity [461], Casimir forces [462][463][464] and torques [465], and gravitational waves [466]. Thanks to the high mechanical Q-factor and tunability of the system parameters, levitated particles are also ideally suited to study nonlinear nanomechanical phenomena [467][468][469], which are important for the development of future nano and micromechanical devices; in fact, levitated particles are the first systems where nonlinear effects have been witnessed already at the level of thermal fluctuations [470].\n\nOptically levitated particles have unique properties not available in traditional nanomechanical resonators. For example, the mechanical properties of a levitated particle can be changed rapidly in situ by simple adjustments in the trapping light. This allows to perform free-fall experiments, which provide new opportunities in force sensing [461,471] and quantum physics [472]. Unlike traditional clamped mechanical resonators, a levitated particle can rotate freely [473][474][475][476]. The rotation can be accelerated to GHz frequencies, faster than any other object [477,478], simply by changing the trapping laser polarization from linear to circular. At these fast rotations, the centrifugal stress is close to the ultimate tensile strength of the material. Hence, levitated particles can be utilized to test material limits on the nanoscale or to study friction at a fundamental level [463,479].\n\n37To distinguish trapping in liquid from trapping in a dilute environment, we refer to the latter as optical levitation.\n\nThe high control over the system parameters, in particular the ability to tune the interaction with the environment via the gas pressure, make levitated nanoparticles also an excellent testbed for studies of single particle thermodynamics [19], a field also known as stochastic thermodynamics [164]. As we have seen in section 4.5.3, many studies in this field have already been conducted in the overdamped regime with optically trapped particles in a liquid medium. In contrast, levitated particles can also operate in the underdamped regime and have enabled the first observation of ballistic Brownian motion [410], the first quantitative measurement of the Kramers turnover [480], experimental tests of fluctuation theorems [481,482], and fundamental tests of thermodynamic laws [483].\n\n\nDynamics in the underdamped regime\n\nWhen transitioning from a viscous medium like water to a dilute gas or vacuum environment, the dynamics of a particle in an optical trap changes fundamentally because of the particle's finite inertia. Including the inertial term, the equation for the center-of-mass motion along a single spatial direction is (equation (8))\nq + γ 0 q + Ω 2 0 1 + ξq 2 q = F L /m + F ext /m,(160)\nwhere Ω 0 = κ/m is the resonance frequency, κ is the stiffness of the optical trap, and m is the mass of the particle. Under most experimental conditions, the fluctuating force F L and damping rate38 γ 0 are due to collisions between the particle and residual gas molecules. We will come back to the stochastic forces and damping contributions in section 5.5. F ext summarizes additional external forces, excluding the optical forces that generate the optical trap, which correspond to the last term on the left hand side. These can be used for example to drive the particle or to apply feedback cooling, as will be discussed in greater detail in section 5.7. The Duffing term ξ ∼ 1/w 2 0 , where w 0 is the width of the optical trap, accounts for deviations from a parabolic trap [470]. The optical potential term also contains nonlinear terms that couple the three orthogonal spatial modes nonlinearly [468,470]. However, these terms become only relevant for sufficiently large oscillation amplitudes and can in most situations be neglected, in particular when feedback cooling is applied. This justifies the treatment of the three-dimensional motion of the trapped particle as three independent harmonic oscillators. Note that, while the modes are uncoupled in a static trap, the two modes that are transverse to the propagation direction can be coupled by modulating the polarization of the trapping light in time. The coupling allows to coherently exchange energy between the modes, since this coupling can be turned on and off on demand [484]. Overall, this platform is very versatile because all system parameters can be controlled to a large degree in the experiment.\n\nIn the underdamped regime, which is defined by the condition γ 0 Ω 0 , the particle undergoes ∼ Q = Ω 0 /γ 0 phase coherent oscillations. This becomes very clear in the PSD\nS qq (ω) = m −2 | χ(ω)| 2 S ff = k B T mπ γ 0 (ω 2 − Ω 2 0 ) 2 + ω 2 γ 2 0 ,(161)\nwhich in the underdamped regime is strongly peaked around the oscillation frequency Ω 0 . Here, χ(ω) = 1/ Ω 2 0 − ω 2 + iωγ 0 ) is the response function of a harmonic oscillator and we assumed thermal noise with a PSD S ff = γ 0 mk B T/π. As we have already seen in section 3.6, in the overdamped regime (Q 1), equation (161) can be approximated by a Lorentzian distribution\nS qq (ω) ≈ k B T 2mπΩ 0 γ 0 /2 (ω − Ω 0 ) 2 + (γ 0 /2) 2 .(162)\n38Note that here we define γ 0 as a rate with units of Hz, in contrast to γ = γ 0 m in equation (8), which has units of [Hz kg]. Fig. 35. From overdamped to underdamped. We plot the measured PSD for the z-axis motion, normalized to the damping rate γ 0 /(2π), at three different pressures for a levitated particle (radius a = 68 nm, laser power P ≈ 150 mW) that is overdamped (blue line), critically damped (orange line), and underdamped (green line). The dashed black lines are least-square fits to equation (162) (1000 mbar) or equation (161) (60 and 2.5 mbar). The colored vertical solid lines indicate the roll-off frequencies Ω c /2π. For the time traces at 60 and 2.5 mbar, the spectra contain leakage signals from the other oscillation axes above 100 kHz. Data from Ref. [485]. Fig. 35 shows experimental data that clearly illustrate the transition from the overdamped to the underdamped regime. In the experiment, the damping γ 0 is changed through the pressure inside the vacuum chamber. At atmospheric pressures, the gas damps the particle motion (γ 0 2Ω 0 , blue) and we observe the spectrum familiar from optical tweezers experiments in liquids (section 3.6), with the maximum at zero frequency. At a pressure of ∼ 60 mbar, the particle is critically damped (γ 0 = 2Ω 0 , orange). At even lower pressures a strong peak evolves at Ω 0 , which is the signature of coherent oscillations. The vertical lines indicate the cut-off frequency Ω c = Ω 2 0 /γ 0 .\n\n\nOptical tweezers setup for vacuum operation\n\nA standard setup for optical levitation in vacuum consists of a vacuum system that allows to control the pressure and thereby the friction the particle experiences, optics that strongly focus a laser beam for trapping, optical detection for real-time readout of the particle motion, and feedback control to stabilize the particle in high vacuum. An example of a nanoparticle optically trapped in a vacuum chamber is shown in Fig. 36. Operating an optical tweezers in vacuum introduces a range of significant changes to the experimental apparatus and also imposes limits to the type of particles that can be trapped. Naturally, there are many ways in which such a setup can be constructed. Here, we follow largely the description given by Hebestreit [485], a schematic picture of whose setup is shown in Fig. 37.\n\n\nParticle types\n\nOne of the great benefits of operating in vacuum is that the particle is very well isolated from the environment. However, this also means that the particle doesn't thermalize with the environment. As a consequence, optical absorption of the trapping laser significantly heats up the internal (bulk) temperature of the particle (section 5.6.5). This limits vacuum trapping to particles that absorb very little of the trapping laser (e.g., silicon and silica) and to wavelengths where optical absorption is minimized (e.g., infrared, the most popular choices being 1064 nm and 1550 nm). There has also been great interest in trapping nanodiamonds, since these can host interesting optical defects such as NV-centers [487]; however, trapping in high vacuum has not been reported yet due to excessive optical absorption from impurities and defects in the diamond [447]. A liquid medium that surrounds the particle not only mitigates heating from optical absorption, it also keeps the particle afloat. Without it, gravity pulls the particles down to the bottom of the chamber, which complicates the loading of individual particles into the trap (section 5.4).\n\nIn addition, without a high-refractive-index medium such as water (n m = 1.33), the refractive index contrast is much larger. Consequently, the scattering force is more prominent in vacuum since the ratio of the scattering force to the gradient force scales as [488,489] \nF scat /F grad ∝ α /α ∝ (ka) 3 ∆(163)\nwhere ∆ = (n 2 p − n 2 m )/(n 2 p + 2n 2 m ) is proportional to the relative index contrast between the particle (n p ) and the surrounding medium (n m ). Therefore, trapping of spherical particles in vacuum with a single beam optical tweezers is limited to radii a ranging from ∼ 30 nm to ∼ 150 nm [488]. However, this is not a fundamental limitation, since the scattering force can be canceled, e.g., with counter-propagating beams [410,490].\n\nAlso non-spherical particles (e.g., dumbells [477], clusters [454], and rods [475]) have been successfully trapped in vacuum. They have quite different scattering properties and therefore experience different optical forces and torques than spherical particles.\n\n\nOptical trap\n\nThe most popular choice for the trapping laser is infrared light at 1064 nm or at 1550 nm. In addition, the laser should have a small relative intensity noise (RIN), low pointing instability, and provide a power at the laser focus on the order of P opt = 70 mW, which requires a laser output power of ≈ 0.5 W. Possible back-reflections into the laser can destabilize the laser output. This can be avoided by introducing a Faraday isolator right after the laser output. A low RIN is important because noise in the intensity at twice the particle oscillation frequency excites its center-of-mass motion [491][492][493] and causes fluctuations of the oscillation frequency, since the particle frequency scales with Ω 0 ∝ P opt .\n\nTo parametrically drive the particle motion, the laser intensity has to be modulated at twice the oscillation frequency, i.e., at up to 400 kHz (section 5.7). Electro-optic modulators (EOMs) or acousto-optic modulators (AOMs) provide the necessary modulation bandwidth. While EOMs have a higher transmission (up to 95%, in contrast to maximally 80% for AOMs) and a higher bandwidth, they feature a much smaller contrast ratio than AOMs [471].\n\nThe trapping laser can also be used to detect the particle motion even during parametric control since the parametric modulation is at a twice the motional frequency and can therefore be filtered out spectrally (section 5.7.2). However, to exclude any cross-talk, it is convenient (albeit not necessary) to have an independent measurement of the particle with a constant laser. Interferences should be avoided, since they can lead to strong fluctuations of the optical trap. This can be achieved with a single laser source that is divided into a high-power branch for trapping and a low-power branch for detection. A polarizing beam splitter (PBS) in combination with a half-wave plate provides a convenient way to cross-polarize and to adjust the power ratio between the two branches. In addition, one branch is frequency-shifted with an AOM, to avoid interference. A second PBS re-combines the two branches, after which they enter the microscope objective colinearly.\n\nA stable three-dimensional trap with a single optical beam requires strong focusing. This can be achieved with a high-NA objective (NA ≥ 0.8). Note that optical absorption in the objective can lead to small shifts of the focus [485]. This can heat the particle motion for large laser intensity modulation, for example in free-fall experiments [471]. Aspheric lenses provide a cheap alternative to more costly microscope objectives. In addition, they usually have a higher transmission and are available with anti-reflection coatings for a large range of wavelengths. This reduces the power requirements and absorption losses. However, they are more sensitive to misalignment than microscope objectives, which makes optical trapping with aspheric lenses much more challenging [485].\n\nBoth AOMs and EOMs distort the laser beam as it passes through. To recover a nice Gaussian beam shape, the laser mode profile is cleaned with a spatial filter before it enters the microscope objective. This increases the quality of the focus and, hence, the trap stiffness. Spatial filtering is achieved either with a pinhole of appropriate size, or more conveniently with a single-mode fiber, which has the advantage of mechanically decoupling different sections of the setup. A polarization-maintaining single-mode fiber maintains the input polarization at the output, while a single-mode fiber with a fiber-based polarization controller allows to create an arbitrary output polarization.\n\nA second microscope objective or aspheric lens inside the vacuum chamber collects and re-collimates the trapping laser and the measurement beam. After the vacuum chamber, the two beams are separated again with another PBS. The trapping beam is dumped, while the probe beam is sent towards the detection setup. Efficient separation of the beams is paramount to avoid detecting the intensity modulation introduced by the EOM, that would then be deleteriously fed into the system via the feedback loop. In fact, mixed polarization states appear in this side of the set-up due to the tight focusing and especially to the mechanical stresses present in the glass windows of the vacuum chamber. To undo these changes in the polarization state, one introduces a half waveplate and a quarter waveplate after the vacuum chamber and minimizes the unwanted trapping light after the PBS while maximizing the probe light, which then goes to the detection setup.\n\n\nDetection\n\nIn liquids, momentum relaxation occurs within a fraction of a microsecond due to the high viscosity of the surrounding medium. This is faster than what most experiments are able to resolve [494]. In contrast, the viscosity of a dilute gas is orders of magnitude smaller. Hence, inertial effects on the dynamics of a particle in an optical tweezers are easily accessible [410]. While optical tweezers experiments in liquids are mostly concerned with the diffusive motion at low frequencies on the order of the corner frequency ω c /(2π) = k /(mγ 0 ) ∼ kHz, the frequency range of interest in experiments under vacuum is centered around the natural oscillation frequency Ω 0 /(2π) = κ/m ∼ 100kHz, where κ is the trap stiffness and γ 0 the damping coefficient.\n\nAs in the case of traditional optical tweezers, the particle motion is measured optically by extracting information contained in the field scattered by the particle. In single-beam optical tweezers, the trapped particle are sub-wavelength scatterers. As such, most of the information about the particle position is contained in the phase of the scattered field. This requires an interferometric measurement to read out the phase.\n\nForward detection provides a convenient way to access the phase, since the non-scattered transmitted light provides a stable reference field without requiring additional optical path stabilization [495]. The signal-to-noise ratio of the detector signal is maximized by increasing the detection efficiency of the scattered photons and by increasing the optical power of the detection beam. Maximum power is provided by using the trapping beam itself, at the expense of introducing cross-talk as mentioned before. Hence, the best signal-to-noise ratio is obtained with a high-NA collection lens and by using all the scattered light to generate the detector signals. In forward scattering, the collected power from the trapping beam amounts to ≈ 50 mW. Hence, optimal detection in forward scattering requires a quadrant photodetector that can handle at least 50 mW optical power, has a high quantum efficiency at the trapping wavelength, has a bandwidth of at least 1 MHz, and is shot-noise limited at power levels above 10 mW per quadrant. Details on a custom detector design that fulfills the above requirements can be found in Refs. [485,493].\n\nForward detection is attractive due to its simplicity. However, the ratio of scattered power to reference power is fixed by the scattering cross section of the particle. Backward detection allows to collect only the scattered photons and add a reference field with variable power. The ultimate position sensitivity to particle motion that is transverse to the optical axis is the same for forward and backward detection. Interestingly, due to interference between incident and scattered photons this is not the case for motion along the optical axis. In the forward direction, destructive interference reduces the sensitivity, with the counterintuitive result that a higher NA collection lens can yield a lower sensitivity. In contrast, constructive interference in the backward direction allows for up to 60% collection efficiency even with a moderate NA = 0.8. That is, more information about the particle position along the optical axis is contained in the backward scattered light than in the forward scattered light [496,497].\n\nInterferometric detection provides the best signal-to-noise ratio. However, it is limited to a small range of particle motion. In contrast, video analysis allows to determine the particle position over a much larger range with a limited signal-to-noise ratio and it requires extremely fast cameras to resolve the relevant timescale ∼ µs in high vacuum [498].\n\n\nFeedback electronics\n\nFeedback cooling allows to extract energy from the center-of-mass motion and stabilizes the trap. The feedback force can be applied either directly (e.g., with electric fields when the particle carries a charge) or parametrically (by modulation of the laser intensity). The electronic circuit that processes the detector signal can be analog [488], digital [415,417], or an off-the-shelf lock-in amplifier [416] (section 5.7). Here, we focus on the latter following the description for parametric feedback based on a phase-locked loop (PLL) given by Hebestreit [471].\n\nThe detector signal for each axis is fed to one of the PLLs. Each PLL detects the frequency and phase of the oscillation along one spatial dimension. The locking range of the PLLs is set to 4 to 8 kHz around the expected center frequency to prevent the PLL from losing lock on the particle oscillation signal. The bandwidth of the loop filter is adjusted to be large enough to follow frequency changes sufficiently fast, and small enough to exclude noise from the feedback signal. In practice, this amounts to a bandwidth of 1 to 16 kHz [471].\n\nThe PLL synthesizes a signal that is phase-locked and at the same frequency or at a harmonic of the input signal with adjustable amplitude and phase. For parametric feedback, one uses the second harmonic and tunes the amplitude and phase to minimize the particle motion. The independent signals for the three oscillation axes are added within the lock-in amplifiers to form the final feedback signal. This feedback signal is applied to the EOM, which in turn modulates the power of the trapping beam. The amplitude of the feedback signals is usually set to a few mV for each axis. For a half-wave voltage of V π = 240 V and amplifier with 400 V/V gain, the optical power is modulated by 2.5% at a feedback signal amplitude of 10 mV [471].\n\nA switch allows to rapidly turn off the feedback signal. This allows to perform relaxation experiments [481]. Additional signals can be added to the feedback signal. This allows to stabilize this oscillation frequency by adjusting the power of the trapping beam with a PID (proportional-integral-derivative) control loop or to switch off the trap for short amounts of time for free-fall experiments [471]. A harmonic signal at twice the particle's motional frequency allows to excite the particle into the nonlinear region of the optical trap [468,469].\n\n\nVacuum system\n\nOptical levitation in high vacuum requires an experimental setup containing a vacuum system to control the gas pressure, optical access for the trapping and detection optics, and an access door on the top if particles are loaded at ambient pressures with a nebulizer (section 5.4.1). A schematic of a vacuum system is shown in Fig. 38. The pressure in the chamber can be monitored continuously over the entire range from ambient pressure down to below 10 −7 mbar with a combination of gauges, e.g., a Pirani gauge for high pressure combined with a hot cathode filament gauge for low pressure.\n\nA set of pumps and valves controls and stabilizes the pressure. The main pump is a turbomolecular pump. A motorized valve between the chamber and the turbo-molecular pump is used to stabilize the pressure below 10 −3 mbar with a PID controller. The turbo-molecular pump is backed by a dry scroll pump via another motorized valve that is mainly used to ensure a smooth evacuation of the chamber without sudden changes in the chamber pressure during the initial pump-down. This is important to prevent loosing a trapped particle during pump-down from ambient pressure after loading. To avoid mechanical vibrations from the vacuum pumps to couple into the optical setup, the pumps are connected to the vacuum chamber through flexible bellows, which decouple the vibrations. Vibration isolation can be further improved by casting the bellows into a sand or concrete block. At pressures below 10 −6 mbar, ion pumps are sufficient to maintain a pressure without introducing vibrations.\n\nM M Fig. 38. Vacuum system. The vacuum system consists of a vacuum chamber with optical vacuum windows, in which the actual trapping takes place. Here, we present a system based on that employed in Ref. [485]. The pressure is monitored using an ion and a Pirani gauge. A turbomolecular pump is connected to the chamber using a flexible bellow to suppress vibrations. A scroll pump supports the turbo pump. The pumping speeds can be regulated manually or with a PID controller using motorized valves (V 1 and V 2 ). The chamber is vented with nitrogen to reduce contamination. O: objective. PID: proportional integral derivative controller.\n\nPressures below 10 −6 mbar require to bake the vacuum chamber to desorb residual water. Many experiments load particles at ambient pressure. Hence, the lowest pressures can be achieved if the chamber is baked in situ while the particle is trapped. Optical access while baking can be provided with heating bands that are wrapped around the vacuum chamber. However, temperatures are limited to ≈ 150 • C to avoid damaging the optical components inside the vacuum chamber and the optical coatings of the windows. The baking process also causes thermal deformation of the optics inside the vacuum chamber. This can then lead to particle loss during baking. Nonetheless, even if particles are not kept in the trap during bake out, baking the chamber in combination with purging with dry air (instead of venting with humid ambient air) reduces the overall water content in the vacuum chamber and leads to improved pump speeds and better vacuum.\n\nHebestreit [471] quantified the gas composition in the vacuum chamber during a trapping experiment with a residual gas analyzer (RGA). The RGA is a mass spectrometer that allows to measure the mass spectrum of a gas and thus to extract the partial pressures of the different gases species. The measured partial pressures of the prominent gas species during a typical pump-down cycle are shown in Fig. 39. The total pressure at the RGA (the sum of all the partial pressures, dashed line) is higher than the pressure measured in the main vacuum chamber with an ion gauge (dashed line), because the pumping speed at the RGA is reduced compared to the main chamber. Initially, the chamber is filled with nitrogen. Interestingly, at low pressures the composition is vastly different. The turbo molecular-pump removes the initial nitrogen very efficiently. Thus, its partial pressure reduces quickly. In contrast, water and hydrogen are continuously desorbed from the walls of the vacuum chamber. As a consequence, the gas in the chamber is dominated by water vapor at pressures below 10 −4 mbar. Baking out the vacuum chamber accelerates the desorption process and thereby helps to get the water and hydrogen out of the chamber. The final gas composition in the pressure range of 10 −6 to 10 −5 mbar is mainly water vapor (58 ± 9%), nitrogen (19 ± 8%), hydrogen (12 ± 1%), oxygen (7 ± 1%), and carbon dioxide (4 ± 1%). Other gases, which contribute with less than 1% are argon, isopropyl alcohol, and oil. The last two probably remain from the solvent of the particle solution and residual machining grease, respectively. As a consequence, the molar mass M ≈ 20 ± 11 · 10 −3 kg mol −1 at low pressure is significantly lower than the molar mass at high pressures, the latter being equal to the molar mass of dry air M air = 28.97 · 10 −3 kg mol −1 . \n\n\nParticle loading\n\nIn contrast to a liquid environment, loading a particle into an optical trap in a dilute gas or vacuum is challenging because the particles will quickly fall down due to gravity. The size of an optical trap is rather small, on the order of the focal volume ∼ λ 3 . Therefore, a particle that passes by the focus at a distance larger than ∼ λ will not be captured. Furthermore, if a particle enters the focal volume at high speed, the low damping in vacuum will not slow the particle down sufficiently for it to get trapped. The maximum speed for successful trapping is v max ∼ λγ 0 , where γ 0 is the damping constant. In water, the viscosity at room temperature is η (water) = 890 µPa s so that, using Stokes formula (equation (6)), γ 0 = 6πηa/m and, therefore, v (water) max = 344 m s −1 for a silica particle of radius a = 75 nm. In air, the viscosity is about two orders of magnitude smaller (η (air) = 18 µPa s) so that v (air) max = 7 m s −1 . The challenge lies, therefore, in finding a technique by which a slow single nanoparticle is brought into the focal volume [488].\n\n\nNebuliser\n\nThe most common approach to particle loading is the nebulizer approach thanks to its simplicity [499]. In this approach, a solution containing silica beads (e.g., Microparticles GmbH, Bangs Laboratories) with radii ranging from a = 50 to 100 nm is sprayed into the open vacuum chamber with a nebulizer (e.g., Omron NE-U22-W) and the particles are trapped at ambient pressure [488] or mild vacuum [500].\n\nThe nebulizer consists essentially of a mesh on top of a piezo element. A little bit of liquid gets into the space between the piezo and the mesh. The motion of the piezo then pushes the liquid through the mesh. The mesh breaks the liquid into little droplets of diameter 2 µm and a nozzle funnels the falling particles to a region near the laser focus (Figs. 40(a-b)). Successful trapping of a particle can be easily noticed observing the laser light scattered by the particle with a camera. However, since the scattered light has a dipole radiation pattern, the polarization of the laser beam should be orthogonal to the direction of observation to see the particle with the camera. The scattered light intensity on the camera scales with the size of the particle as B ∝ a 6 . This allows to distinguish whether the object in the trap is a single nanoparticle or a cluster of multiple particles by comparing the brightness on the camera for several trapping events. For example, in Fig. 40(c), the histogram of the brightness on the camera for ∼ 350 particles shows clearly distinct peaks, which can be associated with the number of particles in the trap [486]. Most of the liquid in the droplet evaporates before or immediately after trapping. However, ∼ 10 layers of water remain physically absorbed on the surface of the particle. These physically absorbed layers evaporate as the gas pressure is reduced down to ∼ 1 mbar. This dehydration process is reversible and water molecules get reabsorbed when the pressure is increased. In the region from 0.1 to 0.5 mbar, one can often observe a sudden decrease in the mechanical frequency and power of the scattered light. This is attributed to a sudden loss of the remaining water from the particle as its internal temperature increases to above ∼ 200 • C. This effect occurs only once and is irreversible when cycling the pressure. Therefore, once the particle has lost the final water layer its water uptake is extremely slow and for all practical purposes the particle size and material composition can be considered pressure independent. For more details on the dehydration process see Refs. [419,486,500].\n\n\nDry approaches\n\nSpraying solvents into a vacuum chamber introduces contaminants that are difficult to pump out, e.g., water molecules. This motivates a dry approach, where particles are released from a substrate instead of being injected from a solution. However, small particles stick to surfaces due to dipole-dipole interactions known as Van der Waals-forces, which scale linearly with particle size; the magnitude of these forces is\nF VdW = 4πaγ s ,(164)\nwhere γ s is the effective surface energy. For example, measurements on silica spheres on glass give F VdW = 176 nN for a = 1 µm [501]. For comparison, the gravitational force for a particle of this size is only ∼ 0.1 pN and, therefore, much too weak to remove the particle. There can also be additional surface forces that lead to even stronger adhesion, for example meniscus forces arising from a thin water layer that is formed on the substrate. Therefore, even larger forces have to be applied to release the particle from the substrate. Two common approaches, which provide dry loading methods and do not require to open the vacuum chamber as in the nebulizer approach, are based on launching particles from surfaces using piezoelectric transducers or laser-induced acoustic desorption (LIAD).\n\nA piezoelectric transducer works really well for micrometer-sized particles [405,502] and allows even for repeated loading of the same particle [503]. When the piezoelectric transducer is driven with oscillation amplitude q p and frequency ω p , the particle feels a force\nF piezo = mω 2 p q p ,(165)\ndue to its inertia, where m = V ρ p is the particle mass with ρ p and V the particle density and volume, respectively. Consequently, the acceleration required to release the particle is\na p = ω 2 p q p = 4πaγ s /m ∝ a −2 .(166)\nHence, for a given maximum acceleration the piezoelectric transducer can provide, there is a minimum particle size that can be shaken off. The smallest particles that have been released with this method have a radius of 150 nm and further details on how to implement this can be found in Ref. [504]. The limited acceleration of the piezo approach can be overcome by generating a shockwave on the surface of the substrate in the LIAD approach. The shockwave can be created by a pulsed frequency-doubled Nd:YAG laser (5 ns pulses of 3 mJ at 532 nm). When focused down to 200 µm on the back side of a thin metallic foil, the laser creates an acoustic wave that launches particles from the front side, which had been previously spin coated with particles [505].\n\nDespite the availability of these techniques, directly loading particles into an optical trap under high vacuum conditions remains an open challenge. With the dry approaches outlined above, particles are still loaded into the trap at ambient pressures or moderate vacuum, where the background gas provides sufficient dissipation because capture requires an additional dissipation mechanism due to the conservative nature of gradient trapping forces and the small size of the trapping region of an optical tweezers. Recently, it was demonstrated that high-vacuum loading into a Paul trap can be achieved [506]. The Paul trap provides a much larger trapping volume and trap depth than an optical tweezers. This allows to turn on the trap exactly when the particle is near the center of the trap. As a consequence, the particle's total energy is given only by its kinetic energy. If the kinetic energy is less than the trap depth, the particle will be stably trapped.\n\nPaul traps are limited to charged particles; however, this is not a strong limitation for loading since the particles generally carry several tens of elementary charges (section 5.6.3).\n\nTransferring particles from a contaminated chamber, where the particle is initially trapped, to a clean science chamber, which always remains in high-vacuum conditions, can be achieved with a mobile optical trap (MobOT) in a load-lock [507]. However, this solution is rather bulky and has a limited throughput at low pressures. Hollow-core photonic crystal fibers provide also a compact solution with high throughput, but have been limited so far to pressures above 10 −2 mbar [508,509].\n\n\nDissipation and noise\n\nA particularly attractive feature of trapping in a dilute or vacuum environment is that the interaction with the environment can be adjusted over several orders of magnitude. The interactions between the particle and its environment result in stochastic forces that excite the particle motion and lead to stochastic dynamics, which are of particular interest when studying thermodynamics of individual small particles and are to be minimized when aiming for the quantum regime.\n\nThe intensity of a stochastic force is characterized by its PSD S noise ff . While the stochastic forces excite the center-of-mass motion of the trapped particle, the interactions with the environment also lead to dissipation at a rate γ noise , which damps the particle motion. There are several sources that contribute to the total damping rate and stochastic noise intensity, which we will discuss in this section.\n\nThe total damping rate and stochastic noise intensity is simply given by the sum of the individual contributions γ 0 = noise γ noise and S ff = noise S noise ff , respectively. Note that for most purposes, the PSD can be considered as frequency independent (white noise) over the bandwidth of the mechanical resonance of the particle, especially when the bandwidth becomes quite narrow in high vacuum. The average energy of each center-of-mass degree of freedom evolves as E = −γ 0 E + Γ where the heating power due to the force noise is Γ = πS ff /m, S ff being the double sided PSD. After a time ≈ 1/γ 0 , the particle reaches an equilibrium, where the ensemble average energy stays constant. This allows to define the effective temperature through the fluctuation-dissipation relation:\nT cm = k −1 B Γ γ 0 = πS ff k B mγ 0 .(167)\nThe effective temperature corresponds to the particle energy averaged over an ensemble of identical experiments. Importantly, since we are concerned here with the energy of a single particle, the energy in each experiment will still fluctuate with a standard deviation ∼ T cm . Generally, one has to be careful to define a temperature for a system out of equilibrium [510]. However, the situation we present here is somewhat simplified because it is steady-state and for many practical purposes the effective bath model that is characterized by an effective damping/temperature gives a good description. However, one can also create situations where this is no longer true. For example, by parametric feedback damping, the temperature alone is not sufficient to give a full description of the bath (section 5.7) For a levitated particle in a dilute gas, the main contribution to the stochastic force noise are collisions with air molecules and radiation damping at pressures below ∼ 10 −7 mbar. In addition, there are technical noise sources that do not pose a fundamental limitation but can impose practical limitations. These are primarily displacement noise of the trap and laser intensity fluctuations. Noise sources can also be introduced intentionally to emulate a different thermal environment without actually changing the real temperature of the environment [316,511]. This can be a useful tool for single particle thermodynamics experiments [19].\n\n\nGas damping\n\nFrom kinetic gas theory, the pressure-dependent gas damping for a spherical particle is [488,512] γ gas\n2π = 3aη ν m 0.619 0.619 + Kn (1 + c K ),(168)\nwhere Kn =l/a is the Knudsen number, c K = 0.31Kn/ 0.785 + 1.152Kn + Kn 2 ,l = πk B T 2m gas η ν P gas is the mean free path, η v is the viscosity, and m gas = M/N A the molecule mass [512], N A being Avogadro's number. For dry air, the molar mass M = 28.97 · 10 −3 kg mol −1 , however the actual gas composition can be different (Fig. 39). For high pressure, the interaction with the gas is so strong that the particle motion is heavily damped and its internal temperature T bulk and centre-of-mass temperature T cm quickly thermalize with the gas temperature T gas . In this regime, the damping does not depend on pressure and is given by Stokes law, i.e., γ gas /(2π) ≈ 3aη v /m. For pressures below P gas ≈ 54.4 mbar (a/µm) −1 , where the mean free pathl is much larger than the radius of the particle a, the damping is linear in the pressure P gas and given by\nγ gas 2π ≈ 0.354 m gas k B T gas P gas ρ p a .(169)\nIn typical experiments, γ gas /2π = c P P gas /a with c P ≈ 50 Hz (µm/mbar). Note that the above holds for a spherical particle. For an anisotropic particle (e.g., a rod), the friction term is different along each of the axes, and depends upon the alignment of the particle [513]. The mean free path of the gas molecules increases with decreasing pressure (e.g.,l ∼ 60 µm at 1 mbar). As a consequence, the particle no longer thermalizes with the gas since the impinging gas molecules no longer carry away enough thermal power to balance the optical absorption from the trapping laser. Due to the increased internal temperature T bulk of the particle, the average energy of the gas molecules after a collision with the particle increases (section 5.6.5). The latter requires that the impinging molecules stick to the surface for a short time before they emerge again from the surface. This type of collision is called diffusive reflection [514,515]. For diffusive reflection, the fluctuating forces originating from the impinging gas molecules are completely uncorrelated. This justifies the treatment of the impinging and the emerging gas molecules as two independent baths with individual bath temperatures, damping and heating rates. The process by which a surface exchanges thermal energy with a gas is called accommodation, which is characterized by the accommodation coefficient\nc acc = T em − T gas T bulk − T gas ,(170)\nwhere T em is the temperature of the gas molecules emitted from the particle surface. The accommodation coefficient quantifies the fraction of the thermal energy that the colliding gas molecule removes from the surface, such that c acc = 1 means that the molecule fully thermalizes with the surface. We do not distinguish between the bulk and the surface temperature, since we assume that they are identical for nanoparticles. Since the mean free path in a dilute gas is long, one can safely assume that an emitted molecule will not interact again with the particle before thermalizing with the environment. Consequently, we can consider the particles that impinge on the particle surface and those that leave it as being in equilibrium with two independent baths [514]. The damping due to the impinging molecules is\nγ im = 8m gas πk B T gas P gas ρ p a(171)\nand the contribution from the emerging molecules is\nγ em = π 8 T em T gas γ im .(172)\nThe total gas damping is then γ gas 2π = γ im 2π + γ em 2π (173) and the total force noise due to the gas is\nS gas ff = mk B π γ im T gas + γ em T em .(174)\nWhen the temperature of the emerging molecules equals the gas temperature (i.e., T em = T gas ), equation (173) reduces to equation (169).\n\n\nRadiation damping and shot noise\n\nOnce the pressure is low enough (i.e., ≤ 10 −7 mbar), gas damping is negligible and the particulate nature of the light field becomes the primary source of dissipation, in the absence of technical noise sources such as laser phase and amplitude fluctuations [416]. Photons arrive at discrete times, where the number of photons arriving per time interval ∆t is given by ∆tP opt ( ω opt ) . The recoil from the fluctuating number of photons impinging on the nanoparticle can be modeled as an effective bath with the characteristics [497,516] γ rad 2π = c dp P scatt 2πmc 2 and S rad ff = c dp\nω L P scatt 2πc 2 ,(175)\nwhere c dp depends on the direction of motion of the particle with respect to the polarization of the laser (in particular, c dp = 1/5 for motion along the direction of polarization, c dp = 2/5 for motion perpendicular to the polarization, and c dp = (2/5 + A 2 ) along the direction of propagation, where A is the radiation pressure contribution defined in Ref. [497]). The scattered power is P scatt = σ scatt I opt , where σ scatt = |α| 2 k 4 opt /(6π 2 0 ) and I opt is the laser intensity. The effective temperature of this bath can be calculated via equation (167) and is ∼ ω L /k B .\n\n\nDisplacement noise\n\nIn the reference frame of the trap (i.e., where the trap appears stationary), acceleration of the trap center a dis appears as a force F dis = ma dis due to the particle's inertia m. Therefore, fluctuations of the trap center with displacement spectral density S dd lead to the effective force spectral density\nS dis ff (ω) = m 2 ω 4 S dd (ω),(176)\nwhere we have used the fact that the acceleration spectral density S aa (ω) = ω 4 S dd (ω). There is no damping associated with displacement noise, since the source of the displacement noise is usually some active component like a pump. Unlike the thermal force spectral density, the displacement noise spectral density is not flat but increases for lower frequencies. However, for the typical narrow resonances at Ω 0 in optical levitation experiments, we can approximate it as white noise, S dis ff ≈ m 2 Ω 4 0 S dd (Ω 0 ). The thermomechanical noise limited acceleration sensitivity S aa = k B T γ 0 /(πm) depends inversely on the particle mass. Hence, heavy particles make particularly good accelerometers [458]. However, since the maximal trap stiffness is limited, heavier particles tend to have lower frequencies at which displacement noise dominates over thermomechanical noise. At the optimal particle size both noise sources contribute equally. Displacement noise is a technical noise source and as such not a fundamental limitation. It is primarily noticeable in setups where the external forces change considerably over the range of movement of the trap center. That is in situations where the force has a strong spatial dependence, for example when the trapped particle interacts with a standing wave of an optical cavity or with the evanescent field from a tapered fiber [492]. If the trap position changes randomly with respect to the cavity or the fiber, the particle experiences fluctuating optical forces from the cavity or evanescent field, respectively.\n\n\nLaser intensity noise\n\nLaser intensity fluctuations are another technical noise source [491,492]. Low frequency drifts of the intensity randomly change the particle's oscillation frequency, since the particle frequency scales with Ω 0 ∝ P opt . This can be a serious limitation in force-sensing experiments, where an external force is transduced into an oscillation amplitude. If the frequency changes, the particle's response to an external force changes even if this force is constant, resulting in a noisy measurement.\n\nIntensity fluctuations at twice the motional frequency 2Ω 0 parametrically excite the particle and thereby raise the particle's energy. If the phase of the particle oscillation is not actively stabilized to the parametric modulation, it self-locks (entrains) to the external modulation [468]. Hence, an intensity modulation at 2Ω 0 without active stabilization always leads to heating as will be discussed in section 5.7. Parametric excitation is a nonlinear process and its strength depends on the motional energy of the particle and therefore can be suppressed by reducing the particle's energy with feedback cooling. In free-space experiments, laser-intensity noise can be reduced sufficiently such that it is smaller than the fundamental photon shot noise [416]. In cavity experiments, however, laser-phase noise is transduced into intensity noise and has been identified as a major obstacle towards reaching the ground state with sideband cooling [421,517].\n\n\nCalibration\n\nPrecise calibration is as important for quantitative measurements in the underdamped regime as it was in the overdamped regime (section 3). As before, the thermal energy k B T serves as a reference to calibrate the detector signal (section 5.6.1). However, in contrast to the underdamped case, where the displacement signal from the particle is very broadband and centered around zero, in the underdamped regime the signal is sharply peaked and centered around a finite frequency. The weak coupling to the thermal bath in high vacuum also allows to use the energy scale of a single motional quantum Ω 0 as a calibration reference. Remarkably, despite the huge difference in energy scale of 7-8 orders of magnitude one can achieve almost perfect agreement [418] (section 5.6.2). In high vacuum charges on the particle are preserved. Section 5.6.3 explores how to control and measure them. The precise knowledge of the charge then allows allows to precisely determine the particle's mass (section 5.6.4). Since a particle in vaccum is well isolated, its internal temperature raises well above the temperature of the environment in an optical tweezers. In section 5.6.6, we will see how to measure the internal temperature by analyzing the particles center of mass motion.\n\n\nDetector calibration\n\nFor the most part optical trapping is concerned with the external degrees of freedom, and in particular with the center-of-mass motion. Quantitative measurements require a precise calibration of the detector signal that relates the measured signal (typically in volts) to the displacements of the particle (in meters). Under the assumption that the detector signal v(t) is proportional to the particle displacement q(t), the calibration factor is defined as the proportionality factor c = v/q with units of [V m −1 ] (or [bit m −1 ] for a digitized detector signal). Using the fact that the expectation value of the potential energy of a harmonic oscillator E pot = 1 2 mΩ 2 0 q(t) 2 = 1 2 k B T, the calibration factor is simply given by\nc = mΩ 2 0 v(t) 2 /k B T ,(177)\nwhich does not depend on γ 0 and thus holds both for the overdamped and underdamped regimes.\n\nIn the underdamped regime, the PSD is highly peaked around the resonance frequencies. This allows to distinguish noise and cross-talk efficiently in the spectral domain. Therefore, it is often preferred to fit the PSD to the single-sided harmonic oscillator PSD\nS vv ( f ) = 2 v(t) 2 π f 2 0 g ( f 2 − f 2 0 ) 2 + f 2 g 2 (178)\nto extract the variance of the particle displacement v(t) 2 from the area under the PSD. In the following, we adopt here the notation from Hebestreit [485], where we convert from angular to ordinary frequencies, i.e., ω = 2π f , γ = 2πg, and S( f ) = 2πS(ω > 0). The experimental single-sided PSD is obtained from the discretely-sampled time trace v(t) as (179) where N is the number of samples, f s is the sampling frequency, and F (v(t)) is the discrete Fourier transform of v(t).\nS vv ( f ) = 2 N f s 1 N F (v(t)) 2 for f > 0,\nFitting the experimental PSD to expression (178) allows to extract the variance v 2 (t) , the damping constant g = γ gas /(2π), and the resonance frequency f 0 = Ω 0 /2π. Then, the damping constant allows to determine the particle radius from the pressure and, therefore, the mass for known mass density (section 5.6.4). Once the mass is known, we can calculate the calibration factor according to equation (177). The biggest uncertainty in determining c stems from the error in the particle mass. However, in many experiments we are only interested in the particle energy and we can define the mass-independent energy calibration factor\nC = c 2 m = Ω 2 0 v 2 k B T .(180)\nThe above calibration method relies on two key assumptions. First, we consider the optical potential to be quadratic such that the particle behaves like a harmonic oscillator. Second, we assume that the detector signal is linear in the particle displacement, that is that the calibration factor does not depend on the particle position. The latter depends significantly on the details of the experiments and will not be futher discussed here. However, it is possible to account for a nonlinear detector response. For calibration of a position dependent calibration factor see [480] and for detector response calibration and discussion see [485].\n\nThe finite potential depth of the optical trap results in anharmonicities in the particle motion for large enough displacements. Due to the symmetry of the optical potential, the lowest nonlinear term is the cubic or Duffing nonlinearity39. The Duffing nonlinearity softens the optical potential at large particle displacement [470]. This distorts and broadens the lineshape of the PSD, and causes an overestimation of the energy when following the calibration procedure described above. To minimize the impact on the shape of the PSD, the calibration is usually carried out at moderate pressures (∼ 10 mbar), where the linewidth (γ gas /(2π) ∼ 10 kHz) is narrow enough to reject noise and cross-talk, but still wide enough that the nonlinear broadening (γ NL /(2π) ∼ 3 kHz) is negligible [485] (Fig. 41a).\n\n39There are some subtelties in the detection along the propagation direction of the trapping laser, where this symmetry is broken by the scattering force, leading to small but finite quadratic nonlinear terms [488]. The energy underestimation is accounted for by considering the kinetic energy instead of the potential energy [518]. The kinetic energy is given by E kin = 1 2 m q 2 = 1 2 k B T even for a nonlinear potential. The variance q 2 can be obtained by integrating over the velocity PSD, which is related to the position-PSD by\nS q q ( f ) = 4π 2 f 2 S qq ( f ).(181)\nHence, we do not have to measure q directly, which is experimentally challenging since it would require sufficient oversampling. Experimentally, we compute the variance by numerical integration over the voltage PSD\nv 2 = ∫ f 0 +b/2 f 0 −b/2 d f 4π 2 f 2 S vv ( f ) = c 2 m k B T = Ck B T .(182)\nThe integration bandwidth b is chosen such that we integrate most of the signal and don't integrate too much of the noise floor. In the measured displacement PSD, we usually have a white noise floor. If we multiply this PSD by f 2 to get the velocity PSD, the noise floor therefore acquires a f 2 dependence as well. Integrating this noise will eventually cause a significant error in the calibration and energy estimate. Drifts in the setup can cause the calibration to change over time. In particular, the calibration procedure detailed above is carried out at moderate pressures (∼ mbar), while most experiments are carried out at much lower pressure many hours or even days after performing the calibration. Hence, it is desirable to check the calibration even at much lower pressure and to correct for potential drift (see Fig. 41b).\n\nHebestreit suggests a method to track changes in calibration at arbitrary pressure [485]. The main idea is to measure the response of the particle to a known force, for example the electrical force from a capacitor around the optical trap. This method assumes that the electrical fields and the charge on the particle do not change through out the experiment. This can be achieved as will be discussed in section 5.6.3. At the initial pressure where the calibration is carried out, a sinusoidal signal is applied to the capacitor. This leads to additional peaks in the response of the particle (Fig. 41). The power in the response peak is\nv 2 e = C 2m 2 Q 2 p E 2 0 (Ω 2 0 − ω 2 d ) 2 + γ 2 ω 2 d ,(183)\nwhere Q p is the total charge on the particle and E 0 is the electric field amplitude in the direction of oscillation at the location of the particle. At a later time (e.g., after pumping down to a lower pressure), we repeat this measurement, yielding v 2 e . At lower pressure, feedback is generally active. Hence, the response function of the particle changes. The new values of Ω 0 and γ can be extracted from a fit to the thermal noise peak. Assuming that Q p E 0 /m remain constant, we find the new calibration factor as\nC = C v 2 e v 2 e (Ω 2 0 − ω 2 d ) 2 + γ 2 ω 2 d (Ω 2 0 − ω 2 d ) 2 + γ 2 ω 2 d .(184)\nWe can also arrive at equation (184) by considering the height of the peak in the PSD instead of the area under the peak. It's value is v 2 e τ, where 2τ is the measurement time (equation (188)). In the previous section we used the thermal fluctuations of the particle to calibrate the detector signal. Similarly, we can use the quantum fluctuations for calibration. However, at ambient temperature, quantum fluctuations of a nanoparticle in an optical tweezers are 8-9 orders of magnitude weaker than the thermal fluctuations. As a consequence, they can only be observed in high vacuum, where the coupling to the thermal bath is sufficiently weak [418].\n\n\nSideband thermometry\n\nWhen light is reflected off the particle, the particle motion is imprinted on the scattered light as a phase modulation, which can be understood as a Doppler shift due to the particle motion. To lowest order, the phase modulation leads to red and blue sidebands at ω L ± Ω 0 , where ω L is the laser frequency (Figs. 42(a-b)). Classically, the power in each sideband is proportional to the variance of the motion x 2 , which in turn is proportional to the temperature of the thermal bath. This is the basis of the calibration procedure described in the previous section.\n\nIn the quantum mechanical formulation, the interaction of light with the trapped particle can be described as a Raman process. The red sideband (Fig. 42(a)) corresponds to Stokes scattering, which raises the population of the mechanical oscillator by a single quantum of mechanical energy, while simultaneously creating a low energy (red) photon. The blue sideband (Fig. 42(b)) corresponds to anti-Stokes scattering, which lowers the oscillator's population by one mechanical quantum, while creating a high energy (blue) photon. The rate at which blue sideband photons are created is proportional ton, just as in the classical case. Importantly, anti-Stokes scattering is impossible by an oscillator in its quantum ground state. In contrast, the rate at which red sidebands are produced is proportional ton + 1.\n\nThe mean occupation number is given by the Bose-Einstein distribution\nn = 1 e Ω 0 k B T − 1(185)\nFor thermal energies large compared to the energy quantum (k B T Ω 0 ), the mean energy of the harmonic oscillator isn Ω 0 = k B T ∝ x 2 , in accordance with classical statistical mechanics and independent of . The ratio of powers in the anti-Stokes and Stokes sidebands is given bȳ (Figs. 42(c)) and vanishes for high temperatures. For thermal energies comparable to a single mechanical quantum excitation (also called phonon), the ratio can serve as a temperature measurement calibrated relative to the quantum of energy of the system [519].\nn /(n + 1) = exp [− Ω 0 /(k B T) ]\nThis so-called sideband thermometry requires to measure the Stokes and anti-Stokes sidebands [418,[520][521][522], which can be achieved with a heterodyne measurement. In a heterodyne measurement the scattered light interferes with a laser beam (local oscillator) with a frequency ω LO which is frequency-detuned from the light that is incident on the particle. This produces a strong beat signal at the difference frequency ∆ LO = ω LO − ω L , with sidebands at ∆ LO ± Ω 0 . For ω LO < ω L , the left sideband corresponds to the red sideband (Stokes) and the right to the blue (anti-Stokes). For ω LO > ω L , the roles are inverted. The difference frequency is chosen such that |∆ LO | Ω 0 , for example |∆ LO | ∼ 1MHz for typical trapping frequencies ∼ 100kHz. The measured sideband asymmetry\nR ∓ = ∫ S r,het vv ( f )d f ∫ S l,het vv ( f )d f(186)\ncan be used to determine the mean occupation number. Here, S l,het vv (S r,het vv ) is the power spectral density of the left (right) sideband and the subscript R − (R + ) is for ω LO < ω L (ω LO > ω L ) (see Fig. 42).\n\nIn principle, R − or R + alone is sufficient to determine the mean occupation number. However, classical artifacts due to a frequency dependent transfer function of the measurement device give rise to an asymmetry that is not related to the quantum dynamics of the particle. Measuring both R − and R + allows to eliminate this artifact and to determine the actual mean occupation number as followsn Fig. 42 shows the measured sidebands of a nanoparticle cooled to an effective temperature of n = 4 [418]. Clearly the area under the red sideband is larger than under the blue sideband. This is the first room temperature measurement of the sideband asymmetry of a massive object and clear signature of its quantum ground state. Remarkably, the inferred occupation number is in very satisfactory agreement with the traditional calibration method detailed in section 5.6.1despite the huge difference in energy scales of the two methods.\n= R − /R + 1 − R − /R + .(187)\n\nCharge calibration\n\nGenerally, particles carry several tens of elementary charges. Under vacuum, the particle is completely isolated from the environment and therefore its charge state is preserved indefinitely. Knowing the charge exactly provides a reliable handle for exerting a known force on the particle by applying an known electric field.\n\nIn optical levitation experiments, particles are loaded at high (∼ 1 atm) pressure to provide enough damping. After the particle has been trapped, the vacuum chamber is evacuated to reduce air damping. As the pressure decreases, residual water desorbs slowly from the particle followed by a sudden change in size, which is attributed to the removal of the final water layers (section 5.4.1). During this final stage, the charge of the particle changes by tens of elementary charges [486], but once the particle has undergone this sudden change, its charge state is preserved indefinitely even when the pressure is increased again because the water is gone.\n\nA relatively simple method to control the charges on the particle is to apply an electrical discharge near the particle. The discharge provides free charges that can be adsorbed by the particle [523]. It can be applied through a bare wire of about 5 cm length at about 5 cm away from the optical trap. The grounded vacuum chamber serves as the counterelectrode. A DC voltage (∼ 7 kV) is sufficient to ionize gas molecules inside the vacuum chamber and to produce a corona discharge at moderate pressures (∼ 1 mbar). Note that an ion pressure gauge can produce also free charges, which lead to unwanted changes in the charge state of the particle, if it is not properly shielded [523]. The ionization process is triggered by random events such as absorption of a ultraviolet photon or cosmic rays that spontaneously ionize neutral air molecules. The free electron and positive ion are accelerated toward opposite directions by the high voltage, which also prevents them to recombine. The electron acquires enough kinetic energy to ionize other molecules, because of its small mass. The secondary electrons ionize more molecules, which then results in an electron avalanche.\n\nEmission of (mainly) UV photons after electron-ion recombination causes the characteristic violet glow of the corona, as shown in Fig. 36. The corona discharge is positive or negative, depending on the polarity of the high DC voltage, which therefore allows to select the ratio of positive-to-negative ionized molecules. Eventually, free charges are adsorbed on the particle surface, thereby changing its net charge monotonically, except for few unfavorable events. These charges can be both electrons and ionized air molecules. Accordingly, by simply turning off the high-voltage source when the particle carries the desired number of charges, the levitated particle can be brought to any desired net-charge state from zero to a few elementary charges [486,523].\n\nThe charge state of the particle is monitored in situ by driving the particle motion with an electric field at frequency ω d . The electric field is created by applying a voltage U(t) = U 0 cos(ω d t) across a capacitor that is centered around the particle. A simple capacitor can be formed by grounding the metal housing of the microscope objective and applying a voltage to the metal holder of the collection lens [523]. This method does not require any additional mechanical components in the setup and therefore does not obstruct access to the particle . Alternatively, dedicated electrodes that fit right at the sides of the optical trap can be beneficial to produce a more homogeneous field at the location of the particle [486], since field inhomogeneities can lead to a finite net charge density. When a voltage U 0 ∼ 10 V is applied to the capacitor at a frequency ω d close to the particle's resonance frequency, the particle's response to the driving field shows as a distinct peak in the single-sided PSD [486] on top of the thermal peak (see Figs. 41 and 43a):\nS qq (ω) = 2m −2 | χ(ω)| 2 S ff + F 2 0 τ 4π sinc 2 [(ω − Ω 0 )τ] ,(188)\nwhere the thermal force spectral density is S ff = mk B T γ 0 /π, χ(ω) = Ω 2 0 − ω 2 + iωγ 0 −1 is the mechanical response function, F 0 = Q p E 0 , E 0 ∼ 500 V m −1 being the amplitude of the electric field modulation at the location of the particle, Q p is the charge on the particle, and 2τ is the duration of the time trace that is used to calculate the position spectral density S qq (ω) = 2τ|q| 2 (equation (179)). The detector signal is demodulated in quadrature with the drive. The demodulated signal can assume both positive and negative values, since the response to the driving field flips phase by π when the polarity of the charge changes. Positive (negative) signal amplitudes can then be associated with positive (negative) net charge, when the transfer functions of the electronics is accounted for. Note that for a small detuning of the drive frequency ω d from the particle's resonance, there is a small phase offset from the response function of the particle.\n\nWhile the corona discharge is active, the demodulated signal features discrete steps, which are the signature of single elementary-charge transfers to and from the nanoparticle (see Fig. 43). Thus, any desired net-charge state from zero to a few elementary charges can be achievd by simply selecting the polarity of the high-voltage source, counting the number of steps, and turning off the high-voltage source when the particle carries the desired number of charges. Importantly, the particle's charge state is preserved when the pressure is reduced and stays indefinitely while the high-voltage source is turned off, even over a period of several days [523].\n\n\nMass Calibration\n\nThe exact value of the particle's inertial mass often plays an important role for precision measurements. For example, the common method for displacement calibration uses the thermal force noise as a reference, where the strength of the force depends on the mass, as we have seen in section 5.6.1. Likewise, nanomechanical mass measurements rely on knowing the mass of the bare resonator [524].\n\nThe manufacturer's size specification gives a good first estimate of the particle size with an error up to ∼ 25%, which stems from the typical size variation (∼ 5%) and the uncertainty in the specified mass density (∼ 10%) of commercially available mono-disperse particles.\n\nThe mass of individual particles can be inferred from kinetic gas theory, which relates the measured damping rate γ 0 to the particle size [512]. In particular, for a spherical particle and at pressures where the mean free path is much larger than the radius of the particle, there is a simple linear relationship [488],\na = 2.223 ρ p m gas k B T P gas γ 0 ,(189)\nwhich relates the pressure to the particle radius a (equation (169)). At room temperature the mean free path is ∼ 60 µm · P −1 gas mbar −1 . Hence, the linear approximation is generally valid for nanoparticles in vacuum. The calibration measurements are generally performed at ∼ 10 mbar, where individual peaks are clearly resolved and nonlinear broadening of the peaks is still negligible. To remove excess water from the particle, which leads to an overestimate of the actual particle size, the pressure should be cycled to below 0.1 mbar and back (section 5.4.1). From the measured radius, one can then infer the mass m = 4 3 πa 3 ρ p with a typical uncertainty of ∼ 35% [486].\n\nThe mass measurement can be significantly improved either by an independent measurement Fig. 44. Mass calibration. Power spectral density S vv (ω) of a thermally and harmonically driven resonator at P = 50 mbar. The broad peak centered at Ω 0 /2π = 125 kHz corresponds to the thermally driven state. We fit it with a Lorentzian function (red) to extract S th vv (ω d ) together with g/(2π) = 31.8 kHz and the corresponding uncertainties. The narrowband peak at ω d /(2π) = 135 kHz, also shown in detail in the inset, depicts the electrical excitation from which we retrieve S d vv (ω d ). Gray data points at the bottom of the plot is the measurement noise, which is ∼ 40 dB below the particle's signal. Figure adapted from Ref. [486]. of the voltage calibration (equation (177)) or by applying a known force. For the former approach, the wavelength of the light provides a well-defined length scale that allows to relate the phase to a displacement in an interferometer. The interferometer is calibrated by changing the relative length of the two interferometer arms and measuring the fringe pattern. For the latter approach, electrical fields can be estimated quite accurately (∼ 1% error) from a numerical simulation of the electrode geometry [486]. Hence, the electrical force on the particle can be estimated with the same accuracy, because the number of charges can be determined exactly (section 5.6.3). Since the thermal force depends on the mass but the electrical force does not, their ratio (equation (188)),\nR S = F 2 0 τ/2 F L = S th vv (ω d ) − S d vv (ω d ) S th vv (ω d ) ,(190)\nprovides a direct method to determine the mass from the measured spectra (Fig. 44). In equation (190), S d vv (ω d ) is the peak of the measured spectrum at the drive frequency ω d when the sinusodial electrical force F el = F 0 cos(ω d t) is applied and S th vv (ω d ) is the thermal background at ω d (Fig. 44). Note that the spectra do not need to be calibrated to apply this procedure. The mass is then simply given by [486] m =\nQ 2 p E 2 0 τ 8k B T γ 0 R S .(191)\n\nInternal temperature\n\nIn optical tweezers experiments, the main focus is on the external degrees of freedom of the particles in the trap (i.e., the translation of the particle and its rotation). However, any particle will absorb some of the trapping light, which raises its internal bulk temperature. The power absorbed by a particle with volume V from a laser beam with intensity I 0 is P abs = 6π I 0 λ VIm n 2 (λ) − 1 n 2 (λ) 2 + 2 .\n\nFor example, for silica, the refractive index is n = 1.45 + i6 · 10 −8 at λ = 1064 nm. Note that absorption can be higher in nanoparticles due to additional absorbers such as defects and impurities. At high pressure, the particle efficiently thermalizes with the surrounding gas by convective cooling with cooling power P con (T bulk ) = −c acc 2 3π (πa 2 )P gas v rms γ sh + 1 γ sh − 1\nT bulk T gas − 1 ,(193)\nwhere v rms = 3k B T gas /m gas is the root mean square velocity of the gas molecules, and γ sh = 7/5 is the specific heat ratio for an ideal diatomic gas. Clearly, as the pressure P gas decreases, convective cooling by the surrounding medium vanishes and the only heat exchange with the environment is radiative. The emitted thermal radiation is P em bb = −P bb (T bulk ), where P bb (T) = 72ζ(5) π 2 V c 3 4 Im n 2 bb − 1 n 2 bb + 2 (k B T) 5 (194) is the the black body radiated power for a dipolar emitter, being ζ(5) ≈ 1.04 the Riemann zeta function [413]. Conversely, the particle also absorbs thermal radiation from the environment at temperature T env with P abs bb = P bb (T env ). The particle bulk temperature can be found from the requirement that at equilibrium all contributions add up to zero, i.e., P abs + P abs bb + P con (T bulk + P em bb (T bulk ) = 0.\n\nThe emitted or incident power due to black-body radiation depends on the dielectric properties of the particle in the entire black-body spectrum, and their calculation therefore involves the integration of the black-body emission rates over all emission frequencies. As an approximation, one assumes a constant permittivity of the particle in the entire spectral range of black-body emission. This can be estimated as the average value of the refractive index over the black-body spectrum. For example, for silica this gives n bb = 1.5 + i0.1.\n\nThe internal temperature of a particle trapped in vacuum can raise significantly above the environmental temperature, to the point where the particle disintegrates if the absorption is too high [448] (Fig. 45). In fact, this problem has prevented optical levitation of nanodiamonds in high vacuum [447]. Even silica particles with low absorption can reach temperatures ∼ 1000 K [525]. As an example, figure 45d shows the internal temperature as a function of pressure. Fig. 45. Heating and temperatures of an optically levitated particle. (a) Damping and (b) heating rate as a function of pressure for feedback γ fb /2π = 20 Hz. (c) Center of mass temperature as a function pressure. At low pressures recoil heating dominates and the rates and the center of mass temperature become pressure independent. (d) Internal temperature as a function of pressure. At a few millibars, there is a sharp transition between convective cooling and radiative cooling via black body radiation and the internal temperature transitions from the ambient temperature to a much higher value. The internal temperature leads to a slightly higher center-of-mass heating rate visible as a small bump in (b). 5.6.6. Coupling between internal temperature and external degrees of freedom As we discussed earlier, the bulk temperature couples to the external degrees of freedom through the residual gas in the vacuum chamber (section 5.5). This coupling can be explained by a simple two-bath model, where the gas molecules that interact with the particle are divided into two families: molecules that impinge on the particle being in thermal equilibrium with the vacuum chamber, and molecules that emerge from the hot surface of the particle at a higher temperature. From equation (174), the thermal force noise increases with increasing bulk temperature. This allows to determine the internal bulk temperature from the fluctuation of the center-of-mass of the particle by solving T cm = T 3/2 gas + π 8 T 3/2 em T gas + π 8 √ T em (196) for T em , where the center-of-mass temperature is defined as T cm = Ω 2 0 v 2 /(Ck B ) , C being the energy calibration factor and v 2 the variance of the voltage fluctuation on the detector (section 5.6.1). Then, from T em one determines the temperature of the particle as T bulk = c −1 acc T em − T gas + T gas .\n\nThe method described above requires the particle's center-of-mass to reach its equilibrium steady state. This is impractical at low pressures where the particle is easily lost without feedback stabilization. At low pressures, the thermal noise force from the interaction with the gas can be determined measuring the heating rate [525]. Thereby, the center-of-mass motion is first reduced with feedback cooling (section 5.7). When the feedback is switched off, the particle energy increases at a rate Γ = γ im T gas + γ em T em = γ im T 3/2 gas + π 8 T 3/2 em T gas . (a) Heating rates as a function of pressure extracted from relaxation measurements without additional heating of the internal temperature (blue) and with heating by a CO 2 laser intensity of 0.47 µW µm −2 (orange). The dashed lines are linear fits to the data points. Error bars are smaller than the marker size. (b) Heating rates at different intensities of the CO 2 laser measured at a pressure of 1 · 10 −5 mbar. The error bars indicate the standard deviation of the measurements. (c) Calculated temperature dependence of oscillation frequency. When the particle is heated with the CO 2 laser, its oscillation frequency increases due to changes in the particle's material properties. This leads to a nearly linear relation between relative frequency change and increase of the internal particle temperature. Adapted from Ref. [525].\n\nWith feedback cooling the oscillation amplitude is low enough that cross-coupling between the degrees of freedom can be neglected. This justifies the approximation where the particle center-of-mass motion is treated as three independent harmonic oscillators. Each degree of motion obeys the following equation of motion (c.f. equation (160)):\nq(t) + γ 0 q(t) + Ω 2 0 q(t) = m −1 [F L (t) + F fb (t)] ,(201)\nwhere F fb (t) is the feedback force (equation (160)). For most purposes, the stochastic contribution F L and the bare damping γ 0 result from interactions with the background gas. Additional contributions are discussed in section 5.5. The feedback damps the particle motion at a rate (γ f b /2π). The additional damping reduces the fluctuations of the position of the particle. As a consequence, feedback cooling reduces the response time to external perturbations and increases the confinement of the particle. Since in a thermal equilibrium the variance of the position is proportional to the temperature of the environment, we define the effective temperature for the center-of-mass motion (equation (167)) as\nT cm = mΩ 2 0 k B q 2 (t) = γ 0 γ 0 + γ fb T,(202)\nwhich assumes that the particle position and energy distributions reflect the distributions of a harmonic oscillator in thermal equilibrium at temperature T cm . It is important to point out, however, that under the action of feedback the particle is not in thermal equilibrium and the assumption of a simple thermal distribution might be violated, in particular for nonlinear feedback schemes (section 5.7.2).\n\n\nForce feedback\n\nFrom an inspection of equation (201), it is apparent that a feedback term proportional to the particle velocity results in damping. Instead, a feedback term proportional to the particle position allows to optimize the transient times [417]. Thus, the feedback force of a linear feedback controller reads\nF fb = −k d q − k p q,(203)\nwhere k d is the derivative feedback gain and k p is the proportional feedback gain. Purely velocity-dependent feedback (i.e., k p = 0) is also called cold damping. Experimentally such a force can be implemented with radiation pressure from a weakly focused beam. This technique was used in Ashkin and Dziedzic's original experiments to stabilize a particle against gravity [403], and later by Li et al. to feedback-cool microspheres in high vacuum [527]. The latter experiment required three laser beams in addition to the trapping laser to exert radiation pressure forces along the three spatial dimensions.\n\nMore recent experiments use electrostatic forces instead [415,417,528]. Electrodes near the optical trap apply a force F Q = Q p E, where Q p is the total charge on the particle and E the electric field at the location of the particle. Often the electrode geometry is chosen such that they form a plate capacitor around the particle so that the field is homogeneous and given by E = (V elec /d elec ) n elec , where V elec is the applied voltage, d elec the distance between the electrodes and n elec a unit vector pointing from one electrode to the other. The electrostatic force along direction e i is then given by F Q = F Q · e i and the electrode geometry can be chosen such that there is a force along all degrees of freedom. Implementation of the velocity-dependent force requires to measure the instantaneous velocity of the particle. In practice, however, this is achieved with a bandpass filter that acts on the position time trace q(t) and produces a time delay such that the signal is in phase with the velocity q(t) [415]. For example if the particle motion is approximately harmonic q(t) ∼ sin(Ω 0 t), a delay τ = π/(2Ω 0 ) leads to a signal ∝ cos(Ω 0 t) that is proportional to the particle velocity q(t).\n\nIn the frequency domain, the equation of motion with linear feedback reads q −ω 2 + iωγ 0 + Ω 2 0 = m −1 F L +f fb ,\n\nwhere the tilde indicates the Fourier transform andf fb (ω) = H(ω) [q +q n ] with filter funcion H(ω) = −ik d ω − k p . Explicitly accounting for the noise in the measurement of the position q n we arrive atq (ω) = m −1 χ fb (ω)F L − χ fb (ω) Ω 2 fb + iωγ fb q n (205) where the susceptibility χ fb (ω) = Ω 2 0 + Ω 2 fb − ω 2 + iω(γ 0 + γ fb ) −1 , and we introduced the feedback damping γ fb = k d /m and frequency shift Ω 2 fb = k p /m. The in-loop detector signal is q IL =q +q n . Hence, the measured in-loop two-sided PSD is [415,417] S IL qq (ω) = | χ fb | 2 m −2 S ff + | χ fb | 2 (Ω 2 0 − ω 2 ) 2 + ω 2 γ 2 0 S nn (206) = m −2 S ff + (Ω 2 0 − ω 2 ) 2 + ω 2 γ 2 0 S nn (Ω 2 0 + Ω 2 fb − ω 2 ) 2 + (γ 0 + γ fb ) 2 ω 2 ,\n\nwhere S qq = |q| 2 and S nn is the noise PSD of the in-loop detector (Fig. 47). The in-loop measurement noise is reintroduced into the feedback loop, which drives the particle. The correlations between the particle's position and the measurement noise lead to noise squashing, where the apparent signal is squashed below the noise floor [529] (Fig. 47b).\n\nIn contrast, an out-of-loop detector provides an independent measurement of the actual particle motionq OoL =q +q ν :\n\nS OoL qq (ω) = | χ fb | 2 m −2 S ff + | χ fb | 2 Ω 4 fb + ω 2 γ 2 fb S nn + S νν (208) = m −2 S ff + Ω 4 fb + ω 2 γ 2 fb S nn (Ω 2 0 + Ω 2 fb − ω 2 ) 2 + (γ 0 + γ fb ) 2 ω 2 + S νν ,\n\nwhere S νν is the noise PSD of the out-of-loop detector (Fig. 47).\n\nIntegration over the PSD yields the position variance40\nq 2 = ∫ ∞ −∞ S OoL qq (ω) − S νν dω = πm −2 S ff + Ω 4 fb S nn (Ω 2 0 + Ω 2 fb )(γ 0 + γ fb ) + πγ 2 fb S nn (γ 0 + γ fb ) ≈ 1 γ fb πS ff m 2 Ω 2 0 + γ fb πS nn .(210)\nIn the approximation we set k p = Ω fb = 0 and γ fb γ 0 and we used S ff = k B T mγ 0 /π. The first term in equation (210) scales with the inverse of the feedback cooling rate. This term resembles the desired action of the feedback, which is to reduce the impact of the heating term given by the fluctuating force. The second term is proportional to the feedback damping rate, which is multiplied by the measurement noise. This term resembles the undesired but inevitable effect of the control-loop heating the particle by feeding back measurement noise. Consequently, there exists an optimum feedback cooling rate γ opt = S ff /S nn /(mΩ 0 ) , where the mode temperature reaches its minimum value q 2 min = 2π √ S ff S nn /(mΩ 0 ) . With cold damping, optically levitated nanoparticles have been cooled to 4 occupational quanta, corresponding to an effective temperature of 11 µK [418] (Fig. 47(b) and Fig. 42).\n\nClassically, the measurement is not perturbative and the position variance can be reduced to arbitrarily low values, because, at least in principle, the noise S nn can be made arbitrarily small (equation (210)). However, quantum mechanics imposes that every measurement is accompanied by a measurement back-action, leading to a finite contribution S nn even in an ideal measurement. In the case of an optical measurement this back action is the random arrival of photons and the resulting random momentum kicks that lead to a stochastic backaction force [416]. Therefore, in a regime where the measurement back-action is the dominant contribution to the force noise, the impression back-action product is S ff S nn = ( /(4π)) 2 /η det , where η det < 1 is the measurement efficiency [415,519]. The impression back-action product is a manifestation of the Heisenberg uncertainty principle and describes the tradeoff between measurement imprecision and measurement back-action. At the optimal feedback gain, the effective phonon occupation number n cm = mΩ 0 q 2 min / − 1/2 depends only on the detection efficiency as n min = (η −1/2 det − 1)/2. At the Heisenberg limit (η det = 1), when the imprecision noise is minimized by optimally detecting all photons scattered by the levitated particle, the particle's center-of-mass motion could, in principle, be brought to its quantum ground state n min = 0. Hence, to reach the ground state with feedback cooling, the experiment should be operated in the photon-recoil-limited regime (P gas ≤ 10 −8 mbar) and the collection of scattered photons needs to be optimized. A promising approach is to collect the scattered photons with a high-finesse optical cavity, which collects a fraction of (η P /(η P + 1)) of the overall scattered power due to the Purcell effect, where η P = (6F c λ 2 )/(π 3 w 2 0 ) is the Purcell factor for a cavity with finesse F c and beam waist w 0 [530].\n\n\nParametric feedback cooling\n\nWhile cooling levitated particles with linear feedback is very powerful, its experimental implementation along all three motional degrees of freedom is challenging. All-optical linear feedback requires three auxiliary laser beams to apply radiation pressure in all directions [527]. Electrical feedback relies on a finite net charge on the levitated object.\n\nThe method of choice to control charge-neutral optically levitated particle is parametric feedback cooling [407], which allows to control all three spatial degrees of freedom with the same laser that is used for trapping, thereby significantly reducing the experimental complexity.\n\nThe basic concept of parametric feedback is to introduce a modulation in time of the intensity of the trapping laser. In particular, applying a modulation δI(t) = I 0 (η para /Ω 0 )q q results in the parametric feedback force F param fb = −k para q 2 q,\n\nwhere k para = Ω 0 mη para is the parametric feedback gain. When the particle moves away from the trap center, the intensity modulation δI(t) stiffens the trap. When the particle falls back to the trap center the feedback softens the trap. Thus, by synchronizing the laser intensity modulation with the particle motion energy is extracted from the center-of-mass motion of the particle [488]. This principle is analogous to the way a child on a swing gains motional energy by modulating its center-of-mass. Constructing the feedback signal as in equation (211) provides exactly the right phase relation, such that the trap stiffens when the particle moves away from the trap center and softens when is falls back, thereby cooling the particle. However, latencies in the experimental implementation require an additional phase shifter to compensate for unwanted phase shifts. With an additional phase shift we can also amplify the particle motion. Note that, in the absence of active feedback, the particle's oscillation self-locks (entrains its phase) to the modulation in such a way that the motion is amplified [531]. Cooling therefore requires active stabilization of the modulation phase.\n\nExperimentally, the detector signal is first frequency-doubled and then phase-shifted, where the right phase is determined empirically by optimizing the cooling performance, i.e., minimizing the position variance q 2 . The QPD or split-detection measurement yields a position signal for each spatial direction. The feedback signal is produced for each axis independently and the three signals are summed together, which allows to effectively cool all spatial degrees of freedom with a single laser beam (section 5.3). For a particle oscillating at frequency Ω 0 with q(t) ∝ cos(Ω 0 t), parametric feedback generates a signal ∝ sin(2Ω 0 ). Thus, the feedback signal appears on the in-loop detector signal predominantly at 2Ω 0 . The contribution to the detector signal at Ω 0 is much smaller, on the order of ∼ η para q 2 and, thus, much smaller than the contribution from the particle motion q 2 . Hence, noise squashing of the in-loop detector signal does not occur and the trapping laser can also be used for particle detection.\n\nDue to the nonlinear nature of the parametric feedback, there is no closed expression for the detector PSD. In particular, the position distribution is not a thermal distribution because large amplitude fluctuations are damped more strongly than small fluctuations [481,532]. However, we can still assign an effective temperature to the center-of-mass motion k B T cm = mΩ 2 0 q 2 . For strong feedback (η para mΩ 0 γ 0 /(k B T) ), the position variance is [481,532] q 2 = 4k B T γ 0 πη para mΩ 3 0 .\n\nIn contrast to linear-feedback cooling, the effective temperature or position variance scales with the square root of the feedback gain and not linearly. This can be understood intuitively. The Fig. 47. Feedback cooling. (a) Single-sided power spectral densitiesS Ool yy of the motion of the nanoparticle measured by the out-of-loop detector for different feedback damping rates γ fb . The solid lines are Lorentzian fits to the data. The gray data points denote the measured shotnoise level on the out-of-loop detector. (b) Power spectral densitiesS IL yy measured by the in-loop detector for the same settings as in (a). In contrast to (a), for a large feedback gain the measured signal drops below the noise floor because of correlations between the detector noise and the feedback signal (noise squashing) . (c) Mode temperature T y derived from the out-of-loop signal as in (a) function of feedback gain γ fb . The red (blue) circles denote the measured values at a pressure of 1.4 · 10 −8 mbar (1.2 · 10 −7 mbar. (d) Steady state under parametric PLL feedback cooling. Mean occupation number along the three principal axes (x, y, z) as a function of gas pressure measured under constant feedback cooling. At low enough pressures, photon recoil becomes the main source of heating and therefore the occupation number remains constant. Adapted from Ref. [415] (a-c) and from Ref. [416] (d). modulation depth and therefore the feedback force is proportional to the energy of the particle. Hence, as the feedback reduces the particle motion it also becomes less effective. Nonetheless, the quantum mechanical theory suggests that ground state-cooling with parametric feedback cooling is possible [533], even though direct feedback seem to be more promising [534].\n\n\nParametric PLL cooling\n\nParametric feedback extracts energy from or pumps energy into the particle center-of-mass motion by modulating the trapping laser and therefore the trap stiffness at twice the particle oscillation frequency. A phase locked loop (PLL) synthesizes an output signal whose phase is related to the phase of a sinusoidal input signal. Thereby, a PLL can track an input frequency and it can generate a frequency that is an integer multiple of the input frequency. Hence, tracking the detector signal with a PLL and feeding the PLL output at the second harmonic into the optical modulator results in a feedback term\nF PLL fb = −k PLL cos(2 [Ω 0 t + φ PLL ])q,(213)\nwhere k PLL = mΩ 2 0 η PLL is the PLL feedback gain and η PLL is the modulation depth. The PLL feedback loop keeps the phase φ PLL fixed with respect to the randomly changing phase of the particle oscillation. Similarly to the case of the q q parametric feedback, a phase of φ PLL = −π/4 reduces the oscillation amplitude. However, in contrast to the previous case, where the modulation depth is proportional to the variance of the particle position, the PLL feedback has a constant amplitude. As a consequence, the position variance with PLL parametric feedback is\nq 2 = k B T mΩ 2 0 1 − η PLL Ω 0 sin(2φ PLL ) 2γ 0 −1 ≈ γ 0 η PLL Ω 0 /2 k B T mΩ 2 0 .(214)\nThe approximation holds for φ PLL = −π/4 and in the limit η PLL Ω 0 /2γ 0 1. Comparing equation (214) with equation (202), we can identify the PLL damping as γ PLL = η PLL Ω 0 /2. Interestingly, even though parametric feedback with a PLL is a nonlinear feedback, we find the same linear scaling as for linear feedback and also the energy distribution follows the Boltzmann distribution associated with a harmonic oscillator at thermal equilibrium with a bath at temperature T cm = (γ 0 /γ PLL )T [532]. Like in the case of linear feedback, noise in the feedback will ultimately limit how far one can cool. However, unlike in the case of linear feedback (equation (210)), there is no analytical solution for the minimum temperature for parametric feedback. Using PLL parametric feedback simultaneous cooling of all three motional degrees of freedom to millikelvin temperatures (n below 100) has been demonstrated [416] (Fig. 47(d)).\n\n\nOutlook\n\nLevitated particles have already opened the door to a wide range of interesting physics in the classical regime and are now entering into the quantum regime. The next challenges are to achieve ground state cooling for all three center-of-mass translational degrees of freedom, to control the librational modes and precession of the particle, and to preparte truely non-classical states.\n\nEven though our discussion has been focused on optical tweezers in high vacuum, their combination with cavity optomechanics will play an important role towards creating more exotic quantum mechanical states beyond the ground state. Once a strong cavity optomechanical interaction has been realized, a range of traditional quantum optics experiments can be realized with these massive particles such as quantum state transfer, quantum squeezing, entanglement, and teleportation [413,432,436].\n\nThe envisioned fundamental tests of quantum mechanics require to control the internal temperature of levitated particles, since high internal temperature will wash out quantum interference effects [535]. The internal temperature can be reduced with optical refrigeration that exploit optical dopants inside the particle [476,536]. Dopants can also be used to directly measure the particle temperature [537]. Experimental data of the internal dynamics of levitated particles could challenge our understanding of thermalization dynamics on the nanoscale [538] and it will be important to measure what is going on inside the particle.\n\nThere is still plenty of room in the classical regime, where levitated particles will allow to test fundamental questions in single particle thermodynamics in the underdamped regime and provide new insights into the nonlinear dynamics of nanomechanical resonators. Most attention has been given to the center-of-mass degrees of freedom and simple Gaussian beams. There remains a lot to be explored about the remaining degrees of freedom and complex light fields, for example beams carrying orbital angular momentum [539]. In addition, new materials, material combinations in the form of doped particles or composite particles and new shapes will enable unprecedented light matter interactions [489]. Optical levitation experiments will hugely benefit from the tools that have been developed for traditional optical tweezers experiments, such as structured light fields [540] and holographic tweezers [43,541]. Those tools will then also allow to study collective particle dynamics, for example optical binding and forces between levitated particles [11]. This will be required the development of new techniques and calibration methods that are able to resolve the fast particle dynamics in high vacuum at a high spatial resolution over a large field of view.\n\nCareful calibration with a solid understanding of all the statistical and systematic error sources will be paramount for precision measurements with levitated particles. In particular, experiments aiming at detecting near field forces at interfaces require to approach a surface with a levitated particle [542]. The surface will scatter the trapping light and, thus, these experiments need detection and calibration techniques that are robust in extreme experimental conditions that might even change over the course of a measurement. We thank Erik Hebestreit, Vijay Jain, Martin Frimmer, Felix Tebbenjohanns, Francesco Ricci for providing us with data and feedback and the Quidant and Novotny groups for stimulating discussions. IPC gratefully thanks Édgar Roldán for illuminating discussions about Carnot cycles and microscopic engines.\n\n\nThermodynamics . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.4 Carnot cycles and Maxwell's demons . . . . . . . . . . . . . . . . . . 84 4.5.5 Other applications to statistical physics and outlook . . . . . . . . . . . 89 4.6 Transport in extended potential landscapes . . . . . . . . . . . . . . . . . . . . 89 4.6.1 Optical forces and potentials in extended light patterns . . . . . . . . . 90 4.6.2 Transport mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.6.3 Other transport mechanisms and applications . . . . . . . . . . . . . . 98 5 Optical tweezers in vacuum 100 5.1 Applications of optical levitation. . . . . . . . . . . . . . . . . . . . . . .. . 100 5.2 Dynamics in the underdamped regime . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Optical tweezers setup for vacuum operation . . . . . . . . . . . . . . . . . . . 103 5.3.1 Particle types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.2 Optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.4 Feedback electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.5 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Particle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.4.1 Nebuliser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.2 Dry approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Dissipation and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.1 Gas damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5.2 Radiation damping and shot noise . . . . . . . . . . . . . . . . . . . . 114 5.5.3 Displacement noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.4 Laser intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.1 Detector calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.2 Sideband thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.6.3 Charge calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6.4 Mass Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6.5 Internal temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.6.6 Coupling between internal temperature and external degrees of freedom 124 5.7 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.7.1 Force feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.7.2 Parametric feedback cooling . . . . . . . . . . . . . . . . . . . . . . . 129 5.7.3 Parametric PLL cooling . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131\n\nFig. 1 .\n1Trappable objects. A broad range of objects have been trapped using optical tweezers, going from single atoms and molecules to microparticles and microorganisms.\n\nFig. 2 .\n2Basic optical tweezers setup. A basic optical tweezers setup is comprised of three parts: the trapping optics, the imaging optics, and the position detection optics. DM: Dichroic mirror. M: mirror. O: microscope objective. C: Condenser. L 1 , L 2 , L 3 , L 4 : lenses. LS: Light source.\n\nFig. 3 .\n3Brownian motion in an optical trap. (a) Trajectory of an optically trapped Brownian particle x(t) and corresponding (b) optical potential and (c) probability density. The equilibrium position is x eq = 0 µm. The experimental parameters are in Tab. 1.\n\nFig. 5 .\n5Polarization-splitting optical tweezers. A typical Mach-Zehnder setup with polarizing beam splitters (PBS 1 and PBS 2 ) can be used for this purpose. Mirrors GM\n\nFig. 7 .\n7Interferometric optical traps. (a) Interferometric optical trap based on a Mach-Zehnder interferometer. (b-c) Different alternatives for the interferometric module. BS: Nonpolarizing beam splitter. RP: Right-angle prism. BB: beam block.\n\nFig. 8 .\n8Holographic optical tweezers. A spatial light modulator (SLM) alters the beam profile of an incoming optical beam This beam is then relayed by a series of lenses to an objective that then generates the desired light pattern at the sample. The SLM can be either placed (a) in a plane conjugated to the back focal plane of the objective, or (b) in a plane conjugated with the image plane of the objective.\n\nFig. 9 .\n9Phase masks and corresponding light intensity. Diffractive optical elements displayed on a phase-only SLM for the generation of: (a) a single optical trap laterally and axially shifted; (b) multiple traps distributed in space (the different brightnesses and sizes of the spots are due to the different positions of the corresponding traps along the vertical z-position); (c) a Laguerre-Gaussian beam of topological charge l = −2 laterally shifted; (d) a Bessel beam of topological charge l = 3.\n\n\n=1 recorded time series of particle's position during the m-th experiment with N samples {x (m) } L =1 uncorrelated time series at time intervals ∆t with ∆t = r∆t and r an integer κ (ex) x experimental estimate of the trap stiffness τ (ex) x experimental estimate of characteristic relaxation time D (ex) , D (th) experimental estimate and theoretical value of diffusion constant γ (ex) , γ (th) experimental estimate and theoretical value of friction coefficient T s total acquisition time for each experiment m\n\nFig. 12 .\n12Autocorrelation function (ACF) analysis. Comparison between the theoretical expression\n\nFig. 14 .\n14Single-molecule pulling experiments: miniTweezers setup. Schematic representations (a) of the miniTweezers setup, (b) of the microfluidics chamber (the flow goes from left to right, and the laser beams propagate perpendicularly to the surface of the chamber), (c) of the assembly procedure for the construction of the chamber, and (d) of the molecular configuration for the DNA hairpin pulling experiments (where the DNA hairpin is held between a bead held by a micropipette and a bead captured by the optical trap, depicted as an harmonic potential). Note that x h2 , x d , x h1 , and x b are distances along the y-axis.\n\n\nAD beads. The protocol for the preparation of the AD beads consists of three steps: (1) exchange the preservation buffer; (2) attach the anti-digoxigenin to the G-protein; (3) exchange the crosslinking buffer. The procedure (∼ 90 min) for the preparation of 1 ml of AD beads (which should be sufficient to perform experiments for several months) is: 1. (The first time the anti-digoxigenin batch is dissolved.) Add 200 µl of PBS (pH 7.4) to dilute the dry anti-digoxigenin (sheep polycolonal anti-dig antibody, Roche 1333 089). 2. Prepare the dimethyl pimelimidate (DMP) crosslinker buffer by mixing 50 mg DMP and 200 µl antibody crosslinker buffer (pH 7.4). The DMP crosslinker buffer needs to be freshly prepared every time the AD beads are synthesized.\n\n\nMolecular buffer.A typical molecular buffer for DNA pulling experiments consists of 10 mM Tris pH 7.513 (Trizma, Sigma Aldrich), 10 mM EDTA14 (EDTA, Sigma Aldrich), 1 M NaCl, and 0.01% NaN 3 (sodium azide, to avoid bacterial growth). To facilitate the molecular buffer preparation, it is convenient to prepare stocks of:1 M Tris pH 7.5 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 6.05 g of Tris-HCl. Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add a solution of 25% HCl until the solution reaches pH 7.5. For longer storage, a final auto-cleavage can be performed. EDTA 0.5 M pH 8.0 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 7.306 g of EDTA. Add NaOH to the solution (the EDTA does not dissolve in water if pH<7.5). Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add NaOH until the solution reaches pH 8.0. 5 M NaCl Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 14.49 g of NaCl. Dissolve using a magnetic mixer and heat the tube to facilitate it. Add Milli-Q water until reaching 50 ml. 1% NaN 3 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 0.5 g of NaN 3 . Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. After the stocks have been prepared, for preparing 50 ml of the molecular buffer: Pour ∼ 30 ml of Milli-Q water in a 50 ml Falcon tube. Add 10 ml of the 5 M NaCl stock, 1 ml of the 0.5 M EDTA pH 8.0 stock, 0.5 ml of the 1 M Tris pH 7.5 stock, and 0.5 ml of the 1% NaN 3 stock. Add Milli-Q water until reaching 50 ml of volume. Mix the molecular buffer and filter it (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International) introducing the filtered solution in a new 50 ml Falcon tube. Incubation of the beads. To finalize the preparation of the beads, incubate in an Eppendorf tube (∼ 1.5 ml) 1 µl of a solution of the biomolecule of interest15 mixed with 14 µl of buffer (where the experiments are going to be performed) and 5 µl of the previously prepared AD beads. After ∼ 25 min have passed, add 1 ml of the molecular buffer.\n\n\nbiotin-AGT TAG TGG TGG AAA CAC AGT GCC AGC GCG CGA GCC ATA AT-3 CD4-HandleB 5 -Phos-CTC ATC TGG AAA CAG ATG AGA TTA TGG CTC GCG ACT TCA CTA ATA CGA CTC ACT ATA GGG A -3 splint 5 -GCG CTG GCA CTG TGT TTC CAC CAC TAA CT-3\n\n6 .\n6Oligonucleotide tailing reaction. Mix all the items in an Eppendorf tube and keep it at 37 • C for 15 min. After the reaction is finished, add 1 to 2 µl of 0.5 M EDTA pH 8.0 to quench the reaction. (*) Included in the Terminal transferase kit from Sigma Aldrich.\n\nFig. 16 .\n16Free-energy of hairpin formation. (a) Hairpin pulling trajectories shown as force-distance curves. The unfolding cycles are plotted in red and the folding cycles in blue. The black dots indicate the initial (L 0 , f min ) and final (L 1 , f max ) points considering for computing the work W FU . Since the L values are obtained from the light position at the position-PSD, the L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero (which corresponds to the center of the position-PSD). (b) Probability density functions (PDFs) of the work, computed using equation(75), for the unfolding (W U , red histograms) and folding (W F , blue histograms) cycles. The value at which the probabilities cross (dashed vertical line) is the free energy difference between the folded and unfolded states ∆G FU . This value is obtained as the crossing point between the lines interpolating the histograms (blue and red solid lines).\n\n\n(b) is obtained by taking L 0 = −140 nm ( f min = 11.35 pN) and L 1 = −60 nm ( f max = 17.16 pN). Note that L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero. The crossing point of the histograms gives ∆G FU = 348.67 k B T (with 1k B T = 4.114 pN nm at T = 25 • C).\n\n\n1 M K phosphate buffer at 7.4 pH is obtained by mixing 40.1 ml of 1 M K 2 HPO 4 and 9.9 ml of 1 M KH 2 PO 4 . The 1 M Na Phosphate buffer at 7.4 pH is obtained by mixing 38.7 ml of 1 M Na 2 HPO 4 and 11.3 ml of 1 M NaH 2 PO 4 . All salts are available from Sigma-Aldrich.\n\n3 .\n3Fill the syringe connected to the dispenser tube selected in the previous point with the bead solution. Suck about half of the Eppendorf tube content with the syringe. 4. Fill the other syringe (upper/lower channel) with RBC solution. Suck about half of the Eppendorf tube content with the syringe.\n\nFig. 17 .\n17Construction of the cellular configuration. Images of different steps to obtain the RBC configuration required for the experiment: (a) the RBC is attached to a bead and partially trapped by the lasers; (b) the bead is trapped, while the RBC is not trapped; (c) the second bead is attached to the other side of the RBC; (d) final experimental configuration at zero reading for the force along the y-direction. The diameter of the beads is 3 µm. The red dot represents the position of the optical trap. The red arrow in (a) and (b) represents the presence and direction of the fluid flow.\n\n18 .\n18Do not turn off the pump flow in the central channel. To minimize the possibility of losing the bead attached to the RBC fixed on the tip of the pipette, do the following steps as quickly as possible. 19. Halve the flow to facilitate the trapping of the second bead. Check that the RBC stays separated from the bead, if this is not the case, increase the flow again to separate the cell from the bead. 20. Move the optical trap to the exit of the bead dispenser tube. 21. Flow the bead solution and try to trap a bead. It will be harder due to the RBC buffer flow on the central channel. To catch a bead with the pump on, it is easier to stay at the exit of the dispenser tube without moving the trap. Wait for the bead to fall into the optical trap instead of actively moving the trap around. 22. Once you have trapped a bead, move the chamber to the working area.\n\nFig. 19 .\n19Force-cellular-extension curves and RBC stiffness. (a) Force-cellular-extension curves for two different RBCs. (b) Stiffness as a function of force obtained from the slope of the force-cellular-extension curves.\n\nFig. 21 .\n21Determination of zero-shear viscosities. (a) Trajectories of particles of distinct diameters trapped in different fluids. From top to bottom: water (2a = 2.73 µm), PNP (2a = 3.25 µm), and aqueous micellar solution of CPyCl/NaSal at 4 mM (2a = 3.25 µm). (b) Autocorrelation functions of the particle position for the cases in water (•), PNP ( ), and micellar solution ( ). The dashed lines correspond to exponential fits. Inset: semilog representrations of the position autocorrelation function for the three cases, normalized by their corresponding variances x(t) 2 = k B T κ .\n\nFig. 22 .\n22Determination of storage and loss moduli. (a) Power spectral density of equilibrium fluctuations of x(t) for a particle trapped by optical tweezers in a wormlike micellar solution of CPyCl/NaSal at 5 mM. Inset: time evolution of x(t) over 10 s. (b) Storage modulus (red •) and loss modulus (blue ) of the wormlike micellar solution determined by passive microrheology. means of\n\nFig. 23 .\n23Active microrheology. (a) Power spectral density of the particle position, driven at different frequencies by a sinusoidal motion of the optical trap. The peaks are located at the imposed driving frequencies f d . Inset: Time evolution of the perturbative force divided by the trap stiffness at f d = 1 Hz (thick solid line) and resulting particle position (thin solid line). (b) Numerical values of the storage modulus (red •) and loss (blue ) obtained by means of equations(90)and(95). The dotted and dashed lines depict the corresponding curves shown inFig. 22(d) obtained by passive microrheology.\n\nFig. 24 .\n24Trajectories and potentials with critical Casimir forces. (a-d) Time evolution of the probability density distribution (coloured background) of the inter-particles distance r(t) obtained from 400 different trajectories after the optical traps are switched off at t = 0 ms for decreasing values of ∆T: (a) ∆T = 456 ± 2 mK, (b) 200 ± 2 mK, (c) 163 ± 2 mK, and (d) 108 ± 2 mK. The solid lines indicate representative individual trajectories. The dashed horizontal line indicates the distance r corresponding to the diameter d of the colloids.\n\nFig. 25 .\n25Parallel and perpendicular diffusion coefficients. Experimental values of the normalized parallel D /D 0 (circles) and perpendicular D ⊥ /D 0 (triangles) diffusion coefficients as functions of the normalized inter-particle distance r/d. The parallel and perpendicular directions refer to the line connecting the centers of the two colloids, and D 0 is the bulk diffusion constant (equation\n\nFig. 26 .\n26Parallel and perpendicular drifts. Drift velocities parallel v and perpendicular v ⊥ to the direction connecting the centers of the two colloids for (a, b) ∆T = 456 ± 2 mK, (c, d) 200 ± 2 mK, (e, f) 163 ± 2 mK and (g, h) 108 ± 2 mK. The symbols with errorbars represent the experimental data, the colored lines represent the corresponding simulation results, and the shaded areas represent the error of the numerical estimates due to the uncertainties in the fit parameters.\n\n\n), v is positive at small values of r/d due to the repulsive electrostatic potential V es , which is dominant in this range and pushes the particles away from each other, while it rapidly vanishes for increasing r/d, because the electrostatic repulsion decays exponentially with the inter-particles distance and no other force is affecting the motion of the colloids. Decreasing ∆T (Figs. 25(c) and 25(e)), v becomes negative within a certain range of values of r/d, due to the arising of attractive critical Casimir forces; however, at smaller values of r/d, V es is dominant and v is still positive. If ∆T is reduced further (T is very close to T c ,Fig. 25(g)), critical Casimir forces became so strong that v presents only negative values because the particles move towards each other until their velocity vanishes at contact; at distances larger than the range of the critical Casimir forces action range, instead, v vanishes and the particles undergo Brownian diffusion.26 InFigs. 25(b,d,f,h), we report the experimental and numerical values for the orthogonal component v ⊥ of the drift velocity (equation\n\nFig. 27 .\n27Kramers' rates. (a) Schematic of a bistable potential generated with a doublebeam optical tweezers. A silica bead is optically trapped with two-orthogonally-polarized and tightly-focused Gaussian beams separated by a distance d = 0.8 µm along the x-direction and focused by a water-immersion objective (NA = 1.2) using an inverted microscope. (b) Trajectory of a Brownian particle (radius r p = 0.48 µm) in an aqueous solution (temperature T = 22 • C) in the bistable potential. The particle spends most of the time near the equilibrium points (A and B, whose separation is measured to be d AB = 0.71 µm), but from time to time it is thermally activated to the saddle point (S) with a mean rate defined by equations(108)and (110). (c) Reconstruction of the force field measured with FORMA (arrows, section 3.8) and of the potential energy measured with the potential analysis (background color, section 3.2). (d-g) Optical potential (d) along the x-direction and (e-f) along the y-direction measured using FORMA. (h-i) Residence time probability in wells (h) A and (i) B. The red curves show the exponential probability function defined by equation (111), with the mean residence times estimated from the experimental data τ A = 3.03 s and τ B = 0.67 s. approximately 10 µm above the bottom coverslip of the sample cell and 90 µm from the top one, so that the hydrodynamic interaction with the walls of the cell is negligible.\n\n\nand the fitted potential parameters are in very good agreement with those directly measured from the escape time of the trajectories trajectories (1/τ (ex) ). equilibrium points are easily identified as the positions where the magnitude of the force field has local minima. The potential profile along the x-direction is shown in Figs. 27(d) and those along the y-direction at the critical points in Figs. 27(e-g).\n\n\n(d) NESS correlation functions involved in the integral form of the generalized FDT (119): C(0) − C(t) (dotted-dashed line), B(t) (dashed line) and C(0) − C(t) − B(t) (solid line). Inset: estimate of the response function R(t) by means of the time derivative of [C(0) − C(t)]/(k B T) (dotted-dashed line), and taking into account the corrective term, [C(0) − C(t) − B(t)]/(k B T) (solid line).\n\nFig. 29 .\n29First Law in stochastic thermodynamics. (a) Upper panel: Portion of the trajectory x(t) (blue solid line) of a colloidal bead (2a = 2.73 µm), dragged by moving an optical tweezers (k = 1.15 pN µm −1 , v = 0.520 µm s −1 ) through water (T = 295 K). The dashed line represents the time evolution of the minimum of the trapping potential: λ(t) = vt + λ(0). Lower panel: time evolution of λ(t) − x(t). (b) Dependence of the mean work W τ (•), mean heat dissipated into the aqueous medium Q τ ( ), and mean variation of potential energy ∆U τ ( ) as a function of the measurement time τ. The solid line represents the values given by equation (133). (c) Dependence of the standard deviation of the work (•), of the heat dissipated into the aqueous medium ( ), and of the variation of potential energy ( ) as a function of τ. The solid line represents the values given by the square roots of equation (134). (d) Distribution of the work (in units of k B T) for different values of the measurement time τ. From inner to outer curves: τ = 0.034 s, 0.069 s, 0.103 s, 0.138 s, 0.172 s, and 0.207 s. (e) Distribution of the dissipated heat (in units of k B T) for different values τ. Same color code as in (d). Inset: verification of the integral fluctuation theorem (139) for the total entropy production. (f) Distribution of the variation of potential energy (in units of k B T) for different values τ. Same color code as in (d).\n\nFig. 30 .\n30Microscopic engines. (a) A sketch of Carnot's cycle in the (S, T)-plane and (b) in the (V, P)-plane. (c) The Stirling cycle in the (V, P)-plane. (d) Implementation of the Stirling cycle using a colloidal particle in an optical trap.\n\nFig. 31 .\n31Energy potentials and trapping positions in a pattern of fringes. The force in a pattern of fringes has a sinusoidal shape with the same period as the fringe pattern. (a) Force amplitude normalized to the average intensity, Q = A/2I 0 , as a function of the period of the fringes, Λ, for two silica particle of different diameters: D p = 2 µm (blue) and 4 µm (red). Negative (positive) values of Q give rise to a potential with energy minima at the brightest (dark) regions of the pattern. (b-c) Optical potentials U(x) for the particles with (b) D p = 2 µm (blue) and (c) D p = 4 µm for a pattern with period Λ x = 2 µm (corresponding to the dashed vertical line in (a)). (d) Variation of Q as a function of the ratio Λ x /D p for microscopic beads made of silica and polystyrene, with respective refractive indices of n p = 1.46 and n p = 1.60 (values defined at λ = 532 nm). The red and blue vertical dashed lines correspond to the values of the ratio Λ x /D p for the examples shown in (b) and (c), respectively. The optical forces and potentials were computed using the ray optics model.\n\nFig. 32\n32Fig. 32. Particle-dependent potential or size effect in asymmetric patterns. Asymmetric pattern of light with period Λ = 4 µm (background) and the resulting optical potentials, U a (x) and U b (x), for two silica beads with diameters D a = 2 µm and D b = 4 µm, respectively. The shape of the optical potential for a given pattern of light can be very different depending on the size of the particle. The potential was computed using the ray optics model.\n\nFig. 33 .\n33Schemes for the generation of a moving pattern of fringes using a Mach-Zehnder interferometer. (a) By exploiting the angular Doppler effect with a λ/2 plate rotating at a constant angular speed Ω. (b) By gradually tilting a glass plate a small angle θ(t). (c) By means of a linear displacement D(t) = vt of a mirror. (d) By a periodic and asymmetric displacement of a mirror, according to a sawtooth function D(t). PBS: polarizing beam splitter. M: mirror. BB: beam block. BS: non-polarizing beam splitter. L: lens.\n\nFig. 34 .\n34Current in an optical sieve. (a) Mean current as a function of the period of the fringes Λ for particles with diameter D p = 2 µm and 5 µm for a fixed optical power P = 130 mW and frequency f = 1 Hz. (b) Mean current as a function of the angular\n\nFig. 36 .\n36Optical trapping in vacuum. Photo of the sample chamber of an optical levitation setup. An optically trapped particle is shown in the inset. Figure adapted from Ref.[486].\n\nFig. 37 .\n37Simplified setup for optical trapping in vacuum. Schematic of a setup for the trapping of particles in vacuum, based on the setup employed in Ref. [485]. EOM: electrooptic modulator. O: objective. PLL: Phase-locked loop. QPD: quadrant photodetector.\n\nFig. 39 .\n39Gas composition in a vaccum chamber. Gas composition in the vacuum chamber at pressures below 10 −3 mbar plotted over time. The solid lines show the partial pressures of different gas species measured with a residual gas analyzer (RGA). The sum of all the partial pressures yields the total pressure at the RGA (dashed line). Due to the configuration of the vacuum system and the reduced pumping speed at the RGA, the pressure at the RGA deviates from the pressure in the main vacuum chamber (dash-dot line). The initial rise in partial pressures after turning on the RGA is attributed to the warm-up process and desorption of gases from the filament of the RGA. Reproduced from Ref.[471].\n\nFig. 40 .\n40Particle loading with a nebulizer. The particle is loaded by spraying a solution of nanoparticles through a nozzle which is placed above the focus. (a) Positioning of the nozzle in the vacuum chamber. (b) Nozzle to funnel the falling particles towards the focus of the trapping laser. (c) Histogram of brightness observed with a camera from the side. The inset shows the brightness over a wider range. Reproduced from Ref. [488] (a,b) and adapted from Ref. [486] (c).\n\nFig. 41 .\n41Calibrated and uncalibrated z-detector signals. (a) Calibrated z-detector signal at 10 ± 1 mbar with electric driving response. The area under that peak corresponds to the thermal temperature k B T. (b) Uncalibrated feedback-cooled oscillator signal with electric drive for recalibration of the signal.\n\nFig. 42 .\n42Sideband thermometry. (a-b) Motional sideband asymmetry measured with heterodyne measurement. The frequency difference ∆ f is measured relative to the (absolute) local oscillator shift of 1 MHz. The gray solid areas show the measurement noise floor, limited by technical laser noise. The vertical dashed lines indicate the integration bandwidth (see equation 186). (c) Mean occupation number as a function of feedback gain. The red diamonds are obtained by integrating the red sideband of the heterodyne spectrum. The black circles show the mean occupation number extracted according to equation (187). The black dashed line corresponds to a parameter-free model (see also section 5.7.1). Figures adapted from Ref.[418].\n\nFig. 43 .\n43Charge calibration. (a) Power spectral density of the motion along the optical axis of a charge-carrying particle at a pressure of 1.9 mbar in the presence of a drive tone f d = ω d /2π applied to the capacitor. The solid line is a Lorentzian function fit to the data. (b) Quadrature component of particle oscillation in response to a driving voltage, demodulated in a bandwidth of 7 Hz. The high-voltage discharge is turned on at t = 0. The oscillation amplitude changes in discrete steps while the high voltage is on. (c) Preparation of charge state. The high voltage is turned off at t = 0, while the particle carries a net charge of 1e. The charge stays constant over the remainder of the measurement. Figure adapted from Ref.[523].\n\nFig. 46 .\n46Particle heating.\n\n\n2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = 2 (ω 2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = [2γ] −1 .\n\nFunding\nJG is supported by the European Commission H2020-MSCA-IF-2014 (SEQOO 655369). IPC acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IA103417. JRGS. acknowledges financial support from DGAPA-UNAM PAPIIT Grant No. IA103320. AVA acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IN111919. KVS acknowledges support from DGAPA-UNAM (grant PAPIIT IN113419 and PASPA grant for sabbatical leave). AVA and GV acknowledge the Swedish Council for Higher Education through the Linnaeus-Palme International Exchange Program (contract 3.3.1.34.10235-2018). GV acknowledges funding from the European Research Council (ERC) Starting Grant ERC-677511 ComplexSwimmers.\n\n\n1.03 µmLaboratory temperature \nT = 20 • C = 293.15 K \n\nSampling frequency \nf s = 10 5 Hz \n\nSampling timestep \n∆t = f −1 \ns = 10 −5 s \nviscosity (water) \nη = 1.00 mPa s \n\nfriction coefficient (Stokes' law (6)) \nγ (th) = 19.4 pN ms µm −1 \n\ndiffusion \ncoefficient \n(Einstein-\nSmoluchowski relation (7)) \n\nD (th) = 0.21 µm 2 s −1 \n\nnumber of experimental trajectories \nM = 5 \n\nnumber data points for each experiment \nN = 10 6 \n\nobjective lens \nOlympus UPLSAPO 60× 1.2 NA water im-\nmersion \n\nlaser wavelength \n1064 nm \n\nlaser power at the sample for experiment I, \nII, and III \n\n2.3 mW, 6.0 mW, and 9.2 mW \n\n\n\nTable 3 .\n3Experimental results. Experimental values for the trap stiffness κ in [pN µm −1 ], the friction coefficient γ in [pN ms µm −1 ], the diffusion constant D in [µm 2 s −1 ], the τ ot, x in [ms], and f in [Hz] inferred using various calibration methods on the data from the experiments I, II and III (Table 1, note, in particular, the expected values γ (th) = 19.4 pN ms µm −1 and D (th) = 0.21 µm 2 s −1 ). The table is divided in two parts corresponding to the methods requiring fitting a function (linear fit / non-linear fit) and to those not requiring it (direct estimation).(ex) \n\n(c, x) \n\nTable 4 .\n4Notation. In this table we summarize the notation related to the acquisition of the time series and that of theoretical and inferred values of the parameters of the optical trap and that of the particle.Given a series of trajectories {x (m) } N =1 with m = 1, . . . M, the experimental distribution ρ (ex) ). The blue markers are the experimental measurements, the solid black lines are the linear fitting, and the dashed red lines are the non-linear fitting. The black circles are the equilibrium distribution and trap potential obtained with the equipartition method (section 3.3). The estimates for the trap stiffness κ x are given inm \n\nx; {x (m) } N \n=1 is constructed as follows: \n\n0 \n\n0.02 \n\n0.04 \n\n0.06 \n\nEquipartition \n\nLinear fiting \n\nNon-linear fitting \n\nExperimental \n\n-50 \n0 \n50 \n\n0 \n\n5 \n\n10 \n\n-50 \n0 \n50 \n-50 \n0 \n50 \n\nFig. 10. Potential analysis. Comparison between theoretical expressions and experimental \nestimates for (a-c) the equilibrium distribution and (d-f) the trap potential for a particle \noptically trapped at three different powers: (a, d) 2.3 mW, (b, e) 6.0 mW, and (c, f) 9.2 mW \n(Table 1\n\nTable\n\n\nTable 8 .\n8Ligation reaction. Mix all the items in an Eppendorf tube and keep them at 16 • C overnight. Afterwards, keep the Eppendorf tube at 65 • C for 10 min to inactivate the ligase.Annealing product \n30 µl \n\n10× T4 DNA ligase buffer \n10 µl \n\n10 mM ATP \n5 µl \n\nT4 DNA ligase (400 U µl −1 ) \n3 µl \n\nTotal Volume \n100 µl \n\n\n\nTable 11 .\n11Extended light patterns.\n. The three channels are drawn in the two parafilm layers using a laser cutter.3. One parafilm layer is attached on the drilled glass coverslip. 4. A glass micropipette with a diameter of ∼ 1 µm is produced by heating and pulling a glass 7Although 100% ethanol can also be used, a 70% solution is preferred because it prevents the sporulation of some microorganisms.\n11Milli-Q water is obtained by filtering the source water (usually distilled water) first through mixed bed ion exchange and organics (activated charcoal) cartridges, and then through a filter which removes any intact organisms. Usually a UV lamp completes the purification process.\n25The same discrepancy can be observed on simulated data considering trajectories sampled with the same time step t s as that used in the experiment. Furthermore, reducing significantly the time step t s in the simulations, the discrepancy with the theoretical line is much less pronounced and eventually disappears as t s → 0.26In proximity of T c , the range of distances larger than the critical Casimir forces action range can actually be explored only via numerical simulations with sufficiently high statistics. In the experiment, instead, the particles almost always stick together.\n30In general, φ(θ) and Q(θ) must not necessarily be the same: while φ(θ) represents the normalized potential energy of the particle, Q(θ) is an arbitrary observable that is periodic with respect to thet a. Since phi(θ) fullfils in particular the required periodicity, in the example described in the main text we choose Q = phi.\nThis is a linear function of the gas pressure (see also equation(171)andFig. 46a). Hence, the slope of the measured heating rates as a function of the gas pressure yields T em from which we can then determine T bulk[485].Note that knowledge of the accommodation factor c acc is required to determine the particle temperature. It can be determined by measuring T em at two different particle temperatures. The particle temperature can be raised by increasing the laser power to which the particle is exposed. Ideally, the additional heating is generated with a weakly focused secondary laser at a wavelength that is strongly absorbed by the particle, e.g., a CO 2 laser for silica particles, which has 6 to 7 orders more absorption than that typical trapping laser. Thereby, the internal temperature of the particle increases without changing the particle response, as would be the case when increasing the power of the trapping laser. (The scattering force is negligible because the intensity is many orders of magnitude smaller.) From the two measurements, we get the difference ∆T em = c acc ∆T bulk . The temperature change ∆T bulk is measured independently from the observation that the particle oscillation frequencydepends on temperature because both the particle refractive index n and its density ρ p are temperature dependent. For silica, this leads to a linear increase of the oscillation frequency with the internal particle temperature. The relative temperature change is given bywith c T = (1.43 ± 0.01) · 10 −5 K −1(Fig. 46c). Using this method, Hebestreit et al.[485,525]measured the accommodation coefficient to be c acc = 0.61±0.07. This allowed them to determine the accommodation coefficient of water on silica for the first time combining this measurement with the measurement of the partial pressures(Fig. 39).Feedback controlOne of the primary goals in optical levitation is to reduce the center-of-mass energy and cool the motional degrees of freedom of a levitated particle to its quantum mechanical ground state (∼ 5 µK for Ω 0 ≈ 100 kHz); the original proposals suggested the use of optical cavities and to cool the particle motion via the optomechanical interaction[412][413][414]. Initial experimental attempts were limited to relatively high pressures due to particle loss, where the interaction with the residual gas is too strong to achieve ground-state cooling[526]. Recently, cavity cooling by coherent scattering was demonstrated as a robust route to three-dimensional cavity cooling primarily limited by fluctuations of the trap center with respect to the cavity, i.e., displacement noise[419,420]. The first demonstration of quantum mechanical motion has been achieved with feedback cooling[418]with a residual occupation ofn = 4, and recently an occupation below n = 1 has been claimed with cavity cooling[424].\nSelect the minimum force from which to start stretching the RBC. 190 pNSelect the minimum force from which to start stretching the RBC (e.g., 0 pN).19\n\nSelect the maximum force at which to stop the RBC stretching. 2020 pNSelect the maximum force at which to stop the RBC stretching (e.g., 20 pN).20\n\n140 nm s −1 ). Select the pulling velocity. Select the pulling velocity (e.g., 140 nm s −1 ).\n\nAcceleration and trapping of particles by radiation pressure. A Ashkin, Phys. Rev. Lett. 24A. Ashkin, \"Acceleration and trapping of particles by radiation pressure,\" Phys. Rev. Lett. 24, 156-159 (1970).\n\nObservation of a single-beam gradient force optical trap for dielectric particles. A Ashkin, J M Dziedzic, J E Bjorkholm, S Chu, Opt. Lett. 11A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, \"Observation of a single-beam gradient force optical trap for dielectric particles,\" Opt. Lett. 11, 288-290 (1986).\n\nOptical trapping and manipulation of neutral particles using lasers: A reprint volume with commentaries. A Ashkin, World ScientificA. Ashkin, Optical trapping and manipulation of neutral particles using lasers: A reprint volume with commentaries (World Scientific, 2006).\n\n. Arthur Ashkin -Nobel Lecture. \"Arthur Ashkin -Nobel Lecture,\" https://www.nobelprize.org/prizes/physics/2018/ashkin/ lecture/. NobelPrize.org. Nobel Media AB 2019. Thu. 4 Jul 2019.\n\nNobel Lecture: The manipulation of neutral particles. S Chu, Rev. Mod. Phys. 70S. Chu, \"Nobel Lecture: The manipulation of neutral particles,\" Rev. Mod. Phys. 70, 685-706 (1998).\n\nForce generation of organelle transport measured in vivo by an infrared laser trap. A Ashkin, K Schütze, J M Dziedzic, U Euteneuer, M Schliwa, Nature. 348A. Ashkin, K. Schütze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, \"Force generation of organelle transport measured in vivo by an infrared laser trap,\" Nature 348, 346-348 (1990).\n\nScanning-force microscope based on an optical trap. L P Ghislain, W W Webb, Opt. Lett. 18L. P. Ghislain and W. W. Webb, \"Scanning-force microscope based on an optical trap,\" Opt. Lett. 18, 1678-1680 (1993).\n\nBead movement by single kinesin molecules studied with optical tweezers. S M Block, L S B Goldstein, B J Schnapp, Nature. 348S. M. Block, L. S. B. Goldstein, and B. J. Schnapp, \"Bead movement by single kinesin molecules studied with optical tweezers,\" Nature 348, 348-352 (1990).\n\nEntropic elasticity of lambda-phage DNA. C Bustamante, J F Marko, E D Siggia, S Smith, Science. 265C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith, \"Entropic elasticity of lambda-phage DNA,\" Science 265, 1599-1599 (1994).\n\nSingle myosin molecule mechanics: Piconewton forces and nanometre steps. J T Finer, R M Simmons, J A Spudich, Nature. 368J. T. Finer, R. M. Simmons, and J. A. Spudich, \"Single myosin molecule mechanics: Piconewton forces and nanometre steps,\" Nature 368, 113-119 (1994).\n\nA revolution in optical manipulation. D G Grier, Nature. 424D. G. Grier, \"A revolution in optical manipulation,\" Nature 424, 810-816 (2003).\n\nColloquium: Gripped by light: Optical binding. K Dholakia, P Zemánek, Rev. Mod. Phys. 82K. Dholakia and P. Zemánek, \"Colloquium: Gripped by light: Optical binding,\" Rev. Mod. Phys. 82, 1767-1791 (2010).\n\nShaping the future of manipulation. K Dholakia, T Čižmár, Nat. Photon. 5K. Dholakia and T. Čižmár, \"Shaping the future of manipulation,\" Nat. Photon. 5, 335-342 (2011).\n\nPlasmon nano-optical tweezers. M L Juan, M Righini, R Quidant, Nat. Photon. 5M. L. Juan, M. Righini, and R. Quidant, \"Plasmon nano-optical tweezers,\" Nat. Photon. 5, 349-356 (2011).\n\nTweezers with a twist. M Padgett, R Bowman, Nat. Photon. 5M. Padgett and R. Bowman, \"Tweezers with a twist,\" Nat. Photon. 5, 343-348 (2011).\n\nHolographic optical tweezers and their relevance to lab on chip devices. M Padgett, R Di Leonardo, Lab Chip. 11M. Padgett and R. Di Leonardo, \"Holographic optical tweezers and their relevance to lab on chip devices,\" Lab Chip 11, 1196-1205 (2011).\n\nOptical trapping and manipulation of nanostructures. O M Maragò, P H Jones, P G Gucciardi, G Volpe, A C Ferrari, Nat. Nanotech. 8O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, \"Optical trapping and manipulation of nanostructures,\" Nat. Nanotech. 8, 807-819 (2013).\n\nRaman spectroscopy of optically trapped particles. D V Petrov, J. Opt. A. Pure. Appl. Op. 9139D. V. Petrov, \"Raman spectroscopy of optically trapped particles,\" J. Opt. A. Pure. Appl. Op. 9, S139 (2007).\n\nColloidal heat engines: A review. I A Martínez, É Roldán, L Dinis, R Rica, Soft Matter. 13I. A. Martínez, É. Roldán, L. Dinis, and R. Rica, \"Colloidal heat engines: A review,\" Soft Matter 13, 22-36 (2017).\n\nLevitated nanoparticles for microscopic thermodynamics: A review. J Gieseler, J Millen, Entropy. 20326J. Gieseler and J. Millen, \"Levitated nanoparticles for microscopic thermodynamics: A review,\" Entropy 20, 326 (2018).\n\nOptical tweezers microrheology: From the basics to advanced techniques and applications. R M Robertson-Anderson, ACS Macro Lett. 7R. M. Robertson-Anderson, \"Optical tweezers microrheology: From the basics to advanced techniques and applications,\" ACS Macro Lett. 7, 968-975 (2018).\n\nOptical tweezers study life under tension. F M Fazal, S M Block, Nat. Photonics. 5F. M. Fazal and S. M. Block, \"Optical tweezers study life under tension,\" Nat. Photonics 5, 318-321 (2011).\n\nP H Jones, O M Maragò, G Volpe, Optical tweezers: Principles and applications. Cambridge University PressP. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: Principles and applications (Cambridge University Press, 2015).\n\nOptical tweezers computational toolbox. T A Nieminen, V L Y Loke, A B Stilgoe, G Knöner, A M Brańczyk, N R Heckenberg, H Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, \"Optical tweezers computational toolbox,\" J. Opt. A: Pure Appl. Opt. 9, S196-S203 (2007).\n\nComputational toolbox for optical tweezers in geometrical optics. A Callegari, M Mijalkov, A B Gököz, G Volpe, J. Opt. Soc. Am. B. 32A. Callegari, M. Mijalkov, A. B. Gököz, and G. Volpe, \"Computational toolbox for optical tweezers in geometrical optics,\" J. Opt. Soc. Am. B 32, B11-B19 (2015).\n\nSimulation of a Brownian particle in an optical trap. G Volpe, G Volpe, Am. J. Phys. 81G. Volpe and G. Volpe, \"Simulation of a Brownian particle in an optical trap,\" Am. J. Phys. 81, 224-230 (2013).\n\nInexpensive optical tweezers for undergraduate laboratories. S P Smith, S R Bhalotra, A L Brody, B L Brown, E K Boyda, M Prentiss, Am. J. Phys. 67S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, \"Inexpensive optical tweezers for undergraduate laboratories,\" Am. J. Phys. 67, 26-35 (1999).\n\nFaster, cheaper, safer optical tweezers for the undergraduate laboratory. J Bechhoefer, S Wilson, Am. J. Phys. 70J. Bechhoefer and S. Wilson, \"Faster, cheaper, safer optical tweezers for the undergraduate laboratory,\" Am. J. Phys. 70, 393-400 (2002).\n\nA simple laser cooling and trapping apparatus for undergraduate laboratories. A S Mellish, A C Wilson, Am. J. Phys. 70A. S. Mellish and A. C. Wilson, \"A simple laser cooling and trapping apparatus for undergraduate laboratories,\" Am. J. Phys. 70, 965-971 (2002).\n\nOptical trapping for undergraduates. D C Appleyard, K Y Vandermeulen, H Lee, M J Lang, Am. J. Phys. 75D. C. Appleyard, K. Y. Vandermeulen, H. Lee, and M. J. Lang, \"Optical trapping for undergraduates,\" Am. J. Phys. 75, 5-14 (2007).\n\nConstruction and calibration of an optical trap on a fluorescence optical microscope. W M Lee, P J Reece, R F Marchington, N K Metzger, K Dholakia, Nat. Prot. 23226W. M. Lee, P. J. Reece, R. F. Marchington, N. K. Metzger, and K. Dholakia, \"Construction and calibration of an optical trap on a fluorescence optical microscope,\" Nat. Prot. 2, 3226 (2007).\n\nMultimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: upgrading a commercial confocal inverted microscope. M Mathew, S I C O Santos, D Zalvidea, P Loza-Alvarez, Rev. Sci. Instrumen. 8073701M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, \"Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: upgrading a commercial confocal inverted microscope,\" Rev. Sci. Instrumen. 80, 073701 (2009).\n\nStep-by-step guide to the realization of advanced optical tweezers. G Pesce, G Volpe, O M Maragó, P H Jones, S Gigan, A Sasso, G Volpe, J. Opt. Soc. Am. B. 32G. Pesce, G. Volpe, O. M. Maragó, P. H. Jones, S. Gigan, A. Sasso, and G. Volpe, \"Step-by-step guide to the realization of advanced optical tweezers,\" J. Opt. Soc. Am. B 32, B84-B98 (2015).\n\nOptical tweezers: A comprehensive tutorial from calibration to applications. L García, A Magazù, G Pesce, J Gieseler, J R Gomez-Solano, I Castillo, M Gironella-Torrent, X Viader-Godoy, F Ritort, A V Arzola, K Volke-Sepulveda, G Volpe, OnlineL. Pérez García, A. Magazù, G. Pesce, J. Gieseler, J. R. Gomez-Solano, I. Castillo, M. Gironella-Torrent, X. Viader- Godoy, F. Ritort, A. V. Arzola, K. Volke-Sepulveda, and G. Volpe, \"Optical tweezers: A comprehensive tutorial from calibration to applications,\" https://github.com/softmatterlab/OpticalTweezersTutorial (2020). Online.\n\nMechanical equivalence of spin and orbital angular momentum of light: an optical spanner. N B Simpson, K Dholakia, L Allen, M J Padgett, Opt. Lett. 22N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, \"Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,\" Opt. Lett. 22, 52-54 (1997).\n\nApplications of laser tweezers in complex fluid rheology. E M Furst, Curr. Opn. Colloid Interf. Sci. 10E. M. Furst, \"Applications of laser tweezers in complex fluid rheology,\" Curr. Opn. Colloid Interf. Sci. 10, 79-86 (2005).\n\nThe optical stretcher: a novel laser tool to micromanipulate cells. J Guck, R Ananthakrishnan, H Mahmood, T J Moon, C C Cunningham, J Käs, Biophys. J. 81J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, \"The optical stretcher: a novel laser tool to micromanipulate cells,\" Biophys. J. 81, 767-784 (2001).\n\nMicrofluidic sorting in an optical lattice. M P Macdonald, G C Spalding, K Dholakia, Nature. 426M. P. MacDonald, G. C. Spalding, and K. Dholakia, \"Microfluidic sorting in an optical lattice,\" Nature 426, 421-424 (2003).\n\nOptical sorting and detection of submicrometer objects in a motional standing wave. T Čižmár, M Šiler, M Šerỳ, P Zemánek, V Garcés-Chávez, K Dholakia, Phys. Rev. B. 7435105T. Čižmár, M. Šiler, M. Šerỳ, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, \"Optical sorting and detection of submicrometer objects in a motional standing wave,\" Phys. Rev. B 74, 035105 (2006).\n\nModulated optical sieve for sorting of polydisperse microparticles. I Ricárdez-Vargas, P Rodríguez-Montero, R Ramos-García, K Volke-Sepúlveda, Appl. Phys. Lett. 88121116I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, \"Modulated optical sieve for sorting of polydisperse microparticles,\" Appl. Phys. Lett. 88, 121116 (2006).\n\nConstruction of multiple-beam optical traps with nanometer-resolution position sensing. K Visscher, S P Gross, S M Block, IEEE J. Sel. Top. Quant. El. 2K. Visscher, S. P. Gross, and S. M. Block, \"Construction of multiple-beam optical traps with nanometer-resolution position sensing,\" IEEE J. Sel. Top. Quant. El. 2, 1066-1076 (1996).\n\nOptical tweezer arrays and optical substrates created with diffractive optics. E R Dufresne, D G Grier, Rev. Sci. Instrumen. 69E. R. Dufresne and D. G. Grier, \"Optical tweezer arrays and optical substrates created with diffractive optics,\" Rev. Sci. Instrumen. 69, 1974-1977 (1998).\n\n3D manipulation of particles into crystal structures using holographic optical tweezers. J Leach, G Sinclair, P Jordan, J Courtial, M J Padgett, J Cooper, Z J Laczik, Opt. Express. 12J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. J. Padgett, J. Cooper, and Z. J. Laczik, \"3D manipulation of particles into crystal structures using holographic optical tweezers,\" Opt. Express 12, 220-226 (2004).\n\nHolographic optical trapping. D G Grier, Y Roichman, Appl. Opt. 45D. G. Grier and Y. Roichman, \"Holographic optical trapping,\" Appl. Opt. 45, 880-887 (2006).\n\nDirect observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity. H He, M E J Friese, N R Heckenberg, H Rubinsztein-Dunlop, Phys. Rev. Lett. 75H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \"Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,\" Phys. Rev. Lett. 75, 826-829 (1995).\n\nOrbital angular momentum of a high-order Bessel light beam. K Volke-Sepulveda, V Garcés-Chávez, S Chávez-Cerda, J Arlt, K Dholakia, J. Opt. B: Quant. Semiclass. Opt. 4K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, \"Orbital angular momentum of a high-order Bessel light beam,\" J. Opt. B: Quant. Semiclass. Opt. 4, S82-S89 (2002).\n\nRotation, oscillation and hydrodynamic synchronization of optically trapped oblate spheroidal microparticles. A V Arzola, P Jákl, L Chvátal, P Zemánek, Opt. Express. 22A. V. Arzola, P. Jákl, L. Chvátal, and P. Zemánek, \"Rotation, oscillation and hydrodynamic synchronization of optically trapped oblate spheroidal microparticles,\" Opt. Express 22, 16207-16221 (2014).\n\nComplex rotational dynamics of multiple spheroidal particles in a circularly polarized, dual beam trap. O Brzobohatỳ, A V Arzola, M Šiler, L Chvátal, P Jákl, S Simpson, P Zemánek, Opt. Express. 23O. Brzobohatỳ, A. V. Arzola, M. Šiler, L. Chvátal, P. Jákl, S. Simpson, and P. Zemánek, \"Complex rotational dynamics of multiple spheroidal particles in a circularly polarized, dual beam trap,\" Opt. Express 23, 7273-7287 (2015).\n\nOptical deformability as an inherent cell marker for testing malignant transformation and metastatic competence. J Guck, S Schinkinger, B Lincoln, F Wottawah, S Ebert, M Romeyke, D Lenz, H M Erickson, R Ananthakrishnan, D Mitchell, J Käs, S Ulvick, C Bilby, Biophys. J. 88J. Guck, S. Schinkinger, B. Lincoln, F. Wottawah, S. Ebert, M. Romeyke, D. Lenz, H. M. Erickson, R. Ananthakrishnan, D. Mitchell, J. Käs, S. Ulvick, and C. Bilby, \"Optical deformability as an inherent cell marker for testing malignant transformation and metastatic competence,\" Biophys. J. 88, 3689-3698 (2005).\n\nLight at work: The use of optical forces for particle manipulation, sorting, and analysis. A Jonáš, P Zemanek, Electrophoresis. 29A. Jonáš and P. Zemanek, \"Light at work: The use of optical forces for particle manipulation, sorting, and analysis,\" Electrophoresis 29, 4813-4851 (2008).\n\nOptical vortex trapping of particles. K T Gahagan, G A Swartzlander, Opt. Lett. 21K. T. Gahagan and G. A. Swartzlander, \"Optical vortex trapping of particles,\" Opt. Lett. 21, 827-829 (1996).\n\nThree-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner. A T O'neil, M J Padgett, Opt. Commun. 185A. T. O'Neil and M. J. Padgett, \"Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,\" Opt. Commun. 185, 139-143 (2000).\n\nTransfer of orbital angular momentum to an optically trapped low-index particle. V Garcés-Chávez, K Volke-Sepulveda, S Chávez-Cerda, W Sibbett, K Dholakia, Phys. Rev. A. 6663402V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, \"Transfer of orbital angular momentum to an optically trapped low-index particle,\" Phys. Rev. A 66, 063402 (2002).\n\nAttractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers. R J Hernández, A Mazzulla, A Pane, K Volke-Sepúlveda, G Cipparrone, Lab Chip. 13R. J. Hernández, A. Mazzulla, A. Pane, K. Volke-Sepúlveda, and G. Cipparrone, \"Attractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers,\" Lab Chip 13, 459-467 (2013).\n\nOptical binding. M M Burns, J.-M Fournier, J A Golovchenko, Phys. Rev. Lett. 63M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, \"Optical binding,\" Phys. Rev. Lett. 63, 1233-1236 (1989).\n\nOptical matter: crystallization and binding in intense optical fields. M M Burns, J.-M Fournier, J A Golovchenko, Science. 249M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, \"Optical matter: crystallization and binding in intense optical fields,\" Science 249, 749-754 (1990).\n\nOptical binding between dielectric particles. S K Mohanty, J T Andrews, P K Gupta, Opt. Express. 12S. K. Mohanty, J. T. Andrews, and P. K. Gupta, \"Optical binding between dielectric particles,\" Opt. Express 12, 2746-2753 (2004).\n\nLong-range one-dimensional longitudinal optical binding. V Karásek, T Čižmár, O Brzobohatỳ, P Zemánek, V Garcés-Chávez, K Dholakia, Phys. Rev. Lett. 101143601V. Karásek, T. Čižmár, O. Brzobohatỳ, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, \"Long-range one-dimensional longitudinal optical binding,\" Phys. Rev. Lett. 101, 143601 (2008).\n\nUltrastrong optical binding of metallic nanoparticles. V Demergis, E.-L Florin, Nanolett. 12V. Demergis and E.-L. Florin, \"Ultrastrong optical binding of metallic nanoparticles,\" Nanolett. 12, 5756-5760 (2012).\n\nLight-controlled assembly of active colloidal molecules. F Schmidt, B Liebchen, H Löwen, G Volpe, J. Chem. Phys. 15094905F. Schmidt, B. Liebchen, H. Löwen, and G. Volpe, \"Light-controlled assembly of active colloidal molecules,\" J. Chem. Phys. 150, 094905 (2019).\n\nOptical thermal ratchet. L P Faucheux, L S Bourdieu, P D Kaplan, A J Libchaber, Phys. Rev. Lett. 74L. P. Faucheux, L. S. Bourdieu, P. D. Kaplan, and A. J. Libchaber, \"Optical thermal ratchet,\" Phys. Rev. Lett. 74, 1504-1507 (1995).\n\nOptical peristalsis. B A Koss, D G Grier, Appl. Phys. Lett. 82B. A. Koss and D. G. Grier, \"Optical peristalsis,\" Appl. Phys. Lett. 82, 3985-3987 (2003).\n\nObservation of flux reversal in a symmetric optical thermal ratchet. S.-H Lee, K Ladavac, M Polin, D G Grier, Phys. Rev. Lett. 94110601S.-H. Lee, K. Ladavac, M. Polin, and D. G. Grier, \"Observation of flux reversal in a symmetric optical thermal ratchet,\" Phys. Rev. Lett. 94, 110601 (2005).\n\nExperimental control of transport and current reversals in a deterministic optical rocking ratchet. A V Arzola, K Volke-Sepúlveda, J L Mateos, Phys. Rev. Lett. 106168104A. V. Arzola, K. Volke-Sepúlveda, and J. L. Mateos, \"Experimental control of transport and current reversals in a deterministic optical rocking ratchet,\" Phys. Rev. Lett. 106, 168104 (2011).\n\nActive Brownian motion tunable by light. I Buttinoni, G Volpe, F Kümmel, G Volpe, C Bechinger, J. Phys.: Condens. Matter. 24284129I. Buttinoni, G. Volpe, F. Kümmel, G. Volpe, and C. Bechinger, \"Active Brownian motion tunable by light,\" J. Phys.: Condens. Matter 24, 284129 (2012).\n\nBrownian motion in a speckle light field: Tunable anomalous diffusion and selective optical manipulation. G Volpe, G Volpe, S Gigan, Sci. Rep. 43936G. Volpe, G. Volpe, and S. Gigan, \"Brownian motion in a speckle light field: Tunable anomalous diffusion and selective optical manipulation,\" Sci. Rep. 4, 3936 (2014).\n\nOmnidirectional transport in fully reconfigurable two dimensional optical ratchets. A V Arzola, M Villasante-Barahona, K Volke-Sepúlveda, P Jákl, P Zemánek, Phys. Rev. Lett. 118138002A. V. Arzola, M. Villasante-Barahona, K. Volke-Sepúlveda, P. Jákl, and P. Zemánek, \"Omnidirectional transport in fully reconfigurable two dimensional optical ratchets,\" Phys. Rev. Lett. 118, 138002 (2017).\n\nSpin to orbital light momentum conversion visualized by particle trajectory. A V Arzola, L Chvátal, P Jákl, P Zemánek, Sci. Rep. 94127A. V. Arzola, L. Chvátal, P. Jákl, and P. Zemánek, \"Spin to orbital light momentum conversion visualized by particle trajectory,\" Sci. Rep. 9, 4127 (2019).\n\nComplex micromachines produced and driven by light. P Galajda, P Ormos, Appl. Phys. Lett. 78P. Galajda and P. Ormos, \"Complex micromachines produced and driven by light,\" Appl. Phys. Lett. 78, 249-251 (2001).\n\nOptically driven micromachine elements. M E J Friese, H Rubinsztein-Dunlop, J Gold, P Hagberg, D Hanstorp, Appl. Phys. Lett. 78M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, \"Optically driven micromachine elements,\" Appl. Phys. Lett. 78, 547-549 (2001).\n\nForce-controllable, optically driven micromachines fabricated by single-step two-photon microstereolithography. S Maruo, K Ikuta, H Korogi, J. Microelectromech. Syst. 12S. Maruo, K. Ikuta, and H. Korogi, \"Force-controllable, optically driven micromachines fabricated by single-step two-photon microstereolithography,\" J. Microelectromech. Syst. 12, 533-539 (2003).\n\nOptical tweezers applied to a microfluidic system. J Enger, M Goksör, K Ramser, P Hagberg, D Hanstorp, Lab Chip. 4J. Enger, M. Goksör, K. Ramser, P. Hagberg, and D. Hanstorp, \"Optical tweezers applied to a microfluidic system,\" Lab Chip 4, 196-200 (2004).\n\nAll-optical control of microfluidic components using form birefringence. S L Neale, M P Macdonald, K Dholakia, T F Krauss, Nat. Mater. 4S. L. Neale, M. P. MacDonald, K. Dholakia, and T. F. Krauss, \"All-optical control of microfluidic components using form birefringence,\" Nat. Mater. 4, 530-533 (2005).\n\nMicrofluidic sorting of mammalian cells by optical force switching. M M Wang, E Tu, D E Raymond, J M Yang, H Zhang, N Hagen, B Dees, E M Mercer, A H Forster, I Kariv, P J Marchand, W F Butler, Nat. Biotechnol. 23M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, \"Microfluidic sorting of mammalian cells by optical force switching,\" Nat. Biotechnol. 23, 83-87 (2005).\n\nAn optically driven pump for microfluidics. J Leach, H Mushfique, R Di Leonardo, M Padgett, J Cooper, Lab Chip. 6J. Leach, H. Mushfique, R. di Leonardo, M. Padgett, and J. Cooper, \"An optically driven pump for microfluidics,\" Lab Chip 6, 735-739 (2006).\n\nA microfluidic system in combination with optical tweezers for analyzing rapid and reversible cytological alterations in single cells upon environmental changes. E Eriksson, J Enger, B Nordlander, N Erjavec, K Ramser, M Goksör, S Hohmann, T Nyström, D Hanstorp, Lab Chip. 7E. Eriksson, J. Enger, B. Nordlander, N. Erjavec, K. Ramser, M. Goksör, S. Hohmann, T. Nyström, and D. Hanstorp, \"A microfluidic system in combination with optical tweezers for analyzing rapid and reversible cytological alterations in single cells upon environmental changes,\" Lab Chip 7, 71-76 (2007).\n\nEnhanced cell sorting and manipulation with combined optical tweezer and microfluidic chip technologies. X Wang, S Chen, M Kong, Z Wang, K D Costa, R A Li, D Sun, Lab Chip. 11X. Wang, S. Chen, M. Kong, Z. Wang, K. D. Costa, R. A. Li, and D. Sun, \"Enhanced cell sorting and manipulation with combined optical tweezer and microfluidic chip technologies,\" Lab Chip 11, 3656-3662 (2011).\n\nOptical alignment and spinning of laser-trapped microscopic particles. M E J Friese, T A Nieminen, N R Heckenberg, H Rubinsztein-Dunlop, Nature. 394M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \"Optical alignment and spinning of laser-trapped microscopic particles,\" Nature 394, 348-350 (1998).\n\nIntrinsic and extrinsic nature of the orbital angular momentum of a light beam. A T O'neil, I Macvicar, L Allen, M J Padgett, Phys. Rev. Lett. 8853601A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, \"Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,\" Phys. Rev. Lett. 88, 053601 (2002).\n\nOrbital angular momentum transfer in helical Mathieu beams. C López-Mariscal, J C Gutiérrez-Vega, G Milne, K Dholakia, Opt. Express. 14C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, \"Orbital angular momentum transfer in helical Mathieu beams,\" Opt. Express 14, 4182-4187 (2006).\n\nForces in optical tweezers with radially and azimuthally polarized trapping beams. T A Nieminen, N R Heckenberg, H Rubinsztein-Dunlop, Opt. Lett. 33T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \"Forces in optical tweezers with radially and azimuthally polarized trapping beams,\" Opt. Lett. 33, 122-124 (2008).\n\nOptically mediated particle clearing using Airy wavepackets. J Baumgartl, M Mazilu, K Dholakia, Nat. Photon. 2J. Baumgartl, M. Mazilu, and K. Dholakia, \"Optically mediated particle clearing using Airy wavepackets,\" Nat. Photon. 2, 675-678 (2008).\n\nDesign for fully steerable dual-trap optical tweezers. E Fällman, O Axner, Appl. Opt. 36E. Fällman and O. Axner, \"Design for fully steerable dual-trap optical tweezers,\" Appl. Opt. 36, 2107-2113 (1997).\n\nPattern formation and flow control of fine particles by laser-scanning micromanipulation. K Sasaki, M Koshioka, H Misawa, N Kitamura, H Masuhara, Opt. Lett. 16K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, \"Pattern formation and flow control of fine particles by laser-scanning micromanipulation,\" Opt. Lett. 16, 1463-1465 (1991).\n\nDesign of a scanning laser optical trap for multiparticle manipulation. C Mio, T Gong, A Terray, D W M Marr, Rev. Sci. Instrumen. 71C. Mio, T. Gong, A. Terray, and D. W. M. Marr, \"Design of a scanning laser optical trap for multiparticle manipulation,\" Rev. Sci. Instrumen. 71, 2196-2200 (2000).\n\nSynchronized laser micromanipulation of multiple targets along each trajectory by single laser. F Arai, K Yoshikawa, T Sakami, T Fukuda, Appl. Phys. Lett. 85F. Arai, K. Yoshikawa, T. Sakami, and T. Fukuda, \"Synchronized laser micromanipulation of multiple targets along each trajectory by single laser,\" Appl. Phys. Lett. 85, 4301-4303 (2004).\n\nPeriodic average structures of colloidal quasicrystals. L Zaidouny, T Bohlein, J Roth, C Bechinger, Soft Matter. 10L. Zaidouny, T. Bohlein, J. Roth, and C. Bechinger, \"Periodic average structures of colloidal quasicrystals,\" Soft Matter 10, 8705-8710 (2014).\n\nInterferometric optical tweezers. A E Chiou, W Wang, G J Sonek, J Hong, M W Berns, Opt. Commun. 133A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, \"Interferometric optical tweezers,\" Opt. Commun. 133, 7-10 (1997).\n\nOptical forces acting on rayleigh particle placed into interference field. P Zemánek, V Karásek, A Sasso, Opt. Commun. 240P. Zemánek, V. Karásek, and A. Sasso, \"Optical forces acting on rayleigh particle placed into interference field,\" Opt. Commun. 240, 401-415 (2004).\n\nCreation and manipulation of three-dimensional optically trapped structures. M P Macdonald, L Paterson, K Volke-Sepulveda, J Arlt, W Sibbett, K Dholakia, Science. 296M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, \"Creation and manipulation of three-dimensional optically trapped structures,\" Science 296, 1101-1103 (2002).\n\nSelective trapping of multiple particles by volume speckle field. V G Shvedov, A V Rode, Y V Izdebskaya, A S Desyatnikov, W Krolikowski, Y S Kivshar, Opt. Express. 18V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, \"Selective trapping of multiple particles by volume speckle field,\" Opt. Express 18, 3137-3142 (2010).\n\nSpeckle optical tweezers: Micromanipulation with random light fields. G Volpe, L Kurz, A Callegari, G Volpe, S Gigan, Opt. Express. 22G. Volpe, L. Kurz, A. Callegari, G. Volpe, and S. Gigan, \"Speckle optical tweezers: Micromanipulation with random light fields,\" Opt. Express 22, 18159-18167 (2014).\n\nMulti-functional optical tweezers using computer-generated holograms. J Liesener, M Reicherter, T Haist, H J Tiziani, Opt. Commun. 185J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, \"Multi-functional optical tweezers using computer-generated holograms,\" Opt. Commun. 185, 77-82 (2000).\n\nDynamic holographic optical tweezers. J E Curtis, B A Koss, D G Grier, Opt. Commun. 207J. E. Curtis, B. A. Koss, and D. G. Grier, \"Dynamic holographic optical tweezers,\" Opt. Commun. 207, 169-175 (2002).\n\nFull phase and amplitude control of holographic optical tweezers with high efficiency. A Jesacher, C Maurer, A Schwaighofer, S Bernet, M Ritsch-Marte, Opt. Express. 16A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, \"Full phase and amplitude control of holographic optical tweezers with high efficiency,\" Opt. Express 16, 4479-4486 (2008).\n\nOptical sorting of nonspherical and living microobjects in moving interference structures. P Jakl, A V Arzola, M Siler, L Chatal, K Volke-Sepulveda, P Zemanek, Opt. Express. 22P. Jakl, A. V. Arzola, M. Siler, L. Chatal, K. Volke-Sepulveda, and P. Zemanek, \"Optical sorting of nonspherical and living microobjects in moving interference structures,\" Opt. Express 22, 29746-29760 (2014).\n\nExact solutions for nondiffracting beams. i. the scalar theory. J Durnin, J. Opt. Soc. Am. A. 4J. Durnin, \"Exact solutions for nondiffracting beams. i. the scalar theory,\" J. Opt. Soc. Am. A 4, 651-654 (1987).\n\nAlternative formulation for invariant optical fields: Mathieu beams. J C Gutiérrez-Vega, M D Iturbe-Castillo, S Chávez-Cerda, Opt. Lett. 25J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, \"Alternative formulation for invariant optical fields: Mathieu beams,\" Opt. Lett. 25, 1493-1495 (2000).\n\nOptical micromanipulation using a Bessel light beam. J Arlt, V Garcés-Chávez, W Sibbett, K Dholakia, Opt. Commun. 197J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, \"Optical micromanipulation using a Bessel light beam,\" Opt. Commun. 197, 239-245 (2001).\n\nM A Bandres, J C Gutiérrez-Vega, S Chávez-Cerda, Parabolic nondiffracting optical wave fields. 29M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, \"Parabolic nondiffracting optical wave fields,\" Opt. Lett. 29, 44-46 (2004).\n\nFresnel axicon. I Golub, Opt. Lett. 31I. Golub, \"Fresnel axicon,\" Opt. Lett. 31, 1890-1892 (2006).\n\nEfficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation. V Arrizón, D Sánchez-De La-Llave, U Ruiz, G Méndez, Opt. Lett. 34V. Arrizón, D. Sánchez-de La-Llave, U. Ruiz, and G. Méndez, \"Efficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation,\" Opt. Lett. 34, 1456-1458 (2009).\n\nExperimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator. R J Hernández-Hernández, R A Terborg, I Ricardez-Vargas, K Volke-Sepúlveda, Appl. Opt. 49R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas, and K. Volke-Sepúlveda, \"Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator,\" Appl. Opt. 49, 6903-6909 (2010).\n\nGeneration of the \"perfect\" optical vortex using a liquid-crystal spatial light modulator. A S Ostrovsky, C Rickenstorff-Parrao, V Arrizón, Opt. Lett. 38A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, \"Generation of the \"perfect\" optical vortex using a liquid-crystal spatial light modulator,\" Opt. Lett. 38, 534-536 (2013).\n\nTheory of optical tweezers. P A M Neto, H M Nussenzveig, Europhys. Lett.). 50EPLP. A. M. Neto and H. M. Nussenzveig, \"Theory of optical tweezers,\" EPL (Europhys. Lett.) 50, 702-710 (2000).\n\nStiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory. A Rohrbach, Phys. Rev. Lett. 95168102A. Rohrbach, \"Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory,\" Phys. Rev. Lett. 95, 168102 (2005).\n\nAbsolute calibration of forces in optical tweezers. R S Dutra, N B Viana, P A M Neto, H M Nussenzveig, Phys. Rev. A. 9013825R. S. Dutra, N. B. Viana, P. A. M. Neto, and H. M. Nussenzveig, \"Absolute calibration of forces in optical tweezers,\" Phys. Rev. A 90, 013825 (2014).\n\nPower spectrum analysis for optical tweezers. K Berg-Sorensen, H Flyvbjerg, Rev. Sci. Instrumen. 75K. Berg-Sorensen and H. Flyvbjerg, \"Power spectrum analysis for optical tweezers,\" Rev. Sci. Instrumen. 75, 594-612 (2004).\n\nHigh-performance reconstruction of microscopic force fields from Brownian trajectories. L García, J Pérez, G Volpe, A V Arzola, G Volpe, Nat. Commun. 95166L. Pérez García, J. Donlucas Pérez, G. Volpe, A. V. Arzola, and G. Volpe, \"High-performance reconstruction of microscopic force fields from Brownian trajectories,\" Nat. Commun. 9, 5166 (2018).\n\nBayesian inference for estimating parameters of Brownian motion. I Castillo, G Volpe, L Pérez García, A V Arzola, preparationI. Pérez Castillo, G. Volpe, L. Pérez García, and A. V. Arzola, \"Bayesian inference for estimating parameters of Brownian motion,\" preparation (2019).\n\nH Risken, The Fokker-Planck equation. SpringerH. Risken, The Fokker-Planck equation (Springer, 1996).\n\nProperties of the noise-induced \"spurious\" drift. I. D Ryter, U Deker, J. Math. Phys. 21D. Ryter and U. Deker, \"Properties of the noise-induced \"spurious\" drift. I.\" J. Math. Phys. 21, 2662-2665 (1980).\n\nDirect measurement of the nonconservative force field generated by optical tweezers. P Wu, R Huang, C Tischer, A Jonas, E.-L Florin, Phys. Rev. Lett. 103108101P. Wu, R. Huang, C. Tischer, A. Jonas, and E.-L. Florin, \"Direct measurement of the nonconservative force field generated by optical tweezers,\" Phys. Rev. Lett. 103, 108101 (2009).\n\nInfluence of noise on force measurements. G Volpe, L Helden, T Brettschneider, J Wehr, C Bechinger, Phys. Rev. Lett. 104170602G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger, \"Influence of noise on force measurements,\" Phys. Rev. Lett. 104, 170602 (2010).\n\nForce measurement in the presence of Brownian noise: Equilibrium-distribution method versus drift method. T Brettschneider, G Volpe, L Helden, J Wehr, C Bechinger, Phys. Rev. E. 8341113T. Brettschneider, G. Volpe, L. Helden, J. Wehr, and C. Bechinger, \"Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus drift method,\" Phys. Rev. E 83, 041113 (2011).\n\nState-dependent diffusion: Thermodynamic consistency and its path integral formulation. A W C Lau, T C Lubensky, Phys. Rev. E. 7611123A. W. C. Lau and T. C. Lubensky, \"State-dependent diffusion: Thermodynamic consistency and its path integral formulation,\" Phys. Rev. E 76, 011123 (2007).\n\nThe Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. S Hottovy, A Mcdaniel, G Volpe, J Wehr, Commun. Math. Phys. 336S. Hottovy, A. McDaniel, G. Volpe, and J. Wehr, \"The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction,\" Commun. Math. Phys. 336, 1259-1283 (2015).\n\nEffective drifts in dynamical systems with multiplicative noise: A review of recent progress. G Volpe, J Wehr, Rep. Prog. Phys. 7953901G. Volpe and J. Wehr, \"Effective drifts in dynamical systems with multiplicative noise: A review of recent progress,\" Rep. Prog. Phys. 79, 053901 (2016).\n\nFast Bayesian inference of the multivariate Ornstein-Uhlenbeck process. R Singh, D Ghosh, R Adhikari, Phys. Rev. E. 9812136R. Singh, D. Ghosh, and R. Adhikari, \"Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process,\" Phys. Rev. E 98, 012136 (2018).\n\nA Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories. S Türkcan, A Alexandrou, J.-B Masson, Biophys. J. 102S. Türkcan, A. Alexandrou, and J.-B. Masson, \"A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories.\" Biophys. J. 102, 2288-2298 (2012).\n\nCalibrating optical tweezers with Bayesian inference. M U Richly, S Türkcan, A Le Gall, N Fiszman, J Masson, N Westbrook, K Perronet, A Alexandrou, Opt. Express. 21M. U. Richly, S. Türkcan, A. Le Gall, N. Fiszman, J. Masson, N. Westbrook, K. Perronet, and A. Alexandrou, \"Calibrating optical tweezers with Bayesian inference,\" Opt. Express 21, 31578-31590 (2013).\n\nA primer on the Bayesian approach to high-density single-molecule trajectories analysis. M El Beheiry, S Türkcan, M U Richly, A Triller, A Alexandrou, M Dahan, J.-B Masson, Biophys. J. 110M. El Beheiry, S. Türkcan, M. U. Richly, A. Triller, A. Alexandrou, M. Dahan, and J.-B. Masson, \"A primer on the Bayesian approach to high-density single-molecule trajectories analysis,\" Biophys. J. 110, 1209-1215 (2016).\n\nFast Bayesian inference of optical trap stiffness and particle diffusion. S Bera, S Paul, R Singh, D Ghosh, A Kundu, A Banerjee, R Adhikari, Sci. Rep. 741638S. Bera, S. Paul, R. Singh, D. Ghosh, A. Kundu, A. Banerjee, and R. Adhikari, \"Fast Bayesian inference of optical trap stiffness and particle diffusion,\" Sci. Rep. 7, 41638 (2017).\n\nTorque detection using Brownian fluctuations. G Volpe, D Petrov, Phys. Rev. Lett. 97210603G. Volpe and D. Petrov, \"Torque detection using Brownian fluctuations,\" Phys. Rev. Lett. 97, 210603 (2006).\n\nBrownian motion in a nonhomogeneous force field and photonic force microscope. G Volpe, G Volpe, D Petrov, Phys. Rev. E. 7661118G. Volpe, G. Volpe, and D. Petrov, \"Brownian motion in a nonhomogeneous force field and photonic force microscope,\" Phys. Rev. E 76, 061118 (2007).\n\nQuantitative assessment of non-conservative radiation forces in an optical trap. G Pesce, G Volpe, A C De Luca, G Rusciano, G Volpe, Europhys. Lett.). 8638002EPLG. Pesce, G. Volpe, A. C. De Luca, G. Rusciano, and G. Volpe, \"Quantitative assessment of non-conservative radiation forces in an optical trap,\" EPL (Europhys. Lett.) 86, 38002 (2009).\n\nQuantitative measurements of force and displacement using an optical trap. R M Simmons, J T Finer, S Chu, J A Spudich, Biophys. J. 70R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, \"Quantitative measurements of force and displacement using an optical trap,\" Biophys. J. 70, 1813-1822 (1996).\n\nCalibration of optical tweezers with positional detection in the back focal plane. S F Tolić-Nørrelykke, E Schäffer, J Howard, F S Pavone, F J , H Flyvbjerg, Rev. Sci. Instrumen. 77103101S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. J., and H. Flyvbjerg, \"Calibration of optical tweezers with positional detection in the back focal plane,\" Rev. Sci. Instrumen. 77, 103101 (2006).\n\nOptical-trap force transducer that operates by direct measurement of light momentum. S B Smith, Y Cui, C Bustamante, Methods in enzymology. 361ElsevierS. B. Smith, Y. Cui, and C. Bustamante, \"Optical-trap force transducer that operates by direct measurement of light momentum,\" in Methods in enzymology, vol. 361 (Elsevier, 2003), pp. 134-162.\n\nA force detection technique for single-beam optical traps based on direct measurement of light momentum changes. A Farré, M Montes-Usategui, Opt. Express. 18A. Farré and M. Montes-Usategui, \"A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,\" Opt. Express 18, 11955-11968 (2010).\n\nDirect measurement of axial optical forces. G Thalhammer, L Obmascher, M Ritsch-Marte, Opt. Express. 23G. Thalhammer, L. Obmascher, and M. Ritsch-Marte, \"Direct measurement of axial optical forces,\" Opt. Express 23, 6112-6129 (2015).\n\nEverything you wanted to know about data analysis and fitting but were afraid to ask. P Young, SpringerP. Young, Everything you wanted to know about data analysis and fitting but were afraid to ask (Springer, 2015).\n\nP J Huber, Robust Statistics. WileyP. J. Huber, Robust Statistics (Wiley, 1981).\n\nFitting a function to time-dependent ensemble averaged data. K Fogelmark, M A Lomholt, A Irbäck, T Ambjörnsson, Sci. Rep. 86984K. Fogelmark, M. A. Lomholt, A. Irbäck, and T. Ambjörnsson, \"Fitting a function to time-dependent ensemble averaged data,\" Sci. Rep. 8, 6984 (2018).\n\nSingle-molecule experiments in biological physics: Methods and applications. F Ritort, J. Phys.: Condens. Matter. 18F. Ritort, \"Single-molecule experiments in biological physics: Methods and applications,\" J. Phys.: Condens. Matter 18, R531-R583 (2006).\n\nSingle-molecule biophysics: At the interface of biology, physics and chemistry. A A Deniz, S Mukhopadhyay, E A Lemke, J. Royal Soc. Interfac. 5A. A. Deniz, S. Mukhopadhyay, and E. A. Lemke, \"Single-molecule biophysics: At the interface of biology, physics and chemistry,\" J. Royal Soc. Interfac. 5, 15-45 (2007).\n\nSingle-molecule micromanipulation techniques. K C Neuman, T Lionnet, J.-F Allemand, Annu. Rev. Mater. Res. 37K. C. Neuman, T. Lionnet, and J.-F. Allemand, \"Single-molecule micromanipulation techniques,\" Annu. Rev. Mater. Res. 37, 33-67 (2007).\n\nRecent advances in optical tweezers. J R Moffitt, Y R Chemla, S B Smith, C Bustamante, Annu. Rev. Biochem. 77J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, \"Recent advances in optical tweezers,\" Annu. Rev. Biochem. 77, 205-228 (2008).\n\nSingle-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy. K C Neuman, A Nagy, Nat. Methods. 5K. C. Neuman and A. Nagy, \"Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy,\" Nat. Methods 5, 491-505 (2008).\n\nHigh-resolution optical tweezers for single-molecule manipulation. X Zhang, L Ma, Y Zhang, Yale J. Biol. Med. 86X. Zhang, L. Ma, and Y. Zhang, \"High-resolution optical tweezers for single-molecule manipulation,\" Yale J. Biol. Med. 86, 367-383 (2013).\n\nInterrogating biology with force: Single molecule high-resolution measurements with optical tweezers. M Capitanio, F S Pavone, Biophys. J. 105M. Capitanio and F. S. Pavone, \"Interrogating biology with force: Single molecule high-resolution measurements with optical tweezers,\" Biophys. J. 105, 1293-1303 (2013).\n\nSingle-molecule techniques in biophysics: A review of the progress in methods and applications. H Miller, Z Zhou, J Shepherd, A J M Wollman, M C Leake, Rep. Prog. Phys. 8124601H. Miller, Z. Zhou, J. Shepherd, A. J. M. Wollman, and M. C. Leake, \"Single-molecule techniques in biophysics: A review of the progress in methods and applications,\" Rep. Prog. Phys. 81, 024601 (2017).\n\nElastic properties of nucleic acids by single-molecule force spectroscopy. J Camunas-Soler, M Ribezzi-Crivellari, F Ritort, Annu. Rev. Biophys. 45J. Camunas-Soler, M. Ribezzi-Crivellari, and F. Ritort, \"Elastic properties of nucleic acids by single-molecule force spectroscopy,\" Annu. Rev. Biophys. 45, 65-84 (2016).\n\nReconstructing folding energy landscapes by single-molecule force spectroscopy. M T Woodside, S M Block, Annu. Rev. Biophys. 43M. T. Woodside and S. M. Block, \"Reconstructing folding energy landscapes by single-molecule force spectroscopy,\" Annu. Rev. Biophys. 43, 19-39 (2014).\n\nThe power of force: Insights into the protein folding process using single-molecule force spectroscopy. J Schönfelder, D De Sancho, R Perez-Jimenez, J. Mol. Biol. 428J. Schönfelder, D. De Sancho, and R. Perez-Jimenez, \"The power of force: Insights into the protein folding process using single-molecule force spectroscopy,\" J. Mol. Biol. 428, 4245-4257 (2016).\n\nSingle molecule high-throughput footprinting of small and large DNA ligands. M Manosas, J Camunas-Soler, V Croquette, F Ritort, Nat. Commun. 8304M. Manosas, J. Camunas-Soler, V. Croquette, and F. Ritort, \"Single molecule high-throughput footprinting of small and large DNA ligands,\" Nat. Commun. 8, 304 (2017).\n\nOptical traps to study properties of molecular motors. J A Spudich, S E Rice, R S Rock, T J Purcell, H M Warrick, Cold Spring Harb. Protoc. 66662J. A. Spudich, S. E. Rice, R. S. Rock, T. J. Purcell, and H. M. Warrick, \"Optical traps to study properties of molecular motors,\" Cold Spring Harb. Protoc. 2011, pdb-top066662 (2011).\n\nNonequilibrium fluctuations in small systems: From physics to biology. F Ritort, Adv. Chem. Phys. 137F. Ritort, \"Nonequilibrium fluctuations in small systems: From physics to biology,\" Adv. Chem. Phys. 137, 31-123 (2008).\n\nExperiments in stochastic thermodynamics: Short history and perspectives. S Ciliberto, Phys. Rev. X. 721051S. Ciliberto, \"Experiments in stochastic thermodynamics: Short history and perspectives,\" Phys. Rev. X 7, 021051 (2017).\n\nA guide to magnetic tweezers and their applications. R Sarkar, V V Rybenkov, Front. Phys. 448R. Sarkar and V. V. Rybenkov, \"A guide to magnetic tweezers and their applications,\" Front. Phys. 4, 48 (2016).\n\nSingle molecule force spectroscopy in biology using the atomic force microscope. J Zlatanova, S M Lindsay, S H Leuba, Prog. Biophys. Mol. Biol. 74J. Zlatanova, S. M. Lindsay, and S. H. Leuba, \"Single molecule force spectroscopy in biology using the atomic force microscope,\" Prog. Biophys. Mol. Biol. 74, 37-61 (2000).\n\nAcoustic force spectroscopy. G Sitters, D Kamsma, G Thalhammer, M Ritsch-Marte, E J G Peterman, G J L Wuite, Nat. Methods. 12G. Sitters, D. Kamsma, G. Thalhammer, M. Ritsch-Marte, E. J. G. Peterman, and G. J. L. Wuite, \"Acoustic force spectroscopy,\" Nat. Methods 12, 47-50 (2015).\n\nSingle-molecule electrometry. F Ruggeri, F Zosel, N Mutter, M Różycka, M Wojtas, A Ożyhar, B Schuler, M Krishnan, Nat. Nanotech. 12F. Ruggeri, F. Zosel, N. Mutter, M. Różycka, M. Wojtas, A. Ożyhar, B. Schuler, and M. Krishnan, \"Single-molecule electrometry,\" Nat. Nanotech. 12, 488-495 (2017).\n\nMicrofluidics for biological measurements with single-molecule resolution. A M Streets, Y Huang, Curr. Opn. Biotechnol. 25A. M. Streets and Y. Huang, \"Microfluidics for biological measurements with single-molecule resolution,\" Curr. Opn. Biotechnol. 25, 69-77 (2014).\n\nSingle-molecule measurements using microneedles. T Yanagida, Y Ishii, A Ishijima, Single Molecule Enzymology. SpringerT. Yanagida, Y. Ishii, and A. Ishijima, \"Single-molecule measurements using microneedles,\" in Single Molecule Enzymology, (Springer, 2011), pp. 143-159.\n\nSensitive force technique to probe molecular adhesion and structural linkages at biological interfaces. E Evans, K Ritchie, R Merkel, Biophys. J. 68E. Evans, K. Ritchie, and R. Merkel, \"Sensitive force technique to probe molecular adhesion and structural linkages at biological interfaces,\" Biophys. J. 68, 2580-2587 (1995).\n\nMassively parallel single-molecule manipulation using centrifugal force. W P Wong, K Halvorsen, Biophys. J. 100W. P. Wong and K. Halvorsen, \"Massively parallel single-molecule manipulation using centrifugal force,\" Biophys. J. 100, L53-L55 (2011).\n\nCounter-propagating dual-trap optical tweezers based on linear momentum conservation. M Ribezzi-Crivellari, J M Huguet, F Ritort, Rev. Sci. Instrumen. 8443104M. Ribezzi-Crivellari, J. M. Huguet, and F. Ritort, \"Counter-propagating dual-trap optical tweezers based on linear momentum conservation,\" Rev. Sci. Instrumen. 84, 043104 (2013).\n\nStatistical and thermodynamic properties of DNA unzipping experiments with optical tweezers. J M Huguet, Universitat de BarcelonaPh.D. thesisJ. M. Huguet, \"Statistical and thermodynamic properties of DNA unzipping experiments with optical tweezers,\" Ph.D. thesis, Universitat de Barcelona (2010).\n\nAssemblies: Pipette Puller. U Di Parma, U. di Parma, \"Assemblies: Pipette Puller,\" http://tweezerslab.unipr.it/cgi-bin/mt/ assemblies.pl/Show?_id=7c1b. Accessed: 2019-12-12.\n\nP Nelson, Biological physics. New YorkWH FreemanP. Nelson, Biological physics (WH Freeman, New York, 2004).\n\nThe structural basis of antibody-antigen recognition. I Sela-Culang, V Kunik, Y Ofran, Front. Immunol. 4302I. Sela-Culang, V. Kunik, and Y. Ofran, \"The structural basis of antibody-antigen recognition,\" Front. Immunol. 4, 302 (2013).\n\nHow the biotin-streptavidin interaction was made even stronger: investigation via crystallography and a chimaeric tetramer. C E Chivers, A L Koner, E D Lowe, M Howarth, Biochem. J. 435C. E. Chivers, A. L. Koner, E. D. Lowe, and M. Howarth, \"How the biotin-streptavidin interaction was made even stronger: investigation via crystallography and a chimaeric tetramer,\" Biochem. J. 435, 55-63 (2011).\n\nThe solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds. F Pincet, J Husson, Biophys. J. 89F. Pincet and J. Husson, \"The solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds,\" Biophys. J. 89, 4374-4381 (2005).\n\nImproving signal/noise resolution in single-molecule experiments using molecular constructs with short handles. N Forns, S Lorenzo, M Manosas, K Hayashi, J M Huguet, F Ritort, Biophys. J. 100N. Forns, S. de Lorenzo, M. Manosas, K. Hayashi, J. M. Huguet, and F. Ritort, \"Improving signal/noise resolution in single-molecule experiments using molecular constructs with short handles,\" Biophys. J. 100, 1765-1774 (2011).\n\nStochastic thermodynamics, fluctuation theorems and molecular machines. U Seifert, Rep. Prog. Phys. 75126001U. Seifert, \"Stochastic thermodynamics, fluctuation theorems and molecular machines,\" Rep. Prog. Phys. 75, 126001 (2012).\n\nStretching DNA. J F Marko, E D Siggia, Macromolecules. 28J. F. Marko and E. D. Siggia, \"Stretching DNA,\" Macromolecules 28, 8759-8770 (1995).\n\nDetermination of the elastic properties of short ssDNA molecules by mechanically folding and unfolding DNA hairpins. A Alemany, F Ritort, Biopolymers. 101A. Alemany and F. Ritort, \"Determination of the elastic properties of short ssDNA molecules by mechanically folding and unfolding DNA hairpins,\" Biopolymers 101, 1193-1199 (2014).\n\n. R R Sinden, DNA Structure and Function. Elsevier ScienceR. R. Sinden, DNA Structure and Function (Elsevier Science, 1994).\n\nA unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics. J Santalucia, Proc. Natl. Acad. Sci. 95J. SantaLucia, \"A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics,\" Proc. Natl. Acad. Sci. 95, 1460-1465 (1998).\n\nHolographic acoustic tweezers. A Marzo, B W Drinkwater, Proc. Natl. Acad. Sci. Natl. Acad. Sci116A. Marzo and B. W. Drinkwater, \"Holographic acoustic tweezers,\" Proc. Natl. Acad. Sci. 116, 84-89 (2019).\n\nCombining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions. A Candelli, G J L Wuite, E J G Peterman, Phys. Chem. Chem. Phys. 13A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, \"Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,\" Phys. Chem. Chem. Phys. 13, 7263-7272 (2011).\n\nCombining single-molecule manipulation and single-molecule detection. J C Cordova, D K Das, H W Manning, M J Lang, Curr. Opn. Struct. Biol. 28J. C. Cordova, D. K. Das, H. W. Manning, and M. J. Lang, \"Combining single-molecule manipulation and single-molecule detection,\" Curr. Opn. Struct. Biol. 28, 142-148 (2014).\n\nHigh-resolutio \"Fleezers\": Dual-trap optical tweezers combined with single-molecule fluorescence detection. K D Whitley, M J Comstock, Y R Chemla, Methods Mol. Biol. pp. K. D. Whitley, M. J. Comstock, and Y. R. Chemla, \"High-resolutio \"Fleezers\": Dual-trap optical tweezers combined with single-molecule fluorescence detection,\" Methods Mol. Biol. pp. 183-256 (2017).\n\nDirect force measurements on DNA in a solid-state nanopore. U F Keyser, B N Koeleman, S Van Dorp, D Krapf, R M M Smeets, S G Lemay, N H Dekker, C Dekker, Nat. Phys. 2U. F. Keyser, B. N. Koeleman, S. Van Dorp, D. Krapf, R. M. M. Smeets, S. G. Lemay, N. H. Dekker, and C. Dekker, \"Direct force measurements on DNA in a solid-state nanopore,\" Nat. Phys. 2, 473-477 (2006).\n\nMeasurement of the position-dependent electrophoretic force on DNA in a glass nanocapillary. R D Bulushev, L J Steinbock, S Khlybov, J F Steinbock, U F Keyser, A Radenovic, Nanolett. 14R. D. Bulushev, L. J. Steinbock, S. Khlybov, J. F. Steinbock, U. F. Keyser, and A. Radenovic, \"Measurement of the position-dependent electrophoretic force on DNA in a glass nanocapillary,\" Nanolett. 14, 6606-6613 (2014).\n\nOptical trapping and manipulation of viruses and bacteria. A Ashkin, J M Dziedzic, Science. 235A. Ashkin and J. M. Dziedzic, \"Optical trapping and manipulation of viruses and bacteria,\" Science 235, 1517-1520 (1987).\n\nOptical trapping and manipulation of single cells using infrared laser beams. A Ashkin, J M Dziedzic, T Yamane, Nature. 330A. Ashkin, J. M. Dziedzic, and T. Yamane, \"Optical trapping and manipulation of single cells using infrared laser beams,\" Nature 330, 769-771 (1987).\n\nCompliance of bacterial flagella measured with optical tweezers. S M Block, D F Blair, H C Berg, Nature. 338S. M. Block, D. F. Blair, and H. C. Berg, \"Compliance of bacterial flagella measured with optical tweezers,\" Nature 338, 514-518 (1989).\n\nQuantifying forces in cell biology. P Roca-Cusachs, V Conte, X Trepat, Nat. Cell Biol. 19P. Roca-Cusachs, V. Conte, and X. Trepat, \"Quantifying forces in cell biology,\" Nat. Cell Biol. 19, 742-751 (2017).\n\nBiomechanical characterization at the cell scale: present and prospects. F Basoli, S M Giannitelli, P Mozetic, M Gori, A Bonfanti, M Trombetta, A Rainer, Front. Physiol. 91449F. Basoli, S. M. Giannitelli, P. Mozetic, M. Gori, A. Bonfanti, M. Trombetta, and A. Rainer, \"Biomechanical characterization at the cell scale: present and prospects,\" Front. Physiol. 9, 1449 (2018).\n\nForce nanoscopy of living cells. D J Müller, Y F Dufrêne, Curr. Biol. 21D. J. Müller and Y. F. Dufrêne, \"Force nanoscopy of living cells,\" Curr. Biol, 21, R212-R216 (2011).\n\nForce nanoscopy of cell mechanics and cell adhesion. Y F Dufrêne, A E Pelling, Nanoscale. 5Y. F. Dufrêne and A. E. Pelling, \"Force nanoscopy of cell mechanics and cell adhesion,\" Nanoscale 5, 4094-4104 (2013).\n\nExploring mechanochemical processes in the cell with optical tweezers. S Hormeno, J R Arias-Gonzalez, Biol. Cell. 98S. Hormeno and J. R. Arias-Gonzalez, \"Exploring mechanochemical processes in the cell with optical tweezers,\" Biol. Cell 98, 679-695 (2006).\n\nOptical trapping: A review of essential concepts. I Verdeny, A Farré, J Soler, C Quesada, E M Badosa, M Montes Usategui, Óptica Pura Apl. 44I. Verdeny, A. Farré, J. Mas Soler, C. López Quesada, E. M. Badosa, and M. Montes Usategui, \"Optical trapping: A review of essential concepts,\" Óptica Pura Apl. 44, 527-551 (2011).\n\nProbing force in living cells with optical tweezers: From single-molecule mechanics to cell mechanotransduction. C Arbore, L Perego, M Sergides, M Capitanio, Biophys. Rev pp. C. Arbore, L. Perego, M. Sergides, and M. Capitanio, \"Probing force in living cells with optical tweezers: From single-molecule mechanics to cell mechanotransduction,\" Biophys. Rev pp. 1-18 (2019).\n\nParticle-tracking microrheology of living cells: Principles and applications. D Wirtz, Annu. Rev. Biophys. 38D. Wirtz, \"Particle-tracking microrheology of living cells: Principles and applications,\" Annu. Rev. Biophys. 38, 301-326 (2009).\n\nSingle particle tracking: From theory to biophysical applications. H Shen, L J Tauzin, R Baiyasi, W Wang, N Moringo, B Shuang, C F Landes, Chem. Rev. 117H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, \"Single particle tracking: From theory to biophysical applications,\" Chem. Rev. 117, 7331-7376 (2017).\n\nTraction force microscopy on soft elastic substrates: A guide to recent computational advances. U S Schwarz, J R D Soiné, Biochimica Biophys. Acta. 1853U. S. Schwarz and J. R. D. Soiné, \"Traction force microscopy on soft elastic substrates: A guide to recent computational advances,\" Biochimica Biophys. Acta 1853, 3095-3104 (2015).\n\nThe future of traction force microscopy. H Colin-York, M Fritzsche, Curr. Opn. Biomed. Eng. 5H. Colin-York and M. Fritzsche, \"The future of traction force microscopy,\" Curr. Opn. Biomed. Eng. 5, 1-5 (2018).\n\nRecent technological advancements in traction force microscopy. A Ferrari, Biophys. Rev. 11A. Ferrari, \"Recent technological advancements in traction force microscopy,\" Biophys. Rev. 11, 679-681 (2019).\n\nMeasurement of cell microrheology by magnetic twisting cytometry with frequency domain demodulation. M Puig-De-Morales, M Grabulosa, J Alcaraz, J Mullol, G N Maksym, J J Fredberg, D Navajas, J. Appl. Physiol. 91M. Puig-De-Morales, M. Grabulosa, J. Alcaraz, J. Mullol, G. N. Maksym, J. J. Fredberg, and D. Navajas, \"Measurement of cell microrheology by magnetic twisting cytometry with frequency domain demodulation,\" J. Appl. Physiol. 91, 1152-1159 (2001).\n\nProbing cellular mechanics with acoustic force spectroscopy. R Sorkin, G Bergamaschi, D Kamsma, G Brand, E Dekel, Y Ofir-Birin, A Rudik, M Gironella, F Ritort, N Regev-Rudzki, W H Roos, G J L Wuite, Mol. Biol. Cell. 29R. Sorkin, G. Bergamaschi, D. Kamsma, G. Brand, E. Dekel, Y. Ofir-Birin, A. Rudik, M. Gironella, F. Ritort, N. Regev-Rudzki, W. H. Roos, and G. J. L. Wuite, \"Probing cellular mechanics with acoustic force spectroscopy,\" Mol. Biol. Cell 29, 2005-2011 (2018).\n\nThe application of micropipette aspiration in molecular mechanics of single cells. L M Lee, A P Liu, J. Nanotech. Eng. Med. 540902L. M. Lee and A. P. Liu, \"The application of micropipette aspiration in molecular mechanics of single cells,\" J. Nanotech. Eng. Med. 5, 040902 (2014).\n\nATP-dependent mechanics of red blood cells. T Betz, M Lenz, J.-F Joanny, C Sykes, Proc. Natl. Acad. Sci. 106T. Betz, M. Lenz, J.-F. Joanny, and C. Sykes, \"ATP-dependent mechanics of red blood cells,\" Proc. Natl. Acad. Sci. 106, 15320-15325 (2009).\n\nEquilibrium physics breakdown reveals the active nature of red blood cell flickering. H Turlier, D A Fedosov, B Audoly, T Auth, N S Gov, C Sykes, J.-F Joanny, G Gompper, T Betz, Nat. Phys. 12H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, \"Equilibrium physics breakdown reveals the active nature of red blood cell flickering,\" Nat. Phys. 12, 513-519 (2016).\n\nElastic properties and secondary structure formation of single-stranded DNA at monovalent and divalent salt conditions. A Bosco, J Camunas-Soler, F Ritort, Nucleic Acids Res. 42A. Bosco, J. Camunas-Soler, and F. Ritort, \"Elastic properties and secondary structure formation of single-stranded DNA at monovalent and divalent salt conditions,\" Nucleic Acids Res. 42, 2064-2074 (2013).\n\nMechanism of red blood cell aging: Relationship of cell density and cell age. S Piomelli, C Seaman, Am. J. Hematol. 42S. Piomelli and C. Seaman, \"Mechanism of red blood cell aging: Relationship of cell density and cell age,\" Am. J. Hematol. 42, 46-52 (1993).\n\nAdvances in the microrheology of complex fluids. T A Waigh, Rep. Prog. Phys. 7974601T. A. Waigh, \"Advances in the microrheology of complex fluids,\" Rep. Prog. Phys. 79, 074601 (2016).\n\nR B Bird, R C Armstrong, O Hassager, Fluid mechanics. New YorkJohn Wiley & Sons1R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of polymeric liquids. Vol. 1: Fluid mechanics (John Wiley & Sons, New York, 1987).\n\nHigh frequency viscoelastic behaviour of low molecular weight hyaluronic acid water solutions. S Fusco, A Borzacchiello, L Miccio, G Pesce, G Rusciano, A Sasso, P A Netti, Biorheol. 44S. Fusco, A. Borzacchiello, L. Miccio, G. Pesce, G. Rusciano, A. Sasso, and P. A. Netti, \"High frequency viscoelastic behaviour of low molecular weight hyaluronic acid water solutions,\" Biorheol. 44, 403-418 (2007).\n\nMicrorheology of complex fluids using optical tweezers: A comparison with macrorheological measurements. G Pesce, A C De Luca, G Rusciano, P A Netti, S Fusco, A Sasso, J. Opt. A Pure Appl. Opt. 1134016G. Pesce, A. C. De Luca, G. Rusciano, P. A. Netti, S. Fusco, and A. Sasso, \"Microrheology of complex fluids using optical tweezers: A comparison with macrorheological measurements,\" J. Opt. A Pure Appl. Opt. 11, 034016 (2009).\n\nFluid mechanics of microrheology. T M Squires, T G Mason, Annu. Rev. Fluid Mech. 42T. M. Squires and T. G. Mason, \"Fluid mechanics of microrheology,\" Annu. Rev. Fluid Mech. 42, 413-438 (2010).\n\nMicrorheology with optical tweezers: data analysis. M Tassieri, R M L Evans, R L Warren, N J Bailey, J M Cooper, New J. Phys. 14115032M. Tassieri, R. M. L. Evans, R. L. Warren, N. J. Bailey, and J. M. Cooper, \"Microrheology with optical tweezers: data analysis,\" New J. Phys. 14, 115032 (2012).\n\nThe fluctuation-dissipation theorem. R Kubo, Rep. Prog. Phys. 29R. Kubo, \"The fluctuation-dissipation theorem,\" Rep. Prog. Phys. 29, 255-284 (1966).\n\nOscillatory rise of bubbles in wormlike micellar fluids with different microstructures. N Z Handzy, A Belmonte, Phys. Rev. Lett. 92124501N. Z. Handzy and A. Belmonte, \"Oscillatory rise of bubbles in wormlike micellar fluids with different microstructures,\" Phys. Rev. Lett. 92, 124501 (2004).\n\nManual of symbols and terminology for physicochemical quantities and units, appendix II: Definitions, terminology and symbols in colloid and surface chemistry. D H Everett, Pure Appl. Chem. 31D. H. Everett, \"Manual of symbols and terminology for physicochemical quantities and units, appendix II: Definitions, terminology and symbols in colloid and surface chemistry,\" Pure Appl. Chem. 31, 577-638 (1972).\n\nStatics and dynamics of worm-like surfactant micelles. M E Cates, S J Candau, J. Phys.: Condens. Matter. 2M. E. Cates and S. J. Candau, \"Statics and dynamics of worm-like surfactant micelles,\" J. Phys.: Condens. Matter 2, 6869-6892 (1990).\n\nMicrorheology with optical tweezers: Measuring the relative viscosity of solutions \"at a glance. M Tassieri, F Giudice, E J Robertson, N Jain, B Fries, R Wilson, A Glidle, F Greco, P A Netti, P L Maffettone, T Bicanic, C J M , Sci. Rep. 58831M. Tassieri, F. Del Giudice, E. J. Robertson, N. Jain, B. Fries, R. Wilson, A. Glidle, F. Greco, P. A. Netti, P. L. Maffettone, T. Bicanic, and C. J. M., \"Microrheology with optical tweezers: Measuring the relative viscosity of solutions \"at a glance\",\" Sci. Rep. 5, 8831 (2015).\n\nR G Larson, The structure and rheology of complex fluids. New YorkOxford University PressR. G. Larson, The structure and rheology of complex fluids (Oxford University Press, New York, 1999).\n\nDetermining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations. B Schnurr, F Gittes, F C Mackintosh, C F Schmidt, Macromolecules. 30B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, \"Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations,\" Macromolecules 30, 7781-7792 (1997).\n\nA symmetrical method to obtain shear moduli from microrheology. K Nishi, M L Kilfoil, C F Schmidt, F C Mackintosh, Soft Matter. 14K. Nishi, M. L. Kilfoil, C. F. Schmidt, and F. C. MacKintosh, \"A symmetrical method to obtain shear moduli from microrheology,\" Soft Matter 14, 3716-3723 (2018).\n\nSmall-world rheology: An introduction to probe-based active microrheology. L G Wilson, W C K Poon, Phys. Chem. Chem. Phys. 13L. G. Wilson and W. C. K. Poon, \"Small-world rheology: An introduction to probe-based active microrheology,\" Phys. Chem. Chem. Phys. 13, 10617-10630 (2011).\n\nActive and passive microrheology in equilibrium and nonequilibrium systems. D Mizuno, D A Head, F C Mackintosh, C F Schmidt, Macromolecules. 41D. Mizuno, D. A. Head, F. C. MacKintosh, and C. F. Schmidt, \"Active and passive microrheology in equilibrium and nonequilibrium systems,\" Macromolecules 41, 7194-7202 (2008).\n\nPower spectrum of out-of-equilibrium forces in living cells: Amplitude and frequency dependence. F Gallet, D Arcizet, P Bohec, A Richert, Soft Matter. 5F. Gallet, D. Arcizet, P. Bohec, and A. Richert, \"Power spectrum of out-of-equilibrium forces in living cells: Amplitude and frequency dependence,\" Soft Matter 5, 2947-2953 (2009).\n\nNucleation and growth of thermoreversible polymer gels. J R Gomez-Solano, V Blickle, C Bechinger, Phys. Rev. E. 8712308J. R. Gomez-Solano, V. Blickle, and C. Bechinger, \"Nucleation and growth of thermoreversible polymer gels,\" Phys. Rev. E 87, 012308 (2013).\n\nExperimental study of out-of-equilibrium fluctuations in a colloidal suspension of laponite using optical traps. P Jop, J R Gomez-Solano, A Petrosyan, S Ciliberto, J. Stat. Mech.: Theo. Exp. 4012P. Jop, J. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, \"Experimental study of out-of-equilibrium fluctuations in a colloidal suspension of laponite using optical traps,\" J. Stat. Mech.: Theo. Exp. 2009, P04012 (2009).\n\nObservation of strongly heterogeneous dynamics at the depinning transition in a colloidal glass. N Şenbil, M Gruber, C Zhang, M Fuchs, F Scheffold, Phys. Rev. Lett. 122108002N. Şenbil, M. Gruber, C. Zhang, M. Fuchs, and F. Scheffold, \"Observation of strongly heterogeneous dynamics at the depinning transition in a colloidal glass,\" Phys. Rev. Lett. 122, 108002 (2019).\n\nBrownian dynamics of a sphere between parallel walls. E R Dufresne, D Altman, D G Grier, Europhys. Lett.). 53EPLE. R. Dufresne, D. Altman, and D. G. Grier, \"Brownian dynamics of a sphere between parallel walls,\" EPL (Europhys. Lett.) 53, 264-270 (2001).\n\nHydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces. G M Wang, R Prabhakar, E M Sevick, Phys. Rev. Lett. 103248303G. M. Wang, R. Prabhakar, and E. M. Sevick, \"Hydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces,\" Phys. Rev. Lett. 103, 248303 (2009).\n\nMeasurement of monolayer viscosity using noncontact microrheology. R Shlomovitz, A A Evans, T Boatwright, M Dennin, A J Levine, Phys. Rev. Lett. 110137802R. Shlomovitz, A. A. Evans, T. Boatwright, M. Dennin, and A. J. Levine, \"Measurement of monolayer viscosity using noncontact microrheology,\" Phys. Rev. Lett. 110, 137802 (2013).\n\nTwo-point microrheology of inhomogeneous soft materials. J C Crocker, M T Valentine, E R Weeks, T Gisler, P D Kaplan, A G Yodh, D A Weitz, Phys. Rev. Lett. 85J. C. Crocker, M. T. Valentine, E. R. Weeks, T. Gisler, P. D. Kaplan, A. G. Yodh, and D. A. Weitz, \"Two-point microrheology of inhomogeneous soft materials,\" Phys. Rev. Lett. 85, 888-891 (2000).\n\nActive microrheology to determine viscoelastic parameters of Stokes-Oldroyd B fluids using optical tweezers. S Paul, A Kundu, A Banerjee, J. Phys. Commun. 335002S. Paul, A. Kundu, and A. Banerjee, \"Active microrheology to determine viscoelastic parameters of Stokes-Oldroyd B fluids using optical tweezers,\" J. Phys. Commun. 3, 035002 (2019).\n\nNonlinear microrheology reveals entanglement-driven molecularlevel viscoelasticity of concentrated DNA. C D Chapman, R M Robertson-Anderson, Phys. Rev. Lett. 11398303C. D. Chapman and R. M. Robertson-Anderson, \"Nonlinear microrheology reveals entanglement-driven molecular- level viscoelasticity of concentrated DNA,\" Phys. Rev. Lett. 113, 098303 (2014).\n\nTransient dynamics of a colloidal particle driven through a viscoelastic fluid. J R Gomez-Solano, C Bechinger, New J. Phys. 17103032J. R. Gomez-Solano and C. Bechinger, \"Transient dynamics of a colloidal particle driven through a viscoelastic fluid,\" New J. Phys. 17, 103032 (2015).\n\nActive and passive microrheology: Theory and simulation. R N Zia, Ann. Rev. Fluid Mech. 50R. N. Zia, \"Active and passive microrheology: Theory and simulation,\" Ann. Rev. Fluid Mech. 50, 371-405 (2018).\n\nAn introduction to dynamics of colloids. J K G Dhont, ElsevierJ. K. G. Dhont, An introduction to dynamics of colloids (Elsevier, 1996).\n\nColloids as big atoms. W Poon, Science. 304W. Poon, \"Colloids as big atoms,\" Science 304, 830-831 (2004).\n\nInsights into phase transition kinetics from colloid science. V J Anderson, H N W Lekkerkerker, Nature. 416V. J. Anderson and H. N. W. Lekkerkerker, \"Insights into phase transition kinetics from colloid science,\" Nature 416, 811-815 (2002).\n\nPhase behaviour of concentrated suspensions of nearly hard colloidal spheres. P N Pusey, W Van Megen, Nature. 320P. N. Pusey and W. Van Megen, \"Phase behaviour of concentrated suspensions of nearly hard colloidal spheres,\" Nature 320, 340-342 (1986).\n\nPrediction of absolute crystal-nucleation rate in hard-sphere colloids. S Auer, D Frenkel, Nature. 409S. Auer and D. Frenkel, \"Prediction of absolute crystal-nucleation rate in hard-sphere colloids,\" Nature 409, 1020-1023 (2001).\n\nDirect observation of pre-critical nuclei in a metastable hard-sphere fluid. M S Elliot, S B Haddon, W C K Poon, J. Phys.: Condens. Matter. 13M. S. Elliot, S. B. Haddon, and W. C. K. Poon, \"Direct observation of pre-critical nuclei in a metastable hard-sphere fluid,\" J. Phys.: Condens. Matter 13, L553-L368 (2001).\n\nReal-space imaging of nucleation and growth in colloidal crystallization. U Gasser, E R Weeks, A Schofield, P N Pusey, D A Weitz, Science. 292U. Gasser, E. R. Weeks, A. Schofield, P. N. Pusey, and D. A. Weitz, \"Real-space imaging of nucleation and growth in colloidal crystallization,\" Science 292, 258-262 (2001).\n\nMultiple glassy states in a simple model system. K N Pham, A M Puertas, J Bergenholtz, S U Egelhaaf, A Moussaıd, P N Pusey, A B Schofield, M E Cates, M Fuchs, W C K Poon, Science. 296K. N. Pham, A. M. Puertas, J. Bergenholtz, S. U. Egelhaaf, A. Moussaıd, P. N. Pusey, A. B. Schofield, M. E. Cates, M. Fuchs, and W. C. K. Poon, \"Multiple glassy states in a simple model system,\" Science 296, 104-106 (2002).\n\nDirect visual observation of thermal capillary waves. D G A L Aarts, M Schmidt, H N W Lekkerkerker, Science. 304D. G. A. L. Aarts, M. Schmidt, and H. N. W. Lekkerkerker, \"Direct visual observation of thermal capillary waves,\" Science 304, 847-850 (2004).\n\nUltrastable particle-stabilized foams. U T Gonzenbach, A R Studart, E Tervoort, L J Gauckler, Ang. Chemie Int. Ed. 45U. T. Gonzenbach, A. R. Studart, E. Tervoort, and L. J. Gauckler, \"Ultrastable particle-stabilized foams,\" Ang. Chemie Int. Ed. 45, 3526-3530 (2006).\n\nUniversal emulsion stabilization from the arrested adsorption of rough particles at liquid-liquid interfaces. M Zanini, C Marschelke, S E Anachkov, E Marini, A Synytska, L Isa, Nat. Commun. 815701M. Zanini, C. Marschelke, S. E. Anachkov, E. Marini, A. Synytska, and L. Isa, \"Universal emulsion stabilization from the arrested adsorption of rough particles at liquid-liquid interfaces,\" Nat. Commun. 8, 15701 (2017).\n\nE Dickinson, D J Mcclements, Advances in food colloids. Springer Science & Business MediaE. Dickinson and D. J. McClements, Advances in food colloids (Springer Science & Business Media, 1995).\n\nChiral colloidal clusters. D Zerrouki, J Baudry, D Pine, P Chaikin, J Bibette, Nature. 455D. Zerrouki, J. Baudry, D. Pine, P. Chaikin, and J. Bibette, \"Chiral colloidal clusters,\" Nature 455, 380-382 (2008).\n\nVan der Waals, Over de Continuiteit van den Gas-en Vloeistoftoestand (Sijthoff, 1873). J , J. D. Van der Waals, Over de Continuiteit van den Gas-en Vloeistoftoestand (Sijthoff, 1873).\n\nVan der Waals forces: a handbook for biologists, chemists, engineers, and physicists. V Parsegian, Cambridge University PressV. Parsegian, Van der Waals forces: a handbook for biologists, chemists, engineers, and physicists (Cambridge University Press, 2005).\n\nDiffuse double-layer models, long-range forces, and ordering in clay colloids. M B Mcbride, P Baveye, Soil Sci. Soc. Am. J. 66M. B. McBride and P. Baveye, \"Diffuse double-layer models, long-range forces, and ordering in clay colloids,\" Soil Sci. Soc. Am. J. 66, 1207-1217 (2002).\n\nImportance of charge regulation in attractive double-layer forces between dissimilar surfaces. I Popa, P Sinha, M Finessi, P Maroni, G Papastavrou, M Borkovec, Phys. Rev. Lett. 104228301I. Popa, P. Sinha, M. Finessi, P. Maroni, G. Papastavrou, and M. Borkovec, \"Importance of charge regulation in attractive double-layer forces between dissimilar surfaces,\" Phys. Rev. Lett. 104, 228301 (2010).\n\nJ N Israelachvili, Intermolecular and surface forces. Academic PressJ. N. Israelachvili, Intermolecular and surface forces (Academic Press, 2015).\n\nW B Russel, W Russel, D A Saville, W R Schowalter, Colloidal dispersions. Cambridge University PressW. B. Russel, W. Russel, D. A. Saville, and W. R. Schowalter, Colloidal dispersions (Cambridge University Press, 1991).\n\nDirect measurement of colloidal forces using an atomic force microscope. W A Ducker, T J Senden, R M Pashley, Nature. 353W. A. Ducker, T. J. Senden, and R. M. Pashley, \"Direct measurement of colloidal forces using an atomic force microscope,\" Nature 353, 239-241 (1991).\n\nFlocculation of protein-stabilized oil-in-water emulsions. E Dickinson, Colloids Surfaces B: Biointerfaces. 81E. Dickinson, \"Flocculation of protein-stabilized oil-in-water emulsions,\" Colloids Surfaces B: Biointerfaces 81, 130-140 (2010).\n\nModern aspects of emulsion science. B P Binks, Royal Society of ChemistryB. P. Binks, Modern aspects of emulsion science (Royal Society of Chemistry, 2007).\n\nCreaming and flocculation in emulsions containing polysaccharide. Y Cao, E Dickinson, D J Wedlock, Food Hydrocoll. 4Y. Cao, E. Dickinson, and D. J. Wedlock, \"Creaming and flocculation in emulsions containing polysaccharide,\" Food Hydrocoll. 4, 185-195 (1990).\n\nCoagulation/flocculation process for dye removal using sludge from water treatment plant: optimization through response surface methodology. S S Moghaddam, M R A Moghaddam, M Arami, J. Hazard. Mater. 175S. S. Moghaddam, M. R. A. Moghaddam, and M. Arami, \"Coagulation/flocculation process for dye removal using sludge from water treatment plant: optimization through response surface methodology,\" J. Hazard. Mater. 175, 651-657 (2010).\n\nCoagulation and flocculation in water and wastewater treatment. J Bratby, IWA publishingJ. Bratby, Coagulation and flocculation in water and wastewater treatment (IWA publishing, 2016).\n\nOn interaction between two bodies immersed in a solution of macromolecules. S Asakura, F Oosawa, J. Chem. Phys. 22S. Asakura and F. Oosawa, \"On interaction between two bodies immersed in a solution of macromolecules,\" J. Chem. Phys. 22, 1255-1256 (1954).\n\nJ.-L Barrat, J.-P Hansen, Basic concepts for simple and complex liquids. Cambridge University PressJ.-L. Barrat and J.-P. Hansen, Basic concepts for simple and complex liquids (Cambridge University Press, 2003).\n\nThe depletion attraction: An underappreciated force driving cellular organization. D Marenduzzo, K Finan, P R Cook, J. Cell. Biol. 175D. Marenduzzo, K. Finan, and P. R. Cook, \"The depletion attraction: An underappreciated force driving cellular organization,\" J. Cell. Biol. 175, 681-686 (2006).\n\nPolymer solutions near an interface. adsorption and depletion layers. P G De Gennes, Macromolecules. 14P. G. de Gennes, \"Polymer solutions near an interface. adsorption and depletion layers,\" Macromolecules 14, 1637-1644 (1981).\n\nSurface tension of high-polymer solutions. F Oosawa, S Asakura, J. Chem. Phys. 22F. Oosawa and S. Asakura, \"Surface tension of high-polymer solutions,\" J. Chem. Phys. 22, 1255-1255 (1954).\n\nPhenomena at the walls in a critical binary mixture. M E Fisher, P G De Gennes, C. R. Acad. Sci. Paris B. 287M. E. Fisher and P. G. de Gennes, \"Phenomena at the walls in a critical binary mixture.\" C. R. Acad. Sci. Paris B 287, 207-209 (1978).\n\nThe Casimir effect: From quantum to critical fluctuations. A Gambassi, J. Phys.: Conf. Ser. 16112037A. Gambassi, \"The Casimir effect: From quantum to critical fluctuations,\" J. Phys.: Conf. Ser. 161, 012037 (2009).\n\nCritical Casimir forces steered by patterned substrates. A Gambassi, S Dietrich, Soft Matter. 7A. Gambassi and S. Dietrich, \"Critical Casimir forces steered by patterned substrates,\" Soft Matter 7, 1247-1253 (2011).\n\nOn the attraction between two perfectly conducting plates. H B G Casimir, Proc. K. Ned. Akad. Wet. 51H. B. G. Casimir, \"On the attraction between two perfectly conducting plates,\" Proc. K. Ned. Akad. Wet 51, 793-795 (1948).\n\nDirect measurement of critical Casimir forces. C Hertlein, L Helden, A Gambassi, S Dietrich, C Bechinger, Nature. 451C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C. Bechinger, \"Direct measurement of critical Casimir forces,\" Nature 451, 172-175 (2008).\n\nNonadditivity of critical Casimir forces. S Paladugu, A Callegari, Y Tuna, L Barth, S Dietrich, A Gambassi, G Volpe, Nat. Commun. 711403S. Paladugu, A. Callegari, Y. Tuna, L. Barth, S. Dietrich, A. Gambassi, and G. Volpe, \"Nonadditivity of critical Casimir forces,\" Nat. Commun. 7, 11403 (2016).\n\nDirect observation of colloidal aggregation by critical Casimir forces. D Bonn, J Otwinowski, S Sacanna, H Guo, G Wegdam, P Schall, Phys. Rev. Lett. 103156101D. Bonn, J. Otwinowski, S. Sacanna, H. Guo, G. Wegdam, and P. Schall, \"Direct observation of colloidal aggregation by critical Casimir forces,\" Phys. Rev. Lett. 103, 156101 (2009).\n\nWhen depletion goes critical. R Piazza, S Buzzaccaro, A Parola, J Colombo, J. Phys.: Condens. Matter. 23194114R. Piazza, S. Buzzaccaro, A. Parola, and J. Colombo, \"When depletion goes critical,\" J. Phys.: Condens. Matter 23, 194114 (2011).\n\nControlling the dynamics of colloidal particles by critical Casimir forces. A Magazzù, A Callegari, J P Staforelli, A Gambassi, S Dietrich, G Volpe, Soft matter. 15A. Magazzù, A. Callegari, J. P. Staforelli, A. Gambassi, S. Dietrich, and G. Volpe, \"Controlling the dynamics of colloidal particles by critical Casimir forces,\" Soft matter 15, 2152-2162 (2019).\n\nCritical Casimir forces in colloidal suspensions on chemically patterned surfaces. F Soyka, O Zvyagolskaya, C Hertlein, L Helden, C Bechinger, Phys. Rev. Lett. 101208301F. Soyka, O. Zvyagolskaya, C. Hertlein, L. Helden, and C. Bechinger, \"Critical Casimir forces in colloidal suspensions on chemically patterned surfaces,\" Phys. Rev. Lett. 101, 208301 (2008).\n\nTunability of critical Casimir interactions by boundary conditions. U Nellen, L Helden, C Bechinger, Europhys. Lett.). 8826001EPLU. Nellen, L. Helden, and C. Bechinger, \"Tunability of critical Casimir interactions by boundary conditions,\" EPL (Europhys. Lett.) 88, 26001 (2009).\n\nOptical tweezers in colloid and interface science. D G Grier, Curr. Opn. Colloid Interface Sci. 2D. G. Grier, \"Optical tweezers in colloid and interface science,\" Curr. Opn. Colloid Interface Sci. 2, 264-270 (1997).\n\nBlinking optical tweezers for microrheology measurements of weak elasticity complex fluids. G Pesce, G Rusciano, A Sasso, Opt. Express. 18G. Pesce, G. Rusciano, and A. Sasso, \"Blinking optical tweezers for microrheology measurements of weak elasticity complex fluids,\" Opt. Express 18, 2116-2126 (2010).\n\nLong-term influence of fluid inertia on the diffusion of a Brownian particle. G Pesce, G Volpe, G Volpe, A Sasso, Phys. Rev. E. 9042309G. Pesce, G. Volpe, G. Volpe, and A. Sasso, \"Long-term influence of fluid inertia on the diffusion of a Brownian particle,\" Phys. Rev. E 90, 042309 (2014).\n\nLower critical solution coexistence curve and physical properties (density, viscosity, surface tension, and interfacial tension) of 2,6-lutidine+water. C A Grattoni, R A Dawe, C Y Seah, J D Gray, J. Chem. Eng. Data. 38C. A. Grattoni, R. A. Dawe, C. Y. Seah, and J. D. Gray, \"Lower critical solution coexistence curve and physical properties (density, viscosity, surface tension, and interfacial tension) of 2,6-lutidine+water,\" J. Chem. Eng. Data 38, 516-519 (1993).\n\nCritical Casimir effect in classical binary liquid mixtures. A Gambassi, A Maciołek, C Hertlein, U Nellen, L Helden, C Bechinger, S Dietrich, Phys. Rev. E. 8061143A. Gambassi, A. Maciołek, C. Hertlein, U. Nellen, L. Helden, C. Bechinger, and S. Dietrich, \"Critical Casimir effect in classical binary liquid mixtures,\" Phys. Rev. E 80, 061143 (2009).\n\nMethods of digital video microscopy for colloidal studies. J C Crocker, D G Grier, J. Colloid Interface Sci. 179J. C. Crocker and D. G. Grier, \"Methods of digital video microscopy for colloidal studies,\" J. Colloid Interface Sci. 179, 298-310 (1996).\n\nMonte Carlo simulation results for critical Casimir forces. O Vasilyev, A Gambassi, A Maciołek, S Dietrich, Europhys. Lett.). 8060009EPLO. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, \"Monte Carlo simulation results for critical Casimir forces,\" EPL (Europhys. Lett.) 80, 60009 (2007).\n\nUniversal scaling functions of critical Casimir forces obtained by Monte Carlo simulations. O Vasilyev, A Gambassi, A Maciołek, S Dietrich, Phys. Rev. E. 7941142O. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, \"Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations,\" Phys. Rev. E 79, 041142 (2009).\n\nThermodynamic Casimir effect: Universality and corrections to scaling. M Hasenbusch, Phys. Rev. B. 85174421M. Hasenbusch, \"Thermodynamic Casimir effect: Universality and corrections to scaling,\" Phys. Rev. B 85, 174421 (2012).\n\nBrownian diffusion of particles with hydrodynamic interaction. G K Batchelor, J. Fluid Mech. 74G. K. Batchelor, \"Brownian diffusion of particles with hydrodynamic interaction,\" J. Fluid Mech. 74, 1-29 (1976).\n\nInterdiffusion coefficient and dynamic viscosity for the mixture 2,6-lutidine+water near the lower consolute point. J C Clunie, J K Baird, Phys. Chem. Liq. 37J. C. Clunie and J. K. Baird, \"Interdiffusion coefficient and dynamic viscosity for the mixture 2,6-lutidine+water near the lower consolute point,\" Phys. Chem. Liq. 37, 357-371 (1999).\n\nAssembling quantum dots via critical Casimir forces. E Marino, T E Kodger, J B Hove, A H Velders, P Schall, Sol. Energy Mater. Sol. Cells. 158E. Marino, T. E. Kodger, J. B. ten Hove, A. H. Velders, and P. Schall, \"Assembling quantum dots via critical Casimir forces,\" Sol. Energy Mater. Sol. Cells 158, 154-159 (2016).\n\nTuning patchy bonds induced by critical Casimir forces. T A Nguyen, A Newton, D J Kraft, P G Bolhuis, P Schall, Materials. 101265T. A. Nguyen, A. Newton, D. J. Kraft, P. G. Bolhuis, and P. Schall, \"Tuning patchy bonds induced by critical Casimir forces,\" Materials 10, 1265 (2017).\n\nMicroscopic engine powered by critical demixing. F Schmidt, A Magazzù, A Callegari, L Biancofiore, F Cichos, G Volpe, Phys. Rev. Lett. 12068004F. Schmidt, A. Magazzù, A. Callegari, L. Biancofiore, F. Cichos, and G. Volpe, \"Microscopic engine powered by critical demixing,\" Phys. Rev. Lett. 120, 068004 (2018).\n\nDirect measurement of the van der Waals interaction between two Rydberg atoms. L Beguin, A Vernier, R Chicireanu, T Lahaye, A Browaeys, Phys. Rev. Lett. 110263201L. Beguin, A. Vernier, R. Chicireanu, T. Lahaye, and A. Browaeys, \"Direct measurement of the van der Waals interaction between two Rydberg atoms,\" Phys. Rev. Lett. 110, 263201 (2013).\n\nMaterials perspective on Casimir and van der Waals interactions. L M Woods, D A R Dalvit, A Tkatchenko, P Rodriguez-Lopez, A W Rodriguez, R Podgornik, Rev. Mod. Phys. 8845003L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. W. Rodriguez, and R. Podgornik, \"Materials perspective on Casimir and van der Waals interactions,\" Rev. Mod. Phys. 88, 045003 (2016).\n\nProgrammable self-assembly. L Cademartiri, K J M Bishop, Nat. Mater. 14L. Cademartiri and K. J. M. Bishop, \"Programmable self-assembly,\" Nat. Mater. 14, 2-9 (2014).\n\nEscape and synchronization of a Brownian particle. A Simon, A Libchaber, Phys. Rev. Lett. 68A. Simon and A. Libchaber, \"Escape and synchronization of a Brownian particle,\" Phys. Rev. Lett. 68, 3375-3379 (1992).\n\nThermally activated transitions in a bistable three-dimensional optical trap. L I Mccann, M Dykman, B Golding, Nature. 402L. I. McCann, M. Dykman, and B. Golding, \"Thermally activated transitions in a bistable three-dimensional optical trap,\" Nature 402, 785-787 (1999).\n\nBrownian motion in a field of force and the diffusion model of chemical reactions. H A Kramers, Physica. 7H. A. Kramers, \"Brownian motion in a field of force and the diffusion model of chemical reactions,\" Physica 7, 284-304 (1940).\n\nFrequency factors in the thermally activated process. R Landauer, J A Swanson, Phys. Rev. 121R. Landauer and J. A. Swanson, \"Frequency factors in the thermally activated process,\" Phys. Rev. 121, 1668-1674 (1961).\n\nDouble nanohole optical trapping: dynamics and protein-antibody co-trapping. A Zehtabi-Oskuie, H Jiang, B R Cyr, D W Rennehan, A A Al-Balushi, R Gordon, Lab Chip. 13A. Zehtabi-Oskuie, H. Jiang, B. R. Cyr, D. W. Rennehan, A. A. Al-Balushi, and R. Gordon, \"Double nanohole optical trapping: dynamics and protein-antibody co-trapping,\" Lab Chip 13, 2563-2568 (2013).\n\nPartial synchronization of stochastic oscillators through hydrodynamic coupling. A Curran, M P Lee, M J Padgett, J M Cooper, R. Di Leonardo, Phys. Rev. Lett. 108240601A. Curran, M. P. Lee, M. J. Padgett, J. M. Cooper, and R. Di Leonardo, \"Partial synchronization of stochastic oscillators through hydrodynamic coupling,\" Phys. Rev. Lett. 108, 240601 (2012).\n\nPhase measurements of barrier crossings in a periodically modulated double-well potential. Y Seol, D L Stein, K Visscher, Phys. Rev. Lett. 10350601Y. Seol, D. L. Stein, and K. Visscher, \"Phase measurements of barrier crossings in a periodically modulated double-well potential,\" Phys. Rev. Lett. 103, 050601 (2009).\n\nIrreversibility and generalized noise. H B Callen, T A Welton, Phys. Rev. 83H. B. Callen and T. A. Welton, \"Irreversibility and generalized noise,\" Phys. Rev. 83, 34-40 (1951).\n\nFluctuation-dissipation: response theory in statistical physics. U M B Marconi, A Puglisi, L Rondoni, A Vulpiani, Phys. Rep. 461U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, \"Fluctuation-dissipation: response theory in statistical physics,\" Phys. Rep. 461, 111-195 (2008).\n\nFluctuation relations in simple examples of non-equilibrium steady states. R Chetrite, G Falkovich, K Gawedzki, J. Stat. Mech.: Theo. Exp. 8005R. Chetrite, G. Falkovich, and K. Gawedzki, \"Fluctuation relations in simple examples of non-equilibrium steady states,\" J. Stat. Mech.: Theo. Exp. 2008, P08005 (2008).\n\nEinstein relation generalized to nonequilibrium. V Blickle, T Speck, C Lutz, U Seifert, C Bechinger, Phys. Rev. Lett. 98210601V. Blickle, T. Speck, C. Lutz, U. Seifert, and C. Bechinger, \"Einstein relation generalized to nonequilibrium,\" Phys. Rev. Lett. 98, 210601 (2007).\n\nExperimental test of a new equality: Measuring heat dissipation in an optically driven colloidal system. S Toyabe, H.-R Jiang, T Nakamura, Y Murayama, M Sano, Phys. Rev. E. 7511122S. Toyabe, H.-R. Jiang, T. Nakamura, Y. Murayama, and M. Sano, \"Experimental test of a new equality: Measuring heat dissipation in an optically driven colloidal system,\" Phys. Rev. E 75, 011122 (2007).\n\nEnergy dissipation of a Brownian particle in a viscoelastic fluid. S Toyabe, M Sano, Phys. Rev. E. 7741403S. Toyabe and M. Sano, \"Energy dissipation of a Brownian particle in a viscoelastic fluid,\" Phys. Rev. E 77, 041403 (2008).\n\nFluctuations, linear response and heat flux of an aging system. J R Gomez-Solano, A Petrosyan, S Ciliberto, Europhys. Lett.). 9810007EPLJ. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, \"Fluctuations, linear response and heat flux of an aging system,\" EPL (Europhys. Lett.) 98, 10007 (2012).\n\nProbing active forces via a fluctuationdissipation relation: Application to living cells. P Bohec, F Gallet, C Maes, S Safaverdi, P Visco, F Van Wijland, Europhys. Lett.). 10250005EPLP. Bohec, F. Gallet, C. Maes, S. Safaverdi, P. Visco, and F. Van Wijland, \"Probing active forces via a fluctuation- dissipation relation: Application to living cells,\" EPL (Europhys. Lett.) 102, 50005 (2013).\n\nPeriodic forcing of a Brownian particle. L P Faucheux, G Stolovitzky, A Libchaber, Phys. Rev. E. 51L. P. Faucheux, G. Stolovitzky, and A. Libchaber, \"Periodic forcing of a Brownian particle,\" Phys. Rev. E 51, 5239-5250 (1995).\n\nExperimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state. J R Gomez-Solano, A Petrosyan, S Ciliberto, R Chetrite, K Gawędzki, Phys. Rev. Lett. 10340601J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, R. Chetrite, and K. Gawędzki, \"Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state,\" Phys. Rev. Lett. 103, 040601 (2009).\n\nFluctuations and response in a non-equilibrium micron-sized system. J R Gomez-Solano, A Petrosyan, S Ciliberto, C Maes, J. Stat. Mech.: Theo. Exp. 1008J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, and C. Maes, \"Fluctuations and response in a non-equilibrium micron-sized system,\" J. Stat. Mech.: Theo. Exp. 2011, P01008 (2011).\n\nCharacterizing potentials by a generalized Boltzmann factor. V Blickle, T Speck, U Seifert, C Bechinger, Phys. Rev. E. 7560101V. Blickle, T. Speck, U. Seifert, and C. Bechinger, \"Characterizing potentials by a generalized Boltzmann factor,\" Phys. Rev. E 75, 060101 (2007).\n\nFluctuations, linear response, and currents in out-of-equilibrium systems. S Ciliberto, R Gomez-Solano, A Petrosyan, Annu. Rev. Condens. Matter Phys. 4S. Ciliberto, R. Gomez-Solano, and A. Petrosyan, \"Fluctuations, linear response, and currents in out-of-equilibrium systems,\" Annu. Rev. Condens. Matter Phys. 4, 235-261 (2013).\n\nLangevin equation and thermodynamics. K Sekimoto, Prog. Theo. Phys. Suppl. 130K. Sekimoto, \"Langevin equation and thermodynamics,\" Prog. Theo. Phys. Suppl. 130, 17-27 (1998).\n\nK Sekimoto, Stochastic energetics. SpringerK. Sekimoto, Stochastic energetics (Springer, 2010).\n\nExperimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. G M Wang, E M Sevick, E Mittag, D J Searles, D J Evans, Phys. Rev. Lett. 8950601G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, \"Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,\" Phys. Rev. Lett. 89, 050601 (2002).\n\nEntropy production along a stochastic trajectory and an integral fluctuation theorem. U Seifert, Phys. Rev. Lett. 9540602U. Seifert, \"Entropy production along a stochastic trajectory and an integral fluctuation theorem,\" Phys. Rev. Lett. 95, 040602 (2005).\n\nNonequilibrium equality for free energy differences. C Jarzynski, Phys. Rev. Lett. 78C. Jarzynski, \"Nonequilibrium equality for free energy differences,\" Phys. Rev. Lett. 78, 2690-2693 (1997).\n\nEntropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. G E Crooks, Phys. Rev. E. 60G. E. Crooks, \"Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences,\" Phys. Rev. E 60, 2721-2726 (1999).\n\nRéflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. S Carnot, Annales scientifiques de l'École Normale Supérieure. 1S. Carnot, \"Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance,\" in Annales scientifiques de l'École Normale Supérieure, vol. 1 (1824), pp. 393-457.\n\nAn inconvenient truth about thermoelectrics. C B Vining, Nat. Mater. 8C. B. Vining, \"An inconvenient truth about thermoelectrics,\" Nat. Mater. 8, 83-85 (2009).\n\nThe efficiency of atomic power stations (a review). I I Novikov, J. Nucl. Energy. 7I. I. Novikov, \"The efficiency of atomic power stations (a review),\" J. Nucl. Energy 7, 125-128 (1958).\n\nEfficiency of a Carnot engine at maximum power output. F L Curzon, B Ahlborn, Am. J. Phys. 43F. L. Curzon and B. Ahlborn, \"Efficiency of a Carnot engine at maximum power output,\" Am. J. Phys. 43, 22-24 (1975).\n\nHeat engines and the Curzon-Ahlborn efficiency. A Hernández, J M M Roco, A Medina, N Sánchez-Salas, Revista Mexicana de Física. 60A. Calvo Hernández, J. M. M. Roco, A. Medina, and N. Sánchez-Salas, \"Heat engines and the Curzon-Ahlborn efficiency,\" Revista Mexicana de Física 60, 384-389 (2014).\n\nRealization of a micrometre-sized stochastic heat engine. V Blickle, C Bechinger, Nat. Phys. 8V. Blickle and C. Bechinger, \"Realization of a micrometre-sized stochastic heat engine,\" Nat. Phys. 8, 143-146 (2012).\n\nA microscopic steam engine implemented in an optical tweezer. P A Quinto-Su, Nat. Commun. 55889P. A. Quinto-Su, \"A microscopic steam engine implemented in an optical tweezer,\" Nat. Commun. 5, 5889 (2014).\n\nEffective heating to several thousand kelvins of an optically trapped sphere in a liquid. I A Martínez, É Roldán, J M R Parrondo, D Petrov, Phys. Rev. E. 8732159I. A. Martínez, É. Roldán, J. M. R. Parrondo, and D. Petrov, \"Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,\" Phys. Rev. E 87, 032159 (2013).\n\nSteady-state fluctuation relations for systems driven by an external random force. J R Gomez-Solano, L Bellon, A Petrosyan, S Ciliberto, Europhys. Lett.). 8960003EPLJ. R. Gomez-Solano, L. Bellon, A. Petrosyan, and S. Ciliberto, \"Steady-state fluctuation relations for systems driven by an external random force,\" EPL (Europhys. Lett.) 89, 60003 (2010).\n\nEnergy flow between two hydrodynamically coupled particles kept at different effective temperatures. A Bérut, A Petrosyan, S Ciliberto, Europhys. Lett.). 10760004EPLA. Bérut, A. Petrosyan, and S. Ciliberto, \"Energy flow between two hydrodynamically coupled particles kept at different effective temperatures,\" EPL (Europhys. Lett.) 107, 60004 (2014).\n\nAdiabatic processes realized with a trapped Brownian particle. I A Martínez, É Roldán, L Dinis, D Petrov, R A Rica, Phys. Rev. Lett. 114120601I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, and R. A. Rica, \"Adiabatic processes realized with a trapped Brownian particle,\" Phys. Rev. Lett. 114, 120601 (2015).\n\nBrownian Carnot engine. I A Martínez, É Roldán, L Dinis, D Petrov, J M R Parrondo, R A Rica, Nat. Phys. 12I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, \"Brownian Carnot engine,\" Nat. Phys. 12, 67-70 (2016).\n\nEfficiency at maximum power: An analytically solvable model for stochastic heat engines. T Schmiedl, U Seifert, Europhys. Lett.). 8120003EPLT. Schmiedl and U. Seifert, \"Efficiency at maximum power: An analytically solvable model for stochastic heat engines,\" EPL (Europhys. Lett.) 81, 20003 (2007).\n\nTheory of Heat, Text-books of science (Longmans, 1871). J C Maxwell, J. C. Maxwell, Theory of Heat, Text-books of science (Longmans, 1871).\n\nÜber die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. L Szilard, Zeit. Physik. 53L. Szilard, \"Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen,\" Zeit. Physik 53, 840-856 (1929).\n\nMaxwell's demon cannot operate: Information and entropy. i. L Brillouin, J. Appl. Phys. 22L. Brillouin, \"Maxwell's demon cannot operate: Information and entropy. i,\" J. Appl. Phys. 22, 334-337 (1951).\n\nIrreversibility and heat generation in the computing process. R Landauer, IBM J. Res. Dev. 5R. Landauer, \"Irreversibility and heat generation in the computing process,\" IBM J. Res. Dev. 5, 183-191 (1961).\n\nThe thermodynamics of computation: A review. C H Bennett, Int. J. Th. Phys. 21C. H. Bennett, \"The thermodynamics of computation: A review,\" Int. J. Th. Phys. 21, 905-940 (1982).\n\nSecond law of thermodynamics with discrete quantum feedback control. T Sagawa, M Ueda, Phys. Rev. Lett. 10080403T. Sagawa and M. Ueda, \"Second law of thermodynamics with discrete quantum feedback control,\" Phys. Rev. Lett. 100, 080403 (2008).\n\nThermodynamics of information processing in small systems. T Sagawa, Prog. Theo. Phys. 127T. Sagawa, \"Thermodynamics of information processing in small systems,\" Prog. Theo. Phys. 127, 1-56 (2012).\n\nExperimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. S Toyabe, T Sagawa, M Ueda, E Muneyuki, M Sano, Nat. Phys. 6S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, \"Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality,\" Nat. Phys. 6, 988-992 (2010).\n\nMinimal energy cost for thermodynamic information processing: measurement and information erasure. T Sagawa, M Ueda, Phys. Rev. Lett. 102250602T. Sagawa and M. Ueda, \"Minimal energy cost for thermodynamic information processing: measurement and information erasure,\" Phys. Rev. Lett. 102, 250602 (2009).\n\nUniversal features in the energetics of symmetry breaking. É Roldán, I A Martinez, J M R Parrondo, D Petrov, Nat. Phys. 10É. Roldán, I. A. Martinez, J. M. R. Parrondo, and D. Petrov, \"Universal features in the energetics of symmetry breaking,\" Nat. Phys. 10, 457-461 (2014).\n\nLarge work extraction and the Landauer limit in a continuous Maxwell demon. M Ribezzi-Crivellari, F Ritort, Nat. Phys. 15M. Ribezzi-Crivellari and F. Ritort, \"Large work extraction and the Landauer limit in a continuous Maxwell demon,\" Nat. Phys. 15, 660-664 (2019).\n\nHydrodynamic synchronization of light driven microrotors. R Di Leonardo, A Búzás, L Kelemen, G Vizsnyiczai, L Oroszi, P Ormos, Phys. Rev. Lett. 10934104R. Di Leonardo, A. Búzás, L. Kelemen, G. Vizsnyiczai, L. Oroszi, and P. Ormos, \"Hydrodynamic synchronization of light driven microrotors,\" Phys. Rev. Lett. 109, 034104 (2012).\n\nCollective synchronization states in arrays of driven colloidal oscillators. R Lhermerout, N Bruot, G M Cicuta, J Kotar, P Cicuta, New J. Phys. 14105023R. Lhermerout, N. Bruot, G. M. Cicuta, J. Kotar, and P. Cicuta, \"Collective synchronization states in arrays of driven colloidal oscillators,\" New J. Phys. 14, 105023 (2012).\n\nStochastic hydrodynamic synchronization in rotating energy landscapes. N Koumakis, R Di Leonardo, Phys. Rev. Lett. 110174103N. Koumakis and R. Di Leonardo, \"Stochastic hydrodynamic synchronization in rotating energy landscapes,\" Phys. Rev. Lett. 110, 174103 (2013).\n\nOptimal hydrodynamic synchronization of colloidal rotors. J Kotar, L Debono, N Bruot, S Box, D Phillips, S Simpson, S Hanna, P Cicuta, Phys. Rev. Lett. 111228103J. Kotar, L. Debono, N. Bruot, S. Box, D. Phillips, S. Simpson, S. Hanna, and P. Cicuta, \"Optimal hydrodynamic synchronization of colloidal rotors,\" Phys. Rev. Lett. 111, 228103 (2013).\n\nActive particles in complex and crowded environments. C Bechinger, R Di Leonardo, H Löwen, C Reichhardt, G Volpe, G Volpe, Rev. Mod. Phys. 8845006C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, \"Active particles in complex and crowded environments,\" Rev. Mod. Phys. 88, 045006 (2016).\n\nMachine learning for active matter. F Cihos, K Gustavsson, B Mehlig, G Volpe, Nat. Mach. Intell. 2F. Cihos, K. Gustavsson, B. Mehlig, and G. Volpe, \"Machine learning for active matter,\" Nat. Mach. Intell. 2, 94-103 (2020).\n\nNon-boltzmann stationary distributions and nonequilibrium relations in active baths. A Argun, A.-R Moradi, E Pinçe, G B Bagci, A Imparato, G Volpe, Phys. Rev. E. 9462150A. Argun, A.-R. Moradi, E. Pinçe, G. B. Bagci, A. Imparato, and G. Volpe, \"Non-boltzmann stationary distributions and nonequilibrium relations in active baths,\" Phys. Rev. E 94, 062150 (2016).\n\nPhase transitions of colloidal monolayers in periodic pinning arrays. K Mangold, P Leiderer, C Bechinger, Phys. Rev. Lett. 90158302K. Mangold, P. Leiderer, and C. Bechinger, \"Phase transitions of colloidal monolayers in periodic pinning arrays,\" Phys. Rev. Lett. 90, 158302 (2003).\n\nTunable colloids: Control of colloidal phase transitions with tunable interactions. A Yethiraj, Soft Matter. 3A. Yethiraj, \"Tunable colloids: Control of colloidal phase transitions with tunable interactions,\" Soft Matter 3, 1099-1115 (2007).\n\nLight-induced phase transitions of colloidal monolayers with crystalline order. L Zaidouny, T Bohlein, R Roth, C Bechinger, Soft Matter. 9L. Zaidouny, T. Bohlein, R. Roth, and C. Bechinger, \"Light-induced phase transitions of colloidal monolayers with crystalline order,\" Soft Matter 9, 9230-9236 (2013).\n\nPhysical grounds for biological effect of laser radiation. A N Rubinov, J. Phys. D: Appl. Phys. 362317A. N. Rubinov, \"Physical grounds for biological effect of laser radiation,\" J. Phys. D: Appl. Phys. 36, 2317 (2003).\n\nTrapping and manipulation of low-index particles in a two-dimensional interferometric optical trap. M P Macdonald, L Paterson, W Sibbett, K Dholakia, P E Bryant, Opt. Lett. 26M. P. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and P. E. Bryant, \"Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,\" Opt. Lett. 26, 863-865 (2001).\n\nDynamical analysis of an optical rocking ratchet: Theory and experiment. A V Arzola, K Volke-Sepulveda, J L Mateos, Phys. Rev. E. 8762910A. V. Arzola, K. Volke-Sepulveda, and J. L. Mateos, \"Dynamical analysis of an optical rocking ratchet: Theory and experiment,\" Phys. Rev. E 87, 062910 (2013).\n\nActive dynamics of colloidal particles in time-varying laser speckle patterns. S Bianchi, R Pruner, G Vizsnyiczai, C Maggi, R. Di Leonardo, Sci. Rep. 627681S. Bianchi, R. Pruner, G. Vizsnyiczai, C. Maggi, and R. Di Leonardo, \"Active dynamics of colloidal particles in time-varying laser speckle patterns,\" Sci. Rep. 6, 27681 (2016).\n\nForce mapping of an extended light pattern in an inclined plane: Deterministic regime. A V Arzola, K Volke-Sepúlveda, J L Mateos, Opt. Express. 17A. V. Arzola, K. Volke-Sepúlveda, and J. L. Mateos, \"Force mapping of an extended light pattern in an inclined plane: Deterministic regime,\" Opt. Express 17, 3429-3440 (2009).\n\nMoving interference patterns created using the angular Doppler-effect. J Arlt, M Macdonald, L Paterson, W Sibbett, K Dholakia, K Volke-Sepulveda, Opt. Express. 10J. Arlt, M. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, \"Moving interference patterns created using the angular Doppler-effect,\" Opt. Express 10, 844-852 (2002).\n\nSurface delivery of a single nanoparticle under moving evanescent standing-wave illumination. M Šiler, T Čižmár, A Jonáš, P Zemánek, New J. Phys. 10113010M. Šiler, T. Čižmár, A. Jonáš, and P. Zemánek, \"Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,\" New J. Phys. 10, 113010 (2008).\n\nPhase behavior of two-dimensional colloidal systems in the presence of periodic light fields. C Bechinger, M Brunner, P Leiderer, Phys. Rev. Lett. 86C. Bechinger, M. Brunner, and P. Leiderer, \"Phase behavior of two-dimensional colloidal systems in the presence of periodic light fields,\" Phys. Rev. Lett. 86, 930-933 (2001).\n\nPhase behavior of colloidal molecular crystals on triangular light lattices. M Brunner, C Bechinger, Phys. Rev. Lett. 88248302M. Brunner and C. Bechinger, \"Phase behavior of colloidal molecular crystals on triangular light lattices,\" Phys. Rev. Lett. 88, 248302 (2002).\n\nTailoring of phononic band structures in colloidal crystals. J Baumgartl, M Zvyagolskaya, C Bechinger, Phys. Rev. Lett. 99205503J. Baumgartl, M. Zvyagolskaya, and C. Bechinger, \"Tailoring of phononic band structures in colloidal crystals,\" Phys. Rev. Lett. 99, 205503 (2007).\n\n3d interferometric optical tweezers using a single spatial light modulator. E Schonbrun, R Piestun, P Jordan, J Cooper, K D Wulff, J Courtial, M Padgett, Opt. Express. 13E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, \"3d interferometric optical tweezers using a single spatial light modulator,\" Opt. Express 13, 3777-3786 (2005).\n\nArchimedean-like tiling on decagonal quasicrystalline surfaces. J Mikhael, J Roth, L Helden, C Bechinger, Nature. 454J. Mikhael, J. Roth, L. Helden, and C. Bechinger, \"Archimedean-like tiling on decagonal quasicrystalline surfaces,\" Nature 454, 501-504 (2008).\n\nProliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields. J Mikhael, M Schmiedeberg, S Rausch, J Roth, H Stark, C Bechinger, Proc. Natl. Acad. Sci. 107J. Mikhael, M. Schmiedeberg, S. Rausch, J. Roth, H. Stark, and C. Bechinger, \"Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields,\" Proc. Natl. Acad. Sci. 107, 7214-7218 (2010).\n\nThree-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles. K Volke-Sepúlveda, S Chávez-Cerda, V Garcés-Chávez, K Dholakia, J. Opt. Soc. Am. B. 21K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, \"Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,\" J. Opt. Soc. Am. B 21, 1749-1757 (2004).\n\nOptical conveyor belt for delivery of submicron objects. T Čižmár, V Garcés-Chávez, K Dholakia, P Zemánek, Appl. Phys. Lett. 86174101T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, \"Optical conveyor belt for delivery of submicron objects,\" Appl. Phys. Lett. 86, 174101 (2005).\n\nTransverse particle dynamics in a Bessel beam. G Milne, K Dholakia, D Mcgloin, K Volke-Sepulveda, P Zemánek, Opt. Express. 15G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, \"Transverse particle dynamics in a Bessel beam,\" Opt. Express 15, 13972-13987 (2007).\n\nLight-induced cell separation in a tailored optical landscape. L Paterson, E Papagiakoumou, G Milne, V Garcés-Chávez, S A Tatarkova, W Sibbett, F J Gunn-Moore, P E Bryant, A C Riches, K Dholakia, Appl. Phys. Lett. 87123901L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, \"Light-induced cell separation in a tailored optical landscape,\" Appl. Phys. Lett. 87, 123901 (2005).\n\nBrownian particle in an optical potential of the washboard type. S A Tatarkova, W Sibbett, K Dholakia, Phys. Rev. Lett. 9138101S. A. Tatarkova, W. Sibbett, and K. Dholakia, \"Brownian particle in an optical potential of the washboard type,\" Phys. Rev. Lett. 91, 038101 (2003).\n\nArtificial Brownian motors: Controlling transport on the nanoscale. P Hänggi, F Marchesoni, Rev. Mod. Phys. 81P. Hänggi and F. Marchesoni, \"Artificial Brownian motors: Controlling transport on the nanoscale,\" Rev. Mod. Phys. 81, 387-442 (2009).\n\nExperimental creation and characterization of random potential-energy landscapes exploiting speckle patterns. J Bewerunge, S U Egelhaaf, Phys. Rev. A. 9313806J. Bewerunge and S. U. Egelhaaf, \"Experimental creation and characterization of random potential-energy landscapes exploiting speckle patterns,\" Phys. Rev. A 93, 013806 (2016).\n\nBrownian motors: Noisy transport far from equilibrium. P Reimann, Phys. Rep. 361P. Reimann, \"Brownian motors: Noisy transport far from equilibrium,\" Phys. Rep. 361, 57-265 (2002).\n\nVariable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate. B A Garetz, S Arnold, Opt. Commun. 31B. A. Garetz and S. Arnold, \"Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,\" Opt. Commun. 31, 1-3 (1979).\n\nAngular Doppler effect. B A Garetz, J. Opt. Soc. Am. 71B. A. Garetz, \"Angular Doppler effect,\" J. Opt. Soc. Am. 71, 609-611 (1981).\n\nControlled rotation of optically trapped microscopic particles. L Paterson, M P Macdonald, J Arlt, W Sibbett, P E Bryant, K Dholakia, Science. 292L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, \"Controlled rotation of optically trapped microscopic particles,\" Science 292, 912-914 (2001).\n\nAn optical nanotrap array movable over a milimetre range. T Čižmár, M Šiler, P Zemánek, Appl. Phys. B. 84T. Čižmár, M. Šiler, and P. Zemánek, \"An optical nanotrap array movable over a milimetre range,\" Appl. Phys. B 84, 197-203 (2006).\n\nThe holographic optical micro-manipulation system based on counter-propagating beams. T Čižmár, O Brzobohatỳ, K Dholakia, P Zemánek, Laser Phys. Lett. 850T. Čižmár, O. Brzobohatỳ, K. Dholakia, and P. Zemánek, \"The holographic optical micro-manipulation system based on counter-propagating beams,\" Laser Phys. Lett. 8, 50 (2010).\n\nOptical conveyors: A class of active tractor beams. D B Ruffner, D G Grier, Phys. Rev. Lett. 109163903D. B. Ruffner and D. G. Grier, \"Optical conveyors: A class of active tractor beams,\" Phys. Rev. Lett. 109, 163903 (2012).\n\nColloidal transport through optical tweezer arrays. Y Roichman, V Wong, D G Grier, Phys. Rev. E. 7511407Y. Roichman, V. Wong, and D. G. Grier, \"Colloidal transport through optical tweezer arrays,\" Phys. Rev. E 75, 011407 (2007).\n\nOptical separation of cells on potential energy landscapes: Enhancement with dielectric tagging. K Dholakia, W M Lee, L Paterson, M P Macdonald, R Mcdonald, I Andreev, P Mthunzi, C T A Brown, R F Marchington, A C Riches, IEEE J. Sel. Top. Quant. El. 13K. Dholakia, W. M. Lee, L. Paterson, M. P. MacDonald, R. McDonald, I. Andreev, P. Mthunzi, C. T. A. Brown, R. F. Marchington, and A. C. Riches, \"Optical separation of cells on potential energy landscapes: Enhancement with dielectric tagging,\" IEEE J. Sel. Top. Quant. El. 13, 1646-1654 (2007).\n\nKinetically locked-in colloidal transport in an array of optical tweezers. P T Korda, M B Taylor, D G Grier, Phys. Rev. Lett. 89128301P. T. Korda, M. B. Taylor, and D. G. Grier, \"Kinetically locked-in colloidal transport in an array of optical tweezers,\" Phys. Rev. Lett. 89, 128301 (2002).\n\nSorting mesoscopic objects with periodic potential landscapes: Optical fractionation. K Ladavac, K Kasza, D G Grier, Phys. Rev. E. 7010901K. Ladavac, K. Kasza, and D. G. Grier, \"Sorting mesoscopic objects with periodic potential landscapes: Optical fractionation,\" Phys. Rev. E 70, 010901 (2004).\n\nMicrofluidic sorting with blinking optical traps. R Dasgupta, R S Verma, P K Gupta, Opt. Lett. 37R. Dasgupta, R. S. Verma, and P. K. Gupta, \"Microfluidic sorting with blinking optical traps,\" Opt. Lett. 37, 1739-1741 (2012).\n\nDynamics, rectification, and fractionation for colloids on flashing substrates. A Libal, C Reichhardt, B Janko, C J O Reichhardt, Phys. Rev. Lett. 96188301A. Libal, C. Reichhardt, B. Janko, and C. J. O. Reichhardt, \"Dynamics, rectification, and fractionation for colloids on flashing substrates,\" Phys. Rev. Lett. 96, 188301 (2006).\n\nOne-dimensional optical thermal ratchets. S.-H Lee, D G Grier, J. Phys. Condens. Matter. 17S.-H. Lee and D. G. Grier, \"One-dimensional optical thermal ratchets,\" J. Phys. Condens. Matter 17, S3685-S3695 (2005).\n\nTwo-dimensional optical thermal ratchets based on Fibonacci spirals. K Xiao, Y Roichman, D G Grier, Phys. Rev. E. 8411131K. Xiao, Y. Roichman, and D. G. Grier, \"Two-dimensional optical thermal ratchets based on Fibonacci spirals,\" Phys. Rev. E 84, 011131 (2011).\n\nNoise-enabled optical ratchets. R León-Montiel, P A Quinto-Su, Sci. Rep. 744287R. León-Montiel and P. A. Quinto-Su, \"Noise-enabled optical ratchets,\" Sci. Rep. 7, 44287 (2017).\n\nOptical chromatography. T Imasaka, Y Kawabata, T Kaneta, Y Ishidzu, Anal. Chem. 67T. Imasaka, Y. Kawabata, T. Kaneta, and Y. Ishidzu, \"Optical chromatography,\" Anal. Chem. 67, 1763-1765 (1995).\n\nRefractive-index-driven separation of colloidal polymer particles using optical chromatography. S J Hart, A V Terray, Appl. Phys. Lett. 83S. J. Hart and A. V. Terray, \"Refractive-index-driven separation of colloidal polymer particles using optical chromatography,\" Appl. Phys. Lett. 83, 5316-5318 (2003).\n\nStatic optical sorting in a laser interference field. P Jákl, T Čižmár, M Šerý, P Zemánek, Appl. Phys. Lett. 92161110P. Jákl, T. Čižmár, M. Šerý, and P. Zemánek, \"Static optical sorting in a laser interference field,\" Appl. Phys. Lett. 92, 161110 (2008).\n\nOptical deflection and sorting of microparticles in a near-field optical geometry. R F Marchington, M Mazilu, S Kuriakose, V Garcés-Chávez, P J Reece, T F Krauss, M Gu, K Dholakia, Opt. Express. 16R. F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P. J. Reece, T. F. Krauss, M. Gu, and K. Dholakia, \"Optical deflection and sorting of microparticles in a near-field optical geometry,\" Opt. Express 16, 3712-3726 (2008).\n\nOptical forces generated by evanescent standing waves and their usage for sub-micron particle delivery. M Šiler, T Čižmár, M Šerỳ, P Zemánek, Appl. Phys. B. 84M. Šiler, T. Čižmár, M. Šerỳ, and P. Zemánek, \"Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,\" Appl. Phys. B 84, 157-165 (2006).\n\nNanomanipulation using near field photonics. D Erickson, X Serey, Y.-F Chen, S Mandal, Lab Chip. 11D. Erickson, X. Serey, Y.-F. Chen, and S. Mandal, \"Nanomanipulation using near field photonics,\" Lab Chip 11, 995-1009 (2011).\n\nExperimental demonstration of optical transport, sorting and self-arrangement using a 'tractor' beam. O Brzobohatỳ, V Karásek, M Šiler, L Chvátal, T Čižmár, P Zemánek, Nat. Photon. 7O. Brzobohatỳ, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, and P. Zemánek, \"Experimental demonstration of optical transport, sorting and self-arrangement using a 'tractor' beam,\" Nat. Photon. 7, 123-127 (2013).\n\nOptofluidic sorting of material chirality by chiral light. G Tkachenko, E Brasselet, Nat. Commun. 53577G. Tkachenko and E. Brasselet, \"Optofluidic sorting of material chirality by chiral light,\" Nat. Commun. 5, 3577 (2014).\n\nLateral chirality-sorting optical forces. A Hayat, J P B Mueller, F Capasso, Proc. Natl. Acad. Sci. 112A. Hayat, J. P. B. Mueller, and F. Capasso, \"Lateral chirality-sorting optical forces,\" Proc. Natl. Acad. Sci. 112, 13190-13194 (2015).\n\nOrientational and directional locking of colloidal clusters driven across periodic surfaces. X Cao, E Panizon, A Vanossi, N Manini, C Bechinger, Nat. Phys. 15X. Cao, E. Panizon, A. Vanossi, N. Manini, and C. Bechinger, \"Orientational and directional locking of colloidal clusters driven across periodic surfaces,\" Nat. Phys. 15, 776-780 (2019).\n\nExperimental observation of the Aubry transition in two-dimensional colloidal monolayers. T Brazda, A Silva, N Manini, A Vanossi, R Guerra, E Tosatti, C Bechinger, Phys. Rev. X. 811050T. Brazda, A. Silva, N. Manini, A. Vanossi, R. Guerra, E. Tosatti, and C. Bechinger, \"Experimental observation of the Aubry transition in two-dimensional colloidal monolayers,\" Phys. Rev. X 8, 011050 (2018).\n\nExperimental observation of directional locking and dynamical ordering of colloidal monolayers driven across quasiperiodic substrates. T Bohlein, C Bechinger, Phys. Rev. Lett. 10958301T. Bohlein and C. Bechinger, \"Experimental observation of directional locking and dynamical ordering of colloidal monolayers driven across quasiperiodic substrates,\" Phys. Rev. Lett. 109, 058301 (2012).\n\nObservation of kinks and antikinks in colloidal monolayers driven across ordered surfaces. T Bohlein, J Mikhael, C Bechinger, Nat. Mater. 11T. Bohlein, J. Mikhael, and C. Bechinger, \"Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces,\" Nat. Mater. 11, 126-130 (2012).\n\nLaser trapping of colloidal metal nanoparticles. A Lehmuskero, P Johansson, H Rubinsztein-Dunlop, L Tong, M Kall, ACS Nano. 9A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, \"Laser trapping of colloidal metal nanoparticles,\" ACS Nano 9, 3453-3469 (2015).\n\nParallel and selective trapping in a patterned plasmonic landscape. M Righini, A S Zelenina, C Girard, R Quidant, Nat. Phys. 3M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, \"Parallel and selective trapping in a patterned plasmonic landscape,\" Nat. Phys. 3, 477-480 (2007).\n\nSurface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range. M Righini, G Volpe, C Girard, D Petrov, R Quidant, Phys. Rev. Lett. 100186804M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, \"Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,\" Phys. Rev. Lett. 100, 186804 (2008).\n\nPlasmonic optical tweezers toward molecular manipulation: tailoring plasmonic nanostructure, light source, and resonant trapping. T Shoji, Y Tsuboi, J. Phys. Chem. Lett. 5T. Shoji and Y. Tsuboi, \"Plasmonic optical tweezers toward molecular manipulation: tailoring plasmonic nanostruc- ture, light source, and resonant trapping,\" J. Phys. Chem. Lett. 5, 2957-2967 (2014).\n\nMassively parallel manipulation of single cells and microparticles using optical images. P Y Chiou, A T Ohta, M C Wu, Nature. 436P. Y. Chiou, A. T. Ohta, and M. C. Wu, \"Massively parallel manipulation of single cells and microparticles using optical images,\" Nature 436, 370-372 (2005).\n\nThermophoretic tweezers for low-power and versatile manipulation of biological cells. L Lin, X Peng, X Wei, Z Mao, C Xie, Y Zheng, ACS Nano. 11L. Lin, X. Peng, X. Wei, Z. Mao, C. Xie, and Y. Zheng, \"Thermophoretic tweezers for low-power and versatile manipulation of biological cells,\" ACS Nano 11, 3147-3154 (2017).\n\nOpto-thermophoretic tweezers and assembly. J Li, L Lin, Y Inoue, Y Zheng, J. Micro Nano-Manufacturing. 640801J. Li, L. Lin, Y. Inoue, and Y. Zheng, \"Opto-thermophoretic tweezers and assembly,\" J. Micro Nano-Manufacturing 6, 040801 (2018).\n\nAcoustic tweezers: patterning cells and microparticles using standing surface acoustic waves (SSAW). J Shi, D Ahmed, X Mao, S.-C S Lin, A Lawit, T J Huang, Lab Chip. 9J. Shi, D. Ahmed, X. Mao, S.-C. S. Lin, A. Lawit, and T. J. Huang, \"Acoustic tweezers: patterning cells and microparticles using standing surface acoustic waves (SSAW),\" Lab Chip 9, 2890-2895 (2009).\n\nMicroscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics. J Friend, L Y Yeo, Rev. Mod. Phys. 83J. Friend and L. Y. Yeo, \"Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics,\" Rev. Mod. Phys. 83, 647-704 (2011).\n\nObservation of a single-beam gradient force acoustical trap for elastic particles: acoustical tweezers. D Baresch, J.-L Thomas, R Marchiano, Phys. Rev. Lett. 11624301D. Baresch, J.-L. Thomas, and R. Marchiano, \"Observation of a single-beam gradient force acoustical trap for elastic particles: acoustical tweezers,\" Phys. Rev. Lett. 116, 024301 (2016).\n\nPerspective on light-induced transport of particles: from optical forces to phoretic motion. P Zemánek, G Volpe, A Jonáš, O Brzobohatỳ, Adv. Opt. Photon. 11P. Zemánek, G. Volpe, A. Jonáš, and O. Brzobohatỳ, \"Perspective on light-induced transport of particles: from optical forces to phoretic motion,\" Adv. Opt. Photon. 11, 577-678 (2019).\n\nFeedback stabilization of optically levitated particles. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 30A. Ashkin and J. M. Dziedzic, \"Feedback stabilization of optically levitated particles,\" Appl. Phys. Lett. 30, 202-204 (1977).\n\nOptical levitation in high vacuum. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 28A. Ashkin and J. M. Dziedzic, \"Optical levitation in high vacuum,\" Appl. Phys. Lett. 28, 333-335 (1976).\n\nOptical levitation by radiation pressure. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 19A. Ashkin and J. M. Dziedzic, \"Optical levitation by radiation pressure,\" Appl. Phys. Lett. 19, 283-285 (1971).\n\nObservation of a single-beam gradient-force optical trap for dielectric particles in air. R Omori, T Kobayashi, A Suzuki, Opt. Lett. 22R. Omori, T. Kobayashi, and A. Suzuki, \"Observation of a single-beam gradient-force optical trap for dielectric particles in air,\" Opt. Lett. 22, 816-818 (1997).\n\nSubkelvin parametric feedback cooling of a laser-trapped nanoparticle. J Gieseler, B Deutsch, R Quidant, L Novotny, Phys. Rev. Lett. 109103603J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, \"Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,\" Phys. Rev. Lett. 109, 103603 (2012).\n\nOptical trapping of nanoparticles by full solid-angle focusing. V Salakhutdinov, M Sondermann, L Carbone, E Giacobino, A Bramati, G Leuchs, Optica. 3V. Salakhutdinov, M. Sondermann, L. Carbone, E. Giacobino, A. Bramati, and G. Leuchs, \"Optical trapping of nanoparticles by full solid-angle focusing,\" Optica 3, 1181-1186 (2016).\n\nExperimental realization of a thermal squeezed state of levitated optomechanics. M Rashid, T Tufarelli, J Bateman, J Vovrosh, D Hempston, M S Kim, H Ulbricht, Phys. Rev. Lett. 117273601M. Rashid, T. Tufarelli, J. Bateman, J. Vovrosh, D. Hempston, M. S. Kim, and H. Ulbricht, \"Experimental realization of a thermal squeezed state of levitated optomechanics,\" Phys. Rev. Lett. 117, 273601 (2016).\n\nMeasurement of the instantaneous velocity of a Brownian particle. T Li, S Kheifets, D Medellin, M G Raizen, Science. 328T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, \"Measurement of the instantaneous velocity of a Brownian particle,\" Science 328, 1673-1675 (2010).\n\nAttonewton force detection using microspheres in a dual-beam optical trap in high vacuum. G Ranjit, D P Atherton, J H Stutz, M Cunningham, A A Geraci, Phys. Rev. A. 9151805G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, \"Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,\" Phys. Rev. A 91, 051805 (2015).\n\nToward quantum superposition of living organisms. O Romero-Isart, M L Juan, R Quidant, J I Cirac, New J. Phys. 1233015O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, \"Toward quantum superposition of living organisms,\" New J. Phys. 12, 033015 (2010).\n\nCavity opto-mechanics using an optically levitated nanosphere. D E Chang, C A Regal, S B Papp, D J Wilson, J Ye, O Painter, H J Kimble, P Zoller, Proc. Natl. Acad. Sci. 107D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, \"Cavity opto-mechanics using an optically levitated nanosphere,\" Proc. Natl. Acad. Sci. 107, 1005-1010 (2010).\n\nCavity cooling of an optically trapped nanoparticle. P F Barker, M N Shneider, Phys. Rev. A. 8123826P. F. Barker and M. N. Shneider, \"Cavity cooling of an optically trapped nanoparticle,\" Phys. Rev. A 81, 023826 (2010).\n\nCold damping of an optically levitated nanoparticle to microkelvin temperatures. F Tebbenjohanns, M Frimmer, A Militaru, V Jain, L Novotny, Phys. Rev. Lett. 122223601F. Tebbenjohanns, M. Frimmer, A. Militaru, V. Jain, and L. Novotny, \"Cold damping of an optically levitated nanoparticle to microkelvin temperatures,\" Phys. Rev. Lett. 122, 223601 (2019).\n\nDirect measurement of photon recoil from a levitated nanoparticle. V Jain, J Gieseler, C Moritz, C Dellago, R Quidant, L Novotny, Phys. Rev. Lett. 116243601V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, \"Direct measurement of photon recoil from a levitated nanoparticle,\" Phys. Rev. Lett. 116, 243601 (2016).\n\nOptimal feedback cooling of a charged levitated nanoparticle with adaptive control. G P Conangla, F Ricci, M T Cuairan, A W Schell, N Meyer, R Quidant, Phys. Rev. Lett. 122223602G. P. Conangla, F. Ricci, M. T. Cuairan, A. W. Schell, N. Meyer, and R. Quidant, \"Optimal feedback cooling of a charged levitated nanoparticle with adaptive control,\" Phys. Rev. Lett. 122, 223602 (2019).\n\nMotional sideband asymmetry of a nanoparticle optically levitated in free space. F Tebbenjohanns, M Frimmer, V Jain, D Windey, L Novotny, arXiv:1908.05079arXiv preprintF. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, \"Motional sideband asymmetry of a nanoparticle optically levitated in free space,\" arXiv preprint arXiv:1908.05079 (2019).\n\nCavity cooling of a levitated nanosphere by coherent scattering. U Delić, M Reisenbauer, D Grass, N Kiesel, V Vuletić, M Aspelmeyer, Phys. Rev. Lett. 122123602U. Delić, M. Reisenbauer, D. Grass, N. Kiesel, V. Vuletić, and M. Aspelmeyer, \"Cavity cooling of a levitated nanosphere by coherent scattering,\" Phys. Rev. Lett. 122, 123602 (2019).\n\nCavity-based 3d cooling of a levitated nanoparticle via coherent scattering. D Windey, C Gonzalez-Ballestero, P Maurer, L Novotny, O Romero-Isart, R Reimann, Phys. Rev. Lett. 122123601D. Windey, C. Gonzalez-Ballestero, P. Maurer, L. Novotny, O. Romero-Isart, and R. Reimann, \"Cavity-based 3d cooling of a levitated nanoparticle via coherent scattering,\" Phys. Rev. Lett. 122, 123601 (2019).\n\nResolved-sideband cooling of a levitated nanoparticle in the presence of laser phase noise. N Meyer, A De Los Rios, P Sommer, J Mestres, V Gieseler, L Jain, R Novotny, Quidant, Phys. Rev. Lett. 123153601N. Meyer, A. de los Rios Sommer, P. Mestres, J. Gieseler, V. Jain, L. Novotny, and R. Quidant, \"Resolved-sideband cooling of a levitated nanoparticle in the presence of laser phase noise,\" Phys. Rev. Lett. 123, 153601 (2019).\n\nSideband cooling of micromechanical motion to the quantum ground state. J D Teufel, T Donner, D Li, J W Harlow, M S Allman, K Cicak, A J Sirois, J D Whittaker, K W Lehnert, R W Simmonds, Nature. 475J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, \"Sideband cooling of micromechanical motion to the quantum ground state,\" Nature 475, 359-405 (2011).\n\nLaser cooling of a nanomechanical oscillator into its quantum ground state. J Chan, T P M Alegre, A H Safavi-Naeini, J T Hill, A Krause, S Gröblacher, M Aspelmeyer, O Painter, Nature. 478J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, \"Laser cooling of a nanomechanical oscillator into its quantum ground state,\" Nature 478, 89-92 (2011).\n\nCooling of a levitated nanoparticle to the motional quantum ground state. U Delić, M Reisenbauer, K Dare, D Grass, V Vuletić, N Kiesel, M Aspelmeyer, Science. 367U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, and M. Aspelmeyer, \"Cooling of a levitated nanoparticle to the motional quantum ground state,\" Science 367, 892-895 (2020).\n\nNobel lecture: Controlling photons in a box and exploring the quantum to classical boundary. S Haroche, Rev. Mod. Phys. 85S. Haroche, \"Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary,\" Rev. Mod. Phys. 85, 1083-1102 (2013).\n\nNobel lecture: Superposition, entanglement, and raising schrödinger's cat. D J Wineland, Rev. Mod. Phys. 851103D. J. Wineland, \"Nobel lecture: Superposition, entanglement, and raising schrödinger's cat,\" Rev. Mod. Phys. 85, 1103 (2013).\n\nQuantum interference distinguishes between constitutional isomers. J Tüxen, S Gerlich, S Eibenberger, M Arndt, M Mayor, Chem. Commun. 46J. Tüxen, S. Gerlich, S. Eibenberger, M. Arndt, and M. Mayor, \"Quantum interference distinguishes between constitutional isomers,\" Chem. Commun. 46, 4145-4147 (2010).\n\nColloquium: Quantum interference of clusters and molecules. K Hornberger, S Gerlich, P Haslinger, S Nimmrichter, M Arndt, Rev. Mod. Phys. 84K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, \"Colloquium: Quantum interference of clusters and molecules,\" Rev. Mod. Phys. 84, 157-173 (2012).\n\nCreating and verifying a quantum superposition in a micro-optomechanical system. D Kleckner, I Pikovski, E Jeffrey, L Ament, E Eliel, J Van Den, D Brink, Bouwmeester, New J. Phys. 1095020D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. Van Den Brink, and D. Bouwmeester, \"Creating and verifying a quantum superposition in a micro-optomechanical system,\" New J. Phys. 10, 095020 (2008).\n\nMacroscopic quantum resonators (maqro). R Kaltenbaek, G Hechenblaikner, N Kiesel, O Romero-Isart, K C Schwab, U Johann, M Aspelmeyer, Exp. Astron. 34R. Kaltenbaek, G. Hechenblaikner, N. Kiesel, O. Romero-Isart, K. C. Schwab, U. Johann, and M. Aspelmeyer, \"Macroscopic quantum resonators (maqro),\" Exp. Astron. 34, 123-164 (2012).\n\nQuantum superposition of massive objects and collapse models. O Romero-Isart, Phys. Rev. A. 8452121O. Romero-Isart, \"Quantum superposition of massive objects and collapse models,\" Phys. Rev. A 84, 052121 (2011).\n\nLarge quantum superpositions and interference of massive nanometer-sized objects. O Romero-Isart, A C Pflanzer, F Blaser, R Kaltenbaek, N Kiesel, M Aspelmeyer, J I Cirac, Phys. Rev. Lett. 10720405O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, \"Large quantum superpositions and interference of massive nanometer-sized objects,\" Phys. Rev. Lett. 107, 020405 (2011).\n\nNear-field interferometry of a free-falling nanoparticle from a point-like source. J Bateman, S Nimmrichter, K Hornberger, H Ulbricht, Nat. Commun. 54788J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, \"Near-field interferometry of a free-falling nanoparticle from a point-like source,\" Nat. Commun. 5, 4788 (2014).\n\nModels of wave-function collapse, underlying theories, and experimental tests. A Bassi, K Lochan, S Satin, T P Singh, H Ulbricht, Rev. Mod. Phys. 85A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, \"Models of wave-function collapse, underlying theories, and experimental tests,\" Rev. Mod. Phys. 85, 471-527 (2013).\n\nOptomechanics of levitated dielectric particles. Z.-Q Yin, A A Geraci, T Li, Int. J. Mod. Phys. B. 271330018Z.-Q. Yin, A. A. Geraci, and T. Li, \"Optomechanics of levitated dielectric particles,\" Int. J. Mod. Phys. B 27, 1330018 (2013).\n\nCavity optomechanics. M Aspelmeyer, T J Kippenberg, F Marquardt, Rev. Mod. Phys. 86M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, \"Cavity optomechanics,\" Rev. Mod. Phys. 86, 1391-1452 (2014).\n\nW P Bowen, G J Milburn, Quantum optomechanics. CRC pressW. P. Bowen and G. J. Milburn, Quantum optomechanics (CRC press, 2015).\n\nClassical characterization of biphoton correlation in waveguide lattices. R Keil, F Dreisow, M Heinrich, A Tünnermann, S Nolte, A Szameit, Phys. Rev. A. 8313808R. Keil, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, \"Classical characterization of biphoton correlation in waveguide lattices,\" Phys. Rev. A 83, 013808 (2011).\n\nHeralded single-phonon preparation, storage, and readout in cavity optomechanics. C Galland, N Sangouard, N Piro, N Gisin, T J Kippenberg, Phys. Rev. Lett. 112143602C. Galland, N. Sangouard, N. Piro, N. Gisin, and T. J. Kippenberg, \"Heralded single-phonon preparation, storage, and readout in cavity optomechanics,\" Phys. Rev. Lett. 112, 143602 (2014).\n\nPhonon counting and intensity interferometry of a nanomechanical resonator. J D Cohen, S M Meenehan, G S Maccabe, S Gröblacher, A H Safavi-Naeini, F Marsili, M D Shaw, O Painter, Nature. 520J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, \"Phonon counting and intensity interferometry of a nanomechanical resonator,\" Nature 520, 522-525 (2015).\n\nHanbury brown and twiss interferometry of single phonons from an optomechanical resonator. S Hong, R Riedinger, I Marinković, A Wallucks, S G Hofer, R A Norte, M Aspelmeyer, S Gröblacher, Science. 358S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, \"Hanbury brown and twiss interferometry of single phonons from an optomechanical resonator,\" Science 358, 203-206 (2017).\n\nCooling of mechanical motion with a two-level system: The high-temperature regime. P , Phys. Rev. B. 82165320P. Rabl, \"Cooling of mechanical motion with a two-level system: The high-temperature regime,\" Phys. Rev. B 82, 165320 (2010).\n\nStrong magnetic coupling between an electronic spin qubit and a mechanical resonator. P Rabl, P Cappellaro, M V G Dutt, L Jiang, J R Maze, M D Lukin, Phys. Rev. B. 7941302P. Rabl, P. Cappellaro, M. V. G. Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, \"Strong magnetic coupling between an electronic spin qubit and a mechanical resonator,\" Phys. Rev. B 79, 041302 (2009).\n\nHybrid opto-mechanical systems with nitrogen-vacancy centers. Z.-Q Yin, N Zhao, T Li, Sci. China Phys., Mech. Astron. 58Z.-Q. Yin, N. Zhao, and T. Li, \"Hybrid opto-mechanical systems with nitrogen-vacancy centers,\" Sci. China Phys., Mech. Astron. 58, 1-12 (2015).\n\nMulti-dimensional single-spin nano-optomechanics with a levitated nanodiamond. L P Neukirch, E Von Haartman, J M Rosenholm, A N Vamivakas, Nat. Photon. 9L. P. Neukirch, E. Von Haartman, J. M. Rosenholm, and A. N. Vamivakas, \"Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,\" Nat. Photon. 9, 653-657 (2015).\n\nObservation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond. L P Neukirch, J Gieseler, R Quidant, L Novotny, A N Vamivakas, Opt. Lett. 38L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. N. Vamivakas, \"Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,\" Opt. Lett. 38, 2976-2979 (2013).\n\nPure nanodiamonds for levitated optomechanics in vacuum. A C Frangeskou, A T M A Rahman, L Gines, S Mandal, O A Williams, P F Barker, G W Morley, New J. Phys. 2043016A. C. Frangeskou, A. T. M. A. Rahman, L. Gines, S. Mandal, O. A. Williams, P. F. Barker, and G. W. Morley, \"Pure nanodiamonds for levitated optomechanics in vacuum,\" New J. Phys. 20, 043016 (2018).\n\nBurning and graphitization of optically levitated nanodiamonds in vacuum. A T M A Rahman, A C Frangeskou, M S Kim, S Bose, G W Morley, P F Barker, Sci. Rep. p. 21633A. T. M. A. Rahman, A. C. Frangeskou, M. S. Kim, S. Bose, G. W. Morley, and P. F. Barker, \"Burning and graphitization of optically levitated nanodiamonds in vacuum,\" Sci. Rep. p. 21633 (2016).\n\nElectron spin control of optically levitated nanodiamonds in vacuum. T M Hoang, J Ahn, J Bang, T Li, Nat. Commun. 712250T. M. Hoang, J. Ahn, J. Bang, and T. Li, \"Electron spin control of optically levitated nanodiamonds in vacuum,\" Nat. Commun. 7, 12250 (2016).\n\nCooling-by-measurement and mechanical state tomography via pulsed optomechanics. M R Vanner, J Hofer, G D Cole, M Aspelmeyer, Nat. Commun. 42295M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, \"Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,\" Nat. Commun. 4, 2295 (2013).\n\nPulsed quantum optomechanics. M R Vanner, I Pikovski, G D Cole, M S Kim, Č Brukner, K Hammerer, G J Milburn, M Aspelmeyer, Proc. Natl. Acad. Sci. Natl. Acad. Sci108M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, \"Pulsed quantum optomechanics,\" Proc. Natl. Acad. Sci. 108, 16182-16187 (2011).\n\nSingle phonon quantum optics -An experimental exploration of a silicon photonics quantum memory. R Riedinger, ViennaPh.D. thesisR. Riedinger, \"Single phonon quantum optics -An experimental exploration of a silicon photonics quantum memory,\" Ph.D. thesis, Vienna (2018).\n\nTorsional optomechanics of a levitated nonspherical nanoparticle. T M Hoang, Y Ma, J Ahn, J Bang, F Robicheaux, Z.-Q Yin, T Li, Phys. Rev. Lett. 117123604T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.-Q. Yin, and T. Li, \"Torsional optomechanics of a levitated nonspherical nanoparticle,\" Phys. Rev. Lett. 117, 123604 (2016).\n\nPrecession motion in levitated optomechanics. M Rashid, M Toroš, A Setter, H Ulbricht, Phys. Rev. Lett. 121253601M. Rashid, M. Toroš, A. Setter, and H. Ulbricht, \"Precession motion in levitated optomechanics,\" Phys. Rev. Lett. 121, 253601 (2018).\n\nZeptonewton force sensing with nanospheres in an optical lattice. G Ranjit, M Cunningham, K Casey, A A Geraci, Phys. Rev. A. 9353801G. Ranjit, M. Cunningham, K. Casey, and A. A. Geraci, \"Zeptonewton force sensing with nanospheres in an optical lattice,\" Phys. Rev. A 93, 053801 (2016).\n\nForce sensing with an optically levitated charged nanoparticle. D Hempston, J Vovrosh, M Toroš, G Winstone, M Rashid, H Ulbricht, Appl. Phys. Lett. 111133111D. Hempston, J. Vovrosh, M. Toroš, G. Winstone, M. Rashid, and H. Ulbricht, \"Force sensing with an optically levitated charged nanoparticle,\" Appl. Phys. Lett. 111, 133111 (2017).\n\nThree-dimensional force-field microscopy with optically levitated microspheres. C P Blakemore, A D Rider, S Roy, Q Wang, A Kawasaki, G Gratta, Phys. Rev. A. 9923816C. P. Blakemore, A. D. Rider, S. Roy, Q. Wang, A. Kawasaki, and G. Gratta, \"Three-dimensional force-field microscopy with optically levitated microspheres,\" Phys. Rev. A 99, 023816 (2019).\n\nOptical levitation of 10-ng spheres with nano-g acceleration sensitivity. F Monteiro, S Ghosh, A G Fine, D C Moore, Phys. Rev. A. 9663841F. Monteiro, S. Ghosh, A. G. Fine, and D. C. Moore, \"Optical levitation of 10-ng spheres with nano-g acceleration sensitivity,\" Phys. Rev. A 96, 063841 (2017).\n\nSearch for screened interactions associated with dark energy below the 100 µ m length scale. A D Rider, D C Moore, C P Blakemore, M Louis, M Lu, G Gratta, Phys. Rev. Lett. 117101101A. D. Rider, D. C. Moore, C. P. Blakemore, M. Louis, M. Lu, and G. Gratta, \"Search for screened interactions associated with dark energy below the 100 µ m length scale,\" Phys. Rev. Lett. 117, 101101 (2016).\n\nSearch for millicharged particles using optically levitated microspheres. D C Moore, A D Rider, G Gratta, Phys. Rev. Lett. 113251801D. C. Moore, A. D. Rider, and G. Gratta, \"Search for millicharged particles using optically levitated microspheres,\" Phys. Rev. Lett. 113, 251801 (2014).\n\nSensing short range forces with a nanosphere matter-wave interferometer. A Geraci, H Goldman, Phys. Rev. D. 9262002A. Geraci and H. Goldman, \"Sensing short range forces with a nanosphere matter-wave interferometer,\" Phys. Rev. D 92, 062002 (2015).\n\nCasimir interaction between a dielectric nanosphere and a metallic plane. A Canaguier-Durand, A Gérardin, R Guérout, P A M Neto, V V Nesvizhevsky, A Y Voronin, A Lambrecht, S Reynaud, Phys. Rev. A. 8332508A. Canaguier-Durand, A. Gérardin, R. Guérout, P. A. M. Neto, V. V. Nesvizhevsky, A. Y. Voronin, A. Lambrecht, and S. Reynaud, \"Casimir interaction between a dielectric nanosphere and a metallic plane,\" Phys. Rev. A 83, 032508 (2011).\n\nLateral Casimir force on a rotating particle near a planar surface. A Manjavacas, F J Rodríguez-Fortuño, F J García De Abajo, A V Zayats, Phys. Rev. Lett. 118133605A. Manjavacas, F. J. Rodríguez-Fortuño, F. J. García de Abajo, and A. V. Zayats, \"Lateral Casimir force on a rotating particle near a planar surface,\" Phys. Rev. Lett. 118, 133605 (2017).\n\nDynamics of a levitated nanosphere by optomechanical coupling and Casimir interaction. W Nie, Y Lan, Y Li, S Zhu, Phys. Rev. A. 8863849W. Nie, Y. Lan, Y. Li, and S. Zhu, \"Dynamics of a levitated nanosphere by optomechanical coupling and Casimir interaction,\" Phys. Rev. A 88, 063849 (2013).\n\nDetecting Casimir torque with an optically levitated nanorod. Z Xu, T Li, Phys. Rev. A. 9633843Z. Xu and T. Li, \"Detecting Casimir torque with an optically levitated nanorod,\" Phys. Rev. A 96, 033843 (2017).\n\nDetecting high-frequency gravitational waves with optically levitated sensors. A Arvanitaki, A A Geraci, Phys. Rev. Lett. 11071105A. Arvanitaki and A. A. Geraci, \"Detecting high-frequency gravitational waves with optically levitated sensors,\" Phys. Rev. Lett. 110, 071105 (2013).\n\nNonlinear dynamics of nanomechanical and micromechanical resonators. R Lifshitz, M C Cross, Rev. Nonlinear Dyn. Complex. pp. R. Lifshitz and M. C. Cross, \"Nonlinear dynamics of nanomechanical and micromechanical resonators,\" Rev. Nonlinear Dyn. Complex. pp. 1-52 (2008).\n\nNonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle. J Gieseler, M Spasenović, L Novotny, R Quidant, Phys. Rev. Lett. 112103603J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, \"Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,\" Phys. Rev. Lett. 112, 103603 (2014).\n\nOptically levitated nanoparticle as a model system for stochastic bistable dynamics. F Ricci, R A Rica, M Spasenović, J Gieseler, L Rondin, L Novotny, R Quidant, Nat. Commun. p. 15141F. Ricci, R. A. Rica, M. Spasenović, J. Gieseler, L. Rondin, L. Novotny, and R. Quidant, \"Optically levitated nanoparticle as a model system for stochastic bistable dynamics,\" Nat. Commun. p. 15141 (2017).\n\nThermal nonlinearities in a nanomechanical oscillator. J Gieseler, L Novotny, R Quidant, Nat. Phys. 9J. Gieseler, L. Novotny, and R. Quidant, \"Thermal nonlinearities in a nanomechanical oscillator,\" Nat. Phys. 9, 806-810 (2013).\n\nSensing static forces with free-falling nanoparticles. E Hebestreit, M Frimmer, R Reimann, L Novotny, Phys. Rev. Lett. 12163602E. Hebestreit, M. Frimmer, R. Reimann, and L. Novotny, \"Sensing static forces with free-falling nanoparticles,\" Phys. Rev. Lett. 121, 063602 (2018).\n\nProbing macroscopic quantum superpositions with nanorotors. B A Stickler, B Papendell, S Kuhn, B Schrinski, J Millen, M Arndt, K Hornberger, New J. Phys. 20122001B. A. Stickler, B. Papendell, S. Kuhn, B. Schrinski, J. Millen, M. Arndt, and K. Hornberger, \"Probing macroscopic quantum superpositions with nanorotors,\" New J. Phys. 20, 122001 (2018).\n\nLaser-induced rotation and cooling of a trapped microgyroscope in vacuum. Y Arita, M Mazilu, K Dholakia, Nat. Commun. 42374Y. Arita, M. Mazilu, and K. Dholakia, \"Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,\" Nat. Commun. 4, 2374 (2013).\n\nCavity-assisted manipulation of freely rotating silicon nanorods in high vacuum. S Kuhn, P Asenbaum, A Kosloff, M Sclafani, B A Stickler, S Nimmrichter, K Hornberger, O Cheshnovsky, F Patolsky, M Arndt, Nano Lett. 15S. Kuhn, P. Asenbaum, A. Kosloff, M. Sclafani, B. A. Stickler, S. Nimmrichter, K. Hornberger, O. Cheshnovsky, F. Patolsky, and M. Arndt, \"Cavity-assisted manipulation of freely rotating silicon nanorods in high vacuum,\" Nano Lett. 15, 5604-5608 (2015).\n\nFull rotational control of levitated silicon nanorods. S Kuhn, A Kosloff, B A Stickler, F Patolsky, K Hornberger, M Arndt, J Millen, Optica. 4S. Kuhn, A. Kosloff, B. A. Stickler, F. Patolsky, K. Hornberger, M. Arndt, and J. Millen, \"Full rotational control of levitated silicon nanorods,\" Optica 4, 356-360 (2017).\n\nLaser refrigeration, alignment and rotation of levitated yb 3+: Ylf nanocrystals. A T M A Rahman, P F Barker, Nat. Photon. 11A. T. M. A. Rahman and P. F. Barker, \"Laser refrigeration, alignment and rotation of levitated yb 3+: Ylf nanocrystals,\" Nat. Photon. 11, 634-639 (2017).\n\nOptically levitated nanodumbbell torsion balance and GHz nanomechanical rotor. J Ahn, Z Xu, J Bang, Y.-H Deng, T M Hoang, Q Han, R.-M Ma, T Li, Phys. Rev. Lett. 12133603J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, \"Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,\" Phys. Rev. Lett. 121, 033603 (2018).\n\nGhz rotation of an optically trapped nanoparticle in vacuum. R Reimann, M Doderer, E Hebestreit, R Diehl, M Frimmer, D Windey, F Tebbenjohanns, L Novotny, Phys. Rev. Lett. 12133602R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, \"Ghz rotation of an optically trapped nanoparticle in vacuum,\" Phys. Rev. Lett. 121, 033602 (2018).\n\nRotational quantum friction. R Zhao, A Manjavacas, F J García De Abajo, J B Pendry, Phys. Rev. Lett. 109123604R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, \"Rotational quantum friction,\" Phys. Rev. Lett. 109, 123604 (2012).\n\nDirect measurement of kramers turnover with a levitated nanoparticle. L Rondin, J Gieseler, F Ricci, R Quidant, C Dellago, L Novotny, Nat. Nanotech. 12L. Rondin, J. Gieseler, F. Ricci, R. Quidant, C. Dellago, and L. Novotny, \"Direct measurement of kramers turnover with a levitated nanoparticle,\" Nat. Nanotech. 12, 1130-1133 (2017).\n\nDynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state. J Gieseler, R Quidant, C Dellago, L Novotny, Nat. Nanotech. 9J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, \"Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state,\" Nat. Nanotech. 9, 358-364 (2014).\n\nExperimental test of the differential fluctuation theorem and a generalized Jarzynski equality for arbitrary initial states. T M Hoang, R Pan, J Ahn, J Bang, H T Quan, T Li, Phys. Rev. Lett. 12080602T. M. Hoang, R. Pan, J. Ahn, J. Bang, H. T. Quan, and T. Li, \"Experimental test of the differential fluctuation theorem and a generalized Jarzynski equality for arbitrary initial states,\" Phys. Rev. Lett. 120, 080602 (2018).\n\nM Debiossac, D Grass, J J Alonso, E Lutz, N Kiesel, arXiv:1904.04889Thermodynamics of continuous non-markovian feedback control. arXiv preprintM. Debiossac, D. Grass, J. J. Alonso, E. Lutz, and N. Kiesel, \"Thermodynamics of continuous non-markovian feedback control,\" arXiv preprint arXiv:1904.04889 (2019).\n\nCooling mechanical oscillators by coherent control. M Frimmer, J Gieseler, L Novotny, Phys. Rev. Lett. 117163601M. Frimmer, J. Gieseler, and L. Novotny, \"Cooling mechanical oscillators by coherent control,\" Phys. Rev. Lett. 117, 163601 (2016).\n\nThermal properties of levitated nanoparticles. E Hebestreit, ETH ZurichPh.D. thesisE. Hebestreit, \"Thermal properties of levitated nanoparticles,\" Ph.D. thesis, ETH Zurich (2017).\n\nLevitodynamics toward force nano-sensors in vacuum. F Ricci, Universitat Politècnica de CatalunyaPh.D. thesisF. Ricci, \"Levitodynamics toward force nano-sensors in vacuum,\" Ph.D. thesis, Universitat Politècnica de Catalunya (2019).\n\nScanning confocal optical microscopy and magnetic resonance on single defect centers. A Gruber, A Dräbenstedt, C Tietz, L Fleury, J Wrachtrup, C Von Borczyskowski, Science. 276A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. Von Borczyskowski, \"Scanning confocal optical microscopy and magnetic resonance on single defect centers,\" Science 276, 2012-2014 (1997).\n\nDynamics of optically levitated nanoparticles in high vacuum. J Gieseler, Universitat Politècnica de CatalunyaPh.D. thesisJ. Gieseler, \"Dynamics of optically levitated nanoparticles in high vacuum,\" Ph.D. thesis, Universitat Politècnica de Catalunya (2014).\n\nTrapping in a material world. S E S Spesyvtseva, K Dholakia, ACS Photon. 3S. E. S. Spesyvtseva and K. Dholakia, \"Trapping in a material world,\" ACS Photon. 3, 719-736 (2016).\n\nCancellation of non-conservative scattering forces in optical traps by counter-propagating beams. S Divitt, L Rondin, L Novotny, Opt. Lett. 40S. Divitt, L. Rondin, and L. Novotny, \"Cancellation of non-conservative scattering forces in optical traps by counter-propagating beams,\" Opt. Lett. 40, 1900-1903 (2015).\n\nDynamics of noise-induced heating in atom traps. M E Gehm, K M O'hara, T A Savard, J E Thomas, Phys. Rev. A. 58M. E. Gehm, K. M. O'hara, T. A. Savard, and J. E. Thomas, \"Dynamics of noise-induced heating in atom traps,\" Phys. Rev. A 58, 3914-3921 (1998).\n\nTheory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach. C Gonzalez-Ballestero, P Maurer, D Windey, L Novotny, R Reimann, O Romero-Isart, arXiv:1902.01282arXiv preprintC. Gonzalez-Ballestero, P. Maurer, D. Windey, L. Novotny, R. Reimann, and O. Romero-Isart, \"Theory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach,\" arXiv preprint arXiv:1902.01282 (2019).\n\nLevitated optomechanics at the photon recoil limit. V Jain, ETH ZurichPh.D. thesisV. Jain, \"Levitated optomechanics at the photon recoil limit,\" Ph.D. thesis, ETH Zurich (2017).\n\nObservation of Brownian motion in liquids at short times: instantaneous velocity and memory loss. S Kheifets, A Simha, K Melin, T Li, M G Raizen, Science. 343S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, \"Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,\" Science 343, 1493-1496 (2014).\n\nThree-dimensional position detection of optically trapped dielectric particles. A Rohrbach, E H K Stelzer, J. Appl. Phys. 91A. Rohrbach and E. H. K. Stelzer, \"Three-dimensional position detection of optically trapped dielectric particles,\" J. Appl. Phys. 91, 5474-5488 (2002).\n\nBackscattering position detection for photonic force microscopy. G Volpe, G Kozyreff, D Petrov, J. Appl. Phys. 10284701G. Volpe, G. Kozyreff, and D. Petrov, \"Backscattering position detection for photonic force microscopy,\" J. Appl. Phys. 102, 084701 (2007).\n\nOptimal position detection of a dipolar scatterer in a focused field. F Tebbenjohanns, M Frimmer, L Novotny, Phys. Rev. A. 10043821F. Tebbenjohanns, M. Frimmer, and L. Novotny, \"Optimal position detection of a dipolar scatterer in a focused field,\" Phys. Rev. A 100, 043821 (2019).\n\nTransverse spin forces and non-equilibrium particle dynamics in a circularly polarized vacuum optical trap. V Svak, O Brzobohatỳ, M Šiler, P Jákl, J Kaňka, P Zemánek, S H Simpson, Nat. Commun. 95453V. Svak, O. Brzobohatỳ, M. Šiler, P. Jákl, J. Kaňka, P. Zemánek, and S. H. Simpson, \"Transverse spin forces and non-equilibrium particle dynamics in a circularly polarized vacuum optical trap,\" Nat. Commun. 9, 5453 (2018).\n\nTrapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps. M D Summers, D R Burnham, D Mcgloin, Opt. Express. 16M. D. Summers, D. R. Burnham, and D. McGloin, \"Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps,\" Opt. Express 16, 7739-7747 (2008).\n\nCavity cooling by coherent scattering of a levitated nanosphere in vacuum. U Delić, ViennaUniversity of ViennaPh.D. thesisU. Delić, \"Cavity cooling by coherent scattering of a levitated nanosphere in vacuum,\" Ph.D. thesis, University of Vienna, Vienna (2019).\n\nAdhesion and friction forces between spherical micrometer-sized particles. L.-O Heim, J Blum, M Preuss, H.-J Butt, Phys. Rev. Lett. 83L.-O. Heim, J. Blum, M. Preuss, and H.-J. Butt, \"Adhesion and friction forces between spherical micrometer-sized particles,\" Phys. Rev. Lett. 83, 3328-3331 (1999).\n\nT Li, Fundamental tests of physics with optically trapped microspheres. Springer Science & Business MediaT. Li, Fundamental tests of physics with optically trapped microspheres (Springer Science & Business Media, 2012).\n\nContact electrification of individual dielectric microparticles measured by optical tweezers in air. H Park, T W Lebrun, ACS Appl. Mater. Interfaces. 8H. Park and T. W. LeBrun, \"Contact electrification of individual dielectric microparticles measured by optical tweezers in air,\" ACS Appl. Mater. Interfaces 8, 34904-34913 (2016).\n\nSensitive force measurements with optically trapped micro-spheres in high vacuum. D P Atherton, RenoUniversity of NevadaPh.D. thesisD. P. Atherton, \"Sensitive force measurements with optically trapped micro-spheres in high vacuum,\" Ph.D. thesis, University of Nevada, Reno (2015).\n\nCavity cooling of free silicon nanoparticles in high vacuum. P Asenbaum, S Kuhn, S Nimmrichter, U Sezer, M Arndt, Nat. Commun. 42743P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, \"Cavity cooling of free silicon nanoparticles in high vacuum,\" Nat. Commun. 4, 2743 (2013).\n\nDirect loading of nanoparticles under high vacuum into a Paul trap for levitodynamical experiments. D S Bykov, P Mestres, L Dania, L Schmöger, T E Northup, arXiv:1905.04204arXiv preprintD. S. Bykov, P. Mestres, L. Dania, L. Schmöger, and T. E. Northup, \"Direct loading of nanoparticles under high vacuum into a Paul trap for levitodynamical experiments,\" arXiv preprint arXiv:1905.04204 (2019).\n\nCooling and manipulation of a levitated nanoparticle with an optical fiber trap. P Mestres, J Berthelot, M Spasenović, J Gieseler, L Novotny, R Quidant, Appl. Phys. Lett. 107151102P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, \"Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,\" Appl. Phys. Lett. 107, 151102 (2015).\n\nFlying particle sensors in hollow-core photonic crystal fibre. D S Bykov, O A Schmidt, T G Euser, P S J Russell, Nat. Photon. 9D. S. Bykov, O. A. Schmidt, T. G. Euser, and P. S. J. Russell, \"Flying particle sensors in hollow-core photonic crystal fibre,\" Nat. Photon. 9, 461-466 (2015).\n\nOptical trapping and control of nanoparticles inside evacuated hollow core photonic crystal fibers. D Grass, J Fesel, S G Hofer, N Kiesel, M Aspelmeyer, Appl. Phys. Lett. 108221103D. Grass, J. Fesel, S. G. Hofer, N. Kiesel, and M. Aspelmeyer, \"Optical trapping and control of nanoparticles inside evacuated hollow core photonic crystal fibers,\" Appl. Phys. Lett. 108, 221103 (2016).\n\nTemperature in non-equilibrium states: A review of open problems and current proposals. J Casas-Vázquez, D Jou, Rep. Prog. Phys. 66J. Casas-Vázquez and D. Jou, \"Temperature in non-equilibrium states: A review of open problems and current proposals,\" Rep. Prog. Phys. 66, 1937-2023 (2003).\n\nRealization of nonequilibrium thermodynamic processes using external colored noise. P Mestres, I A Martinez, A Ortiz-Ambriz, R A Rica, E Roldan, Phys. Rev. E. 9032116P. Mestres, I. A. Martinez, A. Ortiz-Ambriz, R. A. Rica, and E. Roldan, \"Realization of nonequilibrium thermody- namic processes using external colored noise,\" Phys. Rev. E 90, 032116 (2014).\n\nMotion of a spherical particle in a rarefied gas. part 2. drag and thermal polarization. S A Beresnev, V G Chernyak, G A Fomyagin, J. Fluid Mech. 219S. A. Beresnev, V. G. Chernyak, and G. A. Fomyagin, \"Motion of a spherical particle in a rarefied gas. part 2. drag and thermal polarization,\" J. Fluid Mech. 219, 405-421 (1990).\n\nGas-induced friction and diffusion of rigid rotors. L Martinetz, K Hornberger, B A Stickler, Phys. Rev. E. 9752112L. Martinetz, K. Hornberger, and B. A. Stickler, \"Gas-induced friction and diffusion of rigid rotors,\" Phys. Rev. E 97, 052112 (2018).\n\nNanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere. J Millen, T Deesuwan, P Barker, J Anders, Nat. Nanotech. 9J. Millen, T. Deesuwan, P. Barker, and J. Anders, \"Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,\" Nat. Nanotech. 9, 425-429 (2014).\n\nOn the resistance experienced by spheres in their motion through gases. P S Epstein, Phys. Rev. 23P. S. Epstein, \"On the resistance experienced by spheres in their motion through gases,\" Phys. Rev. 23, 710-733 (1924).\n\nRadiation damping of a polarizable particle. L Novotny, Phys. Rev. A. 9632108L. Novotny, \"Radiation damping of a polarizable particle,\" Phys. Rev. A 96, 032108 (2017).\n\nQuantum optomechanics. M Aspelmeyer, P Meystre, K Schwab, Phys. Today. 65M. Aspelmeyer, P. Meystre, and K. Schwab, \"Quantum optomechanics,\" Phys. Today 65, 29-35 (2012).\n\nCalibration and energy measurement of optically levitated nanoparticle sensors. E Hebestreit, M Frimmer, R Reimann, C Dellago, F Ricci, L Novotny, Rev. Sci. Instrumen. 8933111E. Hebestreit, M. Frimmer, R. Reimann, C. Dellago, F. Ricci, and L. Novotny, \"Calibration and energy measurement of optically levitated nanoparticle sensors,\" Rev. Sci. Instrumen. 89, 033111 (2018).\n\nIntroduction to quantum noise, measurement, and amplification. A A Clerk, M H Devoret, S M Girvin, F Marquardt, R J Schoelkopf, Rev. Mod. Phys. 82A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, \"Introduction to quantum noise, measurement, and amplification,\" Rev. Mod. Phys. 82, 1155-1208 (2010).\n\nOptomechanical Raman-ratio thermometry. T P Purdy, P.-L Yu, N S Kampel, R W Peterson, K Cicak, R W Simmonds, C A , Phys. Rev. A. 9231802T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, \"Optomechanical Raman-ratio thermometry,\" Phys. Rev. A 92, 031802 (2015).\n\nObservation of quantum motion of a nanomechanical resonator. A H Safavi-Naeini, J Chan, J T Hill, T P M Alegre, A Krause, O Painter, Phys. Rev. Lett. 10833602A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, \"Observation of quantum motion of a nanomechanical resonator,\" Phys. Rev. Lett. 108, 033602 (2012).\n\nObservation and interpretation of motional sideband asymmetry in a quantum electromechanical device. A J Weinstein, C U Lei, E E Wollman, J Suh, A Metelmann, A A Clerk, K C Schwab, Phys. Rev. X. 441003A. J. Weinstein, C. U. Lei, E. E. Wollman, J. Suh, A. Metelmann, A. A. Clerk, and K. C. Schwab, \"Observation and interpretation of motional sideband asymmetry in a quantum electromechanical device,\" Phys. Rev. X 4, 041003 (2014).\n\nControlling the net charge on a nanoparticle optically levitated in vacuum. M Frimmer, K Luszcz, S Ferreiro, V Jain, E Hebestreit, L Novotny, Phys. Rev. A. 9561801M. Frimmer, K. Luszcz, S. Ferreiro, V. Jain, E. Hebestreit, and L. Novotny, \"Controlling the net charge on a nanoparticle optically levitated in vacuum,\" Phys. Rev. A 95, 061801 (2017).\n\nA nanomechanical mass sensor with yoctogram resolution. J Chaste, A Eichler, J Moser, G Ceballos, R Rurali, A Bachtold, Nat. Nanotech. 7J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, \"A nanomechanical mass sensor with yoctogram resolution,\" Nat. Nanotech. 7, 301-304 (2012).\n\nMeasuring the internal temperature of a levitated nanoparticle in high vacuum. E Hebestreit, R Reimann, M Frimmer, L Novotny, Phys. Rev. A. 9743803E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, \"Measuring the internal temperature of a levitated nanoparticle in high vacuum,\" Phys. Rev. A 97, 043803 (2018).\n\nCavity cooling of an optically levitated submicron particle. N Kiesel, F Blaser, U Delić, D Grass, R Kaltenbaek, M Aspelmeyer, Proc. Natl. Acad. Sci. 110N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, \"Cavity cooling of an optically levitated submicron particle,\" Proc. Natl. Acad. Sci. 110, 14180-14185 (2013).\n\nMillikelvin cooling of an optically trapped microsphere in vacuum. T Li, S Kheifets, M G Raizen, Nat. Phys. 7T. Li, S. Kheifets, and M. G. Raizen, \"Millikelvin cooling of an optically trapped microsphere in vacuum,\" Nat. Phys. 7, 527-530 (2011).\n\nElectric feedback cooling of single charged nanoparticles in an optical trap. M Iwasaki, T Yotsuya, T Naruki, Y Matsuda, M Yoneda, K Aikawa, Phys. Rev. A. 9951401M. Iwasaki, T. Yotsuya, T. Naruki, Y. Matsuda, M. Yoneda, and K. Aikawa, \"Electric feedback cooling of single charged nanoparticles in an optical trap,\" Phys. Rev. A 99, 051401 (2019).\n\nFeedback cooling of a cantilever's fundamental mode below 5 mk. M Poggio, C L Degen, H J Mamin, D Rugar, Phys. Rev. Lett. 9917201M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, \"Feedback cooling of a cantilever's fundamental mode below 5 mk,\" Phys. Rev. Lett. 99, 017201 (2007).\n\nSpontaneous emission probabilities at radio frequencies. E M Purcell, Confined Electrons and Photons. SpringerE. M. Purcell, \"Spontaneous emission probabilities at radio frequencies,\" in Confined Electrons and Photons, (Springer, 1995), pp. 839-839.\n\nA Yariv, Quantum electronics. New YorkJohn Wiley & SonsA. Yariv, Quantum electronics (John Wiley & Sons, New York, 1989).\n\nNon-equilibrium steady state of a driven levitated particle with feedback cooling. J Gieseler, L Novotny, C Moritz, C Dellago, New J. Phys. 1745011J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, \"Non-equilibrium steady state of a driven levitated particle with feedback cooling,\" New J. Phys. 17, 045011 (2015).\n\nQuantum model of cooling and force sensing with an optically trapped nanoparticle. B Rodenburg, L P Neukirch, A N Vamivakas, M Bhattacharya, Optica. 3B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, \"Quantum model of cooling and force sensing with an optically trapped nanoparticle,\" Optica 3, 318-323 (2016).\n\nQuantum calculation of feedback cooling a laser levitated nanoparticle in the shot-noise-dominant regime. C Zhong, T Li, F Robicheaux, arXiv:1708.01203arXiv preprintC. Zhong, T. Li, and F. Robicheaux, \"Quantum calculation of feedback cooling a laser levitated nanoparticle in the shot-noise-dominant regime,\" arXiv preprint arXiv:1708.01203 (2017).\n\nFree nano-object Ramsey interferometry for large quantum superpositions. C Wan, M Scala, G W Morley, A T M A Rahman, H Ulbricht, J Bateman, P F Barker, S Bose, M S Kim, Phys. Rev. Lett. 117143003C. Wan, M. Scala, G. W. Morley, A. T. M. A. Rahman, H. Ulbricht, J. Bateman, P. F. Barker, S. Bose, and M. S. Kim, \"Free nano-object Ramsey interferometry for large quantum superpositions,\" Phys. Rev. Lett. 117, 143003 (2016).\n\nOptical cryocooling of diamond. M Kern, J Jeske, D W M Lau, A D Greentree, F Jelezko, J Twamley, Phys. Rev. B. 95235306M. Kern, J. Jeske, D. W. M. Lau, A. D. Greentree, F. Jelezko, and J. Twamley, \"Optical cryocooling of diamond,\" Phys. Rev. B 95, 235306 (2017).\n\nDiamonds levitating in a Paul trap under vacuum: Measurements of laser-induced heating via NV center thermometry. T Delord, L Nicolas, M Bodini, G Hétet, Appl. Phys. Lett. 11113101T. Delord, L. Nicolas, M. Bodini, and G. Hétet, \"Diamonds levitating in a Paul trap under vacuum: Measurements of laser-induced heating via NV center thermometry,\" Appl. Phys. Lett. 111, 013101 (2017).\n\nInternal quantum dynamics of a nanoparticle in a thermal electromagnetic field: A minimal model. A E R López, C Gonzalez-Ballestero, O Romero-Isart, Phys. Rev. B. 98155405A. E. R. López, C. Gonzalez-Ballestero, and O. Romero-Isart, \"Internal quantum dynamics of a nanoparticle in a thermal electromagnetic field: A minimal model,\" Phys. Rev. B 98, 155405 (2018).\n\nDynamics of a levitated microparticle in vacuum trapped by a perfect vortex beam: three-dimensional motion around a complex optical potential. Y Arita, M Chen, E M Wright, K Dholakia, J. Opt. Soc. Am. B. 34Y. Arita, M. Chen, E. M. Wright, and K. Dholakia, \"Dynamics of a levitated microparticle in vacuum trapped by a perfect vortex beam: three-dimensional motion around a complex optical potential,\" J. Opt. Soc. Am. B 34, C14-C19 (2017).\n\nRoadmap on structured light. H Rubinsztein-Dunlop, A Forbes, M V Berry, M R Dennis, D L Andrews, M Mansuripur, C D , C Alpmann, P Banzer, T Bauer, E Karimi, L Marrucci, M Padgett, M Ritsch-Marte, N M Litchinitser, N P Bigelow, C Rosales-Guzmán, A Belmonte, J P Torres, T W Neely, M Baker, R Gordon, A B Stilgoe, J Romero, A G White, R Fickler, A E Willner, G Xie, B Mcmorran, A M Weiner, J. Opt. 1913001H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. D., C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, \"Roadmap on structured light,\" J. Opt. 19, 013001 (2016).\n\nHolographic optical trapping of aerosol droplets. D R Burnham, D Mcgloin, Opt. Express. 14D. R. Burnham and D. McGloin, \"Holographic optical trapping of aerosol droplets,\" Opt. Express 14, 4175-4181 (2006).\n\nOptical levitation and feedback cooling of a nanoparticle at subwavelength distances from a membrane. R Diehl, E Hebestreit, R Reimann, F Tebbenjohanns, M Frimmer, L Novotny, Phys. Rev. A. 9813851R. Diehl, E. Hebestreit, R. Reimann, F. Tebbenjohanns, M. Frimmer, and L. Novotny, \"Optical levitation and feedback cooling of a nanoparticle at subwavelength distances from a membrane,\" Phys. Rev. A 98, 013851 (2018).\n"},"corpus_id":{"kind":"number","value":240625360,"string":"240,625,360"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2004.05246\",\n \"mag\": 3107944508,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1364/aop.394888\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [],\n \"pdf_sha\": \"051ba960fd2bb0d4a55f78bdc20f53d87acef3cb\",\n \"open_access\": {\n \"license\": \"other-oa\",\n \"url\": \"https://diposit.ub.edu/dspace/bitstream/2445/176402/1/711344.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Optical Tweezers: A Comprehensive Tutorial from Calibration to Applications"},"authors":{"kind":"list like","value":["Jan Gieseler \nICFO -Institut de Ciències Fotòniques\nThe Barcelona Institute of Science and Technology\n08860CastelldefelsBarcelonaSpain\n\nDepartment of Physics\nHarvard University\n17 Oxford StCambridgeMAUSA","Juan Ruben Gomez-Solano \nInstituto de Física\nUniversidad Nacional Autónoma de México\nApdo. Postal 20-364, 01000Cd. MéxicoMexico","Alessandro Magazzù \nDepartment of Physics\nUniversity of Gothenburg\n41296GothenburgSweden","Isaac Pérez Castillo \nInstituto de Física\nUniversidad Nacional Autónoma de México\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\n\nLondon Mathematical Laboratory\n18 Margravine GardensW6 8RHLondonUnited Kingdom","Laura Pérez García \nDepartment of Physics\nUniversity of Gothenburg\n41296GothenburgSweden","Marta Gironella-Torrent \nDepartament de Fsica de la Matèria Condensada\nFacultat de Física\nSmall Biosystems Lab\nUniversitat de Barcelona\n08028BarcelonaSpain","Xavier Viader-Godoy \nDepartament de Fsica de la Matèria Condensada\nFacultat de Física\nSmall Biosystems Lab\nUniversitat de Barcelona\n08028BarcelonaSpain","Felix Ritort \nDepartament de Fsica de la Matèria Condensada\nFacultat de Física\nSmall Biosystems Lab\nUniversitat de Barcelona\n08028BarcelonaSpain","Giuseppe Pesce \nDepartment of Physics\nUniverisità degli di Studi di Napoli \"Federico II\" Complesso universitario monte S. Angelo\nVia Cintia80126NapoliItaly","Alejandro V Arzola \nInstituto de Física\nUniversidad Nacional Autónoma de México\nApdo. Postal 20-364, 01000Cd. MéxicoMexico","Karen Volke-Sepulveda \nInstituto de Física\nUniversidad Nacional Autónoma de México\nApdo. Postal 20-364, 01000Cd. MéxicoMexico","Giovanni Volpe *giovanni.volpe@physics.gu.se \nDepartment of Physics\nUniversity of Gothenburg\n41296GothenburgSweden"],"string":"[\n \"Jan Gieseler \\nICFO -Institut de Ciències Fotòniques\\nThe Barcelona Institute of Science and Technology\\n08860CastelldefelsBarcelonaSpain\\n\\nDepartment of Physics\\nHarvard University\\n17 Oxford StCambridgeMAUSA\",\n \"Juan Ruben Gomez-Solano \\nInstituto de Física\\nUniversidad Nacional Autónoma de México\\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\",\n \"Alessandro Magazzù \\nDepartment of Physics\\nUniversity of Gothenburg\\n41296GothenburgSweden\",\n \"Isaac Pérez Castillo \\nInstituto de Física\\nUniversidad Nacional Autónoma de México\\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\\n\\nLondon Mathematical Laboratory\\n18 Margravine GardensW6 8RHLondonUnited Kingdom\",\n \"Laura Pérez García \\nDepartment of Physics\\nUniversity of Gothenburg\\n41296GothenburgSweden\",\n \"Marta Gironella-Torrent \\nDepartament de Fsica de la Matèria Condensada\\nFacultat de Física\\nSmall Biosystems Lab\\nUniversitat de Barcelona\\n08028BarcelonaSpain\",\n \"Xavier Viader-Godoy \\nDepartament de Fsica de la Matèria Condensada\\nFacultat de Física\\nSmall Biosystems Lab\\nUniversitat de Barcelona\\n08028BarcelonaSpain\",\n \"Felix Ritort \\nDepartament de Fsica de la Matèria Condensada\\nFacultat de Física\\nSmall Biosystems Lab\\nUniversitat de Barcelona\\n08028BarcelonaSpain\",\n \"Giuseppe Pesce \\nDepartment of Physics\\nUniverisità degli di Studi di Napoli \\\"Federico II\\\" Complesso universitario monte S. Angelo\\nVia Cintia80126NapoliItaly\",\n \"Alejandro V Arzola \\nInstituto de Física\\nUniversidad Nacional Autónoma de México\\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\",\n \"Karen Volke-Sepulveda \\nInstituto de Física\\nUniversidad Nacional Autónoma de México\\nApdo. Postal 20-364, 01000Cd. MéxicoMexico\",\n \"Giovanni Volpe *giovanni.volpe@physics.gu.se \\nDepartment of Physics\\nUniversity of Gothenburg\\n41296GothenburgSweden\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Since their invention in 1986 by Arthur Ashkin and colleagues, optical tweezers have become an essential tool in several fields of physics, spectroscopy, biology, nanotechnology, and thermodynamics. In this Tutorial, we provide a primer on how to calibrate optical tweezers and how to use them for advanced applications. After a brief general introduction on optical tweezers, we focus on describing and comparing the various available calibration techniques. Then, we discuss some cutting-edge applications of optical tweezers in a liquid medium, namely to study single-molecule and single-cell mechanics, microrheology, colloidal interactions, statistical physics, and transport phenomena. Finally, we consider optical tweezers in vacuum, where the absence of a viscous medium offers vastly different dynamics and presents new challenges. We conclude with some perspectives for the field and the future application of optical tweezers. This Tutorial provides both a step-by-step guide ideal for non-specialists entering the field and a comprehensive manual of advanced techniques useful for expert practitioners. All the examples are complemented by the sample data and software necessary to reproduce them."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"In 1970, Arthur Ashkin [1] demonstrated that a focused laser beam can accelerate, decelerate, and even stably trap micrometer-sized neutral particles. Later, in 1986, Ashkin and colleagues reported the first realization of an optical tweezers [2,3]: a tightly focused beam of light capable of holding microscopic particles in three dimensions. Thanks to this work, Ashkin was awarded a share of the 2018 Nobel Prize in Physics [4]. Furthermore, one of Ashkin's co-authors, Steven Chu, would go on to use optical tweezing in his work on cooling and trapping atoms; this research earned Chu a share of the 1997 Nobel Prize in Physics [5].\nIn the late 1980s, Ashkin and colleagues used optical tweezers to manipulate biological samples, starting from an individual tobacco mosaic virus and Escherichia coli bacterium [6]. In 1993, Lucien Ghislain and Watt Webb invented photonic force microscopy [?], where an optically trapped particle is used as an extremely sensitive cantilever to probe microscopic force fields ranging from femtonewtons (10 −15 N) to piconewtons (10 −12 N). Since the early 1990s, optical force spectroscopy has been utilized to characterize the mechanical properties of biomolecules and biological motors [7][8][9]. Optical tweezers have also been employed in many fields of physics [10][11][12][13][14][15], nanotechnology [16], spectroscopy [17], nanothermodynamics [18,19], soft matter [20], and biology [21].\nAn optical tweezers consist of a laser beam focused by a high numerical aperture (NA) microscope objective. A microscopic particle (whose refractive index is higher than that of its embedding medium) can then be trapped near the focal spot because of the emergence of scattering and gradient optical forces: the scattering forces are due to the radiation pressure of the light beam and act along the direction of propagation of the beam; and the gradient forces pull the particle towards the high-intensity focal spot. Using optical tweezers, a very broad range of particles has been trapped, as shown in Fig. 1.\nThere are already several resources to build and operate optical tweezers available in the literature. In the 2015 book Optical Tweezers: Principles and Applications [22], Phillip. H. Jones, Onofrio M. Maragó and Giovanni Volpe have discussed in detail the theory behind optical tweezers as well as how to build and operate optical tweezers setups. There are also several other tutorials on specific aspects of optical trapping. In particular, there are computational toolboxes to calculate optical forces using T-matrix methods by Timo Nieminen et al. [23] and geometrical optics by Agnese Callegari et al. [24]. There is a tutorial on how to simulate the Brownian motion of an optically trapped particle [25]. There are several tutorials on how to build optical tweezers setups by Stephen Smith et al. [26], John Bechhoefer and Scott Wilson [27], Angela Mellish and Andrew Wilson [28], David Appleyard et al. [29], Woei Lee et al. [30], Manoj Mathew et al. [31], and Giuseppe Pesce et al. [32].\nIn this Tutorial, we complete and complement these already available resources providing a primer on how to calibrate optical tweezers and how to use them for advanced applications. In particular, after a brief general introduction on optical tweezers (section 2), we focus on describing and comparing the various calibration techniques that are available (section 3). Then, we discuss some of the most exciting cutting-edge applications of optical tweezers in a liquid medium (section 4), namely to study single-molecule and single-cell mechanics, microrheology, colloidal interactions, statistical physics, and transport phenomena. Finally, we consider optical tweezers in vacuum, which features different dynamics than a viscous medium, so that vacuum operation comes with its own experimental challenges and calibration methods (section 5). This Tutorial provides both a step-by-step guide ideal for non-specialists entering the field and a comprehensive manual of advanced techniques useful for expert practitioners. All the examples are complemented by the sample data and the software necessary to reproduce them [33]."},{"title":"Getting started","content":"In this section we provide some essential background information on optical tweezers. First, we will describe a basic optical tweezers setup (section 2.1). Then, we will briefly discuss the Brownian motion of an optically trapped particle and the most basic optical tweezers calibration techniques (section 2.2). Finally, we will give a brief overview of more advanced optical tweezers and of alternative approaches for the trapping and manipulation of microscopic matter (section 2.3)."},{"title":"A simple setup","content":"An optical tweezers is a highly-focused laser beam that can trap microscopic particles. In practice, an optical tweezers can be built by focusing a laser beam through a microscope, which can be either commercial or homemade. As illustrated in Fig. 2, a basic optical tweezers setup is comprised of three parts: trapping, imaging, and position detection.\nTrapping. In most applications, the laser is continuous wave (CW), whose wavelength is in the visible or near-infrared region of the spectrum, and whose power is between 10 mW and 1 W.\nThe laser beam is expanded by a telescope to overfill the back-aperture of a high-NA microscope objective, is directed by a dichroic mirror towards the objective, and finally is focused by the objective to a diffraction-limited spot within the sample holder. This focal spot is the optical tweezers, which can trap a microscopic particle. Usually, the object to be trapped is suspended in an aqueous solution, which is held between a microscope slide and a cover slip; sometimes the particles to be trapped are suspended in other liquids, or in gases, or even in vacuum (section 5).\nImaging. Typically, the trapping and imaging objectives are the same. The light for the imaging is provided by a condenser, which is an integral part of any microscope. It is convenient to image through a dichroic beam splitter, which reflects the trapping light to the microscope, but also transmits some of the imaging light from the condenser to a camera. In both cases, some color filters are typically required in front of the camera to reduce the intensity of the trapping beam to an acceptable level such that the camera is not overexposed."},{"title":"Position detection.","content":"To use the optically trapped particle as a microscopic force transducer, it is necessary to track its position. This can be done by analyzing either the videos of the particle acquired by the camera (digital video microscopy), or the time series acquired through a position detector. In this latter case, the condenser collects the forward scattered light and projects it onto a position detector (typically either a quadrant photodetector or a position sensing detector). This detects the changes in the forward scattered light pattern due to the changes in the particle's position."},{"title":"Optical tweezers calibration","content":"An optically trapped particle can be used as an extremely sensitive microscopic force transducer that can both exert and measure forces ranging from hundreds of piconewtons down to a few femtonewtons.\nAn optically trapped particle experiences a harmonic force that tends to keep it near a stable equilibrium position. Considering only one dimension, for small displacements from the equilibrium position, the force acting on a colloidal particle positioned at x within a trap centered at x eq is\nwhere κ x is the trap stiffness along the x-direction. The resulting trapping potential is harmonic:\nA typical trajectory of an optically trapped particle can be seen in Fig. 3(a). The particle keeps on moving due to the presence of Brownian fluctuations. Therefore, an optically trapped particle is in a dynamic equilibrium between the thermal noise continuously pushing it out of the trap and the optical forces driving it towards the equilibrium position. For the particle to remain within the optical trap, the optical potential well must be sufficiently deep. The depth of the optical potential is typically characterized in units of the thermal energy k B T, where k B is the Boltzmann constant and T is the absolute temperature. This quantity gives a characteristic energy scale for mesoscopic phenomena. The potential well of an optical tweezers should be at least a few k B T deep to be able to confine a particle, as shown in Fig. 3(b).\nThe analysis of the thermal motion of the optically trapped particle in the optical potential provides information about the local forces acting on the particle. For example, since the particle is held in a harmonic trapping potential, its position distribution is Gaussian, as shown in Fig. 3(c), By sampling this probability distribution, it is possible to measure κ x using the equipartition theorem. The equipartition theorem states that for a system at thermodynamic equilibrium at absolute temperature T, the energy associated with each harmonic degree of freedom is equal to 1 2 k B T. Therefore, U(x) = 1 2 κ x (x − x eq ) 2 = 1 2 κ x σ 2 x = 1 2 k B T, where σ 2 x = (x − x eq ) 2 is the variance of the trapped particle position, and\nAs we will see in section 3, there are more sophisticated methods to use the thermal fluctuations of an optically trapped particle to calibrate its stiffness. In particular, it is possible to also exploit the temporal evolution of the particle in the optical trap, for example by calculating its correlation and power spectral density.\nOnce an optical tweezers is calibrated, a constant and homogeneous external force F ext, x shifts the equilibrium position of the trap. The value of the force can be obtained as:\nwhere ∆x eq is the average particle displacement from the original equilibrium position without the external force."},{"title":"Advanced and alternative approaches","content":"Going beyond the basic setup introduced in the previous section, more advanced setups become necessary to extend the optical manipulation toolkit, introducing, e.g., optical spanners [34], optical rheometers [35], optical stretchers [36], and optical sorters [37][38][39]. These tools allow one to manipulate multiple particles at once [40][41][42][43], rotate particles [44][45][46][47], measure the mechanical properties of biological samples [48], sort particles [49], trap particles with refractive index lower than their medium [50][51][52][53], induce optical binding and collective dynamics [11,[54][55][56][57][58][59]. Complex optical potentials, such as periodic and random patterns, provide experimental models to study microscopic and non-linear transport mechanisms [60][61][62][63][64][65][66][67]. Advanced optical manipulation has been employed also to build and study micromachines [68][69][70] and optofluidic devices [37,[71][72][73][74][75][76]. Furthermore, optical manipulation setups have also been successfully employed to investigate the properties of light itself, employing structured light beams with peculiar intensity, phase and polarization distributions [34,44,[77][78][79][80][81].\nIn this section, we will provide a brief overview of the main advanced optical manipulation techniques. We will consider only the overdamped regime (i.e., particles immerse in a viscous medium like water), since the case of underdamped systems involves many different experimental aspects that will be treated in (section 5). Also, unless otherwise stated, we will assume that the refractive index of the particles under consideration is higher than that of the surrounding medium."},{"title":"Beam steering","content":"Most advanced optical micromanipulation setups require to steer the trapping beam to control the position of the optical tweezers. The beam steering device can be as simple as a mirror on a gimbal mount (preferred over conventional kinematic mounts to avoid mirror shifting). For example, this simple approach can be used to move the optical trap within the sample instead of moving the sample with a translation stage while keeping the optical trap fixed. More sophisticated beam-steering devices are also often employed, such as acousto-optic deflectors and spatial light modulators (SLMs).\nWhen implementing beam steering, it is essential to ensure that the available light reaches the back aperture of the microscope objective, irrespective of the beam direction. To this end, a telescope should be added (4f-system) to create conjugate planes between the beam steering device and the back aperture of the objective, as shown in Fig. 4. This telescope can also be used to magnify the beam to slightly overfill the back aperture of the objective (to improve trapping), and a spatial filter can be added at Fourier plane 1 (to remove unwanted higher-order diffraction patterns), which is conjugated with Fourier plane 2. Fig. 4. Beam steering mechanism. The trapping beam is steered by the steering device (e.g., a galvo-mirror, an acousto-optic deflector, or a spatial light modulator). To ensure that the trapping beam reaches the back aperture of the microscope objective (O) while being steered, a telescope is added along the beam path (lenses L 1 and L 2 ) in a 4f-configuration ( f 1 and f 2 are the focal lengths of L 1 and L 2 respectively)."},{"title":"Polarization splitting","content":"The simplest alternative to have two neighboring traps while avoiding the interference between them is to split the original beam into two orthogonally polarized beams [40,82], as shown in Fig. 5. If the main control parameter of a given experiment is the distance between the traps, a single steering device can be enough, keeping one trap fixed while the other one is shifted. This works fine for a small distance between the traps. If the traps need to be moved over a large area, it might be useful to use two steering mirrors to minimize the distortion of each trap. Dynamical control of each trap can be achieved if the gimbal mounts are operated by actuators along the two axes, giving rise to a continuous motion along arbitrary trajectories in two dimensions in the sample plane. Finally, this twin-trap approach can also be used in combination with other advanced techniques, if the aim is to superimpose two complex patterns without interference between them. 1 and GM 2 should be mounted on gimbal mounts, and the distance between each of them and the lens L 1 should be equal to the focal length of L 1 (Fig. 4). For alignment purposes, both traps should be initially superimposed, arriving exactly at the same point in the sample plane. The intensity ratio between the traps can also be controlled, by placing a polarizer followed by a λ/2-retarder at the entrance of the Mach-Zehnder setup. As the retarder is rotated, the proportion of light reflected and transmitted through PBS 1 will change. DM: Dichroic mirror. M: mirror. O: microscope objective. C: Condenser."},{"title":"Time-sharing optical traps","content":"A single beam can be used to generate multiple optical tweezers if it is scanned sufficiently fast, i.e., with a frequency high enough to ensure that an optically trapped particle cannot diffuse away from the optical trap region between consecutive visits of the laser beam. This technique is known as time-sharing or laser-scanning optical trap. It can be performed with different deflecting devices, such as acousto-optic deflectors (AODs), mirror galvanometers (galvo-mirrors), electro-optic deflectors (EODs), and piezo-mirrors [40,[83][84][85][86]. Often, the beam deflecting devices have only one degree of freedom so that, to achieve beam motion over the whole sample plane, it is necessary to use two beam deflecting devices, one for each transverse direction, as illustrated in Fig. 6.\nThe angular range and the scanning frequency of the beam deflecting device determines the number of traps that can be created simultaneously and the extent of the trapping area. There is usually a compromise between the scanning speed and the deflection angular range. The first setup of time-sharing optical tweezers was built with galvo-mirrors [83], which could reach scanning frequencies of the order of 10 Hz. Nowadays, there are similar commercial systems that can reach up to 100 Hz, with a angular range of up to 25 • ; however, the angular range in the optical trapping setup is limited by the maximum acceptance angle of the microscope objective, which is just a few degrees at most. AODs can attain frequencies from tens to hundreds of kilohertz with angular ranges of up to 3 • , which makes them very suitable for trapping applications. For example, these devices have been used in studies of DNA stretching and molecular motors [40]. However, as the AOD working principle is based on the generation of diffraction gratings by means of the propagation of acoustic waves in a glass or quartz cell, resulting in a periodic refractive index modulation, it is worth to keep in mind that the diffraction efficiency can be at Fig. 6. Time-sharing optical tweezers. The trapping beam is time-shared between several optical traps using two beam deflector devices along the xand y-directions (X-BDD and Y-BDD, respectively). Two telescopes are needed (L 1 -L 2 and L 3 -L 4 ) to create conjugate planes between the two deflector devices as well as with the back aperture of the microscope objective O. most around 70%. EODs are even faster than AODs, but with smaller angular ranges of fractions of a degree. Piezo-mirrors are more limited, reaching frequencies of hundreds of Hertz with angular ranges of only a few hundredths of a degree.\nIf the position of a given trap is slightly shifted from one visit to the next one, the bead will follow it, making it possible to describe continuous trajectories within the plane. The average intensity in each trap can also be controlled by means of the number of visits of the scanning beam in a full cycle. For example, if a trapping site is visited twice as much as another one, it will experience twice the average intensity."},{"title":"Interferometric optical traps","content":"Interference patterns can trap multiple particles simultaneously and generate ordered structures [54,55,87]. For example, consider the simplest interference pattern consisting of alternating bright and dark fringes. The dynamics of the particles depend on their size with respect to the characteristic size of the fringes. Particles smaller than the fringes are attracted towards the bright fringes, while remaining free to move along the fringes. However, much more interesting dynamics emerge when the particle size is comparable or even larger than the fringes [39,49,88]. This has been called the size effect and will be explained in more detail in section 4.6.\nTo generate alternating bright and dark fringes, we need two collimated beams travelling with a relative angle: the larger this angle, the smaller the fringe period (the smallest fringe pattern is half wavelength). One possibility is to use a Mach-Zehnder interferometer using non-polarizing beam splitters ( Fig. 7(a)). A 4 f -system is formed by the lens L 1 and the microscope objective that conjugates the mirror on the gimbal mount and the focal plane of the objective (the objective can be replaced by an aspherical lens, since the high-NA requirement may be relaxed in this case). While an extended interference pattern generates strong intensity gradients in the transverse plane, the intensity gradient along the propagation direction is usually not enough to trap in 3D so that the particles are pushed by the scattering force against either the bottom or the top surface of the sample cell.1 Fig. 7(b) shows an alternative configuration for the interferometer module. In both configurations (Figs. 7(a) and 7(b)) a single mirror mounted on a gimbal mount is enough to control the fringe period for small interference angles.\nAlternative configurations are possible, an example of which is illustrated in Fig. 7 two beams are redirected by mirrors M 1 and M 2 towards a right-angle prism, which reflects them towards the objective, which in turn focuses them on the sample. The right-angle prism is mounted on a single-axis translation stage to control the distance between the reflected beams and, therefore, the interference angle. A low NA objective is advisable for this configuration to minimize the changes in the intensity distribution along the z-axis within the sample depth.\nAn extreme case of interference pattern is a speckle light field, which arises from the interference of a multitude of coherent waves with random phases, generated when laser light is, e.g., reflected from a rough surface or transmitted through a turbid medium. Speckle patterns have also been used for optical manipulation [65,89,90]."},{"title":"Holographic optical tweezers","content":"The most versatile optical tweezers setups are arguably based on holographic optical tweezers (HOTs) [41][42][43][91][92][93]. The basic working principle of HOTs is shown in Fig. 8. The wavefront of an incident beam is reshaped by a hologram projected on a spatial light modulator (SLM) and then relayed by a series of lenses to an objective that then generates the desired light pattern at the sample. In practice, there are many possible different HOT configurations. Discussing their details is out of the scope of this Tutorial, but a comprehensive review of the topic (including hologram computing algorithms, a numerical toolbox, and a comparison in terms of the main parameters such as efficiency, average intensity, uniformity, and percentage standard error) can a b c d\nFourier-plane phase mask Image-plane phase mask In the cases (a)-(c), the light pattern is obtained at the Fourier plane ( Fig. 8(a)), while in case (d) it is at the image plane ( Fig. 8(b)).\nbe found in Ref. [22]. Here, we will focus our attention on only two approaches.\nIn the first approach, the desired light pattern at the sample is the Fourier transform of the modulated laser beam right after the SLM. It is necessary to include a 4 f -system between the SLM and the back aperture of the objective to conjugate the two optical planes and to perform spatial filtering (Fourier Plane 1 in Fig. 8(a)). This approach works particularly well to generate multiple focal spots distributed in predetermined locations in space, which serve as individual traps. The simplest case is a single steerable trap ( Fig. 9(a)), and the corresponding hologram is a superposition of a blazed diffraction grating (which determines the lateral position of the focal spot) and a Fresnel lens (which shifts the focal spot along the propagation axis). For the generation of multiple traps, there are different algorithms to calculate the hologram, such as the superposition of gratings and lenses, the random mask encoding, the Gerchberg-Saxton algorithm, and the adaptive-additive algorithm [22]. An example of a phase mask that produces a set of individual traps can be seen in Fig. 9(b). Structured beams (e.g., Laguerre-Gaussian beams, Fig. 9(c)) can also be straightforwardly created with this approach. Furthermore, continuous optical potentials with arbitrary shapes can be produced as well by calculating the corresponding hologram with the aforementioned algorithms [22].\nIn the second approach, a phase profile related to the desired light pattern is directly projected on the SLM and relayed on the objective image plane (this requires an even number of lenses between the two planes, Fig. 8(b)). This configuration is specially suitable when the field of interest is not meant to be tightly focused (low-NA optics), e.g., using phase gratings to split the beam and generate interference fringes at the objective image plane [94]. Also, Bessel beams ( Fig. 9(d)), Mathieu beams, and nondiffracting beams are typically generated at the image plane instead of at the Fourier plane [45,52,79,[95][96][97][98][99][100][101][102].\nTypical switching speeds for nematic liquid crystal SLMs can reach 60 Hz with a large number of discrete phase levels (typically 256), whereas ferroelectric liquid crystal SLMs can switch at a rate of up to 1 kHz, but with only two phase levels (0 and π)."},{"title":"Calibration","content":"As we have seen in section 2.2, an optical trap acts as a microscopic Hookeian spring with a fixed stiffness so that the restoring force is proportional to the displacement of the optically trapped particle from the center of the optical trap. The calibration of an optical tweezers entails the determination of the value of the optical trap stiffness, which in general depends on the properties of both the light beam and the particle. The calibration of an optical tweezers is essential to use it as a microscopic force transducer, i.e., to measure and exert forces in the femtonewton and piconewton range.\nIn principle, it should be possible to determine the value of the stiffness for a given experimental configuration where the properties of the focused laser beam and the optically trapped particle are known. In fact, this has been done in certain cases [67,[103][104][105]. However, this is in general a cumbersome task because an exact description of the focused beam and its interaction with particles depends on several parameters that are very difficult to know accurately and may vary drastically even within the same experiment. For example, just the theoretical description of the focused beam requires precise information about the focusing lens, the thickness of the coverslip, the refractive index of the media, and the position of the focus inside the sample. Moreover, even with all this information at hand, the interaction between a focused beam and a trapped object is very complex, requiring to solve a computationally intensive electromagnetic scattering problem [22]. Therefore, the development and improvement of in-situ calibration methods have been very important to extend the practical applications of optical tweezers.\nThe calibration methods can be classified as either passive or active. Passive methods (sections 3.2 to 3.10) use information about the Brownian trajectory of a particle trapped in a static optical tweezers. Active methods (section 3.11) analyze the response of the particle to a known time-dependent perturbation to the optical trap exerted by an external force, e.g., generated by a periodic movement of the sample cell or of the position of the optical tweezers. The force exerted by the optical tweezers can also be directly determined by measuring how the trapping light is scattered using a digital camera or a photodetector (section 3.12). Most of the mathematical details regarding the non-linear fitting are in section 3.13.\nTo illustrate these calibration methods, we have carried out a series of experiments ( Table 1). The experimental data and the MATLAB implementation of the calibration techniques explained in this section are provided as supplementary materials [33]. An overview and comparison of the calibration methods is in Table 2 and the results obtained on the test datasets is in Table 3.\nWe remark that all methods we are going to describe rely on the knowledge of the particle position in units of length (i.e., meters). However, the detection devices used in optical tweezers setups (e.g., CCD cameras, quadrant photodiodes, and position sensitive detectors) provide signals in pixels or volts. Therefore, it is necessary to determine the conversion factor from pixels or volts to meters. For CCD camera, the pixel-to-meter conversion factor can be determined by standard microscopy techniques (e.g., by using a target with know dimensions) [22,32]. For quadrant photodiodes or position sensitive detectors, the volt-to-meter conversion factor can be determine by scanning an optically trapped particle across a (much weaker) detection beam, in which case one need to have at least two beams. It is also possible to perform this calibration self-consistently using only the position signal of an optically trapped particle acquired by a photodetector, when the diffusion coefficient D of the particle is know (e.g., when the particle radius, the medium viscosity and the temperature are known). The basic idea is that the knowledge of D determines the scaling factor of the functions of the particle position used in the calibration process. The details go beyond this Tutorial and can be found in Ref. [22]."},{"title":"Parameter","content":""},{"title":"Value","content":""},{"title":"Particle","content":"Polystyrene particle, radius a = Table 1. Experimental parameters. A microscopic particle immersed in water is trapped with a tightly-focused laser beam in a standard optical tweezers configuration and tracked using a quadrant photodiode at three different laser powers."},{"title":"Time series acquisition and notation","content":"We consider a spherical particle of radius a immersed in a fluid with viscosity η at an absolute temperature T. According to Stokes' law, the friction coefficient is given by:\nwhile the diffusion constant is given by the Einstein-Smoluchowski relation\nwhere k B = 1.3806 · 10 −5 pN µm K −1 is Boltzmann's constant. The trajectory x(t) of an overdamped Brownian particle is described by the overdamped Langevin equation\nwhere F(x) is a deterministic force exerted on the particle (in our case the optical force) and W x (t) is a white noise with zero mean and Dirac-delta-correlated variance, i.e., W x (t)W x (t ) = δ(t −t ). Experimentally, the particle's position is recorded at discrete times, yielding a time series {x } N =1 , where x ≡ x(t ) at times t = ∆t with = 1, . . . , N. To provide estimates with error bars, we can either repeat the experiment M times or divide the time series of a single experiment of size S into M blocks of N = S/M data points each. Either way, we end up with a set of times series {x (m) } N =1 for m = 1, . . . , M. A summary of the notation is in Table 4.\nAdvantages. Works for any potential (not only harmonic potentials); it does not require fast acquisition but long acquisition times (in a harmonic optical trap, T τ ot, x , where τ ot, x is the optical trap characteristic time). Disadvantages. It does not provide the dynamical properties of the optically trapped particle (e.g., γ or D).\nResult. It infers only κ x .\nAdvantages. The simplest and easiest method to implement; it does not require fast acquisition. Disadvantages. It is restricted to harmonic potentials; it requires large datasets taken for long times T τ ot, x ; the inferred values are very sensitive to errors in the position detection."},{"title":"Mean squared displacement (MSD) analysis (section 3.4)","content":"Result. It infers both κ x and γ. Advantages. it tends to provide accurate values. Disadvantages. It only applies to harmonic potentials; it is very slow due to the large number of operations in estimating MSD; it requires fitting a non-linear function; experimental errors are correlated making estimation from least squares more involved."},{"title":"Autocorrelation function (ACF) analysis (section 3.5)","content":"Result. It infers both κ x and γ. Advantages. Algorithmically faster than MSD. The fitting can be transformed into a linear one. Disadvantages. It only applies to harmonic potentials; it requires fitting a function; experimental errors are correlated making estimation from least squares more involved.\nPower spectral density (PSD) analysis (section 3.6)\nResult. It infers both κ x and γ. Advantages. Algorithmically faster than MSD and ACF; it provides accurate results; it can filter unwanted noise. Disadvantages. It only applies to harmonic potentials; one must fit a function, but this can be transformed into a linear problem. Key reference: [106] Force reconstruction via maximum-likelihood-estimator (FORMA) analysis (section 3.8)\n, κ x /γ, D"},{"title":"Result.","content":"It infers both κ x and γ. Advantages. It is computationally fast to implement; it does not require a large dataset; the estimates are given by simple analytical expressions; it can be extended to any dimension and non-conservative forces. Disadvantages. It works for ∆t τ ot, x , although this can be generalized; estimates for the friction coefficient are sensitive to the exposure time. Key reference: [107] Bayesian inference analysis (section 3.9)\nResult. It infers both κ x and γ. Advantages. It is computationally fast to implement; it can handle small datasets and include information a priori; the estimates are given by simple analytical expressions; it gives the whole probability of the estimates; it can be extended to any dimension and non-conservative forces. Disadvantages. it works for ∆t τ ot, x , although this can be generalized; estimates for the friction coefficient are sensitive to the exposure time. Key reference: [108] Table 2. Overview and comparison of calibration methods in one dimension in the passive configuration. In this table we summarize the practical details of each method, together with advantages, disadvantages and a key reference, whenever possible."},{"title":"Potential analysis","content":"The potential method is fairly easy to implement and works for any conservative force with a potential that enables equilibrium of the particle in an accessible time. For a conservative force\nis the equilibrium potential, and the probability density of finding the particle at a given position x at thermal equilibrium follows the Maxwell-Boltzmann distribution: where ρ 0 is a normalization factor ensuring that ∫ ρ(x)dx = 1. The potential U(x) can then be reconstructed as\nThe expression in equation (9) is an equilibrium distribution that theoretically corresponds to the probability of finding a particle at a given position x from an statistical ensemble of independent particles. In practice, this distribution is usually computed from data sets coming from the trajectory of a particle, consisting of data points not necessarily uncorrelated. For example, in a harmonic potential, two points along the trajectory x(t) and x(t ) are uncorrelated if |t − t | τ ot, x , where τ ot, x = γ/κ x is the characteristic relaxation time of the trap. Thus, to ensure that we have enough independent samples T s ∆t τ ot, x (the second condition ensures that the samples are uncorrelated). An important limitation of the potential methods is that, to sample rare events (e.g., the anharmonicity of the potential), one requires a number of measurements N ∼ 1 ρ(x) , and depending on the point x, N can be extraordinarily large. On the other hand, being an equilibrium method, the potential method does not rely on the knowledge of the viscosity and the radius of the particle to obtain an estimate for the trap stiffness."},{"title":"Notation","content":""},{"title":"Explanation","content":"x ≡ x(t ) position of particle at time t = ∆t Table 3. 1. Determine the common maximum and minimum values of the position variable, denoted x max and x min , respectively.\nAs an example, Fig. 10 illustrates the experimental distributions and potentials for a particle optically trapped at three different powers (corresponding to experiments I, II, and III, Table 1). By means of either a linear (black solid lines) or non-linear fit (dashed red lines), we are able to estimate the value of κ (ex)\nx for the experiments I, II and III (using P = 50 bins and the M = 5 experiments to estimate error bars for each bin). These estimates of κ (ex)\nx are given in Table 3."},{"title":"Equipartition method","content":"The equipartition method is a very simple and easy to implement method, which does not even require fitting a function. The equipartition method is a particular case of the potential method where we assume a harmonic potential:\nso that the particle position distribution corresponds to the Gaussian distribution\nThus, the thermal average of the potential is\nwhich is a restatement of the equipartition theorem, imposing that the energy associated with the harmonic potential degree of freedom is equal to 1 2 k B T. This provides a direct method to estimate κ x as\nwhere we can estimate the equilibrium average (x − x eq ) 2 by a time average. As for the potential method, the trajectory needs to be long enough to probe the equilibrium distribution (T s ∆t τ ot, x ) and work with the uncorrelated subsample {x (m) } L =1 .2 For each trajectory m = 1, . . . , M, an estimate of the ensemble average (x − x eq ) 2 is given by its sample variance\nwhere\n(m) corresponds to the sample mean of the m-th experiment. Thus, for each experiment we obtain\nUsing the results of the M experiments, we calculate its mean value and variance, viz.\nUsing this approach on the experiments I, II and III (Table 1), yields the estimates of κ x reported in Table 3. These estimates are consistent with those obtained using the potential method albeit with larger errors. The harmonic potential corresponding to these estimates of the stiffness is plotted with black circles in Fig. 10 and is also in good agreement with the directly measured potential (blue circles).\nSince the estimation of κ x depends on the inverse of the variance of the position, any random error in the position detection will always lead to an underestimation of the stiffness. On the other hand, if the data position is retrieved after a band-pass filter operation, the variance will diminish and hence the stiffness will be overestimated."},{"title":"Mean squared displacement analysis","content":"Differently from the previous two methods, the mean squared displacement (MSD) analysis takes advantage of the time-correlated properties of the motion of the Brownian particle in the harmonic potential. From the Langevin equation (8) with linear restoring force F(x) = −κ x x, the MSD is:\nwhere (· · · ) denotes time average and τ ot, x = γ κ x is the characteristic relaxation time.3 MSD x (τ) comprises two different time regimes: for τ τ ot,x , MSD x (τ) ≈ 2k B T γ |τ|, the particle exhibits 2It is possible also to use a correlated time series as long as the sampling is done at regular time intervals. 3Following Ref. [109], we first see that the formal solution of the Langevin equation (8) is given by\nFrom here one can find, using the properties of white noise of W x (s), that\nThis result can be used to derive the expression for the MSD x (τ) given by equation (21). free diffusion, while for τ τ ot, x , MSD x (τ) ≈ 2k B T κ x , the particle feels the harmonic potential and we recover the equipartition result given by equation (17).\nIf we measure MSD x (τ) experimentally (blue markers in Figs. 11(a-c)), we can estimate the trap stiffness κ x from a nonlinear fit to equation (21). This requires that ∆t < τ ot, x , such that the time series {x (m) n } N n=1 is correlated for each m = 1, . . . , M. Denoting t j = j∆t for j = 1, . . . , N and τ = ∆t, the experimental estimate for the MSD is:\nFrom the estimates {MSD (ex)\nx,m (τ )} M m=1 , we obtain their experimental mean and covariance matrix:\n} and the theoretical model of the MSD in equation (21), we construct χ 2 according to equation (73) and minimize it numerically to find the estimates κ x and τ ot,x (red shaded lines in Figs. 11(a-c)). There are several aspects to consider when proceeding with the non-linear fitting: First, since the experimental data is time-correlated, we need the covariance matrix when inferring the parameters from the non-linear fitting. Second, as the MSD shows two distinct time-regimes, we must be sure to have a good balance of points in both regimes to avoid overweighting one respect to the other. Third, the experimental error of the MSD increases as the lag time grows since the number of data points used to estimate MSD diminishes. Fourth, errors in position detection have a strong impact particularly in the short time regime, because the MSD is proportional to the position squared. It is good practice to exclude the short times from the MSD fit, up to a fraction of the expected τ ot,x , as we did in Fig. 11.\nFor the experimental data recorded in experiments I, II and III (Table 1), we perform the nonlinear fitting on the interval [0.5τ ot,x , 6τ ot,x ] and estimate the errors as explained in section 3.13. This provides us with the estimates for κ (ex)\nx and τ (ex)\not, x , from which γ = κ x τ ot, x and D is given by the Einstein-Smoluchowski relation (7). All these estimates are reported in Table 3."},{"title":"Experimental MSD","content":"Non-linear fitting Fig. 11. Mean square displacement (MSD) analysis. Comparison between the theoretical expression (21) of the MSD and the experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The red dashed lines are the non-linear fitting and the blue markers with error bars are the experimental estimates. The data are fitted in the interval [0.5τ ot,x , 6τ ot,x ] (the upper limit is shown by the dashed vertical lines). The estimates are given in Table 3.\nThe MSD analysis has a few clear advantages over the equilibrium methods described in the previous two sections. First, the MSD uses more information so that the trap stiffness is estimated more precisely. Second, the MSD analysis can infer also dynamic properties, namely the friction coefficient, which is not directly accessible with the equilibrium methods. However, care must be taken when fitting a function with correlated errors and to have a good balance of data points between the short-time and long-time regimes."},{"title":"Autocorrelation function (ACF)","content":"Starting from the Langevin equation (8) with linear restoring force F(x) = −κ x x, the position autocorrelation function (ACF) is [109]\nFrom a correlated time series sampled at regular time steps ∆t, we can estimate the ACF by using\nwhich gives a set of estimates {C (ex)\nx,m (τ )} M m=1 for each experiment m = 1, . . . , M (blue markers with error bars in Figs. 12(a-c)). From these estimates, we obtain their experimental mean and covariance matrix:\nUsing the the autocorrelation equation (25), we can build χ 2 (κ x , τ ot, x ) according to equation (73) and minimize with respect to the parameters κ x and τ ot, x , solving the resulting set of equations by means of non-linear procedures using iterative solutions (red dashed lines in Figs. 12(a-c)). Another possibility is to use the logarithm of the autocorrelation instead, i.e., Experimental ACF Linear fitting Non-linear fitting (25) of the ACF and the experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The black solid lines and red dashed lines are the linear and non-linear fitting, respectively, while the blue markers with error bars are the experimental estimates. The data are fitted on the interval [0, 2.5τ ot,x ] (the upper limit is shown by the dashed vertical lines). The estimates are given in Table 3. ln C x (τ) = ln(k B T/κ x ) − τ/τ ot, x and perform a linear regression, in which case the experimental dataset is {ln C (ex)\nx (τ )} together with their covariance matrix for the M experiments (black solid lines in Figs. 12(a-c)). As in the MSD analysis, from the estimates for κ x and τ ot, x we can determine estimates for γ = κ x τ ot, x and D = k B T/γ.\nAs in the MSD analysis, the ACF analysis requires ∆t τ ot,x and T s τ ot,x . Furthermore, since the ACF function decays exponentially with time, only the values corresponding to short times, around the relaxation time, are relevant to get the information of the confining potential.\nThe estimates for κ x , τ ot, x , γ and D performing the fittings on the interval [0, 2.5τ ot,x ] are provided in Table 1. The ACF analysis results are comparable to those obtained with the MSD analysis, both using non-linear and analytical linear regression. However, the errors for κ (ex)\nx are bigger than those obtained from the MSD analysis, which can be expected from the expression of the ACF (equation (25)), whose initial value is solely determined by the trap stiffness and not the decaying behavior so that we only have one data point to estimate κ (ex)\nx . An important difference is also the computation time: while the ACF is computed with very efficient algorithms based on the Fast Fourier Transform, the MSD requires a rather inefficient numerical computation."},{"title":"Power spectral density (PSD)","content":"Calibration of an optical tweezer by means of the power spectral density (PSD) is considered to be one of the most reliable methods due to its ability to remove common sources of noise with relative ease since one works in frequency space. In contrast to other methods, the PSD analysis is very robust to unwanted periodic noise in the extracted trajectory of the particle, due to, for example, illumination, electronic noise, fan noise of different electronic components inside the laboratory, or even some resonant vibration that is difficult to isolate with optical tables. The method explained in this section is based on Ref. [106].\nStarting from the Langevin equation (8) with F(x) = −κ x x and performing a Fourier transform, one obtains the PSD, which has a Lorentzian form and is the expected value of the average energy in the frequency domain, Fig. 13. Power spectral density analysis. Comparison between the theoretical expression (29) and experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The blue dots are the experimental estimates |x k | 2 T s given by equation (37), the solid red lines correspond to its expected value P (ex) α = |x k | 2 /T s n w over the n w frequencies belonging to the subdomain α, and the black dashed lines are the result of the linear fitting. The vertical dotted lines indicate the maximum frequency used to perform the fitting; in this example, f max ≡ max{ f α } = f Nyq /4 = f s /8. The estimates are given in Table 3.\nwhere x( f ) is\nand f c, x = κ x /(2πγ) = 1/(2πτ ot, x ) is the corner frequency. To derive equation (29), we have used the fact that the spectrum of the zero-mean white noise, | W x ( f )| 2 , has an exponential distribution with an expected value | W x ( f )| 2 = 1. Experimentally, we have a trajectory sampled with frequency f s = 1/∆t up to time T s , i.e., a time series x j = x(t j ) with t j = j∆t and j = 1, . . . , N, which is used to perform the discrete Fourier transform (DFT):x\nIt is expected that |x k | 2 /T s ≈ |x( f k )| 2 when f s f c, x , so it is important to sample the trajectory at frequencies some orders of magnitude higher than the corner frequency. Furthermore, the discrete PSD will suffer from aliasing. A simple formula for the expected value of the aliased PSD can be derived from the discretized Einstein-Ornstein-Uhlenbeck process [106], yielding\nAs long as we sample at frequencies some orders of magnitude higher than the corner frequency, we can mitigate the aliasing by simply cropping the spectrum at a frequency below the Nyquist frequency f Nyq = f s /2. In principle, the expected value in the PSD can be estimated by averaging the experimental values |x k | 2 /T s over a large set of M experiments. In practice, one would need to acquire hundreds of experiments to have a good estimation. Therefore, it is common to apply a compression process to obtain a good estimation of the expected value with only one or a few experiments. Windowing is the most common compression process: it divides the frequency domain in N w narrow and equidistant subdomains with n w values, whose average represents the expected value at the corresponding mean frequency of the subdomain. This gives rise to the reduced experimental dataset for the PSD P (ex) α = |x k | 2 /T s n w with average frequency f α , with α = 0, 1, 2, . . . , N w − 1. With this procedure, the error in the expected value is σ (ex) α = P (ex) α / √ n w . If multiple realizations of the experiment are available, it is also possible to first compress the experiments and then average the results, giving an error σ (ex) α = P (ex) α / √ Mn w . Once the experimental PSD is estimated, the parameters f c, x and D are estimated by least-square fitting. Using directly the model (29) and the data set\naccording to equation (73) and then apply the non-linear fitting procedure obtaining a numerical solution for the estimates. Alternatively, an analytical linear fitting can be performed using the experimental dataset { f 2 α , 1/P (ex) α } N w −1 α=0 with weights given by W α = 1/(P (ex) −1 α / √ n w ) 2 and the linear model 1/P( f ) = 2π 2 f 2 /D + 2π 2 f 2 c, x /D. Finally, an analytical solution can also be found [106] which minimizes\nThe analytical solutions for the estimates using χ 2 defined by equation (32) are\nwhere\nThe errors for these estimates are\nwhere\nx max −x min arctan 2 x max −x min 1+x max x min , and f min and f max correspond to the minimum and maximum on the list of frequencies { f α }.\nThe blue dots in Figs. 13(a-c) show the values |x k | 2 /T s computed by a fast Fourier transform for the trajectories acquired in experiments I, II, and III (Table 1):\nTheir expected values are obtained by compression using N w = 500 (solid red lines in Figs. 13(ac)) and used to fit the Lorentzian function given by equation (29) using equations (33) and (34) (dashed black lines in Figs. 13(a-c)). To avoid aliasing (which gives rise to the plateau at high frequencies), the fitting was performed only with data corresponding to frequencies smaller than f max = f Nyq /4 = f s /8. From this fitting, we obtain the estimates for f c, x and D and, from these, the estimates for κ x , τ ot,x , and γ. Repeating this procedure for M = 5 experiments, we obtain the mean values of the estimates and their errors. All these estimates are reported in Table 3."},{"title":"Drift method","content":"The drift method [110][111][112][113] relies on the fact that the drag force must be equal to the optical force. From the Langevin equation (8), this implies that\nwhere ∆x x 0 is the local drift, which is the average displacement of a particle when located at initial position x 0 and under the external force F x 0 during a time interval ∆t. From an experimental trajectory the local drift can be estimated as the conditional average\nwhere the S corresponds to the number of times the initial position of the particle is close to x 0 . When applying this method there are several caveats. First, the size of the volume element near x 0 must be small enough to probe correctly the spatial variations of the force field. Second, the position of the particle must be measured with a precision much smaller than that of the volume element. Finally, equation (38) is valid when the diffusion coefficient is constant, while the presence of a space-dependent diffusion coefficient (e.g., due to hydrodynamics) leads to the emergence of spurious drift [112,113].\nIn the presence of a spurious drift, a diffusion gradient term need to be added to equation (38) leading to\nwhere α may take values in the interval [0, 1], depending on the specific stochastic process being modelled. In particular, α = 1 corresponds to a colloidal particle in a thermal bath [114][115][116]."},{"title":"Force reconstruction via maximum-likelihood-estimator analysis (FORMA)","content":"The force reconstruction via maximum-likelihood-estimator analysis (FORMA) was introduced recently to retrieve the force field acting on a Brownian particle from the analysis of its displacements [107]. This method is able to estimate accurately the conservative and nonconservative components of the force field with important advantages over the previously discussed techniques, being parameter-free, requiring ten-fold less data and executing orders-ofmagnitude faster [107]. Let us consider a particle in a harmonic potential and suppose that after a time interval ∆t τ ot, x the Brownian particle is at a position x, having started at position x 0 . Repeating this measurement N times, the likelihood of observing this set of measurements is\nwhich depends on the parameters κ x /γ and D. By maximizing the logarithm of the likelihood given by equation (41) with respect to the parameters κ x /γ and D, one obtains the so-called maximum likelihood estimators (MLEs) for a given time series {x (m) } N =1 :\nFrom these equations and the Einstein-Smoluchowski relation (7), it is straightforward to estimate κ (ex)\nx,m , γ, and D (ex) m . The standard errors for these estimates can be obtained from the mean and standard deviations from repeated experiments: using multiple time series m = 1, . . . , M to obtain the estimates {κ (ex) x,m } M m=1 and {D (ex) m } M m=1 yields the experimental mean and variances:\nFORMA provides also an alternative way to get error estimates for the MLEs given by equation (42) for a single experiment, without having to repeat the experiment. Indeed, if we perform a Taylor expansion of the log-likelihood around the MLEs given by equation (42) up to second order, we obtain a Gaussian approximation for L({x n } N n=1 |{x 0n } N n=1 ). This yields the following standard deviations for κ (ex)\nx and D (ex)\nWhen applying FORMA, ∆t τ ot, x and the exposure time should be much smaller than ∆t (i.e., as close as possible to an instantaneous recording of the particle position).\nWe report the estimates for the experimental data in Table 3, obtained using formulas (42) and (43). Comparing these results with the ones obtained by the other methods, we can see some discrepancies in the diffusion coefficient and hence in the friction coefficient. This is due to the fact that the condition that ∆t τ ot, x and that the exposure time be much smaller than ∆t does not hold true as the laser power is increased, causing the relaxation time to decrease.\nSince FORMA only depends on the local variables of the trajectory, that is the positions and their corresponding displacements, it allows to retrieve the forces of arbitrary optical landscapes, including non-conservative forces, and it does not need a trajectory with regular sampling, commonly required in the other methods. This is particularly useful when the problem is to retrieve the forces of extended potentials, where the particle neither reaches the equilibrium nor is able to explore the whole space by itself, such as for a dual optical tweezer or a speckle pattern [107]. The main drawback in using FORMA is the need for high spatial resolution and fast sampling. The latter can be addressed by generalizing the solution for any arbitrary time interval ∆t, as recently introduced in Ref. [117]."},{"title":"Bayesian inference","content":"While FORMA uses a non-Bayesian linear regression to infer the properties of the trap and the Brownian particle [107], other methods have used Bayesian inference to infer optical forces [117][118][119][120][121]. Recently, we have derived a Bayesian approach using conjugate priors, which renders very simple formulas for the posterior distribution, with known moments at all orders and marginal distributions [108]. Here, we provide an overview of this method.\nLet us suppose that we want to infer D and κ x /γ. Let P 0 (κ x /γ, D) be their a priori probability distribution. Then, given a time series, the posterior probability is given by Bayes' rule:\nwith Z being a normalization factor. Choosing the a-priori probability P 0 (κ x /γ, D) to be a so-called conjugate prior for the model's likelihood,4 we obtain the following marginal posterior distribution for the trap stiffness and the diffusion constant\nwhere InvGamm(x) is the inverse Gamma distribution, and the parameters α N , β N , γ N , and K N 4A conjugate prior is a particular type of a priori distribution, chosen so that the posterior distribution belongs to the same family of distributions of such prior. One may intuitively think of conjugate priors as eigenfunctions of equation (45). are determined by the measurement record:\nThe experimental estimates D (ex) and κ (ex)\nx are the mean values and variances of the posterior distributions P D and P κ x /γ :\nFormulas for higher cumulants can also be calculated analytically from equations (46). The hyper-parameters α 0 , β 0 , γ 0 , and K 0 , determine the a priori information about the model's parameters. For example, α 0 , β 0 , γ 0 , and K 0 can be chosen such that for N = 0 (i.e., without experimental data), we obtain their theoretical values and systematic uncertainties (i.e., D (th) ± σ D (th) and κ (th)\nx /γ ± σ κ (th) x /γ ):\nFrom Table 1, D (th) = 0.21 µm 2 s −1 and, adding an error of 10 %, allows us to fix the values α 0 and β 0 . For the theoretical value of the inverse relaxation time, we are more in the dark, as we do not know the trap stiffness before the calibration procedure; thus, we can use an estimate obtained by another method to determine the values of K 0 and γ 0 . Using Bayesian inference on one of the time series {x (m) n }, we obtain the results reported in Table 3. For the estimates regarding the friction and diffusion coefficients, we notice that the present method has the same issue concerning the exposure time as FORMA.\nThe Bayesian inference method extracts all statistical information from the measured data, so that it provides an unbiased estimator of the experimental parameters. Moreover, it can provide reasonable estimates of the parameters also when little data is available, thanks to the use of prior information. The Bayesian inference method can also be generalized to more than one dimension for any time interval ∆t."},{"title":"Measurement of non-conservative force fields","content":"While the previous methods rely on the forces being conservative, non-conservative forces often play also a role in experiments. The drift method can be straightforwardly applied to measure any force field [111]. Also, the ACF method has been adapted to measure non-conservative forces using cross-correlations [122][123][124]. Here, we briefly show how FORMA can be adapted to measure non-conservative forces [107], while the details for the Bayesian method can be found in [108].\nLet us consider the two-dimensional overdamped Langevin equation\nwhere F(r) is the force field and w is a vector of independent white noise. The Taylor expansion of F(r) around r = 0 yields\nwhere F 0 = F(0) and J 0 = J(0) are the force and the Jacobian at the point r = 0, respectively. We further assume r = 0 to be an equilibrium point so that F 0 = 0. Discretizing the Langevin equation (50), we obtain\nwhere σ = 2Dγ 2 ∆t and w n is a vector of independent Gaussian numbers with zero mean and unit variance. Equation (52) can be understood as a multivariate linear regression model: for a given set of pairs of data points {(r n , f n )} N n=1 , we construct the likelihood for (52), similarly as the expression (41) for the one-dimensional case, and maximize it with respect to J 0 to obtain the corresponding MLE J 0 , which is given by the Moore-Penrose pseudo-inverse:\nwhere R and F are N × 2 matrices given by\nFrom the MLE J 0 , we can estimate the force as F = J 0 · r. This result can be naturally split into conservative and non-conservative parts, corresponding to the symmetric and antisymmetric decompositions of J 0 = J c + J r , respectively. The symmetric part can be rewritten as\nwith R a rotation matrix and κ 1 and κ 2 the trap stiffness along the principal axes; θ represent the orientation of these axes with respect to the Cartesian system of reference. The rotational part is in turn given by\nwith Ω being the angular frequency at which the particle would orbit around the center of the trap in the absence of Brownian noise."},{"title":"Active calibration techniques","content":"All calibration methods discussed so far are passive, i.e., they are based on the determination of the trap stiffness from the stochastic motion of the particle in thermal equilibrium with the surrounding fluid within a harmonic potential. Alternatively, the optical tweezers calibration can be performed by actively applying an external force on the particle and measuring its response. A simple active method consists of exposing an optically trapped particle to an external uniform flow in the x-direction with a known velocity v fluid so that the particle experiences a mean drag force F drag = −γv fluid . Consequently, the particle undergoes a mean displacement ∆x = x − x eq from the trap center x eq so that the restoring optical force balances the viscous drag. The stiffness can then be measured as\nIn practice, a uniform flow can be applied by moving the sample at constant speed −v fluid , e.g., by means of a piezoelectric translation stage, while keeping the particle far away from the cell walls to avoid hydrodynamics interactions which change the value of friction coefficient γ with respect to the value given by Stokes' law [125]. Another active method requires moving sinusoidally the position of the sample stage x stage , i.e.,\nwhile keeping the optical tweezers at fixed x eq = 0 [126]. This method not only provides the value of the trap stiffness, but also the drag coefficient and the spatial conversion factor of the measured particle position x (typically in volts or pixels) into actual distances. The Langevin equation for the motion of the particle in the case of the oscillating stage is\nwhere v stage (t) = 2π f stage A cos(2π f stage t) is the velocity of the stage and W x is a white Gaussian noise. The corresponding PSD of the particle position is\nwhere the thermal contribution is equal to that given by equation (29), while the mechanical response is that of the particle to the excitation frequency f = f stage . Because the measurement time in an experiment is finite, t msr < ∞ , the Dirac-delta peak in equation (60) becomes a spike with finite height P stage = P( f stage ) and width ∆ f . If the total measurement time is chosen to be an integer multiple of the driving frequency of the stage, then the spike reduces to a single datum P J at frequency f J = f stage for some integer J, whereas ∆ f = 2δ f , where δ f corresponds to the frequency resolution of the PSD. In this case, the discrete version of equation (60) for the j-th data point is\nwhere δ j.J represents the Kronecker delta. Therefore, disregarding the single point ( f J , P J ), a fit P fit ( f ) to the one-sided Lorentzian function in equation (61) allows to compute the diffusion coefficient D (1) and the corner frequency f c, x (section 3.6), from which the trap stiffness can be determined as\nIf x has not been yet converted into units of length, the values and units of D (1) and κ (1) are not correct. A further step using the active component of the spectrum in equation (61) allows to get the calibration factor α of x into units of length: x → αx, where α can have units of m/V or m/pixel depending on whether the particle position was recorded with a quadrant photodiode, a position sensitive detector or a camera. The power contained in the triangular spike centered at\n, and the theoretical value given in\n, must be equal. Then, α 2 W exp = W th , from which\nHence, the corrected values with the right units of the trap stiffness and the friction coefficient are\nand\nAn alternative to the aforementioned method is to move the position of the center of the trap, while keeping the sample cell at rest, i.e. x stage = 0 and x eq (t) = A sin(2π f eq t). This can be achieved for example with an acousto-optic deflector or a galvomirror. In this case, unlike the case of an external flow, an effective oscillatory force k x eq (t) = k A sin(2π f eq t) is exerted on the trapped particle and the corresponding Langevin equation is\nThe corresponding power spectral of the particle position is given in this case by\nwhile its discretized form is\nThus, once the PSD of the particle position is computed, the calibration of α, κ and γ proceeds similar to the case of the oscillatory flow created by moving the stage (equations (62)-(65)). Active calibration methods can also be applied to less trivial situations, for instance for particles trapped in viscoleastic fluids. In such a case, the instantaneous drag force on the particle at time t is not simply given by γ dx(t) dt as in equations (59) and (66), but it is related to the frequency-dependent rheological properties of the medium, which are a priori unknown. This will be discussed in detail in section 4.3."},{"title":"Direct optical force measurement","content":"All the calibration methods discussed until now provide an indirect measurement of the optical forces, relying on the properties of the motion of the particle in the optical potential, and therefore depend on the assumption of a spherical particle and a harmonic optical potential. It is also possible to measure directly the optical force [127][128][129].\nDirect measurement of optical forces is based on recording the change of momentum between the ingoing and outgoing light, since the exerted optical force is equal to the change in the momentum flux of the trapping light: let I(θ, ϕ) be the intensity scattered into the direction (θ, ϕ) given in spherical coordinates, then the momentum flux density p of the outgoing light is\nThe optical force acting on the particle corresponds to the difference between the total momentum flux of ingoing and outgoing light, where the incoming momentum flux is given by precisely the same formula but with opposite sign. To correctly apply this method, one needs to detect all of the scattered light. Further details of how this can be achieved experimentally can be found in Refs. [127][128][129]. Strickly speaking this is a rather unfeasible task. However, for most practical applications in which the refractive index of the observed particle is similar to that of the solvent, light scatters only slightly, making its experimental collection possible."},{"title":"Mathematical details on error analysis and fitting","content":"Most of the calibration techniques described above use a physical observable Ψ that depends on a single trajectory. Let us denote its experimental estimate for trajectory m as\n, where α may denote either a time dependence or a spatial one. For example, in the potential method α represents the position associated to the histogram bins, while for the MSD α corresponds to the discrete time. We can then use the M experiments to calculate the experimental mean and the experimental covariance of Ψ. These are:\nImportantly, the experimental covariance matrix ∆ (Ψ) α,β in equation (71) has two terms. The first term corresponds to the experiment-to-experiment fluctuations that arise naturally due to the intrinsic randomness of the physical problem as well as due to the correlation between different points α and β. The second term ( 2 η 1 ,η 2 ,... ) captures the variances due to some known error in some variables (η 1 , η 2 , . . .) that are needed to compute Ψ (ex) q (α) as, for instance, uncertainties δα in the value of α, which can be estimated by standard error propagation techniques.5 5Using propagation of errors, the contribution to the total variance by the errors in the uncorrelated variables\nwhere δ η 1 , δ η 2 , . . . are the errors associated to the corresponding variables [130].\nOnce Ψ (ex) (α) and ∆ (Ψ) α,β have been calculated, they are fitted to the corresponding physical observable Ψ(α; θ) to infer the unknown parameters θ = (θ 1 , . . . , θ q ) of the model. There are several approaches to perform this task. The most popular are based on least-square regressions (LSR), which defines the best estimates of the parameters as those that minimize the mean squared error.6 A fantastic book dealing with these matters is that by Peter Young [130].\nThe general least-square method used in this Tutorial relies on the weighted χ 2 function\nwhere R αβ is a symmetric semi-positive definite matrix that captures the correlation in the experimental dataset. The goal is to find the set of parameters θ ( ) that minimizes χ 2 (θ).\nWhen R is taken to be [∆ (Ψ) ] −1 , this is the so-called generalized least-square method or the correlated χ-squared method. The standard weighted least-square method is recovered when the off-diagonal terms of the experimental covariance are assumed to be zero so that\nPP . If we do not have access to experimental errors, we can perform the standard least-square method by assuming the matrix ∆ (Ψ) to be equal to the identity matrix. In some cases, the inverse of the sample covariance matrix is an ill-posed problem and it is highly recommended to use the standard method of weighted least squares but including the experimental covariance matrix in the error estimation of the parameters [132]. This is the approach we have used to estimate errors for the MSD and ACF methods.\nIf the expression χ 2 (θ) is a simple function of the parameters, the minimization can be carried out analytically. For example, this is the case when Ψ(α; θ) is linear with respect to the model's parameters, that is\nwhere {Φ j (α)} is a set of functions, giving rise to the well established linear least-square fitting.\nIn some cases, a simple redefinition of variables can transform a non-linear fitting into a linear one, as, for instance, in the potential method. When it is not possible to obtain explicit forms for θ ( ) , one must rely on numerical minimizers, being the standard ones the Levenberg-Marquardt-Girard-Wynne-Morrison and the Trust-Region, implemented already in most of the standard computational packages for mathematical analysis, such as Python, MATLAB, Mathematica, and R. However, the computational methods given in these packages do not normally allow to use the experimental covariance matrix to estimate the covariance matrix of the estimated parameters. In this case, it is recommended to use the packages provided in [132], which we also have implemented in this Tutorial."},{"title":"Applications","content":"We will now describe how optical tweezers have been used in several advanced applications. For each field of application, we provide a brief introduction and a detailed description of some paradigmatic experiments. In this section, we focus on experiments with particles optically trapped in a liquid environment, while we discuss experiments with particles optically trapped in air or vacuum in section 5."},{"title":"Single-molecule mechanics","content":"The study of single-molecule mechanics using force spectroscopy techniques has rapidly expanded over the past three decades [133][134][135][136][137][138][139][140]. Many labs worldwide use single-molecule methods 6Less common, but very powerful methods, are the robust least-squares, including the least absolute regression (LAR) and the Bisquare weights [131], both very useful to minimize the effect of outliers.\nto manipulate and track individual molecules. Broadly speaking, single-molecule methods can be classified into two large families depending on whether they manipulate molecules by exerting forces or they visualize molecules in action using appropriate markers. The latter include single-molecule fluorescence used to monitor the time evolution of fluorescently labeled molecular complexes in vitro and in vivo. Manipulation methods are capable of exerting forces from femtonewton to nanonewton. For example, they have been used to study the elastic properties of biopolymers [141], the thermodynamics and kinetics of intramolecular (e.g., molecular folding) [142,143] and intermolecular (e.g., ligand binding) interactions [144], molecular motors [145], and fundamental problems in nonequilibrium physics [146,147]. The field of force spectroscopy includes techniques such as laser optical tweezers [22], magnetic tweezers [148], atomic force microscopy [149], acoustic force spectroscopy [150], electrostatic trapping [151], microfluidics [152], microneedles [153], biomembrane force probe [154], and centrifugal force [155]. These techniques tend to compromise between measurement accuracy and high-throughput capabilities. While magnetic tweezers and acoustic force spectroscopy achieve high throughput, optical tweezers and atomic force microscopy remain the most accurate techniques in the low force (piconewton) and high force (nanonewton) regimes, respectively. Piconewton forces are the most relevant at the molecular level, making optical tweezers an excellent technique to accurately characterize molecular reactions.\nIn this section, we describe in detail how optical tweezers can be used to investigate molecular folding using short DNA hairpins (a double-helix stem of a few tens of base pairs terminated by a loop) as model systems. We will describe how to synthesize a short DNA hairpin, how to unzip it by stretching it while holding it between a micropipette and an optical tweezers, and how to analyze the obtained data. This experiment provides an ideal example of the use of single-molecule mechanics to determine elastic behaviour of biopolymers, to measure the intramolecular weak interactions that stabilize native molecular structures, and to extract the free energy of biomolecule formation from out-of-equilibrium pulling experiments. The experimental setup we use is based on the miniTweezers setup developed by Smith, Cui and Bustamante [127]."},{"title":"Microfluidics chamber","content":"The schematic of the experimental setup is presented in Fig. 14(a). The experiments are performed in a microfluidics chamber placed vertically within the setup. As schematically shown in Figs. 14(b) and 14(c), this microfluidics chamber has three channels: we will refer to the three channels as the upper, central and lower channel. The propagation of both laser beams is perpendicular to the chamber surface (z-axis). The experimental area is restricted to the central channel, where the object of study (biomolecule or cell) is held by two beads, one held by the optical trap and the other held by a glass micropipette, as shown in Fig. 14(d). The upper and lower channels are used to supply the two types of coated beads used in the experiments.\nThe microfluidics chamber is realized by sandwiching two layers of parafilm (Parafilm M, Bemis) between two coverslips (No. 2, dimensions 24 mm × 60 mm × 200 µm), as shown in Fig. 14(c). In detail, the steps to prepare the chamber are:\n1. The entrance and exit holes are drilled in one of the glass coverslips using a laser cutter.\nThe coverslips are cleaned with a solution of 70% ethanol.7 tube (King precision glass, Inc., inside diameter=0.04 mm, outside diameter=0.08 mm, length=6.00 mm, glass type KG-33), as described in section 4.1.3."},{"title":"5.","content":"This glass micropipette is placed on top of the parafilm layer perpendicular to the channels with the tip positioned in the central channel, as shown in Figs. 14(b) and 14(c).\n6. The upper and lower channels are connected to the central one via glass dispenser tubes (King precision glass, Inc., inside diameter=0.04 mm, outside diameter=0.10 mm, length=6.00 mm, glass type KG-33) cut using a diamond-tip cutter to obtain a clean cut.\n7. The second parafilm layer is placed on top and the chamber is closed using the second glass coverslip (previously cleaned with 70% ethanol), as shown in Fig. 14(c).\n8. The chamber is sealed by heating it on a hot plate at 120 • C while exerting a pressure of about 1 Kg in all the chambers' surface (either by placing a weight on top of the microfluidics chamber or, more simply, by exerting pressure by hand). To prevent the glass from breaking, it is recommended to sandwich the chamber between two thicker glass slides to homogenize the pressure applied to their surfaces.\n9. The chamber is placed in the metallic mount, as shown in the inset in Fig. 14(a). The correct alignment between the holes drilled in the glass coverslip and the holes in the mount is critical to ensure the flow in the chamber. It is also very important to tighten the screws to avoid buffer losses between the plastic tubes and the glass chamber, since this would cause flows inside the chamber. Do this carefully, because tightening the screws too much could break the glass coverslip.\n10. The side of the micropipette coming out of the chamber is cut to get rid of the excess tube, leaving only about ∼ 3 cm, and is introduced into a polyethylene tube (polyethylene tubing, Warner Instruments, PE-10/100; outside diameter = 0.61 mm, inside diameter = 0.28 mm). This tube is then fixed to the mount with tape. The pipette is easily breakable, so the tube needs to be placed as straight as possible. The connection is then sealed by using a special glue (Norland, NOA-61; UV Curing Optical Adhesives). The glue is placed between the the glass tube of the pipette and the polyethylene tube to fill the void outer space between tube and pipette via capillarity. After observing that the glue has entered into the tube, the UV-curation is performed by leaving the whole chamber for ∼ 25 min under a UV lamp.\nSince the glue could reach the end of the glass tube of the micropipette and block it, the chamber has to be placed under the UV-radiation as fast as possible. After the curation, a syringe (1 ml Luer Lock, HSW SOFT-JECT U100 Insulin Henke Sass Wolf) is inserted using a needle (BD Microlance 30G × 1/2\" -0.30 mm × 13 mm) at the end of the tube to create suction at the tip of the micropipette.\n11. The holes of the mount are connected with silicone-rubber tubes (Tygon 3350, Saint-Gobain, .031 ID X .093 OD X 50 FT TYGON 335) via nylon socket screws (Nylon set 8 − 32 × 3/8\", Product-Components, previously drilling a hole with a number 45 drill bit, of 0.82\"), as shown in Fig. 14(c). A segment of ∼ 20 cm polyethylene tube (Polyethylene tubing, Warner Instruments, PE-50/100: outside diameter = 0.97 mm, inside diameter = 0.58 mm) is inserted into the silicone-rubber tubes. The three exit tubes are connected to a trash (any small plastic container with a capacity volume ∼ 100 ml), while the three entry channels are connected to a syringe using a polyethylene tube (Polyethylene tubing, Warner Instruments, PE-50/100: outside diameter = 0.97 mm, inside diameter = 0.58 mm) which is inserted into the silicone-rubber tubes on one end, and a needle (HSW FINE-JECT 23G × 1\" -0.6 mm × 25 mm)."},{"title":"The miniTweezers setup","content":"The miniTweezers [127] (Fig. 14(a)) consists of two focused counter-propagating laser beams (P = 200 mW, λ = 845 nm) that create a single optical trap [156]. It employs high-NA objectives (NA = 1.2), but underfills them to be able to collect almost all scattered light to measure the change of light momentum using position sensitive detectors (PSDs), as described in section 3.12.\nBefore the objectives, a pellicle diverts ∼8% of each laser beam to a secondary PSD to determine the optical trap position. The remaining ∼ 92% is collimated by using a lens and it is introduced into the optical axis by using a polarizing beam-splitter that selects the horizontally polarized light. Then the linearly polarized light of each laser is circularly polarized by a quarter waveplate8. The laser beams are focalized by their corresponding objectives and form the optical trap, which can hold dielectric objects with a refraction index higher than the surrounding aqueous medium (e.g., polystyrene beads) and exert forces to them. The exiting light is collected by the opposite objective and is converted to vertically polarized light by another quarter-wave plate. The vertically polarized light can be extracted from the optical path using two polarizing beam-splitters and relay lenses that redirect the light to the PSDs that measure the intensity of the beam (i.e., its exerted force). The miniTweezers has a resolution of 0.1 pN and 1 nm at a 1 kHz acquisition rate. The chamber where the experiments are performed (section 4.1.1) is placed between the two objectives and held with a metallic mount with three stages to permit movement along the x-, yand z-directions. There are two possible ways to manipulate the position of the optical trap with respect to the chamber depending on the precision that is needed. For large displacements (up to hundreds of micrometers), the whole chamber is displaced along the x-, y-and z-directions using stepmotors. For fine displacements (less than a few micrometers, but with a resolution of ∼ 1 nm and rates of displacement up to ∼ 50 nm s −1 ), the optical trap is displaced along the xand y-directions by a 2D piezoelectric motor attached to the tip of the optical fiber (wiggler) used to inject the laser into the optical setup.\nBefore any experiment, the setup needs to be aligned going through the following steps:\n1. Remove any trapped object.\n2. Move the chamber with the stepmotors so that the micropipette is a few micrometers away from the optical trap, in the so-called \"working zone\".\n3. Remove any voltage applied to the piezoelectric motor.\n4. Turn off the LED and remove the light filter, to be able to see the optical trap on the camera."},{"title":"5.","content":"Decrease the laser power until it is possible to distinguish both lasers.\n6. Using the screws of the kinematic mount, move the B laser (the one that has the real image shown on the screen) until it is superposed on the A laser.\n7. Put back on the light filter, turn on the LED, and increase the laser power to its working value.\n8. Set the current values recorded by the PSD as the zero force baseline along the xand y-directions (center of the light spot) and along the z-direction (size of light spot).\n9. Trap a bead with the optical trap by moving the chamber with the stepmotors close to the appropriate dispenser tube. 10. Bring the trapped bead to the working zone.\n11. Fine-tune the alignment of the B laser using the screws of the kinematic mount controlling the wiggler of its fiber so that the xy-force signals from both force-PSDs are as close as zero as possible, i.e., both lasers exert an xy-force as close to zero as possible.\n12. Fine-tune the position of one of the objectives along the z-directions so that the z-force from both force-PSDs is as close to zero as possible.\n8The use of quarter-wave plates ensures that the light coming from the two laser beams do not interact with each other and guarantees that the light reflected from the particle is not returned to the laser but is reflected to the opposite PSD [157]. 13. For both lasers, move the kinematic mount of the mirror that deflects the light that gets towards the position-PSD (between the aspherical lenses and the PSD) until its signal is zeroed (i.e., the light hits the center of the position-PSD)."},{"title":"Pipette making","content":"A micropipette is used to hold a microparticle by air suction, as shown in Fig. 14(b). The tip of this micropipette needs to have an inside diameter ∼ 1 µm, large enough to exert sufficient suction and small enough not to let the microparticles flow inside it. While one might use a commercial pipette puller to produce the pipettes, here, we will explain how a glass tube can be pulled to produce such micropipette using the homemade device shown in Fig. 15 [127,158]. This homemade pipette puller consists of a plastic platform with two parallel metallic bars, which hold a stage and along which a plastic weight can slide up and down. The two ends of the glass tube are fixed with screws to the stage and the weight, respectively. Thus, the glass tube is held in tension due to the pull exerted by gravity on the weight. The central part of the glass tube passes through a platinum wire; the two ends of the wire are connected to an electric supply, which provides a current ramp from 0 to ∼ 6 A in ∼ 8 s. The exact maximum intensity and time of the ramp have to be tuned to get the appropriate diameter and shape of the pipette (in general, longer times and higher intensities correspond to smaller tips). Specifically, the steps required to produce a micropipette are:\n1. Unplug the wire from the electric supply and place the pipette puller horizontally.\n2. Insert the glass tube through the stage, the platinum coil, and the plastic weight. Fig. 15. Pipette puller. First, the glass tube is carefully centered within a coiled platinum wire, as shown in the inset. Then, one end of the tube is attached to the puller, while the other is attached to a weight that pulls the tube down. Finally, an electric current intensity ramp is applied through the platinum wire, heating the adjacent glass tube, which during the melting is pulled down, creating a micropipette. 3. Carefully center the glass tube within the platinum wire. 4. Screw the glass tube at the stage and plastic weight. 5. Place the platform vertically (as shown in Fig. 15), carefully to avoid any sudden hit that may break the glass tube.\n6. Apply the current ramp. The weight will drop into the platform. The geometry of the platinum wire is of a critical importance to obtain the proper shape and size of the micropipette. For that reason, when the weight drops into the platform, we have to ensure that the remaining glass tube of the stage is not touching the platinum wire, otherwise it would get stuck to it.\n7. Unscrew the plastic weight and (with extreme caution and preferably with ethanol-cleaned tweezers) take the micropipette and place it into the chamber being built.9"},{"title":"Preparation of the beads","content":"Biomolecules such as DNA, RNA and proteins are chains whose units are either nucleic bases or aminoacids [159]. These molecules can be conveniently manipulated through beads attached at their ends. Antigen-antibody connections can be used to attach a specific bead at each end, taking advantage of the fact that these connections are extremely specific [160]. In particular, avidin/streptavidin-biotin10 and digoxigenin-anti-digoxigenin connections are often employed in single-molecule force-spectroscopy experiments, because they can hold forces up to ∼ 100 pN at typical optical-tweezers loading rates of ∼ 1 pN s −1 [162]. Each bead is coated with a different molecule, which specifically binds to the cognate tails. Streptavidin-coated (SA) beads can be directly purchased (SPHERO streptavidin -polystyrene particles, 0.5% w/y, 2.17 µm, 5 ml). Anti-digoxigenin-coated (AD) beads are purchased as G-protein-coated polystyrene beads (Kisker Biotechnologies -G-coated polystyrene particles, 0.5% w/y, 3.18 µm, 5 ml), which must then be activated with anti-digoxigenin. The difference in size between these beads permits one to easily distinguish them by microscopy.\nBuffers. The following buffers are required for the preparation of the beads: PBS (pH 7.0) NaCl 0.14 M, KCl 2.7 mM, K 2 HPO 4 · 3H 2 O 61 mM, KH 2 PO 4 39 mM, NaN 3 (sodium azide) 0.02%. To prepare 50 ml: fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water; add 0.406 g of NaCl, 0.01 g of KCl, 0.696 g of K 2 HPO 4 · 3H 2 O, 0.265 g of KH 2 PO 4 , and 1 g of NaN 3 ; dissolve using a magnetic mixer; add Milli-Q water until reaching 50 ml; check the pH and add NaOH until the solution reaches a pH 7.0.\nPBS (pH 7.4) NaCl 0.14 M, KCl 2.7 mM, K 2 HPO 4 · 3H 2 O 80.2 mM, KH 2 PO 4 20 mM, NaN 3 (Sodium azide) 0.02%. For preparing 50 ml: follow the same procedure as for the previous buffer, adjusting to pH 7.4.\nAntibody crosslinker buffer (pH 7.4) Na 2 HPO 4 100 mM, NaCl 100 mM. To prepare 10 ml: fill ∼ 7 ml of a 10 ml Falcon tube with Milli-Q water; add 0.142 g of Na 2 HPO 4 and 0.058 g of NaCl; shake well until the salts have dissolved; add Milli-Q water until the 10 ml of total volume is reached; check the pH and add NaOH until the solution reaches pH 7.4.\n9The micropipette easily breaks or gets blocked by dust particles at the minimum contact of the tip of the glass tube with anything, so it needs to be handled with extreme care.\n10The interaction between streptavidin and biotin is one of the strongest non-covalent interactions in Nature [161].\nAll salts can be acquired from any chemical distributor (e.g., Sigma Aldrich). All products have to be of a biomolecular grade of purity. The water for preparing all buffers has to be Milli-Q water.11 For the RNA experiments, use RNAse-free water for all preparations (and make sure the pHmeter is also cleaned with RNAse-free water)."},{"title":"SA beads.","content":"For the already coated SA beads, the protocol consists in exchanging the buffer where they are preserved with PBS (pH 7.4). The procedure (∼ 30 min) for the preparation of 1 ml of SA beads (which should be sufficient to perform experiments for several months) is:\n1. Homogeneously resuspend the SA beads. To do so, place the container with the purchased particles into a vortex mixer for several seconds. An additional step of several seconds of sonication further improves the resuspension.\n2. Take 1 ml of the resuspended SA beads and place it in a new Eppendorf tube.\n3. Centrifuge the Eppendorf tube at 10000 rpm for 5 min. The dense SA beads will precipitate, while the buffer will float above them.\n4. Extract the overnatant buffer and add 1 ml PBS (pH 7.4). Mix the SA beads and buffer. Sonicate a few seconds and centrifuge at 10000 rpm for 5 min, again. Extract the overnatant, resuspend in 1 ml PBS (pH 7.4) and sonicate during several seconds. This step is performed to exchange the buffer.\n5. Aliquote in 20 tubes (50 µl each). This allows for an optimal sterile preservation of the beads.\n6. Store at 4 • C for up to ∼ 6 months. 3. Homogeneously resuspend the G-coated beads. To do so, place the container with the purchased particles into a vortex mixer for several seconds. An additional step of several seconds of sonication improves the resuspension.\n4. Centrifuge at 5000 rpm for 2 min. The dense beads will precipitate, while the buffer float above them.\n5. Extract the overnatant buffer and add 1 ml antibody crosslinker buffer (pH 7.4). Mix the beads and the buffer, sonicate for a few seconds, and centrifuge at 5000 rpm for 2 min.\n6. Repeat the previous step (washing) and resuspend within 1 ml antibody crosslinker buffer (pH 7.4).\n7. Add 60 µl of anti-DIG antibody and 30 µl of freshly dissolved DMP crosslinker buffer.12\n8. Tumble at room temperature for 60 min.\n9. Centrifuge at 5000 rpm for 2 min.\n10. Wash twice with 1 ml PBS (pH 7.0), resuspend with the same buffer, and sonicate during several seconds.\n11. Aliquote in 20 tubes (50 µl each).\n12. Store at 4 • C for up to ∼ 6 months. For the SA beads, no incubation is required, dilute 1 µl of SA beads in 1 ml of the molecular buffer in an Eppendorf tube.\n12The volume of added anti-DIG antibody will depend on the coating of the beads: it may need to be higher if the vendor supplies the beads with a higher density of coating. Nevertheless, the proportions of anti-DIG and DMP crosslinker buffer have to be preserved (2:1).\n13TRIS is used in the formulation of buffer solutions in the pH range between 7.5 and 8.5.\n14EDTA is widely used for scavenging metallic ions, including divalent ones, which most enzymes need to be active. For this reason, it is widely used as a food preservative or stabilizer. In our case, it inactivates DNAses and RNAses, preventing nucleic acid degradation.\n15Typically, the molecule is concentrated and needs to be diluted 1:10 to 1:100."},{"title":"Synthesis of a short DNA hairpin","content":"As shown in Fig. 14(d), a typical single-molecule experiment is performed using a short DNA hairpin of a few tens of basepairs (bp). A DNA hairpin is a single-stranded DNA molecule that forms a double helix stem ended by a loop. The hairpin is flanked by two double stranded helices that act as handles. Handles are molecular spacers used to manipulate the DNA hairpin. These spacers also avoid non-specific interactions between the DNA hairpin and the beads when performing the experiments. The handles are tagged with biotin and digoxigenin at one end to specifically attach them to the coated beads. The 29-bp handles are chosen because they provide higher rigidity than longer handles and higher signal-to-noise ratio measurements [163].\nTo synthesize short DNA hairpins with 29-bp dsDNA handles, the desired DNA sequence (that we will denote as CD4) is purchased in a series of oligonucleotides that self-assemble into the hairpin and handles [163]. Briefly, a 29-bp sequence is used to make the handles, with the hairpin sequence flanked by the handles at both sides (Table 5). For very short hairpins (≤ 13 bp), the hairpin sequence can be ordered as a single oligonucleotide; however, for longer hairpins, it is useful to split the molecular construct into two oligonucleotides that are annealed and ligated together. Finally, the dsDNA handles are created by annealing a third oligonucleotide that is complementary to the handle region (\"splint\"). The sequences of the oligonucleotides used for the CD4 DNA hairpin used in this section are shown in Table 5. One oligonucleotide ( Table 5, CD4-handleA) is purchased with a biotin at its beginning (5'-biotinylated), whereas the oligonucleotide containing the opposite handle (Table 5, CD4-handleB) is purchased 5'-phosphorylated and then tailed at its 3'-end with multiple digoxigenins. All three oligonucleotides can be purchased from companies such as Eurofins, Fisher Scientific or Sigma Aldrich. Table 5. Oligonucleotides. Oligonucleotides used to synthesize the hairpin by a tailing, annealing and ligation reaction. The splint is used to the create the 29-bp double-stranded DNA handles. The hairpin loop is highlighted in bold and the sequences corresponding to the handles are highlighted in blue.\nTo complete the DNA hairpin synthesis, three main reactions have to be performed:\n1. Tailing reaction: As we have already noted, one oligonucleotide is already purchased with the biotin at its 5' end. However, to be able to stretch the hairpin, digoxigenins need to be added to the other end of the hairpin by performing the tailing reaction described in Table 6.\nAfter the tailing steps, the oligos are purified using the QIAquick Nucleotide Removal Kit (Qiagen) and eluted in 50 µl Tris-Cl 10 mM, giving rise to a final concentration, assuming a 100% efficiency, of ∼ 2 µM."},{"title":"Annealing reaction:","content":"The final construct is then assembled in an equimolar reaction. Since the sequences are complementary, at equal concentration for all the oligonucleotides, the equilibrium structure is the formed hairpin with dsDNA handles (Fig. 14(c)). The annealing is performed in two steps : (a) CD4-handleA + splint and CD4-handleB + splint; (b) mix of the two previous reactions. The annealing protocol is described in Table 7. Table 7. Annealing reaction. Mix the items for each reaction in a separate Eppendorf tube. Keep both reactions at 95 • C for 1 min, 80 • C for 10 min, decrease by 0.5 • C every 10 min down to 40 • C. Hold the temperature, mix both reactions and decrease by 0.5 • C every 20 min down to 10 • C. The decrease in temperature can also be achieved by letting the tubes inside a thermal bath cooling down with the heater off; however, the usage of a thermocycler is recommended, since it allows for a better temperature control.\n3. Ligation: The molecular assembly is almost completed. All the Watson-Crick bonds of the structure have been formed. However, the backbone of the two oligos of the stem of the hairpin are still not covalently bonded. The final step is to connect the phosphorylated 5' end of CD4-handleB oligo with the C3' end of CD4-handleB. To do so, the assembly is then ligated in an overnight reaction using T4 DNA ligase (New England Biolabs) at 16 • C. The ligase is heat inactivated at 65 • C for 10 min, as shown in Table 8."},{"title":"Pulling experiments","content":"Paradigmatic manipulation experiments one can perform with a DNA hairpin are pulling experiments. The DNA hairpin is tethered between the two beads and the force it experiences is cyclically varied between a minimum and a maximum force value. Starting at a low force (typically 0 to 5 pN) with the hairpin folded, the trap is moved away from the pipette at a constant pulling speed and the exerted force is steadily increased. At a given point, the hairpin will reach a force high enough to unravel the double-stranded DNA helix structure reaching the unfolded state. This mechanically induced denaturation process is called unzipping. The stretching of the unfolded hairpin keeps on until a maximum force is reached. Afterwards, the same reverse protocol is applied: the trap is moved backwards at the same pulling speed and the applied force decreased until reaching the minimum force value. In this reverse process (called rezipping) the DNA will refold to the hairpin native state when the applied force is low enough. Once the applied force reaches the initial minimum value a new pulling cycle starts again. The CD4 DNA hairpin is a good molecular marker that can be used for force calibration. It reversibly unfolds and folds at 14.7 ± 0.3 pN at standard conditions (T = 298 K, 1 M NaCl). The detailed procedure to perform the pulling experiment is:\n1. The three channels are filled with the molecular buffer described in section 4.1.4 through the syringes connected to the channels. The buffers have to be kept in the fridge and filtered using a 10 ml syringe and a filter (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International) to prevent microorganism growth.\n2. The chamber (already held in the metallic mount as described in section 4.1.1) is placed between the objectives.\n3. The buffer containing SA beads is flowed in the upper channel and AD beads in the lower channel (the channels can be interchanged).\n4. The microfluidics chamber is moved using the stepper motors to bring the optical trap close to the dispenser tube of the upper supplier channel to capture a SA bead.\n5. After a SA bead is captured by the optical trap, the microfluidics chamber is moved back to position the trapped SA bead on the tip of the micropipette. The syringe is then pulled and the SA bead is held by air suction at the tip of the pipette.\n6. The microfluidics chamber is then moved close to the dispenser tube of the lower channel supplying AD beads.\n7. Once an AD bead is captured (paying special attention not to trap more than a single bead), the AD bead is brought close to the SA bead, i.e., the chamber is moved so that the optical trap is in the working zone.\n8. The tether is formed by poking the SA bead held by the micropipette with the optically trapped AD bead, until a non-zero force is recorded upon retracting the AD from the SA bead. 9. Check that only one molecule has been attached to the bead. To do so, the easiest way is to move the optical trap away from the bead in the pipette and verify that a sudden decrease of the force by ∼ 1.5 pN occurs at ∼ 15pN (red lines in Fig. 16(a)), which indicates the unfolding of the DNA molecule. Then, by moving inwards the optical trap, a similar increase of the force should occur at ∼ 14pN (blue lines in Fig. 16(a)), which indicates that the hairpin is folding back. This equilibrium unzipping-rezipping force has been found to be 14.7 ± 0.3pN at standard conditions (see above).\n10. Now, the pulling experiment can start by cyclically varying the distance L between the pipette-held particle and the center of the trap (as shown in Fig. 14(d)). In this way, repeated unfolding and refolding pulling cycles (∼ 100) are performed over the same molecule while recording the force and the trap position. Some representative data are shown in Fig. 16(a), with a typical pulling speed of 100 nm s −1 . To extract the free energy of formation of the hairpin, as we will see in the next section, it is key that the hairpin is folded at L = L 0 and unfolded at L = L 1 ."},{"title":"Extracting the free-energy of hairpin formation","content":"One of the main applications of single molecule experiments is the possibility to extract free energy differences from irreversible work measurements using fluctuation theorems [164]. From the force-distance curves ( Fig. 16(a)) the work W exerted on the molecule between the starting trap position, L 0 , and the final one, L 1 can be directly measured as the area below the curve:\nFor quasi-static processes (i.e., sufficiently slow pulls) the work equals the free energy difference ∆G FU between the initial L 0 (folded) and final L 1 (unfolded) trap positions. However, this is not true for irreversible pulls which exhibit hysteresis between the stretching (forward) and releasing (reverse) processes. The difference in the area between consecutive unfolding and folding cycles is the work dissipated by the system, which is positive in average according to the second law of thermodynamics. Noteworthy, since we are not in the thermodynamic limit (as we are dealing with small systems), there may be trajectories that violate the second law of thermodynamics. Fluctuation theorems state that the work distributions measured along the forward and reverse processes fulfil a symmetry relation [164]. One of the main consequences of such a relation is that forward and reverse work distributions, despite being different due to the hysteresis, cross each other at the value of ∆G FU , independently of how irreversible the pulling process is. To extract the work distributions we use equation (75) to compute the work for all the forward (stretching) and reverse (releasing) cycles. The probability distributions of the work for the forward and reverse processes are shown in Fig. 16(b) by the red and blue histograms, respectively. The crossing point of the forward and reverse work probability distributions corresponds to the difference in free energy ∆G FU between the folded (L 0 ) and unfolded (L 1 ) states. The work histogram in Fig. 16 The free energy difference between the folded and unfolded states can be written as:\nwhere the term ∆G 0 FU is the free energy of formation of the molecule, which is the quantity we are interested in obtaining, and the other contributions are the reversible work differences related to the different parts of the experimental setup (ssDNA, handles, bead, hairpin dipole, as shown in Fig. 14(d)). These contributions can be computed as follows:\nis the reversible work needed to stretch the unfolded molecule, which is modeled with an inextensible Worm-like chain [165]:\nwhere f ssDNA is the force, x is the molecular extension, n is the number of monomers of the molecule (n = 44 bases for our case), l 0 = 0.59 nm is the contour length per monomer, and p = 1.35 nm is the persistence length (parameters from Ref. [166]). The stretching contribution of this released ssDNA is then computed as\nThe value we obtain is ∆W ssDNA\nis the reversible work needed to stretch the folded molecule, which is modelled as a single dipole of length d = 2 nm [166], corresponding to the double-helix diameter [167]:\nand\nwhere\n3. Finally, the last two contributions to equation (76) can be computed by considering the slope of the force-distance curves just before the rip, where the distance corresponds to the optical trap position (L), as shown in Fig. 16(a). The slope κ eff can be easily obtained by linearly fitting the data points right before the unfolding event. This yields the effective stiffness of two serially connected springs: the trapped bead of stiffness κ B , and the handles stiffness, κ handles :\nAssuming that the hairpin orientation has a much higher stiffness than the handles and the optical trap, the combined work can be written as [166]:\nwhere we have considered that the stiffness of the folded molecule is very large and consequently does not contribute to κ eff . The value we obtain is ∆W handles\nFrom equations (78), (80) and (81), and from the crossing point of the forward and reverse work probability distributions (∆G FU ), we obtain all the terms needed to calculate the free energy of formation of the hairpin, ∆G 0 FU , from equation (76). The free energy of formation at zero force we obtain for this molecule is ∆G 0 FU = 51.9 k B T, which is in agreement with that of the sequence of the CD4 hairpin, ∆G 0 FU = 50.6 k B T, using the values provided in Ref. [168]. Typical errors of ∆G 0 FU are on the order of several k B T, corresponding, in this case, to a 5% to 10% relative error."},{"title":"Other applications to single-molecule mechanics and outlook","content":"\"Take a single DNA molecule and pull from its extremities while recording the force-extension curve until it gets fully straightened.\" This thought experiment, which was just a dream a few decades ago, has now become standard in many research labs worldwide. Since the seminal works on single-molecule manipulation almost 30 years ago [7][8][9], the number of applications of single-molecule mechanics has kept on growing. Applications might be classified in two large sets: those aimed to address specific biological questions (e.g., the mechanics of maintenance and regulatory enzymes in the genome, the energetics and kinetics of DNA hybridization, RNA and protein folding, molecular footprinting of proteins and ligands binding DNA), and those aimed to enlighten fundamental physico-chemical questions (e.g., polymer physics, chemical reactions, nonequilibrium physics) using biomolecular matter as model systems. The reader can have a look at published reviews specifically addressing these fields to discover that the future of the single-molecule field has a long way ahead to run [133][134][135][136][137][138][139][140]. Most optical-tweezers and single-molecule technology is still instrumental craftwork not widely available in research institutes and labs worldwide, yet it is becoming steadily more popular among biologists aiming to enter the field. Future developments in the field will be the parallelization of measurements in optical tweezers making it high-throughput either using holographic techniques [43] (recently extended also for acoustic tweezing [169]), combining single-molecule fluorescence with forcemeasurement techniques [170][171][172], controlled force resistive pulse sensing [173,174], and plasmonic trapping [13]. Finally, a major step would be to apply force spectroscopy techniques to ex-vivo and in-vivo conditions."},{"title":"Single-cell mechanics","content":"Biological microscopic objects such as organelles and living cells have been optically manipulated since the late 1980s [175,176]. In Ref. [177], the swimming motion of optically trapped bacteria was recorded while the trapping forces were measured providing the first example of single-cell optical manipulation. Since then, biomechanics at the cellular scale has kept on growing in terms of techniques and objects of study [178,179]. Techniques such as atomic force microscopy [180,181], optical tweezers [182][183][184], particle tracking microscopy [185,186], traction microscopy [187][188][189], magnetic twisting cytometry [190], acoustic force spectroscopy [191], and micropipette aspiration [192] have become commonly employed to investigate the response of cellular objects to mechanical stresses. Essential cellular processes (e.g., cell division) and collective effects (e.g., cell signaling and migration in tissues) are currently being investigated with such techniques.\nIn this section, we will demonstrate one of the simplest cell manipulation experiments with optical tweezers: the mechanical stretching of a single red blood cell (RBC) to measure its rigidity to an externally applied deformation or strain."},{"title":"RBC buffer preparation","content":"The RBC buffer should mimic the physiological conditions found in human blood. It can be prepared as a solution of 130 mM NaCl, 20 mM K/Na phosphate buffer at 7.4 pH, 10 mM glucose, and 1 mg ml −1 BSA (Bovine Serum Albumin) [193]. It is convenient to prepare higher concentration stocks (1 M) of every solution and store them in the fridge (higher concentration of salts together with low temperature prevent microbial contamination), keeping them in 50 ml Falcon tubes for repeated use. To prepare a 50 ml Falcon tube containing the RBC buffer, follow these steps:\n1. Add 6.5 ml of 1 M NaCl, 1 ml of 1 M K/Na phosphate buffer at 7.4 pH,16 and 0.5 ml of 1 M glucose into an empty 50 ml Falcon tube.\n2. Add 50 g of BSA.\n3. Add 42 ml of milli-Q water.\n4. Mix the RBC buffer and filter it (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International)."},{"title":"Place the filtered solution into a new 50 ml Falcon tube.","content":"All remaining unused RBC buffer must be re-filtered on a daily basis to prevent molecular aggregation and bacterial growth. After three days, it is recommended to dispose of the remaining RBC buffer because the re-filtering will not be enough to ensure proper conservation."},{"title":"Blood extraction","content":"To obtain human blood, the easiest way is to prick the finger of a healthy donor. Use of diabetic testing lancets will ensure sterilization of the pricking device. The extraction is made following these steps:\n1. Disinfect the finger of the healthy donor with ethanol and dry it with handkerchief paper to remove any remaining liquid.\n2. Prick the donor's finger with the lancet and apply some pressure to extract a drop of blood. A single drop of blood will be enough to perform experiments during an entire day.\n3. With a 200 µl pipette take 10 µl of blood from the finger.\n4. Dilute as fast as possible the blood drop in 1 ml of RBC buffer inside a 1.5 ml Eppendorf tube to prevent degradation of the RBCs."},{"title":"Bead preparation","content":"The simplest option to attach the beads to the RBCs is to take advantage of non-specific interactions between polystyrene beads and RBCs.17 To do so:\n1. Add 1 ml RBC buffer in a 1.5 ml Eppendorf tube.\n2. Homogeneously resuspend the polystyrene beads by placing the purchased beads (Microbead NIST Traceable Particle Size Standard, 3.00 µm, Polysciences, Inc.) into a vortex mixer for several seconds. An additional step of several seconds of sonication improves the resuspension.\n3. Add 3 µl of the bead stock solution to the Eppendorf tube.\nIt is recommended to have two Eppendorf tubes with bead solution available to ensure continued supply of the chamber with beads during the experiment."},{"title":"Setup preparation","content":"The experiment can be performed using the same microfluidics chamber and miniTweezers setup presented in section 4.1. In particular, the procedure to build the chamber and to place it into the mount is the same as described in section 4.1.1, the only difference being a longer plastic tube (40 cm) to connect one of the syringes to the inlet of the central channel. Once the chamber is fixed into the mount with the syringes connected through the plastic tubes, the next steps are:\n1. Flow 3 ml of RBC buffer in each of the three channels to produce an homogeneous environment in the chamber.\n2. Identify which dispenser tube (upper/lower) has the most clear exit (i.e., the one displaying the most sharp perpendicular cut in the video image) to the central channel. This will simplify trapping the beads. 5. Introduce the mount with the chamber between the two objectives of the optical tweezers.\n6. Connect the syringe with the larger plastic tube (the one of the central channel) to a pump to apply a controlled flow to the central channel when it is needed. We will use this flow to stretch the RBC-bead dumbbell captured in the optical trap, thereby preventing thermaland photo-damage of the cell."},{"title":"Construction of the cell-bead configuration","content":"To perform cellular pulling experiments, the RBC need to be held between two beads attached to the cell membrane on opposites sides, as shown in Fig. 17(d). One bead is held by the micropipette by air suction, while the other bead is held by the optical trap. The use of beads as handles to manipulate the RBC minimizes the direct exposure of the RBC to the laser light. The steps of the procedure to obtain this cellular configuration are:\n1. Align the lasers without a bead (steps 1-8 described section 4.1.2).\n17There are also several options to treat the beads and RBC to specifically attach them, e.g., using the lectin protein Concanavalin A [191] or biotinylating the RBC [194] 2. Identify the position of the pipette and of the two dispenser tubes that connect the upper and the lower channels with the central one.\n3. Move the chamber using the stepmotors and place the optical trap close to the exit of the dispenser tube that contains the bead solution.\n4. Flow some bead solution pushing the syringe either by hand or using a pump, and trap a bead with the optical trap.\n5. Move the trapped bead next to the pipette tip in the working zone.\n6. Align the lasers with the trapped bead (steps 9-13 described section 4.1.2).\n7. With the bead already trapped, try different flow values with the pump connected to the syringe of the central channel until the force reaches ∼ 10 pN due to the RBC buffer flow. This will be the approximate flow that we will need to keep the optically trapped bead separated from the RBC (Figs. 17a and 17b) after we obtain an attachment.\n8. Turn off the pump.\n9. Move the trapped bead to the area of the blood dispenser tube exit. It is important not to put the bead exactly at the exit of the tube to prevent losing the bead due to the RBC solution flow.\n10. Flow some RBC solution applying a small pressure to the syringe. RBCs will flow into the central channel.\n11. Move the trapped bead close to a RBC and try to make a connection by poking the surface of the RBC with the bead. The connection is formed when the RBC follows the trapped bead when moved.\n12. Move the chamber using the stepmotors to place the bead and the attached RBC next to the micropipette.\n13. Turn on the pump. The configuration should look like Fig. 17(a). The RBC is attached to the bead and partially trapped by the lasers. 14. If the flow is not enough to separate the RBC from the trap and, therefore, to obtain the configuration of Fig. 17(b), it is necessary to move the chamber using the stepmotors along the horizontal direction to exert an extra force and separate the RBC from the optically trapped bead.\n15. Move the bead close to the tip of the pipette. 16. Carefully apply air suction to immobilize the bead. This step is critical because a large enough air suction force could absorb the RBC inside the pipette.\n17. Move the optical trap away from the bead. If the bead is properly fixed to the tip of the pipette, the bead will stay there. If this is not the case, when the optical trap is moved the bead will follow it. 23. Try to make a connection between the trapped bead and the RBC as in Fig. 17(c). Notice that the two beads are on opposite sides of the RBC. 24. Check that the new connection between the bead and the RBC is done by moving the optical trap to the same direction as the pump flow. If the RBC is deformed while moving the trap, the attachment is done. 25. Turn off the pump to remove the flow of the central channel. 26. Nip very carefully the plastic tube that is connected to the entrance of the central channel with a clamp to prevent flows due to the difference of pressure at both ends of the plastic tubes.\n27. Move the stepmotors to obtain the configuration shown in Fig. 17(d), aiming at having a zero reading for the force along the y-direction."},{"title":"Experiments","content":"By cyclically stretching and releasing the RBC, it is possible to obtain the force-extension curves from which the stiffness and deformability of the RBC can be extracted. Starting with a RBC in the configuration described in the previous section ( Fig. 17(d)), it is possible to stretch the cell by moving the optically trapped bead outwards (i.e., away from the micropipette) and release it by moving it inwards (i.e., towards the micropipette).18 Follow these steps to define the stretching-realising protocol:\n18A range for the optical trap displacement of at least ∼ 8 µm is necessary to perform the RBC stretching. Thus, it is recommended to move the optical trap towards the lower limit of its y-displacement, before starting the pulling protocol.\nStarting from the minimum force, the RBC is pulled at the constant velocity until the maximum force. After reaching this maximum force, the RBC is pushed at the same constant velocity to the minimum force. During this process, the displacements and forces along the RBC stretching direction (y-direction) are recorded for both the A-and B-trap in the miniTweezers setup ( Fig. 14(a)); then, the two displacements are averaged to measure the trap displacement, while the two forces are summed to obtain the total force acting on the RBC. The whole process is called pulling cycle. Recording 5 pulling cycles for each RBC is typically enough for the subsequent analysis."},{"title":"Data analysis","content":"The data analysis aims at computing the RBC stiffness knowing the stiffness of the optical trap κ trap . The experiment measures the trap displacement ∆x trap , not the cellular extension ∆x cell . Thus, ∆x cell can be computed from ∆x trap , κ trap , and the measured optical force f trap . For each intermediate trap position from L 0 to L f , we are able to transform L into x cell substracting the 19Depending on the laser power, it could be very hard to reach the 0 pN force as the RBC will be attracted to the optical trap.\n20Above 20 pN, as we have not applied any specific coating to the beads, a very thin membrane tube (theter) forms and alters the force-extension curve."},{"title":"RBC","content":"ΔL Δx trap Fig. 18. Sketch of RBC extension under applied force. The red ellipse represents the RBC, the light blue circles represent the beads, and the small green circle represents the optical trap. The cell extension is ∆x cell , while ∆x trap is the bead displacement respect to the trap center, and ∆L is the displacement of the optical trap. Note that ∆x cell and ∆x trap are distances along the y-axis. trap contribution (Fig. 18), as we show in the following expression:21\nwhere ∆L is the variation of the trap position between that at minimum force and that at maximum force. The resulting force-cellular-extension curve is shown in Fig. 19(a) for the second pulling cycle for two different RBCs. 22 The RBC stiffness can be measured from the slope of the force-cellular-extension curves. As can be observed in Fig. 19(a), the force-cellular-extension curve is not a straight line. This indicates that the stiffness of the RBC depends on the force applied to the cell. For this reason, the slope is computed for five different force windows whose stiffness values are represented in Fig. 19(b). The RBC stiffness increases as we increase the applied force, the stiffness curves of the stretching and releasing cross each other between 6 and 10 pN, and the releasing stiffness is higher than the stretching stiffness at high forces while it is lower at low forces. The expected values for RBC stiffness in this specific geometry are in the range between 3 pN µm −1 and 10 pN µm −1 . The dispersion between different RBCs, in terms of force-cellular-extension curves, can be very large. This can be due to the fact that the RBCs are not separated by density so that the RBCs can have very different ages and, thus, very different physiological properties [196]."},{"title":"Other applications to single-cell mechanics and outlook","content":"Until now, the manipulation of single cells with optical tweezers has not been extensively investigated as single-molecule manipulation. The main reason is the difficulty inherent to the geometry of the pulling assays when working with cells. Indeed, while the specific labeling of molecules makes it possible to design the most diverse pulling geometries, this is more difficult to achieve when manipulating single cells which ultimately require the physical contact between the optically trapped bead and the lipid cell membrane. Additionally, single cell manipulation with optical tweezers is feasible on cells in suspension (such as RBC, lymphocytes, senescent, stem cells), whereas adherent cells are harder to manipulate, often requiring confocal microscopy 21This is the same analysis done to obtain the molecular extension in long nucleic acids (≈ 10 kbp double-stranded nucleic acids or ≈ 1 kbp single-stranded nucleic acid [195]).\n22It is best to analyze the second pulling curve because the first one could exhibit a special transient behavior.\nsetups with optical traps very close to the glass substrate. For the latter class of cells, other techniques (e.g., atomic force microscopy, traction microscopy, magnetic twisting cytometry) turn out to be more informative. Finally, let us mention that while most single cell manipulation studies are carried out in in-vitro conditions (e.g., on the Petri dish), it will be necessary to expand the use of these manipulation techniques to more relevant environments such as ex-vivo tissues.\nIn this case, the mechanical response of a single cell in a tissue-like environment is expected to differ a lot with respect to the isolated cell. Overall, this makes the future of single cell mechanics very promising beyond biology, with implications in physiology and medicine."},{"title":"Microrheology","content":"Rheology investigates the mechanical response of soft materials such as simple liquids, polymer solutions, micellar fluids, colloidal suspensions, liquid crystals, gels, and foams [20,197]. Specifically, it is concerned with flow and deformation of such materials under applied stress or strain. Depending on the specific details of their microstructure, different kinds of rheological behavior can be observed. For instance, for most simple liquids, the rheological response is Newtonian: upon imposing a given strain rate, the resulting stress is proportional to it, thereby dissipating energy instantaneously by viscous resistance. Instead, complex fluids and gels, whose components are usually long macromolecules suspended in a viscous solvent, exhibit viscoelasticity, i.e., both viscous (liquid-like) and elastic (solid-like) responses are possible [198]; therefore, they are able to store and dissipate energy depending on the time-scale of observation.\nThe goal of rheology is to provide quantitative parameters, e.g., viscosities, relaxation moduli and creep compliances, that relate the stress and the strain (or the strain rate) of the material under deformation. To this end, the most straightforward techniques consist in imposing stresses and strains in a controlled manner by means of a rheometer to the material sample of interest and then measuring the corresponding mechanical response. Despite their relative simplicity, these methods require typically milliliter-sized sample volumes, thus being unsuitable for soft matter systems that are expensive or difficult to find in abundance, such as biological fluids and newly synthesized materials. Microrheology overcomes this major drawback by measuring mechanical properties of a soft material sample using colloidal probes directly embedded in it. The rheological information of the material is fully inferred from the motion of the probe, which requires only the detection and tracking of its position over time. Therefore, very small sample amounts, typically ranging from pico-to microliters, are needed. This in turn allows to carry out in-situ rheological measurements which are inaccessible to macroscopic methods, e.g., biofluids within living tissues and cells, soft interfaces and membranes. Furthermore, microrheology offers several advantages over conventional macroscopic rheometry. For instance, unlike bulk rheology, which provides quantities averaged over the entire macroscopic sample, microrheology permits to explore local mechanical properties at the length-scale of the micron-sized probe. This is particularly useful for investigating heterogeneous materials and fluids close to interfaces, where submillimetric spatial variations of the rheological parameters occur. In addition, undesirable artifacts that can arise due to inertial effects when using a macroscopic rheometer are automatically ruled out in microrheology due to the extremely low Reynolds numbers (Re 10 −3 ) of the flows induced by the motion of a micron-sized probe. This expands significantly the available frequency range over which the linear viscoelastic spectrum of the system can be determined without inertial corrections, typically from the range [10 −1 Hz, 10 2 Hz] up to the range [10 −1 Hz, 10 6 Hz]. Besides, conventional rheometers are usually designed for either highly elastic or highly viscous materials, but they are not sensitive enough to measure low viscosities and elastic moduli; then, another advantage of microrheology is its capability to detect very weak viscoelastic parameters. Moreover, the typical forces and energies involved in microrheology, either from the thermal motion of the material molecules or those externally exerted by optical tweezers, are of the order of ∼pN and ∼k B T, respectively. This guarantees that the microstructure of the sample is not irreversibly destroyed by the motion of the probe. 20. Microrheology with an optically trapped particle. (a) Sketch of the general response of the shear stress σ(t) of a viscoelastic material to a step-like shear strain (t)."},{"title":"Basic setup and concepts","content":"For t ≥ 0, the ratio σ(t)/ 0 defines the stress relaxation modulus G(t). (b) Sketch of the typical experimental system used in microrheology: a spherical colloidal bead (radius a), embedded in a soft material, is trapped by optical tweezers. If the position of the optical trap is fixed (x ot (t) ≡ 0, passive microrheology), the particle is only subject to the thermal collisions of the molecules of the surrounding fluid microstructure. If x ot (t) is varied in time, an additional time-dependent force is exerted on the probe particle (active microrheology).\nOptical tweezers provide the ideal tool to manipulate probe particles in microrheology [20,[199][200][201][202], because they permit one to tailor the forces that can be exerted to locally strain the sample.\nTo extract meaningful information by means of standard microrheological methods, some experimental conditions must be fulfilled. First of all, the material must be homogeneous and isotropic at the scale of the probe, which is usually valid when the relevant length-scale of the material, e.g., the mesh size of gels and semidilute polymer solutions, is much smaller than the probe size. In this case, the environment behaves as a continuum and its mechanical response can be fully characterized by a single scalar function, e.g., the stress relaxation modulus G(t). For example, upon applying a step-like shear strain at t = 0 to a material, (t) = 0 Θ(t), where Θ is the Heaviside function, the stress relaxation modulus is given by\nwhere σ(t) is the resulting shear stress ( Fig. 20(a)). For a viscoelastic liquid, G(t) decays to zero over a finite time, while for a viscoelastic solid, it decays to a constant non-zero value. In both cases, the drag force experienced by an embedded particle at time t can be written as\nwhere r(t ) is the instantaneous particle velocity at time t < t and the minus sign indicates that the drag force is opposed to the instantaneous velocity, whereas Γ is a memory function related to G(t), which weights the role of the previous history of the particle motion on its current drag force due to the temporal correlations induced by the surrounding medium. For example, for a spherical particle of radius a, the memory function is Γ(t) = 6πaG(t), which can be regarded as a generalization of the Stokes' law (equation (6)). For a Newtonian fluid with shear viscosity η, the relaxation modulus is G(t) = 2ηδ(t),23 which corresponds to an instantaneous response of the drag force F drag (t) = −6πηa r(t) to the particle velocity r(t). Moreover, for some techniques, it is required that the sample is in thermal equilibrium and that the forces externally applied to the probe are small enough to keep the material in its linear response regime. Under such conditions, the time evolution of a single particle coordinate, x, is described by the generalized Langevin equation [203]\nwhere x ot represents the x-component of the central position of the optical trap, while ζ is a Gaussian noise of zero-mean, ζ(t) = 0, and autocorrelation\nDepending on how optical tweezers are employed to manipulate a colloidal probe embedded in a soft material ( Fig. 20(b)), two kinds of microrheological techniques exist: passive and active [197,201]. In the following sections, we describe how to prepare a prototypical viscoelastic fluid (section 4.3.2) as well as how to determine rheological parameters of simple liquids using passive and active microrheology (sections 4.3.3 and 4.3.4)."},{"title":"Preparation of a viscoelastic fluid and setup","content":"An equimolar wormlike micellar solution made of the surfactant cetylpyridinium chloride (CPyCl) and the salt sodium salicylate (NaSal), both dissolved in water, represents a prototypical viscoelastic fluid. To prepare it, overnight mixing of the two components in deionized water is required at approximately 50 • C, after which a homogenous and isotropic viscoelastic fluid results. 24 To get reproducible rheological parameters of the fluid, the conductivity of water should be kept at most at 10 µS m −1 (milli-Q water is a good option for this purpose). Depending on the surfactant/salt concentration, different kinds of microstructures in the aqueous solvent are formed [204]. For instance, above the first critical micelle concentration but below 4.5 mM, the surfactant molecules aggregate into spherical and cylindrical micelles of radius 2 to 3 nm, which result in a weakly viscoelastic behavior of the dilute solution. Increasing further the concentration up to 10 mM, the surfactant molecules self-organize in long flexible cylindrical micelles, which form, overlap, and deform dynamically in the solvent. Such wormlike micelles have a radius from 2 to 3 nm and a contour length ranging from 100 nm to 1 µm, while their persistence length and mesh size are of order 10 nm for concentrations between 5 and 10 mM. To perform optical-tweezers-based microrheology, a very small amount of micron-sized beads must be suspended in a microliter volume of solution. Special care is needed to avoid air bubbles inside the fluid, which can be easily formed due to the presence of the surfactant. Sonication during a few minutes of the fluid with the dispersed colloidal particles is recommended before being loaded into the sample cell. Such a sample cell consist of a standard chamber (cross-sectional area about 1 cm 2 ) made of a glass slide and a coverslip, separated by a distance of approximately 100 µm by means of, e.g., spacer tape or parafilm strips. The chamber must be laterally sealed with UV-curable optical adhesive after inserting a small volume (∼ 10 µl) of the suspension, in order to avoid evaporation of the fluid and drift. During the measurements, the fluid sample must be kept thermally coupled to a thermostat at constant temperature T within an accuracy range of ±1 K, as the rheological properties of the fluid strongly depend on temperature. It is 23Here, we use the following convention for the Dirac delta ∫ ∞ 0 delt a(t )dt = 1/2. Thus, the factor 2 is needed in the relaxation modulus G(t) to recover the correct friction coefficient.\n24It is recommended to wear gloves and safety googles when handling CPyCl, as direct contact with power can cause skin and eye irritation. crucial to maintain T above the so-called Krafft temperature [205], which is 18 • C for the specific micellar solution described here. Below the Krafft temperature, the surfactant crystallizes, thus suppressing the viscoelasticity of the fluid.\nTo illustrate the microrheological techniques described in the following, two Newtonian fluids with distinct viscosities at room temperature are used. The first one is deoinized water, whose conductivity must be less than 10 µS m −1 in order to facilitate the sample cell preparation. The second Newtonian fluid is propylene glycol n-propyl ether (PNP), also known as 1-propoxy-2propanol, which can be directly purchased with a purity ≈ 99% so that no further purification is needed. On the other hand, two viscoelastic fluids are employed: an equimolar CPyCl/NaSal micellar solution at 4 mM and one at 5 mM. The different fluids and their main properties are summarized in Table 9. Table 9. Microrheology fluids. Fluids used to illustrate microrheology with an optically trapped particle.\nAll the experiments described in this section are performed in a standard optical tweezers setup implemented with an oil-immersion microscope objective (100×, NA = 1.4) which tightly focuses a Gaussian laser beam (λ = 1070 nm) onto the sample. A single spherical particle is trapped and kept at least 10 µ away from any wall of the sample cell in order to avoid hydrodynamic interactions, which can significantly affect the values of the fluid's rheological parameters with respect to those obtained from bulk measurements. Imaging and position detection can be realized by conventional video microscopy. In particular, for the determination of the linear viscoelastic spectrum by passive microrheology, it is required that the acquisition frequency of the camera must be above about 1000 frames per second. Therefore, the use of a position sensitive detector is an alternative for this purpose. Furthermore, beam steering, which is crucial for active microrheology, is achieved here by means of galvomirrors (section 2.3.1)."},{"title":"Passive microrheology","content":"Passive microrhelogy is the simplest approach to study the local mechanical response of soft matter. As its name suggests, the embedded colloidal bead is used as a passive element, whose thermal motion provides all the necessary information to extract, via fluctuation-dissipation relations, the frequency-dependent rheological parameters of the material under investigation. To prevent sedimentation as well as significantly large diffusive displacements of the probe from its initial position, which would otherwise restrict the measurement time of the particle position x(t), a static optical trap can be used to confine the particle motion around a mean fixed position, i.e., x ot (t) = 0, as depicted in Fig. 20(b). Here, we present two direct applications of passive microrheology with optical tweezers for viscous and viscoelastic liquids.\nDetermination of zero-shear viscosities. For a Newtonian liquid, the autocorrelation function of the particle position provides a straightforward way to determine its viscosity η. From equation (85), using the instantaneous memory function Γ(t) = 2γδ(t) for a purely viscous liquid, the expression for its autocorrelation function can be readily derived (section 3.5):\nwhere . . . represents an ensemble average over different realizations of the thermal noise, while τ ot = γ/κ is a decay rate related to the viscous friction coefficient γ of the probe and the trap stiffness κ. In practice, the ensemble average can be replaced by a time average along the stochastic trajectory x(t), provided that the total measurement time of x(t) is much larger than τ ot . For a spherical particle of radius a, the friction coefficient is γ = 6πaη, where η is the fluid viscosity. Then, once the trap stiffness κ is known (e.g., by the equipartition method: κ = k B T x(t) 2 , section 3.3), the experimental autocorrelation function of x(t) can be fitted to the right-hand side of equation (86). Therefore, η can be determined from the fitting parameter τ ot by means of η = κτ ot 6πa . This is illustrated in Fig. 21, where the viscosities at T = 25 • C of two Newtonian fluids, water and propylene glycol n-propyl ether (PNP), are computed by means of this passive method. In Fig. 21(a), we plot the typical trajectories x(t) of particles embedded in these liquids, trapped by optical tweezers at approximately 40 µm away from the cell walls. The corresponding autocorrelation functions of x(t) are shown in Fig. 21(b), where we show that the experimental data (symbols) can be very well fitted to equation (86) (dashed lines). From the equipartition relation, we first determine the values of the trap stiffness for each case: κ = 1.24 pN µm −1 for the particle in water (diameter 2a = 2.73 µm) and κ = 0.90 pN µm −1 for the particle in PNP (diameter 2a = 3.25 µm). Next, from the fitting parameters τ ot , we find the corresponding values of the viscosities of both fluids: 0.9 mPa s for water and 4.5 mPa s for PNP, which agree very well with their bulk values.\nThis technique can be extended to determine steady-state flow properties of viscoelastic fluids, which exhibit both liquid-and solid-like responses at times shorter than a typical time τ. Such a time-scale reflects the stress relaxation of the elastic material microstructure suspended in the solvent. For a viscoelastic micellar solution, τ originates from the continuous formation and breaking of the micelles as well as the reptation modes of the wormlike structures, and can range from a few milliseconds to several seconds in the semidilute regime [206]. Unlike viscoelastic solids (e.g., gels), viscoelastic fluids are able to flow with constant shear rate at sufficiently long times, t τ, after applying constant shear stress at t = 0. Therefore, they are characterized by a zero-shear viscosity η 0 , which in absence of a confining potential, leads to a long-time diffusion of an embedded spherical particle, where the diffusion coefficient is given by k B T 6πaη 0 . In the presence of an optical trap, from equation (85) it can be shown that the position autocorrelation function has the form\nwhere A < 1 is a prefactor that depends mainly on κ and τ. Equation (87) is valid provided that τ ot τ [207], which must be verified a posteriori. To fulfill this condition of large time-scale separation, sufficiently small values of the trap stiffness are needed. In Figs. 21(a) and 21(b), we illustrate this method for a 2a = 3.25 µm silica particle suspended in a dilute micellar solution (concentration 4 mM, T = 295 K), trapped by optical tweezers with stiffness κ = 1.34 pN µm −1 . At this concentration, the viscoelastic fluid is composed of non-overlapping micellar structures, thus resulting in typical relaxation times, τ, of order of milliseconds. We check that equation (87) is a very good approximation to the experimental position autocorrelation function, where only the first experimental point slightly deviates from the fit. The fitting parameter τ ot = 0.16 s yields the value η 0 = 6.9 mPa s, which is also in agreement with reported bulk measurements [204]."},{"title":"Determination of storage and loss moduli. The complex shear modulus","content":", is a quantity used in rheology to characterize linear viscoelasticity of soft matter in the frequency domain [208]. G ( f ) is the storage modulus and G ( f ) is the loss modulus, which account for the elastic energy in phase with the applied strain and the out-of-phase viscous dissipation of the material, respectively. For instance, for a purely viscous liquid (viscosity η), G ( f ) = 0 and G ( f ) = 2π f η, whereas for an elastic solid (elastic modulus G 0 ), G ( f ) = G 0 , and G ( f ) = 0. In general, for viscoelastic materials, both storage and loss moduli are non-zero. G * ( f ) is closely related to the Fourier transform,G( f ), of the stress relaxation modulus, G(t), by\nTherefore, for an applied oscillatory shear strain ( f ) = 0 e 2πi f t with amplitude 0 and frequency f , the resulting shear stress is also oscillatory and can be expressed as G * ( f ) ( f ).\nA very helpful expression for G * ( f ) in terms of the equilibrium fluctuations of the position x(t) of a trapped particle can be derived from the fluctuation-dissipation theorem. The latter relates the linear response function χ(t) of x(t) to a perturbative force F(t), defined as\nwith the equilibrium (F(t) = 0) autocorrelation function of x(t), i.e., x(t)x(0) . In equation (89), the nonequilibrium ensemble average . . . F is defined in presence of the time-dependent perturbation. From equations (85) and (89), the Fourier transform of\nNote that equation (90) shows that the elastic contribution of the optical tweezers stiffness to the mechanical response of the system must be subtracted for a correct determination of the complex shear modulus of the surrounding medium. Moreover, equation (90) involves directly the inverse ofχ( f ), which can be determined by either passive or active microrheology, as will be shown. Therefore, this expression is applicable for both types of microrheological techniques. In particular, in the case of passive microrheology, by making use of the fluctuation-dissipation theorem in frequency domain and the Kramers-Kronig relations, one can find both the real, χ ( f ), and the imaginary, χ ( f ), parts of the response functionχ( f ):\nwhere S( f ) = |x( f )| 2 is the one-sided PSD of x(t) in thermal equilibrium, i.e., the Fourier transform of x(t)x(0) , whereas P denotes the Cauchy principal value of the integral. Thus, once χ ( f ) and χ ( f ) are determined from the experimentally recorded trajectory of the trapped particle, the storage modulus G ( f ) and loss modulus G ( f ) can be computed over the available frequency range [0, f max ] from equation (90). For a given sampling frequency f s of x(t), f max is fixed by the Nyquist frequency: f max = 1 2 f s . This technique needs a rather high sampling frequency of x(t) to avoid a large underestimation of the integral in equation (91).\nIn Fig. 22, we apply the previous method to characterize the linear viscoelasticity of the wormlike micellar solution in the semidilute regime at 5 mM and T = 20 • C. First, we compute the equilibrium PSD, S( f ), of a long trajectory x(t) measured over 25 min at a sampling frequency f s = 2000 Hz. Since any discrete Fourier transform involved in the calculation of S( f ) leads inexorably to a noisy curve, the spectral profile must be smoothed before numerically computing the integral of equation (91). In Fig. 22(a), we show the original noisy profile of S( f ) (blue line) obtained by means of a discrete Fourier transform computed with 2 17 points. A polynomial fit (dashed black line) is performed to smooth the PSD profile. Then, a direct numerical calculation using equations (90)- (92) leads to the values of the storage and loss moduli shown in Fig. 22(b) over the available frequency interval 0 Hz ≤ f ≤ 1000 Hz. At high frequencies approaching 1 2 f s , an abrupt decrease of the numerical values of both G ( f ) and G ( f ) is observed. This is an artifact due to the finite frequency range, which imposes a cut-off in the required Kramers-Kronig integral transformation [209]. Therefore, the physically meaningful values of the components of the complex modulus G * ( f ) are only those at frequencies one decade below the Nyquist frequency [210]."},{"title":"Active microrheology","content":"In active microrheology, the trapped colloidal probe is externally forced through the material to locally strain it, and its mechanical response is directly measured [211]. From this response, G * ( f ) can be determined by means of equation (90). Different kinds of perturbations are possible. For instance, the probe particle can be trapped by static optical tweezers while the whole sample cell is moved by a piezo stage (section 3.11). Here, we will focus on a perturbation created by the oscillatory motion of the position x ot (t) of the optical trap, which exerts a time-dependent force F(t) = κx ot (t) on the particle according to equation (85) (Fig. 20(b)).\nAs active microrheology provides a direct measurement of the linear response function χ(t) defined by equation (89), it does not necessarily require that the material under investigation is in thermal equilibrium. Consequently, this technique is suitable for the investigation of rheological properties of general viscoelastic fluids and solids, including out-of-equilibrium soft matter, such as actin networks [212,213], physical gels [214], and glassy colloidal suspensions [215,216]. It is particularly useful for the determination of the storage G ( f ) and loss G ( f ) modulus at a specific driving frequency f d , for which a single-frequency oscillatory perturbation force\ncan be directly applied to the colloidal probe, where X 0 is the maximum displacement of the optical trap and φ is the initial phase. Therefore, a previous passive calibration of κ is needed to know the applied force (section 3). In order for both the optical trapping and the material microstructure to remain in the linear response regime, X 0 must be sufficiently small. This is generally fulfilled if the typical energy injected by F(t) is at most of the order of ∼ k B T, i.e., X 0 k B T/κ. In Figure 23, we show some results to illustrate the implementation of active microrheology to characterize the rheological properties of a wormlike micellar solution at 5 mM and T = 20 • C (Table 9). In the inset of Figure 23(a), the black line shows the time evolution of F(t)/κ, which results from the sinusoidal motion of the optical trap (equation (93) with X 0 = 200 nm, κ = 1.0 pN µm −1 , f d = 1 Hz). The blue line plots the trajectory x(t) of an actively driven particle (2a = 2 µm), which clearly exhibits an oscillatory behavior with a time delay relative to F(t) due to the resistance of the surrounding viscoelastic fluid. Here, the crucial step is to compute the Fourier transform of the linear response function from the measurement of x(t) and x ot (t). From equation (89), this is given byχ\nwhere the numerator is the value at frequency f of the non-equilibrium ensemble average of the Fourier transform of x(t), in presence of F(t) with the same initial phase φ, whereas the denominator involves the Fourier transform of the trap position. Due to the sinusoidal form of x ot (t), the denominator is zero for all frequencies f different from f d . Therefore, in practice it is customary to compute the inverse of equation (94) and to approximate it at f d by\nwhere the numerator is the Fourier transform of the cross-correlation function between the trap position and the particle position, while the denominator is the value of the PSD of x(t), both quantities at frequency f d , first computed for a given F(t) and then averaged over different realizations of φ. In Fig. 23(a), we plot the PSD of the particle position x(t) in presence of the force F(t) at different driving frequencies f d . The pronounced peaks at f = f d result from the sinusoidal form of F(t)\n, while for f f d , the contribution to S( f ) is only due to thermal noise in the fluid. In Fig. 23(b), we plot as symbols the values of the storage and loss moduli computed by means of equations (90) at the distinct driving frequencies. For comparison, we also represent as dotted and dashed lines the corresponding values determined by passive microrheology, demonstrating the rather good agreement between both methods.\nAlthough more elaborate to implement in experiments, active microrheology is less prone to numerical artifacts than passive microrheology. This is because it is based on a measurement of the mechanical response of the particle position at a single frequency, which does not require any indirect numerical integration over a discrete finite frequency interval as those involved in the Kramers-Kronig relations."},{"title":"Other microrheological applications and outlook","content":"In the previous sections, we have described some basic applications of microrheology with optical tweezers. More advanced methods, which are beyond the scope of this Tutorial, focus on investigating flow and deformation properties of complex materials which do not trivially fulfill the general conditions described above. Some important examples are the following:\nInterfacial microrheology. Near a liquid-solid, a liquid-liquid or a liquid-vapor interface, the viscosity of liquids becomes anisotropic and exhibits a dependence on the closest distance to the interface due to the specific hydrodynamic boundary conditions. Passive microrheology can be applied to determine such a spatial dependence, provided that the three spatial coordinates of the probe particle, embedded in the liquid phase of interest, can be tracked [217][218][219].\nTwo-point microrheology. The motion of pairs of colloidal particles not so far from each other is strongly correlated when suspended in a fluid. Since the flow and strain fields around one of them entrain the second particle, the motion of the latter encodes information of the rheological properties of the fluid at the location of the former. Therefore, a dual-beam optical tweezers can be used to fix their mean separation and perform two-point microrheology. Here, one particle plays the role of a control element, either thermally or externally driven, while the second one is used as a passive probe. This technique is particularly useful to investigate the mechanical response of heterogeneous and anisotropic soft materials, such as gels and polymer solutions [220,221].\nTransient microrheology. Using a combination of both active and passive microrheology, the transient behavior of viscoelastic materials either upon flow startup or cessation can be investigated. For example, during an active period, an optically trapped particle can be driven though the viscoelastic fluid, followed by a sudden release from the tweezers by shutting off the laser, during which its position is recorded. During this non-equilibrium passive period, the particle motion is subject to the recovery of the initially strained fluid microstructure. Consequently, this technique allows to measure in a straightforward manner relaxation times and relaxation moduli of viscoelastic materials [222,223].\nMicrorheology has evolved during the last two decades into a powerful experimental tool to investigate complex soft materials. Most of its current applications rely on a well-established theoretical framework based on equilibrium statistical mechanics, low-Reynolds-number hydrodynamics and linear response theory. Nevertheless, more recent approaches aim to investigate non-linear rheological properties, e.g., shear thinning of polymer solutions, by means of colloidal probes driven at very high velocities or under large-amplitude oscillations [20]. Although the implementation of such microrheological techniques does not represent a big experimental challenge, the interpretation of the data in terms of meaningful parameters and its connection with bulk quantities is not trivial. A complete understanding of such a wealth of information will certainly rely on current theoretical advancements in non-equilibrium soft matter systems [224], which are able to find a direct link between the motion of the probe and the microstructural deformation of the surrounding medium."},{"title":"Colloidal interactions","content":"A colloidal system comprises a continuum medium (the solvent, such as water) and a disperse phase (the solute, such as small colloidal particles from 10 nm to 10 µm). Due to their dimensions, colloidal particles experience thermal fluctuations and undergo Brownian motion [225]. Since their properties can be described using statistical physics ensembles, colloidal suspensions represent an ideal model system for statistical physics and they can be considered as \"big atoms\" [226] mimicking atomic systems at different time and length scales, which make their dynamics accessible by optical experiments. Therefore, colloidal suspensions have been used to investigate phase transitions between gas, liquid, solid, and crystalline phases [227,228], crystal nucleation [229][230][231], glassy states [232], and vapor-liquid interfaces [233]. Colloidal suspensions are also employed in industry to stabilize emulsions (oil in water and water in oil) and foams [234,235]. Colloidal interactions play a key role in all these applications, from food industry [236] to nanotechnology [237]. The main colloidal interactions are:\nVan der Waals forces. Van der Waals forces arise between neutral particles such as atoms, molecules or colloids. They have an electromagnetic origin due to the interaction between the dipoles of the singles particles. They are repulsive at short distance range, due to repulsion between the atoms' electron clouds, and attractive at larger distance. Their intensity quickly decreases with the inter-particle distance following a power law [238,239].\nDouble-layer forces. Double-layer forces are very important for the stability of many biological systems and for the formation of colloidal crystals [240,241]. They result from the spontaneous charging of the particle surfaces when immersed in a liquid, due to ionic adsorption or dissociation. They can be both attractive or repulsive.\nDLVO theory forces. Double-layer forces and van der Waals forces are combined in the DLVO theory (named after Derjaguin, Landau, Verwey und Overbeek) [242] to calculate the interaction between spherical colloidal particles [243,244].\nSteric forces. Steric forces are repulsive nonbonding interactions influencing the spatial conformation and the reactivity of ions and molecules. They can be used to promote or prevent flocculation (steric stabilization) of colloidal solutions. They play an important role in chemistry, biochemistry, pharmacology and food industry, where they are often employed to design the physical properties of food and pharmaceutical products by steric stabilization [236,[245][246][247], as well as to treat waste water and to purify drinkable water by flocculation of metals and airborne particles [248,249].\nDepletion forces. Depletion forces are attractive forces arising in a suspension of large and small particles [250]. Usually, the large particles are colloids (typically, at least a fraction of a micrometer), while the small particles can be smaller colloids, micelles, solvent macromolecules, or dissolved ions [250][251][252][253][254].\nCritical Casimir forces. Critical Casimir forces emerge between objects (e.g., colloidal particles) immersed in a critical mixture close to its critical point [255][256][257]. Althougth these forces were first predicted theoretically in 1978 by M. E. Fisher and P. G. de Gennes [255] in analogy to quantum-electrodynamical (QED) Casimir forces [258], they have been measured directly only recently [259,260] proving their relevance for soft matter and nanotechnological applications thanks to their piconewton strength and nanometric action range [261][262][263].\nHere, we will focus our attention on how to measure critical Casimir forces. In particular, we will describe a typical experiment to quantify the dynamic effects of critical Casimir forces between two colloidal particles."},{"title":"Critical Casimir forces","content":"Critical Casimir forces take place between objects immersed in a critical mixture when they confine the mixture composition fluctuations arising near the critical temperature. These forces are typically in the piconewton and nanometer ranges, they are tunable as a function of temperature, and they feature an exquisite dependence on the surface properties of the involved objects [256,257,264,265]. Attractive critical Casimir forces arise whenever the density fluctuations are confined between objects with the same surface properties (e.g., between two hydrophilic or hydrophobic particles). Repulsive critical Casimir forces take place between objects with opposite surface properties (e.g., between a hydrophilic and a hydrophobic particle)."},{"title":"Experiment outline","content":"In the following sections, we will present in detail how to perform an experiment where the effects of critical Casimir forces on the free dynamic of a pair of colloidal particles are measured using blinking optical tweezers [263,[266][267][268].\nWe employ hydrophilic silica microspheres (diameter d = 2.06 ± 0.05 µm, Microparticles GmbH) dispersed in a water-2.6-lutidine mixture at the critical lutidine mass fraction c c L = 0.286, corresponding to a lower critical point at the temperature T c 34 • C [269,270]. Due to their extremely sensitive dependence on temperature, we have to control and stabilize the sample temperature to within ±2 mK via a feedback controller [260,263].\nUsing two holographic optical traps, we trap two microparticles in the bulk of the critical mixture at a surface-to-surface distance of ≈ 300 nm. This distance is larger than the range of the electrostatic repulsion between the particle surfaces, but still comparable with largest action range of critical Casimir forces. Whenever the two optical potentials are switched off (by chopping the laser beam), the colloids diffuse in the bulk of the critical mixture.\nWhen the temperature T of the solution is sufficiently lower than T c (i.e., for ∆T = T c − T 500 mK) no critical Casimir forces are observed and the particle diffuses freely in the medium. Approaching the critical temperature T c (i.e., for ∆T → 0), density fluctuations start to take place in the critical mixture leading to attractive critical Casimir forces between the two colloids, affecting their diffusion and reducing the average inter-particle distance. To obtain sufficient statistics of the dynamics of the colloids, we need to repeat the entire blinking process for several times (i.e, about 400 times) for each value of ∆T."},{"title":"Experimental setup and feedback temperature controller","content":"The experimental setup requires holographic optical tweezers, digital video microscopy, and feedback temperature control [260,263]. The holographic optical tweezers (section 2.3.5) is realized using a phase-only spatial light modulator, a laser beam with a wavelength of 532 nm, and a oil-immersion objective (100×, NA = 1.30). The resulting traps can be periodically switched on and off by a chopper.\nThe most crucial part of the setup is the temperature controller to stabilize the sample temperature to within ±2 mK. This is realized using a two-stage stabilization protocol. The first temperature stabilization is achieved by keeping constant the temperature of the sample holder to within ±50 mK by a circulating water chiller. The second and finer temperature stabilization to within ±2 mK is achieved by a temperature feedback controller applied through the objective. The controller reads in real time the temperature of the objective and stabilizes it by a Peltier element thermically connected to the objective, which is the closest element to the investigated volume. An important aspect to take into account is the thermal isolation of the sample area as well as of the entire setup to prevent thermal fluctuations due to air flow and to room temperature fluctuations."},{"title":"Potential analysis","content":"The motion of the two colloids is recorded by a camera with acquisition frequency 300 fps and the acquired videos can be analyzed by digital video microscopy [22,271] to obtain the trajectories r 1 (t) and r 2 (t) of the centers of the two particles projected onto the x y-plane, where r l (t) = (x l (t), y l (t)) with l = 1, 2 labelling the particles. Once these trajectories are obtained, it is possible to calculate their relative distance r(t) = |r 2 (t) − r 1 (t)| for different values of T (Fig. 24). The time evolution of the probability density distribution, for all the inter-particle distances r(t) at specific values of ∆T, are reported in Figs. 24(a-d). For ∆T = 456 ± 2 mK, far away from the critical point ( Fig. 24(a)), the particles diffuse freely and there is no critical Casimir forces affecting their behaviour. Increasing the temperature, ∆T = 200 ± 2 mK ( Fig. 24(b)) and ∆T = 163 ± 2 mK (Fig. 24(c)), density fluctuations start to take place in the mixture and critical Casimir forces arise affecting the dynamics of the colloids. We can observe that occasionally they cause adhesion between the colloids as can be inferred from the emergence of a peak in the inter-particle distance probability density at r ≈ 2.16 µm. This peak is due to the gradual emergence of attractive critical Casimir forces between the particles, approaching T c , and indicate that critical Casimir forces are strong enough to produce particle adhesion also in the presence of the optical potentials. The peak position indicates the region where the repulsive electrostatic forces and the attractive critical Casimir forces are balanced [263]. When the temperature T is very close to T c (∆T = 108 ± 2 mK, Fig. 24(d)), strong attractive critical Casimir forces inhibit the free diffusion of the particles, which often adhere to each other so that the values of r lie Sometimes r(t) is smaller than d because a displacement of the colloids along the vertical z-axis causes their projections onto the xy-plane to overlap. This occurs more frequently in the presence of critical Casimir forces and particularly if the particles are temporarily stuck together. (e-h) Equilibrium distribution P eq (r) of the inter-particle distance r(0) (i.e., when the optical tweezers are switched off) for two optically trapped colloids at temperatures (f) ∆T = 456 ± 2 mK, (g) 200 ± 2 mK, (h) 163 ± 2 mK, and (i) 108 ± 2 mK. Each histogram is obtained from 400 different experimental values. The solid black lines are the theoretical distribution of r(0), obtained via Monte Carlo integration (10 6 samples) of two optically trapped particles subjected to the theoretical total potential V(R 1 , R 2 ) (equation (96)). within a small region resulting from the equilibrium between the repulsive electrostatic forces and the attractive critical Casimir forces [263].\nFor each value of ∆T, all the initial inter-particle distances r 1,2 (0) represent the equilibrium distribution of initial positions R l = (x l , y l , z l ) with l = 1, 2 of the two colloids subjected to a total potential including the optical potentials V ot,1 (R 1 ) + V ot,2 (R 2 ), the repulsive electrostatic potential V es (ρ), and eventually the potential of the critical Casimir forces V C (ρ):\nwhere ρ = |R 2 − R 1 | − d is the actual surface-to-surface distance between the two colloids. Note that the projected distance r introduced above is generically smaller than the actual center-to-center distance |R 2 − R 1 |, due to possible displacements of the colloids along the vertical z-direction.\nAssuming that the optical potentials V ot,1 and V ot,2 reported in equation (96) are harmonic, we have:\nwhere the centers R 0,l and the stiffnesses k l of traps l = 1, 2 can be determined experimentally by the separate calibration of the two optical traps. These initial values are fixed in all the experiment and in the following analysis.\nFor the electrostatic repulsion potential V es , we can consider the simple expression [260,270]\nwhere ρ is the surface-to-surface distance between the colloids, ρ es is an effective parameter, which depends on the surface charges, while D is the Debye screening length [260,270]. For the potential V C of the critical Casimir forces, we can consider the theoretical prediction based on the Derjaguin approximation [270]:\nwhere Θ is a universal scaling function [272][273][274] and ξ is the bulk correlation length of the critical fluctuation of the density fluctuations of the critical mixture. Whenever the optical potential is switched on, the particles trajectories evolve under the action of the total potential V(R 1 , R 2 ), and after a sufficiently long time, they reach the equilibrium distribution:\nThe histograms in Figs. 24(e-h) represent the experimental values of the equilibrium distribution P eq and the solid black lines are the corresponding theoretical results obtained from the Monte Carlo integration of equation (100). When the mixure temperature is far away from T c (∆T = 456 ± 2 mK, Fig. 24(e)), the probability distribution P eq (r) is very well approximated by a Gaussian distribution centered at the value r 2.40 µm, corresponding to experimental distance r 0 between the centres of the two optical traps [263]. Reducing ∆T (Figs. 24(e-h)), a peak arises at r 2.16 µm on the left flank of the Gaussian distribution, becoming more dominant at the expense of the Gaussian distribution. Starting from probability distribution P eq (r), we fit its experimental values by Monte Carlo integration of equation (100) based on the theoretical potential V(R 1 , R 2 ). In particular, we use the experimental values of V ot,l (R l ) (equation (97)) and Θ(ρ/ξ) (equation (99)) as input functions, and ρ es , D , and ξ as fitting parameters. In doing this, we assume that the fitting parameters ρ es and D have the same values ( D 13 nm and ρ es 95 nm) for all T, while the correlation length ξ is specific to each T. Once the values of ξ have been obtained, we calculate the real temperature of the solution by fitting the experimental temperature values measured at the objective with the theoretically expected temperature dependence of ξ, according to the method reported in Refs. [260,263]. (105)) fitted to the experimental data. The colors refer to data taken at ∆T = 456 ± 2 mK (blue), 273 ± 2 mK (light blue), 200 ± 2 mK (green), 163 ± 2 mK (yellow), 127 ± 2 mK (orange), and 108 ± 2 mK (red). Errors bars represent the standard deviation of the experimental values. The solid lines represent the theoretical predictions accounting for the effect of the hydrodynamic interaction between the colloids [275]. The nature of the deviations observed in D at short and long distances is discussed in the main text."},{"title":"Drift analysis","content":"The relative position and the distance between the two particles can be used to determine the values of their diffusion coefficient D(r) and their drift velocity v(r) as function of their inter-particle distance r. Due to the hydrodynamic interaction between the two colloids, the particle diffusion coefficient depends on the distance between the particles and is different along the directions parallel and perpendicular to the line connecting the centers of the two particles (see equation (5.5) in Ref. [275]). For this analysis, we consider the particle trajectories as a sequence of values r i , numbered by i and acquired at times it s , where t s is the time between sampling. We can decompose the i-th displacement ∆r (n) i = r i+n − r i into its parallel and perpendicular components:\nand\nwherer i = r i /r i andẑ is the unit vector along the z-direction, which is perpendicular to the x y-plane of observation where the position vectors r i lie. Then, the parallel and perpendicular diffusion coefficients are [263]:\nand D ⊥ (r) = 1 2 where δr represent the amplitude of a small interval around r. The measured normalized values of D ⊥ and D (symbols) are reported in Fig. 25 as functions of the ratio r/d, together with the theoretical prediction obtained in Ref. [275] (solid line) for no-slip boundary conditions. In particular, D ⊥ and D are normalized by the bulk diffusion constant\nwhere η 2 · 10 −3 Ns m −2 is the viscosity of the mixture close to T c [276]. The values of D ⊥ and D in Fig. 25 shows satisfactory agreement, with a systematic discrepancy emerging only in D for r/d 1.05 due to the limited experimental acquisition rate, which does not allow us to resolve times shorter than 3 ms. 25 The parallel and perpendicular drift velocities can be measured using:\nand\nFor large ∆T (Fig. 25(a) (107)). Here, notice that v ⊥ vanishes in all investigated cases and shows no temperature dependence because all the forces at play in this experiment act along the direction which connects the centers of the particles [263]."},{"title":"Other colloidal interactions and outlook","content":"Critical Casimir forces can play an important role in nanoscience and nanotechnology thanks to their piconewton strength, nanometric action range, fine tunability as a function of temperature, and particle surface dependence. They can be employed to manipulate micro and nano-objects (e.g., by controllable periodic deformations of chains), to assemble micro and nano-devices (e.g., via the self-assembly of colloidal molecules [277,278]), and to drive nanomachines (e.g., by powering rotators [279]) at the nano and micro-meter scale.\nBeyond critical Casimir forces, other colloidal interactions have also been studied using optical tweezers. Recently, Van der Waals forces have been directly measured between two isolated optically trapped rubidium atoms [280] opening the path to the atomic modeling of micro-and nano-structured devices controlled by these forces [281]. Double layer forces can also play an important role in nanotechnology due to their ability to tune the transport and deposition rate of micro and nano-sized particles. Steric forces play an important role in tuning the rates and the activation energies of most chemical reactions, crucial in chemistry, biochemistry, and pharmacology. Depletion forces are extensively used to stabilize colloidal solutions by flocculation, to produce the programmable self-assembly of colloidal nanostructures [282], and to drive cellular organization [252]."},{"title":"Statistical physics","content":"The advent of minituarization methods and microscopic manipulation techniques, as those discussed in this Tutorial, has opened the door to understand fundamental concepts and ideas at the foundations of statistical physics, allowing us to explore how macroscopic laws emerge from their microscopic dynamics. In this section, we only discuss experiments performed in a liquid. For related work in vacuum, see section 5 and Ref. [19]. We will start discussing the theory of Kramers transitions and stochastic resonance (section 4.5.1). We will continue exploring non-equilibrium fluctuation-dissipation theorems (section 4.5.2) and introducing stochastic thermodynamics (section 4.5.3). Finally, we will explain how the Carnot's cycle and efficiency of macroscopic engines can be scaled down to the microscopic realm, and conclude with a discussion of the concept of Maxwell's demons and Szilard engines (section 4.5.4)."},{"title":"Kramers' transitions and stochastic resonance","content":"Using optical trapping techniques, Simon et al. [283] and later McCann et al. [284] studied the escape of a Brownian particle from a potential well, visualizing for the first time thermally activated processes and giving in this way a neat experimental proof of Kramers' theory [285].\nKramers' theory in the overdamped limit predicts the mean rate for the activation of processes over an energy barrier ∆U to be\nwhere τ is the mean residence time, also called Kramers' time and τ 0 is the overall relaxation time of the well, which can be expressed in terms of the angular frequencies near the stable\nwith γ the friction coefficient and m the mass of the particle. For the two-dimensional case, the expression for 1/τ 0 takes the following form [286]:\nwhere the superscripts (x) and (y) indicate the parallel and perpendicular directions to the axis joining the stable and unstable critical points, respectively. Fig. 27(a) shows a schematic of the configuration of the optical tweezers to generate a double-well potential.27 A silica bead28 of diameter d p = 0.96 µm is confined in a bistable potential generated by two parallel optical tweezers with orthogonal polarizations and separated by a distance d = 0.8 µm, which are generated by focusing two Gaussian beams with a water immersion objective with NA = 1.2. These two beams are generated with two polarizing beam splitters according to the scheme shown in Fig. 5 (section 2.3.2). The mirrors in the beam splitter are controlled with piezo actuators with 30 nm step resolution leading to a resolution of the position of the optical traps in the sample cell below 20 nm. The trapping region is set 27Alternative experimental configurations have also been used to realize a bistable optical potential. For example, using two nanoholes in a metallic film [287], sculpting the bistable potential with a spatial light modulator [288], or positioning time-shared optical traps in two closely-separated positions using acousto-optical deflectors [289].\n28Interestingly, also 1 µm polystyrene particles can be trapped with this setup, but with these particles we have not been able to observe any bistable behavior. We believe this is because the higher weight and lower refractive index of the silica beads make them more stably trapped along the axial direction of the beams allowing us to explore relatively larger distances between optical tweezers in a range where the bistable potential is neatly observed. A typical trajectory of the Brownian particle in this double-well potential is shown in Fig. 27(b). Along the x-direction joining the equilibrium points, the particle spends most of the time near the equilibrium points A and B, occasionally crossing the saddle point S. Along the perpendicular y-direction, the motion does not appear to be affected by the presence of two neighboring traps. Knowing the friction coefficient of the particle, this trajectory permits us to determine the Kramers' time using equations (108) and (110), where the characteristic frequencies (or equivalently their stiffnesses) at the critical points can be straightforwardly obtained using FORMA (section 3.8). Fig. 27(c) shows the force field in the xy-plane reconstructed using FORMA, from which the Table 10. Characterization of equilibrium points in a bistable potential. The theoretical mean crossing rates (1/τ (th) ) obtained using equations (108) and (110) Table 10 provides the stiffnesses of the equilibrium points (κ x and κ y ), the energy barriers (∆U), and the overall relaxation times (1/τ 0 ), from which we estimate the mean crossing rates (1/τ (th) ) using equations (108) and (110). The average crossing rate can be straightforwardly obtained from the trajectory of the bead by detecting the times at which the particle crosses the saddle point, shown in Fig. 27(f). For the validity of the approximations within Kramers' theory, these crossing events have to be rare, i.e., the time spent by the particle near the stable positions must be always longer than the relaxation time τ 0 defined by the stiffnesses of the critical points, limiting the ideal situations to the cases where the energy barrier is very high in comparison to k B T. This is a homogeneous Poisson process, where each escape event occurs at random with small probability [283]. Therefore, we expect the escape times to be exponentially distributed (red curves in Fig. 27(f)):\nwhere p 0 = 1/τ is the mean crossing rate and τ is the mean residence time. The resulting experimental mean rates are reported in Table 10 (1/τ (ex) ), showing very good agreement with data obtained by means of Kramers' assumptions (1/τ (th) ). Bistable potentials are often employed to study non-linear dynamics. For example, a bistable potential such as that described in this section has been used to demonstrate the emergence of stochastic resonance when the optical tweezers are modulated in time harmonically, but asymmetrically (i.e., one potential well gets deeper while the other one gets shallower) [283]: as the modulation time approaches the Kramers escape time, the escape probability and the activation rate increases. A different process of synchronization between two bistable systems was also demonstrated when two optically generated bistable systems with one bead each in water are set very close to let the particles to interact [288]: when the two bistable systems have slightly different escape rates, the particles synchronize their jumps and lock their rates to an intermediate value between those featured by the two non-interacting systems."},{"title":"Non-equilibrium fluctuation-dissipation relations","content":"In its simplest form the fluctuation-dissipation theorem (FDT) [290] can be expressed as k B T µ = D (Einstein-Smoluchowski equation (7)): it provides a direct relationship between the mobility µ = 1/γ, which measures the response of its velocity to an external force, and its diffusion coefficient D.\nMore broadly, the FDT states that for a system in contact with a heat bath at temperature T, upon applying a small external perturbation h which changes the energy U of the system as U → U − hV, where V is the variable conjugate to h with respect to the energy, the following relation holds for any observable Q [291]:\nwhere\nis the linear response function of Q at time t to the perturbation h at time s, δ δh denotes a functional derivative, i.e.\nand C(t − s) = Q(t)V(s) 0 is the two-time correlation function between the observable Q and V. The brackets . . . h denote a non-equilibrium ensemble average in presence of the perturbation h, whereas . . . 0 represent an equilibrium average for h = 0. The expression (112) of the FDT can be re-written in the integral form as\nwhere χ(t) = ∫ t 0 R(t − s)ds is the integrated response function, which is usually much easier to implement and measure than R(t).\nThe importance of equations (112) and (114) for experiments lies on the fact that linear response functions, which characterize states slightly away from equilibrium, and equilibrium correlation functions can be interchangeably determined by an appropriate choice of the varibles h, Q and V, depending on the specific conditions. For example, for a colloidal particle suspended in water at constant temperature T and trapped by optical tweezers with stiffness κ, the integrated response function of the particle position x with respect to the trap center x OT to an external time-dependent force can be determined by applying a Heaviside-like perturbative displacement of the trap position x OT (s) = x 0 Θ(s) and then by measuring the resulting time evolution of the ratio x(t) h κ x 0 at t ≥ 0. Then, it follows that R(t) = 1\n. Equivalently, it can be indirectly computed by means of the equilibrium quantity 1 k B T ∂ s C(t − s), where t ≥ s, Q = x is the observable of interest, V = x is the variable conjugate to the perturbative force h = κx OT , and C(t − s) = x(t)x(s) 0 , for which no external force is needed. Using the explicit expression for the positional correlation function in thermal equilibrium (section 3.5), x(t)x(s) 0 = k B T κ exp − κ |t−s | γ , one finds R(t) = 1 γ exp − κt γ , t ≥ 0, from which the time evolution of x(t) h in response to any external force h can be determined by means of equation (113)."},{"title":"Fluctuation-dissipation relation for non-equilibrium steady states (NESS).","content":"Optical tweezers can be used to explore some generalizations of equation (112) around non-equilibrium steady states (NESS) for mesoscopic systems such as colloidal particles, molecular motors, and biomolecules [147]. In particular, a 1D model system that has been extensively studied within the context of non-equilibrium statistical physics consists of an overdamped particle with friction coefficient γ, moving on a circle of radius R circle and polar coordinate θ, across a periodic potential U(θ) = U(θ + 2π) under the action of a constant force f 0 and a thermally fluctuating force ξ [164]. The dynamics of the polar coordinate θ is described by the Langevin equation\nwhere φ(θ) = U(θ) max{U(θ)} is the normalized potential profile, A = max{U(θ)} γa 2 is its amplitude, F = 1 γa f 0 is a non-conservative force term,29 and ζ = 1 γa ξ is a Gaussian white noise of zero 29On a circle, any constant force f 0 0 is non-conservative, because it cannot be expressed as the derivative of a potential due to the periodic boundary conditions. mean, i.e., ζ = 0, and autocorrelation ζ(t)ζ(s) = 2D θ δ(t − s), where the angular diffusion coefficient along the circle is given by D θ = D a 2 = k B T γa 2 . Due to the non-conservative force, the system lacks detailed balance: for constant values of F > 0, A > 0 and D θ > 0, and in absence of time-dependent perturbations, θ reaches a NESS, characterized by a distribution ρ NESS (θ), which is different from the equilibrium Boltzmann distribution ρ eq (θ) ∝ exp − Aφ(θ) D θ . Under such conditions, the particle is able to go beyond the potential barrier and explore the whole circle due to the combined effects of thermal fluctuations and the non-conservative force, thereby developing a constant non-vanishing probability current given by\nAround this NESS, the expression (112) of the FDT can be generalized by the following relation [292]\nwhere . . . 0 and . . . h represent now a non-equilibrium ensemble average in the NESS (h = 0) and around the perturbed NESS (h 0), respectively, while the additional term\ntakes into account the extent of the \"violation\" of the conventional FDT due to the non-vanishing current j. Here, the observables Q and V as well as v 0 are functions of θ, while v 0 is called the mean local velocity, and is given by v 0 (θ) = j/ρ NESS (θ). Note that the generalized expression (117) around a NESS reduces to the conventional FDT (112) around thermal equilibrium, when j = 0. The expression (117) of the generalized FDT can also be written in the integral form\nwhere B(t) = ∫ t 0 b(t − s)ds, and represents the generalization of equation (114) for NESS. Similar experimental approaches based on scanning optical tweezers have been followed to investigate fluctuation-response relations for other kinds of non-equilibrium states [293][294][295][296][297]."},{"title":"Experimental implementation of a NESS.","content":"We illustrate the generalization of the FDT around a NESS with optical tweezers. To experimentally realize an overdamped system whose stochastic motion is well-modelled by equation (115), we consider a spherical silica particle (radius a = 1 µm) suspended in water at constant temperature (T = 20 • C, γ = 1.89 · 10 −8 kg s −1 ). The particle is subject to a toroidal optical trap, which can be created by a conventional single trap, whose position is deflected on the plane transverse to the beam axis either by two perpendicular galvanometric mirrors [293,298], or by a pair of perpendicular acousto-optic deflectors (AOD) [299]. For the latter approach, one applies simultaneously sinusoidal frequency modulations to the acoustic waves created inside each AOD X and Y, ∆F X,Y (t) = ∆F sin(2π f R t + α X,Y ), with amplitude ∆F ∼ MHz, scanning frequency f R , and phase difference |α X − α Y | = π 2 . When f R is kept fixed, the incident laser beam is deflected in a periodic manner. In this way the beam focus describes a circular trajectory inside the sample cell, θ OT (t) = 2π f R t + α, of constant radius R circle at frequency f R on the plane perpendicular to the beam propagation. To achieve a periodic force landscape as described by the Langevin model (115), the scanning frequency of the laser beam must be on the order of f R ∼ 10 2 Hz. For a particle with radius a = 1 µm, the optical setup must be built in such away that the final radius of the circular path described by the tweezers is R circle ≈ 2 − 5 µm. Smaller values of R circle lead to a Mexican-hat potential U(θ) in which the particle can spontaneously jump from θ to θ + π, similar to a Kramer process in a double-well potential (section 4.5.1), while larger values of R circle yield very slow dynamics for which it becomes experimentally difficult to sample θ with good statistics. For R circle = 4.12 µm, Fig. 28. Experimental realization of a NESS (a) Typical unwrapped (i.e., defined on the interval [0, ∞)) trajectories θ(t) of a colloidal particle in a NESS. The dashed line represents the mean drift due to the non-zero probability current j induced by the non-conservative term F. Inset: example of wrapped (i.e., defined over [0, 2π)) trajectory θ(t). (b) NESS probability density function of θ, defined over [0, 2π) (bars) and reconstructed potential energy U(θ) (solid line). The arrow indicates the direction of the non-conservative force f 0 , which shifts the maximum of ρ NESS (θ) to the right relative to the minimum of U(θ). Inset: comparison between the NESS distribution ρ NESS (θ) and the equilibrium one ρ eq (θ) (F = 0, sharp peak). (c) Correlation function between the observable Q(θ) = sin θ and the variable V(θ) = sin θ. f R = 200 Hz, and an infrared laser (λ = 1064 nm) with incident power of 30 mW, the resulting scanning speed of the focused beam is v R = 2πR circle f R ≈ 5 mm s −1 , which is so high that it is not able to trap and drag continuously the particle through the fluid because the viscous frictional force due to the surrounding water quickly exceeds the optical trapping force. Consequently, the beam only kicks the particle a small distance δs γv R κ along the polar direction θ at each rotation. During the absence of the beam (≈ 1/ f R = 5 ms) the particle undergoes free diffusion over a length scale l D = D f R 40 nm R circle in the directions radial and perpendicular to the circle. The beam kicks the particle again during the next arrival, thus preventing its escape by diffusion away from the region scanned by the beam focus. Therefore, the particle motion is effectively confined to a torus of major radius R circle and minor radius ∼ l D /2, where the angular position θ is the only relevant dynamical degree of freedom. In addition, a periodic intensity profile can be created along the main circle by sinusoidally modulating in time the laser power P(t) with an amplitude ∆P = 2.1 mW around the mean value of P 0 = 30 mW, at the same frequency as the scanning frequency f R , i.e., P(t) = P 0 + ∆P sin(2π f R t + α X ). These optical conditions result in the desired force landscape: a constant non-conservative force f 0 associated to the mean kick and a static periodic potential U(θ) due to the conservative force exerted by the sinusoidal light intensity profile [299,300].\nThe instantaneous angular coordinate θ of the optically driven particle can be obtained from its 2D trajectories (x, y), by θ = arctan(y/x) ( Fig. 28(a)). The effective parameters F, A and the profile φ(θ) must be determined from an ensemble of independent NESS trajectories θ(t): at least 100 trajectories, each with at least a full rotation on the circle are needed to find the probability current j (dashed line in Fig. 28(a)) and the corresponding nonequilibrium distristriution ρ NESS (θ) (histogram in Fig. 28(b)).\nDepending on the quantity of interest, θ must be defined either on the interval [0, 2π) or [0, ∞). For example, the probability current must be computed as [299]\nwhere θ(t) 0 is the average of an ensemble of independent NESS trajectories θ(t), defined on [0, ∞) ( Fig. 28(a)). This permits us to verify that θ(t) 0 is linear in t, thus leading to a constant j > 0. The quantity j −1 represents the typical time the particle needs to perform a full cycle around the circle, which corresponds to the longest relaxation time of the NESS. For example, for the experiment presented here j −1 = 26 s, which is much larger than the typical equilibrium relaxation time of a particle in water trapped by optical tweezers (∼ ms). Therefore, it is expected that the particle motion in this NESS has very long-lived correlations. In Fig. 28(b), we plot the experimental profile of the NESS distribution ρ NESS (θ), computed from all the data points of the trajectories on [0, 2π). From the experimental profile of ρ NESS (θ) and the value of j, the full force landscape of the particle can be reconstructed [301]. First, taking into account the periodic boundary conditions of the NESS distribution, i.e. ρ NESS (0) = ρ NESS (2π), from equation (116) we get the non-conservative term\nwhereas the potential is given by\nThe experimental values of the parameters considered here yield F = 0.85 rad s −1 and A = 0.87 rad 2 s −1 , which correspond to a non-conservative force f 0 = γaF = 66 fN and an energy potential amplitude max{U(θ)} = γa 2 A = 68.8 k B T, while the normalized potential profile is sinusoidal, φ(θ) = sin θ (red solid line in Fig. 28(b)). In thermal equilibrium (F = 0), the particle motion would be tightly confined around the minimum of such a deep non-linear potential at φ m = 3π/2, with a width ∆θ ≈ k B T γa 2 A = 0.12 rad and a vanishingly small probability to cross the potential barrier of height 2A = 137.6 k B T (inset in Fig. 28(b)). However, under the action of the non-conservative force, the distribution of θ is much broader, with a maximum shifted in the direction of the resulting current j > 0. This reflects the highly non-equilibrium conditions of the system. Indeed, they can be better assessed by means of the generalized FDT relation (119). We focus on the observable Q(θ) = sin θ, which is periodic on the circle and corresponds to the dimensionless potential energy of the particle φ(θ) = U(θ)/max{U(θ)}.30 A variation in the potential amplitude, A → A + δ A translates into the following change in the potential energy: γa 2 Aφ(θ) → γa 2 Aφ(θ) − (−γa 2 δ A)φ(θ). Therefore, by noting that a variation h in a control parameter leads to a change in the potential energy, i.e., U → U − hV, we conclude that in this specific example the integrated response function of Q(θ) to the perturbation h = −γa 2 δ A is given by equation (119), where the variable conjugate to the perturbation is V(θ) = sin θ, whereas ∂ θ V(θ) = cos θ in equation (118). In Fig. 28(c), we plot the NESS correlation function C(t) = sin θ(t) sin θ(0) 0 , revealing an oscillatory behavior with a typical time-scale ≈ 25 s, which corresponds to j −1 . Here, to improve the statistics from the available data, the NESS ensemble average sin θ(t) sin θ(0) 0 is computed first by taking a time average\nalong a given realization of θ(t), where t max > j −1 to span a full cycle of the probability current, and then by performing an additional ensemble average C t max (t) over different independent NESS trajectories θ(t). The same procedure must be carried out for the correlation B(t) defined through equation (118). In Fig. 28(d), we plot both the NESS terms C(0) − C(t) and B(t) involved in the integral generalized FDT (119). We observe that both terms are of the same order of magnitude, thus confirming that the experimental conditions of the system are far from thermal equilibrium. Note that, without the corrective term b defined in equation (118), a determination of the integrated response function χ(t) by means of the equilibrium FDT (114), and consequently R(t), would be wrong. This is shown in the inset of Fig. 28(d) , where we compare the numerical time derivative of 1\n, which shows that without taking into account the corrective term B, the response function R(t) would be highly overestimated around a NESS."},{"title":"Stochastic Thermodynamics","content":"Stochastic thermodynamics is an emerging branch of statistical physics that extends concepts of classical thermodynamics such as heat, work and entropy production to the level of single stochastic trajectories for systems in contact with a heat bath and driven arbitrarily far from thermal equilibrium by means of a well-specified experimental protocol [164]. It provides a number of statistical relationships involving these fluctuating quantities, which generalize the macroscopic laws of thermodynamics and describe the non-equilibrium energetics of many mesoscopic systems of interest in soft matter and biophysics, such as colloidal particles, polymers, biomolecules, molecular motors, biochemical networks, and microelectromechanical systems [147,302]. Thus, stochastic thermodynamics represents nowadays a solid theoretical framework for applications ranging from single-molecule biomechanics (section 4.1) to optimization protocols and the efficiency of microscopic heat engines (section 4.5.4) .\nHere, we present some general notions of stochastic thermodynamics using a paradigmatic system: a colloidal particle embedded in a viscous liquid (friction coefficient γ), in contact with a thermostat at constant temperature T and subject to a possibly time-dependent force. The 1D motion of a single coordinate x of the particle can be described by the Langevin equation\nwhere U(x, λ) is the potential energy which leads to a conservative part −∂U(x, λ)/∂ x in the total force, whereas f (x, λ) represents a non-conservative force, as that described in section 4.5.2. Both forces can be time-dependent through the control paramenter λ, which can be externally varied according to a prescribed protocol from λ(0) to λ(t) during the time interval [0, t]. In addition, in equation (124) µ = 1/γ is the particle mobility, whereas ξ represents a Gaussian white noise with zero-mean and autocorrelation function ξ(t)ξ(s) = 2Dδ(t − s), which accounts for the thermal fluctuations of the particle velocity. Stochastic thermodynamics requires that the properties of the heat bath are not affected by the presence of external driving forces, thus implying that D = k B T µ regardless of the strength of −∂U(x, λ)/∂ x and f (x, λ). Note that a statistically equivalent description of the system modelled by equation (124) can be given by means of the Fokker-Planck equation for the probability density function of x at time t\nwhere the probability current is defined as\nsupplemented with the initial condition ρ(x, t = 0) = ρ 0 (x). Both ρ(x, t) and j(x, t) allow to directly assess how far the system is from thermal equilibrium conditions, for which j = 0 and ρ ∝ exp − U(x) k B T ."},{"title":"First Law.","content":"To generalize classical thermodynamic concepts to the level of an individual stochastic realization of the process x(t), we first note that either a temporal variation of the control parameter λ during the infinitesimal time duration dt or a non-zero non-conservative force, f 0, can be identified as an infinitesimal work increment done on the system:\nThen, from equation (127) an inifinitesimal variation dU in the potential energy during dt can be written as\nThus, by defining the infinitesimal increment of the heat dissipated into the medium as the product of the total force acting on the particle times the infinitesimal displacement dx,\nthe expression (128) can be regarded as a generalization of the First Law of thermodynamics [303] dU = δW − δQ on the level of the stochastic process x(t). From the Langevin equation (124), the work done on the system and the heat dissipated into the medium over the finite time interval [0, τ], are\nrespectively, where the notation [x(t)] highlights the fact that both quantities fluctuate over time and depend on the specific stochastic realization of x(t) up to time τ. The terms which involve dx(t) = dt x(t) in the integrals of equations (130) and (131) must be interpreted in the Stratonovich sense [304]. Note that the definition of the heat given by equations (128) and (131) is physically meaningful, since the first term on the right-hand side of equation (131) is positive definite and represents the energy delivered to the thermostat, while the second term can be either positive or negative and corresponds to the thermal fluctuating energy injected by the medium. Moreover, for systems in thermal equilibrium (i.e., f = 0 and λ = 0), equation (129) consistently reduces to dU = −δQ, where any increment (decrease) in the potential energy of the particle is only due to a fluctuation of the heat taken up from (delivered to) the thermostat, being both on average equal to zero. We analyze the previous concepts in the specific case of a colloidal particle (diameter 2a = 2.73 µm) suspended in water at constant T = 295 K, for which the friction coefficient is γ = 2.44 · 10 −8 kg s −1 . The particle is subject to the harmonic potential of an optical trap, where the position of its minimum plays the role of the control parameter λ in equation (124). In the case where the optical trap is uniformly translated, λ varies linearly over time at constant speed v and, under the initial condition λ(t = 0) = 0, U(x, λ(t)) = 1 2 k (x − vt) 2 , λ(t) = vt, and λ(t) = v. Without loss of generality, we consider the simplest situation where all the forces acting on the system are conservative (i.e., f = 0), thus being the temporal variation of λ the source of non-equilibrium work and dissipated heat. This well-defined protocol can be implemented by means of the deflection of the trapping beam using galvanometric mirrors or acousto-optics deflectors (section 2.3.1). Although with a different thermodyamic interpretation,31 a closely related protocol can be carried out by moving the sample cell at constant speed v by means of a piezoelectric stage (section 3.11) [305]. Fig. 29(a) shows a typical experimental trajectory x(t) of such a colloidal particle, as well as the time evolution of the position of the minimum of the optical trap, λ(t). For times much longer than the trap relaxation time (≈ 21 ms), the dragged particle reaches a steady state, whose statistical properties do not depend on time. In this case, from the definition (130), one can find the work done by the optical tweezers on the trapped particle over a time interval of duration τ along a given stochastic realization of x(t):\nwhere we have dropped the argument [x(t)] for the sake of simplicity. Since for this control parameter and a harmonic potential, equation (124) is linear and ξ is Gaussian, x must also be 31When moving the sample stage at speed v, an external flow is applied to the trapped particle, which results in an extra term in the viscous drag force acting on the particle, i.e. γ( x − v). On the other hand, the potential energy remains unaffected by v, U(x) = κ x 2 /2. This leads to expressions for the work and heat which are different from those obtained when performing a motion of the optical trap.\nGaussian and, thus, W τ is Gaussian as well, with mean\nand variance\nIndeed, using the experimental trajectory x(t) (Fig. 29(a)), we compute numerically W τ by means of equation (132) and check that its mean and standard deviation satisfy equations (133) and (134), as shown by the blue circles in Figs. 29(b) and 29(c), respectively. The distribution of W τ for different values of τ is plotted in Fig. 29(d), where we verify that P(W τ ) is Gaussian. Note that, while W τ is always positive and increases linearly with the observation time τ as the trap drags the particle through the fluid, there can be negative work fluctuations, where the system loses mechanical energy [305]. These rare events, which are at first glance in contradiction with macroscopic intuition, are due to thermal fluctuations, which are not taken into account by classical thermodynamics. The role of such fluctuations, quantified by the ratio [W τ − W τ ] 2 / W τ , is very prominent at short time-scales and become negligible as τ approaches the limit of a macroscopic observation time, i.e., τ γ k . The heat dissipated into the medium due to the motion at constant speed of the trap can be derived from equation (131) as\nIn this case, some care is needed to compute the stochastic integral (132) from the experimental values of x(t). As previously mentioned, Stranotovich integration must be used to get physically meaningful results.32 Specifically, from the collection of N experimental data points x(t j ), defined at each time t j = j−1 N −1 τ over [0, τ], where j = 1, . . . , N, a mid-point discretization of the force −k[x(t) − vt] with respect to the stochastic increment dx(t) must be performed\nIn Fig. 29(e) we show the distribution of Q τ for different values of τ, computed by means of equation (136). In this case, all discrete time steps have a constant value fixed by the inverse of the experimental sampling frequency f s = 145 Hz, i.e., t j+1 − t j = f −1 s = 0.007 s. It should be noted that, although the mean value of Q τ coincides with that of W τ for all τ ≥ 0 in a steady state, as shown in Fig 29(b), their distributions are quite different, see Fig.29(e). In this case, the distribution of the heat, P(Q τ ) has non-Gaussian tails, with standard deviation larger than the variance of the work fluctuations (Fig. 29(c)). In addition, similar to the behavior of the stochastic work, rare events where Q τ is negative are also possible for small values of τ at which thermal fluctuations are significant, i.e., the heat spontaneously flows from the surrounding medium to the particle even if mechanical work is done on it.\nFinally, in Fig. 29(f) we plot the distribution P(∆U τ ) of the variation of the potential energy over a time interval of duration τ:\n32While equation (132) for the work simply requires the product of the deterministic time increment dt with the stochastic particle position relative to the position of the trap, x(t) − vt, the expression for the heat (equation (135)) involves the stochastic increment of the particle position, dx times x(t) − vt.\nIn this case, while the mean value of ∆U τ is equal to 0 for all values of τ ( Fig. 29(b)), consistent with the fact that in a steady state the potential energy of the system remains constant on average, its distribution is strongly non-Gaussian, due to the non-linear dependence of ∆U τ on x(t), with standard deviation which quickly saturates to a constant value, as observed in Fig. 29(c).\nThe statistical properties of W τ , Q τ and ∆U τ for this simple experiment illustrate the main essence of stochastic thermodynamics. Even though the average values of such quantities satisfy the expected thermodynamics behavior, the probability of observing rare events, which are not described by the classical thermodynamics, is not negligible for mesoscopic systems at sufficiently short time scales.\nSecond Law and entropy. Stochastic thermodynamics also offers the possibility to extend the concept of entropy to stochastic trajectories of systems in non-equilibrium states, and provides quantitative relationships that generalize the macroscopic Second Law of thermodynamics. In the case of a system described by equation (124), the variation of the total entropy is defined over a given time interval [0, τ] as\nThe first term on the right-hand side of equation (138) can be identified as a change in entropy due to the heat dissipated into the medium, defined in equation (131), while the second term is called stochastic entropy, and must be determined by solving first the corresponding Fokker-Planck equation (125) and then evaluating the solution ρ(x, t) along the specific stochastic trajectory x(t) with initial condition x(0) at t = 0. Therefore, unlike work, heat and potential energy, the stochastic entropy does not only depend on the particular realization of x(t) but on the ensemble through the quantity ρ(x, t). For example, for the linear motion (λ(t) = vt) of the position of the optical trap considered in the experiment above, the solution of the Fokker-Planck equation (125) is\nunder the condition that the system has already achieved a steady state in the comoving frame of the trap. In this case, the variation of the stochastic entropy over the time interval [0, τ] is given by k 2T\nx(τ) − vτ + γv k 2 − x(0) + γv k 2 , from which the corresponding change of the total entropy can be computed using equations (131) and (138).\nRemarkably, the change in the total entropy defined by equation (138) satisfies the integral fluctuation theorem (IFT) [306] exp − 1 k B ∆S τ = 1 (139) for all measurement times τ and any kind of protocol involving a non-equilibrium temporal variation of the control parameter λ from λ(0) to λ(τ), and possibly including non-conservative forces f 0 that break detailed balance. Using the Jensen's inequality, e a ≥ e a , equation (139) implies that ∆S τ ≥ 0, which corresponds to one of the expressions of the Second Law of thermodynamics. Therefore, the IFT represents a refinement of the Second Law of classical thermodynamics. For the experiment above, we verify the IFT for different values of τ in the inset of Figure 29(e), where the left-hand side of equation (139) is numerically determined using the computed values of ∆S τ . Depending on the specific protocol for the variation of the control parameter λ and on the state of the system (e.g., transient or steady), alternative versions of the Second Law can be derived within the context of stochastic thermodynamics. For example, when driving the system by changing the control parameter from λ(0) to λ(τ), the non-equilibrium work W τ done on the system, defined by equation (130), and the corresponding difference in free energy ∆F = F(λ(τ)) − F(λ(0)), satisfies the relation [307]\nfor any arbitrary non-equilibrium protocol λ(t), 0 ≤ t ≤ τ , provided that the system is initally in thermal equilibrium at t = 0, and all the forces acting on the system are conservative (i.e., f = 0 in equation (124)). Equation (140) is called the Jarzynski relation [307], and derives directly from the IFT (139), in which case the total entropy takes the particular form ∆S τ = (W τ −∆F)/T, where W τ − ∆F is known as dissipated work. A closely related expression, from which the Jarzynski relation (140) can alternatively be derived is the so-called Crooks fluctuation theorem [308]. It quantifies the ratio between the probability density of the work spent in the forward process, ρ(W τ ), when driving the system from thermal equilibrium at λ(0) through an arbitrary nonequilibrium protocol up to λ(τ), and the distribution of the work done in the reversed process, ρ(W τ ), where the control parameter is varied asλ(t) = λ(τ − t), starting in thermal equilibrium atλ(0) = λ(τ). The expression of the Crooks fluctuations theorem is\nThe practical importance of the relations (140) and (141) is that, unlike quasistatic experimental protocols based on classical thermodynamics, they allow to determine free energy differences between two equilibrium states from non-equilibrium measurements of the work performed at an arbitrarily fast rate, for example when studying biomolecules (section 4.1)."},{"title":"Carnot cycles and Maxwell's demons","content":"In this section, we will explore how optical tweezers can be used to get a deeper understanding of Carnot cycles, work efficiency and Maxwell engines. Although the results presented in this section have not been implemented by us in the laboratory directly, we provide sufficient instructions for the readers to be able to readily realize these experiments."},{"title":"Carnot cycles and microscopic engines","content":"In 1824 Sadi Carnot tried to answer basic questions about the efficiency, construction and operation of macroscopic heat engines [309]. This seminal work gave birth to thermodynamics, for which Carnot is considered one of its fathers. It also introduced concepts that later were used by Clausius and Kelvin to define the concept of entropy and introduce the Second Law of thermodynamics. Almost two centuries later, similar questions posed by Carnot on big, chunky, greasy, macroscopic heat engines, are finally being explored in the microscopic realm [18]. Colloquially Carnot's cycle refers to a process in which one extracts work from a system coupled to two reservoirs at different temperatures T c < T h . More precisely, given a system interacting with two thermal baths, Carnot's cycle consists of four steps:\n1. An isothermal expansion at temperature T h , during which an amount of heat Q h is absorbed by the system 2. An adiabatic expansion from T h to T c .\n3. An isothermal contraction at temperature T c , during which an amount of heat Q c is released from the system.\n4. An adiabatic contraction from T c to T h , returning the system back to its original state. The four steps of this cycle are depicted in Fig. 30(a) in the (S, T)-plane, where S refers to the entropy. For a generic fluid with volume V and pressure P, Carnot's cycle in the (V, P)-plane is shown in Fig. 30(b). During one cycle, the extracted work is given by the First Law of thermodynamics (142) and the entropy production is\nThis implies that for a reversible operation (i.e., for a process for which ∆S tot = 0) we obtain the following relationship between the heat exchange and the temperature of both baths:\nCarnot showed that, for a reversible process, the efficiency of Carnot's cycle is\nwhere the efficiency η = −W/Q h is defined as the ratio between the extracted work and the absorbed heat. Ideally, for the process to be reversible, it must be performed infinitely slowly or, more precisely in a time τ much slower than any characteristic relaxation time of the system. This automatically means that the output power, defined as P ≡ −W/τ is zero for an ideal reversible heat engine.\nIn reality, Carnot's cycle must be performed in a finite time τ, which implies a trade-off between output power and efficiency. Due to the Second Law of thermodynamics, , which imposes ∆S tot > 0, we have that η < η C .\nMoreover, practical heat engines have additioanl limitations that further reduce their efficiency [310]. For example, accounting for heat transfer yields a maximum power output given by the Novikov-Curzon-Ahlborn formula [311,312]\nwhich has been confirmed experimentally with different protocols [313]. Another closed thermodynamic cycle, the Stirling cycle, is often used to describe engines that work by compressing and expanding a working fluid at different temperatures.33 This is also composed of four steps, namely:\n1. An isothermal compression during which heat is extracted towards the cold bath T c .\n2. An isochoric heat-addition.\n3. An isothermal expansion during which the system absorbs heat from the hot bath T h ."},{"title":"An isochoric heat-removal.","content":"A sketch of Stirling's cycle can be seen in Fig. 30(c).\nIn the last decade, various research groups have probed the concept of Carnot efficiency in microscopic heat engines, particularly focusing on single-particle or colloidal heat engines. The first of these colloidal heat engines implemented a Stirling engine by using an optically trapped particle [314], as shown in Fig. 30(d). In this design, a colloidal particle plays the role of the working fluid, while the optical trap can be understood as the engine cylinder where the compression and expansion occur. The cycle includes the following phases:\n1. The isothermal compression by increasing the trap stiffness (i.e., increasing the laser output power).\n2. The isochoric head-addition by heating up the colloidal suspension from T c to T h .\n3. The isothermal expansion by decreasing the trap stiffness (i.e., decreasing the laser power)."},{"title":"The isochoric heat-removal by cooling down the colloidal suspension from T h to T c .","content":"Being a microscopic engine, the associated extracted work is a stochastic variable that depends strongly on the cycle duration τ. The first experimental realization of this microscopic Stirling cycle [314] operated between two baths at T c = 22 • C and T h = 90 • C with the maximum mean power −W/τ obtained for τ = 7 s. In these conditions, the average work extracted per cycle was about 4.5 · 10 −22 J, In 2014, a micrometer-sized equivalent of a piston steam engine was realized in an optical tweezer, where a single colloidal microparticle plays the role of the piston. The particle absorbs a small amount of light enabling to locally raise the temperature when the particle is exposed to light. At the beginning of the cycle, the gradient force of an optical tweezers pulls the particle towards the laser focus, giving rise to a dramatic increase in temperature and, consequently, 33The Stirling's cycle has the same efficiency as Carnot's cycle. the explosive vaporization of the surrounding liquid. This causes a bubble, which pushes the particle out of the focal region. As a consequence, the particle cools down and is pulled back again by the optical tweezers [315]. To achieve the sudden bubble formation, this experiment exploited a superheated liquid, which becomes unstable and relaxes back towards equilibrium via the nucleation of a vapor cavitation bubble. The mean extracted work per cycle exceeded that found in Ref. [314] by a striking six to eight orders of magnitude, because both the cycle frequency and maximum displacements were around two orders of magnitude larger.\nThese two approaches to microscopic heat engines rely on laser heating of the solution where the particle is immersed. The heat exchange rates of the heating and cooling processes effectively limit the range of accessible temperatures and the timescales of the experiments. This technical difficulty can be overcome by adding a synthetic noise to the Brownian motion [316]. This permits one to heat without having to raise or lower the physical temperature of the colloidal suspension, allowing fast heating and cooling as well as removing temperature limits, at least in principle. This method is called white-noise technique. Mathematically, the idea works as follows. Consider the underdamped one-dimensional Brownian particle:\nwhere ξ(t) corresponds to the usual thermal noise with\nwhile η(t) is, for example, an electrostatic noise such that\nConsider, in addittion, that these two noise sources are independent. This model is statistically equivalent to an effective model for stochastic thermodynamics,\nwhere ξ eff (t) can be understood as thermal noise with effective temperature34\nThe white-noise techniques allow to reach effective temperatures as high as T = 3300 K. The synthetic noise can be generated by placing the colloidal suspension in an electrophoretic fluid chamber with two electrodes connected to a computer-controlled electric generator that produces a sequence of noisy random electric fields, whose statistical properties closely resemble that of white Gaussian noise [316]. A similar effect of adding a random force applied to the particle can be achieved by modulating the position of the trap by using an acousto-optic deflector [317,318]. A remarkable step forward was achieved in a series of works where the concept of microadiabaticity was introduced [319], and later used for the creation of the first Brownian Carnot engine by concatenating isothermal and microadiabatic protocols [320]. The basic idea is as follows. Recall that, macroscopically, an adiabatic process occurs without exchange of heat (or mass) between the system and its surroundings (i.e., Q(t) = 0). For microscopic systems, the amount of heat is a random variable, so the simplest generalization is to consider those processes for which the average amount of exchanged heat is zero (i.e., Q(t) = 0). Moreover, if we further 34See also section 5.7, where we discuss feedback cooling of trapped particles in vacuum, which can also be discussed in terms of an effective temperature. assume that the process is performed quasi-statically, we also have that the average value of the variation of the total entropy is zero, ∆S tot (t) = 0, and, consequently, the process is isentropic, ∆S sys (t) = 0. In this scenario, a trick to implement a microadiabatic process consists in using protocols which keep the system's entropy constant. Importantly, this process can be quantified fairly precisely. Indeed, since the system's entropy is given by\nand assuming that the system is close to equilibrium, i.e.,\nwhere H(t) is the Hermitian of the system, we obtain that\nThus, a pseudo-adiabatic process corresponds to keeping T 2 (t)/κ(t) constant. Notice that the microscopic Carnot engine was also explored in the work in Ref. [321], but only considering the configurational space and not the phase space. [322] addressed the issue that the Second Law of thermodynamics might be violated by the inclusion of \"intelligent beings\" capable of processing information. In its original version, an ideal gas at temperature T is enclosed in an isolated container divided into two equal parts by a fixed wall. In this wall, there is a trap door operated by a \"demon\" who will open it if an incoming particle from the left (right) is faster (slower) than the average. After a while, the right compartment will have the fastest particles and, consequently, a higher temperature. Therefore, the effect of the demon is to generate a temperature gradient, which can be used to obtain work from the system, without the corresponding energy expenditure, thus violating the Second Law."},{"title":"Maxwell's demons and Szilard engines. The thought experiment known as Maxwell's demon","content":"To better understand the origin of such violation, Leo Szilard proposed his eponymous simplified model. The Szilard engine consists of a single particle within a container divided into two equal partitions separated by a movable, frictionless wall [323]. When the container is put into contact with a single thermal reservoir at temperature T, the movable wall may then be used to extract work (e.g., by lifting a weight), as the particle transfers kinetic energy in successive elastic collisions. The isothermal volume expansion results in a work extraction W ext = k B T ln(V f /V 0 ), where V 0 and V f are the initial and final volumes of the partition that contains the particle. Since the final volume V f is that of the total box, then V 0 /V f = 1/2 is the fraction of the volume that contains the particle. This implies, in turn, that the expected work extracted from the engine is simply k B T ln 2, which is proportional to the Shannon entropy measuring the uncertainty on the initial position of the particle in a symmetric partition.\nLeon Brillouin suggested that the work extraction of the Szilard engine is precisely compensated by the work of measurement and, therefore, no net extraction of work is possible [324]. After a decade, Rolf Landauer introduced a concept of logical irreversibility [325] and Charles H. Bennett explained that the work extraction is precisely compensated by the erasing process of the memory apparatus [326]. Recently, a modified version of the Second Law was derived [327,328], the Jarzynski-Sagawa-Ueda relation, which has also been confirmed experimentally [329]. This latter result claims that the work extraction is compensated by the mutual information between the system and the measurement apparatus which is involved in both the measurement and erasing processes [330]. In a relative recent experiment [331], the authors used a colloidal suspension to implement a Szilard engine to explore how the entropy decreases due to the spontaneous symmetry breaking that is the cornerstone of the Szilard engine and Landauer's principle, and to measure directly this entropic change for a Brownian particle in a bistable potential implemented by two optical traps. Very recently, a continuous versions of the Maxwell's demon, which is capable of extracting arbitrarily large amounts of work per cycle by repeated measurements of the state of a system, was proposed and was experimentally implemented in a single DNA hairpin pulling experiment [332]."},{"title":"Other applications to statistical physics and outlook","content":"In this section, we have presented some well-known applications of optical tweezers to investigate both equilibrium and non-equilibrium properties of model systems in statistical physics. Here, we briefly mention some emerging areas where optical tweezers also represent a powerful tool to study statistical quantities of microscopic systems."},{"title":"Synchronization. Coordinated motion of self-sustained oscillators mediated by means of","content":"hydrodynamic interactions is a fundamental behavior in many biological systems, such as cilia and flagella. Using optically trapped microparticles, it has been possible to elucidate and directly visualize different aspects of the hydrodynamic synchronization of oscillators and rotors. There are different approaches to define these oscillators, for example with light-propelled chiral particles [333], using angular momentum in optical tweezers to propel birefringent or asymmetric particles [46], by means of rotating energy landscapes [334][335][336] or, as we already mentioned in section 4.5.1, by means of bistable potentials sculpted with light [288]. In all these cases, when two of these self-sustained oscillators are close enough, the hydrodynamic interactions perturb and tend to synchronize their phases and their frequencies. Since all of these experiments have been realized at microscopic scale, thermal fluctuations are not negligible and escape transitions play an important role in their dynamics. Active matter. Active particle systems, such as motile microorganisms, are non-equilibrium systems capable of converting autonomously the energy from their surroundings into directed motion. In recent years, also the possibility of developing artificial active matter systems has been extensively explored [337,338]. Optical tweezers can be used to probe such phenomena and their connection to non-equilibrium relations [339]."},{"title":"Phase transitions.","content":"Optical tweezers can be used in a variety of situations to induce phase transitions and to probe critical phenomena [260]. One can, for example, use optical lattices to control colloidal phase transitions via tunable interactions [340][341][342].\nFurther advances in optical trapping could also enable the possibility to address more fundamental questions on statistical aspects of small systems, such as memory effects in non-equilibrium systems. Moreover, most of the studies discussed in this section deal with the behavior of individual particles, but it is natural to wonder how the properties of these systems change in many-body systems. Equally interesting would be to explore how microengines or active baths can be used to perform practical tasks, such as energy harvesting at the microscale."},{"title":"Transport in extended potential landscapes","content":"Microscopic particles in a fluid are constantly diffusing: the smaller the particle, the faster it diffuses, following a power law dependence D ∝ 1/a, where a is the particle radius. As D is a tiny quantity in typical conditions (of the order of µm 2 s −1 ), diffusive transport is very slow. For example, on average a 1 µm particle in water takes approximately 26 days to move by about 1 mm away from its original position. Transport can be greatly enhanced by external forces. These forces generate directed motion, where diffusion generally plays a pernicious role, since the particle will be gradually dispersed away from the path defined otherwise by the force. In Nature, several microorganisms have developed different strategies for exploring their environment in a more efficient and precise way than that of simple diffusion, for example using flagella or cilia.\nMicroscopic objects are often subject to the influence of structured force landscapes, which may either boost or inhibit their motion. For example, some molecular motors inside eukaryotic cells move along the paths defined by microtubules that exhibit a periodic structure of two proteins, acting as a periodic potential. A paradigmatic model that serves as a starting point for studying microscopic dynamics is the motion of a Brownian particle in a corrugated potential landscape under the action of an external force. Being able to study and understand these transport mechanisms is not only important from the fundamental science perspective, but it also opens up the possibility to reproduce them and develop new techniques to perform tasks, such as particle sorting and delivery in lab-on-a-chip platforms.\nAs we will see in detail in this section, colloids are ideal prototypes for the study of microscopic transport phenomena, and extended light patterns provide a powerful and versatile way to create reconfigurable potential energy landscapes in optical micromanipulation systems. We will be primarily concerned with conservative optical forces in this section, but there are also interesting light-induced transport phenomena that emerge in the presence of non-conservative forces (e.g., scattering forces), as we will briefly comment on. Table 11 presents some examples of extended light patterns used in optical manipulation: the images on the left illustrate the optical field of interest generated at the sample (i.e., at the focal plane of the microscope objective or the last lens of the optical trapping setup); the images on the right provide information about how these patterns can be experimentally generated. As explained in section 2.3, there are different ways for creating extended light patterns, but here we will focus our attention on the two most common approaches using interference or diffractive optical elements."},{"title":"Optical forces and potentials in extended light patterns","content":"A subwavelength particle is pulled by the conservative optical gradient forces towards the brightest regions of an extended light pattern. In this case, the particle moves in an extended potential energy landscape mimicking the shape of the intensity pattern, with valleys at maxima and hills at minima of intensity. In this regime, the magnitude of the forces, or the depth of the valleys, increases in proportion to the cube of the particle radius and linearly with optical power. However, for particles whose size is comparable or larger than the wavelength, the shape of the energy landscape might be remarkably different from the intensity distribution.\nImportantly, when working with extended patterns of light, one should keep in mind that the larger the area covered by the pattern, the lower the intensity for a given optical power and, therefore, the magnitude of the optical force decreases. Hence, it is important to estimate the appropriate area of the pattern so that the average intensity is at least of the order of 0.1 mW µm −2 .\nOne-dimensional potentials Let us analyze the simplest case: an extended pattern of fringes obtained from the interference of two collimated beams, as that in the first row of Table 11. At the back aperture of the microscope objective, the Fourier transform of the two beams corresponds to a pair of focused spots, whose separation (2s) determines, together with the focal length of the microscope objective ( f ), the period of the interference pattern at the sample plane: Λ = λ f 2s , where λ is the wavelength of the laser in the background medium. The corresponding intensity distribution is given by"},{"title":"Sinusoidal pattern (periodic in 1D) produced by the interference of two beams (left) and its Fourier spectrum (right).","content":"Any interferometer is suitable to produce this pattern, such as Fresnel biprisms [343] or a Mach-Zehnder interferometer [39,344]. A diffraction grating generated with a SLM can also be used to split the beam [94].\nSawtooth pattern (periodic in 1D) (left) generated by the superposition of two orthogonally polarized sinusoidal patterns (right). A three-armed Mach-Zehnder interferometer can produce two patterns of fringes, one with twice the spatial period of the other, by controlling the polarization of the input beams. The phase difference and intensity ratio determines the asymmetry [63,345]."},{"title":"Oblique lattice (periodic in 2D): Interferometric pattern of four beams (left) and its Fourier spectrum (right).","content":"These patterns can be generated using a tandem of two interferometers or using diffraction gratings designed to split the input beam into multiple beams travelling along the desired directions [344]."},{"title":"Rectangular lattice (periodic in 2D): Diffractive pattern (left) generated with a phase hologram (right).","content":"Holographic phase masks can be designed to produce the target distributions of individual light spots using iterative optimization algorithms [22]. These can be displayed on a SLM and dynamically reconfigured."},{"title":"> >","content":""},{"title":"Quasi-periodic lattice: Interferometric pattern of five equidistant beams (left) and its Fourier spectrum (right).","content":"A pentagon-shaped diffraction grating can be displayed on a phase-only SLM to split the input beam into five beams travelling at the desired angle."},{"title":"Zeroth-order Bessel beam (left) and its annular Fourier spectrum (right).","content":"Bessel beams can be obtained efficiently with a conical lens or axicon. Any nondiffracting beam can also be generated by means of a properly encoded phase mask displayed on a SLM [100,101]."},{"title":"Lattice of asymmetric traps in 2D (left) generated by the superposition of orthogonally polarized identical patterns.","content":"A light lattice generated with a phase hologram can be introduced into a polarization splitting Mach-Zehnder device to produce two identical patterns, whose relative positions and intensity ratio determine the asymmetry of the resulting pattern [66].\nSpeckle pattern (left) generated with a random phase mask (right). Speckle can be generated by passing a laser beam through a diffuser plate [89], by coupling the laser into a multimode optical fiber [65,346], or with a random phase mask displayed on a SLM [107]. where we have assumed that both interfering beams are plane waves with equal intensity, I 0 .\nIn reality, there is always an envelope function modulating the intensity, often with a Gaussian shape I 0 (x, y) = A exp[−2(x 2 + y 2 ) 2 /w 2 0 ], but as long as w 0 Λ, the plane-wave assumption works reasonably well in the central region of the envelope.\nThere are several approaches with different levels of complexity for calculating the optical force exerted by this pattern on a dielectric particle of diameter D p > λ. In all cases, the force is also periodic with the same periodicity as the intensity profile, so it can be written as\nNotice that there is a π/2-phase difference between the expressions for the intensity and that for the force, since both maxima and minima of the intensity function correspond to zeroes of the force, representing either stable or unstable equilibrium positions. The amplitude factor A is proportional to the intensity of light and, more importantly, it depends on the size of the particle and its refractive index, as well as on the period Λ of the pattern [38,39,347], i.e. A = A(D p , n p , Λ).\nTo illustrate this fact, Fig. 31(a) shows the dependence of the normalized amplitude of the force (Q = A/2I 0 ) as a function of the period, for two silica spheres (refractive index n p = 1.46) with diameters D p = 2 µm and 4 µm, respectively, immersed in water (refractive index n m = 1.33). If Λ is larger than the bead size, the particle feels a potential with equilibrium points at the intensity maxima, corresponding to negative values of Q, similar to what happens with subwavelength particles. There is a global minimum in each curve of Fig. 31(a), which represents the optimum period, slightly larger than the diameter of the sphere, that maximizes the magnitude of the force. Now, as the period is progressively reduced, the magnitude of the force decreases until it vanishes, meaning that a light pattern with such a spatial period will have no effect on the sphere. If Λ becomes even shorter, the force changes its sign (Q > 0) and the optical potential reverses, having stable equilibrium points in places where the intensity is minimum. This is schematically illustrated in Figs. 31(b-c), showing each particle at one of its respective equilibrium positions in the same intensity pattern (background), which has the period Λ = 2 µm (marked by the vertical dashed line in Fig. 31(a)). The resulting energy potentials for these particles, U(x), are also plotted, exhibiting a sign reverse with respect to one another.\nIn general, the most important aspect for determining the trapping positions in an extended pattern is the relative size of the particle with respect to the characteristic size of the pattern. In that sense, it is convenient to plot Q in terms of the ratio Λ/D p . Accordingly, Fig. 31(d) depicts the normalized force amplitude as a function of Λ/D p for spheres made of silica n p = 1.46 (solid curve) and polystyrene n p = 1.6 (dashed curve). The red and blue vertical dashed lines indicate the values of the parameter Λ/D p corresponding to the examples in Figs. 31(b) and 31(c), respectively. This kind of pattern has been widely used for optical manipulation of multiple particles and optical sorting [38,39,87,94,344,348,349].\nThe particle-dependent response to the structure of a light pattern is sometimes referred to as the size effect of the optical forces and takes place regardless of the complexity of the pattern. As the optical landscape becomes more intricate, nevertheless, not only the magnitude and the sign of the optical force may change like in our simple example, but also the shape of the potential may be drastically altered depending on the size of the particle. For example, Fig. 32 shows the case of a sawtooth pattern of fringes, also shown in the second row of Table 11. This pattern can be generated by superimposing three beams with linear polarization oriented along the horizontal, vertical and diagonal directions, so that the latter partially interferes with the former two, creating two patterns of fringes orthogonally polarized, one with twice the period of the other [63]. Given the asymmetric intensity in the fringes, a consequent asymmetry could be expected in the potential, and this is true for the smaller particle (D a = 2 µm) shown in Fig. 32. However, a larger particle (D b = 4 µm) experiences a completely different potential, with a slight . Particle-dependent potential or size effect in asymmetric patterns. Asymmetric pattern of light with period Λ = 4 µm (background) and the resulting optical potentials, U a (x) and U b (x), for two silica beads with diameters D a = 2 µm and D b = 4 µm, respectively. The shape of the optical potential for a given pattern of light can be very different depending on the size of the particle. The potential was computed using the ray optics model. asymmetry in the opposite direction. The modification of the potential can be even more radical for other particle sizes. The application of this effect for the transport of particles was analyzed in detail in Refs. [63,345]."},{"title":"Two-dimensional potentials","content":"When the periodicity of the pattern is in 2D the size effect might give rise to different behaviors along each symmetry axis, and thus the equilibrium positions for a given particle might be unpredictable without performing an explicit calculation of the optical potential. Different kinds of light lattices in 2D have been used to study optical binding and colloidal crystallization [54,55], to explore structural phase transitions in colloidal monolayers [340,350,351], to tailor bandgap materials [352], and to optically sort particles in combination with microfluidics [344]. Periodic light lattices in 2D can be produced by the interference of at least three beams. For example, the coherent superposition of three waves, depending on the directions of their wave vectors, can lead to centered-rectangular or hexagonal lattices, whereas the interference of four waves can produce square, oblique or rectangular lattices. The example in the third row of Table 11 corresponds to an oblique or rhombic lattice. An alternative approach to generate 2D-periodic patterns is by means of phase holograms [41], such as the rectangular array of individual spots illustrated in the fourth row of Table 11. In contrast with the multiple interference fields, diffractive patterns can act as a collection of individual traps for 3D confinement, overcoming the size effect and hence facilitating the control of the equilibrium positions of individual particles. A SLM can be used as well to display diffractive gratings and produce multiple interfering beams [94], whilst the 2D-trapping capabilities can be extended to manipulate ordered structures in 3D by taking advantage of the self-imaging Talbot effect [353]. Another interesting case is that of a quasi-periodic pattern arising from the superposition of five waves, which is shown in the fifth row of Table 11. This has been used to study the formation of quasi crystalline structures in colloidal monolayers [354,355].\nA type of extended patterns with circular, elliptical or parabolic symmetry in the transverse plane correspond to Bessel, Mathieu and parabolic nondiffracting beams, respectively [95,96,98]. Their Fourier spectrum at the back aperture of the microscope objective lies on a thin annulus, where the particular beam geometry is determined only by the intensity and phase modulation around the annulus. The optical forces in a Bessel beam, like the pattern depicted in the sixth row of Table 11, exhibit very interesting features due to the size effect [356][357][358], and similar attributes can be expected in the other types of beams. Namely, the multi-ringed structure of the intensity pattern has an envelope that decreases radially outwards. Depending on its size with respect to the pattern, a particle might respond predominantly to the fine structure or to the envelope. This can be used as a passive mechanism to sort particles [359] or to build what is known as a washboard potential [360].\nAnother version of an asymmetric periodic pattern, but now in 2D, is illustrated in the seventh row of Table 11. This is generated by superimposing two identical light lattices, orthogonally polarized, produced with a single phase hologram [66]. By controlling the relative position and the intensity ratio between the two patterns, the asymmetry of the light lattice can be tailored. In general, periodic and asymmetric patterns are very important in the study of the ratchet effect [60,62,63,66], which is the emergence of directed transport in the presence of unbiased external forces due to a spatiotemporal symmetry breaking and, furthermore, is thought to be at the core of the operation mechanism of some biological engines [361].\nRandom light patterns like speckle (bottom row of Table 11) are also very important in the context of particle diffusion and transport. In practice, speckle can be generated by transmitting a laser beam through a glass diffuser, by coupling the laser into a multimode optical fiber, or by encoding a random phase mask on a SLM. A very interesting aspect of the interaction of Brownian particles with a random potential landscape is its major role in diffusion, and how it depends on the several parameters of the potential [65,89,346,362]."},{"title":"Transport mechanisms","content":"So far, we have examined optical forces and potentials in a static situation. Now we will examine the conditions for inducing directed transport in the system, i.e., a net current of particles in a given direction through their interaction with light and, possibly, other external forces. To address this subject, we will start by looking at the general equation of motion for a single particle interacting with an optical potential in the overdamped regime, which is a good description of the micromanipulation systems under consideration. For the sake of simplicity, only one dimension will be examined, this is:\nwhere γ represents an effective drag coefficient, U(x, t) is the optical potential, F(t) denotes an external force, and ξ(t) is the thermal noise, having a correlation function ξ(t)ξ(t ) = 2Dδ(t −t ), with D = k B T/γ being the diffusion coefficient. Although noise usually plays an important role in microscopic-scale dynamics and some transport mechanisms even rely on it, this can be neglected under certain conditions, giving rise to a deterministic dynamics, ξ(t) ≈ 0. For example, Brownian motion of silica or latex spheres immersed in water at room temperature is very subtle if the diameter is D 5 µm, and it becomes practically negligible for D 10 µm.\nIn general terms, we can distinguish between two main transport mechanisms: a dynamic potential and a static potential in the presence of an external driving force. A very simple realization of the former case corresponds to a light pattern of fringes that moves with a constant speed v with respect to the static sample. This corresponds to a travelling periodic potential: U(x, t) = U(x − vt). The dynamics of the particle can be analyzed in a reference frame that moves along with the potential by making the change of variable u = x − vt in equation (157), leading to\nNotice that this is equivalent to adding a constant drag force to a static pattern and thus the following discussion will be valid for the case of a constant external driving force. Whether the particle will follow the pattern or not depends on the relative magnitude between the optical force, F op (u) = −dU/du, and the drag force, F d = −γv, in the moving reference frame. The constant force has the effect of tilting the potential energy landscape, giving rise to what is sometimes referred to as a washboard potential. If |F op | > |F d |, the particle will not be able to escape from the well and, therefore, it will be locked-in to the pattern, moving with the same velocity as the pattern with respect to a fixed reference frame. A particle with this behavior has been called a Brownian surfer [363]. If |F op | < |F d |, the particle will be pulled out from the potential well by the drag force in the moving reference frame or, in the fixed frame, it will not follow the pattern but it will diffuse around its initial position. This is known as a Brownian swimmer [363]. Depending on the thermal noise level, there might be intermediate situations, in which the particle moves in the same direction as the pattern but with a reduced speed v p < v. This happens when |F op | |F d | or, in terms of the potential energy barrier between neighboring wells, when ∆U k B T, so that thermal energy may sometimes help the particle to surmount the potential barrier between consecutive wells. Due to the size effect discussed above, in a polydisperse mixture of particles, some of them may follow the pattern while others may not, depending on their size and refractive index, or even if two types of particles move along with the pattern, their average speeds may be different. Therefore, a moving potential can be used as a mechanism to separate particles of different characteristics, i.e. as an optical sorting device.\nIn practice, there are several ways to move an interference pattern of fringes. Let us consider, for example, an optical micromanipulation setup with a Mach-Zehnder interferometer. We can express the resulting interference pattern at the sample plane as I(x, y) = 4I 0 cos 2 (πx/Λ + ϕ(t)/2), where ϕ(t) represents a time-dependent phase difference introduced in one arm of the interferometer. To generate a travelling potential, ϕ(t) should be a linear function of time. An elegant way to achieve this is by using the angular Doppler effect [364,365], arising when a circularly polarized beam passes through a rotating λ/2-plate, which gives rise to a frequency shift ∆ω = ±2Ω, where Ω is the rotation speed of the λ/2-plate. This can be easily implemented in the Mach-Zehnder interferometer [348], as illustrated in Fig. 33(a). A polarizing beam splitter (PBS) is followed by a λ/4-plate in each arm of the interferometer, producing orthogonal circular polarization states. The rotating λ/2-plate is inserted in one arm, such that it inverts the handedness of the circular polarization at the same time that it introduces an angular Doppler frequency shift. A non-polarizing beam splitter is placed at the output of the interferometer to superimpose the beams. The resulting phase difference between the two beams is ϕ(t) = (∆ω)t = ±2Ωt, where the sign depends on the rotation direction of the λ/2-plate with respect to the polarization handedness [365]. At the sample plane, this will define the direction of motion of the fringes (±x), whereas the speed of motion is v = Ω/k.\nThere are other alternatives for producing motion of the fringes. For example, a very simple option, which can be easily implemented in different types of interferometers, is to introduce a thin glass plate in the path of one of the interfering beams [343]. As illustrated in Fig. 33(b) for the particular case of a Mach-Zehnder interferometer, by gradually tilting the glass plate a small angle θ(t), the optical path length of the corresponding beam is modified, resulting in a relative phase shift between the two beams and the consequent motion of the pattern. A straightforward analysis using geometrical optics yields that, if θ 1, then the change in the optical path length is given by ∆(θ) ≈ T g /2(1 − 1/n) 2 θ 2 , where T g is the thickness of the glass plate and n its refractive index. Therefore, as ∆(θ) is not linear, the phase shift is not linear either: ϕ(t) = k∆(θ(t)). Although this complicates the theoretical description of the system, it is a cheap and accessible choice if the aim is just to make a demonstration of particle transport.\nAnother configuration that provides full control of the phase shift ϕ(t) makes use of a displacement of one of the mirrors in a Mach-Zehnder interferometer. For example, in the first interferometric optical tweezers setup [87], depicted in Fig. 33(c), an extra beam splitter is placed in one arm of the interferometer (BS1), so that mirror M1 retroreflects the beam and, mounted on a motorized translation stage, it is moved with a constant speed: D(t) = vt. This produces a phase shift between the interfering beams of the form ϕ(t) = 2k D(t), leading to a continuous motion of the fringes at the same speed as the mirror (v), while keeping the spatial period of the fringes unaltered. Indeed, the period of the fringes is a function of the lateral distance between the output beams and the focal length of lens L1 (see Fig. 33(c)). Notice, however, that the presence of the beam splitter BS1 yields an additional intensity loss. To equalize the intensity in both arms of the interferometer, two λ/2-plates should be used, one at the entrance of the setup in combination with a polarizing beam splitter (PBS) to adjust the amount of light in each arm, and the other one to set the same polarization in the two beams.\nA variant of this idea, which avoids the extra beam splitter, is shown in Fig. 33(d). A periodic and asymmetric displacement of the mirror M1 might be equivalent to a continuous travelling pattern under certain conditions. This can be done by mounting the mirror on a piezo-stage35 driven with a sawtooth function [39]:\nwhere τ δt, v 1 = D max /τ and v 2 = D max /δt, with D max denoting the maximum displacement of the mirror, which should be small to prevent distortion in the period of the fringes. The frequency of D(t) is f = 1/T, with T = τ + δt the period of the function: D(t + T) = D(t). The motion of the mirror is related to the phase shift of the pattern as ϕ(t) = 2 √ 2k D(t), implying that the fringes at the sample mimic the sawtooth-like displacement, with speeds given by\nThe main idea is that the particle is able to follow the pattern in the direction of the slower motion (u 1 ), but not during the fast motion (u 2 ) in the opposite direction, resulting in a unidirectional transport. Of course, the selection of the parameters36 D max , f , and τ, along with the spatial period of the fringes Λ, is crucial to optimize the current of particles. For example, if D max = mλ/(2"},{"title":"√","content":"2), with m a positive integer, the particle will advance a distance mΛ in one direction and will end up at another minimum of the potential after an entire cycle, provided that Λ is such that optimizes the optical force for the given particle size. Figs. 34(a-b) illustrate the resulting particle current for two sizes of particles as a function of Λ and Ω = 2π f , respectively. An optimum can be clearly identified in each case. This device was used as an optical sieve to separate particles according to their size and it can also be used to sort particles according to their refractive indices [39]. In the latter case, the strategy is to start with a low intensity and gradually increase the optical power up to a value for which only the particles with the highest refractive index contrast with respect to the medium will respond.\nA related approach consists in generating phase gratings with a SLM aimed at splitting an incident beam to produce interference. The advantage is that the angular separation between the beams can be varied by changing the grating, so that the spatial period of the fringes can be tuned and its orientation can be modified as well. Furthermore, the motion of the pattern can be also controlled with the SLM, by adding a continuous phase shift to one of the beams. Such a device has been used to move and sort living objects and particles of different shapes [94].\nIt is worth mentioning that the techniques discussed above can be extended to more complex interference patterns, such as rotating beams, allowing circular motion of beads and structures [?,366]. Similar ideas have been applied as well to build optical conveyors that allow bidirectional transport in 3D. These kinds of devices use coaxial beams, either counter-propagating to generate axial standing waves or co-propagating, and not just Gaussian beams but also other structured fields, like Laguerre-Gaussian and Bessel beams [38,357,[367][368][369], including evanescent fields [349]. The operation principle is basically the same as before: to manage the motion of an interference field by adding a controlled phase shift between the beams.\n35Any programmable motorized translation stage would be useful and, for example, stepper motors are cheaper, at the cost of loosing precision and speed in the motion control.\n36There are alternative choices, but in any case, the dynamic phase ϕ(t) is defined by a set of three independent parameters, set through the motion function of the mirror. frequency Ω of the sawtooth function for the optimum spatial period in each case and fixed power P = 130 mW. The two particle kinds feature different optimal currents as a function of Λ and Ω, enabling to classify them with high throughput. Data from Ref. [39].\nThere have also been different approaches to yield moving patterns based on digital holography. Among the first schemes, we can mention a mechanism called optical peristalsis, in which a lattice of optical traps is created and controlled with a set of phase holograms displayed on a SLM [61]. Following a periodic sequence of three states, the pattern is consecutively displaced by one-third of the lattice constant, reaching the initial state again after an entire cycle. A particle trapped in a given potential well is released when that pattern is turned off and it is immediately attracted towards the nearest shifted well once the next-state pattern is turned on. By repeating this process, the particle is transferred from one well to the neighboring trap after each cycle, experiencing a constant motion in one direction, resembling the operation of a peristaltic pump. The geometry of the lattice can be varied to achieve, for example, a radial current in circular patterns. If the particle's diffusion during the time between consecutive states is negligible, the process is deterministic and the current is given by v = L/T, where L and T are the lattice constant and the temporal period of the sequence, respectively."},{"title":"Other transport mechanisms and applications","content":"As we have already mentioned, a travelling potential is dynamically equivalent to having a particle in a fixed light pattern subject to a constant drag or, more generally, a driving force. The drag force can be provided, for example, by tilting the sample [347], by moving the sample with respect to the pattern [63,370], or by means of a controlled microfluidic flow. The latter option has the important advantage of being suitable to integrate into lab-on-chip platforms [76,371]. In addition, sorting capabilities can be boosted by using 2D light lattices instead of 1D-periodic patterns, offering new control parameters and thus more versatility. Indeed, biological or colloidal assays flowing through two-dimensional patterns may be deflected from the body of the flowing fluid depending on the orientation of the lattice with respect to the flow and on the strength of the optical force, giving rise to discrete angle deviations or fractionation [344,372,373]. Using this approach, a high throughput of sorted samples can be practically and efficiently achieved. Also, microfluidic techniques can be combined with other schemes of optomechanical control, like dynamic or blinking potentials [374].\nA fundamentally different and intriguing mechanism to selectively move objects at the microscopic scale is based on the ratchet effect. In contrast to previous examples where there exist a biased external field to induce a net current of particles, the ratchet effect makes use of non-biased time-varying forces to induce transport. Since these forces are of zero-mean in time and in space, a non-trivial symmetry has to be broken to observe a net current [361]. The light patterns represented in the second and seventh rows of Table 11 are examples of ratchet potentials with a space asymmetry, yet the symmetry breaking may be much more subtle, including variations in time. Actually, the spatial structure and time-modulation of the potential along with the time-dependence of the external force define many different types of ratchets [60,62,63,66,[375][376][377][378]. Moreover, although most of these systems are noise-assisted when particle diffusion is non-negligible, there are also deterministic ratchets [63]. The rich dynamics arising in these systems is due to the delicate interplay among the whole set of encompassed parameters and manifests itself in the diverse phenomena that can be observed, including bidirectional transport depending on size, currents in different directions with respect to the driving, current reversals and chaotic behavior. Interestingly, the ratchet mechanism is ubiquitous, as it appears in many biological, physical and chemical systems, and its study has paved the way to broaden our understanding of transport processes at the micro-and nanoscales [361].\nSo far we have looked mainly at ordered patterns or light lattices, but random patterns have also shown an excellent selectivity for sorting particles [65,89], with the additional advantage of making feasible to sort and select a continuous throughput of chosen constituents in turbid media, common in many biological assays.\nAlthough we did not deal with scattering forces in our previous discussion, it is important to remind that the use of scattering rather than gradient forces was really the first means to transport and separate particles. Indeed, since his earliest experiments in 1970, Ashkin noted that radiation pressure acting on particles of different sizes along the propagation axis of a moderately focused beam was proportional to the size of the particles, so that the beads were guided along the beam axis with different velocities [1]. This idea is the basis of the so-called optical chromatography, in which a Gaussian light beam opposes a fluid flow, causing that particles of various sizes and/or refractive indices find different equilibrium positions along the beam axis where the scattering force is balanced by the force due to the fluid flow [379,380]. Other schemes combine the action of scattering and gradient forces. Consider, for example, the interference pattern of fringes discussed above. In addition to the transverse gradient force, there is usually a scattering force along the propagation direction, which may push the particles selectively according to their relative size with respect to the fringes, so that small particles, attracted to the bright fringes, will be pushed much stronger than larger particles attracted to the dark fringes. Using this interplay between gradient forces and scattering forces, it is possible to activate transport with an all-optical static configuration [381]. These kinds of ideas have been extended further to include the use of evanescent waves generated on the surface of prisms (Kretschmann configuration) or with total internal reflection (TIR) microscope objectives [49,382,383]. An advantage is that the use of gradient and scattering forces arising from evanescent waves in near-field photonic configurations allows one to scale down to tens of nanometers the size of the manipulated objects [49,384].\nAnother puzzling transport mechanism appears in the so-called tractor beam. This is a counterintuitive effect, in which particles move opposite to the direction of the beam propagation due to a pulling scattering force [385]. In the case of a highly nonparaxial wave field, whose angular spectrum components subtend large angles with respect to the main propagation axis, part of the diffracted radiation may indeed have larger wave vector components along the propagation axis than those of the incident field, giving rise to strong forward scattering. The net effect might be a negative radiation pressure (pulling force), depending on the shape of the object, its size, and its interaction with the wave field. It is important to stress though, that this effect is not based on intensity variations and it is not derived from a potential, in contrast with the conservative gradient force acting in conventional optical tweezers. Remarkably, this phenomenon can happen also in rotational dynamics, where the spin angular momentum of light can exert transverse non-conservative forces in the opposite direction than expected in a chiral object [386,387].\nThe simplest description of the dynamical mechanisms previously discussed considers diluted samples, and this is usually a good approximation even if the sample is not well diluted and some interactions exist. When the interactions are strong, a more complete description becomes necessary to fully understand the phenomenology. The origin of the interactions can be manifold, like hydrodynamic, electrical or optical (optical binding), and they may promote collective behaviors such as dynamical ordering, phase transitions or cluster formation [94,342,385,[388][389][390][391].\nUnderstanding and controlling transport from the mesoscale down to the nanoscale is still a key topic in biophysics, nanomedicine and nanotechnology. There is still plenty of room for basic and applied research on the many aspects involved in transport of microscopic objects, including studies on self-propelled particles [337]. Especially in biological environments, the role of light-induced forces has been barely studied so far. On the other hand, many of the ideas and techniques discussed here are being extended to other realms where force fields can be used for contactless manipulation, either light, like in plasmonic [13,16,[392][393][394][395], optoelectronic [396] and some recent thermophoretic devices [397,398], or without light, like in acoustic manipulation [169,[399][400][401]. This range of techniques has broaden the trapping conditions and the accessible range of particle sizes and force magnitudes, going down to femtonewtons for nanoparticle maneuvering and up to micronewtons to trap millimiter-sized beads, and possibly beyond in both directions in the near future. Also, there is a great effort invested in turning the basic concepts into real applications, specially in the development of lab-on-chip platforms, where many modern areas can converge to perform practical tasks, including optomechanics, microfluidics, acoustofluidics, nanoplasmonics, thermoplasmonics, optoelectronics, near-field photonics, spectroscopy, to name but a few. A more extended discussion on light-induced mechanisms of controlled transport can be found in a very recent and comprehensive review [402]."},{"title":"Optical tweezers in vacuum","content":"Even though Ashkin conducted his pioneering experiments on optical forces and radiation pressure in a gas or vacuum environment [1,[403][404][405], for many decades optical forces have been primarily harnessed to conduct optical tweezers experiments in liquids or for trapping and cooling of atoms [5]. Only recently there has been a renewed interest to use optical tweezers in vacuum. The fist single-beam gradient-force optical trap in air was demonstrated in Ref. [406] 10 years after its realization in water [2] and it took another 15 years to demonstrate single beam optical levitation in high vacuum [407].\nThis section will introduce the basic concepts for realizing experiments with optically levitated particles in high vacuum. We will discuss the implementation with a single beam optical trap, even though other geometries for optical trapping such as parabolic mirrors [408,409] or counter-propagating beams [410,411] have also been realized. Specifically, we will focus on the practical aspects such as particle loading, sources of noise, feedback stabilization, calibration, and provide the literature references that will allow the reader to independently implement optical levitation experiments in high vacuum. We begin in section 5.1 with a concise review over the applications of levitated particles in high vacuum."},{"title":"Applications of optical levitation","content":"In vacuum the motion of an optically levitated dielectric nanosphere is well isolated from the environment. This makes it a promising candidate to demonstrate quantum behavior of a macroscopic object at room temperature [412][413][414]. To observe quantum effects, the motional energy of the levitated particle has to be reduced by more than seven orders of magnitude to the level of a single quantum excitation. Ongoing efforts into this direction have already reached impressive cooling rates using feedback cooling [407,[415][416][417][418] and cavity cooling schemes [419][420][421]. While the ultimate cooling limit, i.e., the quantum ground state, has already been reached in a range of nano and micromechanical systems [422,423], ground state cooling of a levitated37 nanoparticle has only be claimed recently [424]. The quantum mechanical ground state is an important milestone towards preparation of these massive objects in non-classical quantum states that are otherwise only known from elementary particles [425], atoms, ions [426], and molecules [427,428].\nInvestigating quantum mechanical states of motion of massive objects allows for experimental tests of fundamental physics in unexplored parameter regimes [429][430][431][432][433][434]. Preparation of nontrivial quantum mechanical states requires either a strong optomechanical interaction [435][436][437] to map non-classical states of light onto mechanical motion [438], a projective quantum measurement such as photon counting [439][440][441], or a quantum nonlinear interaction, which can be provided for example by a defect inside the levitated particle such as a nitrogen-vacancy (NV) center in diamond [442][443][444]. However, the necessary coupling strengths have yet to be demonstrated experimentally and optical levitation experiments with nanodiamonds have been limited by optical absorption [445][446][447][448][449].\nUltimately, levitated particles in the quantum regime might benefit from their simple mechanical mode structure. Conventional mechanical oscillators have a series of normal vibrational modes. Typically, only one mode is considered and the remaining modes are neglected. However, in measurement-based approaches to quantum state preparation [450,451], the displacement of the resonator is measured with a light pulse that is short compared to the oscillation frequency. The light pulse measures over a large bandwidth and thus measures not only the mode of interest but it also measures the displacement due to all the other modes that fall within the bandwidth. As a consequence, these so-called spectator modes add noise, which ultimately limits the efficiency at which the mode of interest can be measured and controlled [452]. Inside nanoparticles, the frequencies of bulk modes are several orders of magnitude higher than those of the center-ofmass motion. As a consequence, one usually only observes three modes corresponding to the center-of-mass motion in each dimension when working with spherical particles. With less symmetrical particles additional torsional [453] and precession [454] modes have been observed.\nThe extreme isolation from the environment also means that the mechanical quality factor (Q-factor) is very large and, hence, the thermo-mechanical noise of these nanomechanical resonators is very low. For this reason, levitated particles can be used as nanomechanical sensors for force [455][456][457], torque [453], and acceleration [458]. Applications of these sensors are envisioned to detect exotic forces [459,460], such as non-Newtonian gravity [461], Casimir forces [462][463][464] and torques [465], and gravitational waves [466]. Thanks to the high mechanical Q-factor and tunability of the system parameters, levitated particles are also ideally suited to study nonlinear nanomechanical phenomena [467][468][469], which are important for the development of future nano and micromechanical devices; in fact, levitated particles are the first systems where nonlinear effects have been witnessed already at the level of thermal fluctuations [470].\nOptically levitated particles have unique properties not available in traditional nanomechanical resonators. For example, the mechanical properties of a levitated particle can be changed rapidly in situ by simple adjustments in the trapping light. This allows to perform free-fall experiments, which provide new opportunities in force sensing [461,471] and quantum physics [472]. Unlike traditional clamped mechanical resonators, a levitated particle can rotate freely [473][474][475][476]. The rotation can be accelerated to GHz frequencies, faster than any other object [477,478], simply by changing the trapping laser polarization from linear to circular. At these fast rotations, the centrifugal stress is close to the ultimate tensile strength of the material. Hence, levitated particles can be utilized to test material limits on the nanoscale or to study friction at a fundamental level [463,479].\n37To distinguish trapping in liquid from trapping in a dilute environment, we refer to the latter as optical levitation.\nThe high control over the system parameters, in particular the ability to tune the interaction with the environment via the gas pressure, make levitated nanoparticles also an excellent testbed for studies of single particle thermodynamics [19], a field also known as stochastic thermodynamics [164]. As we have seen in section 4.5.3, many studies in this field have already been conducted in the overdamped regime with optically trapped particles in a liquid medium. In contrast, levitated particles can also operate in the underdamped regime and have enabled the first observation of ballistic Brownian motion [410], the first quantitative measurement of the Kramers turnover [480], experimental tests of fluctuation theorems [481,482], and fundamental tests of thermodynamic laws [483]."},{"title":"Dynamics in the underdamped regime","content":"When transitioning from a viscous medium like water to a dilute gas or vacuum environment, the dynamics of a particle in an optical trap changes fundamentally because of the particle's finite inertia. Including the inertial term, the equation for the center-of-mass motion along a single spatial direction is (equation (8))\nwhere Ω 0 = κ/m is the resonance frequency, κ is the stiffness of the optical trap, and m is the mass of the particle. Under most experimental conditions, the fluctuating force F L and damping rate38 γ 0 are due to collisions between the particle and residual gas molecules. We will come back to the stochastic forces and damping contributions in section 5.5. F ext summarizes additional external forces, excluding the optical forces that generate the optical trap, which correspond to the last term on the left hand side. These can be used for example to drive the particle or to apply feedback cooling, as will be discussed in greater detail in section 5.7. The Duffing term ξ ∼ 1/w 2 0 , where w 0 is the width of the optical trap, accounts for deviations from a parabolic trap [470]. The optical potential term also contains nonlinear terms that couple the three orthogonal spatial modes nonlinearly [468,470]. However, these terms become only relevant for sufficiently large oscillation amplitudes and can in most situations be neglected, in particular when feedback cooling is applied. This justifies the treatment of the three-dimensional motion of the trapped particle as three independent harmonic oscillators. Note that, while the modes are uncoupled in a static trap, the two modes that are transverse to the propagation direction can be coupled by modulating the polarization of the trapping light in time. The coupling allows to coherently exchange energy between the modes, since this coupling can be turned on and off on demand [484]. Overall, this platform is very versatile because all system parameters can be controlled to a large degree in the experiment.\nIn the underdamped regime, which is defined by the condition γ 0 Ω 0 , the particle undergoes ∼ Q = Ω 0 /γ 0 phase coherent oscillations. This becomes very clear in the PSD\nwhich in the underdamped regime is strongly peaked around the oscillation frequency Ω 0 . Here, χ(ω) = 1/ Ω 2 0 − ω 2 + iωγ 0 ) is the response function of a harmonic oscillator and we assumed thermal noise with a PSD S ff = γ 0 mk B T/π. As we have already seen in section 3.6, in the overdamped regime (Q 1), equation (161) can be approximated by a Lorentzian distribution\n38Note that here we define γ 0 as a rate with units of Hz, in contrast to γ = γ 0 m in equation (8), which has units of [Hz kg]. Fig. 35. From overdamped to underdamped. We plot the measured PSD for the z-axis motion, normalized to the damping rate γ 0 /(2π), at three different pressures for a levitated particle (radius a = 68 nm, laser power P ≈ 150 mW) that is overdamped (blue line), critically damped (orange line), and underdamped (green line). The dashed black lines are least-square fits to equation (162) (1000 mbar) or equation (161) (60 and 2.5 mbar). The colored vertical solid lines indicate the roll-off frequencies Ω c /2π. For the time traces at 60 and 2.5 mbar, the spectra contain leakage signals from the other oscillation axes above 100 kHz. Data from Ref. [485]. Fig. 35 shows experimental data that clearly illustrate the transition from the overdamped to the underdamped regime. In the experiment, the damping γ 0 is changed through the pressure inside the vacuum chamber. At atmospheric pressures, the gas damps the particle motion (γ 0 2Ω 0 , blue) and we observe the spectrum familiar from optical tweezers experiments in liquids (section 3.6), with the maximum at zero frequency. At a pressure of ∼ 60 mbar, the particle is critically damped (γ 0 = 2Ω 0 , orange). At even lower pressures a strong peak evolves at Ω 0 , which is the signature of coherent oscillations. The vertical lines indicate the cut-off frequency Ω c = Ω 2 0 /γ 0 ."},{"title":"Optical tweezers setup for vacuum operation","content":"A standard setup for optical levitation in vacuum consists of a vacuum system that allows to control the pressure and thereby the friction the particle experiences, optics that strongly focus a laser beam for trapping, optical detection for real-time readout of the particle motion, and feedback control to stabilize the particle in high vacuum. An example of a nanoparticle optically trapped in a vacuum chamber is shown in Fig. 36. Operating an optical tweezers in vacuum introduces a range of significant changes to the experimental apparatus and also imposes limits to the type of particles that can be trapped. Naturally, there are many ways in which such a setup can be constructed. Here, we follow largely the description given by Hebestreit [485], a schematic picture of whose setup is shown in Fig. 37."},{"title":"Particle types","content":"One of the great benefits of operating in vacuum is that the particle is very well isolated from the environment. However, this also means that the particle doesn't thermalize with the environment. As a consequence, optical absorption of the trapping laser significantly heats up the internal (bulk) temperature of the particle (section 5.6.5). This limits vacuum trapping to particles that absorb very little of the trapping laser (e.g., silicon and silica) and to wavelengths where optical absorption is minimized (e.g., infrared, the most popular choices being 1064 nm and 1550 nm). There has also been great interest in trapping nanodiamonds, since these can host interesting optical defects such as NV-centers [487]; however, trapping in high vacuum has not been reported yet due to excessive optical absorption from impurities and defects in the diamond [447]. A liquid medium that surrounds the particle not only mitigates heating from optical absorption, it also keeps the particle afloat. Without it, gravity pulls the particles down to the bottom of the chamber, which complicates the loading of individual particles into the trap (section 5.4).\nIn addition, without a high-refractive-index medium such as water (n m = 1.33), the refractive index contrast is much larger. Consequently, the scattering force is more prominent in vacuum since the ratio of the scattering force to the gradient force scales as [488,489]\nwhere ∆ = (n 2 p − n 2 m )/(n 2 p + 2n 2 m ) is proportional to the relative index contrast between the particle (n p ) and the surrounding medium (n m ). Therefore, trapping of spherical particles in vacuum with a single beam optical tweezers is limited to radii a ranging from ∼ 30 nm to ∼ 150 nm [488]. However, this is not a fundamental limitation, since the scattering force can be canceled, e.g., with counter-propagating beams [410,490].\nAlso non-spherical particles (e.g., dumbells [477], clusters [454], and rods [475]) have been successfully trapped in vacuum. They have quite different scattering properties and therefore experience different optical forces and torques than spherical particles."},{"title":"Optical trap","content":"The most popular choice for the trapping laser is infrared light at 1064 nm or at 1550 nm. In addition, the laser should have a small relative intensity noise (RIN), low pointing instability, and provide a power at the laser focus on the order of P opt = 70 mW, which requires a laser output power of ≈ 0.5 W. Possible back-reflections into the laser can destabilize the laser output. This can be avoided by introducing a Faraday isolator right after the laser output. A low RIN is important because noise in the intensity at twice the particle oscillation frequency excites its center-of-mass motion [491][492][493] and causes fluctuations of the oscillation frequency, since the particle frequency scales with Ω 0 ∝ P opt .\nTo parametrically drive the particle motion, the laser intensity has to be modulated at twice the oscillation frequency, i.e., at up to 400 kHz (section 5.7). Electro-optic modulators (EOMs) or acousto-optic modulators (AOMs) provide the necessary modulation bandwidth. While EOMs have a higher transmission (up to 95%, in contrast to maximally 80% for AOMs) and a higher bandwidth, they feature a much smaller contrast ratio than AOMs [471].\nThe trapping laser can also be used to detect the particle motion even during parametric control since the parametric modulation is at a twice the motional frequency and can therefore be filtered out spectrally (section 5.7.2). However, to exclude any cross-talk, it is convenient (albeit not necessary) to have an independent measurement of the particle with a constant laser. Interferences should be avoided, since they can lead to strong fluctuations of the optical trap. This can be achieved with a single laser source that is divided into a high-power branch for trapping and a low-power branch for detection. A polarizing beam splitter (PBS) in combination with a half-wave plate provides a convenient way to cross-polarize and to adjust the power ratio between the two branches. In addition, one branch is frequency-shifted with an AOM, to avoid interference. A second PBS re-combines the two branches, after which they enter the microscope objective colinearly.\nA stable three-dimensional trap with a single optical beam requires strong focusing. This can be achieved with a high-NA objective (NA ≥ 0.8). Note that optical absorption in the objective can lead to small shifts of the focus [485]. This can heat the particle motion for large laser intensity modulation, for example in free-fall experiments [471]. Aspheric lenses provide a cheap alternative to more costly microscope objectives. In addition, they usually have a higher transmission and are available with anti-reflection coatings for a large range of wavelengths. This reduces the power requirements and absorption losses. However, they are more sensitive to misalignment than microscope objectives, which makes optical trapping with aspheric lenses much more challenging [485].\nBoth AOMs and EOMs distort the laser beam as it passes through. To recover a nice Gaussian beam shape, the laser mode profile is cleaned with a spatial filter before it enters the microscope objective. This increases the quality of the focus and, hence, the trap stiffness. Spatial filtering is achieved either with a pinhole of appropriate size, or more conveniently with a single-mode fiber, which has the advantage of mechanically decoupling different sections of the setup. A polarization-maintaining single-mode fiber maintains the input polarization at the output, while a single-mode fiber with a fiber-based polarization controller allows to create an arbitrary output polarization.\nA second microscope objective or aspheric lens inside the vacuum chamber collects and re-collimates the trapping laser and the measurement beam. After the vacuum chamber, the two beams are separated again with another PBS. The trapping beam is dumped, while the probe beam is sent towards the detection setup. Efficient separation of the beams is paramount to avoid detecting the intensity modulation introduced by the EOM, that would then be deleteriously fed into the system via the feedback loop. In fact, mixed polarization states appear in this side of the set-up due to the tight focusing and especially to the mechanical stresses present in the glass windows of the vacuum chamber. To undo these changes in the polarization state, one introduces a half waveplate and a quarter waveplate after the vacuum chamber and minimizes the unwanted trapping light after the PBS while maximizing the probe light, which then goes to the detection setup."},{"title":"Detection","content":"In liquids, momentum relaxation occurs within a fraction of a microsecond due to the high viscosity of the surrounding medium. This is faster than what most experiments are able to resolve [494]. In contrast, the viscosity of a dilute gas is orders of magnitude smaller. Hence, inertial effects on the dynamics of a particle in an optical tweezers are easily accessible [410]. While optical tweezers experiments in liquids are mostly concerned with the diffusive motion at low frequencies on the order of the corner frequency ω c /(2π) = k /(mγ 0 ) ∼ kHz, the frequency range of interest in experiments under vacuum is centered around the natural oscillation frequency Ω 0 /(2π) = κ/m ∼ 100kHz, where κ is the trap stiffness and γ 0 the damping coefficient.\nAs in the case of traditional optical tweezers, the particle motion is measured optically by extracting information contained in the field scattered by the particle. In single-beam optical tweezers, the trapped particle are sub-wavelength scatterers. As such, most of the information about the particle position is contained in the phase of the scattered field. This requires an interferometric measurement to read out the phase.\nForward detection provides a convenient way to access the phase, since the non-scattered transmitted light provides a stable reference field without requiring additional optical path stabilization [495]. The signal-to-noise ratio of the detector signal is maximized by increasing the detection efficiency of the scattered photons and by increasing the optical power of the detection beam. Maximum power is provided by using the trapping beam itself, at the expense of introducing cross-talk as mentioned before. Hence, the best signal-to-noise ratio is obtained with a high-NA collection lens and by using all the scattered light to generate the detector signals. In forward scattering, the collected power from the trapping beam amounts to ≈ 50 mW. Hence, optimal detection in forward scattering requires a quadrant photodetector that can handle at least 50 mW optical power, has a high quantum efficiency at the trapping wavelength, has a bandwidth of at least 1 MHz, and is shot-noise limited at power levels above 10 mW per quadrant. Details on a custom detector design that fulfills the above requirements can be found in Refs. [485,493].\nForward detection is attractive due to its simplicity. However, the ratio of scattered power to reference power is fixed by the scattering cross section of the particle. Backward detection allows to collect only the scattered photons and add a reference field with variable power. The ultimate position sensitivity to particle motion that is transverse to the optical axis is the same for forward and backward detection. Interestingly, due to interference between incident and scattered photons this is not the case for motion along the optical axis. In the forward direction, destructive interference reduces the sensitivity, with the counterintuitive result that a higher NA collection lens can yield a lower sensitivity. In contrast, constructive interference in the backward direction allows for up to 60% collection efficiency even with a moderate NA = 0.8. That is, more information about the particle position along the optical axis is contained in the backward scattered light than in the forward scattered light [496,497].\nInterferometric detection provides the best signal-to-noise ratio. However, it is limited to a small range of particle motion. In contrast, video analysis allows to determine the particle position over a much larger range with a limited signal-to-noise ratio and it requires extremely fast cameras to resolve the relevant timescale ∼ µs in high vacuum [498]."},{"title":"Feedback electronics","content":"Feedback cooling allows to extract energy from the center-of-mass motion and stabilizes the trap. The feedback force can be applied either directly (e.g., with electric fields when the particle carries a charge) or parametrically (by modulation of the laser intensity). The electronic circuit that processes the detector signal can be analog [488], digital [415,417], or an off-the-shelf lock-in amplifier [416] (section 5.7). Here, we focus on the latter following the description for parametric feedback based on a phase-locked loop (PLL) given by Hebestreit [471].\nThe detector signal for each axis is fed to one of the PLLs. Each PLL detects the frequency and phase of the oscillation along one spatial dimension. The locking range of the PLLs is set to 4 to 8 kHz around the expected center frequency to prevent the PLL from losing lock on the particle oscillation signal. The bandwidth of the loop filter is adjusted to be large enough to follow frequency changes sufficiently fast, and small enough to exclude noise from the feedback signal. In practice, this amounts to a bandwidth of 1 to 16 kHz [471].\nThe PLL synthesizes a signal that is phase-locked and at the same frequency or at a harmonic of the input signal with adjustable amplitude and phase. For parametric feedback, one uses the second harmonic and tunes the amplitude and phase to minimize the particle motion. The independent signals for the three oscillation axes are added within the lock-in amplifiers to form the final feedback signal. This feedback signal is applied to the EOM, which in turn modulates the power of the trapping beam. The amplitude of the feedback signals is usually set to a few mV for each axis. For a half-wave voltage of V π = 240 V and amplifier with 400 V/V gain, the optical power is modulated by 2.5% at a feedback signal amplitude of 10 mV [471].\nA switch allows to rapidly turn off the feedback signal. This allows to perform relaxation experiments [481]. Additional signals can be added to the feedback signal. This allows to stabilize this oscillation frequency by adjusting the power of the trapping beam with a PID (proportional-integral-derivative) control loop or to switch off the trap for short amounts of time for free-fall experiments [471]. A harmonic signal at twice the particle's motional frequency allows to excite the particle into the nonlinear region of the optical trap [468,469]."},{"title":"Vacuum system","content":"Optical levitation in high vacuum requires an experimental setup containing a vacuum system to control the gas pressure, optical access for the trapping and detection optics, and an access door on the top if particles are loaded at ambient pressures with a nebulizer (section 5.4.1). A schematic of a vacuum system is shown in Fig. 38. The pressure in the chamber can be monitored continuously over the entire range from ambient pressure down to below 10 −7 mbar with a combination of gauges, e.g., a Pirani gauge for high pressure combined with a hot cathode filament gauge for low pressure.\nA set of pumps and valves controls and stabilizes the pressure. The main pump is a turbomolecular pump. A motorized valve between the chamber and the turbo-molecular pump is used to stabilize the pressure below 10 −3 mbar with a PID controller. The turbo-molecular pump is backed by a dry scroll pump via another motorized valve that is mainly used to ensure a smooth evacuation of the chamber without sudden changes in the chamber pressure during the initial pump-down. This is important to prevent loosing a trapped particle during pump-down from ambient pressure after loading. To avoid mechanical vibrations from the vacuum pumps to couple into the optical setup, the pumps are connected to the vacuum chamber through flexible bellows, which decouple the vibrations. Vibration isolation can be further improved by casting the bellows into a sand or concrete block. At pressures below 10 −6 mbar, ion pumps are sufficient to maintain a pressure without introducing vibrations.\nM M Fig. 38. Vacuum system. The vacuum system consists of a vacuum chamber with optical vacuum windows, in which the actual trapping takes place. Here, we present a system based on that employed in Ref. [485]. The pressure is monitored using an ion and a Pirani gauge. A turbomolecular pump is connected to the chamber using a flexible bellow to suppress vibrations. A scroll pump supports the turbo pump. The pumping speeds can be regulated manually or with a PID controller using motorized valves (V 1 and V 2 ). The chamber is vented with nitrogen to reduce contamination. O: objective. PID: proportional integral derivative controller.\nPressures below 10 −6 mbar require to bake the vacuum chamber to desorb residual water. Many experiments load particles at ambient pressure. Hence, the lowest pressures can be achieved if the chamber is baked in situ while the particle is trapped. Optical access while baking can be provided with heating bands that are wrapped around the vacuum chamber. However, temperatures are limited to ≈ 150 • C to avoid damaging the optical components inside the vacuum chamber and the optical coatings of the windows. The baking process also causes thermal deformation of the optics inside the vacuum chamber. This can then lead to particle loss during baking. Nonetheless, even if particles are not kept in the trap during bake out, baking the chamber in combination with purging with dry air (instead of venting with humid ambient air) reduces the overall water content in the vacuum chamber and leads to improved pump speeds and better vacuum.\nHebestreit [471] quantified the gas composition in the vacuum chamber during a trapping experiment with a residual gas analyzer (RGA). The RGA is a mass spectrometer that allows to measure the mass spectrum of a gas and thus to extract the partial pressures of the different gases species. The measured partial pressures of the prominent gas species during a typical pump-down cycle are shown in Fig. 39. The total pressure at the RGA (the sum of all the partial pressures, dashed line) is higher than the pressure measured in the main vacuum chamber with an ion gauge (dashed line), because the pumping speed at the RGA is reduced compared to the main chamber. Initially, the chamber is filled with nitrogen. Interestingly, at low pressures the composition is vastly different. The turbo molecular-pump removes the initial nitrogen very efficiently. Thus, its partial pressure reduces quickly. In contrast, water and hydrogen are continuously desorbed from the walls of the vacuum chamber. As a consequence, the gas in the chamber is dominated by water vapor at pressures below 10 −4 mbar. Baking out the vacuum chamber accelerates the desorption process and thereby helps to get the water and hydrogen out of the chamber. The final gas composition in the pressure range of 10 −6 to 10 −5 mbar is mainly water vapor (58 ± 9%), nitrogen (19 ± 8%), hydrogen (12 ± 1%), oxygen (7 ± 1%), and carbon dioxide (4 ± 1%). Other gases, which contribute with less than 1% are argon, isopropyl alcohol, and oil. The last two probably remain from the solvent of the particle solution and residual machining grease, respectively. As a consequence, the molar mass M ≈ 20 ± 11 · 10 −3 kg mol −1 at low pressure is significantly lower than the molar mass at high pressures, the latter being equal to the molar mass of dry air M air = 28.97 · 10 −3 kg mol −1 ."},{"title":"Particle loading","content":"In contrast to a liquid environment, loading a particle into an optical trap in a dilute gas or vacuum is challenging because the particles will quickly fall down due to gravity. The size of an optical trap is rather small, on the order of the focal volume ∼ λ 3 . Therefore, a particle that passes by the focus at a distance larger than ∼ λ will not be captured. Furthermore, if a particle enters the focal volume at high speed, the low damping in vacuum will not slow the particle down sufficiently for it to get trapped. The maximum speed for successful trapping is v max ∼ λγ 0 , where γ 0 is the damping constant. In water, the viscosity at room temperature is η (water) = 890 µPa s so that, using Stokes formula (equation (6)), γ 0 = 6πηa/m and, therefore, v (water) max = 344 m s −1 for a silica particle of radius a = 75 nm. In air, the viscosity is about two orders of magnitude smaller (η (air) = 18 µPa s) so that v (air) max = 7 m s −1 . The challenge lies, therefore, in finding a technique by which a slow single nanoparticle is brought into the focal volume [488]."},{"title":"Nebuliser","content":"The most common approach to particle loading is the nebulizer approach thanks to its simplicity [499]. In this approach, a solution containing silica beads (e.g., Microparticles GmbH, Bangs Laboratories) with radii ranging from a = 50 to 100 nm is sprayed into the open vacuum chamber with a nebulizer (e.g., Omron NE-U22-W) and the particles are trapped at ambient pressure [488] or mild vacuum [500].\nThe nebulizer consists essentially of a mesh on top of a piezo element. A little bit of liquid gets into the space between the piezo and the mesh. The motion of the piezo then pushes the liquid through the mesh. The mesh breaks the liquid into little droplets of diameter 2 µm and a nozzle funnels the falling particles to a region near the laser focus (Figs. 40(a-b)). Successful trapping of a particle can be easily noticed observing the laser light scattered by the particle with a camera. However, since the scattered light has a dipole radiation pattern, the polarization of the laser beam should be orthogonal to the direction of observation to see the particle with the camera. The scattered light intensity on the camera scales with the size of the particle as B ∝ a 6 . This allows to distinguish whether the object in the trap is a single nanoparticle or a cluster of multiple particles by comparing the brightness on the camera for several trapping events. For example, in Fig. 40(c), the histogram of the brightness on the camera for ∼ 350 particles shows clearly distinct peaks, which can be associated with the number of particles in the trap [486]. Most of the liquid in the droplet evaporates before or immediately after trapping. However, ∼ 10 layers of water remain physically absorbed on the surface of the particle. These physically absorbed layers evaporate as the gas pressure is reduced down to ∼ 1 mbar. This dehydration process is reversible and water molecules get reabsorbed when the pressure is increased. In the region from 0.1 to 0.5 mbar, one can often observe a sudden decrease in the mechanical frequency and power of the scattered light. This is attributed to a sudden loss of the remaining water from the particle as its internal temperature increases to above ∼ 200 • C. This effect occurs only once and is irreversible when cycling the pressure. Therefore, once the particle has lost the final water layer its water uptake is extremely slow and for all practical purposes the particle size and material composition can be considered pressure independent. For more details on the dehydration process see Refs. [419,486,500]."},{"title":"Dry approaches","content":"Spraying solvents into a vacuum chamber introduces contaminants that are difficult to pump out, e.g., water molecules. This motivates a dry approach, where particles are released from a substrate instead of being injected from a solution. However, small particles stick to surfaces due to dipole-dipole interactions known as Van der Waals-forces, which scale linearly with particle size; the magnitude of these forces is\nwhere γ s is the effective surface energy. For example, measurements on silica spheres on glass give F VdW = 176 nN for a = 1 µm [501]. For comparison, the gravitational force for a particle of this size is only ∼ 0.1 pN and, therefore, much too weak to remove the particle. There can also be additional surface forces that lead to even stronger adhesion, for example meniscus forces arising from a thin water layer that is formed on the substrate. Therefore, even larger forces have to be applied to release the particle from the substrate. Two common approaches, which provide dry loading methods and do not require to open the vacuum chamber as in the nebulizer approach, are based on launching particles from surfaces using piezoelectric transducers or laser-induced acoustic desorption (LIAD).\nA piezoelectric transducer works really well for micrometer-sized particles [405,502] and allows even for repeated loading of the same particle [503]. When the piezoelectric transducer is driven with oscillation amplitude q p and frequency ω p , the particle feels a force\ndue to its inertia, where m = V ρ p is the particle mass with ρ p and V the particle density and volume, respectively. Consequently, the acceleration required to release the particle is\nHence, for a given maximum acceleration the piezoelectric transducer can provide, there is a minimum particle size that can be shaken off. The smallest particles that have been released with this method have a radius of 150 nm and further details on how to implement this can be found in Ref. [504]. The limited acceleration of the piezo approach can be overcome by generating a shockwave on the surface of the substrate in the LIAD approach. The shockwave can be created by a pulsed frequency-doubled Nd:YAG laser (5 ns pulses of 3 mJ at 532 nm). When focused down to 200 µm on the back side of a thin metallic foil, the laser creates an acoustic wave that launches particles from the front side, which had been previously spin coated with particles [505].\nDespite the availability of these techniques, directly loading particles into an optical trap under high vacuum conditions remains an open challenge. With the dry approaches outlined above, particles are still loaded into the trap at ambient pressures or moderate vacuum, where the background gas provides sufficient dissipation because capture requires an additional dissipation mechanism due to the conservative nature of gradient trapping forces and the small size of the trapping region of an optical tweezers. Recently, it was demonstrated that high-vacuum loading into a Paul trap can be achieved [506]. The Paul trap provides a much larger trapping volume and trap depth than an optical tweezers. This allows to turn on the trap exactly when the particle is near the center of the trap. As a consequence, the particle's total energy is given only by its kinetic energy. If the kinetic energy is less than the trap depth, the particle will be stably trapped.\nPaul traps are limited to charged particles; however, this is not a strong limitation for loading since the particles generally carry several tens of elementary charges (section 5.6.3).\nTransferring particles from a contaminated chamber, where the particle is initially trapped, to a clean science chamber, which always remains in high-vacuum conditions, can be achieved with a mobile optical trap (MobOT) in a load-lock [507]. However, this solution is rather bulky and has a limited throughput at low pressures. Hollow-core photonic crystal fibers provide also a compact solution with high throughput, but have been limited so far to pressures above 10 −2 mbar [508,509]."},{"title":"Dissipation and noise","content":"A particularly attractive feature of trapping in a dilute or vacuum environment is that the interaction with the environment can be adjusted over several orders of magnitude. The interactions between the particle and its environment result in stochastic forces that excite the particle motion and lead to stochastic dynamics, which are of particular interest when studying thermodynamics of individual small particles and are to be minimized when aiming for the quantum regime.\nThe intensity of a stochastic force is characterized by its PSD S noise ff . While the stochastic forces excite the center-of-mass motion of the trapped particle, the interactions with the environment also lead to dissipation at a rate γ noise , which damps the particle motion. There are several sources that contribute to the total damping rate and stochastic noise intensity, which we will discuss in this section.\nThe total damping rate and stochastic noise intensity is simply given by the sum of the individual contributions γ 0 = noise γ noise and S ff = noise S noise ff , respectively. Note that for most purposes, the PSD can be considered as frequency independent (white noise) over the bandwidth of the mechanical resonance of the particle, especially when the bandwidth becomes quite narrow in high vacuum. The average energy of each center-of-mass degree of freedom evolves as E = −γ 0 E + Γ where the heating power due to the force noise is Γ = πS ff /m, S ff being the double sided PSD. After a time ≈ 1/γ 0 , the particle reaches an equilibrium, where the ensemble average energy stays constant. This allows to define the effective temperature through the fluctuation-dissipation relation:\nThe effective temperature corresponds to the particle energy averaged over an ensemble of identical experiments. Importantly, since we are concerned here with the energy of a single particle, the energy in each experiment will still fluctuate with a standard deviation ∼ T cm . Generally, one has to be careful to define a temperature for a system out of equilibrium [510]. However, the situation we present here is somewhat simplified because it is steady-state and for many practical purposes the effective bath model that is characterized by an effective damping/temperature gives a good description. However, one can also create situations where this is no longer true. For example, by parametric feedback damping, the temperature alone is not sufficient to give a full description of the bath (section 5.7) For a levitated particle in a dilute gas, the main contribution to the stochastic force noise are collisions with air molecules and radiation damping at pressures below ∼ 10 −7 mbar. In addition, there are technical noise sources that do not pose a fundamental limitation but can impose practical limitations. These are primarily displacement noise of the trap and laser intensity fluctuations. Noise sources can also be introduced intentionally to emulate a different thermal environment without actually changing the real temperature of the environment [316,511]. This can be a useful tool for single particle thermodynamics experiments [19]."},{"title":"Gas damping","content":"From kinetic gas theory, the pressure-dependent gas damping for a spherical particle is [488,512] γ gas\nwhere Kn =l/a is the Knudsen number, c K = 0.31Kn/ 0.785 + 1.152Kn + Kn 2 ,l = πk B T 2m gas η ν P gas is the mean free path, η v is the viscosity, and m gas = M/N A the molecule mass [512], N A being Avogadro's number. For dry air, the molar mass M = 28.97 · 10 −3 kg mol −1 , however the actual gas composition can be different (Fig. 39). For high pressure, the interaction with the gas is so strong that the particle motion is heavily damped and its internal temperature T bulk and centre-of-mass temperature T cm quickly thermalize with the gas temperature T gas . In this regime, the damping does not depend on pressure and is given by Stokes law, i.e., γ gas /(2π) ≈ 3aη v /m. For pressures below P gas ≈ 54.4 mbar (a/µm) −1 , where the mean free pathl is much larger than the radius of the particle a, the damping is linear in the pressure P gas and given by\nIn typical experiments, γ gas /2π = c P P gas /a with c P ≈ 50 Hz (µm/mbar). Note that the above holds for a spherical particle. For an anisotropic particle (e.g., a rod), the friction term is different along each of the axes, and depends upon the alignment of the particle [513]. The mean free path of the gas molecules increases with decreasing pressure (e.g.,l ∼ 60 µm at 1 mbar). As a consequence, the particle no longer thermalizes with the gas since the impinging gas molecules no longer carry away enough thermal power to balance the optical absorption from the trapping laser. Due to the increased internal temperature T bulk of the particle, the average energy of the gas molecules after a collision with the particle increases (section 5.6.5). The latter requires that the impinging molecules stick to the surface for a short time before they emerge again from the surface. This type of collision is called diffusive reflection [514,515]. For diffusive reflection, the fluctuating forces originating from the impinging gas molecules are completely uncorrelated. This justifies the treatment of the impinging and the emerging gas molecules as two independent baths with individual bath temperatures, damping and heating rates. The process by which a surface exchanges thermal energy with a gas is called accommodation, which is characterized by the accommodation coefficient\nwhere T em is the temperature of the gas molecules emitted from the particle surface. The accommodation coefficient quantifies the fraction of the thermal energy that the colliding gas molecule removes from the surface, such that c acc = 1 means that the molecule fully thermalizes with the surface. We do not distinguish between the bulk and the surface temperature, since we assume that they are identical for nanoparticles. Since the mean free path in a dilute gas is long, one can safely assume that an emitted molecule will not interact again with the particle before thermalizing with the environment. Consequently, we can consider the particles that impinge on the particle surface and those that leave it as being in equilibrium with two independent baths [514]. The damping due to the impinging molecules is\nand the contribution from the emerging molecules is\nThe total gas damping is then γ gas 2π = γ im 2π + γ em 2π (173) and the total force noise due to the gas is\nWhen the temperature of the emerging molecules equals the gas temperature (i.e., T em = T gas ), equation (173) reduces to equation (169)."},{"title":"Radiation damping and shot noise","content":"Once the pressure is low enough (i.e., ≤ 10 −7 mbar), gas damping is negligible and the particulate nature of the light field becomes the primary source of dissipation, in the absence of technical noise sources such as laser phase and amplitude fluctuations [416]. Photons arrive at discrete times, where the number of photons arriving per time interval ∆t is given by ∆tP opt ( ω opt ) . The recoil from the fluctuating number of photons impinging on the nanoparticle can be modeled as an effective bath with the characteristics [497,516] γ rad 2π = c dp P scatt 2πmc 2 and S rad ff = c dp\nwhere c dp depends on the direction of motion of the particle with respect to the polarization of the laser (in particular, c dp = 1/5 for motion along the direction of polarization, c dp = 2/5 for motion perpendicular to the polarization, and c dp = (2/5 + A 2 ) along the direction of propagation, where A is the radiation pressure contribution defined in Ref. [497]). The scattered power is P scatt = σ scatt I opt , where σ scatt = |α| 2 k 4 opt /(6π 2 0 ) and I opt is the laser intensity. The effective temperature of this bath can be calculated via equation (167) and is ∼ ω L /k B ."},{"title":"Displacement noise","content":"In the reference frame of the trap (i.e., where the trap appears stationary), acceleration of the trap center a dis appears as a force F dis = ma dis due to the particle's inertia m. Therefore, fluctuations of the trap center with displacement spectral density S dd lead to the effective force spectral density\nwhere we have used the fact that the acceleration spectral density S aa (ω) = ω 4 S dd (ω). There is no damping associated with displacement noise, since the source of the displacement noise is usually some active component like a pump. Unlike the thermal force spectral density, the displacement noise spectral density is not flat but increases for lower frequencies. However, for the typical narrow resonances at Ω 0 in optical levitation experiments, we can approximate it as white noise, S dis ff ≈ m 2 Ω 4 0 S dd (Ω 0 ). The thermomechanical noise limited acceleration sensitivity S aa = k B T γ 0 /(πm) depends inversely on the particle mass. Hence, heavy particles make particularly good accelerometers [458]. However, since the maximal trap stiffness is limited, heavier particles tend to have lower frequencies at which displacement noise dominates over thermomechanical noise. At the optimal particle size both noise sources contribute equally. Displacement noise is a technical noise source and as such not a fundamental limitation. It is primarily noticeable in setups where the external forces change considerably over the range of movement of the trap center. That is in situations where the force has a strong spatial dependence, for example when the trapped particle interacts with a standing wave of an optical cavity or with the evanescent field from a tapered fiber [492]. If the trap position changes randomly with respect to the cavity or the fiber, the particle experiences fluctuating optical forces from the cavity or evanescent field, respectively."},{"title":"Laser intensity noise","content":"Laser intensity fluctuations are another technical noise source [491,492]. Low frequency drifts of the intensity randomly change the particle's oscillation frequency, since the particle frequency scales with Ω 0 ∝ P opt . This can be a serious limitation in force-sensing experiments, where an external force is transduced into an oscillation amplitude. If the frequency changes, the particle's response to an external force changes even if this force is constant, resulting in a noisy measurement.\nIntensity fluctuations at twice the motional frequency 2Ω 0 parametrically excite the particle and thereby raise the particle's energy. If the phase of the particle oscillation is not actively stabilized to the parametric modulation, it self-locks (entrains) to the external modulation [468]. Hence, an intensity modulation at 2Ω 0 without active stabilization always leads to heating as will be discussed in section 5.7. Parametric excitation is a nonlinear process and its strength depends on the motional energy of the particle and therefore can be suppressed by reducing the particle's energy with feedback cooling. In free-space experiments, laser-intensity noise can be reduced sufficiently such that it is smaller than the fundamental photon shot noise [416]. In cavity experiments, however, laser-phase noise is transduced into intensity noise and has been identified as a major obstacle towards reaching the ground state with sideband cooling [421,517]."},{"title":"Calibration","content":"Precise calibration is as important for quantitative measurements in the underdamped regime as it was in the overdamped regime (section 3). As before, the thermal energy k B T serves as a reference to calibrate the detector signal (section 5.6.1). However, in contrast to the underdamped case, where the displacement signal from the particle is very broadband and centered around zero, in the underdamped regime the signal is sharply peaked and centered around a finite frequency. The weak coupling to the thermal bath in high vacuum also allows to use the energy scale of a single motional quantum Ω 0 as a calibration reference. Remarkably, despite the huge difference in energy scale of 7-8 orders of magnitude one can achieve almost perfect agreement [418] (section 5.6.2). In high vacuum charges on the particle are preserved. Section 5.6.3 explores how to control and measure them. The precise knowledge of the charge then allows allows to precisely determine the particle's mass (section 5.6.4). Since a particle in vaccum is well isolated, its internal temperature raises well above the temperature of the environment in an optical tweezers. In section 5.6.6, we will see how to measure the internal temperature by analyzing the particles center of mass motion."},{"title":"Detector calibration","content":"For the most part optical trapping is concerned with the external degrees of freedom, and in particular with the center-of-mass motion. Quantitative measurements require a precise calibration of the detector signal that relates the measured signal (typically in volts) to the displacements of the particle (in meters). Under the assumption that the detector signal v(t) is proportional to the particle displacement q(t), the calibration factor is defined as the proportionality factor c = v/q with units of [V m −1 ] (or [bit m −1 ] for a digitized detector signal). Using the fact that the expectation value of the potential energy of a harmonic oscillator E pot = 1 2 mΩ 2 0 q(t) 2 = 1 2 k B T, the calibration factor is simply given by\nwhich does not depend on γ 0 and thus holds both for the overdamped and underdamped regimes.\nIn the underdamped regime, the PSD is highly peaked around the resonance frequencies. This allows to distinguish noise and cross-talk efficiently in the spectral domain. Therefore, it is often preferred to fit the PSD to the single-sided harmonic oscillator PSD\nto extract the variance of the particle displacement v(t) 2 from the area under the PSD. In the following, we adopt here the notation from Hebestreit [485], where we convert from angular to ordinary frequencies, i.e., ω = 2π f , γ = 2πg, and S( f ) = 2πS(ω > 0). The experimental single-sided PSD is obtained from the discretely-sampled time trace v(t) as (179) where N is the number of samples, f s is the sampling frequency, and F (v(t)) is the discrete Fourier transform of v(t).\nFitting the experimental PSD to expression (178) allows to extract the variance v 2 (t) , the damping constant g = γ gas /(2π), and the resonance frequency f 0 = Ω 0 /2π. Then, the damping constant allows to determine the particle radius from the pressure and, therefore, the mass for known mass density (section 5.6.4). Once the mass is known, we can calculate the calibration factor according to equation (177). The biggest uncertainty in determining c stems from the error in the particle mass. However, in many experiments we are only interested in the particle energy and we can define the mass-independent energy calibration factor\nThe above calibration method relies on two key assumptions. First, we consider the optical potential to be quadratic such that the particle behaves like a harmonic oscillator. Second, we assume that the detector signal is linear in the particle displacement, that is that the calibration factor does not depend on the particle position. The latter depends significantly on the details of the experiments and will not be futher discussed here. However, it is possible to account for a nonlinear detector response. For calibration of a position dependent calibration factor see [480] and for detector response calibration and discussion see [485].\nThe finite potential depth of the optical trap results in anharmonicities in the particle motion for large enough displacements. Due to the symmetry of the optical potential, the lowest nonlinear term is the cubic or Duffing nonlinearity39. The Duffing nonlinearity softens the optical potential at large particle displacement [470]. This distorts and broadens the lineshape of the PSD, and causes an overestimation of the energy when following the calibration procedure described above. To minimize the impact on the shape of the PSD, the calibration is usually carried out at moderate pressures (∼ 10 mbar), where the linewidth (γ gas /(2π) ∼ 10 kHz) is narrow enough to reject noise and cross-talk, but still wide enough that the nonlinear broadening (γ NL /(2π) ∼ 3 kHz) is negligible [485] (Fig. 41a).\n39There are some subtelties in the detection along the propagation direction of the trapping laser, where this symmetry is broken by the scattering force, leading to small but finite quadratic nonlinear terms [488]. The energy underestimation is accounted for by considering the kinetic energy instead of the potential energy [518]. The kinetic energy is given by E kin = 1 2 m q 2 = 1 2 k B T even for a nonlinear potential. The variance q 2 can be obtained by integrating over the velocity PSD, which is related to the position-PSD by\nHence, we do not have to measure q directly, which is experimentally challenging since it would require sufficient oversampling. Experimentally, we compute the variance by numerical integration over the voltage PSD\nThe integration bandwidth b is chosen such that we integrate most of the signal and don't integrate too much of the noise floor. In the measured displacement PSD, we usually have a white noise floor. If we multiply this PSD by f 2 to get the velocity PSD, the noise floor therefore acquires a f 2 dependence as well. Integrating this noise will eventually cause a significant error in the calibration and energy estimate. Drifts in the setup can cause the calibration to change over time. In particular, the calibration procedure detailed above is carried out at moderate pressures (∼ mbar), while most experiments are carried out at much lower pressure many hours or even days after performing the calibration. Hence, it is desirable to check the calibration even at much lower pressure and to correct for potential drift (see Fig. 41b).\nHebestreit suggests a method to track changes in calibration at arbitrary pressure [485]. The main idea is to measure the response of the particle to a known force, for example the electrical force from a capacitor around the optical trap. This method assumes that the electrical fields and the charge on the particle do not change through out the experiment. This can be achieved as will be discussed in section 5.6.3. At the initial pressure where the calibration is carried out, a sinusoidal signal is applied to the capacitor. This leads to additional peaks in the response of the particle (Fig. 41). The power in the response peak is\nwhere Q p is the total charge on the particle and E 0 is the electric field amplitude in the direction of oscillation at the location of the particle. At a later time (e.g., after pumping down to a lower pressure), we repeat this measurement, yielding v 2 e . At lower pressure, feedback is generally active. Hence, the response function of the particle changes. The new values of Ω 0 and γ can be extracted from a fit to the thermal noise peak. Assuming that Q p E 0 /m remain constant, we find the new calibration factor as\nWe can also arrive at equation (184) by considering the height of the peak in the PSD instead of the area under the peak. It's value is v 2 e τ, where 2τ is the measurement time (equation (188)). In the previous section we used the thermal fluctuations of the particle to calibrate the detector signal. Similarly, we can use the quantum fluctuations for calibration. However, at ambient temperature, quantum fluctuations of a nanoparticle in an optical tweezers are 8-9 orders of magnitude weaker than the thermal fluctuations. As a consequence, they can only be observed in high vacuum, where the coupling to the thermal bath is sufficiently weak [418]."},{"title":"Sideband thermometry","content":"When light is reflected off the particle, the particle motion is imprinted on the scattered light as a phase modulation, which can be understood as a Doppler shift due to the particle motion. To lowest order, the phase modulation leads to red and blue sidebands at ω L ± Ω 0 , where ω L is the laser frequency (Figs. 42(a-b)). Classically, the power in each sideband is proportional to the variance of the motion x 2 , which in turn is proportional to the temperature of the thermal bath. This is the basis of the calibration procedure described in the previous section.\nIn the quantum mechanical formulation, the interaction of light with the trapped particle can be described as a Raman process. The red sideband (Fig. 42(a)) corresponds to Stokes scattering, which raises the population of the mechanical oscillator by a single quantum of mechanical energy, while simultaneously creating a low energy (red) photon. The blue sideband (Fig. 42(b)) corresponds to anti-Stokes scattering, which lowers the oscillator's population by one mechanical quantum, while creating a high energy (blue) photon. The rate at which blue sideband photons are created is proportional ton, just as in the classical case. Importantly, anti-Stokes scattering is impossible by an oscillator in its quantum ground state. In contrast, the rate at which red sidebands are produced is proportional ton + 1.\nThe mean occupation number is given by the Bose-Einstein distribution\nFor thermal energies large compared to the energy quantum (k B T Ω 0 ), the mean energy of the harmonic oscillator isn Ω 0 = k B T ∝ x 2 , in accordance with classical statistical mechanics and independent of . The ratio of powers in the anti-Stokes and Stokes sidebands is given bȳ (Figs. 42(c)) and vanishes for high temperatures. For thermal energies comparable to a single mechanical quantum excitation (also called phonon), the ratio can serve as a temperature measurement calibrated relative to the quantum of energy of the system [519].\nThis so-called sideband thermometry requires to measure the Stokes and anti-Stokes sidebands [418,[520][521][522], which can be achieved with a heterodyne measurement. In a heterodyne measurement the scattered light interferes with a laser beam (local oscillator) with a frequency ω LO which is frequency-detuned from the light that is incident on the particle. This produces a strong beat signal at the difference frequency ∆ LO = ω LO − ω L , with sidebands at ∆ LO ± Ω 0 . For ω LO < ω L , the left sideband corresponds to the red sideband (Stokes) and the right to the blue (anti-Stokes). For ω LO > ω L , the roles are inverted. The difference frequency is chosen such that |∆ LO | Ω 0 , for example |∆ LO | ∼ 1MHz for typical trapping frequencies ∼ 100kHz. The measured sideband asymmetry\ncan be used to determine the mean occupation number. Here, S l,het vv (S r,het vv ) is the power spectral density of the left (right) sideband and the subscript R − (R + ) is for ω LO < ω L (ω LO > ω L ) (see Fig. 42).\nIn principle, R − or R + alone is sufficient to determine the mean occupation number. However, classical artifacts due to a frequency dependent transfer function of the measurement device give rise to an asymmetry that is not related to the quantum dynamics of the particle. Measuring both R − and R + allows to eliminate this artifact and to determine the actual mean occupation number as followsn Fig. 42 shows the measured sidebands of a nanoparticle cooled to an effective temperature of n = 4 [418]. Clearly the area under the red sideband is larger than under the blue sideband. This is the first room temperature measurement of the sideband asymmetry of a massive object and clear signature of its quantum ground state. Remarkably, the inferred occupation number is in very satisfactory agreement with the traditional calibration method detailed in section 5.6.1despite the huge difference in energy scales of the two methods."},{"title":"Charge calibration","content":"Generally, particles carry several tens of elementary charges. Under vacuum, the particle is completely isolated from the environment and therefore its charge state is preserved indefinitely. Knowing the charge exactly provides a reliable handle for exerting a known force on the particle by applying an known electric field.\nIn optical levitation experiments, particles are loaded at high (∼ 1 atm) pressure to provide enough damping. After the particle has been trapped, the vacuum chamber is evacuated to reduce air damping. As the pressure decreases, residual water desorbs slowly from the particle followed by a sudden change in size, which is attributed to the removal of the final water layers (section 5.4.1). During this final stage, the charge of the particle changes by tens of elementary charges [486], but once the particle has undergone this sudden change, its charge state is preserved indefinitely even when the pressure is increased again because the water is gone.\nA relatively simple method to control the charges on the particle is to apply an electrical discharge near the particle. The discharge provides free charges that can be adsorbed by the particle [523]. It can be applied through a bare wire of about 5 cm length at about 5 cm away from the optical trap. The grounded vacuum chamber serves as the counterelectrode. A DC voltage (∼ 7 kV) is sufficient to ionize gas molecules inside the vacuum chamber and to produce a corona discharge at moderate pressures (∼ 1 mbar). Note that an ion pressure gauge can produce also free charges, which lead to unwanted changes in the charge state of the particle, if it is not properly shielded [523]. The ionization process is triggered by random events such as absorption of a ultraviolet photon or cosmic rays that spontaneously ionize neutral air molecules. The free electron and positive ion are accelerated toward opposite directions by the high voltage, which also prevents them to recombine. The electron acquires enough kinetic energy to ionize other molecules, because of its small mass. The secondary electrons ionize more molecules, which then results in an electron avalanche.\nEmission of (mainly) UV photons after electron-ion recombination causes the characteristic violet glow of the corona, as shown in Fig. 36. The corona discharge is positive or negative, depending on the polarity of the high DC voltage, which therefore allows to select the ratio of positive-to-negative ionized molecules. Eventually, free charges are adsorbed on the particle surface, thereby changing its net charge monotonically, except for few unfavorable events. These charges can be both electrons and ionized air molecules. Accordingly, by simply turning off the high-voltage source when the particle carries the desired number of charges, the levitated particle can be brought to any desired net-charge state from zero to a few elementary charges [486,523].\nThe charge state of the particle is monitored in situ by driving the particle motion with an electric field at frequency ω d . The electric field is created by applying a voltage U(t) = U 0 cos(ω d t) across a capacitor that is centered around the particle. A simple capacitor can be formed by grounding the metal housing of the microscope objective and applying a voltage to the metal holder of the collection lens [523]. This method does not require any additional mechanical components in the setup and therefore does not obstruct access to the particle . Alternatively, dedicated electrodes that fit right at the sides of the optical trap can be beneficial to produce a more homogeneous field at the location of the particle [486], since field inhomogeneities can lead to a finite net charge density. When a voltage U 0 ∼ 10 V is applied to the capacitor at a frequency ω d close to the particle's resonance frequency, the particle's response to the driving field shows as a distinct peak in the single-sided PSD [486] on top of the thermal peak (see Figs. 41 and 43a):\nwhere the thermal force spectral density is S ff = mk B T γ 0 /π, χ(ω) = Ω 2 0 − ω 2 + iωγ 0 −1 is the mechanical response function, F 0 = Q p E 0 , E 0 ∼ 500 V m −1 being the amplitude of the electric field modulation at the location of the particle, Q p is the charge on the particle, and 2τ is the duration of the time trace that is used to calculate the position spectral density S qq (ω) = 2τ|q| 2 (equation (179)). The detector signal is demodulated in quadrature with the drive. The demodulated signal can assume both positive and negative values, since the response to the driving field flips phase by π when the polarity of the charge changes. Positive (negative) signal amplitudes can then be associated with positive (negative) net charge, when the transfer functions of the electronics is accounted for. Note that for a small detuning of the drive frequency ω d from the particle's resonance, there is a small phase offset from the response function of the particle.\nWhile the corona discharge is active, the demodulated signal features discrete steps, which are the signature of single elementary-charge transfers to and from the nanoparticle (see Fig. 43). Thus, any desired net-charge state from zero to a few elementary charges can be achievd by simply selecting the polarity of the high-voltage source, counting the number of steps, and turning off the high-voltage source when the particle carries the desired number of charges. Importantly, the particle's charge state is preserved when the pressure is reduced and stays indefinitely while the high-voltage source is turned off, even over a period of several days [523]."},{"title":"Mass Calibration","content":"The exact value of the particle's inertial mass often plays an important role for precision measurements. For example, the common method for displacement calibration uses the thermal force noise as a reference, where the strength of the force depends on the mass, as we have seen in section 5.6.1. Likewise, nanomechanical mass measurements rely on knowing the mass of the bare resonator [524].\nThe manufacturer's size specification gives a good first estimate of the particle size with an error up to ∼ 25%, which stems from the typical size variation (∼ 5%) and the uncertainty in the specified mass density (∼ 10%) of commercially available mono-disperse particles.\nThe mass of individual particles can be inferred from kinetic gas theory, which relates the measured damping rate γ 0 to the particle size [512]. In particular, for a spherical particle and at pressures where the mean free path is much larger than the radius of the particle, there is a simple linear relationship [488],\nwhich relates the pressure to the particle radius a (equation (169)). At room temperature the mean free path is ∼ 60 µm · P −1 gas mbar −1 . Hence, the linear approximation is generally valid for nanoparticles in vacuum. The calibration measurements are generally performed at ∼ 10 mbar, where individual peaks are clearly resolved and nonlinear broadening of the peaks is still negligible. To remove excess water from the particle, which leads to an overestimate of the actual particle size, the pressure should be cycled to below 0.1 mbar and back (section 5.4.1). From the measured radius, one can then infer the mass m = 4 3 πa 3 ρ p with a typical uncertainty of ∼ 35% [486].\nThe mass measurement can be significantly improved either by an independent measurement Fig. 44. Mass calibration. Power spectral density S vv (ω) of a thermally and harmonically driven resonator at P = 50 mbar. The broad peak centered at Ω 0 /2π = 125 kHz corresponds to the thermally driven state. We fit it with a Lorentzian function (red) to extract S th vv (ω d ) together with g/(2π) = 31.8 kHz and the corresponding uncertainties. The narrowband peak at ω d /(2π) = 135 kHz, also shown in detail in the inset, depicts the electrical excitation from which we retrieve S d vv (ω d ). Gray data points at the bottom of the plot is the measurement noise, which is ∼ 40 dB below the particle's signal. Figure adapted from Ref. [486]. of the voltage calibration (equation (177)) or by applying a known force. For the former approach, the wavelength of the light provides a well-defined length scale that allows to relate the phase to a displacement in an interferometer. The interferometer is calibrated by changing the relative length of the two interferometer arms and measuring the fringe pattern. For the latter approach, electrical fields can be estimated quite accurately (∼ 1% error) from a numerical simulation of the electrode geometry [486]. Hence, the electrical force on the particle can be estimated with the same accuracy, because the number of charges can be determined exactly (section 5.6.3). Since the thermal force depends on the mass but the electrical force does not, their ratio (equation (188)),\nprovides a direct method to determine the mass from the measured spectra (Fig. 44). In equation (190), S d vv (ω d ) is the peak of the measured spectrum at the drive frequency ω d when the sinusodial electrical force F el = F 0 cos(ω d t) is applied and S th vv (ω d ) is the thermal background at ω d (Fig. 44). Note that the spectra do not need to be calibrated to apply this procedure. The mass is then simply given by [486] m ="},{"title":"Internal temperature","content":"In optical tweezers experiments, the main focus is on the external degrees of freedom of the particles in the trap (i.e., the translation of the particle and its rotation). However, any particle will absorb some of the trapping light, which raises its internal bulk temperature. The power absorbed by a particle with volume V from a laser beam with intensity I 0 is P abs = 6π I 0 λ VIm n 2 (λ) − 1 n 2 (λ) 2 + 2 .\nFor example, for silica, the refractive index is n = 1.45 + i6 · 10 −8 at λ = 1064 nm. Note that absorption can be higher in nanoparticles due to additional absorbers such as defects and impurities. At high pressure, the particle efficiently thermalizes with the surrounding gas by convective cooling with cooling power P con (T bulk ) = −c acc 2 3π (πa 2 )P gas v rms γ sh + 1 γ sh − 1\nwhere v rms = 3k B T gas /m gas is the root mean square velocity of the gas molecules, and γ sh = 7/5 is the specific heat ratio for an ideal diatomic gas. Clearly, as the pressure P gas decreases, convective cooling by the surrounding medium vanishes and the only heat exchange with the environment is radiative. The emitted thermal radiation is P em bb = −P bb (T bulk ), where P bb (T) = 72ζ(5) π 2 V c 3 4 Im n 2 bb − 1 n 2 bb + 2 (k B T) 5 (194) is the the black body radiated power for a dipolar emitter, being ζ(5) ≈ 1.04 the Riemann zeta function [413]. Conversely, the particle also absorbs thermal radiation from the environment at temperature T env with P abs bb = P bb (T env ). The particle bulk temperature can be found from the requirement that at equilibrium all contributions add up to zero, i.e., P abs + P abs bb + P con (T bulk + P em bb (T bulk ) = 0.\nThe emitted or incident power due to black-body radiation depends on the dielectric properties of the particle in the entire black-body spectrum, and their calculation therefore involves the integration of the black-body emission rates over all emission frequencies. As an approximation, one assumes a constant permittivity of the particle in the entire spectral range of black-body emission. This can be estimated as the average value of the refractive index over the black-body spectrum. For example, for silica this gives n bb = 1.5 + i0.1.\nThe internal temperature of a particle trapped in vacuum can raise significantly above the environmental temperature, to the point where the particle disintegrates if the absorption is too high [448] (Fig. 45). In fact, this problem has prevented optical levitation of nanodiamonds in high vacuum [447]. Even silica particles with low absorption can reach temperatures ∼ 1000 K [525]. As an example, figure 45d shows the internal temperature as a function of pressure. Fig. 45. Heating and temperatures of an optically levitated particle. (a) Damping and (b) heating rate as a function of pressure for feedback γ fb /2π = 20 Hz. (c) Center of mass temperature as a function pressure. At low pressures recoil heating dominates and the rates and the center of mass temperature become pressure independent. (d) Internal temperature as a function of pressure. At a few millibars, there is a sharp transition between convective cooling and radiative cooling via black body radiation and the internal temperature transitions from the ambient temperature to a much higher value. The internal temperature leads to a slightly higher center-of-mass heating rate visible as a small bump in (b). 5.6.6. Coupling between internal temperature and external degrees of freedom As we discussed earlier, the bulk temperature couples to the external degrees of freedom through the residual gas in the vacuum chamber (section 5.5). This coupling can be explained by a simple two-bath model, where the gas molecules that interact with the particle are divided into two families: molecules that impinge on the particle being in thermal equilibrium with the vacuum chamber, and molecules that emerge from the hot surface of the particle at a higher temperature. From equation (174), the thermal force noise increases with increasing bulk temperature. This allows to determine the internal bulk temperature from the fluctuation of the center-of-mass of the particle by solving T cm = T 3/2 gas + π 8 T 3/2 em T gas + π 8 √ T em (196) for T em , where the center-of-mass temperature is defined as T cm = Ω 2 0 v 2 /(Ck B ) , C being the energy calibration factor and v 2 the variance of the voltage fluctuation on the detector (section 5.6.1). Then, from T em one determines the temperature of the particle as T bulk = c −1 acc T em − T gas + T gas .\nThe method described above requires the particle's center-of-mass to reach its equilibrium steady state. This is impractical at low pressures where the particle is easily lost without feedback stabilization. At low pressures, the thermal noise force from the interaction with the gas can be determined measuring the heating rate [525]. Thereby, the center-of-mass motion is first reduced with feedback cooling (section 5.7). When the feedback is switched off, the particle energy increases at a rate Γ = γ im T gas + γ em T em = γ im T 3/2 gas + π 8 T 3/2 em T gas . (a) Heating rates as a function of pressure extracted from relaxation measurements without additional heating of the internal temperature (blue) and with heating by a CO 2 laser intensity of 0.47 µW µm −2 (orange). The dashed lines are linear fits to the data points. Error bars are smaller than the marker size. (b) Heating rates at different intensities of the CO 2 laser measured at a pressure of 1 · 10 −5 mbar. The error bars indicate the standard deviation of the measurements. (c) Calculated temperature dependence of oscillation frequency. When the particle is heated with the CO 2 laser, its oscillation frequency increases due to changes in the particle's material properties. This leads to a nearly linear relation between relative frequency change and increase of the internal particle temperature. Adapted from Ref. [525].\nWith feedback cooling the oscillation amplitude is low enough that cross-coupling between the degrees of freedom can be neglected. This justifies the approximation where the particle center-of-mass motion is treated as three independent harmonic oscillators. Each degree of motion obeys the following equation of motion (c.f. equation (160)):\nwhere F fb (t) is the feedback force (equation (160)). For most purposes, the stochastic contribution F L and the bare damping γ 0 result from interactions with the background gas. Additional contributions are discussed in section 5.5. The feedback damps the particle motion at a rate (γ f b /2π). The additional damping reduces the fluctuations of the position of the particle. As a consequence, feedback cooling reduces the response time to external perturbations and increases the confinement of the particle. Since in a thermal equilibrium the variance of the position is proportional to the temperature of the environment, we define the effective temperature for the center-of-mass motion (equation (167)) as\nwhich assumes that the particle position and energy distributions reflect the distributions of a harmonic oscillator in thermal equilibrium at temperature T cm . It is important to point out, however, that under the action of feedback the particle is not in thermal equilibrium and the assumption of a simple thermal distribution might be violated, in particular for nonlinear feedback schemes (section 5.7.2)."},{"title":"Force feedback","content":"From an inspection of equation (201), it is apparent that a feedback term proportional to the particle velocity results in damping. Instead, a feedback term proportional to the particle position allows to optimize the transient times [417]. Thus, the feedback force of a linear feedback controller reads\nwhere k d is the derivative feedback gain and k p is the proportional feedback gain. Purely velocity-dependent feedback (i.e., k p = 0) is also called cold damping. Experimentally such a force can be implemented with radiation pressure from a weakly focused beam. This technique was used in Ashkin and Dziedzic's original experiments to stabilize a particle against gravity [403], and later by Li et al. to feedback-cool microspheres in high vacuum [527]. The latter experiment required three laser beams in addition to the trapping laser to exert radiation pressure forces along the three spatial dimensions.\nMore recent experiments use electrostatic forces instead [415,417,528]. Electrodes near the optical trap apply a force F Q = Q p E, where Q p is the total charge on the particle and E the electric field at the location of the particle. Often the electrode geometry is chosen such that they form a plate capacitor around the particle so that the field is homogeneous and given by E = (V elec /d elec ) n elec , where V elec is the applied voltage, d elec the distance between the electrodes and n elec a unit vector pointing from one electrode to the other. The electrostatic force along direction e i is then given by F Q = F Q · e i and the electrode geometry can be chosen such that there is a force along all degrees of freedom. Implementation of the velocity-dependent force requires to measure the instantaneous velocity of the particle. In practice, however, this is achieved with a bandpass filter that acts on the position time trace q(t) and produces a time delay such that the signal is in phase with the velocity q(t) [415]. For example if the particle motion is approximately harmonic q(t) ∼ sin(Ω 0 t), a delay τ = π/(2Ω 0 ) leads to a signal ∝ cos(Ω 0 t) that is proportional to the particle velocity q(t).\nIn the frequency domain, the equation of motion with linear feedback reads q −ω 2 + iωγ 0 + Ω 2 0 = m −1 F L +f fb ,\nwhere the tilde indicates the Fourier transform andf fb (ω) = H(ω) [q +q n ] with filter funcion H(ω) = −ik d ω − k p . Explicitly accounting for the noise in the measurement of the position q n we arrive atq (ω) = m −1 χ fb (ω)F L − χ fb (ω) Ω 2 fb + iωγ fb q n (205) where the susceptibility χ fb (ω) = Ω 2 0 + Ω 2 fb − ω 2 + iω(γ 0 + γ fb ) −1 , and we introduced the feedback damping γ fb = k d /m and frequency shift Ω 2 fb = k p /m. The in-loop detector signal is q IL =q +q n . Hence, the measured in-loop two-sided PSD is [415,417] S IL qq (ω) = | χ fb | 2 m −2 S ff + | χ fb | 2 (Ω 2 0 − ω 2 ) 2 + ω 2 γ 2 0 S nn (206) = m −2 S ff + (Ω 2 0 − ω 2 ) 2 + ω 2 γ 2 0 S nn (Ω 2 0 + Ω 2 fb − ω 2 ) 2 + (γ 0 + γ fb ) 2 ω 2 ,\nwhere S qq = |q| 2 and S nn is the noise PSD of the in-loop detector (Fig. 47). The in-loop measurement noise is reintroduced into the feedback loop, which drives the particle. The correlations between the particle's position and the measurement noise lead to noise squashing, where the apparent signal is squashed below the noise floor [529] (Fig. 47b).\nIn contrast, an out-of-loop detector provides an independent measurement of the actual particle motionq OoL =q +q ν :\nS OoL qq (ω) = | χ fb | 2 m −2 S ff + | χ fb | 2 Ω 4 fb + ω 2 γ 2 fb S nn + S νν (208) = m −2 S ff + Ω 4 fb + ω 2 γ 2 fb S nn (Ω 2 0 + Ω 2 fb − ω 2 ) 2 + (γ 0 + γ fb ) 2 ω 2 + S νν ,\nwhere S νν is the noise PSD of the out-of-loop detector (Fig. 47).\nIntegration over the PSD yields the position variance40\nIn the approximation we set k p = Ω fb = 0 and γ fb γ 0 and we used S ff = k B T mγ 0 /π. The first term in equation (210) scales with the inverse of the feedback cooling rate. This term resembles the desired action of the feedback, which is to reduce the impact of the heating term given by the fluctuating force. The second term is proportional to the feedback damping rate, which is multiplied by the measurement noise. This term resembles the undesired but inevitable effect of the control-loop heating the particle by feeding back measurement noise. Consequently, there exists an optimum feedback cooling rate γ opt = S ff /S nn /(mΩ 0 ) , where the mode temperature reaches its minimum value q 2 min = 2π √ S ff S nn /(mΩ 0 ) . With cold damping, optically levitated nanoparticles have been cooled to 4 occupational quanta, corresponding to an effective temperature of 11 µK [418] (Fig. 47(b) and Fig. 42).\nClassically, the measurement is not perturbative and the position variance can be reduced to arbitrarily low values, because, at least in principle, the noise S nn can be made arbitrarily small (equation (210)). However, quantum mechanics imposes that every measurement is accompanied by a measurement back-action, leading to a finite contribution S nn even in an ideal measurement. In the case of an optical measurement this back action is the random arrival of photons and the resulting random momentum kicks that lead to a stochastic backaction force [416]. Therefore, in a regime where the measurement back-action is the dominant contribution to the force noise, the impression back-action product is S ff S nn = ( /(4π)) 2 /η det , where η det < 1 is the measurement efficiency [415,519]. The impression back-action product is a manifestation of the Heisenberg uncertainty principle and describes the tradeoff between measurement imprecision and measurement back-action. At the optimal feedback gain, the effective phonon occupation number n cm = mΩ 0 q 2 min / − 1/2 depends only on the detection efficiency as n min = (η −1/2 det − 1)/2. At the Heisenberg limit (η det = 1), when the imprecision noise is minimized by optimally detecting all photons scattered by the levitated particle, the particle's center-of-mass motion could, in principle, be brought to its quantum ground state n min = 0. Hence, to reach the ground state with feedback cooling, the experiment should be operated in the photon-recoil-limited regime (P gas ≤ 10 −8 mbar) and the collection of scattered photons needs to be optimized. A promising approach is to collect the scattered photons with a high-finesse optical cavity, which collects a fraction of (η P /(η P + 1)) of the overall scattered power due to the Purcell effect, where η P = (6F c λ 2 )/(π 3 w 2 0 ) is the Purcell factor for a cavity with finesse F c and beam waist w 0 [530]."},{"title":"Parametric feedback cooling","content":"While cooling levitated particles with linear feedback is very powerful, its experimental implementation along all three motional degrees of freedom is challenging. All-optical linear feedback requires three auxiliary laser beams to apply radiation pressure in all directions [527]. Electrical feedback relies on a finite net charge on the levitated object.\nThe method of choice to control charge-neutral optically levitated particle is parametric feedback cooling [407], which allows to control all three spatial degrees of freedom with the same laser that is used for trapping, thereby significantly reducing the experimental complexity.\nThe basic concept of parametric feedback is to introduce a modulation in time of the intensity of the trapping laser. In particular, applying a modulation δI(t) = I 0 (η para /Ω 0 )q q results in the parametric feedback force F param fb = −k para q 2 q,\nwhere k para = Ω 0 mη para is the parametric feedback gain. When the particle moves away from the trap center, the intensity modulation δI(t) stiffens the trap. When the particle falls back to the trap center the feedback softens the trap. Thus, by synchronizing the laser intensity modulation with the particle motion energy is extracted from the center-of-mass motion of the particle [488]. This principle is analogous to the way a child on a swing gains motional energy by modulating its center-of-mass. Constructing the feedback signal as in equation (211) provides exactly the right phase relation, such that the trap stiffens when the particle moves away from the trap center and softens when is falls back, thereby cooling the particle. However, latencies in the experimental implementation require an additional phase shifter to compensate for unwanted phase shifts. With an additional phase shift we can also amplify the particle motion. Note that, in the absence of active feedback, the particle's oscillation self-locks (entrains its phase) to the modulation in such a way that the motion is amplified [531]. Cooling therefore requires active stabilization of the modulation phase.\nExperimentally, the detector signal is first frequency-doubled and then phase-shifted, where the right phase is determined empirically by optimizing the cooling performance, i.e., minimizing the position variance q 2 . The QPD or split-detection measurement yields a position signal for each spatial direction. The feedback signal is produced for each axis independently and the three signals are summed together, which allows to effectively cool all spatial degrees of freedom with a single laser beam (section 5.3). For a particle oscillating at frequency Ω 0 with q(t) ∝ cos(Ω 0 t), parametric feedback generates a signal ∝ sin(2Ω 0 ). Thus, the feedback signal appears on the in-loop detector signal predominantly at 2Ω 0 . The contribution to the detector signal at Ω 0 is much smaller, on the order of ∼ η para q 2 and, thus, much smaller than the contribution from the particle motion q 2 . Hence, noise squashing of the in-loop detector signal does not occur and the trapping laser can also be used for particle detection.\nDue to the nonlinear nature of the parametric feedback, there is no closed expression for the detector PSD. In particular, the position distribution is not a thermal distribution because large amplitude fluctuations are damped more strongly than small fluctuations [481,532]. However, we can still assign an effective temperature to the center-of-mass motion k B T cm = mΩ 2 0 q 2 . For strong feedback (η para mΩ 0 γ 0 /(k B T) ), the position variance is [481,532] q 2 = 4k B T γ 0 πη para mΩ 3 0 .\nIn contrast to linear-feedback cooling, the effective temperature or position variance scales with the square root of the feedback gain and not linearly. This can be understood intuitively. The Fig. 47. Feedback cooling. (a) Single-sided power spectral densitiesS Ool yy of the motion of the nanoparticle measured by the out-of-loop detector for different feedback damping rates γ fb . The solid lines are Lorentzian fits to the data. The gray data points denote the measured shotnoise level on the out-of-loop detector. (b) Power spectral densitiesS IL yy measured by the in-loop detector for the same settings as in (a). In contrast to (a), for a large feedback gain the measured signal drops below the noise floor because of correlations between the detector noise and the feedback signal (noise squashing) . (c) Mode temperature T y derived from the out-of-loop signal as in (a) function of feedback gain γ fb . The red (blue) circles denote the measured values at a pressure of 1.4 · 10 −8 mbar (1.2 · 10 −7 mbar. (d) Steady state under parametric PLL feedback cooling. Mean occupation number along the three principal axes (x, y, z) as a function of gas pressure measured under constant feedback cooling. At low enough pressures, photon recoil becomes the main source of heating and therefore the occupation number remains constant. Adapted from Ref. [415] (a-c) and from Ref. [416] (d). modulation depth and therefore the feedback force is proportional to the energy of the particle. Hence, as the feedback reduces the particle motion it also becomes less effective. Nonetheless, the quantum mechanical theory suggests that ground state-cooling with parametric feedback cooling is possible [533], even though direct feedback seem to be more promising [534]."},{"title":"Parametric PLL cooling","content":"Parametric feedback extracts energy from or pumps energy into the particle center-of-mass motion by modulating the trapping laser and therefore the trap stiffness at twice the particle oscillation frequency. A phase locked loop (PLL) synthesizes an output signal whose phase is related to the phase of a sinusoidal input signal. Thereby, a PLL can track an input frequency and it can generate a frequency that is an integer multiple of the input frequency. Hence, tracking the detector signal with a PLL and feeding the PLL output at the second harmonic into the optical modulator results in a feedback term\nwhere k PLL = mΩ 2 0 η PLL is the PLL feedback gain and η PLL is the modulation depth. The PLL feedback loop keeps the phase φ PLL fixed with respect to the randomly changing phase of the particle oscillation. Similarly to the case of the q q parametric feedback, a phase of φ PLL = −π/4 reduces the oscillation amplitude. However, in contrast to the previous case, where the modulation depth is proportional to the variance of the particle position, the PLL feedback has a constant amplitude. As a consequence, the position variance with PLL parametric feedback is\nThe approximation holds for φ PLL = −π/4 and in the limit η PLL Ω 0 /2γ 0 1. Comparing equation (214) with equation (202), we can identify the PLL damping as γ PLL = η PLL Ω 0 /2. Interestingly, even though parametric feedback with a PLL is a nonlinear feedback, we find the same linear scaling as for linear feedback and also the energy distribution follows the Boltzmann distribution associated with a harmonic oscillator at thermal equilibrium with a bath at temperature T cm = (γ 0 /γ PLL )T [532]. Like in the case of linear feedback, noise in the feedback will ultimately limit how far one can cool. However, unlike in the case of linear feedback (equation (210)), there is no analytical solution for the minimum temperature for parametric feedback. Using PLL parametric feedback simultaneous cooling of all three motional degrees of freedom to millikelvin temperatures (n below 100) has been demonstrated [416] (Fig. 47(d))."},{"title":"Outlook","content":"Levitated particles have already opened the door to a wide range of interesting physics in the classical regime and are now entering into the quantum regime. The next challenges are to achieve ground state cooling for all three center-of-mass translational degrees of freedom, to control the librational modes and precession of the particle, and to preparte truely non-classical states.\nEven though our discussion has been focused on optical tweezers in high vacuum, their combination with cavity optomechanics will play an important role towards creating more exotic quantum mechanical states beyond the ground state. Once a strong cavity optomechanical interaction has been realized, a range of traditional quantum optics experiments can be realized with these massive particles such as quantum state transfer, quantum squeezing, entanglement, and teleportation [413,432,436].\nThe envisioned fundamental tests of quantum mechanics require to control the internal temperature of levitated particles, since high internal temperature will wash out quantum interference effects [535]. The internal temperature can be reduced with optical refrigeration that exploit optical dopants inside the particle [476,536]. Dopants can also be used to directly measure the particle temperature [537]. Experimental data of the internal dynamics of levitated particles could challenge our understanding of thermalization dynamics on the nanoscale [538] and it will be important to measure what is going on inside the particle.\nThere is still plenty of room in the classical regime, where levitated particles will allow to test fundamental questions in single particle thermodynamics in the underdamped regime and provide new insights into the nonlinear dynamics of nanomechanical resonators. Most attention has been given to the center-of-mass degrees of freedom and simple Gaussian beams. There remains a lot to be explored about the remaining degrees of freedom and complex light fields, for example beams carrying orbital angular momentum [539]. In addition, new materials, material combinations in the form of doped particles or composite particles and new shapes will enable unprecedented light matter interactions [489]. Optical levitation experiments will hugely benefit from the tools that have been developed for traditional optical tweezers experiments, such as structured light fields [540] and holographic tweezers [43,541]. Those tools will then also allow to study collective particle dynamics, for example optical binding and forces between levitated particles [11]. This will be required the development of new techniques and calibration methods that are able to resolve the fast particle dynamics in high vacuum at a high spatial resolution over a large field of view.\nCareful calibration with a solid understanding of all the statistical and systematic error sources will be paramount for precision measurements with levitated particles. In particular, experiments aiming at detecting near field forces at interfaces require to approach a surface with a levitated particle [542]. The surface will scatter the trapping light and, thus, these experiments need detection and calibration techniques that are robust in extreme experimental conditions that might even change over the course of a measurement. We thank Erik Hebestreit, Vijay Jain, Martin Frimmer, Felix Tebbenjohanns, Francesco Ricci for providing us with data and feedback and the Quidant and Novotny groups for stimulating discussions. IPC gratefully thanks Édgar Roldán for illuminating discussions about Carnot cycles and microscopic engines."},{"title":"Fig. 1 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"Fig. 3 .","content":""},{"title":"Fig. 5 .","content":""},{"title":"Fig. 7 .","content":""},{"title":"Fig. 8 .","content":""},{"title":"Fig. 9 .","content":""},{"title":"Fig. 12 .","content":""},{"title":"Fig. 14 .","content":""},{"title":"6 .","content":""},{"title":"Fig. 16 .","content":""},{"title":"3 .","content":""},{"title":"Fig. 17 .","content":""},{"title":"18 .","content":""},{"title":"Fig. 19 .","content":""},{"title":"Fig. 21 .","content":""},{"title":"Fig. 22 .","content":""},{"title":"Fig. 23 .","content":""},{"title":"Fig. 24 .","content":""},{"title":"Fig. 25 .","content":""},{"title":"Fig. 26 .","content":""},{"title":"Fig. 27 .","content":""},{"title":"Fig. 29 .","content":""},{"title":"Fig. 30 .","content":""},{"title":"Fig. 31 .","content":""},{"title":"Fig. 32","content":""},{"title":"Fig. 33 .","content":""},{"title":"Fig. 34 .","content":""},{"title":"Fig. 36 .","content":""},{"title":"Fig. 37 .","content":""},{"title":"Fig. 39 .","content":""},{"title":"Fig. 40 .","content":""},{"title":"Fig. 41 .","content":""},{"title":"Fig. 42 .","content":""},{"title":"Fig. 43 .","content":""},{"title":"Fig. 46 .","content":""},{"title":"Funding","content":""},{"title":"Table 3 .","content":""},{"title":"Table 4 .","content":""},{"title":"Table","content":""},{"title":"Table 8 .","content":""},{"title":"Table 11 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"In 1970, Arthur Ashkin [1] demonstrated that a focused laser beam can accelerate, decelerate, and even stably trap micrometer-sized neutral particles. Later, in 1986, Ashkin and colleagues reported the first realization of an optical tweezers [2,3]: a tightly focused beam of light capable of holding microscopic particles in three dimensions. Thanks to this work, Ashkin was awarded a share of the 2018 Nobel Prize in Physics [4]. Furthermore, one of Ashkin's co-authors, Steven Chu, would go on to use optical tweezing in his work on cooling and trapping atoms; this research earned Chu a share of the 1997 Nobel Prize in Physics [5].\\nIn the late 1980s, Ashkin and colleagues used optical tweezers to manipulate biological samples, starting from an individual tobacco mosaic virus and Escherichia coli bacterium [6]. In 1993, Lucien Ghislain and Watt Webb invented photonic force microscopy [?], where an optically trapped particle is used as an extremely sensitive cantilever to probe microscopic force fields ranging from femtonewtons (10 −15 N) to piconewtons (10 −12 N). Since the early 1990s, optical force spectroscopy has been utilized to characterize the mechanical properties of biomolecules and biological motors [7][8][9]. Optical tweezers have also been employed in many fields of physics [10][11][12][13][14][15], nanotechnology [16], spectroscopy [17], nanothermodynamics [18,19], soft matter [20], and biology [21].\\nAn optical tweezers consist of a laser beam focused by a high numerical aperture (NA) microscope objective. A microscopic particle (whose refractive index is higher than that of its embedding medium) can then be trapped near the focal spot because of the emergence of scattering and gradient optical forces: the scattering forces are due to the radiation pressure of the light beam and act along the direction of propagation of the beam; and the gradient forces pull the particle towards the high-intensity focal spot. Using optical tweezers, a very broad range of particles has been trapped, as shown in Fig. 1.\\nThere are already several resources to build and operate optical tweezers available in the literature. In the 2015 book Optical Tweezers: Principles and Applications [22], Phillip. H. Jones, Onofrio M. Maragó and Giovanni Volpe have discussed in detail the theory behind optical tweezers as well as how to build and operate optical tweezers setups. There are also several other tutorials on specific aspects of optical trapping. In particular, there are computational toolboxes to calculate optical forces using T-matrix methods by Timo Nieminen et al. [23] and geometrical optics by Agnese Callegari et al. [24]. There is a tutorial on how to simulate the Brownian motion of an optically trapped particle [25]. There are several tutorials on how to build optical tweezers setups by Stephen Smith et al. [26], John Bechhoefer and Scott Wilson [27], Angela Mellish and Andrew Wilson [28], David Appleyard et al. [29], Woei Lee et al. [30], Manoj Mathew et al. [31], and Giuseppe Pesce et al. [32].\\nIn this Tutorial, we complete and complement these already available resources providing a primer on how to calibrate optical tweezers and how to use them for advanced applications. In particular, after a brief general introduction on optical tweezers (section 2), we focus on describing and comparing the various calibration techniques that are available (section 3). Then, we discuss some of the most exciting cutting-edge applications of optical tweezers in a liquid medium (section 4), namely to study single-molecule and single-cell mechanics, microrheology, colloidal interactions, statistical physics, and transport phenomena. Finally, we consider optical tweezers in vacuum, which features different dynamics than a viscous medium, so that vacuum operation comes with its own experimental challenges and calibration methods (section 5). This Tutorial provides both a step-by-step guide ideal for non-specialists entering the field and a comprehensive manual of advanced techniques useful for expert practitioners. All the examples are complemented by the sample data and the software necessary to reproduce them [33].\"\n },\n {\n \"title\": \"Getting started\",\n \"content\": \"In this section we provide some essential background information on optical tweezers. First, we will describe a basic optical tweezers setup (section 2.1). Then, we will briefly discuss the Brownian motion of an optically trapped particle and the most basic optical tweezers calibration techniques (section 2.2). Finally, we will give a brief overview of more advanced optical tweezers and of alternative approaches for the trapping and manipulation of microscopic matter (section 2.3).\"\n },\n {\n \"title\": \"A simple setup\",\n \"content\": \"An optical tweezers is a highly-focused laser beam that can trap microscopic particles. In practice, an optical tweezers can be built by focusing a laser beam through a microscope, which can be either commercial or homemade. As illustrated in Fig. 2, a basic optical tweezers setup is comprised of three parts: trapping, imaging, and position detection.\\nTrapping. In most applications, the laser is continuous wave (CW), whose wavelength is in the visible or near-infrared region of the spectrum, and whose power is between 10 mW and 1 W.\\nThe laser beam is expanded by a telescope to overfill the back-aperture of a high-NA microscope objective, is directed by a dichroic mirror towards the objective, and finally is focused by the objective to a diffraction-limited spot within the sample holder. This focal spot is the optical tweezers, which can trap a microscopic particle. Usually, the object to be trapped is suspended in an aqueous solution, which is held between a microscope slide and a cover slip; sometimes the particles to be trapped are suspended in other liquids, or in gases, or even in vacuum (section 5).\\nImaging. Typically, the trapping and imaging objectives are the same. The light for the imaging is provided by a condenser, which is an integral part of any microscope. It is convenient to image through a dichroic beam splitter, which reflects the trapping light to the microscope, but also transmits some of the imaging light from the condenser to a camera. In both cases, some color filters are typically required in front of the camera to reduce the intensity of the trapping beam to an acceptable level such that the camera is not overexposed.\"\n },\n {\n \"title\": \"Position detection.\",\n \"content\": \"To use the optically trapped particle as a microscopic force transducer, it is necessary to track its position. This can be done by analyzing either the videos of the particle acquired by the camera (digital video microscopy), or the time series acquired through a position detector. In this latter case, the condenser collects the forward scattered light and projects it onto a position detector (typically either a quadrant photodetector or a position sensing detector). This detects the changes in the forward scattered light pattern due to the changes in the particle's position.\"\n },\n {\n \"title\": \"Optical tweezers calibration\",\n \"content\": \"An optically trapped particle can be used as an extremely sensitive microscopic force transducer that can both exert and measure forces ranging from hundreds of piconewtons down to a few femtonewtons.\\nAn optically trapped particle experiences a harmonic force that tends to keep it near a stable equilibrium position. Considering only one dimension, for small displacements from the equilibrium position, the force acting on a colloidal particle positioned at x within a trap centered at x eq is\\nwhere κ x is the trap stiffness along the x-direction. The resulting trapping potential is harmonic:\\nA typical trajectory of an optically trapped particle can be seen in Fig. 3(a). The particle keeps on moving due to the presence of Brownian fluctuations. Therefore, an optically trapped particle is in a dynamic equilibrium between the thermal noise continuously pushing it out of the trap and the optical forces driving it towards the equilibrium position. For the particle to remain within the optical trap, the optical potential well must be sufficiently deep. The depth of the optical potential is typically characterized in units of the thermal energy k B T, where k B is the Boltzmann constant and T is the absolute temperature. This quantity gives a characteristic energy scale for mesoscopic phenomena. The potential well of an optical tweezers should be at least a few k B T deep to be able to confine a particle, as shown in Fig. 3(b).\\nThe analysis of the thermal motion of the optically trapped particle in the optical potential provides information about the local forces acting on the particle. For example, since the particle is held in a harmonic trapping potential, its position distribution is Gaussian, as shown in Fig. 3(c), By sampling this probability distribution, it is possible to measure κ x using the equipartition theorem. The equipartition theorem states that for a system at thermodynamic equilibrium at absolute temperature T, the energy associated with each harmonic degree of freedom is equal to 1 2 k B T. Therefore, U(x) = 1 2 κ x (x − x eq ) 2 = 1 2 κ x σ 2 x = 1 2 k B T, where σ 2 x = (x − x eq ) 2 is the variance of the trapped particle position, and\\nAs we will see in section 3, there are more sophisticated methods to use the thermal fluctuations of an optically trapped particle to calibrate its stiffness. In particular, it is possible to also exploit the temporal evolution of the particle in the optical trap, for example by calculating its correlation and power spectral density.\\nOnce an optical tweezers is calibrated, a constant and homogeneous external force F ext, x shifts the equilibrium position of the trap. The value of the force can be obtained as:\\nwhere ∆x eq is the average particle displacement from the original equilibrium position without the external force.\"\n },\n {\n \"title\": \"Advanced and alternative approaches\",\n \"content\": \"Going beyond the basic setup introduced in the previous section, more advanced setups become necessary to extend the optical manipulation toolkit, introducing, e.g., optical spanners [34], optical rheometers [35], optical stretchers [36], and optical sorters [37][38][39]. These tools allow one to manipulate multiple particles at once [40][41][42][43], rotate particles [44][45][46][47], measure the mechanical properties of biological samples [48], sort particles [49], trap particles with refractive index lower than their medium [50][51][52][53], induce optical binding and collective dynamics [11,[54][55][56][57][58][59]. Complex optical potentials, such as periodic and random patterns, provide experimental models to study microscopic and non-linear transport mechanisms [60][61][62][63][64][65][66][67]. Advanced optical manipulation has been employed also to build and study micromachines [68][69][70] and optofluidic devices [37,[71][72][73][74][75][76]. Furthermore, optical manipulation setups have also been successfully employed to investigate the properties of light itself, employing structured light beams with peculiar intensity, phase and polarization distributions [34,44,[77][78][79][80][81].\\nIn this section, we will provide a brief overview of the main advanced optical manipulation techniques. We will consider only the overdamped regime (i.e., particles immerse in a viscous medium like water), since the case of underdamped systems involves many different experimental aspects that will be treated in (section 5). Also, unless otherwise stated, we will assume that the refractive index of the particles under consideration is higher than that of the surrounding medium.\"\n },\n {\n \"title\": \"Beam steering\",\n \"content\": \"Most advanced optical micromanipulation setups require to steer the trapping beam to control the position of the optical tweezers. The beam steering device can be as simple as a mirror on a gimbal mount (preferred over conventional kinematic mounts to avoid mirror shifting). For example, this simple approach can be used to move the optical trap within the sample instead of moving the sample with a translation stage while keeping the optical trap fixed. More sophisticated beam-steering devices are also often employed, such as acousto-optic deflectors and spatial light modulators (SLMs).\\nWhen implementing beam steering, it is essential to ensure that the available light reaches the back aperture of the microscope objective, irrespective of the beam direction. To this end, a telescope should be added (4f-system) to create conjugate planes between the beam steering device and the back aperture of the objective, as shown in Fig. 4. This telescope can also be used to magnify the beam to slightly overfill the back aperture of the objective (to improve trapping), and a spatial filter can be added at Fourier plane 1 (to remove unwanted higher-order diffraction patterns), which is conjugated with Fourier plane 2. Fig. 4. Beam steering mechanism. The trapping beam is steered by the steering device (e.g., a galvo-mirror, an acousto-optic deflector, or a spatial light modulator). To ensure that the trapping beam reaches the back aperture of the microscope objective (O) while being steered, a telescope is added along the beam path (lenses L 1 and L 2 ) in a 4f-configuration ( f 1 and f 2 are the focal lengths of L 1 and L 2 respectively).\"\n },\n {\n \"title\": \"Polarization splitting\",\n \"content\": \"The simplest alternative to have two neighboring traps while avoiding the interference between them is to split the original beam into two orthogonally polarized beams [40,82], as shown in Fig. 5. If the main control parameter of a given experiment is the distance between the traps, a single steering device can be enough, keeping one trap fixed while the other one is shifted. This works fine for a small distance between the traps. If the traps need to be moved over a large area, it might be useful to use two steering mirrors to minimize the distortion of each trap. Dynamical control of each trap can be achieved if the gimbal mounts are operated by actuators along the two axes, giving rise to a continuous motion along arbitrary trajectories in two dimensions in the sample plane. Finally, this twin-trap approach can also be used in combination with other advanced techniques, if the aim is to superimpose two complex patterns without interference between them. 1 and GM 2 should be mounted on gimbal mounts, and the distance between each of them and the lens L 1 should be equal to the focal length of L 1 (Fig. 4). For alignment purposes, both traps should be initially superimposed, arriving exactly at the same point in the sample plane. The intensity ratio between the traps can also be controlled, by placing a polarizer followed by a λ/2-retarder at the entrance of the Mach-Zehnder setup. As the retarder is rotated, the proportion of light reflected and transmitted through PBS 1 will change. DM: Dichroic mirror. M: mirror. O: microscope objective. C: Condenser.\"\n },\n {\n \"title\": \"Time-sharing optical traps\",\n \"content\": \"A single beam can be used to generate multiple optical tweezers if it is scanned sufficiently fast, i.e., with a frequency high enough to ensure that an optically trapped particle cannot diffuse away from the optical trap region between consecutive visits of the laser beam. This technique is known as time-sharing or laser-scanning optical trap. It can be performed with different deflecting devices, such as acousto-optic deflectors (AODs), mirror galvanometers (galvo-mirrors), electro-optic deflectors (EODs), and piezo-mirrors [40,[83][84][85][86]. Often, the beam deflecting devices have only one degree of freedom so that, to achieve beam motion over the whole sample plane, it is necessary to use two beam deflecting devices, one for each transverse direction, as illustrated in Fig. 6.\\nThe angular range and the scanning frequency of the beam deflecting device determines the number of traps that can be created simultaneously and the extent of the trapping area. There is usually a compromise between the scanning speed and the deflection angular range. The first setup of time-sharing optical tweezers was built with galvo-mirrors [83], which could reach scanning frequencies of the order of 10 Hz. Nowadays, there are similar commercial systems that can reach up to 100 Hz, with a angular range of up to 25 • ; however, the angular range in the optical trapping setup is limited by the maximum acceptance angle of the microscope objective, which is just a few degrees at most. AODs can attain frequencies from tens to hundreds of kilohertz with angular ranges of up to 3 • , which makes them very suitable for trapping applications. For example, these devices have been used in studies of DNA stretching and molecular motors [40]. However, as the AOD working principle is based on the generation of diffraction gratings by means of the propagation of acoustic waves in a glass or quartz cell, resulting in a periodic refractive index modulation, it is worth to keep in mind that the diffraction efficiency can be at Fig. 6. Time-sharing optical tweezers. The trapping beam is time-shared between several optical traps using two beam deflector devices along the xand y-directions (X-BDD and Y-BDD, respectively). Two telescopes are needed (L 1 -L 2 and L 3 -L 4 ) to create conjugate planes between the two deflector devices as well as with the back aperture of the microscope objective O. most around 70%. EODs are even faster than AODs, but with smaller angular ranges of fractions of a degree. Piezo-mirrors are more limited, reaching frequencies of hundreds of Hertz with angular ranges of only a few hundredths of a degree.\\nIf the position of a given trap is slightly shifted from one visit to the next one, the bead will follow it, making it possible to describe continuous trajectories within the plane. The average intensity in each trap can also be controlled by means of the number of visits of the scanning beam in a full cycle. For example, if a trapping site is visited twice as much as another one, it will experience twice the average intensity.\"\n },\n {\n \"title\": \"Interferometric optical traps\",\n \"content\": \"Interference patterns can trap multiple particles simultaneously and generate ordered structures [54,55,87]. For example, consider the simplest interference pattern consisting of alternating bright and dark fringes. The dynamics of the particles depend on their size with respect to the characteristic size of the fringes. Particles smaller than the fringes are attracted towards the bright fringes, while remaining free to move along the fringes. However, much more interesting dynamics emerge when the particle size is comparable or even larger than the fringes [39,49,88]. This has been called the size effect and will be explained in more detail in section 4.6.\\nTo generate alternating bright and dark fringes, we need two collimated beams travelling with a relative angle: the larger this angle, the smaller the fringe period (the smallest fringe pattern is half wavelength). One possibility is to use a Mach-Zehnder interferometer using non-polarizing beam splitters ( Fig. 7(a)). A 4 f -system is formed by the lens L 1 and the microscope objective that conjugates the mirror on the gimbal mount and the focal plane of the objective (the objective can be replaced by an aspherical lens, since the high-NA requirement may be relaxed in this case). While an extended interference pattern generates strong intensity gradients in the transverse plane, the intensity gradient along the propagation direction is usually not enough to trap in 3D so that the particles are pushed by the scattering force against either the bottom or the top surface of the sample cell.1 Fig. 7(b) shows an alternative configuration for the interferometer module. In both configurations (Figs. 7(a) and 7(b)) a single mirror mounted on a gimbal mount is enough to control the fringe period for small interference angles.\\nAlternative configurations are possible, an example of which is illustrated in Fig. 7 two beams are redirected by mirrors M 1 and M 2 towards a right-angle prism, which reflects them towards the objective, which in turn focuses them on the sample. The right-angle prism is mounted on a single-axis translation stage to control the distance between the reflected beams and, therefore, the interference angle. A low NA objective is advisable for this configuration to minimize the changes in the intensity distribution along the z-axis within the sample depth.\\nAn extreme case of interference pattern is a speckle light field, which arises from the interference of a multitude of coherent waves with random phases, generated when laser light is, e.g., reflected from a rough surface or transmitted through a turbid medium. Speckle patterns have also been used for optical manipulation [65,89,90].\"\n },\n {\n \"title\": \"Holographic optical tweezers\",\n \"content\": \"The most versatile optical tweezers setups are arguably based on holographic optical tweezers (HOTs) [41][42][43][91][92][93]. The basic working principle of HOTs is shown in Fig. 8. The wavefront of an incident beam is reshaped by a hologram projected on a spatial light modulator (SLM) and then relayed by a series of lenses to an objective that then generates the desired light pattern at the sample. In practice, there are many possible different HOT configurations. Discussing their details is out of the scope of this Tutorial, but a comprehensive review of the topic (including hologram computing algorithms, a numerical toolbox, and a comparison in terms of the main parameters such as efficiency, average intensity, uniformity, and percentage standard error) can a b c d\\nFourier-plane phase mask Image-plane phase mask In the cases (a)-(c), the light pattern is obtained at the Fourier plane ( Fig. 8(a)), while in case (d) it is at the image plane ( Fig. 8(b)).\\nbe found in Ref. [22]. Here, we will focus our attention on only two approaches.\\nIn the first approach, the desired light pattern at the sample is the Fourier transform of the modulated laser beam right after the SLM. It is necessary to include a 4 f -system between the SLM and the back aperture of the objective to conjugate the two optical planes and to perform spatial filtering (Fourier Plane 1 in Fig. 8(a)). This approach works particularly well to generate multiple focal spots distributed in predetermined locations in space, which serve as individual traps. The simplest case is a single steerable trap ( Fig. 9(a)), and the corresponding hologram is a superposition of a blazed diffraction grating (which determines the lateral position of the focal spot) and a Fresnel lens (which shifts the focal spot along the propagation axis). For the generation of multiple traps, there are different algorithms to calculate the hologram, such as the superposition of gratings and lenses, the random mask encoding, the Gerchberg-Saxton algorithm, and the adaptive-additive algorithm [22]. An example of a phase mask that produces a set of individual traps can be seen in Fig. 9(b). Structured beams (e.g., Laguerre-Gaussian beams, Fig. 9(c)) can also be straightforwardly created with this approach. Furthermore, continuous optical potentials with arbitrary shapes can be produced as well by calculating the corresponding hologram with the aforementioned algorithms [22].\\nIn the second approach, a phase profile related to the desired light pattern is directly projected on the SLM and relayed on the objective image plane (this requires an even number of lenses between the two planes, Fig. 8(b)). This configuration is specially suitable when the field of interest is not meant to be tightly focused (low-NA optics), e.g., using phase gratings to split the beam and generate interference fringes at the objective image plane [94]. Also, Bessel beams ( Fig. 9(d)), Mathieu beams, and nondiffracting beams are typically generated at the image plane instead of at the Fourier plane [45,52,79,[95][96][97][98][99][100][101][102].\\nTypical switching speeds for nematic liquid crystal SLMs can reach 60 Hz with a large number of discrete phase levels (typically 256), whereas ferroelectric liquid crystal SLMs can switch at a rate of up to 1 kHz, but with only two phase levels (0 and π).\"\n },\n {\n \"title\": \"Calibration\",\n \"content\": \"As we have seen in section 2.2, an optical trap acts as a microscopic Hookeian spring with a fixed stiffness so that the restoring force is proportional to the displacement of the optically trapped particle from the center of the optical trap. The calibration of an optical tweezers entails the determination of the value of the optical trap stiffness, which in general depends on the properties of both the light beam and the particle. The calibration of an optical tweezers is essential to use it as a microscopic force transducer, i.e., to measure and exert forces in the femtonewton and piconewton range.\\nIn principle, it should be possible to determine the value of the stiffness for a given experimental configuration where the properties of the focused laser beam and the optically trapped particle are known. In fact, this has been done in certain cases [67,[103][104][105]. However, this is in general a cumbersome task because an exact description of the focused beam and its interaction with particles depends on several parameters that are very difficult to know accurately and may vary drastically even within the same experiment. For example, just the theoretical description of the focused beam requires precise information about the focusing lens, the thickness of the coverslip, the refractive index of the media, and the position of the focus inside the sample. Moreover, even with all this information at hand, the interaction between a focused beam and a trapped object is very complex, requiring to solve a computationally intensive electromagnetic scattering problem [22]. Therefore, the development and improvement of in-situ calibration methods have been very important to extend the practical applications of optical tweezers.\\nThe calibration methods can be classified as either passive or active. Passive methods (sections 3.2 to 3.10) use information about the Brownian trajectory of a particle trapped in a static optical tweezers. Active methods (section 3.11) analyze the response of the particle to a known time-dependent perturbation to the optical trap exerted by an external force, e.g., generated by a periodic movement of the sample cell or of the position of the optical tweezers. The force exerted by the optical tweezers can also be directly determined by measuring how the trapping light is scattered using a digital camera or a photodetector (section 3.12). Most of the mathematical details regarding the non-linear fitting are in section 3.13.\\nTo illustrate these calibration methods, we have carried out a series of experiments ( Table 1). The experimental data and the MATLAB implementation of the calibration techniques explained in this section are provided as supplementary materials [33]. An overview and comparison of the calibration methods is in Table 2 and the results obtained on the test datasets is in Table 3.\\nWe remark that all methods we are going to describe rely on the knowledge of the particle position in units of length (i.e., meters). However, the detection devices used in optical tweezers setups (e.g., CCD cameras, quadrant photodiodes, and position sensitive detectors) provide signals in pixels or volts. Therefore, it is necessary to determine the conversion factor from pixels or volts to meters. For CCD camera, the pixel-to-meter conversion factor can be determined by standard microscopy techniques (e.g., by using a target with know dimensions) [22,32]. For quadrant photodiodes or position sensitive detectors, the volt-to-meter conversion factor can be determine by scanning an optically trapped particle across a (much weaker) detection beam, in which case one need to have at least two beams. It is also possible to perform this calibration self-consistently using only the position signal of an optically trapped particle acquired by a photodetector, when the diffusion coefficient D of the particle is know (e.g., when the particle radius, the medium viscosity and the temperature are known). The basic idea is that the knowledge of D determines the scaling factor of the functions of the particle position used in the calibration process. The details go beyond this Tutorial and can be found in Ref. [22].\"\n },\n {\n \"title\": \"Parameter\",\n \"content\": \"\"\n },\n {\n \"title\": \"Value\",\n \"content\": \"\"\n },\n {\n \"title\": \"Particle\",\n \"content\": \"Polystyrene particle, radius a = Table 1. Experimental parameters. A microscopic particle immersed in water is trapped with a tightly-focused laser beam in a standard optical tweezers configuration and tracked using a quadrant photodiode at three different laser powers.\"\n },\n {\n \"title\": \"Time series acquisition and notation\",\n \"content\": \"We consider a spherical particle of radius a immersed in a fluid with viscosity η at an absolute temperature T. According to Stokes' law, the friction coefficient is given by:\\nwhile the diffusion constant is given by the Einstein-Smoluchowski relation\\nwhere k B = 1.3806 · 10 −5 pN µm K −1 is Boltzmann's constant. The trajectory x(t) of an overdamped Brownian particle is described by the overdamped Langevin equation\\nwhere F(x) is a deterministic force exerted on the particle (in our case the optical force) and W x (t) is a white noise with zero mean and Dirac-delta-correlated variance, i.e., W x (t)W x (t ) = δ(t −t ). Experimentally, the particle's position is recorded at discrete times, yielding a time series {x } N =1 , where x ≡ x(t ) at times t = ∆t with = 1, . . . , N. To provide estimates with error bars, we can either repeat the experiment M times or divide the time series of a single experiment of size S into M blocks of N = S/M data points each. Either way, we end up with a set of times series {x (m) } N =1 for m = 1, . . . , M. A summary of the notation is in Table 4.\\nAdvantages. Works for any potential (not only harmonic potentials); it does not require fast acquisition but long acquisition times (in a harmonic optical trap, T τ ot, x , where τ ot, x is the optical trap characteristic time). Disadvantages. It does not provide the dynamical properties of the optically trapped particle (e.g., γ or D).\\nResult. It infers only κ x .\\nAdvantages. The simplest and easiest method to implement; it does not require fast acquisition. Disadvantages. It is restricted to harmonic potentials; it requires large datasets taken for long times T τ ot, x ; the inferred values are very sensitive to errors in the position detection.\"\n },\n {\n \"title\": \"Mean squared displacement (MSD) analysis (section 3.4)\",\n \"content\": \"Result. It infers both κ x and γ. Advantages. it tends to provide accurate values. Disadvantages. It only applies to harmonic potentials; it is very slow due to the large number of operations in estimating MSD; it requires fitting a non-linear function; experimental errors are correlated making estimation from least squares more involved.\"\n },\n {\n \"title\": \"Autocorrelation function (ACF) analysis (section 3.5)\",\n \"content\": \"Result. It infers both κ x and γ. Advantages. Algorithmically faster than MSD. The fitting can be transformed into a linear one. Disadvantages. It only applies to harmonic potentials; it requires fitting a function; experimental errors are correlated making estimation from least squares more involved.\\nPower spectral density (PSD) analysis (section 3.6)\\nResult. It infers both κ x and γ. Advantages. Algorithmically faster than MSD and ACF; it provides accurate results; it can filter unwanted noise. Disadvantages. It only applies to harmonic potentials; one must fit a function, but this can be transformed into a linear problem. Key reference: [106] Force reconstruction via maximum-likelihood-estimator (FORMA) analysis (section 3.8)\\n, κ x /γ, D\"\n },\n {\n \"title\": \"Result.\",\n \"content\": \"It infers both κ x and γ. Advantages. It is computationally fast to implement; it does not require a large dataset; the estimates are given by simple analytical expressions; it can be extended to any dimension and non-conservative forces. Disadvantages. It works for ∆t τ ot, x , although this can be generalized; estimates for the friction coefficient are sensitive to the exposure time. Key reference: [107] Bayesian inference analysis (section 3.9)\\nResult. It infers both κ x and γ. Advantages. It is computationally fast to implement; it can handle small datasets and include information a priori; the estimates are given by simple analytical expressions; it gives the whole probability of the estimates; it can be extended to any dimension and non-conservative forces. Disadvantages. it works for ∆t τ ot, x , although this can be generalized; estimates for the friction coefficient are sensitive to the exposure time. Key reference: [108] Table 2. Overview and comparison of calibration methods in one dimension in the passive configuration. In this table we summarize the practical details of each method, together with advantages, disadvantages and a key reference, whenever possible.\"\n },\n {\n \"title\": \"Potential analysis\",\n \"content\": \"The potential method is fairly easy to implement and works for any conservative force with a potential that enables equilibrium of the particle in an accessible time. For a conservative force\\nis the equilibrium potential, and the probability density of finding the particle at a given position x at thermal equilibrium follows the Maxwell-Boltzmann distribution: where ρ 0 is a normalization factor ensuring that ∫ ρ(x)dx = 1. The potential U(x) can then be reconstructed as\\nThe expression in equation (9) is an equilibrium distribution that theoretically corresponds to the probability of finding a particle at a given position x from an statistical ensemble of independent particles. In practice, this distribution is usually computed from data sets coming from the trajectory of a particle, consisting of data points not necessarily uncorrelated. For example, in a harmonic potential, two points along the trajectory x(t) and x(t ) are uncorrelated if |t − t | τ ot, x , where τ ot, x = γ/κ x is the characteristic relaxation time of the trap. Thus, to ensure that we have enough independent samples T s ∆t τ ot, x (the second condition ensures that the samples are uncorrelated). An important limitation of the potential methods is that, to sample rare events (e.g., the anharmonicity of the potential), one requires a number of measurements N ∼ 1 ρ(x) , and depending on the point x, N can be extraordinarily large. On the other hand, being an equilibrium method, the potential method does not rely on the knowledge of the viscosity and the radius of the particle to obtain an estimate for the trap stiffness.\"\n },\n {\n \"title\": \"Notation\",\n \"content\": \"\"\n },\n {\n \"title\": \"Explanation\",\n \"content\": \"x ≡ x(t ) position of particle at time t = ∆t Table 3. 1. Determine the common maximum and minimum values of the position variable, denoted x max and x min , respectively.\\nAs an example, Fig. 10 illustrates the experimental distributions and potentials for a particle optically trapped at three different powers (corresponding to experiments I, II, and III, Table 1). By means of either a linear (black solid lines) or non-linear fit (dashed red lines), we are able to estimate the value of κ (ex)\\nx for the experiments I, II and III (using P = 50 bins and the M = 5 experiments to estimate error bars for each bin). These estimates of κ (ex)\\nx are given in Table 3.\"\n },\n {\n \"title\": \"Equipartition method\",\n \"content\": \"The equipartition method is a very simple and easy to implement method, which does not even require fitting a function. The equipartition method is a particular case of the potential method where we assume a harmonic potential:\\nso that the particle position distribution corresponds to the Gaussian distribution\\nThus, the thermal average of the potential is\\nwhich is a restatement of the equipartition theorem, imposing that the energy associated with the harmonic potential degree of freedom is equal to 1 2 k B T. This provides a direct method to estimate κ x as\\nwhere we can estimate the equilibrium average (x − x eq ) 2 by a time average. As for the potential method, the trajectory needs to be long enough to probe the equilibrium distribution (T s ∆t τ ot, x ) and work with the uncorrelated subsample {x (m) } L =1 .2 For each trajectory m = 1, . . . , M, an estimate of the ensemble average (x − x eq ) 2 is given by its sample variance\\nwhere\\n(m) corresponds to the sample mean of the m-th experiment. Thus, for each experiment we obtain\\nUsing the results of the M experiments, we calculate its mean value and variance, viz.\\nUsing this approach on the experiments I, II and III (Table 1), yields the estimates of κ x reported in Table 3. These estimates are consistent with those obtained using the potential method albeit with larger errors. The harmonic potential corresponding to these estimates of the stiffness is plotted with black circles in Fig. 10 and is also in good agreement with the directly measured potential (blue circles).\\nSince the estimation of κ x depends on the inverse of the variance of the position, any random error in the position detection will always lead to an underestimation of the stiffness. On the other hand, if the data position is retrieved after a band-pass filter operation, the variance will diminish and hence the stiffness will be overestimated.\"\n },\n {\n \"title\": \"Mean squared displacement analysis\",\n \"content\": \"Differently from the previous two methods, the mean squared displacement (MSD) analysis takes advantage of the time-correlated properties of the motion of the Brownian particle in the harmonic potential. From the Langevin equation (8) with linear restoring force F(x) = −κ x x, the MSD is:\\nwhere (· · · ) denotes time average and τ ot, x = γ κ x is the characteristic relaxation time.3 MSD x (τ) comprises two different time regimes: for τ τ ot,x , MSD x (τ) ≈ 2k B T γ |τ|, the particle exhibits 2It is possible also to use a correlated time series as long as the sampling is done at regular time intervals. 3Following Ref. [109], we first see that the formal solution of the Langevin equation (8) is given by\\nFrom here one can find, using the properties of white noise of W x (s), that\\nThis result can be used to derive the expression for the MSD x (τ) given by equation (21). free diffusion, while for τ τ ot, x , MSD x (τ) ≈ 2k B T κ x , the particle feels the harmonic potential and we recover the equipartition result given by equation (17).\\nIf we measure MSD x (τ) experimentally (blue markers in Figs. 11(a-c)), we can estimate the trap stiffness κ x from a nonlinear fit to equation (21). This requires that ∆t < τ ot, x , such that the time series {x (m) n } N n=1 is correlated for each m = 1, . . . , M. Denoting t j = j∆t for j = 1, . . . , N and τ = ∆t, the experimental estimate for the MSD is:\\nFrom the estimates {MSD (ex)\\nx,m (τ )} M m=1 , we obtain their experimental mean and covariance matrix:\\n} and the theoretical model of the MSD in equation (21), we construct χ 2 according to equation (73) and minimize it numerically to find the estimates κ x and τ ot,x (red shaded lines in Figs. 11(a-c)). There are several aspects to consider when proceeding with the non-linear fitting: First, since the experimental data is time-correlated, we need the covariance matrix when inferring the parameters from the non-linear fitting. Second, as the MSD shows two distinct time-regimes, we must be sure to have a good balance of points in both regimes to avoid overweighting one respect to the other. Third, the experimental error of the MSD increases as the lag time grows since the number of data points used to estimate MSD diminishes. Fourth, errors in position detection have a strong impact particularly in the short time regime, because the MSD is proportional to the position squared. It is good practice to exclude the short times from the MSD fit, up to a fraction of the expected τ ot,x , as we did in Fig. 11.\\nFor the experimental data recorded in experiments I, II and III (Table 1), we perform the nonlinear fitting on the interval [0.5τ ot,x , 6τ ot,x ] and estimate the errors as explained in section 3.13. This provides us with the estimates for κ (ex)\\nx and τ (ex)\\not, x , from which γ = κ x τ ot, x and D is given by the Einstein-Smoluchowski relation (7). All these estimates are reported in Table 3.\"\n },\n {\n \"title\": \"Experimental MSD\",\n \"content\": \"Non-linear fitting Fig. 11. Mean square displacement (MSD) analysis. Comparison between the theoretical expression (21) of the MSD and the experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The red dashed lines are the non-linear fitting and the blue markers with error bars are the experimental estimates. The data are fitted in the interval [0.5τ ot,x , 6τ ot,x ] (the upper limit is shown by the dashed vertical lines). The estimates are given in Table 3.\\nThe MSD analysis has a few clear advantages over the equilibrium methods described in the previous two sections. First, the MSD uses more information so that the trap stiffness is estimated more precisely. Second, the MSD analysis can infer also dynamic properties, namely the friction coefficient, which is not directly accessible with the equilibrium methods. However, care must be taken when fitting a function with correlated errors and to have a good balance of data points between the short-time and long-time regimes.\"\n },\n {\n \"title\": \"Autocorrelation function (ACF)\",\n \"content\": \"Starting from the Langevin equation (8) with linear restoring force F(x) = −κ x x, the position autocorrelation function (ACF) is [109]\\nFrom a correlated time series sampled at regular time steps ∆t, we can estimate the ACF by using\\nwhich gives a set of estimates {C (ex)\\nx,m (τ )} M m=1 for each experiment m = 1, . . . , M (blue markers with error bars in Figs. 12(a-c)). From these estimates, we obtain their experimental mean and covariance matrix:\\nUsing the the autocorrelation equation (25), we can build χ 2 (κ x , τ ot, x ) according to equation (73) and minimize with respect to the parameters κ x and τ ot, x , solving the resulting set of equations by means of non-linear procedures using iterative solutions (red dashed lines in Figs. 12(a-c)). Another possibility is to use the logarithm of the autocorrelation instead, i.e., Experimental ACF Linear fitting Non-linear fitting (25) of the ACF and the experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The black solid lines and red dashed lines are the linear and non-linear fitting, respectively, while the blue markers with error bars are the experimental estimates. The data are fitted on the interval [0, 2.5τ ot,x ] (the upper limit is shown by the dashed vertical lines). The estimates are given in Table 3. ln C x (τ) = ln(k B T/κ x ) − τ/τ ot, x and perform a linear regression, in which case the experimental dataset is {ln C (ex)\\nx (τ )} together with their covariance matrix for the M experiments (black solid lines in Figs. 12(a-c)). As in the MSD analysis, from the estimates for κ x and τ ot, x we can determine estimates for γ = κ x τ ot, x and D = k B T/γ.\\nAs in the MSD analysis, the ACF analysis requires ∆t τ ot,x and T s τ ot,x . Furthermore, since the ACF function decays exponentially with time, only the values corresponding to short times, around the relaxation time, are relevant to get the information of the confining potential.\\nThe estimates for κ x , τ ot, x , γ and D performing the fittings on the interval [0, 2.5τ ot,x ] are provided in Table 1. The ACF analysis results are comparable to those obtained with the MSD analysis, both using non-linear and analytical linear regression. However, the errors for κ (ex)\\nx are bigger than those obtained from the MSD analysis, which can be expected from the expression of the ACF (equation (25)), whose initial value is solely determined by the trap stiffness and not the decaying behavior so that we only have one data point to estimate κ (ex)\\nx . An important difference is also the computation time: while the ACF is computed with very efficient algorithms based on the Fast Fourier Transform, the MSD requires a rather inefficient numerical computation.\"\n },\n {\n \"title\": \"Power spectral density (PSD)\",\n \"content\": \"Calibration of an optical tweezer by means of the power spectral density (PSD) is considered to be one of the most reliable methods due to its ability to remove common sources of noise with relative ease since one works in frequency space. In contrast to other methods, the PSD analysis is very robust to unwanted periodic noise in the extracted trajectory of the particle, due to, for example, illumination, electronic noise, fan noise of different electronic components inside the laboratory, or even some resonant vibration that is difficult to isolate with optical tables. The method explained in this section is based on Ref. [106].\\nStarting from the Langevin equation (8) with F(x) = −κ x x and performing a Fourier transform, one obtains the PSD, which has a Lorentzian form and is the expected value of the average energy in the frequency domain, Fig. 13. Power spectral density analysis. Comparison between the theoretical expression (29) and experimental estimates for a particle optically trapped at (a) 2.3 mW, (b) 6.0 mW, and (c) 9.2 mW ( Table 1). The blue dots are the experimental estimates |x k | 2 T s given by equation (37), the solid red lines correspond to its expected value P (ex) α = |x k | 2 /T s n w over the n w frequencies belonging to the subdomain α, and the black dashed lines are the result of the linear fitting. The vertical dotted lines indicate the maximum frequency used to perform the fitting; in this example, f max ≡ max{ f α } = f Nyq /4 = f s /8. The estimates are given in Table 3.\\nwhere x( f ) is\\nand f c, x = κ x /(2πγ) = 1/(2πτ ot, x ) is the corner frequency. To derive equation (29), we have used the fact that the spectrum of the zero-mean white noise, | W x ( f )| 2 , has an exponential distribution with an expected value | W x ( f )| 2 = 1. Experimentally, we have a trajectory sampled with frequency f s = 1/∆t up to time T s , i.e., a time series x j = x(t j ) with t j = j∆t and j = 1, . . . , N, which is used to perform the discrete Fourier transform (DFT):x\\nIt is expected that |x k | 2 /T s ≈ |x( f k )| 2 when f s f c, x , so it is important to sample the trajectory at frequencies some orders of magnitude higher than the corner frequency. Furthermore, the discrete PSD will suffer from aliasing. A simple formula for the expected value of the aliased PSD can be derived from the discretized Einstein-Ornstein-Uhlenbeck process [106], yielding\\nAs long as we sample at frequencies some orders of magnitude higher than the corner frequency, we can mitigate the aliasing by simply cropping the spectrum at a frequency below the Nyquist frequency f Nyq = f s /2. In principle, the expected value in the PSD can be estimated by averaging the experimental values |x k | 2 /T s over a large set of M experiments. In practice, one would need to acquire hundreds of experiments to have a good estimation. Therefore, it is common to apply a compression process to obtain a good estimation of the expected value with only one or a few experiments. Windowing is the most common compression process: it divides the frequency domain in N w narrow and equidistant subdomains with n w values, whose average represents the expected value at the corresponding mean frequency of the subdomain. This gives rise to the reduced experimental dataset for the PSD P (ex) α = |x k | 2 /T s n w with average frequency f α , with α = 0, 1, 2, . . . , N w − 1. With this procedure, the error in the expected value is σ (ex) α = P (ex) α / √ n w . If multiple realizations of the experiment are available, it is also possible to first compress the experiments and then average the results, giving an error σ (ex) α = P (ex) α / √ Mn w . Once the experimental PSD is estimated, the parameters f c, x and D are estimated by least-square fitting. Using directly the model (29) and the data set\\naccording to equation (73) and then apply the non-linear fitting procedure obtaining a numerical solution for the estimates. Alternatively, an analytical linear fitting can be performed using the experimental dataset { f 2 α , 1/P (ex) α } N w −1 α=0 with weights given by W α = 1/(P (ex) −1 α / √ n w ) 2 and the linear model 1/P( f ) = 2π 2 f 2 /D + 2π 2 f 2 c, x /D. Finally, an analytical solution can also be found [106] which minimizes\\nThe analytical solutions for the estimates using χ 2 defined by equation (32) are\\nwhere\\nThe errors for these estimates are\\nwhere\\nx max −x min arctan 2 x max −x min 1+x max x min , and f min and f max correspond to the minimum and maximum on the list of frequencies { f α }.\\nThe blue dots in Figs. 13(a-c) show the values |x k | 2 /T s computed by a fast Fourier transform for the trajectories acquired in experiments I, II, and III (Table 1):\\nTheir expected values are obtained by compression using N w = 500 (solid red lines in Figs. 13(ac)) and used to fit the Lorentzian function given by equation (29) using equations (33) and (34) (dashed black lines in Figs. 13(a-c)). To avoid aliasing (which gives rise to the plateau at high frequencies), the fitting was performed only with data corresponding to frequencies smaller than f max = f Nyq /4 = f s /8. From this fitting, we obtain the estimates for f c, x and D and, from these, the estimates for κ x , τ ot,x , and γ. Repeating this procedure for M = 5 experiments, we obtain the mean values of the estimates and their errors. All these estimates are reported in Table 3.\"\n },\n {\n \"title\": \"Drift method\",\n \"content\": \"The drift method [110][111][112][113] relies on the fact that the drag force must be equal to the optical force. From the Langevin equation (8), this implies that\\nwhere ∆x x 0 is the local drift, which is the average displacement of a particle when located at initial position x 0 and under the external force F x 0 during a time interval ∆t. From an experimental trajectory the local drift can be estimated as the conditional average\\nwhere the S corresponds to the number of times the initial position of the particle is close to x 0 . When applying this method there are several caveats. First, the size of the volume element near x 0 must be small enough to probe correctly the spatial variations of the force field. Second, the position of the particle must be measured with a precision much smaller than that of the volume element. Finally, equation (38) is valid when the diffusion coefficient is constant, while the presence of a space-dependent diffusion coefficient (e.g., due to hydrodynamics) leads to the emergence of spurious drift [112,113].\\nIn the presence of a spurious drift, a diffusion gradient term need to be added to equation (38) leading to\\nwhere α may take values in the interval [0, 1], depending on the specific stochastic process being modelled. In particular, α = 1 corresponds to a colloidal particle in a thermal bath [114][115][116].\"\n },\n {\n \"title\": \"Force reconstruction via maximum-likelihood-estimator analysis (FORMA)\",\n \"content\": \"The force reconstruction via maximum-likelihood-estimator analysis (FORMA) was introduced recently to retrieve the force field acting on a Brownian particle from the analysis of its displacements [107]. This method is able to estimate accurately the conservative and nonconservative components of the force field with important advantages over the previously discussed techniques, being parameter-free, requiring ten-fold less data and executing orders-ofmagnitude faster [107]. Let us consider a particle in a harmonic potential and suppose that after a time interval ∆t τ ot, x the Brownian particle is at a position x, having started at position x 0 . Repeating this measurement N times, the likelihood of observing this set of measurements is\\nwhich depends on the parameters κ x /γ and D. By maximizing the logarithm of the likelihood given by equation (41) with respect to the parameters κ x /γ and D, one obtains the so-called maximum likelihood estimators (MLEs) for a given time series {x (m) } N =1 :\\nFrom these equations and the Einstein-Smoluchowski relation (7), it is straightforward to estimate κ (ex)\\nx,m , γ, and D (ex) m . The standard errors for these estimates can be obtained from the mean and standard deviations from repeated experiments: using multiple time series m = 1, . . . , M to obtain the estimates {κ (ex) x,m } M m=1 and {D (ex) m } M m=1 yields the experimental mean and variances:\\nFORMA provides also an alternative way to get error estimates for the MLEs given by equation (42) for a single experiment, without having to repeat the experiment. Indeed, if we perform a Taylor expansion of the log-likelihood around the MLEs given by equation (42) up to second order, we obtain a Gaussian approximation for L({x n } N n=1 |{x 0n } N n=1 ). This yields the following standard deviations for κ (ex)\\nx and D (ex)\\nWhen applying FORMA, ∆t τ ot, x and the exposure time should be much smaller than ∆t (i.e., as close as possible to an instantaneous recording of the particle position).\\nWe report the estimates for the experimental data in Table 3, obtained using formulas (42) and (43). Comparing these results with the ones obtained by the other methods, we can see some discrepancies in the diffusion coefficient and hence in the friction coefficient. This is due to the fact that the condition that ∆t τ ot, x and that the exposure time be much smaller than ∆t does not hold true as the laser power is increased, causing the relaxation time to decrease.\\nSince FORMA only depends on the local variables of the trajectory, that is the positions and their corresponding displacements, it allows to retrieve the forces of arbitrary optical landscapes, including non-conservative forces, and it does not need a trajectory with regular sampling, commonly required in the other methods. This is particularly useful when the problem is to retrieve the forces of extended potentials, where the particle neither reaches the equilibrium nor is able to explore the whole space by itself, such as for a dual optical tweezer or a speckle pattern [107]. The main drawback in using FORMA is the need for high spatial resolution and fast sampling. The latter can be addressed by generalizing the solution for any arbitrary time interval ∆t, as recently introduced in Ref. [117].\"\n },\n {\n \"title\": \"Bayesian inference\",\n \"content\": \"While FORMA uses a non-Bayesian linear regression to infer the properties of the trap and the Brownian particle [107], other methods have used Bayesian inference to infer optical forces [117][118][119][120][121]. Recently, we have derived a Bayesian approach using conjugate priors, which renders very simple formulas for the posterior distribution, with known moments at all orders and marginal distributions [108]. Here, we provide an overview of this method.\\nLet us suppose that we want to infer D and κ x /γ. Let P 0 (κ x /γ, D) be their a priori probability distribution. Then, given a time series, the posterior probability is given by Bayes' rule:\\nwith Z being a normalization factor. Choosing the a-priori probability P 0 (κ x /γ, D) to be a so-called conjugate prior for the model's likelihood,4 we obtain the following marginal posterior distribution for the trap stiffness and the diffusion constant\\nwhere InvGamm(x) is the inverse Gamma distribution, and the parameters α N , β N , γ N , and K N 4A conjugate prior is a particular type of a priori distribution, chosen so that the posterior distribution belongs to the same family of distributions of such prior. One may intuitively think of conjugate priors as eigenfunctions of equation (45). are determined by the measurement record:\\nThe experimental estimates D (ex) and κ (ex)\\nx are the mean values and variances of the posterior distributions P D and P κ x /γ :\\nFormulas for higher cumulants can also be calculated analytically from equations (46). The hyper-parameters α 0 , β 0 , γ 0 , and K 0 , determine the a priori information about the model's parameters. For example, α 0 , β 0 , γ 0 , and K 0 can be chosen such that for N = 0 (i.e., without experimental data), we obtain their theoretical values and systematic uncertainties (i.e., D (th) ± σ D (th) and κ (th)\\nx /γ ± σ κ (th) x /γ ):\\nFrom Table 1, D (th) = 0.21 µm 2 s −1 and, adding an error of 10 %, allows us to fix the values α 0 and β 0 . For the theoretical value of the inverse relaxation time, we are more in the dark, as we do not know the trap stiffness before the calibration procedure; thus, we can use an estimate obtained by another method to determine the values of K 0 and γ 0 . Using Bayesian inference on one of the time series {x (m) n }, we obtain the results reported in Table 3. For the estimates regarding the friction and diffusion coefficients, we notice that the present method has the same issue concerning the exposure time as FORMA.\\nThe Bayesian inference method extracts all statistical information from the measured data, so that it provides an unbiased estimator of the experimental parameters. Moreover, it can provide reasonable estimates of the parameters also when little data is available, thanks to the use of prior information. The Bayesian inference method can also be generalized to more than one dimension for any time interval ∆t.\"\n },\n {\n \"title\": \"Measurement of non-conservative force fields\",\n \"content\": \"While the previous methods rely on the forces being conservative, non-conservative forces often play also a role in experiments. The drift method can be straightforwardly applied to measure any force field [111]. Also, the ACF method has been adapted to measure non-conservative forces using cross-correlations [122][123][124]. Here, we briefly show how FORMA can be adapted to measure non-conservative forces [107], while the details for the Bayesian method can be found in [108].\\nLet us consider the two-dimensional overdamped Langevin equation\\nwhere F(r) is the force field and w is a vector of independent white noise. The Taylor expansion of F(r) around r = 0 yields\\nwhere F 0 = F(0) and J 0 = J(0) are the force and the Jacobian at the point r = 0, respectively. We further assume r = 0 to be an equilibrium point so that F 0 = 0. Discretizing the Langevin equation (50), we obtain\\nwhere σ = 2Dγ 2 ∆t and w n is a vector of independent Gaussian numbers with zero mean and unit variance. Equation (52) can be understood as a multivariate linear regression model: for a given set of pairs of data points {(r n , f n )} N n=1 , we construct the likelihood for (52), similarly as the expression (41) for the one-dimensional case, and maximize it with respect to J 0 to obtain the corresponding MLE J 0 , which is given by the Moore-Penrose pseudo-inverse:\\nwhere R and F are N × 2 matrices given by\\nFrom the MLE J 0 , we can estimate the force as F = J 0 · r. This result can be naturally split into conservative and non-conservative parts, corresponding to the symmetric and antisymmetric decompositions of J 0 = J c + J r , respectively. The symmetric part can be rewritten as\\nwith R a rotation matrix and κ 1 and κ 2 the trap stiffness along the principal axes; θ represent the orientation of these axes with respect to the Cartesian system of reference. The rotational part is in turn given by\\nwith Ω being the angular frequency at which the particle would orbit around the center of the trap in the absence of Brownian noise.\"\n },\n {\n \"title\": \"Active calibration techniques\",\n \"content\": \"All calibration methods discussed so far are passive, i.e., they are based on the determination of the trap stiffness from the stochastic motion of the particle in thermal equilibrium with the surrounding fluid within a harmonic potential. Alternatively, the optical tweezers calibration can be performed by actively applying an external force on the particle and measuring its response. A simple active method consists of exposing an optically trapped particle to an external uniform flow in the x-direction with a known velocity v fluid so that the particle experiences a mean drag force F drag = −γv fluid . Consequently, the particle undergoes a mean displacement ∆x = x − x eq from the trap center x eq so that the restoring optical force balances the viscous drag. The stiffness can then be measured as\\nIn practice, a uniform flow can be applied by moving the sample at constant speed −v fluid , e.g., by means of a piezoelectric translation stage, while keeping the particle far away from the cell walls to avoid hydrodynamics interactions which change the value of friction coefficient γ with respect to the value given by Stokes' law [125]. Another active method requires moving sinusoidally the position of the sample stage x stage , i.e.,\\nwhile keeping the optical tweezers at fixed x eq = 0 [126]. This method not only provides the value of the trap stiffness, but also the drag coefficient and the spatial conversion factor of the measured particle position x (typically in volts or pixels) into actual distances. The Langevin equation for the motion of the particle in the case of the oscillating stage is\\nwhere v stage (t) = 2π f stage A cos(2π f stage t) is the velocity of the stage and W x is a white Gaussian noise. The corresponding PSD of the particle position is\\nwhere the thermal contribution is equal to that given by equation (29), while the mechanical response is that of the particle to the excitation frequency f = f stage . Because the measurement time in an experiment is finite, t msr < ∞ , the Dirac-delta peak in equation (60) becomes a spike with finite height P stage = P( f stage ) and width ∆ f . If the total measurement time is chosen to be an integer multiple of the driving frequency of the stage, then the spike reduces to a single datum P J at frequency f J = f stage for some integer J, whereas ∆ f = 2δ f , where δ f corresponds to the frequency resolution of the PSD. In this case, the discrete version of equation (60) for the j-th data point is\\nwhere δ j.J represents the Kronecker delta. Therefore, disregarding the single point ( f J , P J ), a fit P fit ( f ) to the one-sided Lorentzian function in equation (61) allows to compute the diffusion coefficient D (1) and the corner frequency f c, x (section 3.6), from which the trap stiffness can be determined as\\nIf x has not been yet converted into units of length, the values and units of D (1) and κ (1) are not correct. A further step using the active component of the spectrum in equation (61) allows to get the calibration factor α of x into units of length: x → αx, where α can have units of m/V or m/pixel depending on whether the particle position was recorded with a quadrant photodiode, a position sensitive detector or a camera. The power contained in the triangular spike centered at\\n, and the theoretical value given in\\n, must be equal. Then, α 2 W exp = W th , from which\\nHence, the corrected values with the right units of the trap stiffness and the friction coefficient are\\nand\\nAn alternative to the aforementioned method is to move the position of the center of the trap, while keeping the sample cell at rest, i.e. x stage = 0 and x eq (t) = A sin(2π f eq t). This can be achieved for example with an acousto-optic deflector or a galvomirror. In this case, unlike the case of an external flow, an effective oscillatory force k x eq (t) = k A sin(2π f eq t) is exerted on the trapped particle and the corresponding Langevin equation is\\nThe corresponding power spectral of the particle position is given in this case by\\nwhile its discretized form is\\nThus, once the PSD of the particle position is computed, the calibration of α, κ and γ proceeds similar to the case of the oscillatory flow created by moving the stage (equations (62)-(65)). Active calibration methods can also be applied to less trivial situations, for instance for particles trapped in viscoleastic fluids. In such a case, the instantaneous drag force on the particle at time t is not simply given by γ dx(t) dt as in equations (59) and (66), but it is related to the frequency-dependent rheological properties of the medium, which are a priori unknown. This will be discussed in detail in section 4.3.\"\n },\n {\n \"title\": \"Direct optical force measurement\",\n \"content\": \"All the calibration methods discussed until now provide an indirect measurement of the optical forces, relying on the properties of the motion of the particle in the optical potential, and therefore depend on the assumption of a spherical particle and a harmonic optical potential. It is also possible to measure directly the optical force [127][128][129].\\nDirect measurement of optical forces is based on recording the change of momentum between the ingoing and outgoing light, since the exerted optical force is equal to the change in the momentum flux of the trapping light: let I(θ, ϕ) be the intensity scattered into the direction (θ, ϕ) given in spherical coordinates, then the momentum flux density p of the outgoing light is\\nThe optical force acting on the particle corresponds to the difference between the total momentum flux of ingoing and outgoing light, where the incoming momentum flux is given by precisely the same formula but with opposite sign. To correctly apply this method, one needs to detect all of the scattered light. Further details of how this can be achieved experimentally can be found in Refs. [127][128][129]. Strickly speaking this is a rather unfeasible task. However, for most practical applications in which the refractive index of the observed particle is similar to that of the solvent, light scatters only slightly, making its experimental collection possible.\"\n },\n {\n \"title\": \"Mathematical details on error analysis and fitting\",\n \"content\": \"Most of the calibration techniques described above use a physical observable Ψ that depends on a single trajectory. Let us denote its experimental estimate for trajectory m as\\n, where α may denote either a time dependence or a spatial one. For example, in the potential method α represents the position associated to the histogram bins, while for the MSD α corresponds to the discrete time. We can then use the M experiments to calculate the experimental mean and the experimental covariance of Ψ. These are:\\nImportantly, the experimental covariance matrix ∆ (Ψ) α,β in equation (71) has two terms. The first term corresponds to the experiment-to-experiment fluctuations that arise naturally due to the intrinsic randomness of the physical problem as well as due to the correlation between different points α and β. The second term ( 2 η 1 ,η 2 ,... ) captures the variances due to some known error in some variables (η 1 , η 2 , . . .) that are needed to compute Ψ (ex) q (α) as, for instance, uncertainties δα in the value of α, which can be estimated by standard error propagation techniques.5 5Using propagation of errors, the contribution to the total variance by the errors in the uncorrelated variables\\nwhere δ η 1 , δ η 2 , . . . are the errors associated to the corresponding variables [130].\\nOnce Ψ (ex) (α) and ∆ (Ψ) α,β have been calculated, they are fitted to the corresponding physical observable Ψ(α; θ) to infer the unknown parameters θ = (θ 1 , . . . , θ q ) of the model. There are several approaches to perform this task. The most popular are based on least-square regressions (LSR), which defines the best estimates of the parameters as those that minimize the mean squared error.6 A fantastic book dealing with these matters is that by Peter Young [130].\\nThe general least-square method used in this Tutorial relies on the weighted χ 2 function\\nwhere R αβ is a symmetric semi-positive definite matrix that captures the correlation in the experimental dataset. The goal is to find the set of parameters θ ( ) that minimizes χ 2 (θ).\\nWhen R is taken to be [∆ (Ψ) ] −1 , this is the so-called generalized least-square method or the correlated χ-squared method. The standard weighted least-square method is recovered when the off-diagonal terms of the experimental covariance are assumed to be zero so that\\nPP . If we do not have access to experimental errors, we can perform the standard least-square method by assuming the matrix ∆ (Ψ) to be equal to the identity matrix. In some cases, the inverse of the sample covariance matrix is an ill-posed problem and it is highly recommended to use the standard method of weighted least squares but including the experimental covariance matrix in the error estimation of the parameters [132]. This is the approach we have used to estimate errors for the MSD and ACF methods.\\nIf the expression χ 2 (θ) is a simple function of the parameters, the minimization can be carried out analytically. For example, this is the case when Ψ(α; θ) is linear with respect to the model's parameters, that is\\nwhere {Φ j (α)} is a set of functions, giving rise to the well established linear least-square fitting.\\nIn some cases, a simple redefinition of variables can transform a non-linear fitting into a linear one, as, for instance, in the potential method. When it is not possible to obtain explicit forms for θ ( ) , one must rely on numerical minimizers, being the standard ones the Levenberg-Marquardt-Girard-Wynne-Morrison and the Trust-Region, implemented already in most of the standard computational packages for mathematical analysis, such as Python, MATLAB, Mathematica, and R. However, the computational methods given in these packages do not normally allow to use the experimental covariance matrix to estimate the covariance matrix of the estimated parameters. In this case, it is recommended to use the packages provided in [132], which we also have implemented in this Tutorial.\"\n },\n {\n \"title\": \"Applications\",\n \"content\": \"We will now describe how optical tweezers have been used in several advanced applications. For each field of application, we provide a brief introduction and a detailed description of some paradigmatic experiments. In this section, we focus on experiments with particles optically trapped in a liquid environment, while we discuss experiments with particles optically trapped in air or vacuum in section 5.\"\n },\n {\n \"title\": \"Single-molecule mechanics\",\n \"content\": \"The study of single-molecule mechanics using force spectroscopy techniques has rapidly expanded over the past three decades [133][134][135][136][137][138][139][140]. Many labs worldwide use single-molecule methods 6Less common, but very powerful methods, are the robust least-squares, including the least absolute regression (LAR) and the Bisquare weights [131], both very useful to minimize the effect of outliers.\\nto manipulate and track individual molecules. Broadly speaking, single-molecule methods can be classified into two large families depending on whether they manipulate molecules by exerting forces or they visualize molecules in action using appropriate markers. The latter include single-molecule fluorescence used to monitor the time evolution of fluorescently labeled molecular complexes in vitro and in vivo. Manipulation methods are capable of exerting forces from femtonewton to nanonewton. For example, they have been used to study the elastic properties of biopolymers [141], the thermodynamics and kinetics of intramolecular (e.g., molecular folding) [142,143] and intermolecular (e.g., ligand binding) interactions [144], molecular motors [145], and fundamental problems in nonequilibrium physics [146,147]. The field of force spectroscopy includes techniques such as laser optical tweezers [22], magnetic tweezers [148], atomic force microscopy [149], acoustic force spectroscopy [150], electrostatic trapping [151], microfluidics [152], microneedles [153], biomembrane force probe [154], and centrifugal force [155]. These techniques tend to compromise between measurement accuracy and high-throughput capabilities. While magnetic tweezers and acoustic force spectroscopy achieve high throughput, optical tweezers and atomic force microscopy remain the most accurate techniques in the low force (piconewton) and high force (nanonewton) regimes, respectively. Piconewton forces are the most relevant at the molecular level, making optical tweezers an excellent technique to accurately characterize molecular reactions.\\nIn this section, we describe in detail how optical tweezers can be used to investigate molecular folding using short DNA hairpins (a double-helix stem of a few tens of base pairs terminated by a loop) as model systems. We will describe how to synthesize a short DNA hairpin, how to unzip it by stretching it while holding it between a micropipette and an optical tweezers, and how to analyze the obtained data. This experiment provides an ideal example of the use of single-molecule mechanics to determine elastic behaviour of biopolymers, to measure the intramolecular weak interactions that stabilize native molecular structures, and to extract the free energy of biomolecule formation from out-of-equilibrium pulling experiments. The experimental setup we use is based on the miniTweezers setup developed by Smith, Cui and Bustamante [127].\"\n },\n {\n \"title\": \"Microfluidics chamber\",\n \"content\": \"The schematic of the experimental setup is presented in Fig. 14(a). The experiments are performed in a microfluidics chamber placed vertically within the setup. As schematically shown in Figs. 14(b) and 14(c), this microfluidics chamber has three channels: we will refer to the three channels as the upper, central and lower channel. The propagation of both laser beams is perpendicular to the chamber surface (z-axis). The experimental area is restricted to the central channel, where the object of study (biomolecule or cell) is held by two beads, one held by the optical trap and the other held by a glass micropipette, as shown in Fig. 14(d). The upper and lower channels are used to supply the two types of coated beads used in the experiments.\\nThe microfluidics chamber is realized by sandwiching two layers of parafilm (Parafilm M, Bemis) between two coverslips (No. 2, dimensions 24 mm × 60 mm × 200 µm), as shown in Fig. 14(c). In detail, the steps to prepare the chamber are:\\n1. The entrance and exit holes are drilled in one of the glass coverslips using a laser cutter.\\nThe coverslips are cleaned with a solution of 70% ethanol.7 tube (King precision glass, Inc., inside diameter=0.04 mm, outside diameter=0.08 mm, length=6.00 mm, glass type KG-33), as described in section 4.1.3.\"\n },\n {\n \"title\": \"5.\",\n \"content\": \"This glass micropipette is placed on top of the parafilm layer perpendicular to the channels with the tip positioned in the central channel, as shown in Figs. 14(b) and 14(c).\\n6. The upper and lower channels are connected to the central one via glass dispenser tubes (King precision glass, Inc., inside diameter=0.04 mm, outside diameter=0.10 mm, length=6.00 mm, glass type KG-33) cut using a diamond-tip cutter to obtain a clean cut.\\n7. The second parafilm layer is placed on top and the chamber is closed using the second glass coverslip (previously cleaned with 70% ethanol), as shown in Fig. 14(c).\\n8. The chamber is sealed by heating it on a hot plate at 120 • C while exerting a pressure of about 1 Kg in all the chambers' surface (either by placing a weight on top of the microfluidics chamber or, more simply, by exerting pressure by hand). To prevent the glass from breaking, it is recommended to sandwich the chamber between two thicker glass slides to homogenize the pressure applied to their surfaces.\\n9. The chamber is placed in the metallic mount, as shown in the inset in Fig. 14(a). The correct alignment between the holes drilled in the glass coverslip and the holes in the mount is critical to ensure the flow in the chamber. It is also very important to tighten the screws to avoid buffer losses between the plastic tubes and the glass chamber, since this would cause flows inside the chamber. Do this carefully, because tightening the screws too much could break the glass coverslip.\\n10. The side of the micropipette coming out of the chamber is cut to get rid of the excess tube, leaving only about ∼ 3 cm, and is introduced into a polyethylene tube (polyethylene tubing, Warner Instruments, PE-10/100; outside diameter = 0.61 mm, inside diameter = 0.28 mm). This tube is then fixed to the mount with tape. The pipette is easily breakable, so the tube needs to be placed as straight as possible. The connection is then sealed by using a special glue (Norland, NOA-61; UV Curing Optical Adhesives). The glue is placed between the the glass tube of the pipette and the polyethylene tube to fill the void outer space between tube and pipette via capillarity. After observing that the glue has entered into the tube, the UV-curation is performed by leaving the whole chamber for ∼ 25 min under a UV lamp.\\nSince the glue could reach the end of the glass tube of the micropipette and block it, the chamber has to be placed under the UV-radiation as fast as possible. After the curation, a syringe (1 ml Luer Lock, HSW SOFT-JECT U100 Insulin Henke Sass Wolf) is inserted using a needle (BD Microlance 30G × 1/2\\\" -0.30 mm × 13 mm) at the end of the tube to create suction at the tip of the micropipette.\\n11. The holes of the mount are connected with silicone-rubber tubes (Tygon 3350, Saint-Gobain, .031 ID X .093 OD X 50 FT TYGON 335) via nylon socket screws (Nylon set 8 − 32 × 3/8\\\", Product-Components, previously drilling a hole with a number 45 drill bit, of 0.82\\\"), as shown in Fig. 14(c). A segment of ∼ 20 cm polyethylene tube (Polyethylene tubing, Warner Instruments, PE-50/100: outside diameter = 0.97 mm, inside diameter = 0.58 mm) is inserted into the silicone-rubber tubes. The three exit tubes are connected to a trash (any small plastic container with a capacity volume ∼ 100 ml), while the three entry channels are connected to a syringe using a polyethylene tube (Polyethylene tubing, Warner Instruments, PE-50/100: outside diameter = 0.97 mm, inside diameter = 0.58 mm) which is inserted into the silicone-rubber tubes on one end, and a needle (HSW FINE-JECT 23G × 1\\\" -0.6 mm × 25 mm).\"\n },\n {\n \"title\": \"The miniTweezers setup\",\n \"content\": \"The miniTweezers [127] (Fig. 14(a)) consists of two focused counter-propagating laser beams (P = 200 mW, λ = 845 nm) that create a single optical trap [156]. It employs high-NA objectives (NA = 1.2), but underfills them to be able to collect almost all scattered light to measure the change of light momentum using position sensitive detectors (PSDs), as described in section 3.12.\\nBefore the objectives, a pellicle diverts ∼8% of each laser beam to a secondary PSD to determine the optical trap position. The remaining ∼ 92% is collimated by using a lens and it is introduced into the optical axis by using a polarizing beam-splitter that selects the horizontally polarized light. Then the linearly polarized light of each laser is circularly polarized by a quarter waveplate8. The laser beams are focalized by their corresponding objectives and form the optical trap, which can hold dielectric objects with a refraction index higher than the surrounding aqueous medium (e.g., polystyrene beads) and exert forces to them. The exiting light is collected by the opposite objective and is converted to vertically polarized light by another quarter-wave plate. The vertically polarized light can be extracted from the optical path using two polarizing beam-splitters and relay lenses that redirect the light to the PSDs that measure the intensity of the beam (i.e., its exerted force). The miniTweezers has a resolution of 0.1 pN and 1 nm at a 1 kHz acquisition rate. The chamber where the experiments are performed (section 4.1.1) is placed between the two objectives and held with a metallic mount with three stages to permit movement along the x-, yand z-directions. There are two possible ways to manipulate the position of the optical trap with respect to the chamber depending on the precision that is needed. For large displacements (up to hundreds of micrometers), the whole chamber is displaced along the x-, y-and z-directions using stepmotors. For fine displacements (less than a few micrometers, but with a resolution of ∼ 1 nm and rates of displacement up to ∼ 50 nm s −1 ), the optical trap is displaced along the xand y-directions by a 2D piezoelectric motor attached to the tip of the optical fiber (wiggler) used to inject the laser into the optical setup.\\nBefore any experiment, the setup needs to be aligned going through the following steps:\\n1. Remove any trapped object.\\n2. Move the chamber with the stepmotors so that the micropipette is a few micrometers away from the optical trap, in the so-called \\\"working zone\\\".\\n3. Remove any voltage applied to the piezoelectric motor.\\n4. Turn off the LED and remove the light filter, to be able to see the optical trap on the camera.\"\n },\n {\n \"title\": \"5.\",\n \"content\": \"Decrease the laser power until it is possible to distinguish both lasers.\\n6. Using the screws of the kinematic mount, move the B laser (the one that has the real image shown on the screen) until it is superposed on the A laser.\\n7. Put back on the light filter, turn on the LED, and increase the laser power to its working value.\\n8. Set the current values recorded by the PSD as the zero force baseline along the xand y-directions (center of the light spot) and along the z-direction (size of light spot).\\n9. Trap a bead with the optical trap by moving the chamber with the stepmotors close to the appropriate dispenser tube. 10. Bring the trapped bead to the working zone.\\n11. Fine-tune the alignment of the B laser using the screws of the kinematic mount controlling the wiggler of its fiber so that the xy-force signals from both force-PSDs are as close as zero as possible, i.e., both lasers exert an xy-force as close to zero as possible.\\n12. Fine-tune the position of one of the objectives along the z-directions so that the z-force from both force-PSDs is as close to zero as possible.\\n8The use of quarter-wave plates ensures that the light coming from the two laser beams do not interact with each other and guarantees that the light reflected from the particle is not returned to the laser but is reflected to the opposite PSD [157]. 13. For both lasers, move the kinematic mount of the mirror that deflects the light that gets towards the position-PSD (between the aspherical lenses and the PSD) until its signal is zeroed (i.e., the light hits the center of the position-PSD).\"\n },\n {\n \"title\": \"Pipette making\",\n \"content\": \"A micropipette is used to hold a microparticle by air suction, as shown in Fig. 14(b). The tip of this micropipette needs to have an inside diameter ∼ 1 µm, large enough to exert sufficient suction and small enough not to let the microparticles flow inside it. While one might use a commercial pipette puller to produce the pipettes, here, we will explain how a glass tube can be pulled to produce such micropipette using the homemade device shown in Fig. 15 [127,158]. This homemade pipette puller consists of a plastic platform with two parallel metallic bars, which hold a stage and along which a plastic weight can slide up and down. The two ends of the glass tube are fixed with screws to the stage and the weight, respectively. Thus, the glass tube is held in tension due to the pull exerted by gravity on the weight. The central part of the glass tube passes through a platinum wire; the two ends of the wire are connected to an electric supply, which provides a current ramp from 0 to ∼ 6 A in ∼ 8 s. The exact maximum intensity and time of the ramp have to be tuned to get the appropriate diameter and shape of the pipette (in general, longer times and higher intensities correspond to smaller tips). Specifically, the steps required to produce a micropipette are:\\n1. Unplug the wire from the electric supply and place the pipette puller horizontally.\\n2. Insert the glass tube through the stage, the platinum coil, and the plastic weight. Fig. 15. Pipette puller. First, the glass tube is carefully centered within a coiled platinum wire, as shown in the inset. Then, one end of the tube is attached to the puller, while the other is attached to a weight that pulls the tube down. Finally, an electric current intensity ramp is applied through the platinum wire, heating the adjacent glass tube, which during the melting is pulled down, creating a micropipette. 3. Carefully center the glass tube within the platinum wire. 4. Screw the glass tube at the stage and plastic weight. 5. Place the platform vertically (as shown in Fig. 15), carefully to avoid any sudden hit that may break the glass tube.\\n6. Apply the current ramp. The weight will drop into the platform. The geometry of the platinum wire is of a critical importance to obtain the proper shape and size of the micropipette. For that reason, when the weight drops into the platform, we have to ensure that the remaining glass tube of the stage is not touching the platinum wire, otherwise it would get stuck to it.\\n7. Unscrew the plastic weight and (with extreme caution and preferably with ethanol-cleaned tweezers) take the micropipette and place it into the chamber being built.9\"\n },\n {\n \"title\": \"Preparation of the beads\",\n \"content\": \"Biomolecules such as DNA, RNA and proteins are chains whose units are either nucleic bases or aminoacids [159]. These molecules can be conveniently manipulated through beads attached at their ends. Antigen-antibody connections can be used to attach a specific bead at each end, taking advantage of the fact that these connections are extremely specific [160]. In particular, avidin/streptavidin-biotin10 and digoxigenin-anti-digoxigenin connections are often employed in single-molecule force-spectroscopy experiments, because they can hold forces up to ∼ 100 pN at typical optical-tweezers loading rates of ∼ 1 pN s −1 [162]. Each bead is coated with a different molecule, which specifically binds to the cognate tails. Streptavidin-coated (SA) beads can be directly purchased (SPHERO streptavidin -polystyrene particles, 0.5% w/y, 2.17 µm, 5 ml). Anti-digoxigenin-coated (AD) beads are purchased as G-protein-coated polystyrene beads (Kisker Biotechnologies -G-coated polystyrene particles, 0.5% w/y, 3.18 µm, 5 ml), which must then be activated with anti-digoxigenin. The difference in size between these beads permits one to easily distinguish them by microscopy.\\nBuffers. The following buffers are required for the preparation of the beads: PBS (pH 7.0) NaCl 0.14 M, KCl 2.7 mM, K 2 HPO 4 · 3H 2 O 61 mM, KH 2 PO 4 39 mM, NaN 3 (sodium azide) 0.02%. To prepare 50 ml: fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water; add 0.406 g of NaCl, 0.01 g of KCl, 0.696 g of K 2 HPO 4 · 3H 2 O, 0.265 g of KH 2 PO 4 , and 1 g of NaN 3 ; dissolve using a magnetic mixer; add Milli-Q water until reaching 50 ml; check the pH and add NaOH until the solution reaches a pH 7.0.\\nPBS (pH 7.4) NaCl 0.14 M, KCl 2.7 mM, K 2 HPO 4 · 3H 2 O 80.2 mM, KH 2 PO 4 20 mM, NaN 3 (Sodium azide) 0.02%. For preparing 50 ml: follow the same procedure as for the previous buffer, adjusting to pH 7.4.\\nAntibody crosslinker buffer (pH 7.4) Na 2 HPO 4 100 mM, NaCl 100 mM. To prepare 10 ml: fill ∼ 7 ml of a 10 ml Falcon tube with Milli-Q water; add 0.142 g of Na 2 HPO 4 and 0.058 g of NaCl; shake well until the salts have dissolved; add Milli-Q water until the 10 ml of total volume is reached; check the pH and add NaOH until the solution reaches pH 7.4.\\n9The micropipette easily breaks or gets blocked by dust particles at the minimum contact of the tip of the glass tube with anything, so it needs to be handled with extreme care.\\n10The interaction between streptavidin and biotin is one of the strongest non-covalent interactions in Nature [161].\\nAll salts can be acquired from any chemical distributor (e.g., Sigma Aldrich). All products have to be of a biomolecular grade of purity. The water for preparing all buffers has to be Milli-Q water.11 For the RNA experiments, use RNAse-free water for all preparations (and make sure the pHmeter is also cleaned with RNAse-free water).\"\n },\n {\n \"title\": \"SA beads.\",\n \"content\": \"For the already coated SA beads, the protocol consists in exchanging the buffer where they are preserved with PBS (pH 7.4). The procedure (∼ 30 min) for the preparation of 1 ml of SA beads (which should be sufficient to perform experiments for several months) is:\\n1. Homogeneously resuspend the SA beads. To do so, place the container with the purchased particles into a vortex mixer for several seconds. An additional step of several seconds of sonication further improves the resuspension.\\n2. Take 1 ml of the resuspended SA beads and place it in a new Eppendorf tube.\\n3. Centrifuge the Eppendorf tube at 10000 rpm for 5 min. The dense SA beads will precipitate, while the buffer will float above them.\\n4. Extract the overnatant buffer and add 1 ml PBS (pH 7.4). Mix the SA beads and buffer. Sonicate a few seconds and centrifuge at 10000 rpm for 5 min, again. Extract the overnatant, resuspend in 1 ml PBS (pH 7.4) and sonicate during several seconds. This step is performed to exchange the buffer.\\n5. Aliquote in 20 tubes (50 µl each). This allows for an optimal sterile preservation of the beads.\\n6. Store at 4 • C for up to ∼ 6 months. 3. Homogeneously resuspend the G-coated beads. To do so, place the container with the purchased particles into a vortex mixer for several seconds. An additional step of several seconds of sonication improves the resuspension.\\n4. Centrifuge at 5000 rpm for 2 min. The dense beads will precipitate, while the buffer float above them.\\n5. Extract the overnatant buffer and add 1 ml antibody crosslinker buffer (pH 7.4). Mix the beads and the buffer, sonicate for a few seconds, and centrifuge at 5000 rpm for 2 min.\\n6. Repeat the previous step (washing) and resuspend within 1 ml antibody crosslinker buffer (pH 7.4).\\n7. Add 60 µl of anti-DIG antibody and 30 µl of freshly dissolved DMP crosslinker buffer.12\\n8. Tumble at room temperature for 60 min.\\n9. Centrifuge at 5000 rpm for 2 min.\\n10. Wash twice with 1 ml PBS (pH 7.0), resuspend with the same buffer, and sonicate during several seconds.\\n11. Aliquote in 20 tubes (50 µl each).\\n12. Store at 4 • C for up to ∼ 6 months. For the SA beads, no incubation is required, dilute 1 µl of SA beads in 1 ml of the molecular buffer in an Eppendorf tube.\\n12The volume of added anti-DIG antibody will depend on the coating of the beads: it may need to be higher if the vendor supplies the beads with a higher density of coating. Nevertheless, the proportions of anti-DIG and DMP crosslinker buffer have to be preserved (2:1).\\n13TRIS is used in the formulation of buffer solutions in the pH range between 7.5 and 8.5.\\n14EDTA is widely used for scavenging metallic ions, including divalent ones, which most enzymes need to be active. For this reason, it is widely used as a food preservative or stabilizer. In our case, it inactivates DNAses and RNAses, preventing nucleic acid degradation.\\n15Typically, the molecule is concentrated and needs to be diluted 1:10 to 1:100.\"\n },\n {\n \"title\": \"Synthesis of a short DNA hairpin\",\n \"content\": \"As shown in Fig. 14(d), a typical single-molecule experiment is performed using a short DNA hairpin of a few tens of basepairs (bp). A DNA hairpin is a single-stranded DNA molecule that forms a double helix stem ended by a loop. The hairpin is flanked by two double stranded helices that act as handles. Handles are molecular spacers used to manipulate the DNA hairpin. These spacers also avoid non-specific interactions between the DNA hairpin and the beads when performing the experiments. The handles are tagged with biotin and digoxigenin at one end to specifically attach them to the coated beads. The 29-bp handles are chosen because they provide higher rigidity than longer handles and higher signal-to-noise ratio measurements [163].\\nTo synthesize short DNA hairpins with 29-bp dsDNA handles, the desired DNA sequence (that we will denote as CD4) is purchased in a series of oligonucleotides that self-assemble into the hairpin and handles [163]. Briefly, a 29-bp sequence is used to make the handles, with the hairpin sequence flanked by the handles at both sides (Table 5). For very short hairpins (≤ 13 bp), the hairpin sequence can be ordered as a single oligonucleotide; however, for longer hairpins, it is useful to split the molecular construct into two oligonucleotides that are annealed and ligated together. Finally, the dsDNA handles are created by annealing a third oligonucleotide that is complementary to the handle region (\\\"splint\\\"). The sequences of the oligonucleotides used for the CD4 DNA hairpin used in this section are shown in Table 5. One oligonucleotide ( Table 5, CD4-handleA) is purchased with a biotin at its beginning (5'-biotinylated), whereas the oligonucleotide containing the opposite handle (Table 5, CD4-handleB) is purchased 5'-phosphorylated and then tailed at its 3'-end with multiple digoxigenins. All three oligonucleotides can be purchased from companies such as Eurofins, Fisher Scientific or Sigma Aldrich. Table 5. Oligonucleotides. Oligonucleotides used to synthesize the hairpin by a tailing, annealing and ligation reaction. The splint is used to the create the 29-bp double-stranded DNA handles. The hairpin loop is highlighted in bold and the sequences corresponding to the handles are highlighted in blue.\\nTo complete the DNA hairpin synthesis, three main reactions have to be performed:\\n1. Tailing reaction: As we have already noted, one oligonucleotide is already purchased with the biotin at its 5' end. However, to be able to stretch the hairpin, digoxigenins need to be added to the other end of the hairpin by performing the tailing reaction described in Table 6.\\nAfter the tailing steps, the oligos are purified using the QIAquick Nucleotide Removal Kit (Qiagen) and eluted in 50 µl Tris-Cl 10 mM, giving rise to a final concentration, assuming a 100% efficiency, of ∼ 2 µM.\"\n },\n {\n \"title\": \"Annealing reaction:\",\n \"content\": \"The final construct is then assembled in an equimolar reaction. Since the sequences are complementary, at equal concentration for all the oligonucleotides, the equilibrium structure is the formed hairpin with dsDNA handles (Fig. 14(c)). The annealing is performed in two steps : (a) CD4-handleA + splint and CD4-handleB + splint; (b) mix of the two previous reactions. The annealing protocol is described in Table 7. Table 7. Annealing reaction. Mix the items for each reaction in a separate Eppendorf tube. Keep both reactions at 95 • C for 1 min, 80 • C for 10 min, decrease by 0.5 • C every 10 min down to 40 • C. Hold the temperature, mix both reactions and decrease by 0.5 • C every 20 min down to 10 • C. The decrease in temperature can also be achieved by letting the tubes inside a thermal bath cooling down with the heater off; however, the usage of a thermocycler is recommended, since it allows for a better temperature control.\\n3. Ligation: The molecular assembly is almost completed. All the Watson-Crick bonds of the structure have been formed. However, the backbone of the two oligos of the stem of the hairpin are still not covalently bonded. The final step is to connect the phosphorylated 5' end of CD4-handleB oligo with the C3' end of CD4-handleB. To do so, the assembly is then ligated in an overnight reaction using T4 DNA ligase (New England Biolabs) at 16 • C. The ligase is heat inactivated at 65 • C for 10 min, as shown in Table 8.\"\n },\n {\n \"title\": \"Pulling experiments\",\n \"content\": \"Paradigmatic manipulation experiments one can perform with a DNA hairpin are pulling experiments. The DNA hairpin is tethered between the two beads and the force it experiences is cyclically varied between a minimum and a maximum force value. Starting at a low force (typically 0 to 5 pN) with the hairpin folded, the trap is moved away from the pipette at a constant pulling speed and the exerted force is steadily increased. At a given point, the hairpin will reach a force high enough to unravel the double-stranded DNA helix structure reaching the unfolded state. This mechanically induced denaturation process is called unzipping. The stretching of the unfolded hairpin keeps on until a maximum force is reached. Afterwards, the same reverse protocol is applied: the trap is moved backwards at the same pulling speed and the applied force decreased until reaching the minimum force value. In this reverse process (called rezipping) the DNA will refold to the hairpin native state when the applied force is low enough. Once the applied force reaches the initial minimum value a new pulling cycle starts again. The CD4 DNA hairpin is a good molecular marker that can be used for force calibration. It reversibly unfolds and folds at 14.7 ± 0.3 pN at standard conditions (T = 298 K, 1 M NaCl). The detailed procedure to perform the pulling experiment is:\\n1. The three channels are filled with the molecular buffer described in section 4.1.4 through the syringes connected to the channels. The buffers have to be kept in the fridge and filtered using a 10 ml syringe and a filter (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International) to prevent microorganism growth.\\n2. The chamber (already held in the metallic mount as described in section 4.1.1) is placed between the objectives.\\n3. The buffer containing SA beads is flowed in the upper channel and AD beads in the lower channel (the channels can be interchanged).\\n4. The microfluidics chamber is moved using the stepper motors to bring the optical trap close to the dispenser tube of the upper supplier channel to capture a SA bead.\\n5. After a SA bead is captured by the optical trap, the microfluidics chamber is moved back to position the trapped SA bead on the tip of the micropipette. The syringe is then pulled and the SA bead is held by air suction at the tip of the pipette.\\n6. The microfluidics chamber is then moved close to the dispenser tube of the lower channel supplying AD beads.\\n7. Once an AD bead is captured (paying special attention not to trap more than a single bead), the AD bead is brought close to the SA bead, i.e., the chamber is moved so that the optical trap is in the working zone.\\n8. The tether is formed by poking the SA bead held by the micropipette with the optically trapped AD bead, until a non-zero force is recorded upon retracting the AD from the SA bead. 9. Check that only one molecule has been attached to the bead. To do so, the easiest way is to move the optical trap away from the bead in the pipette and verify that a sudden decrease of the force by ∼ 1.5 pN occurs at ∼ 15pN (red lines in Fig. 16(a)), which indicates the unfolding of the DNA molecule. Then, by moving inwards the optical trap, a similar increase of the force should occur at ∼ 14pN (blue lines in Fig. 16(a)), which indicates that the hairpin is folding back. This equilibrium unzipping-rezipping force has been found to be 14.7 ± 0.3pN at standard conditions (see above).\\n10. Now, the pulling experiment can start by cyclically varying the distance L between the pipette-held particle and the center of the trap (as shown in Fig. 14(d)). In this way, repeated unfolding and refolding pulling cycles (∼ 100) are performed over the same molecule while recording the force and the trap position. Some representative data are shown in Fig. 16(a), with a typical pulling speed of 100 nm s −1 . To extract the free energy of formation of the hairpin, as we will see in the next section, it is key that the hairpin is folded at L = L 0 and unfolded at L = L 1 .\"\n },\n {\n \"title\": \"Extracting the free-energy of hairpin formation\",\n \"content\": \"One of the main applications of single molecule experiments is the possibility to extract free energy differences from irreversible work measurements using fluctuation theorems [164]. From the force-distance curves ( Fig. 16(a)) the work W exerted on the molecule between the starting trap position, L 0 , and the final one, L 1 can be directly measured as the area below the curve:\\nFor quasi-static processes (i.e., sufficiently slow pulls) the work equals the free energy difference ∆G FU between the initial L 0 (folded) and final L 1 (unfolded) trap positions. However, this is not true for irreversible pulls which exhibit hysteresis between the stretching (forward) and releasing (reverse) processes. The difference in the area between consecutive unfolding and folding cycles is the work dissipated by the system, which is positive in average according to the second law of thermodynamics. Noteworthy, since we are not in the thermodynamic limit (as we are dealing with small systems), there may be trajectories that violate the second law of thermodynamics. Fluctuation theorems state that the work distributions measured along the forward and reverse processes fulfil a symmetry relation [164]. One of the main consequences of such a relation is that forward and reverse work distributions, despite being different due to the hysteresis, cross each other at the value of ∆G FU , independently of how irreversible the pulling process is. To extract the work distributions we use equation (75) to compute the work for all the forward (stretching) and reverse (releasing) cycles. The probability distributions of the work for the forward and reverse processes are shown in Fig. 16(b) by the red and blue histograms, respectively. The crossing point of the forward and reverse work probability distributions corresponds to the difference in free energy ∆G FU between the folded (L 0 ) and unfolded (L 1 ) states. The work histogram in Fig. 16 The free energy difference between the folded and unfolded states can be written as:\\nwhere the term ∆G 0 FU is the free energy of formation of the molecule, which is the quantity we are interested in obtaining, and the other contributions are the reversible work differences related to the different parts of the experimental setup (ssDNA, handles, bead, hairpin dipole, as shown in Fig. 14(d)). These contributions can be computed as follows:\\nis the reversible work needed to stretch the unfolded molecule, which is modeled with an inextensible Worm-like chain [165]:\\nwhere f ssDNA is the force, x is the molecular extension, n is the number of monomers of the molecule (n = 44 bases for our case), l 0 = 0.59 nm is the contour length per monomer, and p = 1.35 nm is the persistence length (parameters from Ref. [166]). The stretching contribution of this released ssDNA is then computed as\\nThe value we obtain is ∆W ssDNA\\nis the reversible work needed to stretch the folded molecule, which is modelled as a single dipole of length d = 2 nm [166], corresponding to the double-helix diameter [167]:\\nand\\nwhere\\n3. Finally, the last two contributions to equation (76) can be computed by considering the slope of the force-distance curves just before the rip, where the distance corresponds to the optical trap position (L), as shown in Fig. 16(a). The slope κ eff can be easily obtained by linearly fitting the data points right before the unfolding event. This yields the effective stiffness of two serially connected springs: the trapped bead of stiffness κ B , and the handles stiffness, κ handles :\\nAssuming that the hairpin orientation has a much higher stiffness than the handles and the optical trap, the combined work can be written as [166]:\\nwhere we have considered that the stiffness of the folded molecule is very large and consequently does not contribute to κ eff . The value we obtain is ∆W handles\\nFrom equations (78), (80) and (81), and from the crossing point of the forward and reverse work probability distributions (∆G FU ), we obtain all the terms needed to calculate the free energy of formation of the hairpin, ∆G 0 FU , from equation (76). The free energy of formation at zero force we obtain for this molecule is ∆G 0 FU = 51.9 k B T, which is in agreement with that of the sequence of the CD4 hairpin, ∆G 0 FU = 50.6 k B T, using the values provided in Ref. [168]. Typical errors of ∆G 0 FU are on the order of several k B T, corresponding, in this case, to a 5% to 10% relative error.\"\n },\n {\n \"title\": \"Other applications to single-molecule mechanics and outlook\",\n \"content\": \"\\\"Take a single DNA molecule and pull from its extremities while recording the force-extension curve until it gets fully straightened.\\\" This thought experiment, which was just a dream a few decades ago, has now become standard in many research labs worldwide. Since the seminal works on single-molecule manipulation almost 30 years ago [7][8][9], the number of applications of single-molecule mechanics has kept on growing. Applications might be classified in two large sets: those aimed to address specific biological questions (e.g., the mechanics of maintenance and regulatory enzymes in the genome, the energetics and kinetics of DNA hybridization, RNA and protein folding, molecular footprinting of proteins and ligands binding DNA), and those aimed to enlighten fundamental physico-chemical questions (e.g., polymer physics, chemical reactions, nonequilibrium physics) using biomolecular matter as model systems. The reader can have a look at published reviews specifically addressing these fields to discover that the future of the single-molecule field has a long way ahead to run [133][134][135][136][137][138][139][140]. Most optical-tweezers and single-molecule technology is still instrumental craftwork not widely available in research institutes and labs worldwide, yet it is becoming steadily more popular among biologists aiming to enter the field. Future developments in the field will be the parallelization of measurements in optical tweezers making it high-throughput either using holographic techniques [43] (recently extended also for acoustic tweezing [169]), combining single-molecule fluorescence with forcemeasurement techniques [170][171][172], controlled force resistive pulse sensing [173,174], and plasmonic trapping [13]. Finally, a major step would be to apply force spectroscopy techniques to ex-vivo and in-vivo conditions.\"\n },\n {\n \"title\": \"Single-cell mechanics\",\n \"content\": \"Biological microscopic objects such as organelles and living cells have been optically manipulated since the late 1980s [175,176]. In Ref. [177], the swimming motion of optically trapped bacteria was recorded while the trapping forces were measured providing the first example of single-cell optical manipulation. Since then, biomechanics at the cellular scale has kept on growing in terms of techniques and objects of study [178,179]. Techniques such as atomic force microscopy [180,181], optical tweezers [182][183][184], particle tracking microscopy [185,186], traction microscopy [187][188][189], magnetic twisting cytometry [190], acoustic force spectroscopy [191], and micropipette aspiration [192] have become commonly employed to investigate the response of cellular objects to mechanical stresses. Essential cellular processes (e.g., cell division) and collective effects (e.g., cell signaling and migration in tissues) are currently being investigated with such techniques.\\nIn this section, we will demonstrate one of the simplest cell manipulation experiments with optical tweezers: the mechanical stretching of a single red blood cell (RBC) to measure its rigidity to an externally applied deformation or strain.\"\n },\n {\n \"title\": \"RBC buffer preparation\",\n \"content\": \"The RBC buffer should mimic the physiological conditions found in human blood. It can be prepared as a solution of 130 mM NaCl, 20 mM K/Na phosphate buffer at 7.4 pH, 10 mM glucose, and 1 mg ml −1 BSA (Bovine Serum Albumin) [193]. It is convenient to prepare higher concentration stocks (1 M) of every solution and store them in the fridge (higher concentration of salts together with low temperature prevent microbial contamination), keeping them in 50 ml Falcon tubes for repeated use. To prepare a 50 ml Falcon tube containing the RBC buffer, follow these steps:\\n1. Add 6.5 ml of 1 M NaCl, 1 ml of 1 M K/Na phosphate buffer at 7.4 pH,16 and 0.5 ml of 1 M glucose into an empty 50 ml Falcon tube.\\n2. Add 50 g of BSA.\\n3. Add 42 ml of milli-Q water.\\n4. Mix the RBC buffer and filter it (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International).\"\n },\n {\n \"title\": \"Place the filtered solution into a new 50 ml Falcon tube.\",\n \"content\": \"All remaining unused RBC buffer must be re-filtered on a daily basis to prevent molecular aggregation and bacterial growth. After three days, it is recommended to dispose of the remaining RBC buffer because the re-filtering will not be enough to ensure proper conservation.\"\n },\n {\n \"title\": \"Blood extraction\",\n \"content\": \"To obtain human blood, the easiest way is to prick the finger of a healthy donor. Use of diabetic testing lancets will ensure sterilization of the pricking device. The extraction is made following these steps:\\n1. Disinfect the finger of the healthy donor with ethanol and dry it with handkerchief paper to remove any remaining liquid.\\n2. Prick the donor's finger with the lancet and apply some pressure to extract a drop of blood. A single drop of blood will be enough to perform experiments during an entire day.\\n3. With a 200 µl pipette take 10 µl of blood from the finger.\\n4. Dilute as fast as possible the blood drop in 1 ml of RBC buffer inside a 1.5 ml Eppendorf tube to prevent degradation of the RBCs.\"\n },\n {\n \"title\": \"Bead preparation\",\n \"content\": \"The simplest option to attach the beads to the RBCs is to take advantage of non-specific interactions between polystyrene beads and RBCs.17 To do so:\\n1. Add 1 ml RBC buffer in a 1.5 ml Eppendorf tube.\\n2. Homogeneously resuspend the polystyrene beads by placing the purchased beads (Microbead NIST Traceable Particle Size Standard, 3.00 µm, Polysciences, Inc.) into a vortex mixer for several seconds. An additional step of several seconds of sonication improves the resuspension.\\n3. Add 3 µl of the bead stock solution to the Eppendorf tube.\\nIt is recommended to have two Eppendorf tubes with bead solution available to ensure continued supply of the chamber with beads during the experiment.\"\n },\n {\n \"title\": \"Setup preparation\",\n \"content\": \"The experiment can be performed using the same microfluidics chamber and miniTweezers setup presented in section 4.1. In particular, the procedure to build the chamber and to place it into the mount is the same as described in section 4.1.1, the only difference being a longer plastic tube (40 cm) to connect one of the syringes to the inlet of the central channel. Once the chamber is fixed into the mount with the syringes connected through the plastic tubes, the next steps are:\\n1. Flow 3 ml of RBC buffer in each of the three channels to produce an homogeneous environment in the chamber.\\n2. Identify which dispenser tube (upper/lower) has the most clear exit (i.e., the one displaying the most sharp perpendicular cut in the video image) to the central channel. This will simplify trapping the beads. 5. Introduce the mount with the chamber between the two objectives of the optical tweezers.\\n6. Connect the syringe with the larger plastic tube (the one of the central channel) to a pump to apply a controlled flow to the central channel when it is needed. We will use this flow to stretch the RBC-bead dumbbell captured in the optical trap, thereby preventing thermaland photo-damage of the cell.\"\n },\n {\n \"title\": \"Construction of the cell-bead configuration\",\n \"content\": \"To perform cellular pulling experiments, the RBC need to be held between two beads attached to the cell membrane on opposites sides, as shown in Fig. 17(d). One bead is held by the micropipette by air suction, while the other bead is held by the optical trap. The use of beads as handles to manipulate the RBC minimizes the direct exposure of the RBC to the laser light. The steps of the procedure to obtain this cellular configuration are:\\n1. Align the lasers without a bead (steps 1-8 described section 4.1.2).\\n17There are also several options to treat the beads and RBC to specifically attach them, e.g., using the lectin protein Concanavalin A [191] or biotinylating the RBC [194] 2. Identify the position of the pipette and of the two dispenser tubes that connect the upper and the lower channels with the central one.\\n3. Move the chamber using the stepmotors and place the optical trap close to the exit of the dispenser tube that contains the bead solution.\\n4. Flow some bead solution pushing the syringe either by hand or using a pump, and trap a bead with the optical trap.\\n5. Move the trapped bead next to the pipette tip in the working zone.\\n6. Align the lasers with the trapped bead (steps 9-13 described section 4.1.2).\\n7. With the bead already trapped, try different flow values with the pump connected to the syringe of the central channel until the force reaches ∼ 10 pN due to the RBC buffer flow. This will be the approximate flow that we will need to keep the optically trapped bead separated from the RBC (Figs. 17a and 17b) after we obtain an attachment.\\n8. Turn off the pump.\\n9. Move the trapped bead to the area of the blood dispenser tube exit. It is important not to put the bead exactly at the exit of the tube to prevent losing the bead due to the RBC solution flow.\\n10. Flow some RBC solution applying a small pressure to the syringe. RBCs will flow into the central channel.\\n11. Move the trapped bead close to a RBC and try to make a connection by poking the surface of the RBC with the bead. The connection is formed when the RBC follows the trapped bead when moved.\\n12. Move the chamber using the stepmotors to place the bead and the attached RBC next to the micropipette.\\n13. Turn on the pump. The configuration should look like Fig. 17(a). The RBC is attached to the bead and partially trapped by the lasers. 14. If the flow is not enough to separate the RBC from the trap and, therefore, to obtain the configuration of Fig. 17(b), it is necessary to move the chamber using the stepmotors along the horizontal direction to exert an extra force and separate the RBC from the optically trapped bead.\\n15. Move the bead close to the tip of the pipette. 16. Carefully apply air suction to immobilize the bead. This step is critical because a large enough air suction force could absorb the RBC inside the pipette.\\n17. Move the optical trap away from the bead. If the bead is properly fixed to the tip of the pipette, the bead will stay there. If this is not the case, when the optical trap is moved the bead will follow it. 23. Try to make a connection between the trapped bead and the RBC as in Fig. 17(c). Notice that the two beads are on opposite sides of the RBC. 24. Check that the new connection between the bead and the RBC is done by moving the optical trap to the same direction as the pump flow. If the RBC is deformed while moving the trap, the attachment is done. 25. Turn off the pump to remove the flow of the central channel. 26. Nip very carefully the plastic tube that is connected to the entrance of the central channel with a clamp to prevent flows due to the difference of pressure at both ends of the plastic tubes.\\n27. Move the stepmotors to obtain the configuration shown in Fig. 17(d), aiming at having a zero reading for the force along the y-direction.\"\n },\n {\n \"title\": \"Experiments\",\n \"content\": \"By cyclically stretching and releasing the RBC, it is possible to obtain the force-extension curves from which the stiffness and deformability of the RBC can be extracted. Starting with a RBC in the configuration described in the previous section ( Fig. 17(d)), it is possible to stretch the cell by moving the optically trapped bead outwards (i.e., away from the micropipette) and release it by moving it inwards (i.e., towards the micropipette).18 Follow these steps to define the stretching-realising protocol:\\n18A range for the optical trap displacement of at least ∼ 8 µm is necessary to perform the RBC stretching. Thus, it is recommended to move the optical trap towards the lower limit of its y-displacement, before starting the pulling protocol.\\nStarting from the minimum force, the RBC is pulled at the constant velocity until the maximum force. After reaching this maximum force, the RBC is pushed at the same constant velocity to the minimum force. During this process, the displacements and forces along the RBC stretching direction (y-direction) are recorded for both the A-and B-trap in the miniTweezers setup ( Fig. 14(a)); then, the two displacements are averaged to measure the trap displacement, while the two forces are summed to obtain the total force acting on the RBC. The whole process is called pulling cycle. Recording 5 pulling cycles for each RBC is typically enough for the subsequent analysis.\"\n },\n {\n \"title\": \"Data analysis\",\n \"content\": \"The data analysis aims at computing the RBC stiffness knowing the stiffness of the optical trap κ trap . The experiment measures the trap displacement ∆x trap , not the cellular extension ∆x cell . Thus, ∆x cell can be computed from ∆x trap , κ trap , and the measured optical force f trap . For each intermediate trap position from L 0 to L f , we are able to transform L into x cell substracting the 19Depending on the laser power, it could be very hard to reach the 0 pN force as the RBC will be attracted to the optical trap.\\n20Above 20 pN, as we have not applied any specific coating to the beads, a very thin membrane tube (theter) forms and alters the force-extension curve.\"\n },\n {\n \"title\": \"RBC\",\n \"content\": \"ΔL Δx trap Fig. 18. Sketch of RBC extension under applied force. The red ellipse represents the RBC, the light blue circles represent the beads, and the small green circle represents the optical trap. The cell extension is ∆x cell , while ∆x trap is the bead displacement respect to the trap center, and ∆L is the displacement of the optical trap. Note that ∆x cell and ∆x trap are distances along the y-axis. trap contribution (Fig. 18), as we show in the following expression:21\\nwhere ∆L is the variation of the trap position between that at minimum force and that at maximum force. The resulting force-cellular-extension curve is shown in Fig. 19(a) for the second pulling cycle for two different RBCs. 22 The RBC stiffness can be measured from the slope of the force-cellular-extension curves. As can be observed in Fig. 19(a), the force-cellular-extension curve is not a straight line. This indicates that the stiffness of the RBC depends on the force applied to the cell. For this reason, the slope is computed for five different force windows whose stiffness values are represented in Fig. 19(b). The RBC stiffness increases as we increase the applied force, the stiffness curves of the stretching and releasing cross each other between 6 and 10 pN, and the releasing stiffness is higher than the stretching stiffness at high forces while it is lower at low forces. The expected values for RBC stiffness in this specific geometry are in the range between 3 pN µm −1 and 10 pN µm −1 . The dispersion between different RBCs, in terms of force-cellular-extension curves, can be very large. This can be due to the fact that the RBCs are not separated by density so that the RBCs can have very different ages and, thus, very different physiological properties [196].\"\n },\n {\n \"title\": \"Other applications to single-cell mechanics and outlook\",\n \"content\": \"Until now, the manipulation of single cells with optical tweezers has not been extensively investigated as single-molecule manipulation. The main reason is the difficulty inherent to the geometry of the pulling assays when working with cells. Indeed, while the specific labeling of molecules makes it possible to design the most diverse pulling geometries, this is more difficult to achieve when manipulating single cells which ultimately require the physical contact between the optically trapped bead and the lipid cell membrane. Additionally, single cell manipulation with optical tweezers is feasible on cells in suspension (such as RBC, lymphocytes, senescent, stem cells), whereas adherent cells are harder to manipulate, often requiring confocal microscopy 21This is the same analysis done to obtain the molecular extension in long nucleic acids (≈ 10 kbp double-stranded nucleic acids or ≈ 1 kbp single-stranded nucleic acid [195]).\\n22It is best to analyze the second pulling curve because the first one could exhibit a special transient behavior.\\nsetups with optical traps very close to the glass substrate. For the latter class of cells, other techniques (e.g., atomic force microscopy, traction microscopy, magnetic twisting cytometry) turn out to be more informative. Finally, let us mention that while most single cell manipulation studies are carried out in in-vitro conditions (e.g., on the Petri dish), it will be necessary to expand the use of these manipulation techniques to more relevant environments such as ex-vivo tissues.\\nIn this case, the mechanical response of a single cell in a tissue-like environment is expected to differ a lot with respect to the isolated cell. Overall, this makes the future of single cell mechanics very promising beyond biology, with implications in physiology and medicine.\"\n },\n {\n \"title\": \"Microrheology\",\n \"content\": \"Rheology investigates the mechanical response of soft materials such as simple liquids, polymer solutions, micellar fluids, colloidal suspensions, liquid crystals, gels, and foams [20,197]. Specifically, it is concerned with flow and deformation of such materials under applied stress or strain. Depending on the specific details of their microstructure, different kinds of rheological behavior can be observed. For instance, for most simple liquids, the rheological response is Newtonian: upon imposing a given strain rate, the resulting stress is proportional to it, thereby dissipating energy instantaneously by viscous resistance. Instead, complex fluids and gels, whose components are usually long macromolecules suspended in a viscous solvent, exhibit viscoelasticity, i.e., both viscous (liquid-like) and elastic (solid-like) responses are possible [198]; therefore, they are able to store and dissipate energy depending on the time-scale of observation.\\nThe goal of rheology is to provide quantitative parameters, e.g., viscosities, relaxation moduli and creep compliances, that relate the stress and the strain (or the strain rate) of the material under deformation. To this end, the most straightforward techniques consist in imposing stresses and strains in a controlled manner by means of a rheometer to the material sample of interest and then measuring the corresponding mechanical response. Despite their relative simplicity, these methods require typically milliliter-sized sample volumes, thus being unsuitable for soft matter systems that are expensive or difficult to find in abundance, such as biological fluids and newly synthesized materials. Microrheology overcomes this major drawback by measuring mechanical properties of a soft material sample using colloidal probes directly embedded in it. The rheological information of the material is fully inferred from the motion of the probe, which requires only the detection and tracking of its position over time. Therefore, very small sample amounts, typically ranging from pico-to microliters, are needed. This in turn allows to carry out in-situ rheological measurements which are inaccessible to macroscopic methods, e.g., biofluids within living tissues and cells, soft interfaces and membranes. Furthermore, microrheology offers several advantages over conventional macroscopic rheometry. For instance, unlike bulk rheology, which provides quantities averaged over the entire macroscopic sample, microrheology permits to explore local mechanical properties at the length-scale of the micron-sized probe. This is particularly useful for investigating heterogeneous materials and fluids close to interfaces, where submillimetric spatial variations of the rheological parameters occur. In addition, undesirable artifacts that can arise due to inertial effects when using a macroscopic rheometer are automatically ruled out in microrheology due to the extremely low Reynolds numbers (Re 10 −3 ) of the flows induced by the motion of a micron-sized probe. This expands significantly the available frequency range over which the linear viscoelastic spectrum of the system can be determined without inertial corrections, typically from the range [10 −1 Hz, 10 2 Hz] up to the range [10 −1 Hz, 10 6 Hz]. Besides, conventional rheometers are usually designed for either highly elastic or highly viscous materials, but they are not sensitive enough to measure low viscosities and elastic moduli; then, another advantage of microrheology is its capability to detect very weak viscoelastic parameters. Moreover, the typical forces and energies involved in microrheology, either from the thermal motion of the material molecules or those externally exerted by optical tweezers, are of the order of ∼pN and ∼k B T, respectively. This guarantees that the microstructure of the sample is not irreversibly destroyed by the motion of the probe. 20. Microrheology with an optically trapped particle. (a) Sketch of the general response of the shear stress σ(t) of a viscoelastic material to a step-like shear strain (t).\"\n },\n {\n \"title\": \"Basic setup and concepts\",\n \"content\": \"For t ≥ 0, the ratio σ(t)/ 0 defines the stress relaxation modulus G(t). (b) Sketch of the typical experimental system used in microrheology: a spherical colloidal bead (radius a), embedded in a soft material, is trapped by optical tweezers. If the position of the optical trap is fixed (x ot (t) ≡ 0, passive microrheology), the particle is only subject to the thermal collisions of the molecules of the surrounding fluid microstructure. If x ot (t) is varied in time, an additional time-dependent force is exerted on the probe particle (active microrheology).\\nOptical tweezers provide the ideal tool to manipulate probe particles in microrheology [20,[199][200][201][202], because they permit one to tailor the forces that can be exerted to locally strain the sample.\\nTo extract meaningful information by means of standard microrheological methods, some experimental conditions must be fulfilled. First of all, the material must be homogeneous and isotropic at the scale of the probe, which is usually valid when the relevant length-scale of the material, e.g., the mesh size of gels and semidilute polymer solutions, is much smaller than the probe size. In this case, the environment behaves as a continuum and its mechanical response can be fully characterized by a single scalar function, e.g., the stress relaxation modulus G(t). For example, upon applying a step-like shear strain at t = 0 to a material, (t) = 0 Θ(t), where Θ is the Heaviside function, the stress relaxation modulus is given by\\nwhere σ(t) is the resulting shear stress ( Fig. 20(a)). For a viscoelastic liquid, G(t) decays to zero over a finite time, while for a viscoelastic solid, it decays to a constant non-zero value. In both cases, the drag force experienced by an embedded particle at time t can be written as\\nwhere r(t ) is the instantaneous particle velocity at time t < t and the minus sign indicates that the drag force is opposed to the instantaneous velocity, whereas Γ is a memory function related to G(t), which weights the role of the previous history of the particle motion on its current drag force due to the temporal correlations induced by the surrounding medium. For example, for a spherical particle of radius a, the memory function is Γ(t) = 6πaG(t), which can be regarded as a generalization of the Stokes' law (equation (6)). For a Newtonian fluid with shear viscosity η, the relaxation modulus is G(t) = 2ηδ(t),23 which corresponds to an instantaneous response of the drag force F drag (t) = −6πηa r(t) to the particle velocity r(t). Moreover, for some techniques, it is required that the sample is in thermal equilibrium and that the forces externally applied to the probe are small enough to keep the material in its linear response regime. Under such conditions, the time evolution of a single particle coordinate, x, is described by the generalized Langevin equation [203]\\nwhere x ot represents the x-component of the central position of the optical trap, while ζ is a Gaussian noise of zero-mean, ζ(t) = 0, and autocorrelation\\nDepending on how optical tweezers are employed to manipulate a colloidal probe embedded in a soft material ( Fig. 20(b)), two kinds of microrheological techniques exist: passive and active [197,201]. In the following sections, we describe how to prepare a prototypical viscoelastic fluid (section 4.3.2) as well as how to determine rheological parameters of simple liquids using passive and active microrheology (sections 4.3.3 and 4.3.4).\"\n },\n {\n \"title\": \"Preparation of a viscoelastic fluid and setup\",\n \"content\": \"An equimolar wormlike micellar solution made of the surfactant cetylpyridinium chloride (CPyCl) and the salt sodium salicylate (NaSal), both dissolved in water, represents a prototypical viscoelastic fluid. To prepare it, overnight mixing of the two components in deionized water is required at approximately 50 • C, after which a homogenous and isotropic viscoelastic fluid results. 24 To get reproducible rheological parameters of the fluid, the conductivity of water should be kept at most at 10 µS m −1 (milli-Q water is a good option for this purpose). Depending on the surfactant/salt concentration, different kinds of microstructures in the aqueous solvent are formed [204]. For instance, above the first critical micelle concentration but below 4.5 mM, the surfactant molecules aggregate into spherical and cylindrical micelles of radius 2 to 3 nm, which result in a weakly viscoelastic behavior of the dilute solution. Increasing further the concentration up to 10 mM, the surfactant molecules self-organize in long flexible cylindrical micelles, which form, overlap, and deform dynamically in the solvent. Such wormlike micelles have a radius from 2 to 3 nm and a contour length ranging from 100 nm to 1 µm, while their persistence length and mesh size are of order 10 nm for concentrations between 5 and 10 mM. To perform optical-tweezers-based microrheology, a very small amount of micron-sized beads must be suspended in a microliter volume of solution. Special care is needed to avoid air bubbles inside the fluid, which can be easily formed due to the presence of the surfactant. Sonication during a few minutes of the fluid with the dispersed colloidal particles is recommended before being loaded into the sample cell. Such a sample cell consist of a standard chamber (cross-sectional area about 1 cm 2 ) made of a glass slide and a coverslip, separated by a distance of approximately 100 µm by means of, e.g., spacer tape or parafilm strips. The chamber must be laterally sealed with UV-curable optical adhesive after inserting a small volume (∼ 10 µl) of the suspension, in order to avoid evaporation of the fluid and drift. During the measurements, the fluid sample must be kept thermally coupled to a thermostat at constant temperature T within an accuracy range of ±1 K, as the rheological properties of the fluid strongly depend on temperature. It is 23Here, we use the following convention for the Dirac delta ∫ ∞ 0 delt a(t )dt = 1/2. Thus, the factor 2 is needed in the relaxation modulus G(t) to recover the correct friction coefficient.\\n24It is recommended to wear gloves and safety googles when handling CPyCl, as direct contact with power can cause skin and eye irritation. crucial to maintain T above the so-called Krafft temperature [205], which is 18 • C for the specific micellar solution described here. Below the Krafft temperature, the surfactant crystallizes, thus suppressing the viscoelasticity of the fluid.\\nTo illustrate the microrheological techniques described in the following, two Newtonian fluids with distinct viscosities at room temperature are used. The first one is deoinized water, whose conductivity must be less than 10 µS m −1 in order to facilitate the sample cell preparation. The second Newtonian fluid is propylene glycol n-propyl ether (PNP), also known as 1-propoxy-2propanol, which can be directly purchased with a purity ≈ 99% so that no further purification is needed. On the other hand, two viscoelastic fluids are employed: an equimolar CPyCl/NaSal micellar solution at 4 mM and one at 5 mM. The different fluids and their main properties are summarized in Table 9. Table 9. Microrheology fluids. Fluids used to illustrate microrheology with an optically trapped particle.\\nAll the experiments described in this section are performed in a standard optical tweezers setup implemented with an oil-immersion microscope objective (100×, NA = 1.4) which tightly focuses a Gaussian laser beam (λ = 1070 nm) onto the sample. A single spherical particle is trapped and kept at least 10 µ away from any wall of the sample cell in order to avoid hydrodynamic interactions, which can significantly affect the values of the fluid's rheological parameters with respect to those obtained from bulk measurements. Imaging and position detection can be realized by conventional video microscopy. In particular, for the determination of the linear viscoelastic spectrum by passive microrheology, it is required that the acquisition frequency of the camera must be above about 1000 frames per second. Therefore, the use of a position sensitive detector is an alternative for this purpose. Furthermore, beam steering, which is crucial for active microrheology, is achieved here by means of galvomirrors (section 2.3.1).\"\n },\n {\n \"title\": \"Passive microrheology\",\n \"content\": \"Passive microrhelogy is the simplest approach to study the local mechanical response of soft matter. As its name suggests, the embedded colloidal bead is used as a passive element, whose thermal motion provides all the necessary information to extract, via fluctuation-dissipation relations, the frequency-dependent rheological parameters of the material under investigation. To prevent sedimentation as well as significantly large diffusive displacements of the probe from its initial position, which would otherwise restrict the measurement time of the particle position x(t), a static optical trap can be used to confine the particle motion around a mean fixed position, i.e., x ot (t) = 0, as depicted in Fig. 20(b). Here, we present two direct applications of passive microrheology with optical tweezers for viscous and viscoelastic liquids.\\nDetermination of zero-shear viscosities. For a Newtonian liquid, the autocorrelation function of the particle position provides a straightforward way to determine its viscosity η. From equation (85), using the instantaneous memory function Γ(t) = 2γδ(t) for a purely viscous liquid, the expression for its autocorrelation function can be readily derived (section 3.5):\\nwhere . . . represents an ensemble average over different realizations of the thermal noise, while τ ot = γ/κ is a decay rate related to the viscous friction coefficient γ of the probe and the trap stiffness κ. In practice, the ensemble average can be replaced by a time average along the stochastic trajectory x(t), provided that the total measurement time of x(t) is much larger than τ ot . For a spherical particle of radius a, the friction coefficient is γ = 6πaη, where η is the fluid viscosity. Then, once the trap stiffness κ is known (e.g., by the equipartition method: κ = k B T x(t) 2 , section 3.3), the experimental autocorrelation function of x(t) can be fitted to the right-hand side of equation (86). Therefore, η can be determined from the fitting parameter τ ot by means of η = κτ ot 6πa . This is illustrated in Fig. 21, where the viscosities at T = 25 • C of two Newtonian fluids, water and propylene glycol n-propyl ether (PNP), are computed by means of this passive method. In Fig. 21(a), we plot the typical trajectories x(t) of particles embedded in these liquids, trapped by optical tweezers at approximately 40 µm away from the cell walls. The corresponding autocorrelation functions of x(t) are shown in Fig. 21(b), where we show that the experimental data (symbols) can be very well fitted to equation (86) (dashed lines). From the equipartition relation, we first determine the values of the trap stiffness for each case: κ = 1.24 pN µm −1 for the particle in water (diameter 2a = 2.73 µm) and κ = 0.90 pN µm −1 for the particle in PNP (diameter 2a = 3.25 µm). Next, from the fitting parameters τ ot , we find the corresponding values of the viscosities of both fluids: 0.9 mPa s for water and 4.5 mPa s for PNP, which agree very well with their bulk values.\\nThis technique can be extended to determine steady-state flow properties of viscoelastic fluids, which exhibit both liquid-and solid-like responses at times shorter than a typical time τ. Such a time-scale reflects the stress relaxation of the elastic material microstructure suspended in the solvent. For a viscoelastic micellar solution, τ originates from the continuous formation and breaking of the micelles as well as the reptation modes of the wormlike structures, and can range from a few milliseconds to several seconds in the semidilute regime [206]. Unlike viscoelastic solids (e.g., gels), viscoelastic fluids are able to flow with constant shear rate at sufficiently long times, t τ, after applying constant shear stress at t = 0. Therefore, they are characterized by a zero-shear viscosity η 0 , which in absence of a confining potential, leads to a long-time diffusion of an embedded spherical particle, where the diffusion coefficient is given by k B T 6πaη 0 . In the presence of an optical trap, from equation (85) it can be shown that the position autocorrelation function has the form\\nwhere A < 1 is a prefactor that depends mainly on κ and τ. Equation (87) is valid provided that τ ot τ [207], which must be verified a posteriori. To fulfill this condition of large time-scale separation, sufficiently small values of the trap stiffness are needed. In Figs. 21(a) and 21(b), we illustrate this method for a 2a = 3.25 µm silica particle suspended in a dilute micellar solution (concentration 4 mM, T = 295 K), trapped by optical tweezers with stiffness κ = 1.34 pN µm −1 . At this concentration, the viscoelastic fluid is composed of non-overlapping micellar structures, thus resulting in typical relaxation times, τ, of order of milliseconds. We check that equation (87) is a very good approximation to the experimental position autocorrelation function, where only the first experimental point slightly deviates from the fit. The fitting parameter τ ot = 0.16 s yields the value η 0 = 6.9 mPa s, which is also in agreement with reported bulk measurements [204].\"\n },\n {\n \"title\": \"Determination of storage and loss moduli. The complex shear modulus\",\n \"content\": \", is a quantity used in rheology to characterize linear viscoelasticity of soft matter in the frequency domain [208]. G ( f ) is the storage modulus and G ( f ) is the loss modulus, which account for the elastic energy in phase with the applied strain and the out-of-phase viscous dissipation of the material, respectively. For instance, for a purely viscous liquid (viscosity η), G ( f ) = 0 and G ( f ) = 2π f η, whereas for an elastic solid (elastic modulus G 0 ), G ( f ) = G 0 , and G ( f ) = 0. In general, for viscoelastic materials, both storage and loss moduli are non-zero. G * ( f ) is closely related to the Fourier transform,G( f ), of the stress relaxation modulus, G(t), by\\nTherefore, for an applied oscillatory shear strain ( f ) = 0 e 2πi f t with amplitude 0 and frequency f , the resulting shear stress is also oscillatory and can be expressed as G * ( f ) ( f ).\\nA very helpful expression for G * ( f ) in terms of the equilibrium fluctuations of the position x(t) of a trapped particle can be derived from the fluctuation-dissipation theorem. The latter relates the linear response function χ(t) of x(t) to a perturbative force F(t), defined as\\nwith the equilibrium (F(t) = 0) autocorrelation function of x(t), i.e., x(t)x(0) . In equation (89), the nonequilibrium ensemble average . . . F is defined in presence of the time-dependent perturbation. From equations (85) and (89), the Fourier transform of\\nNote that equation (90) shows that the elastic contribution of the optical tweezers stiffness to the mechanical response of the system must be subtracted for a correct determination of the complex shear modulus of the surrounding medium. Moreover, equation (90) involves directly the inverse ofχ( f ), which can be determined by either passive or active microrheology, as will be shown. Therefore, this expression is applicable for both types of microrheological techniques. In particular, in the case of passive microrheology, by making use of the fluctuation-dissipation theorem in frequency domain and the Kramers-Kronig relations, one can find both the real, χ ( f ), and the imaginary, χ ( f ), parts of the response functionχ( f ):\\nwhere S( f ) = |x( f )| 2 is the one-sided PSD of x(t) in thermal equilibrium, i.e., the Fourier transform of x(t)x(0) , whereas P denotes the Cauchy principal value of the integral. Thus, once χ ( f ) and χ ( f ) are determined from the experimentally recorded trajectory of the trapped particle, the storage modulus G ( f ) and loss modulus G ( f ) can be computed over the available frequency range [0, f max ] from equation (90). For a given sampling frequency f s of x(t), f max is fixed by the Nyquist frequency: f max = 1 2 f s . This technique needs a rather high sampling frequency of x(t) to avoid a large underestimation of the integral in equation (91).\\nIn Fig. 22, we apply the previous method to characterize the linear viscoelasticity of the wormlike micellar solution in the semidilute regime at 5 mM and T = 20 • C. First, we compute the equilibrium PSD, S( f ), of a long trajectory x(t) measured over 25 min at a sampling frequency f s = 2000 Hz. Since any discrete Fourier transform involved in the calculation of S( f ) leads inexorably to a noisy curve, the spectral profile must be smoothed before numerically computing the integral of equation (91). In Fig. 22(a), we show the original noisy profile of S( f ) (blue line) obtained by means of a discrete Fourier transform computed with 2 17 points. A polynomial fit (dashed black line) is performed to smooth the PSD profile. Then, a direct numerical calculation using equations (90)- (92) leads to the values of the storage and loss moduli shown in Fig. 22(b) over the available frequency interval 0 Hz ≤ f ≤ 1000 Hz. At high frequencies approaching 1 2 f s , an abrupt decrease of the numerical values of both G ( f ) and G ( f ) is observed. This is an artifact due to the finite frequency range, which imposes a cut-off in the required Kramers-Kronig integral transformation [209]. Therefore, the physically meaningful values of the components of the complex modulus G * ( f ) are only those at frequencies one decade below the Nyquist frequency [210].\"\n },\n {\n \"title\": \"Active microrheology\",\n \"content\": \"In active microrheology, the trapped colloidal probe is externally forced through the material to locally strain it, and its mechanical response is directly measured [211]. From this response, G * ( f ) can be determined by means of equation (90). Different kinds of perturbations are possible. For instance, the probe particle can be trapped by static optical tweezers while the whole sample cell is moved by a piezo stage (section 3.11). Here, we will focus on a perturbation created by the oscillatory motion of the position x ot (t) of the optical trap, which exerts a time-dependent force F(t) = κx ot (t) on the particle according to equation (85) (Fig. 20(b)).\\nAs active microrheology provides a direct measurement of the linear response function χ(t) defined by equation (89), it does not necessarily require that the material under investigation is in thermal equilibrium. Consequently, this technique is suitable for the investigation of rheological properties of general viscoelastic fluids and solids, including out-of-equilibrium soft matter, such as actin networks [212,213], physical gels [214], and glassy colloidal suspensions [215,216]. It is particularly useful for the determination of the storage G ( f ) and loss G ( f ) modulus at a specific driving frequency f d , for which a single-frequency oscillatory perturbation force\\ncan be directly applied to the colloidal probe, where X 0 is the maximum displacement of the optical trap and φ is the initial phase. Therefore, a previous passive calibration of κ is needed to know the applied force (section 3). In order for both the optical trapping and the material microstructure to remain in the linear response regime, X 0 must be sufficiently small. This is generally fulfilled if the typical energy injected by F(t) is at most of the order of ∼ k B T, i.e., X 0 k B T/κ. In Figure 23, we show some results to illustrate the implementation of active microrheology to characterize the rheological properties of a wormlike micellar solution at 5 mM and T = 20 • C (Table 9). In the inset of Figure 23(a), the black line shows the time evolution of F(t)/κ, which results from the sinusoidal motion of the optical trap (equation (93) with X 0 = 200 nm, κ = 1.0 pN µm −1 , f d = 1 Hz). The blue line plots the trajectory x(t) of an actively driven particle (2a = 2 µm), which clearly exhibits an oscillatory behavior with a time delay relative to F(t) due to the resistance of the surrounding viscoelastic fluid. Here, the crucial step is to compute the Fourier transform of the linear response function from the measurement of x(t) and x ot (t). From equation (89), this is given byχ\\nwhere the numerator is the value at frequency f of the non-equilibrium ensemble average of the Fourier transform of x(t), in presence of F(t) with the same initial phase φ, whereas the denominator involves the Fourier transform of the trap position. Due to the sinusoidal form of x ot (t), the denominator is zero for all frequencies f different from f d . Therefore, in practice it is customary to compute the inverse of equation (94) and to approximate it at f d by\\nwhere the numerator is the Fourier transform of the cross-correlation function between the trap position and the particle position, while the denominator is the value of the PSD of x(t), both quantities at frequency f d , first computed for a given F(t) and then averaged over different realizations of φ. In Fig. 23(a), we plot the PSD of the particle position x(t) in presence of the force F(t) at different driving frequencies f d . The pronounced peaks at f = f d result from the sinusoidal form of F(t)\\n, while for f f d , the contribution to S( f ) is only due to thermal noise in the fluid. In Fig. 23(b), we plot as symbols the values of the storage and loss moduli computed by means of equations (90) at the distinct driving frequencies. For comparison, we also represent as dotted and dashed lines the corresponding values determined by passive microrheology, demonstrating the rather good agreement between both methods.\\nAlthough more elaborate to implement in experiments, active microrheology is less prone to numerical artifacts than passive microrheology. This is because it is based on a measurement of the mechanical response of the particle position at a single frequency, which does not require any indirect numerical integration over a discrete finite frequency interval as those involved in the Kramers-Kronig relations.\"\n },\n {\n \"title\": \"Other microrheological applications and outlook\",\n \"content\": \"In the previous sections, we have described some basic applications of microrheology with optical tweezers. More advanced methods, which are beyond the scope of this Tutorial, focus on investigating flow and deformation properties of complex materials which do not trivially fulfill the general conditions described above. Some important examples are the following:\\nInterfacial microrheology. Near a liquid-solid, a liquid-liquid or a liquid-vapor interface, the viscosity of liquids becomes anisotropic and exhibits a dependence on the closest distance to the interface due to the specific hydrodynamic boundary conditions. Passive microrheology can be applied to determine such a spatial dependence, provided that the three spatial coordinates of the probe particle, embedded in the liquid phase of interest, can be tracked [217][218][219].\\nTwo-point microrheology. The motion of pairs of colloidal particles not so far from each other is strongly correlated when suspended in a fluid. Since the flow and strain fields around one of them entrain the second particle, the motion of the latter encodes information of the rheological properties of the fluid at the location of the former. Therefore, a dual-beam optical tweezers can be used to fix their mean separation and perform two-point microrheology. Here, one particle plays the role of a control element, either thermally or externally driven, while the second one is used as a passive probe. This technique is particularly useful to investigate the mechanical response of heterogeneous and anisotropic soft materials, such as gels and polymer solutions [220,221].\\nTransient microrheology. Using a combination of both active and passive microrheology, the transient behavior of viscoelastic materials either upon flow startup or cessation can be investigated. For example, during an active period, an optically trapped particle can be driven though the viscoelastic fluid, followed by a sudden release from the tweezers by shutting off the laser, during which its position is recorded. During this non-equilibrium passive period, the particle motion is subject to the recovery of the initially strained fluid microstructure. Consequently, this technique allows to measure in a straightforward manner relaxation times and relaxation moduli of viscoelastic materials [222,223].\\nMicrorheology has evolved during the last two decades into a powerful experimental tool to investigate complex soft materials. Most of its current applications rely on a well-established theoretical framework based on equilibrium statistical mechanics, low-Reynolds-number hydrodynamics and linear response theory. Nevertheless, more recent approaches aim to investigate non-linear rheological properties, e.g., shear thinning of polymer solutions, by means of colloidal probes driven at very high velocities or under large-amplitude oscillations [20]. Although the implementation of such microrheological techniques does not represent a big experimental challenge, the interpretation of the data in terms of meaningful parameters and its connection with bulk quantities is not trivial. A complete understanding of such a wealth of information will certainly rely on current theoretical advancements in non-equilibrium soft matter systems [224], which are able to find a direct link between the motion of the probe and the microstructural deformation of the surrounding medium.\"\n },\n {\n \"title\": \"Colloidal interactions\",\n \"content\": \"A colloidal system comprises a continuum medium (the solvent, such as water) and a disperse phase (the solute, such as small colloidal particles from 10 nm to 10 µm). Due to their dimensions, colloidal particles experience thermal fluctuations and undergo Brownian motion [225]. Since their properties can be described using statistical physics ensembles, colloidal suspensions represent an ideal model system for statistical physics and they can be considered as \\\"big atoms\\\" [226] mimicking atomic systems at different time and length scales, which make their dynamics accessible by optical experiments. Therefore, colloidal suspensions have been used to investigate phase transitions between gas, liquid, solid, and crystalline phases [227,228], crystal nucleation [229][230][231], glassy states [232], and vapor-liquid interfaces [233]. Colloidal suspensions are also employed in industry to stabilize emulsions (oil in water and water in oil) and foams [234,235]. Colloidal interactions play a key role in all these applications, from food industry [236] to nanotechnology [237]. The main colloidal interactions are:\\nVan der Waals forces. Van der Waals forces arise between neutral particles such as atoms, molecules or colloids. They have an electromagnetic origin due to the interaction between the dipoles of the singles particles. They are repulsive at short distance range, due to repulsion between the atoms' electron clouds, and attractive at larger distance. Their intensity quickly decreases with the inter-particle distance following a power law [238,239].\\nDouble-layer forces. Double-layer forces are very important for the stability of many biological systems and for the formation of colloidal crystals [240,241]. They result from the spontaneous charging of the particle surfaces when immersed in a liquid, due to ionic adsorption or dissociation. They can be both attractive or repulsive.\\nDLVO theory forces. Double-layer forces and van der Waals forces are combined in the DLVO theory (named after Derjaguin, Landau, Verwey und Overbeek) [242] to calculate the interaction between spherical colloidal particles [243,244].\\nSteric forces. Steric forces are repulsive nonbonding interactions influencing the spatial conformation and the reactivity of ions and molecules. They can be used to promote or prevent flocculation (steric stabilization) of colloidal solutions. They play an important role in chemistry, biochemistry, pharmacology and food industry, where they are often employed to design the physical properties of food and pharmaceutical products by steric stabilization [236,[245][246][247], as well as to treat waste water and to purify drinkable water by flocculation of metals and airborne particles [248,249].\\nDepletion forces. Depletion forces are attractive forces arising in a suspension of large and small particles [250]. Usually, the large particles are colloids (typically, at least a fraction of a micrometer), while the small particles can be smaller colloids, micelles, solvent macromolecules, or dissolved ions [250][251][252][253][254].\\nCritical Casimir forces. Critical Casimir forces emerge between objects (e.g., colloidal particles) immersed in a critical mixture close to its critical point [255][256][257]. Althougth these forces were first predicted theoretically in 1978 by M. E. Fisher and P. G. de Gennes [255] in analogy to quantum-electrodynamical (QED) Casimir forces [258], they have been measured directly only recently [259,260] proving their relevance for soft matter and nanotechnological applications thanks to their piconewton strength and nanometric action range [261][262][263].\\nHere, we will focus our attention on how to measure critical Casimir forces. In particular, we will describe a typical experiment to quantify the dynamic effects of critical Casimir forces between two colloidal particles.\"\n },\n {\n \"title\": \"Critical Casimir forces\",\n \"content\": \"Critical Casimir forces take place between objects immersed in a critical mixture when they confine the mixture composition fluctuations arising near the critical temperature. These forces are typically in the piconewton and nanometer ranges, they are tunable as a function of temperature, and they feature an exquisite dependence on the surface properties of the involved objects [256,257,264,265]. Attractive critical Casimir forces arise whenever the density fluctuations are confined between objects with the same surface properties (e.g., between two hydrophilic or hydrophobic particles). Repulsive critical Casimir forces take place between objects with opposite surface properties (e.g., between a hydrophilic and a hydrophobic particle).\"\n },\n {\n \"title\": \"Experiment outline\",\n \"content\": \"In the following sections, we will present in detail how to perform an experiment where the effects of critical Casimir forces on the free dynamic of a pair of colloidal particles are measured using blinking optical tweezers [263,[266][267][268].\\nWe employ hydrophilic silica microspheres (diameter d = 2.06 ± 0.05 µm, Microparticles GmbH) dispersed in a water-2.6-lutidine mixture at the critical lutidine mass fraction c c L = 0.286, corresponding to a lower critical point at the temperature T c 34 • C [269,270]. Due to their extremely sensitive dependence on temperature, we have to control and stabilize the sample temperature to within ±2 mK via a feedback controller [260,263].\\nUsing two holographic optical traps, we trap two microparticles in the bulk of the critical mixture at a surface-to-surface distance of ≈ 300 nm. This distance is larger than the range of the electrostatic repulsion between the particle surfaces, but still comparable with largest action range of critical Casimir forces. Whenever the two optical potentials are switched off (by chopping the laser beam), the colloids diffuse in the bulk of the critical mixture.\\nWhen the temperature T of the solution is sufficiently lower than T c (i.e., for ∆T = T c − T 500 mK) no critical Casimir forces are observed and the particle diffuses freely in the medium. Approaching the critical temperature T c (i.e., for ∆T → 0), density fluctuations start to take place in the critical mixture leading to attractive critical Casimir forces between the two colloids, affecting their diffusion and reducing the average inter-particle distance. To obtain sufficient statistics of the dynamics of the colloids, we need to repeat the entire blinking process for several times (i.e, about 400 times) for each value of ∆T.\"\n },\n {\n \"title\": \"Experimental setup and feedback temperature controller\",\n \"content\": \"The experimental setup requires holographic optical tweezers, digital video microscopy, and feedback temperature control [260,263]. The holographic optical tweezers (section 2.3.5) is realized using a phase-only spatial light modulator, a laser beam with a wavelength of 532 nm, and a oil-immersion objective (100×, NA = 1.30). The resulting traps can be periodically switched on and off by a chopper.\\nThe most crucial part of the setup is the temperature controller to stabilize the sample temperature to within ±2 mK. This is realized using a two-stage stabilization protocol. The first temperature stabilization is achieved by keeping constant the temperature of the sample holder to within ±50 mK by a circulating water chiller. The second and finer temperature stabilization to within ±2 mK is achieved by a temperature feedback controller applied through the objective. The controller reads in real time the temperature of the objective and stabilizes it by a Peltier element thermically connected to the objective, which is the closest element to the investigated volume. An important aspect to take into account is the thermal isolation of the sample area as well as of the entire setup to prevent thermal fluctuations due to air flow and to room temperature fluctuations.\"\n },\n {\n \"title\": \"Potential analysis\",\n \"content\": \"The motion of the two colloids is recorded by a camera with acquisition frequency 300 fps and the acquired videos can be analyzed by digital video microscopy [22,271] to obtain the trajectories r 1 (t) and r 2 (t) of the centers of the two particles projected onto the x y-plane, where r l (t) = (x l (t), y l (t)) with l = 1, 2 labelling the particles. Once these trajectories are obtained, it is possible to calculate their relative distance r(t) = |r 2 (t) − r 1 (t)| for different values of T (Fig. 24). The time evolution of the probability density distribution, for all the inter-particle distances r(t) at specific values of ∆T, are reported in Figs. 24(a-d). For ∆T = 456 ± 2 mK, far away from the critical point ( Fig. 24(a)), the particles diffuse freely and there is no critical Casimir forces affecting their behaviour. Increasing the temperature, ∆T = 200 ± 2 mK ( Fig. 24(b)) and ∆T = 163 ± 2 mK (Fig. 24(c)), density fluctuations start to take place in the mixture and critical Casimir forces arise affecting the dynamics of the colloids. We can observe that occasionally they cause adhesion between the colloids as can be inferred from the emergence of a peak in the inter-particle distance probability density at r ≈ 2.16 µm. This peak is due to the gradual emergence of attractive critical Casimir forces between the particles, approaching T c , and indicate that critical Casimir forces are strong enough to produce particle adhesion also in the presence of the optical potentials. The peak position indicates the region where the repulsive electrostatic forces and the attractive critical Casimir forces are balanced [263]. When the temperature T is very close to T c (∆T = 108 ± 2 mK, Fig. 24(d)), strong attractive critical Casimir forces inhibit the free diffusion of the particles, which often adhere to each other so that the values of r lie Sometimes r(t) is smaller than d because a displacement of the colloids along the vertical z-axis causes their projections onto the xy-plane to overlap. This occurs more frequently in the presence of critical Casimir forces and particularly if the particles are temporarily stuck together. (e-h) Equilibrium distribution P eq (r) of the inter-particle distance r(0) (i.e., when the optical tweezers are switched off) for two optically trapped colloids at temperatures (f) ∆T = 456 ± 2 mK, (g) 200 ± 2 mK, (h) 163 ± 2 mK, and (i) 108 ± 2 mK. Each histogram is obtained from 400 different experimental values. The solid black lines are the theoretical distribution of r(0), obtained via Monte Carlo integration (10 6 samples) of two optically trapped particles subjected to the theoretical total potential V(R 1 , R 2 ) (equation (96)). within a small region resulting from the equilibrium between the repulsive electrostatic forces and the attractive critical Casimir forces [263].\\nFor each value of ∆T, all the initial inter-particle distances r 1,2 (0) represent the equilibrium distribution of initial positions R l = (x l , y l , z l ) with l = 1, 2 of the two colloids subjected to a total potential including the optical potentials V ot,1 (R 1 ) + V ot,2 (R 2 ), the repulsive electrostatic potential V es (ρ), and eventually the potential of the critical Casimir forces V C (ρ):\\nwhere ρ = |R 2 − R 1 | − d is the actual surface-to-surface distance between the two colloids. Note that the projected distance r introduced above is generically smaller than the actual center-to-center distance |R 2 − R 1 |, due to possible displacements of the colloids along the vertical z-direction.\\nAssuming that the optical potentials V ot,1 and V ot,2 reported in equation (96) are harmonic, we have:\\nwhere the centers R 0,l and the stiffnesses k l of traps l = 1, 2 can be determined experimentally by the separate calibration of the two optical traps. These initial values are fixed in all the experiment and in the following analysis.\\nFor the electrostatic repulsion potential V es , we can consider the simple expression [260,270]\\nwhere ρ is the surface-to-surface distance between the colloids, ρ es is an effective parameter, which depends on the surface charges, while D is the Debye screening length [260,270]. For the potential V C of the critical Casimir forces, we can consider the theoretical prediction based on the Derjaguin approximation [270]:\\nwhere Θ is a universal scaling function [272][273][274] and ξ is the bulk correlation length of the critical fluctuation of the density fluctuations of the critical mixture. Whenever the optical potential is switched on, the particles trajectories evolve under the action of the total potential V(R 1 , R 2 ), and after a sufficiently long time, they reach the equilibrium distribution:\\nThe histograms in Figs. 24(e-h) represent the experimental values of the equilibrium distribution P eq and the solid black lines are the corresponding theoretical results obtained from the Monte Carlo integration of equation (100). When the mixure temperature is far away from T c (∆T = 456 ± 2 mK, Fig. 24(e)), the probability distribution P eq (r) is very well approximated by a Gaussian distribution centered at the value r 2.40 µm, corresponding to experimental distance r 0 between the centres of the two optical traps [263]. Reducing ∆T (Figs. 24(e-h)), a peak arises at r 2.16 µm on the left flank of the Gaussian distribution, becoming more dominant at the expense of the Gaussian distribution. Starting from probability distribution P eq (r), we fit its experimental values by Monte Carlo integration of equation (100) based on the theoretical potential V(R 1 , R 2 ). In particular, we use the experimental values of V ot,l (R l ) (equation (97)) and Θ(ρ/ξ) (equation (99)) as input functions, and ρ es , D , and ξ as fitting parameters. In doing this, we assume that the fitting parameters ρ es and D have the same values ( D 13 nm and ρ es 95 nm) for all T, while the correlation length ξ is specific to each T. Once the values of ξ have been obtained, we calculate the real temperature of the solution by fitting the experimental temperature values measured at the objective with the theoretically expected temperature dependence of ξ, according to the method reported in Refs. [260,263]. (105)) fitted to the experimental data. The colors refer to data taken at ∆T = 456 ± 2 mK (blue), 273 ± 2 mK (light blue), 200 ± 2 mK (green), 163 ± 2 mK (yellow), 127 ± 2 mK (orange), and 108 ± 2 mK (red). Errors bars represent the standard deviation of the experimental values. The solid lines represent the theoretical predictions accounting for the effect of the hydrodynamic interaction between the colloids [275]. The nature of the deviations observed in D at short and long distances is discussed in the main text.\"\n },\n {\n \"title\": \"Drift analysis\",\n \"content\": \"The relative position and the distance between the two particles can be used to determine the values of their diffusion coefficient D(r) and their drift velocity v(r) as function of their inter-particle distance r. Due to the hydrodynamic interaction between the two colloids, the particle diffusion coefficient depends on the distance between the particles and is different along the directions parallel and perpendicular to the line connecting the centers of the two particles (see equation (5.5) in Ref. [275]). For this analysis, we consider the particle trajectories as a sequence of values r i , numbered by i and acquired at times it s , where t s is the time between sampling. We can decompose the i-th displacement ∆r (n) i = r i+n − r i into its parallel and perpendicular components:\\nand\\nwherer i = r i /r i andẑ is the unit vector along the z-direction, which is perpendicular to the x y-plane of observation where the position vectors r i lie. Then, the parallel and perpendicular diffusion coefficients are [263]:\\nand D ⊥ (r) = 1 2 where δr represent the amplitude of a small interval around r. The measured normalized values of D ⊥ and D (symbols) are reported in Fig. 25 as functions of the ratio r/d, together with the theoretical prediction obtained in Ref. [275] (solid line) for no-slip boundary conditions. In particular, D ⊥ and D are normalized by the bulk diffusion constant\\nwhere η 2 · 10 −3 Ns m −2 is the viscosity of the mixture close to T c [276]. The values of D ⊥ and D in Fig. 25 shows satisfactory agreement, with a systematic discrepancy emerging only in D for r/d 1.05 due to the limited experimental acquisition rate, which does not allow us to resolve times shorter than 3 ms. 25 The parallel and perpendicular drift velocities can be measured using:\\nand\\nFor large ∆T (Fig. 25(a) (107)). Here, notice that v ⊥ vanishes in all investigated cases and shows no temperature dependence because all the forces at play in this experiment act along the direction which connects the centers of the particles [263].\"\n },\n {\n \"title\": \"Other colloidal interactions and outlook\",\n \"content\": \"Critical Casimir forces can play an important role in nanoscience and nanotechnology thanks to their piconewton strength, nanometric action range, fine tunability as a function of temperature, and particle surface dependence. They can be employed to manipulate micro and nano-objects (e.g., by controllable periodic deformations of chains), to assemble micro and nano-devices (e.g., via the self-assembly of colloidal molecules [277,278]), and to drive nanomachines (e.g., by powering rotators [279]) at the nano and micro-meter scale.\\nBeyond critical Casimir forces, other colloidal interactions have also been studied using optical tweezers. Recently, Van der Waals forces have been directly measured between two isolated optically trapped rubidium atoms [280] opening the path to the atomic modeling of micro-and nano-structured devices controlled by these forces [281]. Double layer forces can also play an important role in nanotechnology due to their ability to tune the transport and deposition rate of micro and nano-sized particles. Steric forces play an important role in tuning the rates and the activation energies of most chemical reactions, crucial in chemistry, biochemistry, and pharmacology. Depletion forces are extensively used to stabilize colloidal solutions by flocculation, to produce the programmable self-assembly of colloidal nanostructures [282], and to drive cellular organization [252].\"\n },\n {\n \"title\": \"Statistical physics\",\n \"content\": \"The advent of minituarization methods and microscopic manipulation techniques, as those discussed in this Tutorial, has opened the door to understand fundamental concepts and ideas at the foundations of statistical physics, allowing us to explore how macroscopic laws emerge from their microscopic dynamics. In this section, we only discuss experiments performed in a liquid. For related work in vacuum, see section 5 and Ref. [19]. We will start discussing the theory of Kramers transitions and stochastic resonance (section 4.5.1). We will continue exploring non-equilibrium fluctuation-dissipation theorems (section 4.5.2) and introducing stochastic thermodynamics (section 4.5.3). Finally, we will explain how the Carnot's cycle and efficiency of macroscopic engines can be scaled down to the microscopic realm, and conclude with a discussion of the concept of Maxwell's demons and Szilard engines (section 4.5.4).\"\n },\n {\n \"title\": \"Kramers' transitions and stochastic resonance\",\n \"content\": \"Using optical trapping techniques, Simon et al. [283] and later McCann et al. [284] studied the escape of a Brownian particle from a potential well, visualizing for the first time thermally activated processes and giving in this way a neat experimental proof of Kramers' theory [285].\\nKramers' theory in the overdamped limit predicts the mean rate for the activation of processes over an energy barrier ∆U to be\\nwhere τ is the mean residence time, also called Kramers' time and τ 0 is the overall relaxation time of the well, which can be expressed in terms of the angular frequencies near the stable\\nwith γ the friction coefficient and m the mass of the particle. For the two-dimensional case, the expression for 1/τ 0 takes the following form [286]:\\nwhere the superscripts (x) and (y) indicate the parallel and perpendicular directions to the axis joining the stable and unstable critical points, respectively. Fig. 27(a) shows a schematic of the configuration of the optical tweezers to generate a double-well potential.27 A silica bead28 of diameter d p = 0.96 µm is confined in a bistable potential generated by two parallel optical tweezers with orthogonal polarizations and separated by a distance d = 0.8 µm, which are generated by focusing two Gaussian beams with a water immersion objective with NA = 1.2. These two beams are generated with two polarizing beam splitters according to the scheme shown in Fig. 5 (section 2.3.2). The mirrors in the beam splitter are controlled with piezo actuators with 30 nm step resolution leading to a resolution of the position of the optical traps in the sample cell below 20 nm. The trapping region is set 27Alternative experimental configurations have also been used to realize a bistable optical potential. For example, using two nanoholes in a metallic film [287], sculpting the bistable potential with a spatial light modulator [288], or positioning time-shared optical traps in two closely-separated positions using acousto-optical deflectors [289].\\n28Interestingly, also 1 µm polystyrene particles can be trapped with this setup, but with these particles we have not been able to observe any bistable behavior. We believe this is because the higher weight and lower refractive index of the silica beads make them more stably trapped along the axial direction of the beams allowing us to explore relatively larger distances between optical tweezers in a range where the bistable potential is neatly observed. A typical trajectory of the Brownian particle in this double-well potential is shown in Fig. 27(b). Along the x-direction joining the equilibrium points, the particle spends most of the time near the equilibrium points A and B, occasionally crossing the saddle point S. Along the perpendicular y-direction, the motion does not appear to be affected by the presence of two neighboring traps. Knowing the friction coefficient of the particle, this trajectory permits us to determine the Kramers' time using equations (108) and (110), where the characteristic frequencies (or equivalently their stiffnesses) at the critical points can be straightforwardly obtained using FORMA (section 3.8). Fig. 27(c) shows the force field in the xy-plane reconstructed using FORMA, from which the Table 10. Characterization of equilibrium points in a bistable potential. The theoretical mean crossing rates (1/τ (th) ) obtained using equations (108) and (110) Table 10 provides the stiffnesses of the equilibrium points (κ x and κ y ), the energy barriers (∆U), and the overall relaxation times (1/τ 0 ), from which we estimate the mean crossing rates (1/τ (th) ) using equations (108) and (110). The average crossing rate can be straightforwardly obtained from the trajectory of the bead by detecting the times at which the particle crosses the saddle point, shown in Fig. 27(f). For the validity of the approximations within Kramers' theory, these crossing events have to be rare, i.e., the time spent by the particle near the stable positions must be always longer than the relaxation time τ 0 defined by the stiffnesses of the critical points, limiting the ideal situations to the cases where the energy barrier is very high in comparison to k B T. This is a homogeneous Poisson process, where each escape event occurs at random with small probability [283]. Therefore, we expect the escape times to be exponentially distributed (red curves in Fig. 27(f)):\\nwhere p 0 = 1/τ is the mean crossing rate and τ is the mean residence time. The resulting experimental mean rates are reported in Table 10 (1/τ (ex) ), showing very good agreement with data obtained by means of Kramers' assumptions (1/τ (th) ). Bistable potentials are often employed to study non-linear dynamics. For example, a bistable potential such as that described in this section has been used to demonstrate the emergence of stochastic resonance when the optical tweezers are modulated in time harmonically, but asymmetrically (i.e., one potential well gets deeper while the other one gets shallower) [283]: as the modulation time approaches the Kramers escape time, the escape probability and the activation rate increases. A different process of synchronization between two bistable systems was also demonstrated when two optically generated bistable systems with one bead each in water are set very close to let the particles to interact [288]: when the two bistable systems have slightly different escape rates, the particles synchronize their jumps and lock their rates to an intermediate value between those featured by the two non-interacting systems.\"\n },\n {\n \"title\": \"Non-equilibrium fluctuation-dissipation relations\",\n \"content\": \"In its simplest form the fluctuation-dissipation theorem (FDT) [290] can be expressed as k B T µ = D (Einstein-Smoluchowski equation (7)): it provides a direct relationship between the mobility µ = 1/γ, which measures the response of its velocity to an external force, and its diffusion coefficient D.\\nMore broadly, the FDT states that for a system in contact with a heat bath at temperature T, upon applying a small external perturbation h which changes the energy U of the system as U → U − hV, where V is the variable conjugate to h with respect to the energy, the following relation holds for any observable Q [291]:\\nwhere\\nis the linear response function of Q at time t to the perturbation h at time s, δ δh denotes a functional derivative, i.e.\\nand C(t − s) = Q(t)V(s) 0 is the two-time correlation function between the observable Q and V. The brackets . . . h denote a non-equilibrium ensemble average in presence of the perturbation h, whereas . . . 0 represent an equilibrium average for h = 0. The expression (112) of the FDT can be re-written in the integral form as\\nwhere χ(t) = ∫ t 0 R(t − s)ds is the integrated response function, which is usually much easier to implement and measure than R(t).\\nThe importance of equations (112) and (114) for experiments lies on the fact that linear response functions, which characterize states slightly away from equilibrium, and equilibrium correlation functions can be interchangeably determined by an appropriate choice of the varibles h, Q and V, depending on the specific conditions. For example, for a colloidal particle suspended in water at constant temperature T and trapped by optical tweezers with stiffness κ, the integrated response function of the particle position x with respect to the trap center x OT to an external time-dependent force can be determined by applying a Heaviside-like perturbative displacement of the trap position x OT (s) = x 0 Θ(s) and then by measuring the resulting time evolution of the ratio x(t) h κ x 0 at t ≥ 0. Then, it follows that R(t) = 1\\n. Equivalently, it can be indirectly computed by means of the equilibrium quantity 1 k B T ∂ s C(t − s), where t ≥ s, Q = x is the observable of interest, V = x is the variable conjugate to the perturbative force h = κx OT , and C(t − s) = x(t)x(s) 0 , for which no external force is needed. Using the explicit expression for the positional correlation function in thermal equilibrium (section 3.5), x(t)x(s) 0 = k B T κ exp − κ |t−s | γ , one finds R(t) = 1 γ exp − κt γ , t ≥ 0, from which the time evolution of x(t) h in response to any external force h can be determined by means of equation (113).\"\n },\n {\n \"title\": \"Fluctuation-dissipation relation for non-equilibrium steady states (NESS).\",\n \"content\": \"Optical tweezers can be used to explore some generalizations of equation (112) around non-equilibrium steady states (NESS) for mesoscopic systems such as colloidal particles, molecular motors, and biomolecules [147]. In particular, a 1D model system that has been extensively studied within the context of non-equilibrium statistical physics consists of an overdamped particle with friction coefficient γ, moving on a circle of radius R circle and polar coordinate θ, across a periodic potential U(θ) = U(θ + 2π) under the action of a constant force f 0 and a thermally fluctuating force ξ [164]. The dynamics of the polar coordinate θ is described by the Langevin equation\\nwhere φ(θ) = U(θ) max{U(θ)} is the normalized potential profile, A = max{U(θ)} γa 2 is its amplitude, F = 1 γa f 0 is a non-conservative force term,29 and ζ = 1 γa ξ is a Gaussian white noise of zero 29On a circle, any constant force f 0 0 is non-conservative, because it cannot be expressed as the derivative of a potential due to the periodic boundary conditions. mean, i.e., ζ = 0, and autocorrelation ζ(t)ζ(s) = 2D θ δ(t − s), where the angular diffusion coefficient along the circle is given by D θ = D a 2 = k B T γa 2 . Due to the non-conservative force, the system lacks detailed balance: for constant values of F > 0, A > 0 and D θ > 0, and in absence of time-dependent perturbations, θ reaches a NESS, characterized by a distribution ρ NESS (θ), which is different from the equilibrium Boltzmann distribution ρ eq (θ) ∝ exp − Aφ(θ) D θ . Under such conditions, the particle is able to go beyond the potential barrier and explore the whole circle due to the combined effects of thermal fluctuations and the non-conservative force, thereby developing a constant non-vanishing probability current given by\\nAround this NESS, the expression (112) of the FDT can be generalized by the following relation [292]\\nwhere . . . 0 and . . . h represent now a non-equilibrium ensemble average in the NESS (h = 0) and around the perturbed NESS (h 0), respectively, while the additional term\\ntakes into account the extent of the \\\"violation\\\" of the conventional FDT due to the non-vanishing current j. Here, the observables Q and V as well as v 0 are functions of θ, while v 0 is called the mean local velocity, and is given by v 0 (θ) = j/ρ NESS (θ). Note that the generalized expression (117) around a NESS reduces to the conventional FDT (112) around thermal equilibrium, when j = 0. The expression (117) of the generalized FDT can also be written in the integral form\\nwhere B(t) = ∫ t 0 b(t − s)ds, and represents the generalization of equation (114) for NESS. Similar experimental approaches based on scanning optical tweezers have been followed to investigate fluctuation-response relations for other kinds of non-equilibrium states [293][294][295][296][297].\"\n },\n {\n \"title\": \"Experimental implementation of a NESS.\",\n \"content\": \"We illustrate the generalization of the FDT around a NESS with optical tweezers. To experimentally realize an overdamped system whose stochastic motion is well-modelled by equation (115), we consider a spherical silica particle (radius a = 1 µm) suspended in water at constant temperature (T = 20 • C, γ = 1.89 · 10 −8 kg s −1 ). The particle is subject to a toroidal optical trap, which can be created by a conventional single trap, whose position is deflected on the plane transverse to the beam axis either by two perpendicular galvanometric mirrors [293,298], or by a pair of perpendicular acousto-optic deflectors (AOD) [299]. For the latter approach, one applies simultaneously sinusoidal frequency modulations to the acoustic waves created inside each AOD X and Y, ∆F X,Y (t) = ∆F sin(2π f R t + α X,Y ), with amplitude ∆F ∼ MHz, scanning frequency f R , and phase difference |α X − α Y | = π 2 . When f R is kept fixed, the incident laser beam is deflected in a periodic manner. In this way the beam focus describes a circular trajectory inside the sample cell, θ OT (t) = 2π f R t + α, of constant radius R circle at frequency f R on the plane perpendicular to the beam propagation. To achieve a periodic force landscape as described by the Langevin model (115), the scanning frequency of the laser beam must be on the order of f R ∼ 10 2 Hz. For a particle with radius a = 1 µm, the optical setup must be built in such away that the final radius of the circular path described by the tweezers is R circle ≈ 2 − 5 µm. Smaller values of R circle lead to a Mexican-hat potential U(θ) in which the particle can spontaneously jump from θ to θ + π, similar to a Kramer process in a double-well potential (section 4.5.1), while larger values of R circle yield very slow dynamics for which it becomes experimentally difficult to sample θ with good statistics. For R circle = 4.12 µm, Fig. 28. Experimental realization of a NESS (a) Typical unwrapped (i.e., defined on the interval [0, ∞)) trajectories θ(t) of a colloidal particle in a NESS. The dashed line represents the mean drift due to the non-zero probability current j induced by the non-conservative term F. Inset: example of wrapped (i.e., defined over [0, 2π)) trajectory θ(t). (b) NESS probability density function of θ, defined over [0, 2π) (bars) and reconstructed potential energy U(θ) (solid line). The arrow indicates the direction of the non-conservative force f 0 , which shifts the maximum of ρ NESS (θ) to the right relative to the minimum of U(θ). Inset: comparison between the NESS distribution ρ NESS (θ) and the equilibrium one ρ eq (θ) (F = 0, sharp peak). (c) Correlation function between the observable Q(θ) = sin θ and the variable V(θ) = sin θ. f R = 200 Hz, and an infrared laser (λ = 1064 nm) with incident power of 30 mW, the resulting scanning speed of the focused beam is v R = 2πR circle f R ≈ 5 mm s −1 , which is so high that it is not able to trap and drag continuously the particle through the fluid because the viscous frictional force due to the surrounding water quickly exceeds the optical trapping force. Consequently, the beam only kicks the particle a small distance δs γv R κ along the polar direction θ at each rotation. During the absence of the beam (≈ 1/ f R = 5 ms) the particle undergoes free diffusion over a length scale l D = D f R 40 nm R circle in the directions radial and perpendicular to the circle. The beam kicks the particle again during the next arrival, thus preventing its escape by diffusion away from the region scanned by the beam focus. Therefore, the particle motion is effectively confined to a torus of major radius R circle and minor radius ∼ l D /2, where the angular position θ is the only relevant dynamical degree of freedom. In addition, a periodic intensity profile can be created along the main circle by sinusoidally modulating in time the laser power P(t) with an amplitude ∆P = 2.1 mW around the mean value of P 0 = 30 mW, at the same frequency as the scanning frequency f R , i.e., P(t) = P 0 + ∆P sin(2π f R t + α X ). These optical conditions result in the desired force landscape: a constant non-conservative force f 0 associated to the mean kick and a static periodic potential U(θ) due to the conservative force exerted by the sinusoidal light intensity profile [299,300].\\nThe instantaneous angular coordinate θ of the optically driven particle can be obtained from its 2D trajectories (x, y), by θ = arctan(y/x) ( Fig. 28(a)). The effective parameters F, A and the profile φ(θ) must be determined from an ensemble of independent NESS trajectories θ(t): at least 100 trajectories, each with at least a full rotation on the circle are needed to find the probability current j (dashed line in Fig. 28(a)) and the corresponding nonequilibrium distristriution ρ NESS (θ) (histogram in Fig. 28(b)).\\nDepending on the quantity of interest, θ must be defined either on the interval [0, 2π) or [0, ∞). For example, the probability current must be computed as [299]\\nwhere θ(t) 0 is the average of an ensemble of independent NESS trajectories θ(t), defined on [0, ∞) ( Fig. 28(a)). This permits us to verify that θ(t) 0 is linear in t, thus leading to a constant j > 0. The quantity j −1 represents the typical time the particle needs to perform a full cycle around the circle, which corresponds to the longest relaxation time of the NESS. For example, for the experiment presented here j −1 = 26 s, which is much larger than the typical equilibrium relaxation time of a particle in water trapped by optical tweezers (∼ ms). Therefore, it is expected that the particle motion in this NESS has very long-lived correlations. In Fig. 28(b), we plot the experimental profile of the NESS distribution ρ NESS (θ), computed from all the data points of the trajectories on [0, 2π). From the experimental profile of ρ NESS (θ) and the value of j, the full force landscape of the particle can be reconstructed [301]. First, taking into account the periodic boundary conditions of the NESS distribution, i.e. ρ NESS (0) = ρ NESS (2π), from equation (116) we get the non-conservative term\\nwhereas the potential is given by\\nThe experimental values of the parameters considered here yield F = 0.85 rad s −1 and A = 0.87 rad 2 s −1 , which correspond to a non-conservative force f 0 = γaF = 66 fN and an energy potential amplitude max{U(θ)} = γa 2 A = 68.8 k B T, while the normalized potential profile is sinusoidal, φ(θ) = sin θ (red solid line in Fig. 28(b)). In thermal equilibrium (F = 0), the particle motion would be tightly confined around the minimum of such a deep non-linear potential at φ m = 3π/2, with a width ∆θ ≈ k B T γa 2 A = 0.12 rad and a vanishingly small probability to cross the potential barrier of height 2A = 137.6 k B T (inset in Fig. 28(b)). However, under the action of the non-conservative force, the distribution of θ is much broader, with a maximum shifted in the direction of the resulting current j > 0. This reflects the highly non-equilibrium conditions of the system. Indeed, they can be better assessed by means of the generalized FDT relation (119). We focus on the observable Q(θ) = sin θ, which is periodic on the circle and corresponds to the dimensionless potential energy of the particle φ(θ) = U(θ)/max{U(θ)}.30 A variation in the potential amplitude, A → A + δ A translates into the following change in the potential energy: γa 2 Aφ(θ) → γa 2 Aφ(θ) − (−γa 2 δ A)φ(θ). Therefore, by noting that a variation h in a control parameter leads to a change in the potential energy, i.e., U → U − hV, we conclude that in this specific example the integrated response function of Q(θ) to the perturbation h = −γa 2 δ A is given by equation (119), where the variable conjugate to the perturbation is V(θ) = sin θ, whereas ∂ θ V(θ) = cos θ in equation (118). In Fig. 28(c), we plot the NESS correlation function C(t) = sin θ(t) sin θ(0) 0 , revealing an oscillatory behavior with a typical time-scale ≈ 25 s, which corresponds to j −1 . Here, to improve the statistics from the available data, the NESS ensemble average sin θ(t) sin θ(0) 0 is computed first by taking a time average\\nalong a given realization of θ(t), where t max > j −1 to span a full cycle of the probability current, and then by performing an additional ensemble average C t max (t) over different independent NESS trajectories θ(t). The same procedure must be carried out for the correlation B(t) defined through equation (118). In Fig. 28(d), we plot both the NESS terms C(0) − C(t) and B(t) involved in the integral generalized FDT (119). We observe that both terms are of the same order of magnitude, thus confirming that the experimental conditions of the system are far from thermal equilibrium. Note that, without the corrective term b defined in equation (118), a determination of the integrated response function χ(t) by means of the equilibrium FDT (114), and consequently R(t), would be wrong. This is shown in the inset of Fig. 28(d) , where we compare the numerical time derivative of 1\\n, which shows that without taking into account the corrective term B, the response function R(t) would be highly overestimated around a NESS.\"\n },\n {\n \"title\": \"Stochastic Thermodynamics\",\n \"content\": \"Stochastic thermodynamics is an emerging branch of statistical physics that extends concepts of classical thermodynamics such as heat, work and entropy production to the level of single stochastic trajectories for systems in contact with a heat bath and driven arbitrarily far from thermal equilibrium by means of a well-specified experimental protocol [164]. It provides a number of statistical relationships involving these fluctuating quantities, which generalize the macroscopic laws of thermodynamics and describe the non-equilibrium energetics of many mesoscopic systems of interest in soft matter and biophysics, such as colloidal particles, polymers, biomolecules, molecular motors, biochemical networks, and microelectromechanical systems [147,302]. Thus, stochastic thermodynamics represents nowadays a solid theoretical framework for applications ranging from single-molecule biomechanics (section 4.1) to optimization protocols and the efficiency of microscopic heat engines (section 4.5.4) .\\nHere, we present some general notions of stochastic thermodynamics using a paradigmatic system: a colloidal particle embedded in a viscous liquid (friction coefficient γ), in contact with a thermostat at constant temperature T and subject to a possibly time-dependent force. The 1D motion of a single coordinate x of the particle can be described by the Langevin equation\\nwhere U(x, λ) is the potential energy which leads to a conservative part −∂U(x, λ)/∂ x in the total force, whereas f (x, λ) represents a non-conservative force, as that described in section 4.5.2. Both forces can be time-dependent through the control paramenter λ, which can be externally varied according to a prescribed protocol from λ(0) to λ(t) during the time interval [0, t]. In addition, in equation (124) µ = 1/γ is the particle mobility, whereas ξ represents a Gaussian white noise with zero-mean and autocorrelation function ξ(t)ξ(s) = 2Dδ(t − s), which accounts for the thermal fluctuations of the particle velocity. Stochastic thermodynamics requires that the properties of the heat bath are not affected by the presence of external driving forces, thus implying that D = k B T µ regardless of the strength of −∂U(x, λ)/∂ x and f (x, λ). Note that a statistically equivalent description of the system modelled by equation (124) can be given by means of the Fokker-Planck equation for the probability density function of x at time t\\nwhere the probability current is defined as\\nsupplemented with the initial condition ρ(x, t = 0) = ρ 0 (x). Both ρ(x, t) and j(x, t) allow to directly assess how far the system is from thermal equilibrium conditions, for which j = 0 and ρ ∝ exp − U(x) k B T .\"\n },\n {\n \"title\": \"First Law.\",\n \"content\": \"To generalize classical thermodynamic concepts to the level of an individual stochastic realization of the process x(t), we first note that either a temporal variation of the control parameter λ during the infinitesimal time duration dt or a non-zero non-conservative force, f 0, can be identified as an infinitesimal work increment done on the system:\\nThen, from equation (127) an inifinitesimal variation dU in the potential energy during dt can be written as\\nThus, by defining the infinitesimal increment of the heat dissipated into the medium as the product of the total force acting on the particle times the infinitesimal displacement dx,\\nthe expression (128) can be regarded as a generalization of the First Law of thermodynamics [303] dU = δW − δQ on the level of the stochastic process x(t). From the Langevin equation (124), the work done on the system and the heat dissipated into the medium over the finite time interval [0, τ], are\\nrespectively, where the notation [x(t)] highlights the fact that both quantities fluctuate over time and depend on the specific stochastic realization of x(t) up to time τ. The terms which involve dx(t) = dt x(t) in the integrals of equations (130) and (131) must be interpreted in the Stratonovich sense [304]. Note that the definition of the heat given by equations (128) and (131) is physically meaningful, since the first term on the right-hand side of equation (131) is positive definite and represents the energy delivered to the thermostat, while the second term can be either positive or negative and corresponds to the thermal fluctuating energy injected by the medium. Moreover, for systems in thermal equilibrium (i.e., f = 0 and λ = 0), equation (129) consistently reduces to dU = −δQ, where any increment (decrease) in the potential energy of the particle is only due to a fluctuation of the heat taken up from (delivered to) the thermostat, being both on average equal to zero. We analyze the previous concepts in the specific case of a colloidal particle (diameter 2a = 2.73 µm) suspended in water at constant T = 295 K, for which the friction coefficient is γ = 2.44 · 10 −8 kg s −1 . The particle is subject to the harmonic potential of an optical trap, where the position of its minimum plays the role of the control parameter λ in equation (124). In the case where the optical trap is uniformly translated, λ varies linearly over time at constant speed v and, under the initial condition λ(t = 0) = 0, U(x, λ(t)) = 1 2 k (x − vt) 2 , λ(t) = vt, and λ(t) = v. Without loss of generality, we consider the simplest situation where all the forces acting on the system are conservative (i.e., f = 0), thus being the temporal variation of λ the source of non-equilibrium work and dissipated heat. This well-defined protocol can be implemented by means of the deflection of the trapping beam using galvanometric mirrors or acousto-optics deflectors (section 2.3.1). Although with a different thermodyamic interpretation,31 a closely related protocol can be carried out by moving the sample cell at constant speed v by means of a piezoelectric stage (section 3.11) [305]. Fig. 29(a) shows a typical experimental trajectory x(t) of such a colloidal particle, as well as the time evolution of the position of the minimum of the optical trap, λ(t). For times much longer than the trap relaxation time (≈ 21 ms), the dragged particle reaches a steady state, whose statistical properties do not depend on time. In this case, from the definition (130), one can find the work done by the optical tweezers on the trapped particle over a time interval of duration τ along a given stochastic realization of x(t):\\nwhere we have dropped the argument [x(t)] for the sake of simplicity. Since for this control parameter and a harmonic potential, equation (124) is linear and ξ is Gaussian, x must also be 31When moving the sample stage at speed v, an external flow is applied to the trapped particle, which results in an extra term in the viscous drag force acting on the particle, i.e. γ( x − v). On the other hand, the potential energy remains unaffected by v, U(x) = κ x 2 /2. This leads to expressions for the work and heat which are different from those obtained when performing a motion of the optical trap.\\nGaussian and, thus, W τ is Gaussian as well, with mean\\nand variance\\nIndeed, using the experimental trajectory x(t) (Fig. 29(a)), we compute numerically W τ by means of equation (132) and check that its mean and standard deviation satisfy equations (133) and (134), as shown by the blue circles in Figs. 29(b) and 29(c), respectively. The distribution of W τ for different values of τ is plotted in Fig. 29(d), where we verify that P(W τ ) is Gaussian. Note that, while W τ is always positive and increases linearly with the observation time τ as the trap drags the particle through the fluid, there can be negative work fluctuations, where the system loses mechanical energy [305]. These rare events, which are at first glance in contradiction with macroscopic intuition, are due to thermal fluctuations, which are not taken into account by classical thermodynamics. The role of such fluctuations, quantified by the ratio [W τ − W τ ] 2 / W τ , is very prominent at short time-scales and become negligible as τ approaches the limit of a macroscopic observation time, i.e., τ γ k . The heat dissipated into the medium due to the motion at constant speed of the trap can be derived from equation (131) as\\nIn this case, some care is needed to compute the stochastic integral (132) from the experimental values of x(t). As previously mentioned, Stranotovich integration must be used to get physically meaningful results.32 Specifically, from the collection of N experimental data points x(t j ), defined at each time t j = j−1 N −1 τ over [0, τ], where j = 1, . . . , N, a mid-point discretization of the force −k[x(t) − vt] with respect to the stochastic increment dx(t) must be performed\\nIn Fig. 29(e) we show the distribution of Q τ for different values of τ, computed by means of equation (136). In this case, all discrete time steps have a constant value fixed by the inverse of the experimental sampling frequency f s = 145 Hz, i.e., t j+1 − t j = f −1 s = 0.007 s. It should be noted that, although the mean value of Q τ coincides with that of W τ for all τ ≥ 0 in a steady state, as shown in Fig 29(b), their distributions are quite different, see Fig.29(e). In this case, the distribution of the heat, P(Q τ ) has non-Gaussian tails, with standard deviation larger than the variance of the work fluctuations (Fig. 29(c)). In addition, similar to the behavior of the stochastic work, rare events where Q τ is negative are also possible for small values of τ at which thermal fluctuations are significant, i.e., the heat spontaneously flows from the surrounding medium to the particle even if mechanical work is done on it.\\nFinally, in Fig. 29(f) we plot the distribution P(∆U τ ) of the variation of the potential energy over a time interval of duration τ:\\n32While equation (132) for the work simply requires the product of the deterministic time increment dt with the stochastic particle position relative to the position of the trap, x(t) − vt, the expression for the heat (equation (135)) involves the stochastic increment of the particle position, dx times x(t) − vt.\\nIn this case, while the mean value of ∆U τ is equal to 0 for all values of τ ( Fig. 29(b)), consistent with the fact that in a steady state the potential energy of the system remains constant on average, its distribution is strongly non-Gaussian, due to the non-linear dependence of ∆U τ on x(t), with standard deviation which quickly saturates to a constant value, as observed in Fig. 29(c).\\nThe statistical properties of W τ , Q τ and ∆U τ for this simple experiment illustrate the main essence of stochastic thermodynamics. Even though the average values of such quantities satisfy the expected thermodynamics behavior, the probability of observing rare events, which are not described by the classical thermodynamics, is not negligible for mesoscopic systems at sufficiently short time scales.\\nSecond Law and entropy. Stochastic thermodynamics also offers the possibility to extend the concept of entropy to stochastic trajectories of systems in non-equilibrium states, and provides quantitative relationships that generalize the macroscopic Second Law of thermodynamics. In the case of a system described by equation (124), the variation of the total entropy is defined over a given time interval [0, τ] as\\nThe first term on the right-hand side of equation (138) can be identified as a change in entropy due to the heat dissipated into the medium, defined in equation (131), while the second term is called stochastic entropy, and must be determined by solving first the corresponding Fokker-Planck equation (125) and then evaluating the solution ρ(x, t) along the specific stochastic trajectory x(t) with initial condition x(0) at t = 0. Therefore, unlike work, heat and potential energy, the stochastic entropy does not only depend on the particular realization of x(t) but on the ensemble through the quantity ρ(x, t). For example, for the linear motion (λ(t) = vt) of the position of the optical trap considered in the experiment above, the solution of the Fokker-Planck equation (125) is\\nunder the condition that the system has already achieved a steady state in the comoving frame of the trap. In this case, the variation of the stochastic entropy over the time interval [0, τ] is given by k 2T\\nx(τ) − vτ + γv k 2 − x(0) + γv k 2 , from which the corresponding change of the total entropy can be computed using equations (131) and (138).\\nRemarkably, the change in the total entropy defined by equation (138) satisfies the integral fluctuation theorem (IFT) [306] exp − 1 k B ∆S τ = 1 (139) for all measurement times τ and any kind of protocol involving a non-equilibrium temporal variation of the control parameter λ from λ(0) to λ(τ), and possibly including non-conservative forces f 0 that break detailed balance. Using the Jensen's inequality, e a ≥ e a , equation (139) implies that ∆S τ ≥ 0, which corresponds to one of the expressions of the Second Law of thermodynamics. Therefore, the IFT represents a refinement of the Second Law of classical thermodynamics. For the experiment above, we verify the IFT for different values of τ in the inset of Figure 29(e), where the left-hand side of equation (139) is numerically determined using the computed values of ∆S τ . Depending on the specific protocol for the variation of the control parameter λ and on the state of the system (e.g., transient or steady), alternative versions of the Second Law can be derived within the context of stochastic thermodynamics. For example, when driving the system by changing the control parameter from λ(0) to λ(τ), the non-equilibrium work W τ done on the system, defined by equation (130), and the corresponding difference in free energy ∆F = F(λ(τ)) − F(λ(0)), satisfies the relation [307]\\nfor any arbitrary non-equilibrium protocol λ(t), 0 ≤ t ≤ τ , provided that the system is initally in thermal equilibrium at t = 0, and all the forces acting on the system are conservative (i.e., f = 0 in equation (124)). Equation (140) is called the Jarzynski relation [307], and derives directly from the IFT (139), in which case the total entropy takes the particular form ∆S τ = (W τ −∆F)/T, where W τ − ∆F is known as dissipated work. A closely related expression, from which the Jarzynski relation (140) can alternatively be derived is the so-called Crooks fluctuation theorem [308]. It quantifies the ratio between the probability density of the work spent in the forward process, ρ(W τ ), when driving the system from thermal equilibrium at λ(0) through an arbitrary nonequilibrium protocol up to λ(τ), and the distribution of the work done in the reversed process, ρ(W τ ), where the control parameter is varied asλ(t) = λ(τ − t), starting in thermal equilibrium atλ(0) = λ(τ). The expression of the Crooks fluctuations theorem is\\nThe practical importance of the relations (140) and (141) is that, unlike quasistatic experimental protocols based on classical thermodynamics, they allow to determine free energy differences between two equilibrium states from non-equilibrium measurements of the work performed at an arbitrarily fast rate, for example when studying biomolecules (section 4.1).\"\n },\n {\n \"title\": \"Carnot cycles and Maxwell's demons\",\n \"content\": \"In this section, we will explore how optical tweezers can be used to get a deeper understanding of Carnot cycles, work efficiency and Maxwell engines. Although the results presented in this section have not been implemented by us in the laboratory directly, we provide sufficient instructions for the readers to be able to readily realize these experiments.\"\n },\n {\n \"title\": \"Carnot cycles and microscopic engines\",\n \"content\": \"In 1824 Sadi Carnot tried to answer basic questions about the efficiency, construction and operation of macroscopic heat engines [309]. This seminal work gave birth to thermodynamics, for which Carnot is considered one of its fathers. It also introduced concepts that later were used by Clausius and Kelvin to define the concept of entropy and introduce the Second Law of thermodynamics. Almost two centuries later, similar questions posed by Carnot on big, chunky, greasy, macroscopic heat engines, are finally being explored in the microscopic realm [18]. Colloquially Carnot's cycle refers to a process in which one extracts work from a system coupled to two reservoirs at different temperatures T c < T h . More precisely, given a system interacting with two thermal baths, Carnot's cycle consists of four steps:\\n1. An isothermal expansion at temperature T h , during which an amount of heat Q h is absorbed by the system 2. An adiabatic expansion from T h to T c .\\n3. An isothermal contraction at temperature T c , during which an amount of heat Q c is released from the system.\\n4. An adiabatic contraction from T c to T h , returning the system back to its original state. The four steps of this cycle are depicted in Fig. 30(a) in the (S, T)-plane, where S refers to the entropy. For a generic fluid with volume V and pressure P, Carnot's cycle in the (V, P)-plane is shown in Fig. 30(b). During one cycle, the extracted work is given by the First Law of thermodynamics (142) and the entropy production is\\nThis implies that for a reversible operation (i.e., for a process for which ∆S tot = 0) we obtain the following relationship between the heat exchange and the temperature of both baths:\\nCarnot showed that, for a reversible process, the efficiency of Carnot's cycle is\\nwhere the efficiency η = −W/Q h is defined as the ratio between the extracted work and the absorbed heat. Ideally, for the process to be reversible, it must be performed infinitely slowly or, more precisely in a time τ much slower than any characteristic relaxation time of the system. This automatically means that the output power, defined as P ≡ −W/τ is zero for an ideal reversible heat engine.\\nIn reality, Carnot's cycle must be performed in a finite time τ, which implies a trade-off between output power and efficiency. Due to the Second Law of thermodynamics, , which imposes ∆S tot > 0, we have that η < η C .\\nMoreover, practical heat engines have additioanl limitations that further reduce their efficiency [310]. For example, accounting for heat transfer yields a maximum power output given by the Novikov-Curzon-Ahlborn formula [311,312]\\nwhich has been confirmed experimentally with different protocols [313]. Another closed thermodynamic cycle, the Stirling cycle, is often used to describe engines that work by compressing and expanding a working fluid at different temperatures.33 This is also composed of four steps, namely:\\n1. An isothermal compression during which heat is extracted towards the cold bath T c .\\n2. An isochoric heat-addition.\\n3. An isothermal expansion during which the system absorbs heat from the hot bath T h .\"\n },\n {\n \"title\": \"An isochoric heat-removal.\",\n \"content\": \"A sketch of Stirling's cycle can be seen in Fig. 30(c).\\nIn the last decade, various research groups have probed the concept of Carnot efficiency in microscopic heat engines, particularly focusing on single-particle or colloidal heat engines. The first of these colloidal heat engines implemented a Stirling engine by using an optically trapped particle [314], as shown in Fig. 30(d). In this design, a colloidal particle plays the role of the working fluid, while the optical trap can be understood as the engine cylinder where the compression and expansion occur. The cycle includes the following phases:\\n1. The isothermal compression by increasing the trap stiffness (i.e., increasing the laser output power).\\n2. The isochoric head-addition by heating up the colloidal suspension from T c to T h .\\n3. The isothermal expansion by decreasing the trap stiffness (i.e., decreasing the laser power).\"\n },\n {\n \"title\": \"The isochoric heat-removal by cooling down the colloidal suspension from T h to T c .\",\n \"content\": \"Being a microscopic engine, the associated extracted work is a stochastic variable that depends strongly on the cycle duration τ. The first experimental realization of this microscopic Stirling cycle [314] operated between two baths at T c = 22 • C and T h = 90 • C with the maximum mean power −W/τ obtained for τ = 7 s. In these conditions, the average work extracted per cycle was about 4.5 · 10 −22 J, In 2014, a micrometer-sized equivalent of a piston steam engine was realized in an optical tweezer, where a single colloidal microparticle plays the role of the piston. The particle absorbs a small amount of light enabling to locally raise the temperature when the particle is exposed to light. At the beginning of the cycle, the gradient force of an optical tweezers pulls the particle towards the laser focus, giving rise to a dramatic increase in temperature and, consequently, 33The Stirling's cycle has the same efficiency as Carnot's cycle. the explosive vaporization of the surrounding liquid. This causes a bubble, which pushes the particle out of the focal region. As a consequence, the particle cools down and is pulled back again by the optical tweezers [315]. To achieve the sudden bubble formation, this experiment exploited a superheated liquid, which becomes unstable and relaxes back towards equilibrium via the nucleation of a vapor cavitation bubble. The mean extracted work per cycle exceeded that found in Ref. [314] by a striking six to eight orders of magnitude, because both the cycle frequency and maximum displacements were around two orders of magnitude larger.\\nThese two approaches to microscopic heat engines rely on laser heating of the solution where the particle is immersed. The heat exchange rates of the heating and cooling processes effectively limit the range of accessible temperatures and the timescales of the experiments. This technical difficulty can be overcome by adding a synthetic noise to the Brownian motion [316]. This permits one to heat without having to raise or lower the physical temperature of the colloidal suspension, allowing fast heating and cooling as well as removing temperature limits, at least in principle. This method is called white-noise technique. Mathematically, the idea works as follows. Consider the underdamped one-dimensional Brownian particle:\\nwhere ξ(t) corresponds to the usual thermal noise with\\nwhile η(t) is, for example, an electrostatic noise such that\\nConsider, in addittion, that these two noise sources are independent. This model is statistically equivalent to an effective model for stochastic thermodynamics,\\nwhere ξ eff (t) can be understood as thermal noise with effective temperature34\\nThe white-noise techniques allow to reach effective temperatures as high as T = 3300 K. The synthetic noise can be generated by placing the colloidal suspension in an electrophoretic fluid chamber with two electrodes connected to a computer-controlled electric generator that produces a sequence of noisy random electric fields, whose statistical properties closely resemble that of white Gaussian noise [316]. A similar effect of adding a random force applied to the particle can be achieved by modulating the position of the trap by using an acousto-optic deflector [317,318]. A remarkable step forward was achieved in a series of works where the concept of microadiabaticity was introduced [319], and later used for the creation of the first Brownian Carnot engine by concatenating isothermal and microadiabatic protocols [320]. The basic idea is as follows. Recall that, macroscopically, an adiabatic process occurs without exchange of heat (or mass) between the system and its surroundings (i.e., Q(t) = 0). For microscopic systems, the amount of heat is a random variable, so the simplest generalization is to consider those processes for which the average amount of exchanged heat is zero (i.e., Q(t) = 0). Moreover, if we further 34See also section 5.7, where we discuss feedback cooling of trapped particles in vacuum, which can also be discussed in terms of an effective temperature. assume that the process is performed quasi-statically, we also have that the average value of the variation of the total entropy is zero, ∆S tot (t) = 0, and, consequently, the process is isentropic, ∆S sys (t) = 0. In this scenario, a trick to implement a microadiabatic process consists in using protocols which keep the system's entropy constant. Importantly, this process can be quantified fairly precisely. Indeed, since the system's entropy is given by\\nand assuming that the system is close to equilibrium, i.e.,\\nwhere H(t) is the Hermitian of the system, we obtain that\\nThus, a pseudo-adiabatic process corresponds to keeping T 2 (t)/κ(t) constant. Notice that the microscopic Carnot engine was also explored in the work in Ref. [321], but only considering the configurational space and not the phase space. [322] addressed the issue that the Second Law of thermodynamics might be violated by the inclusion of \\\"intelligent beings\\\" capable of processing information. In its original version, an ideal gas at temperature T is enclosed in an isolated container divided into two equal parts by a fixed wall. In this wall, there is a trap door operated by a \\\"demon\\\" who will open it if an incoming particle from the left (right) is faster (slower) than the average. After a while, the right compartment will have the fastest particles and, consequently, a higher temperature. Therefore, the effect of the demon is to generate a temperature gradient, which can be used to obtain work from the system, without the corresponding energy expenditure, thus violating the Second Law.\"\n },\n {\n \"title\": \"Maxwell's demons and Szilard engines. The thought experiment known as Maxwell's demon\",\n \"content\": \"To better understand the origin of such violation, Leo Szilard proposed his eponymous simplified model. The Szilard engine consists of a single particle within a container divided into two equal partitions separated by a movable, frictionless wall [323]. When the container is put into contact with a single thermal reservoir at temperature T, the movable wall may then be used to extract work (e.g., by lifting a weight), as the particle transfers kinetic energy in successive elastic collisions. The isothermal volume expansion results in a work extraction W ext = k B T ln(V f /V 0 ), where V 0 and V f are the initial and final volumes of the partition that contains the particle. Since the final volume V f is that of the total box, then V 0 /V f = 1/2 is the fraction of the volume that contains the particle. This implies, in turn, that the expected work extracted from the engine is simply k B T ln 2, which is proportional to the Shannon entropy measuring the uncertainty on the initial position of the particle in a symmetric partition.\\nLeon Brillouin suggested that the work extraction of the Szilard engine is precisely compensated by the work of measurement and, therefore, no net extraction of work is possible [324]. After a decade, Rolf Landauer introduced a concept of logical irreversibility [325] and Charles H. Bennett explained that the work extraction is precisely compensated by the erasing process of the memory apparatus [326]. Recently, a modified version of the Second Law was derived [327,328], the Jarzynski-Sagawa-Ueda relation, which has also been confirmed experimentally [329]. This latter result claims that the work extraction is compensated by the mutual information between the system and the measurement apparatus which is involved in both the measurement and erasing processes [330]. In a relative recent experiment [331], the authors used a colloidal suspension to implement a Szilard engine to explore how the entropy decreases due to the spontaneous symmetry breaking that is the cornerstone of the Szilard engine and Landauer's principle, and to measure directly this entropic change for a Brownian particle in a bistable potential implemented by two optical traps. Very recently, a continuous versions of the Maxwell's demon, which is capable of extracting arbitrarily large amounts of work per cycle by repeated measurements of the state of a system, was proposed and was experimentally implemented in a single DNA hairpin pulling experiment [332].\"\n },\n {\n \"title\": \"Other applications to statistical physics and outlook\",\n \"content\": \"In this section, we have presented some well-known applications of optical tweezers to investigate both equilibrium and non-equilibrium properties of model systems in statistical physics. Here, we briefly mention some emerging areas where optical tweezers also represent a powerful tool to study statistical quantities of microscopic systems.\"\n },\n {\n \"title\": \"Synchronization. Coordinated motion of self-sustained oscillators mediated by means of\",\n \"content\": \"hydrodynamic interactions is a fundamental behavior in many biological systems, such as cilia and flagella. Using optically trapped microparticles, it has been possible to elucidate and directly visualize different aspects of the hydrodynamic synchronization of oscillators and rotors. There are different approaches to define these oscillators, for example with light-propelled chiral particles [333], using angular momentum in optical tweezers to propel birefringent or asymmetric particles [46], by means of rotating energy landscapes [334][335][336] or, as we already mentioned in section 4.5.1, by means of bistable potentials sculpted with light [288]. In all these cases, when two of these self-sustained oscillators are close enough, the hydrodynamic interactions perturb and tend to synchronize their phases and their frequencies. Since all of these experiments have been realized at microscopic scale, thermal fluctuations are not negligible and escape transitions play an important role in their dynamics. Active matter. Active particle systems, such as motile microorganisms, are non-equilibrium systems capable of converting autonomously the energy from their surroundings into directed motion. In recent years, also the possibility of developing artificial active matter systems has been extensively explored [337,338]. Optical tweezers can be used to probe such phenomena and their connection to non-equilibrium relations [339].\"\n },\n {\n \"title\": \"Phase transitions.\",\n \"content\": \"Optical tweezers can be used in a variety of situations to induce phase transitions and to probe critical phenomena [260]. One can, for example, use optical lattices to control colloidal phase transitions via tunable interactions [340][341][342].\\nFurther advances in optical trapping could also enable the possibility to address more fundamental questions on statistical aspects of small systems, such as memory effects in non-equilibrium systems. Moreover, most of the studies discussed in this section deal with the behavior of individual particles, but it is natural to wonder how the properties of these systems change in many-body systems. Equally interesting would be to explore how microengines or active baths can be used to perform practical tasks, such as energy harvesting at the microscale.\"\n },\n {\n \"title\": \"Transport in extended potential landscapes\",\n \"content\": \"Microscopic particles in a fluid are constantly diffusing: the smaller the particle, the faster it diffuses, following a power law dependence D ∝ 1/a, where a is the particle radius. As D is a tiny quantity in typical conditions (of the order of µm 2 s −1 ), diffusive transport is very slow. For example, on average a 1 µm particle in water takes approximately 26 days to move by about 1 mm away from its original position. Transport can be greatly enhanced by external forces. These forces generate directed motion, where diffusion generally plays a pernicious role, since the particle will be gradually dispersed away from the path defined otherwise by the force. In Nature, several microorganisms have developed different strategies for exploring their environment in a more efficient and precise way than that of simple diffusion, for example using flagella or cilia.\\nMicroscopic objects are often subject to the influence of structured force landscapes, which may either boost or inhibit their motion. For example, some molecular motors inside eukaryotic cells move along the paths defined by microtubules that exhibit a periodic structure of two proteins, acting as a periodic potential. A paradigmatic model that serves as a starting point for studying microscopic dynamics is the motion of a Brownian particle in a corrugated potential landscape under the action of an external force. Being able to study and understand these transport mechanisms is not only important from the fundamental science perspective, but it also opens up the possibility to reproduce them and develop new techniques to perform tasks, such as particle sorting and delivery in lab-on-a-chip platforms.\\nAs we will see in detail in this section, colloids are ideal prototypes for the study of microscopic transport phenomena, and extended light patterns provide a powerful and versatile way to create reconfigurable potential energy landscapes in optical micromanipulation systems. We will be primarily concerned with conservative optical forces in this section, but there are also interesting light-induced transport phenomena that emerge in the presence of non-conservative forces (e.g., scattering forces), as we will briefly comment on. Table 11 presents some examples of extended light patterns used in optical manipulation: the images on the left illustrate the optical field of interest generated at the sample (i.e., at the focal plane of the microscope objective or the last lens of the optical trapping setup); the images on the right provide information about how these patterns can be experimentally generated. As explained in section 2.3, there are different ways for creating extended light patterns, but here we will focus our attention on the two most common approaches using interference or diffractive optical elements.\"\n },\n {\n \"title\": \"Optical forces and potentials in extended light patterns\",\n \"content\": \"A subwavelength particle is pulled by the conservative optical gradient forces towards the brightest regions of an extended light pattern. In this case, the particle moves in an extended potential energy landscape mimicking the shape of the intensity pattern, with valleys at maxima and hills at minima of intensity. In this regime, the magnitude of the forces, or the depth of the valleys, increases in proportion to the cube of the particle radius and linearly with optical power. However, for particles whose size is comparable or larger than the wavelength, the shape of the energy landscape might be remarkably different from the intensity distribution.\\nImportantly, when working with extended patterns of light, one should keep in mind that the larger the area covered by the pattern, the lower the intensity for a given optical power and, therefore, the magnitude of the optical force decreases. Hence, it is important to estimate the appropriate area of the pattern so that the average intensity is at least of the order of 0.1 mW µm −2 .\\nOne-dimensional potentials Let us analyze the simplest case: an extended pattern of fringes obtained from the interference of two collimated beams, as that in the first row of Table 11. At the back aperture of the microscope objective, the Fourier transform of the two beams corresponds to a pair of focused spots, whose separation (2s) determines, together with the focal length of the microscope objective ( f ), the period of the interference pattern at the sample plane: Λ = λ f 2s , where λ is the wavelength of the laser in the background medium. The corresponding intensity distribution is given by\"\n },\n {\n \"title\": \"Sinusoidal pattern (periodic in 1D) produced by the interference of two beams (left) and its Fourier spectrum (right).\",\n \"content\": \"Any interferometer is suitable to produce this pattern, such as Fresnel biprisms [343] or a Mach-Zehnder interferometer [39,344]. A diffraction grating generated with a SLM can also be used to split the beam [94].\\nSawtooth pattern (periodic in 1D) (left) generated by the superposition of two orthogonally polarized sinusoidal patterns (right). A three-armed Mach-Zehnder interferometer can produce two patterns of fringes, one with twice the spatial period of the other, by controlling the polarization of the input beams. The phase difference and intensity ratio determines the asymmetry [63,345].\"\n },\n {\n \"title\": \"Oblique lattice (periodic in 2D): Interferometric pattern of four beams (left) and its Fourier spectrum (right).\",\n \"content\": \"These patterns can be generated using a tandem of two interferometers or using diffraction gratings designed to split the input beam into multiple beams travelling along the desired directions [344].\"\n },\n {\n \"title\": \"Rectangular lattice (periodic in 2D): Diffractive pattern (left) generated with a phase hologram (right).\",\n \"content\": \"Holographic phase masks can be designed to produce the target distributions of individual light spots using iterative optimization algorithms [22]. These can be displayed on a SLM and dynamically reconfigured.\"\n },\n {\n \"title\": \"> >\",\n \"content\": \"\"\n },\n {\n \"title\": \"Quasi-periodic lattice: Interferometric pattern of five equidistant beams (left) and its Fourier spectrum (right).\",\n \"content\": \"A pentagon-shaped diffraction grating can be displayed on a phase-only SLM to split the input beam into five beams travelling at the desired angle.\"\n },\n {\n \"title\": \"Zeroth-order Bessel beam (left) and its annular Fourier spectrum (right).\",\n \"content\": \"Bessel beams can be obtained efficiently with a conical lens or axicon. Any nondiffracting beam can also be generated by means of a properly encoded phase mask displayed on a SLM [100,101].\"\n },\n {\n \"title\": \"Lattice of asymmetric traps in 2D (left) generated by the superposition of orthogonally polarized identical patterns.\",\n \"content\": \"A light lattice generated with a phase hologram can be introduced into a polarization splitting Mach-Zehnder device to produce two identical patterns, whose relative positions and intensity ratio determine the asymmetry of the resulting pattern [66].\\nSpeckle pattern (left) generated with a random phase mask (right). Speckle can be generated by passing a laser beam through a diffuser plate [89], by coupling the laser into a multimode optical fiber [65,346], or with a random phase mask displayed on a SLM [107]. where we have assumed that both interfering beams are plane waves with equal intensity, I 0 .\\nIn reality, there is always an envelope function modulating the intensity, often with a Gaussian shape I 0 (x, y) = A exp[−2(x 2 + y 2 ) 2 /w 2 0 ], but as long as w 0 Λ, the plane-wave assumption works reasonably well in the central region of the envelope.\\nThere are several approaches with different levels of complexity for calculating the optical force exerted by this pattern on a dielectric particle of diameter D p > λ. In all cases, the force is also periodic with the same periodicity as the intensity profile, so it can be written as\\nNotice that there is a π/2-phase difference between the expressions for the intensity and that for the force, since both maxima and minima of the intensity function correspond to zeroes of the force, representing either stable or unstable equilibrium positions. The amplitude factor A is proportional to the intensity of light and, more importantly, it depends on the size of the particle and its refractive index, as well as on the period Λ of the pattern [38,39,347], i.e. A = A(D p , n p , Λ).\\nTo illustrate this fact, Fig. 31(a) shows the dependence of the normalized amplitude of the force (Q = A/2I 0 ) as a function of the period, for two silica spheres (refractive index n p = 1.46) with diameters D p = 2 µm and 4 µm, respectively, immersed in water (refractive index n m = 1.33). If Λ is larger than the bead size, the particle feels a potential with equilibrium points at the intensity maxima, corresponding to negative values of Q, similar to what happens with subwavelength particles. There is a global minimum in each curve of Fig. 31(a), which represents the optimum period, slightly larger than the diameter of the sphere, that maximizes the magnitude of the force. Now, as the period is progressively reduced, the magnitude of the force decreases until it vanishes, meaning that a light pattern with such a spatial period will have no effect on the sphere. If Λ becomes even shorter, the force changes its sign (Q > 0) and the optical potential reverses, having stable equilibrium points in places where the intensity is minimum. This is schematically illustrated in Figs. 31(b-c), showing each particle at one of its respective equilibrium positions in the same intensity pattern (background), which has the period Λ = 2 µm (marked by the vertical dashed line in Fig. 31(a)). The resulting energy potentials for these particles, U(x), are also plotted, exhibiting a sign reverse with respect to one another.\\nIn general, the most important aspect for determining the trapping positions in an extended pattern is the relative size of the particle with respect to the characteristic size of the pattern. In that sense, it is convenient to plot Q in terms of the ratio Λ/D p . Accordingly, Fig. 31(d) depicts the normalized force amplitude as a function of Λ/D p for spheres made of silica n p = 1.46 (solid curve) and polystyrene n p = 1.6 (dashed curve). The red and blue vertical dashed lines indicate the values of the parameter Λ/D p corresponding to the examples in Figs. 31(b) and 31(c), respectively. This kind of pattern has been widely used for optical manipulation of multiple particles and optical sorting [38,39,87,94,344,348,349].\\nThe particle-dependent response to the structure of a light pattern is sometimes referred to as the size effect of the optical forces and takes place regardless of the complexity of the pattern. As the optical landscape becomes more intricate, nevertheless, not only the magnitude and the sign of the optical force may change like in our simple example, but also the shape of the potential may be drastically altered depending on the size of the particle. For example, Fig. 32 shows the case of a sawtooth pattern of fringes, also shown in the second row of Table 11. This pattern can be generated by superimposing three beams with linear polarization oriented along the horizontal, vertical and diagonal directions, so that the latter partially interferes with the former two, creating two patterns of fringes orthogonally polarized, one with twice the period of the other [63]. Given the asymmetric intensity in the fringes, a consequent asymmetry could be expected in the potential, and this is true for the smaller particle (D a = 2 µm) shown in Fig. 32. However, a larger particle (D b = 4 µm) experiences a completely different potential, with a slight . Particle-dependent potential or size effect in asymmetric patterns. Asymmetric pattern of light with period Λ = 4 µm (background) and the resulting optical potentials, U a (x) and U b (x), for two silica beads with diameters D a = 2 µm and D b = 4 µm, respectively. The shape of the optical potential for a given pattern of light can be very different depending on the size of the particle. The potential was computed using the ray optics model. asymmetry in the opposite direction. The modification of the potential can be even more radical for other particle sizes. The application of this effect for the transport of particles was analyzed in detail in Refs. [63,345].\"\n },\n {\n \"title\": \"Two-dimensional potentials\",\n \"content\": \"When the periodicity of the pattern is in 2D the size effect might give rise to different behaviors along each symmetry axis, and thus the equilibrium positions for a given particle might be unpredictable without performing an explicit calculation of the optical potential. Different kinds of light lattices in 2D have been used to study optical binding and colloidal crystallization [54,55], to explore structural phase transitions in colloidal monolayers [340,350,351], to tailor bandgap materials [352], and to optically sort particles in combination with microfluidics [344]. Periodic light lattices in 2D can be produced by the interference of at least three beams. For example, the coherent superposition of three waves, depending on the directions of their wave vectors, can lead to centered-rectangular or hexagonal lattices, whereas the interference of four waves can produce square, oblique or rectangular lattices. The example in the third row of Table 11 corresponds to an oblique or rhombic lattice. An alternative approach to generate 2D-periodic patterns is by means of phase holograms [41], such as the rectangular array of individual spots illustrated in the fourth row of Table 11. In contrast with the multiple interference fields, diffractive patterns can act as a collection of individual traps for 3D confinement, overcoming the size effect and hence facilitating the control of the equilibrium positions of individual particles. A SLM can be used as well to display diffractive gratings and produce multiple interfering beams [94], whilst the 2D-trapping capabilities can be extended to manipulate ordered structures in 3D by taking advantage of the self-imaging Talbot effect [353]. Another interesting case is that of a quasi-periodic pattern arising from the superposition of five waves, which is shown in the fifth row of Table 11. This has been used to study the formation of quasi crystalline structures in colloidal monolayers [354,355].\\nA type of extended patterns with circular, elliptical or parabolic symmetry in the transverse plane correspond to Bessel, Mathieu and parabolic nondiffracting beams, respectively [95,96,98]. Their Fourier spectrum at the back aperture of the microscope objective lies on a thin annulus, where the particular beam geometry is determined only by the intensity and phase modulation around the annulus. The optical forces in a Bessel beam, like the pattern depicted in the sixth row of Table 11, exhibit very interesting features due to the size effect [356][357][358], and similar attributes can be expected in the other types of beams. Namely, the multi-ringed structure of the intensity pattern has an envelope that decreases radially outwards. Depending on its size with respect to the pattern, a particle might respond predominantly to the fine structure or to the envelope. This can be used as a passive mechanism to sort particles [359] or to build what is known as a washboard potential [360].\\nAnother version of an asymmetric periodic pattern, but now in 2D, is illustrated in the seventh row of Table 11. This is generated by superimposing two identical light lattices, orthogonally polarized, produced with a single phase hologram [66]. By controlling the relative position and the intensity ratio between the two patterns, the asymmetry of the light lattice can be tailored. In general, periodic and asymmetric patterns are very important in the study of the ratchet effect [60,62,63,66], which is the emergence of directed transport in the presence of unbiased external forces due to a spatiotemporal symmetry breaking and, furthermore, is thought to be at the core of the operation mechanism of some biological engines [361].\\nRandom light patterns like speckle (bottom row of Table 11) are also very important in the context of particle diffusion and transport. In practice, speckle can be generated by transmitting a laser beam through a glass diffuser, by coupling the laser into a multimode optical fiber, or by encoding a random phase mask on a SLM. A very interesting aspect of the interaction of Brownian particles with a random potential landscape is its major role in diffusion, and how it depends on the several parameters of the potential [65,89,346,362].\"\n },\n {\n \"title\": \"Transport mechanisms\",\n \"content\": \"So far, we have examined optical forces and potentials in a static situation. Now we will examine the conditions for inducing directed transport in the system, i.e., a net current of particles in a given direction through their interaction with light and, possibly, other external forces. To address this subject, we will start by looking at the general equation of motion for a single particle interacting with an optical potential in the overdamped regime, which is a good description of the micromanipulation systems under consideration. For the sake of simplicity, only one dimension will be examined, this is:\\nwhere γ represents an effective drag coefficient, U(x, t) is the optical potential, F(t) denotes an external force, and ξ(t) is the thermal noise, having a correlation function ξ(t)ξ(t ) = 2Dδ(t −t ), with D = k B T/γ being the diffusion coefficient. Although noise usually plays an important role in microscopic-scale dynamics and some transport mechanisms even rely on it, this can be neglected under certain conditions, giving rise to a deterministic dynamics, ξ(t) ≈ 0. For example, Brownian motion of silica or latex spheres immersed in water at room temperature is very subtle if the diameter is D 5 µm, and it becomes practically negligible for D 10 µm.\\nIn general terms, we can distinguish between two main transport mechanisms: a dynamic potential and a static potential in the presence of an external driving force. A very simple realization of the former case corresponds to a light pattern of fringes that moves with a constant speed v with respect to the static sample. This corresponds to a travelling periodic potential: U(x, t) = U(x − vt). The dynamics of the particle can be analyzed in a reference frame that moves along with the potential by making the change of variable u = x − vt in equation (157), leading to\\nNotice that this is equivalent to adding a constant drag force to a static pattern and thus the following discussion will be valid for the case of a constant external driving force. Whether the particle will follow the pattern or not depends on the relative magnitude between the optical force, F op (u) = −dU/du, and the drag force, F d = −γv, in the moving reference frame. The constant force has the effect of tilting the potential energy landscape, giving rise to what is sometimes referred to as a washboard potential. If |F op | > |F d |, the particle will not be able to escape from the well and, therefore, it will be locked-in to the pattern, moving with the same velocity as the pattern with respect to a fixed reference frame. A particle with this behavior has been called a Brownian surfer [363]. If |F op | < |F d |, the particle will be pulled out from the potential well by the drag force in the moving reference frame or, in the fixed frame, it will not follow the pattern but it will diffuse around its initial position. This is known as a Brownian swimmer [363]. Depending on the thermal noise level, there might be intermediate situations, in which the particle moves in the same direction as the pattern but with a reduced speed v p < v. This happens when |F op | |F d | or, in terms of the potential energy barrier between neighboring wells, when ∆U k B T, so that thermal energy may sometimes help the particle to surmount the potential barrier between consecutive wells. Due to the size effect discussed above, in a polydisperse mixture of particles, some of them may follow the pattern while others may not, depending on their size and refractive index, or even if two types of particles move along with the pattern, their average speeds may be different. Therefore, a moving potential can be used as a mechanism to separate particles of different characteristics, i.e. as an optical sorting device.\\nIn practice, there are several ways to move an interference pattern of fringes. Let us consider, for example, an optical micromanipulation setup with a Mach-Zehnder interferometer. We can express the resulting interference pattern at the sample plane as I(x, y) = 4I 0 cos 2 (πx/Λ + ϕ(t)/2), where ϕ(t) represents a time-dependent phase difference introduced in one arm of the interferometer. To generate a travelling potential, ϕ(t) should be a linear function of time. An elegant way to achieve this is by using the angular Doppler effect [364,365], arising when a circularly polarized beam passes through a rotating λ/2-plate, which gives rise to a frequency shift ∆ω = ±2Ω, where Ω is the rotation speed of the λ/2-plate. This can be easily implemented in the Mach-Zehnder interferometer [348], as illustrated in Fig. 33(a). A polarizing beam splitter (PBS) is followed by a λ/4-plate in each arm of the interferometer, producing orthogonal circular polarization states. The rotating λ/2-plate is inserted in one arm, such that it inverts the handedness of the circular polarization at the same time that it introduces an angular Doppler frequency shift. A non-polarizing beam splitter is placed at the output of the interferometer to superimpose the beams. The resulting phase difference between the two beams is ϕ(t) = (∆ω)t = ±2Ωt, where the sign depends on the rotation direction of the λ/2-plate with respect to the polarization handedness [365]. At the sample plane, this will define the direction of motion of the fringes (±x), whereas the speed of motion is v = Ω/k.\\nThere are other alternatives for producing motion of the fringes. For example, a very simple option, which can be easily implemented in different types of interferometers, is to introduce a thin glass plate in the path of one of the interfering beams [343]. As illustrated in Fig. 33(b) for the particular case of a Mach-Zehnder interferometer, by gradually tilting the glass plate a small angle θ(t), the optical path length of the corresponding beam is modified, resulting in a relative phase shift between the two beams and the consequent motion of the pattern. A straightforward analysis using geometrical optics yields that, if θ 1, then the change in the optical path length is given by ∆(θ) ≈ T g /2(1 − 1/n) 2 θ 2 , where T g is the thickness of the glass plate and n its refractive index. Therefore, as ∆(θ) is not linear, the phase shift is not linear either: ϕ(t) = k∆(θ(t)). Although this complicates the theoretical description of the system, it is a cheap and accessible choice if the aim is just to make a demonstration of particle transport.\\nAnother configuration that provides full control of the phase shift ϕ(t) makes use of a displacement of one of the mirrors in a Mach-Zehnder interferometer. For example, in the first interferometric optical tweezers setup [87], depicted in Fig. 33(c), an extra beam splitter is placed in one arm of the interferometer (BS1), so that mirror M1 retroreflects the beam and, mounted on a motorized translation stage, it is moved with a constant speed: D(t) = vt. This produces a phase shift between the interfering beams of the form ϕ(t) = 2k D(t), leading to a continuous motion of the fringes at the same speed as the mirror (v), while keeping the spatial period of the fringes unaltered. Indeed, the period of the fringes is a function of the lateral distance between the output beams and the focal length of lens L1 (see Fig. 33(c)). Notice, however, that the presence of the beam splitter BS1 yields an additional intensity loss. To equalize the intensity in both arms of the interferometer, two λ/2-plates should be used, one at the entrance of the setup in combination with a polarizing beam splitter (PBS) to adjust the amount of light in each arm, and the other one to set the same polarization in the two beams.\\nA variant of this idea, which avoids the extra beam splitter, is shown in Fig. 33(d). A periodic and asymmetric displacement of the mirror M1 might be equivalent to a continuous travelling pattern under certain conditions. This can be done by mounting the mirror on a piezo-stage35 driven with a sawtooth function [39]:\\nwhere τ δt, v 1 = D max /τ and v 2 = D max /δt, with D max denoting the maximum displacement of the mirror, which should be small to prevent distortion in the period of the fringes. The frequency of D(t) is f = 1/T, with T = τ + δt the period of the function: D(t + T) = D(t). The motion of the mirror is related to the phase shift of the pattern as ϕ(t) = 2 √ 2k D(t), implying that the fringes at the sample mimic the sawtooth-like displacement, with speeds given by\\nThe main idea is that the particle is able to follow the pattern in the direction of the slower motion (u 1 ), but not during the fast motion (u 2 ) in the opposite direction, resulting in a unidirectional transport. Of course, the selection of the parameters36 D max , f , and τ, along with the spatial period of the fringes Λ, is crucial to optimize the current of particles. For example, if D max = mλ/(2\"\n },\n {\n \"title\": \"√\",\n \"content\": \"2), with m a positive integer, the particle will advance a distance mΛ in one direction and will end up at another minimum of the potential after an entire cycle, provided that Λ is such that optimizes the optical force for the given particle size. Figs. 34(a-b) illustrate the resulting particle current for two sizes of particles as a function of Λ and Ω = 2π f , respectively. An optimum can be clearly identified in each case. This device was used as an optical sieve to separate particles according to their size and it can also be used to sort particles according to their refractive indices [39]. In the latter case, the strategy is to start with a low intensity and gradually increase the optical power up to a value for which only the particles with the highest refractive index contrast with respect to the medium will respond.\\nA related approach consists in generating phase gratings with a SLM aimed at splitting an incident beam to produce interference. The advantage is that the angular separation between the beams can be varied by changing the grating, so that the spatial period of the fringes can be tuned and its orientation can be modified as well. Furthermore, the motion of the pattern can be also controlled with the SLM, by adding a continuous phase shift to one of the beams. Such a device has been used to move and sort living objects and particles of different shapes [94].\\nIt is worth mentioning that the techniques discussed above can be extended to more complex interference patterns, such as rotating beams, allowing circular motion of beads and structures [?,366]. Similar ideas have been applied as well to build optical conveyors that allow bidirectional transport in 3D. These kinds of devices use coaxial beams, either counter-propagating to generate axial standing waves or co-propagating, and not just Gaussian beams but also other structured fields, like Laguerre-Gaussian and Bessel beams [38,357,[367][368][369], including evanescent fields [349]. The operation principle is basically the same as before: to manage the motion of an interference field by adding a controlled phase shift between the beams.\\n35Any programmable motorized translation stage would be useful and, for example, stepper motors are cheaper, at the cost of loosing precision and speed in the motion control.\\n36There are alternative choices, but in any case, the dynamic phase ϕ(t) is defined by a set of three independent parameters, set through the motion function of the mirror. frequency Ω of the sawtooth function for the optimum spatial period in each case and fixed power P = 130 mW. The two particle kinds feature different optimal currents as a function of Λ and Ω, enabling to classify them with high throughput. Data from Ref. [39].\\nThere have also been different approaches to yield moving patterns based on digital holography. Among the first schemes, we can mention a mechanism called optical peristalsis, in which a lattice of optical traps is created and controlled with a set of phase holograms displayed on a SLM [61]. Following a periodic sequence of three states, the pattern is consecutively displaced by one-third of the lattice constant, reaching the initial state again after an entire cycle. A particle trapped in a given potential well is released when that pattern is turned off and it is immediately attracted towards the nearest shifted well once the next-state pattern is turned on. By repeating this process, the particle is transferred from one well to the neighboring trap after each cycle, experiencing a constant motion in one direction, resembling the operation of a peristaltic pump. The geometry of the lattice can be varied to achieve, for example, a radial current in circular patterns. If the particle's diffusion during the time between consecutive states is negligible, the process is deterministic and the current is given by v = L/T, where L and T are the lattice constant and the temporal period of the sequence, respectively.\"\n },\n {\n \"title\": \"Other transport mechanisms and applications\",\n \"content\": \"As we have already mentioned, a travelling potential is dynamically equivalent to having a particle in a fixed light pattern subject to a constant drag or, more generally, a driving force. The drag force can be provided, for example, by tilting the sample [347], by moving the sample with respect to the pattern [63,370], or by means of a controlled microfluidic flow. The latter option has the important advantage of being suitable to integrate into lab-on-chip platforms [76,371]. In addition, sorting capabilities can be boosted by using 2D light lattices instead of 1D-periodic patterns, offering new control parameters and thus more versatility. Indeed, biological or colloidal assays flowing through two-dimensional patterns may be deflected from the body of the flowing fluid depending on the orientation of the lattice with respect to the flow and on the strength of the optical force, giving rise to discrete angle deviations or fractionation [344,372,373]. Using this approach, a high throughput of sorted samples can be practically and efficiently achieved. Also, microfluidic techniques can be combined with other schemes of optomechanical control, like dynamic or blinking potentials [374].\\nA fundamentally different and intriguing mechanism to selectively move objects at the microscopic scale is based on the ratchet effect. In contrast to previous examples where there exist a biased external field to induce a net current of particles, the ratchet effect makes use of non-biased time-varying forces to induce transport. Since these forces are of zero-mean in time and in space, a non-trivial symmetry has to be broken to observe a net current [361]. The light patterns represented in the second and seventh rows of Table 11 are examples of ratchet potentials with a space asymmetry, yet the symmetry breaking may be much more subtle, including variations in time. Actually, the spatial structure and time-modulation of the potential along with the time-dependence of the external force define many different types of ratchets [60,62,63,66,[375][376][377][378]. Moreover, although most of these systems are noise-assisted when particle diffusion is non-negligible, there are also deterministic ratchets [63]. The rich dynamics arising in these systems is due to the delicate interplay among the whole set of encompassed parameters and manifests itself in the diverse phenomena that can be observed, including bidirectional transport depending on size, currents in different directions with respect to the driving, current reversals and chaotic behavior. Interestingly, the ratchet mechanism is ubiquitous, as it appears in many biological, physical and chemical systems, and its study has paved the way to broaden our understanding of transport processes at the micro-and nanoscales [361].\\nSo far we have looked mainly at ordered patterns or light lattices, but random patterns have also shown an excellent selectivity for sorting particles [65,89], with the additional advantage of making feasible to sort and select a continuous throughput of chosen constituents in turbid media, common in many biological assays.\\nAlthough we did not deal with scattering forces in our previous discussion, it is important to remind that the use of scattering rather than gradient forces was really the first means to transport and separate particles. Indeed, since his earliest experiments in 1970, Ashkin noted that radiation pressure acting on particles of different sizes along the propagation axis of a moderately focused beam was proportional to the size of the particles, so that the beads were guided along the beam axis with different velocities [1]. This idea is the basis of the so-called optical chromatography, in which a Gaussian light beam opposes a fluid flow, causing that particles of various sizes and/or refractive indices find different equilibrium positions along the beam axis where the scattering force is balanced by the force due to the fluid flow [379,380]. Other schemes combine the action of scattering and gradient forces. Consider, for example, the interference pattern of fringes discussed above. In addition to the transverse gradient force, there is usually a scattering force along the propagation direction, which may push the particles selectively according to their relative size with respect to the fringes, so that small particles, attracted to the bright fringes, will be pushed much stronger than larger particles attracted to the dark fringes. Using this interplay between gradient forces and scattering forces, it is possible to activate transport with an all-optical static configuration [381]. These kinds of ideas have been extended further to include the use of evanescent waves generated on the surface of prisms (Kretschmann configuration) or with total internal reflection (TIR) microscope objectives [49,382,383]. An advantage is that the use of gradient and scattering forces arising from evanescent waves in near-field photonic configurations allows one to scale down to tens of nanometers the size of the manipulated objects [49,384].\\nAnother puzzling transport mechanism appears in the so-called tractor beam. This is a counterintuitive effect, in which particles move opposite to the direction of the beam propagation due to a pulling scattering force [385]. In the case of a highly nonparaxial wave field, whose angular spectrum components subtend large angles with respect to the main propagation axis, part of the diffracted radiation may indeed have larger wave vector components along the propagation axis than those of the incident field, giving rise to strong forward scattering. The net effect might be a negative radiation pressure (pulling force), depending on the shape of the object, its size, and its interaction with the wave field. It is important to stress though, that this effect is not based on intensity variations and it is not derived from a potential, in contrast with the conservative gradient force acting in conventional optical tweezers. Remarkably, this phenomenon can happen also in rotational dynamics, where the spin angular momentum of light can exert transverse non-conservative forces in the opposite direction than expected in a chiral object [386,387].\\nThe simplest description of the dynamical mechanisms previously discussed considers diluted samples, and this is usually a good approximation even if the sample is not well diluted and some interactions exist. When the interactions are strong, a more complete description becomes necessary to fully understand the phenomenology. The origin of the interactions can be manifold, like hydrodynamic, electrical or optical (optical binding), and they may promote collective behaviors such as dynamical ordering, phase transitions or cluster formation [94,342,385,[388][389][390][391].\\nUnderstanding and controlling transport from the mesoscale down to the nanoscale is still a key topic in biophysics, nanomedicine and nanotechnology. There is still plenty of room for basic and applied research on the many aspects involved in transport of microscopic objects, including studies on self-propelled particles [337]. Especially in biological environments, the role of light-induced forces has been barely studied so far. On the other hand, many of the ideas and techniques discussed here are being extended to other realms where force fields can be used for contactless manipulation, either light, like in plasmonic [13,16,[392][393][394][395], optoelectronic [396] and some recent thermophoretic devices [397,398], or without light, like in acoustic manipulation [169,[399][400][401]. This range of techniques has broaden the trapping conditions and the accessible range of particle sizes and force magnitudes, going down to femtonewtons for nanoparticle maneuvering and up to micronewtons to trap millimiter-sized beads, and possibly beyond in both directions in the near future. Also, there is a great effort invested in turning the basic concepts into real applications, specially in the development of lab-on-chip platforms, where many modern areas can converge to perform practical tasks, including optomechanics, microfluidics, acoustofluidics, nanoplasmonics, thermoplasmonics, optoelectronics, near-field photonics, spectroscopy, to name but a few. A more extended discussion on light-induced mechanisms of controlled transport can be found in a very recent and comprehensive review [402].\"\n },\n {\n \"title\": \"Optical tweezers in vacuum\",\n \"content\": \"Even though Ashkin conducted his pioneering experiments on optical forces and radiation pressure in a gas or vacuum environment [1,[403][404][405], for many decades optical forces have been primarily harnessed to conduct optical tweezers experiments in liquids or for trapping and cooling of atoms [5]. Only recently there has been a renewed interest to use optical tweezers in vacuum. The fist single-beam gradient-force optical trap in air was demonstrated in Ref. [406] 10 years after its realization in water [2] and it took another 15 years to demonstrate single beam optical levitation in high vacuum [407].\\nThis section will introduce the basic concepts for realizing experiments with optically levitated particles in high vacuum. We will discuss the implementation with a single beam optical trap, even though other geometries for optical trapping such as parabolic mirrors [408,409] or counter-propagating beams [410,411] have also been realized. Specifically, we will focus on the practical aspects such as particle loading, sources of noise, feedback stabilization, calibration, and provide the literature references that will allow the reader to independently implement optical levitation experiments in high vacuum. We begin in section 5.1 with a concise review over the applications of levitated particles in high vacuum.\"\n },\n {\n \"title\": \"Applications of optical levitation\",\n \"content\": \"In vacuum the motion of an optically levitated dielectric nanosphere is well isolated from the environment. This makes it a promising candidate to demonstrate quantum behavior of a macroscopic object at room temperature [412][413][414]. To observe quantum effects, the motional energy of the levitated particle has to be reduced by more than seven orders of magnitude to the level of a single quantum excitation. Ongoing efforts into this direction have already reached impressive cooling rates using feedback cooling [407,[415][416][417][418] and cavity cooling schemes [419][420][421]. While the ultimate cooling limit, i.e., the quantum ground state, has already been reached in a range of nano and micromechanical systems [422,423], ground state cooling of a levitated37 nanoparticle has only be claimed recently [424]. The quantum mechanical ground state is an important milestone towards preparation of these massive objects in non-classical quantum states that are otherwise only known from elementary particles [425], atoms, ions [426], and molecules [427,428].\\nInvestigating quantum mechanical states of motion of massive objects allows for experimental tests of fundamental physics in unexplored parameter regimes [429][430][431][432][433][434]. Preparation of nontrivial quantum mechanical states requires either a strong optomechanical interaction [435][436][437] to map non-classical states of light onto mechanical motion [438], a projective quantum measurement such as photon counting [439][440][441], or a quantum nonlinear interaction, which can be provided for example by a defect inside the levitated particle such as a nitrogen-vacancy (NV) center in diamond [442][443][444]. However, the necessary coupling strengths have yet to be demonstrated experimentally and optical levitation experiments with nanodiamonds have been limited by optical absorption [445][446][447][448][449].\\nUltimately, levitated particles in the quantum regime might benefit from their simple mechanical mode structure. Conventional mechanical oscillators have a series of normal vibrational modes. Typically, only one mode is considered and the remaining modes are neglected. However, in measurement-based approaches to quantum state preparation [450,451], the displacement of the resonator is measured with a light pulse that is short compared to the oscillation frequency. The light pulse measures over a large bandwidth and thus measures not only the mode of interest but it also measures the displacement due to all the other modes that fall within the bandwidth. As a consequence, these so-called spectator modes add noise, which ultimately limits the efficiency at which the mode of interest can be measured and controlled [452]. Inside nanoparticles, the frequencies of bulk modes are several orders of magnitude higher than those of the center-ofmass motion. As a consequence, one usually only observes three modes corresponding to the center-of-mass motion in each dimension when working with spherical particles. With less symmetrical particles additional torsional [453] and precession [454] modes have been observed.\\nThe extreme isolation from the environment also means that the mechanical quality factor (Q-factor) is very large and, hence, the thermo-mechanical noise of these nanomechanical resonators is very low. For this reason, levitated particles can be used as nanomechanical sensors for force [455][456][457], torque [453], and acceleration [458]. Applications of these sensors are envisioned to detect exotic forces [459,460], such as non-Newtonian gravity [461], Casimir forces [462][463][464] and torques [465], and gravitational waves [466]. Thanks to the high mechanical Q-factor and tunability of the system parameters, levitated particles are also ideally suited to study nonlinear nanomechanical phenomena [467][468][469], which are important for the development of future nano and micromechanical devices; in fact, levitated particles are the first systems where nonlinear effects have been witnessed already at the level of thermal fluctuations [470].\\nOptically levitated particles have unique properties not available in traditional nanomechanical resonators. For example, the mechanical properties of a levitated particle can be changed rapidly in situ by simple adjustments in the trapping light. This allows to perform free-fall experiments, which provide new opportunities in force sensing [461,471] and quantum physics [472]. Unlike traditional clamped mechanical resonators, a levitated particle can rotate freely [473][474][475][476]. The rotation can be accelerated to GHz frequencies, faster than any other object [477,478], simply by changing the trapping laser polarization from linear to circular. At these fast rotations, the centrifugal stress is close to the ultimate tensile strength of the material. Hence, levitated particles can be utilized to test material limits on the nanoscale or to study friction at a fundamental level [463,479].\\n37To distinguish trapping in liquid from trapping in a dilute environment, we refer to the latter as optical levitation.\\nThe high control over the system parameters, in particular the ability to tune the interaction with the environment via the gas pressure, make levitated nanoparticles also an excellent testbed for studies of single particle thermodynamics [19], a field also known as stochastic thermodynamics [164]. As we have seen in section 4.5.3, many studies in this field have already been conducted in the overdamped regime with optically trapped particles in a liquid medium. In contrast, levitated particles can also operate in the underdamped regime and have enabled the first observation of ballistic Brownian motion [410], the first quantitative measurement of the Kramers turnover [480], experimental tests of fluctuation theorems [481,482], and fundamental tests of thermodynamic laws [483].\"\n },\n {\n \"title\": \"Dynamics in the underdamped regime\",\n \"content\": \"When transitioning from a viscous medium like water to a dilute gas or vacuum environment, the dynamics of a particle in an optical trap changes fundamentally because of the particle's finite inertia. Including the inertial term, the equation for the center-of-mass motion along a single spatial direction is (equation (8))\\nwhere Ω 0 = κ/m is the resonance frequency, κ is the stiffness of the optical trap, and m is the mass of the particle. Under most experimental conditions, the fluctuating force F L and damping rate38 γ 0 are due to collisions between the particle and residual gas molecules. We will come back to the stochastic forces and damping contributions in section 5.5. F ext summarizes additional external forces, excluding the optical forces that generate the optical trap, which correspond to the last term on the left hand side. These can be used for example to drive the particle or to apply feedback cooling, as will be discussed in greater detail in section 5.7. The Duffing term ξ ∼ 1/w 2 0 , where w 0 is the width of the optical trap, accounts for deviations from a parabolic trap [470]. The optical potential term also contains nonlinear terms that couple the three orthogonal spatial modes nonlinearly [468,470]. However, these terms become only relevant for sufficiently large oscillation amplitudes and can in most situations be neglected, in particular when feedback cooling is applied. This justifies the treatment of the three-dimensional motion of the trapped particle as three independent harmonic oscillators. Note that, while the modes are uncoupled in a static trap, the two modes that are transverse to the propagation direction can be coupled by modulating the polarization of the trapping light in time. The coupling allows to coherently exchange energy between the modes, since this coupling can be turned on and off on demand [484]. Overall, this platform is very versatile because all system parameters can be controlled to a large degree in the experiment.\\nIn the underdamped regime, which is defined by the condition γ 0 Ω 0 , the particle undergoes ∼ Q = Ω 0 /γ 0 phase coherent oscillations. This becomes very clear in the PSD\\nwhich in the underdamped regime is strongly peaked around the oscillation frequency Ω 0 . Here, χ(ω) = 1/ Ω 2 0 − ω 2 + iωγ 0 ) is the response function of a harmonic oscillator and we assumed thermal noise with a PSD S ff = γ 0 mk B T/π. As we have already seen in section 3.6, in the overdamped regime (Q 1), equation (161) can be approximated by a Lorentzian distribution\\n38Note that here we define γ 0 as a rate with units of Hz, in contrast to γ = γ 0 m in equation (8), which has units of [Hz kg]. Fig. 35. From overdamped to underdamped. We plot the measured PSD for the z-axis motion, normalized to the damping rate γ 0 /(2π), at three different pressures for a levitated particle (radius a = 68 nm, laser power P ≈ 150 mW) that is overdamped (blue line), critically damped (orange line), and underdamped (green line). The dashed black lines are least-square fits to equation (162) (1000 mbar) or equation (161) (60 and 2.5 mbar). The colored vertical solid lines indicate the roll-off frequencies Ω c /2π. For the time traces at 60 and 2.5 mbar, the spectra contain leakage signals from the other oscillation axes above 100 kHz. Data from Ref. [485]. Fig. 35 shows experimental data that clearly illustrate the transition from the overdamped to the underdamped regime. In the experiment, the damping γ 0 is changed through the pressure inside the vacuum chamber. At atmospheric pressures, the gas damps the particle motion (γ 0 2Ω 0 , blue) and we observe the spectrum familiar from optical tweezers experiments in liquids (section 3.6), with the maximum at zero frequency. At a pressure of ∼ 60 mbar, the particle is critically damped (γ 0 = 2Ω 0 , orange). At even lower pressures a strong peak evolves at Ω 0 , which is the signature of coherent oscillations. The vertical lines indicate the cut-off frequency Ω c = Ω 2 0 /γ 0 .\"\n },\n {\n \"title\": \"Optical tweezers setup for vacuum operation\",\n \"content\": \"A standard setup for optical levitation in vacuum consists of a vacuum system that allows to control the pressure and thereby the friction the particle experiences, optics that strongly focus a laser beam for trapping, optical detection for real-time readout of the particle motion, and feedback control to stabilize the particle in high vacuum. An example of a nanoparticle optically trapped in a vacuum chamber is shown in Fig. 36. Operating an optical tweezers in vacuum introduces a range of significant changes to the experimental apparatus and also imposes limits to the type of particles that can be trapped. Naturally, there are many ways in which such a setup can be constructed. Here, we follow largely the description given by Hebestreit [485], a schematic picture of whose setup is shown in Fig. 37.\"\n },\n {\n \"title\": \"Particle types\",\n \"content\": \"One of the great benefits of operating in vacuum is that the particle is very well isolated from the environment. However, this also means that the particle doesn't thermalize with the environment. As a consequence, optical absorption of the trapping laser significantly heats up the internal (bulk) temperature of the particle (section 5.6.5). This limits vacuum trapping to particles that absorb very little of the trapping laser (e.g., silicon and silica) and to wavelengths where optical absorption is minimized (e.g., infrared, the most popular choices being 1064 nm and 1550 nm). There has also been great interest in trapping nanodiamonds, since these can host interesting optical defects such as NV-centers [487]; however, trapping in high vacuum has not been reported yet due to excessive optical absorption from impurities and defects in the diamond [447]. A liquid medium that surrounds the particle not only mitigates heating from optical absorption, it also keeps the particle afloat. Without it, gravity pulls the particles down to the bottom of the chamber, which complicates the loading of individual particles into the trap (section 5.4).\\nIn addition, without a high-refractive-index medium such as water (n m = 1.33), the refractive index contrast is much larger. Consequently, the scattering force is more prominent in vacuum since the ratio of the scattering force to the gradient force scales as [488,489]\\nwhere ∆ = (n 2 p − n 2 m )/(n 2 p + 2n 2 m ) is proportional to the relative index contrast between the particle (n p ) and the surrounding medium (n m ). Therefore, trapping of spherical particles in vacuum with a single beam optical tweezers is limited to radii a ranging from ∼ 30 nm to ∼ 150 nm [488]. However, this is not a fundamental limitation, since the scattering force can be canceled, e.g., with counter-propagating beams [410,490].\\nAlso non-spherical particles (e.g., dumbells [477], clusters [454], and rods [475]) have been successfully trapped in vacuum. They have quite different scattering properties and therefore experience different optical forces and torques than spherical particles.\"\n },\n {\n \"title\": \"Optical trap\",\n \"content\": \"The most popular choice for the trapping laser is infrared light at 1064 nm or at 1550 nm. In addition, the laser should have a small relative intensity noise (RIN), low pointing instability, and provide a power at the laser focus on the order of P opt = 70 mW, which requires a laser output power of ≈ 0.5 W. Possible back-reflections into the laser can destabilize the laser output. This can be avoided by introducing a Faraday isolator right after the laser output. A low RIN is important because noise in the intensity at twice the particle oscillation frequency excites its center-of-mass motion [491][492][493] and causes fluctuations of the oscillation frequency, since the particle frequency scales with Ω 0 ∝ P opt .\\nTo parametrically drive the particle motion, the laser intensity has to be modulated at twice the oscillation frequency, i.e., at up to 400 kHz (section 5.7). Electro-optic modulators (EOMs) or acousto-optic modulators (AOMs) provide the necessary modulation bandwidth. While EOMs have a higher transmission (up to 95%, in contrast to maximally 80% for AOMs) and a higher bandwidth, they feature a much smaller contrast ratio than AOMs [471].\\nThe trapping laser can also be used to detect the particle motion even during parametric control since the parametric modulation is at a twice the motional frequency and can therefore be filtered out spectrally (section 5.7.2). However, to exclude any cross-talk, it is convenient (albeit not necessary) to have an independent measurement of the particle with a constant laser. Interferences should be avoided, since they can lead to strong fluctuations of the optical trap. This can be achieved with a single laser source that is divided into a high-power branch for trapping and a low-power branch for detection. A polarizing beam splitter (PBS) in combination with a half-wave plate provides a convenient way to cross-polarize and to adjust the power ratio between the two branches. In addition, one branch is frequency-shifted with an AOM, to avoid interference. A second PBS re-combines the two branches, after which they enter the microscope objective colinearly.\\nA stable three-dimensional trap with a single optical beam requires strong focusing. This can be achieved with a high-NA objective (NA ≥ 0.8). Note that optical absorption in the objective can lead to small shifts of the focus [485]. This can heat the particle motion for large laser intensity modulation, for example in free-fall experiments [471]. Aspheric lenses provide a cheap alternative to more costly microscope objectives. In addition, they usually have a higher transmission and are available with anti-reflection coatings for a large range of wavelengths. This reduces the power requirements and absorption losses. However, they are more sensitive to misalignment than microscope objectives, which makes optical trapping with aspheric lenses much more challenging [485].\\nBoth AOMs and EOMs distort the laser beam as it passes through. To recover a nice Gaussian beam shape, the laser mode profile is cleaned with a spatial filter before it enters the microscope objective. This increases the quality of the focus and, hence, the trap stiffness. Spatial filtering is achieved either with a pinhole of appropriate size, or more conveniently with a single-mode fiber, which has the advantage of mechanically decoupling different sections of the setup. A polarization-maintaining single-mode fiber maintains the input polarization at the output, while a single-mode fiber with a fiber-based polarization controller allows to create an arbitrary output polarization.\\nA second microscope objective or aspheric lens inside the vacuum chamber collects and re-collimates the trapping laser and the measurement beam. After the vacuum chamber, the two beams are separated again with another PBS. The trapping beam is dumped, while the probe beam is sent towards the detection setup. Efficient separation of the beams is paramount to avoid detecting the intensity modulation introduced by the EOM, that would then be deleteriously fed into the system via the feedback loop. In fact, mixed polarization states appear in this side of the set-up due to the tight focusing and especially to the mechanical stresses present in the glass windows of the vacuum chamber. To undo these changes in the polarization state, one introduces a half waveplate and a quarter waveplate after the vacuum chamber and minimizes the unwanted trapping light after the PBS while maximizing the probe light, which then goes to the detection setup.\"\n },\n {\n \"title\": \"Detection\",\n \"content\": \"In liquids, momentum relaxation occurs within a fraction of a microsecond due to the high viscosity of the surrounding medium. This is faster than what most experiments are able to resolve [494]. In contrast, the viscosity of a dilute gas is orders of magnitude smaller. Hence, inertial effects on the dynamics of a particle in an optical tweezers are easily accessible [410]. While optical tweezers experiments in liquids are mostly concerned with the diffusive motion at low frequencies on the order of the corner frequency ω c /(2π) = k /(mγ 0 ) ∼ kHz, the frequency range of interest in experiments under vacuum is centered around the natural oscillation frequency Ω 0 /(2π) = κ/m ∼ 100kHz, where κ is the trap stiffness and γ 0 the damping coefficient.\\nAs in the case of traditional optical tweezers, the particle motion is measured optically by extracting information contained in the field scattered by the particle. In single-beam optical tweezers, the trapped particle are sub-wavelength scatterers. As such, most of the information about the particle position is contained in the phase of the scattered field. This requires an interferometric measurement to read out the phase.\\nForward detection provides a convenient way to access the phase, since the non-scattered transmitted light provides a stable reference field without requiring additional optical path stabilization [495]. The signal-to-noise ratio of the detector signal is maximized by increasing the detection efficiency of the scattered photons and by increasing the optical power of the detection beam. Maximum power is provided by using the trapping beam itself, at the expense of introducing cross-talk as mentioned before. Hence, the best signal-to-noise ratio is obtained with a high-NA collection lens and by using all the scattered light to generate the detector signals. In forward scattering, the collected power from the trapping beam amounts to ≈ 50 mW. Hence, optimal detection in forward scattering requires a quadrant photodetector that can handle at least 50 mW optical power, has a high quantum efficiency at the trapping wavelength, has a bandwidth of at least 1 MHz, and is shot-noise limited at power levels above 10 mW per quadrant. Details on a custom detector design that fulfills the above requirements can be found in Refs. [485,493].\\nForward detection is attractive due to its simplicity. However, the ratio of scattered power to reference power is fixed by the scattering cross section of the particle. Backward detection allows to collect only the scattered photons and add a reference field with variable power. The ultimate position sensitivity to particle motion that is transverse to the optical axis is the same for forward and backward detection. Interestingly, due to interference between incident and scattered photons this is not the case for motion along the optical axis. In the forward direction, destructive interference reduces the sensitivity, with the counterintuitive result that a higher NA collection lens can yield a lower sensitivity. In contrast, constructive interference in the backward direction allows for up to 60% collection efficiency even with a moderate NA = 0.8. That is, more information about the particle position along the optical axis is contained in the backward scattered light than in the forward scattered light [496,497].\\nInterferometric detection provides the best signal-to-noise ratio. However, it is limited to a small range of particle motion. In contrast, video analysis allows to determine the particle position over a much larger range with a limited signal-to-noise ratio and it requires extremely fast cameras to resolve the relevant timescale ∼ µs in high vacuum [498].\"\n },\n {\n \"title\": \"Feedback electronics\",\n \"content\": \"Feedback cooling allows to extract energy from the center-of-mass motion and stabilizes the trap. The feedback force can be applied either directly (e.g., with electric fields when the particle carries a charge) or parametrically (by modulation of the laser intensity). The electronic circuit that processes the detector signal can be analog [488], digital [415,417], or an off-the-shelf lock-in amplifier [416] (section 5.7). Here, we focus on the latter following the description for parametric feedback based on a phase-locked loop (PLL) given by Hebestreit [471].\\nThe detector signal for each axis is fed to one of the PLLs. Each PLL detects the frequency and phase of the oscillation along one spatial dimension. The locking range of the PLLs is set to 4 to 8 kHz around the expected center frequency to prevent the PLL from losing lock on the particle oscillation signal. The bandwidth of the loop filter is adjusted to be large enough to follow frequency changes sufficiently fast, and small enough to exclude noise from the feedback signal. In practice, this amounts to a bandwidth of 1 to 16 kHz [471].\\nThe PLL synthesizes a signal that is phase-locked and at the same frequency or at a harmonic of the input signal with adjustable amplitude and phase. For parametric feedback, one uses the second harmonic and tunes the amplitude and phase to minimize the particle motion. The independent signals for the three oscillation axes are added within the lock-in amplifiers to form the final feedback signal. This feedback signal is applied to the EOM, which in turn modulates the power of the trapping beam. The amplitude of the feedback signals is usually set to a few mV for each axis. For a half-wave voltage of V π = 240 V and amplifier with 400 V/V gain, the optical power is modulated by 2.5% at a feedback signal amplitude of 10 mV [471].\\nA switch allows to rapidly turn off the feedback signal. This allows to perform relaxation experiments [481]. Additional signals can be added to the feedback signal. This allows to stabilize this oscillation frequency by adjusting the power of the trapping beam with a PID (proportional-integral-derivative) control loop or to switch off the trap for short amounts of time for free-fall experiments [471]. A harmonic signal at twice the particle's motional frequency allows to excite the particle into the nonlinear region of the optical trap [468,469].\"\n },\n {\n \"title\": \"Vacuum system\",\n \"content\": \"Optical levitation in high vacuum requires an experimental setup containing a vacuum system to control the gas pressure, optical access for the trapping and detection optics, and an access door on the top if particles are loaded at ambient pressures with a nebulizer (section 5.4.1). A schematic of a vacuum system is shown in Fig. 38. The pressure in the chamber can be monitored continuously over the entire range from ambient pressure down to below 10 −7 mbar with a combination of gauges, e.g., a Pirani gauge for high pressure combined with a hot cathode filament gauge for low pressure.\\nA set of pumps and valves controls and stabilizes the pressure. The main pump is a turbomolecular pump. A motorized valve between the chamber and the turbo-molecular pump is used to stabilize the pressure below 10 −3 mbar with a PID controller. The turbo-molecular pump is backed by a dry scroll pump via another motorized valve that is mainly used to ensure a smooth evacuation of the chamber without sudden changes in the chamber pressure during the initial pump-down. This is important to prevent loosing a trapped particle during pump-down from ambient pressure after loading. To avoid mechanical vibrations from the vacuum pumps to couple into the optical setup, the pumps are connected to the vacuum chamber through flexible bellows, which decouple the vibrations. Vibration isolation can be further improved by casting the bellows into a sand or concrete block. At pressures below 10 −6 mbar, ion pumps are sufficient to maintain a pressure without introducing vibrations.\\nM M Fig. 38. Vacuum system. The vacuum system consists of a vacuum chamber with optical vacuum windows, in which the actual trapping takes place. Here, we present a system based on that employed in Ref. [485]. The pressure is monitored using an ion and a Pirani gauge. A turbomolecular pump is connected to the chamber using a flexible bellow to suppress vibrations. A scroll pump supports the turbo pump. The pumping speeds can be regulated manually or with a PID controller using motorized valves (V 1 and V 2 ). The chamber is vented with nitrogen to reduce contamination. O: objective. PID: proportional integral derivative controller.\\nPressures below 10 −6 mbar require to bake the vacuum chamber to desorb residual water. Many experiments load particles at ambient pressure. Hence, the lowest pressures can be achieved if the chamber is baked in situ while the particle is trapped. Optical access while baking can be provided with heating bands that are wrapped around the vacuum chamber. However, temperatures are limited to ≈ 150 • C to avoid damaging the optical components inside the vacuum chamber and the optical coatings of the windows. The baking process also causes thermal deformation of the optics inside the vacuum chamber. This can then lead to particle loss during baking. Nonetheless, even if particles are not kept in the trap during bake out, baking the chamber in combination with purging with dry air (instead of venting with humid ambient air) reduces the overall water content in the vacuum chamber and leads to improved pump speeds and better vacuum.\\nHebestreit [471] quantified the gas composition in the vacuum chamber during a trapping experiment with a residual gas analyzer (RGA). The RGA is a mass spectrometer that allows to measure the mass spectrum of a gas and thus to extract the partial pressures of the different gases species. The measured partial pressures of the prominent gas species during a typical pump-down cycle are shown in Fig. 39. The total pressure at the RGA (the sum of all the partial pressures, dashed line) is higher than the pressure measured in the main vacuum chamber with an ion gauge (dashed line), because the pumping speed at the RGA is reduced compared to the main chamber. Initially, the chamber is filled with nitrogen. Interestingly, at low pressures the composition is vastly different. The turbo molecular-pump removes the initial nitrogen very efficiently. Thus, its partial pressure reduces quickly. In contrast, water and hydrogen are continuously desorbed from the walls of the vacuum chamber. As a consequence, the gas in the chamber is dominated by water vapor at pressures below 10 −4 mbar. Baking out the vacuum chamber accelerates the desorption process and thereby helps to get the water and hydrogen out of the chamber. The final gas composition in the pressure range of 10 −6 to 10 −5 mbar is mainly water vapor (58 ± 9%), nitrogen (19 ± 8%), hydrogen (12 ± 1%), oxygen (7 ± 1%), and carbon dioxide (4 ± 1%). Other gases, which contribute with less than 1% are argon, isopropyl alcohol, and oil. The last two probably remain from the solvent of the particle solution and residual machining grease, respectively. As a consequence, the molar mass M ≈ 20 ± 11 · 10 −3 kg mol −1 at low pressure is significantly lower than the molar mass at high pressures, the latter being equal to the molar mass of dry air M air = 28.97 · 10 −3 kg mol −1 .\"\n },\n {\n \"title\": \"Particle loading\",\n \"content\": \"In contrast to a liquid environment, loading a particle into an optical trap in a dilute gas or vacuum is challenging because the particles will quickly fall down due to gravity. The size of an optical trap is rather small, on the order of the focal volume ∼ λ 3 . Therefore, a particle that passes by the focus at a distance larger than ∼ λ will not be captured. Furthermore, if a particle enters the focal volume at high speed, the low damping in vacuum will not slow the particle down sufficiently for it to get trapped. The maximum speed for successful trapping is v max ∼ λγ 0 , where γ 0 is the damping constant. In water, the viscosity at room temperature is η (water) = 890 µPa s so that, using Stokes formula (equation (6)), γ 0 = 6πηa/m and, therefore, v (water) max = 344 m s −1 for a silica particle of radius a = 75 nm. In air, the viscosity is about two orders of magnitude smaller (η (air) = 18 µPa s) so that v (air) max = 7 m s −1 . The challenge lies, therefore, in finding a technique by which a slow single nanoparticle is brought into the focal volume [488].\"\n },\n {\n \"title\": \"Nebuliser\",\n \"content\": \"The most common approach to particle loading is the nebulizer approach thanks to its simplicity [499]. In this approach, a solution containing silica beads (e.g., Microparticles GmbH, Bangs Laboratories) with radii ranging from a = 50 to 100 nm is sprayed into the open vacuum chamber with a nebulizer (e.g., Omron NE-U22-W) and the particles are trapped at ambient pressure [488] or mild vacuum [500].\\nThe nebulizer consists essentially of a mesh on top of a piezo element. A little bit of liquid gets into the space between the piezo and the mesh. The motion of the piezo then pushes the liquid through the mesh. The mesh breaks the liquid into little droplets of diameter 2 µm and a nozzle funnels the falling particles to a region near the laser focus (Figs. 40(a-b)). Successful trapping of a particle can be easily noticed observing the laser light scattered by the particle with a camera. However, since the scattered light has a dipole radiation pattern, the polarization of the laser beam should be orthogonal to the direction of observation to see the particle with the camera. The scattered light intensity on the camera scales with the size of the particle as B ∝ a 6 . This allows to distinguish whether the object in the trap is a single nanoparticle or a cluster of multiple particles by comparing the brightness on the camera for several trapping events. For example, in Fig. 40(c), the histogram of the brightness on the camera for ∼ 350 particles shows clearly distinct peaks, which can be associated with the number of particles in the trap [486]. Most of the liquid in the droplet evaporates before or immediately after trapping. However, ∼ 10 layers of water remain physically absorbed on the surface of the particle. These physically absorbed layers evaporate as the gas pressure is reduced down to ∼ 1 mbar. This dehydration process is reversible and water molecules get reabsorbed when the pressure is increased. In the region from 0.1 to 0.5 mbar, one can often observe a sudden decrease in the mechanical frequency and power of the scattered light. This is attributed to a sudden loss of the remaining water from the particle as its internal temperature increases to above ∼ 200 • C. This effect occurs only once and is irreversible when cycling the pressure. Therefore, once the particle has lost the final water layer its water uptake is extremely slow and for all practical purposes the particle size and material composition can be considered pressure independent. For more details on the dehydration process see Refs. [419,486,500].\"\n },\n {\n \"title\": \"Dry approaches\",\n \"content\": \"Spraying solvents into a vacuum chamber introduces contaminants that are difficult to pump out, e.g., water molecules. This motivates a dry approach, where particles are released from a substrate instead of being injected from a solution. However, small particles stick to surfaces due to dipole-dipole interactions known as Van der Waals-forces, which scale linearly with particle size; the magnitude of these forces is\\nwhere γ s is the effective surface energy. For example, measurements on silica spheres on glass give F VdW = 176 nN for a = 1 µm [501]. For comparison, the gravitational force for a particle of this size is only ∼ 0.1 pN and, therefore, much too weak to remove the particle. There can also be additional surface forces that lead to even stronger adhesion, for example meniscus forces arising from a thin water layer that is formed on the substrate. Therefore, even larger forces have to be applied to release the particle from the substrate. Two common approaches, which provide dry loading methods and do not require to open the vacuum chamber as in the nebulizer approach, are based on launching particles from surfaces using piezoelectric transducers or laser-induced acoustic desorption (LIAD).\\nA piezoelectric transducer works really well for micrometer-sized particles [405,502] and allows even for repeated loading of the same particle [503]. When the piezoelectric transducer is driven with oscillation amplitude q p and frequency ω p , the particle feels a force\\ndue to its inertia, where m = V ρ p is the particle mass with ρ p and V the particle density and volume, respectively. Consequently, the acceleration required to release the particle is\\nHence, for a given maximum acceleration the piezoelectric transducer can provide, there is a minimum particle size that can be shaken off. The smallest particles that have been released with this method have a radius of 150 nm and further details on how to implement this can be found in Ref. [504]. The limited acceleration of the piezo approach can be overcome by generating a shockwave on the surface of the substrate in the LIAD approach. The shockwave can be created by a pulsed frequency-doubled Nd:YAG laser (5 ns pulses of 3 mJ at 532 nm). When focused down to 200 µm on the back side of a thin metallic foil, the laser creates an acoustic wave that launches particles from the front side, which had been previously spin coated with particles [505].\\nDespite the availability of these techniques, directly loading particles into an optical trap under high vacuum conditions remains an open challenge. With the dry approaches outlined above, particles are still loaded into the trap at ambient pressures or moderate vacuum, where the background gas provides sufficient dissipation because capture requires an additional dissipation mechanism due to the conservative nature of gradient trapping forces and the small size of the trapping region of an optical tweezers. Recently, it was demonstrated that high-vacuum loading into a Paul trap can be achieved [506]. The Paul trap provides a much larger trapping volume and trap depth than an optical tweezers. This allows to turn on the trap exactly when the particle is near the center of the trap. As a consequence, the particle's total energy is given only by its kinetic energy. If the kinetic energy is less than the trap depth, the particle will be stably trapped.\\nPaul traps are limited to charged particles; however, this is not a strong limitation for loading since the particles generally carry several tens of elementary charges (section 5.6.3).\\nTransferring particles from a contaminated chamber, where the particle is initially trapped, to a clean science chamber, which always remains in high-vacuum conditions, can be achieved with a mobile optical trap (MobOT) in a load-lock [507]. However, this solution is rather bulky and has a limited throughput at low pressures. Hollow-core photonic crystal fibers provide also a compact solution with high throughput, but have been limited so far to pressures above 10 −2 mbar [508,509].\"\n },\n {\n \"title\": \"Dissipation and noise\",\n \"content\": \"A particularly attractive feature of trapping in a dilute or vacuum environment is that the interaction with the environment can be adjusted over several orders of magnitude. The interactions between the particle and its environment result in stochastic forces that excite the particle motion and lead to stochastic dynamics, which are of particular interest when studying thermodynamics of individual small particles and are to be minimized when aiming for the quantum regime.\\nThe intensity of a stochastic force is characterized by its PSD S noise ff . While the stochastic forces excite the center-of-mass motion of the trapped particle, the interactions with the environment also lead to dissipation at a rate γ noise , which damps the particle motion. There are several sources that contribute to the total damping rate and stochastic noise intensity, which we will discuss in this section.\\nThe total damping rate and stochastic noise intensity is simply given by the sum of the individual contributions γ 0 = noise γ noise and S ff = noise S noise ff , respectively. Note that for most purposes, the PSD can be considered as frequency independent (white noise) over the bandwidth of the mechanical resonance of the particle, especially when the bandwidth becomes quite narrow in high vacuum. The average energy of each center-of-mass degree of freedom evolves as E = −γ 0 E + Γ where the heating power due to the force noise is Γ = πS ff /m, S ff being the double sided PSD. After a time ≈ 1/γ 0 , the particle reaches an equilibrium, where the ensemble average energy stays constant. This allows to define the effective temperature through the fluctuation-dissipation relation:\\nThe effective temperature corresponds to the particle energy averaged over an ensemble of identical experiments. Importantly, since we are concerned here with the energy of a single particle, the energy in each experiment will still fluctuate with a standard deviation ∼ T cm . Generally, one has to be careful to define a temperature for a system out of equilibrium [510]. However, the situation we present here is somewhat simplified because it is steady-state and for many practical purposes the effective bath model that is characterized by an effective damping/temperature gives a good description. However, one can also create situations where this is no longer true. For example, by parametric feedback damping, the temperature alone is not sufficient to give a full description of the bath (section 5.7) For a levitated particle in a dilute gas, the main contribution to the stochastic force noise are collisions with air molecules and radiation damping at pressures below ∼ 10 −7 mbar. In addition, there are technical noise sources that do not pose a fundamental limitation but can impose practical limitations. These are primarily displacement noise of the trap and laser intensity fluctuations. Noise sources can also be introduced intentionally to emulate a different thermal environment without actually changing the real temperature of the environment [316,511]. This can be a useful tool for single particle thermodynamics experiments [19].\"\n },\n {\n \"title\": \"Gas damping\",\n \"content\": \"From kinetic gas theory, the pressure-dependent gas damping for a spherical particle is [488,512] γ gas\\nwhere Kn =l/a is the Knudsen number, c K = 0.31Kn/ 0.785 + 1.152Kn + Kn 2 ,l = πk B T 2m gas η ν P gas is the mean free path, η v is the viscosity, and m gas = M/N A the molecule mass [512], N A being Avogadro's number. For dry air, the molar mass M = 28.97 · 10 −3 kg mol −1 , however the actual gas composition can be different (Fig. 39). For high pressure, the interaction with the gas is so strong that the particle motion is heavily damped and its internal temperature T bulk and centre-of-mass temperature T cm quickly thermalize with the gas temperature T gas . In this regime, the damping does not depend on pressure and is given by Stokes law, i.e., γ gas /(2π) ≈ 3aη v /m. For pressures below P gas ≈ 54.4 mbar (a/µm) −1 , where the mean free pathl is much larger than the radius of the particle a, the damping is linear in the pressure P gas and given by\\nIn typical experiments, γ gas /2π = c P P gas /a with c P ≈ 50 Hz (µm/mbar). Note that the above holds for a spherical particle. For an anisotropic particle (e.g., a rod), the friction term is different along each of the axes, and depends upon the alignment of the particle [513]. The mean free path of the gas molecules increases with decreasing pressure (e.g.,l ∼ 60 µm at 1 mbar). As a consequence, the particle no longer thermalizes with the gas since the impinging gas molecules no longer carry away enough thermal power to balance the optical absorption from the trapping laser. Due to the increased internal temperature T bulk of the particle, the average energy of the gas molecules after a collision with the particle increases (section 5.6.5). The latter requires that the impinging molecules stick to the surface for a short time before they emerge again from the surface. This type of collision is called diffusive reflection [514,515]. For diffusive reflection, the fluctuating forces originating from the impinging gas molecules are completely uncorrelated. This justifies the treatment of the impinging and the emerging gas molecules as two independent baths with individual bath temperatures, damping and heating rates. The process by which a surface exchanges thermal energy with a gas is called accommodation, which is characterized by the accommodation coefficient\\nwhere T em is the temperature of the gas molecules emitted from the particle surface. The accommodation coefficient quantifies the fraction of the thermal energy that the colliding gas molecule removes from the surface, such that c acc = 1 means that the molecule fully thermalizes with the surface. We do not distinguish between the bulk and the surface temperature, since we assume that they are identical for nanoparticles. Since the mean free path in a dilute gas is long, one can safely assume that an emitted molecule will not interact again with the particle before thermalizing with the environment. Consequently, we can consider the particles that impinge on the particle surface and those that leave it as being in equilibrium with two independent baths [514]. The damping due to the impinging molecules is\\nand the contribution from the emerging molecules is\\nThe total gas damping is then γ gas 2π = γ im 2π + γ em 2π (173) and the total force noise due to the gas is\\nWhen the temperature of the emerging molecules equals the gas temperature (i.e., T em = T gas ), equation (173) reduces to equation (169).\"\n },\n {\n \"title\": \"Radiation damping and shot noise\",\n \"content\": \"Once the pressure is low enough (i.e., ≤ 10 −7 mbar), gas damping is negligible and the particulate nature of the light field becomes the primary source of dissipation, in the absence of technical noise sources such as laser phase and amplitude fluctuations [416]. Photons arrive at discrete times, where the number of photons arriving per time interval ∆t is given by ∆tP opt ( ω opt ) . The recoil from the fluctuating number of photons impinging on the nanoparticle can be modeled as an effective bath with the characteristics [497,516] γ rad 2π = c dp P scatt 2πmc 2 and S rad ff = c dp\\nwhere c dp depends on the direction of motion of the particle with respect to the polarization of the laser (in particular, c dp = 1/5 for motion along the direction of polarization, c dp = 2/5 for motion perpendicular to the polarization, and c dp = (2/5 + A 2 ) along the direction of propagation, where A is the radiation pressure contribution defined in Ref. [497]). The scattered power is P scatt = σ scatt I opt , where σ scatt = |α| 2 k 4 opt /(6π 2 0 ) and I opt is the laser intensity. The effective temperature of this bath can be calculated via equation (167) and is ∼ ω L /k B .\"\n },\n {\n \"title\": \"Displacement noise\",\n \"content\": \"In the reference frame of the trap (i.e., where the trap appears stationary), acceleration of the trap center a dis appears as a force F dis = ma dis due to the particle's inertia m. Therefore, fluctuations of the trap center with displacement spectral density S dd lead to the effective force spectral density\\nwhere we have used the fact that the acceleration spectral density S aa (ω) = ω 4 S dd (ω). There is no damping associated with displacement noise, since the source of the displacement noise is usually some active component like a pump. Unlike the thermal force spectral density, the displacement noise spectral density is not flat but increases for lower frequencies. However, for the typical narrow resonances at Ω 0 in optical levitation experiments, we can approximate it as white noise, S dis ff ≈ m 2 Ω 4 0 S dd (Ω 0 ). The thermomechanical noise limited acceleration sensitivity S aa = k B T γ 0 /(πm) depends inversely on the particle mass. Hence, heavy particles make particularly good accelerometers [458]. However, since the maximal trap stiffness is limited, heavier particles tend to have lower frequencies at which displacement noise dominates over thermomechanical noise. At the optimal particle size both noise sources contribute equally. Displacement noise is a technical noise source and as such not a fundamental limitation. It is primarily noticeable in setups where the external forces change considerably over the range of movement of the trap center. That is in situations where the force has a strong spatial dependence, for example when the trapped particle interacts with a standing wave of an optical cavity or with the evanescent field from a tapered fiber [492]. If the trap position changes randomly with respect to the cavity or the fiber, the particle experiences fluctuating optical forces from the cavity or evanescent field, respectively.\"\n },\n {\n \"title\": \"Laser intensity noise\",\n \"content\": \"Laser intensity fluctuations are another technical noise source [491,492]. Low frequency drifts of the intensity randomly change the particle's oscillation frequency, since the particle frequency scales with Ω 0 ∝ P opt . This can be a serious limitation in force-sensing experiments, where an external force is transduced into an oscillation amplitude. If the frequency changes, the particle's response to an external force changes even if this force is constant, resulting in a noisy measurement.\\nIntensity fluctuations at twice the motional frequency 2Ω 0 parametrically excite the particle and thereby raise the particle's energy. If the phase of the particle oscillation is not actively stabilized to the parametric modulation, it self-locks (entrains) to the external modulation [468]. Hence, an intensity modulation at 2Ω 0 without active stabilization always leads to heating as will be discussed in section 5.7. Parametric excitation is a nonlinear process and its strength depends on the motional energy of the particle and therefore can be suppressed by reducing the particle's energy with feedback cooling. In free-space experiments, laser-intensity noise can be reduced sufficiently such that it is smaller than the fundamental photon shot noise [416]. In cavity experiments, however, laser-phase noise is transduced into intensity noise and has been identified as a major obstacle towards reaching the ground state with sideband cooling [421,517].\"\n },\n {\n \"title\": \"Calibration\",\n \"content\": \"Precise calibration is as important for quantitative measurements in the underdamped regime as it was in the overdamped regime (section 3). As before, the thermal energy k B T serves as a reference to calibrate the detector signal (section 5.6.1). However, in contrast to the underdamped case, where the displacement signal from the particle is very broadband and centered around zero, in the underdamped regime the signal is sharply peaked and centered around a finite frequency. The weak coupling to the thermal bath in high vacuum also allows to use the energy scale of a single motional quantum Ω 0 as a calibration reference. Remarkably, despite the huge difference in energy scale of 7-8 orders of magnitude one can achieve almost perfect agreement [418] (section 5.6.2). In high vacuum charges on the particle are preserved. Section 5.6.3 explores how to control and measure them. The precise knowledge of the charge then allows allows to precisely determine the particle's mass (section 5.6.4). Since a particle in vaccum is well isolated, its internal temperature raises well above the temperature of the environment in an optical tweezers. In section 5.6.6, we will see how to measure the internal temperature by analyzing the particles center of mass motion.\"\n },\n {\n \"title\": \"Detector calibration\",\n \"content\": \"For the most part optical trapping is concerned with the external degrees of freedom, and in particular with the center-of-mass motion. Quantitative measurements require a precise calibration of the detector signal that relates the measured signal (typically in volts) to the displacements of the particle (in meters). Under the assumption that the detector signal v(t) is proportional to the particle displacement q(t), the calibration factor is defined as the proportionality factor c = v/q with units of [V m −1 ] (or [bit m −1 ] for a digitized detector signal). Using the fact that the expectation value of the potential energy of a harmonic oscillator E pot = 1 2 mΩ 2 0 q(t) 2 = 1 2 k B T, the calibration factor is simply given by\\nwhich does not depend on γ 0 and thus holds both for the overdamped and underdamped regimes.\\nIn the underdamped regime, the PSD is highly peaked around the resonance frequencies. This allows to distinguish noise and cross-talk efficiently in the spectral domain. Therefore, it is often preferred to fit the PSD to the single-sided harmonic oscillator PSD\\nto extract the variance of the particle displacement v(t) 2 from the area under the PSD. In the following, we adopt here the notation from Hebestreit [485], where we convert from angular to ordinary frequencies, i.e., ω = 2π f , γ = 2πg, and S( f ) = 2πS(ω > 0). The experimental single-sided PSD is obtained from the discretely-sampled time trace v(t) as (179) where N is the number of samples, f s is the sampling frequency, and F (v(t)) is the discrete Fourier transform of v(t).\\nFitting the experimental PSD to expression (178) allows to extract the variance v 2 (t) , the damping constant g = γ gas /(2π), and the resonance frequency f 0 = Ω 0 /2π. Then, the damping constant allows to determine the particle radius from the pressure and, therefore, the mass for known mass density (section 5.6.4). Once the mass is known, we can calculate the calibration factor according to equation (177). The biggest uncertainty in determining c stems from the error in the particle mass. However, in many experiments we are only interested in the particle energy and we can define the mass-independent energy calibration factor\\nThe above calibration method relies on two key assumptions. First, we consider the optical potential to be quadratic such that the particle behaves like a harmonic oscillator. Second, we assume that the detector signal is linear in the particle displacement, that is that the calibration factor does not depend on the particle position. The latter depends significantly on the details of the experiments and will not be futher discussed here. However, it is possible to account for a nonlinear detector response. For calibration of a position dependent calibration factor see [480] and for detector response calibration and discussion see [485].\\nThe finite potential depth of the optical trap results in anharmonicities in the particle motion for large enough displacements. Due to the symmetry of the optical potential, the lowest nonlinear term is the cubic or Duffing nonlinearity39. The Duffing nonlinearity softens the optical potential at large particle displacement [470]. This distorts and broadens the lineshape of the PSD, and causes an overestimation of the energy when following the calibration procedure described above. To minimize the impact on the shape of the PSD, the calibration is usually carried out at moderate pressures (∼ 10 mbar), where the linewidth (γ gas /(2π) ∼ 10 kHz) is narrow enough to reject noise and cross-talk, but still wide enough that the nonlinear broadening (γ NL /(2π) ∼ 3 kHz) is negligible [485] (Fig. 41a).\\n39There are some subtelties in the detection along the propagation direction of the trapping laser, where this symmetry is broken by the scattering force, leading to small but finite quadratic nonlinear terms [488]. The energy underestimation is accounted for by considering the kinetic energy instead of the potential energy [518]. The kinetic energy is given by E kin = 1 2 m q 2 = 1 2 k B T even for a nonlinear potential. The variance q 2 can be obtained by integrating over the velocity PSD, which is related to the position-PSD by\\nHence, we do not have to measure q directly, which is experimentally challenging since it would require sufficient oversampling. Experimentally, we compute the variance by numerical integration over the voltage PSD\\nThe integration bandwidth b is chosen such that we integrate most of the signal and don't integrate too much of the noise floor. In the measured displacement PSD, we usually have a white noise floor. If we multiply this PSD by f 2 to get the velocity PSD, the noise floor therefore acquires a f 2 dependence as well. Integrating this noise will eventually cause a significant error in the calibration and energy estimate. Drifts in the setup can cause the calibration to change over time. In particular, the calibration procedure detailed above is carried out at moderate pressures (∼ mbar), while most experiments are carried out at much lower pressure many hours or even days after performing the calibration. Hence, it is desirable to check the calibration even at much lower pressure and to correct for potential drift (see Fig. 41b).\\nHebestreit suggests a method to track changes in calibration at arbitrary pressure [485]. The main idea is to measure the response of the particle to a known force, for example the electrical force from a capacitor around the optical trap. This method assumes that the electrical fields and the charge on the particle do not change through out the experiment. This can be achieved as will be discussed in section 5.6.3. At the initial pressure where the calibration is carried out, a sinusoidal signal is applied to the capacitor. This leads to additional peaks in the response of the particle (Fig. 41). The power in the response peak is\\nwhere Q p is the total charge on the particle and E 0 is the electric field amplitude in the direction of oscillation at the location of the particle. At a later time (e.g., after pumping down to a lower pressure), we repeat this measurement, yielding v 2 e . At lower pressure, feedback is generally active. Hence, the response function of the particle changes. The new values of Ω 0 and γ can be extracted from a fit to the thermal noise peak. Assuming that Q p E 0 /m remain constant, we find the new calibration factor as\\nWe can also arrive at equation (184) by considering the height of the peak in the PSD instead of the area under the peak. It's value is v 2 e τ, where 2τ is the measurement time (equation (188)). In the previous section we used the thermal fluctuations of the particle to calibrate the detector signal. Similarly, we can use the quantum fluctuations for calibration. However, at ambient temperature, quantum fluctuations of a nanoparticle in an optical tweezers are 8-9 orders of magnitude weaker than the thermal fluctuations. As a consequence, they can only be observed in high vacuum, where the coupling to the thermal bath is sufficiently weak [418].\"\n },\n {\n \"title\": \"Sideband thermometry\",\n \"content\": \"When light is reflected off the particle, the particle motion is imprinted on the scattered light as a phase modulation, which can be understood as a Doppler shift due to the particle motion. To lowest order, the phase modulation leads to red and blue sidebands at ω L ± Ω 0 , where ω L is the laser frequency (Figs. 42(a-b)). Classically, the power in each sideband is proportional to the variance of the motion x 2 , which in turn is proportional to the temperature of the thermal bath. This is the basis of the calibration procedure described in the previous section.\\nIn the quantum mechanical formulation, the interaction of light with the trapped particle can be described as a Raman process. The red sideband (Fig. 42(a)) corresponds to Stokes scattering, which raises the population of the mechanical oscillator by a single quantum of mechanical energy, while simultaneously creating a low energy (red) photon. The blue sideband (Fig. 42(b)) corresponds to anti-Stokes scattering, which lowers the oscillator's population by one mechanical quantum, while creating a high energy (blue) photon. The rate at which blue sideband photons are created is proportional ton, just as in the classical case. Importantly, anti-Stokes scattering is impossible by an oscillator in its quantum ground state. In contrast, the rate at which red sidebands are produced is proportional ton + 1.\\nThe mean occupation number is given by the Bose-Einstein distribution\\nFor thermal energies large compared to the energy quantum (k B T Ω 0 ), the mean energy of the harmonic oscillator isn Ω 0 = k B T ∝ x 2 , in accordance with classical statistical mechanics and independent of . The ratio of powers in the anti-Stokes and Stokes sidebands is given bȳ (Figs. 42(c)) and vanishes for high temperatures. For thermal energies comparable to a single mechanical quantum excitation (also called phonon), the ratio can serve as a temperature measurement calibrated relative to the quantum of energy of the system [519].\\nThis so-called sideband thermometry requires to measure the Stokes and anti-Stokes sidebands [418,[520][521][522], which can be achieved with a heterodyne measurement. In a heterodyne measurement the scattered light interferes with a laser beam (local oscillator) with a frequency ω LO which is frequency-detuned from the light that is incident on the particle. This produces a strong beat signal at the difference frequency ∆ LO = ω LO − ω L , with sidebands at ∆ LO ± Ω 0 . For ω LO < ω L , the left sideband corresponds to the red sideband (Stokes) and the right to the blue (anti-Stokes). For ω LO > ω L , the roles are inverted. The difference frequency is chosen such that |∆ LO | Ω 0 , for example |∆ LO | ∼ 1MHz for typical trapping frequencies ∼ 100kHz. The measured sideband asymmetry\\ncan be used to determine the mean occupation number. Here, S l,het vv (S r,het vv ) is the power spectral density of the left (right) sideband and the subscript R − (R + ) is for ω LO < ω L (ω LO > ω L ) (see Fig. 42).\\nIn principle, R − or R + alone is sufficient to determine the mean occupation number. However, classical artifacts due to a frequency dependent transfer function of the measurement device give rise to an asymmetry that is not related to the quantum dynamics of the particle. Measuring both R − and R + allows to eliminate this artifact and to determine the actual mean occupation number as followsn Fig. 42 shows the measured sidebands of a nanoparticle cooled to an effective temperature of n = 4 [418]. Clearly the area under the red sideband is larger than under the blue sideband. This is the first room temperature measurement of the sideband asymmetry of a massive object and clear signature of its quantum ground state. Remarkably, the inferred occupation number is in very satisfactory agreement with the traditional calibration method detailed in section 5.6.1despite the huge difference in energy scales of the two methods.\"\n },\n {\n \"title\": \"Charge calibration\",\n \"content\": \"Generally, particles carry several tens of elementary charges. Under vacuum, the particle is completely isolated from the environment and therefore its charge state is preserved indefinitely. Knowing the charge exactly provides a reliable handle for exerting a known force on the particle by applying an known electric field.\\nIn optical levitation experiments, particles are loaded at high (∼ 1 atm) pressure to provide enough damping. After the particle has been trapped, the vacuum chamber is evacuated to reduce air damping. As the pressure decreases, residual water desorbs slowly from the particle followed by a sudden change in size, which is attributed to the removal of the final water layers (section 5.4.1). During this final stage, the charge of the particle changes by tens of elementary charges [486], but once the particle has undergone this sudden change, its charge state is preserved indefinitely even when the pressure is increased again because the water is gone.\\nA relatively simple method to control the charges on the particle is to apply an electrical discharge near the particle. The discharge provides free charges that can be adsorbed by the particle [523]. It can be applied through a bare wire of about 5 cm length at about 5 cm away from the optical trap. The grounded vacuum chamber serves as the counterelectrode. A DC voltage (∼ 7 kV) is sufficient to ionize gas molecules inside the vacuum chamber and to produce a corona discharge at moderate pressures (∼ 1 mbar). Note that an ion pressure gauge can produce also free charges, which lead to unwanted changes in the charge state of the particle, if it is not properly shielded [523]. The ionization process is triggered by random events such as absorption of a ultraviolet photon or cosmic rays that spontaneously ionize neutral air molecules. The free electron and positive ion are accelerated toward opposite directions by the high voltage, which also prevents them to recombine. The electron acquires enough kinetic energy to ionize other molecules, because of its small mass. The secondary electrons ionize more molecules, which then results in an electron avalanche.\\nEmission of (mainly) UV photons after electron-ion recombination causes the characteristic violet glow of the corona, as shown in Fig. 36. The corona discharge is positive or negative, depending on the polarity of the high DC voltage, which therefore allows to select the ratio of positive-to-negative ionized molecules. Eventually, free charges are adsorbed on the particle surface, thereby changing its net charge monotonically, except for few unfavorable events. These charges can be both electrons and ionized air molecules. Accordingly, by simply turning off the high-voltage source when the particle carries the desired number of charges, the levitated particle can be brought to any desired net-charge state from zero to a few elementary charges [486,523].\\nThe charge state of the particle is monitored in situ by driving the particle motion with an electric field at frequency ω d . The electric field is created by applying a voltage U(t) = U 0 cos(ω d t) across a capacitor that is centered around the particle. A simple capacitor can be formed by grounding the metal housing of the microscope objective and applying a voltage to the metal holder of the collection lens [523]. This method does not require any additional mechanical components in the setup and therefore does not obstruct access to the particle . Alternatively, dedicated electrodes that fit right at the sides of the optical trap can be beneficial to produce a more homogeneous field at the location of the particle [486], since field inhomogeneities can lead to a finite net charge density. When a voltage U 0 ∼ 10 V is applied to the capacitor at a frequency ω d close to the particle's resonance frequency, the particle's response to the driving field shows as a distinct peak in the single-sided PSD [486] on top of the thermal peak (see Figs. 41 and 43a):\\nwhere the thermal force spectral density is S ff = mk B T γ 0 /π, χ(ω) = Ω 2 0 − ω 2 + iωγ 0 −1 is the mechanical response function, F 0 = Q p E 0 , E 0 ∼ 500 V m −1 being the amplitude of the electric field modulation at the location of the particle, Q p is the charge on the particle, and 2τ is the duration of the time trace that is used to calculate the position spectral density S qq (ω) = 2τ|q| 2 (equation (179)). The detector signal is demodulated in quadrature with the drive. The demodulated signal can assume both positive and negative values, since the response to the driving field flips phase by π when the polarity of the charge changes. Positive (negative) signal amplitudes can then be associated with positive (negative) net charge, when the transfer functions of the electronics is accounted for. Note that for a small detuning of the drive frequency ω d from the particle's resonance, there is a small phase offset from the response function of the particle.\\nWhile the corona discharge is active, the demodulated signal features discrete steps, which are the signature of single elementary-charge transfers to and from the nanoparticle (see Fig. 43). Thus, any desired net-charge state from zero to a few elementary charges can be achievd by simply selecting the polarity of the high-voltage source, counting the number of steps, and turning off the high-voltage source when the particle carries the desired number of charges. Importantly, the particle's charge state is preserved when the pressure is reduced and stays indefinitely while the high-voltage source is turned off, even over a period of several days [523].\"\n },\n {\n \"title\": \"Mass Calibration\",\n \"content\": \"The exact value of the particle's inertial mass often plays an important role for precision measurements. For example, the common method for displacement calibration uses the thermal force noise as a reference, where the strength of the force depends on the mass, as we have seen in section 5.6.1. Likewise, nanomechanical mass measurements rely on knowing the mass of the bare resonator [524].\\nThe manufacturer's size specification gives a good first estimate of the particle size with an error up to ∼ 25%, which stems from the typical size variation (∼ 5%) and the uncertainty in the specified mass density (∼ 10%) of commercially available mono-disperse particles.\\nThe mass of individual particles can be inferred from kinetic gas theory, which relates the measured damping rate γ 0 to the particle size [512]. In particular, for a spherical particle and at pressures where the mean free path is much larger than the radius of the particle, there is a simple linear relationship [488],\\nwhich relates the pressure to the particle radius a (equation (169)). At room temperature the mean free path is ∼ 60 µm · P −1 gas mbar −1 . Hence, the linear approximation is generally valid for nanoparticles in vacuum. The calibration measurements are generally performed at ∼ 10 mbar, where individual peaks are clearly resolved and nonlinear broadening of the peaks is still negligible. To remove excess water from the particle, which leads to an overestimate of the actual particle size, the pressure should be cycled to below 0.1 mbar and back (section 5.4.1). From the measured radius, one can then infer the mass m = 4 3 πa 3 ρ p with a typical uncertainty of ∼ 35% [486].\\nThe mass measurement can be significantly improved either by an independent measurement Fig. 44. Mass calibration. Power spectral density S vv (ω) of a thermally and harmonically driven resonator at P = 50 mbar. The broad peak centered at Ω 0 /2π = 125 kHz corresponds to the thermally driven state. We fit it with a Lorentzian function (red) to extract S th vv (ω d ) together with g/(2π) = 31.8 kHz and the corresponding uncertainties. The narrowband peak at ω d /(2π) = 135 kHz, also shown in detail in the inset, depicts the electrical excitation from which we retrieve S d vv (ω d ). Gray data points at the bottom of the plot is the measurement noise, which is ∼ 40 dB below the particle's signal. Figure adapted from Ref. [486]. of the voltage calibration (equation (177)) or by applying a known force. For the former approach, the wavelength of the light provides a well-defined length scale that allows to relate the phase to a displacement in an interferometer. The interferometer is calibrated by changing the relative length of the two interferometer arms and measuring the fringe pattern. For the latter approach, electrical fields can be estimated quite accurately (∼ 1% error) from a numerical simulation of the electrode geometry [486]. Hence, the electrical force on the particle can be estimated with the same accuracy, because the number of charges can be determined exactly (section 5.6.3). Since the thermal force depends on the mass but the electrical force does not, their ratio (equation (188)),\\nprovides a direct method to determine the mass from the measured spectra (Fig. 44). In equation (190), S d vv (ω d ) is the peak of the measured spectrum at the drive frequency ω d when the sinusodial electrical force F el = F 0 cos(ω d t) is applied and S th vv (ω d ) is the thermal background at ω d (Fig. 44). Note that the spectra do not need to be calibrated to apply this procedure. The mass is then simply given by [486] m =\"\n },\n {\n \"title\": \"Internal temperature\",\n \"content\": \"In optical tweezers experiments, the main focus is on the external degrees of freedom of the particles in the trap (i.e., the translation of the particle and its rotation). However, any particle will absorb some of the trapping light, which raises its internal bulk temperature. The power absorbed by a particle with volume V from a laser beam with intensity I 0 is P abs = 6π I 0 λ VIm n 2 (λ) − 1 n 2 (λ) 2 + 2 .\\nFor example, for silica, the refractive index is n = 1.45 + i6 · 10 −8 at λ = 1064 nm. Note that absorption can be higher in nanoparticles due to additional absorbers such as defects and impurities. At high pressure, the particle efficiently thermalizes with the surrounding gas by convective cooling with cooling power P con (T bulk ) = −c acc 2 3π (πa 2 )P gas v rms γ sh + 1 γ sh − 1\\nwhere v rms = 3k B T gas /m gas is the root mean square velocity of the gas molecules, and γ sh = 7/5 is the specific heat ratio for an ideal diatomic gas. Clearly, as the pressure P gas decreases, convective cooling by the surrounding medium vanishes and the only heat exchange with the environment is radiative. The emitted thermal radiation is P em bb = −P bb (T bulk ), where P bb (T) = 72ζ(5) π 2 V c 3 4 Im n 2 bb − 1 n 2 bb + 2 (k B T) 5 (194) is the the black body radiated power for a dipolar emitter, being ζ(5) ≈ 1.04 the Riemann zeta function [413]. Conversely, the particle also absorbs thermal radiation from the environment at temperature T env with P abs bb = P bb (T env ). The particle bulk temperature can be found from the requirement that at equilibrium all contributions add up to zero, i.e., P abs + P abs bb + P con (T bulk + P em bb (T bulk ) = 0.\\nThe emitted or incident power due to black-body radiation depends on the dielectric properties of the particle in the entire black-body spectrum, and their calculation therefore involves the integration of the black-body emission rates over all emission frequencies. As an approximation, one assumes a constant permittivity of the particle in the entire spectral range of black-body emission. This can be estimated as the average value of the refractive index over the black-body spectrum. For example, for silica this gives n bb = 1.5 + i0.1.\\nThe internal temperature of a particle trapped in vacuum can raise significantly above the environmental temperature, to the point where the particle disintegrates if the absorption is too high [448] (Fig. 45). In fact, this problem has prevented optical levitation of nanodiamonds in high vacuum [447]. Even silica particles with low absorption can reach temperatures ∼ 1000 K [525]. As an example, figure 45d shows the internal temperature as a function of pressure. Fig. 45. Heating and temperatures of an optically levitated particle. (a) Damping and (b) heating rate as a function of pressure for feedback γ fb /2π = 20 Hz. (c) Center of mass temperature as a function pressure. At low pressures recoil heating dominates and the rates and the center of mass temperature become pressure independent. (d) Internal temperature as a function of pressure. At a few millibars, there is a sharp transition between convective cooling and radiative cooling via black body radiation and the internal temperature transitions from the ambient temperature to a much higher value. The internal temperature leads to a slightly higher center-of-mass heating rate visible as a small bump in (b). 5.6.6. Coupling between internal temperature and external degrees of freedom As we discussed earlier, the bulk temperature couples to the external degrees of freedom through the residual gas in the vacuum chamber (section 5.5). This coupling can be explained by a simple two-bath model, where the gas molecules that interact with the particle are divided into two families: molecules that impinge on the particle being in thermal equilibrium with the vacuum chamber, and molecules that emerge from the hot surface of the particle at a higher temperature. From equation (174), the thermal force noise increases with increasing bulk temperature. This allows to determine the internal bulk temperature from the fluctuation of the center-of-mass of the particle by solving T cm = T 3/2 gas + π 8 T 3/2 em T gas + π 8 √ T em (196) for T em , where the center-of-mass temperature is defined as T cm = Ω 2 0 v 2 /(Ck B ) , C being the energy calibration factor and v 2 the variance of the voltage fluctuation on the detector (section 5.6.1). Then, from T em one determines the temperature of the particle as T bulk = c −1 acc T em − T gas + T gas .\\nThe method described above requires the particle's center-of-mass to reach its equilibrium steady state. This is impractical at low pressures where the particle is easily lost without feedback stabilization. At low pressures, the thermal noise force from the interaction with the gas can be determined measuring the heating rate [525]. Thereby, the center-of-mass motion is first reduced with feedback cooling (section 5.7). When the feedback is switched off, the particle energy increases at a rate Γ = γ im T gas + γ em T em = γ im T 3/2 gas + π 8 T 3/2 em T gas . (a) Heating rates as a function of pressure extracted from relaxation measurements without additional heating of the internal temperature (blue) and with heating by a CO 2 laser intensity of 0.47 µW µm −2 (orange). The dashed lines are linear fits to the data points. Error bars are smaller than the marker size. (b) Heating rates at different intensities of the CO 2 laser measured at a pressure of 1 · 10 −5 mbar. The error bars indicate the standard deviation of the measurements. (c) Calculated temperature dependence of oscillation frequency. When the particle is heated with the CO 2 laser, its oscillation frequency increases due to changes in the particle's material properties. This leads to a nearly linear relation between relative frequency change and increase of the internal particle temperature. Adapted from Ref. [525].\\nWith feedback cooling the oscillation amplitude is low enough that cross-coupling between the degrees of freedom can be neglected. This justifies the approximation where the particle center-of-mass motion is treated as three independent harmonic oscillators. Each degree of motion obeys the following equation of motion (c.f. equation (160)):\\nwhere F fb (t) is the feedback force (equation (160)). For most purposes, the stochastic contribution F L and the bare damping γ 0 result from interactions with the background gas. Additional contributions are discussed in section 5.5. The feedback damps the particle motion at a rate (γ f b /2π). The additional damping reduces the fluctuations of the position of the particle. As a consequence, feedback cooling reduces the response time to external perturbations and increases the confinement of the particle. Since in a thermal equilibrium the variance of the position is proportional to the temperature of the environment, we define the effective temperature for the center-of-mass motion (equation (167)) as\\nwhich assumes that the particle position and energy distributions reflect the distributions of a harmonic oscillator in thermal equilibrium at temperature T cm . It is important to point out, however, that under the action of feedback the particle is not in thermal equilibrium and the assumption of a simple thermal distribution might be violated, in particular for nonlinear feedback schemes (section 5.7.2).\"\n },\n {\n \"title\": \"Force feedback\",\n \"content\": \"From an inspection of equation (201), it is apparent that a feedback term proportional to the particle velocity results in damping. Instead, a feedback term proportional to the particle position allows to optimize the transient times [417]. Thus, the feedback force of a linear feedback controller reads\\nwhere k d is the derivative feedback gain and k p is the proportional feedback gain. Purely velocity-dependent feedback (i.e., k p = 0) is also called cold damping. Experimentally such a force can be implemented with radiation pressure from a weakly focused beam. This technique was used in Ashkin and Dziedzic's original experiments to stabilize a particle against gravity [403], and later by Li et al. to feedback-cool microspheres in high vacuum [527]. The latter experiment required three laser beams in addition to the trapping laser to exert radiation pressure forces along the three spatial dimensions.\\nMore recent experiments use electrostatic forces instead [415,417,528]. Electrodes near the optical trap apply a force F Q = Q p E, where Q p is the total charge on the particle and E the electric field at the location of the particle. Often the electrode geometry is chosen such that they form a plate capacitor around the particle so that the field is homogeneous and given by E = (V elec /d elec ) n elec , where V elec is the applied voltage, d elec the distance between the electrodes and n elec a unit vector pointing from one electrode to the other. The electrostatic force along direction e i is then given by F Q = F Q · e i and the electrode geometry can be chosen such that there is a force along all degrees of freedom. Implementation of the velocity-dependent force requires to measure the instantaneous velocity of the particle. In practice, however, this is achieved with a bandpass filter that acts on the position time trace q(t) and produces a time delay such that the signal is in phase with the velocity q(t) [415]. For example if the particle motion is approximately harmonic q(t) ∼ sin(Ω 0 t), a delay τ = π/(2Ω 0 ) leads to a signal ∝ cos(Ω 0 t) that is proportional to the particle velocity q(t).\\nIn the frequency domain, the equation of motion with linear feedback reads q −ω 2 + iωγ 0 + Ω 2 0 = m −1 F L +f fb ,\\nwhere the tilde indicates the Fourier transform andf fb (ω) = H(ω) [q +q n ] with filter funcion H(ω) = −ik d ω − k p . Explicitly accounting for the noise in the measurement of the position q n we arrive atq (ω) = m −1 χ fb (ω)F L − χ fb (ω) Ω 2 fb + iωγ fb q n (205) where the susceptibility χ fb (ω) = Ω 2 0 + Ω 2 fb − ω 2 + iω(γ 0 + γ fb ) −1 , and we introduced the feedback damping γ fb = k d /m and frequency shift Ω 2 fb = k p /m. The in-loop detector signal is q IL =q +q n . Hence, the measured in-loop two-sided PSD is [415,417] S IL qq (ω) = | χ fb | 2 m −2 S ff + | χ fb | 2 (Ω 2 0 − ω 2 ) 2 + ω 2 γ 2 0 S nn (206) = m −2 S ff + (Ω 2 0 − ω 2 ) 2 + ω 2 γ 2 0 S nn (Ω 2 0 + Ω 2 fb − ω 2 ) 2 + (γ 0 + γ fb ) 2 ω 2 ,\\nwhere S qq = |q| 2 and S nn is the noise PSD of the in-loop detector (Fig. 47). The in-loop measurement noise is reintroduced into the feedback loop, which drives the particle. The correlations between the particle's position and the measurement noise lead to noise squashing, where the apparent signal is squashed below the noise floor [529] (Fig. 47b).\\nIn contrast, an out-of-loop detector provides an independent measurement of the actual particle motionq OoL =q +q ν :\\nS OoL qq (ω) = | χ fb | 2 m −2 S ff + | χ fb | 2 Ω 4 fb + ω 2 γ 2 fb S nn + S νν (208) = m −2 S ff + Ω 4 fb + ω 2 γ 2 fb S nn (Ω 2 0 + Ω 2 fb − ω 2 ) 2 + (γ 0 + γ fb ) 2 ω 2 + S νν ,\\nwhere S νν is the noise PSD of the out-of-loop detector (Fig. 47).\\nIntegration over the PSD yields the position variance40\\nIn the approximation we set k p = Ω fb = 0 and γ fb γ 0 and we used S ff = k B T mγ 0 /π. The first term in equation (210) scales with the inverse of the feedback cooling rate. This term resembles the desired action of the feedback, which is to reduce the impact of the heating term given by the fluctuating force. The second term is proportional to the feedback damping rate, which is multiplied by the measurement noise. This term resembles the undesired but inevitable effect of the control-loop heating the particle by feeding back measurement noise. Consequently, there exists an optimum feedback cooling rate γ opt = S ff /S nn /(mΩ 0 ) , where the mode temperature reaches its minimum value q 2 min = 2π √ S ff S nn /(mΩ 0 ) . With cold damping, optically levitated nanoparticles have been cooled to 4 occupational quanta, corresponding to an effective temperature of 11 µK [418] (Fig. 47(b) and Fig. 42).\\nClassically, the measurement is not perturbative and the position variance can be reduced to arbitrarily low values, because, at least in principle, the noise S nn can be made arbitrarily small (equation (210)). However, quantum mechanics imposes that every measurement is accompanied by a measurement back-action, leading to a finite contribution S nn even in an ideal measurement. In the case of an optical measurement this back action is the random arrival of photons and the resulting random momentum kicks that lead to a stochastic backaction force [416]. Therefore, in a regime where the measurement back-action is the dominant contribution to the force noise, the impression back-action product is S ff S nn = ( /(4π)) 2 /η det , where η det < 1 is the measurement efficiency [415,519]. The impression back-action product is a manifestation of the Heisenberg uncertainty principle and describes the tradeoff between measurement imprecision and measurement back-action. At the optimal feedback gain, the effective phonon occupation number n cm = mΩ 0 q 2 min / − 1/2 depends only on the detection efficiency as n min = (η −1/2 det − 1)/2. At the Heisenberg limit (η det = 1), when the imprecision noise is minimized by optimally detecting all photons scattered by the levitated particle, the particle's center-of-mass motion could, in principle, be brought to its quantum ground state n min = 0. Hence, to reach the ground state with feedback cooling, the experiment should be operated in the photon-recoil-limited regime (P gas ≤ 10 −8 mbar) and the collection of scattered photons needs to be optimized. A promising approach is to collect the scattered photons with a high-finesse optical cavity, which collects a fraction of (η P /(η P + 1)) of the overall scattered power due to the Purcell effect, where η P = (6F c λ 2 )/(π 3 w 2 0 ) is the Purcell factor for a cavity with finesse F c and beam waist w 0 [530].\"\n },\n {\n \"title\": \"Parametric feedback cooling\",\n \"content\": \"While cooling levitated particles with linear feedback is very powerful, its experimental implementation along all three motional degrees of freedom is challenging. All-optical linear feedback requires three auxiliary laser beams to apply radiation pressure in all directions [527]. Electrical feedback relies on a finite net charge on the levitated object.\\nThe method of choice to control charge-neutral optically levitated particle is parametric feedback cooling [407], which allows to control all three spatial degrees of freedom with the same laser that is used for trapping, thereby significantly reducing the experimental complexity.\\nThe basic concept of parametric feedback is to introduce a modulation in time of the intensity of the trapping laser. In particular, applying a modulation δI(t) = I 0 (η para /Ω 0 )q q results in the parametric feedback force F param fb = −k para q 2 q,\\nwhere k para = Ω 0 mη para is the parametric feedback gain. When the particle moves away from the trap center, the intensity modulation δI(t) stiffens the trap. When the particle falls back to the trap center the feedback softens the trap. Thus, by synchronizing the laser intensity modulation with the particle motion energy is extracted from the center-of-mass motion of the particle [488]. This principle is analogous to the way a child on a swing gains motional energy by modulating its center-of-mass. Constructing the feedback signal as in equation (211) provides exactly the right phase relation, such that the trap stiffens when the particle moves away from the trap center and softens when is falls back, thereby cooling the particle. However, latencies in the experimental implementation require an additional phase shifter to compensate for unwanted phase shifts. With an additional phase shift we can also amplify the particle motion. Note that, in the absence of active feedback, the particle's oscillation self-locks (entrains its phase) to the modulation in such a way that the motion is amplified [531]. Cooling therefore requires active stabilization of the modulation phase.\\nExperimentally, the detector signal is first frequency-doubled and then phase-shifted, where the right phase is determined empirically by optimizing the cooling performance, i.e., minimizing the position variance q 2 . The QPD or split-detection measurement yields a position signal for each spatial direction. The feedback signal is produced for each axis independently and the three signals are summed together, which allows to effectively cool all spatial degrees of freedom with a single laser beam (section 5.3). For a particle oscillating at frequency Ω 0 with q(t) ∝ cos(Ω 0 t), parametric feedback generates a signal ∝ sin(2Ω 0 ). Thus, the feedback signal appears on the in-loop detector signal predominantly at 2Ω 0 . The contribution to the detector signal at Ω 0 is much smaller, on the order of ∼ η para q 2 and, thus, much smaller than the contribution from the particle motion q 2 . Hence, noise squashing of the in-loop detector signal does not occur and the trapping laser can also be used for particle detection.\\nDue to the nonlinear nature of the parametric feedback, there is no closed expression for the detector PSD. In particular, the position distribution is not a thermal distribution because large amplitude fluctuations are damped more strongly than small fluctuations [481,532]. However, we can still assign an effective temperature to the center-of-mass motion k B T cm = mΩ 2 0 q 2 . For strong feedback (η para mΩ 0 γ 0 /(k B T) ), the position variance is [481,532] q 2 = 4k B T γ 0 πη para mΩ 3 0 .\\nIn contrast to linear-feedback cooling, the effective temperature or position variance scales with the square root of the feedback gain and not linearly. This can be understood intuitively. The Fig. 47. Feedback cooling. (a) Single-sided power spectral densitiesS Ool yy of the motion of the nanoparticle measured by the out-of-loop detector for different feedback damping rates γ fb . The solid lines are Lorentzian fits to the data. The gray data points denote the measured shotnoise level on the out-of-loop detector. (b) Power spectral densitiesS IL yy measured by the in-loop detector for the same settings as in (a). In contrast to (a), for a large feedback gain the measured signal drops below the noise floor because of correlations between the detector noise and the feedback signal (noise squashing) . (c) Mode temperature T y derived from the out-of-loop signal as in (a) function of feedback gain γ fb . The red (blue) circles denote the measured values at a pressure of 1.4 · 10 −8 mbar (1.2 · 10 −7 mbar. (d) Steady state under parametric PLL feedback cooling. Mean occupation number along the three principal axes (x, y, z) as a function of gas pressure measured under constant feedback cooling. At low enough pressures, photon recoil becomes the main source of heating and therefore the occupation number remains constant. Adapted from Ref. [415] (a-c) and from Ref. [416] (d). modulation depth and therefore the feedback force is proportional to the energy of the particle. Hence, as the feedback reduces the particle motion it also becomes less effective. Nonetheless, the quantum mechanical theory suggests that ground state-cooling with parametric feedback cooling is possible [533], even though direct feedback seem to be more promising [534].\"\n },\n {\n \"title\": \"Parametric PLL cooling\",\n \"content\": \"Parametric feedback extracts energy from or pumps energy into the particle center-of-mass motion by modulating the trapping laser and therefore the trap stiffness at twice the particle oscillation frequency. A phase locked loop (PLL) synthesizes an output signal whose phase is related to the phase of a sinusoidal input signal. Thereby, a PLL can track an input frequency and it can generate a frequency that is an integer multiple of the input frequency. Hence, tracking the detector signal with a PLL and feeding the PLL output at the second harmonic into the optical modulator results in a feedback term\\nwhere k PLL = mΩ 2 0 η PLL is the PLL feedback gain and η PLL is the modulation depth. The PLL feedback loop keeps the phase φ PLL fixed with respect to the randomly changing phase of the particle oscillation. Similarly to the case of the q q parametric feedback, a phase of φ PLL = −π/4 reduces the oscillation amplitude. However, in contrast to the previous case, where the modulation depth is proportional to the variance of the particle position, the PLL feedback has a constant amplitude. As a consequence, the position variance with PLL parametric feedback is\\nThe approximation holds for φ PLL = −π/4 and in the limit η PLL Ω 0 /2γ 0 1. Comparing equation (214) with equation (202), we can identify the PLL damping as γ PLL = η PLL Ω 0 /2. Interestingly, even though parametric feedback with a PLL is a nonlinear feedback, we find the same linear scaling as for linear feedback and also the energy distribution follows the Boltzmann distribution associated with a harmonic oscillator at thermal equilibrium with a bath at temperature T cm = (γ 0 /γ PLL )T [532]. Like in the case of linear feedback, noise in the feedback will ultimately limit how far one can cool. However, unlike in the case of linear feedback (equation (210)), there is no analytical solution for the minimum temperature for parametric feedback. Using PLL parametric feedback simultaneous cooling of all three motional degrees of freedom to millikelvin temperatures (n below 100) has been demonstrated [416] (Fig. 47(d)).\"\n },\n {\n \"title\": \"Outlook\",\n \"content\": \"Levitated particles have already opened the door to a wide range of interesting physics in the classical regime and are now entering into the quantum regime. The next challenges are to achieve ground state cooling for all three center-of-mass translational degrees of freedom, to control the librational modes and precession of the particle, and to preparte truely non-classical states.\\nEven though our discussion has been focused on optical tweezers in high vacuum, their combination with cavity optomechanics will play an important role towards creating more exotic quantum mechanical states beyond the ground state. Once a strong cavity optomechanical interaction has been realized, a range of traditional quantum optics experiments can be realized with these massive particles such as quantum state transfer, quantum squeezing, entanglement, and teleportation [413,432,436].\\nThe envisioned fundamental tests of quantum mechanics require to control the internal temperature of levitated particles, since high internal temperature will wash out quantum interference effects [535]. The internal temperature can be reduced with optical refrigeration that exploit optical dopants inside the particle [476,536]. Dopants can also be used to directly measure the particle temperature [537]. Experimental data of the internal dynamics of levitated particles could challenge our understanding of thermalization dynamics on the nanoscale [538] and it will be important to measure what is going on inside the particle.\\nThere is still plenty of room in the classical regime, where levitated particles will allow to test fundamental questions in single particle thermodynamics in the underdamped regime and provide new insights into the nonlinear dynamics of nanomechanical resonators. Most attention has been given to the center-of-mass degrees of freedom and simple Gaussian beams. There remains a lot to be explored about the remaining degrees of freedom and complex light fields, for example beams carrying orbital angular momentum [539]. In addition, new materials, material combinations in the form of doped particles or composite particles and new shapes will enable unprecedented light matter interactions [489]. Optical levitation experiments will hugely benefit from the tools that have been developed for traditional optical tweezers experiments, such as structured light fields [540] and holographic tweezers [43,541]. Those tools will then also allow to study collective particle dynamics, for example optical binding and forces between levitated particles [11]. This will be required the development of new techniques and calibration methods that are able to resolve the fast particle dynamics in high vacuum at a high spatial resolution over a large field of view.\\nCareful calibration with a solid understanding of all the statistical and systematic error sources will be paramount for precision measurements with levitated particles. In particular, experiments aiming at detecting near field forces at interfaces require to approach a surface with a levitated particle [542]. The surface will scatter the trapping light and, thus, these experiments need detection and calibration techniques that are robust in extreme experimental conditions that might even change over the course of a measurement. We thank Erik Hebestreit, Vijay Jain, Martin Frimmer, Felix Tebbenjohanns, Francesco Ricci for providing us with data and feedback and the Quidant and Novotny groups for stimulating discussions. IPC gratefully thanks Édgar Roldán for illuminating discussions about Carnot cycles and microscopic engines.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 9 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 12 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 14 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 16 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 17 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"18 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 19 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 21 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 22 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 23 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 24 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 25 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 26 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 27 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 29 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 30 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 31 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 32\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 33 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 34 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 36 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 37 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 39 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 40 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 41 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 42 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 43 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 46 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Funding\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 11 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.4 Carnot cycles and Maxwell's demons . . . . . . . . . . . . . . . . . . 84 4.5.5 Other applications to statistical physics and outlook . . . . . . . . . . . 89 4.6 Transport in extended potential landscapes . . . . . . . . . . . . . . . . . . . . 89 4.6.1 Optical forces and potentials in extended light patterns . . . . . . . . . 90 4.6.2 Transport mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.6.3 Other transport mechanisms and applications . . . . . . . . . . . . . . 98 5 Optical tweezers in vacuum 100 5.1 Applications of optical levitation. . . . . . . . . . . . . . . . . . . . . . .. . 100 5.2 Dynamics in the underdamped regime . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Optical tweezers setup for vacuum operation . . . . . . . . . . . . . . . . . . . 103 5.3.1 Particle types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.2 Optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.4 Feedback electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.5 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Particle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.4.1 Nebuliser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.2 Dry approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Dissipation and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.1 Gas damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5.2 Radiation damping and shot noise . . . . . . . . . . . . . . . . . . . . 114 5.5.3 Displacement noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.4 Laser intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.1 Detector calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.2 Sideband thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.6.3 Charge calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6.4 Mass Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6.5 Internal temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.6.6 Coupling between internal temperature and external degrees of freedom 124 5.7 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.7.1 Force feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.7.2 Parametric feedback cooling . . . . . . . . . . . . . . . . . . . . . . . 129 5.7.3 Parametric PLL cooling . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131","content":"Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.4 Carnot cycles and Maxwell's demons . . . . . . . . . . . . . . . . . . 84 4.5.5 Other applications to statistical physics and outlook . . . . . . . . . . . 89 4.6 Transport in extended potential landscapes . . . . . . . . . . . . . . . . . . . . 89 4.6.1 Optical forces and potentials in extended light patterns . . . . . . . . . 90 4.6.2 Transport mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.6.3 Other transport mechanisms and applications . . . . . . . . . . . . . . 98 5 Optical tweezers in vacuum 100 5.1 Applications of optical levitation. . . . . . . . . . . . . . . . . . . . . . .. . 100 5.2 Dynamics in the underdamped regime . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Optical tweezers setup for vacuum operation . . . . . . . . . . . . . . . . . . . 103 5.3.1 Particle types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.2 Optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.4 Feedback electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.5 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Particle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.4.1 Nebuliser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.2 Dry approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Dissipation and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.1 Gas damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5.2 Radiation damping and shot noise . . . . . . . . . . . . . . . . . . . . 114 5.5.3 Displacement noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.4 Laser intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.1 Detector calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.2 Sideband thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.6.3 Charge calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6.4 Mass Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6.5 Internal temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.6.6 Coupling between internal temperature and external degrees of freedom 124 5.7 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.7.1 Force feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.7.2 Parametric feedback cooling . . . . . . . . . . . . . . . . . . . . . . . 129 5.7.3 Parametric PLL cooling . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131"},{"id":"","caption":"Trappable objects. A broad range of objects have been trapped using optical tweezers, going from single atoms and molecules to microparticles and microorganisms.","content":"Fig. 1 .\n1Trappable objects. A broad range of objects have been trapped using optical tweezers, going from single atoms and molecules to microparticles and microorganisms."},{"id":"","caption":"Basic optical tweezers setup. A basic optical tweezers setup is comprised of three parts: the trapping optics, the imaging optics, and the position detection optics. DM: Dichroic mirror. M: mirror. O: microscope objective. C: Condenser. L 1 , L 2 , L 3 , L 4 : lenses. LS: Light source.","content":"Fig. 2 .\n2Basic optical tweezers setup. A basic optical tweezers setup is comprised of three parts: the trapping optics, the imaging optics, and the position detection optics. DM: Dichroic mirror. M: mirror. O: microscope objective. C: Condenser. L 1 , L 2 , L 3 , L 4 : lenses. LS: Light source."},{"id":"","caption":"Brownian motion in an optical trap. (a) Trajectory of an optically trapped Brownian particle x(t) and corresponding (b) optical potential and (c) probability density. The equilibrium position is x eq = 0 µm. The experimental parameters are in Tab. 1.","content":"Fig. 3 .\n3Brownian motion in an optical trap. (a) Trajectory of an optically trapped Brownian particle x(t) and corresponding (b) optical potential and (c) probability density. The equilibrium position is x eq = 0 µm. The experimental parameters are in Tab. 1."},{"id":"","caption":"Polarization-splitting optical tweezers. A typical Mach-Zehnder setup with polarizing beam splitters (PBS 1 and PBS 2 ) can be used for this purpose. Mirrors GM","content":"Fig. 5 .\n5Polarization-splitting optical tweezers. A typical Mach-Zehnder setup with polarizing beam splitters (PBS 1 and PBS 2 ) can be used for this purpose. Mirrors GM"},{"id":"","caption":"Interferometric optical traps. (a) Interferometric optical trap based on a Mach-Zehnder interferometer. (b-c) Different alternatives for the interferometric module. BS: Nonpolarizing beam splitter. RP: Right-angle prism. BB: beam block.","content":"Fig. 7 .\n7Interferometric optical traps. (a) Interferometric optical trap based on a Mach-Zehnder interferometer. (b-c) Different alternatives for the interferometric module. BS: Nonpolarizing beam splitter. RP: Right-angle prism. BB: beam block."},{"id":"","caption":"Holographic optical tweezers. A spatial light modulator (SLM) alters the beam profile of an incoming optical beam This beam is then relayed by a series of lenses to an objective that then generates the desired light pattern at the sample. The SLM can be either placed (a) in a plane conjugated to the back focal plane of the objective, or (b) in a plane conjugated with the image plane of the objective.","content":"Fig. 8 .\n8Holographic optical tweezers. A spatial light modulator (SLM) alters the beam profile of an incoming optical beam This beam is then relayed by a series of lenses to an objective that then generates the desired light pattern at the sample. The SLM can be either placed (a) in a plane conjugated to the back focal plane of the objective, or (b) in a plane conjugated with the image plane of the objective."},{"id":"","caption":"Phase masks and corresponding light intensity. Diffractive optical elements displayed on a phase-only SLM for the generation of: (a) a single optical trap laterally and axially shifted; (b) multiple traps distributed in space (the different brightnesses and sizes of the spots are due to the different positions of the corresponding traps along the vertical z-position); (c) a Laguerre-Gaussian beam of topological charge l = −2 laterally shifted; (d) a Bessel beam of topological charge l = 3.","content":"Fig. 9 .\n9Phase masks and corresponding light intensity. Diffractive optical elements displayed on a phase-only SLM for the generation of: (a) a single optical trap laterally and axially shifted; (b) multiple traps distributed in space (the different brightnesses and sizes of the spots are due to the different positions of the corresponding traps along the vertical z-position); (c) a Laguerre-Gaussian beam of topological charge l = −2 laterally shifted; (d) a Bessel beam of topological charge l = 3."},{"id":"","caption":"=1 recorded time series of particle's position during the m-th experiment with N samples {x (m) } L =1 uncorrelated time series at time intervals ∆t with ∆t = r∆t and r an integer κ (ex) x experimental estimate of the trap stiffness τ (ex) x experimental estimate of characteristic relaxation time D (ex) , D (th) experimental estimate and theoretical value of diffusion constant γ (ex) , γ (th) experimental estimate and theoretical value of friction coefficient T s total acquisition time for each experiment m","content":"=1 recorded time series of particle's position during the m-th experiment with N samples {x (m) } L =1 uncorrelated time series at time intervals ∆t with ∆t = r∆t and r an integer κ (ex) x experimental estimate of the trap stiffness τ (ex) x experimental estimate of characteristic relaxation time D (ex) , D (th) experimental estimate and theoretical value of diffusion constant γ (ex) , γ (th) experimental estimate and theoretical value of friction coefficient T s total acquisition time for each experiment m"},{"id":"","caption":"Autocorrelation function (ACF) analysis. Comparison between the theoretical expression","content":"Fig. 12 .\n12Autocorrelation function (ACF) analysis. Comparison between the theoretical expression"},{"id":"","caption":"Single-molecule pulling experiments: miniTweezers setup. Schematic representations (a) of the miniTweezers setup, (b) of the microfluidics chamber (the flow goes from left to right, and the laser beams propagate perpendicularly to the surface of the chamber), (c) of the assembly procedure for the construction of the chamber, and (d) of the molecular configuration for the DNA hairpin pulling experiments (where the DNA hairpin is held between a bead held by a micropipette and a bead captured by the optical trap, depicted as an harmonic potential). Note that x h2 , x d , x h1 , and x b are distances along the y-axis.","content":"Fig. 14 .\n14Single-molecule pulling experiments: miniTweezers setup. Schematic representations (a) of the miniTweezers setup, (b) of the microfluidics chamber (the flow goes from left to right, and the laser beams propagate perpendicularly to the surface of the chamber), (c) of the assembly procedure for the construction of the chamber, and (d) of the molecular configuration for the DNA hairpin pulling experiments (where the DNA hairpin is held between a bead held by a micropipette and a bead captured by the optical trap, depicted as an harmonic potential). Note that x h2 , x d , x h1 , and x b are distances along the y-axis."},{"id":"","caption":"AD beads. The protocol for the preparation of the AD beads consists of three steps: (1) exchange the preservation buffer; (2) attach the anti-digoxigenin to the G-protein; (3) exchange the crosslinking buffer. The procedure (∼ 90 min) for the preparation of 1 ml of AD beads (which should be sufficient to perform experiments for several months) is: 1. (The first time the anti-digoxigenin batch is dissolved.) Add 200 µl of PBS (pH 7.4) to dilute the dry anti-digoxigenin (sheep polycolonal anti-dig antibody, Roche 1333 089). 2. Prepare the dimethyl pimelimidate (DMP) crosslinker buffer by mixing 50 mg DMP and 200 µl antibody crosslinker buffer (pH 7.4). The DMP crosslinker buffer needs to be freshly prepared every time the AD beads are synthesized.","content":"AD beads. The protocol for the preparation of the AD beads consists of three steps: (1) exchange the preservation buffer; (2) attach the anti-digoxigenin to the G-protein; (3) exchange the crosslinking buffer. The procedure (∼ 90 min) for the preparation of 1 ml of AD beads (which should be sufficient to perform experiments for several months) is: 1. (The first time the anti-digoxigenin batch is dissolved.) Add 200 µl of PBS (pH 7.4) to dilute the dry anti-digoxigenin (sheep polycolonal anti-dig antibody, Roche 1333 089). 2. Prepare the dimethyl pimelimidate (DMP) crosslinker buffer by mixing 50 mg DMP and 200 µl antibody crosslinker buffer (pH 7.4). The DMP crosslinker buffer needs to be freshly prepared every time the AD beads are synthesized."},{"id":"","caption":"Molecular buffer.A typical molecular buffer for DNA pulling experiments consists of 10 mM Tris pH 7.513 (Trizma, Sigma Aldrich), 10 mM EDTA14 (EDTA, Sigma Aldrich), 1 M NaCl, and 0.01% NaN 3 (sodium azide, to avoid bacterial growth). To facilitate the molecular buffer preparation, it is convenient to prepare stocks of:1 M Tris pH 7.5 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 6.05 g of Tris-HCl. Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add a solution of 25% HCl until the solution reaches pH 7.5. For longer storage, a final auto-cleavage can be performed. EDTA 0.5 M pH 8.0 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 7.306 g of EDTA. Add NaOH to the solution (the EDTA does not dissolve in water if pH<7.5). Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add NaOH until the solution reaches pH 8.0. 5 M NaCl Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 14.49 g of NaCl. Dissolve using a magnetic mixer and heat the tube to facilitate it. Add Milli-Q water until reaching 50 ml. 1% NaN 3 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 0.5 g of NaN 3 . Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. After the stocks have been prepared, for preparing 50 ml of the molecular buffer: Pour ∼ 30 ml of Milli-Q water in a 50 ml Falcon tube. Add 10 ml of the 5 M NaCl stock, 1 ml of the 0.5 M EDTA pH 8.0 stock, 0.5 ml of the 1 M Tris pH 7.5 stock, and 0.5 ml of the 1% NaN 3 stock. Add Milli-Q water until reaching 50 ml of volume. Mix the molecular buffer and filter it (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International) introducing the filtered solution in a new 50 ml Falcon tube. Incubation of the beads. To finalize the preparation of the beads, incubate in an Eppendorf tube (∼ 1.5 ml) 1 µl of a solution of the biomolecule of interest15 mixed with 14 µl of buffer (where the experiments are going to be performed) and 5 µl of the previously prepared AD beads. After ∼ 25 min have passed, add 1 ml of the molecular buffer.","content":"Molecular buffer.A typical molecular buffer for DNA pulling experiments consists of 10 mM Tris pH 7.513 (Trizma, Sigma Aldrich), 10 mM EDTA14 (EDTA, Sigma Aldrich), 1 M NaCl, and 0.01% NaN 3 (sodium azide, to avoid bacterial growth). To facilitate the molecular buffer preparation, it is convenient to prepare stocks of:1 M Tris pH 7.5 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 6.05 g of Tris-HCl. Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add a solution of 25% HCl until the solution reaches pH 7.5. For longer storage, a final auto-cleavage can be performed. EDTA 0.5 M pH 8.0 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 7.306 g of EDTA. Add NaOH to the solution (the EDTA does not dissolve in water if pH<7.5). Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add NaOH until the solution reaches pH 8.0. 5 M NaCl Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 14.49 g of NaCl. Dissolve using a magnetic mixer and heat the tube to facilitate it. Add Milli-Q water until reaching 50 ml. 1% NaN 3 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 0.5 g of NaN 3 . Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. After the stocks have been prepared, for preparing 50 ml of the molecular buffer: Pour ∼ 30 ml of Milli-Q water in a 50 ml Falcon tube. Add 10 ml of the 5 M NaCl stock, 1 ml of the 0.5 M EDTA pH 8.0 stock, 0.5 ml of the 1 M Tris pH 7.5 stock, and 0.5 ml of the 1% NaN 3 stock. Add Milli-Q water until reaching 50 ml of volume. Mix the molecular buffer and filter it (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International) introducing the filtered solution in a new 50 ml Falcon tube. Incubation of the beads. To finalize the preparation of the beads, incubate in an Eppendorf tube (∼ 1.5 ml) 1 µl of a solution of the biomolecule of interest15 mixed with 14 µl of buffer (where the experiments are going to be performed) and 5 µl of the previously prepared AD beads. After ∼ 25 min have passed, add 1 ml of the molecular buffer."},{"id":"","caption":"biotin-AGT TAG TGG TGG AAA CAC AGT GCC AGC GCG CGA GCC ATA AT-3 CD4-HandleB 5 -Phos-CTC ATC TGG AAA CAG ATG AGA TTA TGG CTC GCG ACT TCA CTA ATA CGA CTC ACT ATA GGG A -3 splint 5 -GCG CTG GCA CTG TGT TTC CAC CAC TAA CT-3","content":"biotin-AGT TAG TGG TGG AAA CAC AGT GCC AGC GCG CGA GCC ATA AT-3 CD4-HandleB 5 -Phos-CTC ATC TGG AAA CAG ATG AGA TTA TGG CTC GCG ACT TCA CTA ATA CGA CTC ACT ATA GGG A -3 splint 5 -GCG CTG GCA CTG TGT TTC CAC CAC TAA CT-3"},{"id":"","caption":"Oligonucleotide tailing reaction. Mix all the items in an Eppendorf tube and keep it at 37 • C for 15 min. After the reaction is finished, add 1 to 2 µl of 0.5 M EDTA pH 8.0 to quench the reaction. (*) Included in the Terminal transferase kit from Sigma Aldrich.","content":"6 .\n6Oligonucleotide tailing reaction. Mix all the items in an Eppendorf tube and keep it at 37 • C for 15 min. After the reaction is finished, add 1 to 2 µl of 0.5 M EDTA pH 8.0 to quench the reaction. (*) Included in the Terminal transferase kit from Sigma Aldrich."},{"id":"","caption":"Free-energy of hairpin formation. (a) Hairpin pulling trajectories shown as force-distance curves. The unfolding cycles are plotted in red and the folding cycles in blue. The black dots indicate the initial (L 0 , f min ) and final (L 1 , f max ) points considering for computing the work W FU . Since the L values are obtained from the light position at the position-PSD, the L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero (which corresponds to the center of the position-PSD). (b) Probability density functions (PDFs) of the work, computed using equation(75), for the unfolding (W U , red histograms) and folding (W F , blue histograms) cycles. The value at which the probabilities cross (dashed vertical line) is the free energy difference between the folded and unfolded states ∆G FU . This value is obtained as the crossing point between the lines interpolating the histograms (blue and red solid lines).","content":"Fig. 16 .\n16Free-energy of hairpin formation. (a) Hairpin pulling trajectories shown as force-distance curves. The unfolding cycles are plotted in red and the folding cycles in blue. The black dots indicate the initial (L 0 , f min ) and final (L 1 , f max ) points considering for computing the work W FU . Since the L values are obtained from the light position at the position-PSD, the L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero (which corresponds to the center of the position-PSD). (b) Probability density functions (PDFs) of the work, computed using equation(75), for the unfolding (W U , red histograms) and folding (W F , blue histograms) cycles. The value at which the probabilities cross (dashed vertical line) is the free energy difference between the folded and unfolded states ∆G FU . This value is obtained as the crossing point between the lines interpolating the histograms (blue and red solid lines)."},{"id":"","caption":"(b) is obtained by taking L 0 = −140 nm ( f min = 11.35 pN) and L 1 = −60 nm ( f max = 17.16 pN). Note that L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero. The crossing point of the histograms gives ∆G FU = 348.67 k B T (with 1k B T = 4.114 pN nm at T = 25 • C).","content":"(b) is obtained by taking L 0 = −140 nm ( f min = 11.35 pN) and L 1 = −60 nm ( f max = 17.16 pN). Note that L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero. The crossing point of the histograms gives ∆G FU = 348.67 k B T (with 1k B T = 4.114 pN nm at T = 25 • C)."},{"id":"","caption":"1 M K phosphate buffer at 7.4 pH is obtained by mixing 40.1 ml of 1 M K 2 HPO 4 and 9.9 ml of 1 M KH 2 PO 4 . The 1 M Na Phosphate buffer at 7.4 pH is obtained by mixing 38.7 ml of 1 M Na 2 HPO 4 and 11.3 ml of 1 M NaH 2 PO 4 . All salts are available from Sigma-Aldrich.","content":"1 M K phosphate buffer at 7.4 pH is obtained by mixing 40.1 ml of 1 M K 2 HPO 4 and 9.9 ml of 1 M KH 2 PO 4 . The 1 M Na Phosphate buffer at 7.4 pH is obtained by mixing 38.7 ml of 1 M Na 2 HPO 4 and 11.3 ml of 1 M NaH 2 PO 4 . All salts are available from Sigma-Aldrich."},{"id":"","caption":"Fill the syringe connected to the dispenser tube selected in the previous point with the bead solution. Suck about half of the Eppendorf tube content with the syringe. 4. Fill the other syringe (upper/lower channel) with RBC solution. Suck about half of the Eppendorf tube content with the syringe.","content":"3 .\n3Fill the syringe connected to the dispenser tube selected in the previous point with the bead solution. Suck about half of the Eppendorf tube content with the syringe. 4. Fill the other syringe (upper/lower channel) with RBC solution. Suck about half of the Eppendorf tube content with the syringe."},{"id":"","caption":"Construction of the cellular configuration. Images of different steps to obtain the RBC configuration required for the experiment: (a) the RBC is attached to a bead and partially trapped by the lasers; (b) the bead is trapped, while the RBC is not trapped; (c) the second bead is attached to the other side of the RBC; (d) final experimental configuration at zero reading for the force along the y-direction. The diameter of the beads is 3 µm. The red dot represents the position of the optical trap. The red arrow in (a) and (b) represents the presence and direction of the fluid flow.","content":"Fig. 17 .\n17Construction of the cellular configuration. Images of different steps to obtain the RBC configuration required for the experiment: (a) the RBC is attached to a bead and partially trapped by the lasers; (b) the bead is trapped, while the RBC is not trapped; (c) the second bead is attached to the other side of the RBC; (d) final experimental configuration at zero reading for the force along the y-direction. The diameter of the beads is 3 µm. The red dot represents the position of the optical trap. The red arrow in (a) and (b) represents the presence and direction of the fluid flow."},{"id":"","caption":"Do not turn off the pump flow in the central channel. To minimize the possibility of losing the bead attached to the RBC fixed on the tip of the pipette, do the following steps as quickly as possible. 19. Halve the flow to facilitate the trapping of the second bead. Check that the RBC stays separated from the bead, if this is not the case, increase the flow again to separate the cell from the bead. 20. Move the optical trap to the exit of the bead dispenser tube. 21. Flow the bead solution and try to trap a bead. It will be harder due to the RBC buffer flow on the central channel. To catch a bead with the pump on, it is easier to stay at the exit of the dispenser tube without moving the trap. Wait for the bead to fall into the optical trap instead of actively moving the trap around. 22. Once you have trapped a bead, move the chamber to the working area.","content":"18 .\n18Do not turn off the pump flow in the central channel. To minimize the possibility of losing the bead attached to the RBC fixed on the tip of the pipette, do the following steps as quickly as possible. 19. Halve the flow to facilitate the trapping of the second bead. Check that the RBC stays separated from the bead, if this is not the case, increase the flow again to separate the cell from the bead. 20. Move the optical trap to the exit of the bead dispenser tube. 21. Flow the bead solution and try to trap a bead. It will be harder due to the RBC buffer flow on the central channel. To catch a bead with the pump on, it is easier to stay at the exit of the dispenser tube without moving the trap. Wait for the bead to fall into the optical trap instead of actively moving the trap around. 22. Once you have trapped a bead, move the chamber to the working area."},{"id":"","caption":"Force-cellular-extension curves and RBC stiffness. (a) Force-cellular-extension curves for two different RBCs. (b) Stiffness as a function of force obtained from the slope of the force-cellular-extension curves.","content":"Fig. 19 .\n19Force-cellular-extension curves and RBC stiffness. (a) Force-cellular-extension curves for two different RBCs. (b) Stiffness as a function of force obtained from the slope of the force-cellular-extension curves."},{"id":"","caption":"Determination of zero-shear viscosities. (a) Trajectories of particles of distinct diameters trapped in different fluids. From top to bottom: water (2a = 2.73 µm), PNP (2a = 3.25 µm), and aqueous micellar solution of CPyCl/NaSal at 4 mM (2a = 3.25 µm). (b) Autocorrelation functions of the particle position for the cases in water (•), PNP ( ), and micellar solution ( ). The dashed lines correspond to exponential fits. Inset: semilog representrations of the position autocorrelation function for the three cases, normalized by their corresponding variances x(t) 2 = k B T κ .","content":"Fig. 21 .\n21Determination of zero-shear viscosities. (a) Trajectories of particles of distinct diameters trapped in different fluids. From top to bottom: water (2a = 2.73 µm), PNP (2a = 3.25 µm), and aqueous micellar solution of CPyCl/NaSal at 4 mM (2a = 3.25 µm). (b) Autocorrelation functions of the particle position for the cases in water (•), PNP ( ), and micellar solution ( ). The dashed lines correspond to exponential fits. Inset: semilog representrations of the position autocorrelation function for the three cases, normalized by their corresponding variances x(t) 2 = k B T κ ."},{"id":"","caption":"Determination of storage and loss moduli. (a) Power spectral density of equilibrium fluctuations of x(t) for a particle trapped by optical tweezers in a wormlike micellar solution of CPyCl/NaSal at 5 mM. Inset: time evolution of x(t) over 10 s. (b) Storage modulus (red •) and loss modulus (blue ) of the wormlike micellar solution determined by passive microrheology. means of","content":"Fig. 22 .\n22Determination of storage and loss moduli. (a) Power spectral density of equilibrium fluctuations of x(t) for a particle trapped by optical tweezers in a wormlike micellar solution of CPyCl/NaSal at 5 mM. Inset: time evolution of x(t) over 10 s. (b) Storage modulus (red •) and loss modulus (blue ) of the wormlike micellar solution determined by passive microrheology. means of"},{"id":"","caption":"Active microrheology. (a) Power spectral density of the particle position, driven at different frequencies by a sinusoidal motion of the optical trap. The peaks are located at the imposed driving frequencies f d . Inset: Time evolution of the perturbative force divided by the trap stiffness at f d = 1 Hz (thick solid line) and resulting particle position (thin solid line). (b) Numerical values of the storage modulus (red •) and loss (blue ) obtained by means of equations(90)and(95). The dotted and dashed lines depict the corresponding curves shown inFig. 22(d) obtained by passive microrheology.","content":"Fig. 23 .\n23Active microrheology. (a) Power spectral density of the particle position, driven at different frequencies by a sinusoidal motion of the optical trap. The peaks are located at the imposed driving frequencies f d . Inset: Time evolution of the perturbative force divided by the trap stiffness at f d = 1 Hz (thick solid line) and resulting particle position (thin solid line). (b) Numerical values of the storage modulus (red •) and loss (blue ) obtained by means of equations(90)and(95). The dotted and dashed lines depict the corresponding curves shown inFig. 22(d) obtained by passive microrheology."},{"id":"","caption":"Trajectories and potentials with critical Casimir forces. (a-d) Time evolution of the probability density distribution (coloured background) of the inter-particles distance r(t) obtained from 400 different trajectories after the optical traps are switched off at t = 0 ms for decreasing values of ∆T: (a) ∆T = 456 ± 2 mK, (b) 200 ± 2 mK, (c) 163 ± 2 mK, and (d) 108 ± 2 mK. The solid lines indicate representative individual trajectories. The dashed horizontal line indicates the distance r corresponding to the diameter d of the colloids.","content":"Fig. 24 .\n24Trajectories and potentials with critical Casimir forces. (a-d) Time evolution of the probability density distribution (coloured background) of the inter-particles distance r(t) obtained from 400 different trajectories after the optical traps are switched off at t = 0 ms for decreasing values of ∆T: (a) ∆T = 456 ± 2 mK, (b) 200 ± 2 mK, (c) 163 ± 2 mK, and (d) 108 ± 2 mK. The solid lines indicate representative individual trajectories. The dashed horizontal line indicates the distance r corresponding to the diameter d of the colloids."},{"id":"","caption":"Parallel and perpendicular diffusion coefficients. Experimental values of the normalized parallel D /D 0 (circles) and perpendicular D ⊥ /D 0 (triangles) diffusion coefficients as functions of the normalized inter-particle distance r/d. The parallel and perpendicular directions refer to the line connecting the centers of the two colloids, and D 0 is the bulk diffusion constant (equation","content":"Fig. 25 .\n25Parallel and perpendicular diffusion coefficients. Experimental values of the normalized parallel D /D 0 (circles) and perpendicular D ⊥ /D 0 (triangles) diffusion coefficients as functions of the normalized inter-particle distance r/d. The parallel and perpendicular directions refer to the line connecting the centers of the two colloids, and D 0 is the bulk diffusion constant (equation"},{"id":"","caption":"Parallel and perpendicular drifts. Drift velocities parallel v and perpendicular v ⊥ to the direction connecting the centers of the two colloids for (a, b) ∆T = 456 ± 2 mK, (c, d) 200 ± 2 mK, (e, f) 163 ± 2 mK and (g, h) 108 ± 2 mK. The symbols with errorbars represent the experimental data, the colored lines represent the corresponding simulation results, and the shaded areas represent the error of the numerical estimates due to the uncertainties in the fit parameters.","content":"Fig. 26 .\n26Parallel and perpendicular drifts. Drift velocities parallel v and perpendicular v ⊥ to the direction connecting the centers of the two colloids for (a, b) ∆T = 456 ± 2 mK, (c, d) 200 ± 2 mK, (e, f) 163 ± 2 mK and (g, h) 108 ± 2 mK. The symbols with errorbars represent the experimental data, the colored lines represent the corresponding simulation results, and the shaded areas represent the error of the numerical estimates due to the uncertainties in the fit parameters."},{"id":"","caption":"), v is positive at small values of r/d due to the repulsive electrostatic potential V es , which is dominant in this range and pushes the particles away from each other, while it rapidly vanishes for increasing r/d, because the electrostatic repulsion decays exponentially with the inter-particles distance and no other force is affecting the motion of the colloids. Decreasing ∆T (Figs. 25(c) and 25(e)), v becomes negative within a certain range of values of r/d, due to the arising of attractive critical Casimir forces; however, at smaller values of r/d, V es is dominant and v is still positive. If ∆T is reduced further (T is very close to T c ,Fig. 25(g)), critical Casimir forces became so strong that v presents only negative values because the particles move towards each other until their velocity vanishes at contact; at distances larger than the range of the critical Casimir forces action range, instead, v vanishes and the particles undergo Brownian diffusion.26 InFigs. 25(b,d,f,h), we report the experimental and numerical values for the orthogonal component v ⊥ of the drift velocity (equation","content":"), v is positive at small values of r/d due to the repulsive electrostatic potential V es , which is dominant in this range and pushes the particles away from each other, while it rapidly vanishes for increasing r/d, because the electrostatic repulsion decays exponentially with the inter-particles distance and no other force is affecting the motion of the colloids. Decreasing ∆T (Figs. 25(c) and 25(e)), v becomes negative within a certain range of values of r/d, due to the arising of attractive critical Casimir forces; however, at smaller values of r/d, V es is dominant and v is still positive. If ∆T is reduced further (T is very close to T c ,Fig. 25(g)), critical Casimir forces became so strong that v presents only negative values because the particles move towards each other until their velocity vanishes at contact; at distances larger than the range of the critical Casimir forces action range, instead, v vanishes and the particles undergo Brownian diffusion.26 InFigs. 25(b,d,f,h), we report the experimental and numerical values for the orthogonal component v ⊥ of the drift velocity (equation"},{"id":"","caption":"Kramers' rates. (a) Schematic of a bistable potential generated with a doublebeam optical tweezers. A silica bead is optically trapped with two-orthogonally-polarized and tightly-focused Gaussian beams separated by a distance d = 0.8 µm along the x-direction and focused by a water-immersion objective (NA = 1.2) using an inverted microscope. (b) Trajectory of a Brownian particle (radius r p = 0.48 µm) in an aqueous solution (temperature T = 22 • C) in the bistable potential. The particle spends most of the time near the equilibrium points (A and B, whose separation is measured to be d AB = 0.71 µm), but from time to time it is thermally activated to the saddle point (S) with a mean rate defined by equations(108)and (110). (c) Reconstruction of the force field measured with FORMA (arrows, section 3.8) and of the potential energy measured with the potential analysis (background color, section 3.2). (d-g) Optical potential (d) along the x-direction and (e-f) along the y-direction measured using FORMA. (h-i) Residence time probability in wells (h) A and (i) B. The red curves show the exponential probability function defined by equation (111), with the mean residence times estimated from the experimental data τ A = 3.03 s and τ B = 0.67 s. approximately 10 µm above the bottom coverslip of the sample cell and 90 µm from the top one, so that the hydrodynamic interaction with the walls of the cell is negligible.","content":"Fig. 27 .\n27Kramers' rates. (a) Schematic of a bistable potential generated with a doublebeam optical tweezers. A silica bead is optically trapped with two-orthogonally-polarized and tightly-focused Gaussian beams separated by a distance d = 0.8 µm along the x-direction and focused by a water-immersion objective (NA = 1.2) using an inverted microscope. (b) Trajectory of a Brownian particle (radius r p = 0.48 µm) in an aqueous solution (temperature T = 22 • C) in the bistable potential. The particle spends most of the time near the equilibrium points (A and B, whose separation is measured to be d AB = 0.71 µm), but from time to time it is thermally activated to the saddle point (S) with a mean rate defined by equations(108)and (110). (c) Reconstruction of the force field measured with FORMA (arrows, section 3.8) and of the potential energy measured with the potential analysis (background color, section 3.2). (d-g) Optical potential (d) along the x-direction and (e-f) along the y-direction measured using FORMA. (h-i) Residence time probability in wells (h) A and (i) B. The red curves show the exponential probability function defined by equation (111), with the mean residence times estimated from the experimental data τ A = 3.03 s and τ B = 0.67 s. approximately 10 µm above the bottom coverslip of the sample cell and 90 µm from the top one, so that the hydrodynamic interaction with the walls of the cell is negligible."},{"id":"","caption":"and the fitted potential parameters are in very good agreement with those directly measured from the escape time of the trajectories trajectories (1/τ (ex) ). equilibrium points are easily identified as the positions where the magnitude of the force field has local minima. The potential profile along the x-direction is shown in Figs. 27(d) and those along the y-direction at the critical points in Figs. 27(e-g).","content":"and the fitted potential parameters are in very good agreement with those directly measured from the escape time of the trajectories trajectories (1/τ (ex) ). equilibrium points are easily identified as the positions where the magnitude of the force field has local minima. The potential profile along the x-direction is shown in Figs. 27(d) and those along the y-direction at the critical points in Figs. 27(e-g)."},{"id":"","caption":"(d) NESS correlation functions involved in the integral form of the generalized FDT (119): C(0) − C(t) (dotted-dashed line), B(t) (dashed line) and C(0) − C(t) − B(t) (solid line). Inset: estimate of the response function R(t) by means of the time derivative of [C(0) − C(t)]/(k B T) (dotted-dashed line), and taking into account the corrective term, [C(0) − C(t) − B(t)]/(k B T) (solid line).","content":"(d) NESS correlation functions involved in the integral form of the generalized FDT (119): C(0) − C(t) (dotted-dashed line), B(t) (dashed line) and C(0) − C(t) − B(t) (solid line). Inset: estimate of the response function R(t) by means of the time derivative of [C(0) − C(t)]/(k B T) (dotted-dashed line), and taking into account the corrective term, [C(0) − C(t) − B(t)]/(k B T) (solid line)."},{"id":"","caption":"First Law in stochastic thermodynamics. (a) Upper panel: Portion of the trajectory x(t) (blue solid line) of a colloidal bead (2a = 2.73 µm), dragged by moving an optical tweezers (k = 1.15 pN µm −1 , v = 0.520 µm s −1 ) through water (T = 295 K). The dashed line represents the time evolution of the minimum of the trapping potential: λ(t) = vt + λ(0). Lower panel: time evolution of λ(t) − x(t). (b) Dependence of the mean work W τ (•), mean heat dissipated into the aqueous medium Q τ ( ), and mean variation of potential energy ∆U τ ( ) as a function of the measurement time τ. The solid line represents the values given by equation (133). (c) Dependence of the standard deviation of the work (•), of the heat dissipated into the aqueous medium ( ), and of the variation of potential energy ( ) as a function of τ. The solid line represents the values given by the square roots of equation (134). (d) Distribution of the work (in units of k B T) for different values of the measurement time τ. From inner to outer curves: τ = 0.034 s, 0.069 s, 0.103 s, 0.138 s, 0.172 s, and 0.207 s. (e) Distribution of the dissipated heat (in units of k B T) for different values τ. Same color code as in (d). Inset: verification of the integral fluctuation theorem (139) for the total entropy production. (f) Distribution of the variation of potential energy (in units of k B T) for different values τ. Same color code as in (d).","content":"Fig. 29 .\n29First Law in stochastic thermodynamics. (a) Upper panel: Portion of the trajectory x(t) (blue solid line) of a colloidal bead (2a = 2.73 µm), dragged by moving an optical tweezers (k = 1.15 pN µm −1 , v = 0.520 µm s −1 ) through water (T = 295 K). The dashed line represents the time evolution of the minimum of the trapping potential: λ(t) = vt + λ(0). Lower panel: time evolution of λ(t) − x(t). (b) Dependence of the mean work W τ (•), mean heat dissipated into the aqueous medium Q τ ( ), and mean variation of potential energy ∆U τ ( ) as a function of the measurement time τ. The solid line represents the values given by equation (133). (c) Dependence of the standard deviation of the work (•), of the heat dissipated into the aqueous medium ( ), and of the variation of potential energy ( ) as a function of τ. The solid line represents the values given by the square roots of equation (134). (d) Distribution of the work (in units of k B T) for different values of the measurement time τ. From inner to outer curves: τ = 0.034 s, 0.069 s, 0.103 s, 0.138 s, 0.172 s, and 0.207 s. (e) Distribution of the dissipated heat (in units of k B T) for different values τ. Same color code as in (d). Inset: verification of the integral fluctuation theorem (139) for the total entropy production. (f) Distribution of the variation of potential energy (in units of k B T) for different values τ. Same color code as in (d)."},{"id":"","caption":"Microscopic engines. (a) A sketch of Carnot's cycle in the (S, T)-plane and (b) in the (V, P)-plane. (c) The Stirling cycle in the (V, P)-plane. (d) Implementation of the Stirling cycle using a colloidal particle in an optical trap.","content":"Fig. 30 .\n30Microscopic engines. (a) A sketch of Carnot's cycle in the (S, T)-plane and (b) in the (V, P)-plane. (c) The Stirling cycle in the (V, P)-plane. (d) Implementation of the Stirling cycle using a colloidal particle in an optical trap."},{"id":"","caption":"Energy potentials and trapping positions in a pattern of fringes. The force in a pattern of fringes has a sinusoidal shape with the same period as the fringe pattern. (a) Force amplitude normalized to the average intensity, Q = A/2I 0 , as a function of the period of the fringes, Λ, for two silica particle of different diameters: D p = 2 µm (blue) and 4 µm (red). Negative (positive) values of Q give rise to a potential with energy minima at the brightest (dark) regions of the pattern. (b-c) Optical potentials U(x) for the particles with (b) D p = 2 µm (blue) and (c) D p = 4 µm for a pattern with period Λ x = 2 µm (corresponding to the dashed vertical line in (a)). (d) Variation of Q as a function of the ratio Λ x /D p for microscopic beads made of silica and polystyrene, with respective refractive indices of n p = 1.46 and n p = 1.60 (values defined at λ = 532 nm). The red and blue vertical dashed lines correspond to the values of the ratio Λ x /D p for the examples shown in (b) and (c), respectively. The optical forces and potentials were computed using the ray optics model.","content":"Fig. 31 .\n31Energy potentials and trapping positions in a pattern of fringes. The force in a pattern of fringes has a sinusoidal shape with the same period as the fringe pattern. (a) Force amplitude normalized to the average intensity, Q = A/2I 0 , as a function of the period of the fringes, Λ, for two silica particle of different diameters: D p = 2 µm (blue) and 4 µm (red). Negative (positive) values of Q give rise to a potential with energy minima at the brightest (dark) regions of the pattern. (b-c) Optical potentials U(x) for the particles with (b) D p = 2 µm (blue) and (c) D p = 4 µm for a pattern with period Λ x = 2 µm (corresponding to the dashed vertical line in (a)). (d) Variation of Q as a function of the ratio Λ x /D p for microscopic beads made of silica and polystyrene, with respective refractive indices of n p = 1.46 and n p = 1.60 (values defined at λ = 532 nm). The red and blue vertical dashed lines correspond to the values of the ratio Λ x /D p for the examples shown in (b) and (c), respectively. The optical forces and potentials were computed using the ray optics model."},{"id":"","caption":"Fig. 32. Particle-dependent potential or size effect in asymmetric patterns. Asymmetric pattern of light with period Λ = 4 µm (background) and the resulting optical potentials, U a (x) and U b (x), for two silica beads with diameters D a = 2 µm and D b = 4 µm, respectively. The shape of the optical potential for a given pattern of light can be very different depending on the size of the particle. The potential was computed using the ray optics model.","content":"Fig. 32\n32Fig. 32. Particle-dependent potential or size effect in asymmetric patterns. Asymmetric pattern of light with period Λ = 4 µm (background) and the resulting optical potentials, U a (x) and U b (x), for two silica beads with diameters D a = 2 µm and D b = 4 µm, respectively. The shape of the optical potential for a given pattern of light can be very different depending on the size of the particle. The potential was computed using the ray optics model."},{"id":"","caption":"Schemes for the generation of a moving pattern of fringes using a Mach-Zehnder interferometer. (a) By exploiting the angular Doppler effect with a λ/2 plate rotating at a constant angular speed Ω. (b) By gradually tilting a glass plate a small angle θ(t). (c) By means of a linear displacement D(t) = vt of a mirror. (d) By a periodic and asymmetric displacement of a mirror, according to a sawtooth function D(t). PBS: polarizing beam splitter. M: mirror. BB: beam block. BS: non-polarizing beam splitter. L: lens.","content":"Fig. 33 .\n33Schemes for the generation of a moving pattern of fringes using a Mach-Zehnder interferometer. (a) By exploiting the angular Doppler effect with a λ/2 plate rotating at a constant angular speed Ω. (b) By gradually tilting a glass plate a small angle θ(t). (c) By means of a linear displacement D(t) = vt of a mirror. (d) By a periodic and asymmetric displacement of a mirror, according to a sawtooth function D(t). PBS: polarizing beam splitter. M: mirror. BB: beam block. BS: non-polarizing beam splitter. L: lens."},{"id":"","caption":"Current in an optical sieve. (a) Mean current as a function of the period of the fringes Λ for particles with diameter D p = 2 µm and 5 µm for a fixed optical power P = 130 mW and frequency f = 1 Hz. (b) Mean current as a function of the angular","content":"Fig. 34 .\n34Current in an optical sieve. (a) Mean current as a function of the period of the fringes Λ for particles with diameter D p = 2 µm and 5 µm for a fixed optical power P = 130 mW and frequency f = 1 Hz. (b) Mean current as a function of the angular"},{"id":"","caption":"Optical trapping in vacuum. Photo of the sample chamber of an optical levitation setup. An optically trapped particle is shown in the inset. Figure adapted from Ref.[486].","content":"Fig. 36 .\n36Optical trapping in vacuum. Photo of the sample chamber of an optical levitation setup. An optically trapped particle is shown in the inset. Figure adapted from Ref.[486]."},{"id":"","caption":"Simplified setup for optical trapping in vacuum. Schematic of a setup for the trapping of particles in vacuum, based on the setup employed in Ref. [485]. EOM: electrooptic modulator. O: objective. PLL: Phase-locked loop. QPD: quadrant photodetector.","content":"Fig. 37 .\n37Simplified setup for optical trapping in vacuum. Schematic of a setup for the trapping of particles in vacuum, based on the setup employed in Ref. [485]. EOM: electrooptic modulator. O: objective. PLL: Phase-locked loop. QPD: quadrant photodetector."},{"id":"","caption":"Gas composition in a vaccum chamber. Gas composition in the vacuum chamber at pressures below 10 −3 mbar plotted over time. The solid lines show the partial pressures of different gas species measured with a residual gas analyzer (RGA). The sum of all the partial pressures yields the total pressure at the RGA (dashed line). Due to the configuration of the vacuum system and the reduced pumping speed at the RGA, the pressure at the RGA deviates from the pressure in the main vacuum chamber (dash-dot line). The initial rise in partial pressures after turning on the RGA is attributed to the warm-up process and desorption of gases from the filament of the RGA. Reproduced from Ref.[471].","content":"Fig. 39 .\n39Gas composition in a vaccum chamber. Gas composition in the vacuum chamber at pressures below 10 −3 mbar plotted over time. The solid lines show the partial pressures of different gas species measured with a residual gas analyzer (RGA). The sum of all the partial pressures yields the total pressure at the RGA (dashed line). Due to the configuration of the vacuum system and the reduced pumping speed at the RGA, the pressure at the RGA deviates from the pressure in the main vacuum chamber (dash-dot line). The initial rise in partial pressures after turning on the RGA is attributed to the warm-up process and desorption of gases from the filament of the RGA. Reproduced from Ref.[471]."},{"id":"","caption":"Particle loading with a nebulizer. The particle is loaded by spraying a solution of nanoparticles through a nozzle which is placed above the focus. (a) Positioning of the nozzle in the vacuum chamber. (b) Nozzle to funnel the falling particles towards the focus of the trapping laser. (c) Histogram of brightness observed with a camera from the side. The inset shows the brightness over a wider range. Reproduced from Ref. [488] (a,b) and adapted from Ref. [486] (c).","content":"Fig. 40 .\n40Particle loading with a nebulizer. The particle is loaded by spraying a solution of nanoparticles through a nozzle which is placed above the focus. (a) Positioning of the nozzle in the vacuum chamber. (b) Nozzle to funnel the falling particles towards the focus of the trapping laser. (c) Histogram of brightness observed with a camera from the side. The inset shows the brightness over a wider range. Reproduced from Ref. [488] (a,b) and adapted from Ref. [486] (c)."},{"id":"","caption":"Calibrated and uncalibrated z-detector signals. (a) Calibrated z-detector signal at 10 ± 1 mbar with electric driving response. The area under that peak corresponds to the thermal temperature k B T. (b) Uncalibrated feedback-cooled oscillator signal with electric drive for recalibration of the signal.","content":"Fig. 41 .\n41Calibrated and uncalibrated z-detector signals. (a) Calibrated z-detector signal at 10 ± 1 mbar with electric driving response. The area under that peak corresponds to the thermal temperature k B T. (b) Uncalibrated feedback-cooled oscillator signal with electric drive for recalibration of the signal."},{"id":"","caption":"Sideband thermometry. (a-b) Motional sideband asymmetry measured with heterodyne measurement. The frequency difference ∆ f is measured relative to the (absolute) local oscillator shift of 1 MHz. The gray solid areas show the measurement noise floor, limited by technical laser noise. The vertical dashed lines indicate the integration bandwidth (see equation 186). (c) Mean occupation number as a function of feedback gain. The red diamonds are obtained by integrating the red sideband of the heterodyne spectrum. The black circles show the mean occupation number extracted according to equation (187). The black dashed line corresponds to a parameter-free model (see also section 5.7.1). Figures adapted from Ref.[418].","content":"Fig. 42 .\n42Sideband thermometry. (a-b) Motional sideband asymmetry measured with heterodyne measurement. The frequency difference ∆ f is measured relative to the (absolute) local oscillator shift of 1 MHz. The gray solid areas show the measurement noise floor, limited by technical laser noise. The vertical dashed lines indicate the integration bandwidth (see equation 186). (c) Mean occupation number as a function of feedback gain. The red diamonds are obtained by integrating the red sideband of the heterodyne spectrum. The black circles show the mean occupation number extracted according to equation (187). The black dashed line corresponds to a parameter-free model (see also section 5.7.1). Figures adapted from Ref.[418]."},{"id":"","caption":"Charge calibration. (a) Power spectral density of the motion along the optical axis of a charge-carrying particle at a pressure of 1.9 mbar in the presence of a drive tone f d = ω d /2π applied to the capacitor. The solid line is a Lorentzian function fit to the data. (b) Quadrature component of particle oscillation in response to a driving voltage, demodulated in a bandwidth of 7 Hz. The high-voltage discharge is turned on at t = 0. The oscillation amplitude changes in discrete steps while the high voltage is on. (c) Preparation of charge state. The high voltage is turned off at t = 0, while the particle carries a net charge of 1e. The charge stays constant over the remainder of the measurement. Figure adapted from Ref.[523].","content":"Fig. 43 .\n43Charge calibration. (a) Power spectral density of the motion along the optical axis of a charge-carrying particle at a pressure of 1.9 mbar in the presence of a drive tone f d = ω d /2π applied to the capacitor. The solid line is a Lorentzian function fit to the data. (b) Quadrature component of particle oscillation in response to a driving voltage, demodulated in a bandwidth of 7 Hz. The high-voltage discharge is turned on at t = 0. The oscillation amplitude changes in discrete steps while the high voltage is on. (c) Preparation of charge state. The high voltage is turned off at t = 0, while the particle carries a net charge of 1e. The charge stays constant over the remainder of the measurement. Figure adapted from Ref.[523]."},{"id":"","caption":"Particle heating.","content":"Fig. 46 .\n46Particle heating."},{"id":"","caption":"2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = 2 (ω 2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = [2γ] −1 .","content":"2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = 2 (ω 2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = [2γ] −1 ."},{"id":"","caption":"JG is supported by the European Commission H2020-MSCA-IF-2014 (SEQOO 655369). IPC acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IA103417. JRGS. acknowledges financial support from DGAPA-UNAM PAPIIT Grant No. IA103320. AVA acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IN111919. KVS acknowledges support from DGAPA-UNAM (grant PAPIIT IN113419 and PASPA grant for sabbatical leave). AVA and GV acknowledge the Swedish Council for Higher Education through the Linnaeus-Palme International Exchange Program (contract 3.3.1.34.10235-2018). GV acknowledges funding from the European Research Council (ERC) Starting Grant ERC-677511 ComplexSwimmers.","content":"Funding\nJG is supported by the European Commission H2020-MSCA-IF-2014 (SEQOO 655369). IPC acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IA103417. JRGS. acknowledges financial support from DGAPA-UNAM PAPIIT Grant No. IA103320. AVA acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IN111919. KVS acknowledges support from DGAPA-UNAM (grant PAPIIT IN113419 and PASPA grant for sabbatical leave). AVA and GV acknowledge the Swedish Council for Higher Education through the Linnaeus-Palme International Exchange Program (contract 3.3.1.34.10235-2018). GV acknowledges funding from the European Research Council (ERC) Starting Grant ERC-677511 ComplexSwimmers."},{"id":"","caption":"1.03 µm","content":"1.03 µmLaboratory temperature \nT = 20 • C = 293.15 K \n\nSampling frequency \nf s = 10 5 Hz \n\nSampling timestep \n∆t = f −1 \ns = 10 −5 s \nviscosity (water) \nη = 1.00 mPa s \n\nfriction coefficient (Stokes' law (6)) \nγ (th) = 19.4 pN ms µm −1 \n\ndiffusion \ncoefficient \n(Einstein-\nSmoluchowski relation (7)) \n\nD (th) = 0.21 µm 2 s −1 \n\nnumber of experimental trajectories \nM = 5 \n\nnumber data points for each experiment \nN = 10 6 \n\nobjective lens \nOlympus UPLSAPO 60× 1.2 NA water im-\nmersion \n\nlaser wavelength \n1064 nm \n\nlaser power at the sample for experiment I, \nII, and III \n\n2.3 mW, 6.0 mW, and 9.2 mW"},{"id":"","caption":"Experimental results. Experimental values for the trap stiffness κ in [pN µm −1 ], the friction coefficient γ in [pN ms µm −1 ], the diffusion constant D in [µm 2 s −1 ], the τ ot, x in [ms], and f in [Hz] inferred using various calibration methods on the data from the experiments I, II and III (Table 1, note, in particular, the expected values γ (th) = 19.4 pN ms µm −1 and D (th) = 0.21 µm 2 s −1 ). The table is divided in two parts corresponding to the methods requiring fitting a function (linear fit / non-linear fit) and to those not requiring it (direct estimation).","content":"Table 3 .\n3Experimental results. Experimental values for the trap stiffness κ in [pN µm −1 ], the friction coefficient γ in [pN ms µm −1 ], the diffusion constant D in [µm 2 s −1 ], the τ ot, x in [ms], and f in [Hz] inferred using various calibration methods on the data from the experiments I, II and III (Table 1, note, in particular, the expected values γ (th) = 19.4 pN ms µm −1 and D (th) = 0.21 µm 2 s −1 ). The table is divided in two parts corresponding to the methods requiring fitting a function (linear fit / non-linear fit) and to those not requiring it (direct estimation).(ex) \n\n(c, x)"},{"id":"","caption":"Notation. In this table we summarize the notation related to the acquisition of the time series and that of theoretical and inferred values of the parameters of the optical trap and that of the particle.Given a series of trajectories {x (m) } N =1 with m = 1, . . . M, the experimental distribution ρ (ex) ). The blue markers are the experimental measurements, the solid black lines are the linear fitting, and the dashed red lines are the non-linear fitting. The black circles are the equilibrium distribution and trap potential obtained with the equipartition method (section 3.3). The estimates for the trap stiffness κ x are given in","content":"Table 4 .\n4Notation. In this table we summarize the notation related to the acquisition of the time series and that of theoretical and inferred values of the parameters of the optical trap and that of the particle.Given a series of trajectories {x (m) } N =1 with m = 1, . . . M, the experimental distribution ρ (ex) ). The blue markers are the experimental measurements, the solid black lines are the linear fitting, and the dashed red lines are the non-linear fitting. The black circles are the equilibrium distribution and trap potential obtained with the equipartition method (section 3.3). The estimates for the trap stiffness κ x are given inm \n\nx; {x (m) } N \n=1 is constructed as follows: \n\n0 \n\n0.02 \n\n0.04 \n\n0.06 \n\nEquipartition \n\nLinear fiting \n\nNon-linear fitting \n\nExperimental \n\n-50 \n0 \n50 \n\n0 \n\n5 \n\n10 \n\n-50 \n0 \n50 \n-50 \n0 \n50 \n\nFig. 10. Potential analysis. Comparison between theoretical expressions and experimental \nestimates for (a-c) the equilibrium distribution and (d-f) the trap potential for a particle \noptically trapped at three different powers: (a, d) 2.3 mW, (b, e) 6.0 mW, and (c, f) 9.2 mW \n(Table 1"},{"id":"","caption":"Ligation reaction. Mix all the items in an Eppendorf tube and keep them at 16 • C overnight. Afterwards, keep the Eppendorf tube at 65 • C for 10 min to inactivate the ligase.","content":"Table"},{"id":"","caption":"Extended light patterns.","content":"Table 8 .\n8Ligation reaction. Mix all the items in an Eppendorf tube and keep them at 16 • C overnight. Afterwards, keep the Eppendorf tube at 65 • C for 10 min to inactivate the ligase.Annealing product \n30 µl \n\n10× T4 DNA ligase buffer \n10 µl \n\n10 mM ATP \n5 µl \n\nT4 DNA ligase (400 U µl −1 ) \n3 µl \n\nTotal Volume \n100 µl"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.4 Carnot cycles and Maxwell's demons . . . . . . . . . . . . . . . . . . 84 4.5.5 Other applications to statistical physics and outlook . . . . . . . . . . . 89 4.6 Transport in extended potential landscapes . . . . . . . . . . . . . . . . . . . . 89 4.6.1 Optical forces and potentials in extended light patterns . . . . . . . . . 90 4.6.2 Transport mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.6.3 Other transport mechanisms and applications . . . . . . . . . . . . . . 98 5 Optical tweezers in vacuum 100 5.1 Applications of optical levitation. . . . . . . . . . . . . . . . . . . . . . .. . 100 5.2 Dynamics in the underdamped regime . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Optical tweezers setup for vacuum operation . . . . . . . . . . . . . . . . . . . 103 5.3.1 Particle types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.2 Optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.4 Feedback electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.5 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Particle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.4.1 Nebuliser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.2 Dry approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Dissipation and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.1 Gas damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5.2 Radiation damping and shot noise . . . . . . . . . . . . . . . . . . . . 114 5.5.3 Displacement noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.4 Laser intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.1 Detector calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.2 Sideband thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.6.3 Charge calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6.4 Mass Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6.5 Internal temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.6.6 Coupling between internal temperature and external degrees of freedom 124 5.7 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.7.1 Force feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.7.2 Parametric feedback cooling . . . . . . . . . . . . . . . . . . . . . . . 129 5.7.3 Parametric PLL cooling . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131\",\n \"content\": \"Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 78 4.5.4 Carnot cycles and Maxwell's demons . . . . . . . . . . . . . . . . . . 84 4.5.5 Other applications to statistical physics and outlook . . . . . . . . . . . 89 4.6 Transport in extended potential landscapes . . . . . . . . . . . . . . . . . . . . 89 4.6.1 Optical forces and potentials in extended light patterns . . . . . . . . . 90 4.6.2 Transport mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.6.3 Other transport mechanisms and applications . . . . . . . . . . . . . . 98 5 Optical tweezers in vacuum 100 5.1 Applications of optical levitation. . . . . . . . . . . . . . . . . . . . . . .. . 100 5.2 Dynamics in the underdamped regime . . . . . . . . . . . . . . . . . . . . . . 102 5.3 Optical tweezers setup for vacuum operation . . . . . . . . . . . . . . . . . . . 103 5.3.1 Particle types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.2 Optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.3 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3.4 Feedback electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.5 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Particle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.4.1 Nebuliser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.2 Dry approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.5 Dissipation and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.1 Gas damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5.2 Radiation damping and shot noise . . . . . . . . . . . . . . . . . . . . 114 5.5.3 Displacement noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.4 Laser intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.1 Detector calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.6.2 Sideband thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.6.3 Charge calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6.4 Mass Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6.5 Internal temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.6.6 Coupling between internal temperature and external degrees of freedom 124 5.7 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.7.1 Force feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.7.2 Parametric feedback cooling . . . . . . . . . . . . . . . . . . . . . . . 129 5.7.3 Parametric PLL cooling . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Trappable objects. A broad range of objects have been trapped using optical tweezers, going from single atoms and molecules to microparticles and microorganisms.\",\n \"content\": \"Fig. 1 .\\n1Trappable objects. A broad range of objects have been trapped using optical tweezers, going from single atoms and molecules to microparticles and microorganisms.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Basic optical tweezers setup. A basic optical tweezers setup is comprised of three parts: the trapping optics, the imaging optics, and the position detection optics. DM: Dichroic mirror. M: mirror. O: microscope objective. C: Condenser. L 1 , L 2 , L 3 , L 4 : lenses. LS: Light source.\",\n \"content\": \"Fig. 2 .\\n2Basic optical tweezers setup. A basic optical tweezers setup is comprised of three parts: the trapping optics, the imaging optics, and the position detection optics. DM: Dichroic mirror. M: mirror. O: microscope objective. C: Condenser. L 1 , L 2 , L 3 , L 4 : lenses. LS: Light source.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Brownian motion in an optical trap. (a) Trajectory of an optically trapped Brownian particle x(t) and corresponding (b) optical potential and (c) probability density. The equilibrium position is x eq = 0 µm. The experimental parameters are in Tab. 1.\",\n \"content\": \"Fig. 3 .\\n3Brownian motion in an optical trap. (a) Trajectory of an optically trapped Brownian particle x(t) and corresponding (b) optical potential and (c) probability density. The equilibrium position is x eq = 0 µm. The experimental parameters are in Tab. 1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Polarization-splitting optical tweezers. A typical Mach-Zehnder setup with polarizing beam splitters (PBS 1 and PBS 2 ) can be used for this purpose. Mirrors GM\",\n \"content\": \"Fig. 5 .\\n5Polarization-splitting optical tweezers. A typical Mach-Zehnder setup with polarizing beam splitters (PBS 1 and PBS 2 ) can be used for this purpose. Mirrors GM\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Interferometric optical traps. (a) Interferometric optical trap based on a Mach-Zehnder interferometer. (b-c) Different alternatives for the interferometric module. BS: Nonpolarizing beam splitter. RP: Right-angle prism. BB: beam block.\",\n \"content\": \"Fig. 7 .\\n7Interferometric optical traps. (a) Interferometric optical trap based on a Mach-Zehnder interferometer. (b-c) Different alternatives for the interferometric module. BS: Nonpolarizing beam splitter. RP: Right-angle prism. BB: beam block.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Holographic optical tweezers. A spatial light modulator (SLM) alters the beam profile of an incoming optical beam This beam is then relayed by a series of lenses to an objective that then generates the desired light pattern at the sample. The SLM can be either placed (a) in a plane conjugated to the back focal plane of the objective, or (b) in a plane conjugated with the image plane of the objective.\",\n \"content\": \"Fig. 8 .\\n8Holographic optical tweezers. A spatial light modulator (SLM) alters the beam profile of an incoming optical beam This beam is then relayed by a series of lenses to an objective that then generates the desired light pattern at the sample. The SLM can be either placed (a) in a plane conjugated to the back focal plane of the objective, or (b) in a plane conjugated with the image plane of the objective.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Phase masks and corresponding light intensity. Diffractive optical elements displayed on a phase-only SLM for the generation of: (a) a single optical trap laterally and axially shifted; (b) multiple traps distributed in space (the different brightnesses and sizes of the spots are due to the different positions of the corresponding traps along the vertical z-position); (c) a Laguerre-Gaussian beam of topological charge l = −2 laterally shifted; (d) a Bessel beam of topological charge l = 3.\",\n \"content\": \"Fig. 9 .\\n9Phase masks and corresponding light intensity. Diffractive optical elements displayed on a phase-only SLM for the generation of: (a) a single optical trap laterally and axially shifted; (b) multiple traps distributed in space (the different brightnesses and sizes of the spots are due to the different positions of the corresponding traps along the vertical z-position); (c) a Laguerre-Gaussian beam of topological charge l = −2 laterally shifted; (d) a Bessel beam of topological charge l = 3.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"=1 recorded time series of particle's position during the m-th experiment with N samples {x (m) } L =1 uncorrelated time series at time intervals ∆t with ∆t = r∆t and r an integer κ (ex) x experimental estimate of the trap stiffness τ (ex) x experimental estimate of characteristic relaxation time D (ex) , D (th) experimental estimate and theoretical value of diffusion constant γ (ex) , γ (th) experimental estimate and theoretical value of friction coefficient T s total acquisition time for each experiment m\",\n \"content\": \"=1 recorded time series of particle's position during the m-th experiment with N samples {x (m) } L =1 uncorrelated time series at time intervals ∆t with ∆t = r∆t and r an integer κ (ex) x experimental estimate of the trap stiffness τ (ex) x experimental estimate of characteristic relaxation time D (ex) , D (th) experimental estimate and theoretical value of diffusion constant γ (ex) , γ (th) experimental estimate and theoretical value of friction coefficient T s total acquisition time for each experiment m\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Autocorrelation function (ACF) analysis. Comparison between the theoretical expression\",\n \"content\": \"Fig. 12 .\\n12Autocorrelation function (ACF) analysis. Comparison between the theoretical expression\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Single-molecule pulling experiments: miniTweezers setup. Schematic representations (a) of the miniTweezers setup, (b) of the microfluidics chamber (the flow goes from left to right, and the laser beams propagate perpendicularly to the surface of the chamber), (c) of the assembly procedure for the construction of the chamber, and (d) of the molecular configuration for the DNA hairpin pulling experiments (where the DNA hairpin is held between a bead held by a micropipette and a bead captured by the optical trap, depicted as an harmonic potential). Note that x h2 , x d , x h1 , and x b are distances along the y-axis.\",\n \"content\": \"Fig. 14 .\\n14Single-molecule pulling experiments: miniTweezers setup. Schematic representations (a) of the miniTweezers setup, (b) of the microfluidics chamber (the flow goes from left to right, and the laser beams propagate perpendicularly to the surface of the chamber), (c) of the assembly procedure for the construction of the chamber, and (d) of the molecular configuration for the DNA hairpin pulling experiments (where the DNA hairpin is held between a bead held by a micropipette and a bead captured by the optical trap, depicted as an harmonic potential). Note that x h2 , x d , x h1 , and x b are distances along the y-axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"AD beads. The protocol for the preparation of the AD beads consists of three steps: (1) exchange the preservation buffer; (2) attach the anti-digoxigenin to the G-protein; (3) exchange the crosslinking buffer. The procedure (∼ 90 min) for the preparation of 1 ml of AD beads (which should be sufficient to perform experiments for several months) is: 1. (The first time the anti-digoxigenin batch is dissolved.) Add 200 µl of PBS (pH 7.4) to dilute the dry anti-digoxigenin (sheep polycolonal anti-dig antibody, Roche 1333 089). 2. Prepare the dimethyl pimelimidate (DMP) crosslinker buffer by mixing 50 mg DMP and 200 µl antibody crosslinker buffer (pH 7.4). The DMP crosslinker buffer needs to be freshly prepared every time the AD beads are synthesized.\",\n \"content\": \"AD beads. The protocol for the preparation of the AD beads consists of three steps: (1) exchange the preservation buffer; (2) attach the anti-digoxigenin to the G-protein; (3) exchange the crosslinking buffer. The procedure (∼ 90 min) for the preparation of 1 ml of AD beads (which should be sufficient to perform experiments for several months) is: 1. (The first time the anti-digoxigenin batch is dissolved.) Add 200 µl of PBS (pH 7.4) to dilute the dry anti-digoxigenin (sheep polycolonal anti-dig antibody, Roche 1333 089). 2. Prepare the dimethyl pimelimidate (DMP) crosslinker buffer by mixing 50 mg DMP and 200 µl antibody crosslinker buffer (pH 7.4). The DMP crosslinker buffer needs to be freshly prepared every time the AD beads are synthesized.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular buffer.A typical molecular buffer for DNA pulling experiments consists of 10 mM Tris pH 7.513 (Trizma, Sigma Aldrich), 10 mM EDTA14 (EDTA, Sigma Aldrich), 1 M NaCl, and 0.01% NaN 3 (sodium azide, to avoid bacterial growth). To facilitate the molecular buffer preparation, it is convenient to prepare stocks of:1 M Tris pH 7.5 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 6.05 g of Tris-HCl. Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add a solution of 25% HCl until the solution reaches pH 7.5. For longer storage, a final auto-cleavage can be performed. EDTA 0.5 M pH 8.0 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 7.306 g of EDTA. Add NaOH to the solution (the EDTA does not dissolve in water if pH<7.5). Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add NaOH until the solution reaches pH 8.0. 5 M NaCl Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 14.49 g of NaCl. Dissolve using a magnetic mixer and heat the tube to facilitate it. Add Milli-Q water until reaching 50 ml. 1% NaN 3 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 0.5 g of NaN 3 . Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. After the stocks have been prepared, for preparing 50 ml of the molecular buffer: Pour ∼ 30 ml of Milli-Q water in a 50 ml Falcon tube. Add 10 ml of the 5 M NaCl stock, 1 ml of the 0.5 M EDTA pH 8.0 stock, 0.5 ml of the 1 M Tris pH 7.5 stock, and 0.5 ml of the 1% NaN 3 stock. Add Milli-Q water until reaching 50 ml of volume. Mix the molecular buffer and filter it (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International) introducing the filtered solution in a new 50 ml Falcon tube. Incubation of the beads. To finalize the preparation of the beads, incubate in an Eppendorf tube (∼ 1.5 ml) 1 µl of a solution of the biomolecule of interest15 mixed with 14 µl of buffer (where the experiments are going to be performed) and 5 µl of the previously prepared AD beads. After ∼ 25 min have passed, add 1 ml of the molecular buffer.\",\n \"content\": \"Molecular buffer.A typical molecular buffer for DNA pulling experiments consists of 10 mM Tris pH 7.513 (Trizma, Sigma Aldrich), 10 mM EDTA14 (EDTA, Sigma Aldrich), 1 M NaCl, and 0.01% NaN 3 (sodium azide, to avoid bacterial growth). To facilitate the molecular buffer preparation, it is convenient to prepare stocks of:1 M Tris pH 7.5 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 6.05 g of Tris-HCl. Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add a solution of 25% HCl until the solution reaches pH 7.5. For longer storage, a final auto-cleavage can be performed. EDTA 0.5 M pH 8.0 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 7.306 g of EDTA. Add NaOH to the solution (the EDTA does not dissolve in water if pH<7.5). Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. Check the pH and add NaOH until the solution reaches pH 8.0. 5 M NaCl Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 14.49 g of NaCl. Dissolve using a magnetic mixer and heat the tube to facilitate it. Add Milli-Q water until reaching 50 ml. 1% NaN 3 Fill ∼ 40 ml of a 50 ml Falcon tube with Milli-Q water. Add 0.5 g of NaN 3 . Dissolve using a magnetic mixer. Add Milli-Q water until reaching 50 ml. After the stocks have been prepared, for preparing 50 ml of the molecular buffer: Pour ∼ 30 ml of Milli-Q water in a 50 ml Falcon tube. Add 10 ml of the 5 M NaCl stock, 1 ml of the 0.5 M EDTA pH 8.0 stock, 0.5 ml of the 1 M Tris pH 7.5 stock, and 0.5 ml of the 1% NaN 3 stock. Add Milli-Q water until reaching 50 ml of volume. Mix the molecular buffer and filter it (Sterile Syringe Filter, w/0.2 µm Cellulose, Acetate Membrane, VWR International) introducing the filtered solution in a new 50 ml Falcon tube. Incubation of the beads. To finalize the preparation of the beads, incubate in an Eppendorf tube (∼ 1.5 ml) 1 µl of a solution of the biomolecule of interest15 mixed with 14 µl of buffer (where the experiments are going to be performed) and 5 µl of the previously prepared AD beads. After ∼ 25 min have passed, add 1 ml of the molecular buffer.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"biotin-AGT TAG TGG TGG AAA CAC AGT GCC AGC GCG CGA GCC ATA AT-3 CD4-HandleB 5 -Phos-CTC ATC TGG AAA CAG ATG AGA TTA TGG CTC GCG ACT TCA CTA ATA CGA CTC ACT ATA GGG A -3 splint 5 -GCG CTG GCA CTG TGT TTC CAC CAC TAA CT-3\",\n \"content\": \"biotin-AGT TAG TGG TGG AAA CAC AGT GCC AGC GCG CGA GCC ATA AT-3 CD4-HandleB 5 -Phos-CTC ATC TGG AAA CAG ATG AGA TTA TGG CTC GCG ACT TCA CTA ATA CGA CTC ACT ATA GGG A -3 splint 5 -GCG CTG GCA CTG TGT TTC CAC CAC TAA CT-3\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Oligonucleotide tailing reaction. Mix all the items in an Eppendorf tube and keep it at 37 • C for 15 min. After the reaction is finished, add 1 to 2 µl of 0.5 M EDTA pH 8.0 to quench the reaction. (*) Included in the Terminal transferase kit from Sigma Aldrich.\",\n \"content\": \"6 .\\n6Oligonucleotide tailing reaction. Mix all the items in an Eppendorf tube and keep it at 37 • C for 15 min. After the reaction is finished, add 1 to 2 µl of 0.5 M EDTA pH 8.0 to quench the reaction. (*) Included in the Terminal transferase kit from Sigma Aldrich.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Free-energy of hairpin formation. (a) Hairpin pulling trajectories shown as force-distance curves. The unfolding cycles are plotted in red and the folding cycles in blue. The black dots indicate the initial (L 0 , f min ) and final (L 1 , f max ) points considering for computing the work W FU . Since the L values are obtained from the light position at the position-PSD, the L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero (which corresponds to the center of the position-PSD). (b) Probability density functions (PDFs) of the work, computed using equation(75), for the unfolding (W U , red histograms) and folding (W F , blue histograms) cycles. The value at which the probabilities cross (dashed vertical line) is the free energy difference between the folded and unfolded states ∆G FU . This value is obtained as the crossing point between the lines interpolating the histograms (blue and red solid lines).\",\n \"content\": \"Fig. 16 .\\n16Free-energy of hairpin formation. (a) Hairpin pulling trajectories shown as force-distance curves. The unfolding cycles are plotted in red and the folding cycles in blue. The black dots indicate the initial (L 0 , f min ) and final (L 1 , f max ) points considering for computing the work W FU . Since the L values are obtained from the light position at the position-PSD, the L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero (which corresponds to the center of the position-PSD). (b) Probability density functions (PDFs) of the work, computed using equation(75), for the unfolding (W U , red histograms) and folding (W F , blue histograms) cycles. The value at which the probabilities cross (dashed vertical line) is the free energy difference between the folded and unfolded states ∆G FU . This value is obtained as the crossing point between the lines interpolating the histograms (blue and red solid lines).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(b) is obtained by taking L 0 = −140 nm ( f min = 11.35 pN) and L 1 = −60 nm ( f max = 17.16 pN). Note that L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero. The crossing point of the histograms gives ∆G FU = 348.67 k B T (with 1k B T = 4.114 pN nm at T = 25 • C).\",\n \"content\": \"(b) is obtained by taking L 0 = −140 nm ( f min = 11.35 pN) and L 1 = −60 nm ( f max = 17.16 pN). Note that L 0 and L 1 stand as relative trap positions with respect to an arbitrary zero. The crossing point of the histograms gives ∆G FU = 348.67 k B T (with 1k B T = 4.114 pN nm at T = 25 • C).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"1 M K phosphate buffer at 7.4 pH is obtained by mixing 40.1 ml of 1 M K 2 HPO 4 and 9.9 ml of 1 M KH 2 PO 4 . The 1 M Na Phosphate buffer at 7.4 pH is obtained by mixing 38.7 ml of 1 M Na 2 HPO 4 and 11.3 ml of 1 M NaH 2 PO 4 . All salts are available from Sigma-Aldrich.\",\n \"content\": \"1 M K phosphate buffer at 7.4 pH is obtained by mixing 40.1 ml of 1 M K 2 HPO 4 and 9.9 ml of 1 M KH 2 PO 4 . The 1 M Na Phosphate buffer at 7.4 pH is obtained by mixing 38.7 ml of 1 M Na 2 HPO 4 and 11.3 ml of 1 M NaH 2 PO 4 . All salts are available from Sigma-Aldrich.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fill the syringe connected to the dispenser tube selected in the previous point with the bead solution. Suck about half of the Eppendorf tube content with the syringe. 4. Fill the other syringe (upper/lower channel) with RBC solution. Suck about half of the Eppendorf tube content with the syringe.\",\n \"content\": \"3 .\\n3Fill the syringe connected to the dispenser tube selected in the previous point with the bead solution. Suck about half of the Eppendorf tube content with the syringe. 4. Fill the other syringe (upper/lower channel) with RBC solution. Suck about half of the Eppendorf tube content with the syringe.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Construction of the cellular configuration. Images of different steps to obtain the RBC configuration required for the experiment: (a) the RBC is attached to a bead and partially trapped by the lasers; (b) the bead is trapped, while the RBC is not trapped; (c) the second bead is attached to the other side of the RBC; (d) final experimental configuration at zero reading for the force along the y-direction. The diameter of the beads is 3 µm. The red dot represents the position of the optical trap. The red arrow in (a) and (b) represents the presence and direction of the fluid flow.\",\n \"content\": \"Fig. 17 .\\n17Construction of the cellular configuration. Images of different steps to obtain the RBC configuration required for the experiment: (a) the RBC is attached to a bead and partially trapped by the lasers; (b) the bead is trapped, while the RBC is not trapped; (c) the second bead is attached to the other side of the RBC; (d) final experimental configuration at zero reading for the force along the y-direction. The diameter of the beads is 3 µm. The red dot represents the position of the optical trap. The red arrow in (a) and (b) represents the presence and direction of the fluid flow.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Do not turn off the pump flow in the central channel. To minimize the possibility of losing the bead attached to the RBC fixed on the tip of the pipette, do the following steps as quickly as possible. 19. Halve the flow to facilitate the trapping of the second bead. Check that the RBC stays separated from the bead, if this is not the case, increase the flow again to separate the cell from the bead. 20. Move the optical trap to the exit of the bead dispenser tube. 21. Flow the bead solution and try to trap a bead. It will be harder due to the RBC buffer flow on the central channel. To catch a bead with the pump on, it is easier to stay at the exit of the dispenser tube without moving the trap. Wait for the bead to fall into the optical trap instead of actively moving the trap around. 22. Once you have trapped a bead, move the chamber to the working area.\",\n \"content\": \"18 .\\n18Do not turn off the pump flow in the central channel. To minimize the possibility of losing the bead attached to the RBC fixed on the tip of the pipette, do the following steps as quickly as possible. 19. Halve the flow to facilitate the trapping of the second bead. Check that the RBC stays separated from the bead, if this is not the case, increase the flow again to separate the cell from the bead. 20. Move the optical trap to the exit of the bead dispenser tube. 21. Flow the bead solution and try to trap a bead. It will be harder due to the RBC buffer flow on the central channel. To catch a bead with the pump on, it is easier to stay at the exit of the dispenser tube without moving the trap. Wait for the bead to fall into the optical trap instead of actively moving the trap around. 22. Once you have trapped a bead, move the chamber to the working area.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Force-cellular-extension curves and RBC stiffness. (a) Force-cellular-extension curves for two different RBCs. (b) Stiffness as a function of force obtained from the slope of the force-cellular-extension curves.\",\n \"content\": \"Fig. 19 .\\n19Force-cellular-extension curves and RBC stiffness. (a) Force-cellular-extension curves for two different RBCs. (b) Stiffness as a function of force obtained from the slope of the force-cellular-extension curves.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Determination of zero-shear viscosities. (a) Trajectories of particles of distinct diameters trapped in different fluids. From top to bottom: water (2a = 2.73 µm), PNP (2a = 3.25 µm), and aqueous micellar solution of CPyCl/NaSal at 4 mM (2a = 3.25 µm). (b) Autocorrelation functions of the particle position for the cases in water (•), PNP ( ), and micellar solution ( ). The dashed lines correspond to exponential fits. Inset: semilog representrations of the position autocorrelation function for the three cases, normalized by their corresponding variances x(t) 2 = k B T κ .\",\n \"content\": \"Fig. 21 .\\n21Determination of zero-shear viscosities. (a) Trajectories of particles of distinct diameters trapped in different fluids. From top to bottom: water (2a = 2.73 µm), PNP (2a = 3.25 µm), and aqueous micellar solution of CPyCl/NaSal at 4 mM (2a = 3.25 µm). (b) Autocorrelation functions of the particle position for the cases in water (•), PNP ( ), and micellar solution ( ). The dashed lines correspond to exponential fits. Inset: semilog representrations of the position autocorrelation function for the three cases, normalized by their corresponding variances x(t) 2 = k B T κ .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Determination of storage and loss moduli. (a) Power spectral density of equilibrium fluctuations of x(t) for a particle trapped by optical tweezers in a wormlike micellar solution of CPyCl/NaSal at 5 mM. Inset: time evolution of x(t) over 10 s. (b) Storage modulus (red •) and loss modulus (blue ) of the wormlike micellar solution determined by passive microrheology. means of\",\n \"content\": \"Fig. 22 .\\n22Determination of storage and loss moduli. (a) Power spectral density of equilibrium fluctuations of x(t) for a particle trapped by optical tweezers in a wormlike micellar solution of CPyCl/NaSal at 5 mM. Inset: time evolution of x(t) over 10 s. (b) Storage modulus (red •) and loss modulus (blue ) of the wormlike micellar solution determined by passive microrheology. means of\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Active microrheology. (a) Power spectral density of the particle position, driven at different frequencies by a sinusoidal motion of the optical trap. The peaks are located at the imposed driving frequencies f d . Inset: Time evolution of the perturbative force divided by the trap stiffness at f d = 1 Hz (thick solid line) and resulting particle position (thin solid line). (b) Numerical values of the storage modulus (red •) and loss (blue ) obtained by means of equations(90)and(95). The dotted and dashed lines depict the corresponding curves shown inFig. 22(d) obtained by passive microrheology.\",\n \"content\": \"Fig. 23 .\\n23Active microrheology. (a) Power spectral density of the particle position, driven at different frequencies by a sinusoidal motion of the optical trap. The peaks are located at the imposed driving frequencies f d . Inset: Time evolution of the perturbative force divided by the trap stiffness at f d = 1 Hz (thick solid line) and resulting particle position (thin solid line). (b) Numerical values of the storage modulus (red •) and loss (blue ) obtained by means of equations(90)and(95). The dotted and dashed lines depict the corresponding curves shown inFig. 22(d) obtained by passive microrheology.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Trajectories and potentials with critical Casimir forces. (a-d) Time evolution of the probability density distribution (coloured background) of the inter-particles distance r(t) obtained from 400 different trajectories after the optical traps are switched off at t = 0 ms for decreasing values of ∆T: (a) ∆T = 456 ± 2 mK, (b) 200 ± 2 mK, (c) 163 ± 2 mK, and (d) 108 ± 2 mK. The solid lines indicate representative individual trajectories. The dashed horizontal line indicates the distance r corresponding to the diameter d of the colloids.\",\n \"content\": \"Fig. 24 .\\n24Trajectories and potentials with critical Casimir forces. (a-d) Time evolution of the probability density distribution (coloured background) of the inter-particles distance r(t) obtained from 400 different trajectories after the optical traps are switched off at t = 0 ms for decreasing values of ∆T: (a) ∆T = 456 ± 2 mK, (b) 200 ± 2 mK, (c) 163 ± 2 mK, and (d) 108 ± 2 mK. The solid lines indicate representative individual trajectories. The dashed horizontal line indicates the distance r corresponding to the diameter d of the colloids.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Parallel and perpendicular diffusion coefficients. Experimental values of the normalized parallel D /D 0 (circles) and perpendicular D ⊥ /D 0 (triangles) diffusion coefficients as functions of the normalized inter-particle distance r/d. The parallel and perpendicular directions refer to the line connecting the centers of the two colloids, and D 0 is the bulk diffusion constant (equation\",\n \"content\": \"Fig. 25 .\\n25Parallel and perpendicular diffusion coefficients. Experimental values of the normalized parallel D /D 0 (circles) and perpendicular D ⊥ /D 0 (triangles) diffusion coefficients as functions of the normalized inter-particle distance r/d. The parallel and perpendicular directions refer to the line connecting the centers of the two colloids, and D 0 is the bulk diffusion constant (equation\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Parallel and perpendicular drifts. Drift velocities parallel v and perpendicular v ⊥ to the direction connecting the centers of the two colloids for (a, b) ∆T = 456 ± 2 mK, (c, d) 200 ± 2 mK, (e, f) 163 ± 2 mK and (g, h) 108 ± 2 mK. The symbols with errorbars represent the experimental data, the colored lines represent the corresponding simulation results, and the shaded areas represent the error of the numerical estimates due to the uncertainties in the fit parameters.\",\n \"content\": \"Fig. 26 .\\n26Parallel and perpendicular drifts. Drift velocities parallel v and perpendicular v ⊥ to the direction connecting the centers of the two colloids for (a, b) ∆T = 456 ± 2 mK, (c, d) 200 ± 2 mK, (e, f) 163 ± 2 mK and (g, h) 108 ± 2 mK. The symbols with errorbars represent the experimental data, the colored lines represent the corresponding simulation results, and the shaded areas represent the error of the numerical estimates due to the uncertainties in the fit parameters.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"), v is positive at small values of r/d due to the repulsive electrostatic potential V es , which is dominant in this range and pushes the particles away from each other, while it rapidly vanishes for increasing r/d, because the electrostatic repulsion decays exponentially with the inter-particles distance and no other force is affecting the motion of the colloids. Decreasing ∆T (Figs. 25(c) and 25(e)), v becomes negative within a certain range of values of r/d, due to the arising of attractive critical Casimir forces; however, at smaller values of r/d, V es is dominant and v is still positive. If ∆T is reduced further (T is very close to T c ,Fig. 25(g)), critical Casimir forces became so strong that v presents only negative values because the particles move towards each other until their velocity vanishes at contact; at distances larger than the range of the critical Casimir forces action range, instead, v vanishes and the particles undergo Brownian diffusion.26 InFigs. 25(b,d,f,h), we report the experimental and numerical values for the orthogonal component v ⊥ of the drift velocity (equation\",\n \"content\": \"), v is positive at small values of r/d due to the repulsive electrostatic potential V es , which is dominant in this range and pushes the particles away from each other, while it rapidly vanishes for increasing r/d, because the electrostatic repulsion decays exponentially with the inter-particles distance and no other force is affecting the motion of the colloids. Decreasing ∆T (Figs. 25(c) and 25(e)), v becomes negative within a certain range of values of r/d, due to the arising of attractive critical Casimir forces; however, at smaller values of r/d, V es is dominant and v is still positive. If ∆T is reduced further (T is very close to T c ,Fig. 25(g)), critical Casimir forces became so strong that v presents only negative values because the particles move towards each other until their velocity vanishes at contact; at distances larger than the range of the critical Casimir forces action range, instead, v vanishes and the particles undergo Brownian diffusion.26 InFigs. 25(b,d,f,h), we report the experimental and numerical values for the orthogonal component v ⊥ of the drift velocity (equation\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Kramers' rates. (a) Schematic of a bistable potential generated with a doublebeam optical tweezers. A silica bead is optically trapped with two-orthogonally-polarized and tightly-focused Gaussian beams separated by a distance d = 0.8 µm along the x-direction and focused by a water-immersion objective (NA = 1.2) using an inverted microscope. (b) Trajectory of a Brownian particle (radius r p = 0.48 µm) in an aqueous solution (temperature T = 22 • C) in the bistable potential. The particle spends most of the time near the equilibrium points (A and B, whose separation is measured to be d AB = 0.71 µm), but from time to time it is thermally activated to the saddle point (S) with a mean rate defined by equations(108)and (110). (c) Reconstruction of the force field measured with FORMA (arrows, section 3.8) and of the potential energy measured with the potential analysis (background color, section 3.2). (d-g) Optical potential (d) along the x-direction and (e-f) along the y-direction measured using FORMA. (h-i) Residence time probability in wells (h) A and (i) B. The red curves show the exponential probability function defined by equation (111), with the mean residence times estimated from the experimental data τ A = 3.03 s and τ B = 0.67 s. approximately 10 µm above the bottom coverslip of the sample cell and 90 µm from the top one, so that the hydrodynamic interaction with the walls of the cell is negligible.\",\n \"content\": \"Fig. 27 .\\n27Kramers' rates. (a) Schematic of a bistable potential generated with a doublebeam optical tweezers. A silica bead is optically trapped with two-orthogonally-polarized and tightly-focused Gaussian beams separated by a distance d = 0.8 µm along the x-direction and focused by a water-immersion objective (NA = 1.2) using an inverted microscope. (b) Trajectory of a Brownian particle (radius r p = 0.48 µm) in an aqueous solution (temperature T = 22 • C) in the bistable potential. The particle spends most of the time near the equilibrium points (A and B, whose separation is measured to be d AB = 0.71 µm), but from time to time it is thermally activated to the saddle point (S) with a mean rate defined by equations(108)and (110). (c) Reconstruction of the force field measured with FORMA (arrows, section 3.8) and of the potential energy measured with the potential analysis (background color, section 3.2). (d-g) Optical potential (d) along the x-direction and (e-f) along the y-direction measured using FORMA. (h-i) Residence time probability in wells (h) A and (i) B. The red curves show the exponential probability function defined by equation (111), with the mean residence times estimated from the experimental data τ A = 3.03 s and τ B = 0.67 s. approximately 10 µm above the bottom coverslip of the sample cell and 90 µm from the top one, so that the hydrodynamic interaction with the walls of the cell is negligible.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"and the fitted potential parameters are in very good agreement with those directly measured from the escape time of the trajectories trajectories (1/τ (ex) ). equilibrium points are easily identified as the positions where the magnitude of the force field has local minima. The potential profile along the x-direction is shown in Figs. 27(d) and those along the y-direction at the critical points in Figs. 27(e-g).\",\n \"content\": \"and the fitted potential parameters are in very good agreement with those directly measured from the escape time of the trajectories trajectories (1/τ (ex) ). equilibrium points are easily identified as the positions where the magnitude of the force field has local minima. The potential profile along the x-direction is shown in Figs. 27(d) and those along the y-direction at the critical points in Figs. 27(e-g).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(d) NESS correlation functions involved in the integral form of the generalized FDT (119): C(0) − C(t) (dotted-dashed line), B(t) (dashed line) and C(0) − C(t) − B(t) (solid line). Inset: estimate of the response function R(t) by means of the time derivative of [C(0) − C(t)]/(k B T) (dotted-dashed line), and taking into account the corrective term, [C(0) − C(t) − B(t)]/(k B T) (solid line).\",\n \"content\": \"(d) NESS correlation functions involved in the integral form of the generalized FDT (119): C(0) − C(t) (dotted-dashed line), B(t) (dashed line) and C(0) − C(t) − B(t) (solid line). Inset: estimate of the response function R(t) by means of the time derivative of [C(0) − C(t)]/(k B T) (dotted-dashed line), and taking into account the corrective term, [C(0) − C(t) − B(t)]/(k B T) (solid line).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"First Law in stochastic thermodynamics. (a) Upper panel: Portion of the trajectory x(t) (blue solid line) of a colloidal bead (2a = 2.73 µm), dragged by moving an optical tweezers (k = 1.15 pN µm −1 , v = 0.520 µm s −1 ) through water (T = 295 K). The dashed line represents the time evolution of the minimum of the trapping potential: λ(t) = vt + λ(0). Lower panel: time evolution of λ(t) − x(t). (b) Dependence of the mean work W τ (•), mean heat dissipated into the aqueous medium Q τ ( ), and mean variation of potential energy ∆U τ ( ) as a function of the measurement time τ. The solid line represents the values given by equation (133). (c) Dependence of the standard deviation of the work (•), of the heat dissipated into the aqueous medium ( ), and of the variation of potential energy ( ) as a function of τ. The solid line represents the values given by the square roots of equation (134). (d) Distribution of the work (in units of k B T) for different values of the measurement time τ. From inner to outer curves: τ = 0.034 s, 0.069 s, 0.103 s, 0.138 s, 0.172 s, and 0.207 s. (e) Distribution of the dissipated heat (in units of k B T) for different values τ. Same color code as in (d). Inset: verification of the integral fluctuation theorem (139) for the total entropy production. (f) Distribution of the variation of potential energy (in units of k B T) for different values τ. Same color code as in (d).\",\n \"content\": \"Fig. 29 .\\n29First Law in stochastic thermodynamics. (a) Upper panel: Portion of the trajectory x(t) (blue solid line) of a colloidal bead (2a = 2.73 µm), dragged by moving an optical tweezers (k = 1.15 pN µm −1 , v = 0.520 µm s −1 ) through water (T = 295 K). The dashed line represents the time evolution of the minimum of the trapping potential: λ(t) = vt + λ(0). Lower panel: time evolution of λ(t) − x(t). (b) Dependence of the mean work W τ (•), mean heat dissipated into the aqueous medium Q τ ( ), and mean variation of potential energy ∆U τ ( ) as a function of the measurement time τ. The solid line represents the values given by equation (133). (c) Dependence of the standard deviation of the work (•), of the heat dissipated into the aqueous medium ( ), and of the variation of potential energy ( ) as a function of τ. The solid line represents the values given by the square roots of equation (134). (d) Distribution of the work (in units of k B T) for different values of the measurement time τ. From inner to outer curves: τ = 0.034 s, 0.069 s, 0.103 s, 0.138 s, 0.172 s, and 0.207 s. (e) Distribution of the dissipated heat (in units of k B T) for different values τ. Same color code as in (d). Inset: verification of the integral fluctuation theorem (139) for the total entropy production. (f) Distribution of the variation of potential energy (in units of k B T) for different values τ. Same color code as in (d).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Microscopic engines. (a) A sketch of Carnot's cycle in the (S, T)-plane and (b) in the (V, P)-plane. (c) The Stirling cycle in the (V, P)-plane. (d) Implementation of the Stirling cycle using a colloidal particle in an optical trap.\",\n \"content\": \"Fig. 30 .\\n30Microscopic engines. (a) A sketch of Carnot's cycle in the (S, T)-plane and (b) in the (V, P)-plane. (c) The Stirling cycle in the (V, P)-plane. (d) Implementation of the Stirling cycle using a colloidal particle in an optical trap.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Energy potentials and trapping positions in a pattern of fringes. The force in a pattern of fringes has a sinusoidal shape with the same period as the fringe pattern. (a) Force amplitude normalized to the average intensity, Q = A/2I 0 , as a function of the period of the fringes, Λ, for two silica particle of different diameters: D p = 2 µm (blue) and 4 µm (red). Negative (positive) values of Q give rise to a potential with energy minima at the brightest (dark) regions of the pattern. (b-c) Optical potentials U(x) for the particles with (b) D p = 2 µm (blue) and (c) D p = 4 µm for a pattern with period Λ x = 2 µm (corresponding to the dashed vertical line in (a)). (d) Variation of Q as a function of the ratio Λ x /D p for microscopic beads made of silica and polystyrene, with respective refractive indices of n p = 1.46 and n p = 1.60 (values defined at λ = 532 nm). The red and blue vertical dashed lines correspond to the values of the ratio Λ x /D p for the examples shown in (b) and (c), respectively. The optical forces and potentials were computed using the ray optics model.\",\n \"content\": \"Fig. 31 .\\n31Energy potentials and trapping positions in a pattern of fringes. The force in a pattern of fringes has a sinusoidal shape with the same period as the fringe pattern. (a) Force amplitude normalized to the average intensity, Q = A/2I 0 , as a function of the period of the fringes, Λ, for two silica particle of different diameters: D p = 2 µm (blue) and 4 µm (red). Negative (positive) values of Q give rise to a potential with energy minima at the brightest (dark) regions of the pattern. (b-c) Optical potentials U(x) for the particles with (b) D p = 2 µm (blue) and (c) D p = 4 µm for a pattern with period Λ x = 2 µm (corresponding to the dashed vertical line in (a)). (d) Variation of Q as a function of the ratio Λ x /D p for microscopic beads made of silica and polystyrene, with respective refractive indices of n p = 1.46 and n p = 1.60 (values defined at λ = 532 nm). The red and blue vertical dashed lines correspond to the values of the ratio Λ x /D p for the examples shown in (b) and (c), respectively. The optical forces and potentials were computed using the ray optics model.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 32. Particle-dependent potential or size effect in asymmetric patterns. Asymmetric pattern of light with period Λ = 4 µm (background) and the resulting optical potentials, U a (x) and U b (x), for two silica beads with diameters D a = 2 µm and D b = 4 µm, respectively. The shape of the optical potential for a given pattern of light can be very different depending on the size of the particle. The potential was computed using the ray optics model.\",\n \"content\": \"Fig. 32\\n32Fig. 32. Particle-dependent potential or size effect in asymmetric patterns. Asymmetric pattern of light with period Λ = 4 µm (background) and the resulting optical potentials, U a (x) and U b (x), for two silica beads with diameters D a = 2 µm and D b = 4 µm, respectively. The shape of the optical potential for a given pattern of light can be very different depending on the size of the particle. The potential was computed using the ray optics model.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Schemes for the generation of a moving pattern of fringes using a Mach-Zehnder interferometer. (a) By exploiting the angular Doppler effect with a λ/2 plate rotating at a constant angular speed Ω. (b) By gradually tilting a glass plate a small angle θ(t). (c) By means of a linear displacement D(t) = vt of a mirror. (d) By a periodic and asymmetric displacement of a mirror, according to a sawtooth function D(t). PBS: polarizing beam splitter. M: mirror. BB: beam block. BS: non-polarizing beam splitter. L: lens.\",\n \"content\": \"Fig. 33 .\\n33Schemes for the generation of a moving pattern of fringes using a Mach-Zehnder interferometer. (a) By exploiting the angular Doppler effect with a λ/2 plate rotating at a constant angular speed Ω. (b) By gradually tilting a glass plate a small angle θ(t). (c) By means of a linear displacement D(t) = vt of a mirror. (d) By a periodic and asymmetric displacement of a mirror, according to a sawtooth function D(t). PBS: polarizing beam splitter. M: mirror. BB: beam block. BS: non-polarizing beam splitter. L: lens.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Current in an optical sieve. (a) Mean current as a function of the period of the fringes Λ for particles with diameter D p = 2 µm and 5 µm for a fixed optical power P = 130 mW and frequency f = 1 Hz. (b) Mean current as a function of the angular\",\n \"content\": \"Fig. 34 .\\n34Current in an optical sieve. (a) Mean current as a function of the period of the fringes Λ for particles with diameter D p = 2 µm and 5 µm for a fixed optical power P = 130 mW and frequency f = 1 Hz. (b) Mean current as a function of the angular\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Optical trapping in vacuum. Photo of the sample chamber of an optical levitation setup. An optically trapped particle is shown in the inset. Figure adapted from Ref.[486].\",\n \"content\": \"Fig. 36 .\\n36Optical trapping in vacuum. Photo of the sample chamber of an optical levitation setup. An optically trapped particle is shown in the inset. Figure adapted from Ref.[486].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Simplified setup for optical trapping in vacuum. Schematic of a setup for the trapping of particles in vacuum, based on the setup employed in Ref. [485]. EOM: electrooptic modulator. O: objective. PLL: Phase-locked loop. QPD: quadrant photodetector.\",\n \"content\": \"Fig. 37 .\\n37Simplified setup for optical trapping in vacuum. Schematic of a setup for the trapping of particles in vacuum, based on the setup employed in Ref. [485]. EOM: electrooptic modulator. O: objective. PLL: Phase-locked loop. QPD: quadrant photodetector.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Gas composition in a vaccum chamber. Gas composition in the vacuum chamber at pressures below 10 −3 mbar plotted over time. The solid lines show the partial pressures of different gas species measured with a residual gas analyzer (RGA). The sum of all the partial pressures yields the total pressure at the RGA (dashed line). Due to the configuration of the vacuum system and the reduced pumping speed at the RGA, the pressure at the RGA deviates from the pressure in the main vacuum chamber (dash-dot line). The initial rise in partial pressures after turning on the RGA is attributed to the warm-up process and desorption of gases from the filament of the RGA. Reproduced from Ref.[471].\",\n \"content\": \"Fig. 39 .\\n39Gas composition in a vaccum chamber. Gas composition in the vacuum chamber at pressures below 10 −3 mbar plotted over time. The solid lines show the partial pressures of different gas species measured with a residual gas analyzer (RGA). The sum of all the partial pressures yields the total pressure at the RGA (dashed line). Due to the configuration of the vacuum system and the reduced pumping speed at the RGA, the pressure at the RGA deviates from the pressure in the main vacuum chamber (dash-dot line). The initial rise in partial pressures after turning on the RGA is attributed to the warm-up process and desorption of gases from the filament of the RGA. Reproduced from Ref.[471].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Particle loading with a nebulizer. The particle is loaded by spraying a solution of nanoparticles through a nozzle which is placed above the focus. (a) Positioning of the nozzle in the vacuum chamber. (b) Nozzle to funnel the falling particles towards the focus of the trapping laser. (c) Histogram of brightness observed with a camera from the side. The inset shows the brightness over a wider range. Reproduced from Ref. [488] (a,b) and adapted from Ref. [486] (c).\",\n \"content\": \"Fig. 40 .\\n40Particle loading with a nebulizer. The particle is loaded by spraying a solution of nanoparticles through a nozzle which is placed above the focus. (a) Positioning of the nozzle in the vacuum chamber. (b) Nozzle to funnel the falling particles towards the focus of the trapping laser. (c) Histogram of brightness observed with a camera from the side. The inset shows the brightness over a wider range. Reproduced from Ref. [488] (a,b) and adapted from Ref. [486] (c).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Calibrated and uncalibrated z-detector signals. (a) Calibrated z-detector signal at 10 ± 1 mbar with electric driving response. The area under that peak corresponds to the thermal temperature k B T. (b) Uncalibrated feedback-cooled oscillator signal with electric drive for recalibration of the signal.\",\n \"content\": \"Fig. 41 .\\n41Calibrated and uncalibrated z-detector signals. (a) Calibrated z-detector signal at 10 ± 1 mbar with electric driving response. The area under that peak corresponds to the thermal temperature k B T. (b) Uncalibrated feedback-cooled oscillator signal with electric drive for recalibration of the signal.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Sideband thermometry. (a-b) Motional sideband asymmetry measured with heterodyne measurement. The frequency difference ∆ f is measured relative to the (absolute) local oscillator shift of 1 MHz. The gray solid areas show the measurement noise floor, limited by technical laser noise. The vertical dashed lines indicate the integration bandwidth (see equation 186). (c) Mean occupation number as a function of feedback gain. The red diamonds are obtained by integrating the red sideband of the heterodyne spectrum. The black circles show the mean occupation number extracted according to equation (187). The black dashed line corresponds to a parameter-free model (see also section 5.7.1). Figures adapted from Ref.[418].\",\n \"content\": \"Fig. 42 .\\n42Sideband thermometry. (a-b) Motional sideband asymmetry measured with heterodyne measurement. The frequency difference ∆ f is measured relative to the (absolute) local oscillator shift of 1 MHz. The gray solid areas show the measurement noise floor, limited by technical laser noise. The vertical dashed lines indicate the integration bandwidth (see equation 186). (c) Mean occupation number as a function of feedback gain. The red diamonds are obtained by integrating the red sideband of the heterodyne spectrum. The black circles show the mean occupation number extracted according to equation (187). The black dashed line corresponds to a parameter-free model (see also section 5.7.1). Figures adapted from Ref.[418].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Charge calibration. (a) Power spectral density of the motion along the optical axis of a charge-carrying particle at a pressure of 1.9 mbar in the presence of a drive tone f d = ω d /2π applied to the capacitor. The solid line is a Lorentzian function fit to the data. (b) Quadrature component of particle oscillation in response to a driving voltage, demodulated in a bandwidth of 7 Hz. The high-voltage discharge is turned on at t = 0. The oscillation amplitude changes in discrete steps while the high voltage is on. (c) Preparation of charge state. The high voltage is turned off at t = 0, while the particle carries a net charge of 1e. The charge stays constant over the remainder of the measurement. Figure adapted from Ref.[523].\",\n \"content\": \"Fig. 43 .\\n43Charge calibration. (a) Power spectral density of the motion along the optical axis of a charge-carrying particle at a pressure of 1.9 mbar in the presence of a drive tone f d = ω d /2π applied to the capacitor. The solid line is a Lorentzian function fit to the data. (b) Quadrature component of particle oscillation in response to a driving voltage, demodulated in a bandwidth of 7 Hz. The high-voltage discharge is turned on at t = 0. The oscillation amplitude changes in discrete steps while the high voltage is on. (c) Preparation of charge state. The high voltage is turned off at t = 0, while the particle carries a net charge of 1e. The charge stays constant over the remainder of the measurement. Figure adapted from Ref.[523].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Particle heating.\",\n \"content\": \"Fig. 46 .\\n46Particle heating.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = 2 (ω 2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = [2γ] −1 .\",\n \"content\": \"2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = 2 (ω 2 − ω 2 0 ) 2 + γ 2 ω 2 −1 dω = [2γ] −1 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"JG is supported by the European Commission H2020-MSCA-IF-2014 (SEQOO 655369). IPC acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IA103417. JRGS. acknowledges financial support from DGAPA-UNAM PAPIIT Grant No. IA103320. AVA acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IN111919. KVS acknowledges support from DGAPA-UNAM (grant PAPIIT IN113419 and PASPA grant for sabbatical leave). AVA and GV acknowledge the Swedish Council for Higher Education through the Linnaeus-Palme International Exchange Program (contract 3.3.1.34.10235-2018). GV acknowledges funding from the European Research Council (ERC) Starting Grant ERC-677511 ComplexSwimmers.\",\n \"content\": \"Funding\\nJG is supported by the European Commission H2020-MSCA-IF-2014 (SEQOO 655369). IPC acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IA103417. JRGS. acknowledges financial support from DGAPA-UNAM PAPIIT Grant No. IA103320. AVA acknowledges financial support from the funding UNAM-DGAPA-PAPIIT-IN111919. KVS acknowledges support from DGAPA-UNAM (grant PAPIIT IN113419 and PASPA grant for sabbatical leave). AVA and GV acknowledge the Swedish Council for Higher Education through the Linnaeus-Palme International Exchange Program (contract 3.3.1.34.10235-2018). GV acknowledges funding from the European Research Council (ERC) Starting Grant ERC-677511 ComplexSwimmers.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"1.03 µm\",\n \"content\": \"1.03 µmLaboratory temperature \\nT = 20 • C = 293.15 K \\n\\nSampling frequency \\nf s = 10 5 Hz \\n\\nSampling timestep \\n∆t = f −1 \\ns = 10 −5 s \\nviscosity (water) \\nη = 1.00 mPa s \\n\\nfriction coefficient (Stokes' law (6)) \\nγ (th) = 19.4 pN ms µm −1 \\n\\ndiffusion \\ncoefficient \\n(Einstein-\\nSmoluchowski relation (7)) \\n\\nD (th) = 0.21 µm 2 s −1 \\n\\nnumber of experimental trajectories \\nM = 5 \\n\\nnumber data points for each experiment \\nN = 10 6 \\n\\nobjective lens \\nOlympus UPLSAPO 60× 1.2 NA water im-\\nmersion \\n\\nlaser wavelength \\n1064 nm \\n\\nlaser power at the sample for experiment I, \\nII, and III \\n\\n2.3 mW, 6.0 mW, and 9.2 mW\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Experimental results. Experimental values for the trap stiffness κ in [pN µm −1 ], the friction coefficient γ in [pN ms µm −1 ], the diffusion constant D in [µm 2 s −1 ], the τ ot, x in [ms], and f in [Hz] inferred using various calibration methods on the data from the experiments I, II and III (Table 1, note, in particular, the expected values γ (th) = 19.4 pN ms µm −1 and D (th) = 0.21 µm 2 s −1 ). The table is divided in two parts corresponding to the methods requiring fitting a function (linear fit / non-linear fit) and to those not requiring it (direct estimation).\",\n \"content\": \"Table 3 .\\n3Experimental results. Experimental values for the trap stiffness κ in [pN µm −1 ], the friction coefficient γ in [pN ms µm −1 ], the diffusion constant D in [µm 2 s −1 ], the τ ot, x in [ms], and f in [Hz] inferred using various calibration methods on the data from the experiments I, II and III (Table 1, note, in particular, the expected values γ (th) = 19.4 pN ms µm −1 and D (th) = 0.21 µm 2 s −1 ). The table is divided in two parts corresponding to the methods requiring fitting a function (linear fit / non-linear fit) and to those not requiring it (direct estimation).(ex) \\n\\n(c, x)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Notation. In this table we summarize the notation related to the acquisition of the time series and that of theoretical and inferred values of the parameters of the optical trap and that of the particle.Given a series of trajectories {x (m) } N =1 with m = 1, . . . M, the experimental distribution ρ (ex) ). The blue markers are the experimental measurements, the solid black lines are the linear fitting, and the dashed red lines are the non-linear fitting. The black circles are the equilibrium distribution and trap potential obtained with the equipartition method (section 3.3). The estimates for the trap stiffness κ x are given in\",\n \"content\": \"Table 4 .\\n4Notation. In this table we summarize the notation related to the acquisition of the time series and that of theoretical and inferred values of the parameters of the optical trap and that of the particle.Given a series of trajectories {x (m) } N =1 with m = 1, . . . M, the experimental distribution ρ (ex) ). The blue markers are the experimental measurements, the solid black lines are the linear fitting, and the dashed red lines are the non-linear fitting. The black circles are the equilibrium distribution and trap potential obtained with the equipartition method (section 3.3). The estimates for the trap stiffness κ x are given inm \\n\\nx; {x (m) } N \\n=1 is constructed as follows: \\n\\n0 \\n\\n0.02 \\n\\n0.04 \\n\\n0.06 \\n\\nEquipartition \\n\\nLinear fiting \\n\\nNon-linear fitting \\n\\nExperimental \\n\\n-50 \\n0 \\n50 \\n\\n0 \\n\\n5 \\n\\n10 \\n\\n-50 \\n0 \\n50 \\n-50 \\n0 \\n50 \\n\\nFig. 10. Potential analysis. Comparison between theoretical expressions and experimental \\nestimates for (a-c) the equilibrium distribution and (d-f) the trap potential for a particle \\noptically trapped at three different powers: (a, d) 2.3 mW, (b, e) 6.0 mW, and (c, f) 9.2 mW \\n(Table 1\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ligation reaction. Mix all the items in an Eppendorf tube and keep them at 16 • C overnight. Afterwards, keep the Eppendorf tube at 65 • C for 10 min to inactivate the ligase.\",\n \"content\": \"Table\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Extended light patterns.\",\n \"content\": \"Table 8 .\\n8Ligation reaction. Mix all the items in an Eppendorf tube and keep them at 16 • C overnight. Afterwards, keep the Eppendorf tube at 65 • C for 10 min to inactivate the ligase.Annealing product \\n30 µl \\n\\n10× T4 DNA ligase buffer \\n10 µl \\n\\n10 mM ATP \\n5 µl \\n\\nT4 DNA ligase (400 U µl −1 ) \\n3 µl \\n\\nTotal Volume \\n100 µl\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Select the minimum force from which to start stretching the RBC. 190 pNSelect the minimum force from which to start stretching the RBC (e.g., 0 pN).19","doi":""},{"text":"Select the maximum force at which to stop the RBC stretching. 2020 pNSelect the maximum force at which to stop the RBC stretching (e.g., 20 pN).20","doi":""},{"text":"140 nm s −1 ). Select the pulling velocity. Select the pulling velocity (e.g., 140 nm s −1 ).","doi":""},{"text":"Acceleration and trapping of particles by radiation pressure. A Ashkin, Phys. Rev. Lett. 24A. Ashkin, \"Acceleration and trapping of particles by radiation pressure,\" Phys. Rev. Lett. 24, 156-159 (1970).","doi":""},{"text":"Observation of a single-beam gradient force optical trap for dielectric particles. A Ashkin, J M Dziedzic, J E Bjorkholm, S Chu, Opt. Lett. 11A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, \"Observation of a single-beam gradient force optical trap for dielectric particles,\" Opt. Lett. 11, 288-290 (1986).","doi":""},{"text":"Optical trapping and manipulation of neutral particles using lasers: A reprint volume with commentaries. A Ashkin, World ScientificA. Ashkin, Optical trapping and manipulation of neutral particles using lasers: A reprint volume with commentaries (World Scientific, 2006).","doi":""},{"text":". Arthur Ashkin -Nobel Lecture. \"Arthur Ashkin -Nobel Lecture,\" https://www.nobelprize.org/prizes/physics/2018/ashkin/ lecture/. NobelPrize.org. Nobel Media AB 2019. Thu. 4 Jul 2019.","doi":""},{"text":"Nobel Lecture: The manipulation of neutral particles. S Chu, Rev. Mod. Phys. 70S. Chu, \"Nobel Lecture: The manipulation of neutral particles,\" Rev. Mod. Phys. 70, 685-706 (1998).","doi":""},{"text":"Force generation of organelle transport measured in vivo by an infrared laser trap. A Ashkin, K Schütze, J M Dziedzic, U Euteneuer, M Schliwa, Nature. 348A. Ashkin, K. Schütze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, \"Force generation of organelle transport measured in vivo by an infrared laser trap,\" Nature 348, 346-348 (1990).","doi":""},{"text":"Scanning-force microscope based on an optical trap. L P Ghislain, W W Webb, Opt. Lett. 18L. P. Ghislain and W. W. Webb, \"Scanning-force microscope based on an optical trap,\" Opt. Lett. 18, 1678-1680 (1993).","doi":""},{"text":"Bead movement by single kinesin molecules studied with optical tweezers. S M Block, L S B Goldstein, B J Schnapp, Nature. 348S. M. Block, L. S. B. Goldstein, and B. J. Schnapp, \"Bead movement by single kinesin molecules studied with optical tweezers,\" Nature 348, 348-352 (1990).","doi":""},{"text":"Entropic elasticity of lambda-phage DNA. C Bustamante, J F Marko, E D Siggia, S Smith, Science. 265C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith, \"Entropic elasticity of lambda-phage DNA,\" Science 265, 1599-1599 (1994).","doi":""},{"text":"Single myosin molecule mechanics: Piconewton forces and nanometre steps. J T Finer, R M Simmons, J A Spudich, Nature. 368J. T. Finer, R. M. Simmons, and J. A. Spudich, \"Single myosin molecule mechanics: Piconewton forces and nanometre steps,\" Nature 368, 113-119 (1994).","doi":""},{"text":"A revolution in optical manipulation. D G Grier, Nature. 424D. G. Grier, \"A revolution in optical manipulation,\" Nature 424, 810-816 (2003).","doi":""},{"text":"Colloquium: Gripped by light: Optical binding. K Dholakia, P Zemánek, Rev. Mod. Phys. 82K. Dholakia and P. Zemánek, \"Colloquium: Gripped by light: Optical binding,\" Rev. Mod. Phys. 82, 1767-1791 (2010).","doi":""},{"text":"Shaping the future of manipulation. K Dholakia, T Čižmár, Nat. Photon. 5K. Dholakia and T. Čižmár, \"Shaping the future of manipulation,\" Nat. Photon. 5, 335-342 (2011).","doi":""},{"text":"Plasmon nano-optical tweezers. M L Juan, M Righini, R Quidant, Nat. Photon. 5M. L. Juan, M. Righini, and R. Quidant, \"Plasmon nano-optical tweezers,\" Nat. Photon. 5, 349-356 (2011).","doi":""},{"text":"Tweezers with a twist. M Padgett, R Bowman, Nat. Photon. 5M. Padgett and R. Bowman, \"Tweezers with a twist,\" Nat. Photon. 5, 343-348 (2011).","doi":""},{"text":"Holographic optical tweezers and their relevance to lab on chip devices. M Padgett, R Di Leonardo, Lab Chip. 11M. Padgett and R. Di Leonardo, \"Holographic optical tweezers and their relevance to lab on chip devices,\" Lab Chip 11, 1196-1205 (2011).","doi":""},{"text":"Optical trapping and manipulation of nanostructures. O M Maragò, P H Jones, P G Gucciardi, G Volpe, A C Ferrari, Nat. Nanotech. 8O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, \"Optical trapping and manipulation of nanostructures,\" Nat. Nanotech. 8, 807-819 (2013).","doi":""},{"text":"Raman spectroscopy of optically trapped particles. D V Petrov, J. Opt. A. Pure. Appl. Op. 9139D. V. Petrov, \"Raman spectroscopy of optically trapped particles,\" J. Opt. A. Pure. Appl. Op. 9, S139 (2007).","doi":""},{"text":"Colloidal heat engines: A review. I A Martínez, É Roldán, L Dinis, R Rica, Soft Matter. 13I. A. Martínez, É. Roldán, L. Dinis, and R. Rica, \"Colloidal heat engines: A review,\" Soft Matter 13, 22-36 (2017).","doi":""},{"text":"Levitated nanoparticles for microscopic thermodynamics: A review. J Gieseler, J Millen, Entropy. 20326J. Gieseler and J. Millen, \"Levitated nanoparticles for microscopic thermodynamics: A review,\" Entropy 20, 326 (2018).","doi":""},{"text":"Optical tweezers microrheology: From the basics to advanced techniques and applications. R M Robertson-Anderson, ACS Macro Lett. 7R. M. Robertson-Anderson, \"Optical tweezers microrheology: From the basics to advanced techniques and applications,\" ACS Macro Lett. 7, 968-975 (2018).","doi":""},{"text":"Optical tweezers study life under tension. F M Fazal, S M Block, Nat. Photonics. 5F. M. Fazal and S. M. Block, \"Optical tweezers study life under tension,\" Nat. Photonics 5, 318-321 (2011).","doi":""},{"text":"P H Jones, O M Maragò, G Volpe, Optical tweezers: Principles and applications. Cambridge University PressP. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: Principles and applications (Cambridge University Press, 2015).","doi":""},{"text":"Optical tweezers computational toolbox. T A Nieminen, V L Y Loke, A B Stilgoe, G Knöner, A M Brańczyk, N R Heckenberg, H Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, \"Optical tweezers computational toolbox,\" J. Opt. A: Pure Appl. Opt. 9, S196-S203 (2007).","doi":""},{"text":"Computational toolbox for optical tweezers in geometrical optics. A Callegari, M Mijalkov, A B Gököz, G Volpe, J. Opt. Soc. Am. B. 32A. Callegari, M. Mijalkov, A. B. Gököz, and G. Volpe, \"Computational toolbox for optical tweezers in geometrical optics,\" J. Opt. Soc. Am. B 32, B11-B19 (2015).","doi":""},{"text":"Simulation of a Brownian particle in an optical trap. G Volpe, G Volpe, Am. J. Phys. 81G. Volpe and G. Volpe, \"Simulation of a Brownian particle in an optical trap,\" Am. J. Phys. 81, 224-230 (2013).","doi":""},{"text":"Inexpensive optical tweezers for undergraduate laboratories. S P Smith, S R Bhalotra, A L Brody, B L Brown, E K Boyda, M Prentiss, Am. J. Phys. 67S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, \"Inexpensive optical tweezers for undergraduate laboratories,\" Am. J. Phys. 67, 26-35 (1999).","doi":""},{"text":"Faster, cheaper, safer optical tweezers for the undergraduate laboratory. J Bechhoefer, S Wilson, Am. J. Phys. 70J. Bechhoefer and S. Wilson, \"Faster, cheaper, safer optical tweezers for the undergraduate laboratory,\" Am. J. Phys. 70, 393-400 (2002).","doi":""},{"text":"A simple laser cooling and trapping apparatus for undergraduate laboratories. A S Mellish, A C Wilson, Am. J. Phys. 70A. S. Mellish and A. C. Wilson, \"A simple laser cooling and trapping apparatus for undergraduate laboratories,\" Am. J. Phys. 70, 965-971 (2002).","doi":""},{"text":"Optical trapping for undergraduates. D C Appleyard, K Y Vandermeulen, H Lee, M J Lang, Am. J. Phys. 75D. C. Appleyard, K. Y. Vandermeulen, H. Lee, and M. J. Lang, \"Optical trapping for undergraduates,\" Am. J. Phys. 75, 5-14 (2007).","doi":""},{"text":"Construction and calibration of an optical trap on a fluorescence optical microscope. W M Lee, P J Reece, R F Marchington, N K Metzger, K Dholakia, Nat. Prot. 23226W. M. Lee, P. J. Reece, R. F. Marchington, N. K. Metzger, and K. Dholakia, \"Construction and calibration of an optical trap on a fluorescence optical microscope,\" Nat. Prot. 2, 3226 (2007).","doi":""},{"text":"Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: upgrading a commercial confocal inverted microscope. M Mathew, S I C O Santos, D Zalvidea, P Loza-Alvarez, Rev. Sci. Instrumen. 8073701M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, \"Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: upgrading a commercial confocal inverted microscope,\" Rev. Sci. Instrumen. 80, 073701 (2009).","doi":""},{"text":"Step-by-step guide to the realization of advanced optical tweezers. G Pesce, G Volpe, O M Maragó, P H Jones, S Gigan, A Sasso, G Volpe, J. Opt. Soc. Am. B. 32G. Pesce, G. Volpe, O. M. Maragó, P. H. Jones, S. Gigan, A. Sasso, and G. Volpe, \"Step-by-step guide to the realization of advanced optical tweezers,\" J. Opt. Soc. Am. B 32, B84-B98 (2015).","doi":""},{"text":"Optical tweezers: A comprehensive tutorial from calibration to applications. L García, A Magazù, G Pesce, J Gieseler, J R Gomez-Solano, I Castillo, M Gironella-Torrent, X Viader-Godoy, F Ritort, A V Arzola, K Volke-Sepulveda, G Volpe, OnlineL. Pérez García, A. Magazù, G. Pesce, J. Gieseler, J. R. Gomez-Solano, I. Castillo, M. Gironella-Torrent, X. Viader- Godoy, F. Ritort, A. V. Arzola, K. Volke-Sepulveda, and G. Volpe, \"Optical tweezers: A comprehensive tutorial from calibration to applications,\" https://github.com/softmatterlab/OpticalTweezersTutorial (2020). Online.","doi":""},{"text":"Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner. N B Simpson, K Dholakia, L Allen, M J Padgett, Opt. Lett. 22N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, \"Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,\" Opt. Lett. 22, 52-54 (1997).","doi":""},{"text":"Applications of laser tweezers in complex fluid rheology. E M Furst, Curr. Opn. Colloid Interf. Sci. 10E. M. Furst, \"Applications of laser tweezers in complex fluid rheology,\" Curr. Opn. Colloid Interf. Sci. 10, 79-86 (2005).","doi":""},{"text":"The optical stretcher: a novel laser tool to micromanipulate cells. J Guck, R Ananthakrishnan, H Mahmood, T J Moon, C C Cunningham, J Käs, Biophys. J. 81J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, \"The optical stretcher: a novel laser tool to micromanipulate cells,\" Biophys. J. 81, 767-784 (2001).","doi":""},{"text":"Microfluidic sorting in an optical lattice. M P Macdonald, G C Spalding, K Dholakia, Nature. 426M. P. MacDonald, G. C. Spalding, and K. Dholakia, \"Microfluidic sorting in an optical lattice,\" Nature 426, 421-424 (2003).","doi":""},{"text":"Optical sorting and detection of submicrometer objects in a motional standing wave. T Čižmár, M Šiler, M Šerỳ, P Zemánek, V Garcés-Chávez, K Dholakia, Phys. Rev. B. 7435105T. Čižmár, M. Šiler, M. Šerỳ, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, \"Optical sorting and detection of submicrometer objects in a motional standing wave,\" Phys. Rev. B 74, 035105 (2006).","doi":""},{"text":"Modulated optical sieve for sorting of polydisperse microparticles. I Ricárdez-Vargas, P Rodríguez-Montero, R Ramos-García, K Volke-Sepúlveda, Appl. Phys. Lett. 88121116I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, \"Modulated optical sieve for sorting of polydisperse microparticles,\" Appl. Phys. Lett. 88, 121116 (2006).","doi":""},{"text":"Construction of multiple-beam optical traps with nanometer-resolution position sensing. K Visscher, S P Gross, S M Block, IEEE J. Sel. Top. Quant. El. 2K. Visscher, S. P. Gross, and S. M. Block, \"Construction of multiple-beam optical traps with nanometer-resolution position sensing,\" IEEE J. Sel. Top. Quant. El. 2, 1066-1076 (1996).","doi":""},{"text":"Optical tweezer arrays and optical substrates created with diffractive optics. E R Dufresne, D G Grier, Rev. Sci. Instrumen. 69E. R. Dufresne and D. G. Grier, \"Optical tweezer arrays and optical substrates created with diffractive optics,\" Rev. Sci. Instrumen. 69, 1974-1977 (1998).","doi":""},{"text":"3D manipulation of particles into crystal structures using holographic optical tweezers. J Leach, G Sinclair, P Jordan, J Courtial, M J Padgett, J Cooper, Z J Laczik, Opt. Express. 12J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. J. Padgett, J. Cooper, and Z. J. Laczik, \"3D manipulation of particles into crystal structures using holographic optical tweezers,\" Opt. Express 12, 220-226 (2004).","doi":""},{"text":"Holographic optical trapping. D G Grier, Y Roichman, Appl. Opt. 45D. G. Grier and Y. Roichman, \"Holographic optical trapping,\" Appl. Opt. 45, 880-887 (2006).","doi":""},{"text":"Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity. H He, M E J Friese, N R Heckenberg, H Rubinsztein-Dunlop, Phys. Rev. Lett. 75H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \"Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,\" Phys. Rev. Lett. 75, 826-829 (1995).","doi":""},{"text":"Orbital angular momentum of a high-order Bessel light beam. K Volke-Sepulveda, V Garcés-Chávez, S Chávez-Cerda, J Arlt, K Dholakia, J. Opt. B: Quant. Semiclass. Opt. 4K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, \"Orbital angular momentum of a high-order Bessel light beam,\" J. Opt. B: Quant. Semiclass. Opt. 4, S82-S89 (2002).","doi":""},{"text":"Rotation, oscillation and hydrodynamic synchronization of optically trapped oblate spheroidal microparticles. A V Arzola, P Jákl, L Chvátal, P Zemánek, Opt. Express. 22A. V. Arzola, P. Jákl, L. Chvátal, and P. Zemánek, \"Rotation, oscillation and hydrodynamic synchronization of optically trapped oblate spheroidal microparticles,\" Opt. Express 22, 16207-16221 (2014).","doi":""},{"text":"Complex rotational dynamics of multiple spheroidal particles in a circularly polarized, dual beam trap. O Brzobohatỳ, A V Arzola, M Šiler, L Chvátal, P Jákl, S Simpson, P Zemánek, Opt. Express. 23O. Brzobohatỳ, A. V. Arzola, M. Šiler, L. Chvátal, P. Jákl, S. Simpson, and P. Zemánek, \"Complex rotational dynamics of multiple spheroidal particles in a circularly polarized, dual beam trap,\" Opt. Express 23, 7273-7287 (2015).","doi":""},{"text":"Optical deformability as an inherent cell marker for testing malignant transformation and metastatic competence. J Guck, S Schinkinger, B Lincoln, F Wottawah, S Ebert, M Romeyke, D Lenz, H M Erickson, R Ananthakrishnan, D Mitchell, J Käs, S Ulvick, C Bilby, Biophys. J. 88J. Guck, S. Schinkinger, B. Lincoln, F. Wottawah, S. Ebert, M. Romeyke, D. Lenz, H. M. Erickson, R. Ananthakrishnan, D. Mitchell, J. Käs, S. Ulvick, and C. Bilby, \"Optical deformability as an inherent cell marker for testing malignant transformation and metastatic competence,\" Biophys. J. 88, 3689-3698 (2005).","doi":""},{"text":"Light at work: The use of optical forces for particle manipulation, sorting, and analysis. A Jonáš, P Zemanek, Electrophoresis. 29A. Jonáš and P. Zemanek, \"Light at work: The use of optical forces for particle manipulation, sorting, and analysis,\" Electrophoresis 29, 4813-4851 (2008).","doi":""},{"text":"Optical vortex trapping of particles. K T Gahagan, G A Swartzlander, Opt. Lett. 21K. T. Gahagan and G. A. Swartzlander, \"Optical vortex trapping of particles,\" Opt. Lett. 21, 827-829 (1996).","doi":""},{"text":"Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner. A T O'neil, M J Padgett, Opt. Commun. 185A. T. O'Neil and M. J. Padgett, \"Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,\" Opt. Commun. 185, 139-143 (2000).","doi":""},{"text":"Transfer of orbital angular momentum to an optically trapped low-index particle. V Garcés-Chávez, K Volke-Sepulveda, S Chávez-Cerda, W Sibbett, K Dholakia, Phys. Rev. A. 6663402V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, \"Transfer of orbital angular momentum to an optically trapped low-index particle,\" Phys. Rev. A 66, 063402 (2002).","doi":""},{"text":"Attractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers. R J Hernández, A Mazzulla, A Pane, K Volke-Sepúlveda, G Cipparrone, Lab Chip. 13R. J. Hernández, A. Mazzulla, A. Pane, K. Volke-Sepúlveda, and G. Cipparrone, \"Attractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers,\" Lab Chip 13, 459-467 (2013).","doi":""},{"text":"Optical binding. M M Burns, J.-M Fournier, J A Golovchenko, Phys. Rev. Lett. 63M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, \"Optical binding,\" Phys. Rev. Lett. 63, 1233-1236 (1989).","doi":""},{"text":"Optical matter: crystallization and binding in intense optical fields. M M Burns, J.-M Fournier, J A Golovchenko, Science. 249M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, \"Optical matter: crystallization and binding in intense optical fields,\" Science 249, 749-754 (1990).","doi":""},{"text":"Optical binding between dielectric particles. S K Mohanty, J T Andrews, P K Gupta, Opt. Express. 12S. K. Mohanty, J. T. Andrews, and P. K. Gupta, \"Optical binding between dielectric particles,\" Opt. Express 12, 2746-2753 (2004).","doi":""},{"text":"Long-range one-dimensional longitudinal optical binding. V Karásek, T Čižmár, O Brzobohatỳ, P Zemánek, V Garcés-Chávez, K Dholakia, Phys. Rev. Lett. 101143601V. Karásek, T. Čižmár, O. Brzobohatỳ, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, \"Long-range one-dimensional longitudinal optical binding,\" Phys. Rev. Lett. 101, 143601 (2008).","doi":""},{"text":"Ultrastrong optical binding of metallic nanoparticles. V Demergis, E.-L Florin, Nanolett. 12V. Demergis and E.-L. Florin, \"Ultrastrong optical binding of metallic nanoparticles,\" Nanolett. 12, 5756-5760 (2012).","doi":""},{"text":"Light-controlled assembly of active colloidal molecules. F Schmidt, B Liebchen, H Löwen, G Volpe, J. Chem. Phys. 15094905F. Schmidt, B. Liebchen, H. Löwen, and G. Volpe, \"Light-controlled assembly of active colloidal molecules,\" J. Chem. Phys. 150, 094905 (2019).","doi":""},{"text":"Optical thermal ratchet. L P Faucheux, L S Bourdieu, P D Kaplan, A J Libchaber, Phys. Rev. Lett. 74L. P. Faucheux, L. S. Bourdieu, P. D. Kaplan, and A. J. Libchaber, \"Optical thermal ratchet,\" Phys. Rev. Lett. 74, 1504-1507 (1995).","doi":""},{"text":"Optical peristalsis. B A Koss, D G Grier, Appl. Phys. Lett. 82B. A. Koss and D. G. Grier, \"Optical peristalsis,\" Appl. Phys. Lett. 82, 3985-3987 (2003).","doi":""},{"text":"Observation of flux reversal in a symmetric optical thermal ratchet. S.-H Lee, K Ladavac, M Polin, D G Grier, Phys. Rev. Lett. 94110601S.-H. Lee, K. Ladavac, M. Polin, and D. G. Grier, \"Observation of flux reversal in a symmetric optical thermal ratchet,\" Phys. Rev. Lett. 94, 110601 (2005).","doi":""},{"text":"Experimental control of transport and current reversals in a deterministic optical rocking ratchet. A V Arzola, K Volke-Sepúlveda, J L Mateos, Phys. Rev. Lett. 106168104A. V. Arzola, K. Volke-Sepúlveda, and J. L. Mateos, \"Experimental control of transport and current reversals in a deterministic optical rocking ratchet,\" Phys. Rev. Lett. 106, 168104 (2011).","doi":""},{"text":"Active Brownian motion tunable by light. I Buttinoni, G Volpe, F Kümmel, G Volpe, C Bechinger, J. Phys.: Condens. Matter. 24284129I. Buttinoni, G. Volpe, F. Kümmel, G. Volpe, and C. Bechinger, \"Active Brownian motion tunable by light,\" J. Phys.: Condens. Matter 24, 284129 (2012).","doi":""},{"text":"Brownian motion in a speckle light field: Tunable anomalous diffusion and selective optical manipulation. G Volpe, G Volpe, S Gigan, Sci. Rep. 43936G. Volpe, G. Volpe, and S. Gigan, \"Brownian motion in a speckle light field: Tunable anomalous diffusion and selective optical manipulation,\" Sci. Rep. 4, 3936 (2014).","doi":""},{"text":"Omnidirectional transport in fully reconfigurable two dimensional optical ratchets. A V Arzola, M Villasante-Barahona, K Volke-Sepúlveda, P Jákl, P Zemánek, Phys. Rev. Lett. 118138002A. V. Arzola, M. Villasante-Barahona, K. Volke-Sepúlveda, P. Jákl, and P. Zemánek, \"Omnidirectional transport in fully reconfigurable two dimensional optical ratchets,\" Phys. Rev. Lett. 118, 138002 (2017).","doi":""},{"text":"Spin to orbital light momentum conversion visualized by particle trajectory. A V Arzola, L Chvátal, P Jákl, P Zemánek, Sci. Rep. 94127A. V. Arzola, L. Chvátal, P. Jákl, and P. Zemánek, \"Spin to orbital light momentum conversion visualized by particle trajectory,\" Sci. Rep. 9, 4127 (2019).","doi":""},{"text":"Complex micromachines produced and driven by light. P Galajda, P Ormos, Appl. Phys. Lett. 78P. Galajda and P. Ormos, \"Complex micromachines produced and driven by light,\" Appl. Phys. Lett. 78, 249-251 (2001).","doi":""},{"text":"Optically driven micromachine elements. M E J Friese, H Rubinsztein-Dunlop, J Gold, P Hagberg, D Hanstorp, Appl. Phys. Lett. 78M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, \"Optically driven micromachine elements,\" Appl. Phys. Lett. 78, 547-549 (2001).","doi":""},{"text":"Force-controllable, optically driven micromachines fabricated by single-step two-photon microstereolithography. S Maruo, K Ikuta, H Korogi, J. Microelectromech. Syst. 12S. Maruo, K. Ikuta, and H. Korogi, \"Force-controllable, optically driven micromachines fabricated by single-step two-photon microstereolithography,\" J. Microelectromech. Syst. 12, 533-539 (2003).","doi":""},{"text":"Optical tweezers applied to a microfluidic system. J Enger, M Goksör, K Ramser, P Hagberg, D Hanstorp, Lab Chip. 4J. Enger, M. Goksör, K. Ramser, P. Hagberg, and D. Hanstorp, \"Optical tweezers applied to a microfluidic system,\" Lab Chip 4, 196-200 (2004).","doi":""},{"text":"All-optical control of microfluidic components using form birefringence. S L Neale, M P Macdonald, K Dholakia, T F Krauss, Nat. Mater. 4S. L. Neale, M. P. MacDonald, K. Dholakia, and T. F. Krauss, \"All-optical control of microfluidic components using form birefringence,\" Nat. Mater. 4, 530-533 (2005).","doi":""},{"text":"Microfluidic sorting of mammalian cells by optical force switching. M M Wang, E Tu, D E Raymond, J M Yang, H Zhang, N Hagen, B Dees, E M Mercer, A H Forster, I Kariv, P J Marchand, W F Butler, Nat. Biotechnol. 23M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, \"Microfluidic sorting of mammalian cells by optical force switching,\" Nat. Biotechnol. 23, 83-87 (2005).","doi":""},{"text":"An optically driven pump for microfluidics. J Leach, H Mushfique, R Di Leonardo, M Padgett, J Cooper, Lab Chip. 6J. Leach, H. Mushfique, R. di Leonardo, M. Padgett, and J. Cooper, \"An optically driven pump for microfluidics,\" Lab Chip 6, 735-739 (2006).","doi":""},{"text":"A microfluidic system in combination with optical tweezers for analyzing rapid and reversible cytological alterations in single cells upon environmental changes. E Eriksson, J Enger, B Nordlander, N Erjavec, K Ramser, M Goksör, S Hohmann, T Nyström, D Hanstorp, Lab Chip. 7E. Eriksson, J. Enger, B. Nordlander, N. Erjavec, K. Ramser, M. Goksör, S. Hohmann, T. Nyström, and D. Hanstorp, \"A microfluidic system in combination with optical tweezers for analyzing rapid and reversible cytological alterations in single cells upon environmental changes,\" Lab Chip 7, 71-76 (2007).","doi":""},{"text":"Enhanced cell sorting and manipulation with combined optical tweezer and microfluidic chip technologies. X Wang, S Chen, M Kong, Z Wang, K D Costa, R A Li, D Sun, Lab Chip. 11X. Wang, S. Chen, M. Kong, Z. Wang, K. D. Costa, R. A. Li, and D. Sun, \"Enhanced cell sorting and manipulation with combined optical tweezer and microfluidic chip technologies,\" Lab Chip 11, 3656-3662 (2011).","doi":""},{"text":"Optical alignment and spinning of laser-trapped microscopic particles. M E J Friese, T A Nieminen, N R Heckenberg, H Rubinsztein-Dunlop, Nature. 394M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \"Optical alignment and spinning of laser-trapped microscopic particles,\" Nature 394, 348-350 (1998).","doi":""},{"text":"Intrinsic and extrinsic nature of the orbital angular momentum of a light beam. A T O'neil, I Macvicar, L Allen, M J Padgett, Phys. Rev. Lett. 8853601A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, \"Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,\" Phys. Rev. Lett. 88, 053601 (2002).","doi":""},{"text":"Orbital angular momentum transfer in helical Mathieu beams. C López-Mariscal, J C Gutiérrez-Vega, G Milne, K Dholakia, Opt. Express. 14C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, \"Orbital angular momentum transfer in helical Mathieu beams,\" Opt. Express 14, 4182-4187 (2006).","doi":""},{"text":"Forces in optical tweezers with radially and azimuthally polarized trapping beams. T A Nieminen, N R Heckenberg, H Rubinsztein-Dunlop, Opt. Lett. 33T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \"Forces in optical tweezers with radially and azimuthally polarized trapping beams,\" Opt. Lett. 33, 122-124 (2008).","doi":""},{"text":"Optically mediated particle clearing using Airy wavepackets. J Baumgartl, M Mazilu, K Dholakia, Nat. Photon. 2J. Baumgartl, M. Mazilu, and K. Dholakia, \"Optically mediated particle clearing using Airy wavepackets,\" Nat. Photon. 2, 675-678 (2008).","doi":""},{"text":"Design for fully steerable dual-trap optical tweezers. E Fällman, O Axner, Appl. Opt. 36E. Fällman and O. Axner, \"Design for fully steerable dual-trap optical tweezers,\" Appl. Opt. 36, 2107-2113 (1997).","doi":""},{"text":"Pattern formation and flow control of fine particles by laser-scanning micromanipulation. K Sasaki, M Koshioka, H Misawa, N Kitamura, H Masuhara, Opt. Lett. 16K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, \"Pattern formation and flow control of fine particles by laser-scanning micromanipulation,\" Opt. Lett. 16, 1463-1465 (1991).","doi":""},{"text":"Design of a scanning laser optical trap for multiparticle manipulation. C Mio, T Gong, A Terray, D W M Marr, Rev. Sci. Instrumen. 71C. Mio, T. Gong, A. Terray, and D. W. M. Marr, \"Design of a scanning laser optical trap for multiparticle manipulation,\" Rev. Sci. Instrumen. 71, 2196-2200 (2000).","doi":""},{"text":"Synchronized laser micromanipulation of multiple targets along each trajectory by single laser. F Arai, K Yoshikawa, T Sakami, T Fukuda, Appl. Phys. Lett. 85F. Arai, K. Yoshikawa, T. Sakami, and T. Fukuda, \"Synchronized laser micromanipulation of multiple targets along each trajectory by single laser,\" Appl. Phys. Lett. 85, 4301-4303 (2004).","doi":""},{"text":"Periodic average structures of colloidal quasicrystals. L Zaidouny, T Bohlein, J Roth, C Bechinger, Soft Matter. 10L. Zaidouny, T. Bohlein, J. Roth, and C. Bechinger, \"Periodic average structures of colloidal quasicrystals,\" Soft Matter 10, 8705-8710 (2014).","doi":""},{"text":"Interferometric optical tweezers. A E Chiou, W Wang, G J Sonek, J Hong, M W Berns, Opt. Commun. 133A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, \"Interferometric optical tweezers,\" Opt. Commun. 133, 7-10 (1997).","doi":""},{"text":"Optical forces acting on rayleigh particle placed into interference field. P Zemánek, V Karásek, A Sasso, Opt. Commun. 240P. Zemánek, V. Karásek, and A. Sasso, \"Optical forces acting on rayleigh particle placed into interference field,\" Opt. Commun. 240, 401-415 (2004).","doi":""},{"text":"Creation and manipulation of three-dimensional optically trapped structures. M P Macdonald, L Paterson, K Volke-Sepulveda, J Arlt, W Sibbett, K Dholakia, Science. 296M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, \"Creation and manipulation of three-dimensional optically trapped structures,\" Science 296, 1101-1103 (2002).","doi":""},{"text":"Selective trapping of multiple particles by volume speckle field. V G Shvedov, A V Rode, Y V Izdebskaya, A S Desyatnikov, W Krolikowski, Y S Kivshar, Opt. Express. 18V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, \"Selective trapping of multiple particles by volume speckle field,\" Opt. Express 18, 3137-3142 (2010).","doi":""},{"text":"Speckle optical tweezers: Micromanipulation with random light fields. G Volpe, L Kurz, A Callegari, G Volpe, S Gigan, Opt. Express. 22G. Volpe, L. Kurz, A. Callegari, G. Volpe, and S. Gigan, \"Speckle optical tweezers: Micromanipulation with random light fields,\" Opt. Express 22, 18159-18167 (2014).","doi":""},{"text":"Multi-functional optical tweezers using computer-generated holograms. J Liesener, M Reicherter, T Haist, H J Tiziani, Opt. Commun. 185J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, \"Multi-functional optical tweezers using computer-generated holograms,\" Opt. Commun. 185, 77-82 (2000).","doi":""},{"text":"Dynamic holographic optical tweezers. J E Curtis, B A Koss, D G Grier, Opt. Commun. 207J. E. Curtis, B. A. Koss, and D. G. Grier, \"Dynamic holographic optical tweezers,\" Opt. Commun. 207, 169-175 (2002).","doi":""},{"text":"Full phase and amplitude control of holographic optical tweezers with high efficiency. A Jesacher, C Maurer, A Schwaighofer, S Bernet, M Ritsch-Marte, Opt. Express. 16A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, \"Full phase and amplitude control of holographic optical tweezers with high efficiency,\" Opt. Express 16, 4479-4486 (2008).","doi":""},{"text":"Optical sorting of nonspherical and living microobjects in moving interference structures. P Jakl, A V Arzola, M Siler, L Chatal, K Volke-Sepulveda, P Zemanek, Opt. Express. 22P. Jakl, A. V. Arzola, M. Siler, L. Chatal, K. Volke-Sepulveda, and P. Zemanek, \"Optical sorting of nonspherical and living microobjects in moving interference structures,\" Opt. Express 22, 29746-29760 (2014).","doi":""},{"text":"Exact solutions for nondiffracting beams. i. the scalar theory. J Durnin, J. Opt. Soc. Am. A. 4J. Durnin, \"Exact solutions for nondiffracting beams. i. the scalar theory,\" J. Opt. Soc. Am. A 4, 651-654 (1987).","doi":""},{"text":"Alternative formulation for invariant optical fields: Mathieu beams. J C Gutiérrez-Vega, M D Iturbe-Castillo, S Chávez-Cerda, Opt. Lett. 25J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, \"Alternative formulation for invariant optical fields: Mathieu beams,\" Opt. Lett. 25, 1493-1495 (2000).","doi":""},{"text":"Optical micromanipulation using a Bessel light beam. J Arlt, V Garcés-Chávez, W Sibbett, K Dholakia, Opt. Commun. 197J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, \"Optical micromanipulation using a Bessel light beam,\" Opt. Commun. 197, 239-245 (2001).","doi":""},{"text":"M A Bandres, J C Gutiérrez-Vega, S Chávez-Cerda, Parabolic nondiffracting optical wave fields. 29M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, \"Parabolic nondiffracting optical wave fields,\" Opt. Lett. 29, 44-46 (2004).","doi":""},{"text":"Fresnel axicon. I Golub, Opt. Lett. 31I. Golub, \"Fresnel axicon,\" Opt. Lett. 31, 1890-1892 (2006).","doi":""},{"text":"Efficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation. V Arrizón, D Sánchez-De La-Llave, U Ruiz, G Méndez, Opt. Lett. 34V. Arrizón, D. Sánchez-de La-Llave, U. Ruiz, and G. Méndez, \"Efficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation,\" Opt. Lett. 34, 1456-1458 (2009).","doi":""},{"text":"Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator. R J Hernández-Hernández, R A Terborg, I Ricardez-Vargas, K Volke-Sepúlveda, Appl. Opt. 49R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas, and K. Volke-Sepúlveda, \"Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator,\" Appl. Opt. 49, 6903-6909 (2010).","doi":""},{"text":"Generation of the \"perfect\" optical vortex using a liquid-crystal spatial light modulator. A S Ostrovsky, C Rickenstorff-Parrao, V Arrizón, Opt. Lett. 38A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, \"Generation of the \"perfect\" optical vortex using a liquid-crystal spatial light modulator,\" Opt. Lett. 38, 534-536 (2013).","doi":""},{"text":"Theory of optical tweezers. P A M Neto, H M Nussenzveig, Europhys. Lett.). 50EPLP. A. M. Neto and H. M. Nussenzveig, \"Theory of optical tweezers,\" EPL (Europhys. Lett.) 50, 702-710 (2000).","doi":""},{"text":"Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory. A Rohrbach, Phys. Rev. Lett. 95168102A. Rohrbach, \"Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory,\" Phys. Rev. Lett. 95, 168102 (2005).","doi":""},{"text":"Absolute calibration of forces in optical tweezers. R S Dutra, N B Viana, P A M Neto, H M Nussenzveig, Phys. Rev. A. 9013825R. S. Dutra, N. B. Viana, P. A. M. Neto, and H. M. Nussenzveig, \"Absolute calibration of forces in optical tweezers,\" Phys. Rev. A 90, 013825 (2014).","doi":""},{"text":"Power spectrum analysis for optical tweezers. K Berg-Sorensen, H Flyvbjerg, Rev. Sci. Instrumen. 75K. Berg-Sorensen and H. Flyvbjerg, \"Power spectrum analysis for optical tweezers,\" Rev. Sci. Instrumen. 75, 594-612 (2004).","doi":""},{"text":"High-performance reconstruction of microscopic force fields from Brownian trajectories. L García, J Pérez, G Volpe, A V Arzola, G Volpe, Nat. Commun. 95166L. Pérez García, J. Donlucas Pérez, G. Volpe, A. V. Arzola, and G. Volpe, \"High-performance reconstruction of microscopic force fields from Brownian trajectories,\" Nat. Commun. 9, 5166 (2018).","doi":""},{"text":"Bayesian inference for estimating parameters of Brownian motion. I Castillo, G Volpe, L Pérez García, A V Arzola, preparationI. Pérez Castillo, G. Volpe, L. Pérez García, and A. V. Arzola, \"Bayesian inference for estimating parameters of Brownian motion,\" preparation (2019).","doi":""},{"text":"H Risken, The Fokker-Planck equation. SpringerH. Risken, The Fokker-Planck equation (Springer, 1996).","doi":""},{"text":"Properties of the noise-induced \"spurious\" drift. I. D Ryter, U Deker, J. Math. Phys. 21D. Ryter and U. Deker, \"Properties of the noise-induced \"spurious\" drift. I.\" J. Math. Phys. 21, 2662-2665 (1980).","doi":""},{"text":"Direct measurement of the nonconservative force field generated by optical tweezers. P Wu, R Huang, C Tischer, A Jonas, E.-L Florin, Phys. Rev. Lett. 103108101P. Wu, R. Huang, C. Tischer, A. Jonas, and E.-L. Florin, \"Direct measurement of the nonconservative force field generated by optical tweezers,\" Phys. Rev. Lett. 103, 108101 (2009).","doi":""},{"text":"Influence of noise on force measurements. G Volpe, L Helden, T Brettschneider, J Wehr, C Bechinger, Phys. Rev. Lett. 104170602G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger, \"Influence of noise on force measurements,\" Phys. Rev. Lett. 104, 170602 (2010).","doi":""},{"text":"Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus drift method. T Brettschneider, G Volpe, L Helden, J Wehr, C Bechinger, Phys. Rev. E. 8341113T. Brettschneider, G. Volpe, L. Helden, J. Wehr, and C. Bechinger, \"Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus drift method,\" Phys. Rev. E 83, 041113 (2011).","doi":""},{"text":"State-dependent diffusion: Thermodynamic consistency and its path integral formulation. A W C Lau, T C Lubensky, Phys. Rev. E. 7611123A. W. C. Lau and T. C. Lubensky, \"State-dependent diffusion: Thermodynamic consistency and its path integral formulation,\" Phys. Rev. E 76, 011123 (2007).","doi":""},{"text":"The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. S Hottovy, A Mcdaniel, G Volpe, J Wehr, Commun. Math. Phys. 336S. Hottovy, A. McDaniel, G. Volpe, and J. Wehr, \"The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction,\" Commun. Math. Phys. 336, 1259-1283 (2015).","doi":""},{"text":"Effective drifts in dynamical systems with multiplicative noise: A review of recent progress. G Volpe, J Wehr, Rep. Prog. Phys. 7953901G. Volpe and J. Wehr, \"Effective drifts in dynamical systems with multiplicative noise: A review of recent progress,\" Rep. Prog. Phys. 79, 053901 (2016).","doi":""},{"text":"Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process. R Singh, D Ghosh, R Adhikari, Phys. Rev. E. 9812136R. Singh, D. Ghosh, and R. Adhikari, \"Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process,\" Phys. Rev. E 98, 012136 (2018).","doi":""},{"text":"A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories. S Türkcan, A Alexandrou, J.-B Masson, Biophys. J. 102S. Türkcan, A. Alexandrou, and J.-B. Masson, \"A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories.\" Biophys. J. 102, 2288-2298 (2012).","doi":""},{"text":"Calibrating optical tweezers with Bayesian inference. M U Richly, S Türkcan, A Le Gall, N Fiszman, J Masson, N Westbrook, K Perronet, A Alexandrou, Opt. Express. 21M. U. Richly, S. Türkcan, A. Le Gall, N. Fiszman, J. Masson, N. Westbrook, K. Perronet, and A. Alexandrou, \"Calibrating optical tweezers with Bayesian inference,\" Opt. Express 21, 31578-31590 (2013).","doi":""},{"text":"A primer on the Bayesian approach to high-density single-molecule trajectories analysis. M El Beheiry, S Türkcan, M U Richly, A Triller, A Alexandrou, M Dahan, J.-B Masson, Biophys. J. 110M. El Beheiry, S. Türkcan, M. U. Richly, A. Triller, A. Alexandrou, M. Dahan, and J.-B. Masson, \"A primer on the Bayesian approach to high-density single-molecule trajectories analysis,\" Biophys. J. 110, 1209-1215 (2016).","doi":""},{"text":"Fast Bayesian inference of optical trap stiffness and particle diffusion. S Bera, S Paul, R Singh, D Ghosh, A Kundu, A Banerjee, R Adhikari, Sci. Rep. 741638S. Bera, S. Paul, R. Singh, D. Ghosh, A. Kundu, A. Banerjee, and R. Adhikari, \"Fast Bayesian inference of optical trap stiffness and particle diffusion,\" Sci. Rep. 7, 41638 (2017).","doi":""},{"text":"Torque detection using Brownian fluctuations. G Volpe, D Petrov, Phys. Rev. Lett. 97210603G. Volpe and D. Petrov, \"Torque detection using Brownian fluctuations,\" Phys. Rev. Lett. 97, 210603 (2006).","doi":""},{"text":"Brownian motion in a nonhomogeneous force field and photonic force microscope. G Volpe, G Volpe, D Petrov, Phys. Rev. E. 7661118G. Volpe, G. Volpe, and D. Petrov, \"Brownian motion in a nonhomogeneous force field and photonic force microscope,\" Phys. Rev. E 76, 061118 (2007).","doi":""},{"text":"Quantitative assessment of non-conservative radiation forces in an optical trap. G Pesce, G Volpe, A C De Luca, G Rusciano, G Volpe, Europhys. Lett.). 8638002EPLG. Pesce, G. Volpe, A. C. De Luca, G. Rusciano, and G. Volpe, \"Quantitative assessment of non-conservative radiation forces in an optical trap,\" EPL (Europhys. Lett.) 86, 38002 (2009).","doi":""},{"text":"Quantitative measurements of force and displacement using an optical trap. R M Simmons, J T Finer, S Chu, J A Spudich, Biophys. J. 70R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, \"Quantitative measurements of force and displacement using an optical trap,\" Biophys. J. 70, 1813-1822 (1996).","doi":""},{"text":"Calibration of optical tweezers with positional detection in the back focal plane. S F Tolić-Nørrelykke, E Schäffer, J Howard, F S Pavone, F J , H Flyvbjerg, Rev. Sci. Instrumen. 77103101S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. J., and H. Flyvbjerg, \"Calibration of optical tweezers with positional detection in the back focal plane,\" Rev. Sci. Instrumen. 77, 103101 (2006).","doi":""},{"text":"Optical-trap force transducer that operates by direct measurement of light momentum. S B Smith, Y Cui, C Bustamante, Methods in enzymology. 361ElsevierS. B. Smith, Y. Cui, and C. Bustamante, \"Optical-trap force transducer that operates by direct measurement of light momentum,\" in Methods in enzymology, vol. 361 (Elsevier, 2003), pp. 134-162.","doi":""},{"text":"A force detection technique for single-beam optical traps based on direct measurement of light momentum changes. A Farré, M Montes-Usategui, Opt. Express. 18A. Farré and M. Montes-Usategui, \"A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,\" Opt. Express 18, 11955-11968 (2010).","doi":""},{"text":"Direct measurement of axial optical forces. G Thalhammer, L Obmascher, M Ritsch-Marte, Opt. Express. 23G. Thalhammer, L. Obmascher, and M. Ritsch-Marte, \"Direct measurement of axial optical forces,\" Opt. Express 23, 6112-6129 (2015).","doi":""},{"text":"Everything you wanted to know about data analysis and fitting but were afraid to ask. P Young, SpringerP. Young, Everything you wanted to know about data analysis and fitting but were afraid to ask (Springer, 2015).","doi":""},{"text":"P J Huber, Robust Statistics. WileyP. J. Huber, Robust Statistics (Wiley, 1981).","doi":""},{"text":"Fitting a function to time-dependent ensemble averaged data. K Fogelmark, M A Lomholt, A Irbäck, T Ambjörnsson, Sci. Rep. 86984K. Fogelmark, M. A. Lomholt, A. Irbäck, and T. Ambjörnsson, \"Fitting a function to time-dependent ensemble averaged data,\" Sci. Rep. 8, 6984 (2018).","doi":""},{"text":"Single-molecule experiments in biological physics: Methods and applications. F Ritort, J. Phys.: Condens. Matter. 18F. Ritort, \"Single-molecule experiments in biological physics: Methods and applications,\" J. Phys.: Condens. Matter 18, R531-R583 (2006).","doi":""},{"text":"Single-molecule biophysics: At the interface of biology, physics and chemistry. A A Deniz, S Mukhopadhyay, E A Lemke, J. Royal Soc. Interfac. 5A. A. Deniz, S. Mukhopadhyay, and E. A. Lemke, \"Single-molecule biophysics: At the interface of biology, physics and chemistry,\" J. Royal Soc. Interfac. 5, 15-45 (2007).","doi":""},{"text":"Single-molecule micromanipulation techniques. K C Neuman, T Lionnet, J.-F Allemand, Annu. Rev. Mater. Res. 37K. C. Neuman, T. Lionnet, and J.-F. Allemand, \"Single-molecule micromanipulation techniques,\" Annu. Rev. Mater. Res. 37, 33-67 (2007).","doi":""},{"text":"Recent advances in optical tweezers. J R Moffitt, Y R Chemla, S B Smith, C Bustamante, Annu. Rev. Biochem. 77J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, \"Recent advances in optical tweezers,\" Annu. Rev. Biochem. 77, 205-228 (2008).","doi":""},{"text":"Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy. K C Neuman, A Nagy, Nat. Methods. 5K. C. Neuman and A. Nagy, \"Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy,\" Nat. Methods 5, 491-505 (2008).","doi":""},{"text":"High-resolution optical tweezers for single-molecule manipulation. X Zhang, L Ma, Y Zhang, Yale J. Biol. Med. 86X. Zhang, L. Ma, and Y. Zhang, \"High-resolution optical tweezers for single-molecule manipulation,\" Yale J. Biol. Med. 86, 367-383 (2013).","doi":""},{"text":"Interrogating biology with force: Single molecule high-resolution measurements with optical tweezers. M Capitanio, F S Pavone, Biophys. J. 105M. Capitanio and F. S. Pavone, \"Interrogating biology with force: Single molecule high-resolution measurements with optical tweezers,\" Biophys. J. 105, 1293-1303 (2013).","doi":""},{"text":"Single-molecule techniques in biophysics: A review of the progress in methods and applications. H Miller, Z Zhou, J Shepherd, A J M Wollman, M C Leake, Rep. Prog. Phys. 8124601H. Miller, Z. Zhou, J. Shepherd, A. J. M. Wollman, and M. C. Leake, \"Single-molecule techniques in biophysics: A review of the progress in methods and applications,\" Rep. Prog. Phys. 81, 024601 (2017).","doi":""},{"text":"Elastic properties of nucleic acids by single-molecule force spectroscopy. J Camunas-Soler, M Ribezzi-Crivellari, F Ritort, Annu. Rev. Biophys. 45J. Camunas-Soler, M. Ribezzi-Crivellari, and F. Ritort, \"Elastic properties of nucleic acids by single-molecule force spectroscopy,\" Annu. Rev. Biophys. 45, 65-84 (2016).","doi":""},{"text":"Reconstructing folding energy landscapes by single-molecule force spectroscopy. M T Woodside, S M Block, Annu. Rev. Biophys. 43M. T. Woodside and S. M. Block, \"Reconstructing folding energy landscapes by single-molecule force spectroscopy,\" Annu. Rev. Biophys. 43, 19-39 (2014).","doi":""},{"text":"The power of force: Insights into the protein folding process using single-molecule force spectroscopy. J Schönfelder, D De Sancho, R Perez-Jimenez, J. Mol. Biol. 428J. Schönfelder, D. De Sancho, and R. Perez-Jimenez, \"The power of force: Insights into the protein folding process using single-molecule force spectroscopy,\" J. Mol. Biol. 428, 4245-4257 (2016).","doi":""},{"text":"Single molecule high-throughput footprinting of small and large DNA ligands. M Manosas, J Camunas-Soler, V Croquette, F Ritort, Nat. Commun. 8304M. Manosas, J. Camunas-Soler, V. Croquette, and F. Ritort, \"Single molecule high-throughput footprinting of small and large DNA ligands,\" Nat. Commun. 8, 304 (2017).","doi":""},{"text":"Optical traps to study properties of molecular motors. J A Spudich, S E Rice, R S Rock, T J Purcell, H M Warrick, Cold Spring Harb. Protoc. 66662J. A. Spudich, S. E. Rice, R. S. Rock, T. J. Purcell, and H. M. Warrick, \"Optical traps to study properties of molecular motors,\" Cold Spring Harb. Protoc. 2011, pdb-top066662 (2011).","doi":""},{"text":"Nonequilibrium fluctuations in small systems: From physics to biology. F Ritort, Adv. Chem. Phys. 137F. Ritort, \"Nonequilibrium fluctuations in small systems: From physics to biology,\" Adv. Chem. Phys. 137, 31-123 (2008).","doi":""},{"text":"Experiments in stochastic thermodynamics: Short history and perspectives. S Ciliberto, Phys. Rev. X. 721051S. Ciliberto, \"Experiments in stochastic thermodynamics: Short history and perspectives,\" Phys. Rev. X 7, 021051 (2017).","doi":""},{"text":"A guide to magnetic tweezers and their applications. R Sarkar, V V Rybenkov, Front. Phys. 448R. Sarkar and V. V. Rybenkov, \"A guide to magnetic tweezers and their applications,\" Front. Phys. 4, 48 (2016).","doi":""},{"text":"Single molecule force spectroscopy in biology using the atomic force microscope. J Zlatanova, S M Lindsay, S H Leuba, Prog. Biophys. Mol. Biol. 74J. Zlatanova, S. M. Lindsay, and S. H. Leuba, \"Single molecule force spectroscopy in biology using the atomic force microscope,\" Prog. Biophys. Mol. Biol. 74, 37-61 (2000).","doi":""},{"text":"Acoustic force spectroscopy. G Sitters, D Kamsma, G Thalhammer, M Ritsch-Marte, E J G Peterman, G J L Wuite, Nat. Methods. 12G. Sitters, D. Kamsma, G. Thalhammer, M. Ritsch-Marte, E. J. G. Peterman, and G. J. L. Wuite, \"Acoustic force spectroscopy,\" Nat. Methods 12, 47-50 (2015).","doi":""},{"text":"Single-molecule electrometry. F Ruggeri, F Zosel, N Mutter, M Różycka, M Wojtas, A Ożyhar, B Schuler, M Krishnan, Nat. Nanotech. 12F. Ruggeri, F. Zosel, N. Mutter, M. Różycka, M. Wojtas, A. Ożyhar, B. Schuler, and M. Krishnan, \"Single-molecule electrometry,\" Nat. Nanotech. 12, 488-495 (2017).","doi":""},{"text":"Microfluidics for biological measurements with single-molecule resolution. A M Streets, Y Huang, Curr. Opn. Biotechnol. 25A. M. Streets and Y. Huang, \"Microfluidics for biological measurements with single-molecule resolution,\" Curr. Opn. Biotechnol. 25, 69-77 (2014).","doi":""},{"text":"Single-molecule measurements using microneedles. T Yanagida, Y Ishii, A Ishijima, Single Molecule Enzymology. SpringerT. Yanagida, Y. Ishii, and A. Ishijima, \"Single-molecule measurements using microneedles,\" in Single Molecule Enzymology, (Springer, 2011), pp. 143-159.","doi":""},{"text":"Sensitive force technique to probe molecular adhesion and structural linkages at biological interfaces. E Evans, K Ritchie, R Merkel, Biophys. J. 68E. Evans, K. Ritchie, and R. Merkel, \"Sensitive force technique to probe molecular adhesion and structural linkages at biological interfaces,\" Biophys. J. 68, 2580-2587 (1995).","doi":""},{"text":"Massively parallel single-molecule manipulation using centrifugal force. W P Wong, K Halvorsen, Biophys. J. 100W. P. Wong and K. Halvorsen, \"Massively parallel single-molecule manipulation using centrifugal force,\" Biophys. J. 100, L53-L55 (2011).","doi":""},{"text":"Counter-propagating dual-trap optical tweezers based on linear momentum conservation. M Ribezzi-Crivellari, J M Huguet, F Ritort, Rev. Sci. Instrumen. 8443104M. Ribezzi-Crivellari, J. M. Huguet, and F. Ritort, \"Counter-propagating dual-trap optical tweezers based on linear momentum conservation,\" Rev. Sci. Instrumen. 84, 043104 (2013).","doi":""},{"text":"Statistical and thermodynamic properties of DNA unzipping experiments with optical tweezers. J M Huguet, Universitat de BarcelonaPh.D. thesisJ. M. Huguet, \"Statistical and thermodynamic properties of DNA unzipping experiments with optical tweezers,\" Ph.D. thesis, Universitat de Barcelona (2010).","doi":""},{"text":"Assemblies: Pipette Puller. U Di Parma, U. di Parma, \"Assemblies: Pipette Puller,\" http://tweezerslab.unipr.it/cgi-bin/mt/ assemblies.pl/Show?_id=7c1b. Accessed: 2019-12-12.","doi":""},{"text":"P Nelson, Biological physics. New YorkWH FreemanP. Nelson, Biological physics (WH Freeman, New York, 2004).","doi":""},{"text":"The structural basis of antibody-antigen recognition. I Sela-Culang, V Kunik, Y Ofran, Front. Immunol. 4302I. Sela-Culang, V. Kunik, and Y. Ofran, \"The structural basis of antibody-antigen recognition,\" Front. Immunol. 4, 302 (2013).","doi":""},{"text":"How the biotin-streptavidin interaction was made even stronger: investigation via crystallography and a chimaeric tetramer. C E Chivers, A L Koner, E D Lowe, M Howarth, Biochem. J. 435C. E. Chivers, A. L. Koner, E. D. Lowe, and M. Howarth, \"How the biotin-streptavidin interaction was made even stronger: investigation via crystallography and a chimaeric tetramer,\" Biochem. J. 435, 55-63 (2011).","doi":""},{"text":"The solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds. F Pincet, J Husson, Biophys. J. 89F. Pincet and J. Husson, \"The solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds,\" Biophys. J. 89, 4374-4381 (2005).","doi":""},{"text":"Improving signal/noise resolution in single-molecule experiments using molecular constructs with short handles. N Forns, S Lorenzo, M Manosas, K Hayashi, J M Huguet, F Ritort, Biophys. J. 100N. Forns, S. de Lorenzo, M. Manosas, K. Hayashi, J. M. Huguet, and F. Ritort, \"Improving signal/noise resolution in single-molecule experiments using molecular constructs with short handles,\" Biophys. J. 100, 1765-1774 (2011).","doi":""},{"text":"Stochastic thermodynamics, fluctuation theorems and molecular machines. U Seifert, Rep. Prog. Phys. 75126001U. Seifert, \"Stochastic thermodynamics, fluctuation theorems and molecular machines,\" Rep. Prog. Phys. 75, 126001 (2012).","doi":""},{"text":"Stretching DNA. J F Marko, E D Siggia, Macromolecules. 28J. F. Marko and E. D. Siggia, \"Stretching DNA,\" Macromolecules 28, 8759-8770 (1995).","doi":""},{"text":"Determination of the elastic properties of short ssDNA molecules by mechanically folding and unfolding DNA hairpins. A Alemany, F Ritort, Biopolymers. 101A. Alemany and F. Ritort, \"Determination of the elastic properties of short ssDNA molecules by mechanically folding and unfolding DNA hairpins,\" Biopolymers 101, 1193-1199 (2014).","doi":""},{"text":". R R Sinden, DNA Structure and Function. Elsevier ScienceR. R. Sinden, DNA Structure and Function (Elsevier Science, 1994).","doi":""},{"text":"A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics. J Santalucia, Proc. Natl. Acad. Sci. 95J. SantaLucia, \"A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics,\" Proc. Natl. Acad. Sci. 95, 1460-1465 (1998).","doi":""},{"text":"Holographic acoustic tweezers. A Marzo, B W Drinkwater, Proc. Natl. Acad. Sci. Natl. Acad. Sci116A. Marzo and B. W. Drinkwater, \"Holographic acoustic tweezers,\" Proc. Natl. Acad. Sci. 116, 84-89 (2019).","doi":""},{"text":"Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions. A Candelli, G J L Wuite, E J G Peterman, Phys. Chem. Chem. Phys. 13A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, \"Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,\" Phys. Chem. Chem. Phys. 13, 7263-7272 (2011).","doi":""},{"text":"Combining single-molecule manipulation and single-molecule detection. J C Cordova, D K Das, H W Manning, M J Lang, Curr. Opn. Struct. Biol. 28J. C. Cordova, D. K. Das, H. W. Manning, and M. J. Lang, \"Combining single-molecule manipulation and single-molecule detection,\" Curr. Opn. Struct. Biol. 28, 142-148 (2014).","doi":""},{"text":"High-resolutio \"Fleezers\": Dual-trap optical tweezers combined with single-molecule fluorescence detection. K D Whitley, M J Comstock, Y R Chemla, Methods Mol. Biol. pp. K. D. Whitley, M. J. Comstock, and Y. R. Chemla, \"High-resolutio \"Fleezers\": Dual-trap optical tweezers combined with single-molecule fluorescence detection,\" Methods Mol. Biol. pp. 183-256 (2017).","doi":""},{"text":"Direct force measurements on DNA in a solid-state nanopore. U F Keyser, B N Koeleman, S Van Dorp, D Krapf, R M M Smeets, S G Lemay, N H Dekker, C Dekker, Nat. Phys. 2U. F. Keyser, B. N. Koeleman, S. Van Dorp, D. Krapf, R. M. M. Smeets, S. G. Lemay, N. H. Dekker, and C. Dekker, \"Direct force measurements on DNA in a solid-state nanopore,\" Nat. Phys. 2, 473-477 (2006).","doi":""},{"text":"Measurement of the position-dependent electrophoretic force on DNA in a glass nanocapillary. R D Bulushev, L J Steinbock, S Khlybov, J F Steinbock, U F Keyser, A Radenovic, Nanolett. 14R. D. Bulushev, L. J. Steinbock, S. Khlybov, J. F. Steinbock, U. F. Keyser, and A. Radenovic, \"Measurement of the position-dependent electrophoretic force on DNA in a glass nanocapillary,\" Nanolett. 14, 6606-6613 (2014).","doi":""},{"text":"Optical trapping and manipulation of viruses and bacteria. A Ashkin, J M Dziedzic, Science. 235A. Ashkin and J. M. Dziedzic, \"Optical trapping and manipulation of viruses and bacteria,\" Science 235, 1517-1520 (1987).","doi":""},{"text":"Optical trapping and manipulation of single cells using infrared laser beams. A Ashkin, J M Dziedzic, T Yamane, Nature. 330A. Ashkin, J. M. Dziedzic, and T. Yamane, \"Optical trapping and manipulation of single cells using infrared laser beams,\" Nature 330, 769-771 (1987).","doi":""},{"text":"Compliance of bacterial flagella measured with optical tweezers. S M Block, D F Blair, H C Berg, Nature. 338S. M. Block, D. F. Blair, and H. C. Berg, \"Compliance of bacterial flagella measured with optical tweezers,\" Nature 338, 514-518 (1989).","doi":""},{"text":"Quantifying forces in cell biology. P Roca-Cusachs, V Conte, X Trepat, Nat. Cell Biol. 19P. Roca-Cusachs, V. Conte, and X. Trepat, \"Quantifying forces in cell biology,\" Nat. Cell Biol. 19, 742-751 (2017).","doi":""},{"text":"Biomechanical characterization at the cell scale: present and prospects. F Basoli, S M Giannitelli, P Mozetic, M Gori, A Bonfanti, M Trombetta, A Rainer, Front. Physiol. 91449F. Basoli, S. M. Giannitelli, P. Mozetic, M. Gori, A. Bonfanti, M. Trombetta, and A. Rainer, \"Biomechanical characterization at the cell scale: present and prospects,\" Front. Physiol. 9, 1449 (2018).","doi":""},{"text":"Force nanoscopy of living cells. D J Müller, Y F Dufrêne, Curr. Biol. 21D. J. Müller and Y. F. Dufrêne, \"Force nanoscopy of living cells,\" Curr. Biol, 21, R212-R216 (2011).","doi":""},{"text":"Force nanoscopy of cell mechanics and cell adhesion. Y F Dufrêne, A E Pelling, Nanoscale. 5Y. F. Dufrêne and A. E. Pelling, \"Force nanoscopy of cell mechanics and cell adhesion,\" Nanoscale 5, 4094-4104 (2013).","doi":""},{"text":"Exploring mechanochemical processes in the cell with optical tweezers. S Hormeno, J R Arias-Gonzalez, Biol. Cell. 98S. Hormeno and J. R. Arias-Gonzalez, \"Exploring mechanochemical processes in the cell with optical tweezers,\" Biol. Cell 98, 679-695 (2006).","doi":""},{"text":"Optical trapping: A review of essential concepts. I Verdeny, A Farré, J Soler, C Quesada, E M Badosa, M Montes Usategui, Óptica Pura Apl. 44I. Verdeny, A. Farré, J. Mas Soler, C. López Quesada, E. M. Badosa, and M. Montes Usategui, \"Optical trapping: A review of essential concepts,\" Óptica Pura Apl. 44, 527-551 (2011).","doi":""},{"text":"Probing force in living cells with optical tweezers: From single-molecule mechanics to cell mechanotransduction. C Arbore, L Perego, M Sergides, M Capitanio, Biophys. Rev pp. C. Arbore, L. Perego, M. Sergides, and M. Capitanio, \"Probing force in living cells with optical tweezers: From single-molecule mechanics to cell mechanotransduction,\" Biophys. Rev pp. 1-18 (2019).","doi":""},{"text":"Particle-tracking microrheology of living cells: Principles and applications. D Wirtz, Annu. Rev. Biophys. 38D. Wirtz, \"Particle-tracking microrheology of living cells: Principles and applications,\" Annu. Rev. Biophys. 38, 301-326 (2009).","doi":""},{"text":"Single particle tracking: From theory to biophysical applications. H Shen, L J Tauzin, R Baiyasi, W Wang, N Moringo, B Shuang, C F Landes, Chem. Rev. 117H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, \"Single particle tracking: From theory to biophysical applications,\" Chem. Rev. 117, 7331-7376 (2017).","doi":""},{"text":"Traction force microscopy on soft elastic substrates: A guide to recent computational advances. U S Schwarz, J R D Soiné, Biochimica Biophys. Acta. 1853U. S. Schwarz and J. R. D. Soiné, \"Traction force microscopy on soft elastic substrates: A guide to recent computational advances,\" Biochimica Biophys. Acta 1853, 3095-3104 (2015).","doi":""},{"text":"The future of traction force microscopy. H Colin-York, M Fritzsche, Curr. Opn. Biomed. Eng. 5H. Colin-York and M. Fritzsche, \"The future of traction force microscopy,\" Curr. Opn. Biomed. Eng. 5, 1-5 (2018).","doi":""},{"text":"Recent technological advancements in traction force microscopy. A Ferrari, Biophys. Rev. 11A. Ferrari, \"Recent technological advancements in traction force microscopy,\" Biophys. Rev. 11, 679-681 (2019).","doi":""},{"text":"Measurement of cell microrheology by magnetic twisting cytometry with frequency domain demodulation. M Puig-De-Morales, M Grabulosa, J Alcaraz, J Mullol, G N Maksym, J J Fredberg, D Navajas, J. Appl. Physiol. 91M. Puig-De-Morales, M. Grabulosa, J. Alcaraz, J. Mullol, G. N. Maksym, J. J. Fredberg, and D. Navajas, \"Measurement of cell microrheology by magnetic twisting cytometry with frequency domain demodulation,\" J. Appl. Physiol. 91, 1152-1159 (2001).","doi":""},{"text":"Probing cellular mechanics with acoustic force spectroscopy. R Sorkin, G Bergamaschi, D Kamsma, G Brand, E Dekel, Y Ofir-Birin, A Rudik, M Gironella, F Ritort, N Regev-Rudzki, W H Roos, G J L Wuite, Mol. Biol. Cell. 29R. Sorkin, G. Bergamaschi, D. Kamsma, G. Brand, E. Dekel, Y. Ofir-Birin, A. Rudik, M. Gironella, F. Ritort, N. Regev-Rudzki, W. H. Roos, and G. J. L. Wuite, \"Probing cellular mechanics with acoustic force spectroscopy,\" Mol. Biol. Cell 29, 2005-2011 (2018).","doi":""},{"text":"The application of micropipette aspiration in molecular mechanics of single cells. L M Lee, A P Liu, J. Nanotech. Eng. Med. 540902L. M. Lee and A. P. Liu, \"The application of micropipette aspiration in molecular mechanics of single cells,\" J. Nanotech. Eng. Med. 5, 040902 (2014).","doi":""},{"text":"ATP-dependent mechanics of red blood cells. T Betz, M Lenz, J.-F Joanny, C Sykes, Proc. Natl. Acad. Sci. 106T. Betz, M. Lenz, J.-F. Joanny, and C. Sykes, \"ATP-dependent mechanics of red blood cells,\" Proc. Natl. Acad. Sci. 106, 15320-15325 (2009).","doi":""},{"text":"Equilibrium physics breakdown reveals the active nature of red blood cell flickering. H Turlier, D A Fedosov, B Audoly, T Auth, N S Gov, C Sykes, J.-F Joanny, G Gompper, T Betz, Nat. Phys. 12H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, \"Equilibrium physics breakdown reveals the active nature of red blood cell flickering,\" Nat. Phys. 12, 513-519 (2016).","doi":""},{"text":"Elastic properties and secondary structure formation of single-stranded DNA at monovalent and divalent salt conditions. A Bosco, J Camunas-Soler, F Ritort, Nucleic Acids Res. 42A. Bosco, J. Camunas-Soler, and F. Ritort, \"Elastic properties and secondary structure formation of single-stranded DNA at monovalent and divalent salt conditions,\" Nucleic Acids Res. 42, 2064-2074 (2013).","doi":""},{"text":"Mechanism of red blood cell aging: Relationship of cell density and cell age. S Piomelli, C Seaman, Am. J. Hematol. 42S. Piomelli and C. Seaman, \"Mechanism of red blood cell aging: Relationship of cell density and cell age,\" Am. J. Hematol. 42, 46-52 (1993).","doi":""},{"text":"Advances in the microrheology of complex fluids. T A Waigh, Rep. Prog. Phys. 7974601T. A. Waigh, \"Advances in the microrheology of complex fluids,\" Rep. Prog. Phys. 79, 074601 (2016).","doi":""},{"text":"R B Bird, R C Armstrong, O Hassager, Fluid mechanics. New YorkJohn Wiley & Sons1R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of polymeric liquids. Vol. 1: Fluid mechanics (John Wiley & Sons, New York, 1987).","doi":""},{"text":"High frequency viscoelastic behaviour of low molecular weight hyaluronic acid water solutions. S Fusco, A Borzacchiello, L Miccio, G Pesce, G Rusciano, A Sasso, P A Netti, Biorheol. 44S. Fusco, A. Borzacchiello, L. Miccio, G. Pesce, G. Rusciano, A. Sasso, and P. A. Netti, \"High frequency viscoelastic behaviour of low molecular weight hyaluronic acid water solutions,\" Biorheol. 44, 403-418 (2007).","doi":""},{"text":"Microrheology of complex fluids using optical tweezers: A comparison with macrorheological measurements. G Pesce, A C De Luca, G Rusciano, P A Netti, S Fusco, A Sasso, J. Opt. A Pure Appl. Opt. 1134016G. Pesce, A. C. De Luca, G. Rusciano, P. A. Netti, S. Fusco, and A. Sasso, \"Microrheology of complex fluids using optical tweezers: A comparison with macrorheological measurements,\" J. Opt. A Pure Appl. Opt. 11, 034016 (2009).","doi":""},{"text":"Fluid mechanics of microrheology. T M Squires, T G Mason, Annu. Rev. Fluid Mech. 42T. M. Squires and T. G. Mason, \"Fluid mechanics of microrheology,\" Annu. Rev. Fluid Mech. 42, 413-438 (2010).","doi":""},{"text":"Microrheology with optical tweezers: data analysis. M Tassieri, R M L Evans, R L Warren, N J Bailey, J M Cooper, New J. Phys. 14115032M. Tassieri, R. M. L. Evans, R. L. Warren, N. J. Bailey, and J. M. Cooper, \"Microrheology with optical tweezers: data analysis,\" New J. Phys. 14, 115032 (2012).","doi":""},{"text":"The fluctuation-dissipation theorem. R Kubo, Rep. Prog. Phys. 29R. Kubo, \"The fluctuation-dissipation theorem,\" Rep. Prog. Phys. 29, 255-284 (1966).","doi":""},{"text":"Oscillatory rise of bubbles in wormlike micellar fluids with different microstructures. N Z Handzy, A Belmonte, Phys. Rev. Lett. 92124501N. Z. Handzy and A. Belmonte, \"Oscillatory rise of bubbles in wormlike micellar fluids with different microstructures,\" Phys. Rev. Lett. 92, 124501 (2004).","doi":""},{"text":"Manual of symbols and terminology for physicochemical quantities and units, appendix II: Definitions, terminology and symbols in colloid and surface chemistry. D H Everett, Pure Appl. Chem. 31D. H. Everett, \"Manual of symbols and terminology for physicochemical quantities and units, appendix II: Definitions, terminology and symbols in colloid and surface chemistry,\" Pure Appl. Chem. 31, 577-638 (1972).","doi":""},{"text":"Statics and dynamics of worm-like surfactant micelles. M E Cates, S J Candau, J. Phys.: Condens. Matter. 2M. E. Cates and S. J. Candau, \"Statics and dynamics of worm-like surfactant micelles,\" J. Phys.: Condens. Matter 2, 6869-6892 (1990).","doi":""},{"text":"Microrheology with optical tweezers: Measuring the relative viscosity of solutions \"at a glance. M Tassieri, F Giudice, E J Robertson, N Jain, B Fries, R Wilson, A Glidle, F Greco, P A Netti, P L Maffettone, T Bicanic, C J M , Sci. Rep. 58831M. Tassieri, F. Del Giudice, E. J. Robertson, N. Jain, B. Fries, R. Wilson, A. Glidle, F. Greco, P. A. Netti, P. L. Maffettone, T. Bicanic, and C. J. M., \"Microrheology with optical tweezers: Measuring the relative viscosity of solutions \"at a glance\",\" Sci. Rep. 5, 8831 (2015).","doi":""},{"text":"R G Larson, The structure and rheology of complex fluids. New YorkOxford University PressR. G. Larson, The structure and rheology of complex fluids (Oxford University Press, New York, 1999).","doi":""},{"text":"Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations. B Schnurr, F Gittes, F C Mackintosh, C F Schmidt, Macromolecules. 30B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, \"Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations,\" Macromolecules 30, 7781-7792 (1997).","doi":""},{"text":"A symmetrical method to obtain shear moduli from microrheology. K Nishi, M L Kilfoil, C F Schmidt, F C Mackintosh, Soft Matter. 14K. Nishi, M. L. Kilfoil, C. F. Schmidt, and F. C. MacKintosh, \"A symmetrical method to obtain shear moduli from microrheology,\" Soft Matter 14, 3716-3723 (2018).","doi":""},{"text":"Small-world rheology: An introduction to probe-based active microrheology. L G Wilson, W C K Poon, Phys. Chem. Chem. Phys. 13L. G. Wilson and W. C. K. Poon, \"Small-world rheology: An introduction to probe-based active microrheology,\" Phys. Chem. Chem. Phys. 13, 10617-10630 (2011).","doi":""},{"text":"Active and passive microrheology in equilibrium and nonequilibrium systems. D Mizuno, D A Head, F C Mackintosh, C F Schmidt, Macromolecules. 41D. Mizuno, D. A. Head, F. C. MacKintosh, and C. F. Schmidt, \"Active and passive microrheology in equilibrium and nonequilibrium systems,\" Macromolecules 41, 7194-7202 (2008).","doi":""},{"text":"Power spectrum of out-of-equilibrium forces in living cells: Amplitude and frequency dependence. F Gallet, D Arcizet, P Bohec, A Richert, Soft Matter. 5F. Gallet, D. Arcizet, P. Bohec, and A. Richert, \"Power spectrum of out-of-equilibrium forces in living cells: Amplitude and frequency dependence,\" Soft Matter 5, 2947-2953 (2009).","doi":""},{"text":"Nucleation and growth of thermoreversible polymer gels. J R Gomez-Solano, V Blickle, C Bechinger, Phys. Rev. E. 8712308J. R. Gomez-Solano, V. Blickle, and C. Bechinger, \"Nucleation and growth of thermoreversible polymer gels,\" Phys. Rev. E 87, 012308 (2013).","doi":""},{"text":"Experimental study of out-of-equilibrium fluctuations in a colloidal suspension of laponite using optical traps. P Jop, J R Gomez-Solano, A Petrosyan, S Ciliberto, J. Stat. Mech.: Theo. Exp. 4012P. Jop, J. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, \"Experimental study of out-of-equilibrium fluctuations in a colloidal suspension of laponite using optical traps,\" J. Stat. Mech.: Theo. Exp. 2009, P04012 (2009).","doi":""},{"text":"Observation of strongly heterogeneous dynamics at the depinning transition in a colloidal glass. N Şenbil, M Gruber, C Zhang, M Fuchs, F Scheffold, Phys. Rev. Lett. 122108002N. Şenbil, M. Gruber, C. Zhang, M. Fuchs, and F. Scheffold, \"Observation of strongly heterogeneous dynamics at the depinning transition in a colloidal glass,\" Phys. Rev. Lett. 122, 108002 (2019).","doi":""},{"text":"Brownian dynamics of a sphere between parallel walls. E R Dufresne, D Altman, D G Grier, Europhys. Lett.). 53EPLE. R. Dufresne, D. Altman, and D. G. Grier, \"Brownian dynamics of a sphere between parallel walls,\" EPL (Europhys. Lett.) 53, 264-270 (2001).","doi":""},{"text":"Hydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces. G M Wang, R Prabhakar, E M Sevick, Phys. Rev. Lett. 103248303G. M. Wang, R. Prabhakar, and E. M. Sevick, \"Hydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces,\" Phys. Rev. Lett. 103, 248303 (2009).","doi":""},{"text":"Measurement of monolayer viscosity using noncontact microrheology. R Shlomovitz, A A Evans, T Boatwright, M Dennin, A J Levine, Phys. Rev. Lett. 110137802R. Shlomovitz, A. A. Evans, T. Boatwright, M. Dennin, and A. J. Levine, \"Measurement of monolayer viscosity using noncontact microrheology,\" Phys. Rev. Lett. 110, 137802 (2013).","doi":""},{"text":"Two-point microrheology of inhomogeneous soft materials. J C Crocker, M T Valentine, E R Weeks, T Gisler, P D Kaplan, A G Yodh, D A Weitz, Phys. Rev. Lett. 85J. C. Crocker, M. T. Valentine, E. R. Weeks, T. Gisler, P. D. Kaplan, A. G. Yodh, and D. A. Weitz, \"Two-point microrheology of inhomogeneous soft materials,\" Phys. Rev. Lett. 85, 888-891 (2000).","doi":""},{"text":"Active microrheology to determine viscoelastic parameters of Stokes-Oldroyd B fluids using optical tweezers. S Paul, A Kundu, A Banerjee, J. Phys. Commun. 335002S. Paul, A. Kundu, and A. Banerjee, \"Active microrheology to determine viscoelastic parameters of Stokes-Oldroyd B fluids using optical tweezers,\" J. Phys. Commun. 3, 035002 (2019).","doi":""},{"text":"Nonlinear microrheology reveals entanglement-driven molecularlevel viscoelasticity of concentrated DNA. C D Chapman, R M Robertson-Anderson, Phys. Rev. Lett. 11398303C. D. Chapman and R. M. Robertson-Anderson, \"Nonlinear microrheology reveals entanglement-driven molecular- level viscoelasticity of concentrated DNA,\" Phys. Rev. Lett. 113, 098303 (2014).","doi":""},{"text":"Transient dynamics of a colloidal particle driven through a viscoelastic fluid. J R Gomez-Solano, C Bechinger, New J. Phys. 17103032J. R. Gomez-Solano and C. Bechinger, \"Transient dynamics of a colloidal particle driven through a viscoelastic fluid,\" New J. Phys. 17, 103032 (2015).","doi":""},{"text":"Active and passive microrheology: Theory and simulation. R N Zia, Ann. Rev. Fluid Mech. 50R. N. Zia, \"Active and passive microrheology: Theory and simulation,\" Ann. Rev. Fluid Mech. 50, 371-405 (2018).","doi":""},{"text":"An introduction to dynamics of colloids. J K G Dhont, ElsevierJ. K. G. Dhont, An introduction to dynamics of colloids (Elsevier, 1996).","doi":""},{"text":"Colloids as big atoms. W Poon, Science. 304W. Poon, \"Colloids as big atoms,\" Science 304, 830-831 (2004).","doi":""},{"text":"Insights into phase transition kinetics from colloid science. V J Anderson, H N W Lekkerkerker, Nature. 416V. J. Anderson and H. N. W. Lekkerkerker, \"Insights into phase transition kinetics from colloid science,\" Nature 416, 811-815 (2002).","doi":""},{"text":"Phase behaviour of concentrated suspensions of nearly hard colloidal spheres. P N Pusey, W Van Megen, Nature. 320P. N. Pusey and W. Van Megen, \"Phase behaviour of concentrated suspensions of nearly hard colloidal spheres,\" Nature 320, 340-342 (1986).","doi":""},{"text":"Prediction of absolute crystal-nucleation rate in hard-sphere colloids. S Auer, D Frenkel, Nature. 409S. Auer and D. Frenkel, \"Prediction of absolute crystal-nucleation rate in hard-sphere colloids,\" Nature 409, 1020-1023 (2001).","doi":""},{"text":"Direct observation of pre-critical nuclei in a metastable hard-sphere fluid. M S Elliot, S B Haddon, W C K Poon, J. Phys.: Condens. Matter. 13M. S. Elliot, S. B. Haddon, and W. C. K. Poon, \"Direct observation of pre-critical nuclei in a metastable hard-sphere fluid,\" J. Phys.: Condens. Matter 13, L553-L368 (2001).","doi":""},{"text":"Real-space imaging of nucleation and growth in colloidal crystallization. U Gasser, E R Weeks, A Schofield, P N Pusey, D A Weitz, Science. 292U. Gasser, E. R. Weeks, A. Schofield, P. N. Pusey, and D. A. Weitz, \"Real-space imaging of nucleation and growth in colloidal crystallization,\" Science 292, 258-262 (2001).","doi":""},{"text":"Multiple glassy states in a simple model system. K N Pham, A M Puertas, J Bergenholtz, S U Egelhaaf, A Moussaıd, P N Pusey, A B Schofield, M E Cates, M Fuchs, W C K Poon, Science. 296K. N. Pham, A. M. Puertas, J. Bergenholtz, S. U. Egelhaaf, A. Moussaıd, P. N. Pusey, A. B. Schofield, M. E. Cates, M. Fuchs, and W. C. K. Poon, \"Multiple glassy states in a simple model system,\" Science 296, 104-106 (2002).","doi":""},{"text":"Direct visual observation of thermal capillary waves. D G A L Aarts, M Schmidt, H N W Lekkerkerker, Science. 304D. G. A. L. Aarts, M. Schmidt, and H. N. W. Lekkerkerker, \"Direct visual observation of thermal capillary waves,\" Science 304, 847-850 (2004).","doi":""},{"text":"Ultrastable particle-stabilized foams. U T Gonzenbach, A R Studart, E Tervoort, L J Gauckler, Ang. Chemie Int. Ed. 45U. T. Gonzenbach, A. R. Studart, E. Tervoort, and L. J. Gauckler, \"Ultrastable particle-stabilized foams,\" Ang. Chemie Int. Ed. 45, 3526-3530 (2006).","doi":""},{"text":"Universal emulsion stabilization from the arrested adsorption of rough particles at liquid-liquid interfaces. M Zanini, C Marschelke, S E Anachkov, E Marini, A Synytska, L Isa, Nat. Commun. 815701M. Zanini, C. Marschelke, S. E. Anachkov, E. Marini, A. Synytska, and L. Isa, \"Universal emulsion stabilization from the arrested adsorption of rough particles at liquid-liquid interfaces,\" Nat. Commun. 8, 15701 (2017).","doi":""},{"text":"E Dickinson, D J Mcclements, Advances in food colloids. Springer Science & Business MediaE. Dickinson and D. J. McClements, Advances in food colloids (Springer Science & Business Media, 1995).","doi":""},{"text":"Chiral colloidal clusters. D Zerrouki, J Baudry, D Pine, P Chaikin, J Bibette, Nature. 455D. Zerrouki, J. Baudry, D. Pine, P. Chaikin, and J. Bibette, \"Chiral colloidal clusters,\" Nature 455, 380-382 (2008).","doi":""},{"text":"Van der Waals, Over de Continuiteit van den Gas-en Vloeistoftoestand (Sijthoff, 1873). J , J. D. Van der Waals, Over de Continuiteit van den Gas-en Vloeistoftoestand (Sijthoff, 1873).","doi":""},{"text":"Van der Waals forces: a handbook for biologists, chemists, engineers, and physicists. V Parsegian, Cambridge University PressV. Parsegian, Van der Waals forces: a handbook for biologists, chemists, engineers, and physicists (Cambridge University Press, 2005).","doi":""},{"text":"Diffuse double-layer models, long-range forces, and ordering in clay colloids. M B Mcbride, P Baveye, Soil Sci. Soc. Am. J. 66M. B. McBride and P. Baveye, \"Diffuse double-layer models, long-range forces, and ordering in clay colloids,\" Soil Sci. Soc. Am. J. 66, 1207-1217 (2002).","doi":""},{"text":"Importance of charge regulation in attractive double-layer forces between dissimilar surfaces. I Popa, P Sinha, M Finessi, P Maroni, G Papastavrou, M Borkovec, Phys. Rev. Lett. 104228301I. Popa, P. Sinha, M. Finessi, P. Maroni, G. Papastavrou, and M. Borkovec, \"Importance of charge regulation in attractive double-layer forces between dissimilar surfaces,\" Phys. Rev. Lett. 104, 228301 (2010).","doi":""},{"text":"J N Israelachvili, Intermolecular and surface forces. Academic PressJ. N. Israelachvili, Intermolecular and surface forces (Academic Press, 2015).","doi":""},{"text":"W B Russel, W Russel, D A Saville, W R Schowalter, Colloidal dispersions. Cambridge University PressW. B. Russel, W. Russel, D. A. Saville, and W. R. Schowalter, Colloidal dispersions (Cambridge University Press, 1991).","doi":""},{"text":"Direct measurement of colloidal forces using an atomic force microscope. W A Ducker, T J Senden, R M Pashley, Nature. 353W. A. Ducker, T. J. Senden, and R. M. Pashley, \"Direct measurement of colloidal forces using an atomic force microscope,\" Nature 353, 239-241 (1991).","doi":""},{"text":"Flocculation of protein-stabilized oil-in-water emulsions. E Dickinson, Colloids Surfaces B: Biointerfaces. 81E. Dickinson, \"Flocculation of protein-stabilized oil-in-water emulsions,\" Colloids Surfaces B: Biointerfaces 81, 130-140 (2010).","doi":""},{"text":"Modern aspects of emulsion science. B P Binks, Royal Society of ChemistryB. P. Binks, Modern aspects of emulsion science (Royal Society of Chemistry, 2007).","doi":""},{"text":"Creaming and flocculation in emulsions containing polysaccharide. Y Cao, E Dickinson, D J Wedlock, Food Hydrocoll. 4Y. Cao, E. Dickinson, and D. J. Wedlock, \"Creaming and flocculation in emulsions containing polysaccharide,\" Food Hydrocoll. 4, 185-195 (1990).","doi":""},{"text":"Coagulation/flocculation process for dye removal using sludge from water treatment plant: optimization through response surface methodology. S S Moghaddam, M R A Moghaddam, M Arami, J. Hazard. Mater. 175S. S. Moghaddam, M. R. A. Moghaddam, and M. Arami, \"Coagulation/flocculation process for dye removal using sludge from water treatment plant: optimization through response surface methodology,\" J. Hazard. Mater. 175, 651-657 (2010).","doi":""},{"text":"Coagulation and flocculation in water and wastewater treatment. J Bratby, IWA publishingJ. Bratby, Coagulation and flocculation in water and wastewater treatment (IWA publishing, 2016).","doi":""},{"text":"On interaction between two bodies immersed in a solution of macromolecules. S Asakura, F Oosawa, J. Chem. Phys. 22S. Asakura and F. Oosawa, \"On interaction between two bodies immersed in a solution of macromolecules,\" J. Chem. Phys. 22, 1255-1256 (1954).","doi":""},{"text":"J.-L Barrat, J.-P Hansen, Basic concepts for simple and complex liquids. Cambridge University PressJ.-L. Barrat and J.-P. Hansen, Basic concepts for simple and complex liquids (Cambridge University Press, 2003).","doi":""},{"text":"The depletion attraction: An underappreciated force driving cellular organization. D Marenduzzo, K Finan, P R Cook, J. Cell. Biol. 175D. Marenduzzo, K. Finan, and P. R. Cook, \"The depletion attraction: An underappreciated force driving cellular organization,\" J. Cell. Biol. 175, 681-686 (2006).","doi":""},{"text":"Polymer solutions near an interface. adsorption and depletion layers. P G De Gennes, Macromolecules. 14P. G. de Gennes, \"Polymer solutions near an interface. adsorption and depletion layers,\" Macromolecules 14, 1637-1644 (1981).","doi":""},{"text":"Surface tension of high-polymer solutions. F Oosawa, S Asakura, J. Chem. Phys. 22F. Oosawa and S. Asakura, \"Surface tension of high-polymer solutions,\" J. Chem. Phys. 22, 1255-1255 (1954).","doi":""},{"text":"Phenomena at the walls in a critical binary mixture. M E Fisher, P G De Gennes, C. R. Acad. Sci. Paris B. 287M. E. Fisher and P. G. de Gennes, \"Phenomena at the walls in a critical binary mixture.\" C. R. Acad. Sci. Paris B 287, 207-209 (1978).","doi":""},{"text":"The Casimir effect: From quantum to critical fluctuations. A Gambassi, J. Phys.: Conf. Ser. 16112037A. Gambassi, \"The Casimir effect: From quantum to critical fluctuations,\" J. Phys.: Conf. Ser. 161, 012037 (2009).","doi":""},{"text":"Critical Casimir forces steered by patterned substrates. A Gambassi, S Dietrich, Soft Matter. 7A. Gambassi and S. Dietrich, \"Critical Casimir forces steered by patterned substrates,\" Soft Matter 7, 1247-1253 (2011).","doi":""},{"text":"On the attraction between two perfectly conducting plates. H B G Casimir, Proc. K. Ned. Akad. Wet. 51H. B. G. Casimir, \"On the attraction between two perfectly conducting plates,\" Proc. K. Ned. Akad. Wet 51, 793-795 (1948).","doi":""},{"text":"Direct measurement of critical Casimir forces. C Hertlein, L Helden, A Gambassi, S Dietrich, C Bechinger, Nature. 451C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C. Bechinger, \"Direct measurement of critical Casimir forces,\" Nature 451, 172-175 (2008).","doi":""},{"text":"Nonadditivity of critical Casimir forces. S Paladugu, A Callegari, Y Tuna, L Barth, S Dietrich, A Gambassi, G Volpe, Nat. Commun. 711403S. Paladugu, A. Callegari, Y. Tuna, L. Barth, S. Dietrich, A. Gambassi, and G. Volpe, \"Nonadditivity of critical Casimir forces,\" Nat. Commun. 7, 11403 (2016).","doi":""},{"text":"Direct observation of colloidal aggregation by critical Casimir forces. D Bonn, J Otwinowski, S Sacanna, H Guo, G Wegdam, P Schall, Phys. Rev. Lett. 103156101D. Bonn, J. Otwinowski, S. Sacanna, H. Guo, G. Wegdam, and P. Schall, \"Direct observation of colloidal aggregation by critical Casimir forces,\" Phys. Rev. Lett. 103, 156101 (2009).","doi":""},{"text":"When depletion goes critical. R Piazza, S Buzzaccaro, A Parola, J Colombo, J. Phys.: Condens. Matter. 23194114R. Piazza, S. Buzzaccaro, A. Parola, and J. Colombo, \"When depletion goes critical,\" J. Phys.: Condens. Matter 23, 194114 (2011).","doi":""},{"text":"Controlling the dynamics of colloidal particles by critical Casimir forces. A Magazzù, A Callegari, J P Staforelli, A Gambassi, S Dietrich, G Volpe, Soft matter. 15A. Magazzù, A. Callegari, J. P. Staforelli, A. Gambassi, S. Dietrich, and G. Volpe, \"Controlling the dynamics of colloidal particles by critical Casimir forces,\" Soft matter 15, 2152-2162 (2019).","doi":""},{"text":"Critical Casimir forces in colloidal suspensions on chemically patterned surfaces. F Soyka, O Zvyagolskaya, C Hertlein, L Helden, C Bechinger, Phys. Rev. Lett. 101208301F. Soyka, O. Zvyagolskaya, C. Hertlein, L. Helden, and C. Bechinger, \"Critical Casimir forces in colloidal suspensions on chemically patterned surfaces,\" Phys. Rev. Lett. 101, 208301 (2008).","doi":""},{"text":"Tunability of critical Casimir interactions by boundary conditions. U Nellen, L Helden, C Bechinger, Europhys. Lett.). 8826001EPLU. Nellen, L. Helden, and C. Bechinger, \"Tunability of critical Casimir interactions by boundary conditions,\" EPL (Europhys. Lett.) 88, 26001 (2009).","doi":""},{"text":"Optical tweezers in colloid and interface science. D G Grier, Curr. Opn. Colloid Interface Sci. 2D. G. Grier, \"Optical tweezers in colloid and interface science,\" Curr. Opn. Colloid Interface Sci. 2, 264-270 (1997).","doi":""},{"text":"Blinking optical tweezers for microrheology measurements of weak elasticity complex fluids. G Pesce, G Rusciano, A Sasso, Opt. Express. 18G. Pesce, G. Rusciano, and A. Sasso, \"Blinking optical tweezers for microrheology measurements of weak elasticity complex fluids,\" Opt. Express 18, 2116-2126 (2010).","doi":""},{"text":"Long-term influence of fluid inertia on the diffusion of a Brownian particle. G Pesce, G Volpe, G Volpe, A Sasso, Phys. Rev. E. 9042309G. Pesce, G. Volpe, G. Volpe, and A. Sasso, \"Long-term influence of fluid inertia on the diffusion of a Brownian particle,\" Phys. Rev. E 90, 042309 (2014).","doi":""},{"text":"Lower critical solution coexistence curve and physical properties (density, viscosity, surface tension, and interfacial tension) of 2,6-lutidine+water. C A Grattoni, R A Dawe, C Y Seah, J D Gray, J. Chem. Eng. Data. 38C. A. Grattoni, R. A. Dawe, C. Y. Seah, and J. D. Gray, \"Lower critical solution coexistence curve and physical properties (density, viscosity, surface tension, and interfacial tension) of 2,6-lutidine+water,\" J. Chem. Eng. Data 38, 516-519 (1993).","doi":""},{"text":"Critical Casimir effect in classical binary liquid mixtures. A Gambassi, A Maciołek, C Hertlein, U Nellen, L Helden, C Bechinger, S Dietrich, Phys. Rev. E. 8061143A. Gambassi, A. Maciołek, C. Hertlein, U. Nellen, L. Helden, C. Bechinger, and S. Dietrich, \"Critical Casimir effect in classical binary liquid mixtures,\" Phys. Rev. E 80, 061143 (2009).","doi":""},{"text":"Methods of digital video microscopy for colloidal studies. J C Crocker, D G Grier, J. Colloid Interface Sci. 179J. C. Crocker and D. G. Grier, \"Methods of digital video microscopy for colloidal studies,\" J. Colloid Interface Sci. 179, 298-310 (1996).","doi":""},{"text":"Monte Carlo simulation results for critical Casimir forces. O Vasilyev, A Gambassi, A Maciołek, S Dietrich, Europhys. Lett.). 8060009EPLO. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, \"Monte Carlo simulation results for critical Casimir forces,\" EPL (Europhys. Lett.) 80, 60009 (2007).","doi":""},{"text":"Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations. O Vasilyev, A Gambassi, A Maciołek, S Dietrich, Phys. Rev. E. 7941142O. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, \"Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations,\" Phys. Rev. E 79, 041142 (2009).","doi":""},{"text":"Thermodynamic Casimir effect: Universality and corrections to scaling. M Hasenbusch, Phys. Rev. B. 85174421M. Hasenbusch, \"Thermodynamic Casimir effect: Universality and corrections to scaling,\" Phys. Rev. B 85, 174421 (2012).","doi":""},{"text":"Brownian diffusion of particles with hydrodynamic interaction. G K Batchelor, J. Fluid Mech. 74G. K. Batchelor, \"Brownian diffusion of particles with hydrodynamic interaction,\" J. Fluid Mech. 74, 1-29 (1976).","doi":""},{"text":"Interdiffusion coefficient and dynamic viscosity for the mixture 2,6-lutidine+water near the lower consolute point. J C Clunie, J K Baird, Phys. Chem. Liq. 37J. C. Clunie and J. K. Baird, \"Interdiffusion coefficient and dynamic viscosity for the mixture 2,6-lutidine+water near the lower consolute point,\" Phys. Chem. Liq. 37, 357-371 (1999).","doi":""},{"text":"Assembling quantum dots via critical Casimir forces. E Marino, T E Kodger, J B Hove, A H Velders, P Schall, Sol. Energy Mater. Sol. Cells. 158E. Marino, T. E. Kodger, J. B. ten Hove, A. H. Velders, and P. Schall, \"Assembling quantum dots via critical Casimir forces,\" Sol. Energy Mater. Sol. Cells 158, 154-159 (2016).","doi":""},{"text":"Tuning patchy bonds induced by critical Casimir forces. T A Nguyen, A Newton, D J Kraft, P G Bolhuis, P Schall, Materials. 101265T. A. Nguyen, A. Newton, D. J. Kraft, P. G. Bolhuis, and P. Schall, \"Tuning patchy bonds induced by critical Casimir forces,\" Materials 10, 1265 (2017).","doi":""},{"text":"Microscopic engine powered by critical demixing. F Schmidt, A Magazzù, A Callegari, L Biancofiore, F Cichos, G Volpe, Phys. Rev. Lett. 12068004F. Schmidt, A. Magazzù, A. Callegari, L. Biancofiore, F. Cichos, and G. Volpe, \"Microscopic engine powered by critical demixing,\" Phys. Rev. Lett. 120, 068004 (2018).","doi":""},{"text":"Direct measurement of the van der Waals interaction between two Rydberg atoms. L Beguin, A Vernier, R Chicireanu, T Lahaye, A Browaeys, Phys. Rev. Lett. 110263201L. Beguin, A. Vernier, R. Chicireanu, T. Lahaye, and A. Browaeys, \"Direct measurement of the van der Waals interaction between two Rydberg atoms,\" Phys. Rev. Lett. 110, 263201 (2013).","doi":""},{"text":"Materials perspective on Casimir and van der Waals interactions. L M Woods, D A R Dalvit, A Tkatchenko, P Rodriguez-Lopez, A W Rodriguez, R Podgornik, Rev. Mod. Phys. 8845003L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. W. Rodriguez, and R. Podgornik, \"Materials perspective on Casimir and van der Waals interactions,\" Rev. Mod. Phys. 88, 045003 (2016).","doi":""},{"text":"Programmable self-assembly. L Cademartiri, K J M Bishop, Nat. Mater. 14L. Cademartiri and K. J. M. Bishop, \"Programmable self-assembly,\" Nat. Mater. 14, 2-9 (2014).","doi":""},{"text":"Escape and synchronization of a Brownian particle. A Simon, A Libchaber, Phys. Rev. Lett. 68A. Simon and A. Libchaber, \"Escape and synchronization of a Brownian particle,\" Phys. Rev. Lett. 68, 3375-3379 (1992).","doi":""},{"text":"Thermally activated transitions in a bistable three-dimensional optical trap. L I Mccann, M Dykman, B Golding, Nature. 402L. I. McCann, M. Dykman, and B. Golding, \"Thermally activated transitions in a bistable three-dimensional optical trap,\" Nature 402, 785-787 (1999).","doi":""},{"text":"Brownian motion in a field of force and the diffusion model of chemical reactions. H A Kramers, Physica. 7H. A. Kramers, \"Brownian motion in a field of force and the diffusion model of chemical reactions,\" Physica 7, 284-304 (1940).","doi":""},{"text":"Frequency factors in the thermally activated process. R Landauer, J A Swanson, Phys. Rev. 121R. Landauer and J. A. Swanson, \"Frequency factors in the thermally activated process,\" Phys. Rev. 121, 1668-1674 (1961).","doi":""},{"text":"Double nanohole optical trapping: dynamics and protein-antibody co-trapping. A Zehtabi-Oskuie, H Jiang, B R Cyr, D W Rennehan, A A Al-Balushi, R Gordon, Lab Chip. 13A. Zehtabi-Oskuie, H. Jiang, B. R. Cyr, D. W. Rennehan, A. A. Al-Balushi, and R. Gordon, \"Double nanohole optical trapping: dynamics and protein-antibody co-trapping,\" Lab Chip 13, 2563-2568 (2013).","doi":""},{"text":"Partial synchronization of stochastic oscillators through hydrodynamic coupling. A Curran, M P Lee, M J Padgett, J M Cooper, R. Di Leonardo, Phys. Rev. Lett. 108240601A. Curran, M. P. Lee, M. J. Padgett, J. M. Cooper, and R. Di Leonardo, \"Partial synchronization of stochastic oscillators through hydrodynamic coupling,\" Phys. Rev. Lett. 108, 240601 (2012).","doi":""},{"text":"Phase measurements of barrier crossings in a periodically modulated double-well potential. Y Seol, D L Stein, K Visscher, Phys. Rev. Lett. 10350601Y. Seol, D. L. Stein, and K. Visscher, \"Phase measurements of barrier crossings in a periodically modulated double-well potential,\" Phys. Rev. Lett. 103, 050601 (2009).","doi":""},{"text":"Irreversibility and generalized noise. H B Callen, T A Welton, Phys. Rev. 83H. B. Callen and T. A. Welton, \"Irreversibility and generalized noise,\" Phys. Rev. 83, 34-40 (1951).","doi":""},{"text":"Fluctuation-dissipation: response theory in statistical physics. U M B Marconi, A Puglisi, L Rondoni, A Vulpiani, Phys. Rep. 461U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, \"Fluctuation-dissipation: response theory in statistical physics,\" Phys. Rep. 461, 111-195 (2008).","doi":""},{"text":"Fluctuation relations in simple examples of non-equilibrium steady states. R Chetrite, G Falkovich, K Gawedzki, J. Stat. Mech.: Theo. Exp. 8005R. Chetrite, G. Falkovich, and K. Gawedzki, \"Fluctuation relations in simple examples of non-equilibrium steady states,\" J. Stat. Mech.: Theo. Exp. 2008, P08005 (2008).","doi":""},{"text":"Einstein relation generalized to nonequilibrium. V Blickle, T Speck, C Lutz, U Seifert, C Bechinger, Phys. Rev. Lett. 98210601V. Blickle, T. Speck, C. Lutz, U. Seifert, and C. Bechinger, \"Einstein relation generalized to nonequilibrium,\" Phys. Rev. Lett. 98, 210601 (2007).","doi":""},{"text":"Experimental test of a new equality: Measuring heat dissipation in an optically driven colloidal system. S Toyabe, H.-R Jiang, T Nakamura, Y Murayama, M Sano, Phys. Rev. E. 7511122S. Toyabe, H.-R. Jiang, T. Nakamura, Y. Murayama, and M. Sano, \"Experimental test of a new equality: Measuring heat dissipation in an optically driven colloidal system,\" Phys. Rev. E 75, 011122 (2007).","doi":""},{"text":"Energy dissipation of a Brownian particle in a viscoelastic fluid. S Toyabe, M Sano, Phys. Rev. E. 7741403S. Toyabe and M. Sano, \"Energy dissipation of a Brownian particle in a viscoelastic fluid,\" Phys. Rev. E 77, 041403 (2008).","doi":""},{"text":"Fluctuations, linear response and heat flux of an aging system. J R Gomez-Solano, A Petrosyan, S Ciliberto, Europhys. Lett.). 9810007EPLJ. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, \"Fluctuations, linear response and heat flux of an aging system,\" EPL (Europhys. Lett.) 98, 10007 (2012).","doi":""},{"text":"Probing active forces via a fluctuationdissipation relation: Application to living cells. P Bohec, F Gallet, C Maes, S Safaverdi, P Visco, F Van Wijland, Europhys. Lett.). 10250005EPLP. Bohec, F. Gallet, C. Maes, S. Safaverdi, P. Visco, and F. Van Wijland, \"Probing active forces via a fluctuation- dissipation relation: Application to living cells,\" EPL (Europhys. Lett.) 102, 50005 (2013).","doi":""},{"text":"Periodic forcing of a Brownian particle. L P Faucheux, G Stolovitzky, A Libchaber, Phys. Rev. E. 51L. P. Faucheux, G. Stolovitzky, and A. Libchaber, \"Periodic forcing of a Brownian particle,\" Phys. Rev. E 51, 5239-5250 (1995).","doi":""},{"text":"Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state. J R Gomez-Solano, A Petrosyan, S Ciliberto, R Chetrite, K Gawędzki, Phys. Rev. Lett. 10340601J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, R. Chetrite, and K. Gawędzki, \"Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state,\" Phys. Rev. Lett. 103, 040601 (2009).","doi":""},{"text":"Fluctuations and response in a non-equilibrium micron-sized system. J R Gomez-Solano, A Petrosyan, S Ciliberto, C Maes, J. Stat. Mech.: Theo. Exp. 1008J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, and C. Maes, \"Fluctuations and response in a non-equilibrium micron-sized system,\" J. Stat. Mech.: Theo. Exp. 2011, P01008 (2011).","doi":""},{"text":"Characterizing potentials by a generalized Boltzmann factor. V Blickle, T Speck, U Seifert, C Bechinger, Phys. Rev. E. 7560101V. Blickle, T. Speck, U. Seifert, and C. Bechinger, \"Characterizing potentials by a generalized Boltzmann factor,\" Phys. Rev. E 75, 060101 (2007).","doi":""},{"text":"Fluctuations, linear response, and currents in out-of-equilibrium systems. S Ciliberto, R Gomez-Solano, A Petrosyan, Annu. Rev. Condens. Matter Phys. 4S. Ciliberto, R. Gomez-Solano, and A. Petrosyan, \"Fluctuations, linear response, and currents in out-of-equilibrium systems,\" Annu. Rev. Condens. Matter Phys. 4, 235-261 (2013).","doi":""},{"text":"Langevin equation and thermodynamics. K Sekimoto, Prog. Theo. Phys. Suppl. 130K. Sekimoto, \"Langevin equation and thermodynamics,\" Prog. Theo. Phys. Suppl. 130, 17-27 (1998).","doi":""},{"text":"K Sekimoto, Stochastic energetics. SpringerK. Sekimoto, Stochastic energetics (Springer, 2010).","doi":""},{"text":"Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. G M Wang, E M Sevick, E Mittag, D J Searles, D J Evans, Phys. Rev. Lett. 8950601G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, \"Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,\" Phys. Rev. Lett. 89, 050601 (2002).","doi":""},{"text":"Entropy production along a stochastic trajectory and an integral fluctuation theorem. U Seifert, Phys. Rev. Lett. 9540602U. Seifert, \"Entropy production along a stochastic trajectory and an integral fluctuation theorem,\" Phys. Rev. Lett. 95, 040602 (2005).","doi":""},{"text":"Nonequilibrium equality for free energy differences. C Jarzynski, Phys. Rev. Lett. 78C. Jarzynski, \"Nonequilibrium equality for free energy differences,\" Phys. Rev. Lett. 78, 2690-2693 (1997).","doi":""},{"text":"Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. G E Crooks, Phys. Rev. E. 60G. E. Crooks, \"Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences,\" Phys. Rev. E 60, 2721-2726 (1999).","doi":""},{"text":"Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. S Carnot, Annales scientifiques de l'École Normale Supérieure. 1S. Carnot, \"Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance,\" in Annales scientifiques de l'École Normale Supérieure, vol. 1 (1824), pp. 393-457.","doi":""},{"text":"An inconvenient truth about thermoelectrics. C B Vining, Nat. Mater. 8C. B. Vining, \"An inconvenient truth about thermoelectrics,\" Nat. Mater. 8, 83-85 (2009).","doi":""},{"text":"The efficiency of atomic power stations (a review). I I Novikov, J. Nucl. Energy. 7I. I. Novikov, \"The efficiency of atomic power stations (a review),\" J. Nucl. Energy 7, 125-128 (1958).","doi":""},{"text":"Efficiency of a Carnot engine at maximum power output. F L Curzon, B Ahlborn, Am. J. Phys. 43F. L. Curzon and B. Ahlborn, \"Efficiency of a Carnot engine at maximum power output,\" Am. J. Phys. 43, 22-24 (1975).","doi":""},{"text":"Heat engines and the Curzon-Ahlborn efficiency. A Hernández, J M M Roco, A Medina, N Sánchez-Salas, Revista Mexicana de Física. 60A. Calvo Hernández, J. M. M. Roco, A. Medina, and N. Sánchez-Salas, \"Heat engines and the Curzon-Ahlborn efficiency,\" Revista Mexicana de Física 60, 384-389 (2014).","doi":""},{"text":"Realization of a micrometre-sized stochastic heat engine. V Blickle, C Bechinger, Nat. Phys. 8V. Blickle and C. Bechinger, \"Realization of a micrometre-sized stochastic heat engine,\" Nat. Phys. 8, 143-146 (2012).","doi":""},{"text":"A microscopic steam engine implemented in an optical tweezer. P A Quinto-Su, Nat. Commun. 55889P. A. Quinto-Su, \"A microscopic steam engine implemented in an optical tweezer,\" Nat. Commun. 5, 5889 (2014).","doi":""},{"text":"Effective heating to several thousand kelvins of an optically trapped sphere in a liquid. I A Martínez, É Roldán, J M R Parrondo, D Petrov, Phys. Rev. E. 8732159I. A. Martínez, É. Roldán, J. M. R. Parrondo, and D. Petrov, \"Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,\" Phys. Rev. E 87, 032159 (2013).","doi":""},{"text":"Steady-state fluctuation relations for systems driven by an external random force. J R Gomez-Solano, L Bellon, A Petrosyan, S Ciliberto, Europhys. Lett.). 8960003EPLJ. R. Gomez-Solano, L. Bellon, A. Petrosyan, and S. Ciliberto, \"Steady-state fluctuation relations for systems driven by an external random force,\" EPL (Europhys. Lett.) 89, 60003 (2010).","doi":""},{"text":"Energy flow between two hydrodynamically coupled particles kept at different effective temperatures. A Bérut, A Petrosyan, S Ciliberto, Europhys. Lett.). 10760004EPLA. Bérut, A. Petrosyan, and S. Ciliberto, \"Energy flow between two hydrodynamically coupled particles kept at different effective temperatures,\" EPL (Europhys. Lett.) 107, 60004 (2014).","doi":""},{"text":"Adiabatic processes realized with a trapped Brownian particle. I A Martínez, É Roldán, L Dinis, D Petrov, R A Rica, Phys. Rev. Lett. 114120601I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, and R. A. Rica, \"Adiabatic processes realized with a trapped Brownian particle,\" Phys. Rev. Lett. 114, 120601 (2015).","doi":""},{"text":"Brownian Carnot engine. I A Martínez, É Roldán, L Dinis, D Petrov, J M R Parrondo, R A Rica, Nat. Phys. 12I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, \"Brownian Carnot engine,\" Nat. Phys. 12, 67-70 (2016).","doi":""},{"text":"Efficiency at maximum power: An analytically solvable model for stochastic heat engines. T Schmiedl, U Seifert, Europhys. Lett.). 8120003EPLT. Schmiedl and U. Seifert, \"Efficiency at maximum power: An analytically solvable model for stochastic heat engines,\" EPL (Europhys. Lett.) 81, 20003 (2007).","doi":""},{"text":"Theory of Heat, Text-books of science (Longmans, 1871). J C Maxwell, J. C. Maxwell, Theory of Heat, Text-books of science (Longmans, 1871).","doi":""},{"text":"Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. L Szilard, Zeit. Physik. 53L. Szilard, \"Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen,\" Zeit. Physik 53, 840-856 (1929).","doi":""},{"text":"Maxwell's demon cannot operate: Information and entropy. i. L Brillouin, J. Appl. Phys. 22L. Brillouin, \"Maxwell's demon cannot operate: Information and entropy. i,\" J. Appl. Phys. 22, 334-337 (1951).","doi":""},{"text":"Irreversibility and heat generation in the computing process. R Landauer, IBM J. Res. Dev. 5R. Landauer, \"Irreversibility and heat generation in the computing process,\" IBM J. Res. Dev. 5, 183-191 (1961).","doi":""},{"text":"The thermodynamics of computation: A review. C H Bennett, Int. J. Th. Phys. 21C. H. Bennett, \"The thermodynamics of computation: A review,\" Int. J. Th. Phys. 21, 905-940 (1982).","doi":""},{"text":"Second law of thermodynamics with discrete quantum feedback control. T Sagawa, M Ueda, Phys. Rev. Lett. 10080403T. Sagawa and M. Ueda, \"Second law of thermodynamics with discrete quantum feedback control,\" Phys. Rev. Lett. 100, 080403 (2008).","doi":""},{"text":"Thermodynamics of information processing in small systems. T Sagawa, Prog. Theo. Phys. 127T. Sagawa, \"Thermodynamics of information processing in small systems,\" Prog. Theo. Phys. 127, 1-56 (2012).","doi":""},{"text":"Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. S Toyabe, T Sagawa, M Ueda, E Muneyuki, M Sano, Nat. Phys. 6S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, \"Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality,\" Nat. Phys. 6, 988-992 (2010).","doi":""},{"text":"Minimal energy cost for thermodynamic information processing: measurement and information erasure. T Sagawa, M Ueda, Phys. Rev. Lett. 102250602T. Sagawa and M. Ueda, \"Minimal energy cost for thermodynamic information processing: measurement and information erasure,\" Phys. Rev. Lett. 102, 250602 (2009).","doi":""},{"text":"Universal features in the energetics of symmetry breaking. É Roldán, I A Martinez, J M R Parrondo, D Petrov, Nat. Phys. 10É. Roldán, I. A. Martinez, J. M. R. Parrondo, and D. Petrov, \"Universal features in the energetics of symmetry breaking,\" Nat. Phys. 10, 457-461 (2014).","doi":""},{"text":"Large work extraction and the Landauer limit in a continuous Maxwell demon. M Ribezzi-Crivellari, F Ritort, Nat. Phys. 15M. Ribezzi-Crivellari and F. Ritort, \"Large work extraction and the Landauer limit in a continuous Maxwell demon,\" Nat. Phys. 15, 660-664 (2019).","doi":""},{"text":"Hydrodynamic synchronization of light driven microrotors. R Di Leonardo, A Búzás, L Kelemen, G Vizsnyiczai, L Oroszi, P Ormos, Phys. Rev. Lett. 10934104R. Di Leonardo, A. Búzás, L. Kelemen, G. Vizsnyiczai, L. Oroszi, and P. Ormos, \"Hydrodynamic synchronization of light driven microrotors,\" Phys. Rev. Lett. 109, 034104 (2012).","doi":""},{"text":"Collective synchronization states in arrays of driven colloidal oscillators. R Lhermerout, N Bruot, G M Cicuta, J Kotar, P Cicuta, New J. Phys. 14105023R. Lhermerout, N. Bruot, G. M. Cicuta, J. Kotar, and P. Cicuta, \"Collective synchronization states in arrays of driven colloidal oscillators,\" New J. Phys. 14, 105023 (2012).","doi":""},{"text":"Stochastic hydrodynamic synchronization in rotating energy landscapes. N Koumakis, R Di Leonardo, Phys. Rev. Lett. 110174103N. Koumakis and R. Di Leonardo, \"Stochastic hydrodynamic synchronization in rotating energy landscapes,\" Phys. Rev. Lett. 110, 174103 (2013).","doi":""},{"text":"Optimal hydrodynamic synchronization of colloidal rotors. J Kotar, L Debono, N Bruot, S Box, D Phillips, S Simpson, S Hanna, P Cicuta, Phys. Rev. Lett. 111228103J. Kotar, L. Debono, N. Bruot, S. Box, D. Phillips, S. Simpson, S. Hanna, and P. Cicuta, \"Optimal hydrodynamic synchronization of colloidal rotors,\" Phys. Rev. Lett. 111, 228103 (2013).","doi":""},{"text":"Active particles in complex and crowded environments. C Bechinger, R Di Leonardo, H Löwen, C Reichhardt, G Volpe, G Volpe, Rev. Mod. Phys. 8845006C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, \"Active particles in complex and crowded environments,\" Rev. Mod. Phys. 88, 045006 (2016).","doi":""},{"text":"Machine learning for active matter. F Cihos, K Gustavsson, B Mehlig, G Volpe, Nat. Mach. Intell. 2F. Cihos, K. Gustavsson, B. Mehlig, and G. Volpe, \"Machine learning for active matter,\" Nat. Mach. Intell. 2, 94-103 (2020).","doi":""},{"text":"Non-boltzmann stationary distributions and nonequilibrium relations in active baths. A Argun, A.-R Moradi, E Pinçe, G B Bagci, A Imparato, G Volpe, Phys. Rev. E. 9462150A. Argun, A.-R. Moradi, E. Pinçe, G. B. Bagci, A. Imparato, and G. Volpe, \"Non-boltzmann stationary distributions and nonequilibrium relations in active baths,\" Phys. Rev. E 94, 062150 (2016).","doi":""},{"text":"Phase transitions of colloidal monolayers in periodic pinning arrays. K Mangold, P Leiderer, C Bechinger, Phys. Rev. Lett. 90158302K. Mangold, P. Leiderer, and C. Bechinger, \"Phase transitions of colloidal monolayers in periodic pinning arrays,\" Phys. Rev. Lett. 90, 158302 (2003).","doi":""},{"text":"Tunable colloids: Control of colloidal phase transitions with tunable interactions. A Yethiraj, Soft Matter. 3A. Yethiraj, \"Tunable colloids: Control of colloidal phase transitions with tunable interactions,\" Soft Matter 3, 1099-1115 (2007).","doi":""},{"text":"Light-induced phase transitions of colloidal monolayers with crystalline order. L Zaidouny, T Bohlein, R Roth, C Bechinger, Soft Matter. 9L. Zaidouny, T. Bohlein, R. Roth, and C. Bechinger, \"Light-induced phase transitions of colloidal monolayers with crystalline order,\" Soft Matter 9, 9230-9236 (2013).","doi":""},{"text":"Physical grounds for biological effect of laser radiation. A N Rubinov, J. Phys. D: Appl. Phys. 362317A. N. Rubinov, \"Physical grounds for biological effect of laser radiation,\" J. Phys. D: Appl. Phys. 36, 2317 (2003).","doi":""},{"text":"Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap. M P Macdonald, L Paterson, W Sibbett, K Dholakia, P E Bryant, Opt. Lett. 26M. P. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and P. E. Bryant, \"Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,\" Opt. Lett. 26, 863-865 (2001).","doi":""},{"text":"Dynamical analysis of an optical rocking ratchet: Theory and experiment. A V Arzola, K Volke-Sepulveda, J L Mateos, Phys. Rev. E. 8762910A. V. Arzola, K. Volke-Sepulveda, and J. L. Mateos, \"Dynamical analysis of an optical rocking ratchet: Theory and experiment,\" Phys. Rev. E 87, 062910 (2013).","doi":""},{"text":"Active dynamics of colloidal particles in time-varying laser speckle patterns. S Bianchi, R Pruner, G Vizsnyiczai, C Maggi, R. Di Leonardo, Sci. Rep. 627681S. Bianchi, R. Pruner, G. Vizsnyiczai, C. Maggi, and R. Di Leonardo, \"Active dynamics of colloidal particles in time-varying laser speckle patterns,\" Sci. Rep. 6, 27681 (2016).","doi":""},{"text":"Force mapping of an extended light pattern in an inclined plane: Deterministic regime. A V Arzola, K Volke-Sepúlveda, J L Mateos, Opt. Express. 17A. V. Arzola, K. Volke-Sepúlveda, and J. L. Mateos, \"Force mapping of an extended light pattern in an inclined plane: Deterministic regime,\" Opt. Express 17, 3429-3440 (2009).","doi":""},{"text":"Moving interference patterns created using the angular Doppler-effect. J Arlt, M Macdonald, L Paterson, W Sibbett, K Dholakia, K Volke-Sepulveda, Opt. Express. 10J. Arlt, M. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, \"Moving interference patterns created using the angular Doppler-effect,\" Opt. Express 10, 844-852 (2002).","doi":""},{"text":"Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination. M Šiler, T Čižmár, A Jonáš, P Zemánek, New J. Phys. 10113010M. Šiler, T. Čižmár, A. Jonáš, and P. Zemánek, \"Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,\" New J. Phys. 10, 113010 (2008).","doi":""},{"text":"Phase behavior of two-dimensional colloidal systems in the presence of periodic light fields. C Bechinger, M Brunner, P Leiderer, Phys. Rev. Lett. 86C. Bechinger, M. Brunner, and P. Leiderer, \"Phase behavior of two-dimensional colloidal systems in the presence of periodic light fields,\" Phys. Rev. Lett. 86, 930-933 (2001).","doi":""},{"text":"Phase behavior of colloidal molecular crystals on triangular light lattices. M Brunner, C Bechinger, Phys. Rev. Lett. 88248302M. Brunner and C. Bechinger, \"Phase behavior of colloidal molecular crystals on triangular light lattices,\" Phys. Rev. Lett. 88, 248302 (2002).","doi":""},{"text":"Tailoring of phononic band structures in colloidal crystals. J Baumgartl, M Zvyagolskaya, C Bechinger, Phys. Rev. Lett. 99205503J. Baumgartl, M. Zvyagolskaya, and C. Bechinger, \"Tailoring of phononic band structures in colloidal crystals,\" Phys. Rev. Lett. 99, 205503 (2007).","doi":""},{"text":"3d interferometric optical tweezers using a single spatial light modulator. E Schonbrun, R Piestun, P Jordan, J Cooper, K D Wulff, J Courtial, M Padgett, Opt. Express. 13E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, \"3d interferometric optical tweezers using a single spatial light modulator,\" Opt. Express 13, 3777-3786 (2005).","doi":""},{"text":"Archimedean-like tiling on decagonal quasicrystalline surfaces. J Mikhael, J Roth, L Helden, C Bechinger, Nature. 454J. Mikhael, J. Roth, L. Helden, and C. Bechinger, \"Archimedean-like tiling on decagonal quasicrystalline surfaces,\" Nature 454, 501-504 (2008).","doi":""},{"text":"Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields. J Mikhael, M Schmiedeberg, S Rausch, J Roth, H Stark, C Bechinger, Proc. Natl. Acad. Sci. 107J. Mikhael, M. Schmiedeberg, S. Rausch, J. Roth, H. Stark, and C. Bechinger, \"Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields,\" Proc. Natl. Acad. Sci. 107, 7214-7218 (2010).","doi":""},{"text":"Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles. K Volke-Sepúlveda, S Chávez-Cerda, V Garcés-Chávez, K Dholakia, J. Opt. Soc. Am. B. 21K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, \"Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,\" J. Opt. Soc. Am. B 21, 1749-1757 (2004).","doi":""},{"text":"Optical conveyor belt for delivery of submicron objects. T Čižmár, V Garcés-Chávez, K Dholakia, P Zemánek, Appl. Phys. Lett. 86174101T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, \"Optical conveyor belt for delivery of submicron objects,\" Appl. Phys. Lett. 86, 174101 (2005).","doi":""},{"text":"Transverse particle dynamics in a Bessel beam. G Milne, K Dholakia, D Mcgloin, K Volke-Sepulveda, P Zemánek, Opt. Express. 15G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, \"Transverse particle dynamics in a Bessel beam,\" Opt. Express 15, 13972-13987 (2007).","doi":""},{"text":"Light-induced cell separation in a tailored optical landscape. L Paterson, E Papagiakoumou, G Milne, V Garcés-Chávez, S A Tatarkova, W Sibbett, F J Gunn-Moore, P E Bryant, A C Riches, K Dholakia, Appl. Phys. Lett. 87123901L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, \"Light-induced cell separation in a tailored optical landscape,\" Appl. Phys. Lett. 87, 123901 (2005).","doi":""},{"text":"Brownian particle in an optical potential of the washboard type. S A Tatarkova, W Sibbett, K Dholakia, Phys. Rev. Lett. 9138101S. A. Tatarkova, W. Sibbett, and K. Dholakia, \"Brownian particle in an optical potential of the washboard type,\" Phys. Rev. Lett. 91, 038101 (2003).","doi":""},{"text":"Artificial Brownian motors: Controlling transport on the nanoscale. P Hänggi, F Marchesoni, Rev. Mod. Phys. 81P. Hänggi and F. Marchesoni, \"Artificial Brownian motors: Controlling transport on the nanoscale,\" Rev. Mod. Phys. 81, 387-442 (2009).","doi":""},{"text":"Experimental creation and characterization of random potential-energy landscapes exploiting speckle patterns. J Bewerunge, S U Egelhaaf, Phys. Rev. A. 9313806J. Bewerunge and S. U. Egelhaaf, \"Experimental creation and characterization of random potential-energy landscapes exploiting speckle patterns,\" Phys. Rev. A 93, 013806 (2016).","doi":""},{"text":"Brownian motors: Noisy transport far from equilibrium. P Reimann, Phys. Rep. 361P. Reimann, \"Brownian motors: Noisy transport far from equilibrium,\" Phys. Rep. 361, 57-265 (2002).","doi":""},{"text":"Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate. B A Garetz, S Arnold, Opt. Commun. 31B. A. Garetz and S. Arnold, \"Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,\" Opt. Commun. 31, 1-3 (1979).","doi":""},{"text":"Angular Doppler effect. B A Garetz, J. Opt. Soc. Am. 71B. A. Garetz, \"Angular Doppler effect,\" J. Opt. Soc. Am. 71, 609-611 (1981).","doi":""},{"text":"Controlled rotation of optically trapped microscopic particles. L Paterson, M P Macdonald, J Arlt, W Sibbett, P E Bryant, K Dholakia, Science. 292L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, \"Controlled rotation of optically trapped microscopic particles,\" Science 292, 912-914 (2001).","doi":""},{"text":"An optical nanotrap array movable over a milimetre range. T Čižmár, M Šiler, P Zemánek, Appl. Phys. B. 84T. Čižmár, M. Šiler, and P. Zemánek, \"An optical nanotrap array movable over a milimetre range,\" Appl. Phys. B 84, 197-203 (2006).","doi":""},{"text":"The holographic optical micro-manipulation system based on counter-propagating beams. T Čižmár, O Brzobohatỳ, K Dholakia, P Zemánek, Laser Phys. Lett. 850T. Čižmár, O. Brzobohatỳ, K. Dholakia, and P. Zemánek, \"The holographic optical micro-manipulation system based on counter-propagating beams,\" Laser Phys. Lett. 8, 50 (2010).","doi":""},{"text":"Optical conveyors: A class of active tractor beams. D B Ruffner, D G Grier, Phys. Rev. Lett. 109163903D. B. Ruffner and D. G. Grier, \"Optical conveyors: A class of active tractor beams,\" Phys. Rev. Lett. 109, 163903 (2012).","doi":""},{"text":"Colloidal transport through optical tweezer arrays. Y Roichman, V Wong, D G Grier, Phys. Rev. E. 7511407Y. Roichman, V. Wong, and D. G. Grier, \"Colloidal transport through optical tweezer arrays,\" Phys. Rev. E 75, 011407 (2007).","doi":""},{"text":"Optical separation of cells on potential energy landscapes: Enhancement with dielectric tagging. K Dholakia, W M Lee, L Paterson, M P Macdonald, R Mcdonald, I Andreev, P Mthunzi, C T A Brown, R F Marchington, A C Riches, IEEE J. Sel. Top. Quant. El. 13K. Dholakia, W. M. Lee, L. Paterson, M. P. MacDonald, R. McDonald, I. Andreev, P. Mthunzi, C. T. A. Brown, R. F. Marchington, and A. C. Riches, \"Optical separation of cells on potential energy landscapes: Enhancement with dielectric tagging,\" IEEE J. Sel. Top. Quant. El. 13, 1646-1654 (2007).","doi":""},{"text":"Kinetically locked-in colloidal transport in an array of optical tweezers. P T Korda, M B Taylor, D G Grier, Phys. Rev. Lett. 89128301P. T. Korda, M. B. Taylor, and D. G. Grier, \"Kinetically locked-in colloidal transport in an array of optical tweezers,\" Phys. Rev. Lett. 89, 128301 (2002).","doi":""},{"text":"Sorting mesoscopic objects with periodic potential landscapes: Optical fractionation. K Ladavac, K Kasza, D G Grier, Phys. Rev. E. 7010901K. Ladavac, K. Kasza, and D. G. Grier, \"Sorting mesoscopic objects with periodic potential landscapes: Optical fractionation,\" Phys. Rev. E 70, 010901 (2004).","doi":""},{"text":"Microfluidic sorting with blinking optical traps. R Dasgupta, R S Verma, P K Gupta, Opt. Lett. 37R. Dasgupta, R. S. Verma, and P. K. Gupta, \"Microfluidic sorting with blinking optical traps,\" Opt. Lett. 37, 1739-1741 (2012).","doi":""},{"text":"Dynamics, rectification, and fractionation for colloids on flashing substrates. A Libal, C Reichhardt, B Janko, C J O Reichhardt, Phys. Rev. Lett. 96188301A. Libal, C. Reichhardt, B. Janko, and C. J. O. Reichhardt, \"Dynamics, rectification, and fractionation for colloids on flashing substrates,\" Phys. Rev. Lett. 96, 188301 (2006).","doi":""},{"text":"One-dimensional optical thermal ratchets. S.-H Lee, D G Grier, J. Phys. Condens. Matter. 17S.-H. Lee and D. G. Grier, \"One-dimensional optical thermal ratchets,\" J. Phys. Condens. Matter 17, S3685-S3695 (2005).","doi":""},{"text":"Two-dimensional optical thermal ratchets based on Fibonacci spirals. K Xiao, Y Roichman, D G Grier, Phys. Rev. E. 8411131K. Xiao, Y. Roichman, and D. G. Grier, \"Two-dimensional optical thermal ratchets based on Fibonacci spirals,\" Phys. Rev. E 84, 011131 (2011).","doi":""},{"text":"Noise-enabled optical ratchets. R León-Montiel, P A Quinto-Su, Sci. Rep. 744287R. León-Montiel and P. A. Quinto-Su, \"Noise-enabled optical ratchets,\" Sci. Rep. 7, 44287 (2017).","doi":""},{"text":"Optical chromatography. T Imasaka, Y Kawabata, T Kaneta, Y Ishidzu, Anal. Chem. 67T. Imasaka, Y. Kawabata, T. Kaneta, and Y. Ishidzu, \"Optical chromatography,\" Anal. Chem. 67, 1763-1765 (1995).","doi":""},{"text":"Refractive-index-driven separation of colloidal polymer particles using optical chromatography. S J Hart, A V Terray, Appl. Phys. Lett. 83S. J. Hart and A. V. Terray, \"Refractive-index-driven separation of colloidal polymer particles using optical chromatography,\" Appl. Phys. Lett. 83, 5316-5318 (2003).","doi":""},{"text":"Static optical sorting in a laser interference field. P Jákl, T Čižmár, M Šerý, P Zemánek, Appl. Phys. Lett. 92161110P. Jákl, T. Čižmár, M. Šerý, and P. Zemánek, \"Static optical sorting in a laser interference field,\" Appl. Phys. Lett. 92, 161110 (2008).","doi":""},{"text":"Optical deflection and sorting of microparticles in a near-field optical geometry. R F Marchington, M Mazilu, S Kuriakose, V Garcés-Chávez, P J Reece, T F Krauss, M Gu, K Dholakia, Opt. Express. 16R. F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P. J. Reece, T. F. Krauss, M. Gu, and K. Dholakia, \"Optical deflection and sorting of microparticles in a near-field optical geometry,\" Opt. Express 16, 3712-3726 (2008).","doi":""},{"text":"Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery. M Šiler, T Čižmár, M Šerỳ, P Zemánek, Appl. Phys. B. 84M. Šiler, T. Čižmár, M. Šerỳ, and P. Zemánek, \"Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,\" Appl. Phys. B 84, 157-165 (2006).","doi":""},{"text":"Nanomanipulation using near field photonics. D Erickson, X Serey, Y.-F Chen, S Mandal, Lab Chip. 11D. Erickson, X. Serey, Y.-F. Chen, and S. Mandal, \"Nanomanipulation using near field photonics,\" Lab Chip 11, 995-1009 (2011).","doi":""},{"text":"Experimental demonstration of optical transport, sorting and self-arrangement using a 'tractor' beam. O Brzobohatỳ, V Karásek, M Šiler, L Chvátal, T Čižmár, P Zemánek, Nat. Photon. 7O. Brzobohatỳ, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, and P. Zemánek, \"Experimental demonstration of optical transport, sorting and self-arrangement using a 'tractor' beam,\" Nat. Photon. 7, 123-127 (2013).","doi":""},{"text":"Optofluidic sorting of material chirality by chiral light. G Tkachenko, E Brasselet, Nat. Commun. 53577G. Tkachenko and E. Brasselet, \"Optofluidic sorting of material chirality by chiral light,\" Nat. Commun. 5, 3577 (2014).","doi":""},{"text":"Lateral chirality-sorting optical forces. A Hayat, J P B Mueller, F Capasso, Proc. Natl. Acad. Sci. 112A. Hayat, J. P. B. Mueller, and F. Capasso, \"Lateral chirality-sorting optical forces,\" Proc. Natl. Acad. Sci. 112, 13190-13194 (2015).","doi":""},{"text":"Orientational and directional locking of colloidal clusters driven across periodic surfaces. X Cao, E Panizon, A Vanossi, N Manini, C Bechinger, Nat. Phys. 15X. Cao, E. Panizon, A. Vanossi, N. Manini, and C. Bechinger, \"Orientational and directional locking of colloidal clusters driven across periodic surfaces,\" Nat. Phys. 15, 776-780 (2019).","doi":""},{"text":"Experimental observation of the Aubry transition in two-dimensional colloidal monolayers. T Brazda, A Silva, N Manini, A Vanossi, R Guerra, E Tosatti, C Bechinger, Phys. Rev. X. 811050T. Brazda, A. Silva, N. Manini, A. Vanossi, R. Guerra, E. Tosatti, and C. Bechinger, \"Experimental observation of the Aubry transition in two-dimensional colloidal monolayers,\" Phys. Rev. X 8, 011050 (2018).","doi":""},{"text":"Experimental observation of directional locking and dynamical ordering of colloidal monolayers driven across quasiperiodic substrates. T Bohlein, C Bechinger, Phys. Rev. Lett. 10958301T. Bohlein and C. Bechinger, \"Experimental observation of directional locking and dynamical ordering of colloidal monolayers driven across quasiperiodic substrates,\" Phys. Rev. Lett. 109, 058301 (2012).","doi":""},{"text":"Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces. T Bohlein, J Mikhael, C Bechinger, Nat. Mater. 11T. Bohlein, J. Mikhael, and C. Bechinger, \"Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces,\" Nat. Mater. 11, 126-130 (2012).","doi":""},{"text":"Laser trapping of colloidal metal nanoparticles. A Lehmuskero, P Johansson, H Rubinsztein-Dunlop, L Tong, M Kall, ACS Nano. 9A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, \"Laser trapping of colloidal metal nanoparticles,\" ACS Nano 9, 3453-3469 (2015).","doi":""},{"text":"Parallel and selective trapping in a patterned plasmonic landscape. M Righini, A S Zelenina, C Girard, R Quidant, Nat. Phys. 3M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, \"Parallel and selective trapping in a patterned plasmonic landscape,\" Nat. Phys. 3, 477-480 (2007).","doi":""},{"text":"Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range. M Righini, G Volpe, C Girard, D Petrov, R Quidant, Phys. Rev. Lett. 100186804M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, \"Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,\" Phys. Rev. Lett. 100, 186804 (2008).","doi":""},{"text":"Plasmonic optical tweezers toward molecular manipulation: tailoring plasmonic nanostructure, light source, and resonant trapping. T Shoji, Y Tsuboi, J. Phys. Chem. Lett. 5T. Shoji and Y. Tsuboi, \"Plasmonic optical tweezers toward molecular manipulation: tailoring plasmonic nanostruc- ture, light source, and resonant trapping,\" J. Phys. Chem. Lett. 5, 2957-2967 (2014).","doi":""},{"text":"Massively parallel manipulation of single cells and microparticles using optical images. P Y Chiou, A T Ohta, M C Wu, Nature. 436P. Y. Chiou, A. T. Ohta, and M. C. Wu, \"Massively parallel manipulation of single cells and microparticles using optical images,\" Nature 436, 370-372 (2005).","doi":""},{"text":"Thermophoretic tweezers for low-power and versatile manipulation of biological cells. L Lin, X Peng, X Wei, Z Mao, C Xie, Y Zheng, ACS Nano. 11L. Lin, X. Peng, X. Wei, Z. Mao, C. Xie, and Y. Zheng, \"Thermophoretic tweezers for low-power and versatile manipulation of biological cells,\" ACS Nano 11, 3147-3154 (2017).","doi":""},{"text":"Opto-thermophoretic tweezers and assembly. J Li, L Lin, Y Inoue, Y Zheng, J. Micro Nano-Manufacturing. 640801J. Li, L. Lin, Y. Inoue, and Y. Zheng, \"Opto-thermophoretic tweezers and assembly,\" J. Micro Nano-Manufacturing 6, 040801 (2018).","doi":""},{"text":"Acoustic tweezers: patterning cells and microparticles using standing surface acoustic waves (SSAW). J Shi, D Ahmed, X Mao, S.-C S Lin, A Lawit, T J Huang, Lab Chip. 9J. Shi, D. Ahmed, X. Mao, S.-C. S. Lin, A. Lawit, and T. J. Huang, \"Acoustic tweezers: patterning cells and microparticles using standing surface acoustic waves (SSAW),\" Lab Chip 9, 2890-2895 (2009).","doi":""},{"text":"Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics. J Friend, L Y Yeo, Rev. Mod. Phys. 83J. Friend and L. Y. Yeo, \"Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics,\" Rev. Mod. Phys. 83, 647-704 (2011).","doi":""},{"text":"Observation of a single-beam gradient force acoustical trap for elastic particles: acoustical tweezers. D Baresch, J.-L Thomas, R Marchiano, Phys. Rev. Lett. 11624301D. Baresch, J.-L. Thomas, and R. Marchiano, \"Observation of a single-beam gradient force acoustical trap for elastic particles: acoustical tweezers,\" Phys. Rev. Lett. 116, 024301 (2016).","doi":""},{"text":"Perspective on light-induced transport of particles: from optical forces to phoretic motion. P Zemánek, G Volpe, A Jonáš, O Brzobohatỳ, Adv. Opt. Photon. 11P. Zemánek, G. Volpe, A. Jonáš, and O. Brzobohatỳ, \"Perspective on light-induced transport of particles: from optical forces to phoretic motion,\" Adv. Opt. Photon. 11, 577-678 (2019).","doi":""},{"text":"Feedback stabilization of optically levitated particles. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 30A. Ashkin and J. M. Dziedzic, \"Feedback stabilization of optically levitated particles,\" Appl. Phys. Lett. 30, 202-204 (1977).","doi":""},{"text":"Optical levitation in high vacuum. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 28A. Ashkin and J. M. Dziedzic, \"Optical levitation in high vacuum,\" Appl. Phys. Lett. 28, 333-335 (1976).","doi":""},{"text":"Optical levitation by radiation pressure. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 19A. Ashkin and J. M. Dziedzic, \"Optical levitation by radiation pressure,\" Appl. Phys. Lett. 19, 283-285 (1971).","doi":""},{"text":"Observation of a single-beam gradient-force optical trap for dielectric particles in air. R Omori, T Kobayashi, A Suzuki, Opt. Lett. 22R. Omori, T. Kobayashi, and A. Suzuki, \"Observation of a single-beam gradient-force optical trap for dielectric particles in air,\" Opt. Lett. 22, 816-818 (1997).","doi":""},{"text":"Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. J Gieseler, B Deutsch, R Quidant, L Novotny, Phys. Rev. Lett. 109103603J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, \"Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,\" Phys. Rev. Lett. 109, 103603 (2012).","doi":""},{"text":"Optical trapping of nanoparticles by full solid-angle focusing. V Salakhutdinov, M Sondermann, L Carbone, E Giacobino, A Bramati, G Leuchs, Optica. 3V. Salakhutdinov, M. Sondermann, L. Carbone, E. Giacobino, A. Bramati, and G. Leuchs, \"Optical trapping of nanoparticles by full solid-angle focusing,\" Optica 3, 1181-1186 (2016).","doi":""},{"text":"Experimental realization of a thermal squeezed state of levitated optomechanics. M Rashid, T Tufarelli, J Bateman, J Vovrosh, D Hempston, M S Kim, H Ulbricht, Phys. Rev. Lett. 117273601M. Rashid, T. Tufarelli, J. Bateman, J. Vovrosh, D. Hempston, M. S. Kim, and H. Ulbricht, \"Experimental realization of a thermal squeezed state of levitated optomechanics,\" Phys. Rev. Lett. 117, 273601 (2016).","doi":""},{"text":"Measurement of the instantaneous velocity of a Brownian particle. T Li, S Kheifets, D Medellin, M G Raizen, Science. 328T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, \"Measurement of the instantaneous velocity of a Brownian particle,\" Science 328, 1673-1675 (2010).","doi":""},{"text":"Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum. G Ranjit, D P Atherton, J H Stutz, M Cunningham, A A Geraci, Phys. Rev. A. 9151805G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, \"Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,\" Phys. Rev. A 91, 051805 (2015).","doi":""},{"text":"Toward quantum superposition of living organisms. O Romero-Isart, M L Juan, R Quidant, J I Cirac, New J. Phys. 1233015O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, \"Toward quantum superposition of living organisms,\" New J. Phys. 12, 033015 (2010).","doi":""},{"text":"Cavity opto-mechanics using an optically levitated nanosphere. D E Chang, C A Regal, S B Papp, D J Wilson, J Ye, O Painter, H J Kimble, P Zoller, Proc. Natl. Acad. Sci. 107D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, \"Cavity opto-mechanics using an optically levitated nanosphere,\" Proc. Natl. Acad. Sci. 107, 1005-1010 (2010).","doi":""},{"text":"Cavity cooling of an optically trapped nanoparticle. P F Barker, M N Shneider, Phys. Rev. A. 8123826P. F. Barker and M. N. Shneider, \"Cavity cooling of an optically trapped nanoparticle,\" Phys. Rev. A 81, 023826 (2010).","doi":""},{"text":"Cold damping of an optically levitated nanoparticle to microkelvin temperatures. F Tebbenjohanns, M Frimmer, A Militaru, V Jain, L Novotny, Phys. Rev. Lett. 122223601F. Tebbenjohanns, M. Frimmer, A. Militaru, V. Jain, and L. Novotny, \"Cold damping of an optically levitated nanoparticle to microkelvin temperatures,\" Phys. Rev. Lett. 122, 223601 (2019).","doi":""},{"text":"Direct measurement of photon recoil from a levitated nanoparticle. V Jain, J Gieseler, C Moritz, C Dellago, R Quidant, L Novotny, Phys. Rev. Lett. 116243601V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, \"Direct measurement of photon recoil from a levitated nanoparticle,\" Phys. Rev. Lett. 116, 243601 (2016).","doi":""},{"text":"Optimal feedback cooling of a charged levitated nanoparticle with adaptive control. G P Conangla, F Ricci, M T Cuairan, A W Schell, N Meyer, R Quidant, Phys. Rev. Lett. 122223602G. P. Conangla, F. Ricci, M. T. Cuairan, A. W. Schell, N. Meyer, and R. Quidant, \"Optimal feedback cooling of a charged levitated nanoparticle with adaptive control,\" Phys. Rev. Lett. 122, 223602 (2019).","doi":""},{"text":"Motional sideband asymmetry of a nanoparticle optically levitated in free space. F Tebbenjohanns, M Frimmer, V Jain, D Windey, L Novotny, arXiv:1908.05079arXiv preprintF. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, \"Motional sideband asymmetry of a nanoparticle optically levitated in free space,\" arXiv preprint arXiv:1908.05079 (2019).","doi":""},{"text":"Cavity cooling of a levitated nanosphere by coherent scattering. U Delić, M Reisenbauer, D Grass, N Kiesel, V Vuletić, M Aspelmeyer, Phys. Rev. Lett. 122123602U. Delić, M. Reisenbauer, D. Grass, N. Kiesel, V. Vuletić, and M. Aspelmeyer, \"Cavity cooling of a levitated nanosphere by coherent scattering,\" Phys. Rev. Lett. 122, 123602 (2019).","doi":""},{"text":"Cavity-based 3d cooling of a levitated nanoparticle via coherent scattering. D Windey, C Gonzalez-Ballestero, P Maurer, L Novotny, O Romero-Isart, R Reimann, Phys. Rev. Lett. 122123601D. Windey, C. Gonzalez-Ballestero, P. Maurer, L. Novotny, O. Romero-Isart, and R. Reimann, \"Cavity-based 3d cooling of a levitated nanoparticle via coherent scattering,\" Phys. Rev. Lett. 122, 123601 (2019).","doi":""},{"text":"Resolved-sideband cooling of a levitated nanoparticle in the presence of laser phase noise. N Meyer, A De Los Rios, P Sommer, J Mestres, V Gieseler, L Jain, R Novotny, Quidant, Phys. Rev. Lett. 123153601N. Meyer, A. de los Rios Sommer, P. Mestres, J. Gieseler, V. Jain, L. Novotny, and R. Quidant, \"Resolved-sideband cooling of a levitated nanoparticle in the presence of laser phase noise,\" Phys. Rev. Lett. 123, 153601 (2019).","doi":""},{"text":"Sideband cooling of micromechanical motion to the quantum ground state. J D Teufel, T Donner, D Li, J W Harlow, M S Allman, K Cicak, A J Sirois, J D Whittaker, K W Lehnert, R W Simmonds, Nature. 475J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, \"Sideband cooling of micromechanical motion to the quantum ground state,\" Nature 475, 359-405 (2011).","doi":""},{"text":"Laser cooling of a nanomechanical oscillator into its quantum ground state. J Chan, T P M Alegre, A H Safavi-Naeini, J T Hill, A Krause, S Gröblacher, M Aspelmeyer, O Painter, Nature. 478J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, \"Laser cooling of a nanomechanical oscillator into its quantum ground state,\" Nature 478, 89-92 (2011).","doi":""},{"text":"Cooling of a levitated nanoparticle to the motional quantum ground state. U Delić, M Reisenbauer, K Dare, D Grass, V Vuletić, N Kiesel, M Aspelmeyer, Science. 367U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, and M. Aspelmeyer, \"Cooling of a levitated nanoparticle to the motional quantum ground state,\" Science 367, 892-895 (2020).","doi":""},{"text":"Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary. S Haroche, Rev. Mod. Phys. 85S. Haroche, \"Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary,\" Rev. Mod. Phys. 85, 1083-1102 (2013).","doi":""},{"text":"Nobel lecture: Superposition, entanglement, and raising schrödinger's cat. D J Wineland, Rev. Mod. Phys. 851103D. J. Wineland, \"Nobel lecture: Superposition, entanglement, and raising schrödinger's cat,\" Rev. Mod. Phys. 85, 1103 (2013).","doi":""},{"text":"Quantum interference distinguishes between constitutional isomers. J Tüxen, S Gerlich, S Eibenberger, M Arndt, M Mayor, Chem. Commun. 46J. Tüxen, S. Gerlich, S. Eibenberger, M. Arndt, and M. Mayor, \"Quantum interference distinguishes between constitutional isomers,\" Chem. Commun. 46, 4145-4147 (2010).","doi":""},{"text":"Colloquium: Quantum interference of clusters and molecules. K Hornberger, S Gerlich, P Haslinger, S Nimmrichter, M Arndt, Rev. Mod. Phys. 84K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, \"Colloquium: Quantum interference of clusters and molecules,\" Rev. Mod. Phys. 84, 157-173 (2012).","doi":""},{"text":"Creating and verifying a quantum superposition in a micro-optomechanical system. D Kleckner, I Pikovski, E Jeffrey, L Ament, E Eliel, J Van Den, D Brink, Bouwmeester, New J. Phys. 1095020D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. Van Den Brink, and D. Bouwmeester, \"Creating and verifying a quantum superposition in a micro-optomechanical system,\" New J. Phys. 10, 095020 (2008).","doi":""},{"text":"Macroscopic quantum resonators (maqro). R Kaltenbaek, G Hechenblaikner, N Kiesel, O Romero-Isart, K C Schwab, U Johann, M Aspelmeyer, Exp. Astron. 34R. Kaltenbaek, G. Hechenblaikner, N. Kiesel, O. Romero-Isart, K. C. Schwab, U. Johann, and M. Aspelmeyer, \"Macroscopic quantum resonators (maqro),\" Exp. Astron. 34, 123-164 (2012).","doi":""},{"text":"Quantum superposition of massive objects and collapse models. O Romero-Isart, Phys. Rev. A. 8452121O. Romero-Isart, \"Quantum superposition of massive objects and collapse models,\" Phys. Rev. A 84, 052121 (2011).","doi":""},{"text":"Large quantum superpositions and interference of massive nanometer-sized objects. O Romero-Isart, A C Pflanzer, F Blaser, R Kaltenbaek, N Kiesel, M Aspelmeyer, J I Cirac, Phys. Rev. Lett. 10720405O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, \"Large quantum superpositions and interference of massive nanometer-sized objects,\" Phys. Rev. Lett. 107, 020405 (2011).","doi":""},{"text":"Near-field interferometry of a free-falling nanoparticle from a point-like source. J Bateman, S Nimmrichter, K Hornberger, H Ulbricht, Nat. Commun. 54788J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, \"Near-field interferometry of a free-falling nanoparticle from a point-like source,\" Nat. Commun. 5, 4788 (2014).","doi":""},{"text":"Models of wave-function collapse, underlying theories, and experimental tests. A Bassi, K Lochan, S Satin, T P Singh, H Ulbricht, Rev. Mod. Phys. 85A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, \"Models of wave-function collapse, underlying theories, and experimental tests,\" Rev. Mod. Phys. 85, 471-527 (2013).","doi":""},{"text":"Optomechanics of levitated dielectric particles. Z.-Q Yin, A A Geraci, T Li, Int. J. Mod. Phys. B. 271330018Z.-Q. Yin, A. A. Geraci, and T. Li, \"Optomechanics of levitated dielectric particles,\" Int. J. Mod. Phys. B 27, 1330018 (2013).","doi":""},{"text":"Cavity optomechanics. M Aspelmeyer, T J Kippenberg, F Marquardt, Rev. Mod. Phys. 86M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, \"Cavity optomechanics,\" Rev. Mod. Phys. 86, 1391-1452 (2014).","doi":""},{"text":"W P Bowen, G J Milburn, Quantum optomechanics. CRC pressW. P. Bowen and G. J. Milburn, Quantum optomechanics (CRC press, 2015).","doi":""},{"text":"Classical characterization of biphoton correlation in waveguide lattices. R Keil, F Dreisow, M Heinrich, A Tünnermann, S Nolte, A Szameit, Phys. Rev. A. 8313808R. Keil, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, \"Classical characterization of biphoton correlation in waveguide lattices,\" Phys. Rev. A 83, 013808 (2011).","doi":""},{"text":"Heralded single-phonon preparation, storage, and readout in cavity optomechanics. C Galland, N Sangouard, N Piro, N Gisin, T J Kippenberg, Phys. Rev. Lett. 112143602C. Galland, N. Sangouard, N. Piro, N. Gisin, and T. J. Kippenberg, \"Heralded single-phonon preparation, storage, and readout in cavity optomechanics,\" Phys. Rev. Lett. 112, 143602 (2014).","doi":""},{"text":"Phonon counting and intensity interferometry of a nanomechanical resonator. J D Cohen, S M Meenehan, G S Maccabe, S Gröblacher, A H Safavi-Naeini, F Marsili, M D Shaw, O Painter, Nature. 520J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, \"Phonon counting and intensity interferometry of a nanomechanical resonator,\" Nature 520, 522-525 (2015).","doi":""},{"text":"Hanbury brown and twiss interferometry of single phonons from an optomechanical resonator. S Hong, R Riedinger, I Marinković, A Wallucks, S G Hofer, R A Norte, M Aspelmeyer, S Gröblacher, Science. 358S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, \"Hanbury brown and twiss interferometry of single phonons from an optomechanical resonator,\" Science 358, 203-206 (2017).","doi":""},{"text":"Cooling of mechanical motion with a two-level system: The high-temperature regime. P , Phys. Rev. B. 82165320P. Rabl, \"Cooling of mechanical motion with a two-level system: The high-temperature regime,\" Phys. Rev. B 82, 165320 (2010).","doi":""},{"text":"Strong magnetic coupling between an electronic spin qubit and a mechanical resonator. P Rabl, P Cappellaro, M V G Dutt, L Jiang, J R Maze, M D Lukin, Phys. Rev. B. 7941302P. Rabl, P. Cappellaro, M. V. G. Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, \"Strong magnetic coupling between an electronic spin qubit and a mechanical resonator,\" Phys. Rev. B 79, 041302 (2009).","doi":""},{"text":"Hybrid opto-mechanical systems with nitrogen-vacancy centers. Z.-Q Yin, N Zhao, T Li, Sci. China Phys., Mech. Astron. 58Z.-Q. Yin, N. Zhao, and T. Li, \"Hybrid opto-mechanical systems with nitrogen-vacancy centers,\" Sci. China Phys., Mech. Astron. 58, 1-12 (2015).","doi":""},{"text":"Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond. L P Neukirch, E Von Haartman, J M Rosenholm, A N Vamivakas, Nat. Photon. 9L. P. Neukirch, E. Von Haartman, J. M. Rosenholm, and A. N. Vamivakas, \"Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,\" Nat. Photon. 9, 653-657 (2015).","doi":""},{"text":"Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond. L P Neukirch, J Gieseler, R Quidant, L Novotny, A N Vamivakas, Opt. Lett. 38L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. N. Vamivakas, \"Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,\" Opt. Lett. 38, 2976-2979 (2013).","doi":""},{"text":"Pure nanodiamonds for levitated optomechanics in vacuum. A C Frangeskou, A T M A Rahman, L Gines, S Mandal, O A Williams, P F Barker, G W Morley, New J. Phys. 2043016A. C. Frangeskou, A. T. M. A. Rahman, L. Gines, S. Mandal, O. A. Williams, P. F. Barker, and G. W. Morley, \"Pure nanodiamonds for levitated optomechanics in vacuum,\" New J. Phys. 20, 043016 (2018).","doi":""},{"text":"Burning and graphitization of optically levitated nanodiamonds in vacuum. A T M A Rahman, A C Frangeskou, M S Kim, S Bose, G W Morley, P F Barker, Sci. Rep. p. 21633A. T. M. A. Rahman, A. C. Frangeskou, M. S. Kim, S. Bose, G. W. Morley, and P. F. Barker, \"Burning and graphitization of optically levitated nanodiamonds in vacuum,\" Sci. Rep. p. 21633 (2016).","doi":""},{"text":"Electron spin control of optically levitated nanodiamonds in vacuum. T M Hoang, J Ahn, J Bang, T Li, Nat. Commun. 712250T. M. Hoang, J. Ahn, J. Bang, and T. Li, \"Electron spin control of optically levitated nanodiamonds in vacuum,\" Nat. Commun. 7, 12250 (2016).","doi":""},{"text":"Cooling-by-measurement and mechanical state tomography via pulsed optomechanics. M R Vanner, J Hofer, G D Cole, M Aspelmeyer, Nat. Commun. 42295M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, \"Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,\" Nat. Commun. 4, 2295 (2013).","doi":""},{"text":"Pulsed quantum optomechanics. M R Vanner, I Pikovski, G D Cole, M S Kim, Č Brukner, K Hammerer, G J Milburn, M Aspelmeyer, Proc. Natl. Acad. Sci. Natl. Acad. Sci108M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, \"Pulsed quantum optomechanics,\" Proc. Natl. Acad. Sci. 108, 16182-16187 (2011).","doi":""},{"text":"Single phonon quantum optics -An experimental exploration of a silicon photonics quantum memory. R Riedinger, ViennaPh.D. thesisR. Riedinger, \"Single phonon quantum optics -An experimental exploration of a silicon photonics quantum memory,\" Ph.D. thesis, Vienna (2018).","doi":""},{"text":"Torsional optomechanics of a levitated nonspherical nanoparticle. T M Hoang, Y Ma, J Ahn, J Bang, F Robicheaux, Z.-Q Yin, T Li, Phys. Rev. Lett. 117123604T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.-Q. Yin, and T. Li, \"Torsional optomechanics of a levitated nonspherical nanoparticle,\" Phys. Rev. Lett. 117, 123604 (2016).","doi":""},{"text":"Precession motion in levitated optomechanics. M Rashid, M Toroš, A Setter, H Ulbricht, Phys. Rev. Lett. 121253601M. Rashid, M. Toroš, A. Setter, and H. Ulbricht, \"Precession motion in levitated optomechanics,\" Phys. Rev. Lett. 121, 253601 (2018).","doi":""},{"text":"Zeptonewton force sensing with nanospheres in an optical lattice. G Ranjit, M Cunningham, K Casey, A A Geraci, Phys. Rev. A. 9353801G. Ranjit, M. Cunningham, K. Casey, and A. A. Geraci, \"Zeptonewton force sensing with nanospheres in an optical lattice,\" Phys. Rev. A 93, 053801 (2016).","doi":""},{"text":"Force sensing with an optically levitated charged nanoparticle. D Hempston, J Vovrosh, M Toroš, G Winstone, M Rashid, H Ulbricht, Appl. Phys. Lett. 111133111D. Hempston, J. Vovrosh, M. Toroš, G. Winstone, M. Rashid, and H. Ulbricht, \"Force sensing with an optically levitated charged nanoparticle,\" Appl. Phys. Lett. 111, 133111 (2017).","doi":""},{"text":"Three-dimensional force-field microscopy with optically levitated microspheres. C P Blakemore, A D Rider, S Roy, Q Wang, A Kawasaki, G Gratta, Phys. Rev. A. 9923816C. P. Blakemore, A. D. Rider, S. Roy, Q. Wang, A. Kawasaki, and G. Gratta, \"Three-dimensional force-field microscopy with optically levitated microspheres,\" Phys. Rev. A 99, 023816 (2019).","doi":""},{"text":"Optical levitation of 10-ng spheres with nano-g acceleration sensitivity. F Monteiro, S Ghosh, A G Fine, D C Moore, Phys. Rev. A. 9663841F. Monteiro, S. Ghosh, A. G. Fine, and D. C. Moore, \"Optical levitation of 10-ng spheres with nano-g acceleration sensitivity,\" Phys. Rev. A 96, 063841 (2017).","doi":""},{"text":"Search for screened interactions associated with dark energy below the 100 µ m length scale. A D Rider, D C Moore, C P Blakemore, M Louis, M Lu, G Gratta, Phys. Rev. Lett. 117101101A. D. Rider, D. C. Moore, C. P. Blakemore, M. Louis, M. Lu, and G. Gratta, \"Search for screened interactions associated with dark energy below the 100 µ m length scale,\" Phys. Rev. Lett. 117, 101101 (2016).","doi":""},{"text":"Search for millicharged particles using optically levitated microspheres. D C Moore, A D Rider, G Gratta, Phys. Rev. Lett. 113251801D. C. Moore, A. D. Rider, and G. Gratta, \"Search for millicharged particles using optically levitated microspheres,\" Phys. Rev. Lett. 113, 251801 (2014).","doi":""},{"text":"Sensing short range forces with a nanosphere matter-wave interferometer. A Geraci, H Goldman, Phys. Rev. D. 9262002A. Geraci and H. Goldman, \"Sensing short range forces with a nanosphere matter-wave interferometer,\" Phys. Rev. D 92, 062002 (2015).","doi":""},{"text":"Casimir interaction between a dielectric nanosphere and a metallic plane. A Canaguier-Durand, A Gérardin, R Guérout, P A M Neto, V V Nesvizhevsky, A Y Voronin, A Lambrecht, S Reynaud, Phys. Rev. A. 8332508A. Canaguier-Durand, A. Gérardin, R. Guérout, P. A. M. Neto, V. V. Nesvizhevsky, A. Y. Voronin, A. Lambrecht, and S. Reynaud, \"Casimir interaction between a dielectric nanosphere and a metallic plane,\" Phys. Rev. A 83, 032508 (2011).","doi":""},{"text":"Lateral Casimir force on a rotating particle near a planar surface. A Manjavacas, F J Rodríguez-Fortuño, F J García De Abajo, A V Zayats, Phys. Rev. Lett. 118133605A. Manjavacas, F. J. Rodríguez-Fortuño, F. J. García de Abajo, and A. V. Zayats, \"Lateral Casimir force on a rotating particle near a planar surface,\" Phys. Rev. Lett. 118, 133605 (2017).","doi":""},{"text":"Dynamics of a levitated nanosphere by optomechanical coupling and Casimir interaction. W Nie, Y Lan, Y Li, S Zhu, Phys. Rev. A. 8863849W. Nie, Y. Lan, Y. Li, and S. Zhu, \"Dynamics of a levitated nanosphere by optomechanical coupling and Casimir interaction,\" Phys. Rev. A 88, 063849 (2013).","doi":""},{"text":"Detecting Casimir torque with an optically levitated nanorod. Z Xu, T Li, Phys. Rev. A. 9633843Z. Xu and T. Li, \"Detecting Casimir torque with an optically levitated nanorod,\" Phys. Rev. A 96, 033843 (2017).","doi":""},{"text":"Detecting high-frequency gravitational waves with optically levitated sensors. A Arvanitaki, A A Geraci, Phys. Rev. Lett. 11071105A. Arvanitaki and A. A. Geraci, \"Detecting high-frequency gravitational waves with optically levitated sensors,\" Phys. Rev. Lett. 110, 071105 (2013).","doi":""},{"text":"Nonlinear dynamics of nanomechanical and micromechanical resonators. R Lifshitz, M C Cross, Rev. Nonlinear Dyn. Complex. pp. R. Lifshitz and M. C. Cross, \"Nonlinear dynamics of nanomechanical and micromechanical resonators,\" Rev. Nonlinear Dyn. Complex. pp. 1-52 (2008).","doi":""},{"text":"Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle. J Gieseler, M Spasenović, L Novotny, R Quidant, Phys. Rev. Lett. 112103603J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, \"Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,\" Phys. Rev. Lett. 112, 103603 (2014).","doi":""},{"text":"Optically levitated nanoparticle as a model system for stochastic bistable dynamics. F Ricci, R A Rica, M Spasenović, J Gieseler, L Rondin, L Novotny, R Quidant, Nat. Commun. p. 15141F. Ricci, R. A. Rica, M. Spasenović, J. Gieseler, L. Rondin, L. Novotny, and R. Quidant, \"Optically levitated nanoparticle as a model system for stochastic bistable dynamics,\" Nat. Commun. p. 15141 (2017).","doi":""},{"text":"Thermal nonlinearities in a nanomechanical oscillator. J Gieseler, L Novotny, R Quidant, Nat. Phys. 9J. Gieseler, L. Novotny, and R. Quidant, \"Thermal nonlinearities in a nanomechanical oscillator,\" Nat. Phys. 9, 806-810 (2013).","doi":""},{"text":"Sensing static forces with free-falling nanoparticles. E Hebestreit, M Frimmer, R Reimann, L Novotny, Phys. Rev. Lett. 12163602E. Hebestreit, M. Frimmer, R. Reimann, and L. Novotny, \"Sensing static forces with free-falling nanoparticles,\" Phys. Rev. Lett. 121, 063602 (2018).","doi":""},{"text":"Probing macroscopic quantum superpositions with nanorotors. B A Stickler, B Papendell, S Kuhn, B Schrinski, J Millen, M Arndt, K Hornberger, New J. Phys. 20122001B. A. Stickler, B. Papendell, S. Kuhn, B. Schrinski, J. Millen, M. Arndt, and K. Hornberger, \"Probing macroscopic quantum superpositions with nanorotors,\" New J. Phys. 20, 122001 (2018).","doi":""},{"text":"Laser-induced rotation and cooling of a trapped microgyroscope in vacuum. Y Arita, M Mazilu, K Dholakia, Nat. Commun. 42374Y. Arita, M. Mazilu, and K. Dholakia, \"Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,\" Nat. Commun. 4, 2374 (2013).","doi":""},{"text":"Cavity-assisted manipulation of freely rotating silicon nanorods in high vacuum. S Kuhn, P Asenbaum, A Kosloff, M Sclafani, B A Stickler, S Nimmrichter, K Hornberger, O Cheshnovsky, F Patolsky, M Arndt, Nano Lett. 15S. Kuhn, P. Asenbaum, A. Kosloff, M. Sclafani, B. A. Stickler, S. Nimmrichter, K. Hornberger, O. Cheshnovsky, F. Patolsky, and M. Arndt, \"Cavity-assisted manipulation of freely rotating silicon nanorods in high vacuum,\" Nano Lett. 15, 5604-5608 (2015).","doi":""},{"text":"Full rotational control of levitated silicon nanorods. S Kuhn, A Kosloff, B A Stickler, F Patolsky, K Hornberger, M Arndt, J Millen, Optica. 4S. Kuhn, A. Kosloff, B. A. Stickler, F. Patolsky, K. Hornberger, M. Arndt, and J. Millen, \"Full rotational control of levitated silicon nanorods,\" Optica 4, 356-360 (2017).","doi":""},{"text":"Laser refrigeration, alignment and rotation of levitated yb 3+: Ylf nanocrystals. A T M A Rahman, P F Barker, Nat. Photon. 11A. T. M. A. Rahman and P. F. Barker, \"Laser refrigeration, alignment and rotation of levitated yb 3+: Ylf nanocrystals,\" Nat. Photon. 11, 634-639 (2017).","doi":""},{"text":"Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor. J Ahn, Z Xu, J Bang, Y.-H Deng, T M Hoang, Q Han, R.-M Ma, T Li, Phys. Rev. Lett. 12133603J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, \"Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,\" Phys. Rev. Lett. 121, 033603 (2018).","doi":""},{"text":"Ghz rotation of an optically trapped nanoparticle in vacuum. R Reimann, M Doderer, E Hebestreit, R Diehl, M Frimmer, D Windey, F Tebbenjohanns, L Novotny, Phys. Rev. Lett. 12133602R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, \"Ghz rotation of an optically trapped nanoparticle in vacuum,\" Phys. Rev. Lett. 121, 033602 (2018).","doi":""},{"text":"Rotational quantum friction. R Zhao, A Manjavacas, F J García De Abajo, J B Pendry, Phys. Rev. Lett. 109123604R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, \"Rotational quantum friction,\" Phys. Rev. Lett. 109, 123604 (2012).","doi":""},{"text":"Direct measurement of kramers turnover with a levitated nanoparticle. L Rondin, J Gieseler, F Ricci, R Quidant, C Dellago, L Novotny, Nat. Nanotech. 12L. Rondin, J. Gieseler, F. Ricci, R. Quidant, C. Dellago, and L. Novotny, \"Direct measurement of kramers turnover with a levitated nanoparticle,\" Nat. Nanotech. 12, 1130-1133 (2017).","doi":""},{"text":"Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state. J Gieseler, R Quidant, C Dellago, L Novotny, Nat. Nanotech. 9J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, \"Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state,\" Nat. Nanotech. 9, 358-364 (2014).","doi":""},{"text":"Experimental test of the differential fluctuation theorem and a generalized Jarzynski equality for arbitrary initial states. T M Hoang, R Pan, J Ahn, J Bang, H T Quan, T Li, Phys. Rev. Lett. 12080602T. M. Hoang, R. Pan, J. Ahn, J. Bang, H. T. Quan, and T. Li, \"Experimental test of the differential fluctuation theorem and a generalized Jarzynski equality for arbitrary initial states,\" Phys. Rev. Lett. 120, 080602 (2018).","doi":""},{"text":"M Debiossac, D Grass, J J Alonso, E Lutz, N Kiesel, arXiv:1904.04889Thermodynamics of continuous non-markovian feedback control. arXiv preprintM. Debiossac, D. Grass, J. J. Alonso, E. Lutz, and N. Kiesel, \"Thermodynamics of continuous non-markovian feedback control,\" arXiv preprint arXiv:1904.04889 (2019).","doi":""},{"text":"Cooling mechanical oscillators by coherent control. M Frimmer, J Gieseler, L Novotny, Phys. Rev. Lett. 117163601M. Frimmer, J. Gieseler, and L. Novotny, \"Cooling mechanical oscillators by coherent control,\" Phys. Rev. Lett. 117, 163601 (2016).","doi":""},{"text":"Thermal properties of levitated nanoparticles. E Hebestreit, ETH ZurichPh.D. thesisE. Hebestreit, \"Thermal properties of levitated nanoparticles,\" Ph.D. thesis, ETH Zurich (2017).","doi":""},{"text":"Levitodynamics toward force nano-sensors in vacuum. F Ricci, Universitat Politècnica de CatalunyaPh.D. thesisF. Ricci, \"Levitodynamics toward force nano-sensors in vacuum,\" Ph.D. thesis, Universitat Politècnica de Catalunya (2019).","doi":""},{"text":"Scanning confocal optical microscopy and magnetic resonance on single defect centers. A Gruber, A Dräbenstedt, C Tietz, L Fleury, J Wrachtrup, C Von Borczyskowski, Science. 276A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. Von Borczyskowski, \"Scanning confocal optical microscopy and magnetic resonance on single defect centers,\" Science 276, 2012-2014 (1997).","doi":""},{"text":"Dynamics of optically levitated nanoparticles in high vacuum. J Gieseler, Universitat Politècnica de CatalunyaPh.D. thesisJ. Gieseler, \"Dynamics of optically levitated nanoparticles in high vacuum,\" Ph.D. thesis, Universitat Politècnica de Catalunya (2014).","doi":""},{"text":"Trapping in a material world. S E S Spesyvtseva, K Dholakia, ACS Photon. 3S. E. S. Spesyvtseva and K. Dholakia, \"Trapping in a material world,\" ACS Photon. 3, 719-736 (2016).","doi":""},{"text":"Cancellation of non-conservative scattering forces in optical traps by counter-propagating beams. S Divitt, L Rondin, L Novotny, Opt. Lett. 40S. Divitt, L. Rondin, and L. Novotny, \"Cancellation of non-conservative scattering forces in optical traps by counter-propagating beams,\" Opt. Lett. 40, 1900-1903 (2015).","doi":""},{"text":"Dynamics of noise-induced heating in atom traps. M E Gehm, K M O'hara, T A Savard, J E Thomas, Phys. Rev. A. 58M. E. Gehm, K. M. O'hara, T. A. Savard, and J. E. Thomas, \"Dynamics of noise-induced heating in atom traps,\" Phys. Rev. A 58, 3914-3921 (1998).","doi":""},{"text":"Theory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach. C Gonzalez-Ballestero, P Maurer, D Windey, L Novotny, R Reimann, O Romero-Isart, arXiv:1902.01282arXiv preprintC. Gonzalez-Ballestero, P. Maurer, D. Windey, L. Novotny, R. Reimann, and O. Romero-Isart, \"Theory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach,\" arXiv preprint arXiv:1902.01282 (2019).","doi":""},{"text":"Levitated optomechanics at the photon recoil limit. V Jain, ETH ZurichPh.D. thesisV. Jain, \"Levitated optomechanics at the photon recoil limit,\" Ph.D. thesis, ETH Zurich (2017).","doi":""},{"text":"Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss. S Kheifets, A Simha, K Melin, T Li, M G Raizen, Science. 343S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, \"Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,\" Science 343, 1493-1496 (2014).","doi":""},{"text":"Three-dimensional position detection of optically trapped dielectric particles. A Rohrbach, E H K Stelzer, J. Appl. Phys. 91A. Rohrbach and E. H. K. Stelzer, \"Three-dimensional position detection of optically trapped dielectric particles,\" J. Appl. Phys. 91, 5474-5488 (2002).","doi":""},{"text":"Backscattering position detection for photonic force microscopy. G Volpe, G Kozyreff, D Petrov, J. Appl. Phys. 10284701G. Volpe, G. Kozyreff, and D. Petrov, \"Backscattering position detection for photonic force microscopy,\" J. Appl. Phys. 102, 084701 (2007).","doi":""},{"text":"Optimal position detection of a dipolar scatterer in a focused field. F Tebbenjohanns, M Frimmer, L Novotny, Phys. Rev. A. 10043821F. Tebbenjohanns, M. Frimmer, and L. Novotny, \"Optimal position detection of a dipolar scatterer in a focused field,\" Phys. Rev. A 100, 043821 (2019).","doi":""},{"text":"Transverse spin forces and non-equilibrium particle dynamics in a circularly polarized vacuum optical trap. V Svak, O Brzobohatỳ, M Šiler, P Jákl, J Kaňka, P Zemánek, S H Simpson, Nat. Commun. 95453V. Svak, O. Brzobohatỳ, M. Šiler, P. Jákl, J. Kaňka, P. Zemánek, and S. H. Simpson, \"Transverse spin forces and non-equilibrium particle dynamics in a circularly polarized vacuum optical trap,\" Nat. Commun. 9, 5453 (2018).","doi":""},{"text":"Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps. M D Summers, D R Burnham, D Mcgloin, Opt. Express. 16M. D. Summers, D. R. Burnham, and D. McGloin, \"Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps,\" Opt. Express 16, 7739-7747 (2008).","doi":""},{"text":"Cavity cooling by coherent scattering of a levitated nanosphere in vacuum. U Delić, ViennaUniversity of ViennaPh.D. thesisU. Delić, \"Cavity cooling by coherent scattering of a levitated nanosphere in vacuum,\" Ph.D. thesis, University of Vienna, Vienna (2019).","doi":""},{"text":"Adhesion and friction forces between spherical micrometer-sized particles. L.-O Heim, J Blum, M Preuss, H.-J Butt, Phys. Rev. Lett. 83L.-O. Heim, J. Blum, M. Preuss, and H.-J. Butt, \"Adhesion and friction forces between spherical micrometer-sized particles,\" Phys. Rev. Lett. 83, 3328-3331 (1999).","doi":""},{"text":"T Li, Fundamental tests of physics with optically trapped microspheres. Springer Science & Business MediaT. Li, Fundamental tests of physics with optically trapped microspheres (Springer Science & Business Media, 2012).","doi":""},{"text":"Contact electrification of individual dielectric microparticles measured by optical tweezers in air. H Park, T W Lebrun, ACS Appl. Mater. Interfaces. 8H. Park and T. W. LeBrun, \"Contact electrification of individual dielectric microparticles measured by optical tweezers in air,\" ACS Appl. Mater. Interfaces 8, 34904-34913 (2016).","doi":""},{"text":"Sensitive force measurements with optically trapped micro-spheres in high vacuum. D P Atherton, RenoUniversity of NevadaPh.D. thesisD. P. Atherton, \"Sensitive force measurements with optically trapped micro-spheres in high vacuum,\" Ph.D. thesis, University of Nevada, Reno (2015).","doi":""},{"text":"Cavity cooling of free silicon nanoparticles in high vacuum. P Asenbaum, S Kuhn, S Nimmrichter, U Sezer, M Arndt, Nat. Commun. 42743P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, \"Cavity cooling of free silicon nanoparticles in high vacuum,\" Nat. Commun. 4, 2743 (2013).","doi":""},{"text":"Direct loading of nanoparticles under high vacuum into a Paul trap for levitodynamical experiments. D S Bykov, P Mestres, L Dania, L Schmöger, T E Northup, arXiv:1905.04204arXiv preprintD. S. Bykov, P. Mestres, L. Dania, L. Schmöger, and T. E. Northup, \"Direct loading of nanoparticles under high vacuum into a Paul trap for levitodynamical experiments,\" arXiv preprint arXiv:1905.04204 (2019).","doi":""},{"text":"Cooling and manipulation of a levitated nanoparticle with an optical fiber trap. P Mestres, J Berthelot, M Spasenović, J Gieseler, L Novotny, R Quidant, Appl. Phys. Lett. 107151102P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, \"Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,\" Appl. Phys. Lett. 107, 151102 (2015).","doi":""},{"text":"Flying particle sensors in hollow-core photonic crystal fibre. D S Bykov, O A Schmidt, T G Euser, P S J Russell, Nat. Photon. 9D. S. Bykov, O. A. Schmidt, T. G. Euser, and P. S. J. Russell, \"Flying particle sensors in hollow-core photonic crystal fibre,\" Nat. Photon. 9, 461-466 (2015).","doi":""},{"text":"Optical trapping and control of nanoparticles inside evacuated hollow core photonic crystal fibers. D Grass, J Fesel, S G Hofer, N Kiesel, M Aspelmeyer, Appl. Phys. Lett. 108221103D. Grass, J. Fesel, S. G. Hofer, N. Kiesel, and M. Aspelmeyer, \"Optical trapping and control of nanoparticles inside evacuated hollow core photonic crystal fibers,\" Appl. Phys. Lett. 108, 221103 (2016).","doi":""},{"text":"Temperature in non-equilibrium states: A review of open problems and current proposals. J Casas-Vázquez, D Jou, Rep. Prog. Phys. 66J. Casas-Vázquez and D. Jou, \"Temperature in non-equilibrium states: A review of open problems and current proposals,\" Rep. Prog. Phys. 66, 1937-2023 (2003).","doi":""},{"text":"Realization of nonequilibrium thermodynamic processes using external colored noise. P Mestres, I A Martinez, A Ortiz-Ambriz, R A Rica, E Roldan, Phys. Rev. E. 9032116P. Mestres, I. A. Martinez, A. Ortiz-Ambriz, R. A. Rica, and E. Roldan, \"Realization of nonequilibrium thermody- namic processes using external colored noise,\" Phys. Rev. E 90, 032116 (2014).","doi":""},{"text":"Motion of a spherical particle in a rarefied gas. part 2. drag and thermal polarization. S A Beresnev, V G Chernyak, G A Fomyagin, J. Fluid Mech. 219S. A. Beresnev, V. G. Chernyak, and G. A. Fomyagin, \"Motion of a spherical particle in a rarefied gas. part 2. drag and thermal polarization,\" J. Fluid Mech. 219, 405-421 (1990).","doi":""},{"text":"Gas-induced friction and diffusion of rigid rotors. L Martinetz, K Hornberger, B A Stickler, Phys. Rev. E. 9752112L. Martinetz, K. Hornberger, and B. A. Stickler, \"Gas-induced friction and diffusion of rigid rotors,\" Phys. Rev. E 97, 052112 (2018).","doi":""},{"text":"Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere. J Millen, T Deesuwan, P Barker, J Anders, Nat. Nanotech. 9J. Millen, T. Deesuwan, P. Barker, and J. Anders, \"Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,\" Nat. Nanotech. 9, 425-429 (2014).","doi":""},{"text":"On the resistance experienced by spheres in their motion through gases. P S Epstein, Phys. Rev. 23P. S. Epstein, \"On the resistance experienced by spheres in their motion through gases,\" Phys. Rev. 23, 710-733 (1924).","doi":""},{"text":"Radiation damping of a polarizable particle. L Novotny, Phys. Rev. A. 9632108L. Novotny, \"Radiation damping of a polarizable particle,\" Phys. Rev. A 96, 032108 (2017).","doi":""},{"text":"Quantum optomechanics. M Aspelmeyer, P Meystre, K Schwab, Phys. Today. 65M. Aspelmeyer, P. Meystre, and K. Schwab, \"Quantum optomechanics,\" Phys. Today 65, 29-35 (2012).","doi":""},{"text":"Calibration and energy measurement of optically levitated nanoparticle sensors. E Hebestreit, M Frimmer, R Reimann, C Dellago, F Ricci, L Novotny, Rev. Sci. Instrumen. 8933111E. Hebestreit, M. Frimmer, R. Reimann, C. Dellago, F. Ricci, and L. Novotny, \"Calibration and energy measurement of optically levitated nanoparticle sensors,\" Rev. Sci. Instrumen. 89, 033111 (2018).","doi":""},{"text":"Introduction to quantum noise, measurement, and amplification. A A Clerk, M H Devoret, S M Girvin, F Marquardt, R J Schoelkopf, Rev. Mod. Phys. 82A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, \"Introduction to quantum noise, measurement, and amplification,\" Rev. Mod. Phys. 82, 1155-1208 (2010).","doi":""},{"text":"Optomechanical Raman-ratio thermometry. T P Purdy, P.-L Yu, N S Kampel, R W Peterson, K Cicak, R W Simmonds, C A , Phys. Rev. A. 9231802T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, \"Optomechanical Raman-ratio thermometry,\" Phys. Rev. A 92, 031802 (2015).","doi":""},{"text":"Observation of quantum motion of a nanomechanical resonator. A H Safavi-Naeini, J Chan, J T Hill, T P M Alegre, A Krause, O Painter, Phys. Rev. Lett. 10833602A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, \"Observation of quantum motion of a nanomechanical resonator,\" Phys. Rev. Lett. 108, 033602 (2012).","doi":""},{"text":"Observation and interpretation of motional sideband asymmetry in a quantum electromechanical device. A J Weinstein, C U Lei, E E Wollman, J Suh, A Metelmann, A A Clerk, K C Schwab, Phys. Rev. X. 441003A. J. Weinstein, C. U. Lei, E. E. Wollman, J. Suh, A. Metelmann, A. A. Clerk, and K. C. Schwab, \"Observation and interpretation of motional sideband asymmetry in a quantum electromechanical device,\" Phys. Rev. X 4, 041003 (2014).","doi":""},{"text":"Controlling the net charge on a nanoparticle optically levitated in vacuum. M Frimmer, K Luszcz, S Ferreiro, V Jain, E Hebestreit, L Novotny, Phys. Rev. A. 9561801M. Frimmer, K. Luszcz, S. Ferreiro, V. Jain, E. Hebestreit, and L. Novotny, \"Controlling the net charge on a nanoparticle optically levitated in vacuum,\" Phys. Rev. A 95, 061801 (2017).","doi":""},{"text":"A nanomechanical mass sensor with yoctogram resolution. J Chaste, A Eichler, J Moser, G Ceballos, R Rurali, A Bachtold, Nat. Nanotech. 7J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, \"A nanomechanical mass sensor with yoctogram resolution,\" Nat. Nanotech. 7, 301-304 (2012).","doi":""},{"text":"Measuring the internal temperature of a levitated nanoparticle in high vacuum. E Hebestreit, R Reimann, M Frimmer, L Novotny, Phys. Rev. A. 9743803E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, \"Measuring the internal temperature of a levitated nanoparticle in high vacuum,\" Phys. Rev. A 97, 043803 (2018).","doi":""},{"text":"Cavity cooling of an optically levitated submicron particle. N Kiesel, F Blaser, U Delić, D Grass, R Kaltenbaek, M Aspelmeyer, Proc. Natl. Acad. Sci. 110N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, \"Cavity cooling of an optically levitated submicron particle,\" Proc. Natl. Acad. Sci. 110, 14180-14185 (2013).","doi":""},{"text":"Millikelvin cooling of an optically trapped microsphere in vacuum. T Li, S Kheifets, M G Raizen, Nat. Phys. 7T. Li, S. Kheifets, and M. G. Raizen, \"Millikelvin cooling of an optically trapped microsphere in vacuum,\" Nat. Phys. 7, 527-530 (2011).","doi":""},{"text":"Electric feedback cooling of single charged nanoparticles in an optical trap. M Iwasaki, T Yotsuya, T Naruki, Y Matsuda, M Yoneda, K Aikawa, Phys. Rev. A. 9951401M. Iwasaki, T. Yotsuya, T. Naruki, Y. Matsuda, M. Yoneda, and K. Aikawa, \"Electric feedback cooling of single charged nanoparticles in an optical trap,\" Phys. Rev. A 99, 051401 (2019).","doi":""},{"text":"Feedback cooling of a cantilever's fundamental mode below 5 mk. M Poggio, C L Degen, H J Mamin, D Rugar, Phys. Rev. Lett. 9917201M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, \"Feedback cooling of a cantilever's fundamental mode below 5 mk,\" Phys. Rev. Lett. 99, 017201 (2007).","doi":""},{"text":"Spontaneous emission probabilities at radio frequencies. E M Purcell, Confined Electrons and Photons. SpringerE. M. Purcell, \"Spontaneous emission probabilities at radio frequencies,\" in Confined Electrons and Photons, (Springer, 1995), pp. 839-839.","doi":""},{"text":"A Yariv, Quantum electronics. New YorkJohn Wiley & SonsA. Yariv, Quantum electronics (John Wiley & Sons, New York, 1989).","doi":""},{"text":"Non-equilibrium steady state of a driven levitated particle with feedback cooling. J Gieseler, L Novotny, C Moritz, C Dellago, New J. Phys. 1745011J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, \"Non-equilibrium steady state of a driven levitated particle with feedback cooling,\" New J. Phys. 17, 045011 (2015).","doi":""},{"text":"Quantum model of cooling and force sensing with an optically trapped nanoparticle. B Rodenburg, L P Neukirch, A N Vamivakas, M Bhattacharya, Optica. 3B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, \"Quantum model of cooling and force sensing with an optically trapped nanoparticle,\" Optica 3, 318-323 (2016).","doi":""},{"text":"Quantum calculation of feedback cooling a laser levitated nanoparticle in the shot-noise-dominant regime. C Zhong, T Li, F Robicheaux, arXiv:1708.01203arXiv preprintC. Zhong, T. Li, and F. Robicheaux, \"Quantum calculation of feedback cooling a laser levitated nanoparticle in the shot-noise-dominant regime,\" arXiv preprint arXiv:1708.01203 (2017).","doi":""},{"text":"Free nano-object Ramsey interferometry for large quantum superpositions. C Wan, M Scala, G W Morley, A T M A Rahman, H Ulbricht, J Bateman, P F Barker, S Bose, M S Kim, Phys. Rev. Lett. 117143003C. Wan, M. Scala, G. W. Morley, A. T. M. A. Rahman, H. Ulbricht, J. Bateman, P. F. Barker, S. Bose, and M. S. Kim, \"Free nano-object Ramsey interferometry for large quantum superpositions,\" Phys. Rev. Lett. 117, 143003 (2016).","doi":""},{"text":"Optical cryocooling of diamond. M Kern, J Jeske, D W M Lau, A D Greentree, F Jelezko, J Twamley, Phys. Rev. B. 95235306M. Kern, J. Jeske, D. W. M. Lau, A. D. Greentree, F. Jelezko, and J. Twamley, \"Optical cryocooling of diamond,\" Phys. Rev. B 95, 235306 (2017).","doi":""},{"text":"Diamonds levitating in a Paul trap under vacuum: Measurements of laser-induced heating via NV center thermometry. T Delord, L Nicolas, M Bodini, G Hétet, Appl. Phys. Lett. 11113101T. Delord, L. Nicolas, M. Bodini, and G. Hétet, \"Diamonds levitating in a Paul trap under vacuum: Measurements of laser-induced heating via NV center thermometry,\" Appl. Phys. Lett. 111, 013101 (2017).","doi":""},{"text":"Internal quantum dynamics of a nanoparticle in a thermal electromagnetic field: A minimal model. A E R López, C Gonzalez-Ballestero, O Romero-Isart, Phys. Rev. B. 98155405A. E. R. López, C. Gonzalez-Ballestero, and O. Romero-Isart, \"Internal quantum dynamics of a nanoparticle in a thermal electromagnetic field: A minimal model,\" Phys. Rev. B 98, 155405 (2018).","doi":""},{"text":"Dynamics of a levitated microparticle in vacuum trapped by a perfect vortex beam: three-dimensional motion around a complex optical potential. Y Arita, M Chen, E M Wright, K Dholakia, J. Opt. Soc. Am. B. 34Y. Arita, M. Chen, E. M. Wright, and K. Dholakia, \"Dynamics of a levitated microparticle in vacuum trapped by a perfect vortex beam: three-dimensional motion around a complex optical potential,\" J. Opt. Soc. Am. B 34, C14-C19 (2017).","doi":""},{"text":"Roadmap on structured light. H Rubinsztein-Dunlop, A Forbes, M V Berry, M R Dennis, D L Andrews, M Mansuripur, C D , C Alpmann, P Banzer, T Bauer, E Karimi, L Marrucci, M Padgett, M Ritsch-Marte, N M Litchinitser, N P Bigelow, C Rosales-Guzmán, A Belmonte, J P Torres, T W Neely, M Baker, R Gordon, A B Stilgoe, J Romero, A G White, R Fickler, A E Willner, G Xie, B Mcmorran, A M Weiner, J. Opt. 1913001H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. D., C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, \"Roadmap on structured light,\" J. Opt. 19, 013001 (2016).","doi":""},{"text":"Holographic optical trapping of aerosol droplets. D R Burnham, D Mcgloin, Opt. Express. 14D. R. Burnham and D. McGloin, \"Holographic optical trapping of aerosol droplets,\" Opt. Express 14, 4175-4181 (2006).","doi":""},{"text":"Optical levitation and feedback cooling of a nanoparticle at subwavelength distances from a membrane. R Diehl, E Hebestreit, R Reimann, F Tebbenjohanns, M Frimmer, L Novotny, Phys. Rev. A. 9813851R. Diehl, E. Hebestreit, R. Reimann, F. Tebbenjohanns, M. Frimmer, and L. Novotny, \"Optical levitation and feedback cooling of a nanoparticle at subwavelength distances from a membrane,\" Phys. Rev. A 98, 013851 (2018).","doi":""}],"string":"[\n {\n \"text\": \"Select the minimum force from which to start stretching the RBC. 190 pNSelect the minimum force from which to start stretching the RBC (e.g., 0 pN).19\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Select the maximum force at which to stop the RBC stretching. 2020 pNSelect the maximum force at which to stop the RBC stretching (e.g., 20 pN).20\",\n \"doi\": \"\"\n },\n {\n \"text\": \"140 nm s −1 ). Select the pulling velocity. Select the pulling velocity (e.g., 140 nm s −1 ).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Acceleration and trapping of particles by radiation pressure. A Ashkin, Phys. Rev. Lett. 24A. Ashkin, \\\"Acceleration and trapping of particles by radiation pressure,\\\" Phys. Rev. Lett. 24, 156-159 (1970).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of a single-beam gradient force optical trap for dielectric particles. A Ashkin, J M Dziedzic, J E Bjorkholm, S Chu, Opt. Lett. 11A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, \\\"Observation of a single-beam gradient force optical trap for dielectric particles,\\\" Opt. Lett. 11, 288-290 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical trapping and manipulation of neutral particles using lasers: A reprint volume with commentaries. A Ashkin, World ScientificA. Ashkin, Optical trapping and manipulation of neutral particles using lasers: A reprint volume with commentaries (World Scientific, 2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Arthur Ashkin -Nobel Lecture. \\\"Arthur Ashkin -Nobel Lecture,\\\" https://www.nobelprize.org/prizes/physics/2018/ashkin/ lecture/. NobelPrize.org. Nobel Media AB 2019. Thu. 4 Jul 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nobel Lecture: The manipulation of neutral particles. S Chu, Rev. Mod. Phys. 70S. Chu, \\\"Nobel Lecture: The manipulation of neutral particles,\\\" Rev. Mod. Phys. 70, 685-706 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Force generation of organelle transport measured in vivo by an infrared laser trap. A Ashkin, K Schütze, J M Dziedzic, U Euteneuer, M Schliwa, Nature. 348A. Ashkin, K. Schütze, J. M. Dziedzic, U. Euteneuer, and M. Schliwa, \\\"Force generation of organelle transport measured in vivo by an infrared laser trap,\\\" Nature 348, 346-348 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scanning-force microscope based on an optical trap. L P Ghislain, W W Webb, Opt. Lett. 18L. P. Ghislain and W. W. Webb, \\\"Scanning-force microscope based on an optical trap,\\\" Opt. Lett. 18, 1678-1680 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bead movement by single kinesin molecules studied with optical tweezers. S M Block, L S B Goldstein, B J Schnapp, Nature. 348S. M. Block, L. S. B. Goldstein, and B. J. Schnapp, \\\"Bead movement by single kinesin molecules studied with optical tweezers,\\\" Nature 348, 348-352 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Entropic elasticity of lambda-phage DNA. C Bustamante, J F Marko, E D Siggia, S Smith, Science. 265C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith, \\\"Entropic elasticity of lambda-phage DNA,\\\" Science 265, 1599-1599 (1994).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single myosin molecule mechanics: Piconewton forces and nanometre steps. J T Finer, R M Simmons, J A Spudich, Nature. 368J. T. Finer, R. M. Simmons, and J. A. Spudich, \\\"Single myosin molecule mechanics: Piconewton forces and nanometre steps,\\\" Nature 368, 113-119 (1994).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A revolution in optical manipulation. D G Grier, Nature. 424D. G. Grier, \\\"A revolution in optical manipulation,\\\" Nature 424, 810-816 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Colloquium: Gripped by light: Optical binding. K Dholakia, P Zemánek, Rev. Mod. Phys. 82K. Dholakia and P. Zemánek, \\\"Colloquium: Gripped by light: Optical binding,\\\" Rev. Mod. Phys. 82, 1767-1791 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Shaping the future of manipulation. K Dholakia, T Čižmár, Nat. Photon. 5K. Dholakia and T. Čižmár, \\\"Shaping the future of manipulation,\\\" Nat. Photon. 5, 335-342 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasmon nano-optical tweezers. M L Juan, M Righini, R Quidant, Nat. Photon. 5M. L. Juan, M. Righini, and R. Quidant, \\\"Plasmon nano-optical tweezers,\\\" Nat. Photon. 5, 349-356 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tweezers with a twist. M Padgett, R Bowman, Nat. Photon. 5M. Padgett and R. Bowman, \\\"Tweezers with a twist,\\\" Nat. Photon. 5, 343-348 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Holographic optical tweezers and their relevance to lab on chip devices. M Padgett, R Di Leonardo, Lab Chip. 11M. Padgett and R. Di Leonardo, \\\"Holographic optical tweezers and their relevance to lab on chip devices,\\\" Lab Chip 11, 1196-1205 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical trapping and manipulation of nanostructures. O M Maragò, P H Jones, P G Gucciardi, G Volpe, A C Ferrari, Nat. Nanotech. 8O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, \\\"Optical trapping and manipulation of nanostructures,\\\" Nat. Nanotech. 8, 807-819 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Raman spectroscopy of optically trapped particles. D V Petrov, J. Opt. A. Pure. Appl. Op. 9139D. V. Petrov, \\\"Raman spectroscopy of optically trapped particles,\\\" J. Opt. A. Pure. Appl. Op. 9, S139 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Colloidal heat engines: A review. I A Martínez, É Roldán, L Dinis, R Rica, Soft Matter. 13I. A. Martínez, É. Roldán, L. Dinis, and R. Rica, \\\"Colloidal heat engines: A review,\\\" Soft Matter 13, 22-36 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Levitated nanoparticles for microscopic thermodynamics: A review. J Gieseler, J Millen, Entropy. 20326J. Gieseler and J. Millen, \\\"Levitated nanoparticles for microscopic thermodynamics: A review,\\\" Entropy 20, 326 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical tweezers microrheology: From the basics to advanced techniques and applications. R M Robertson-Anderson, ACS Macro Lett. 7R. M. Robertson-Anderson, \\\"Optical tweezers microrheology: From the basics to advanced techniques and applications,\\\" ACS Macro Lett. 7, 968-975 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical tweezers study life under tension. F M Fazal, S M Block, Nat. Photonics. 5F. M. Fazal and S. M. Block, \\\"Optical tweezers study life under tension,\\\" Nat. Photonics 5, 318-321 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P H Jones, O M Maragò, G Volpe, Optical tweezers: Principles and applications. Cambridge University PressP. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: Principles and applications (Cambridge University Press, 2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical tweezers computational toolbox. T A Nieminen, V L Y Loke, A B Stilgoe, G Knöner, A M Brańczyk, N R Heckenberg, H Rubinsztein-Dunlop, J. Opt. A: Pure Appl. Opt. 9T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein- Dunlop, \\\"Optical tweezers computational toolbox,\\\" J. Opt. A: Pure Appl. Opt. 9, S196-S203 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Computational toolbox for optical tweezers in geometrical optics. A Callegari, M Mijalkov, A B Gököz, G Volpe, J. Opt. Soc. Am. B. 32A. Callegari, M. Mijalkov, A. B. Gököz, and G. Volpe, \\\"Computational toolbox for optical tweezers in geometrical optics,\\\" J. Opt. Soc. Am. B 32, B11-B19 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Simulation of a Brownian particle in an optical trap. G Volpe, G Volpe, Am. J. Phys. 81G. Volpe and G. Volpe, \\\"Simulation of a Brownian particle in an optical trap,\\\" Am. J. Phys. 81, 224-230 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Inexpensive optical tweezers for undergraduate laboratories. S P Smith, S R Bhalotra, A L Brody, B L Brown, E K Boyda, M Prentiss, Am. J. Phys. 67S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and M. Prentiss, \\\"Inexpensive optical tweezers for undergraduate laboratories,\\\" Am. J. Phys. 67, 26-35 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Faster, cheaper, safer optical tweezers for the undergraduate laboratory. J Bechhoefer, S Wilson, Am. J. Phys. 70J. Bechhoefer and S. Wilson, \\\"Faster, cheaper, safer optical tweezers for the undergraduate laboratory,\\\" Am. J. Phys. 70, 393-400 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A simple laser cooling and trapping apparatus for undergraduate laboratories. A S Mellish, A C Wilson, Am. J. Phys. 70A. S. Mellish and A. C. Wilson, \\\"A simple laser cooling and trapping apparatus for undergraduate laboratories,\\\" Am. J. Phys. 70, 965-971 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical trapping for undergraduates. D C Appleyard, K Y Vandermeulen, H Lee, M J Lang, Am. J. Phys. 75D. C. Appleyard, K. Y. Vandermeulen, H. Lee, and M. J. Lang, \\\"Optical trapping for undergraduates,\\\" Am. J. Phys. 75, 5-14 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Construction and calibration of an optical trap on a fluorescence optical microscope. W M Lee, P J Reece, R F Marchington, N K Metzger, K Dholakia, Nat. Prot. 23226W. M. Lee, P. J. Reece, R. F. Marchington, N. K. Metzger, and K. Dholakia, \\\"Construction and calibration of an optical trap on a fluorescence optical microscope,\\\" Nat. Prot. 2, 3226 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: upgrading a commercial confocal inverted microscope. M Mathew, S I C O Santos, D Zalvidea, P Loza-Alvarez, Rev. Sci. Instrumen. 8073701M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, \\\"Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: upgrading a commercial confocal inverted microscope,\\\" Rev. Sci. Instrumen. 80, 073701 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Step-by-step guide to the realization of advanced optical tweezers. G Pesce, G Volpe, O M Maragó, P H Jones, S Gigan, A Sasso, G Volpe, J. Opt. Soc. Am. B. 32G. Pesce, G. Volpe, O. M. Maragó, P. H. Jones, S. Gigan, A. Sasso, and G. Volpe, \\\"Step-by-step guide to the realization of advanced optical tweezers,\\\" J. Opt. Soc. Am. B 32, B84-B98 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical tweezers: A comprehensive tutorial from calibration to applications. L García, A Magazù, G Pesce, J Gieseler, J R Gomez-Solano, I Castillo, M Gironella-Torrent, X Viader-Godoy, F Ritort, A V Arzola, K Volke-Sepulveda, G Volpe, OnlineL. Pérez García, A. Magazù, G. Pesce, J. Gieseler, J. R. Gomez-Solano, I. Castillo, M. Gironella-Torrent, X. Viader- Godoy, F. Ritort, A. V. Arzola, K. Volke-Sepulveda, and G. Volpe, \\\"Optical tweezers: A comprehensive tutorial from calibration to applications,\\\" https://github.com/softmatterlab/OpticalTweezersTutorial (2020). Online.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner. N B Simpson, K Dholakia, L Allen, M J Padgett, Opt. Lett. 22N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, \\\"Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,\\\" Opt. Lett. 22, 52-54 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Applications of laser tweezers in complex fluid rheology. E M Furst, Curr. Opn. Colloid Interf. Sci. 10E. M. Furst, \\\"Applications of laser tweezers in complex fluid rheology,\\\" Curr. Opn. Colloid Interf. Sci. 10, 79-86 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The optical stretcher: a novel laser tool to micromanipulate cells. J Guck, R Ananthakrishnan, H Mahmood, T J Moon, C C Cunningham, J Käs, Biophys. J. 81J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C. C. Cunningham, and J. Käs, \\\"The optical stretcher: a novel laser tool to micromanipulate cells,\\\" Biophys. J. 81, 767-784 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microfluidic sorting in an optical lattice. M P Macdonald, G C Spalding, K Dholakia, Nature. 426M. P. MacDonald, G. C. Spalding, and K. Dholakia, \\\"Microfluidic sorting in an optical lattice,\\\" Nature 426, 421-424 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical sorting and detection of submicrometer objects in a motional standing wave. T Čižmár, M Šiler, M Šerỳ, P Zemánek, V Garcés-Chávez, K Dholakia, Phys. Rev. B. 7435105T. Čižmár, M. Šiler, M. Šerỳ, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, \\\"Optical sorting and detection of submicrometer objects in a motional standing wave,\\\" Phys. Rev. B 74, 035105 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Modulated optical sieve for sorting of polydisperse microparticles. I Ricárdez-Vargas, P Rodríguez-Montero, R Ramos-García, K Volke-Sepúlveda, Appl. Phys. Lett. 88121116I. Ricárdez-Vargas, P. Rodríguez-Montero, R. Ramos-García, and K. Volke-Sepúlveda, \\\"Modulated optical sieve for sorting of polydisperse microparticles,\\\" Appl. Phys. Lett. 88, 121116 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Construction of multiple-beam optical traps with nanometer-resolution position sensing. K Visscher, S P Gross, S M Block, IEEE J. Sel. Top. Quant. El. 2K. Visscher, S. P. Gross, and S. M. Block, \\\"Construction of multiple-beam optical traps with nanometer-resolution position sensing,\\\" IEEE J. Sel. Top. Quant. El. 2, 1066-1076 (1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical tweezer arrays and optical substrates created with diffractive optics. E R Dufresne, D G Grier, Rev. Sci. Instrumen. 69E. R. Dufresne and D. G. Grier, \\\"Optical tweezer arrays and optical substrates created with diffractive optics,\\\" Rev. Sci. Instrumen. 69, 1974-1977 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"3D manipulation of particles into crystal structures using holographic optical tweezers. J Leach, G Sinclair, P Jordan, J Courtial, M J Padgett, J Cooper, Z J Laczik, Opt. Express. 12J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. J. Padgett, J. Cooper, and Z. J. Laczik, \\\"3D manipulation of particles into crystal structures using holographic optical tweezers,\\\" Opt. Express 12, 220-226 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Holographic optical trapping. D G Grier, Y Roichman, Appl. Opt. 45D. G. Grier and Y. Roichman, \\\"Holographic optical trapping,\\\" Appl. Opt. 45, 880-887 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity. H He, M E J Friese, N R Heckenberg, H Rubinsztein-Dunlop, Phys. Rev. Lett. 75H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \\\"Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,\\\" Phys. Rev. Lett. 75, 826-829 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Orbital angular momentum of a high-order Bessel light beam. K Volke-Sepulveda, V Garcés-Chávez, S Chávez-Cerda, J Arlt, K Dholakia, J. Opt. B: Quant. Semiclass. Opt. 4K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, \\\"Orbital angular momentum of a high-order Bessel light beam,\\\" J. Opt. B: Quant. Semiclass. Opt. 4, S82-S89 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rotation, oscillation and hydrodynamic synchronization of optically trapped oblate spheroidal microparticles. A V Arzola, P Jákl, L Chvátal, P Zemánek, Opt. Express. 22A. V. Arzola, P. Jákl, L. Chvátal, and P. Zemánek, \\\"Rotation, oscillation and hydrodynamic synchronization of optically trapped oblate spheroidal microparticles,\\\" Opt. Express 22, 16207-16221 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Complex rotational dynamics of multiple spheroidal particles in a circularly polarized, dual beam trap. O Brzobohatỳ, A V Arzola, M Šiler, L Chvátal, P Jákl, S Simpson, P Zemánek, Opt. Express. 23O. Brzobohatỳ, A. V. Arzola, M. Šiler, L. Chvátal, P. Jákl, S. Simpson, and P. Zemánek, \\\"Complex rotational dynamics of multiple spheroidal particles in a circularly polarized, dual beam trap,\\\" Opt. Express 23, 7273-7287 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical deformability as an inherent cell marker for testing malignant transformation and metastatic competence. J Guck, S Schinkinger, B Lincoln, F Wottawah, S Ebert, M Romeyke, D Lenz, H M Erickson, R Ananthakrishnan, D Mitchell, J Käs, S Ulvick, C Bilby, Biophys. J. 88J. Guck, S. Schinkinger, B. Lincoln, F. Wottawah, S. Ebert, M. Romeyke, D. Lenz, H. M. Erickson, R. Ananthakrishnan, D. Mitchell, J. Käs, S. Ulvick, and C. Bilby, \\\"Optical deformability as an inherent cell marker for testing malignant transformation and metastatic competence,\\\" Biophys. J. 88, 3689-3698 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Light at work: The use of optical forces for particle manipulation, sorting, and analysis. A Jonáš, P Zemanek, Electrophoresis. 29A. Jonáš and P. Zemanek, \\\"Light at work: The use of optical forces for particle manipulation, sorting, and analysis,\\\" Electrophoresis 29, 4813-4851 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical vortex trapping of particles. K T Gahagan, G A Swartzlander, Opt. Lett. 21K. T. Gahagan and G. A. Swartzlander, \\\"Optical vortex trapping of particles,\\\" Opt. Lett. 21, 827-829 (1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner. A T O'neil, M J Padgett, Opt. Commun. 185A. T. O'Neil and M. J. Padgett, \\\"Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,\\\" Opt. Commun. 185, 139-143 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transfer of orbital angular momentum to an optically trapped low-index particle. V Garcés-Chávez, K Volke-Sepulveda, S Chávez-Cerda, W Sibbett, K Dholakia, Phys. Rev. A. 6663402V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, \\\"Transfer of orbital angular momentum to an optically trapped low-index particle,\\\" Phys. Rev. A 66, 063402 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Attractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers. R J Hernández, A Mazzulla, A Pane, K Volke-Sepúlveda, G Cipparrone, Lab Chip. 13R. J. Hernández, A. Mazzulla, A. Pane, K. Volke-Sepúlveda, and G. Cipparrone, \\\"Attractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers,\\\" Lab Chip 13, 459-467 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical binding. M M Burns, J.-M Fournier, J A Golovchenko, Phys. Rev. Lett. 63M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, \\\"Optical binding,\\\" Phys. Rev. Lett. 63, 1233-1236 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical matter: crystallization and binding in intense optical fields. M M Burns, J.-M Fournier, J A Golovchenko, Science. 249M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, \\\"Optical matter: crystallization and binding in intense optical fields,\\\" Science 249, 749-754 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical binding between dielectric particles. S K Mohanty, J T Andrews, P K Gupta, Opt. Express. 12S. K. Mohanty, J. T. Andrews, and P. K. Gupta, \\\"Optical binding between dielectric particles,\\\" Opt. Express 12, 2746-2753 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Long-range one-dimensional longitudinal optical binding. V Karásek, T Čižmár, O Brzobohatỳ, P Zemánek, V Garcés-Chávez, K Dholakia, Phys. Rev. Lett. 101143601V. Karásek, T. Čižmár, O. Brzobohatỳ, P. Zemánek, V. Garcés-Chávez, and K. Dholakia, \\\"Long-range one-dimensional longitudinal optical binding,\\\" Phys. Rev. Lett. 101, 143601 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ultrastrong optical binding of metallic nanoparticles. V Demergis, E.-L Florin, Nanolett. 12V. Demergis and E.-L. Florin, \\\"Ultrastrong optical binding of metallic nanoparticles,\\\" Nanolett. 12, 5756-5760 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Light-controlled assembly of active colloidal molecules. F Schmidt, B Liebchen, H Löwen, G Volpe, J. Chem. Phys. 15094905F. Schmidt, B. Liebchen, H. Löwen, and G. Volpe, \\\"Light-controlled assembly of active colloidal molecules,\\\" J. Chem. Phys. 150, 094905 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical thermal ratchet. L P Faucheux, L S Bourdieu, P D Kaplan, A J Libchaber, Phys. Rev. Lett. 74L. P. Faucheux, L. S. Bourdieu, P. D. Kaplan, and A. J. Libchaber, \\\"Optical thermal ratchet,\\\" Phys. Rev. Lett. 74, 1504-1507 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical peristalsis. B A Koss, D G Grier, Appl. Phys. Lett. 82B. A. Koss and D. G. Grier, \\\"Optical peristalsis,\\\" Appl. Phys. Lett. 82, 3985-3987 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of flux reversal in a symmetric optical thermal ratchet. S.-H Lee, K Ladavac, M Polin, D G Grier, Phys. Rev. Lett. 94110601S.-H. Lee, K. Ladavac, M. Polin, and D. G. Grier, \\\"Observation of flux reversal in a symmetric optical thermal ratchet,\\\" Phys. Rev. Lett. 94, 110601 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental control of transport and current reversals in a deterministic optical rocking ratchet. A V Arzola, K Volke-Sepúlveda, J L Mateos, Phys. Rev. Lett. 106168104A. V. Arzola, K. Volke-Sepúlveda, and J. L. Mateos, \\\"Experimental control of transport and current reversals in a deterministic optical rocking ratchet,\\\" Phys. Rev. Lett. 106, 168104 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Active Brownian motion tunable by light. I Buttinoni, G Volpe, F Kümmel, G Volpe, C Bechinger, J. Phys.: Condens. Matter. 24284129I. Buttinoni, G. Volpe, F. Kümmel, G. Volpe, and C. Bechinger, \\\"Active Brownian motion tunable by light,\\\" J. Phys.: Condens. Matter 24, 284129 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian motion in a speckle light field: Tunable anomalous diffusion and selective optical manipulation. G Volpe, G Volpe, S Gigan, Sci. Rep. 43936G. Volpe, G. Volpe, and S. Gigan, \\\"Brownian motion in a speckle light field: Tunable anomalous diffusion and selective optical manipulation,\\\" Sci. Rep. 4, 3936 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Omnidirectional transport in fully reconfigurable two dimensional optical ratchets. A V Arzola, M Villasante-Barahona, K Volke-Sepúlveda, P Jákl, P Zemánek, Phys. Rev. Lett. 118138002A. V. Arzola, M. Villasante-Barahona, K. Volke-Sepúlveda, P. Jákl, and P. Zemánek, \\\"Omnidirectional transport in fully reconfigurable two dimensional optical ratchets,\\\" Phys. Rev. Lett. 118, 138002 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spin to orbital light momentum conversion visualized by particle trajectory. A V Arzola, L Chvátal, P Jákl, P Zemánek, Sci. Rep. 94127A. V. Arzola, L. Chvátal, P. Jákl, and P. Zemánek, \\\"Spin to orbital light momentum conversion visualized by particle trajectory,\\\" Sci. Rep. 9, 4127 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Complex micromachines produced and driven by light. P Galajda, P Ormos, Appl. Phys. Lett. 78P. Galajda and P. Ormos, \\\"Complex micromachines produced and driven by light,\\\" Appl. Phys. Lett. 78, 249-251 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optically driven micromachine elements. M E J Friese, H Rubinsztein-Dunlop, J Gold, P Hagberg, D Hanstorp, Appl. Phys. Lett. 78M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, \\\"Optically driven micromachine elements,\\\" Appl. Phys. Lett. 78, 547-549 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Force-controllable, optically driven micromachines fabricated by single-step two-photon microstereolithography. S Maruo, K Ikuta, H Korogi, J. Microelectromech. Syst. 12S. Maruo, K. Ikuta, and H. Korogi, \\\"Force-controllable, optically driven micromachines fabricated by single-step two-photon microstereolithography,\\\" J. Microelectromech. Syst. 12, 533-539 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical tweezers applied to a microfluidic system. J Enger, M Goksör, K Ramser, P Hagberg, D Hanstorp, Lab Chip. 4J. Enger, M. Goksör, K. Ramser, P. Hagberg, and D. Hanstorp, \\\"Optical tweezers applied to a microfluidic system,\\\" Lab Chip 4, 196-200 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"All-optical control of microfluidic components using form birefringence. S L Neale, M P Macdonald, K Dholakia, T F Krauss, Nat. Mater. 4S. L. Neale, M. P. MacDonald, K. Dholakia, and T. F. Krauss, \\\"All-optical control of microfluidic components using form birefringence,\\\" Nat. Mater. 4, 530-533 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microfluidic sorting of mammalian cells by optical force switching. M M Wang, E Tu, D E Raymond, J M Yang, H Zhang, N Hagen, B Dees, E M Mercer, A H Forster, I Kariv, P J Marchand, W F Butler, Nat. Biotechnol. 23M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, \\\"Microfluidic sorting of mammalian cells by optical force switching,\\\" Nat. Biotechnol. 23, 83-87 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An optically driven pump for microfluidics. J Leach, H Mushfique, R Di Leonardo, M Padgett, J Cooper, Lab Chip. 6J. Leach, H. Mushfique, R. di Leonardo, M. Padgett, and J. Cooper, \\\"An optically driven pump for microfluidics,\\\" Lab Chip 6, 735-739 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A microfluidic system in combination with optical tweezers for analyzing rapid and reversible cytological alterations in single cells upon environmental changes. E Eriksson, J Enger, B Nordlander, N Erjavec, K Ramser, M Goksör, S Hohmann, T Nyström, D Hanstorp, Lab Chip. 7E. Eriksson, J. Enger, B. Nordlander, N. Erjavec, K. Ramser, M. Goksör, S. Hohmann, T. Nyström, and D. Hanstorp, \\\"A microfluidic system in combination with optical tweezers for analyzing rapid and reversible cytological alterations in single cells upon environmental changes,\\\" Lab Chip 7, 71-76 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Enhanced cell sorting and manipulation with combined optical tweezer and microfluidic chip technologies. X Wang, S Chen, M Kong, Z Wang, K D Costa, R A Li, D Sun, Lab Chip. 11X. Wang, S. Chen, M. Kong, Z. Wang, K. D. Costa, R. A. Li, and D. Sun, \\\"Enhanced cell sorting and manipulation with combined optical tweezer and microfluidic chip technologies,\\\" Lab Chip 11, 3656-3662 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical alignment and spinning of laser-trapped microscopic particles. M E J Friese, T A Nieminen, N R Heckenberg, H Rubinsztein-Dunlop, Nature. 394M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \\\"Optical alignment and spinning of laser-trapped microscopic particles,\\\" Nature 394, 348-350 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Intrinsic and extrinsic nature of the orbital angular momentum of a light beam. A T O'neil, I Macvicar, L Allen, M J Padgett, Phys. Rev. Lett. 8853601A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, \\\"Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,\\\" Phys. Rev. Lett. 88, 053601 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Orbital angular momentum transfer in helical Mathieu beams. C López-Mariscal, J C Gutiérrez-Vega, G Milne, K Dholakia, Opt. Express. 14C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, \\\"Orbital angular momentum transfer in helical Mathieu beams,\\\" Opt. Express 14, 4182-4187 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Forces in optical tweezers with radially and azimuthally polarized trapping beams. T A Nieminen, N R Heckenberg, H Rubinsztein-Dunlop, Opt. Lett. 33T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, \\\"Forces in optical tweezers with radially and azimuthally polarized trapping beams,\\\" Opt. Lett. 33, 122-124 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optically mediated particle clearing using Airy wavepackets. J Baumgartl, M Mazilu, K Dholakia, Nat. Photon. 2J. Baumgartl, M. Mazilu, and K. Dholakia, \\\"Optically mediated particle clearing using Airy wavepackets,\\\" Nat. Photon. 2, 675-678 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Design for fully steerable dual-trap optical tweezers. E Fällman, O Axner, Appl. Opt. 36E. Fällman and O. Axner, \\\"Design for fully steerable dual-trap optical tweezers,\\\" Appl. Opt. 36, 2107-2113 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pattern formation and flow control of fine particles by laser-scanning micromanipulation. K Sasaki, M Koshioka, H Misawa, N Kitamura, H Masuhara, Opt. Lett. 16K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, and H. Masuhara, \\\"Pattern formation and flow control of fine particles by laser-scanning micromanipulation,\\\" Opt. Lett. 16, 1463-1465 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Design of a scanning laser optical trap for multiparticle manipulation. C Mio, T Gong, A Terray, D W M Marr, Rev. Sci. Instrumen. 71C. Mio, T. Gong, A. Terray, and D. W. M. Marr, \\\"Design of a scanning laser optical trap for multiparticle manipulation,\\\" Rev. Sci. Instrumen. 71, 2196-2200 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Synchronized laser micromanipulation of multiple targets along each trajectory by single laser. F Arai, K Yoshikawa, T Sakami, T Fukuda, Appl. Phys. Lett. 85F. Arai, K. Yoshikawa, T. Sakami, and T. Fukuda, \\\"Synchronized laser micromanipulation of multiple targets along each trajectory by single laser,\\\" Appl. Phys. Lett. 85, 4301-4303 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Periodic average structures of colloidal quasicrystals. L Zaidouny, T Bohlein, J Roth, C Bechinger, Soft Matter. 10L. Zaidouny, T. Bohlein, J. Roth, and C. Bechinger, \\\"Periodic average structures of colloidal quasicrystals,\\\" Soft Matter 10, 8705-8710 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interferometric optical tweezers. A E Chiou, W Wang, G J Sonek, J Hong, M W Berns, Opt. Commun. 133A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, and M. W. Berns, \\\"Interferometric optical tweezers,\\\" Opt. Commun. 133, 7-10 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical forces acting on rayleigh particle placed into interference field. P Zemánek, V Karásek, A Sasso, Opt. Commun. 240P. Zemánek, V. Karásek, and A. Sasso, \\\"Optical forces acting on rayleigh particle placed into interference field,\\\" Opt. Commun. 240, 401-415 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Creation and manipulation of three-dimensional optically trapped structures. M P Macdonald, L Paterson, K Volke-Sepulveda, J Arlt, W Sibbett, K Dholakia, Science. 296M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, \\\"Creation and manipulation of three-dimensional optically trapped structures,\\\" Science 296, 1101-1103 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Selective trapping of multiple particles by volume speckle field. V G Shvedov, A V Rode, Y V Izdebskaya, A S Desyatnikov, W Krolikowski, Y S Kivshar, Opt. Express. 18V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, \\\"Selective trapping of multiple particles by volume speckle field,\\\" Opt. Express 18, 3137-3142 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Speckle optical tweezers: Micromanipulation with random light fields. G Volpe, L Kurz, A Callegari, G Volpe, S Gigan, Opt. Express. 22G. Volpe, L. Kurz, A. Callegari, G. Volpe, and S. Gigan, \\\"Speckle optical tweezers: Micromanipulation with random light fields,\\\" Opt. Express 22, 18159-18167 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multi-functional optical tweezers using computer-generated holograms. J Liesener, M Reicherter, T Haist, H J Tiziani, Opt. Commun. 185J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, \\\"Multi-functional optical tweezers using computer-generated holograms,\\\" Opt. Commun. 185, 77-82 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamic holographic optical tweezers. J E Curtis, B A Koss, D G Grier, Opt. Commun. 207J. E. Curtis, B. A. Koss, and D. G. Grier, \\\"Dynamic holographic optical tweezers,\\\" Opt. Commun. 207, 169-175 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Full phase and amplitude control of holographic optical tweezers with high efficiency. A Jesacher, C Maurer, A Schwaighofer, S Bernet, M Ritsch-Marte, Opt. Express. 16A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, \\\"Full phase and amplitude control of holographic optical tweezers with high efficiency,\\\" Opt. Express 16, 4479-4486 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical sorting of nonspherical and living microobjects in moving interference structures. P Jakl, A V Arzola, M Siler, L Chatal, K Volke-Sepulveda, P Zemanek, Opt. Express. 22P. Jakl, A. V. Arzola, M. Siler, L. Chatal, K. Volke-Sepulveda, and P. Zemanek, \\\"Optical sorting of nonspherical and living microobjects in moving interference structures,\\\" Opt. Express 22, 29746-29760 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Exact solutions for nondiffracting beams. i. the scalar theory. J Durnin, J. Opt. Soc. Am. A. 4J. Durnin, \\\"Exact solutions for nondiffracting beams. i. the scalar theory,\\\" J. Opt. Soc. Am. A 4, 651-654 (1987).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Alternative formulation for invariant optical fields: Mathieu beams. J C Gutiérrez-Vega, M D Iturbe-Castillo, S Chávez-Cerda, Opt. Lett. 25J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, \\\"Alternative formulation for invariant optical fields: Mathieu beams,\\\" Opt. Lett. 25, 1493-1495 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical micromanipulation using a Bessel light beam. J Arlt, V Garcés-Chávez, W Sibbett, K Dholakia, Opt. Commun. 197J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, \\\"Optical micromanipulation using a Bessel light beam,\\\" Opt. Commun. 197, 239-245 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M A Bandres, J C Gutiérrez-Vega, S Chávez-Cerda, Parabolic nondiffracting optical wave fields. 29M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, \\\"Parabolic nondiffracting optical wave fields,\\\" Opt. Lett. 29, 44-46 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fresnel axicon. I Golub, Opt. Lett. 31I. Golub, \\\"Fresnel axicon,\\\" Opt. Lett. 31, 1890-1892 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Efficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation. V Arrizón, D Sánchez-De La-Llave, U Ruiz, G Méndez, Opt. Lett. 34V. Arrizón, D. Sánchez-de La-Llave, U. Ruiz, and G. Méndez, \\\"Efficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation,\\\" Opt. Lett. 34, 1456-1458 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator. R J Hernández-Hernández, R A Terborg, I Ricardez-Vargas, K Volke-Sepúlveda, Appl. Opt. 49R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas, and K. Volke-Sepúlveda, \\\"Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator,\\\" Appl. Opt. 49, 6903-6909 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Generation of the \\\"perfect\\\" optical vortex using a liquid-crystal spatial light modulator. A S Ostrovsky, C Rickenstorff-Parrao, V Arrizón, Opt. Lett. 38A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, \\\"Generation of the \\\"perfect\\\" optical vortex using a liquid-crystal spatial light modulator,\\\" Opt. Lett. 38, 534-536 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Theory of optical tweezers. P A M Neto, H M Nussenzveig, Europhys. Lett.). 50EPLP. A. M. Neto and H. M. Nussenzveig, \\\"Theory of optical tweezers,\\\" EPL (Europhys. Lett.) 50, 702-710 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory. A Rohrbach, Phys. Rev. Lett. 95168102A. Rohrbach, \\\"Stiffness of optical traps: Quantitative agreement between experiment and electromagnetic theory,\\\" Phys. Rev. Lett. 95, 168102 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Absolute calibration of forces in optical tweezers. R S Dutra, N B Viana, P A M Neto, H M Nussenzveig, Phys. Rev. A. 9013825R. S. Dutra, N. B. Viana, P. A. M. Neto, and H. M. Nussenzveig, \\\"Absolute calibration of forces in optical tweezers,\\\" Phys. Rev. A 90, 013825 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Power spectrum analysis for optical tweezers. K Berg-Sorensen, H Flyvbjerg, Rev. Sci. Instrumen. 75K. Berg-Sorensen and H. Flyvbjerg, \\\"Power spectrum analysis for optical tweezers,\\\" Rev. Sci. Instrumen. 75, 594-612 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High-performance reconstruction of microscopic force fields from Brownian trajectories. L García, J Pérez, G Volpe, A V Arzola, G Volpe, Nat. Commun. 95166L. Pérez García, J. Donlucas Pérez, G. Volpe, A. V. Arzola, and G. Volpe, \\\"High-performance reconstruction of microscopic force fields from Brownian trajectories,\\\" Nat. Commun. 9, 5166 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bayesian inference for estimating parameters of Brownian motion. I Castillo, G Volpe, L Pérez García, A V Arzola, preparationI. Pérez Castillo, G. Volpe, L. Pérez García, and A. V. Arzola, \\\"Bayesian inference for estimating parameters of Brownian motion,\\\" preparation (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"H Risken, The Fokker-Planck equation. SpringerH. Risken, The Fokker-Planck equation (Springer, 1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Properties of the noise-induced \\\"spurious\\\" drift. I. D Ryter, U Deker, J. Math. Phys. 21D. Ryter and U. Deker, \\\"Properties of the noise-induced \\\"spurious\\\" drift. I.\\\" J. Math. Phys. 21, 2662-2665 (1980).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct measurement of the nonconservative force field generated by optical tweezers. P Wu, R Huang, C Tischer, A Jonas, E.-L Florin, Phys. Rev. Lett. 103108101P. Wu, R. Huang, C. Tischer, A. Jonas, and E.-L. Florin, \\\"Direct measurement of the nonconservative force field generated by optical tweezers,\\\" Phys. Rev. Lett. 103, 108101 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Influence of noise on force measurements. G Volpe, L Helden, T Brettschneider, J Wehr, C Bechinger, Phys. Rev. Lett. 104170602G. Volpe, L. Helden, T. Brettschneider, J. Wehr, and C. Bechinger, \\\"Influence of noise on force measurements,\\\" Phys. Rev. Lett. 104, 170602 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus drift method. T Brettschneider, G Volpe, L Helden, J Wehr, C Bechinger, Phys. Rev. E. 8341113T. Brettschneider, G. Volpe, L. Helden, J. Wehr, and C. Bechinger, \\\"Force measurement in the presence of Brownian noise: Equilibrium-distribution method versus drift method,\\\" Phys. Rev. E 83, 041113 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"State-dependent diffusion: Thermodynamic consistency and its path integral formulation. A W C Lau, T C Lubensky, Phys. Rev. E. 7611123A. W. C. Lau and T. C. Lubensky, \\\"State-dependent diffusion: Thermodynamic consistency and its path integral formulation,\\\" Phys. Rev. E 76, 011123 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. S Hottovy, A Mcdaniel, G Volpe, J Wehr, Commun. Math. Phys. 336S. Hottovy, A. McDaniel, G. Volpe, and J. Wehr, \\\"The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction,\\\" Commun. Math. Phys. 336, 1259-1283 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Effective drifts in dynamical systems with multiplicative noise: A review of recent progress. G Volpe, J Wehr, Rep. Prog. Phys. 7953901G. Volpe and J. Wehr, \\\"Effective drifts in dynamical systems with multiplicative noise: A review of recent progress,\\\" Rep. Prog. Phys. 79, 053901 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process. R Singh, D Ghosh, R Adhikari, Phys. Rev. E. 9812136R. Singh, D. Ghosh, and R. Adhikari, \\\"Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process,\\\" Phys. Rev. E 98, 012136 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories. S Türkcan, A Alexandrou, J.-B Masson, Biophys. J. 102S. Türkcan, A. Alexandrou, and J.-B. Masson, \\\"A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories.\\\" Biophys. J. 102, 2288-2298 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Calibrating optical tweezers with Bayesian inference. M U Richly, S Türkcan, A Le Gall, N Fiszman, J Masson, N Westbrook, K Perronet, A Alexandrou, Opt. Express. 21M. U. Richly, S. Türkcan, A. Le Gall, N. Fiszman, J. Masson, N. Westbrook, K. Perronet, and A. Alexandrou, \\\"Calibrating optical tweezers with Bayesian inference,\\\" Opt. Express 21, 31578-31590 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A primer on the Bayesian approach to high-density single-molecule trajectories analysis. M El Beheiry, S Türkcan, M U Richly, A Triller, A Alexandrou, M Dahan, J.-B Masson, Biophys. J. 110M. El Beheiry, S. Türkcan, M. U. Richly, A. Triller, A. Alexandrou, M. Dahan, and J.-B. Masson, \\\"A primer on the Bayesian approach to high-density single-molecule trajectories analysis,\\\" Biophys. J. 110, 1209-1215 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fast Bayesian inference of optical trap stiffness and particle diffusion. S Bera, S Paul, R Singh, D Ghosh, A Kundu, A Banerjee, R Adhikari, Sci. Rep. 741638S. Bera, S. Paul, R. Singh, D. Ghosh, A. Kundu, A. Banerjee, and R. Adhikari, \\\"Fast Bayesian inference of optical trap stiffness and particle diffusion,\\\" Sci. Rep. 7, 41638 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Torque detection using Brownian fluctuations. G Volpe, D Petrov, Phys. Rev. Lett. 97210603G. Volpe and D. Petrov, \\\"Torque detection using Brownian fluctuations,\\\" Phys. Rev. Lett. 97, 210603 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian motion in a nonhomogeneous force field and photonic force microscope. G Volpe, G Volpe, D Petrov, Phys. Rev. E. 7661118G. Volpe, G. Volpe, and D. Petrov, \\\"Brownian motion in a nonhomogeneous force field and photonic force microscope,\\\" Phys. Rev. E 76, 061118 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantitative assessment of non-conservative radiation forces in an optical trap. G Pesce, G Volpe, A C De Luca, G Rusciano, G Volpe, Europhys. Lett.). 8638002EPLG. Pesce, G. Volpe, A. C. De Luca, G. Rusciano, and G. Volpe, \\\"Quantitative assessment of non-conservative radiation forces in an optical trap,\\\" EPL (Europhys. Lett.) 86, 38002 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantitative measurements of force and displacement using an optical trap. R M Simmons, J T Finer, S Chu, J A Spudich, Biophys. J. 70R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, \\\"Quantitative measurements of force and displacement using an optical trap,\\\" Biophys. J. 70, 1813-1822 (1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Calibration of optical tweezers with positional detection in the back focal plane. S F Tolić-Nørrelykke, E Schäffer, J Howard, F S Pavone, F J , H Flyvbjerg, Rev. Sci. Instrumen. 77103101S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. J., and H. Flyvbjerg, \\\"Calibration of optical tweezers with positional detection in the back focal plane,\\\" Rev. Sci. Instrumen. 77, 103101 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical-trap force transducer that operates by direct measurement of light momentum. S B Smith, Y Cui, C Bustamante, Methods in enzymology. 361ElsevierS. B. Smith, Y. Cui, and C. Bustamante, \\\"Optical-trap force transducer that operates by direct measurement of light momentum,\\\" in Methods in enzymology, vol. 361 (Elsevier, 2003), pp. 134-162.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A force detection technique for single-beam optical traps based on direct measurement of light momentum changes. A Farré, M Montes-Usategui, Opt. Express. 18A. Farré and M. Montes-Usategui, \\\"A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,\\\" Opt. Express 18, 11955-11968 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct measurement of axial optical forces. G Thalhammer, L Obmascher, M Ritsch-Marte, Opt. Express. 23G. Thalhammer, L. Obmascher, and M. Ritsch-Marte, \\\"Direct measurement of axial optical forces,\\\" Opt. Express 23, 6112-6129 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Everything you wanted to know about data analysis and fitting but were afraid to ask. P Young, SpringerP. Young, Everything you wanted to know about data analysis and fitting but were afraid to ask (Springer, 2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P J Huber, Robust Statistics. WileyP. J. Huber, Robust Statistics (Wiley, 1981).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fitting a function to time-dependent ensemble averaged data. K Fogelmark, M A Lomholt, A Irbäck, T Ambjörnsson, Sci. Rep. 86984K. Fogelmark, M. A. Lomholt, A. Irbäck, and T. Ambjörnsson, \\\"Fitting a function to time-dependent ensemble averaged data,\\\" Sci. Rep. 8, 6984 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single-molecule experiments in biological physics: Methods and applications. F Ritort, J. Phys.: Condens. Matter. 18F. Ritort, \\\"Single-molecule experiments in biological physics: Methods and applications,\\\" J. Phys.: Condens. Matter 18, R531-R583 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single-molecule biophysics: At the interface of biology, physics and chemistry. A A Deniz, S Mukhopadhyay, E A Lemke, J. Royal Soc. Interfac. 5A. A. Deniz, S. Mukhopadhyay, and E. A. Lemke, \\\"Single-molecule biophysics: At the interface of biology, physics and chemistry,\\\" J. Royal Soc. Interfac. 5, 15-45 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single-molecule micromanipulation techniques. K C Neuman, T Lionnet, J.-F Allemand, Annu. Rev. Mater. Res. 37K. C. Neuman, T. Lionnet, and J.-F. Allemand, \\\"Single-molecule micromanipulation techniques,\\\" Annu. Rev. Mater. Res. 37, 33-67 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Recent advances in optical tweezers. J R Moffitt, Y R Chemla, S B Smith, C Bustamante, Annu. Rev. Biochem. 77J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, \\\"Recent advances in optical tweezers,\\\" Annu. Rev. Biochem. 77, 205-228 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy. K C Neuman, A Nagy, Nat. Methods. 5K. C. Neuman and A. Nagy, \\\"Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy,\\\" Nat. Methods 5, 491-505 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High-resolution optical tweezers for single-molecule manipulation. X Zhang, L Ma, Y Zhang, Yale J. Biol. Med. 86X. Zhang, L. Ma, and Y. Zhang, \\\"High-resolution optical tweezers for single-molecule manipulation,\\\" Yale J. Biol. Med. 86, 367-383 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interrogating biology with force: Single molecule high-resolution measurements with optical tweezers. M Capitanio, F S Pavone, Biophys. J. 105M. Capitanio and F. S. Pavone, \\\"Interrogating biology with force: Single molecule high-resolution measurements with optical tweezers,\\\" Biophys. J. 105, 1293-1303 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single-molecule techniques in biophysics: A review of the progress in methods and applications. H Miller, Z Zhou, J Shepherd, A J M Wollman, M C Leake, Rep. Prog. Phys. 8124601H. Miller, Z. Zhou, J. Shepherd, A. J. M. Wollman, and M. C. Leake, \\\"Single-molecule techniques in biophysics: A review of the progress in methods and applications,\\\" Rep. Prog. Phys. 81, 024601 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Elastic properties of nucleic acids by single-molecule force spectroscopy. J Camunas-Soler, M Ribezzi-Crivellari, F Ritort, Annu. Rev. Biophys. 45J. Camunas-Soler, M. Ribezzi-Crivellari, and F. Ritort, \\\"Elastic properties of nucleic acids by single-molecule force spectroscopy,\\\" Annu. Rev. Biophys. 45, 65-84 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reconstructing folding energy landscapes by single-molecule force spectroscopy. M T Woodside, S M Block, Annu. Rev. Biophys. 43M. T. Woodside and S. M. Block, \\\"Reconstructing folding energy landscapes by single-molecule force spectroscopy,\\\" Annu. Rev. Biophys. 43, 19-39 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The power of force: Insights into the protein folding process using single-molecule force spectroscopy. J Schönfelder, D De Sancho, R Perez-Jimenez, J. Mol. Biol. 428J. Schönfelder, D. De Sancho, and R. Perez-Jimenez, \\\"The power of force: Insights into the protein folding process using single-molecule force spectroscopy,\\\" J. Mol. Biol. 428, 4245-4257 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single molecule high-throughput footprinting of small and large DNA ligands. M Manosas, J Camunas-Soler, V Croquette, F Ritort, Nat. Commun. 8304M. Manosas, J. Camunas-Soler, V. Croquette, and F. Ritort, \\\"Single molecule high-throughput footprinting of small and large DNA ligands,\\\" Nat. Commun. 8, 304 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical traps to study properties of molecular motors. J A Spudich, S E Rice, R S Rock, T J Purcell, H M Warrick, Cold Spring Harb. Protoc. 66662J. A. Spudich, S. E. Rice, R. S. Rock, T. J. Purcell, and H. M. Warrick, \\\"Optical traps to study properties of molecular motors,\\\" Cold Spring Harb. Protoc. 2011, pdb-top066662 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonequilibrium fluctuations in small systems: From physics to biology. F Ritort, Adv. Chem. Phys. 137F. Ritort, \\\"Nonequilibrium fluctuations in small systems: From physics to biology,\\\" Adv. Chem. Phys. 137, 31-123 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experiments in stochastic thermodynamics: Short history and perspectives. S Ciliberto, Phys. Rev. X. 721051S. Ciliberto, \\\"Experiments in stochastic thermodynamics: Short history and perspectives,\\\" Phys. Rev. X 7, 021051 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A guide to magnetic tweezers and their applications. R Sarkar, V V Rybenkov, Front. Phys. 448R. Sarkar and V. V. Rybenkov, \\\"A guide to magnetic tweezers and their applications,\\\" Front. Phys. 4, 48 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single molecule force spectroscopy in biology using the atomic force microscope. J Zlatanova, S M Lindsay, S H Leuba, Prog. Biophys. Mol. Biol. 74J. Zlatanova, S. M. Lindsay, and S. H. Leuba, \\\"Single molecule force spectroscopy in biology using the atomic force microscope,\\\" Prog. Biophys. Mol. Biol. 74, 37-61 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Acoustic force spectroscopy. G Sitters, D Kamsma, G Thalhammer, M Ritsch-Marte, E J G Peterman, G J L Wuite, Nat. Methods. 12G. Sitters, D. Kamsma, G. Thalhammer, M. Ritsch-Marte, E. J. G. Peterman, and G. J. L. Wuite, \\\"Acoustic force spectroscopy,\\\" Nat. Methods 12, 47-50 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single-molecule electrometry. F Ruggeri, F Zosel, N Mutter, M Różycka, M Wojtas, A Ożyhar, B Schuler, M Krishnan, Nat. Nanotech. 12F. Ruggeri, F. Zosel, N. Mutter, M. Różycka, M. Wojtas, A. Ożyhar, B. Schuler, and M. Krishnan, \\\"Single-molecule electrometry,\\\" Nat. Nanotech. 12, 488-495 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microfluidics for biological measurements with single-molecule resolution. A M Streets, Y Huang, Curr. Opn. Biotechnol. 25A. M. Streets and Y. Huang, \\\"Microfluidics for biological measurements with single-molecule resolution,\\\" Curr. Opn. Biotechnol. 25, 69-77 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single-molecule measurements using microneedles. T Yanagida, Y Ishii, A Ishijima, Single Molecule Enzymology. SpringerT. Yanagida, Y. Ishii, and A. Ishijima, \\\"Single-molecule measurements using microneedles,\\\" in Single Molecule Enzymology, (Springer, 2011), pp. 143-159.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sensitive force technique to probe molecular adhesion and structural linkages at biological interfaces. E Evans, K Ritchie, R Merkel, Biophys. J. 68E. Evans, K. Ritchie, and R. Merkel, \\\"Sensitive force technique to probe molecular adhesion and structural linkages at biological interfaces,\\\" Biophys. J. 68, 2580-2587 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Massively parallel single-molecule manipulation using centrifugal force. W P Wong, K Halvorsen, Biophys. J. 100W. P. Wong and K. Halvorsen, \\\"Massively parallel single-molecule manipulation using centrifugal force,\\\" Biophys. J. 100, L53-L55 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Counter-propagating dual-trap optical tweezers based on linear momentum conservation. M Ribezzi-Crivellari, J M Huguet, F Ritort, Rev. Sci. Instrumen. 8443104M. Ribezzi-Crivellari, J. M. Huguet, and F. Ritort, \\\"Counter-propagating dual-trap optical tweezers based on linear momentum conservation,\\\" Rev. Sci. Instrumen. 84, 043104 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statistical and thermodynamic properties of DNA unzipping experiments with optical tweezers. J M Huguet, Universitat de BarcelonaPh.D. thesisJ. M. Huguet, \\\"Statistical and thermodynamic properties of DNA unzipping experiments with optical tweezers,\\\" Ph.D. thesis, Universitat de Barcelona (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Assemblies: Pipette Puller. U Di Parma, U. di Parma, \\\"Assemblies: Pipette Puller,\\\" http://tweezerslab.unipr.it/cgi-bin/mt/ assemblies.pl/Show?_id=7c1b. Accessed: 2019-12-12.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P Nelson, Biological physics. New YorkWH FreemanP. Nelson, Biological physics (WH Freeman, New York, 2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The structural basis of antibody-antigen recognition. I Sela-Culang, V Kunik, Y Ofran, Front. Immunol. 4302I. Sela-Culang, V. Kunik, and Y. Ofran, \\\"The structural basis of antibody-antigen recognition,\\\" Front. Immunol. 4, 302 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"How the biotin-streptavidin interaction was made even stronger: investigation via crystallography and a chimaeric tetramer. C E Chivers, A L Koner, E D Lowe, M Howarth, Biochem. J. 435C. E. Chivers, A. L. Koner, E. D. Lowe, and M. Howarth, \\\"How the biotin-streptavidin interaction was made even stronger: investigation via crystallography and a chimaeric tetramer,\\\" Biochem. J. 435, 55-63 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds. F Pincet, J Husson, Biophys. J. 89F. Pincet and J. Husson, \\\"The solution to the streptavidin-biotin paradox: the influence of history on the strength of single molecular bonds,\\\" Biophys. J. 89, 4374-4381 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Improving signal/noise resolution in single-molecule experiments using molecular constructs with short handles. N Forns, S Lorenzo, M Manosas, K Hayashi, J M Huguet, F Ritort, Biophys. J. 100N. Forns, S. de Lorenzo, M. Manosas, K. Hayashi, J. M. Huguet, and F. Ritort, \\\"Improving signal/noise resolution in single-molecule experiments using molecular constructs with short handles,\\\" Biophys. J. 100, 1765-1774 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stochastic thermodynamics, fluctuation theorems and molecular machines. U Seifert, Rep. Prog. Phys. 75126001U. Seifert, \\\"Stochastic thermodynamics, fluctuation theorems and molecular machines,\\\" Rep. Prog. Phys. 75, 126001 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stretching DNA. J F Marko, E D Siggia, Macromolecules. 28J. F. Marko and E. D. Siggia, \\\"Stretching DNA,\\\" Macromolecules 28, 8759-8770 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Determination of the elastic properties of short ssDNA molecules by mechanically folding and unfolding DNA hairpins. A Alemany, F Ritort, Biopolymers. 101A. Alemany and F. Ritort, \\\"Determination of the elastic properties of short ssDNA molecules by mechanically folding and unfolding DNA hairpins,\\\" Biopolymers 101, 1193-1199 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R R Sinden, DNA Structure and Function. Elsevier ScienceR. R. Sinden, DNA Structure and Function (Elsevier Science, 1994).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics. J Santalucia, Proc. Natl. Acad. Sci. 95J. SantaLucia, \\\"A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics,\\\" Proc. Natl. Acad. Sci. 95, 1460-1465 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Holographic acoustic tweezers. A Marzo, B W Drinkwater, Proc. Natl. Acad. Sci. Natl. Acad. Sci116A. Marzo and B. W. Drinkwater, \\\"Holographic acoustic tweezers,\\\" Proc. Natl. Acad. Sci. 116, 84-89 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions. A Candelli, G J L Wuite, E J G Peterman, Phys. Chem. Chem. Phys. 13A. Candelli, G. J. L. Wuite, and E. J. G. Peterman, \\\"Combining optical trapping, fluorescence microscopy and micro-fluidics for single molecule studies of DNA-protein interactions,\\\" Phys. Chem. Chem. Phys. 13, 7263-7272 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Combining single-molecule manipulation and single-molecule detection. J C Cordova, D K Das, H W Manning, M J Lang, Curr. Opn. Struct. Biol. 28J. C. Cordova, D. K. Das, H. W. Manning, and M. J. Lang, \\\"Combining single-molecule manipulation and single-molecule detection,\\\" Curr. Opn. Struct. Biol. 28, 142-148 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High-resolutio \\\"Fleezers\\\": Dual-trap optical tweezers combined with single-molecule fluorescence detection. K D Whitley, M J Comstock, Y R Chemla, Methods Mol. Biol. pp. K. D. Whitley, M. J. Comstock, and Y. R. Chemla, \\\"High-resolutio \\\"Fleezers\\\": Dual-trap optical tweezers combined with single-molecule fluorescence detection,\\\" Methods Mol. Biol. pp. 183-256 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct force measurements on DNA in a solid-state nanopore. U F Keyser, B N Koeleman, S Van Dorp, D Krapf, R M M Smeets, S G Lemay, N H Dekker, C Dekker, Nat. Phys. 2U. F. Keyser, B. N. Koeleman, S. Van Dorp, D. Krapf, R. M. M. Smeets, S. G. Lemay, N. H. Dekker, and C. Dekker, \\\"Direct force measurements on DNA in a solid-state nanopore,\\\" Nat. Phys. 2, 473-477 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Measurement of the position-dependent electrophoretic force on DNA in a glass nanocapillary. R D Bulushev, L J Steinbock, S Khlybov, J F Steinbock, U F Keyser, A Radenovic, Nanolett. 14R. D. Bulushev, L. J. Steinbock, S. Khlybov, J. F. Steinbock, U. F. Keyser, and A. Radenovic, \\\"Measurement of the position-dependent electrophoretic force on DNA in a glass nanocapillary,\\\" Nanolett. 14, 6606-6613 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical trapping and manipulation of viruses and bacteria. A Ashkin, J M Dziedzic, Science. 235A. Ashkin and J. M. Dziedzic, \\\"Optical trapping and manipulation of viruses and bacteria,\\\" Science 235, 1517-1520 (1987).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical trapping and manipulation of single cells using infrared laser beams. A Ashkin, J M Dziedzic, T Yamane, Nature. 330A. Ashkin, J. M. Dziedzic, and T. Yamane, \\\"Optical trapping and manipulation of single cells using infrared laser beams,\\\" Nature 330, 769-771 (1987).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Compliance of bacterial flagella measured with optical tweezers. S M Block, D F Blair, H C Berg, Nature. 338S. M. Block, D. F. Blair, and H. C. Berg, \\\"Compliance of bacterial flagella measured with optical tweezers,\\\" Nature 338, 514-518 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantifying forces in cell biology. P Roca-Cusachs, V Conte, X Trepat, Nat. Cell Biol. 19P. Roca-Cusachs, V. Conte, and X. Trepat, \\\"Quantifying forces in cell biology,\\\" Nat. Cell Biol. 19, 742-751 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Biomechanical characterization at the cell scale: present and prospects. F Basoli, S M Giannitelli, P Mozetic, M Gori, A Bonfanti, M Trombetta, A Rainer, Front. Physiol. 91449F. Basoli, S. M. Giannitelli, P. Mozetic, M. Gori, A. Bonfanti, M. Trombetta, and A. Rainer, \\\"Biomechanical characterization at the cell scale: present and prospects,\\\" Front. Physiol. 9, 1449 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Force nanoscopy of living cells. D J Müller, Y F Dufrêne, Curr. Biol. 21D. J. Müller and Y. F. Dufrêne, \\\"Force nanoscopy of living cells,\\\" Curr. Biol, 21, R212-R216 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Force nanoscopy of cell mechanics and cell adhesion. Y F Dufrêne, A E Pelling, Nanoscale. 5Y. F. Dufrêne and A. E. Pelling, \\\"Force nanoscopy of cell mechanics and cell adhesion,\\\" Nanoscale 5, 4094-4104 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Exploring mechanochemical processes in the cell with optical tweezers. S Hormeno, J R Arias-Gonzalez, Biol. Cell. 98S. Hormeno and J. R. Arias-Gonzalez, \\\"Exploring mechanochemical processes in the cell with optical tweezers,\\\" Biol. Cell 98, 679-695 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical trapping: A review of essential concepts. I Verdeny, A Farré, J Soler, C Quesada, E M Badosa, M Montes Usategui, Óptica Pura Apl. 44I. Verdeny, A. Farré, J. Mas Soler, C. López Quesada, E. M. Badosa, and M. Montes Usategui, \\\"Optical trapping: A review of essential concepts,\\\" Óptica Pura Apl. 44, 527-551 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Probing force in living cells with optical tweezers: From single-molecule mechanics to cell mechanotransduction. C Arbore, L Perego, M Sergides, M Capitanio, Biophys. Rev pp. C. Arbore, L. Perego, M. Sergides, and M. Capitanio, \\\"Probing force in living cells with optical tweezers: From single-molecule mechanics to cell mechanotransduction,\\\" Biophys. Rev pp. 1-18 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Particle-tracking microrheology of living cells: Principles and applications. D Wirtz, Annu. Rev. Biophys. 38D. Wirtz, \\\"Particle-tracking microrheology of living cells: Principles and applications,\\\" Annu. Rev. Biophys. 38, 301-326 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single particle tracking: From theory to biophysical applications. H Shen, L J Tauzin, R Baiyasi, W Wang, N Moringo, B Shuang, C F Landes, Chem. Rev. 117H. Shen, L. J. Tauzin, R. Baiyasi, W. Wang, N. Moringo, B. Shuang, and C. F. Landes, \\\"Single particle tracking: From theory to biophysical applications,\\\" Chem. Rev. 117, 7331-7376 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Traction force microscopy on soft elastic substrates: A guide to recent computational advances. U S Schwarz, J R D Soiné, Biochimica Biophys. Acta. 1853U. S. Schwarz and J. R. D. Soiné, \\\"Traction force microscopy on soft elastic substrates: A guide to recent computational advances,\\\" Biochimica Biophys. Acta 1853, 3095-3104 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The future of traction force microscopy. H Colin-York, M Fritzsche, Curr. Opn. Biomed. Eng. 5H. Colin-York and M. Fritzsche, \\\"The future of traction force microscopy,\\\" Curr. Opn. Biomed. Eng. 5, 1-5 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Recent technological advancements in traction force microscopy. A Ferrari, Biophys. Rev. 11A. Ferrari, \\\"Recent technological advancements in traction force microscopy,\\\" Biophys. Rev. 11, 679-681 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Measurement of cell microrheology by magnetic twisting cytometry with frequency domain demodulation. M Puig-De-Morales, M Grabulosa, J Alcaraz, J Mullol, G N Maksym, J J Fredberg, D Navajas, J. Appl. Physiol. 91M. Puig-De-Morales, M. Grabulosa, J. Alcaraz, J. Mullol, G. N. Maksym, J. J. Fredberg, and D. Navajas, \\\"Measurement of cell microrheology by magnetic twisting cytometry with frequency domain demodulation,\\\" J. Appl. Physiol. 91, 1152-1159 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Probing cellular mechanics with acoustic force spectroscopy. R Sorkin, G Bergamaschi, D Kamsma, G Brand, E Dekel, Y Ofir-Birin, A Rudik, M Gironella, F Ritort, N Regev-Rudzki, W H Roos, G J L Wuite, Mol. Biol. Cell. 29R. Sorkin, G. Bergamaschi, D. Kamsma, G. Brand, E. Dekel, Y. Ofir-Birin, A. Rudik, M. Gironella, F. Ritort, N. Regev-Rudzki, W. H. Roos, and G. J. L. Wuite, \\\"Probing cellular mechanics with acoustic force spectroscopy,\\\" Mol. Biol. Cell 29, 2005-2011 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The application of micropipette aspiration in molecular mechanics of single cells. L M Lee, A P Liu, J. Nanotech. Eng. Med. 540902L. M. Lee and A. P. Liu, \\\"The application of micropipette aspiration in molecular mechanics of single cells,\\\" J. Nanotech. Eng. Med. 5, 040902 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ATP-dependent mechanics of red blood cells. T Betz, M Lenz, J.-F Joanny, C Sykes, Proc. Natl. Acad. Sci. 106T. Betz, M. Lenz, J.-F. Joanny, and C. Sykes, \\\"ATP-dependent mechanics of red blood cells,\\\" Proc. Natl. Acad. Sci. 106, 15320-15325 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Equilibrium physics breakdown reveals the active nature of red blood cell flickering. H Turlier, D A Fedosov, B Audoly, T Auth, N S Gov, C Sykes, J.-F Joanny, G Gompper, T Betz, Nat. Phys. 12H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, \\\"Equilibrium physics breakdown reveals the active nature of red blood cell flickering,\\\" Nat. Phys. 12, 513-519 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Elastic properties and secondary structure formation of single-stranded DNA at monovalent and divalent salt conditions. A Bosco, J Camunas-Soler, F Ritort, Nucleic Acids Res. 42A. Bosco, J. Camunas-Soler, and F. Ritort, \\\"Elastic properties and secondary structure formation of single-stranded DNA at monovalent and divalent salt conditions,\\\" Nucleic Acids Res. 42, 2064-2074 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mechanism of red blood cell aging: Relationship of cell density and cell age. S Piomelli, C Seaman, Am. J. Hematol. 42S. Piomelli and C. Seaman, \\\"Mechanism of red blood cell aging: Relationship of cell density and cell age,\\\" Am. J. Hematol. 42, 46-52 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Advances in the microrheology of complex fluids. T A Waigh, Rep. Prog. Phys. 7974601T. A. Waigh, \\\"Advances in the microrheology of complex fluids,\\\" Rep. Prog. Phys. 79, 074601 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R B Bird, R C Armstrong, O Hassager, Fluid mechanics. New YorkJohn Wiley & Sons1R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of polymeric liquids. Vol. 1: Fluid mechanics (John Wiley & Sons, New York, 1987).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High frequency viscoelastic behaviour of low molecular weight hyaluronic acid water solutions. S Fusco, A Borzacchiello, L Miccio, G Pesce, G Rusciano, A Sasso, P A Netti, Biorheol. 44S. Fusco, A. Borzacchiello, L. Miccio, G. Pesce, G. Rusciano, A. Sasso, and P. A. Netti, \\\"High frequency viscoelastic behaviour of low molecular weight hyaluronic acid water solutions,\\\" Biorheol. 44, 403-418 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microrheology of complex fluids using optical tweezers: A comparison with macrorheological measurements. G Pesce, A C De Luca, G Rusciano, P A Netti, S Fusco, A Sasso, J. Opt. A Pure Appl. Opt. 1134016G. Pesce, A. C. De Luca, G. Rusciano, P. A. Netti, S. Fusco, and A. Sasso, \\\"Microrheology of complex fluids using optical tweezers: A comparison with macrorheological measurements,\\\" J. Opt. A Pure Appl. Opt. 11, 034016 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluid mechanics of microrheology. T M Squires, T G Mason, Annu. Rev. Fluid Mech. 42T. M. Squires and T. G. Mason, \\\"Fluid mechanics of microrheology,\\\" Annu. Rev. Fluid Mech. 42, 413-438 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microrheology with optical tweezers: data analysis. M Tassieri, R M L Evans, R L Warren, N J Bailey, J M Cooper, New J. Phys. 14115032M. Tassieri, R. M. L. Evans, R. L. Warren, N. J. Bailey, and J. M. Cooper, \\\"Microrheology with optical tweezers: data analysis,\\\" New J. Phys. 14, 115032 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The fluctuation-dissipation theorem. R Kubo, Rep. Prog. Phys. 29R. Kubo, \\\"The fluctuation-dissipation theorem,\\\" Rep. Prog. Phys. 29, 255-284 (1966).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Oscillatory rise of bubbles in wormlike micellar fluids with different microstructures. N Z Handzy, A Belmonte, Phys. Rev. Lett. 92124501N. Z. Handzy and A. Belmonte, \\\"Oscillatory rise of bubbles in wormlike micellar fluids with different microstructures,\\\" Phys. Rev. Lett. 92, 124501 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Manual of symbols and terminology for physicochemical quantities and units, appendix II: Definitions, terminology and symbols in colloid and surface chemistry. D H Everett, Pure Appl. Chem. 31D. H. Everett, \\\"Manual of symbols and terminology for physicochemical quantities and units, appendix II: Definitions, terminology and symbols in colloid and surface chemistry,\\\" Pure Appl. Chem. 31, 577-638 (1972).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statics and dynamics of worm-like surfactant micelles. M E Cates, S J Candau, J. Phys.: Condens. Matter. 2M. E. Cates and S. J. Candau, \\\"Statics and dynamics of worm-like surfactant micelles,\\\" J. Phys.: Condens. Matter 2, 6869-6892 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microrheology with optical tweezers: Measuring the relative viscosity of solutions \\\"at a glance. M Tassieri, F Giudice, E J Robertson, N Jain, B Fries, R Wilson, A Glidle, F Greco, P A Netti, P L Maffettone, T Bicanic, C J M , Sci. Rep. 58831M. Tassieri, F. Del Giudice, E. J. Robertson, N. Jain, B. Fries, R. Wilson, A. Glidle, F. Greco, P. A. Netti, P. L. Maffettone, T. Bicanic, and C. J. M., \\\"Microrheology with optical tweezers: Measuring the relative viscosity of solutions \\\"at a glance\\\",\\\" Sci. Rep. 5, 8831 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R G Larson, The structure and rheology of complex fluids. New YorkOxford University PressR. G. Larson, The structure and rheology of complex fluids (Oxford University Press, New York, 1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations. B Schnurr, F Gittes, F C Mackintosh, C F Schmidt, Macromolecules. 30B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt, \\\"Determining microscopic viscoelasticity in flexible and semiflexible polymer networks from thermal fluctuations,\\\" Macromolecules 30, 7781-7792 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A symmetrical method to obtain shear moduli from microrheology. K Nishi, M L Kilfoil, C F Schmidt, F C Mackintosh, Soft Matter. 14K. Nishi, M. L. Kilfoil, C. F. Schmidt, and F. C. MacKintosh, \\\"A symmetrical method to obtain shear moduli from microrheology,\\\" Soft Matter 14, 3716-3723 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Small-world rheology: An introduction to probe-based active microrheology. L G Wilson, W C K Poon, Phys. Chem. Chem. Phys. 13L. G. Wilson and W. C. K. Poon, \\\"Small-world rheology: An introduction to probe-based active microrheology,\\\" Phys. Chem. Chem. Phys. 13, 10617-10630 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Active and passive microrheology in equilibrium and nonequilibrium systems. D Mizuno, D A Head, F C Mackintosh, C F Schmidt, Macromolecules. 41D. Mizuno, D. A. Head, F. C. MacKintosh, and C. F. Schmidt, \\\"Active and passive microrheology in equilibrium and nonequilibrium systems,\\\" Macromolecules 41, 7194-7202 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Power spectrum of out-of-equilibrium forces in living cells: Amplitude and frequency dependence. F Gallet, D Arcizet, P Bohec, A Richert, Soft Matter. 5F. Gallet, D. Arcizet, P. Bohec, and A. Richert, \\\"Power spectrum of out-of-equilibrium forces in living cells: Amplitude and frequency dependence,\\\" Soft Matter 5, 2947-2953 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nucleation and growth of thermoreversible polymer gels. J R Gomez-Solano, V Blickle, C Bechinger, Phys. Rev. E. 8712308J. R. Gomez-Solano, V. Blickle, and C. Bechinger, \\\"Nucleation and growth of thermoreversible polymer gels,\\\" Phys. Rev. E 87, 012308 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental study of out-of-equilibrium fluctuations in a colloidal suspension of laponite using optical traps. P Jop, J R Gomez-Solano, A Petrosyan, S Ciliberto, J. Stat. Mech.: Theo. Exp. 4012P. Jop, J. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, \\\"Experimental study of out-of-equilibrium fluctuations in a colloidal suspension of laponite using optical traps,\\\" J. Stat. Mech.: Theo. Exp. 2009, P04012 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of strongly heterogeneous dynamics at the depinning transition in a colloidal glass. N Şenbil, M Gruber, C Zhang, M Fuchs, F Scheffold, Phys. Rev. Lett. 122108002N. Şenbil, M. Gruber, C. Zhang, M. Fuchs, and F. Scheffold, \\\"Observation of strongly heterogeneous dynamics at the depinning transition in a colloidal glass,\\\" Phys. Rev. Lett. 122, 108002 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian dynamics of a sphere between parallel walls. E R Dufresne, D Altman, D G Grier, Europhys. Lett.). 53EPLE. R. Dufresne, D. Altman, and D. G. Grier, \\\"Brownian dynamics of a sphere between parallel walls,\\\" EPL (Europhys. Lett.) 53, 264-270 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces. G M Wang, R Prabhakar, E M Sevick, Phys. Rev. Lett. 103248303G. M. Wang, R. Prabhakar, and E. M. Sevick, \\\"Hydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces,\\\" Phys. Rev. Lett. 103, 248303 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Measurement of monolayer viscosity using noncontact microrheology. R Shlomovitz, A A Evans, T Boatwright, M Dennin, A J Levine, Phys. Rev. Lett. 110137802R. Shlomovitz, A. A. Evans, T. Boatwright, M. Dennin, and A. J. Levine, \\\"Measurement of monolayer viscosity using noncontact microrheology,\\\" Phys. Rev. Lett. 110, 137802 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Two-point microrheology of inhomogeneous soft materials. J C Crocker, M T Valentine, E R Weeks, T Gisler, P D Kaplan, A G Yodh, D A Weitz, Phys. Rev. Lett. 85J. C. Crocker, M. T. Valentine, E. R. Weeks, T. Gisler, P. D. Kaplan, A. G. Yodh, and D. A. Weitz, \\\"Two-point microrheology of inhomogeneous soft materials,\\\" Phys. Rev. Lett. 85, 888-891 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Active microrheology to determine viscoelastic parameters of Stokes-Oldroyd B fluids using optical tweezers. S Paul, A Kundu, A Banerjee, J. Phys. Commun. 335002S. Paul, A. Kundu, and A. Banerjee, \\\"Active microrheology to determine viscoelastic parameters of Stokes-Oldroyd B fluids using optical tweezers,\\\" J. Phys. Commun. 3, 035002 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonlinear microrheology reveals entanglement-driven molecularlevel viscoelasticity of concentrated DNA. C D Chapman, R M Robertson-Anderson, Phys. Rev. Lett. 11398303C. D. Chapman and R. M. Robertson-Anderson, \\\"Nonlinear microrheology reveals entanglement-driven molecular- level viscoelasticity of concentrated DNA,\\\" Phys. Rev. Lett. 113, 098303 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transient dynamics of a colloidal particle driven through a viscoelastic fluid. J R Gomez-Solano, C Bechinger, New J. Phys. 17103032J. R. Gomez-Solano and C. Bechinger, \\\"Transient dynamics of a colloidal particle driven through a viscoelastic fluid,\\\" New J. Phys. 17, 103032 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Active and passive microrheology: Theory and simulation. R N Zia, Ann. Rev. Fluid Mech. 50R. N. Zia, \\\"Active and passive microrheology: Theory and simulation,\\\" Ann. Rev. Fluid Mech. 50, 371-405 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An introduction to dynamics of colloids. J K G Dhont, ElsevierJ. K. G. Dhont, An introduction to dynamics of colloids (Elsevier, 1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Colloids as big atoms. W Poon, Science. 304W. Poon, \\\"Colloids as big atoms,\\\" Science 304, 830-831 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Insights into phase transition kinetics from colloid science. V J Anderson, H N W Lekkerkerker, Nature. 416V. J. Anderson and H. N. W. Lekkerkerker, \\\"Insights into phase transition kinetics from colloid science,\\\" Nature 416, 811-815 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Phase behaviour of concentrated suspensions of nearly hard colloidal spheres. P N Pusey, W Van Megen, Nature. 320P. N. Pusey and W. Van Megen, \\\"Phase behaviour of concentrated suspensions of nearly hard colloidal spheres,\\\" Nature 320, 340-342 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Prediction of absolute crystal-nucleation rate in hard-sphere colloids. S Auer, D Frenkel, Nature. 409S. Auer and D. Frenkel, \\\"Prediction of absolute crystal-nucleation rate in hard-sphere colloids,\\\" Nature 409, 1020-1023 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct observation of pre-critical nuclei in a metastable hard-sphere fluid. M S Elliot, S B Haddon, W C K Poon, J. Phys.: Condens. Matter. 13M. S. Elliot, S. B. Haddon, and W. C. K. Poon, \\\"Direct observation of pre-critical nuclei in a metastable hard-sphere fluid,\\\" J. Phys.: Condens. Matter 13, L553-L368 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Real-space imaging of nucleation and growth in colloidal crystallization. U Gasser, E R Weeks, A Schofield, P N Pusey, D A Weitz, Science. 292U. Gasser, E. R. Weeks, A. Schofield, P. N. Pusey, and D. A. Weitz, \\\"Real-space imaging of nucleation and growth in colloidal crystallization,\\\" Science 292, 258-262 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multiple glassy states in a simple model system. K N Pham, A M Puertas, J Bergenholtz, S U Egelhaaf, A Moussaıd, P N Pusey, A B Schofield, M E Cates, M Fuchs, W C K Poon, Science. 296K. N. Pham, A. M. Puertas, J. Bergenholtz, S. U. Egelhaaf, A. Moussaıd, P. N. Pusey, A. B. Schofield, M. E. Cates, M. Fuchs, and W. C. K. Poon, \\\"Multiple glassy states in a simple model system,\\\" Science 296, 104-106 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct visual observation of thermal capillary waves. D G A L Aarts, M Schmidt, H N W Lekkerkerker, Science. 304D. G. A. L. Aarts, M. Schmidt, and H. N. W. Lekkerkerker, \\\"Direct visual observation of thermal capillary waves,\\\" Science 304, 847-850 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ultrastable particle-stabilized foams. U T Gonzenbach, A R Studart, E Tervoort, L J Gauckler, Ang. Chemie Int. Ed. 45U. T. Gonzenbach, A. R. Studart, E. Tervoort, and L. J. Gauckler, \\\"Ultrastable particle-stabilized foams,\\\" Ang. Chemie Int. Ed. 45, 3526-3530 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universal emulsion stabilization from the arrested adsorption of rough particles at liquid-liquid interfaces. M Zanini, C Marschelke, S E Anachkov, E Marini, A Synytska, L Isa, Nat. Commun. 815701M. Zanini, C. Marschelke, S. E. Anachkov, E. Marini, A. Synytska, and L. Isa, \\\"Universal emulsion stabilization from the arrested adsorption of rough particles at liquid-liquid interfaces,\\\" Nat. Commun. 8, 15701 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"E Dickinson, D J Mcclements, Advances in food colloids. Springer Science & Business MediaE. Dickinson and D. J. McClements, Advances in food colloids (Springer Science & Business Media, 1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Chiral colloidal clusters. D Zerrouki, J Baudry, D Pine, P Chaikin, J Bibette, Nature. 455D. Zerrouki, J. Baudry, D. Pine, P. Chaikin, and J. Bibette, \\\"Chiral colloidal clusters,\\\" Nature 455, 380-382 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Van der Waals, Over de Continuiteit van den Gas-en Vloeistoftoestand (Sijthoff, 1873). J , J. D. Van der Waals, Over de Continuiteit van den Gas-en Vloeistoftoestand (Sijthoff, 1873).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Van der Waals forces: a handbook for biologists, chemists, engineers, and physicists. V Parsegian, Cambridge University PressV. Parsegian, Van der Waals forces: a handbook for biologists, chemists, engineers, and physicists (Cambridge University Press, 2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Diffuse double-layer models, long-range forces, and ordering in clay colloids. M B Mcbride, P Baveye, Soil Sci. Soc. Am. J. 66M. B. McBride and P. Baveye, \\\"Diffuse double-layer models, long-range forces, and ordering in clay colloids,\\\" Soil Sci. Soc. Am. J. 66, 1207-1217 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Importance of charge regulation in attractive double-layer forces between dissimilar surfaces. I Popa, P Sinha, M Finessi, P Maroni, G Papastavrou, M Borkovec, Phys. Rev. Lett. 104228301I. Popa, P. Sinha, M. Finessi, P. Maroni, G. Papastavrou, and M. Borkovec, \\\"Importance of charge regulation in attractive double-layer forces between dissimilar surfaces,\\\" Phys. Rev. Lett. 104, 228301 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J N Israelachvili, Intermolecular and surface forces. Academic PressJ. N. Israelachvili, Intermolecular and surface forces (Academic Press, 2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W B Russel, W Russel, D A Saville, W R Schowalter, Colloidal dispersions. Cambridge University PressW. B. Russel, W. Russel, D. A. Saville, and W. R. Schowalter, Colloidal dispersions (Cambridge University Press, 1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct measurement of colloidal forces using an atomic force microscope. W A Ducker, T J Senden, R M Pashley, Nature. 353W. A. Ducker, T. J. Senden, and R. M. Pashley, \\\"Direct measurement of colloidal forces using an atomic force microscope,\\\" Nature 353, 239-241 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Flocculation of protein-stabilized oil-in-water emulsions. E Dickinson, Colloids Surfaces B: Biointerfaces. 81E. Dickinson, \\\"Flocculation of protein-stabilized oil-in-water emulsions,\\\" Colloids Surfaces B: Biointerfaces 81, 130-140 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Modern aspects of emulsion science. B P Binks, Royal Society of ChemistryB. P. Binks, Modern aspects of emulsion science (Royal Society of Chemistry, 2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Creaming and flocculation in emulsions containing polysaccharide. Y Cao, E Dickinson, D J Wedlock, Food Hydrocoll. 4Y. Cao, E. Dickinson, and D. J. Wedlock, \\\"Creaming and flocculation in emulsions containing polysaccharide,\\\" Food Hydrocoll. 4, 185-195 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Coagulation/flocculation process for dye removal using sludge from water treatment plant: optimization through response surface methodology. S S Moghaddam, M R A Moghaddam, M Arami, J. Hazard. Mater. 175S. S. Moghaddam, M. R. A. Moghaddam, and M. Arami, \\\"Coagulation/flocculation process for dye removal using sludge from water treatment plant: optimization through response surface methodology,\\\" J. Hazard. Mater. 175, 651-657 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Coagulation and flocculation in water and wastewater treatment. J Bratby, IWA publishingJ. Bratby, Coagulation and flocculation in water and wastewater treatment (IWA publishing, 2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On interaction between two bodies immersed in a solution of macromolecules. S Asakura, F Oosawa, J. Chem. Phys. 22S. Asakura and F. Oosawa, \\\"On interaction between two bodies immersed in a solution of macromolecules,\\\" J. Chem. Phys. 22, 1255-1256 (1954).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J.-L Barrat, J.-P Hansen, Basic concepts for simple and complex liquids. Cambridge University PressJ.-L. Barrat and J.-P. Hansen, Basic concepts for simple and complex liquids (Cambridge University Press, 2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The depletion attraction: An underappreciated force driving cellular organization. D Marenduzzo, K Finan, P R Cook, J. Cell. Biol. 175D. Marenduzzo, K. Finan, and P. R. Cook, \\\"The depletion attraction: An underappreciated force driving cellular organization,\\\" J. Cell. Biol. 175, 681-686 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polymer solutions near an interface. adsorption and depletion layers. P G De Gennes, Macromolecules. 14P. G. de Gennes, \\\"Polymer solutions near an interface. adsorption and depletion layers,\\\" Macromolecules 14, 1637-1644 (1981).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Surface tension of high-polymer solutions. F Oosawa, S Asakura, J. Chem. Phys. 22F. Oosawa and S. Asakura, \\\"Surface tension of high-polymer solutions,\\\" J. Chem. Phys. 22, 1255-1255 (1954).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Phenomena at the walls in a critical binary mixture. M E Fisher, P G De Gennes, C. R. Acad. Sci. Paris B. 287M. E. Fisher and P. G. de Gennes, \\\"Phenomena at the walls in a critical binary mixture.\\\" C. R. Acad. Sci. Paris B 287, 207-209 (1978).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Casimir effect: From quantum to critical fluctuations. A Gambassi, J. Phys.: Conf. Ser. 16112037A. Gambassi, \\\"The Casimir effect: From quantum to critical fluctuations,\\\" J. Phys.: Conf. Ser. 161, 012037 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Critical Casimir forces steered by patterned substrates. A Gambassi, S Dietrich, Soft Matter. 7A. Gambassi and S. Dietrich, \\\"Critical Casimir forces steered by patterned substrates,\\\" Soft Matter 7, 1247-1253 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the attraction between two perfectly conducting plates. H B G Casimir, Proc. K. Ned. Akad. Wet. 51H. B. G. Casimir, \\\"On the attraction between two perfectly conducting plates,\\\" Proc. K. Ned. Akad. Wet 51, 793-795 (1948).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct measurement of critical Casimir forces. C Hertlein, L Helden, A Gambassi, S Dietrich, C Bechinger, Nature. 451C. Hertlein, L. Helden, A. Gambassi, S. Dietrich, and C. Bechinger, \\\"Direct measurement of critical Casimir forces,\\\" Nature 451, 172-175 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonadditivity of critical Casimir forces. S Paladugu, A Callegari, Y Tuna, L Barth, S Dietrich, A Gambassi, G Volpe, Nat. Commun. 711403S. Paladugu, A. Callegari, Y. Tuna, L. Barth, S. Dietrich, A. Gambassi, and G. Volpe, \\\"Nonadditivity of critical Casimir forces,\\\" Nat. Commun. 7, 11403 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct observation of colloidal aggregation by critical Casimir forces. D Bonn, J Otwinowski, S Sacanna, H Guo, G Wegdam, P Schall, Phys. Rev. Lett. 103156101D. Bonn, J. Otwinowski, S. Sacanna, H. Guo, G. Wegdam, and P. Schall, \\\"Direct observation of colloidal aggregation by critical Casimir forces,\\\" Phys. Rev. Lett. 103, 156101 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"When depletion goes critical. R Piazza, S Buzzaccaro, A Parola, J Colombo, J. Phys.: Condens. Matter. 23194114R. Piazza, S. Buzzaccaro, A. Parola, and J. Colombo, \\\"When depletion goes critical,\\\" J. Phys.: Condens. Matter 23, 194114 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Controlling the dynamics of colloidal particles by critical Casimir forces. A Magazzù, A Callegari, J P Staforelli, A Gambassi, S Dietrich, G Volpe, Soft matter. 15A. Magazzù, A. Callegari, J. P. Staforelli, A. Gambassi, S. Dietrich, and G. Volpe, \\\"Controlling the dynamics of colloidal particles by critical Casimir forces,\\\" Soft matter 15, 2152-2162 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Critical Casimir forces in colloidal suspensions on chemically patterned surfaces. F Soyka, O Zvyagolskaya, C Hertlein, L Helden, C Bechinger, Phys. Rev. Lett. 101208301F. Soyka, O. Zvyagolskaya, C. Hertlein, L. Helden, and C. Bechinger, \\\"Critical Casimir forces in colloidal suspensions on chemically patterned surfaces,\\\" Phys. Rev. Lett. 101, 208301 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tunability of critical Casimir interactions by boundary conditions. U Nellen, L Helden, C Bechinger, Europhys. Lett.). 8826001EPLU. Nellen, L. Helden, and C. Bechinger, \\\"Tunability of critical Casimir interactions by boundary conditions,\\\" EPL (Europhys. Lett.) 88, 26001 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical tweezers in colloid and interface science. D G Grier, Curr. Opn. Colloid Interface Sci. 2D. G. Grier, \\\"Optical tweezers in colloid and interface science,\\\" Curr. Opn. Colloid Interface Sci. 2, 264-270 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Blinking optical tweezers for microrheology measurements of weak elasticity complex fluids. G Pesce, G Rusciano, A Sasso, Opt. Express. 18G. Pesce, G. Rusciano, and A. Sasso, \\\"Blinking optical tweezers for microrheology measurements of weak elasticity complex fluids,\\\" Opt. Express 18, 2116-2126 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Long-term influence of fluid inertia on the diffusion of a Brownian particle. G Pesce, G Volpe, G Volpe, A Sasso, Phys. Rev. E. 9042309G. Pesce, G. Volpe, G. Volpe, and A. Sasso, \\\"Long-term influence of fluid inertia on the diffusion of a Brownian particle,\\\" Phys. Rev. E 90, 042309 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lower critical solution coexistence curve and physical properties (density, viscosity, surface tension, and interfacial tension) of 2,6-lutidine+water. C A Grattoni, R A Dawe, C Y Seah, J D Gray, J. Chem. Eng. Data. 38C. A. Grattoni, R. A. Dawe, C. Y. Seah, and J. D. Gray, \\\"Lower critical solution coexistence curve and physical properties (density, viscosity, surface tension, and interfacial tension) of 2,6-lutidine+water,\\\" J. Chem. Eng. Data 38, 516-519 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Critical Casimir effect in classical binary liquid mixtures. A Gambassi, A Maciołek, C Hertlein, U Nellen, L Helden, C Bechinger, S Dietrich, Phys. Rev. E. 8061143A. Gambassi, A. Maciołek, C. Hertlein, U. Nellen, L. Helden, C. Bechinger, and S. Dietrich, \\\"Critical Casimir effect in classical binary liquid mixtures,\\\" Phys. Rev. E 80, 061143 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Methods of digital video microscopy for colloidal studies. J C Crocker, D G Grier, J. Colloid Interface Sci. 179J. C. Crocker and D. G. Grier, \\\"Methods of digital video microscopy for colloidal studies,\\\" J. Colloid Interface Sci. 179, 298-310 (1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Monte Carlo simulation results for critical Casimir forces. O Vasilyev, A Gambassi, A Maciołek, S Dietrich, Europhys. Lett.). 8060009EPLO. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, \\\"Monte Carlo simulation results for critical Casimir forces,\\\" EPL (Europhys. Lett.) 80, 60009 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations. O Vasilyev, A Gambassi, A Maciołek, S Dietrich, Phys. Rev. E. 7941142O. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, \\\"Universal scaling functions of critical Casimir forces obtained by Monte Carlo simulations,\\\" Phys. Rev. E 79, 041142 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Thermodynamic Casimir effect: Universality and corrections to scaling. M Hasenbusch, Phys. Rev. B. 85174421M. Hasenbusch, \\\"Thermodynamic Casimir effect: Universality and corrections to scaling,\\\" Phys. Rev. B 85, 174421 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian diffusion of particles with hydrodynamic interaction. G K Batchelor, J. Fluid Mech. 74G. K. Batchelor, \\\"Brownian diffusion of particles with hydrodynamic interaction,\\\" J. Fluid Mech. 74, 1-29 (1976).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interdiffusion coefficient and dynamic viscosity for the mixture 2,6-lutidine+water near the lower consolute point. J C Clunie, J K Baird, Phys. Chem. Liq. 37J. C. Clunie and J. K. Baird, \\\"Interdiffusion coefficient and dynamic viscosity for the mixture 2,6-lutidine+water near the lower consolute point,\\\" Phys. Chem. Liq. 37, 357-371 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Assembling quantum dots via critical Casimir forces. E Marino, T E Kodger, J B Hove, A H Velders, P Schall, Sol. Energy Mater. Sol. Cells. 158E. Marino, T. E. Kodger, J. B. ten Hove, A. H. Velders, and P. Schall, \\\"Assembling quantum dots via critical Casimir forces,\\\" Sol. Energy Mater. Sol. Cells 158, 154-159 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tuning patchy bonds induced by critical Casimir forces. T A Nguyen, A Newton, D J Kraft, P G Bolhuis, P Schall, Materials. 101265T. A. Nguyen, A. Newton, D. J. Kraft, P. G. Bolhuis, and P. Schall, \\\"Tuning patchy bonds induced by critical Casimir forces,\\\" Materials 10, 1265 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microscopic engine powered by critical demixing. F Schmidt, A Magazzù, A Callegari, L Biancofiore, F Cichos, G Volpe, Phys. Rev. Lett. 12068004F. Schmidt, A. Magazzù, A. Callegari, L. Biancofiore, F. Cichos, and G. Volpe, \\\"Microscopic engine powered by critical demixing,\\\" Phys. Rev. Lett. 120, 068004 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct measurement of the van der Waals interaction between two Rydberg atoms. L Beguin, A Vernier, R Chicireanu, T Lahaye, A Browaeys, Phys. Rev. Lett. 110263201L. Beguin, A. Vernier, R. Chicireanu, T. Lahaye, and A. Browaeys, \\\"Direct measurement of the van der Waals interaction between two Rydberg atoms,\\\" Phys. Rev. Lett. 110, 263201 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Materials perspective on Casimir and van der Waals interactions. L M Woods, D A R Dalvit, A Tkatchenko, P Rodriguez-Lopez, A W Rodriguez, R Podgornik, Rev. Mod. Phys. 8845003L. M. Woods, D. A. R. Dalvit, A. Tkatchenko, P. Rodriguez-Lopez, A. W. Rodriguez, and R. Podgornik, \\\"Materials perspective on Casimir and van der Waals interactions,\\\" Rev. Mod. Phys. 88, 045003 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Programmable self-assembly. L Cademartiri, K J M Bishop, Nat. Mater. 14L. Cademartiri and K. J. M. Bishop, \\\"Programmable self-assembly,\\\" Nat. Mater. 14, 2-9 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Escape and synchronization of a Brownian particle. A Simon, A Libchaber, Phys. Rev. Lett. 68A. Simon and A. Libchaber, \\\"Escape and synchronization of a Brownian particle,\\\" Phys. Rev. Lett. 68, 3375-3379 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Thermally activated transitions in a bistable three-dimensional optical trap. L I Mccann, M Dykman, B Golding, Nature. 402L. I. McCann, M. Dykman, and B. Golding, \\\"Thermally activated transitions in a bistable three-dimensional optical trap,\\\" Nature 402, 785-787 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian motion in a field of force and the diffusion model of chemical reactions. H A Kramers, Physica. 7H. A. Kramers, \\\"Brownian motion in a field of force and the diffusion model of chemical reactions,\\\" Physica 7, 284-304 (1940).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Frequency factors in the thermally activated process. R Landauer, J A Swanson, Phys. Rev. 121R. Landauer and J. A. Swanson, \\\"Frequency factors in the thermally activated process,\\\" Phys. Rev. 121, 1668-1674 (1961).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Double nanohole optical trapping: dynamics and protein-antibody co-trapping. A Zehtabi-Oskuie, H Jiang, B R Cyr, D W Rennehan, A A Al-Balushi, R Gordon, Lab Chip. 13A. Zehtabi-Oskuie, H. Jiang, B. R. Cyr, D. W. Rennehan, A. A. Al-Balushi, and R. Gordon, \\\"Double nanohole optical trapping: dynamics and protein-antibody co-trapping,\\\" Lab Chip 13, 2563-2568 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Partial synchronization of stochastic oscillators through hydrodynamic coupling. A Curran, M P Lee, M J Padgett, J M Cooper, R. Di Leonardo, Phys. Rev. Lett. 108240601A. Curran, M. P. Lee, M. J. Padgett, J. M. Cooper, and R. Di Leonardo, \\\"Partial synchronization of stochastic oscillators through hydrodynamic coupling,\\\" Phys. Rev. Lett. 108, 240601 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Phase measurements of barrier crossings in a periodically modulated double-well potential. Y Seol, D L Stein, K Visscher, Phys. Rev. Lett. 10350601Y. Seol, D. L. Stein, and K. Visscher, \\\"Phase measurements of barrier crossings in a periodically modulated double-well potential,\\\" Phys. Rev. Lett. 103, 050601 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Irreversibility and generalized noise. H B Callen, T A Welton, Phys. Rev. 83H. B. Callen and T. A. Welton, \\\"Irreversibility and generalized noise,\\\" Phys. Rev. 83, 34-40 (1951).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluctuation-dissipation: response theory in statistical physics. U M B Marconi, A Puglisi, L Rondoni, A Vulpiani, Phys. Rep. 461U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, \\\"Fluctuation-dissipation: response theory in statistical physics,\\\" Phys. Rep. 461, 111-195 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluctuation relations in simple examples of non-equilibrium steady states. R Chetrite, G Falkovich, K Gawedzki, J. Stat. Mech.: Theo. Exp. 8005R. Chetrite, G. Falkovich, and K. Gawedzki, \\\"Fluctuation relations in simple examples of non-equilibrium steady states,\\\" J. Stat. Mech.: Theo. Exp. 2008, P08005 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Einstein relation generalized to nonequilibrium. V Blickle, T Speck, C Lutz, U Seifert, C Bechinger, Phys. Rev. Lett. 98210601V. Blickle, T. Speck, C. Lutz, U. Seifert, and C. Bechinger, \\\"Einstein relation generalized to nonequilibrium,\\\" Phys. Rev. Lett. 98, 210601 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental test of a new equality: Measuring heat dissipation in an optically driven colloidal system. S Toyabe, H.-R Jiang, T Nakamura, Y Murayama, M Sano, Phys. Rev. E. 7511122S. Toyabe, H.-R. Jiang, T. Nakamura, Y. Murayama, and M. Sano, \\\"Experimental test of a new equality: Measuring heat dissipation in an optically driven colloidal system,\\\" Phys. Rev. E 75, 011122 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Energy dissipation of a Brownian particle in a viscoelastic fluid. S Toyabe, M Sano, Phys. Rev. E. 7741403S. Toyabe and M. Sano, \\\"Energy dissipation of a Brownian particle in a viscoelastic fluid,\\\" Phys. Rev. E 77, 041403 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluctuations, linear response and heat flux of an aging system. J R Gomez-Solano, A Petrosyan, S Ciliberto, Europhys. Lett.). 9810007EPLJ. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, \\\"Fluctuations, linear response and heat flux of an aging system,\\\" EPL (Europhys. Lett.) 98, 10007 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Probing active forces via a fluctuationdissipation relation: Application to living cells. P Bohec, F Gallet, C Maes, S Safaverdi, P Visco, F Van Wijland, Europhys. Lett.). 10250005EPLP. Bohec, F. Gallet, C. Maes, S. Safaverdi, P. Visco, and F. Van Wijland, \\\"Probing active forces via a fluctuation- dissipation relation: Application to living cells,\\\" EPL (Europhys. Lett.) 102, 50005 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Periodic forcing of a Brownian particle. L P Faucheux, G Stolovitzky, A Libchaber, Phys. Rev. E. 51L. P. Faucheux, G. Stolovitzky, and A. Libchaber, \\\"Periodic forcing of a Brownian particle,\\\" Phys. Rev. E 51, 5239-5250 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state. J R Gomez-Solano, A Petrosyan, S Ciliberto, R Chetrite, K Gawędzki, Phys. Rev. Lett. 10340601J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, R. Chetrite, and K. Gawędzki, \\\"Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state,\\\" Phys. Rev. Lett. 103, 040601 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluctuations and response in a non-equilibrium micron-sized system. J R Gomez-Solano, A Petrosyan, S Ciliberto, C Maes, J. Stat. Mech.: Theo. Exp. 1008J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, and C. Maes, \\\"Fluctuations and response in a non-equilibrium micron-sized system,\\\" J. Stat. Mech.: Theo. Exp. 2011, P01008 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Characterizing potentials by a generalized Boltzmann factor. V Blickle, T Speck, U Seifert, C Bechinger, Phys. Rev. E. 7560101V. Blickle, T. Speck, U. Seifert, and C. Bechinger, \\\"Characterizing potentials by a generalized Boltzmann factor,\\\" Phys. Rev. E 75, 060101 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluctuations, linear response, and currents in out-of-equilibrium systems. S Ciliberto, R Gomez-Solano, A Petrosyan, Annu. Rev. Condens. Matter Phys. 4S. Ciliberto, R. Gomez-Solano, and A. Petrosyan, \\\"Fluctuations, linear response, and currents in out-of-equilibrium systems,\\\" Annu. Rev. Condens. Matter Phys. 4, 235-261 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Langevin equation and thermodynamics. K Sekimoto, Prog. Theo. Phys. Suppl. 130K. Sekimoto, \\\"Langevin equation and thermodynamics,\\\" Prog. Theo. Phys. Suppl. 130, 17-27 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"K Sekimoto, Stochastic energetics. SpringerK. Sekimoto, Stochastic energetics (Springer, 2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. G M Wang, E M Sevick, E Mittag, D J Searles, D J Evans, Phys. Rev. Lett. 8950601G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, \\\"Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,\\\" Phys. Rev. Lett. 89, 050601 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Entropy production along a stochastic trajectory and an integral fluctuation theorem. U Seifert, Phys. Rev. Lett. 9540602U. Seifert, \\\"Entropy production along a stochastic trajectory and an integral fluctuation theorem,\\\" Phys. Rev. Lett. 95, 040602 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonequilibrium equality for free energy differences. C Jarzynski, Phys. Rev. Lett. 78C. Jarzynski, \\\"Nonequilibrium equality for free energy differences,\\\" Phys. Rev. Lett. 78, 2690-2693 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. G E Crooks, Phys. Rev. E. 60G. E. Crooks, \\\"Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences,\\\" Phys. Rev. E 60, 2721-2726 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. S Carnot, Annales scientifiques de l'École Normale Supérieure. 1S. Carnot, \\\"Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance,\\\" in Annales scientifiques de l'École Normale Supérieure, vol. 1 (1824), pp. 393-457.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An inconvenient truth about thermoelectrics. C B Vining, Nat. Mater. 8C. B. Vining, \\\"An inconvenient truth about thermoelectrics,\\\" Nat. Mater. 8, 83-85 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The efficiency of atomic power stations (a review). I I Novikov, J. Nucl. Energy. 7I. I. Novikov, \\\"The efficiency of atomic power stations (a review),\\\" J. Nucl. Energy 7, 125-128 (1958).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Efficiency of a Carnot engine at maximum power output. F L Curzon, B Ahlborn, Am. J. Phys. 43F. L. Curzon and B. Ahlborn, \\\"Efficiency of a Carnot engine at maximum power output,\\\" Am. J. Phys. 43, 22-24 (1975).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Heat engines and the Curzon-Ahlborn efficiency. A Hernández, J M M Roco, A Medina, N Sánchez-Salas, Revista Mexicana de Física. 60A. Calvo Hernández, J. M. M. Roco, A. Medina, and N. Sánchez-Salas, \\\"Heat engines and the Curzon-Ahlborn efficiency,\\\" Revista Mexicana de Física 60, 384-389 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Realization of a micrometre-sized stochastic heat engine. V Blickle, C Bechinger, Nat. Phys. 8V. Blickle and C. Bechinger, \\\"Realization of a micrometre-sized stochastic heat engine,\\\" Nat. Phys. 8, 143-146 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A microscopic steam engine implemented in an optical tweezer. P A Quinto-Su, Nat. Commun. 55889P. A. Quinto-Su, \\\"A microscopic steam engine implemented in an optical tweezer,\\\" Nat. Commun. 5, 5889 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Effective heating to several thousand kelvins of an optically trapped sphere in a liquid. I A Martínez, É Roldán, J M R Parrondo, D Petrov, Phys. Rev. E. 8732159I. A. Martínez, É. Roldán, J. M. R. Parrondo, and D. Petrov, \\\"Effective heating to several thousand kelvins of an optically trapped sphere in a liquid,\\\" Phys. Rev. E 87, 032159 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Steady-state fluctuation relations for systems driven by an external random force. J R Gomez-Solano, L Bellon, A Petrosyan, S Ciliberto, Europhys. Lett.). 8960003EPLJ. R. Gomez-Solano, L. Bellon, A. Petrosyan, and S. Ciliberto, \\\"Steady-state fluctuation relations for systems driven by an external random force,\\\" EPL (Europhys. Lett.) 89, 60003 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Energy flow between two hydrodynamically coupled particles kept at different effective temperatures. A Bérut, A Petrosyan, S Ciliberto, Europhys. Lett.). 10760004EPLA. Bérut, A. Petrosyan, and S. Ciliberto, \\\"Energy flow between two hydrodynamically coupled particles kept at different effective temperatures,\\\" EPL (Europhys. Lett.) 107, 60004 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Adiabatic processes realized with a trapped Brownian particle. I A Martínez, É Roldán, L Dinis, D Petrov, R A Rica, Phys. Rev. Lett. 114120601I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, and R. A. Rica, \\\"Adiabatic processes realized with a trapped Brownian particle,\\\" Phys. Rev. Lett. 114, 120601 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian Carnot engine. I A Martínez, É Roldán, L Dinis, D Petrov, J M R Parrondo, R A Rica, Nat. Phys. 12I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, \\\"Brownian Carnot engine,\\\" Nat. Phys. 12, 67-70 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Efficiency at maximum power: An analytically solvable model for stochastic heat engines. T Schmiedl, U Seifert, Europhys. Lett.). 8120003EPLT. Schmiedl and U. Seifert, \\\"Efficiency at maximum power: An analytically solvable model for stochastic heat engines,\\\" EPL (Europhys. Lett.) 81, 20003 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Theory of Heat, Text-books of science (Longmans, 1871). J C Maxwell, J. C. Maxwell, Theory of Heat, Text-books of science (Longmans, 1871).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. L Szilard, Zeit. Physik. 53L. Szilard, \\\"Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen,\\\" Zeit. Physik 53, 840-856 (1929).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Maxwell's demon cannot operate: Information and entropy. i. L Brillouin, J. Appl. Phys. 22L. Brillouin, \\\"Maxwell's demon cannot operate: Information and entropy. i,\\\" J. Appl. Phys. 22, 334-337 (1951).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Irreversibility and heat generation in the computing process. R Landauer, IBM J. Res. Dev. 5R. Landauer, \\\"Irreversibility and heat generation in the computing process,\\\" IBM J. Res. Dev. 5, 183-191 (1961).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The thermodynamics of computation: A review. C H Bennett, Int. J. Th. Phys. 21C. H. Bennett, \\\"The thermodynamics of computation: A review,\\\" Int. J. Th. Phys. 21, 905-940 (1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Second law of thermodynamics with discrete quantum feedback control. T Sagawa, M Ueda, Phys. Rev. Lett. 10080403T. Sagawa and M. Ueda, \\\"Second law of thermodynamics with discrete quantum feedback control,\\\" Phys. Rev. Lett. 100, 080403 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Thermodynamics of information processing in small systems. T Sagawa, Prog. Theo. Phys. 127T. Sagawa, \\\"Thermodynamics of information processing in small systems,\\\" Prog. Theo. Phys. 127, 1-56 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. S Toyabe, T Sagawa, M Ueda, E Muneyuki, M Sano, Nat. Phys. 6S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, \\\"Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality,\\\" Nat. Phys. 6, 988-992 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Minimal energy cost for thermodynamic information processing: measurement and information erasure. T Sagawa, M Ueda, Phys. Rev. Lett. 102250602T. Sagawa and M. Ueda, \\\"Minimal energy cost for thermodynamic information processing: measurement and information erasure,\\\" Phys. Rev. Lett. 102, 250602 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universal features in the energetics of symmetry breaking. É Roldán, I A Martinez, J M R Parrondo, D Petrov, Nat. Phys. 10É. Roldán, I. A. Martinez, J. M. R. Parrondo, and D. Petrov, \\\"Universal features in the energetics of symmetry breaking,\\\" Nat. Phys. 10, 457-461 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Large work extraction and the Landauer limit in a continuous Maxwell demon. M Ribezzi-Crivellari, F Ritort, Nat. Phys. 15M. Ribezzi-Crivellari and F. Ritort, \\\"Large work extraction and the Landauer limit in a continuous Maxwell demon,\\\" Nat. Phys. 15, 660-664 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hydrodynamic synchronization of light driven microrotors. R Di Leonardo, A Búzás, L Kelemen, G Vizsnyiczai, L Oroszi, P Ormos, Phys. Rev. Lett. 10934104R. Di Leonardo, A. Búzás, L. Kelemen, G. Vizsnyiczai, L. Oroszi, and P. Ormos, \\\"Hydrodynamic synchronization of light driven microrotors,\\\" Phys. Rev. Lett. 109, 034104 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Collective synchronization states in arrays of driven colloidal oscillators. R Lhermerout, N Bruot, G M Cicuta, J Kotar, P Cicuta, New J. Phys. 14105023R. Lhermerout, N. Bruot, G. M. Cicuta, J. Kotar, and P. Cicuta, \\\"Collective synchronization states in arrays of driven colloidal oscillators,\\\" New J. Phys. 14, 105023 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stochastic hydrodynamic synchronization in rotating energy landscapes. N Koumakis, R Di Leonardo, Phys. Rev. Lett. 110174103N. Koumakis and R. Di Leonardo, \\\"Stochastic hydrodynamic synchronization in rotating energy landscapes,\\\" Phys. Rev. Lett. 110, 174103 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal hydrodynamic synchronization of colloidal rotors. J Kotar, L Debono, N Bruot, S Box, D Phillips, S Simpson, S Hanna, P Cicuta, Phys. Rev. Lett. 111228103J. Kotar, L. Debono, N. Bruot, S. Box, D. Phillips, S. Simpson, S. Hanna, and P. Cicuta, \\\"Optimal hydrodynamic synchronization of colloidal rotors,\\\" Phys. Rev. Lett. 111, 228103 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Active particles in complex and crowded environments. C Bechinger, R Di Leonardo, H Löwen, C Reichhardt, G Volpe, G Volpe, Rev. Mod. Phys. 8845006C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, \\\"Active particles in complex and crowded environments,\\\" Rev. Mod. Phys. 88, 045006 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Machine learning for active matter. F Cihos, K Gustavsson, B Mehlig, G Volpe, Nat. Mach. Intell. 2F. Cihos, K. Gustavsson, B. Mehlig, and G. Volpe, \\\"Machine learning for active matter,\\\" Nat. Mach. Intell. 2, 94-103 (2020).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Non-boltzmann stationary distributions and nonequilibrium relations in active baths. A Argun, A.-R Moradi, E Pinçe, G B Bagci, A Imparato, G Volpe, Phys. Rev. E. 9462150A. Argun, A.-R. Moradi, E. Pinçe, G. B. Bagci, A. Imparato, and G. Volpe, \\\"Non-boltzmann stationary distributions and nonequilibrium relations in active baths,\\\" Phys. Rev. E 94, 062150 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Phase transitions of colloidal monolayers in periodic pinning arrays. K Mangold, P Leiderer, C Bechinger, Phys. Rev. Lett. 90158302K. Mangold, P. Leiderer, and C. Bechinger, \\\"Phase transitions of colloidal monolayers in periodic pinning arrays,\\\" Phys. Rev. Lett. 90, 158302 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tunable colloids: Control of colloidal phase transitions with tunable interactions. A Yethiraj, Soft Matter. 3A. Yethiraj, \\\"Tunable colloids: Control of colloidal phase transitions with tunable interactions,\\\" Soft Matter 3, 1099-1115 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Light-induced phase transitions of colloidal monolayers with crystalline order. L Zaidouny, T Bohlein, R Roth, C Bechinger, Soft Matter. 9L. Zaidouny, T. Bohlein, R. Roth, and C. Bechinger, \\\"Light-induced phase transitions of colloidal monolayers with crystalline order,\\\" Soft Matter 9, 9230-9236 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Physical grounds for biological effect of laser radiation. A N Rubinov, J. Phys. D: Appl. Phys. 362317A. N. Rubinov, \\\"Physical grounds for biological effect of laser radiation,\\\" J. Phys. D: Appl. Phys. 36, 2317 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap. M P Macdonald, L Paterson, W Sibbett, K Dholakia, P E Bryant, Opt. Lett. 26M. P. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and P. E. Bryant, \\\"Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,\\\" Opt. Lett. 26, 863-865 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamical analysis of an optical rocking ratchet: Theory and experiment. A V Arzola, K Volke-Sepulveda, J L Mateos, Phys. Rev. E. 8762910A. V. Arzola, K. Volke-Sepulveda, and J. L. Mateos, \\\"Dynamical analysis of an optical rocking ratchet: Theory and experiment,\\\" Phys. Rev. E 87, 062910 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Active dynamics of colloidal particles in time-varying laser speckle patterns. S Bianchi, R Pruner, G Vizsnyiczai, C Maggi, R. Di Leonardo, Sci. Rep. 627681S. Bianchi, R. Pruner, G. Vizsnyiczai, C. Maggi, and R. Di Leonardo, \\\"Active dynamics of colloidal particles in time-varying laser speckle patterns,\\\" Sci. Rep. 6, 27681 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Force mapping of an extended light pattern in an inclined plane: Deterministic regime. A V Arzola, K Volke-Sepúlveda, J L Mateos, Opt. Express. 17A. V. Arzola, K. Volke-Sepúlveda, and J. L. Mateos, \\\"Force mapping of an extended light pattern in an inclined plane: Deterministic regime,\\\" Opt. Express 17, 3429-3440 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Moving interference patterns created using the angular Doppler-effect. J Arlt, M Macdonald, L Paterson, W Sibbett, K Dholakia, K Volke-Sepulveda, Opt. Express. 10J. Arlt, M. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and K. Volke-Sepulveda, \\\"Moving interference patterns created using the angular Doppler-effect,\\\" Opt. Express 10, 844-852 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination. M Šiler, T Čižmár, A Jonáš, P Zemánek, New J. Phys. 10113010M. Šiler, T. Čižmár, A. Jonáš, and P. Zemánek, \\\"Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,\\\" New J. Phys. 10, 113010 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Phase behavior of two-dimensional colloidal systems in the presence of periodic light fields. C Bechinger, M Brunner, P Leiderer, Phys. Rev. Lett. 86C. Bechinger, M. Brunner, and P. Leiderer, \\\"Phase behavior of two-dimensional colloidal systems in the presence of periodic light fields,\\\" Phys. Rev. Lett. 86, 930-933 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Phase behavior of colloidal molecular crystals on triangular light lattices. M Brunner, C Bechinger, Phys. Rev. Lett. 88248302M. Brunner and C. Bechinger, \\\"Phase behavior of colloidal molecular crystals on triangular light lattices,\\\" Phys. Rev. Lett. 88, 248302 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tailoring of phononic band structures in colloidal crystals. J Baumgartl, M Zvyagolskaya, C Bechinger, Phys. Rev. Lett. 99205503J. Baumgartl, M. Zvyagolskaya, and C. Bechinger, \\\"Tailoring of phononic band structures in colloidal crystals,\\\" Phys. Rev. Lett. 99, 205503 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"3d interferometric optical tweezers using a single spatial light modulator. E Schonbrun, R Piestun, P Jordan, J Cooper, K D Wulff, J Courtial, M Padgett, Opt. Express. 13E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, \\\"3d interferometric optical tweezers using a single spatial light modulator,\\\" Opt. Express 13, 3777-3786 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Archimedean-like tiling on decagonal quasicrystalline surfaces. J Mikhael, J Roth, L Helden, C Bechinger, Nature. 454J. Mikhael, J. Roth, L. Helden, and C. Bechinger, \\\"Archimedean-like tiling on decagonal quasicrystalline surfaces,\\\" Nature 454, 501-504 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields. J Mikhael, M Schmiedeberg, S Rausch, J Roth, H Stark, C Bechinger, Proc. Natl. Acad. Sci. 107J. Mikhael, M. Schmiedeberg, S. Rausch, J. Roth, H. Stark, and C. Bechinger, \\\"Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields,\\\" Proc. Natl. Acad. Sci. 107, 7214-7218 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles. K Volke-Sepúlveda, S Chávez-Cerda, V Garcés-Chávez, K Dholakia, J. Opt. Soc. Am. B. 21K. Volke-Sepúlveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, \\\"Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,\\\" J. Opt. Soc. Am. B 21, 1749-1757 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical conveyor belt for delivery of submicron objects. T Čižmár, V Garcés-Chávez, K Dholakia, P Zemánek, Appl. Phys. Lett. 86174101T. Čižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, \\\"Optical conveyor belt for delivery of submicron objects,\\\" Appl. Phys. Lett. 86, 174101 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transverse particle dynamics in a Bessel beam. G Milne, K Dholakia, D Mcgloin, K Volke-Sepulveda, P Zemánek, Opt. Express. 15G. Milne, K. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemánek, \\\"Transverse particle dynamics in a Bessel beam,\\\" Opt. Express 15, 13972-13987 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Light-induced cell separation in a tailored optical landscape. L Paterson, E Papagiakoumou, G Milne, V Garcés-Chávez, S A Tatarkova, W Sibbett, F J Gunn-Moore, P E Bryant, A C Riches, K Dholakia, Appl. Phys. Lett. 87123901L. Paterson, E. Papagiakoumou, G. Milne, V. Garcés-Chávez, S. A. Tatarkova, W. Sibbett, F. J. Gunn-Moore, P. E. Bryant, A. C. Riches, and K. Dholakia, \\\"Light-induced cell separation in a tailored optical landscape,\\\" Appl. Phys. Lett. 87, 123901 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian particle in an optical potential of the washboard type. S A Tatarkova, W Sibbett, K Dholakia, Phys. Rev. Lett. 9138101S. A. Tatarkova, W. Sibbett, and K. Dholakia, \\\"Brownian particle in an optical potential of the washboard type,\\\" Phys. Rev. Lett. 91, 038101 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Artificial Brownian motors: Controlling transport on the nanoscale. P Hänggi, F Marchesoni, Rev. Mod. Phys. 81P. Hänggi and F. Marchesoni, \\\"Artificial Brownian motors: Controlling transport on the nanoscale,\\\" Rev. Mod. Phys. 81, 387-442 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental creation and characterization of random potential-energy landscapes exploiting speckle patterns. J Bewerunge, S U Egelhaaf, Phys. Rev. A. 9313806J. Bewerunge and S. U. Egelhaaf, \\\"Experimental creation and characterization of random potential-energy landscapes exploiting speckle patterns,\\\" Phys. Rev. A 93, 013806 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian motors: Noisy transport far from equilibrium. P Reimann, Phys. Rep. 361P. Reimann, \\\"Brownian motors: Noisy transport far from equilibrium,\\\" Phys. Rep. 361, 57-265 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate. B A Garetz, S Arnold, Opt. Commun. 31B. A. Garetz and S. Arnold, \\\"Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate,\\\" Opt. Commun. 31, 1-3 (1979).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Angular Doppler effect. B A Garetz, J. Opt. Soc. Am. 71B. A. Garetz, \\\"Angular Doppler effect,\\\" J. Opt. Soc. Am. 71, 609-611 (1981).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Controlled rotation of optically trapped microscopic particles. L Paterson, M P Macdonald, J Arlt, W Sibbett, P E Bryant, K Dholakia, Science. 292L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, \\\"Controlled rotation of optically trapped microscopic particles,\\\" Science 292, 912-914 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An optical nanotrap array movable over a milimetre range. T Čižmár, M Šiler, P Zemánek, Appl. Phys. B. 84T. Čižmár, M. Šiler, and P. Zemánek, \\\"An optical nanotrap array movable over a milimetre range,\\\" Appl. Phys. B 84, 197-203 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The holographic optical micro-manipulation system based on counter-propagating beams. T Čižmár, O Brzobohatỳ, K Dholakia, P Zemánek, Laser Phys. Lett. 850T. Čižmár, O. Brzobohatỳ, K. Dholakia, and P. Zemánek, \\\"The holographic optical micro-manipulation system based on counter-propagating beams,\\\" Laser Phys. Lett. 8, 50 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical conveyors: A class of active tractor beams. D B Ruffner, D G Grier, Phys. Rev. Lett. 109163903D. B. Ruffner and D. G. Grier, \\\"Optical conveyors: A class of active tractor beams,\\\" Phys. Rev. Lett. 109, 163903 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Colloidal transport through optical tweezer arrays. Y Roichman, V Wong, D G Grier, Phys. Rev. E. 7511407Y. Roichman, V. Wong, and D. G. Grier, \\\"Colloidal transport through optical tweezer arrays,\\\" Phys. Rev. E 75, 011407 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical separation of cells on potential energy landscapes: Enhancement with dielectric tagging. K Dholakia, W M Lee, L Paterson, M P Macdonald, R Mcdonald, I Andreev, P Mthunzi, C T A Brown, R F Marchington, A C Riches, IEEE J. Sel. Top. Quant. El. 13K. Dholakia, W. M. Lee, L. Paterson, M. P. MacDonald, R. McDonald, I. Andreev, P. Mthunzi, C. T. A. Brown, R. F. Marchington, and A. C. Riches, \\\"Optical separation of cells on potential energy landscapes: Enhancement with dielectric tagging,\\\" IEEE J. Sel. Top. Quant. El. 13, 1646-1654 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Kinetically locked-in colloidal transport in an array of optical tweezers. P T Korda, M B Taylor, D G Grier, Phys. Rev. Lett. 89128301P. T. Korda, M. B. Taylor, and D. G. Grier, \\\"Kinetically locked-in colloidal transport in an array of optical tweezers,\\\" Phys. Rev. Lett. 89, 128301 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sorting mesoscopic objects with periodic potential landscapes: Optical fractionation. K Ladavac, K Kasza, D G Grier, Phys. Rev. E. 7010901K. Ladavac, K. Kasza, and D. G. Grier, \\\"Sorting mesoscopic objects with periodic potential landscapes: Optical fractionation,\\\" Phys. Rev. E 70, 010901 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microfluidic sorting with blinking optical traps. R Dasgupta, R S Verma, P K Gupta, Opt. Lett. 37R. Dasgupta, R. S. Verma, and P. K. Gupta, \\\"Microfluidic sorting with blinking optical traps,\\\" Opt. Lett. 37, 1739-1741 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamics, rectification, and fractionation for colloids on flashing substrates. A Libal, C Reichhardt, B Janko, C J O Reichhardt, Phys. Rev. Lett. 96188301A. Libal, C. Reichhardt, B. Janko, and C. J. O. Reichhardt, \\\"Dynamics, rectification, and fractionation for colloids on flashing substrates,\\\" Phys. Rev. Lett. 96, 188301 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"One-dimensional optical thermal ratchets. S.-H Lee, D G Grier, J. Phys. Condens. Matter. 17S.-H. Lee and D. G. Grier, \\\"One-dimensional optical thermal ratchets,\\\" J. Phys. Condens. Matter 17, S3685-S3695 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Two-dimensional optical thermal ratchets based on Fibonacci spirals. K Xiao, Y Roichman, D G Grier, Phys. Rev. E. 8411131K. Xiao, Y. Roichman, and D. G. Grier, \\\"Two-dimensional optical thermal ratchets based on Fibonacci spirals,\\\" Phys. Rev. E 84, 011131 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Noise-enabled optical ratchets. R León-Montiel, P A Quinto-Su, Sci. Rep. 744287R. León-Montiel and P. A. Quinto-Su, \\\"Noise-enabled optical ratchets,\\\" Sci. Rep. 7, 44287 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical chromatography. T Imasaka, Y Kawabata, T Kaneta, Y Ishidzu, Anal. Chem. 67T. Imasaka, Y. Kawabata, T. Kaneta, and Y. Ishidzu, \\\"Optical chromatography,\\\" Anal. Chem. 67, 1763-1765 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Refractive-index-driven separation of colloidal polymer particles using optical chromatography. S J Hart, A V Terray, Appl. Phys. Lett. 83S. J. Hart and A. V. Terray, \\\"Refractive-index-driven separation of colloidal polymer particles using optical chromatography,\\\" Appl. Phys. Lett. 83, 5316-5318 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Static optical sorting in a laser interference field. P Jákl, T Čižmár, M Šerý, P Zemánek, Appl. Phys. Lett. 92161110P. Jákl, T. Čižmár, M. Šerý, and P. Zemánek, \\\"Static optical sorting in a laser interference field,\\\" Appl. Phys. Lett. 92, 161110 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical deflection and sorting of microparticles in a near-field optical geometry. R F Marchington, M Mazilu, S Kuriakose, V Garcés-Chávez, P J Reece, T F Krauss, M Gu, K Dholakia, Opt. Express. 16R. F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P. J. Reece, T. F. Krauss, M. Gu, and K. Dholakia, \\\"Optical deflection and sorting of microparticles in a near-field optical geometry,\\\" Opt. Express 16, 3712-3726 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery. M Šiler, T Čižmár, M Šerỳ, P Zemánek, Appl. Phys. B. 84M. Šiler, T. Čižmár, M. Šerỳ, and P. Zemánek, \\\"Optical forces generated by evanescent standing waves and their usage for sub-micron particle delivery,\\\" Appl. Phys. B 84, 157-165 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nanomanipulation using near field photonics. D Erickson, X Serey, Y.-F Chen, S Mandal, Lab Chip. 11D. Erickson, X. Serey, Y.-F. Chen, and S. Mandal, \\\"Nanomanipulation using near field photonics,\\\" Lab Chip 11, 995-1009 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental demonstration of optical transport, sorting and self-arrangement using a 'tractor' beam. O Brzobohatỳ, V Karásek, M Šiler, L Chvátal, T Čižmár, P Zemánek, Nat. Photon. 7O. Brzobohatỳ, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, and P. Zemánek, \\\"Experimental demonstration of optical transport, sorting and self-arrangement using a 'tractor' beam,\\\" Nat. Photon. 7, 123-127 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optofluidic sorting of material chirality by chiral light. G Tkachenko, E Brasselet, Nat. Commun. 53577G. Tkachenko and E. Brasselet, \\\"Optofluidic sorting of material chirality by chiral light,\\\" Nat. Commun. 5, 3577 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lateral chirality-sorting optical forces. A Hayat, J P B Mueller, F Capasso, Proc. Natl. Acad. Sci. 112A. Hayat, J. P. B. Mueller, and F. Capasso, \\\"Lateral chirality-sorting optical forces,\\\" Proc. Natl. Acad. Sci. 112, 13190-13194 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Orientational and directional locking of colloidal clusters driven across periodic surfaces. X Cao, E Panizon, A Vanossi, N Manini, C Bechinger, Nat. Phys. 15X. Cao, E. Panizon, A. Vanossi, N. Manini, and C. Bechinger, \\\"Orientational and directional locking of colloidal clusters driven across periodic surfaces,\\\" Nat. Phys. 15, 776-780 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental observation of the Aubry transition in two-dimensional colloidal monolayers. T Brazda, A Silva, N Manini, A Vanossi, R Guerra, E Tosatti, C Bechinger, Phys. Rev. X. 811050T. Brazda, A. Silva, N. Manini, A. Vanossi, R. Guerra, E. Tosatti, and C. Bechinger, \\\"Experimental observation of the Aubry transition in two-dimensional colloidal monolayers,\\\" Phys. Rev. X 8, 011050 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental observation of directional locking and dynamical ordering of colloidal monolayers driven across quasiperiodic substrates. T Bohlein, C Bechinger, Phys. Rev. Lett. 10958301T. Bohlein and C. Bechinger, \\\"Experimental observation of directional locking and dynamical ordering of colloidal monolayers driven across quasiperiodic substrates,\\\" Phys. Rev. Lett. 109, 058301 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces. T Bohlein, J Mikhael, C Bechinger, Nat. Mater. 11T. Bohlein, J. Mikhael, and C. Bechinger, \\\"Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces,\\\" Nat. Mater. 11, 126-130 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Laser trapping of colloidal metal nanoparticles. A Lehmuskero, P Johansson, H Rubinsztein-Dunlop, L Tong, M Kall, ACS Nano. 9A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong, and M. Kall, \\\"Laser trapping of colloidal metal nanoparticles,\\\" ACS Nano 9, 3453-3469 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Parallel and selective trapping in a patterned plasmonic landscape. M Righini, A S Zelenina, C Girard, R Quidant, Nat. Phys. 3M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, \\\"Parallel and selective trapping in a patterned plasmonic landscape,\\\" Nat. Phys. 3, 477-480 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range. M Righini, G Volpe, C Girard, D Petrov, R Quidant, Phys. Rev. Lett. 100186804M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, \\\"Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,\\\" Phys. Rev. Lett. 100, 186804 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasmonic optical tweezers toward molecular manipulation: tailoring plasmonic nanostructure, light source, and resonant trapping. T Shoji, Y Tsuboi, J. Phys. Chem. Lett. 5T. Shoji and Y. Tsuboi, \\\"Plasmonic optical tweezers toward molecular manipulation: tailoring plasmonic nanostruc- ture, light source, and resonant trapping,\\\" J. Phys. Chem. Lett. 5, 2957-2967 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Massively parallel manipulation of single cells and microparticles using optical images. P Y Chiou, A T Ohta, M C Wu, Nature. 436P. Y. Chiou, A. T. Ohta, and M. C. Wu, \\\"Massively parallel manipulation of single cells and microparticles using optical images,\\\" Nature 436, 370-372 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Thermophoretic tweezers for low-power and versatile manipulation of biological cells. L Lin, X Peng, X Wei, Z Mao, C Xie, Y Zheng, ACS Nano. 11L. Lin, X. Peng, X. Wei, Z. Mao, C. Xie, and Y. Zheng, \\\"Thermophoretic tweezers for low-power and versatile manipulation of biological cells,\\\" ACS Nano 11, 3147-3154 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Opto-thermophoretic tweezers and assembly. J Li, L Lin, Y Inoue, Y Zheng, J. Micro Nano-Manufacturing. 640801J. Li, L. Lin, Y. Inoue, and Y. Zheng, \\\"Opto-thermophoretic tweezers and assembly,\\\" J. Micro Nano-Manufacturing 6, 040801 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Acoustic tweezers: patterning cells and microparticles using standing surface acoustic waves (SSAW). J Shi, D Ahmed, X Mao, S.-C S Lin, A Lawit, T J Huang, Lab Chip. 9J. Shi, D. Ahmed, X. Mao, S.-C. S. Lin, A. Lawit, and T. J. Huang, \\\"Acoustic tweezers: patterning cells and microparticles using standing surface acoustic waves (SSAW),\\\" Lab Chip 9, 2890-2895 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics. J Friend, L Y Yeo, Rev. Mod. Phys. 83J. Friend and L. Y. Yeo, \\\"Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics,\\\" Rev. Mod. Phys. 83, 647-704 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of a single-beam gradient force acoustical trap for elastic particles: acoustical tweezers. D Baresch, J.-L Thomas, R Marchiano, Phys. Rev. Lett. 11624301D. Baresch, J.-L. Thomas, and R. Marchiano, \\\"Observation of a single-beam gradient force acoustical trap for elastic particles: acoustical tweezers,\\\" Phys. Rev. Lett. 116, 024301 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Perspective on light-induced transport of particles: from optical forces to phoretic motion. P Zemánek, G Volpe, A Jonáš, O Brzobohatỳ, Adv. Opt. Photon. 11P. Zemánek, G. Volpe, A. Jonáš, and O. Brzobohatỳ, \\\"Perspective on light-induced transport of particles: from optical forces to phoretic motion,\\\" Adv. Opt. Photon. 11, 577-678 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Feedback stabilization of optically levitated particles. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 30A. Ashkin and J. M. Dziedzic, \\\"Feedback stabilization of optically levitated particles,\\\" Appl. Phys. Lett. 30, 202-204 (1977).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical levitation in high vacuum. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 28A. Ashkin and J. M. Dziedzic, \\\"Optical levitation in high vacuum,\\\" Appl. Phys. Lett. 28, 333-335 (1976).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical levitation by radiation pressure. A Ashkin, J M Dziedzic, Appl. Phys. Lett. 19A. Ashkin and J. M. Dziedzic, \\\"Optical levitation by radiation pressure,\\\" Appl. Phys. Lett. 19, 283-285 (1971).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of a single-beam gradient-force optical trap for dielectric particles in air. R Omori, T Kobayashi, A Suzuki, Opt. Lett. 22R. Omori, T. Kobayashi, and A. Suzuki, \\\"Observation of a single-beam gradient-force optical trap for dielectric particles in air,\\\" Opt. Lett. 22, 816-818 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. J Gieseler, B Deutsch, R Quidant, L Novotny, Phys. Rev. Lett. 109103603J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, \\\"Subkelvin parametric feedback cooling of a laser-trapped nanoparticle,\\\" Phys. Rev. Lett. 109, 103603 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical trapping of nanoparticles by full solid-angle focusing. V Salakhutdinov, M Sondermann, L Carbone, E Giacobino, A Bramati, G Leuchs, Optica. 3V. Salakhutdinov, M. Sondermann, L. Carbone, E. Giacobino, A. Bramati, and G. Leuchs, \\\"Optical trapping of nanoparticles by full solid-angle focusing,\\\" Optica 3, 1181-1186 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental realization of a thermal squeezed state of levitated optomechanics. M Rashid, T Tufarelli, J Bateman, J Vovrosh, D Hempston, M S Kim, H Ulbricht, Phys. Rev. Lett. 117273601M. Rashid, T. Tufarelli, J. Bateman, J. Vovrosh, D. Hempston, M. S. Kim, and H. Ulbricht, \\\"Experimental realization of a thermal squeezed state of levitated optomechanics,\\\" Phys. Rev. Lett. 117, 273601 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Measurement of the instantaneous velocity of a Brownian particle. T Li, S Kheifets, D Medellin, M G Raizen, Science. 328T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, \\\"Measurement of the instantaneous velocity of a Brownian particle,\\\" Science 328, 1673-1675 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum. G Ranjit, D P Atherton, J H Stutz, M Cunningham, A A Geraci, Phys. Rev. A. 9151805G. Ranjit, D. P. Atherton, J. H. Stutz, M. Cunningham, and A. A. Geraci, \\\"Attonewton force detection using microspheres in a dual-beam optical trap in high vacuum,\\\" Phys. Rev. A 91, 051805 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Toward quantum superposition of living organisms. O Romero-Isart, M L Juan, R Quidant, J I Cirac, New J. Phys. 1233015O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, \\\"Toward quantum superposition of living organisms,\\\" New J. Phys. 12, 033015 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity opto-mechanics using an optically levitated nanosphere. D E Chang, C A Regal, S B Papp, D J Wilson, J Ye, O Painter, H J Kimble, P Zoller, Proc. Natl. Acad. Sci. 107D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, \\\"Cavity opto-mechanics using an optically levitated nanosphere,\\\" Proc. Natl. Acad. Sci. 107, 1005-1010 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity cooling of an optically trapped nanoparticle. P F Barker, M N Shneider, Phys. Rev. A. 8123826P. F. Barker and M. N. Shneider, \\\"Cavity cooling of an optically trapped nanoparticle,\\\" Phys. Rev. A 81, 023826 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cold damping of an optically levitated nanoparticle to microkelvin temperatures. F Tebbenjohanns, M Frimmer, A Militaru, V Jain, L Novotny, Phys. Rev. Lett. 122223601F. Tebbenjohanns, M. Frimmer, A. Militaru, V. Jain, and L. Novotny, \\\"Cold damping of an optically levitated nanoparticle to microkelvin temperatures,\\\" Phys. Rev. Lett. 122, 223601 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct measurement of photon recoil from a levitated nanoparticle. V Jain, J Gieseler, C Moritz, C Dellago, R Quidant, L Novotny, Phys. Rev. Lett. 116243601V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, and L. Novotny, \\\"Direct measurement of photon recoil from a levitated nanoparticle,\\\" Phys. Rev. Lett. 116, 243601 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal feedback cooling of a charged levitated nanoparticle with adaptive control. G P Conangla, F Ricci, M T Cuairan, A W Schell, N Meyer, R Quidant, Phys. Rev. Lett. 122223602G. P. Conangla, F. Ricci, M. T. Cuairan, A. W. Schell, N. Meyer, and R. Quidant, \\\"Optimal feedback cooling of a charged levitated nanoparticle with adaptive control,\\\" Phys. Rev. Lett. 122, 223602 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Motional sideband asymmetry of a nanoparticle optically levitated in free space. F Tebbenjohanns, M Frimmer, V Jain, D Windey, L Novotny, arXiv:1908.05079arXiv preprintF. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, and L. Novotny, \\\"Motional sideband asymmetry of a nanoparticle optically levitated in free space,\\\" arXiv preprint arXiv:1908.05079 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity cooling of a levitated nanosphere by coherent scattering. U Delić, M Reisenbauer, D Grass, N Kiesel, V Vuletić, M Aspelmeyer, Phys. Rev. Lett. 122123602U. Delić, M. Reisenbauer, D. Grass, N. Kiesel, V. Vuletić, and M. Aspelmeyer, \\\"Cavity cooling of a levitated nanosphere by coherent scattering,\\\" Phys. Rev. Lett. 122, 123602 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity-based 3d cooling of a levitated nanoparticle via coherent scattering. D Windey, C Gonzalez-Ballestero, P Maurer, L Novotny, O Romero-Isart, R Reimann, Phys. Rev. Lett. 122123601D. Windey, C. Gonzalez-Ballestero, P. Maurer, L. Novotny, O. Romero-Isart, and R. Reimann, \\\"Cavity-based 3d cooling of a levitated nanoparticle via coherent scattering,\\\" Phys. Rev. Lett. 122, 123601 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Resolved-sideband cooling of a levitated nanoparticle in the presence of laser phase noise. N Meyer, A De Los Rios, P Sommer, J Mestres, V Gieseler, L Jain, R Novotny, Quidant, Phys. Rev. Lett. 123153601N. Meyer, A. de los Rios Sommer, P. Mestres, J. Gieseler, V. Jain, L. Novotny, and R. Quidant, \\\"Resolved-sideband cooling of a levitated nanoparticle in the presence of laser phase noise,\\\" Phys. Rev. Lett. 123, 153601 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sideband cooling of micromechanical motion to the quantum ground state. J D Teufel, T Donner, D Li, J W Harlow, M S Allman, K Cicak, A J Sirois, J D Whittaker, K W Lehnert, R W Simmonds, Nature. 475J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, \\\"Sideband cooling of micromechanical motion to the quantum ground state,\\\" Nature 475, 359-405 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Laser cooling of a nanomechanical oscillator into its quantum ground state. J Chan, T P M Alegre, A H Safavi-Naeini, J T Hill, A Krause, S Gröblacher, M Aspelmeyer, O Painter, Nature. 478J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, \\\"Laser cooling of a nanomechanical oscillator into its quantum ground state,\\\" Nature 478, 89-92 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cooling of a levitated nanoparticle to the motional quantum ground state. U Delić, M Reisenbauer, K Dare, D Grass, V Vuletić, N Kiesel, M Aspelmeyer, Science. 367U. Delić, M. Reisenbauer, K. Dare, D. Grass, V. Vuletić, N. Kiesel, and M. Aspelmeyer, \\\"Cooling of a levitated nanoparticle to the motional quantum ground state,\\\" Science 367, 892-895 (2020).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary. S Haroche, Rev. Mod. Phys. 85S. Haroche, \\\"Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary,\\\" Rev. Mod. Phys. 85, 1083-1102 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nobel lecture: Superposition, entanglement, and raising schrödinger's cat. D J Wineland, Rev. Mod. Phys. 851103D. J. Wineland, \\\"Nobel lecture: Superposition, entanglement, and raising schrödinger's cat,\\\" Rev. Mod. Phys. 85, 1103 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum interference distinguishes between constitutional isomers. J Tüxen, S Gerlich, S Eibenberger, M Arndt, M Mayor, Chem. Commun. 46J. Tüxen, S. Gerlich, S. Eibenberger, M. Arndt, and M. Mayor, \\\"Quantum interference distinguishes between constitutional isomers,\\\" Chem. Commun. 46, 4145-4147 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Colloquium: Quantum interference of clusters and molecules. K Hornberger, S Gerlich, P Haslinger, S Nimmrichter, M Arndt, Rev. Mod. Phys. 84K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, \\\"Colloquium: Quantum interference of clusters and molecules,\\\" Rev. Mod. Phys. 84, 157-173 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Creating and verifying a quantum superposition in a micro-optomechanical system. D Kleckner, I Pikovski, E Jeffrey, L Ament, E Eliel, J Van Den, D Brink, Bouwmeester, New J. Phys. 1095020D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. Van Den Brink, and D. Bouwmeester, \\\"Creating and verifying a quantum superposition in a micro-optomechanical system,\\\" New J. Phys. 10, 095020 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Macroscopic quantum resonators (maqro). R Kaltenbaek, G Hechenblaikner, N Kiesel, O Romero-Isart, K C Schwab, U Johann, M Aspelmeyer, Exp. Astron. 34R. Kaltenbaek, G. Hechenblaikner, N. Kiesel, O. Romero-Isart, K. C. Schwab, U. Johann, and M. Aspelmeyer, \\\"Macroscopic quantum resonators (maqro),\\\" Exp. Astron. 34, 123-164 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum superposition of massive objects and collapse models. O Romero-Isart, Phys. Rev. A. 8452121O. Romero-Isart, \\\"Quantum superposition of massive objects and collapse models,\\\" Phys. Rev. A 84, 052121 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Large quantum superpositions and interference of massive nanometer-sized objects. O Romero-Isart, A C Pflanzer, F Blaser, R Kaltenbaek, N Kiesel, M Aspelmeyer, J I Cirac, Phys. Rev. Lett. 10720405O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, \\\"Large quantum superpositions and interference of massive nanometer-sized objects,\\\" Phys. Rev. Lett. 107, 020405 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Near-field interferometry of a free-falling nanoparticle from a point-like source. J Bateman, S Nimmrichter, K Hornberger, H Ulbricht, Nat. Commun. 54788J. Bateman, S. Nimmrichter, K. Hornberger, and H. Ulbricht, \\\"Near-field interferometry of a free-falling nanoparticle from a point-like source,\\\" Nat. Commun. 5, 4788 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Models of wave-function collapse, underlying theories, and experimental tests. A Bassi, K Lochan, S Satin, T P Singh, H Ulbricht, Rev. Mod. Phys. 85A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, \\\"Models of wave-function collapse, underlying theories, and experimental tests,\\\" Rev. Mod. Phys. 85, 471-527 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optomechanics of levitated dielectric particles. Z.-Q Yin, A A Geraci, T Li, Int. J. Mod. Phys. B. 271330018Z.-Q. Yin, A. A. Geraci, and T. Li, \\\"Optomechanics of levitated dielectric particles,\\\" Int. J. Mod. Phys. B 27, 1330018 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity optomechanics. M Aspelmeyer, T J Kippenberg, F Marquardt, Rev. Mod. Phys. 86M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, \\\"Cavity optomechanics,\\\" Rev. Mod. Phys. 86, 1391-1452 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W P Bowen, G J Milburn, Quantum optomechanics. CRC pressW. P. Bowen and G. J. Milburn, Quantum optomechanics (CRC press, 2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Classical characterization of biphoton correlation in waveguide lattices. R Keil, F Dreisow, M Heinrich, A Tünnermann, S Nolte, A Szameit, Phys. Rev. A. 8313808R. Keil, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, \\\"Classical characterization of biphoton correlation in waveguide lattices,\\\" Phys. Rev. A 83, 013808 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Heralded single-phonon preparation, storage, and readout in cavity optomechanics. C Galland, N Sangouard, N Piro, N Gisin, T J Kippenberg, Phys. Rev. Lett. 112143602C. Galland, N. Sangouard, N. Piro, N. Gisin, and T. J. Kippenberg, \\\"Heralded single-phonon preparation, storage, and readout in cavity optomechanics,\\\" Phys. Rev. Lett. 112, 143602 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Phonon counting and intensity interferometry of a nanomechanical resonator. J D Cohen, S M Meenehan, G S Maccabe, S Gröblacher, A H Safavi-Naeini, F Marsili, M D Shaw, O Painter, Nature. 520J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, \\\"Phonon counting and intensity interferometry of a nanomechanical resonator,\\\" Nature 520, 522-525 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hanbury brown and twiss interferometry of single phonons from an optomechanical resonator. S Hong, R Riedinger, I Marinković, A Wallucks, S G Hofer, R A Norte, M Aspelmeyer, S Gröblacher, Science. 358S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, \\\"Hanbury brown and twiss interferometry of single phonons from an optomechanical resonator,\\\" Science 358, 203-206 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cooling of mechanical motion with a two-level system: The high-temperature regime. P , Phys. Rev. B. 82165320P. Rabl, \\\"Cooling of mechanical motion with a two-level system: The high-temperature regime,\\\" Phys. Rev. B 82, 165320 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Strong magnetic coupling between an electronic spin qubit and a mechanical resonator. P Rabl, P Cappellaro, M V G Dutt, L Jiang, J R Maze, M D Lukin, Phys. Rev. B. 7941302P. Rabl, P. Cappellaro, M. V. G. Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, \\\"Strong magnetic coupling between an electronic spin qubit and a mechanical resonator,\\\" Phys. Rev. B 79, 041302 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hybrid opto-mechanical systems with nitrogen-vacancy centers. Z.-Q Yin, N Zhao, T Li, Sci. China Phys., Mech. Astron. 58Z.-Q. Yin, N. Zhao, and T. Li, \\\"Hybrid opto-mechanical systems with nitrogen-vacancy centers,\\\" Sci. China Phys., Mech. Astron. 58, 1-12 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond. L P Neukirch, E Von Haartman, J M Rosenholm, A N Vamivakas, Nat. Photon. 9L. P. Neukirch, E. Von Haartman, J. M. Rosenholm, and A. N. Vamivakas, \\\"Multi-dimensional single-spin nano-optomechanics with a levitated nanodiamond,\\\" Nat. Photon. 9, 653-657 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond. L P Neukirch, J Gieseler, R Quidant, L Novotny, A N Vamivakas, Opt. Lett. 38L. P. Neukirch, J. Gieseler, R. Quidant, L. Novotny, and A. N. Vamivakas, \\\"Observation of nitrogen vacancy photoluminescence from an optically levitated nanodiamond,\\\" Opt. Lett. 38, 2976-2979 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pure nanodiamonds for levitated optomechanics in vacuum. A C Frangeskou, A T M A Rahman, L Gines, S Mandal, O A Williams, P F Barker, G W Morley, New J. Phys. 2043016A. C. Frangeskou, A. T. M. A. Rahman, L. Gines, S. Mandal, O. A. Williams, P. F. Barker, and G. W. Morley, \\\"Pure nanodiamonds for levitated optomechanics in vacuum,\\\" New J. Phys. 20, 043016 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Burning and graphitization of optically levitated nanodiamonds in vacuum. A T M A Rahman, A C Frangeskou, M S Kim, S Bose, G W Morley, P F Barker, Sci. Rep. p. 21633A. T. M. A. Rahman, A. C. Frangeskou, M. S. Kim, S. Bose, G. W. Morley, and P. F. Barker, \\\"Burning and graphitization of optically levitated nanodiamonds in vacuum,\\\" Sci. Rep. p. 21633 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Electron spin control of optically levitated nanodiamonds in vacuum. T M Hoang, J Ahn, J Bang, T Li, Nat. Commun. 712250T. M. Hoang, J. Ahn, J. Bang, and T. Li, \\\"Electron spin control of optically levitated nanodiamonds in vacuum,\\\" Nat. Commun. 7, 12250 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cooling-by-measurement and mechanical state tomography via pulsed optomechanics. M R Vanner, J Hofer, G D Cole, M Aspelmeyer, Nat. Commun. 42295M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, \\\"Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,\\\" Nat. Commun. 4, 2295 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pulsed quantum optomechanics. M R Vanner, I Pikovski, G D Cole, M S Kim, Č Brukner, K Hammerer, G J Milburn, M Aspelmeyer, Proc. Natl. Acad. Sci. Natl. Acad. Sci108M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, \\\"Pulsed quantum optomechanics,\\\" Proc. Natl. Acad. Sci. 108, 16182-16187 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Single phonon quantum optics -An experimental exploration of a silicon photonics quantum memory. R Riedinger, ViennaPh.D. thesisR. Riedinger, \\\"Single phonon quantum optics -An experimental exploration of a silicon photonics quantum memory,\\\" Ph.D. thesis, Vienna (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Torsional optomechanics of a levitated nonspherical nanoparticle. T M Hoang, Y Ma, J Ahn, J Bang, F Robicheaux, Z.-Q Yin, T Li, Phys. Rev. Lett. 117123604T. M. Hoang, Y. Ma, J. Ahn, J. Bang, F. Robicheaux, Z.-Q. Yin, and T. Li, \\\"Torsional optomechanics of a levitated nonspherical nanoparticle,\\\" Phys. Rev. Lett. 117, 123604 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Precession motion in levitated optomechanics. M Rashid, M Toroš, A Setter, H Ulbricht, Phys. Rev. Lett. 121253601M. Rashid, M. Toroš, A. Setter, and H. Ulbricht, \\\"Precession motion in levitated optomechanics,\\\" Phys. Rev. Lett. 121, 253601 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Zeptonewton force sensing with nanospheres in an optical lattice. G Ranjit, M Cunningham, K Casey, A A Geraci, Phys. Rev. A. 9353801G. Ranjit, M. Cunningham, K. Casey, and A. A. Geraci, \\\"Zeptonewton force sensing with nanospheres in an optical lattice,\\\" Phys. Rev. A 93, 053801 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Force sensing with an optically levitated charged nanoparticle. D Hempston, J Vovrosh, M Toroš, G Winstone, M Rashid, H Ulbricht, Appl. Phys. Lett. 111133111D. Hempston, J. Vovrosh, M. Toroš, G. Winstone, M. Rashid, and H. Ulbricht, \\\"Force sensing with an optically levitated charged nanoparticle,\\\" Appl. Phys. Lett. 111, 133111 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Three-dimensional force-field microscopy with optically levitated microspheres. C P Blakemore, A D Rider, S Roy, Q Wang, A Kawasaki, G Gratta, Phys. Rev. A. 9923816C. P. Blakemore, A. D. Rider, S. Roy, Q. Wang, A. Kawasaki, and G. Gratta, \\\"Three-dimensional force-field microscopy with optically levitated microspheres,\\\" Phys. Rev. A 99, 023816 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical levitation of 10-ng spheres with nano-g acceleration sensitivity. F Monteiro, S Ghosh, A G Fine, D C Moore, Phys. Rev. A. 9663841F. Monteiro, S. Ghosh, A. G. Fine, and D. C. Moore, \\\"Optical levitation of 10-ng spheres with nano-g acceleration sensitivity,\\\" Phys. Rev. A 96, 063841 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for screened interactions associated with dark energy below the 100 µ m length scale. A D Rider, D C Moore, C P Blakemore, M Louis, M Lu, G Gratta, Phys. Rev. Lett. 117101101A. D. Rider, D. C. Moore, C. P. Blakemore, M. Louis, M. Lu, and G. Gratta, \\\"Search for screened interactions associated with dark energy below the 100 µ m length scale,\\\" Phys. Rev. Lett. 117, 101101 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for millicharged particles using optically levitated microspheres. D C Moore, A D Rider, G Gratta, Phys. Rev. Lett. 113251801D. C. Moore, A. D. Rider, and G. Gratta, \\\"Search for millicharged particles using optically levitated microspheres,\\\" Phys. Rev. Lett. 113, 251801 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sensing short range forces with a nanosphere matter-wave interferometer. A Geraci, H Goldman, Phys. Rev. D. 9262002A. Geraci and H. Goldman, \\\"Sensing short range forces with a nanosphere matter-wave interferometer,\\\" Phys. Rev. D 92, 062002 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Casimir interaction between a dielectric nanosphere and a metallic plane. A Canaguier-Durand, A Gérardin, R Guérout, P A M Neto, V V Nesvizhevsky, A Y Voronin, A Lambrecht, S Reynaud, Phys. Rev. A. 8332508A. Canaguier-Durand, A. Gérardin, R. Guérout, P. A. M. Neto, V. V. Nesvizhevsky, A. Y. Voronin, A. Lambrecht, and S. Reynaud, \\\"Casimir interaction between a dielectric nanosphere and a metallic plane,\\\" Phys. Rev. A 83, 032508 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lateral Casimir force on a rotating particle near a planar surface. A Manjavacas, F J Rodríguez-Fortuño, F J García De Abajo, A V Zayats, Phys. Rev. Lett. 118133605A. Manjavacas, F. J. Rodríguez-Fortuño, F. J. García de Abajo, and A. V. Zayats, \\\"Lateral Casimir force on a rotating particle near a planar surface,\\\" Phys. Rev. Lett. 118, 133605 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamics of a levitated nanosphere by optomechanical coupling and Casimir interaction. W Nie, Y Lan, Y Li, S Zhu, Phys. Rev. A. 8863849W. Nie, Y. Lan, Y. Li, and S. Zhu, \\\"Dynamics of a levitated nanosphere by optomechanical coupling and Casimir interaction,\\\" Phys. Rev. A 88, 063849 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Detecting Casimir torque with an optically levitated nanorod. Z Xu, T Li, Phys. Rev. A. 9633843Z. Xu and T. Li, \\\"Detecting Casimir torque with an optically levitated nanorod,\\\" Phys. Rev. A 96, 033843 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Detecting high-frequency gravitational waves with optically levitated sensors. A Arvanitaki, A A Geraci, Phys. Rev. Lett. 11071105A. Arvanitaki and A. A. Geraci, \\\"Detecting high-frequency gravitational waves with optically levitated sensors,\\\" Phys. Rev. Lett. 110, 071105 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonlinear dynamics of nanomechanical and micromechanical resonators. R Lifshitz, M C Cross, Rev. Nonlinear Dyn. Complex. pp. R. Lifshitz and M. C. Cross, \\\"Nonlinear dynamics of nanomechanical and micromechanical resonators,\\\" Rev. Nonlinear Dyn. Complex. pp. 1-52 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle. J Gieseler, M Spasenović, L Novotny, R Quidant, Phys. Rev. Lett. 112103603J. Gieseler, M. Spasenović, L. Novotny, and R. Quidant, \\\"Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle,\\\" Phys. Rev. Lett. 112, 103603 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optically levitated nanoparticle as a model system for stochastic bistable dynamics. F Ricci, R A Rica, M Spasenović, J Gieseler, L Rondin, L Novotny, R Quidant, Nat. Commun. p. 15141F. Ricci, R. A. Rica, M. Spasenović, J. Gieseler, L. Rondin, L. Novotny, and R. Quidant, \\\"Optically levitated nanoparticle as a model system for stochastic bistable dynamics,\\\" Nat. Commun. p. 15141 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Thermal nonlinearities in a nanomechanical oscillator. J Gieseler, L Novotny, R Quidant, Nat. Phys. 9J. Gieseler, L. Novotny, and R. Quidant, \\\"Thermal nonlinearities in a nanomechanical oscillator,\\\" Nat. Phys. 9, 806-810 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sensing static forces with free-falling nanoparticles. E Hebestreit, M Frimmer, R Reimann, L Novotny, Phys. Rev. Lett. 12163602E. Hebestreit, M. Frimmer, R. Reimann, and L. Novotny, \\\"Sensing static forces with free-falling nanoparticles,\\\" Phys. Rev. Lett. 121, 063602 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Probing macroscopic quantum superpositions with nanorotors. B A Stickler, B Papendell, S Kuhn, B Schrinski, J Millen, M Arndt, K Hornberger, New J. Phys. 20122001B. A. Stickler, B. Papendell, S. Kuhn, B. Schrinski, J. Millen, M. Arndt, and K. Hornberger, \\\"Probing macroscopic quantum superpositions with nanorotors,\\\" New J. Phys. 20, 122001 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Laser-induced rotation and cooling of a trapped microgyroscope in vacuum. Y Arita, M Mazilu, K Dholakia, Nat. Commun. 42374Y. Arita, M. Mazilu, and K. Dholakia, \\\"Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,\\\" Nat. Commun. 4, 2374 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity-assisted manipulation of freely rotating silicon nanorods in high vacuum. S Kuhn, P Asenbaum, A Kosloff, M Sclafani, B A Stickler, S Nimmrichter, K Hornberger, O Cheshnovsky, F Patolsky, M Arndt, Nano Lett. 15S. Kuhn, P. Asenbaum, A. Kosloff, M. Sclafani, B. A. Stickler, S. Nimmrichter, K. Hornberger, O. Cheshnovsky, F. Patolsky, and M. Arndt, \\\"Cavity-assisted manipulation of freely rotating silicon nanorods in high vacuum,\\\" Nano Lett. 15, 5604-5608 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Full rotational control of levitated silicon nanorods. S Kuhn, A Kosloff, B A Stickler, F Patolsky, K Hornberger, M Arndt, J Millen, Optica. 4S. Kuhn, A. Kosloff, B. A. Stickler, F. Patolsky, K. Hornberger, M. Arndt, and J. Millen, \\\"Full rotational control of levitated silicon nanorods,\\\" Optica 4, 356-360 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Laser refrigeration, alignment and rotation of levitated yb 3+: Ylf nanocrystals. A T M A Rahman, P F Barker, Nat. Photon. 11A. T. M. A. Rahman and P. F. Barker, \\\"Laser refrigeration, alignment and rotation of levitated yb 3+: Ylf nanocrystals,\\\" Nat. Photon. 11, 634-639 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor. J Ahn, Z Xu, J Bang, Y.-H Deng, T M Hoang, Q Han, R.-M Ma, T Li, Phys. Rev. Lett. 12133603J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, \\\"Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,\\\" Phys. Rev. Lett. 121, 033603 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ghz rotation of an optically trapped nanoparticle in vacuum. R Reimann, M Doderer, E Hebestreit, R Diehl, M Frimmer, D Windey, F Tebbenjohanns, L Novotny, Phys. Rev. Lett. 12133602R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, \\\"Ghz rotation of an optically trapped nanoparticle in vacuum,\\\" Phys. Rev. Lett. 121, 033602 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rotational quantum friction. R Zhao, A Manjavacas, F J García De Abajo, J B Pendry, Phys. Rev. Lett. 109123604R. Zhao, A. Manjavacas, F. J. García de Abajo, and J. B. Pendry, \\\"Rotational quantum friction,\\\" Phys. Rev. Lett. 109, 123604 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct measurement of kramers turnover with a levitated nanoparticle. L Rondin, J Gieseler, F Ricci, R Quidant, C Dellago, L Novotny, Nat. Nanotech. 12L. Rondin, J. Gieseler, F. Ricci, R. Quidant, C. Dellago, and L. Novotny, \\\"Direct measurement of kramers turnover with a levitated nanoparticle,\\\" Nat. Nanotech. 12, 1130-1133 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state. J Gieseler, R Quidant, C Dellago, L Novotny, Nat. Nanotech. 9J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, \\\"Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state,\\\" Nat. Nanotech. 9, 358-364 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental test of the differential fluctuation theorem and a generalized Jarzynski equality for arbitrary initial states. T M Hoang, R Pan, J Ahn, J Bang, H T Quan, T Li, Phys. Rev. Lett. 12080602T. M. Hoang, R. Pan, J. Ahn, J. Bang, H. T. Quan, and T. Li, \\\"Experimental test of the differential fluctuation theorem and a generalized Jarzynski equality for arbitrary initial states,\\\" Phys. Rev. Lett. 120, 080602 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Debiossac, D Grass, J J Alonso, E Lutz, N Kiesel, arXiv:1904.04889Thermodynamics of continuous non-markovian feedback control. arXiv preprintM. Debiossac, D. Grass, J. J. Alonso, E. Lutz, and N. Kiesel, \\\"Thermodynamics of continuous non-markovian feedback control,\\\" arXiv preprint arXiv:1904.04889 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cooling mechanical oscillators by coherent control. M Frimmer, J Gieseler, L Novotny, Phys. Rev. Lett. 117163601M. Frimmer, J. Gieseler, and L. Novotny, \\\"Cooling mechanical oscillators by coherent control,\\\" Phys. Rev. Lett. 117, 163601 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Thermal properties of levitated nanoparticles. E Hebestreit, ETH ZurichPh.D. thesisE. Hebestreit, \\\"Thermal properties of levitated nanoparticles,\\\" Ph.D. thesis, ETH Zurich (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Levitodynamics toward force nano-sensors in vacuum. F Ricci, Universitat Politècnica de CatalunyaPh.D. thesisF. Ricci, \\\"Levitodynamics toward force nano-sensors in vacuum,\\\" Ph.D. thesis, Universitat Politècnica de Catalunya (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scanning confocal optical microscopy and magnetic resonance on single defect centers. A Gruber, A Dräbenstedt, C Tietz, L Fleury, J Wrachtrup, C Von Borczyskowski, Science. 276A. Gruber, A. Dräbenstedt, C. Tietz, L. Fleury, J. Wrachtrup, and C. Von Borczyskowski, \\\"Scanning confocal optical microscopy and magnetic resonance on single defect centers,\\\" Science 276, 2012-2014 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamics of optically levitated nanoparticles in high vacuum. J Gieseler, Universitat Politècnica de CatalunyaPh.D. thesisJ. Gieseler, \\\"Dynamics of optically levitated nanoparticles in high vacuum,\\\" Ph.D. thesis, Universitat Politècnica de Catalunya (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Trapping in a material world. S E S Spesyvtseva, K Dholakia, ACS Photon. 3S. E. S. Spesyvtseva and K. Dholakia, \\\"Trapping in a material world,\\\" ACS Photon. 3, 719-736 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cancellation of non-conservative scattering forces in optical traps by counter-propagating beams. S Divitt, L Rondin, L Novotny, Opt. Lett. 40S. Divitt, L. Rondin, and L. Novotny, \\\"Cancellation of non-conservative scattering forces in optical traps by counter-propagating beams,\\\" Opt. Lett. 40, 1900-1903 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamics of noise-induced heating in atom traps. M E Gehm, K M O'hara, T A Savard, J E Thomas, Phys. Rev. A. 58M. E. Gehm, K. M. O'hara, T. A. Savard, and J. E. Thomas, \\\"Dynamics of noise-induced heating in atom traps,\\\" Phys. Rev. A 58, 3914-3921 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Theory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach. C Gonzalez-Ballestero, P Maurer, D Windey, L Novotny, R Reimann, O Romero-Isart, arXiv:1902.01282arXiv preprintC. Gonzalez-Ballestero, P. Maurer, D. Windey, L. Novotny, R. Reimann, and O. Romero-Isart, \\\"Theory for cavity cooling of levitated nanoparticles via coherent scattering: Master equation approach,\\\" arXiv preprint arXiv:1902.01282 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Levitated optomechanics at the photon recoil limit. V Jain, ETH ZurichPh.D. thesisV. Jain, \\\"Levitated optomechanics at the photon recoil limit,\\\" Ph.D. thesis, ETH Zurich (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss. S Kheifets, A Simha, K Melin, T Li, M G Raizen, Science. 343S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, \\\"Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,\\\" Science 343, 1493-1496 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Three-dimensional position detection of optically trapped dielectric particles. A Rohrbach, E H K Stelzer, J. Appl. Phys. 91A. Rohrbach and E. H. K. Stelzer, \\\"Three-dimensional position detection of optically trapped dielectric particles,\\\" J. Appl. Phys. 91, 5474-5488 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Backscattering position detection for photonic force microscopy. G Volpe, G Kozyreff, D Petrov, J. Appl. Phys. 10284701G. Volpe, G. Kozyreff, and D. Petrov, \\\"Backscattering position detection for photonic force microscopy,\\\" J. Appl. Phys. 102, 084701 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimal position detection of a dipolar scatterer in a focused field. F Tebbenjohanns, M Frimmer, L Novotny, Phys. Rev. A. 10043821F. Tebbenjohanns, M. Frimmer, and L. Novotny, \\\"Optimal position detection of a dipolar scatterer in a focused field,\\\" Phys. Rev. A 100, 043821 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transverse spin forces and non-equilibrium particle dynamics in a circularly polarized vacuum optical trap. V Svak, O Brzobohatỳ, M Šiler, P Jákl, J Kaňka, P Zemánek, S H Simpson, Nat. Commun. 95453V. Svak, O. Brzobohatỳ, M. Šiler, P. Jákl, J. Kaňka, P. Zemánek, and S. H. Simpson, \\\"Transverse spin forces and non-equilibrium particle dynamics in a circularly polarized vacuum optical trap,\\\" Nat. Commun. 9, 5453 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps. M D Summers, D R Burnham, D Mcgloin, Opt. Express. 16M. D. Summers, D. R. Burnham, and D. McGloin, \\\"Trapping solid aerosols with optical tweezers: A comparison between gas and liquid phase optical traps,\\\" Opt. Express 16, 7739-7747 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity cooling by coherent scattering of a levitated nanosphere in vacuum. U Delić, ViennaUniversity of ViennaPh.D. thesisU. Delić, \\\"Cavity cooling by coherent scattering of a levitated nanosphere in vacuum,\\\" Ph.D. thesis, University of Vienna, Vienna (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Adhesion and friction forces between spherical micrometer-sized particles. L.-O Heim, J Blum, M Preuss, H.-J Butt, Phys. Rev. Lett. 83L.-O. Heim, J. Blum, M. Preuss, and H.-J. Butt, \\\"Adhesion and friction forces between spherical micrometer-sized particles,\\\" Phys. Rev. Lett. 83, 3328-3331 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"T Li, Fundamental tests of physics with optically trapped microspheres. Springer Science & Business MediaT. Li, Fundamental tests of physics with optically trapped microspheres (Springer Science & Business Media, 2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Contact electrification of individual dielectric microparticles measured by optical tweezers in air. H Park, T W Lebrun, ACS Appl. Mater. Interfaces. 8H. Park and T. W. LeBrun, \\\"Contact electrification of individual dielectric microparticles measured by optical tweezers in air,\\\" ACS Appl. Mater. Interfaces 8, 34904-34913 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sensitive force measurements with optically trapped micro-spheres in high vacuum. D P Atherton, RenoUniversity of NevadaPh.D. thesisD. P. Atherton, \\\"Sensitive force measurements with optically trapped micro-spheres in high vacuum,\\\" Ph.D. thesis, University of Nevada, Reno (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity cooling of free silicon nanoparticles in high vacuum. P Asenbaum, S Kuhn, S Nimmrichter, U Sezer, M Arndt, Nat. Commun. 42743P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, and M. Arndt, \\\"Cavity cooling of free silicon nanoparticles in high vacuum,\\\" Nat. Commun. 4, 2743 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct loading of nanoparticles under high vacuum into a Paul trap for levitodynamical experiments. D S Bykov, P Mestres, L Dania, L Schmöger, T E Northup, arXiv:1905.04204arXiv preprintD. S. Bykov, P. Mestres, L. Dania, L. Schmöger, and T. E. Northup, \\\"Direct loading of nanoparticles under high vacuum into a Paul trap for levitodynamical experiments,\\\" arXiv preprint arXiv:1905.04204 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cooling and manipulation of a levitated nanoparticle with an optical fiber trap. P Mestres, J Berthelot, M Spasenović, J Gieseler, L Novotny, R Quidant, Appl. Phys. Lett. 107151102P. Mestres, J. Berthelot, M. Spasenović, J. Gieseler, L. Novotny, and R. Quidant, \\\"Cooling and manipulation of a levitated nanoparticle with an optical fiber trap,\\\" Appl. Phys. Lett. 107, 151102 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Flying particle sensors in hollow-core photonic crystal fibre. D S Bykov, O A Schmidt, T G Euser, P S J Russell, Nat. Photon. 9D. S. Bykov, O. A. Schmidt, T. G. Euser, and P. S. J. Russell, \\\"Flying particle sensors in hollow-core photonic crystal fibre,\\\" Nat. Photon. 9, 461-466 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical trapping and control of nanoparticles inside evacuated hollow core photonic crystal fibers. D Grass, J Fesel, S G Hofer, N Kiesel, M Aspelmeyer, Appl. Phys. Lett. 108221103D. Grass, J. Fesel, S. G. Hofer, N. Kiesel, and M. Aspelmeyer, \\\"Optical trapping and control of nanoparticles inside evacuated hollow core photonic crystal fibers,\\\" Appl. Phys. Lett. 108, 221103 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Temperature in non-equilibrium states: A review of open problems and current proposals. J Casas-Vázquez, D Jou, Rep. Prog. Phys. 66J. Casas-Vázquez and D. Jou, \\\"Temperature in non-equilibrium states: A review of open problems and current proposals,\\\" Rep. Prog. Phys. 66, 1937-2023 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Realization of nonequilibrium thermodynamic processes using external colored noise. P Mestres, I A Martinez, A Ortiz-Ambriz, R A Rica, E Roldan, Phys. Rev. E. 9032116P. Mestres, I. A. Martinez, A. Ortiz-Ambriz, R. A. Rica, and E. Roldan, \\\"Realization of nonequilibrium thermody- namic processes using external colored noise,\\\" Phys. Rev. E 90, 032116 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Motion of a spherical particle in a rarefied gas. part 2. drag and thermal polarization. S A Beresnev, V G Chernyak, G A Fomyagin, J. Fluid Mech. 219S. A. Beresnev, V. G. Chernyak, and G. A. Fomyagin, \\\"Motion of a spherical particle in a rarefied gas. part 2. drag and thermal polarization,\\\" J. Fluid Mech. 219, 405-421 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gas-induced friction and diffusion of rigid rotors. L Martinetz, K Hornberger, B A Stickler, Phys. Rev. E. 9752112L. Martinetz, K. Hornberger, and B. A. Stickler, \\\"Gas-induced friction and diffusion of rigid rotors,\\\" Phys. Rev. E 97, 052112 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere. J Millen, T Deesuwan, P Barker, J Anders, Nat. Nanotech. 9J. Millen, T. Deesuwan, P. Barker, and J. Anders, \\\"Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,\\\" Nat. Nanotech. 9, 425-429 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the resistance experienced by spheres in their motion through gases. P S Epstein, Phys. Rev. 23P. S. Epstein, \\\"On the resistance experienced by spheres in their motion through gases,\\\" Phys. Rev. 23, 710-733 (1924).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Radiation damping of a polarizable particle. L Novotny, Phys. Rev. A. 9632108L. Novotny, \\\"Radiation damping of a polarizable particle,\\\" Phys. Rev. A 96, 032108 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum optomechanics. M Aspelmeyer, P Meystre, K Schwab, Phys. Today. 65M. Aspelmeyer, P. Meystre, and K. Schwab, \\\"Quantum optomechanics,\\\" Phys. Today 65, 29-35 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Calibration and energy measurement of optically levitated nanoparticle sensors. E Hebestreit, M Frimmer, R Reimann, C Dellago, F Ricci, L Novotny, Rev. Sci. Instrumen. 8933111E. Hebestreit, M. Frimmer, R. Reimann, C. Dellago, F. Ricci, and L. Novotny, \\\"Calibration and energy measurement of optically levitated nanoparticle sensors,\\\" Rev. Sci. Instrumen. 89, 033111 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to quantum noise, measurement, and amplification. A A Clerk, M H Devoret, S M Girvin, F Marquardt, R J Schoelkopf, Rev. Mod. Phys. 82A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, \\\"Introduction to quantum noise, measurement, and amplification,\\\" Rev. Mod. Phys. 82, 1155-1208 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optomechanical Raman-ratio thermometry. T P Purdy, P.-L Yu, N S Kampel, R W Peterson, K Cicak, R W Simmonds, C A , Phys. Rev. A. 9231802T. P. Purdy, P.-L. Yu, N. S. Kampel, R. W. Peterson, K. Cicak, R. W. Simmonds, and C. A. Regal, \\\"Optomechanical Raman-ratio thermometry,\\\" Phys. Rev. A 92, 031802 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation of quantum motion of a nanomechanical resonator. A H Safavi-Naeini, J Chan, J T Hill, T P M Alegre, A Krause, O Painter, Phys. Rev. Lett. 10833602A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter, \\\"Observation of quantum motion of a nanomechanical resonator,\\\" Phys. Rev. Lett. 108, 033602 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Observation and interpretation of motional sideband asymmetry in a quantum electromechanical device. A J Weinstein, C U Lei, E E Wollman, J Suh, A Metelmann, A A Clerk, K C Schwab, Phys. Rev. X. 441003A. J. Weinstein, C. U. Lei, E. E. Wollman, J. Suh, A. Metelmann, A. A. Clerk, and K. C. Schwab, \\\"Observation and interpretation of motional sideband asymmetry in a quantum electromechanical device,\\\" Phys. Rev. X 4, 041003 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Controlling the net charge on a nanoparticle optically levitated in vacuum. M Frimmer, K Luszcz, S Ferreiro, V Jain, E Hebestreit, L Novotny, Phys. Rev. A. 9561801M. Frimmer, K. Luszcz, S. Ferreiro, V. Jain, E. Hebestreit, and L. Novotny, \\\"Controlling the net charge on a nanoparticle optically levitated in vacuum,\\\" Phys. Rev. A 95, 061801 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A nanomechanical mass sensor with yoctogram resolution. J Chaste, A Eichler, J Moser, G Ceballos, R Rurali, A Bachtold, Nat. Nanotech. 7J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, and A. Bachtold, \\\"A nanomechanical mass sensor with yoctogram resolution,\\\" Nat. Nanotech. 7, 301-304 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Measuring the internal temperature of a levitated nanoparticle in high vacuum. E Hebestreit, R Reimann, M Frimmer, L Novotny, Phys. Rev. A. 9743803E. Hebestreit, R. Reimann, M. Frimmer, and L. Novotny, \\\"Measuring the internal temperature of a levitated nanoparticle in high vacuum,\\\" Phys. Rev. A 97, 043803 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cavity cooling of an optically levitated submicron particle. N Kiesel, F Blaser, U Delić, D Grass, R Kaltenbaek, M Aspelmeyer, Proc. Natl. Acad. Sci. 110N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, \\\"Cavity cooling of an optically levitated submicron particle,\\\" Proc. Natl. Acad. Sci. 110, 14180-14185 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Millikelvin cooling of an optically trapped microsphere in vacuum. T Li, S Kheifets, M G Raizen, Nat. Phys. 7T. Li, S. Kheifets, and M. G. Raizen, \\\"Millikelvin cooling of an optically trapped microsphere in vacuum,\\\" Nat. Phys. 7, 527-530 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Electric feedback cooling of single charged nanoparticles in an optical trap. M Iwasaki, T Yotsuya, T Naruki, Y Matsuda, M Yoneda, K Aikawa, Phys. Rev. A. 9951401M. Iwasaki, T. Yotsuya, T. Naruki, Y. Matsuda, M. Yoneda, and K. Aikawa, \\\"Electric feedback cooling of single charged nanoparticles in an optical trap,\\\" Phys. Rev. A 99, 051401 (2019).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Feedback cooling of a cantilever's fundamental mode below 5 mk. M Poggio, C L Degen, H J Mamin, D Rugar, Phys. Rev. Lett. 9917201M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, \\\"Feedback cooling of a cantilever's fundamental mode below 5 mk,\\\" Phys. Rev. Lett. 99, 017201 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spontaneous emission probabilities at radio frequencies. E M Purcell, Confined Electrons and Photons. SpringerE. M. Purcell, \\\"Spontaneous emission probabilities at radio frequencies,\\\" in Confined Electrons and Photons, (Springer, 1995), pp. 839-839.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Yariv, Quantum electronics. New YorkJohn Wiley & SonsA. Yariv, Quantum electronics (John Wiley & Sons, New York, 1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Non-equilibrium steady state of a driven levitated particle with feedback cooling. J Gieseler, L Novotny, C Moritz, C Dellago, New J. Phys. 1745011J. Gieseler, L. Novotny, C. Moritz, and C. Dellago, \\\"Non-equilibrium steady state of a driven levitated particle with feedback cooling,\\\" New J. Phys. 17, 045011 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum model of cooling and force sensing with an optically trapped nanoparticle. B Rodenburg, L P Neukirch, A N Vamivakas, M Bhattacharya, Optica. 3B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, \\\"Quantum model of cooling and force sensing with an optically trapped nanoparticle,\\\" Optica 3, 318-323 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum calculation of feedback cooling a laser levitated nanoparticle in the shot-noise-dominant regime. C Zhong, T Li, F Robicheaux, arXiv:1708.01203arXiv preprintC. Zhong, T. Li, and F. Robicheaux, \\\"Quantum calculation of feedback cooling a laser levitated nanoparticle in the shot-noise-dominant regime,\\\" arXiv preprint arXiv:1708.01203 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Free nano-object Ramsey interferometry for large quantum superpositions. C Wan, M Scala, G W Morley, A T M A Rahman, H Ulbricht, J Bateman, P F Barker, S Bose, M S Kim, Phys. Rev. Lett. 117143003C. Wan, M. Scala, G. W. Morley, A. T. M. A. Rahman, H. Ulbricht, J. Bateman, P. F. Barker, S. Bose, and M. S. Kim, \\\"Free nano-object Ramsey interferometry for large quantum superpositions,\\\" Phys. Rev. Lett. 117, 143003 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical cryocooling of diamond. M Kern, J Jeske, D W M Lau, A D Greentree, F Jelezko, J Twamley, Phys. Rev. B. 95235306M. Kern, J. Jeske, D. W. M. Lau, A. D. Greentree, F. Jelezko, and J. Twamley, \\\"Optical cryocooling of diamond,\\\" Phys. Rev. B 95, 235306 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Diamonds levitating in a Paul trap under vacuum: Measurements of laser-induced heating via NV center thermometry. T Delord, L Nicolas, M Bodini, G Hétet, Appl. Phys. Lett. 11113101T. Delord, L. Nicolas, M. Bodini, and G. Hétet, \\\"Diamonds levitating in a Paul trap under vacuum: Measurements of laser-induced heating via NV center thermometry,\\\" Appl. Phys. Lett. 111, 013101 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Internal quantum dynamics of a nanoparticle in a thermal electromagnetic field: A minimal model. A E R López, C Gonzalez-Ballestero, O Romero-Isart, Phys. Rev. B. 98155405A. E. R. López, C. Gonzalez-Ballestero, and O. Romero-Isart, \\\"Internal quantum dynamics of a nanoparticle in a thermal electromagnetic field: A minimal model,\\\" Phys. Rev. B 98, 155405 (2018).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamics of a levitated microparticle in vacuum trapped by a perfect vortex beam: three-dimensional motion around a complex optical potential. Y Arita, M Chen, E M Wright, K Dholakia, J. Opt. Soc. Am. B. 34Y. Arita, M. Chen, E. M. Wright, and K. Dholakia, \\\"Dynamics of a levitated microparticle in vacuum trapped by a perfect vortex beam: three-dimensional motion around a complex optical potential,\\\" J. Opt. Soc. Am. B 34, C14-C19 (2017).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Roadmap on structured light. H Rubinsztein-Dunlop, A Forbes, M V Berry, M R Dennis, D L Andrews, M Mansuripur, C D , C Alpmann, P Banzer, T Bauer, E Karimi, L Marrucci, M Padgett, M Ritsch-Marte, N M Litchinitser, N P Bigelow, C Rosales-Guzmán, A Belmonte, J P Torres, T W Neely, M Baker, R Gordon, A B Stilgoe, J Romero, A G White, R Fickler, A E Willner, G Xie, B Mcmorran, A M Weiner, J. Opt. 1913001H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. D., C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, \\\"Roadmap on structured light,\\\" J. Opt. 19, 013001 (2016).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Holographic optical trapping of aerosol droplets. D R Burnham, D Mcgloin, Opt. Express. 14D. R. Burnham and D. McGloin, \\\"Holographic optical trapping of aerosol droplets,\\\" Opt. Express 14, 4175-4181 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical levitation and feedback cooling of a nanoparticle at subwavelength distances from a membrane. R Diehl, E Hebestreit, R Reimann, F Tebbenjohanns, M Frimmer, L Novotny, Phys. Rev. A. 9813851R. Diehl, E. Hebestreit, R. Reimann, F. Tebbenjohanns, M. Frimmer, and L. Novotny, \\\"Optical levitation and feedback cooling of a nanoparticle at subwavelength distances from a membrane,\\\" Phys. Rev. A 98, 013851 (2018).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":30,"cells":{"text":{"kind":"string","value":"\nGAFA Geometric And Functional Analysis TOPOLOGICAL PROPERTIES OF POSITIVELY CURVED MANIFOLDS WITH SYMMETRY\n2014\n\nManuel Amann \nLee Kennard \nGAFA Geometric And Functional Analysis TOPOLOGICAL PROPERTIES OF POSITIVELY CURVED MANIFOLDS WITH SYMMETRY\n\nGeom. Funct. Anal\n24201410.1007/s00039-014-0300-9\nManifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry. 1378 M. AMANN AND L. KENNARD GAFA Many topological classification results of varying strengths (diffeomorphism, homeomorphism, homotopy, etc.) have been proven under the assumption that the symmetry rank is sufficiently large. The classification theorems of Grove-Searle (Theorem 2.1) and Wilking (Theorem 2.2) are prototypical examples. For related results and context, we refer the reader to the comprehensive surveys of Grove [GRO09] and Wilking [WI07]. Our first result concerns the Euler characteristic of positively curved manifolds with symmetry. The topological classifications of Grove et al. imply complete calculations of the Euler characteristic under linear symmetry rank assumptions. In [KEN13], the second author proved that χ(M ) > 0 for all positively curved (4k)dimensional manifolds M with symmetry rank at least 2 log 2 (4k)−2. Under a slightly larger logarithmic symmetry assumption, we prove the following upper bound for χ(M ). (See Theorem 4.1 for a stronger, but more involved, statement.) Theorem A. If M n is a closed Riemannian manifold with positive sectional curvature and symmetry rank at least log 4/3 (n), then χ(M ) < 2 3(log 2 n) 2 .We would like to highlight connections to some conjectured upper bounds for the Betti numbers of non-negatively curved manifolds. Since χ(M ) ≤ b i (M ), such bounds immediately imply bounds for the Euler characteristic.Gromov's Betti number estimate provides one such bound. Specifically, Gromov proved that there exists a constant C(n) such that b i (M ) ≤ C(n) for all n-manifolds M that admit a non-negatively curved metric (see[GRO81]). It is conjectured that C(n) can be replaced by 2 n , the sum of the Betti numbers of the n-dimensional torus.The second conjectured upper bound follows from a combination of the Bott-Grove-Halperin conjecture and the Hopf conjecture. Bott-Grove-Halperin conjectured that manifolds admitting non-negative curvature are rationally elliptic (see Grove [GRO02, Section 5]), and Hopf conjectured that even-dimensional manifolds admitting positive curvature have positive Euler characteristic. Given these conjectures, it follows that an oriented, closed, even-dimensional manifold M n that admits positive curvature has Euler characteristic at most 2 n/2 . Since the estimate in Theorem A is asymptotically smaller than both 2 n and 2 n/2 , even a verification of these conjectures would not imply our result.Our second result provides a stronger bound on χ(M ) by assuming an asymptotically larger symmetry assumption than in Theorem A. Specifically, the symmetry assumption is linear in the dimension, however the slope is allowed to be arbitrarily small. Theorem B. Let δ > 0. There exists a constant c = c(δ) such that the following holds: If a closed Riemannian manifold M n has positive sectional curvature and an effective, isometric action by a torus T with dim(T ) ≥ δn, then the fixed-point set M T has at most cn components and χ(M ) < cn log 2 n. GAFA POSITIVE CURVATURE UNDER SYMMETRY 1379\n\nIntroduction\n\nThe question of whether a given smooth manifold admits a Riemannian metric with positive sectional curvature is nearly as old as the subject of Riemannian geometry itself. However, there are not many known examples of manifolds admitting positive sectional curvature. In addition to spheres and projectives spaces, only a short list of families arise (see Ziller [ZIL07] for a survey of examples, and see Dearricott [DEA11], Grove et al. [GVZ11], and Petersen and Wilhelm [PET00] for two new examples in dimension seven).\n\nThe complementary task of proving obstructions to positive curvature has proven to be just as difficult. In fact, for simply connected closed manifolds, every known topological obstruction to positive sectional curvature is already an obstruction to non-negative sectional curvature.\n\nIn the 1990s, Karsten Grove initiated a research program that has been breathing new life into these problems for two decades. The idea is to focus on positively curved metrics that admit large isometry groups. The measure of symmetry we will consider in this paper is the symmetry rank, which is the rank of the isometry group. We recall that the symmetry rank of a Riemannian manifold is at least r if and only if there exists an effective, isometric action a torus of dimension r.\n\nKeywords and phrases: Positive curvature, Symmetry, Euler characteristic, Betti numbers, Hopf conjecture, Elliptic genus, Symmetric spaces Mathematics Subject Classification Primary 53C20, Secondary 57N65 The first author was supported by a grant of the German Research Foundation. The second author was partially supported by National Science Foundation Grant DMS-1045292. See Theorem 5.1 for a more elaborate statement containing the concretely stated constant c(δ). To be specific in just one case, we mention the following consequence of Theorem 5.1: If M 2n is a closed, positively curved Riemannian manifold, and if M admits an effective, isometric action by T r with r ≥ (2n) 8 + 2 log 2 (2n) + 1, then the fixed-point set of the torus action has fewer than 3 2 χ(CP n ) components and χ(M ) < χ(CP n )(1 + log 2 n).\n\nLet us state two corollaries of Theorem A. They relate to Hopf's conjecture that no positively curved metric exists on S 2 ×S 2 . More generally, it is conjectured that no positively curved metric exists on a non-trivial product of simply connected closed manifolds or on a compact symmetric space of rank greater than one.\n\nCorollary C (Stable Hopf conjecture with symmetry). Let M n be a closed, oneconnected Riemannian manifold with even dimension, positive sectional curvature, and symmetry rank at least log 4/3 n.\n• If M = N ×k = N × N × · · · × N , then k < 3(log 2 n) 2 . • If M = N #k = N #N # · · · #N with χ(N ) = 2, then k < 2 3(log 2 n) 2 .\nWe remark on the name of the corollary: In the first conclusion, we may start with a manifold N m and consider positively curved metrics on N ×k with symmetry rank log 4/3 (dim(N ×k )). The conclusion in this case is that k < 3(log 2 (km)) 2 , which again cannot be true for arbitrarily large k.\n\nWe also remark that, in some cases, we can replace N ×k by more general products or by total spaces of iterated fibrations. See Section 7 for related statements.\n\nCorollary D. Assume M 2n is a positively curved Riemannian manifold with the rational homotopy type of a simply connected, compact symmetric space N . If M has symmetry rank at least log 4/3 n + 7, then N has the form Q × S 2n1 × · · · × S 2ns for some 0 ≤ s < 3(log 2 n) 2 and some\nQ ∈ {S 2q , CP q , HP q } ∪ {SO(p + q)/SO(p) × SO(q) | 2 ≤ p ≤ 3}.\nThis improves Theorem A in [KEN00] by bounding the number of spherical factors. The proofs of Corollaries C and D are contained in Section 7.\n\nOur next result concerns the positivity of the Euler characteristic. In particular, we require it to deduce Corollaries C and D.\n\nTheorem E. Let M 2n be a one-connected, closed Riemannian manifold with positive sectional curvature and symmetry rank at least 2 log 2 (2n) + 2. If the second, third, or fourth Betti number of M vanishes, then χ(M ) ≥ 2.\n\nThe proof follows from a collection of previous results. It does however extend Theorem A of [KEN13], at least for manifolds with symmetry rank r ≥ 2 log 2 n + 2. For a discussion of related results, see Section 6.\n\nWe take a moment here to summarize the techniques involved in the proofs, as well as to describe the layout of this article. Perhaps most importantly, Wilking's connectedness lemma and the resulting induction machinery and cohomological periodicity play a large role. We summarize these and other required results in Section 2. Building upon this, we prove three new, convenient sufficient conditions for a manifold to have 4-periodic rational cohomology in Section 3 (see Propositions 3.1, 3.2, and 3.3).\n\nAnother key aspect of the proofs involves proving bounds on the topology of the fixed-point set M T of the torus action. For Theorem A, we accomplish this by adapting ideas from the first named author's Ph.D. thesis [AMA09], where positive quaternionic Kähler manifolds with symmetry are studied. For Theorem B, we analyze the direct sum of isotropy representations over multiple fixed points. The latter idea is a useful observation that does not seem to have been made before. It was inspired by a conversation with Burkhard Wilking (see Proposition 3.4, and the discussion in Section 3).\n\nSections 4 and 5 contain the proofs of Theorems A and B, respectively. We wish to emphasize that the proofs are independent of each other. In particular, among the four propositions we prove in Section 3, only Propositions 3.1 and 3.2 are used for Theorem A, while Theorem 1 requires only Propositions 3.2, 3.3, and 3.4.\n\nFinally, the proof of Theorem E only requires results in Section 2, as well as a few additional arguments. It is contained in Section 6. Corollaries C and D follow from Theorems A, B, and E. The proofs are contained in Section 7.\n\n\nPreliminaries\n\nIn this section, we summarize a number of previous results that will be used in the proofs in this article. We begin with two important results referenced in the introduction. ).\n\nNext, we state a number of results whose importance to the Grove research program is already well established. Theorem 2.3 (Conner [CON57]). If T acts smoothly on a closed manifold M , then\nχ(M T ) = χ(M ). Moreover, (1) b 2i (M T ) ≤ b 2i (M ), and (2) b 2i+1 (M T ) ≤ b 2i+1 (M ).\nThe first statement will allow us to pull some information about M T up to M , while the second and third results will allow us to push information about M or its submanifolds down to M T . [BER66,GS94]). Suppose T is a torus acting by isometries on a closed, positively curved manifold M n . If n is even, then the fixed-point set M T is nonempty, and if n is odd, then a codimension one subtorus has nonempty fixed-point set.\n\n\nTheorem 2.4 (Berger\n\nTheorem 2.5 (Frankel [FRA61]). Let M n be a closed, compact Riemannian manifold with positive sectional curvature. If N n1 1 and N n2 2 are totally geodesic submanifolds of M with n 1 + n 2 ≥ n, then N 1 ∩ N 2 is nonempty.\n\nThe bound on the dimensions of N 1 and N 2 is equivalent to the bound n − k 1 − k 2 ≥ 0, where k i is the codimension of N i . This result was greatly improved by Wilking [WIL03]: Theorem 2.6 (Connectedness lemma). Suppose M n is a closed Riemannian manifold with positive sectional curvature. Given a highly connected inclusion of closed, orientable manifolds, applying Poincaré duality to each manifold produces a certain periodicity in cohomology (again see [WIL03]): This periodicity is especially strong when l = 0. For example, because M is connected, x i generates H ik (M ; Z) for 0 < ik < n. We call this property periodicity (see [KEN13]): Definition 2.8. For an integer k > 0, a ring R, and a connected space M , we say that\n(1) If N n−k is a closed, embedded, totally geodesic submanifold of M , then N → M is (n − 2k + 1)-connected. (2) If N n−k1 1 and N n−k2 2 are closed, embedded, totally geodesic submanifolds of M with k 1 ≤ k 2 , then N 1 ∩ N 2 → N 2 is (n − k 1 − k 2 )-connected. Recall that an inclusion N → M is called h-connected if π i (M, N ) = 0 for i ≤ h.H * (M ; R) is k-periodic if there exists x ∈ H k (M ; R) such that the map H i (M ; R) → H i+k (M ; R) induced by multiplication by x is surjective for 0 ≤ i < n − k and injective for 0 < i ≤ n − k.\nExamples of manifolds with periodic cohomology include spheres, projective spaces, and some products such as S 2 × HP m or N × CaP 2 for any manifold N of dimension less than eight.\n\nIn practice, we conclude periodicity from the connectedness lemma and Theorem 2.7. Specifically, if N 1 and N 2 intersect transversely as in part (2) of the connectedness lemma, then H * (N 2 ; Z) is k 1 -periodic. It follows that the rational cohomology of N 2 is also k 1 -periodic. The following refinement of this rational periodicity is proved in [KEN13]: While these connectedness and periodicity theorems have no symmetry assumptions, their main applications have been to the study of positively curved manifolds with large symmetry. Indeed, fixed point sets of isometries are totally geodesic, and with enough isometries, one can guarantee the existence of totally geodesic submanifolds with small codimension and pairs of submanifolds that intersect transversely. In the next section, we illustrate how this works by considering three situations in which one can use the periodicity theorem to conclude that a manifold has 4-periodic rational cohomology.\n\nAnother consequence of the connectedness and periodicity theorems is the following, which we use in the proofs of Proposition 3.3 and Theorem E. \ndim(T ) ≥ 2 log 2 n + c 2 − 1.\nFor\nall x ∈ M T , there exists H ⊆ T such that N = M H x satisfies (1) H * (N ; Q) is 4-periodic, (2) N ⊆ M is c-connected, (3) dim(N ) ≡ n mod 4, and (4) dim(N ) ≥ c + 4. In particular, if dim(N ) ≥ 8, there exists x ∈ H 4 (M ; Q) such that the maps H i (M ; Q) → H i+4 (M ; Q) induced by multiplication by x are surjective for 0 ≤ i ≤ c − 4 and injective for 0 < i ≤ c − 4.\n\nNew Tools\n\nIn this section, we prove four propositions that provide the essential tools for the proofs of the main theorems. The first three (Propositions 3.1, 3.2, and 3.3) use the connectedness and periodicity theorems. They provide convenient sufficient conditions for a manifold to have 4-periodic rational cohomology. Since such manifolds have Euler characteristic and even Betti numbers bounded above by those of complex projective space, these results provide important information required for the proofs of Theorems A and B.\n\nThe final proposition (Proposition 3.4) builds on a technique of Wilking whereby one applies the theory of error correcting codes to isotropy representations. Inspired by a suggestion of Wilking, we extend this idea to the direct sum of isotropy representations. The result is that we obtain components of fixed-point sets of involutions that contain a large portion of the fixed-point set of the torus action. Proposition 3.4 is crucial to the proof of Theorem B. \nx ∈ M T . If there exists a subgroup Z j 2 ⊆ T with j ≥ log 2 n −1 such that every τ ∈ Z j 2 satisfies cod(M τ x ) ≤ n/4, then H * (M ; Q) is 4-periodic.\nProof. Inductively choose a j algebraically independent involutions τ 1 , . . . , τ j ∈ Z j 2 such that\nk i = cod ((N i−1 ) τi x ⊆ N i−1 ) ≥ cod ((N i−1 ) τ x ⊆ N i−1 ) for all τ ∈ Z i 2 \\ τ 1 , . . . , τ i−1 , where N h = M τ1,...,τh x\nfor all h ≥ 1 and N 0 = M . By maximality of k i , we have both of the estimates\nk i ≥ cod ((N i−1 ) τi+1 x ⊆ N i−1 ) = k i+1 + l and k i ≥ cod ((N i−1 ) τiτi+1 x ⊆ N i−1 ) = k i+1 + (k i − l),\nwhere l is the dimension of the intersection of the (−1)-eigenspaces of the actions of τ i+1 and τ i τ i+1 on the normal space of N i−1 . Geometrically, l is the codimension of\n(N i−1 ) τi,τi+1 x inside (N i−1 ) τiτi+1 x .\nTogether, these two estimates imply that k i ≥ 2k i+1 for all i ≥ 1. We draw two conclusions from this. First,\nk j ≤ k j−1 /2 ≤ · · · ≤ k 1 /2 j−1 ≤ n/2 j+1 ≤ n/2 log 2 n < 2.\nSince k j is even, we conclude that k j = 0. Second, for all 1 ≤ h ≤ j,\n4k h + (k h−1 + · · · + k 1 ) ≤ n 2 + n 2 h ≤ n, so we can conclude both of the estimates 2k h ≤ dim (N h−1 ) and dim (N h−1 ) − 2 cod (N h ⊆ N h−1 ) + 1 > 1 2 dim (N h−1 ) .\nWe will use the first estimate in a moment. The second estimate implies that the\ninclusion N h → N h−1 is c-connected with c > 1 2 dim N h−1 . As a consequence, if H * (N h ; Q) is 4-periodic for some h, then H * (M ; Q)\nwould be 4-periodic as well. We will use this fact later. The second sufficient condition for rational 4-periodicity is an immediate consequence of the periodicity theorem. If there exists an involution σ ∈ T such that M σ x has dim(M σ x ) ≥ n 2 and is fixed by another involution in T , then M has 4-periodic rational cohomology. In particular, this applies if M σ x is fixed by a two-dimensional torus in T .\nLet 2 ≤ i ≤ j be the the minimal index such that k i = 0. For 1 ≤ h < i, let l h denote the dimension of the intersection of the (−1)-eigenspaces of the action of τ h and τ i on T x (N h−1 ). Geometrically l h is the codimension of (N h−1 ) τh,τi x inside (N h−1 ) τhτi x . By replacing τ i by τ τ i for some τ ∈ τ 1 , . . . , τ i−1 , we may assume that l h ≤ k h /2 for all 1 ≤ h < i. Let 1 ≤ h < i be the maximal index such that l h > 0. Observe that some l h > 0 since the subgroup Z j 2 acts effectively on M . It follows that N h is the transverse intersection of (N h−1 ) τi x and (N h−1 ) τiτh inside N h−1 . Since 2 cod ((N h−1 ) τi x ⊆ N h−1 ) + 2 cod ((N h−1 ) τiτh x ⊆ N h−1 ) is equal to 2l h + 2(k h − l h ) = 2k h ≤ dim (N h−1 ), we conclude that H * (N h−1 ; Q)\n\nProof. Choose a non-trivial involution\nτ ∈ T not equal to σ. Since M σ x is fixed by τ , it is contained in M τ x . Moreover, both inclusions M σ x ⊆ M τ x ⊆ M are strict since the action of T is effective. Similarly, M σ x ⊆ M στ x ⊆ M with both inclusions strict. From the isotropy repre- sentation of Z 2 2 = σ, τ ⊆ T at the fixed point x, it follows that M σ x is the transverse intersection of M τ x and M στ x . Since 2 cod(M τ x ) + 2 cod(M στ x ) = 2 cod(M σ x ) ≤ n,\nthe periodicity theorem implies that H * (M ; Q) is 4-periodic.\n\nThe third sufficient condition for rational 4-periodicity is the following. Its proof is a simple combination of the connectedness lemma and Theorems 2.7 and 2.10. Proof. By 2.7, the existence of a codimension k totally geodesic submanifold implies that there exists y ∈ H k (M ; Q) such that the maps H i (M ; Q) → H i+k (M ; Q) given by multiplication by y are injective for k − 1 < i ≤ n − 2k + 1 and surjective for k − 1 ≤ i < n − 2k + 1. Next by 2.10 with c = k +4, the symmetry rank assumption implies the existence of\nx ∈ H 4 (M ; Q) such that the maps H i (M ; Q) → H i+4 (M ; Q) are surjective for 0 ≤ i ≤ k and injective for 0 < i ≤ k. In particular, y = ax k−m 4\nfor some 0 ≤ m < 4 and a ∈ H m (M ; Q). We claim that x induces periodicity in H * (M ; Q).\n\nWe first show that it suffices to show injectivity of the multiplication maps\nH i (M ; Q) → H i+4 (M ; Q) for 0 < i ≤ n − 4.\nIndeed, Poincaré duality would imply that the Betti numbers satisfy\nb i ≤ b i+4 = b n−i−4 ≤ b n−i = b i for all 0 < i < n − 4, hence the injective maps H i (M ; Q) → H i+4 (M ; Q) must be isomorphisms for dimensional reasons. Since the map H 0 (M ; Q) → H 4 (M ; Q)\nis already known to be surjective by our choice of x, the proof would be complete.\n\nWe now prove that multiplication maps\nH i (M ; Q) → H i+4 (M ; Q) are injective for 0 < i ≤ n − 4. Since k ≤ n+2\n4 , the four cases that follow constitute a proof:\n• If 0 < i ≤ k, injectivity follows by our choice of x. • If k ≤ i ≤ n − 2k + 1, injectivity follows since x is a factor of y and since H i (M ; Q) ·y −→ H i+k (M ; Q) is injective for these values of i. • If 2k ≤ i ≤ n − k − 3,\n\ninjectivity follows from the fact that\nH i−k (M ; Q) ·x −→ H i−k+4 (M ; Q) ·y −→ H i+4 (M ; Q)\nis a composition of injective maps that equals the composition\nH i−k (M ; Q) ·y −→ H i (M ; Q) ·x −→ H i+4 (M ; Q),\nthe first map of which is an isomorphism.\n\n• If n − k − 4 < i ≤ n − 4, injectivity follows from a direct argument. Let z ∈ H i (M ; Q) be a nonzero element. By Poincaré duality, there exists w ∈ H n−i such that zw = 0. The multiplicative property of x implies that w = xw for some w ∈ H n−i−4 . Hence\n0 = zw = z(xw ) = (zx)w ,\nwhich implies xz = 0, completing the proof.\n\nThe final tool we require in the proofs involves an application of the theory of error-correcting codes. This idea was used for the first time in the subject in Wilking [WIL03] to produce fixed-point sets of isometries of torus actions that have small codimension. In the setting of positively curved manifolds, the inclusion maps of these totally geodesic fixed-point sets are highly connected by the connectedness lemma, hence one can construct inductive arguments over dimension as in [WIL03]. In addition, one obtains periodicity results as discussed in Section 2.\n\nWe briefly summarize how this technique works. First, one requires a fixed point x ∈ M T of a torus action on M . By restricting attention to the subgroup of involutions in T , one associates to the isotropy represention a linear error-correcting code. One can then apply algebraic bounds from the theory of error-correcting codes to make conclusions about the Hamming weights of the code. Given this, one translates this algebraic information into geometric data about the fixed-point sets of involutions in the torus.\n\nHere, inspired by a suggestion of Wilking, we consider an extension of this idea. The starting point is to simultaneously consider the isotropy representations at multiple fixed points x i ∈ M T . Specifically, one observes that the direct sum of isotropy representions, when restricted to the subgroup of involutions, yields another errorcorrecting code. Hence one can implement the strategy outlined above. The translation of algebraic information into geometric data is less immediate. However, if M is positively curved, one can apply Frankel's theorem to prove that the chosen components of fixed-point sets of involutions have, in fact, a large number of components of M T contained in them. The specific result we will use is the following. 2 r involutions, together with Frankel's theorem, will be the key step in the proof of Theorem A. The conclusion for general t is really only used to prove Theorem B, but for this it is crucial. The basic idea for the proof of the first conclusion of Theorem B is that, by iterated use of this proposition, one can find involutions ι 1 , . . . , ι h ∈ T r and components N i ⊆ M ιi such that\n• h = log 2 (t + 1) , • cod(N i ) ≤ n−c 2 for all i, and • N 1 ∪ N 2 ∪ · · · ∪ N h contains x i for all 1 ≤ i ≤ t.\nA variation of this idea is used again in the proof of the second conclusion of Theorem B.\n\nProof. Choose a basis for each T xi M so that the isotropy representation T r → SO(T xi M ) factors through U(n/2) ⊆ SO(T xi M ). It follows that the Z r 2 ⊆ T r maps via the isotropy representation to a copy of Z \nZ r 2 → t i=1 Z n/2 2 ∼ = Z tn/2 2 .\nLet π i denote the projection of the direct sum onto the i-th summand of Z n/2 2 . The Hamming weight of the image of ι ∈ Z r 2 under the composition of the isotropy representation and π i is half the codimension k i of M ι xi . Set k = k i . We claim that a nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Assuming this for a moment, we conclude the proof. Relabel the x i so that the codimensions k i = cod(M ι xi ) are increasing in i. It follows that, for all 1 < j ≤ t+1 2 ,\nt 2 (k 1 + k j ) ≤ t i=1 k i ≤ t(n − c) 2 ≤ tn 2 .\nUsing Frankel's theorem, we conclude that the components M ι x1 and M ι xj intersect and hence coincide. Since we also have\ntk 1 ≤ t i=1 k i ≤ t(n − c) 2 ,\nwe conclude the component M ι x1 of M ι satisfies cod(M ι x1 ) ≤ n−c 2 and contains at least t+1\n\nWe proceed by contradiction to show that some nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Suppose therefore that k ≥ t(n−c)+2 2 for all nontrivial ι ∈ Z r 2 . Applying Griesmer's bound, we obtain\ntn 2 ≥ r−1 i=0 t(n − c) + 2 2 i+2 .\nWe estimate the sum on the right-hand side as follows:\ntn 2 ≥ r− tc 2 −1 i=0 tn − tc + 2 2 i+2 + r−1 i=r− tc 2 1 = tn − tc + 2 2 − tn − tc + 2 2 r−tc/2+1 + tc 2 .\nRearranging, we conclude that\nr ≤ tc 2 − 1 + log 2 (tn − tc + 2) .\nBut this contradicts the assumed bound on r, so the result follows.\n\n\nProof of Theorem A\n\nWe first state Theorem 4.1, which is the most general version of Theorem A we can prove. It provides an upper bound on the Euler characteristic of the form χ(M ) ≤ f 0 (n). We then deduce Theorem 1 from Theorem 4.1 by proving the estimate f 0 (n) < 2 3(log 2 n) 2 . The heart of this section is the proof of Theorem A. At the end of the section, we comment on the quality of the approximation provided by Theorem 1 and illustrate the result graphically. In order to state Theorem 4.1, we require two definitions. Set s(n) = log 2 n + log 2 (n + 2) − 2, for n > 0, set f 0 (n) = n 2 + 1 for n ≤ 52, and define f 0 (n) for n ≥ 54 by the recursive formula \nf 0 (n) = 2 s(n) − 1 f 0 2 3n − 4 8 .χ(M ) ≤ b 2i (M T ) ≤ f 0 (n).\n\nGAFA POSITIVE CURVATURE UNDER SYMMETRY 1389\n\nWe briefly provide some explanation for the definition of f 0 (n). We suggest to see its definition as being motivated by two facts, the first providing the 2 s(n) − 1 factor, and the second yielding the argument 2 3n−4 8 ≈ 3 4 n. The latter corresponds to the base of 4/3 of the logarithms we use.\n\nMore specifically, in the proof of Case 2 below, we will prove that M T is contained in the union of submanifolds N j . It follows that b 2i (M T ) is bounded above by j i b 2i (N T j ). Now the N j correspond to unique, non-trivial involutions in a subgroup Z s(n) 2 of T , hence there are at most 2 s(n) − 1 of them. In addition, we show by an inductive argument that b 2i (N T j ) ≤ f 0 (dim N j ) and dim N j < 3 4 n for all N j . Putting this together, we deduce that b 2i (M T ) ≤ f 0 (n) by using the recursive formula for f 0 (n).\n\nFinally note that, below dimension 54, our symmetry assumption is sufficiently large that the classification results of Section 2 imply immediately that the upper bound of the Euler characteristic of M n is provided by χ(CP n/2 ) = n 2 + 1 = f 0 (n). Proof of Theorem A. By Theorem 4.1, it suffices to show that f 0 (n) < 2 3(log 2 n) 2 . This clearly holds for n = 2, so it suffices to prove it for n ≥ 4. In this range, we actually prove the following, stronger estimate:\nf 0 (n) ≤ n 2 + 1 1+log 4/3 ( n 2 +1)\n.\n\nFor n ≤ 52, this is obvious since f 0 (n) = n 2 + 1. Proceeding inductively, for n ≥ 54, we have .\n\nWe proceed to the proof of Theorem 4.1. Recall that M n is a closed, positively curved manifold that admits an effective, isometric action by a torus T with dim(T ) ≥ log 4/3 n.\n\nFirst, if n < 22, the bound on the dimension of T implies that dim(T ) > n 2 . As this contradicts the maximal possible symmetry rank (Theorem 2.1), the theorem is vacuously true in these dimensions. Moreover, the theorem continues to hold for dimensions n ≤ 52 by Wilking's homotopy classification theorem (Theorem 2.2) and the definition of f 0 . We proceed by induction to prove the result for n ≥ 54.\n\nSet s = s(n), and choose any subgroup Z s 2 ⊆ T of involutions. We split the proof of the induction step into two cases. GAFA Case 1 There exists x ∈ M T such that every ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n 2 actually has cod (M ι x ) ≤ n 4 . Case 2 For all x ∈ M T , there exists ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and n 4 < cod (M ι x ) ≤ n 2 . If Case 1 occurs, we prove that M has 4-periodic rational cohomology. By Conner's theorem, we obtain the desired estimate\nb 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1 ≤ f 0 (n).\n\nProof (Proof in Case 1).\n\nFix some x ∈ M T as in the statement of Case 1. Let 0 ≤ j ≤ log 2 n − 1 be maximal such that there exists a Z j 2 ⊆ Z s 2 with the property that every σ ∈ Z j 2 has dim ker T | M σ x ≤ 1 and cod (M σ x ) ≤ n/4. We claim that j = log 2 n − 1. By Proposition 3.1 and the comments above, this would suffice to show that M is rationally 4-periodic.\n\nSuppose\ninstead that j ≤ log 2 n − 2. Choose a Z s−j 2 ⊆ Z s 2 so that Z j 2 ∩ Z s−j 2 = {id}. Observe that s − j ≥ s(n) − ( log 2 n − 2) = log 2 (n + 2) ,\nso Proposition 3.4 applies with t = 1 and c = 0 (alternatively, one can apply [KEN00,\nLemma 1.8]). Choose a nontrivial τ ∈ Z s−j 2 with cod (M τ x ) ≤ n 2 . If dim ker T | M τ x ≥ 2, then Proposition 3.2 implies that H * (M ; Q) is 4-periodic.\nSince this is our claim, we may assume that dim ker T | M τ x ≤ 1 and hence, by the assumption in Case 1, that cod (M τ x ) ≤ n 4 . Fix for a moment any σ ∈ Z j 2 . Observe that M στ\nx ⊇ M σ x ∩ M τ x , so cod (M στ x ) ≤ cod (M σ x ∩ M τ x ) ≤ cod (M σ x ) + cod (M τ x ) ≤ n/2.\nBy Proposition 3.2 again, we can conclude without loss of generality that dim ker T | M στ x ≤ 1. But now the assumption in Case 1 implies that cod (M στ x ) ≤ n/4. Hence every ι in the Z j+1 2 generated by Z j 2 and τ satisfies dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n/4. This contradicts the maximality of j and hence concludes the proof that j ≥ log 2 n − 1.\n\nTo conclude the proof of Theorem 4.1, it suffices to prove that b 2i (M T ) ≤ f 0 (n) in Case 2. In this case, we cannot calculate H * (M ; Q). Instead, we cover M T by at most 2 s(n) −1 submanifolds, each of which satisfies the induction hypothesis. Adding together the estimates for each of these submanifolds and applying the recursive definition of f 0 , we conclude the desired bound.\n\nProof (Proof in Case 2). We use the notation b even (X) = b 2i (X) for the sum of the even Betti numbers of a space X.\n\nSuppose The T -action on M restricts to a T action on each M ιj xj . After dividing by the kernel of the induced action, we obtain an effective action by a torus T j on M ιj xj with dim(T j ) ≥ dim(T ) − 1. Since dim(T ) ≥ log 4/3 n and n > 4 3 n j , we see that dim (T j ) ≥ log 4/3 (n j ). It follows from the induction hypothesis that This finishes the proof of Theorem 4.1 and hence of Theorem A.\n\nRemark 4.2. We remark that our Case 1, Case 2 approach directly yields vanishing theorems for the elliptic genus as proven in [WEI00].\n\nWe conclude this section with a remark on the size of f 0 (n). See Table 1 for some approximate values of n i and κ i . Figure 1. Comparison of the functions f 0 (n) (blue) and κ 1 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) By an argument similar to the one above that shows f 0 (n) < 2 3(log 2 n) 2 , one could similarly show the following, better estimates for f 0 (n): For all i ≥ 1 and n ≥ n i ,\nf 0 (n) ≤ κ i n 2 + 1 1+log 4/3 ( n 2 +1)\n.\n\nWe will not pursue this here. However, we compare the function f 0 with the approximation given in Theorem A against the background of an exponential bound. For this note that it is easily possible to modify the arguments we provide in order to obtain the exponential function we draw in the pictures below as an alternative upper bound. From the pictures already it should be obvious why we did not spend more effort on this.\n\nAlready from the graphs in Figures 1 and 2, one sees that the functions grow strictly subexponentially. As for the asymptotic behavior, the exponential reference function may easily be chosen such that the power of two is a genuine fraction of n.\n\n\nProof of Theorem B\n\nFor integers α ≥ 3 and n > 0, define s α (n) = n 2α + 2 log 2 n 2α + α + 3.\n\nGAFA POSITIVE CURVATURE UNDER SYMMETRY 1393 Figure 2. Comparison of the functions f 0 (n) (blue) and κ 6 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) Theorem B will follow from the following theorem.\n\nTheorem 5.1. Let α ≥ 3 be an integer, and let M n be a closed, simply connected Riemannian manifold with positive sectional curvature. Assume T is a torus acting isometrically and effectively on M with dim(T ) ≥ s α (n). The following hold:\n(1) b 0 (M T ) ≤ a α n + 1, where a α = 3 · 2 α−4 /α. (2) χ(M ) ≤ b 2i (M T ) ≤ n 2 + 1 1 + log bα n 2 + 1 when α ≥ 4, where b α = 2 α−3 2 α−3 −1 .\nWhen α = 3, the upper bound is n 2 + 1.\n\nWe show now that this theorem implies Theorem B.\n\nProof of Theorem B (Proof of Theorem 1). Let δ > 0, and assume M n is a closed, positively curved Riemannian manifold with symmetry rank r > δn.\n\nChoose an integer α ≥ 3 with δ > 1 2α , and choose another integer n 0 such that n ≥ n 0 implies δn > s α (n). Finally, choose a constant c such that all of the following hold:\n\n• For all m ≤ n 0 , every non-negatively curved m-manifold N has b i (N ) ≤ c. • a α n + 1 ≤ cn for all n ≥ n 0 .\n• 2 log bα (2) ≤ c.\nObserve that the first is possible by Gromov's Betti number estimate.\n\nIt now follows from either Gromov's theorem (if n ≤ n 0 ) or Theorem 5.1 (if n > n 0 ) that b 0 (M T ) ≤ cn and χ(M ) < cn log 2 n.\n\nWe spend the rest of this section proving Theorem 5.1. Our task is threefold. First we prove Lemma 5.2, which implies the α = 3 case of Theorem 5.1. Using this as the base case for an induction over α, we prove the two parts of Theorem 5.1 in Theorems 5.5 and 5.6, respectively. Note that the conclusion is the same as in Wilking's cohomology classification (Theorem 2.2) for n ≥ 6000. Using Wilking's proof, we obtain the same conclusion in all dimensions using the slightly larger symmetry rank bound.\n\nProof of Lemma 5.2. We induct over dimension. For n ≤ 3, the theorem is trivial. For 4 ≤ n ≤ 16, dim(T ) ≥ n/2, hence the result follows from the Grove and Searle's diffeomorphism classification (Theorem 2.1). For 17 ≤ n ≤ 120, the symmetry rank is at least dim(T ) ≥ n 4 + 1, so the theorem follows from Wilking's homotopy classification (Theorem 2.2).\n\nSuppose now that n > 120. The assumption on T implies dim(T ) ≥ max n 6 + 1, n 8 + 14 .\n\nBy Proposition 8.1 in [WIL03], if there exists an involution ι ∈ T with cod(M ι ) ≤ 7n 24 , then the conclusion of the lemma holds. It therefore suffices to prove the existence of such an involution.\n\nTo see that such an involution exists, we adapt the proof of Corollary 3.2.a in [WIL03]. Let T denote the torus acting effectively and isometrically on M . By Berger's theorem, there exists T r ⊆ T such that T r has a fixed point x ∈ M , where r = dim(T ) − δ and where δ = 0 if n is even and δ = 1 if n is odd. We consider the isotropy representation where m = n/2 . The original proof concludes that this is the case for r ≥ m 3 + 1 and m ≥ 2700, and in particular for r ≥ n 6 + 1 and n ≥ 6000. In our case, the assumption on r in terms of m is that\nr = dim(T ) − δ ≥ m 3 + 2 log 2 (m) − 5 − δ.\nExcept for n = 126 and n = 131, this estimate holds for all n > 120. For n = 126 and n = 131, one can instead apply the Griesmer bound to find the desired involution.\n\nIn all cases therefore, if n > 120, the involution ι with cod(M ι x ) ≤ 7n/24 exists. This concludes the proof of the lemma.\n\nWith the base case (α = 3) complete, we pause for a moment to prove a couple of lemmas that will facilitate the induction over α. The third estimate of Lemma 5.3 together with Lemma 5.4 already suggest how this will work. Lemma 5.3. For α ≥ 4, s α (n) satisfies the following:\n(1) If n ≤ α(α − 1)/3, then s α (n) ≥ s α−1 (n). (2) If n > α(α − 1)/3, then s α (n) ≥ log 2 (a α n 2 + 2n + 2). (3) If k ≥ n/α, then s α (n) − 1 ≥ s α−1 (n − k). Proof. The estimate s α (n) ≥ s α−1 (n) is equivalent to 1 ≥ n 2α(α − 1) + 2 log 2 α α − 1 .\nThe logarithm term is decreasing in α, so it is at most 2 log 2 4 3 < 5 6 . In particular, s α (n) ≥ s α−1 (n) holds if 1 ≥ n 2α(α−1) + 5 6 , which is to say if n ≤ α(α − 1)/3. For the second estimate, suppose that n > α(α − 1)/3. Observe that α ≥ 4 implies\n2 n 2α +2 > 2 α−1 6 +2 > α, hence 2 sα(n) = 2 n 2α +α+3 n 2α 2 > α · 2 α+1 n 2α 2 = 2 α−1 n 2 α .\nOn the other hand, α ≥ 4 and n > α(α − 1)/3 imply that\n2n + 2 < α(α − 1) 3 n + 1 4 α(α − 1) 3 2 < 5 4 n 2 ≤ 5 8 2 α−1 n 2 α .\nCombining this estimate with the one above for s α (n), we conclude the proof of the second estimate as follows:\n\na α n 2 + 2n + 2 = 3 8\n2 α−1 n 2 α + (2n + 2) < 2 α−1 n 2 α < 2 sα(n) .\nFinally, if k ≥ n/α, then n − k ≤ α−1 α n. Hence we conclude the third estimate as follows: Here, we are using the notation from Section 4 that dim ker T | M ι x denotes the dimension of the kernel of the induced T -action on M ι x . Proof. Choose ι and x as in the assumption of the lemma. If k = cod(M ι x ) is at most n/α, then Proposition 3.3 implies that M has 4-periodic rational cohomology. Indeed, the proposition applies since α ≥ 4 implies that α + 3 − 2 log 2 (2α) ≥ 1 and hence that\ns α−1 (n − k) = n − k 2(α − 1) + 2 log 2 n − k 2(α − 1) + (α − 1) + 3 ≤ n 2α + 2 log 2 n 2α + α + 3 − 1 = s α (n) − 1.dim(T ) ≥ s α (n) ≥ k 2 + 2 log 2 (n) + 1.\nOn the other hand, if k ≤ n/2 and dim ker T | M ι x ≥ 2, then we are in the situation of Proposition 3.2, so we conclude again that H * (M ; Q) is 4-periodic.\n\nWe are ready to prove the first part of Theorem 5.1: Proof. By Lemma 5.2, the theorem follows in the base case of α = 3. We proceed by induction the prove the theorem for α ≥ 4. By Lemma 5.3 and the induction hypothesis, we may we may assume that n > α(α − 1)/3. In particular, by Lemma 5.3 again, we may assume that s α (n) ≥ log 2 (a α n 2 + 2n + 2).\n\nNext observe that if M has 4-periodic rational cohomology, then the even Betti numbers of M are at most one. Hence we can conclude the theorem using Conner's theorem:\nb 0 (M T ) ≤ b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1.\nIn particular, by Lemma 5.4, we may assume that, for all nontrivial involutions ι ∈ T and for all x ∈ M ι , if cod M ι x ≤ n/2, then cod M ι x > n/α and dim ker T | M ι x ≤ 1. We proceed by contradiction. Assume there exist t = a α n + 2 distinct components of M T . The estimate on s α (n) established at the beginning of the proof implies that dim(T ) ≥ s α (n) ≥ log 2 (t n + 2). By Proposition 3.4 (with c = 0), there exists a nontrivial involution ι ∈ T and a component N ⊆ M ι with cod(N ) ≤ n 2 such that N contains at least t +1 2 of the t components of M T . As established above, we may assume that cod(N ) > n/α and dim ker T | M ι x ≤ 1. Let T = T / ker(T | N ), and observe that T acts effectively and isometrically on N . Moreover, the components of N T are the components of M T that are contained in N . In particular, the number of components of N T is at least t +1 2 . On the other hand, since\ndim(T ) ≥ dim(T ) − 1 ≥ s α (n) − 1 ≥ s α−1 (dim N )\nby Lemma 5.3, the induction hypothesis implies that the number of components of N T is at most a α−1 dim(N ) + 1. Putting these estimates together, we conclude that\nt + 1 2 ≤ a α−1 dim(N ) + 1 ≤ 3 · 2 α−5 α − 1 α − 1 α n + 1,\nand hence that t ≤ a α n + 1. This contradicts our definition of t , so the proof is complete.\n\nWith the proof of Theorem 5.5 complete, it suffices to prove the following, which is the the second part of Theorem 5.1. where b α = 2 α−3 /(2 α−3 − 1).\n\nThe first step of the proof is to find submanifolds that have 4-periodic cohomology and that cover multiple components of M T . The second step is to use these submanifolds together with Conner's theorem to conclude the upper bound on b even (M T ). To quantify the first step, we prove the following: Taking h = log bα ((b α − 1)t + 1) , we see that we can cover M T by rationally 4periodic submanifolds N 1 , . . . , N h ⊆ M . Applying this fact together with Conner's theorem, we have b even (M T ) =\nF ⊆M T b even (F ) ≤ h i=1 b even (N T i ) ≤ h i=1 b even (N i ).\nRecall that we have b even (N i ) ≤ n 2 + 1 for all i, hence, by using the first part of the theorem, we can conclude the theorem by estimating h as follows:\n\nh ≤ log bα ((b α − 1)(a α n + 1) + 1) = log bα 1 2 α−3 − 1 3 · 2 α−4 α n + b α = 1 + log bα 3n 2α + 1 < 1 + log bα n 2 + 1 .\n\n\nOn Bounding the Euler Characteristic from Below\n\nIn this section, we prove the following, which contains Theorem E: We remark that the assumption dim(T ) ≥ 2 log 2 (n) − 2 suffices in the first case. Indeed, it follows from [KEN13, Theorem A] that b 2i+1 (M T ) = 0, hence the argument below implies that χ(M ) ≥ 2. We proceed to the proof, which will require the rest of this section.\n\nFirst, the result on equivariant formality follows directly from the properties above it. We note that, if we make the slightly stronger assumption that dim(T ) ≥ log 4/3 (n), then this theorem and Theorem A imply two-sided bounds on the Betti numbers of M .\n\nSecond, we use the following theorem (see [ \n\n\nAs a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h.\n\nTheorem 2. 7\n7Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l.\n\nTheorem 2. 9 (\n9Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i.\n\n\nTheorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that\n\nProposition 3 . 1 .\n31Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let\n\n\nis 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic.\n\nProposition 3. 2 .\n2Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M .\n\nProposition 3 . 3 .\n33Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic.\n\nProposition 3. 4 .\n4Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387\n\n⊆\nSO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism\n\nTheorem 4. 1 .\n1Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then\n\n\nM T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s\n\nTjF.\n≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n).\n\nRemark 4. 3 .•\n3Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )\n\nLemma 5 . 2 .( 1 )\n521Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1.\n\n\nT r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0.\n\nLemma 5. 4 .\n4Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology.\n\nTheorem 5. 5 [\n5Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α.\n\nTheorem 5. 6 [\n6Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )\n\nLemma 5 . 7 (\n57Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) .\n\nTheorem 6. 1 .\n1Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M ).\n\n\nTheorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP2 \n\nGAFA \n\nPOSITIVE CURVATURE UNDER SYMMETRY \n1381 \n\n(2) M has the integral cohomology of HP n/4 , \n(3) n ≡ 2 mod 4 n−2 \n\n4 ; F), \nor \n(4) n ≡ 3 mod 4 n−3 \n\n4 \n\nTable 1\n1Some approximated valuesi = \nn i = \nκ i = \n\n0 \n0 \n1 \n1 \n5 4 \n3 .14823 × 10 −15 \n2 \n7 4 \n1 .45259 × 10 −15 \n3 \n100 \n4.80780 × 10 −16 \n4 \n135 \n3.40869 × 10 −16 \n5 \n183 \n1.10871 × 10 −16 \n6 \n247 \n2.15684 × 10 −17 \n\n\n\n\nBRE72, Corollary IV.2.3, p. 178]):\nof the x i . This concludes the proof given the existence of ι ∈ Z r as above.\nAcknowledgmentsWe are grateful to Burkhard Wilking for a motivating discussion, as well as to Anand Dessai, Nicolas Weisskopf, and Wolfgang Ziller for commenting on earlier versions. We also wish to thank the referees for several suggestions to improve the presentation of the article.(1) every M i contains t i components of M T for some t i ≥ t−12 i + 1, (2) T acts on every M i and dim (T / ker(T | Mi )) ≥ s α−i (n i ), and (3) M j has 4-periodic rational cohomology.Proof ofLemma 5.7. Clearly the chain M = M 0 of length 0 satisfies properties(1)and(2). Let M = M n0 0 ⊇ · · · ⊇ M nj j denote a maximal chain satisfying properties (1) and(2).First note that the maximality of j implies n j > (α − j)(α − j − 1)/3. Indeed if this were not the case, then we could define M j+1 = M j and conclude from Lemma 5.3 that the chain M 0 ⊇ · · · ⊇ M j+1 satisfies properties (1) and(2).Next note that, if j ≥ α − 3, then the subchain M = M 0 ⊇ · · · ⊇ M α−3 has properties (1)-(3) sincewhich implies that M α−3 has 4-periodic rational cohomology by Lemma 5.2.We may assume therefore that j ≤ α − 4. We show in this case that M j has 4-periodic rational cohomology. We will do this by choosing an involution ι ∈ T and a component M j+1 of M ι j , using the maximality of j to conclude that M j+1 does not satisfy properties(1)and(2), then using this information to conclude that M j has 4-periodic rational cohomology.Our first task is to choose the involution ι. Set T = T / ker(T | Mj ). Note that T is a torus acting effectively on M j with dimension at least s α−j (n j ) by property(2). By property (1) together with the first part of Theorem 5.5, we haveIn particular,where the last inequality holds by Lemma 5.3 and our assumption that n j > (α − j)(α − j − 1)/3. This estimate allows us to apply Proposition 3.4 (with t = t j and c = 0). Choose a nontrivial involution ι ∈ T and a component M j+1 of M ι j with cod(M j+1 ⊆ M j ) ≤ nj 2 such that M j+1 covers at least tj+1 2 of the t j components of M T j = M T j . In particular, property (1) and our choice of M j+1 imply that M j+1 covers at leastof the components of M T , hence the chain M = M 0 ⊇ · · · ⊇ M j+1 satisfies property (1). By maximality of j, this chain must fail to satisfy property(2). It follows thatIf cod(M j+1 ⊆ M j ) < dim Mj α−j , then M j has 4-periodic rational cohomology by Proposition 3.3, and if dim(T / ker(T | Mj+1 )) ≤ dim(T / ker(T | Mj )) − 2, then it follows as in the proof of Proposition 3.2 that M j is 4-periodic. This concludes the proof of the existence of a chain M = M 0 ⊇ · · · ⊇ M j satisfying properties (1), (2), and (3).With the first step complete, the second step of the proof of Theorem 5.6 is to cover M T by chains M = M 0 ⊇ · · · ⊇ M j of varying lengths j ≤ α − 3 satisfying properties (1), (2), and (3). We will see that at most 1 + log bα n 2 + 1 chains are required. Since the M j in each such chain has 4-periodic rational cohomology, it satisfiesCombining these facts, Theorem 5.6-and hence Theorem 5.1-follows. We proceed with the details.Proof of Theorem 5.6. First, by the claim, there exists a submanifold N 1 ⊆ M with 4-periodic rational cohomology that covers at least t−1Next, we apply the claim to the remaining u 1 components of M T . This yields a submanifold N 2 ⊆ M with 4-periodic rational cohomology such that N 1 ∪ N 2 covers all but u 2 components of M T , whereContinuing in this way, we claim that we obtain a cover of M T by rationally 4-Indeed, after h steps of this process, there are u h components of M T not yet covered, andIf the right-hand side is <1, then we have covered M T by the submanifolds N 1 , . . ., N h . Moreover, the right-hand side is less than one if and only ifTheorem 6.2. If X is a compact, closed, orientable manifold, then no torus action on X can have exactly one fixed point.Assume for a moment that each of Assumptions(hence it suffices to show that N has vanishing odd Betti numbers. Moreover, by (a) and (b), N is simply connected and has 4-periodic Betti numbers, so it suffices to prove that b 3 (N ) = 0. We prove this in three cases, which together cover Assumptions (1-3). First, assume n ≡ 0 mod 4. By (c), dim(N ) is also divisible by four, hence fourperiodicity and Poincaré duality imply Finally, assume n ≡ 2 mod 4, and assume that the Bott-Grove-Halperin conjecture holds. Note that, by the previous case, we may assume that b 2 (N ) = b 4 (N ) = 1. By (e), N has positive curvature, so the Bott-Grove-Halperin conjecture implies that N is rationally elliptic. In particular, χ(N ) ≥ 0. Using four-periodicity and the condition that dim(N ) ≡ n ≡ 2 mod 4, we can calculate χ(N ) in terms of the b 3 (N ):By (d), this inequality implies b 3 (N ) ≤ 3. But b 3 (N ) must be even by Poincaré duality and the graded commutativity of the cup product, hence b 3 (N ) ≤ 2. Returning to the expression above for χ(N ), we conclude that χ(N ) > 0, and returning to the assumption of rational ellipticity, we conclude that b 3 (N ) = 0, as required.Proofs of Corollaries C and DIn this section we prove Corollary C and comment on some consequences of it. We then state and prove a more elaborate version of Corollary D.Proof of Corollary C. Recall that we have a closed, one-connected manifold M n with positive sectional curvature and symmetry rank log 4/3 n. We assume that M is a kfold Cartesian product or a k-fold connected sum of a manifold N . We claim that k < 3(log 2 n) 2 or k < 2 3(log 2 n) 2 , respectively. We prove the two cases simultaneously.Observe that log 4/3 n ≥ 2 log 2 n, hence b 2 (M ) ≤ 1 by Theorem 2.10. Since b 2 (M ) = kb 2 (N ) in both cases, and since we may assume without loss of generality that k ≥ 2, we conclude that b 2 (M ) = 0. This implies χ(M ) ≥ 2 by Theorem E.Theorem A implies that χ(M ) ≤ 2 3(log 2 n) 2 . Using the multiplicativity and additivity formulas for the Euler characteristic, we obtain eitherin the respective cases.These inequalities imply eitherThe first is a contradiction for k ≥ 3(log 2 n) 2 , while the second implies χ(N ) = 2 for k ≥ 2 3(log 2 n) 2 , another contradiction.GAFA POSITIVE CURVATURE UNDER SYMMETRY 1403Remark 7.1.• Note that if there exists a non-negatively curved metric on M , also the product metric is non-negatively curved. No similar result on the existence of nonnegatively curved metrics on connected sums seems to be known unless it is the sum of two rank one symmetric spaces (see[CHE73]). Thus the result on the Cartesian power, i.e. the first assertion in the stable Hopf conjecture with symmetry (see Corollary C), appears to be stronger than the second one. • As for the assertion on the iterated fibrations, the proof is similar. Since the F i = G i /H i are compact homogeneous spaces of equal rank Lie groups, they have positive Euler characteristic and their Betti numbers are concentrated in even degrees. Hence χ(F i ) ≥ 2 for all i by Poincaré duality, and the arguments in the proof of Corollary C apply equally.Example 7.2. Let us make the result provided by Corollary C more precise in the classical case of a product with factors S 2 . For example, the estimates in Theorem 1 imply that there is no metric of positive curvature on any of (S 2 ) ×124 , (S 2 ) ×125 , . . . , (S 2 ) ×314 compatible with the isometric action of a 20-torus.We now proceed to Corollary D.Proof of Corollary D. We have log 4/3 (2n)+7 ≥ 2 log 2 (2n)+7 for n ≥ 1. By Theorem 3.3 in[KEN00], there exists a product of spheres S = S n1 × · · · × S nt with n i ≥ 16 such that We leave it to the interested reader to vary this result using the factors κ i from Section 4. Besides, a variation of this result using Theorem B is obvious. Also, the presented result is certainly only interesting for large n, since in small dimensions dimension estimates will be better.\n. N = S, N = S,\n\nN = S × R Where R ∈ {cp M, SO(m + 2)/SO(m) × SO. N = S × R where R ∈ {CP m , SO(m + 2)/SO(m) × SO(2)},\n\nN = S × R Where R ∈ {hp M, SO(m + 3)/SO(m) × SO. N = S × R where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)},\n\nN = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 2 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}, or\n\nN = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 3 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}.\n\nThis implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. Note that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. ), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and. in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 sNote that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. In Cases (1), (2), and (3), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and, in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 s .\n\nSince χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors. Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors\n\nPositive Quaternionic Kähler Manifolds. M Amann, WWU MünsterPhD thesisM. Amann. Positive Quaternionic Kähler Manifolds. PhD thesis, WWU Münster, (2009).\n\nTrois remarques sur les variétés riemanniennesà courbure positive. M Berger, C. R. Math. Acad. Sci. 263M. Berger. Trois remarques sur les variétés riemanniennesà courbure positive. C. R. Math. Acad. Sci. Paris, 263 (1966), A76-A78.\n\nIntroduction to Compact Transformation Groups. G Bredon, Academic PressG. Bredon. Introduction to Compact Transformation Groups. Academic Press (1972).\n\nSome examples of manifolds of nonnegative curvature. J Cheeger, J. Differential Geom. 8J. Cheeger. Some examples of manifolds of nonnegative curvature. J. Differential Geom., 8(1973), 623-628.\n\nOn the action of the circle group. P Conner, Michigan Math. J. 4P. Conner. On the action of the circle group. Michigan Math. J., 4 (1957), 241-247.\n\nA 7-manifold with positive curvature. O Dearricott, Duke Math. J. 2O. Dearricott. A 7-manifold with positive curvature. Duke Math. J., (2)158 (2011), 307-346.\n\nManifolds with positive curvature. T Frankel, Pacific J. Math. 11T. Frankel. Manifolds with positive curvature. Pacific J. Math., 11 (1961), 165-174.\n\nCurvature, diameter and Betti numbers. M Gromov, Comment. Math. Helv. 1M. Gromov. Curvature, diameter and Betti numbers. Comment. Math. Helv., (1)56 (1981), 179-195.\n\nGeometry of, and via, symmetries. K Grove, Conformal, Riemannian and Lagrangian geometry. Knoxville, TN; Providence, RI27K. Grove. Geometry of, and via, symmetries. In Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), vol. 27 of Univ. Lecture Ser. Amer. Math. Soc., Providence, RI (2002), pp. 31-53.\n\nGAFA POSITIVE CURVATURE UNDER SYMMETRY 1405. GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405\n\nXIII. Geometry, analysis, and algebraic geometry: Forty years of the. K Grove, Surveys in Differential Geometry. Developments around positive sectional curvatureK. Grove. Developments around positive sectional curvature. Surveys in Dif- ferential Geometry, Vol. XIII. Geometry, analysis, and algebraic geometry: Forty years of the Journal of Differential Geometry (2009), pp. 117-133.\n\nPositively curved manifolds with maximal symmetry rank. K Grove, C Searle, J. Pure Appl. Algebra. 1K. Grove and C. Searle. Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Algebra, (1)91 (1994), 137-142.\n\nAn exotic T 1 S 4 with positive curvature. K Grove, L Verdiani, W Ziller, Geom. Funct. Anal. 3K. Grove, L. Verdiani, and W. Ziller. An exotic T 1 S 4 with positive curva- ture. Geom. Funct. Anal., (3)21 (2011), 499-524.\n\nOn the Hopf conjecture with symmetry. L Kennard, Geom. Topol. 17L. Kennard. On the Hopf conjecture with symmetry. Geom. Topol., 17 (2013), 563-593.\n\nPositively curved Riemannian metrics with logarithmic symmetry rank bounds. L Kennard, arXiv:1209.4627v1Comment. Math. Helv. to appearL. Kennard. Positively curved Riemannian metrics with logarithmic symmetry rank bounds. Comment. Math. Helv., (to appear), arXiv:1209.4627v1.\n\nAn exotic sphere with positive curvature. P Petersen, F Wilhelm, arXiv:0805.0812v3preprintP. Petersen and F. Wilhelm. An exotic sphere with positive curvature. preprint, arXiv:0805.0812v3.\n\nPositive curvature and the elliptic genus. N Weisskopf, arXiv:1305.5175v1preprintN. Weisskopf. Positive curvature and the elliptic genus. preprint, arXiv:1305.5175v1.\n\nTorus actions on manifolds of positive sectional curvature. B Wilking, Acta Math. 2B. Wilking. Torus actions on manifolds of positive sectional curvature. Acta Math., (2)191 (2003), 259-297.\n\nNonnegatively and Positively Curved Manifolds. B Wilking, Surveys in Differential Geometry. Metric and Comparison GeometryB. Wilking. Nonnegatively and Positively Curved Manifolds. Surveys in Differ- ential Geometry, Vol. XI. Metric and Comparison Geometry, (2007) pp. 25-62.\n\nExamples of Riemannian manifolds with non-negative sectional curvature. W Ziller, Surveys in Differential Geometry. W. Ziller. Examples of Riemannian manifolds with non-negative sectional cur- vature. Surveys in Differential Geometry, Vol. XI. Metric and Comparison Geom- etry (2007), pp. 63-102.\n\nInstitut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit. Manuel Amann, Fakultät Für Mathematik, Manuel Amann, Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit.edu URL: http://topology.math.kit.edu/21 54.php\n"},"corpus_id":{"kind":"number","value":56089315,"string":"56,089,315"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1302.6709\",\n \"mag\": 2950395298,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1007/s00039-014-0300-9\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [],\n \"pdf_sha\": \"577e2b91e9514ac56f9d4a1971505db71ea327b3\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1302.6709.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"GAFA Geometric And Functional Analysis TOPOLOGICAL PROPERTIES OF POSITIVELY CURVED MANIFOLDS WITH SYMMETRY"},"authors":{"kind":"list like","value":["Manuel Amann","Lee Kennard","Manuel Amann","Lee Kennard"],"string":"[\n \"Manuel Amann\",\n \"Lee Kennard\",\n \"Manuel Amann\",\n \"Lee Kennard\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry. 1378 M. AMANN AND L. KENNARD GAFA Many topological classification results of varying strengths (diffeomorphism, homeomorphism, homotopy, etc.) have been proven under the assumption that the symmetry rank is sufficiently large. The classification theorems of Grove-Searle (Theorem 2.1) and Wilking (Theorem 2.2) are prototypical examples. For related results and context, we refer the reader to the comprehensive surveys of Grove [GRO09] and Wilking [WI07]. Our first result concerns the Euler characteristic of positively curved manifolds with symmetry. The topological classifications of Grove et al. imply complete calculations of the Euler characteristic under linear symmetry rank assumptions. In [KEN13], the second author proved that χ(M ) > 0 for all positively curved (4k)dimensional manifolds M with symmetry rank at least 2 log 2 (4k)−2. Under a slightly larger logarithmic symmetry assumption, we prove the following upper bound for χ(M ). (See Theorem 4.1 for a stronger, but more involved, statement.) Theorem A. If M n is a closed Riemannian manifold with positive sectional curvature and symmetry rank at least log 4/3 (n), then χ(M ) < 2 3(log 2 n) 2 .We would like to highlight connections to some conjectured upper bounds for the Betti numbers of non-negatively curved manifolds. Since χ(M ) ≤ b i (M ), such bounds immediately imply bounds for the Euler characteristic.Gromov's Betti number estimate provides one such bound. Specifically, Gromov proved that there exists a constant C(n) such that b i (M ) ≤ C(n) for all n-manifolds M that admit a non-negatively curved metric (see[GRO81]). It is conjectured that C(n) can be replaced by 2 n , the sum of the Betti numbers of the n-dimensional torus.The second conjectured upper bound follows from a combination of the Bott-Grove-Halperin conjecture and the Hopf conjecture. Bott-Grove-Halperin conjectured that manifolds admitting non-negative curvature are rationally elliptic (see Grove [GRO02, Section 5]), and Hopf conjectured that even-dimensional manifolds admitting positive curvature have positive Euler characteristic. Given these conjectures, it follows that an oriented, closed, even-dimensional manifold M n that admits positive curvature has Euler characteristic at most 2 n/2 . Since the estimate in Theorem A is asymptotically smaller than both 2 n and 2 n/2 , even a verification of these conjectures would not imply our result.Our second result provides a stronger bound on χ(M ) by assuming an asymptotically larger symmetry assumption than in Theorem A. Specifically, the symmetry assumption is linear in the dimension, however the slope is allowed to be arbitrarily small. Theorem B. Let δ > 0. There exists a constant c = c(δ) such that the following holds: If a closed Riemannian manifold M n has positive sectional curvature and an effective, isometric action by a torus T with dim(T ) ≥ δn, then the fixed-point set M T has at most cn components and χ(M ) < cn log 2 n. GAFA POSITIVE CURVATURE UNDER SYMMETRY 1379"},"sections":{"kind":"list like","value":[{"title":"Introduction","content":""},{"title":"Introduction","content":"The question of whether a given smooth manifold admits a Riemannian metric with positive sectional curvature is nearly as old as the subject of Riemannian geometry itself. However, there are not many known examples of manifolds admitting positive sectional curvature. In addition to spheres and projectives spaces, only a short list of families arise (see Ziller [ZIL07] for a survey of examples, and see Dearricott [DEA11], Grove et al. [GVZ11], and Petersen and Wilhelm [PET00] for two new examples in dimension seven).\nThe question of whether a given smooth manifold admits a Riemannian metric with positive sectional curvature is nearly as old as the subject of Riemannian geometry itself. However, there are not many known examples of manifolds admitting positive sectional curvature. In addition to spheres and projectives spaces, only a short list of families arise (see Ziller [ZIL07] for a survey of examples, and see Dearricott [DEA11], Grove et al. [GVZ11], and Petersen and Wilhelm [PET00] for two new examples in dimension seven).\nThe complementary task of proving obstructions to positive curvature has proven to be just as difficult. In fact, for simply connected closed manifolds, every known topological obstruction to positive sectional curvature is already an obstruction to non-negative sectional curvature.\nThe complementary task of proving obstructions to positive curvature has proven to be just as difficult. In fact, for simply connected closed manifolds, every known topological obstruction to positive sectional curvature is already an obstruction to non-negative sectional curvature.\nIn the 1990s, Karsten Grove initiated a research program that has been breathing new life into these problems for two decades. The idea is to focus on positively curved metrics that admit large isometry groups. The measure of symmetry we will consider in this paper is the symmetry rank, which is the rank of the isometry group. We recall that the symmetry rank of a Riemannian manifold is at least r if and only if there exists an effective, isometric action a torus of dimension r.\nIn the 1990s, Karsten Grove initiated a research program that has been breathing new life into these problems for two decades. The idea is to focus on positively curved metrics that admit large isometry groups. The measure of symmetry we will consider in this paper is the symmetry rank, which is the rank of the isometry group. We recall that the symmetry rank of a Riemannian manifold is at least r if and only if there exists an effective, isometric action a torus of dimension r.\nKeywords and phrases: Positive curvature, Symmetry, Euler characteristic, Betti numbers, Hopf conjecture, Elliptic genus, Symmetric spaces Mathematics Subject Classification Primary 53C20, Secondary 57N65 The first author was supported by a grant of the German Research Foundation. The second author was partially supported by National Science Foundation Grant DMS-1045292. See Theorem 5.1 for a more elaborate statement containing the concretely stated constant c(δ). To be specific in just one case, we mention the following consequence of Theorem 5.1: If M 2n is a closed, positively curved Riemannian manifold, and if M admits an effective, isometric action by T r with r ≥ (2n) 8 + 2 log 2 (2n) + 1, then the fixed-point set of the torus action has fewer than 3 2 χ(CP n ) components and χ(M ) < χ(CP n )(1 + log 2 n).\nKeywords and phrases: Positive curvature, Symmetry, Euler characteristic, Betti numbers, Hopf conjecture, Elliptic genus, Symmetric spaces Mathematics Subject Classification Primary 53C20, Secondary 57N65 The first author was supported by a grant of the German Research Foundation. The second author was partially supported by National Science Foundation Grant DMS-1045292. See Theorem 5.1 for a more elaborate statement containing the concretely stated constant c(δ). To be specific in just one case, we mention the following consequence of Theorem 5.1: If M 2n is a closed, positively curved Riemannian manifold, and if M admits an effective, isometric action by T r with r ≥ (2n) 8 + 2 log 2 (2n) + 1, then the fixed-point set of the torus action has fewer than 3 2 χ(CP n ) components and χ(M ) < χ(CP n )(1 + log 2 n).\nLet us state two corollaries of Theorem A. They relate to Hopf's conjecture that no positively curved metric exists on S 2 ×S 2 . More generally, it is conjectured that no positively curved metric exists on a non-trivial product of simply connected closed manifolds or on a compact symmetric space of rank greater than one.\nLet us state two corollaries of Theorem A. They relate to Hopf's conjecture that no positively curved metric exists on S 2 ×S 2 . More generally, it is conjectured that no positively curved metric exists on a non-trivial product of simply connected closed manifolds or on a compact symmetric space of rank greater than one.\nCorollary C (Stable Hopf conjecture with symmetry). Let M n be a closed, oneconnected Riemannian manifold with even dimension, positive sectional curvature, and symmetry rank at least log 4/3 n.\nCorollary C (Stable Hopf conjecture with symmetry). Let M n be a closed, oneconnected Riemannian manifold with even dimension, positive sectional curvature, and symmetry rank at least log 4/3 n.\nWe remark on the name of the corollary: In the first conclusion, we may start with a manifold N m and consider positively curved metrics on N ×k with symmetry rank log 4/3 (dim(N ×k )). The conclusion in this case is that k < 3(log 2 (km)) 2 , which again cannot be true for arbitrarily large k.\nWe remark on the name of the corollary: In the first conclusion, we may start with a manifold N m and consider positively curved metrics on N ×k with symmetry rank log 4/3 (dim(N ×k )). The conclusion in this case is that k < 3(log 2 (km)) 2 , which again cannot be true for arbitrarily large k.\nWe also remark that, in some cases, we can replace N ×k by more general products or by total spaces of iterated fibrations. See Section 7 for related statements.\nWe also remark that, in some cases, we can replace N ×k by more general products or by total spaces of iterated fibrations. See Section 7 for related statements.\nCorollary D. Assume M 2n is a positively curved Riemannian manifold with the rational homotopy type of a simply connected, compact symmetric space N . If M has symmetry rank at least log 4/3 n + 7, then N has the form Q × S 2n1 × · · · × S 2ns for some 0 ≤ s < 3(log 2 n) 2 and some\nCorollary D. Assume M 2n is a positively curved Riemannian manifold with the rational homotopy type of a simply connected, compact symmetric space N . If M has symmetry rank at least log 4/3 n + 7, then N has the form Q × S 2n1 × · · · × S 2ns for some 0 ≤ s < 3(log 2 n) 2 and some\nThis improves Theorem A in [KEN00] by bounding the number of spherical factors. The proofs of Corollaries C and D are contained in Section 7.\nThis improves Theorem A in [KEN00] by bounding the number of spherical factors. The proofs of Corollaries C and D are contained in Section 7.\nOur next result concerns the positivity of the Euler characteristic. In particular, we require it to deduce Corollaries C and D.\nOur next result concerns the positivity of the Euler characteristic. In particular, we require it to deduce Corollaries C and D.\nTheorem E. Let M 2n be a one-connected, closed Riemannian manifold with positive sectional curvature and symmetry rank at least 2 log 2 (2n) + 2. If the second, third, or fourth Betti number of M vanishes, then χ(M ) ≥ 2.\nTheorem E. Let M 2n be a one-connected, closed Riemannian manifold with positive sectional curvature and symmetry rank at least 2 log 2 (2n) + 2. If the second, third, or fourth Betti number of M vanishes, then χ(M ) ≥ 2.\nThe proof follows from a collection of previous results. It does however extend Theorem A of [KEN13], at least for manifolds with symmetry rank r ≥ 2 log 2 n + 2. For a discussion of related results, see Section 6.\nThe proof follows from a collection of previous results. It does however extend Theorem A of [KEN13], at least for manifolds with symmetry rank r ≥ 2 log 2 n + 2. For a discussion of related results, see Section 6.\nWe take a moment here to summarize the techniques involved in the proofs, as well as to describe the layout of this article. Perhaps most importantly, Wilking's connectedness lemma and the resulting induction machinery and cohomological periodicity play a large role. We summarize these and other required results in Section 2. Building upon this, we prove three new, convenient sufficient conditions for a manifold to have 4-periodic rational cohomology in Section 3 (see Propositions 3.1, 3.2, and 3.3).\nWe take a moment here to summarize the techniques involved in the proofs, as well as to describe the layout of this article. Perhaps most importantly, Wilking's connectedness lemma and the resulting induction machinery and cohomological periodicity play a large role. We summarize these and other required results in Section 2. Building upon this, we prove three new, convenient sufficient conditions for a manifold to have 4-periodic rational cohomology in Section 3 (see Propositions 3.1, 3.2, and 3.3).\nAnother key aspect of the proofs involves proving bounds on the topology of the fixed-point set M T of the torus action. For Theorem A, we accomplish this by adapting ideas from the first named author's Ph.D. thesis [AMA09], where positive quaternionic Kähler manifolds with symmetry are studied. For Theorem B, we analyze the direct sum of isotropy representations over multiple fixed points. The latter idea is a useful observation that does not seem to have been made before. It was inspired by a conversation with Burkhard Wilking (see Proposition 3.4, and the discussion in Section 3).\nAnother key aspect of the proofs involves proving bounds on the topology of the fixed-point set M T of the torus action. For Theorem A, we accomplish this by adapting ideas from the first named author's Ph.D. thesis [AMA09], where positive quaternionic Kähler manifolds with symmetry are studied. For Theorem B, we analyze the direct sum of isotropy representations over multiple fixed points. The latter idea is a useful observation that does not seem to have been made before. It was inspired by a conversation with Burkhard Wilking (see Proposition 3.4, and the discussion in Section 3).\nSections 4 and 5 contain the proofs of Theorems A and B, respectively. We wish to emphasize that the proofs are independent of each other. In particular, among the four propositions we prove in Section 3, only Propositions 3.1 and 3.2 are used for Theorem A, while Theorem 1 requires only Propositions 3.2, 3.3, and 3.4.\nSections 4 and 5 contain the proofs of Theorems A and B, respectively. We wish to emphasize that the proofs are independent of each other. In particular, among the four propositions we prove in Section 3, only Propositions 3.1 and 3.2 are used for Theorem A, while Theorem 1 requires only Propositions 3.2, 3.3, and 3.4.\nFinally, the proof of Theorem E only requires results in Section 2, as well as a few additional arguments. It is contained in Section 6. Corollaries C and D follow from Theorems A, B, and E. The proofs are contained in Section 7.\nFinally, the proof of Theorem E only requires results in Section 2, as well as a few additional arguments. It is contained in Section 6. Corollaries C and D follow from Theorems A, B, and E. The proofs are contained in Section 7."},{"title":"Preliminaries","content":""},{"title":"Preliminaries","content":"In this section, we summarize a number of previous results that will be used in the proofs in this article. We begin with two important results referenced in the introduction. ).\nIn this section, we summarize a number of previous results that will be used in the proofs in this article. We begin with two important results referenced in the introduction. ).\nNext, we state a number of results whose importance to the Grove research program is already well established. Theorem 2.3 (Conner [CON57]). If T acts smoothly on a closed manifold M , then\nNext, we state a number of results whose importance to the Grove research program is already well established. Theorem 2.3 (Conner [CON57]). If T acts smoothly on a closed manifold M , then\nThe first statement will allow us to pull some information about M T up to M , while the second and third results will allow us to push information about M or its submanifolds down to M T . [BER66,GS94]). Suppose T is a torus acting by isometries on a closed, positively curved manifold M n . If n is even, then the fixed-point set M T is nonempty, and if n is odd, then a codimension one subtorus has nonempty fixed-point set.\nThe first statement will allow us to pull some information about M T up to M , while the second and third results will allow us to push information about M or its submanifolds down to M T . [BER66,GS94]). Suppose T is a torus acting by isometries on a closed, positively curved manifold M n . If n is even, then the fixed-point set M T is nonempty, and if n is odd, then a codimension one subtorus has nonempty fixed-point set."},{"title":"Theorem 2.4 (Berger","content":""},{"title":"Theorem 2.4 (Berger","content":"Theorem 2.5 (Frankel [FRA61]). Let M n be a closed, compact Riemannian manifold with positive sectional curvature. If N n1 1 and N n2 2 are totally geodesic submanifolds of M with n 1 + n 2 ≥ n, then N 1 ∩ N 2 is nonempty.\nTheorem 2.5 (Frankel [FRA61]). Let M n be a closed, compact Riemannian manifold with positive sectional curvature. If N n1 1 and N n2 2 are totally geodesic submanifolds of M with n 1 + n 2 ≥ n, then N 1 ∩ N 2 is nonempty.\nThe bound on the dimensions of N 1 and N 2 is equivalent to the bound n − k 1 − k 2 ≥ 0, where k i is the codimension of N i . This result was greatly improved by Wilking [WIL03]: Theorem 2.6 (Connectedness lemma). Suppose M n is a closed Riemannian manifold with positive sectional curvature. Given a highly connected inclusion of closed, orientable manifolds, applying Poincaré duality to each manifold produces a certain periodicity in cohomology (again see [WIL03]): This periodicity is especially strong when l = 0. For example, because M is connected, x i generates H ik (M ; Z) for 0 < ik < n. We call this property periodicity (see [KEN13]): Definition 2.8. For an integer k > 0, a ring R, and a connected space M , we say that\nThe bound on the dimensions of N 1 and N 2 is equivalent to the bound n − k 1 − k 2 ≥ 0, where k i is the codimension of N i . This result was greatly improved by Wilking [WIL03]: Theorem 2.6 (Connectedness lemma). Suppose M n is a closed Riemannian manifold with positive sectional curvature. Given a highly connected inclusion of closed, orientable manifolds, applying Poincaré duality to each manifold produces a certain periodicity in cohomology (again see [WIL03]): This periodicity is especially strong when l = 0. For example, because M is connected, x i generates H ik (M ; Z) for 0 < ik < n. We call this property periodicity (see [KEN13]): Definition 2.8. For an integer k > 0, a ring R, and a connected space M , we say that\nExamples of manifolds with periodic cohomology include spheres, projective spaces, and some products such as S 2 × HP m or N × CaP 2 for any manifold N of dimension less than eight.\nExamples of manifolds with periodic cohomology include spheres, projective spaces, and some products such as S 2 × HP m or N × CaP 2 for any manifold N of dimension less than eight.\nIn practice, we conclude periodicity from the connectedness lemma and Theorem 2.7. Specifically, if N 1 and N 2 intersect transversely as in part (2) of the connectedness lemma, then H * (N 2 ; Z) is k 1 -periodic. It follows that the rational cohomology of N 2 is also k 1 -periodic. The following refinement of this rational periodicity is proved in [KEN13]: While these connectedness and periodicity theorems have no symmetry assumptions, their main applications have been to the study of positively curved manifolds with large symmetry. Indeed, fixed point sets of isometries are totally geodesic, and with enough isometries, one can guarantee the existence of totally geodesic submanifolds with small codimension and pairs of submanifolds that intersect transversely. In the next section, we illustrate how this works by considering three situations in which one can use the periodicity theorem to conclude that a manifold has 4-periodic rational cohomology.\nIn practice, we conclude periodicity from the connectedness lemma and Theorem 2.7. Specifically, if N 1 and N 2 intersect transversely as in part (2) of the connectedness lemma, then H * (N 2 ; Z) is k 1 -periodic. It follows that the rational cohomology of N 2 is also k 1 -periodic. The following refinement of this rational periodicity is proved in [KEN13]: While these connectedness and periodicity theorems have no symmetry assumptions, their main applications have been to the study of positively curved manifolds with large symmetry. Indeed, fixed point sets of isometries are totally geodesic, and with enough isometries, one can guarantee the existence of totally geodesic submanifolds with small codimension and pairs of submanifolds that intersect transversely. In the next section, we illustrate how this works by considering three situations in which one can use the periodicity theorem to conclude that a manifold has 4-periodic rational cohomology.\nAnother consequence of the connectedness and periodicity theorems is the following, which we use in the proofs of Proposition 3.3 and Theorem E.\nAnother consequence of the connectedness and periodicity theorems is the following, which we use in the proofs of Proposition 3.3 and Theorem E.\nFor\nFor"},{"title":"New Tools","content":""},{"title":"New Tools","content":"In this section, we prove four propositions that provide the essential tools for the proofs of the main theorems. The first three (Propositions 3.1, 3.2, and 3.3) use the connectedness and periodicity theorems. They provide convenient sufficient conditions for a manifold to have 4-periodic rational cohomology. Since such manifolds have Euler characteristic and even Betti numbers bounded above by those of complex projective space, these results provide important information required for the proofs of Theorems A and B.\nIn this section, we prove four propositions that provide the essential tools for the proofs of the main theorems. The first three (Propositions 3.1, 3.2, and 3.3) use the connectedness and periodicity theorems. They provide convenient sufficient conditions for a manifold to have 4-periodic rational cohomology. Since such manifolds have Euler characteristic and even Betti numbers bounded above by those of complex projective space, these results provide important information required for the proofs of Theorems A and B.\nThe final proposition (Proposition 3.4) builds on a technique of Wilking whereby one applies the theory of error correcting codes to isotropy representations. Inspired by a suggestion of Wilking, we extend this idea to the direct sum of isotropy representations. The result is that we obtain components of fixed-point sets of involutions that contain a large portion of the fixed-point set of the torus action. Proposition 3.4 is crucial to the proof of Theorem B.\nThe final proposition (Proposition 3.4) builds on a technique of Wilking whereby one applies the theory of error correcting codes to isotropy representations. Inspired by a suggestion of Wilking, we extend this idea to the direct sum of isotropy representations. The result is that we obtain components of fixed-point sets of involutions that contain a large portion of the fixed-point set of the torus action. Proposition 3.4 is crucial to the proof of Theorem B.\nProof. Inductively choose a j algebraically independent involutions τ 1 , . . . , τ j ∈ Z j 2 such that\nProof. Inductively choose a j algebraically independent involutions τ 1 , . . . , τ j ∈ Z j 2 such that\nfor all h ≥ 1 and N 0 = M . By maximality of k i , we have both of the estimates\nfor all h ≥ 1 and N 0 = M . By maximality of k i , we have both of the estimates\nwhere l is the dimension of the intersection of the (−1)-eigenspaces of the actions of τ i+1 and τ i τ i+1 on the normal space of N i−1 . Geometrically, l is the codimension of\nwhere l is the dimension of the intersection of the (−1)-eigenspaces of the actions of τ i+1 and τ i τ i+1 on the normal space of N i−1 . Geometrically, l is the codimension of\nTogether, these two estimates imply that k i ≥ 2k i+1 for all i ≥ 1. We draw two conclusions from this. First,\nTogether, these two estimates imply that k i ≥ 2k i+1 for all i ≥ 1. We draw two conclusions from this. First,\nSince k j is even, we conclude that k j = 0. Second, for all 1 ≤ h ≤ j,\nSince k j is even, we conclude that k j = 0. Second, for all 1 ≤ h ≤ j,\nWe will use the first estimate in a moment. The second estimate implies that the\nWe will use the first estimate in a moment. The second estimate implies that the\nwould be 4-periodic as well. We will use this fact later. The second sufficient condition for rational 4-periodicity is an immediate consequence of the periodicity theorem. If there exists an involution σ ∈ T such that M σ x has dim(M σ x ) ≥ n 2 and is fixed by another involution in T , then M has 4-periodic rational cohomology. In particular, this applies if M σ x is fixed by a two-dimensional torus in T .\nwould be 4-periodic as well. We will use this fact later. The second sufficient condition for rational 4-periodicity is an immediate consequence of the periodicity theorem. If there exists an involution σ ∈ T such that M σ x has dim(M σ x ) ≥ n 2 and is fixed by another involution in T , then M has 4-periodic rational cohomology. In particular, this applies if M σ x is fixed by a two-dimensional torus in T ."},{"title":"Proof. Choose a non-trivial involution","content":""},{"title":"Proof. Choose a non-trivial involution","content":"the periodicity theorem implies that H * (M ; Q) is 4-periodic.\nthe periodicity theorem implies that H * (M ; Q) is 4-periodic.\nThe third sufficient condition for rational 4-periodicity is the following. Its proof is a simple combination of the connectedness lemma and Theorems 2.7 and 2.10. Proof. By 2.7, the existence of a codimension k totally geodesic submanifold implies that there exists y ∈ H k (M ; Q) such that the maps H i (M ; Q) → H i+k (M ; Q) given by multiplication by y are injective for k − 1 < i ≤ n − 2k + 1 and surjective for k − 1 ≤ i < n − 2k + 1. Next by 2.10 with c = k +4, the symmetry rank assumption implies the existence of\nThe third sufficient condition for rational 4-periodicity is the following. Its proof is a simple combination of the connectedness lemma and Theorems 2.7 and 2.10. Proof. By 2.7, the existence of a codimension k totally geodesic submanifold implies that there exists y ∈ H k (M ; Q) such that the maps H i (M ; Q) → H i+k (M ; Q) given by multiplication by y are injective for k − 1 < i ≤ n − 2k + 1 and surjective for k − 1 ≤ i < n − 2k + 1. Next by 2.10 with c = k +4, the symmetry rank assumption implies the existence of\nfor some 0 ≤ m < 4 and a ∈ H m (M ; Q). We claim that x induces periodicity in H * (M ; Q).\nfor some 0 ≤ m < 4 and a ∈ H m (M ; Q). We claim that x induces periodicity in H * (M ; Q).\nWe first show that it suffices to show injectivity of the multiplication maps\nWe first show that it suffices to show injectivity of the multiplication maps\nIndeed, Poincaré duality would imply that the Betti numbers satisfy\nIndeed, Poincaré duality would imply that the Betti numbers satisfy\nis already known to be surjective by our choice of x, the proof would be complete.\nis already known to be surjective by our choice of x, the proof would be complete.\nWe now prove that multiplication maps\nWe now prove that multiplication maps\n4 , the four cases that follow constitute a proof:\n4 , the four cases that follow constitute a proof:"},{"title":"injectivity follows from the fact that","content":""},{"title":"injectivity follows from the fact that","content":"is a composition of injective maps that equals the composition\nis a composition of injective maps that equals the composition\nthe first map of which is an isomorphism.\nthe first map of which is an isomorphism.\n• If n − k − 4 < i ≤ n − 4, injectivity follows from a direct argument. Let z ∈ H i (M ; Q) be a nonzero element. By Poincaré duality, there exists w ∈ H n−i such that zw = 0. The multiplicative property of x implies that w = xw for some w ∈ H n−i−4 . Hence\n• If n − k − 4 < i ≤ n − 4, injectivity follows from a direct argument. Let z ∈ H i (M ; Q) be a nonzero element. By Poincaré duality, there exists w ∈ H n−i such that zw = 0. The multiplicative property of x implies that w = xw for some w ∈ H n−i−4 . Hence\nwhich implies xz = 0, completing the proof.\nwhich implies xz = 0, completing the proof.\nThe final tool we require in the proofs involves an application of the theory of error-correcting codes. This idea was used for the first time in the subject in Wilking [WIL03] to produce fixed-point sets of isometries of torus actions that have small codimension. In the setting of positively curved manifolds, the inclusion maps of these totally geodesic fixed-point sets are highly connected by the connectedness lemma, hence one can construct inductive arguments over dimension as in [WIL03]. In addition, one obtains periodicity results as discussed in Section 2.\nThe final tool we require in the proofs involves an application of the theory of error-correcting codes. This idea was used for the first time in the subject in Wilking [WIL03] to produce fixed-point sets of isometries of torus actions that have small codimension. In the setting of positively curved manifolds, the inclusion maps of these totally geodesic fixed-point sets are highly connected by the connectedness lemma, hence one can construct inductive arguments over dimension as in [WIL03]. In addition, one obtains periodicity results as discussed in Section 2.\nWe briefly summarize how this technique works. First, one requires a fixed point x ∈ M T of a torus action on M . By restricting attention to the subgroup of involutions in T , one associates to the isotropy represention a linear error-correcting code. One can then apply algebraic bounds from the theory of error-correcting codes to make conclusions about the Hamming weights of the code. Given this, one translates this algebraic information into geometric data about the fixed-point sets of involutions in the torus.\nWe briefly summarize how this technique works. First, one requires a fixed point x ∈ M T of a torus action on M . By restricting attention to the subgroup of involutions in T , one associates to the isotropy represention a linear error-correcting code. One can then apply algebraic bounds from the theory of error-correcting codes to make conclusions about the Hamming weights of the code. Given this, one translates this algebraic information into geometric data about the fixed-point sets of involutions in the torus.\nHere, inspired by a suggestion of Wilking, we consider an extension of this idea. The starting point is to simultaneously consider the isotropy representations at multiple fixed points x i ∈ M T . Specifically, one observes that the direct sum of isotropy representions, when restricted to the subgroup of involutions, yields another errorcorrecting code. Hence one can implement the strategy outlined above. The translation of algebraic information into geometric data is less immediate. However, if M is positively curved, one can apply Frankel's theorem to prove that the chosen components of fixed-point sets of involutions have, in fact, a large number of components of M T contained in them. The specific result we will use is the following. 2 r involutions, together with Frankel's theorem, will be the key step in the proof of Theorem A. The conclusion for general t is really only used to prove Theorem B, but for this it is crucial. The basic idea for the proof of the first conclusion of Theorem B is that, by iterated use of this proposition, one can find involutions ι 1 , . . . , ι h ∈ T r and components N i ⊆ M ιi such that\nHere, inspired by a suggestion of Wilking, we consider an extension of this idea. The starting point is to simultaneously consider the isotropy representations at multiple fixed points x i ∈ M T . Specifically, one observes that the direct sum of isotropy representions, when restricted to the subgroup of involutions, yields another errorcorrecting code. Hence one can implement the strategy outlined above. The translation of algebraic information into geometric data is less immediate. However, if M is positively curved, one can apply Frankel's theorem to prove that the chosen components of fixed-point sets of involutions have, in fact, a large number of components of M T contained in them. The specific result we will use is the following. 2 r involutions, together with Frankel's theorem, will be the key step in the proof of Theorem A. The conclusion for general t is really only used to prove Theorem B, but for this it is crucial. The basic idea for the proof of the first conclusion of Theorem B is that, by iterated use of this proposition, one can find involutions ι 1 , . . . , ι h ∈ T r and components N i ⊆ M ιi such that\nA variation of this idea is used again in the proof of the second conclusion of Theorem B.\nA variation of this idea is used again in the proof of the second conclusion of Theorem B.\nProof. Choose a basis for each T xi M so that the isotropy representation T r → SO(T xi M ) factors through U(n/2) ⊆ SO(T xi M ). It follows that the Z r 2 ⊆ T r maps via the isotropy representation to a copy of Z\nProof. Choose a basis for each T xi M so that the isotropy representation T r → SO(T xi M ) factors through U(n/2) ⊆ SO(T xi M ). It follows that the Z r 2 ⊆ T r maps via the isotropy representation to a copy of Z\nLet π i denote the projection of the direct sum onto the i-th summand of Z n/2 2 . The Hamming weight of the image of ι ∈ Z r 2 under the composition of the isotropy representation and π i is half the codimension k i of M ι xi . Set k = k i . We claim that a nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Assuming this for a moment, we conclude the proof. Relabel the x i so that the codimensions k i = cod(M ι xi ) are increasing in i. It follows that, for all 1 < j ≤ t+1 2 ,\nLet π i denote the projection of the direct sum onto the i-th summand of Z n/2 2 . The Hamming weight of the image of ι ∈ Z r 2 under the composition of the isotropy representation and π i is half the codimension k i of M ι xi . Set k = k i . We claim that a nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Assuming this for a moment, we conclude the proof. Relabel the x i so that the codimensions k i = cod(M ι xi ) are increasing in i. It follows that, for all 1 < j ≤ t+1 2 ,\nUsing Frankel's theorem, we conclude that the components M ι x1 and M ι xj intersect and hence coincide. Since we also have\nUsing Frankel's theorem, we conclude that the components M ι x1 and M ι xj intersect and hence coincide. Since we also have\nwe conclude the component M ι x1 of M ι satisfies cod(M ι x1 ) ≤ n−c 2 and contains at least t+1\nwe conclude the component M ι x1 of M ι satisfies cod(M ι x1 ) ≤ n−c 2 and contains at least t+1\nWe proceed by contradiction to show that some nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Suppose therefore that k ≥ t(n−c)+2 2 for all nontrivial ι ∈ Z r 2 . Applying Griesmer's bound, we obtain\nWe proceed by contradiction to show that some nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Suppose therefore that k ≥ t(n−c)+2 2 for all nontrivial ι ∈ Z r 2 . Applying Griesmer's bound, we obtain\nWe estimate the sum on the right-hand side as follows:\nWe estimate the sum on the right-hand side as follows:\nRearranging, we conclude that\nRearranging, we conclude that\nBut this contradicts the assumed bound on r, so the result follows.\nBut this contradicts the assumed bound on r, so the result follows."},{"title":"Proof of Theorem A","content":""},{"title":"Proof of Theorem A","content":"We first state Theorem 4.1, which is the most general version of Theorem A we can prove. It provides an upper bound on the Euler characteristic of the form χ(M ) ≤ f 0 (n). We then deduce Theorem 1 from Theorem 4.1 by proving the estimate f 0 (n) < 2 3(log 2 n) 2 . The heart of this section is the proof of Theorem A. At the end of the section, we comment on the quality of the approximation provided by Theorem 1 and illustrate the result graphically. In order to state Theorem 4.1, we require two definitions. Set s(n) = log 2 n + log 2 (n + 2) − 2, for n > 0, set f 0 (n) = n 2 + 1 for n ≤ 52, and define f 0 (n) for n ≥ 54 by the recursive formula\nWe first state Theorem 4.1, which is the most general version of Theorem A we can prove. It provides an upper bound on the Euler characteristic of the form χ(M ) ≤ f 0 (n). We then deduce Theorem 1 from Theorem 4.1 by proving the estimate f 0 (n) < 2 3(log 2 n) 2 . The heart of this section is the proof of Theorem A. At the end of the section, we comment on the quality of the approximation provided by Theorem 1 and illustrate the result graphically. In order to state Theorem 4.1, we require two definitions. Set s(n) = log 2 n + log 2 (n + 2) − 2, for n > 0, set f 0 (n) = n 2 + 1 for n ≤ 52, and define f 0 (n) for n ≥ 54 by the recursive formula"},{"title":"GAFA POSITIVE CURVATURE UNDER SYMMETRY 1389","content":""},{"title":"GAFA POSITIVE CURVATURE UNDER SYMMETRY 1389","content":"We briefly provide some explanation for the definition of f 0 (n). We suggest to see its definition as being motivated by two facts, the first providing the 2 s(n) − 1 factor, and the second yielding the argument 2 3n−4 8 ≈ 3 4 n. The latter corresponds to the base of 4/3 of the logarithms we use.\nWe briefly provide some explanation for the definition of f 0 (n). We suggest to see its definition as being motivated by two facts, the first providing the 2 s(n) − 1 factor, and the second yielding the argument 2 3n−4 8 ≈ 3 4 n. The latter corresponds to the base of 4/3 of the logarithms we use.\nMore specifically, in the proof of Case 2 below, we will prove that M T is contained in the union of submanifolds N j . It follows that b 2i (M T ) is bounded above by j i b 2i (N T j ). Now the N j correspond to unique, non-trivial involutions in a subgroup Z s(n) 2 of T , hence there are at most 2 s(n) − 1 of them. In addition, we show by an inductive argument that b 2i (N T j ) ≤ f 0 (dim N j ) and dim N j < 3 4 n for all N j . Putting this together, we deduce that b 2i (M T ) ≤ f 0 (n) by using the recursive formula for f 0 (n).\nMore specifically, in the proof of Case 2 below, we will prove that M T is contained in the union of submanifolds N j . It follows that b 2i (M T ) is bounded above by j i b 2i (N T j ). Now the N j correspond to unique, non-trivial involutions in a subgroup Z s(n) 2 of T , hence there are at most 2 s(n) − 1 of them. In addition, we show by an inductive argument that b 2i (N T j ) ≤ f 0 (dim N j ) and dim N j < 3 4 n for all N j . Putting this together, we deduce that b 2i (M T ) ≤ f 0 (n) by using the recursive formula for f 0 (n).\nFinally note that, below dimension 54, our symmetry assumption is sufficiently large that the classification results of Section 2 imply immediately that the upper bound of the Euler characteristic of M n is provided by χ(CP n/2 ) = n 2 + 1 = f 0 (n). Proof of Theorem A. By Theorem 4.1, it suffices to show that f 0 (n) < 2 3(log 2 n) 2 . This clearly holds for n = 2, so it suffices to prove it for n ≥ 4. In this range, we actually prove the following, stronger estimate:\nFinally note that, below dimension 54, our symmetry assumption is sufficiently large that the classification results of Section 2 imply immediately that the upper bound of the Euler characteristic of M n is provided by χ(CP n/2 ) = n 2 + 1 = f 0 (n). Proof of Theorem A. By Theorem 4.1, it suffices to show that f 0 (n) < 2 3(log 2 n) 2 . This clearly holds for n = 2, so it suffices to prove it for n ≥ 4. In this range, we actually prove the following, stronger estimate:\n.\n.\nFor n ≤ 52, this is obvious since f 0 (n) = n 2 + 1. Proceeding inductively, for n ≥ 54, we have .\nFor n ≤ 52, this is obvious since f 0 (n) = n 2 + 1. Proceeding inductively, for n ≥ 54, we have .\nWe proceed to the proof of Theorem 4.1. Recall that M n is a closed, positively curved manifold that admits an effective, isometric action by a torus T with dim(T ) ≥ log 4/3 n.\nWe proceed to the proof of Theorem 4.1. Recall that M n is a closed, positively curved manifold that admits an effective, isometric action by a torus T with dim(T ) ≥ log 4/3 n.\nFirst, if n < 22, the bound on the dimension of T implies that dim(T ) > n 2 . As this contradicts the maximal possible symmetry rank (Theorem 2.1), the theorem is vacuously true in these dimensions. Moreover, the theorem continues to hold for dimensions n ≤ 52 by Wilking's homotopy classification theorem (Theorem 2.2) and the definition of f 0 . We proceed by induction to prove the result for n ≥ 54.\nFirst, if n < 22, the bound on the dimension of T implies that dim(T ) > n 2 . As this contradicts the maximal possible symmetry rank (Theorem 2.1), the theorem is vacuously true in these dimensions. Moreover, the theorem continues to hold for dimensions n ≤ 52 by Wilking's homotopy classification theorem (Theorem 2.2) and the definition of f 0 . We proceed by induction to prove the result for n ≥ 54.\nSet s = s(n), and choose any subgroup Z s 2 ⊆ T of involutions. We split the proof of the induction step into two cases. GAFA Case 1 There exists x ∈ M T such that every ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n 2 actually has cod (M ι x ) ≤ n 4 . Case 2 For all x ∈ M T , there exists ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and n 4 < cod (M ι x ) ≤ n 2 . If Case 1 occurs, we prove that M has 4-periodic rational cohomology. By Conner's theorem, we obtain the desired estimate\nSet s = s(n), and choose any subgroup Z s 2 ⊆ T of involutions. We split the proof of the induction step into two cases. GAFA Case 1 There exists x ∈ M T such that every ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n 2 actually has cod (M ι x ) ≤ n 4 . Case 2 For all x ∈ M T , there exists ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and n 4 < cod (M ι x ) ≤ n 2 . If Case 1 occurs, we prove that M has 4-periodic rational cohomology. By Conner's theorem, we obtain the desired estimate"},{"title":"Proof (Proof in Case 1).","content":""},{"title":"Proof (Proof in Case 1).","content":"Fix some x ∈ M T as in the statement of Case 1. Let 0 ≤ j ≤ log 2 n − 1 be maximal such that there exists a Z j 2 ⊆ Z s 2 with the property that every σ ∈ Z j 2 has dim ker T | M σ x ≤ 1 and cod (M σ x ) ≤ n/4. We claim that j = log 2 n − 1. By Proposition 3.1 and the comments above, this would suffice to show that M is rationally 4-periodic.\nFix some x ∈ M T as in the statement of Case 1. Let 0 ≤ j ≤ log 2 n − 1 be maximal such that there exists a Z j 2 ⊆ Z s 2 with the property that every σ ∈ Z j 2 has dim ker T | M σ x ≤ 1 and cod (M σ x ) ≤ n/4. We claim that j = log 2 n − 1. By Proposition 3.1 and the comments above, this would suffice to show that M is rationally 4-periodic.\nSuppose\nSuppose\nso Proposition 3.4 applies with t = 1 and c = 0 (alternatively, one can apply [KEN00,\nso Proposition 3.4 applies with t = 1 and c = 0 (alternatively, one can apply [KEN00,\nSince this is our claim, we may assume that dim ker T | M τ x ≤ 1 and hence, by the assumption in Case 1, that cod (M τ x ) ≤ n 4 . Fix for a moment any σ ∈ Z j 2 . Observe that M στ\nSince this is our claim, we may assume that dim ker T | M τ x ≤ 1 and hence, by the assumption in Case 1, that cod (M τ x ) ≤ n 4 . Fix for a moment any σ ∈ Z j 2 . Observe that M στ\nBy Proposition 3.2 again, we can conclude without loss of generality that dim ker T | M στ x ≤ 1. But now the assumption in Case 1 implies that cod (M στ x ) ≤ n/4. Hence every ι in the Z j+1 2 generated by Z j 2 and τ satisfies dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n/4. This contradicts the maximality of j and hence concludes the proof that j ≥ log 2 n − 1.\nBy Proposition 3.2 again, we can conclude without loss of generality that dim ker T | M στ x ≤ 1. But now the assumption in Case 1 implies that cod (M στ x ) ≤ n/4. Hence every ι in the Z j+1 2 generated by Z j 2 and τ satisfies dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n/4. This contradicts the maximality of j and hence concludes the proof that j ≥ log 2 n − 1.\nTo conclude the proof of Theorem 4.1, it suffices to prove that b 2i (M T ) ≤ f 0 (n) in Case 2. In this case, we cannot calculate H * (M ; Q). Instead, we cover M T by at most 2 s(n) −1 submanifolds, each of which satisfies the induction hypothesis. Adding together the estimates for each of these submanifolds and applying the recursive definition of f 0 , we conclude the desired bound.\nTo conclude the proof of Theorem 4.1, it suffices to prove that b 2i (M T ) ≤ f 0 (n) in Case 2. In this case, we cannot calculate H * (M ; Q). Instead, we cover M T by at most 2 s(n) −1 submanifolds, each of which satisfies the induction hypothesis. Adding together the estimates for each of these submanifolds and applying the recursive definition of f 0 , we conclude the desired bound.\nProof (Proof in Case 2). We use the notation b even (X) = b 2i (X) for the sum of the even Betti numbers of a space X.\nProof (Proof in Case 2). We use the notation b even (X) = b 2i (X) for the sum of the even Betti numbers of a space X.\nSuppose The T -action on M restricts to a T action on each M ιj xj . After dividing by the kernel of the induced action, we obtain an effective action by a torus T j on M ιj xj with dim(T j ) ≥ dim(T ) − 1. Since dim(T ) ≥ log 4/3 n and n > 4 3 n j , we see that dim (T j ) ≥ log 4/3 (n j ). It follows from the induction hypothesis that This finishes the proof of Theorem 4.1 and hence of Theorem A.\nSuppose The T -action on M restricts to a T action on each M ιj xj . After dividing by the kernel of the induced action, we obtain an effective action by a torus T j on M ιj xj with dim(T j ) ≥ dim(T ) − 1. Since dim(T ) ≥ log 4/3 n and n > 4 3 n j , we see that dim (T j ) ≥ log 4/3 (n j ). It follows from the induction hypothesis that This finishes the proof of Theorem 4.1 and hence of Theorem A.\nRemark 4.2. We remark that our Case 1, Case 2 approach directly yields vanishing theorems for the elliptic genus as proven in [WEI00].\nRemark 4.2. We remark that our Case 1, Case 2 approach directly yields vanishing theorems for the elliptic genus as proven in [WEI00].\nWe conclude this section with a remark on the size of f 0 (n). See Table 1 for some approximate values of n i and κ i . Figure 1. Comparison of the functions f 0 (n) (blue) and κ 1 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) By an argument similar to the one above that shows f 0 (n) < 2 3(log 2 n) 2 , one could similarly show the following, better estimates for f 0 (n): For all i ≥ 1 and n ≥ n i ,\nWe conclude this section with a remark on the size of f 0 (n). See Table 1 for some approximate values of n i and κ i . Figure 1. Comparison of the functions f 0 (n) (blue) and κ 1 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) By an argument similar to the one above that shows f 0 (n) < 2 3(log 2 n) 2 , one could similarly show the following, better estimates for f 0 (n): For all i ≥ 1 and n ≥ n i ,\n.\n.\nWe will not pursue this here. However, we compare the function f 0 with the approximation given in Theorem A against the background of an exponential bound. For this note that it is easily possible to modify the arguments we provide in order to obtain the exponential function we draw in the pictures below as an alternative upper bound. From the pictures already it should be obvious why we did not spend more effort on this.\nWe will not pursue this here. However, we compare the function f 0 with the approximation given in Theorem A against the background of an exponential bound. For this note that it is easily possible to modify the arguments we provide in order to obtain the exponential function we draw in the pictures below as an alternative upper bound. From the pictures already it should be obvious why we did not spend more effort on this.\nAlready from the graphs in Figures 1 and 2, one sees that the functions grow strictly subexponentially. As for the asymptotic behavior, the exponential reference function may easily be chosen such that the power of two is a genuine fraction of n.\nAlready from the graphs in Figures 1 and 2, one sees that the functions grow strictly subexponentially. As for the asymptotic behavior, the exponential reference function may easily be chosen such that the power of two is a genuine fraction of n."},{"title":"Proof of Theorem B","content":""},{"title":"Proof of Theorem B","content":"For integers α ≥ 3 and n > 0, define s α (n) = n 2α + 2 log 2 n 2α + α + 3.\nFor integers α ≥ 3 and n > 0, define s α (n) = n 2α + 2 log 2 n 2α + α + 3.\nGAFA POSITIVE CURVATURE UNDER SYMMETRY 1393 Figure 2. Comparison of the functions f 0 (n) (blue) and κ 6 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) Theorem B will follow from the following theorem.\nGAFA POSITIVE CURVATURE UNDER SYMMETRY 1393 Figure 2. Comparison of the functions f 0 (n) (blue) and κ 6 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) Theorem B will follow from the following theorem.\nTheorem 5.1. Let α ≥ 3 be an integer, and let M n be a closed, simply connected Riemannian manifold with positive sectional curvature. Assume T is a torus acting isometrically and effectively on M with dim(T ) ≥ s α (n). The following hold:\nTheorem 5.1. Let α ≥ 3 be an integer, and let M n be a closed, simply connected Riemannian manifold with positive sectional curvature. Assume T is a torus acting isometrically and effectively on M with dim(T ) ≥ s α (n). The following hold:\nWhen α = 3, the upper bound is n 2 + 1.\nWhen α = 3, the upper bound is n 2 + 1.\nWe show now that this theorem implies Theorem B.\nWe show now that this theorem implies Theorem B.\nProof of Theorem B (Proof of Theorem 1). Let δ > 0, and assume M n is a closed, positively curved Riemannian manifold with symmetry rank r > δn.\nProof of Theorem B (Proof of Theorem 1). Let δ > 0, and assume M n is a closed, positively curved Riemannian manifold with symmetry rank r > δn.\nChoose an integer α ≥ 3 with δ > 1 2α , and choose another integer n 0 such that n ≥ n 0 implies δn > s α (n). Finally, choose a constant c such that all of the following hold:\nChoose an integer α ≥ 3 with δ > 1 2α , and choose another integer n 0 such that n ≥ n 0 implies δn > s α (n). Finally, choose a constant c such that all of the following hold:\n• For all m ≤ n 0 , every non-negatively curved m-manifold N has b i (N ) ≤ c. • a α n + 1 ≤ cn for all n ≥ n 0 .\n• For all m ≤ n 0 , every non-negatively curved m-manifold N has b i (N ) ≤ c. • a α n + 1 ≤ cn for all n ≥ n 0 .\nObserve that the first is possible by Gromov's Betti number estimate.\nObserve that the first is possible by Gromov's Betti number estimate.\nIt now follows from either Gromov's theorem (if n ≤ n 0 ) or Theorem 5.1 (if n > n 0 ) that b 0 (M T ) ≤ cn and χ(M ) < cn log 2 n.\nIt now follows from either Gromov's theorem (if n ≤ n 0 ) or Theorem 5.1 (if n > n 0 ) that b 0 (M T ) ≤ cn and χ(M ) < cn log 2 n.\nWe spend the rest of this section proving Theorem 5.1. Our task is threefold. First we prove Lemma 5.2, which implies the α = 3 case of Theorem 5.1. Using this as the base case for an induction over α, we prove the two parts of Theorem 5.1 in Theorems 5.5 and 5.6, respectively. Note that the conclusion is the same as in Wilking's cohomology classification (Theorem 2.2) for n ≥ 6000. Using Wilking's proof, we obtain the same conclusion in all dimensions using the slightly larger symmetry rank bound.\nWe spend the rest of this section proving Theorem 5.1. Our task is threefold. First we prove Lemma 5.2, which implies the α = 3 case of Theorem 5.1. Using this as the base case for an induction over α, we prove the two parts of Theorem 5.1 in Theorems 5.5 and 5.6, respectively. Note that the conclusion is the same as in Wilking's cohomology classification (Theorem 2.2) for n ≥ 6000. Using Wilking's proof, we obtain the same conclusion in all dimensions using the slightly larger symmetry rank bound.\nProof of Lemma 5.2. We induct over dimension. For n ≤ 3, the theorem is trivial. For 4 ≤ n ≤ 16, dim(T ) ≥ n/2, hence the result follows from the Grove and Searle's diffeomorphism classification (Theorem 2.1). For 17 ≤ n ≤ 120, the symmetry rank is at least dim(T ) ≥ n 4 + 1, so the theorem follows from Wilking's homotopy classification (Theorem 2.2).\nProof of Lemma 5.2. We induct over dimension. For n ≤ 3, the theorem is trivial. For 4 ≤ n ≤ 16, dim(T ) ≥ n/2, hence the result follows from the Grove and Searle's diffeomorphism classification (Theorem 2.1). For 17 ≤ n ≤ 120, the symmetry rank is at least dim(T ) ≥ n 4 + 1, so the theorem follows from Wilking's homotopy classification (Theorem 2.2).\nSuppose now that n > 120. The assumption on T implies dim(T ) ≥ max n 6 + 1, n 8 + 14 .\nSuppose now that n > 120. The assumption on T implies dim(T ) ≥ max n 6 + 1, n 8 + 14 .\nBy Proposition 8.1 in [WIL03], if there exists an involution ι ∈ T with cod(M ι ) ≤ 7n 24 , then the conclusion of the lemma holds. It therefore suffices to prove the existence of such an involution.\nBy Proposition 8.1 in [WIL03], if there exists an involution ι ∈ T with cod(M ι ) ≤ 7n 24 , then the conclusion of the lemma holds. It therefore suffices to prove the existence of such an involution.\nTo see that such an involution exists, we adapt the proof of Corollary 3.2.a in [WIL03]. Let T denote the torus acting effectively and isometrically on M . By Berger's theorem, there exists T r ⊆ T such that T r has a fixed point x ∈ M , where r = dim(T ) − δ and where δ = 0 if n is even and δ = 1 if n is odd. We consider the isotropy representation where m = n/2 . The original proof concludes that this is the case for r ≥ m 3 + 1 and m ≥ 2700, and in particular for r ≥ n 6 + 1 and n ≥ 6000. In our case, the assumption on r in terms of m is that\nTo see that such an involution exists, we adapt the proof of Corollary 3.2.a in [WIL03]. Let T denote the torus acting effectively and isometrically on M . By Berger's theorem, there exists T r ⊆ T such that T r has a fixed point x ∈ M , where r = dim(T ) − δ and where δ = 0 if n is even and δ = 1 if n is odd. We consider the isotropy representation where m = n/2 . The original proof concludes that this is the case for r ≥ m 3 + 1 and m ≥ 2700, and in particular for r ≥ n 6 + 1 and n ≥ 6000. In our case, the assumption on r in terms of m is that\nExcept for n = 126 and n = 131, this estimate holds for all n > 120. For n = 126 and n = 131, one can instead apply the Griesmer bound to find the desired involution.\nExcept for n = 126 and n = 131, this estimate holds for all n > 120. For n = 126 and n = 131, one can instead apply the Griesmer bound to find the desired involution.\nIn all cases therefore, if n > 120, the involution ι with cod(M ι x ) ≤ 7n/24 exists. This concludes the proof of the lemma.\nIn all cases therefore, if n > 120, the involution ι with cod(M ι x ) ≤ 7n/24 exists. This concludes the proof of the lemma.\nWith the base case (α = 3) complete, we pause for a moment to prove a couple of lemmas that will facilitate the induction over α. The third estimate of Lemma 5.3 together with Lemma 5.4 already suggest how this will work. Lemma 5.3. For α ≥ 4, s α (n) satisfies the following:\nWith the base case (α = 3) complete, we pause for a moment to prove a couple of lemmas that will facilitate the induction over α. The third estimate of Lemma 5.3 together with Lemma 5.4 already suggest how this will work. Lemma 5.3. For α ≥ 4, s α (n) satisfies the following:\nThe logarithm term is decreasing in α, so it is at most 2 log 2 4 3 < 5 6 . In particular, s α (n) ≥ s α−1 (n) holds if 1 ≥ n 2α(α−1) + 5 6 , which is to say if n ≤ α(α − 1)/3. For the second estimate, suppose that n > α(α − 1)/3. Observe that α ≥ 4 implies\nThe logarithm term is decreasing in α, so it is at most 2 log 2 4 3 < 5 6 . In particular, s α (n) ≥ s α−1 (n) holds if 1 ≥ n 2α(α−1) + 5 6 , which is to say if n ≤ α(α − 1)/3. For the second estimate, suppose that n > α(α − 1)/3. Observe that α ≥ 4 implies\nOn the other hand, α ≥ 4 and n > α(α − 1)/3 imply that\nOn the other hand, α ≥ 4 and n > α(α − 1)/3 imply that\nCombining this estimate with the one above for s α (n), we conclude the proof of the second estimate as follows:\nCombining this estimate with the one above for s α (n), we conclude the proof of the second estimate as follows:\na α n 2 + 2n + 2 = 3 8\na α n 2 + 2n + 2 = 3 8\nFinally, if k ≥ n/α, then n − k ≤ α−1 α n. Hence we conclude the third estimate as follows: Here, we are using the notation from Section 4 that dim ker T | M ι x denotes the dimension of the kernel of the induced T -action on M ι x . Proof. Choose ι and x as in the assumption of the lemma. If k = cod(M ι x ) is at most n/α, then Proposition 3.3 implies that M has 4-periodic rational cohomology. Indeed, the proposition applies since α ≥ 4 implies that α + 3 − 2 log 2 (2α) ≥ 1 and hence that\nFinally, if k ≥ n/α, then n − k ≤ α−1 α n. Hence we conclude the third estimate as follows: Here, we are using the notation from Section 4 that dim ker T | M ι x denotes the dimension of the kernel of the induced T -action on M ι x . Proof. Choose ι and x as in the assumption of the lemma. If k = cod(M ι x ) is at most n/α, then Proposition 3.3 implies that M has 4-periodic rational cohomology. Indeed, the proposition applies since α ≥ 4 implies that α + 3 − 2 log 2 (2α) ≥ 1 and hence that\nOn the other hand, if k ≤ n/2 and dim ker T | M ι x ≥ 2, then we are in the situation of Proposition 3.2, so we conclude again that H * (M ; Q) is 4-periodic.\nOn the other hand, if k ≤ n/2 and dim ker T | M ι x ≥ 2, then we are in the situation of Proposition 3.2, so we conclude again that H * (M ; Q) is 4-periodic.\nWe are ready to prove the first part of Theorem 5.1: Proof. By Lemma 5.2, the theorem follows in the base case of α = 3. We proceed by induction the prove the theorem for α ≥ 4. By Lemma 5.3 and the induction hypothesis, we may we may assume that n > α(α − 1)/3. In particular, by Lemma 5.3 again, we may assume that s α (n) ≥ log 2 (a α n 2 + 2n + 2).\nWe are ready to prove the first part of Theorem 5.1: Proof. By Lemma 5.2, the theorem follows in the base case of α = 3. We proceed by induction the prove the theorem for α ≥ 4. By Lemma 5.3 and the induction hypothesis, we may we may assume that n > α(α − 1)/3. In particular, by Lemma 5.3 again, we may assume that s α (n) ≥ log 2 (a α n 2 + 2n + 2).\nNext observe that if M has 4-periodic rational cohomology, then the even Betti numbers of M are at most one. Hence we can conclude the theorem using Conner's theorem:\nNext observe that if M has 4-periodic rational cohomology, then the even Betti numbers of M are at most one. Hence we can conclude the theorem using Conner's theorem:\nIn particular, by Lemma 5.4, we may assume that, for all nontrivial involutions ι ∈ T and for all x ∈ M ι , if cod M ι x ≤ n/2, then cod M ι x > n/α and dim ker T | M ι x ≤ 1. We proceed by contradiction. Assume there exist t = a α n + 2 distinct components of M T . The estimate on s α (n) established at the beginning of the proof implies that dim(T ) ≥ s α (n) ≥ log 2 (t n + 2). By Proposition 3.4 (with c = 0), there exists a nontrivial involution ι ∈ T and a component N ⊆ M ι with cod(N ) ≤ n 2 such that N contains at least t +1 2 of the t components of M T . As established above, we may assume that cod(N ) > n/α and dim ker T | M ι x ≤ 1. Let T = T / ker(T | N ), and observe that T acts effectively and isometrically on N . Moreover, the components of N T are the components of M T that are contained in N . In particular, the number of components of N T is at least t +1 2 . On the other hand, since\nIn particular, by Lemma 5.4, we may assume that, for all nontrivial involutions ι ∈ T and for all x ∈ M ι , if cod M ι x ≤ n/2, then cod M ι x > n/α and dim ker T | M ι x ≤ 1. We proceed by contradiction. Assume there exist t = a α n + 2 distinct components of M T . The estimate on s α (n) established at the beginning of the proof implies that dim(T ) ≥ s α (n) ≥ log 2 (t n + 2). By Proposition 3.4 (with c = 0), there exists a nontrivial involution ι ∈ T and a component N ⊆ M ι with cod(N ) ≤ n 2 such that N contains at least t +1 2 of the t components of M T . As established above, we may assume that cod(N ) > n/α and dim ker T | M ι x ≤ 1. Let T = T / ker(T | N ), and observe that T acts effectively and isometrically on N . Moreover, the components of N T are the components of M T that are contained in N . In particular, the number of components of N T is at least t +1 2 . On the other hand, since\nby Lemma 5.3, the induction hypothesis implies that the number of components of N T is at most a α−1 dim(N ) + 1. Putting these estimates together, we conclude that\nby Lemma 5.3, the induction hypothesis implies that the number of components of N T is at most a α−1 dim(N ) + 1. Putting these estimates together, we conclude that\nand hence that t ≤ a α n + 1. This contradicts our definition of t , so the proof is complete.\nand hence that t ≤ a α n + 1. This contradicts our definition of t , so the proof is complete.\nWith the proof of Theorem 5.5 complete, it suffices to prove the following, which is the the second part of Theorem 5.1. where b α = 2 α−3 /(2 α−3 − 1).\nWith the proof of Theorem 5.5 complete, it suffices to prove the following, which is the the second part of Theorem 5.1. where b α = 2 α−3 /(2 α−3 − 1).\nThe first step of the proof is to find submanifolds that have 4-periodic cohomology and that cover multiple components of M T . The second step is to use these submanifolds together with Conner's theorem to conclude the upper bound on b even (M T ). To quantify the first step, we prove the following: Taking h = log bα ((b α − 1)t + 1) , we see that we can cover M T by rationally 4periodic submanifolds N 1 , . . . , N h ⊆ M . Applying this fact together with Conner's theorem, we have b even (M T ) =\nThe first step of the proof is to find submanifolds that have 4-periodic cohomology and that cover multiple components of M T . The second step is to use these submanifolds together with Conner's theorem to conclude the upper bound on b even (M T ). To quantify the first step, we prove the following: Taking h = log bα ((b α − 1)t + 1) , we see that we can cover M T by rationally 4periodic submanifolds N 1 , . . . , N h ⊆ M . Applying this fact together with Conner's theorem, we have b even (M T ) =\nRecall that we have b even (N i ) ≤ n 2 + 1 for all i, hence, by using the first part of the theorem, we can conclude the theorem by estimating h as follows:\nRecall that we have b even (N i ) ≤ n 2 + 1 for all i, hence, by using the first part of the theorem, we can conclude the theorem by estimating h as follows:\nh ≤ log bα ((b α − 1)(a α n + 1) + 1) = log bα 1 2 α−3 − 1 3 · 2 α−4 α n + b α = 1 + log bα 3n 2α + 1 < 1 + log bα n 2 + 1 .\nh ≤ log bα ((b α − 1)(a α n + 1) + 1) = log bα 1 2 α−3 − 1 3 · 2 α−4 α n + b α = 1 + log bα 3n 2α + 1 < 1 + log bα n 2 + 1 ."},{"title":"On Bounding the Euler Characteristic from Below","content":""},{"title":"On Bounding the Euler Characteristic from Below","content":"In this section, we prove the following, which contains Theorem E: We remark that the assumption dim(T ) ≥ 2 log 2 (n) − 2 suffices in the first case. Indeed, it follows from [KEN13, Theorem A] that b 2i+1 (M T ) = 0, hence the argument below implies that χ(M ) ≥ 2. We proceed to the proof, which will require the rest of this section.\nIn this section, we prove the following, which contains Theorem E: We remark that the assumption dim(T ) ≥ 2 log 2 (n) − 2 suffices in the first case. Indeed, it follows from [KEN13, Theorem A] that b 2i+1 (M T ) = 0, hence the argument below implies that χ(M ) ≥ 2. We proceed to the proof, which will require the rest of this section.\nFirst, the result on equivariant formality follows directly from the properties above it. We note that, if we make the slightly stronger assumption that dim(T ) ≥ log 4/3 (n), then this theorem and Theorem A imply two-sided bounds on the Betti numbers of M .\nFirst, the result on equivariant formality follows directly from the properties above it. We note that, if we make the slightly stronger assumption that dim(T ) ≥ log 4/3 (n), then this theorem and Theorem A imply two-sided bounds on the Betti numbers of M .\nSecond, we use the following theorem (see [\nSecond, we use the following theorem (see ["},{"title":"Theorem 2. 7","content":""},{"title":"Theorem 2. 7","content":""},{"title":"Theorem 2. 9 (","content":""},{"title":"Theorem 2. 9 (","content":""},{"title":"Proposition 3 . 1 .","content":""},{"title":"Proposition 3 . 1 .","content":""},{"title":"Proposition 3. 2 .","content":""},{"title":"Proposition 3. 2 .","content":""},{"title":"Proposition 3 . 3 .","content":""},{"title":"Proposition 3 . 3 .","content":""},{"title":"Proposition 3. 4 .","content":""},{"title":"Proposition 3. 4 .","content":""},{"title":"⊆","content":""},{"title":"⊆","content":""},{"title":"Theorem 4. 1 .","content":""},{"title":"Theorem 4. 1 .","content":""},{"title":"TjF.","content":""},{"title":"TjF.","content":""},{"title":"Remark 4. 3 .•","content":""},{"title":"Remark 4. 3 .•","content":""},{"title":"Lemma 5 . 2 .( 1 )","content":""},{"title":"Lemma 5 . 2 .( 1 )","content":""},{"title":"Lemma 5. 4 .","content":""},{"title":"Lemma 5. 4 .","content":""},{"title":"Theorem 5. 5 [","content":""},{"title":"Theorem 5. 5 [","content":""},{"title":"Theorem 5. 6 [","content":""},{"title":"Theorem 5. 6 [","content":""},{"title":"Lemma 5 . 7 (","content":""},{"title":"Lemma 5 . 7 (","content":""},{"title":"Theorem 6. 1 .","content":""},{"title":"Theorem 6. 1 .","content":""},{"title":"Table 1","content":""},{"title":"Table 1","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"\"\n },\n {\n \"title\": \"Introduction\",\n \"content\": \"The question of whether a given smooth manifold admits a Riemannian metric with positive sectional curvature is nearly as old as the subject of Riemannian geometry itself. However, there are not many known examples of manifolds admitting positive sectional curvature. In addition to spheres and projectives spaces, only a short list of families arise (see Ziller [ZIL07] for a survey of examples, and see Dearricott [DEA11], Grove et al. [GVZ11], and Petersen and Wilhelm [PET00] for two new examples in dimension seven).\\nThe question of whether a given smooth manifold admits a Riemannian metric with positive sectional curvature is nearly as old as the subject of Riemannian geometry itself. However, there are not many known examples of manifolds admitting positive sectional curvature. In addition to spheres and projectives spaces, only a short list of families arise (see Ziller [ZIL07] for a survey of examples, and see Dearricott [DEA11], Grove et al. [GVZ11], and Petersen and Wilhelm [PET00] for two new examples in dimension seven).\\nThe complementary task of proving obstructions to positive curvature has proven to be just as difficult. In fact, for simply connected closed manifolds, every known topological obstruction to positive sectional curvature is already an obstruction to non-negative sectional curvature.\\nThe complementary task of proving obstructions to positive curvature has proven to be just as difficult. In fact, for simply connected closed manifolds, every known topological obstruction to positive sectional curvature is already an obstruction to non-negative sectional curvature.\\nIn the 1990s, Karsten Grove initiated a research program that has been breathing new life into these problems for two decades. The idea is to focus on positively curved metrics that admit large isometry groups. The measure of symmetry we will consider in this paper is the symmetry rank, which is the rank of the isometry group. We recall that the symmetry rank of a Riemannian manifold is at least r if and only if there exists an effective, isometric action a torus of dimension r.\\nIn the 1990s, Karsten Grove initiated a research program that has been breathing new life into these problems for two decades. The idea is to focus on positively curved metrics that admit large isometry groups. The measure of symmetry we will consider in this paper is the symmetry rank, which is the rank of the isometry group. We recall that the symmetry rank of a Riemannian manifold is at least r if and only if there exists an effective, isometric action a torus of dimension r.\\nKeywords and phrases: Positive curvature, Symmetry, Euler characteristic, Betti numbers, Hopf conjecture, Elliptic genus, Symmetric spaces Mathematics Subject Classification Primary 53C20, Secondary 57N65 The first author was supported by a grant of the German Research Foundation. The second author was partially supported by National Science Foundation Grant DMS-1045292. See Theorem 5.1 for a more elaborate statement containing the concretely stated constant c(δ). To be specific in just one case, we mention the following consequence of Theorem 5.1: If M 2n is a closed, positively curved Riemannian manifold, and if M admits an effective, isometric action by T r with r ≥ (2n) 8 + 2 log 2 (2n) + 1, then the fixed-point set of the torus action has fewer than 3 2 χ(CP n ) components and χ(M ) < χ(CP n )(1 + log 2 n).\\nKeywords and phrases: Positive curvature, Symmetry, Euler characteristic, Betti numbers, Hopf conjecture, Elliptic genus, Symmetric spaces Mathematics Subject Classification Primary 53C20, Secondary 57N65 The first author was supported by a grant of the German Research Foundation. The second author was partially supported by National Science Foundation Grant DMS-1045292. See Theorem 5.1 for a more elaborate statement containing the concretely stated constant c(δ). To be specific in just one case, we mention the following consequence of Theorem 5.1: If M 2n is a closed, positively curved Riemannian manifold, and if M admits an effective, isometric action by T r with r ≥ (2n) 8 + 2 log 2 (2n) + 1, then the fixed-point set of the torus action has fewer than 3 2 χ(CP n ) components and χ(M ) < χ(CP n )(1 + log 2 n).\\nLet us state two corollaries of Theorem A. They relate to Hopf's conjecture that no positively curved metric exists on S 2 ×S 2 . More generally, it is conjectured that no positively curved metric exists on a non-trivial product of simply connected closed manifolds or on a compact symmetric space of rank greater than one.\\nLet us state two corollaries of Theorem A. They relate to Hopf's conjecture that no positively curved metric exists on S 2 ×S 2 . More generally, it is conjectured that no positively curved metric exists on a non-trivial product of simply connected closed manifolds or on a compact symmetric space of rank greater than one.\\nCorollary C (Stable Hopf conjecture with symmetry). Let M n be a closed, oneconnected Riemannian manifold with even dimension, positive sectional curvature, and symmetry rank at least log 4/3 n.\\nCorollary C (Stable Hopf conjecture with symmetry). Let M n be a closed, oneconnected Riemannian manifold with even dimension, positive sectional curvature, and symmetry rank at least log 4/3 n.\\nWe remark on the name of the corollary: In the first conclusion, we may start with a manifold N m and consider positively curved metrics on N ×k with symmetry rank log 4/3 (dim(N ×k )). The conclusion in this case is that k < 3(log 2 (km)) 2 , which again cannot be true for arbitrarily large k.\\nWe remark on the name of the corollary: In the first conclusion, we may start with a manifold N m and consider positively curved metrics on N ×k with symmetry rank log 4/3 (dim(N ×k )). The conclusion in this case is that k < 3(log 2 (km)) 2 , which again cannot be true for arbitrarily large k.\\nWe also remark that, in some cases, we can replace N ×k by more general products or by total spaces of iterated fibrations. See Section 7 for related statements.\\nWe also remark that, in some cases, we can replace N ×k by more general products or by total spaces of iterated fibrations. See Section 7 for related statements.\\nCorollary D. Assume M 2n is a positively curved Riemannian manifold with the rational homotopy type of a simply connected, compact symmetric space N . If M has symmetry rank at least log 4/3 n + 7, then N has the form Q × S 2n1 × · · · × S 2ns for some 0 ≤ s < 3(log 2 n) 2 and some\\nCorollary D. Assume M 2n is a positively curved Riemannian manifold with the rational homotopy type of a simply connected, compact symmetric space N . If M has symmetry rank at least log 4/3 n + 7, then N has the form Q × S 2n1 × · · · × S 2ns for some 0 ≤ s < 3(log 2 n) 2 and some\\nThis improves Theorem A in [KEN00] by bounding the number of spherical factors. The proofs of Corollaries C and D are contained in Section 7.\\nThis improves Theorem A in [KEN00] by bounding the number of spherical factors. The proofs of Corollaries C and D are contained in Section 7.\\nOur next result concerns the positivity of the Euler characteristic. In particular, we require it to deduce Corollaries C and D.\\nOur next result concerns the positivity of the Euler characteristic. In particular, we require it to deduce Corollaries C and D.\\nTheorem E. Let M 2n be a one-connected, closed Riemannian manifold with positive sectional curvature and symmetry rank at least 2 log 2 (2n) + 2. If the second, third, or fourth Betti number of M vanishes, then χ(M ) ≥ 2.\\nTheorem E. Let M 2n be a one-connected, closed Riemannian manifold with positive sectional curvature and symmetry rank at least 2 log 2 (2n) + 2. If the second, third, or fourth Betti number of M vanishes, then χ(M ) ≥ 2.\\nThe proof follows from a collection of previous results. It does however extend Theorem A of [KEN13], at least for manifolds with symmetry rank r ≥ 2 log 2 n + 2. For a discussion of related results, see Section 6.\\nThe proof follows from a collection of previous results. It does however extend Theorem A of [KEN13], at least for manifolds with symmetry rank r ≥ 2 log 2 n + 2. For a discussion of related results, see Section 6.\\nWe take a moment here to summarize the techniques involved in the proofs, as well as to describe the layout of this article. Perhaps most importantly, Wilking's connectedness lemma and the resulting induction machinery and cohomological periodicity play a large role. We summarize these and other required results in Section 2. Building upon this, we prove three new, convenient sufficient conditions for a manifold to have 4-periodic rational cohomology in Section 3 (see Propositions 3.1, 3.2, and 3.3).\\nWe take a moment here to summarize the techniques involved in the proofs, as well as to describe the layout of this article. Perhaps most importantly, Wilking's connectedness lemma and the resulting induction machinery and cohomological periodicity play a large role. We summarize these and other required results in Section 2. Building upon this, we prove three new, convenient sufficient conditions for a manifold to have 4-periodic rational cohomology in Section 3 (see Propositions 3.1, 3.2, and 3.3).\\nAnother key aspect of the proofs involves proving bounds on the topology of the fixed-point set M T of the torus action. For Theorem A, we accomplish this by adapting ideas from the first named author's Ph.D. thesis [AMA09], where positive quaternionic Kähler manifolds with symmetry are studied. For Theorem B, we analyze the direct sum of isotropy representations over multiple fixed points. The latter idea is a useful observation that does not seem to have been made before. It was inspired by a conversation with Burkhard Wilking (see Proposition 3.4, and the discussion in Section 3).\\nAnother key aspect of the proofs involves proving bounds on the topology of the fixed-point set M T of the torus action. For Theorem A, we accomplish this by adapting ideas from the first named author's Ph.D. thesis [AMA09], where positive quaternionic Kähler manifolds with symmetry are studied. For Theorem B, we analyze the direct sum of isotropy representations over multiple fixed points. The latter idea is a useful observation that does not seem to have been made before. It was inspired by a conversation with Burkhard Wilking (see Proposition 3.4, and the discussion in Section 3).\\nSections 4 and 5 contain the proofs of Theorems A and B, respectively. We wish to emphasize that the proofs are independent of each other. In particular, among the four propositions we prove in Section 3, only Propositions 3.1 and 3.2 are used for Theorem A, while Theorem 1 requires only Propositions 3.2, 3.3, and 3.4.\\nSections 4 and 5 contain the proofs of Theorems A and B, respectively. We wish to emphasize that the proofs are independent of each other. In particular, among the four propositions we prove in Section 3, only Propositions 3.1 and 3.2 are used for Theorem A, while Theorem 1 requires only Propositions 3.2, 3.3, and 3.4.\\nFinally, the proof of Theorem E only requires results in Section 2, as well as a few additional arguments. It is contained in Section 6. Corollaries C and D follow from Theorems A, B, and E. The proofs are contained in Section 7.\\nFinally, the proof of Theorem E only requires results in Section 2, as well as a few additional arguments. It is contained in Section 6. Corollaries C and D follow from Theorems A, B, and E. The proofs are contained in Section 7.\"\n },\n {\n \"title\": \"Preliminaries\",\n \"content\": \"\"\n },\n {\n \"title\": \"Preliminaries\",\n \"content\": \"In this section, we summarize a number of previous results that will be used in the proofs in this article. We begin with two important results referenced in the introduction. ).\\nIn this section, we summarize a number of previous results that will be used in the proofs in this article. We begin with two important results referenced in the introduction. ).\\nNext, we state a number of results whose importance to the Grove research program is already well established. Theorem 2.3 (Conner [CON57]). If T acts smoothly on a closed manifold M , then\\nNext, we state a number of results whose importance to the Grove research program is already well established. Theorem 2.3 (Conner [CON57]). If T acts smoothly on a closed manifold M , then\\nThe first statement will allow us to pull some information about M T up to M , while the second and third results will allow us to push information about M or its submanifolds down to M T . [BER66,GS94]). Suppose T is a torus acting by isometries on a closed, positively curved manifold M n . If n is even, then the fixed-point set M T is nonempty, and if n is odd, then a codimension one subtorus has nonempty fixed-point set.\\nThe first statement will allow us to pull some information about M T up to M , while the second and third results will allow us to push information about M or its submanifolds down to M T . [BER66,GS94]). Suppose T is a torus acting by isometries on a closed, positively curved manifold M n . If n is even, then the fixed-point set M T is nonempty, and if n is odd, then a codimension one subtorus has nonempty fixed-point set.\"\n },\n {\n \"title\": \"Theorem 2.4 (Berger\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2.4 (Berger\",\n \"content\": \"Theorem 2.5 (Frankel [FRA61]). Let M n be a closed, compact Riemannian manifold with positive sectional curvature. If N n1 1 and N n2 2 are totally geodesic submanifolds of M with n 1 + n 2 ≥ n, then N 1 ∩ N 2 is nonempty.\\nTheorem 2.5 (Frankel [FRA61]). Let M n be a closed, compact Riemannian manifold with positive sectional curvature. If N n1 1 and N n2 2 are totally geodesic submanifolds of M with n 1 + n 2 ≥ n, then N 1 ∩ N 2 is nonempty.\\nThe bound on the dimensions of N 1 and N 2 is equivalent to the bound n − k 1 − k 2 ≥ 0, where k i is the codimension of N i . This result was greatly improved by Wilking [WIL03]: Theorem 2.6 (Connectedness lemma). Suppose M n is a closed Riemannian manifold with positive sectional curvature. Given a highly connected inclusion of closed, orientable manifolds, applying Poincaré duality to each manifold produces a certain periodicity in cohomology (again see [WIL03]): This periodicity is especially strong when l = 0. For example, because M is connected, x i generates H ik (M ; Z) for 0 < ik < n. We call this property periodicity (see [KEN13]): Definition 2.8. For an integer k > 0, a ring R, and a connected space M , we say that\\nThe bound on the dimensions of N 1 and N 2 is equivalent to the bound n − k 1 − k 2 ≥ 0, where k i is the codimension of N i . This result was greatly improved by Wilking [WIL03]: Theorem 2.6 (Connectedness lemma). Suppose M n is a closed Riemannian manifold with positive sectional curvature. Given a highly connected inclusion of closed, orientable manifolds, applying Poincaré duality to each manifold produces a certain periodicity in cohomology (again see [WIL03]): This periodicity is especially strong when l = 0. For example, because M is connected, x i generates H ik (M ; Z) for 0 < ik < n. We call this property periodicity (see [KEN13]): Definition 2.8. For an integer k > 0, a ring R, and a connected space M , we say that\\nExamples of manifolds with periodic cohomology include spheres, projective spaces, and some products such as S 2 × HP m or N × CaP 2 for any manifold N of dimension less than eight.\\nExamples of manifolds with periodic cohomology include spheres, projective spaces, and some products such as S 2 × HP m or N × CaP 2 for any manifold N of dimension less than eight.\\nIn practice, we conclude periodicity from the connectedness lemma and Theorem 2.7. Specifically, if N 1 and N 2 intersect transversely as in part (2) of the connectedness lemma, then H * (N 2 ; Z) is k 1 -periodic. It follows that the rational cohomology of N 2 is also k 1 -periodic. The following refinement of this rational periodicity is proved in [KEN13]: While these connectedness and periodicity theorems have no symmetry assumptions, their main applications have been to the study of positively curved manifolds with large symmetry. Indeed, fixed point sets of isometries are totally geodesic, and with enough isometries, one can guarantee the existence of totally geodesic submanifolds with small codimension and pairs of submanifolds that intersect transversely. In the next section, we illustrate how this works by considering three situations in which one can use the periodicity theorem to conclude that a manifold has 4-periodic rational cohomology.\\nIn practice, we conclude periodicity from the connectedness lemma and Theorem 2.7. Specifically, if N 1 and N 2 intersect transversely as in part (2) of the connectedness lemma, then H * (N 2 ; Z) is k 1 -periodic. It follows that the rational cohomology of N 2 is also k 1 -periodic. The following refinement of this rational periodicity is proved in [KEN13]: While these connectedness and periodicity theorems have no symmetry assumptions, their main applications have been to the study of positively curved manifolds with large symmetry. Indeed, fixed point sets of isometries are totally geodesic, and with enough isometries, one can guarantee the existence of totally geodesic submanifolds with small codimension and pairs of submanifolds that intersect transversely. In the next section, we illustrate how this works by considering three situations in which one can use the periodicity theorem to conclude that a manifold has 4-periodic rational cohomology.\\nAnother consequence of the connectedness and periodicity theorems is the following, which we use in the proofs of Proposition 3.3 and Theorem E.\\nAnother consequence of the connectedness and periodicity theorems is the following, which we use in the proofs of Proposition 3.3 and Theorem E.\\nFor\\nFor\"\n },\n {\n \"title\": \"New Tools\",\n \"content\": \"\"\n },\n {\n \"title\": \"New Tools\",\n \"content\": \"In this section, we prove four propositions that provide the essential tools for the proofs of the main theorems. The first three (Propositions 3.1, 3.2, and 3.3) use the connectedness and periodicity theorems. They provide convenient sufficient conditions for a manifold to have 4-periodic rational cohomology. Since such manifolds have Euler characteristic and even Betti numbers bounded above by those of complex projective space, these results provide important information required for the proofs of Theorems A and B.\\nIn this section, we prove four propositions that provide the essential tools for the proofs of the main theorems. The first three (Propositions 3.1, 3.2, and 3.3) use the connectedness and periodicity theorems. They provide convenient sufficient conditions for a manifold to have 4-periodic rational cohomology. Since such manifolds have Euler characteristic and even Betti numbers bounded above by those of complex projective space, these results provide important information required for the proofs of Theorems A and B.\\nThe final proposition (Proposition 3.4) builds on a technique of Wilking whereby one applies the theory of error correcting codes to isotropy representations. Inspired by a suggestion of Wilking, we extend this idea to the direct sum of isotropy representations. The result is that we obtain components of fixed-point sets of involutions that contain a large portion of the fixed-point set of the torus action. Proposition 3.4 is crucial to the proof of Theorem B.\\nThe final proposition (Proposition 3.4) builds on a technique of Wilking whereby one applies the theory of error correcting codes to isotropy representations. Inspired by a suggestion of Wilking, we extend this idea to the direct sum of isotropy representations. The result is that we obtain components of fixed-point sets of involutions that contain a large portion of the fixed-point set of the torus action. Proposition 3.4 is crucial to the proof of Theorem B.\\nProof. Inductively choose a j algebraically independent involutions τ 1 , . . . , τ j ∈ Z j 2 such that\\nProof. Inductively choose a j algebraically independent involutions τ 1 , . . . , τ j ∈ Z j 2 such that\\nfor all h ≥ 1 and N 0 = M . By maximality of k i , we have both of the estimates\\nfor all h ≥ 1 and N 0 = M . By maximality of k i , we have both of the estimates\\nwhere l is the dimension of the intersection of the (−1)-eigenspaces of the actions of τ i+1 and τ i τ i+1 on the normal space of N i−1 . Geometrically, l is the codimension of\\nwhere l is the dimension of the intersection of the (−1)-eigenspaces of the actions of τ i+1 and τ i τ i+1 on the normal space of N i−1 . Geometrically, l is the codimension of\\nTogether, these two estimates imply that k i ≥ 2k i+1 for all i ≥ 1. We draw two conclusions from this. First,\\nTogether, these two estimates imply that k i ≥ 2k i+1 for all i ≥ 1. We draw two conclusions from this. First,\\nSince k j is even, we conclude that k j = 0. Second, for all 1 ≤ h ≤ j,\\nSince k j is even, we conclude that k j = 0. Second, for all 1 ≤ h ≤ j,\\nWe will use the first estimate in a moment. The second estimate implies that the\\nWe will use the first estimate in a moment. The second estimate implies that the\\nwould be 4-periodic as well. We will use this fact later. The second sufficient condition for rational 4-periodicity is an immediate consequence of the periodicity theorem. If there exists an involution σ ∈ T such that M σ x has dim(M σ x ) ≥ n 2 and is fixed by another involution in T , then M has 4-periodic rational cohomology. In particular, this applies if M σ x is fixed by a two-dimensional torus in T .\\nwould be 4-periodic as well. We will use this fact later. The second sufficient condition for rational 4-periodicity is an immediate consequence of the periodicity theorem. If there exists an involution σ ∈ T such that M σ x has dim(M σ x ) ≥ n 2 and is fixed by another involution in T , then M has 4-periodic rational cohomology. In particular, this applies if M σ x is fixed by a two-dimensional torus in T .\"\n },\n {\n \"title\": \"Proof. Choose a non-trivial involution\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proof. Choose a non-trivial involution\",\n \"content\": \"the periodicity theorem implies that H * (M ; Q) is 4-periodic.\\nthe periodicity theorem implies that H * (M ; Q) is 4-periodic.\\nThe third sufficient condition for rational 4-periodicity is the following. Its proof is a simple combination of the connectedness lemma and Theorems 2.7 and 2.10. Proof. By 2.7, the existence of a codimension k totally geodesic submanifold implies that there exists y ∈ H k (M ; Q) such that the maps H i (M ; Q) → H i+k (M ; Q) given by multiplication by y are injective for k − 1 < i ≤ n − 2k + 1 and surjective for k − 1 ≤ i < n − 2k + 1. Next by 2.10 with c = k +4, the symmetry rank assumption implies the existence of\\nThe third sufficient condition for rational 4-periodicity is the following. Its proof is a simple combination of the connectedness lemma and Theorems 2.7 and 2.10. Proof. By 2.7, the existence of a codimension k totally geodesic submanifold implies that there exists y ∈ H k (M ; Q) such that the maps H i (M ; Q) → H i+k (M ; Q) given by multiplication by y are injective for k − 1 < i ≤ n − 2k + 1 and surjective for k − 1 ≤ i < n − 2k + 1. Next by 2.10 with c = k +4, the symmetry rank assumption implies the existence of\\nfor some 0 ≤ m < 4 and a ∈ H m (M ; Q). We claim that x induces periodicity in H * (M ; Q).\\nfor some 0 ≤ m < 4 and a ∈ H m (M ; Q). We claim that x induces periodicity in H * (M ; Q).\\nWe first show that it suffices to show injectivity of the multiplication maps\\nWe first show that it suffices to show injectivity of the multiplication maps\\nIndeed, Poincaré duality would imply that the Betti numbers satisfy\\nIndeed, Poincaré duality would imply that the Betti numbers satisfy\\nis already known to be surjective by our choice of x, the proof would be complete.\\nis already known to be surjective by our choice of x, the proof would be complete.\\nWe now prove that multiplication maps\\nWe now prove that multiplication maps\\n4 , the four cases that follow constitute a proof:\\n4 , the four cases that follow constitute a proof:\"\n },\n {\n \"title\": \"injectivity follows from the fact that\",\n \"content\": \"\"\n },\n {\n \"title\": \"injectivity follows from the fact that\",\n \"content\": \"is a composition of injective maps that equals the composition\\nis a composition of injective maps that equals the composition\\nthe first map of which is an isomorphism.\\nthe first map of which is an isomorphism.\\n• If n − k − 4 < i ≤ n − 4, injectivity follows from a direct argument. Let z ∈ H i (M ; Q) be a nonzero element. By Poincaré duality, there exists w ∈ H n−i such that zw = 0. The multiplicative property of x implies that w = xw for some w ∈ H n−i−4 . Hence\\n• If n − k − 4 < i ≤ n − 4, injectivity follows from a direct argument. Let z ∈ H i (M ; Q) be a nonzero element. By Poincaré duality, there exists w ∈ H n−i such that zw = 0. The multiplicative property of x implies that w = xw for some w ∈ H n−i−4 . Hence\\nwhich implies xz = 0, completing the proof.\\nwhich implies xz = 0, completing the proof.\\nThe final tool we require in the proofs involves an application of the theory of error-correcting codes. This idea was used for the first time in the subject in Wilking [WIL03] to produce fixed-point sets of isometries of torus actions that have small codimension. In the setting of positively curved manifolds, the inclusion maps of these totally geodesic fixed-point sets are highly connected by the connectedness lemma, hence one can construct inductive arguments over dimension as in [WIL03]. In addition, one obtains periodicity results as discussed in Section 2.\\nThe final tool we require in the proofs involves an application of the theory of error-correcting codes. This idea was used for the first time in the subject in Wilking [WIL03] to produce fixed-point sets of isometries of torus actions that have small codimension. In the setting of positively curved manifolds, the inclusion maps of these totally geodesic fixed-point sets are highly connected by the connectedness lemma, hence one can construct inductive arguments over dimension as in [WIL03]. In addition, one obtains periodicity results as discussed in Section 2.\\nWe briefly summarize how this technique works. First, one requires a fixed point x ∈ M T of a torus action on M . By restricting attention to the subgroup of involutions in T , one associates to the isotropy represention a linear error-correcting code. One can then apply algebraic bounds from the theory of error-correcting codes to make conclusions about the Hamming weights of the code. Given this, one translates this algebraic information into geometric data about the fixed-point sets of involutions in the torus.\\nWe briefly summarize how this technique works. First, one requires a fixed point x ∈ M T of a torus action on M . By restricting attention to the subgroup of involutions in T , one associates to the isotropy represention a linear error-correcting code. One can then apply algebraic bounds from the theory of error-correcting codes to make conclusions about the Hamming weights of the code. Given this, one translates this algebraic information into geometric data about the fixed-point sets of involutions in the torus.\\nHere, inspired by a suggestion of Wilking, we consider an extension of this idea. The starting point is to simultaneously consider the isotropy representations at multiple fixed points x i ∈ M T . Specifically, one observes that the direct sum of isotropy representions, when restricted to the subgroup of involutions, yields another errorcorrecting code. Hence one can implement the strategy outlined above. The translation of algebraic information into geometric data is less immediate. However, if M is positively curved, one can apply Frankel's theorem to prove that the chosen components of fixed-point sets of involutions have, in fact, a large number of components of M T contained in them. The specific result we will use is the following. 2 r involutions, together with Frankel's theorem, will be the key step in the proof of Theorem A. The conclusion for general t is really only used to prove Theorem B, but for this it is crucial. The basic idea for the proof of the first conclusion of Theorem B is that, by iterated use of this proposition, one can find involutions ι 1 , . . . , ι h ∈ T r and components N i ⊆ M ιi such that\\nHere, inspired by a suggestion of Wilking, we consider an extension of this idea. The starting point is to simultaneously consider the isotropy representations at multiple fixed points x i ∈ M T . Specifically, one observes that the direct sum of isotropy representions, when restricted to the subgroup of involutions, yields another errorcorrecting code. Hence one can implement the strategy outlined above. The translation of algebraic information into geometric data is less immediate. However, if M is positively curved, one can apply Frankel's theorem to prove that the chosen components of fixed-point sets of involutions have, in fact, a large number of components of M T contained in them. The specific result we will use is the following. 2 r involutions, together with Frankel's theorem, will be the key step in the proof of Theorem A. The conclusion for general t is really only used to prove Theorem B, but for this it is crucial. The basic idea for the proof of the first conclusion of Theorem B is that, by iterated use of this proposition, one can find involutions ι 1 , . . . , ι h ∈ T r and components N i ⊆ M ιi such that\\nA variation of this idea is used again in the proof of the second conclusion of Theorem B.\\nA variation of this idea is used again in the proof of the second conclusion of Theorem B.\\nProof. Choose a basis for each T xi M so that the isotropy representation T r → SO(T xi M ) factors through U(n/2) ⊆ SO(T xi M ). It follows that the Z r 2 ⊆ T r maps via the isotropy representation to a copy of Z\\nProof. Choose a basis for each T xi M so that the isotropy representation T r → SO(T xi M ) factors through U(n/2) ⊆ SO(T xi M ). It follows that the Z r 2 ⊆ T r maps via the isotropy representation to a copy of Z\\nLet π i denote the projection of the direct sum onto the i-th summand of Z n/2 2 . The Hamming weight of the image of ι ∈ Z r 2 under the composition of the isotropy representation and π i is half the codimension k i of M ι xi . Set k = k i . We claim that a nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Assuming this for a moment, we conclude the proof. Relabel the x i so that the codimensions k i = cod(M ι xi ) are increasing in i. It follows that, for all 1 < j ≤ t+1 2 ,\\nLet π i denote the projection of the direct sum onto the i-th summand of Z n/2 2 . The Hamming weight of the image of ι ∈ Z r 2 under the composition of the isotropy representation and π i is half the codimension k i of M ι xi . Set k = k i . We claim that a nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Assuming this for a moment, we conclude the proof. Relabel the x i so that the codimensions k i = cod(M ι xi ) are increasing in i. It follows that, for all 1 < j ≤ t+1 2 ,\\nUsing Frankel's theorem, we conclude that the components M ι x1 and M ι xj intersect and hence coincide. Since we also have\\nUsing Frankel's theorem, we conclude that the components M ι x1 and M ι xj intersect and hence coincide. Since we also have\\nwe conclude the component M ι x1 of M ι satisfies cod(M ι x1 ) ≤ n−c 2 and contains at least t+1\\nwe conclude the component M ι x1 of M ι satisfies cod(M ι x1 ) ≤ n−c 2 and contains at least t+1\\nWe proceed by contradiction to show that some nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Suppose therefore that k ≥ t(n−c)+2 2 for all nontrivial ι ∈ Z r 2 . Applying Griesmer's bound, we obtain\\nWe proceed by contradiction to show that some nontrivial ι ∈ Z r exists with k ≤ t(n−c) 2 . Suppose therefore that k ≥ t(n−c)+2 2 for all nontrivial ι ∈ Z r 2 . Applying Griesmer's bound, we obtain\\nWe estimate the sum on the right-hand side as follows:\\nWe estimate the sum on the right-hand side as follows:\\nRearranging, we conclude that\\nRearranging, we conclude that\\nBut this contradicts the assumed bound on r, so the result follows.\\nBut this contradicts the assumed bound on r, so the result follows.\"\n },\n {\n \"title\": \"Proof of Theorem A\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proof of Theorem A\",\n \"content\": \"We first state Theorem 4.1, which is the most general version of Theorem A we can prove. It provides an upper bound on the Euler characteristic of the form χ(M ) ≤ f 0 (n). We then deduce Theorem 1 from Theorem 4.1 by proving the estimate f 0 (n) < 2 3(log 2 n) 2 . The heart of this section is the proof of Theorem A. At the end of the section, we comment on the quality of the approximation provided by Theorem 1 and illustrate the result graphically. In order to state Theorem 4.1, we require two definitions. Set s(n) = log 2 n + log 2 (n + 2) − 2, for n > 0, set f 0 (n) = n 2 + 1 for n ≤ 52, and define f 0 (n) for n ≥ 54 by the recursive formula\\nWe first state Theorem 4.1, which is the most general version of Theorem A we can prove. It provides an upper bound on the Euler characteristic of the form χ(M ) ≤ f 0 (n). We then deduce Theorem 1 from Theorem 4.1 by proving the estimate f 0 (n) < 2 3(log 2 n) 2 . The heart of this section is the proof of Theorem A. At the end of the section, we comment on the quality of the approximation provided by Theorem 1 and illustrate the result graphically. In order to state Theorem 4.1, we require two definitions. Set s(n) = log 2 n + log 2 (n + 2) − 2, for n > 0, set f 0 (n) = n 2 + 1 for n ≤ 52, and define f 0 (n) for n ≥ 54 by the recursive formula\"\n },\n {\n \"title\": \"GAFA POSITIVE CURVATURE UNDER SYMMETRY 1389\",\n \"content\": \"\"\n },\n {\n \"title\": \"GAFA POSITIVE CURVATURE UNDER SYMMETRY 1389\",\n \"content\": \"We briefly provide some explanation for the definition of f 0 (n). We suggest to see its definition as being motivated by two facts, the first providing the 2 s(n) − 1 factor, and the second yielding the argument 2 3n−4 8 ≈ 3 4 n. The latter corresponds to the base of 4/3 of the logarithms we use.\\nWe briefly provide some explanation for the definition of f 0 (n). We suggest to see its definition as being motivated by two facts, the first providing the 2 s(n) − 1 factor, and the second yielding the argument 2 3n−4 8 ≈ 3 4 n. The latter corresponds to the base of 4/3 of the logarithms we use.\\nMore specifically, in the proof of Case 2 below, we will prove that M T is contained in the union of submanifolds N j . It follows that b 2i (M T ) is bounded above by j i b 2i (N T j ). Now the N j correspond to unique, non-trivial involutions in a subgroup Z s(n) 2 of T , hence there are at most 2 s(n) − 1 of them. In addition, we show by an inductive argument that b 2i (N T j ) ≤ f 0 (dim N j ) and dim N j < 3 4 n for all N j . Putting this together, we deduce that b 2i (M T ) ≤ f 0 (n) by using the recursive formula for f 0 (n).\\nMore specifically, in the proof of Case 2 below, we will prove that M T is contained in the union of submanifolds N j . It follows that b 2i (M T ) is bounded above by j i b 2i (N T j ). Now the N j correspond to unique, non-trivial involutions in a subgroup Z s(n) 2 of T , hence there are at most 2 s(n) − 1 of them. In addition, we show by an inductive argument that b 2i (N T j ) ≤ f 0 (dim N j ) and dim N j < 3 4 n for all N j . Putting this together, we deduce that b 2i (M T ) ≤ f 0 (n) by using the recursive formula for f 0 (n).\\nFinally note that, below dimension 54, our symmetry assumption is sufficiently large that the classification results of Section 2 imply immediately that the upper bound of the Euler characteristic of M n is provided by χ(CP n/2 ) = n 2 + 1 = f 0 (n). Proof of Theorem A. By Theorem 4.1, it suffices to show that f 0 (n) < 2 3(log 2 n) 2 . This clearly holds for n = 2, so it suffices to prove it for n ≥ 4. In this range, we actually prove the following, stronger estimate:\\nFinally note that, below dimension 54, our symmetry assumption is sufficiently large that the classification results of Section 2 imply immediately that the upper bound of the Euler characteristic of M n is provided by χ(CP n/2 ) = n 2 + 1 = f 0 (n). Proof of Theorem A. By Theorem 4.1, it suffices to show that f 0 (n) < 2 3(log 2 n) 2 . This clearly holds for n = 2, so it suffices to prove it for n ≥ 4. In this range, we actually prove the following, stronger estimate:\\n.\\n.\\nFor n ≤ 52, this is obvious since f 0 (n) = n 2 + 1. Proceeding inductively, for n ≥ 54, we have .\\nFor n ≤ 52, this is obvious since f 0 (n) = n 2 + 1. Proceeding inductively, for n ≥ 54, we have .\\nWe proceed to the proof of Theorem 4.1. Recall that M n is a closed, positively curved manifold that admits an effective, isometric action by a torus T with dim(T ) ≥ log 4/3 n.\\nWe proceed to the proof of Theorem 4.1. Recall that M n is a closed, positively curved manifold that admits an effective, isometric action by a torus T with dim(T ) ≥ log 4/3 n.\\nFirst, if n < 22, the bound on the dimension of T implies that dim(T ) > n 2 . As this contradicts the maximal possible symmetry rank (Theorem 2.1), the theorem is vacuously true in these dimensions. Moreover, the theorem continues to hold for dimensions n ≤ 52 by Wilking's homotopy classification theorem (Theorem 2.2) and the definition of f 0 . We proceed by induction to prove the result for n ≥ 54.\\nFirst, if n < 22, the bound on the dimension of T implies that dim(T ) > n 2 . As this contradicts the maximal possible symmetry rank (Theorem 2.1), the theorem is vacuously true in these dimensions. Moreover, the theorem continues to hold for dimensions n ≤ 52 by Wilking's homotopy classification theorem (Theorem 2.2) and the definition of f 0 . We proceed by induction to prove the result for n ≥ 54.\\nSet s = s(n), and choose any subgroup Z s 2 ⊆ T of involutions. We split the proof of the induction step into two cases. GAFA Case 1 There exists x ∈ M T such that every ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n 2 actually has cod (M ι x ) ≤ n 4 . Case 2 For all x ∈ M T , there exists ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and n 4 < cod (M ι x ) ≤ n 2 . If Case 1 occurs, we prove that M has 4-periodic rational cohomology. By Conner's theorem, we obtain the desired estimate\\nSet s = s(n), and choose any subgroup Z s 2 ⊆ T of involutions. We split the proof of the induction step into two cases. GAFA Case 1 There exists x ∈ M T such that every ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n 2 actually has cod (M ι x ) ≤ n 4 . Case 2 For all x ∈ M T , there exists ι ∈ Z s 2 with dim ker T | M ι x ≤ 1 and n 4 < cod (M ι x ) ≤ n 2 . If Case 1 occurs, we prove that M has 4-periodic rational cohomology. By Conner's theorem, we obtain the desired estimate\"\n },\n {\n \"title\": \"Proof (Proof in Case 1).\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proof (Proof in Case 1).\",\n \"content\": \"Fix some x ∈ M T as in the statement of Case 1. Let 0 ≤ j ≤ log 2 n − 1 be maximal such that there exists a Z j 2 ⊆ Z s 2 with the property that every σ ∈ Z j 2 has dim ker T | M σ x ≤ 1 and cod (M σ x ) ≤ n/4. We claim that j = log 2 n − 1. By Proposition 3.1 and the comments above, this would suffice to show that M is rationally 4-periodic.\\nFix some x ∈ M T as in the statement of Case 1. Let 0 ≤ j ≤ log 2 n − 1 be maximal such that there exists a Z j 2 ⊆ Z s 2 with the property that every σ ∈ Z j 2 has dim ker T | M σ x ≤ 1 and cod (M σ x ) ≤ n/4. We claim that j = log 2 n − 1. By Proposition 3.1 and the comments above, this would suffice to show that M is rationally 4-periodic.\\nSuppose\\nSuppose\\nso Proposition 3.4 applies with t = 1 and c = 0 (alternatively, one can apply [KEN00,\\nso Proposition 3.4 applies with t = 1 and c = 0 (alternatively, one can apply [KEN00,\\nSince this is our claim, we may assume that dim ker T | M τ x ≤ 1 and hence, by the assumption in Case 1, that cod (M τ x ) ≤ n 4 . Fix for a moment any σ ∈ Z j 2 . Observe that M στ\\nSince this is our claim, we may assume that dim ker T | M τ x ≤ 1 and hence, by the assumption in Case 1, that cod (M τ x ) ≤ n 4 . Fix for a moment any σ ∈ Z j 2 . Observe that M στ\\nBy Proposition 3.2 again, we can conclude without loss of generality that dim ker T | M στ x ≤ 1. But now the assumption in Case 1 implies that cod (M στ x ) ≤ n/4. Hence every ι in the Z j+1 2 generated by Z j 2 and τ satisfies dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n/4. This contradicts the maximality of j and hence concludes the proof that j ≥ log 2 n − 1.\\nBy Proposition 3.2 again, we can conclude without loss of generality that dim ker T | M στ x ≤ 1. But now the assumption in Case 1 implies that cod (M στ x ) ≤ n/4. Hence every ι in the Z j+1 2 generated by Z j 2 and τ satisfies dim ker T | M ι x ≤ 1 and cod (M ι x ) ≤ n/4. This contradicts the maximality of j and hence concludes the proof that j ≥ log 2 n − 1.\\nTo conclude the proof of Theorem 4.1, it suffices to prove that b 2i (M T ) ≤ f 0 (n) in Case 2. In this case, we cannot calculate H * (M ; Q). Instead, we cover M T by at most 2 s(n) −1 submanifolds, each of which satisfies the induction hypothesis. Adding together the estimates for each of these submanifolds and applying the recursive definition of f 0 , we conclude the desired bound.\\nTo conclude the proof of Theorem 4.1, it suffices to prove that b 2i (M T ) ≤ f 0 (n) in Case 2. In this case, we cannot calculate H * (M ; Q). Instead, we cover M T by at most 2 s(n) −1 submanifolds, each of which satisfies the induction hypothesis. Adding together the estimates for each of these submanifolds and applying the recursive definition of f 0 , we conclude the desired bound.\\nProof (Proof in Case 2). We use the notation b even (X) = b 2i (X) for the sum of the even Betti numbers of a space X.\\nProof (Proof in Case 2). We use the notation b even (X) = b 2i (X) for the sum of the even Betti numbers of a space X.\\nSuppose The T -action on M restricts to a T action on each M ιj xj . After dividing by the kernel of the induced action, we obtain an effective action by a torus T j on M ιj xj with dim(T j ) ≥ dim(T ) − 1. Since dim(T ) ≥ log 4/3 n and n > 4 3 n j , we see that dim (T j ) ≥ log 4/3 (n j ). It follows from the induction hypothesis that This finishes the proof of Theorem 4.1 and hence of Theorem A.\\nSuppose The T -action on M restricts to a T action on each M ιj xj . After dividing by the kernel of the induced action, we obtain an effective action by a torus T j on M ιj xj with dim(T j ) ≥ dim(T ) − 1. Since dim(T ) ≥ log 4/3 n and n > 4 3 n j , we see that dim (T j ) ≥ log 4/3 (n j ). It follows from the induction hypothesis that This finishes the proof of Theorem 4.1 and hence of Theorem A.\\nRemark 4.2. We remark that our Case 1, Case 2 approach directly yields vanishing theorems for the elliptic genus as proven in [WEI00].\\nRemark 4.2. We remark that our Case 1, Case 2 approach directly yields vanishing theorems for the elliptic genus as proven in [WEI00].\\nWe conclude this section with a remark on the size of f 0 (n). See Table 1 for some approximate values of n i and κ i . Figure 1. Comparison of the functions f 0 (n) (blue) and κ 1 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) By an argument similar to the one above that shows f 0 (n) < 2 3(log 2 n) 2 , one could similarly show the following, better estimates for f 0 (n): For all i ≥ 1 and n ≥ n i ,\\nWe conclude this section with a remark on the size of f 0 (n). See Table 1 for some approximate values of n i and κ i . Figure 1. Comparison of the functions f 0 (n) (blue) and κ 1 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) By an argument similar to the one above that shows f 0 (n) < 2 3(log 2 n) 2 , one could similarly show the following, better estimates for f 0 (n): For all i ≥ 1 and n ≥ n i ,\\n.\\n.\\nWe will not pursue this here. However, we compare the function f 0 with the approximation given in Theorem A against the background of an exponential bound. For this note that it is easily possible to modify the arguments we provide in order to obtain the exponential function we draw in the pictures below as an alternative upper bound. From the pictures already it should be obvious why we did not spend more effort on this.\\nWe will not pursue this here. However, we compare the function f 0 with the approximation given in Theorem A against the background of an exponential bound. For this note that it is easily possible to modify the arguments we provide in order to obtain the exponential function we draw in the pictures below as an alternative upper bound. From the pictures already it should be obvious why we did not spend more effort on this.\\nAlready from the graphs in Figures 1 and 2, one sees that the functions grow strictly subexponentially. As for the asymptotic behavior, the exponential reference function may easily be chosen such that the power of two is a genuine fraction of n.\\nAlready from the graphs in Figures 1 and 2, one sees that the functions grow strictly subexponentially. As for the asymptotic behavior, the exponential reference function may easily be chosen such that the power of two is a genuine fraction of n.\"\n },\n {\n \"title\": \"Proof of Theorem B\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proof of Theorem B\",\n \"content\": \"For integers α ≥ 3 and n > 0, define s α (n) = n 2α + 2 log 2 n 2α + α + 3.\\nFor integers α ≥ 3 and n > 0, define s α (n) = n 2α + 2 log 2 n 2α + α + 3.\\nGAFA POSITIVE CURVATURE UNDER SYMMETRY 1393 Figure 2. Comparison of the functions f 0 (n) (blue) and κ 6 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) Theorem B will follow from the following theorem.\\nGAFA POSITIVE CURVATURE UNDER SYMMETRY 1393 Figure 2. Comparison of the functions f 0 (n) (blue) and κ 6 · ( n 2 + 1) 1+log 4/3 ( n 2 +1) (red) under a logarithmically scaled y-axis respectively both axes being logarithmically scaled. As a reference the function 1.13576 × 10 −12 · 2 n , which also bounds f 0 , (black) is added (color figure online) Theorem B will follow from the following theorem.\\nTheorem 5.1. Let α ≥ 3 be an integer, and let M n be a closed, simply connected Riemannian manifold with positive sectional curvature. Assume T is a torus acting isometrically and effectively on M with dim(T ) ≥ s α (n). The following hold:\\nTheorem 5.1. Let α ≥ 3 be an integer, and let M n be a closed, simply connected Riemannian manifold with positive sectional curvature. Assume T is a torus acting isometrically and effectively on M with dim(T ) ≥ s α (n). The following hold:\\nWhen α = 3, the upper bound is n 2 + 1.\\nWhen α = 3, the upper bound is n 2 + 1.\\nWe show now that this theorem implies Theorem B.\\nWe show now that this theorem implies Theorem B.\\nProof of Theorem B (Proof of Theorem 1). Let δ > 0, and assume M n is a closed, positively curved Riemannian manifold with symmetry rank r > δn.\\nProof of Theorem B (Proof of Theorem 1). Let δ > 0, and assume M n is a closed, positively curved Riemannian manifold with symmetry rank r > δn.\\nChoose an integer α ≥ 3 with δ > 1 2α , and choose another integer n 0 such that n ≥ n 0 implies δn > s α (n). Finally, choose a constant c such that all of the following hold:\\nChoose an integer α ≥ 3 with δ > 1 2α , and choose another integer n 0 such that n ≥ n 0 implies δn > s α (n). Finally, choose a constant c such that all of the following hold:\\n• For all m ≤ n 0 , every non-negatively curved m-manifold N has b i (N ) ≤ c. • a α n + 1 ≤ cn for all n ≥ n 0 .\\n• For all m ≤ n 0 , every non-negatively curved m-manifold N has b i (N ) ≤ c. • a α n + 1 ≤ cn for all n ≥ n 0 .\\nObserve that the first is possible by Gromov's Betti number estimate.\\nObserve that the first is possible by Gromov's Betti number estimate.\\nIt now follows from either Gromov's theorem (if n ≤ n 0 ) or Theorem 5.1 (if n > n 0 ) that b 0 (M T ) ≤ cn and χ(M ) < cn log 2 n.\\nIt now follows from either Gromov's theorem (if n ≤ n 0 ) or Theorem 5.1 (if n > n 0 ) that b 0 (M T ) ≤ cn and χ(M ) < cn log 2 n.\\nWe spend the rest of this section proving Theorem 5.1. Our task is threefold. First we prove Lemma 5.2, which implies the α = 3 case of Theorem 5.1. Using this as the base case for an induction over α, we prove the two parts of Theorem 5.1 in Theorems 5.5 and 5.6, respectively. Note that the conclusion is the same as in Wilking's cohomology classification (Theorem 2.2) for n ≥ 6000. Using Wilking's proof, we obtain the same conclusion in all dimensions using the slightly larger symmetry rank bound.\\nWe spend the rest of this section proving Theorem 5.1. Our task is threefold. First we prove Lemma 5.2, which implies the α = 3 case of Theorem 5.1. Using this as the base case for an induction over α, we prove the two parts of Theorem 5.1 in Theorems 5.5 and 5.6, respectively. Note that the conclusion is the same as in Wilking's cohomology classification (Theorem 2.2) for n ≥ 6000. Using Wilking's proof, we obtain the same conclusion in all dimensions using the slightly larger symmetry rank bound.\\nProof of Lemma 5.2. We induct over dimension. For n ≤ 3, the theorem is trivial. For 4 ≤ n ≤ 16, dim(T ) ≥ n/2, hence the result follows from the Grove and Searle's diffeomorphism classification (Theorem 2.1). For 17 ≤ n ≤ 120, the symmetry rank is at least dim(T ) ≥ n 4 + 1, so the theorem follows from Wilking's homotopy classification (Theorem 2.2).\\nProof of Lemma 5.2. We induct over dimension. For n ≤ 3, the theorem is trivial. For 4 ≤ n ≤ 16, dim(T ) ≥ n/2, hence the result follows from the Grove and Searle's diffeomorphism classification (Theorem 2.1). For 17 ≤ n ≤ 120, the symmetry rank is at least dim(T ) ≥ n 4 + 1, so the theorem follows from Wilking's homotopy classification (Theorem 2.2).\\nSuppose now that n > 120. The assumption on T implies dim(T ) ≥ max n 6 + 1, n 8 + 14 .\\nSuppose now that n > 120. The assumption on T implies dim(T ) ≥ max n 6 + 1, n 8 + 14 .\\nBy Proposition 8.1 in [WIL03], if there exists an involution ι ∈ T with cod(M ι ) ≤ 7n 24 , then the conclusion of the lemma holds. It therefore suffices to prove the existence of such an involution.\\nBy Proposition 8.1 in [WIL03], if there exists an involution ι ∈ T with cod(M ι ) ≤ 7n 24 , then the conclusion of the lemma holds. It therefore suffices to prove the existence of such an involution.\\nTo see that such an involution exists, we adapt the proof of Corollary 3.2.a in [WIL03]. Let T denote the torus acting effectively and isometrically on M . By Berger's theorem, there exists T r ⊆ T such that T r has a fixed point x ∈ M , where r = dim(T ) − δ and where δ = 0 if n is even and δ = 1 if n is odd. We consider the isotropy representation where m = n/2 . The original proof concludes that this is the case for r ≥ m 3 + 1 and m ≥ 2700, and in particular for r ≥ n 6 + 1 and n ≥ 6000. In our case, the assumption on r in terms of m is that\\nTo see that such an involution exists, we adapt the proof of Corollary 3.2.a in [WIL03]. Let T denote the torus acting effectively and isometrically on M . By Berger's theorem, there exists T r ⊆ T such that T r has a fixed point x ∈ M , where r = dim(T ) − δ and where δ = 0 if n is even and δ = 1 if n is odd. We consider the isotropy representation where m = n/2 . The original proof concludes that this is the case for r ≥ m 3 + 1 and m ≥ 2700, and in particular for r ≥ n 6 + 1 and n ≥ 6000. In our case, the assumption on r in terms of m is that\\nExcept for n = 126 and n = 131, this estimate holds for all n > 120. For n = 126 and n = 131, one can instead apply the Griesmer bound to find the desired involution.\\nExcept for n = 126 and n = 131, this estimate holds for all n > 120. For n = 126 and n = 131, one can instead apply the Griesmer bound to find the desired involution.\\nIn all cases therefore, if n > 120, the involution ι with cod(M ι x ) ≤ 7n/24 exists. This concludes the proof of the lemma.\\nIn all cases therefore, if n > 120, the involution ι with cod(M ι x ) ≤ 7n/24 exists. This concludes the proof of the lemma.\\nWith the base case (α = 3) complete, we pause for a moment to prove a couple of lemmas that will facilitate the induction over α. The third estimate of Lemma 5.3 together with Lemma 5.4 already suggest how this will work. Lemma 5.3. For α ≥ 4, s α (n) satisfies the following:\\nWith the base case (α = 3) complete, we pause for a moment to prove a couple of lemmas that will facilitate the induction over α. The third estimate of Lemma 5.3 together with Lemma 5.4 already suggest how this will work. Lemma 5.3. For α ≥ 4, s α (n) satisfies the following:\\nThe logarithm term is decreasing in α, so it is at most 2 log 2 4 3 < 5 6 . In particular, s α (n) ≥ s α−1 (n) holds if 1 ≥ n 2α(α−1) + 5 6 , which is to say if n ≤ α(α − 1)/3. For the second estimate, suppose that n > α(α − 1)/3. Observe that α ≥ 4 implies\\nThe logarithm term is decreasing in α, so it is at most 2 log 2 4 3 < 5 6 . In particular, s α (n) ≥ s α−1 (n) holds if 1 ≥ n 2α(α−1) + 5 6 , which is to say if n ≤ α(α − 1)/3. For the second estimate, suppose that n > α(α − 1)/3. Observe that α ≥ 4 implies\\nOn the other hand, α ≥ 4 and n > α(α − 1)/3 imply that\\nOn the other hand, α ≥ 4 and n > α(α − 1)/3 imply that\\nCombining this estimate with the one above for s α (n), we conclude the proof of the second estimate as follows:\\nCombining this estimate with the one above for s α (n), we conclude the proof of the second estimate as follows:\\na α n 2 + 2n + 2 = 3 8\\na α n 2 + 2n + 2 = 3 8\\nFinally, if k ≥ n/α, then n − k ≤ α−1 α n. Hence we conclude the third estimate as follows: Here, we are using the notation from Section 4 that dim ker T | M ι x denotes the dimension of the kernel of the induced T -action on M ι x . Proof. Choose ι and x as in the assumption of the lemma. If k = cod(M ι x ) is at most n/α, then Proposition 3.3 implies that M has 4-periodic rational cohomology. Indeed, the proposition applies since α ≥ 4 implies that α + 3 − 2 log 2 (2α) ≥ 1 and hence that\\nFinally, if k ≥ n/α, then n − k ≤ α−1 α n. Hence we conclude the third estimate as follows: Here, we are using the notation from Section 4 that dim ker T | M ι x denotes the dimension of the kernel of the induced T -action on M ι x . Proof. Choose ι and x as in the assumption of the lemma. If k = cod(M ι x ) is at most n/α, then Proposition 3.3 implies that M has 4-periodic rational cohomology. Indeed, the proposition applies since α ≥ 4 implies that α + 3 − 2 log 2 (2α) ≥ 1 and hence that\\nOn the other hand, if k ≤ n/2 and dim ker T | M ι x ≥ 2, then we are in the situation of Proposition 3.2, so we conclude again that H * (M ; Q) is 4-periodic.\\nOn the other hand, if k ≤ n/2 and dim ker T | M ι x ≥ 2, then we are in the situation of Proposition 3.2, so we conclude again that H * (M ; Q) is 4-periodic.\\nWe are ready to prove the first part of Theorem 5.1: Proof. By Lemma 5.2, the theorem follows in the base case of α = 3. We proceed by induction the prove the theorem for α ≥ 4. By Lemma 5.3 and the induction hypothesis, we may we may assume that n > α(α − 1)/3. In particular, by Lemma 5.3 again, we may assume that s α (n) ≥ log 2 (a α n 2 + 2n + 2).\\nWe are ready to prove the first part of Theorem 5.1: Proof. By Lemma 5.2, the theorem follows in the base case of α = 3. We proceed by induction the prove the theorem for α ≥ 4. By Lemma 5.3 and the induction hypothesis, we may we may assume that n > α(α − 1)/3. In particular, by Lemma 5.3 again, we may assume that s α (n) ≥ log 2 (a α n 2 + 2n + 2).\\nNext observe that if M has 4-periodic rational cohomology, then the even Betti numbers of M are at most one. Hence we can conclude the theorem using Conner's theorem:\\nNext observe that if M has 4-periodic rational cohomology, then the even Betti numbers of M are at most one. Hence we can conclude the theorem using Conner's theorem:\\nIn particular, by Lemma 5.4, we may assume that, for all nontrivial involutions ι ∈ T and for all x ∈ M ι , if cod M ι x ≤ n/2, then cod M ι x > n/α and dim ker T | M ι x ≤ 1. We proceed by contradiction. Assume there exist t = a α n + 2 distinct components of M T . The estimate on s α (n) established at the beginning of the proof implies that dim(T ) ≥ s α (n) ≥ log 2 (t n + 2). By Proposition 3.4 (with c = 0), there exists a nontrivial involution ι ∈ T and a component N ⊆ M ι with cod(N ) ≤ n 2 such that N contains at least t +1 2 of the t components of M T . As established above, we may assume that cod(N ) > n/α and dim ker T | M ι x ≤ 1. Let T = T / ker(T | N ), and observe that T acts effectively and isometrically on N . Moreover, the components of N T are the components of M T that are contained in N . In particular, the number of components of N T is at least t +1 2 . On the other hand, since\\nIn particular, by Lemma 5.4, we may assume that, for all nontrivial involutions ι ∈ T and for all x ∈ M ι , if cod M ι x ≤ n/2, then cod M ι x > n/α and dim ker T | M ι x ≤ 1. We proceed by contradiction. Assume there exist t = a α n + 2 distinct components of M T . The estimate on s α (n) established at the beginning of the proof implies that dim(T ) ≥ s α (n) ≥ log 2 (t n + 2). By Proposition 3.4 (with c = 0), there exists a nontrivial involution ι ∈ T and a component N ⊆ M ι with cod(N ) ≤ n 2 such that N contains at least t +1 2 of the t components of M T . As established above, we may assume that cod(N ) > n/α and dim ker T | M ι x ≤ 1. Let T = T / ker(T | N ), and observe that T acts effectively and isometrically on N . Moreover, the components of N T are the components of M T that are contained in N . In particular, the number of components of N T is at least t +1 2 . On the other hand, since\\nby Lemma 5.3, the induction hypothesis implies that the number of components of N T is at most a α−1 dim(N ) + 1. Putting these estimates together, we conclude that\\nby Lemma 5.3, the induction hypothesis implies that the number of components of N T is at most a α−1 dim(N ) + 1. Putting these estimates together, we conclude that\\nand hence that t ≤ a α n + 1. This contradicts our definition of t , so the proof is complete.\\nand hence that t ≤ a α n + 1. This contradicts our definition of t , so the proof is complete.\\nWith the proof of Theorem 5.5 complete, it suffices to prove the following, which is the the second part of Theorem 5.1. where b α = 2 α−3 /(2 α−3 − 1).\\nWith the proof of Theorem 5.5 complete, it suffices to prove the following, which is the the second part of Theorem 5.1. where b α = 2 α−3 /(2 α−3 − 1).\\nThe first step of the proof is to find submanifolds that have 4-periodic cohomology and that cover multiple components of M T . The second step is to use these submanifolds together with Conner's theorem to conclude the upper bound on b even (M T ). To quantify the first step, we prove the following: Taking h = log bα ((b α − 1)t + 1) , we see that we can cover M T by rationally 4periodic submanifolds N 1 , . . . , N h ⊆ M . Applying this fact together with Conner's theorem, we have b even (M T ) =\\nThe first step of the proof is to find submanifolds that have 4-periodic cohomology and that cover multiple components of M T . The second step is to use these submanifolds together with Conner's theorem to conclude the upper bound on b even (M T ). To quantify the first step, we prove the following: Taking h = log bα ((b α − 1)t + 1) , we see that we can cover M T by rationally 4periodic submanifolds N 1 , . . . , N h ⊆ M . Applying this fact together with Conner's theorem, we have b even (M T ) =\\nRecall that we have b even (N i ) ≤ n 2 + 1 for all i, hence, by using the first part of the theorem, we can conclude the theorem by estimating h as follows:\\nRecall that we have b even (N i ) ≤ n 2 + 1 for all i, hence, by using the first part of the theorem, we can conclude the theorem by estimating h as follows:\\nh ≤ log bα ((b α − 1)(a α n + 1) + 1) = log bα 1 2 α−3 − 1 3 · 2 α−4 α n + b α = 1 + log bα 3n 2α + 1 < 1 + log bα n 2 + 1 .\\nh ≤ log bα ((b α − 1)(a α n + 1) + 1) = log bα 1 2 α−3 − 1 3 · 2 α−4 α n + b α = 1 + log bα 3n 2α + 1 < 1 + log bα n 2 + 1 .\"\n },\n {\n \"title\": \"On Bounding the Euler Characteristic from Below\",\n \"content\": \"\"\n },\n {\n \"title\": \"On Bounding the Euler Characteristic from Below\",\n \"content\": \"In this section, we prove the following, which contains Theorem E: We remark that the assumption dim(T ) ≥ 2 log 2 (n) − 2 suffices in the first case. Indeed, it follows from [KEN13, Theorem A] that b 2i+1 (M T ) = 0, hence the argument below implies that χ(M ) ≥ 2. We proceed to the proof, which will require the rest of this section.\\nIn this section, we prove the following, which contains Theorem E: We remark that the assumption dim(T ) ≥ 2 log 2 (n) − 2 suffices in the first case. Indeed, it follows from [KEN13, Theorem A] that b 2i+1 (M T ) = 0, hence the argument below implies that χ(M ) ≥ 2. We proceed to the proof, which will require the rest of this section.\\nFirst, the result on equivariant formality follows directly from the properties above it. We note that, if we make the slightly stronger assumption that dim(T ) ≥ log 4/3 (n), then this theorem and Theorem A imply two-sided bounds on the Betti numbers of M .\\nFirst, the result on equivariant formality follows directly from the properties above it. We note that, if we make the slightly stronger assumption that dim(T ) ≥ log 4/3 (n), then this theorem and Theorem A imply two-sided bounds on the Betti numbers of M .\\nSecond, we use the following theorem (see [\\nSecond, we use the following theorem (see [\"\n },\n {\n \"title\": \"Theorem 2. 7\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2. 7\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2. 9 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2. 9 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3 . 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3 . 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 3. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"⊆\",\n \"content\": \"\"\n },\n {\n \"title\": \"⊆\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"TjF.\",\n \"content\": \"\"\n },\n {\n \"title\": \"TjF.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Remark 4. 3 .•\",\n \"content\": \"\"\n },\n {\n \"title\": \"Remark 4. 3 .•\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5 . 2 .( 1 )\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5 . 2 .( 1 )\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 5. 5 [\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 5. 5 [\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 5. 6 [\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 5. 6 [\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5 . 7 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5 . 7 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 6. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 6. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"As a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h.","content":"As a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h."},{"id":"","caption":"As a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h.","content":"As a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h."},{"id":"","caption":"Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l.","content":"Theorem 2. 7\n7Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l."},{"id":"","caption":"Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l.","content":"Theorem 2. 7\n7Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l."},{"id":"","caption":"Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i.","content":"Theorem 2. 9 (\n9Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i."},{"id":"","caption":"Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i.","content":"Theorem 2. 9 (\n9Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i."},{"id":"","caption":"Theorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that","content":"Theorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that"},{"id":"","caption":"Theorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that","content":"Theorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that"},{"id":"","caption":"Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let","content":"Proposition 3 . 1 .\n31Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let"},{"id":"","caption":"Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let","content":"Proposition 3 . 1 .\n31Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let"},{"id":"","caption":"is 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic.","content":"is 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic."},{"id":"","caption":"is 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic.","content":"is 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic."},{"id":"","caption":"Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M .","content":"Proposition 3. 2 .\n2Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M ."},{"id":"","caption":"Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M .","content":"Proposition 3. 2 .\n2Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M ."},{"id":"","caption":"Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic.","content":"Proposition 3 . 3 .\n33Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic."},{"id":"","caption":"Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic.","content":"Proposition 3 . 3 .\n33Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic."},{"id":"","caption":"Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387","content":"Proposition 3. 4 .\n4Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387"},{"id":"","caption":"Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387","content":"Proposition 3. 4 .\n4Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387"},{"id":"","caption":"SO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism","content":"⊆\nSO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism"},{"id":"","caption":"SO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism","content":"⊆\nSO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism"},{"id":"","caption":"Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then","content":"Theorem 4. 1 .\n1Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then"},{"id":"","caption":"Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then","content":"Theorem 4. 1 .\n1Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then"},{"id":"","caption":"M T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s","content":"M T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s"},{"id":"","caption":"M T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s","content":"M T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s"},{"id":"","caption":"≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n).","content":"TjF.\n≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n)."},{"id":"","caption":"≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n).","content":"TjF.\n≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n)."},{"id":"","caption":"Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )","content":"Remark 4. 3 .•\n3Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )"},{"id":"","caption":"Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )","content":"Remark 4. 3 .•\n3Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )"},{"id":"","caption":"Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1.","content":"Lemma 5 . 2 .( 1 )\n521Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1."},{"id":"","caption":"Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1.","content":"Lemma 5 . 2 .( 1 )\n521Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1."},{"id":"","caption":"T r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0.","content":"T r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0."},{"id":"","caption":"T r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0.","content":"T r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0."},{"id":"","caption":"Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology.","content":"Lemma 5. 4 .\n4Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology."},{"id":"","caption":"Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology.","content":"Lemma 5. 4 .\n4Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology."},{"id":"","caption":"Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α.","content":"Theorem 5. 5 [\n5Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α."},{"id":"","caption":"Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α.","content":"Theorem 5. 5 [\n5Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α."},{"id":"","caption":"Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )","content":"Theorem 5. 6 [\n6Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )"},{"id":"","caption":"Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )","content":"Theorem 5. 6 [\n6Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )"},{"id":"","caption":"Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) .","content":"Lemma 5 . 7 (\n57Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) ."},{"id":"","caption":"Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) .","content":"Lemma 5 . 7 (\n57Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) ."},{"id":"","caption":"Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M ).","content":"Theorem 6. 1 .\n1Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M )."},{"id":"","caption":"Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M ).","content":"Theorem 6. 1 .\n1Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M )."},{"id":"","caption":"Theorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP","content":"Theorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP2 \n\nGAFA \n\nPOSITIVE CURVATURE UNDER SYMMETRY \n1381 \n\n(2) M has the integral cohomology of HP n/4 , \n(3) n ≡ 2 mod 4 n−2 \n\n4 ; F), \nor \n(4) n ≡ 3 mod 4 n−3 \n\n4"},{"id":"","caption":"Theorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP","content":"Theorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP2 \n\nGAFA \n\nPOSITIVE CURVATURE UNDER SYMMETRY \n1381 \n\n(2) M has the integral cohomology of HP n/4 , \n(3) n ≡ 2 mod 4 n−2 \n\n4 ; F), \nor \n(4) n ≡ 3 mod 4 n−3 \n\n4"},{"id":"","caption":"Some approximated values","content":"Table 1\n1Some approximated valuesi = \nn i = \nκ i = \n\n0 \n0 \n1 \n1 \n5 4 \n3 .14823 × 10 −15 \n2 \n7 4 \n1 .45259 × 10 −15 \n3 \n100 \n4.80780 × 10 −16 \n4 \n135 \n3.40869 × 10 −16 \n5 \n183 \n1.10871 × 10 −16 \n6 \n247 \n2.15684 × 10 −17"},{"id":"","caption":"Some approximated values","content":"Table 1\n1Some approximated valuesi = \nn i = \nκ i = \n\n0 \n0 \n1 \n1 \n5 4 \n3 .14823 × 10 −15 \n2 \n7 4 \n1 .45259 × 10 −15 \n3 \n100 \n4.80780 × 10 −16 \n4 \n135 \n3.40869 × 10 −16 \n5 \n183 \n1.10871 × 10 −16 \n6 \n247 \n2.15684 × 10 −17"},{"id":"","caption":"BRE72, Corollary IV.2.3, p. 178]):","content":"BRE72, Corollary IV.2.3, p. 178]):"},{"id":"","caption":"BRE72, Corollary IV.2.3, p. 178]):","content":"BRE72, Corollary IV.2.3, p. 178]):"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"As a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h.\",\n \"content\": \"As a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h.\",\n \"content\": \"As a consequence, the map H i (N ; Z) → H i (M ; Z) induced by inclusion is an isomorphism for i < h and a surjection for i = h, and the map H i (M ; Z) → H i (N ; Z) induced by inclusion is an isomorphisms for i < h and an injection for i = h.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l.\",\n \"content\": \"Theorem 2. 7\\n7Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l.\",\n \"content\": \"Theorem 2. 7\\n7Let M n and N n−k be closed, connected, oriented manifolds. If the inclusion N → M is (n − k − l)-connected, then there exists x ∈ H k (M ; Z) such that the map H i (M ; Z) → H i+k (M ; Z) induced by multiplication by x is a surjection for l ≤ i < n − k − l and an injection for l < i ≤ n − k − l.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i.\",\n \"content\": \"Theorem 2. 9 (\\n9Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i.\",\n \"content\": \"Theorem 2. 9 (\\n9Periodicity theorem). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature. Let N n−k1 1 and N n−k2 2 be connected, closed, embedded, totally geodesic submanifolds that intersect transversely. If 2k 1 + 2k 2 ≤ n, the rational cohomology rings of M , N 1 , N 2 , and N 1 ∩ N 2 are 4-periodic. In particular, b 2i (M ) ≤ 1 for all i.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that\",\n \"content\": \"Theorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that\",\n \"content\": \"Theorem 2.10 (Kennard[KEN00], Theorem 2.2). Let n ≥ c ≥ 2 be even integers, and let M n be a closed, one-connected manifold with positive sectional curvature. Assume T is a torus acting effectively by isometries on M such that\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let\",\n \"content\": \"Proposition 3 . 1 .\\n31Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let\",\n \"content\": \"Proposition 3 . 1 .\\n31Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and let T be a torus acting effectively by isometries on M . Assume M T is nonempty, and let\"\n },\n {\n \"id\": \"\",\n \"caption\": \"is 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic.\",\n \"content\": \"is 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"is 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic.\",\n \"content\": \"is 4-periodic by the periodicity theorem. As already established, this implies that H * (M ; Q) is 4-periodic.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M .\",\n \"content\": \"Proposition 3. 2 .\\n2Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M .\",\n \"content\": \"Proposition 3. 2 .\\n2Let M n be a closed, one-connected, positively curved Riemannian manifold, let T be a torus acting isometrically and effectively on M , and let T ⊆ T denote a subtorus that fixes a point x ∈ M .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic.\",\n \"content\": \"Proposition 3 . 3 .\\n33Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic.\",\n \"content\": \"Proposition 3 . 3 .\\n33Let M n be a closed, one-connected manifold with positive sectional curvature. If M both contains a totally geodesic submanifold of codimension k ≤ n+2 4 and has symmetry rank at least 2 log 2 n+ k 2 +1, then H * (M ; Q) is 4-periodic.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387\",\n \"content\": \"Proposition 3. 4 .\\n4Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387\",\n \"content\": \"Proposition 3. 4 .\\n4Let n ≥ c ≥ 0 be even. Suppose a torus T r acts effectively and isometrically on a closed, connected, positively curved manifold M n . If x 1 , . . . , x t ∈ M are fixed points of the T r -action on M , and if r ≥ tc 2 + log 2 (tn − tc + 2) , then there exists a nontrivial involution ι ∈ T r and a component N ⊆ M ι with cod(N ) ≤ n−c 2 such that N contains t+1 2 of the x i . The t = 1 statement is essentially Lemma 1.8 in [KEN00], and it implies that we can cover each fixed-point component of M T by a component of a fixed-point set of an involution with dimension at least n/2. An argument using the fact T r has only GAFA POSITIVE CURVATURE UNDER SYMMETRY 1387\"\n },\n {\n \"id\": \"\",\n \"caption\": \"SO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism\",\n \"content\": \"⊆\\nSO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism\"\n },\n {\n \"id\": \"\",\n \"caption\": \"SO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism\",\n \"content\": \"⊆\\nSO(T xi M ). Moreover, the map Z r 2 → Z n/2 2 has the property that the Hamming weight of the image of ι ∈ Z r 2 is half of the codimension of M ι xi . Consider the direct sum T r → SO t i=1 T xi M of the isotropy representations at x 1 , . . . , x t . Restricting to the involutions in T r , we obtain a homomorphism\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then\",\n \"content\": \"Theorem 4. 1 .\\n1Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then\",\n \"content\": \"Theorem 4. 1 .\\n1Let M n be a connected, closed, even-dimensional Riemannian manifold with positive sectional curvature. If a torus T of dimension at least log 4/3 (n) acts effectively and isometrically on M , then\"\n },\n {\n \"id\": \"\",\n \"caption\": \"M T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s\",\n \"content\": \"M T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s\"\n },\n {\n \"id\": \"\",\n \"caption\": \"M T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s\",\n \"content\": \"M T has t components, and choose one point x i in each component. The assumption in Case 2 implies that there exist nontrivial involutions ι 1 , . . . , ι t ∈ Z s\"\n },\n {\n \"id\": \"\",\n \"caption\": \"≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\\\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n).\",\n \"content\": \"TjF.\\n≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\\\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\\\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n).\",\n \"content\": \"TjF.\\n≤ f 0 (n j ). But the components of M ιj xj Tj are precisely the components of M T that also lie in M ιj xj . Moreover, n j < 3n 4 , so n j ≤ 2 We will use this estimate in a moment. The key observation is that the set {ι 1 , . . . , ι t } ⊆ Z s 2 \\\\ {id} has at most 2 s − 1 = 2 s(n) − 1 elements. Moreover, since each n j ≥ n/2, we can conclude from Frankel's theorem that the set {M ι1 x1 , . . . , M ιt xt } also has at most 2 s(n) − 1 elements. Let J ⊆ {1, . . . , t} denote an index set in one-to-one correspondence with the set of M ιj xj . Since every component F ⊆ M T is contained in some M ιj xj T with j ∈ J, we have the estimate b even (M T ) = ⊆M Applying the estimate above and the recursive definition of f 0 , we see that b even (M T ) ≤ 2 s(n) − 1 f 0 2 3n − 4 8 = f 0 (n).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )\",\n \"content\": \"Remark 4. 3 .•\\n3Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )\",\n \"content\": \"Remark 4. 3 .•\\n3Make the following definitions.• Set n 0 = 0, n 1 = 54 and, for i ≥ 2, let n i be the minimal integer satisfying n i−1 ≤ 2 Set κ 0 = 1 and, for i ≥ 0, set κ i = f 0 (n i )\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1.\",\n \"content\": \"Lemma 5 . 2 .( 1 )\\n521Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1.\",\n \"content\": \"Lemma 5 . 2 .( 1 )\\n521Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with M is homotopy equivalent to S n or CP n/2 , (2) M has the integral cohomology of HP n/4 , (3) n ≡ 2 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP n−2 4 ; F), or (4) n ≡ 3 mod 4 and, for all fields F, the cohomology of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP In particular, if dim(T ) ≥ s 3 (n), then M has 4-periodic cohomology and b 2i (M T ) ≤ b 2i (M ) ≤ n 2 + 1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"T r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0.\",\n \"content\": \"T r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"T r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0.\",\n \"content\": \"T r → SO(T x M ) = SO(n) at x. The dimension of the (−1)-eigenspace of the image of an involution ι ∈ T r in SO(n) is equal to the codimension of M ι x . The proof of Corollary 3.2.a in[WIL03] states that such an nontrivial involution ι ∈ T r exists with cod(M ι x ) ≤ 7n/24 if r ≥ 0.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology.\",\n \"content\": \"Lemma 5. 4 .\\n4Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology.\",\n \"content\": \"Lemma 5. 4 .\\n4Assume M n is a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M with dim(T ) ≥ s α (n) and α ≥ 4. If there exists a nontrivial involution ι ∈ T and a point x ∈ M ι such that cod(M ι x ) ≤ n α or such that cod(M ι x ) ≤ n 2 and dim ker T | M ι x ≥ 2, then M has 4-periodic rational cohomology.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α.\",\n \"content\": \"Theorem 5. 5 [\\n5Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α.\",\n \"content\": \"Theorem 5. 5 [\\n5Theorem 5.1, Part 1]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), then b 0 (M T ) ≤ a α n + 1 where a α = 3 · 2 α−4 /α.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )\",\n \"content\": \"Theorem 5. 6 [\\n6Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )\",\n \"content\": \"Theorem 5. 6 [\\n6Theorem 5.1, Part 2]. Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively by isometries on M . If α ≥ 3 and dim(T ) ≥ s α (n), thenχ(M ) ≤ b 2i (M T )\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) .\",\n \"content\": \"Lemma 5 . 7 (\\n57Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) .\",\n \"content\": \"Lemma 5 . 7 (\\n57Chains with periodic tails). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature, and assume T is a torus acting effectively and isometrically on M . If α ≥ 3 and dim(T ) ≥ s α (n), then there exist 0 ≤ j ≤ α − 3 and a chain M = M n0 0 ⊇ · · · ⊇ M nj j of submanifolds satisfying which in turn holds if and only if h ≥ log bα ((b α − 1)t + 1) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M ).\",\n \"content\": \"Theorem 6. 1 .\\n1Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M ).\",\n \"content\": \"Theorem 6. 1 .\\n1Let M n be an even-dimensional, one-connected, closed Riemannian manifold with positive sectional curvature. If a torus T acts effectively by isometries on M with dim(T ) ≥ 2 log 2 (n) + 2, then b 2i+1 (M T ) = 0 and χ(M ) ≥ 2 if any one of the following holds: (1) n ≡ 0 mod 4. (2) b 2 (M ), b 3 (M ), or b 4 (M ) is zero. (3) the Bott-Grove-Halperin conjecture holds. If, in addition, the torus acts equivariantly formally, then the odd Betti numbers of M vanish and b 2i (M ) = χ(M ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP\",\n \"content\": \"Theorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP2 \\n\\nGAFA \\n\\nPOSITIVE CURVATURE UNDER SYMMETRY \\n1381 \\n\\n(2) M has the integral cohomology of HP n/4 , \\n(3) n ≡ 2 mod 4 n−2 \\n\\n4 ; F), \\nor \\n(4) n ≡ 3 mod 4 n−3 \\n\\n4\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP\",\n \"content\": \"Theorem 2.1 (Maximal symmetry rank classification, Grove and Searle[GS94]). If T r acts effectively by isometries on a closed, one-connected, positively curved Riemannian manifold, then r ≤ n+1 with equality only if M is diffeomorphic to S n or CP n/2 .Theorem 2.2 (Wilking[WIL03]). Let M n be a closed, one-connected Riemannian manifold with positive sectional curvature and symmetry rank r.• (Homotopy classification) If r ≥ n 4 +1 and n ≥ 10, then M is homotopy equivalent to S n , CP n/2 , HP n/4 , or CaP 2 .• (Cohomology classification) If r ≥ n 6 + 1 and n ≥ 6000, then (1) M is homotopy equivalent to S n or CP n/2 , and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (CP n/2 ; F) or H * (S 2 × HP and, for all fields F, the cohomology algebra of M with coefficients in F is isomorphic to either H * (S n ; F) or H * (S 3 × HP2 \\n\\nGAFA \\n\\nPOSITIVE CURVATURE UNDER SYMMETRY \\n1381 \\n\\n(2) M has the integral cohomology of HP n/4 , \\n(3) n ≡ 2 mod 4 n−2 \\n\\n4 ; F), \\nor \\n(4) n ≡ 3 mod 4 n−3 \\n\\n4\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Some approximated values\",\n \"content\": \"Table 1\\n1Some approximated valuesi = \\nn i = \\nκ i = \\n\\n0 \\n0 \\n1 \\n1 \\n5 4 \\n3 .14823 × 10 −15 \\n2 \\n7 4 \\n1 .45259 × 10 −15 \\n3 \\n100 \\n4.80780 × 10 −16 \\n4 \\n135 \\n3.40869 × 10 −16 \\n5 \\n183 \\n1.10871 × 10 −16 \\n6 \\n247 \\n2.15684 × 10 −17\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Some approximated values\",\n \"content\": \"Table 1\\n1Some approximated valuesi = \\nn i = \\nκ i = \\n\\n0 \\n0 \\n1 \\n1 \\n5 4 \\n3 .14823 × 10 −15 \\n2 \\n7 4 \\n1 .45259 × 10 −15 \\n3 \\n100 \\n4.80780 × 10 −16 \\n4 \\n135 \\n3.40869 × 10 −16 \\n5 \\n183 \\n1.10871 × 10 −16 \\n6 \\n247 \\n2.15684 × 10 −17\"\n },\n {\n \"id\": \"\",\n \"caption\": \"BRE72, Corollary IV.2.3, p. 178]):\",\n \"content\": \"BRE72, Corollary IV.2.3, p. 178]):\"\n },\n {\n \"id\": \"\",\n \"caption\": \"BRE72, Corollary IV.2.3, p. 178]):\",\n \"content\": \"BRE72, Corollary IV.2.3, p. 178]):\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". N = S, N = S,","doi":""},{"text":"N = S × R Where R ∈ {cp M, SO(m + 2)/SO(m) × SO. N = S × R where R ∈ {CP m , SO(m + 2)/SO(m) × SO(2)},","doi":""},{"text":"N = S × R Where R ∈ {hp M, SO(m + 3)/SO(m) × SO. N = S × R where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)},","doi":""},{"text":"N = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 2 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}, or","doi":""},{"text":"N = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 3 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}.","doi":""},{"text":"This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. Note that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. ), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and. in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 sNote that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. In Cases (1), (2), and (3), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and, in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 s .","doi":""},{"text":"Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors. Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors","doi":""},{"text":"Positive Quaternionic Kähler Manifolds. M Amann, WWU MünsterPhD thesisM. Amann. Positive Quaternionic Kähler Manifolds. PhD thesis, WWU Münster, (2009).","doi":""},{"text":"Trois remarques sur les variétés riemanniennesà courbure positive. M Berger, C. R. Math. Acad. Sci. 263M. Berger. Trois remarques sur les variétés riemanniennesà courbure positive. C. R. Math. Acad. Sci. Paris, 263 (1966), A76-A78.","doi":""},{"text":"Introduction to Compact Transformation Groups. G Bredon, Academic PressG. Bredon. Introduction to Compact Transformation Groups. Academic Press (1972).","doi":""},{"text":"Some examples of manifolds of nonnegative curvature. J Cheeger, J. Differential Geom. 8J. Cheeger. Some examples of manifolds of nonnegative curvature. J. Differential Geom., 8(1973), 623-628.","doi":""},{"text":"On the action of the circle group. P Conner, Michigan Math. J. 4P. Conner. On the action of the circle group. Michigan Math. J., 4 (1957), 241-247.","doi":""},{"text":"A 7-manifold with positive curvature. O Dearricott, Duke Math. J. 2O. Dearricott. A 7-manifold with positive curvature. Duke Math. J., (2)158 (2011), 307-346.","doi":""},{"text":"Manifolds with positive curvature. T Frankel, Pacific J. Math. 11T. Frankel. Manifolds with positive curvature. Pacific J. Math., 11 (1961), 165-174.","doi":""},{"text":"Curvature, diameter and Betti numbers. M Gromov, Comment. Math. Helv. 1M. Gromov. Curvature, diameter and Betti numbers. Comment. Math. Helv., (1)56 (1981), 179-195.","doi":""},{"text":"Geometry of, and via, symmetries. K Grove, Conformal, Riemannian and Lagrangian geometry. Knoxville, TN; Providence, RI27K. Grove. Geometry of, and via, symmetries. In Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), vol. 27 of Univ. Lecture Ser. Amer. Math. Soc., Providence, RI (2002), pp. 31-53.","doi":""},{"text":"GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405. GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405","doi":""},{"text":"XIII. Geometry, analysis, and algebraic geometry: Forty years of the. K Grove, Surveys in Differential Geometry. Developments around positive sectional curvatureK. Grove. Developments around positive sectional curvature. Surveys in Dif- ferential Geometry, Vol. XIII. Geometry, analysis, and algebraic geometry: Forty years of the Journal of Differential Geometry (2009), pp. 117-133.","doi":""},{"text":"Positively curved manifolds with maximal symmetry rank. K Grove, C Searle, J. Pure Appl. Algebra. 1K. Grove and C. Searle. Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Algebra, (1)91 (1994), 137-142.","doi":""},{"text":"An exotic T 1 S 4 with positive curvature. K Grove, L Verdiani, W Ziller, Geom. Funct. Anal. 3K. Grove, L. Verdiani, and W. Ziller. An exotic T 1 S 4 with positive curva- ture. Geom. Funct. Anal., (3)21 (2011), 499-524.","doi":""},{"text":"On the Hopf conjecture with symmetry. L Kennard, Geom. Topol. 17L. Kennard. On the Hopf conjecture with symmetry. Geom. Topol., 17 (2013), 563-593.","doi":""},{"text":"Positively curved Riemannian metrics with logarithmic symmetry rank bounds. L Kennard, arXiv:1209.4627v1Comment. Math. Helv. to appearL. Kennard. Positively curved Riemannian metrics with logarithmic symmetry rank bounds. Comment. Math. Helv., (to appear), arXiv:1209.4627v1.","doi":"arXiv:1209.4627v1"},{"text":"An exotic sphere with positive curvature. P Petersen, F Wilhelm, arXiv:0805.0812v3preprintP. Petersen and F. Wilhelm. An exotic sphere with positive curvature. preprint, arXiv:0805.0812v3.","doi":"arXiv:0805.0812v3"},{"text":"Positive curvature and the elliptic genus. N Weisskopf, arXiv:1305.5175v1preprintN. Weisskopf. Positive curvature and the elliptic genus. preprint, arXiv:1305.5175v1.","doi":"arXiv:1305.5175v1"},{"text":"Torus actions on manifolds of positive sectional curvature. B Wilking, Acta Math. 2B. Wilking. Torus actions on manifolds of positive sectional curvature. Acta Math., (2)191 (2003), 259-297.","doi":""},{"text":"Nonnegatively and Positively Curved Manifolds. B Wilking, Surveys in Differential Geometry. Metric and Comparison GeometryB. Wilking. Nonnegatively and Positively Curved Manifolds. Surveys in Differ- ential Geometry, Vol. XI. Metric and Comparison Geometry, (2007) pp. 25-62.","doi":""},{"text":"Examples of Riemannian manifolds with non-negative sectional curvature. W Ziller, Surveys in Differential Geometry. W. Ziller. Examples of Riemannian manifolds with non-negative sectional cur- vature. Surveys in Differential Geometry, Vol. XI. Metric and Comparison Geom- etry (2007), pp. 63-102.","doi":""},{"text":"Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit. Manuel Amann, Fakultät Für Mathematik, Manuel Amann, Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit.edu URL: http://topology.math.kit.edu/21 54.php","doi":""},{"text":". N = S, N = S,","doi":""},{"text":"N = S × R Where R ∈ {cp M, SO(m + 2)/SO(m) × SO. N = S × R where R ∈ {CP m , SO(m + 2)/SO(m) × SO(2)},","doi":""},{"text":"N = S × R Where R ∈ {hp M, SO(m + 3)/SO(m) × SO. N = S × R where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)},","doi":""},{"text":"N = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 2 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}, or","doi":""},{"text":"N = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 3 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}.","doi":""},{"text":"This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. Note that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. ), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and. in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 sNote that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. In Cases (1), (2), and (3), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and, in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 s .","doi":""},{"text":"Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors. Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors","doi":""},{"text":"Positive Quaternionic Kähler Manifolds. M Amann, WWU MünsterPhD thesisM. Amann. Positive Quaternionic Kähler Manifolds. PhD thesis, WWU Münster, (2009).","doi":""},{"text":"Trois remarques sur les variétés riemanniennesà courbure positive. M Berger, C. R. Math. Acad. Sci. 263M. Berger. Trois remarques sur les variétés riemanniennesà courbure positive. C. R. Math. Acad. Sci. Paris, 263 (1966), A76-A78.","doi":""},{"text":"Introduction to Compact Transformation Groups. G Bredon, Academic PressG. Bredon. Introduction to Compact Transformation Groups. Academic Press (1972).","doi":""},{"text":"Some examples of manifolds of nonnegative curvature. J Cheeger, J. Differential Geom. 8J. Cheeger. Some examples of manifolds of nonnegative curvature. J. Differential Geom., 8(1973), 623-628.","doi":""},{"text":"On the action of the circle group. P Conner, Michigan Math. J. 4P. Conner. On the action of the circle group. Michigan Math. J., 4 (1957), 241-247.","doi":""},{"text":"A 7-manifold with positive curvature. O Dearricott, Duke Math. J. 2O. Dearricott. A 7-manifold with positive curvature. Duke Math. J., (2)158 (2011), 307-346.","doi":""},{"text":"Manifolds with positive curvature. T Frankel, Pacific J. Math. 11T. Frankel. Manifolds with positive curvature. Pacific J. Math., 11 (1961), 165-174.","doi":""},{"text":"Curvature, diameter and Betti numbers. M Gromov, Comment. Math. Helv. 1M. Gromov. Curvature, diameter and Betti numbers. Comment. Math. Helv., (1)56 (1981), 179-195.","doi":""},{"text":"Geometry of, and via, symmetries. K Grove, Conformal, Riemannian and Lagrangian geometry. Knoxville, TN; Providence, RI27K. Grove. Geometry of, and via, symmetries. In Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), vol. 27 of Univ. Lecture Ser. Amer. Math. Soc., Providence, RI (2002), pp. 31-53.","doi":""},{"text":"GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405. GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405","doi":""},{"text":"XIII. Geometry, analysis, and algebraic geometry: Forty years of the. K Grove, Surveys in Differential Geometry. Developments around positive sectional curvatureK. Grove. Developments around positive sectional curvature. Surveys in Dif- ferential Geometry, Vol. XIII. Geometry, analysis, and algebraic geometry: Forty years of the Journal of Differential Geometry (2009), pp. 117-133.","doi":""},{"text":"Positively curved manifolds with maximal symmetry rank. K Grove, C Searle, J. Pure Appl. Algebra. 1K. Grove and C. Searle. Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Algebra, (1)91 (1994), 137-142.","doi":""},{"text":"An exotic T 1 S 4 with positive curvature. K Grove, L Verdiani, W Ziller, Geom. Funct. Anal. 3K. Grove, L. Verdiani, and W. Ziller. An exotic T 1 S 4 with positive curva- ture. Geom. Funct. Anal., (3)21 (2011), 499-524.","doi":""},{"text":"On the Hopf conjecture with symmetry. L Kennard, Geom. Topol. 17L. Kennard. On the Hopf conjecture with symmetry. Geom. Topol., 17 (2013), 563-593.","doi":""},{"text":"Positively curved Riemannian metrics with logarithmic symmetry rank bounds. L Kennard, arXiv:1209.4627v1Comment. Math. Helv. to appearL. Kennard. Positively curved Riemannian metrics with logarithmic symmetry rank bounds. Comment. Math. Helv., (to appear), arXiv:1209.4627v1.","doi":"arXiv:1209.4627v1"},{"text":"An exotic sphere with positive curvature. P Petersen, F Wilhelm, arXiv:0805.0812v3preprintP. Petersen and F. Wilhelm. An exotic sphere with positive curvature. preprint, arXiv:0805.0812v3.","doi":"arXiv:0805.0812v3"},{"text":"Positive curvature and the elliptic genus. N Weisskopf, arXiv:1305.5175v1preprintN. Weisskopf. Positive curvature and the elliptic genus. preprint, arXiv:1305.5175v1.","doi":"arXiv:1305.5175v1"},{"text":"Torus actions on manifolds of positive sectional curvature. B Wilking, Acta Math. 2B. Wilking. Torus actions on manifolds of positive sectional curvature. Acta Math., (2)191 (2003), 259-297.","doi":""},{"text":"Nonnegatively and Positively Curved Manifolds. B Wilking, Surveys in Differential Geometry. Metric and Comparison GeometryB. Wilking. Nonnegatively and Positively Curved Manifolds. Surveys in Differ- ential Geometry, Vol. XI. Metric and Comparison Geometry, (2007) pp. 25-62.","doi":""},{"text":"Examples of Riemannian manifolds with non-negative sectional curvature. W Ziller, Surveys in Differential Geometry. W. Ziller. Examples of Riemannian manifolds with non-negative sectional cur- vature. Surveys in Differential Geometry, Vol. XI. Metric and Comparison Geom- etry (2007), pp. 63-102.","doi":""},{"text":"Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit. Manuel Amann, Fakultät Für Mathematik, Manuel Amann, Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit.edu URL: http://topology.math.kit.edu/21 54.php","doi":""}],"string":"[\n {\n \"text\": \". N = S, N = S,\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N = S × R Where R ∈ {cp M, SO(m + 2)/SO(m) × SO. N = S × R where R ∈ {CP m , SO(m + 2)/SO(m) × SO(2)},\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N = S × R Where R ∈ {hp M, SO(m + 3)/SO(m) × SO. N = S × R where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)},\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 2 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}, or\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 3 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. Note that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. ), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and. in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 sNote that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. In Cases (1), (2), and (3), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and, in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 s .\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors. Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Positive Quaternionic Kähler Manifolds. M Amann, WWU MünsterPhD thesisM. Amann. Positive Quaternionic Kähler Manifolds. PhD thesis, WWU Münster, (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Trois remarques sur les variétés riemanniennesà courbure positive. M Berger, C. R. Math. Acad. Sci. 263M. Berger. Trois remarques sur les variétés riemanniennesà courbure positive. C. R. Math. Acad. Sci. Paris, 263 (1966), A76-A78.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to Compact Transformation Groups. G Bredon, Academic PressG. Bredon. Introduction to Compact Transformation Groups. Academic Press (1972).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Some examples of manifolds of nonnegative curvature. J Cheeger, J. Differential Geom. 8J. Cheeger. Some examples of manifolds of nonnegative curvature. J. Differential Geom., 8(1973), 623-628.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the action of the circle group. P Conner, Michigan Math. J. 4P. Conner. On the action of the circle group. Michigan Math. J., 4 (1957), 241-247.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A 7-manifold with positive curvature. O Dearricott, Duke Math. J. 2O. Dearricott. A 7-manifold with positive curvature. Duke Math. J., (2)158 (2011), 307-346.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Manifolds with positive curvature. T Frankel, Pacific J. Math. 11T. Frankel. Manifolds with positive curvature. Pacific J. Math., 11 (1961), 165-174.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Curvature, diameter and Betti numbers. M Gromov, Comment. Math. Helv. 1M. Gromov. Curvature, diameter and Betti numbers. Comment. Math. Helv., (1)56 (1981), 179-195.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geometry of, and via, symmetries. K Grove, Conformal, Riemannian and Lagrangian geometry. Knoxville, TN; Providence, RI27K. Grove. Geometry of, and via, symmetries. In Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), vol. 27 of Univ. Lecture Ser. Amer. Math. Soc., Providence, RI (2002), pp. 31-53.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405. GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405\",\n \"doi\": \"\"\n },\n {\n \"text\": \"XIII. Geometry, analysis, and algebraic geometry: Forty years of the. K Grove, Surveys in Differential Geometry. Developments around positive sectional curvatureK. Grove. Developments around positive sectional curvature. Surveys in Dif- ferential Geometry, Vol. XIII. Geometry, analysis, and algebraic geometry: Forty years of the Journal of Differential Geometry (2009), pp. 117-133.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Positively curved manifolds with maximal symmetry rank. K Grove, C Searle, J. Pure Appl. Algebra. 1K. Grove and C. Searle. Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Algebra, (1)91 (1994), 137-142.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An exotic T 1 S 4 with positive curvature. K Grove, L Verdiani, W Ziller, Geom. Funct. Anal. 3K. Grove, L. Verdiani, and W. Ziller. An exotic T 1 S 4 with positive curva- ture. Geom. Funct. Anal., (3)21 (2011), 499-524.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Hopf conjecture with symmetry. L Kennard, Geom. Topol. 17L. Kennard. On the Hopf conjecture with symmetry. Geom. Topol., 17 (2013), 563-593.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Positively curved Riemannian metrics with logarithmic symmetry rank bounds. L Kennard, arXiv:1209.4627v1Comment. Math. Helv. to appearL. Kennard. Positively curved Riemannian metrics with logarithmic symmetry rank bounds. Comment. Math. Helv., (to appear), arXiv:1209.4627v1.\",\n \"doi\": \"arXiv:1209.4627v1\"\n },\n {\n \"text\": \"An exotic sphere with positive curvature. P Petersen, F Wilhelm, arXiv:0805.0812v3preprintP. Petersen and F. Wilhelm. An exotic sphere with positive curvature. preprint, arXiv:0805.0812v3.\",\n \"doi\": \"arXiv:0805.0812v3\"\n },\n {\n \"text\": \"Positive curvature and the elliptic genus. N Weisskopf, arXiv:1305.5175v1preprintN. Weisskopf. Positive curvature and the elliptic genus. preprint, arXiv:1305.5175v1.\",\n \"doi\": \"arXiv:1305.5175v1\"\n },\n {\n \"text\": \"Torus actions on manifolds of positive sectional curvature. B Wilking, Acta Math. 2B. Wilking. Torus actions on manifolds of positive sectional curvature. Acta Math., (2)191 (2003), 259-297.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonnegatively and Positively Curved Manifolds. B Wilking, Surveys in Differential Geometry. Metric and Comparison GeometryB. Wilking. Nonnegatively and Positively Curved Manifolds. Surveys in Differ- ential Geometry, Vol. XI. Metric and Comparison Geometry, (2007) pp. 25-62.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Examples of Riemannian manifolds with non-negative sectional curvature. W Ziller, Surveys in Differential Geometry. W. Ziller. Examples of Riemannian manifolds with non-negative sectional cur- vature. Surveys in Differential Geometry, Vol. XI. Metric and Comparison Geom- etry (2007), pp. 63-102.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit. Manuel Amann, Fakultät Für Mathematik, Manuel Amann, Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit.edu URL: http://topology.math.kit.edu/21 54.php\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N = S, N = S,\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N = S × R Where R ∈ {cp M, SO(m + 2)/SO(m) × SO. N = S × R where R ∈ {CP m , SO(m + 2)/SO(m) × SO(2)},\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N = S × R Where R ∈ {hp M, SO(m + 3)/SO(m) × SO. N = S × R where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)},\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 2 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}, or\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N = S × R × S, where R ∈ {HP m , SO(m + 3)/SO(m) × SO. N = S × R × S 3 where R ∈ {HP m , SO(m + 3)/SO(m) × SO(3)}.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. Note that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. ), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and. in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 sNote that the number of spherical factors s (as in the statement of the corollary) is t in the first three cases and is t + 1 in the last two cases. It follows that b 2 (M ) = 0 or b 3 (M ) = 0. In particular, χ(M ) > 0 by Theorem E. This implies that each n i is even, that Case (5) cannot occur, and that R = SO(m + 3)/SO(m) × SO(3) can only occur if m is even. In Cases (1), (2), and (3), we estimate the Euler characteristic by χ(N ) ≥ χ(S) = 2 t = 2 s , and, in Case (4), we estimate χ(N ) ≥ χ(S)χ(S 2 ) = 2 t+1 = 2 s .\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors. Since χ(N ) = χ(M ) < 2 3(log 2 n) 2 , we conclude that the number of spherical factors\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Positive Quaternionic Kähler Manifolds. M Amann, WWU MünsterPhD thesisM. Amann. Positive Quaternionic Kähler Manifolds. PhD thesis, WWU Münster, (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Trois remarques sur les variétés riemanniennesà courbure positive. M Berger, C. R. Math. Acad. Sci. 263M. Berger. Trois remarques sur les variétés riemanniennesà courbure positive. C. R. Math. Acad. Sci. Paris, 263 (1966), A76-A78.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to Compact Transformation Groups. G Bredon, Academic PressG. Bredon. Introduction to Compact Transformation Groups. Academic Press (1972).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Some examples of manifolds of nonnegative curvature. J Cheeger, J. Differential Geom. 8J. Cheeger. Some examples of manifolds of nonnegative curvature. J. Differential Geom., 8(1973), 623-628.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the action of the circle group. P Conner, Michigan Math. J. 4P. Conner. On the action of the circle group. Michigan Math. J., 4 (1957), 241-247.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A 7-manifold with positive curvature. O Dearricott, Duke Math. J. 2O. Dearricott. A 7-manifold with positive curvature. Duke Math. J., (2)158 (2011), 307-346.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Manifolds with positive curvature. T Frankel, Pacific J. Math. 11T. Frankel. Manifolds with positive curvature. Pacific J. Math., 11 (1961), 165-174.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Curvature, diameter and Betti numbers. M Gromov, Comment. Math. Helv. 1M. Gromov. Curvature, diameter and Betti numbers. Comment. Math. Helv., (1)56 (1981), 179-195.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geometry of, and via, symmetries. K Grove, Conformal, Riemannian and Lagrangian geometry. Knoxville, TN; Providence, RI27K. Grove. Geometry of, and via, symmetries. In Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), vol. 27 of Univ. Lecture Ser. Amer. Math. Soc., Providence, RI (2002), pp. 31-53.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405. GAFA POSITIVE CURVATURE UNDER SYMMETRY 1405\",\n \"doi\": \"\"\n },\n {\n \"text\": \"XIII. Geometry, analysis, and algebraic geometry: Forty years of the. K Grove, Surveys in Differential Geometry. Developments around positive sectional curvatureK. Grove. Developments around positive sectional curvature. Surveys in Dif- ferential Geometry, Vol. XIII. Geometry, analysis, and algebraic geometry: Forty years of the Journal of Differential Geometry (2009), pp. 117-133.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Positively curved manifolds with maximal symmetry rank. K Grove, C Searle, J. Pure Appl. Algebra. 1K. Grove and C. Searle. Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Algebra, (1)91 (1994), 137-142.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An exotic T 1 S 4 with positive curvature. K Grove, L Verdiani, W Ziller, Geom. Funct. Anal. 3K. Grove, L. Verdiani, and W. Ziller. An exotic T 1 S 4 with positive curva- ture. Geom. Funct. Anal., (3)21 (2011), 499-524.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Hopf conjecture with symmetry. L Kennard, Geom. Topol. 17L. Kennard. On the Hopf conjecture with symmetry. Geom. Topol., 17 (2013), 563-593.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Positively curved Riemannian metrics with logarithmic symmetry rank bounds. L Kennard, arXiv:1209.4627v1Comment. Math. Helv. to appearL. Kennard. Positively curved Riemannian metrics with logarithmic symmetry rank bounds. Comment. Math. Helv., (to appear), arXiv:1209.4627v1.\",\n \"doi\": \"arXiv:1209.4627v1\"\n },\n {\n \"text\": \"An exotic sphere with positive curvature. P Petersen, F Wilhelm, arXiv:0805.0812v3preprintP. Petersen and F. Wilhelm. An exotic sphere with positive curvature. preprint, arXiv:0805.0812v3.\",\n \"doi\": \"arXiv:0805.0812v3\"\n },\n {\n \"text\": \"Positive curvature and the elliptic genus. N Weisskopf, arXiv:1305.5175v1preprintN. Weisskopf. Positive curvature and the elliptic genus. preprint, arXiv:1305.5175v1.\",\n \"doi\": \"arXiv:1305.5175v1\"\n },\n {\n \"text\": \"Torus actions on manifolds of positive sectional curvature. B Wilking, Acta Math. 2B. Wilking. Torus actions on manifolds of positive sectional curvature. Acta Math., (2)191 (2003), 259-297.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonnegatively and Positively Curved Manifolds. B Wilking, Surveys in Differential Geometry. Metric and Comparison GeometryB. Wilking. Nonnegatively and Positively Curved Manifolds. Surveys in Differ- ential Geometry, Vol. XI. Metric and Comparison Geometry, (2007) pp. 25-62.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Examples of Riemannian manifolds with non-negative sectional curvature. W Ziller, Surveys in Differential Geometry. W. Ziller. Examples of Riemannian manifolds with non-negative sectional cur- vature. Surveys in Differential Geometry, Vol. XI. Metric and Comparison Geom- etry (2007), pp. 63-102.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit. Manuel Amann, Fakultät Für Mathematik, Manuel Amann, Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstraße 89-93, 76133 Karlsruhe, Germany manuel.amann@kit.edu URL: http://topology.math.kit.edu/21 54.php\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":31,"cells":{"text":{"kind":"string","value":"\nOPERATOR ALGEBRAS OF FUNCTIONS\n29 Jul 2009\n\nMeghna Mittal \nVern I Paulsen \nOPERATOR ALGEBRAS OF FUNCTIONS\n29 Jul 2009\nWe present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.\n\nIntroduction\n\nA concrete operator algebra A is just a subalgebra of B(H), the bounded operators on a Hilbert space H. The operator norm on B(H) gives rise to a norm on A. Moreover, the identification M n (A) ∼ = A ⊗ M n ⊆ B(H ⊗ C n ) ∼ = B(H n ) endows the matrices over A with a family of norms in a natural way, where M n denotes the algebra of n × n matrices. It is common practice to identify two operator algebras A and B as being the \"same\" if and only if there exists an algebra isomorphism π : A → B that is not only an isometry, but which also preserves all the matrix norms, that is such that (π(a i,j )) Mn(B) = (a i,j ) Mn(A) , for every n and every element (a i,j ) ∈ M n (A). Such a map π is called a completely isometric isomorphism. In [13] an abstract characterization of operator algebras, in the above sense, was given and since that time a theory of such algebras has evolved. For more details on the abstract theory of operator algebras, see [11], [23] or [27].\n\nIn this note we present a theory for a special class of abstract abelian operator algebras which contains many important examples arising in function theoretic operator theory, including the Schur-Agler and the Arveson-Drury-Popescu algebras. This theory will allow us to answer certain types of questions about such algebras in a unified manner. We will prove that our hypotheses give an abstract characterization of operator algebras that are completely isometrically isomorphic to multiplier algebras of vector-valued reproducing kernel Hilbert spaces.\n\nOur results will show that under certain mild hypotheses, operator algebra norms which are defined by taking the supremum of certain families of operators on Hilbert spaces of arbitrary dimensions can be obtained by restricting the family of operators to finite dimensional Hilbert spaces. Thus, in a certain sense, which will be explained later, our results give conditions that guarantee that an algebra is residually finite dimensional.\n\nFinally, we apply our results to study \"quantized function theories\" on various domains. Our work in this direction should be compared to that of Ambrozie-Timotin [4], Ball-Bolotnikov [10] and Kalyuzhnyi-Verbovetzkii [20]. These authors obtain the same type of factorization theorem via unitary colligation methods as we obtain via operator algebra methods, but they assume somewhat different(and in many cases stronger) hypotheses. We, also, obtain a bit more information about the algebras themselves, including the fact that in many cases their norms can be obtained as supremums over matrices instead of operators.\n\nWe now give the relevant definitions. Recall that given any set X the set of all complex-valued functions on X is an algebra over the field of complex numbers.\n\nDefinition 1.1. We call A an operator algebra of functions on a set X provided:\n\n(1) A is a subalgebra of the algebra of functions on X, (2) A separates the points of X and contains the constant functions, (3) for each n M n (A) is equipped with a norm . Mn(A) , such that the set of norms satisfy the BRS axioms [13] to be an abstract operator algebra, (4) for each x ∈ X, the evaluation functional, π x : A → C, given by π x (f ) = f (x) is bounded.\n\nA few remarks and observations are in order. First note that if A is an operator algebra of functions on X and B ⊆ A is any subalgebra, which contains the constant functions and still separates points, then B, equipped with the norms that M n (B) inherits as a subspace of M n (A) is still an operator algebra of functions.\n\nThe basic example of an operator algebra of functions is ℓ ∞ (X), the algebra of all bounded functions on X. If for (f i,j ) ∈ M n (ℓ ∞ (X)) we set\n(f i,j ) Mn(ℓ ∞ (X)) = (f i,j ) ∞ ≡ sup{ (f i,j (x)) Mn : x ∈ X},\nwhere · Mn is the norm on M n obtained via the identification M n = B(C n ), then it readily follows that properties (1)-(4) of the above definition are satisfied. Thus, ℓ ∞ (X) is an operator algebra of functions on X in our sense and any subalgebra of ℓ ∞ (X) that contains the constants and separates points will be an operator algebra of functions on X when equipped with the subspace norms. Proof. Since π x : A → C is bounded and the norm is sub-multiplicative, we have that for any f ∈ A, |f (x)| n = |π x (f n )| ≤ π x f n ≤ π x f n . Taking the n-th root of each side of this inequality and letting n → +∞, yields |f (x)| ≤ f , and hence, f ∈ ℓ ∞ (X). Note also that π x = 1. Finally, since every bounded, linear functional on an operator space is completely bounded and the norm and the cb-norm are equal, we have that π x cb = π x = 1. Thus, for (f i,j ) ∈ M n (A), we have (f i,j (x)) Mn = (π x (f i,j )) ≤ (f i,j ) Mn(A) .\n\nGiven an operator algebra of functions on a set X, A and F = {x 1 , ..., x k } a set of distinct points in X, we define I F = {f ∈ A : f (z) = 0 ∀ z ∈ F } and note that M n (I F ) = {f ∈ M n (A) : f (z) = 0 ∀ z ∈ F } ∀ n. The quotient space A/I F has a natural set of matrix norms given by defining (f i,j + I F ) = inf{ (f i,j + g i,j ) Mn(A) : g i,j ∈ I F }. Alternatively, this is the norm on M n (A/I F ) that comes via the identification, M n (A/I F ) = M n (A)/M n (I F ), where the latter space is given its quotient norm. It is easily checked that this family of matrix norms satisfies the BRS conditions and so gives A/I F the structure of an abstract operator algebra.\n\nWe let π F (f ) = f + I F denote the quotient map π F : A → A/I F so that for each n, π Since A is an algebra which separates points on X and contains constant functions, it follows that there exists functions f 1 , ..., f k ∈ A, such that f i (x j ) = δ i,j , where δ i,j denotes the Dirac delta function. If we set E j = π F (f j ), then it is easily seen that whenever f ∈ A and f (x i ) = λ i , i = 1, ..., k, then π F (f ) = λ 1 E 1 + · · · + λ k E k . Moreover, E i E j = δ i,j E i , and E 1 + · · · + E k = 1, where 1 denotes the identity of the algebra A/I F . Thus, A/I F = span{E 1 , ..., E k }, is a unital algebra spanned by k commuting idempotents. Such algebras were called k-idempotent operator algebras in [24] and we will use a number of results from that paper.\n\n\nDefinition 1.3. An operator algebra of functions A on a set X, is called a local operator algebra of functions if it satisfies\nsup F π (n)\nF ((f ij )) = (f ij ) ∀ (f ij ) ∈ M n (A) and for every n, where the supremum is taken over all finite subsets of X.\n\nThe following result shows that every operator algebra of functions can be re-normed so that it becomes local. Proposition 1.4. Let A be an operator algebra of functions on X, let A L = A and define a family of matrix norms on A L , by setting (f i,j ) Mn(A L ) = sup F (π F (f i,j )) Mn(A/I F ) . Then A L is a local operator algebra of functions on X and the identity map, id : A → A L , is completely contractive.\n\nProof. It is clear from the definition of the norms on A L that the identity map is completely contractive and it is readily checked that A L is an operator algebra of functions on X.\n\nLetπ F :\nA L → A L /I F , denote the quotient map, so that π F (f ) = inf{ f + g L : g ∈ I F } ≤ inf{ f + g : g ∈ I F } = π F (f ) , since f + g L ≤ f + g .\nWe claim that for any f ∈ A, and any finite subset F ⊆ X, we have that π F (f ) = π F (f ) . To see the other inequality note that for g ∈ I F , and G ⊆ X a finite set, we have f + g L = sup G π G (f + g) ≥ π F (f + g) = π F (f ) . Hence, π F (f ) ≥ π F (f ) , and equality follows. A similar calculation shows that (π F (f i,j )) = (π F (f i,j )) , for any matrix of functions.\n\nNow it easily follows that A L is local, since\nsup F (π F (f i,j )) = sup F (π F (f i,j )) = (f i,j ) Mn(A L ) .\nWe letà denote the set of functions that are BPW limits of bounded nets of functions in A. Definition 1.5. Given an operator algebra of functions A on X we say that f : X → C is a BPW limit of A if there exists a uniformly bounded net (f λ ) λ ∈ A that converges pointwise on X to f. We say that A is BPW complete if it contains the set of functions that are BPW limits of bounded nets of functions in A. Letà denote the set of such functions, that is,\nA = {f : ∃ a bounded net f λ ⊆ A such that f (z) = lim λ f λ (z) ∀ z ∈ X} Given (f i,j ) ∈ M n (Ã), we set (f i,j ) Mn(Ã) = inf {C : (f i,j (x)) = lim λ (f λ i,j (x)) and (f λ i,j ) ≤ C}\nIt is easily checked that for each n, the above formula defines a norm on M n (Ã). It is also easily checked that a matrix-valued function, (f i,j ) : X → M n is the pointwise limit of a uniformly bounded net (f λ i,j ) ∈ M n (A) if and only if f i,j ∈à for every i, j. Lemma 1.6. Let A be an operator algebra of functions on X and let (f i,j ) ∈ M n (Ã). Then\n(f i,j ) Mn(Ã) = inf {C : ∀F ⊆ X finite ∃ g F i,j ∈ A with (f i,j | F ) = (g F i,j | F ), (g F i,j ) ≤ C}. Proof.\nThe collection of finite subsets of X determines a directed set, ordered by inclusion. If we choose for each finite set F, functions (g F i,j ) satisfying the conditions of the right hand set, then these functions define a net that converges BPW to (f i,j ) and hence, the right hand side is larger than the left. Conversely, given a net (f λ i,j ) that converges pointwise to (f i,j ) and satisfies (f λ i,j ) ≤ C and any finite set\nF = {x 1 , ..., x k }, choose functions in A such that f i (x j ) = δ i,j . If we let A λ l = (f i,j (x l )) − (f λ i,j (x l )), then (g λ i,j ) = (f λ i,j ) + A λ 1 f 1 + · · · A λ k f k ∈ M n (A) and is equal to (f i,j ) on F. Moreover, (g λ i,j ) ≤ (f λ i,j ) + A λ 1 Mn f 1 A + · · · + A λ k Mn f k A .\nThus, given ǫ > 0, since the functions f 1 , ..., f k depend only on F, we may choose λ so that (g λ i,j ) < C + ǫ. This shows the other inequality. Lemma 1.7. Let A be an operator algebra of functions on the set X, theñ A equipped with the collection of norms on M n (Ã) given in Definition 1.5 is an operator algebra.\n\nProof. It is clear from the definition ofà that it is an algebra. Thus, it is enough to check that the axioms of BRS are satisfied by the algebraà equipped with the matrix norms given in the Definition 1.5.\n\nIf L and M are scalar matrices of appropriate sizes and G ∈ M n (Ã), then\nfor ǫ > 0 there exists G λ ∈ M n (A) such that lim λ G λ (x) = G(x) ∀ x ∈ X and sup λ G λ Mn(A) ≤ G Mn(Ã) +ǫ. Since A is an operator space, LG λ M ∈ M n (A) and LG λ M Mn(A) ≤ L G λ Mn(A) M . Note that it follows that LGM Mn(Ã) ≤ L G Mn(Ã) M , since LG λ M → LGM pointwise and sup λ LG λ M Mn(A) ≤ L ( G Mn(Ã) + ǫ) M ∀ ǫ > 0. If G, H ∈ M n (Ã), then for every ǫ > 0 there exists G λ , H λ ∈ M n (A) such that lim λ G λ (x) = lim λ G(x) and lim λ H λ (x) = H(x) ∀x ∈ X. Also, we have that sup λ G λ Mn(A) ≤ G Mn(Ã) + ǫ and sup λ H λ Mn(A) ≤ H Mn(Ã) + ǫ. Let L = GH and L λ = G λ H λ . Since A is matrix normed algebra, L λ ∈ M n (A) and L λ Mn(A) ≤ G λ Mn(A) H λ Mn(A) for every λ. ⇒ lim λ L λ (x) = L(x) and sup λ L λ Mn(A) ≤ sup λ G λ Mn(A) sup λ H λ Mn(A) .\nThis yields L Mn(Ã) ≤ G Mn(Ã) H Mn(Ã) , and so the multiplication is completely contractive.\n\nFinally, to see that the L ∞ conditions are met, let G ∈ M n (Ã) and\nH ∈ M m (Ã). Given ǫ > 0, ∃ G λ ∈ M n (A) and H λ ∈ M m (A) such that lim λ G λ (x) = G(x) , lim λ H λ (x) = H(x) and sup λ G λ Mn(A) ≤ G Mn(Ã) + ǫ, sup λ H λ Mn(A) ≤ H Mn(Ã) + ǫ. Note that G λ ⊕H λ ∈ M n+m (A) and G λ ⊕H λ = max{ G λ Mn(A) , H λ Mn(A) } for every λ which implies that G ⊕ H ∈ M n+m (Ã), and G ⊕ H M n+m (Ã) ≤ sup λ G λ ⊕ H λ = sup λ [max{ G λ Mn(A) , H λ Mm(A) }] = max{sup λ G λ Mn(A) , sup λ H λ Mm(A) } ≤ max{ G Mn(Ã) + ǫ, H Mm(Ã) + ǫ} ∀ ǫ > 0.\nThis shows that G⊕ H M n+m (Ã) ≤ max{ G Mn(Ã) , H Mm(Ã) }, and so the L ∞ condition follows. This completes the proof of the result. Proof. It is clear from the definition of the norms onà that the identity map from A toà is completely contractive and thus A ⊆à as sets. This indeed shows thatà separates points of X and contains constant functions.\nLet (f ij ) ∈ M n (Ã) and ǫ > 0, then ∃ a net (f λ ij ) ∈ M n (A) such that lim λ (f λ ij (x)) = (f ij (x)) ∀ x ∈ X and sup λ (f λ ij ) Mn(A) ≤ (f ij ) Mn(Ã) + ǫ.\nSince A is an operator algebra of functions on the set X, we have that\n(f λ ij ) ∞ ≤ (f λ ij ) Mn(A) ∀ λ =⇒ sup λ (f λ ij ) ∞ ≤ (f ij ) Mn(Ã) + ǫ. Fix z ∈ X, then (f ij (z)) = lim λ (f λ ij (z)) ≤ sup λ (f λ ij ) ∞ ≤ (f ij ) Mn(Ã) + ǫ ∀ ǫ > 0.\nBy letting ǫ → 0 and supping over z ∈ X, we get that\n(f ij ) ∞ ≤ (f ij ) Mn(Ã) .\nHence,à is an operator algebra of functions on the set X.\nDenoteĨ F = {f ∈à : f | F ≡ 0} and let (f ij ) ∈ M n (Ã). Then, clearly sup F (f ij +Ĩ F ) Mn(Ã/Ĩ F ) = (f ij ) Mn(Ã)\n. To see the other inequality, as-\nsume that sup F (f ij +Ĩ F ) < 1, then ∀ finite F ⊆ X ∃ (h F ij ) ∈ M n (Ã) such that (h F ij )| F = (f F ij )| F and sup F h F ij ≤ 1. Fix a set F ⊆ X and (h F ij ) ∈ M n (Ã) then ∀ finite F ′ ⊆ X ∃ (k F ′ ij ) ∈ M n (A) such that (k F ′ ij )| F ′ = (h F ij )| F ′ and sup F ′ k F ′ ij ≤ 1. In particular, let F ′ = F then (k F ij )| F = (h F ij )| F = (f ij )| F and sup F k F ij ≤ 1. ⇒ (f ij ) Mn(Ã) ≤ 1, and hence (f ij ) Mn(Ã) ≤ sup F (f ij +Ĩ F ) Mn(Ã/Ĩ F ) . Finally, given that (f ij ) ∈ M n (A), (f ij ) Mn(Ã) = sup F (f ij +Ĩ F ) Mn(Ã/Ĩ F ) . Note that ∀ F ⊆ X we have (f ij +Ĩ F ) Mn(Ã/Ĩ F ) ≤ (f ij + I F ) Mn(A/I F ) ,\nsince I F ⊆Ĩ F . We claim that for any (f ij ) ∈ M n (A), and for any finite subset F ⊆ X, we have that (f ij + I F ) = (f ij +Ĩ F ) . To see the other inequality, let (g ij ) ∈ M n (Ĩ F ). Then ∀ ǫ > 0, and G ⊆ X,\n∃ (h G ij ) ∈ M n (A) such that (h G ij )| G = (f ij + g ij )| G and sup G (h G ij ) ≤ (f ij + g ij ) + ǫ. Hence, (f ij + I F ) = (h F ij + I F ) ≤ (h F ij ) ≤ (f ij + g ij ) + ǫ ∀ ǫ > 0\n, and the equality follows. Now it is immediate to see that,\n(f ij ) Mn(Ã) = sup F (f ij + I F ) = (f ij ) Mn(A L )\nCorollary 1.9. If A is a BPW complete operator algebra then A L =à completely isometrically.\n\nProof. Since A is BPW complete, A =à as sets. But by Lemma 1.6, the norm defined on A L agrees with the norm defined onÃ. Remark 1.10. In the view of above corollary, we denote the the norm oñ A by . L . Lemma 1.11. If A is an operator algebra of functions on X, then Ball(A L ) is BPW dense in Ball(Ã) andà is BPW complete, i.e.,à =Ã.\n\nProof. It can be easily checked that above is equivalent to showing that A L is BPW dense inÃ. We'll only prove that A L BP W ⊆Ã, since the other containment follows immediately by the definition ofÃ. Let {f λ } be a net in A L such that f λ → f pointwise and sup λ f λ A L < C. Then for fixed F ⊆ X and ǫ > 0,\n∃ λ F such that |f λ F (z) − f (z)| < ǫ ∀z ∈ F. Also since sup λ f λ < C, there exists g λ F ∈ I F such that f λ F + g λ F < C. Note that the function h F = f λ F + g λ F ∈ A such that h F A < C, and h F → f pointwise. Hence, f ∈Ã. Thus, A L is BPW dense inÃ.\nFinally, note that the argument similar to the above readily yields that A is BPW complete.\n\nAll the above lemmas can be summarized as the following theorem. Theorem 1.12. If A is an operator algebra of functions on X, thenà is a BPW complete local operator algebra of functions on X which contains A L completely isometrically as a BPW dense subalgebra. Definition 1.13. Given an operator algebra of functions A on X, we call A the BPW completion of A.\n\nWe now present a few examples to illustrate these concepts. We will delay the main family of examples to the next section. Example 1.14. If A is a uniform algebra, then there exists a compact, Hausdorff space X, such that A can be represented as a subalgebra of C(X) that separates points. If we endow A with the matrix-normed structure that it inherits as a subalgebra of C(X), namely, (f i,j ) = (f i,j ) ∞ ≡ sup{ (f i,j ) Mn : x ∈ X}, then A is a local operator algebra of functions on X. Indeed, to achieve the norm, it is sufficient to take the supremum over all finite subsets consisting of one point. In this case the BPW completionà is completely isometrically isomorphic to the subalgebra of ℓ ∞ (X) consisting of functions that are bounded, pointwise limits of functions in A. Example 1. 16. Let X = ǫD, 0 < ǫ < 1 and A = {f ∈ H ∞ (D) : f : X → C}. If we endow A with the matrix-normed structure on H ∞ (D), then A is an operator algebra of functions on X. Also, it can be verified that A is a local operator algebra of functions and that A =Ã. \nIndeed, if F = (f ij ) ∈ M n (A) with (f ij + I Y ) ∞ < 1 ∀ finite subset Y ⊆ X, then ∃ H Y ∈ M n (A) such that H Y ∞ ≤ 1 and H Y → F pointwise on X. Note by Montel's theorem ∃ a subnet H Y ′ and G ∈ M n (H ∞ (D)) such that G ∞ ≤ 1 and H Y ′ → Gand set F = max{ F ∞ , F ( 0 b 0 0 ) }, F ∈ M n (A)\n. It can be easily verified that A is a BPW complete operator algebra of functions. However, we also claim that A is local. To prove this we proceed by contradiction.\nSuppose ∃ F = (f ij ) ∈ M n (H ∞ (D)) such that F > 1 > c, where c = sup Y (f ij + I Y ) .\nIn this case,\nF = F ( 0 b 0 0 ) , since (f ij + I Y ) = F (λ) when Y = {λ}. Let ǫ = 1−c 4b and Y = {0, ǫ} ⊆ D, then ∃ G ∈ M n (H ∞ (D)\n) such that G| Y = 0 and F + G < 1+c 2 . Thus, we can write B Y (z) = z−ǫ 1−ǭz , so that we can write G(z) = zB Y (z)H(z), for some H ∈ M n (H ∞ ). It follows that H ∞ < 2, since G ∞ < 2. We now consider\n1 < F ( 0 b 0 0 ) ≤ (F + G)( 0 b 0 0 ) + G( 0 b 0 0 ) ≤ 1 + c 2 + 0 bG ′ (0) 0 0 = 1 + c 2 + b|B Y (0)| H(0) ≤ 1 + c 2 + 2bǫ = 1 + c 2 + 2b 1 − c 4b = 1 contradiction.\nExample 1.18. This is an example of a non-local algebra that arises from boundary behavior.\nSuppose A = {f ∈ A(D) ⊆ C(D − ) : f : D → C} equipped with the family of matrix norms F = max{ F ∞ , F ( 1 1 0 −1 ) }, F ∈ M n (A).\nThen it is easy to check that A is an operator algebra of functions on the set D. Also, it can be verified that A is not local. To see this, note\nthat z = 1 1 0 −1 . For each Y = {z 1 , z 2 , . . . , z n }, we define B Y (z) = Π n i=1 ( z−z i 1−z i z ) and choose h ∈ A such that h(1) = −B Y (1) and h(−1) = B Y (−1). Let g(z) = z +B Y (z)h(z)α, where α = z −1 h ∞+ z > 0. Then g ∈ A and g| Y = z| Y =⇒ π Y (z) = π Y (g) ≤ g ≤ max{1 + α h ∞ , (1 − α) z } < z .\nThus, sup Y ⊆D π Y (z) < z and hence A is not local. Example 1. 19. This example shows that one can easily build non-local algebras by adding \"values\" outside of the set X. Let A be the algebra of polynomials regarded as functions on the set X = D. Then A endowed with the matrix-normed structure as (p ij ) = max{ (p ij ) ∞ , (p ij (2)) }, is an operator algebra of functions on the set X. To see that A is not local, let\np ∈ A be such that p ∞ < |p(2)|. For each finite subset Y = {z 1 , . . . , z n } of X, let h Y (z) = Π n i=1 (z − z i ) and g Y (z) = p(z) − αh Y (z)p(2), where α = |p(2)|− p ∞ 2|p(2)| h Y ∞ > 0. Note that g Y ≤ (1 − α)|p(2)| and g Y | Y = p| Y =⇒ π Y (p) = π Y (g Y ) ≤ g Y ≤ (1 − α)|p(2)| < p . Hence, it follows that A is not local. Finally, observe that in this case A cannot be BPW complete. For example, if we take p n = 1 3 n i=0 ( z 3 ) i ∈ A then p n (z) → f (z) = 1 3−z ∀ z ∈ D and p n < f ⇒ A L à .\nExample 1.20. It is still an open problem as to whether or not every unital contractive, homomorphism ρ :\nH ∞ (D) → B(H) is completely contractive.\nFor a recent discussion of this problem see [25]. Let's assume that ρ is a contractive homomorphism that is not completely contractive. Let B = H ∞ (D), but endow it with the family of matrix-norms given by,\n|||(f i,j )||| = max{ (f i,j ) ∞ , (ρ(f i,j )) }. Note that |||f ||| = f ∞ , for f ∈ B.\nIt is easily checked that B is a BPW complete operator algebra of functions on D. However, since every contactive homomorphism of A(D) is completely contractive, we have that for\n(f i,j ) ∈ M n (A(D)), |||(f i,j )||| = (f i,j ) ∞ . If Y = {x 1 , ..., x k } is a finite subset of D and F = (f i,j ) ∈ M n (B), then there is G = (g i,j ) ∈ M n (A(D)), such that F (x) = G(x) for all x ∈ Y, and G ∞ = F ∞ . Hence, π (n) Y (F ) ≤ F ∞ . Thus, sup Y π (n) Y (F ) = F ∞ .\nIt follows that B is not local and thatB = B L = H ∞ (D), with its usual supremum norm operator algebra structure.\n\nIn particular, if there does exist a contractive but not completely contractive representation of H ∞ (D), then we have constructed an example of a non-local BPW complete operator algebra of functions on D.\n\n\nA Characterization of Local Operator Algebras of Functions\n\nThe main goal of this section is to prove that every BPW complete local operator algebra of functions is completely isometrically isomorphic to the algebra of multipliers on a reproducing kernel Hilbert space of vector-valued functions. Moreover, we wil show that every such algebra is a dual operator algebra in the precise sense of [11]. We will then prove that for such BPW algebras, weak*-convergence and BPW convergence coincide on bounded balls.\n\nGiven a set X and a Hilbert space H, then by a reproducing kernel Hilbert space of H-valued functions, we mean a vector space L of H-valued functions that is equipped with a norm and an inner product that makes it a Hilbert space and which has the property that for every x ∈ X, the evaluation map E x : L → H, is a bounded, linear map. Recall that given a Hilbert space H, a matrix of operators,\nT = (T i,j ) ∈ M k (B(H)) is regarded as an operator on the Hilbert space H (k) ≡ H ⊗ C k , which is the direct sum of k copies of H. A function K : X × X → B(H),\nwhere H is a Hilbert space, is called a positive definite operator-valued function on X, provided that for every finite set of (distinct)points {x 1 , ..., x k } in X, the operator-valued matrix, (K(x i , x j )) is positive semidefinite. Given a reproducing kernel Hilbert space of H-valued functions, if we set K(x, y) = E x E * y , then K is positive definite and is called the reproducing kernel of L. There is a converse to this fact, generally called Moore's theorem, which states that given any positive definite operatorvalued function K : X × X → B(H), then there exists a unique reproducing kernel Hilbert space of H-valued functions on X, such that K(x, y) = E x E * y . We will denote this space by L(K, H). Finally, given any reproducing kernel Hilbert space L of H-valued functions with reproducing kernel K, a function f : X → C is called a multiplier provided that for every g ∈ L, the function f g ∈ F. In this case it follows by an application of the closed graph theorem that the map M f : L → L, defined by M f (g) = f g, is a bounded, linear map. The set of all multipliers is denoted M(K) and is easily seen to be an algebra of functions on X and a subalgebra of B(L). The reader can find proofs of the above facts in [14] [3]. Also, we refer to the fundamental work of Pedrick [26] for further treatment of vector-valued reproducing kernel Hilbert spaces. Proof. It is enough to show that the unit ball is weak*-closed by the Krein-Smulian theorem. So let {M f λ } be a net of multipliers in the unit ball of B(L) that converges in the weak*-topology to an operator T. We must show that T is a multiplier.\n\nLet x ∈ X be fixed and assume that there exists g ∈ L, with g(\nx) = h = 0. Then T g, E * x h L = lim λ M f λ g, E * x h L = lim λ E x (M f λ g), h H = lim λ f λ (x) h 2 .\nThis shows that at every such x the net {f λ (x)} converges to some value. Set f (x) equal to this limit and for all other x's set f (x) = 0.\n\nWe claim that f is a multiplier and that T = M f .\n\nNote that if g(x) = 0 for every g ∈ L, then E x = E * x = 0. Thus, we have that for any g ∈ L and any h ∈ H,\nE x (T g), h H = lim λ E x (M f λ g), h H = lim λ f λ (x) g(x), h H = f (x) g(x)\n, h H . Since this holds for every h ∈ H, we have that E x (T g) = f (x)g(x), and so T = M f and f is a multiplier.\n\nEvery weak*-closed subspace of V ⊆ B(H) has a predual and it is the operator space dual of this predual. Also, if an abstract operator algebra is the dual of an operator space, then it can be represented completely isometrically and weak*-continuously as a weak*-closed subalgebra of the bounded operators on some Hilbert space. For this reason an operator algebra that has a predual as an operator space is called a dual operator algebra. See the book of [11] for the proofs of these facts. Thus, in summary, the above lemma shows that every multiplier algebra is a dual operator algebra. \n\n\nProof. The multiplier norm of a given matrix-valued function\nF = (f i,j ) ∈ M n (M(K)) is the least constant C such that ((C 2 I n − F (x i )F (x j ) * ) ⊗ K(x i , x j )) ≥ 0, for all sets of finitely many points, Y = {x 1 , ..., x k } ⊆ X.\nApplying this fact to a set consisting of a single point, we have that\n(C 2 I n −F (x)F (x) * )⊗K(x, x) ≥ 0, and it follows that C 2 I n −F (x)F (x) * ≥ 0.\nThus, F (x) ≤ C = F and we have that point evaluations are completely contractive on M(K). Since M(K) contains the constants and separates points by hypothesis, it is an operator algebra of functions on X.\n\nSuppose that M(K) was not local, then there would exist F ∈ M n (M(K)), and a real number C, such that sup Y π (n) Y < C < F . Then for each finite set Y = {x 1 , ..., x k } we could choose G ∈ M n (M(K)), with G < C, and G(x) = F (x), for every x ∈ Y. But then we would have that\n((C 2 I n −F (x i )F (x j ) * )⊗K(x i , x j )) = ((C 2 I n −G(x i )G(x j ) * )⊗K(x i , x j )) ≥ 0, and since Y was arbitrary, F ≤ C, a contradiction. Thus, M(K) is local. Finally, assume that f λ ∈ M(K), is a net in M(K), with f λ ≤ C, and lim λ (x) = f (x), pointwise. If g ∈ L with g L = M, then (M C) 2 K(x, y) − f λ (x)g(x), f λ (y)g(y) is positive definite. By taking pointwise limits, we obtain that (M C) 2 K(x, y) − f (x)g(x), f (y)g(y)\nis positive definite. From the characterization of functions and their norms in a reproducing kernel Hilbert space, this implies that\nf g ∈ L, with f g L ≤ M C. Hence, f ∈ M(K) with M f ≤ C. Thus, M(K) is BPW complete.\nIn general, M(K) need not separate points on X. In fact, it is possible that L does not separate points and if g(x 1 ) = g(x 2 ), for every g ∈ L, then necessarily f (x 1 ) = f (x 2 ) for every f ∈ M(K).\n\nFollowing [24], by a k-idempotent operator algebra, C, we mean that we are given k operators,\n{E 1 , ..., E k } on some Hilbert space H, such that E i E j = E j E i = δ i,j E i , I = E 1 + · · · + E k and C = span{E 1 , ..., E k }. Proposition 2.3. Let C = span{E 1 , ..., E k } be a k-idempotent operator algebra on the Hilbert space H, let Y = {x 1 , ..., x k } be a set of k distinct points and define K : Y × Y → B(H) by K(x i , x j ) = E i E * j . Then K is positive definite and C is completely isometrically isomorphic to M(K) via the map that sends a 1 E 1 + · · · + a k E k to the multiplier f (x i ) = a i .\nProof. It is easily checked that K is positive definite. We first prove that the map is an isometry.\nGiven B = k i=1 a i ⊗ E i ∈ C, let f : Y → C be defined by f (x i ) = a i . We have that f ∈ M(K) with f ≤ C if and only if P = ((C 2 − f (x i )f (x j ) * )K(x i , x j )) is positive semidefinite in B(H (k) ). Let v = e 1 ⊗ v 1 + · · · e k ⊗ v k ∈ H (k) , let h = k j=1 E * j v j and note that E * j h = E * j v j . Finally, set h = k i=1 h i . Thus, P v, v = k i,j=1 (C 2 − a iāj ) E i E * j v j , v i = k i,j=1 (C 2 − a iāj ) E * j h, E * i h = C 2 h 2 − B * h, B * h = C 2 h 2 − B * h 2 .\nHence, B ≤ C implies that P is positive and so M f ≤ B . For the converse, given any\nh let v = k j=1 e j ⊗ E * j h, and note that P v, v ≥ 0, implies that B * h ≤ C, and so B ≤ M f .\nThe proof of the complete isometry is similar but notationally cumbersome. Proof. Let Y be a finite subset of X. Since A/I Y is a |Y |-idempotent operator algebra, by the above lemma, there exists a vector valued kernel\nK Y such that A/I Y = M(K Y ) completely isometrically. Define K Y (x, y) = K Y (x, y) when (x, y) ∈ Y × Y, 0 when (x, y) ∈ Y × Y. and set K = Y ⊕ K Y ,\nwhere the direct sum is over all finite subsets of X.\n\nThen it is easily checked that K is positive definite.\nLet f ∈ M n (M(K)) with M f ≤ 1. This is equivalent to ((I − f (x i )f (x j ) * ) ⊗ K(x i , x j )) ≥ 0 ⇐⇒ ((I − f (x i )f (x j ) * ) ⊗ K F (x i , x j )) ≥ 0 ∀ F ⊆ X.\nThis last condition is equivalent to the existence for each F of some\nf F ∈ M n (A) such that π F (f F ) ≤ 1 and f F = f on F. The net of functions {f F } then converges BPW to f. Hence, f ∈à with f L ≤ 1.\nThis proves that M n (M(K)) = M n (Ã) isometrically.\n\n\nCorollary 2.5. Every BPW complete local operator algebra of functions is a dual operator algebra.\n\nProof. In this case we have that A =à = M(K) completely isometrically. By Lemma 2.1, this latter algebra is a dual operator algebra.\n\nThe above theorem gives a weak*-topology to a local operator algebra of function, A by using the identification A ⊆à = M(K) and taking the weak*-topology of M(K). The following proposition proves that convergence of bounded nets in this weak*-topology on A is same as BPW convergence. Proof. Let H denote the reproducing kernel Hilbert space of C-valued functions on X with kernel K for whichà = M(K). Recall that if E x : H → C, is the linear map given by evaluation at\nx, then K(x, y) = E x E * y . Also, if v ∈ C, and h ∈ H, then h, E * x v H = h(x), v C . First assume that the net (f λ ) λ ∈ Ball(A) converges to f in the weak*- topology. Using the identification ofà = M(K), we have that the operators M f λ of multiplication by f λ , converge in the weak*-topology of B(C) to M f . Then for any x ∈ X, h ∈ H, v ∈ C, we have that f λ (x) h(x), v C = f (x)h(x), v C = M f λ h, E * x v H → M f h, E * x v H = f (x) h(x), v C . Thus, if there is a vector in C and a vector in H such that h(x), v C = 0, then we have that f λ (x) → f (x)\n. It is readily seen that such vectors exist if and only if E x = 0, or equivalently, K(x, x) = 0. But this follows from the construction of K as a direct sum of positive definite functions over all finite subsets of X. For fixed x ∈ X and the one element subset Y 0 = {x}, we have that the 1-idempotent algebra A/I Y 0 = 0 and so K Y 0 (x, x) = 0, which is one term in the direct sum for K(x, x).\n\nConversely, assume that f λ < K, ∀λ and f λ → f pointwise on X. We must prove that M f λ → M f in the weak*-topology on B(H). But since this is a bounded net of operators, it will be enough to show convergence in the weak operator topology and arbitrary vectors can be replaced by vectors from a spanning set. Thus, it will be enough to show that for v 1 , v 2 ∈ C and\nx 1 , x 2 ∈ X, we have that M f λ E * x 1 v 1 , E * x 2 v 2 H → M f E * x 1 v 1 , E * x 2 v 2 H . But we have, M f λ E * x 1 v 1 , E * x 2 v 2 H = E x 2 (M f λ E * x 1 v 1 ), v 2 C = f λ (x 2 ) K(x 2 , x 1 )v 1 , v 2 C → f (x 2 ) K(x 2 , x 1 )v 1 , v 2 C = M f E * x 1 v 1 , E 8 x 2 v 2 H ,\nand the result follows.\n\nCorollary 2.7. The ball of a local operator algebra of functions is weak*dense in the ball of its BPW completion.\n\n\nResidually Finite Dimensional Operator Algebras\n\nA C*-algebra B is called residually finite-dimensional (RFD) if it has a separating family of finite-dimensional representations. Note that when B is separable and RFD it has a seperating sequence of finite dimensional representations. Since every one-to-one * -homomorphism of a C*-algebra is completely isometric, a C*-algebra B is RFD if and only if for all n, and for every (b i,j ) ∈ M n (B), we have that (b i,j ) Mn(B) = sup{ (π(b i,j )) } where the suprema is taken over all * -homomorphisms, π : B → M k ∀ k. RFD C*algebras have been studied in [10], [15], [5], [19].\n\nWe know from BRS that any operator algebra can be represented on a Hilbert space, i.e., given an abstract unital operator algebra A there exists an unital completely isometric homomorphism π : A → B(H). Since a representation of a C*-algebra is a *-homomorphism if and only if it is completely contractive, the following definition gives us a natural way of extending the notion of RFD to operator algebras. \nb i,j ) ∈ M n (B), (b i,j ) = sup{ (π(b i,j )) },\nwhere the suprema is taken over all completely contractive homomorphisms, π : B → M k ∀ k. A dual operator algebra B is called weak*-RFD if this last equality holds when the completely contractive homomorphisms are also required to be weak*continuous.\n\nThe following result is implicitly contained in [24], but the precise statement that we shall need does not appear there. Thus, we refer the reader to [24] to be able to fully understand the proof since we have used some of the definitions and results from [24] without stating them. Proof. From the Corollary 3.3 of [24] we have that the Schur ideal S(B * B) is non-trivial and bounded. Thus, we can define the algebra A(S(B * B)) = span{E 1 , · · · , E k }, where E i = n Q∈S −1 n (B * B) ⊕Q 1/2 (I n ⊗ E ii )Q −1/2 is the idempotent operator that live on n Q∈S −1 n ⊕M k (M n ). By using Theorem 3.2 of [24] we get that S(A(S(B * B))A(S(B * B)) * ) = S(B * B) This further imples that A(S(B * B))A(S(B * B)) * = B * B completely order isomorphically under the map which sends E * i E j to F * i F j . Finally, by restricting the same map to A we get a map which sends E i to F i completely isometrically. Hence, the result follows. Proof. Let A be an abstract k-idempotent operator algebra. Note that A is a dual operator algebra being a finite dimensional operator algebra.\n\nFrom this it follows that there exists a Hilbert space, H and a weak*continuous completely isometric homomorphism, π : A → B(H). Note that B = π(A) is a concrete k-idempotent algebra generated by the idempotents, B = span{F 1 , F 2 , ..., F k } contained in B(H). Thus, from the above lemma B = A(S(B * B)) completely isometrically. For each n ∈ N, Q ∈ S −1 n , we define π Q n : B → M k (M n ) via π Q n (F i ) = Q 1/2 (I n ⊗ E ii )Q −1/2 Assume for the moment that we have proven that π Q n is a weak*-continuous completely contractive homomorphism. Then for every (b ij ) ∈ M k (B) we must have that sup n,Q∈S −1\nn (π Q n (b ij )) = (b ij ) , and hence (b ij ) = sup m { (ρ(b ij )) } where supremum is taken over all weak*-continuous cc ho- momorphism ρ : B → M m ∀ m.\nFinally, note that for any (a ij ) ∈ M k (A) we have that (a ij ) M k (A) = sup m { (ρ(π(a ij ))) } where supremum is taken over all weak*-continuous cc homomorphisms ρ : B * B → M m which follows from the fact that π is a complete isometry. Since ρ • π is a weak*-continuous cc homomorphism on A for every weak*-continuous cc homomorphism, ρ on B * B. This implies that (a ij ) M k (A) = sup{ (γ(a ij )) } where supremum is taken over all finite dimensional weak*-continuous cc homomorphisms, and hence the result.\n\nThus, it remains to show that π Q n is a weak*-continuous cc homomorphism. Note that it is easy to check that it is a completely contractive homomorphism and sup n,Q∈S −1\nn (π Q n (b ij )) = (b ij ) ∀ (b ij ) ∈ M k (B * B)\n. Finally, to see that it is weak*-continuous, let f λ be a bounded net in B * B which converges in the weak*-topology. Note that we can write\nf λ = k i=1 α k 1 E i and f = k i=1 α 1 E i . We define for each i, ψ i : B * B → C via ψ i (g) = α g i where g = k i=1 α g i E i .\nNote that for each i, ψ i is a bounded linear functional on B * B, since the linear maps on a finite dimensional normed space are bounded. Thus, for\ngiven ǫ > 0, ∃ λ 0 such that |α λ i − α i | < ǫ max 1≤i≤k {π Q n (E i )} ∀λ ≥ λ 0 .\nFinally, to see that π Q n (f λ ) → π Q n (f ) in the weak*-topology, fix unit vectors h, k ∈ H and consider\n| π Q n (f λ )h, k − π Q n (f )h, k | = | k i=1 (α λ i − α i ) π Q n (E i )h, k | ≤ k i=1 |α λ i − α i || π Q n (E i )h, k | < ǫ ∀λ ≥ λ 0 .\nThis shows that the map π Q n is a weak*-continuous and cc homomorphism for every Q ∈ S −1 n and n. Theorem 3.4. Every BPW complete local operator algebra of functions is waek*-RFD.\n\nProof. Let A be a BPW complete local operator algebra of functions on the set X and F be a finite subset of X, A/I F is a |F |-idempotent operator algebra. Thus, it follows from the above lemma that A/I F is weak*-\nRFD, i.e., ∀ ([f ij ]) ∈ M k (A/I F ) we have ([f ij ]) = sup n F { (ρ F ([f ij ])) }\nwhere supremum is taken over all weak*-continuous cc homomorphisms ρ F : A/I F → M n F and all integers n F .\nLet (f ij ) ∈ M k (A), then (f ij ) M k (A) = sup F ([f ij ]) since A is local.\nRecall, that the weak*-topology on A requires all the quotient maps of the form π F : A → A/I F , π F (f ) = [f ] to be weak*-continuous. Thus for each finite subset F ⊆ X, π F is a weak*-continuous cc homomorphism. Let ρ F : A/I F → M n F be a weak*-continuous cc homomorphism, then define δ F = π F • ρ F which is indeed a weak*-continuous cc homomorphism. Consider\n(ρ F ([f ij ])) = (ρ F (π F (f ij ))) = (δ F (f ij )) ≤ sup n { (π(f ij )) : π : A → M n weak*-cont. cc homo} ≤ (f ij ) M k (A) . Finally, note that ([f ij ]) M k (A/I F ) = sup n F { (ρ F ([f ij ])) },\nwhere supremum is taken over all weak*-continuous cc homomorphisms, ρ F and positive integers n F . Hence, we obtain the result by taking supremum over all the finite subsets F ⊆ X. Proof. The proof of this follows immediately since every local operator algebra is contained in some BPW complete local operator algebra completely isometrically.\n\n\nQuantized Function Theory on Domains\n\nWhenever one replaces scalar variables by operator variables in a problem or definition, then this process is often referred to as quantization. It is in this sense that we would like to quantize the function theory on a family of complex domains. In some sense this process has already been carried out for balls in the work of Drury [18], Popescu [28], Arveson [6], and Davidson and Pitts [17] and for polydisks in the work of Agler [1], [2], and Ball and Trent [8]. Furthermore, the idea of \"quantizing\" other domains defined by inequalities occurs in [4], [10], and [20]. We approach these same ideas via operator algebra methods. We will show that in many cases this process yields local operator algebras of functions to which the results of the earlier sections can be applied.\n\nWe begin by defining a family of open sets for which our techniques will apply. ,n k , k ∈ I, whose components are analytic functions on G, and G = {z ∈ C N : F k (z) < 1, k ∈ I}, then we call G an analytically presented domain and we call the set of functions R = {F k : G − → M m k ,n k : k ∈ I} an analytic presentation of G. The subalgebra A of the algebra of functions on G generated by the component functions {f k,i,j : 1 ≤ i ≤ m k , 1 ≤ j ≤ n k , k ∈ I} and the constant function is called the algebra of the presentation. We say that R is a separating analytic presentation provided that the algebra A separates points on G.\n\nRemark 4.2. An analytic presentation of G by a finite set of matrix-valued functions, F k : G → M m k ,n k , 1 ≤ k ≤ K, can always be replaced by the single block diagonal matrix-valued function, F (z) = F 1 (z) ⊕ · · · ⊕ F K (z) into M m,n with m = m 1 + · · · m K , n = n 1 + · · · n K and we will sometimes do this to simplify proofs. But it is often convenient to think in terms of the set, especially since this will explain the sums that occur in the Agler's factorization formula.\n\nNote that when we have a analytically presented domain, then every function in the algebra of the presentation is an analytic function on G. H m ). We call the homomorphism π an admissible strict representation when these inequalities are all strictly less than 1. Given (g i,j ) ∈ M n (A) we set (g i,j ) u = sup{ (π(g i,j )) }, where the supremum is taken over all admissible representations π of A. We let (g i,j ) u 0 denote the supremum that is obtained when we restrict to admissible strict representations.\n\nThe theory of [4] and [10] studies domains defined as above with the additional restrictions that the set of defining functions is a finite set of polynomials, but they do not need their polynomials to separate points. Our results should be compared to theirs.\n\n\nProposition 4.4. Let G have a separating analytical presentation and let\n\nA be the algebra of the presentation. Then A endowed with either of the family of norms · u or · u 0 is an operator algebra of functions on G.\n\nProof. It is clear that it is an operator algebra and by definition it is an algebra of functions on G. It follows from the hypotheses that it separates points of G. Finally, for every λ = (λ 1 , ..., λ N ) ∈ G, we have a representation of A on the one-dimensional Hilbert space given by π λ (f ) = f (λ). Hence, |f (λ)| ≤ f u and so A is an operator algebra of functions on G.\n\nIt will be convenient to say that matrices, A 1 , . . . A m are of compatible sizes if the product, A 1 · · · A m exists, that is, provided that each A i is an n i × n i+1 matrix.\n\nGiven an analytically presented domain G, we include one extra function, F k 0 which denotes the constant function 1. By an admissible block diagonal matrix over G we mean a block diagonal matrix-valued function of the form D(z) = diag(F k 1 , ..., F km ) where k i ∈ I ∪ k 0 for 0 ≤ i ≤ m. Thus, we are allowing blocks of 1's in D(z). Finally, given a matrix B we let B (q) = diag(B, ...., B) denote the block diagonal matrix that repeats B q times. (i) P u < 1, (ii) there exists an integer l, matrices of scalars C j , 1 ≤ j ≤ l with C j < 1 and admissible block diagonal matrices D j (z), 1 ≤ j ≤ l, which are of compatible sizes and are such that\nP (z) = C 1 D 1 (z) · · · C l D l (z).\n(iii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 , P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is a finite set, such that\nI m − P (z)P (w) * = R + P 0 (z)P 0 (w) * + k∈K P k (z)(I − F k (z)F k (w) * ) (q k ) P k (w) * where r k = q k m k and z = (z 1 , ..., z N ), w = (w 1 , ..., w N ) ∈ G.\nProof. Although we will not logically need it, we first show that (ii) implies (i), since this is the easiest implication and helps to illustrate some ideas. Note that if π : A → B(H) is any admissible representation, then the norm of π of any admissible block diagonal matrix is at most 1. Thus, if P has the form of (ii), then for any admissible π, we will have (π(p i,j )) expressed as a product of scalar matrices and operator matrices all of norm at most one and hence, (π(P i,j )) ≤ C 1 · · · C l < 1. Thus, P u ≤ C 1 · · · C l < 1. We now prove that (i) implies (ii). The ideas of the proof are similar to [23, Corollary 18.2], [12, Corollary 2.11] and [21, Theorem 1] and use in an essential way the abstract characterization of operator algebras. For each m, n ∈ N, one proves that P m,n := inf { C 1 . . . C l }, defines a norm on M m,n (A), where the infimum is taken over all l and all ways to factor P (z) = C 1 D 1 (z i 1 ) · · · C l D l (z i l ) as a product of matrices of compatible sizes with scalar matrices C j , 1 ≤ j ≤ l and admissible block diagonal matrices D j , 1 ≤ j ≤ l.\n\nMoreover, one can verify that M m,n (A) with this family { . m,n } m,n of norms satisfies the axioms for an abstract unital operator algebra as given in [13] and hence by the Blecher-Ruan-Sinclair representation theorem [13](see also [23]) there exists a Hilbert space H and a unital completely isometric isomorphism π : A −→ B(H). Thus, for every m, n ∈ N and for every P = (p ij ) ∈ M m,n (A), we have P m,n = (π(p ij )) . However, π (m k ,n k ) (F k ) = (π(f k,i,j )) ≤ 1 for 1 ≤ i ≤ K, and so, π is an admissible representation. Thus, P m,n = (π(p ij )) ≤ P u . Hence, if P u < 1, then P m,n < 1 which implies that such a factorization exists. This completes the proof that (i) implies (ii).\n\nWe will now prove that (ii) implies (iii). Suppose that P has a factorization as in (ii). Let K ⊆ I be the finite subset of all indices that appear in the block-diagonal matrices appearing in the factorization of P. We will use induction on l to prove that (iii) holds.\n\nFirst, assume that l = 1 so that P (z) = C 1 D 1 (z). Then,\nI m − P (z)P (w) * = I m − (C 1 D 1 (z))(C 1 D 1 (w)) * = (I m − C 1 C * 1 ) + C 1 (I − D 1 (z)D 1 (w) * ) C * 1 . Since D 1 (z) is an admissible block diagonal matrix the (i, i)-th block diago- nal entry of I − D 1 (z)D 1 (w) * is I − F k i (z)F k i (w) * for some finite collection, k i .\nLet E k be the diagonal matrix that has 1's wherever F k appears(so E k = 0 when there is no F k term in D 1 ). Hence,\nC 1 (I − D 1 (z)D 1 (w) * )C * 1 = k C 1 E k (I − F k (z)F k (w) * )E k C * 1 .\nTherefore, gathering terms for common values of i,\nI m − P (z)P (w) * = R 0 + k∈K P k (I − F k (z)F k (w) * )P * k ,\nwhere R 0 = I m − C 1 C * 1 is a positive, invertible matrix and P i is, in this case a constant. Thus, the form (iii) holds, when l = 1.\n\nWe now assume that the form (iii) holds for any R(z) that has a factorization of length at most l − 1, and assume that\nP (z) = C 1 D 1 (z) · · · D l−1 (z)C l D l (z) = C 1 D 1 (z)R(z),\nwhere R(z) has a factorization of length l − 1.\n\nNote that a sum of expressions such as on the right hand side of (iii) is again such an expression. This follows by using the fact that given any two expressions A(z), B(z), we can write\nA(z)A(w) * + B(z)B(w) * = C(z)C(w) * where C(z) = (A(z), B(z)).\nThus, it will be sufficient to show that I m − P (z)P (w) * is a sum of expressions as above. To this end we have that,\nI m − P (z)P (w) * = (I m − C 1 D 1 (z)D 1 (w) * C * 1 ) + (C 1 D 1 (z))(I − R(z)R(w) * )(D 1 (w) * C * 1 )\n. The first term of the above equation is of the form as on the right hand side of (iii) by case l = 1. Also, the quantity (I − R(z)R(w) * ) = R 0 R * 0 + R 0 (z)R 0 (w) * + k∈K R k (z)(I − F k (z)F k (w) * ) (q k ) R k (w) * by the inductive hypothesis. Hence,\nC 1 D 1 (z)(I − R(z)R(w) * )D 1 (w) * C * 1 = (C 1 D 1 (z)R 0 )(C 1 D(w)R 0 ) * + [C 1 D 1 (z)R 0 (z)][C 1 D 1 (w)R 0 (w)] * + k∈K [C 1 D 1 (z)R k (z)](I − F k (z)F k (w) * ) (q k ) [C 1 D 1 (w)R k (w)] * .\nThus, we have expressed (I − P (z)P (w) * ) as a sum of two terms both of which can be written in the form desired. Using again our remark that the sum of two such expressions is again such an expression, we have the required form.\n\nFinally, we will prove (iii) implies (i). Let π : A → B(H) be an admissible representation and let P = (p i,j ) ∈ M m,n (A) have a factorization as in (iii). To avoid far too many superscripts we write simplify π (m,n) to Π. Now observe that\nI m − Π(P )Π(P ) * = Π(R) + Π(P 0 )Π(P 0 ) * + k∈K Π(P k )(I − Π(F k )Π(F k ) * ) (q k ) (Π(P k )) * .\nClearly the first two terms of the sum are positive. But since π is an admissible representation, Π(F k ) ≤ 1 and hence, (I − Π(F k )Π(F k ) * ) ≥ 0. Hence, each term on the right hand side of the above inequality is positive and since R is strictly positive, say R ≥ δI m for some scalar δ > 0, we have that I m − Π(P )Π(P ) * ≥ δI m . Therefore, Π(P ) ≤ √ 1 − δ. Thus, since π was an arbitrary admissible representation, P u ≤ √ 1 − δ < 1, which proves (i).\n\nWhen we require the functions in the presentation to be row vectorvalued, then the above theory simplifies somewhat and begins to look more familiar. Let G be an analytically presented domain with presentation F k : G → M 1,n k , k ∈ I. We identify M 1,n with the Hilbert space C n so\nthat 1 − F k (z)F k (w) * = 1 − F k (z), F k (w) ,\nwhere the inner product is in C n . In this case we shall say that G is presented by vector-valued functions. (i) P u < 1, (ii) there exists an integer l, matrices of scalars C j , 1 ≤ j ≤ l with C j < 1 and admissible block diagonal matrices D j (z), 1 ≤ j ≤ l, which are of compatible sizes and are such that P (z) = C 1 D 1 (z) · · · C l D l (z).\n\n(iii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 ∈ M m,r 0 (A), P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is finite, such that\nI m − P (z)P (w) * = R + P 0 (z)P 0 (w) * + k∈K (1 − F k (z), F k (w) )P k (z)P k (w) * where z = (z 1 , ..., z N ), w = (w 1 , ..., w N ) ∈ G.\nThe following result gives us a Nevanlinna-type result for the algebra of presentation.\n\nTheorem 4.7. Let Y be a finite subset of an analytically presented domain G with separating analytic presentation F k = (f k,i,j ) : G → M m k ,n k , k ∈ I, let A be the algebra of the presentation and let P be a M m,n -valued function defined on a finite subset Y = {x 1 , · · · , x l } of G. Then the following are equivalent:\n\n(i) there existsP ∈ M mn (A) such thatP | Y = P and P u < 1.\n\n(ii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 ∈ M m,r 0 (A), P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is a finite set, such that\nI m − P (z)P (w) * = R + P 0 (z)P 0 (w) * + k∈K P k (z)(I − F k (z)F k (w) * ) (q k ) P k (w) * where r k = q k m k and z = (z 1 , ..., z N ), w = (w 1 , ..., w N ) ∈ Y.\nProof. Note that (i) ⇒ (ii) follows immediately as a corollary of Theorem 4.5. Thus it only remains to show that (ii) ⇒ (i). Since A is an operator algebra of functions, therefore, A/I Y is a finite dimensional operator algebra of idempotents and A/I Y = span{E 1 , · · · , E l } for some l = |Y |. Thus there exists a Hilbert space, H Y and a completely isometric representation π of A/I Y . By Theorem 2.4, there exists a kernel K Y such that π(A/I Y ) = M(K Y ) completely isometrically under the map ρ :\nπ(A/I Y ) → M(K Y ) which sends π(B) to M f , where B = l i=1 a i π(E i ) and f : Y → C is a func- tion defined by f (x i ) = a i . Note that ((I − F k (x i )F k (x j )) ⊗ K Y (x i , x j )) = ((I − π(F k + I Y )(x i )π(F k + I Y )(x j ) * ) ⊗ K Y (x i , x j )) ij ≥ 0 ∀ k ∈ I since π(F k + I Y ) ≤ F k u ≤ 1 ∀ k ∈ I. From this it follows that ((I m − (π(P + I Y ))(x i )(π(P +I Y ))(x j ) * )⊗K Y (x i , x j )) ij ≥ 0.\nUsing that R > 0, we get that π(P + I Y ) < 1. This shows that there existsP ∈ A such thatP | Y = P and P u < 1. This completes the proof.\n\nWe now turn towards defining quantized versions of the bounded analytic functions on these domains. For this we need to recall that the joint Taylor spectrum of a commuting N -tuple of operators T = (T 1 , ..., T N ), is a compact set, σ(T ) ⊆ C N and that there is an analytic functional calculus defined for any function that is holomorphic in a neighborhood of σ(T ). \nQ(G) = {T = (T 1 , T 2 , . . . , T N ) ∈ B(H) : σ(T ) ⊆ G and F k (T ) ≤ 1, ∀k ∈ I},\nwhere H is an arbitrary Hilbert space. We set\nQ 0,0 (G) = {T = (T 1 , T 2 , . . . , T N ) ∈ M n : σ(T ) ⊆ G and F k (T ) ≤ 1, ∀k ∈ I},\nwhere n is an arbitrary positive integer.\n\nNote that if we identify a point (λ 1 , ..., λ N ) ∈ C N with an N-tuple of commuting operators on a one-dimensional Hilbert space, then we have that G ⊆ Q(G).\n\nIf T = (T 1 , ..., T N ) ∈ Q(G), is a commuting N-tuple of operators on the Hilbert space H, then since the joint Taylor spectrum of T is contained in G, we have that if f is analytic on G, then there is an operator f (T ) defined and the map π : Hol(G) → B(H) is a homomorphism, where Hol(G) denotes the algebra of analytic functions on G.\nDefinition 4.9. Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}. We define H ∞ R (G) to be the set of functions f ∈ Hol(G), such that f R ≡ sup{ f (T ) : T ∈ Q(G)} is finite. Given (f i,j ) ∈ M n (H ∞ R (G)), we set (f i,j ) R = sup{ (f i,j (T )) : T ∈ Q(G)}.\nWe are interested in determining when H ∞ R (G) =Ã, completely isometrically and whether or not the · R norm is attained on the smaller set Q 0,0 (G).\n\nNote that since each point in G ⊆ Q(G), we have that\nH ∞ R (G) ⊆ H ∞ (G), and f ∞ ≤ f R . Also, we have that A ⊆ H ∞ R (G) and for (f i,j ) ∈ M n (A), (f i,j ) R ≤ (f i,j ) u 0 ≤ (f i,j ) u .\nThus, the inclusion of A into H ∞ R (G) might not even be isometric. Theorem 4.10. Let G be an analytically presented domain with a separating presentation R = {F k : G → M m k ,n k : k ∈ I}, let A be the algebra of the presentation and let A be the BPW-completion of A. Then\n(i) A = H ∞ R (G), completely isometrically, (ii) H ∞ R (G) is a local, weak*-RFD, dual operator algebra,\nProof. Let f ∈ M n ( A), with f L < 1. Then there exists a net of functions, f λ ∈ M n (A), such that f λ u < 1 and lim λ f λ (z) = f (z) for every z ∈ G.\n\nSince f λ ∞ < 1, by Montel's Theorem, there is a subsequence {f n } of this net that converges to f uniformly on compact sets. Hence, if T ∈ Q(G), then lim n f (T ) − f n (T ) = 0 and so f (T ) ≤ sup n f n (T ) ≤ 1. Thus, we have that f ∈ M n (H ∞ R (G)), with f R ≤ 1. This proves that A ⊆ H ∞ R (G) and that f L ≤ f R .\n\nConversely, let g ∈ M n (H ∞ R (G) with g R < 1. For any finite set Y = {y 1 , ..., y t } ⊆ G, let A/I Y = span{E 1 , ..., E t } be the corresponding t-idempotent algebra and let π Y : A → A/I Y denote the quotient map. Write y i = (y i,1 , ..., y i,N ), 1 ≤ i ≤ t and let T j = y 1,j E 1 + · · · + y t,j E t , 1 ≤ j ≤ N so that T = (T 1 , ..., T N ) is a commuting N -tuple of opertors with σ(T ) = Y. For k ∈ I, we have that F k (T ) = F k (y 1 ) ⊗ E 1 + · · · + F k (y t ) ⊗ E t = π Y (F k ) ≤ F k u = 1. Thus, T ∈ Q(G), and so, g(T ) = g(y 1 ) ⊗ E 1 + · · · + g(y t ) ⊗ E t ≤ g R < 1. Since A separates points, we may pick f ∈ M n (A) such that f = g on Y. Hence, π Y (f ) = f (T ) = g(T ) and π Y (f ) < 1. Thus, we may pick f Y ∈ M n (A), such that π Y (f Y ) = π Y (f ) and f Y u < 1. This net of functions, {f Y } converges to g pointwise and is bounded. Therefore, g ∈ M n ( A) and g L ≤ 1. This proves that H ∞ R (G) ⊆ A and that g L ≤ g R .\n\nThus, A = H ∞ R (G) and the two matrix norms are equal for matrices of all sizes. The rest of the conclusions follow from the results on BPWcompletions.\n\n\nRemark 4.11. The above result yields that for every\nf ∈ H ∞ R (G), f R = sup{ π(f ) } where supremum is taken over all finite dimensional weak*- continuous representations, π : H ∞ R (G) → M n ∀ n. Under additional hy- potheses, we can show that f R = sup{ f (T ) : T ∈ Q 00 (G)} ∀f ∈ H ∞ R (G)\n. Also, we can verify these hypotheses are met for most of the algebras given in the example section. In particular, for Examples 5. 1, 5.2, 5.3, 5.4, 5.6, 5.7, and 5.8\n\n\n. It would be interesting to know if this can be done in general.\n\nWe now seek other characterisations of the functions in H ∞ R (G). In particular, we wish to obtain analogues of Agler's factorization theorem and of the results in [4] and [10]. By Theorem 2.4, if we are given an analytically presented domain G ⊆ C N , with presentation R = {F k : G → M m k ,n k , k ∈ I}, then there exists a Hilbert space H and a positive definite function, K : G × G → B(H) such that A = M(K). We shall denote any kernel satisfying this property by K R . (\ni) f ∈ M m (H ∞ R (G)) and f R ≤ 1, (ii) (I m − f (z)f (w) * ) ⊗ K R (z, w) is positive definite, (iii) I m − f (z)f (w) * is an R-limit.\nIn the case when the presentation contains only finitely many functions we can say considerably more about R-limits. \nI m − P (z)P (w) * = R+ P 0 (z)P 0 (w) * + K k=1 P k (z)(I − F k (z)F k (w) * ) (q k ) P k (w) * , then P k (z) ≤ C ∀k, ∀z ∈ S..\nProof. By the continuity of the functions, there is a constant δ > 0, such that F k (z) ≤ 1 − δ, ∀k, ∀z ∈ S. Thus, we have that I − F k (z)F k (z) * ≥ δI, ∀ k, ∀z ∈ S. Also, we have that Proof. We identify (I m − F (z)F (w) * ) (q λ ) = (I m − F (z)F (w) * ) ⊗ I C q λ , and the p × mq λ matrix-valued function P λ as an analytic function, P λ : G → B(C m ⊗ C q λ , C p ). Writing C m ⊗ C q λ = C q λ ⊕ · · · C q λ (m times) allows us to write P λ (z) = [P λ 1 (z), . . . , P λ m (z)] where each P λ i (z) is p × q λ . Also, if we let f 1 (z), ..., f m (z) be the (1, n) vectors that represent the rows of the matrix F, then we have that F (z)F (w) * = m i,j=1 f i (z)f j (w) * E i,j . Finally, we have that\nI m ≥ I m −P (z)P (z) * ≥ P k (z)(I−F k (z)F k (z) * ) (q k ) P k (z) * ≥ δP k (z)P k (z) * This shows that P k (z) ≤ 1/δ ∀k, ∀ z ∈ S.H λ (z, w) = m i=1 P λ i (z)P λ i (w) * − m i,j=1 f i (z)f j (w) * P λ i (z)P λ j (w). Let K λ (z, w) = (P λ i (z)P λ j (w) * ), so that K λ : G × G → M m (M p ) = B(C m ⊗ C p )\n, is a positive definite function that is analytic in z and coanalytic in w. By dropping to a subnet, if necessary, we may assume that K λ converges uniformly on compact subsets of G to K = (K i,j ) : G × G → M m (M p ). Note that this implies that P λ i (z)P λ j (w) * → K i,j (z, w) for all i, j and that K is a positive definite function that is analytic in z and coanalytic in w.\n\nThe positive definite K gives rise to a reproducing kernel Hilbert space H of analytic C m ⊗ C p -valued functions on G. If we let E(z) : H → C m ⊗ C p , be the evaluation functional, then K(z, w) = E(z)E(w) * and E : G → B(H, C m ⊗ C p ) is analytic. Identifying C m ⊗ C p = C p ⊕ · · · ⊕ C p (m times), yields analytic functions,\nE i : G → B(H, C p ), i = 1, ..., m, such that (K i,j (z, w)) = K(z, w) = E(z)E(w) * = (E i (z)E j (w) * ).\nDefine an analytic map R : G → B(H ⊗ C m , C p ) by identifying H ⊗ C m = H ⊕ · · · ⊕ H(m times) and setting R(z)(h 1 ⊕ · · · ⊕ h m ) = E 1 (z)h 1 + · · · + E m (z)h m . Thus, we have that\nR(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * = m i=1 E i (z)E i (w) * − m i,j=1 f i (z)f j (w) * E i (z)E j (w) * = m i=1 K i,i (z, w) − m i,j=1 f i (z)f j (w) * K i,j (z, w) = lim λ m i=1 P λ i (z)P λ i (w) * − m i,j=1 f i (z)f j (w) * P λ i (z)P λ j (w) * = H(z, w),\nand the proof is complete. (1) f ∈ M mn (H ∞ R (G)) and f R ≤ 1, (2) there exists an analytic operator-valued function R 0 :\nG → B(H 0 , C m ) and F k -limits, H k : G × G → M m , such that I − f (z)f (w) * = R 0 (z)R 0 (w) * + K k=1 H k (z, w) ∀ z, w ∈ G, (3) there exist F k -limits, H k (z, w), such that I − f (z)f (w) * = K k=1 H k (z, w) ∀z, w ∈ G.\nProof. Recall thatà = H ∞ R (G). Let us first assume that f ∈ M mn (Ã) and f Mmn(Ã) < 1. Then for each finite set Y , there exists f Y ∈ M mm (A) such that f Y converges to f pointwise and f Y u ≤ 1. We may assume that\nf Y u < 1 by replacing f Y by f Y 1+1/|Y | ,\nwhere |Y | denotes the cardinality of the set Y .\n\nThus by Theorem 4.5 there exists a positive, invertible matrix R Y ∈ M m and matrices P Y k ∈ M m,r k Y (A), 0 ≤ k ≤ K, such that\nI m −f Y (z)f Y (w) * = R Y +P Y 0 (z)P Y 0 (w) * + K k=1 P Y k (z)(I−F k (z)F k (w) * ) (q k Y ) P Y k (w) *\nwhere r k Y = q k Y m k and z, w ∈ G. If we define a map F 0 : G → M m 0 ,n 0 as the zero map then the above factorization can be written as\nI m − f Y (z)f Y (w) * = R Y + K k=0 P Y k (z)(I − F k (z)F k (w) * ) (q k Y ) P Y k (w) * where r k Y = q k Y m k and z, w ∈ G.\nNote that the net R Y is uniformly bounded above by 1, thus there exists R ∈ M m and a subnet R Ys which converges to R.\n\nFinally, since the net f Y converges to f pointwise we have that the net\nK k=1 P Y k (z)(I − F k (z)F k (w) * ) (q k Y ) P Y k (w) * converges pointwise on G.\nAlso note that for each k, < P Y k > is the net of vector-valued holomorphic function and is uniformly bounded on compact subsets of G by Proposition 4.14.\n\nThus by Proposition 4.15 there exists F k -limit for each 0 ≤ k ≤ K, that is, there exists K+1 Hilbert spaces H k and K+1 analytic function,\nR k : G → B(H k ⊗ C M , C p ) such that H k (z, w) = R k (z)[(I m − F k (z)F k (w) * ) ⊗ I H k ]R k (w) * and the corresponding subnet of the net K k=0 P Y k (z)(I − F k (z)F k (w) * ) (q k Y ) P Y k (w) * converges to K k=0 H k (z, w) ∀ z, w ∈ G.\nThis completes the proof that (1) implies (2). To show the converse, assume that there exists an analytic operatorvalued function R 0 : G → B(H 0 , C m ) and K analytic functions,\nR k : G → B(H k ⊗ C M , C p ) on some Hilbert space H k such that I − f (z)f (w) * = R 0 (z)R 0 (w) * + K k=1 R k (z)(I −F k (z)F k (w) * ) (q k ) R k (w) * ∀ z, w ∈ G.\nBy using Theorem 2.4 there exists a vector-valued Kernel K such that M n (M(K)) = M n (Ã) completely isometrically for every n. It is easy to see that ((I − f (z)f (w) * )⊗K(z, w)) ≥ 0 ∀z, w ∈ G. This is equivalent to f ∈ M m (M(K)) and M f ≤ 1 which in turn is equivalent to (1). Thus, (1) and (2) are equivalent.\n\nClearly, (3) implies (2). The arguement for why (2) implies (3) is contained in [10] and we recall it. If we fix any k 0 , then since F k 0 (z) < 1 on G, we have that |f k 0 ,1,1 (z)| 2 + · · · + |f ko,1,m (z)| 2 < 1 on G. From this it follows that H(z, w) = (1 − f k 0 ,1,1 (z)f k 0 ,1,1 (w) − · · · − f k 0 ,1,m (z)f k 0 ,1,m (w)) is an F k 0 -limit and that H −1 (z, w) is positive definite. Now we have that R 0 (z)R 0 (w) * H −1 (z, w) is positive definite and so we may write, R 0 (z)R 0 (w) * H −1 (z, w) = G 0 (z)G 0 (w) * and we have that R 0 (z)R 0 (w) * = G 0 (z)H(z, w)G(w) . This shows that R 0 (z)R 0 (w) * is an F k -limit and so it may be absorbed into the sum.\n\n\nExamples and Applications\n\nIn this section we present a few examples to illustrate the above definitions and results. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T n ) such that T 1 T * 1 + · · · + T N T * N ≤ I, which is commonly called a row contraction and an admissible strict representation is given when T 1 T * 1 + · · · + T N T * N < I, which is generally referred to as a strict row contraction. In this case one can again easily see that · u = · u 0 by using the same r < 1 argument as in the last example and that f ∈ H ∞ R (B N ) if and only if f R = sup{ f (T ) : T 1 T * 1 + · · · + T N T * N < I} < +∞ These are the norms on polynomials considered by Drury [18], Popescu [28], Arveson [6], and Davidson and Pitts [17]. By Theorem 4.18 we will have for i.e., such that T * 1 T 1 + · · · + T * N T N ≤ I, which is generally referred to as a column contraction. This time the norm on H ∞ R (B N ) will be defined by taking suprema over all strict column contractions and we will have that f R ≤ 1 if and only if\nI m − f (z)f (w) * = N i=1 (1 − z iwi )K i (z, w),f ∈ M m,n (H ∞ R (B N )) that f R ≤ 1 if and only if I m − f (z)f (w) * = (1 − z, w )K(z, w), where K : B N × B N → MI m − f (z)f (w) * = R 1 (z)[(I N − (z iwj )) ⊗ I H ]R 1 (w) * for some R 1 : B N → B(C m , H), analytic. Example 5.4. Let G = B N as above, let F 1 (z) = (z 1 , ..., z N ) : B − N → M 1,N and F 2 (z) = (z 1 , ..., z N ) t : B N → M N,1 .\nAgain this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T N ) such that T 1 T * 1 + · · · + T N T * N ≤ I and T * 1 T 1 + · · · + T * N T N ≤ I, that is, which is both a row and column contraction. This time the norm on H ∞ R (B N ) is defined as the supremum over all commuting N-tuples that are both strict row and column contractions. We will have that f ∈ M m,n (H ∞ R (B N )) with f R ≤ 1 if and only if\nI m − f (z)f (w) * = (1 − z, w )K 1 (z, w) + R 1 (z)[(I N − (z iwj )) ⊗ I H ]R 1 (w) * ,\nwhere K 1 and R 1 are as before.\n\nThe last three examples illustrate that it is possible to have multiple rational representations of G, all with the same algebra, but which give rise to (possibly) different operator algebra norms on A. Thus, the operator algebra norm depends not just on G, but also on the particular presentation of G that one has chosen. We have surpressed this dependence on R to keep our notation simplified. and it can be seen that H ∞ R (D) is defined by\nf R = sup{ f (T ) : T 2 < 1, T 3 < 1} < +∞.\nIn this case we have that f ∈ M m,n (H ∞ R (D) and f R ≤ 1 if and only if\nI m − f (z)f (w) * = (1 − z 2w 2 )K 1 (z, w) + (1 − z 3w 3 )K 2 (z, w).\nExample 5.6. Let L = {z ∈ C : |z − a| < 1, |z − b| < 1}, where |a − b| < 1, then the functions f 1 (z) = z −a, f 2 (z) = z −b give a polynomial presentation of this \"lens\". The algebra of this presentation is again the algebra of polynomials. An admissible representation of this algebra is defined by choosing any operator satisfying T − aI ≤ 1 and T − bI ≤ 1, with strict inequalities for the admissible strict representations. In this case we easily see that · u = · u 0 , since given any operator T satisfying T − aI ≤ 1 and T − bI ≤ 1, and r < 1, S r = rT + (1 − r)(a + b) corresponds to the admissible strict representations and for any matrix of polynomials (p i,j (T )) = lim r→1 (p ij (S r )) . This algebra with this norm was studied in [9]. Their work shows that this norm is completely boundedly equivalent to the usual supremum norm. Our results imply that f ∈ M m,n (H ∞ R (L)) and f R ≤ 1 if and only if \nI m −f (z)f (w) * = (1−(z −a)(w − b))K 1 (z, w)+(1−(z −b)(w − b))K 2 (z, w).I m − f (z)f (w) * = R 1 (z)[(I M − (z i,j )(w i,j ) * ) ⊗ I H ]R 1 (w) * ,\nfor some appropriatley chosen R 1 .\n\nAll of the above examples are also covered by the theory of [4] and [10], except that their definition of the norm is slightly different. We address this difference in a later remark. We now turn to some examples that are not covered by their theory.\n\nExample 5.8. Let 0 < r < 1 be fixed and let A r = {z ∈ C : r < |z| < 1} be an annulus. Then this has a rational presentation given by F 1 (z) = z and F 2 (z) = rz −1 , and the algebra of this presentation is just the Laurent polynomials. Admissible representations of this algebra are given by selecting any invertible operator T satisfying T ≤ 1 and T −1 ≤ r −1 . Admissible strict representations are given by invertible operators satisfying T < 1 and T −1 < r −1 . It is no longer quite so clear that · u = · u 0 . However, this algebra with these norms is studied by the first author in [22] and among other results the equality of these norms was shown. In [17], it was shown that · u is completely boundedly equivalent to the usual supremum norm. In this case one can see that the · R is achieved by taking suprema over all T satisfying T < 1 and T −1 < r −1 . The formula for the norm is given by f R ≤ 1 if and only if I m − f (z)f (w) * = (1 − zw)K 1 (z, w) + (1 − r 2 z −1 w −1 )K 2 (z, w). Our results imply that f R ≤ 1 if and only if\n1 − f (z)f (w) * = (1 − φ(z)φ(w) * )K 1 (z, w)\nIn particular, if we take φ(z) = z−1 z+1 then it maps the halfplane H = {z : Re(z) > 0} to the unit disk. For this particular φ, we have that Q R = {T : σ(T ) ⊆ H and Re(T ) ≥ 0}. where w(T ) denotes the numerical radius of T. Thus, H ∞ R (D) becomes the \"universal\" operator algebra that one obtains by substituting an operator of numerical radius less than one for the variable z and we have a quite different quantization of the unit disk. Our results give a formula for this norm, but only in terms of R-limits, so further work would need to be done to make it explicit.\n\nExample 5.12. There is a second way that one can quantize many convex sets. Let G = {z : |z − a k | < r k , k ∈ I} ⊆ C be an open, bounded convex set that can be expressed as an intersection of a possibly infinite set of open disks. For example, any convex set with a smooth boundary with uniformly bounded curvature can be expressed in such a fashion. Then G has a rational presentation given by F k (z) = r −1 k (z − a k ), k ∈ I the algebra of the presentation is just the polynomial algebra and an admissible representation is given by selecting any operator T satisfying, T − a k I ≤ r k , k ∈ I. Thus, we again a factorization result, but only in terms of R-limits.\n\nThe above definitions allow one to consider many other examples. For example, one could fix 0 < r < 1 and let G = {z ∈ B N : r < |z 1 |}, with rational presentation f 1 (z) = (z 1 , ..., z N ) ∈ M 1,N , and f 2 (z) = rz −1\n\n1 . An admissible representation would then correspond to a commuting row contraction with T 1 invertible and T −1 1 ≤ r −1 .\n\nWe now compare and contrast some of our hypotheses with those of [4] and [10].\n\nRemark 5.13. Let G = {z ∈ C N : F k (z) < 1 ∀ k = 1, · · · , K} where the F k 's are matrix-valued polynomials defined on G. Then for f ∈ Hol(G), [4] and [10] really study a norm given by f s = sup{ f (T ) } where the supremum is taken over all commuting N -tuples of operators T with F k (T ) < 1 ∀k. We wish to contrast this norm with our f R . In [4] it is shown that the hypotheses F k (T ) < 1, k = 1, ..., K implies that σ(T ) ⊆ G. Thus, we have that f s ≤ f R . In fact, we have that f s = f R . This can be seen by the fact that they obtain identical factorization theorems to ours. This can also be seen directly in some cases where the algebra A contains the polynomials and when it can be seen that · R is attained by taking the supremum over matrices(see Remark 4.11). Indeed, if f R is attained as the supremum over commuting N -tuples of finite matrices T = (T 1 , ..., T N ) satisfying σ(T ) ⊆ G and F k (T ) ≤ 1 then such an N -tuple of commuting matrices, can be conjugated by a unitary to be simultaneously put in upper triangular form. Now it is easily argued that the strictly upper triangular entries can be shrunk slightly so that one obtains new N -tuples T ǫ = (T 1,ǫ , ..., T N,ǫ ) satisfying, F k (T ǫ ) < 1, k = 1, ..., K and T i − T i,ǫ < ǫ. But we do not have a simple direct argument that works in all cases.\n\nRemark 5.14. We do not know how generally it is the case that · u is a local norm. That is, we do not know if f u = f R for f ∈ M n (A). In particular, we do not know if this is the case for Example 5.5. In this case, the algebra of the of the presentation is A = span{z n : n ≥ 0, n = 1}. If we write a polynomial p ∈ A in terms of its even and odd decomposition, p = p e + p o , then p e (z) = q(z 2 ) and p o = z 3 r(z 2 ) for some polynomials q, r. \n\nProposition 1 . 2 .\n12Let A be an operator algebra of functions on X, then A ⊆ ℓ ∞ (X), and for every n and every(f i,j ) ∈ M n (A), we have (f ij ) ∞ ≤ (f ij ) Mn(A) .\n\nF\n: M n (A) → M n (A/I F ) ∼ = M n (A)/M n (I F ).\n\nLemma 1. 8 .\n8If A is an operator algebra of functions on the set X, thenà equipped with the norms of Definition 1.5 is a local operator algebra of functions on X. Moreover, for (f i,j ) ∈ M n (A), (f i,j ) Mn(Ã) = (f i,j ) Mn(A L ) .\n\nExample 1. 15 .\n15Let A = A(D) ⊆ C(D − ) be the subalgebra of the algebra of continuous functions on the closed disk consisting of the functions that are analytic on the open disk D. Identifying M n (A(D)) ⊆ M n (C(D − )) as a subalgebra of the algebra of continuous functions from the closed disk to the matrices, equipped with the supremum norm, gives A(D) it's usual operator algebra structure. With this structure it can be regarded as a local operator algebra of functions on D or on D − . If we regard it as a local operator algebra of functions on D − , then A(D) A(D). To see that the containment is strict, note that f (z) = (1 + z)/2 ∈ A(D) and f n (z) → χ {1} , the characteristic function of the singleton {1}.\n\n\nHowever, if we regard A(D) as a local operator algebra of functions on D, then its BPW completion A(D) = H ∞ (D), the bounded analytic functions on the disk, with its usual operator structure.\n\n\nuniformly on compact subsets of D. Thus, by the identity theorem F ≡ G on D. Hence, A is a local operator algebra. Finally, by using Lemma 1.11 and a similar argument, one can also show thatà = H ∞ (D).\n\nExample 1 . 17 .\n117Let A = H ∞ (D) but endowed with a new norm. Fix b > 1,\n\nLemma 2 . 1 .\n21Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H). Then M(K) ⊆ B(L) is a weak*-closed subalgebra.\n\nTheorem 2 . 2 .\n22Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X ×X → B(H) and let M(K) ⊆ B(L) denote the multiplier algebra, endowed with the operator algebra structure that it inherits as a subalgebra. If K(x, x) = 0, for every x ∈ X and M(K) separates points on X, then M(K) is a BPW complete local dual operator algebra of functions on X.\n\nTheorem 2 . 4 .\n24Let A be an operator algebra of functions on the set X then there exists a Hilbert space, C and a positive definite function K : X × X → B(C) such that M(K) =à completely isometrically.\n\nProposition 2. 6 .\n6Let A be a local operator algebra of functions on the set X. Then the net (f λ ) λ ∈ Ball(A) converges in the weak*-topology if and only if it converges pointwise on X.\n\nDefinition 3. 1 .\n1An operator algebra, B is called RFD if for every n and for every (\n\nLemma 3. 2 .\n2Let B be a concrete k-idempotent operator algebra. Then S(B * B)) is a Schur ideal affiliated with B, i.e., B = A(S(B * B)) completely isometrically.\n\nTheorem 3. 3 .\n3Every k-idempotent operator algebra is weak*-RFD.\n\nCorollary 3. 5 .\n5Every local operator algebra of functions is RFD.\n\nDefinition 4 . 1 .\n41Let G ⊆ C N be an open set. If there exists a set of matrixvalued functions, F k = (f k,i,j ) : G − → M m k\n\nDefinition 4 . 3 .\n43Let G ⊆ C N be an analytically presented domain with presentation F = (f i,j ) : G → M m,n , and let H be a Hilbert space. A homomorphism π : A → B(H) of the algebra of the presentation is called an admissible representation provided that (π(f i,j )) ≤ 1 in M m,n (B(H) = B(H n ,\n\nTheorem 4. 5 .\n5Let G be an analytically presented domain with presentationR = {F k = (f k,i,j ) : G → M m k ,n k , k ∈ I},let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A), where m, n are arbitrary. Then the following are equivalent:\n\nCorollary 4. 6 .\n6Let G be presented by vector-valued functions, F k = (f k,j ) : G → M 1,n k , k ∈ I, let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A). Then the following are equivalent:\n\nDefinition 4 . 8 .\n48Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}. We define the quantized version of G to be the collection of all commuting N-tuples of operators,\n\nDefinition 4. 12 .\n12Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}.We shall call a function H : G × G → M m an R-limit, provided that H is the pointwise limit of a net of functions H λ : G × G → M m of the form given byTheorem 4.5(iii).\n\nCorollary 4. 13 .\n13Let G ⊆ C N be an analytically presented domain, with a separating presentation R = {F k : G − → M m k ,n k , k ∈ I}. Then the following are equivalent:\n\nProposition 4 . 14 .\n414Let G be an analytically presented domain with a finite presentation R= {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}.For each compact subset S ⊆ G, there exists a constant C, depending only on S, such that given a factorization of the form,\n\n\nThe proof of the following result is essentially contained in[10, Lemma 3.3].\n\nProposition 4 . 15 .\n415Let G be a bounded domain in C N and let F = (f i,j ) : G → M m,n be analytic with F (z) < 1 for z ∈ G. If H : G × G → M p is analytic in the first variables, coanalytic in the second variables and there exists a net of matrix-valued functions P λ ∈ M p,r λ (Hol(G)) which are uniformly bounded on compact subsets of G, such that H(z, w) is the pointwise limit of H λ (z, w) = P λ (z)(I m − F (z)F (w) * ) (q λ ) P λ (w) * where r λ = q λ m k , then there exists a Hilbert space H and an analytic function, R : G → B(H ⊗ C M , C p ) such that H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * .\n\nRemark 4 . 16 .\n416Conversely, any function that can be written in the formH(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * can beexpressed as a limit of a net as above by considering the directed set of all finite dimensional subspaces of H and for each finite dimensional subspace setting H F (z, w) = R F (z)[(I m − F (z)F (w) * ) ⊗ I F ]R F (w) * , where R F (z) = R(z)(P F ⊗ I m ) with P F the orthogonal projection onto F.\n\nDefinition 4 . 17 .\n417We shall refer to a function H : G × G → M m (M p ) that can be expressed as H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ H]R(w) * for some Hilbert space H and some analytic function R : G → B(H ⊗ C m , C p ), as an F-limit. Theorem 4.18. Let G be an analytically presented domain with a finite separating presentation R = {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}, let f = (f ij ) be a M m,n -valued function defined on G. Then the following are equivalent:\n\nExample 5 . 1 .\n51Let G = D N , be the polydisk which has a presentation given by the coordinate functions F i (z) = z i , 1 ≤ i ≤ N. Then the algebra of this presentation is the algebra of polynomials and an admissible representation is given by any choice of N commuting contractions, (T 1 , ..., T N ) on a Hilbert space. Given a matrix of polynomials,(p i,j ) u = sup (p i,j (T 1 , ..., T N ))where the supremum is taken over all N-tuples of commuting contractions. This is the norm considered by Agler in[1], which is sometimes called the Schur-Agler norm[21].Our Q(D N ) = {T = (T 1 , ..., T N ) : σ(T ) ⊆ D N and T i ≤ 1}. Note that if we replace such a T by rT = (rT 1 , ..., rT N ) then rT i < 1, rT ∈ Q R (D N ) and taking suprema over all T ∈ Q R (D N ) will be the same as taking a suprema over this smaller set. Thus, the algebra H ∞ R (D N ) consists of those analytic functions f such that f R = sup{ f (T 1 , ..., T N ) : T i < 1, i = 1, ..., N } < +∞. By Theorem 4.18 for f ∈ M m,n (H ∞ R (D N ), we have that f R ≤ 1 if and only if\n\n\nfor some analytic-coanalytic positive definite functions, K i : D N × D N → M m . Example 5.2. Let G = B N denote the unit Euclidean ball in C N . If we let F 1 (z) = (z 1 , ..., z N ) : B − N → M 1,N , then this gives us a polynomial presentation. Again the algebra of the presentation is the polynomial algebra.\n\n\nm is an analytic-coanalytic positive definite function.Example 5.3. Let G = B N as above and let F 1 (z) = (z 1 , ..., z N ) t : B N → M N,1 .Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T N ) such that (T 1 , ..., T N ) t ≤ 1,\n\nExample 5. 5 .\n5Let G = D the open unit disk in the complex plane and let F 1 (z) = z 2 , F 2 (z) = z 3 . It is easy to check that the algebra A of this presentation is generated by the component functions and the constant function is the span of the monomials, {1, z n : n ≥ 2}, and that A separates the points of G. In this case an (strict)admissible representation, π : A → B(H), is given by any choice of a pair of commuting (strict)contractions, A = π(z 2 ), B = π(z 3 ), satisfying A 3 = B 2 . Again, it is easy to see that . u = . u 0 . On the other hand Q(D) = {T : σ(T ) ⊆ D and T 2 ≤ 1, T 3 ≤ 1}\n\nExample 5 . 7 .\n57Let G = {(z i,j ) ∈ M M,N : (z i,j ) < 1} and let F : G → M M,N be the identity map F (z) = (z i,j ). Then this is a polynomial presentation of G and the algebra of the presentation is the algebra of polynomials in the M N variables {z i,j }. An admissible representation of this algebra is given by any choice of M N commuting operators {T i,j } on a Hilbert space H, such that (T i,j ) ≤ 1 in M M,N (B(H)) and as above, one can show that · R is acheived by taking suprema over all commuting M N -tuples for which (T i,j ) < 1. We have that f ∈ M m,n (H ∞ R (G) and f R ≤ 1 if and only if\n\nExample 5 . 9 .\n59Let G be a simply connected domain in C and φ : G → D be a biholomorphic map. Then G = {z ∈ C : |φ(z)| < 1} and the Q(G) = {T : σ(T ) ⊆ G and φ(T ) ≤ 1} where R = {φ}. In this case the algebra A of the presentation is just the algebra of all polynomials in φ. Thus, an admissible representation of this algebra is defined by choosing an operator B ∈ B(H) that satisfies B ≤ 1 and defining π : A → B(H) via π(p(φ)) = p(B), where p is a polynomial. A strict admissible representation is defined similarly by first choosing a strict contraction. In this case, it is immediate that . u = . u 0 and that f ∈ H ∞ R (G) if and only if f ∈ Hol(G) andf R = sup{ f (T ) : T ∈ Q R (G)} < +∞.\n\nExample 5 . 10 .\n510Similarly, if we let G = {z ∈ C N : |φ i (z)| < 1, i = 1, ..., N } where φ i (z) = z i −1 z i +1 , then G is an intersection of half planes and Q R consists of all commuting N -tuples of operators, (T 1 , ..., T N ) such that σ(T i ) ⊆ H and Re(T i ) ≥ 0 for all i. Applying our results, we obtain a factorization result for half planes. These algebras have been studied by D.Kalyuzhnyi-Verbovetzkii in[20].Example 5.11. Let G ⊆ C be an open convex set and represent it as an intersection of half planes H θ . Each half plane can be expressed as {z : |F θ (z)| < 1} for some family of linear fractional maps. If we let R = {F θ }, then Q R (G) = {T : σ(T ) ⊆ G and F θ (T ) ≤ 1 ∀θ}. Moreover, each inequality F θ (T ) ≤ 1 can be re-written as an operator inequality for the real part of some translate and rotation of T. For example, when G = D, we may take F θ (z) = z z−2e iθ , for 0 ≤ θ < 2π. In this case, one checks that F θ (T ) ≤ 1 if and only if Re(e iθ T ) ≤ I. Thus, it follows that Q(D) = {T : σ(T ) ⊆ D and w(T ) ≤ 1},\n\n\nIn this case it is easily seen thatp u = sup{ q(A) + Br(A) : A ≤ 1, B ≤ 1, AB = BA, A 3 = B 2 }, while p L = p R = sup{ p(T ) : T 2 ≤ 1, T 3 ≤ 1}.\nM. MITTAL AND V. I. PAULSEN\n\nSome interpolation theorems of Nevanlinna-Pick type. J Agler, PreprintJ. Agler, Some interpolation theorems of Nevanlinna-Pick type, Preprint, 1988.\n\nOn the Representation of Certain Holomorphic Function defined on a Polydisk. J Agler, Topics in Operator Theory: Ernst D. Hellinger Memorial Volume. J. Agler, On the Representation of Certain Holomorphic Function defined on a Poly- disk, Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, 48 (1990),47- 66.\n\nD Alpay, A Dijksma, J Rovnyak, H De Snoo, Schur Functions, Operator Colligation, and Reproducing Kernel Pontryagin Spaces, Operator Theory Advances and Applications. BaselBirkhauser96D. Alpay, A. Dijksma, J. Rovnyak and H. de Snoo, Schur Functions, Operator Colli- gation, and Reproducing Kernel Pontryagin Spaces, Operator Theory Advances and Applications Vol. 96, Birkhauser,Basel, pp. 1-13.\n\nA von Neumann type inequality for certain domains in C n. C G Ambrozie, D Timotin, Proc. Amer. Math. Soc. 131C.G. Ambrozie and D. Timotin, A von Neumann type inequality for certain domains in C n , Proc. Amer. Math. Soc., 131(2003), 859-869.\n\nOn residually finite-dimensional C*-algebras. R J Archbold, Proc. Amer. Math. Soc. 123946089MRR. J. Archbold, On residually finite-dimensional C*-algebras, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2935-2937. MR 95m:46089\n\nSubalgebras of C * -algebras. III. Multivariable operator theory. W Arveson, Acta Math. 1812W. Arveson, Subalgebras of C * -algebras. III. Multivariable operator theory, Acta Math. 181(1998), no. 2, 159-228.\n\nRealization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining functions in C n. J A Ball, V Bolotnikov, Jour. Func. Anal. J.A. Ball and V. Bolotnikov, Realization and interpolation for Schur-Agler-class func- tions on domains with matrix polynomial defining functions in C n , Jour. Func. Anal.,\n\nUnitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables. J A Ball, T T Trent, J. Funct. Anal. 197J.A. Ball and T.T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 197 (1998), 1-61.\n\nA lenticular version of a von Neumann inequality. B Beckermann, M Crouzeix, Archiv der Mathematik. 86B. Beckermann, M. Crouzeix : A lenticular version of a von Neumann inequality. Archiv der Mathematik, vol. 86, 2006, p. 352-355.\n\nBlackadar operator algebra. B , B. Blackadar operator algebra.\n\nOperator Algebras and Their Modules-an Operator Space Approach. D P Blecher, C Le Merdy, Oxford University Press30D.P. Blecher and C. Le Merdy, Operator Algebras and Their Modules-an Operator Space Approach, London Mathematical Society Monographs, Vol 30, Oxford Univer- sity Press, 2004.\n\nExplicit construction of universal operator algebras and an application to polynomial factorization. D P Blecher, V I Paulsen, Proc. Amer. Math. Soc. 112D.P. Blecher and V.I. Paulsen, Explicit construction of universal operator algebras and an application to polynomial factorization, Proc. Amer. Math. Soc. 112(1991), 839-850.\n\nA characterization of operator algebras. D P Blecher, Z.-J Ruan, A M Sinclair, J. Funct. Anal. 89D.P. Blecher, Z.-J. Ruan and A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89(1990), 188-201.\n\nBanach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy. J Burbea, P Masani, Research Notes in Mathematics. 90Pitman Advanced Publishing ProgramJ. Burbea and P. Masani, Banach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy, Research Notes in Mathematics 90, Boston-London-Melbourne, Pitman Advanced Publishing Program (1984)\n\nResidually finite dimensional C* algebras. M Dadarlat, Operator algebras and operator theory: International Conference on Operator Algebras and Operator Theory ,By Liming Ge. Shanghai, ChinaM. Dadarlat, Residually finite dimensional C* algebras ,45-50, in Operator algebras and operator theory: International Conference on Operator Algebras and Operator Theory ,By Liming Ge, July 4-9, 1997, Shanghai, China.\n\nNevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras. K R Davidson, D Pitts, Integral Equations Operator Theory. 313K.R. Davidson and D. Pitts, Nevanlinna-Pick interpolation for non-commutative ana- lytic Toeplitz algebras, Integral Equations Operator Theory 31(1998), no. 3, 321-337.\n\nCompletely bounded maps and hypo-Dirichlet algebras. R G Douglas, V I Paulsen, Acta Sci. Math. (Szeged). 501-2R.G. Douglas and V.I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged) 50(1986), no. 1-2, 143-157.\n\nA generalization of von Neumann's inequality to the complex ball. S W Drury, Proc. Amer. Math. Soc. 683S.W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc. 68(1978), no. 3, 300-304.\n\nFree and residually finite-dimensional C*-algebras. K R Goodearl, P , J. Funct. Anal. 90246078MRK. R. Goodearl and P. Menal, Free and residually finite-dimensional C*-algebras, J. Funct. Anal. 90 (1990), no. 2, 391-410. MR 91f:46078\n\nOn the Bessmertnyi class of homogeneous positive holomorphic functions of several variables. D Kalyuzhnyi-Verbovetzkii, J.A. Ball, J.W. Helton, M. Klaus, LD. Kalyuzhnyi-Verbovetzkii, On the Bessmertnyi class of homogeneous positive holo- morphic functions of several variables, in: J.A. Ball, J.W. Helton, M. Klaus, L.\n\nRodman, Current trends in Operator Theory and its Applications. BaselBirkhauser-Verlag149Rodman (Eds.) Current trends in Operator Theory and its Applications, Vol. OT 149, Birkhauser-Verlag, Basel, 2004, 255-289.\n\nAn operator algebraic proof of Agler's factorization theorem. S Lata, M Mittal, V I Paulsen, Proc. Amer. Math. Soc. to appearS. Lata, M. Mittal, V.I. Paulsen, An operator algebraic proof of Agler's factorization theorem, Proc. Amer. Math. Soc., to appear.\n\n. M , University of HoustonPhD ThesisM. Mittal, PhD Thesis, University of Houston.\n\nCompletely Bounded Maps and Operator Algebras. V I Paulsen, Cambridge studies in Advanced Mathematics. 78V.I. Paulsen , Completely Bounded Maps and Operator Algebras, Cambridge studies in Advanced Mathematics, 78 (2002).\n\nOperator Algebra of Idempotents. V I Paulsen, J. Funct. Anal. 181V.I. Paulsen, Operator Algebra of Idempotents, J. Funct. Anal. 181, 209-226 (2001).\n\nV I Paulsen, M Raghupathi, Representations of Logmodular Algebras, Proc. of the AMS. to appearV.I. Paulsen and M. Raghupathi, Representations of Logmodular Algebras, Proc. of the AMS, to appear.\n\nTheory of Reproducing kernels in Hilbert spaces of vector-valued functions. G B Pedrick, Univ.of Kansas Tech. Rep. 19G.B. Pedrick, Theory of Reproducing kernels in Hilbert spaces of vector-valued func- tions, Univ.of Kansas Tech. Rep. 19, Lawrence, 1957.\n\nIntroduction to Operator Space Theory. G Pisier, Lecture note series. Cambridge University Press294G. Pisier, Introduction to Operator Space Theory, London Mathematical Society Lec- ture note series 294, Cambridge University Press, (2003).\n\nInterpolation problems in several variables. (English summary). G Popescu, J. Math. Anal. Appl. 2271G. Popescu, Interpolation problems in several variables. (English summary) J. Math. Anal. Appl. 227 (1998), no. 1, 227-250.\n\nA joint spectrum for several commuting operators. J L Taylor, J. Funct. Anal. 6J.L. Taylor, A joint spectrum for several commuting operators J. Funct. Anal. 6 (1970), 172-191.\n\nThe analytic functional calculus for several commuting contractions. J L Taylor, Acta Math. 125J.L. Taylor, The analytic functional calculus for several commuting contractions, Acta Math. 125 (1970), 1-38.\n\nAnalytic Functional Calculus and Spectral Decompositions. J L Taylor, Math. and its Appl., East European Series. 1D. Reidel Publishing CoDepartment of Mathematics, University of Houston ; Department of Mathematics, University of HoustonE-mail address: mittal@math.uh. A. E-mail address: vern@math.uh.eduJ.L. Taylor, Analytic Functional Calculus and Spectral Decompositions, Math. and its Appl., East European Series, vol. 1, D. Reidel Publishing Co. Dordrecht, Holland, 1982. Department of Mathematics, University of Houston, Houston, Texas 77204- 3476, U.S.A. E-mail address: mittal@math.uh.edu Department of Mathematics, University of Houston, Houston, Texas 77204- 3476, U.S.A. E-mail address: vern@math.uh.edu\n"},"corpus_id":{"kind":"number","value":14935653,"string":"14,935,653"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0907.5184\",\n \"mag\": 2951663843,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1016/j.jfa.2010.01.006\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0907.5184v1.pdf\"\n ],\n \"pdf_sha\": \"94f2a4cc8e9edae9e647817a53eab299bdbe9c3a\",\n \"open_access\": {\n \"license\": \"elsevier-specific: oa user license\",\n \"url\": \"https://doi.org/10.1016/j.jfa.2010.01.006\",\n \"status\": \"BRONZE\"\n }\n}"},"title":{"kind":"string","value":"OPERATOR ALGEBRAS OF FUNCTIONS"},"authors":{"kind":"list like","value":["Meghna Mittal","Vern I Paulsen"],"string":"[\n \"Meghna Mittal\",\n \"Vern I Paulsen\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"A concrete operator algebra A is just a subalgebra of B(H), the bounded operators on a Hilbert space H. The operator norm on B(H) gives rise to a norm on A. Moreover, the identification M n (A) ∼ = A ⊗ M n ⊆ B(H ⊗ C n ) ∼ = B(H n ) endows the matrices over A with a family of norms in a natural way, where M n denotes the algebra of n × n matrices. It is common practice to identify two operator algebras A and B as being the \"same\" if and only if there exists an algebra isomorphism π : A → B that is not only an isometry, but which also preserves all the matrix norms, that is such that (π(a i,j )) Mn(B) = (a i,j ) Mn(A) , for every n and every element (a i,j ) ∈ M n (A). Such a map π is called a completely isometric isomorphism. In [13] an abstract characterization of operator algebras, in the above sense, was given and since that time a theory of such algebras has evolved. For more details on the abstract theory of operator algebras, see [11], [23] or [27].\nIn this note we present a theory for a special class of abstract abelian operator algebras which contains many important examples arising in function theoretic operator theory, including the Schur-Agler and the Arveson-Drury-Popescu algebras. This theory will allow us to answer certain types of questions about such algebras in a unified manner. We will prove that our hypotheses give an abstract characterization of operator algebras that are completely isometrically isomorphic to multiplier algebras of vector-valued reproducing kernel Hilbert spaces.\nOur results will show that under certain mild hypotheses, operator algebra norms which are defined by taking the supremum of certain families of operators on Hilbert spaces of arbitrary dimensions can be obtained by restricting the family of operators to finite dimensional Hilbert spaces. Thus, in a certain sense, which will be explained later, our results give conditions that guarantee that an algebra is residually finite dimensional.\nFinally, we apply our results to study \"quantized function theories\" on various domains. Our work in this direction should be compared to that of Ambrozie-Timotin [4], Ball-Bolotnikov [10] and Kalyuzhnyi-Verbovetzkii [20]. These authors obtain the same type of factorization theorem via unitary colligation methods as we obtain via operator algebra methods, but they assume somewhat different(and in many cases stronger) hypotheses. We, also, obtain a bit more information about the algebras themselves, including the fact that in many cases their norms can be obtained as supremums over matrices instead of operators.\nWe now give the relevant definitions. Recall that given any set X the set of all complex-valued functions on X is an algebra over the field of complex numbers.\nDefinition 1.1. We call A an operator algebra of functions on a set X provided:\n(1) A is a subalgebra of the algebra of functions on X, (2) A separates the points of X and contains the constant functions, (3) for each n M n (A) is equipped with a norm . Mn(A) , such that the set of norms satisfy the BRS axioms [13] to be an abstract operator algebra, (4) for each x ∈ X, the evaluation functional, π x : A → C, given by π x (f ) = f (x) is bounded.\nA few remarks and observations are in order. First note that if A is an operator algebra of functions on X and B ⊆ A is any subalgebra, which contains the constant functions and still separates points, then B, equipped with the norms that M n (B) inherits as a subspace of M n (A) is still an operator algebra of functions.\nThe basic example of an operator algebra of functions is ℓ ∞ (X), the algebra of all bounded functions on X. If for (f i,j ) ∈ M n (ℓ ∞ (X)) we set\nwhere · Mn is the norm on M n obtained via the identification M n = B(C n ), then it readily follows that properties (1)-(4) of the above definition are satisfied. Thus, ℓ ∞ (X) is an operator algebra of functions on X in our sense and any subalgebra of ℓ ∞ (X) that contains the constants and separates points will be an operator algebra of functions on X when equipped with the subspace norms. Proof. Since π x : A → C is bounded and the norm is sub-multiplicative, we have that for any f ∈ A, |f (x)| n = |π x (f n )| ≤ π x f n ≤ π x f n . Taking the n-th root of each side of this inequality and letting n → +∞, yields |f (x)| ≤ f , and hence, f ∈ ℓ ∞ (X). Note also that π x = 1. Finally, since every bounded, linear functional on an operator space is completely bounded and the norm and the cb-norm are equal, we have that π x cb = π x = 1. Thus, for (f i,j ) ∈ M n (A), we have (f i,j (x)) Mn = (π x (f i,j )) ≤ (f i,j ) Mn(A) .\nGiven an operator algebra of functions on a set X, A and F = {x 1 , ..., x k } a set of distinct points in X, we define I F = {f ∈ A : f (z) = 0 ∀ z ∈ F } and note that M n (I F ) = {f ∈ M n (A) : f (z) = 0 ∀ z ∈ F } ∀ n. The quotient space A/I F has a natural set of matrix norms given by defining (f i,j + I F ) = inf{ (f i,j + g i,j ) Mn(A) : g i,j ∈ I F }. Alternatively, this is the norm on M n (A/I F ) that comes via the identification, M n (A/I F ) = M n (A)/M n (I F ), where the latter space is given its quotient norm. It is easily checked that this family of matrix norms satisfies the BRS conditions and so gives A/I F the structure of an abstract operator algebra.\nWe let π F (f ) = f + I F denote the quotient map π F : A → A/I F so that for each n, π Since A is an algebra which separates points on X and contains constant functions, it follows that there exists functions f 1 , ..., f k ∈ A, such that f i (x j ) = δ i,j , where δ i,j denotes the Dirac delta function. If we set E j = π F (f j ), then it is easily seen that whenever f ∈ A and f (x i ) = λ i , i = 1, ..., k, then π F (f ) = λ 1 E 1 + · · · + λ k E k . Moreover, E i E j = δ i,j E i , and E 1 + · · · + E k = 1, where 1 denotes the identity of the algebra A/I F . Thus, A/I F = span{E 1 , ..., E k }, is a unital algebra spanned by k commuting idempotents. Such algebras were called k-idempotent operator algebras in [24] and we will use a number of results from that paper."},{"title":"Definition 1.3. An operator algebra of functions A on a set X, is called a local operator algebra of functions if it satisfies","content":"F ((f ij )) = (f ij ) ∀ (f ij ) ∈ M n (A) and for every n, where the supremum is taken over all finite subsets of X.\nThe following result shows that every operator algebra of functions can be re-normed so that it becomes local. Proposition 1.4. Let A be an operator algebra of functions on X, let A L = A and define a family of matrix norms on A L , by setting (f i,j ) Mn(A L ) = sup F (π F (f i,j )) Mn(A/I F ) . Then A L is a local operator algebra of functions on X and the identity map, id : A → A L , is completely contractive.\nProof. It is clear from the definition of the norms on A L that the identity map is completely contractive and it is readily checked that A L is an operator algebra of functions on X.\nLetπ F :\nWe claim that for any f ∈ A, and any finite subset F ⊆ X, we have that π F (f ) = π F (f ) . To see the other inequality note that for g ∈ I F , and G ⊆ X a finite set, we have f + g L = sup G π G (f + g) ≥ π F (f + g) = π F (f ) . Hence, π F (f ) ≥ π F (f ) , and equality follows. A similar calculation shows that (π F (f i,j )) = (π F (f i,j )) , for any matrix of functions.\nNow it easily follows that A L is local, since\nWe letà denote the set of functions that are BPW limits of bounded nets of functions in A. Definition 1.5. Given an operator algebra of functions A on X we say that f : X → C is a BPW limit of A if there exists a uniformly bounded net (f λ ) λ ∈ A that converges pointwise on X to f. We say that A is BPW complete if it contains the set of functions that are BPW limits of bounded nets of functions in A. Letà denote the set of such functions, that is,\nIt is easily checked that for each n, the above formula defines a norm on M n (Ã). It is also easily checked that a matrix-valued function, (f i,j ) : X → M n is the pointwise limit of a uniformly bounded net (f λ i,j ) ∈ M n (A) if and only if f i,j ∈à for every i, j. Lemma 1.6. Let A be an operator algebra of functions on X and let (f i,j ) ∈ M n (Ã). Then\nThe collection of finite subsets of X determines a directed set, ordered by inclusion. If we choose for each finite set F, functions (g F i,j ) satisfying the conditions of the right hand set, then these functions define a net that converges BPW to (f i,j ) and hence, the right hand side is larger than the left. Conversely, given a net (f λ i,j ) that converges pointwise to (f i,j ) and satisfies (f λ i,j ) ≤ C and any finite set\nThus, given ǫ > 0, since the functions f 1 , ..., f k depend only on F, we may choose λ so that (g λ i,j ) < C + ǫ. This shows the other inequality. Lemma 1.7. Let A be an operator algebra of functions on the set X, theñ A equipped with the collection of norms on M n (Ã) given in Definition 1.5 is an operator algebra.\nProof. It is clear from the definition ofà that it is an algebra. Thus, it is enough to check that the axioms of BRS are satisfied by the algebraà equipped with the matrix norms given in the Definition 1.5.\nIf L and M are scalar matrices of appropriate sizes and G ∈ M n (Ã), then\nThis yields L Mn(Ã) ≤ G Mn(Ã) H Mn(Ã) , and so the multiplication is completely contractive.\nFinally, to see that the L ∞ conditions are met, let G ∈ M n (Ã) and\nThis shows that G⊕ H M n+m (Ã) ≤ max{ G Mn(Ã) , H Mm(Ã) }, and so the L ∞ condition follows. This completes the proof of the result. Proof. It is clear from the definition of the norms onà that the identity map from A toà is completely contractive and thus A ⊆à as sets. This indeed shows thatà separates points of X and contains constant functions.\nSince A is an operator algebra of functions on the set X, we have that\nBy letting ǫ → 0 and supping over z ∈ X, we get that\nHence,à is an operator algebra of functions on the set X.\n. To see the other inequality, as-\nsince I F ⊆Ĩ F . We claim that for any (f ij ) ∈ M n (A), and for any finite subset F ⊆ X, we have that (f ij + I F ) = (f ij +Ĩ F ) . To see the other inequality, let (g ij ) ∈ M n (Ĩ F ). Then ∀ ǫ > 0, and G ⊆ X,\n, and the equality follows. Now it is immediate to see that,\nCorollary 1.9. If A is a BPW complete operator algebra then A L =à completely isometrically.\nProof. Since A is BPW complete, A =à as sets. But by Lemma 1.6, the norm defined on A L agrees with the norm defined onÃ. Remark 1.10. In the view of above corollary, we denote the the norm oñ A by . L . Lemma 1.11. If A is an operator algebra of functions on X, then Ball(A L ) is BPW dense in Ball(Ã) andà is BPW complete, i.e.,à =Ã.\nProof. It can be easily checked that above is equivalent to showing that A L is BPW dense inÃ. We'll only prove that A L BP W ⊆Ã, since the other containment follows immediately by the definition ofÃ. Let {f λ } be a net in A L such that f λ → f pointwise and sup λ f λ A L < C. Then for fixed F ⊆ X and ǫ > 0,\nFinally, note that the argument similar to the above readily yields that A is BPW complete.\nAll the above lemmas can be summarized as the following theorem. Theorem 1.12. If A is an operator algebra of functions on X, thenà is a BPW complete local operator algebra of functions on X which contains A L completely isometrically as a BPW dense subalgebra. Definition 1.13. Given an operator algebra of functions A on X, we call A the BPW completion of A.\nWe now present a few examples to illustrate these concepts. We will delay the main family of examples to the next section. Example 1.14. If A is a uniform algebra, then there exists a compact, Hausdorff space X, such that A can be represented as a subalgebra of C(X) that separates points. If we endow A with the matrix-normed structure that it inherits as a subalgebra of C(X), namely, (f i,j ) = (f i,j ) ∞ ≡ sup{ (f i,j ) Mn : x ∈ X}, then A is a local operator algebra of functions on X. Indeed, to achieve the norm, it is sufficient to take the supremum over all finite subsets consisting of one point. In this case the BPW completionà is completely isometrically isomorphic to the subalgebra of ℓ ∞ (X) consisting of functions that are bounded, pointwise limits of functions in A. Example 1. 16. Let X = ǫD, 0 < ǫ < 1 and A = {f ∈ H ∞ (D) : f : X → C}. If we endow A with the matrix-normed structure on H ∞ (D), then A is an operator algebra of functions on X. Also, it can be verified that A is a local operator algebra of functions and that A =Ã.\n. It can be easily verified that A is a BPW complete operator algebra of functions. However, we also claim that A is local. To prove this we proceed by contradiction.\nIn this case,\n) such that G| Y = 0 and F + G < 1+c 2 . Thus, we can write B Y (z) = z−ǫ 1−ǭz , so that we can write G(z) = zB Y (z)H(z), for some H ∈ M n (H ∞ ). It follows that H ∞ < 2, since G ∞ < 2. We now consider\nExample 1.18. This is an example of a non-local algebra that arises from boundary behavior.\nThen it is easy to check that A is an operator algebra of functions on the set D. Also, it can be verified that A is not local. To see this, note\nThus, sup Y ⊆D π Y (z) < z and hence A is not local. Example 1. 19. This example shows that one can easily build non-local algebras by adding \"values\" outside of the set X. Let A be the algebra of polynomials regarded as functions on the set X = D. Then A endowed with the matrix-normed structure as (p ij ) = max{ (p ij ) ∞ , (p ij (2)) }, is an operator algebra of functions on the set X. To see that A is not local, let\nExample 1.20. It is still an open problem as to whether or not every unital contractive, homomorphism ρ :\nFor a recent discussion of this problem see [25]. Let's assume that ρ is a contractive homomorphism that is not completely contractive. Let B = H ∞ (D), but endow it with the family of matrix-norms given by,\nIt is easily checked that B is a BPW complete operator algebra of functions on D. However, since every contactive homomorphism of A(D) is completely contractive, we have that for\nIt follows that B is not local and thatB = B L = H ∞ (D), with its usual supremum norm operator algebra structure.\nIn particular, if there does exist a contractive but not completely contractive representation of H ∞ (D), then we have constructed an example of a non-local BPW complete operator algebra of functions on D."},{"title":"A Characterization of Local Operator Algebras of Functions","content":"The main goal of this section is to prove that every BPW complete local operator algebra of functions is completely isometrically isomorphic to the algebra of multipliers on a reproducing kernel Hilbert space of vector-valued functions. Moreover, we wil show that every such algebra is a dual operator algebra in the precise sense of [11]. We will then prove that for such BPW algebras, weak*-convergence and BPW convergence coincide on bounded balls.\nGiven a set X and a Hilbert space H, then by a reproducing kernel Hilbert space of H-valued functions, we mean a vector space L of H-valued functions that is equipped with a norm and an inner product that makes it a Hilbert space and which has the property that for every x ∈ X, the evaluation map E x : L → H, is a bounded, linear map. Recall that given a Hilbert space H, a matrix of operators,\nwhere H is a Hilbert space, is called a positive definite operator-valued function on X, provided that for every finite set of (distinct)points {x 1 , ..., x k } in X, the operator-valued matrix, (K(x i , x j )) is positive semidefinite. Given a reproducing kernel Hilbert space of H-valued functions, if we set K(x, y) = E x E * y , then K is positive definite and is called the reproducing kernel of L. There is a converse to this fact, generally called Moore's theorem, which states that given any positive definite operatorvalued function K : X × X → B(H), then there exists a unique reproducing kernel Hilbert space of H-valued functions on X, such that K(x, y) = E x E * y . We will denote this space by L(K, H). Finally, given any reproducing kernel Hilbert space L of H-valued functions with reproducing kernel K, a function f : X → C is called a multiplier provided that for every g ∈ L, the function f g ∈ F. In this case it follows by an application of the closed graph theorem that the map M f : L → L, defined by M f (g) = f g, is a bounded, linear map. The set of all multipliers is denoted M(K) and is easily seen to be an algebra of functions on X and a subalgebra of B(L). The reader can find proofs of the above facts in [14] [3]. Also, we refer to the fundamental work of Pedrick [26] for further treatment of vector-valued reproducing kernel Hilbert spaces. Proof. It is enough to show that the unit ball is weak*-closed by the Krein-Smulian theorem. So let {M f λ } be a net of multipliers in the unit ball of B(L) that converges in the weak*-topology to an operator T. We must show that T is a multiplier.\nLet x ∈ X be fixed and assume that there exists g ∈ L, with g(\nThis shows that at every such x the net {f λ (x)} converges to some value. Set f (x) equal to this limit and for all other x's set f (x) = 0.\nWe claim that f is a multiplier and that T = M f .\nNote that if g(x) = 0 for every g ∈ L, then E x = E * x = 0. Thus, we have that for any g ∈ L and any h ∈ H,\n, h H . Since this holds for every h ∈ H, we have that E x (T g) = f (x)g(x), and so T = M f and f is a multiplier.\nEvery weak*-closed subspace of V ⊆ B(H) has a predual and it is the operator space dual of this predual. Also, if an abstract operator algebra is the dual of an operator space, then it can be represented completely isometrically and weak*-continuously as a weak*-closed subalgebra of the bounded operators on some Hilbert space. For this reason an operator algebra that has a predual as an operator space is called a dual operator algebra. See the book of [11] for the proofs of these facts. Thus, in summary, the above lemma shows that every multiplier algebra is a dual operator algebra."},{"title":"Proof. The multiplier norm of a given matrix-valued function","content":"Applying this fact to a set consisting of a single point, we have that\nThus, F (x) ≤ C = F and we have that point evaluations are completely contractive on M(K). Since M(K) contains the constants and separates points by hypothesis, it is an operator algebra of functions on X.\nSuppose that M(K) was not local, then there would exist F ∈ M n (M(K)), and a real number C, such that sup Y π (n) Y < C < F . Then for each finite set Y = {x 1 , ..., x k } we could choose G ∈ M n (M(K)), with G < C, and G(x) = F (x), for every x ∈ Y. But then we would have that\nis positive definite. From the characterization of functions and their norms in a reproducing kernel Hilbert space, this implies that\nIn general, M(K) need not separate points on X. In fact, it is possible that L does not separate points and if g(x 1 ) = g(x 2 ), for every g ∈ L, then necessarily f (x 1 ) = f (x 2 ) for every f ∈ M(K).\nFollowing [24], by a k-idempotent operator algebra, C, we mean that we are given k operators,\nProof. It is easily checked that K is positive definite. We first prove that the map is an isometry.\nHence, B ≤ C implies that P is positive and so M f ≤ B . For the converse, given any\nThe proof of the complete isometry is similar but notationally cumbersome. Proof. Let Y be a finite subset of X. Since A/I Y is a |Y |-idempotent operator algebra, by the above lemma, there exists a vector valued kernel\nwhere the direct sum is over all finite subsets of X.\nThen it is easily checked that K is positive definite.\nThis last condition is equivalent to the existence for each F of some\nThis proves that M n (M(K)) = M n (Ã) isometrically."},{"title":"Corollary 2.5. Every BPW complete local operator algebra of functions is a dual operator algebra.","content":"Proof. In this case we have that A =à = M(K) completely isometrically. By Lemma 2.1, this latter algebra is a dual operator algebra.\nThe above theorem gives a weak*-topology to a local operator algebra of function, A by using the identification A ⊆à = M(K) and taking the weak*-topology of M(K). The following proposition proves that convergence of bounded nets in this weak*-topology on A is same as BPW convergence. Proof. Let H denote the reproducing kernel Hilbert space of C-valued functions on X with kernel K for whichà = M(K). Recall that if E x : H → C, is the linear map given by evaluation at\n. It is readily seen that such vectors exist if and only if E x = 0, or equivalently, K(x, x) = 0. But this follows from the construction of K as a direct sum of positive definite functions over all finite subsets of X. For fixed x ∈ X and the one element subset Y 0 = {x}, we have that the 1-idempotent algebra A/I Y 0 = 0 and so K Y 0 (x, x) = 0, which is one term in the direct sum for K(x, x).\nConversely, assume that f λ < K, ∀λ and f λ → f pointwise on X. We must prove that M f λ → M f in the weak*-topology on B(H). But since this is a bounded net of operators, it will be enough to show convergence in the weak operator topology and arbitrary vectors can be replaced by vectors from a spanning set. Thus, it will be enough to show that for v 1 , v 2 ∈ C and\nand the result follows.\nCorollary 2.7. The ball of a local operator algebra of functions is weak*dense in the ball of its BPW completion."},{"title":"Residually Finite Dimensional Operator Algebras","content":"A C*-algebra B is called residually finite-dimensional (RFD) if it has a separating family of finite-dimensional representations. Note that when B is separable and RFD it has a seperating sequence of finite dimensional representations. Since every one-to-one * -homomorphism of a C*-algebra is completely isometric, a C*-algebra B is RFD if and only if for all n, and for every (b i,j ) ∈ M n (B), we have that (b i,j ) Mn(B) = sup{ (π(b i,j )) } where the suprema is taken over all * -homomorphisms, π : B → M k ∀ k. RFD C*algebras have been studied in [10], [15], [5], [19].\nWe know from BRS that any operator algebra can be represented on a Hilbert space, i.e., given an abstract unital operator algebra A there exists an unital completely isometric homomorphism π : A → B(H). Since a representation of a C*-algebra is a *-homomorphism if and only if it is completely contractive, the following definition gives us a natural way of extending the notion of RFD to operator algebras.\nwhere the suprema is taken over all completely contractive homomorphisms, π : B → M k ∀ k. A dual operator algebra B is called weak*-RFD if this last equality holds when the completely contractive homomorphisms are also required to be weak*continuous.\nThe following result is implicitly contained in [24], but the precise statement that we shall need does not appear there. Thus, we refer the reader to [24] to be able to fully understand the proof since we have used some of the definitions and results from [24] without stating them. Proof. From the Corollary 3.3 of [24] we have that the Schur ideal S(B * B) is non-trivial and bounded. Thus, we can define the algebra A(S(B * B)) = span{E 1 , · · · , E k }, where E i = n Q∈S −1 n (B * B) ⊕Q 1/2 (I n ⊗ E ii )Q −1/2 is the idempotent operator that live on n Q∈S −1 n ⊕M k (M n ). By using Theorem 3.2 of [24] we get that S(A(S(B * B))A(S(B * B)) * ) = S(B * B) This further imples that A(S(B * B))A(S(B * B)) * = B * B completely order isomorphically under the map which sends E * i E j to F * i F j . Finally, by restricting the same map to A we get a map which sends E i to F i completely isometrically. Hence, the result follows. Proof. Let A be an abstract k-idempotent operator algebra. Note that A is a dual operator algebra being a finite dimensional operator algebra.\nFrom this it follows that there exists a Hilbert space, H and a weak*continuous completely isometric homomorphism, π : A → B(H). Note that B = π(A) is a concrete k-idempotent algebra generated by the idempotents, B = span{F 1 , F 2 , ..., F k } contained in B(H). Thus, from the above lemma B = A(S(B * B)) completely isometrically. For each n ∈ N, Q ∈ S −1 n , we define π Q n : B → M k (M n ) via π Q n (F i ) = Q 1/2 (I n ⊗ E ii )Q −1/2 Assume for the moment that we have proven that π Q n is a weak*-continuous completely contractive homomorphism. Then for every (b ij ) ∈ M k (B) we must have that sup n,Q∈S −1\nFinally, note that for any (a ij ) ∈ M k (A) we have that (a ij ) M k (A) = sup m { (ρ(π(a ij ))) } where supremum is taken over all weak*-continuous cc homomorphisms ρ : B * B → M m which follows from the fact that π is a complete isometry. Since ρ • π is a weak*-continuous cc homomorphism on A for every weak*-continuous cc homomorphism, ρ on B * B. This implies that (a ij ) M k (A) = sup{ (γ(a ij )) } where supremum is taken over all finite dimensional weak*-continuous cc homomorphisms, and hence the result.\nThus, it remains to show that π Q n is a weak*-continuous cc homomorphism. Note that it is easy to check that it is a completely contractive homomorphism and sup n,Q∈S −1\n. Finally, to see that it is weak*-continuous, let f λ be a bounded net in B * B which converges in the weak*-topology. Note that we can write\nNote that for each i, ψ i is a bounded linear functional on B * B, since the linear maps on a finite dimensional normed space are bounded. Thus, for\nFinally, to see that π Q n (f λ ) → π Q n (f ) in the weak*-topology, fix unit vectors h, k ∈ H and consider\nThis shows that the map π Q n is a weak*-continuous and cc homomorphism for every Q ∈ S −1 n and n. Theorem 3.4. Every BPW complete local operator algebra of functions is waek*-RFD.\nProof. Let A be a BPW complete local operator algebra of functions on the set X and F be a finite subset of X, A/I F is a |F |-idempotent operator algebra. Thus, it follows from the above lemma that A/I F is weak*-\nwhere supremum is taken over all weak*-continuous cc homomorphisms ρ F : A/I F → M n F and all integers n F .\nRecall, that the weak*-topology on A requires all the quotient maps of the form π F : A → A/I F , π F (f ) = [f ] to be weak*-continuous. Thus for each finite subset F ⊆ X, π F is a weak*-continuous cc homomorphism. Let ρ F : A/I F → M n F be a weak*-continuous cc homomorphism, then define δ F = π F • ρ F which is indeed a weak*-continuous cc homomorphism. Consider\nwhere supremum is taken over all weak*-continuous cc homomorphisms, ρ F and positive integers n F . Hence, we obtain the result by taking supremum over all the finite subsets F ⊆ X. Proof. The proof of this follows immediately since every local operator algebra is contained in some BPW complete local operator algebra completely isometrically."},{"title":"Quantized Function Theory on Domains","content":"Whenever one replaces scalar variables by operator variables in a problem or definition, then this process is often referred to as quantization. It is in this sense that we would like to quantize the function theory on a family of complex domains. In some sense this process has already been carried out for balls in the work of Drury [18], Popescu [28], Arveson [6], and Davidson and Pitts [17] and for polydisks in the work of Agler [1], [2], and Ball and Trent [8]. Furthermore, the idea of \"quantizing\" other domains defined by inequalities occurs in [4], [10], and [20]. We approach these same ideas via operator algebra methods. We will show that in many cases this process yields local operator algebras of functions to which the results of the earlier sections can be applied.\nWe begin by defining a family of open sets for which our techniques will apply. ,n k , k ∈ I, whose components are analytic functions on G, and G = {z ∈ C N : F k (z) < 1, k ∈ I}, then we call G an analytically presented domain and we call the set of functions R = {F k : G − → M m k ,n k : k ∈ I} an analytic presentation of G. The subalgebra A of the algebra of functions on G generated by the component functions {f k,i,j : 1 ≤ i ≤ m k , 1 ≤ j ≤ n k , k ∈ I} and the constant function is called the algebra of the presentation. We say that R is a separating analytic presentation provided that the algebra A separates points on G.\nRemark 4.2. An analytic presentation of G by a finite set of matrix-valued functions, F k : G → M m k ,n k , 1 ≤ k ≤ K, can always be replaced by the single block diagonal matrix-valued function, F (z) = F 1 (z) ⊕ · · · ⊕ F K (z) into M m,n with m = m 1 + · · · m K , n = n 1 + · · · n K and we will sometimes do this to simplify proofs. But it is often convenient to think in terms of the set, especially since this will explain the sums that occur in the Agler's factorization formula.\nNote that when we have a analytically presented domain, then every function in the algebra of the presentation is an analytic function on G. H m ). We call the homomorphism π an admissible strict representation when these inequalities are all strictly less than 1. Given (g i,j ) ∈ M n (A) we set (g i,j ) u = sup{ (π(g i,j )) }, where the supremum is taken over all admissible representations π of A. We let (g i,j ) u 0 denote the supremum that is obtained when we restrict to admissible strict representations.\nThe theory of [4] and [10] studies domains defined as above with the additional restrictions that the set of defining functions is a finite set of polynomials, but they do not need their polynomials to separate points. Our results should be compared to theirs."},{"title":"Proposition 4.4. Let G have a separating analytical presentation and let","content":"A be the algebra of the presentation. Then A endowed with either of the family of norms · u or · u 0 is an operator algebra of functions on G.\nProof. It is clear that it is an operator algebra and by definition it is an algebra of functions on G. It follows from the hypotheses that it separates points of G. Finally, for every λ = (λ 1 , ..., λ N ) ∈ G, we have a representation of A on the one-dimensional Hilbert space given by π λ (f ) = f (λ). Hence, |f (λ)| ≤ f u and so A is an operator algebra of functions on G.\nIt will be convenient to say that matrices, A 1 , . . . A m are of compatible sizes if the product, A 1 · · · A m exists, that is, provided that each A i is an n i × n i+1 matrix.\nGiven an analytically presented domain G, we include one extra function, F k 0 which denotes the constant function 1. By an admissible block diagonal matrix over G we mean a block diagonal matrix-valued function of the form D(z) = diag(F k 1 , ..., F km ) where k i ∈ I ∪ k 0 for 0 ≤ i ≤ m. Thus, we are allowing blocks of 1's in D(z). Finally, given a matrix B we let B (q) = diag(B, ...., B) denote the block diagonal matrix that repeats B q times. (i) P u < 1, (ii) there exists an integer l, matrices of scalars C j , 1 ≤ j ≤ l with C j < 1 and admissible block diagonal matrices D j (z), 1 ≤ j ≤ l, which are of compatible sizes and are such that\n(iii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 , P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is a finite set, such that\nProof. Although we will not logically need it, we first show that (ii) implies (i), since this is the easiest implication and helps to illustrate some ideas. Note that if π : A → B(H) is any admissible representation, then the norm of π of any admissible block diagonal matrix is at most 1. Thus, if P has the form of (ii), then for any admissible π, we will have (π(p i,j )) expressed as a product of scalar matrices and operator matrices all of norm at most one and hence, (π(P i,j )) ≤ C 1 · · · C l < 1. Thus, P u ≤ C 1 · · · C l < 1. We now prove that (i) implies (ii). The ideas of the proof are similar to [23, Corollary 18.2], [12, Corollary 2.11] and [21, Theorem 1] and use in an essential way the abstract characterization of operator algebras. For each m, n ∈ N, one proves that P m,n := inf { C 1 . . . C l }, defines a norm on M m,n (A), where the infimum is taken over all l and all ways to factor P (z) = C 1 D 1 (z i 1 ) · · · C l D l (z i l ) as a product of matrices of compatible sizes with scalar matrices C j , 1 ≤ j ≤ l and admissible block diagonal matrices D j , 1 ≤ j ≤ l.\nMoreover, one can verify that M m,n (A) with this family { . m,n } m,n of norms satisfies the axioms for an abstract unital operator algebra as given in [13] and hence by the Blecher-Ruan-Sinclair representation theorem [13](see also [23]) there exists a Hilbert space H and a unital completely isometric isomorphism π : A −→ B(H). Thus, for every m, n ∈ N and for every P = (p ij ) ∈ M m,n (A), we have P m,n = (π(p ij )) . However, π (m k ,n k ) (F k ) = (π(f k,i,j )) ≤ 1 for 1 ≤ i ≤ K, and so, π is an admissible representation. Thus, P m,n = (π(p ij )) ≤ P u . Hence, if P u < 1, then P m,n < 1 which implies that such a factorization exists. This completes the proof that (i) implies (ii).\nWe will now prove that (ii) implies (iii). Suppose that P has a factorization as in (ii). Let K ⊆ I be the finite subset of all indices that appear in the block-diagonal matrices appearing in the factorization of P. We will use induction on l to prove that (iii) holds.\nFirst, assume that l = 1 so that P (z) = C 1 D 1 (z). Then,\nLet E k be the diagonal matrix that has 1's wherever F k appears(so E k = 0 when there is no F k term in D 1 ). Hence,\nTherefore, gathering terms for common values of i,\nwhere R 0 = I m − C 1 C * 1 is a positive, invertible matrix and P i is, in this case a constant. Thus, the form (iii) holds, when l = 1.\nWe now assume that the form (iii) holds for any R(z) that has a factorization of length at most l − 1, and assume that\nwhere R(z) has a factorization of length l − 1.\nNote that a sum of expressions such as on the right hand side of (iii) is again such an expression. This follows by using the fact that given any two expressions A(z), B(z), we can write\nThus, it will be sufficient to show that I m − P (z)P (w) * is a sum of expressions as above. To this end we have that,\n. The first term of the above equation is of the form as on the right hand side of (iii) by case l = 1. Also, the quantity (I − R(z)R(w) * ) = R 0 R * 0 + R 0 (z)R 0 (w) * + k∈K R k (z)(I − F k (z)F k (w) * ) (q k ) R k (w) * by the inductive hypothesis. Hence,\nThus, we have expressed (I − P (z)P (w) * ) as a sum of two terms both of which can be written in the form desired. Using again our remark that the sum of two such expressions is again such an expression, we have the required form.\nFinally, we will prove (iii) implies (i). Let π : A → B(H) be an admissible representation and let P = (p i,j ) ∈ M m,n (A) have a factorization as in (iii). To avoid far too many superscripts we write simplify π (m,n) to Π. Now observe that\nClearly the first two terms of the sum are positive. But since π is an admissible representation, Π(F k ) ≤ 1 and hence, (I − Π(F k )Π(F k ) * ) ≥ 0. Hence, each term on the right hand side of the above inequality is positive and since R is strictly positive, say R ≥ δI m for some scalar δ > 0, we have that I m − Π(P )Π(P ) * ≥ δI m . Therefore, Π(P ) ≤ √ 1 − δ. Thus, since π was an arbitrary admissible representation, P u ≤ √ 1 − δ < 1, which proves (i).\nWhen we require the functions in the presentation to be row vectorvalued, then the above theory simplifies somewhat and begins to look more familiar. Let G be an analytically presented domain with presentation F k : G → M 1,n k , k ∈ I. We identify M 1,n with the Hilbert space C n so\nwhere the inner product is in C n . In this case we shall say that G is presented by vector-valued functions. (i) P u < 1, (ii) there exists an integer l, matrices of scalars C j , 1 ≤ j ≤ l with C j < 1 and admissible block diagonal matrices D j (z), 1 ≤ j ≤ l, which are of compatible sizes and are such that P (z) = C 1 D 1 (z) · · · C l D l (z).\n(iii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 ∈ M m,r 0 (A), P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is finite, such that\nThe following result gives us a Nevanlinna-type result for the algebra of presentation.\nTheorem 4.7. Let Y be a finite subset of an analytically presented domain G with separating analytic presentation F k = (f k,i,j ) : G → M m k ,n k , k ∈ I, let A be the algebra of the presentation and let P be a M m,n -valued function defined on a finite subset Y = {x 1 , · · · , x l } of G. Then the following are equivalent:\n(i) there existsP ∈ M mn (A) such thatP | Y = P and P u < 1.\n(ii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 ∈ M m,r 0 (A), P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is a finite set, such that\nProof. Note that (i) ⇒ (ii) follows immediately as a corollary of Theorem 4.5. Thus it only remains to show that (ii) ⇒ (i). Since A is an operator algebra of functions, therefore, A/I Y is a finite dimensional operator algebra of idempotents and A/I Y = span{E 1 , · · · , E l } for some l = |Y |. Thus there exists a Hilbert space, H Y and a completely isometric representation π of A/I Y . By Theorem 2.4, there exists a kernel K Y such that π(A/I Y ) = M(K Y ) completely isometrically under the map ρ :\nUsing that R > 0, we get that π(P + I Y ) < 1. This shows that there existsP ∈ A such thatP | Y = P and P u < 1. This completes the proof.\nWe now turn towards defining quantized versions of the bounded analytic functions on these domains. For this we need to recall that the joint Taylor spectrum of a commuting N -tuple of operators T = (T 1 , ..., T N ), is a compact set, σ(T ) ⊆ C N and that there is an analytic functional calculus defined for any function that is holomorphic in a neighborhood of σ(T ).\nwhere H is an arbitrary Hilbert space. We set\nwhere n is an arbitrary positive integer.\nNote that if we identify a point (λ 1 , ..., λ N ) ∈ C N with an N-tuple of commuting operators on a one-dimensional Hilbert space, then we have that G ⊆ Q(G).\nIf T = (T 1 , ..., T N ) ∈ Q(G), is a commuting N-tuple of operators on the Hilbert space H, then since the joint Taylor spectrum of T is contained in G, we have that if f is analytic on G, then there is an operator f (T ) defined and the map π : Hol(G) → B(H) is a homomorphism, where Hol(G) denotes the algebra of analytic functions on G.\nWe are interested in determining when H ∞ R (G) =Ã, completely isometrically and whether or not the · R norm is attained on the smaller set Q 0,0 (G).\nNote that since each point in G ⊆ Q(G), we have that\nThus, the inclusion of A into H ∞ R (G) might not even be isometric. Theorem 4.10. Let G be an analytically presented domain with a separating presentation R = {F k : G → M m k ,n k : k ∈ I}, let A be the algebra of the presentation and let A be the BPW-completion of A. Then\nProof. Let f ∈ M n ( A), with f L < 1. Then there exists a net of functions, f λ ∈ M n (A), such that f λ u < 1 and lim λ f λ (z) = f (z) for every z ∈ G.\nSince f λ ∞ < 1, by Montel's Theorem, there is a subsequence {f n } of this net that converges to f uniformly on compact sets. Hence, if T ∈ Q(G), then lim n f (T ) − f n (T ) = 0 and so f (T ) ≤ sup n f n (T ) ≤ 1. Thus, we have that f ∈ M n (H ∞ R (G)), with f R ≤ 1. This proves that A ⊆ H ∞ R (G) and that f L ≤ f R .\nConversely, let g ∈ M n (H ∞ R (G) with g R < 1. For any finite set Y = {y 1 , ..., y t } ⊆ G, let A/I Y = span{E 1 , ..., E t } be the corresponding t-idempotent algebra and let π Y : A → A/I Y denote the quotient map. Write y i = (y i,1 , ..., y i,N ), 1 ≤ i ≤ t and let T j = y 1,j E 1 + · · · + y t,j E t , 1 ≤ j ≤ N so that T = (T 1 , ..., T N ) is a commuting N -tuple of opertors with σ(T ) = Y. For k ∈ I, we have that F k (T ) = F k (y 1 ) ⊗ E 1 + · · · + F k (y t ) ⊗ E t = π Y (F k ) ≤ F k u = 1. Thus, T ∈ Q(G), and so, g(T ) = g(y 1 ) ⊗ E 1 + · · · + g(y t ) ⊗ E t ≤ g R < 1. Since A separates points, we may pick f ∈ M n (A) such that f = g on Y. Hence, π Y (f ) = f (T ) = g(T ) and π Y (f ) < 1. Thus, we may pick f Y ∈ M n (A), such that π Y (f Y ) = π Y (f ) and f Y u < 1. This net of functions, {f Y } converges to g pointwise and is bounded. Therefore, g ∈ M n ( A) and g L ≤ 1. This proves that H ∞ R (G) ⊆ A and that g L ≤ g R .\nThus, A = H ∞ R (G) and the two matrix norms are equal for matrices of all sizes. The rest of the conclusions follow from the results on BPWcompletions."},{"title":"Remark 4.11. The above result yields that for every","content":". Also, we can verify these hypotheses are met for most of the algebras given in the example section. In particular, for Examples 5. 1, 5.2, 5.3, 5.4, 5.6, 5.7, and 5.8"},{"title":". It would be interesting to know if this can be done in general.","content":"We now seek other characterisations of the functions in H ∞ R (G). In particular, we wish to obtain analogues of Agler's factorization theorem and of the results in [4] and [10]. By Theorem 2.4, if we are given an analytically presented domain G ⊆ C N , with presentation R = {F k : G → M m k ,n k , k ∈ I}, then there exists a Hilbert space H and a positive definite function, K : G × G → B(H) such that A = M(K). We shall denote any kernel satisfying this property by K R . (\nIn the case when the presentation contains only finitely many functions we can say considerably more about R-limits.\nProof. By the continuity of the functions, there is a constant δ > 0, such that F k (z) ≤ 1 − δ, ∀k, ∀z ∈ S. Thus, we have that I − F k (z)F k (z) * ≥ δI, ∀ k, ∀z ∈ S. Also, we have that Proof. We identify (I m − F (z)F (w) * ) (q λ ) = (I m − F (z)F (w) * ) ⊗ I C q λ , and the p × mq λ matrix-valued function P λ as an analytic function, P λ : G → B(C m ⊗ C q λ , C p ). Writing C m ⊗ C q λ = C q λ ⊕ · · · C q λ (m times) allows us to write P λ (z) = [P λ 1 (z), . . . , P λ m (z)] where each P λ i (z) is p × q λ . Also, if we let f 1 (z), ..., f m (z) be the (1, n) vectors that represent the rows of the matrix F, then we have that F (z)F (w) * = m i,j=1 f i (z)f j (w) * E i,j . Finally, we have that\n, is a positive definite function that is analytic in z and coanalytic in w. By dropping to a subnet, if necessary, we may assume that K λ converges uniformly on compact subsets of G to K = (K i,j ) : G × G → M m (M p ). Note that this implies that P λ i (z)P λ j (w) * → K i,j (z, w) for all i, j and that K is a positive definite function that is analytic in z and coanalytic in w.\nThe positive definite K gives rise to a reproducing kernel Hilbert space H of analytic C m ⊗ C p -valued functions on G. If we let E(z) : H → C m ⊗ C p , be the evaluation functional, then K(z, w) = E(z)E(w) * and E : G → B(H, C m ⊗ C p ) is analytic. Identifying C m ⊗ C p = C p ⊕ · · · ⊕ C p (m times), yields analytic functions,\nDefine an analytic map R : G → B(H ⊗ C m , C p ) by identifying H ⊗ C m = H ⊕ · · · ⊕ H(m times) and setting R(z)(h 1 ⊕ · · · ⊕ h m ) = E 1 (z)h 1 + · · · + E m (z)h m . Thus, we have that\nand the proof is complete. (1) f ∈ M mn (H ∞ R (G)) and f R ≤ 1, (2) there exists an analytic operator-valued function R 0 :\nProof. Recall thatà = H ∞ R (G). Let us first assume that f ∈ M mn (Ã) and f Mmn(Ã) < 1. Then for each finite set Y , there exists f Y ∈ M mm (A) such that f Y converges to f pointwise and f Y u ≤ 1. We may assume that\nwhere |Y | denotes the cardinality of the set Y .\nThus by Theorem 4.5 there exists a positive, invertible matrix R Y ∈ M m and matrices P Y k ∈ M m,r k Y (A), 0 ≤ k ≤ K, such that\nwhere r k Y = q k Y m k and z, w ∈ G. If we define a map F 0 : G → M m 0 ,n 0 as the zero map then the above factorization can be written as\nNote that the net R Y is uniformly bounded above by 1, thus there exists R ∈ M m and a subnet R Ys which converges to R.\nFinally, since the net f Y converges to f pointwise we have that the net\nAlso note that for each k, < P Y k > is the net of vector-valued holomorphic function and is uniformly bounded on compact subsets of G by Proposition 4.14.\nThus by Proposition 4.15 there exists F k -limit for each 0 ≤ k ≤ K, that is, there exists K+1 Hilbert spaces H k and K+1 analytic function,\nThis completes the proof that (1) implies (2). To show the converse, assume that there exists an analytic operatorvalued function R 0 : G → B(H 0 , C m ) and K analytic functions,\nBy using Theorem 2.4 there exists a vector-valued Kernel K such that M n (M(K)) = M n (Ã) completely isometrically for every n. It is easy to see that ((I − f (z)f (w) * )⊗K(z, w)) ≥ 0 ∀z, w ∈ G. This is equivalent to f ∈ M m (M(K)) and M f ≤ 1 which in turn is equivalent to (1). Thus, (1) and (2) are equivalent.\nClearly, (3) implies (2). The arguement for why (2) implies (3) is contained in [10] and we recall it. If we fix any k 0 , then since F k 0 (z) < 1 on G, we have that |f k 0 ,1,1 (z)| 2 + · · · + |f ko,1,m (z)| 2 < 1 on G. From this it follows that H(z, w) = (1 − f k 0 ,1,1 (z)f k 0 ,1,1 (w) − · · · − f k 0 ,1,m (z)f k 0 ,1,m (w)) is an F k 0 -limit and that H −1 (z, w) is positive definite. Now we have that R 0 (z)R 0 (w) * H −1 (z, w) is positive definite and so we may write, R 0 (z)R 0 (w) * H −1 (z, w) = G 0 (z)G 0 (w) * and we have that R 0 (z)R 0 (w) * = G 0 (z)H(z, w)G(w) . This shows that R 0 (z)R 0 (w) * is an F k -limit and so it may be absorbed into the sum."},{"title":"Examples and Applications","content":"In this section we present a few examples to illustrate the above definitions and results. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T n ) such that T 1 T * 1 + · · · + T N T * N ≤ I, which is commonly called a row contraction and an admissible strict representation is given when T 1 T * 1 + · · · + T N T * N < I, which is generally referred to as a strict row contraction. In this case one can again easily see that · u = · u 0 by using the same r < 1 argument as in the last example and that f ∈ H ∞ R (B N ) if and only if f R = sup{ f (T ) : T 1 T * 1 + · · · + T N T * N < I} < +∞ These are the norms on polynomials considered by Drury [18], Popescu [28], Arveson [6], and Davidson and Pitts [17]. By Theorem 4.18 we will have for i.e., such that T * 1 T 1 + · · · + T * N T N ≤ I, which is generally referred to as a column contraction. This time the norm on H ∞ R (B N ) will be defined by taking suprema over all strict column contractions and we will have that f R ≤ 1 if and only if\nAgain this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T N ) such that T 1 T * 1 + · · · + T N T * N ≤ I and T * 1 T 1 + · · · + T * N T N ≤ I, that is, which is both a row and column contraction. This time the norm on H ∞ R (B N ) is defined as the supremum over all commuting N-tuples that are both strict row and column contractions. We will have that f ∈ M m,n (H ∞ R (B N )) with f R ≤ 1 if and only if\nwhere K 1 and R 1 are as before.\nThe last three examples illustrate that it is possible to have multiple rational representations of G, all with the same algebra, but which give rise to (possibly) different operator algebra norms on A. Thus, the operator algebra norm depends not just on G, but also on the particular presentation of G that one has chosen. We have surpressed this dependence on R to keep our notation simplified. and it can be seen that H ∞ R (D) is defined by\nIn this case we have that f ∈ M m,n (H ∞ R (D) and f R ≤ 1 if and only if\nExample 5.6. Let L = {z ∈ C : |z − a| < 1, |z − b| < 1}, where |a − b| < 1, then the functions f 1 (z) = z −a, f 2 (z) = z −b give a polynomial presentation of this \"lens\". The algebra of this presentation is again the algebra of polynomials. An admissible representation of this algebra is defined by choosing any operator satisfying T − aI ≤ 1 and T − bI ≤ 1, with strict inequalities for the admissible strict representations. In this case we easily see that · u = · u 0 , since given any operator T satisfying T − aI ≤ 1 and T − bI ≤ 1, and r < 1, S r = rT + (1 − r)(a + b) corresponds to the admissible strict representations and for any matrix of polynomials (p i,j (T )) = lim r→1 (p ij (S r )) . This algebra with this norm was studied in [9]. Their work shows that this norm is completely boundedly equivalent to the usual supremum norm. Our results imply that f ∈ M m,n (H ∞ R (L)) and f R ≤ 1 if and only if\nfor some appropriatley chosen R 1 .\nAll of the above examples are also covered by the theory of [4] and [10], except that their definition of the norm is slightly different. We address this difference in a later remark. We now turn to some examples that are not covered by their theory.\nExample 5.8. Let 0 < r < 1 be fixed and let A r = {z ∈ C : r < |z| < 1} be an annulus. Then this has a rational presentation given by F 1 (z) = z and F 2 (z) = rz −1 , and the algebra of this presentation is just the Laurent polynomials. Admissible representations of this algebra are given by selecting any invertible operator T satisfying T ≤ 1 and T −1 ≤ r −1 . Admissible strict representations are given by invertible operators satisfying T < 1 and T −1 < r −1 . It is no longer quite so clear that · u = · u 0 . However, this algebra with these norms is studied by the first author in [22] and among other results the equality of these norms was shown. In [17], it was shown that · u is completely boundedly equivalent to the usual supremum norm. In this case one can see that the · R is achieved by taking suprema over all T satisfying T < 1 and T −1 < r −1 . The formula for the norm is given by f R ≤ 1 if and only if I m − f (z)f (w) * = (1 − zw)K 1 (z, w) + (1 − r 2 z −1 w −1 )K 2 (z, w). Our results imply that f R ≤ 1 if and only if\nIn particular, if we take φ(z) = z−1 z+1 then it maps the halfplane H = {z : Re(z) > 0} to the unit disk. For this particular φ, we have that Q R = {T : σ(T ) ⊆ H and Re(T ) ≥ 0}. where w(T ) denotes the numerical radius of T. Thus, H ∞ R (D) becomes the \"universal\" operator algebra that one obtains by substituting an operator of numerical radius less than one for the variable z and we have a quite different quantization of the unit disk. Our results give a formula for this norm, but only in terms of R-limits, so further work would need to be done to make it explicit.\nExample 5.12. There is a second way that one can quantize many convex sets. Let G = {z : |z − a k | < r k , k ∈ I} ⊆ C be an open, bounded convex set that can be expressed as an intersection of a possibly infinite set of open disks. For example, any convex set with a smooth boundary with uniformly bounded curvature can be expressed in such a fashion. Then G has a rational presentation given by F k (z) = r −1 k (z − a k ), k ∈ I the algebra of the presentation is just the polynomial algebra and an admissible representation is given by selecting any operator T satisfying, T − a k I ≤ r k , k ∈ I. Thus, we again a factorization result, but only in terms of R-limits.\nThe above definitions allow one to consider many other examples. For example, one could fix 0 < r < 1 and let G = {z ∈ B N : r < |z 1 |}, with rational presentation f 1 (z) = (z 1 , ..., z N ) ∈ M 1,N , and f 2 (z) = rz −1\n1 . An admissible representation would then correspond to a commuting row contraction with T 1 invertible and T −1 1 ≤ r −1 .\nWe now compare and contrast some of our hypotheses with those of [4] and [10].\nRemark 5.13. Let G = {z ∈ C N : F k (z) < 1 ∀ k = 1, · · · , K} where the F k 's are matrix-valued polynomials defined on G. Then for f ∈ Hol(G), [4] and [10] really study a norm given by f s = sup{ f (T ) } where the supremum is taken over all commuting N -tuples of operators T with F k (T ) < 1 ∀k. We wish to contrast this norm with our f R . In [4] it is shown that the hypotheses F k (T ) < 1, k = 1, ..., K implies that σ(T ) ⊆ G. Thus, we have that f s ≤ f R . In fact, we have that f s = f R . This can be seen by the fact that they obtain identical factorization theorems to ours. This can also be seen directly in some cases where the algebra A contains the polynomials and when it can be seen that · R is attained by taking the supremum over matrices(see Remark 4.11). Indeed, if f R is attained as the supremum over commuting N -tuples of finite matrices T = (T 1 , ..., T N ) satisfying σ(T ) ⊆ G and F k (T ) ≤ 1 then such an N -tuple of commuting matrices, can be conjugated by a unitary to be simultaneously put in upper triangular form. Now it is easily argued that the strictly upper triangular entries can be shrunk slightly so that one obtains new N -tuples T ǫ = (T 1,ǫ , ..., T N,ǫ ) satisfying, F k (T ǫ ) < 1, k = 1, ..., K and T i − T i,ǫ < ǫ. But we do not have a simple direct argument that works in all cases.\nRemark 5.14. We do not know how generally it is the case that · u is a local norm. That is, we do not know if f u = f R for f ∈ M n (A). In particular, we do not know if this is the case for Example 5.5. In this case, the algebra of the of the presentation is A = span{z n : n ≥ 0, n = 1}. If we write a polynomial p ∈ A in terms of its even and odd decomposition, p = p e + p o , then p e (z) = q(z 2 ) and p o = z 3 r(z 2 ) for some polynomials q, r."},{"title":"Proposition 1 . 2 .","content":""},{"title":"F","content":""},{"title":"Lemma 1. 8 .","content":""},{"title":"Example 1. 15 .","content":""},{"title":"Example 1 . 17 .","content":""},{"title":"Lemma 2 . 1 .","content":""},{"title":"Theorem 2 . 2 .","content":""},{"title":"Theorem 2 . 4 .","content":""},{"title":"Proposition 2. 6 .","content":""},{"title":"Definition 3. 1 .","content":""},{"title":"Lemma 3. 2 .","content":""},{"title":"Theorem 3. 3 .","content":""},{"title":"Corollary 3. 5 .","content":""},{"title":"Definition 4 . 1 .","content":""},{"title":"Definition 4 . 3 .","content":""},{"title":"Theorem 4. 5 .","content":""},{"title":"Corollary 4. 6 .","content":""},{"title":"Definition 4 . 8 .","content":""},{"title":"Definition 4. 12 .","content":""},{"title":"Corollary 4. 13 .","content":""},{"title":"Proposition 4 . 14 .","content":""},{"title":"Proposition 4 . 15 .","content":""},{"title":"Remark 4 . 16 .","content":""},{"title":"Definition 4 . 17 .","content":""},{"title":"Example 5 . 1 .","content":""},{"title":"Example 5. 5 .","content":""},{"title":"Example 5 . 7 .","content":""},{"title":"Example 5 . 9 .","content":""},{"title":"Example 5 . 10 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"A concrete operator algebra A is just a subalgebra of B(H), the bounded operators on a Hilbert space H. The operator norm on B(H) gives rise to a norm on A. Moreover, the identification M n (A) ∼ = A ⊗ M n ⊆ B(H ⊗ C n ) ∼ = B(H n ) endows the matrices over A with a family of norms in a natural way, where M n denotes the algebra of n × n matrices. It is common practice to identify two operator algebras A and B as being the \\\"same\\\" if and only if there exists an algebra isomorphism π : A → B that is not only an isometry, but which also preserves all the matrix norms, that is such that (π(a i,j )) Mn(B) = (a i,j ) Mn(A) , for every n and every element (a i,j ) ∈ M n (A). Such a map π is called a completely isometric isomorphism. In [13] an abstract characterization of operator algebras, in the above sense, was given and since that time a theory of such algebras has evolved. For more details on the abstract theory of operator algebras, see [11], [23] or [27].\\nIn this note we present a theory for a special class of abstract abelian operator algebras which contains many important examples arising in function theoretic operator theory, including the Schur-Agler and the Arveson-Drury-Popescu algebras. This theory will allow us to answer certain types of questions about such algebras in a unified manner. We will prove that our hypotheses give an abstract characterization of operator algebras that are completely isometrically isomorphic to multiplier algebras of vector-valued reproducing kernel Hilbert spaces.\\nOur results will show that under certain mild hypotheses, operator algebra norms which are defined by taking the supremum of certain families of operators on Hilbert spaces of arbitrary dimensions can be obtained by restricting the family of operators to finite dimensional Hilbert spaces. Thus, in a certain sense, which will be explained later, our results give conditions that guarantee that an algebra is residually finite dimensional.\\nFinally, we apply our results to study \\\"quantized function theories\\\" on various domains. Our work in this direction should be compared to that of Ambrozie-Timotin [4], Ball-Bolotnikov [10] and Kalyuzhnyi-Verbovetzkii [20]. These authors obtain the same type of factorization theorem via unitary colligation methods as we obtain via operator algebra methods, but they assume somewhat different(and in many cases stronger) hypotheses. We, also, obtain a bit more information about the algebras themselves, including the fact that in many cases their norms can be obtained as supremums over matrices instead of operators.\\nWe now give the relevant definitions. Recall that given any set X the set of all complex-valued functions on X is an algebra over the field of complex numbers.\\nDefinition 1.1. We call A an operator algebra of functions on a set X provided:\\n(1) A is a subalgebra of the algebra of functions on X, (2) A separates the points of X and contains the constant functions, (3) for each n M n (A) is equipped with a norm . Mn(A) , such that the set of norms satisfy the BRS axioms [13] to be an abstract operator algebra, (4) for each x ∈ X, the evaluation functional, π x : A → C, given by π x (f ) = f (x) is bounded.\\nA few remarks and observations are in order. First note that if A is an operator algebra of functions on X and B ⊆ A is any subalgebra, which contains the constant functions and still separates points, then B, equipped with the norms that M n (B) inherits as a subspace of M n (A) is still an operator algebra of functions.\\nThe basic example of an operator algebra of functions is ℓ ∞ (X), the algebra of all bounded functions on X. If for (f i,j ) ∈ M n (ℓ ∞ (X)) we set\\nwhere · Mn is the norm on M n obtained via the identification M n = B(C n ), then it readily follows that properties (1)-(4) of the above definition are satisfied. Thus, ℓ ∞ (X) is an operator algebra of functions on X in our sense and any subalgebra of ℓ ∞ (X) that contains the constants and separates points will be an operator algebra of functions on X when equipped with the subspace norms. Proof. Since π x : A → C is bounded and the norm is sub-multiplicative, we have that for any f ∈ A, |f (x)| n = |π x (f n )| ≤ π x f n ≤ π x f n . Taking the n-th root of each side of this inequality and letting n → +∞, yields |f (x)| ≤ f , and hence, f ∈ ℓ ∞ (X). Note also that π x = 1. Finally, since every bounded, linear functional on an operator space is completely bounded and the norm and the cb-norm are equal, we have that π x cb = π x = 1. Thus, for (f i,j ) ∈ M n (A), we have (f i,j (x)) Mn = (π x (f i,j )) ≤ (f i,j ) Mn(A) .\\nGiven an operator algebra of functions on a set X, A and F = {x 1 , ..., x k } a set of distinct points in X, we define I F = {f ∈ A : f (z) = 0 ∀ z ∈ F } and note that M n (I F ) = {f ∈ M n (A) : f (z) = 0 ∀ z ∈ F } ∀ n. The quotient space A/I F has a natural set of matrix norms given by defining (f i,j + I F ) = inf{ (f i,j + g i,j ) Mn(A) : g i,j ∈ I F }. Alternatively, this is the norm on M n (A/I F ) that comes via the identification, M n (A/I F ) = M n (A)/M n (I F ), where the latter space is given its quotient norm. It is easily checked that this family of matrix norms satisfies the BRS conditions and so gives A/I F the structure of an abstract operator algebra.\\nWe let π F (f ) = f + I F denote the quotient map π F : A → A/I F so that for each n, π Since A is an algebra which separates points on X and contains constant functions, it follows that there exists functions f 1 , ..., f k ∈ A, such that f i (x j ) = δ i,j , where δ i,j denotes the Dirac delta function. If we set E j = π F (f j ), then it is easily seen that whenever f ∈ A and f (x i ) = λ i , i = 1, ..., k, then π F (f ) = λ 1 E 1 + · · · + λ k E k . Moreover, E i E j = δ i,j E i , and E 1 + · · · + E k = 1, where 1 denotes the identity of the algebra A/I F . Thus, A/I F = span{E 1 , ..., E k }, is a unital algebra spanned by k commuting idempotents. Such algebras were called k-idempotent operator algebras in [24] and we will use a number of results from that paper.\"\n },\n {\n \"title\": \"Definition 1.3. An operator algebra of functions A on a set X, is called a local operator algebra of functions if it satisfies\",\n \"content\": \"F ((f ij )) = (f ij ) ∀ (f ij ) ∈ M n (A) and for every n, where the supremum is taken over all finite subsets of X.\\nThe following result shows that every operator algebra of functions can be re-normed so that it becomes local. Proposition 1.4. Let A be an operator algebra of functions on X, let A L = A and define a family of matrix norms on A L , by setting (f i,j ) Mn(A L ) = sup F (π F (f i,j )) Mn(A/I F ) . Then A L is a local operator algebra of functions on X and the identity map, id : A → A L , is completely contractive.\\nProof. It is clear from the definition of the norms on A L that the identity map is completely contractive and it is readily checked that A L is an operator algebra of functions on X.\\nLetπ F :\\nWe claim that for any f ∈ A, and any finite subset F ⊆ X, we have that π F (f ) = π F (f ) . To see the other inequality note that for g ∈ I F , and G ⊆ X a finite set, we have f + g L = sup G π G (f + g) ≥ π F (f + g) = π F (f ) . Hence, π F (f ) ≥ π F (f ) , and equality follows. A similar calculation shows that (π F (f i,j )) = (π F (f i,j )) , for any matrix of functions.\\nNow it easily follows that A L is local, since\\nWe letà denote the set of functions that are BPW limits of bounded nets of functions in A. Definition 1.5. Given an operator algebra of functions A on X we say that f : X → C is a BPW limit of A if there exists a uniformly bounded net (f λ ) λ ∈ A that converges pointwise on X to f. We say that A is BPW complete if it contains the set of functions that are BPW limits of bounded nets of functions in A. Letà denote the set of such functions, that is,\\nIt is easily checked that for each n, the above formula defines a norm on M n (Ã). It is also easily checked that a matrix-valued function, (f i,j ) : X → M n is the pointwise limit of a uniformly bounded net (f λ i,j ) ∈ M n (A) if and only if f i,j ∈à for every i, j. Lemma 1.6. Let A be an operator algebra of functions on X and let (f i,j ) ∈ M n (Ã). Then\\nThe collection of finite subsets of X determines a directed set, ordered by inclusion. If we choose for each finite set F, functions (g F i,j ) satisfying the conditions of the right hand set, then these functions define a net that converges BPW to (f i,j ) and hence, the right hand side is larger than the left. Conversely, given a net (f λ i,j ) that converges pointwise to (f i,j ) and satisfies (f λ i,j ) ≤ C and any finite set\\nThus, given ǫ > 0, since the functions f 1 , ..., f k depend only on F, we may choose λ so that (g λ i,j ) < C + ǫ. This shows the other inequality. Lemma 1.7. Let A be an operator algebra of functions on the set X, theñ A equipped with the collection of norms on M n (Ã) given in Definition 1.5 is an operator algebra.\\nProof. It is clear from the definition ofà that it is an algebra. Thus, it is enough to check that the axioms of BRS are satisfied by the algebraà equipped with the matrix norms given in the Definition 1.5.\\nIf L and M are scalar matrices of appropriate sizes and G ∈ M n (Ã), then\\nThis yields L Mn(Ã) ≤ G Mn(Ã) H Mn(Ã) , and so the multiplication is completely contractive.\\nFinally, to see that the L ∞ conditions are met, let G ∈ M n (Ã) and\\nThis shows that G⊕ H M n+m (Ã) ≤ max{ G Mn(Ã) , H Mm(Ã) }, and so the L ∞ condition follows. This completes the proof of the result. Proof. It is clear from the definition of the norms onà that the identity map from A toà is completely contractive and thus A ⊆à as sets. This indeed shows thatà separates points of X and contains constant functions.\\nSince A is an operator algebra of functions on the set X, we have that\\nBy letting ǫ → 0 and supping over z ∈ X, we get that\\nHence,à is an operator algebra of functions on the set X.\\n. To see the other inequality, as-\\nsince I F ⊆Ĩ F . We claim that for any (f ij ) ∈ M n (A), and for any finite subset F ⊆ X, we have that (f ij + I F ) = (f ij +Ĩ F ) . To see the other inequality, let (g ij ) ∈ M n (Ĩ F ). Then ∀ ǫ > 0, and G ⊆ X,\\n, and the equality follows. Now it is immediate to see that,\\nCorollary 1.9. If A is a BPW complete operator algebra then A L =à completely isometrically.\\nProof. Since A is BPW complete, A =à as sets. But by Lemma 1.6, the norm defined on A L agrees with the norm defined onÃ. Remark 1.10. In the view of above corollary, we denote the the norm oñ A by . L . Lemma 1.11. If A is an operator algebra of functions on X, then Ball(A L ) is BPW dense in Ball(Ã) andà is BPW complete, i.e.,à =Ã.\\nProof. It can be easily checked that above is equivalent to showing that A L is BPW dense inÃ. We'll only prove that A L BP W ⊆Ã, since the other containment follows immediately by the definition ofÃ. Let {f λ } be a net in A L such that f λ → f pointwise and sup λ f λ A L < C. Then for fixed F ⊆ X and ǫ > 0,\\nFinally, note that the argument similar to the above readily yields that A is BPW complete.\\nAll the above lemmas can be summarized as the following theorem. Theorem 1.12. If A is an operator algebra of functions on X, thenà is a BPW complete local operator algebra of functions on X which contains A L completely isometrically as a BPW dense subalgebra. Definition 1.13. Given an operator algebra of functions A on X, we call A the BPW completion of A.\\nWe now present a few examples to illustrate these concepts. We will delay the main family of examples to the next section. Example 1.14. If A is a uniform algebra, then there exists a compact, Hausdorff space X, such that A can be represented as a subalgebra of C(X) that separates points. If we endow A with the matrix-normed structure that it inherits as a subalgebra of C(X), namely, (f i,j ) = (f i,j ) ∞ ≡ sup{ (f i,j ) Mn : x ∈ X}, then A is a local operator algebra of functions on X. Indeed, to achieve the norm, it is sufficient to take the supremum over all finite subsets consisting of one point. In this case the BPW completionà is completely isometrically isomorphic to the subalgebra of ℓ ∞ (X) consisting of functions that are bounded, pointwise limits of functions in A. Example 1. 16. Let X = ǫD, 0 < ǫ < 1 and A = {f ∈ H ∞ (D) : f : X → C}. If we endow A with the matrix-normed structure on H ∞ (D), then A is an operator algebra of functions on X. Also, it can be verified that A is a local operator algebra of functions and that A =Ã.\\n. It can be easily verified that A is a BPW complete operator algebra of functions. However, we also claim that A is local. To prove this we proceed by contradiction.\\nIn this case,\\n) such that G| Y = 0 and F + G < 1+c 2 . Thus, we can write B Y (z) = z−ǫ 1−ǭz , so that we can write G(z) = zB Y (z)H(z), for some H ∈ M n (H ∞ ). It follows that H ∞ < 2, since G ∞ < 2. We now consider\\nExample 1.18. This is an example of a non-local algebra that arises from boundary behavior.\\nThen it is easy to check that A is an operator algebra of functions on the set D. Also, it can be verified that A is not local. To see this, note\\nThus, sup Y ⊆D π Y (z) < z and hence A is not local. Example 1. 19. This example shows that one can easily build non-local algebras by adding \\\"values\\\" outside of the set X. Let A be the algebra of polynomials regarded as functions on the set X = D. Then A endowed with the matrix-normed structure as (p ij ) = max{ (p ij ) ∞ , (p ij (2)) }, is an operator algebra of functions on the set X. To see that A is not local, let\\nExample 1.20. It is still an open problem as to whether or not every unital contractive, homomorphism ρ :\\nFor a recent discussion of this problem see [25]. Let's assume that ρ is a contractive homomorphism that is not completely contractive. Let B = H ∞ (D), but endow it with the family of matrix-norms given by,\\nIt is easily checked that B is a BPW complete operator algebra of functions on D. However, since every contactive homomorphism of A(D) is completely contractive, we have that for\\nIt follows that B is not local and thatB = B L = H ∞ (D), with its usual supremum norm operator algebra structure.\\nIn particular, if there does exist a contractive but not completely contractive representation of H ∞ (D), then we have constructed an example of a non-local BPW complete operator algebra of functions on D.\"\n },\n {\n \"title\": \"A Characterization of Local Operator Algebras of Functions\",\n \"content\": \"The main goal of this section is to prove that every BPW complete local operator algebra of functions is completely isometrically isomorphic to the algebra of multipliers on a reproducing kernel Hilbert space of vector-valued functions. Moreover, we wil show that every such algebra is a dual operator algebra in the precise sense of [11]. We will then prove that for such BPW algebras, weak*-convergence and BPW convergence coincide on bounded balls.\\nGiven a set X and a Hilbert space H, then by a reproducing kernel Hilbert space of H-valued functions, we mean a vector space L of H-valued functions that is equipped with a norm and an inner product that makes it a Hilbert space and which has the property that for every x ∈ X, the evaluation map E x : L → H, is a bounded, linear map. Recall that given a Hilbert space H, a matrix of operators,\\nwhere H is a Hilbert space, is called a positive definite operator-valued function on X, provided that for every finite set of (distinct)points {x 1 , ..., x k } in X, the operator-valued matrix, (K(x i , x j )) is positive semidefinite. Given a reproducing kernel Hilbert space of H-valued functions, if we set K(x, y) = E x E * y , then K is positive definite and is called the reproducing kernel of L. There is a converse to this fact, generally called Moore's theorem, which states that given any positive definite operatorvalued function K : X × X → B(H), then there exists a unique reproducing kernel Hilbert space of H-valued functions on X, such that K(x, y) = E x E * y . We will denote this space by L(K, H). Finally, given any reproducing kernel Hilbert space L of H-valued functions with reproducing kernel K, a function f : X → C is called a multiplier provided that for every g ∈ L, the function f g ∈ F. In this case it follows by an application of the closed graph theorem that the map M f : L → L, defined by M f (g) = f g, is a bounded, linear map. The set of all multipliers is denoted M(K) and is easily seen to be an algebra of functions on X and a subalgebra of B(L). The reader can find proofs of the above facts in [14] [3]. Also, we refer to the fundamental work of Pedrick [26] for further treatment of vector-valued reproducing kernel Hilbert spaces. Proof. It is enough to show that the unit ball is weak*-closed by the Krein-Smulian theorem. So let {M f λ } be a net of multipliers in the unit ball of B(L) that converges in the weak*-topology to an operator T. We must show that T is a multiplier.\\nLet x ∈ X be fixed and assume that there exists g ∈ L, with g(\\nThis shows that at every such x the net {f λ (x)} converges to some value. Set f (x) equal to this limit and for all other x's set f (x) = 0.\\nWe claim that f is a multiplier and that T = M f .\\nNote that if g(x) = 0 for every g ∈ L, then E x = E * x = 0. Thus, we have that for any g ∈ L and any h ∈ H,\\n, h H . Since this holds for every h ∈ H, we have that E x (T g) = f (x)g(x), and so T = M f and f is a multiplier.\\nEvery weak*-closed subspace of V ⊆ B(H) has a predual and it is the operator space dual of this predual. Also, if an abstract operator algebra is the dual of an operator space, then it can be represented completely isometrically and weak*-continuously as a weak*-closed subalgebra of the bounded operators on some Hilbert space. For this reason an operator algebra that has a predual as an operator space is called a dual operator algebra. See the book of [11] for the proofs of these facts. Thus, in summary, the above lemma shows that every multiplier algebra is a dual operator algebra.\"\n },\n {\n \"title\": \"Proof. The multiplier norm of a given matrix-valued function\",\n \"content\": \"Applying this fact to a set consisting of a single point, we have that\\nThus, F (x) ≤ C = F and we have that point evaluations are completely contractive on M(K). Since M(K) contains the constants and separates points by hypothesis, it is an operator algebra of functions on X.\\nSuppose that M(K) was not local, then there would exist F ∈ M n (M(K)), and a real number C, such that sup Y π (n) Y < C < F . Then for each finite set Y = {x 1 , ..., x k } we could choose G ∈ M n (M(K)), with G < C, and G(x) = F (x), for every x ∈ Y. But then we would have that\\nis positive definite. From the characterization of functions and their norms in a reproducing kernel Hilbert space, this implies that\\nIn general, M(K) need not separate points on X. In fact, it is possible that L does not separate points and if g(x 1 ) = g(x 2 ), for every g ∈ L, then necessarily f (x 1 ) = f (x 2 ) for every f ∈ M(K).\\nFollowing [24], by a k-idempotent operator algebra, C, we mean that we are given k operators,\\nProof. It is easily checked that K is positive definite. We first prove that the map is an isometry.\\nHence, B ≤ C implies that P is positive and so M f ≤ B . For the converse, given any\\nThe proof of the complete isometry is similar but notationally cumbersome. Proof. Let Y be a finite subset of X. Since A/I Y is a |Y |-idempotent operator algebra, by the above lemma, there exists a vector valued kernel\\nwhere the direct sum is over all finite subsets of X.\\nThen it is easily checked that K is positive definite.\\nThis last condition is equivalent to the existence for each F of some\\nThis proves that M n (M(K)) = M n (Ã) isometrically.\"\n },\n {\n \"title\": \"Corollary 2.5. Every BPW complete local operator algebra of functions is a dual operator algebra.\",\n \"content\": \"Proof. In this case we have that A =à = M(K) completely isometrically. By Lemma 2.1, this latter algebra is a dual operator algebra.\\nThe above theorem gives a weak*-topology to a local operator algebra of function, A by using the identification A ⊆à = M(K) and taking the weak*-topology of M(K). The following proposition proves that convergence of bounded nets in this weak*-topology on A is same as BPW convergence. Proof. Let H denote the reproducing kernel Hilbert space of C-valued functions on X with kernel K for whichà = M(K). Recall that if E x : H → C, is the linear map given by evaluation at\\n. It is readily seen that such vectors exist if and only if E x = 0, or equivalently, K(x, x) = 0. But this follows from the construction of K as a direct sum of positive definite functions over all finite subsets of X. For fixed x ∈ X and the one element subset Y 0 = {x}, we have that the 1-idempotent algebra A/I Y 0 = 0 and so K Y 0 (x, x) = 0, which is one term in the direct sum for K(x, x).\\nConversely, assume that f λ < K, ∀λ and f λ → f pointwise on X. We must prove that M f λ → M f in the weak*-topology on B(H). But since this is a bounded net of operators, it will be enough to show convergence in the weak operator topology and arbitrary vectors can be replaced by vectors from a spanning set. Thus, it will be enough to show that for v 1 , v 2 ∈ C and\\nand the result follows.\\nCorollary 2.7. The ball of a local operator algebra of functions is weak*dense in the ball of its BPW completion.\"\n },\n {\n \"title\": \"Residually Finite Dimensional Operator Algebras\",\n \"content\": \"A C*-algebra B is called residually finite-dimensional (RFD) if it has a separating family of finite-dimensional representations. Note that when B is separable and RFD it has a seperating sequence of finite dimensional representations. Since every one-to-one * -homomorphism of a C*-algebra is completely isometric, a C*-algebra B is RFD if and only if for all n, and for every (b i,j ) ∈ M n (B), we have that (b i,j ) Mn(B) = sup{ (π(b i,j )) } where the suprema is taken over all * -homomorphisms, π : B → M k ∀ k. RFD C*algebras have been studied in [10], [15], [5], [19].\\nWe know from BRS that any operator algebra can be represented on a Hilbert space, i.e., given an abstract unital operator algebra A there exists an unital completely isometric homomorphism π : A → B(H). Since a representation of a C*-algebra is a *-homomorphism if and only if it is completely contractive, the following definition gives us a natural way of extending the notion of RFD to operator algebras.\\nwhere the suprema is taken over all completely contractive homomorphisms, π : B → M k ∀ k. A dual operator algebra B is called weak*-RFD if this last equality holds when the completely contractive homomorphisms are also required to be weak*continuous.\\nThe following result is implicitly contained in [24], but the precise statement that we shall need does not appear there. Thus, we refer the reader to [24] to be able to fully understand the proof since we have used some of the definitions and results from [24] without stating them. Proof. From the Corollary 3.3 of [24] we have that the Schur ideal S(B * B) is non-trivial and bounded. Thus, we can define the algebra A(S(B * B)) = span{E 1 , · · · , E k }, where E i = n Q∈S −1 n (B * B) ⊕Q 1/2 (I n ⊗ E ii )Q −1/2 is the idempotent operator that live on n Q∈S −1 n ⊕M k (M n ). By using Theorem 3.2 of [24] we get that S(A(S(B * B))A(S(B * B)) * ) = S(B * B) This further imples that A(S(B * B))A(S(B * B)) * = B * B completely order isomorphically under the map which sends E * i E j to F * i F j . Finally, by restricting the same map to A we get a map which sends E i to F i completely isometrically. Hence, the result follows. Proof. Let A be an abstract k-idempotent operator algebra. Note that A is a dual operator algebra being a finite dimensional operator algebra.\\nFrom this it follows that there exists a Hilbert space, H and a weak*continuous completely isometric homomorphism, π : A → B(H). Note that B = π(A) is a concrete k-idempotent algebra generated by the idempotents, B = span{F 1 , F 2 , ..., F k } contained in B(H). Thus, from the above lemma B = A(S(B * B)) completely isometrically. For each n ∈ N, Q ∈ S −1 n , we define π Q n : B → M k (M n ) via π Q n (F i ) = Q 1/2 (I n ⊗ E ii )Q −1/2 Assume for the moment that we have proven that π Q n is a weak*-continuous completely contractive homomorphism. Then for every (b ij ) ∈ M k (B) we must have that sup n,Q∈S −1\\nFinally, note that for any (a ij ) ∈ M k (A) we have that (a ij ) M k (A) = sup m { (ρ(π(a ij ))) } where supremum is taken over all weak*-continuous cc homomorphisms ρ : B * B → M m which follows from the fact that π is a complete isometry. Since ρ • π is a weak*-continuous cc homomorphism on A for every weak*-continuous cc homomorphism, ρ on B * B. This implies that (a ij ) M k (A) = sup{ (γ(a ij )) } where supremum is taken over all finite dimensional weak*-continuous cc homomorphisms, and hence the result.\\nThus, it remains to show that π Q n is a weak*-continuous cc homomorphism. Note that it is easy to check that it is a completely contractive homomorphism and sup n,Q∈S −1\\n. Finally, to see that it is weak*-continuous, let f λ be a bounded net in B * B which converges in the weak*-topology. Note that we can write\\nNote that for each i, ψ i is a bounded linear functional on B * B, since the linear maps on a finite dimensional normed space are bounded. Thus, for\\nFinally, to see that π Q n (f λ ) → π Q n (f ) in the weak*-topology, fix unit vectors h, k ∈ H and consider\\nThis shows that the map π Q n is a weak*-continuous and cc homomorphism for every Q ∈ S −1 n and n. Theorem 3.4. Every BPW complete local operator algebra of functions is waek*-RFD.\\nProof. Let A be a BPW complete local operator algebra of functions on the set X and F be a finite subset of X, A/I F is a |F |-idempotent operator algebra. Thus, it follows from the above lemma that A/I F is weak*-\\nwhere supremum is taken over all weak*-continuous cc homomorphisms ρ F : A/I F → M n F and all integers n F .\\nRecall, that the weak*-topology on A requires all the quotient maps of the form π F : A → A/I F , π F (f ) = [f ] to be weak*-continuous. Thus for each finite subset F ⊆ X, π F is a weak*-continuous cc homomorphism. Let ρ F : A/I F → M n F be a weak*-continuous cc homomorphism, then define δ F = π F • ρ F which is indeed a weak*-continuous cc homomorphism. Consider\\nwhere supremum is taken over all weak*-continuous cc homomorphisms, ρ F and positive integers n F . Hence, we obtain the result by taking supremum over all the finite subsets F ⊆ X. Proof. The proof of this follows immediately since every local operator algebra is contained in some BPW complete local operator algebra completely isometrically.\"\n },\n {\n \"title\": \"Quantized Function Theory on Domains\",\n \"content\": \"Whenever one replaces scalar variables by operator variables in a problem or definition, then this process is often referred to as quantization. It is in this sense that we would like to quantize the function theory on a family of complex domains. In some sense this process has already been carried out for balls in the work of Drury [18], Popescu [28], Arveson [6], and Davidson and Pitts [17] and for polydisks in the work of Agler [1], [2], and Ball and Trent [8]. Furthermore, the idea of \\\"quantizing\\\" other domains defined by inequalities occurs in [4], [10], and [20]. We approach these same ideas via operator algebra methods. We will show that in many cases this process yields local operator algebras of functions to which the results of the earlier sections can be applied.\\nWe begin by defining a family of open sets for which our techniques will apply. ,n k , k ∈ I, whose components are analytic functions on G, and G = {z ∈ C N : F k (z) < 1, k ∈ I}, then we call G an analytically presented domain and we call the set of functions R = {F k : G − → M m k ,n k : k ∈ I} an analytic presentation of G. The subalgebra A of the algebra of functions on G generated by the component functions {f k,i,j : 1 ≤ i ≤ m k , 1 ≤ j ≤ n k , k ∈ I} and the constant function is called the algebra of the presentation. We say that R is a separating analytic presentation provided that the algebra A separates points on G.\\nRemark 4.2. An analytic presentation of G by a finite set of matrix-valued functions, F k : G → M m k ,n k , 1 ≤ k ≤ K, can always be replaced by the single block diagonal matrix-valued function, F (z) = F 1 (z) ⊕ · · · ⊕ F K (z) into M m,n with m = m 1 + · · · m K , n = n 1 + · · · n K and we will sometimes do this to simplify proofs. But it is often convenient to think in terms of the set, especially since this will explain the sums that occur in the Agler's factorization formula.\\nNote that when we have a analytically presented domain, then every function in the algebra of the presentation is an analytic function on G. H m ). We call the homomorphism π an admissible strict representation when these inequalities are all strictly less than 1. Given (g i,j ) ∈ M n (A) we set (g i,j ) u = sup{ (π(g i,j )) }, where the supremum is taken over all admissible representations π of A. We let (g i,j ) u 0 denote the supremum that is obtained when we restrict to admissible strict representations.\\nThe theory of [4] and [10] studies domains defined as above with the additional restrictions that the set of defining functions is a finite set of polynomials, but they do not need their polynomials to separate points. Our results should be compared to theirs.\"\n },\n {\n \"title\": \"Proposition 4.4. Let G have a separating analytical presentation and let\",\n \"content\": \"A be the algebra of the presentation. Then A endowed with either of the family of norms · u or · u 0 is an operator algebra of functions on G.\\nProof. It is clear that it is an operator algebra and by definition it is an algebra of functions on G. It follows from the hypotheses that it separates points of G. Finally, for every λ = (λ 1 , ..., λ N ) ∈ G, we have a representation of A on the one-dimensional Hilbert space given by π λ (f ) = f (λ). Hence, |f (λ)| ≤ f u and so A is an operator algebra of functions on G.\\nIt will be convenient to say that matrices, A 1 , . . . A m are of compatible sizes if the product, A 1 · · · A m exists, that is, provided that each A i is an n i × n i+1 matrix.\\nGiven an analytically presented domain G, we include one extra function, F k 0 which denotes the constant function 1. By an admissible block diagonal matrix over G we mean a block diagonal matrix-valued function of the form D(z) = diag(F k 1 , ..., F km ) where k i ∈ I ∪ k 0 for 0 ≤ i ≤ m. Thus, we are allowing blocks of 1's in D(z). Finally, given a matrix B we let B (q) = diag(B, ...., B) denote the block diagonal matrix that repeats B q times. (i) P u < 1, (ii) there exists an integer l, matrices of scalars C j , 1 ≤ j ≤ l with C j < 1 and admissible block diagonal matrices D j (z), 1 ≤ j ≤ l, which are of compatible sizes and are such that\\n(iii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 , P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is a finite set, such that\\nProof. Although we will not logically need it, we first show that (ii) implies (i), since this is the easiest implication and helps to illustrate some ideas. Note that if π : A → B(H) is any admissible representation, then the norm of π of any admissible block diagonal matrix is at most 1. Thus, if P has the form of (ii), then for any admissible π, we will have (π(p i,j )) expressed as a product of scalar matrices and operator matrices all of norm at most one and hence, (π(P i,j )) ≤ C 1 · · · C l < 1. Thus, P u ≤ C 1 · · · C l < 1. We now prove that (i) implies (ii). The ideas of the proof are similar to [23, Corollary 18.2], [12, Corollary 2.11] and [21, Theorem 1] and use in an essential way the abstract characterization of operator algebras. For each m, n ∈ N, one proves that P m,n := inf { C 1 . . . C l }, defines a norm on M m,n (A), where the infimum is taken over all l and all ways to factor P (z) = C 1 D 1 (z i 1 ) · · · C l D l (z i l ) as a product of matrices of compatible sizes with scalar matrices C j , 1 ≤ j ≤ l and admissible block diagonal matrices D j , 1 ≤ j ≤ l.\\nMoreover, one can verify that M m,n (A) with this family { . m,n } m,n of norms satisfies the axioms for an abstract unital operator algebra as given in [13] and hence by the Blecher-Ruan-Sinclair representation theorem [13](see also [23]) there exists a Hilbert space H and a unital completely isometric isomorphism π : A −→ B(H). Thus, for every m, n ∈ N and for every P = (p ij ) ∈ M m,n (A), we have P m,n = (π(p ij )) . However, π (m k ,n k ) (F k ) = (π(f k,i,j )) ≤ 1 for 1 ≤ i ≤ K, and so, π is an admissible representation. Thus, P m,n = (π(p ij )) ≤ P u . Hence, if P u < 1, then P m,n < 1 which implies that such a factorization exists. This completes the proof that (i) implies (ii).\\nWe will now prove that (ii) implies (iii). Suppose that P has a factorization as in (ii). Let K ⊆ I be the finite subset of all indices that appear in the block-diagonal matrices appearing in the factorization of P. We will use induction on l to prove that (iii) holds.\\nFirst, assume that l = 1 so that P (z) = C 1 D 1 (z). Then,\\nLet E k be the diagonal matrix that has 1's wherever F k appears(so E k = 0 when there is no F k term in D 1 ). Hence,\\nTherefore, gathering terms for common values of i,\\nwhere R 0 = I m − C 1 C * 1 is a positive, invertible matrix and P i is, in this case a constant. Thus, the form (iii) holds, when l = 1.\\nWe now assume that the form (iii) holds for any R(z) that has a factorization of length at most l − 1, and assume that\\nwhere R(z) has a factorization of length l − 1.\\nNote that a sum of expressions such as on the right hand side of (iii) is again such an expression. This follows by using the fact that given any two expressions A(z), B(z), we can write\\nThus, it will be sufficient to show that I m − P (z)P (w) * is a sum of expressions as above. To this end we have that,\\n. The first term of the above equation is of the form as on the right hand side of (iii) by case l = 1. Also, the quantity (I − R(z)R(w) * ) = R 0 R * 0 + R 0 (z)R 0 (w) * + k∈K R k (z)(I − F k (z)F k (w) * ) (q k ) R k (w) * by the inductive hypothesis. Hence,\\nThus, we have expressed (I − P (z)P (w) * ) as a sum of two terms both of which can be written in the form desired. Using again our remark that the sum of two such expressions is again such an expression, we have the required form.\\nFinally, we will prove (iii) implies (i). Let π : A → B(H) be an admissible representation and let P = (p i,j ) ∈ M m,n (A) have a factorization as in (iii). To avoid far too many superscripts we write simplify π (m,n) to Π. Now observe that\\nClearly the first two terms of the sum are positive. But since π is an admissible representation, Π(F k ) ≤ 1 and hence, (I − Π(F k )Π(F k ) * ) ≥ 0. Hence, each term on the right hand side of the above inequality is positive and since R is strictly positive, say R ≥ δI m for some scalar δ > 0, we have that I m − Π(P )Π(P ) * ≥ δI m . Therefore, Π(P ) ≤ √ 1 − δ. Thus, since π was an arbitrary admissible representation, P u ≤ √ 1 − δ < 1, which proves (i).\\nWhen we require the functions in the presentation to be row vectorvalued, then the above theory simplifies somewhat and begins to look more familiar. Let G be an analytically presented domain with presentation F k : G → M 1,n k , k ∈ I. We identify M 1,n with the Hilbert space C n so\\nwhere the inner product is in C n . In this case we shall say that G is presented by vector-valued functions. (i) P u < 1, (ii) there exists an integer l, matrices of scalars C j , 1 ≤ j ≤ l with C j < 1 and admissible block diagonal matrices D j (z), 1 ≤ j ≤ l, which are of compatible sizes and are such that P (z) = C 1 D 1 (z) · · · C l D l (z).\\n(iii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 ∈ M m,r 0 (A), P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is finite, such that\\nThe following result gives us a Nevanlinna-type result for the algebra of presentation.\\nTheorem 4.7. Let Y be a finite subset of an analytically presented domain G with separating analytic presentation F k = (f k,i,j ) : G → M m k ,n k , k ∈ I, let A be the algebra of the presentation and let P be a M m,n -valued function defined on a finite subset Y = {x 1 , · · · , x l } of G. Then the following are equivalent:\\n(i) there existsP ∈ M mn (A) such thatP | Y = P and P u < 1.\\n(ii) there exists a positive, invertible matrix R ∈ M m and matrices P 0 ∈ M m,r 0 (A), P k ∈ M m,r k (A), k ∈ K, where K ⊆ I is a finite set, such that\\nProof. Note that (i) ⇒ (ii) follows immediately as a corollary of Theorem 4.5. Thus it only remains to show that (ii) ⇒ (i). Since A is an operator algebra of functions, therefore, A/I Y is a finite dimensional operator algebra of idempotents and A/I Y = span{E 1 , · · · , E l } for some l = |Y |. Thus there exists a Hilbert space, H Y and a completely isometric representation π of A/I Y . By Theorem 2.4, there exists a kernel K Y such that π(A/I Y ) = M(K Y ) completely isometrically under the map ρ :\\nUsing that R > 0, we get that π(P + I Y ) < 1. This shows that there existsP ∈ A such thatP | Y = P and P u < 1. This completes the proof.\\nWe now turn towards defining quantized versions of the bounded analytic functions on these domains. For this we need to recall that the joint Taylor spectrum of a commuting N -tuple of operators T = (T 1 , ..., T N ), is a compact set, σ(T ) ⊆ C N and that there is an analytic functional calculus defined for any function that is holomorphic in a neighborhood of σ(T ).\\nwhere H is an arbitrary Hilbert space. We set\\nwhere n is an arbitrary positive integer.\\nNote that if we identify a point (λ 1 , ..., λ N ) ∈ C N with an N-tuple of commuting operators on a one-dimensional Hilbert space, then we have that G ⊆ Q(G).\\nIf T = (T 1 , ..., T N ) ∈ Q(G), is a commuting N-tuple of operators on the Hilbert space H, then since the joint Taylor spectrum of T is contained in G, we have that if f is analytic on G, then there is an operator f (T ) defined and the map π : Hol(G) → B(H) is a homomorphism, where Hol(G) denotes the algebra of analytic functions on G.\\nWe are interested in determining when H ∞ R (G) =Ã, completely isometrically and whether or not the · R norm is attained on the smaller set Q 0,0 (G).\\nNote that since each point in G ⊆ Q(G), we have that\\nThus, the inclusion of A into H ∞ R (G) might not even be isometric. Theorem 4.10. Let G be an analytically presented domain with a separating presentation R = {F k : G → M m k ,n k : k ∈ I}, let A be the algebra of the presentation and let A be the BPW-completion of A. Then\\nProof. Let f ∈ M n ( A), with f L < 1. Then there exists a net of functions, f λ ∈ M n (A), such that f λ u < 1 and lim λ f λ (z) = f (z) for every z ∈ G.\\nSince f λ ∞ < 1, by Montel's Theorem, there is a subsequence {f n } of this net that converges to f uniformly on compact sets. Hence, if T ∈ Q(G), then lim n f (T ) − f n (T ) = 0 and so f (T ) ≤ sup n f n (T ) ≤ 1. Thus, we have that f ∈ M n (H ∞ R (G)), with f R ≤ 1. This proves that A ⊆ H ∞ R (G) and that f L ≤ f R .\\nConversely, let g ∈ M n (H ∞ R (G) with g R < 1. For any finite set Y = {y 1 , ..., y t } ⊆ G, let A/I Y = span{E 1 , ..., E t } be the corresponding t-idempotent algebra and let π Y : A → A/I Y denote the quotient map. Write y i = (y i,1 , ..., y i,N ), 1 ≤ i ≤ t and let T j = y 1,j E 1 + · · · + y t,j E t , 1 ≤ j ≤ N so that T = (T 1 , ..., T N ) is a commuting N -tuple of opertors with σ(T ) = Y. For k ∈ I, we have that F k (T ) = F k (y 1 ) ⊗ E 1 + · · · + F k (y t ) ⊗ E t = π Y (F k ) ≤ F k u = 1. Thus, T ∈ Q(G), and so, g(T ) = g(y 1 ) ⊗ E 1 + · · · + g(y t ) ⊗ E t ≤ g R < 1. Since A separates points, we may pick f ∈ M n (A) such that f = g on Y. Hence, π Y (f ) = f (T ) = g(T ) and π Y (f ) < 1. Thus, we may pick f Y ∈ M n (A), such that π Y (f Y ) = π Y (f ) and f Y u < 1. This net of functions, {f Y } converges to g pointwise and is bounded. Therefore, g ∈ M n ( A) and g L ≤ 1. This proves that H ∞ R (G) ⊆ A and that g L ≤ g R .\\nThus, A = H ∞ R (G) and the two matrix norms are equal for matrices of all sizes. The rest of the conclusions follow from the results on BPWcompletions.\"\n },\n {\n \"title\": \"Remark 4.11. The above result yields that for every\",\n \"content\": \". Also, we can verify these hypotheses are met for most of the algebras given in the example section. In particular, for Examples 5. 1, 5.2, 5.3, 5.4, 5.6, 5.7, and 5.8\"\n },\n {\n \"title\": \". It would be interesting to know if this can be done in general.\",\n \"content\": \"We now seek other characterisations of the functions in H ∞ R (G). In particular, we wish to obtain analogues of Agler's factorization theorem and of the results in [4] and [10]. By Theorem 2.4, if we are given an analytically presented domain G ⊆ C N , with presentation R = {F k : G → M m k ,n k , k ∈ I}, then there exists a Hilbert space H and a positive definite function, K : G × G → B(H) such that A = M(K). We shall denote any kernel satisfying this property by K R . (\\nIn the case when the presentation contains only finitely many functions we can say considerably more about R-limits.\\nProof. By the continuity of the functions, there is a constant δ > 0, such that F k (z) ≤ 1 − δ, ∀k, ∀z ∈ S. Thus, we have that I − F k (z)F k (z) * ≥ δI, ∀ k, ∀z ∈ S. Also, we have that Proof. We identify (I m − F (z)F (w) * ) (q λ ) = (I m − F (z)F (w) * ) ⊗ I C q λ , and the p × mq λ matrix-valued function P λ as an analytic function, P λ : G → B(C m ⊗ C q λ , C p ). Writing C m ⊗ C q λ = C q λ ⊕ · · · C q λ (m times) allows us to write P λ (z) = [P λ 1 (z), . . . , P λ m (z)] where each P λ i (z) is p × q λ . Also, if we let f 1 (z), ..., f m (z) be the (1, n) vectors that represent the rows of the matrix F, then we have that F (z)F (w) * = m i,j=1 f i (z)f j (w) * E i,j . Finally, we have that\\n, is a positive definite function that is analytic in z and coanalytic in w. By dropping to a subnet, if necessary, we may assume that K λ converges uniformly on compact subsets of G to K = (K i,j ) : G × G → M m (M p ). Note that this implies that P λ i (z)P λ j (w) * → K i,j (z, w) for all i, j and that K is a positive definite function that is analytic in z and coanalytic in w.\\nThe positive definite K gives rise to a reproducing kernel Hilbert space H of analytic C m ⊗ C p -valued functions on G. If we let E(z) : H → C m ⊗ C p , be the evaluation functional, then K(z, w) = E(z)E(w) * and E : G → B(H, C m ⊗ C p ) is analytic. Identifying C m ⊗ C p = C p ⊕ · · · ⊕ C p (m times), yields analytic functions,\\nDefine an analytic map R : G → B(H ⊗ C m , C p ) by identifying H ⊗ C m = H ⊕ · · · ⊕ H(m times) and setting R(z)(h 1 ⊕ · · · ⊕ h m ) = E 1 (z)h 1 + · · · + E m (z)h m . Thus, we have that\\nand the proof is complete. (1) f ∈ M mn (H ∞ R (G)) and f R ≤ 1, (2) there exists an analytic operator-valued function R 0 :\\nProof. Recall thatà = H ∞ R (G). Let us first assume that f ∈ M mn (Ã) and f Mmn(Ã) < 1. Then for each finite set Y , there exists f Y ∈ M mm (A) such that f Y converges to f pointwise and f Y u ≤ 1. We may assume that\\nwhere |Y | denotes the cardinality of the set Y .\\nThus by Theorem 4.5 there exists a positive, invertible matrix R Y ∈ M m and matrices P Y k ∈ M m,r k Y (A), 0 ≤ k ≤ K, such that\\nwhere r k Y = q k Y m k and z, w ∈ G. If we define a map F 0 : G → M m 0 ,n 0 as the zero map then the above factorization can be written as\\nNote that the net R Y is uniformly bounded above by 1, thus there exists R ∈ M m and a subnet R Ys which converges to R.\\nFinally, since the net f Y converges to f pointwise we have that the net\\nAlso note that for each k, < P Y k > is the net of vector-valued holomorphic function and is uniformly bounded on compact subsets of G by Proposition 4.14.\\nThus by Proposition 4.15 there exists F k -limit for each 0 ≤ k ≤ K, that is, there exists K+1 Hilbert spaces H k and K+1 analytic function,\\nThis completes the proof that (1) implies (2). To show the converse, assume that there exists an analytic operatorvalued function R 0 : G → B(H 0 , C m ) and K analytic functions,\\nBy using Theorem 2.4 there exists a vector-valued Kernel K such that M n (M(K)) = M n (Ã) completely isometrically for every n. It is easy to see that ((I − f (z)f (w) * )⊗K(z, w)) ≥ 0 ∀z, w ∈ G. This is equivalent to f ∈ M m (M(K)) and M f ≤ 1 which in turn is equivalent to (1). Thus, (1) and (2) are equivalent.\\nClearly, (3) implies (2). The arguement for why (2) implies (3) is contained in [10] and we recall it. If we fix any k 0 , then since F k 0 (z) < 1 on G, we have that |f k 0 ,1,1 (z)| 2 + · · · + |f ko,1,m (z)| 2 < 1 on G. From this it follows that H(z, w) = (1 − f k 0 ,1,1 (z)f k 0 ,1,1 (w) − · · · − f k 0 ,1,m (z)f k 0 ,1,m (w)) is an F k 0 -limit and that H −1 (z, w) is positive definite. Now we have that R 0 (z)R 0 (w) * H −1 (z, w) is positive definite and so we may write, R 0 (z)R 0 (w) * H −1 (z, w) = G 0 (z)G 0 (w) * and we have that R 0 (z)R 0 (w) * = G 0 (z)H(z, w)G(w) . This shows that R 0 (z)R 0 (w) * is an F k -limit and so it may be absorbed into the sum.\"\n },\n {\n \"title\": \"Examples and Applications\",\n \"content\": \"In this section we present a few examples to illustrate the above definitions and results. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T n ) such that T 1 T * 1 + · · · + T N T * N ≤ I, which is commonly called a row contraction and an admissible strict representation is given when T 1 T * 1 + · · · + T N T * N < I, which is generally referred to as a strict row contraction. In this case one can again easily see that · u = · u 0 by using the same r < 1 argument as in the last example and that f ∈ H ∞ R (B N ) if and only if f R = sup{ f (T ) : T 1 T * 1 + · · · + T N T * N < I} < +∞ These are the norms on polynomials considered by Drury [18], Popescu [28], Arveson [6], and Davidson and Pitts [17]. By Theorem 4.18 we will have for i.e., such that T * 1 T 1 + · · · + T * N T N ≤ I, which is generally referred to as a column contraction. This time the norm on H ∞ R (B N ) will be defined by taking suprema over all strict column contractions and we will have that f R ≤ 1 if and only if\\nAgain this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T N ) such that T 1 T * 1 + · · · + T N T * N ≤ I and T * 1 T 1 + · · · + T * N T N ≤ I, that is, which is both a row and column contraction. This time the norm on H ∞ R (B N ) is defined as the supremum over all commuting N-tuples that are both strict row and column contractions. We will have that f ∈ M m,n (H ∞ R (B N )) with f R ≤ 1 if and only if\\nwhere K 1 and R 1 are as before.\\nThe last three examples illustrate that it is possible to have multiple rational representations of G, all with the same algebra, but which give rise to (possibly) different operator algebra norms on A. Thus, the operator algebra norm depends not just on G, but also on the particular presentation of G that one has chosen. We have surpressed this dependence on R to keep our notation simplified. and it can be seen that H ∞ R (D) is defined by\\nIn this case we have that f ∈ M m,n (H ∞ R (D) and f R ≤ 1 if and only if\\nExample 5.6. Let L = {z ∈ C : |z − a| < 1, |z − b| < 1}, where |a − b| < 1, then the functions f 1 (z) = z −a, f 2 (z) = z −b give a polynomial presentation of this \\\"lens\\\". The algebra of this presentation is again the algebra of polynomials. An admissible representation of this algebra is defined by choosing any operator satisfying T − aI ≤ 1 and T − bI ≤ 1, with strict inequalities for the admissible strict representations. In this case we easily see that · u = · u 0 , since given any operator T satisfying T − aI ≤ 1 and T − bI ≤ 1, and r < 1, S r = rT + (1 − r)(a + b) corresponds to the admissible strict representations and for any matrix of polynomials (p i,j (T )) = lim r→1 (p ij (S r )) . This algebra with this norm was studied in [9]. Their work shows that this norm is completely boundedly equivalent to the usual supremum norm. Our results imply that f ∈ M m,n (H ∞ R (L)) and f R ≤ 1 if and only if\\nfor some appropriatley chosen R 1 .\\nAll of the above examples are also covered by the theory of [4] and [10], except that their definition of the norm is slightly different. We address this difference in a later remark. We now turn to some examples that are not covered by their theory.\\nExample 5.8. Let 0 < r < 1 be fixed and let A r = {z ∈ C : r < |z| < 1} be an annulus. Then this has a rational presentation given by F 1 (z) = z and F 2 (z) = rz −1 , and the algebra of this presentation is just the Laurent polynomials. Admissible representations of this algebra are given by selecting any invertible operator T satisfying T ≤ 1 and T −1 ≤ r −1 . Admissible strict representations are given by invertible operators satisfying T < 1 and T −1 < r −1 . It is no longer quite so clear that · u = · u 0 . However, this algebra with these norms is studied by the first author in [22] and among other results the equality of these norms was shown. In [17], it was shown that · u is completely boundedly equivalent to the usual supremum norm. In this case one can see that the · R is achieved by taking suprema over all T satisfying T < 1 and T −1 < r −1 . The formula for the norm is given by f R ≤ 1 if and only if I m − f (z)f (w) * = (1 − zw)K 1 (z, w) + (1 − r 2 z −1 w −1 )K 2 (z, w). Our results imply that f R ≤ 1 if and only if\\nIn particular, if we take φ(z) = z−1 z+1 then it maps the halfplane H = {z : Re(z) > 0} to the unit disk. For this particular φ, we have that Q R = {T : σ(T ) ⊆ H and Re(T ) ≥ 0}. where w(T ) denotes the numerical radius of T. Thus, H ∞ R (D) becomes the \\\"universal\\\" operator algebra that one obtains by substituting an operator of numerical radius less than one for the variable z and we have a quite different quantization of the unit disk. Our results give a formula for this norm, but only in terms of R-limits, so further work would need to be done to make it explicit.\\nExample 5.12. There is a second way that one can quantize many convex sets. Let G = {z : |z − a k | < r k , k ∈ I} ⊆ C be an open, bounded convex set that can be expressed as an intersection of a possibly infinite set of open disks. For example, any convex set with a smooth boundary with uniformly bounded curvature can be expressed in such a fashion. Then G has a rational presentation given by F k (z) = r −1 k (z − a k ), k ∈ I the algebra of the presentation is just the polynomial algebra and an admissible representation is given by selecting any operator T satisfying, T − a k I ≤ r k , k ∈ I. Thus, we again a factorization result, but only in terms of R-limits.\\nThe above definitions allow one to consider many other examples. For example, one could fix 0 < r < 1 and let G = {z ∈ B N : r < |z 1 |}, with rational presentation f 1 (z) = (z 1 , ..., z N ) ∈ M 1,N , and f 2 (z) = rz −1\\n1 . An admissible representation would then correspond to a commuting row contraction with T 1 invertible and T −1 1 ≤ r −1 .\\nWe now compare and contrast some of our hypotheses with those of [4] and [10].\\nRemark 5.13. Let G = {z ∈ C N : F k (z) < 1 ∀ k = 1, · · · , K} where the F k 's are matrix-valued polynomials defined on G. Then for f ∈ Hol(G), [4] and [10] really study a norm given by f s = sup{ f (T ) } where the supremum is taken over all commuting N -tuples of operators T with F k (T ) < 1 ∀k. We wish to contrast this norm with our f R . In [4] it is shown that the hypotheses F k (T ) < 1, k = 1, ..., K implies that σ(T ) ⊆ G. Thus, we have that f s ≤ f R . In fact, we have that f s = f R . This can be seen by the fact that they obtain identical factorization theorems to ours. This can also be seen directly in some cases where the algebra A contains the polynomials and when it can be seen that · R is attained by taking the supremum over matrices(see Remark 4.11). Indeed, if f R is attained as the supremum over commuting N -tuples of finite matrices T = (T 1 , ..., T N ) satisfying σ(T ) ⊆ G and F k (T ) ≤ 1 then such an N -tuple of commuting matrices, can be conjugated by a unitary to be simultaneously put in upper triangular form. Now it is easily argued that the strictly upper triangular entries can be shrunk slightly so that one obtains new N -tuples T ǫ = (T 1,ǫ , ..., T N,ǫ ) satisfying, F k (T ǫ ) < 1, k = 1, ..., K and T i − T i,ǫ < ǫ. But we do not have a simple direct argument that works in all cases.\\nRemark 5.14. We do not know how generally it is the case that · u is a local norm. That is, we do not know if f u = f R for f ∈ M n (A). In particular, we do not know if this is the case for Example 5.5. In this case, the algebra of the of the presentation is A = span{z n : n ≥ 0, n = 1}. If we write a polynomial p ∈ A in terms of its even and odd decomposition, p = p e + p o , then p e (z) = q(z 2 ) and p o = z 3 r(z 2 ) for some polynomials q, r.\"\n },\n {\n \"title\": \"Proposition 1 . 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"F\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 1. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Example 1. 15 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Example 1 . 17 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 2 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2 . 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2 . 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 2. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Definition 3. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 3. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 3. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 3. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Definition 4 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Definition 4 . 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 4. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Definition 4 . 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Definition 4. 12 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 4. 13 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 4 . 14 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 4 . 15 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Remark 4 . 16 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Definition 4 . 17 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Example 5 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Example 5. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Example 5 . 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Example 5 . 9 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Example 5 . 10 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Let A be an operator algebra of functions on X, then A ⊆ ℓ ∞ (X), and for every n and every(f i,j ) ∈ M n (A), we have (f ij ) ∞ ≤ (f ij ) Mn(A) .","content":"Proposition 1 . 2 .\n12Let A be an operator algebra of functions on X, then A ⊆ ℓ ∞ (X), and for every n and every(f i,j ) ∈ M n (A), we have (f ij ) ∞ ≤ (f ij ) Mn(A) ."},{"id":"","caption":": M n (A) → M n (A/I F ) ∼ = M n (A)/M n (I F ).","content":"F\n: M n (A) → M n (A/I F ) ∼ = M n (A)/M n (I F )."},{"id":"","caption":"If A is an operator algebra of functions on the set X, thenà equipped with the norms of Definition 1.5 is a local operator algebra of functions on X. Moreover, for (f i,j ) ∈ M n (A), (f i,j ) Mn(Ã) = (f i,j ) Mn(A L ) .","content":"Lemma 1. 8 .\n8If A is an operator algebra of functions on the set X, thenà equipped with the norms of Definition 1.5 is a local operator algebra of functions on X. Moreover, for (f i,j ) ∈ M n (A), (f i,j ) Mn(Ã) = (f i,j ) Mn(A L ) ."},{"id":"","caption":"Let A = A(D) ⊆ C(D − ) be the subalgebra of the algebra of continuous functions on the closed disk consisting of the functions that are analytic on the open disk D. Identifying M n (A(D)) ⊆ M n (C(D − )) as a subalgebra of the algebra of continuous functions from the closed disk to the matrices, equipped with the supremum norm, gives A(D) it's usual operator algebra structure. With this structure it can be regarded as a local operator algebra of functions on D or on D − . If we regard it as a local operator algebra of functions on D − , then A(D) A(D). To see that the containment is strict, note that f (z) = (1 + z)/2 ∈ A(D) and f n (z) → χ {1} , the characteristic function of the singleton {1}.","content":"Example 1. 15 .\n15Let A = A(D) ⊆ C(D − ) be the subalgebra of the algebra of continuous functions on the closed disk consisting of the functions that are analytic on the open disk D. Identifying M n (A(D)) ⊆ M n (C(D − )) as a subalgebra of the algebra of continuous functions from the closed disk to the matrices, equipped with the supremum norm, gives A(D) it's usual operator algebra structure. With this structure it can be regarded as a local operator algebra of functions on D or on D − . If we regard it as a local operator algebra of functions on D − , then A(D) A(D). To see that the containment is strict, note that f (z) = (1 + z)/2 ∈ A(D) and f n (z) → χ {1} , the characteristic function of the singleton {1}."},{"id":"","caption":"However, if we regard A(D) as a local operator algebra of functions on D, then its BPW completion A(D) = H ∞ (D), the bounded analytic functions on the disk, with its usual operator structure.","content":"However, if we regard A(D) as a local operator algebra of functions on D, then its BPW completion A(D) = H ∞ (D), the bounded analytic functions on the disk, with its usual operator structure."},{"id":"","caption":"uniformly on compact subsets of D. Thus, by the identity theorem F ≡ G on D. Hence, A is a local operator algebra. Finally, by using Lemma 1.11 and a similar argument, one can also show thatà = H ∞ (D).","content":"uniformly on compact subsets of D. Thus, by the identity theorem F ≡ G on D. Hence, A is a local operator algebra. Finally, by using Lemma 1.11 and a similar argument, one can also show thatà = H ∞ (D)."},{"id":"","caption":"Let A = H ∞ (D) but endowed with a new norm. Fix b > 1,","content":"Example 1 . 17 .\n117Let A = H ∞ (D) but endowed with a new norm. Fix b > 1,"},{"id":"","caption":"Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H). Then M(K) ⊆ B(L) is a weak*-closed subalgebra.","content":"Lemma 2 . 1 .\n21Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H). Then M(K) ⊆ B(L) is a weak*-closed subalgebra."},{"id":"","caption":"Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X ×X → B(H) and let M(K) ⊆ B(L) denote the multiplier algebra, endowed with the operator algebra structure that it inherits as a subalgebra. If K(x, x) = 0, for every x ∈ X and M(K) separates points on X, then M(K) is a BPW complete local dual operator algebra of functions on X.","content":"Theorem 2 . 2 .\n22Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X ×X → B(H) and let M(K) ⊆ B(L) denote the multiplier algebra, endowed with the operator algebra structure that it inherits as a subalgebra. If K(x, x) = 0, for every x ∈ X and M(K) separates points on X, then M(K) is a BPW complete local dual operator algebra of functions on X."},{"id":"","caption":"Let A be an operator algebra of functions on the set X then there exists a Hilbert space, C and a positive definite function K : X × X → B(C) such that M(K) =à completely isometrically.","content":"Theorem 2 . 4 .\n24Let A be an operator algebra of functions on the set X then there exists a Hilbert space, C and a positive definite function K : X × X → B(C) such that M(K) =à completely isometrically."},{"id":"","caption":"Let A be a local operator algebra of functions on the set X. Then the net (f λ ) λ ∈ Ball(A) converges in the weak*-topology if and only if it converges pointwise on X.","content":"Proposition 2. 6 .\n6Let A be a local operator algebra of functions on the set X. Then the net (f λ ) λ ∈ Ball(A) converges in the weak*-topology if and only if it converges pointwise on X."},{"id":"","caption":"An operator algebra, B is called RFD if for every n and for every (","content":"Definition 3. 1 .\n1An operator algebra, B is called RFD if for every n and for every ("},{"id":"","caption":"Let B be a concrete k-idempotent operator algebra. Then S(B * B)) is a Schur ideal affiliated with B, i.e., B = A(S(B * B)) completely isometrically.","content":"Lemma 3. 2 .\n2Let B be a concrete k-idempotent operator algebra. Then S(B * B)) is a Schur ideal affiliated with B, i.e., B = A(S(B * B)) completely isometrically."},{"id":"","caption":"Every k-idempotent operator algebra is weak*-RFD.","content":"Theorem 3. 3 .\n3Every k-idempotent operator algebra is weak*-RFD."},{"id":"","caption":"Every local operator algebra of functions is RFD.","content":"Corollary 3. 5 .\n5Every local operator algebra of functions is RFD."},{"id":"","caption":"Let G ⊆ C N be an open set. If there exists a set of matrixvalued functions, F k = (f k,i,j ) : G − → M m k","content":"Definition 4 . 1 .\n41Let G ⊆ C N be an open set. If there exists a set of matrixvalued functions, F k = (f k,i,j ) : G − → M m k"},{"id":"","caption":"Let G ⊆ C N be an analytically presented domain with presentation F = (f i,j ) : G → M m,n , and let H be a Hilbert space. A homomorphism π : A → B(H) of the algebra of the presentation is called an admissible representation provided that (π(f i,j )) ≤ 1 in M m,n (B(H) = B(H n ,","content":"Definition 4 . 3 .\n43Let G ⊆ C N be an analytically presented domain with presentation F = (f i,j ) : G → M m,n , and let H be a Hilbert space. A homomorphism π : A → B(H) of the algebra of the presentation is called an admissible representation provided that (π(f i,j )) ≤ 1 in M m,n (B(H) = B(H n ,"},{"id":"","caption":"Let G be an analytically presented domain with presentationR = {F k = (f k,i,j ) : G → M m k ,n k , k ∈ I},let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A), where m, n are arbitrary. Then the following are equivalent:","content":"Theorem 4. 5 .\n5Let G be an analytically presented domain with presentationR = {F k = (f k,i,j ) : G → M m k ,n k , k ∈ I},let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A), where m, n are arbitrary. Then the following are equivalent:"},{"id":"","caption":"Let G be presented by vector-valued functions, F k = (f k,j ) : G → M 1,n k , k ∈ I, let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A). Then the following are equivalent:","content":"Corollary 4. 6 .\n6Let G be presented by vector-valued functions, F k = (f k,j ) : G → M 1,n k , k ∈ I, let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A). Then the following are equivalent:"},{"id":"","caption":"Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}. We define the quantized version of G to be the collection of all commuting N-tuples of operators,","content":"Definition 4 . 8 .\n48Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}. We define the quantized version of G to be the collection of all commuting N-tuples of operators,"},{"id":"","caption":"Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}.We shall call a function H : G × G → M m an R-limit, provided that H is the pointwise limit of a net of functions H λ : G × G → M m of the form given byTheorem 4.5(iii).","content":"Definition 4. 12 .\n12Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}.We shall call a function H : G × G → M m an R-limit, provided that H is the pointwise limit of a net of functions H λ : G × G → M m of the form given byTheorem 4.5(iii)."},{"id":"","caption":"Let G ⊆ C N be an analytically presented domain, with a separating presentation R = {F k : G − → M m k ,n k , k ∈ I}. Then the following are equivalent:","content":"Corollary 4. 13 .\n13Let G ⊆ C N be an analytically presented domain, with a separating presentation R = {F k : G − → M m k ,n k , k ∈ I}. Then the following are equivalent:"},{"id":"","caption":"Let G be an analytically presented domain with a finite presentation R= {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}.For each compact subset S ⊆ G, there exists a constant C, depending only on S, such that given a factorization of the form,","content":"Proposition 4 . 14 .\n414Let G be an analytically presented domain with a finite presentation R= {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}.For each compact subset S ⊆ G, there exists a constant C, depending only on S, such that given a factorization of the form,"},{"id":"","caption":"The proof of the following result is essentially contained in[10, Lemma 3.3].","content":"The proof of the following result is essentially contained in[10, Lemma 3.3]."},{"id":"","caption":"Let G be a bounded domain in C N and let F = (f i,j ) : G → M m,n be analytic with F (z) < 1 for z ∈ G. If H : G × G → M p is analytic in the first variables, coanalytic in the second variables and there exists a net of matrix-valued functions P λ ∈ M p,r λ (Hol(G)) which are uniformly bounded on compact subsets of G, such that H(z, w) is the pointwise limit of H λ (z, w) = P λ (z)(I m − F (z)F (w) * ) (q λ ) P λ (w) * where r λ = q λ m k , then there exists a Hilbert space H and an analytic function, R : G → B(H ⊗ C M , C p ) such that H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * .","content":"Proposition 4 . 15 .\n415Let G be a bounded domain in C N and let F = (f i,j ) : G → M m,n be analytic with F (z) < 1 for z ∈ G. If H : G × G → M p is analytic in the first variables, coanalytic in the second variables and there exists a net of matrix-valued functions P λ ∈ M p,r λ (Hol(G)) which are uniformly bounded on compact subsets of G, such that H(z, w) is the pointwise limit of H λ (z, w) = P λ (z)(I m − F (z)F (w) * ) (q λ ) P λ (w) * where r λ = q λ m k , then there exists a Hilbert space H and an analytic function, R : G → B(H ⊗ C M , C p ) such that H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * ."},{"id":"","caption":"Conversely, any function that can be written in the formH(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * can beexpressed as a limit of a net as above by considering the directed set of all finite dimensional subspaces of H and for each finite dimensional subspace setting H F (z, w) = R F (z)[(I m − F (z)F (w) * ) ⊗ I F ]R F (w) * , where R F (z) = R(z)(P F ⊗ I m ) with P F the orthogonal projection onto F.","content":"Remark 4 . 16 .\n416Conversely, any function that can be written in the formH(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * can beexpressed as a limit of a net as above by considering the directed set of all finite dimensional subspaces of H and for each finite dimensional subspace setting H F (z, w) = R F (z)[(I m − F (z)F (w) * ) ⊗ I F ]R F (w) * , where R F (z) = R(z)(P F ⊗ I m ) with P F the orthogonal projection onto F."},{"id":"","caption":"We shall refer to a function H : G × G → M m (M p ) that can be expressed as H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ H]R(w) * for some Hilbert space H and some analytic function R : G → B(H ⊗ C m , C p ), as an F-limit. Theorem 4.18. Let G be an analytically presented domain with a finite separating presentation R = {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}, let f = (f ij ) be a M m,n -valued function defined on G. Then the following are equivalent:","content":"Definition 4 . 17 .\n417We shall refer to a function H : G × G → M m (M p ) that can be expressed as H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ H]R(w) * for some Hilbert space H and some analytic function R : G → B(H ⊗ C m , C p ), as an F-limit. Theorem 4.18. Let G be an analytically presented domain with a finite separating presentation R = {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}, let f = (f ij ) be a M m,n -valued function defined on G. Then the following are equivalent:"},{"id":"","caption":"Let G = D N , be the polydisk which has a presentation given by the coordinate functions F i (z) = z i , 1 ≤ i ≤ N. Then the algebra of this presentation is the algebra of polynomials and an admissible representation is given by any choice of N commuting contractions, (T 1 , ..., T N ) on a Hilbert space. Given a matrix of polynomials,(p i,j ) u = sup (p i,j (T 1 , ..., T N ))where the supremum is taken over all N-tuples of commuting contractions. This is the norm considered by Agler in[1], which is sometimes called the Schur-Agler norm[21].Our Q(D N ) = {T = (T 1 , ..., T N ) : σ(T ) ⊆ D N and T i ≤ 1}. Note that if we replace such a T by rT = (rT 1 , ..., rT N ) then rT i < 1, rT ∈ Q R (D N ) and taking suprema over all T ∈ Q R (D N ) will be the same as taking a suprema over this smaller set. Thus, the algebra H ∞ R (D N ) consists of those analytic functions f such that f R = sup{ f (T 1 , ..., T N ) : T i < 1, i = 1, ..., N } < +∞. By Theorem 4.18 for f ∈ M m,n (H ∞ R (D N ), we have that f R ≤ 1 if and only if","content":"Example 5 . 1 .\n51Let G = D N , be the polydisk which has a presentation given by the coordinate functions F i (z) = z i , 1 ≤ i ≤ N. Then the algebra of this presentation is the algebra of polynomials and an admissible representation is given by any choice of N commuting contractions, (T 1 , ..., T N ) on a Hilbert space. Given a matrix of polynomials,(p i,j ) u = sup (p i,j (T 1 , ..., T N ))where the supremum is taken over all N-tuples of commuting contractions. This is the norm considered by Agler in[1], which is sometimes called the Schur-Agler norm[21].Our Q(D N ) = {T = (T 1 , ..., T N ) : σ(T ) ⊆ D N and T i ≤ 1}. Note that if we replace such a T by rT = (rT 1 , ..., rT N ) then rT i < 1, rT ∈ Q R (D N ) and taking suprema over all T ∈ Q R (D N ) will be the same as taking a suprema over this smaller set. Thus, the algebra H ∞ R (D N ) consists of those analytic functions f such that f R = sup{ f (T 1 , ..., T N ) : T i < 1, i = 1, ..., N } < +∞. By Theorem 4.18 for f ∈ M m,n (H ∞ R (D N ), we have that f R ≤ 1 if and only if"},{"id":"","caption":"for some analytic-coanalytic positive definite functions, K i : D N × D N → M m . Example 5.2. Let G = B N denote the unit Euclidean ball in C N . If we let F 1 (z) = (z 1 , ..., z N ) : B − N → M 1,N , then this gives us a polynomial presentation. Again the algebra of the presentation is the polynomial algebra.","content":"for some analytic-coanalytic positive definite functions, K i : D N × D N → M m . Example 5.2. Let G = B N denote the unit Euclidean ball in C N . If we let F 1 (z) = (z 1 , ..., z N ) : B − N → M 1,N , then this gives us a polynomial presentation. Again the algebra of the presentation is the polynomial algebra."},{"id":"","caption":"m is an analytic-coanalytic positive definite function.Example 5.3. Let G = B N as above and let F 1 (z) = (z 1 , ..., z N ) t : B N → M N,1 .Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T N ) such that (T 1 , ..., T N ) t ≤ 1,","content":"m is an analytic-coanalytic positive definite function.Example 5.3. Let G = B N as above and let F 1 (z) = (z 1 , ..., z N ) t : B N → M N,1 .Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T N ) such that (T 1 , ..., T N ) t ≤ 1,"},{"id":"","caption":"Let G = D the open unit disk in the complex plane and let F 1 (z) = z 2 , F 2 (z) = z 3 . It is easy to check that the algebra A of this presentation is generated by the component functions and the constant function is the span of the monomials, {1, z n : n ≥ 2}, and that A separates the points of G. In this case an (strict)admissible representation, π : A → B(H), is given by any choice of a pair of commuting (strict)contractions, A = π(z 2 ), B = π(z 3 ), satisfying A 3 = B 2 . Again, it is easy to see that . u = . u 0 . On the other hand Q(D) = {T : σ(T ) ⊆ D and T 2 ≤ 1, T 3 ≤ 1}","content":"Example 5. 5 .\n5Let G = D the open unit disk in the complex plane and let F 1 (z) = z 2 , F 2 (z) = z 3 . It is easy to check that the algebra A of this presentation is generated by the component functions and the constant function is the span of the monomials, {1, z n : n ≥ 2}, and that A separates the points of G. In this case an (strict)admissible representation, π : A → B(H), is given by any choice of a pair of commuting (strict)contractions, A = π(z 2 ), B = π(z 3 ), satisfying A 3 = B 2 . Again, it is easy to see that . u = . u 0 . On the other hand Q(D) = {T : σ(T ) ⊆ D and T 2 ≤ 1, T 3 ≤ 1}"},{"id":"","caption":"Let G = {(z i,j ) ∈ M M,N : (z i,j ) < 1} and let F : G → M M,N be the identity map F (z) = (z i,j ). Then this is a polynomial presentation of G and the algebra of the presentation is the algebra of polynomials in the M N variables {z i,j }. An admissible representation of this algebra is given by any choice of M N commuting operators {T i,j } on a Hilbert space H, such that (T i,j ) ≤ 1 in M M,N (B(H)) and as above, one can show that · R is acheived by taking suprema over all commuting M N -tuples for which (T i,j ) < 1. We have that f ∈ M m,n (H ∞ R (G) and f R ≤ 1 if and only if","content":"Example 5 . 7 .\n57Let G = {(z i,j ) ∈ M M,N : (z i,j ) < 1} and let F : G → M M,N be the identity map F (z) = (z i,j ). Then this is a polynomial presentation of G and the algebra of the presentation is the algebra of polynomials in the M N variables {z i,j }. An admissible representation of this algebra is given by any choice of M N commuting operators {T i,j } on a Hilbert space H, such that (T i,j ) ≤ 1 in M M,N (B(H)) and as above, one can show that · R is acheived by taking suprema over all commuting M N -tuples for which (T i,j ) < 1. We have that f ∈ M m,n (H ∞ R (G) and f R ≤ 1 if and only if"},{"id":"","caption":"Let G be a simply connected domain in C and φ : G → D be a biholomorphic map. Then G = {z ∈ C : |φ(z)| < 1} and the Q(G) = {T : σ(T ) ⊆ G and φ(T ) ≤ 1} where R = {φ}. In this case the algebra A of the presentation is just the algebra of all polynomials in φ. Thus, an admissible representation of this algebra is defined by choosing an operator B ∈ B(H) that satisfies B ≤ 1 and defining π : A → B(H) via π(p(φ)) = p(B), where p is a polynomial. A strict admissible representation is defined similarly by first choosing a strict contraction. In this case, it is immediate that . u = . u 0 and that f ∈ H ∞ R (G) if and only if f ∈ Hol(G) andf R = sup{ f (T ) : T ∈ Q R (G)} < +∞.","content":"Example 5 . 9 .\n59Let G be a simply connected domain in C and φ : G → D be a biholomorphic map. Then G = {z ∈ C : |φ(z)| < 1} and the Q(G) = {T : σ(T ) ⊆ G and φ(T ) ≤ 1} where R = {φ}. In this case the algebra A of the presentation is just the algebra of all polynomials in φ. Thus, an admissible representation of this algebra is defined by choosing an operator B ∈ B(H) that satisfies B ≤ 1 and defining π : A → B(H) via π(p(φ)) = p(B), where p is a polynomial. A strict admissible representation is defined similarly by first choosing a strict contraction. In this case, it is immediate that . u = . u 0 and that f ∈ H ∞ R (G) if and only if f ∈ Hol(G) andf R = sup{ f (T ) : T ∈ Q R (G)} < +∞."},{"id":"","caption":"Similarly, if we let G = {z ∈ C N : |φ i (z)| < 1, i = 1, ..., N } where φ i (z) = z i −1 z i +1 , then G is an intersection of half planes and Q R consists of all commuting N -tuples of operators, (T 1 , ..., T N ) such that σ(T i ) ⊆ H and Re(T i ) ≥ 0 for all i. Applying our results, we obtain a factorization result for half planes. These algebras have been studied by D.Kalyuzhnyi-Verbovetzkii in[20].Example 5.11. Let G ⊆ C be an open convex set and represent it as an intersection of half planes H θ . Each half plane can be expressed as {z : |F θ (z)| < 1} for some family of linear fractional maps. If we let R = {F θ }, then Q R (G) = {T : σ(T ) ⊆ G and F θ (T ) ≤ 1 ∀θ}. Moreover, each inequality F θ (T ) ≤ 1 can be re-written as an operator inequality for the real part of some translate and rotation of T. For example, when G = D, we may take F θ (z) = z z−2e iθ , for 0 ≤ θ < 2π. In this case, one checks that F θ (T ) ≤ 1 if and only if Re(e iθ T ) ≤ I. Thus, it follows that Q(D) = {T : σ(T ) ⊆ D and w(T ) ≤ 1},","content":"Example 5 . 10 .\n510Similarly, if we let G = {z ∈ C N : |φ i (z)| < 1, i = 1, ..., N } where φ i (z) = z i −1 z i +1 , then G is an intersection of half planes and Q R consists of all commuting N -tuples of operators, (T 1 , ..., T N ) such that σ(T i ) ⊆ H and Re(T i ) ≥ 0 for all i. Applying our results, we obtain a factorization result for half planes. These algebras have been studied by D.Kalyuzhnyi-Verbovetzkii in[20].Example 5.11. Let G ⊆ C be an open convex set and represent it as an intersection of half planes H θ . Each half plane can be expressed as {z : |F θ (z)| < 1} for some family of linear fractional maps. If we let R = {F θ }, then Q R (G) = {T : σ(T ) ⊆ G and F θ (T ) ≤ 1 ∀θ}. Moreover, each inequality F θ (T ) ≤ 1 can be re-written as an operator inequality for the real part of some translate and rotation of T. For example, when G = D, we may take F θ (z) = z z−2e iθ , for 0 ≤ θ < 2π. In this case, one checks that F θ (T ) ≤ 1 if and only if Re(e iθ T ) ≤ I. Thus, it follows that Q(D) = {T : σ(T ) ⊆ D and w(T ) ≤ 1},"},{"id":"","caption":"In this case it is easily seen thatp u = sup{ q(A) + Br(A) : A ≤ 1, B ≤ 1, AB = BA, A 3 = B 2 }, while p L = p R = sup{ p(T ) : T 2 ≤ 1, T 3 ≤ 1}.","content":"In this case it is easily seen thatp u = sup{ q(A) + Br(A) : A ≤ 1, B ≤ 1, AB = BA, A 3 = B 2 }, while p L = p R = sup{ p(T ) : T 2 ≤ 1, T 3 ≤ 1}."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Let A be an operator algebra of functions on X, then A ⊆ ℓ ∞ (X), and for every n and every(f i,j ) ∈ M n (A), we have (f ij ) ∞ ≤ (f ij ) Mn(A) .\",\n \"content\": \"Proposition 1 . 2 .\\n12Let A be an operator algebra of functions on X, then A ⊆ ℓ ∞ (X), and for every n and every(f i,j ) ∈ M n (A), we have (f ij ) ∞ ≤ (f ij ) Mn(A) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \": M n (A) → M n (A/I F ) ∼ = M n (A)/M n (I F ).\",\n \"content\": \"F\\n: M n (A) → M n (A/I F ) ∼ = M n (A)/M n (I F ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"If A is an operator algebra of functions on the set X, thenà equipped with the norms of Definition 1.5 is a local operator algebra of functions on X. Moreover, for (f i,j ) ∈ M n (A), (f i,j ) Mn(Ã) = (f i,j ) Mn(A L ) .\",\n \"content\": \"Lemma 1. 8 .\\n8If A is an operator algebra of functions on the set X, thenà equipped with the norms of Definition 1.5 is a local operator algebra of functions on X. Moreover, for (f i,j ) ∈ M n (A), (f i,j ) Mn(Ã) = (f i,j ) Mn(A L ) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let A = A(D) ⊆ C(D − ) be the subalgebra of the algebra of continuous functions on the closed disk consisting of the functions that are analytic on the open disk D. Identifying M n (A(D)) ⊆ M n (C(D − )) as a subalgebra of the algebra of continuous functions from the closed disk to the matrices, equipped with the supremum norm, gives A(D) it's usual operator algebra structure. With this structure it can be regarded as a local operator algebra of functions on D or on D − . If we regard it as a local operator algebra of functions on D − , then A(D) A(D). To see that the containment is strict, note that f (z) = (1 + z)/2 ∈ A(D) and f n (z) → χ {1} , the characteristic function of the singleton {1}.\",\n \"content\": \"Example 1. 15 .\\n15Let A = A(D) ⊆ C(D − ) be the subalgebra of the algebra of continuous functions on the closed disk consisting of the functions that are analytic on the open disk D. Identifying M n (A(D)) ⊆ M n (C(D − )) as a subalgebra of the algebra of continuous functions from the closed disk to the matrices, equipped with the supremum norm, gives A(D) it's usual operator algebra structure. With this structure it can be regarded as a local operator algebra of functions on D or on D − . If we regard it as a local operator algebra of functions on D − , then A(D) A(D). To see that the containment is strict, note that f (z) = (1 + z)/2 ∈ A(D) and f n (z) → χ {1} , the characteristic function of the singleton {1}.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"However, if we regard A(D) as a local operator algebra of functions on D, then its BPW completion A(D) = H ∞ (D), the bounded analytic functions on the disk, with its usual operator structure.\",\n \"content\": \"However, if we regard A(D) as a local operator algebra of functions on D, then its BPW completion A(D) = H ∞ (D), the bounded analytic functions on the disk, with its usual operator structure.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"uniformly on compact subsets of D. Thus, by the identity theorem F ≡ G on D. Hence, A is a local operator algebra. Finally, by using Lemma 1.11 and a similar argument, one can also show thatà = H ∞ (D).\",\n \"content\": \"uniformly on compact subsets of D. Thus, by the identity theorem F ≡ G on D. Hence, A is a local operator algebra. Finally, by using Lemma 1.11 and a similar argument, one can also show thatà = H ∞ (D).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let A = H ∞ (D) but endowed with a new norm. Fix b > 1,\",\n \"content\": \"Example 1 . 17 .\\n117Let A = H ∞ (D) but endowed with a new norm. Fix b > 1,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H). Then M(K) ⊆ B(L) is a weak*-closed subalgebra.\",\n \"content\": \"Lemma 2 . 1 .\\n21Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X × X → B(H). Then M(K) ⊆ B(L) is a weak*-closed subalgebra.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X ×X → B(H) and let M(K) ⊆ B(L) denote the multiplier algebra, endowed with the operator algebra structure that it inherits as a subalgebra. If K(x, x) = 0, for every x ∈ X and M(K) separates points on X, then M(K) is a BPW complete local dual operator algebra of functions on X.\",\n \"content\": \"Theorem 2 . 2 .\\n22Let L be a reproducing kernel Hilbert space of H-valued functions with reproducing kernel K : X ×X → B(H) and let M(K) ⊆ B(L) denote the multiplier algebra, endowed with the operator algebra structure that it inherits as a subalgebra. If K(x, x) = 0, for every x ∈ X and M(K) separates points on X, then M(K) is a BPW complete local dual operator algebra of functions on X.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let A be an operator algebra of functions on the set X then there exists a Hilbert space, C and a positive definite function K : X × X → B(C) such that M(K) =à completely isometrically.\",\n \"content\": \"Theorem 2 . 4 .\\n24Let A be an operator algebra of functions on the set X then there exists a Hilbert space, C and a positive definite function K : X × X → B(C) such that M(K) =à completely isometrically.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let A be a local operator algebra of functions on the set X. Then the net (f λ ) λ ∈ Ball(A) converges in the weak*-topology if and only if it converges pointwise on X.\",\n \"content\": \"Proposition 2. 6 .\\n6Let A be a local operator algebra of functions on the set X. Then the net (f λ ) λ ∈ Ball(A) converges in the weak*-topology if and only if it converges pointwise on X.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"An operator algebra, B is called RFD if for every n and for every (\",\n \"content\": \"Definition 3. 1 .\\n1An operator algebra, B is called RFD if for every n and for every (\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let B be a concrete k-idempotent operator algebra. Then S(B * B)) is a Schur ideal affiliated with B, i.e., B = A(S(B * B)) completely isometrically.\",\n \"content\": \"Lemma 3. 2 .\\n2Let B be a concrete k-idempotent operator algebra. Then S(B * B)) is a Schur ideal affiliated with B, i.e., B = A(S(B * B)) completely isometrically.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Every k-idempotent operator algebra is weak*-RFD.\",\n \"content\": \"Theorem 3. 3 .\\n3Every k-idempotent operator algebra is weak*-RFD.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Every local operator algebra of functions is RFD.\",\n \"content\": \"Corollary 3. 5 .\\n5Every local operator algebra of functions is RFD.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G ⊆ C N be an open set. If there exists a set of matrixvalued functions, F k = (f k,i,j ) : G − → M m k\",\n \"content\": \"Definition 4 . 1 .\\n41Let G ⊆ C N be an open set. If there exists a set of matrixvalued functions, F k = (f k,i,j ) : G − → M m k\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G ⊆ C N be an analytically presented domain with presentation F = (f i,j ) : G → M m,n , and let H be a Hilbert space. A homomorphism π : A → B(H) of the algebra of the presentation is called an admissible representation provided that (π(f i,j )) ≤ 1 in M m,n (B(H) = B(H n ,\",\n \"content\": \"Definition 4 . 3 .\\n43Let G ⊆ C N be an analytically presented domain with presentation F = (f i,j ) : G → M m,n , and let H be a Hilbert space. A homomorphism π : A → B(H) of the algebra of the presentation is called an admissible representation provided that (π(f i,j )) ≤ 1 in M m,n (B(H) = B(H n ,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G be an analytically presented domain with presentationR = {F k = (f k,i,j ) : G → M m k ,n k , k ∈ I},let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A), where m, n are arbitrary. Then the following are equivalent:\",\n \"content\": \"Theorem 4. 5 .\\n5Let G be an analytically presented domain with presentationR = {F k = (f k,i,j ) : G → M m k ,n k , k ∈ I},let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A), where m, n are arbitrary. Then the following are equivalent:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G be presented by vector-valued functions, F k = (f k,j ) : G → M 1,n k , k ∈ I, let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A). Then the following are equivalent:\",\n \"content\": \"Corollary 4. 6 .\\n6Let G be presented by vector-valued functions, F k = (f k,j ) : G → M 1,n k , k ∈ I, let A be the algebra of the presentation and let P = (p ij ) ∈ M m,n (A). Then the following are equivalent:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}. We define the quantized version of G to be the collection of all commuting N-tuples of operators,\",\n \"content\": \"Definition 4 . 8 .\\n48Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}. We define the quantized version of G to be the collection of all commuting N-tuples of operators,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}.We shall call a function H : G × G → M m an R-limit, provided that H is the pointwise limit of a net of functions H λ : G × G → M m of the form given byTheorem 4.5(iii).\",\n \"content\": \"Definition 4. 12 .\\n12Let G ⊆ C N be an analytically presented domain, with presentation R = {F k : G − → M m k ,n k , k ∈ I}.We shall call a function H : G × G → M m an R-limit, provided that H is the pointwise limit of a net of functions H λ : G × G → M m of the form given byTheorem 4.5(iii).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G ⊆ C N be an analytically presented domain, with a separating presentation R = {F k : G − → M m k ,n k , k ∈ I}. Then the following are equivalent:\",\n \"content\": \"Corollary 4. 13 .\\n13Let G ⊆ C N be an analytically presented domain, with a separating presentation R = {F k : G − → M m k ,n k , k ∈ I}. Then the following are equivalent:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G be an analytically presented domain with a finite presentation R= {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}.For each compact subset S ⊆ G, there exists a constant C, depending only on S, such that given a factorization of the form,\",\n \"content\": \"Proposition 4 . 14 .\\n414Let G be an analytically presented domain with a finite presentation R= {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}.For each compact subset S ⊆ G, there exists a constant C, depending only on S, such that given a factorization of the form,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The proof of the following result is essentially contained in[10, Lemma 3.3].\",\n \"content\": \"The proof of the following result is essentially contained in[10, Lemma 3.3].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G be a bounded domain in C N and let F = (f i,j ) : G → M m,n be analytic with F (z) < 1 for z ∈ G. If H : G × G → M p is analytic in the first variables, coanalytic in the second variables and there exists a net of matrix-valued functions P λ ∈ M p,r λ (Hol(G)) which are uniformly bounded on compact subsets of G, such that H(z, w) is the pointwise limit of H λ (z, w) = P λ (z)(I m − F (z)F (w) * ) (q λ ) P λ (w) * where r λ = q λ m k , then there exists a Hilbert space H and an analytic function, R : G → B(H ⊗ C M , C p ) such that H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * .\",\n \"content\": \"Proposition 4 . 15 .\\n415Let G be a bounded domain in C N and let F = (f i,j ) : G → M m,n be analytic with F (z) < 1 for z ∈ G. If H : G × G → M p is analytic in the first variables, coanalytic in the second variables and there exists a net of matrix-valued functions P λ ∈ M p,r λ (Hol(G)) which are uniformly bounded on compact subsets of G, such that H(z, w) is the pointwise limit of H λ (z, w) = P λ (z)(I m − F (z)F (w) * ) (q λ ) P λ (w) * where r λ = q λ m k , then there exists a Hilbert space H and an analytic function, R : G → B(H ⊗ C M , C p ) such that H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Conversely, any function that can be written in the formH(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * can beexpressed as a limit of a net as above by considering the directed set of all finite dimensional subspaces of H and for each finite dimensional subspace setting H F (z, w) = R F (z)[(I m − F (z)F (w) * ) ⊗ I F ]R F (w) * , where R F (z) = R(z)(P F ⊗ I m ) with P F the orthogonal projection onto F.\",\n \"content\": \"Remark 4 . 16 .\\n416Conversely, any function that can be written in the formH(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ I H ]R(w) * can beexpressed as a limit of a net as above by considering the directed set of all finite dimensional subspaces of H and for each finite dimensional subspace setting H F (z, w) = R F (z)[(I m − F (z)F (w) * ) ⊗ I F ]R F (w) * , where R F (z) = R(z)(P F ⊗ I m ) with P F the orthogonal projection onto F.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"We shall refer to a function H : G × G → M m (M p ) that can be expressed as H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ H]R(w) * for some Hilbert space H and some analytic function R : G → B(H ⊗ C m , C p ), as an F-limit. Theorem 4.18. Let G be an analytically presented domain with a finite separating presentation R = {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}, let f = (f ij ) be a M m,n -valued function defined on G. Then the following are equivalent:\",\n \"content\": \"Definition 4 . 17 .\\n417We shall refer to a function H : G × G → M m (M p ) that can be expressed as H(z, w) = R(z)[(I m − F (z)F (w) * ) ⊗ H]R(w) * for some Hilbert space H and some analytic function R : G → B(H ⊗ C m , C p ), as an F-limit. Theorem 4.18. Let G be an analytically presented domain with a finite separating presentation R = {F k = (f k,i,j ) : G → M m k ,n k , 1 ≤ k ≤ K}, let f = (f ij ) be a M m,n -valued function defined on G. Then the following are equivalent:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G = D N , be the polydisk which has a presentation given by the coordinate functions F i (z) = z i , 1 ≤ i ≤ N. Then the algebra of this presentation is the algebra of polynomials and an admissible representation is given by any choice of N commuting contractions, (T 1 , ..., T N ) on a Hilbert space. Given a matrix of polynomials,(p i,j ) u = sup (p i,j (T 1 , ..., T N ))where the supremum is taken over all N-tuples of commuting contractions. This is the norm considered by Agler in[1], which is sometimes called the Schur-Agler norm[21].Our Q(D N ) = {T = (T 1 , ..., T N ) : σ(T ) ⊆ D N and T i ≤ 1}. Note that if we replace such a T by rT = (rT 1 , ..., rT N ) then rT i < 1, rT ∈ Q R (D N ) and taking suprema over all T ∈ Q R (D N ) will be the same as taking a suprema over this smaller set. Thus, the algebra H ∞ R (D N ) consists of those analytic functions f such that f R = sup{ f (T 1 , ..., T N ) : T i < 1, i = 1, ..., N } < +∞. By Theorem 4.18 for f ∈ M m,n (H ∞ R (D N ), we have that f R ≤ 1 if and only if\",\n \"content\": \"Example 5 . 1 .\\n51Let G = D N , be the polydisk which has a presentation given by the coordinate functions F i (z) = z i , 1 ≤ i ≤ N. Then the algebra of this presentation is the algebra of polynomials and an admissible representation is given by any choice of N commuting contractions, (T 1 , ..., T N ) on a Hilbert space. Given a matrix of polynomials,(p i,j ) u = sup (p i,j (T 1 , ..., T N ))where the supremum is taken over all N-tuples of commuting contractions. This is the norm considered by Agler in[1], which is sometimes called the Schur-Agler norm[21].Our Q(D N ) = {T = (T 1 , ..., T N ) : σ(T ) ⊆ D N and T i ≤ 1}. Note that if we replace such a T by rT = (rT 1 , ..., rT N ) then rT i < 1, rT ∈ Q R (D N ) and taking suprema over all T ∈ Q R (D N ) will be the same as taking a suprema over this smaller set. Thus, the algebra H ∞ R (D N ) consists of those analytic functions f such that f R = sup{ f (T 1 , ..., T N ) : T i < 1, i = 1, ..., N } < +∞. By Theorem 4.18 for f ∈ M m,n (H ∞ R (D N ), we have that f R ≤ 1 if and only if\"\n },\n {\n \"id\": \"\",\n \"caption\": \"for some analytic-coanalytic positive definite functions, K i : D N × D N → M m . Example 5.2. Let G = B N denote the unit Euclidean ball in C N . If we let F 1 (z) = (z 1 , ..., z N ) : B − N → M 1,N , then this gives us a polynomial presentation. Again the algebra of the presentation is the polynomial algebra.\",\n \"content\": \"for some analytic-coanalytic positive definite functions, K i : D N × D N → M m . Example 5.2. Let G = B N denote the unit Euclidean ball in C N . If we let F 1 (z) = (z 1 , ..., z N ) : B − N → M 1,N , then this gives us a polynomial presentation. Again the algebra of the presentation is the polynomial algebra.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"m is an analytic-coanalytic positive definite function.Example 5.3. Let G = B N as above and let F 1 (z) = (z 1 , ..., z N ) t : B N → M N,1 .Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T N ) such that (T 1 , ..., T N ) t ≤ 1,\",\n \"content\": \"m is an analytic-coanalytic positive definite function.Example 5.3. Let G = B N as above and let F 1 (z) = (z 1 , ..., z N ) t : B N → M N,1 .Again this is a rational presentation of G and the algebra of the presentation is the polynomials. An admissible representation corresponds to an N-tuple of commuting operators (T 1 , ..., T N ) such that (T 1 , ..., T N ) t ≤ 1,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G = D the open unit disk in the complex plane and let F 1 (z) = z 2 , F 2 (z) = z 3 . It is easy to check that the algebra A of this presentation is generated by the component functions and the constant function is the span of the monomials, {1, z n : n ≥ 2}, and that A separates the points of G. In this case an (strict)admissible representation, π : A → B(H), is given by any choice of a pair of commuting (strict)contractions, A = π(z 2 ), B = π(z 3 ), satisfying A 3 = B 2 . Again, it is easy to see that . u = . u 0 . On the other hand Q(D) = {T : σ(T ) ⊆ D and T 2 ≤ 1, T 3 ≤ 1}\",\n \"content\": \"Example 5. 5 .\\n5Let G = D the open unit disk in the complex plane and let F 1 (z) = z 2 , F 2 (z) = z 3 . It is easy to check that the algebra A of this presentation is generated by the component functions and the constant function is the span of the monomials, {1, z n : n ≥ 2}, and that A separates the points of G. In this case an (strict)admissible representation, π : A → B(H), is given by any choice of a pair of commuting (strict)contractions, A = π(z 2 ), B = π(z 3 ), satisfying A 3 = B 2 . Again, it is easy to see that . u = . u 0 . On the other hand Q(D) = {T : σ(T ) ⊆ D and T 2 ≤ 1, T 3 ≤ 1}\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G = {(z i,j ) ∈ M M,N : (z i,j ) < 1} and let F : G → M M,N be the identity map F (z) = (z i,j ). Then this is a polynomial presentation of G and the algebra of the presentation is the algebra of polynomials in the M N variables {z i,j }. An admissible representation of this algebra is given by any choice of M N commuting operators {T i,j } on a Hilbert space H, such that (T i,j ) ≤ 1 in M M,N (B(H)) and as above, one can show that · R is acheived by taking suprema over all commuting M N -tuples for which (T i,j ) < 1. We have that f ∈ M m,n (H ∞ R (G) and f R ≤ 1 if and only if\",\n \"content\": \"Example 5 . 7 .\\n57Let G = {(z i,j ) ∈ M M,N : (z i,j ) < 1} and let F : G → M M,N be the identity map F (z) = (z i,j ). Then this is a polynomial presentation of G and the algebra of the presentation is the algebra of polynomials in the M N variables {z i,j }. An admissible representation of this algebra is given by any choice of M N commuting operators {T i,j } on a Hilbert space H, such that (T i,j ) ≤ 1 in M M,N (B(H)) and as above, one can show that · R is acheived by taking suprema over all commuting M N -tuples for which (T i,j ) < 1. We have that f ∈ M m,n (H ∞ R (G) and f R ≤ 1 if and only if\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let G be a simply connected domain in C and φ : G → D be a biholomorphic map. Then G = {z ∈ C : |φ(z)| < 1} and the Q(G) = {T : σ(T ) ⊆ G and φ(T ) ≤ 1} where R = {φ}. In this case the algebra A of the presentation is just the algebra of all polynomials in φ. Thus, an admissible representation of this algebra is defined by choosing an operator B ∈ B(H) that satisfies B ≤ 1 and defining π : A → B(H) via π(p(φ)) = p(B), where p is a polynomial. A strict admissible representation is defined similarly by first choosing a strict contraction. In this case, it is immediate that . u = . u 0 and that f ∈ H ∞ R (G) if and only if f ∈ Hol(G) andf R = sup{ f (T ) : T ∈ Q R (G)} < +∞.\",\n \"content\": \"Example 5 . 9 .\\n59Let G be a simply connected domain in C and φ : G → D be a biholomorphic map. Then G = {z ∈ C : |φ(z)| < 1} and the Q(G) = {T : σ(T ) ⊆ G and φ(T ) ≤ 1} where R = {φ}. In this case the algebra A of the presentation is just the algebra of all polynomials in φ. Thus, an admissible representation of this algebra is defined by choosing an operator B ∈ B(H) that satisfies B ≤ 1 and defining π : A → B(H) via π(p(φ)) = p(B), where p is a polynomial. A strict admissible representation is defined similarly by first choosing a strict contraction. In this case, it is immediate that . u = . u 0 and that f ∈ H ∞ R (G) if and only if f ∈ Hol(G) andf R = sup{ f (T ) : T ∈ Q R (G)} < +∞.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Similarly, if we let G = {z ∈ C N : |φ i (z)| < 1, i = 1, ..., N } where φ i (z) = z i −1 z i +1 , then G is an intersection of half planes and Q R consists of all commuting N -tuples of operators, (T 1 , ..., T N ) such that σ(T i ) ⊆ H and Re(T i ) ≥ 0 for all i. Applying our results, we obtain a factorization result for half planes. These algebras have been studied by D.Kalyuzhnyi-Verbovetzkii in[20].Example 5.11. Let G ⊆ C be an open convex set and represent it as an intersection of half planes H θ . Each half plane can be expressed as {z : |F θ (z)| < 1} for some family of linear fractional maps. If we let R = {F θ }, then Q R (G) = {T : σ(T ) ⊆ G and F θ (T ) ≤ 1 ∀θ}. Moreover, each inequality F θ (T ) ≤ 1 can be re-written as an operator inequality for the real part of some translate and rotation of T. For example, when G = D, we may take F θ (z) = z z−2e iθ , for 0 ≤ θ < 2π. In this case, one checks that F θ (T ) ≤ 1 if and only if Re(e iθ T ) ≤ I. Thus, it follows that Q(D) = {T : σ(T ) ⊆ D and w(T ) ≤ 1},\",\n \"content\": \"Example 5 . 10 .\\n510Similarly, if we let G = {z ∈ C N : |φ i (z)| < 1, i = 1, ..., N } where φ i (z) = z i −1 z i +1 , then G is an intersection of half planes and Q R consists of all commuting N -tuples of operators, (T 1 , ..., T N ) such that σ(T i ) ⊆ H and Re(T i ) ≥ 0 for all i. Applying our results, we obtain a factorization result for half planes. These algebras have been studied by D.Kalyuzhnyi-Verbovetzkii in[20].Example 5.11. Let G ⊆ C be an open convex set and represent it as an intersection of half planes H θ . Each half plane can be expressed as {z : |F θ (z)| < 1} for some family of linear fractional maps. If we let R = {F θ }, then Q R (G) = {T : σ(T ) ⊆ G and F θ (T ) ≤ 1 ∀θ}. Moreover, each inequality F θ (T ) ≤ 1 can be re-written as an operator inequality for the real part of some translate and rotation of T. For example, when G = D, we may take F θ (z) = z z−2e iθ , for 0 ≤ θ < 2π. In this case, one checks that F θ (T ) ≤ 1 if and only if Re(e iθ T ) ≤ I. Thus, it follows that Q(D) = {T : σ(T ) ⊆ D and w(T ) ≤ 1},\"\n },\n {\n \"id\": \"\",\n \"caption\": \"In this case it is easily seen thatp u = sup{ q(A) + Br(A) : A ≤ 1, B ≤ 1, AB = BA, A 3 = B 2 }, while p L = p R = sup{ p(T ) : T 2 ≤ 1, T 3 ≤ 1}.\",\n \"content\": \"In this case it is easily seen thatp u = sup{ q(A) + Br(A) : A ≤ 1, B ≤ 1, AB = BA, A 3 = B 2 }, while p L = p R = sup{ p(T ) : T 2 ≤ 1, T 3 ≤ 1}.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Some interpolation theorems of Nevanlinna-Pick type. J Agler, PreprintJ. Agler, Some interpolation theorems of Nevanlinna-Pick type, Preprint, 1988.","doi":""},{"text":"On the Representation of Certain Holomorphic Function defined on a Polydisk. J Agler, Topics in Operator Theory: Ernst D. Hellinger Memorial Volume. J. Agler, On the Representation of Certain Holomorphic Function defined on a Poly- disk, Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, 48 (1990),47- 66.","doi":""},{"text":"D Alpay, A Dijksma, J Rovnyak, H De Snoo, Schur Functions, Operator Colligation, and Reproducing Kernel Pontryagin Spaces, Operator Theory Advances and Applications. BaselBirkhauser96D. Alpay, A. Dijksma, J. Rovnyak and H. de Snoo, Schur Functions, Operator Colli- gation, and Reproducing Kernel Pontryagin Spaces, Operator Theory Advances and Applications Vol. 96, Birkhauser,Basel, pp. 1-13.","doi":""},{"text":"A von Neumann type inequality for certain domains in C n. C G Ambrozie, D Timotin, Proc. Amer. Math. Soc. 131C.G. Ambrozie and D. Timotin, A von Neumann type inequality for certain domains in C n , Proc. Amer. Math. Soc., 131(2003), 859-869.","doi":""},{"text":"On residually finite-dimensional C*-algebras. R J Archbold, Proc. Amer. Math. Soc. 123946089MRR. J. Archbold, On residually finite-dimensional C*-algebras, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2935-2937. MR 95m:46089","doi":""},{"text":"Subalgebras of C * -algebras. III. Multivariable operator theory. W Arveson, Acta Math. 1812W. Arveson, Subalgebras of C * -algebras. III. Multivariable operator theory, Acta Math. 181(1998), no. 2, 159-228.","doi":""},{"text":"Realization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining functions in C n. J A Ball, V Bolotnikov, Jour. Func. Anal. J.A. Ball and V. Bolotnikov, Realization and interpolation for Schur-Agler-class func- tions on domains with matrix polynomial defining functions in C n , Jour. Func. Anal.,","doi":""},{"text":"Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables. J A Ball, T T Trent, J. Funct. Anal. 197J.A. Ball and T.T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 197 (1998), 1-61.","doi":""},{"text":"A lenticular version of a von Neumann inequality. B Beckermann, M Crouzeix, Archiv der Mathematik. 86B. Beckermann, M. Crouzeix : A lenticular version of a von Neumann inequality. Archiv der Mathematik, vol. 86, 2006, p. 352-355.","doi":""},{"text":"Blackadar operator algebra. B , B. Blackadar operator algebra.","doi":""},{"text":"Operator Algebras and Their Modules-an Operator Space Approach. D P Blecher, C Le Merdy, Oxford University Press30D.P. Blecher and C. Le Merdy, Operator Algebras and Their Modules-an Operator Space Approach, London Mathematical Society Monographs, Vol 30, Oxford Univer- sity Press, 2004.","doi":""},{"text":"Explicit construction of universal operator algebras and an application to polynomial factorization. D P Blecher, V I Paulsen, Proc. Amer. Math. Soc. 112D.P. Blecher and V.I. Paulsen, Explicit construction of universal operator algebras and an application to polynomial factorization, Proc. Amer. Math. Soc. 112(1991), 839-850.","doi":""},{"text":"A characterization of operator algebras. D P Blecher, Z.-J Ruan, A M Sinclair, J. Funct. Anal. 89D.P. Blecher, Z.-J. Ruan and A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89(1990), 188-201.","doi":""},{"text":"Banach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy. J Burbea, P Masani, Research Notes in Mathematics. 90Pitman Advanced Publishing ProgramJ. Burbea and P. Masani, Banach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy, Research Notes in Mathematics 90, Boston-London-Melbourne, Pitman Advanced Publishing Program (1984)","doi":""},{"text":"Residually finite dimensional C* algebras. M Dadarlat, Operator algebras and operator theory: International Conference on Operator Algebras and Operator Theory ,By Liming Ge. Shanghai, ChinaM. Dadarlat, Residually finite dimensional C* algebras ,45-50, in Operator algebras and operator theory: International Conference on Operator Algebras and Operator Theory ,By Liming Ge, July 4-9, 1997, Shanghai, China.","doi":""},{"text":"Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras. K R Davidson, D Pitts, Integral Equations Operator Theory. 313K.R. Davidson and D. Pitts, Nevanlinna-Pick interpolation for non-commutative ana- lytic Toeplitz algebras, Integral Equations Operator Theory 31(1998), no. 3, 321-337.","doi":""},{"text":"Completely bounded maps and hypo-Dirichlet algebras. R G Douglas, V I Paulsen, Acta Sci. Math. (Szeged). 501-2R.G. Douglas and V.I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged) 50(1986), no. 1-2, 143-157.","doi":""},{"text":"A generalization of von Neumann's inequality to the complex ball. S W Drury, Proc. Amer. Math. Soc. 683S.W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc. 68(1978), no. 3, 300-304.","doi":""},{"text":"Free and residually finite-dimensional C*-algebras. K R Goodearl, P , J. Funct. Anal. 90246078MRK. R. Goodearl and P. Menal, Free and residually finite-dimensional C*-algebras, J. Funct. Anal. 90 (1990), no. 2, 391-410. MR 91f:46078","doi":""},{"text":"On the Bessmertnyi class of homogeneous positive holomorphic functions of several variables. D Kalyuzhnyi-Verbovetzkii, J.A. Ball, J.W. Helton, M. Klaus, LD. Kalyuzhnyi-Verbovetzkii, On the Bessmertnyi class of homogeneous positive holo- morphic functions of several variables, in: J.A. Ball, J.W. Helton, M. Klaus, L.","doi":""},{"text":"Rodman, Current trends in Operator Theory and its Applications. BaselBirkhauser-Verlag149Rodman (Eds.) Current trends in Operator Theory and its Applications, Vol. OT 149, Birkhauser-Verlag, Basel, 2004, 255-289.","doi":""},{"text":"An operator algebraic proof of Agler's factorization theorem. S Lata, M Mittal, V I Paulsen, Proc. Amer. Math. Soc. to appearS. Lata, M. Mittal, V.I. Paulsen, An operator algebraic proof of Agler's factorization theorem, Proc. Amer. Math. Soc., to appear.","doi":""},{"text":". M , University of HoustonPhD ThesisM. Mittal, PhD Thesis, University of Houston.","doi":""},{"text":"Completely Bounded Maps and Operator Algebras. V I Paulsen, Cambridge studies in Advanced Mathematics. 78V.I. Paulsen , Completely Bounded Maps and Operator Algebras, Cambridge studies in Advanced Mathematics, 78 (2002).","doi":""},{"text":"Operator Algebra of Idempotents. V I Paulsen, J. Funct. Anal. 181V.I. Paulsen, Operator Algebra of Idempotents, J. Funct. Anal. 181, 209-226 (2001).","doi":""},{"text":"V I Paulsen, M Raghupathi, Representations of Logmodular Algebras, Proc. of the AMS. to appearV.I. Paulsen and M. Raghupathi, Representations of Logmodular Algebras, Proc. of the AMS, to appear.","doi":""},{"text":"Theory of Reproducing kernels in Hilbert spaces of vector-valued functions. G B Pedrick, Univ.of Kansas Tech. Rep. 19G.B. Pedrick, Theory of Reproducing kernels in Hilbert spaces of vector-valued func- tions, Univ.of Kansas Tech. Rep. 19, Lawrence, 1957.","doi":""},{"text":"Introduction to Operator Space Theory. G Pisier, Lecture note series. Cambridge University Press294G. Pisier, Introduction to Operator Space Theory, London Mathematical Society Lec- ture note series 294, Cambridge University Press, (2003).","doi":""},{"text":"Interpolation problems in several variables. (English summary). G Popescu, J. Math. Anal. Appl. 2271G. Popescu, Interpolation problems in several variables. (English summary) J. Math. Anal. Appl. 227 (1998), no. 1, 227-250.","doi":""},{"text":"A joint spectrum for several commuting operators. J L Taylor, J. Funct. Anal. 6J.L. Taylor, A joint spectrum for several commuting operators J. Funct. Anal. 6 (1970), 172-191.","doi":""},{"text":"The analytic functional calculus for several commuting contractions. J L Taylor, Acta Math. 125J.L. Taylor, The analytic functional calculus for several commuting contractions, Acta Math. 125 (1970), 1-38.","doi":""},{"text":"Analytic Functional Calculus and Spectral Decompositions. J L Taylor, Math. and its Appl., East European Series. 1D. Reidel Publishing CoDepartment of Mathematics, University of Houston ; Department of Mathematics, University of HoustonE-mail address: mittal@math.uh. A. E-mail address: vern@math.uh.eduJ.L. Taylor, Analytic Functional Calculus and Spectral Decompositions, Math. and its Appl., East European Series, vol. 1, D. Reidel Publishing Co. Dordrecht, Holland, 1982. Department of Mathematics, University of Houston, Houston, Texas 77204- 3476, U.S.A. E-mail address: mittal@math.uh.edu Department of Mathematics, University of Houston, Houston, Texas 77204- 3476, U.S.A. E-mail address: vern@math.uh.edu","doi":""}],"string":"[\n {\n \"text\": \"Some interpolation theorems of Nevanlinna-Pick type. J Agler, PreprintJ. Agler, Some interpolation theorems of Nevanlinna-Pick type, Preprint, 1988.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Representation of Certain Holomorphic Function defined on a Polydisk. J Agler, Topics in Operator Theory: Ernst D. Hellinger Memorial Volume. J. Agler, On the Representation of Certain Holomorphic Function defined on a Poly- disk, Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, 48 (1990),47- 66.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Alpay, A Dijksma, J Rovnyak, H De Snoo, Schur Functions, Operator Colligation, and Reproducing Kernel Pontryagin Spaces, Operator Theory Advances and Applications. BaselBirkhauser96D. Alpay, A. Dijksma, J. Rovnyak and H. de Snoo, Schur Functions, Operator Colli- gation, and Reproducing Kernel Pontryagin Spaces, Operator Theory Advances and Applications Vol. 96, Birkhauser,Basel, pp. 1-13.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A von Neumann type inequality for certain domains in C n. C G Ambrozie, D Timotin, Proc. Amer. Math. Soc. 131C.G. Ambrozie and D. Timotin, A von Neumann type inequality for certain domains in C n , Proc. Amer. Math. Soc., 131(2003), 859-869.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On residually finite-dimensional C*-algebras. R J Archbold, Proc. Amer. Math. Soc. 123946089MRR. J. Archbold, On residually finite-dimensional C*-algebras, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2935-2937. MR 95m:46089\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Subalgebras of C * -algebras. III. Multivariable operator theory. W Arveson, Acta Math. 1812W. Arveson, Subalgebras of C * -algebras. III. Multivariable operator theory, Acta Math. 181(1998), no. 2, 159-228.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Realization and interpolation for Schur-Agler-class functions on domains with matrix polynomial defining functions in C n. J A Ball, V Bolotnikov, Jour. Func. Anal. J.A. Ball and V. Bolotnikov, Realization and interpolation for Schur-Agler-class func- tions on domains with matrix polynomial defining functions in C n , Jour. Func. Anal.,\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables. J A Ball, T T Trent, J. Funct. Anal. 197J.A. Ball and T.T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 197 (1998), 1-61.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A lenticular version of a von Neumann inequality. B Beckermann, M Crouzeix, Archiv der Mathematik. 86B. Beckermann, M. Crouzeix : A lenticular version of a von Neumann inequality. Archiv der Mathematik, vol. 86, 2006, p. 352-355.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Blackadar operator algebra. B , B. Blackadar operator algebra.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Operator Algebras and Their Modules-an Operator Space Approach. D P Blecher, C Le Merdy, Oxford University Press30D.P. Blecher and C. Le Merdy, Operator Algebras and Their Modules-an Operator Space Approach, London Mathematical Society Monographs, Vol 30, Oxford Univer- sity Press, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Explicit construction of universal operator algebras and an application to polynomial factorization. D P Blecher, V I Paulsen, Proc. Amer. Math. Soc. 112D.P. Blecher and V.I. Paulsen, Explicit construction of universal operator algebras and an application to polynomial factorization, Proc. Amer. Math. Soc. 112(1991), 839-850.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A characterization of operator algebras. D P Blecher, Z.-J Ruan, A M Sinclair, J. Funct. Anal. 89D.P. Blecher, Z.-J. Ruan and A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89(1990), 188-201.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Banach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy. J Burbea, P Masani, Research Notes in Mathematics. 90Pitman Advanced Publishing ProgramJ. Burbea and P. Masani, Banach and Hilbert spaces of vector-valued functions. Their general theory and applications to holomorphy, Research Notes in Mathematics 90, Boston-London-Melbourne, Pitman Advanced Publishing Program (1984)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Residually finite dimensional C* algebras. M Dadarlat, Operator algebras and operator theory: International Conference on Operator Algebras and Operator Theory ,By Liming Ge. Shanghai, ChinaM. Dadarlat, Residually finite dimensional C* algebras ,45-50, in Operator algebras and operator theory: International Conference on Operator Algebras and Operator Theory ,By Liming Ge, July 4-9, 1997, Shanghai, China.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras. K R Davidson, D Pitts, Integral Equations Operator Theory. 313K.R. Davidson and D. Pitts, Nevanlinna-Pick interpolation for non-commutative ana- lytic Toeplitz algebras, Integral Equations Operator Theory 31(1998), no. 3, 321-337.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Completely bounded maps and hypo-Dirichlet algebras. R G Douglas, V I Paulsen, Acta Sci. Math. (Szeged). 501-2R.G. Douglas and V.I. Paulsen, Completely bounded maps and hypo-Dirichlet algebras, Acta Sci. Math. (Szeged) 50(1986), no. 1-2, 143-157.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A generalization of von Neumann's inequality to the complex ball. S W Drury, Proc. Amer. Math. Soc. 683S.W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc. 68(1978), no. 3, 300-304.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Free and residually finite-dimensional C*-algebras. K R Goodearl, P , J. Funct. Anal. 90246078MRK. R. Goodearl and P. Menal, Free and residually finite-dimensional C*-algebras, J. Funct. Anal. 90 (1990), no. 2, 391-410. MR 91f:46078\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Bessmertnyi class of homogeneous positive holomorphic functions of several variables. D Kalyuzhnyi-Verbovetzkii, J.A. Ball, J.W. Helton, M. Klaus, LD. Kalyuzhnyi-Verbovetzkii, On the Bessmertnyi class of homogeneous positive holo- morphic functions of several variables, in: J.A. Ball, J.W. Helton, M. Klaus, L.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rodman, Current trends in Operator Theory and its Applications. BaselBirkhauser-Verlag149Rodman (Eds.) Current trends in Operator Theory and its Applications, Vol. OT 149, Birkhauser-Verlag, Basel, 2004, 255-289.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An operator algebraic proof of Agler's factorization theorem. S Lata, M Mittal, V I Paulsen, Proc. Amer. Math. Soc. to appearS. Lata, M. Mittal, V.I. Paulsen, An operator algebraic proof of Agler's factorization theorem, Proc. Amer. Math. Soc., to appear.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M , University of HoustonPhD ThesisM. Mittal, PhD Thesis, University of Houston.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Completely Bounded Maps and Operator Algebras. V I Paulsen, Cambridge studies in Advanced Mathematics. 78V.I. Paulsen , Completely Bounded Maps and Operator Algebras, Cambridge studies in Advanced Mathematics, 78 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Operator Algebra of Idempotents. V I Paulsen, J. Funct. Anal. 181V.I. Paulsen, Operator Algebra of Idempotents, J. Funct. Anal. 181, 209-226 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V I Paulsen, M Raghupathi, Representations of Logmodular Algebras, Proc. of the AMS. to appearV.I. Paulsen and M. Raghupathi, Representations of Logmodular Algebras, Proc. of the AMS, to appear.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Theory of Reproducing kernels in Hilbert spaces of vector-valued functions. G B Pedrick, Univ.of Kansas Tech. Rep. 19G.B. Pedrick, Theory of Reproducing kernels in Hilbert spaces of vector-valued func- tions, Univ.of Kansas Tech. Rep. 19, Lawrence, 1957.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to Operator Space Theory. G Pisier, Lecture note series. Cambridge University Press294G. Pisier, Introduction to Operator Space Theory, London Mathematical Society Lec- ture note series 294, Cambridge University Press, (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interpolation problems in several variables. (English summary). G Popescu, J. Math. Anal. Appl. 2271G. Popescu, Interpolation problems in several variables. (English summary) J. Math. Anal. Appl. 227 (1998), no. 1, 227-250.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A joint spectrum for several commuting operators. J L Taylor, J. Funct. Anal. 6J.L. Taylor, A joint spectrum for several commuting operators J. Funct. Anal. 6 (1970), 172-191.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The analytic functional calculus for several commuting contractions. J L Taylor, Acta Math. 125J.L. Taylor, The analytic functional calculus for several commuting contractions, Acta Math. 125 (1970), 1-38.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Analytic Functional Calculus and Spectral Decompositions. J L Taylor, Math. and its Appl., East European Series. 1D. Reidel Publishing CoDepartment of Mathematics, University of Houston ; Department of Mathematics, University of HoustonE-mail address: mittal@math.uh. A. E-mail address: vern@math.uh.eduJ.L. Taylor, Analytic Functional Calculus and Spectral Decompositions, Math. and its Appl., East European Series, vol. 1, D. Reidel Publishing Co. Dordrecht, Holland, 1982. Department of Mathematics, University of Houston, Houston, Texas 77204- 3476, U.S.A. E-mail address: mittal@math.uh.edu Department of Mathematics, University of Houston, Houston, Texas 77204- 3476, U.S.A. E-mail address: vern@math.uh.edu\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":32,"cells":{"text":{"kind":"string","value":"\nA new and homogeneous metallicity scale for Galactic classical Cepheids II. The abundance of iron and α elements\nFebruary 17, 2022\n\nR Da Silva \nJ Crestani \nG Bono \nV F Braga \nV D'orazi \nB Lemasle \nM Bergemann \nM Dall'ora \nG Fiorentino \nP François \nM A T Groenewegen \nL Inno \nV Kovtyukh \nR.-P Kudritzki \nN Matsunaga \nM Monelli \nA Pietrinferni \nL Porcelli \nJ Storm \nM Tantalo \nF Thévénin \nA new and homogeneous metallicity scale for Galactic classical Cepheids II. The abundance of iron and α elements\nFebruary 17, 2022Astronomy & Astrophysics manuscript no. daSilvaetal2021 (Affiliations can be found after the references) Received ...; accepted ...Galaxy: disk -stars: abundances -stars: fundamental parameters -stars: variables: Cepheids\nContext. Classical Cepheids are the most popular distance indicators and tracers of young stellar populations. The key advantage is that they are bright and they can be easily identified in Local Group and Local Volume galaxies. Their evolutionary and pulsation properties depend on their chemical abundances. Aims. The main aim of this investigation is to perform a new and accurate abundance analysis of two tens of calibrating Galactic Cepheids using high spectral resolution (R∼40,000-115,000) and high S/N spectra (∼400) covering the entire pulsation cycle. Methods. We focus our attention on possible systematics affecting the estimate of atmospheric parameters and elemental abundances along the pulsation cycle. We cleaned the line list by using atomic transition parameters based on laboratory measurements and by removing lines that are either blended or display abundance variations along the pulsation cycle. Results. The spectroscopic approach that we developed brings forward small dispersions in the variation of the atmospheric parameters (σ(T eff )∼50 K, σ(log g)∼0.2 dex, and σ(ξ)∼0.2 km s −1 ) and in the abundance of both iron ( 0.05 dex) and α elements ( 0.10 dex) over the entire pulsation cycle. We also provide new and accurate effective temperature templates by splitting the calibrating Cepheids into four different period bins, ranging from short to long periods. For each period bin, we performed an analytical fit with Fourier series providing θ = 5040 /T eff as a function of the pulsation phase. Conclusions. The current findings are a good viaticum to trace the chemical enrichment of the Galactic thin disk by using classical Cepheids and a fundamental stepping stone for further investigations into the more metal-poor regime typical of Magellanic Cepheids.\n\nIntroduction\n\nThe current empirical evidence indicates that the estimates of the Hubble constant based on Cosmic Microwave Background (CMB) and on the standard cosmological model (Λ Cold Dark Matter, ΛCDM) are at odds with the direct estimates based on primary (classical Cepheids) and secondary (type Ia Supernovae) distance indicators (Planck Collaboration et al. 2020;Riess et al. 2019). The current tension between early and late Universe measurements is at a 5-σ level. It is not clear whether it might be explained by unknown systematics or whether this disagreement is opening the path for new physics beyond the standard cosmological model (see, e.g., Niedermann & Sloth 2020, and references therein). In this context, it is worth mentioning Partly based on data obtained with the STELLA robotic telescopes in Tenerife, a facility of The Leibniz Institute for Astrophysics Potsdam (AIP) jointly operated by the AIP and by the Instituto de Astrofísica de Canarias (IAC).\n\nTables 2, 3, 4, 8, and 9 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/. that the use of the Tip of the Red Giant Branch (TRGB) as primary distance indicators gives values of the Hubble constant that are intermediate between those based on the CMB and those based on the Cepheid distance scale (Freedman et al. 2020).\n\nThere are solid reasons to believe that the difference between the TRGB and the Cepheid distance scale is caused by the dependence of the metal content on the zero-point and/or the slope of the adopted diagnostics (Bono et al. 2010;Pietrzyński et al. 2013). The dependence of optical and near-infrared diagnostics on metallicity is still a controversial issue for both theoretical (Marconi et al. 2005;Bono et al. 2010) and empirical (Groenewegen 2018(Groenewegen , 2020Ripepi et al. 2019Ripepi et al. , 2021Breuval et al. 2020Breuval et al. , 2021 approaches. However, the improved accuracy of geometrical distances from Gaia DR3, and the homogeneity of spectroscopic abundances will provide solid constraints on this longstanding problem (Ripepi et al. 2022, and references therein).\n\nClassical Cepheids are bright (−2 ≤ M V ≤ −7 mag) radially pulsating stars with periods ranging from roughly one day to more than one hundred days (see, e.g., Skowron et al. 2019, and references therein). Their iron abundances range from solar values in the solar vicinity to ∼0.5 dex more metal-rich in the inner Galactic disk and to −0.5 dex more metal-poor in the outer disk (Inno et al. 2019). Cepheids in the Large Magellanic Cloud (LMC) are on average a factor of two more metal- Notes. The first four columns give the star name, the right ascension and declination, and the pulsation mode (0: fundamental; 1: first overtone; 0B: fundamental with bump). The Cols. 5 and 6 give the iron abundance from literature and the corresponding references. Columns from 7 to 10 show the number of optical spectra used for each spectrograph: N F : FEROS; N H : HARPS; N U : UVES; N S : STELLA. The last column lists the total number of spectra per target.\n\nReferences. (a) 1: Genovali et al. (2014); for T Vul, FF Aql, δ Cep, and S Sge we assumed a typical error of 0.1 dex since they were not provided by the original authors; 2: Proxauf et al. (2018), from which the uncertainty adopted is the largest value between σ and std.\n\npoor (Romaniello et al. 2021), and those in the Small Magellanic Cloud (SMC) a factor of four more metal-poor than solar (Lemasle et al. 2017). Cepheids are intermediate-mass stars crossing the instability strip during central helium burning, the so-called blue loops, and their progenitor mass ranges from ∼3 to ∼10 M -the exact limit depends on the chemical composition (Bono et al. 2000b;De Somma et al. 2021). Classical Cepheids are solid stellar beacons to trace young stellar populations even in the far side of the Galactic thin disk (Minniti et al. 2020(Minniti et al. , 2021 and in stellar systems experiencing recent star formation events such as dwarf irregulars and star forming galaxies (Neeley et al. 2021). Their period-luminosity relations (Breuval et al. 2020;Ripepi et al. 2020) provide accurate (at the level of ∼1-2%) measurements of individual distances. Individual distances together with detailed information concerning their elemental abundances (iron peak, α, neutron capture), provide the unique opportunity to investigate the radial gradients and, in turn, the recent (t ≤ 200-300 Myr) chemical enrichment history of the thin disk and nearby stellar systems (Romaniello et al. 2008;Lemasle et al. 2013;Genovali et al. 2015;da Silva et al. 2016). Furthermore, classical Cepheids are fundamental laboratories for investigating not only the advanced evolutionary phases of intermediate-mass stars (Bono et al. 2000a;Neilson et al. 2011;Prada Moroni et al. 2012;De Somma et al. 2021), but also for constraining the physical mechanisms driving their variability (Bono et al. 1999;Marconi et al. 2013).\n\nAlthough classical Cepheids are the cross-road of several recent and long-standing astrophysical problems, we still lack accurate investigations concerning the variation of their physical properties along the pulsation cycle. The current studies are mainly based on both optical and near-infrared (NIR) light curves, but we still lack detailed spectroscopic analysis concerning the variation of effective temperature, surface gravity, and microturbulent velocity (see, e.g., Lemasle et al. 2020;Kovtyukh 2007;Proxauf et al. 2018, and references therein).\n\nPioneering investigations based on multiple (from a few to almost two dozen) spectra along the pulsation cycle were provided more than ten years ago by Luck & Andrievsky (2004), Kovtyukh et al. (2005), Andrievsky et al. (2005), and Luck et al. (2008), in which they evaluated the variation of the atmospheric parameters and the abundances in both short-and long-period Cepheids. More recently, we performed a detailed investigation, based on a large dataset of high-resolution spectra, for a sample of 14 calibrating Cepheids (Proxauf et al. 2018). In that work, the number of spectra per object ranges from five to more than 120, and the spectra covered a significant portion of the pulsation cycle. In the same investigation we also used a new and more homogeneous list of Line Depth Ratios (LDR) to estimate the effective temperature directly from the spectra, an approach that is independent of the photometric properties. The reader interested in a more detailed discussion is referred to Kovtyukh (2007, and references therein). However, all these investigations were hampered by three limitations: a) Line list -We compiled a list of iron lines using line lists available in the literature. The sources of atomic transition parameters were inhomogeneous, because they did not pass a detailed scrutiny. b) Phase coverage -In spite of the large number of spectra adopted in the spectroscopic analysis, several objects had spectra collected at similar phases and, consequently, they had a modest phase coverage. c) Intrinsic variation -The inhomogeneity of the atomic transition parameters introduces uncertainties in the spectroscopic surface gravity and in the microturbulent velocity. This caused an increase in the standard deviation at fixed The vertical red lines indicate the position in the rest-frame wavelength of atomic lines either only used for the abundance determination or also adopted by the LDR method. For the sake of clarity the continuum of each spectrum was shifted to arbitrary levels.\n\npulsation phase, and in turn, some limitation in the analysis of their variation along the pulsation cycle.\n\nTo overcome these limitations, the current investigation is based on a new approach including five steps: i) We updated the atomic transition parameters of our line lists with more recent and accurate laboratory measurements available in the literature; ii) We collected new high signal-to-noise ratio (S/N) and highresolution spectra for our sample of calibrating Cepheids, and we also performed a detailed search for similar quality spectra available in online science archives. This translates into an increase in the number of calibrating Cepheids by almost 50% (from 14 to 20). Moreover, we also added new spectra to Cepheids already included in the former sample. The current sample includes 19 stars pulsating in the fundamental mode and, for the first time used by us as a calibrating Cepheid, one star (FF Aql) pulsating in the first overtone mode, and their metallicities range from −0.08 (β Dor) to 0.19 dex (VY Sgr); iii) We performed a detailed visual check of hundreds of lines along the pulsation cycle to create a new homogeneous line list. Special attention was paid in the identification of lines showing blends (or poor definition of the continuum), or having any correlation with the effective temperature (due to possible NLTE effects) or with the equivalent widths (due to saturated lines), or presenting any systematic over-or under-abundance; iv) We improved the algorithm that we use for the estimate of the atmospheric parameters, and in turn, for the estimate of the elemental abundances; v) In addition to the atmospheric parameters and iron abundances, the current study also includes the abundances of five α elements: Mg, Si, S, Ca, and Ti.\n\nWe notice that the star AV Sgr, which was one of the calibrating Cepheids in Proxauf et al. (2018), is not included in the current work. The reason is that, after the revision of the Fe line list, we were not able to derive accurate estimates of the atmospheric parameters for many of the spectra of this variable. The number of spectra with accurate measurements was not enough to provide a good coverage of the pulsation cycle, and therefore, it was removed from the list of calibrating Cepheids. The parameters published in our previous paper can still be used, but keeping in mind that they are based on a different version of the line list.\n\nThe structure of the paper is the following: in Sect. 2 we discuss the spectroscopic dataset and provide detailed information concerning the current sample of calibrating Cepheids. In Sect. 3 we discuss the data reduction process and introduce the atomic line lists used. Sect. 4 includes a description of the radial-velocity and photometric data used. In Sect. 5 we explain how we derived the atmospheric parameters and their variation along the pulsation cycle. The iron and α-element abundances and their variation along the pulsation cycle are discussed in Sect. 6. A summary of the results and some final remarks concerning this project are given in Sect. 7.\n\n\nSpectroscopic datasets\n\nThe spectroscopic datasets analyzed in the current paper are spectra collected using several different instruments. Three of them are mounted on telescopes of the European Southern Observatory (ESO): the High Accuracy Radial velocity Planet Searcher (HARPS; Mayor et al. 2003) at the 3.6 m, the Fiberfed Extended Range Optical Spectrograph (FEROS; Kaufer et al. 1999) at the 2.2 m MPG/ESO, and the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) at the Very Large Telescope. The other instrument is the STELLA Echelle Spectrograph (SES; Strassmeier et al. 2004Strassmeier et al. , 2010 located at the Izãna Observatory on Tenerife in the Canary islands.\n\nThe spectral resolution of the quoted spectrographs for the instrument settings used are R∼40 000 (UVES), R∼115 000 (HARPS), R∼48 000 (FEROS), and R∼55 000 (STELLA). For details on the wavelength ranges of our spectra we refer the reader to Sect. 2 of the papers Proxauf et al. (2018) and Crestani et al. (2021).\n\nThe entire spectroscopic sample includes reduced spectra downloaded from the ESO and the STELLA archives, for a total of 1383 spectra of 285 stars: 199 HARPS spectra of 9 stars, 419 FEROS spectra of 151 stars, 450 STELLA spectra of 68 stars, and 315 UVES spectra of 123 stars. Their S/N ranges from a few tens to more than 400 (about 70% of our spectra have S/N of at least 100). From this sample we selected the spectra of 20 targets that cover either a significant part (about half or more) or the entire pulsation cycle. From now on we refer to this subsample as our sample of 20 calibrating Cepheids. Details on the number of spectra are given in Table 1.\n\n\nData reduction and analysis\n\nAs mentioned in Proxauf et al. (2018), the spectra from UVES and HARPS (Phase 3) were already pre-reduced (i.e., reduced up to the wavelength calibration step) by their own pipeline. As previously done, the FEROS spectra were reduced using a modified version of the Data Reduction System (DRS) pipeline. A few tens of FEROS spectra collected before 2004, which initially could not be reduced, were later on reduced and delivered by one of the Phase 3 Data Releases (DR3 and DR3.1). However, they are all low-quality spectra that were not included in our analysis.\n\nAfter the pre-reduction steps, all the selected spectra were normalized to the continuum using the Image Reduction and Analysis Facility (IRAF 1 ) by fitting cubic spline functions to a set of continuum windows. An exception is the case of the STELLA spectra, which were reduced by a dedicated pipeline (Weber et al. 2012) based on IRAF routines and that includes several reduction steps up to the continuum normalization.\n\nFor more details about the quality of the spectra we refer the reader to Sect. 3 of Proxauf et al. (2018), which also includes a figure (their Fig. 1) showing a few examples of HARPS, FEROS, UVES, and STELLA spectra of different metallicities and S/N. In our Fig. 1, we plot an example of HARPS spectra for β Dor and of STELLA spectra for ζ Gem in different pulsation phases, showing how the profile of the absorption lines may change (in depth and in shape) along the pulsation cycle. Studies of line asymmetry changes over the pulsation cycle of Cepheid stars were also reported by Nardetto et al. (2006) and Nardetto et al. (2008), in a series of papers on high-resolution spectroscopy, for Fe i and Hα lines in the optical and, more recently, by Nardetto et al. (2018) for the Na i line at 2208.969 nm.\n\n\nLine lists and equivalent widths\n\nSimilarly to what we did in our previous works, here we use three different line lists: The Cepheids variables are bright stars and thus we have at our disposal very high S/N, broad wavelength range spectra. The precision of the atmospheric parameters and of the chemical abundances, however, depends not only on the quality of the spectra, but also on the quality of the adopted line list. With this in mind, a large portion of the present work was dedicated to finding the best atomic transitions for iron and α elements that can be detected in Cepheids. In practice, this means i) collecting the most precise transition parameters, ii) removing absorption lines that are blended with other lines, and iii) eliminating lines that for any reason deviate significantly from the average value for their chemical species, or that display a dependency on any quantity that changes across the pulsation cycle, such as the effective temperature.\n\nIn order to address the first point, we departed from a list of hundreds of atomic transitions for iron and α elements commonly employed in the literature. We updated their transition parameters whenever possible with the updated laboratory measurements from Ruffoni Bottom: same as in the top panel but for the V-band light curve, its PLOESS fit and its mean value. We note that, within the uncertainty, the light curve is co-phased with the RV curve.\n\nFe ii, Lawler et al. (2013) for Ti i, and Wood et al. (2013) for Ti ii. We also made ample use of the astrophysical, but homogeneous and precise compilation of Meléndez & Barbuy (2009) for Fe ii. For the remaining lines, we adopted the transition parameters collected and updated by the National Institute of Standards and Technology (NIST) Atomic Spectra Database (Kramida et al. 2020). If a line is not available in any of these sources, it is eliminated from the preliminary list and not used for the computation of atmospheric parameters nor chemical abundances.\n\nWith this preliminary list at hand, we addressed the second point mentioned above by removing any blended lines, using as reference the Solar spectrum table by Moore et al. (1966) and synthetic spectra. Then we measured their equivalent width (EW) using the Automatic Routine for line Equivalent widths in stellar Spectra (ARES, Sousa et al. 2007Sousa et al. , 2015.\n\nFinally, we addressed the third point by removing deviant lines. In a first iteration, we estimated initial values for the atmospheric parameters and the chemical abundances. Lines are considered deviant if they result in abundances diverging by at least 3 σ from the mean for a given element, or if their behavior across the pulsation cycle is irregular. This latter situation is investigated by plotting all line-by-line measurements for each chemical species and individual exposure of our calibrating stars versus phase, effective temperature, and equivalent width, and removing lines that show strong trends in any of those planes. The calibrating stars mostly used are those with the largest number of individual spectra covering the whole pulsation cycle, such as β Dor, ζ Gem, and FF Aql. Tables 2 and 3 with quality flag 1, were used to compute the final atmospheric parameters and chemical abundances for the whole sample. The remaining lines, Fig. 4. Photometric amplitude as a function of the logarithmic period (Bailey diagram). Our calibrating Cepheids are represented by colored symbols whereas the Cepheids from literature Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006) are shown in light gray circles.\n\n\nThe final clean lists, shown in\n\nshown in the same tables with quality flag 0, were measured only for comparison purposes.\n\nAn example of a good Fe line is shown in Fig. 2 for β Dor and ζ Gem. Examples and a description of the typical behavior of irregular (less reliable) lines are found in the Appendix A, in the plots of Figs. A.1-A.5. These same irregular lines, however, might provide good results for other stars, for instance, of different spectral types, or having spectra of different quality or different resolution. This is the reason why we preferred to exclude some lines only for this particular analysis, and keep the initial line lists as they are to be used in future studies. It is worth mentioning that we only excluded lines that were clearly having problems in most of the stars in our sample.\n\n\nRadial velocity and photometric data\n\nThe radial velocity (RV) measurements for our sample were done using the following methods: i) for HARPS and UVES spectra, we used IRAF packages to cross-correlate the target spectrum with a solar template spectrum (Solar flux atlas from 296 to 1300 nm, Kurucz et al. 1984) degraded to the UVES resolution; ii) for FEROS spectra we adopted the RVs derived by ARES, the routine we used for the equivalent width measurements (see Sect. 3.1); iii) the STELLA radial velocities are written in the header of the FITS files and they are based on crosscorrelations with a template spectrum, performed by a dedicated reduction pipeline also using IRAF. In case of the Cepheids, a G-type dwarf template is used.\n\nThe uncertainties adopted for these RV estimates depend on the method used: for HARPS, UVES, and STELLA spectra, the uncertainties are estimated by IRAF during the cross-correlation; for FEROS spectra, they are estimated by ARES using spectral windows free of lines. Typical values for our HARPS, UVES, and FEROS spectra range from a few m/s in the best cases to a few km s −1 , depending on the spectral type. For the STELLA spectra, we adopted an uncertainty of 0.2 km s −1 . This is a conservative estimate considering that typical formal errors for high S/N spectra of Cepheids stars is about 50 m s −1 and that the RV zero point for this spectrograph has RMS variations of 30 to 150 m s −1 over two years. The RV estimates and the corresponding uncertainties for our sample of calibrating Cepheids are listed in Table 4.\n\nIn addition to our own RV measurements, we adopted literature RV curves and light curves to provide accurate ephemerides, meaning reference epoch (T 0 ) plus pulsation period (P) for these Table 4. Excerpt from the list of atmospheric parameters, Fe abundances, and radial velocities for each spectrum of the 20 calibrating Cepheids. Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. Columns 4, 5, and 6 give, respectively, the effective temperature and its standard deviation, surface gravity, and microturbulent velocity. The Cols. 7-8 and 9-10 list both the Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used (for the Fe ii abundances, a minimal uncertainty of 0.11 dex was adopt when computing the mean abundances, as described in the text). The last column gives the radial velocities with their uncertainties. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the mean magnitude in visual bands, the magnitude amplitude, the mean radial velocity, the RV amplitude, the light curve source, the pulsation period and zero-phase reference epoch of mean magnitude (or mean radial velocity), and the Ephemerides source. For the variables VY Sgr, RZ Vel, and WZ Sgr, our RV measurements and those by Groenewegen (2008, and references therein) show a shift in phase of −0.05, +0.03, and −0.03, respectively. To co-phase the two datasets, the latter sample of radial velocities plotted in Fig variables. More specifically, we collected RV curves and V-band light curves from Pel (1976) and Groenewegen (2008, and references therein). For UZ Sct, we adopted the V-band light curve from the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappee et al. 2014;Kochanek et al. 2017) because it is the only well-sampled light curve that can be co-phased with our RV time series.\nName Dataset MJD [d] T eff ± σ [K] log g υ t [km s −1 ] Fe i ± σ N Fe i Fe ii ± σ N Fe ii RV ± σ [km s −1 ] R\nWe estimated the pulsation periods of our targets by using our own interactive method ) based on the Lomb-Scargle periodogram (Scargle 1982). Historically, the most commonly used T 0 is the time of maximum light (T max ), which matches the time of minimum velocity. Starting from Inno et al. (2015), we have been promoting the use of another reference for both Cepheids and RR Lyrae, that is, the epoch of the mean magnitude on the rising branch of the light curve (T opt mean ) or, equivalently, the epoch of mean velocity on the decreasing branch of the RV curve (T RV mean ). See Fig. 3 for a visualization of this reference epoch on both the light and RV curves. A detailed procedure to derive T opt mean is described in Braga et al. (2021). The advantages of the T opt mean reference epoch over T max are wellexplained in Inno et al. (2015) and quantitatively discussed in Braga et al. (2021). Name Notes. Columns from 1 to 5 give, respectively, the star name, the mean effective temperature and mean θ, the θ amplitude (see Section 5.2), and the number of spectra used to compute these mean values. Columns from 6 to 9 lists the surface gravity, microturbulent velocity, and iron abundances from neutral and ionized lines. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 4. Column 10 gives the weighted iron abundance and its standard error, calculated from Cols. 8 and 9. The standard deviation calculated using all individual measurements from both Fe i and Fe ii lines is also shown within parentheses. The last column gives the number of spectra used to derive log g, υ t , and the iron abundances.\nT eff ± σ [K] θ ± σ ∆θ ± σ N spec (T eff ) log g ± σ υ t ± σ [km s −1 ] [Fe i/H] ± σ [Fe ii/H] ± σ [Fe/H] ± σ (std) N spec R\nThe light and radial-velocity curves for all the sample stars are shown in the Appendix B, in Fig. B.1. The estimated ephemerides of our targets are listed in Table 5 together with the mean magnitudes and magnitude amplitudes in the visual band, and the mean radial velocities and RV amplitudes. We noticed that for the stars VY Sgr, RZ Vel, and WZ Sgr, for which the new period and T 0 values are based on the radial velocities derived by Groenewegen (2008, and references therein), the new ephemerides caused a shift in phase between the two datasets shown in Fig. B.1. To overcome this problem, we applied a shift of −0.05, +0.03, and −0.03, respectively, to the literature data plotted in these panels. Figure 4 shows the distribution of the current calibrating Cepheids (colored symbols), together with Galactic Cepheids available in the literature (light gray circles), in the so-called Bailey diagram (luminosity amplitude as a function of the logarithmic period). The plotted data display the classical \"V\" shape in which the local minimum around ten days is mainly caused by the Hertzsprung progression (Bono et al. 2000b). By having a look also at Fig. B.1, we see a subgroup of classical Cepheids showing a well defined bump on the light curve. The phase of such bumps moves from the decreasing branch for periods between ∼6 and ∼9 days, passing across the maximum for periods between ∼9 and ∼12 days, and along the rising branch for periods between ∼12 and ∼16 days. At the center of the Hertzsprung progression (P∼10 days) the light curve is flat topped and the luminosity amplitude attains a well defined minimum (Bono et al. 2000b(Bono et al. , 2002. These variables are called Bump Cepheids for avoiding to be mixed-up with the Beat Cepheids, that is, classical Cepheids pulsating simultaneously in two or more modes. The calibrating Cepheids cover both the short-and the longperiod range, and in particular, the current sample includes seven Cepheids across the Hertzsprung progression.\n\n\nAtmospheric parameters and effective temperature curve templates\n\n\nEffective temperature, surface gravity, and microturbulent velocity\n\nThe approach we adopted to estimate the atmospheric parameters (i.e., the effective temperature, the surface gravity, and the microturbulent velocity) was already discussed in detail by Proxauf et al. (2018). Here we only recap the key points. The T eff along the pulsation cycle was estimated by using the LDR method, which relies on the correlation between the line depth ratios of pairs of absorption lines in the spectra of different stars and the effective temperature of the same stars. The surface gravity was derived through the ionization equilibrium of Fe i and Fe ii lines, and the microturbulent velocity was obtained by minimizing the dependence of the abundances provided by single Fe i lines on their EWs. During this procedure, the effective temperature was kept fixed, whereas log g and ξ were iteratively changed until convergence. The metallicity used as input by our algorithm is updated in each step, and the adopted value is [Fe i/H], which is the mean iron abundances provided by individual Fe i lines. The estimates of T eff , log g, and ξ for the individual spectrum of our sample are listed in Table 4. The uncertainties on T eff are the standard deviation calculated using the LDR method, that is, from the effective temperatures provided by the individual pairs of lines. The uncertainties on the individual estimates of log g and ξ are assumed to be of the order of 0.3 dex and 0.5 km s −1 , respectively (see Genovali et al. 2014, for a detailed discussion). The weighted mean values of the atmospheric parameters are listed in Table 6 together with the corresponding standard errors calculated for each calibrating Cepheid.\n\nIn the Appendix B, Figs. B.2, B.3, and B.4 show the variation of the atmospheric parameters along the pulsation cycle. To provide a quantitative estimate of the differences between the current approach and our previous measurements, we fitted Fourier series to the effective temperature, surface gravity, and microturbulent velocity curves in order to evaluate their amplitudes, A&A proofs: manuscript no. daSilvaetal2021 The panels show the phase-folded curves separating the Cepheids in different bins of pulsation period (values within parentheses): from 3 to 5 days, from 5 to 7 days, from 7 to 9.5 days, and from 9.5 to 10.5 days. For the variable S Sge, the two outliers shown in Fig. B.2 were not used in the construction of the T eff curve templates. The curves display the fit of the Fourier series to the data (see Table 7). The standard deviations of the fit are also shown. similarly to what we did in Proxauf et al. (2018, their Figs. 3 and 4). We found that, on average, the errors on the derived amplitudes for the effective temperature curves are a factor 2 smaller than our previous estimates, whereas for the surface gravity the differences are even more significant, with typical errors almost 3 times smaller. For the microturbulent velocity variation, the errors on the current amplitude estimates are more than twice smaller (the dispersions are ∼50 K for T eff , ∼0.2 dex for log g, and ∼0.2 km s −1 for ξ). This means that the variation of the atmospheric parameters along the pulsation cycle is now much better constrained, thanks not only to the larger number of spectra but also to the better determination of such parameters. The improvement in the estimate of log g and ξ is mainly due to the improvement in the line list, which reduced the number of spurious abundance values provided by unreliable iron lines. As described in the Appendix A, we removed lines that are blended, lines that display a well defined trend with effective temperature or with equivalent width, among others. The estimate of T eff is based on the same line list that we previously used, therefore the improvement is mainly due to the larger number of spectra analyzed in the current work. All in all, these changes increased the accuracy in the atmospheric parameters. The reader interested in detailed discussion concerning the estimate of the intrinsic parameters of classical Cepheids is referred to Vasilyev et al. (2018) and Lemasle et al. (2020).\n\nWe note in passing that the improvement in the accuracy and in the sampling of the pulsation cycle allow us to state that the minimum in the effective temperature is often approached in the same phases in which the minimum in surface gravity is also approached. Moreover, and even more importantly, these minimum phases anticipate the increase in microturbulent velocity. This parameter, as expected, attains it maximum just before the maximum in effective temperature, which can be clearly observed by comparing several panels of Figs. B.2 and B.4. \n\n\nEffective temperature curve templates\n\nBy taking advantage of the substantial phase coverage of our sample of calibrating Cepheids, we used the effective temperature measurements to compute new templates for different bins of pulsation periods. However, instead of using the T eff curves directly, we preferred to provide templates for the theta parameter -defined as θ = 5040/T eff -given its linear dependency on the Cousins R-I color index (Taylor 1994).\n\nThe approach is similar to the NIR light-curve templates provided by Inno et al. (2015): first, we adopted the ephemerides in Table 5 to fold the θ curves (θCs). Subsequently, we normalized the folded θCs by subtracting their average and dividing by their amplitudes. Our θCs are well sampled, meaning that we have enough phase points to separate the Cepheids into different period bins and to provide analytical relations for the θ curve templates in each bin. We adopted the same period thresholds introduced by Inno et al. (2015). Therefore, based on their Table 1, we generated four cumulative and normalized theta curves for the bins 2 (3-5 days, three Cepheids), 3 (5-7 days, three Cepheids), 4 (7-9.5 days, two Cepheids), and 5 (9.5-10.5 days, two Cepheids). By adopting the same period bins we are able to provide θC templates that are homogeneous with those from the NIR light curves. The reader interested in a more detailed and quantitative discussion concerning the use of cumulative and normalized curves to derive the analytical fits, together with the adopted thresholds for the different period bins, is referred to Inno et al. (2015).\n\nThe four normalized θ curves are separately shown in Fig. 5, and in Fig. 6 we compare the Fourier series fitted to the data of each bin. The corresponding coefficients of the fitted functions are listed in Table 7. The separation of the Cepheids into different period bins is a mandatory step because not only the amplitude of the theta curves (listed in Table 6), but also their shape changes with period (as shown in Figs. 5 and 6). Although we do have data for Cepheids with periods outside of the selected pe- riod bins, they are either not as well sampled as the data for the other Cepheids, or the theta curves do not overlap well because the number of measurements is too limited. More accurate and homogeneous data are required to overcome this limitation.\n\nIn order to allow the reader to use these curve templates to derive the mean effective temperature by using the ephemerides, the luminosity amplitude (∆V), and a single spectroscopic measurement of the T eff , we also provide a linear relation between ∆V and ∆θ: ∆θ = (0.184 ± 0.014)∆V + (0.000 ± 0.011); (σ = 0.013)\n\n(1)\n\nThe estimate of the ∆θ amplitude has to be applied to the normalized template as a multiplicative factor. Only after this rescaling operation the template can be anchored to the empirical data and used to derive the mean T eff . Optical and NIR light curves of the calibrating Cepheids, together with effective temperature and radial velocity curves, will be adopted by our group to perform a detailed comparison with nonlinear, convective hydrodynamical models of classical Cepheids. Dating back to Natale et al. (2008) it has been found that the simultaneous fit of both luminosity and radial-velocity variations provides solid constraints on the physical assumptions adopted to build pulsation models (Marconi et al. 2013). Moreover, the use of the effective temperature curves (shapes and amplitudes), covering a broad range of pulsation periods, brings forward the opportunity to constrain, on a quantitative basis, the efficiency of the convective transport over the entire pulsation cycle.\n\nArticle number, page 9 of 31 A&A proofs: manuscript no. daSilvaetal2021 Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. The other columns give the abundances from both neutral and ionized lines, their standard deviations, and the number of lines used. For each element X, the weighted mean of X i and X ii abundances (weighted by 1/σ 2 ) and its intrinsic error is also shown. Magnesium and Sulfur abundances are from neutral lines only. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the abundances from both neutral and ionized lines, their uncertainties, and the number of spectra used. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 8. The standard deviation calculated using all individual abundances from both neutral and ionized lines is also shown. The complete table is available at the CDS. Fig. 9. Atmospheric parameters derived by keeping fixed the effective temperature (previously obtained from the LDR method) in comparison with the same parameters derived with T eff as a free parameter. The panels show the comparison for the β Dor and ζ Gem stars.\n\n\nIron and α-element abundances\n\n\nIron abundance\n\nThe determination of the atmospheric parameters was done with the Python wrapper 2 of the MOOG code (Sneden 2002). This is a LTE radiative code that we applied to model atmospheres interpolated on the grids of Castelli & Kurucz (2004). Once a convergence is achieved in the determination of T eff , log g, ξ, and [Fe/H], as described in Sect. 5.1, MOOG also provides the final iron abundance derived from individual Fe i and Fe ii lines. Such lines are those included in the input line list and that passed the selection criteria that we applied to choose the best atomic transitions. Table 4 lists, for each spectrum in our sample, the mean Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used. The mean Fe i and Fe ii abundances derived for each star from the individual spectra, weighted by the standard deviations aforementioned, and the corresponding standard errors are shown in Table 6. Column 8 gives our final determination for the stellar metallicity, which is the weighted mean calculated from both the [Fe i/H] and the [Fe ii/H] abundances. The adopted uncertainties are the largest values between the standard error computed from the weighted mean (σ) and the standard deviation (std). Figure B.5 shows the derived metallicities as a function of the pulsation phase. Data plotted in this figure show that the iron abundances are quite stable along the pulsation cycle. This outcome applies to short-/long-period Cepheids and to Bump Cepheids. We note that the current approach improves the results obtained by Proxauf et al. (2018), since in their iron abundances was still present a mild variation along the rising branch of large amplitude Cepheids (in our current results the dispersions are in most cases smaller than 0.05 dex). The improvement is mainly due to the very careful selection of the lines adopted to estimate the iron abundance. The difference is soundly supported by the distribution of the standard deviations displayed in Fig. 7 for the Cepheids in common. The current standard deviations for both [Fe i/H] (violet) and [Fe ii/H] (green) lines (bottom panel) are smaller by 30%-40% when compared with our previous investigation. Moreover and even more importantly, they also attain similar values.\n\nTo further constrain the difference between the current and our previous investigation, Fig. 8 shows the comparison for both the atmospheric parameters and the iron abundance. The agreement is remarkable over the entire temperature range. The mean and the standard deviations for the atmospheric parameters are well within the current uncertainties. The new iron abundances are, on average, ∼0.1 dex more metal-poor. Owing to the similarity in the atmospheric parameters, this difference seems an obvious consequence of the new line list.\n\nSeveral different approaches have been suggested in the literature to estimate the atmospheric parameters of variable stars (Fukue et al. 2015;Jian et al. 2020;Lemasle et al. 2020;Matsunaga et al. 2021;Taniguchi et al. 2021;Romaniello et al. 2021). In the current investigation, the effective temperature is firstly derived using the LDR method, then the other atmospheric parameters are estimated at fixed effective temperature. The advantages of this approach have already been discussed in several papers. To provide a more quantitative estimate of the differences compared to another method normally used, in which the minimization process also includes T eff as a free parameter, we performed a new and independent determination of the three atmospheric parameters plus abundances simultaneously for two variables in our sample (β Dor and ζ Gem). The differences between our canonical approach and the literature approach are shown in Fig. 9. The overall agreement is quite good over the entire temperature range. Indeed, the mean and the standard deviations attain tiny values. The mean difference in effective temperature is 55 K, and for the surface gravity it is 0.12 dex only, while the microturbulent velocities attain almost identical values. The quoted differences cause a small increase of 0.04 dex in the iron abundance when T eff is a free parameter. We note that we adopted the same line list in the tests performed either with fixed or with free T eff .\n\nIn closing this subsection, we also investigate whether the current iron abundances display any clear trend with the Galactocentric distance (R G ). The top panel of Fig. 10 shows the iron abundance as a function of R G for the calibrating Cepheids (blue circles) and for Cepheids available in the literature (gray circles). We performed a linear fit over the entire sample and we found the following relation:\n\n[Fe/H] = (−0.055 ± 0.003)R G + (0.43 ± 0.03)\n\nThe current slope agrees quite well with similar estimates available in the literature and, in particular, with the recent estimate of the iron radial gradient (dashed line) provided by Ripepi et al. (2022). We note that, to overcome variations in the zero-point mainly introduced by innermost and outermost Cepheids, the gradient from Ripepi et al. (2022) was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The bottom panel of Fig. 10 shows the iron abundance as a function of the logarithmic pulsation period. Data plotted in this panel agree quite well with similar estimates available in the literature. Moreover, they do not show any significant variation when moving from young (long-period) to less young (shortperiod) classical Cepheids.\n\n\nα-element abundances\n\nThe abundances of the α elements were derived using pyMOOGi as well (except sulfur, for which the abundances were estimated using the spectral synthesis method -see Sect. 6.2.1). We used the ab f ind driver, which requires as input i) the model atmospheres, calculated from the atmospheric parameters obtained for each star, and ii) a list of lines containing the wavelength, the atomic number, the lower-level excitation potential, the log g f , and the measured equivalent widths. Table 8 lists the individual α-element abundances obtained for each spectrum together with the standard deviations and the number of lines used for each element. The mean abundances, weighted by the standard deviations, and the corresponding standard errors are shown in Table 9. The same mean data are plotted in Figs. 11,12,and B.6 for our 20 calibrating Cepheids (blue circles) and for similar data available in the literature (gray circles).\n\nFor cases in which the abundance of a given element at a given phase is based on only one or two lines, or if the estimated error is lower than a typical value for each element/species, the uncertainty is set to that typical value, namely: σ( Table 8 for spectra having at least 2 lines of the same element/species. The adopted standard solar abundances for both iron and α elements are from Asplund et al. (2009), namely: A(Fe) = 7.50, A(Mg) = 7.60, A(Si) = 7.51, A(S) = 7.12, A(Ca) = 6.34, and A(Ti) = 4.95, with A representing the typical logarithmic notation where H is defined to be A(H) = 12.00. These values are in good agreement with the very recent determinations done by Asplund et al. (2021), which were obtained using the most up-to-date atomic and molecular data.\n\nFor both iron and α abundances, we evaluated the systematic differences for stars in common between the current investigation and measurements from literature: ), Luck & Lambert (2011, Lemasle et al. (2013), andYong et al. (2006) for iron and α elements, and Proxauf et al. (2018) for iron only. The differences for all the investigated elements are listed in Table 10. In order to perform a direct comparison, we applied these zero-point differences to the literature datasets by putting the iron and the α-element abundances in our metallicity scale. The spectroscopic samples for which there are no variables in common between our investigation and those available in the literature, the homogenization of the abundances was performed in two steps. To be more specific, we have no variables in common with Yong et al. (2006), therefore, we first scaled their abundances to match those from Luck & Lambert (2011), and then we scaled them to match our metallicity scale. Data plotted in Fig. B.6 display, once again, the remarkable agreement of the α-element abundances over the entire pulsation cycle (the dispersions in most cases are smaller than 0.10 dex). This outcome applies to all the investigated α elements (except sulfur, for which we have abundances at a single phase only), and to both neutral and ionized species. We note in passing that this finding seems more relevant when compared with the variation of the iron abundances along the pulsation cycle. Indeed, the αelement abundances are only based on a modest number or, in some cases, just a few lines. Therefore, they are more prone to possible systematics.\n\nTo further investigate the internal consistency of the current α-element abundances with similar estimates available in the literature, we checked the behavior of the [X/Fe] abundance ratios as a function of the stellar metallicity. Data plotted in the left panels of Fig. 11 show that the current results agree, within the errors, quite well with similar determinations done in previous investigations (Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006). The key difference seems to be the smaller dispersion at fixed iron abundance, thus suggesting that a fraction of the spread currently observed might still be caused by intrinsic errors. This finding is further supported by the evidence that the current estimates do not show any clear trend when plotted against the pulsation period (see the right panels of Fig. 11). We note that classical Cepheids, when moving from short to long periods, become systematically cooler and more expanded (lower surface gravity).\n\n\nSulfur abundances\n\nTo determine the sulfur abundances, we carried out spectral synthesis calculations by exploiting the multiplet at 6757 Å, which is proven to be reliable and not affected by NLTE effects (see, e.g., Takeda et al. 2016;Duffau et al. 2017, and references therein). Synthetic spectra were computed using the synth driver of MOOG (2019 version), the same grid of model atmospheres by Castelli & Kurucz (2004) used for the other elements, and atomic parameters from Caffau et al. (2007). We synthesized a 30 Å wide region around the S i multiplet in order to determine the broadening of the spectral lines. Then we degraded the synthetic spectra to match the resolution of the observational datasets. The best fit was found with χ-square minimization methods, as routinely done in the literature. Finally, we estimated the internal errors, with a mean value of σ([S i/H]) = 0.10 dex. These errors are due to continuum placement (related to the S/N, quality of the spectra, best fitting procedure) and due to errors on the atmospheric parameters (this was done in the standard way, which is by varying one parameter at a time and inspecting the corresponding variations). We refer the reader to D'Orazi et al. (2017D'Orazi et al. ( , 2020 for details on error computation.\n\nThe reasons why we focused our attention on S i abundances for our sample of calibrating Cepheids are manifold:\n\ni)-Sulfur is a volatile element that is not locked in dust grains, which means that no correction is required to compare the current sulfur abundances with predictions based on chemical evolution models (Chiappini et al. 2001;Cescutti et al. 2007). Therefore, sulfur is a more solid tracer of α-element abundances than Si and Ca. These three α elements are produced during the oxygen burning and their abundances should show similar chemical enrichment histories (Limongi & Chieffi 2003).\n\nii)-Sulfur brings forward several key advantages when compared with other α elements, since it shows several lines at both optical and NIR wavelengths. The most popular sulfur lines are the multiplets at 6757 Å and at 8694 Å, but they are weak and they are typically measured on stars more metal-rich than [Fe/H] ∼ −1.5 dex. More recently, the multiplet at 9237 Å has also been used because it is very strong and it can be measured even in very metal-poor stars (Nissen et al. 2004;Caffau et al. 2005Caffau et al. , 2007. However, these lines are also affected by NLTE effects, making them even stronger, and the sulfur abundances need to be properly treated. Even more recently, high-resolution NIR spectrographs also provided the opportunity to use the S i multiplet at 1045 nm for both dwarf and giant stars (Caffau et al. 2016).\n\niii)-Possibility of comparison with the S abundances in external galaxies, such as Blue Compact Galaxies, Damped Lyman Alpha systems (Dessauges-Zavadsky et al. 2007), and Active Galactic Nuclei (Liu et al. 2015, Mizumoto et al., submitted to ApJ). Indeed, thanks to strong emission lines, the S abundances can be measured even at large redshifts.\n\niv)-In our previous investigation focused on α-element abundances in classical Cepheids (Genovali et al. 2015) we neglected sulfur. In the current study, we perform accurate measurements of the multiplet at 6757 Å. The multiplet at 9237 Å is only available in a small subsets of spectra (UVES), but the lines are too strong (EW ≥ 300 mÅ) and it was neglected. Data plotted in Fig. 12 show the [X/H] (left panels) and the [X/Fe] (right panels) α-element abundances as a function of the Galactocentric distances, estimated by Genovali et al. (2014) using NIR mean magnitudes and predicted Period-Luminosity relations (Inno et al. 2013). The radial gradients plotted in the left panels indicate that sulfur is the α element with the steeper slope (see labeled values). Indeed, its slope is at least a factor of two steeper than for Ca (0.076 vs. 0.031 dex kpc −1 ) and ∼50% steeper than those for Mg and Si. In passing, we also note that the radial gradient of the abundance ratios plotted in the right panels of the same figure display, as expected (see Genovali et al. 2015), either a flat trend (Si) or a marginal positive slope (Mg, Ti) across the thin disk. There are two exceptions. Ca is steadily becoming overabundant (positive slope) when moving from the innermost (more metal-rich) to the outermost (more metal-poor) disk regions. This is a consequence of the fact that the slope of its radial gradient is shallower than the iron gradient (0.31 vs. 0.55 dex kpc −1 ). S shows a negative slope, because it is the only α-element with a slope in the radial gradient steeper than the iron gradient (0.76 vs. 0.55 dex kpc −1 ).\n\n\nSummary and final remarks\n\nWe performed a new spectroscopic analysis of almost two dozen of northern and southern calibrating Cepheids. As a whole, we collected more than 500 high-resolution (R∼40 000-115 000), high S/N (∼50-370) optical spectra with four different spectrographs (STELLA, FEROS, HARPS, UVES) covering either a substantial portion or the entire pulsation cycle. The main aim of this investigation is twofold: i) provide new and accurate constraints on possible systematics affecting the estimate of atmospheric parameters and elemental abundances along the pulsation cycle; ii) develop new approaches and/or diagnostics to provide accurate estimates of both atmospheric parameters and chemical abundances. To accomplish the former goal, we assembled three different line lists. By using different line lists available in the literature, we compiled a very detailed list of both Fe i and Fe ii lines. Moreover, we also collected a sizable sample of lines for α-elements: Mg, Si, S, Ca, and Ti. Special care was also paid to the line list of both iron-peak and α-elements used for the estimate of the effective temperature using the LDR method.\n\nIn order to provide homogeneous and accurate estimates of both atmospheric parameters and elemental abundances, we undertook a lengthy and detailed analysis of the quoted line lists. Our approach moved along three different paths: i) we collected the three most precise atomic transition parameters available in the literature; ii) we removed absorption lines that are blended with other lines; iii) we removed lines that either show relevant abundance variations from the average value of their chemical species or that display a steady variation along the pulsation cycle with atmospheric parameters. Concerning the transition parameters, we gave the priority to recent laboratory measurements, then to homogeneous astrophysical estimates and, for the remaining lines, to the transition parameters available on the NIST Atomic Spectra Database. To identify blended lines, we adopted both the solar spectrum by Moore et al. (1966) and a grid of synthetic spectra. Finally, we applied an iterative approach to remove lines showing significant variations. This latter step was done based on a preliminary estimate of the atmospheric parameters and the abundances. We were then able to identify lines that show either a variation in abundance at the 3-σ level from the mean value of the specific element and/or a steady variation along the pulsation cycle. In more detail, to properly identify robust lines, we plotted all the lines available in our initial line lists, for each chemical species, individual exposure, and calibrating Cepheid, as a function of the pulsation phase, effective temperature, and equivalent width. Lastly, we performed a new estimate of the atmospheric parameters and the individual elemental abundances, using an improved version of our algorithm and using only the line lists based on optimal lines.\n\nThe comparison between the current atmospheric parameters and similar estimated provided by our group, using a similar approach but old line lists, show smoother variation along the pulsation cycle and, in particular, smaller standard deviations (∼50 K for the effective temperature, ∼0.2 dex for the surface gravity, and ∼0.2 km s −1 for the microturbulent velocity). The estimate of both iron and α-element abundances is also significantly improved, since they display an almost constant value as a function of the pulsation phase. The dispersions are 0.05 dex for iron and 0.10 dex for the α elements. We note that this is the first time that α-element abundances are critically investigated for possible variations along the pulsation cycle.\n\nThis finding is further supported by the comparison with similar abundance estimates available in the literature. We found that the current calibrating Cepheids display, at fixed either iron abundance or pulsation period, smaller standard deviations. This result suggests that a good fraction of the spread normally observed in elemental abundances is mainly caused by the adopted line list and by the algorithm used to estimate the atmospheric parameters.\n\nMoreover and even more importantly, we also derived for the first time new and accurate effective temperature templates. Following Inno et al. (2015), the calibrating Cepheids were split into four different period bins, ranging from 3 to 10.5 days in period. For each period bin, we performed an analytical fit with Fourier series providing θ = 5040 /T eff as a function of the pulsation phase. A similar approach was already applied to fundamental RR Lyrae by For & Sneden (2010) and by Magurno et al. (2018), but the variations along the pulsation cycle are, at fixed pulsation phase, larger due to the larger luminosity amplitudes and probably to the possible occurrence of nonlinear phenomena (shocks). The current findings appear quite promising, because the new templates can provide accurate estimates of the effective temperature once the pulsation period and the phase (and in turn the reference epoch) of a Cepheid are known. We note that this information is crucial in the abundance analysis of NIR spectra, because in the metal-poor regimes only a modest number of ionized and neutral iron lines are present.\n\nThe current investigation was focused on Galactic Cepheids with iron abundances close to the solar value. In a forthcoming paper we plan to apply the same approach to more metal-poor Magellanic Cloud Cepheids. The panels of Fig. B.1 show the phase-folded light and radialvelocity curves for our sample of 20 calibrating Cepheids. The stellar magnitude in visual bands were taken from literature, as explained in Sect. 4. The radial velocities were either measured from our spectra or taken from literature. Only FF Aql and β Dor have RV values derived from our spectra only.\n\nOur RV measurements agree quite well with the literature values. Significant differences are seen only for two stars, T Vul and S Sge, which might be related to instrumental drifts caused by their long baseline observations (for both stars, the two datasets, ours and from literature, are separated by almost 17 years) or, most likely, to the binarity of these objects (Evans 1992;Evans et al. 1993). Table 5, for T Vul we derived two different values of reference epoch (T 0 ), one from our own radial velocities and another one from the light curve. For S Sge we have independent measurements of T 0 and pulsation period derived from both the RV datasets and from the light curve. Another particular case is β Dor, one of our bump Cepheids (see Table 1), for which we have independent values of T 0 and period derived from our RV curve and from the literature light curve.\n\n\nAs detailed in\n\nThe main criteria we adopted for selecting the light curves available in the literature were: i) well-sampled photometric data; ii) minimization of the difference in epoch between photometric and spectroscopic data. It goes without saying that wellsampled light curves from Gaia DR3 will allow us to provide accurate and homogeneous ephemerides for the majority of classical Cepheids.\n\n\nPartly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra.\n\nFig. 1 .\n1Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel).\n\nFig. 2 .\n2Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\n\n\na) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA.\n\n\net al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for\n\nFig. 3 .\n3Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled.\n\n\n. B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve.\n\nFig. 5 .\n5Normalized θ (5040/T eff ) as a function of the pulsation phase.\n\nFig. 6 .\n6Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation.\n\nFig. 7 .\n7Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018).\n\nFig. 8 .\n8Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values.\n\nFig. 10 .\n10Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show.\n\nFig. 11 .\n11[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors.\n\n\n[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in\n\nFig. 12 .\n12[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)\n\nFig. A. 1 .\n1Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\n\nFig. A. 2 .\n2Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths.\n\nFig. A. 3 .\n3Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends.\n\nFig. A. 4 .\n4Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects.\n\nFig. A. 5 .\n5Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves\n\n\nArticle number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.Name \nα ICRS \nδ ICRS \nMode [Fe/H] lit ± σ Ref. a \nN F N H N U N S N tot \n\nR TrA \n15:19:45.712 −66:29:45.742 \n0 \n−0.01 ± 0.03 \n2 \n1 \n14 \n... \n... \n15 \nT Vul \n20:51:28.238 +28:15:01.817 \n0 \n0.07 ± 0.10 \n1 \n... \n... \n... \n26 \n26 \nFF Aql \n18:58:14.748 +17:21:39.296 \n1 \n0.10 ± 0.10 \n1 \n... \n... \n... \n27 \n27 \nS Cru \n12:54:21.997 −58:25:50.215 \n0 \n0.09 ± 0.04 \n2 \n... \n... \n... \n13 \n13 \nδ Cep \n22:29:10.265 +58:24:54.714 \n0 \n0.09 ± 0.10 \n1 \n... \n... \n... \n18 \n18 \nY Sgr \n18:21:22.986 −18:51:36.002 \n0 \n0.00 ± 0.06 \n2 \n... \n20 \n... \n4 \n24 \nXX Sgr \n18:24:44.500 −16:47:49.820 \n0 \n0.06 ± 0.04 \n2 \n4 \n... \n5 \n3 \n12 \nη Aql \n19:52:28.368 +01:00:20.370 \n0B \n0.24 ± 0.09 \n2 \n... \n... \n... \n11 \n11 \nS Sge \n19:56:01.261 +16:38:05.236 \n0B \n0.14 ± 0.10 \n1 \n... \n... \n... \n21 \n21 \nV500 Sco 17:48:37.501 −30:28:33.461 \n0B \n−0.07 ± 0.08 \n1 \n... \n... \n4 \n3 \n7 \nβ Dor \n05:33:37.512 −62:29:23.323 \n0B \n−0.03 ± 0.05 \n2 \n1 \n46 \n... \n... \n47 \nζ Gem \n07:04:06.530 +20:34:13.074 \n0B \n0.16 ± 0.05 \n2 \n... \n47 \n... \n81 128 \nVY Sgr \n18:12:04.566 −20:42:14.465 \n0B \n0.25 ± 0.08 \n2 \n29 \n... \n4 \n... \n33 \nUZ Sct \n18:31:22.367 −12:55:43.339 \n0B \n0.11 ± 0.09 \n2 \n28 \n... \n6 \n... \n34 \nY Oph \n17:52:38.702 −06:08:36.875 \n0 \n0.08 ± 0.05 \n2 \n... \n8 \n... \n... \n8 \nRY Sco \n17:50:52.345 −33:42:20.411 \n0 \n0.01 ± 0.06 \n1 \n... \n... \n5 \n3 \n8 \nRZ Vel \n08:37:01.303 −44:06:52.844 \n0 \n0.08 ± 0.06 \n2 \n1 \n11 \n... \n... \n12 \nV340 Ara 16:45:19.112 −51:20:33.394 \n0 \n0.23 ± 0.07 \n2 \n26 \n... \n6 \n... \n32 \nWZ Sgr \n18:16:59.716 −19:04:32.989 \n0 \n0.28 ± 0.08 \n1 \n1 \n... \n5 \n3 \n9 \nRS Pup \n08:13:04.216 −34:34:42.692 \n0 \n0.14 ± 0.07 \n2 \n... \n15 \n... \n3 \n18 \n\n\n\nTable 2 .\n2Excerpt from the list of Fe i and Fe ii atomic lines.λ [Å] \nSpecies ξ [eV] log g f \nQuality \nflag \nRef. \n\n3763.789 \nFe i \n0.989 −0.220 \n1 \n1 \n3787.880 \nFe i \n1.010 −0.840 \n1 \n1 \n... \n... \n... \n... \n... \n7655.488 \nFe ii \n3.892 −3.560 \n1 \n2 \n7711.724 \nFe ii \n3.903 −2.500 \n1 \n2 \n\nNotes. The quality flag column indicates if the line was used (1) or not \nused (0) in the current study. The complete table is available at the CDS. \n\nReferences. 1: O'Brian et al. (1991); 2: Meléndez & Barbuy (2009); 3: \nNIST: Kramida et al. (2020). \n\n\n\nTable 3 .\n3The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.λ [Å] \nSpecies ξ [eV] log g f \nQuality \nflag \nRef. \n\n3829.355 \nMg i \n2.709 −0.227 \n0 \n3 \n4571.096 \nMg i \n0.000 −5.620 \n1 \n3 \n... \n... \n... \n... \n... \n6606.956 \nTi ii \n2.061 −2.790 \n1 \n3 \n7214.729 \nTi ii \n2.590 −1.750 \n1 \n3 \n\n\n\nTable 5 .\n5Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.Name \nV \n[mag] \n\n∆V \n[mag] \n\nγ \n[km/s] \n\n∆RV \n[km/s] \n\nLC \nsource a \n\nPeriod \n[days] \n\nT 0 − 2 400 000 \n[days] \n\nEphemerides \nsource b \n\nR TrA \n6.66 ± 0.02 0.56 ± 0.02 -12.44 ± 2.54 28.1 ± 2.2 \n1 \n3.3892582 \n44 423.59377 \n1 \nT Vul \n5.75 ± 0.03 0.64 ± 0.04 \n0.85 ± 0.11 34.5 ± 3.5 \n1 \n4.435462 \n57 632.18950 \n0 \n50 735.20310 \n2 \nFF Aql \n5.19 ± 0.01 0.33 ± 0.03 -11.36 ± 0.10 16.1 ± 1.5 \n2 \n4.470916 \n56 888.66207 \n0 \nS Cru \n6.60 ± 0.02 0.73 ± 0.03 \n-4.70 ± 0.28 33.8 ± 4.3 \n1 \n4.68973 \n55 279.67331 \n0 \nδ Cep \n3.94 ± 0.01 0.84 ± 0.04 -16.32 ± 0.20 38.9 ± 4.9 \n1 \n5.3663 \n56 917.54900 \n0 \nY Sgr \n5.74 ± 0.02 0.72 ± 0.03 \n-2.84 ± 3.02 34.2 ± 3.0 \n1 \n5.7733866 \n54 946.78504 \n1 \nXX Sgr \n8.86 ± 0.02 0.87 ± 0.04 \n12.30 ± 0.46 41.6 ± 1.2 \n1 \n6.4243013 \n54 946.67631 \n1 \nη Aql \n3.89 ± 0.02 0.78 ± 0.03 -14.47 ± 0.63 42.1 ± 2.2 \n1 \n7.1768572 \n50 724.43545 \n1 \nS Sge \n5.61 ± 0.03 0.72 ± 0.03 -16.40 ± 0.25 39.3 ± 4.7 \n1 \n8.3445707 \n57 636.28926 \n0 \n48 746.52000 \n3 \n8.3823514 \n50 339.17710 \n4 \nV500 Sco \n8.74 ± 0.01 0.71 ± 0.05 \n-7.61 ± 3.37 32.2 ± 5.4 \n1 \n9.317 \n44 789.61621 \n1 \nβ Dor \n5.25 ± 0.01 0.63 ± 0.06 \n8.17 ± 0.11 32.9 ± 2.3 \n2 \n9.84308 \n53 284.79472 \n0 \n9.8527371 \n41 149.27150 \n4 \nζ Gem \n3.89 ± 0.03 0.49 ± 0.03 \n6.78 ± 0.42 26.5 ± 1.0 \n1 \n10.149816 \n54 805.78476 \n1 \nVY Sgr \n11.46 ± 0.02 1.14 ± 0.05 \n14.23 ± 1.58 50.4 ± 4.4 \n1 \n13.55845 \n54 171.32722 \n1 \nUZ Sct \n11.25 ± 0.03 0.75 ± 0.05 \n39.52 ± 0.18 44.7 ± 3.5 \n0 \n14.744162 \n56 171.62381 \n0 \nY Oph \n6.15 ± 0.02 0.49 ± 0.03 \n-7.56 ± 2.01 16.5 ± 2.0 \n1 \n17.12633896 53 209.96732 \n1 \nRY Sco \n8.02 ± 0.02 0.83 ± 0.03 -17.73 ± 2.02 30.1 ± 3.0 \n1 \n20.321538 \n45 077.36644 \n1 \nRZ Vel \n7.09 ± 0.04 1.17 ± 0.05 \n24.11 ± 4.52 54.6 ± 5.4 \n1 \n20.399727 \n50 776.13608 \n1 \nV340 Ara 10.20 ± 0.06 1.08 ± 0.07 -76.25 ± 4.19 47.7 ± 9.3 \n1 \n20.811386 \n56 138.97597 \n0 \nWZ Sgr \n8.03 ± 0.03 1.08 ± 0.05 -17.78 ± 1.17 54.2 ± 2.9 \n1 \n21.849708 \n50 691.28534 \n1 \nRS Pup \n6.99 ± 0.03 1.11 ± 0.04 \n25.80 ± 1.91 48.5 ± 2.1 \n1 \n41.488634 \n53 092.94483 \n1 \n\n\n\nTable 6 .\n6Mean atmospheric parameters derived for the 20 calibrating Cepheids.\n\nTable 7 .\n7Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.3 < P < 5 \n[days] \n\n5 < P < 7 \n[days] \n\n7 < P < 9.5 \n[days] \n\n9.5 < P < 10.5 \n[days] \n\nc 0 \n0.44 ± 0.21 0.44 ± 0.23 \n0.48 ± 0.26 0.47 ± 0.12 \nc 1 \n2.1 ± 0.5 \n2.1 ± 0.5 \n−4.4 ± 0.6 \n−4.4 ± 0.2 \nc 2 \n0.13 ± 0.22 0.15 ± 0.24 \n0.14 ± 0.28 0.04 ± 0.12 \nc 3 −4.5 ± 1.4 \n2.2 ± 1.4 \n2.6 ± 2.0 \n4.2 ± 2.9 \nc 4 \n0.06 ± 0.21 0.06 ± 0.23 \n0.05 ± 0.27 0.04 ± 0.11 \nc 5 \n1.4 ± 3.6 \n2.6 ± 3.7 \n2.2 ± 5.6 \n4.4 ± 2.7 \nc 6 \n0.03 ± 0.21 0.03 ± 0.22 \n0.03 ± 0.28 0.02 ± 0.11 \nc 7 \n1.3 ± 8.3 \n2.7 ± 7.6 \n−0.93 ± 9.56 \n4.2 ± 6.9 \nc 8 \n0.01 ± 0.19 \n... \n... \n... \nc 9 \n2.8 ± 16.7 \n... \n... \n... \n\n\n\nTable 8 .\n8Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.Name \nDataset \nMJD \n[d] \n[Mg/H] ± σ N Mg ... \nTi i ± σ \nN Ti i \nTi ii ± σ \nN Ti ii \n[Ti/H] ± σ \n\nR TrA \nHARPS 53150.1453265 0.20 ± 0.11 \n1 \n... −0.21 ± 0.15 \n8 \n0.05 ± 0.10 \n2 \n−0.12 ± 0.19 \nR TrA \nHARPS 53150.1353086 0.17 ± 0.11 \n1 \n... −0.17 ± 0.27 \n12 \n0.01 ± 0.10 \n3 \n−0.14 ± 0.25 \n... \n... \n... \n... \n... \n... \n... \n... \n... \n... \n... \nRS Pup STELLA 57708.2167938 0.24 ± 0.11 \n2 \n... −0.06 ± 0.28 \n15 \n0.22 ± 0.10 \n4 \n0.09 ± 0.19 \nRS Pup STELLA 57713.2628277 \n... \n... \n... \n... \n... \n... \n... \n... \n\n\n\nTable 9 .\n9Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.Name \n[Mg/H] ± σ (std) N spec ... [Ti i/H] ± σ \n[Ti ii/H] ± σ \n[Ti/H] ± σ (std) \nN spec \n\nR TrA \n0.10 ± 0.03 (0.08) \n15 \n... −0.19 ± 0.04 −0.09 ± 0.03 −0.18 ± 0.04 (0.10) \n15 \nT Vul \n0.07 ± 0.03 (0.15) \n26 \n... −0.11 ± 0.03 −0.07 ± 0.02 −0.10 ± 0.03 (0.10) \n26 \n... \n... \n... \n... \n... \n... \n... \n... \nWZ Sgr 0.12 ± 0.06 (0.03) \n4 \n... \n0.08 ± 0.07 \n0.01 ± 0.05 \n0.06 ± 0.07 (0.12) \n7 \nRS Pup 0.09 ± 0.03 (0.14) \n10 \n... \n0.02 ± 0.04 \n0.16 ± 0.03 \n0.04 ± 0.04 (0.14) \n16 \n\n\n\nTable 10 .\n10Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).Abundance \nratio \nDatasets 1 \nZero-point \ndifference \nN Common \n\n[Fe/H] \nP18−TS \n0.07 ± 0.06 \n13 \n[Fe/H] \nLII−TS \n0.05 ± 0.07 \n20 \n[Fe/H] \nLIII−TS \n0.11 ± 0.08 \n6 \n[Fe/H] \nLII−LEM \n0.07 ± 0.12 \n55 \n[Fe/H] \nYON−LIII −0.33 ± 0.15 \n21 \n[Mg/H] \nLII−TS \n−0.14 ± 0.11 \n16 \n[Mg/H] \nLIII−TS \n0.11 ± 0.16 \n5 \n[Mg/H] \nLII−LEM −0.26 ± 0.24 \n35 \n[Mg/H] \nYON−LIII −0.09 ± 0.14 \n16 \n[Si/H] \nLII−TS \n−0.13 ± 0.07 \n20 \n[Si/H] \nLIII−TS \n−0.01 ± 0.09 \n6 \n[Si/H] \nLII−LEM −0.06 ± 0.11 \n55 \n[Si/H] \nYON−LIII −0.12 ± 0.07 \n20 \n[S/H] \nLII−TS \n0.11 ± 0.13 \n17 \n[S/H] \nLII−LEM −0.13 ± 0.19 \n45 \n[Ca/H] \nLII−TS \n−0.07 ± 0.10 \n20 \n[Ca/H] \nLIII−TS \n−0.01 ± 0.10 \n6 \n[Ca/H] \nLII−LEM −0.06 ± 0.17 \n55 \n[Ca/H] \nYON−LIII −0.16 ± 0.11 \n21 \n[Ti/H] \nLII−TS \n0.12 ± 0.09 \n20 \n[Ti/H] \nLIII−TS \n0.17 ± 0.12 \n6 \n[Ti/H] \nYON−LIII \n0.33 ± 0.20 \n20 \n\n\npyMOOGi: https://github.com/madamow/pymoogi Article number, page 10 of 31 R. da Silva et al.: A new and homogeneous metallicity scale for Galactic classical Cepheids\nINAF -Osservatorio Astronomico di Roma, via Frascati 33, 00078 Monte Porzio Catone, Italy 2 Agenzia Spaziale Italiana, Space Science Data Center, via del Politecnico snc, 00133 Rome, Italy 3 Dipartimento di Fisica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy\nAcknowledgements. It is a real pleasure to thank the anonymous referee for her/his positive appreciation of the current investigation and for her/his pertinent suggestions that improved the content and the readability of the paper. It is also a pleasure to thank F. Matteucci and E. Spitoni for several detailed discussions concerning the role of α elements in constraining chemical evolution models. R.P.K. gratefully acknowledges support by the Munich Excellence Cluster OriginsAppendix A: Cleaning from irregular atomic linesMany of the atomic lines present in our initial line lists of iron and α elements turned out to be not reliable enough. The abundances provided for our spectra by such irregular lines exhibit at least one of the following problems:Fig. A.5. These problems are probably not caused by saturation but by a wrong value of log g f , or by a wrong continuum identification, which is not always an easy task in regions with too many lines. For several cases we were able to adjust the log g f values using a high-resolution spectrum of Arcturus as reference. For other unsolved cases the lines were simply not used in our analysis.Appendix B.2: Atmospheric parameters and metallicityThe variation of the atmospheric parameters (T eff , log g, and ξ) and the iron abundances along the pulsation cycle are shown in the panels of figures from B.2 to B.5. It is worth mentioning that the number of points in the T eff panels are larger than in the other panels. This is because the determination of the effective temperature and of the other parameters are based on different and independent methods. For many spectra we were only able to derive the effective temperature (using the LDR method), whereas the derivation of the other parameters did not converge when we tried to force the ionization equilibrium of Fe i and Fe ii lines and the independence of the abundances on the equivalent widths (very often due to a limited number of good lines). The variable S Sge shows two outliers in the T eff curve plotted in the panel ofFig. B.2. However, these values might be correct because their shift with respect to the other spectra could be due to a limited accuracy in their phase determination, caused either by uncertainties in the period estimate or by any variation in period.Appendix B.3: Abundances of α elementsFigures B.6 shows the [α/H] abundance ratios for our sample as a function of the pulsation phase, in which we also compare the abundances from neutral (colored symbols) and ionized (light gray symbols) elements, when available. The agreement between the two species is normally within 2 sigma, with only a few exceptions for the Ca abundances (the Ca ii abundances are\n. S M Andrievsky, R E Luck, V V Kovtyukh, AJ. 1301880Andrievsky, S. M., Luck, R. E., & Kovtyukh, V. V. 2005, AJ, 130, 1880\n\n. M Asplund, A M Amarsi, N Grevesse, A&A. 653141Asplund, M., Amarsi, A. M., & Grevesse, N. 2021, A&A, 653, A141\n\n. M Asplund, N Grevesse, A J Sauval, P Scott, ARA&A. 47481Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481\n\n. M T Belmonte, J C Pickering, M P Ruffoni, ApJ. 848125Belmonte, M. T., Pickering, J. C., Ruffoni, M. P., et al. 2017, ApJ, 848, 125\n\n. G Bono, F Caputo, S Cassisi, ApJ. 543955Bono, G., Caputo, F., Cassisi, S., et al. 2000a, ApJ, 543, 955\n\n. G Bono, F Caputo, M Marconi, I Musella, ApJ. 715277Bono, G., Caputo, F., Marconi, M., & Musella, I. 2010, ApJ, 715, 277\n\n. G Bono, V Castellani, M Marconi, ApJ. 56583Bono, G., Castellani, V., & Marconi, M. 2002, ApJ, 565, L83\n\n. G Bono, M Marconi, R F Stellingwerf, ApJS. 122167Bono, G., Marconi, M., & Stellingwerf, R. F. 1999, ApJS, 122, 167\n\n. G Bono, M Marconi, R F Stellingwerf, A&A. 360245Bono, G., Marconi, M., & Stellingwerf, R. F. 2000b, A&A, 360, 245\n\n. V F Braga, J Crestani, M Fabrizio, ApJ. 91985Braga, V. F., Crestani, J., Fabrizio, M., et al. 2021, ApJ, 919, 85\n\n. V F Braga, P B Stetson, G Bono, AJ. 152170Braga, V. F., Stetson, P. B., Bono, G., et al. 2016, AJ, 152, 170\n\n. L Breuval, P Kervella, R I Anderson, A&A. 643115Breuval, L., Kervella, P., Anderson, R. I., et al. 2020, A&A, 643, A115\n\n. L Breuval, P Kervella, P Wielgórski, ApJ. 91338Breuval, L., Kervella, P., Wielgórski, P., et al. 2021, ApJ, 913, 38\n\n. E Caffau, S Andrievsky, S Korotin, A&A. 58516Caffau, E., Andrievsky, S., Korotin, S., et al. 2016, A&A, 585, A16\n\n. E Caffau, P Bonifacio, R Faraggiana, A&A. 441533Caffau, E., Bonifacio, P., Faraggiana, R., et al. 2005, A&A, 441, 533\n\n. E Caffau, R Faraggiana, P Bonifacio, H G Ludwig, M Steffen, A&A. 470699Caffau, E., Faraggiana, R., Bonifacio, P., Ludwig, H. G., & Steffen, M. 2007, A&A, 470, 699\n\nArXiv Astrophysics e-prints. F Castelli, R L Kurucz, astro-ph/0405087Castelli, F. & Kurucz, R. L. 2004, ArXiv Astrophysics e-prints [astro-ph/0405087]\n\n. G Cescutti, F Matteucci, P François, C Chiappini, A&A. 462943Cescutti, G., Matteucci, F., François, P., & Chiappini, C. 2007, A&A, 462, 943\n\n. C Chiappini, F Matteucci, D Romano, ApJ. 5541044Chiappini, C., Matteucci, F., & Romano, D. 2001, ApJ, 554, 1044\n\n. J Crestani, V F Braga, M Fabrizio, ApJ. 91410125A&ACrestani, J., Braga, V. F., Fabrizio, M., et al. 2021, ApJ, 914, 10 da Silva, R., Lemasle, B., Bono, G., et al. 2016, A&A, 586, A125\n\n. G De Somma, M Marconi, S Cassisi, MNRAS. 5081473De Somma, G., Marconi, M., Cassisi, S., et al. 2021, MNRAS, 508, 1473\n\nOptical and IR Telescope Instrumentation and Detectors. H Dekker, S D'odorico, A Kaufer, B Delabre, H Kotzlowski, Proc. SPIE. M. Iye & A. F. MoorwoodSPIE4008Dekker, H., D'Odorico, S., Kaufer, A., Delabre, B., & Kotzlowski, H. 2000, in Proc. SPIE, Vol. 4008, Optical and IR Telescope Instrumentation and Detec- tors, ed. M. Iye & A. F. Moorwood, 534-545\n\n. E A Den Hartog, J E Lawler, C Sneden, J J Cowan, A Brukhovesky, ApJS. 24333Den Hartog, E. A., Lawler, J. E., Sneden, C., Cowan, J. J., & Brukhovesky, A. 2019, ApJS, 243, 33\n\n. E A Den Hartog, M P Ruffoni, J E Lawler, ApJS. 21523Den Hartog, E. A., Ruffoni, M. P., Lawler, J. E., et al. 2014, ApJS, 215, 23\n\n. M Dessauges-Zavadsky, F Calura, J X Prochaska, S D'odorico, F Matteucci, A&A. 470431Dessauges-Zavadsky, M., Calura, F., Prochaska, J. X., D'Odorico, S., & Mat- teucci, F. 2007, A&A, 470, 431\n\n. V D'orazi, S Desidera, R G Gratton, A&A. 59819D'Orazi, V., Desidera, S., Gratton, R. G., et al. 2017, A&A, 598, A19\n\n. V D'orazi, E Oliva, A Bragaglia, A&A. 63338D'Orazi, V., Oliva, E., Bragaglia, A., et al. 2020, A&A, 633, A38\n\n. S Duffau, E Caffau, L Sbordone, A&A. 604128Duffau, S., Caffau, E., Sbordone, L., et al. 2017, A&A, 604, A128\n\n. N R Evans, AJ. 104216Evans, N. R. 1992, AJ, 104, 216\n\n. N R Evans, D L Welch, M H Slovak, Thomas G Barnes, I Moffett, T J , AJ. 1061599Evans, N. R., Welch, D. L., Slovak, M. H., Barnes, Thomas G., I., & Moffett, T. J. 1993, AJ, 106, 1599\n\n. B.-Q For, C Sneden, AJ. 1401694For, B.-Q. & Sneden, C. 2010, AJ, 140, 1694\n\n. W L Freedman, B F Madore, T Hoyt, ApJ. 89157Freedman, W. L., Madore, B. F., Hoyt, T., et al. 2020, ApJ, 891, 57\n\n. K Fukue, N Matsunaga, R Yamamoto, ApJ. 81264Fukue, K., Matsunaga, N., Yamamoto, R., et al. 2015, ApJ, 812, 64\n\n. K Genovali, B Lemasle, G Bono, A&A. 56637Genovali, K., Lemasle, B., Bono, G., et al. 2014, A&A, 566, A37\n\n. K Genovali, B Lemasle, G Bono, A&A. 554132Genovali, K., Lemasle, B., Bono, G., et al. 2013, A&A, 554, A132\n\n. K Genovali, B Lemasle, R Da Silva, A&A. 58017Genovali, K., Lemasle, B., da Silva, R., et al. 2015, A&A, 580, A17\n\n. G Gilmore, S Randich, M Asplund, The Messenger. 14725Gilmore, G., Randich, S., Asplund, M., et al. 2012, The Messenger, 147, 25\n\n. M A T Groenewegen, A&A. 48825Groenewegen, M. A. T. 2008, A&A, 488, 25\n\n. M A Groenewegen, A&A. 6198Groenewegen, M. A. T. 2018, A&A, 619, A8\n\n. M A T Groenewegen, A&A. 640113Groenewegen, M. A. T. 2020, A&A, 640, A113\n\n. L Inno, N Matsunaga, G Bono, ApJ. 76484Inno, L., Matsunaga, N., Bono, G., et al. 2013, ApJ, 764, 84\n\n. L Inno, N Matsunaga, M Romaniello, A&A. 57630Inno, L., Matsunaga, N., Romaniello, M., et al. 2015, A&A, 576, A30\n\n. L Inno, M A Urbaneja, N Matsunaga, MNRAS. 48283Inno, L., Urbaneja, M. A., Matsunaga, N., et al. 2019, MNRAS, 482, 83\n\n. M Jian, D Taniguchi, N Matsunaga, MNRAS. 4941724Jian, M., Taniguchi, D., Matsunaga, N., et al. 2020, MNRAS, 494, 1724\n\n. A Kaufer, O Stahl, S Tubbesing, The Messenger. 958Kaufer, A., Stahl, O., Tubbesing, S., et al. 1999, The Messenger, 95, 8\n\n. C S Kochanek, B J Shappee, K Z Stanek, PASP. 129104502Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, PASP, 129, 104502\n\n. V V Kovtyukh, MNRAS. 378617Kovtyukh, V. V. 2007, MNRAS, 378, 617\n\n. V V Kovtyukh, S M Andrievsky, S I Belik, R E Luck, AJ. 129433Kovtyukh, V. V., Andrievsky, S. M., Belik, S. I., & Luck, R. E. 2005, AJ, 129, 433\n\nA Kramida, R Yu, J Reader, & the NIST ASD Team. 2020, NIST Atomic Spectra Database. version 5.8Kramida, A., Yu., R., Reader, J., & the NIST ASD Team. 2020, NIST Atomic Spectra Database, version 5.8\n\nR L Kurucz, I Furenlid, J Brault, L Testerman, Solar flux atlas from 296 to 1300 nm. Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, Solar flux atlas from 296 to 1300 nm\n\n. J E Lawler, A Guzman, M P Wood, C Sneden, J J Cowan, ApJS. 20511Lawler, J. E., Guzman, A., Wood, M. P., Sneden, C., & Cowan, J. J. 2013, ApJS, 205, 11\n\n. B Lemasle, P François, K Genovali, A&A. 55831Lemasle, B., François, P., Genovali, K., et al. 2013, A&A, 558, A31\n\n. B Lemasle, M A T Groenewegen, E K Grebel, A&A. 60885Lemasle, B., Groenewegen, M. A. T., Grebel, E. K., et al. 2017, A&A, 608, A85\n\n. B Lemasle, M Hanke, J Storm, G Bono, E K Grebel, A&A. 64171Lemasle, B., Hanke, M., Storm, J., Bono, G., & Grebel, E. K. 2020, A&A, 641, A71\n\n. M Limongi, A Chieffi, ApJ. 592404Limongi, M. & Chieffi, A. 2003, ApJ, 592, 404\n\n. G Liu, N Arav, D S N Rupke, ApJS. 2219Liu, G., Arav, N., & Rupke, D. S. N. 2015, ApJS, 221, 9\n\n. R E Luck, S M Andrievsky, AJ. 128343Luck, R. E. & Andrievsky, S. M. 2004, AJ, 128, 343\n\n. R E Luck, S M Andrievsky, A Fokin, V V Kovtyukh, AJ. 13698Luck, R. E., Andrievsky, S. M., Fokin, A., & Kovtyukh, V. V. 2008, AJ, 136, 98\n\n. R E Luck, S M Andrievsky, V V Kovtyukh, W Gieren, D Graczyk, AJ. 14251Luck, R. E., Andrievsky, S. M., Kovtyukh, V. V., Gieren, W., & Graczyk, D. 2011, AJ, 142, 51\n\n. R E Luck, D L Lambert, AJ. 142136Luck, R. E. & Lambert, D. L. 2011, AJ, 142, 136\n\n. D Magurno, C Sneden, V F Braga, ApJ. 86457Magurno, D., Sneden, C., Braga, V. F., et al. 2018, ApJ, 864, 57\n\n. M Marconi, R Molinaro, G Bono, ApJ. 7686Marconi, M., Molinaro, R., Bono, G., et al. 2013, ApJ, 768, L6\n\n. M Marconi, I Musella, G Fiorentino, ApJ. 632590Marconi, M., Musella, I., & Fiorentino, G. 2005, ApJ, 632, 590\n\n. N Matsunaga, M Jian, D Taniguchi, S S Elgueta, MNRAS. 5061031Matsunaga, N., Jian, M., Taniguchi, D., & Elgueta, S. S. 2021, MNRAS, 506, 1031\n\n. M Mayor, F Pepe, D Queloz, The Messenger. 11420Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20\n\n. A Mcwilliam, G Wallerstein, M Mottini, ApJ. 778149McWilliam, A., Wallerstein, G., & Mottini, M. 2013, ApJ, 778, 149\n\n. J Meléndez, B Barbuy, A&A. 497611Meléndez, J. & Barbuy, B. 2009, A&A, 497, 611\n\nA&A proofs: manuscript no. daSilvaetal2021. A&A proofs: manuscript no. daSilvaetal2021\n\n. J H Minniti, L Sbordone, A Rojas-Arriagada, A&A. 64092Minniti, J. H., Sbordone, L., Rojas-Arriagada, A., et al. 2020, A&A, 640, A92\n\n. J H Minniti, M Zoccali, A Rojas-Arriagada, A&A. 654138Minniti, J. H., Zoccali, M., Rojas-Arriagada, A., et al. 2021, A&A, 654, A138\n\nThe Solar Spectrum 2935 Å to 8770 Å. C E Moore, M Minnaert, J Houtgast, Moore, C. E., Minnaert, M., & Houtgast, J. 1966, The Solar Spectrum 2935 Å to 8770 Å\n\n. N Nardetto, J H Groh, S Kraus, F Millour, D Gillet, A&A. 4891263Nardetto, N., Groh, J. H., Kraus, S., Millour, F., & Gillet, D. 2008, A&A, 489, 1263\n\n. N Nardetto, D Mourard, P Kervella, A&A. 453309Nardetto, N., Mourard, D., Kervella, P., et al. 2006, A&A, 453, 309\n\n. N Nardetto, E Poretti, A Gallenne, A&A. 61692Nardetto, N., Poretti, E., Gallenne, A., et al. 2018, A&A, 616, A92\n\n. G Natale, M Marconi, G Bono, ApJ. 67493Natale, G., Marconi, M., & Bono, G. 2008, ApJ, 674, L93\n\n. J R Neeley, M Monelli, M Marengo, ApJ. 920152Neeley, J. R., Monelli, M., Marengo, M., et al. 2021, ApJ, 920, 152\n\n. H R Neilson, M Cantiello, N Langer, A&A. 5299Neilson, H. R., Cantiello, M., & Langer, N. 2011, A&A, 529, L9\n\n. F Niedermann, M S Sloth, Phys. Rev. D. 10263527Niedermann, F. & Sloth, M. S. 2020, Phys. Rev. D, 102, 063527\n\n. P E Nissen, Y Q Chen, M Asplund, M Pettini, A&A. 415993Nissen, P. E., Chen, Y. Q., Asplund, M., & Pettini, M. 2004, A&A, 415, 993\n\n. T R O'brian, M E Wickliffe, J E Lawler, W Whaling, J W Brault, Journal of the Optical Society of America B Optical Physics. 81185O'Brian, T. R., Wickliffe, M. E., Lawler, J. E., Whaling, W., & Brault, J. W. 1991, Journal of the Optical Society of America B Optical Physics, 8, 1185\n\n. J W Pel, A&AS. 24413Pel, J. W. 1976, A&AS, 24, 413\n\n. G Pietrzyński, D Graczyk, W Gieren, Nature. 49576Pietrzyński, G., Graczyk, D., Gieren, W., et al. 2013, Nature, 495, 76\n\n. N Aghanim, Planck CollaborationY Akrami, Planck CollaborationA&A. 6416Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6\n\n. P G Prada Moroni, M Gennaro, G Bono, ApJ. 749108Prada Moroni, P. G., Gennaro, M., Bono, G., et al. 2012, ApJ, 749, 108\n\n. B Proxauf, R Da Silva, V V Kovtyukh, A&A. 61682Proxauf, B., da Silva, R., Kovtyukh, V. V., et al. 2018, A&A, 616, A82\n\n. S Randich, G Gilmore, -Eso Gaia, Consortium, The Messenger. 15447Randich, S., Gilmore, G., & Gaia-ESO Consortium. 2013, The Messenger, 154, 47\n\n. A G Riess, S Casertano, W Yuan, L M Macri, D Scolnic, ApJ. 87685Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ, 876, 85\n\n. V Ripepi, G Catanzaro, G Clementini, arXiv:2201.01126arXiv e-printsRipepi, V., Catanzaro, G., Clementini, G., et al. 2022, arXiv e-prints, arXiv:2201.01126\n\n. V Ripepi, G Catanzaro, R Molinaro, MNRAS. 5084047Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2021, MNRAS, 508, 4047\n\n. V Ripepi, G Catanzaro, R Molinaro, A&A. 642230Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2020, A&A, 642, A230\n\n. V Ripepi, R Molinaro, I Musella, A&A. 62514Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14\n\n. M Romaniello, F Primas, M Mottini, A&A. 488731Romaniello, M., Primas, F., Mottini, M., et al. 2008, A&A, 488, 731\n\n. M Romaniello, A Riess, S Mancino, arXiv:2110.08860arXiv e-printsRomaniello, M., Riess, A., Mancino, S., et al. 2021, arXiv e-prints, arXiv:2110.08860\n\n. M P Ruffoni, E A Hartog, J E Lawler, MNRAS. 4413127Ruffoni, M. P., Den Hartog, E. A., Lawler, J. E., et al. 2014, MNRAS, 441, 3127\n\n. J D Scargle, ApJ. 263835Scargle, J. D. 1982, ApJ, 263, 835\n\n. B J Shappee, J L Prieto, D Grupe, ApJ. 78848Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48\n\n. D M Skowron, J Skowron, P Mróz, Acta Astron. 69305Skowron, D. M., Skowron, J., Mróz, P., et al. 2019, Acta Astron., 69, 305\n\nC Sneden, The MOOG code. Sneden, C. 2002, The MOOG code, http://www.as.utexas.edu/~chris/ moog.html\n\n. S G Sousa, N C Santos, V Adibekyan, E Delgado-Mena, G Israelian, A&A. 57767Sousa, S. G., Santos, N. C., Adibekyan, V., Delgado-Mena, E., & Israelian, G. 2015, A&A, 577, A67\n\n. S G Sousa, N C Santos, G Israelian, M Mayor, M J P F G Monteiro, A&A. 469783Sousa, S. G., Santos, N. C., Israelian, G., Mayor, M., & Monteiro, M. J. P. F. G. 2007, A&A, 469, 783\n\n. K G Strassmeier, T Granzer, M Weber, Astronomische Nachrichten. 325527Strassmeier, K. G., Granzer, T., Weber, M., et al. 2004, Astronomische Nachrichten, 325, 527\n\n. K G Strassmeier, T Granzer, M Weber, Advances in Astronomy. 970306Strassmeier, K. G., Granzer, T., Weber, M., et al. 2010, Advances in Astronomy, 2010, 970306\n\n. Y Takeda, M Omiya, H Harakawa, B Sato, PASJ. 6881Takeda, Y., Omiya, M., Harakawa, H., & Sato, B. 2016, PASJ, 68, 81\n\n. D Taniguchi, N Matsunaga, M Jian, MNRAS. 5024210Taniguchi, D., Matsunaga, N., Jian, M., et al. 2021, MNRAS, 502, 4210\n\n. B J Taylor, PASP. 106452Taylor, B. J. 1994, PASP, 106, 452\n\n. V Vasilyev, H G Ludwig, B Freytag, B Lemasle, M Marconi, A&A. 61119Vasilyev, V., Ludwig, H. G., Freytag, B., Lemasle, B., & Marconi, M. 2018, A&A, 611, A19\n\n. K A Venn, M D Shetrone, M J Irwin, ApJ. 751102Venn, K. A., Shetrone, M. D., Irwin, M. J., et al. 2012, ApJ, 751, 102\n\nM Weber, T Granzer, K G Strassmeier, Proc. SPIE. SPIE845184510Weber, M., Granzer, T., & Strassmeier, K. G. 2012, in Proc. SPIE, Vol. 8451, Software and Cyberinfrastructure for Astronomy II, 84510K\n\n. M P Wood, J E Lawler, C Sneden, J J Cowan, ApJS. 20827Wood, M. P., Lawler, J. E., Sneden, C., & Cowan, J. J. 2013, ApJS, 208, 27\n\n. D Yong, B W Carney, M L Teixera De Almeida, B L Pohl, AJ. 1312256Yong, D., Carney, B. W., Teixera de Almeida, M. L., & Pohl, B. L. 2006, AJ, 131, 2256\n\n. Inaf -Osservatorio Astronomico Di Padova, 535122Padova, Italy4 INAF -Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy\n\nZentrum für Astronomie der Universität Heidelberg. Heidelberg, Germany69120Astronomisches Rechen-Institut, Zentrum für Astronomie der Uni- versität Heidelberg, Mönchhofstr. 12-14, 69120, Heidelberg, Ger- many\n\nDK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte. Blegdamsvej. 1761Niels Bohr International Academy, Niels Bohr Institute ; PSL Research UniversityCNRSNiels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte, Napoli, Italy 8 GEPI, Observatoire de Paris, PSL Research University, CNRS, 61\n\nAvenue de l'Observatoire. 75014Paris, FranceAvenue de l'Observatoire, 75014, Paris, France\n\nRinglaan 3, 1180, Brussels, Belgium 11 Science and Technology Department. Koninklijke Sterrenwacht Van België, Naples, ItalyParthenope University of NaplesKoninklijke Sterrenwacht van België, Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department, Parthenope University of Naples, Naples, Italy\n\nGermany 14 Institute for Astronomy. München, 2680University of Hawaii at ManoaMünchen, Germany 14 Institute for Astronomy, University of Hawaii at Manoa, 2680\n\nTokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH). Woodlawn Drive, Honolulu, Hongo, Bunkyo-Ku, E- 38200Japan 17 Instituto de Astrofísica de Canarias (IAC). Motoyama, Kamigamo, Kita-ku, Kyoto; La Laguna; La Laguna, Spain; Teramo, Italy19USA 15 Department of Astronomy, School of Science, The University of Tokyo ; Kyoto Sangyo University ; Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULLE-38205Koyama Astronomical Observatory. Via M. Maggini, s/n, I-64100Woodlawn Drive, Honolulu, HI 96822, USA 15 Department of Astronomy, School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH), Koyama Astronomical Observatory, Kyoto Sangyo University, Mo- toyama, Kamigamo, Kita-ku, Kyoto 603-8555, Japan 17 Instituto de Astrofísica de Canarias (IAC), La Laguna, E-38205, Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULL), E- 38200, La Laguna, Spain 19 INAF-Osservatorio Astronomico d'Abruzzo, Via M. Maggini, s/n, I-64100, Teramo, Italy\n\nAn der Sternwarte 16. Astrophysik Leibniz-Institut Für, Potsdam, AIPPotsdam, GermanyLeibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany\n\n. Nice Cedex. 7293Université Côte d'Azur, Observatoire de la CôteCSUniversité Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, La- grange UMR 7293, CS 34229, 06304, Nice Cedex 4, France\n"},"corpus_id":{"kind":"number","value":247177925,"string":"247,177,925"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2202.07945\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1051/0004-6361/202142957\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2202.07945v1.pdf\"\n ],\n \"pdf_sha\": \"09b13c4d0069eac62e09bc3b666ed7c567813bcd\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://www.aanda.org/articles/aa/pdf/forth/aa42957-21.pdf\",\n \"status\": \"BRONZE\"\n }\n}"},"title":{"kind":"string","value":"A new and homogeneous metallicity scale for Galactic classical Cepheids II. The abundance of iron and α elements"},"authors":{"kind":"list like","value":["R Da Silva","J Crestani","G Bono","V F Braga","V D'orazi","B Lemasle","M Bergemann","M Dall'ora","G Fiorentino","P François","M A T Groenewegen","L Inno","V Kovtyukh","R.-P Kudritzki","N Matsunaga","M Monelli","A Pietrinferni","L Porcelli","J Storm","M Tantalo","F Thévénin","R Da Silva","J Crestani","G Bono","V F Braga","V D'orazi","B Lemasle","M Bergemann","M Dall'ora","G Fiorentino","P François","M A T Groenewegen","L Inno","V Kovtyukh","R.-P Kudritzki","N Matsunaga","M Monelli","A Pietrinferni","L Porcelli","J Storm","M Tantalo","F Thévénin"],"string":"[\n \"R Da Silva\",\n \"J Crestani\",\n \"G Bono\",\n \"V F Braga\",\n \"V D'orazi\",\n \"B Lemasle\",\n \"M Bergemann\",\n \"M Dall'ora\",\n \"G Fiorentino\",\n \"P François\",\n \"M A T Groenewegen\",\n \"L Inno\",\n \"V Kovtyukh\",\n \"R.-P Kudritzki\",\n \"N Matsunaga\",\n \"M Monelli\",\n \"A Pietrinferni\",\n \"L Porcelli\",\n \"J Storm\",\n \"M Tantalo\",\n \"F Thévénin\",\n \"R Da Silva\",\n \"J Crestani\",\n \"G Bono\",\n \"V F Braga\",\n \"V D'orazi\",\n \"B Lemasle\",\n \"M Bergemann\",\n \"M Dall'ora\",\n \"G Fiorentino\",\n \"P François\",\n \"M A T Groenewegen\",\n \"L Inno\",\n \"V Kovtyukh\",\n \"R.-P Kudritzki\",\n \"N Matsunaga\",\n \"M Monelli\",\n \"A Pietrinferni\",\n \"L Porcelli\",\n \"J Storm\",\n \"M Tantalo\",\n \"F Thévénin\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Context. Classical Cepheids are the most popular distance indicators and tracers of young stellar populations. The key advantage is that they are bright and they can be easily identified in Local Group and Local Volume galaxies. Their evolutionary and pulsation properties depend on their chemical abundances. Aims. The main aim of this investigation is to perform a new and accurate abundance analysis of two tens of calibrating Galactic Cepheids using high spectral resolution (R∼40,000-115,000) and high S/N spectra (∼400) covering the entire pulsation cycle. Methods. We focus our attention on possible systematics affecting the estimate of atmospheric parameters and elemental abundances along the pulsation cycle. We cleaned the line list by using atomic transition parameters based on laboratory measurements and by removing lines that are either blended or display abundance variations along the pulsation cycle. Results. The spectroscopic approach that we developed brings forward small dispersions in the variation of the atmospheric parameters (σ(T eff )∼50 K, σ(log g)∼0.2 dex, and σ(ξ)∼0.2 km s −1 ) and in the abundance of both iron ( 0.05 dex) and α elements ( 0.10 dex) over the entire pulsation cycle. We also provide new and accurate effective temperature templates by splitting the calibrating Cepheids into four different period bins, ranging from short to long periods. For each period bin, we performed an analytical fit with Fourier series providing θ = 5040 /T eff as a function of the pulsation phase. Conclusions. The current findings are a good viaticum to trace the chemical enrichment of the Galactic thin disk by using classical Cepheids and a fundamental stepping stone for further investigations into the more metal-poor regime typical of Magellanic Cepheids."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":""},{"title":"Introduction","content":"The current empirical evidence indicates that the estimates of the Hubble constant based on Cosmic Microwave Background (CMB) and on the standard cosmological model (Λ Cold Dark Matter, ΛCDM) are at odds with the direct estimates based on primary (classical Cepheids) and secondary (type Ia Supernovae) distance indicators (Planck Collaboration et al. 2020;Riess et al. 2019). The current tension between early and late Universe measurements is at a 5-σ level. It is not clear whether it might be explained by unknown systematics or whether this disagreement is opening the path for new physics beyond the standard cosmological model (see, e.g., Niedermann & Sloth 2020, and references therein). In this context, it is worth mentioning Partly based on data obtained with the STELLA robotic telescopes in Tenerife, a facility of The Leibniz Institute for Astrophysics Potsdam (AIP) jointly operated by the AIP and by the Instituto de Astrofísica de Canarias (IAC).\nThe current empirical evidence indicates that the estimates of the Hubble constant based on Cosmic Microwave Background (CMB) and on the standard cosmological model (Λ Cold Dark Matter, ΛCDM) are at odds with the direct estimates based on primary (classical Cepheids) and secondary (type Ia Supernovae) distance indicators (Planck Collaboration et al. 2020;Riess et al. 2019). The current tension between early and late Universe measurements is at a 5-σ level. It is not clear whether it might be explained by unknown systematics or whether this disagreement is opening the path for new physics beyond the standard cosmological model (see, e.g., Niedermann & Sloth 2020, and references therein). In this context, it is worth mentioning Partly based on data obtained with the STELLA robotic telescopes in Tenerife, a facility of The Leibniz Institute for Astrophysics Potsdam (AIP) jointly operated by the AIP and by the Instituto de Astrofísica de Canarias (IAC).\nTables 2, 3, 4, 8, and 9 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/. that the use of the Tip of the Red Giant Branch (TRGB) as primary distance indicators gives values of the Hubble constant that are intermediate between those based on the CMB and those based on the Cepheid distance scale (Freedman et al. 2020).\nTables 2, 3, 4, 8, and 9 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/. that the use of the Tip of the Red Giant Branch (TRGB) as primary distance indicators gives values of the Hubble constant that are intermediate between those based on the CMB and those based on the Cepheid distance scale (Freedman et al. 2020).\nThere are solid reasons to believe that the difference between the TRGB and the Cepheid distance scale is caused by the dependence of the metal content on the zero-point and/or the slope of the adopted diagnostics (Bono et al. 2010;Pietrzyński et al. 2013). The dependence of optical and near-infrared diagnostics on metallicity is still a controversial issue for both theoretical (Marconi et al. 2005;Bono et al. 2010) and empirical (Groenewegen 2018(Groenewegen , 2020Ripepi et al. 2019Ripepi et al. , 2021Breuval et al. 2020Breuval et al. , 2021 approaches. However, the improved accuracy of geometrical distances from Gaia DR3, and the homogeneity of spectroscopic abundances will provide solid constraints on this longstanding problem (Ripepi et al. 2022, and references therein).\nThere are solid reasons to believe that the difference between the TRGB and the Cepheid distance scale is caused by the dependence of the metal content on the zero-point and/or the slope of the adopted diagnostics (Bono et al. 2010;Pietrzyński et al. 2013). The dependence of optical and near-infrared diagnostics on metallicity is still a controversial issue for both theoretical (Marconi et al. 2005;Bono et al. 2010) and empirical (Groenewegen 2018(Groenewegen , 2020Ripepi et al. 2019Ripepi et al. , 2021Breuval et al. 2020Breuval et al. , 2021 approaches. However, the improved accuracy of geometrical distances from Gaia DR3, and the homogeneity of spectroscopic abundances will provide solid constraints on this longstanding problem (Ripepi et al. 2022, and references therein).\nClassical Cepheids are bright (−2 ≤ M V ≤ −7 mag) radially pulsating stars with periods ranging from roughly one day to more than one hundred days (see, e.g., Skowron et al. 2019, and references therein). Their iron abundances range from solar values in the solar vicinity to ∼0.5 dex more metal-rich in the inner Galactic disk and to −0.5 dex more metal-poor in the outer disk (Inno et al. 2019). Cepheids in the Large Magellanic Cloud (LMC) are on average a factor of two more metal- Notes. The first four columns give the star name, the right ascension and declination, and the pulsation mode (0: fundamental; 1: first overtone; 0B: fundamental with bump). The Cols. 5 and 6 give the iron abundance from literature and the corresponding references. Columns from 7 to 10 show the number of optical spectra used for each spectrograph: N F : FEROS; N H : HARPS; N U : UVES; N S : STELLA. The last column lists the total number of spectra per target.\nClassical Cepheids are bright (−2 ≤ M V ≤ −7 mag) radially pulsating stars with periods ranging from roughly one day to more than one hundred days (see, e.g., Skowron et al. 2019, and references therein). Their iron abundances range from solar values in the solar vicinity to ∼0.5 dex more metal-rich in the inner Galactic disk and to −0.5 dex more metal-poor in the outer disk (Inno et al. 2019). Cepheids in the Large Magellanic Cloud (LMC) are on average a factor of two more metal- Notes. The first four columns give the star name, the right ascension and declination, and the pulsation mode (0: fundamental; 1: first overtone; 0B: fundamental with bump). The Cols. 5 and 6 give the iron abundance from literature and the corresponding references. Columns from 7 to 10 show the number of optical spectra used for each spectrograph: N F : FEROS; N H : HARPS; N U : UVES; N S : STELLA. The last column lists the total number of spectra per target.\nReferences. (a) 1: Genovali et al. (2014); for T Vul, FF Aql, δ Cep, and S Sge we assumed a typical error of 0.1 dex since they were not provided by the original authors; 2: Proxauf et al. (2018), from which the uncertainty adopted is the largest value between σ and std.\nReferences. (a) 1: Genovali et al. (2014); for T Vul, FF Aql, δ Cep, and S Sge we assumed a typical error of 0.1 dex since they were not provided by the original authors; 2: Proxauf et al. (2018), from which the uncertainty adopted is the largest value between σ and std.\npoor (Romaniello et al. 2021), and those in the Small Magellanic Cloud (SMC) a factor of four more metal-poor than solar (Lemasle et al. 2017). Cepheids are intermediate-mass stars crossing the instability strip during central helium burning, the so-called blue loops, and their progenitor mass ranges from ∼3 to ∼10 M -the exact limit depends on the chemical composition (Bono et al. 2000b;De Somma et al. 2021). Classical Cepheids are solid stellar beacons to trace young stellar populations even in the far side of the Galactic thin disk (Minniti et al. 2020(Minniti et al. , 2021 and in stellar systems experiencing recent star formation events such as dwarf irregulars and star forming galaxies (Neeley et al. 2021). Their period-luminosity relations (Breuval et al. 2020;Ripepi et al. 2020) provide accurate (at the level of ∼1-2%) measurements of individual distances. Individual distances together with detailed information concerning their elemental abundances (iron peak, α, neutron capture), provide the unique opportunity to investigate the radial gradients and, in turn, the recent (t ≤ 200-300 Myr) chemical enrichment history of the thin disk and nearby stellar systems (Romaniello et al. 2008;Lemasle et al. 2013;Genovali et al. 2015;da Silva et al. 2016). Furthermore, classical Cepheids are fundamental laboratories for investigating not only the advanced evolutionary phases of intermediate-mass stars (Bono et al. 2000a;Neilson et al. 2011;Prada Moroni et al. 2012;De Somma et al. 2021), but also for constraining the physical mechanisms driving their variability (Bono et al. 1999;Marconi et al. 2013).\npoor (Romaniello et al. 2021), and those in the Small Magellanic Cloud (SMC) a factor of four more metal-poor than solar (Lemasle et al. 2017). Cepheids are intermediate-mass stars crossing the instability strip during central helium burning, the so-called blue loops, and their progenitor mass ranges from ∼3 to ∼10 M -the exact limit depends on the chemical composition (Bono et al. 2000b;De Somma et al. 2021). Classical Cepheids are solid stellar beacons to trace young stellar populations even in the far side of the Galactic thin disk (Minniti et al. 2020(Minniti et al. , 2021 and in stellar systems experiencing recent star formation events such as dwarf irregulars and star forming galaxies (Neeley et al. 2021). Their period-luminosity relations (Breuval et al. 2020;Ripepi et al. 2020) provide accurate (at the level of ∼1-2%) measurements of individual distances. Individual distances together with detailed information concerning their elemental abundances (iron peak, α, neutron capture), provide the unique opportunity to investigate the radial gradients and, in turn, the recent (t ≤ 200-300 Myr) chemical enrichment history of the thin disk and nearby stellar systems (Romaniello et al. 2008;Lemasle et al. 2013;Genovali et al. 2015;da Silva et al. 2016). Furthermore, classical Cepheids are fundamental laboratories for investigating not only the advanced evolutionary phases of intermediate-mass stars (Bono et al. 2000a;Neilson et al. 2011;Prada Moroni et al. 2012;De Somma et al. 2021), but also for constraining the physical mechanisms driving their variability (Bono et al. 1999;Marconi et al. 2013).\nAlthough classical Cepheids are the cross-road of several recent and long-standing astrophysical problems, we still lack accurate investigations concerning the variation of their physical properties along the pulsation cycle. The current studies are mainly based on both optical and near-infrared (NIR) light curves, but we still lack detailed spectroscopic analysis concerning the variation of effective temperature, surface gravity, and microturbulent velocity (see, e.g., Lemasle et al. 2020;Kovtyukh 2007;Proxauf et al. 2018, and references therein).\nAlthough classical Cepheids are the cross-road of several recent and long-standing astrophysical problems, we still lack accurate investigations concerning the variation of their physical properties along the pulsation cycle. The current studies are mainly based on both optical and near-infrared (NIR) light curves, but we still lack detailed spectroscopic analysis concerning the variation of effective temperature, surface gravity, and microturbulent velocity (see, e.g., Lemasle et al. 2020;Kovtyukh 2007;Proxauf et al. 2018, and references therein).\nPioneering investigations based on multiple (from a few to almost two dozen) spectra along the pulsation cycle were provided more than ten years ago by Luck & Andrievsky (2004), Kovtyukh et al. (2005), Andrievsky et al. (2005), and Luck et al. (2008), in which they evaluated the variation of the atmospheric parameters and the abundances in both short-and long-period Cepheids. More recently, we performed a detailed investigation, based on a large dataset of high-resolution spectra, for a sample of 14 calibrating Cepheids (Proxauf et al. 2018). In that work, the number of spectra per object ranges from five to more than 120, and the spectra covered a significant portion of the pulsation cycle. In the same investigation we also used a new and more homogeneous list of Line Depth Ratios (LDR) to estimate the effective temperature directly from the spectra, an approach that is independent of the photometric properties. The reader interested in a more detailed discussion is referred to Kovtyukh (2007, and references therein). However, all these investigations were hampered by three limitations: a) Line list -We compiled a list of iron lines using line lists available in the literature. The sources of atomic transition parameters were inhomogeneous, because they did not pass a detailed scrutiny. b) Phase coverage -In spite of the large number of spectra adopted in the spectroscopic analysis, several objects had spectra collected at similar phases and, consequently, they had a modest phase coverage. c) Intrinsic variation -The inhomogeneity of the atomic transition parameters introduces uncertainties in the spectroscopic surface gravity and in the microturbulent velocity. This caused an increase in the standard deviation at fixed The vertical red lines indicate the position in the rest-frame wavelength of atomic lines either only used for the abundance determination or also adopted by the LDR method. For the sake of clarity the continuum of each spectrum was shifted to arbitrary levels.\nPioneering investigations based on multiple (from a few to almost two dozen) spectra along the pulsation cycle were provided more than ten years ago by Luck & Andrievsky (2004), Kovtyukh et al. (2005), Andrievsky et al. (2005), and Luck et al. (2008), in which they evaluated the variation of the atmospheric parameters and the abundances in both short-and long-period Cepheids. More recently, we performed a detailed investigation, based on a large dataset of high-resolution spectra, for a sample of 14 calibrating Cepheids (Proxauf et al. 2018). In that work, the number of spectra per object ranges from five to more than 120, and the spectra covered a significant portion of the pulsation cycle. In the same investigation we also used a new and more homogeneous list of Line Depth Ratios (LDR) to estimate the effective temperature directly from the spectra, an approach that is independent of the photometric properties. The reader interested in a more detailed discussion is referred to Kovtyukh (2007, and references therein). However, all these investigations were hampered by three limitations: a) Line list -We compiled a list of iron lines using line lists available in the literature. The sources of atomic transition parameters were inhomogeneous, because they did not pass a detailed scrutiny. b) Phase coverage -In spite of the large number of spectra adopted in the spectroscopic analysis, several objects had spectra collected at similar phases and, consequently, they had a modest phase coverage. c) Intrinsic variation -The inhomogeneity of the atomic transition parameters introduces uncertainties in the spectroscopic surface gravity and in the microturbulent velocity. This caused an increase in the standard deviation at fixed The vertical red lines indicate the position in the rest-frame wavelength of atomic lines either only used for the abundance determination or also adopted by the LDR method. For the sake of clarity the continuum of each spectrum was shifted to arbitrary levels.\npulsation phase, and in turn, some limitation in the analysis of their variation along the pulsation cycle.\npulsation phase, and in turn, some limitation in the analysis of their variation along the pulsation cycle.\nTo overcome these limitations, the current investigation is based on a new approach including five steps: i) We updated the atomic transition parameters of our line lists with more recent and accurate laboratory measurements available in the literature; ii) We collected new high signal-to-noise ratio (S/N) and highresolution spectra for our sample of calibrating Cepheids, and we also performed a detailed search for similar quality spectra available in online science archives. This translates into an increase in the number of calibrating Cepheids by almost 50% (from 14 to 20). Moreover, we also added new spectra to Cepheids already included in the former sample. The current sample includes 19 stars pulsating in the fundamental mode and, for the first time used by us as a calibrating Cepheid, one star (FF Aql) pulsating in the first overtone mode, and their metallicities range from −0.08 (β Dor) to 0.19 dex (VY Sgr); iii) We performed a detailed visual check of hundreds of lines along the pulsation cycle to create a new homogeneous line list. Special attention was paid in the identification of lines showing blends (or poor definition of the continuum), or having any correlation with the effective temperature (due to possible NLTE effects) or with the equivalent widths (due to saturated lines), or presenting any systematic over-or under-abundance; iv) We improved the algorithm that we use for the estimate of the atmospheric parameters, and in turn, for the estimate of the elemental abundances; v) In addition to the atmospheric parameters and iron abundances, the current study also includes the abundances of five α elements: Mg, Si, S, Ca, and Ti.\nTo overcome these limitations, the current investigation is based on a new approach including five steps: i) We updated the atomic transition parameters of our line lists with more recent and accurate laboratory measurements available in the literature; ii) We collected new high signal-to-noise ratio (S/N) and highresolution spectra for our sample of calibrating Cepheids, and we also performed a detailed search for similar quality spectra available in online science archives. This translates into an increase in the number of calibrating Cepheids by almost 50% (from 14 to 20). Moreover, we also added new spectra to Cepheids already included in the former sample. The current sample includes 19 stars pulsating in the fundamental mode and, for the first time used by us as a calibrating Cepheid, one star (FF Aql) pulsating in the first overtone mode, and their metallicities range from −0.08 (β Dor) to 0.19 dex (VY Sgr); iii) We performed a detailed visual check of hundreds of lines along the pulsation cycle to create a new homogeneous line list. Special attention was paid in the identification of lines showing blends (or poor definition of the continuum), or having any correlation with the effective temperature (due to possible NLTE effects) or with the equivalent widths (due to saturated lines), or presenting any systematic over-or under-abundance; iv) We improved the algorithm that we use for the estimate of the atmospheric parameters, and in turn, for the estimate of the elemental abundances; v) In addition to the atmospheric parameters and iron abundances, the current study also includes the abundances of five α elements: Mg, Si, S, Ca, and Ti.\nWe notice that the star AV Sgr, which was one of the calibrating Cepheids in Proxauf et al. (2018), is not included in the current work. The reason is that, after the revision of the Fe line list, we were not able to derive accurate estimates of the atmospheric parameters for many of the spectra of this variable. The number of spectra with accurate measurements was not enough to provide a good coverage of the pulsation cycle, and therefore, it was removed from the list of calibrating Cepheids. The parameters published in our previous paper can still be used, but keeping in mind that they are based on a different version of the line list.\nWe notice that the star AV Sgr, which was one of the calibrating Cepheids in Proxauf et al. (2018), is not included in the current work. The reason is that, after the revision of the Fe line list, we were not able to derive accurate estimates of the atmospheric parameters for many of the spectra of this variable. The number of spectra with accurate measurements was not enough to provide a good coverage of the pulsation cycle, and therefore, it was removed from the list of calibrating Cepheids. The parameters published in our previous paper can still be used, but keeping in mind that they are based on a different version of the line list.\nThe structure of the paper is the following: in Sect. 2 we discuss the spectroscopic dataset and provide detailed information concerning the current sample of calibrating Cepheids. In Sect. 3 we discuss the data reduction process and introduce the atomic line lists used. Sect. 4 includes a description of the radial-velocity and photometric data used. In Sect. 5 we explain how we derived the atmospheric parameters and their variation along the pulsation cycle. The iron and α-element abundances and their variation along the pulsation cycle are discussed in Sect. 6. A summary of the results and some final remarks concerning this project are given in Sect. 7.\nThe structure of the paper is the following: in Sect. 2 we discuss the spectroscopic dataset and provide detailed information concerning the current sample of calibrating Cepheids. In Sect. 3 we discuss the data reduction process and introduce the atomic line lists used. Sect. 4 includes a description of the radial-velocity and photometric data used. In Sect. 5 we explain how we derived the atmospheric parameters and their variation along the pulsation cycle. The iron and α-element abundances and their variation along the pulsation cycle are discussed in Sect. 6. A summary of the results and some final remarks concerning this project are given in Sect. 7."},{"title":"Spectroscopic datasets","content":""},{"title":"Spectroscopic datasets","content":"The spectroscopic datasets analyzed in the current paper are spectra collected using several different instruments. Three of them are mounted on telescopes of the European Southern Observatory (ESO): the High Accuracy Radial velocity Planet Searcher (HARPS; Mayor et al. 2003) at the 3.6 m, the Fiberfed Extended Range Optical Spectrograph (FEROS; Kaufer et al. 1999) at the 2.2 m MPG/ESO, and the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) at the Very Large Telescope. The other instrument is the STELLA Echelle Spectrograph (SES; Strassmeier et al. 2004Strassmeier et al. , 2010 located at the Izãna Observatory on Tenerife in the Canary islands.\nThe spectroscopic datasets analyzed in the current paper are spectra collected using several different instruments. Three of them are mounted on telescopes of the European Southern Observatory (ESO): the High Accuracy Radial velocity Planet Searcher (HARPS; Mayor et al. 2003) at the 3.6 m, the Fiberfed Extended Range Optical Spectrograph (FEROS; Kaufer et al. 1999) at the 2.2 m MPG/ESO, and the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) at the Very Large Telescope. The other instrument is the STELLA Echelle Spectrograph (SES; Strassmeier et al. 2004Strassmeier et al. , 2010 located at the Izãna Observatory on Tenerife in the Canary islands.\nThe spectral resolution of the quoted spectrographs for the instrument settings used are R∼40 000 (UVES), R∼115 000 (HARPS), R∼48 000 (FEROS), and R∼55 000 (STELLA). For details on the wavelength ranges of our spectra we refer the reader to Sect. 2 of the papers Proxauf et al. (2018) and Crestani et al. (2021).\nThe spectral resolution of the quoted spectrographs for the instrument settings used are R∼40 000 (UVES), R∼115 000 (HARPS), R∼48 000 (FEROS), and R∼55 000 (STELLA). For details on the wavelength ranges of our spectra we refer the reader to Sect. 2 of the papers Proxauf et al. (2018) and Crestani et al. (2021).\nThe entire spectroscopic sample includes reduced spectra downloaded from the ESO and the STELLA archives, for a total of 1383 spectra of 285 stars: 199 HARPS spectra of 9 stars, 419 FEROS spectra of 151 stars, 450 STELLA spectra of 68 stars, and 315 UVES spectra of 123 stars. Their S/N ranges from a few tens to more than 400 (about 70% of our spectra have S/N of at least 100). From this sample we selected the spectra of 20 targets that cover either a significant part (about half or more) or the entire pulsation cycle. From now on we refer to this subsample as our sample of 20 calibrating Cepheids. Details on the number of spectra are given in Table 1.\nThe entire spectroscopic sample includes reduced spectra downloaded from the ESO and the STELLA archives, for a total of 1383 spectra of 285 stars: 199 HARPS spectra of 9 stars, 419 FEROS spectra of 151 stars, 450 STELLA spectra of 68 stars, and 315 UVES spectra of 123 stars. Their S/N ranges from a few tens to more than 400 (about 70% of our spectra have S/N of at least 100). From this sample we selected the spectra of 20 targets that cover either a significant part (about half or more) or the entire pulsation cycle. From now on we refer to this subsample as our sample of 20 calibrating Cepheids. Details on the number of spectra are given in Table 1."},{"title":"Data reduction and analysis","content":""},{"title":"Data reduction and analysis","content":"As mentioned in Proxauf et al. (2018), the spectra from UVES and HARPS (Phase 3) were already pre-reduced (i.e., reduced up to the wavelength calibration step) by their own pipeline. As previously done, the FEROS spectra were reduced using a modified version of the Data Reduction System (DRS) pipeline. A few tens of FEROS spectra collected before 2004, which initially could not be reduced, were later on reduced and delivered by one of the Phase 3 Data Releases (DR3 and DR3.1). However, they are all low-quality spectra that were not included in our analysis.\nAs mentioned in Proxauf et al. (2018), the spectra from UVES and HARPS (Phase 3) were already pre-reduced (i.e., reduced up to the wavelength calibration step) by their own pipeline. As previously done, the FEROS spectra were reduced using a modified version of the Data Reduction System (DRS) pipeline. A few tens of FEROS spectra collected before 2004, which initially could not be reduced, were later on reduced and delivered by one of the Phase 3 Data Releases (DR3 and DR3.1). However, they are all low-quality spectra that were not included in our analysis.\nAfter the pre-reduction steps, all the selected spectra were normalized to the continuum using the Image Reduction and Analysis Facility (IRAF 1 ) by fitting cubic spline functions to a set of continuum windows. An exception is the case of the STELLA spectra, which were reduced by a dedicated pipeline (Weber et al. 2012) based on IRAF routines and that includes several reduction steps up to the continuum normalization.\nAfter the pre-reduction steps, all the selected spectra were normalized to the continuum using the Image Reduction and Analysis Facility (IRAF 1 ) by fitting cubic spline functions to a set of continuum windows. An exception is the case of the STELLA spectra, which were reduced by a dedicated pipeline (Weber et al. 2012) based on IRAF routines and that includes several reduction steps up to the continuum normalization.\nFor more details about the quality of the spectra we refer the reader to Sect. 3 of Proxauf et al. (2018), which also includes a figure (their Fig. 1) showing a few examples of HARPS, FEROS, UVES, and STELLA spectra of different metallicities and S/N. In our Fig. 1, we plot an example of HARPS spectra for β Dor and of STELLA spectra for ζ Gem in different pulsation phases, showing how the profile of the absorption lines may change (in depth and in shape) along the pulsation cycle. Studies of line asymmetry changes over the pulsation cycle of Cepheid stars were also reported by Nardetto et al. (2006) and Nardetto et al. (2008), in a series of papers on high-resolution spectroscopy, for Fe i and Hα lines in the optical and, more recently, by Nardetto et al. (2018) for the Na i line at 2208.969 nm.\nFor more details about the quality of the spectra we refer the reader to Sect. 3 of Proxauf et al. (2018), which also includes a figure (their Fig. 1) showing a few examples of HARPS, FEROS, UVES, and STELLA spectra of different metallicities and S/N. In our Fig. 1, we plot an example of HARPS spectra for β Dor and of STELLA spectra for ζ Gem in different pulsation phases, showing how the profile of the absorption lines may change (in depth and in shape) along the pulsation cycle. Studies of line asymmetry changes over the pulsation cycle of Cepheid stars were also reported by Nardetto et al. (2006) and Nardetto et al. (2008), in a series of papers on high-resolution spectroscopy, for Fe i and Hα lines in the optical and, more recently, by Nardetto et al. (2018) for the Na i line at 2208.969 nm."},{"title":"Line lists and equivalent widths","content":""},{"title":"Line lists and equivalent widths","content":"Similarly to what we did in our previous works, here we use three different line lists: The Cepheids variables are bright stars and thus we have at our disposal very high S/N, broad wavelength range spectra. The precision of the atmospheric parameters and of the chemical abundances, however, depends not only on the quality of the spectra, but also on the quality of the adopted line list. With this in mind, a large portion of the present work was dedicated to finding the best atomic transitions for iron and α elements that can be detected in Cepheids. In practice, this means i) collecting the most precise transition parameters, ii) removing absorption lines that are blended with other lines, and iii) eliminating lines that for any reason deviate significantly from the average value for their chemical species, or that display a dependency on any quantity that changes across the pulsation cycle, such as the effective temperature.\nSimilarly to what we did in our previous works, here we use three different line lists: The Cepheids variables are bright stars and thus we have at our disposal very high S/N, broad wavelength range spectra. The precision of the atmospheric parameters and of the chemical abundances, however, depends not only on the quality of the spectra, but also on the quality of the adopted line list. With this in mind, a large portion of the present work was dedicated to finding the best atomic transitions for iron and α elements that can be detected in Cepheids. In practice, this means i) collecting the most precise transition parameters, ii) removing absorption lines that are blended with other lines, and iii) eliminating lines that for any reason deviate significantly from the average value for their chemical species, or that display a dependency on any quantity that changes across the pulsation cycle, such as the effective temperature.\nIn order to address the first point, we departed from a list of hundreds of atomic transitions for iron and α elements commonly employed in the literature. We updated their transition parameters whenever possible with the updated laboratory measurements from Ruffoni Bottom: same as in the top panel but for the V-band light curve, its PLOESS fit and its mean value. We note that, within the uncertainty, the light curve is co-phased with the RV curve.\nIn order to address the first point, we departed from a list of hundreds of atomic transitions for iron and α elements commonly employed in the literature. We updated their transition parameters whenever possible with the updated laboratory measurements from Ruffoni Bottom: same as in the top panel but for the V-band light curve, its PLOESS fit and its mean value. We note that, within the uncertainty, the light curve is co-phased with the RV curve.\nFe ii, Lawler et al. (2013) for Ti i, and Wood et al. (2013) for Ti ii. We also made ample use of the astrophysical, but homogeneous and precise compilation of Meléndez & Barbuy (2009) for Fe ii. For the remaining lines, we adopted the transition parameters collected and updated by the National Institute of Standards and Technology (NIST) Atomic Spectra Database (Kramida et al. 2020). If a line is not available in any of these sources, it is eliminated from the preliminary list and not used for the computation of atmospheric parameters nor chemical abundances.\nFe ii, Lawler et al. (2013) for Ti i, and Wood et al. (2013) for Ti ii. We also made ample use of the astrophysical, but homogeneous and precise compilation of Meléndez & Barbuy (2009) for Fe ii. For the remaining lines, we adopted the transition parameters collected and updated by the National Institute of Standards and Technology (NIST) Atomic Spectra Database (Kramida et al. 2020). If a line is not available in any of these sources, it is eliminated from the preliminary list and not used for the computation of atmospheric parameters nor chemical abundances.\nWith this preliminary list at hand, we addressed the second point mentioned above by removing any blended lines, using as reference the Solar spectrum table by Moore et al. (1966) and synthetic spectra. Then we measured their equivalent width (EW) using the Automatic Routine for line Equivalent widths in stellar Spectra (ARES, Sousa et al. 2007Sousa et al. , 2015.\nWith this preliminary list at hand, we addressed the second point mentioned above by removing any blended lines, using as reference the Solar spectrum table by Moore et al. (1966) and synthetic spectra. Then we measured their equivalent width (EW) using the Automatic Routine for line Equivalent widths in stellar Spectra (ARES, Sousa et al. 2007Sousa et al. , 2015.\nFinally, we addressed the third point by removing deviant lines. In a first iteration, we estimated initial values for the atmospheric parameters and the chemical abundances. Lines are considered deviant if they result in abundances diverging by at least 3 σ from the mean for a given element, or if their behavior across the pulsation cycle is irregular. This latter situation is investigated by plotting all line-by-line measurements for each chemical species and individual exposure of our calibrating stars versus phase, effective temperature, and equivalent width, and removing lines that show strong trends in any of those planes. The calibrating stars mostly used are those with the largest number of individual spectra covering the whole pulsation cycle, such as β Dor, ζ Gem, and FF Aql. Tables 2 and 3 with quality flag 1, were used to compute the final atmospheric parameters and chemical abundances for the whole sample. The remaining lines, Fig. 4. Photometric amplitude as a function of the logarithmic period (Bailey diagram). Our calibrating Cepheids are represented by colored symbols whereas the Cepheids from literature Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006) are shown in light gray circles.\nFinally, we addressed the third point by removing deviant lines. In a first iteration, we estimated initial values for the atmospheric parameters and the chemical abundances. Lines are considered deviant if they result in abundances diverging by at least 3 σ from the mean for a given element, or if their behavior across the pulsation cycle is irregular. This latter situation is investigated by plotting all line-by-line measurements for each chemical species and individual exposure of our calibrating stars versus phase, effective temperature, and equivalent width, and removing lines that show strong trends in any of those planes. The calibrating stars mostly used are those with the largest number of individual spectra covering the whole pulsation cycle, such as β Dor, ζ Gem, and FF Aql. Tables 2 and 3 with quality flag 1, were used to compute the final atmospheric parameters and chemical abundances for the whole sample. The remaining lines, Fig. 4. Photometric amplitude as a function of the logarithmic period (Bailey diagram). Our calibrating Cepheids are represented by colored symbols whereas the Cepheids from literature Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006) are shown in light gray circles."},{"title":"The final clean lists, shown in","content":""},{"title":"The final clean lists, shown in","content":"shown in the same tables with quality flag 0, were measured only for comparison purposes.\nshown in the same tables with quality flag 0, were measured only for comparison purposes.\nAn example of a good Fe line is shown in Fig. 2 for β Dor and ζ Gem. Examples and a description of the typical behavior of irregular (less reliable) lines are found in the Appendix A, in the plots of Figs. A.1-A.5. These same irregular lines, however, might provide good results for other stars, for instance, of different spectral types, or having spectra of different quality or different resolution. This is the reason why we preferred to exclude some lines only for this particular analysis, and keep the initial line lists as they are to be used in future studies. It is worth mentioning that we only excluded lines that were clearly having problems in most of the stars in our sample.\nAn example of a good Fe line is shown in Fig. 2 for β Dor and ζ Gem. Examples and a description of the typical behavior of irregular (less reliable) lines are found in the Appendix A, in the plots of Figs. A.1-A.5. These same irregular lines, however, might provide good results for other stars, for instance, of different spectral types, or having spectra of different quality or different resolution. This is the reason why we preferred to exclude some lines only for this particular analysis, and keep the initial line lists as they are to be used in future studies. It is worth mentioning that we only excluded lines that were clearly having problems in most of the stars in our sample."},{"title":"Radial velocity and photometric data","content":""},{"title":"Radial velocity and photometric data","content":"The radial velocity (RV) measurements for our sample were done using the following methods: i) for HARPS and UVES spectra, we used IRAF packages to cross-correlate the target spectrum with a solar template spectrum (Solar flux atlas from 296 to 1300 nm, Kurucz et al. 1984) degraded to the UVES resolution; ii) for FEROS spectra we adopted the RVs derived by ARES, the routine we used for the equivalent width measurements (see Sect. 3.1); iii) the STELLA radial velocities are written in the header of the FITS files and they are based on crosscorrelations with a template spectrum, performed by a dedicated reduction pipeline also using IRAF. In case of the Cepheids, a G-type dwarf template is used.\nThe radial velocity (RV) measurements for our sample were done using the following methods: i) for HARPS and UVES spectra, we used IRAF packages to cross-correlate the target spectrum with a solar template spectrum (Solar flux atlas from 296 to 1300 nm, Kurucz et al. 1984) degraded to the UVES resolution; ii) for FEROS spectra we adopted the RVs derived by ARES, the routine we used for the equivalent width measurements (see Sect. 3.1); iii) the STELLA radial velocities are written in the header of the FITS files and they are based on crosscorrelations with a template spectrum, performed by a dedicated reduction pipeline also using IRAF. In case of the Cepheids, a G-type dwarf template is used.\nThe uncertainties adopted for these RV estimates depend on the method used: for HARPS, UVES, and STELLA spectra, the uncertainties are estimated by IRAF during the cross-correlation; for FEROS spectra, they are estimated by ARES using spectral windows free of lines. Typical values for our HARPS, UVES, and FEROS spectra range from a few m/s in the best cases to a few km s −1 , depending on the spectral type. For the STELLA spectra, we adopted an uncertainty of 0.2 km s −1 . This is a conservative estimate considering that typical formal errors for high S/N spectra of Cepheids stars is about 50 m s −1 and that the RV zero point for this spectrograph has RMS variations of 30 to 150 m s −1 over two years. The RV estimates and the corresponding uncertainties for our sample of calibrating Cepheids are listed in Table 4.\nThe uncertainties adopted for these RV estimates depend on the method used: for HARPS, UVES, and STELLA spectra, the uncertainties are estimated by IRAF during the cross-correlation; for FEROS spectra, they are estimated by ARES using spectral windows free of lines. Typical values for our HARPS, UVES, and FEROS spectra range from a few m/s in the best cases to a few km s −1 , depending on the spectral type. For the STELLA spectra, we adopted an uncertainty of 0.2 km s −1 . This is a conservative estimate considering that typical formal errors for high S/N spectra of Cepheids stars is about 50 m s −1 and that the RV zero point for this spectrograph has RMS variations of 30 to 150 m s −1 over two years. The RV estimates and the corresponding uncertainties for our sample of calibrating Cepheids are listed in Table 4.\nIn addition to our own RV measurements, we adopted literature RV curves and light curves to provide accurate ephemerides, meaning reference epoch (T 0 ) plus pulsation period (P) for these Table 4. Excerpt from the list of atmospheric parameters, Fe abundances, and radial velocities for each spectrum of the 20 calibrating Cepheids. Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. Columns 4, 5, and 6 give, respectively, the effective temperature and its standard deviation, surface gravity, and microturbulent velocity. The Cols. 7-8 and 9-10 list both the Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used (for the Fe ii abundances, a minimal uncertainty of 0.11 dex was adopt when computing the mean abundances, as described in the text). The last column gives the radial velocities with their uncertainties. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the mean magnitude in visual bands, the magnitude amplitude, the mean radial velocity, the RV amplitude, the light curve source, the pulsation period and zero-phase reference epoch of mean magnitude (or mean radial velocity), and the Ephemerides source. For the variables VY Sgr, RZ Vel, and WZ Sgr, our RV measurements and those by Groenewegen (2008, and references therein) show a shift in phase of −0.05, +0.03, and −0.03, respectively. To co-phase the two datasets, the latter sample of radial velocities plotted in Fig variables. More specifically, we collected RV curves and V-band light curves from Pel (1976) and Groenewegen (2008, and references therein). For UZ Sct, we adopted the V-band light curve from the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappee et al. 2014;Kochanek et al. 2017) because it is the only well-sampled light curve that can be co-phased with our RV time series.\nIn addition to our own RV measurements, we adopted literature RV curves and light curves to provide accurate ephemerides, meaning reference epoch (T 0 ) plus pulsation period (P) for these Table 4. Excerpt from the list of atmospheric parameters, Fe abundances, and radial velocities for each spectrum of the 20 calibrating Cepheids. Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. Columns 4, 5, and 6 give, respectively, the effective temperature and its standard deviation, surface gravity, and microturbulent velocity. The Cols. 7-8 and 9-10 list both the Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used (for the Fe ii abundances, a minimal uncertainty of 0.11 dex was adopt when computing the mean abundances, as described in the text). The last column gives the radial velocities with their uncertainties. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the mean magnitude in visual bands, the magnitude amplitude, the mean radial velocity, the RV amplitude, the light curve source, the pulsation period and zero-phase reference epoch of mean magnitude (or mean radial velocity), and the Ephemerides source. For the variables VY Sgr, RZ Vel, and WZ Sgr, our RV measurements and those by Groenewegen (2008, and references therein) show a shift in phase of −0.05, +0.03, and −0.03, respectively. To co-phase the two datasets, the latter sample of radial velocities plotted in Fig variables. More specifically, we collected RV curves and V-band light curves from Pel (1976) and Groenewegen (2008, and references therein). For UZ Sct, we adopted the V-band light curve from the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappee et al. 2014;Kochanek et al. 2017) because it is the only well-sampled light curve that can be co-phased with our RV time series.\nWe estimated the pulsation periods of our targets by using our own interactive method ) based on the Lomb-Scargle periodogram (Scargle 1982). Historically, the most commonly used T 0 is the time of maximum light (T max ), which matches the time of minimum velocity. Starting from Inno et al. (2015), we have been promoting the use of another reference for both Cepheids and RR Lyrae, that is, the epoch of the mean magnitude on the rising branch of the light curve (T opt mean ) or, equivalently, the epoch of mean velocity on the decreasing branch of the RV curve (T RV mean ). See Fig. 3 for a visualization of this reference epoch on both the light and RV curves. A detailed procedure to derive T opt mean is described in Braga et al. (2021). The advantages of the T opt mean reference epoch over T max are wellexplained in Inno et al. (2015) and quantitatively discussed in Braga et al. (2021). Name Notes. Columns from 1 to 5 give, respectively, the star name, the mean effective temperature and mean θ, the θ amplitude (see Section 5.2), and the number of spectra used to compute these mean values. Columns from 6 to 9 lists the surface gravity, microturbulent velocity, and iron abundances from neutral and ionized lines. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 4. Column 10 gives the weighted iron abundance and its standard error, calculated from Cols. 8 and 9. The standard deviation calculated using all individual measurements from both Fe i and Fe ii lines is also shown within parentheses. The last column gives the number of spectra used to derive log g, υ t , and the iron abundances.\nWe estimated the pulsation periods of our targets by using our own interactive method ) based on the Lomb-Scargle periodogram (Scargle 1982). Historically, the most commonly used T 0 is the time of maximum light (T max ), which matches the time of minimum velocity. Starting from Inno et al. (2015), we have been promoting the use of another reference for both Cepheids and RR Lyrae, that is, the epoch of the mean magnitude on the rising branch of the light curve (T opt mean ) or, equivalently, the epoch of mean velocity on the decreasing branch of the RV curve (T RV mean ). See Fig. 3 for a visualization of this reference epoch on both the light and RV curves. A detailed procedure to derive T opt mean is described in Braga et al. (2021). The advantages of the T opt mean reference epoch over T max are wellexplained in Inno et al. (2015) and quantitatively discussed in Braga et al. (2021). Name Notes. Columns from 1 to 5 give, respectively, the star name, the mean effective temperature and mean θ, the θ amplitude (see Section 5.2), and the number of spectra used to compute these mean values. Columns from 6 to 9 lists the surface gravity, microturbulent velocity, and iron abundances from neutral and ionized lines. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 4. Column 10 gives the weighted iron abundance and its standard error, calculated from Cols. 8 and 9. The standard deviation calculated using all individual measurements from both Fe i and Fe ii lines is also shown within parentheses. The last column gives the number of spectra used to derive log g, υ t , and the iron abundances.\nThe light and radial-velocity curves for all the sample stars are shown in the Appendix B, in Fig. B.1. The estimated ephemerides of our targets are listed in Table 5 together with the mean magnitudes and magnitude amplitudes in the visual band, and the mean radial velocities and RV amplitudes. We noticed that for the stars VY Sgr, RZ Vel, and WZ Sgr, for which the new period and T 0 values are based on the radial velocities derived by Groenewegen (2008, and references therein), the new ephemerides caused a shift in phase between the two datasets shown in Fig. B.1. To overcome this problem, we applied a shift of −0.05, +0.03, and −0.03, respectively, to the literature data plotted in these panels. Figure 4 shows the distribution of the current calibrating Cepheids (colored symbols), together with Galactic Cepheids available in the literature (light gray circles), in the so-called Bailey diagram (luminosity amplitude as a function of the logarithmic period). The plotted data display the classical \"V\" shape in which the local minimum around ten days is mainly caused by the Hertzsprung progression (Bono et al. 2000b). By having a look also at Fig. B.1, we see a subgroup of classical Cepheids showing a well defined bump on the light curve. The phase of such bumps moves from the decreasing branch for periods between ∼6 and ∼9 days, passing across the maximum for periods between ∼9 and ∼12 days, and along the rising branch for periods between ∼12 and ∼16 days. At the center of the Hertzsprung progression (P∼10 days) the light curve is flat topped and the luminosity amplitude attains a well defined minimum (Bono et al. 2000b(Bono et al. , 2002. These variables are called Bump Cepheids for avoiding to be mixed-up with the Beat Cepheids, that is, classical Cepheids pulsating simultaneously in two or more modes. The calibrating Cepheids cover both the short-and the longperiod range, and in particular, the current sample includes seven Cepheids across the Hertzsprung progression.\nThe light and radial-velocity curves for all the sample stars are shown in the Appendix B, in Fig. B.1. The estimated ephemerides of our targets are listed in Table 5 together with the mean magnitudes and magnitude amplitudes in the visual band, and the mean radial velocities and RV amplitudes. We noticed that for the stars VY Sgr, RZ Vel, and WZ Sgr, for which the new period and T 0 values are based on the radial velocities derived by Groenewegen (2008, and references therein), the new ephemerides caused a shift in phase between the two datasets shown in Fig. B.1. To overcome this problem, we applied a shift of −0.05, +0.03, and −0.03, respectively, to the literature data plotted in these panels. Figure 4 shows the distribution of the current calibrating Cepheids (colored symbols), together with Galactic Cepheids available in the literature (light gray circles), in the so-called Bailey diagram (luminosity amplitude as a function of the logarithmic period). The plotted data display the classical \"V\" shape in which the local minimum around ten days is mainly caused by the Hertzsprung progression (Bono et al. 2000b). By having a look also at Fig. B.1, we see a subgroup of classical Cepheids showing a well defined bump on the light curve. The phase of such bumps moves from the decreasing branch for periods between ∼6 and ∼9 days, passing across the maximum for periods between ∼9 and ∼12 days, and along the rising branch for periods between ∼12 and ∼16 days. At the center of the Hertzsprung progression (P∼10 days) the light curve is flat topped and the luminosity amplitude attains a well defined minimum (Bono et al. 2000b(Bono et al. , 2002. These variables are called Bump Cepheids for avoiding to be mixed-up with the Beat Cepheids, that is, classical Cepheids pulsating simultaneously in two or more modes. The calibrating Cepheids cover both the short-and the longperiod range, and in particular, the current sample includes seven Cepheids across the Hertzsprung progression."},{"title":"Atmospheric parameters and effective temperature curve templates","content":""},{"title":"Atmospheric parameters and effective temperature curve templates","content":""},{"title":"Effective temperature, surface gravity, and microturbulent velocity","content":""},{"title":"Effective temperature, surface gravity, and microturbulent velocity","content":"The approach we adopted to estimate the atmospheric parameters (i.e., the effective temperature, the surface gravity, and the microturbulent velocity) was already discussed in detail by Proxauf et al. (2018). Here we only recap the key points. The T eff along the pulsation cycle was estimated by using the LDR method, which relies on the correlation between the line depth ratios of pairs of absorption lines in the spectra of different stars and the effective temperature of the same stars. The surface gravity was derived through the ionization equilibrium of Fe i and Fe ii lines, and the microturbulent velocity was obtained by minimizing the dependence of the abundances provided by single Fe i lines on their EWs. During this procedure, the effective temperature was kept fixed, whereas log g and ξ were iteratively changed until convergence. The metallicity used as input by our algorithm is updated in each step, and the adopted value is [Fe i/H], which is the mean iron abundances provided by individual Fe i lines. The estimates of T eff , log g, and ξ for the individual spectrum of our sample are listed in Table 4. The uncertainties on T eff are the standard deviation calculated using the LDR method, that is, from the effective temperatures provided by the individual pairs of lines. The uncertainties on the individual estimates of log g and ξ are assumed to be of the order of 0.3 dex and 0.5 km s −1 , respectively (see Genovali et al. 2014, for a detailed discussion). The weighted mean values of the atmospheric parameters are listed in Table 6 together with the corresponding standard errors calculated for each calibrating Cepheid.\nThe approach we adopted to estimate the atmospheric parameters (i.e., the effective temperature, the surface gravity, and the microturbulent velocity) was already discussed in detail by Proxauf et al. (2018). Here we only recap the key points. The T eff along the pulsation cycle was estimated by using the LDR method, which relies on the correlation between the line depth ratios of pairs of absorption lines in the spectra of different stars and the effective temperature of the same stars. The surface gravity was derived through the ionization equilibrium of Fe i and Fe ii lines, and the microturbulent velocity was obtained by minimizing the dependence of the abundances provided by single Fe i lines on their EWs. During this procedure, the effective temperature was kept fixed, whereas log g and ξ were iteratively changed until convergence. The metallicity used as input by our algorithm is updated in each step, and the adopted value is [Fe i/H], which is the mean iron abundances provided by individual Fe i lines. The estimates of T eff , log g, and ξ for the individual spectrum of our sample are listed in Table 4. The uncertainties on T eff are the standard deviation calculated using the LDR method, that is, from the effective temperatures provided by the individual pairs of lines. The uncertainties on the individual estimates of log g and ξ are assumed to be of the order of 0.3 dex and 0.5 km s −1 , respectively (see Genovali et al. 2014, for a detailed discussion). The weighted mean values of the atmospheric parameters are listed in Table 6 together with the corresponding standard errors calculated for each calibrating Cepheid.\nIn the Appendix B, Figs. B.2, B.3, and B.4 show the variation of the atmospheric parameters along the pulsation cycle. To provide a quantitative estimate of the differences between the current approach and our previous measurements, we fitted Fourier series to the effective temperature, surface gravity, and microturbulent velocity curves in order to evaluate their amplitudes, A&A proofs: manuscript no. daSilvaetal2021 The panels show the phase-folded curves separating the Cepheids in different bins of pulsation period (values within parentheses): from 3 to 5 days, from 5 to 7 days, from 7 to 9.5 days, and from 9.5 to 10.5 days. For the variable S Sge, the two outliers shown in Fig. B.2 were not used in the construction of the T eff curve templates. The curves display the fit of the Fourier series to the data (see Table 7). The standard deviations of the fit are also shown. similarly to what we did in Proxauf et al. (2018, their Figs. 3 and 4). We found that, on average, the errors on the derived amplitudes for the effective temperature curves are a factor 2 smaller than our previous estimates, whereas for the surface gravity the differences are even more significant, with typical errors almost 3 times smaller. For the microturbulent velocity variation, the errors on the current amplitude estimates are more than twice smaller (the dispersions are ∼50 K for T eff , ∼0.2 dex for log g, and ∼0.2 km s −1 for ξ). This means that the variation of the atmospheric parameters along the pulsation cycle is now much better constrained, thanks not only to the larger number of spectra but also to the better determination of such parameters. The improvement in the estimate of log g and ξ is mainly due to the improvement in the line list, which reduced the number of spurious abundance values provided by unreliable iron lines. As described in the Appendix A, we removed lines that are blended, lines that display a well defined trend with effective temperature or with equivalent width, among others. The estimate of T eff is based on the same line list that we previously used, therefore the improvement is mainly due to the larger number of spectra analyzed in the current work. All in all, these changes increased the accuracy in the atmospheric parameters. The reader interested in detailed discussion concerning the estimate of the intrinsic parameters of classical Cepheids is referred to Vasilyev et al. (2018) and Lemasle et al. (2020).\nIn the Appendix B, Figs. B.2, B.3, and B.4 show the variation of the atmospheric parameters along the pulsation cycle. To provide a quantitative estimate of the differences between the current approach and our previous measurements, we fitted Fourier series to the effective temperature, surface gravity, and microturbulent velocity curves in order to evaluate their amplitudes, A&A proofs: manuscript no. daSilvaetal2021 The panels show the phase-folded curves separating the Cepheids in different bins of pulsation period (values within parentheses): from 3 to 5 days, from 5 to 7 days, from 7 to 9.5 days, and from 9.5 to 10.5 days. For the variable S Sge, the two outliers shown in Fig. B.2 were not used in the construction of the T eff curve templates. The curves display the fit of the Fourier series to the data (see Table 7). The standard deviations of the fit are also shown. similarly to what we did in Proxauf et al. (2018, their Figs. 3 and 4). We found that, on average, the errors on the derived amplitudes for the effective temperature curves are a factor 2 smaller than our previous estimates, whereas for the surface gravity the differences are even more significant, with typical errors almost 3 times smaller. For the microturbulent velocity variation, the errors on the current amplitude estimates are more than twice smaller (the dispersions are ∼50 K for T eff , ∼0.2 dex for log g, and ∼0.2 km s −1 for ξ). This means that the variation of the atmospheric parameters along the pulsation cycle is now much better constrained, thanks not only to the larger number of spectra but also to the better determination of such parameters. The improvement in the estimate of log g and ξ is mainly due to the improvement in the line list, which reduced the number of spurious abundance values provided by unreliable iron lines. As described in the Appendix A, we removed lines that are blended, lines that display a well defined trend with effective temperature or with equivalent width, among others. The estimate of T eff is based on the same line list that we previously used, therefore the improvement is mainly due to the larger number of spectra analyzed in the current work. All in all, these changes increased the accuracy in the atmospheric parameters. The reader interested in detailed discussion concerning the estimate of the intrinsic parameters of classical Cepheids is referred to Vasilyev et al. (2018) and Lemasle et al. (2020).\nWe note in passing that the improvement in the accuracy and in the sampling of the pulsation cycle allow us to state that the minimum in the effective temperature is often approached in the same phases in which the minimum in surface gravity is also approached. Moreover, and even more importantly, these minimum phases anticipate the increase in microturbulent velocity. This parameter, as expected, attains it maximum just before the maximum in effective temperature, which can be clearly observed by comparing several panels of Figs. B.2 and B.4.\nWe note in passing that the improvement in the accuracy and in the sampling of the pulsation cycle allow us to state that the minimum in the effective temperature is often approached in the same phases in which the minimum in surface gravity is also approached. Moreover, and even more importantly, these minimum phases anticipate the increase in microturbulent velocity. This parameter, as expected, attains it maximum just before the maximum in effective temperature, which can be clearly observed by comparing several panels of Figs. B.2 and B.4."},{"title":"Effective temperature curve templates","content":""},{"title":"Effective temperature curve templates","content":"By taking advantage of the substantial phase coverage of our sample of calibrating Cepheids, we used the effective temperature measurements to compute new templates for different bins of pulsation periods. However, instead of using the T eff curves directly, we preferred to provide templates for the theta parameter -defined as θ = 5040/T eff -given its linear dependency on the Cousins R-I color index (Taylor 1994).\nBy taking advantage of the substantial phase coverage of our sample of calibrating Cepheids, we used the effective temperature measurements to compute new templates for different bins of pulsation periods. However, instead of using the T eff curves directly, we preferred to provide templates for the theta parameter -defined as θ = 5040/T eff -given its linear dependency on the Cousins R-I color index (Taylor 1994).\nThe approach is similar to the NIR light-curve templates provided by Inno et al. (2015): first, we adopted the ephemerides in Table 5 to fold the θ curves (θCs). Subsequently, we normalized the folded θCs by subtracting their average and dividing by their amplitudes. Our θCs are well sampled, meaning that we have enough phase points to separate the Cepheids into different period bins and to provide analytical relations for the θ curve templates in each bin. We adopted the same period thresholds introduced by Inno et al. (2015). Therefore, based on their Table 1, we generated four cumulative and normalized theta curves for the bins 2 (3-5 days, three Cepheids), 3 (5-7 days, three Cepheids), 4 (7-9.5 days, two Cepheids), and 5 (9.5-10.5 days, two Cepheids). By adopting the same period bins we are able to provide θC templates that are homogeneous with those from the NIR light curves. The reader interested in a more detailed and quantitative discussion concerning the use of cumulative and normalized curves to derive the analytical fits, together with the adopted thresholds for the different period bins, is referred to Inno et al. (2015).\nThe approach is similar to the NIR light-curve templates provided by Inno et al. (2015): first, we adopted the ephemerides in Table 5 to fold the θ curves (θCs). Subsequently, we normalized the folded θCs by subtracting their average and dividing by their amplitudes. Our θCs are well sampled, meaning that we have enough phase points to separate the Cepheids into different period bins and to provide analytical relations for the θ curve templates in each bin. We adopted the same period thresholds introduced by Inno et al. (2015). Therefore, based on their Table 1, we generated four cumulative and normalized theta curves for the bins 2 (3-5 days, three Cepheids), 3 (5-7 days, three Cepheids), 4 (7-9.5 days, two Cepheids), and 5 (9.5-10.5 days, two Cepheids). By adopting the same period bins we are able to provide θC templates that are homogeneous with those from the NIR light curves. The reader interested in a more detailed and quantitative discussion concerning the use of cumulative and normalized curves to derive the analytical fits, together with the adopted thresholds for the different period bins, is referred to Inno et al. (2015).\nThe four normalized θ curves are separately shown in Fig. 5, and in Fig. 6 we compare the Fourier series fitted to the data of each bin. The corresponding coefficients of the fitted functions are listed in Table 7. The separation of the Cepheids into different period bins is a mandatory step because not only the amplitude of the theta curves (listed in Table 6), but also their shape changes with period (as shown in Figs. 5 and 6). Although we do have data for Cepheids with periods outside of the selected pe- riod bins, they are either not as well sampled as the data for the other Cepheids, or the theta curves do not overlap well because the number of measurements is too limited. More accurate and homogeneous data are required to overcome this limitation.\nThe four normalized θ curves are separately shown in Fig. 5, and in Fig. 6 we compare the Fourier series fitted to the data of each bin. The corresponding coefficients of the fitted functions are listed in Table 7. The separation of the Cepheids into different period bins is a mandatory step because not only the amplitude of the theta curves (listed in Table 6), but also their shape changes with period (as shown in Figs. 5 and 6). Although we do have data for Cepheids with periods outside of the selected pe- riod bins, they are either not as well sampled as the data for the other Cepheids, or the theta curves do not overlap well because the number of measurements is too limited. More accurate and homogeneous data are required to overcome this limitation.\nIn order to allow the reader to use these curve templates to derive the mean effective temperature by using the ephemerides, the luminosity amplitude (∆V), and a single spectroscopic measurement of the T eff , we also provide a linear relation between ∆V and ∆θ: ∆θ = (0.184 ± 0.014)∆V + (0.000 ± 0.011); (σ = 0.013)\nIn order to allow the reader to use these curve templates to derive the mean effective temperature by using the ephemerides, the luminosity amplitude (∆V), and a single spectroscopic measurement of the T eff , we also provide a linear relation between ∆V and ∆θ: ∆θ = (0.184 ± 0.014)∆V + (0.000 ± 0.011); (σ = 0.013)\n(1)\n(1)\nThe estimate of the ∆θ amplitude has to be applied to the normalized template as a multiplicative factor. Only after this rescaling operation the template can be anchored to the empirical data and used to derive the mean T eff . Optical and NIR light curves of the calibrating Cepheids, together with effective temperature and radial velocity curves, will be adopted by our group to perform a detailed comparison with nonlinear, convective hydrodynamical models of classical Cepheids. Dating back to Natale et al. (2008) it has been found that the simultaneous fit of both luminosity and radial-velocity variations provides solid constraints on the physical assumptions adopted to build pulsation models (Marconi et al. 2013). Moreover, the use of the effective temperature curves (shapes and amplitudes), covering a broad range of pulsation periods, brings forward the opportunity to constrain, on a quantitative basis, the efficiency of the convective transport over the entire pulsation cycle.\nThe estimate of the ∆θ amplitude has to be applied to the normalized template as a multiplicative factor. Only after this rescaling operation the template can be anchored to the empirical data and used to derive the mean T eff . Optical and NIR light curves of the calibrating Cepheids, together with effective temperature and radial velocity curves, will be adopted by our group to perform a detailed comparison with nonlinear, convective hydrodynamical models of classical Cepheids. Dating back to Natale et al. (2008) it has been found that the simultaneous fit of both luminosity and radial-velocity variations provides solid constraints on the physical assumptions adopted to build pulsation models (Marconi et al. 2013). Moreover, the use of the effective temperature curves (shapes and amplitudes), covering a broad range of pulsation periods, brings forward the opportunity to constrain, on a quantitative basis, the efficiency of the convective transport over the entire pulsation cycle.\nArticle number, page 9 of 31 A&A proofs: manuscript no. daSilvaetal2021 Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. The other columns give the abundances from both neutral and ionized lines, their standard deviations, and the number of lines used. For each element X, the weighted mean of X i and X ii abundances (weighted by 1/σ 2 ) and its intrinsic error is also shown. Magnesium and Sulfur abundances are from neutral lines only. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the abundances from both neutral and ionized lines, their uncertainties, and the number of spectra used. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 8. The standard deviation calculated using all individual abundances from both neutral and ionized lines is also shown. The complete table is available at the CDS. Fig. 9. Atmospheric parameters derived by keeping fixed the effective temperature (previously obtained from the LDR method) in comparison with the same parameters derived with T eff as a free parameter. The panels show the comparison for the β Dor and ζ Gem stars.\nArticle number, page 9 of 31 A&A proofs: manuscript no. daSilvaetal2021 Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. The other columns give the abundances from both neutral and ionized lines, their standard deviations, and the number of lines used. For each element X, the weighted mean of X i and X ii abundances (weighted by 1/σ 2 ) and its intrinsic error is also shown. Magnesium and Sulfur abundances are from neutral lines only. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the abundances from both neutral and ionized lines, their uncertainties, and the number of spectra used. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 8. The standard deviation calculated using all individual abundances from both neutral and ionized lines is also shown. The complete table is available at the CDS. Fig. 9. Atmospheric parameters derived by keeping fixed the effective temperature (previously obtained from the LDR method) in comparison with the same parameters derived with T eff as a free parameter. The panels show the comparison for the β Dor and ζ Gem stars."},{"title":"Iron and α-element abundances","content":""},{"title":"Iron and α-element abundances","content":""},{"title":"Iron abundance","content":""},{"title":"Iron abundance","content":"The determination of the atmospheric parameters was done with the Python wrapper 2 of the MOOG code (Sneden 2002). This is a LTE radiative code that we applied to model atmospheres interpolated on the grids of Castelli & Kurucz (2004). Once a convergence is achieved in the determination of T eff , log g, ξ, and [Fe/H], as described in Sect. 5.1, MOOG also provides the final iron abundance derived from individual Fe i and Fe ii lines. Such lines are those included in the input line list and that passed the selection criteria that we applied to choose the best atomic transitions. Table 4 lists, for each spectrum in our sample, the mean Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used. The mean Fe i and Fe ii abundances derived for each star from the individual spectra, weighted by the standard deviations aforementioned, and the corresponding standard errors are shown in Table 6. Column 8 gives our final determination for the stellar metallicity, which is the weighted mean calculated from both the [Fe i/H] and the [Fe ii/H] abundances. The adopted uncertainties are the largest values between the standard error computed from the weighted mean (σ) and the standard deviation (std). Figure B.5 shows the derived metallicities as a function of the pulsation phase. Data plotted in this figure show that the iron abundances are quite stable along the pulsation cycle. This outcome applies to short-/long-period Cepheids and to Bump Cepheids. We note that the current approach improves the results obtained by Proxauf et al. (2018), since in their iron abundances was still present a mild variation along the rising branch of large amplitude Cepheids (in our current results the dispersions are in most cases smaller than 0.05 dex). The improvement is mainly due to the very careful selection of the lines adopted to estimate the iron abundance. The difference is soundly supported by the distribution of the standard deviations displayed in Fig. 7 for the Cepheids in common. The current standard deviations for both [Fe i/H] (violet) and [Fe ii/H] (green) lines (bottom panel) are smaller by 30%-40% when compared with our previous investigation. Moreover and even more importantly, they also attain similar values.\nThe determination of the atmospheric parameters was done with the Python wrapper 2 of the MOOG code (Sneden 2002). This is a LTE radiative code that we applied to model atmospheres interpolated on the grids of Castelli & Kurucz (2004). Once a convergence is achieved in the determination of T eff , log g, ξ, and [Fe/H], as described in Sect. 5.1, MOOG also provides the final iron abundance derived from individual Fe i and Fe ii lines. Such lines are those included in the input line list and that passed the selection criteria that we applied to choose the best atomic transitions. Table 4 lists, for each spectrum in our sample, the mean Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used. The mean Fe i and Fe ii abundances derived for each star from the individual spectra, weighted by the standard deviations aforementioned, and the corresponding standard errors are shown in Table 6. Column 8 gives our final determination for the stellar metallicity, which is the weighted mean calculated from both the [Fe i/H] and the [Fe ii/H] abundances. The adopted uncertainties are the largest values between the standard error computed from the weighted mean (σ) and the standard deviation (std). Figure B.5 shows the derived metallicities as a function of the pulsation phase. Data plotted in this figure show that the iron abundances are quite stable along the pulsation cycle. This outcome applies to short-/long-period Cepheids and to Bump Cepheids. We note that the current approach improves the results obtained by Proxauf et al. (2018), since in their iron abundances was still present a mild variation along the rising branch of large amplitude Cepheids (in our current results the dispersions are in most cases smaller than 0.05 dex). The improvement is mainly due to the very careful selection of the lines adopted to estimate the iron abundance. The difference is soundly supported by the distribution of the standard deviations displayed in Fig. 7 for the Cepheids in common. The current standard deviations for both [Fe i/H] (violet) and [Fe ii/H] (green) lines (bottom panel) are smaller by 30%-40% when compared with our previous investigation. Moreover and even more importantly, they also attain similar values.\nTo further constrain the difference between the current and our previous investigation, Fig. 8 shows the comparison for both the atmospheric parameters and the iron abundance. The agreement is remarkable over the entire temperature range. The mean and the standard deviations for the atmospheric parameters are well within the current uncertainties. The new iron abundances are, on average, ∼0.1 dex more metal-poor. Owing to the similarity in the atmospheric parameters, this difference seems an obvious consequence of the new line list.\nTo further constrain the difference between the current and our previous investigation, Fig. 8 shows the comparison for both the atmospheric parameters and the iron abundance. The agreement is remarkable over the entire temperature range. The mean and the standard deviations for the atmospheric parameters are well within the current uncertainties. The new iron abundances are, on average, ∼0.1 dex more metal-poor. Owing to the similarity in the atmospheric parameters, this difference seems an obvious consequence of the new line list.\nSeveral different approaches have been suggested in the literature to estimate the atmospheric parameters of variable stars (Fukue et al. 2015;Jian et al. 2020;Lemasle et al. 2020;Matsunaga et al. 2021;Taniguchi et al. 2021;Romaniello et al. 2021). In the current investigation, the effective temperature is firstly derived using the LDR method, then the other atmospheric parameters are estimated at fixed effective temperature. The advantages of this approach have already been discussed in several papers. To provide a more quantitative estimate of the differences compared to another method normally used, in which the minimization process also includes T eff as a free parameter, we performed a new and independent determination of the three atmospheric parameters plus abundances simultaneously for two variables in our sample (β Dor and ζ Gem). The differences between our canonical approach and the literature approach are shown in Fig. 9. The overall agreement is quite good over the entire temperature range. Indeed, the mean and the standard deviations attain tiny values. The mean difference in effective temperature is 55 K, and for the surface gravity it is 0.12 dex only, while the microturbulent velocities attain almost identical values. The quoted differences cause a small increase of 0.04 dex in the iron abundance when T eff is a free parameter. We note that we adopted the same line list in the tests performed either with fixed or with free T eff .\nSeveral different approaches have been suggested in the literature to estimate the atmospheric parameters of variable stars (Fukue et al. 2015;Jian et al. 2020;Lemasle et al. 2020;Matsunaga et al. 2021;Taniguchi et al. 2021;Romaniello et al. 2021). In the current investigation, the effective temperature is firstly derived using the LDR method, then the other atmospheric parameters are estimated at fixed effective temperature. The advantages of this approach have already been discussed in several papers. To provide a more quantitative estimate of the differences compared to another method normally used, in which the minimization process also includes T eff as a free parameter, we performed a new and independent determination of the three atmospheric parameters plus abundances simultaneously for two variables in our sample (β Dor and ζ Gem). The differences between our canonical approach and the literature approach are shown in Fig. 9. The overall agreement is quite good over the entire temperature range. Indeed, the mean and the standard deviations attain tiny values. The mean difference in effective temperature is 55 K, and for the surface gravity it is 0.12 dex only, while the microturbulent velocities attain almost identical values. The quoted differences cause a small increase of 0.04 dex in the iron abundance when T eff is a free parameter. We note that we adopted the same line list in the tests performed either with fixed or with free T eff .\nIn closing this subsection, we also investigate whether the current iron abundances display any clear trend with the Galactocentric distance (R G ). The top panel of Fig. 10 shows the iron abundance as a function of R G for the calibrating Cepheids (blue circles) and for Cepheids available in the literature (gray circles). We performed a linear fit over the entire sample and we found the following relation:\nIn closing this subsection, we also investigate whether the current iron abundances display any clear trend with the Galactocentric distance (R G ). The top panel of Fig. 10 shows the iron abundance as a function of R G for the calibrating Cepheids (blue circles) and for Cepheids available in the literature (gray circles). We performed a linear fit over the entire sample and we found the following relation:\n[Fe/H] = (−0.055 ± 0.003)R G + (0.43 ± 0.03)\n[Fe/H] = (−0.055 ± 0.003)R G + (0.43 ± 0.03)\nThe current slope agrees quite well with similar estimates available in the literature and, in particular, with the recent estimate of the iron radial gradient (dashed line) provided by Ripepi et al. (2022). We note that, to overcome variations in the zero-point mainly introduced by innermost and outermost Cepheids, the gradient from Ripepi et al. (2022) was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The bottom panel of Fig. 10 shows the iron abundance as a function of the logarithmic pulsation period. Data plotted in this panel agree quite well with similar estimates available in the literature. Moreover, they do not show any significant variation when moving from young (long-period) to less young (shortperiod) classical Cepheids.\nThe current slope agrees quite well with similar estimates available in the literature and, in particular, with the recent estimate of the iron radial gradient (dashed line) provided by Ripepi et al. (2022). We note that, to overcome variations in the zero-point mainly introduced by innermost and outermost Cepheids, the gradient from Ripepi et al. (2022) was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The bottom panel of Fig. 10 shows the iron abundance as a function of the logarithmic pulsation period. Data plotted in this panel agree quite well with similar estimates available in the literature. Moreover, they do not show any significant variation when moving from young (long-period) to less young (shortperiod) classical Cepheids."},{"title":"α-element abundances","content":""},{"title":"α-element abundances","content":"The abundances of the α elements were derived using pyMOOGi as well (except sulfur, for which the abundances were estimated using the spectral synthesis method -see Sect. 6.2.1). We used the ab f ind driver, which requires as input i) the model atmospheres, calculated from the atmospheric parameters obtained for each star, and ii) a list of lines containing the wavelength, the atomic number, the lower-level excitation potential, the log g f , and the measured equivalent widths. Table 8 lists the individual α-element abundances obtained for each spectrum together with the standard deviations and the number of lines used for each element. The mean abundances, weighted by the standard deviations, and the corresponding standard errors are shown in Table 9. The same mean data are plotted in Figs. 11,12,and B.6 for our 20 calibrating Cepheids (blue circles) and for similar data available in the literature (gray circles).\nThe abundances of the α elements were derived using pyMOOGi as well (except sulfur, for which the abundances were estimated using the spectral synthesis method -see Sect. 6.2.1). We used the ab f ind driver, which requires as input i) the model atmospheres, calculated from the atmospheric parameters obtained for each star, and ii) a list of lines containing the wavelength, the atomic number, the lower-level excitation potential, the log g f , and the measured equivalent widths. Table 8 lists the individual α-element abundances obtained for each spectrum together with the standard deviations and the number of lines used for each element. The mean abundances, weighted by the standard deviations, and the corresponding standard errors are shown in Table 9. The same mean data are plotted in Figs. 11,12,and B.6 for our 20 calibrating Cepheids (blue circles) and for similar data available in the literature (gray circles).\nFor cases in which the abundance of a given element at a given phase is based on only one or two lines, or if the estimated error is lower than a typical value for each element/species, the uncertainty is set to that typical value, namely: σ( Table 8 for spectra having at least 2 lines of the same element/species. The adopted standard solar abundances for both iron and α elements are from Asplund et al. (2009), namely: A(Fe) = 7.50, A(Mg) = 7.60, A(Si) = 7.51, A(S) = 7.12, A(Ca) = 6.34, and A(Ti) = 4.95, with A representing the typical logarithmic notation where H is defined to be A(H) = 12.00. These values are in good agreement with the very recent determinations done by Asplund et al. (2021), which were obtained using the most up-to-date atomic and molecular data.\nFor cases in which the abundance of a given element at a given phase is based on only one or two lines, or if the estimated error is lower than a typical value for each element/species, the uncertainty is set to that typical value, namely: σ( Table 8 for spectra having at least 2 lines of the same element/species. The adopted standard solar abundances for both iron and α elements are from Asplund et al. (2009), namely: A(Fe) = 7.50, A(Mg) = 7.60, A(Si) = 7.51, A(S) = 7.12, A(Ca) = 6.34, and A(Ti) = 4.95, with A representing the typical logarithmic notation where H is defined to be A(H) = 12.00. These values are in good agreement with the very recent determinations done by Asplund et al. (2021), which were obtained using the most up-to-date atomic and molecular data.\nFor both iron and α abundances, we evaluated the systematic differences for stars in common between the current investigation and measurements from literature: ), Luck & Lambert (2011, Lemasle et al. (2013), andYong et al. (2006) for iron and α elements, and Proxauf et al. (2018) for iron only. The differences for all the investigated elements are listed in Table 10. In order to perform a direct comparison, we applied these zero-point differences to the literature datasets by putting the iron and the α-element abundances in our metallicity scale. The spectroscopic samples for which there are no variables in common between our investigation and those available in the literature, the homogenization of the abundances was performed in two steps. To be more specific, we have no variables in common with Yong et al. (2006), therefore, we first scaled their abundances to match those from Luck & Lambert (2011), and then we scaled them to match our metallicity scale. Data plotted in Fig. B.6 display, once again, the remarkable agreement of the α-element abundances over the entire pulsation cycle (the dispersions in most cases are smaller than 0.10 dex). This outcome applies to all the investigated α elements (except sulfur, for which we have abundances at a single phase only), and to both neutral and ionized species. We note in passing that this finding seems more relevant when compared with the variation of the iron abundances along the pulsation cycle. Indeed, the αelement abundances are only based on a modest number or, in some cases, just a few lines. Therefore, they are more prone to possible systematics.\nFor both iron and α abundances, we evaluated the systematic differences for stars in common between the current investigation and measurements from literature: ), Luck & Lambert (2011, Lemasle et al. (2013), andYong et al. (2006) for iron and α elements, and Proxauf et al. (2018) for iron only. The differences for all the investigated elements are listed in Table 10. In order to perform a direct comparison, we applied these zero-point differences to the literature datasets by putting the iron and the α-element abundances in our metallicity scale. The spectroscopic samples for which there are no variables in common between our investigation and those available in the literature, the homogenization of the abundances was performed in two steps. To be more specific, we have no variables in common with Yong et al. (2006), therefore, we first scaled their abundances to match those from Luck & Lambert (2011), and then we scaled them to match our metallicity scale. Data plotted in Fig. B.6 display, once again, the remarkable agreement of the α-element abundances over the entire pulsation cycle (the dispersions in most cases are smaller than 0.10 dex). This outcome applies to all the investigated α elements (except sulfur, for which we have abundances at a single phase only), and to both neutral and ionized species. We note in passing that this finding seems more relevant when compared with the variation of the iron abundances along the pulsation cycle. Indeed, the αelement abundances are only based on a modest number or, in some cases, just a few lines. Therefore, they are more prone to possible systematics.\nTo further investigate the internal consistency of the current α-element abundances with similar estimates available in the literature, we checked the behavior of the [X/Fe] abundance ratios as a function of the stellar metallicity. Data plotted in the left panels of Fig. 11 show that the current results agree, within the errors, quite well with similar determinations done in previous investigations (Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006). The key difference seems to be the smaller dispersion at fixed iron abundance, thus suggesting that a fraction of the spread currently observed might still be caused by intrinsic errors. This finding is further supported by the evidence that the current estimates do not show any clear trend when plotted against the pulsation period (see the right panels of Fig. 11). We note that classical Cepheids, when moving from short to long periods, become systematically cooler and more expanded (lower surface gravity).\nTo further investigate the internal consistency of the current α-element abundances with similar estimates available in the literature, we checked the behavior of the [X/Fe] abundance ratios as a function of the stellar metallicity. Data plotted in the left panels of Fig. 11 show that the current results agree, within the errors, quite well with similar determinations done in previous investigations (Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006). The key difference seems to be the smaller dispersion at fixed iron abundance, thus suggesting that a fraction of the spread currently observed might still be caused by intrinsic errors. This finding is further supported by the evidence that the current estimates do not show any clear trend when plotted against the pulsation period (see the right panels of Fig. 11). We note that classical Cepheids, when moving from short to long periods, become systematically cooler and more expanded (lower surface gravity)."},{"title":"Sulfur abundances","content":""},{"title":"Sulfur abundances","content":"To determine the sulfur abundances, we carried out spectral synthesis calculations by exploiting the multiplet at 6757 Å, which is proven to be reliable and not affected by NLTE effects (see, e.g., Takeda et al. 2016;Duffau et al. 2017, and references therein). Synthetic spectra were computed using the synth driver of MOOG (2019 version), the same grid of model atmospheres by Castelli & Kurucz (2004) used for the other elements, and atomic parameters from Caffau et al. (2007). We synthesized a 30 Å wide region around the S i multiplet in order to determine the broadening of the spectral lines. Then we degraded the synthetic spectra to match the resolution of the observational datasets. The best fit was found with χ-square minimization methods, as routinely done in the literature. Finally, we estimated the internal errors, with a mean value of σ([S i/H]) = 0.10 dex. These errors are due to continuum placement (related to the S/N, quality of the spectra, best fitting procedure) and due to errors on the atmospheric parameters (this was done in the standard way, which is by varying one parameter at a time and inspecting the corresponding variations). We refer the reader to D'Orazi et al. (2017D'Orazi et al. ( , 2020 for details on error computation.\nTo determine the sulfur abundances, we carried out spectral synthesis calculations by exploiting the multiplet at 6757 Å, which is proven to be reliable and not affected by NLTE effects (see, e.g., Takeda et al. 2016;Duffau et al. 2017, and references therein). Synthetic spectra were computed using the synth driver of MOOG (2019 version), the same grid of model atmospheres by Castelli & Kurucz (2004) used for the other elements, and atomic parameters from Caffau et al. (2007). We synthesized a 30 Å wide region around the S i multiplet in order to determine the broadening of the spectral lines. Then we degraded the synthetic spectra to match the resolution of the observational datasets. The best fit was found with χ-square minimization methods, as routinely done in the literature. Finally, we estimated the internal errors, with a mean value of σ([S i/H]) = 0.10 dex. These errors are due to continuum placement (related to the S/N, quality of the spectra, best fitting procedure) and due to errors on the atmospheric parameters (this was done in the standard way, which is by varying one parameter at a time and inspecting the corresponding variations). We refer the reader to D'Orazi et al. (2017D'Orazi et al. ( , 2020 for details on error computation.\nThe reasons why we focused our attention on S i abundances for our sample of calibrating Cepheids are manifold:\nThe reasons why we focused our attention on S i abundances for our sample of calibrating Cepheids are manifold:\ni)-Sulfur is a volatile element that is not locked in dust grains, which means that no correction is required to compare the current sulfur abundances with predictions based on chemical evolution models (Chiappini et al. 2001;Cescutti et al. 2007). Therefore, sulfur is a more solid tracer of α-element abundances than Si and Ca. These three α elements are produced during the oxygen burning and their abundances should show similar chemical enrichment histories (Limongi & Chieffi 2003).\ni)-Sulfur is a volatile element that is not locked in dust grains, which means that no correction is required to compare the current sulfur abundances with predictions based on chemical evolution models (Chiappini et al. 2001;Cescutti et al. 2007). Therefore, sulfur is a more solid tracer of α-element abundances than Si and Ca. These three α elements are produced during the oxygen burning and their abundances should show similar chemical enrichment histories (Limongi & Chieffi 2003).\nii)-Sulfur brings forward several key advantages when compared with other α elements, since it shows several lines at both optical and NIR wavelengths. The most popular sulfur lines are the multiplets at 6757 Å and at 8694 Å, but they are weak and they are typically measured on stars more metal-rich than [Fe/H] ∼ −1.5 dex. More recently, the multiplet at 9237 Å has also been used because it is very strong and it can be measured even in very metal-poor stars (Nissen et al. 2004;Caffau et al. 2005Caffau et al. , 2007. However, these lines are also affected by NLTE effects, making them even stronger, and the sulfur abundances need to be properly treated. Even more recently, high-resolution NIR spectrographs also provided the opportunity to use the S i multiplet at 1045 nm for both dwarf and giant stars (Caffau et al. 2016).\nii)-Sulfur brings forward several key advantages when compared with other α elements, since it shows several lines at both optical and NIR wavelengths. The most popular sulfur lines are the multiplets at 6757 Å and at 8694 Å, but they are weak and they are typically measured on stars more metal-rich than [Fe/H] ∼ −1.5 dex. More recently, the multiplet at 9237 Å has also been used because it is very strong and it can be measured even in very metal-poor stars (Nissen et al. 2004;Caffau et al. 2005Caffau et al. , 2007. However, these lines are also affected by NLTE effects, making them even stronger, and the sulfur abundances need to be properly treated. Even more recently, high-resolution NIR spectrographs also provided the opportunity to use the S i multiplet at 1045 nm for both dwarf and giant stars (Caffau et al. 2016).\niii)-Possibility of comparison with the S abundances in external galaxies, such as Blue Compact Galaxies, Damped Lyman Alpha systems (Dessauges-Zavadsky et al. 2007), and Active Galactic Nuclei (Liu et al. 2015, Mizumoto et al., submitted to ApJ). Indeed, thanks to strong emission lines, the S abundances can be measured even at large redshifts.\niii)-Possibility of comparison with the S abundances in external galaxies, such as Blue Compact Galaxies, Damped Lyman Alpha systems (Dessauges-Zavadsky et al. 2007), and Active Galactic Nuclei (Liu et al. 2015, Mizumoto et al., submitted to ApJ). Indeed, thanks to strong emission lines, the S abundances can be measured even at large redshifts.\niv)-In our previous investigation focused on α-element abundances in classical Cepheids (Genovali et al. 2015) we neglected sulfur. In the current study, we perform accurate measurements of the multiplet at 6757 Å. The multiplet at 9237 Å is only available in a small subsets of spectra (UVES), but the lines are too strong (EW ≥ 300 mÅ) and it was neglected. Data plotted in Fig. 12 show the [X/H] (left panels) and the [X/Fe] (right panels) α-element abundances as a function of the Galactocentric distances, estimated by Genovali et al. (2014) using NIR mean magnitudes and predicted Period-Luminosity relations (Inno et al. 2013). The radial gradients plotted in the left panels indicate that sulfur is the α element with the steeper slope (see labeled values). Indeed, its slope is at least a factor of two steeper than for Ca (0.076 vs. 0.031 dex kpc −1 ) and ∼50% steeper than those for Mg and Si. In passing, we also note that the radial gradient of the abundance ratios plotted in the right panels of the same figure display, as expected (see Genovali et al. 2015), either a flat trend (Si) or a marginal positive slope (Mg, Ti) across the thin disk. There are two exceptions. Ca is steadily becoming overabundant (positive slope) when moving from the innermost (more metal-rich) to the outermost (more metal-poor) disk regions. This is a consequence of the fact that the slope of its radial gradient is shallower than the iron gradient (0.31 vs. 0.55 dex kpc −1 ). S shows a negative slope, because it is the only α-element with a slope in the radial gradient steeper than the iron gradient (0.76 vs. 0.55 dex kpc −1 ).\niv)-In our previous investigation focused on α-element abundances in classical Cepheids (Genovali et al. 2015) we neglected sulfur. In the current study, we perform accurate measurements of the multiplet at 6757 Å. The multiplet at 9237 Å is only available in a small subsets of spectra (UVES), but the lines are too strong (EW ≥ 300 mÅ) and it was neglected. Data plotted in Fig. 12 show the [X/H] (left panels) and the [X/Fe] (right panels) α-element abundances as a function of the Galactocentric distances, estimated by Genovali et al. (2014) using NIR mean magnitudes and predicted Period-Luminosity relations (Inno et al. 2013). The radial gradients plotted in the left panels indicate that sulfur is the α element with the steeper slope (see labeled values). Indeed, its slope is at least a factor of two steeper than for Ca (0.076 vs. 0.031 dex kpc −1 ) and ∼50% steeper than those for Mg and Si. In passing, we also note that the radial gradient of the abundance ratios plotted in the right panels of the same figure display, as expected (see Genovali et al. 2015), either a flat trend (Si) or a marginal positive slope (Mg, Ti) across the thin disk. There are two exceptions. Ca is steadily becoming overabundant (positive slope) when moving from the innermost (more metal-rich) to the outermost (more metal-poor) disk regions. This is a consequence of the fact that the slope of its radial gradient is shallower than the iron gradient (0.31 vs. 0.55 dex kpc −1 ). S shows a negative slope, because it is the only α-element with a slope in the radial gradient steeper than the iron gradient (0.76 vs. 0.55 dex kpc −1 )."},{"title":"Summary and final remarks","content":""},{"title":"Summary and final remarks","content":"We performed a new spectroscopic analysis of almost two dozen of northern and southern calibrating Cepheids. As a whole, we collected more than 500 high-resolution (R∼40 000-115 000), high S/N (∼50-370) optical spectra with four different spectrographs (STELLA, FEROS, HARPS, UVES) covering either a substantial portion or the entire pulsation cycle. The main aim of this investigation is twofold: i) provide new and accurate constraints on possible systematics affecting the estimate of atmospheric parameters and elemental abundances along the pulsation cycle; ii) develop new approaches and/or diagnostics to provide accurate estimates of both atmospheric parameters and chemical abundances. To accomplish the former goal, we assembled three different line lists. By using different line lists available in the literature, we compiled a very detailed list of both Fe i and Fe ii lines. Moreover, we also collected a sizable sample of lines for α-elements: Mg, Si, S, Ca, and Ti. Special care was also paid to the line list of both iron-peak and α-elements used for the estimate of the effective temperature using the LDR method.\nWe performed a new spectroscopic analysis of almost two dozen of northern and southern calibrating Cepheids. As a whole, we collected more than 500 high-resolution (R∼40 000-115 000), high S/N (∼50-370) optical spectra with four different spectrographs (STELLA, FEROS, HARPS, UVES) covering either a substantial portion or the entire pulsation cycle. The main aim of this investigation is twofold: i) provide new and accurate constraints on possible systematics affecting the estimate of atmospheric parameters and elemental abundances along the pulsation cycle; ii) develop new approaches and/or diagnostics to provide accurate estimates of both atmospheric parameters and chemical abundances. To accomplish the former goal, we assembled three different line lists. By using different line lists available in the literature, we compiled a very detailed list of both Fe i and Fe ii lines. Moreover, we also collected a sizable sample of lines for α-elements: Mg, Si, S, Ca, and Ti. Special care was also paid to the line list of both iron-peak and α-elements used for the estimate of the effective temperature using the LDR method.\nIn order to provide homogeneous and accurate estimates of both atmospheric parameters and elemental abundances, we undertook a lengthy and detailed analysis of the quoted line lists. Our approach moved along three different paths: i) we collected the three most precise atomic transition parameters available in the literature; ii) we removed absorption lines that are blended with other lines; iii) we removed lines that either show relevant abundance variations from the average value of their chemical species or that display a steady variation along the pulsation cycle with atmospheric parameters. Concerning the transition parameters, we gave the priority to recent laboratory measurements, then to homogeneous astrophysical estimates and, for the remaining lines, to the transition parameters available on the NIST Atomic Spectra Database. To identify blended lines, we adopted both the solar spectrum by Moore et al. (1966) and a grid of synthetic spectra. Finally, we applied an iterative approach to remove lines showing significant variations. This latter step was done based on a preliminary estimate of the atmospheric parameters and the abundances. We were then able to identify lines that show either a variation in abundance at the 3-σ level from the mean value of the specific element and/or a steady variation along the pulsation cycle. In more detail, to properly identify robust lines, we plotted all the lines available in our initial line lists, for each chemical species, individual exposure, and calibrating Cepheid, as a function of the pulsation phase, effective temperature, and equivalent width. Lastly, we performed a new estimate of the atmospheric parameters and the individual elemental abundances, using an improved version of our algorithm and using only the line lists based on optimal lines.\nIn order to provide homogeneous and accurate estimates of both atmospheric parameters and elemental abundances, we undertook a lengthy and detailed analysis of the quoted line lists. Our approach moved along three different paths: i) we collected the three most precise atomic transition parameters available in the literature; ii) we removed absorption lines that are blended with other lines; iii) we removed lines that either show relevant abundance variations from the average value of their chemical species or that display a steady variation along the pulsation cycle with atmospheric parameters. Concerning the transition parameters, we gave the priority to recent laboratory measurements, then to homogeneous astrophysical estimates and, for the remaining lines, to the transition parameters available on the NIST Atomic Spectra Database. To identify blended lines, we adopted both the solar spectrum by Moore et al. (1966) and a grid of synthetic spectra. Finally, we applied an iterative approach to remove lines showing significant variations. This latter step was done based on a preliminary estimate of the atmospheric parameters and the abundances. We were then able to identify lines that show either a variation in abundance at the 3-σ level from the mean value of the specific element and/or a steady variation along the pulsation cycle. In more detail, to properly identify robust lines, we plotted all the lines available in our initial line lists, for each chemical species, individual exposure, and calibrating Cepheid, as a function of the pulsation phase, effective temperature, and equivalent width. Lastly, we performed a new estimate of the atmospheric parameters and the individual elemental abundances, using an improved version of our algorithm and using only the line lists based on optimal lines.\nThe comparison between the current atmospheric parameters and similar estimated provided by our group, using a similar approach but old line lists, show smoother variation along the pulsation cycle and, in particular, smaller standard deviations (∼50 K for the effective temperature, ∼0.2 dex for the surface gravity, and ∼0.2 km s −1 for the microturbulent velocity). The estimate of both iron and α-element abundances is also significantly improved, since they display an almost constant value as a function of the pulsation phase. The dispersions are 0.05 dex for iron and 0.10 dex for the α elements. We note that this is the first time that α-element abundances are critically investigated for possible variations along the pulsation cycle.\nThe comparison between the current atmospheric parameters and similar estimated provided by our group, using a similar approach but old line lists, show smoother variation along the pulsation cycle and, in particular, smaller standard deviations (∼50 K for the effective temperature, ∼0.2 dex for the surface gravity, and ∼0.2 km s −1 for the microturbulent velocity). The estimate of both iron and α-element abundances is also significantly improved, since they display an almost constant value as a function of the pulsation phase. The dispersions are 0.05 dex for iron and 0.10 dex for the α elements. We note that this is the first time that α-element abundances are critically investigated for possible variations along the pulsation cycle.\nThis finding is further supported by the comparison with similar abundance estimates available in the literature. We found that the current calibrating Cepheids display, at fixed either iron abundance or pulsation period, smaller standard deviations. This result suggests that a good fraction of the spread normally observed in elemental abundances is mainly caused by the adopted line list and by the algorithm used to estimate the atmospheric parameters.\nThis finding is further supported by the comparison with similar abundance estimates available in the literature. We found that the current calibrating Cepheids display, at fixed either iron abundance or pulsation period, smaller standard deviations. This result suggests that a good fraction of the spread normally observed in elemental abundances is mainly caused by the adopted line list and by the algorithm used to estimate the atmospheric parameters.\nMoreover and even more importantly, we also derived for the first time new and accurate effective temperature templates. Following Inno et al. (2015), the calibrating Cepheids were split into four different period bins, ranging from 3 to 10.5 days in period. For each period bin, we performed an analytical fit with Fourier series providing θ = 5040 /T eff as a function of the pulsation phase. A similar approach was already applied to fundamental RR Lyrae by For & Sneden (2010) and by Magurno et al. (2018), but the variations along the pulsation cycle are, at fixed pulsation phase, larger due to the larger luminosity amplitudes and probably to the possible occurrence of nonlinear phenomena (shocks). The current findings appear quite promising, because the new templates can provide accurate estimates of the effective temperature once the pulsation period and the phase (and in turn the reference epoch) of a Cepheid are known. We note that this information is crucial in the abundance analysis of NIR spectra, because in the metal-poor regimes only a modest number of ionized and neutral iron lines are present.\nMoreover and even more importantly, we also derived for the first time new and accurate effective temperature templates. Following Inno et al. (2015), the calibrating Cepheids were split into four different period bins, ranging from 3 to 10.5 days in period. For each period bin, we performed an analytical fit with Fourier series providing θ = 5040 /T eff as a function of the pulsation phase. A similar approach was already applied to fundamental RR Lyrae by For & Sneden (2010) and by Magurno et al. (2018), but the variations along the pulsation cycle are, at fixed pulsation phase, larger due to the larger luminosity amplitudes and probably to the possible occurrence of nonlinear phenomena (shocks). The current findings appear quite promising, because the new templates can provide accurate estimates of the effective temperature once the pulsation period and the phase (and in turn the reference epoch) of a Cepheid are known. We note that this information is crucial in the abundance analysis of NIR spectra, because in the metal-poor regimes only a modest number of ionized and neutral iron lines are present.\nThe current investigation was focused on Galactic Cepheids with iron abundances close to the solar value. In a forthcoming paper we plan to apply the same approach to more metal-poor Magellanic Cloud Cepheids. The panels of Fig. B.1 show the phase-folded light and radialvelocity curves for our sample of 20 calibrating Cepheids. The stellar magnitude in visual bands were taken from literature, as explained in Sect. 4. The radial velocities were either measured from our spectra or taken from literature. Only FF Aql and β Dor have RV values derived from our spectra only.\nThe current investigation was focused on Galactic Cepheids with iron abundances close to the solar value. In a forthcoming paper we plan to apply the same approach to more metal-poor Magellanic Cloud Cepheids. The panels of Fig. B.1 show the phase-folded light and radialvelocity curves for our sample of 20 calibrating Cepheids. The stellar magnitude in visual bands were taken from literature, as explained in Sect. 4. The radial velocities were either measured from our spectra or taken from literature. Only FF Aql and β Dor have RV values derived from our spectra only.\nOur RV measurements agree quite well with the literature values. Significant differences are seen only for two stars, T Vul and S Sge, which might be related to instrumental drifts caused by their long baseline observations (for both stars, the two datasets, ours and from literature, are separated by almost 17 years) or, most likely, to the binarity of these objects (Evans 1992;Evans et al. 1993). Table 5, for T Vul we derived two different values of reference epoch (T 0 ), one from our own radial velocities and another one from the light curve. For S Sge we have independent measurements of T 0 and pulsation period derived from both the RV datasets and from the light curve. Another particular case is β Dor, one of our bump Cepheids (see Table 1), for which we have independent values of T 0 and period derived from our RV curve and from the literature light curve.\nOur RV measurements agree quite well with the literature values. Significant differences are seen only for two stars, T Vul and S Sge, which might be related to instrumental drifts caused by their long baseline observations (for both stars, the two datasets, ours and from literature, are separated by almost 17 years) or, most likely, to the binarity of these objects (Evans 1992;Evans et al. 1993). Table 5, for T Vul we derived two different values of reference epoch (T 0 ), one from our own radial velocities and another one from the light curve. For S Sge we have independent measurements of T 0 and pulsation period derived from both the RV datasets and from the light curve. Another particular case is β Dor, one of our bump Cepheids (see Table 1), for which we have independent values of T 0 and period derived from our RV curve and from the literature light curve."},{"title":"As detailed in","content":""},{"title":"As detailed in","content":"The main criteria we adopted for selecting the light curves available in the literature were: i) well-sampled photometric data; ii) minimization of the difference in epoch between photometric and spectroscopic data. It goes without saying that wellsampled light curves from Gaia DR3 will allow us to provide accurate and homogeneous ephemerides for the majority of classical Cepheids.\nThe main criteria we adopted for selecting the light curves available in the literature were: i) well-sampled photometric data; ii) minimization of the difference in epoch between photometric and spectroscopic data. It goes without saying that wellsampled light curves from Gaia DR3 will allow us to provide accurate and homogeneous ephemerides for the majority of classical Cepheids."},{"title":"Fig. 1 .","content":""},{"title":"Fig. 1 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"Fig. 3 .","content":""},{"title":"Fig. 3 .","content":""},{"title":"Fig. 5 .","content":""},{"title":"Fig. 5 .","content":""},{"title":"Fig. 6 .","content":""},{"title":"Fig. 6 .","content":""},{"title":"Fig. 7 .","content":""},{"title":"Fig. 7 .","content":""},{"title":"Fig. 8 .","content":""},{"title":"Fig. 8 .","content":""},{"title":"Fig. 10 .","content":""},{"title":"Fig. 10 .","content":""},{"title":"Fig. 11 .","content":""},{"title":"Fig. 11 .","content":""},{"title":"Fig. 12 .","content":""},{"title":"Fig. 12 .","content":""},{"title":"Fig. A. 1 .","content":""},{"title":"Fig. A. 1 .","content":""},{"title":"Fig. A. 2 .","content":""},{"title":"Fig. A. 2 .","content":""},{"title":"Fig. A. 3 .","content":""},{"title":"Fig. A. 3 .","content":""},{"title":"Fig. A. 4 .","content":""},{"title":"Fig. A. 4 .","content":""},{"title":"Fig. A. 5 .","content":""},{"title":"Fig. A. 5 .","content":""},{"title":"Table 2 .","content":""},{"title":"Table 2 .","content":""},{"title":"Table 3 .","content":""},{"title":"Table 3 .","content":""},{"title":"Table 5 .","content":""},{"title":"Table 5 .","content":""},{"title":"Table 6 .","content":""},{"title":"Table 6 .","content":""},{"title":"Table 7 .","content":""},{"title":"Table 7 .","content":""},{"title":"Table 8 .","content":""},{"title":"Table 8 .","content":""},{"title":"Table 9 .","content":""},{"title":"Table 9 .","content":""},{"title":"Table 10 .","content":""},{"title":"Table 10 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"\"\n },\n {\n \"title\": \"Introduction\",\n \"content\": \"The current empirical evidence indicates that the estimates of the Hubble constant based on Cosmic Microwave Background (CMB) and on the standard cosmological model (Λ Cold Dark Matter, ΛCDM) are at odds with the direct estimates based on primary (classical Cepheids) and secondary (type Ia Supernovae) distance indicators (Planck Collaboration et al. 2020;Riess et al. 2019). The current tension between early and late Universe measurements is at a 5-σ level. It is not clear whether it might be explained by unknown systematics or whether this disagreement is opening the path for new physics beyond the standard cosmological model (see, e.g., Niedermann & Sloth 2020, and references therein). In this context, it is worth mentioning Partly based on data obtained with the STELLA robotic telescopes in Tenerife, a facility of The Leibniz Institute for Astrophysics Potsdam (AIP) jointly operated by the AIP and by the Instituto de Astrofísica de Canarias (IAC).\\nThe current empirical evidence indicates that the estimates of the Hubble constant based on Cosmic Microwave Background (CMB) and on the standard cosmological model (Λ Cold Dark Matter, ΛCDM) are at odds with the direct estimates based on primary (classical Cepheids) and secondary (type Ia Supernovae) distance indicators (Planck Collaboration et al. 2020;Riess et al. 2019). The current tension between early and late Universe measurements is at a 5-σ level. It is not clear whether it might be explained by unknown systematics or whether this disagreement is opening the path for new physics beyond the standard cosmological model (see, e.g., Niedermann & Sloth 2020, and references therein). In this context, it is worth mentioning Partly based on data obtained with the STELLA robotic telescopes in Tenerife, a facility of The Leibniz Institute for Astrophysics Potsdam (AIP) jointly operated by the AIP and by the Instituto de Astrofísica de Canarias (IAC).\\nTables 2, 3, 4, 8, and 9 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/. that the use of the Tip of the Red Giant Branch (TRGB) as primary distance indicators gives values of the Hubble constant that are intermediate between those based on the CMB and those based on the Cepheid distance scale (Freedman et al. 2020).\\nTables 2, 3, 4, 8, and 9 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/. that the use of the Tip of the Red Giant Branch (TRGB) as primary distance indicators gives values of the Hubble constant that are intermediate between those based on the CMB and those based on the Cepheid distance scale (Freedman et al. 2020).\\nThere are solid reasons to believe that the difference between the TRGB and the Cepheid distance scale is caused by the dependence of the metal content on the zero-point and/or the slope of the adopted diagnostics (Bono et al. 2010;Pietrzyński et al. 2013). The dependence of optical and near-infrared diagnostics on metallicity is still a controversial issue for both theoretical (Marconi et al. 2005;Bono et al. 2010) and empirical (Groenewegen 2018(Groenewegen , 2020Ripepi et al. 2019Ripepi et al. , 2021Breuval et al. 2020Breuval et al. , 2021 approaches. However, the improved accuracy of geometrical distances from Gaia DR3, and the homogeneity of spectroscopic abundances will provide solid constraints on this longstanding problem (Ripepi et al. 2022, and references therein).\\nThere are solid reasons to believe that the difference between the TRGB and the Cepheid distance scale is caused by the dependence of the metal content on the zero-point and/or the slope of the adopted diagnostics (Bono et al. 2010;Pietrzyński et al. 2013). The dependence of optical and near-infrared diagnostics on metallicity is still a controversial issue for both theoretical (Marconi et al. 2005;Bono et al. 2010) and empirical (Groenewegen 2018(Groenewegen , 2020Ripepi et al. 2019Ripepi et al. , 2021Breuval et al. 2020Breuval et al. , 2021 approaches. However, the improved accuracy of geometrical distances from Gaia DR3, and the homogeneity of spectroscopic abundances will provide solid constraints on this longstanding problem (Ripepi et al. 2022, and references therein).\\nClassical Cepheids are bright (−2 ≤ M V ≤ −7 mag) radially pulsating stars with periods ranging from roughly one day to more than one hundred days (see, e.g., Skowron et al. 2019, and references therein). Their iron abundances range from solar values in the solar vicinity to ∼0.5 dex more metal-rich in the inner Galactic disk and to −0.5 dex more metal-poor in the outer disk (Inno et al. 2019). Cepheids in the Large Magellanic Cloud (LMC) are on average a factor of two more metal- Notes. The first four columns give the star name, the right ascension and declination, and the pulsation mode (0: fundamental; 1: first overtone; 0B: fundamental with bump). The Cols. 5 and 6 give the iron abundance from literature and the corresponding references. Columns from 7 to 10 show the number of optical spectra used for each spectrograph: N F : FEROS; N H : HARPS; N U : UVES; N S : STELLA. The last column lists the total number of spectra per target.\\nClassical Cepheids are bright (−2 ≤ M V ≤ −7 mag) radially pulsating stars with periods ranging from roughly one day to more than one hundred days (see, e.g., Skowron et al. 2019, and references therein). Their iron abundances range from solar values in the solar vicinity to ∼0.5 dex more metal-rich in the inner Galactic disk and to −0.5 dex more metal-poor in the outer disk (Inno et al. 2019). Cepheids in the Large Magellanic Cloud (LMC) are on average a factor of two more metal- Notes. The first four columns give the star name, the right ascension and declination, and the pulsation mode (0: fundamental; 1: first overtone; 0B: fundamental with bump). The Cols. 5 and 6 give the iron abundance from literature and the corresponding references. Columns from 7 to 10 show the number of optical spectra used for each spectrograph: N F : FEROS; N H : HARPS; N U : UVES; N S : STELLA. The last column lists the total number of spectra per target.\\nReferences. (a) 1: Genovali et al. (2014); for T Vul, FF Aql, δ Cep, and S Sge we assumed a typical error of 0.1 dex since they were not provided by the original authors; 2: Proxauf et al. (2018), from which the uncertainty adopted is the largest value between σ and std.\\nReferences. (a) 1: Genovali et al. (2014); for T Vul, FF Aql, δ Cep, and S Sge we assumed a typical error of 0.1 dex since they were not provided by the original authors; 2: Proxauf et al. (2018), from which the uncertainty adopted is the largest value between σ and std.\\npoor (Romaniello et al. 2021), and those in the Small Magellanic Cloud (SMC) a factor of four more metal-poor than solar (Lemasle et al. 2017). Cepheids are intermediate-mass stars crossing the instability strip during central helium burning, the so-called blue loops, and their progenitor mass ranges from ∼3 to ∼10 M -the exact limit depends on the chemical composition (Bono et al. 2000b;De Somma et al. 2021). Classical Cepheids are solid stellar beacons to trace young stellar populations even in the far side of the Galactic thin disk (Minniti et al. 2020(Minniti et al. , 2021 and in stellar systems experiencing recent star formation events such as dwarf irregulars and star forming galaxies (Neeley et al. 2021). Their period-luminosity relations (Breuval et al. 2020;Ripepi et al. 2020) provide accurate (at the level of ∼1-2%) measurements of individual distances. Individual distances together with detailed information concerning their elemental abundances (iron peak, α, neutron capture), provide the unique opportunity to investigate the radial gradients and, in turn, the recent (t ≤ 200-300 Myr) chemical enrichment history of the thin disk and nearby stellar systems (Romaniello et al. 2008;Lemasle et al. 2013;Genovali et al. 2015;da Silva et al. 2016). Furthermore, classical Cepheids are fundamental laboratories for investigating not only the advanced evolutionary phases of intermediate-mass stars (Bono et al. 2000a;Neilson et al. 2011;Prada Moroni et al. 2012;De Somma et al. 2021), but also for constraining the physical mechanisms driving their variability (Bono et al. 1999;Marconi et al. 2013).\\npoor (Romaniello et al. 2021), and those in the Small Magellanic Cloud (SMC) a factor of four more metal-poor than solar (Lemasle et al. 2017). Cepheids are intermediate-mass stars crossing the instability strip during central helium burning, the so-called blue loops, and their progenitor mass ranges from ∼3 to ∼10 M -the exact limit depends on the chemical composition (Bono et al. 2000b;De Somma et al. 2021). Classical Cepheids are solid stellar beacons to trace young stellar populations even in the far side of the Galactic thin disk (Minniti et al. 2020(Minniti et al. , 2021 and in stellar systems experiencing recent star formation events such as dwarf irregulars and star forming galaxies (Neeley et al. 2021). Their period-luminosity relations (Breuval et al. 2020;Ripepi et al. 2020) provide accurate (at the level of ∼1-2%) measurements of individual distances. Individual distances together with detailed information concerning their elemental abundances (iron peak, α, neutron capture), provide the unique opportunity to investigate the radial gradients and, in turn, the recent (t ≤ 200-300 Myr) chemical enrichment history of the thin disk and nearby stellar systems (Romaniello et al. 2008;Lemasle et al. 2013;Genovali et al. 2015;da Silva et al. 2016). Furthermore, classical Cepheids are fundamental laboratories for investigating not only the advanced evolutionary phases of intermediate-mass stars (Bono et al. 2000a;Neilson et al. 2011;Prada Moroni et al. 2012;De Somma et al. 2021), but also for constraining the physical mechanisms driving their variability (Bono et al. 1999;Marconi et al. 2013).\\nAlthough classical Cepheids are the cross-road of several recent and long-standing astrophysical problems, we still lack accurate investigations concerning the variation of their physical properties along the pulsation cycle. The current studies are mainly based on both optical and near-infrared (NIR) light curves, but we still lack detailed spectroscopic analysis concerning the variation of effective temperature, surface gravity, and microturbulent velocity (see, e.g., Lemasle et al. 2020;Kovtyukh 2007;Proxauf et al. 2018, and references therein).\\nAlthough classical Cepheids are the cross-road of several recent and long-standing astrophysical problems, we still lack accurate investigations concerning the variation of their physical properties along the pulsation cycle. The current studies are mainly based on both optical and near-infrared (NIR) light curves, but we still lack detailed spectroscopic analysis concerning the variation of effective temperature, surface gravity, and microturbulent velocity (see, e.g., Lemasle et al. 2020;Kovtyukh 2007;Proxauf et al. 2018, and references therein).\\nPioneering investigations based on multiple (from a few to almost two dozen) spectra along the pulsation cycle were provided more than ten years ago by Luck & Andrievsky (2004), Kovtyukh et al. (2005), Andrievsky et al. (2005), and Luck et al. (2008), in which they evaluated the variation of the atmospheric parameters and the abundances in both short-and long-period Cepheids. More recently, we performed a detailed investigation, based on a large dataset of high-resolution spectra, for a sample of 14 calibrating Cepheids (Proxauf et al. 2018). In that work, the number of spectra per object ranges from five to more than 120, and the spectra covered a significant portion of the pulsation cycle. In the same investigation we also used a new and more homogeneous list of Line Depth Ratios (LDR) to estimate the effective temperature directly from the spectra, an approach that is independent of the photometric properties. The reader interested in a more detailed discussion is referred to Kovtyukh (2007, and references therein). However, all these investigations were hampered by three limitations: a) Line list -We compiled a list of iron lines using line lists available in the literature. The sources of atomic transition parameters were inhomogeneous, because they did not pass a detailed scrutiny. b) Phase coverage -In spite of the large number of spectra adopted in the spectroscopic analysis, several objects had spectra collected at similar phases and, consequently, they had a modest phase coverage. c) Intrinsic variation -The inhomogeneity of the atomic transition parameters introduces uncertainties in the spectroscopic surface gravity and in the microturbulent velocity. This caused an increase in the standard deviation at fixed The vertical red lines indicate the position in the rest-frame wavelength of atomic lines either only used for the abundance determination or also adopted by the LDR method. For the sake of clarity the continuum of each spectrum was shifted to arbitrary levels.\\nPioneering investigations based on multiple (from a few to almost two dozen) spectra along the pulsation cycle were provided more than ten years ago by Luck & Andrievsky (2004), Kovtyukh et al. (2005), Andrievsky et al. (2005), and Luck et al. (2008), in which they evaluated the variation of the atmospheric parameters and the abundances in both short-and long-period Cepheids. More recently, we performed a detailed investigation, based on a large dataset of high-resolution spectra, for a sample of 14 calibrating Cepheids (Proxauf et al. 2018). In that work, the number of spectra per object ranges from five to more than 120, and the spectra covered a significant portion of the pulsation cycle. In the same investigation we also used a new and more homogeneous list of Line Depth Ratios (LDR) to estimate the effective temperature directly from the spectra, an approach that is independent of the photometric properties. The reader interested in a more detailed discussion is referred to Kovtyukh (2007, and references therein). However, all these investigations were hampered by three limitations: a) Line list -We compiled a list of iron lines using line lists available in the literature. The sources of atomic transition parameters were inhomogeneous, because they did not pass a detailed scrutiny. b) Phase coverage -In spite of the large number of spectra adopted in the spectroscopic analysis, several objects had spectra collected at similar phases and, consequently, they had a modest phase coverage. c) Intrinsic variation -The inhomogeneity of the atomic transition parameters introduces uncertainties in the spectroscopic surface gravity and in the microturbulent velocity. This caused an increase in the standard deviation at fixed The vertical red lines indicate the position in the rest-frame wavelength of atomic lines either only used for the abundance determination or also adopted by the LDR method. For the sake of clarity the continuum of each spectrum was shifted to arbitrary levels.\\npulsation phase, and in turn, some limitation in the analysis of their variation along the pulsation cycle.\\npulsation phase, and in turn, some limitation in the analysis of their variation along the pulsation cycle.\\nTo overcome these limitations, the current investigation is based on a new approach including five steps: i) We updated the atomic transition parameters of our line lists with more recent and accurate laboratory measurements available in the literature; ii) We collected new high signal-to-noise ratio (S/N) and highresolution spectra for our sample of calibrating Cepheids, and we also performed a detailed search for similar quality spectra available in online science archives. This translates into an increase in the number of calibrating Cepheids by almost 50% (from 14 to 20). Moreover, we also added new spectra to Cepheids already included in the former sample. The current sample includes 19 stars pulsating in the fundamental mode and, for the first time used by us as a calibrating Cepheid, one star (FF Aql) pulsating in the first overtone mode, and their metallicities range from −0.08 (β Dor) to 0.19 dex (VY Sgr); iii) We performed a detailed visual check of hundreds of lines along the pulsation cycle to create a new homogeneous line list. Special attention was paid in the identification of lines showing blends (or poor definition of the continuum), or having any correlation with the effective temperature (due to possible NLTE effects) or with the equivalent widths (due to saturated lines), or presenting any systematic over-or under-abundance; iv) We improved the algorithm that we use for the estimate of the atmospheric parameters, and in turn, for the estimate of the elemental abundances; v) In addition to the atmospheric parameters and iron abundances, the current study also includes the abundances of five α elements: Mg, Si, S, Ca, and Ti.\\nTo overcome these limitations, the current investigation is based on a new approach including five steps: i) We updated the atomic transition parameters of our line lists with more recent and accurate laboratory measurements available in the literature; ii) We collected new high signal-to-noise ratio (S/N) and highresolution spectra for our sample of calibrating Cepheids, and we also performed a detailed search for similar quality spectra available in online science archives. This translates into an increase in the number of calibrating Cepheids by almost 50% (from 14 to 20). Moreover, we also added new spectra to Cepheids already included in the former sample. The current sample includes 19 stars pulsating in the fundamental mode and, for the first time used by us as a calibrating Cepheid, one star (FF Aql) pulsating in the first overtone mode, and their metallicities range from −0.08 (β Dor) to 0.19 dex (VY Sgr); iii) We performed a detailed visual check of hundreds of lines along the pulsation cycle to create a new homogeneous line list. Special attention was paid in the identification of lines showing blends (or poor definition of the continuum), or having any correlation with the effective temperature (due to possible NLTE effects) or with the equivalent widths (due to saturated lines), or presenting any systematic over-or under-abundance; iv) We improved the algorithm that we use for the estimate of the atmospheric parameters, and in turn, for the estimate of the elemental abundances; v) In addition to the atmospheric parameters and iron abundances, the current study also includes the abundances of five α elements: Mg, Si, S, Ca, and Ti.\\nWe notice that the star AV Sgr, which was one of the calibrating Cepheids in Proxauf et al. (2018), is not included in the current work. The reason is that, after the revision of the Fe line list, we were not able to derive accurate estimates of the atmospheric parameters for many of the spectra of this variable. The number of spectra with accurate measurements was not enough to provide a good coverage of the pulsation cycle, and therefore, it was removed from the list of calibrating Cepheids. The parameters published in our previous paper can still be used, but keeping in mind that they are based on a different version of the line list.\\nWe notice that the star AV Sgr, which was one of the calibrating Cepheids in Proxauf et al. (2018), is not included in the current work. The reason is that, after the revision of the Fe line list, we were not able to derive accurate estimates of the atmospheric parameters for many of the spectra of this variable. The number of spectra with accurate measurements was not enough to provide a good coverage of the pulsation cycle, and therefore, it was removed from the list of calibrating Cepheids. The parameters published in our previous paper can still be used, but keeping in mind that they are based on a different version of the line list.\\nThe structure of the paper is the following: in Sect. 2 we discuss the spectroscopic dataset and provide detailed information concerning the current sample of calibrating Cepheids. In Sect. 3 we discuss the data reduction process and introduce the atomic line lists used. Sect. 4 includes a description of the radial-velocity and photometric data used. In Sect. 5 we explain how we derived the atmospheric parameters and their variation along the pulsation cycle. The iron and α-element abundances and their variation along the pulsation cycle are discussed in Sect. 6. A summary of the results and some final remarks concerning this project are given in Sect. 7.\\nThe structure of the paper is the following: in Sect. 2 we discuss the spectroscopic dataset and provide detailed information concerning the current sample of calibrating Cepheids. In Sect. 3 we discuss the data reduction process and introduce the atomic line lists used. Sect. 4 includes a description of the radial-velocity and photometric data used. In Sect. 5 we explain how we derived the atmospheric parameters and their variation along the pulsation cycle. The iron and α-element abundances and their variation along the pulsation cycle are discussed in Sect. 6. A summary of the results and some final remarks concerning this project are given in Sect. 7.\"\n },\n {\n \"title\": \"Spectroscopic datasets\",\n \"content\": \"\"\n },\n {\n \"title\": \"Spectroscopic datasets\",\n \"content\": \"The spectroscopic datasets analyzed in the current paper are spectra collected using several different instruments. Three of them are mounted on telescopes of the European Southern Observatory (ESO): the High Accuracy Radial velocity Planet Searcher (HARPS; Mayor et al. 2003) at the 3.6 m, the Fiberfed Extended Range Optical Spectrograph (FEROS; Kaufer et al. 1999) at the 2.2 m MPG/ESO, and the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) at the Very Large Telescope. The other instrument is the STELLA Echelle Spectrograph (SES; Strassmeier et al. 2004Strassmeier et al. , 2010 located at the Izãna Observatory on Tenerife in the Canary islands.\\nThe spectroscopic datasets analyzed in the current paper are spectra collected using several different instruments. Three of them are mounted on telescopes of the European Southern Observatory (ESO): the High Accuracy Radial velocity Planet Searcher (HARPS; Mayor et al. 2003) at the 3.6 m, the Fiberfed Extended Range Optical Spectrograph (FEROS; Kaufer et al. 1999) at the 2.2 m MPG/ESO, and the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) at the Very Large Telescope. The other instrument is the STELLA Echelle Spectrograph (SES; Strassmeier et al. 2004Strassmeier et al. , 2010 located at the Izãna Observatory on Tenerife in the Canary islands.\\nThe spectral resolution of the quoted spectrographs for the instrument settings used are R∼40 000 (UVES), R∼115 000 (HARPS), R∼48 000 (FEROS), and R∼55 000 (STELLA). For details on the wavelength ranges of our spectra we refer the reader to Sect. 2 of the papers Proxauf et al. (2018) and Crestani et al. (2021).\\nThe spectral resolution of the quoted spectrographs for the instrument settings used are R∼40 000 (UVES), R∼115 000 (HARPS), R∼48 000 (FEROS), and R∼55 000 (STELLA). For details on the wavelength ranges of our spectra we refer the reader to Sect. 2 of the papers Proxauf et al. (2018) and Crestani et al. (2021).\\nThe entire spectroscopic sample includes reduced spectra downloaded from the ESO and the STELLA archives, for a total of 1383 spectra of 285 stars: 199 HARPS spectra of 9 stars, 419 FEROS spectra of 151 stars, 450 STELLA spectra of 68 stars, and 315 UVES spectra of 123 stars. Their S/N ranges from a few tens to more than 400 (about 70% of our spectra have S/N of at least 100). From this sample we selected the spectra of 20 targets that cover either a significant part (about half or more) or the entire pulsation cycle. From now on we refer to this subsample as our sample of 20 calibrating Cepheids. Details on the number of spectra are given in Table 1.\\nThe entire spectroscopic sample includes reduced spectra downloaded from the ESO and the STELLA archives, for a total of 1383 spectra of 285 stars: 199 HARPS spectra of 9 stars, 419 FEROS spectra of 151 stars, 450 STELLA spectra of 68 stars, and 315 UVES spectra of 123 stars. Their S/N ranges from a few tens to more than 400 (about 70% of our spectra have S/N of at least 100). From this sample we selected the spectra of 20 targets that cover either a significant part (about half or more) or the entire pulsation cycle. From now on we refer to this subsample as our sample of 20 calibrating Cepheids. Details on the number of spectra are given in Table 1.\"\n },\n {\n \"title\": \"Data reduction and analysis\",\n \"content\": \"\"\n },\n {\n \"title\": \"Data reduction and analysis\",\n \"content\": \"As mentioned in Proxauf et al. (2018), the spectra from UVES and HARPS (Phase 3) were already pre-reduced (i.e., reduced up to the wavelength calibration step) by their own pipeline. As previously done, the FEROS spectra were reduced using a modified version of the Data Reduction System (DRS) pipeline. A few tens of FEROS spectra collected before 2004, which initially could not be reduced, were later on reduced and delivered by one of the Phase 3 Data Releases (DR3 and DR3.1). However, they are all low-quality spectra that were not included in our analysis.\\nAs mentioned in Proxauf et al. (2018), the spectra from UVES and HARPS (Phase 3) were already pre-reduced (i.e., reduced up to the wavelength calibration step) by their own pipeline. As previously done, the FEROS spectra were reduced using a modified version of the Data Reduction System (DRS) pipeline. A few tens of FEROS spectra collected before 2004, which initially could not be reduced, were later on reduced and delivered by one of the Phase 3 Data Releases (DR3 and DR3.1). However, they are all low-quality spectra that were not included in our analysis.\\nAfter the pre-reduction steps, all the selected spectra were normalized to the continuum using the Image Reduction and Analysis Facility (IRAF 1 ) by fitting cubic spline functions to a set of continuum windows. An exception is the case of the STELLA spectra, which were reduced by a dedicated pipeline (Weber et al. 2012) based on IRAF routines and that includes several reduction steps up to the continuum normalization.\\nAfter the pre-reduction steps, all the selected spectra were normalized to the continuum using the Image Reduction and Analysis Facility (IRAF 1 ) by fitting cubic spline functions to a set of continuum windows. An exception is the case of the STELLA spectra, which were reduced by a dedicated pipeline (Weber et al. 2012) based on IRAF routines and that includes several reduction steps up to the continuum normalization.\\nFor more details about the quality of the spectra we refer the reader to Sect. 3 of Proxauf et al. (2018), which also includes a figure (their Fig. 1) showing a few examples of HARPS, FEROS, UVES, and STELLA spectra of different metallicities and S/N. In our Fig. 1, we plot an example of HARPS spectra for β Dor and of STELLA spectra for ζ Gem in different pulsation phases, showing how the profile of the absorption lines may change (in depth and in shape) along the pulsation cycle. Studies of line asymmetry changes over the pulsation cycle of Cepheid stars were also reported by Nardetto et al. (2006) and Nardetto et al. (2008), in a series of papers on high-resolution spectroscopy, for Fe i and Hα lines in the optical and, more recently, by Nardetto et al. (2018) for the Na i line at 2208.969 nm.\\nFor more details about the quality of the spectra we refer the reader to Sect. 3 of Proxauf et al. (2018), which also includes a figure (their Fig. 1) showing a few examples of HARPS, FEROS, UVES, and STELLA spectra of different metallicities and S/N. In our Fig. 1, we plot an example of HARPS spectra for β Dor and of STELLA spectra for ζ Gem in different pulsation phases, showing how the profile of the absorption lines may change (in depth and in shape) along the pulsation cycle. Studies of line asymmetry changes over the pulsation cycle of Cepheid stars were also reported by Nardetto et al. (2006) and Nardetto et al. (2008), in a series of papers on high-resolution spectroscopy, for Fe i and Hα lines in the optical and, more recently, by Nardetto et al. (2018) for the Na i line at 2208.969 nm.\"\n },\n {\n \"title\": \"Line lists and equivalent widths\",\n \"content\": \"\"\n },\n {\n \"title\": \"Line lists and equivalent widths\",\n \"content\": \"Similarly to what we did in our previous works, here we use three different line lists: The Cepheids variables are bright stars and thus we have at our disposal very high S/N, broad wavelength range spectra. The precision of the atmospheric parameters and of the chemical abundances, however, depends not only on the quality of the spectra, but also on the quality of the adopted line list. With this in mind, a large portion of the present work was dedicated to finding the best atomic transitions for iron and α elements that can be detected in Cepheids. In practice, this means i) collecting the most precise transition parameters, ii) removing absorption lines that are blended with other lines, and iii) eliminating lines that for any reason deviate significantly from the average value for their chemical species, or that display a dependency on any quantity that changes across the pulsation cycle, such as the effective temperature.\\nSimilarly to what we did in our previous works, here we use three different line lists: The Cepheids variables are bright stars and thus we have at our disposal very high S/N, broad wavelength range spectra. The precision of the atmospheric parameters and of the chemical abundances, however, depends not only on the quality of the spectra, but also on the quality of the adopted line list. With this in mind, a large portion of the present work was dedicated to finding the best atomic transitions for iron and α elements that can be detected in Cepheids. In practice, this means i) collecting the most precise transition parameters, ii) removing absorption lines that are blended with other lines, and iii) eliminating lines that for any reason deviate significantly from the average value for their chemical species, or that display a dependency on any quantity that changes across the pulsation cycle, such as the effective temperature.\\nIn order to address the first point, we departed from a list of hundreds of atomic transitions for iron and α elements commonly employed in the literature. We updated their transition parameters whenever possible with the updated laboratory measurements from Ruffoni Bottom: same as in the top panel but for the V-band light curve, its PLOESS fit and its mean value. We note that, within the uncertainty, the light curve is co-phased with the RV curve.\\nIn order to address the first point, we departed from a list of hundreds of atomic transitions for iron and α elements commonly employed in the literature. We updated their transition parameters whenever possible with the updated laboratory measurements from Ruffoni Bottom: same as in the top panel but for the V-band light curve, its PLOESS fit and its mean value. We note that, within the uncertainty, the light curve is co-phased with the RV curve.\\nFe ii, Lawler et al. (2013) for Ti i, and Wood et al. (2013) for Ti ii. We also made ample use of the astrophysical, but homogeneous and precise compilation of Meléndez & Barbuy (2009) for Fe ii. For the remaining lines, we adopted the transition parameters collected and updated by the National Institute of Standards and Technology (NIST) Atomic Spectra Database (Kramida et al. 2020). If a line is not available in any of these sources, it is eliminated from the preliminary list and not used for the computation of atmospheric parameters nor chemical abundances.\\nFe ii, Lawler et al. (2013) for Ti i, and Wood et al. (2013) for Ti ii. We also made ample use of the astrophysical, but homogeneous and precise compilation of Meléndez & Barbuy (2009) for Fe ii. For the remaining lines, we adopted the transition parameters collected and updated by the National Institute of Standards and Technology (NIST) Atomic Spectra Database (Kramida et al. 2020). If a line is not available in any of these sources, it is eliminated from the preliminary list and not used for the computation of atmospheric parameters nor chemical abundances.\\nWith this preliminary list at hand, we addressed the second point mentioned above by removing any blended lines, using as reference the Solar spectrum table by Moore et al. (1966) and synthetic spectra. Then we measured their equivalent width (EW) using the Automatic Routine for line Equivalent widths in stellar Spectra (ARES, Sousa et al. 2007Sousa et al. , 2015.\\nWith this preliminary list at hand, we addressed the second point mentioned above by removing any blended lines, using as reference the Solar spectrum table by Moore et al. (1966) and synthetic spectra. Then we measured their equivalent width (EW) using the Automatic Routine for line Equivalent widths in stellar Spectra (ARES, Sousa et al. 2007Sousa et al. , 2015.\\nFinally, we addressed the third point by removing deviant lines. In a first iteration, we estimated initial values for the atmospheric parameters and the chemical abundances. Lines are considered deviant if they result in abundances diverging by at least 3 σ from the mean for a given element, or if their behavior across the pulsation cycle is irregular. This latter situation is investigated by plotting all line-by-line measurements for each chemical species and individual exposure of our calibrating stars versus phase, effective temperature, and equivalent width, and removing lines that show strong trends in any of those planes. The calibrating stars mostly used are those with the largest number of individual spectra covering the whole pulsation cycle, such as β Dor, ζ Gem, and FF Aql. Tables 2 and 3 with quality flag 1, were used to compute the final atmospheric parameters and chemical abundances for the whole sample. The remaining lines, Fig. 4. Photometric amplitude as a function of the logarithmic period (Bailey diagram). Our calibrating Cepheids are represented by colored symbols whereas the Cepheids from literature Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006) are shown in light gray circles.\\nFinally, we addressed the third point by removing deviant lines. In a first iteration, we estimated initial values for the atmospheric parameters and the chemical abundances. Lines are considered deviant if they result in abundances diverging by at least 3 σ from the mean for a given element, or if their behavior across the pulsation cycle is irregular. This latter situation is investigated by plotting all line-by-line measurements for each chemical species and individual exposure of our calibrating stars versus phase, effective temperature, and equivalent width, and removing lines that show strong trends in any of those planes. The calibrating stars mostly used are those with the largest number of individual spectra covering the whole pulsation cycle, such as β Dor, ζ Gem, and FF Aql. Tables 2 and 3 with quality flag 1, were used to compute the final atmospheric parameters and chemical abundances for the whole sample. The remaining lines, Fig. 4. Photometric amplitude as a function of the logarithmic period (Bailey diagram). Our calibrating Cepheids are represented by colored symbols whereas the Cepheids from literature Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006) are shown in light gray circles.\"\n },\n {\n \"title\": \"The final clean lists, shown in\",\n \"content\": \"\"\n },\n {\n \"title\": \"The final clean lists, shown in\",\n \"content\": \"shown in the same tables with quality flag 0, were measured only for comparison purposes.\\nshown in the same tables with quality flag 0, were measured only for comparison purposes.\\nAn example of a good Fe line is shown in Fig. 2 for β Dor and ζ Gem. Examples and a description of the typical behavior of irregular (less reliable) lines are found in the Appendix A, in the plots of Figs. A.1-A.5. These same irregular lines, however, might provide good results for other stars, for instance, of different spectral types, or having spectra of different quality or different resolution. This is the reason why we preferred to exclude some lines only for this particular analysis, and keep the initial line lists as they are to be used in future studies. It is worth mentioning that we only excluded lines that were clearly having problems in most of the stars in our sample.\\nAn example of a good Fe line is shown in Fig. 2 for β Dor and ζ Gem. Examples and a description of the typical behavior of irregular (less reliable) lines are found in the Appendix A, in the plots of Figs. A.1-A.5. These same irregular lines, however, might provide good results for other stars, for instance, of different spectral types, or having spectra of different quality or different resolution. This is the reason why we preferred to exclude some lines only for this particular analysis, and keep the initial line lists as they are to be used in future studies. It is worth mentioning that we only excluded lines that were clearly having problems in most of the stars in our sample.\"\n },\n {\n \"title\": \"Radial velocity and photometric data\",\n \"content\": \"\"\n },\n {\n \"title\": \"Radial velocity and photometric data\",\n \"content\": \"The radial velocity (RV) measurements for our sample were done using the following methods: i) for HARPS and UVES spectra, we used IRAF packages to cross-correlate the target spectrum with a solar template spectrum (Solar flux atlas from 296 to 1300 nm, Kurucz et al. 1984) degraded to the UVES resolution; ii) for FEROS spectra we adopted the RVs derived by ARES, the routine we used for the equivalent width measurements (see Sect. 3.1); iii) the STELLA radial velocities are written in the header of the FITS files and they are based on crosscorrelations with a template spectrum, performed by a dedicated reduction pipeline also using IRAF. In case of the Cepheids, a G-type dwarf template is used.\\nThe radial velocity (RV) measurements for our sample were done using the following methods: i) for HARPS and UVES spectra, we used IRAF packages to cross-correlate the target spectrum with a solar template spectrum (Solar flux atlas from 296 to 1300 nm, Kurucz et al. 1984) degraded to the UVES resolution; ii) for FEROS spectra we adopted the RVs derived by ARES, the routine we used for the equivalent width measurements (see Sect. 3.1); iii) the STELLA radial velocities are written in the header of the FITS files and they are based on crosscorrelations with a template spectrum, performed by a dedicated reduction pipeline also using IRAF. In case of the Cepheids, a G-type dwarf template is used.\\nThe uncertainties adopted for these RV estimates depend on the method used: for HARPS, UVES, and STELLA spectra, the uncertainties are estimated by IRAF during the cross-correlation; for FEROS spectra, they are estimated by ARES using spectral windows free of lines. Typical values for our HARPS, UVES, and FEROS spectra range from a few m/s in the best cases to a few km s −1 , depending on the spectral type. For the STELLA spectra, we adopted an uncertainty of 0.2 km s −1 . This is a conservative estimate considering that typical formal errors for high S/N spectra of Cepheids stars is about 50 m s −1 and that the RV zero point for this spectrograph has RMS variations of 30 to 150 m s −1 over two years. The RV estimates and the corresponding uncertainties for our sample of calibrating Cepheids are listed in Table 4.\\nThe uncertainties adopted for these RV estimates depend on the method used: for HARPS, UVES, and STELLA spectra, the uncertainties are estimated by IRAF during the cross-correlation; for FEROS spectra, they are estimated by ARES using spectral windows free of lines. Typical values for our HARPS, UVES, and FEROS spectra range from a few m/s in the best cases to a few km s −1 , depending on the spectral type. For the STELLA spectra, we adopted an uncertainty of 0.2 km s −1 . This is a conservative estimate considering that typical formal errors for high S/N spectra of Cepheids stars is about 50 m s −1 and that the RV zero point for this spectrograph has RMS variations of 30 to 150 m s −1 over two years. The RV estimates and the corresponding uncertainties for our sample of calibrating Cepheids are listed in Table 4.\\nIn addition to our own RV measurements, we adopted literature RV curves and light curves to provide accurate ephemerides, meaning reference epoch (T 0 ) plus pulsation period (P) for these Table 4. Excerpt from the list of atmospheric parameters, Fe abundances, and radial velocities for each spectrum of the 20 calibrating Cepheids. Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. Columns 4, 5, and 6 give, respectively, the effective temperature and its standard deviation, surface gravity, and microturbulent velocity. The Cols. 7-8 and 9-10 list both the Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used (for the Fe ii abundances, a minimal uncertainty of 0.11 dex was adopt when computing the mean abundances, as described in the text). The last column gives the radial velocities with their uncertainties. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the mean magnitude in visual bands, the magnitude amplitude, the mean radial velocity, the RV amplitude, the light curve source, the pulsation period and zero-phase reference epoch of mean magnitude (or mean radial velocity), and the Ephemerides source. For the variables VY Sgr, RZ Vel, and WZ Sgr, our RV measurements and those by Groenewegen (2008, and references therein) show a shift in phase of −0.05, +0.03, and −0.03, respectively. To co-phase the two datasets, the latter sample of radial velocities plotted in Fig variables. More specifically, we collected RV curves and V-band light curves from Pel (1976) and Groenewegen (2008, and references therein). For UZ Sct, we adopted the V-band light curve from the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappee et al. 2014;Kochanek et al. 2017) because it is the only well-sampled light curve that can be co-phased with our RV time series.\\nIn addition to our own RV measurements, we adopted literature RV curves and light curves to provide accurate ephemerides, meaning reference epoch (T 0 ) plus pulsation period (P) for these Table 4. Excerpt from the list of atmospheric parameters, Fe abundances, and radial velocities for each spectrum of the 20 calibrating Cepheids. Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. Columns 4, 5, and 6 give, respectively, the effective temperature and its standard deviation, surface gravity, and microturbulent velocity. The Cols. 7-8 and 9-10 list both the Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used (for the Fe ii abundances, a minimal uncertainty of 0.11 dex was adopt when computing the mean abundances, as described in the text). The last column gives the radial velocities with their uncertainties. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the mean magnitude in visual bands, the magnitude amplitude, the mean radial velocity, the RV amplitude, the light curve source, the pulsation period and zero-phase reference epoch of mean magnitude (or mean radial velocity), and the Ephemerides source. For the variables VY Sgr, RZ Vel, and WZ Sgr, our RV measurements and those by Groenewegen (2008, and references therein) show a shift in phase of −0.05, +0.03, and −0.03, respectively. To co-phase the two datasets, the latter sample of radial velocities plotted in Fig variables. More specifically, we collected RV curves and V-band light curves from Pel (1976) and Groenewegen (2008, and references therein). For UZ Sct, we adopted the V-band light curve from the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappee et al. 2014;Kochanek et al. 2017) because it is the only well-sampled light curve that can be co-phased with our RV time series.\\nWe estimated the pulsation periods of our targets by using our own interactive method ) based on the Lomb-Scargle periodogram (Scargle 1982). Historically, the most commonly used T 0 is the time of maximum light (T max ), which matches the time of minimum velocity. Starting from Inno et al. (2015), we have been promoting the use of another reference for both Cepheids and RR Lyrae, that is, the epoch of the mean magnitude on the rising branch of the light curve (T opt mean ) or, equivalently, the epoch of mean velocity on the decreasing branch of the RV curve (T RV mean ). See Fig. 3 for a visualization of this reference epoch on both the light and RV curves. A detailed procedure to derive T opt mean is described in Braga et al. (2021). The advantages of the T opt mean reference epoch over T max are wellexplained in Inno et al. (2015) and quantitatively discussed in Braga et al. (2021). Name Notes. Columns from 1 to 5 give, respectively, the star name, the mean effective temperature and mean θ, the θ amplitude (see Section 5.2), and the number of spectra used to compute these mean values. Columns from 6 to 9 lists the surface gravity, microturbulent velocity, and iron abundances from neutral and ionized lines. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 4. Column 10 gives the weighted iron abundance and its standard error, calculated from Cols. 8 and 9. The standard deviation calculated using all individual measurements from both Fe i and Fe ii lines is also shown within parentheses. The last column gives the number of spectra used to derive log g, υ t , and the iron abundances.\\nWe estimated the pulsation periods of our targets by using our own interactive method ) based on the Lomb-Scargle periodogram (Scargle 1982). Historically, the most commonly used T 0 is the time of maximum light (T max ), which matches the time of minimum velocity. Starting from Inno et al. (2015), we have been promoting the use of another reference for both Cepheids and RR Lyrae, that is, the epoch of the mean magnitude on the rising branch of the light curve (T opt mean ) or, equivalently, the epoch of mean velocity on the decreasing branch of the RV curve (T RV mean ). See Fig. 3 for a visualization of this reference epoch on both the light and RV curves. A detailed procedure to derive T opt mean is described in Braga et al. (2021). The advantages of the T opt mean reference epoch over T max are wellexplained in Inno et al. (2015) and quantitatively discussed in Braga et al. (2021). Name Notes. Columns from 1 to 5 give, respectively, the star name, the mean effective temperature and mean θ, the θ amplitude (see Section 5.2), and the number of spectra used to compute these mean values. Columns from 6 to 9 lists the surface gravity, microturbulent velocity, and iron abundances from neutral and ionized lines. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 4. Column 10 gives the weighted iron abundance and its standard error, calculated from Cols. 8 and 9. The standard deviation calculated using all individual measurements from both Fe i and Fe ii lines is also shown within parentheses. The last column gives the number of spectra used to derive log g, υ t , and the iron abundances.\\nThe light and radial-velocity curves for all the sample stars are shown in the Appendix B, in Fig. B.1. The estimated ephemerides of our targets are listed in Table 5 together with the mean magnitudes and magnitude amplitudes in the visual band, and the mean radial velocities and RV amplitudes. We noticed that for the stars VY Sgr, RZ Vel, and WZ Sgr, for which the new period and T 0 values are based on the radial velocities derived by Groenewegen (2008, and references therein), the new ephemerides caused a shift in phase between the two datasets shown in Fig. B.1. To overcome this problem, we applied a shift of −0.05, +0.03, and −0.03, respectively, to the literature data plotted in these panels. Figure 4 shows the distribution of the current calibrating Cepheids (colored symbols), together with Galactic Cepheids available in the literature (light gray circles), in the so-called Bailey diagram (luminosity amplitude as a function of the logarithmic period). The plotted data display the classical \\\"V\\\" shape in which the local minimum around ten days is mainly caused by the Hertzsprung progression (Bono et al. 2000b). By having a look also at Fig. B.1, we see a subgroup of classical Cepheids showing a well defined bump on the light curve. The phase of such bumps moves from the decreasing branch for periods between ∼6 and ∼9 days, passing across the maximum for periods between ∼9 and ∼12 days, and along the rising branch for periods between ∼12 and ∼16 days. At the center of the Hertzsprung progression (P∼10 days) the light curve is flat topped and the luminosity amplitude attains a well defined minimum (Bono et al. 2000b(Bono et al. , 2002. These variables are called Bump Cepheids for avoiding to be mixed-up with the Beat Cepheids, that is, classical Cepheids pulsating simultaneously in two or more modes. The calibrating Cepheids cover both the short-and the longperiod range, and in particular, the current sample includes seven Cepheids across the Hertzsprung progression.\\nThe light and radial-velocity curves for all the sample stars are shown in the Appendix B, in Fig. B.1. The estimated ephemerides of our targets are listed in Table 5 together with the mean magnitudes and magnitude amplitudes in the visual band, and the mean radial velocities and RV amplitudes. We noticed that for the stars VY Sgr, RZ Vel, and WZ Sgr, for which the new period and T 0 values are based on the radial velocities derived by Groenewegen (2008, and references therein), the new ephemerides caused a shift in phase between the two datasets shown in Fig. B.1. To overcome this problem, we applied a shift of −0.05, +0.03, and −0.03, respectively, to the literature data plotted in these panels. Figure 4 shows the distribution of the current calibrating Cepheids (colored symbols), together with Galactic Cepheids available in the literature (light gray circles), in the so-called Bailey diagram (luminosity amplitude as a function of the logarithmic period). The plotted data display the classical \\\"V\\\" shape in which the local minimum around ten days is mainly caused by the Hertzsprung progression (Bono et al. 2000b). By having a look also at Fig. B.1, we see a subgroup of classical Cepheids showing a well defined bump on the light curve. The phase of such bumps moves from the decreasing branch for periods between ∼6 and ∼9 days, passing across the maximum for periods between ∼9 and ∼12 days, and along the rising branch for periods between ∼12 and ∼16 days. At the center of the Hertzsprung progression (P∼10 days) the light curve is flat topped and the luminosity amplitude attains a well defined minimum (Bono et al. 2000b(Bono et al. , 2002. These variables are called Bump Cepheids for avoiding to be mixed-up with the Beat Cepheids, that is, classical Cepheids pulsating simultaneously in two or more modes. The calibrating Cepheids cover both the short-and the longperiod range, and in particular, the current sample includes seven Cepheids across the Hertzsprung progression.\"\n },\n {\n \"title\": \"Atmospheric parameters and effective temperature curve templates\",\n \"content\": \"\"\n },\n {\n \"title\": \"Atmospheric parameters and effective temperature curve templates\",\n \"content\": \"\"\n },\n {\n \"title\": \"Effective temperature, surface gravity, and microturbulent velocity\",\n \"content\": \"\"\n },\n {\n \"title\": \"Effective temperature, surface gravity, and microturbulent velocity\",\n \"content\": \"The approach we adopted to estimate the atmospheric parameters (i.e., the effective temperature, the surface gravity, and the microturbulent velocity) was already discussed in detail by Proxauf et al. (2018). Here we only recap the key points. The T eff along the pulsation cycle was estimated by using the LDR method, which relies on the correlation between the line depth ratios of pairs of absorption lines in the spectra of different stars and the effective temperature of the same stars. The surface gravity was derived through the ionization equilibrium of Fe i and Fe ii lines, and the microturbulent velocity was obtained by minimizing the dependence of the abundances provided by single Fe i lines on their EWs. During this procedure, the effective temperature was kept fixed, whereas log g and ξ were iteratively changed until convergence. The metallicity used as input by our algorithm is updated in each step, and the adopted value is [Fe i/H], which is the mean iron abundances provided by individual Fe i lines. The estimates of T eff , log g, and ξ for the individual spectrum of our sample are listed in Table 4. The uncertainties on T eff are the standard deviation calculated using the LDR method, that is, from the effective temperatures provided by the individual pairs of lines. The uncertainties on the individual estimates of log g and ξ are assumed to be of the order of 0.3 dex and 0.5 km s −1 , respectively (see Genovali et al. 2014, for a detailed discussion). The weighted mean values of the atmospheric parameters are listed in Table 6 together with the corresponding standard errors calculated for each calibrating Cepheid.\\nThe approach we adopted to estimate the atmospheric parameters (i.e., the effective temperature, the surface gravity, and the microturbulent velocity) was already discussed in detail by Proxauf et al. (2018). Here we only recap the key points. The T eff along the pulsation cycle was estimated by using the LDR method, which relies on the correlation between the line depth ratios of pairs of absorption lines in the spectra of different stars and the effective temperature of the same stars. The surface gravity was derived through the ionization equilibrium of Fe i and Fe ii lines, and the microturbulent velocity was obtained by minimizing the dependence of the abundances provided by single Fe i lines on their EWs. During this procedure, the effective temperature was kept fixed, whereas log g and ξ were iteratively changed until convergence. The metallicity used as input by our algorithm is updated in each step, and the adopted value is [Fe i/H], which is the mean iron abundances provided by individual Fe i lines. The estimates of T eff , log g, and ξ for the individual spectrum of our sample are listed in Table 4. The uncertainties on T eff are the standard deviation calculated using the LDR method, that is, from the effective temperatures provided by the individual pairs of lines. The uncertainties on the individual estimates of log g and ξ are assumed to be of the order of 0.3 dex and 0.5 km s −1 , respectively (see Genovali et al. 2014, for a detailed discussion). The weighted mean values of the atmospheric parameters are listed in Table 6 together with the corresponding standard errors calculated for each calibrating Cepheid.\\nIn the Appendix B, Figs. B.2, B.3, and B.4 show the variation of the atmospheric parameters along the pulsation cycle. To provide a quantitative estimate of the differences between the current approach and our previous measurements, we fitted Fourier series to the effective temperature, surface gravity, and microturbulent velocity curves in order to evaluate their amplitudes, A&A proofs: manuscript no. daSilvaetal2021 The panels show the phase-folded curves separating the Cepheids in different bins of pulsation period (values within parentheses): from 3 to 5 days, from 5 to 7 days, from 7 to 9.5 days, and from 9.5 to 10.5 days. For the variable S Sge, the two outliers shown in Fig. B.2 were not used in the construction of the T eff curve templates. The curves display the fit of the Fourier series to the data (see Table 7). The standard deviations of the fit are also shown. similarly to what we did in Proxauf et al. (2018, their Figs. 3 and 4). We found that, on average, the errors on the derived amplitudes for the effective temperature curves are a factor 2 smaller than our previous estimates, whereas for the surface gravity the differences are even more significant, with typical errors almost 3 times smaller. For the microturbulent velocity variation, the errors on the current amplitude estimates are more than twice smaller (the dispersions are ∼50 K for T eff , ∼0.2 dex for log g, and ∼0.2 km s −1 for ξ). This means that the variation of the atmospheric parameters along the pulsation cycle is now much better constrained, thanks not only to the larger number of spectra but also to the better determination of such parameters. The improvement in the estimate of log g and ξ is mainly due to the improvement in the line list, which reduced the number of spurious abundance values provided by unreliable iron lines. As described in the Appendix A, we removed lines that are blended, lines that display a well defined trend with effective temperature or with equivalent width, among others. The estimate of T eff is based on the same line list that we previously used, therefore the improvement is mainly due to the larger number of spectra analyzed in the current work. All in all, these changes increased the accuracy in the atmospheric parameters. The reader interested in detailed discussion concerning the estimate of the intrinsic parameters of classical Cepheids is referred to Vasilyev et al. (2018) and Lemasle et al. (2020).\\nIn the Appendix B, Figs. B.2, B.3, and B.4 show the variation of the atmospheric parameters along the pulsation cycle. To provide a quantitative estimate of the differences between the current approach and our previous measurements, we fitted Fourier series to the effective temperature, surface gravity, and microturbulent velocity curves in order to evaluate their amplitudes, A&A proofs: manuscript no. daSilvaetal2021 The panels show the phase-folded curves separating the Cepheids in different bins of pulsation period (values within parentheses): from 3 to 5 days, from 5 to 7 days, from 7 to 9.5 days, and from 9.5 to 10.5 days. For the variable S Sge, the two outliers shown in Fig. B.2 were not used in the construction of the T eff curve templates. The curves display the fit of the Fourier series to the data (see Table 7). The standard deviations of the fit are also shown. similarly to what we did in Proxauf et al. (2018, their Figs. 3 and 4). We found that, on average, the errors on the derived amplitudes for the effective temperature curves are a factor 2 smaller than our previous estimates, whereas for the surface gravity the differences are even more significant, with typical errors almost 3 times smaller. For the microturbulent velocity variation, the errors on the current amplitude estimates are more than twice smaller (the dispersions are ∼50 K for T eff , ∼0.2 dex for log g, and ∼0.2 km s −1 for ξ). This means that the variation of the atmospheric parameters along the pulsation cycle is now much better constrained, thanks not only to the larger number of spectra but also to the better determination of such parameters. The improvement in the estimate of log g and ξ is mainly due to the improvement in the line list, which reduced the number of spurious abundance values provided by unreliable iron lines. As described in the Appendix A, we removed lines that are blended, lines that display a well defined trend with effective temperature or with equivalent width, among others. The estimate of T eff is based on the same line list that we previously used, therefore the improvement is mainly due to the larger number of spectra analyzed in the current work. All in all, these changes increased the accuracy in the atmospheric parameters. The reader interested in detailed discussion concerning the estimate of the intrinsic parameters of classical Cepheids is referred to Vasilyev et al. (2018) and Lemasle et al. (2020).\\nWe note in passing that the improvement in the accuracy and in the sampling of the pulsation cycle allow us to state that the minimum in the effective temperature is often approached in the same phases in which the minimum in surface gravity is also approached. Moreover, and even more importantly, these minimum phases anticipate the increase in microturbulent velocity. This parameter, as expected, attains it maximum just before the maximum in effective temperature, which can be clearly observed by comparing several panels of Figs. B.2 and B.4.\\nWe note in passing that the improvement in the accuracy and in the sampling of the pulsation cycle allow us to state that the minimum in the effective temperature is often approached in the same phases in which the minimum in surface gravity is also approached. Moreover, and even more importantly, these minimum phases anticipate the increase in microturbulent velocity. This parameter, as expected, attains it maximum just before the maximum in effective temperature, which can be clearly observed by comparing several panels of Figs. B.2 and B.4.\"\n },\n {\n \"title\": \"Effective temperature curve templates\",\n \"content\": \"\"\n },\n {\n \"title\": \"Effective temperature curve templates\",\n \"content\": \"By taking advantage of the substantial phase coverage of our sample of calibrating Cepheids, we used the effective temperature measurements to compute new templates for different bins of pulsation periods. However, instead of using the T eff curves directly, we preferred to provide templates for the theta parameter -defined as θ = 5040/T eff -given its linear dependency on the Cousins R-I color index (Taylor 1994).\\nBy taking advantage of the substantial phase coverage of our sample of calibrating Cepheids, we used the effective temperature measurements to compute new templates for different bins of pulsation periods. However, instead of using the T eff curves directly, we preferred to provide templates for the theta parameter -defined as θ = 5040/T eff -given its linear dependency on the Cousins R-I color index (Taylor 1994).\\nThe approach is similar to the NIR light-curve templates provided by Inno et al. (2015): first, we adopted the ephemerides in Table 5 to fold the θ curves (θCs). Subsequently, we normalized the folded θCs by subtracting their average and dividing by their amplitudes. Our θCs are well sampled, meaning that we have enough phase points to separate the Cepheids into different period bins and to provide analytical relations for the θ curve templates in each bin. We adopted the same period thresholds introduced by Inno et al. (2015). Therefore, based on their Table 1, we generated four cumulative and normalized theta curves for the bins 2 (3-5 days, three Cepheids), 3 (5-7 days, three Cepheids), 4 (7-9.5 days, two Cepheids), and 5 (9.5-10.5 days, two Cepheids). By adopting the same period bins we are able to provide θC templates that are homogeneous with those from the NIR light curves. The reader interested in a more detailed and quantitative discussion concerning the use of cumulative and normalized curves to derive the analytical fits, together with the adopted thresholds for the different period bins, is referred to Inno et al. (2015).\\nThe approach is similar to the NIR light-curve templates provided by Inno et al. (2015): first, we adopted the ephemerides in Table 5 to fold the θ curves (θCs). Subsequently, we normalized the folded θCs by subtracting their average and dividing by their amplitudes. Our θCs are well sampled, meaning that we have enough phase points to separate the Cepheids into different period bins and to provide analytical relations for the θ curve templates in each bin. We adopted the same period thresholds introduced by Inno et al. (2015). Therefore, based on their Table 1, we generated four cumulative and normalized theta curves for the bins 2 (3-5 days, three Cepheids), 3 (5-7 days, three Cepheids), 4 (7-9.5 days, two Cepheids), and 5 (9.5-10.5 days, two Cepheids). By adopting the same period bins we are able to provide θC templates that are homogeneous with those from the NIR light curves. The reader interested in a more detailed and quantitative discussion concerning the use of cumulative and normalized curves to derive the analytical fits, together with the adopted thresholds for the different period bins, is referred to Inno et al. (2015).\\nThe four normalized θ curves are separately shown in Fig. 5, and in Fig. 6 we compare the Fourier series fitted to the data of each bin. The corresponding coefficients of the fitted functions are listed in Table 7. The separation of the Cepheids into different period bins is a mandatory step because not only the amplitude of the theta curves (listed in Table 6), but also their shape changes with period (as shown in Figs. 5 and 6). Although we do have data for Cepheids with periods outside of the selected pe- riod bins, they are either not as well sampled as the data for the other Cepheids, or the theta curves do not overlap well because the number of measurements is too limited. More accurate and homogeneous data are required to overcome this limitation.\\nThe four normalized θ curves are separately shown in Fig. 5, and in Fig. 6 we compare the Fourier series fitted to the data of each bin. The corresponding coefficients of the fitted functions are listed in Table 7. The separation of the Cepheids into different period bins is a mandatory step because not only the amplitude of the theta curves (listed in Table 6), but also their shape changes with period (as shown in Figs. 5 and 6). Although we do have data for Cepheids with periods outside of the selected pe- riod bins, they are either not as well sampled as the data for the other Cepheids, or the theta curves do not overlap well because the number of measurements is too limited. More accurate and homogeneous data are required to overcome this limitation.\\nIn order to allow the reader to use these curve templates to derive the mean effective temperature by using the ephemerides, the luminosity amplitude (∆V), and a single spectroscopic measurement of the T eff , we also provide a linear relation between ∆V and ∆θ: ∆θ = (0.184 ± 0.014)∆V + (0.000 ± 0.011); (σ = 0.013)\\nIn order to allow the reader to use these curve templates to derive the mean effective temperature by using the ephemerides, the luminosity amplitude (∆V), and a single spectroscopic measurement of the T eff , we also provide a linear relation between ∆V and ∆θ: ∆θ = (0.184 ± 0.014)∆V + (0.000 ± 0.011); (σ = 0.013)\\n(1)\\n(1)\\nThe estimate of the ∆θ amplitude has to be applied to the normalized template as a multiplicative factor. Only after this rescaling operation the template can be anchored to the empirical data and used to derive the mean T eff . Optical and NIR light curves of the calibrating Cepheids, together with effective temperature and radial velocity curves, will be adopted by our group to perform a detailed comparison with nonlinear, convective hydrodynamical models of classical Cepheids. Dating back to Natale et al. (2008) it has been found that the simultaneous fit of both luminosity and radial-velocity variations provides solid constraints on the physical assumptions adopted to build pulsation models (Marconi et al. 2013). Moreover, the use of the effective temperature curves (shapes and amplitudes), covering a broad range of pulsation periods, brings forward the opportunity to constrain, on a quantitative basis, the efficiency of the convective transport over the entire pulsation cycle.\\nThe estimate of the ∆θ amplitude has to be applied to the normalized template as a multiplicative factor. Only after this rescaling operation the template can be anchored to the empirical data and used to derive the mean T eff . Optical and NIR light curves of the calibrating Cepheids, together with effective temperature and radial velocity curves, will be adopted by our group to perform a detailed comparison with nonlinear, convective hydrodynamical models of classical Cepheids. Dating back to Natale et al. (2008) it has been found that the simultaneous fit of both luminosity and radial-velocity variations provides solid constraints on the physical assumptions adopted to build pulsation models (Marconi et al. 2013). Moreover, the use of the effective temperature curves (shapes and amplitudes), covering a broad range of pulsation periods, brings forward the opportunity to constrain, on a quantitative basis, the efficiency of the convective transport over the entire pulsation cycle.\\nArticle number, page 9 of 31 A&A proofs: manuscript no. daSilvaetal2021 Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. The other columns give the abundances from both neutral and ionized lines, their standard deviations, and the number of lines used. For each element X, the weighted mean of X i and X ii abundances (weighted by 1/σ 2 ) and its intrinsic error is also shown. Magnesium and Sulfur abundances are from neutral lines only. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the abundances from both neutral and ionized lines, their uncertainties, and the number of spectra used. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 8. The standard deviation calculated using all individual abundances from both neutral and ionized lines is also shown. The complete table is available at the CDS. Fig. 9. Atmospheric parameters derived by keeping fixed the effective temperature (previously obtained from the LDR method) in comparison with the same parameters derived with T eff as a free parameter. The panels show the comparison for the β Dor and ζ Gem stars.\\nArticle number, page 9 of 31 A&A proofs: manuscript no. daSilvaetal2021 Notes. The first three columns give the target name, spectroscopic dataset, and Modified Julian Date at which the spectrum was collected. The other columns give the abundances from both neutral and ionized lines, their standard deviations, and the number of lines used. For each element X, the weighted mean of X i and X ii abundances (weighted by 1/σ 2 ) and its intrinsic error is also shown. Magnesium and Sulfur abundances are from neutral lines only. The complete table is available at the CDS. Notes. From left to right the columns give the star name, the abundances from both neutral and ionized lines, their uncertainties, and the number of spectra used. These are the weighted mean values and their standard errors computed from the individual measurements listed in Table 8. The standard deviation calculated using all individual abundances from both neutral and ionized lines is also shown. The complete table is available at the CDS. Fig. 9. Atmospheric parameters derived by keeping fixed the effective temperature (previously obtained from the LDR method) in comparison with the same parameters derived with T eff as a free parameter. The panels show the comparison for the β Dor and ζ Gem stars.\"\n },\n {\n \"title\": \"Iron and α-element abundances\",\n \"content\": \"\"\n },\n {\n \"title\": \"Iron and α-element abundances\",\n \"content\": \"\"\n },\n {\n \"title\": \"Iron abundance\",\n \"content\": \"\"\n },\n {\n \"title\": \"Iron abundance\",\n \"content\": \"The determination of the atmospheric parameters was done with the Python wrapper 2 of the MOOG code (Sneden 2002). This is a LTE radiative code that we applied to model atmospheres interpolated on the grids of Castelli & Kurucz (2004). Once a convergence is achieved in the determination of T eff , log g, ξ, and [Fe/H], as described in Sect. 5.1, MOOG also provides the final iron abundance derived from individual Fe i and Fe ii lines. Such lines are those included in the input line list and that passed the selection criteria that we applied to choose the best atomic transitions. Table 4 lists, for each spectrum in our sample, the mean Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used. The mean Fe i and Fe ii abundances derived for each star from the individual spectra, weighted by the standard deviations aforementioned, and the corresponding standard errors are shown in Table 6. Column 8 gives our final determination for the stellar metallicity, which is the weighted mean calculated from both the [Fe i/H] and the [Fe ii/H] abundances. The adopted uncertainties are the largest values between the standard error computed from the weighted mean (σ) and the standard deviation (std). Figure B.5 shows the derived metallicities as a function of the pulsation phase. Data plotted in this figure show that the iron abundances are quite stable along the pulsation cycle. This outcome applies to short-/long-period Cepheids and to Bump Cepheids. We note that the current approach improves the results obtained by Proxauf et al. (2018), since in their iron abundances was still present a mild variation along the rising branch of large amplitude Cepheids (in our current results the dispersions are in most cases smaller than 0.05 dex). The improvement is mainly due to the very careful selection of the lines adopted to estimate the iron abundance. The difference is soundly supported by the distribution of the standard deviations displayed in Fig. 7 for the Cepheids in common. The current standard deviations for both [Fe i/H] (violet) and [Fe ii/H] (green) lines (bottom panel) are smaller by 30%-40% when compared with our previous investigation. Moreover and even more importantly, they also attain similar values.\\nThe determination of the atmospheric parameters was done with the Python wrapper 2 of the MOOG code (Sneden 2002). This is a LTE radiative code that we applied to model atmospheres interpolated on the grids of Castelli & Kurucz (2004). Once a convergence is achieved in the determination of T eff , log g, ξ, and [Fe/H], as described in Sect. 5.1, MOOG also provides the final iron abundance derived from individual Fe i and Fe ii lines. Such lines are those included in the input line list and that passed the selection criteria that we applied to choose the best atomic transitions. Table 4 lists, for each spectrum in our sample, the mean Fe i and Fe ii abundances derived from individual lines together with the standard deviations and the number of lines used. The mean Fe i and Fe ii abundances derived for each star from the individual spectra, weighted by the standard deviations aforementioned, and the corresponding standard errors are shown in Table 6. Column 8 gives our final determination for the stellar metallicity, which is the weighted mean calculated from both the [Fe i/H] and the [Fe ii/H] abundances. The adopted uncertainties are the largest values between the standard error computed from the weighted mean (σ) and the standard deviation (std). Figure B.5 shows the derived metallicities as a function of the pulsation phase. Data plotted in this figure show that the iron abundances are quite stable along the pulsation cycle. This outcome applies to short-/long-period Cepheids and to Bump Cepheids. We note that the current approach improves the results obtained by Proxauf et al. (2018), since in their iron abundances was still present a mild variation along the rising branch of large amplitude Cepheids (in our current results the dispersions are in most cases smaller than 0.05 dex). The improvement is mainly due to the very careful selection of the lines adopted to estimate the iron abundance. The difference is soundly supported by the distribution of the standard deviations displayed in Fig. 7 for the Cepheids in common. The current standard deviations for both [Fe i/H] (violet) and [Fe ii/H] (green) lines (bottom panel) are smaller by 30%-40% when compared with our previous investigation. Moreover and even more importantly, they also attain similar values.\\nTo further constrain the difference between the current and our previous investigation, Fig. 8 shows the comparison for both the atmospheric parameters and the iron abundance. The agreement is remarkable over the entire temperature range. The mean and the standard deviations for the atmospheric parameters are well within the current uncertainties. The new iron abundances are, on average, ∼0.1 dex more metal-poor. Owing to the similarity in the atmospheric parameters, this difference seems an obvious consequence of the new line list.\\nTo further constrain the difference between the current and our previous investigation, Fig. 8 shows the comparison for both the atmospheric parameters and the iron abundance. The agreement is remarkable over the entire temperature range. The mean and the standard deviations for the atmospheric parameters are well within the current uncertainties. The new iron abundances are, on average, ∼0.1 dex more metal-poor. Owing to the similarity in the atmospheric parameters, this difference seems an obvious consequence of the new line list.\\nSeveral different approaches have been suggested in the literature to estimate the atmospheric parameters of variable stars (Fukue et al. 2015;Jian et al. 2020;Lemasle et al. 2020;Matsunaga et al. 2021;Taniguchi et al. 2021;Romaniello et al. 2021). In the current investigation, the effective temperature is firstly derived using the LDR method, then the other atmospheric parameters are estimated at fixed effective temperature. The advantages of this approach have already been discussed in several papers. To provide a more quantitative estimate of the differences compared to another method normally used, in which the minimization process also includes T eff as a free parameter, we performed a new and independent determination of the three atmospheric parameters plus abundances simultaneously for two variables in our sample (β Dor and ζ Gem). The differences between our canonical approach and the literature approach are shown in Fig. 9. The overall agreement is quite good over the entire temperature range. Indeed, the mean and the standard deviations attain tiny values. The mean difference in effective temperature is 55 K, and for the surface gravity it is 0.12 dex only, while the microturbulent velocities attain almost identical values. The quoted differences cause a small increase of 0.04 dex in the iron abundance when T eff is a free parameter. We note that we adopted the same line list in the tests performed either with fixed or with free T eff .\\nSeveral different approaches have been suggested in the literature to estimate the atmospheric parameters of variable stars (Fukue et al. 2015;Jian et al. 2020;Lemasle et al. 2020;Matsunaga et al. 2021;Taniguchi et al. 2021;Romaniello et al. 2021). In the current investigation, the effective temperature is firstly derived using the LDR method, then the other atmospheric parameters are estimated at fixed effective temperature. The advantages of this approach have already been discussed in several papers. To provide a more quantitative estimate of the differences compared to another method normally used, in which the minimization process also includes T eff as a free parameter, we performed a new and independent determination of the three atmospheric parameters plus abundances simultaneously for two variables in our sample (β Dor and ζ Gem). The differences between our canonical approach and the literature approach are shown in Fig. 9. The overall agreement is quite good over the entire temperature range. Indeed, the mean and the standard deviations attain tiny values. The mean difference in effective temperature is 55 K, and for the surface gravity it is 0.12 dex only, while the microturbulent velocities attain almost identical values. The quoted differences cause a small increase of 0.04 dex in the iron abundance when T eff is a free parameter. We note that we adopted the same line list in the tests performed either with fixed or with free T eff .\\nIn closing this subsection, we also investigate whether the current iron abundances display any clear trend with the Galactocentric distance (R G ). The top panel of Fig. 10 shows the iron abundance as a function of R G for the calibrating Cepheids (blue circles) and for Cepheids available in the literature (gray circles). We performed a linear fit over the entire sample and we found the following relation:\\nIn closing this subsection, we also investigate whether the current iron abundances display any clear trend with the Galactocentric distance (R G ). The top panel of Fig. 10 shows the iron abundance as a function of R G for the calibrating Cepheids (blue circles) and for Cepheids available in the literature (gray circles). We performed a linear fit over the entire sample and we found the following relation:\\n[Fe/H] = (−0.055 ± 0.003)R G + (0.43 ± 0.03)\\n[Fe/H] = (−0.055 ± 0.003)R G + (0.43 ± 0.03)\\nThe current slope agrees quite well with similar estimates available in the literature and, in particular, with the recent estimate of the iron radial gradient (dashed line) provided by Ripepi et al. (2022). We note that, to overcome variations in the zero-point mainly introduced by innermost and outermost Cepheids, the gradient from Ripepi et al. (2022) was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The bottom panel of Fig. 10 shows the iron abundance as a function of the logarithmic pulsation period. Data plotted in this panel agree quite well with similar estimates available in the literature. Moreover, they do not show any significant variation when moving from young (long-period) to less young (shortperiod) classical Cepheids.\\nThe current slope agrees quite well with similar estimates available in the literature and, in particular, with the recent estimate of the iron radial gradient (dashed line) provided by Ripepi et al. (2022). We note that, to overcome variations in the zero-point mainly introduced by innermost and outermost Cepheids, the gradient from Ripepi et al. (2022) was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The bottom panel of Fig. 10 shows the iron abundance as a function of the logarithmic pulsation period. Data plotted in this panel agree quite well with similar estimates available in the literature. Moreover, they do not show any significant variation when moving from young (long-period) to less young (shortperiod) classical Cepheids.\"\n },\n {\n \"title\": \"α-element abundances\",\n \"content\": \"\"\n },\n {\n \"title\": \"α-element abundances\",\n \"content\": \"The abundances of the α elements were derived using pyMOOGi as well (except sulfur, for which the abundances were estimated using the spectral synthesis method -see Sect. 6.2.1). We used the ab f ind driver, which requires as input i) the model atmospheres, calculated from the atmospheric parameters obtained for each star, and ii) a list of lines containing the wavelength, the atomic number, the lower-level excitation potential, the log g f , and the measured equivalent widths. Table 8 lists the individual α-element abundances obtained for each spectrum together with the standard deviations and the number of lines used for each element. The mean abundances, weighted by the standard deviations, and the corresponding standard errors are shown in Table 9. The same mean data are plotted in Figs. 11,12,and B.6 for our 20 calibrating Cepheids (blue circles) and for similar data available in the literature (gray circles).\\nThe abundances of the α elements were derived using pyMOOGi as well (except sulfur, for which the abundances were estimated using the spectral synthesis method -see Sect. 6.2.1). We used the ab f ind driver, which requires as input i) the model atmospheres, calculated from the atmospheric parameters obtained for each star, and ii) a list of lines containing the wavelength, the atomic number, the lower-level excitation potential, the log g f , and the measured equivalent widths. Table 8 lists the individual α-element abundances obtained for each spectrum together with the standard deviations and the number of lines used for each element. The mean abundances, weighted by the standard deviations, and the corresponding standard errors are shown in Table 9. The same mean data are plotted in Figs. 11,12,and B.6 for our 20 calibrating Cepheids (blue circles) and for similar data available in the literature (gray circles).\\nFor cases in which the abundance of a given element at a given phase is based on only one or two lines, or if the estimated error is lower than a typical value for each element/species, the uncertainty is set to that typical value, namely: σ( Table 8 for spectra having at least 2 lines of the same element/species. The adopted standard solar abundances for both iron and α elements are from Asplund et al. (2009), namely: A(Fe) = 7.50, A(Mg) = 7.60, A(Si) = 7.51, A(S) = 7.12, A(Ca) = 6.34, and A(Ti) = 4.95, with A representing the typical logarithmic notation where H is defined to be A(H) = 12.00. These values are in good agreement with the very recent determinations done by Asplund et al. (2021), which were obtained using the most up-to-date atomic and molecular data.\\nFor cases in which the abundance of a given element at a given phase is based on only one or two lines, or if the estimated error is lower than a typical value for each element/species, the uncertainty is set to that typical value, namely: σ( Table 8 for spectra having at least 2 lines of the same element/species. The adopted standard solar abundances for both iron and α elements are from Asplund et al. (2009), namely: A(Fe) = 7.50, A(Mg) = 7.60, A(Si) = 7.51, A(S) = 7.12, A(Ca) = 6.34, and A(Ti) = 4.95, with A representing the typical logarithmic notation where H is defined to be A(H) = 12.00. These values are in good agreement with the very recent determinations done by Asplund et al. (2021), which were obtained using the most up-to-date atomic and molecular data.\\nFor both iron and α abundances, we evaluated the systematic differences for stars in common between the current investigation and measurements from literature: ), Luck & Lambert (2011, Lemasle et al. (2013), andYong et al. (2006) for iron and α elements, and Proxauf et al. (2018) for iron only. The differences for all the investigated elements are listed in Table 10. In order to perform a direct comparison, we applied these zero-point differences to the literature datasets by putting the iron and the α-element abundances in our metallicity scale. The spectroscopic samples for which there are no variables in common between our investigation and those available in the literature, the homogenization of the abundances was performed in two steps. To be more specific, we have no variables in common with Yong et al. (2006), therefore, we first scaled their abundances to match those from Luck & Lambert (2011), and then we scaled them to match our metallicity scale. Data plotted in Fig. B.6 display, once again, the remarkable agreement of the α-element abundances over the entire pulsation cycle (the dispersions in most cases are smaller than 0.10 dex). This outcome applies to all the investigated α elements (except sulfur, for which we have abundances at a single phase only), and to both neutral and ionized species. We note in passing that this finding seems more relevant when compared with the variation of the iron abundances along the pulsation cycle. Indeed, the αelement abundances are only based on a modest number or, in some cases, just a few lines. Therefore, they are more prone to possible systematics.\\nFor both iron and α abundances, we evaluated the systematic differences for stars in common between the current investigation and measurements from literature: ), Luck & Lambert (2011, Lemasle et al. (2013), andYong et al. (2006) for iron and α elements, and Proxauf et al. (2018) for iron only. The differences for all the investigated elements are listed in Table 10. In order to perform a direct comparison, we applied these zero-point differences to the literature datasets by putting the iron and the α-element abundances in our metallicity scale. The spectroscopic samples for which there are no variables in common between our investigation and those available in the literature, the homogenization of the abundances was performed in two steps. To be more specific, we have no variables in common with Yong et al. (2006), therefore, we first scaled their abundances to match those from Luck & Lambert (2011), and then we scaled them to match our metallicity scale. Data plotted in Fig. B.6 display, once again, the remarkable agreement of the α-element abundances over the entire pulsation cycle (the dispersions in most cases are smaller than 0.10 dex). This outcome applies to all the investigated α elements (except sulfur, for which we have abundances at a single phase only), and to both neutral and ionized species. We note in passing that this finding seems more relevant when compared with the variation of the iron abundances along the pulsation cycle. Indeed, the αelement abundances are only based on a modest number or, in some cases, just a few lines. Therefore, they are more prone to possible systematics.\\nTo further investigate the internal consistency of the current α-element abundances with similar estimates available in the literature, we checked the behavior of the [X/Fe] abundance ratios as a function of the stellar metallicity. Data plotted in the left panels of Fig. 11 show that the current results agree, within the errors, quite well with similar determinations done in previous investigations (Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006). The key difference seems to be the smaller dispersion at fixed iron abundance, thus suggesting that a fraction of the spread currently observed might still be caused by intrinsic errors. This finding is further supported by the evidence that the current estimates do not show any clear trend when plotted against the pulsation period (see the right panels of Fig. 11). We note that classical Cepheids, when moving from short to long periods, become systematically cooler and more expanded (lower surface gravity).\\nTo further investigate the internal consistency of the current α-element abundances with similar estimates available in the literature, we checked the behavior of the [X/Fe] abundance ratios as a function of the stellar metallicity. Data plotted in the left panels of Fig. 11 show that the current results agree, within the errors, quite well with similar determinations done in previous investigations (Luck & Lambert 2011;Lemasle et al. 2013;Yong et al. 2006). The key difference seems to be the smaller dispersion at fixed iron abundance, thus suggesting that a fraction of the spread currently observed might still be caused by intrinsic errors. This finding is further supported by the evidence that the current estimates do not show any clear trend when plotted against the pulsation period (see the right panels of Fig. 11). We note that classical Cepheids, when moving from short to long periods, become systematically cooler and more expanded (lower surface gravity).\"\n },\n {\n \"title\": \"Sulfur abundances\",\n \"content\": \"\"\n },\n {\n \"title\": \"Sulfur abundances\",\n \"content\": \"To determine the sulfur abundances, we carried out spectral synthesis calculations by exploiting the multiplet at 6757 Å, which is proven to be reliable and not affected by NLTE effects (see, e.g., Takeda et al. 2016;Duffau et al. 2017, and references therein). Synthetic spectra were computed using the synth driver of MOOG (2019 version), the same grid of model atmospheres by Castelli & Kurucz (2004) used for the other elements, and atomic parameters from Caffau et al. (2007). We synthesized a 30 Å wide region around the S i multiplet in order to determine the broadening of the spectral lines. Then we degraded the synthetic spectra to match the resolution of the observational datasets. The best fit was found with χ-square minimization methods, as routinely done in the literature. Finally, we estimated the internal errors, with a mean value of σ([S i/H]) = 0.10 dex. These errors are due to continuum placement (related to the S/N, quality of the spectra, best fitting procedure) and due to errors on the atmospheric parameters (this was done in the standard way, which is by varying one parameter at a time and inspecting the corresponding variations). We refer the reader to D'Orazi et al. (2017D'Orazi et al. ( , 2020 for details on error computation.\\nTo determine the sulfur abundances, we carried out spectral synthesis calculations by exploiting the multiplet at 6757 Å, which is proven to be reliable and not affected by NLTE effects (see, e.g., Takeda et al. 2016;Duffau et al. 2017, and references therein). Synthetic spectra were computed using the synth driver of MOOG (2019 version), the same grid of model atmospheres by Castelli & Kurucz (2004) used for the other elements, and atomic parameters from Caffau et al. (2007). We synthesized a 30 Å wide region around the S i multiplet in order to determine the broadening of the spectral lines. Then we degraded the synthetic spectra to match the resolution of the observational datasets. The best fit was found with χ-square minimization methods, as routinely done in the literature. Finally, we estimated the internal errors, with a mean value of σ([S i/H]) = 0.10 dex. These errors are due to continuum placement (related to the S/N, quality of the spectra, best fitting procedure) and due to errors on the atmospheric parameters (this was done in the standard way, which is by varying one parameter at a time and inspecting the corresponding variations). We refer the reader to D'Orazi et al. (2017D'Orazi et al. ( , 2020 for details on error computation.\\nThe reasons why we focused our attention on S i abundances for our sample of calibrating Cepheids are manifold:\\nThe reasons why we focused our attention on S i abundances for our sample of calibrating Cepheids are manifold:\\ni)-Sulfur is a volatile element that is not locked in dust grains, which means that no correction is required to compare the current sulfur abundances with predictions based on chemical evolution models (Chiappini et al. 2001;Cescutti et al. 2007). Therefore, sulfur is a more solid tracer of α-element abundances than Si and Ca. These three α elements are produced during the oxygen burning and their abundances should show similar chemical enrichment histories (Limongi & Chieffi 2003).\\ni)-Sulfur is a volatile element that is not locked in dust grains, which means that no correction is required to compare the current sulfur abundances with predictions based on chemical evolution models (Chiappini et al. 2001;Cescutti et al. 2007). Therefore, sulfur is a more solid tracer of α-element abundances than Si and Ca. These three α elements are produced during the oxygen burning and their abundances should show similar chemical enrichment histories (Limongi & Chieffi 2003).\\nii)-Sulfur brings forward several key advantages when compared with other α elements, since it shows several lines at both optical and NIR wavelengths. The most popular sulfur lines are the multiplets at 6757 Å and at 8694 Å, but they are weak and they are typically measured on stars more metal-rich than [Fe/H] ∼ −1.5 dex. More recently, the multiplet at 9237 Å has also been used because it is very strong and it can be measured even in very metal-poor stars (Nissen et al. 2004;Caffau et al. 2005Caffau et al. , 2007. However, these lines are also affected by NLTE effects, making them even stronger, and the sulfur abundances need to be properly treated. Even more recently, high-resolution NIR spectrographs also provided the opportunity to use the S i multiplet at 1045 nm for both dwarf and giant stars (Caffau et al. 2016).\\nii)-Sulfur brings forward several key advantages when compared with other α elements, since it shows several lines at both optical and NIR wavelengths. The most popular sulfur lines are the multiplets at 6757 Å and at 8694 Å, but they are weak and they are typically measured on stars more metal-rich than [Fe/H] ∼ −1.5 dex. More recently, the multiplet at 9237 Å has also been used because it is very strong and it can be measured even in very metal-poor stars (Nissen et al. 2004;Caffau et al. 2005Caffau et al. , 2007. However, these lines are also affected by NLTE effects, making them even stronger, and the sulfur abundances need to be properly treated. Even more recently, high-resolution NIR spectrographs also provided the opportunity to use the S i multiplet at 1045 nm for both dwarf and giant stars (Caffau et al. 2016).\\niii)-Possibility of comparison with the S abundances in external galaxies, such as Blue Compact Galaxies, Damped Lyman Alpha systems (Dessauges-Zavadsky et al. 2007), and Active Galactic Nuclei (Liu et al. 2015, Mizumoto et al., submitted to ApJ). Indeed, thanks to strong emission lines, the S abundances can be measured even at large redshifts.\\niii)-Possibility of comparison with the S abundances in external galaxies, such as Blue Compact Galaxies, Damped Lyman Alpha systems (Dessauges-Zavadsky et al. 2007), and Active Galactic Nuclei (Liu et al. 2015, Mizumoto et al., submitted to ApJ). Indeed, thanks to strong emission lines, the S abundances can be measured even at large redshifts.\\niv)-In our previous investigation focused on α-element abundances in classical Cepheids (Genovali et al. 2015) we neglected sulfur. In the current study, we perform accurate measurements of the multiplet at 6757 Å. The multiplet at 9237 Å is only available in a small subsets of spectra (UVES), but the lines are too strong (EW ≥ 300 mÅ) and it was neglected. Data plotted in Fig. 12 show the [X/H] (left panels) and the [X/Fe] (right panels) α-element abundances as a function of the Galactocentric distances, estimated by Genovali et al. (2014) using NIR mean magnitudes and predicted Period-Luminosity relations (Inno et al. 2013). The radial gradients plotted in the left panels indicate that sulfur is the α element with the steeper slope (see labeled values). Indeed, its slope is at least a factor of two steeper than for Ca (0.076 vs. 0.031 dex kpc −1 ) and ∼50% steeper than those for Mg and Si. In passing, we also note that the radial gradient of the abundance ratios plotted in the right panels of the same figure display, as expected (see Genovali et al. 2015), either a flat trend (Si) or a marginal positive slope (Mg, Ti) across the thin disk. There are two exceptions. Ca is steadily becoming overabundant (positive slope) when moving from the innermost (more metal-rich) to the outermost (more metal-poor) disk regions. This is a consequence of the fact that the slope of its radial gradient is shallower than the iron gradient (0.31 vs. 0.55 dex kpc −1 ). S shows a negative slope, because it is the only α-element with a slope in the radial gradient steeper than the iron gradient (0.76 vs. 0.55 dex kpc −1 ).\\niv)-In our previous investigation focused on α-element abundances in classical Cepheids (Genovali et al. 2015) we neglected sulfur. In the current study, we perform accurate measurements of the multiplet at 6757 Å. The multiplet at 9237 Å is only available in a small subsets of spectra (UVES), but the lines are too strong (EW ≥ 300 mÅ) and it was neglected. Data plotted in Fig. 12 show the [X/H] (left panels) and the [X/Fe] (right panels) α-element abundances as a function of the Galactocentric distances, estimated by Genovali et al. (2014) using NIR mean magnitudes and predicted Period-Luminosity relations (Inno et al. 2013). The radial gradients plotted in the left panels indicate that sulfur is the α element with the steeper slope (see labeled values). Indeed, its slope is at least a factor of two steeper than for Ca (0.076 vs. 0.031 dex kpc −1 ) and ∼50% steeper than those for Mg and Si. In passing, we also note that the radial gradient of the abundance ratios plotted in the right panels of the same figure display, as expected (see Genovali et al. 2015), either a flat trend (Si) or a marginal positive slope (Mg, Ti) across the thin disk. There are two exceptions. Ca is steadily becoming overabundant (positive slope) when moving from the innermost (more metal-rich) to the outermost (more metal-poor) disk regions. This is a consequence of the fact that the slope of its radial gradient is shallower than the iron gradient (0.31 vs. 0.55 dex kpc −1 ). S shows a negative slope, because it is the only α-element with a slope in the radial gradient steeper than the iron gradient (0.76 vs. 0.55 dex kpc −1 ).\"\n },\n {\n \"title\": \"Summary and final remarks\",\n \"content\": \"\"\n },\n {\n \"title\": \"Summary and final remarks\",\n \"content\": \"We performed a new spectroscopic analysis of almost two dozen of northern and southern calibrating Cepheids. As a whole, we collected more than 500 high-resolution (R∼40 000-115 000), high S/N (∼50-370) optical spectra with four different spectrographs (STELLA, FEROS, HARPS, UVES) covering either a substantial portion or the entire pulsation cycle. The main aim of this investigation is twofold: i) provide new and accurate constraints on possible systematics affecting the estimate of atmospheric parameters and elemental abundances along the pulsation cycle; ii) develop new approaches and/or diagnostics to provide accurate estimates of both atmospheric parameters and chemical abundances. To accomplish the former goal, we assembled three different line lists. By using different line lists available in the literature, we compiled a very detailed list of both Fe i and Fe ii lines. Moreover, we also collected a sizable sample of lines for α-elements: Mg, Si, S, Ca, and Ti. Special care was also paid to the line list of both iron-peak and α-elements used for the estimate of the effective temperature using the LDR method.\\nWe performed a new spectroscopic analysis of almost two dozen of northern and southern calibrating Cepheids. As a whole, we collected more than 500 high-resolution (R∼40 000-115 000), high S/N (∼50-370) optical spectra with four different spectrographs (STELLA, FEROS, HARPS, UVES) covering either a substantial portion or the entire pulsation cycle. The main aim of this investigation is twofold: i) provide new and accurate constraints on possible systematics affecting the estimate of atmospheric parameters and elemental abundances along the pulsation cycle; ii) develop new approaches and/or diagnostics to provide accurate estimates of both atmospheric parameters and chemical abundances. To accomplish the former goal, we assembled three different line lists. By using different line lists available in the literature, we compiled a very detailed list of both Fe i and Fe ii lines. Moreover, we also collected a sizable sample of lines for α-elements: Mg, Si, S, Ca, and Ti. Special care was also paid to the line list of both iron-peak and α-elements used for the estimate of the effective temperature using the LDR method.\\nIn order to provide homogeneous and accurate estimates of both atmospheric parameters and elemental abundances, we undertook a lengthy and detailed analysis of the quoted line lists. Our approach moved along three different paths: i) we collected the three most precise atomic transition parameters available in the literature; ii) we removed absorption lines that are blended with other lines; iii) we removed lines that either show relevant abundance variations from the average value of their chemical species or that display a steady variation along the pulsation cycle with atmospheric parameters. Concerning the transition parameters, we gave the priority to recent laboratory measurements, then to homogeneous astrophysical estimates and, for the remaining lines, to the transition parameters available on the NIST Atomic Spectra Database. To identify blended lines, we adopted both the solar spectrum by Moore et al. (1966) and a grid of synthetic spectra. Finally, we applied an iterative approach to remove lines showing significant variations. This latter step was done based on a preliminary estimate of the atmospheric parameters and the abundances. We were then able to identify lines that show either a variation in abundance at the 3-σ level from the mean value of the specific element and/or a steady variation along the pulsation cycle. In more detail, to properly identify robust lines, we plotted all the lines available in our initial line lists, for each chemical species, individual exposure, and calibrating Cepheid, as a function of the pulsation phase, effective temperature, and equivalent width. Lastly, we performed a new estimate of the atmospheric parameters and the individual elemental abundances, using an improved version of our algorithm and using only the line lists based on optimal lines.\\nIn order to provide homogeneous and accurate estimates of both atmospheric parameters and elemental abundances, we undertook a lengthy and detailed analysis of the quoted line lists. Our approach moved along three different paths: i) we collected the three most precise atomic transition parameters available in the literature; ii) we removed absorption lines that are blended with other lines; iii) we removed lines that either show relevant abundance variations from the average value of their chemical species or that display a steady variation along the pulsation cycle with atmospheric parameters. Concerning the transition parameters, we gave the priority to recent laboratory measurements, then to homogeneous astrophysical estimates and, for the remaining lines, to the transition parameters available on the NIST Atomic Spectra Database. To identify blended lines, we adopted both the solar spectrum by Moore et al. (1966) and a grid of synthetic spectra. Finally, we applied an iterative approach to remove lines showing significant variations. This latter step was done based on a preliminary estimate of the atmospheric parameters and the abundances. We were then able to identify lines that show either a variation in abundance at the 3-σ level from the mean value of the specific element and/or a steady variation along the pulsation cycle. In more detail, to properly identify robust lines, we plotted all the lines available in our initial line lists, for each chemical species, individual exposure, and calibrating Cepheid, as a function of the pulsation phase, effective temperature, and equivalent width. Lastly, we performed a new estimate of the atmospheric parameters and the individual elemental abundances, using an improved version of our algorithm and using only the line lists based on optimal lines.\\nThe comparison between the current atmospheric parameters and similar estimated provided by our group, using a similar approach but old line lists, show smoother variation along the pulsation cycle and, in particular, smaller standard deviations (∼50 K for the effective temperature, ∼0.2 dex for the surface gravity, and ∼0.2 km s −1 for the microturbulent velocity). The estimate of both iron and α-element abundances is also significantly improved, since they display an almost constant value as a function of the pulsation phase. The dispersions are 0.05 dex for iron and 0.10 dex for the α elements. We note that this is the first time that α-element abundances are critically investigated for possible variations along the pulsation cycle.\\nThe comparison between the current atmospheric parameters and similar estimated provided by our group, using a similar approach but old line lists, show smoother variation along the pulsation cycle and, in particular, smaller standard deviations (∼50 K for the effective temperature, ∼0.2 dex for the surface gravity, and ∼0.2 km s −1 for the microturbulent velocity). The estimate of both iron and α-element abundances is also significantly improved, since they display an almost constant value as a function of the pulsation phase. The dispersions are 0.05 dex for iron and 0.10 dex for the α elements. We note that this is the first time that α-element abundances are critically investigated for possible variations along the pulsation cycle.\\nThis finding is further supported by the comparison with similar abundance estimates available in the literature. We found that the current calibrating Cepheids display, at fixed either iron abundance or pulsation period, smaller standard deviations. This result suggests that a good fraction of the spread normally observed in elemental abundances is mainly caused by the adopted line list and by the algorithm used to estimate the atmospheric parameters.\\nThis finding is further supported by the comparison with similar abundance estimates available in the literature. We found that the current calibrating Cepheids display, at fixed either iron abundance or pulsation period, smaller standard deviations. This result suggests that a good fraction of the spread normally observed in elemental abundances is mainly caused by the adopted line list and by the algorithm used to estimate the atmospheric parameters.\\nMoreover and even more importantly, we also derived for the first time new and accurate effective temperature templates. Following Inno et al. (2015), the calibrating Cepheids were split into four different period bins, ranging from 3 to 10.5 days in period. For each period bin, we performed an analytical fit with Fourier series providing θ = 5040 /T eff as a function of the pulsation phase. A similar approach was already applied to fundamental RR Lyrae by For & Sneden (2010) and by Magurno et al. (2018), but the variations along the pulsation cycle are, at fixed pulsation phase, larger due to the larger luminosity amplitudes and probably to the possible occurrence of nonlinear phenomena (shocks). The current findings appear quite promising, because the new templates can provide accurate estimates of the effective temperature once the pulsation period and the phase (and in turn the reference epoch) of a Cepheid are known. We note that this information is crucial in the abundance analysis of NIR spectra, because in the metal-poor regimes only a modest number of ionized and neutral iron lines are present.\\nMoreover and even more importantly, we also derived for the first time new and accurate effective temperature templates. Following Inno et al. (2015), the calibrating Cepheids were split into four different period bins, ranging from 3 to 10.5 days in period. For each period bin, we performed an analytical fit with Fourier series providing θ = 5040 /T eff as a function of the pulsation phase. A similar approach was already applied to fundamental RR Lyrae by For & Sneden (2010) and by Magurno et al. (2018), but the variations along the pulsation cycle are, at fixed pulsation phase, larger due to the larger luminosity amplitudes and probably to the possible occurrence of nonlinear phenomena (shocks). The current findings appear quite promising, because the new templates can provide accurate estimates of the effective temperature once the pulsation period and the phase (and in turn the reference epoch) of a Cepheid are known. We note that this information is crucial in the abundance analysis of NIR spectra, because in the metal-poor regimes only a modest number of ionized and neutral iron lines are present.\\nThe current investigation was focused on Galactic Cepheids with iron abundances close to the solar value. In a forthcoming paper we plan to apply the same approach to more metal-poor Magellanic Cloud Cepheids. The panels of Fig. B.1 show the phase-folded light and radialvelocity curves for our sample of 20 calibrating Cepheids. The stellar magnitude in visual bands were taken from literature, as explained in Sect. 4. The radial velocities were either measured from our spectra or taken from literature. Only FF Aql and β Dor have RV values derived from our spectra only.\\nThe current investigation was focused on Galactic Cepheids with iron abundances close to the solar value. In a forthcoming paper we plan to apply the same approach to more metal-poor Magellanic Cloud Cepheids. The panels of Fig. B.1 show the phase-folded light and radialvelocity curves for our sample of 20 calibrating Cepheids. The stellar magnitude in visual bands were taken from literature, as explained in Sect. 4. The radial velocities were either measured from our spectra or taken from literature. Only FF Aql and β Dor have RV values derived from our spectra only.\\nOur RV measurements agree quite well with the literature values. Significant differences are seen only for two stars, T Vul and S Sge, which might be related to instrumental drifts caused by their long baseline observations (for both stars, the two datasets, ours and from literature, are separated by almost 17 years) or, most likely, to the binarity of these objects (Evans 1992;Evans et al. 1993). Table 5, for T Vul we derived two different values of reference epoch (T 0 ), one from our own radial velocities and another one from the light curve. For S Sge we have independent measurements of T 0 and pulsation period derived from both the RV datasets and from the light curve. Another particular case is β Dor, one of our bump Cepheids (see Table 1), for which we have independent values of T 0 and period derived from our RV curve and from the literature light curve.\\nOur RV measurements agree quite well with the literature values. Significant differences are seen only for two stars, T Vul and S Sge, which might be related to instrumental drifts caused by their long baseline observations (for both stars, the two datasets, ours and from literature, are separated by almost 17 years) or, most likely, to the binarity of these objects (Evans 1992;Evans et al. 1993). Table 5, for T Vul we derived two different values of reference epoch (T 0 ), one from our own radial velocities and another one from the light curve. For S Sge we have independent measurements of T 0 and pulsation period derived from both the RV datasets and from the light curve. Another particular case is β Dor, one of our bump Cepheids (see Table 1), for which we have independent values of T 0 and period derived from our RV curve and from the literature light curve.\"\n },\n {\n \"title\": \"As detailed in\",\n \"content\": \"\"\n },\n {\n \"title\": \"As detailed in\",\n \"content\": \"The main criteria we adopted for selecting the light curves available in the literature were: i) well-sampled photometric data; ii) minimization of the difference in epoch between photometric and spectroscopic data. It goes without saying that wellsampled light curves from Gaia DR3 will allow us to provide accurate and homogeneous ephemerides for the majority of classical Cepheids.\\nThe main criteria we adopted for selecting the light curves available in the literature were: i) well-sampled photometric data; ii) minimization of the difference in epoch between photometric and spectroscopic data. It goes without saying that wellsampled light curves from Gaia DR3 will allow us to provide accurate and homogeneous ephemerides for the majority of classical Cepheids.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 10 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 10 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 11 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 11 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 12 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 12 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. A. 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 9 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 9 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 10 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 10 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Partly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra.","content":"Partly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra."},{"id":"","caption":"Partly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra.","content":"Partly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra."},{"id":"","caption":"Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel).","content":"Fig. 1 .\n1Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel)."},{"id":"","caption":"Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel).","content":"Fig. 1 .\n1Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel)."},{"id":"","caption":"Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.","content":"Fig. 2 .\n2Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean."},{"id":"","caption":"Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.","content":"Fig. 2 .\n2Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean."},{"id":"","caption":"a) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA.","content":"a) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA."},{"id":"","caption":"a) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA.","content":"a) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA."},{"id":"","caption":"et al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for","content":"et al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for"},{"id":"","caption":"et al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for","content":"et al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for"},{"id":"","caption":"Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled.","content":"Fig. 3 .\n3Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled."},{"id":"","caption":"Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled.","content":"Fig. 3 .\n3Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled."},{"id":"","caption":". B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve.","content":". B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve."},{"id":"","caption":". B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve.","content":". B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve."},{"id":"","caption":"Normalized θ (5040/T eff ) as a function of the pulsation phase.","content":"Fig. 5 .\n5Normalized θ (5040/T eff ) as a function of the pulsation phase."},{"id":"","caption":"Normalized θ (5040/T eff ) as a function of the pulsation phase.","content":"Fig. 5 .\n5Normalized θ (5040/T eff ) as a function of the pulsation phase."},{"id":"","caption":"Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation.","content":"Fig. 6 .\n6Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation."},{"id":"","caption":"Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation.","content":"Fig. 6 .\n6Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation."},{"id":"","caption":"Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018).","content":"Fig. 7 .\n7Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018)."},{"id":"","caption":"Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018).","content":"Fig. 7 .\n7Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018)."},{"id":"","caption":"Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values.","content":"Fig. 8 .\n8Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values."},{"id":"","caption":"Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values.","content":"Fig. 8 .\n8Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values."},{"id":"","caption":"Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show.","content":"Fig. 10 .\n10Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show."},{"id":"","caption":"Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show.","content":"Fig. 10 .\n10Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show."},{"id":"","caption":"[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors.","content":"Fig. 11 .\n11[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors."},{"id":"","caption":"[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors.","content":"Fig. 11 .\n11[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors."},{"id":"","caption":"[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in","content":"[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in"},{"id":"","caption":"[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in","content":"[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in"},{"id":"","caption":"[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)","content":"Fig. 12 .\n12[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)"},{"id":"","caption":"[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)","content":"Fig. 12 .\n12[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)"},{"id":"","caption":"Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.","content":"Fig. A. 1 .\n1Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean."},{"id":"","caption":"Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.","content":"Fig. A. 1 .\n1Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean."},{"id":"","caption":"Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths.","content":"Fig. A. 2 .\n2Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths."},{"id":"","caption":"Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths.","content":"Fig. A. 2 .\n2Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths."},{"id":"","caption":"Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends.","content":"Fig. A. 3 .\n3Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends."},{"id":"","caption":"Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends.","content":"Fig. A. 3 .\n3Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends."},{"id":"","caption":"Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects.","content":"Fig. A. 4 .\n4Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects."},{"id":"","caption":"Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects.","content":"Fig. A. 4 .\n4Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects."},{"id":"","caption":"Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves","content":"Fig. A. 5 .\n5Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves"},{"id":"","caption":"Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves","content":"Fig. A. 5 .\n5Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves"},{"id":"","caption":"Article number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.","content":"Article number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.Name \nα ICRS \nδ ICRS \nMode [Fe/H] lit ± σ Ref. a \nN F N H N U N S N tot \n\nR TrA \n15:19:45.712 −66:29:45.742 \n0 \n−0.01 ± 0.03 \n2 \n1 \n14 \n... \n... \n15 \nT Vul \n20:51:28.238 +28:15:01.817 \n0 \n0.07 ± 0.10 \n1 \n... \n... \n... \n26 \n26 \nFF Aql \n18:58:14.748 +17:21:39.296 \n1 \n0.10 ± 0.10 \n1 \n... \n... \n... \n27 \n27 \nS Cru \n12:54:21.997 −58:25:50.215 \n0 \n0.09 ± 0.04 \n2 \n... \n... \n... \n13 \n13 \nδ Cep \n22:29:10.265 +58:24:54.714 \n0 \n0.09 ± 0.10 \n1 \n... \n... \n... \n18 \n18 \nY Sgr \n18:21:22.986 −18:51:36.002 \n0 \n0.00 ± 0.06 \n2 \n... \n20 \n... \n4 \n24 \nXX Sgr \n18:24:44.500 −16:47:49.820 \n0 \n0.06 ± 0.04 \n2 \n4 \n... \n5 \n3 \n12 \nη Aql \n19:52:28.368 +01:00:20.370 \n0B \n0.24 ± 0.09 \n2 \n... \n... \n... \n11 \n11 \nS Sge \n19:56:01.261 +16:38:05.236 \n0B \n0.14 ± 0.10 \n1 \n... \n... \n... \n21 \n21 \nV500 Sco 17:48:37.501 −30:28:33.461 \n0B \n−0.07 ± 0.08 \n1 \n... \n... \n4 \n3 \n7 \nβ Dor \n05:33:37.512 −62:29:23.323 \n0B \n−0.03 ± 0.05 \n2 \n1 \n46 \n... \n... \n47 \nζ Gem \n07:04:06.530 +20:34:13.074 \n0B \n0.16 ± 0.05 \n2 \n... \n47 \n... \n81 128 \nVY Sgr \n18:12:04.566 −20:42:14.465 \n0B \n0.25 ± 0.08 \n2 \n29 \n... \n4 \n... \n33 \nUZ Sct \n18:31:22.367 −12:55:43.339 \n0B \n0.11 ± 0.09 \n2 \n28 \n... \n6 \n... \n34 \nY Oph \n17:52:38.702 −06:08:36.875 \n0 \n0.08 ± 0.05 \n2 \n... \n8 \n... \n... \n8 \nRY Sco \n17:50:52.345 −33:42:20.411 \n0 \n0.01 ± 0.06 \n1 \n... \n... \n5 \n3 \n8 \nRZ Vel \n08:37:01.303 −44:06:52.844 \n0 \n0.08 ± 0.06 \n2 \n1 \n11 \n... \n... \n12 \nV340 Ara 16:45:19.112 −51:20:33.394 \n0 \n0.23 ± 0.07 \n2 \n26 \n... \n6 \n... \n32 \nWZ Sgr \n18:16:59.716 −19:04:32.989 \n0 \n0.28 ± 0.08 \n1 \n1 \n... \n5 \n3 \n9 \nRS Pup \n08:13:04.216 −34:34:42.692 \n0 \n0.14 ± 0.07 \n2 \n... \n15 \n... \n3 \n18"},{"id":"","caption":"Article number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.","content":"Article number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.Name \nα ICRS \nδ ICRS \nMode [Fe/H] lit ± σ Ref. a \nN F N H N U N S N tot \n\nR TrA \n15:19:45.712 −66:29:45.742 \n0 \n−0.01 ± 0.03 \n2 \n1 \n14 \n... \n... \n15 \nT Vul \n20:51:28.238 +28:15:01.817 \n0 \n0.07 ± 0.10 \n1 \n... \n... \n... \n26 \n26 \nFF Aql \n18:58:14.748 +17:21:39.296 \n1 \n0.10 ± 0.10 \n1 \n... \n... \n... \n27 \n27 \nS Cru \n12:54:21.997 −58:25:50.215 \n0 \n0.09 ± 0.04 \n2 \n... \n... \n... \n13 \n13 \nδ Cep \n22:29:10.265 +58:24:54.714 \n0 \n0.09 ± 0.10 \n1 \n... \n... \n... \n18 \n18 \nY Sgr \n18:21:22.986 −18:51:36.002 \n0 \n0.00 ± 0.06 \n2 \n... \n20 \n... \n4 \n24 \nXX Sgr \n18:24:44.500 −16:47:49.820 \n0 \n0.06 ± 0.04 \n2 \n4 \n... \n5 \n3 \n12 \nη Aql \n19:52:28.368 +01:00:20.370 \n0B \n0.24 ± 0.09 \n2 \n... \n... \n... \n11 \n11 \nS Sge \n19:56:01.261 +16:38:05.236 \n0B \n0.14 ± 0.10 \n1 \n... \n... \n... \n21 \n21 \nV500 Sco 17:48:37.501 −30:28:33.461 \n0B \n−0.07 ± 0.08 \n1 \n... \n... \n4 \n3 \n7 \nβ Dor \n05:33:37.512 −62:29:23.323 \n0B \n−0.03 ± 0.05 \n2 \n1 \n46 \n... \n... \n47 \nζ Gem \n07:04:06.530 +20:34:13.074 \n0B \n0.16 ± 0.05 \n2 \n... \n47 \n... \n81 128 \nVY Sgr \n18:12:04.566 −20:42:14.465 \n0B \n0.25 ± 0.08 \n2 \n29 \n... \n4 \n... \n33 \nUZ Sct \n18:31:22.367 −12:55:43.339 \n0B \n0.11 ± 0.09 \n2 \n28 \n... \n6 \n... \n34 \nY Oph \n17:52:38.702 −06:08:36.875 \n0 \n0.08 ± 0.05 \n2 \n... \n8 \n... \n... \n8 \nRY Sco \n17:50:52.345 −33:42:20.411 \n0 \n0.01 ± 0.06 \n1 \n... \n... \n5 \n3 \n8 \nRZ Vel \n08:37:01.303 −44:06:52.844 \n0 \n0.08 ± 0.06 \n2 \n1 \n11 \n... \n... \n12 \nV340 Ara 16:45:19.112 −51:20:33.394 \n0 \n0.23 ± 0.07 \n2 \n26 \n... \n6 \n... \n32 \nWZ Sgr \n18:16:59.716 −19:04:32.989 \n0 \n0.28 ± 0.08 \n1 \n1 \n... \n5 \n3 \n9 \nRS Pup \n08:13:04.216 −34:34:42.692 \n0 \n0.14 ± 0.07 \n2 \n... \n15 \n... \n3 \n18"},{"id":"","caption":"Excerpt from the list of Fe i and Fe ii atomic lines.","content":"Table 2 .\n2Excerpt from the list of Fe i and Fe ii atomic lines.λ [Å] \nSpecies ξ [eV] log g f \nQuality \nflag \nRef. \n\n3763.789 \nFe i \n0.989 −0.220 \n1 \n1 \n3787.880 \nFe i \n1.010 −0.840 \n1 \n1 \n... \n... \n... \n... \n... \n7655.488 \nFe ii \n3.892 −3.560 \n1 \n2 \n7711.724 \nFe ii \n3.903 −2.500 \n1 \n2 \n\nNotes. The quality flag column indicates if the line was used (1) or not \nused (0) in the current study. The complete table is available at the CDS. \n\nReferences. 1: O'Brian et al. (1991); 2: Meléndez & Barbuy (2009); 3: \nNIST: Kramida et al. (2020)."},{"id":"","caption":"Excerpt from the list of Fe i and Fe ii atomic lines.","content":"Table 2 .\n2Excerpt from the list of Fe i and Fe ii atomic lines.λ [Å] \nSpecies ξ [eV] log g f \nQuality \nflag \nRef. \n\n3763.789 \nFe i \n0.989 −0.220 \n1 \n1 \n3787.880 \nFe i \n1.010 −0.840 \n1 \n1 \n... \n... \n... \n... \n... \n7655.488 \nFe ii \n3.892 −3.560 \n1 \n2 \n7711.724 \nFe ii \n3.903 −2.500 \n1 \n2 \n\nNotes. The quality flag column indicates if the line was used (1) or not \nused (0) in the current study. The complete table is available at the CDS. \n\nReferences. 1: O'Brian et al. (1991); 2: Meléndez & Barbuy (2009); 3: \nNIST: Kramida et al. (2020)."},{"id":"","caption":"The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.","content":"Table 3 .\n3The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.λ [Å] \nSpecies ξ [eV] log g f \nQuality \nflag \nRef. \n\n3829.355 \nMg i \n2.709 −0.227 \n0 \n3 \n4571.096 \nMg i \n0.000 −5.620 \n1 \n3 \n... \n... \n... \n... \n... \n6606.956 \nTi ii \n2.061 −2.790 \n1 \n3 \n7214.729 \nTi ii \n2.590 −1.750 \n1 \n3"},{"id":"","caption":"The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.","content":"Table 3 .\n3The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.λ [Å] \nSpecies ξ [eV] log g f \nQuality \nflag \nRef. \n\n3829.355 \nMg i \n2.709 −0.227 \n0 \n3 \n4571.096 \nMg i \n0.000 −5.620 \n1 \n3 \n... \n... \n... \n... \n... \n6606.956 \nTi ii \n2.061 −2.790 \n1 \n3 \n7214.729 \nTi ii \n2.590 −1.750 \n1 \n3"},{"id":"","caption":"Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.","content":"Table 5 .\n5Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.Name \nV \n[mag] \n\n∆V \n[mag] \n\nγ \n[km/s] \n\n∆RV \n[km/s] \n\nLC \nsource a \n\nPeriod \n[days] \n\nT 0 − 2 400 000 \n[days] \n\nEphemerides \nsource b \n\nR TrA \n6.66 ± 0.02 0.56 ± 0.02 -12.44 ± 2.54 28.1 ± 2.2 \n1 \n3.3892582 \n44 423.59377 \n1 \nT Vul \n5.75 ± 0.03 0.64 ± 0.04 \n0.85 ± 0.11 34.5 ± 3.5 \n1 \n4.435462 \n57 632.18950 \n0 \n50 735.20310 \n2 \nFF Aql \n5.19 ± 0.01 0.33 ± 0.03 -11.36 ± 0.10 16.1 ± 1.5 \n2 \n4.470916 \n56 888.66207 \n0 \nS Cru \n6.60 ± 0.02 0.73 ± 0.03 \n-4.70 ± 0.28 33.8 ± 4.3 \n1 \n4.68973 \n55 279.67331 \n0 \nδ Cep \n3.94 ± 0.01 0.84 ± 0.04 -16.32 ± 0.20 38.9 ± 4.9 \n1 \n5.3663 \n56 917.54900 \n0 \nY Sgr \n5.74 ± 0.02 0.72 ± 0.03 \n-2.84 ± 3.02 34.2 ± 3.0 \n1 \n5.7733866 \n54 946.78504 \n1 \nXX Sgr \n8.86 ± 0.02 0.87 ± 0.04 \n12.30 ± 0.46 41.6 ± 1.2 \n1 \n6.4243013 \n54 946.67631 \n1 \nη Aql \n3.89 ± 0.02 0.78 ± 0.03 -14.47 ± 0.63 42.1 ± 2.2 \n1 \n7.1768572 \n50 724.43545 \n1 \nS Sge \n5.61 ± 0.03 0.72 ± 0.03 -16.40 ± 0.25 39.3 ± 4.7 \n1 \n8.3445707 \n57 636.28926 \n0 \n48 746.52000 \n3 \n8.3823514 \n50 339.17710 \n4 \nV500 Sco \n8.74 ± 0.01 0.71 ± 0.05 \n-7.61 ± 3.37 32.2 ± 5.4 \n1 \n9.317 \n44 789.61621 \n1 \nβ Dor \n5.25 ± 0.01 0.63 ± 0.06 \n8.17 ± 0.11 32.9 ± 2.3 \n2 \n9.84308 \n53 284.79472 \n0 \n9.8527371 \n41 149.27150 \n4 \nζ Gem \n3.89 ± 0.03 0.49 ± 0.03 \n6.78 ± 0.42 26.5 ± 1.0 \n1 \n10.149816 \n54 805.78476 \n1 \nVY Sgr \n11.46 ± 0.02 1.14 ± 0.05 \n14.23 ± 1.58 50.4 ± 4.4 \n1 \n13.55845 \n54 171.32722 \n1 \nUZ Sct \n11.25 ± 0.03 0.75 ± 0.05 \n39.52 ± 0.18 44.7 ± 3.5 \n0 \n14.744162 \n56 171.62381 \n0 \nY Oph \n6.15 ± 0.02 0.49 ± 0.03 \n-7.56 ± 2.01 16.5 ± 2.0 \n1 \n17.12633896 53 209.96732 \n1 \nRY Sco \n8.02 ± 0.02 0.83 ± 0.03 -17.73 ± 2.02 30.1 ± 3.0 \n1 \n20.321538 \n45 077.36644 \n1 \nRZ Vel \n7.09 ± 0.04 1.17 ± 0.05 \n24.11 ± 4.52 54.6 ± 5.4 \n1 \n20.399727 \n50 776.13608 \n1 \nV340 Ara 10.20 ± 0.06 1.08 ± 0.07 -76.25 ± 4.19 47.7 ± 9.3 \n1 \n20.811386 \n56 138.97597 \n0 \nWZ Sgr \n8.03 ± 0.03 1.08 ± 0.05 -17.78 ± 1.17 54.2 ± 2.9 \n1 \n21.849708 \n50 691.28534 \n1 \nRS Pup \n6.99 ± 0.03 1.11 ± 0.04 \n25.80 ± 1.91 48.5 ± 2.1 \n1 \n41.488634 \n53 092.94483 \n1"},{"id":"","caption":"Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.","content":"Table 5 .\n5Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.Name \nV \n[mag] \n\n∆V \n[mag] \n\nγ \n[km/s] \n\n∆RV \n[km/s] \n\nLC \nsource a \n\nPeriod \n[days] \n\nT 0 − 2 400 000 \n[days] \n\nEphemerides \nsource b \n\nR TrA \n6.66 ± 0.02 0.56 ± 0.02 -12.44 ± 2.54 28.1 ± 2.2 \n1 \n3.3892582 \n44 423.59377 \n1 \nT Vul \n5.75 ± 0.03 0.64 ± 0.04 \n0.85 ± 0.11 34.5 ± 3.5 \n1 \n4.435462 \n57 632.18950 \n0 \n50 735.20310 \n2 \nFF Aql \n5.19 ± 0.01 0.33 ± 0.03 -11.36 ± 0.10 16.1 ± 1.5 \n2 \n4.470916 \n56 888.66207 \n0 \nS Cru \n6.60 ± 0.02 0.73 ± 0.03 \n-4.70 ± 0.28 33.8 ± 4.3 \n1 \n4.68973 \n55 279.67331 \n0 \nδ Cep \n3.94 ± 0.01 0.84 ± 0.04 -16.32 ± 0.20 38.9 ± 4.9 \n1 \n5.3663 \n56 917.54900 \n0 \nY Sgr \n5.74 ± 0.02 0.72 ± 0.03 \n-2.84 ± 3.02 34.2 ± 3.0 \n1 \n5.7733866 \n54 946.78504 \n1 \nXX Sgr \n8.86 ± 0.02 0.87 ± 0.04 \n12.30 ± 0.46 41.6 ± 1.2 \n1 \n6.4243013 \n54 946.67631 \n1 \nη Aql \n3.89 ± 0.02 0.78 ± 0.03 -14.47 ± 0.63 42.1 ± 2.2 \n1 \n7.1768572 \n50 724.43545 \n1 \nS Sge \n5.61 ± 0.03 0.72 ± 0.03 -16.40 ± 0.25 39.3 ± 4.7 \n1 \n8.3445707 \n57 636.28926 \n0 \n48 746.52000 \n3 \n8.3823514 \n50 339.17710 \n4 \nV500 Sco \n8.74 ± 0.01 0.71 ± 0.05 \n-7.61 ± 3.37 32.2 ± 5.4 \n1 \n9.317 \n44 789.61621 \n1 \nβ Dor \n5.25 ± 0.01 0.63 ± 0.06 \n8.17 ± 0.11 32.9 ± 2.3 \n2 \n9.84308 \n53 284.79472 \n0 \n9.8527371 \n41 149.27150 \n4 \nζ Gem \n3.89 ± 0.03 0.49 ± 0.03 \n6.78 ± 0.42 26.5 ± 1.0 \n1 \n10.149816 \n54 805.78476 \n1 \nVY Sgr \n11.46 ± 0.02 1.14 ± 0.05 \n14.23 ± 1.58 50.4 ± 4.4 \n1 \n13.55845 \n54 171.32722 \n1 \nUZ Sct \n11.25 ± 0.03 0.75 ± 0.05 \n39.52 ± 0.18 44.7 ± 3.5 \n0 \n14.744162 \n56 171.62381 \n0 \nY Oph \n6.15 ± 0.02 0.49 ± 0.03 \n-7.56 ± 2.01 16.5 ± 2.0 \n1 \n17.12633896 53 209.96732 \n1 \nRY Sco \n8.02 ± 0.02 0.83 ± 0.03 -17.73 ± 2.02 30.1 ± 3.0 \n1 \n20.321538 \n45 077.36644 \n1 \nRZ Vel \n7.09 ± 0.04 1.17 ± 0.05 \n24.11 ± 4.52 54.6 ± 5.4 \n1 \n20.399727 \n50 776.13608 \n1 \nV340 Ara 10.20 ± 0.06 1.08 ± 0.07 -76.25 ± 4.19 47.7 ± 9.3 \n1 \n20.811386 \n56 138.97597 \n0 \nWZ Sgr \n8.03 ± 0.03 1.08 ± 0.05 -17.78 ± 1.17 54.2 ± 2.9 \n1 \n21.849708 \n50 691.28534 \n1 \nRS Pup \n6.99 ± 0.03 1.11 ± 0.04 \n25.80 ± 1.91 48.5 ± 2.1 \n1 \n41.488634 \n53 092.94483 \n1"},{"id":"","caption":"Mean atmospheric parameters derived for the 20 calibrating Cepheids.","content":"Table 6 .\n6Mean atmospheric parameters derived for the 20 calibrating Cepheids."},{"id":"","caption":"Mean atmospheric parameters derived for the 20 calibrating Cepheids.","content":"Table 6 .\n6Mean atmospheric parameters derived for the 20 calibrating Cepheids."},{"id":"","caption":"Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.","content":"Table 7 .\n7Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.3 < P < 5 \n[days] \n\n5 < P < 7 \n[days] \n\n7 < P < 9.5 \n[days] \n\n9.5 < P < 10.5 \n[days] \n\nc 0 \n0.44 ± 0.21 0.44 ± 0.23 \n0.48 ± 0.26 0.47 ± 0.12 \nc 1 \n2.1 ± 0.5 \n2.1 ± 0.5 \n−4.4 ± 0.6 \n−4.4 ± 0.2 \nc 2 \n0.13 ± 0.22 0.15 ± 0.24 \n0.14 ± 0.28 0.04 ± 0.12 \nc 3 −4.5 ± 1.4 \n2.2 ± 1.4 \n2.6 ± 2.0 \n4.2 ± 2.9 \nc 4 \n0.06 ± 0.21 0.06 ± 0.23 \n0.05 ± 0.27 0.04 ± 0.11 \nc 5 \n1.4 ± 3.6 \n2.6 ± 3.7 \n2.2 ± 5.6 \n4.4 ± 2.7 \nc 6 \n0.03 ± 0.21 0.03 ± 0.22 \n0.03 ± 0.28 0.02 ± 0.11 \nc 7 \n1.3 ± 8.3 \n2.7 ± 7.6 \n−0.93 ± 9.56 \n4.2 ± 6.9 \nc 8 \n0.01 ± 0.19 \n... \n... \n... \nc 9 \n2.8 ± 16.7 \n... \n... \n..."},{"id":"","caption":"Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.","content":"Table 7 .\n7Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.3 < P < 5 \n[days] \n\n5 < P < 7 \n[days] \n\n7 < P < 9.5 \n[days] \n\n9.5 < P < 10.5 \n[days] \n\nc 0 \n0.44 ± 0.21 0.44 ± 0.23 \n0.48 ± 0.26 0.47 ± 0.12 \nc 1 \n2.1 ± 0.5 \n2.1 ± 0.5 \n−4.4 ± 0.6 \n−4.4 ± 0.2 \nc 2 \n0.13 ± 0.22 0.15 ± 0.24 \n0.14 ± 0.28 0.04 ± 0.12 \nc 3 −4.5 ± 1.4 \n2.2 ± 1.4 \n2.6 ± 2.0 \n4.2 ± 2.9 \nc 4 \n0.06 ± 0.21 0.06 ± 0.23 \n0.05 ± 0.27 0.04 ± 0.11 \nc 5 \n1.4 ± 3.6 \n2.6 ± 3.7 \n2.2 ± 5.6 \n4.4 ± 2.7 \nc 6 \n0.03 ± 0.21 0.03 ± 0.22 \n0.03 ± 0.28 0.02 ± 0.11 \nc 7 \n1.3 ± 8.3 \n2.7 ± 7.6 \n−0.93 ± 9.56 \n4.2 ± 6.9 \nc 8 \n0.01 ± 0.19 \n... \n... \n... \nc 9 \n2.8 ± 16.7 \n... \n... \n..."},{"id":"","caption":"Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.","content":"Table 8 .\n8Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.Name \nDataset \nMJD \n[d] \n[Mg/H] ± σ N Mg ... \nTi i ± σ \nN Ti i \nTi ii ± σ \nN Ti ii \n[Ti/H] ± σ \n\nR TrA \nHARPS 53150.1453265 0.20 ± 0.11 \n1 \n... −0.21 ± 0.15 \n8 \n0.05 ± 0.10 \n2 \n−0.12 ± 0.19 \nR TrA \nHARPS 53150.1353086 0.17 ± 0.11 \n1 \n... −0.17 ± 0.27 \n12 \n0.01 ± 0.10 \n3 \n−0.14 ± 0.25 \n... \n... \n... \n... \n... \n... \n... \n... \n... \n... \n... \nRS Pup STELLA 57708.2167938 0.24 ± 0.11 \n2 \n... −0.06 ± 0.28 \n15 \n0.22 ± 0.10 \n4 \n0.09 ± 0.19 \nRS Pup STELLA 57713.2628277 \n... \n... \n... \n... \n... \n... \n... \n..."},{"id":"","caption":"Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.","content":"Table 8 .\n8Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.Name \nDataset \nMJD \n[d] \n[Mg/H] ± σ N Mg ... \nTi i ± σ \nN Ti i \nTi ii ± σ \nN Ti ii \n[Ti/H] ± σ \n\nR TrA \nHARPS 53150.1453265 0.20 ± 0.11 \n1 \n... −0.21 ± 0.15 \n8 \n0.05 ± 0.10 \n2 \n−0.12 ± 0.19 \nR TrA \nHARPS 53150.1353086 0.17 ± 0.11 \n1 \n... −0.17 ± 0.27 \n12 \n0.01 ± 0.10 \n3 \n−0.14 ± 0.25 \n... \n... \n... \n... \n... \n... \n... \n... \n... \n... \n... \nRS Pup STELLA 57708.2167938 0.24 ± 0.11 \n2 \n... −0.06 ± 0.28 \n15 \n0.22 ± 0.10 \n4 \n0.09 ± 0.19 \nRS Pup STELLA 57713.2628277 \n... \n... \n... \n... \n... \n... \n... \n..."},{"id":"","caption":"Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.","content":"Table 9 .\n9Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.Name \n[Mg/H] ± σ (std) N spec ... [Ti i/H] ± σ \n[Ti ii/H] ± σ \n[Ti/H] ± σ (std) \nN spec \n\nR TrA \n0.10 ± 0.03 (0.08) \n15 \n... −0.19 ± 0.04 −0.09 ± 0.03 −0.18 ± 0.04 (0.10) \n15 \nT Vul \n0.07 ± 0.03 (0.15) \n26 \n... −0.11 ± 0.03 −0.07 ± 0.02 −0.10 ± 0.03 (0.10) \n26 \n... \n... \n... \n... \n... \n... \n... \n... \nWZ Sgr 0.12 ± 0.06 (0.03) \n4 \n... \n0.08 ± 0.07 \n0.01 ± 0.05 \n0.06 ± 0.07 (0.12) \n7 \nRS Pup 0.09 ± 0.03 (0.14) \n10 \n... \n0.02 ± 0.04 \n0.16 ± 0.03 \n0.04 ± 0.04 (0.14) \n16"},{"id":"","caption":"Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.","content":"Table 9 .\n9Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.Name \n[Mg/H] ± σ (std) N spec ... [Ti i/H] ± σ \n[Ti ii/H] ± σ \n[Ti/H] ± σ (std) \nN spec \n\nR TrA \n0.10 ± 0.03 (0.08) \n15 \n... −0.19 ± 0.04 −0.09 ± 0.03 −0.18 ± 0.04 (0.10) \n15 \nT Vul \n0.07 ± 0.03 (0.15) \n26 \n... −0.11 ± 0.03 −0.07 ± 0.02 −0.10 ± 0.03 (0.10) \n26 \n... \n... \n... \n... \n... \n... \n... \n... \nWZ Sgr 0.12 ± 0.06 (0.03) \n4 \n... \n0.08 ± 0.07 \n0.01 ± 0.05 \n0.06 ± 0.07 (0.12) \n7 \nRS Pup 0.09 ± 0.03 (0.14) \n10 \n... \n0.02 ± 0.04 \n0.16 ± 0.03 \n0.04 ± 0.04 (0.14) \n16"},{"id":"","caption":"Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).","content":"Table 10 .\n10Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).Abundance \nratio \nDatasets 1 \nZero-point \ndifference \nN Common \n\n[Fe/H] \nP18−TS \n0.07 ± 0.06 \n13 \n[Fe/H] \nLII−TS \n0.05 ± 0.07 \n20 \n[Fe/H] \nLIII−TS \n0.11 ± 0.08 \n6 \n[Fe/H] \nLII−LEM \n0.07 ± 0.12 \n55 \n[Fe/H] \nYON−LIII −0.33 ± 0.15 \n21 \n[Mg/H] \nLII−TS \n−0.14 ± 0.11 \n16 \n[Mg/H] \nLIII−TS \n0.11 ± 0.16 \n5 \n[Mg/H] \nLII−LEM −0.26 ± 0.24 \n35 \n[Mg/H] \nYON−LIII −0.09 ± 0.14 \n16 \n[Si/H] \nLII−TS \n−0.13 ± 0.07 \n20 \n[Si/H] \nLIII−TS \n−0.01 ± 0.09 \n6 \n[Si/H] \nLII−LEM −0.06 ± 0.11 \n55 \n[Si/H] \nYON−LIII −0.12 ± 0.07 \n20 \n[S/H] \nLII−TS \n0.11 ± 0.13 \n17 \n[S/H] \nLII−LEM −0.13 ± 0.19 \n45 \n[Ca/H] \nLII−TS \n−0.07 ± 0.10 \n20 \n[Ca/H] \nLIII−TS \n−0.01 ± 0.10 \n6 \n[Ca/H] \nLII−LEM −0.06 ± 0.17 \n55 \n[Ca/H] \nYON−LIII −0.16 ± 0.11 \n21 \n[Ti/H] \nLII−TS \n0.12 ± 0.09 \n20 \n[Ti/H] \nLIII−TS \n0.17 ± 0.12 \n6 \n[Ti/H] \nYON−LIII \n0.33 ± 0.20 \n20"},{"id":"","caption":"Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).","content":"Table 10 .\n10Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).Abundance \nratio \nDatasets 1 \nZero-point \ndifference \nN Common \n\n[Fe/H] \nP18−TS \n0.07 ± 0.06 \n13 \n[Fe/H] \nLII−TS \n0.05 ± 0.07 \n20 \n[Fe/H] \nLIII−TS \n0.11 ± 0.08 \n6 \n[Fe/H] \nLII−LEM \n0.07 ± 0.12 \n55 \n[Fe/H] \nYON−LIII −0.33 ± 0.15 \n21 \n[Mg/H] \nLII−TS \n−0.14 ± 0.11 \n16 \n[Mg/H] \nLIII−TS \n0.11 ± 0.16 \n5 \n[Mg/H] \nLII−LEM −0.26 ± 0.24 \n35 \n[Mg/H] \nYON−LIII −0.09 ± 0.14 \n16 \n[Si/H] \nLII−TS \n−0.13 ± 0.07 \n20 \n[Si/H] \nLIII−TS \n−0.01 ± 0.09 \n6 \n[Si/H] \nLII−LEM −0.06 ± 0.11 \n55 \n[Si/H] \nYON−LIII −0.12 ± 0.07 \n20 \n[S/H] \nLII−TS \n0.11 ± 0.13 \n17 \n[S/H] \nLII−LEM −0.13 ± 0.19 \n45 \n[Ca/H] \nLII−TS \n−0.07 ± 0.10 \n20 \n[Ca/H] \nLIII−TS \n−0.01 ± 0.10 \n6 \n[Ca/H] \nLII−LEM −0.06 ± 0.17 \n55 \n[Ca/H] \nYON−LIII −0.16 ± 0.11 \n21 \n[Ti/H] \nLII−TS \n0.12 ± 0.09 \n20 \n[Ti/H] \nLIII−TS \n0.17 ± 0.12 \n6 \n[Ti/H] \nYON−LIII \n0.33 ± 0.20 \n20"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Partly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra.\",\n \"content\": \"Partly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Partly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra.\",\n \"content\": \"Partly based on observations made with ESO Telescopes at the La Silla/Paranal Observatories under program IDs: 072.D-0419, 073.D-0136, and 190.D-0237 for HARPS spectra; 084.B-0029, 087.A-9013, 074.D-0008, 075.D-0676, and 60.A-9120 for FEROS spectra; 081.D-0928, 082.D-0901, 089.D-0767, and 093.D-0816 for UVES spectra.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel).\",\n \"content\": \"Fig. 1 .\\n1Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel).\",\n \"content\": \"Fig. 1 .\\n1Examples of high-resolution spectra for the stars β Dor and ζ Gem in different pulsation phases (values on the right side of each panel).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\",\n \"content\": \"Fig. 2 .\\n2Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\",\n \"content\": \"Fig. 2 .\\n2Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width. These are examples of good lines that were kept in our initial line list for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA.\",\n \"content\": \"a) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA.\",\n \"content\": \"a) one containing atomic lines of N, Si, S, Ca, Ti, V, Cr, Mn, Fe, Co, and Ni, created from a combination of four line lists received from V. Kovtyukh (for a total of 153 atomic lines). These lines are used to derive the effective temperature (T eff ) of our sample stars according to the procedure described in Sect. 4.2 of Proxauf et al. (2018); b) one containing 424 Fe i and 97 Fe ii lines, created from a combination of lines from Genovali et al. (2013) with lines from the Gaia-ESO Survey (GES, Gilmore et al. 2012; Randich et al. 2013). It is used as a reference line list in the determination of both the stellar surface gravity (log g), as described in Sect. 5.1, and the stellar metallicity, as discussed in Sect. 6; c) one containing lines of α elements (Mg: 9 lines; Si: 11 lines; S: 1 line; Ca: 36 lines; and Ti: 72 lines), based on line lists from For & Sneden (2010), Venn et al. (2012), Lemasle et al. (2013), and McWilliam et al. (2013, and references therein). 1 The IRAF package is distributed by the National Optical Astronomy Observatories (NOAO), USA.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"et al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for\",\n \"content\": \"et al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for\"\n },\n {\n \"id\": \"\",\n \"caption\": \"et al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for\",\n \"content\": \"et al. (2014), Den Hartog et al. (2014), and Belmonte et al. (2017) for Fe i, Den Hartog et al. (2019) for\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled.\",\n \"content\": \"Fig. 3 .\\n3Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled.\",\n \"content\": \"Fig. 3 .\\n3Top: RV curve of the fundamental pulsator S Cru. The red solid line displays the PLOESS fit to the RV curve. The horizontal red dashed and dotted lines display the mean RV ( RV ) and its uncertainty. The vertical red dotted line displays the phase where RV equals RV on the rising branch. The name, RV , period and reference epoch are labelled.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve.\",\n \"content\": \". B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve.\",\n \"content\": \". B.1 were shifted.References. (a) 0: ASAS-SN(Shappee et al. 2014;Kochanek et al. 2017); 1: Groenewegen (2008, and references therein); 2:Pel (1976). (b) 0: Period and T 0 derived from our own radial velocity curves; 1: Period and T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 2: T 0 derived from the light curve; 3: T 0 derived from the radial velocities byGroenewegen (2008, and references therein); 4: Period and T 0 derived from the light curve.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Normalized θ (5040/T eff ) as a function of the pulsation phase.\",\n \"content\": \"Fig. 5 .\\n5Normalized θ (5040/T eff ) as a function of the pulsation phase.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Normalized θ (5040/T eff ) as a function of the pulsation phase.\",\n \"content\": \"Fig. 5 .\\n5Normalized θ (5040/T eff ) as a function of the pulsation phase.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation.\",\n \"content\": \"Fig. 6 .\\n6Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation.\",\n \"content\": \"Fig. 6 .\\n6Normalized θ (5040/T eff ) as a function of the pulsation phase showing the fits of the Fourier series to the data ofFig. 5. The shaded areas display the 1-σ standard deviation.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018).\",\n \"content\": \"Fig. 7 .\\n7Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018).\",\n \"content\": \"Fig. 7 .\\n7Distribution of standard deviations computed from individual abundances of Fe i and Fe ii lines. The panels show the histogram for spectra of stars in common with Proxauf et al. (2018).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values.\",\n \"content\": \"Fig. 8 .\\n8Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values.\",\n \"content\": \"Fig. 8 .\\n8Atmospheric parameters differences for stars in common withProxauf et al. (2018). Each panel shows our previous determinations subtracted by the current values.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show.\",\n \"content\": \"Fig. 10 .\\n10Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show.\",\n \"content\": \"Fig. 10 .\\n10Stellar metallicity measured for our sample of 20 calibrating Cepheids compared with results from literature. Top panel: metallicity as a function of the Galactocentric distance from the current work (blue circles) and from literature (gray circles): Luck et al. (2011, LII); Luck & Lambert (2011, LIII); Lemasle et al. (2013, LEM); Yong et al. (2006, YON). A linear regression (solid black line plus equation) fitted to the entire sample is compared with the radial gradient provided by Ripepi et al. (2022) (dashed magenta line). The latter was artificially shifted to coincide with the current radial gradient at R G = 10 kpc. The R G values are from Genovali et al. (2014). Bottom panel: same as the top, but as a function of the logarithmic pulsation period. A linear regression (solid blue line plus equation) fitted to the current sample is also show.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors.\",\n \"content\": \"Fig. 11 .\\n11[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors.\",\n \"content\": \"Fig. 11 .\\n11[X/Fe] abundances as a function of metallicity (left panels) and of the logarithmic pulsation period (right panels) comparing our 20 calibrating Cepheids with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in\",\n \"content\": \"[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in\",\n \"content\": \"[Mg/H]) = 0.11, σ([Si i/H]) = 0.10, σ([Ca i/H]) = 0.18, σ([Ca ii/H]) = 0.18, σ([Ti i/H]) = 0.14, and σ([Ti ii/H]) = 0.10 dex (for the sulfur abundances, the uncertainties come from the spectral synthesis). The estimate of these typical errors was done by calculating the median of the standard deviations listed in\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)\",\n \"content\": \"Fig. 12 .\\n12[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)\",\n \"content\": \"Fig. 12 .\\n12[X/H] and [X/Fe] abundance ratios as a function of the Galactocentric distance (R G ) for our 20 calibrating Cepheids compared with stars from literature: LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006). The error bars indicate our typical errors. Linear regressions fitted to the entire sample are also shown (solid lines)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\",\n \"content\": \"Fig. A. 1 .\\n1Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\",\n \"content\": \"Fig. A. 1 .\\n1Iron abundances as a function of the pulsation phase, effective temperature, and equivalent width for lines excluded from the initial line list. These examples, for HARPS (green circles), FEROS (blue triangle), and STELLA (red stars) spectra of β Dor and ζ Gem, show a dependence of the abundances on the equivalent widths possibly caused by blends. Lines too weak or too strong (probably saturated) are represented by light gray symbols. The hatched regions around the dashed lines indicate the 1-σ uncertainty around the mean.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths.\",\n \"content\": \"Fig. A. 2 .\\n2Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths.\",\n \"content\": \"Fig. A. 2 .\\n2Same as in Fig. A.1 but showing an example of a line excluded because of having a bump around pulsation phases 0.8-0.9, not caused by saturation or by any correlation with the equivalent widths.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends.\",\n \"content\": \"Fig. A. 3 .\\n3Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends.\",\n \"content\": \"Fig. A. 3 .\\n3Same as in Fig. A.1 but showing examples of lines excluded either because of being completely saturated or because of having a chaotic distribution, probably caused by blends.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects.\",\n \"content\": \"Fig. A. 4 .\\n4Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects.\",\n \"content\": \"Fig. A. 4 .\\n4Same as in Fig. A.1 but showing an example of a line excluded because of having some correlation with the effective temperature, possibly caused by NLTE effects.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves\",\n \"content\": \"Fig. A. 5 .\\n5Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves\",\n \"content\": \"Fig. A. 5 .\\n5Same as in Fig. A.1 but showing an example of a line that, even if the dispersion in abundance is very small, was excluded from the initial line list because of having an unexpected over-abundance in metallicity, possibly caused by a wrong log g f value. Appendix B: Variations along the pulsation cycle Appendix B.1: Light and radial-velocity curves\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Article number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.\",\n \"content\": \"Article number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.Name \\nα ICRS \\nδ ICRS \\nMode [Fe/H] lit ± σ Ref. a \\nN F N H N U N S N tot \\n\\nR TrA \\n15:19:45.712 −66:29:45.742 \\n0 \\n−0.01 ± 0.03 \\n2 \\n1 \\n14 \\n... \\n... \\n15 \\nT Vul \\n20:51:28.238 +28:15:01.817 \\n0 \\n0.07 ± 0.10 \\n1 \\n... \\n... \\n... \\n26 \\n26 \\nFF Aql \\n18:58:14.748 +17:21:39.296 \\n1 \\n0.10 ± 0.10 \\n1 \\n... \\n... \\n... \\n27 \\n27 \\nS Cru \\n12:54:21.997 −58:25:50.215 \\n0 \\n0.09 ± 0.04 \\n2 \\n... \\n... \\n... \\n13 \\n13 \\nδ Cep \\n22:29:10.265 +58:24:54.714 \\n0 \\n0.09 ± 0.10 \\n1 \\n... \\n... \\n... \\n18 \\n18 \\nY Sgr \\n18:21:22.986 −18:51:36.002 \\n0 \\n0.00 ± 0.06 \\n2 \\n... \\n20 \\n... \\n4 \\n24 \\nXX Sgr \\n18:24:44.500 −16:47:49.820 \\n0 \\n0.06 ± 0.04 \\n2 \\n4 \\n... \\n5 \\n3 \\n12 \\nη Aql \\n19:52:28.368 +01:00:20.370 \\n0B \\n0.24 ± 0.09 \\n2 \\n... \\n... \\n... \\n11 \\n11 \\nS Sge \\n19:56:01.261 +16:38:05.236 \\n0B \\n0.14 ± 0.10 \\n1 \\n... \\n... \\n... \\n21 \\n21 \\nV500 Sco 17:48:37.501 −30:28:33.461 \\n0B \\n−0.07 ± 0.08 \\n1 \\n... \\n... \\n4 \\n3 \\n7 \\nβ Dor \\n05:33:37.512 −62:29:23.323 \\n0B \\n−0.03 ± 0.05 \\n2 \\n1 \\n46 \\n... \\n... \\n47 \\nζ Gem \\n07:04:06.530 +20:34:13.074 \\n0B \\n0.16 ± 0.05 \\n2 \\n... \\n47 \\n... \\n81 128 \\nVY Sgr \\n18:12:04.566 −20:42:14.465 \\n0B \\n0.25 ± 0.08 \\n2 \\n29 \\n... \\n4 \\n... \\n33 \\nUZ Sct \\n18:31:22.367 −12:55:43.339 \\n0B \\n0.11 ± 0.09 \\n2 \\n28 \\n... \\n6 \\n... \\n34 \\nY Oph \\n17:52:38.702 −06:08:36.875 \\n0 \\n0.08 ± 0.05 \\n2 \\n... \\n8 \\n... \\n... \\n8 \\nRY Sco \\n17:50:52.345 −33:42:20.411 \\n0 \\n0.01 ± 0.06 \\n1 \\n... \\n... \\n5 \\n3 \\n8 \\nRZ Vel \\n08:37:01.303 −44:06:52.844 \\n0 \\n0.08 ± 0.06 \\n2 \\n1 \\n11 \\n... \\n... \\n12 \\nV340 Ara 16:45:19.112 −51:20:33.394 \\n0 \\n0.23 ± 0.07 \\n2 \\n26 \\n... \\n6 \\n... \\n32 \\nWZ Sgr \\n18:16:59.716 −19:04:32.989 \\n0 \\n0.28 ± 0.08 \\n1 \\n1 \\n... \\n5 \\n3 \\n9 \\nRS Pup \\n08:13:04.216 −34:34:42.692 \\n0 \\n0.14 ± 0.07 \\n2 \\n... \\n15 \\n... \\n3 \\n18\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Article number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.\",\n \"content\": \"Article number, page 1 of 31 arXiv:2202.07945v1 [astro-ph.SR] 16 Feb 2022 A&A proofs: manuscript no. daSilvaetal2021Table 1. Sample of 20 calibrating Cepheids with high-resolution spectra covering either a substantial portion or the entire pulsation cycle.Name \\nα ICRS \\nδ ICRS \\nMode [Fe/H] lit ± σ Ref. a \\nN F N H N U N S N tot \\n\\nR TrA \\n15:19:45.712 −66:29:45.742 \\n0 \\n−0.01 ± 0.03 \\n2 \\n1 \\n14 \\n... \\n... \\n15 \\nT Vul \\n20:51:28.238 +28:15:01.817 \\n0 \\n0.07 ± 0.10 \\n1 \\n... \\n... \\n... \\n26 \\n26 \\nFF Aql \\n18:58:14.748 +17:21:39.296 \\n1 \\n0.10 ± 0.10 \\n1 \\n... \\n... \\n... \\n27 \\n27 \\nS Cru \\n12:54:21.997 −58:25:50.215 \\n0 \\n0.09 ± 0.04 \\n2 \\n... \\n... \\n... \\n13 \\n13 \\nδ Cep \\n22:29:10.265 +58:24:54.714 \\n0 \\n0.09 ± 0.10 \\n1 \\n... \\n... \\n... \\n18 \\n18 \\nY Sgr \\n18:21:22.986 −18:51:36.002 \\n0 \\n0.00 ± 0.06 \\n2 \\n... \\n20 \\n... \\n4 \\n24 \\nXX Sgr \\n18:24:44.500 −16:47:49.820 \\n0 \\n0.06 ± 0.04 \\n2 \\n4 \\n... \\n5 \\n3 \\n12 \\nη Aql \\n19:52:28.368 +01:00:20.370 \\n0B \\n0.24 ± 0.09 \\n2 \\n... \\n... \\n... \\n11 \\n11 \\nS Sge \\n19:56:01.261 +16:38:05.236 \\n0B \\n0.14 ± 0.10 \\n1 \\n... \\n... \\n... \\n21 \\n21 \\nV500 Sco 17:48:37.501 −30:28:33.461 \\n0B \\n−0.07 ± 0.08 \\n1 \\n... \\n... \\n4 \\n3 \\n7 \\nβ Dor \\n05:33:37.512 −62:29:23.323 \\n0B \\n−0.03 ± 0.05 \\n2 \\n1 \\n46 \\n... \\n... \\n47 \\nζ Gem \\n07:04:06.530 +20:34:13.074 \\n0B \\n0.16 ± 0.05 \\n2 \\n... \\n47 \\n... \\n81 128 \\nVY Sgr \\n18:12:04.566 −20:42:14.465 \\n0B \\n0.25 ± 0.08 \\n2 \\n29 \\n... \\n4 \\n... \\n33 \\nUZ Sct \\n18:31:22.367 −12:55:43.339 \\n0B \\n0.11 ± 0.09 \\n2 \\n28 \\n... \\n6 \\n... \\n34 \\nY Oph \\n17:52:38.702 −06:08:36.875 \\n0 \\n0.08 ± 0.05 \\n2 \\n... \\n8 \\n... \\n... \\n8 \\nRY Sco \\n17:50:52.345 −33:42:20.411 \\n0 \\n0.01 ± 0.06 \\n1 \\n... \\n... \\n5 \\n3 \\n8 \\nRZ Vel \\n08:37:01.303 −44:06:52.844 \\n0 \\n0.08 ± 0.06 \\n2 \\n1 \\n11 \\n... \\n... \\n12 \\nV340 Ara 16:45:19.112 −51:20:33.394 \\n0 \\n0.23 ± 0.07 \\n2 \\n26 \\n... \\n6 \\n... \\n32 \\nWZ Sgr \\n18:16:59.716 −19:04:32.989 \\n0 \\n0.28 ± 0.08 \\n1 \\n1 \\n... \\n5 \\n3 \\n9 \\nRS Pup \\n08:13:04.216 −34:34:42.692 \\n0 \\n0.14 ± 0.07 \\n2 \\n... \\n15 \\n... \\n3 \\n18\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Excerpt from the list of Fe i and Fe ii atomic lines.\",\n \"content\": \"Table 2 .\\n2Excerpt from the list of Fe i and Fe ii atomic lines.λ [Å] \\nSpecies ξ [eV] log g f \\nQuality \\nflag \\nRef. \\n\\n3763.789 \\nFe i \\n0.989 −0.220 \\n1 \\n1 \\n3787.880 \\nFe i \\n1.010 −0.840 \\n1 \\n1 \\n... \\n... \\n... \\n... \\n... \\n7655.488 \\nFe ii \\n3.892 −3.560 \\n1 \\n2 \\n7711.724 \\nFe ii \\n3.903 −2.500 \\n1 \\n2 \\n\\nNotes. The quality flag column indicates if the line was used (1) or not \\nused (0) in the current study. The complete table is available at the CDS. \\n\\nReferences. 1: O'Brian et al. (1991); 2: Meléndez & Barbuy (2009); 3: \\nNIST: Kramida et al. (2020).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Excerpt from the list of Fe i and Fe ii atomic lines.\",\n \"content\": \"Table 2 .\\n2Excerpt from the list of Fe i and Fe ii atomic lines.λ [Å] \\nSpecies ξ [eV] log g f \\nQuality \\nflag \\nRef. \\n\\n3763.789 \\nFe i \\n0.989 −0.220 \\n1 \\n1 \\n3787.880 \\nFe i \\n1.010 −0.840 \\n1 \\n1 \\n... \\n... \\n... \\n... \\n... \\n7655.488 \\nFe ii \\n3.892 −3.560 \\n1 \\n2 \\n7711.724 \\nFe ii \\n3.903 −2.500 \\n1 \\n2 \\n\\nNotes. The quality flag column indicates if the line was used (1) or not \\nused (0) in the current study. The complete table is available at the CDS. \\n\\nReferences. 1: O'Brian et al. (1991); 2: Meléndez & Barbuy (2009); 3: \\nNIST: Kramida et al. (2020).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.\",\n \"content\": \"Table 3 .\\n3The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.λ [Å] \\nSpecies ξ [eV] log g f \\nQuality \\nflag \\nRef. \\n\\n3829.355 \\nMg i \\n2.709 −0.227 \\n0 \\n3 \\n4571.096 \\nMg i \\n0.000 −5.620 \\n1 \\n3 \\n... \\n... \\n... \\n... \\n... \\n6606.956 \\nTi ii \\n2.061 −2.790 \\n1 \\n3 \\n7214.729 \\nTi ii \\n2.590 −1.750 \\n1 \\n3\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.\",\n \"content\": \"Table 3 .\\n3The same as inTable 2but showing an excerpt of atomic lines from the list of α elements.λ [Å] \\nSpecies ξ [eV] log g f \\nQuality \\nflag \\nRef. \\n\\n3829.355 \\nMg i \\n2.709 −0.227 \\n0 \\n3 \\n4571.096 \\nMg i \\n0.000 −5.620 \\n1 \\n3 \\n... \\n... \\n... \\n... \\n... \\n6606.956 \\nTi ii \\n2.061 −2.790 \\n1 \\n3 \\n7214.729 \\nTi ii \\n2.590 −1.750 \\n1 \\n3\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.\",\n \"content\": \"Table 5 .\\n5Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.Name \\nV \\n[mag] \\n\\n∆V \\n[mag] \\n\\nγ \\n[km/s] \\n\\n∆RV \\n[km/s] \\n\\nLC \\nsource a \\n\\nPeriod \\n[days] \\n\\nT 0 − 2 400 000 \\n[days] \\n\\nEphemerides \\nsource b \\n\\nR TrA \\n6.66 ± 0.02 0.56 ± 0.02 -12.44 ± 2.54 28.1 ± 2.2 \\n1 \\n3.3892582 \\n44 423.59377 \\n1 \\nT Vul \\n5.75 ± 0.03 0.64 ± 0.04 \\n0.85 ± 0.11 34.5 ± 3.5 \\n1 \\n4.435462 \\n57 632.18950 \\n0 \\n50 735.20310 \\n2 \\nFF Aql \\n5.19 ± 0.01 0.33 ± 0.03 -11.36 ± 0.10 16.1 ± 1.5 \\n2 \\n4.470916 \\n56 888.66207 \\n0 \\nS Cru \\n6.60 ± 0.02 0.73 ± 0.03 \\n-4.70 ± 0.28 33.8 ± 4.3 \\n1 \\n4.68973 \\n55 279.67331 \\n0 \\nδ Cep \\n3.94 ± 0.01 0.84 ± 0.04 -16.32 ± 0.20 38.9 ± 4.9 \\n1 \\n5.3663 \\n56 917.54900 \\n0 \\nY Sgr \\n5.74 ± 0.02 0.72 ± 0.03 \\n-2.84 ± 3.02 34.2 ± 3.0 \\n1 \\n5.7733866 \\n54 946.78504 \\n1 \\nXX Sgr \\n8.86 ± 0.02 0.87 ± 0.04 \\n12.30 ± 0.46 41.6 ± 1.2 \\n1 \\n6.4243013 \\n54 946.67631 \\n1 \\nη Aql \\n3.89 ± 0.02 0.78 ± 0.03 -14.47 ± 0.63 42.1 ± 2.2 \\n1 \\n7.1768572 \\n50 724.43545 \\n1 \\nS Sge \\n5.61 ± 0.03 0.72 ± 0.03 -16.40 ± 0.25 39.3 ± 4.7 \\n1 \\n8.3445707 \\n57 636.28926 \\n0 \\n48 746.52000 \\n3 \\n8.3823514 \\n50 339.17710 \\n4 \\nV500 Sco \\n8.74 ± 0.01 0.71 ± 0.05 \\n-7.61 ± 3.37 32.2 ± 5.4 \\n1 \\n9.317 \\n44 789.61621 \\n1 \\nβ Dor \\n5.25 ± 0.01 0.63 ± 0.06 \\n8.17 ± 0.11 32.9 ± 2.3 \\n2 \\n9.84308 \\n53 284.79472 \\n0 \\n9.8527371 \\n41 149.27150 \\n4 \\nζ Gem \\n3.89 ± 0.03 0.49 ± 0.03 \\n6.78 ± 0.42 26.5 ± 1.0 \\n1 \\n10.149816 \\n54 805.78476 \\n1 \\nVY Sgr \\n11.46 ± 0.02 1.14 ± 0.05 \\n14.23 ± 1.58 50.4 ± 4.4 \\n1 \\n13.55845 \\n54 171.32722 \\n1 \\nUZ Sct \\n11.25 ± 0.03 0.75 ± 0.05 \\n39.52 ± 0.18 44.7 ± 3.5 \\n0 \\n14.744162 \\n56 171.62381 \\n0 \\nY Oph \\n6.15 ± 0.02 0.49 ± 0.03 \\n-7.56 ± 2.01 16.5 ± 2.0 \\n1 \\n17.12633896 53 209.96732 \\n1 \\nRY Sco \\n8.02 ± 0.02 0.83 ± 0.03 -17.73 ± 2.02 30.1 ± 3.0 \\n1 \\n20.321538 \\n45 077.36644 \\n1 \\nRZ Vel \\n7.09 ± 0.04 1.17 ± 0.05 \\n24.11 ± 4.52 54.6 ± 5.4 \\n1 \\n20.399727 \\n50 776.13608 \\n1 \\nV340 Ara 10.20 ± 0.06 1.08 ± 0.07 -76.25 ± 4.19 47.7 ± 9.3 \\n1 \\n20.811386 \\n56 138.97597 \\n0 \\nWZ Sgr \\n8.03 ± 0.03 1.08 ± 0.05 -17.78 ± 1.17 54.2 ± 2.9 \\n1 \\n21.849708 \\n50 691.28534 \\n1 \\nRS Pup \\n6.99 ± 0.03 1.11 ± 0.04 \\n25.80 ± 1.91 48.5 ± 2.1 \\n1 \\n41.488634 \\n53 092.94483 \\n1\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.\",\n \"content\": \"Table 5 .\\n5Photometric and radial-velocity parameters for our sample of 20 calibrating Cepheids.Name \\nV \\n[mag] \\n\\n∆V \\n[mag] \\n\\nγ \\n[km/s] \\n\\n∆RV \\n[km/s] \\n\\nLC \\nsource a \\n\\nPeriod \\n[days] \\n\\nT 0 − 2 400 000 \\n[days] \\n\\nEphemerides \\nsource b \\n\\nR TrA \\n6.66 ± 0.02 0.56 ± 0.02 -12.44 ± 2.54 28.1 ± 2.2 \\n1 \\n3.3892582 \\n44 423.59377 \\n1 \\nT Vul \\n5.75 ± 0.03 0.64 ± 0.04 \\n0.85 ± 0.11 34.5 ± 3.5 \\n1 \\n4.435462 \\n57 632.18950 \\n0 \\n50 735.20310 \\n2 \\nFF Aql \\n5.19 ± 0.01 0.33 ± 0.03 -11.36 ± 0.10 16.1 ± 1.5 \\n2 \\n4.470916 \\n56 888.66207 \\n0 \\nS Cru \\n6.60 ± 0.02 0.73 ± 0.03 \\n-4.70 ± 0.28 33.8 ± 4.3 \\n1 \\n4.68973 \\n55 279.67331 \\n0 \\nδ Cep \\n3.94 ± 0.01 0.84 ± 0.04 -16.32 ± 0.20 38.9 ± 4.9 \\n1 \\n5.3663 \\n56 917.54900 \\n0 \\nY Sgr \\n5.74 ± 0.02 0.72 ± 0.03 \\n-2.84 ± 3.02 34.2 ± 3.0 \\n1 \\n5.7733866 \\n54 946.78504 \\n1 \\nXX Sgr \\n8.86 ± 0.02 0.87 ± 0.04 \\n12.30 ± 0.46 41.6 ± 1.2 \\n1 \\n6.4243013 \\n54 946.67631 \\n1 \\nη Aql \\n3.89 ± 0.02 0.78 ± 0.03 -14.47 ± 0.63 42.1 ± 2.2 \\n1 \\n7.1768572 \\n50 724.43545 \\n1 \\nS Sge \\n5.61 ± 0.03 0.72 ± 0.03 -16.40 ± 0.25 39.3 ± 4.7 \\n1 \\n8.3445707 \\n57 636.28926 \\n0 \\n48 746.52000 \\n3 \\n8.3823514 \\n50 339.17710 \\n4 \\nV500 Sco \\n8.74 ± 0.01 0.71 ± 0.05 \\n-7.61 ± 3.37 32.2 ± 5.4 \\n1 \\n9.317 \\n44 789.61621 \\n1 \\nβ Dor \\n5.25 ± 0.01 0.63 ± 0.06 \\n8.17 ± 0.11 32.9 ± 2.3 \\n2 \\n9.84308 \\n53 284.79472 \\n0 \\n9.8527371 \\n41 149.27150 \\n4 \\nζ Gem \\n3.89 ± 0.03 0.49 ± 0.03 \\n6.78 ± 0.42 26.5 ± 1.0 \\n1 \\n10.149816 \\n54 805.78476 \\n1 \\nVY Sgr \\n11.46 ± 0.02 1.14 ± 0.05 \\n14.23 ± 1.58 50.4 ± 4.4 \\n1 \\n13.55845 \\n54 171.32722 \\n1 \\nUZ Sct \\n11.25 ± 0.03 0.75 ± 0.05 \\n39.52 ± 0.18 44.7 ± 3.5 \\n0 \\n14.744162 \\n56 171.62381 \\n0 \\nY Oph \\n6.15 ± 0.02 0.49 ± 0.03 \\n-7.56 ± 2.01 16.5 ± 2.0 \\n1 \\n17.12633896 53 209.96732 \\n1 \\nRY Sco \\n8.02 ± 0.02 0.83 ± 0.03 -17.73 ± 2.02 30.1 ± 3.0 \\n1 \\n20.321538 \\n45 077.36644 \\n1 \\nRZ Vel \\n7.09 ± 0.04 1.17 ± 0.05 \\n24.11 ± 4.52 54.6 ± 5.4 \\n1 \\n20.399727 \\n50 776.13608 \\n1 \\nV340 Ara 10.20 ± 0.06 1.08 ± 0.07 -76.25 ± 4.19 47.7 ± 9.3 \\n1 \\n20.811386 \\n56 138.97597 \\n0 \\nWZ Sgr \\n8.03 ± 0.03 1.08 ± 0.05 -17.78 ± 1.17 54.2 ± 2.9 \\n1 \\n21.849708 \\n50 691.28534 \\n1 \\nRS Pup \\n6.99 ± 0.03 1.11 ± 0.04 \\n25.80 ± 1.91 48.5 ± 2.1 \\n1 \\n41.488634 \\n53 092.94483 \\n1\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Mean atmospheric parameters derived for the 20 calibrating Cepheids.\",\n \"content\": \"Table 6 .\\n6Mean atmospheric parameters derived for the 20 calibrating Cepheids.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Mean atmospheric parameters derived for the 20 calibrating Cepheids.\",\n \"content\": \"Table 6 .\\n6Mean atmospheric parameters derived for the 20 calibrating Cepheids.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.\",\n \"content\": \"Table 7 .\\n7Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.3 < P < 5 \\n[days] \\n\\n5 < P < 7 \\n[days] \\n\\n7 < P < 9.5 \\n[days] \\n\\n9.5 < P < 10.5 \\n[days] \\n\\nc 0 \\n0.44 ± 0.21 0.44 ± 0.23 \\n0.48 ± 0.26 0.47 ± 0.12 \\nc 1 \\n2.1 ± 0.5 \\n2.1 ± 0.5 \\n−4.4 ± 0.6 \\n−4.4 ± 0.2 \\nc 2 \\n0.13 ± 0.22 0.15 ± 0.24 \\n0.14 ± 0.28 0.04 ± 0.12 \\nc 3 −4.5 ± 1.4 \\n2.2 ± 1.4 \\n2.6 ± 2.0 \\n4.2 ± 2.9 \\nc 4 \\n0.06 ± 0.21 0.06 ± 0.23 \\n0.05 ± 0.27 0.04 ± 0.11 \\nc 5 \\n1.4 ± 3.6 \\n2.6 ± 3.7 \\n2.2 ± 5.6 \\n4.4 ± 2.7 \\nc 6 \\n0.03 ± 0.21 0.03 ± 0.22 \\n0.03 ± 0.28 0.02 ± 0.11 \\nc 7 \\n1.3 ± 8.3 \\n2.7 ± 7.6 \\n−0.93 ± 9.56 \\n4.2 ± 6.9 \\nc 8 \\n0.01 ± 0.19 \\n... \\n... \\n... \\nc 9 \\n2.8 ± 16.7 \\n... \\n... \\n...\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.\",\n \"content\": \"Table 7 .\\n7Coefficients of the Fourier series fitted to the data shown inFig. 5for different bins of pulsation period.Notes. The Fourier series are of the form c 2i cos [2(i + 1)πx + c 2i+1 ] for i = 0, 1, . . . , N − 1, where N is the number of terms.3 < P < 5 \\n[days] \\n\\n5 < P < 7 \\n[days] \\n\\n7 < P < 9.5 \\n[days] \\n\\n9.5 < P < 10.5 \\n[days] \\n\\nc 0 \\n0.44 ± 0.21 0.44 ± 0.23 \\n0.48 ± 0.26 0.47 ± 0.12 \\nc 1 \\n2.1 ± 0.5 \\n2.1 ± 0.5 \\n−4.4 ± 0.6 \\n−4.4 ± 0.2 \\nc 2 \\n0.13 ± 0.22 0.15 ± 0.24 \\n0.14 ± 0.28 0.04 ± 0.12 \\nc 3 −4.5 ± 1.4 \\n2.2 ± 1.4 \\n2.6 ± 2.0 \\n4.2 ± 2.9 \\nc 4 \\n0.06 ± 0.21 0.06 ± 0.23 \\n0.05 ± 0.27 0.04 ± 0.11 \\nc 5 \\n1.4 ± 3.6 \\n2.6 ± 3.7 \\n2.2 ± 5.6 \\n4.4 ± 2.7 \\nc 6 \\n0.03 ± 0.21 0.03 ± 0.22 \\n0.03 ± 0.28 0.02 ± 0.11 \\nc 7 \\n1.3 ± 8.3 \\n2.7 ± 7.6 \\n−0.93 ± 9.56 \\n4.2 ± 6.9 \\nc 8 \\n0.01 ± 0.19 \\n... \\n... \\n... \\nc 9 \\n2.8 ± 16.7 \\n... \\n... \\n...\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.\",\n \"content\": \"Table 8 .\\n8Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.Name \\nDataset \\nMJD \\n[d] \\n[Mg/H] ± σ N Mg ... \\nTi i ± σ \\nN Ti i \\nTi ii ± σ \\nN Ti ii \\n[Ti/H] ± σ \\n\\nR TrA \\nHARPS 53150.1453265 0.20 ± 0.11 \\n1 \\n... −0.21 ± 0.15 \\n8 \\n0.05 ± 0.10 \\n2 \\n−0.12 ± 0.19 \\nR TrA \\nHARPS 53150.1353086 0.17 ± 0.11 \\n1 \\n... −0.17 ± 0.27 \\n12 \\n0.01 ± 0.10 \\n3 \\n−0.14 ± 0.25 \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\nRS Pup STELLA 57708.2167938 0.24 ± 0.11 \\n2 \\n... −0.06 ± 0.28 \\n15 \\n0.22 ± 0.10 \\n4 \\n0.09 ± 0.19 \\nRS Pup STELLA 57713.2628277 \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n...\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.\",\n \"content\": \"Table 8 .\\n8Excerpt from the list of abundances of α elements for each spectrum of the 20 calibrating Cepheids.Name \\nDataset \\nMJD \\n[d] \\n[Mg/H] ± σ N Mg ... \\nTi i ± σ \\nN Ti i \\nTi ii ± σ \\nN Ti ii \\n[Ti/H] ± σ \\n\\nR TrA \\nHARPS 53150.1453265 0.20 ± 0.11 \\n1 \\n... −0.21 ± 0.15 \\n8 \\n0.05 ± 0.10 \\n2 \\n−0.12 ± 0.19 \\nR TrA \\nHARPS 53150.1353086 0.17 ± 0.11 \\n1 \\n... −0.17 ± 0.27 \\n12 \\n0.01 ± 0.10 \\n3 \\n−0.14 ± 0.25 \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\nRS Pup STELLA 57708.2167938 0.24 ± 0.11 \\n2 \\n... −0.06 ± 0.28 \\n15 \\n0.22 ± 0.10 \\n4 \\n0.09 ± 0.19 \\nRS Pup STELLA 57713.2628277 \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n...\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.\",\n \"content\": \"Table 9 .\\n9Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.Name \\n[Mg/H] ± σ (std) N spec ... [Ti i/H] ± σ \\n[Ti ii/H] ± σ \\n[Ti/H] ± σ (std) \\nN spec \\n\\nR TrA \\n0.10 ± 0.03 (0.08) \\n15 \\n... −0.19 ± 0.04 −0.09 ± 0.03 −0.18 ± 0.04 (0.10) \\n15 \\nT Vul \\n0.07 ± 0.03 (0.15) \\n26 \\n... −0.11 ± 0.03 −0.07 ± 0.02 −0.10 ± 0.03 (0.10) \\n26 \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\nWZ Sgr 0.12 ± 0.06 (0.03) \\n4 \\n... \\n0.08 ± 0.07 \\n0.01 ± 0.05 \\n0.06 ± 0.07 (0.12) \\n7 \\nRS Pup 0.09 ± 0.03 (0.14) \\n10 \\n... \\n0.02 ± 0.04 \\n0.16 ± 0.03 \\n0.04 ± 0.04 (0.14) \\n16\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.\",\n \"content\": \"Table 9 .\\n9Excerpt from the list of mean abundances of α elements derived for the 20 calibrating Cepheids.Name \\n[Mg/H] ± σ (std) N spec ... [Ti i/H] ± σ \\n[Ti ii/H] ± σ \\n[Ti/H] ± σ (std) \\nN spec \\n\\nR TrA \\n0.10 ± 0.03 (0.08) \\n15 \\n... −0.19 ± 0.04 −0.09 ± 0.03 −0.18 ± 0.04 (0.10) \\n15 \\nT Vul \\n0.07 ± 0.03 (0.15) \\n26 \\n... −0.11 ± 0.03 −0.07 ± 0.02 −0.10 ± 0.03 (0.10) \\n26 \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\n... \\nWZ Sgr 0.12 ± 0.06 (0.03) \\n4 \\n... \\n0.08 ± 0.07 \\n0.01 ± 0.05 \\n0.06 ± 0.07 (0.12) \\n7 \\nRS Pup 0.09 ± 0.03 (0.14) \\n10 \\n... \\n0.02 ± 0.04 \\n0.16 ± 0.03 \\n0.04 ± 0.04 (0.14) \\n16\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).\",\n \"content\": \"Table 10 .\\n10Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).Abundance \\nratio \\nDatasets 1 \\nZero-point \\ndifference \\nN Common \\n\\n[Fe/H] \\nP18−TS \\n0.07 ± 0.06 \\n13 \\n[Fe/H] \\nLII−TS \\n0.05 ± 0.07 \\n20 \\n[Fe/H] \\nLIII−TS \\n0.11 ± 0.08 \\n6 \\n[Fe/H] \\nLII−LEM \\n0.07 ± 0.12 \\n55 \\n[Fe/H] \\nYON−LIII −0.33 ± 0.15 \\n21 \\n[Mg/H] \\nLII−TS \\n−0.14 ± 0.11 \\n16 \\n[Mg/H] \\nLIII−TS \\n0.11 ± 0.16 \\n5 \\n[Mg/H] \\nLII−LEM −0.26 ± 0.24 \\n35 \\n[Mg/H] \\nYON−LIII −0.09 ± 0.14 \\n16 \\n[Si/H] \\nLII−TS \\n−0.13 ± 0.07 \\n20 \\n[Si/H] \\nLIII−TS \\n−0.01 ± 0.09 \\n6 \\n[Si/H] \\nLII−LEM −0.06 ± 0.11 \\n55 \\n[Si/H] \\nYON−LIII −0.12 ± 0.07 \\n20 \\n[S/H] \\nLII−TS \\n0.11 ± 0.13 \\n17 \\n[S/H] \\nLII−LEM −0.13 ± 0.19 \\n45 \\n[Ca/H] \\nLII−TS \\n−0.07 ± 0.10 \\n20 \\n[Ca/H] \\nLIII−TS \\n−0.01 ± 0.10 \\n6 \\n[Ca/H] \\nLII−LEM −0.06 ± 0.17 \\n55 \\n[Ca/H] \\nYON−LIII −0.16 ± 0.11 \\n21 \\n[Ti/H] \\nLII−TS \\n0.12 ± 0.09 \\n20 \\n[Ti/H] \\nLIII−TS \\n0.17 ± 0.12 \\n6 \\n[Ti/H] \\nYON−LIII \\n0.33 ± 0.20 \\n20\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).\",\n \"content\": \"Table 10 .\\n10Mean abundance differences for stars in common between the current sample of calibrating Cepheids and other datasets.Notes. (1) TS: this study; P18: Proxauf et al. (2018); LII: Luck et al. (2011); LIII: Luck & Lambert (2011); LEM: Lemasle et al. (2013); YON: Yong et al. (2006).Abundance \\nratio \\nDatasets 1 \\nZero-point \\ndifference \\nN Common \\n\\n[Fe/H] \\nP18−TS \\n0.07 ± 0.06 \\n13 \\n[Fe/H] \\nLII−TS \\n0.05 ± 0.07 \\n20 \\n[Fe/H] \\nLIII−TS \\n0.11 ± 0.08 \\n6 \\n[Fe/H] \\nLII−LEM \\n0.07 ± 0.12 \\n55 \\n[Fe/H] \\nYON−LIII −0.33 ± 0.15 \\n21 \\n[Mg/H] \\nLII−TS \\n−0.14 ± 0.11 \\n16 \\n[Mg/H] \\nLIII−TS \\n0.11 ± 0.16 \\n5 \\n[Mg/H] \\nLII−LEM −0.26 ± 0.24 \\n35 \\n[Mg/H] \\nYON−LIII −0.09 ± 0.14 \\n16 \\n[Si/H] \\nLII−TS \\n−0.13 ± 0.07 \\n20 \\n[Si/H] \\nLIII−TS \\n−0.01 ± 0.09 \\n6 \\n[Si/H] \\nLII−LEM −0.06 ± 0.11 \\n55 \\n[Si/H] \\nYON−LIII −0.12 ± 0.07 \\n20 \\n[S/H] \\nLII−TS \\n0.11 ± 0.13 \\n17 \\n[S/H] \\nLII−LEM −0.13 ± 0.19 \\n45 \\n[Ca/H] \\nLII−TS \\n−0.07 ± 0.10 \\n20 \\n[Ca/H] \\nLIII−TS \\n−0.01 ± 0.10 \\n6 \\n[Ca/H] \\nLII−LEM −0.06 ± 0.17 \\n55 \\n[Ca/H] \\nYON−LIII −0.16 ± 0.11 \\n21 \\n[Ti/H] \\nLII−TS \\n0.12 ± 0.09 \\n20 \\n[Ti/H] \\nLIII−TS \\n0.17 ± 0.12 \\n6 \\n[Ti/H] \\nYON−LIII \\n0.33 ± 0.20 \\n20\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". S M Andrievsky, R E Luck, V V Kovtyukh, AJ. 1301880Andrievsky, S. M., Luck, R. E., & Kovtyukh, V. V. 2005, AJ, 130, 1880","doi":""},{"text":". M Asplund, A M Amarsi, N Grevesse, A&A. 653141Asplund, M., Amarsi, A. M., & Grevesse, N. 2021, A&A, 653, A141","doi":""},{"text":". M Asplund, N Grevesse, A J Sauval, P Scott, ARA&A. 47481Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481","doi":""},{"text":". M T Belmonte, J C Pickering, M P Ruffoni, ApJ. 848125Belmonte, M. T., Pickering, J. C., Ruffoni, M. P., et al. 2017, ApJ, 848, 125","doi":""},{"text":". G Bono, F Caputo, S Cassisi, ApJ. 543955Bono, G., Caputo, F., Cassisi, S., et al. 2000a, ApJ, 543, 955","doi":""},{"text":". G Bono, F Caputo, M Marconi, I Musella, ApJ. 715277Bono, G., Caputo, F., Marconi, M., & Musella, I. 2010, ApJ, 715, 277","doi":""},{"text":". G Bono, V Castellani, M Marconi, ApJ. 56583Bono, G., Castellani, V., & Marconi, M. 2002, ApJ, 565, L83","doi":""},{"text":". G Bono, M Marconi, R F Stellingwerf, ApJS. 122167Bono, G., Marconi, M., & Stellingwerf, R. F. 1999, ApJS, 122, 167","doi":""},{"text":". G Bono, M Marconi, R F Stellingwerf, A&A. 360245Bono, G., Marconi, M., & Stellingwerf, R. F. 2000b, A&A, 360, 245","doi":""},{"text":". V F Braga, J Crestani, M Fabrizio, ApJ. 91985Braga, V. F., Crestani, J., Fabrizio, M., et al. 2021, ApJ, 919, 85","doi":""},{"text":". V F Braga, P B Stetson, G Bono, AJ. 152170Braga, V. F., Stetson, P. B., Bono, G., et al. 2016, AJ, 152, 170","doi":""},{"text":". L Breuval, P Kervella, R I Anderson, A&A. 643115Breuval, L., Kervella, P., Anderson, R. I., et al. 2020, A&A, 643, A115","doi":""},{"text":". L Breuval, P Kervella, P Wielgórski, ApJ. 91338Breuval, L., Kervella, P., Wielgórski, P., et al. 2021, ApJ, 913, 38","doi":""},{"text":". E Caffau, S Andrievsky, S Korotin, A&A. 58516Caffau, E., Andrievsky, S., Korotin, S., et al. 2016, A&A, 585, A16","doi":""},{"text":". E Caffau, P Bonifacio, R Faraggiana, A&A. 441533Caffau, E., Bonifacio, P., Faraggiana, R., et al. 2005, A&A, 441, 533","doi":""},{"text":". E Caffau, R Faraggiana, P Bonifacio, H G Ludwig, M Steffen, A&A. 470699Caffau, E., Faraggiana, R., Bonifacio, P., Ludwig, H. G., & Steffen, M. 2007, A&A, 470, 699","doi":""},{"text":"ArXiv Astrophysics e-prints. F Castelli, R L Kurucz, astro-ph/0405087Castelli, F. & Kurucz, R. L. 2004, ArXiv Astrophysics e-prints [astro-ph/0405087]","doi":"astro-ph/0405087"},{"text":". G Cescutti, F Matteucci, P François, C Chiappini, A&A. 462943Cescutti, G., Matteucci, F., François, P., & Chiappini, C. 2007, A&A, 462, 943","doi":""},{"text":". C Chiappini, F Matteucci, D Romano, ApJ. 5541044Chiappini, C., Matteucci, F., & Romano, D. 2001, ApJ, 554, 1044","doi":""},{"text":". J Crestani, V F Braga, M Fabrizio, ApJ. 91410125A&ACrestani, J., Braga, V. F., Fabrizio, M., et al. 2021, ApJ, 914, 10 da Silva, R., Lemasle, B., Bono, G., et al. 2016, A&A, 586, A125","doi":""},{"text":". G De Somma, M Marconi, S Cassisi, MNRAS. 5081473De Somma, G., Marconi, M., Cassisi, S., et al. 2021, MNRAS, 508, 1473","doi":""},{"text":"Optical and IR Telescope Instrumentation and Detectors. H Dekker, S D'odorico, A Kaufer, B Delabre, H Kotzlowski, Proc. SPIE. M. Iye & A. F. MoorwoodSPIE4008Dekker, H., D'Odorico, S., Kaufer, A., Delabre, B., & Kotzlowski, H. 2000, in Proc. SPIE, Vol. 4008, Optical and IR Telescope Instrumentation and Detec- tors, ed. M. Iye & A. F. Moorwood, 534-545","doi":""},{"text":". E A Den Hartog, J E Lawler, C Sneden, J J Cowan, A Brukhovesky, ApJS. 24333Den Hartog, E. A., Lawler, J. E., Sneden, C., Cowan, J. J., & Brukhovesky, A. 2019, ApJS, 243, 33","doi":""},{"text":". E A Den Hartog, M P Ruffoni, J E Lawler, ApJS. 21523Den Hartog, E. A., Ruffoni, M. P., Lawler, J. E., et al. 2014, ApJS, 215, 23","doi":""},{"text":". M Dessauges-Zavadsky, F Calura, J X Prochaska, S D'odorico, F Matteucci, A&A. 470431Dessauges-Zavadsky, M., Calura, F., Prochaska, J. X., D'Odorico, S., & Mat- teucci, F. 2007, A&A, 470, 431","doi":""},{"text":". V D'orazi, S Desidera, R G Gratton, A&A. 59819D'Orazi, V., Desidera, S., Gratton, R. G., et al. 2017, A&A, 598, A19","doi":""},{"text":". V D'orazi, E Oliva, A Bragaglia, A&A. 63338D'Orazi, V., Oliva, E., Bragaglia, A., et al. 2020, A&A, 633, A38","doi":""},{"text":". S Duffau, E Caffau, L Sbordone, A&A. 604128Duffau, S., Caffau, E., Sbordone, L., et al. 2017, A&A, 604, A128","doi":""},{"text":". N R Evans, AJ. 104216Evans, N. R. 1992, AJ, 104, 216","doi":""},{"text":". N R Evans, D L Welch, M H Slovak, Thomas G Barnes, I Moffett, T J , AJ. 1061599Evans, N. R., Welch, D. L., Slovak, M. H., Barnes, Thomas G., I., & Moffett, T. J. 1993, AJ, 106, 1599","doi":""},{"text":". B.-Q For, C Sneden, AJ. 1401694For, B.-Q. & Sneden, C. 2010, AJ, 140, 1694","doi":""},{"text":". W L Freedman, B F Madore, T Hoyt, ApJ. 89157Freedman, W. L., Madore, B. F., Hoyt, T., et al. 2020, ApJ, 891, 57","doi":""},{"text":". K Fukue, N Matsunaga, R Yamamoto, ApJ. 81264Fukue, K., Matsunaga, N., Yamamoto, R., et al. 2015, ApJ, 812, 64","doi":""},{"text":". K Genovali, B Lemasle, G Bono, A&A. 56637Genovali, K., Lemasle, B., Bono, G., et al. 2014, A&A, 566, A37","doi":""},{"text":". K Genovali, B Lemasle, G Bono, A&A. 554132Genovali, K., Lemasle, B., Bono, G., et al. 2013, A&A, 554, A132","doi":""},{"text":". K Genovali, B Lemasle, R Da Silva, A&A. 58017Genovali, K., Lemasle, B., da Silva, R., et al. 2015, A&A, 580, A17","doi":""},{"text":". G Gilmore, S Randich, M Asplund, The Messenger. 14725Gilmore, G., Randich, S., Asplund, M., et al. 2012, The Messenger, 147, 25","doi":""},{"text":". M A T Groenewegen, A&A. 48825Groenewegen, M. A. T. 2008, A&A, 488, 25","doi":""},{"text":". M A Groenewegen, A&A. 6198Groenewegen, M. A. T. 2018, A&A, 619, A8","doi":""},{"text":". M A T Groenewegen, A&A. 640113Groenewegen, M. A. T. 2020, A&A, 640, A113","doi":""},{"text":". L Inno, N Matsunaga, G Bono, ApJ. 76484Inno, L., Matsunaga, N., Bono, G., et al. 2013, ApJ, 764, 84","doi":""},{"text":". L Inno, N Matsunaga, M Romaniello, A&A. 57630Inno, L., Matsunaga, N., Romaniello, M., et al. 2015, A&A, 576, A30","doi":""},{"text":". L Inno, M A Urbaneja, N Matsunaga, MNRAS. 48283Inno, L., Urbaneja, M. A., Matsunaga, N., et al. 2019, MNRAS, 482, 83","doi":""},{"text":". M Jian, D Taniguchi, N Matsunaga, MNRAS. 4941724Jian, M., Taniguchi, D., Matsunaga, N., et al. 2020, MNRAS, 494, 1724","doi":""},{"text":". A Kaufer, O Stahl, S Tubbesing, The Messenger. 958Kaufer, A., Stahl, O., Tubbesing, S., et al. 1999, The Messenger, 95, 8","doi":""},{"text":". C S Kochanek, B J Shappee, K Z Stanek, PASP. 129104502Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, PASP, 129, 104502","doi":""},{"text":". V V Kovtyukh, MNRAS. 378617Kovtyukh, V. V. 2007, MNRAS, 378, 617","doi":""},{"text":". V V Kovtyukh, S M Andrievsky, S I Belik, R E Luck, AJ. 129433Kovtyukh, V. V., Andrievsky, S. M., Belik, S. I., & Luck, R. E. 2005, AJ, 129, 433","doi":""},{"text":"A Kramida, R Yu, J Reader, & the NIST ASD Team. 2020, NIST Atomic Spectra Database. version 5.8Kramida, A., Yu., R., Reader, J., & the NIST ASD Team. 2020, NIST Atomic Spectra Database, version 5.8","doi":""},{"text":"R L Kurucz, I Furenlid, J Brault, L Testerman, Solar flux atlas from 296 to 1300 nm. Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, Solar flux atlas from 296 to 1300 nm","doi":""},{"text":". J E Lawler, A Guzman, M P Wood, C Sneden, J J Cowan, ApJS. 20511Lawler, J. E., Guzman, A., Wood, M. P., Sneden, C., & Cowan, J. J. 2013, ApJS, 205, 11","doi":""},{"text":". B Lemasle, P François, K Genovali, A&A. 55831Lemasle, B., François, P., Genovali, K., et al. 2013, A&A, 558, A31","doi":""},{"text":". B Lemasle, M A T Groenewegen, E K Grebel, A&A. 60885Lemasle, B., Groenewegen, M. A. T., Grebel, E. K., et al. 2017, A&A, 608, A85","doi":""},{"text":". B Lemasle, M Hanke, J Storm, G Bono, E K Grebel, A&A. 64171Lemasle, B., Hanke, M., Storm, J., Bono, G., & Grebel, E. K. 2020, A&A, 641, A71","doi":""},{"text":". M Limongi, A Chieffi, ApJ. 592404Limongi, M. & Chieffi, A. 2003, ApJ, 592, 404","doi":""},{"text":". G Liu, N Arav, D S N Rupke, ApJS. 2219Liu, G., Arav, N., & Rupke, D. S. N. 2015, ApJS, 221, 9","doi":""},{"text":". R E Luck, S M Andrievsky, AJ. 128343Luck, R. E. & Andrievsky, S. M. 2004, AJ, 128, 343","doi":""},{"text":". R E Luck, S M Andrievsky, A Fokin, V V Kovtyukh, AJ. 13698Luck, R. E., Andrievsky, S. M., Fokin, A., & Kovtyukh, V. V. 2008, AJ, 136, 98","doi":""},{"text":". R E Luck, S M Andrievsky, V V Kovtyukh, W Gieren, D Graczyk, AJ. 14251Luck, R. E., Andrievsky, S. M., Kovtyukh, V. V., Gieren, W., & Graczyk, D. 2011, AJ, 142, 51","doi":""},{"text":". R E Luck, D L Lambert, AJ. 142136Luck, R. E. & Lambert, D. L. 2011, AJ, 142, 136","doi":""},{"text":". D Magurno, C Sneden, V F Braga, ApJ. 86457Magurno, D., Sneden, C., Braga, V. F., et al. 2018, ApJ, 864, 57","doi":""},{"text":". M Marconi, R Molinaro, G Bono, ApJ. 7686Marconi, M., Molinaro, R., Bono, G., et al. 2013, ApJ, 768, L6","doi":""},{"text":". M Marconi, I Musella, G Fiorentino, ApJ. 632590Marconi, M., Musella, I., & Fiorentino, G. 2005, ApJ, 632, 590","doi":""},{"text":". N Matsunaga, M Jian, D Taniguchi, S S Elgueta, MNRAS. 5061031Matsunaga, N., Jian, M., Taniguchi, D., & Elgueta, S. S. 2021, MNRAS, 506, 1031","doi":""},{"text":". M Mayor, F Pepe, D Queloz, The Messenger. 11420Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20","doi":""},{"text":". A Mcwilliam, G Wallerstein, M Mottini, ApJ. 778149McWilliam, A., Wallerstein, G., & Mottini, M. 2013, ApJ, 778, 149","doi":""},{"text":". J Meléndez, B Barbuy, A&A. 497611Meléndez, J. & Barbuy, B. 2009, A&A, 497, 611","doi":""},{"text":"A&A proofs: manuscript no. daSilvaetal2021. A&A proofs: manuscript no. daSilvaetal2021","doi":""},{"text":". J H Minniti, L Sbordone, A Rojas-Arriagada, A&A. 64092Minniti, J. H., Sbordone, L., Rojas-Arriagada, A., et al. 2020, A&A, 640, A92","doi":""},{"text":". J H Minniti, M Zoccali, A Rojas-Arriagada, A&A. 654138Minniti, J. H., Zoccali, M., Rojas-Arriagada, A., et al. 2021, A&A, 654, A138","doi":""},{"text":"The Solar Spectrum 2935 Å to 8770 Å. C E Moore, M Minnaert, J Houtgast, Moore, C. E., Minnaert, M., & Houtgast, J. 1966, The Solar Spectrum 2935 Å to 8770 Å","doi":""},{"text":". N Nardetto, J H Groh, S Kraus, F Millour, D Gillet, A&A. 4891263Nardetto, N., Groh, J. H., Kraus, S., Millour, F., & Gillet, D. 2008, A&A, 489, 1263","doi":""},{"text":". N Nardetto, D Mourard, P Kervella, A&A. 453309Nardetto, N., Mourard, D., Kervella, P., et al. 2006, A&A, 453, 309","doi":""},{"text":". N Nardetto, E Poretti, A Gallenne, A&A. 61692Nardetto, N., Poretti, E., Gallenne, A., et al. 2018, A&A, 616, A92","doi":""},{"text":". G Natale, M Marconi, G Bono, ApJ. 67493Natale, G., Marconi, M., & Bono, G. 2008, ApJ, 674, L93","doi":""},{"text":". J R Neeley, M Monelli, M Marengo, ApJ. 920152Neeley, J. R., Monelli, M., Marengo, M., et al. 2021, ApJ, 920, 152","doi":""},{"text":". H R Neilson, M Cantiello, N Langer, A&A. 5299Neilson, H. R., Cantiello, M., & Langer, N. 2011, A&A, 529, L9","doi":""},{"text":". F Niedermann, M S Sloth, Phys. Rev. D. 10263527Niedermann, F. & Sloth, M. S. 2020, Phys. Rev. D, 102, 063527","doi":""},{"text":". P E Nissen, Y Q Chen, M Asplund, M Pettini, A&A. 415993Nissen, P. E., Chen, Y. Q., Asplund, M., & Pettini, M. 2004, A&A, 415, 993","doi":""},{"text":". T R O'brian, M E Wickliffe, J E Lawler, W Whaling, J W Brault, Journal of the Optical Society of America B Optical Physics. 81185O'Brian, T. R., Wickliffe, M. E., Lawler, J. E., Whaling, W., & Brault, J. W. 1991, Journal of the Optical Society of America B Optical Physics, 8, 1185","doi":""},{"text":". J W Pel, A&AS. 24413Pel, J. W. 1976, A&AS, 24, 413","doi":""},{"text":". G Pietrzyński, D Graczyk, W Gieren, Nature. 49576Pietrzyński, G., Graczyk, D., Gieren, W., et al. 2013, Nature, 495, 76","doi":""},{"text":". N Aghanim, Planck CollaborationY Akrami, Planck CollaborationA&A. 6416Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6","doi":""},{"text":". P G Prada Moroni, M Gennaro, G Bono, ApJ. 749108Prada Moroni, P. G., Gennaro, M., Bono, G., et al. 2012, ApJ, 749, 108","doi":""},{"text":". B Proxauf, R Da Silva, V V Kovtyukh, A&A. 61682Proxauf, B., da Silva, R., Kovtyukh, V. V., et al. 2018, A&A, 616, A82","doi":""},{"text":". S Randich, G Gilmore, -Eso Gaia, Consortium, The Messenger. 15447Randich, S., Gilmore, G., & Gaia-ESO Consortium. 2013, The Messenger, 154, 47","doi":""},{"text":". A G Riess, S Casertano, W Yuan, L M Macri, D Scolnic, ApJ. 87685Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ, 876, 85","doi":""},{"text":". V Ripepi, G Catanzaro, G Clementini, arXiv:2201.01126arXiv e-printsRipepi, V., Catanzaro, G., Clementini, G., et al. 2022, arXiv e-prints, arXiv:2201.01126","doi":"arXiv:2201.01126"},{"text":". V Ripepi, G Catanzaro, R Molinaro, MNRAS. 5084047Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2021, MNRAS, 508, 4047","doi":""},{"text":". V Ripepi, G Catanzaro, R Molinaro, A&A. 642230Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2020, A&A, 642, A230","doi":""},{"text":". V Ripepi, R Molinaro, I Musella, A&A. 62514Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14","doi":""},{"text":". M Romaniello, F Primas, M Mottini, A&A. 488731Romaniello, M., Primas, F., Mottini, M., et al. 2008, A&A, 488, 731","doi":""},{"text":". M Romaniello, A Riess, S Mancino, arXiv:2110.08860arXiv e-printsRomaniello, M., Riess, A., Mancino, S., et al. 2021, arXiv e-prints, arXiv:2110.08860","doi":"arXiv:2110.08860"},{"text":". M P Ruffoni, E A Hartog, J E Lawler, MNRAS. 4413127Ruffoni, M. P., Den Hartog, E. A., Lawler, J. E., et al. 2014, MNRAS, 441, 3127","doi":""},{"text":". J D Scargle, ApJ. 263835Scargle, J. D. 1982, ApJ, 263, 835","doi":""},{"text":". B J Shappee, J L Prieto, D Grupe, ApJ. 78848Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48","doi":""},{"text":". D M Skowron, J Skowron, P Mróz, Acta Astron. 69305Skowron, D. M., Skowron, J., Mróz, P., et al. 2019, Acta Astron., 69, 305","doi":""},{"text":"C Sneden, The MOOG code. Sneden, C. 2002, The MOOG code, http://www.as.utexas.edu/~chris/ moog.html","doi":""},{"text":". S G Sousa, N C Santos, V Adibekyan, E Delgado-Mena, G Israelian, A&A. 57767Sousa, S. G., Santos, N. C., Adibekyan, V., Delgado-Mena, E., & Israelian, G. 2015, A&A, 577, A67","doi":""},{"text":". S G Sousa, N C Santos, G Israelian, M Mayor, M J P F G Monteiro, A&A. 469783Sousa, S. G., Santos, N. C., Israelian, G., Mayor, M., & Monteiro, M. J. P. F. G. 2007, A&A, 469, 783","doi":""},{"text":". K G Strassmeier, T Granzer, M Weber, Astronomische Nachrichten. 325527Strassmeier, K. G., Granzer, T., Weber, M., et al. 2004, Astronomische Nachrichten, 325, 527","doi":""},{"text":". K G Strassmeier, T Granzer, M Weber, Advances in Astronomy. 970306Strassmeier, K. G., Granzer, T., Weber, M., et al. 2010, Advances in Astronomy, 2010, 970306","doi":""},{"text":". Y Takeda, M Omiya, H Harakawa, B Sato, PASJ. 6881Takeda, Y., Omiya, M., Harakawa, H., & Sato, B. 2016, PASJ, 68, 81","doi":""},{"text":". D Taniguchi, N Matsunaga, M Jian, MNRAS. 5024210Taniguchi, D., Matsunaga, N., Jian, M., et al. 2021, MNRAS, 502, 4210","doi":""},{"text":". B J Taylor, PASP. 106452Taylor, B. J. 1994, PASP, 106, 452","doi":""},{"text":". V Vasilyev, H G Ludwig, B Freytag, B Lemasle, M Marconi, A&A. 61119Vasilyev, V., Ludwig, H. G., Freytag, B., Lemasle, B., & Marconi, M. 2018, A&A, 611, A19","doi":""},{"text":". K A Venn, M D Shetrone, M J Irwin, ApJ. 751102Venn, K. A., Shetrone, M. D., Irwin, M. J., et al. 2012, ApJ, 751, 102","doi":""},{"text":"M Weber, T Granzer, K G Strassmeier, Proc. SPIE. SPIE845184510Weber, M., Granzer, T., & Strassmeier, K. G. 2012, in Proc. SPIE, Vol. 8451, Software and Cyberinfrastructure for Astronomy II, 84510K","doi":""},{"text":". M P Wood, J E Lawler, C Sneden, J J Cowan, ApJS. 20827Wood, M. P., Lawler, J. E., Sneden, C., & Cowan, J. J. 2013, ApJS, 208, 27","doi":""},{"text":". D Yong, B W Carney, M L Teixera De Almeida, B L Pohl, AJ. 1312256Yong, D., Carney, B. W., Teixera de Almeida, M. L., & Pohl, B. L. 2006, AJ, 131, 2256","doi":""},{"text":". Inaf -Osservatorio Astronomico Di Padova, 535122Padova, Italy4 INAF -Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy","doi":""},{"text":"Zentrum für Astronomie der Universität Heidelberg. Heidelberg, Germany69120Astronomisches Rechen-Institut, Zentrum für Astronomie der Uni- versität Heidelberg, Mönchhofstr. 12-14, 69120, Heidelberg, Ger- many","doi":""},{"text":"DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte. Blegdamsvej. 1761Niels Bohr International Academy, Niels Bohr Institute ; PSL Research UniversityCNRSNiels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte, Napoli, Italy 8 GEPI, Observatoire de Paris, PSL Research University, CNRS, 61","doi":""},{"text":"Avenue de l'Observatoire. 75014Paris, FranceAvenue de l'Observatoire, 75014, Paris, France","doi":""},{"text":"Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department. Koninklijke Sterrenwacht Van België, Naples, ItalyParthenope University of NaplesKoninklijke Sterrenwacht van België, Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department, Parthenope University of Naples, Naples, Italy","doi":""},{"text":"Germany 14 Institute for Astronomy. München, 2680University of Hawaii at ManoaMünchen, Germany 14 Institute for Astronomy, University of Hawaii at Manoa, 2680","doi":""},{"text":"Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH). Woodlawn Drive, Honolulu, Hongo, Bunkyo-Ku, E- 38200Japan 17 Instituto de Astrofísica de Canarias (IAC). Motoyama, Kamigamo, Kita-ku, Kyoto; La Laguna; La Laguna, Spain; Teramo, Italy19USA 15 Department of Astronomy, School of Science, The University of Tokyo ; Kyoto Sangyo University ; Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULLE-38205Koyama Astronomical Observatory. Via M. Maggini, s/n, I-64100Woodlawn Drive, Honolulu, HI 96822, USA 15 Department of Astronomy, School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH), Koyama Astronomical Observatory, Kyoto Sangyo University, Mo- toyama, Kamigamo, Kita-ku, Kyoto 603-8555, Japan 17 Instituto de Astrofísica de Canarias (IAC), La Laguna, E-38205, Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULL), E- 38200, La Laguna, Spain 19 INAF-Osservatorio Astronomico d'Abruzzo, Via M. Maggini, s/n, I-64100, Teramo, Italy","doi":"E- 38200"},{"text":"An der Sternwarte 16. Astrophysik Leibniz-Institut Für, Potsdam, AIPPotsdam, GermanyLeibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany","doi":""},{"text":". Nice Cedex. 7293Université Côte d'Azur, Observatoire de la CôteCSUniversité Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, La- grange UMR 7293, CS 34229, 06304, Nice Cedex 4, France","doi":""},{"text":". S M Andrievsky, R E Luck, V V Kovtyukh, AJ. 1301880Andrievsky, S. M., Luck, R. E., & Kovtyukh, V. V. 2005, AJ, 130, 1880","doi":""},{"text":". M Asplund, A M Amarsi, N Grevesse, A&A. 653141Asplund, M., Amarsi, A. M., & Grevesse, N. 2021, A&A, 653, A141","doi":""},{"text":". M Asplund, N Grevesse, A J Sauval, P Scott, ARA&A. 47481Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481","doi":""},{"text":". M T Belmonte, J C Pickering, M P Ruffoni, ApJ. 848125Belmonte, M. T., Pickering, J. C., Ruffoni, M. P., et al. 2017, ApJ, 848, 125","doi":""},{"text":". G Bono, F Caputo, S Cassisi, ApJ. 543955Bono, G., Caputo, F., Cassisi, S., et al. 2000a, ApJ, 543, 955","doi":""},{"text":". G Bono, F Caputo, M Marconi, I Musella, ApJ. 715277Bono, G., Caputo, F., Marconi, M., & Musella, I. 2010, ApJ, 715, 277","doi":""},{"text":". G Bono, V Castellani, M Marconi, ApJ. 56583Bono, G., Castellani, V., & Marconi, M. 2002, ApJ, 565, L83","doi":""},{"text":". G Bono, M Marconi, R F Stellingwerf, ApJS. 122167Bono, G., Marconi, M., & Stellingwerf, R. F. 1999, ApJS, 122, 167","doi":""},{"text":". G Bono, M Marconi, R F Stellingwerf, A&A. 360245Bono, G., Marconi, M., & Stellingwerf, R. F. 2000b, A&A, 360, 245","doi":""},{"text":". V F Braga, J Crestani, M Fabrizio, ApJ. 91985Braga, V. F., Crestani, J., Fabrizio, M., et al. 2021, ApJ, 919, 85","doi":""},{"text":". V F Braga, P B Stetson, G Bono, AJ. 152170Braga, V. F., Stetson, P. B., Bono, G., et al. 2016, AJ, 152, 170","doi":""},{"text":". L Breuval, P Kervella, R I Anderson, A&A. 643115Breuval, L., Kervella, P., Anderson, R. I., et al. 2020, A&A, 643, A115","doi":""},{"text":". L Breuval, P Kervella, P Wielgórski, ApJ. 91338Breuval, L., Kervella, P., Wielgórski, P., et al. 2021, ApJ, 913, 38","doi":""},{"text":". E Caffau, S Andrievsky, S Korotin, A&A. 58516Caffau, E., Andrievsky, S., Korotin, S., et al. 2016, A&A, 585, A16","doi":""},{"text":". E Caffau, P Bonifacio, R Faraggiana, A&A. 441533Caffau, E., Bonifacio, P., Faraggiana, R., et al. 2005, A&A, 441, 533","doi":""},{"text":". E Caffau, R Faraggiana, P Bonifacio, H G Ludwig, M Steffen, A&A. 470699Caffau, E., Faraggiana, R., Bonifacio, P., Ludwig, H. G., & Steffen, M. 2007, A&A, 470, 699","doi":""},{"text":"ArXiv Astrophysics e-prints. F Castelli, R L Kurucz, astro-ph/0405087Castelli, F. & Kurucz, R. L. 2004, ArXiv Astrophysics e-prints [astro-ph/0405087]","doi":"astro-ph/0405087"},{"text":". G Cescutti, F Matteucci, P François, C Chiappini, A&A. 462943Cescutti, G., Matteucci, F., François, P., & Chiappini, C. 2007, A&A, 462, 943","doi":""},{"text":". C Chiappini, F Matteucci, D Romano, ApJ. 5541044Chiappini, C., Matteucci, F., & Romano, D. 2001, ApJ, 554, 1044","doi":""},{"text":". J Crestani, V F Braga, M Fabrizio, ApJ. 91410125A&ACrestani, J., Braga, V. F., Fabrizio, M., et al. 2021, ApJ, 914, 10 da Silva, R., Lemasle, B., Bono, G., et al. 2016, A&A, 586, A125","doi":""},{"text":". G De Somma, M Marconi, S Cassisi, MNRAS. 5081473De Somma, G., Marconi, M., Cassisi, S., et al. 2021, MNRAS, 508, 1473","doi":""},{"text":"Optical and IR Telescope Instrumentation and Detectors. H Dekker, S D'odorico, A Kaufer, B Delabre, H Kotzlowski, Proc. SPIE. M. Iye & A. F. MoorwoodSPIE4008Dekker, H., D'Odorico, S., Kaufer, A., Delabre, B., & Kotzlowski, H. 2000, in Proc. SPIE, Vol. 4008, Optical and IR Telescope Instrumentation and Detec- tors, ed. M. Iye & A. F. Moorwood, 534-545","doi":""},{"text":". E A Den Hartog, J E Lawler, C Sneden, J J Cowan, A Brukhovesky, ApJS. 24333Den Hartog, E. A., Lawler, J. E., Sneden, C., Cowan, J. J., & Brukhovesky, A. 2019, ApJS, 243, 33","doi":""},{"text":". E A Den Hartog, M P Ruffoni, J E Lawler, ApJS. 21523Den Hartog, E. A., Ruffoni, M. P., Lawler, J. E., et al. 2014, ApJS, 215, 23","doi":""},{"text":". M Dessauges-Zavadsky, F Calura, J X Prochaska, S D'odorico, F Matteucci, A&A. 470431Dessauges-Zavadsky, M., Calura, F., Prochaska, J. X., D'Odorico, S., & Mat- teucci, F. 2007, A&A, 470, 431","doi":""},{"text":". V D'orazi, S Desidera, R G Gratton, A&A. 59819D'Orazi, V., Desidera, S., Gratton, R. G., et al. 2017, A&A, 598, A19","doi":""},{"text":". V D'orazi, E Oliva, A Bragaglia, A&A. 63338D'Orazi, V., Oliva, E., Bragaglia, A., et al. 2020, A&A, 633, A38","doi":""},{"text":". S Duffau, E Caffau, L Sbordone, A&A. 604128Duffau, S., Caffau, E., Sbordone, L., et al. 2017, A&A, 604, A128","doi":""},{"text":". N R Evans, AJ. 104216Evans, N. R. 1992, AJ, 104, 216","doi":""},{"text":". N R Evans, D L Welch, M H Slovak, Thomas G Barnes, I Moffett, T J , AJ. 1061599Evans, N. R., Welch, D. L., Slovak, M. H., Barnes, Thomas G., I., & Moffett, T. J. 1993, AJ, 106, 1599","doi":""},{"text":". B.-Q For, C Sneden, AJ. 1401694For, B.-Q. & Sneden, C. 2010, AJ, 140, 1694","doi":""},{"text":". W L Freedman, B F Madore, T Hoyt, ApJ. 89157Freedman, W. L., Madore, B. F., Hoyt, T., et al. 2020, ApJ, 891, 57","doi":""},{"text":". K Fukue, N Matsunaga, R Yamamoto, ApJ. 81264Fukue, K., Matsunaga, N., Yamamoto, R., et al. 2015, ApJ, 812, 64","doi":""},{"text":". K Genovali, B Lemasle, G Bono, A&A. 56637Genovali, K., Lemasle, B., Bono, G., et al. 2014, A&A, 566, A37","doi":""},{"text":". K Genovali, B Lemasle, G Bono, A&A. 554132Genovali, K., Lemasle, B., Bono, G., et al. 2013, A&A, 554, A132","doi":""},{"text":". K Genovali, B Lemasle, R Da Silva, A&A. 58017Genovali, K., Lemasle, B., da Silva, R., et al. 2015, A&A, 580, A17","doi":""},{"text":". G Gilmore, S Randich, M Asplund, The Messenger. 14725Gilmore, G., Randich, S., Asplund, M., et al. 2012, The Messenger, 147, 25","doi":""},{"text":". M A T Groenewegen, A&A. 48825Groenewegen, M. A. T. 2008, A&A, 488, 25","doi":""},{"text":". M A Groenewegen, A&A. 6198Groenewegen, M. A. T. 2018, A&A, 619, A8","doi":""},{"text":". M A T Groenewegen, A&A. 640113Groenewegen, M. A. T. 2020, A&A, 640, A113","doi":""},{"text":". L Inno, N Matsunaga, G Bono, ApJ. 76484Inno, L., Matsunaga, N., Bono, G., et al. 2013, ApJ, 764, 84","doi":""},{"text":". L Inno, N Matsunaga, M Romaniello, A&A. 57630Inno, L., Matsunaga, N., Romaniello, M., et al. 2015, A&A, 576, A30","doi":""},{"text":". L Inno, M A Urbaneja, N Matsunaga, MNRAS. 48283Inno, L., Urbaneja, M. A., Matsunaga, N., et al. 2019, MNRAS, 482, 83","doi":""},{"text":". M Jian, D Taniguchi, N Matsunaga, MNRAS. 4941724Jian, M., Taniguchi, D., Matsunaga, N., et al. 2020, MNRAS, 494, 1724","doi":""},{"text":". A Kaufer, O Stahl, S Tubbesing, The Messenger. 958Kaufer, A., Stahl, O., Tubbesing, S., et al. 1999, The Messenger, 95, 8","doi":""},{"text":". C S Kochanek, B J Shappee, K Z Stanek, PASP. 129104502Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, PASP, 129, 104502","doi":""},{"text":". V V Kovtyukh, MNRAS. 378617Kovtyukh, V. V. 2007, MNRAS, 378, 617","doi":""},{"text":". V V Kovtyukh, S M Andrievsky, S I Belik, R E Luck, AJ. 129433Kovtyukh, V. V., Andrievsky, S. M., Belik, S. I., & Luck, R. E. 2005, AJ, 129, 433","doi":""},{"text":"A Kramida, R Yu, J Reader, & the NIST ASD Team. 2020, NIST Atomic Spectra Database. version 5.8Kramida, A., Yu., R., Reader, J., & the NIST ASD Team. 2020, NIST Atomic Spectra Database, version 5.8","doi":""},{"text":"R L Kurucz, I Furenlid, J Brault, L Testerman, Solar flux atlas from 296 to 1300 nm. Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, Solar flux atlas from 296 to 1300 nm","doi":""},{"text":". J E Lawler, A Guzman, M P Wood, C Sneden, J J Cowan, ApJS. 20511Lawler, J. E., Guzman, A., Wood, M. P., Sneden, C., & Cowan, J. J. 2013, ApJS, 205, 11","doi":""},{"text":". B Lemasle, P François, K Genovali, A&A. 55831Lemasle, B., François, P., Genovali, K., et al. 2013, A&A, 558, A31","doi":""},{"text":". B Lemasle, M A T Groenewegen, E K Grebel, A&A. 60885Lemasle, B., Groenewegen, M. A. T., Grebel, E. K., et al. 2017, A&A, 608, A85","doi":""},{"text":". B Lemasle, M Hanke, J Storm, G Bono, E K Grebel, A&A. 64171Lemasle, B., Hanke, M., Storm, J., Bono, G., & Grebel, E. K. 2020, A&A, 641, A71","doi":""},{"text":". M Limongi, A Chieffi, ApJ. 592404Limongi, M. & Chieffi, A. 2003, ApJ, 592, 404","doi":""},{"text":". G Liu, N Arav, D S N Rupke, ApJS. 2219Liu, G., Arav, N., & Rupke, D. S. N. 2015, ApJS, 221, 9","doi":""},{"text":". R E Luck, S M Andrievsky, AJ. 128343Luck, R. E. & Andrievsky, S. M. 2004, AJ, 128, 343","doi":""},{"text":". R E Luck, S M Andrievsky, A Fokin, V V Kovtyukh, AJ. 13698Luck, R. E., Andrievsky, S. M., Fokin, A., & Kovtyukh, V. V. 2008, AJ, 136, 98","doi":""},{"text":". R E Luck, S M Andrievsky, V V Kovtyukh, W Gieren, D Graczyk, AJ. 14251Luck, R. E., Andrievsky, S. M., Kovtyukh, V. V., Gieren, W., & Graczyk, D. 2011, AJ, 142, 51","doi":""},{"text":". R E Luck, D L Lambert, AJ. 142136Luck, R. E. & Lambert, D. L. 2011, AJ, 142, 136","doi":""},{"text":". D Magurno, C Sneden, V F Braga, ApJ. 86457Magurno, D., Sneden, C., Braga, V. F., et al. 2018, ApJ, 864, 57","doi":""},{"text":". M Marconi, R Molinaro, G Bono, ApJ. 7686Marconi, M., Molinaro, R., Bono, G., et al. 2013, ApJ, 768, L6","doi":""},{"text":". M Marconi, I Musella, G Fiorentino, ApJ. 632590Marconi, M., Musella, I., & Fiorentino, G. 2005, ApJ, 632, 590","doi":""},{"text":". N Matsunaga, M Jian, D Taniguchi, S S Elgueta, MNRAS. 5061031Matsunaga, N., Jian, M., Taniguchi, D., & Elgueta, S. S. 2021, MNRAS, 506, 1031","doi":""},{"text":". M Mayor, F Pepe, D Queloz, The Messenger. 11420Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20","doi":""},{"text":". A Mcwilliam, G Wallerstein, M Mottini, ApJ. 778149McWilliam, A., Wallerstein, G., & Mottini, M. 2013, ApJ, 778, 149","doi":""},{"text":". J Meléndez, B Barbuy, A&A. 497611Meléndez, J. & Barbuy, B. 2009, A&A, 497, 611","doi":""},{"text":"A&A proofs: manuscript no. daSilvaetal2021. A&A proofs: manuscript no. daSilvaetal2021","doi":""},{"text":". J H Minniti, L Sbordone, A Rojas-Arriagada, A&A. 64092Minniti, J. H., Sbordone, L., Rojas-Arriagada, A., et al. 2020, A&A, 640, A92","doi":""},{"text":". J H Minniti, M Zoccali, A Rojas-Arriagada, A&A. 654138Minniti, J. H., Zoccali, M., Rojas-Arriagada, A., et al. 2021, A&A, 654, A138","doi":""},{"text":"The Solar Spectrum 2935 Å to 8770 Å. C E Moore, M Minnaert, J Houtgast, Moore, C. E., Minnaert, M., & Houtgast, J. 1966, The Solar Spectrum 2935 Å to 8770 Å","doi":""},{"text":". N Nardetto, J H Groh, S Kraus, F Millour, D Gillet, A&A. 4891263Nardetto, N., Groh, J. H., Kraus, S., Millour, F., & Gillet, D. 2008, A&A, 489, 1263","doi":""},{"text":". N Nardetto, D Mourard, P Kervella, A&A. 453309Nardetto, N., Mourard, D., Kervella, P., et al. 2006, A&A, 453, 309","doi":""},{"text":". N Nardetto, E Poretti, A Gallenne, A&A. 61692Nardetto, N., Poretti, E., Gallenne, A., et al. 2018, A&A, 616, A92","doi":""},{"text":". G Natale, M Marconi, G Bono, ApJ. 67493Natale, G., Marconi, M., & Bono, G. 2008, ApJ, 674, L93","doi":""},{"text":". J R Neeley, M Monelli, M Marengo, ApJ. 920152Neeley, J. R., Monelli, M., Marengo, M., et al. 2021, ApJ, 920, 152","doi":""},{"text":". H R Neilson, M Cantiello, N Langer, A&A. 5299Neilson, H. R., Cantiello, M., & Langer, N. 2011, A&A, 529, L9","doi":""},{"text":". F Niedermann, M S Sloth, Phys. Rev. D. 10263527Niedermann, F. & Sloth, M. S. 2020, Phys. Rev. D, 102, 063527","doi":""},{"text":". P E Nissen, Y Q Chen, M Asplund, M Pettini, A&A. 415993Nissen, P. E., Chen, Y. Q., Asplund, M., & Pettini, M. 2004, A&A, 415, 993","doi":""},{"text":". T R O'brian, M E Wickliffe, J E Lawler, W Whaling, J W Brault, Journal of the Optical Society of America B Optical Physics. 81185O'Brian, T. R., Wickliffe, M. E., Lawler, J. E., Whaling, W., & Brault, J. W. 1991, Journal of the Optical Society of America B Optical Physics, 8, 1185","doi":""},{"text":". J W Pel, A&AS. 24413Pel, J. W. 1976, A&AS, 24, 413","doi":""},{"text":". G Pietrzyński, D Graczyk, W Gieren, Nature. 49576Pietrzyński, G., Graczyk, D., Gieren, W., et al. 2013, Nature, 495, 76","doi":""},{"text":". N Aghanim, Planck CollaborationY Akrami, Planck CollaborationA&A. 6416Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6","doi":""},{"text":". P G Prada Moroni, M Gennaro, G Bono, ApJ. 749108Prada Moroni, P. G., Gennaro, M., Bono, G., et al. 2012, ApJ, 749, 108","doi":""},{"text":". B Proxauf, R Da Silva, V V Kovtyukh, A&A. 61682Proxauf, B., da Silva, R., Kovtyukh, V. V., et al. 2018, A&A, 616, A82","doi":""},{"text":". S Randich, G Gilmore, -Eso Gaia, Consortium, The Messenger. 15447Randich, S., Gilmore, G., & Gaia-ESO Consortium. 2013, The Messenger, 154, 47","doi":""},{"text":". A G Riess, S Casertano, W Yuan, L M Macri, D Scolnic, ApJ. 87685Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ, 876, 85","doi":""},{"text":". V Ripepi, G Catanzaro, G Clementini, arXiv:2201.01126arXiv e-printsRipepi, V., Catanzaro, G., Clementini, G., et al. 2022, arXiv e-prints, arXiv:2201.01126","doi":"arXiv:2201.01126"},{"text":". V Ripepi, G Catanzaro, R Molinaro, MNRAS. 5084047Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2021, MNRAS, 508, 4047","doi":""},{"text":". V Ripepi, G Catanzaro, R Molinaro, A&A. 642230Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2020, A&A, 642, A230","doi":""},{"text":". V Ripepi, R Molinaro, I Musella, A&A. 62514Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14","doi":""},{"text":". M Romaniello, F Primas, M Mottini, A&A. 488731Romaniello, M., Primas, F., Mottini, M., et al. 2008, A&A, 488, 731","doi":""},{"text":". M Romaniello, A Riess, S Mancino, arXiv:2110.08860arXiv e-printsRomaniello, M., Riess, A., Mancino, S., et al. 2021, arXiv e-prints, arXiv:2110.08860","doi":"arXiv:2110.08860"},{"text":". M P Ruffoni, E A Hartog, J E Lawler, MNRAS. 4413127Ruffoni, M. P., Den Hartog, E. A., Lawler, J. E., et al. 2014, MNRAS, 441, 3127","doi":""},{"text":". J D Scargle, ApJ. 263835Scargle, J. D. 1982, ApJ, 263, 835","doi":""},{"text":". B J Shappee, J L Prieto, D Grupe, ApJ. 78848Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48","doi":""},{"text":". D M Skowron, J Skowron, P Mróz, Acta Astron. 69305Skowron, D. M., Skowron, J., Mróz, P., et al. 2019, Acta Astron., 69, 305","doi":""},{"text":"C Sneden, The MOOG code. Sneden, C. 2002, The MOOG code, http://www.as.utexas.edu/~chris/ moog.html","doi":""},{"text":". S G Sousa, N C Santos, V Adibekyan, E Delgado-Mena, G Israelian, A&A. 57767Sousa, S. G., Santos, N. C., Adibekyan, V., Delgado-Mena, E., & Israelian, G. 2015, A&A, 577, A67","doi":""},{"text":". S G Sousa, N C Santos, G Israelian, M Mayor, M J P F G Monteiro, A&A. 469783Sousa, S. G., Santos, N. C., Israelian, G., Mayor, M., & Monteiro, M. J. P. F. G. 2007, A&A, 469, 783","doi":""},{"text":". K G Strassmeier, T Granzer, M Weber, Astronomische Nachrichten. 325527Strassmeier, K. G., Granzer, T., Weber, M., et al. 2004, Astronomische Nachrichten, 325, 527","doi":""},{"text":". K G Strassmeier, T Granzer, M Weber, Advances in Astronomy. 970306Strassmeier, K. G., Granzer, T., Weber, M., et al. 2010, Advances in Astronomy, 2010, 970306","doi":""},{"text":". Y Takeda, M Omiya, H Harakawa, B Sato, PASJ. 6881Takeda, Y., Omiya, M., Harakawa, H., & Sato, B. 2016, PASJ, 68, 81","doi":""},{"text":". D Taniguchi, N Matsunaga, M Jian, MNRAS. 5024210Taniguchi, D., Matsunaga, N., Jian, M., et al. 2021, MNRAS, 502, 4210","doi":""},{"text":". B J Taylor, PASP. 106452Taylor, B. J. 1994, PASP, 106, 452","doi":""},{"text":". V Vasilyev, H G Ludwig, B Freytag, B Lemasle, M Marconi, A&A. 61119Vasilyev, V., Ludwig, H. G., Freytag, B., Lemasle, B., & Marconi, M. 2018, A&A, 611, A19","doi":""},{"text":". K A Venn, M D Shetrone, M J Irwin, ApJ. 751102Venn, K. A., Shetrone, M. D., Irwin, M. J., et al. 2012, ApJ, 751, 102","doi":""},{"text":"M Weber, T Granzer, K G Strassmeier, Proc. SPIE. SPIE845184510Weber, M., Granzer, T., & Strassmeier, K. G. 2012, in Proc. SPIE, Vol. 8451, Software and Cyberinfrastructure for Astronomy II, 84510K","doi":""},{"text":". M P Wood, J E Lawler, C Sneden, J J Cowan, ApJS. 20827Wood, M. P., Lawler, J. E., Sneden, C., & Cowan, J. J. 2013, ApJS, 208, 27","doi":""},{"text":". D Yong, B W Carney, M L Teixera De Almeida, B L Pohl, AJ. 1312256Yong, D., Carney, B. W., Teixera de Almeida, M. L., & Pohl, B. L. 2006, AJ, 131, 2256","doi":""},{"text":". Inaf -Osservatorio Astronomico Di Padova, 535122Padova, Italy4 INAF -Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy","doi":""},{"text":"Zentrum für Astronomie der Universität Heidelberg. Heidelberg, Germany69120Astronomisches Rechen-Institut, Zentrum für Astronomie der Uni- versität Heidelberg, Mönchhofstr. 12-14, 69120, Heidelberg, Ger- many","doi":""},{"text":"DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte. Blegdamsvej. 1761Niels Bohr International Academy, Niels Bohr Institute ; PSL Research UniversityCNRSNiels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte, Napoli, Italy 8 GEPI, Observatoire de Paris, PSL Research University, CNRS, 61","doi":""},{"text":"Avenue de l'Observatoire. 75014Paris, FranceAvenue de l'Observatoire, 75014, Paris, France","doi":""},{"text":"Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department. Koninklijke Sterrenwacht Van België, Naples, ItalyParthenope University of NaplesKoninklijke Sterrenwacht van België, Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department, Parthenope University of Naples, Naples, Italy","doi":""},{"text":"Germany 14 Institute for Astronomy. München, 2680University of Hawaii at ManoaMünchen, Germany 14 Institute for Astronomy, University of Hawaii at Manoa, 2680","doi":""},{"text":"Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH). Woodlawn Drive, Honolulu, Hongo, Bunkyo-Ku, E- 38200Japan 17 Instituto de Astrofísica de Canarias (IAC). Motoyama, Kamigamo, Kita-ku, Kyoto; La Laguna; La Laguna, Spain; Teramo, Italy19USA 15 Department of Astronomy, School of Science, The University of Tokyo ; Kyoto Sangyo University ; Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULLE-38205Koyama Astronomical Observatory. Via M. Maggini, s/n, I-64100Woodlawn Drive, Honolulu, HI 96822, USA 15 Department of Astronomy, School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH), Koyama Astronomical Observatory, Kyoto Sangyo University, Mo- toyama, Kamigamo, Kita-ku, Kyoto 603-8555, Japan 17 Instituto de Astrofísica de Canarias (IAC), La Laguna, E-38205, Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULL), E- 38200, La Laguna, Spain 19 INAF-Osservatorio Astronomico d'Abruzzo, Via M. Maggini, s/n, I-64100, Teramo, Italy","doi":"E- 38200"},{"text":"An der Sternwarte 16. Astrophysik Leibniz-Institut Für, Potsdam, AIPPotsdam, GermanyLeibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany","doi":""},{"text":". Nice Cedex. 7293Université Côte d'Azur, Observatoire de la CôteCSUniversité Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, La- grange UMR 7293, CS 34229, 06304, Nice Cedex 4, France","doi":""}],"string":"[\n {\n \"text\": \". S M Andrievsky, R E Luck, V V Kovtyukh, AJ. 1301880Andrievsky, S. M., Luck, R. E., & Kovtyukh, V. V. 2005, AJ, 130, 1880\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Asplund, A M Amarsi, N Grevesse, A&A. 653141Asplund, M., Amarsi, A. M., & Grevesse, N. 2021, A&A, 653, A141\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Asplund, N Grevesse, A J Sauval, P Scott, ARA&A. 47481Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M T Belmonte, J C Pickering, M P Ruffoni, ApJ. 848125Belmonte, M. T., Pickering, J. C., Ruffoni, M. P., et al. 2017, ApJ, 848, 125\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, F Caputo, S Cassisi, ApJ. 543955Bono, G., Caputo, F., Cassisi, S., et al. 2000a, ApJ, 543, 955\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, F Caputo, M Marconi, I Musella, ApJ. 715277Bono, G., Caputo, F., Marconi, M., & Musella, I. 2010, ApJ, 715, 277\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, V Castellani, M Marconi, ApJ. 56583Bono, G., Castellani, V., & Marconi, M. 2002, ApJ, 565, L83\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, M Marconi, R F Stellingwerf, ApJS. 122167Bono, G., Marconi, M., & Stellingwerf, R. F. 1999, ApJS, 122, 167\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, M Marconi, R F Stellingwerf, A&A. 360245Bono, G., Marconi, M., & Stellingwerf, R. F. 2000b, A&A, 360, 245\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V F Braga, J Crestani, M Fabrizio, ApJ. 91985Braga, V. F., Crestani, J., Fabrizio, M., et al. 2021, ApJ, 919, 85\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V F Braga, P B Stetson, G Bono, AJ. 152170Braga, V. F., Stetson, P. B., Bono, G., et al. 2016, AJ, 152, 170\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Breuval, P Kervella, R I Anderson, A&A. 643115Breuval, L., Kervella, P., Anderson, R. I., et al. 2020, A&A, 643, A115\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Breuval, P Kervella, P Wielgórski, ApJ. 91338Breuval, L., Kervella, P., Wielgórski, P., et al. 2021, ApJ, 913, 38\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Caffau, S Andrievsky, S Korotin, A&A. 58516Caffau, E., Andrievsky, S., Korotin, S., et al. 2016, A&A, 585, A16\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Caffau, P Bonifacio, R Faraggiana, A&A. 441533Caffau, E., Bonifacio, P., Faraggiana, R., et al. 2005, A&A, 441, 533\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Caffau, R Faraggiana, P Bonifacio, H G Ludwig, M Steffen, A&A. 470699Caffau, E., Faraggiana, R., Bonifacio, P., Ludwig, H. G., & Steffen, M. 2007, A&A, 470, 699\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ArXiv Astrophysics e-prints. F Castelli, R L Kurucz, astro-ph/0405087Castelli, F. & Kurucz, R. L. 2004, ArXiv Astrophysics e-prints [astro-ph/0405087]\",\n \"doi\": \"astro-ph/0405087\"\n },\n {\n \"text\": \". G Cescutti, F Matteucci, P François, C Chiappini, A&A. 462943Cescutti, G., Matteucci, F., François, P., & Chiappini, C. 2007, A&A, 462, 943\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Chiappini, F Matteucci, D Romano, ApJ. 5541044Chiappini, C., Matteucci, F., & Romano, D. 2001, ApJ, 554, 1044\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Crestani, V F Braga, M Fabrizio, ApJ. 91410125A&ACrestani, J., Braga, V. F., Fabrizio, M., et al. 2021, ApJ, 914, 10 da Silva, R., Lemasle, B., Bono, G., et al. 2016, A&A, 586, A125\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G De Somma, M Marconi, S Cassisi, MNRAS. 5081473De Somma, G., Marconi, M., Cassisi, S., et al. 2021, MNRAS, 508, 1473\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical and IR Telescope Instrumentation and Detectors. H Dekker, S D'odorico, A Kaufer, B Delabre, H Kotzlowski, Proc. SPIE. M. Iye & A. F. MoorwoodSPIE4008Dekker, H., D'Odorico, S., Kaufer, A., Delabre, B., & Kotzlowski, H. 2000, in Proc. SPIE, Vol. 4008, Optical and IR Telescope Instrumentation and Detec- tors, ed. M. Iye & A. F. Moorwood, 534-545\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E A Den Hartog, J E Lawler, C Sneden, J J Cowan, A Brukhovesky, ApJS. 24333Den Hartog, E. A., Lawler, J. E., Sneden, C., Cowan, J. J., & Brukhovesky, A. 2019, ApJS, 243, 33\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E A Den Hartog, M P Ruffoni, J E Lawler, ApJS. 21523Den Hartog, E. A., Ruffoni, M. P., Lawler, J. E., et al. 2014, ApJS, 215, 23\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Dessauges-Zavadsky, F Calura, J X Prochaska, S D'odorico, F Matteucci, A&A. 470431Dessauges-Zavadsky, M., Calura, F., Prochaska, J. X., D'Odorico, S., & Mat- teucci, F. 2007, A&A, 470, 431\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V D'orazi, S Desidera, R G Gratton, A&A. 59819D'Orazi, V., Desidera, S., Gratton, R. G., et al. 2017, A&A, 598, A19\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V D'orazi, E Oliva, A Bragaglia, A&A. 63338D'Orazi, V., Oliva, E., Bragaglia, A., et al. 2020, A&A, 633, A38\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Duffau, E Caffau, L Sbordone, A&A. 604128Duffau, S., Caffau, E., Sbordone, L., et al. 2017, A&A, 604, A128\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N R Evans, AJ. 104216Evans, N. R. 1992, AJ, 104, 216\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N R Evans, D L Welch, M H Slovak, Thomas G Barnes, I Moffett, T J , AJ. 1061599Evans, N. R., Welch, D. L., Slovak, M. H., Barnes, Thomas G., I., & Moffett, T. J. 1993, AJ, 106, 1599\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B.-Q For, C Sneden, AJ. 1401694For, B.-Q. & Sneden, C. 2010, AJ, 140, 1694\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W L Freedman, B F Madore, T Hoyt, ApJ. 89157Freedman, W. L., Madore, B. F., Hoyt, T., et al. 2020, ApJ, 891, 57\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Fukue, N Matsunaga, R Yamamoto, ApJ. 81264Fukue, K., Matsunaga, N., Yamamoto, R., et al. 2015, ApJ, 812, 64\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Genovali, B Lemasle, G Bono, A&A. 56637Genovali, K., Lemasle, B., Bono, G., et al. 2014, A&A, 566, A37\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Genovali, B Lemasle, G Bono, A&A. 554132Genovali, K., Lemasle, B., Bono, G., et al. 2013, A&A, 554, A132\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Genovali, B Lemasle, R Da Silva, A&A. 58017Genovali, K., Lemasle, B., da Silva, R., et al. 2015, A&A, 580, A17\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Gilmore, S Randich, M Asplund, The Messenger. 14725Gilmore, G., Randich, S., Asplund, M., et al. 2012, The Messenger, 147, 25\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M A T Groenewegen, A&A. 48825Groenewegen, M. A. T. 2008, A&A, 488, 25\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M A Groenewegen, A&A. 6198Groenewegen, M. A. T. 2018, A&A, 619, A8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M A T Groenewegen, A&A. 640113Groenewegen, M. A. T. 2020, A&A, 640, A113\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Inno, N Matsunaga, G Bono, ApJ. 76484Inno, L., Matsunaga, N., Bono, G., et al. 2013, ApJ, 764, 84\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Inno, N Matsunaga, M Romaniello, A&A. 57630Inno, L., Matsunaga, N., Romaniello, M., et al. 2015, A&A, 576, A30\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Inno, M A Urbaneja, N Matsunaga, MNRAS. 48283Inno, L., Urbaneja, M. A., Matsunaga, N., et al. 2019, MNRAS, 482, 83\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Jian, D Taniguchi, N Matsunaga, MNRAS. 4941724Jian, M., Taniguchi, D., Matsunaga, N., et al. 2020, MNRAS, 494, 1724\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Kaufer, O Stahl, S Tubbesing, The Messenger. 958Kaufer, A., Stahl, O., Tubbesing, S., et al. 1999, The Messenger, 95, 8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C S Kochanek, B J Shappee, K Z Stanek, PASP. 129104502Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, PASP, 129, 104502\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V V Kovtyukh, MNRAS. 378617Kovtyukh, V. V. 2007, MNRAS, 378, 617\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V V Kovtyukh, S M Andrievsky, S I Belik, R E Luck, AJ. 129433Kovtyukh, V. V., Andrievsky, S. M., Belik, S. I., & Luck, R. E. 2005, AJ, 129, 433\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Kramida, R Yu, J Reader, & the NIST ASD Team. 2020, NIST Atomic Spectra Database. version 5.8Kramida, A., Yu., R., Reader, J., & the NIST ASD Team. 2020, NIST Atomic Spectra Database, version 5.8\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R L Kurucz, I Furenlid, J Brault, L Testerman, Solar flux atlas from 296 to 1300 nm. Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, Solar flux atlas from 296 to 1300 nm\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J E Lawler, A Guzman, M P Wood, C Sneden, J J Cowan, ApJS. 20511Lawler, J. E., Guzman, A., Wood, M. P., Sneden, C., & Cowan, J. J. 2013, ApJS, 205, 11\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Lemasle, P François, K Genovali, A&A. 55831Lemasle, B., François, P., Genovali, K., et al. 2013, A&A, 558, A31\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Lemasle, M A T Groenewegen, E K Grebel, A&A. 60885Lemasle, B., Groenewegen, M. A. T., Grebel, E. K., et al. 2017, A&A, 608, A85\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Lemasle, M Hanke, J Storm, G Bono, E K Grebel, A&A. 64171Lemasle, B., Hanke, M., Storm, J., Bono, G., & Grebel, E. K. 2020, A&A, 641, A71\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Limongi, A Chieffi, ApJ. 592404Limongi, M. & Chieffi, A. 2003, ApJ, 592, 404\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Liu, N Arav, D S N Rupke, ApJS. 2219Liu, G., Arav, N., & Rupke, D. S. N. 2015, ApJS, 221, 9\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E Luck, S M Andrievsky, AJ. 128343Luck, R. E. & Andrievsky, S. M. 2004, AJ, 128, 343\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E Luck, S M Andrievsky, A Fokin, V V Kovtyukh, AJ. 13698Luck, R. E., Andrievsky, S. M., Fokin, A., & Kovtyukh, V. V. 2008, AJ, 136, 98\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E Luck, S M Andrievsky, V V Kovtyukh, W Gieren, D Graczyk, AJ. 14251Luck, R. E., Andrievsky, S. M., Kovtyukh, V. V., Gieren, W., & Graczyk, D. 2011, AJ, 142, 51\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E Luck, D L Lambert, AJ. 142136Luck, R. E. & Lambert, D. L. 2011, AJ, 142, 136\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Magurno, C Sneden, V F Braga, ApJ. 86457Magurno, D., Sneden, C., Braga, V. F., et al. 2018, ApJ, 864, 57\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Marconi, R Molinaro, G Bono, ApJ. 7686Marconi, M., Molinaro, R., Bono, G., et al. 2013, ApJ, 768, L6\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Marconi, I Musella, G Fiorentino, ApJ. 632590Marconi, M., Musella, I., & Fiorentino, G. 2005, ApJ, 632, 590\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Matsunaga, M Jian, D Taniguchi, S S Elgueta, MNRAS. 5061031Matsunaga, N., Jian, M., Taniguchi, D., & Elgueta, S. S. 2021, MNRAS, 506, 1031\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Mayor, F Pepe, D Queloz, The Messenger. 11420Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Mcwilliam, G Wallerstein, M Mottini, ApJ. 778149McWilliam, A., Wallerstein, G., & Mottini, M. 2013, ApJ, 778, 149\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Meléndez, B Barbuy, A&A. 497611Meléndez, J. & Barbuy, B. 2009, A&A, 497, 611\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A&A proofs: manuscript no. daSilvaetal2021. A&A proofs: manuscript no. daSilvaetal2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J H Minniti, L Sbordone, A Rojas-Arriagada, A&A. 64092Minniti, J. H., Sbordone, L., Rojas-Arriagada, A., et al. 2020, A&A, 640, A92\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J H Minniti, M Zoccali, A Rojas-Arriagada, A&A. 654138Minniti, J. H., Zoccali, M., Rojas-Arriagada, A., et al. 2021, A&A, 654, A138\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Solar Spectrum 2935 Å to 8770 Å. C E Moore, M Minnaert, J Houtgast, Moore, C. E., Minnaert, M., & Houtgast, J. 1966, The Solar Spectrum 2935 Å to 8770 Å\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Nardetto, J H Groh, S Kraus, F Millour, D Gillet, A&A. 4891263Nardetto, N., Groh, J. H., Kraus, S., Millour, F., & Gillet, D. 2008, A&A, 489, 1263\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Nardetto, D Mourard, P Kervella, A&A. 453309Nardetto, N., Mourard, D., Kervella, P., et al. 2006, A&A, 453, 309\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Nardetto, E Poretti, A Gallenne, A&A. 61692Nardetto, N., Poretti, E., Gallenne, A., et al. 2018, A&A, 616, A92\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Natale, M Marconi, G Bono, ApJ. 67493Natale, G., Marconi, M., & Bono, G. 2008, ApJ, 674, L93\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J R Neeley, M Monelli, M Marengo, ApJ. 920152Neeley, J. R., Monelli, M., Marengo, M., et al. 2021, ApJ, 920, 152\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H R Neilson, M Cantiello, N Langer, A&A. 5299Neilson, H. R., Cantiello, M., & Langer, N. 2011, A&A, 529, L9\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Niedermann, M S Sloth, Phys. Rev. D. 10263527Niedermann, F. & Sloth, M. S. 2020, Phys. Rev. D, 102, 063527\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P E Nissen, Y Q Chen, M Asplund, M Pettini, A&A. 415993Nissen, P. E., Chen, Y. Q., Asplund, M., & Pettini, M. 2004, A&A, 415, 993\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R O'brian, M E Wickliffe, J E Lawler, W Whaling, J W Brault, Journal of the Optical Society of America B Optical Physics. 81185O'Brian, T. R., Wickliffe, M. E., Lawler, J. E., Whaling, W., & Brault, J. W. 1991, Journal of the Optical Society of America B Optical Physics, 8, 1185\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J W Pel, A&AS. 24413Pel, J. W. 1976, A&AS, 24, 413\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Pietrzyński, D Graczyk, W Gieren, Nature. 49576Pietrzyński, G., Graczyk, D., Gieren, W., et al. 2013, Nature, 495, 76\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Aghanim, Planck CollaborationY Akrami, Planck CollaborationA&A. 6416Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P G Prada Moroni, M Gennaro, G Bono, ApJ. 749108Prada Moroni, P. G., Gennaro, M., Bono, G., et al. 2012, ApJ, 749, 108\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Proxauf, R Da Silva, V V Kovtyukh, A&A. 61682Proxauf, B., da Silva, R., Kovtyukh, V. V., et al. 2018, A&A, 616, A82\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Randich, G Gilmore, -Eso Gaia, Consortium, The Messenger. 15447Randich, S., Gilmore, G., & Gaia-ESO Consortium. 2013, The Messenger, 154, 47\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A G Riess, S Casertano, W Yuan, L M Macri, D Scolnic, ApJ. 87685Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ, 876, 85\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Ripepi, G Catanzaro, G Clementini, arXiv:2201.01126arXiv e-printsRipepi, V., Catanzaro, G., Clementini, G., et al. 2022, arXiv e-prints, arXiv:2201.01126\",\n \"doi\": \"arXiv:2201.01126\"\n },\n {\n \"text\": \". V Ripepi, G Catanzaro, R Molinaro, MNRAS. 5084047Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2021, MNRAS, 508, 4047\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Ripepi, G Catanzaro, R Molinaro, A&A. 642230Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2020, A&A, 642, A230\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Ripepi, R Molinaro, I Musella, A&A. 62514Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Romaniello, F Primas, M Mottini, A&A. 488731Romaniello, M., Primas, F., Mottini, M., et al. 2008, A&A, 488, 731\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Romaniello, A Riess, S Mancino, arXiv:2110.08860arXiv e-printsRomaniello, M., Riess, A., Mancino, S., et al. 2021, arXiv e-prints, arXiv:2110.08860\",\n \"doi\": \"arXiv:2110.08860\"\n },\n {\n \"text\": \". M P Ruffoni, E A Hartog, J E Lawler, MNRAS. 4413127Ruffoni, M. P., Den Hartog, E. A., Lawler, J. E., et al. 2014, MNRAS, 441, 3127\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Scargle, ApJ. 263835Scargle, J. D. 1982, ApJ, 263, 835\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B J Shappee, J L Prieto, D Grupe, ApJ. 78848Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D M Skowron, J Skowron, P Mróz, Acta Astron. 69305Skowron, D. M., Skowron, J., Mróz, P., et al. 2019, Acta Astron., 69, 305\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Sneden, The MOOG code. Sneden, C. 2002, The MOOG code, http://www.as.utexas.edu/~chris/ moog.html\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S G Sousa, N C Santos, V Adibekyan, E Delgado-Mena, G Israelian, A&A. 57767Sousa, S. G., Santos, N. C., Adibekyan, V., Delgado-Mena, E., & Israelian, G. 2015, A&A, 577, A67\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S G Sousa, N C Santos, G Israelian, M Mayor, M J P F G Monteiro, A&A. 469783Sousa, S. G., Santos, N. C., Israelian, G., Mayor, M., & Monteiro, M. J. P. F. G. 2007, A&A, 469, 783\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K G Strassmeier, T Granzer, M Weber, Astronomische Nachrichten. 325527Strassmeier, K. G., Granzer, T., Weber, M., et al. 2004, Astronomische Nachrichten, 325, 527\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K G Strassmeier, T Granzer, M Weber, Advances in Astronomy. 970306Strassmeier, K. G., Granzer, T., Weber, M., et al. 2010, Advances in Astronomy, 2010, 970306\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Takeda, M Omiya, H Harakawa, B Sato, PASJ. 6881Takeda, Y., Omiya, M., Harakawa, H., & Sato, B. 2016, PASJ, 68, 81\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Taniguchi, N Matsunaga, M Jian, MNRAS. 5024210Taniguchi, D., Matsunaga, N., Jian, M., et al. 2021, MNRAS, 502, 4210\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B J Taylor, PASP. 106452Taylor, B. J. 1994, PASP, 106, 452\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Vasilyev, H G Ludwig, B Freytag, B Lemasle, M Marconi, A&A. 61119Vasilyev, V., Ludwig, H. G., Freytag, B., Lemasle, B., & Marconi, M. 2018, A&A, 611, A19\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K A Venn, M D Shetrone, M J Irwin, ApJ. 751102Venn, K. A., Shetrone, M. D., Irwin, M. J., et al. 2012, ApJ, 751, 102\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Weber, T Granzer, K G Strassmeier, Proc. SPIE. SPIE845184510Weber, M., Granzer, T., & Strassmeier, K. G. 2012, in Proc. SPIE, Vol. 8451, Software and Cyberinfrastructure for Astronomy II, 84510K\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M P Wood, J E Lawler, C Sneden, J J Cowan, ApJS. 20827Wood, M. P., Lawler, J. E., Sneden, C., & Cowan, J. J. 2013, ApJS, 208, 27\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Yong, B W Carney, M L Teixera De Almeida, B L Pohl, AJ. 1312256Yong, D., Carney, B. W., Teixera de Almeida, M. L., & Pohl, B. L. 2006, AJ, 131, 2256\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Inaf -Osservatorio Astronomico Di Padova, 535122Padova, Italy4 INAF -Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Zentrum für Astronomie der Universität Heidelberg. Heidelberg, Germany69120Astronomisches Rechen-Institut, Zentrum für Astronomie der Uni- versität Heidelberg, Mönchhofstr. 12-14, 69120, Heidelberg, Ger- many\",\n \"doi\": \"\"\n },\n {\n \"text\": \"DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte. Blegdamsvej. 1761Niels Bohr International Academy, Niels Bohr Institute ; PSL Research UniversityCNRSNiels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte, Napoli, Italy 8 GEPI, Observatoire de Paris, PSL Research University, CNRS, 61\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Avenue de l'Observatoire. 75014Paris, FranceAvenue de l'Observatoire, 75014, Paris, France\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department. Koninklijke Sterrenwacht Van België, Naples, ItalyParthenope University of NaplesKoninklijke Sterrenwacht van België, Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department, Parthenope University of Naples, Naples, Italy\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Germany 14 Institute for Astronomy. München, 2680University of Hawaii at ManoaMünchen, Germany 14 Institute for Astronomy, University of Hawaii at Manoa, 2680\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH). Woodlawn Drive, Honolulu, Hongo, Bunkyo-Ku, E- 38200Japan 17 Instituto de Astrofísica de Canarias (IAC). Motoyama, Kamigamo, Kita-ku, Kyoto; La Laguna; La Laguna, Spain; Teramo, Italy19USA 15 Department of Astronomy, School of Science, The University of Tokyo ; Kyoto Sangyo University ; Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULLE-38205Koyama Astronomical Observatory. Via M. Maggini, s/n, I-64100Woodlawn Drive, Honolulu, HI 96822, USA 15 Department of Astronomy, School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH), Koyama Astronomical Observatory, Kyoto Sangyo University, Mo- toyama, Kamigamo, Kita-ku, Kyoto 603-8555, Japan 17 Instituto de Astrofísica de Canarias (IAC), La Laguna, E-38205, Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULL), E- 38200, La Laguna, Spain 19 INAF-Osservatorio Astronomico d'Abruzzo, Via M. Maggini, s/n, I-64100, Teramo, Italy\",\n \"doi\": \"E- 38200\"\n },\n {\n \"text\": \"An der Sternwarte 16. Astrophysik Leibniz-Institut Für, Potsdam, AIPPotsdam, GermanyLeibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Nice Cedex. 7293Université Côte d'Azur, Observatoire de la CôteCSUniversité Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, La- grange UMR 7293, CS 34229, 06304, Nice Cedex 4, France\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S M Andrievsky, R E Luck, V V Kovtyukh, AJ. 1301880Andrievsky, S. M., Luck, R. E., & Kovtyukh, V. V. 2005, AJ, 130, 1880\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Asplund, A M Amarsi, N Grevesse, A&A. 653141Asplund, M., Amarsi, A. M., & Grevesse, N. 2021, A&A, 653, A141\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Asplund, N Grevesse, A J Sauval, P Scott, ARA&A. 47481Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M T Belmonte, J C Pickering, M P Ruffoni, ApJ. 848125Belmonte, M. T., Pickering, J. C., Ruffoni, M. P., et al. 2017, ApJ, 848, 125\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, F Caputo, S Cassisi, ApJ. 543955Bono, G., Caputo, F., Cassisi, S., et al. 2000a, ApJ, 543, 955\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, F Caputo, M Marconi, I Musella, ApJ. 715277Bono, G., Caputo, F., Marconi, M., & Musella, I. 2010, ApJ, 715, 277\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, V Castellani, M Marconi, ApJ. 56583Bono, G., Castellani, V., & Marconi, M. 2002, ApJ, 565, L83\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, M Marconi, R F Stellingwerf, ApJS. 122167Bono, G., Marconi, M., & Stellingwerf, R. F. 1999, ApJS, 122, 167\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Bono, M Marconi, R F Stellingwerf, A&A. 360245Bono, G., Marconi, M., & Stellingwerf, R. F. 2000b, A&A, 360, 245\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V F Braga, J Crestani, M Fabrizio, ApJ. 91985Braga, V. F., Crestani, J., Fabrizio, M., et al. 2021, ApJ, 919, 85\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V F Braga, P B Stetson, G Bono, AJ. 152170Braga, V. F., Stetson, P. B., Bono, G., et al. 2016, AJ, 152, 170\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Breuval, P Kervella, R I Anderson, A&A. 643115Breuval, L., Kervella, P., Anderson, R. I., et al. 2020, A&A, 643, A115\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Breuval, P Kervella, P Wielgórski, ApJ. 91338Breuval, L., Kervella, P., Wielgórski, P., et al. 2021, ApJ, 913, 38\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Caffau, S Andrievsky, S Korotin, A&A. 58516Caffau, E., Andrievsky, S., Korotin, S., et al. 2016, A&A, 585, A16\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Caffau, P Bonifacio, R Faraggiana, A&A. 441533Caffau, E., Bonifacio, P., Faraggiana, R., et al. 2005, A&A, 441, 533\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Caffau, R Faraggiana, P Bonifacio, H G Ludwig, M Steffen, A&A. 470699Caffau, E., Faraggiana, R., Bonifacio, P., Ludwig, H. G., & Steffen, M. 2007, A&A, 470, 699\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ArXiv Astrophysics e-prints. F Castelli, R L Kurucz, astro-ph/0405087Castelli, F. & Kurucz, R. L. 2004, ArXiv Astrophysics e-prints [astro-ph/0405087]\",\n \"doi\": \"astro-ph/0405087\"\n },\n {\n \"text\": \". G Cescutti, F Matteucci, P François, C Chiappini, A&A. 462943Cescutti, G., Matteucci, F., François, P., & Chiappini, C. 2007, A&A, 462, 943\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Chiappini, F Matteucci, D Romano, ApJ. 5541044Chiappini, C., Matteucci, F., & Romano, D. 2001, ApJ, 554, 1044\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Crestani, V F Braga, M Fabrizio, ApJ. 91410125A&ACrestani, J., Braga, V. F., Fabrizio, M., et al. 2021, ApJ, 914, 10 da Silva, R., Lemasle, B., Bono, G., et al. 2016, A&A, 586, A125\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G De Somma, M Marconi, S Cassisi, MNRAS. 5081473De Somma, G., Marconi, M., Cassisi, S., et al. 2021, MNRAS, 508, 1473\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical and IR Telescope Instrumentation and Detectors. H Dekker, S D'odorico, A Kaufer, B Delabre, H Kotzlowski, Proc. SPIE. M. Iye & A. F. MoorwoodSPIE4008Dekker, H., D'Odorico, S., Kaufer, A., Delabre, B., & Kotzlowski, H. 2000, in Proc. SPIE, Vol. 4008, Optical and IR Telescope Instrumentation and Detec- tors, ed. M. Iye & A. F. Moorwood, 534-545\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E A Den Hartog, J E Lawler, C Sneden, J J Cowan, A Brukhovesky, ApJS. 24333Den Hartog, E. A., Lawler, J. E., Sneden, C., Cowan, J. J., & Brukhovesky, A. 2019, ApJS, 243, 33\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E A Den Hartog, M P Ruffoni, J E Lawler, ApJS. 21523Den Hartog, E. A., Ruffoni, M. P., Lawler, J. E., et al. 2014, ApJS, 215, 23\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Dessauges-Zavadsky, F Calura, J X Prochaska, S D'odorico, F Matteucci, A&A. 470431Dessauges-Zavadsky, M., Calura, F., Prochaska, J. X., D'Odorico, S., & Mat- teucci, F. 2007, A&A, 470, 431\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V D'orazi, S Desidera, R G Gratton, A&A. 59819D'Orazi, V., Desidera, S., Gratton, R. G., et al. 2017, A&A, 598, A19\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V D'orazi, E Oliva, A Bragaglia, A&A. 63338D'Orazi, V., Oliva, E., Bragaglia, A., et al. 2020, A&A, 633, A38\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Duffau, E Caffau, L Sbordone, A&A. 604128Duffau, S., Caffau, E., Sbordone, L., et al. 2017, A&A, 604, A128\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N R Evans, AJ. 104216Evans, N. R. 1992, AJ, 104, 216\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N R Evans, D L Welch, M H Slovak, Thomas G Barnes, I Moffett, T J , AJ. 1061599Evans, N. R., Welch, D. L., Slovak, M. H., Barnes, Thomas G., I., & Moffett, T. J. 1993, AJ, 106, 1599\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B.-Q For, C Sneden, AJ. 1401694For, B.-Q. & Sneden, C. 2010, AJ, 140, 1694\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W L Freedman, B F Madore, T Hoyt, ApJ. 89157Freedman, W. L., Madore, B. F., Hoyt, T., et al. 2020, ApJ, 891, 57\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Fukue, N Matsunaga, R Yamamoto, ApJ. 81264Fukue, K., Matsunaga, N., Yamamoto, R., et al. 2015, ApJ, 812, 64\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Genovali, B Lemasle, G Bono, A&A. 56637Genovali, K., Lemasle, B., Bono, G., et al. 2014, A&A, 566, A37\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Genovali, B Lemasle, G Bono, A&A. 554132Genovali, K., Lemasle, B., Bono, G., et al. 2013, A&A, 554, A132\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Genovali, B Lemasle, R Da Silva, A&A. 58017Genovali, K., Lemasle, B., da Silva, R., et al. 2015, A&A, 580, A17\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Gilmore, S Randich, M Asplund, The Messenger. 14725Gilmore, G., Randich, S., Asplund, M., et al. 2012, The Messenger, 147, 25\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M A T Groenewegen, A&A. 48825Groenewegen, M. A. T. 2008, A&A, 488, 25\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M A Groenewegen, A&A. 6198Groenewegen, M. A. T. 2018, A&A, 619, A8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M A T Groenewegen, A&A. 640113Groenewegen, M. A. T. 2020, A&A, 640, A113\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Inno, N Matsunaga, G Bono, ApJ. 76484Inno, L., Matsunaga, N., Bono, G., et al. 2013, ApJ, 764, 84\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Inno, N Matsunaga, M Romaniello, A&A. 57630Inno, L., Matsunaga, N., Romaniello, M., et al. 2015, A&A, 576, A30\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Inno, M A Urbaneja, N Matsunaga, MNRAS. 48283Inno, L., Urbaneja, M. A., Matsunaga, N., et al. 2019, MNRAS, 482, 83\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Jian, D Taniguchi, N Matsunaga, MNRAS. 4941724Jian, M., Taniguchi, D., Matsunaga, N., et al. 2020, MNRAS, 494, 1724\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Kaufer, O Stahl, S Tubbesing, The Messenger. 958Kaufer, A., Stahl, O., Tubbesing, S., et al. 1999, The Messenger, 95, 8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C S Kochanek, B J Shappee, K Z Stanek, PASP. 129104502Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, PASP, 129, 104502\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V V Kovtyukh, MNRAS. 378617Kovtyukh, V. V. 2007, MNRAS, 378, 617\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V V Kovtyukh, S M Andrievsky, S I Belik, R E Luck, AJ. 129433Kovtyukh, V. V., Andrievsky, S. M., Belik, S. I., & Luck, R. E. 2005, AJ, 129, 433\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Kramida, R Yu, J Reader, & the NIST ASD Team. 2020, NIST Atomic Spectra Database. version 5.8Kramida, A., Yu., R., Reader, J., & the NIST ASD Team. 2020, NIST Atomic Spectra Database, version 5.8\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R L Kurucz, I Furenlid, J Brault, L Testerman, Solar flux atlas from 296 to 1300 nm. Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, Solar flux atlas from 296 to 1300 nm\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J E Lawler, A Guzman, M P Wood, C Sneden, J J Cowan, ApJS. 20511Lawler, J. E., Guzman, A., Wood, M. P., Sneden, C., & Cowan, J. J. 2013, ApJS, 205, 11\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Lemasle, P François, K Genovali, A&A. 55831Lemasle, B., François, P., Genovali, K., et al. 2013, A&A, 558, A31\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Lemasle, M A T Groenewegen, E K Grebel, A&A. 60885Lemasle, B., Groenewegen, M. A. T., Grebel, E. K., et al. 2017, A&A, 608, A85\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Lemasle, M Hanke, J Storm, G Bono, E K Grebel, A&A. 64171Lemasle, B., Hanke, M., Storm, J., Bono, G., & Grebel, E. K. 2020, A&A, 641, A71\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Limongi, A Chieffi, ApJ. 592404Limongi, M. & Chieffi, A. 2003, ApJ, 592, 404\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Liu, N Arav, D S N Rupke, ApJS. 2219Liu, G., Arav, N., & Rupke, D. S. N. 2015, ApJS, 221, 9\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E Luck, S M Andrievsky, AJ. 128343Luck, R. E. & Andrievsky, S. M. 2004, AJ, 128, 343\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E Luck, S M Andrievsky, A Fokin, V V Kovtyukh, AJ. 13698Luck, R. E., Andrievsky, S. M., Fokin, A., & Kovtyukh, V. V. 2008, AJ, 136, 98\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E Luck, S M Andrievsky, V V Kovtyukh, W Gieren, D Graczyk, AJ. 14251Luck, R. E., Andrievsky, S. M., Kovtyukh, V. V., Gieren, W., & Graczyk, D. 2011, AJ, 142, 51\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R E Luck, D L Lambert, AJ. 142136Luck, R. E. & Lambert, D. L. 2011, AJ, 142, 136\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Magurno, C Sneden, V F Braga, ApJ. 86457Magurno, D., Sneden, C., Braga, V. F., et al. 2018, ApJ, 864, 57\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Marconi, R Molinaro, G Bono, ApJ. 7686Marconi, M., Molinaro, R., Bono, G., et al. 2013, ApJ, 768, L6\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Marconi, I Musella, G Fiorentino, ApJ. 632590Marconi, M., Musella, I., & Fiorentino, G. 2005, ApJ, 632, 590\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Matsunaga, M Jian, D Taniguchi, S S Elgueta, MNRAS. 5061031Matsunaga, N., Jian, M., Taniguchi, D., & Elgueta, S. S. 2021, MNRAS, 506, 1031\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Mayor, F Pepe, D Queloz, The Messenger. 11420Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Mcwilliam, G Wallerstein, M Mottini, ApJ. 778149McWilliam, A., Wallerstein, G., & Mottini, M. 2013, ApJ, 778, 149\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Meléndez, B Barbuy, A&A. 497611Meléndez, J. & Barbuy, B. 2009, A&A, 497, 611\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A&A proofs: manuscript no. daSilvaetal2021. A&A proofs: manuscript no. daSilvaetal2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J H Minniti, L Sbordone, A Rojas-Arriagada, A&A. 64092Minniti, J. H., Sbordone, L., Rojas-Arriagada, A., et al. 2020, A&A, 640, A92\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J H Minniti, M Zoccali, A Rojas-Arriagada, A&A. 654138Minniti, J. H., Zoccali, M., Rojas-Arriagada, A., et al. 2021, A&A, 654, A138\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Solar Spectrum 2935 Å to 8770 Å. C E Moore, M Minnaert, J Houtgast, Moore, C. E., Minnaert, M., & Houtgast, J. 1966, The Solar Spectrum 2935 Å to 8770 Å\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Nardetto, J H Groh, S Kraus, F Millour, D Gillet, A&A. 4891263Nardetto, N., Groh, J. H., Kraus, S., Millour, F., & Gillet, D. 2008, A&A, 489, 1263\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Nardetto, D Mourard, P Kervella, A&A. 453309Nardetto, N., Mourard, D., Kervella, P., et al. 2006, A&A, 453, 309\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Nardetto, E Poretti, A Gallenne, A&A. 61692Nardetto, N., Poretti, E., Gallenne, A., et al. 2018, A&A, 616, A92\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Natale, M Marconi, G Bono, ApJ. 67493Natale, G., Marconi, M., & Bono, G. 2008, ApJ, 674, L93\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J R Neeley, M Monelli, M Marengo, ApJ. 920152Neeley, J. R., Monelli, M., Marengo, M., et al. 2021, ApJ, 920, 152\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H R Neilson, M Cantiello, N Langer, A&A. 5299Neilson, H. R., Cantiello, M., & Langer, N. 2011, A&A, 529, L9\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Niedermann, M S Sloth, Phys. Rev. D. 10263527Niedermann, F. & Sloth, M. S. 2020, Phys. Rev. D, 102, 063527\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P E Nissen, Y Q Chen, M Asplund, M Pettini, A&A. 415993Nissen, P. E., Chen, Y. Q., Asplund, M., & Pettini, M. 2004, A&A, 415, 993\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T R O'brian, M E Wickliffe, J E Lawler, W Whaling, J W Brault, Journal of the Optical Society of America B Optical Physics. 81185O'Brian, T. R., Wickliffe, M. E., Lawler, J. E., Whaling, W., & Brault, J. W. 1991, Journal of the Optical Society of America B Optical Physics, 8, 1185\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J W Pel, A&AS. 24413Pel, J. W. 1976, A&AS, 24, 413\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Pietrzyński, D Graczyk, W Gieren, Nature. 49576Pietrzyński, G., Graczyk, D., Gieren, W., et al. 2013, Nature, 495, 76\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Aghanim, Planck CollaborationY Akrami, Planck CollaborationA&A. 6416Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A6\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P G Prada Moroni, M Gennaro, G Bono, ApJ. 749108Prada Moroni, P. G., Gennaro, M., Bono, G., et al. 2012, ApJ, 749, 108\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Proxauf, R Da Silva, V V Kovtyukh, A&A. 61682Proxauf, B., da Silva, R., Kovtyukh, V. V., et al. 2018, A&A, 616, A82\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Randich, G Gilmore, -Eso Gaia, Consortium, The Messenger. 15447Randich, S., Gilmore, G., & Gaia-ESO Consortium. 2013, The Messenger, 154, 47\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A G Riess, S Casertano, W Yuan, L M Macri, D Scolnic, ApJ. 87685Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ, 876, 85\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Ripepi, G Catanzaro, G Clementini, arXiv:2201.01126arXiv e-printsRipepi, V., Catanzaro, G., Clementini, G., et al. 2022, arXiv e-prints, arXiv:2201.01126\",\n \"doi\": \"arXiv:2201.01126\"\n },\n {\n \"text\": \". V Ripepi, G Catanzaro, R Molinaro, MNRAS. 5084047Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2021, MNRAS, 508, 4047\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Ripepi, G Catanzaro, R Molinaro, A&A. 642230Ripepi, V., Catanzaro, G., Molinaro, R., et al. 2020, A&A, 642, A230\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Ripepi, R Molinaro, I Musella, A&A. 62514Ripepi, V., Molinaro, R., Musella, I., et al. 2019, A&A, 625, A14\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Romaniello, F Primas, M Mottini, A&A. 488731Romaniello, M., Primas, F., Mottini, M., et al. 2008, A&A, 488, 731\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Romaniello, A Riess, S Mancino, arXiv:2110.08860arXiv e-printsRomaniello, M., Riess, A., Mancino, S., et al. 2021, arXiv e-prints, arXiv:2110.08860\",\n \"doi\": \"arXiv:2110.08860\"\n },\n {\n \"text\": \". M P Ruffoni, E A Hartog, J E Lawler, MNRAS. 4413127Ruffoni, M. P., Den Hartog, E. A., Lawler, J. E., et al. 2014, MNRAS, 441, 3127\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Scargle, ApJ. 263835Scargle, J. D. 1982, ApJ, 263, 835\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B J Shappee, J L Prieto, D Grupe, ApJ. 78848Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D M Skowron, J Skowron, P Mróz, Acta Astron. 69305Skowron, D. M., Skowron, J., Mróz, P., et al. 2019, Acta Astron., 69, 305\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Sneden, The MOOG code. Sneden, C. 2002, The MOOG code, http://www.as.utexas.edu/~chris/ moog.html\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S G Sousa, N C Santos, V Adibekyan, E Delgado-Mena, G Israelian, A&A. 57767Sousa, S. G., Santos, N. C., Adibekyan, V., Delgado-Mena, E., & Israelian, G. 2015, A&A, 577, A67\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S G Sousa, N C Santos, G Israelian, M Mayor, M J P F G Monteiro, A&A. 469783Sousa, S. G., Santos, N. C., Israelian, G., Mayor, M., & Monteiro, M. J. P. F. G. 2007, A&A, 469, 783\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K G Strassmeier, T Granzer, M Weber, Astronomische Nachrichten. 325527Strassmeier, K. G., Granzer, T., Weber, M., et al. 2004, Astronomische Nachrichten, 325, 527\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K G Strassmeier, T Granzer, M Weber, Advances in Astronomy. 970306Strassmeier, K. G., Granzer, T., Weber, M., et al. 2010, Advances in Astronomy, 2010, 970306\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Takeda, M Omiya, H Harakawa, B Sato, PASJ. 6881Takeda, Y., Omiya, M., Harakawa, H., & Sato, B. 2016, PASJ, 68, 81\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Taniguchi, N Matsunaga, M Jian, MNRAS. 5024210Taniguchi, D., Matsunaga, N., Jian, M., et al. 2021, MNRAS, 502, 4210\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B J Taylor, PASP. 106452Taylor, B. J. 1994, PASP, 106, 452\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Vasilyev, H G Ludwig, B Freytag, B Lemasle, M Marconi, A&A. 61119Vasilyev, V., Ludwig, H. G., Freytag, B., Lemasle, B., & Marconi, M. 2018, A&A, 611, A19\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K A Venn, M D Shetrone, M J Irwin, ApJ. 751102Venn, K. A., Shetrone, M. D., Irwin, M. J., et al. 2012, ApJ, 751, 102\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Weber, T Granzer, K G Strassmeier, Proc. SPIE. SPIE845184510Weber, M., Granzer, T., & Strassmeier, K. G. 2012, in Proc. SPIE, Vol. 8451, Software and Cyberinfrastructure for Astronomy II, 84510K\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M P Wood, J E Lawler, C Sneden, J J Cowan, ApJS. 20827Wood, M. P., Lawler, J. E., Sneden, C., & Cowan, J. J. 2013, ApJS, 208, 27\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Yong, B W Carney, M L Teixera De Almeida, B L Pohl, AJ. 1312256Yong, D., Carney, B. W., Teixera de Almeida, M. L., & Pohl, B. L. 2006, AJ, 131, 2256\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Inaf -Osservatorio Astronomico Di Padova, 535122Padova, Italy4 INAF -Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Zentrum für Astronomie der Universität Heidelberg. Heidelberg, Germany69120Astronomisches Rechen-Institut, Zentrum für Astronomie der Uni- versität Heidelberg, Mönchhofstr. 12-14, 69120, Heidelberg, Ger- many\",\n \"doi\": \"\"\n },\n {\n \"text\": \"DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte. Blegdamsvej. 1761Niels Bohr International Academy, Niels Bohr Institute ; PSL Research UniversityCNRSNiels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 7 INAF -Osservatorio Astronomico di Capodimonte, Napoli, Italy 8 GEPI, Observatoire de Paris, PSL Research University, CNRS, 61\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Avenue de l'Observatoire. 75014Paris, FranceAvenue de l'Observatoire, 75014, Paris, France\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department. Koninklijke Sterrenwacht Van België, Naples, ItalyParthenope University of NaplesKoninklijke Sterrenwacht van België, Ringlaan 3, 1180, Brussels, Belgium 11 Science and Technology Department, Parthenope University of Naples, Naples, Italy\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Germany 14 Institute for Astronomy. München, 2680University of Hawaii at ManoaMünchen, Germany 14 Institute for Astronomy, University of Hawaii at Manoa, 2680\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH). Woodlawn Drive, Honolulu, Hongo, Bunkyo-Ku, E- 38200Japan 17 Instituto de Astrofísica de Canarias (IAC). Motoyama, Kamigamo, Kita-ku, Kyoto; La Laguna; La Laguna, Spain; Teramo, Italy19USA 15 Department of Astronomy, School of Science, The University of Tokyo ; Kyoto Sangyo University ; Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULLE-38205Koyama Astronomical Observatory. Via M. Maggini, s/n, I-64100Woodlawn Drive, Honolulu, HI 96822, USA 15 Department of Astronomy, School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 16 Laboratory of Infrared High-Resolution spectroscopy (LiH), Koyama Astronomical Observatory, Kyoto Sangyo University, Mo- toyama, Kamigamo, Kita-ku, Kyoto 603-8555, Japan 17 Instituto de Astrofísica de Canarias (IAC), La Laguna, E-38205, Spain 18 Departamento de Astrofísica, Universidad de La Laguna (ULL), E- 38200, La Laguna, Spain 19 INAF-Osservatorio Astronomico d'Abruzzo, Via M. Maggini, s/n, I-64100, Teramo, Italy\",\n \"doi\": \"E- 38200\"\n },\n {\n \"text\": \"An der Sternwarte 16. Astrophysik Leibniz-Institut Für, Potsdam, AIPPotsdam, GermanyLeibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482, Potsdam, Germany\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Nice Cedex. 7293Université Côte d'Azur, Observatoire de la CôteCSUniversité Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, La- grange UMR 7293, CS 34229, 06304, Nice Cedex 4, France\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":33,"cells":{"text":{"kind":"string","value":"\nPOLAR CODES' SIMPLICITY, RANDOM CODES' DURABILITY\n\n\nHsin-Po Wang \nIwan Duursma \nPOLAR CODES' SIMPLICITY, RANDOM CODES' DURABILITY\n\nOver any discrete memoryless channel, we build codes such that: for one, their block error probabilities and code rates scale like random codes'; and for two, their encoding and decoding complexities scale like polar codes'. Quantitatively, for any constants π, ρ > 0 such that π + 2ρ < 1, we construct a sequence of error correction codes with block length N approaching infinity, block error probability exp(−N π ), code rate N −ρ less than the Shannon capacity, and encoding and decoding complexity O(N log N ) per code block. The putative codes take uniform ς-ary messages for sender's choice of prime ς. The putative codes are optimal in the following manner: Should π + 2ρ > 1, no such codes exist for generic channels regardless of alphabet and complexity.\n\nIntroduction\n\nRichard W. Hamming is one of the first few people who had the idea that by grouping information in blocks with redundancies, a calculating machine can correct errors by its own and proceed to the next command instead of halting. Their solution, now called Hamming codes, is found in [Ham50]. Claude E. Shannon, a colleague of Hamming in Bell labs, theorized the communication channels and showed that a channel associates to a number called capacity, which represents the ultimate limit of the efficiency of communications over that channel.\n\nTo brief the rest of the history, we follow the analogy [AW14] used. Shannon's eternal result, noisy channel coding theorem [Sha48], is considered the analog of the law of large numbers (LLN). The theorem implies that there exists a sequence of longer and longer block codes whose block error probabilities approach 0 and code rates approach the capacity, which is analogous to that the empirical average of random variables is close to the mean with high probability. Robert G. Gallager, Shannon, Robert M. Fano, and followers extended the LLN result by looking at how the block error probability P e scales when the code rate R is fixed. They showed that the error probability P e scales like exp(−E r (R)N ). Here N is the block length, E r (R) is a constant depending on R. This paradigm is considered the analog of the large deviations principle (LDP). See [Fan61,Gal65,SGB67,Gal68,Gal73,Bla74,BF02,iFLM11,DZF16]. Meanwhile, a series of works fix the error probability P e and looked at how the code rate R scales [Wol57,WEI60,Dob61,Str62,BKB04,Hay09,PPV10]. They showed that the code rate R scales like I − Q −1 (P e ) V /N for I the capacity, Q −1 the inverse of the standard Q-function, and V an intrinsic parameter of the channel. The parameter V is called the dispersion or varentropy by different authors. It is the \"variance\" of the channel while I is the \"mean\" of the channel. This turns out to be more than an analog-the random variable log(W (Y | X)/W out (Y )) called information density or information spectrum has mean I and variance V . This paradigm is considered the analog of the central limit They showed that the quantity N (I − R) 2 /|log P e | converges to 2V , twice the very dispersion appearing in the CLT paradigm. This paradigm is considered the analog of the moderate deviations principle (MDP). On a parallel track, the engineering aspects of the communication theory thrive. Codes with excellent practicality are proposed. To name a few, Reed-Muller (1964), trellis modulation (1970s), turbo (1990s), low-density parity-check (1963,1996), Repeat-accumulate (1998), Fountain (1998), and polar (2009).\n\nAmong the long list of inventions, only trellis modulation, low-density paritycheck, and polar achieve the LLN paradigm over nontrivial channels-they are capacity-achieving. Among these three, polar stands out as the only code that achieves the CLT paradigm (optimally), the only code that achieves the LDP paradigm (optimally), and the only code that achieves the MDP paradigm (suboptimally). If only polar code achieves the optimal MDP paradigm. We brief the history of polar codes below. Unless stated otherwise, I means the symmetric capacity in the next three paragraphs.\n\nErdal Arıkan's original works on channel polarization [Ari08,Ari09] established the foundation of polar codes, placing polar codes in the LLN paradigm on day one. Arıkan and Telatar [AT09] characterized the LDP behavior of polar codes, showing that P e scales like exp(− √ N ) when an R < I is fixed. Later, Korada-Şaşoglu-Urbanke [KSU10] generalized polar codes from Arikan's kernel [ 1 1 0 1 ] to any invertible -by-matrix G, granted that 2 and G is not column-equivalent to a lower triangular matrix. And then they showed that the LDP behavior is P e ≈ exp(−N Ec(G) ) where E c (G) is a constant depending on the kernel matrix G. The notation E c (G) is meant to resemble Gallager's error exponent E r (R) but the former is at this level exp(−N • ) while the latter is at this level exp(−•N ). The LDP behavior of polar codes is then refined in [HMTU13]. Therein, P e is approximated by exp(− E ) where E = E c (G)n − V c (G)nQ −1 (R/I) + o( √ n) is a more accurate exponent, is the matrix dimension, n is the depth of the code, and V c (G) is another constant depending on G. The notation V c (G) is meant to resemble the channel dispersion V . Appearing to be a CLT behavior, this result lies in the corner of the LDP paradigm that touches the MDP paradigm. Finally, Mori-Tanaka [MT14] generalized everything above to channels of prime power input size. Over arbitrary input alphabets, [ŞTA09a,Sas11] Note that results utilizing different kernels over various channels are mixed. The higher ρ, π, the better performance. The curve part of [WD18] is ρ = 1 − h 2 (π). [Ari09,AT09]. Over binary but asymmetric channels, [SRDR12,HY13] showed the counterpart of [Ari09,AT09] with I being the Shannon capacity. No further result on the LDP side, e.g. over non-binary asymmetric channels, is known. The present work fills the gap.\n\nThe CLT behavior of polar codes turns out to be difficult to characterize. It was Korada-Montanari-Telatar-Urbanke [KMTU10] who came up with the idea that approximating an eigenfunction tightly bounds the eigenvalue −ρ . Here ρ > 0 is a number such that R scales like I − N −ρ with a fixed P e . They had 0.2669 ρ 0.2841 over binary erasure channels (BECs). The upper bound was brought down to 3.553ρ 1 [GHU12]. Hassini-Alishahi-Urbanke [HAU14] lifted the lower bound to 3.627ρ 1 over BECs and proved a lower bound 6ρ 1 over binaryinput discrete-output memoryless channels (BDMCs). The latter is suboptimal so [GB14,MHU16] improved the bound to 5.702ρ 1 and to 4.714ρ 1. Additive white Gaussian noise channles (AWGNCs) have continuous output alphabet, but [FT17] show that they have 4.714ρ 1 too. Over BECs particularly, [FV14,YFV19] examined a series of larger kernels; the current record is a 64-by-64 kernel believed to have 2.9ρ 1. Near the end of the road to 2ρ < 1, [PU16] showed that by allowing q → ∞, Reed-Solomon kernels achieve 2ρ < 1 over q-ary channels. This does not really prove that polar codes achieve 2ρ < 1 over any specific channel, but gave hopes. Fazeli-Hassani-Mondelli-Vardy [FHMV17,FHMV18], eventually, showed that large random kernels achieve 2ρ < 1 over BECs, breaking the barrier. Guruswami-Riazanov-Ye [GRY19] extended their result to all BDMCs utilizing the dynamic kernel technique. Over the remaining channels, the present work fills the gap.\n\nBetween LDP and CLT is polar codes' MDP behavior. Guruswami-Xia [GX13] showed that there exists ρ > 0 such that P e scales like exp(−N 0.49 ) while R scales like I − N −ρ over BDMCs. This raised a question about what are the possible pairs (π, ρ) such that (P e , R) scales like (exp(−N π ), I − N −ρ ). Mondelli-Hassani-Urbanke [MHU16] answered this, partially, in the same paper they bounded ρ. They showed that under a certain curve connecting (0, 1/5.714) and (1/2, 0) all (π, ρ) are achievable over BDMCs. For BECs the upper left corner is (0, 1/4.627). A straightforward generalization to AWGNCs was also given in [FT17]. We in [WD18] improved their result, suggesting that via a combinatorial trick the upper left corner of the curve is (0, ρ) for any ρ that is valid in the CLT regime. The same trick also implicated that over BECs all (π, ρ) such that π + 2ρ < 1 are achievable, which is mainly owing to [FHMV17]'s result that 2ρ < 1 over BECs is achievable. Meanwhile, [BGN + 18] made the first step to investigate the general kernel matrices over general prime-ary channels. They showed that it is possible to achieve ρ > 0 with P e ≈ N −Ω(1) . This is, strictly speaking, \"only\" a CLT behavior as the desired error probability in the MDP world is exp(−N π ). Later, B lasiok-Guruswami-Sudan [BGS18] were able to show that for all π < E c (G) there exists ρ > 0 such that (π, ρ) is achievable. This makes it a direct generalization of [GX13] to all polarizing kernel matrices G over all prime-ary channels. Over the remaining channels, the present work fills the gap.\n\nThe following works, though not counting as predecessors of ours, have impact on us through their insights on the essence of the channel polarization: [Kor09, HKU09, KU10, Ari10, CK10, Mor10, SP11, Sas12b, Sas12a, TV13, Has13, PB13, TV15, Mon16, Nas16,Nas17].\n\nI resumes to be the Shannon capacity. Readers are now prepared to be presented the main theorem.\n\nTheorem 1 (the main theorem-polar codes' simplicity, random codes' durability). Let W be any discrete memoryless channel. Fix a prime ς 2. Fix constants π, ρ > 0 such that π + 2ρ < 1. There exists a sequence of block codes with encoding and decoding algorithms such that: (cs) the codes accept uniform ς-ary messages. (cn) the block length N approaches infinity; (cp) the block error probability falls below exp(−N π ); (cr) the code rate exceeds I − N −ρ ; and (cc) the encoding and decoding complexity is O(N log N ) per code block.\n\nThe proof of the main theorem spans over Sections 2 to 8, lemmas continuing in Appendices A to C. The entry points are Sections 2.1 for (cs), 2.2 for (cn), 3.2 for (cc), 4.2 for (cp), and 6.3 for (cr). The main theorem is optimal in the following manner.\n\nProposition 2 (optimality). Fix π, ρ > 0 such that π + 2ρ > 1. Assume V > 0. For the rest of the section, we outline the ideas to prove Theorem 1. The proof is a straightforward remix of polar coding techniques and random coding techniques if it were not for a few hurdles.\n\nHurdle of input alphabet size: The majority of the polar coding theory assumes that the input alphabet of the underlying channel is binary, of prime size, or, less likely, of prime power size. But the main theorem aims for arbitrary finite alphabets.\n\nFinite alphabets do possess polarization behavior but the speed of polarization has room for improvement [Sas11,Theorem 3.5]. We will overcome this by adding \"dummy symbols\" into the input alphabet to make it a prime power.\n\nHurdle of asymmetric channel: Although asymmetric channels do polarize, the input distributions do not automatically become the uniform distribution. Precomposing a source coding machinery helps generate the desired distribution and has been proposed before [ŞTA09a, Section III.D] [Ari10, Section IV]. On the other hand Honda-Yamanoto [HY13] showed that one polar code can do both source coding and noisy channel coding at once. We borrow their idea.\n\nHurdle of kernel selection: Judging and identifying the best-behaved kernel gets harder as we need finer descriptions of the performance of the code. The good result for the BEC case depends heavily on the erasure nature of the channels (that they are ordered by their capacities). Other general results are not strong enough to meet our goal. To overcome, we borrow a technique called dynamic kernels from [YB15]. The idea is to prepare more than one polarizing kernel and apply the proper one on a channel-by-channel basis. This makes a paradigm shift from one kernel fits all channels to every channel deserves a tailor-made kernel. We will, once per channel, apply the random coding theory to show the existence of a proper kernel.\n\nHurdle of output alphabet size: Even with the great freedom to choose one kernel for each and every channel, there lies the difficulty that some performance bounds are proven with one fixed channel in mind to favor the big-O notations. Those bounds are prone to depend on the size of the output alphabet, which grows to infinity as the channel transformations take place. Meanwhile, some universal bounds are proven that depend only on the size of the input alphabet, which is invariant under channel transformations. We will borrow a bound derived in [CS07,DCS14].\n\n1.1. Organization. Section 2 reviews channels and entropy notations; Section 2.1 explains how to overcome the hurdle of arbitrary input alphabet size. Section 3 reviews the channel transformations; Section 3.1 designs the decoder; Section 3.2 analyzes its complexity; Section 3.3 designs the encoder, overcoming the hurdle of asymmetric channel. Section 4 reviews the channel parameters such as the Bhattacharyya parameter; Section 4.2 shows how to control the block error probability. Section 5 reviews the channel processes; Section 5.1 argues that the global MDP behaviors of H(W n ) and H(V n ) imply the main theorem. The main theorem is thus reduced to the behavior of certain channel processes. Section 6 proves that the global MDP behavior we want holds granted that the local LDP and CLT behaviors hold, effectively boiling the main theorem down to the local behaviors. Section 6.2 introduces the random kernel trick and Section 6.3 introduces the dynamic kernel trick to overcome the hurdle of kernel selection. Section 7 confirms the local LDP behavior. The proof distills properties of the weight distribution of random codes. Sections 7.1 and 7.2 proves the two fundamental theorems of polar coding. Section 8 confirms the local CLT behavior. Contributions from Gallager and Hayashi are utilized. Section 8.2 invokes Chang-Sahai's universal bound, overcoming the hurdle of output alphabet size.\n\n1.2. Three families of randomnesses. The randomnesses from the sender's message, the channel, and the randomized rounding constitute the first family. Typeset in Roman font are random variables (U, X, Y, . . . ), probability measures (P, Q, W ), entropies (H, I), and other parameters (P e , Z, T, S . . . ) in this family. The randomness from the channel process, one main technique in the polar coding literature, is the second family. Typeset in sans serif font are stochastic processes (K n , W n , H n , Z n , . . . ), probability measure (P), and expectation (E ) in this family. The randomness from random kernel ensembles, the main technique in the random coding literature, is the third family. Typeset in blackboard bold font are random variables (G, X, K), probability measure (P, with exceptions), expectation (E), and Kullback-Leibler divergence (D, with exceptions) in this family.\n\n\nChannel and Entropy Preliminaries\n\nA discrete memoryless channel is a Markov chain W : X → Y. Here X is a finite set of input alphabet; Y is a finite set of output alphabet; and W is an array of transition probabilities W (y | x) ∈ [0, 1] for all x ∈ X and y ∈ Y. The numbers satisfy y∈Y W (y | x) = 1 for all x ∈ X , which represents the fact that each x must be transitioned to some unique y. When X and Y are clear from the context, we call W a channel. Although the input distribution is not part of the channel data, we write W in (x) to denote the input distribution. When W in (x) is understood from the context, we write W (x, y) to denote the joint distribution W (y | x)W in (x), write W out (y) to denote the output distribution, and write W (x | y) to denote the a posteriori probability W (x, y)/W out (y). (Thus the interpretation of W (• | •) depends on the arguments and the context.) A tuple of inputs (x i , x i+1 , . . . , x j ) is abbreviated as x j i . Same for y j i for tuple of outputs, and for u j i for general variables. We assume memoryless channels, and write W (y 1 | x 1 ) to denote the product measure i=1 W (y i | x i ) for consecutive usages. We write W in (x 1 ), W out (y 1 ), W (x 1 , y 1 ), and W (x 1 | y 1 ) to denote the input, output, joint, and a posteriori probabilities.\n\nLet X, Y be two r.v.s (random variables). Let H(X), H(X | Y ), and I(X ; Y ) be the standard entropy, conditional entropy, and mutual information. The base of the logarithm will be assigned later. When X is the input fed into some channel W : X → Y and Y is the corresponding output, we say H(W ) and I(W ) to mean H(X | Y ) and I(X ; Y ). When the distribution of X (the input distribution) is chosen to maximize I(W ), it is called the capacity-achieving input distribution and I(W ) is called the (Shannon) capacity of the channel W : X → Y. Unless stated otherwise, the input distributions will be capacity-achieving.\n\n2.1. Reduce input size to prime power. Immediately after we declared what channels are concerned (those with finite input and output alphabets), we show that it suffices to consider input alphabets of prime power size. Let W : X → Y be a channel. Let the input alphabet X be of size s. Let q be any prime power greater than or equal to s. Degrade the channel W as follows: Let symbols in X be ξ 1 , ξ 2 , . . . , ξ s . Let ξ s+1 , ξ s+2 , . . . , ξ q be q − s extra symbols. Let X be X ∪ {ξ s+1 , ξ s+2 , . . . , ξ q }; this is the extended alphabet. Define a dummy channel : X → X by letting (ξ min(i,s) | ξ i ) be 1 for all i = 1, 2, . . . , , q. That is, all extra symbols collapse to ξ s while the old symbols remain. The composition of the two channels\nW • : X −→ X W −→ Y\nforms a degraded channel with prime power input size. By the data processing inequality, the Shannon capacity of the degraded channel W • is no greater than W 's Shannon capacity. Meanwhile, it is clear that the degraded channel W • achieves W 's capacity by the same input distribution, ignoring extra symbols. In other words, I(W • ) = I(W ). This constitutes the input size reduction.\n\nHereafter, we assume the size of the input alphabet X is q, where q is a prime power. If the sender wants to send uniform binary messages, let q be a power of 2. If the sender wants to send uniform quaternary messages, let q be a power of 3. In case the sender wishes to send uniform quaternary messages but does not want to split an information bit over two channel symbols, let q be a power of 4. Bonus: should the sender want to send uniform senary messages, choose q 2 a power of 2 and q 3 a power of 3 such that q 2 q 3 is a power of 6; then alternate between q = q 2 and q = q 3 . That is, the sender breaks every senary bit into a binary component and a ternary component, sends the binary component through the q = q 2 code block, and send the ternary component through the q = q 3 code block. For other message alphabets, apply the fundamental theorem of arithmetic.\n\nFix a q. Let F q be the finite field of order q (with the addition and multiplication structure). Identifying X with F q , we will use them interchangeably. We say that W is a q-ary channel when the variables X and Y are remotely relevant. It is worth keeping in mind that for inequalities in this work, q = 2 is the most difficult case and q 2 will be used silently.\n\nWe clarified (cs), there are (cn), (cc), (cp), and (cr) to go.\n\n2.2. On the message alphabet and the block length. The fact that we have some freedom to choose q blurs the meaning of the block length N since, say, a q 2 -bit bears twice as much message as a q-bit does. Notwithstanding, we would like to remind readers that multiplication and division of N by any constant do not alter the semantics of the main theorem. This is because O(N log N ) can absorb any constant; exp(−N π ) and N −ρ too can by fluctuating π and ρ a bit. A more series aftereffect is caused by mixing code blocks with distinct q. When the sender attempts to send uniform 30-ary messages, they choose q 2 , q 3 , q 5 s and switch among the three block codes. The q = q 2 blocks have their own block length N 2 just like the other blocks have N 3 or N 5 as block lengths. The de facto block length N , the minimal number of the channel usages before the receiver can decode everything sent so far, is thus three times the least common multiple of N 2 , N 3 , and N 5 . We claim without a proof (but it will be clear once we prove the rest of the main theorem) that it is possible to make N 2 = N 3 = N 5 and consequently N = 3N 2 . Again, increasing N by three-fold does not make any difference. For numbers with more prime factors, a similar reasoning applies.\n\nWe recommend readers not to worry about the message alphabet as there exists a powerful solution-to pre-compose another code that re-encodes an arbitrary finite message distribution (not necessarily uniform) to a uniform prime power-ary input distribution. The existence of such code, by duality, is tightly bonded to the existence of a error-correction code that carries uniform prime power-ary messages over channels of arbitrary arity. The latter is exactly what the main theorem concerns.\n\nWe clarified (cs) and (cn) in this section; there are (cc), (cp), and (cr) to go. We continue proving the main theorem in the next section.\n\n\nChannel Transformation\n\nLet 2 be an integer. This will be the dimension of the kernel matrices. But for now, let us introduce a flexible framework. Fix a q-ary channel W : X → Y. Let U 1 , U 2 , . . . , U be r.v.s taking values in X . For 1 i j , let U j i be the joint r.v. U i U i+1 · · · U j . Let g W : X → X be a bijective map; that is, H(U 1 | g W (U 1 )) = 0. We now feed X 1 := g W (U 1 ) into i.i.d. (independent and identically distributed) copies of the channel W . Let Y 1 ∈ Y be the corresponding output. The chain rule of conditional entropy reads\nH(U 1 | Y 1 ) = H(U | U −1 1 Y 1 ) + H(U −1 | U −2 1 Y 1 ) + · · · + H(U 1 | Y 1 ).\n(1)\n\nInterpretation: to estimate U 1 given Y 1 , we first estimate U 1 given Y 1 ; and then use the estimateÛ 1 to further estimate U 2 ; afterward, we estimate U 3 givenÛ 1 , U 2 , and Y 1 ; and so on. To achieve W 's capacity, g W (U 1 ) must follow a certain capacity-achieving distribution. Since g W is bijective, this induces a distribution of U 1 . (Remark: we imply nothing about whether U 1 , U 2 , . . . , U are i.i.d or not.) Fix this distribution, then\nI(U 1 ; Y 1 ) = I(U ; Y 1 | U −1 1 ) + I(U −1 ; Y 1 | U −2 1 ) + · · · + I(U 1 ; Y 1 ).\nThese two chain rules motivate the channel transformation:\nLet [ ] be the set of integers {1, 2, . . . , }. For each i ∈ [ ], let W (i) : X → X i−1 × Y be a channel where W (i) (u i−1 1 y 1 | u i ) is the probability that U i−1 1 Y 1 = u i−1 1 y 1 conditioned on U i = u i . A more lengthy but exact form reads W (i) (u i−1 1 y 1 | u i ) := u i+1 W (g W (u 1 ), y 1 ) ÷ u i−1 1 u i+1 W in (g W (u 1 )) .\n\nIts input distribution W\n\n(i) in (u i ) is determined by that of U i . It may sound weird that W (i) will tell the receiver the input of W (1) , W (2) , . . . , W (i−1) for free. But in reality, W (i) acts as an interactive device where the receiver (not the sender) needs to input what U i−1 1 is and the device will output something that looks like U i−1 1 Y 1 ; only when the receiver inputs the correct U i−1 1 does the device return the correct U i−1 1 Y 1 . Under this interpretation, the de facto capability of W (i) is thus\nI(U i ; Y 1 | U i−1 1 ) instead of I(U i ; U i−1 1\nY 1 ), which justifies the chain rule of the mutual information. To avoid confusion, we prefer H(W (i) ) over I(W (i) ) in calculations.\n\nWhat makes the idea of channel transformation powerful is that the transformations apply recursively. The precise formulation is as below: Fix any i ∈ [ ]. Let (X (i) ) 1 , (X (i) ) 2 , . . . , (X (i) ) ∈ X be i.i.d. copies of the capacity-achieving input of W (i) ; let (Y (i) ) 1 , (Y (i) ) 2 , . . . , (Y (i) ) ∈ X i−1 ×Y be the corresponding outputs. Let g W (i) : X → X be a bijection. Define a tuple of r.v.s (U (i) ) 1 := (g W (i) ) −1 ((X (i) ) 1 ); that is to say, g W (i) ((U (i) ) 1 ) = (X (i) ) 1 . For each j ∈ [ ], we define a depth-2 channel\n(W (i) ) (j) : X → X j−1 ×(X i−1 ×Y ) where (W (i) ) (j) ((u (i) ) i−1 1 (y (i) ) 1 | (u (i) ) j ) is the probability that (U (i) ) j−1 1 (Y (i) ) 1 = (u (i) ) j−1 1 (y (i) ) 1 conditioned on (U (i) ) j = (u (i) ) j .\nTo sum up, we can define (W (i) ) (1) , (W (i) ) (2) , . . . , (W (i) ) ( ) out of W (i) for any i ∈ [ ] in the same way we define W (1) , W (2) , . . . , W ( ) out of W . For i, j ∈ [ ], each (W (i) ) (j) is again a channel, so the transformations apply to generate depth-3 channels. In the setup of the classical polar coding, a fixed bijection g is used to define W (i) , (W (i) ) (j) , ((W (i) ) (j) ) (k) , et seq. To reach the optimal MDP paradigm, we\n(Step 1-a) output (W (1) (u 1 | y 1 ) : u 1 ∈ X ). (Step 1-b) inputû 1 . (Step 0) input (W (x i | y i ) : x i ∈ X ) for all i ∈ [ ]. (\nStep 4) output g W (û 1 ).\n\n(Step 2-a) output (W (2) (u 2 |û 1 y 1 ) : u 2 ∈ X ). (Step 2-b) input:û 2 .\n\n(Step 3-a) output (W (3) (u 3 |û 2 1 y 1 ) : u 3 ∈ X ). (Step 3-b) inputû 3 . Figure 2. A DU with = 3 and its I/Os. allow g W to depend on the channel W . That is to say, we need (presumably distinct) bijections g W (i) : X → X for every i ∈ [ ] when we want to define (W (i) ) (j) out of W (i) . Similarly, we need yet another 2 bijections g (W (i) ) (j) : X → X for every i, j ∈ [ ] in defining depth-3 channels. And the recursion goes on ad infinitum. 3.1. Design of the decoder. To implement channel transformations, we define a DU (decoding unit) to be an automata as follows: It is a box with pins on the left and pins on the right. Each pin is connected to another DU, a CH, an FH, or an IH (to be defined later). Each pin may take inputs or output but not at the same moment. A DU works as follows: Let W : X → Y be the channel it is to transform. (Step 0) For all i ∈ [ ], the i-th pin on the left takes the input y i . The input is passed in the form of the a posteriori distribution (W (x i | y i ) : x i ∈ X ). This is what Arıkan calls α-representation [Ari15b, Section II.A]. (Step 1-a) It computes the a posteriori distribution of U 1 given y 1 ; that is, (W (1) (u 1 | y 1 ) : u 1 ∈ X ). And then it outputs this tuple of probabilities to the first pin on the right. (Step 1b) At a later moment, it will receive an estimateû 1 of U 1 from the first pin on the right. Note thatû 1 is a hard symbol in X , not a soft tuple of probabilities. (Step 2-a) It computes the a posteriori distribution of U 2 givenû 1 y 1 ; that is to say, it pretends that U 1 happens to beû 1 and computes (W (2) (u 2 |û 1 y 1 ) : u 2 ∈ X ) accordingly. And then it outputs this tuple of probabilities to the second pin on the right. (Step 2-b) At a later moment, it will receive an estimateû 2 of U 2 from the second pin on the right. (Step i-a) In general, it computes W (i) (u i |û i−1 1 y 1 ) for all u i ∈ X and then output the tuple to the i-th pin on the right. (Step i-b) After a while, it will receiveû i . (Step + 1) Once it receivesû from the last pin on the right, it computesŷ 1 := g W (û 1 ), and then outputŷ i to the i-th pin on the left for all i ∈ [ ]. See Figures 2 to 4 for illustrations.\n\n\nDU\n\nThe general rule to arrange the DUs is as follows: For a depth-n construction, put DUs in an n−1 -by-n array. Each DU is indexed by (k 1 , k 2 , . . . , k n−1 ; m) where k 1 , k 2 , . . . , k n−1 ∈ [ ] and m ∈ [n]. For all m ∈ [n − 1] and all k 1 , k 2 , . . . , k n ∈ [ ], connect the k m -th pin on the right of the (k 1 , . . . , k m−1 , k m+1 , . . . , k n ; m)-th DU to the k m+1 -th pin on the left of the (k 1 , . . . , k m , k m+2 , . . . , k n ; m + 1)-th DU. Here, the (k 1 , k 2 , . . . , k n−1 ; m)-th DU is to transform the channel (· · · (W (k1) ) · · · ) (km) (1,1) W\n\n(1,1) Figure 3. 6 DUs are chained together to implement (W (1) ) (1) , . . . , (W (3) ) (3) . Boxes marked \"W \" are channels. Boxes next to channels are CHs; the labels are their indexes. Boxes at the rightmost column are either FHs or IHs; the labels are their indexes. Note that DUs in the first column use the same g W . DUs in the second column use g W (1) , g W (2) , and g W (3) , respectively.\n(1,2) W (1,2) (1,3) W (1,3) (2,1) W (2,1) (2,2) W (2,2) (2,3) W (2,3) (3,1) W (3,1) (3,2) W (3,2) (3,3) W (3,3)\ninto (· · · (W (k1) ) · · · ) (km+1) . The k 1 -th pin on the left of the (k 2 , . . . , k n ; 1)-th DU connects to a CH (channel helper) indexed by (k 1 , k 2 , . . . , k n ). Each CH then connects to the output of a copy of the channel W . The k n -th pin on the right of the (k 1 , . . . , k n−1 ; n)-th DU connects to either an FH (frozen bit helper) or an IH (information bit helper); in either case, the connected helper is indexed by (k 1 , k 2 , . . . , k n ). Let I ⊂ [ ] n be the set of indexes (k 1 , k 2 , . . . , k n ) such that the k n -th pin on the right of the (k 1 , . . . , k n−1 ; n)-th DU connects to an IH. Then [ ] n \\ I is the set of indexes where the pin connects to an FH. On the left hand side of the DU array, the task of the (k 1 , k 2 , . . . , k n )-th CH is to receive the channel output Y (k1,k2,...,kn) ∈ Y and then forward the a posteriori distribution (W (x (k1,k2,...,kn) | Y (k1,k2,...,kn) ) : x (k1,k2,...,kn) ∈ X ) to the DU array. On the right hand side, FHs correspond to what Arıkan called frozen bits-bits that do not carry information and the receiver knows their values as part of the communication protocol. The task of the (k 1 , k 2 , . . . , k n )-th FH is to receive the a posteriori distribution of the (k 1 , k 2 , . . . , k n )-th frozen bit and then return the correct symbol U (k1,k2,...,kn) ∈ X back to the DU array. IHs correspond to information bits that carry the sender's messages. The task of the (k 1 , k 2 , . . . , k n )-th IH is to receive the a posteriori distribution of the (k 1 , k 2 , . . . , k n )-th information bit and then return the most probable symbolÛ (k1,k2,...,kn) ∈ X back to the DU array. When all IHs are activated once, a code block completes. The most probable symbols they returned to the DU array form the decoded messageÛ I , meaning the tuple (Û (k1,k2,...,kn) : (k 1 , k 2 , . . . , k n ) ∈ I).\n\nWhat we just established is the successive cancellation decoder of polar codes that could be found in most works that implement polar codes. For instance, [Ari09, Section VIII], [Kor09, Section 3.2], and [HY13, Section III], and [EKMF + 17, Section Vi.B]. See especially [GRY19, Section 9] for an almost identical construction albeit they had q = 2 in mind. We replicate the whole story to demonstrate that each DU may use a unique bijection \"g\" without changing the overall structure too much. Whether or not this construction can transmit information reliably is . 12 DUs (with = 2) are chained together to implement ((W (1) ) (1) ) (1) , . . . , ((W (2) ) (2) ) (2) . DUs in the first column use g W ; DUs in the second column use g W (1) and g W (2) ; DUs in the third column use g (W (1) ) (1) , g (W (1) ) (2) , g (W (2) ) (1) , and g (W (2) ) (2) . discussed in Section 4.2. There, we will also clarify how to arrange FHs and IHs. The complexity can be estimated prior to further specification.\n\n3.2. Complexity of the decoder. There are various models that measure the complexity of a structure. The polar coding community uses a variant of the circuit complexity where the arithmetic of real numbers costs O(1) and passing probabilities between DUs costs O(1). The complexity of the DU array is thus the number of the DUs multiplied by the complexity of a single DU. The number of DUs is n−1 n. The complexity of a DU depends on how a DU computes the a posteriori probabilities W (j) (u i | u i−1 1 y 1 ) out of W (x i | y i ). The naïve approach is to exhaust all possible inputs u 1 ∈ X and compute the a posteriori probabilities using Bayesian formulas. This costs O( 10 q +10 ) (here 10 is an overestimate). Hence the overall complexity is O( n−1 n 10 q +10 ). In our setup, however, q is fixed, will be chosen upon knowing π, ρ, and n goes to infinity afterwards. So we advertise that the complexity is O( n n), or O(N log N ). Here N := n is the block length, equal to the number of copies of the channel W attached to the DU array. The complexities of the CHs, FHs, and IHs can be computed similarly. They are all bounded by O( n+10 q 10 ). Thus the decoder as a whole costs O(N log N ).\n\nWe claim that the encoder has the same complexity O(N log N ) although we have not defined the encoder yet. The encoder is essentially a special decoder and is the subject of the next subsection.\n\n3.3. Design of the encoder. The encoder will be an exact copy of the decoder except that CHs and IHs will behave differently. In greater detail: Let there be an n−1 -by-n array of DUs indexed and connected in the same way described in Section 3.1. Each DU executes the exact same task described in Section 3.1. The left pins of the DUs in the first column each connect to a CH. The right pins of the DUs in the last column each connect to the same type of device (an IH or an FH) as its twin-DU in the decoder does. Here, as part of the encoder, a CH will output the capacity-achieving input distribution (W in (x) for all x ∈ X ) into the DU array. For each (k 1 , k 2 , . . . , k n ) ∈ I, the (k 1 , k 2 , . . . , k n )-th IH will receive a recommended distribution of the (k 1 , k 2 , . . . , k n )-th information bit and then return the message symbol U (k1,k2,...,kn) ∈ X the sender wants to send back to the DU array. For each (k 1 , k 2 , . . . , k n ) ∈ [ ] n \\ I, the (k 1 , k 2 , . . . , k n )-th FH will receive a recommended distribution of the (k 1 , k 2 , . . . , k n )-th frozen bit and then return a r.v. U (k1,k2,...,kn) ∈ X that follows that distribution back to the DU array. This r.v. is simulated by a pseudo random number generator shared between the encoder and the decoder. The twin-FH in the decoder, regardless what distribution it receives, will return the exact same symbol U (k1,k2,...,kn) back to the DU array. This step is called randomized rounding and is found in [Kor09, Section 3.3], [KU10, Section III], [KT10, Section II], and [HY13, Section III.A]. After all IHs return the sender's messages and all FHs returns randomly rounded bits to the DU array, the CHs will each get a codeword symbol X (k1,k2,...,kn) ∈ X from the DU array. And then each CH will forward that symbol to an i.i.d. copy of the channel W .\n\nThis design is a copy of [HY13]'s encoder explained in our terminology. It is clear that the encoding complexity will be O(N log N ), too. Alongside the decoder, the encoder creates its own channel transformations. Let W : X → Y be a q-ary channel and X be a capacity-achieving input. Define a flattening channel W : X → {η} that erases all information. Then the encoder is effectively synthesizing depth-1 channels \nW (i) : X → X i−1 × {η} for each i ∈ [ ], depth-2 channels (W (i) ) (j) : X → X j−1 × (X i−1 × {η} ) for each j ∈ [ ], depth-3 channels ((W (i) ) (j) ) (k) : X → X k−1 × (X j−1 × (X i−1 × {η} ) ) for each k ∈ [ ],W (i) (u i−1 1 y 1 | u i ) is the probability that U i−1 1 = u i−1 1 conditioned on U i = u i , or equally u i+1 W in (g W (u 1 )) ÷ u i−1 1 u i+1 W in (g W (u 1 )) .\n\nMoreover, H(W ) = H(X) and H(W\n(i) ) = H(U i | U i−1 1 ).\nNo \"Y \" plays any role here since they are constant. The fact that a channel as boring as W is helpful to our main theorem will be covered later, in Section 4.2.\n\nWe clarified (cs), (cn), and (cc) up to this section; there are (cp) and (cr) to go.\n\n\nChannel Parameters\n\nLet W : X → Y be a q-ary channel. Let X be a capacity-achieving input and Y be the corresponding output. Besides H and I, there are several channel parameters that capture the qualities of channels. Here is a list of parameters extracted from the work [MT14] of Mori and Tanaka.\n\nBoth H(X | Y ) and H(W ) are the base-q conditional entropy, the base chosen such that 0\nH(X | Y ) H(X) 1. Both I(X | Y ) and I(W ) are the base-q mutual information, and hence 0 I(X | Y ) H(X) 1. P e (X | Y )\nis the error probability of the maximum a posteriori (MAP) decoder. The MAP decoder looks at an output y ∈ Y and chooses a symbolx ∈ X that maximizes W (x | y). When the output is Y = y, the probability that the MAP decoder does not choose X asx is 1 − max x∈X W (x | y). Therefore, P e (X | Y ) = y∈Y W out (y)(1 − max x∈X W (x | y)). In a channel-centric narrative, we also write P e (W ) for P e (X | Y ).\n\nZ(X | Y ) is the rescaled sum of Bhattacharyya coefficients of the transition distribution W (y | x) for the uniform input. For non-uniform inputs, a modification is made to generalize the definition and the properties that used to hold. Intuitively speaking, a MAP decoder seeing y is \"confident\" if W (x | y) is small for all but one\n\nx, or equivalently, if the product W (x, y)W (x , y) is small for all distinct x, x ∈ X . The Bhattacharyya parameter measures the \"confidence\" by\nZ(X | Y ) := 1 q − 1 x,x ∈Fq x =x y∈Y » W (x, y)W (x , y).\nIn addition, define\nZ mad (X | Y ) := max 0 =d∈Fq x∈Fq y∈Y » W (x, y)W (x + d, y).\nWe also write Z(W ), and Z mad (W ) for these quantities. Remarks: The rescaling T (X | Y ) is the weighted average of the total variation distances from the a posteriori distributions (W (x | y) : x ∈ X ) to the uniform noise (1/q, 1/q, . . . , 1/q). More formally, it is defined to be y∈Y W out (y) x∈X |W (x | y) − 1/q|. We also write T (W ) for this quantity.\nis such that 0 Z Z mad (q − 1)Z q − 1. Our definition of Z mad is different from the Z max in [MT14],\nS(X | Y ) is the weighted average of the L 1 -norms of the Fourier coefficients of the a posteriori distributions. The formal definition is as follows. Let tr : F q → F p be the field trace, where F q = X and F p is the prime subfield. Let χ : F q → C be an additive character defined as χ(x) := exp(2πi tr(x)/p), where 2πi is temporarily the period of exp. Define the Fourier coefficient\nM (w | y) := z∈Fq W (z | y)χ(wz).\nDefine the S-parameters\nS(X | Y ) := 1 q − 1 0 =w∈Fq y∈Y W out (y) · M (w | y) , and S max (X | Y ) := max 0 =w∈Fq y∈Y W out (y) · M (w | y) .\nWe also write S(W ) and S max (W ) for these quantities. Remarks: The rescaling is such that 0 S S max (q − 1)S q − 1. An interpretation is as follows: Fix a y. When W (x | y) is roughly equal to 1/q for all x ∈ F q , the Fourier coefficient M (w | y) = z∈Fq W (z | y)χ(wz) should be roughly z∈Fq χ(wz)/q = 0. The S-parameter measures how far those coefficients are from zero.\n\n\n4.1.\n\nRelations among channel parameters. The following is a series of lemmas we extract from existing works. They characterize the relations among H, I, P e , Z, T , and S.\nLemma 3. [MT14, Lemma 22 with k = 1] For any q-ary channel W , q − 1 q 2 » 1 + (q − 1)Z(W ) − » 1 − Z(W ) 2 P e (W ) q − 1 2 Z(W ).\nLemma 4. [MT14, Lemma 23 with k = q − 1] For any q-ary channel W ,\nq − 1 q − P e (W ) T (W ) 2 q − 1 q − 1 q (q − 1)qP e (W ) − (q − 1)(q − 2) . Lemma 5. [MT14, Lemma 26 with k = q − 1] For any q-ary channel W , 1 − q q − 1 P e (W ) S(W ) (q − 1)q q − 1 q − P e (W ) 1 − q q − 1 q − 2 q − 1 .\nLemma 6. [FM94, Theorem 1] For any q-ary channel W ,\nh 2 (P e (W )) + P e (W ) log 2 (q − 1) H(W ) log 2 q 2P e (W )\nand\nH(W ) log 2 q (q − 1)q log 2 q q − 1 P e (W ) − q − 2 q − 1 + log 2 (q − 1).\nHere, h 2 is the binary entropy function; h 2 (1/2) = 1. The upper bound is Fano's inequality. The first lower bound fits when H(W ) and P e (W ) are small; the second lower bound fits when H(W ) and P e (W ) are close to 1.\n\nThe above lemmas inspire the following characterization: Let A and B be two channel parameters, we say A, B are bi-Hölder at . This notion generalizes to tuples of more parameters. Now we can summarize Lemmas 3 to 6 in a more concise statement.\n\nLemma 7 (implicit bi-Hölder tolls). Parameters H, P e , Z, Z mad are bi-Hölder at (0, 0, 0, 0). Parameters H, P e , T, S, S max are bi-Hölder at (1, 1 − 1/q, 0, 0, 0).\n\nProof. Z, Z mad are bi-Hölder at (0, 0) since Z Z mad (q − 1)Z. Lemma 3 implies that P e , Z are bi-Hölder at (0, 0). Lemma 6 (with the first lower bound) implies that P e , H are bi-Hölder at (0, 0). Now apply the transitivity to conclude the first statement. For the second statement, S, S max are bi-Hölder at (0, 0) since S S max (q − 1)S. Lemma 5 implies that P e , S are bi-Hölder at (1 − 1/q, 0). Lemma 4 implies that P e , T are bi-Hölder at (1 − 1/q, 0). Lemma 6 (with the second lower bound) implies that P e , H are bi-Hölder at (1 − 1/q, 1). Now apply the transitivity to conclude.\n\nSee also [MT14,Corollary 28] for what inspired us. They use notation A e ∼ B to mean A, B are bi-Hölder at (0, 0) and at (1, 1). For some very technical details on the way toward the main theorem, we need explicit Hölder relations among H, Z mad , and S max . We claim them here. The proof is nothing but looking closer into Lemmas 3, 5, and 6. A written-out proof is in Appendix A.\n\nLemma 8 (explicit Hölder tolls). log is natural. For all q-ary channels W , the following hold:\nZ mad (W ) q » H(W ) log 4 q,(2)H(W ) » e(q − 1)Z mad (W )/2,(3)S max (W ) (q − 1)q » (1 − H(W )) log(q)/2, and(4)\n4.2. Control of the block error probability. Let W be the channel we want to communicate over; and let X be any input. In the classical theory of polar coding, the second last step of the construction of the block code is to determine a subset I ⊂ [ ] n of indexes that points to the depth-n channels that transmit information bits. When decoding this code, a block error happens if the successive cancellation decoder fails to decode any information bit. Let E (k1,k2,...,kn) be the event that the first error occurs when the decoder is solving for the input to (· · · (W (k1) ) · · · ) (kn) , i.e., whenÛ (k1,k2,...,kn) = U (k1,k2,...,kn) and the equality holds for lexicographically earlier indexes. Then the event's probability measure P (E (k1,k2,...,kn) ) is no more than the bit error probability P e (· · · (W (k1) ) · · · ) (kn) . By the union bound, the block error probability of the decoder is bounded from above by a sum\nP {Û I = U I } (k1,k2,...,kn)∈I P e (· · · (W (k1) ) · · · ) (kn) .\nWith this observation, we may define I to be the set of indexes (k 1 , k 2 , . . . , k n ) ∈ [ ] n such that H (· · · (W (k1) ) · · · ) (kn) < θ n for some clever choice of the threshold θ n > 0. This immediately implies P e (· · · (W (k1) ) · · · ) (kn) < cθ d n for some c, d > 0 by Lemma 7. Let θ n be exp(− πn n). The sum of P e is less than n cθ d n < exp(− πn ) for sufficiently large n, which is the block error probability we claimed. Remark: Arıkan used a different criterion Z < θ n . It still implies P e < cθ d n and that the sum of P e is less than n cθ d n < exp( πn ) for large n. The benefit of controlling P e using other parameters is that some parameters are easier to control (because Theorems 9 and 10 exist).\n\nFor the main theorem where the channel W is asymmetric, we want to control both the decoder block error and the encoder block error. Here, the encoder block error is not the encoder's failure to encode a message, but rather its failure to generate the capacity-achieving input distribution of W . To penalize, imagine that we employ an oracle that claims an encoder block error whenever the generated codeword should have been another word to fit the ideal distribution. That way, the actual block error probability will not exceed the sum of the encoder and decoder block error probabilities. More rigorously, let P be the probability measure assuming the ideal distribution of U I and Q be the probability measure assuming the actual U I generated by the encoder. Then the overall block error probability can be bounded by\nQ{Û I = U I } P {Û I = U I } + P − Q .\nP {Û I = U I } as the decoder block error probability is bounded before. The encoder block error probability is represented by P − Q , the total variation distance from P to Q. There is a telescoping argument similar to how we control the decoder error-classifying events by the first input bit where the oracle disagrees with the encoder [Kor09, Lemma 3.5] [KU10, Lemma 4] [KT10, Lemma 2] [HY13, Lemma 1]. It yields that the encoder block error probability is bounded from above by the sum\nP − Q (k1,k2,...,kn)∈I T (· · · (W (k1) ) · · · ) (kn) .\nIn controlling the encoder bit error probability, we strengthen the policy of collecting indexes (k 1 , k 2 , . . . , k n ) ∈ [ ] n for I by asking for H (· · · (W (k1) ) · · · ) (kn) > 1 − θ n .\n\nThe latter immediately implies T (· · · (W (k1) ) · · · ) (kn) < cθ d n by Lemma 7. As a consequence, the overall block error probability is controlled by Q{Û I = U I } P {Û I = U I } + P − Q < 2 n cθ d n < exp(− πn ) for n large. The preceding argument is a paraphrase of the proof of [HY13, Theorem 13]; Inequalities (59) and (57) therein are the keys. So far the block length, the complexity, and the error aspects of the main theorem are covered, it remains to control the code rate |I|/ n . In other words, we are to compute the cardinality of I given that I ⊂ [ ] n is the set of indexes such that H (· · · (W (k1) ) · · · ) (kn) < θ n and 1 − H (· · · (W (k1) ) · · · ) (kn) < θ n , where θ n := exp(− πn n).\n\n\n4.3.\n\nBefore and after channel transformations. Alongside the relations among different parameters applied to the same channel, there are also relations between the same parameter applied to the original and the transformed channels. That\ni=1 H(W (i) ) = H(W ) is one.\nThere are two more that are pivotal in the theory of polar coding but require more prerequisites. Assume that g W : X → X is a linear isomorphism given by the multiplication of an invertible matrix G from the right-g W (u 1 ) := u 1 G. The following framework extends to nonlinear bijections but we do not need that much. (There is also the paradigm that random linear codes perform better than random codebooks for that a bad linear code tends to hoard a lot of short codewords at once, effectively removing them from the ensemble pool. So there is a good reason to stick to the linear case.) Let 0 i−1 1 1 i u i+1 ∈ F q be a tuple of i − 1 many 0 followed by a 1 and − i arbitrary symbols. A coset code is a subset of codewords of the form\n{0 i−1 1 1 i u i+1 G : u i+1 ∈ F −i q } ⊂ F q .\nThe coset codes have weight distributions just like every other code does. Let wt(x 1 ) be the hamming weight of x 1 . The weight enumerator of the i-th coset code is defined to be a one-variable polynomial over the integers\nf (i) GZ (z) := u i+1 z wt(0 i−1 1 1iu i+1 G) ∈ Z[z].\nWe can now state the second relation. This is considered the main cause of why polar coding ever exists/works. \nZ mad (W (i) ) f (i) GZ (Z mad (W ))\n. The proof is postponed until Section 7.1. The fundamental theorems come as a pair. Let u i−1 1 1 i 0 i+1 ∈ F q be a tuple of i − 1 arbitrary symbols followed by a 1 and − i many 0. Let G − be the inverse transpose of G. The weight enumerator of the i-th dual coset code is defined to be this one-variable polynomial over the integers\nf (i) GS (s) := u i−1 1 s wt(u i−1 1 1i0 i+1 G − ) ∈ Z[s].\nWe can now state the third relation, the dual of the second. The proof is postponed until Section 7.2. \nS max (W (i) ) f (i) GS (S max (W )).\nRemark: These two bounds are not tight-the equality does not hold for BECs. In detail, Arıkan's original bound reads Z mad (W (1) ) 2Z mad (W )−Z mad (W ) 2 while our bound turns into Z mad (W (1) ) 2Z mad (W ), the subtraction term missing. We are simply not able to prove a version that degenerates to an equality over erasure channels, nor does any prior work seem to. This causes a serious aftermath that Z mad (W n ) (to be defined later) is no longer a supermartingale. Nonetheless, this bound is strong enough to collaborate with the random coding theory. See, for example, how we compensate in Appendix C.1.\n\nWe clarified (cs), (cn), (cc) and (cp) up to this section; there is (cr) to go.\n\n\nChannel Processes\nLet K 1 , K 2 , K 3 , . . . be i.i.d. uniform r.v.s on [ ], where [ ] is the set of integers {1, 2, . . . , }. Let W be the q-ary channel we want to communicate over. Let W 0 be W . For each nonnegative integer n, let W n+1 be (W n ) (Kn+1) , which means (· · · (W (K1) ) · · · ) (Kn+1) in full. Recall that at the end of Section 3.3 we defined W , W (i) , (W (i) ) (j) , ((W (i) ) (j) ) (k) , et seq. all with the same series of input dis- tributions and bijections. Let V 0 be W ; let V n+1 be (V n ) (Kn+1) , which means (· · · (W (K1) ) · · · ) (Kn+1) in full.\nThese r.v.s provide a new family of randomness that does not appear in the encoding and decoding algorithms, but they help us understand the code rate |I|/ n in this manner: Counting how many indexes are in I is nothing more than measuring the probability P{(K 1 , K 2 , . . . , K n ) ∈ I}. With the processes W n and V n thus defined, it is further equivalent to measuring the probability P{H(W n ) < θ n and 1 − H(V n ) < θ n }, where θ n := exp(− πn n). Moreover, it suffices to know how H(W n ) and H(V n ) behave as stochastic processes taking values in [0, 1] without comprehending W n and V n themselves. The general fact is that H(W n ) is either very small (channel is reliable) or very close to 1 (channel is noisy). Arıkan called this phenomenon channel polarization. The following claim generalizes channel polarization and implies the main theorem.\n\nClaim 11. Fix any π, ρ > 0 such that π + 2ρ < 1. We will choose an and a series of bijections of F q -namely,\ng W , g W (i) , g (W (i) ) (j) , g ((W (i) ) (j) ) (k) , et seq.-such that P{H(W n ) < exp(− πn n)} > 1 − H(W ) − −ρn+o(n) , P{1 − H(W n ) < exp(− πn n)} > H(W ) − −ρn+o(n) , P{H(V n ) < exp(− πn n)} > 1 − H(W ) − −ρn+o(n) , and P{1 − H(V n ) < exp(− πn n)} > H(W ) − −ρn+o(n) .\nHere, o(n) is the little-o function in n; it is such that o(n)/n → 0 as n → ∞.\n\nFor polar codes over symmetric channels, the first inequality in Claim 11 alone implies that the code rate is 1 − H(W ) − −ρn+o(n) = I(W ) − N −ρ+o(1) . The first two inequalities imply the polarization behavior that channels become either satisfactorily reliable (low H(W n )) or desperately noisy (high H(W n )). For asymmetric channels, however, we need to characterize H(V n ) alongside H(W n ). The last two inequalities in Claim 11 show that the same series of bijections polarize W at the same time they polarize W . While W contains no randomness form the channel W , what is polarized is that each input bit U (k1,k2,...,kn) either depends heavily on lexicographically earlier input bits (low H(V n )) or behaves like a free r.v. conditioned on earlier bits (high H(V n )). We then categorize the fate of indexes in [ ] n into the following three types. (A) Free and reliable: These are indexes that will be in I; they point to channels that transmit information bits. (B) Free but noisy: The sender can feed information into these channels only to find that the decoder will almost always make some mistakes. The sender should, instead, feed some pseudo random numbers shared with the receiver. (C) Dependent and reliable. The input of these channels depends on previous inputs. Their main purpose is to shape the capacity-achieving input distribution. (D) Dependent but noisy is not possible because H(V n ) H(W n ). This is the key to [HY13, Theorem 1]. We reproduce their proof in the next subsection. 5.1. Claim 11 implies the main theorem. As mentioned, H(V n ) H(W n ) so (D) dependent but noisy is not possible. Let A n be the intersection event of\nimpossible (B) H(X | Y ) (C) 1 − H(X) (A) I(W ) 1 (noisy) 0 (reliable) 0 (dependent) 1 (free) H(V n ) H(W n )free {1 − H(V n ) < exp(− πn n)} and reliable {H(W n ) < exp(− πn n)}. Let B n be the intersection event of free and noisy {1 − H(W n ) < exp(− πn n)}. Let C n be the intersection event of dependent {H(V n ) < exp(− πn n)} and reliable. Since noisy implies free, P(B n ) > H(W ) − −ρn+o(n) follows the second inequality in Claim 11. Also since dependent implies reliable, P(C n ) > 1 − H(W ) − −ρn+o(n)\nfollows the third inequality in Claim 11. Note that A n or B n implies free but not \"neither reliable nor noisy\"; that is, (free ∧ reliable) ∨ (free ∧ noisy) → free ∧ ¬(¬reliable ∨ ¬noisy). We deduce that P\n(A n ) + P(B n ) > H(W ) − −ρn+o(n) − 2 −ρn+o(n) .\nSimilarly, since A n or C n implies reliable but not \"neither free nor dependent,\" we deduce that P\n(A n ) + P(C n ) > 1 − H(W ) − −ρn+o(n) − 2 −ρn+o(n) .\nIn summary, we derive that\nP(A n ) (P(A n ) + P(B n )) + (P(A n ) + P(C n )) − 1 > (H(W ) − 3 −ρn+o(n) ) + (1 − H(W ) − 3 −ρn+o(n) ) − 1 = H(X) − H(X | Y ) − 6 −ρn+o(n) = I(W ) − −ρn+o(n)\nFinally, recall that I collects free and reliable indexes, so the code rate is |I|/ n = P(A n ) > I(W ) − −ρn+o(n) . We almost finish the proof of the main theorem except that we claimed I(W ) − N −ρ = I(W ) − −ρn , without the little-o term. It can be fixed by finding a slightly larger > ρ such that π + 2 < 1 still holds, and then rerunning the whole argument again with the new . The conclusion becomes that the code rate is at least I(W ) − − n+o(n) . Since − n + o(n) < −ρn for sufficiently large n, this completes the proof of the main theorem. It remains to show that Claim 11 can be achieved.\n\n\nGlobal MDP Behavior Modulo Local Behaviors\n\nIn this section, we put constraints on an abstract process {H n } and show that they imply inequalities of the form P{H n < threshold} > limit measure − decaying gap as those in Claim 11. Let F n be the sigma-algebra generated by K 1 , K 2 , . . . , K n for each n. Then F 0 ⊂ F 1 ⊂ F 2 ⊂ · · · form a filtration of sigma-algebras. Let {H n }, {Z n }, and {S n } be three stochastic processes adapted to {F n } (meaning K 1 , K 2 , . . . , K n determine H n , Z n , S n ). The following assumptions are easy to verify when we reveal what those processes are: (cb) 0 H n , Z n , S n and H n 1;\n(cm) {H n } is a martingale, i.e., E [H n+1 | F n ] = H n ; (ct) H n q 3 √ Z n and Z n q 3 √ H n along with 1 − H n q 3 √ S n as well as S n q 3 √ 1 − H n for all n.\nFurthermore, assume large kernels: (cl) max(e 4 , q 5 , 3 q ). Let α := log(log )/ log be a small number shrinking as increases. Define the potential function h α : [0, 1] → [0, 1] to be h α (z) := min(z, 1 − z) α . (Remark: h 2 is not a special case of h α for α = 2; we expect α 1 in practice.) Here are the difficult but sufficient criteria for the main theorem.\n\nLemma 12. (calculus machinery for global MDP) Assume criteria (cb), (cm), (ct), and (cl). Assume the local LDP behavior: Z n+1 exp(qZ n )(qZ n ) K 2 n+1 /3 and S n+1 exp(qS n )(qS n ) ( +1−Kn+1) 2 /3 . Assume the local CLT behavior:\nE [h α (H n+1 ) | F n ] < 4 −1/2+α .\nThen, for any constants π, ρ > 0 such that\nπ + 2ρ 1 − 8α,(6)\nthe following holds:\nP{H n < exp(− πn n)} > 1 − H 0 − −ρn+o(n) .(7)\nWe defer the proof until Appendix B. The term K 2 n+1 /3 in the lemma is to control the local LDP behavior of the process {H n }-the behavior of H n+1 when H n is close to 0 and the behavior that is closely related to the LDP behavior of polar codes. The term is chosen in a way such that h 2 ((k 2 /3 )/ ) < k/ and such that k (k 2 /3 ) t is easy to handle. In [FHMV17, Theorem 7], a similar criterion is stated and is annotated as faster polarization at the tails. In [BGS18, Definition 2.4], a similar criterion is stated and is annotated as strong suction at the low end. The eigenfunction h α in the lemma is to control the local CLT behavior of the process {H n }-the behavior of H n when it is away from 0 and the behavior that is closely related to the CLT behavior of polar codes. In [FHMV17, Theorem 7], a similar criterion is annotated as near optimal polarization in the middle with h FHMV (z) := (z(1 − z)) α for positive but small α at most log(log )/ log . In [BGS18, Definition 2.3], a similar criterion is annotated as variance in the middle with h BGS (z) := min(z, 1 − z). Note how our choice of h α (z) := min(z, 1 − z) α resembles theirs. In both cases, the criteria are local because they refer to a small slice of the process, focusing on how H n+1 (or Z n+1 ) behaves in terms of H n (or Z n ). This perspective frees [FHMV17,BGS18] from considering the (global ) process {H n } as a whole and simplifies the analysis. We specifically benefiti from the fact that we can choose the bijection g (···(W (k 1 ) )··· ) (kn ) solely according to the channels (· · · (W (k1) ) · · · ) (kn) and (· · · (W (k1) ) · · · ) (kn) instead of the complete channel family-tree. This is also the approach taken in [GRY19].\n\n6.1. Lemma 12 helps achieve Claim 11. To do so, one advantage is that the two desired behaviors are local. They only involve how H n+1 , Z n+1 and S n+1 behave conditioned on the history F n . A potential tedious aspect is that for each candidate of the bijection g (···(W (k 1 ) )··· ) (kn ) , we have to verify the two behaviors four times, once for each of the four triples of channel parameters. Luckily, within random coding theory, we are in the situation that to choose an object that satisfies multiple criteria, it suffices to choose the object from an ensemble and compute the probabilities that each criterion fails; as long as the sum of failing probabilities is small, most objects satisfy. Even more luckily, when we choose a bijection g (···(W (k 1 ) )··· ) (kn ) from some ensemble, we only have to compute the probability that the local CLT or LDP behavior fails for {H n }, {Z n }, and {S n } being {H(W n )}, {Z mad (W n )}, and {S max (W n )} but not the other three triples. This is because other three triples are the special case and/or the dual of this triple. Elaboration: Since V n are q-ary channels just like W n are, inequalities hold true for arbitrary W n should hold true for any V n . Also, since Z mad and S max are in duality, inequalities hold true for H, Z mad , S max hold true for 1 − H, S max , Z mad . The duality is due to the duality between FTPCZ and FTPCS, within the explicit Hölder tolls, and within the ensemble of bijections we are to choose g (···(W (k 1 ) )··· ) (kn ) from.\n\n6.2. Random linear isomorphisms as bijections. Fix q and . Let GL( , q) be the group of -by-invertible matrices over F q together with the ordinary matrix multiplication. Select an element G ∈ GL( , q) uniformly at random. Let g W : F q → F q be the multiplication of G from the right, namely g W (u 1 ) := u 1 G. This map is bijective since G is invertible. Let W be a q-ary channel. Recall that we defined q-ary channels W (1) , W (2) , . . . , W ( ) in Section 3. To emphasis that these imaginary channels depend heavily on the randomness source G, we call \nZ mad (W (i) G ) exp(qZ mad (W ) )(qZ mad (W )) i 2 /3 , and(8)S max (W (i) G ) exp(qS max (W ) )(qS max (W )) ( +1−i) 2 /3 .(9)\nLemma 14 (local CLT behavior). Fix an 20. Recall α := log(log )/ log and h α (z) := min(z, 1 − z) α . Let G vary; with probability less than 2 − log( )/20 , this fails:\n1 i=1 h α (H(W (i) G )) < 4 −1/2+α .(10)\n6.3. Local behaviors imply Claim 11 (and hence the main theorem). We now can see how Lemmas 12 to 14 imply that Claim 11 is achievable for the right choice of and bijections g W , g W (i) , et seq.: For any given q-ary channel W , let be max(e 4 , q 5 , 3 q ). For any given π, ρ > 0 such that π + 2ρ < 1, enlarge such that Inequality (6) holds, given α := log(log )/ log . Consider a random kernel G as a candidate of the bijection g W . Increase further so that the failing probabilities-Lemma 13's 3q − √ /13 and Lemma 14's 2 − log( )/20 -amount to 1/3 or less. Recall the flattening channel W . The probability that any of the inequalities in Lemmas 13 and 14 fails for W is less than 1/3, too. Invoke the union bound; 1/3 + 1/3 < 1. Hence there exists a solid choice of g W as the multiplication of some proper instance of G from the right. With this g W determined, we define W (i) and W (i) for all i ∈ [ ]. Consider first i = 1, anything that has been done to W now applies to W (i) . That is, let g W (i) be the multiplication of a random kernel G from the right. With W , i, and W (i) G replaced by W (i) , j, and (W (i) ) (j) G , the probabilities that inequalities in Lemmas 13 and 14 fail add up to 1/3 or less. So is the flattening ( ) counterpart. Hence there is a solid choice of g W (i) . Repeat this for every other i = 2, 3, . . . , . Once finished, proceed to choosing g (W (i) ) (j) for all i, j ∈ [ ]. And so on and so forth for cases beyond depth-2. Notice that we always make a solid choice of a bijection before we proceed to the next level of channels, hence the failing probabilities of Lemmas 13 and 14 do not accumulate as the depth increases. By how we select bijections, the criteria in Lemma 12 hold for ({H n }, {Z n }, {S n }) being these four triples: In this section, we will first prove the two fundamental theorems of polar coding in Section 7.1 (for the Z-end) and in Section 7.2 (for the S-end). And then we will target that the following inequalities hold with high probability:\n({H(W n )}, {Z mad (W n )}, {S(W n )}) and ({1 − H(W n )}, {S max (W n )}, {Z mad (W n )}) along with ({H(V n )}, {Z mad (V n )}, {S max (V n )}) as well as ({1 − H(V n )}, {S max (V n )}, {Z mad (V n )})Z mad (W (i) G )\nexp(qZ mad (W ) )(qZ mad (W )) i 2 /3 , and (8's copy)\nS max (W ( +1−i) G ) exp(qS max (W ) )(qS max (W )) i 2 /3 . (9's copy)\nBy the duality between the two fundamental theorems and between the two targeted inequalities, it is not hard to see that it suffices to prove the Z mad -case and the S maxcase follows immediately. We will prove that the first targeted inequality, for each i ∈ [ ], holds with probability 1 − 3q − √ /13 in Section 7.4, closing this section.\n\n7.1. Proof of FTPCZ (Theorem 9). As is promised in Section 4.3, we prove the two fundamental theorems of polar coding. We first go for the Z-end.\nRecall that f (i) GZ (z) := u i+1 z wt(0 i−1 1 1iu i+1 G)\nis the weight enumerator of the i-th coset code. Theorem 9 claims that Z mad (W (i) ) f \n(i) ) is max 0 =di∈Fq ui∈Fq u i−1 1 y 1 ∈F i q ×Y » W (i) (u i , u i−1 1 y 1 )W (i) (u i + d i , u i−1 1 y 1 ).\nBy the nature of max 0 =di∈Fq , it suffices to show that the double sum is at most f (i) GZ (Z mad (W )) for arbitrary nonzero d i . In the upcoming argument, tuple concatenation takes precedence over vector-matrix multiplication and vector addition. Fix a d i , we argue that\nui∈Fq u i−1 1 y 1 ∈F i q ×Y » W (i) (u i , u i−1 1 y 1 )W (i) (u i + d i , u i−1 1 y 1 ) = u i 1 y 1 » W (i) (u i , u i−1 1 y 1 )W (i) (u i + d i , u i−1 1 y 1 ) = u i 1 y 1 u i+1 ∈F −i q W (u i 1 u i+1 G, y 1 ) v i+1 ∈F −i q W (u i−1 1 (u i + d i )v i+1 G, y 1 ) u i 1 y 1 u i+1 v i+1 » W (u i 1 u i+1 G, y 1 )W (u i−1 1 (u i + d i )v i+1 G, y 1 ) = y 1 u 1 d i+1 ∈F −i q » W (u 1 G, y 1 )W (u i−1 1 (u i + d i )G, y 1 ) = y 1 x 1 ∈F q d i+1 » W (x 1 , y 1 )W (x 1 + 0 i−1 1 d i G, y 1 ) = d i+1 y 1 x 1 » W (x 1 , y 1 )W (x 1 + e 1 , y 1 ) = d i+1 y 1 x 1 j∈[ ] » W (x j , y j )W (x j + e j , y j ) = d i+1 y 1 x 1 j∈J » W (x j , y j )W (x j + e j , y j ) k / ∈J W (x k , y k ) = d i+1 j∈J xj yj » W (x j , y j )W (x j + e j , y j ) k / ∈J x k y k W (x k , y k ) = d i+1 j∈J xj yj » W (x j , y j )W (x j + e j , y j ) d i+1 j∈J max 0 =ej ∈Fq xj yj » W (x j , y j )W (x j + e j , y j ) = d i+1 j∈J Z mad (W ) = d i+1 Z mad (W ) |J| = d i+1 Z mad (W ) wt(0 i−1 1 did i+1 G) = d i+1 Z mad (W ) wt(0 i−1 1 1id i+1 G) = f (i) GZ (Z mad (W )).\nThe first equality abbreviates the summation. The next equality expands W (i) by the very definition, where u i+1 and v i+1 are free variables in F q . The next inequality is by the sub-additivity of the square root. In the next equality we define d i+1 := v i+1 − u i+1 ; so summing over v i+1 is equivalent to summing over d i+1 . In the next equality we define x 1 := u 1 G; so summing over u 1 is equivalent to summing over x 1 as G is invertible. In the next equality we substitute e 1 := 0 i−1 1 d i G and reorder the summation. The next equality expands the product of the memoryless channels. The next equality classifies indexes into two classes-j ∈ J are those such that e j = 0 and k / ∈ J are such that e k = 0. The next equality is the distributive law ax + ay + bx + by = (a + b)(x + y). The next equality uses the fact that the W (x, y) sum to 1. In the next inequality we replace e j by a nonzero element that maximizes the sum in the parentheses. In the next equality we realize that the maximum is the Bhattacharyya parameter (surprisingly). The second last equality uses the fact that multiplying a vector by a scalar preserves its hamming weight. And quod erat demonstrandum. Experienced readers may find that all but the last inequality follows the proof strategy [KSU10, Lemma 10]. 7.2. Proof of FTPCS (Theorem 10). We now go for the S-end of the fundamental theorem of polar coding. Recall the character χ(x) := exp(2πi tr(x)/p). We need the following properties: (pa) χ(0) = 1; (pb) |χ(x)| = 1 for all x ∈ F q ; (pc) χ(x)χ(z) = χ(x + z) for all x, z ∈ F q ; (pd) x∈Fq χ(x) = 0. See also [MT14, Definition 24] or a dedicated book [Ter99]. To prove the theorem, we first verify that Fourier coefficients recover the origin: Let M (w, y)\n:= W out (y)M (w | y) = z∈Fq W (z, y)χ(wz), then w∈Fq M (w, y)χ(−xw) = w∈Fq z∈Fq W (z, y)χ(wz)χ(−xw) = z∈Fq W (z, y) w∈Fq χ(w(z − x)) = z∈Fq W (z, y)qI{z − x = 0} = qW (x, y).\nThe first equality expands M (w, y) by the definition. The next equality uses that χ is an additive character (pc), and reorders the summation. The next equality uses w∈Fq χ(w) = 0 (pd) and w∈Fq χ(0) = q (pa); and I is the indicator function.\n\nKnowing that W (x j , y j ) = q −1 wj ∈Fq M (w j , y j )χ(−x j w j ), we proceed to\nW (i) (u i , u i−1 1 y 1 ) = u i+1 W (u 1 G, y 1 ) = u i+1 ∈F −i q W (x 1 , y 1 ) = u i+1 j∈[ ] W (x j , y j ) = u i+1 j∈[ ] q −1 wj ∈Fq M (w j , y j )χ(−x j w j ) = q − u i+1 w 1 j∈[ ] M (w j , y j )χ(−x j w j ) = q − u i+1 w 1 χ(−x 1 (w 1 ) ) j∈[ ] M (w j , y j ) = q − u i+1 w 1 χ(−x 1 (w 1 ) )M (w 1 , y 1 ) = q − u i+1 w 1 χ(−u 1 G(w 1 ) )M (w 1 , y 1 ) = q − u i+1 w 1 χ(−u 1 (w 1 G ) )M (w 1 , y 1 ) = q − u i+1 v 1 χ(−u 1 (v 1 ) )M (v 1 G − , y 1 ) = q − v 1 χ(−u i 1 (v i 1 ) )M (v 1 G − , y 1 ) u i+1 χ(−u i+1 (v i+1 ) ) = q − v 1 χ(−u i 1 (v i 1 ) )M (v 1 G − , y 1 )q −i I{v i+1 = 0} = q −i v i 1 χ(−u i 1 (v i 1 ) )M (v i 1 0 i+1 G − , y 1 ).\nThe first equality expands the definition of W (i) . In the next equality, we substitute x 1 := u 1 G. The next equality expand the definition of W down to W . The next two equalities Fourier expand W and reorder the operators. The next equality merges all χ(−x j w j ) into one term by additivity (pc). In the next equality we define M (w 1 , y 1 ) to be the product of all M (w j , y j ). The next two equalities use x 1 (w 1 ) = u 1 G(w 1 ) = u 1 (w 1 G ) . In the next equality we define v 1 := w 1 G ; so summing over w 1 is equivalent to summing over v 1 . (Recall that G − is the notation of the inverse transpose of G.) The next three equalities sum over u i+1 to force v i+1 = 0.\n\nHaving\nW (i) (u i , u i−1 1 y 1 ) = q −i v i 1 χ(−u i 1 (v i 1 ) )M (v i 1 0 i+1 G − , y 1 ) in mind, we move on to M (i) (ω i , u i−1 1 y 1 ) := zi∈Fq W (i) (z i , u i−1 1 y 1 )χ(ω i z i ) = zi∈Fq q −i v i 1 χ(−u i−1 1 z i (v i 1 ) )M (v i 1 0 i+1 G − , y 1 )χ(ω i z i ) = q −i v i 1 χ(−u i−1 1 (v i−1 1 ) )M (v i 1 0 i+1 G − , y 1 ) zi∈Fq χ(z i (ω i − v i )) = q −i v i 1 χ(−u i−1 1 (v i−1 1 ) )M (v i 1 0 i+1 G − , y 1 )qI{ω i = v i } = q 1−i v i−1 1 χ(−u i−1 1 (v i−1 1 ) )M (v i−1 1 ω i 0 i+1 G − , y 1 ).\nIn the first line we let M (i) be the Fourier coefficient of W (i) . The next equality plugs in what we have about W (i) in mind. The next three equalities sum over z i to force\nv i = ω i . With M (i) (ω i , u i−1 1 y 1 ) = q 1−i v i−1 1 χ(−u i−1 1 (v i−1 1 ) )M (v i−1 1 ω i 0 i+1 G − , y 1 ) in place, we obtain that with arbitrary 0 = ω i ∈ F q , u i−1 1 y 1 ∈F i−1 ×Y |M (i) (ω i , u i−1 1 y 1 )| (11) = u i−1 1 y 1 q 1−i v i−1 1 χ(−u i−1 1 (v i−1 1 ) )M (v i−1 1 ω i 0 i+1 G − , y 1 ) u i−1 1 y 1 q 1−i v i−1 1 |M (v i−1 1 ω i 0 i+1 G − , y 1 )| = y 1 v i−1 1 |M (v i−1 1 ω i 0 i+1 G − , y 1 )| = y 1 v i−1 1 j∈[ ] |M (w j , y j )| = y 1 v i−1 1 j∈J |M (w j , y j )| k / ∈J |M (w k , y k )| = v i−1 1 j∈J yj |M (w j , y j )| k / ∈J y k |M (w k , y k )| = v i−1 1 j∈J yj |M (w j , y j )| v i−1 1 j∈J S max (W ) = v i−1 1 S max (W ) |J| = v i−1 1 S max (W ) wt(v i−1 1 ωi0 i+1 G − ) = v i−1 1 S max (W ) wt(v i−1 1 1i0 i+1 G − ) = f (i) GS (S max (W )).\nThe first inequality expands the Fourier coefficient. The next inequality is triangle plus (pb). The next equality cancels the summation over u i−1 1 with q 1−i . In the next equality we substitute w 1 := v i−1 1 ω i 0 i+1 G − ; slightly different from the w 1 above, they are now restricted to a proper subspace. The next equality classifies indexes into two classes-j ∈ J are those such that w j = 0 and k / ∈ J are such that w k = 0. The next two equalities reorder the operators and simplify y k |M (0, y k )| = y k W out (y k ) = 1. The next inequality replaces w j by the one that maximizes y k |M (w j , y k )|. The rest is trivial. Theorem 10 claims that S max (W (i) ) f (i)\nGS (S max (W )), where f (i)\nGS is the weight enumerator of the i-th dual coset code. Since S max (W (i) ) is just the maximum of Formula (11) over 0 = ω i ∈ F q , we arrive at S max (W (i) ) f \nh 2 (z) √ ez.\nSee the left half of Figure 6 for evidence. More generally, for all prime powers q,\n1 − 1 log q D z q − 1 q = −z log q z q − 1 − (1 − z) log q (1 − z) √ ez\nfor all z ∈ [0, 1], where D is the Kullback-Leibler divergence. This falls back to the h 2 case when q = 2. See the right half of Figure 6 for evidences for q = 3, 4, 5, 7. It can be observed that as q → ∞ the function tends to a line connecting (0, 0) and (1, 1), hence the upper bound should hold. Taking derivative in q shows that the left hand side decreases as q increases and x < 1/2. where z := Z mad (W ) for short. This inequality is in fact a consequence of\nf (i) GZ (z) (1 + (q − 1)z) − i 2 /3 ((q − 1)z) i 2 /3(12)\nbecause (1 + a) b < exp(ab). We will show the last inequality. Now divide i into two cases: 1 i √ 3 and √ 3 < i . For i = 1, 2, . . . , √ 3 , the exponent i 2 /3 is simply 1, so the inequality to be proven reads f For i = √ 3 + 1, √ 3 + 2, . . . , , let k := − i and let d := i 2 /3 . These variables resemble the dimension and the minimal distance of a linear block code as in the notation an [ , k, d]-code in classical (algebraic) coding theory. To make Inequality (12) hold, we execute a two-phase procedure to avoid all codewords of weight less than d and to eliminate kernels with poor overall scores. In further detail, we will reject a kernel G if there exists u i+1 such that wt(0 i−1 1 1u i+1 G) < d and call it phase I. Afterwards, among surviving kernels with only heavy codewords, we will reject a kernel if its overall score f (i) GZ (z) is too low and call it phase II. The failing probability 3q − √ /13 is the price we pay for rejecting. Up to this point, two things remain to be analyzed: how much probability we pay for rejecting light codewords in phase I (answer: q − √ /13 ), and what is the Markov cutoff that makes Inequality (12) in phase II (answer: 2q − √ /13 ). Phase I analysis is as follows: Fix u i+1 and vary G ∈ GL( , q); the codeword\nX 1 := 0 i−1 1 1 i u i+1 G is a nonzero vector distributed uniformly on F q \\ {0 1 }.\nThis distribution is almost identical to the uniform distribution on F q . Assume X 1 follows the latter; this makes X 1 lighter, which is compatible with the direction of the inequalities we want. Then the probability that X 1 has weight less than d is the probability that Bernoulli trials-X j being \"zero\" with probability 1/q and \"nonzero\" with probability (q −1)/q-result in less than d \"nonzero\"s. By the large deviations theory [DZ10, Exercise 2.2.23(b)], wt(X 1 ) < d holds with probability less than\nexp − D d 1 2 = 2 − (1−h2(d/ ))\nfor the q = 2 case, where D is the Kullback-Leibler divergence. For general q, similarly, wt(X 1 ) < d holds with probability less than\nexp − D d 1 − 1 q = exp −d log d/ 1 − 1/q − ( − d) log 1 − d/ 1/q .\nThis quantity is less than q − (1−h2(d/ )) by Figure 6 (meaning that q = 2 is the most difficult case). By Figure 6, h 2 (d/ ) < ed/ = ei 2 /3 2 = ( e/3)i/ < 0.952i/ . So the rejecting probability is less than q − (1−h2(d/ )) < q − +0.952i . Take into account that there are q −i possibilities of u i+1 . The union bound yields q −i q − +0.952i = q −0.048i < q −0.048 √ 3 < q − √ /13 . Hence the rejecting probability q − √ /13 . Phase I ends here. Phase II analysis is as follows: After we reject some G in phase I, some codewords will disappear; particularly, this includes all codewords of low weights. Therefore, the expectation of f (i) GZ (z) is bounded by the weight enumerator of all heavy codewords rescaled by the number of codewords. In detail, start from\nE[f (i) GZ (z) | G survives phase I] = E[f (i) GZ (z)I{G survives}]/P{G survives} E[f (i) GZ (z)I{G survives}]/(1 − q − √ /13 ). (13)\nI is the indicator function. In the denominator, 1 − q − √ /13 > 1/4 as 30. Put that aside and redefine d := i 2 /3 . The expected value part is bounded from above by\nE[f (i) GZ (z)I{G survives}] = E u i+1 z wt(u i+1 G) I{G survives} E u i+1 z wt(u i+1 G) I{wt(u i+1 G) d} = u i+1 E[z wt(u i+1 G) I{wt(u i+1 G) d}] q −i E[z wt(X 1 ) I{wt(X 1 ) d}] = q −i q − x 1 z wt(x 1 ) I{wt(x 1 ) d} = q −i w d Ä w ä z w (q − 1) w q −i w d Ä d äÄ −d w−d ä z w (q − 1) w = q −i Ä d ä w d Ä −d w−d ä z w−d (q − 1) w−d ((q − 1)z) d . = q −i Ä d ä (1 + (q − 1)z) −d ((q − 1)z) d (overestimate the scalar q −i Ä d ä ) (q − √ /13 /2)(1 + (q − 1)z) −d ((q − 1)z) d .\nThe first equality expands the definition. The next inequality replaces G surviving phase I by a weaker condition. The next equality switches E and . The next inequality replaces the ensemble of u i+1 G by a uniform X 1 ∈ F q . The next equality expands the definition of the expectation over X 1 . The next equality counts codewords. The next inequality selects w positions by first selecting d and then selecting w − d. The next two equalities factor and apply the binomial theorem. The rest is by a series of inequalities that overestimate the scalar:\nq −i d = q −i i 2 /4 < q −i i 2 /4 /2 < q −i 2 h2(i 2 /4 2 ) /2 q −i+ h2(i 2 /4 2 ) /2. Similar to the end of phase I, the exponent part is −i + h 2 (i 2 /3 2 ) < −i + ei 2 /3 2 = −i + i e/3 < −0.048i < −0.048 √ 3 < − √ /13\n. Hence the scalar part is less than q − √ /13 /2. Put 1 − q − √ /13 > 1/4 back to the denominator as in Inequality (13);\nE[f (i) GZ (z) | G survives phase I] has an upper bound of 2q − √ /13 (1 + (q − 1)z) −d ((q − 1)z) d .\nBy Markov's inequality, Inequality (12) holds with probability 1−2q − √ /13 , i.e., the rejecting probability is 2q − √ /13 . Phase II ends here. The sum of the two rejecting probabilities is 3q − √ /13 as claimed in Lemma 13, hence the lemma settled. 7.5. Bibliographic remarks. Concerning the fundamental theorems: Nonlinear g W is not taken into consideration for that it is hard to imagine how MacWilliams duality works then. Also the S-parameter does not generalize to non-field input alphabet. Concerning random linear codes: [BF02, Section II.C] portrays a clear picture of the weight distribution of binary random linear codes. Section 7.4 accommodates and extends their argument to general prime power q. Concerning the LDP behavior: [KSU10, Theorem 22] showed that π < 1 can be arbitrary close to 1 over binary alphabet utilizing the Bose-Chaudhuri-Hocquenghem codes. Our Lemma 13 on the other hand, implies that almost all kernels make π close to 1.\n\nIt remains to prove Lemmas 12 and 14.\n\n\nLocal CLT Behavior (Proof of Lemma 14)\n\nWe are to prove that the following inequality holds with high probability:\ni=1 h α (H(W (i) G )) < 4 1/2+α . (10's copy)\nThe target inequality is the sum of the following three inequalities:\ni= H(W ) + 1/2+α +1 h α (H(W (i) G )) < 1/2+α ,(14)H(W ) + 1/2+α i= H(W ) − 1/2+α +1 h α (H(W (i) G )) < 2 1/2+α , and H(W ) − 1/2+α i=1 h α (H(W (i) G )) < 1/2+α .(15)\nThe second one is trivial as h α (z) (1/2) α . The first one will be proven in Section 8.3 with failing probability − log( )/20 . The third one will be proven in Section 8.4 with failing probability − log( )/20 . Before the main proofs, we devote Section 8.1 to introduce the symmetrization trick, which will reduce our proof to the case of symmetric q-ary channels. A channel W being symmetric means that for any affine shifting ξ ∈ F q , there exists an permutation σ on Y such that W (y | ξ + x) = W (σ(y) | x) holds for all x ∈ F q and y ∈ Y. It also means that the uniform input achieves the Shannon capacity. This justifies the usage of linear codes. In Section 8.2, we invoke some universal bound on entropies and exponents from Chang, Draper, and Sahai's works. Finally, we will be abusing the theory of random linear codes in Section 8.3 for noisy channel coding and in Section 8.4 for secrecy over wiretap channels. 8.1. Symmetrize channel and uniformize input. Let W : F q → Y be any q-ary channel; let X and Y be some input and the corresponding output. Symmetrize the channel as follows: Let Ξ ∈ F q be a uniform r.v. independent of X, Y . Let W : F q × (F q × Y) → [0, 1] be the probability mass function of this combination of r.v.s (Ξ + X, (X, Y )) ∈ F q × (F q × Y). ThisW behaves like a channel such that, quote, unquote,W ((x, y) | z) = W (x, y)/q for all inputs z ∈ F q and outputs (x, y) ∈ F q × Y. Despite that this channel might be properly simulated by a symmetric channel with feedback to the sender, all that matters is that the biased input X is neutralized by the uniform r.v. Ξ, and becomes uniform. Let g W be the multiplication of an invertible matrix G from the right. LetW (i) (u i , u i−1 1 x 1 y 1 ) be the probability mass function of the tuple (U i , U i−1 1 X 1 Y 1 ), where U 1 G = Ξ 1 + X 1 . This definition is compatible with the channel transformation ofW as ifW was an actual channel in the first place. Let H(W (i) ) be H(U i | U i−1 1 X 1 Y 1 ); this is also compatible. The following lemma justifies whyW is useful in theory.\n\nLemma 15 (channel symmetrization).W is a symmetric q-ary channel,\nH(W ) = H(W ), and H(W (i) ) = H(W (i) ) for all i ∈ [ ].\nThis lemma is by [MT14, Definition 6 and Lemmas 7 and 8], plus the arguments in between. See also [HY13,Theorem 2] where they cared about whether Z(W (i) ) = Z(W (i) ). One could also expand all definitions to verify the identities.\n\nThe consequence of this lemma is thatW behaves like a shadow copy of W , but is symmetric. All inequalities involving entropies of W and W (i) are reduced to inequalities involving entropies ofW andW (i) . Subsequently, passing statements toW is effectively assuming that the channel W is symmetric with the uniform input to begin with. In the upcoming subsections, we will prove that the targeted inequalities, (14) and (15), hold for any symmetric q-ary channel W with the uniform input with high probability. We conjecture that the symmetrization technique is optional as it seems like a wrapper of complicated Bayesian formulas. 8.2. Chang-Sahai's universal quadratic bound. This and the next two subsections contain the most convoluted part of the proof of Lemma 14. This subsection prepares a universal upper bound on Gallager's E-null function, which ultimately evolves into a universal lower bound on Gallager's error exponent.\n\nLet W : X → Y be a q-ary channel. Symmetry is not required in this subsection but it is in the next two. Assume the uniform input distribution W in (x) = 1/q for all x ∈ X . Define Gallager's E-null function and its complement [CS07, Formula (1)]:\nE 0 (t) := − log y∈Y x∈X W in (x)W (y | x) 1/(1+t) 1+t\n, and\nE 0 (t) := log y∈Y x∈X W (x, y) 1/(1+t) 1+t .\nBy complement we mean that under the uniform input,Ē 0 (t) degenerates tō\nE 0 (t) = log y∈Y x∈X (q −1 W (y | x)) 1/(1+t) 1+t = t log q − log y∈Y x∈X (q −1 W (y | x)) 1/(1+t) 1+t = t log q − E 0 (t).\nEquivalently, E 0 (t) +Ē 0 (t) = t log q. Do not confuse W with W t , the latter W is tilted. When t = 0, the tilted falls back to its italic origin W 0 (x, y) = W (x, y). These measures can be interpreted as follows: W t behaves like a channel with a dedicated input distribution. The first fraction in the definition specifies the output distribution W t out (y). The second fraction specifies the a posteriori distribution W t (x | y) when y is known. As W t is not an actual channel, it is not meaningful to alter the input distribution and ask for the corresponding output. Like the symmetrization trick, all that matters is that we can compute entropies, and what not, as if they were real channels. Quantities we are interested in are listed below: Let H e be the base-e entropy. Let H e (W t ) be H e (X t | Y t ) where (X t , Y t ) is a tuple r.v. that follows W t . Let H e (X t y) be the entropy of the a posteriori distribution of X t given Y t = y; to be specific,\nH e (X t y) = x∈X W t (x | y) logW t (x | y).\n[CS07, Formula (13) and (19)] have that the following hold for t ∈ [0, 1]:\nd dtĒ 0 (t) = H e (W t ), and d 2 dt 2Ē 0 (t) = d dt H e (W t ) = 1 1 + t y∈Y W t out (y) x∈X W t (x | y) log(W t (x | y)) 2 + + t 1 + t y∈Y W t out (y)H e (X t y) 2 − H e (W t ) 2 .(16)\nCareful readers may verify them by hand or follow [CS07,Formulas (13) to (19)] and [DCS14, Lemmas 9 and 10]. Similar computations are also carried out by [AW10,AW14]. Notice thatĒ 0 (t), H e (W t ), and every other term in Equation (16) are all holomorphic functions in t on the half-plane Re t > −1 (there is a singularity at 1/(1 + t) = ∞). By the identity theorem in complex analysis [BMPS02,Corollary 8.16], Equation (16) holds for all t ∈ [−2/5, 1]. Dropping the nonpositive square, we deduce an upper bound for each t ∈ [−2/5, 1]:\nd 2 dt 2Ē 0 (t) 1 1 + t y∈Y W t out (y) x∈X W t (x | y) log(W t (x | y)) 2 + + max(0, t) 1 + t y∈Y W t out (y)H e (X t y) 2 .(17)\nThis upper bound onĒ 0 (t) is a linear combination of\nx∈X W t (x | y) log(W t (x | y)) 2\nand H e (X t y) 2 parametrized by y ∈ Y, so it remains to bound them separately. For the second kind of constituents, the entropy cannot exceed log q so H e (X t y) 2 log(q) 2 . For the first kind of constituents, the following lemma adapted from [CS07, Lemma 1] helps.\n\nLemma 16 (second moment). If w 1 , w 2 , , . . . , w q are positive numbers of sum 1, then i w i log(w i ) 2 ß log(q) 2 for q 3 0.563 for q = 2 ™ 1.2 log(q) 2 .\n\nWith the lemma, we do have x∈X W t (x | y)(logW t (x | y)) 2 1.2 log(q) 2 . Now Inequality (17) becomes\nd 2 dt 2Ē 0 (t) 1 1 + t y∈Y W t out (y) · 1.2 log(q) 2 + max(0, t) 1 + t y∈Y W t out (y) log(q) 2 1 1 + t · 1.2 log(q) 2 + max(0, t) 1 + t log(q) 2 2 log(q) 2\nfor all t ∈ [−2/5, 1]. Since E 0 (t) is a linear function t log q minusĒ 0 (t), their first derivatives sum to log q while their second derivatives are opposite. Hence the following lemma. \nh α (H(W (i) G )) ( − j)h α 1 − j i=j+1 H(W (i) G )\nwhere j := H(W ) + 1/2+α for short. It suffices to prove that the right hand side is less than −1/2+α . In the spirit of the motivational Chain Rule (1), the sum of the chain of H(W (i) G ) on the right hand side is H(U j+1 | U j 1 Y 1 ). In order to prove Inequality (14), we will show\n( − j)h α 1 − j H(U j+1 | U j 1 Y 1 ) < −1/2+α .(18)\nBut what is H(U j+1 | U j 1 Y 1 )? It measures the equivocation at Bob's end when U j 1 is known to Bob. In other words, we may as well pretend that there is a random rectangular full-rank matrix G with columns and only k := − j = − H(W ) − 1/2+α rows, that Alice computes and sends X 1 := U k 1 G to Bob, and that Bob attempts to decodeÛ k 1 upon receiving Y 1 using the MAP decoder. The equivocation is thus, by Fano's inequality, bounded in terms of the probability that Bob fails to decode U k 1 :\nH(U j+1 | U j 1 Y 1 ) −P e log q P e − (1 − P e ) log q (1 − P e ) + P e log q (q k − 1)\n−P e log q P e + P e log q + P e · k = P e · 1 − log P e log q + k . (19) Here P e is the probability that Bob fails to decode,Û k 1 = U k 1 . The following is how to compute Bob's decoder block error probability. The generator matrix G Alice uses is selected uniformly from the ensemble of full-rank k-by-matrices. The difference of every pair of codewords distributes uniformly on F q \\ {0 1 }. Over symmetric channels, the difference alone determines the output's joint distribution because W (y 1 | ξ 1 + x 1 ) = W (σ 1 (y 1 ) | x 1 ) for some componentwise permutation σ 1 on Y depending on ξ 1 . Gallager's bound applies. To elaborate, let t ∈ [0, 1]. Bob's average error probability satisfies [Gal68, Inequalities (5.6.2) to (5.6.14)]\nEP {Bob fails to decode U k 1 given G } = E u k 1 q −k y 1 W (y 1 | u k 1 G )P {Bob hasÛ k 1 = u k 1 given G , u k 1 , y 1 } = E y 1 W (y 1 | 0 1 )P {Bob hasÛ k 1 = 0 k 1 given G , 0 k 1 , y 1 } E y 1 W (y 1 | 0 1 ) v k 1 =0 k 1 P {Bob prefers v k 1 over 0 k 1 given G } t E y 1 W (y 1 | 0 1 ) v k 1 =0 k 1 W (y 1 | v k 1 G ) 1/(1+t) W (y 1 | 0 1 ) 1/(1+t) t = E y 1 W (y 1 | 0 1 ) 1/(1+t) v k 1 =0 k 1 W (y 1 | v k 1 G ) 1/(1+t) t y 1 W (y 1 | 0 1 ) 1/(1+t) E v k 1 =0 k 1 W (y 1 | v k 1 G ) 1/(1+t) t = y 1 W (y 1 | 0 1 ) 1/(1+t) x 1 =0 1 q k − 1 q − 1 · W (y 1 | x 1 ) 1/(1+t) t q −kt y 1 W (y 1 | 0 1 ) 1/(1+t) x 1 =0 1 q − W (y 1 | x 1 ) 1/(1+t) t q −kt y 1 W (y 1 | 0 1 ) 1/(1+t) x 1 q − W (y 1 | x 1 ) 1/(1+t) t = q −kt y 1 x 1 q − W (y 1 | x 1 ) 1/(1+t) x 1 q − W (y 1 | x 1 ) 1/(1+t) t = q −kt y 1 x 1 q − W (y 1 | x 1 ) 1/(1+t) 1+t = q −kt y 1 x 1 W in (x 1 )W (y 1 | x 1 ) 1/(1+t) 1+t\n= exp(kt log q − (the E-null function of W )(t)) = exp(kt log q − E 0 (t)).\n\nIn summary, EP {Bob fails to decode U k 1 given G } exp(kt log q − E 0 (t)) whenever 0 t 1. Recall the universal quadratic bound E 0 (t) I(W )t log q−t 2 log(q) 2 derived in Lemma 17. We obtain that the exponent is\nkt log q − E 0 (t) = ( − H(W ) − 1/2+α )t log q − E 0 (t) ( − H(W ) − 1/2+α )t log q − (I(W )t log q − t 2 log(q) 2 ) = ( t log q − 1/2+α )t log q\n(redeem the supremum at t = −1/2+α /2 log q)\n→ ( −1/2+α /2 − 1/2+α ) −1/2+α /2 = − 2α /4 = − 2 log(log )/ log /4 = − log( ) 2 /4.\nSo far we obtain that the average error probability is less than exp(− log( ) 2 /4) = − log( )/4 . Run Markov's inequality with cutoff − log( )/20 . That is, we reject kernels such that P {Bob fails to decode U k 1 given G } − log( )/5 . Then the rejecting probability is − log( )/20 because 1/20 + 1/5 = 1/4. An upper bound on Bob's error probability being P e < − log( )/5 , an upper bound on Bob's equivocation is − log( )/5 1 − log − log( )/5 log q + k = − log( )/5 1 + log( ) 2 /5 log q + k by Inequality (19). Plugging the latter in kh α (here/k), we derive that the left hand side of Inequality (18) is less than kh α − log( )/5 k 1 + log( ) 2 /5 log q + k = k − log( )/5 1 + log( ) 2 /5 k log q + 1 α = −α log( )/5 k 1 + log( ) 2 /5 k log q + 1 α < −α log( )/5 1 + log( ) 2 /5 log q + 1 α < −α log( )/5 · · 2 α = 2 α log( ) − log( )/5 .\n\nThe first inequality uses that the left hand side increases monotonically in k and k := − j = − H(W ) − 1/2+α < . The second inequality uses the assumption 2. In any regard, the quantity at the end of the inequalities decays to 0 as → ∞, so eventually it becomes less than 1/2+α , the right hand side of Inequality (18). This proves that Inequality (14) holds with failing probability − log( )/20 as soon as is large enough. The lower bound on in the statement of Lemma 14 is large enough, hence the first half of Lemma 14 settled. 8.4. Hayashi's argument at Eve's end. This subsection contains the very last ingredient of the proof of Lemma 14. We dealt with Inequality (14) in the last subsection. We now deal with\nH(W ) − 1/2+α i=1 h α (H(W (i) G )) < 1/2+α .\n(15's copy)\n\nSimilar to how we motivated Inequality (18), we apply Jensen's inequality and the chain rule of conditional entropy to simplify Inequality (15). The left hand side becomes jh α (H(U j 1 | Y 1 )/j) where j := H(W ) − 1/2+α for short. (This is not the same j as in the last subsection.) The input uniform, the argument of h α is H(U j 1 | Y 1 )/j = 1 − I(U j 1 ; Y 1 )/j, which can be replaced by I(U j 1 | Y 1 )/j thanks to the evenness h α (1 − z) = h α (z). We will show\njh α 1 j I(U j 1 ; Y 1 ) < 1/2+α .(20)\nBut what is I(U j 1 ; Y 1 )? It is the amount of information Eve learns from wiretapping Y 1 if they know that U j+1 are junk. In other words, we may pretend that Alice transmits X 1 := U j 1 V j+1 G with confidential bits U j 1 and obfuscating bits V j+1 , Bob receives X 1 in full, and Eve learns Y 1 . This context falls back to (a special case of) the traditional setup of wiretap channels [Wyn75] where various bounds are studied, some in terms of Gallager's E-null function.\n\nHere are some preliminaries to control the information leaked to Eve. We follow the blueprint of how Hayashi derived the secrecy exponent in [Hay06,Inequality (21)]. Consider the communication protocol depicted in Figure 7: Karl fixes a kernel G ∈ GL( , q) and everyone knows G. Alice chooses the confidential message U 1 . Vincent chooses the obfuscating bits V j+1 . Charlie generates Y 1 by plugging X 1 := U j 1 V j+1 G into a simulator of W . Eve learns Y 1 and is interested in knowing U j 1 alone. So the channel on topic is the composition of Vincent and Charlie. Notation: Running out of symbols, we all use P with proper subscriptions to indicate the corresponding probability measures. That said, indexes in the subscription will be omitted. As Eve is interested in the relation between U j 1 and Y 1 , let Y 1 Gu j 1 be the\nKarl chooses G Alice chooses U j 1 Vincent chooses V j+1 X 1 let to be U j 1 V j+1 G Charlie generates Y 1 by X 1 Eve learns G, Y 1\nThe channel Eve cares about Figure 7. A finer setup for Hayashi's secrecy exponent. Charlie generates Y 1 such that X 1 := U j 1 V j+1 G and Y 1 follow W . Despite of the seemingly sequential structure, Karl, Alice, and Vincent work independently.\n\nr.v. that follows the a posteriori distribution of Y 1 given G = G and U j 1 = u j 1 . More formally, P Y Gu (y 1 ) = P Y |GU (y 1 | G, u 1 ) = P GU Y (G, u j 1 , y 1 )/P GU (G, u j 1 ). We could have defined Y 1 G to be the a posteriori distribution of Y 1 given G = G; but it is simply the same distribution as Y 1 since U j 1 V j+1 G traverses all inputs uniformly regardless of the choice of G. That is, P Y |G (y 1 | G) = P Y (y 1 ) for all y 1 ∈ Y .\n\nFix G as an instance of G. Let I e be the base-e mutual information. The channel Eve cares about leaks information of this amount:\nI e (U j 1 ; Y 1 | G) = u j 1 y 1 P U Y |G (u j 1 , y 1 | G) log P Y |GU (y 1 | G, u j 1 ) P Y |G (y 1 | G) = u j 1 P U (u j 1 ) y 1 P Y |GU (y 1 | G, u j 1 ) log P Y |GU (y 1 | G, u j 1 ) P Y |G (y 1 | G) = u j 1 P U (u j 1 ) y 1 P Y Gu (y 1 ) log P Y Gu (y 1 ) P Y (y 1 ) = u j 1 P U (u j 1 )D(Y 1 Gu j 1 Y 1 ). (21) D(Y 1 Gu j 1 Y 1 )\nis the Kullback-Leibler divergence from the a posteriori distribution of Y 1 given G, u j 1 to the coarsest distribution Y 1 . We are to take expectation over G to find the average information leak since we are interested in Markov's inequality. Equality (21) yields\nEI e (U j 1 ; Y 1 | G) = G P G (G) u j 1 P U (u j 1 )D(Y 1 Gu j 1 Y 1 ).(22)\nWe now discover that there are redundancies in traversing all G and u 1 : After all,\nX j 1 is u j 1 V j+1 G = u j 1 0 j+1 G + 0 j 1 V j+1 G,\nwhich is a fixed linear combination of the first j rows plus a random vector from the span of the bottom − j rows. When V 1 varies, the track of X 1 forms an affine subspace of F q , a coset code as in the context of the fundamental theorems. So what matters is the distribution of this coset code. In this regard, we replace the uniform ensemble of (G, U j 1 ) by the uniform ensemble of K a rank-( − j) affine subspace of F q , where j := H(W ) − 1/2+α . Karl and Alice together choose K uniformly. Vincent chooses X 1 ∈ K uniformly. Charlie generates Y 1 by throwing X 1 into a simulator of W . See Figure 8 for the Karl and Alice choose K\nVincent chooses X 1 ∈ K Charlie generates Y 1 by X 1 Eve learns Y 1\nThe modified channel Figure 8. A simplified setup for Hayashi's secrecy exponent. Charlie generates Y 1 such that X 1 and Y 1 follow W .\n\ndepiction of the new scheme. Hence Equality (22) becomes\nEI e (U j 1 ; Y 1 | G) = K P K (K)D(Y 1 K Y 1 )\nwhere Y 1 K is the a posteriori distribution of Y 1 given K = K. Suddenly, the quantity EI e (U j 1 ; Y 1 | G) we are interested in turns into the mutual information I e (K ; Y 1 ) between K and Y 1 as K replaces the role of U j 1 in Formula (21). Recall that in Lemma 17 the mutual information is the derivative of Gallager's E-null function. We exploit this. Define the double-stroke E-null function for (K, Y 1 ) as follows\nE 0 (t) := − log y 1 K P K (K)P Y |K (y 1 | K) 1/(1+t) 1+t .\nThen E 0 (0) = I e (K ; Y 1 ) = EI e (U j 1 ; Y 1 | G). Owing to the concavity of the E-null function, E 0 (0) E 0 (t)/t whenever −2/5 t < 0. Recap: To bound the average leaked information EI e (U j 1 ; Y 1 | G) it suffices to bound I e (K ; Y 1 ), which is then morphing to bounding E 0 (0) from above and to bounding E 0 (t) from below.\n\nThe double-stroke E-null function is bounded as below. Assume −2/5 t < 0. Let s be −t/(1 + t); so 0 < s 2/3 and (1 + s)(1 + t) = 1. For any fixed K and fixed x 1 ∈ K, the base of the (1 + t)-th root in the definition of the double-stroke E-null function is\nP Y |K (y 1 | K) = ξ 1 ∈K P X|K (ξ 1 | K)P Y |X (y 1 | ξ 1 ) = ξ 1 ∈K q j P X (ξ 1 )P Y |X (y 1 | ξ 1 ) = ξ 1 ∈K q j P XY (ξ 1 , y 1 ) = q j P XY (x 1 , y 1 ) + q j x 1 =ξ 1 ∈K P XY (ξ 1 , y 1 ) = q j P XY (x 1 , y 1 ) + q j P XY (K\\x 1 , y 1 ).\nHere P XY (K\\x 1 , y 1 ) is a temporary shorthand for the summation of P XY (ξ 1 , y 1 ) over ξ 1 ∈ K that excludes x 1 . Raise it to the power of 1/(1 + t) = (1 + s):\nP Y |K (y 1 | K) 1+s = P Y |K (y 1 | K)P Y |K (y 1 | K) s = x 1 ∈K q j P XY (x 1 , y 1 )P Y |K (y 1 | K) s =\nx 1 ∈K q j P XY (x 1 , y 1 ) q j P XY (x 1 , y 1 ) + q j P XY (K\\x 1 , y 1 ) s x 1 ∈K q j P XY (x 1 , y 1 ) q js P XY (x 1 , y 1 ) s + q js P XY (K\\x 1 , y 1 ) s = q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (x 1 , y 1 ) s + q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1 ) s = q j+s x 1 ∈K P XY (x 1 , y 1 ) 1+s + q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1 \n) s .\nThe inequality uses that the s-th power is sub-additive. Then the inner sum of the E-null function morphs as follows\nK P K (K)P Y |K (y 1 | K) 1+s K P K (K) q j+js x 1 ∈K P XY (x 1 , y 1 ) 1+s + q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1 ) s = q j+js K P K (K) x 1 ∈K P XY (x 1 , y 1 ) 1+s + (major arc) + q j+js K P K (K)\nx 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1 ) s .\n\n\n(minor arc)\n\nDivide and conquer-the inner sum of the double-stroke E-null function is split into two arcs as shown. The major arc is exactly\nq j+js K P K (K) x 1 ∈K P XY (x 1 , y 1 ) 1+s = q j+js q −j x 1 ∈F q P XY (x 1 , y 1 ) 1+s = q js x 1 ∈F q P X (x 1 ) 1+s P Y |X (y 1 | x 1 ) 1+s = q js− s x 1 ∈F q P X (x 1 )P Y |X (y 1 | x 1 ) 1+s .\nThe minor arc is loosen to q j+js K P K (K)\n\nx 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1 ) s = q j+js\nx 1 ∈F q P XY (x 1 , y 1 ) K x 1 P K (K)P XY (K\\x 1 , y 1 ) s = q j+js x 1 ∈F q P XY (x 1 , y 1 )q −j K x 1 P K|X (K|x 1 )P XY (K\\x 1 , y 1 ) s q j+js x 1 ∈F q P XY (x 1 , y 1 )q −j K x 1 P K|X (K|x 1 )P XY (K\\x 1 , y 1 ) s = q j+js x 1 ∈F q P XY (x 1 , y 1 )q −j K x 1 P K|X (K|x 1 ) x 1 =ξ 1 ∈K P XY (ξ 1 , y 1 ) s = q j+js x 1 ∈F q P XY (x 1 , y 1 )q −j q −j − 1 q − 1 x 1 =ξ 1 ∈F q P XY (ξ 1 , y 1 ) s x 1 ∈F q P XY (x 1 , y 1 ) x 1 =ξ 1 ∈F q P XY (ξ 1 , y 1 ) s x 1 ∈F q P XY (x 1 , y 1 ) ξ 1 ∈F q P XY (ξ 1 , y 1 ) s = x 1 ∈F q P XY (x 1 , y 1 )P Y (y 1 ) s = P Y (y 1 )P Y (y 1 ) s = P Y (y 1 ) 1+s .\nBoth major and minor arcs conquered, merge them and raise to the (1+t)-th power. The summand for any fixed y 1 in the definition of the double-stroke E-null function is\nK P K (K)P Y |K (y 1 | K) 1/(1+t) 1+t (minor + major) 1+t minor 1+t + major 1+t P Y (y 1 ) 1+s 1+t + q js− s x 1 ∈F q P X (x 1 )P Y |X (y 1 | x 1 ) 1+s 1+t = P Y (y 1 ) + q t−jt x 1 ∈F q P X (x 1 )P Y |X (y 1 | x 1 ) 1+s 1+t\nWe can finally bound the double-stroke E-null function per se:\nexp(−E 0 (t)) = y 1 K P K (K)P Y |K (y 1 | K) 1/(1+t) 1+t y 1 P Y (y 1 ) + q t−jt x 1 ∈F q P X (x 1 )P Y |X (y 1 | x 1 ) 1+s 1+t = 1 + q t−jt y 1 x 1 ∈F q P X (x 1 )P Y |X (y 1 | x 1 ) 1+s 1+t = 1 + q t−jt exp(−(the E-null function of W )(t)) = 1 + q t−jt exp(− E 0 (t)).\nAll efforts we spent on bounding I e (U j 1 ; Y 1 ) are for three creeds: First, we see Gallager's bound possessing innate elegance. Second, it fits the paradigm that solving the primary and the dual problems as a whole is easier than solving the primary problem alone. Third, the universal quadratic bound is waiting ahead for the E-null function. We infer that\nEI e (U j 1 ; Y 1 | G) = I e (K ; Y 1 ) = E 0 (0) 1 t E 0 (t) = 1 −t log exp(−E 0 (t)) 1 −t log 1 + q t−jt exp(− E 0 (t)) 1 −t q t−jt exp(− E 0 (t)) = exp(− log(−t) + ( − j)t log q − E 0 (t)).\nRecall the universal quadratic bound E 0 (t) I(W )t log q − t 2 log(q) 2 as stated in Lemma 17 and used in the previous subsection. But this time −2/5 t < 0. We obtain that the exponent is\n− log(−t) + ( − j)t log q − E 0 (t) = − log(−t) + ( − H(W ) + 1/2+α )t log q − E 0 (t) − log(−t) + ( − H(W ) + 1/2+α )t log q − (I(W )t log q − t 2 log(q) 2 ) = − log(−t) + ( t log q + 1/2+α )t log q (redeem at t = − −1/2+α /2 log q) → − log( −1/2+α /2 log q) − (− −1/2+α /2 + 1/2+α ) −1/2+α /2\n= log( )/2 − α log + log 2 + log log q − 2α /4 = log( )/2 − log log + log 2 + log log q − 2 log(log )/ log /4 < log( )/2 + log log q − log( ) 2 /4.\n\nThe first inequality uses − j = − H(W ) + 1/2+α . The last inequality uses the assumption e 2 . With the last line we conclude that EI e (U j 1 ; Y 1 | G) < exp(log( )/2 + log log q − log( ) 2 /4) = 1/2−log( )/4 log q. Switch back to the base-q mutual information EI(U j 1 ; Y 1 | G) < 1/2−log( )/4 . We now reject kernels such that I(U j 1 ; Y 1 | G) 1/2−log( )/5 . By Markov's inequality, the opposite direction (<) holds with probability 1 − − log( )/20 because 1/5 + 1/20 = 1/4. Plug this upper bound into h α . The left hand side of Inequality (20) is less than\njh α 1 j 1/2−log( )/5 = jj −α α/2−α log( )/5 < 1−α α/2−α log( )/5 = 1−α/2−α log( )/5 = log( ) −1/2−log( )/5 .\nThe inequality uses that the left hand side increases monotonically in j and j := H(W ) − 1/2+α < . In any regard, the quantity at the end of the inequalities decays to 0 as → ∞, so eventually it becomes less than 1/2+α , the right hand side of Inequality (20 [Ooh02], although no formal proof was found there. See [HM11,BTM17] for alternative descriptions and approaches on the same topic. For readers who took Lemma 12 as granted or went through Appendix B in advance, this is the last sentence of the proof of the main theorem-polar codes' simplicity, random codes' durability.\n\n\nConclusions\n\nShannon introduced what we now understand as discrete memoryless channels seventy-two years ago. In the beginning, Shannon had no tool but developed their own theory of typical set, proved the noisy channel coding theory, and justified the notion of capacity. Gallager brought in error exponents. Capacities and error exponents quantify first and second order terms in the asymptotic performance of codes. Only in 2010 we are revealed the complete second order term. It was around the time that polar coding as a graceful instrument to explore the limits at low cost was discovered when Arıkan experimented with the channel transformation and with error exponents. Another ten years it took to grow variants and proof techniques of polar coding. Ultimately, it is feasible, and done by us coincidentally, to piece the puzzle together to show the mere possibility to achieve the second order limits at low cost. An overall comparison is integrated in Table 2. Columns are classes of channels; from left to right: (BEC) binary erasure channels; (BDMC) binary-input discreteoutput memoryless channels; (p-ary) channels of prime input size; (q-ary) channels of prime power input size; (finite) channels of discrete input. Columns to the right are wider than columns to the left. The last column is exceptional; (asym.) is about whether we can achieve the true Shannon capacity, instead of the symmetric capacity. Rows are goals; from top to bottom: (LLN) to achieve (symmetric) capacity; (wLDP) there exists π > 0 such that P e < exp(−N π ); (wCLT) there exists ρ > 0 such that R > I − N −ρ ; (wMDP) there exists π, ρ > 0 such that P e < exp(−N π ) and R > I − N −ρ at once; (LDP) the π in (wLDP) can be arbitrarily close to 1; (CLT) the ρ in (wCLT) can be arbitrarily close to 1/2; (MDP) the (π, ρ)-pair can be arbitrarily close to π + 2ρ = 1. Row (MDP) implies every other row; row (CLT) implies (wCLT); row (LDP) implies (wLDP); and every other row implies row (LLN). Rows (LDP) and (CLT) together almost imply (MDP) (need the partial distance profile). Cells represent how various goals are achieved over various channels. The greenish background means it is possible using Arıkan's kernel [ 1 1 0 1 ]. The purplish background means it is possible using other kernels. The orangish background means it is only possible using dynamic kernels.\n\nThe following works made critical progresses but our classification fails to include them: Rate-dependent result in LDP paradigm [HMTU13]. Optimal relations among channel parameters [MT14]. First family of (π, ρ) pairs in MDP paradigm [MHU16]. AWGNCs intersecting MDP [FT17].\n\nWe did our best to excavate the archive but throughout the course of manuscript preparation we found ourselves underestimating early works multiple times so the record kept updating. We sincerely hope to hear about possible references to add to the table.\n\nPotential improvements include but are not limited to the following: (Tolls) Tighten the explicit Hölder tolls. The current toll between any pair of parameters H, P e , Z, Z mad , T , S, and S max is roughly the sum of tolls collected when traveling through the spanning tree illustrated in Lemma 7. Some improvements potentially tighten the bounds in Lemma 12. (FTPC) Tighten the two fundamental theorems such that they degenerate to equalities over erasure channels. Once done, {Z n } is a supermartingale and Appendix C.1 is obsolete. (Symmetry) generalize the arguments in Sections 8.2 to 8.4 to asymmetric channels. Once done, Section 8.1 is obsolete. Note that the proof of the fundamental theorems applies to asymmetric channels. (Bijection) Early works on polar coding over arbitrary alphabets introduced arbitrary bijections g W . Generalize the two fundamental theorems to include arbitrary bijections. (Dynamic) Achieve the main theorem with a large, but fixed, kernel. This does not immediately make the code practical. But the answer should shed light on our understanding of coding. (Alphabet) Achieve the main theorem without the reduction to prime power alphabets. This is currently not an option because linear codes are barely defined over non-fields. Plus the S-parameter-and thus FTPCS-would just break. (Dispersion) Recall Proposition 2. Weird things happens when the channel dispersion vanishes V = 0. Can we describe those channels better? One example of such channels is this:\n 1/2 1/2 0 1/2 0 1/2 0 1/2 1/2 .\nWe look forward to generalizations of the main theorem to non-identical channels (i.e., non-stationary) [Mah17], non-independent channels (i.e., with memory) [WHY + 15, ST16], deletion channels [TPFV19,LT19], channels with restrictions on input distributions (e.g., due to energy constrain) [FT16], wiretap channels [ŞV13], rate-distortion problem [HKU09], Wyner-Ziv problem [HKU09], Slepian-Wolf problem [Abb15], broadcast channels [GAG15,MHSU15], and multiple access channels [AT12,NT16]. We focus on noisy channel coding in this work for its historical significance.\n\nAppendix A. Explicit Hölder Tolls (Proof of Lemma 8)\n\nAs is promised in Section 4.1, we prove the explicit Hölder toll. Let W be a q-ary channel. In the upcoming arguments, H, P e , Z, Z mad , S, and S max mean H(W ), P e (W ), Z(W ), Z mad (W ), S(W ), and S max (W ), respectively. Also q means q − 1, and q means q − 2. Furthermore, lg means the base-2 logarithm; this is handy when we jump back and forth between nats, bits, and q-bits.\n\nFirst we show Z mad q H log 4 q.\n\n(2's copy)\n\nStart from Z mad : By the definition Z mad q Z. Move on to Z: By Lemma 3,\nq q −2 ( √ 1 + q Z − √ 1 − Z) 2 P e so √ 1 + q Z − √ 1 − Z q P e /q . Multiplying by the conjugate yields (1 + q Z) − (1 − Z) q P e /q ( √ 1 + q Z + √ 1 − Z).\nThe left hand side is qZ; in the right hand side √ 1 + q z + √ 1 − z has maximum q/ √ q at z = q /q by calculus. So Z P e /q (q/ √ q ) = q √ P e /q . Move on to P e : By Lemma 6 (the first lower bound), 2P e H lg q or equivalently P e H log 4 q. Now we chain the inequalities Z mad q Z q √ P e q H log 4 q. This completes Inequality (2). That being proven, we use the weaker form Z mad q 3 √ H in the calculus machinery for global MDP. Second we show H » eq Z mad /2.\n\n(3's copy) Start from H: By Lemma 6 (the upper bound, Fano's inequality), H lg q h 2 (P e )+ P e lg q . By Figure 6, h 2 (P e ) + P e lg q √ eP e + P e lg q = √ P e ( √ e + √ P e lg q ). What is inside parentheses is less than √ e + q /q lg q . Hence H √ P e ( √ e + q /q lg q )/ lg q. Focus on the scalar part-( √ e + q /q lg q )/ lg q has maximum √ e at q = 2 (remember that q 2). So H √ eP e . Move on to P e : By Lemma 3, P e q Z/2. Move on to Z: By definition Z Z mad . Now we chain the inequalities H √ eP e eq Z/2 eq Z mad /2. This completes Inequality (3). That being proven, we use the weaker form H q 3 Z mad in the calculus machinery for global MDP.\n\nThird we show (notice the logarithm is natural)\nS max q q » (1 − H) log(q)/2.\n(4's copy)\n\nStart from S max : By definition S max q S. Move on to S: By Lemma 5, S q q(q /q − P e ) » 1 − q q q q . The square root simplifies to 1/(q ) 2 = 1/q as qq = (q ) 2 − 1. So S q − qP e . Move on to q − qP e : By Lemma 6 (the upper bound, Fano's inequality), H lg q h 2 (P e ) + P e lg q . We claim that h 2 (P e ) + P e lg q lg q − 2(q /q − P e ) 2 / log 2. To prove the claim, Taylor expand both sides at P e = q /q. Verify that both evaluate to lg q at P e = q /q; verify that both have derivative 0 at P e = q /q; and verify that the acceleration of the left hand side, −1/(P e (1 − P e ) log 2), is more negative than the acceleration of the right hand side, −4/ log 2. By Taylor's theorem, mean value theorem, or Euler method, the function with greater acceleration is greater; hence the claim. See also [FM94, Fig. 1]; the Φ-curve seems parabolic at the upper right corner. Now we have H lg q lg q−2(q /q−P e ) 2 / log 2, which is equivalent to 2(q /q−P e ) 2 / log q 1−H and to q − qP e q (1 − H) log(q)/2. Now we chain the inequalities S max q S q (q − qP e ) q q (1 − H) log(q)/2. This completes Inequality (4). That being proven, we use the weaker form\nS max q 3 √ 1 − H\nin the calculus machinery for global MDP. Fourth we show 1 − H q S max / log q.\n\n(5's copy) Start from 1 − H: By Lemma 6 (the second lower bound), H lg q q q lg(q/q ) × (P e − q /q ) + lg q . The right hand side is lg q − q lg(q/q )(q − qP e ) by matching the (rational) coefficients of P e lg q, P e lg q , lg q, and lg q , respectively. As H lg q lg q − q lg(q/q )(q − qP e ) we bound lg(q/q ) = lg(1 + 1/q ) 1/q by the tangent line at 1/q = 0. So H lg q lg q − (q − qP e ) and hence 1 − H (q − qP e )/ lg q. Move on to q − qP e : By Lemma 5, 1 − qP e /q S so q − qP e q S. Move on to S: By definition S S max . Now we chain the inequalities 1 − H (q − qP e )/ lg q q S/ lg q q S max / lg q. This completes Inequality (5). That being proven, we use the weaker form 1 − H q 3 S max in the calculus machinery for global MDP. This is end of the proof of Lemma 8. The proof of Lemma 7 follows the same logic, only shorter.\n\nAppendix B. Calculus Machinery for Global MDP (Proof of Lemma 12)\n\nWe are to prove that\nP{H n < exp(− πn n)} > 1 − H 0 − −ρn+o(n) (7's copy)\ngiven criteria (cb), (cm), (ct), and (cl), the local LDP behavior, the local CLT behavior, and that π + 2ρ 1 − 8α. The proof is split into several stepping stones. We will prove each of the following inequalities (including two equalities) in each of the upcoming subsections. This will be proven in Appendix B.1: The eigen behavior reads\nE [h α (H n+1 ) | F n ] 4 −1/2+3α h α (H n ).(23)\nThis will be proven in Appendix B.2: As a lemma, {H n } and {Z n } converges to 0 with probability 1 − H 0 , i.e.,\nP{Z n → 0} = P{H n → 0} = 1 − H 0 .(24)\nThis will be proven in Appendix B.3: The en23 behavior reads\nP{Z n < exp(−n 2/3 )} > H 0 − (−1/2+4α)n+o(n) .(25)\nThis will be proven in Appendix C.1: As a lemma, {min( −2 , 4 √ Z n )} is a supermartingale, i.e.,\nE [min( −2 , 4 Z n+1 ) | F n ] min( −2 , 4 Z n ).(26)\nThis will be proven in Appendix C.2: As a lemma, the following holds when Z 0 < −8 :\nZ n+1 Z K 2 n+1 /3 ·3/4 n and\nE [( K 2 n+1 /3 · 3/4) −1/2 | F n ] < −1/2+2α . (27) This will be proven in Appendix C.3: The een13 behavior reads\nP{Z n < exp(−e n 1/3 )} > 1 − H 0 − (−1/2+4α)n+o(n) .(28)\nThis will be proven in Appendix C.4: The elpin behavior reads for any constants π, ρ > 0 such that π + ρ 1 − 8α,\nP{Z n < exp(− πn n 2 )} > 1 − H 0 − −ρn+o(n) .(29)\nThe last inequality is a bi-Hölder toll away from P{H n < exp(− πn n)} > 1 − H 0 − −ρn+o(n) , (7's copy) our destination. This finishes the proof of Lemma 12. The eigen, en23, een13, and elpin behaviors are intermediate checkpoints pinned in a way that moving from one to the next is easy while skipping any of them makes the next unreachable. Their entire purpose is to form a chain that connects the local LDP and CLT behaviors to the global MDP behavior and we do not specify if any of them falls inside the LDP, CLT or MDP paradigm. B.1. The eigen behavior. We want to prove Inequality (23), E [h α (H n+1 ) | F n ] 4 −1/2+3α h α (H n ), given the local LDP behavior and the local CLT behavior. The idea is that, for H n that is close to 1/2, the local CLT behavior provides a measurement of the dichotomy/bifurcation behavior of H n+1 . For H n that are close to 0, the Z -part of the local LDP behavior provides a measurement of the attraction toward 0. For H n that is close to 1, the S-part handles it dually. The formal proof is below.\n\nInequality (23) is a local statement so we may assume n = 0. To prove that The last case-the reliable case: when H 0 < −2 , we further split it into two subcases per how K 1 compares to k := 1/2+5α/2 . For the small K 1 subcase, the martingale property fits. For the large K 1 subcase, the local LDP behavior fits:\nE [h α (H 1 )] 4 −1/2+3α h α (H 0 ),E [h α (H 1 )] = E [h α (H 1 ) | K 1 k]k/ + E [h α (H 1 ) | K 1 > k](1 − k/ ) h α (E [H 1 | K 1 k])k/ + h α (E [H 1 | K 1 > k])(1 − k/ ).(30)\nFor the K 1 k := 1/2+5α/2 subcase, the martingale property\nE [H 1 ] = H 0 implies that E [H 1 | K 1 k] H 0 /k. Thus h α (E [H 1 | K 1 k])k/ h α (H 0 /k)k/ = h α (H 0 ) α k −α k −1 = h α (H 0 ) α −α/2−5α 2 /2 1/2+5α/2 −1 −1/2+3α h α (H 0 )\n. And the K 1 k subcase is closed. For the K 1 > k := 1/2+5α/2 subcase, pay the explicit Hölder toll: Z 0 q 3 √ H 0 < q 3 / < 1. Invoke the local LDP behavior:\nE [Z 1 | K 1 > k] E [ exp(qZ 0 )(qZ 0 ) K 2 1 /3 | K 1 > k]\nexp(qZ 0 )(qZ 0 ) k 2 /3 exp(q 4 )(q 4 √ H 0 ) k 2 /3 = exp(q 4 )(q 8 H 0 ) log( ) 5 /6 . Pay the return-trip toll: H 1 q 3 √ Z 1 q 3 1/2 exp(q 4 /2)(q 8 H 0 ) log( ) 5 /12 . Now we claim and prove that the following quantity is less than 1: (there is nothing to show if h α (H 0 ) = 0) (h α (H 1 )/ −1/2+3α h α (H 0 )) 12/α = H 12 The inequality involving 1.6 uses q 5 . The inequality involving 0.3 uses max(41, 3 q ). The inequality involving 15 uses max(22, q 5 ). We just showed that h α (H 1 )/ −1/2+3α h α (H 0 ) is less than 1, with and hence without the 12/α-th power. Thus E [h α (H 1 ) | K 1 > k] −1/2+3α h α (H 0 ). And the K 1 > k := 1/2+5α/2 subcase is closed. To sum up the reliable case: We bound separately the two terms in Formula (30). They are both at most −1/2+3α h α (H 1 ), hence their sum is at most 2 −1/2+3α h α (H 1 ). Since Inequality (23) wants 4 instead of 2, The reliable case is closed. And the proof of the eigen behavior, Inequality (23), is sound when combining the three cases.\n\nBibliographic remarks: [FHMV17, Theorem 7] also cut the cases at −2 and 1 − −2 . In contrast, [GRY19, Theorem 5.1] cut at −4 and 1 − −4 − ε. A potential improvement is, when −2 H 0 < −1 , Inequality (15) will simply evaporate. Similarly, Inequality (14) evaporates when 1 − −1 < H 0 1 − −2 . They tighten the right hand side of Inequality (10). The lesson here is that the hard transition between local LDP and CLT behaviors weakens the bounds.\n\nB.2. Polarization in mean. We want to prove Equality (24), P{Z 0 → 0} = P{H 0 → 0} = 1 − H 0 , given the martingale property and the eigen behavior. The idea is that the eigen behavior expels H n from being close to 1/2, so the only reasonable limits are 0 and 1. The formal proof is below.\n\nAs B.3. The en23 behavior. We want to prove P{Z n < exp(−n 2/3 )} < 1 − H 0 − (−1/2+4α)n+o(n) , namely Inequality (25), given the eigen behavior and the polarization in mean. The idea is to read off the behavior of {H n } from the behavior of {h α (H n )} in the eigen behavior. The formal proof is below.\nE [h α (H n+1 ) | F n ] −1/2+4α h α (H n ) by e 4 .\nThis simplifies the eigenvalue. Without loss of generality, we rescale h α such that h α (H 0 ) = 1. Let ε n be exp(−n 3/4 ); note that ε n H 0 1 − ε n for n large enough. Owing to h α 's concavity, that h α (0) = h(1) = 0, and that h α (H 0 ) = 1, we deduce that h α (z) ε n whenever ε n z 1 − ε n . Consider these three events as a partition: let A n be {H n < ε n }; let B n be {ε n H n 1 − ε n }; let C n be {1 − ε n < H n }. Note that B n implies h α (H n ) ε n .\n\nNext we show P(B n ) < (−1/2+4α)n+o(n) : Telescoping leads to E[h α (H n )] h a (H 0 ) (−1/2+4α)n = (−1/2+4α)n . Markov's inequality leads to P{h(H n ) ε n } E[h(H n )]/ε n (−1/2+4α)n /ε n = −(−1/2+4α)n+O(n 3/4 ) < (−1/2+4α)n+o(n) . Therefore P(B n ) P{h(H n ) ε n } < (−1/2+4α)n+o(n) , as desired. Moreover, summing the geometric series leads to m n P(B m ) < (−1/2+4α)n+o(n) .\n\nNext we show 1 − H 0 − P(A n ) < (−1/2+4α)n+o(n) : The left hand side is at most the probability measure of {H ∞ = 0} \\ A n . That is the probability that H n was not small (not in A n ) but H n+1 , H n+1 , . . . will end up converging to 0. Note that being a martingale causes 1 − H n+1\n\n(1 − H n ), which forbids H n jumping from C n directly into A n+1 -it must pass by B m for some m n before ever landing in A m+1 . From the summation of P(B m ) over m n we know that very few descendants of H n can do that; the probability measure of {H ∞ = 0} \\ A n is less than (−1/2+4α)n+o(n) . Therefore 1 − H 0 − P(A n ) < (−1/2+4α)n+o(n) and hence P{H n < exp(−n 3/4 )} = P(A n ) > 1 − H 0 − (−1/2+4α)n+o(n) . Pay the bi-Hölder toll P{Z n < exp(−n 2/3 )} > 1 − H 0 − (−1/2+4α)n+o(n) . And the proof of the en23 behavior, Inequality (25), is sound.\n\nhere is an aesthetic choice; min( −2 , 2+ε √ Z n ) is also a supermartingale but only for astronomic (depending on q). For any non-random kernel, a small enough power works provided that the kernel polarizes channels in the first place.\n\nC.2. A Cramér-Chernoff gadget. Let D n+1 be K 2 n+1 /3 · 3/4. We want to prove inequalities in (27), that Z n < −8 implies Z n+1 Z Dn+1 n and E [D −1/2 n+1 | F n ] < −1/2+2α , given the local LDP behavior. The motivation is to reformat the local inequalities so that it is easy to telescope for future reference. The formal proof is below.\n\nThey are both local statements so we may assume n = 0. When Z 0 < −8 , invoke the local LDP behavior Z 1 exp(qZ 0 )(qZ 0 ) K 2 \n1 /3 e(qZ 0 ) K 2 1 /3 e(q 4 Z 0 ) K 2 1 /3 /4 Z K 2 1 /3 ·3/4 0 e( −7 ) K 2 1 /3 /4 Z K 2 1 /3 ·3/4 0 Z K 2 1 /3 ·3/4 0 .k 2 4 −1/2 < 1 √ 3 2 √ 3 + 1 √ 4 √ 3 dk k = 2 −1/2 + 2 −1/2 log k √ 3\n< 2 −1/2 + 2 −1/2 log = 2 −1/2 + 2 −1/2+α < 4 −1/2+α < −1/2+2α .\n\nThe last inequality uses e 4 . This validates the second inequality in (27). And the proof of inequalities in (27) is sound.\n\nC.3. The een13 Behavior. We want to prove P{Z n < exp(−e n 1/3 )} > 1 − H 0 − (−1/2+4α)n+o(n) , namely Inequality (28), given the en23 behavior, the supermartingale property, and the Cramér-Chernoff gadget. The idea is to apply the gadget consecutively to show that Z n becomes smaller and smaller as n increases. To reach the goal exp(−e n 1/3 ), we apply √ n times to avoid losing too much code rate. /a m from above.) Conditioning on A m , we want to estimate the probability that Z k −8 for some k m, which is equal to the probability that min( −2 , 4 √ Z k ) −2 for some k m. Recall that min( −2 , 4 √ Z k ) was made a supermartingale. Hence by Doob's optional stopping theorem [Dur19, Exercise 4.8.2], P{min( −2 , 4 √ Z k ) −2 for some k m | A m } min( −2 , 4 √ Z m ) 2 < exp(−m 2/3 /4) 2 . This is an upper bound on b m /a m and will be summoned in Formula (32).\n\n(Bound c m /a m from above.) We want to estimate how often does Inequality (31) happen. It is the probability of (D m+1 D m+2 · · · D m+ \n\nThe first three equalities are by the definitions of g m and E m 0 . The next equality is simple algebra. The next two inequalities are by 0 e m /a m 1. The next inequality is by the definition of E m . The last inequality summons upper bounds derived in the last few paragraphs. The last line contains two terms in the big parentheses. Between them (−1/2+3α) happens. More precisely, we attempt to bound Z m+ √ n when E m happens for each m = √ n, 2 √ n, . . . , n − √ n. When E m happens, its superevent A m happens, so we know that Z m < exp(−m 2/3 ). But B m does not happen, so Z k < −8 for all k m. This implies that Z k+1 Z D k+1 k for those k. Telescope; Z m+ √ n is less than Z m raised to the power of D m+1 D m+2 · · · D m+ √ n . But C m does not happen, so the product is greater than 2α eqZ k for all k m + √ n so long as Z k stays below −8 , which it does because B m is excluded. Then telescope again; Z n ( eq) n−m− √ n Z m+ √ n < ( eq) n exp(−m 2/3 2α √ n ) < exp(−e n 1/3 ) provided that n is sufficiently large. In other words, E n− √ n 0 implies Z n < exp(−e n 1/3 ).\n\n(Summary.) Now we conclude that P{Z n < exp(−e n 1/3 )} P(E n− √ n 0 ) = e n 0 > 1 − H 0 − (−1/2+4α)n+o(n) . And hence the proof of the een13 behavior, Inequality (28), is sound.\n\nThis subsection is parallel to [WD18, Section V]. Do not confuse this subsection with the next. The subtlety is explained in [WD18, Section III].\n\nC.4. The elpin behavior. Recall π, ρ > 0 is such that π + 2ρ 1 − 8α. We want to prove P{Z n < exp(− πn n 2 )} > 1 − H 0 − −ρn+o(n) , namely Inequality (29), given the een13 behavior, the supermartingale property, and the Cramér-Chernoff gadget. The idea is to apply the gadget consecutively to show that Z n becomes smaller and smaller as n increases. To reach the goal exp(− πn ), we apply as many times as possible before we run out of depth n. (Bound c m /a m from above.) We want to estimate how often does Inequality (33) happen. It is the probability of (D m+1 D m+2 · · · D n ) −1/2 −πn/2 . By Markov's inequality, this probability is at most E [D −1/2 1 ] n−m πn/2 < (−1/2+2α)(n−m) πn/2 = (1/2−2α)m−(1/2−2α−π/2)n (1/2−2α)m−(ρ+2α)n . The last inequality uses Inequality (6), π + 2ρ 1 − 8α. This is an upper bound on c m /a m and will be summoned in Formula (34).\n\n(Bound f + m from above.) The definition of f m reads 1 − H 0 − a m 0 . Here a m 0 is the probability measure of A m 0 , and A m 0 is a superevent of A m by how the former is defined. Event A m 0 must contain {Z m < exp(−e m 1/3 )} by how A m was defined. By the een13 behavior, P{Z m < exp(−e m 1/3 )} > 1 − H 0 − (−1/2+4α)m+o(m) . Chaining all inequalities together, we infer that f m < (−1/2+4α)m+o(m) . Let f + m be max(0, f m+ √ n ) so we can write f + m < (−1/2+4α)m+o(m) . This upper bound will be summoned in Formula (34).\n\n(Bound e n 0 from below.) We start rewriting g m − f + m with (f m− happens. More precisely, we attempt to bound Z n when E m happens for each m = √ n, 2 √ n, . . . , n − √ n. When E m happens, its superevent A m happens, so we know that Z m < exp(−e m 1/3 ). But B m does not happen, so Z k < −8 for all k m. This implies Z k+1 Z D k+1 k for those k. Telescope; Z n is less than Z m raised to the power of D m+1 D m+2 · · · D n . But C m does not happen, so the product is greater than πn . Jointly we have Z n Z πn m < exp(−e m 1/3 πn ) < exp(− πn n 2 ). In other words, E n− √ n 0 implies Z n < exp(− πn n 2 ). (Summary.) Now we conclude that P{Z n < exp(− πn n 2 )} P(E n− √ n 0 ) = e n 0 > 1 − H 0 − −ρn+o(n) . And hence the proof of the elpin behavior, Inequality (29), is sound.\n\nThis subsection is parallel to [WD18, Section VI]. Do not confuse this subsection with the previous. The subtlety is explained in [WD18, Section III].\n\nAs we finish proving Inequalities (23) to (29), we finish the proof of Lemma 12. Lemmas 12 to 14 are all finished. This is the last sentence of the proof of the main theorem.\n\n\nAppendix D. Constants Dependence Summary\n\nGiven a discrete memoryless channel W . The sender chooses the message alphabet size ς 2. Depending on the factorization of ς, we choose q to be a certain prime power or alternate between q 2 , q 3 , q 5 , . . . (a finite list depending on ς). Fix a q. Given π, ρ > 0 such that π + 2ρ < 1; fix them. Choose ; this also determines α := log(log )/ log . The choice of is such that π + 2ρ 1 − 8α and such that the failing probabilities in Lemmas 13 and 14 do not sum to one. It depends on q, π, ρ. Once is fixed, the complexity is a function in n (or in N = n ). The asymptotic complexity O(N log N ) hides the scalar term that is determined by q and . The decaying gap −ρn+o(n) in Claim 11 and Lemma 12 hides two things: A scalar term in front of determined by q and alongside with a O(n 1−ε ) term determined by the choice of en23 and een13 checkpoints. This ε is fixed throughout the paper and is irrespective of ς, π, ρ, q, .\n\nFigure 1 .\n1Recent works on polar coding arranged on a ρ-π plot.\n\n\nConditions (cn), (cp), and (cr) cannot hold simultaneously.Proof. If so, N (I − R) 2 /|log P e | −N N −2ρ /N π = N 1−2ρ−π → 0 as N → ∞. This contradicts lim inf N →∞ N (I − R) 2 /|log P e | 2V > 0 [AW10,Theorem 2.2] [PV10, Theorem 6] [AW14, Theorem 2]. Remark: For V = 0 channels, the correct threshold seems to be π + ρ = 1 [Pol10, Inequality (3.354)] [AW14, Remark 1].\n\nFigure 4\n4Figure 4. 12 DUs (with = 2) are chained together to implement ((W (1) ) (1) ) (1) , . . . , ((W (2) ) (2) ) (2) . DUs in the first column use g W ; DUs in the second column use g W (1) and g W (2) ; DUs in the third column use g (W (1) ) (1) , g (W (1) ) (2) , g (W (2) ) (1) , and g (W (2) ) (2) .\n\n\nbut rather a mixture of Z max and Z d therein. That said, the definitions of other parameters-H, I, P e , Z, T , S, and S max -match [MT14]'s. Cf. [Sas11, Section 3.C].\n\n\n(a, b) if there exists c, d > 0 such that |A(W )−a| < c|B(W )−b| d and |B(W )−b| < c|A(W )−a| d for all q-ary channels W . The notion of bi-Hölder is an equivalence relation. In particular, if A, B are bi-Hölder at (a, b) and (B, C) are bi-Hölder at (b, c), then (A, C) are bi-Hölder at (a, c). In this case, it makes sense to say A, B, C are bi-Hölder at (a, b, c)\n\n\nTheorem 10 (fundamental theorem of polar coding-the S-end, FTPCS). [Kor09, Lemma 5.7] [KU10, Theorem 19] [KT10, Lemma 6] [MT14, Lemma 34] [GRY19, Inequalities (74) and (75)]\n\nFigure 5 .\n5The trichotomy of the fates of synthetic channels. Label (A) marks the corner of the free and reliable channels. Label I(W ) beneath (A) is the limit of the probability measure P(A n ) = P{W n is free and reliable} as n → ∞. Labels (B) and (C) and the numbers beneath marks the corresponding fates and probability measures.\n\n\n. The following two lemmas help verify the two criteria in Lemma 12. Proofs are given in upcoming sections, 7 and 8.Lemma 13 (local LDP behavior). Fix an 30. Let G vary; with probability less than 3q − √ /13 , each of the following fails for each i ∈ [ ]:\n\n\nHence the process {H n } satisfies Inequality (7) for the four processes: {H(W n )} and {1 − H(W n )} along with {H(V n )} as well as {1 − H(V n )}. This results in the four inequalities in Claim 11. And we are done. It remains to prove Lemmas 12 to 14 in order to prove the main theorem. 7. Local LDP Behavior (Proof of Lemma 13)\n\n\nGZ (Z mad (W )). By the definition of W (i) and the definition of the Bhattacharyya parameter, Z mad (W\n\n\nGS (S max (W )). And quod erat demonstrandum. Experienced readers may find that all but the last inequality is a duplicate of [MT14, Lemma 34]. 7.3. An upper bound on entropy functions. For all z ∈ [0, 1],\n\nFigure 6 .\n6To the left: Binary entropy function h 2 (z) and an upper bound of √ ez. To the right: 1 − D(z 1 − 1/q)/ log q for q = 3, 4, 5, 7 and an upper bound of √ ez.\n\n7. 4 .\n4On the weight distribution of random linear codes. This subsection contains the nontrivial part of the proof of Lemma 13. Fix any i ∈ [ ]. We want to prove that when G ∈ GL( , q) is selected uniformly at random, the inequalityZ mad (W (i) G )exp(qZ mad (W ) )(qZ mad (W )) . In bounding the left hand side, the fundamental theorem of polar coding-Z-end reads Z mad (W\n\n\nthe weight enumerator of codewords of the form 0 i−1 1 1u i+1 G. Thus it remains to show the inequality with the left hand side replaced f (i) GZ (z) exp(qz )(qz) i 2 /3\n\n\n(q −1)z) −1 (q −1)z. The right hand side over counts all nonzero codewords by choosing a nonzero position ( ), assigning a nonzero symbol ((q − 1)z), and filling in the rest of − 1 blanks arbitrarily ((1 + (q − 1)z) −1 ). On the left hand side, f (i) GZ enumerates only codewords of the form 0 i−1 1 1 i u i+1 G, which are all nonzero as G is invertible. Hence Inequality (12) holds for i √ 3 and nonnegative z regardless of what kernel G is in effect.\n\n\nFor non-uniform inputs, W in (x) does not penetrate the summations. The E-null function and its complement deeply associate to the following family of measures: For any t ∈ [−2/5, 1], define the t-tilted probability mass function W t : X × Y → [0, 1] as in [CS07, Definition 1] W t (x, y) := ξ∈X W (ξ, y) 1/(1+t) 1+t η∈Y ξ∈X W (ξ, η) 1/(1+t) 1+t × W (x, y) 1/(1+t) ξ∈X W (ξ, y) 1/(1+t) .\n\nLemma 17 (\n17universal quadratic bound). [CS07, Theorem 2]. Cf. [Gal65, Theorem 5.6.3]. Let W be a q-ary channel. Assume the uniform input distribution. Then Gallager's E-null function satisfies E 0 (0) = 0, E 0 (0) = I(W ) log q, and E 0 (t) −2 log(q) 2for all t ∈ [−2/5, 1]. In particular, it satisfies E 0 (t) I(W )t log q − t 2 log(q) 2 .\n\n\nwe divide it into three cases per how H 0 compares to −2 and 1 − −2 . The mediocre case:if −2 H 0 1 − −2 then h α (H 0 ) −2α . The local CLT behavior implies E [h α (H 1 )] < 4 −1/2+α = 4 −1/2+3α −2α 4 −1/2+3α h α (H 0 )and we are done with this case. The noisy case: if H 0 > 1 − −2 , we replace (H, Z , S) by (1 − H, S, Z ) to dual it to the reliable case dealt below and we are done with this case. (This is the only place in the proof where we ever mentioned S explicitly. Nevertheless, every statement concerning Z concerns S by duality.)\n\n\n6 exp(6q 4 )(q 8 H 0 ) log( ) 5 6 log −30 H −12 0 = q 36+8 log( ) 5 e 6q 4 6 log −24 H log( ) 5 −12 0 < q 36+8 log( ) 5 e 6q 4 6 log −24 −2 log( ) 5 +24 = q 36+8 log( ) 5 e 6q 4 6 log −1.6 log( ) 5 −0.4 log( ) 5 q 36+8 log( ) 5 e 6q 4 6 log −8 log(q) log( ) 4 −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log( ) 6 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log(41) 2 (q log 3) 4 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 6.02q 4 < q 36 6 log −0.1 log( ) 5 = exp(36 log q + 6 log( ) 2 − log( ) 6 /30 − log( ) 6 /15) = exp(36 log q + 6 log( ) 2 − 5 log(q) log(22) 5 /30 − log(22) 4 log( ) 2 /15) < exp(0) 1.\n\n.∪\n(Define events.) Let E 0 0 be the empty event. For every m = √ n, 2 √ n, . . . , n− √ n, we define five series of events A m , B m , C m , E m , and E m 0 inductively as below: LetA m be {Z m < exp(−m 2/3 )} \\ E Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D m+ m be A m \\ (B m ∪ C m ). Let E m 0 be E E m . Let a m , b m , c m ,e m , and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let g m be 1 − H 0 − e m 0 . (Bound b m\n\n+∪\nMarkov's inequality. This is an upper bound on c m /a m and will be summoned in Formula (32).(Bound (g m−√ n − a m ) + from above.) By definition, a m ). The definition of A m forces it to be disjoint from E A m . This union event must contain the event {Z m < exp(−m 2/3 )} by how A m was defined. From the en23 behavior P{Z m < exp(−m 2/3 )} > 1 − H 0 − (−1/2+4α)m . Chaining all inequalities together, we deduce that g m− √ n − a m < (−1/2+4α)m+o(m) . Let (g m− √ n − a m ) + be max(0, g m− √ n − a m ) so we can write (g m− √ n − a m ) + < (−1/2+4α)m+o(m) . This upper bound will be summoned in Formula (32).(Bound e n 0 from below.) We start rewriting g m with g + m being max(0, g m ):g m = 1 − H 0 − e m 0 = 1 − H 0\n\n∪\nwe define six series of events A m , A m 0 , B m , C m , E m , and E m 0 inductively as follows:Let A m be {Z m < exp(−e m 1/3 )} \\ A A m . Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D n πn . (33) Let E m be A m \\(B m ∪C m ). Let E m 0 be E m− √ n 0 ∪E m . Let a m , a m 0 , b m , c m , e m ,and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let f m be 1 − H 0 − a m 0 and let g m be 1 − H 0 − e m 0 . (Bound b m /a m from above.) Conditioning on A m , we want to estimate the probability that Z k −8 for some k m, which is equal to the probability that min( −2 , 4 √ Z k ) −2 for some k m. Recall that min( −2 , 4 √ Z k ) was made a supermartingale. Hence by Doob's optional stopping theorem [Dur19, Exercise 4.8.2], P{min( −2 , 4 √ Z k ) −2 for some k m | A m } min( −2 , 4 √ Z m ) 2 < exp(−e m 1/3 /4) 2 . This is an upper bound on b m /a m and will be summoned in Formula (34).\n\n\na m ):g m − f + m = 1 − H 0 − e m − (1 − H 0 − a m ) + = g m− √ n − e m − (f m− √ n − a m ) + g m− √ n − e m −e m a m (f m− √ n − a m ) + g m− √ n − e m − two equalities are by the definitions of g m and f m . The next inequality is by 0 e m /a m 1. The next inequality is by max(0, f − a) = max(a, f ) − a max(0, f ) − a. The next equality is simple algebra. The next inequality is by the definition of E m . The last inequality summons upper bounds derived in the last few paragraphs. Now the last line contains two terms in the big parentheses. Between them, (1/2−2α)m−(ρ+2α)n dominates exp(−e m 1/3 /4) 2 once n → ∞.\n\n\nshowed the equivalence of1/2 \n\n0 \n0 \n1 \n\nconjectured boundary \nfor V = 0 \n\n[FHMV17, GRY19] \n\nThm. 1 \n[YFV19] \n[HAU14] \n\n[WD18] \n\n[MHU16] \n\n[KSU10, MT14] \n[GX13] \n\n[BGN + 18] \n\n[BGS18] \n[AT09, Sas11, HY13] \n\n\n\n\nPrudent readers are invited to check [ŞTA09a, STA09b, the paragraph before Section III] [YB15, PSL16, EKMF + 17, WD18, GRY19] for a list of inhomogeneous configurations of kernels. See [ŞTA09a, STA09b, MT10, Sas11] for how nonlinear bijections are similar to (or different from) linear bijections.\n\n\net seq. utilizing the same input distributions and series of bijections. For instance,\n\n\nTheorem 9 (fundamental theorem of polar coding-the Z-end, FTPCZ). [Ari09, Proposition 5] [KSU10, Lemma 10] [Sas11, Lemma 3.5] [FHMV17, Section 4.1] [MT14, Lemma 33]\n\n\nThe formulation and the choice of the variables make it clear how Lemma 12 will be applied to support Claim 11. For instance, if we let {H n }, {Z n } and {S n } be {H(W n )}, {Z mad (W n )}, and {S max (W n )}, respectively, then Lemma 12 supports the first of the four inequalities in Claim 11. Moreover, if we let {H n }, {Z n }, and {S n } be {1 − H(W n )}, {S max (W n )}, and {Z mad (W n )}, then Lemma 12 supports the second inequality of Claim 11. If {H n }, {Z n }, and {S n } are let to be {H(V n )}, {Z mad (V n )}, and {S max (V n )}, that supports the third inequality. If {H n }, {Z n }, and {S n } are let to be {1 − H(V n )}, {S max (V n )}, and {Z mad (V n )}, that supports the forth inequality. The criteria (cb), (cm), (ct), and (cl) listed above Lemma 12 are easy to verify; for instance, Lemma 8 implies (ct) for all four cases. It remains to show that for each of the four triples of processes, the local LDP behavior and the local CLT behavior hold.\n\n\n8.3. Gallager's argument at Bob's end. This subsection take advantage of the universal bound developed three lines ago and starts actually proving Lemma 14.This subsection deals with \n\ni= H(W ) + 1/2+α +1 \n\nh α (H(W \n\n(i) \n\nG )) < −1/2+α \n(14's copy) \n\nby passing it to an inequality that captures the performance of noisy channel coding. \nOwing to h α 's concavity, the left hand side of Inequality (14) is \n\ni=j+1 \n\n\n\n\na bounded martingale {H n } converges to an r.v.-which we call H ∞ -a.s. (almost surely). This is Doob's martingale convergence theorem [Dur19, Theorem 4.2.11]. Owing to h α 's continuity, h α (H n ) → h α (H ∞ ) a.s. Point-wise convergence and (uniform) boundedness imply convergence in L 1 , i.e., E [h α (H n )] → E [h α (H ∞ )] as n → ∞. This is Lebesgue's dominated convergence theorem [Dur19, Theorem 1.6.7]. By the eigen behavior, E [h α (H n )] decays toward 0 by a constant factor every time n increases, thus E [h α (H ∞ )] is 0. This forces h a (H ∞ ) = 0 a.s. and hence H ∞ ∈ {0, 1} a.s. Since H ∞ is Bernoulli P{H ∞= 0} = E [I{H ∞ = 0}] = E [1 − H ∞ ] ← E [1 − H n ] = 1 − H 0 . SoP{H n → 0} = 1 − H 0 . By the implicit bi-Hölder toll, H n → 0 if and only if Z n → 0, thus the latter has the same probability measure. And the proof of Equality (24) is sound. Bibliographic remarks: The statement H n → H ∞ ∈ {0, 1} is usually referred to as channel polarization in spite of that it does not guarantee the corresponding codes to be capacity-achieving. See also [Ari09, Proposition 10] [MT14, Definition 3] [Sas11, Lemma 3.8]. This lemma should have been bestowed upon the fundamental theorem but it is not mandatory if some sort of CLT behavior is present. See [MHU16, Lemma 1] [FHMV17, Lemma 4] [GRY19, Lemma 9.5]. Recently, Reed-Muller codes' channels are shown to polarize [AY19]; Reed-Muller codes achieving capacity is not a consequence, but a different story.\n\n\nThe fourth inequality uses q 4 . That validates the first inequality in(27). For the second inequality,E [D \n\n−1/2 \n1 \n\n] = \n1 \n\nk=1 \n\nk 2 \n3 \n· \n3 \n4 \n\n−1/2 \n\n< \n1 \n\n√ \n3 \n\nk=1 \n\n3 \n4 \n\n−1/2 \n\n+ \n1 \n\nk= \n√ \n3 +1 \n\n\n− H(W ) (q − 1)S max (W )/ log q.(5)\nAppendix C. The een13 and elpin BehaviorsIn this section, we continue proving Lemma 12. The previous section covers (23) to (25). We are left with (26) to (29).C.1. A Supermartingale. We want to show that a certain monotonic function in Z n is a supermartingale so we can control how frequently does Z n stay in the turf where the local LDP behavior dominates. Making it a supermartingale, we are able to cite Doob's optional stopping theorem [Dur19, Theorem 4.8.4 and Exercise 4.8.2] later. The formal proof is below.Inequality (26) is a local statement so we may assume n = 0. To prove that√ Z 0 , we may assume Z 0 < −8 or the inequality becomes trivial. Invoke the local LDP behavior Z 1 exp(qZ 0 )(qZ 0 ) K 2 1 /3 2 (qZ 0 ) K 2 1 /3 . The last inequality uses q . When K 1 √ 3 , we do nothing but apply the last-resort exponent 1:When K 1 > √ 3 , the stronger exponent applies:Combining the two cases that are cut per how K 1 compares to √ 3 , we infer thatThe last inequality uses max(50, q 5 ). And the proof of Inequality (26) is sound. Bibliographic remarks: This lemma is inspired by[Ari09,Proposition 9]. In [HAU14, Lemma 22] Arıkan's lemma is overlooked and another is reinvented that serves the same purpose. The latter lemma also served in [MHU16, Theorem 3]. We generalized the idea to non-binary cases in [WD18, Lemma 1]. The quartic root\nRandomness and dependencies extraction via polarization, with applications to slepian-wolf coding and secrecy. E , Abbe , IEEE Transactions on Information Theory. 615E. Abbe. Randomness and dependencies extraction via polarization, with applica- tions to slepian-wolf coding and secrecy. IEEE Transactions on Information Theory, 61(5):2388-2398, May 2015.\n\nChannel polarization: A method for constructing capacity-achieving codes. E Arikan, IEEE International Symposium on Information Theory. E. Arikan. Channel polarization: A method for constructing capacity-achieving codes. In 2008 IEEE International Symposium on Information Theory, pages 1173- 1177, July 2008.\n\nChannel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. E Arikan, IEEE Transactions on Information Theory. 557E. Arikan. Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Transactions on Informa- tion Theory, 55(7):3051-3073, July 2009.\n\nSource polarization. E Arikan, 2010 IEEE International Symposium on Information Theory. E. Arikan. Source polarization. In 2010 IEEE International Symposium on Infor- mation Theory, pages 899-903, June 2010.\n\nA packing lemma for polar codes. E Arikan, 2015 IEEE International Symposium on Information Theory (ISIT). E. Arikan. A packing lemma for polar codes. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 2441-2445, June 2015.\n\nVariance of entropy decreases under the polar transform. Erdal Arikan, abs/1501.05005CoRRErdal Arikan. Variance of entropy decreases under the polar transform. CoRR, abs/1501.05005, 2015.\n\nOn the rate of channel polarization. E Arikan, E Telatar, IEEE International Symposium on Information Theory. E. Arikan and E. Telatar. On the rate of channel polarization. In 2009 IEEE Inter- national Symposium on Information Theory, pages 1493-1495, June 2009.\n\nPolar codes for the m-user multiple access channel. E Abbe, E Telatar, IEEE Transactions on Information Theory. 588E. Abbe and E. Telatar. Polar codes for the m-user multiple access channel. IEEE Transactions on Information Theory, 58(8):5437-5448, Aug 2012.\n\nModerate deviation analysis of channel coding: Discrete memoryless case. Y Altug, A B Wagner, 2010 IEEE International Symposium on Information Theory. Y. Altug and A. B. Wagner. Moderate deviation analysis of channel coding: Discrete memoryless case. In 2010 IEEE International Symposium on Information Theory, pages 265-269, June 2010.\n\nModerate deviations in channel coding. Y Altug, A B Wagner, IEEE Transactions on Information Theory. 608Y. Altug and A. B. Wagner. Moderate deviations in channel coding. IEEE Transac- tions on Information Theory, 60(8):4417-4426, Aug 2014.\n\nReed-muller codes polarize. CoRR, abs. Emmanuel Abbe, Min Ye, Emmanuel Abbe and Min Ye. Reed-muller codes polarize. CoRR, abs/1901.11533, 2019.\n\nRandom codes: minimum distances and error exponents. A Barg, G D Forney, IEEE Transactions on Information Theory. 489A. Barg and G. D. Forney. Random codes: minimum distances and error exponents. IEEE Transactions on Information Theory, 48(9):2568-2573, Sep. 2002.\n\nGeneral strong polarization. CoRR. + 18] Jaroslaw, Venkatesan Blasiok, Preetum Guruswami, Atri Nakkiran, Madhu Rudra, Sudan, abs/1802.02718+ 18] Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, Atri Rudra, and Madhu Sudan. General strong polarization. CoRR, abs/1802.02718, 2018.\n\nPolar codes with exponentially small error at finite block length. CoRR. Jaroslaw Blasiok, Venkatesan Guruswami, Madhu Sudan, abs/1810.04298Jaroslaw Blasiok, Venkatesan Guruswami, and Madhu Sudan. Polar codes with ex- ponentially small error at finite block length. CoRR, abs/1810.04298, 2018.\n\nHow quickly can we approach channel capacity?. D Baron, M A Khojastepour, R G Baraniuk, Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers. 1D. Baron, M. A. Khojastepour, and R. G. Baraniuk. How quickly can we approach channel capacity? In Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, 2004., volume 1, pages 1096-1100 Vol.1, Nov 2004.\n\nHypothesis testing and information theory. R Blahut, IEEE Transactions on Information Theory. 204R. Blahut. Hypothesis testing and information theory. IEEE Transactions on Infor- mation Theory, 20(4):405-417, July 1974.\n\nA first course in complex analysis. Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka, Department of Mathematics, San Francisco State UniversityMatthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka. A first course in complex analysis. Department of Mathematics, San Francisco State University, 2002.\n\nExact random coding secrecy exponents for the wiretap channel. M Bastani Parizi, E Telatar, N Merhav, IEEE Transactions on Information Theory. 631M. Bastani Parizi, E. Telatar, and N. Merhav. Exact random coding secrecy expo- nents for the wiretap channel. IEEE Transactions on Information Theory, 63(1):509- 531, Jan 2017.\n\nLossless source coding with polar codes. H S Cronie, S B Korada, 2010 IEEE International Symposium on Information Theory. H. S. Cronie and S. B. Korada. Lossless source coding with polar codes. In 2010 IEEE International Symposium on Information Theory, pages 904-908, June 2010.\n\nUniversal quadratic lower bounds on source coding error exponents. C Chang, A Sahai, 41st Annual Conference on Information Sciences and Systems. C. Chang and A. Sahai. Universal quadratic lower bounds on source coding error exponents. In 2007 41st Annual Conference on Information Sciences and Systems, pages 714-719, March 2007.\n\nLossless coding for distributed streaming sources. S C Draper, C Chang, A Sahai, IEEE Transactions on Information Theory. 603S. C. Draper, C. Chang, and A. Sahai. Lossless coding for distributed streaming sources. IEEE Transactions on Information Theory, 60(3):1447-1474, March 2014.\n\nMathematical problems in the shannon theory of optimal coding of information. R L Dobrushin, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. the Fourth Berkeley Symposium on Mathematical Statistics and ProbabilityBerkeley, CalifUniversity of California Press1Contributions to the Theory of StatisticsR. L. Dobrushin. Mathematical problems in the shannon theory of optimal coding of information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, pages 211-252, Berkeley, Calif., 1961. University of California Press.\n\nRick Durrett, Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press5 editionRick Durrett. Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 5 edition, 2019.\n\nAmir Dembo, Ofer Zeitouni, Large deviations techniques and applications. BerlinSpringer-VerlagStochastic Modelling and Applied Probability. Corrected reprint of the second (1998) editionAmir Dembo and Ofer Zeitouni. Large deviations techniques and applications, vol- ume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition.\n\nThe random coding bound is tight for the average linear code or lattice. Y Domb, R Zamir, M Feder, IEEE Transactions on Information Theory. 621Y. Domb, R. Zamir, and M. Feder. The random coding bound is tight for the average linear code or lattice. IEEE Transactions on Information Theory, 62(1):121-130, Jan 2016.\n\nRelaxed polar codes. M El-Khamy, H Mahdavifar, G Feygin, J Lee, I Kang, IEEE Transactions on Information Theory. 634+ 17] M. El-Khamy, H. Mahdavifar, G. Feygin, J. Lee, and I. Kang. Relaxed polar codes. IEEE Transactions on Information Theory, 63(4):1986-2000, April 2017.\n\nR M Fano, Transmission of Information: A Statistical Theory of Communications. M.I.T. PressR.M. Fano. Transmission of Information: A Statistical Theory of Communications. M.I.T. Press, 1961.\n\nBinary linear codes with optimal scaling and quasi-linear complexity. Arman Fazeli, S Hamed Hassani, Marco Mondelli, Alexander Vardy, abs/1711.01339CoRRArman Fazeli, S. Hamed Hassani, Marco Mondelli, and Alexander Vardy. Binary lin- ear codes with optimal scaling and quasi-linear complexity. CoRR, abs/1711.01339, 2017.\n\nBinary linear codes with optimal scaling: Polar codes with large kernels. A Fazeli, H Hassani, M Mondelli, A Vardy, 2018 IEEE Information Theory Workshop (ITW). A. Fazeli, H. Hassani, M. Mondelli, and A. Vardy. Binary linear codes with optimal scaling: Polar codes with large kernels. In 2018 IEEE Information Theory Workshop (ITW), pages 1-5, Nov 2018.\n\nRelations between entropy and error probability. M Feder, N Merhav, IEEE Transactions on Information Theory. 401M. Feder and N. Merhav. Relations between entropy and error probability. IEEE Transactions on Information Theory, 40(1):259-266, Jan 1994.\n\nOn the scaling exponent of polar codes for binary-input energy-harvesting channels. S L Fong, V Y F Tan, IEEE Journal on Selected Areas in Communications. 3412S. L. Fong and V. Y. F. Tan. On the scaling exponent of polar codes for binary-input energy-harvesting channels. IEEE Journal on Selected Areas in Communications, 34(12):3540-3551, Dec 2016.\n\nScaling exponent and moderate deviations asymptotics of polar codes for the awgn channel. L Silas, Fong, Y F Vincent, Tan, Entropy. 197Silas L. Fong and Vincent Y. F. Tan. Scaling exponent and moderate deviations asymptotics of polar codes for the awgn channel. Entropy, 19(7), 2017.\n\nOn the scaling exponent of binary polarization kernels. A Fazeli, A Vardy, 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton). A. Fazeli and A. Vardy. On the scaling exponent of binary polarization kernels. In 2014 52nd Annual Allerton Conference on Communication, Control, and Comput- ing (Allerton), pages 797-804, Sept 2014.\n\nPolar codes for broadcast channels. N Goela, E Abbe, M Gastpar, IEEE Transactions on Information Theory. 612N. Goela, E. Abbe, and M. Gastpar. Polar codes for broadcast channels. IEEE Transactions on Information Theory, 61(2):758-782, Feb 2015.\n\nA simple derivation of the coding theorem and some applications. R Gallager, IEEE Transactions on Information Theory. 111R. Gallager. A simple derivation of the coding theorem and some applications. IEEE Transactions on Information Theory, 11(1):3-18, January 1965.\n\nInformation Theory and Reliable Communication. Robert G Gallager, John Wiley & Sons, IncNew York, NY, USARobert G. Gallager. Information Theory and Reliable Communication. John Wiley & Sons, Inc., New York, NY, USA, 1968.\n\nThe random coding bound is tight for the average code (corresp.). R Gallager, IEEE Transactions on Information Theory. 192R. Gallager. The random coding bound is tight for the average code (corresp.). IEEE Transactions on Information Theory, 19(2):244-246, March 1973.\n\nImproved bounds on the finite length scaling of polar codes. D Goldin, D Burshtein, IEEE Transactions on Information Theory. 6011D. Goldin and D. Burshtein. Improved bounds on the finite length scaling of polar codes. IEEE Transactions on Information Theory, 60(11):6966-6978, Nov 2014.\n\nUniversal bounds on the scaling behavior of polar codes. A Goli, S H Hassani, R Urbanke, 2012 IEEE International Symposium on Information Theory Proceedings. A. Goli, S. H. Hassani, and R. Urbanke. Universal bounds on the scaling behav- ior of polar codes. In 2012 IEEE International Symposium on Information Theory Proceedings, pages 1957-1961, July 2012.\n\nArıkan meets Shannon: Polar codes with near-optimal convergence to channel capacity. Venkatesan Guruswami, Andrii Riazanov, Min Ye, arXiv:1911.03858arXiv e-printsVenkatesan Guruswami, Andrii Riazanov, and Min Ye. Arıkan meets Shannon: Po- lar codes with near-optimal convergence to channel capacity. arXiv e-prints, page arXiv:1911.03858, Nov 2019.\n\nPolar codes: Speed of polarization and polynomial gap to capacity. V Guruswami, P Xia, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science. V. Guruswami and P. Xia. Polar codes: Speed of polarization and polynomial gap to capacity. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 310-319, Oct 2013.\n\nError detecting and error correcting codes. The Bell System Technical. R W Hamming, Journal. 292R. W. Hamming. Error detecting and error correcting codes. The Bell System Tech- nical Journal, 29(2):147-160, April 1950.\n\nPolarization and spatial coupling two techniques to boost performance. Hassani Seyed Hamed, Seyed Hamed Hassani. Polarization and spatial coupling two techniques to boost performance. 2013.\n\nFinite-length scaling for polar codes. S H Hassani, K Alishahi, R L Urbanke, IEEE Transactions on Information Theory. 6010S. H. Hassani, K. Alishahi, and R. L. Urbanke. Finite-length scaling for polar codes. IEEE Transactions on Information Theory, 60(10):5875-5898, Oct 2014.\n\nGeneral nonasymptotic and asymptotic formulas in channel resolvability and identification capacity and their application to the wiretap channel. M Hayashi, IEEE Transactions on Information Theory. 524M. Hayashi. General nonasymptotic and asymptotic formulas in channel resolvabil- ity and identification capacity and their application to the wiretap channel. IEEE Transactions on Information Theory, 52(4):1562-1575, April 2006.\n\nInformation spectrum approach to second-order coding rate in channel coding. M Hayashi, IEEE Transactions on Information Theory. 5511M. Hayashi. Information spectrum approach to second-order coding rate in channel coding. IEEE Transactions on Information Theory, 55(11):4947-4966, Nov 2009.\n\nPerformance of polar codes for channel and source coding. N Hussami, S B Korada, R Urbanke, 2009 IEEE International Symposium on Information Theory. N. Hussami, S. B. Korada, and R. Urbanke. Performance of polar codes for channel and source coding. In 2009 IEEE International Symposium on Information Theory, pages 1488-1492, June 2009.\n\nUniversally attainable error and information exponents, and equivocation rate for the broadcast channels with confidential messages. M Hayashi, R Matsumoto, 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton). M. Hayashi and R. Matsumoto. Universally attainable error and information expo- nents, and equivocation rate for the broadcast channels with confidential messages. In 2011 49th Annual Allerton Conference on Communication, Control, and Com- puting (Allerton), pages 439-444, Sep. 2011.\n\nRate-dependent analysis of the asymptotic behavior of channel polarization. S H Hassani, R Mori, T Tanaka, R L Urbanke, IEEE Transactions on Information Theory. 594S. H. Hassani, R. Mori, T. Tanaka, and R. L. Urbanke. Rate-dependent analysis of the asymptotic behavior of channel polarization. IEEE Transactions on Information Theory, 59(4):2267-2276, April 2013.\n\nErasure and undetected error probabilities in the moderate deviations regime. M Hayashi, V Y F Tan, 2015 IEEE International Symposium on Information Theory (ISIT). M. Hayashi and V. Y. F. Tan. Erasure and undetected error probabilities in the moderate deviations regime. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 1821-1825, June 2015.\n\nPolar coding without alphabet extension for asymmetric models. J Honda, H Yamamoto, IEEE Transactions on Information Theory. 5912J. Honda and H. Yamamoto. Polar coding without alphabet extension for asymmetric models. IEEE Transactions on Information Theory, 59(12):7829-7838, Dec 2013.\n\nExtremes of random coding error exponents. A G Fàbregas, I Land, A Martinez, 2011 IEEE International Symposium on Information Theory Proceedings. A. G. i. Fàbregas, I. Land, and A. Martinez. Extremes of random coding error expo- nents. In 2011 IEEE International Symposium on Information Theory Proceedings, pages 2896-2898, July 2011.\n\nReed-muller codes achieve capacity on erasure channels. ] S Kkm + 17, S Kudekar, M Kumar, H D Mondelli, E Pfister, R L Şaşoǧlu, Urbanke, IEEE Transactions on Information Theory. 637KKM + 17] S. Kudekar, S. Kumar, M. Mondelli, H. D. Pfister, E. Şaşoǧlu, and R. L. Urbanke. Reed-muller codes achieve capacity on erasure channels. IEEE Transactions on In- formation Theory, 63(7):4298-4316, July 2017.\n\nAn empirical scaling law for polar codes. S B Korada, A Montanari, E Telatar, R Urbanke, 2010 IEEE International Symposium on Information Theory. S. B. Korada, A. Montanari, E. Telatar, and R. Urbanke. An empirical scaling law for polar codes. In 2010 IEEE International Symposium on Information Theory, pages 884-888, June 2010.\n\nPolar codes for channel and source coding. Korada Satish Babu, 181Satish Babu Korada. Polar codes for channel and source coding. page 181, 2009.\n\nPolar codes: Characterization of exponent, bounds, and constructions. S B Korada, E Sasoglu, R Urbanke, IEEE Transactions on Information Theory. 5612S. B. Korada, E. Sasoglu, and R. Urbanke. Polar codes: Characterization of ex- ponent, bounds, and constructions. IEEE Transactions on Information Theory, 56(12):6253-6264, Dec 2010.\n\nPolar codes for q-ary source coding. M Karzand, E Telatar, 2010 IEEE International Symposium on Information Theory. M. Karzand and E. Telatar. Polar codes for q-ary source coding. In 2010 IEEE International Symposium on Information Theory, pages 909-912, June 2010.\n\nPolar codes are optimal for lossy source coding. S B Korada, R L Urbanke, IEEE Transactions on Information Theory. 564S. B. Korada and R. L. Urbanke. Polar codes are optimal for lossy source coding. IEEE Transactions on Information Theory, 56(4):1751-1768, April 2010.\n\nFixed-length lossy compression in the finite blocklength regime. V Kostina, S Verdu, IEEE Transactions on Information Theory. 586V. Kostina and S. Verdu. Fixed-length lossy compression in the finite blocklength regime. IEEE Transactions on Information Theory, 58(6):3309-3338, June 2012.\n\nLossy joint source-channel coding in the finite blocklength regime. V Kostina, S Verdú, IEEE Transactions on Information Theory. 595V. Kostina and S. Verdú. Lossy joint source-channel coding in the finite blocklength regime. IEEE Transactions on Information Theory, 59(5):2545-2575, May 2013.\n\nOn the capacity of channels with deletions and states. Yonglong Li, Y F Vincent, Tan, Yonglong Li and Vincent Y. F. Tan. On the capacity of channels with deletions and states, 2019.\n\nFast polarization for non-stationary channels. H Mahdavifar, 2017 IEEE International Symposium on Information Theory (ISIT). H. Mahdavifar. Fast polarization for non-stationary channels. In 2017 IEEE Inter- national Symposium on Information Theory (ISIT), pages 849-853, June 2017.\n\nAchieving marton's region for broadcast channels using polar codes. M Mondelli, S H Hassani, I Sason, R L Urbanke, IEEE Transactions on Information Theory. 612M. Mondelli, S. H. Hassani, I. Sason, and R. L. Urbanke. Achieving marton's region for broadcast channels using polar codes. IEEE Transactions on Information Theory, 61(2):783-800, Feb 2015.\n\nUnified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors. M Mondelli, S H Hassani, R L Urbanke, IEEE Transactions on Information Theory. 6212M. Mondelli, S. H. Hassani, and R. L. Urbanke. Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors. IEEE Transac- tions on Information Theory, 62(12):6698-6712, Dec 2016.\n\nFrom polar to reed-muller codes unified scaling, non-standard channels, and a proven conjecture. Marco Mondelli, 230Marco Mondelli. From polar to reed-muller codes unified scaling, non-standard chan- nels, and a proven conjecture. page 230, 2016.\n\nProperties and construction of polar codes. Ryuhei Mori, abs/1002.3521CoRRRyuhei Mori. Properties and construction of polar codes. CoRR, abs/1002.3521, 2010.\n\nChannel polarization on q-ary discrete memoryless channels by arbitrary kernels. R Mori, T Tanaka, 2010 IEEE International Symposium on Information Theory. R. Mori and T. Tanaka. Channel polarization on q-ary discrete memoryless chan- nels by arbitrary kernels. In 2010 IEEE International Symposium on Information Theory, pages 894-898, June 2010.\n\nSource and channel polarization over finite fields and reedsolomon matrices. R Mori, T Tanaka, IEEE Transactions on Information Theory. 605R. Mori and T. Tanaka. Source and channel polarization over finite fields and reed- solomon matrices. IEEE Transactions on Information Theory, 60(5):2720-2736, May 2014.\n\nAn ergodic theory of binary operations-part i: Key properties. R Nasser, IEEE Transactions on Information Theory. 6212R. Nasser. An ergodic theory of binary operations-part i: Key properties. IEEE Transactions on Information Theory, 62(12):6931-6952, Dec 2016.\n\nAn ergodic theory of binary operations-part ii: Applications to polarization. R Nasser, IEEE Transactions on Information Theory. 632R. Nasser. An ergodic theory of binary operations-part ii: Applications to polar- ization. IEEE Transactions on Information Theory, 63(2):1063-1083, Feb 2017.\n\nPolar codes for arbitrary dmcs and arbitrary macs. R Nasser, E Telatar, IEEE Transactions on Information Theory. 626R. Nasser and E. Telatar. Polar codes for arbitrary dmcs and arbitrary macs. IEEE Transactions on Information Theory, 62(6):2917-2936, June 2016.\n\nError probability of identification via channels at rates above capacity. Y Oohama, Proceedings IEEE International Symposium on Information Theory. IEEE International Symposium on Information Theory26Y. Oohama. Error probability of identification via channels at rates above capacity. In Proceedings IEEE International Symposium on Information Theory,, pages 26-, June 2002.\n\nPolar codes for q-ary channels, q = 2 r. W Park, A Barg, IEEE Transactions on Information Theory. 592W. Park and A. Barg. Polar codes for q-ary channels, q = 2 r . IEEE Transactions on Information Theory, 59(2):955-969, Feb 2013.\n\nChannel coding: Non-asymptotic fundamental limits. Yury Polyanskiy, Princeton UniversityPhD thesisYury Polyanskiy. Channel coding: Non-asymptotic fundamental limits. PhD thesis, Princeton University, 2010.\n\nChannel coding rate in the finite blocklength regime. Y Polyanskiy, H V Poor, S Verdu, IEEE Transactions on Information Theory. 565Y. Polyanskiy, H. V. Poor, and S. Verdu. Channel coding rate in the finite blocklength regime. IEEE Transactions on Information Theory, 56(5):2307-2359, May 2010.\n\nMixed-kernels constructions of polar codes. N Presman, O Shapira, S Litsyn, IEEE Journal on Selected Areas in Communications. 342N. Presman, O. Shapira, and S. Litsyn. Mixed-kernels constructions of polar codes. IEEE Journal on Selected Areas in Communications, 34(2):239-253, Feb 2016.\n\nNear-optimal finite-length scaling for polar codes over large alphabets. H D Pfister, R Urbanke, 2016 IEEE International Symposium on Information Theory (ISIT). H. D. Pfister and R. Urbanke. Near-optimal finite-length scaling for polar codes over large alphabets. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 215-219, July 2016.\n\nChannel dispersion and moderate deviations limits for memoryless channels. Y Polyanskiy, S Verdú, 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton). Y. Polyanskiy and S. Verdú. Channel dispersion and moderate deviations limits for memoryless channels. In 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1334-1339, Sept 2010.\n\nPolar coding theorems for discrete systems. Eren Sasoglu, 104Eren Sasoglu. Polar coding theorems for discrete systems. page 104, 2011.\n\nPolar codes for discrete alphabets. Eren Sasoglu, 2012 IEEE International Symposium on Information Theory Proceedings. Eren Sasoglu. Polar codes for discrete alphabets. In 2012 IEEE International Sym- posium on Information Theory Proceedings, pages 2137-2141, July 2012.\n\nPolarization and polar codes. Eren Sasoglu, Foundations and Trends R in Communications and Information Theory. 84Eren Sasoglu. Polarization and polar codes. Foundations and Trends R in Commu- nications and Information Theory, 8(4):259-381, 2012.\n\nLower bounds to error probability for coding on discrete memoryless channels. i. Information and Control. C E Shannon, R G Gallager, E R Berlekamp, 10C.E. Shannon, R.G. Gallager, and E.R. Berlekamp. Lower bounds to error probabil- ity for coding on discrete memoryless channels. i. Information and Control, 10(1):65 -103, 1967.\n\nA mathematical theory of communication. C E Shannon, Bell System Tech. J. 27C. E. Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379-423, 623-656, 1948.\n\nMultilevel polarization of polar codes over arbitrary discrete memoryless channels. A G Sahebi, S S Pradhan, 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton). A. G. Sahebi and S. S. Pradhan. Multilevel polarization of polar codes over arbi- trary discrete memoryless channels. In 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1718-1725, Sep. 2011.\n\nAchieving the capacity of any dmc using only polar codes. D Sutter, J M Renes, F Dupuis, R Renner, 2012 IEEE Information Theory Workshop. D. Sutter, J. M. Renes, F. Dupuis, and R. Renner. Achieving the capacity of any dmc using only polar codes. In 2012 IEEE Information Theory Workshop, pages 114-118, Sep. 2012.\n\nPolar coding for processes with memory. Eren Sasoglu, Ido Tal, abs/1602.01870CoRREren Sasoglu and Ido Tal. Polar coding for processes with memory. CoRR, abs/1602.01870, 2016.\n\nPolarization for arbitrary discrete memoryless channels. E Şaşoglu, E Telatar, E Arikan, IEEE Information Theory Workshop. E. Şaşoglu, E. Telatar, and E. Arikan. Polarization for arbitrary discrete memoryless channels. In 2009 IEEE Information Theory Workshop, pages 144-148, Oct 2009.\n\nPolarization for arbitrary discrete memoryless channels. Eren Sasoglu, Emre Telatar, Erdal Arikan, abs/0908.0302CoRREren Sasoglu, Emre Telatar, and Erdal Arikan. Polarization for arbitrary discrete memoryless channels. CoRR, abs/0908.0302, 2009.\n\nAsymptotische abschätzungen in shannons informationstheorie. V Strassen, Transactions of the Third Prague Conference on Information Theory. Publishing House of the Czechoslovak Academy of SciencesV. Strassen. Asymptotische abschätzungen in shannons informationstheorie. In Transactions of the Third Prague Conference on Information Theory, pages 689- 723. Publishing House of the Czechoslovak Academy of Sciences, 1962.\n\nA new polar coding scheme for strong security on wiretap channels. E Şaşoglu, A Vardy, 2013 IEEE International Symposium on Information Theory. E. Şaşoglu and A. Vardy. A new polar coding scheme for strong security on wiretap channels. In 2013 IEEE International Symposium on Information Theory, pages 1117-1121, July 2013.\n\nFourier analysis on finite groups and applications, volume 43 of London Mathematical Society Student Texts. Audrey Terras, Cambridge University PressCambridgeAudrey Terras. Fourier analysis on finite groups and applications, volume 43 of Lon- don Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999.\n\nPolar codes for the deletion channel: Weak and strong polarization. Ido Tal, Henry D Pfister, Arman Fazeli, Alexander Vardy, abs/1904.13385CoRR. Ido Tal, Henry D. Pfister, Arman Fazeli, and Alexander Vardy. Polar codes for the deletion channel: Weak and strong polarization. CoRR, abs/1904.13385, 2019.\n\nHow to construct polar codes. I Tal, A Vardy, IEEE Transactions on Information Theory. 5910I. Tal and A. Vardy. How to construct polar codes. IEEE Transactions on Informa- tion Theory, 59(10):6562-6582, Oct 2013.\n\nList decoding of polar codes. I Tal, A Vardy, IEEE Transactions on Information Theory. 615I. Tal and A. Vardy. List decoding of polar codes. IEEE Transactions on Information Theory, 61(5):2213-2226, May 2015.\n\nPolar-like codes and asymptotic tradeoff among block length, code rate, and error probability. Hsin-Po Wang, Iwan M Duursma, abs/1812.08112CoRR. Hsin-Po Wang and Iwan M. Duursma. Polar-like codes and asymptotic tradeoff among block length, code rate, and error probability. CoRR, abs/1812.08112, 2018.\n\nOn the strong converse of the coding theorem for symmetric channels without memory. Lionel Weiss. ; R, J Wang, H Honda, R Yamamoto, Y Liu, Hou, IEEE Information Theory Workshop -Fall (ITW). 18Construction of polar codes for channels with memoryLIONEL WEISS. On the strong converse of the coding theorem for symmetric chan- nels without memory. Quarterly of Applied Mathematics, 18(3):209-214, 1960. [WHY + 15] R. Wang, J. Honda, H. Yamamoto, R. Liu, and Y. Hou. Construction of polar codes for channels with memory. In 2015 IEEE Information Theory Workshop - Fall (ITW), pages 187-191, Oct 2015.\n\nThe coding of messages subject to chance errors. J Wolfowitz, Illinois J. Math. 141957J. Wolfowitz. The coding of messages subject to chance errors. Illinois J. Math., 1(4):591-606, 12 1957.\n\nThe wire-tap channel. A D Wyner, Bell System Technical Journal. 548A. D. Wyner. The wire-tap channel. Bell System Technical Journal, 54(8):1355-1387, 1975.\n\nPolar codes using dynamic kernels. M Ye, A Barg, 2015 IEEE International Symposium on Information Theory (ISIT). M. Ye and A. Barg. Polar codes using dynamic kernels. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 231-235, June 2015.\n\nExplicit polar codes with small scaling exponent. Hanwen Yao, Arman Fazeli, Alexander Vardy, abs/1901.08186CoRRHanwen Yao, Arman Fazeli, and Alexander Vardy. Explicit polar codes with small scaling exponent. CoRR, abs/1901.08186, 2019.\n"},"corpus_id":{"kind":"number","value":209415033,"string":"209,415,033"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1912.08995\",\n \"mag\": 3108984347,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/tit/WangD21\",\n \"doi\": \"10.1109/tit.2020.3041570\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1912.08995v1.pdf\"\n ],\n \"pdf_sha\": \"bb41b289d2e4bdf964737788d9cc28170c91b43a\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://ieeexplore.ieee.org/ielx7/18/9356438/09274521.pdf\",\n \"status\": \"HYBRID\"\n }\n}"},"title":{"kind":"string","value":"POLAR CODES' SIMPLICITY, RANDOM CODES' DURABILITY"},"authors":{"kind":"list like","value":["Hsin-Po Wang","Iwan Duursma"],"string":"[\n \"Hsin-Po Wang\",\n \"Iwan Duursma\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Over any discrete memoryless channel, we build codes such that: for one, their block error probabilities and code rates scale like random codes'; and for two, their encoding and decoding complexities scale like polar codes'. Quantitatively, for any constants π, ρ > 0 such that π + 2ρ < 1, we construct a sequence of error correction codes with block length N approaching infinity, block error probability exp(−N π ), code rate N −ρ less than the Shannon capacity, and encoding and decoding complexity O(N log N ) per code block. The putative codes take uniform ς-ary messages for sender's choice of prime ς. The putative codes are optimal in the following manner: Should π + 2ρ > 1, no such codes exist for generic channels regardless of alphabet and complexity."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Richard W. Hamming is one of the first few people who had the idea that by grouping information in blocks with redundancies, a calculating machine can correct errors by its own and proceed to the next command instead of halting. Their solution, now called Hamming codes, is found in [Ham50]. Claude E. Shannon, a colleague of Hamming in Bell labs, theorized the communication channels and showed that a channel associates to a number called capacity, which represents the ultimate limit of the efficiency of communications over that channel.\nTo brief the rest of the history, we follow the analogy [AW14] used. Shannon's eternal result, noisy channel coding theorem [Sha48], is considered the analog of the law of large numbers (LLN). The theorem implies that there exists a sequence of longer and longer block codes whose block error probabilities approach 0 and code rates approach the capacity, which is analogous to that the empirical average of random variables is close to the mean with high probability. Robert G. Gallager, Shannon, Robert M. Fano, and followers extended the LLN result by looking at how the block error probability P e scales when the code rate R is fixed. They showed that the error probability P e scales like exp(−E r (R)N ). Here N is the block length, E r (R) is a constant depending on R. This paradigm is considered the analog of the large deviations principle (LDP). See [Fan61,Gal65,SGB67,Gal68,Gal73,Bla74,BF02,iFLM11,DZF16]. Meanwhile, a series of works fix the error probability P e and looked at how the code rate R scales [Wol57,WEI60,Dob61,Str62,BKB04,Hay09,PPV10]. They showed that the code rate R scales like I − Q −1 (P e ) V /N for I the capacity, Q −1 the inverse of the standard Q-function, and V an intrinsic parameter of the channel. The parameter V is called the dispersion or varentropy by different authors. It is the \"variance\" of the channel while I is the \"mean\" of the channel. This turns out to be more than an analog-the random variable log(W (Y | X)/W out (Y )) called information density or information spectrum has mean I and variance V . This paradigm is considered the analog of the central limit They showed that the quantity N (I − R) 2 /|log P e | converges to 2V , twice the very dispersion appearing in the CLT paradigm. This paradigm is considered the analog of the moderate deviations principle (MDP). On a parallel track, the engineering aspects of the communication theory thrive. Codes with excellent practicality are proposed. To name a few, Reed-Muller (1964), trellis modulation (1970s), turbo (1990s), low-density parity-check (1963,1996), Repeat-accumulate (1998), Fountain (1998), and polar (2009).\nAmong the long list of inventions, only trellis modulation, low-density paritycheck, and polar achieve the LLN paradigm over nontrivial channels-they are capacity-achieving. Among these three, polar stands out as the only code that achieves the CLT paradigm (optimally), the only code that achieves the LDP paradigm (optimally), and the only code that achieves the MDP paradigm (suboptimally). If only polar code achieves the optimal MDP paradigm. We brief the history of polar codes below. Unless stated otherwise, I means the symmetric capacity in the next three paragraphs.\nErdal Arıkan's original works on channel polarization [Ari08,Ari09] established the foundation of polar codes, placing polar codes in the LLN paradigm on day one. Arıkan and Telatar [AT09] characterized the LDP behavior of polar codes, showing that P e scales like exp(− √ N ) when an R < I is fixed. Later, Korada-Şaşoglu-Urbanke [KSU10] generalized polar codes from Arikan's kernel [ 1 1 0 1 ] to any invertible -by-matrix G, granted that 2 and G is not column-equivalent to a lower triangular matrix. And then they showed that the LDP behavior is P e ≈ exp(−N Ec(G) ) where E c (G) is a constant depending on the kernel matrix G. The notation E c (G) is meant to resemble Gallager's error exponent E r (R) but the former is at this level exp(−N • ) while the latter is at this level exp(−•N ). The LDP behavior of polar codes is then refined in [HMTU13]. Therein, P e is approximated by exp(− E ) where E = E c (G)n − V c (G)nQ −1 (R/I) + o( √ n) is a more accurate exponent, is the matrix dimension, n is the depth of the code, and V c (G) is another constant depending on G. The notation V c (G) is meant to resemble the channel dispersion V . Appearing to be a CLT behavior, this result lies in the corner of the LDP paradigm that touches the MDP paradigm. Finally, Mori-Tanaka [MT14] generalized everything above to channels of prime power input size. Over arbitrary input alphabets, [ŞTA09a,Sas11] Note that results utilizing different kernels over various channels are mixed. The higher ρ, π, the better performance. The curve part of [WD18] is ρ = 1 − h 2 (π). [Ari09,AT09]. Over binary but asymmetric channels, [SRDR12,HY13] showed the counterpart of [Ari09,AT09] with I being the Shannon capacity. No further result on the LDP side, e.g. over non-binary asymmetric channels, is known. The present work fills the gap.\nThe CLT behavior of polar codes turns out to be difficult to characterize. It was Korada-Montanari-Telatar-Urbanke [KMTU10] who came up with the idea that approximating an eigenfunction tightly bounds the eigenvalue −ρ . Here ρ > 0 is a number such that R scales like I − N −ρ with a fixed P e . They had 0.2669 ρ 0.2841 over binary erasure channels (BECs). The upper bound was brought down to 3.553ρ 1 [GHU12]. Hassini-Alishahi-Urbanke [HAU14] lifted the lower bound to 3.627ρ 1 over BECs and proved a lower bound 6ρ 1 over binaryinput discrete-output memoryless channels (BDMCs). The latter is suboptimal so [GB14,MHU16] improved the bound to 5.702ρ 1 and to 4.714ρ 1. Additive white Gaussian noise channles (AWGNCs) have continuous output alphabet, but [FT17] show that they have 4.714ρ 1 too. Over BECs particularly, [FV14,YFV19] examined a series of larger kernels; the current record is a 64-by-64 kernel believed to have 2.9ρ 1. Near the end of the road to 2ρ < 1, [PU16] showed that by allowing q → ∞, Reed-Solomon kernels achieve 2ρ < 1 over q-ary channels. This does not really prove that polar codes achieve 2ρ < 1 over any specific channel, but gave hopes. Fazeli-Hassani-Mondelli-Vardy [FHMV17,FHMV18], eventually, showed that large random kernels achieve 2ρ < 1 over BECs, breaking the barrier. Guruswami-Riazanov-Ye [GRY19] extended their result to all BDMCs utilizing the dynamic kernel technique. Over the remaining channels, the present work fills the gap.\nBetween LDP and CLT is polar codes' MDP behavior. Guruswami-Xia [GX13] showed that there exists ρ > 0 such that P e scales like exp(−N 0.49 ) while R scales like I − N −ρ over BDMCs. This raised a question about what are the possible pairs (π, ρ) such that (P e , R) scales like (exp(−N π ), I − N −ρ ). Mondelli-Hassani-Urbanke [MHU16] answered this, partially, in the same paper they bounded ρ. They showed that under a certain curve connecting (0, 1/5.714) and (1/2, 0) all (π, ρ) are achievable over BDMCs. For BECs the upper left corner is (0, 1/4.627). A straightforward generalization to AWGNCs was also given in [FT17]. We in [WD18] improved their result, suggesting that via a combinatorial trick the upper left corner of the curve is (0, ρ) for any ρ that is valid in the CLT regime. The same trick also implicated that over BECs all (π, ρ) such that π + 2ρ < 1 are achievable, which is mainly owing to [FHMV17]'s result that 2ρ < 1 over BECs is achievable. Meanwhile, [BGN + 18] made the first step to investigate the general kernel matrices over general prime-ary channels. They showed that it is possible to achieve ρ > 0 with P e ≈ N −Ω(1) . This is, strictly speaking, \"only\" a CLT behavior as the desired error probability in the MDP world is exp(−N π ). Later, B lasiok-Guruswami-Sudan [BGS18] were able to show that for all π < E c (G) there exists ρ > 0 such that (π, ρ) is achievable. This makes it a direct generalization of [GX13] to all polarizing kernel matrices G over all prime-ary channels. Over the remaining channels, the present work fills the gap.\nThe following works, though not counting as predecessors of ours, have impact on us through their insights on the essence of the channel polarization: [Kor09, HKU09, KU10, Ari10, CK10, Mor10, SP11, Sas12b, Sas12a, TV13, Has13, PB13, TV15, Mon16, Nas16,Nas17].\nI resumes to be the Shannon capacity. Readers are now prepared to be presented the main theorem.\nTheorem 1 (the main theorem-polar codes' simplicity, random codes' durability). Let W be any discrete memoryless channel. Fix a prime ς 2. Fix constants π, ρ > 0 such that π + 2ρ < 1. There exists a sequence of block codes with encoding and decoding algorithms such that: (cs) the codes accept uniform ς-ary messages. (cn) the block length N approaches infinity; (cp) the block error probability falls below exp(−N π ); (cr) the code rate exceeds I − N −ρ ; and (cc) the encoding and decoding complexity is O(N log N ) per code block.\nThe proof of the main theorem spans over Sections 2 to 8, lemmas continuing in Appendices A to C. The entry points are Sections 2.1 for (cs), 2.2 for (cn), 3.2 for (cc), 4.2 for (cp), and 6.3 for (cr). The main theorem is optimal in the following manner.\nProposition 2 (optimality). Fix π, ρ > 0 such that π + 2ρ > 1. Assume V > 0. For the rest of the section, we outline the ideas to prove Theorem 1. The proof is a straightforward remix of polar coding techniques and random coding techniques if it were not for a few hurdles.\nHurdle of input alphabet size: The majority of the polar coding theory assumes that the input alphabet of the underlying channel is binary, of prime size, or, less likely, of prime power size. But the main theorem aims for arbitrary finite alphabets.\nFinite alphabets do possess polarization behavior but the speed of polarization has room for improvement [Sas11,Theorem 3.5]. We will overcome this by adding \"dummy symbols\" into the input alphabet to make it a prime power.\nHurdle of asymmetric channel: Although asymmetric channels do polarize, the input distributions do not automatically become the uniform distribution. Precomposing a source coding machinery helps generate the desired distribution and has been proposed before [ŞTA09a, Section III.D] [Ari10, Section IV]. On the other hand Honda-Yamanoto [HY13] showed that one polar code can do both source coding and noisy channel coding at once. We borrow their idea.\nHurdle of kernel selection: Judging and identifying the best-behaved kernel gets harder as we need finer descriptions of the performance of the code. The good result for the BEC case depends heavily on the erasure nature of the channels (that they are ordered by their capacities). Other general results are not strong enough to meet our goal. To overcome, we borrow a technique called dynamic kernels from [YB15]. The idea is to prepare more than one polarizing kernel and apply the proper one on a channel-by-channel basis. This makes a paradigm shift from one kernel fits all channels to every channel deserves a tailor-made kernel. We will, once per channel, apply the random coding theory to show the existence of a proper kernel.\nHurdle of output alphabet size: Even with the great freedom to choose one kernel for each and every channel, there lies the difficulty that some performance bounds are proven with one fixed channel in mind to favor the big-O notations. Those bounds are prone to depend on the size of the output alphabet, which grows to infinity as the channel transformations take place. Meanwhile, some universal bounds are proven that depend only on the size of the input alphabet, which is invariant under channel transformations. We will borrow a bound derived in [CS07,DCS14].\n1.1. Organization. Section 2 reviews channels and entropy notations; Section 2.1 explains how to overcome the hurdle of arbitrary input alphabet size. Section 3 reviews the channel transformations; Section 3.1 designs the decoder; Section 3.2 analyzes its complexity; Section 3.3 designs the encoder, overcoming the hurdle of asymmetric channel. Section 4 reviews the channel parameters such as the Bhattacharyya parameter; Section 4.2 shows how to control the block error probability. Section 5 reviews the channel processes; Section 5.1 argues that the global MDP behaviors of H(W n ) and H(V n ) imply the main theorem. The main theorem is thus reduced to the behavior of certain channel processes. Section 6 proves that the global MDP behavior we want holds granted that the local LDP and CLT behaviors hold, effectively boiling the main theorem down to the local behaviors. Section 6.2 introduces the random kernel trick and Section 6.3 introduces the dynamic kernel trick to overcome the hurdle of kernel selection. Section 7 confirms the local LDP behavior. The proof distills properties of the weight distribution of random codes. Sections 7.1 and 7.2 proves the two fundamental theorems of polar coding. Section 8 confirms the local CLT behavior. Contributions from Gallager and Hayashi are utilized. Section 8.2 invokes Chang-Sahai's universal bound, overcoming the hurdle of output alphabet size.\n1.2. Three families of randomnesses. The randomnesses from the sender's message, the channel, and the randomized rounding constitute the first family. Typeset in Roman font are random variables (U, X, Y, . . . ), probability measures (P, Q, W ), entropies (H, I), and other parameters (P e , Z, T, S . . . ) in this family. The randomness from the channel process, one main technique in the polar coding literature, is the second family. Typeset in sans serif font are stochastic processes (K n , W n , H n , Z n , . . . ), probability measure (P), and expectation (E ) in this family. The randomness from random kernel ensembles, the main technique in the random coding literature, is the third family. Typeset in blackboard bold font are random variables (G, X, K), probability measure (P, with exceptions), expectation (E), and Kullback-Leibler divergence (D, with exceptions) in this family."},{"title":"Channel and Entropy Preliminaries","content":"A discrete memoryless channel is a Markov chain W : X → Y. Here X is a finite set of input alphabet; Y is a finite set of output alphabet; and W is an array of transition probabilities W (y | x) ∈ [0, 1] for all x ∈ X and y ∈ Y. The numbers satisfy y∈Y W (y | x) = 1 for all x ∈ X , which represents the fact that each x must be transitioned to some unique y. When X and Y are clear from the context, we call W a channel. Although the input distribution is not part of the channel data, we write W in (x) to denote the input distribution. When W in (x) is understood from the context, we write W (x, y) to denote the joint distribution W (y | x)W in (x), write W out (y) to denote the output distribution, and write W (x | y) to denote the a posteriori probability W (x, y)/W out (y). (Thus the interpretation of W (• | •) depends on the arguments and the context.) A tuple of inputs (x i , x i+1 , . . . , x j ) is abbreviated as x j i . Same for y j i for tuple of outputs, and for u j i for general variables. We assume memoryless channels, and write W (y 1 | x 1 ) to denote the product measure i=1 W (y i | x i ) for consecutive usages. We write W in (x 1 ), W out (y 1 ), W (x 1 , y 1 ), and W (x 1 | y 1 ) to denote the input, output, joint, and a posteriori probabilities.\nLet X, Y be two r.v.s (random variables). Let H(X), H(X | Y ), and I(X ; Y ) be the standard entropy, conditional entropy, and mutual information. The base of the logarithm will be assigned later. When X is the input fed into some channel W : X → Y and Y is the corresponding output, we say H(W ) and I(W ) to mean H(X | Y ) and I(X ; Y ). When the distribution of X (the input distribution) is chosen to maximize I(W ), it is called the capacity-achieving input distribution and I(W ) is called the (Shannon) capacity of the channel W : X → Y. Unless stated otherwise, the input distributions will be capacity-achieving.\n2.1. Reduce input size to prime power. Immediately after we declared what channels are concerned (those with finite input and output alphabets), we show that it suffices to consider input alphabets of prime power size. Let W : X → Y be a channel. Let the input alphabet X be of size s. Let q be any prime power greater than or equal to s. Degrade the channel W as follows: Let symbols in X be ξ 1 , ξ 2 , . . . , ξ s . Let ξ s+1 , ξ s+2 , . . . , ξ q be q − s extra symbols. Let X be X ∪ {ξ s+1 , ξ s+2 , . . . , ξ q }; this is the extended alphabet. Define a dummy channel : X → X by letting (ξ min(i,s) | ξ i ) be 1 for all i = 1, 2, . . . , , q. That is, all extra symbols collapse to ξ s while the old symbols remain. The composition of the two channels\nforms a degraded channel with prime power input size. By the data processing inequality, the Shannon capacity of the degraded channel W • is no greater than W 's Shannon capacity. Meanwhile, it is clear that the degraded channel W • achieves W 's capacity by the same input distribution, ignoring extra symbols. In other words, I(W • ) = I(W ). This constitutes the input size reduction.\nHereafter, we assume the size of the input alphabet X is q, where q is a prime power. If the sender wants to send uniform binary messages, let q be a power of 2. If the sender wants to send uniform quaternary messages, let q be a power of 3. In case the sender wishes to send uniform quaternary messages but does not want to split an information bit over two channel symbols, let q be a power of 4. Bonus: should the sender want to send uniform senary messages, choose q 2 a power of 2 and q 3 a power of 3 such that q 2 q 3 is a power of 6; then alternate between q = q 2 and q = q 3 . That is, the sender breaks every senary bit into a binary component and a ternary component, sends the binary component through the q = q 2 code block, and send the ternary component through the q = q 3 code block. For other message alphabets, apply the fundamental theorem of arithmetic.\nFix a q. Let F q be the finite field of order q (with the addition and multiplication structure). Identifying X with F q , we will use them interchangeably. We say that W is a q-ary channel when the variables X and Y are remotely relevant. It is worth keeping in mind that for inequalities in this work, q = 2 is the most difficult case and q 2 will be used silently.\nWe clarified (cs), there are (cn), (cc), (cp), and (cr) to go.\n2.2. On the message alphabet and the block length. The fact that we have some freedom to choose q blurs the meaning of the block length N since, say, a q 2 -bit bears twice as much message as a q-bit does. Notwithstanding, we would like to remind readers that multiplication and division of N by any constant do not alter the semantics of the main theorem. This is because O(N log N ) can absorb any constant; exp(−N π ) and N −ρ too can by fluctuating π and ρ a bit. A more series aftereffect is caused by mixing code blocks with distinct q. When the sender attempts to send uniform 30-ary messages, they choose q 2 , q 3 , q 5 s and switch among the three block codes. The q = q 2 blocks have their own block length N 2 just like the other blocks have N 3 or N 5 as block lengths. The de facto block length N , the minimal number of the channel usages before the receiver can decode everything sent so far, is thus three times the least common multiple of N 2 , N 3 , and N 5 . We claim without a proof (but it will be clear once we prove the rest of the main theorem) that it is possible to make N 2 = N 3 = N 5 and consequently N = 3N 2 . Again, increasing N by three-fold does not make any difference. For numbers with more prime factors, a similar reasoning applies.\nWe recommend readers not to worry about the message alphabet as there exists a powerful solution-to pre-compose another code that re-encodes an arbitrary finite message distribution (not necessarily uniform) to a uniform prime power-ary input distribution. The existence of such code, by duality, is tightly bonded to the existence of a error-correction code that carries uniform prime power-ary messages over channels of arbitrary arity. The latter is exactly what the main theorem concerns.\nWe clarified (cs) and (cn) in this section; there are (cc), (cp), and (cr) to go. We continue proving the main theorem in the next section."},{"title":"Channel Transformation","content":"Let 2 be an integer. This will be the dimension of the kernel matrices. But for now, let us introduce a flexible framework. Fix a q-ary channel W : X → Y. Let U 1 , U 2 , . . . , U be r.v.s taking values in X . For 1 i j , let U j i be the joint r.v. U i U i+1 · · · U j . Let g W : X → X be a bijective map; that is, H(U 1 | g W (U 1 )) = 0. We now feed X 1 := g W (U 1 ) into i.i.d. (independent and identically distributed) copies of the channel W . Let Y 1 ∈ Y be the corresponding output. The chain rule of conditional entropy reads\n(1)\nInterpretation: to estimate U 1 given Y 1 , we first estimate U 1 given Y 1 ; and then use the estimateÛ 1 to further estimate U 2 ; afterward, we estimate U 3 givenÛ 1 , U 2 , and Y 1 ; and so on. To achieve W 's capacity, g W (U 1 ) must follow a certain capacity-achieving distribution. Since g W is bijective, this induces a distribution of U 1 . (Remark: we imply nothing about whether U 1 , U 2 , . . . , U are i.i.d or not.) Fix this distribution, then\nThese two chain rules motivate the channel transformation:"},{"title":"Its input distribution W","content":"(i) in (u i ) is determined by that of U i . It may sound weird that W (i) will tell the receiver the input of W (1) , W (2) , . . . , W (i−1) for free. But in reality, W (i) acts as an interactive device where the receiver (not the sender) needs to input what U i−1 1 is and the device will output something that looks like U i−1 1 Y 1 ; only when the receiver inputs the correct U i−1 1 does the device return the correct U i−1 1 Y 1 . Under this interpretation, the de facto capability of W (i) is thus\nY 1 ), which justifies the chain rule of the mutual information. To avoid confusion, we prefer H(W (i) ) over I(W (i) ) in calculations.\nWhat makes the idea of channel transformation powerful is that the transformations apply recursively. The precise formulation is as below: Fix any i ∈ [ ]. Let (X (i) ) 1 , (X (i) ) 2 , . . . , (X (i) ) ∈ X be i.i.d. copies of the capacity-achieving input of W (i) ; let (Y (i) ) 1 , (Y (i) ) 2 , . . . , (Y (i) ) ∈ X i−1 ×Y be the corresponding outputs. Let g W (i) : X → X be a bijection. Define a tuple of r.v.s (U (i) ) 1 := (g W (i) ) −1 ((X (i) ) 1 ); that is to say, g W (i) ((U (i) ) 1 ) = (X (i) ) 1 . For each j ∈ [ ], we define a depth-2 channel\nTo sum up, we can define (W (i) ) (1) , (W (i) ) (2) , . . . , (W (i) ) ( ) out of W (i) for any i ∈ [ ] in the same way we define W (1) , W (2) , . . . , W ( ) out of W . For i, j ∈ [ ], each (W (i) ) (j) is again a channel, so the transformations apply to generate depth-3 channels. In the setup of the classical polar coding, a fixed bijection g is used to define W (i) , (W (i) ) (j) , ((W (i) ) (j) ) (k) , et seq. To reach the optimal MDP paradigm, we\nStep 4) output g W (û 1 ).\n(Step 2-a) output (W (2) (u 2 |û 1 y 1 ) : u 2 ∈ X ). (Step 2-b) input:û 2 .\n(Step 3-a) output (W (3) (u 3 |û 2 1 y 1 ) : u 3 ∈ X ). (Step 3-b) inputû 3 . Figure 2. A DU with = 3 and its I/Os. allow g W to depend on the channel W . That is to say, we need (presumably distinct) bijections g W (i) : X → X for every i ∈ [ ] when we want to define (W (i) ) (j) out of W (i) . Similarly, we need yet another 2 bijections g (W (i) ) (j) : X → X for every i, j ∈ [ ] in defining depth-3 channels. And the recursion goes on ad infinitum. 3.1. Design of the decoder. To implement channel transformations, we define a DU (decoding unit) to be an automata as follows: It is a box with pins on the left and pins on the right. Each pin is connected to another DU, a CH, an FH, or an IH (to be defined later). Each pin may take inputs or output but not at the same moment. A DU works as follows: Let W : X → Y be the channel it is to transform. (Step 0) For all i ∈ [ ], the i-th pin on the left takes the input y i . The input is passed in the form of the a posteriori distribution (W (x i | y i ) : x i ∈ X ). This is what Arıkan calls α-representation [Ari15b, Section II.A]. (Step 1-a) It computes the a posteriori distribution of U 1 given y 1 ; that is, (W (1) (u 1 | y 1 ) : u 1 ∈ X ). And then it outputs this tuple of probabilities to the first pin on the right. (Step 1b) At a later moment, it will receive an estimateû 1 of U 1 from the first pin on the right. Note thatû 1 is a hard symbol in X , not a soft tuple of probabilities. (Step 2-a) It computes the a posteriori distribution of U 2 givenû 1 y 1 ; that is to say, it pretends that U 1 happens to beû 1 and computes (W (2) (u 2 |û 1 y 1 ) : u 2 ∈ X ) accordingly. And then it outputs this tuple of probabilities to the second pin on the right. (Step 2-b) At a later moment, it will receive an estimateû 2 of U 2 from the second pin on the right. (Step i-a) In general, it computes W (i) (u i |û i−1 1 y 1 ) for all u i ∈ X and then output the tuple to the i-th pin on the right. (Step i-b) After a while, it will receiveû i . (Step + 1) Once it receivesû from the last pin on the right, it computesŷ 1 := g W (û 1 ), and then outputŷ i to the i-th pin on the left for all i ∈ [ ]. See Figures 2 to 4 for illustrations."},{"title":"DU","content":"The general rule to arrange the DUs is as follows: For a depth-n construction, put DUs in an n−1 -by-n array. Each DU is indexed by (k 1 , k 2 , . . . , k n−1 ; m) where k 1 , k 2 , . . . , k n−1 ∈ [ ] and m ∈ [n]. For all m ∈ [n − 1] and all k 1 , k 2 , . . . , k n ∈ [ ], connect the k m -th pin on the right of the (k 1 , . . . , k m−1 , k m+1 , . . . , k n ; m)-th DU to the k m+1 -th pin on the left of the (k 1 , . . . , k m , k m+2 , . . . , k n ; m + 1)-th DU. Here, the (k 1 , k 2 , . . . , k n−1 ; m)-th DU is to transform the channel (· · · (W (k1) ) · · · ) (km) (1,1) W\n(1,1) Figure 3. 6 DUs are chained together to implement (W (1) ) (1) , . . . , (W (3) ) (3) . Boxes marked \"W \" are channels. Boxes next to channels are CHs; the labels are their indexes. Boxes at the rightmost column are either FHs or IHs; the labels are their indexes. Note that DUs in the first column use the same g W . DUs in the second column use g W (1) , g W (2) , and g W (3) , respectively.\ninto (· · · (W (k1) ) · · · ) (km+1) . The k 1 -th pin on the left of the (k 2 , . . . , k n ; 1)-th DU connects to a CH (channel helper) indexed by (k 1 , k 2 , . . . , k n ). Each CH then connects to the output of a copy of the channel W . The k n -th pin on the right of the (k 1 , . . . , k n−1 ; n)-th DU connects to either an FH (frozen bit helper) or an IH (information bit helper); in either case, the connected helper is indexed by (k 1 , k 2 , . . . , k n ). Let I ⊂ [ ] n be the set of indexes (k 1 , k 2 , . . . , k n ) such that the k n -th pin on the right of the (k 1 , . . . , k n−1 ; n)-th DU connects to an IH. Then [ ] n \\ I is the set of indexes where the pin connects to an FH. On the left hand side of the DU array, the task of the (k 1 , k 2 , . . . , k n )-th CH is to receive the channel output Y (k1,k2,...,kn) ∈ Y and then forward the a posteriori distribution (W (x (k1,k2,...,kn) | Y (k1,k2,...,kn) ) : x (k1,k2,...,kn) ∈ X ) to the DU array. On the right hand side, FHs correspond to what Arıkan called frozen bits-bits that do not carry information and the receiver knows their values as part of the communication protocol. The task of the (k 1 , k 2 , . . . , k n )-th FH is to receive the a posteriori distribution of the (k 1 , k 2 , . . . , k n )-th frozen bit and then return the correct symbol U (k1,k2,...,kn) ∈ X back to the DU array. IHs correspond to information bits that carry the sender's messages. The task of the (k 1 , k 2 , . . . , k n )-th IH is to receive the a posteriori distribution of the (k 1 , k 2 , . . . , k n )-th information bit and then return the most probable symbolÛ (k1,k2,...,kn) ∈ X back to the DU array. When all IHs are activated once, a code block completes. The most probable symbols they returned to the DU array form the decoded messageÛ I , meaning the tuple (Û (k1,k2,...,kn) : (k 1 , k 2 , . . . , k n ) ∈ I).\nWhat we just established is the successive cancellation decoder of polar codes that could be found in most works that implement polar codes. For instance, [Ari09, Section VIII], [Kor09, Section 3.2], and [HY13, Section III], and [EKMF + 17, Section Vi.B]. See especially [GRY19, Section 9] for an almost identical construction albeit they had q = 2 in mind. We replicate the whole story to demonstrate that each DU may use a unique bijection \"g\" without changing the overall structure too much. Whether or not this construction can transmit information reliably is . 12 DUs (with = 2) are chained together to implement ((W (1) ) (1) ) (1) , . . . , ((W (2) ) (2) ) (2) . DUs in the first column use g W ; DUs in the second column use g W (1) and g W (2) ; DUs in the third column use g (W (1) ) (1) , g (W (1) ) (2) , g (W (2) ) (1) , and g (W (2) ) (2) . discussed in Section 4.2. There, we will also clarify how to arrange FHs and IHs. The complexity can be estimated prior to further specification.\n3.2. Complexity of the decoder. There are various models that measure the complexity of a structure. The polar coding community uses a variant of the circuit complexity where the arithmetic of real numbers costs O(1) and passing probabilities between DUs costs O(1). The complexity of the DU array is thus the number of the DUs multiplied by the complexity of a single DU. The number of DUs is n−1 n. The complexity of a DU depends on how a DU computes the a posteriori probabilities W (j) (u i | u i−1 1 y 1 ) out of W (x i | y i ). The naïve approach is to exhaust all possible inputs u 1 ∈ X and compute the a posteriori probabilities using Bayesian formulas. This costs O( 10 q +10 ) (here 10 is an overestimate). Hence the overall complexity is O( n−1 n 10 q +10 ). In our setup, however, q is fixed, will be chosen upon knowing π, ρ, and n goes to infinity afterwards. So we advertise that the complexity is O( n n), or O(N log N ). Here N := n is the block length, equal to the number of copies of the channel W attached to the DU array. The complexities of the CHs, FHs, and IHs can be computed similarly. They are all bounded by O( n+10 q 10 ). Thus the decoder as a whole costs O(N log N ).\nWe claim that the encoder has the same complexity O(N log N ) although we have not defined the encoder yet. The encoder is essentially a special decoder and is the subject of the next subsection.\n3.3. Design of the encoder. The encoder will be an exact copy of the decoder except that CHs and IHs will behave differently. In greater detail: Let there be an n−1 -by-n array of DUs indexed and connected in the same way described in Section 3.1. Each DU executes the exact same task described in Section 3.1. The left pins of the DUs in the first column each connect to a CH. The right pins of the DUs in the last column each connect to the same type of device (an IH or an FH) as its twin-DU in the decoder does. Here, as part of the encoder, a CH will output the capacity-achieving input distribution (W in (x) for all x ∈ X ) into the DU array. For each (k 1 , k 2 , . . . , k n ) ∈ I, the (k 1 , k 2 , . . . , k n )-th IH will receive a recommended distribution of the (k 1 , k 2 , . . . , k n )-th information bit and then return the message symbol U (k1,k2,...,kn) ∈ X the sender wants to send back to the DU array. For each (k 1 , k 2 , . . . , k n ) ∈ [ ] n \\ I, the (k 1 , k 2 , . . . , k n )-th FH will receive a recommended distribution of the (k 1 , k 2 , . . . , k n )-th frozen bit and then return a r.v. U (k1,k2,...,kn) ∈ X that follows that distribution back to the DU array. This r.v. is simulated by a pseudo random number generator shared between the encoder and the decoder. The twin-FH in the decoder, regardless what distribution it receives, will return the exact same symbol U (k1,k2,...,kn) back to the DU array. This step is called randomized rounding and is found in [Kor09, Section 3.3], [KU10, Section III], [KT10, Section II], and [HY13, Section III.A]. After all IHs return the sender's messages and all FHs returns randomly rounded bits to the DU array, the CHs will each get a codeword symbol X (k1,k2,...,kn) ∈ X from the DU array. And then each CH will forward that symbol to an i.i.d. copy of the channel W .\nThis design is a copy of [HY13]'s encoder explained in our terminology. It is clear that the encoding complexity will be O(N log N ), too. Alongside the decoder, the encoder creates its own channel transformations. Let W : X → Y be a q-ary channel and X be a capacity-achieving input. Define a flattening channel W : X → {η} that erases all information. Then the encoder is effectively synthesizing depth-1 channels"},{"title":"Moreover, H(W ) = H(X) and H(W","content":"No \"Y \" plays any role here since they are constant. The fact that a channel as boring as W is helpful to our main theorem will be covered later, in Section 4.2.\nWe clarified (cs), (cn), and (cc) up to this section; there are (cp) and (cr) to go."},{"title":"Channel Parameters","content":"Let W : X → Y be a q-ary channel. Let X be a capacity-achieving input and Y be the corresponding output. Besides H and I, there are several channel parameters that capture the qualities of channels. Here is a list of parameters extracted from the work [MT14] of Mori and Tanaka.\nBoth H(X | Y ) and H(W ) are the base-q conditional entropy, the base chosen such that 0\nis the error probability of the maximum a posteriori (MAP) decoder. The MAP decoder looks at an output y ∈ Y and chooses a symbolx ∈ X that maximizes W (x | y). When the output is Y = y, the probability that the MAP decoder does not choose X asx is 1 − max x∈X W (x | y). Therefore, P e (X | Y ) = y∈Y W out (y)(1 − max x∈X W (x | y)). In a channel-centric narrative, we also write P e (W ) for P e (X | Y ).\nZ(X | Y ) is the rescaled sum of Bhattacharyya coefficients of the transition distribution W (y | x) for the uniform input. For non-uniform inputs, a modification is made to generalize the definition and the properties that used to hold. Intuitively speaking, a MAP decoder seeing y is \"confident\" if W (x | y) is small for all but one\nx, or equivalently, if the product W (x, y)W (x , y) is small for all distinct x, x ∈ X . The Bhattacharyya parameter measures the \"confidence\" by\nIn addition, define\nWe also write Z(W ), and Z mad (W ) for these quantities. Remarks: The rescaling T (X | Y ) is the weighted average of the total variation distances from the a posteriori distributions (W (x | y) : x ∈ X ) to the uniform noise (1/q, 1/q, . . . , 1/q). More formally, it is defined to be y∈Y W out (y) x∈X |W (x | y) − 1/q|. We also write T (W ) for this quantity.\nS(X | Y ) is the weighted average of the L 1 -norms of the Fourier coefficients of the a posteriori distributions. The formal definition is as follows. Let tr : F q → F p be the field trace, where F q = X and F p is the prime subfield. Let χ : F q → C be an additive character defined as χ(x) := exp(2πi tr(x)/p), where 2πi is temporarily the period of exp. Define the Fourier coefficient\nDefine the S-parameters\nWe also write S(W ) and S max (W ) for these quantities. Remarks: The rescaling is such that 0 S S max (q − 1)S q − 1. An interpretation is as follows: Fix a y. When W (x | y) is roughly equal to 1/q for all x ∈ F q , the Fourier coefficient M (w | y) = z∈Fq W (z | y)χ(wz) should be roughly z∈Fq χ(wz)/q = 0. The S-parameter measures how far those coefficients are from zero."},{"title":"4.1.","content":"Relations among channel parameters. The following is a series of lemmas we extract from existing works. They characterize the relations among H, I, P e , Z, T , and S.\nLemma 4. [MT14, Lemma 23 with k = q − 1] For any q-ary channel W ,\nLemma 6. [FM94, Theorem 1] For any q-ary channel W ,\nand\nHere, h 2 is the binary entropy function; h 2 (1/2) = 1. The upper bound is Fano's inequality. The first lower bound fits when H(W ) and P e (W ) are small; the second lower bound fits when H(W ) and P e (W ) are close to 1.\nThe above lemmas inspire the following characterization: Let A and B be two channel parameters, we say A, B are bi-Hölder at . This notion generalizes to tuples of more parameters. Now we can summarize Lemmas 3 to 6 in a more concise statement.\nLemma 7 (implicit bi-Hölder tolls). Parameters H, P e , Z, Z mad are bi-Hölder at (0, 0, 0, 0). Parameters H, P e , T, S, S max are bi-Hölder at (1, 1 − 1/q, 0, 0, 0).\nProof. Z, Z mad are bi-Hölder at (0, 0) since Z Z mad (q − 1)Z. Lemma 3 implies that P e , Z are bi-Hölder at (0, 0). Lemma 6 (with the first lower bound) implies that P e , H are bi-Hölder at (0, 0). Now apply the transitivity to conclude the first statement. For the second statement, S, S max are bi-Hölder at (0, 0) since S S max (q − 1)S. Lemma 5 implies that P e , S are bi-Hölder at (1 − 1/q, 0). Lemma 4 implies that P e , T are bi-Hölder at (1 − 1/q, 0). Lemma 6 (with the second lower bound) implies that P e , H are bi-Hölder at (1 − 1/q, 1). Now apply the transitivity to conclude.\nSee also [MT14,Corollary 28] for what inspired us. They use notation A e ∼ B to mean A, B are bi-Hölder at (0, 0) and at (1, 1). For some very technical details on the way toward the main theorem, we need explicit Hölder relations among H, Z mad , and S max . We claim them here. The proof is nothing but looking closer into Lemmas 3, 5, and 6. A written-out proof is in Appendix A.\nLemma 8 (explicit Hölder tolls). log is natural. For all q-ary channels W , the following hold:\n4.2. Control of the block error probability. Let W be the channel we want to communicate over; and let X be any input. In the classical theory of polar coding, the second last step of the construction of the block code is to determine a subset I ⊂ [ ] n of indexes that points to the depth-n channels that transmit information bits. When decoding this code, a block error happens if the successive cancellation decoder fails to decode any information bit. Let E (k1,k2,...,kn) be the event that the first error occurs when the decoder is solving for the input to (· · · (W (k1) ) · · · ) (kn) , i.e., whenÛ (k1,k2,...,kn) = U (k1,k2,...,kn) and the equality holds for lexicographically earlier indexes. Then the event's probability measure P (E (k1,k2,...,kn) ) is no more than the bit error probability P e (· · · (W (k1) ) · · · ) (kn) . By the union bound, the block error probability of the decoder is bounded from above by a sum\nWith this observation, we may define I to be the set of indexes (k 1 , k 2 , . . . , k n ) ∈ [ ] n such that H (· · · (W (k1) ) · · · ) (kn) < θ n for some clever choice of the threshold θ n > 0. This immediately implies P e (· · · (W (k1) ) · · · ) (kn) < cθ d n for some c, d > 0 by Lemma 7. Let θ n be exp(− πn n). The sum of P e is less than n cθ d n < exp(− πn ) for sufficiently large n, which is the block error probability we claimed. Remark: Arıkan used a different criterion Z < θ n . It still implies P e < cθ d n and that the sum of P e is less than n cθ d n < exp( πn ) for large n. The benefit of controlling P e using other parameters is that some parameters are easier to control (because Theorems 9 and 10 exist).\nFor the main theorem where the channel W is asymmetric, we want to control both the decoder block error and the encoder block error. Here, the encoder block error is not the encoder's failure to encode a message, but rather its failure to generate the capacity-achieving input distribution of W . To penalize, imagine that we employ an oracle that claims an encoder block error whenever the generated codeword should have been another word to fit the ideal distribution. That way, the actual block error probability will not exceed the sum of the encoder and decoder block error probabilities. More rigorously, let P be the probability measure assuming the ideal distribution of U I and Q be the probability measure assuming the actual U I generated by the encoder. Then the overall block error probability can be bounded by\nP {Û I = U I } as the decoder block error probability is bounded before. The encoder block error probability is represented by P − Q , the total variation distance from P to Q. There is a telescoping argument similar to how we control the decoder error-classifying events by the first input bit where the oracle disagrees with the encoder [Kor09, Lemma 3.5] [KU10, Lemma 4] [KT10, Lemma 2] [HY13, Lemma 1]. It yields that the encoder block error probability is bounded from above by the sum\nIn controlling the encoder bit error probability, we strengthen the policy of collecting indexes (k 1 , k 2 , . . . , k n ) ∈ [ ] n for I by asking for H (· · · (W (k1) ) · · · ) (kn) > 1 − θ n .\nThe latter immediately implies T (· · · (W (k1) ) · · · ) (kn) < cθ d n by Lemma 7. As a consequence, the overall block error probability is controlled by Q{Û I = U I } P {Û I = U I } + P − Q < 2 n cθ d n < exp(− πn ) for n large. The preceding argument is a paraphrase of the proof of [HY13, Theorem 13]; Inequalities (59) and (57) therein are the keys. So far the block length, the complexity, and the error aspects of the main theorem are covered, it remains to control the code rate |I|/ n . In other words, we are to compute the cardinality of I given that I ⊂ [ ] n is the set of indexes such that H (· · · (W (k1) ) · · · ) (kn) < θ n and 1 − H (· · · (W (k1) ) · · · ) (kn) < θ n , where θ n := exp(− πn n)."},{"title":"4.3.","content":"Before and after channel transformations. Alongside the relations among different parameters applied to the same channel, there are also relations between the same parameter applied to the original and the transformed channels. That\nThere are two more that are pivotal in the theory of polar coding but require more prerequisites. Assume that g W : X → X is a linear isomorphism given by the multiplication of an invertible matrix G from the right-g W (u 1 ) := u 1 G. The following framework extends to nonlinear bijections but we do not need that much. (There is also the paradigm that random linear codes perform better than random codebooks for that a bad linear code tends to hoard a lot of short codewords at once, effectively removing them from the ensemble pool. So there is a good reason to stick to the linear case.) Let 0 i−1 1 1 i u i+1 ∈ F q be a tuple of i − 1 many 0 followed by a 1 and − i arbitrary symbols. A coset code is a subset of codewords of the form\nThe coset codes have weight distributions just like every other code does. Let wt(x 1 ) be the hamming weight of x 1 . The weight enumerator of the i-th coset code is defined to be a one-variable polynomial over the integers\nWe can now state the second relation. This is considered the main cause of why polar coding ever exists/works.\n. The proof is postponed until Section 7.1. The fundamental theorems come as a pair. Let u i−1 1 1 i 0 i+1 ∈ F q be a tuple of i − 1 arbitrary symbols followed by a 1 and − i many 0. Let G − be the inverse transpose of G. The weight enumerator of the i-th dual coset code is defined to be this one-variable polynomial over the integers\nWe can now state the third relation, the dual of the second. The proof is postponed until Section 7.2.\nRemark: These two bounds are not tight-the equality does not hold for BECs. In detail, Arıkan's original bound reads Z mad (W (1) ) 2Z mad (W )−Z mad (W ) 2 while our bound turns into Z mad (W (1) ) 2Z mad (W ), the subtraction term missing. We are simply not able to prove a version that degenerates to an equality over erasure channels, nor does any prior work seem to. This causes a serious aftermath that Z mad (W n ) (to be defined later) is no longer a supermartingale. Nonetheless, this bound is strong enough to collaborate with the random coding theory. See, for example, how we compensate in Appendix C.1.\nWe clarified (cs), (cn), (cc) and (cp) up to this section; there is (cr) to go."},{"title":"Channel Processes","content":"These r.v.s provide a new family of randomness that does not appear in the encoding and decoding algorithms, but they help us understand the code rate |I|/ n in this manner: Counting how many indexes are in I is nothing more than measuring the probability P{(K 1 , K 2 , . . . , K n ) ∈ I}. With the processes W n and V n thus defined, it is further equivalent to measuring the probability P{H(W n ) < θ n and 1 − H(V n ) < θ n }, where θ n := exp(− πn n). Moreover, it suffices to know how H(W n ) and H(V n ) behave as stochastic processes taking values in [0, 1] without comprehending W n and V n themselves. The general fact is that H(W n ) is either very small (channel is reliable) or very close to 1 (channel is noisy). Arıkan called this phenomenon channel polarization. The following claim generalizes channel polarization and implies the main theorem.\nClaim 11. Fix any π, ρ > 0 such that π + 2ρ < 1. We will choose an and a series of bijections of F q -namely,\nHere, o(n) is the little-o function in n; it is such that o(n)/n → 0 as n → ∞.\nFor polar codes over symmetric channels, the first inequality in Claim 11 alone implies that the code rate is 1 − H(W ) − −ρn+o(n) = I(W ) − N −ρ+o(1) . The first two inequalities imply the polarization behavior that channels become either satisfactorily reliable (low H(W n )) or desperately noisy (high H(W n )). For asymmetric channels, however, we need to characterize H(V n ) alongside H(W n ). The last two inequalities in Claim 11 show that the same series of bijections polarize W at the same time they polarize W . While W contains no randomness form the channel W , what is polarized is that each input bit U (k1,k2,...,kn) either depends heavily on lexicographically earlier input bits (low H(V n )) or behaves like a free r.v. conditioned on earlier bits (high H(V n )). We then categorize the fate of indexes in [ ] n into the following three types. (A) Free and reliable: These are indexes that will be in I; they point to channels that transmit information bits. (B) Free but noisy: The sender can feed information into these channels only to find that the decoder will almost always make some mistakes. The sender should, instead, feed some pseudo random numbers shared with the receiver. (C) Dependent and reliable. The input of these channels depends on previous inputs. Their main purpose is to shape the capacity-achieving input distribution. (D) Dependent but noisy is not possible because H(V n ) H(W n ). This is the key to [HY13, Theorem 1]. We reproduce their proof in the next subsection. 5.1. Claim 11 implies the main theorem. As mentioned, H(V n ) H(W n ) so (D) dependent but noisy is not possible. Let A n be the intersection event of\nfollows the third inequality in Claim 11. Note that A n or B n implies free but not \"neither reliable nor noisy\"; that is, (free ∧ reliable) ∨ (free ∧ noisy) → free ∧ ¬(¬reliable ∨ ¬noisy). We deduce that P\nSimilarly, since A n or C n implies reliable but not \"neither free nor dependent,\" we deduce that P\nIn summary, we derive that\nFinally, recall that I collects free and reliable indexes, so the code rate is |I|/ n = P(A n ) > I(W ) − −ρn+o(n) . We almost finish the proof of the main theorem except that we claimed I(W ) − N −ρ = I(W ) − −ρn , without the little-o term. It can be fixed by finding a slightly larger > ρ such that π + 2 < 1 still holds, and then rerunning the whole argument again with the new . The conclusion becomes that the code rate is at least I(W ) − − n+o(n) . Since − n + o(n) < −ρn for sufficiently large n, this completes the proof of the main theorem. It remains to show that Claim 11 can be achieved."},{"title":"Global MDP Behavior Modulo Local Behaviors","content":"In this section, we put constraints on an abstract process {H n } and show that they imply inequalities of the form P{H n < threshold} > limit measure − decaying gap as those in Claim 11. Let F n be the sigma-algebra generated by K 1 , K 2 , . . . , K n for each n. Then F 0 ⊂ F 1 ⊂ F 2 ⊂ · · · form a filtration of sigma-algebras. Let {H n }, {Z n }, and {S n } be three stochastic processes adapted to {F n } (meaning K 1 , K 2 , . . . , K n determine H n , Z n , S n ). The following assumptions are easy to verify when we reveal what those processes are: (cb) 0 H n , Z n , S n and H n 1;\nFurthermore, assume large kernels: (cl) max(e 4 , q 5 , 3 q ). Let α := log(log )/ log be a small number shrinking as increases. Define the potential function h α : [0, 1] → [0, 1] to be h α (z) := min(z, 1 − z) α . (Remark: h 2 is not a special case of h α for α = 2; we expect α 1 in practice.) Here are the difficult but sufficient criteria for the main theorem.\nLemma 12. (calculus machinery for global MDP) Assume criteria (cb), (cm), (ct), and (cl). Assume the local LDP behavior: Z n+1 exp(qZ n )(qZ n ) K 2 n+1 /3 and S n+1 exp(qS n )(qS n ) ( +1−Kn+1) 2 /3 . Assume the local CLT behavior:\nThen, for any constants π, ρ > 0 such that\nthe following holds:\nWe defer the proof until Appendix B. The term K 2 n+1 /3 in the lemma is to control the local LDP behavior of the process {H n }-the behavior of H n+1 when H n is close to 0 and the behavior that is closely related to the LDP behavior of polar codes. The term is chosen in a way such that h 2 ((k 2 /3 )/ ) < k/ and such that k (k 2 /3 ) t is easy to handle. In [FHMV17, Theorem 7], a similar criterion is stated and is annotated as faster polarization at the tails. In [BGS18, Definition 2.4], a similar criterion is stated and is annotated as strong suction at the low end. The eigenfunction h α in the lemma is to control the local CLT behavior of the process {H n }-the behavior of H n when it is away from 0 and the behavior that is closely related to the CLT behavior of polar codes. In [FHMV17, Theorem 7], a similar criterion is annotated as near optimal polarization in the middle with h FHMV (z) := (z(1 − z)) α for positive but small α at most log(log )/ log . In [BGS18, Definition 2.3], a similar criterion is annotated as variance in the middle with h BGS (z) := min(z, 1 − z). Note how our choice of h α (z) := min(z, 1 − z) α resembles theirs. In both cases, the criteria are local because they refer to a small slice of the process, focusing on how H n+1 (or Z n+1 ) behaves in terms of H n (or Z n ). This perspective frees [FHMV17,BGS18] from considering the (global ) process {H n } as a whole and simplifies the analysis. We specifically benefiti from the fact that we can choose the bijection g (···(W (k 1 ) )··· ) (kn ) solely according to the channels (· · · (W (k1) ) · · · ) (kn) and (· · · (W (k1) ) · · · ) (kn) instead of the complete channel family-tree. This is also the approach taken in [GRY19].\n6.1. Lemma 12 helps achieve Claim 11. To do so, one advantage is that the two desired behaviors are local. They only involve how H n+1 , Z n+1 and S n+1 behave conditioned on the history F n . A potential tedious aspect is that for each candidate of the bijection g (···(W (k 1 ) )··· ) (kn ) , we have to verify the two behaviors four times, once for each of the four triples of channel parameters. Luckily, within random coding theory, we are in the situation that to choose an object that satisfies multiple criteria, it suffices to choose the object from an ensemble and compute the probabilities that each criterion fails; as long as the sum of failing probabilities is small, most objects satisfy. Even more luckily, when we choose a bijection g (···(W (k 1 ) )··· ) (kn ) from some ensemble, we only have to compute the probability that the local CLT or LDP behavior fails for {H n }, {Z n }, and {S n } being {H(W n )}, {Z mad (W n )}, and {S max (W n )} but not the other three triples. This is because other three triples are the special case and/or the dual of this triple. Elaboration: Since V n are q-ary channels just like W n are, inequalities hold true for arbitrary W n should hold true for any V n . Also, since Z mad and S max are in duality, inequalities hold true for H, Z mad , S max hold true for 1 − H, S max , Z mad . The duality is due to the duality between FTPCZ and FTPCS, within the explicit Hölder tolls, and within the ensemble of bijections we are to choose g (···(W (k 1 ) )··· ) (kn ) from.\n6.2. Random linear isomorphisms as bijections. Fix q and . Let GL( , q) be the group of -by-invertible matrices over F q together with the ordinary matrix multiplication. Select an element G ∈ GL( , q) uniformly at random. Let g W : F q → F q be the multiplication of G from the right, namely g W (u 1 ) := u 1 G. This map is bijective since G is invertible. Let W be a q-ary channel. Recall that we defined q-ary channels W (1) , W (2) , . . . , W ( ) in Section 3. To emphasis that these imaginary channels depend heavily on the randomness source G, we call\nLemma 14 (local CLT behavior). Fix an 20. Recall α := log(log )/ log and h α (z) := min(z, 1 − z) α . Let G vary; with probability less than 2 − log( )/20 , this fails:\n6.3. Local behaviors imply Claim 11 (and hence the main theorem). We now can see how Lemmas 12 to 14 imply that Claim 11 is achievable for the right choice of and bijections g W , g W (i) , et seq.: For any given q-ary channel W , let be max(e 4 , q 5 , 3 q ). For any given π, ρ > 0 such that π + 2ρ < 1, enlarge such that Inequality (6) holds, given α := log(log )/ log . Consider a random kernel G as a candidate of the bijection g W . Increase further so that the failing probabilities-Lemma 13's 3q − √ /13 and Lemma 14's 2 − log( )/20 -amount to 1/3 or less. Recall the flattening channel W . The probability that any of the inequalities in Lemmas 13 and 14 fails for W is less than 1/3, too. Invoke the union bound; 1/3 + 1/3 < 1. Hence there exists a solid choice of g W as the multiplication of some proper instance of G from the right. With this g W determined, we define W (i) and W (i) for all i ∈ [ ]. Consider first i = 1, anything that has been done to W now applies to W (i) . That is, let g W (i) be the multiplication of a random kernel G from the right. With W , i, and W (i) G replaced by W (i) , j, and (W (i) ) (j) G , the probabilities that inequalities in Lemmas 13 and 14 fail add up to 1/3 or less. So is the flattening ( ) counterpart. Hence there is a solid choice of g W (i) . Repeat this for every other i = 2, 3, . . . , . Once finished, proceed to choosing g (W (i) ) (j) for all i, j ∈ [ ]. And so on and so forth for cases beyond depth-2. Notice that we always make a solid choice of a bijection before we proceed to the next level of channels, hence the failing probabilities of Lemmas 13 and 14 do not accumulate as the depth increases. By how we select bijections, the criteria in Lemma 12 hold for ({H n }, {Z n }, {S n }) being these four triples: In this section, we will first prove the two fundamental theorems of polar coding in Section 7.1 (for the Z-end) and in Section 7.2 (for the S-end). And then we will target that the following inequalities hold with high probability:\nexp(qZ mad (W ) )(qZ mad (W )) i 2 /3 , and (8's copy)\nBy the duality between the two fundamental theorems and between the two targeted inequalities, it is not hard to see that it suffices to prove the Z mad -case and the S maxcase follows immediately. We will prove that the first targeted inequality, for each i ∈ [ ], holds with probability 1 − 3q − √ /13 in Section 7.4, closing this section.\n7.1. Proof of FTPCZ (Theorem 9). As is promised in Section 4.3, we prove the two fundamental theorems of polar coding. We first go for the Z-end.\nis the weight enumerator of the i-th coset code. Theorem 9 claims that Z mad (W (i) ) f\nBy the nature of max 0 =di∈Fq , it suffices to show that the double sum is at most f (i) GZ (Z mad (W )) for arbitrary nonzero d i . In the upcoming argument, tuple concatenation takes precedence over vector-matrix multiplication and vector addition. Fix a d i , we argue that\nThe first equality abbreviates the summation. The next equality expands W (i) by the very definition, where u i+1 and v i+1 are free variables in F q . The next inequality is by the sub-additivity of the square root. In the next equality we define d i+1 := v i+1 − u i+1 ; so summing over v i+1 is equivalent to summing over d i+1 . In the next equality we define x 1 := u 1 G; so summing over u 1 is equivalent to summing over x 1 as G is invertible. In the next equality we substitute e 1 := 0 i−1 1 d i G and reorder the summation. The next equality expands the product of the memoryless channels. The next equality classifies indexes into two classes-j ∈ J are those such that e j = 0 and k / ∈ J are such that e k = 0. The next equality is the distributive law ax + ay + bx + by = (a + b)(x + y). The next equality uses the fact that the W (x, y) sum to 1. In the next inequality we replace e j by a nonzero element that maximizes the sum in the parentheses. In the next equality we realize that the maximum is the Bhattacharyya parameter (surprisingly). The second last equality uses the fact that multiplying a vector by a scalar preserves its hamming weight. And quod erat demonstrandum. Experienced readers may find that all but the last inequality follows the proof strategy [KSU10, Lemma 10]. 7.2. Proof of FTPCS (Theorem 10). We now go for the S-end of the fundamental theorem of polar coding. Recall the character χ(x) := exp(2πi tr(x)/p). We need the following properties: (pa) χ(0) = 1; (pb) |χ(x)| = 1 for all x ∈ F q ; (pc) χ(x)χ(z) = χ(x + z) for all x, z ∈ F q ; (pd) x∈Fq χ(x) = 0. See also [MT14, Definition 24] or a dedicated book [Ter99]. To prove the theorem, we first verify that Fourier coefficients recover the origin: Let M (w, y)\nThe first equality expands M (w, y) by the definition. The next equality uses that χ is an additive character (pc), and reorders the summation. The next equality uses w∈Fq χ(w) = 0 (pd) and w∈Fq χ(0) = q (pa); and I is the indicator function.\nKnowing that W (x j , y j ) = q −1 wj ∈Fq M (w j , y j )χ(−x j w j ), we proceed to\nThe first equality expands the definition of W (i) . In the next equality, we substitute x 1 := u 1 G. The next equality expand the definition of W down to W . The next two equalities Fourier expand W and reorder the operators. The next equality merges all χ(−x j w j ) into one term by additivity (pc). In the next equality we define M (w 1 , y 1 ) to be the product of all M (w j , y j ). The next two equalities use x 1 (w 1 ) = u 1 G(w 1 ) = u 1 (w 1 G ) . In the next equality we define v 1 := w 1 G ; so summing over w 1 is equivalent to summing over v 1 . (Recall that G − is the notation of the inverse transpose of G.) The next three equalities sum over u i+1 to force v i+1 = 0.\nHaving\nIn the first line we let M (i) be the Fourier coefficient of W (i) . The next equality plugs in what we have about W (i) in mind. The next three equalities sum over z i to force\nThe first inequality expands the Fourier coefficient. The next inequality is triangle plus (pb). The next equality cancels the summation over u i−1 1 with q 1−i . In the next equality we substitute w 1 := v i−1 1 ω i 0 i+1 G − ; slightly different from the w 1 above, they are now restricted to a proper subspace. The next equality classifies indexes into two classes-j ∈ J are those such that w j = 0 and k / ∈ J are such that w k = 0. The next two equalities reorder the operators and simplify y k |M (0, y k )| = y k W out (y k ) = 1. The next inequality replaces w j by the one that maximizes y k |M (w j , y k )|. The rest is trivial. Theorem 10 claims that S max (W (i) ) f (i)\nGS is the weight enumerator of the i-th dual coset code. Since S max (W (i) ) is just the maximum of Formula (11) over 0 = ω i ∈ F q , we arrive at S max (W (i) ) f\nSee the left half of Figure 6 for evidence. More generally, for all prime powers q,\nfor all z ∈ [0, 1], where D is the Kullback-Leibler divergence. This falls back to the h 2 case when q = 2. See the right half of Figure 6 for evidences for q = 3, 4, 5, 7. It can be observed that as q → ∞ the function tends to a line connecting (0, 0) and (1, 1), hence the upper bound should hold. Taking derivative in q shows that the left hand side decreases as q increases and x < 1/2. where z := Z mad (W ) for short. This inequality is in fact a consequence of\nbecause (1 + a) b < exp(ab). We will show the last inequality. Now divide i into two cases: 1 i √ 3 and √ 3 < i . For i = 1, 2, . . . , √ 3 , the exponent i 2 /3 is simply 1, so the inequality to be proven reads f For i = √ 3 + 1, √ 3 + 2, . . . , , let k := − i and let d := i 2 /3 . These variables resemble the dimension and the minimal distance of a linear block code as in the notation an [ , k, d]-code in classical (algebraic) coding theory. To make Inequality (12) hold, we execute a two-phase procedure to avoid all codewords of weight less than d and to eliminate kernels with poor overall scores. In further detail, we will reject a kernel G if there exists u i+1 such that wt(0 i−1 1 1u i+1 G) < d and call it phase I. Afterwards, among surviving kernels with only heavy codewords, we will reject a kernel if its overall score f (i) GZ (z) is too low and call it phase II. The failing probability 3q − √ /13 is the price we pay for rejecting. Up to this point, two things remain to be analyzed: how much probability we pay for rejecting light codewords in phase I (answer: q − √ /13 ), and what is the Markov cutoff that makes Inequality (12) in phase II (answer: 2q − √ /13 ). Phase I analysis is as follows: Fix u i+1 and vary G ∈ GL( , q); the codeword\nThis distribution is almost identical to the uniform distribution on F q . Assume X 1 follows the latter; this makes X 1 lighter, which is compatible with the direction of the inequalities we want. Then the probability that X 1 has weight less than d is the probability that Bernoulli trials-X j being \"zero\" with probability 1/q and \"nonzero\" with probability (q −1)/q-result in less than d \"nonzero\"s. By the large deviations theory [DZ10, Exercise 2.2.23(b)], wt(X 1 ) < d holds with probability less than\nfor the q = 2 case, where D is the Kullback-Leibler divergence. For general q, similarly, wt(X 1 ) < d holds with probability less than\nThis quantity is less than q − (1−h2(d/ )) by Figure 6 (meaning that q = 2 is the most difficult case). By Figure 6, h 2 (d/ ) < ed/ = ei 2 /3 2 = ( e/3)i/ < 0.952i/ . So the rejecting probability is less than q − (1−h2(d/ )) < q − +0.952i . Take into account that there are q −i possibilities of u i+1 . The union bound yields q −i q − +0.952i = q −0.048i < q −0.048 √ 3 < q − √ /13 . Hence the rejecting probability q − √ /13 . Phase I ends here. Phase II analysis is as follows: After we reject some G in phase I, some codewords will disappear; particularly, this includes all codewords of low weights. Therefore, the expectation of f (i) GZ (z) is bounded by the weight enumerator of all heavy codewords rescaled by the number of codewords. In detail, start from\nI is the indicator function. In the denominator, 1 − q − √ /13 > 1/4 as 30. Put that aside and redefine d := i 2 /3 . The expected value part is bounded from above by\nThe first equality expands the definition. The next inequality replaces G surviving phase I by a weaker condition. The next equality switches E and . The next inequality replaces the ensemble of u i+1 G by a uniform X 1 ∈ F q . The next equality expands the definition of the expectation over X 1 . The next equality counts codewords. The next inequality selects w positions by first selecting d and then selecting w − d. The next two equalities factor and apply the binomial theorem. The rest is by a series of inequalities that overestimate the scalar:\n. Hence the scalar part is less than q − √ /13 /2. Put 1 − q − √ /13 > 1/4 back to the denominator as in Inequality (13);\nBy Markov's inequality, Inequality (12) holds with probability 1−2q − √ /13 , i.e., the rejecting probability is 2q − √ /13 . Phase II ends here. The sum of the two rejecting probabilities is 3q − √ /13 as claimed in Lemma 13, hence the lemma settled. 7.5. Bibliographic remarks. Concerning the fundamental theorems: Nonlinear g W is not taken into consideration for that it is hard to imagine how MacWilliams duality works then. Also the S-parameter does not generalize to non-field input alphabet. Concerning random linear codes: [BF02, Section II.C] portrays a clear picture of the weight distribution of binary random linear codes. Section 7.4 accommodates and extends their argument to general prime power q. Concerning the LDP behavior: [KSU10, Theorem 22] showed that π < 1 can be arbitrary close to 1 over binary alphabet utilizing the Bose-Chaudhuri-Hocquenghem codes. Our Lemma 13 on the other hand, implies that almost all kernels make π close to 1.\nIt remains to prove Lemmas 12 and 14."},{"title":"Local CLT Behavior (Proof of Lemma 14)","content":"We are to prove that the following inequality holds with high probability:\nThe target inequality is the sum of the following three inequalities:\nThe second one is trivial as h α (z) (1/2) α . The first one will be proven in Section 8.3 with failing probability − log( )/20 . The third one will be proven in Section 8.4 with failing probability − log( )/20 . Before the main proofs, we devote Section 8.1 to introduce the symmetrization trick, which will reduce our proof to the case of symmetric q-ary channels. A channel W being symmetric means that for any affine shifting ξ ∈ F q , there exists an permutation σ on Y such that W (y | ξ + x) = W (σ(y) | x) holds for all x ∈ F q and y ∈ Y. It also means that the uniform input achieves the Shannon capacity. This justifies the usage of linear codes. In Section 8.2, we invoke some universal bound on entropies and exponents from Chang, Draper, and Sahai's works. Finally, we will be abusing the theory of random linear codes in Section 8.3 for noisy channel coding and in Section 8.4 for secrecy over wiretap channels. 8.1. Symmetrize channel and uniformize input. Let W : F q → Y be any q-ary channel; let X and Y be some input and the corresponding output. Symmetrize the channel as follows: Let Ξ ∈ F q be a uniform r.v. independent of X, Y . Let W : F q × (F q × Y) → [0, 1] be the probability mass function of this combination of r.v.s (Ξ + X, (X, Y )) ∈ F q × (F q × Y). ThisW behaves like a channel such that, quote, unquote,W ((x, y) | z) = W (x, y)/q for all inputs z ∈ F q and outputs (x, y) ∈ F q × Y. Despite that this channel might be properly simulated by a symmetric channel with feedback to the sender, all that matters is that the biased input X is neutralized by the uniform r.v. Ξ, and becomes uniform. Let g W be the multiplication of an invertible matrix G from the right. LetW (i) (u i , u i−1 1 x 1 y 1 ) be the probability mass function of the tuple (U i , U i−1 1 X 1 Y 1 ), where U 1 G = Ξ 1 + X 1 . This definition is compatible with the channel transformation ofW as ifW was an actual channel in the first place. Let H(W (i) ) be H(U i | U i−1 1 X 1 Y 1 ); this is also compatible. The following lemma justifies whyW is useful in theory.\nLemma 15 (channel symmetrization).W is a symmetric q-ary channel,\nThis lemma is by [MT14, Definition 6 and Lemmas 7 and 8], plus the arguments in between. See also [HY13,Theorem 2] where they cared about whether Z(W (i) ) = Z(W (i) ). One could also expand all definitions to verify the identities.\nThe consequence of this lemma is thatW behaves like a shadow copy of W , but is symmetric. All inequalities involving entropies of W and W (i) are reduced to inequalities involving entropies ofW andW (i) . Subsequently, passing statements toW is effectively assuming that the channel W is symmetric with the uniform input to begin with. In the upcoming subsections, we will prove that the targeted inequalities, (14) and (15), hold for any symmetric q-ary channel W with the uniform input with high probability. We conjecture that the symmetrization technique is optional as it seems like a wrapper of complicated Bayesian formulas. 8.2. Chang-Sahai's universal quadratic bound. This and the next two subsections contain the most convoluted part of the proof of Lemma 14. This subsection prepares a universal upper bound on Gallager's E-null function, which ultimately evolves into a universal lower bound on Gallager's error exponent.\nLet W : X → Y be a q-ary channel. Symmetry is not required in this subsection but it is in the next two. Assume the uniform input distribution W in (x) = 1/q for all x ∈ X . Define Gallager's E-null function and its complement [CS07, Formula (1)]:\n, and\nBy complement we mean that under the uniform input,Ē 0 (t) degenerates tō\nEquivalently, E 0 (t) +Ē 0 (t) = t log q. Do not confuse W with W t , the latter W is tilted. When t = 0, the tilted falls back to its italic origin W 0 (x, y) = W (x, y). These measures can be interpreted as follows: W t behaves like a channel with a dedicated input distribution. The first fraction in the definition specifies the output distribution W t out (y). The second fraction specifies the a posteriori distribution W t (x | y) when y is known. As W t is not an actual channel, it is not meaningful to alter the input distribution and ask for the corresponding output. Like the symmetrization trick, all that matters is that we can compute entropies, and what not, as if they were real channels. Quantities we are interested in are listed below: Let H e be the base-e entropy. Let H e (W t ) be H e (X t | Y t ) where (X t , Y t ) is a tuple r.v. that follows W t . Let H e (X t y) be the entropy of the a posteriori distribution of X t given Y t = y; to be specific,\n[CS07, Formula (13) and (19)] have that the following hold for t ∈ [0, 1]:\nCareful readers may verify them by hand or follow [CS07,Formulas (13) to (19)] and [DCS14, Lemmas 9 and 10]. Similar computations are also carried out by [AW10,AW14]. Notice thatĒ 0 (t), H e (W t ), and every other term in Equation (16) are all holomorphic functions in t on the half-plane Re t > −1 (there is a singularity at 1/(1 + t) = ∞). By the identity theorem in complex analysis [BMPS02,Corollary 8.16], Equation (16) holds for all t ∈ [−2/5, 1]. Dropping the nonpositive square, we deduce an upper bound for each t ∈ [−2/5, 1]:\nThis upper bound onĒ 0 (t) is a linear combination of\nand H e (X t y) 2 parametrized by y ∈ Y, so it remains to bound them separately. For the second kind of constituents, the entropy cannot exceed log q so H e (X t y) 2 log(q) 2 . For the first kind of constituents, the following lemma adapted from [CS07, Lemma 1] helps.\nLemma 16 (second moment). If w 1 , w 2 , , . . . , w q are positive numbers of sum 1, then i w i log(w i ) 2 ß log(q) 2 for q 3 0.563 for q = 2 ™ 1.2 log(q) 2 .\nWith the lemma, we do have x∈X W t (x | y)(logW t (x | y)) 2 1.2 log(q) 2 . Now Inequality (17) becomes\nfor all t ∈ [−2/5, 1]. Since E 0 (t) is a linear function t log q minusĒ 0 (t), their first derivatives sum to log q while their second derivatives are opposite. Hence the following lemma.\nwhere j := H(W ) + 1/2+α for short. It suffices to prove that the right hand side is less than −1/2+α . In the spirit of the motivational Chain Rule (1), the sum of the chain of H(W (i) G ) on the right hand side is H(U j+1 | U j 1 Y 1 ). In order to prove Inequality (14), we will show\nBut what is H(U j+1 | U j 1 Y 1 )? It measures the equivocation at Bob's end when U j 1 is known to Bob. In other words, we may as well pretend that there is a random rectangular full-rank matrix G with columns and only k := − j = − H(W ) − 1/2+α rows, that Alice computes and sends X 1 := U k 1 G to Bob, and that Bob attempts to decodeÛ k 1 upon receiving Y 1 using the MAP decoder. The equivocation is thus, by Fano's inequality, bounded in terms of the probability that Bob fails to decode U k 1 :\n−P e log q P e + P e log q + P e · k = P e · 1 − log P e log q + k . (19) Here P e is the probability that Bob fails to decode,Û k 1 = U k 1 . The following is how to compute Bob's decoder block error probability. The generator matrix G Alice uses is selected uniformly from the ensemble of full-rank k-by-matrices. The difference of every pair of codewords distributes uniformly on F q \\ {0 1 }. Over symmetric channels, the difference alone determines the output's joint distribution because W (y 1 | ξ 1 + x 1 ) = W (σ 1 (y 1 ) | x 1 ) for some componentwise permutation σ 1 on Y depending on ξ 1 . Gallager's bound applies. To elaborate, let t ∈ [0, 1]. Bob's average error probability satisfies [Gal68, Inequalities (5.6.2) to (5.6.14)]\n= exp(kt log q − (the E-null function of W )(t)) = exp(kt log q − E 0 (t)).\nIn summary, EP {Bob fails to decode U k 1 given G } exp(kt log q − E 0 (t)) whenever 0 t 1. Recall the universal quadratic bound E 0 (t) I(W )t log q−t 2 log(q) 2 derived in Lemma 17. We obtain that the exponent is\n(redeem the supremum at t = −1/2+α /2 log q)\nSo far we obtain that the average error probability is less than exp(− log( ) 2 /4) = − log( )/4 . Run Markov's inequality with cutoff − log( )/20 . That is, we reject kernels such that P {Bob fails to decode U k 1 given G } − log( )/5 . Then the rejecting probability is − log( )/20 because 1/20 + 1/5 = 1/4. An upper bound on Bob's error probability being P e < − log( )/5 , an upper bound on Bob's equivocation is − log( )/5 1 − log − log( )/5 log q + k = − log( )/5 1 + log( ) 2 /5 log q + k by Inequality (19). Plugging the latter in kh α (here/k), we derive that the left hand side of Inequality (18) is less than kh α − log( )/5 k 1 + log( ) 2 /5 log q + k = k − log( )/5 1 + log( ) 2 /5 k log q + 1 α = −α log( )/5 k 1 + log( ) 2 /5 k log q + 1 α < −α log( )/5 1 + log( ) 2 /5 log q + 1 α < −α log( )/5 · · 2 α = 2 α log( ) − log( )/5 .\nThe first inequality uses that the left hand side increases monotonically in k and k := − j = − H(W ) − 1/2+α < . The second inequality uses the assumption 2. In any regard, the quantity at the end of the inequalities decays to 0 as → ∞, so eventually it becomes less than 1/2+α , the right hand side of Inequality (18). This proves that Inequality (14) holds with failing probability − log( )/20 as soon as is large enough. The lower bound on in the statement of Lemma 14 is large enough, hence the first half of Lemma 14 settled. 8.4. Hayashi's argument at Eve's end. This subsection contains the very last ingredient of the proof of Lemma 14. We dealt with Inequality (14) in the last subsection. We now deal with\n(15's copy)\nSimilar to how we motivated Inequality (18), we apply Jensen's inequality and the chain rule of conditional entropy to simplify Inequality (15). The left hand side becomes jh α (H(U j 1 | Y 1 )/j) where j := H(W ) − 1/2+α for short. (This is not the same j as in the last subsection.) The input uniform, the argument of h α is H(U j 1 | Y 1 )/j = 1 − I(U j 1 ; Y 1 )/j, which can be replaced by I(U j 1 | Y 1 )/j thanks to the evenness h α (1 − z) = h α (z). We will show\nBut what is I(U j 1 ; Y 1 )? It is the amount of information Eve learns from wiretapping Y 1 if they know that U j+1 are junk. In other words, we may pretend that Alice transmits X 1 := U j 1 V j+1 G with confidential bits U j 1 and obfuscating bits V j+1 , Bob receives X 1 in full, and Eve learns Y 1 . This context falls back to (a special case of) the traditional setup of wiretap channels [Wyn75] where various bounds are studied, some in terms of Gallager's E-null function.\nHere are some preliminaries to control the information leaked to Eve. We follow the blueprint of how Hayashi derived the secrecy exponent in [Hay06,Inequality (21)]. Consider the communication protocol depicted in Figure 7: Karl fixes a kernel G ∈ GL( , q) and everyone knows G. Alice chooses the confidential message U 1 . Vincent chooses the obfuscating bits V j+1 . Charlie generates Y 1 by plugging X 1 := U j 1 V j+1 G into a simulator of W . Eve learns Y 1 and is interested in knowing U j 1 alone. So the channel on topic is the composition of Vincent and Charlie. Notation: Running out of symbols, we all use P with proper subscriptions to indicate the corresponding probability measures. That said, indexes in the subscription will be omitted. As Eve is interested in the relation between U j 1 and Y 1 , let Y 1 Gu j 1 be the\nThe channel Eve cares about Figure 7. A finer setup for Hayashi's secrecy exponent. Charlie generates Y 1 such that X 1 := U j 1 V j+1 G and Y 1 follow W . Despite of the seemingly sequential structure, Karl, Alice, and Vincent work independently.\nr.v. that follows the a posteriori distribution of Y 1 given G = G and U j 1 = u j 1 . More formally, P Y Gu (y 1 ) = P Y |GU (y 1 | G, u 1 ) = P GU Y (G, u j 1 , y 1 )/P GU (G, u j 1 ). We could have defined Y 1 G to be the a posteriori distribution of Y 1 given G = G; but it is simply the same distribution as Y 1 since U j 1 V j+1 G traverses all inputs uniformly regardless of the choice of G. That is, P Y |G (y 1 | G) = P Y (y 1 ) for all y 1 ∈ Y .\nFix G as an instance of G. Let I e be the base-e mutual information. The channel Eve cares about leaks information of this amount:\nis the Kullback-Leibler divergence from the a posteriori distribution of Y 1 given G, u j 1 to the coarsest distribution Y 1 . We are to take expectation over G to find the average information leak since we are interested in Markov's inequality. Equality (21) yields\nWe now discover that there are redundancies in traversing all G and u 1 : After all,\nwhich is a fixed linear combination of the first j rows plus a random vector from the span of the bottom − j rows. When V 1 varies, the track of X 1 forms an affine subspace of F q , a coset code as in the context of the fundamental theorems. So what matters is the distribution of this coset code. In this regard, we replace the uniform ensemble of (G, U j 1 ) by the uniform ensemble of K a rank-( − j) affine subspace of F q , where j := H(W ) − 1/2+α . Karl and Alice together choose K uniformly. Vincent chooses X 1 ∈ K uniformly. Charlie generates Y 1 by throwing X 1 into a simulator of W . See Figure 8 for the Karl and Alice choose K\nThe modified channel Figure 8. A simplified setup for Hayashi's secrecy exponent. Charlie generates Y 1 such that X 1 and Y 1 follow W .\ndepiction of the new scheme. Hence Equality (22) becomes\nwhere Y 1 K is the a posteriori distribution of Y 1 given K = K. Suddenly, the quantity EI e (U j 1 ; Y 1 | G) we are interested in turns into the mutual information I e (K ; Y 1 ) between K and Y 1 as K replaces the role of U j 1 in Formula (21). Recall that in Lemma 17 the mutual information is the derivative of Gallager's E-null function. We exploit this. Define the double-stroke E-null function for (K, Y 1 ) as follows\nThen E 0 (0) = I e (K ; Y 1 ) = EI e (U j 1 ; Y 1 | G). Owing to the concavity of the E-null function, E 0 (0) E 0 (t)/t whenever −2/5 t < 0. Recap: To bound the average leaked information EI e (U j 1 ; Y 1 | G) it suffices to bound I e (K ; Y 1 ), which is then morphing to bounding E 0 (0) from above and to bounding E 0 (t) from below.\nThe double-stroke E-null function is bounded as below. Assume −2/5 t < 0. Let s be −t/(1 + t); so 0 < s 2/3 and (1 + s)(1 + t) = 1. For any fixed K and fixed x 1 ∈ K, the base of the (1 + t)-th root in the definition of the double-stroke E-null function is\nHere P XY (K\\x 1 , y 1 ) is a temporary shorthand for the summation of P XY (ξ 1 , y 1 ) over ξ 1 ∈ K that excludes x 1 . Raise it to the power of 1/(1 + t) = (1 + s):\nx 1 ∈K q j P XY (x 1 , y 1 ) q j P XY (x 1 , y 1 ) + q j P XY (K\\x 1 , y 1 ) s x 1 ∈K q j P XY (x 1 , y 1 ) q js P XY (x 1 , y 1 ) s + q js P XY (K\\x 1 , y 1 ) s = q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (x 1 , y 1 ) s + q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1 ) s = q j+s x 1 ∈K P XY (x 1 , y 1 ) 1+s + q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1\nThe inequality uses that the s-th power is sub-additive. Then the inner sum of the E-null function morphs as follows\nx 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1 ) s ."},{"title":"(minor arc)","content":"Divide and conquer-the inner sum of the double-stroke E-null function is split into two arcs as shown. The major arc is exactly\nThe minor arc is loosen to q j+js K P K (K)\nx 1 ∈K P XY (x 1 , y 1 )P XY (K\\x 1 , y 1 ) s = q j+js\nBoth major and minor arcs conquered, merge them and raise to the (1+t)-th power. The summand for any fixed y 1 in the definition of the double-stroke E-null function is\nWe can finally bound the double-stroke E-null function per se:\nAll efforts we spent on bounding I e (U j 1 ; Y 1 ) are for three creeds: First, we see Gallager's bound possessing innate elegance. Second, it fits the paradigm that solving the primary and the dual problems as a whole is easier than solving the primary problem alone. Third, the universal quadratic bound is waiting ahead for the E-null function. We infer that\nRecall the universal quadratic bound E 0 (t) I(W )t log q − t 2 log(q) 2 as stated in Lemma 17 and used in the previous subsection. But this time −2/5 t < 0. We obtain that the exponent is\n= log( )/2 − α log + log 2 + log log q − 2α /4 = log( )/2 − log log + log 2 + log log q − 2 log(log )/ log /4 < log( )/2 + log log q − log( ) 2 /4.\nThe first inequality uses − j = − H(W ) + 1/2+α . The last inequality uses the assumption e 2 . With the last line we conclude that EI e (U j 1 ; Y 1 | G) < exp(log( )/2 + log log q − log( ) 2 /4) = 1/2−log( )/4 log q. Switch back to the base-q mutual information EI(U j 1 ; Y 1 | G) < 1/2−log( )/4 . We now reject kernels such that I(U j 1 ; Y 1 | G) 1/2−log( )/5 . By Markov's inequality, the opposite direction (<) holds with probability 1 − − log( )/20 because 1/5 + 1/20 = 1/4. Plug this upper bound into h α . The left hand side of Inequality (20) is less than\nThe inequality uses that the left hand side increases monotonically in j and j := H(W ) − 1/2+α < . In any regard, the quantity at the end of the inequalities decays to 0 as → ∞, so eventually it becomes less than 1/2+α , the right hand side of Inequality (20 [Ooh02], although no formal proof was found there. See [HM11,BTM17] for alternative descriptions and approaches on the same topic. For readers who took Lemma 12 as granted or went through Appendix B in advance, this is the last sentence of the proof of the main theorem-polar codes' simplicity, random codes' durability."},{"title":"Conclusions","content":"Shannon introduced what we now understand as discrete memoryless channels seventy-two years ago. In the beginning, Shannon had no tool but developed their own theory of typical set, proved the noisy channel coding theory, and justified the notion of capacity. Gallager brought in error exponents. Capacities and error exponents quantify first and second order terms in the asymptotic performance of codes. Only in 2010 we are revealed the complete second order term. It was around the time that polar coding as a graceful instrument to explore the limits at low cost was discovered when Arıkan experimented with the channel transformation and with error exponents. Another ten years it took to grow variants and proof techniques of polar coding. Ultimately, it is feasible, and done by us coincidentally, to piece the puzzle together to show the mere possibility to achieve the second order limits at low cost. An overall comparison is integrated in Table 2. Columns are classes of channels; from left to right: (BEC) binary erasure channels; (BDMC) binary-input discreteoutput memoryless channels; (p-ary) channels of prime input size; (q-ary) channels of prime power input size; (finite) channels of discrete input. Columns to the right are wider than columns to the left. The last column is exceptional; (asym.) is about whether we can achieve the true Shannon capacity, instead of the symmetric capacity. Rows are goals; from top to bottom: (LLN) to achieve (symmetric) capacity; (wLDP) there exists π > 0 such that P e < exp(−N π ); (wCLT) there exists ρ > 0 such that R > I − N −ρ ; (wMDP) there exists π, ρ > 0 such that P e < exp(−N π ) and R > I − N −ρ at once; (LDP) the π in (wLDP) can be arbitrarily close to 1; (CLT) the ρ in (wCLT) can be arbitrarily close to 1/2; (MDP) the (π, ρ)-pair can be arbitrarily close to π + 2ρ = 1. Row (MDP) implies every other row; row (CLT) implies (wCLT); row (LDP) implies (wLDP); and every other row implies row (LLN). Rows (LDP) and (CLT) together almost imply (MDP) (need the partial distance profile). Cells represent how various goals are achieved over various channels. The greenish background means it is possible using Arıkan's kernel [ 1 1 0 1 ]. The purplish background means it is possible using other kernels. The orangish background means it is only possible using dynamic kernels.\nThe following works made critical progresses but our classification fails to include them: Rate-dependent result in LDP paradigm [HMTU13]. Optimal relations among channel parameters [MT14]. First family of (π, ρ) pairs in MDP paradigm [MHU16]. AWGNCs intersecting MDP [FT17].\nWe did our best to excavate the archive but throughout the course of manuscript preparation we found ourselves underestimating early works multiple times so the record kept updating. We sincerely hope to hear about possible references to add to the table.\nPotential improvements include but are not limited to the following: (Tolls) Tighten the explicit Hölder tolls. The current toll between any pair of parameters H, P e , Z, Z mad , T , S, and S max is roughly the sum of tolls collected when traveling through the spanning tree illustrated in Lemma 7. Some improvements potentially tighten the bounds in Lemma 12. (FTPC) Tighten the two fundamental theorems such that they degenerate to equalities over erasure channels. Once done, {Z n } is a supermartingale and Appendix C.1 is obsolete. (Symmetry) generalize the arguments in Sections 8.2 to 8.4 to asymmetric channels. Once done, Section 8.1 is obsolete. Note that the proof of the fundamental theorems applies to asymmetric channels. (Bijection) Early works on polar coding over arbitrary alphabets introduced arbitrary bijections g W . Generalize the two fundamental theorems to include arbitrary bijections. (Dynamic) Achieve the main theorem with a large, but fixed, kernel. This does not immediately make the code practical. But the answer should shed light on our understanding of coding. (Alphabet) Achieve the main theorem without the reduction to prime power alphabets. This is currently not an option because linear codes are barely defined over non-fields. Plus the S-parameter-and thus FTPCS-would just break. (Dispersion) Recall Proposition 2. Weird things happens when the channel dispersion vanishes V = 0. Can we describe those channels better? One example of such channels is this:\nWe look forward to generalizations of the main theorem to non-identical channels (i.e., non-stationary) [Mah17], non-independent channels (i.e., with memory) [WHY + 15, ST16], deletion channels [TPFV19,LT19], channels with restrictions on input distributions (e.g., due to energy constrain) [FT16], wiretap channels [ŞV13], rate-distortion problem [HKU09], Wyner-Ziv problem [HKU09], Slepian-Wolf problem [Abb15], broadcast channels [GAG15,MHSU15], and multiple access channels [AT12,NT16]. We focus on noisy channel coding in this work for its historical significance.\nAppendix A. Explicit Hölder Tolls (Proof of Lemma 8)\nAs is promised in Section 4.1, we prove the explicit Hölder toll. Let W be a q-ary channel. In the upcoming arguments, H, P e , Z, Z mad , S, and S max mean H(W ), P e (W ), Z(W ), Z mad (W ), S(W ), and S max (W ), respectively. Also q means q − 1, and q means q − 2. Furthermore, lg means the base-2 logarithm; this is handy when we jump back and forth between nats, bits, and q-bits.\nFirst we show Z mad q H log 4 q.\n(2's copy)\nStart from Z mad : By the definition Z mad q Z. Move on to Z: By Lemma 3,\nThe left hand side is qZ; in the right hand side √ 1 + q z + √ 1 − z has maximum q/ √ q at z = q /q by calculus. So Z P e /q (q/ √ q ) = q √ P e /q . Move on to P e : By Lemma 6 (the first lower bound), 2P e H lg q or equivalently P e H log 4 q. Now we chain the inequalities Z mad q Z q √ P e q H log 4 q. This completes Inequality (2). That being proven, we use the weaker form Z mad q 3 √ H in the calculus machinery for global MDP. Second we show H » eq Z mad /2.\n(3's copy) Start from H: By Lemma 6 (the upper bound, Fano's inequality), H lg q h 2 (P e )+ P e lg q . By Figure 6, h 2 (P e ) + P e lg q √ eP e + P e lg q = √ P e ( √ e + √ P e lg q ). What is inside parentheses is less than √ e + q /q lg q . Hence H √ P e ( √ e + q /q lg q )/ lg q. Focus on the scalar part-( √ e + q /q lg q )/ lg q has maximum √ e at q = 2 (remember that q 2). So H √ eP e . Move on to P e : By Lemma 3, P e q Z/2. Move on to Z: By definition Z Z mad . Now we chain the inequalities H √ eP e eq Z/2 eq Z mad /2. This completes Inequality (3). That being proven, we use the weaker form H q 3 Z mad in the calculus machinery for global MDP.\nThird we show (notice the logarithm is natural)\n(4's copy)\nStart from S max : By definition S max q S. Move on to S: By Lemma 5, S q q(q /q − P e ) » 1 − q q q q . The square root simplifies to 1/(q ) 2 = 1/q as qq = (q ) 2 − 1. So S q − qP e . Move on to q − qP e : By Lemma 6 (the upper bound, Fano's inequality), H lg q h 2 (P e ) + P e lg q . We claim that h 2 (P e ) + P e lg q lg q − 2(q /q − P e ) 2 / log 2. To prove the claim, Taylor expand both sides at P e = q /q. Verify that both evaluate to lg q at P e = q /q; verify that both have derivative 0 at P e = q /q; and verify that the acceleration of the left hand side, −1/(P e (1 − P e ) log 2), is more negative than the acceleration of the right hand side, −4/ log 2. By Taylor's theorem, mean value theorem, or Euler method, the function with greater acceleration is greater; hence the claim. See also [FM94, Fig. 1]; the Φ-curve seems parabolic at the upper right corner. Now we have H lg q lg q−2(q /q−P e ) 2 / log 2, which is equivalent to 2(q /q−P e ) 2 / log q 1−H and to q − qP e q (1 − H) log(q)/2. Now we chain the inequalities S max q S q (q − qP e ) q q (1 − H) log(q)/2. This completes Inequality (4). That being proven, we use the weaker form\nin the calculus machinery for global MDP. Fourth we show 1 − H q S max / log q.\n(5's copy) Start from 1 − H: By Lemma 6 (the second lower bound), H lg q q q lg(q/q ) × (P e − q /q ) + lg q . The right hand side is lg q − q lg(q/q )(q − qP e ) by matching the (rational) coefficients of P e lg q, P e lg q , lg q, and lg q , respectively. As H lg q lg q − q lg(q/q )(q − qP e ) we bound lg(q/q ) = lg(1 + 1/q ) 1/q by the tangent line at 1/q = 0. So H lg q lg q − (q − qP e ) and hence 1 − H (q − qP e )/ lg q. Move on to q − qP e : By Lemma 5, 1 − qP e /q S so q − qP e q S. Move on to S: By definition S S max . Now we chain the inequalities 1 − H (q − qP e )/ lg q q S/ lg q q S max / lg q. This completes Inequality (5). That being proven, we use the weaker form 1 − H q 3 S max in the calculus machinery for global MDP. This is end of the proof of Lemma 8. The proof of Lemma 7 follows the same logic, only shorter.\nAppendix B. Calculus Machinery for Global MDP (Proof of Lemma 12)\nWe are to prove that\ngiven criteria (cb), (cm), (ct), and (cl), the local LDP behavior, the local CLT behavior, and that π + 2ρ 1 − 8α. The proof is split into several stepping stones. We will prove each of the following inequalities (including two equalities) in each of the upcoming subsections. This will be proven in Appendix B.1: The eigen behavior reads\nThis will be proven in Appendix B.2: As a lemma, {H n } and {Z n } converges to 0 with probability 1 − H 0 , i.e.,\nThis will be proven in Appendix B.3: The en23 behavior reads\nThis will be proven in Appendix C.1: As a lemma, {min( −2 , 4 √ Z n )} is a supermartingale, i.e.,\nThis will be proven in Appendix C.2: As a lemma, the following holds when Z 0 < −8 :\nE [( K 2 n+1 /3 · 3/4) −1/2 | F n ] < −1/2+2α . (27) This will be proven in Appendix C.3: The een13 behavior reads\nThis will be proven in Appendix C.4: The elpin behavior reads for any constants π, ρ > 0 such that π + ρ 1 − 8α,\nThe last inequality is a bi-Hölder toll away from P{H n < exp(− πn n)} > 1 − H 0 − −ρn+o(n) , (7's copy) our destination. This finishes the proof of Lemma 12. The eigen, en23, een13, and elpin behaviors are intermediate checkpoints pinned in a way that moving from one to the next is easy while skipping any of them makes the next unreachable. Their entire purpose is to form a chain that connects the local LDP and CLT behaviors to the global MDP behavior and we do not specify if any of them falls inside the LDP, CLT or MDP paradigm. B.1. The eigen behavior. We want to prove Inequality (23), E [h α (H n+1 ) | F n ] 4 −1/2+3α h α (H n ), given the local LDP behavior and the local CLT behavior. The idea is that, for H n that is close to 1/2, the local CLT behavior provides a measurement of the dichotomy/bifurcation behavior of H n+1 . For H n that are close to 0, the Z -part of the local LDP behavior provides a measurement of the attraction toward 0. For H n that is close to 1, the S-part handles it dually. The formal proof is below.\nInequality (23) is a local statement so we may assume n = 0. To prove that The last case-the reliable case: when H 0 < −2 , we further split it into two subcases per how K 1 compares to k := 1/2+5α/2 . For the small K 1 subcase, the martingale property fits. For the large K 1 subcase, the local LDP behavior fits:\nFor the K 1 k := 1/2+5α/2 subcase, the martingale property\n. And the K 1 k subcase is closed. For the K 1 > k := 1/2+5α/2 subcase, pay the explicit Hölder toll: Z 0 q 3 √ H 0 < q 3 / < 1. Invoke the local LDP behavior:\nexp(qZ 0 )(qZ 0 ) k 2 /3 exp(q 4 )(q 4 √ H 0 ) k 2 /3 = exp(q 4 )(q 8 H 0 ) log( ) 5 /6 . Pay the return-trip toll: H 1 q 3 √ Z 1 q 3 1/2 exp(q 4 /2)(q 8 H 0 ) log( ) 5 /12 . Now we claim and prove that the following quantity is less than 1: (there is nothing to show if h α (H 0 ) = 0) (h α (H 1 )/ −1/2+3α h α (H 0 )) 12/α = H 12 The inequality involving 1.6 uses q 5 . The inequality involving 0.3 uses max(41, 3 q ). The inequality involving 15 uses max(22, q 5 ). We just showed that h α (H 1 )/ −1/2+3α h α (H 0 ) is less than 1, with and hence without the 12/α-th power. Thus E [h α (H 1 ) | K 1 > k] −1/2+3α h α (H 0 ). And the K 1 > k := 1/2+5α/2 subcase is closed. To sum up the reliable case: We bound separately the two terms in Formula (30). They are both at most −1/2+3α h α (H 1 ), hence their sum is at most 2 −1/2+3α h α (H 1 ). Since Inequality (23) wants 4 instead of 2, The reliable case is closed. And the proof of the eigen behavior, Inequality (23), is sound when combining the three cases.\nBibliographic remarks: [FHMV17, Theorem 7] also cut the cases at −2 and 1 − −2 . In contrast, [GRY19, Theorem 5.1] cut at −4 and 1 − −4 − ε. A potential improvement is, when −2 H 0 < −1 , Inequality (15) will simply evaporate. Similarly, Inequality (14) evaporates when 1 − −1 < H 0 1 − −2 . They tighten the right hand side of Inequality (10). The lesson here is that the hard transition between local LDP and CLT behaviors weakens the bounds.\nB.2. Polarization in mean. We want to prove Equality (24), P{Z 0 → 0} = P{H 0 → 0} = 1 − H 0 , given the martingale property and the eigen behavior. The idea is that the eigen behavior expels H n from being close to 1/2, so the only reasonable limits are 0 and 1. The formal proof is below.\nAs B.3. The en23 behavior. We want to prove P{Z n < exp(−n 2/3 )} < 1 − H 0 − (−1/2+4α)n+o(n) , namely Inequality (25), given the eigen behavior and the polarization in mean. The idea is to read off the behavior of {H n } from the behavior of {h α (H n )} in the eigen behavior. The formal proof is below.\nThis simplifies the eigenvalue. Without loss of generality, we rescale h α such that h α (H 0 ) = 1. Let ε n be exp(−n 3/4 ); note that ε n H 0 1 − ε n for n large enough. Owing to h α 's concavity, that h α (0) = h(1) = 0, and that h α (H 0 ) = 1, we deduce that h α (z) ε n whenever ε n z 1 − ε n . Consider these three events as a partition: let A n be {H n < ε n }; let B n be {ε n H n 1 − ε n }; let C n be {1 − ε n < H n }. Note that B n implies h α (H n ) ε n .\nNext we show P(B n ) < (−1/2+4α)n+o(n) : Telescoping leads to E[h α (H n )] h a (H 0 ) (−1/2+4α)n = (−1/2+4α)n . Markov's inequality leads to P{h(H n ) ε n } E[h(H n )]/ε n (−1/2+4α)n /ε n = −(−1/2+4α)n+O(n 3/4 ) < (−1/2+4α)n+o(n) . Therefore P(B n ) P{h(H n ) ε n } < (−1/2+4α)n+o(n) , as desired. Moreover, summing the geometric series leads to m n P(B m ) < (−1/2+4α)n+o(n) .\nNext we show 1 − H 0 − P(A n ) < (−1/2+4α)n+o(n) : The left hand side is at most the probability measure of {H ∞ = 0} \\ A n . That is the probability that H n was not small (not in A n ) but H n+1 , H n+1 , . . . will end up converging to 0. Note that being a martingale causes 1 − H n+1\n(1 − H n ), which forbids H n jumping from C n directly into A n+1 -it must pass by B m for some m n before ever landing in A m+1 . From the summation of P(B m ) over m n we know that very few descendants of H n can do that; the probability measure of {H ∞ = 0} \\ A n is less than (−1/2+4α)n+o(n) . Therefore 1 − H 0 − P(A n ) < (−1/2+4α)n+o(n) and hence P{H n < exp(−n 3/4 )} = P(A n ) > 1 − H 0 − (−1/2+4α)n+o(n) . Pay the bi-Hölder toll P{Z n < exp(−n 2/3 )} > 1 − H 0 − (−1/2+4α)n+o(n) . And the proof of the en23 behavior, Inequality (25), is sound.\nhere is an aesthetic choice; min( −2 , 2+ε √ Z n ) is also a supermartingale but only for astronomic (depending on q). For any non-random kernel, a small enough power works provided that the kernel polarizes channels in the first place.\nC.2. A Cramér-Chernoff gadget. Let D n+1 be K 2 n+1 /3 · 3/4. We want to prove inequalities in (27), that Z n < −8 implies Z n+1 Z Dn+1 n and E [D −1/2 n+1 | F n ] < −1/2+2α , given the local LDP behavior. The motivation is to reformat the local inequalities so that it is easy to telescope for future reference. The formal proof is below.\nThey are both local statements so we may assume n = 0. When Z 0 < −8 , invoke the local LDP behavior Z 1 exp(qZ 0 )(qZ 0 ) K 2\n< 2 −1/2 + 2 −1/2 log = 2 −1/2 + 2 −1/2+α < 4 −1/2+α < −1/2+2α .\nThe last inequality uses e 4 . This validates the second inequality in (27). And the proof of inequalities in (27) is sound.\nC.3. The een13 Behavior. We want to prove P{Z n < exp(−e n 1/3 )} > 1 − H 0 − (−1/2+4α)n+o(n) , namely Inequality (28), given the en23 behavior, the supermartingale property, and the Cramér-Chernoff gadget. The idea is to apply the gadget consecutively to show that Z n becomes smaller and smaller as n increases. To reach the goal exp(−e n 1/3 ), we apply √ n times to avoid losing too much code rate. /a m from above.) Conditioning on A m , we want to estimate the probability that Z k −8 for some k m, which is equal to the probability that min( −2 , 4 √ Z k ) −2 for some k m. Recall that min( −2 , 4 √ Z k ) was made a supermartingale. Hence by Doob's optional stopping theorem [Dur19, Exercise 4.8.2], P{min( −2 , 4 √ Z k ) −2 for some k m | A m } min( −2 , 4 √ Z m ) 2 < exp(−m 2/3 /4) 2 . This is an upper bound on b m /a m and will be summoned in Formula (32).\n(Bound c m /a m from above.) We want to estimate how often does Inequality (31) happen. It is the probability of (D m+1 D m+2 · · · D m+\nThe first three equalities are by the definitions of g m and E m 0 . The next equality is simple algebra. The next two inequalities are by 0 e m /a m 1. The next inequality is by the definition of E m . The last inequality summons upper bounds derived in the last few paragraphs. The last line contains two terms in the big parentheses. Between them (−1/2+3α) happens. More precisely, we attempt to bound Z m+ √ n when E m happens for each m = √ n, 2 √ n, . . . , n − √ n. When E m happens, its superevent A m happens, so we know that Z m < exp(−m 2/3 ). But B m does not happen, so Z k < −8 for all k m. This implies that Z k+1 Z D k+1 k for those k. Telescope; Z m+ √ n is less than Z m raised to the power of D m+1 D m+2 · · · D m+ √ n . But C m does not happen, so the product is greater than 2α eqZ k for all k m + √ n so long as Z k stays below −8 , which it does because B m is excluded. Then telescope again; Z n ( eq) n−m− √ n Z m+ √ n < ( eq) n exp(−m 2/3 2α √ n ) < exp(−e n 1/3 ) provided that n is sufficiently large. In other words, E n− √ n 0 implies Z n < exp(−e n 1/3 ).\n(Summary.) Now we conclude that P{Z n < exp(−e n 1/3 )} P(E n− √ n 0 ) = e n 0 > 1 − H 0 − (−1/2+4α)n+o(n) . And hence the proof of the een13 behavior, Inequality (28), is sound.\nThis subsection is parallel to [WD18, Section V]. Do not confuse this subsection with the next. The subtlety is explained in [WD18, Section III].\nC.4. The elpin behavior. Recall π, ρ > 0 is such that π + 2ρ 1 − 8α. We want to prove P{Z n < exp(− πn n 2 )} > 1 − H 0 − −ρn+o(n) , namely Inequality (29), given the een13 behavior, the supermartingale property, and the Cramér-Chernoff gadget. The idea is to apply the gadget consecutively to show that Z n becomes smaller and smaller as n increases. To reach the goal exp(− πn ), we apply as many times as possible before we run out of depth n. (Bound c m /a m from above.) We want to estimate how often does Inequality (33) happen. It is the probability of (D m+1 D m+2 · · · D n ) −1/2 −πn/2 . By Markov's inequality, this probability is at most E [D −1/2 1 ] n−m πn/2 < (−1/2+2α)(n−m) πn/2 = (1/2−2α)m−(1/2−2α−π/2)n (1/2−2α)m−(ρ+2α)n . The last inequality uses Inequality (6), π + 2ρ 1 − 8α. This is an upper bound on c m /a m and will be summoned in Formula (34).\n(Bound f + m from above.) The definition of f m reads 1 − H 0 − a m 0 . Here a m 0 is the probability measure of A m 0 , and A m 0 is a superevent of A m by how the former is defined. Event A m 0 must contain {Z m < exp(−e m 1/3 )} by how A m was defined. By the een13 behavior, P{Z m < exp(−e m 1/3 )} > 1 − H 0 − (−1/2+4α)m+o(m) . Chaining all inequalities together, we infer that f m < (−1/2+4α)m+o(m) . Let f + m be max(0, f m+ √ n ) so we can write f + m < (−1/2+4α)m+o(m) . This upper bound will be summoned in Formula (34).\n(Bound e n 0 from below.) We start rewriting g m − f + m with (f m− happens. More precisely, we attempt to bound Z n when E m happens for each m = √ n, 2 √ n, . . . , n − √ n. When E m happens, its superevent A m happens, so we know that Z m < exp(−e m 1/3 ). But B m does not happen, so Z k < −8 for all k m. This implies Z k+1 Z D k+1 k for those k. Telescope; Z n is less than Z m raised to the power of D m+1 D m+2 · · · D n . But C m does not happen, so the product is greater than πn . Jointly we have Z n Z πn m < exp(−e m 1/3 πn ) < exp(− πn n 2 ). In other words, E n− √ n 0 implies Z n < exp(− πn n 2 ). (Summary.) Now we conclude that P{Z n < exp(− πn n 2 )} P(E n− √ n 0 ) = e n 0 > 1 − H 0 − −ρn+o(n) . And hence the proof of the elpin behavior, Inequality (29), is sound.\nThis subsection is parallel to [WD18, Section VI]. Do not confuse this subsection with the previous. The subtlety is explained in [WD18, Section III].\nAs we finish proving Inequalities (23) to (29), we finish the proof of Lemma 12. Lemmas 12 to 14 are all finished. This is the last sentence of the proof of the main theorem."},{"title":"Appendix D. Constants Dependence Summary","content":"Given a discrete memoryless channel W . The sender chooses the message alphabet size ς 2. Depending on the factorization of ς, we choose q to be a certain prime power or alternate between q 2 , q 3 , q 5 , . . . (a finite list depending on ς). Fix a q. Given π, ρ > 0 such that π + 2ρ < 1; fix them. Choose ; this also determines α := log(log )/ log . The choice of is such that π + 2ρ 1 − 8α and such that the failing probabilities in Lemmas 13 and 14 do not sum to one. It depends on q, π, ρ. Once is fixed, the complexity is a function in n (or in N = n ). The asymptotic complexity O(N log N ) hides the scalar term that is determined by q and . The decaying gap −ρn+o(n) in Claim 11 and Lemma 12 hides two things: A scalar term in front of determined by q and alongside with a O(n 1−ε ) term determined by the choice of en23 and een13 checkpoints. This ε is fixed throughout the paper and is irrespective of ς, π, ρ, q, ."},{"title":"Figure 1 .","content":""},{"title":"Figure 4","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 6 .","content":""},{"title":"7. 4 .","content":""},{"title":"Lemma 17 (","content":""},{"title":".∪","content":""},{"title":"+∪","content":""},{"title":"∪","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Richard W. Hamming is one of the first few people who had the idea that by grouping information in blocks with redundancies, a calculating machine can correct errors by its own and proceed to the next command instead of halting. Their solution, now called Hamming codes, is found in [Ham50]. Claude E. Shannon, a colleague of Hamming in Bell labs, theorized the communication channels and showed that a channel associates to a number called capacity, which represents the ultimate limit of the efficiency of communications over that channel.\\nTo brief the rest of the history, we follow the analogy [AW14] used. Shannon's eternal result, noisy channel coding theorem [Sha48], is considered the analog of the law of large numbers (LLN). The theorem implies that there exists a sequence of longer and longer block codes whose block error probabilities approach 0 and code rates approach the capacity, which is analogous to that the empirical average of random variables is close to the mean with high probability. Robert G. Gallager, Shannon, Robert M. Fano, and followers extended the LLN result by looking at how the block error probability P e scales when the code rate R is fixed. They showed that the error probability P e scales like exp(−E r (R)N ). Here N is the block length, E r (R) is a constant depending on R. This paradigm is considered the analog of the large deviations principle (LDP). See [Fan61,Gal65,SGB67,Gal68,Gal73,Bla74,BF02,iFLM11,DZF16]. Meanwhile, a series of works fix the error probability P e and looked at how the code rate R scales [Wol57,WEI60,Dob61,Str62,BKB04,Hay09,PPV10]. They showed that the code rate R scales like I − Q −1 (P e ) V /N for I the capacity, Q −1 the inverse of the standard Q-function, and V an intrinsic parameter of the channel. The parameter V is called the dispersion or varentropy by different authors. It is the \\\"variance\\\" of the channel while I is the \\\"mean\\\" of the channel. This turns out to be more than an analog-the random variable log(W (Y | X)/W out (Y )) called information density or information spectrum has mean I and variance V . This paradigm is considered the analog of the central limit They showed that the quantity N (I − R) 2 /|log P e | converges to 2V , twice the very dispersion appearing in the CLT paradigm. This paradigm is considered the analog of the moderate deviations principle (MDP). On a parallel track, the engineering aspects of the communication theory thrive. Codes with excellent practicality are proposed. To name a few, Reed-Muller (1964), trellis modulation (1970s), turbo (1990s), low-density parity-check (1963,1996), Repeat-accumulate (1998), Fountain (1998), and polar (2009).\\nAmong the long list of inventions, only trellis modulation, low-density paritycheck, and polar achieve the LLN paradigm over nontrivial channels-they are capacity-achieving. Among these three, polar stands out as the only code that achieves the CLT paradigm (optimally), the only code that achieves the LDP paradigm (optimally), and the only code that achieves the MDP paradigm (suboptimally). If only polar code achieves the optimal MDP paradigm. We brief the history of polar codes below. Unless stated otherwise, I means the symmetric capacity in the next three paragraphs.\\nErdal Arıkan's original works on channel polarization [Ari08,Ari09] established the foundation of polar codes, placing polar codes in the LLN paradigm on day one. Arıkan and Telatar [AT09] characterized the LDP behavior of polar codes, showing that P e scales like exp(− √ N ) when an R < I is fixed. Later, Korada-Şaşoglu-Urbanke [KSU10] generalized polar codes from Arikan's kernel [ 1 1 0 1 ] to any invertible -by-matrix G, granted that 2 and G is not column-equivalent to a lower triangular matrix. And then they showed that the LDP behavior is P e ≈ exp(−N Ec(G) ) where E c (G) is a constant depending on the kernel matrix G. The notation E c (G) is meant to resemble Gallager's error exponent E r (R) but the former is at this level exp(−N • ) while the latter is at this level exp(−•N ). The LDP behavior of polar codes is then refined in [HMTU13]. Therein, P e is approximated by exp(− E ) where E = E c (G)n − V c (G)nQ −1 (R/I) + o( √ n) is a more accurate exponent, is the matrix dimension, n is the depth of the code, and V c (G) is another constant depending on G. The notation V c (G) is meant to resemble the channel dispersion V . Appearing to be a CLT behavior, this result lies in the corner of the LDP paradigm that touches the MDP paradigm. Finally, Mori-Tanaka [MT14] generalized everything above to channels of prime power input size. Over arbitrary input alphabets, [ŞTA09a,Sas11] Note that results utilizing different kernels over various channels are mixed. The higher ρ, π, the better performance. The curve part of [WD18] is ρ = 1 − h 2 (π). [Ari09,AT09]. Over binary but asymmetric channels, [SRDR12,HY13] showed the counterpart of [Ari09,AT09] with I being the Shannon capacity. No further result on the LDP side, e.g. over non-binary asymmetric channels, is known. The present work fills the gap.\\nThe CLT behavior of polar codes turns out to be difficult to characterize. It was Korada-Montanari-Telatar-Urbanke [KMTU10] who came up with the idea that approximating an eigenfunction tightly bounds the eigenvalue −ρ . Here ρ > 0 is a number such that R scales like I − N −ρ with a fixed P e . They had 0.2669 ρ 0.2841 over binary erasure channels (BECs). The upper bound was brought down to 3.553ρ 1 [GHU12]. Hassini-Alishahi-Urbanke [HAU14] lifted the lower bound to 3.627ρ 1 over BECs and proved a lower bound 6ρ 1 over binaryinput discrete-output memoryless channels (BDMCs). The latter is suboptimal so [GB14,MHU16] improved the bound to 5.702ρ 1 and to 4.714ρ 1. Additive white Gaussian noise channles (AWGNCs) have continuous output alphabet, but [FT17] show that they have 4.714ρ 1 too. Over BECs particularly, [FV14,YFV19] examined a series of larger kernels; the current record is a 64-by-64 kernel believed to have 2.9ρ 1. Near the end of the road to 2ρ < 1, [PU16] showed that by allowing q → ∞, Reed-Solomon kernels achieve 2ρ < 1 over q-ary channels. This does not really prove that polar codes achieve 2ρ < 1 over any specific channel, but gave hopes. Fazeli-Hassani-Mondelli-Vardy [FHMV17,FHMV18], eventually, showed that large random kernels achieve 2ρ < 1 over BECs, breaking the barrier. Guruswami-Riazanov-Ye [GRY19] extended their result to all BDMCs utilizing the dynamic kernel technique. Over the remaining channels, the present work fills the gap.\\nBetween LDP and CLT is polar codes' MDP behavior. Guruswami-Xia [GX13] showed that there exists ρ > 0 such that P e scales like exp(−N 0.49 ) while R scales like I − N −ρ over BDMCs. This raised a question about what are the possible pairs (π, ρ) such that (P e , R) scales like (exp(−N π ), I − N −ρ ). Mondelli-Hassani-Urbanke [MHU16] answered this, partially, in the same paper they bounded ρ. They showed that under a certain curve connecting (0, 1/5.714) and (1/2, 0) all (π, ρ) are achievable over BDMCs. For BECs the upper left corner is (0, 1/4.627). A straightforward generalization to AWGNCs was also given in [FT17]. We in [WD18] improved their result, suggesting that via a combinatorial trick the upper left corner of the curve is (0, ρ) for any ρ that is valid in the CLT regime. The same trick also implicated that over BECs all (π, ρ) such that π + 2ρ < 1 are achievable, which is mainly owing to [FHMV17]'s result that 2ρ < 1 over BECs is achievable. Meanwhile, [BGN + 18] made the first step to investigate the general kernel matrices over general prime-ary channels. They showed that it is possible to achieve ρ > 0 with P e ≈ N −Ω(1) . This is, strictly speaking, \\\"only\\\" a CLT behavior as the desired error probability in the MDP world is exp(−N π ). Later, B lasiok-Guruswami-Sudan [BGS18] were able to show that for all π < E c (G) there exists ρ > 0 such that (π, ρ) is achievable. This makes it a direct generalization of [GX13] to all polarizing kernel matrices G over all prime-ary channels. Over the remaining channels, the present work fills the gap.\\nThe following works, though not counting as predecessors of ours, have impact on us through their insights on the essence of the channel polarization: [Kor09, HKU09, KU10, Ari10, CK10, Mor10, SP11, Sas12b, Sas12a, TV13, Has13, PB13, TV15, Mon16, Nas16,Nas17].\\nI resumes to be the Shannon capacity. Readers are now prepared to be presented the main theorem.\\nTheorem 1 (the main theorem-polar codes' simplicity, random codes' durability). Let W be any discrete memoryless channel. Fix a prime ς 2. Fix constants π, ρ > 0 such that π + 2ρ < 1. There exists a sequence of block codes with encoding and decoding algorithms such that: (cs) the codes accept uniform ς-ary messages. (cn) the block length N approaches infinity; (cp) the block error probability falls below exp(−N π ); (cr) the code rate exceeds I − N −ρ ; and (cc) the encoding and decoding complexity is O(N log N ) per code block.\\nThe proof of the main theorem spans over Sections 2 to 8, lemmas continuing in Appendices A to C. The entry points are Sections 2.1 for (cs), 2.2 for (cn), 3.2 for (cc), 4.2 for (cp), and 6.3 for (cr). The main theorem is optimal in the following manner.\\nProposition 2 (optimality). Fix π, ρ > 0 such that π + 2ρ > 1. Assume V > 0. For the rest of the section, we outline the ideas to prove Theorem 1. The proof is a straightforward remix of polar coding techniques and random coding techniques if it were not for a few hurdles.\\nHurdle of input alphabet size: The majority of the polar coding theory assumes that the input alphabet of the underlying channel is binary, of prime size, or, less likely, of prime power size. But the main theorem aims for arbitrary finite alphabets.\\nFinite alphabets do possess polarization behavior but the speed of polarization has room for improvement [Sas11,Theorem 3.5]. We will overcome this by adding \\\"dummy symbols\\\" into the input alphabet to make it a prime power.\\nHurdle of asymmetric channel: Although asymmetric channels do polarize, the input distributions do not automatically become the uniform distribution. Precomposing a source coding machinery helps generate the desired distribution and has been proposed before [ŞTA09a, Section III.D] [Ari10, Section IV]. On the other hand Honda-Yamanoto [HY13] showed that one polar code can do both source coding and noisy channel coding at once. We borrow their idea.\\nHurdle of kernel selection: Judging and identifying the best-behaved kernel gets harder as we need finer descriptions of the performance of the code. The good result for the BEC case depends heavily on the erasure nature of the channels (that they are ordered by their capacities). Other general results are not strong enough to meet our goal. To overcome, we borrow a technique called dynamic kernels from [YB15]. The idea is to prepare more than one polarizing kernel and apply the proper one on a channel-by-channel basis. This makes a paradigm shift from one kernel fits all channels to every channel deserves a tailor-made kernel. We will, once per channel, apply the random coding theory to show the existence of a proper kernel.\\nHurdle of output alphabet size: Even with the great freedom to choose one kernel for each and every channel, there lies the difficulty that some performance bounds are proven with one fixed channel in mind to favor the big-O notations. Those bounds are prone to depend on the size of the output alphabet, which grows to infinity as the channel transformations take place. Meanwhile, some universal bounds are proven that depend only on the size of the input alphabet, which is invariant under channel transformations. We will borrow a bound derived in [CS07,DCS14].\\n1.1. Organization. Section 2 reviews channels and entropy notations; Section 2.1 explains how to overcome the hurdle of arbitrary input alphabet size. Section 3 reviews the channel transformations; Section 3.1 designs the decoder; Section 3.2 analyzes its complexity; Section 3.3 designs the encoder, overcoming the hurdle of asymmetric channel. Section 4 reviews the channel parameters such as the Bhattacharyya parameter; Section 4.2 shows how to control the block error probability. Section 5 reviews the channel processes; Section 5.1 argues that the global MDP behaviors of H(W n ) and H(V n ) imply the main theorem. The main theorem is thus reduced to the behavior of certain channel processes. Section 6 proves that the global MDP behavior we want holds granted that the local LDP and CLT behaviors hold, effectively boiling the main theorem down to the local behaviors. Section 6.2 introduces the random kernel trick and Section 6.3 introduces the dynamic kernel trick to overcome the hurdle of kernel selection. Section 7 confirms the local LDP behavior. The proof distills properties of the weight distribution of random codes. Sections 7.1 and 7.2 proves the two fundamental theorems of polar coding. Section 8 confirms the local CLT behavior. Contributions from Gallager and Hayashi are utilized. Section 8.2 invokes Chang-Sahai's universal bound, overcoming the hurdle of output alphabet size.\\n1.2. Three families of randomnesses. The randomnesses from the sender's message, the channel, and the randomized rounding constitute the first family. Typeset in Roman font are random variables (U, X, Y, . . . ), probability measures (P, Q, W ), entropies (H, I), and other parameters (P e , Z, T, S . . . ) in this family. The randomness from the channel process, one main technique in the polar coding literature, is the second family. Typeset in sans serif font are stochastic processes (K n , W n , H n , Z n , . . . ), probability measure (P), and expectation (E ) in this family. The randomness from random kernel ensembles, the main technique in the random coding literature, is the third family. Typeset in blackboard bold font are random variables (G, X, K), probability measure (P, with exceptions), expectation (E), and Kullback-Leibler divergence (D, with exceptions) in this family.\"\n },\n {\n \"title\": \"Channel and Entropy Preliminaries\",\n \"content\": \"A discrete memoryless channel is a Markov chain W : X → Y. Here X is a finite set of input alphabet; Y is a finite set of output alphabet; and W is an array of transition probabilities W (y | x) ∈ [0, 1] for all x ∈ X and y ∈ Y. The numbers satisfy y∈Y W (y | x) = 1 for all x ∈ X , which represents the fact that each x must be transitioned to some unique y. When X and Y are clear from the context, we call W a channel. Although the input distribution is not part of the channel data, we write W in (x) to denote the input distribution. When W in (x) is understood from the context, we write W (x, y) to denote the joint distribution W (y | x)W in (x), write W out (y) to denote the output distribution, and write W (x | y) to denote the a posteriori probability W (x, y)/W out (y). (Thus the interpretation of W (• | •) depends on the arguments and the context.) A tuple of inputs (x i , x i+1 , . . . , x j ) is abbreviated as x j i . Same for y j i for tuple of outputs, and for u j i for general variables. We assume memoryless channels, and write W (y 1 | x 1 ) to denote the product measure i=1 W (y i | x i ) for consecutive usages. We write W in (x 1 ), W out (y 1 ), W (x 1 , y 1 ), and W (x 1 | y 1 ) to denote the input, output, joint, and a posteriori probabilities.\\nLet X, Y be two r.v.s (random variables). Let H(X), H(X | Y ), and I(X ; Y ) be the standard entropy, conditional entropy, and mutual information. The base of the logarithm will be assigned later. When X is the input fed into some channel W : X → Y and Y is the corresponding output, we say H(W ) and I(W ) to mean H(X | Y ) and I(X ; Y ). When the distribution of X (the input distribution) is chosen to maximize I(W ), it is called the capacity-achieving input distribution and I(W ) is called the (Shannon) capacity of the channel W : X → Y. Unless stated otherwise, the input distributions will be capacity-achieving.\\n2.1. Reduce input size to prime power. Immediately after we declared what channels are concerned (those with finite input and output alphabets), we show that it suffices to consider input alphabets of prime power size. Let W : X → Y be a channel. Let the input alphabet X be of size s. Let q be any prime power greater than or equal to s. Degrade the channel W as follows: Let symbols in X be ξ 1 , ξ 2 , . . . , ξ s . Let ξ s+1 , ξ s+2 , . . . , ξ q be q − s extra symbols. Let X be X ∪ {ξ s+1 , ξ s+2 , . . . , ξ q }; this is the extended alphabet. Define a dummy channel : X → X by letting (ξ min(i,s) | ξ i ) be 1 for all i = 1, 2, . . . , , q. That is, all extra symbols collapse to ξ s while the old symbols remain. The composition of the two channels\\nforms a degraded channel with prime power input size. By the data processing inequality, the Shannon capacity of the degraded channel W • is no greater than W 's Shannon capacity. Meanwhile, it is clear that the degraded channel W • achieves W 's capacity by the same input distribution, ignoring extra symbols. In other words, I(W • ) = I(W ). This constitutes the input size reduction.\\nHereafter, we assume the size of the input alphabet X is q, where q is a prime power. If the sender wants to send uniform binary messages, let q be a power of 2. If the sender wants to send uniform quaternary messages, let q be a power of 3. In case the sender wishes to send uniform quaternary messages but does not want to split an information bit over two channel symbols, let q be a power of 4. Bonus: should the sender want to send uniform senary messages, choose q 2 a power of 2 and q 3 a power of 3 such that q 2 q 3 is a power of 6; then alternate between q = q 2 and q = q 3 . That is, the sender breaks every senary bit into a binary component and a ternary component, sends the binary component through the q = q 2 code block, and send the ternary component through the q = q 3 code block. For other message alphabets, apply the fundamental theorem of arithmetic.\\nFix a q. Let F q be the finite field of order q (with the addition and multiplication structure). Identifying X with F q , we will use them interchangeably. We say that W is a q-ary channel when the variables X and Y are remotely relevant. It is worth keeping in mind that for inequalities in this work, q = 2 is the most difficult case and q 2 will be used silently.\\nWe clarified (cs), there are (cn), (cc), (cp), and (cr) to go.\\n2.2. On the message alphabet and the block length. The fact that we have some freedom to choose q blurs the meaning of the block length N since, say, a q 2 -bit bears twice as much message as a q-bit does. Notwithstanding, we would like to remind readers that multiplication and division of N by any constant do not alter the semantics of the main theorem. This is because O(N log N ) can absorb any constant; exp(−N π ) and N −ρ too can by fluctuating π and ρ a bit. A more series aftereffect is caused by mixing code blocks with distinct q. When the sender attempts to send uniform 30-ary messages, they choose q 2 , q 3 , q 5 s and switch among the three block codes. The q = q 2 blocks have their own block length N 2 just like the other blocks have N 3 or N 5 as block lengths. The de facto block length N , the minimal number of the channel usages before the receiver can decode everything sent so far, is thus three times the least common multiple of N 2 , N 3 , and N 5 . We claim without a proof (but it will be clear once we prove the rest of the main theorem) that it is possible to make N 2 = N 3 = N 5 and consequently N = 3N 2 . Again, increasing N by three-fold does not make any difference. For numbers with more prime factors, a similar reasoning applies.\\nWe recommend readers not to worry about the message alphabet as there exists a powerful solution-to pre-compose another code that re-encodes an arbitrary finite message distribution (not necessarily uniform) to a uniform prime power-ary input distribution. The existence of such code, by duality, is tightly bonded to the existence of a error-correction code that carries uniform prime power-ary messages over channels of arbitrary arity. The latter is exactly what the main theorem concerns.\\nWe clarified (cs) and (cn) in this section; there are (cc), (cp), and (cr) to go. We continue proving the main theorem in the next section.\"\n },\n {\n \"title\": \"Channel Transformation\",\n \"content\": \"Let 2 be an integer. This will be the dimension of the kernel matrices. But for now, let us introduce a flexible framework. Fix a q-ary channel W : X → Y. Let U 1 , U 2 , . . . , U be r.v.s taking values in X . For 1 i j , let U j i be the joint r.v. U i U i+1 · · · U j . Let g W : X → X be a bijective map; that is, H(U 1 | g W (U 1 )) = 0. We now feed X 1 := g W (U 1 ) into i.i.d. (independent and identically distributed) copies of the channel W . Let Y 1 ∈ Y be the corresponding output. The chain rule of conditional entropy reads\\n(1)\\nInterpretation: to estimate U 1 given Y 1 , we first estimate U 1 given Y 1 ; and then use the estimateÛ 1 to further estimate U 2 ; afterward, we estimate U 3 givenÛ 1 , U 2 , and Y 1 ; and so on. To achieve W 's capacity, g W (U 1 ) must follow a certain capacity-achieving distribution. Since g W is bijective, this induces a distribution of U 1 . (Remark: we imply nothing about whether U 1 , U 2 , . . . , U are i.i.d or not.) Fix this distribution, then\\nThese two chain rules motivate the channel transformation:\"\n },\n {\n \"title\": \"Its input distribution W\",\n \"content\": \"(i) in (u i ) is determined by that of U i . It may sound weird that W (i) will tell the receiver the input of W (1) , W (2) , . . . , W (i−1) for free. But in reality, W (i) acts as an interactive device where the receiver (not the sender) needs to input what U i−1 1 is and the device will output something that looks like U i−1 1 Y 1 ; only when the receiver inputs the correct U i−1 1 does the device return the correct U i−1 1 Y 1 . Under this interpretation, the de facto capability of W (i) is thus\\nY 1 ), which justifies the chain rule of the mutual information. To avoid confusion, we prefer H(W (i) ) over I(W (i) ) in calculations.\\nWhat makes the idea of channel transformation powerful is that the transformations apply recursively. The precise formulation is as below: Fix any i ∈ [ ]. Let (X (i) ) 1 , (X (i) ) 2 , . . . , (X (i) ) ∈ X be i.i.d. copies of the capacity-achieving input of W (i) ; let (Y (i) ) 1 , (Y (i) ) 2 , . . . , (Y (i) ) ∈ X i−1 ×Y be the corresponding outputs. Let g W (i) : X → X be a bijection. Define a tuple of r.v.s (U (i) ) 1 := (g W (i) ) −1 ((X (i) ) 1 ); that is to say, g W (i) ((U (i) ) 1 ) = (X (i) ) 1 . For each j ∈ [ ], we define a depth-2 channel\\nTo sum up, we can define (W (i) ) (1) , (W (i) ) (2) , . . . , (W (i) ) ( ) out of W (i) for any i ∈ [ ] in the same way we define W (1) , W (2) , . . . , W ( ) out of W . For i, j ∈ [ ], each (W (i) ) (j) is again a channel, so the transformations apply to generate depth-3 channels. In the setup of the classical polar coding, a fixed bijection g is used to define W (i) , (W (i) ) (j) , ((W (i) ) (j) ) (k) , et seq. To reach the optimal MDP paradigm, we\\nStep 4) output g W (û 1 ).\\n(Step 2-a) output (W (2) (u 2 |û 1 y 1 ) : u 2 ∈ X ). (Step 2-b) input:û 2 .\\n(Step 3-a) output (W (3) (u 3 |û 2 1 y 1 ) : u 3 ∈ X ). (Step 3-b) inputû 3 . Figure 2. A DU with = 3 and its I/Os. allow g W to depend on the channel W . That is to say, we need (presumably distinct) bijections g W (i) : X → X for every i ∈ [ ] when we want to define (W (i) ) (j) out of W (i) . Similarly, we need yet another 2 bijections g (W (i) ) (j) : X → X for every i, j ∈ [ ] in defining depth-3 channels. And the recursion goes on ad infinitum. 3.1. Design of the decoder. To implement channel transformations, we define a DU (decoding unit) to be an automata as follows: It is a box with pins on the left and pins on the right. Each pin is connected to another DU, a CH, an FH, or an IH (to be defined later). Each pin may take inputs or output but not at the same moment. A DU works as follows: Let W : X → Y be the channel it is to transform. (Step 0) For all i ∈ [ ], the i-th pin on the left takes the input y i . The input is passed in the form of the a posteriori distribution (W (x i | y i ) : x i ∈ X ). This is what Arıkan calls α-representation [Ari15b, Section II.A]. (Step 1-a) It computes the a posteriori distribution of U 1 given y 1 ; that is, (W (1) (u 1 | y 1 ) : u 1 ∈ X ). And then it outputs this tuple of probabilities to the first pin on the right. (Step 1b) At a later moment, it will receive an estimateû 1 of U 1 from the first pin on the right. Note thatû 1 is a hard symbol in X , not a soft tuple of probabilities. (Step 2-a) It computes the a posteriori distribution of U 2 givenû 1 y 1 ; that is to say, it pretends that U 1 happens to beû 1 and computes (W (2) (u 2 |û 1 y 1 ) : u 2 ∈ X ) accordingly. And then it outputs this tuple of probabilities to the second pin on the right. (Step 2-b) At a later moment, it will receive an estimateû 2 of U 2 from the second pin on the right. (Step i-a) In general, it computes W (i) (u i |û i−1 1 y 1 ) for all u i ∈ X and then output the tuple to the i-th pin on the right. (Step i-b) After a while, it will receiveû i . (Step + 1) Once it receivesû from the last pin on the right, it computesŷ 1 := g W (û 1 ), and then outputŷ i to the i-th pin on the left for all i ∈ [ ]. See Figures 2 to 4 for illustrations.\"\n },\n {\n \"title\": \"DU\",\n \"content\": \"The general rule to arrange the DUs is as follows: For a depth-n construction, put DUs in an n−1 -by-n array. Each DU is indexed by (k 1 , k 2 , . . . , k n−1 ; m) where k 1 , k 2 , . . . , k n−1 ∈ [ ] and m ∈ [n]. For all m ∈ [n − 1] and all k 1 , k 2 , . . . , k n ∈ [ ], connect the k m -th pin on the right of the (k 1 , . . . , k m−1 , k m+1 , . . . , k n ; m)-th DU to the k m+1 -th pin on the left of the (k 1 , . . . , k m , k m+2 , . . . , k n ; m + 1)-th DU. Here, the (k 1 , k 2 , . . . , k n−1 ; m)-th DU is to transform the channel (· · · (W (k1) ) · · · ) (km) (1,1) W\\n(1,1) Figure 3. 6 DUs are chained together to implement (W (1) ) (1) , . . . , (W (3) ) (3) . Boxes marked \\\"W \\\" are channels. Boxes next to channels are CHs; the labels are their indexes. Boxes at the rightmost column are either FHs or IHs; the labels are their indexes. Note that DUs in the first column use the same g W . DUs in the second column use g W (1) , g W (2) , and g W (3) , respectively.\\ninto (· · · (W (k1) ) · · · ) (km+1) . The k 1 -th pin on the left of the (k 2 , . . . , k n ; 1)-th DU connects to a CH (channel helper) indexed by (k 1 , k 2 , . . . , k n ). Each CH then connects to the output of a copy of the channel W . The k n -th pin on the right of the (k 1 , . . . , k n−1 ; n)-th DU connects to either an FH (frozen bit helper) or an IH (information bit helper); in either case, the connected helper is indexed by (k 1 , k 2 , . . . , k n ). Let I ⊂ [ ] n be the set of indexes (k 1 , k 2 , . . . , k n ) such that the k n -th pin on the right of the (k 1 , . . . , k n−1 ; n)-th DU connects to an IH. Then [ ] n \\\\ I is the set of indexes where the pin connects to an FH. On the left hand side of the DU array, the task of the (k 1 , k 2 , . . . , k n )-th CH is to receive the channel output Y (k1,k2,...,kn) ∈ Y and then forward the a posteriori distribution (W (x (k1,k2,...,kn) | Y (k1,k2,...,kn) ) : x (k1,k2,...,kn) ∈ X ) to the DU array. On the right hand side, FHs correspond to what Arıkan called frozen bits-bits that do not carry information and the receiver knows their values as part of the communication protocol. The task of the (k 1 , k 2 , . . . , k n )-th FH is to receive the a posteriori distribution of the (k 1 , k 2 , . . . , k n )-th frozen bit and then return the correct symbol U (k1,k2,...,kn) ∈ X back to the DU array. IHs correspond to information bits that carry the sender's messages. The task of the (k 1 , k 2 , . . . , k n )-th IH is to receive the a posteriori distribution of the (k 1 , k 2 , . . . , k n )-th information bit and then return the most probable symbolÛ (k1,k2,...,kn) ∈ X back to the DU array. When all IHs are activated once, a code block completes. The most probable symbols they returned to the DU array form the decoded messageÛ I , meaning the tuple (Û (k1,k2,...,kn) : (k 1 , k 2 , . . . , k n ) ∈ I).\\nWhat we just established is the successive cancellation decoder of polar codes that could be found in most works that implement polar codes. For instance, [Ari09, Section VIII], [Kor09, Section 3.2], and [HY13, Section III], and [EKMF + 17, Section Vi.B]. See especially [GRY19, Section 9] for an almost identical construction albeit they had q = 2 in mind. We replicate the whole story to demonstrate that each DU may use a unique bijection \\\"g\\\" without changing the overall structure too much. Whether or not this construction can transmit information reliably is . 12 DUs (with = 2) are chained together to implement ((W (1) ) (1) ) (1) , . . . , ((W (2) ) (2) ) (2) . DUs in the first column use g W ; DUs in the second column use g W (1) and g W (2) ; DUs in the third column use g (W (1) ) (1) , g (W (1) ) (2) , g (W (2) ) (1) , and g (W (2) ) (2) . discussed in Section 4.2. There, we will also clarify how to arrange FHs and IHs. The complexity can be estimated prior to further specification.\\n3.2. Complexity of the decoder. There are various models that measure the complexity of a structure. The polar coding community uses a variant of the circuit complexity where the arithmetic of real numbers costs O(1) and passing probabilities between DUs costs O(1). The complexity of the DU array is thus the number of the DUs multiplied by the complexity of a single DU. The number of DUs is n−1 n. The complexity of a DU depends on how a DU computes the a posteriori probabilities W (j) (u i | u i−1 1 y 1 ) out of W (x i | y i ). The naïve approach is to exhaust all possible inputs u 1 ∈ X and compute the a posteriori probabilities using Bayesian formulas. This costs O( 10 q +10 ) (here 10 is an overestimate). Hence the overall complexity is O( n−1 n 10 q +10 ). In our setup, however, q is fixed, will be chosen upon knowing π, ρ, and n goes to infinity afterwards. So we advertise that the complexity is O( n n), or O(N log N ). Here N := n is the block length, equal to the number of copies of the channel W attached to the DU array. The complexities of the CHs, FHs, and IHs can be computed similarly. They are all bounded by O( n+10 q 10 ). Thus the decoder as a whole costs O(N log N ).\\nWe claim that the encoder has the same complexity O(N log N ) although we have not defined the encoder yet. The encoder is essentially a special decoder and is the subject of the next subsection.\\n3.3. Design of the encoder. The encoder will be an exact copy of the decoder except that CHs and IHs will behave differently. In greater detail: Let there be an n−1 -by-n array of DUs indexed and connected in the same way described in Section 3.1. Each DU executes the exact same task described in Section 3.1. The left pins of the DUs in the first column each connect to a CH. The right pins of the DUs in the last column each connect to the same type of device (an IH or an FH) as its twin-DU in the decoder does. Here, as part of the encoder, a CH will output the capacity-achieving input distribution (W in (x) for all x ∈ X ) into the DU array. For each (k 1 , k 2 , . . . , k n ) ∈ I, the (k 1 , k 2 , . . . , k n )-th IH will receive a recommended distribution of the (k 1 , k 2 , . . . , k n )-th information bit and then return the message symbol U (k1,k2,...,kn) ∈ X the sender wants to send back to the DU array. For each (k 1 , k 2 , . . . , k n ) ∈ [ ] n \\\\ I, the (k 1 , k 2 , . . . , k n )-th FH will receive a recommended distribution of the (k 1 , k 2 , . . . , k n )-th frozen bit and then return a r.v. U (k1,k2,...,kn) ∈ X that follows that distribution back to the DU array. This r.v. is simulated by a pseudo random number generator shared between the encoder and the decoder. The twin-FH in the decoder, regardless what distribution it receives, will return the exact same symbol U (k1,k2,...,kn) back to the DU array. This step is called randomized rounding and is found in [Kor09, Section 3.3], [KU10, Section III], [KT10, Section II], and [HY13, Section III.A]. After all IHs return the sender's messages and all FHs returns randomly rounded bits to the DU array, the CHs will each get a codeword symbol X (k1,k2,...,kn) ∈ X from the DU array. And then each CH will forward that symbol to an i.i.d. copy of the channel W .\\nThis design is a copy of [HY13]'s encoder explained in our terminology. It is clear that the encoding complexity will be O(N log N ), too. Alongside the decoder, the encoder creates its own channel transformations. Let W : X → Y be a q-ary channel and X be a capacity-achieving input. Define a flattening channel W : X → {η} that erases all information. Then the encoder is effectively synthesizing depth-1 channels\"\n },\n {\n \"title\": \"Moreover, H(W ) = H(X) and H(W\",\n \"content\": \"No \\\"Y \\\" plays any role here since they are constant. The fact that a channel as boring as W is helpful to our main theorem will be covered later, in Section 4.2.\\nWe clarified (cs), (cn), and (cc) up to this section; there are (cp) and (cr) to go.\"\n },\n {\n \"title\": \"Channel Parameters\",\n \"content\": \"Let W : X → Y be a q-ary channel. Let X be a capacity-achieving input and Y be the corresponding output. Besides H and I, there are several channel parameters that capture the qualities of channels. Here is a list of parameters extracted from the work [MT14] of Mori and Tanaka.\\nBoth H(X | Y ) and H(W ) are the base-q conditional entropy, the base chosen such that 0\\nis the error probability of the maximum a posteriori (MAP) decoder. The MAP decoder looks at an output y ∈ Y and chooses a symbolx ∈ X that maximizes W (x | y). When the output is Y = y, the probability that the MAP decoder does not choose X asx is 1 − max x∈X W (x | y). Therefore, P e (X | Y ) = y∈Y W out (y)(1 − max x∈X W (x | y)). In a channel-centric narrative, we also write P e (W ) for P e (X | Y ).\\nZ(X | Y ) is the rescaled sum of Bhattacharyya coefficients of the transition distribution W (y | x) for the uniform input. For non-uniform inputs, a modification is made to generalize the definition and the properties that used to hold. Intuitively speaking, a MAP decoder seeing y is \\\"confident\\\" if W (x | y) is small for all but one\\nx, or equivalently, if the product W (x, y)W (x , y) is small for all distinct x, x ∈ X . The Bhattacharyya parameter measures the \\\"confidence\\\" by\\nIn addition, define\\nWe also write Z(W ), and Z mad (W ) for these quantities. Remarks: The rescaling T (X | Y ) is the weighted average of the total variation distances from the a posteriori distributions (W (x | y) : x ∈ X ) to the uniform noise (1/q, 1/q, . . . , 1/q). More formally, it is defined to be y∈Y W out (y) x∈X |W (x | y) − 1/q|. We also write T (W ) for this quantity.\\nS(X | Y ) is the weighted average of the L 1 -norms of the Fourier coefficients of the a posteriori distributions. The formal definition is as follows. Let tr : F q → F p be the field trace, where F q = X and F p is the prime subfield. Let χ : F q → C be an additive character defined as χ(x) := exp(2πi tr(x)/p), where 2πi is temporarily the period of exp. Define the Fourier coefficient\\nDefine the S-parameters\\nWe also write S(W ) and S max (W ) for these quantities. Remarks: The rescaling is such that 0 S S max (q − 1)S q − 1. An interpretation is as follows: Fix a y. When W (x | y) is roughly equal to 1/q for all x ∈ F q , the Fourier coefficient M (w | y) = z∈Fq W (z | y)χ(wz) should be roughly z∈Fq χ(wz)/q = 0. The S-parameter measures how far those coefficients are from zero.\"\n },\n {\n \"title\": \"4.1.\",\n \"content\": \"Relations among channel parameters. The following is a series of lemmas we extract from existing works. They characterize the relations among H, I, P e , Z, T , and S.\\nLemma 4. [MT14, Lemma 23 with k = q − 1] For any q-ary channel W ,\\nLemma 6. [FM94, Theorem 1] For any q-ary channel W ,\\nand\\nHere, h 2 is the binary entropy function; h 2 (1/2) = 1. The upper bound is Fano's inequality. The first lower bound fits when H(W ) and P e (W ) are small; the second lower bound fits when H(W ) and P e (W ) are close to 1.\\nThe above lemmas inspire the following characterization: Let A and B be two channel parameters, we say A, B are bi-Hölder at . This notion generalizes to tuples of more parameters. Now we can summarize Lemmas 3 to 6 in a more concise statement.\\nLemma 7 (implicit bi-Hölder tolls). Parameters H, P e , Z, Z mad are bi-Hölder at (0, 0, 0, 0). Parameters H, P e , T, S, S max are bi-Hölder at (1, 1 − 1/q, 0, 0, 0).\\nProof. Z, Z mad are bi-Hölder at (0, 0) since Z Z mad (q − 1)Z. Lemma 3 implies that P e , Z are bi-Hölder at (0, 0). Lemma 6 (with the first lower bound) implies that P e , H are bi-Hölder at (0, 0). Now apply the transitivity to conclude the first statement. For the second statement, S, S max are bi-Hölder at (0, 0) since S S max (q − 1)S. Lemma 5 implies that P e , S are bi-Hölder at (1 − 1/q, 0). Lemma 4 implies that P e , T are bi-Hölder at (1 − 1/q, 0). Lemma 6 (with the second lower bound) implies that P e , H are bi-Hölder at (1 − 1/q, 1). Now apply the transitivity to conclude.\\nSee also [MT14,Corollary 28] for what inspired us. They use notation A e ∼ B to mean A, B are bi-Hölder at (0, 0) and at (1, 1). For some very technical details on the way toward the main theorem, we need explicit Hölder relations among H, Z mad , and S max . We claim them here. The proof is nothing but looking closer into Lemmas 3, 5, and 6. A written-out proof is in Appendix A.\\nLemma 8 (explicit Hölder tolls). log is natural. For all q-ary channels W , the following hold:\\n4.2. Control of the block error probability. Let W be the channel we want to communicate over; and let X be any input. In the classical theory of polar coding, the second last step of the construction of the block code is to determine a subset I ⊂ [ ] n of indexes that points to the depth-n channels that transmit information bits. When decoding this code, a block error happens if the successive cancellation decoder fails to decode any information bit. Let E (k1,k2,...,kn) be the event that the first error occurs when the decoder is solving for the input to (· · · (W (k1) ) · · · ) (kn) , i.e., whenÛ (k1,k2,...,kn) = U (k1,k2,...,kn) and the equality holds for lexicographically earlier indexes. Then the event's probability measure P (E (k1,k2,...,kn) ) is no more than the bit error probability P e (· · · (W (k1) ) · · · ) (kn) . By the union bound, the block error probability of the decoder is bounded from above by a sum\\nWith this observation, we may define I to be the set of indexes (k 1 , k 2 , . . . , k n ) ∈ [ ] n such that H (· · · (W (k1) ) · · · ) (kn) < θ n for some clever choice of the threshold θ n > 0. This immediately implies P e (· · · (W (k1) ) · · · ) (kn) < cθ d n for some c, d > 0 by Lemma 7. Let θ n be exp(− πn n). The sum of P e is less than n cθ d n < exp(− πn ) for sufficiently large n, which is the block error probability we claimed. Remark: Arıkan used a different criterion Z < θ n . It still implies P e < cθ d n and that the sum of P e is less than n cθ d n < exp( πn ) for large n. The benefit of controlling P e using other parameters is that some parameters are easier to control (because Theorems 9 and 10 exist).\\nFor the main theorem where the channel W is asymmetric, we want to control both the decoder block error and the encoder block error. Here, the encoder block error is not the encoder's failure to encode a message, but rather its failure to generate the capacity-achieving input distribution of W . To penalize, imagine that we employ an oracle that claims an encoder block error whenever the generated codeword should have been another word to fit the ideal distribution. That way, the actual block error probability will not exceed the sum of the encoder and decoder block error probabilities. More rigorously, let P be the probability measure assuming the ideal distribution of U I and Q be the probability measure assuming the actual U I generated by the encoder. Then the overall block error probability can be bounded by\\nP {Û I = U I } as the decoder block error probability is bounded before. The encoder block error probability is represented by P − Q , the total variation distance from P to Q. There is a telescoping argument similar to how we control the decoder error-classifying events by the first input bit where the oracle disagrees with the encoder [Kor09, Lemma 3.5] [KU10, Lemma 4] [KT10, Lemma 2] [HY13, Lemma 1]. It yields that the encoder block error probability is bounded from above by the sum\\nIn controlling the encoder bit error probability, we strengthen the policy of collecting indexes (k 1 , k 2 , . . . , k n ) ∈ [ ] n for I by asking for H (· · · (W (k1) ) · · · ) (kn) > 1 − θ n .\\nThe latter immediately implies T (· · · (W (k1) ) · · · ) (kn) < cθ d n by Lemma 7. As a consequence, the overall block error probability is controlled by Q{Û I = U I } P {Û I = U I } + P − Q < 2 n cθ d n < exp(− πn ) for n large. The preceding argument is a paraphrase of the proof of [HY13, Theorem 13]; Inequalities (59) and (57) therein are the keys. So far the block length, the complexity, and the error aspects of the main theorem are covered, it remains to control the code rate |I|/ n . In other words, we are to compute the cardinality of I given that I ⊂ [ ] n is the set of indexes such that H (· · · (W (k1) ) · · · ) (kn) < θ n and 1 − H (· · · (W (k1) ) · · · ) (kn) < θ n , where θ n := exp(− πn n).\"\n },\n {\n \"title\": \"4.3.\",\n \"content\": \"Before and after channel transformations. Alongside the relations among different parameters applied to the same channel, there are also relations between the same parameter applied to the original and the transformed channels. That\\nThere are two more that are pivotal in the theory of polar coding but require more prerequisites. Assume that g W : X → X is a linear isomorphism given by the multiplication of an invertible matrix G from the right-g W (u 1 ) := u 1 G. The following framework extends to nonlinear bijections but we do not need that much. (There is also the paradigm that random linear codes perform better than random codebooks for that a bad linear code tends to hoard a lot of short codewords at once, effectively removing them from the ensemble pool. So there is a good reason to stick to the linear case.) Let 0 i−1 1 1 i u i+1 ∈ F q be a tuple of i − 1 many 0 followed by a 1 and − i arbitrary symbols. A coset code is a subset of codewords of the form\\nThe coset codes have weight distributions just like every other code does. Let wt(x 1 ) be the hamming weight of x 1 . The weight enumerator of the i-th coset code is defined to be a one-variable polynomial over the integers\\nWe can now state the second relation. This is considered the main cause of why polar coding ever exists/works.\\n. The proof is postponed until Section 7.1. The fundamental theorems come as a pair. Let u i−1 1 1 i 0 i+1 ∈ F q be a tuple of i − 1 arbitrary symbols followed by a 1 and − i many 0. Let G − be the inverse transpose of G. The weight enumerator of the i-th dual coset code is defined to be this one-variable polynomial over the integers\\nWe can now state the third relation, the dual of the second. The proof is postponed until Section 7.2.\\nRemark: These two bounds are not tight-the equality does not hold for BECs. In detail, Arıkan's original bound reads Z mad (W (1) ) 2Z mad (W )−Z mad (W ) 2 while our bound turns into Z mad (W (1) ) 2Z mad (W ), the subtraction term missing. We are simply not able to prove a version that degenerates to an equality over erasure channels, nor does any prior work seem to. This causes a serious aftermath that Z mad (W n ) (to be defined later) is no longer a supermartingale. Nonetheless, this bound is strong enough to collaborate with the random coding theory. See, for example, how we compensate in Appendix C.1.\\nWe clarified (cs), (cn), (cc) and (cp) up to this section; there is (cr) to go.\"\n },\n {\n \"title\": \"Channel Processes\",\n \"content\": \"These r.v.s provide a new family of randomness that does not appear in the encoding and decoding algorithms, but they help us understand the code rate |I|/ n in this manner: Counting how many indexes are in I is nothing more than measuring the probability P{(K 1 , K 2 , . . . , K n ) ∈ I}. With the processes W n and V n thus defined, it is further equivalent to measuring the probability P{H(W n ) < θ n and 1 − H(V n ) < θ n }, where θ n := exp(− πn n). Moreover, it suffices to know how H(W n ) and H(V n ) behave as stochastic processes taking values in [0, 1] without comprehending W n and V n themselves. The general fact is that H(W n ) is either very small (channel is reliable) or very close to 1 (channel is noisy). Arıkan called this phenomenon channel polarization. The following claim generalizes channel polarization and implies the main theorem.\\nClaim 11. Fix any π, ρ > 0 such that π + 2ρ < 1. We will choose an and a series of bijections of F q -namely,\\nHere, o(n) is the little-o function in n; it is such that o(n)/n → 0 as n → ∞.\\nFor polar codes over symmetric channels, the first inequality in Claim 11 alone implies that the code rate is 1 − H(W ) − −ρn+o(n) = I(W ) − N −ρ+o(1) . The first two inequalities imply the polarization behavior that channels become either satisfactorily reliable (low H(W n )) or desperately noisy (high H(W n )). For asymmetric channels, however, we need to characterize H(V n ) alongside H(W n ). The last two inequalities in Claim 11 show that the same series of bijections polarize W at the same time they polarize W . While W contains no randomness form the channel W , what is polarized is that each input bit U (k1,k2,...,kn) either depends heavily on lexicographically earlier input bits (low H(V n )) or behaves like a free r.v. conditioned on earlier bits (high H(V n )). We then categorize the fate of indexes in [ ] n into the following three types. (A) Free and reliable: These are indexes that will be in I; they point to channels that transmit information bits. (B) Free but noisy: The sender can feed information into these channels only to find that the decoder will almost always make some mistakes. The sender should, instead, feed some pseudo random numbers shared with the receiver. (C) Dependent and reliable. The input of these channels depends on previous inputs. Their main purpose is to shape the capacity-achieving input distribution. (D) Dependent but noisy is not possible because H(V n ) H(W n ). This is the key to [HY13, Theorem 1]. We reproduce their proof in the next subsection. 5.1. Claim 11 implies the main theorem. As mentioned, H(V n ) H(W n ) so (D) dependent but noisy is not possible. Let A n be the intersection event of\\nfollows the third inequality in Claim 11. Note that A n or B n implies free but not \\\"neither reliable nor noisy\\\"; that is, (free ∧ reliable) ∨ (free ∧ noisy) → free ∧ ¬(¬reliable ∨ ¬noisy). We deduce that P\\nSimilarly, since A n or C n implies reliable but not \\\"neither free nor dependent,\\\" we deduce that P\\nIn summary, we derive that\\nFinally, recall that I collects free and reliable indexes, so the code rate is |I|/ n = P(A n ) > I(W ) − −ρn+o(n) . We almost finish the proof of the main theorem except that we claimed I(W ) − N −ρ = I(W ) − −ρn , without the little-o term. It can be fixed by finding a slightly larger > ρ such that π + 2 < 1 still holds, and then rerunning the whole argument again with the new . The conclusion becomes that the code rate is at least I(W ) − − n+o(n) . Since − n + o(n) < −ρn for sufficiently large n, this completes the proof of the main theorem. It remains to show that Claim 11 can be achieved.\"\n },\n {\n \"title\": \"Global MDP Behavior Modulo Local Behaviors\",\n \"content\": \"In this section, we put constraints on an abstract process {H n } and show that they imply inequalities of the form P{H n < threshold} > limit measure − decaying gap as those in Claim 11. Let F n be the sigma-algebra generated by K 1 , K 2 , . . . , K n for each n. Then F 0 ⊂ F 1 ⊂ F 2 ⊂ · · · form a filtration of sigma-algebras. Let {H n }, {Z n }, and {S n } be three stochastic processes adapted to {F n } (meaning K 1 , K 2 , . . . , K n determine H n , Z n , S n ). The following assumptions are easy to verify when we reveal what those processes are: (cb) 0 H n , Z n , S n and H n 1;\\nFurthermore, assume large kernels: (cl) max(e 4 , q 5 , 3 q ). Let α := log(log )/ log be a small number shrinking as increases. Define the potential function h α : [0, 1] → [0, 1] to be h α (z) := min(z, 1 − z) α . (Remark: h 2 is not a special case of h α for α = 2; we expect α 1 in practice.) Here are the difficult but sufficient criteria for the main theorem.\\nLemma 12. (calculus machinery for global MDP) Assume criteria (cb), (cm), (ct), and (cl). Assume the local LDP behavior: Z n+1 exp(qZ n )(qZ n ) K 2 n+1 /3 and S n+1 exp(qS n )(qS n ) ( +1−Kn+1) 2 /3 . Assume the local CLT behavior:\\nThen, for any constants π, ρ > 0 such that\\nthe following holds:\\nWe defer the proof until Appendix B. The term K 2 n+1 /3 in the lemma is to control the local LDP behavior of the process {H n }-the behavior of H n+1 when H n is close to 0 and the behavior that is closely related to the LDP behavior of polar codes. The term is chosen in a way such that h 2 ((k 2 /3 )/ ) < k/ and such that k (k 2 /3 ) t is easy to handle. In [FHMV17, Theorem 7], a similar criterion is stated and is annotated as faster polarization at the tails. In [BGS18, Definition 2.4], a similar criterion is stated and is annotated as strong suction at the low end. The eigenfunction h α in the lemma is to control the local CLT behavior of the process {H n }-the behavior of H n when it is away from 0 and the behavior that is closely related to the CLT behavior of polar codes. In [FHMV17, Theorem 7], a similar criterion is annotated as near optimal polarization in the middle with h FHMV (z) := (z(1 − z)) α for positive but small α at most log(log )/ log . In [BGS18, Definition 2.3], a similar criterion is annotated as variance in the middle with h BGS (z) := min(z, 1 − z). Note how our choice of h α (z) := min(z, 1 − z) α resembles theirs. In both cases, the criteria are local because they refer to a small slice of the process, focusing on how H n+1 (or Z n+1 ) behaves in terms of H n (or Z n ). This perspective frees [FHMV17,BGS18] from considering the (global ) process {H n } as a whole and simplifies the analysis. We specifically benefiti from the fact that we can choose the bijection g (···(W (k 1 ) )··· ) (kn ) solely according to the channels (· · · (W (k1) ) · · · ) (kn) and (· · · (W (k1) ) · · · ) (kn) instead of the complete channel family-tree. This is also the approach taken in [GRY19].\\n6.1. Lemma 12 helps achieve Claim 11. To do so, one advantage is that the two desired behaviors are local. They only involve how H n+1 , Z n+1 and S n+1 behave conditioned on the history F n . A potential tedious aspect is that for each candidate of the bijection g (···(W (k 1 ) )··· ) (kn ) , we have to verify the two behaviors four times, once for each of the four triples of channel parameters. Luckily, within random coding theory, we are in the situation that to choose an object that satisfies multiple criteria, it suffices to choose the object from an ensemble and compute the probabilities that each criterion fails; as long as the sum of failing probabilities is small, most objects satisfy. Even more luckily, when we choose a bijection g (···(W (k 1 ) )··· ) (kn ) from some ensemble, we only have to compute the probability that the local CLT or LDP behavior fails for {H n }, {Z n }, and {S n } being {H(W n )}, {Z mad (W n )}, and {S max (W n )} but not the other three triples. This is because other three triples are the special case and/or the dual of this triple. Elaboration: Since V n are q-ary channels just like W n are, inequalities hold true for arbitrary W n should hold true for any V n . Also, since Z mad and S max are in duality, inequalities hold true for H, Z mad , S max hold true for 1 − H, S max , Z mad . The duality is due to the duality between FTPCZ and FTPCS, within the explicit Hölder tolls, and within the ensemble of bijections we are to choose g (···(W (k 1 ) )··· ) (kn ) from.\\n6.2. Random linear isomorphisms as bijections. Fix q and . Let GL( , q) be the group of -by-invertible matrices over F q together with the ordinary matrix multiplication. Select an element G ∈ GL( , q) uniformly at random. Let g W : F q → F q be the multiplication of G from the right, namely g W (u 1 ) := u 1 G. This map is bijective since G is invertible. Let W be a q-ary channel. Recall that we defined q-ary channels W (1) , W (2) , . . . , W ( ) in Section 3. To emphasis that these imaginary channels depend heavily on the randomness source G, we call\\nLemma 14 (local CLT behavior). Fix an 20. Recall α := log(log )/ log and h α (z) := min(z, 1 − z) α . Let G vary; with probability less than 2 − log( )/20 , this fails:\\n6.3. Local behaviors imply Claim 11 (and hence the main theorem). We now can see how Lemmas 12 to 14 imply that Claim 11 is achievable for the right choice of and bijections g W , g W (i) , et seq.: For any given q-ary channel W , let be max(e 4 , q 5 , 3 q ). For any given π, ρ > 0 such that π + 2ρ < 1, enlarge such that Inequality (6) holds, given α := log(log )/ log . Consider a random kernel G as a candidate of the bijection g W . Increase further so that the failing probabilities-Lemma 13's 3q − √ /13 and Lemma 14's 2 − log( )/20 -amount to 1/3 or less. Recall the flattening channel W . The probability that any of the inequalities in Lemmas 13 and 14 fails for W is less than 1/3, too. Invoke the union bound; 1/3 + 1/3 < 1. Hence there exists a solid choice of g W as the multiplication of some proper instance of G from the right. With this g W determined, we define W (i) and W (i) for all i ∈ [ ]. Consider first i = 1, anything that has been done to W now applies to W (i) . That is, let g W (i) be the multiplication of a random kernel G from the right. With W , i, and W (i) G replaced by W (i) , j, and (W (i) ) (j) G , the probabilities that inequalities in Lemmas 13 and 14 fail add up to 1/3 or less. So is the flattening ( ) counterpart. Hence there is a solid choice of g W (i) . Repeat this for every other i = 2, 3, . . . , . Once finished, proceed to choosing g (W (i) ) (j) for all i, j ∈ [ ]. And so on and so forth for cases beyond depth-2. Notice that we always make a solid choice of a bijection before we proceed to the next level of channels, hence the failing probabilities of Lemmas 13 and 14 do not accumulate as the depth increases. By how we select bijections, the criteria in Lemma 12 hold for ({H n }, {Z n }, {S n }) being these four triples: In this section, we will first prove the two fundamental theorems of polar coding in Section 7.1 (for the Z-end) and in Section 7.2 (for the S-end). And then we will target that the following inequalities hold with high probability:\\nexp(qZ mad (W ) )(qZ mad (W )) i 2 /3 , and (8's copy)\\nBy the duality between the two fundamental theorems and between the two targeted inequalities, it is not hard to see that it suffices to prove the Z mad -case and the S maxcase follows immediately. We will prove that the first targeted inequality, for each i ∈ [ ], holds with probability 1 − 3q − √ /13 in Section 7.4, closing this section.\\n7.1. Proof of FTPCZ (Theorem 9). As is promised in Section 4.3, we prove the two fundamental theorems of polar coding. We first go for the Z-end.\\nis the weight enumerator of the i-th coset code. Theorem 9 claims that Z mad (W (i) ) f\\nBy the nature of max 0 =di∈Fq , it suffices to show that the double sum is at most f (i) GZ (Z mad (W )) for arbitrary nonzero d i . In the upcoming argument, tuple concatenation takes precedence over vector-matrix multiplication and vector addition. Fix a d i , we argue that\\nThe first equality abbreviates the summation. The next equality expands W (i) by the very definition, where u i+1 and v i+1 are free variables in F q . The next inequality is by the sub-additivity of the square root. In the next equality we define d i+1 := v i+1 − u i+1 ; so summing over v i+1 is equivalent to summing over d i+1 . In the next equality we define x 1 := u 1 G; so summing over u 1 is equivalent to summing over x 1 as G is invertible. In the next equality we substitute e 1 := 0 i−1 1 d i G and reorder the summation. The next equality expands the product of the memoryless channels. The next equality classifies indexes into two classes-j ∈ J are those such that e j = 0 and k / ∈ J are such that e k = 0. The next equality is the distributive law ax + ay + bx + by = (a + b)(x + y). The next equality uses the fact that the W (x, y) sum to 1. In the next inequality we replace e j by a nonzero element that maximizes the sum in the parentheses. In the next equality we realize that the maximum is the Bhattacharyya parameter (surprisingly). The second last equality uses the fact that multiplying a vector by a scalar preserves its hamming weight. And quod erat demonstrandum. Experienced readers may find that all but the last inequality follows the proof strategy [KSU10, Lemma 10]. 7.2. Proof of FTPCS (Theorem 10). We now go for the S-end of the fundamental theorem of polar coding. Recall the character χ(x) := exp(2πi tr(x)/p). We need the following properties: (pa) χ(0) = 1; (pb) |χ(x)| = 1 for all x ∈ F q ; (pc) χ(x)χ(z) = χ(x + z) for all x, z ∈ F q ; (pd) x∈Fq χ(x) = 0. See also [MT14, Definition 24] or a dedicated book [Ter99]. To prove the theorem, we first verify that Fourier coefficients recover the origin: Let M (w, y)\\nThe first equality expands M (w, y) by the definition. The next equality uses that χ is an additive character (pc), and reorders the summation. The next equality uses w∈Fq χ(w) = 0 (pd) and w∈Fq χ(0) = q (pa); and I is the indicator function.\\nKnowing that W (x j , y j ) = q −1 wj ∈Fq M (w j , y j )χ(−x j w j ), we proceed to\\nThe first equality expands the definition of W (i) . In the next equality, we substitute x 1 := u 1 G. The next equality expand the definition of W down to W . The next two equalities Fourier expand W and reorder the operators. The next equality merges all χ(−x j w j ) into one term by additivity (pc). In the next equality we define M (w 1 , y 1 ) to be the product of all M (w j , y j ). The next two equalities use x 1 (w 1 ) = u 1 G(w 1 ) = u 1 (w 1 G ) . In the next equality we define v 1 := w 1 G ; so summing over w 1 is equivalent to summing over v 1 . (Recall that G − is the notation of the inverse transpose of G.) The next three equalities sum over u i+1 to force v i+1 = 0.\\nHaving\\nIn the first line we let M (i) be the Fourier coefficient of W (i) . The next equality plugs in what we have about W (i) in mind. The next three equalities sum over z i to force\\nThe first inequality expands the Fourier coefficient. The next inequality is triangle plus (pb). The next equality cancels the summation over u i−1 1 with q 1−i . In the next equality we substitute w 1 := v i−1 1 ω i 0 i+1 G − ; slightly different from the w 1 above, they are now restricted to a proper subspace. The next equality classifies indexes into two classes-j ∈ J are those such that w j = 0 and k / ∈ J are such that w k = 0. The next two equalities reorder the operators and simplify y k |M (0, y k )| = y k W out (y k ) = 1. The next inequality replaces w j by the one that maximizes y k |M (w j , y k )|. The rest is trivial. Theorem 10 claims that S max (W (i) ) f (i)\\nGS is the weight enumerator of the i-th dual coset code. Since S max (W (i) ) is just the maximum of Formula (11) over 0 = ω i ∈ F q , we arrive at S max (W (i) ) f\\nSee the left half of Figure 6 for evidence. More generally, for all prime powers q,\\nfor all z ∈ [0, 1], where D is the Kullback-Leibler divergence. This falls back to the h 2 case when q = 2. See the right half of Figure 6 for evidences for q = 3, 4, 5, 7. It can be observed that as q → ∞ the function tends to a line connecting (0, 0) and (1, 1), hence the upper bound should hold. Taking derivative in q shows that the left hand side decreases as q increases and x < 1/2. where z := Z mad (W ) for short. This inequality is in fact a consequence of\\nbecause (1 + a) b < exp(ab). We will show the last inequality. Now divide i into two cases: 1 i √ 3 and √ 3 < i . For i = 1, 2, . . . , √ 3 , the exponent i 2 /3 is simply 1, so the inequality to be proven reads f For i = √ 3 + 1, √ 3 + 2, . . . , , let k := − i and let d := i 2 /3 . These variables resemble the dimension and the minimal distance of a linear block code as in the notation an [ , k, d]-code in classical (algebraic) coding theory. To make Inequality (12) hold, we execute a two-phase procedure to avoid all codewords of weight less than d and to eliminate kernels with poor overall scores. In further detail, we will reject a kernel G if there exists u i+1 such that wt(0 i−1 1 1u i+1 G) < d and call it phase I. Afterwards, among surviving kernels with only heavy codewords, we will reject a kernel if its overall score f (i) GZ (z) is too low and call it phase II. The failing probability 3q − √ /13 is the price we pay for rejecting. Up to this point, two things remain to be analyzed: how much probability we pay for rejecting light codewords in phase I (answer: q − √ /13 ), and what is the Markov cutoff that makes Inequality (12) in phase II (answer: 2q − √ /13 ). Phase I analysis is as follows: Fix u i+1 and vary G ∈ GL( , q); the codeword\\nThis distribution is almost identical to the uniform distribution on F q . Assume X 1 follows the latter; this makes X 1 lighter, which is compatible with the direction of the inequalities we want. Then the probability that X 1 has weight less than d is the probability that Bernoulli trials-X j being \\\"zero\\\" with probability 1/q and \\\"nonzero\\\" with probability (q −1)/q-result in less than d \\\"nonzero\\\"s. By the large deviations theory [DZ10, Exercise 2.2.23(b)], wt(X 1 ) < d holds with probability less than\\nfor the q = 2 case, where D is the Kullback-Leibler divergence. For general q, similarly, wt(X 1 ) < d holds with probability less than\\nThis quantity is less than q − (1−h2(d/ )) by Figure 6 (meaning that q = 2 is the most difficult case). By Figure 6, h 2 (d/ ) < ed/ = ei 2 /3 2 = ( e/3)i/ < 0.952i/ . So the rejecting probability is less than q − (1−h2(d/ )) < q − +0.952i . Take into account that there are q −i possibilities of u i+1 . The union bound yields q −i q − +0.952i = q −0.048i < q −0.048 √ 3 < q − √ /13 . Hence the rejecting probability q − √ /13 . Phase I ends here. Phase II analysis is as follows: After we reject some G in phase I, some codewords will disappear; particularly, this includes all codewords of low weights. Therefore, the expectation of f (i) GZ (z) is bounded by the weight enumerator of all heavy codewords rescaled by the number of codewords. In detail, start from\\nI is the indicator function. In the denominator, 1 − q − √ /13 > 1/4 as 30. Put that aside and redefine d := i 2 /3 . The expected value part is bounded from above by\\nThe first equality expands the definition. The next inequality replaces G surviving phase I by a weaker condition. The next equality switches E and . The next inequality replaces the ensemble of u i+1 G by a uniform X 1 ∈ F q . The next equality expands the definition of the expectation over X 1 . The next equality counts codewords. The next inequality selects w positions by first selecting d and then selecting w − d. The next two equalities factor and apply the binomial theorem. The rest is by a series of inequalities that overestimate the scalar:\\n. Hence the scalar part is less than q − √ /13 /2. Put 1 − q − √ /13 > 1/4 back to the denominator as in Inequality (13);\\nBy Markov's inequality, Inequality (12) holds with probability 1−2q − √ /13 , i.e., the rejecting probability is 2q − √ /13 . Phase II ends here. The sum of the two rejecting probabilities is 3q − √ /13 as claimed in Lemma 13, hence the lemma settled. 7.5. Bibliographic remarks. Concerning the fundamental theorems: Nonlinear g W is not taken into consideration for that it is hard to imagine how MacWilliams duality works then. Also the S-parameter does not generalize to non-field input alphabet. Concerning random linear codes: [BF02, Section II.C] portrays a clear picture of the weight distribution of binary random linear codes. Section 7.4 accommodates and extends their argument to general prime power q. Concerning the LDP behavior: [KSU10, Theorem 22] showed that π < 1 can be arbitrary close to 1 over binary alphabet utilizing the Bose-Chaudhuri-Hocquenghem codes. Our Lemma 13 on the other hand, implies that almost all kernels make π close to 1.\\nIt remains to prove Lemmas 12 and 14.\"\n },\n {\n \"title\": \"Local CLT Behavior (Proof of Lemma 14)\",\n \"content\": \"We are to prove that the following inequality holds with high probability:\\nThe target inequality is the sum of the following three inequalities:\\nThe second one is trivial as h α (z) (1/2) α . The first one will be proven in Section 8.3 with failing probability − log( )/20 . The third one will be proven in Section 8.4 with failing probability − log( )/20 . Before the main proofs, we devote Section 8.1 to introduce the symmetrization trick, which will reduce our proof to the case of symmetric q-ary channels. A channel W being symmetric means that for any affine shifting ξ ∈ F q , there exists an permutation σ on Y such that W (y | ξ + x) = W (σ(y) | x) holds for all x ∈ F q and y ∈ Y. It also means that the uniform input achieves the Shannon capacity. This justifies the usage of linear codes. In Section 8.2, we invoke some universal bound on entropies and exponents from Chang, Draper, and Sahai's works. Finally, we will be abusing the theory of random linear codes in Section 8.3 for noisy channel coding and in Section 8.4 for secrecy over wiretap channels. 8.1. Symmetrize channel and uniformize input. Let W : F q → Y be any q-ary channel; let X and Y be some input and the corresponding output. Symmetrize the channel as follows: Let Ξ ∈ F q be a uniform r.v. independent of X, Y . Let W : F q × (F q × Y) → [0, 1] be the probability mass function of this combination of r.v.s (Ξ + X, (X, Y )) ∈ F q × (F q × Y). ThisW behaves like a channel such that, quote, unquote,W ((x, y) | z) = W (x, y)/q for all inputs z ∈ F q and outputs (x, y) ∈ F q × Y. Despite that this channel might be properly simulated by a symmetric channel with feedback to the sender, all that matters is that the biased input X is neutralized by the uniform r.v. Ξ, and becomes uniform. Let g W be the multiplication of an invertible matrix G from the right. LetW (i) (u i , u i−1 1 x 1 y 1 ) be the probability mass function of the tuple (U i , U i−1 1 X 1 Y 1 ), where U 1 G = Ξ 1 + X 1 . This definition is compatible with the channel transformation ofW as ifW was an actual channel in the first place. Let H(W (i) ) be H(U i | U i−1 1 X 1 Y 1 ); this is also compatible. The following lemma justifies whyW is useful in theory.\\nLemma 15 (channel symmetrization).W is a symmetric q-ary channel,\\nThis lemma is by [MT14, Definition 6 and Lemmas 7 and 8], plus the arguments in between. See also [HY13,Theorem 2] where they cared about whether Z(W (i) ) = Z(W (i) ). One could also expand all definitions to verify the identities.\\nThe consequence of this lemma is thatW behaves like a shadow copy of W , but is symmetric. All inequalities involving entropies of W and W (i) are reduced to inequalities involving entropies ofW andW (i) . Subsequently, passing statements toW is effectively assuming that the channel W is symmetric with the uniform input to begin with. In the upcoming subsections, we will prove that the targeted inequalities, (14) and (15), hold for any symmetric q-ary channel W with the uniform input with high probability. We conjecture that the symmetrization technique is optional as it seems like a wrapper of complicated Bayesian formulas. 8.2. Chang-Sahai's universal quadratic bound. This and the next two subsections contain the most convoluted part of the proof of Lemma 14. This subsection prepares a universal upper bound on Gallager's E-null function, which ultimately evolves into a universal lower bound on Gallager's error exponent.\\nLet W : X → Y be a q-ary channel. Symmetry is not required in this subsection but it is in the next two. Assume the uniform input distribution W in (x) = 1/q for all x ∈ X . Define Gallager's E-null function and its complement [CS07, Formula (1)]:\\n, and\\nBy complement we mean that under the uniform input,Ē 0 (t) degenerates tō\\nEquivalently, E 0 (t) +Ē 0 (t) = t log q. Do not confuse W with W t , the latter W is tilted. When t = 0, the tilted falls back to its italic origin W 0 (x, y) = W (x, y). These measures can be interpreted as follows: W t behaves like a channel with a dedicated input distribution. The first fraction in the definition specifies the output distribution W t out (y). The second fraction specifies the a posteriori distribution W t (x | y) when y is known. As W t is not an actual channel, it is not meaningful to alter the input distribution and ask for the corresponding output. Like the symmetrization trick, all that matters is that we can compute entropies, and what not, as if they were real channels. Quantities we are interested in are listed below: Let H e be the base-e entropy. Let H e (W t ) be H e (X t | Y t ) where (X t , Y t ) is a tuple r.v. that follows W t . Let H e (X t y) be the entropy of the a posteriori distribution of X t given Y t = y; to be specific,\\n[CS07, Formula (13) and (19)] have that the following hold for t ∈ [0, 1]:\\nCareful readers may verify them by hand or follow [CS07,Formulas (13) to (19)] and [DCS14, Lemmas 9 and 10]. Similar computations are also carried out by [AW10,AW14]. Notice thatĒ 0 (t), H e (W t ), and every other term in Equation (16) are all holomorphic functions in t on the half-plane Re t > −1 (there is a singularity at 1/(1 + t) = ∞). By the identity theorem in complex analysis [BMPS02,Corollary 8.16], Equation (16) holds for all t ∈ [−2/5, 1]. Dropping the nonpositive square, we deduce an upper bound for each t ∈ [−2/5, 1]:\\nThis upper bound onĒ 0 (t) is a linear combination of\\nand H e (X t y) 2 parametrized by y ∈ Y, so it remains to bound them separately. For the second kind of constituents, the entropy cannot exceed log q so H e (X t y) 2 log(q) 2 . For the first kind of constituents, the following lemma adapted from [CS07, Lemma 1] helps.\\nLemma 16 (second moment). If w 1 , w 2 , , . . . , w q are positive numbers of sum 1, then i w i log(w i ) 2 ß log(q) 2 for q 3 0.563 for q = 2 ™ 1.2 log(q) 2 .\\nWith the lemma, we do have x∈X W t (x | y)(logW t (x | y)) 2 1.2 log(q) 2 . Now Inequality (17) becomes\\nfor all t ∈ [−2/5, 1]. Since E 0 (t) is a linear function t log q minusĒ 0 (t), their first derivatives sum to log q while their second derivatives are opposite. Hence the following lemma.\\nwhere j := H(W ) + 1/2+α for short. It suffices to prove that the right hand side is less than −1/2+α . In the spirit of the motivational Chain Rule (1), the sum of the chain of H(W (i) G ) on the right hand side is H(U j+1 | U j 1 Y 1 ). In order to prove Inequality (14), we will show\\nBut what is H(U j+1 | U j 1 Y 1 )? It measures the equivocation at Bob's end when U j 1 is known to Bob. In other words, we may as well pretend that there is a random rectangular full-rank matrix G with columns and only k := − j = − H(W ) − 1/2+α rows, that Alice computes and sends X 1 := U k 1 G to Bob, and that Bob attempts to decodeÛ k 1 upon receiving Y 1 using the MAP decoder. The equivocation is thus, by Fano's inequality, bounded in terms of the probability that Bob fails to decode U k 1 :\\n−P e log q P e + P e log q + P e · k = P e · 1 − log P e log q + k . (19) Here P e is the probability that Bob fails to decode,Û k 1 = U k 1 . The following is how to compute Bob's decoder block error probability. The generator matrix G Alice uses is selected uniformly from the ensemble of full-rank k-by-matrices. The difference of every pair of codewords distributes uniformly on F q \\\\ {0 1 }. Over symmetric channels, the difference alone determines the output's joint distribution because W (y 1 | ξ 1 + x 1 ) = W (σ 1 (y 1 ) | x 1 ) for some componentwise permutation σ 1 on Y depending on ξ 1 . Gallager's bound applies. To elaborate, let t ∈ [0, 1]. Bob's average error probability satisfies [Gal68, Inequalities (5.6.2) to (5.6.14)]\\n= exp(kt log q − (the E-null function of W )(t)) = exp(kt log q − E 0 (t)).\\nIn summary, EP {Bob fails to decode U k 1 given G } exp(kt log q − E 0 (t)) whenever 0 t 1. Recall the universal quadratic bound E 0 (t) I(W )t log q−t 2 log(q) 2 derived in Lemma 17. We obtain that the exponent is\\n(redeem the supremum at t = −1/2+α /2 log q)\\nSo far we obtain that the average error probability is less than exp(− log( ) 2 /4) = − log( )/4 . Run Markov's inequality with cutoff − log( )/20 . That is, we reject kernels such that P {Bob fails to decode U k 1 given G } − log( )/5 . Then the rejecting probability is − log( )/20 because 1/20 + 1/5 = 1/4. An upper bound on Bob's error probability being P e < − log( )/5 , an upper bound on Bob's equivocation is − log( )/5 1 − log − log( )/5 log q + k = − log( )/5 1 + log( ) 2 /5 log q + k by Inequality (19). Plugging the latter in kh α (here/k), we derive that the left hand side of Inequality (18) is less than kh α − log( )/5 k 1 + log( ) 2 /5 log q + k = k − log( )/5 1 + log( ) 2 /5 k log q + 1 α = −α log( )/5 k 1 + log( ) 2 /5 k log q + 1 α < −α log( )/5 1 + log( ) 2 /5 log q + 1 α < −α log( )/5 · · 2 α = 2 α log( ) − log( )/5 .\\nThe first inequality uses that the left hand side increases monotonically in k and k := − j = − H(W ) − 1/2+α < . The second inequality uses the assumption 2. In any regard, the quantity at the end of the inequalities decays to 0 as → ∞, so eventually it becomes less than 1/2+α , the right hand side of Inequality (18). This proves that Inequality (14) holds with failing probability − log( )/20 as soon as is large enough. The lower bound on in the statement of Lemma 14 is large enough, hence the first half of Lemma 14 settled. 8.4. Hayashi's argument at Eve's end. This subsection contains the very last ingredient of the proof of Lemma 14. We dealt with Inequality (14) in the last subsection. We now deal with\\n(15's copy)\\nSimilar to how we motivated Inequality (18), we apply Jensen's inequality and the chain rule of conditional entropy to simplify Inequality (15). The left hand side becomes jh α (H(U j 1 | Y 1 )/j) where j := H(W ) − 1/2+α for short. (This is not the same j as in the last subsection.) The input uniform, the argument of h α is H(U j 1 | Y 1 )/j = 1 − I(U j 1 ; Y 1 )/j, which can be replaced by I(U j 1 | Y 1 )/j thanks to the evenness h α (1 − z) = h α (z). We will show\\nBut what is I(U j 1 ; Y 1 )? It is the amount of information Eve learns from wiretapping Y 1 if they know that U j+1 are junk. In other words, we may pretend that Alice transmits X 1 := U j 1 V j+1 G with confidential bits U j 1 and obfuscating bits V j+1 , Bob receives X 1 in full, and Eve learns Y 1 . This context falls back to (a special case of) the traditional setup of wiretap channels [Wyn75] where various bounds are studied, some in terms of Gallager's E-null function.\\nHere are some preliminaries to control the information leaked to Eve. We follow the blueprint of how Hayashi derived the secrecy exponent in [Hay06,Inequality (21)]. Consider the communication protocol depicted in Figure 7: Karl fixes a kernel G ∈ GL( , q) and everyone knows G. Alice chooses the confidential message U 1 . Vincent chooses the obfuscating bits V j+1 . Charlie generates Y 1 by plugging X 1 := U j 1 V j+1 G into a simulator of W . Eve learns Y 1 and is interested in knowing U j 1 alone. So the channel on topic is the composition of Vincent and Charlie. Notation: Running out of symbols, we all use P with proper subscriptions to indicate the corresponding probability measures. That said, indexes in the subscription will be omitted. As Eve is interested in the relation between U j 1 and Y 1 , let Y 1 Gu j 1 be the\\nThe channel Eve cares about Figure 7. A finer setup for Hayashi's secrecy exponent. Charlie generates Y 1 such that X 1 := U j 1 V j+1 G and Y 1 follow W . Despite of the seemingly sequential structure, Karl, Alice, and Vincent work independently.\\nr.v. that follows the a posteriori distribution of Y 1 given G = G and U j 1 = u j 1 . More formally, P Y Gu (y 1 ) = P Y |GU (y 1 | G, u 1 ) = P GU Y (G, u j 1 , y 1 )/P GU (G, u j 1 ). We could have defined Y 1 G to be the a posteriori distribution of Y 1 given G = G; but it is simply the same distribution as Y 1 since U j 1 V j+1 G traverses all inputs uniformly regardless of the choice of G. That is, P Y |G (y 1 | G) = P Y (y 1 ) for all y 1 ∈ Y .\\nFix G as an instance of G. Let I e be the base-e mutual information. The channel Eve cares about leaks information of this amount:\\nis the Kullback-Leibler divergence from the a posteriori distribution of Y 1 given G, u j 1 to the coarsest distribution Y 1 . We are to take expectation over G to find the average information leak since we are interested in Markov's inequality. Equality (21) yields\\nWe now discover that there are redundancies in traversing all G and u 1 : After all,\\nwhich is a fixed linear combination of the first j rows plus a random vector from the span of the bottom − j rows. When V 1 varies, the track of X 1 forms an affine subspace of F q , a coset code as in the context of the fundamental theorems. So what matters is the distribution of this coset code. In this regard, we replace the uniform ensemble of (G, U j 1 ) by the uniform ensemble of K a rank-( − j) affine subspace of F q , where j := H(W ) − 1/2+α . Karl and Alice together choose K uniformly. Vincent chooses X 1 ∈ K uniformly. Charlie generates Y 1 by throwing X 1 into a simulator of W . See Figure 8 for the Karl and Alice choose K\\nThe modified channel Figure 8. A simplified setup for Hayashi's secrecy exponent. Charlie generates Y 1 such that X 1 and Y 1 follow W .\\ndepiction of the new scheme. Hence Equality (22) becomes\\nwhere Y 1 K is the a posteriori distribution of Y 1 given K = K. Suddenly, the quantity EI e (U j 1 ; Y 1 | G) we are interested in turns into the mutual information I e (K ; Y 1 ) between K and Y 1 as K replaces the role of U j 1 in Formula (21). Recall that in Lemma 17 the mutual information is the derivative of Gallager's E-null function. We exploit this. Define the double-stroke E-null function for (K, Y 1 ) as follows\\nThen E 0 (0) = I e (K ; Y 1 ) = EI e (U j 1 ; Y 1 | G). Owing to the concavity of the E-null function, E 0 (0) E 0 (t)/t whenever −2/5 t < 0. Recap: To bound the average leaked information EI e (U j 1 ; Y 1 | G) it suffices to bound I e (K ; Y 1 ), which is then morphing to bounding E 0 (0) from above and to bounding E 0 (t) from below.\\nThe double-stroke E-null function is bounded as below. Assume −2/5 t < 0. Let s be −t/(1 + t); so 0 < s 2/3 and (1 + s)(1 + t) = 1. For any fixed K and fixed x 1 ∈ K, the base of the (1 + t)-th root in the definition of the double-stroke E-null function is\\nHere P XY (K\\\\x 1 , y 1 ) is a temporary shorthand for the summation of P XY (ξ 1 , y 1 ) over ξ 1 ∈ K that excludes x 1 . Raise it to the power of 1/(1 + t) = (1 + s):\\nx 1 ∈K q j P XY (x 1 , y 1 ) q j P XY (x 1 , y 1 ) + q j P XY (K\\\\x 1 , y 1 ) s x 1 ∈K q j P XY (x 1 , y 1 ) q js P XY (x 1 , y 1 ) s + q js P XY (K\\\\x 1 , y 1 ) s = q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (x 1 , y 1 ) s + q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (K\\\\x 1 , y 1 ) s = q j+s x 1 ∈K P XY (x 1 , y 1 ) 1+s + q j+js x 1 ∈K P XY (x 1 , y 1 )P XY (K\\\\x 1 , y 1\\nThe inequality uses that the s-th power is sub-additive. Then the inner sum of the E-null function morphs as follows\\nx 1 ∈K P XY (x 1 , y 1 )P XY (K\\\\x 1 , y 1 ) s .\"\n },\n {\n \"title\": \"(minor arc)\",\n \"content\": \"Divide and conquer-the inner sum of the double-stroke E-null function is split into two arcs as shown. The major arc is exactly\\nThe minor arc is loosen to q j+js K P K (K)\\nx 1 ∈K P XY (x 1 , y 1 )P XY (K\\\\x 1 , y 1 ) s = q j+js\\nBoth major and minor arcs conquered, merge them and raise to the (1+t)-th power. The summand for any fixed y 1 in the definition of the double-stroke E-null function is\\nWe can finally bound the double-stroke E-null function per se:\\nAll efforts we spent on bounding I e (U j 1 ; Y 1 ) are for three creeds: First, we see Gallager's bound possessing innate elegance. Second, it fits the paradigm that solving the primary and the dual problems as a whole is easier than solving the primary problem alone. Third, the universal quadratic bound is waiting ahead for the E-null function. We infer that\\nRecall the universal quadratic bound E 0 (t) I(W )t log q − t 2 log(q) 2 as stated in Lemma 17 and used in the previous subsection. But this time −2/5 t < 0. We obtain that the exponent is\\n= log( )/2 − α log + log 2 + log log q − 2α /4 = log( )/2 − log log + log 2 + log log q − 2 log(log )/ log /4 < log( )/2 + log log q − log( ) 2 /4.\\nThe first inequality uses − j = − H(W ) + 1/2+α . The last inequality uses the assumption e 2 . With the last line we conclude that EI e (U j 1 ; Y 1 | G) < exp(log( )/2 + log log q − log( ) 2 /4) = 1/2−log( )/4 log q. Switch back to the base-q mutual information EI(U j 1 ; Y 1 | G) < 1/2−log( )/4 . We now reject kernels such that I(U j 1 ; Y 1 | G) 1/2−log( )/5 . By Markov's inequality, the opposite direction (<) holds with probability 1 − − log( )/20 because 1/5 + 1/20 = 1/4. Plug this upper bound into h α . The left hand side of Inequality (20) is less than\\nThe inequality uses that the left hand side increases monotonically in j and j := H(W ) − 1/2+α < . In any regard, the quantity at the end of the inequalities decays to 0 as → ∞, so eventually it becomes less than 1/2+α , the right hand side of Inequality (20 [Ooh02], although no formal proof was found there. See [HM11,BTM17] for alternative descriptions and approaches on the same topic. For readers who took Lemma 12 as granted or went through Appendix B in advance, this is the last sentence of the proof of the main theorem-polar codes' simplicity, random codes' durability.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"Shannon introduced what we now understand as discrete memoryless channels seventy-two years ago. In the beginning, Shannon had no tool but developed their own theory of typical set, proved the noisy channel coding theory, and justified the notion of capacity. Gallager brought in error exponents. Capacities and error exponents quantify first and second order terms in the asymptotic performance of codes. Only in 2010 we are revealed the complete second order term. It was around the time that polar coding as a graceful instrument to explore the limits at low cost was discovered when Arıkan experimented with the channel transformation and with error exponents. Another ten years it took to grow variants and proof techniques of polar coding. Ultimately, it is feasible, and done by us coincidentally, to piece the puzzle together to show the mere possibility to achieve the second order limits at low cost. An overall comparison is integrated in Table 2. Columns are classes of channels; from left to right: (BEC) binary erasure channels; (BDMC) binary-input discreteoutput memoryless channels; (p-ary) channels of prime input size; (q-ary) channels of prime power input size; (finite) channels of discrete input. Columns to the right are wider than columns to the left. The last column is exceptional; (asym.) is about whether we can achieve the true Shannon capacity, instead of the symmetric capacity. Rows are goals; from top to bottom: (LLN) to achieve (symmetric) capacity; (wLDP) there exists π > 0 such that P e < exp(−N π ); (wCLT) there exists ρ > 0 such that R > I − N −ρ ; (wMDP) there exists π, ρ > 0 such that P e < exp(−N π ) and R > I − N −ρ at once; (LDP) the π in (wLDP) can be arbitrarily close to 1; (CLT) the ρ in (wCLT) can be arbitrarily close to 1/2; (MDP) the (π, ρ)-pair can be arbitrarily close to π + 2ρ = 1. Row (MDP) implies every other row; row (CLT) implies (wCLT); row (LDP) implies (wLDP); and every other row implies row (LLN). Rows (LDP) and (CLT) together almost imply (MDP) (need the partial distance profile). Cells represent how various goals are achieved over various channels. The greenish background means it is possible using Arıkan's kernel [ 1 1 0 1 ]. The purplish background means it is possible using other kernels. The orangish background means it is only possible using dynamic kernels.\\nThe following works made critical progresses but our classification fails to include them: Rate-dependent result in LDP paradigm [HMTU13]. Optimal relations among channel parameters [MT14]. First family of (π, ρ) pairs in MDP paradigm [MHU16]. AWGNCs intersecting MDP [FT17].\\nWe did our best to excavate the archive but throughout the course of manuscript preparation we found ourselves underestimating early works multiple times so the record kept updating. We sincerely hope to hear about possible references to add to the table.\\nPotential improvements include but are not limited to the following: (Tolls) Tighten the explicit Hölder tolls. The current toll between any pair of parameters H, P e , Z, Z mad , T , S, and S max is roughly the sum of tolls collected when traveling through the spanning tree illustrated in Lemma 7. Some improvements potentially tighten the bounds in Lemma 12. (FTPC) Tighten the two fundamental theorems such that they degenerate to equalities over erasure channels. Once done, {Z n } is a supermartingale and Appendix C.1 is obsolete. (Symmetry) generalize the arguments in Sections 8.2 to 8.4 to asymmetric channels. Once done, Section 8.1 is obsolete. Note that the proof of the fundamental theorems applies to asymmetric channels. (Bijection) Early works on polar coding over arbitrary alphabets introduced arbitrary bijections g W . Generalize the two fundamental theorems to include arbitrary bijections. (Dynamic) Achieve the main theorem with a large, but fixed, kernel. This does not immediately make the code practical. But the answer should shed light on our understanding of coding. (Alphabet) Achieve the main theorem without the reduction to prime power alphabets. This is currently not an option because linear codes are barely defined over non-fields. Plus the S-parameter-and thus FTPCS-would just break. (Dispersion) Recall Proposition 2. Weird things happens when the channel dispersion vanishes V = 0. Can we describe those channels better? One example of such channels is this:\\nWe look forward to generalizations of the main theorem to non-identical channels (i.e., non-stationary) [Mah17], non-independent channels (i.e., with memory) [WHY + 15, ST16], deletion channels [TPFV19,LT19], channels with restrictions on input distributions (e.g., due to energy constrain) [FT16], wiretap channels [ŞV13], rate-distortion problem [HKU09], Wyner-Ziv problem [HKU09], Slepian-Wolf problem [Abb15], broadcast channels [GAG15,MHSU15], and multiple access channels [AT12,NT16]. We focus on noisy channel coding in this work for its historical significance.\\nAppendix A. Explicit Hölder Tolls (Proof of Lemma 8)\\nAs is promised in Section 4.1, we prove the explicit Hölder toll. Let W be a q-ary channel. In the upcoming arguments, H, P e , Z, Z mad , S, and S max mean H(W ), P e (W ), Z(W ), Z mad (W ), S(W ), and S max (W ), respectively. Also q means q − 1, and q means q − 2. Furthermore, lg means the base-2 logarithm; this is handy when we jump back and forth between nats, bits, and q-bits.\\nFirst we show Z mad q H log 4 q.\\n(2's copy)\\nStart from Z mad : By the definition Z mad q Z. Move on to Z: By Lemma 3,\\nThe left hand side is qZ; in the right hand side √ 1 + q z + √ 1 − z has maximum q/ √ q at z = q /q by calculus. So Z P e /q (q/ √ q ) = q √ P e /q . Move on to P e : By Lemma 6 (the first lower bound), 2P e H lg q or equivalently P e H log 4 q. Now we chain the inequalities Z mad q Z q √ P e q H log 4 q. This completes Inequality (2). That being proven, we use the weaker form Z mad q 3 √ H in the calculus machinery for global MDP. Second we show H » eq Z mad /2.\\n(3's copy) Start from H: By Lemma 6 (the upper bound, Fano's inequality), H lg q h 2 (P e )+ P e lg q . By Figure 6, h 2 (P e ) + P e lg q √ eP e + P e lg q = √ P e ( √ e + √ P e lg q ). What is inside parentheses is less than √ e + q /q lg q . Hence H √ P e ( √ e + q /q lg q )/ lg q. Focus on the scalar part-( √ e + q /q lg q )/ lg q has maximum √ e at q = 2 (remember that q 2). So H √ eP e . Move on to P e : By Lemma 3, P e q Z/2. Move on to Z: By definition Z Z mad . Now we chain the inequalities H √ eP e eq Z/2 eq Z mad /2. This completes Inequality (3). That being proven, we use the weaker form H q 3 Z mad in the calculus machinery for global MDP.\\nThird we show (notice the logarithm is natural)\\n(4's copy)\\nStart from S max : By definition S max q S. Move on to S: By Lemma 5, S q q(q /q − P e ) » 1 − q q q q . The square root simplifies to 1/(q ) 2 = 1/q as qq = (q ) 2 − 1. So S q − qP e . Move on to q − qP e : By Lemma 6 (the upper bound, Fano's inequality), H lg q h 2 (P e ) + P e lg q . We claim that h 2 (P e ) + P e lg q lg q − 2(q /q − P e ) 2 / log 2. To prove the claim, Taylor expand both sides at P e = q /q. Verify that both evaluate to lg q at P e = q /q; verify that both have derivative 0 at P e = q /q; and verify that the acceleration of the left hand side, −1/(P e (1 − P e ) log 2), is more negative than the acceleration of the right hand side, −4/ log 2. By Taylor's theorem, mean value theorem, or Euler method, the function with greater acceleration is greater; hence the claim. See also [FM94, Fig. 1]; the Φ-curve seems parabolic at the upper right corner. Now we have H lg q lg q−2(q /q−P e ) 2 / log 2, which is equivalent to 2(q /q−P e ) 2 / log q 1−H and to q − qP e q (1 − H) log(q)/2. Now we chain the inequalities S max q S q (q − qP e ) q q (1 − H) log(q)/2. This completes Inequality (4). That being proven, we use the weaker form\\nin the calculus machinery for global MDP. Fourth we show 1 − H q S max / log q.\\n(5's copy) Start from 1 − H: By Lemma 6 (the second lower bound), H lg q q q lg(q/q ) × (P e − q /q ) + lg q . The right hand side is lg q − q lg(q/q )(q − qP e ) by matching the (rational) coefficients of P e lg q, P e lg q , lg q, and lg q , respectively. As H lg q lg q − q lg(q/q )(q − qP e ) we bound lg(q/q ) = lg(1 + 1/q ) 1/q by the tangent line at 1/q = 0. So H lg q lg q − (q − qP e ) and hence 1 − H (q − qP e )/ lg q. Move on to q − qP e : By Lemma 5, 1 − qP e /q S so q − qP e q S. Move on to S: By definition S S max . Now we chain the inequalities 1 − H (q − qP e )/ lg q q S/ lg q q S max / lg q. This completes Inequality (5). That being proven, we use the weaker form 1 − H q 3 S max in the calculus machinery for global MDP. This is end of the proof of Lemma 8. The proof of Lemma 7 follows the same logic, only shorter.\\nAppendix B. Calculus Machinery for Global MDP (Proof of Lemma 12)\\nWe are to prove that\\ngiven criteria (cb), (cm), (ct), and (cl), the local LDP behavior, the local CLT behavior, and that π + 2ρ 1 − 8α. The proof is split into several stepping stones. We will prove each of the following inequalities (including two equalities) in each of the upcoming subsections. This will be proven in Appendix B.1: The eigen behavior reads\\nThis will be proven in Appendix B.2: As a lemma, {H n } and {Z n } converges to 0 with probability 1 − H 0 , i.e.,\\nThis will be proven in Appendix B.3: The en23 behavior reads\\nThis will be proven in Appendix C.1: As a lemma, {min( −2 , 4 √ Z n )} is a supermartingale, i.e.,\\nThis will be proven in Appendix C.2: As a lemma, the following holds when Z 0 < −8 :\\nE [( K 2 n+1 /3 · 3/4) −1/2 | F n ] < −1/2+2α . (27) This will be proven in Appendix C.3: The een13 behavior reads\\nThis will be proven in Appendix C.4: The elpin behavior reads for any constants π, ρ > 0 such that π + ρ 1 − 8α,\\nThe last inequality is a bi-Hölder toll away from P{H n < exp(− πn n)} > 1 − H 0 − −ρn+o(n) , (7's copy) our destination. This finishes the proof of Lemma 12. The eigen, en23, een13, and elpin behaviors are intermediate checkpoints pinned in a way that moving from one to the next is easy while skipping any of them makes the next unreachable. Their entire purpose is to form a chain that connects the local LDP and CLT behaviors to the global MDP behavior and we do not specify if any of them falls inside the LDP, CLT or MDP paradigm. B.1. The eigen behavior. We want to prove Inequality (23), E [h α (H n+1 ) | F n ] 4 −1/2+3α h α (H n ), given the local LDP behavior and the local CLT behavior. The idea is that, for H n that is close to 1/2, the local CLT behavior provides a measurement of the dichotomy/bifurcation behavior of H n+1 . For H n that are close to 0, the Z -part of the local LDP behavior provides a measurement of the attraction toward 0. For H n that is close to 1, the S-part handles it dually. The formal proof is below.\\nInequality (23) is a local statement so we may assume n = 0. To prove that The last case-the reliable case: when H 0 < −2 , we further split it into two subcases per how K 1 compares to k := 1/2+5α/2 . For the small K 1 subcase, the martingale property fits. For the large K 1 subcase, the local LDP behavior fits:\\nFor the K 1 k := 1/2+5α/2 subcase, the martingale property\\n. And the K 1 k subcase is closed. For the K 1 > k := 1/2+5α/2 subcase, pay the explicit Hölder toll: Z 0 q 3 √ H 0 < q 3 / < 1. Invoke the local LDP behavior:\\nexp(qZ 0 )(qZ 0 ) k 2 /3 exp(q 4 )(q 4 √ H 0 ) k 2 /3 = exp(q 4 )(q 8 H 0 ) log( ) 5 /6 . Pay the return-trip toll: H 1 q 3 √ Z 1 q 3 1/2 exp(q 4 /2)(q 8 H 0 ) log( ) 5 /12 . Now we claim and prove that the following quantity is less than 1: (there is nothing to show if h α (H 0 ) = 0) (h α (H 1 )/ −1/2+3α h α (H 0 )) 12/α = H 12 The inequality involving 1.6 uses q 5 . The inequality involving 0.3 uses max(41, 3 q ). The inequality involving 15 uses max(22, q 5 ). We just showed that h α (H 1 )/ −1/2+3α h α (H 0 ) is less than 1, with and hence without the 12/α-th power. Thus E [h α (H 1 ) | K 1 > k] −1/2+3α h α (H 0 ). And the K 1 > k := 1/2+5α/2 subcase is closed. To sum up the reliable case: We bound separately the two terms in Formula (30). They are both at most −1/2+3α h α (H 1 ), hence their sum is at most 2 −1/2+3α h α (H 1 ). Since Inequality (23) wants 4 instead of 2, The reliable case is closed. And the proof of the eigen behavior, Inequality (23), is sound when combining the three cases.\\nBibliographic remarks: [FHMV17, Theorem 7] also cut the cases at −2 and 1 − −2 . In contrast, [GRY19, Theorem 5.1] cut at −4 and 1 − −4 − ε. A potential improvement is, when −2 H 0 < −1 , Inequality (15) will simply evaporate. Similarly, Inequality (14) evaporates when 1 − −1 < H 0 1 − −2 . They tighten the right hand side of Inequality (10). The lesson here is that the hard transition between local LDP and CLT behaviors weakens the bounds.\\nB.2. Polarization in mean. We want to prove Equality (24), P{Z 0 → 0} = P{H 0 → 0} = 1 − H 0 , given the martingale property and the eigen behavior. The idea is that the eigen behavior expels H n from being close to 1/2, so the only reasonable limits are 0 and 1. The formal proof is below.\\nAs B.3. The en23 behavior. We want to prove P{Z n < exp(−n 2/3 )} < 1 − H 0 − (−1/2+4α)n+o(n) , namely Inequality (25), given the eigen behavior and the polarization in mean. The idea is to read off the behavior of {H n } from the behavior of {h α (H n )} in the eigen behavior. The formal proof is below.\\nThis simplifies the eigenvalue. Without loss of generality, we rescale h α such that h α (H 0 ) = 1. Let ε n be exp(−n 3/4 ); note that ε n H 0 1 − ε n for n large enough. Owing to h α 's concavity, that h α (0) = h(1) = 0, and that h α (H 0 ) = 1, we deduce that h α (z) ε n whenever ε n z 1 − ε n . Consider these three events as a partition: let A n be {H n < ε n }; let B n be {ε n H n 1 − ε n }; let C n be {1 − ε n < H n }. Note that B n implies h α (H n ) ε n .\\nNext we show P(B n ) < (−1/2+4α)n+o(n) : Telescoping leads to E[h α (H n )] h a (H 0 ) (−1/2+4α)n = (−1/2+4α)n . Markov's inequality leads to P{h(H n ) ε n } E[h(H n )]/ε n (−1/2+4α)n /ε n = −(−1/2+4α)n+O(n 3/4 ) < (−1/2+4α)n+o(n) . Therefore P(B n ) P{h(H n ) ε n } < (−1/2+4α)n+o(n) , as desired. Moreover, summing the geometric series leads to m n P(B m ) < (−1/2+4α)n+o(n) .\\nNext we show 1 − H 0 − P(A n ) < (−1/2+4α)n+o(n) : The left hand side is at most the probability measure of {H ∞ = 0} \\\\ A n . That is the probability that H n was not small (not in A n ) but H n+1 , H n+1 , . . . will end up converging to 0. Note that being a martingale causes 1 − H n+1\\n(1 − H n ), which forbids H n jumping from C n directly into A n+1 -it must pass by B m for some m n before ever landing in A m+1 . From the summation of P(B m ) over m n we know that very few descendants of H n can do that; the probability measure of {H ∞ = 0} \\\\ A n is less than (−1/2+4α)n+o(n) . Therefore 1 − H 0 − P(A n ) < (−1/2+4α)n+o(n) and hence P{H n < exp(−n 3/4 )} = P(A n ) > 1 − H 0 − (−1/2+4α)n+o(n) . Pay the bi-Hölder toll P{Z n < exp(−n 2/3 )} > 1 − H 0 − (−1/2+4α)n+o(n) . And the proof of the en23 behavior, Inequality (25), is sound.\\nhere is an aesthetic choice; min( −2 , 2+ε √ Z n ) is also a supermartingale but only for astronomic (depending on q). For any non-random kernel, a small enough power works provided that the kernel polarizes channels in the first place.\\nC.2. A Cramér-Chernoff gadget. Let D n+1 be K 2 n+1 /3 · 3/4. We want to prove inequalities in (27), that Z n < −8 implies Z n+1 Z Dn+1 n and E [D −1/2 n+1 | F n ] < −1/2+2α , given the local LDP behavior. The motivation is to reformat the local inequalities so that it is easy to telescope for future reference. The formal proof is below.\\nThey are both local statements so we may assume n = 0. When Z 0 < −8 , invoke the local LDP behavior Z 1 exp(qZ 0 )(qZ 0 ) K 2\\n< 2 −1/2 + 2 −1/2 log = 2 −1/2 + 2 −1/2+α < 4 −1/2+α < −1/2+2α .\\nThe last inequality uses e 4 . This validates the second inequality in (27). And the proof of inequalities in (27) is sound.\\nC.3. The een13 Behavior. We want to prove P{Z n < exp(−e n 1/3 )} > 1 − H 0 − (−1/2+4α)n+o(n) , namely Inequality (28), given the en23 behavior, the supermartingale property, and the Cramér-Chernoff gadget. The idea is to apply the gadget consecutively to show that Z n becomes smaller and smaller as n increases. To reach the goal exp(−e n 1/3 ), we apply √ n times to avoid losing too much code rate. /a m from above.) Conditioning on A m , we want to estimate the probability that Z k −8 for some k m, which is equal to the probability that min( −2 , 4 √ Z k ) −2 for some k m. Recall that min( −2 , 4 √ Z k ) was made a supermartingale. Hence by Doob's optional stopping theorem [Dur19, Exercise 4.8.2], P{min( −2 , 4 √ Z k ) −2 for some k m | A m } min( −2 , 4 √ Z m ) 2 < exp(−m 2/3 /4) 2 . This is an upper bound on b m /a m and will be summoned in Formula (32).\\n(Bound c m /a m from above.) We want to estimate how often does Inequality (31) happen. It is the probability of (D m+1 D m+2 · · · D m+\\nThe first three equalities are by the definitions of g m and E m 0 . The next equality is simple algebra. The next two inequalities are by 0 e m /a m 1. The next inequality is by the definition of E m . The last inequality summons upper bounds derived in the last few paragraphs. The last line contains two terms in the big parentheses. Between them (−1/2+3α) happens. More precisely, we attempt to bound Z m+ √ n when E m happens for each m = √ n, 2 √ n, . . . , n − √ n. When E m happens, its superevent A m happens, so we know that Z m < exp(−m 2/3 ). But B m does not happen, so Z k < −8 for all k m. This implies that Z k+1 Z D k+1 k for those k. Telescope; Z m+ √ n is less than Z m raised to the power of D m+1 D m+2 · · · D m+ √ n . But C m does not happen, so the product is greater than 2α eqZ k for all k m + √ n so long as Z k stays below −8 , which it does because B m is excluded. Then telescope again; Z n ( eq) n−m− √ n Z m+ √ n < ( eq) n exp(−m 2/3 2α √ n ) < exp(−e n 1/3 ) provided that n is sufficiently large. In other words, E n− √ n 0 implies Z n < exp(−e n 1/3 ).\\n(Summary.) Now we conclude that P{Z n < exp(−e n 1/3 )} P(E n− √ n 0 ) = e n 0 > 1 − H 0 − (−1/2+4α)n+o(n) . And hence the proof of the een13 behavior, Inequality (28), is sound.\\nThis subsection is parallel to [WD18, Section V]. Do not confuse this subsection with the next. The subtlety is explained in [WD18, Section III].\\nC.4. The elpin behavior. Recall π, ρ > 0 is such that π + 2ρ 1 − 8α. We want to prove P{Z n < exp(− πn n 2 )} > 1 − H 0 − −ρn+o(n) , namely Inequality (29), given the een13 behavior, the supermartingale property, and the Cramér-Chernoff gadget. The idea is to apply the gadget consecutively to show that Z n becomes smaller and smaller as n increases. To reach the goal exp(− πn ), we apply as many times as possible before we run out of depth n. (Bound c m /a m from above.) We want to estimate how often does Inequality (33) happen. It is the probability of (D m+1 D m+2 · · · D n ) −1/2 −πn/2 . By Markov's inequality, this probability is at most E [D −1/2 1 ] n−m πn/2 < (−1/2+2α)(n−m) πn/2 = (1/2−2α)m−(1/2−2α−π/2)n (1/2−2α)m−(ρ+2α)n . The last inequality uses Inequality (6), π + 2ρ 1 − 8α. This is an upper bound on c m /a m and will be summoned in Formula (34).\\n(Bound f + m from above.) The definition of f m reads 1 − H 0 − a m 0 . Here a m 0 is the probability measure of A m 0 , and A m 0 is a superevent of A m by how the former is defined. Event A m 0 must contain {Z m < exp(−e m 1/3 )} by how A m was defined. By the een13 behavior, P{Z m < exp(−e m 1/3 )} > 1 − H 0 − (−1/2+4α)m+o(m) . Chaining all inequalities together, we infer that f m < (−1/2+4α)m+o(m) . Let f + m be max(0, f m+ √ n ) so we can write f + m < (−1/2+4α)m+o(m) . This upper bound will be summoned in Formula (34).\\n(Bound e n 0 from below.) We start rewriting g m − f + m with (f m− happens. More precisely, we attempt to bound Z n when E m happens for each m = √ n, 2 √ n, . . . , n − √ n. When E m happens, its superevent A m happens, so we know that Z m < exp(−e m 1/3 ). But B m does not happen, so Z k < −8 for all k m. This implies Z k+1 Z D k+1 k for those k. Telescope; Z n is less than Z m raised to the power of D m+1 D m+2 · · · D n . But C m does not happen, so the product is greater than πn . Jointly we have Z n Z πn m < exp(−e m 1/3 πn ) < exp(− πn n 2 ). In other words, E n− √ n 0 implies Z n < exp(− πn n 2 ). (Summary.) Now we conclude that P{Z n < exp(− πn n 2 )} P(E n− √ n 0 ) = e n 0 > 1 − H 0 − −ρn+o(n) . And hence the proof of the elpin behavior, Inequality (29), is sound.\\nThis subsection is parallel to [WD18, Section VI]. Do not confuse this subsection with the previous. The subtlety is explained in [WD18, Section III].\\nAs we finish proving Inequalities (23) to (29), we finish the proof of Lemma 12. Lemmas 12 to 14 are all finished. This is the last sentence of the proof of the main theorem.\"\n },\n {\n \"title\": \"Appendix D. Constants Dependence Summary\",\n \"content\": \"Given a discrete memoryless channel W . The sender chooses the message alphabet size ς 2. Depending on the factorization of ς, we choose q to be a certain prime power or alternate between q 2 , q 3 , q 5 , . . . (a finite list depending on ς). Fix a q. Given π, ρ > 0 such that π + 2ρ < 1; fix them. Choose ; this also determines α := log(log )/ log . The choice of is such that π + 2ρ 1 − 8α and such that the failing probabilities in Lemmas 13 and 14 do not sum to one. It depends on q, π, ρ. Once is fixed, the complexity is a function in n (or in N = n ). The asymptotic complexity O(N log N ) hides the scalar term that is determined by q and . The decaying gap −ρn+o(n) in Claim 11 and Lemma 12 hides two things: A scalar term in front of determined by q and alongside with a O(n 1−ε ) term determined by the choice of en23 and een13 checkpoints. This ε is fixed throughout the paper and is irrespective of ς, π, ρ, q, .\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"7. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 17 (\",\n \"content\": \"\"\n },\n {\n \"title\": \".∪\",\n \"content\": \"\"\n },\n {\n \"title\": \"+∪\",\n \"content\": \"\"\n },\n {\n \"title\": \"∪\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Recent works on polar coding arranged on a ρ-π plot.","content":"Figure 1 .\n1Recent works on polar coding arranged on a ρ-π plot."},{"id":"","caption":"Conditions (cn), (cp), and (cr) cannot hold simultaneously.Proof. If so, N (I − R) 2 /|log P e | −N N −2ρ /N π = N 1−2ρ−π → 0 as N → ∞. This contradicts lim inf N →∞ N (I − R) 2 /|log P e | 2V > 0 [AW10,Theorem 2.2] [PV10, Theorem 6] [AW14, Theorem 2]. Remark: For V = 0 channels, the correct threshold seems to be π + ρ = 1 [Pol10, Inequality (3.354)] [AW14, Remark 1].","content":"Conditions (cn), (cp), and (cr) cannot hold simultaneously.Proof. If so, N (I − R) 2 /|log P e | −N N −2ρ /N π = N 1−2ρ−π → 0 as N → ∞. This contradicts lim inf N →∞ N (I − R) 2 /|log P e | 2V > 0 [AW10,Theorem 2.2] [PV10, Theorem 6] [AW14, Theorem 2]. Remark: For V = 0 channels, the correct threshold seems to be π + ρ = 1 [Pol10, Inequality (3.354)] [AW14, Remark 1]."},{"id":"","caption":"Figure 4. 12 DUs (with = 2) are chained together to implement ((W (1) ) (1) ) (1) , . . . , ((W (2) ) (2) ) (2) . DUs in the first column use g W ; DUs in the second column use g W (1) and g W (2) ; DUs in the third column use g (W (1) ) (1) , g (W (1) ) (2) , g (W (2) ) (1) , and g (W (2) ) (2) .","content":"Figure 4\n4Figure 4. 12 DUs (with = 2) are chained together to implement ((W (1) ) (1) ) (1) , . . . , ((W (2) ) (2) ) (2) . DUs in the first column use g W ; DUs in the second column use g W (1) and g W (2) ; DUs in the third column use g (W (1) ) (1) , g (W (1) ) (2) , g (W (2) ) (1) , and g (W (2) ) (2) ."},{"id":"","caption":"but rather a mixture of Z max and Z d therein. That said, the definitions of other parameters-H, I, P e , Z, T , S, and S max -match [MT14]'s. Cf. [Sas11, Section 3.C].","content":"but rather a mixture of Z max and Z d therein. That said, the definitions of other parameters-H, I, P e , Z, T , S, and S max -match [MT14]'s. Cf. [Sas11, Section 3.C]."},{"id":"","caption":"(a, b) if there exists c, d > 0 such that |A(W )−a| < c|B(W )−b| d and |B(W )−b| < c|A(W )−a| d for all q-ary channels W . The notion of bi-Hölder is an equivalence relation. In particular, if A, B are bi-Hölder at (a, b) and (B, C) are bi-Hölder at (b, c), then (A, C) are bi-Hölder at (a, c). In this case, it makes sense to say A, B, C are bi-Hölder at (a, b, c)","content":"(a, b) if there exists c, d > 0 such that |A(W )−a| < c|B(W )−b| d and |B(W )−b| < c|A(W )−a| d for all q-ary channels W . The notion of bi-Hölder is an equivalence relation. In particular, if A, B are bi-Hölder at (a, b) and (B, C) are bi-Hölder at (b, c), then (A, C) are bi-Hölder at (a, c). In this case, it makes sense to say A, B, C are bi-Hölder at (a, b, c)"},{"id":"","caption":"Theorem 10 (fundamental theorem of polar coding-the S-end, FTPCS). [Kor09, Lemma 5.7] [KU10, Theorem 19] [KT10, Lemma 6] [MT14, Lemma 34] [GRY19, Inequalities (74) and (75)]","content":"Theorem 10 (fundamental theorem of polar coding-the S-end, FTPCS). [Kor09, Lemma 5.7] [KU10, Theorem 19] [KT10, Lemma 6] [MT14, Lemma 34] [GRY19, Inequalities (74) and (75)]"},{"id":"","caption":"The trichotomy of the fates of synthetic channels. Label (A) marks the corner of the free and reliable channels. Label I(W ) beneath (A) is the limit of the probability measure P(A n ) = P{W n is free and reliable} as n → ∞. Labels (B) and (C) and the numbers beneath marks the corresponding fates and probability measures.","content":"Figure 5 .\n5The trichotomy of the fates of synthetic channels. Label (A) marks the corner of the free and reliable channels. Label I(W ) beneath (A) is the limit of the probability measure P(A n ) = P{W n is free and reliable} as n → ∞. Labels (B) and (C) and the numbers beneath marks the corresponding fates and probability measures."},{"id":"","caption":". The following two lemmas help verify the two criteria in Lemma 12. Proofs are given in upcoming sections, 7 and 8.Lemma 13 (local LDP behavior). Fix an 30. Let G vary; with probability less than 3q − √ /13 , each of the following fails for each i ∈ [ ]:","content":". The following two lemmas help verify the two criteria in Lemma 12. Proofs are given in upcoming sections, 7 and 8.Lemma 13 (local LDP behavior). Fix an 30. Let G vary; with probability less than 3q − √ /13 , each of the following fails for each i ∈ [ ]:"},{"id":"","caption":"Hence the process {H n } satisfies Inequality (7) for the four processes: {H(W n )} and {1 − H(W n )} along with {H(V n )} as well as {1 − H(V n )}. This results in the four inequalities in Claim 11. And we are done. It remains to prove Lemmas 12 to 14 in order to prove the main theorem. 7. Local LDP Behavior (Proof of Lemma 13)","content":"Hence the process {H n } satisfies Inequality (7) for the four processes: {H(W n )} and {1 − H(W n )} along with {H(V n )} as well as {1 − H(V n )}. This results in the four inequalities in Claim 11. And we are done. It remains to prove Lemmas 12 to 14 in order to prove the main theorem. 7. Local LDP Behavior (Proof of Lemma 13)"},{"id":"","caption":"GZ (Z mad (W )). By the definition of W (i) and the definition of the Bhattacharyya parameter, Z mad (W","content":"GZ (Z mad (W )). By the definition of W (i) and the definition of the Bhattacharyya parameter, Z mad (W"},{"id":"","caption":"GS (S max (W )). And quod erat demonstrandum. Experienced readers may find that all but the last inequality is a duplicate of [MT14, Lemma 34]. 7.3. An upper bound on entropy functions. For all z ∈ [0, 1],","content":"GS (S max (W )). And quod erat demonstrandum. Experienced readers may find that all but the last inequality is a duplicate of [MT14, Lemma 34]. 7.3. An upper bound on entropy functions. For all z ∈ [0, 1],"},{"id":"","caption":"To the left: Binary entropy function h 2 (z) and an upper bound of √ ez. To the right: 1 − D(z 1 − 1/q)/ log q for q = 3, 4, 5, 7 and an upper bound of √ ez.","content":"Figure 6 .\n6To the left: Binary entropy function h 2 (z) and an upper bound of √ ez. To the right: 1 − D(z 1 − 1/q)/ log q for q = 3, 4, 5, 7 and an upper bound of √ ez."},{"id":"","caption":"On the weight distribution of random linear codes. This subsection contains the nontrivial part of the proof of Lemma 13. Fix any i ∈ [ ]. We want to prove that when G ∈ GL( , q) is selected uniformly at random, the inequalityZ mad (W (i) G )exp(qZ mad (W ) )(qZ mad (W )) . In bounding the left hand side, the fundamental theorem of polar coding-Z-end reads Z mad (W","content":"7. 4 .\n4On the weight distribution of random linear codes. This subsection contains the nontrivial part of the proof of Lemma 13. Fix any i ∈ [ ]. We want to prove that when G ∈ GL( , q) is selected uniformly at random, the inequalityZ mad (W (i) G )exp(qZ mad (W ) )(qZ mad (W )) . In bounding the left hand side, the fundamental theorem of polar coding-Z-end reads Z mad (W"},{"id":"","caption":"the weight enumerator of codewords of the form 0 i−1 1 1u i+1 G. Thus it remains to show the inequality with the left hand side replaced f (i) GZ (z) exp(qz )(qz) i 2 /3","content":"the weight enumerator of codewords of the form 0 i−1 1 1u i+1 G. Thus it remains to show the inequality with the left hand side replaced f (i) GZ (z) exp(qz )(qz) i 2 /3"},{"id":"","caption":"(q −1)z) −1 (q −1)z. The right hand side over counts all nonzero codewords by choosing a nonzero position ( ), assigning a nonzero symbol ((q − 1)z), and filling in the rest of − 1 blanks arbitrarily ((1 + (q − 1)z) −1 ). On the left hand side, f (i) GZ enumerates only codewords of the form 0 i−1 1 1 i u i+1 G, which are all nonzero as G is invertible. Hence Inequality (12) holds for i √ 3 and nonnegative z regardless of what kernel G is in effect.","content":"(q −1)z) −1 (q −1)z. The right hand side over counts all nonzero codewords by choosing a nonzero position ( ), assigning a nonzero symbol ((q − 1)z), and filling in the rest of − 1 blanks arbitrarily ((1 + (q − 1)z) −1 ). On the left hand side, f (i) GZ enumerates only codewords of the form 0 i−1 1 1 i u i+1 G, which are all nonzero as G is invertible. Hence Inequality (12) holds for i √ 3 and nonnegative z regardless of what kernel G is in effect."},{"id":"","caption":"For non-uniform inputs, W in (x) does not penetrate the summations. The E-null function and its complement deeply associate to the following family of measures: For any t ∈ [−2/5, 1], define the t-tilted probability mass function W t : X × Y → [0, 1] as in [CS07, Definition 1] W t (x, y) := ξ∈X W (ξ, y) 1/(1+t) 1+t η∈Y ξ∈X W (ξ, η) 1/(1+t) 1+t × W (x, y) 1/(1+t) ξ∈X W (ξ, y) 1/(1+t) .","content":"For non-uniform inputs, W in (x) does not penetrate the summations. The E-null function and its complement deeply associate to the following family of measures: For any t ∈ [−2/5, 1], define the t-tilted probability mass function W t : X × Y → [0, 1] as in [CS07, Definition 1] W t (x, y) := ξ∈X W (ξ, y) 1/(1+t) 1+t η∈Y ξ∈X W (ξ, η) 1/(1+t) 1+t × W (x, y) 1/(1+t) ξ∈X W (ξ, y) 1/(1+t) ."},{"id":"","caption":"universal quadratic bound). [CS07, Theorem 2]. Cf. [Gal65, Theorem 5.6.3]. Let W be a q-ary channel. Assume the uniform input distribution. Then Gallager's E-null function satisfies E 0 (0) = 0, E 0 (0) = I(W ) log q, and E 0 (t) −2 log(q) 2for all t ∈ [−2/5, 1]. In particular, it satisfies E 0 (t) I(W )t log q − t 2 log(q) 2 .","content":"Lemma 17 (\n17universal quadratic bound). [CS07, Theorem 2]. Cf. [Gal65, Theorem 5.6.3]. Let W be a q-ary channel. Assume the uniform input distribution. Then Gallager's E-null function satisfies E 0 (0) = 0, E 0 (0) = I(W ) log q, and E 0 (t) −2 log(q) 2for all t ∈ [−2/5, 1]. In particular, it satisfies E 0 (t) I(W )t log q − t 2 log(q) 2 ."},{"id":"","caption":"we divide it into three cases per how H 0 compares to −2 and 1 − −2 . The mediocre case:if −2 H 0 1 − −2 then h α (H 0 ) −2α . The local CLT behavior implies E [h α (H 1 )] < 4 −1/2+α = 4 −1/2+3α −2α 4 −1/2+3α h α (H 0 )and we are done with this case. The noisy case: if H 0 > 1 − −2 , we replace (H, Z , S) by (1 − H, S, Z ) to dual it to the reliable case dealt below and we are done with this case. (This is the only place in the proof where we ever mentioned S explicitly. Nevertheless, every statement concerning Z concerns S by duality.)","content":"we divide it into three cases per how H 0 compares to −2 and 1 − −2 . The mediocre case:if −2 H 0 1 − −2 then h α (H 0 ) −2α . The local CLT behavior implies E [h α (H 1 )] < 4 −1/2+α = 4 −1/2+3α −2α 4 −1/2+3α h α (H 0 )and we are done with this case. The noisy case: if H 0 > 1 − −2 , we replace (H, Z , S) by (1 − H, S, Z ) to dual it to the reliable case dealt below and we are done with this case. (This is the only place in the proof where we ever mentioned S explicitly. Nevertheless, every statement concerning Z concerns S by duality.)"},{"id":"","caption":"6 exp(6q 4 )(q 8 H 0 ) log( ) 5 6 log −30 H −12 0 = q 36+8 log( ) 5 e 6q 4 6 log −24 H log( ) 5 −12 0 < q 36+8 log( ) 5 e 6q 4 6 log −24 −2 log( ) 5 +24 = q 36+8 log( ) 5 e 6q 4 6 log −1.6 log( ) 5 −0.4 log( ) 5 q 36+8 log( ) 5 e 6q 4 6 log −8 log(q) log( ) 4 −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log( ) 6 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log(41) 2 (q log 3) 4 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 6.02q 4 < q 36 6 log −0.1 log( ) 5 = exp(36 log q + 6 log( ) 2 − log( ) 6 /30 − log( ) 6 /15) = exp(36 log q + 6 log( ) 2 − 5 log(q) log(22) 5 /30 − log(22) 4 log( ) 2 /15) < exp(0) 1.","content":"6 exp(6q 4 )(q 8 H 0 ) log( ) 5 6 log −30 H −12 0 = q 36+8 log( ) 5 e 6q 4 6 log −24 H log( ) 5 −12 0 < q 36+8 log( ) 5 e 6q 4 6 log −24 −2 log( ) 5 +24 = q 36+8 log( ) 5 e 6q 4 6 log −1.6 log( ) 5 −0.4 log( ) 5 q 36+8 log( ) 5 e 6q 4 6 log −8 log(q) log( ) 4 −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log( ) 6 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log(41) 2 (q log 3) 4 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 6.02q 4 < q 36 6 log −0.1 log( ) 5 = exp(36 log q + 6 log( ) 2 − log( ) 6 /30 − log( ) 6 /15) = exp(36 log q + 6 log( ) 2 − 5 log(q) log(22) 5 /30 − log(22) 4 log( ) 2 /15) < exp(0) 1."},{"id":"","caption":"(Define events.) Let E 0 0 be the empty event. For every m = √ n, 2 √ n, . . . , n− √ n, we define five series of events A m , B m , C m , E m , and E m 0 inductively as below: LetA m be {Z m < exp(−m 2/3 )} \\ E Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D m+ m be A m \\ (B m ∪ C m ). Let E m 0 be E E m . Let a m , b m , c m ,e m , and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let g m be 1 − H 0 − e m 0 . (Bound b m","content":".∪\n(Define events.) Let E 0 0 be the empty event. For every m = √ n, 2 √ n, . . . , n− √ n, we define five series of events A m , B m , C m , E m , and E m 0 inductively as below: LetA m be {Z m < exp(−m 2/3 )} \\ E Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D m+ m be A m \\ (B m ∪ C m ). Let E m 0 be E E m . Let a m , b m , c m ,e m , and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let g m be 1 − H 0 − e m 0 . (Bound b m"},{"id":"","caption":"Markov's inequality. This is an upper bound on c m /a m and will be summoned in Formula (32).(Bound (g m−√ n − a m ) + from above.) By definition, a m ). The definition of A m forces it to be disjoint from E A m . This union event must contain the event {Z m < exp(−m 2/3 )} by how A m was defined. From the en23 behavior P{Z m < exp(−m 2/3 )} > 1 − H 0 − (−1/2+4α)m . Chaining all inequalities together, we deduce that g m− √ n − a m < (−1/2+4α)m+o(m) . Let (g m− √ n − a m ) + be max(0, g m− √ n − a m ) so we can write (g m− √ n − a m ) + < (−1/2+4α)m+o(m) . This upper bound will be summoned in Formula (32).(Bound e n 0 from below.) We start rewriting g m with g + m being max(0, g m ):g m = 1 − H 0 − e m 0 = 1 − H 0","content":"+∪\nMarkov's inequality. This is an upper bound on c m /a m and will be summoned in Formula (32).(Bound (g m−√ n − a m ) + from above.) By definition, a m ). The definition of A m forces it to be disjoint from E A m . This union event must contain the event {Z m < exp(−m 2/3 )} by how A m was defined. From the en23 behavior P{Z m < exp(−m 2/3 )} > 1 − H 0 − (−1/2+4α)m . Chaining all inequalities together, we deduce that g m− √ n − a m < (−1/2+4α)m+o(m) . Let (g m− √ n − a m ) + be max(0, g m− √ n − a m ) so we can write (g m− √ n − a m ) + < (−1/2+4α)m+o(m) . This upper bound will be summoned in Formula (32).(Bound e n 0 from below.) We start rewriting g m with g + m being max(0, g m ):g m = 1 − H 0 − e m 0 = 1 − H 0"},{"id":"","caption":"we define six series of events A m , A m 0 , B m , C m , E m , and E m 0 inductively as follows:Let A m be {Z m < exp(−e m 1/3 )} \\ A A m . Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D n πn . (33) Let E m be A m \\(B m ∪C m ). Let E m 0 be E m− √ n 0 ∪E m . Let a m , a m 0 , b m , c m , e m ,and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let f m be 1 − H 0 − a m 0 and let g m be 1 − H 0 − e m 0 . (Bound b m /a m from above.) Conditioning on A m , we want to estimate the probability that Z k −8 for some k m, which is equal to the probability that min( −2 , 4 √ Z k ) −2 for some k m. Recall that min( −2 , 4 √ Z k ) was made a supermartingale. Hence by Doob's optional stopping theorem [Dur19, Exercise 4.8.2], P{min( −2 , 4 √ Z k ) −2 for some k m | A m } min( −2 , 4 √ Z m ) 2 < exp(−e m 1/3 /4) 2 . This is an upper bound on b m /a m and will be summoned in Formula (34).","content":"∪\nwe define six series of events A m , A m 0 , B m , C m , E m , and E m 0 inductively as follows:Let A m be {Z m < exp(−e m 1/3 )} \\ A A m . Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D n πn . (33) Let E m be A m \\(B m ∪C m ). Let E m 0 be E m− √ n 0 ∪E m . Let a m , a m 0 , b m , c m , e m ,and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let f m be 1 − H 0 − a m 0 and let g m be 1 − H 0 − e m 0 . (Bound b m /a m from above.) Conditioning on A m , we want to estimate the probability that Z k −8 for some k m, which is equal to the probability that min( −2 , 4 √ Z k ) −2 for some k m. Recall that min( −2 , 4 √ Z k ) was made a supermartingale. Hence by Doob's optional stopping theorem [Dur19, Exercise 4.8.2], P{min( −2 , 4 √ Z k ) −2 for some k m | A m } min( −2 , 4 √ Z m ) 2 < exp(−e m 1/3 /4) 2 . This is an upper bound on b m /a m and will be summoned in Formula (34)."},{"id":"","caption":"a m ):g m − f + m = 1 − H 0 − e m − (1 − H 0 − a m ) + = g m− √ n − e m − (f m− √ n − a m ) + g m− √ n − e m −e m a m (f m− √ n − a m ) + g m− √ n − e m − two equalities are by the definitions of g m and f m . The next inequality is by 0 e m /a m 1. The next inequality is by max(0, f − a) = max(a, f ) − a max(0, f ) − a. The next equality is simple algebra. The next inequality is by the definition of E m . The last inequality summons upper bounds derived in the last few paragraphs. Now the last line contains two terms in the big parentheses. Between them, (1/2−2α)m−(ρ+2α)n dominates exp(−e m 1/3 /4) 2 once n → ∞.","content":"a m ):g m − f + m = 1 − H 0 − e m − (1 − H 0 − a m ) + = g m− √ n − e m − (f m− √ n − a m ) + g m− √ n − e m −e m a m (f m− √ n − a m ) + g m− √ n − e m − two equalities are by the definitions of g m and f m . The next inequality is by 0 e m /a m 1. The next inequality is by max(0, f − a) = max(a, f ) − a max(0, f ) − a. The next equality is simple algebra. The next inequality is by the definition of E m . The last inequality summons upper bounds derived in the last few paragraphs. Now the last line contains two terms in the big parentheses. Between them, (1/2−2α)m−(ρ+2α)n dominates exp(−e m 1/3 /4) 2 once n → ∞."},{"id":"","caption":"showed the equivalence of","content":"showed the equivalence of1/2 \n\n0 \n0 \n1 \n\nconjectured boundary \nfor V = 0 \n\n[FHMV17, GRY19] \n\nThm. 1 \n[YFV19] \n[HAU14] \n\n[WD18] \n\n[MHU16] \n\n[KSU10, MT14] \n[GX13] \n\n[BGN + 18] \n\n[BGS18] \n[AT09, Sas11, HY13]"},{"id":"","caption":"Prudent readers are invited to check [ŞTA09a, STA09b, the paragraph before Section III] [YB15, PSL16, EKMF + 17, WD18, GRY19] for a list of inhomogeneous configurations of kernels. See [ŞTA09a, STA09b, MT10, Sas11] for how nonlinear bijections are similar to (or different from) linear bijections.","content":"Prudent readers are invited to check [ŞTA09a, STA09b, the paragraph before Section III] [YB15, PSL16, EKMF + 17, WD18, GRY19] for a list of inhomogeneous configurations of kernels. See [ŞTA09a, STA09b, MT10, Sas11] for how nonlinear bijections are similar to (or different from) linear bijections."},{"id":"","caption":"et seq. utilizing the same input distributions and series of bijections. For instance,","content":"et seq. utilizing the same input distributions and series of bijections. For instance,"},{"id":"","caption":"Theorem 9 (fundamental theorem of polar coding-the Z-end, FTPCZ). [Ari09, Proposition 5] [KSU10, Lemma 10] [Sas11, Lemma 3.5] [FHMV17, Section 4.1] [MT14, Lemma 33]","content":"Theorem 9 (fundamental theorem of polar coding-the Z-end, FTPCZ). [Ari09, Proposition 5] [KSU10, Lemma 10] [Sas11, Lemma 3.5] [FHMV17, Section 4.1] [MT14, Lemma 33]"},{"id":"","caption":"The formulation and the choice of the variables make it clear how Lemma 12 will be applied to support Claim 11. For instance, if we let {H n }, {Z n } and {S n } be {H(W n )}, {Z mad (W n )}, and {S max (W n )}, respectively, then Lemma 12 supports the first of the four inequalities in Claim 11. Moreover, if we let {H n }, {Z n }, and {S n } be {1 − H(W n )}, {S max (W n )}, and {Z mad (W n )}, then Lemma 12 supports the second inequality of Claim 11. If {H n }, {Z n }, and {S n } are let to be {H(V n )}, {Z mad (V n )}, and {S max (V n )}, that supports the third inequality. If {H n }, {Z n }, and {S n } are let to be {1 − H(V n )}, {S max (V n )}, and {Z mad (V n )}, that supports the forth inequality. The criteria (cb), (cm), (ct), and (cl) listed above Lemma 12 are easy to verify; for instance, Lemma 8 implies (ct) for all four cases. It remains to show that for each of the four triples of processes, the local LDP behavior and the local CLT behavior hold.","content":"The formulation and the choice of the variables make it clear how Lemma 12 will be applied to support Claim 11. For instance, if we let {H n }, {Z n } and {S n } be {H(W n )}, {Z mad (W n )}, and {S max (W n )}, respectively, then Lemma 12 supports the first of the four inequalities in Claim 11. Moreover, if we let {H n }, {Z n }, and {S n } be {1 − H(W n )}, {S max (W n )}, and {Z mad (W n )}, then Lemma 12 supports the second inequality of Claim 11. If {H n }, {Z n }, and {S n } are let to be {H(V n )}, {Z mad (V n )}, and {S max (V n )}, that supports the third inequality. If {H n }, {Z n }, and {S n } are let to be {1 − H(V n )}, {S max (V n )}, and {Z mad (V n )}, that supports the forth inequality. The criteria (cb), (cm), (ct), and (cl) listed above Lemma 12 are easy to verify; for instance, Lemma 8 implies (ct) for all four cases. It remains to show that for each of the four triples of processes, the local LDP behavior and the local CLT behavior hold."},{"id":"","caption":"8.3. Gallager's argument at Bob's end. This subsection take advantage of the universal bound developed three lines ago and starts actually proving Lemma 14.","content":"8.3. Gallager's argument at Bob's end. This subsection take advantage of the universal bound developed three lines ago and starts actually proving Lemma 14.This subsection deals with \n\ni= H(W ) + 1/2+α +1 \n\nh α (H(W \n\n(i) \n\nG )) < −1/2+α \n(14's copy) \n\nby passing it to an inequality that captures the performance of noisy channel coding. \nOwing to h α 's concavity, the left hand side of Inequality (14) is \n\ni=j+1"},{"id":"","caption":"a bounded martingale {H n } converges to an r.v.-which we call H ∞ -a.s. (almost surely). This is Doob's martingale convergence theorem [Dur19, Theorem 4.2.11]. Owing to h α 's continuity, h α (H n ) → h α (H ∞ ) a.s. Point-wise convergence and (uniform) boundedness imply convergence in L 1 , i.e., E [h α (H n )] → E [h α (H ∞ )] as n → ∞. This is Lebesgue's dominated convergence theorem [Dur19, Theorem 1.6.7]. By the eigen behavior, E [h α (H n )] decays toward 0 by a constant factor every time n increases, thus E [h α (H ∞ )] is 0. This forces h a (H ∞ ) = 0 a.s. and hence H ∞ ∈ {0, 1} a.s. Since H ∞ is Bernoulli P{H ∞= 0} = E [I{H ∞ = 0}] = E [1 − H ∞ ] ← E [1 − H n ] = 1 − H 0 . SoP{H n → 0} = 1 − H 0 . By the implicit bi-Hölder toll, H n → 0 if and only if Z n → 0, thus the latter has the same probability measure. And the proof of Equality (24) is sound. Bibliographic remarks: The statement H n → H ∞ ∈ {0, 1} is usually referred to as channel polarization in spite of that it does not guarantee the corresponding codes to be capacity-achieving. See also [Ari09, Proposition 10] [MT14, Definition 3] [Sas11, Lemma 3.8]. This lemma should have been bestowed upon the fundamental theorem but it is not mandatory if some sort of CLT behavior is present. See [MHU16, Lemma 1] [FHMV17, Lemma 4] [GRY19, Lemma 9.5]. Recently, Reed-Muller codes' channels are shown to polarize [AY19]; Reed-Muller codes achieving capacity is not a consequence, but a different story.","content":"a bounded martingale {H n } converges to an r.v.-which we call H ∞ -a.s. (almost surely). This is Doob's martingale convergence theorem [Dur19, Theorem 4.2.11]. Owing to h α 's continuity, h α (H n ) → h α (H ∞ ) a.s. Point-wise convergence and (uniform) boundedness imply convergence in L 1 , i.e., E [h α (H n )] → E [h α (H ∞ )] as n → ∞. This is Lebesgue's dominated convergence theorem [Dur19, Theorem 1.6.7]. By the eigen behavior, E [h α (H n )] decays toward 0 by a constant factor every time n increases, thus E [h α (H ∞ )] is 0. This forces h a (H ∞ ) = 0 a.s. and hence H ∞ ∈ {0, 1} a.s. Since H ∞ is Bernoulli P{H ∞= 0} = E [I{H ∞ = 0}] = E [1 − H ∞ ] ← E [1 − H n ] = 1 − H 0 . SoP{H n → 0} = 1 − H 0 . By the implicit bi-Hölder toll, H n → 0 if and only if Z n → 0, thus the latter has the same probability measure. And the proof of Equality (24) is sound. Bibliographic remarks: The statement H n → H ∞ ∈ {0, 1} is usually referred to as channel polarization in spite of that it does not guarantee the corresponding codes to be capacity-achieving. See also [Ari09, Proposition 10] [MT14, Definition 3] [Sas11, Lemma 3.8]. This lemma should have been bestowed upon the fundamental theorem but it is not mandatory if some sort of CLT behavior is present. See [MHU16, Lemma 1] [FHMV17, Lemma 4] [GRY19, Lemma 9.5]. Recently, Reed-Muller codes' channels are shown to polarize [AY19]; Reed-Muller codes achieving capacity is not a consequence, but a different story."},{"id":"","caption":"The fourth inequality uses q 4 . That validates the first inequality in(27). For the second inequality,","content":"The fourth inequality uses q 4 . That validates the first inequality in(27). For the second inequality,E [D \n\n−1/2 \n1 \n\n] = \n1 \n\nk=1 \n\nk 2 \n3 \n· \n3 \n4 \n\n−1/2 \n\n< \n1 \n\n√ \n3 \n\nk=1 \n\n3 \n4 \n\n−1/2 \n\n+ \n1 \n\nk= \n√ \n3 +1"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Recent works on polar coding arranged on a ρ-π plot.\",\n \"content\": \"Figure 1 .\\n1Recent works on polar coding arranged on a ρ-π plot.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Conditions (cn), (cp), and (cr) cannot hold simultaneously.Proof. If so, N (I − R) 2 /|log P e | −N N −2ρ /N π = N 1−2ρ−π → 0 as N → ∞. This contradicts lim inf N →∞ N (I − R) 2 /|log P e | 2V > 0 [AW10,Theorem 2.2] [PV10, Theorem 6] [AW14, Theorem 2]. Remark: For V = 0 channels, the correct threshold seems to be π + ρ = 1 [Pol10, Inequality (3.354)] [AW14, Remark 1].\",\n \"content\": \"Conditions (cn), (cp), and (cr) cannot hold simultaneously.Proof. If so, N (I − R) 2 /|log P e | −N N −2ρ /N π = N 1−2ρ−π → 0 as N → ∞. This contradicts lim inf N →∞ N (I − R) 2 /|log P e | 2V > 0 [AW10,Theorem 2.2] [PV10, Theorem 6] [AW14, Theorem 2]. Remark: For V = 0 channels, the correct threshold seems to be π + ρ = 1 [Pol10, Inequality (3.354)] [AW14, Remark 1].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 4. 12 DUs (with = 2) are chained together to implement ((W (1) ) (1) ) (1) , . . . , ((W (2) ) (2) ) (2) . DUs in the first column use g W ; DUs in the second column use g W (1) and g W (2) ; DUs in the third column use g (W (1) ) (1) , g (W (1) ) (2) , g (W (2) ) (1) , and g (W (2) ) (2) .\",\n \"content\": \"Figure 4\\n4Figure 4. 12 DUs (with = 2) are chained together to implement ((W (1) ) (1) ) (1) , . . . , ((W (2) ) (2) ) (2) . DUs in the first column use g W ; DUs in the second column use g W (1) and g W (2) ; DUs in the third column use g (W (1) ) (1) , g (W (1) ) (2) , g (W (2) ) (1) , and g (W (2) ) (2) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"but rather a mixture of Z max and Z d therein. That said, the definitions of other parameters-H, I, P e , Z, T , S, and S max -match [MT14]'s. Cf. [Sas11, Section 3.C].\",\n \"content\": \"but rather a mixture of Z max and Z d therein. That said, the definitions of other parameters-H, I, P e , Z, T , S, and S max -match [MT14]'s. Cf. [Sas11, Section 3.C].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a, b) if there exists c, d > 0 such that |A(W )−a| < c|B(W )−b| d and |B(W )−b| < c|A(W )−a| d for all q-ary channels W . The notion of bi-Hölder is an equivalence relation. In particular, if A, B are bi-Hölder at (a, b) and (B, C) are bi-Hölder at (b, c), then (A, C) are bi-Hölder at (a, c). In this case, it makes sense to say A, B, C are bi-Hölder at (a, b, c)\",\n \"content\": \"(a, b) if there exists c, d > 0 such that |A(W )−a| < c|B(W )−b| d and |B(W )−b| < c|A(W )−a| d for all q-ary channels W . The notion of bi-Hölder is an equivalence relation. In particular, if A, B are bi-Hölder at (a, b) and (B, C) are bi-Hölder at (b, c), then (A, C) are bi-Hölder at (a, c). In this case, it makes sense to say A, B, C are bi-Hölder at (a, b, c)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 10 (fundamental theorem of polar coding-the S-end, FTPCS). [Kor09, Lemma 5.7] [KU10, Theorem 19] [KT10, Lemma 6] [MT14, Lemma 34] [GRY19, Inequalities (74) and (75)]\",\n \"content\": \"Theorem 10 (fundamental theorem of polar coding-the S-end, FTPCS). [Kor09, Lemma 5.7] [KU10, Theorem 19] [KT10, Lemma 6] [MT14, Lemma 34] [GRY19, Inequalities (74) and (75)]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The trichotomy of the fates of synthetic channels. Label (A) marks the corner of the free and reliable channels. Label I(W ) beneath (A) is the limit of the probability measure P(A n ) = P{W n is free and reliable} as n → ∞. Labels (B) and (C) and the numbers beneath marks the corresponding fates and probability measures.\",\n \"content\": \"Figure 5 .\\n5The trichotomy of the fates of synthetic channels. Label (A) marks the corner of the free and reliable channels. Label I(W ) beneath (A) is the limit of the probability measure P(A n ) = P{W n is free and reliable} as n → ∞. Labels (B) and (C) and the numbers beneath marks the corresponding fates and probability measures.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". The following two lemmas help verify the two criteria in Lemma 12. Proofs are given in upcoming sections, 7 and 8.Lemma 13 (local LDP behavior). Fix an 30. Let G vary; with probability less than 3q − √ /13 , each of the following fails for each i ∈ [ ]:\",\n \"content\": \". The following two lemmas help verify the two criteria in Lemma 12. Proofs are given in upcoming sections, 7 and 8.Lemma 13 (local LDP behavior). Fix an 30. Let G vary; with probability less than 3q − √ /13 , each of the following fails for each i ∈ [ ]:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Hence the process {H n } satisfies Inequality (7) for the four processes: {H(W n )} and {1 − H(W n )} along with {H(V n )} as well as {1 − H(V n )}. This results in the four inequalities in Claim 11. And we are done. It remains to prove Lemmas 12 to 14 in order to prove the main theorem. 7. Local LDP Behavior (Proof of Lemma 13)\",\n \"content\": \"Hence the process {H n } satisfies Inequality (7) for the four processes: {H(W n )} and {1 − H(W n )} along with {H(V n )} as well as {1 − H(V n )}. This results in the four inequalities in Claim 11. And we are done. It remains to prove Lemmas 12 to 14 in order to prove the main theorem. 7. Local LDP Behavior (Proof of Lemma 13)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"GZ (Z mad (W )). By the definition of W (i) and the definition of the Bhattacharyya parameter, Z mad (W\",\n \"content\": \"GZ (Z mad (W )). By the definition of W (i) and the definition of the Bhattacharyya parameter, Z mad (W\"\n },\n {\n \"id\": \"\",\n \"caption\": \"GS (S max (W )). And quod erat demonstrandum. Experienced readers may find that all but the last inequality is a duplicate of [MT14, Lemma 34]. 7.3. An upper bound on entropy functions. For all z ∈ [0, 1],\",\n \"content\": \"GS (S max (W )). And quod erat demonstrandum. Experienced readers may find that all but the last inequality is a duplicate of [MT14, Lemma 34]. 7.3. An upper bound on entropy functions. For all z ∈ [0, 1],\"\n },\n {\n \"id\": \"\",\n \"caption\": \"To the left: Binary entropy function h 2 (z) and an upper bound of √ ez. To the right: 1 − D(z 1 − 1/q)/ log q for q = 3, 4, 5, 7 and an upper bound of √ ez.\",\n \"content\": \"Figure 6 .\\n6To the left: Binary entropy function h 2 (z) and an upper bound of √ ez. To the right: 1 − D(z 1 − 1/q)/ log q for q = 3, 4, 5, 7 and an upper bound of √ ez.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"On the weight distribution of random linear codes. This subsection contains the nontrivial part of the proof of Lemma 13. Fix any i ∈ [ ]. We want to prove that when G ∈ GL( , q) is selected uniformly at random, the inequalityZ mad (W (i) G )exp(qZ mad (W ) )(qZ mad (W )) . In bounding the left hand side, the fundamental theorem of polar coding-Z-end reads Z mad (W\",\n \"content\": \"7. 4 .\\n4On the weight distribution of random linear codes. This subsection contains the nontrivial part of the proof of Lemma 13. Fix any i ∈ [ ]. We want to prove that when G ∈ GL( , q) is selected uniformly at random, the inequalityZ mad (W (i) G )exp(qZ mad (W ) )(qZ mad (W )) . In bounding the left hand side, the fundamental theorem of polar coding-Z-end reads Z mad (W\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the weight enumerator of codewords of the form 0 i−1 1 1u i+1 G. Thus it remains to show the inequality with the left hand side replaced f (i) GZ (z) exp(qz )(qz) i 2 /3\",\n \"content\": \"the weight enumerator of codewords of the form 0 i−1 1 1u i+1 G. Thus it remains to show the inequality with the left hand side replaced f (i) GZ (z) exp(qz )(qz) i 2 /3\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(q −1)z) −1 (q −1)z. The right hand side over counts all nonzero codewords by choosing a nonzero position ( ), assigning a nonzero symbol ((q − 1)z), and filling in the rest of − 1 blanks arbitrarily ((1 + (q − 1)z) −1 ). On the left hand side, f (i) GZ enumerates only codewords of the form 0 i−1 1 1 i u i+1 G, which are all nonzero as G is invertible. Hence Inequality (12) holds for i √ 3 and nonnegative z regardless of what kernel G is in effect.\",\n \"content\": \"(q −1)z) −1 (q −1)z. The right hand side over counts all nonzero codewords by choosing a nonzero position ( ), assigning a nonzero symbol ((q − 1)z), and filling in the rest of − 1 blanks arbitrarily ((1 + (q − 1)z) −1 ). On the left hand side, f (i) GZ enumerates only codewords of the form 0 i−1 1 1 i u i+1 G, which are all nonzero as G is invertible. Hence Inequality (12) holds for i √ 3 and nonnegative z regardless of what kernel G is in effect.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"For non-uniform inputs, W in (x) does not penetrate the summations. The E-null function and its complement deeply associate to the following family of measures: For any t ∈ [−2/5, 1], define the t-tilted probability mass function W t : X × Y → [0, 1] as in [CS07, Definition 1] W t (x, y) := ξ∈X W (ξ, y) 1/(1+t) 1+t η∈Y ξ∈X W (ξ, η) 1/(1+t) 1+t × W (x, y) 1/(1+t) ξ∈X W (ξ, y) 1/(1+t) .\",\n \"content\": \"For non-uniform inputs, W in (x) does not penetrate the summations. The E-null function and its complement deeply associate to the following family of measures: For any t ∈ [−2/5, 1], define the t-tilted probability mass function W t : X × Y → [0, 1] as in [CS07, Definition 1] W t (x, y) := ξ∈X W (ξ, y) 1/(1+t) 1+t η∈Y ξ∈X W (ξ, η) 1/(1+t) 1+t × W (x, y) 1/(1+t) ξ∈X W (ξ, y) 1/(1+t) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"universal quadratic bound). [CS07, Theorem 2]. Cf. [Gal65, Theorem 5.6.3]. Let W be a q-ary channel. Assume the uniform input distribution. Then Gallager's E-null function satisfies E 0 (0) = 0, E 0 (0) = I(W ) log q, and E 0 (t) −2 log(q) 2for all t ∈ [−2/5, 1]. In particular, it satisfies E 0 (t) I(W )t log q − t 2 log(q) 2 .\",\n \"content\": \"Lemma 17 (\\n17universal quadratic bound). [CS07, Theorem 2]. Cf. [Gal65, Theorem 5.6.3]. Let W be a q-ary channel. Assume the uniform input distribution. Then Gallager's E-null function satisfies E 0 (0) = 0, E 0 (0) = I(W ) log q, and E 0 (t) −2 log(q) 2for all t ∈ [−2/5, 1]. In particular, it satisfies E 0 (t) I(W )t log q − t 2 log(q) 2 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"we divide it into three cases per how H 0 compares to −2 and 1 − −2 . The mediocre case:if −2 H 0 1 − −2 then h α (H 0 ) −2α . The local CLT behavior implies E [h α (H 1 )] < 4 −1/2+α = 4 −1/2+3α −2α 4 −1/2+3α h α (H 0 )and we are done with this case. The noisy case: if H 0 > 1 − −2 , we replace (H, Z , S) by (1 − H, S, Z ) to dual it to the reliable case dealt below and we are done with this case. (This is the only place in the proof where we ever mentioned S explicitly. Nevertheless, every statement concerning Z concerns S by duality.)\",\n \"content\": \"we divide it into three cases per how H 0 compares to −2 and 1 − −2 . The mediocre case:if −2 H 0 1 − −2 then h α (H 0 ) −2α . The local CLT behavior implies E [h α (H 1 )] < 4 −1/2+α = 4 −1/2+3α −2α 4 −1/2+3α h α (H 0 )and we are done with this case. The noisy case: if H 0 > 1 − −2 , we replace (H, Z , S) by (1 − H, S, Z ) to dual it to the reliable case dealt below and we are done with this case. (This is the only place in the proof where we ever mentioned S explicitly. Nevertheless, every statement concerning Z concerns S by duality.)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"6 exp(6q 4 )(q 8 H 0 ) log( ) 5 6 log −30 H −12 0 = q 36+8 log( ) 5 e 6q 4 6 log −24 H log( ) 5 −12 0 < q 36+8 log( ) 5 e 6q 4 6 log −24 −2 log( ) 5 +24 = q 36+8 log( ) 5 e 6q 4 6 log −1.6 log( ) 5 −0.4 log( ) 5 q 36+8 log( ) 5 e 6q 4 6 log −8 log(q) log( ) 4 −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log( ) 6 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log(41) 2 (q log 3) 4 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 6.02q 4 < q 36 6 log −0.1 log( ) 5 = exp(36 log q + 6 log( ) 2 − log( ) 6 /30 − log( ) 6 /15) = exp(36 log q + 6 log( ) 2 − 5 log(q) log(22) 5 /30 − log(22) 4 log( ) 2 /15) < exp(0) 1.\",\n \"content\": \"6 exp(6q 4 )(q 8 H 0 ) log( ) 5 6 log −30 H −12 0 = q 36+8 log( ) 5 e 6q 4 6 log −24 H log( ) 5 −12 0 < q 36+8 log( ) 5 e 6q 4 6 log −24 −2 log( ) 5 +24 = q 36+8 log( ) 5 e 6q 4 6 log −1.6 log( ) 5 −0.4 log( ) 5 q 36+8 log( ) 5 e 6q 4 6 log −8 log(q) log( ) 4 −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.4 log( ) 5 = q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log( ) 6 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 0.3 log(41) 2 (q log 3) 4 < q 36 e 6q 4 6 log −0.1 log( ) 5 e 6.02q 4 < q 36 6 log −0.1 log( ) 5 = exp(36 log q + 6 log( ) 2 − log( ) 6 /30 − log( ) 6 /15) = exp(36 log q + 6 log( ) 2 − 5 log(q) log(22) 5 /30 − log(22) 4 log( ) 2 /15) < exp(0) 1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(Define events.) Let E 0 0 be the empty event. For every m = √ n, 2 √ n, . . . , n− √ n, we define five series of events A m , B m , C m , E m , and E m 0 inductively as below: LetA m be {Z m < exp(−m 2/3 )} \\\\ E Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D m+ m be A m \\\\ (B m ∪ C m ). Let E m 0 be E E m . Let a m , b m , c m ,e m , and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let g m be 1 − H 0 − e m 0 . (Bound b m\",\n \"content\": \".∪\\n(Define events.) Let E 0 0 be the empty event. For every m = √ n, 2 √ n, . . . , n− √ n, we define five series of events A m , B m , C m , E m , and E m 0 inductively as below: LetA m be {Z m < exp(−m 2/3 )} \\\\ E Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D m+ m be A m \\\\ (B m ∪ C m ). Let E m 0 be E E m . Let a m , b m , c m ,e m , and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let g m be 1 − H 0 − e m 0 . (Bound b m\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Markov's inequality. This is an upper bound on c m /a m and will be summoned in Formula (32).(Bound (g m−√ n − a m ) + from above.) By definition, a m ). The definition of A m forces it to be disjoint from E A m . This union event must contain the event {Z m < exp(−m 2/3 )} by how A m was defined. From the en23 behavior P{Z m < exp(−m 2/3 )} > 1 − H 0 − (−1/2+4α)m . Chaining all inequalities together, we deduce that g m− √ n − a m < (−1/2+4α)m+o(m) . Let (g m− √ n − a m ) + be max(0, g m− √ n − a m ) so we can write (g m− √ n − a m ) + < (−1/2+4α)m+o(m) . This upper bound will be summoned in Formula (32).(Bound e n 0 from below.) We start rewriting g m with g + m being max(0, g m ):g m = 1 − H 0 − e m 0 = 1 − H 0\",\n \"content\": \"+∪\\nMarkov's inequality. This is an upper bound on c m /a m and will be summoned in Formula (32).(Bound (g m−√ n − a m ) + from above.) By definition, a m ). The definition of A m forces it to be disjoint from E A m . This union event must contain the event {Z m < exp(−m 2/3 )} by how A m was defined. From the en23 behavior P{Z m < exp(−m 2/3 )} > 1 − H 0 − (−1/2+4α)m . Chaining all inequalities together, we deduce that g m− √ n − a m < (−1/2+4α)m+o(m) . Let (g m− √ n − a m ) + be max(0, g m− √ n − a m ) so we can write (g m− √ n − a m ) + < (−1/2+4α)m+o(m) . This upper bound will be summoned in Formula (32).(Bound e n 0 from below.) We start rewriting g m with g + m being max(0, g m ):g m = 1 − H 0 − e m 0 = 1 − H 0\"\n },\n {\n \"id\": \"\",\n \"caption\": \"we define six series of events A m , A m 0 , B m , C m , E m , and E m 0 inductively as follows:Let A m be {Z m < exp(−e m 1/3 )} \\\\ A A m . Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D n πn . (33) Let E m be A m \\\\(B m ∪C m ). Let E m 0 be E m− √ n 0 ∪E m . Let a m , a m 0 , b m , c m , e m ,and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let f m be 1 − H 0 − a m 0 and let g m be 1 − H 0 − e m 0 . (Bound b m /a m from above.) Conditioning on A m , we want to estimate the probability that Z k −8 for some k m, which is equal to the probability that min( −2 , 4 √ Z k ) −2 for some k m. Recall that min( −2 , 4 √ Z k ) was made a supermartingale. Hence by Doob's optional stopping theorem [Dur19, Exercise 4.8.2], P{min( −2 , 4 √ Z k ) −2 for some k m | A m } min( −2 , 4 √ Z m ) 2 < exp(−e m 1/3 /4) 2 . This is an upper bound on b m /a m and will be summoned in Formula (34).\",\n \"content\": \"∪\\nwe define six series of events A m , A m 0 , B m , C m , E m , and E m 0 inductively as follows:Let A m be {Z m < exp(−e m 1/3 )} \\\\ A A m . Let B m be a subevent of A m where Z k −8 for some k m. Let C m a subevent of A m where D m+1 D m+2 · · · D n πn . (33) Let E m be A m \\\\(B m ∪C m ). Let E m 0 be E m− √ n 0 ∪E m . Let a m , a m 0 , b m , c m , e m ,and e m 0 be the probability measures of the corresponding capital letter events. Moreover, let f m be 1 − H 0 − a m 0 and let g m be 1 − H 0 − e m 0 . (Bound b m /a m from above.) Conditioning on A m , we want to estimate the probability that Z k −8 for some k m, which is equal to the probability that min( −2 , 4 √ Z k ) −2 for some k m. Recall that min( −2 , 4 √ Z k ) was made a supermartingale. Hence by Doob's optional stopping theorem [Dur19, Exercise 4.8.2], P{min( −2 , 4 √ Z k ) −2 for some k m | A m } min( −2 , 4 √ Z m ) 2 < exp(−e m 1/3 /4) 2 . This is an upper bound on b m /a m and will be summoned in Formula (34).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a m ):g m − f + m = 1 − H 0 − e m − (1 − H 0 − a m ) + = g m− √ n − e m − (f m− √ n − a m ) + g m− √ n − e m −e m a m (f m− √ n − a m ) + g m− √ n − e m − two equalities are by the definitions of g m and f m . The next inequality is by 0 e m /a m 1. The next inequality is by max(0, f − a) = max(a, f ) − a max(0, f ) − a. The next equality is simple algebra. The next inequality is by the definition of E m . The last inequality summons upper bounds derived in the last few paragraphs. Now the last line contains two terms in the big parentheses. Between them, (1/2−2α)m−(ρ+2α)n dominates exp(−e m 1/3 /4) 2 once n → ∞.\",\n \"content\": \"a m ):g m − f + m = 1 − H 0 − e m − (1 − H 0 − a m ) + = g m− √ n − e m − (f m− √ n − a m ) + g m− √ n − e m −e m a m (f m− √ n − a m ) + g m− √ n − e m − two equalities are by the definitions of g m and f m . The next inequality is by 0 e m /a m 1. The next inequality is by max(0, f − a) = max(a, f ) − a max(0, f ) − a. The next equality is simple algebra. The next inequality is by the definition of E m . The last inequality summons upper bounds derived in the last few paragraphs. Now the last line contains two terms in the big parentheses. Between them, (1/2−2α)m−(ρ+2α)n dominates exp(−e m 1/3 /4) 2 once n → ∞.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"showed the equivalence of\",\n \"content\": \"showed the equivalence of1/2 \\n\\n0 \\n0 \\n1 \\n\\nconjectured boundary \\nfor V = 0 \\n\\n[FHMV17, GRY19] \\n\\nThm. 1 \\n[YFV19] \\n[HAU14] \\n\\n[WD18] \\n\\n[MHU16] \\n\\n[KSU10, MT14] \\n[GX13] \\n\\n[BGN + 18] \\n\\n[BGS18] \\n[AT09, Sas11, HY13]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Prudent readers are invited to check [ŞTA09a, STA09b, the paragraph before Section III] [YB15, PSL16, EKMF + 17, WD18, GRY19] for a list of inhomogeneous configurations of kernels. See [ŞTA09a, STA09b, MT10, Sas11] for how nonlinear bijections are similar to (or different from) linear bijections.\",\n \"content\": \"Prudent readers are invited to check [ŞTA09a, STA09b, the paragraph before Section III] [YB15, PSL16, EKMF + 17, WD18, GRY19] for a list of inhomogeneous configurations of kernels. See [ŞTA09a, STA09b, MT10, Sas11] for how nonlinear bijections are similar to (or different from) linear bijections.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"et seq. utilizing the same input distributions and series of bijections. For instance,\",\n \"content\": \"et seq. utilizing the same input distributions and series of bijections. For instance,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 9 (fundamental theorem of polar coding-the Z-end, FTPCZ). [Ari09, Proposition 5] [KSU10, Lemma 10] [Sas11, Lemma 3.5] [FHMV17, Section 4.1] [MT14, Lemma 33]\",\n \"content\": \"Theorem 9 (fundamental theorem of polar coding-the Z-end, FTPCZ). [Ari09, Proposition 5] [KSU10, Lemma 10] [Sas11, Lemma 3.5] [FHMV17, Section 4.1] [MT14, Lemma 33]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The formulation and the choice of the variables make it clear how Lemma 12 will be applied to support Claim 11. For instance, if we let {H n }, {Z n } and {S n } be {H(W n )}, {Z mad (W n )}, and {S max (W n )}, respectively, then Lemma 12 supports the first of the four inequalities in Claim 11. Moreover, if we let {H n }, {Z n }, and {S n } be {1 − H(W n )}, {S max (W n )}, and {Z mad (W n )}, then Lemma 12 supports the second inequality of Claim 11. If {H n }, {Z n }, and {S n } are let to be {H(V n )}, {Z mad (V n )}, and {S max (V n )}, that supports the third inequality. If {H n }, {Z n }, and {S n } are let to be {1 − H(V n )}, {S max (V n )}, and {Z mad (V n )}, that supports the forth inequality. The criteria (cb), (cm), (ct), and (cl) listed above Lemma 12 are easy to verify; for instance, Lemma 8 implies (ct) for all four cases. It remains to show that for each of the four triples of processes, the local LDP behavior and the local CLT behavior hold.\",\n \"content\": \"The formulation and the choice of the variables make it clear how Lemma 12 will be applied to support Claim 11. For instance, if we let {H n }, {Z n } and {S n } be {H(W n )}, {Z mad (W n )}, and {S max (W n )}, respectively, then Lemma 12 supports the first of the four inequalities in Claim 11. Moreover, if we let {H n }, {Z n }, and {S n } be {1 − H(W n )}, {S max (W n )}, and {Z mad (W n )}, then Lemma 12 supports the second inequality of Claim 11. If {H n }, {Z n }, and {S n } are let to be {H(V n )}, {Z mad (V n )}, and {S max (V n )}, that supports the third inequality. If {H n }, {Z n }, and {S n } are let to be {1 − H(V n )}, {S max (V n )}, and {Z mad (V n )}, that supports the forth inequality. The criteria (cb), (cm), (ct), and (cl) listed above Lemma 12 are easy to verify; for instance, Lemma 8 implies (ct) for all four cases. It remains to show that for each of the four triples of processes, the local LDP behavior and the local CLT behavior hold.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"8.3. Gallager's argument at Bob's end. This subsection take advantage of the universal bound developed three lines ago and starts actually proving Lemma 14.\",\n \"content\": \"8.3. Gallager's argument at Bob's end. This subsection take advantage of the universal bound developed three lines ago and starts actually proving Lemma 14.This subsection deals with \\n\\ni= H(W ) + 1/2+α +1 \\n\\nh α (H(W \\n\\n(i) \\n\\nG )) < −1/2+α \\n(14's copy) \\n\\nby passing it to an inequality that captures the performance of noisy channel coding. \\nOwing to h α 's concavity, the left hand side of Inequality (14) is \\n\\ni=j+1\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a bounded martingale {H n } converges to an r.v.-which we call H ∞ -a.s. (almost surely). This is Doob's martingale convergence theorem [Dur19, Theorem 4.2.11]. Owing to h α 's continuity, h α (H n ) → h α (H ∞ ) a.s. Point-wise convergence and (uniform) boundedness imply convergence in L 1 , i.e., E [h α (H n )] → E [h α (H ∞ )] as n → ∞. This is Lebesgue's dominated convergence theorem [Dur19, Theorem 1.6.7]. By the eigen behavior, E [h α (H n )] decays toward 0 by a constant factor every time n increases, thus E [h α (H ∞ )] is 0. This forces h a (H ∞ ) = 0 a.s. and hence H ∞ ∈ {0, 1} a.s. Since H ∞ is Bernoulli P{H ∞= 0} = E [I{H ∞ = 0}] = E [1 − H ∞ ] ← E [1 − H n ] = 1 − H 0 . SoP{H n → 0} = 1 − H 0 . By the implicit bi-Hölder toll, H n → 0 if and only if Z n → 0, thus the latter has the same probability measure. And the proof of Equality (24) is sound. Bibliographic remarks: The statement H n → H ∞ ∈ {0, 1} is usually referred to as channel polarization in spite of that it does not guarantee the corresponding codes to be capacity-achieving. See also [Ari09, Proposition 10] [MT14, Definition 3] [Sas11, Lemma 3.8]. This lemma should have been bestowed upon the fundamental theorem but it is not mandatory if some sort of CLT behavior is present. See [MHU16, Lemma 1] [FHMV17, Lemma 4] [GRY19, Lemma 9.5]. Recently, Reed-Muller codes' channels are shown to polarize [AY19]; Reed-Muller codes achieving capacity is not a consequence, but a different story.\",\n \"content\": \"a bounded martingale {H n } converges to an r.v.-which we call H ∞ -a.s. (almost surely). This is Doob's martingale convergence theorem [Dur19, Theorem 4.2.11]. Owing to h α 's continuity, h α (H n ) → h α (H ∞ ) a.s. Point-wise convergence and (uniform) boundedness imply convergence in L 1 , i.e., E [h α (H n )] → E [h α (H ∞ )] as n → ∞. This is Lebesgue's dominated convergence theorem [Dur19, Theorem 1.6.7]. By the eigen behavior, E [h α (H n )] decays toward 0 by a constant factor every time n increases, thus E [h α (H ∞ )] is 0. This forces h a (H ∞ ) = 0 a.s. and hence H ∞ ∈ {0, 1} a.s. Since H ∞ is Bernoulli P{H ∞= 0} = E [I{H ∞ = 0}] = E [1 − H ∞ ] ← E [1 − H n ] = 1 − H 0 . SoP{H n → 0} = 1 − H 0 . By the implicit bi-Hölder toll, H n → 0 if and only if Z n → 0, thus the latter has the same probability measure. And the proof of Equality (24) is sound. Bibliographic remarks: The statement H n → H ∞ ∈ {0, 1} is usually referred to as channel polarization in spite of that it does not guarantee the corresponding codes to be capacity-achieving. See also [Ari09, Proposition 10] [MT14, Definition 3] [Sas11, Lemma 3.8]. This lemma should have been bestowed upon the fundamental theorem but it is not mandatory if some sort of CLT behavior is present. See [MHU16, Lemma 1] [FHMV17, Lemma 4] [GRY19, Lemma 9.5]. Recently, Reed-Muller codes' channels are shown to polarize [AY19]; Reed-Muller codes achieving capacity is not a consequence, but a different story.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The fourth inequality uses q 4 . That validates the first inequality in(27). For the second inequality,\",\n \"content\": \"The fourth inequality uses q 4 . That validates the first inequality in(27). For the second inequality,E [D \\n\\n−1/2 \\n1 \\n\\n] = \\n1 \\n\\nk=1 \\n\\nk 2 \\n3 \\n· \\n3 \\n4 \\n\\n−1/2 \\n\\n< \\n1 \\n\\n√ \\n3 \\n\\nk=1 \\n\\n3 \\n4 \\n\\n−1/2 \\n\\n+ \\n1 \\n\\nk= \\n√ \\n3 +1\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Randomness and dependencies extraction via polarization, with applications to slepian-wolf coding and secrecy. E , Abbe , IEEE Transactions on Information Theory. 615E. Abbe. Randomness and dependencies extraction via polarization, with applica- tions to slepian-wolf coding and secrecy. IEEE Transactions on Information Theory, 61(5):2388-2398, May 2015.","doi":""},{"text":"Channel polarization: A method for constructing capacity-achieving codes. E Arikan, IEEE International Symposium on Information Theory. E. Arikan. Channel polarization: A method for constructing capacity-achieving codes. In 2008 IEEE International Symposium on Information Theory, pages 1173- 1177, July 2008.","doi":""},{"text":"Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. E Arikan, IEEE Transactions on Information Theory. 557E. Arikan. Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Transactions on Informa- tion Theory, 55(7):3051-3073, July 2009.","doi":""},{"text":"Source polarization. E Arikan, 2010 IEEE International Symposium on Information Theory. E. Arikan. Source polarization. In 2010 IEEE International Symposium on Infor- mation Theory, pages 899-903, June 2010.","doi":""},{"text":"A packing lemma for polar codes. E Arikan, 2015 IEEE International Symposium on Information Theory (ISIT). E. Arikan. A packing lemma for polar codes. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 2441-2445, June 2015.","doi":""},{"text":"Variance of entropy decreases under the polar transform. Erdal Arikan, abs/1501.05005CoRRErdal Arikan. Variance of entropy decreases under the polar transform. CoRR, abs/1501.05005, 2015.","doi":""},{"text":"On the rate of channel polarization. E Arikan, E Telatar, IEEE International Symposium on Information Theory. E. Arikan and E. Telatar. On the rate of channel polarization. In 2009 IEEE Inter- national Symposium on Information Theory, pages 1493-1495, June 2009.","doi":""},{"text":"Polar codes for the m-user multiple access channel. E Abbe, E Telatar, IEEE Transactions on Information Theory. 588E. Abbe and E. Telatar. Polar codes for the m-user multiple access channel. IEEE Transactions on Information Theory, 58(8):5437-5448, Aug 2012.","doi":""},{"text":"Moderate deviation analysis of channel coding: Discrete memoryless case. Y Altug, A B Wagner, 2010 IEEE International Symposium on Information Theory. Y. Altug and A. B. Wagner. Moderate deviation analysis of channel coding: Discrete memoryless case. In 2010 IEEE International Symposium on Information Theory, pages 265-269, June 2010.","doi":""},{"text":"Moderate deviations in channel coding. Y Altug, A B Wagner, IEEE Transactions on Information Theory. 608Y. Altug and A. B. Wagner. Moderate deviations in channel coding. IEEE Transac- tions on Information Theory, 60(8):4417-4426, Aug 2014.","doi":""},{"text":"Reed-muller codes polarize. CoRR, abs. Emmanuel Abbe, Min Ye, Emmanuel Abbe and Min Ye. Reed-muller codes polarize. CoRR, abs/1901.11533, 2019.","doi":""},{"text":"Random codes: minimum distances and error exponents. A Barg, G D Forney, IEEE Transactions on Information Theory. 489A. Barg and G. D. Forney. Random codes: minimum distances and error exponents. IEEE Transactions on Information Theory, 48(9):2568-2573, Sep. 2002.","doi":""},{"text":"General strong polarization. CoRR. + 18] Jaroslaw, Venkatesan Blasiok, Preetum Guruswami, Atri Nakkiran, Madhu Rudra, Sudan, abs/1802.02718+ 18] Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, Atri Rudra, and Madhu Sudan. General strong polarization. CoRR, abs/1802.02718, 2018.","doi":""},{"text":"Polar codes with exponentially small error at finite block length. CoRR. Jaroslaw Blasiok, Venkatesan Guruswami, Madhu Sudan, abs/1810.04298Jaroslaw Blasiok, Venkatesan Guruswami, and Madhu Sudan. Polar codes with ex- ponentially small error at finite block length. CoRR, abs/1810.04298, 2018.","doi":""},{"text":"How quickly can we approach channel capacity?. D Baron, M A Khojastepour, R G Baraniuk, Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers. 1D. Baron, M. A. Khojastepour, and R. G. Baraniuk. How quickly can we approach channel capacity? In Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, 2004., volume 1, pages 1096-1100 Vol.1, Nov 2004.","doi":""},{"text":"Hypothesis testing and information theory. R Blahut, IEEE Transactions on Information Theory. 204R. Blahut. Hypothesis testing and information theory. IEEE Transactions on Infor- mation Theory, 20(4):405-417, July 1974.","doi":""},{"text":"A first course in complex analysis. Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka, Department of Mathematics, San Francisco State UniversityMatthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka. A first course in complex analysis. Department of Mathematics, San Francisco State University, 2002.","doi":""},{"text":"Exact random coding secrecy exponents for the wiretap channel. M Bastani Parizi, E Telatar, N Merhav, IEEE Transactions on Information Theory. 631M. Bastani Parizi, E. Telatar, and N. Merhav. Exact random coding secrecy expo- nents for the wiretap channel. IEEE Transactions on Information Theory, 63(1):509- 531, Jan 2017.","doi":""},{"text":"Lossless source coding with polar codes. H S Cronie, S B Korada, 2010 IEEE International Symposium on Information Theory. H. S. Cronie and S. B. Korada. Lossless source coding with polar codes. In 2010 IEEE International Symposium on Information Theory, pages 904-908, June 2010.","doi":""},{"text":"Universal quadratic lower bounds on source coding error exponents. C Chang, A Sahai, 41st Annual Conference on Information Sciences and Systems. C. Chang and A. Sahai. Universal quadratic lower bounds on source coding error exponents. In 2007 41st Annual Conference on Information Sciences and Systems, pages 714-719, March 2007.","doi":""},{"text":"Lossless coding for distributed streaming sources. S C Draper, C Chang, A Sahai, IEEE Transactions on Information Theory. 603S. C. Draper, C. Chang, and A. Sahai. Lossless coding for distributed streaming sources. IEEE Transactions on Information Theory, 60(3):1447-1474, March 2014.","doi":""},{"text":"Mathematical problems in the shannon theory of optimal coding of information. R L Dobrushin, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. the Fourth Berkeley Symposium on Mathematical Statistics and ProbabilityBerkeley, CalifUniversity of California Press1Contributions to the Theory of StatisticsR. L. Dobrushin. Mathematical problems in the shannon theory of optimal coding of information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, pages 211-252, Berkeley, Calif., 1961. University of California Press.","doi":""},{"text":"Rick Durrett, Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press5 editionRick Durrett. Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 5 edition, 2019.","doi":""},{"text":"Amir Dembo, Ofer Zeitouni, Large deviations techniques and applications. BerlinSpringer-VerlagStochastic Modelling and Applied Probability. Corrected reprint of the second (1998) editionAmir Dembo and Ofer Zeitouni. Large deviations techniques and applications, vol- ume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition.","doi":""},{"text":"The random coding bound is tight for the average linear code or lattice. Y Domb, R Zamir, M Feder, IEEE Transactions on Information Theory. 621Y. Domb, R. Zamir, and M. Feder. The random coding bound is tight for the average linear code or lattice. IEEE Transactions on Information Theory, 62(1):121-130, Jan 2016.","doi":""},{"text":"Relaxed polar codes. M El-Khamy, H Mahdavifar, G Feygin, J Lee, I Kang, IEEE Transactions on Information Theory. 634+ 17] M. El-Khamy, H. Mahdavifar, G. Feygin, J. Lee, and I. Kang. Relaxed polar codes. IEEE Transactions on Information Theory, 63(4):1986-2000, April 2017.","doi":""},{"text":"R M Fano, Transmission of Information: A Statistical Theory of Communications. M.I.T. PressR.M. Fano. Transmission of Information: A Statistical Theory of Communications. M.I.T. Press, 1961.","doi":""},{"text":"Binary linear codes with optimal scaling and quasi-linear complexity. Arman Fazeli, S Hamed Hassani, Marco Mondelli, Alexander Vardy, abs/1711.01339CoRRArman Fazeli, S. Hamed Hassani, Marco Mondelli, and Alexander Vardy. Binary lin- ear codes with optimal scaling and quasi-linear complexity. CoRR, abs/1711.01339, 2017.","doi":""},{"text":"Binary linear codes with optimal scaling: Polar codes with large kernels. A Fazeli, H Hassani, M Mondelli, A Vardy, 2018 IEEE Information Theory Workshop (ITW). A. Fazeli, H. Hassani, M. Mondelli, and A. Vardy. Binary linear codes with optimal scaling: Polar codes with large kernels. In 2018 IEEE Information Theory Workshop (ITW), pages 1-5, Nov 2018.","doi":""},{"text":"Relations between entropy and error probability. M Feder, N Merhav, IEEE Transactions on Information Theory. 401M. Feder and N. Merhav. Relations between entropy and error probability. IEEE Transactions on Information Theory, 40(1):259-266, Jan 1994.","doi":""},{"text":"On the scaling exponent of polar codes for binary-input energy-harvesting channels. S L Fong, V Y F Tan, IEEE Journal on Selected Areas in Communications. 3412S. L. Fong and V. Y. F. Tan. On the scaling exponent of polar codes for binary-input energy-harvesting channels. IEEE Journal on Selected Areas in Communications, 34(12):3540-3551, Dec 2016.","doi":""},{"text":"Scaling exponent and moderate deviations asymptotics of polar codes for the awgn channel. L Silas, Fong, Y F Vincent, Tan, Entropy. 197Silas L. Fong and Vincent Y. F. Tan. Scaling exponent and moderate deviations asymptotics of polar codes for the awgn channel. Entropy, 19(7), 2017.","doi":""},{"text":"On the scaling exponent of binary polarization kernels. A Fazeli, A Vardy, 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton). A. Fazeli and A. Vardy. On the scaling exponent of binary polarization kernels. In 2014 52nd Annual Allerton Conference on Communication, Control, and Comput- ing (Allerton), pages 797-804, Sept 2014.","doi":""},{"text":"Polar codes for broadcast channels. N Goela, E Abbe, M Gastpar, IEEE Transactions on Information Theory. 612N. Goela, E. Abbe, and M. Gastpar. Polar codes for broadcast channels. IEEE Transactions on Information Theory, 61(2):758-782, Feb 2015.","doi":""},{"text":"A simple derivation of the coding theorem and some applications. R Gallager, IEEE Transactions on Information Theory. 111R. Gallager. A simple derivation of the coding theorem and some applications. IEEE Transactions on Information Theory, 11(1):3-18, January 1965.","doi":""},{"text":"Information Theory and Reliable Communication. Robert G Gallager, John Wiley & Sons, IncNew York, NY, USARobert G. Gallager. Information Theory and Reliable Communication. John Wiley & Sons, Inc., New York, NY, USA, 1968.","doi":""},{"text":"The random coding bound is tight for the average code (corresp.). R Gallager, IEEE Transactions on Information Theory. 192R. Gallager. The random coding bound is tight for the average code (corresp.). IEEE Transactions on Information Theory, 19(2):244-246, March 1973.","doi":""},{"text":"Improved bounds on the finite length scaling of polar codes. D Goldin, D Burshtein, IEEE Transactions on Information Theory. 6011D. Goldin and D. Burshtein. Improved bounds on the finite length scaling of polar codes. IEEE Transactions on Information Theory, 60(11):6966-6978, Nov 2014.","doi":""},{"text":"Universal bounds on the scaling behavior of polar codes. A Goli, S H Hassani, R Urbanke, 2012 IEEE International Symposium on Information Theory Proceedings. A. Goli, S. H. Hassani, and R. Urbanke. Universal bounds on the scaling behav- ior of polar codes. In 2012 IEEE International Symposium on Information Theory Proceedings, pages 1957-1961, July 2012.","doi":""},{"text":"Arıkan meets Shannon: Polar codes with near-optimal convergence to channel capacity. Venkatesan Guruswami, Andrii Riazanov, Min Ye, arXiv:1911.03858arXiv e-printsVenkatesan Guruswami, Andrii Riazanov, and Min Ye. Arıkan meets Shannon: Po- lar codes with near-optimal convergence to channel capacity. arXiv e-prints, page arXiv:1911.03858, Nov 2019.","doi":""},{"text":"Polar codes: Speed of polarization and polynomial gap to capacity. V Guruswami, P Xia, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science. V. Guruswami and P. Xia. Polar codes: Speed of polarization and polynomial gap to capacity. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 310-319, Oct 2013.","doi":""},{"text":"Error detecting and error correcting codes. The Bell System Technical. R W Hamming, Journal. 292R. W. Hamming. Error detecting and error correcting codes. The Bell System Tech- nical Journal, 29(2):147-160, April 1950.","doi":""},{"text":"Polarization and spatial coupling two techniques to boost performance. Hassani Seyed Hamed, Seyed Hamed Hassani. Polarization and spatial coupling two techniques to boost performance. 2013.","doi":""},{"text":"Finite-length scaling for polar codes. S H Hassani, K Alishahi, R L Urbanke, IEEE Transactions on Information Theory. 6010S. H. Hassani, K. Alishahi, and R. L. Urbanke. Finite-length scaling for polar codes. IEEE Transactions on Information Theory, 60(10):5875-5898, Oct 2014.","doi":""},{"text":"General nonasymptotic and asymptotic formulas in channel resolvability and identification capacity and their application to the wiretap channel. M Hayashi, IEEE Transactions on Information Theory. 524M. Hayashi. General nonasymptotic and asymptotic formulas in channel resolvabil- ity and identification capacity and their application to the wiretap channel. IEEE Transactions on Information Theory, 52(4):1562-1575, April 2006.","doi":""},{"text":"Information spectrum approach to second-order coding rate in channel coding. M Hayashi, IEEE Transactions on Information Theory. 5511M. Hayashi. Information spectrum approach to second-order coding rate in channel coding. IEEE Transactions on Information Theory, 55(11):4947-4966, Nov 2009.","doi":""},{"text":"Performance of polar codes for channel and source coding. N Hussami, S B Korada, R Urbanke, 2009 IEEE International Symposium on Information Theory. N. Hussami, S. B. Korada, and R. Urbanke. Performance of polar codes for channel and source coding. In 2009 IEEE International Symposium on Information Theory, pages 1488-1492, June 2009.","doi":""},{"text":"Universally attainable error and information exponents, and equivocation rate for the broadcast channels with confidential messages. M Hayashi, R Matsumoto, 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton). M. Hayashi and R. Matsumoto. Universally attainable error and information expo- nents, and equivocation rate for the broadcast channels with confidential messages. In 2011 49th Annual Allerton Conference on Communication, Control, and Com- puting (Allerton), pages 439-444, Sep. 2011.","doi":""},{"text":"Rate-dependent analysis of the asymptotic behavior of channel polarization. S H Hassani, R Mori, T Tanaka, R L Urbanke, IEEE Transactions on Information Theory. 594S. H. Hassani, R. Mori, T. Tanaka, and R. L. Urbanke. Rate-dependent analysis of the asymptotic behavior of channel polarization. IEEE Transactions on Information Theory, 59(4):2267-2276, April 2013.","doi":""},{"text":"Erasure and undetected error probabilities in the moderate deviations regime. M Hayashi, V Y F Tan, 2015 IEEE International Symposium on Information Theory (ISIT). M. Hayashi and V. Y. F. Tan. Erasure and undetected error probabilities in the moderate deviations regime. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 1821-1825, June 2015.","doi":""},{"text":"Polar coding without alphabet extension for asymmetric models. J Honda, H Yamamoto, IEEE Transactions on Information Theory. 5912J. Honda and H. Yamamoto. Polar coding without alphabet extension for asymmetric models. IEEE Transactions on Information Theory, 59(12):7829-7838, Dec 2013.","doi":""},{"text":"Extremes of random coding error exponents. A G Fàbregas, I Land, A Martinez, 2011 IEEE International Symposium on Information Theory Proceedings. A. G. i. Fàbregas, I. Land, and A. Martinez. Extremes of random coding error expo- nents. In 2011 IEEE International Symposium on Information Theory Proceedings, pages 2896-2898, July 2011.","doi":""},{"text":"Reed-muller codes achieve capacity on erasure channels. ] S Kkm + 17, S Kudekar, M Kumar, H D Mondelli, E Pfister, R L Şaşoǧlu, Urbanke, IEEE Transactions on Information Theory. 637KKM + 17] S. Kudekar, S. Kumar, M. Mondelli, H. D. Pfister, E. Şaşoǧlu, and R. L. Urbanke. Reed-muller codes achieve capacity on erasure channels. IEEE Transactions on In- formation Theory, 63(7):4298-4316, July 2017.","doi":""},{"text":"An empirical scaling law for polar codes. S B Korada, A Montanari, E Telatar, R Urbanke, 2010 IEEE International Symposium on Information Theory. S. B. Korada, A. Montanari, E. Telatar, and R. Urbanke. An empirical scaling law for polar codes. In 2010 IEEE International Symposium on Information Theory, pages 884-888, June 2010.","doi":""},{"text":"Polar codes for channel and source coding. Korada Satish Babu, 181Satish Babu Korada. Polar codes for channel and source coding. page 181, 2009.","doi":""},{"text":"Polar codes: Characterization of exponent, bounds, and constructions. S B Korada, E Sasoglu, R Urbanke, IEEE Transactions on Information Theory. 5612S. B. Korada, E. Sasoglu, and R. Urbanke. Polar codes: Characterization of ex- ponent, bounds, and constructions. IEEE Transactions on Information Theory, 56(12):6253-6264, Dec 2010.","doi":""},{"text":"Polar codes for q-ary source coding. M Karzand, E Telatar, 2010 IEEE International Symposium on Information Theory. M. Karzand and E. Telatar. Polar codes for q-ary source coding. In 2010 IEEE International Symposium on Information Theory, pages 909-912, June 2010.","doi":""},{"text":"Polar codes are optimal for lossy source coding. S B Korada, R L Urbanke, IEEE Transactions on Information Theory. 564S. B. Korada and R. L. Urbanke. Polar codes are optimal for lossy source coding. IEEE Transactions on Information Theory, 56(4):1751-1768, April 2010.","doi":""},{"text":"Fixed-length lossy compression in the finite blocklength regime. V Kostina, S Verdu, IEEE Transactions on Information Theory. 586V. Kostina and S. Verdu. Fixed-length lossy compression in the finite blocklength regime. IEEE Transactions on Information Theory, 58(6):3309-3338, June 2012.","doi":""},{"text":"Lossy joint source-channel coding in the finite blocklength regime. V Kostina, S Verdú, IEEE Transactions on Information Theory. 595V. Kostina and S. Verdú. Lossy joint source-channel coding in the finite blocklength regime. IEEE Transactions on Information Theory, 59(5):2545-2575, May 2013.","doi":""},{"text":"On the capacity of channels with deletions and states. Yonglong Li, Y F Vincent, Tan, Yonglong Li and Vincent Y. F. Tan. On the capacity of channels with deletions and states, 2019.","doi":""},{"text":"Fast polarization for non-stationary channels. H Mahdavifar, 2017 IEEE International Symposium on Information Theory (ISIT). H. Mahdavifar. Fast polarization for non-stationary channels. In 2017 IEEE Inter- national Symposium on Information Theory (ISIT), pages 849-853, June 2017.","doi":""},{"text":"Achieving marton's region for broadcast channels using polar codes. M Mondelli, S H Hassani, I Sason, R L Urbanke, IEEE Transactions on Information Theory. 612M. Mondelli, S. H. Hassani, I. Sason, and R. L. Urbanke. Achieving marton's region for broadcast channels using polar codes. IEEE Transactions on Information Theory, 61(2):783-800, Feb 2015.","doi":""},{"text":"Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors. M Mondelli, S H Hassani, R L Urbanke, IEEE Transactions on Information Theory. 6212M. Mondelli, S. H. Hassani, and R. L. Urbanke. Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors. IEEE Transac- tions on Information Theory, 62(12):6698-6712, Dec 2016.","doi":""},{"text":"From polar to reed-muller codes unified scaling, non-standard channels, and a proven conjecture. Marco Mondelli, 230Marco Mondelli. From polar to reed-muller codes unified scaling, non-standard chan- nels, and a proven conjecture. page 230, 2016.","doi":""},{"text":"Properties and construction of polar codes. Ryuhei Mori, abs/1002.3521CoRRRyuhei Mori. Properties and construction of polar codes. CoRR, abs/1002.3521, 2010.","doi":""},{"text":"Channel polarization on q-ary discrete memoryless channels by arbitrary kernels. R Mori, T Tanaka, 2010 IEEE International Symposium on Information Theory. R. Mori and T. Tanaka. Channel polarization on q-ary discrete memoryless chan- nels by arbitrary kernels. In 2010 IEEE International Symposium on Information Theory, pages 894-898, June 2010.","doi":""},{"text":"Source and channel polarization over finite fields and reedsolomon matrices. R Mori, T Tanaka, IEEE Transactions on Information Theory. 605R. Mori and T. Tanaka. Source and channel polarization over finite fields and reed- solomon matrices. IEEE Transactions on Information Theory, 60(5):2720-2736, May 2014.","doi":""},{"text":"An ergodic theory of binary operations-part i: Key properties. R Nasser, IEEE Transactions on Information Theory. 6212R. Nasser. An ergodic theory of binary operations-part i: Key properties. IEEE Transactions on Information Theory, 62(12):6931-6952, Dec 2016.","doi":""},{"text":"An ergodic theory of binary operations-part ii: Applications to polarization. R Nasser, IEEE Transactions on Information Theory. 632R. Nasser. An ergodic theory of binary operations-part ii: Applications to polar- ization. IEEE Transactions on Information Theory, 63(2):1063-1083, Feb 2017.","doi":""},{"text":"Polar codes for arbitrary dmcs and arbitrary macs. R Nasser, E Telatar, IEEE Transactions on Information Theory. 626R. Nasser and E. Telatar. Polar codes for arbitrary dmcs and arbitrary macs. IEEE Transactions on Information Theory, 62(6):2917-2936, June 2016.","doi":""},{"text":"Error probability of identification via channels at rates above capacity. Y Oohama, Proceedings IEEE International Symposium on Information Theory. IEEE International Symposium on Information Theory26Y. Oohama. Error probability of identification via channels at rates above capacity. In Proceedings IEEE International Symposium on Information Theory,, pages 26-, June 2002.","doi":""},{"text":"Polar codes for q-ary channels, q = 2 r. W Park, A Barg, IEEE Transactions on Information Theory. 592W. Park and A. Barg. Polar codes for q-ary channels, q = 2 r . IEEE Transactions on Information Theory, 59(2):955-969, Feb 2013.","doi":""},{"text":"Channel coding: Non-asymptotic fundamental limits. Yury Polyanskiy, Princeton UniversityPhD thesisYury Polyanskiy. Channel coding: Non-asymptotic fundamental limits. PhD thesis, Princeton University, 2010.","doi":""},{"text":"Channel coding rate in the finite blocklength regime. Y Polyanskiy, H V Poor, S Verdu, IEEE Transactions on Information Theory. 565Y. Polyanskiy, H. V. Poor, and S. Verdu. Channel coding rate in the finite blocklength regime. IEEE Transactions on Information Theory, 56(5):2307-2359, May 2010.","doi":""},{"text":"Mixed-kernels constructions of polar codes. N Presman, O Shapira, S Litsyn, IEEE Journal on Selected Areas in Communications. 342N. Presman, O. Shapira, and S. Litsyn. Mixed-kernels constructions of polar codes. IEEE Journal on Selected Areas in Communications, 34(2):239-253, Feb 2016.","doi":""},{"text":"Near-optimal finite-length scaling for polar codes over large alphabets. H D Pfister, R Urbanke, 2016 IEEE International Symposium on Information Theory (ISIT). H. D. Pfister and R. Urbanke. Near-optimal finite-length scaling for polar codes over large alphabets. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 215-219, July 2016.","doi":""},{"text":"Channel dispersion and moderate deviations limits for memoryless channels. Y Polyanskiy, S Verdú, 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton). Y. Polyanskiy and S. Verdú. Channel dispersion and moderate deviations limits for memoryless channels. In 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1334-1339, Sept 2010.","doi":""},{"text":"Polar coding theorems for discrete systems. Eren Sasoglu, 104Eren Sasoglu. Polar coding theorems for discrete systems. page 104, 2011.","doi":""},{"text":"Polar codes for discrete alphabets. Eren Sasoglu, 2012 IEEE International Symposium on Information Theory Proceedings. Eren Sasoglu. Polar codes for discrete alphabets. In 2012 IEEE International Sym- posium on Information Theory Proceedings, pages 2137-2141, July 2012.","doi":""},{"text":"Polarization and polar codes. Eren Sasoglu, Foundations and Trends R in Communications and Information Theory. 84Eren Sasoglu. Polarization and polar codes. Foundations and Trends R in Commu- nications and Information Theory, 8(4):259-381, 2012.","doi":""},{"text":"Lower bounds to error probability for coding on discrete memoryless channels. i. Information and Control. C E Shannon, R G Gallager, E R Berlekamp, 10C.E. Shannon, R.G. Gallager, and E.R. Berlekamp. Lower bounds to error probabil- ity for coding on discrete memoryless channels. i. Information and Control, 10(1):65 -103, 1967.","doi":""},{"text":"A mathematical theory of communication. C E Shannon, Bell System Tech. J. 27C. E. Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379-423, 623-656, 1948.","doi":""},{"text":"Multilevel polarization of polar codes over arbitrary discrete memoryless channels. A G Sahebi, S S Pradhan, 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton). A. G. Sahebi and S. S. Pradhan. Multilevel polarization of polar codes over arbi- trary discrete memoryless channels. In 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1718-1725, Sep. 2011.","doi":""},{"text":"Achieving the capacity of any dmc using only polar codes. D Sutter, J M Renes, F Dupuis, R Renner, 2012 IEEE Information Theory Workshop. D. Sutter, J. M. Renes, F. Dupuis, and R. Renner. Achieving the capacity of any dmc using only polar codes. In 2012 IEEE Information Theory Workshop, pages 114-118, Sep. 2012.","doi":""},{"text":"Polar coding for processes with memory. Eren Sasoglu, Ido Tal, abs/1602.01870CoRREren Sasoglu and Ido Tal. Polar coding for processes with memory. CoRR, abs/1602.01870, 2016.","doi":""},{"text":"Polarization for arbitrary discrete memoryless channels. E Şaşoglu, E Telatar, E Arikan, IEEE Information Theory Workshop. E. Şaşoglu, E. Telatar, and E. Arikan. Polarization for arbitrary discrete memoryless channels. In 2009 IEEE Information Theory Workshop, pages 144-148, Oct 2009.","doi":""},{"text":"Polarization for arbitrary discrete memoryless channels. Eren Sasoglu, Emre Telatar, Erdal Arikan, abs/0908.0302CoRREren Sasoglu, Emre Telatar, and Erdal Arikan. Polarization for arbitrary discrete memoryless channels. CoRR, abs/0908.0302, 2009.","doi":""},{"text":"Asymptotische abschätzungen in shannons informationstheorie. V Strassen, Transactions of the Third Prague Conference on Information Theory. Publishing House of the Czechoslovak Academy of SciencesV. Strassen. Asymptotische abschätzungen in shannons informationstheorie. In Transactions of the Third Prague Conference on Information Theory, pages 689- 723. Publishing House of the Czechoslovak Academy of Sciences, 1962.","doi":""},{"text":"A new polar coding scheme for strong security on wiretap channels. E Şaşoglu, A Vardy, 2013 IEEE International Symposium on Information Theory. E. Şaşoglu and A. Vardy. A new polar coding scheme for strong security on wiretap channels. In 2013 IEEE International Symposium on Information Theory, pages 1117-1121, July 2013.","doi":""},{"text":"Fourier analysis on finite groups and applications, volume 43 of London Mathematical Society Student Texts. Audrey Terras, Cambridge University PressCambridgeAudrey Terras. Fourier analysis on finite groups and applications, volume 43 of Lon- don Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999.","doi":""},{"text":"Polar codes for the deletion channel: Weak and strong polarization. Ido Tal, Henry D Pfister, Arman Fazeli, Alexander Vardy, abs/1904.13385CoRR. Ido Tal, Henry D. Pfister, Arman Fazeli, and Alexander Vardy. Polar codes for the deletion channel: Weak and strong polarization. CoRR, abs/1904.13385, 2019.","doi":""},{"text":"How to construct polar codes. I Tal, A Vardy, IEEE Transactions on Information Theory. 5910I. Tal and A. Vardy. How to construct polar codes. IEEE Transactions on Informa- tion Theory, 59(10):6562-6582, Oct 2013.","doi":""},{"text":"List decoding of polar codes. I Tal, A Vardy, IEEE Transactions on Information Theory. 615I. Tal and A. Vardy. List decoding of polar codes. IEEE Transactions on Information Theory, 61(5):2213-2226, May 2015.","doi":""},{"text":"Polar-like codes and asymptotic tradeoff among block length, code rate, and error probability. Hsin-Po Wang, Iwan M Duursma, abs/1812.08112CoRR. Hsin-Po Wang and Iwan M. Duursma. Polar-like codes and asymptotic tradeoff among block length, code rate, and error probability. CoRR, abs/1812.08112, 2018.","doi":""},{"text":"On the strong converse of the coding theorem for symmetric channels without memory. Lionel Weiss. ; R, J Wang, H Honda, R Yamamoto, Y Liu, Hou, IEEE Information Theory Workshop -Fall (ITW). 18Construction of polar codes for channels with memoryLIONEL WEISS. On the strong converse of the coding theorem for symmetric chan- nels without memory. Quarterly of Applied Mathematics, 18(3):209-214, 1960. [WHY + 15] R. Wang, J. Honda, H. Yamamoto, R. Liu, and Y. Hou. Construction of polar codes for channels with memory. In 2015 IEEE Information Theory Workshop - Fall (ITW), pages 187-191, Oct 2015.","doi":""},{"text":"The coding of messages subject to chance errors. J Wolfowitz, Illinois J. Math. 141957J. Wolfowitz. The coding of messages subject to chance errors. Illinois J. Math., 1(4):591-606, 12 1957.","doi":""},{"text":"The wire-tap channel. A D Wyner, Bell System Technical Journal. 548A. D. Wyner. The wire-tap channel. Bell System Technical Journal, 54(8):1355-1387, 1975.","doi":""},{"text":"Polar codes using dynamic kernels. M Ye, A Barg, 2015 IEEE International Symposium on Information Theory (ISIT). M. Ye and A. Barg. Polar codes using dynamic kernels. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 231-235, June 2015.","doi":""},{"text":"Explicit polar codes with small scaling exponent. Hanwen Yao, Arman Fazeli, Alexander Vardy, abs/1901.08186CoRRHanwen Yao, Arman Fazeli, and Alexander Vardy. Explicit polar codes with small scaling exponent. CoRR, abs/1901.08186, 2019.","doi":""}],"string":"[\n {\n \"text\": \"Randomness and dependencies extraction via polarization, with applications to slepian-wolf coding and secrecy. E , Abbe , IEEE Transactions on Information Theory. 615E. Abbe. Randomness and dependencies extraction via polarization, with applica- tions to slepian-wolf coding and secrecy. IEEE Transactions on Information Theory, 61(5):2388-2398, May 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Channel polarization: A method for constructing capacity-achieving codes. E Arikan, IEEE International Symposium on Information Theory. E. Arikan. Channel polarization: A method for constructing capacity-achieving codes. In 2008 IEEE International Symposium on Information Theory, pages 1173- 1177, July 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. E Arikan, IEEE Transactions on Information Theory. 557E. Arikan. Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Transactions on Informa- tion Theory, 55(7):3051-3073, July 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Source polarization. E Arikan, 2010 IEEE International Symposium on Information Theory. E. Arikan. Source polarization. In 2010 IEEE International Symposium on Infor- mation Theory, pages 899-903, June 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A packing lemma for polar codes. E Arikan, 2015 IEEE International Symposium on Information Theory (ISIT). E. Arikan. A packing lemma for polar codes. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 2441-2445, June 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Variance of entropy decreases under the polar transform. Erdal Arikan, abs/1501.05005CoRRErdal Arikan. Variance of entropy decreases under the polar transform. CoRR, abs/1501.05005, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the rate of channel polarization. E Arikan, E Telatar, IEEE International Symposium on Information Theory. E. Arikan and E. Telatar. On the rate of channel polarization. In 2009 IEEE Inter- national Symposium on Information Theory, pages 1493-1495, June 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes for the m-user multiple access channel. E Abbe, E Telatar, IEEE Transactions on Information Theory. 588E. Abbe and E. Telatar. Polar codes for the m-user multiple access channel. IEEE Transactions on Information Theory, 58(8):5437-5448, Aug 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Moderate deviation analysis of channel coding: Discrete memoryless case. Y Altug, A B Wagner, 2010 IEEE International Symposium on Information Theory. Y. Altug and A. B. Wagner. Moderate deviation analysis of channel coding: Discrete memoryless case. In 2010 IEEE International Symposium on Information Theory, pages 265-269, June 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Moderate deviations in channel coding. Y Altug, A B Wagner, IEEE Transactions on Information Theory. 608Y. Altug and A. B. Wagner. Moderate deviations in channel coding. IEEE Transac- tions on Information Theory, 60(8):4417-4426, Aug 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reed-muller codes polarize. CoRR, abs. Emmanuel Abbe, Min Ye, Emmanuel Abbe and Min Ye. Reed-muller codes polarize. CoRR, abs/1901.11533, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Random codes: minimum distances and error exponents. A Barg, G D Forney, IEEE Transactions on Information Theory. 489A. Barg and G. D. Forney. Random codes: minimum distances and error exponents. IEEE Transactions on Information Theory, 48(9):2568-2573, Sep. 2002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"General strong polarization. CoRR. + 18] Jaroslaw, Venkatesan Blasiok, Preetum Guruswami, Atri Nakkiran, Madhu Rudra, Sudan, abs/1802.02718+ 18] Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, Atri Rudra, and Madhu Sudan. General strong polarization. CoRR, abs/1802.02718, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes with exponentially small error at finite block length. CoRR. Jaroslaw Blasiok, Venkatesan Guruswami, Madhu Sudan, abs/1810.04298Jaroslaw Blasiok, Venkatesan Guruswami, and Madhu Sudan. Polar codes with ex- ponentially small error at finite block length. CoRR, abs/1810.04298, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"How quickly can we approach channel capacity?. D Baron, M A Khojastepour, R G Baraniuk, Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers. 1D. Baron, M. A. Khojastepour, and R. G. Baraniuk. How quickly can we approach channel capacity? In Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, 2004., volume 1, pages 1096-1100 Vol.1, Nov 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hypothesis testing and information theory. R Blahut, IEEE Transactions on Information Theory. 204R. Blahut. Hypothesis testing and information theory. IEEE Transactions on Infor- mation Theory, 20(4):405-417, July 1974.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A first course in complex analysis. Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka, Department of Mathematics, San Francisco State UniversityMatthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka. A first course in complex analysis. Department of Mathematics, San Francisco State University, 2002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Exact random coding secrecy exponents for the wiretap channel. M Bastani Parizi, E Telatar, N Merhav, IEEE Transactions on Information Theory. 631M. Bastani Parizi, E. Telatar, and N. Merhav. Exact random coding secrecy expo- nents for the wiretap channel. IEEE Transactions on Information Theory, 63(1):509- 531, Jan 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lossless source coding with polar codes. H S Cronie, S B Korada, 2010 IEEE International Symposium on Information Theory. H. S. Cronie and S. B. Korada. Lossless source coding with polar codes. In 2010 IEEE International Symposium on Information Theory, pages 904-908, June 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universal quadratic lower bounds on source coding error exponents. C Chang, A Sahai, 41st Annual Conference on Information Sciences and Systems. C. Chang and A. Sahai. Universal quadratic lower bounds on source coding error exponents. In 2007 41st Annual Conference on Information Sciences and Systems, pages 714-719, March 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lossless coding for distributed streaming sources. S C Draper, C Chang, A Sahai, IEEE Transactions on Information Theory. 603S. C. Draper, C. Chang, and A. Sahai. Lossless coding for distributed streaming sources. IEEE Transactions on Information Theory, 60(3):1447-1474, March 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mathematical problems in the shannon theory of optimal coding of information. R L Dobrushin, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. the Fourth Berkeley Symposium on Mathematical Statistics and ProbabilityBerkeley, CalifUniversity of California Press1Contributions to the Theory of StatisticsR. L. Dobrushin. Mathematical problems in the shannon theory of optimal coding of information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, pages 211-252, Berkeley, Calif., 1961. University of California Press.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rick Durrett, Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press5 editionRick Durrett. Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 5 edition, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Amir Dembo, Ofer Zeitouni, Large deviations techniques and applications. BerlinSpringer-VerlagStochastic Modelling and Applied Probability. Corrected reprint of the second (1998) editionAmir Dembo and Ofer Zeitouni. Large deviations techniques and applications, vol- ume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The random coding bound is tight for the average linear code or lattice. Y Domb, R Zamir, M Feder, IEEE Transactions on Information Theory. 621Y. Domb, R. Zamir, and M. Feder. The random coding bound is tight for the average linear code or lattice. IEEE Transactions on Information Theory, 62(1):121-130, Jan 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Relaxed polar codes. M El-Khamy, H Mahdavifar, G Feygin, J Lee, I Kang, IEEE Transactions on Information Theory. 634+ 17] M. El-Khamy, H. Mahdavifar, G. Feygin, J. Lee, and I. Kang. Relaxed polar codes. IEEE Transactions on Information Theory, 63(4):1986-2000, April 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R M Fano, Transmission of Information: A Statistical Theory of Communications. M.I.T. PressR.M. Fano. Transmission of Information: A Statistical Theory of Communications. M.I.T. Press, 1961.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Binary linear codes with optimal scaling and quasi-linear complexity. Arman Fazeli, S Hamed Hassani, Marco Mondelli, Alexander Vardy, abs/1711.01339CoRRArman Fazeli, S. Hamed Hassani, Marco Mondelli, and Alexander Vardy. Binary lin- ear codes with optimal scaling and quasi-linear complexity. CoRR, abs/1711.01339, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Binary linear codes with optimal scaling: Polar codes with large kernels. A Fazeli, H Hassani, M Mondelli, A Vardy, 2018 IEEE Information Theory Workshop (ITW). A. Fazeli, H. Hassani, M. Mondelli, and A. Vardy. Binary linear codes with optimal scaling: Polar codes with large kernels. In 2018 IEEE Information Theory Workshop (ITW), pages 1-5, Nov 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Relations between entropy and error probability. M Feder, N Merhav, IEEE Transactions on Information Theory. 401M. Feder and N. Merhav. Relations between entropy and error probability. IEEE Transactions on Information Theory, 40(1):259-266, Jan 1994.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the scaling exponent of polar codes for binary-input energy-harvesting channels. S L Fong, V Y F Tan, IEEE Journal on Selected Areas in Communications. 3412S. L. Fong and V. Y. F. Tan. On the scaling exponent of polar codes for binary-input energy-harvesting channels. IEEE Journal on Selected Areas in Communications, 34(12):3540-3551, Dec 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scaling exponent and moderate deviations asymptotics of polar codes for the awgn channel. L Silas, Fong, Y F Vincent, Tan, Entropy. 197Silas L. Fong and Vincent Y. F. Tan. Scaling exponent and moderate deviations asymptotics of polar codes for the awgn channel. Entropy, 19(7), 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the scaling exponent of binary polarization kernels. A Fazeli, A Vardy, 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton). A. Fazeli and A. Vardy. On the scaling exponent of binary polarization kernels. In 2014 52nd Annual Allerton Conference on Communication, Control, and Comput- ing (Allerton), pages 797-804, Sept 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes for broadcast channels. N Goela, E Abbe, M Gastpar, IEEE Transactions on Information Theory. 612N. Goela, E. Abbe, and M. Gastpar. Polar codes for broadcast channels. IEEE Transactions on Information Theory, 61(2):758-782, Feb 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A simple derivation of the coding theorem and some applications. R Gallager, IEEE Transactions on Information Theory. 111R. Gallager. A simple derivation of the coding theorem and some applications. IEEE Transactions on Information Theory, 11(1):3-18, January 1965.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Information Theory and Reliable Communication. Robert G Gallager, John Wiley & Sons, IncNew York, NY, USARobert G. Gallager. Information Theory and Reliable Communication. John Wiley & Sons, Inc., New York, NY, USA, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The random coding bound is tight for the average code (corresp.). R Gallager, IEEE Transactions on Information Theory. 192R. Gallager. The random coding bound is tight for the average code (corresp.). IEEE Transactions on Information Theory, 19(2):244-246, March 1973.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Improved bounds on the finite length scaling of polar codes. D Goldin, D Burshtein, IEEE Transactions on Information Theory. 6011D. Goldin and D. Burshtein. Improved bounds on the finite length scaling of polar codes. IEEE Transactions on Information Theory, 60(11):6966-6978, Nov 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universal bounds on the scaling behavior of polar codes. A Goli, S H Hassani, R Urbanke, 2012 IEEE International Symposium on Information Theory Proceedings. A. Goli, S. H. Hassani, and R. Urbanke. Universal bounds on the scaling behav- ior of polar codes. In 2012 IEEE International Symposium on Information Theory Proceedings, pages 1957-1961, July 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Arıkan meets Shannon: Polar codes with near-optimal convergence to channel capacity. Venkatesan Guruswami, Andrii Riazanov, Min Ye, arXiv:1911.03858arXiv e-printsVenkatesan Guruswami, Andrii Riazanov, and Min Ye. Arıkan meets Shannon: Po- lar codes with near-optimal convergence to channel capacity. arXiv e-prints, page arXiv:1911.03858, Nov 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes: Speed of polarization and polynomial gap to capacity. V Guruswami, P Xia, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science. V. Guruswami and P. Xia. Polar codes: Speed of polarization and polynomial gap to capacity. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 310-319, Oct 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Error detecting and error correcting codes. The Bell System Technical. R W Hamming, Journal. 292R. W. Hamming. Error detecting and error correcting codes. The Bell System Tech- nical Journal, 29(2):147-160, April 1950.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarization and spatial coupling two techniques to boost performance. Hassani Seyed Hamed, Seyed Hamed Hassani. Polarization and spatial coupling two techniques to boost performance. 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Finite-length scaling for polar codes. S H Hassani, K Alishahi, R L Urbanke, IEEE Transactions on Information Theory. 6010S. H. Hassani, K. Alishahi, and R. L. Urbanke. Finite-length scaling for polar codes. IEEE Transactions on Information Theory, 60(10):5875-5898, Oct 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"General nonasymptotic and asymptotic formulas in channel resolvability and identification capacity and their application to the wiretap channel. M Hayashi, IEEE Transactions on Information Theory. 524M. Hayashi. General nonasymptotic and asymptotic formulas in channel resolvabil- ity and identification capacity and their application to the wiretap channel. IEEE Transactions on Information Theory, 52(4):1562-1575, April 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Information spectrum approach to second-order coding rate in channel coding. M Hayashi, IEEE Transactions on Information Theory. 5511M. Hayashi. Information spectrum approach to second-order coding rate in channel coding. IEEE Transactions on Information Theory, 55(11):4947-4966, Nov 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Performance of polar codes for channel and source coding. N Hussami, S B Korada, R Urbanke, 2009 IEEE International Symposium on Information Theory. N. Hussami, S. B. Korada, and R. Urbanke. Performance of polar codes for channel and source coding. In 2009 IEEE International Symposium on Information Theory, pages 1488-1492, June 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universally attainable error and information exponents, and equivocation rate for the broadcast channels with confidential messages. M Hayashi, R Matsumoto, 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton). M. Hayashi and R. Matsumoto. Universally attainable error and information expo- nents, and equivocation rate for the broadcast channels with confidential messages. In 2011 49th Annual Allerton Conference on Communication, Control, and Com- puting (Allerton), pages 439-444, Sep. 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rate-dependent analysis of the asymptotic behavior of channel polarization. S H Hassani, R Mori, T Tanaka, R L Urbanke, IEEE Transactions on Information Theory. 594S. H. Hassani, R. Mori, T. Tanaka, and R. L. Urbanke. Rate-dependent analysis of the asymptotic behavior of channel polarization. IEEE Transactions on Information Theory, 59(4):2267-2276, April 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Erasure and undetected error probabilities in the moderate deviations regime. M Hayashi, V Y F Tan, 2015 IEEE International Symposium on Information Theory (ISIT). M. Hayashi and V. Y. F. Tan. Erasure and undetected error probabilities in the moderate deviations regime. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 1821-1825, June 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar coding without alphabet extension for asymmetric models. J Honda, H Yamamoto, IEEE Transactions on Information Theory. 5912J. Honda and H. Yamamoto. Polar coding without alphabet extension for asymmetric models. IEEE Transactions on Information Theory, 59(12):7829-7838, Dec 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Extremes of random coding error exponents. A G Fàbregas, I Land, A Martinez, 2011 IEEE International Symposium on Information Theory Proceedings. A. G. i. Fàbregas, I. Land, and A. Martinez. Extremes of random coding error expo- nents. In 2011 IEEE International Symposium on Information Theory Proceedings, pages 2896-2898, July 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reed-muller codes achieve capacity on erasure channels. ] S Kkm + 17, S Kudekar, M Kumar, H D Mondelli, E Pfister, R L Şaşoǧlu, Urbanke, IEEE Transactions on Information Theory. 637KKM + 17] S. Kudekar, S. Kumar, M. Mondelli, H. D. Pfister, E. Şaşoǧlu, and R. L. Urbanke. Reed-muller codes achieve capacity on erasure channels. IEEE Transactions on In- formation Theory, 63(7):4298-4316, July 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An empirical scaling law for polar codes. S B Korada, A Montanari, E Telatar, R Urbanke, 2010 IEEE International Symposium on Information Theory. S. B. Korada, A. Montanari, E. Telatar, and R. Urbanke. An empirical scaling law for polar codes. In 2010 IEEE International Symposium on Information Theory, pages 884-888, June 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes for channel and source coding. Korada Satish Babu, 181Satish Babu Korada. Polar codes for channel and source coding. page 181, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes: Characterization of exponent, bounds, and constructions. S B Korada, E Sasoglu, R Urbanke, IEEE Transactions on Information Theory. 5612S. B. Korada, E. Sasoglu, and R. Urbanke. Polar codes: Characterization of ex- ponent, bounds, and constructions. IEEE Transactions on Information Theory, 56(12):6253-6264, Dec 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes for q-ary source coding. M Karzand, E Telatar, 2010 IEEE International Symposium on Information Theory. M. Karzand and E. Telatar. Polar codes for q-ary source coding. In 2010 IEEE International Symposium on Information Theory, pages 909-912, June 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes are optimal for lossy source coding. S B Korada, R L Urbanke, IEEE Transactions on Information Theory. 564S. B. Korada and R. L. Urbanke. Polar codes are optimal for lossy source coding. IEEE Transactions on Information Theory, 56(4):1751-1768, April 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fixed-length lossy compression in the finite blocklength regime. V Kostina, S Verdu, IEEE Transactions on Information Theory. 586V. Kostina and S. Verdu. Fixed-length lossy compression in the finite blocklength regime. IEEE Transactions on Information Theory, 58(6):3309-3338, June 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lossy joint source-channel coding in the finite blocklength regime. V Kostina, S Verdú, IEEE Transactions on Information Theory. 595V. Kostina and S. Verdú. Lossy joint source-channel coding in the finite blocklength regime. IEEE Transactions on Information Theory, 59(5):2545-2575, May 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the capacity of channels with deletions and states. Yonglong Li, Y F Vincent, Tan, Yonglong Li and Vincent Y. F. Tan. On the capacity of channels with deletions and states, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fast polarization for non-stationary channels. H Mahdavifar, 2017 IEEE International Symposium on Information Theory (ISIT). H. Mahdavifar. Fast polarization for non-stationary channels. In 2017 IEEE Inter- national Symposium on Information Theory (ISIT), pages 849-853, June 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Achieving marton's region for broadcast channels using polar codes. M Mondelli, S H Hassani, I Sason, R L Urbanke, IEEE Transactions on Information Theory. 612M. Mondelli, S. H. Hassani, I. Sason, and R. L. Urbanke. Achieving marton's region for broadcast channels using polar codes. IEEE Transactions on Information Theory, 61(2):783-800, Feb 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors. M Mondelli, S H Hassani, R L Urbanke, IEEE Transactions on Information Theory. 6212M. Mondelli, S. H. Hassani, and R. L. Urbanke. Unified scaling of polar codes: Error exponent, scaling exponent, moderate deviations, and error floors. IEEE Transac- tions on Information Theory, 62(12):6698-6712, Dec 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"From polar to reed-muller codes unified scaling, non-standard channels, and a proven conjecture. Marco Mondelli, 230Marco Mondelli. From polar to reed-muller codes unified scaling, non-standard chan- nels, and a proven conjecture. page 230, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Properties and construction of polar codes. Ryuhei Mori, abs/1002.3521CoRRRyuhei Mori. Properties and construction of polar codes. CoRR, abs/1002.3521, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Channel polarization on q-ary discrete memoryless channels by arbitrary kernels. R Mori, T Tanaka, 2010 IEEE International Symposium on Information Theory. R. Mori and T. Tanaka. Channel polarization on q-ary discrete memoryless chan- nels by arbitrary kernels. In 2010 IEEE International Symposium on Information Theory, pages 894-898, June 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Source and channel polarization over finite fields and reedsolomon matrices. R Mori, T Tanaka, IEEE Transactions on Information Theory. 605R. Mori and T. Tanaka. Source and channel polarization over finite fields and reed- solomon matrices. IEEE Transactions on Information Theory, 60(5):2720-2736, May 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An ergodic theory of binary operations-part i: Key properties. R Nasser, IEEE Transactions on Information Theory. 6212R. Nasser. An ergodic theory of binary operations-part i: Key properties. IEEE Transactions on Information Theory, 62(12):6931-6952, Dec 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An ergodic theory of binary operations-part ii: Applications to polarization. R Nasser, IEEE Transactions on Information Theory. 632R. Nasser. An ergodic theory of binary operations-part ii: Applications to polar- ization. IEEE Transactions on Information Theory, 63(2):1063-1083, Feb 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes for arbitrary dmcs and arbitrary macs. R Nasser, E Telatar, IEEE Transactions on Information Theory. 626R. Nasser and E. Telatar. Polar codes for arbitrary dmcs and arbitrary macs. IEEE Transactions on Information Theory, 62(6):2917-2936, June 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Error probability of identification via channels at rates above capacity. Y Oohama, Proceedings IEEE International Symposium on Information Theory. IEEE International Symposium on Information Theory26Y. Oohama. Error probability of identification via channels at rates above capacity. In Proceedings IEEE International Symposium on Information Theory,, pages 26-, June 2002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes for q-ary channels, q = 2 r. W Park, A Barg, IEEE Transactions on Information Theory. 592W. Park and A. Barg. Polar codes for q-ary channels, q = 2 r . IEEE Transactions on Information Theory, 59(2):955-969, Feb 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Channel coding: Non-asymptotic fundamental limits. Yury Polyanskiy, Princeton UniversityPhD thesisYury Polyanskiy. Channel coding: Non-asymptotic fundamental limits. PhD thesis, Princeton University, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Channel coding rate in the finite blocklength regime. Y Polyanskiy, H V Poor, S Verdu, IEEE Transactions on Information Theory. 565Y. Polyanskiy, H. V. Poor, and S. Verdu. Channel coding rate in the finite blocklength regime. IEEE Transactions on Information Theory, 56(5):2307-2359, May 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mixed-kernels constructions of polar codes. N Presman, O Shapira, S Litsyn, IEEE Journal on Selected Areas in Communications. 342N. Presman, O. Shapira, and S. Litsyn. Mixed-kernels constructions of polar codes. IEEE Journal on Selected Areas in Communications, 34(2):239-253, Feb 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Near-optimal finite-length scaling for polar codes over large alphabets. H D Pfister, R Urbanke, 2016 IEEE International Symposium on Information Theory (ISIT). H. D. Pfister and R. Urbanke. Near-optimal finite-length scaling for polar codes over large alphabets. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 215-219, July 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Channel dispersion and moderate deviations limits for memoryless channels. Y Polyanskiy, S Verdú, 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton). Y. Polyanskiy and S. Verdú. Channel dispersion and moderate deviations limits for memoryless channels. In 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1334-1339, Sept 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar coding theorems for discrete systems. Eren Sasoglu, 104Eren Sasoglu. Polar coding theorems for discrete systems. page 104, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes for discrete alphabets. Eren Sasoglu, 2012 IEEE International Symposium on Information Theory Proceedings. Eren Sasoglu. Polar codes for discrete alphabets. In 2012 IEEE International Sym- posium on Information Theory Proceedings, pages 2137-2141, July 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarization and polar codes. Eren Sasoglu, Foundations and Trends R in Communications and Information Theory. 84Eren Sasoglu. Polarization and polar codes. Foundations and Trends R in Commu- nications and Information Theory, 8(4):259-381, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lower bounds to error probability for coding on discrete memoryless channels. i. Information and Control. C E Shannon, R G Gallager, E R Berlekamp, 10C.E. Shannon, R.G. Gallager, and E.R. Berlekamp. Lower bounds to error probabil- ity for coding on discrete memoryless channels. i. Information and Control, 10(1):65 -103, 1967.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A mathematical theory of communication. C E Shannon, Bell System Tech. J. 27C. E. Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379-423, 623-656, 1948.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multilevel polarization of polar codes over arbitrary discrete memoryless channels. A G Sahebi, S S Pradhan, 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton). A. G. Sahebi and S. S. Pradhan. Multilevel polarization of polar codes over arbi- trary discrete memoryless channels. In 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1718-1725, Sep. 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Achieving the capacity of any dmc using only polar codes. D Sutter, J M Renes, F Dupuis, R Renner, 2012 IEEE Information Theory Workshop. D. Sutter, J. M. Renes, F. Dupuis, and R. Renner. Achieving the capacity of any dmc using only polar codes. In 2012 IEEE Information Theory Workshop, pages 114-118, Sep. 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar coding for processes with memory. Eren Sasoglu, Ido Tal, abs/1602.01870CoRREren Sasoglu and Ido Tal. Polar coding for processes with memory. CoRR, abs/1602.01870, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarization for arbitrary discrete memoryless channels. E Şaşoglu, E Telatar, E Arikan, IEEE Information Theory Workshop. E. Şaşoglu, E. Telatar, and E. Arikan. Polarization for arbitrary discrete memoryless channels. In 2009 IEEE Information Theory Workshop, pages 144-148, Oct 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarization for arbitrary discrete memoryless channels. Eren Sasoglu, Emre Telatar, Erdal Arikan, abs/0908.0302CoRREren Sasoglu, Emre Telatar, and Erdal Arikan. Polarization for arbitrary discrete memoryless channels. CoRR, abs/0908.0302, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Asymptotische abschätzungen in shannons informationstheorie. V Strassen, Transactions of the Third Prague Conference on Information Theory. Publishing House of the Czechoslovak Academy of SciencesV. Strassen. Asymptotische abschätzungen in shannons informationstheorie. In Transactions of the Third Prague Conference on Information Theory, pages 689- 723. Publishing House of the Czechoslovak Academy of Sciences, 1962.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A new polar coding scheme for strong security on wiretap channels. E Şaşoglu, A Vardy, 2013 IEEE International Symposium on Information Theory. E. Şaşoglu and A. Vardy. A new polar coding scheme for strong security on wiretap channels. In 2013 IEEE International Symposium on Information Theory, pages 1117-1121, July 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fourier analysis on finite groups and applications, volume 43 of London Mathematical Society Student Texts. Audrey Terras, Cambridge University PressCambridgeAudrey Terras. Fourier analysis on finite groups and applications, volume 43 of Lon- don Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes for the deletion channel: Weak and strong polarization. Ido Tal, Henry D Pfister, Arman Fazeli, Alexander Vardy, abs/1904.13385CoRR. Ido Tal, Henry D. Pfister, Arman Fazeli, and Alexander Vardy. Polar codes for the deletion channel: Weak and strong polarization. CoRR, abs/1904.13385, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"How to construct polar codes. I Tal, A Vardy, IEEE Transactions on Information Theory. 5910I. Tal and A. Vardy. How to construct polar codes. IEEE Transactions on Informa- tion Theory, 59(10):6562-6582, Oct 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"List decoding of polar codes. I Tal, A Vardy, IEEE Transactions on Information Theory. 615I. Tal and A. Vardy. List decoding of polar codes. IEEE Transactions on Information Theory, 61(5):2213-2226, May 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar-like codes and asymptotic tradeoff among block length, code rate, and error probability. Hsin-Po Wang, Iwan M Duursma, abs/1812.08112CoRR. Hsin-Po Wang and Iwan M. Duursma. Polar-like codes and asymptotic tradeoff among block length, code rate, and error probability. CoRR, abs/1812.08112, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the strong converse of the coding theorem for symmetric channels without memory. Lionel Weiss. ; R, J Wang, H Honda, R Yamamoto, Y Liu, Hou, IEEE Information Theory Workshop -Fall (ITW). 18Construction of polar codes for channels with memoryLIONEL WEISS. On the strong converse of the coding theorem for symmetric chan- nels without memory. Quarterly of Applied Mathematics, 18(3):209-214, 1960. [WHY + 15] R. Wang, J. Honda, H. Yamamoto, R. Liu, and Y. Hou. Construction of polar codes for channels with memory. In 2015 IEEE Information Theory Workshop - Fall (ITW), pages 187-191, Oct 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The coding of messages subject to chance errors. J Wolfowitz, Illinois J. Math. 141957J. Wolfowitz. The coding of messages subject to chance errors. Illinois J. Math., 1(4):591-606, 12 1957.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The wire-tap channel. A D Wyner, Bell System Technical Journal. 548A. D. Wyner. The wire-tap channel. Bell System Technical Journal, 54(8):1355-1387, 1975.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polar codes using dynamic kernels. M Ye, A Barg, 2015 IEEE International Symposium on Information Theory (ISIT). M. Ye and A. Barg. Polar codes using dynamic kernels. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 231-235, June 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Explicit polar codes with small scaling exponent. Hanwen Yao, Arman Fazeli, Alexander Vardy, abs/1901.08186CoRRHanwen Yao, Arman Fazeli, and Alexander Vardy. Explicit polar codes with small scaling exponent. CoRR, abs/1901.08186, 2019.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":34,"cells":{"text":{"kind":"string","value":"\nLocal times of multifractional Brownian sheets\n2008\n\nMark Meerschaert \nDepartment of Statistics and Probability\nMichigan State University\n48824East LansingMIUSA\n\nDongsheng Wu dongsheng.wu@uah.edu \nDepartment of Mathematical Sciences\nUniversity of Alabama in Huntsville\n35899HuntsvilleALUSA\n\nYimin Xiao **xiao@stt.msu.edu \nDepartment of Statistics and Probability\nMichigan State University\n48824East LansingMIUSA\n\nLocal times of multifractional Brownian sheets\n\nBernoulli\n143200810.3150/08-BEJ126Hausdorff dimensionlevel setslocal timesmultifractional Brownian sheetsone-sided sectorial local non-determinism\nDenote by H(t) = (H1(t), . . . , HN (t)) a function in tUnder some regularity conditions on the function H(t), we prove the existence, joint continuity and the Hölder regularity of the local times of {B H(t) (t)}. We also determine the Hausdorff dimensions of the level sets of {B H(t) (t)}. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets.\n\nIntroduction\n\nA one-dimensional fractional Brownian motion (fBm) ξ α = {ξ α (t), t ∈ R} with Hurst index α ∈ (0, 1) is a real-valued, centered Gaussian process with covariance function given by E[ξ α (s)ξ α (t)] = 1 2 [|s| 2α + |t| 2α − |t − s| 2α ] ∀s, t ∈ R.\n\n(1.1)\n\nIt was introduced, as a moving average Gaussian process, by Mandelbrot and Van Ness [31]. Fractional Brownian motion has interesting properties such as self-similarity of order α ∈ (0, 1), stationary increments and long-range dependence (when α > 1/2), which make it a good candidate for modeling different phenomena in, for example, finance and telecommunication. However, this model may be restrictive, due to the fact that all of its regularity and fractal properties are governed by the single Hurst parameter α. To model phenomena whose regularity evolves in time, for example, Internet traffic or image processing, Lévy-Véhel and Peltier [28] and Benassi, Jaffard and Roux [7] independently introduced multifractional Brownian motion (mfBm) in terms of moving average This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2008, Vol. 14, No. 3, 865-898. This reprint differs from the original in pagination and typographic detail.\n\nrepresentation and harmonizable representation, respectively. Multifractional Brownian motion is governed by a Hurst functional α(t) with certain regularity in place of the constant Hurst parameter α in fBm.\n\nSeveral authors have studied sample path and statistical properties of mfBm. For example, Benassi, Jaffard and Roux [7] considered the sample path Hölder regularity of mfBm and determined the Hausdorff dimension of its graph. Ayache, Cohen and Lévy-Véhel [2] and Herbin [22] studied the covariance structure of mfBm from its harmonisable representation. Recently, Boufoussi, Dozzi and Guerbaz [13,14] studied the existence, joint continuity and Hölder regularity of the local time of mfBm and established Chung's law of the iterated logarithm for mfBm. The main tool they applied to derive their results is the property of one-sided local non-determinism of mfBm.\n\nThere are multiparameter extensions of fBm, among which, two typical examples are multiparameter Lévy fBm ξ α = {ξ α (t), t ∈ R N } (whose covariance function is given by (1.1) with | · | being the Euclidean norm in R N ) and fractional Brownian sheets, where the former is isotropic, while the latter are anisotropic in general. Since they were introduced by Kamont [24] (see also Ayache, Léger and Pontier [4]), fractional Brownian sheets (fBs) have been studied extensively as a representative of anisotropic Gaussian random fields in recent years. See, for example, Dunker [18], Mason and Shi [32], Øksendal and Zhang [33], Xiao and Zhang [45], Ayache and Xiao [6], Ayache, Wu and Xiao [5], Wu and Xiao [39] and the references therein for further information. Still, the regularity of fBs does not evolve in the N -dimensional 'time' parameter t ∈ R N + . To model anisotropic Gaussian random fields whose regularity evolves in time, such as images, Ayache and Léger [3] and Herbin [22] introduced so-called multifractional Brownian sheets (mfBs) in terms of their moving average representations and harmonisable representations, where the constant Hurst vector of fBs is substituted by Hurst functionals. Furthermore, they showed that mfBs have a continuous modification and determined the pointwise and local Hölder exponent of mfBs. They also proved that mfBs are locally self-similar. We refer to Ayache and Léger [3] and Herbin [22] for the definitions of the corresponding concepts and results.\n\nIn studying anisotropic random fields, Xiao [44] suggested that it is more convenient to use the following metric ρ K on R N :\nρ K (s, t) = N ℓ=1 |s ℓ − t ℓ | K ℓ ∀s, t ∈ R N ,(1.2)\nwhere K = (K 1 , . . . , K N ) ∈ (0, 1) N is a fixed vector. Denote by H(t) = (H 1 (t), . . . , H N (t)) a function in t ∈ R N + with values in (0, 1) N . We say that H(t) satisfies Condition A if there exist a positive number α ∈ (0, 1) and a vector (K 1 , . . . , K N ) ∈ (0, 1) N such that A.1. for every ℓ ∈ {1, . . . , N }, α ≤ H ℓ (t) ≤ K ℓ for all t ∈ R N + ; A.2. H ℓ (t) (ℓ = 1, . . . , N ) satisfies a ρ K -Lipschitz condition on every compact set, that is, for every compact subset I ⊆ R N + , there exist positive constants c ℓ = c ℓ (I) and δ such that\n|H ℓ (t) − H ℓ (s)| ≤ c ℓ ρ K (s, t)\n∀s, t ∈ I with |s − t| < δ. Now, we are ready to define multifractional Brownian sheets via their moving average representations. \nB H(t) 0 (t) = R N N ℓ=1 [(t ℓ − u ℓ ) H ℓ (t)−1/2 + − (−u ℓ ) H ℓ (t)−1/2 + ]W (du) ∀t ∈ R N + ,(1.\n\n3)\n\nwhere s + = max{s, 0}, and where W = {W (s), s ∈ R N } is a standard real-valued Brownian sheet.\n\nRemark 1.2. Our Definition 1.1 generalizes the definition in Ayache and Léger [3], where they define H ℓ (·) as a function of t ℓ ∈ R + (ℓ = 1, . . . , N ). Herbin [22] defines multifractional Brownian sheets by using the following moving average representation (cf. [22] Definition 2, page 1261):\nB ′ 0 H(t) (t) = R N N ℓ=1 [|t ℓ − u ℓ | H ℓ (t)−1/2 − |u ℓ | H ℓ (t)−1/2 ]W (du) ∀t ∈ R N + . (1.4)\nBased on Dobrić and Ojeda [16] (see also Stoev and Taqqu [35]), we know that the multifractional Brownian sheet defined by (1.3) and the multifractional Brownian sheet defined by (1.4) have different correlation structures in general, even if N = 1. Our definition is more convenient to use than that of Herbin's when we derive the one-sided sectorial local non-determinism for multifractional Brownian sheets in Section 2. The form (1.3) is preferred in some applications in the one-dimensional case because it is easier to separate the future from the past.\n\nDefine the Gaussian random field\n{B H(t) (t)} = {B H(t) (t) : t ∈ R N + } with values in R d by B H(t) (t) = (B H(t) 1 (t), . . . , B H(t) d (t)) ∀t ∈ R N + ,(1.5)\nwhere {B\nH(t) 1 (t)}, . . . , {B H(t) d (t)} are d independent copies of {B H(t) 0 (t)}. Then {B H(t) (t), t ∈ R N + } is called an (N, d)-multifractional Brownian sheet with functional Hurst index H(t). Note that if N = 1, then {B H(t) (t)} is a multifractional Brownian motion in R d with Hurst index H 1 (t) ∈ (0, 1); if N > 1 and H 1 (t) ≡ H 1 , . . . , H N (t) ≡ H N , then {B H(t) (t)} is an (N, d)-fractional Brownian sheet with Hurst index H = (H 1 , . . . , H N ), which will be denoted {B H (t), t ∈ R N + }.\nIn this paper, we will study the existence and regularity of the local times of multifractional Brownian sheets. Our main technical tool is the property of one-sided sectorial local non-determinism. This property is more general than the sectorial local non-determinism which was first introduced by Khoshnevisan and Xiao [27] for the Brownian sheet and then extended by Wu and Xiao [39] to fractional Brownian sheets. See Section 2 for more information.\n\nOur results show that multifractional Brownian sheets are similar to fractional Brownian sheets in many ways. They admit jointly continuous local times when the indices H i (t) stay in the range for which the fractional Brownian sheet has jointly continuous local times. We also establish a Hausdorff dimension result and, essentially, the dimension of the level set is the same as for the constant parameter case, except that we take the supremum of the constant parameter formula. We also show that the supremum can be taken locally, to establish that the fractal dimension of the random field varies in space. Hence, multifractional Brownian sheets are useful in applications such as composite materials, or porous media flow, when the material properties vary in space. They may also find useful applications in image processing.\n\nThe rest of this paper is organized as follows. In Section 2, we prove some basic results on mfBs that will be useful to our arguments. In Section 3, we provide a sufficient condition for the existence of L 2 -local times of the (N, d) mfBs and prove that the condition also implies the joint continuity of the local times. We prove the Hölder regularity of the local times in Section 4. Finally, we derive the local Hausdorff dimensions of the level sets of {B H(t) (t)} in Section 5. Our results extend the results of Ayache and Xiao [6] and Ayache, Wu and Xiao [5] for fractional Brownian sheets and Boufoussi, Dozzi and Guerbaz [13,14] for multifractional Brownian motion to multifractional Brownian sheets.\n\nWe end the Introduction with some notation. Throughout this paper, the underlying parameter space is R N or R N + = [0, ∞) N . A parameter t ∈ R N is written as t = (t 1 , . . . , t N ), or as c , if t 1 = · · · = t N = c. For any s, t ∈ R N such that s j < t j (j = 1, . . . , N ) (we denote this by s ≺ t), we define the closed interval (or rectangle) [s, t] = N j=1 [s j , t j ]. We use A to represent the collection of closed intervals [s, t] with s, t ∈ [ε, Λ] N for some fixed positive numbers ε and Λ. For any integer m ≥ 1, we always write λ m for the Lebesgue measure on R m and use ·, · and | · | to denote the ordinary scalar product and the Euclidean norm in R m , respectively.\n\nThroughout this paper, an unspecified positive and finite constant will be denoted by c, which may not be the same in each occurrence. More specific constants in Section i are numbered as c i,1 , c i,2 , . . . .\n\n\nPreliminaries\n\nIn this section, we will provide some lemmas that are useful for proving our main results. Lemma 2.1 is an extension of Boufoussi et al. [13], Lemma 3.1 from fractional Brownian motion to fractional Brownian sheets.\n\nLemma 2.1. Let 0 < ε < Λ and 0 < α < γ < 1 be fixed constants. Let {Z κ 0 (t), (t, κ) ∈ R N + × [α, γ] N } be a real-valued Gaussian random field defined by equation (1.3) with\nH(t) ≡ κ. There then exists a constant c 2,1 = c(α, γ, ε, Λ, N ) > 0 such that E[Z β 0 (t) − Z β ′ 0 (t)] 2 ≤ c 2,1 |β − β ′ | 2 (2.1) for all t ∈ [ε, Λ] N and all β, β ′ ∈ [α, γ] N . Proof. For any β, β ′ ∈ [α, γ] N , we define κ 0 = β and κ j = (β ′ 1 , . . . , β ′ j , β j+1 , . . . , β N ) for j = 1, . . . , N . Clearly, κ N = β ′ . Since E[Z β 0 (t) − Z β ′ 0 (t)] 2 ≤ N N j=1 E[Z κ j−1 0 (t) − Z κ j 0 (t)] 2 ,(2.2)\nit suffices for us to prove that for j ∈ {1, . . . , N } fixed,\nE[Z κ j−1 0 (t) − Z κ j 0 (t)] 2 ≤ c(α, γ, ε, Λ, N )|β j − β ′ j | 2 . (2.3)\nBy using the moving average representation of fBs (equation (1.3)), we have that\nE[Z κ j−1 0 (t) − Z κ j 0 (t)] 2 = R N −1 j−1 ℓ=1 [(t ℓ − u ℓ ) β ′ ℓ −1/2 + − (−u ℓ ) β ′ ℓ −1/2 + ] 2 × N ℓ=j+1 [(t ℓ − u ℓ ) β ℓ −1/2 + − (−u ℓ ) β ℓ −1/2 + ] 2 dǔ j (2.4) × R [(t j − u j ) βj −1/2 + − (−u j ) βj −1/2 + − ((t j − u j ) β ′ j −1/2 + − (−u j ) β ′ j −1/2 + )] 2 du j := I 1 × I 2 ,\nwhereǔ j = (u 1 , . . . , u j−1 , u j+1 , . . . , u N ). A change of variables shows that\nI 1 = c j−1 ℓ=1 t 2β ℓ ℓ · N ℓ=j+1 t 2β ′ ℓ ℓ ,(2.5)\nwhich is bounded for all t ∈ [ε, Λ] N and β, β ′ ∈ [α, γ] N . Next, we estimate I 2 . Without loss of generality, we may assume β j > β ′ j . We rewrite I 2 as the following summation:\nI 2 = tj 0 [(t j − u j ) βj −1/2 − (t j − u j ) β ′ j −1/2 ] 2 du j + 0 −∞ [((t j − u j ) βj −1/2 − (−u j ) βj −1/2 ) (2.6) − ((t j − u j ) β ′ j −1/2 − (−u j ) β ′ j −1/2 )] 2 du j := II 1 + II 2 .\nBy the mean value theorem, we have that for some β ′ j ≤ η j1 ≤ β j (η j1 may depend on t j ),\nII 1 = tj 0 [(t j − u j ) ηj1−1/2 ln(t j − u j )] 2 |β j − β ′ j | 2 du j (2.7) ≤ c 2,2 |β j − β ′ j | 2\nfor all t ∈ [ε, Λ] N and β, β ′ ∈ [α, γ] N . In the above, the last inequality follows from a change of variables. Similarly, we have that for some β ′ j ≤ η j2 ≤ β j , (t)} be a multifractional Brownian sheet in R. There exist positive constants δ > 0, c 2,4 and c 2,5 such that for all s, t ∈ [ε, Λ] N with |s − t| < δ, for any\nII 2 = 0 −∞ [(t j − u j ) ηj2−1/2 ln(t j − u j ) − (−u j ) ηj2−1/2 ln(−u j )] 2 |β j − β ′ j | 2 du j (2.8) ≤ c 2,3 |β j − β ′ j | 2 for all t ∈ [u ∈ N ℓ=1 [s ℓ ∧ t ℓ , s ℓ ∨ t ℓ ], we have c 2,4 N ℓ=1 |t ℓ − s ℓ | 2H ℓ (u) ≤ E[B H(t) 0 (t) − B H(s) 0 (s)] 2 ≤ c 2,5 N ℓ=1 |t ℓ − s ℓ | 2H ℓ (u) . (2.9)\nProof. By the elementary inequalities\n3(a 2 + b 2 + c 2 ) ≥ (a + b + c) 2 ≥ 1 2 a 2 − 4b 2 − 4c 2 , we have that 3(E[B H(u) 0 (t) − B H(u) 0 (s)] 2 + E[B H(t) 0 (t) − B H(u) 0 (t)] 2 + E[B H(s) 0 (s) − B H(u) 0 (s)] 2 ) ≥ E[B H(t) 0 (t) − B H(s) 0 (s)] 2 (2.10) ≥ 1 2 E[B H(u) 0 (t) − B H(u) 0 (s)] 2 − 4E[B H(t) 0 (t) − B H(u) 0 (t)] 2 − 4E[B H(u) 0 (s) − B H(s) 0 (s)] 2 .\nThe first term on the right-hand side of (2.10) is the variance of the increment of a fractional Brownian sheet with Hurst index H(u). By Lemma 8 in Ayache and Xiao [6], there exist positive constants c 2,6 and c 2,7 , depending only on α, γ = max{K 1 , . . . , K N }, ε and N , such that\nc 2,6 N ℓ=1 |t ℓ − s ℓ | 2H ℓ (u) ≤ E[B H(u) 0 (t) − B H(u) 0 (s)] 2 ≤ c 2,7 N ℓ=1 |t ℓ − s ℓ | 2H ℓ (u) . (2.11)\nMeanwhile, by Condition A and Lemma 2.1, there exists δ > 0 small such that for all s, t ∈ [ε, Λ] N with |s − t| < δ, we have |u − s| < δ, which implies\nE[B H(u) 0 (s) − B H(s) 0 (s)] 2 ≤ c 2,1 |H(u) − H(s)| 2 (2.12) = c 2,1 N ℓ=1 |H ℓ (u) − H ℓ (s)| 2 ≤ c 2,8 N ℓ=1 |t ℓ − s ℓ | 2K ℓ\nand, similarly,\nE[B H(u) 0 (t) − B H(t) 0 (t)] 2 ≤ c 2,9 N ℓ=1 |t ℓ − s ℓ | 2K ℓ . (2.13)\nCombining (2.10) with (2.11), (2.12) and (2.13), and noting that sup t ℓ H ℓ (t ℓ ) ≤ K ℓ (still by Definition 1.1), we can see that there exists δ > 0, which depends only on α, γ, ε, Λ and N , such that for |s − t| < δ, (2.9) holds. This completes the proof of Lemma 2.2. Remark 2.3. For any fixed t ∈ [ε, Λ] N and any fixed unit vector η ∈ R N , let us consider the increment\nY η (t) = B H(t+η) 0 (t + η) − B H(t) 0 (t).\nIn light of Lemma 2.2, it would be interesting to compute the bounds on the correlation function\nr t,η (s) = E[Y η (t + s)Y η (t)] E[Y η (t + s) 2 ]E[Y η (t) 2 ]\n(2.14)\n\nand its time-and direction-varying spectral density (or Wigner-Ville distribution)\nf t,η (ξ) = R N e −i s,ξ r t,η (s) ds (2.15)\nsince those are the properties of the random field that are usually used to model natural phenomena.\n\nAs we mentioned in the Introduction, the main technical tool of this paper is the property of sectorial local non-determinism. Hence, it will be helpful to say a few words about the history of various forms of local non-determinism and briefly recall some of the definitions. The concept of local non-determinism, which provides a powerful tool for dealing with the complex dependence structure of Gaussian processes, was first introduced by Berman [11] to unify and extend his methods for studying local times of real-valued Gaussian processes, and then extended by Pitt [34] to Gaussian random fields. The notion of strong local non-determinism was later introduced by Cuzick and DuPreez [15] for Gaussian processes (i.e., N = 1) and can be extended in various ways to Gaussian random fields (see Xiao [43]). The simplest is the following. Let X = {X(t), t ∈ R N } be a real-valued Gaussian random field with 0 < E[X(t) 2 ] < ∞ for all t ∈ I, where I ∈ A is an interval. Let φ be a given function such that φ(0) = 0 and φ(r) > 0 for r > 0. Then X is said to be strongly locally φ-non-deterministic (SLφND) on I if there exist positive constants c 2,10 and r 0 such that for all t ∈ I and all 0 < r ≤ min{|t|, r 0 }, Var(X(t)|X(s) : s ∈ I, r ≤ |s − t| ≤ r 0 ) ≥ c 2,10 φ(r).\n\n(2.16)\n\nHere, Var(X(t)|X(s), s ∈ S) denotes the conditional variance of X(t), given X(s), s ∈ S. Pitt [34] proved that fractional Brownian motion ξ α is SLφND with φ(r) = r 2α . When N = 1 and Var(X(t)|X(s) : s ∈ I, r ≤ t − s ≤ r 0 ) ≥ c 2,10 φ(r), X is said to satisfy one-sided strong local φ-non-determinism. The properties of strong local non-determinism have played important roles in studying the regularity of local times, small ball probabilities, uniform Hausdorff dimension results and other sample path properties of Gaussian processes and Gaussian random fields. We refer to Xiao [43,44] for more information on applications of SLφND, as well as other forms of local non-determinism. It is well known that the Brownian sheet does not satisfy the property of local nondeterminism in the sense of Berman and Pitt (hence, it does not satisfy (2.16) either). Recently, Khoshnevisan and Xiao [27] introduced the concept of sectorial local nondeterminism and proved that the Brownian sheet has the property of sectorial local nondeterminism. Wu and Xiao [39] extended the result of Khoshnevisan and Xiao [27] and proved that the fractional Brownian sheet {B H 0 (t), t ∈ R N + }, where H = (H 1 , . . . , H N ) ∈ (0, 1), also has the property of sectorial local non-determinism. Namely, for any fixed positive number ε ∈ (0, 1), there exists a positive constant c 2,11 , depending only on ε, H and N , such that for all positive integers n ≥ 2, and all t 1 , . . . , t n ∈ [ε, ∞) N , we have\nVar(B H 0 (t n )|B H 0 (t 1 ), . . . , B H 0 (t n−1 )) ≥ c 2,11 N j=1 min 0≤k≤n−1 |t n j − t k j | 2Hj ,(2.17)\nwhere t 0 = 0. The concept of sectorial local non-determinism opens the door to unifying the previously different treatments for the Brownian sheet and fractional Brownian motion, and can also be used to prove new results about fractional Brownian motion. See, for example, Khoshnevisan, Wu and Xiao [26], Wu and Xiao [39] and Ayache, Wu and Xiao [5] for studies on sample path properties of the Brownian sheet and fractional Brownian sheets by using the sectorial local non-determinism property.\n\nIt is not known whether the multifractional Brownian sheet {B\nH(t) 0\n(t)} is sectorially locally non-deterministic. To study the local times of (N, d)-mfBs {B H(t) (t)}, we will work with multifractional Liouville sheets (mfLs) at first and prove that they can be well approximated by Gaussian processes which have the property of one-sided strong local non-determinism (see (2.22) and (2.23) below). For a function H(t) = (H 1 (t), . . . , H N (t)) satisfying Condition A, the real-valued, centered Gaussian random field {X\nH(t) 0 (t)} = {X H(t) 0 (t), t ∈ R N + } defined by X H(t) 0 (t) = [0,t] N ℓ=1 (t ℓ − s ℓ ) H ℓ (t)−1/2 W (ds), t ∈ R N + , (2.18)\nis called a multifractional Liouville sheet with functional Hurst index H(t). One parameter mfLs were first introduced by Lim and Muniandy [30] as an extension of fBm; see Lim [29] for more properties on one parameter mfLs.\nIt follows from (1.3) that for every t ∈ R N + , B H(t) 0 (t) = X H(t) 0 (t) + (−∞,t]\\[0,t] N ℓ=1 g ℓ (t ℓ , s ℓ )W (ds), (2.19) where g ℓ (t ℓ , s ℓ ) = ((t ℓ − s ℓ ) + ) H ℓ (t)−1/2 − ((−s ℓ ) + ) H ℓ (t)−1/2\nand the two fields on the right-hand side of (2.19) are independent. We will show that in studying the regularity properties of the local times of {B\nH(t) 0 (t)}, the Liouville sheet {X H(t) 0\n(t)} plays a crucial role and the second field in (2.19) can be neglected. More precisely, we will make use of the following property. For all integers n ≥ 2, t 1 , . . . , t n ∈ R N + and u 1 , . . . , u n ∈ R, we have\nVar n j=1 u j B H(t j ) 0 (t j ) ≥ Var n j=1 u j X H(t j ) 0 (t j ) .\n(2.20)\n\nNext, we use an argument in Ayache and Xiao [6] to provide a useful decomposition for {X\nH(t) 0 (t)}. For every t ∈ [ε, Λ] N ,\nwe decompose the rectangle [0, t] into the following disjoint union of subrectangles:\n[0, t] = [0, ε] N ∪ N ℓ=1 R ℓ (t) ∪ ∆(ε, t), (2.21) where R ℓ (t) = {r ∈ [0, Λ] N : 0 ≤ r i ≤ ε if i = ℓ, ε < r ℓ ≤ t ℓ } and ∆(ε, t)\ncan be written as a union of 2 N − N − 1 subrectangles of [0, t]. Denote the integrand in (2.18) by g(t, r). It\nfollows from (2.21) that for every t ∈ [ε, ∞) N , X H(t) 0 (t) = [0,ε] N g(t, r)W (dr) + N ℓ=1 R ℓ (t) g(t, r)W (dr) + ∆(ε,t) g(t, r)W (dr) := X(ε, t) + N ℓ=1 Y ℓ (t) + Z(ε, t). (2.22)\nSince the processes X(ε, t), Y ℓ (t) (1 ≤ ℓ ≤ N ) and Z(ε, t) are defined by the stochastic integrals with respect to W over disjoint sets, they are independent Gaussian random fields.\n\nThe following lemma shows that every process Y ℓ (t) has the property of strong local non-determinism along the ℓth direction. It will be essential to our proofs.\n\nLemma 2.4. Let I ∈ A and let ℓ ∈ {1, 2, . . ., N } be fixed. For all integers n ≥ 2 and t 1 , . . . , t n ∈ I such that\nt 1 ℓ ≤ t 2 ℓ ≤ · · · ≤ t n ℓ , we have Var(Y ℓ (t n )|Y ℓ (t j ) : 0 ≤ j ≤ n − 1) ≥ c 2,12 |t n ℓ − t n−1 ℓ | 2H ℓ (t n ) , (2.23)\nwhere t 0 ℓ = 0 and c 2,12 > 0 is a constant depending only on ε and I.\n\nProof. The proof is in the same spirit as the proof of Lemma 2.1 of Ayache, Wu and Xiao [5]. Working in the Hilbert space setting, the conditional variance in (2.23) is the square of the L 2 (P)-distance of Y ℓ (t n ) from the subspace generated by Y ℓ (t j )(0 ≤ j ≤ n − 1). Hence, it is sufficient to show that there exists a constant c 2,12 such that\nE Y ℓ (t n ) − n−1 j=1 a j Y ℓ (t j ) 2 ≥ c 2,12 |t n ℓ − t n−1 ℓ | 2H ℓ (t n ) (2.24)\nfor all a j ∈ R (j = 1, . . . , n − 1). However, by splitting R ℓ (t n ) into two disjoint parts and using the independence, we derive that\nE Y ℓ (t n ) − n−1 j=0 a j Y ℓ (t j ) 2 ≥ E R ℓ (t n )\\R ℓ (t n−1 ) g(t n , r)W (dr) 2 ≥ ε 0 · · · t n ℓ t n−1 ℓ · · · ε 0 N k=1 (t n k − r k ) 2H k (t n )−1 dr (2.25) ≥ c 2,12 |t n ℓ − t n−1 ℓ | 2H ℓ (t n ) .\nThis proves (2.24) and hence Lemma 2.4.\n\nCombining Lemma 2.2 and Lemma 2.4 with the proofs of Lemma 2.1 and Lemma 8.1 in Berman [11] (see also Boufoussi, Dozzi and Guerbaz [13], Theorem 3.3), we have the following.\n\nProposition 2.5. For every integer n ≥ 2, there exist positive constants C n and δ (both of these may depend on n) such that for every ℓ = 1, . . . , N ,\nVar n j=1 u j [Y ℓ (t j ) − Y ℓ (t j−1 )] ≥ C n n j=1 u 2 j Var[Y ℓ (t j ) − Y ℓ (t j−1 )] (2.26)\nfor all (u 1 , . . . , u n ) ∈ R n and all points t 1 , . . . , t n ∈ I satisfying t 1 ℓ < · · · < t n ℓ with t n ℓ − t 1 ℓ < δ.\n\nThe following lemma relates the multifractional Brownian sheet {B H(t) 0 (t)} to the independent Gaussian random fields Y ℓ (ℓ = 1, . . . , N ), which is a direct extension of Ayache, Wu and Xiao [5], Lemma 2.2.\n\nLemma 2.6. Let I ∈ A. For all integers n ≥ 2, t 1 , . . . , t n ∈ I and u 1 , . . . , u n ∈ R, we have\nVar n j=1 u j B H(t j ) 0 (t j ) ≥ N ℓ=1 Var n j=1 u j Y ℓ (t j ) . (2.27)\nConsequently, for all k ∈ {1, . . . , N } and positive numbers p 1 , . . . , p k ≥ 1 satisfying \nk ℓ=1 p −1 ℓ = 1, we have 1 [det Cov(B H(t 1 ) 0 (t 1 ), . . . , B H(t n ) 0 (t n ))] 1/2 ≤ k ℓ=1 c n 2,13 [det Cov(Y ℓ (t 1 ), . . . , Y ℓ (t n ))] 1/(2p ℓ ) ,(2.H(t) 0\n(t)} satisfies the following property: for all integers n ≥ 2 and t 1 , . . . , t n ∈ I such that t j ≺ t n for every j ≤ n − 1,\nVar(B H 0 (t n )|B H 0 (t 1 ), . . . , B H 0 (t n−1 )) ≥ c 2,12 N j=1 min 0≤k≤n−1 (t n j − t k j ) 2Hj (t n ) . (2.29)\nThis property is weaker than (2.17) and will be referred to as one-sided sectorial local non-determinism.\n\nWe will also make use of the following technical lemmas, among which Lemma 2.8 is from Xiao and Zhang [45], Lemma 2.9 is proved in Ayache and Xiao [6] and Lemma 2.10 and Lemma 2.11 are from Ayache, Wu and Xiao [5]. \nJ := J(A, B) = 1 0 dt (A + t α ) β (B + t) η . (2.31)\nThere then exist finite constants c 2,15 and c 2,16 , depending only on α, β and η, such that the following hold for all reals A, B > 0 satisfying A 1/α ≤ c 2,15 B:\n(i) if αβ > 1, then J ≤ c 2,16 1 A β−α −1 B η ; (2.32) (ii) if αβ = 1, then J ≤ c 2,16 1 B η log(1 + BA −1/α ); (2.33) (iii) if 0 < αβ < 1 and αβ + η = 1, then J ≤ c 2,16 1 B αβ+η−1 + 1 . (2.34) Lemma 2.10. Let (ρ 1 , . . . , ρ N ) ⊆ (0, 1) N . For any q ∈ [0, N ℓ=1 ρ −1 ℓ ), let τ ∈ {1, . . . , N } be the integer such that τ −1 ℓ=1 1 ρ ℓ ≤ q < τ ℓ=1 1 ρ ℓ , (2.35)\nwith the convention that 0\nℓ=1 1 ρ ℓ := 0.\nThere then exists a positive constant ∆ τ ≤ 1, depending only on (ρ 1 , . . . , ρ N ), such that for every ∆ ∈ (0, ∆ τ ), we can find τ real numbers\np ℓ ≥ 1 (1 ≤ ℓ ≤ τ ) satisfying the following properties: τ ℓ=1 1 p ℓ = 1, ρ ℓ q p ℓ < 1 ∀ℓ = 1, . . . , τ (2.36)\nand\n(1 − ∆) τ ℓ=1 ρ ℓ q p ℓ ≤ ρ τ q + τ − τ ℓ=1 ρ τ ρ ℓ . (2.37) Furthermore, if we let α τ := τ ℓ=1 1 ρ ℓ − q > 0, then for any positive number ρ ∈ (0, α τ 2τ ), there exists an ℓ 0 ∈ {1, . . . , τ } such that ρ ℓ 0 q p ℓ0 + 2ρ ℓ 0 ρ < 1. (2.38)\nLemma 2.11. For all integers n ≥ 1, positive numbers a, r, 0 < b j < 1 and an arbitrary\ns 0 ∈ [0, a/2], a≤s1≤···≤sn≤a+r n j=1 (s j − s j−1 ) −bj ds 1 · · · ds n ≤ c n 2,17 (n!) (1/n) n j=1 bj −1 r n− n j=2 bj , (2.39) where c 2,17 > 0 is a constant depending only on a and the b j 's. In particular, if b j = α for all j = 1, . . . , n, then a≤s1≤···≤sn≤a+r n j=1 (s j − s j−1 ) −α ds 1 · · · ds n ≤ c n 2,17 (n!) α−1 r n(1−(1−1/n)α) . (2.40)\nFinally, we conclude this section by briefly recalling some aspects of the theory of local times. For excellent surveys on local times of random and deterministic vector fields, we refer to Geman and Horowitz [20] and Dozzi [17].\n\nLet X(t) be a Borel vector field on R N with values in R d . For any Borel set I ⊆ R N , the occupation measure of X on I is defined as the following measure on R d :\nµ I (•) = λ N {t ∈ I : X(t) ∈ •}.\nIf µ I is absolutely continuous with respect to λ d , we say that X(t) has local times on I and define its local times, L(•, I), as the Radon-Nikodým derivative of µ I with respect to λ d , that is,\nL(x, I) = dµ I dλ d (x) ∀x ∈ R d .\nIn the above, x is the so-called space variable and I is the time variable. Sometimes, we write L(x, t) in place of L(x, [0, t]). Note that if X has local times on I, then for every Borel set J ⊆ I, L(x, J) also exists. By standard martingale and monotone class arguments, one can deduce that the local times have a measurable modification that satisfies the following occupation density formula (see Geman and Horowitz [20], Theorem 6.4): for every Borel set I ⊆ R N and every measurable function f :\nR d → R + , I f (X(t)) dt = R d f (x)L(x, I) dx. (2.41) Suppose we fix a rectangle I = N i=1 [a i , a i + h i ].\nThen, whenever we can choose a version of the local time, still denoted by\nL(x, N i=1 [a i , a i +t i ]), such that it is a continuous function of (x, t 1 , . . . , t N ) ∈ R d × N i=1 [0, h i ],\nX is said to have a jointly continuous local time on I. When a local time is jointly continuous, L(x, •) can be extended to be a finite Borel measure supported on the level set\nX −1 I (x) = {t ∈ I : X(t) = x}; (2.42)\nsee Adler [1] for details. In other words, local times often act as a natural measure on the level sets of X. Hence, they are useful in studying the various fractal properties of level sets and inverse images of the vector field X. In this regard, we refer to Berman [10], Ehm [19] and Xiao [41]. It follows from Geman and Horowitz [20], ((25.5) and (25.7)) (see also Pitt [34]) that for all x, y ∈ R d , I ∈ A and all integers n ≥ 1,\nE[L(x, I) n ] = (2π) −nd I n R nd exp −i n j=1 u j , x × E exp i n j=1 u j , X(t j ) du dt (2.43)\nand for all even integers n ≥ 2, where u = (u 1 , . . . , u n ), t = (t 1 , . . . , t n ) and each u j ∈ R d , t j ∈ I ⊆ (0, ∞) N . In the coordinate notation, we then write u j = (u j 1 , . . . , u j d ).\n\n\nLocal times: existence and joint continuity\n\nIn this section, we consider the existence and regularity of local times of mfBs. The index H depends on I, but, for simplicity, we have deleted I from the notation. Proof. Without loss of generality, we may, and will, assume that δ = diam(I) is sufficiently small that (2.9) holds for all s, t ∈ I. In particular, we assume δ ≤ 1.\n\nTo prove the existence of local time on I, by Geman and Horowitz [20] Theorem 21.9, it suffices to prove that\nI I (E[B H(t) 0 (t) − B H(s) 0 (s)] 2 ) −d/2 ds dt < ∞. (3.2)\nIt follows from (2.9) and a change of variables that\nI I (E[B H(t) 0 (t) − B H(s) 0 (s)] 2 ) −d/2 ds dt ≤ c I I N ℓ=1 |t ℓ − s ℓ | 2H ℓ (s) −d/2 ds dt (3.3) ≤ c 3,1 I I N ℓ=1 |t ℓ − s ℓ | 2H ℓ −d/2 ds dt. Since d < N ℓ=1 1 H ℓ\n, we can estimate the last integral in a way similar to the proof of Xiao and Zhang [45] Theorem 3.6. Namely, by applying Lemma 2.8 repeatedly, we derive that the last integral is finite. This proves (3.2) and hence Theorem 3.1.\n\nThe following is a consequence of Theorem 3.1 which gives a more natural condition for the existence of local times of mfBs. H ℓ (t) for all t ∈ I, but the equality only holds for t in a set of Lebesgue measure 0, the existence of local times is rather subtle and requires the imposition of further assumptions on (H 1 (t), . . . , H N (t)). Hence, it will not be discussed here.\n\nWe now consider the joint continuity of the local times of {B H(t) (t)}. For convenience, we first prove that under the same condition as in Theorem 3.1, the local time of {B H(t) (t)} has a version that is jointly continuous in both space and time variables. We then apply the same argument as in the proof of Corollary 3.2 to show that the same conclusion holds, provided\nN ℓ=1 1 H ℓ (t) > d for all t ∈ I.\nOur results extend those of Ehm [19] for the Brownian sheet and of Ayache, Wu and Xiao [5] for fractional Brownian sheets. The main idea for proving Theorem 3.4 is similar to those in Ehm [19], Xiao [41] and Ayache, Wu and Xiao [5]. That is, we first apply the Fourier analytic arguments to derive estimates for the moments of the local times and then apply a multiparameter version of the Kolmogorov continuity theorem (cf. Khoshnevisan [25]). As in Ayache, Wu and Xiao [5], the 'one-sided' sectorial local non-deterministic properties of multifractional Liouville sheets proved in Section 2 (see Lemma 2.4 and Proposition 2.5) will play important roles in deriving moment estimates in Lemmas 3.5 and 3.7 below. However, due to the nonstationarity and the lack of two-sided local non-determinism of multifractional Brownian sheets, we need to make several modifications in our proofs.\nTheorem 3.4. Let {B H(t) (t)} = {B H(t) (t), t ∈ R N + } be\nFor convenience, we further assume 0 < H 1 ≤ · · · ≤ H N < 1. \nE[L(x, T ) n ] ≤ c n 3,2 (n!) N −βτ h nβτ , (3.6) where β τ = N − τ − H τ d + τ ℓ=1 H τ /H ℓ .\nProof. For later use, we will start with an arbitrary closed interval T = N ℓ=1 [a ℓ , a ℓ + h ℓ ] ⊆ I. It follows from (2.43) and the fact that {B\nH(t) 1 (t)}, . . . , {B H(t) d (t)} are indepen- dent copies of {B H(t) 0\n(t)} that for all integers n ≥ 1,\nE[L(x, T ) n ] ≤ (2π) −nd T n d k=1 R n exp − 1 2 Var n j=1 u j k B H(t j ) 0 (t j ) dU k dt, (3.7)\nwhere U k = (u 1 k , . . . , u n k ) ∈ R n . Fix k = 1, . . . , d and denote the inner integral in (3.7) by N k . Then by Lemma 2.6, we have\nN k ≤ R n exp − 1 2 N ℓ=1 Var n j=1 u j k Y ℓ (t j ) dU k (3.8) ≤ R n exp − 1 2 τ ℓ=1 Var n j=1 u j k Y ℓ (t j ) dU k .\nSince (3.5) holds, we apply Lemma 2.10 with ∆ = n −1 and q = d to obtain τ positive numbers p 1 , . . . , p τ ≥ 1 satisfying (2.36) and (2.37).\n\nApplying the generalized Hölder inequality (Hardy [21] page 140), we derive that\nN k ≤ τ ℓ=1 R n exp − p ℓ 2 Var n j=1 u j k Y ℓ (t j ) dU k 1/p ℓ = c n 3,3 τ ℓ=1 [det Cov(Y ℓ (t 1 ), . . . , Y ℓ (t n ))] −1/(2p ℓ ) ,(3.9)\nwhere the last equality follows from the fact that (Y ℓ (t 1 ), . . . , Y ℓ (t n )) is a Gaussian vector with mean 0. Hence, it follows from (3.7) and (3.9) that\nE[L(x, T ) n ] ≤ c n 3,3 T n τ ℓ=1 [det Cov(Y ℓ (t 1 ), . . . , Y ℓ (t n ))] −d/(2p ℓ ) dt. (3.10)\nTo evaluate the integral in (3.10), we will first integrate [dt 1 ℓ · · · dt n ℓ ] for ℓ = 1, . . . , τ . To this end, we use the following fact about multivariate normal distributions: for any Gaussian random vector (Z 1 , . . . , Z n ), det Cov(Z 1 , . . . , Z n ) = Var(Z 1 ) n j=2 Var(Z j |Z 1 , . . . , Z j−1 ).\n\n(3.11)\n\nBy the above fact and Lemma 2.4, we can derive that for every ℓ ∈ {1, . . . , τ } and for all\nt 1 , . . . , t n ∈ T = N ℓ=1 [a ℓ , a ℓ + h ℓ ] satisfying a ℓ ≤ t π ℓ (1) ℓ ≤ t π ℓ (2) ℓ ≤ · · · ≤ t π ℓ (n) ℓ ≤ a ℓ + h ℓ (3.12)\nfor some permutation π ℓ of {1, . . . , N }, we have\ndet Cov(Y ℓ (t 1 ), . . . , Y ℓ (t n )) ≥ c n 3,4 n j=1 (t π ℓ (j) ℓ − t π ℓ (j−1) ℓ ) 2H ℓ (t π ℓ (j) ) (3.13) ≥ c n 3,5 n j=1 (t π ℓ (j) ℓ − t π ℓ (j−1) ℓ ) 2H ℓ ,\nwhere t π ℓ (0) ℓ := ε (recall the decomposition (2.22)). We choose ε < 1 2 min{a ℓ , 1 ≤ ℓ ≤ N } so that Lemma 2.11 is applicable. It follows from (3.12) and (3.13) that\n[a ℓ ,a ℓ +h ℓ ] n [det Cov(Y ℓ (t 1 ), . . . , Y ℓ (t n ))] −d/(2p ℓ ) dt 1 ℓ · · · dt n ℓ ≤ π ℓ c n a ℓ ≤t π ℓ (1) ℓ ≤···≤t π ℓ (n) ℓ ≤a ℓ +h ℓ n j=1 1 (t π ℓ (j) ℓ − t π ℓ (j−1) ℓ ) H ℓ d/p ℓ dt 1 ℓ · · · dt n ℓ (3.14) ≤ c n 3,6 (n!) H ℓ d/p ℓ h n(1−(1−1/n)H ℓ d/p ℓ ) ℓ .\nIn the above, the last inequality follows from (2.40) (recall that H ℓ d/p ℓ < 1). Combining (3.10) and (3.14), we have\nE[L(x, T ) n ] ≤ c n 3,7 (n!) τ ℓ=1 H ℓ d/p ℓ τ ℓ=1 h n(1−(1−1/n)H ℓ d/p ℓ ) ℓ · N ℓ=τ +1 h n ℓ . (3.15)\nWe now consider the special case when T = [a, a + h ], that is, h 1 = · · · = h N = h with h < δ, (3.15) and (2.37) with ∆ = n −1 and q = d together yield\nE[L(x, T ) n ] ≤ c n 3,8 (n!) τ ℓ=1 H ℓ d/p ℓ h n(N −(1−n −1 ) τ ℓ=1 H ℓ d/p ℓ ) (3.16) ≤ c n 3,9 (n!) N −βτ h nβτ .\nThis proves (3.6).\n\nRemark 3.6. In the proof of Lemma 3.5, if we apply the generalized Hölder inequality to the first integral in (3.8) with N positive numbers p 1 , . . . , p N defined by\np ℓ = N i=1 H ℓ H i (ℓ = 1, . . . , N ),\nthen the above proof shows that if T ⊆ I ∈ A, then, similarly to (3.15), the following inequality holds:\nE[L(x, T ) n ] ≤ c n 3,10 (n!) N ν λ N (T ) n(1−ν) ,(3.17)\nwhere ν = d/( N ℓ=1 1 H ℓ ) ∈ (0, 1). We will apply this inequality in the proof of Theorem 3.4 below. \nwhere α τ = τ ℓ=1 1 H ℓ − d.\nProof. Let γ ∈ (0, 1 ∧ ατ 2τ ) be a constant. Note that by the elementary inequalities\n|e iu − 1| ≤ 2 1−γ |u| γ for all u ∈ R (3.19)\nand |u + v| γ ≤ |u| γ + |v| γ , we see that for all u 1 , . . . , u n , x, y ∈ R d ,\nn j=1 |e −i u j ,x − e −i u j ,y | ≤ 2 (1−γ)n |x − y| nγ ′ n j=1 |u j kj | γ , (3.20)\nwhere the summation ′ is taken over all the sequences (k 1 , . . . , k n ) ∈ {1, . . . , d} n . It follows from (2.44), (3.20) and Lemma 2.6 that for every even integer n ≥ 2,\nE[(L(x, T ) − L(y, T )) n ] ≤ |x − y| nγ ′ T n dt R nd n m=1 |u m km | γ exp − 1 2 Var n j=1 u j , B H(t j ) (t j ) du = |x − y| nγ ′ T n dt R nd n m=1 |u m km | γ d k=1 exp − 1 2 Var n j=1 u j k B H(t j ) k (t j ) du (3.21) ≤ |x − y| nγ ′ T n dt R nd n m=1 |u m km | γ d k=1 exp − 1 2 N ℓ=1 Var n j=1 u j k Y ℓ (t j ) du ≤ |x − y| nγ ′ T n dt R nd n m=1 |u m km | γ d k=1 exp − 1 2 τ ℓ=1 Var n j=1 u j k Y ℓ (t j ) du = |x − y| nγ ′ T n dt d k=1 R n n j=1 |u j k | γη j k exp − 1 2 τ ℓ=1 Var n j=1 u j k Y ℓ (t j ) dU k ,\nwhere η j k = 1 if k = k j and η j k = 0 otherwise. Note that for every j ∈ {1, . . . , n}, we have d k=1 η j k = 1. We take β ℓ = H ℓ (1 ≤ ℓ ≤ N ), ∆ = 1/n and q = d in Lemma 2.10 and let p ℓ (ℓ = 1, . . . , τ ) be the constants satisfying (2.36) and (2.37). Observe that since γ ∈ (0, ατ 2τ ), there exists\nan ℓ 0 ∈ {1, . . . , τ } such that H ℓ0 d p ℓ0 + 2H ℓ0 γ < 1. (3.22)\nCombining (3.21) with the generalized Hölder inequality, we have that\nE[(L(x, T ) − L(y, T )) n ] ≤ |x − y| nγ ′ T n dt (3.23) × d k=1 R n n j=1 |u j k | γη j k p ℓ 0 exp − 1 2 Var n j=1 u j k Y ℓ0 (t j ) dU k 1/p ℓ 0 × τ ℓ =ℓ0 R n exp − 1 2 Var n j=1 u j k Y ℓ (t j ) dU k 1/p ℓ .\nFor any n points t 1 , . . . , t n ∈ T , let π 1 , . . . , π N be N permutations of {1, 2, . . . , n} such that for every 1 ≤ ℓ ≤ N ,\nt π ℓ (1) ℓ ≤ t π ℓ (2) ℓ ≤ · · · ≤ t π ℓ (n) ℓ . (3.24) Let M ℓ0 = R n n j=1 |u j k | γη j k p ℓ 0 exp − 1 2 Var n j=1 u j k Y ℓ0 (t j ) dU k . (3.25)\nBy changing the variables of the above integral by means of the transformation\nu π ℓ 0 (j) k = v j k − v j+1 k , j = 1, . . . , n; u π ℓ 0 (n) k = v n k , we have that n j=1 u j k Y ℓ0 (t j ) = n j=1 v j k (Y ℓ0 (t π ℓ 0 (j) ) − Y ℓ0 (t π ℓ 0 (j−1) )),\nwhere t π ℓ 0 (0) = 0. Furthermore, by the elementary inequality that for\nξ > 0, |a − b| ξ ≤ c ξ (|a| ξ + |b| ξ ), where c ξ = 2 ξ−1 if ξ > 1 and 1 if ξ ≤ 1, we have that n j=1 |u π ℓ 0 (j) k | γη j k p ℓ 0 = n−1 j=1 |v j k − v j+1 k | γη j k p ℓ 0 |v n k | γη n k p ℓ 0 (3.26) ≤ c n n−1 j=1 (|v j k | γη j k p ℓ 0 + |v j+1 k | γη j k p ℓ 0 )|v n k | γη n k p ℓ 0 .\nMoreover, the last product is equal to a finite sum of terms, each of the form n j=1 |v j k | γη j k p ℓ 0 εj , where ε j = 0, 1, or 2 and\nn j=1 d k=1 η j k ε j = n. Let σ 2 ℓ0,j = E[(Y ℓ0 (t π ℓ 0 (j) ) − Y ℓ0 (t π ℓ 0 (j−1) )) 2 ]\n. By Proposition 2.5, we know that M ℓ0 is dominated by the sum over all possible choices of (ε 1 , . . . , ε n ) ∈ {0, 1, 2} n of the following terms\nR n n j=1 |v j k | γη j k p ℓ 0 εj exp − C n 2 n j=1 (v j k ) 2 σ 2 ℓ0,j dV k , (3.27)\nwhere V k = (v 1 k , . . . , v n k ) ∈ R n . By another change of variable, w j k = σ ℓ0,j v j k , the integral in (3.27) can be represented by\nn j=1 σ −1−γη j k p ℓ 0 εj ℓ0,j R n n j=1 |w j k | γη j k p ℓ 0 εj exp − C n 2 n j=1 (w j k ) 2 dW k (3.28) := C n,1 n j=1 σ −1−γη j k p ℓ 0 εj ℓ0,j , where C n,1 = R n n j=1 |w j k | γη j k p ℓ 0 εj exp − C n 2 n j=1 (w j k ) 2 dW k\nis a constant depending on n. Thus, we have obtained that\nM ℓ0 ≤ c n n j=1 σ −1−γη j k p ℓ 0 εj ℓ0,j .\n(3.29)\n\nThe other integrals for ℓ = ℓ 0 in (3.23) are easier and can be estimated similarly. Combining (3.23) with Lemma 2.4 (which gives lower bounds for σ 2 ℓ,j ), (3.28), (3.29) and the definition of T , we have\nE[(L(x, T ) − L(y, T )) n ] ≤ c n |x − y| nγ ′ Π ℓ 0 n j=1 (t π ℓ 0 (j) ℓ0 − t π ℓ 0 (j−1) ℓ0 ) −(H ℓ 0 d/p ℓ 0 )−γH ℓ 0 εj dt 1 ℓ0 · · · dt n ℓ0 (3.30) × τ ℓ =ℓ0 Π ℓ n j=1 1 (t π ℓ (j) ℓ − t π ℓ (j−1) ℓ ) Hℓd/p ℓ dt 1 ℓ · · · dt n ℓ × N ℓ=τ +1 h n ℓ .\nIn the above,\nΠ ℓ = {a ℓ ≤ t π ℓ (1) ℓ ≤ · · · ≤ t π ℓ (n) ℓ ≤ a ℓ + h ℓ } for every 1 ≤ ℓ ≤ τ .\nWe now take h 1 = · · · = h N = h < δ in (3.30). Then by Lemma 2.11 and noting that (3.22) holds, we have\nE[(L(x, T ) − L(y, T )) n ] ≤ c n |x − y| nγ h n(N −(1− 1 n ) τ ℓ=1 H ℓ d/p ℓ −H ℓ 0 γ) (3.31) ≤ c n |x − y| nγ h n(βτ −Hτ γ) ,\nwhere the last inequality follows from Lemma 2.10 and the fact that H ℓ0 ≤ H τ . Proof Theorem 3.4. Let I = [a, b] ∈ A be fixed and assume that (3.5) holds. It follows from Lemma 3.7 and the multiparameter version of Kolmogorov's continuity theorem (cf. Khoshnevisan [25]) that for every T ⊆ I, the mfBs {B H(t) } has almost surely a local time L(x, T ) that is continuous for all x ∈ R d .\n\nTo prove the joint continuity, observe that for all x, y ∈ R d and s, t ∈ I with |s − t| < δ, where δ > 0 is the same as in Lemma 2.2, we have\nE[(L(x, [a, s]) − L(y, [a, t])) n ] (3.32) ≤ 2 n−1 {E[(L(x, [a, s]) − L(x, [a, t])) n ] + E[(L(x, [a, t]) − L(y, [a, t])) n ]}.\nSince the difference L(x, [a, s]) − L(x, [a, t]) can be written as a sum of a finite number (which only depends on N ) of terms of the form L(x, T j ), where each T j ∈ A is a closed subinterval of I with at least one edge length ≤ |s − t|, we can use Lemma 3.5 and Remark 3.6 to bound the first term in (3.32). On the other hand, the second term in (3.32) can be dealt with using Lemma 3.7, as above. Consequently, for some small γ ∈ (0, 1), the righthand side of (3.32) is bounded by c n 3,11 (|x − y| + |s − t|) nγ , where n ≥ 2 is an arbitrary even integer. Therefore, the joint continuity of the local times on I again follows from the multiparameter version of Kolmogorov's continuity theorem. This finishes the proof of Theorem 3.4.\n\nSimilarly to the proof of Corollary 3.2, we derive from Theorem 3.4 the following, more general, result. The next corollary is a direct consequence of Corollary 3.8 and the continuity of the Hurst functionals. We state it to emphasize that the joint continuity of the local time is a local property depending on H(t). (t), t ∈ R N + } be a real-valued multifractional Brownian sheet defined by the harmonizable representation (cf. Herbin [22])\nB H(t) 0 (t) = R N N ℓ=1 e it ℓ λ ℓ − 1 |λ ℓ | H ℓ (t)+1/2 W (dλ), ∀t ∈ R N + ,(3.33)\nwhere W is the complex-valued Gaussian random measure in R N with the Lebesgue measure Λ N as its control measure. Based on Dobríc and Ojeda [16] (see also Stoev and Taqqu \n\n\nHölder conditions for L(x, •)\n\nIn this section, we investigate the local and global asymptotic behavior of the local time L(x, ·) at x as a random measure on R N + . Results in this section carry information about fractal properties of the sample functions of mfBs; see Section 5.\n\nBy applying Lemma 3.5, we can prove the following technical lemmas, which will be useful in this section. \nE[L(x + B H(a) (a), T ) n ] ≤ c n 4,1 (n!) N −βτ r nβτ . (4.1)\nProof. The proof is similar to the proof of Ayache, Wu and Xiao [5] Lemma 3.11 and we include it here for completeness. For each fixed a ∈ I, we define the Gaussian random\nfield Y = {Y (t), t ∈ R N + } with values in R d by Y (t) = B H(t) (t) − B H(a) (a)\n. It follows from (2.41) that if {B H(t) (t)} has a local time L(x, S) on any Borel set S, then Y also has a local timeL(x, S) on S and, moreover, L(x + B H(a) (a), S) =L(x, S). With a little modification, the proof of Lemma 3.5 works for the Gaussian field Y . Hence, we derive that (4.1) holds.\n\nThe following lemma is a consequence of Lemma 4.1 and Chebyshev's inequality. N , d, I and H) such that for all a ∈ I, T = [a, a + r ] with r ∈ (0, δ), where δ > 0 is the same as in Lemma 2.2, x ∈ R d and u > 1 large enough, we have\nP{L(x + B H(a) (a), T ) ≥ c 4,2 r βτ u N −βτ } ≤ exp(−bu). (4.2)\nLet U (t, r) be the open ball centered at t with radius r, let H U be the vector defined by (3.1) with I = U (t, r) and let τ (U ) be the positive integer satisfying the corresponding condition (3.5). By applying Lemma 3.5 and the Borel-Cantelli lemma, one can easily derive the following law of the iterated logarithm for the local time L(x, ·): there exists a positive constant c 4,3 such that for every x ∈ R d and t ∈ (0, ∞) N ,\nlim sup r→0 L(x, U (t, r)) ϕ U (r) ≤ c 4,3 , (4.3) where ϕ U (r) = r β τ (U ) (log log(1/r)) N −β τ (U ) with β τ (U) = N − τ (U ) − H U τ (U) d + τ (U) ℓ=1 H U τ (U) H U ℓ .\nBecause of the continuity of H ℓ (t) (1 ≤ ℓ ≤ N ), it can be verified that τ (U ) → τ (t) and β τ (U) → β τ (t) as r → 0, (4.4) where τ (t) is the unique integer satisfying\nτ (t)−1 ℓ=1 1 H ℓ (t) ≤ d < τ (t) ℓ=1 1 H ℓ (t) (4.5)\nand\nβ τ (t) = N − τ (t) − H τ (t) (t)d + τ (t) ℓ=1 H τ (t) (t) H ℓ (t) . (4.6)\nIt follows from Fubini's theorem that with probability one, (4.3) holds for almost all t ∈ (0, ∞) N . We now prove a stronger version of this result, which is useful in determining the Hausdorff dimension of the level set. where ϕ t (r) = r β τ (t) (log log(1/r)) N −β τ (t) and where β τ (t) is defined by (4.6).\n\nProof. For every integer k > 0, we consider the random measure L k (x, •) on the Borel subsets C of I defined by\nL k (x, C) = C (2πk) d/2 exp − k|B H(t) (t) − x| 2 2 dt (4.8) = C R d exp − |ξ| 2 2k + i ξ, B H(t) (t) − x dξ dt.\nThen, by the occupation density formula (2.41) and the continuity of the function y → L(y, C), one can verify that almost surely L k (x, C) → L(x, C) as k → ∞ for every Borel set C ⊆ I. For every integer m ≥ 1, let f m (t) = L(x, U (t, 2 −m )). From the proof of Theorem 3.4, we can see that almost surely the functions f m (t) are continuous and bounded. Hence, we have, almost surely, for all integers m, n ≥ 1,\nI [f m (t)] n L(x, dt) = lim k→∞ I [f m (t)] n L k (x, dt). (4.9)\nIt follows from (4.9), (4.8) and the proof of Pitt [34], Proposition 3.1 that for every positive integer n ≥ 1,\nE I [f m (t)] n L(x, dt) = 1 2π (n+1)d I U(t,2 −m ) n R (n+1)d exp −i n+1 j=1\nx, u j (4.10)\n\n× E exp i Let γ > 0 be a constant, the value of which will be determined later. We consider the random set\nI m (ω) = {t ∈ I : f m (t) ≥ γϕ U (2 −m )}.\nDenote by µ ω the restriction of the random measure L(x, ·) to I, that is, µ ω (E) = L(x, E ∩ I) for all Borel sets E ∈ R N + . We now take n = ⌊log m⌋, where ⌊y⌋ denotes the integer part of y. Then by applying (4.11) and by Stirling's formula, we have\nEµ ω (I m ) ≤ E I [f m (t)] n L(x, dt) [γϕ U (2 −m )] n (4.12) ≤ c n 4,4 (n!) N −β τ (U ) 2 −mnβ τ (U ) γ n 2 −mnβ τ (U ) (log m) n(N −β τ (U ) ) ≤ m −2 ,\nprovided γ > 0 is chosen large enough, say, γ ≥ c 4,2 . This implies that\nE ∞ m=1 µ ω (I m ) < ∞.\nTherefore, with probability 1 for µ ω -almost all t ∈ I, we have (4.14)\nlim sup m→∞ L(x, U (t, 2 −m )) ϕ U (2 −m ) ≤ c 4,2 .\nFinally, for any r > 0 small enough, there exists an integer m such that 2 −m ≤ r < 2 −m+1 and (4.13) is applicable. Since ϕ t is increasing near 0, (4.7) follows from (4.13) and a monotonicity argument.\n\nRecall that the pointwise Hölder exponent of a random field {X(t), t ∈ R N } at a point t ∈ R N is defined by We remark that the corresponding problem of finding the Hausdorff dimensions of the level sets of multifractional Brownian motion has been investigated by Boufoussi et al. [13] and the Hausdorff dimensions of the level sets of fractional Brownian sheets were studied in Ayache and Xiao [6]. As shown by the following theorem, the fractal structure of Γ x is much richer than the level sets of multifractional Brownian motion and is locally reminiscent of the level sets of a fractional Brownian sheet.\nα X (t) = sup α : lim |h|→0 X(t + h) − X(t) |h| α = 0 .Theorem 5.1. Let {B H(t) (t)} = {B H(t) (t), t ∈ R N + } be an (N, d)-multifractional\nBrownian sheet with Hurst functionals H ℓ (t) (ℓ = 1, . . . , N ). For any interval I ∈ A, let t * ∈ I be a point satisfying\nN ℓ=1 1 H ℓ (t * ) = max t∈I N ℓ=1 1 H ℓ (t) and 0 < H 1 (t * ) ≤ · · · ≤ H N (t * ) < 1. If N ℓ=1 1 H ℓ (t * ) < d, then for every x ∈ R d , we have Γ x ∩ I = ∅ a.s. If N ℓ=1 1\nH ℓ (t * ) > d, then for any x ∈ R d , with positive probability,\ndim H (Γ x ∩ I) = β τ (t * ) , (5.1) where β τ (t * ) = min k ℓ=1 H k (t * ) H ℓ (t * ) + N − k − H k (t * )d, 1 ≤ k ≤ N .\nRemark 5.2. It can be verified that if (4.5) holds for t = t * , then β τ (t * ) is the same as in (4.6) with t replaced by t * . In the special case where H(t) = H is a constant, β τ (t * ) reduces to the form derived in Ayache and Xiao [6] for the Hausdorff dimension of the level sets of a fractional Brownian sheet. When\nN ℓ=1 1\nH ℓ (t * ) = d, we believe that for every x ∈ R d , Γ x ∩ I = ∅ a.s. However, the method of this paper is not sufficient for proving this statement.\n\nProof of Theorem 5.1. We will follow the proof of Ayache and Xiao [6] Theorem 5.\n\nFirst, we prove\ndim H (Γ x ∩ I) ≤ min k ℓ=1 H k (t * ) H ℓ (t * ) + N − k − H k (t * )d, 1 ≤ k ≤ N a.s. (5.2)\nand Γ x ∩ I = ∅ a.s. when the right-hand side of (5.2) is negative. Without loss of any generality, we may assume that I = [a, a + h ] and h is small so that Lemma 2.2 is applicable. For an integer n ≥ 1, divide the interval I into m n ≤ n N ℓ=1 (H ℓ (t * )) −1 subrectangles R n,j of side lengths n −1/H ℓ (t * ) h (ℓ = 1, . . . , N ). Let 0 < ̺ < 1 be fixed and let κ n,j be the lower-left vertex of R n,j . Define ρ(s, t) = (E[B H(t) (t) − B H(s) (s)] 2 ) 1/2 , denote by N ρ (R n,j , ε) the smallest number of ρ-balls of radius ε needed to cover R n,j and denote by D the diameter of R n,j , that is, D := sup{ρ(s, t) : s, t ∈ R n,j }.\nBy Lemma 2.2, we have D 0 log N ρ (R n,j , ε) dε ≤ cn −1 .\nCombining the above inequality with Talagrand [36], Lemma 2.1 gives\nP max s,t∈Rn,j |B H(s) (s) − B H(t) (t)| > n −(1−̺) ≤ e −cn 2̺ . (5.3)\nThen for n sufficiently large, we have, for any fixed x ∈ I,\nP{x ∈ {B H(u) (u), u ∈ R n,j }} ≤ P max s,t∈Rn,j |B H(s) (s) − B H(t) (t)| ≤ n −(1−̺) ; x ∈ {B H(u) (u), u ∈ R n,j } + P max s,t∈Rn,j |B H(s) (s) − B H(t) (t)| > n −(1−̺) (5.4) ≤ P{|B H(κn,j ) (κ n,j ) − x| ≤ n −(1−̺) } + e −cn 2̺ ≤ c 5,1 n −(1−̺)d .\nIn the above, we have applied the fact that Var(B H(t) 0 (t)) ≥ c for all t ∈ I to derive the last inequality. If\nN ℓ=1 1 H ℓ (t * ) < d, we choose ̺ > 0 such that (1 − ̺)d > N ℓ=1 1 H ℓ (t * )\n. Let N n be the number of rectangles R n,j such that x ∈ {B H(u) (u), u ∈ R n,j }. It follows from (5.4) that\nE(N n ) ≤ c 5,1 n N ℓ=1 (H ℓ (t * )) −1 n −(1−̺)d → 0 as n → ∞. (5.5)\nSince the random variables N n are integer-valued, (5.5) and Fatou's lemma imply that a.s. N n = 0 for infinitely many n. Therefore, Γ x ∩ I = ∅ a.s. We now assume\nN ℓ=1 1 H ℓ (t * ) > d. For every k ∈ {1, 2, . . . , N }, define η k = k ℓ=1 H k (t * ) H ℓ (t * ) + N − k − H k (t * )d.\nBy Ayache and Xiao [6], Lemma 7, we have η k > 0. Define a covering {R ′ n,j } of Γ x ∩ I by R ′ n,j = R n,j if x ∈ {B H(u) (u), u ∈ R n,j } and R ′ n,j = ∅ otherwise. R ′ n,j can be covered by n N ℓ=k+1 (H k (t * ) −1 −H ℓ (t * ) −1 ) cubes of side length n −H k (t * ) −1 h. Thus, we can cover the level set Γ x ∩ I by a sequence of cubes of side length n −H k (t * ) −1 h. Let ̺ ∈ (0, 1) be small and let\nη k (̺) = k ℓ=1 H k (t * ) H ℓ (t * ) + N − k − H k (t * )(1 − ̺)d.\nClearly, η k (̺) > η k > 0. In the following, we prove that the Hausdorff dimension of Γ x ∩ I is bounded above by η k (̺) almost surely. To this end, by (5.4), we have\nE mn j=1 n N ℓ=k+1 (H k (t * ) −1 −H ℓ (t * ) −1 ) (n −H k (t * ) −1 ) η k (̺) ½ {x∈{B H(u) (u),u∈Rn,j}} (5.6) ≤ c 5,2 n N ℓ=1 H ℓ (t * ) −1 + N ℓ=k+1 (H k (t * ) −1 −H ℓ (t * ) −1 )−η k (̺)H k (t * ) −1 −(1−̺)d = c 5,2 .\nFatou's lemma implies that the η k (̺)-dimensional Hausdorff measure of Γ x ∩I is finite a.s. and thus dim H (Γ x ∩ I) ≤ η k (̺) almost surely. Letting ̺ ↓ 0 along the rational numbers, we obtain dim H (Γ x ∩ I) ≤ η k and therefore (5.2).\n\nTo prove the lower bound in (5.1), we assume τ (t * ) = k, that is,\nk−1 ℓ=1 1 H ℓ (t * ) ≤ d < k ℓ=1 1 H ℓ (t * )\n.\n\nBy Condition A, there exists a positive number ς such that for all t ∈ I ς := [t * − ς , t * + ς ] ∩ I, we have\nk−1 ℓ=1 1 H ℓ ≤ d < k ℓ=1 1 H ℓ ,(5.7)\nwhere H ℓ (1 ≤ ℓ ≤ N ) are defined as in (3.1) with I ς in place of I. Note that (5.7) and Ayache and Xiao [6], Lemma 3.3, imply that\nk ℓ=1 H k H ℓ + N − k − H k d ∈ (N − k, N − k + 1].\nThus, we can choose ̺ > 0 such that\nγ := k ℓ=1 H k H ℓ + N − k − H k (1 + ̺)d > N − k. (5.8)\nIt is sufficient to prove that there is a constant c 5,3 > 0 such that\nP{dim H (Γ x ∩ I ς ) ≥ γ} ≥ c 5,3 . (5.9)\nOur proof of (5.9) is based on the capacity argument due to Kahane [23]. Similar methods have been used by Adler [1], Testard [37] and Xiao [40], to mention just a few. Let M + γ be the space of all non-negative measures on R N with finite γ-energy. It is known (cf. Adler [1]) that M + γ is a complete metric space under the metric\nµ γ = R N R N µ(dt)µ(ds) |t − s| γ . (5.10)\nWe define a sequence of random positive measures µ n on the Borel sets C ⊆ I ς by\nµ n (C) = C (2πn) d/2 exp − n|B H(t) (t) − x| 2 2 dt (5.11) = C R d exp − |ξ| 2 2n + i ξ, B H(t) (t) − x dξ dt.\nIt follows from Kahane [23] or Testard [37] that if there are positive and finite constants c 5,4 and c 5,5 , independent of ̺ and such that E( µ n ) ≥ c 5,4 , E( µ n 2 ) ≤ c 5,5 , (5.12) E( µ n γ ) < +∞, (5.13) where µ n = µ n (I ς ), then there is a subsequence of {µ n }, say {µ n k }, such that µ n k → µ in M + γ and µ is strictly positive with probability ≥ c 2 5,4 /(2c 5,5 ). It follows from (4.8) that µ has its support in Γ x ∩ I ς almost surely. Hence, Frostman's theorem yields (5.9).\n\nIt remains to verify (5.12) and (5.13). Let σ 2 (t) = Var(B H(t) (t)). By Fubini's theorem, we have \nE( µ n ) = Iς R d e −i ξ,x exp − |ξ| 2 2n E exp(i ξ, B H(t) (t) ) dξ dt = Iς R d e −i ξ( N ℓ=1 t H ℓ ℓ ) d ( N ℓ=1 t ℓ ) γ dt 1 ,\nwhere the two inequalities follow from (5.16) and a change of variables. By using Lemma 2.9 in the same way, we see that E( µ n γ ) < +∞ for any γ defined in (5.8). This proves (5.13). Finally, by letting ̺ ↓ 0, the lower bound for the Hausdorff dimension in (5.1) follows from (5.9) and we have therefore proven Theorem 5.1.\n\nThe proof of Theorem 5.1 suggests that we can consider the Hausdorff dimension of the level set in any neighborhood of a point t ∈ (0, ∞), provided N ℓ=1 1 H ℓ (t) > d. However, in order to obtain an almost sure result, we have to consider Γ x at a random level x = B H(t) (t). The following corollary can be considered as a local Hausdorff dimension result for the level sets of mfBs. where β τ (t 0 ) is defined by (4.6) with t 0 in place of t.\n\nProof. For any t 0 ∈ (0, ∞) N such that On the other hand, by using an argument similar to the one used in the proof of Berman [9] Theorem 2.1 (see also the proof of Xiao [42] Theorem 1.1), we can show that for every ε > 0 and r ∈ (0, r 0 ) small enough, P dim H (Γ B H(t) (t) ∩ U (t 0 , r)) ≥ max s∈U(t 0 ,r) β τ (s) − ε a.e. t ∈ U (t 0 , r 0 ) = 1.\n\n(5.23)\n\nBy letting r ↓ 0 and ε ↓ 0 along the rational numbers, we see that (5.19) follows from (5.22) and (5.23).\n\nRemark 5.4. Corollary 5.3 shows the explicit way in which the fractal properties of the multifractional Brownian sheet vary in space. In short, the local Hausdorff dimension is derived from the constant parameter formula. Fractional Brownian sheets are essentially fractional integrals of Brownian sheets (cf. Benson et al. [8] and Biermé, Meerschaert and Scheffler [12]). The term \"multifractional\" indicates that the order of fractional integration varies in space. It would be interesting to explore the connection between multifractional Brownian sheets and multifractals. For example, are the level sets of the mfBs multifractals and, if so, how do their structure functions depend on the Hurst index function H(t)?\n\nDefinition 1 . 1 .\n11Let H(t) = (H 1 (t), . . . , H N (t)) be a function in t ∈ R N + with values in (0, 1) N satisfying Condition A. A real-valued multifractional Brownian sheet {B ), t ∈ R N + } with functional Hurst index H(t)is defined as the following moving average Wiener integral:\n\nLemma 2. 8 ..\n8Let 0 < h < 1 be a constant. Then for any δ > 2h, M > 0 and β > 0, there exists a positive and finite constant c 2,14 , depending only on δ, ε, β and M , such that for all 0 < a ≤ M , |s − r| 2h ] −β ds ≤ c 2,14 (a −(Let α, β and η be positive constants. For A > 0 and B > 0, let\n\nE\n[(L(x, I) − L(y, I)) n ] = (2π) −nd I n R nd n j=1 e −i u j ,x − e −i u j ,y j , X(t j ) du dt,\n\nTheorem 3. 1 .mfBs\n1Let I ∈ A and H be the vector defined in (3.1). If d < {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.\n\nCorollary 3. 2 .\n2Let I ∈ A be fixed. If N ℓ=1 1H ℓ (t) > d for all t ∈ I, then mfBs {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.Proof. Since the functions H 1 (t), . . . , H N (t) are uniformly continuous on I, we can divide I into subintervals {I p } such that for each I p , we haveN ℓ=1 1 H ℓ (p) > d, where H ℓ (p) = max t∈Ip H ℓ (t). It follows from Theorem 3.1 that on every I p , mfBs {B H(t) (t)} has an L 2 -integrable local time L(·, I p ) almost surely. This implies that {B H(t) (t)} has an L 2 -integrable local time on I, which concludes the proof of Corollary 3.2.\n\nRemark 3. 3 .\n3The condition in Corollary 3.2 is almost the best possible, in the sense that, if d > N ℓ=1 1 H ℓ (t) for some t ∈ I, then it can be proven using Geman and Horowitz [20] Theorem 21.9 that {B H(t) (t)} has no L 2 (R d × Ω)-integrable local times on I. In the case when d ≤ N ℓ=1 1\n\n. 5 .\n5Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying There then exists a positive constant c 3,2 , depending only on N , d, H and I, such that for all x ∈ R d , all subintervals T = [a, a + h ] ⊆ I with h > 0 small and all integers n ≥ 1,\n\nLemma 3. 7 .\n7Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying (3.5). There then exists a positive and finite constant c n , depending on N, d, H, I and n, such that for all subintervals T = [a, a + h ] ⊆ I with h > 0 small, x, y ∈ R d with |x − y| ≤ 1, all even integers n ≥ 1 and all γ ∈ (0, 1 ∧ ατ 2τ ), E[(L(x, T ) − L(y, T )) n ] ≤ c n |x − y| nγ h n(βτ −Hτ γ) , (3.18)\n\nCorollary 3. 8 .\n8Let I ∈ A be fixed. If N ℓ=1 1 H ℓ (t) > d for all t ∈ I,then mfBs {B H(t) (t)} has a jointly continuous local time on I almost surely.\n\nCorollary 3 . 9 .\n39Let t 0 ∈ [ε, Λ] N be fixed. If\n\nH ℓ (t 0 )\n0> d, then mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ) for some r 0 > 0 almost surely, where U (t 0 , r 0 ) is the open ball centered at t 0 with radius r 0 .\n\n\nwe can prove a similar result as in Lemma 2.2 (cf. Boufoussi, Dozzi and Guerbaz[13], Wu[38]). Hence, the existence of local times of the (N, d)-multifractional Brownian sheet { B H(t) is (even one-sided) sectorial locally non-deterministic (since the Fourier analytic technique employed in Wu and Xiao[39] for proving that fractional Brownian sheets are sectorial local non-deterministic fails when H(t) varies in t).It is an open problem to establish the joint continuity of the local times of { B H(t) (t)}.\n\nLemma 4 . 1 .\n41Under the conditions of Lemma 3.5, there exists a positive and finite constant c 4,1 , depending only on N, d, H and I, such that for all a ∈ I and hypercubes T = [a, a + r ] ⊆ I with r < δ, where δ > 0 is the same as in Lemma 2.2, x ∈ R d and all integers n ≥ 1,\n\nLemma 4. 2 .\n2Under the conditions of Lemma 3.5, there exist positive constants c 4,2 , b (depending only on\n\nTheorem 4 . 3 ..\n43Let I ∈ A be a fixed interval and assume that d < For any fixed x ∈ R d , let L(x, ·) be the local time of {B H(t) (t)} at x which is a random measure supported on the level set. There then exists a positive and finite constant c 4,4 , independent of x, such that with probability 1, the following holds for L(x, ·)-almost all t ∈ I: lim sup r→0 L(x, U (t, r)) ϕ t (r) ≤ c 4,4 , (4.7)\n\n\n, B H(s j ) (s j ) du ds, where u = (u 1 , . . . , u n+1 ) ∈ R (n+1)d and s = (t, s 1 , . . . , s n ). Similarly to the proof of (3.6), for sufficiently large m, we have that the right-hand side of equation (4.10) is at most n!) N −β τ (U ) 2 −mnβ τ (U ) , where c 4,4 is a positive finite constant depending only on N, d, H, and I.\n\n\n, we can see that for m sufficiently large, there exists a constant c 4,5 > 0 such that ϕ U (2 −m ) ≤ c 4,5 ϕ t (2 −m ). Therefore, we have lim sup m→∞ L(x, U (t, 2 −m )) ϕ t (2 −m ) ≤ c 4,6 .\n\n\n4.3, and noting that L(·, t) vanishes outside some compact set U (depending on ω), we have the following corollary.\n\nCorollary 4. 4 .\n4For every x ∈ R d , the pointwise Hölder exponent α L of L(x, t) at t satisfiesα L (t) ≥ β τ (t)a.s.(4.16) 5. Hausdorff dimensions of the level sets For x ∈ R d , let Γ x = {t ∈ (0, ∞) N : B H(t) (t) = x} be the level set of the multifractional Brownian sheet {B H(t) (t)}. In this section, we determine the Hausdorff dimension of Γ x .\n\n\nDenote by I 2d the identity matrix of order 2d, by Cov(B H(s) (s), B H(t) (t)) the covariance matrix of (B H(s) (s), B H(t) (t)), let Σ = n −1 I 2d + Cov(B H(s) (s), B H(t) (t)) and denote by (ξ, η) ′ the transpose of the row vector (ξ, η). Lemma 2.4 and the regularity of the H(·), it can be proven that fors, t ∈ I ς , ℓ − t ℓ | 2H ℓ (t) ≥ c 5,6 N ℓ=1 |s ℓ − t ℓ | 2H ℓ . ℓ − t ℓ | 2H ℓ ] d/2 ds dt := c 5,5 < ∞. |s ℓ − t ℓ |) γ ( N ℓ=1 |s ℓ − t ℓ | 2H ℓ )\n\nCorollary 5. 3 .\n3Let {B H(t) (t)} = {B H(t) (t), t ∈ R N + } be an (N, d)-multifractional Brownian sheet with Hurst functionals H ℓ (t) (ℓ = 1, . . . , N ). If t 0 ∈ (0, ∞) N satisfies N ℓ=1 1 H ℓ (t 0 ) > d, then there exists r 0 > 0 such that P lim r→0 dim H (Γ B H(t) (t) ∩ U (t 0 , r)) = β τ (t 0 )for a.e. t ∈ U (t 0 , r 0 ) = 1,(5.19) \n\nH\nℓ (t 0 ) > d, there exists a positive number r 0 such that for all s ∈ U (t 0 , r 0 ), we haveN ℓ=1 1H ℓ (s) > d. By Corollary 3.9, mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ). By (5.2), we have that for every 0 < r < r 0 and x ∈ R d ,dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) 5.20)and Fubini's theorem, we see thatP dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) β τ (s) , a.e. x ∈ R d = 1. (5.21)Since {B H(t) (t)} has a local time on U (t 0 , r 0 ), the occupation density formula (2.41) and (5.21) together imply thatP dim H (Γ B H(t) (t) ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r)β τ (s) a.e. t ∈ U (t 0 , r 0 ) = 1.(5.22) \n\n\n28) where det Cov(Z 1 , . . . , Z n ) denotes the determinant of the covariance matrix of the Gaussian random vector (Z 1 , . . . , Z n ).Remark 2.7. By (2.27) and Lemma 2.4, we derive that {B\n\n\na multifractional Brownian sheet with values in R d . Let I ∈ A and let H be the vector defined in (3.1). If d <N \nℓ=1 \n1 \nH ℓ \n\n, \n\nthen {B H(t) (t)} has a jointly continuous local time on I. \n\n\n\n\nThe proof of Theorem 3.4 is similar to the proofs of Xiao and Zhang[45] Theorem 4.1 and Ayache, Wu and Xiao [5] Theorem 3.1; we include it here for completeness.\nAcknowledgementsWe thank the anonymous referees for their many valuable comments and suggestions. The research of Mark Meerschaert is supported in part by NSF Grant DMS-07-06440. The research of Yimin Xiao is supported in part by NSF Grant DMS-07-06728.References\nThe Geometry of Random Fields. R J Adler, Wiley611857New YorkAdler, R.J. (1981). The Geometry of Random Fields. New York: Wiley. MR0611857\n\nThe covariance structure of multifractional Brownian motion, with application to long range dependence. Proceeding of ICASSP. A Ayache, S Cohen, J Lévy-Véhel, IstambulAyache, A., Cohen, S. and Lévy-Véhel, J. (2002). The covariance structure of multifractional Brownian motion, with application to long range dependence. Proceeding of ICASSP. Istambul.\n\nFractional and multifractional Brownian sheet. A Ayache, S Léger, PreprintAyache, A. and Léger, S. (2000). Fractional and multifractional Brownian sheet. Preprint.\n\nA Ayache, S Léger, M Pontier, Drap Brownien fractionnaire. Potential Theory. 171906407Ayache, A., Léger, S. and Pontier, M. (2002). Drap Brownien fractionnaire. Potential The- ory 17 31-43. MR1906407\n\nJoint continuity of the local times of fractional Brownian sheets. A Ayache, D Wu, Y Xiao, Ann. Inst. H. Poincaré Probab. Statist. To appearAyache, A., Wu, D. and Xiao, Y. (2008). Joint continuity of the local times of fractional Brownian sheets. Ann. Inst. H. Poincaré Probab. Statist. To appear.\n\nAsymptotic properties and Hausdorff dimensions of fractional Brownian sheets. A Ayache, Y Xiao, J. Fourier Anal. Appl. 11Ayache, A. and Xiao, Y. (2005). Asymptotic properties and Hausdorff dimensions of frac- tional Brownian sheets. J. Fourier Anal. Appl. 11 407-439. MR2169474\n\nElliptic Gaussian random processes. A Benassi, S Jaffard, D Roux, Rev. Mat. Iberoamericana. 131462329Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 19-90. MR1462329\n\nAquifer Operator-Scaling and the effect on solute mixing and dispersion. D Benson, M M M Meerschaert, B Bäumer, H P Scheffler, Water Resour. Res. 42Benson, D. Meerschaert, M.M.M., Bäumer, B. and Scheffler, H.P. (2006). Aquifer Operator- Scaling and the effect on solute mixing and dispersion Water Resour. Res. 42 1-18.\n\nGaussian processes with stationary increments: Local times and sample function properties. S M Berman, Ann. Math. Statist. 41272035Berman, S.M. (1970). Gaussian processes with stationary increments: Local times and sample function properties. Ann. Math. Statist. 41 1260-1272. MR0272035\n\nGaussian sample function: Uniform dimension and Hölder conditions nowhere. S M Berman, Nagoya Math. J. 46Berman, S.M. (1972). Gaussian sample function: Uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46 63-86. MR0307320\n\nLocal non-determinism and local times of Gaussian processes. S M Berman, MR0317397Indiana Univ. Math. J. 23Berman, S.M. (1973). Local non-determinism and local times of Gaussian processes. Indiana Univ. Math. J. 23 69-94. MR0317397\n\nOperator scaling stable random fields. H Biermé, M M Meerschaert, H.-P Scheffler, Stochastic. Process. Appl. 1172290879Biermé, H., Meerschaert, M.M. and Scheffler, H.-P. (2007). Operator scaling stable random fields. Stochastic. Process. Appl. 117 312-332. MR2290879\n\nOn the local time of the multifractional Brownian motion. B Boufoussi, M Dozzi, R Guerbaz, Stochastics. 78Boufoussi, B., Dozzi, M. and Guerbaz, R. (2006). On the local time of the multifractional Brownian motion. Stochastics 78 33-49. MR2219711\n\nSample path properties of the local time of multifractional Brownian motion. B Boufoussi, M Dozzi, R Guerbaz, Bernoulli. 13Boufoussi, B., Dozzi, M. and Guerbaz, R. (2007). Sample path properties of the local time of multifractional Brownian motion. Bernoulli 13 849-867. MR2348754\n\nJoint continuity of Gaussian local times. J Cuzick, J Dupreez, Ann. Probab. 10Cuzick, J. and DuPreez, J. (1982). Joint continuity of Gaussian local times. Ann. Probab. 10 810-817. MR0659550\n\nV Dobrić, F M Ojeda, Fractional Brownian fields, duality, and martingales. IMS Lecture Notes: High Dimensional Probability. Beachwood, OHInstitute of Mathematical Statistics51Dobrić, V. and Ojeda, F.M. (2006). Fractional Brownian fields, duality, and martingales. IMS Lecture Notes: High Dimensional Probability 51 77-95. Beachwood, OH: Institute of Mathematical Statistics.\n\nOccupation density and sample path properties of N -parameter processes. M Dozzi, Topics in Spatial Stochastic Processes. Martina Franca; BerlinSpringer18021975519Dozzi, M. (2002). Occupation density and sample path properties of N -parameter processes. In Topics in Spatial Stochastic Processes (Martina Franca, 2001) 127-166. Lecture Notes in Math. 1802. Berlin: Springer. MR1975519\n\nEstimates for the small ball probabilities of the fractional Brownian sheet. T Dunker, J. Theoret. Probab. 131777539Dunker, T. (2000). Estimates for the small ball probabilities of the fractional Brownian sheet. J. Theoret. Probab. 13 357-382. MR1777539\n\nSample function properties of multi-parameter stable processes. W Ehm, Z. Wahrsch. Verw. Gebiete. 56618272Ehm, W. (1981). Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw. Gebiete 56 195-228. MR0618272\n\nOccupation densities. D Geman, J Horowitz, Ann. Probab. 8Geman, D. and Horowitz, J. (1980). Occupation densities. Ann. Probab. 8 1-67. MR0556414\n\nInequalities. G H Hardy, Cambridge Univ. PressCambridgeHardy, G.H. (1934). Inequalities. Cambridge: Cambridge Univ. Press.\n\nFrom N parameter fractional Brownian motions to N parameter multifractional Brownian motions. Rocky Mount. E Herbin, J. Math. 362274895Herbin, E. (2006). From N parameter fractional Brownian motions to N parameter multi- fractional Brownian motions. Rocky Mount. J. Math. 36 1249-1284. MR2274895\n\nSome Random Series of Functions. J.-P Kahane, Cambridge Univ. Press8330732nd ed. CambridgeKahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge: Cambridge Univ. Press. MR0833073\n\nOn the fractional anisotropic Wiener field. A Kamont, Probab. Math. Statist. 161407935Kamont, A. (1996). On the fractional anisotropic Wiener field. Probab. Math. Statist. 16 85-98. MR1407935\n\nMultiparameter Processes: An Introduction to Random Fields. D Khoshnevisan, Springer1914748New YorkKhoshnevisan, D. (2002). Multiparameter Processes: An Introduction to Random Fields. New York: Springer. MR1914748\n\nSectorial local non-determinism and geometric properties of the Brownian sheet. D Khoshnevisan, D Wu, Y Xiao, Electronic J. Probab. 11Khoshnevisan, D., Wu, D. and Xiao, Y. (2006). Sectorial local non-determinism and geo- metric properties of the Brownian sheet, Electronic J. Probab. 11 817-843. MR2261054\n\nImages of the Brownian sheet. D Khoshnevisan, Y Xiao, Trans. Amer. Math. Soc. 3592299449Khoshnevisan, D. and Xiao, Y. (2007). Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 3125-3151. MR2299449\n\nMultifractional Brownian motion: definition and preliminary results. J Lévy-Véhel, R F Peltier, RR-2645Technical ReportINRIALévy-Véhel, J. and Peltier, R.F. (1996). Multifractional Brownian motion: definition and preliminary results. Technical Report RR-2645, INRIA.\n\nFractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type. S C Lim, J. Phys. A: Math. Gen. 341819933Lim, S.C. (2001). Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type. J. Phys. A: Math. Gen. 34 1301-1310. MR1819933\n\nOn some possible generalizations of fractional Brownian motion. S C Lim, S V Muniandy, Phys. Lett. A. 266Lim, S.C. and Muniandy, S.V. (2000). On some possible generalizations of fractional Brow- nian motion. Phys. Lett. A 266 140-145. MR1741314\n\nFractional Brownian motion fractional noises and applications. B Mandelbrot, J Van Ness, SIAM Rev. 10242239Mandelbrot, B. and Van Ness, J. (1968). Fractional Brownian motion fractional noises and applications. SIAM Rev. 10 422-437. MR0242239\n\nSmall deviations for some multi-parameter Gaussian processes. D M Mason, Z Shi, J. Theoret. Probab. 141822902Mason, D.M. and Shi, Z. (2001). Small deviations for some multi-parameter Gaussian pro- cesses. J. Theoret. Probab. 14 213-239. MR1822902\n\nMultiparameter fractional Brownian motion and quasilinear stochastic partial differential equations. B Øksendal, T Zhang, MR1922562Stochastics Stochastics Rep. 71Øksendal, B. and Zhang, T. (2000). Multiparameter fractional Brownian motion and quasi- linear stochastic partial differential equations. Stochastics Stochastics Rep. 71 141-163. MR1922562\n\nLocal times for Gaussian vector fields. L D Pitt, 309- 330. MR0471055Indiana Univ. Math. J. 27Pitt, L.D. (1978). Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 309- 330. MR0471055\n\nHow rich is the class of multifractional Brownian motions? Stochastic Process. S Stoev, M S Taqqu, Appl. 116Stoev, S. and Taqqu, M.S. (2006). How rich is the class of multifractional Brownian motions? Stochastic Process. Appl. 116 200-221. MR2197974\n\nHausdorff measure of trajectories of multiparameter fractional Brownian motion. M Talagrand, Ann. Probab. 231334170Talagrand, M. (1995). Hausdorff measure of trajectories of multiparameter fractional Brow- nian motion. Ann. Probab. 23 767-775. MR1334170\n\nPolarité, points multiples et géométrie de certain processus gaussiens. F Testard, U.P.S. ToulousePubl. du Laboratoire de Statistique et Probabilités de lTestard, F. (1986). Polarité, points multiples et géométrie de certain processus gaussiens. Publ. du Laboratoire de Statistique et Probabilités de l'U.P.S. Toulouse 01-86.\n\nDimension results for multifractional Brownian sheets. D Wu, J. Math. Phys. 482337692Paper 073511Wu, D. (2007). Dimension results for multifractional Brownian sheets. J. Math. Phys. 48 Paper 073511. MR2337692\n\nGeometric properties of fractional Brownian sheets. D Wu, Y Xiao, J. Fourier Anal. Appl. 13Wu, D. and Xiao, Y. (2007). Geometric properties of fractional Brownian sheets. J. Fourier Anal. Appl. 13 1-37. MR2296726\n\nDimension results for Gaussian vector fields and index-α stable fields. Y Xiao, Ann. Probab. 231330771Xiao, Y. (1995). Dimension results for Gaussian vector fields and index-α stable fields. Ann. Probab. 23 273-291. MR1330771\n\nHölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Y Xiao, Probab. Theory Related Fields. 1091469923Xiao, Y. (1997). Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields 109 129-157. MR1469923\n\nThe Hausdorff dimension of the level sets of stable processes in random scenery. Y Xiao, Acta. Sci. Math. (Szeged). 651702175Xiao, Y. (1999). The Hausdorff dimension of the level sets of stable processes in random scenery. Acta. Sci. Math. (Szeged) 65 373-383. MR1702175\n\nProperties of local non-determinism of Gaussian and stable random fields and their applications. Y Xiao, 157-193. MR2225751Ann. Fac. Sci. Toulouse Math. Xiao, Y. (2006). Properties of local non-determinism of Gaussian and stable random fields and their applications. Ann. Fac. Sci. Toulouse Math. XV 157-193. MR2225751\n\nSample path properties of anisotropic Gaussian random fields. Y Xiao, PreprintXiao, Y. (2007). Sample path properties of anisotropic Gaussian random fields. Preprint.\n\nLocal times of fractional Brownian sheets. Y Xiao, T Zhang, Probab. Theory Related Fields. 124Xiao, Y. and Zhang, T. (2002). Local times of fractional Brownian sheets. Probab. Theory Related Fields 124 204-226. MR1936017\n"},"corpus_id":{"kind":"number","value":18276475,"string":"18,276,475"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0810.4438\",\n \"mag\": 2120890892,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.3150/08-bej126\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0810.4438v1.pdf\"\n ],\n \"pdf_sha\": \"b0655011c4dd7e485aa284a3c2ef7ad59a791be7\",\n \"open_access\": {\n \"license\": \"implied-oa\",\n \"url\": \"https://doi.org/10.3150/08-bej126\",\n \"status\": \"HYBRID\"\n }\n}"},"title":{"kind":"string","value":"Local times of multifractional Brownian sheets"},"authors":{"kind":"list like","value":["Mark Meerschaert \nDepartment of Statistics and Probability\nMichigan State University\n48824East LansingMIUSA","Dongsheng Wu dongsheng.wu@uah.edu \nDepartment of Mathematical Sciences\nUniversity of Alabama in Huntsville\n35899HuntsvilleALUSA","Yimin Xiao **xiao@stt.msu.edu \nDepartment of Statistics and Probability\nMichigan State University\n48824East LansingMIUSA"],"string":"[\n \"Mark Meerschaert \\nDepartment of Statistics and Probability\\nMichigan State University\\n48824East LansingMIUSA\",\n \"Dongsheng Wu dongsheng.wu@uah.edu \\nDepartment of Mathematical Sciences\\nUniversity of Alabama in Huntsville\\n35899HuntsvilleALUSA\",\n \"Yimin Xiao **xiao@stt.msu.edu \\nDepartment of Statistics and Probability\\nMichigan State University\\n48824East LansingMIUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Denote by H(t) = (H1(t), . . . , HN (t)) a function in tUnder some regularity conditions on the function H(t), we prove the existence, joint continuity and the Hölder regularity of the local times of {B H(t) (t)}. We also determine the Hausdorff dimensions of the level sets of {B H(t) (t)}. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"A one-dimensional fractional Brownian motion (fBm) ξ α = {ξ α (t), t ∈ R} with Hurst index α ∈ (0, 1) is a real-valued, centered Gaussian process with covariance function given by E[ξ α (s)ξ α (t)] = 1 2 [|s| 2α + |t| 2α − |t − s| 2α ] ∀s, t ∈ R.\n(1.1)\nIt was introduced, as a moving average Gaussian process, by Mandelbrot and Van Ness [31]. Fractional Brownian motion has interesting properties such as self-similarity of order α ∈ (0, 1), stationary increments and long-range dependence (when α > 1/2), which make it a good candidate for modeling different phenomena in, for example, finance and telecommunication. However, this model may be restrictive, due to the fact that all of its regularity and fractal properties are governed by the single Hurst parameter α. To model phenomena whose regularity evolves in time, for example, Internet traffic or image processing, Lévy-Véhel and Peltier [28] and Benassi, Jaffard and Roux [7] independently introduced multifractional Brownian motion (mfBm) in terms of moving average This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2008, Vol. 14, No. 3, 865-898. This reprint differs from the original in pagination and typographic detail.\nrepresentation and harmonizable representation, respectively. Multifractional Brownian motion is governed by a Hurst functional α(t) with certain regularity in place of the constant Hurst parameter α in fBm.\nSeveral authors have studied sample path and statistical properties of mfBm. For example, Benassi, Jaffard and Roux [7] considered the sample path Hölder regularity of mfBm and determined the Hausdorff dimension of its graph. Ayache, Cohen and Lévy-Véhel [2] and Herbin [22] studied the covariance structure of mfBm from its harmonisable representation. Recently, Boufoussi, Dozzi and Guerbaz [13,14] studied the existence, joint continuity and Hölder regularity of the local time of mfBm and established Chung's law of the iterated logarithm for mfBm. The main tool they applied to derive their results is the property of one-sided local non-determinism of mfBm.\nThere are multiparameter extensions of fBm, among which, two typical examples are multiparameter Lévy fBm ξ α = {ξ α (t), t ∈ R N } (whose covariance function is given by (1.1) with | · | being the Euclidean norm in R N ) and fractional Brownian sheets, where the former is isotropic, while the latter are anisotropic in general. Since they were introduced by Kamont [24] (see also Ayache, Léger and Pontier [4]), fractional Brownian sheets (fBs) have been studied extensively as a representative of anisotropic Gaussian random fields in recent years. See, for example, Dunker [18], Mason and Shi [32], Øksendal and Zhang [33], Xiao and Zhang [45], Ayache and Xiao [6], Ayache, Wu and Xiao [5], Wu and Xiao [39] and the references therein for further information. Still, the regularity of fBs does not evolve in the N -dimensional 'time' parameter t ∈ R N + . To model anisotropic Gaussian random fields whose regularity evolves in time, such as images, Ayache and Léger [3] and Herbin [22] introduced so-called multifractional Brownian sheets (mfBs) in terms of their moving average representations and harmonisable representations, where the constant Hurst vector of fBs is substituted by Hurst functionals. Furthermore, they showed that mfBs have a continuous modification and determined the pointwise and local Hölder exponent of mfBs. They also proved that mfBs are locally self-similar. We refer to Ayache and Léger [3] and Herbin [22] for the definitions of the corresponding concepts and results.\nIn studying anisotropic random fields, Xiao [44] suggested that it is more convenient to use the following metric ρ K on R N :\nwhere K = (K 1 , . . . , K N ) ∈ (0, 1) N is a fixed vector. Denote by H(t) = (H 1 (t), . . . , H N (t)) a function in t ∈ R N + with values in (0, 1) N . We say that H(t) satisfies Condition A if there exist a positive number α ∈ (0, 1) and a vector (K 1 , . . . , K N ) ∈ (0, 1) N such that A.1. for every ℓ ∈ {1, . . . , N }, α ≤ H ℓ (t) ≤ K ℓ for all t ∈ R N + ; A.2. H ℓ (t) (ℓ = 1, . . . , N ) satisfies a ρ K -Lipschitz condition on every compact set, that is, for every compact subset I ⊆ R N + , there exist positive constants c ℓ = c ℓ (I) and δ such that\n∀s, t ∈ I with |s − t| < δ. Now, we are ready to define multifractional Brownian sheets via their moving average representations."},{"title":"3)","content":"where s + = max{s, 0}, and where W = {W (s), s ∈ R N } is a standard real-valued Brownian sheet.\nRemark 1.2. Our Definition 1.1 generalizes the definition in Ayache and Léger [3], where they define H ℓ (·) as a function of t ℓ ∈ R + (ℓ = 1, . . . , N ). Herbin [22] defines multifractional Brownian sheets by using the following moving average representation (cf. [22] Definition 2, page 1261):\nBased on Dobrić and Ojeda [16] (see also Stoev and Taqqu [35]), we know that the multifractional Brownian sheet defined by (1.3) and the multifractional Brownian sheet defined by (1.4) have different correlation structures in general, even if N = 1. Our definition is more convenient to use than that of Herbin's when we derive the one-sided sectorial local non-determinism for multifractional Brownian sheets in Section 2. The form (1.3) is preferred in some applications in the one-dimensional case because it is easier to separate the future from the past.\nDefine the Gaussian random field\nwhere {B\nIn this paper, we will study the existence and regularity of the local times of multifractional Brownian sheets. Our main technical tool is the property of one-sided sectorial local non-determinism. This property is more general than the sectorial local non-determinism which was first introduced by Khoshnevisan and Xiao [27] for the Brownian sheet and then extended by Wu and Xiao [39] to fractional Brownian sheets. See Section 2 for more information.\nOur results show that multifractional Brownian sheets are similar to fractional Brownian sheets in many ways. They admit jointly continuous local times when the indices H i (t) stay in the range for which the fractional Brownian sheet has jointly continuous local times. We also establish a Hausdorff dimension result and, essentially, the dimension of the level set is the same as for the constant parameter case, except that we take the supremum of the constant parameter formula. We also show that the supremum can be taken locally, to establish that the fractal dimension of the random field varies in space. Hence, multifractional Brownian sheets are useful in applications such as composite materials, or porous media flow, when the material properties vary in space. They may also find useful applications in image processing.\nThe rest of this paper is organized as follows. In Section 2, we prove some basic results on mfBs that will be useful to our arguments. In Section 3, we provide a sufficient condition for the existence of L 2 -local times of the (N, d) mfBs and prove that the condition also implies the joint continuity of the local times. We prove the Hölder regularity of the local times in Section 4. Finally, we derive the local Hausdorff dimensions of the level sets of {B H(t) (t)} in Section 5. Our results extend the results of Ayache and Xiao [6] and Ayache, Wu and Xiao [5] for fractional Brownian sheets and Boufoussi, Dozzi and Guerbaz [13,14] for multifractional Brownian motion to multifractional Brownian sheets.\nWe end the Introduction with some notation. Throughout this paper, the underlying parameter space is R N or R N + = [0, ∞) N . A parameter t ∈ R N is written as t = (t 1 , . . . , t N ), or as c , if t 1 = · · · = t N = c. For any s, t ∈ R N such that s j < t j (j = 1, . . . , N ) (we denote this by s ≺ t), we define the closed interval (or rectangle) [s, t] = N j=1 [s j , t j ]. We use A to represent the collection of closed intervals [s, t] with s, t ∈ [ε, Λ] N for some fixed positive numbers ε and Λ. For any integer m ≥ 1, we always write λ m for the Lebesgue measure on R m and use ·, · and | · | to denote the ordinary scalar product and the Euclidean norm in R m , respectively.\nThroughout this paper, an unspecified positive and finite constant will be denoted by c, which may not be the same in each occurrence. More specific constants in Section i are numbered as c i,1 , c i,2 , . . . ."},{"title":"Preliminaries","content":"In this section, we will provide some lemmas that are useful for proving our main results. Lemma 2.1 is an extension of Boufoussi et al. [13], Lemma 3.1 from fractional Brownian motion to fractional Brownian sheets.\nLemma 2.1. Let 0 < ε < Λ and 0 < α < γ < 1 be fixed constants. Let {Z κ 0 (t), (t, κ) ∈ R N + × [α, γ] N } be a real-valued Gaussian random field defined by equation (1.3) with\nit suffices for us to prove that for j ∈ {1, . . . , N } fixed,\nBy using the moving average representation of fBs (equation (1.3)), we have that\nwhereǔ j = (u 1 , . . . , u j−1 , u j+1 , . . . , u N ). A change of variables shows that\nwhich is bounded for all t ∈ [ε, Λ] N and β, β ′ ∈ [α, γ] N . Next, we estimate I 2 . Without loss of generality, we may assume β j > β ′ j . We rewrite I 2 as the following summation:\nBy the mean value theorem, we have that for some β ′ j ≤ η j1 ≤ β j (η j1 may depend on t j ),\nfor all t ∈ [ε, Λ] N and β, β ′ ∈ [α, γ] N . In the above, the last inequality follows from a change of variables. Similarly, we have that for some β ′ j ≤ η j2 ≤ β j , (t)} be a multifractional Brownian sheet in R. There exist positive constants δ > 0, c 2,4 and c 2,5 such that for all s, t ∈ [ε, Λ] N with |s − t| < δ, for any\nProof. By the elementary inequalities\nThe first term on the right-hand side of (2.10) is the variance of the increment of a fractional Brownian sheet with Hurst index H(u). By Lemma 8 in Ayache and Xiao [6], there exist positive constants c 2,6 and c 2,7 , depending only on α, γ = max{K 1 , . . . , K N }, ε and N , such that\nMeanwhile, by Condition A and Lemma 2.1, there exists δ > 0 small such that for all s, t ∈ [ε, Λ] N with |s − t| < δ, we have |u − s| < δ, which implies\nand, similarly,\nCombining (2.10) with (2.11), (2.12) and (2.13), and noting that sup t ℓ H ℓ (t ℓ ) ≤ K ℓ (still by Definition 1.1), we can see that there exists δ > 0, which depends only on α, γ, ε, Λ and N , such that for |s − t| < δ, (2.9) holds. This completes the proof of Lemma 2.2. Remark 2.3. For any fixed t ∈ [ε, Λ] N and any fixed unit vector η ∈ R N , let us consider the increment\nIn light of Lemma 2.2, it would be interesting to compute the bounds on the correlation function\n(2.14)\nand its time-and direction-varying spectral density (or Wigner-Ville distribution)\nsince those are the properties of the random field that are usually used to model natural phenomena.\nAs we mentioned in the Introduction, the main technical tool of this paper is the property of sectorial local non-determinism. Hence, it will be helpful to say a few words about the history of various forms of local non-determinism and briefly recall some of the definitions. The concept of local non-determinism, which provides a powerful tool for dealing with the complex dependence structure of Gaussian processes, was first introduced by Berman [11] to unify and extend his methods for studying local times of real-valued Gaussian processes, and then extended by Pitt [34] to Gaussian random fields. The notion of strong local non-determinism was later introduced by Cuzick and DuPreez [15] for Gaussian processes (i.e., N = 1) and can be extended in various ways to Gaussian random fields (see Xiao [43]). The simplest is the following. Let X = {X(t), t ∈ R N } be a real-valued Gaussian random field with 0 < E[X(t) 2 ] < ∞ for all t ∈ I, where I ∈ A is an interval. Let φ be a given function such that φ(0) = 0 and φ(r) > 0 for r > 0. Then X is said to be strongly locally φ-non-deterministic (SLφND) on I if there exist positive constants c 2,10 and r 0 such that for all t ∈ I and all 0 < r ≤ min{|t|, r 0 }, Var(X(t)|X(s) : s ∈ I, r ≤ |s − t| ≤ r 0 ) ≥ c 2,10 φ(r).\n(2.16)\nHere, Var(X(t)|X(s), s ∈ S) denotes the conditional variance of X(t), given X(s), s ∈ S. Pitt [34] proved that fractional Brownian motion ξ α is SLφND with φ(r) = r 2α . When N = 1 and Var(X(t)|X(s) : s ∈ I, r ≤ t − s ≤ r 0 ) ≥ c 2,10 φ(r), X is said to satisfy one-sided strong local φ-non-determinism. The properties of strong local non-determinism have played important roles in studying the regularity of local times, small ball probabilities, uniform Hausdorff dimension results and other sample path properties of Gaussian processes and Gaussian random fields. We refer to Xiao [43,44] for more information on applications of SLφND, as well as other forms of local non-determinism. It is well known that the Brownian sheet does not satisfy the property of local nondeterminism in the sense of Berman and Pitt (hence, it does not satisfy (2.16) either). Recently, Khoshnevisan and Xiao [27] introduced the concept of sectorial local nondeterminism and proved that the Brownian sheet has the property of sectorial local nondeterminism. Wu and Xiao [39] extended the result of Khoshnevisan and Xiao [27] and proved that the fractional Brownian sheet {B H 0 (t), t ∈ R N + }, where H = (H 1 , . . . , H N ) ∈ (0, 1), also has the property of sectorial local non-determinism. Namely, for any fixed positive number ε ∈ (0, 1), there exists a positive constant c 2,11 , depending only on ε, H and N , such that for all positive integers n ≥ 2, and all t 1 , . . . , t n ∈ [ε, ∞) N , we have\nwhere t 0 = 0. The concept of sectorial local non-determinism opens the door to unifying the previously different treatments for the Brownian sheet and fractional Brownian motion, and can also be used to prove new results about fractional Brownian motion. See, for example, Khoshnevisan, Wu and Xiao [26], Wu and Xiao [39] and Ayache, Wu and Xiao [5] for studies on sample path properties of the Brownian sheet and fractional Brownian sheets by using the sectorial local non-determinism property.\nIt is not known whether the multifractional Brownian sheet {B\n(t)} is sectorially locally non-deterministic. To study the local times of (N, d)-mfBs {B H(t) (t)}, we will work with multifractional Liouville sheets (mfLs) at first and prove that they can be well approximated by Gaussian processes which have the property of one-sided strong local non-determinism (see (2.22) and (2.23) below). For a function H(t) = (H 1 (t), . . . , H N (t)) satisfying Condition A, the real-valued, centered Gaussian random field {X\nis called a multifractional Liouville sheet with functional Hurst index H(t). One parameter mfLs were first introduced by Lim and Muniandy [30] as an extension of fBm; see Lim [29] for more properties on one parameter mfLs.\nand the two fields on the right-hand side of (2.19) are independent. We will show that in studying the regularity properties of the local times of {B\n(t)} plays a crucial role and the second field in (2.19) can be neglected. More precisely, we will make use of the following property. For all integers n ≥ 2, t 1 , . . . , t n ∈ R N + and u 1 , . . . , u n ∈ R, we have\n(2.20)\nNext, we use an argument in Ayache and Xiao [6] to provide a useful decomposition for {X\nwe decompose the rectangle [0, t] into the following disjoint union of subrectangles:\ncan be written as a union of 2 N − N − 1 subrectangles of [0, t]. Denote the integrand in (2.18) by g(t, r). It\nSince the processes X(ε, t), Y ℓ (t) (1 ≤ ℓ ≤ N ) and Z(ε, t) are defined by the stochastic integrals with respect to W over disjoint sets, they are independent Gaussian random fields.\nThe following lemma shows that every process Y ℓ (t) has the property of strong local non-determinism along the ℓth direction. It will be essential to our proofs.\nLemma 2.4. Let I ∈ A and let ℓ ∈ {1, 2, . . ., N } be fixed. For all integers n ≥ 2 and t 1 , . . . , t n ∈ I such that\nwhere t 0 ℓ = 0 and c 2,12 > 0 is a constant depending only on ε and I.\nProof. The proof is in the same spirit as the proof of Lemma 2.1 of Ayache, Wu and Xiao [5]. Working in the Hilbert space setting, the conditional variance in (2.23) is the square of the L 2 (P)-distance of Y ℓ (t n ) from the subspace generated by Y ℓ (t j )(0 ≤ j ≤ n − 1). Hence, it is sufficient to show that there exists a constant c 2,12 such that\nfor all a j ∈ R (j = 1, . . . , n − 1). However, by splitting R ℓ (t n ) into two disjoint parts and using the independence, we derive that\nThis proves (2.24) and hence Lemma 2.4.\nCombining Lemma 2.2 and Lemma 2.4 with the proofs of Lemma 2.1 and Lemma 8.1 in Berman [11] (see also Boufoussi, Dozzi and Guerbaz [13], Theorem 3.3), we have the following.\nProposition 2.5. For every integer n ≥ 2, there exist positive constants C n and δ (both of these may depend on n) such that for every ℓ = 1, . . . , N ,\nfor all (u 1 , . . . , u n ) ∈ R n and all points t 1 , . . . , t n ∈ I satisfying t 1 ℓ < · · · < t n ℓ with t n ℓ − t 1 ℓ < δ.\nThe following lemma relates the multifractional Brownian sheet {B H(t) 0 (t)} to the independent Gaussian random fields Y ℓ (ℓ = 1, . . . , N ), which is a direct extension of Ayache, Wu and Xiao [5], Lemma 2.2.\nLemma 2.6. Let I ∈ A. For all integers n ≥ 2, t 1 , . . . , t n ∈ I and u 1 , . . . , u n ∈ R, we have\nConsequently, for all k ∈ {1, . . . , N } and positive numbers p 1 , . . . , p k ≥ 1 satisfying\n(t)} satisfies the following property: for all integers n ≥ 2 and t 1 , . . . , t n ∈ I such that t j ≺ t n for every j ≤ n − 1,\nThis property is weaker than (2.17) and will be referred to as one-sided sectorial local non-determinism.\nWe will also make use of the following technical lemmas, among which Lemma 2.8 is from Xiao and Zhang [45], Lemma 2.9 is proved in Ayache and Xiao [6] and Lemma 2.10 and Lemma 2.11 are from Ayache, Wu and Xiao [5].\nThere then exist finite constants c 2,15 and c 2,16 , depending only on α, β and η, such that the following hold for all reals A, B > 0 satisfying A 1/α ≤ c 2,15 B:\nwith the convention that 0\nThere then exists a positive constant ∆ τ ≤ 1, depending only on (ρ 1 , . . . , ρ N ), such that for every ∆ ∈ (0, ∆ τ ), we can find τ real numbers\nand\nLemma 2.11. For all integers n ≥ 1, positive numbers a, r, 0 < b j < 1 and an arbitrary\nFinally, we conclude this section by briefly recalling some aspects of the theory of local times. For excellent surveys on local times of random and deterministic vector fields, we refer to Geman and Horowitz [20] and Dozzi [17].\nLet X(t) be a Borel vector field on R N with values in R d . For any Borel set I ⊆ R N , the occupation measure of X on I is defined as the following measure on R d :\nIf µ I is absolutely continuous with respect to λ d , we say that X(t) has local times on I and define its local times, L(•, I), as the Radon-Nikodým derivative of µ I with respect to λ d , that is,\nIn the above, x is the so-called space variable and I is the time variable. Sometimes, we write L(x, t) in place of L(x, [0, t]). Note that if X has local times on I, then for every Borel set J ⊆ I, L(x, J) also exists. By standard martingale and monotone class arguments, one can deduce that the local times have a measurable modification that satisfies the following occupation density formula (see Geman and Horowitz [20], Theorem 6.4): for every Borel set I ⊆ R N and every measurable function f :\nThen, whenever we can choose a version of the local time, still denoted by\nX is said to have a jointly continuous local time on I. When a local time is jointly continuous, L(x, •) can be extended to be a finite Borel measure supported on the level set\nsee Adler [1] for details. In other words, local times often act as a natural measure on the level sets of X. Hence, they are useful in studying the various fractal properties of level sets and inverse images of the vector field X. In this regard, we refer to Berman [10], Ehm [19] and Xiao [41]. It follows from Geman and Horowitz [20], ((25.5) and (25.7)) (see also Pitt [34]) that for all x, y ∈ R d , I ∈ A and all integers n ≥ 1,\nand for all even integers n ≥ 2, where u = (u 1 , . . . , u n ), t = (t 1 , . . . , t n ) and each u j ∈ R d , t j ∈ I ⊆ (0, ∞) N . In the coordinate notation, we then write u j = (u j 1 , . . . , u j d )."},{"title":"Local times: existence and joint continuity","content":"In this section, we consider the existence and regularity of local times of mfBs. The index H depends on I, but, for simplicity, we have deleted I from the notation. Proof. Without loss of generality, we may, and will, assume that δ = diam(I) is sufficiently small that (2.9) holds for all s, t ∈ I. In particular, we assume δ ≤ 1.\nTo prove the existence of local time on I, by Geman and Horowitz [20] Theorem 21.9, it suffices to prove that\nIt follows from (2.9) and a change of variables that\n, we can estimate the last integral in a way similar to the proof of Xiao and Zhang [45] Theorem 3.6. Namely, by applying Lemma 2.8 repeatedly, we derive that the last integral is finite. This proves (3.2) and hence Theorem 3.1.\nThe following is a consequence of Theorem 3.1 which gives a more natural condition for the existence of local times of mfBs. H ℓ (t) for all t ∈ I, but the equality only holds for t in a set of Lebesgue measure 0, the existence of local times is rather subtle and requires the imposition of further assumptions on (H 1 (t), . . . , H N (t)). Hence, it will not be discussed here.\nWe now consider the joint continuity of the local times of {B H(t) (t)}. For convenience, we first prove that under the same condition as in Theorem 3.1, the local time of {B H(t) (t)} has a version that is jointly continuous in both space and time variables. We then apply the same argument as in the proof of Corollary 3.2 to show that the same conclusion holds, provided\nOur results extend those of Ehm [19] for the Brownian sheet and of Ayache, Wu and Xiao [5] for fractional Brownian sheets. The main idea for proving Theorem 3.4 is similar to those in Ehm [19], Xiao [41] and Ayache, Wu and Xiao [5]. That is, we first apply the Fourier analytic arguments to derive estimates for the moments of the local times and then apply a multiparameter version of the Kolmogorov continuity theorem (cf. Khoshnevisan [25]). As in Ayache, Wu and Xiao [5], the 'one-sided' sectorial local non-deterministic properties of multifractional Liouville sheets proved in Section 2 (see Lemma 2.4 and Proposition 2.5) will play important roles in deriving moment estimates in Lemmas 3.5 and 3.7 below. However, due to the nonstationarity and the lack of two-sided local non-determinism of multifractional Brownian sheets, we need to make several modifications in our proofs.\nFor convenience, we further assume 0 < H 1 ≤ · · · ≤ H N < 1.\nProof. For later use, we will start with an arbitrary closed interval T = N ℓ=1 [a ℓ , a ℓ + h ℓ ] ⊆ I. It follows from (2.43) and the fact that {B\n(t)} that for all integers n ≥ 1,\nwhere U k = (u 1 k , . . . , u n k ) ∈ R n . Fix k = 1, . . . , d and denote the inner integral in (3.7) by N k . Then by Lemma 2.6, we have\nSince (3.5) holds, we apply Lemma 2.10 with ∆ = n −1 and q = d to obtain τ positive numbers p 1 , . . . , p τ ≥ 1 satisfying (2.36) and (2.37).\nApplying the generalized Hölder inequality (Hardy [21] page 140), we derive that\nwhere the last equality follows from the fact that (Y ℓ (t 1 ), . . . , Y ℓ (t n )) is a Gaussian vector with mean 0. Hence, it follows from (3.7) and (3.9) that\nTo evaluate the integral in (3.10), we will first integrate [dt 1 ℓ · · · dt n ℓ ] for ℓ = 1, . . . , τ . To this end, we use the following fact about multivariate normal distributions: for any Gaussian random vector (Z 1 , . . . , Z n ), det Cov(Z 1 , . . . , Z n ) = Var(Z 1 ) n j=2 Var(Z j |Z 1 , . . . , Z j−1 ).\n(3.11)\nBy the above fact and Lemma 2.4, we can derive that for every ℓ ∈ {1, . . . , τ } and for all\nfor some permutation π ℓ of {1, . . . , N }, we have\nwhere t π ℓ (0) ℓ := ε (recall the decomposition (2.22)). We choose ε < 1 2 min{a ℓ , 1 ≤ ℓ ≤ N } so that Lemma 2.11 is applicable. It follows from (3.12) and (3.13) that\nIn the above, the last inequality follows from (2.40) (recall that H ℓ d/p ℓ < 1). Combining (3.10) and (3.14), we have\nWe now consider the special case when T = [a, a + h ], that is, h 1 = · · · = h N = h with h < δ, (3.15) and (2.37) with ∆ = n −1 and q = d together yield\nThis proves (3.6).\nRemark 3.6. In the proof of Lemma 3.5, if we apply the generalized Hölder inequality to the first integral in (3.8) with N positive numbers p 1 , . . . , p N defined by\nthen the above proof shows that if T ⊆ I ∈ A, then, similarly to (3.15), the following inequality holds:\nwhere ν = d/( N ℓ=1 1 H ℓ ) ∈ (0, 1). We will apply this inequality in the proof of Theorem 3.4 below.\nProof. Let γ ∈ (0, 1 ∧ ατ 2τ ) be a constant. Note that by the elementary inequalities\nand |u + v| γ ≤ |u| γ + |v| γ , we see that for all u 1 , . . . , u n , x, y ∈ R d ,\nwhere the summation ′ is taken over all the sequences (k 1 , . . . , k n ) ∈ {1, . . . , d} n . It follows from (2.44), (3.20) and Lemma 2.6 that for every even integer n ≥ 2,\nwhere η j k = 1 if k = k j and η j k = 0 otherwise. Note that for every j ∈ {1, . . . , n}, we have d k=1 η j k = 1. We take β ℓ = H ℓ (1 ≤ ℓ ≤ N ), ∆ = 1/n and q = d in Lemma 2.10 and let p ℓ (ℓ = 1, . . . , τ ) be the constants satisfying (2.36) and (2.37). Observe that since γ ∈ (0, ατ 2τ ), there exists\nCombining (3.21) with the generalized Hölder inequality, we have that\nFor any n points t 1 , . . . , t n ∈ T , let π 1 , . . . , π N be N permutations of {1, 2, . . . , n} such that for every 1 ≤ ℓ ≤ N ,\nBy changing the variables of the above integral by means of the transformation\nwhere t π ℓ 0 (0) = 0. Furthermore, by the elementary inequality that for\nMoreover, the last product is equal to a finite sum of terms, each of the form n j=1 |v j k | γη j k p ℓ 0 εj , where ε j = 0, 1, or 2 and\n. By Proposition 2.5, we know that M ℓ0 is dominated by the sum over all possible choices of (ε 1 , . . . , ε n ) ∈ {0, 1, 2} n of the following terms\nwhere V k = (v 1 k , . . . , v n k ) ∈ R n . By another change of variable, w j k = σ ℓ0,j v j k , the integral in (3.27) can be represented by\nis a constant depending on n. Thus, we have obtained that\n(3.29)\nThe other integrals for ℓ = ℓ 0 in (3.23) are easier and can be estimated similarly. Combining (3.23) with Lemma 2.4 (which gives lower bounds for σ 2 ℓ,j ), (3.28), (3.29) and the definition of T , we have\nIn the above,\nWe now take h 1 = · · · = h N = h < δ in (3.30). Then by Lemma 2.11 and noting that (3.22) holds, we have\nwhere the last inequality follows from Lemma 2.10 and the fact that H ℓ0 ≤ H τ . Proof Theorem 3.4. Let I = [a, b] ∈ A be fixed and assume that (3.5) holds. It follows from Lemma 3.7 and the multiparameter version of Kolmogorov's continuity theorem (cf. Khoshnevisan [25]) that for every T ⊆ I, the mfBs {B H(t) } has almost surely a local time L(x, T ) that is continuous for all x ∈ R d .\nTo prove the joint continuity, observe that for all x, y ∈ R d and s, t ∈ I with |s − t| < δ, where δ > 0 is the same as in Lemma 2.2, we have\nSince the difference L(x, [a, s]) − L(x, [a, t]) can be written as a sum of a finite number (which only depends on N ) of terms of the form L(x, T j ), where each T j ∈ A is a closed subinterval of I with at least one edge length ≤ |s − t|, we can use Lemma 3.5 and Remark 3.6 to bound the first term in (3.32). On the other hand, the second term in (3.32) can be dealt with using Lemma 3.7, as above. Consequently, for some small γ ∈ (0, 1), the righthand side of (3.32) is bounded by c n 3,11 (|x − y| + |s − t|) nγ , where n ≥ 2 is an arbitrary even integer. Therefore, the joint continuity of the local times on I again follows from the multiparameter version of Kolmogorov's continuity theorem. This finishes the proof of Theorem 3.4.\nSimilarly to the proof of Corollary 3.2, we derive from Theorem 3.4 the following, more general, result. The next corollary is a direct consequence of Corollary 3.8 and the continuity of the Hurst functionals. We state it to emphasize that the joint continuity of the local time is a local property depending on H(t). (t), t ∈ R N + } be a real-valued multifractional Brownian sheet defined by the harmonizable representation (cf. Herbin [22])\nwhere W is the complex-valued Gaussian random measure in R N with the Lebesgue measure Λ N as its control measure. Based on Dobríc and Ojeda [16] (see also Stoev and Taqqu"},{"title":"Hölder conditions for L(x, •)","content":"In this section, we investigate the local and global asymptotic behavior of the local time L(x, ·) at x as a random measure on R N + . Results in this section carry information about fractal properties of the sample functions of mfBs; see Section 5.\nBy applying Lemma 3.5, we can prove the following technical lemmas, which will be useful in this section.\nProof. The proof is similar to the proof of Ayache, Wu and Xiao [5] Lemma 3.11 and we include it here for completeness. For each fixed a ∈ I, we define the Gaussian random\n. It follows from (2.41) that if {B H(t) (t)} has a local time L(x, S) on any Borel set S, then Y also has a local timeL(x, S) on S and, moreover, L(x + B H(a) (a), S) =L(x, S). With a little modification, the proof of Lemma 3.5 works for the Gaussian field Y . Hence, we derive that (4.1) holds.\nThe following lemma is a consequence of Lemma 4.1 and Chebyshev's inequality. N , d, I and H) such that for all a ∈ I, T = [a, a + r ] with r ∈ (0, δ), where δ > 0 is the same as in Lemma 2.2, x ∈ R d and u > 1 large enough, we have\nLet U (t, r) be the open ball centered at t with radius r, let H U be the vector defined by (3.1) with I = U (t, r) and let τ (U ) be the positive integer satisfying the corresponding condition (3.5). By applying Lemma 3.5 and the Borel-Cantelli lemma, one can easily derive the following law of the iterated logarithm for the local time L(x, ·): there exists a positive constant c 4,3 such that for every x ∈ R d and t ∈ (0, ∞) N ,\nBecause of the continuity of H ℓ (t) (1 ≤ ℓ ≤ N ), it can be verified that τ (U ) → τ (t) and β τ (U) → β τ (t) as r → 0, (4.4) where τ (t) is the unique integer satisfying\nand\nIt follows from Fubini's theorem that with probability one, (4.3) holds for almost all t ∈ (0, ∞) N . We now prove a stronger version of this result, which is useful in determining the Hausdorff dimension of the level set. where ϕ t (r) = r β τ (t) (log log(1/r)) N −β τ (t) and where β τ (t) is defined by (4.6).\nProof. For every integer k > 0, we consider the random measure L k (x, •) on the Borel subsets C of I defined by\nThen, by the occupation density formula (2.41) and the continuity of the function y → L(y, C), one can verify that almost surely L k (x, C) → L(x, C) as k → ∞ for every Borel set C ⊆ I. For every integer m ≥ 1, let f m (t) = L(x, U (t, 2 −m )). From the proof of Theorem 3.4, we can see that almost surely the functions f m (t) are continuous and bounded. Hence, we have, almost surely, for all integers m, n ≥ 1,\nIt follows from (4.9), (4.8) and the proof of Pitt [34], Proposition 3.1 that for every positive integer n ≥ 1,\nx, u j (4.10)\n× E exp i Let γ > 0 be a constant, the value of which will be determined later. We consider the random set\nDenote by µ ω the restriction of the random measure L(x, ·) to I, that is, µ ω (E) = L(x, E ∩ I) for all Borel sets E ∈ R N + . We now take n = ⌊log m⌋, where ⌊y⌋ denotes the integer part of y. Then by applying (4.11) and by Stirling's formula, we have\nprovided γ > 0 is chosen large enough, say, γ ≥ c 4,2 . This implies that\nTherefore, with probability 1 for µ ω -almost all t ∈ I, we have (4.14)\nFinally, for any r > 0 small enough, there exists an integer m such that 2 −m ≤ r < 2 −m+1 and (4.13) is applicable. Since ϕ t is increasing near 0, (4.7) follows from (4.13) and a monotonicity argument.\nRecall that the pointwise Hölder exponent of a random field {X(t), t ∈ R N } at a point t ∈ R N is defined by We remark that the corresponding problem of finding the Hausdorff dimensions of the level sets of multifractional Brownian motion has been investigated by Boufoussi et al. [13] and the Hausdorff dimensions of the level sets of fractional Brownian sheets were studied in Ayache and Xiao [6]. As shown by the following theorem, the fractal structure of Γ x is much richer than the level sets of multifractional Brownian motion and is locally reminiscent of the level sets of a fractional Brownian sheet.\nBrownian sheet with Hurst functionals H ℓ (t) (ℓ = 1, . . . , N ). For any interval I ∈ A, let t * ∈ I be a point satisfying\nH ℓ (t * ) > d, then for any x ∈ R d , with positive probability,\nRemark 5.2. It can be verified that if (4.5) holds for t = t * , then β τ (t * ) is the same as in (4.6) with t replaced by t * . In the special case where H(t) = H is a constant, β τ (t * ) reduces to the form derived in Ayache and Xiao [6] for the Hausdorff dimension of the level sets of a fractional Brownian sheet. When\nH ℓ (t * ) = d, we believe that for every x ∈ R d , Γ x ∩ I = ∅ a.s. However, the method of this paper is not sufficient for proving this statement.\nProof of Theorem 5.1. We will follow the proof of Ayache and Xiao [6] Theorem 5.\nFirst, we prove\nand Γ x ∩ I = ∅ a.s. when the right-hand side of (5.2) is negative. Without loss of any generality, we may assume that I = [a, a + h ] and h is small so that Lemma 2.2 is applicable. For an integer n ≥ 1, divide the interval I into m n ≤ n N ℓ=1 (H ℓ (t * )) −1 subrectangles R n,j of side lengths n −1/H ℓ (t * ) h (ℓ = 1, . . . , N ). Let 0 < ̺ < 1 be fixed and let κ n,j be the lower-left vertex of R n,j . Define ρ(s, t) = (E[B H(t) (t) − B H(s) (s)] 2 ) 1/2 , denote by N ρ (R n,j , ε) the smallest number of ρ-balls of radius ε needed to cover R n,j and denote by D the diameter of R n,j , that is, D := sup{ρ(s, t) : s, t ∈ R n,j }.\nCombining the above inequality with Talagrand [36], Lemma 2.1 gives\nThen for n sufficiently large, we have, for any fixed x ∈ I,\nIn the above, we have applied the fact that Var(B H(t) 0 (t)) ≥ c for all t ∈ I to derive the last inequality. If\n. Let N n be the number of rectangles R n,j such that x ∈ {B H(u) (u), u ∈ R n,j }. It follows from (5.4) that\nSince the random variables N n are integer-valued, (5.5) and Fatou's lemma imply that a.s. N n = 0 for infinitely many n. Therefore, Γ x ∩ I = ∅ a.s. We now assume\nBy Ayache and Xiao [6], Lemma 7, we have η k > 0. Define a covering {R ′ n,j } of Γ x ∩ I by R ′ n,j = R n,j if x ∈ {B H(u) (u), u ∈ R n,j } and R ′ n,j = ∅ otherwise. R ′ n,j can be covered by n N ℓ=k+1 (H k (t * ) −1 −H ℓ (t * ) −1 ) cubes of side length n −H k (t * ) −1 h. Thus, we can cover the level set Γ x ∩ I by a sequence of cubes of side length n −H k (t * ) −1 h. Let ̺ ∈ (0, 1) be small and let\nClearly, η k (̺) > η k > 0. In the following, we prove that the Hausdorff dimension of Γ x ∩ I is bounded above by η k (̺) almost surely. To this end, by (5.4), we have\nFatou's lemma implies that the η k (̺)-dimensional Hausdorff measure of Γ x ∩I is finite a.s. and thus dim H (Γ x ∩ I) ≤ η k (̺) almost surely. Letting ̺ ↓ 0 along the rational numbers, we obtain dim H (Γ x ∩ I) ≤ η k and therefore (5.2).\nTo prove the lower bound in (5.1), we assume τ (t * ) = k, that is,\n.\nBy Condition A, there exists a positive number ς such that for all t ∈ I ς := [t * − ς , t * + ς ] ∩ I, we have\nwhere H ℓ (1 ≤ ℓ ≤ N ) are defined as in (3.1) with I ς in place of I. Note that (5.7) and Ayache and Xiao [6], Lemma 3.3, imply that\nThus, we can choose ̺ > 0 such that\nIt is sufficient to prove that there is a constant c 5,3 > 0 such that\nOur proof of (5.9) is based on the capacity argument due to Kahane [23]. Similar methods have been used by Adler [1], Testard [37] and Xiao [40], to mention just a few. Let M + γ be the space of all non-negative measures on R N with finite γ-energy. It is known (cf. Adler [1]) that M + γ is a complete metric space under the metric\nWe define a sequence of random positive measures µ n on the Borel sets C ⊆ I ς by\nIt follows from Kahane [23] or Testard [37] that if there are positive and finite constants c 5,4 and c 5,5 , independent of ̺ and such that E( µ n ) ≥ c 5,4 , E( µ n 2 ) ≤ c 5,5 , (5.12) E( µ n γ ) < +∞, (5.13) where µ n = µ n (I ς ), then there is a subsequence of {µ n }, say {µ n k }, such that µ n k → µ in M + γ and µ is strictly positive with probability ≥ c 2 5,4 /(2c 5,5 ). It follows from (4.8) that µ has its support in Γ x ∩ I ς almost surely. Hence, Frostman's theorem yields (5.9).\nIt remains to verify (5.12) and (5.13). Let σ 2 (t) = Var(B H(t) (t)). By Fubini's theorem, we have\nwhere the two inequalities follow from (5.16) and a change of variables. By using Lemma 2.9 in the same way, we see that E( µ n γ ) < +∞ for any γ defined in (5.8). This proves (5.13). Finally, by letting ̺ ↓ 0, the lower bound for the Hausdorff dimension in (5.1) follows from (5.9) and we have therefore proven Theorem 5.1.\nThe proof of Theorem 5.1 suggests that we can consider the Hausdorff dimension of the level set in any neighborhood of a point t ∈ (0, ∞), provided N ℓ=1 1 H ℓ (t) > d. However, in order to obtain an almost sure result, we have to consider Γ x at a random level x = B H(t) (t). The following corollary can be considered as a local Hausdorff dimension result for the level sets of mfBs. where β τ (t 0 ) is defined by (4.6) with t 0 in place of t.\nProof. For any t 0 ∈ (0, ∞) N such that On the other hand, by using an argument similar to the one used in the proof of Berman [9] Theorem 2.1 (see also the proof of Xiao [42] Theorem 1.1), we can show that for every ε > 0 and r ∈ (0, r 0 ) small enough, P dim H (Γ B H(t) (t) ∩ U (t 0 , r)) ≥ max s∈U(t 0 ,r) β τ (s) − ε a.e. t ∈ U (t 0 , r 0 ) = 1.\n(5.23)\nBy letting r ↓ 0 and ε ↓ 0 along the rational numbers, we see that (5.19) follows from (5.22) and (5.23).\nRemark 5.4. Corollary 5.3 shows the explicit way in which the fractal properties of the multifractional Brownian sheet vary in space. In short, the local Hausdorff dimension is derived from the constant parameter formula. Fractional Brownian sheets are essentially fractional integrals of Brownian sheets (cf. Benson et al. [8] and Biermé, Meerschaert and Scheffler [12]). The term \"multifractional\" indicates that the order of fractional integration varies in space. It would be interesting to explore the connection between multifractional Brownian sheets and multifractals. For example, are the level sets of the mfBs multifractals and, if so, how do their structure functions depend on the Hurst index function H(t)?"},{"title":"Definition 1 . 1 .","content":""},{"title":"Lemma 2. 8 ..","content":""},{"title":"E","content":""},{"title":"Theorem 3. 1 .mfBs","content":""},{"title":"Corollary 3. 2 .","content":""},{"title":"Remark 3. 3 .","content":""},{"title":". 5 .","content":""},{"title":"Lemma 3. 7 .","content":""},{"title":"Corollary 3. 8 .","content":""},{"title":"Corollary 3 . 9 .","content":""},{"title":"H ℓ (t 0 )","content":""},{"title":"Lemma 4 . 1 .","content":""},{"title":"Lemma 4. 2 .","content":""},{"title":"Theorem 4 . 3 ..","content":""},{"title":"Corollary 4. 4 .","content":""},{"title":"Corollary 5. 3 .","content":""},{"title":"H","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"A one-dimensional fractional Brownian motion (fBm) ξ α = {ξ α (t), t ∈ R} with Hurst index α ∈ (0, 1) is a real-valued, centered Gaussian process with covariance function given by E[ξ α (s)ξ α (t)] = 1 2 [|s| 2α + |t| 2α − |t − s| 2α ] ∀s, t ∈ R.\\n(1.1)\\nIt was introduced, as a moving average Gaussian process, by Mandelbrot and Van Ness [31]. Fractional Brownian motion has interesting properties such as self-similarity of order α ∈ (0, 1), stationary increments and long-range dependence (when α > 1/2), which make it a good candidate for modeling different phenomena in, for example, finance and telecommunication. However, this model may be restrictive, due to the fact that all of its regularity and fractal properties are governed by the single Hurst parameter α. To model phenomena whose regularity evolves in time, for example, Internet traffic or image processing, Lévy-Véhel and Peltier [28] and Benassi, Jaffard and Roux [7] independently introduced multifractional Brownian motion (mfBm) in terms of moving average This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2008, Vol. 14, No. 3, 865-898. This reprint differs from the original in pagination and typographic detail.\\nrepresentation and harmonizable representation, respectively. Multifractional Brownian motion is governed by a Hurst functional α(t) with certain regularity in place of the constant Hurst parameter α in fBm.\\nSeveral authors have studied sample path and statistical properties of mfBm. For example, Benassi, Jaffard and Roux [7] considered the sample path Hölder regularity of mfBm and determined the Hausdorff dimension of its graph. Ayache, Cohen and Lévy-Véhel [2] and Herbin [22] studied the covariance structure of mfBm from its harmonisable representation. Recently, Boufoussi, Dozzi and Guerbaz [13,14] studied the existence, joint continuity and Hölder regularity of the local time of mfBm and established Chung's law of the iterated logarithm for mfBm. The main tool they applied to derive their results is the property of one-sided local non-determinism of mfBm.\\nThere are multiparameter extensions of fBm, among which, two typical examples are multiparameter Lévy fBm ξ α = {ξ α (t), t ∈ R N } (whose covariance function is given by (1.1) with | · | being the Euclidean norm in R N ) and fractional Brownian sheets, where the former is isotropic, while the latter are anisotropic in general. Since they were introduced by Kamont [24] (see also Ayache, Léger and Pontier [4]), fractional Brownian sheets (fBs) have been studied extensively as a representative of anisotropic Gaussian random fields in recent years. See, for example, Dunker [18], Mason and Shi [32], Øksendal and Zhang [33], Xiao and Zhang [45], Ayache and Xiao [6], Ayache, Wu and Xiao [5], Wu and Xiao [39] and the references therein for further information. Still, the regularity of fBs does not evolve in the N -dimensional 'time' parameter t ∈ R N + . To model anisotropic Gaussian random fields whose regularity evolves in time, such as images, Ayache and Léger [3] and Herbin [22] introduced so-called multifractional Brownian sheets (mfBs) in terms of their moving average representations and harmonisable representations, where the constant Hurst vector of fBs is substituted by Hurst functionals. Furthermore, they showed that mfBs have a continuous modification and determined the pointwise and local Hölder exponent of mfBs. They also proved that mfBs are locally self-similar. We refer to Ayache and Léger [3] and Herbin [22] for the definitions of the corresponding concepts and results.\\nIn studying anisotropic random fields, Xiao [44] suggested that it is more convenient to use the following metric ρ K on R N :\\nwhere K = (K 1 , . . . , K N ) ∈ (0, 1) N is a fixed vector. Denote by H(t) = (H 1 (t), . . . , H N (t)) a function in t ∈ R N + with values in (0, 1) N . We say that H(t) satisfies Condition A if there exist a positive number α ∈ (0, 1) and a vector (K 1 , . . . , K N ) ∈ (0, 1) N such that A.1. for every ℓ ∈ {1, . . . , N }, α ≤ H ℓ (t) ≤ K ℓ for all t ∈ R N + ; A.2. H ℓ (t) (ℓ = 1, . . . , N ) satisfies a ρ K -Lipschitz condition on every compact set, that is, for every compact subset I ⊆ R N + , there exist positive constants c ℓ = c ℓ (I) and δ such that\\n∀s, t ∈ I with |s − t| < δ. Now, we are ready to define multifractional Brownian sheets via their moving average representations.\"\n },\n {\n \"title\": \"3)\",\n \"content\": \"where s + = max{s, 0}, and where W = {W (s), s ∈ R N } is a standard real-valued Brownian sheet.\\nRemark 1.2. Our Definition 1.1 generalizes the definition in Ayache and Léger [3], where they define H ℓ (·) as a function of t ℓ ∈ R + (ℓ = 1, . . . , N ). Herbin [22] defines multifractional Brownian sheets by using the following moving average representation (cf. [22] Definition 2, page 1261):\\nBased on Dobrić and Ojeda [16] (see also Stoev and Taqqu [35]), we know that the multifractional Brownian sheet defined by (1.3) and the multifractional Brownian sheet defined by (1.4) have different correlation structures in general, even if N = 1. Our definition is more convenient to use than that of Herbin's when we derive the one-sided sectorial local non-determinism for multifractional Brownian sheets in Section 2. The form (1.3) is preferred in some applications in the one-dimensional case because it is easier to separate the future from the past.\\nDefine the Gaussian random field\\nwhere {B\\nIn this paper, we will study the existence and regularity of the local times of multifractional Brownian sheets. Our main technical tool is the property of one-sided sectorial local non-determinism. This property is more general than the sectorial local non-determinism which was first introduced by Khoshnevisan and Xiao [27] for the Brownian sheet and then extended by Wu and Xiao [39] to fractional Brownian sheets. See Section 2 for more information.\\nOur results show that multifractional Brownian sheets are similar to fractional Brownian sheets in many ways. They admit jointly continuous local times when the indices H i (t) stay in the range for which the fractional Brownian sheet has jointly continuous local times. We also establish a Hausdorff dimension result and, essentially, the dimension of the level set is the same as for the constant parameter case, except that we take the supremum of the constant parameter formula. We also show that the supremum can be taken locally, to establish that the fractal dimension of the random field varies in space. Hence, multifractional Brownian sheets are useful in applications such as composite materials, or porous media flow, when the material properties vary in space. They may also find useful applications in image processing.\\nThe rest of this paper is organized as follows. In Section 2, we prove some basic results on mfBs that will be useful to our arguments. In Section 3, we provide a sufficient condition for the existence of L 2 -local times of the (N, d) mfBs and prove that the condition also implies the joint continuity of the local times. We prove the Hölder regularity of the local times in Section 4. Finally, we derive the local Hausdorff dimensions of the level sets of {B H(t) (t)} in Section 5. Our results extend the results of Ayache and Xiao [6] and Ayache, Wu and Xiao [5] for fractional Brownian sheets and Boufoussi, Dozzi and Guerbaz [13,14] for multifractional Brownian motion to multifractional Brownian sheets.\\nWe end the Introduction with some notation. Throughout this paper, the underlying parameter space is R N or R N + = [0, ∞) N . A parameter t ∈ R N is written as t = (t 1 , . . . , t N ), or as c , if t 1 = · · · = t N = c. For any s, t ∈ R N such that s j < t j (j = 1, . . . , N ) (we denote this by s ≺ t), we define the closed interval (or rectangle) [s, t] = N j=1 [s j , t j ]. We use A to represent the collection of closed intervals [s, t] with s, t ∈ [ε, Λ] N for some fixed positive numbers ε and Λ. For any integer m ≥ 1, we always write λ m for the Lebesgue measure on R m and use ·, · and | · | to denote the ordinary scalar product and the Euclidean norm in R m , respectively.\\nThroughout this paper, an unspecified positive and finite constant will be denoted by c, which may not be the same in each occurrence. More specific constants in Section i are numbered as c i,1 , c i,2 , . . . .\"\n },\n {\n \"title\": \"Preliminaries\",\n \"content\": \"In this section, we will provide some lemmas that are useful for proving our main results. Lemma 2.1 is an extension of Boufoussi et al. [13], Lemma 3.1 from fractional Brownian motion to fractional Brownian sheets.\\nLemma 2.1. Let 0 < ε < Λ and 0 < α < γ < 1 be fixed constants. Let {Z κ 0 (t), (t, κ) ∈ R N + × [α, γ] N } be a real-valued Gaussian random field defined by equation (1.3) with\\nit suffices for us to prove that for j ∈ {1, . . . , N } fixed,\\nBy using the moving average representation of fBs (equation (1.3)), we have that\\nwhereǔ j = (u 1 , . . . , u j−1 , u j+1 , . . . , u N ). A change of variables shows that\\nwhich is bounded for all t ∈ [ε, Λ] N and β, β ′ ∈ [α, γ] N . Next, we estimate I 2 . Without loss of generality, we may assume β j > β ′ j . We rewrite I 2 as the following summation:\\nBy the mean value theorem, we have that for some β ′ j ≤ η j1 ≤ β j (η j1 may depend on t j ),\\nfor all t ∈ [ε, Λ] N and β, β ′ ∈ [α, γ] N . In the above, the last inequality follows from a change of variables. Similarly, we have that for some β ′ j ≤ η j2 ≤ β j , (t)} be a multifractional Brownian sheet in R. There exist positive constants δ > 0, c 2,4 and c 2,5 such that for all s, t ∈ [ε, Λ] N with |s − t| < δ, for any\\nProof. By the elementary inequalities\\nThe first term on the right-hand side of (2.10) is the variance of the increment of a fractional Brownian sheet with Hurst index H(u). By Lemma 8 in Ayache and Xiao [6], there exist positive constants c 2,6 and c 2,7 , depending only on α, γ = max{K 1 , . . . , K N }, ε and N , such that\\nMeanwhile, by Condition A and Lemma 2.1, there exists δ > 0 small such that for all s, t ∈ [ε, Λ] N with |s − t| < δ, we have |u − s| < δ, which implies\\nand, similarly,\\nCombining (2.10) with (2.11), (2.12) and (2.13), and noting that sup t ℓ H ℓ (t ℓ ) ≤ K ℓ (still by Definition 1.1), we can see that there exists δ > 0, which depends only on α, γ, ε, Λ and N , such that for |s − t| < δ, (2.9) holds. This completes the proof of Lemma 2.2. Remark 2.3. For any fixed t ∈ [ε, Λ] N and any fixed unit vector η ∈ R N , let us consider the increment\\nIn light of Lemma 2.2, it would be interesting to compute the bounds on the correlation function\\n(2.14)\\nand its time-and direction-varying spectral density (or Wigner-Ville distribution)\\nsince those are the properties of the random field that are usually used to model natural phenomena.\\nAs we mentioned in the Introduction, the main technical tool of this paper is the property of sectorial local non-determinism. Hence, it will be helpful to say a few words about the history of various forms of local non-determinism and briefly recall some of the definitions. The concept of local non-determinism, which provides a powerful tool for dealing with the complex dependence structure of Gaussian processes, was first introduced by Berman [11] to unify and extend his methods for studying local times of real-valued Gaussian processes, and then extended by Pitt [34] to Gaussian random fields. The notion of strong local non-determinism was later introduced by Cuzick and DuPreez [15] for Gaussian processes (i.e., N = 1) and can be extended in various ways to Gaussian random fields (see Xiao [43]). The simplest is the following. Let X = {X(t), t ∈ R N } be a real-valued Gaussian random field with 0 < E[X(t) 2 ] < ∞ for all t ∈ I, where I ∈ A is an interval. Let φ be a given function such that φ(0) = 0 and φ(r) > 0 for r > 0. Then X is said to be strongly locally φ-non-deterministic (SLφND) on I if there exist positive constants c 2,10 and r 0 such that for all t ∈ I and all 0 < r ≤ min{|t|, r 0 }, Var(X(t)|X(s) : s ∈ I, r ≤ |s − t| ≤ r 0 ) ≥ c 2,10 φ(r).\\n(2.16)\\nHere, Var(X(t)|X(s), s ∈ S) denotes the conditional variance of X(t), given X(s), s ∈ S. Pitt [34] proved that fractional Brownian motion ξ α is SLφND with φ(r) = r 2α . When N = 1 and Var(X(t)|X(s) : s ∈ I, r ≤ t − s ≤ r 0 ) ≥ c 2,10 φ(r), X is said to satisfy one-sided strong local φ-non-determinism. The properties of strong local non-determinism have played important roles in studying the regularity of local times, small ball probabilities, uniform Hausdorff dimension results and other sample path properties of Gaussian processes and Gaussian random fields. We refer to Xiao [43,44] for more information on applications of SLφND, as well as other forms of local non-determinism. It is well known that the Brownian sheet does not satisfy the property of local nondeterminism in the sense of Berman and Pitt (hence, it does not satisfy (2.16) either). Recently, Khoshnevisan and Xiao [27] introduced the concept of sectorial local nondeterminism and proved that the Brownian sheet has the property of sectorial local nondeterminism. Wu and Xiao [39] extended the result of Khoshnevisan and Xiao [27] and proved that the fractional Brownian sheet {B H 0 (t), t ∈ R N + }, where H = (H 1 , . . . , H N ) ∈ (0, 1), also has the property of sectorial local non-determinism. Namely, for any fixed positive number ε ∈ (0, 1), there exists a positive constant c 2,11 , depending only on ε, H and N , such that for all positive integers n ≥ 2, and all t 1 , . . . , t n ∈ [ε, ∞) N , we have\\nwhere t 0 = 0. The concept of sectorial local non-determinism opens the door to unifying the previously different treatments for the Brownian sheet and fractional Brownian motion, and can also be used to prove new results about fractional Brownian motion. See, for example, Khoshnevisan, Wu and Xiao [26], Wu and Xiao [39] and Ayache, Wu and Xiao [5] for studies on sample path properties of the Brownian sheet and fractional Brownian sheets by using the sectorial local non-determinism property.\\nIt is not known whether the multifractional Brownian sheet {B\\n(t)} is sectorially locally non-deterministic. To study the local times of (N, d)-mfBs {B H(t) (t)}, we will work with multifractional Liouville sheets (mfLs) at first and prove that they can be well approximated by Gaussian processes which have the property of one-sided strong local non-determinism (see (2.22) and (2.23) below). For a function H(t) = (H 1 (t), . . . , H N (t)) satisfying Condition A, the real-valued, centered Gaussian random field {X\\nis called a multifractional Liouville sheet with functional Hurst index H(t). One parameter mfLs were first introduced by Lim and Muniandy [30] as an extension of fBm; see Lim [29] for more properties on one parameter mfLs.\\nand the two fields on the right-hand side of (2.19) are independent. We will show that in studying the regularity properties of the local times of {B\\n(t)} plays a crucial role and the second field in (2.19) can be neglected. More precisely, we will make use of the following property. For all integers n ≥ 2, t 1 , . . . , t n ∈ R N + and u 1 , . . . , u n ∈ R, we have\\n(2.20)\\nNext, we use an argument in Ayache and Xiao [6] to provide a useful decomposition for {X\\nwe decompose the rectangle [0, t] into the following disjoint union of subrectangles:\\ncan be written as a union of 2 N − N − 1 subrectangles of [0, t]. Denote the integrand in (2.18) by g(t, r). It\\nSince the processes X(ε, t), Y ℓ (t) (1 ≤ ℓ ≤ N ) and Z(ε, t) are defined by the stochastic integrals with respect to W over disjoint sets, they are independent Gaussian random fields.\\nThe following lemma shows that every process Y ℓ (t) has the property of strong local non-determinism along the ℓth direction. It will be essential to our proofs.\\nLemma 2.4. Let I ∈ A and let ℓ ∈ {1, 2, . . ., N } be fixed. For all integers n ≥ 2 and t 1 , . . . , t n ∈ I such that\\nwhere t 0 ℓ = 0 and c 2,12 > 0 is a constant depending only on ε and I.\\nProof. The proof is in the same spirit as the proof of Lemma 2.1 of Ayache, Wu and Xiao [5]. Working in the Hilbert space setting, the conditional variance in (2.23) is the square of the L 2 (P)-distance of Y ℓ (t n ) from the subspace generated by Y ℓ (t j )(0 ≤ j ≤ n − 1). Hence, it is sufficient to show that there exists a constant c 2,12 such that\\nfor all a j ∈ R (j = 1, . . . , n − 1). However, by splitting R ℓ (t n ) into two disjoint parts and using the independence, we derive that\\nThis proves (2.24) and hence Lemma 2.4.\\nCombining Lemma 2.2 and Lemma 2.4 with the proofs of Lemma 2.1 and Lemma 8.1 in Berman [11] (see also Boufoussi, Dozzi and Guerbaz [13], Theorem 3.3), we have the following.\\nProposition 2.5. For every integer n ≥ 2, there exist positive constants C n and δ (both of these may depend on n) such that for every ℓ = 1, . . . , N ,\\nfor all (u 1 , . . . , u n ) ∈ R n and all points t 1 , . . . , t n ∈ I satisfying t 1 ℓ < · · · < t n ℓ with t n ℓ − t 1 ℓ < δ.\\nThe following lemma relates the multifractional Brownian sheet {B H(t) 0 (t)} to the independent Gaussian random fields Y ℓ (ℓ = 1, . . . , N ), which is a direct extension of Ayache, Wu and Xiao [5], Lemma 2.2.\\nLemma 2.6. Let I ∈ A. For all integers n ≥ 2, t 1 , . . . , t n ∈ I and u 1 , . . . , u n ∈ R, we have\\nConsequently, for all k ∈ {1, . . . , N } and positive numbers p 1 , . . . , p k ≥ 1 satisfying\\n(t)} satisfies the following property: for all integers n ≥ 2 and t 1 , . . . , t n ∈ I such that t j ≺ t n for every j ≤ n − 1,\\nThis property is weaker than (2.17) and will be referred to as one-sided sectorial local non-determinism.\\nWe will also make use of the following technical lemmas, among which Lemma 2.8 is from Xiao and Zhang [45], Lemma 2.9 is proved in Ayache and Xiao [6] and Lemma 2.10 and Lemma 2.11 are from Ayache, Wu and Xiao [5].\\nThere then exist finite constants c 2,15 and c 2,16 , depending only on α, β and η, such that the following hold for all reals A, B > 0 satisfying A 1/α ≤ c 2,15 B:\\nwith the convention that 0\\nThere then exists a positive constant ∆ τ ≤ 1, depending only on (ρ 1 , . . . , ρ N ), such that for every ∆ ∈ (0, ∆ τ ), we can find τ real numbers\\nand\\nLemma 2.11. For all integers n ≥ 1, positive numbers a, r, 0 < b j < 1 and an arbitrary\\nFinally, we conclude this section by briefly recalling some aspects of the theory of local times. For excellent surveys on local times of random and deterministic vector fields, we refer to Geman and Horowitz [20] and Dozzi [17].\\nLet X(t) be a Borel vector field on R N with values in R d . For any Borel set I ⊆ R N , the occupation measure of X on I is defined as the following measure on R d :\\nIf µ I is absolutely continuous with respect to λ d , we say that X(t) has local times on I and define its local times, L(•, I), as the Radon-Nikodým derivative of µ I with respect to λ d , that is,\\nIn the above, x is the so-called space variable and I is the time variable. Sometimes, we write L(x, t) in place of L(x, [0, t]). Note that if X has local times on I, then for every Borel set J ⊆ I, L(x, J) also exists. By standard martingale and monotone class arguments, one can deduce that the local times have a measurable modification that satisfies the following occupation density formula (see Geman and Horowitz [20], Theorem 6.4): for every Borel set I ⊆ R N and every measurable function f :\\nThen, whenever we can choose a version of the local time, still denoted by\\nX is said to have a jointly continuous local time on I. When a local time is jointly continuous, L(x, •) can be extended to be a finite Borel measure supported on the level set\\nsee Adler [1] for details. In other words, local times often act as a natural measure on the level sets of X. Hence, they are useful in studying the various fractal properties of level sets and inverse images of the vector field X. In this regard, we refer to Berman [10], Ehm [19] and Xiao [41]. It follows from Geman and Horowitz [20], ((25.5) and (25.7)) (see also Pitt [34]) that for all x, y ∈ R d , I ∈ A and all integers n ≥ 1,\\nand for all even integers n ≥ 2, where u = (u 1 , . . . , u n ), t = (t 1 , . . . , t n ) and each u j ∈ R d , t j ∈ I ⊆ (0, ∞) N . In the coordinate notation, we then write u j = (u j 1 , . . . , u j d ).\"\n },\n {\n \"title\": \"Local times: existence and joint continuity\",\n \"content\": \"In this section, we consider the existence and regularity of local times of mfBs. The index H depends on I, but, for simplicity, we have deleted I from the notation. Proof. Without loss of generality, we may, and will, assume that δ = diam(I) is sufficiently small that (2.9) holds for all s, t ∈ I. In particular, we assume δ ≤ 1.\\nTo prove the existence of local time on I, by Geman and Horowitz [20] Theorem 21.9, it suffices to prove that\\nIt follows from (2.9) and a change of variables that\\n, we can estimate the last integral in a way similar to the proof of Xiao and Zhang [45] Theorem 3.6. Namely, by applying Lemma 2.8 repeatedly, we derive that the last integral is finite. This proves (3.2) and hence Theorem 3.1.\\nThe following is a consequence of Theorem 3.1 which gives a more natural condition for the existence of local times of mfBs. H ℓ (t) for all t ∈ I, but the equality only holds for t in a set of Lebesgue measure 0, the existence of local times is rather subtle and requires the imposition of further assumptions on (H 1 (t), . . . , H N (t)). Hence, it will not be discussed here.\\nWe now consider the joint continuity of the local times of {B H(t) (t)}. For convenience, we first prove that under the same condition as in Theorem 3.1, the local time of {B H(t) (t)} has a version that is jointly continuous in both space and time variables. We then apply the same argument as in the proof of Corollary 3.2 to show that the same conclusion holds, provided\\nOur results extend those of Ehm [19] for the Brownian sheet and of Ayache, Wu and Xiao [5] for fractional Brownian sheets. The main idea for proving Theorem 3.4 is similar to those in Ehm [19], Xiao [41] and Ayache, Wu and Xiao [5]. That is, we first apply the Fourier analytic arguments to derive estimates for the moments of the local times and then apply a multiparameter version of the Kolmogorov continuity theorem (cf. Khoshnevisan [25]). As in Ayache, Wu and Xiao [5], the 'one-sided' sectorial local non-deterministic properties of multifractional Liouville sheets proved in Section 2 (see Lemma 2.4 and Proposition 2.5) will play important roles in deriving moment estimates in Lemmas 3.5 and 3.7 below. However, due to the nonstationarity and the lack of two-sided local non-determinism of multifractional Brownian sheets, we need to make several modifications in our proofs.\\nFor convenience, we further assume 0 < H 1 ≤ · · · ≤ H N < 1.\\nProof. For later use, we will start with an arbitrary closed interval T = N ℓ=1 [a ℓ , a ℓ + h ℓ ] ⊆ I. It follows from (2.43) and the fact that {B\\n(t)} that for all integers n ≥ 1,\\nwhere U k = (u 1 k , . . . , u n k ) ∈ R n . Fix k = 1, . . . , d and denote the inner integral in (3.7) by N k . Then by Lemma 2.6, we have\\nSince (3.5) holds, we apply Lemma 2.10 with ∆ = n −1 and q = d to obtain τ positive numbers p 1 , . . . , p τ ≥ 1 satisfying (2.36) and (2.37).\\nApplying the generalized Hölder inequality (Hardy [21] page 140), we derive that\\nwhere the last equality follows from the fact that (Y ℓ (t 1 ), . . . , Y ℓ (t n )) is a Gaussian vector with mean 0. Hence, it follows from (3.7) and (3.9) that\\nTo evaluate the integral in (3.10), we will first integrate [dt 1 ℓ · · · dt n ℓ ] for ℓ = 1, . . . , τ . To this end, we use the following fact about multivariate normal distributions: for any Gaussian random vector (Z 1 , . . . , Z n ), det Cov(Z 1 , . . . , Z n ) = Var(Z 1 ) n j=2 Var(Z j |Z 1 , . . . , Z j−1 ).\\n(3.11)\\nBy the above fact and Lemma 2.4, we can derive that for every ℓ ∈ {1, . . . , τ } and for all\\nfor some permutation π ℓ of {1, . . . , N }, we have\\nwhere t π ℓ (0) ℓ := ε (recall the decomposition (2.22)). We choose ε < 1 2 min{a ℓ , 1 ≤ ℓ ≤ N } so that Lemma 2.11 is applicable. It follows from (3.12) and (3.13) that\\nIn the above, the last inequality follows from (2.40) (recall that H ℓ d/p ℓ < 1). Combining (3.10) and (3.14), we have\\nWe now consider the special case when T = [a, a + h ], that is, h 1 = · · · = h N = h with h < δ, (3.15) and (2.37) with ∆ = n −1 and q = d together yield\\nThis proves (3.6).\\nRemark 3.6. In the proof of Lemma 3.5, if we apply the generalized Hölder inequality to the first integral in (3.8) with N positive numbers p 1 , . . . , p N defined by\\nthen the above proof shows that if T ⊆ I ∈ A, then, similarly to (3.15), the following inequality holds:\\nwhere ν = d/( N ℓ=1 1 H ℓ ) ∈ (0, 1). We will apply this inequality in the proof of Theorem 3.4 below.\\nProof. Let γ ∈ (0, 1 ∧ ατ 2τ ) be a constant. Note that by the elementary inequalities\\nand |u + v| γ ≤ |u| γ + |v| γ , we see that for all u 1 , . . . , u n , x, y ∈ R d ,\\nwhere the summation ′ is taken over all the sequences (k 1 , . . . , k n ) ∈ {1, . . . , d} n . It follows from (2.44), (3.20) and Lemma 2.6 that for every even integer n ≥ 2,\\nwhere η j k = 1 if k = k j and η j k = 0 otherwise. Note that for every j ∈ {1, . . . , n}, we have d k=1 η j k = 1. We take β ℓ = H ℓ (1 ≤ ℓ ≤ N ), ∆ = 1/n and q = d in Lemma 2.10 and let p ℓ (ℓ = 1, . . . , τ ) be the constants satisfying (2.36) and (2.37). Observe that since γ ∈ (0, ατ 2τ ), there exists\\nCombining (3.21) with the generalized Hölder inequality, we have that\\nFor any n points t 1 , . . . , t n ∈ T , let π 1 , . . . , π N be N permutations of {1, 2, . . . , n} such that for every 1 ≤ ℓ ≤ N ,\\nBy changing the variables of the above integral by means of the transformation\\nwhere t π ℓ 0 (0) = 0. Furthermore, by the elementary inequality that for\\nMoreover, the last product is equal to a finite sum of terms, each of the form n j=1 |v j k | γη j k p ℓ 0 εj , where ε j = 0, 1, or 2 and\\n. By Proposition 2.5, we know that M ℓ0 is dominated by the sum over all possible choices of (ε 1 , . . . , ε n ) ∈ {0, 1, 2} n of the following terms\\nwhere V k = (v 1 k , . . . , v n k ) ∈ R n . By another change of variable, w j k = σ ℓ0,j v j k , the integral in (3.27) can be represented by\\nis a constant depending on n. Thus, we have obtained that\\n(3.29)\\nThe other integrals for ℓ = ℓ 0 in (3.23) are easier and can be estimated similarly. Combining (3.23) with Lemma 2.4 (which gives lower bounds for σ 2 ℓ,j ), (3.28), (3.29) and the definition of T , we have\\nIn the above,\\nWe now take h 1 = · · · = h N = h < δ in (3.30). Then by Lemma 2.11 and noting that (3.22) holds, we have\\nwhere the last inequality follows from Lemma 2.10 and the fact that H ℓ0 ≤ H τ . Proof Theorem 3.4. Let I = [a, b] ∈ A be fixed and assume that (3.5) holds. It follows from Lemma 3.7 and the multiparameter version of Kolmogorov's continuity theorem (cf. Khoshnevisan [25]) that for every T ⊆ I, the mfBs {B H(t) } has almost surely a local time L(x, T ) that is continuous for all x ∈ R d .\\nTo prove the joint continuity, observe that for all x, y ∈ R d and s, t ∈ I with |s − t| < δ, where δ > 0 is the same as in Lemma 2.2, we have\\nSince the difference L(x, [a, s]) − L(x, [a, t]) can be written as a sum of a finite number (which only depends on N ) of terms of the form L(x, T j ), where each T j ∈ A is a closed subinterval of I with at least one edge length ≤ |s − t|, we can use Lemma 3.5 and Remark 3.6 to bound the first term in (3.32). On the other hand, the second term in (3.32) can be dealt with using Lemma 3.7, as above. Consequently, for some small γ ∈ (0, 1), the righthand side of (3.32) is bounded by c n 3,11 (|x − y| + |s − t|) nγ , where n ≥ 2 is an arbitrary even integer. Therefore, the joint continuity of the local times on I again follows from the multiparameter version of Kolmogorov's continuity theorem. This finishes the proof of Theorem 3.4.\\nSimilarly to the proof of Corollary 3.2, we derive from Theorem 3.4 the following, more general, result. The next corollary is a direct consequence of Corollary 3.8 and the continuity of the Hurst functionals. We state it to emphasize that the joint continuity of the local time is a local property depending on H(t). (t), t ∈ R N + } be a real-valued multifractional Brownian sheet defined by the harmonizable representation (cf. Herbin [22])\\nwhere W is the complex-valued Gaussian random measure in R N with the Lebesgue measure Λ N as its control measure. Based on Dobríc and Ojeda [16] (see also Stoev and Taqqu\"\n },\n {\n \"title\": \"Hölder conditions for L(x, •)\",\n \"content\": \"In this section, we investigate the local and global asymptotic behavior of the local time L(x, ·) at x as a random measure on R N + . Results in this section carry information about fractal properties of the sample functions of mfBs; see Section 5.\\nBy applying Lemma 3.5, we can prove the following technical lemmas, which will be useful in this section.\\nProof. The proof is similar to the proof of Ayache, Wu and Xiao [5] Lemma 3.11 and we include it here for completeness. For each fixed a ∈ I, we define the Gaussian random\\n. It follows from (2.41) that if {B H(t) (t)} has a local time L(x, S) on any Borel set S, then Y also has a local timeL(x, S) on S and, moreover, L(x + B H(a) (a), S) =L(x, S). With a little modification, the proof of Lemma 3.5 works for the Gaussian field Y . Hence, we derive that (4.1) holds.\\nThe following lemma is a consequence of Lemma 4.1 and Chebyshev's inequality. N , d, I and H) such that for all a ∈ I, T = [a, a + r ] with r ∈ (0, δ), where δ > 0 is the same as in Lemma 2.2, x ∈ R d and u > 1 large enough, we have\\nLet U (t, r) be the open ball centered at t with radius r, let H U be the vector defined by (3.1) with I = U (t, r) and let τ (U ) be the positive integer satisfying the corresponding condition (3.5). By applying Lemma 3.5 and the Borel-Cantelli lemma, one can easily derive the following law of the iterated logarithm for the local time L(x, ·): there exists a positive constant c 4,3 such that for every x ∈ R d and t ∈ (0, ∞) N ,\\nBecause of the continuity of H ℓ (t) (1 ≤ ℓ ≤ N ), it can be verified that τ (U ) → τ (t) and β τ (U) → β τ (t) as r → 0, (4.4) where τ (t) is the unique integer satisfying\\nand\\nIt follows from Fubini's theorem that with probability one, (4.3) holds for almost all t ∈ (0, ∞) N . We now prove a stronger version of this result, which is useful in determining the Hausdorff dimension of the level set. where ϕ t (r) = r β τ (t) (log log(1/r)) N −β τ (t) and where β τ (t) is defined by (4.6).\\nProof. For every integer k > 0, we consider the random measure L k (x, •) on the Borel subsets C of I defined by\\nThen, by the occupation density formula (2.41) and the continuity of the function y → L(y, C), one can verify that almost surely L k (x, C) → L(x, C) as k → ∞ for every Borel set C ⊆ I. For every integer m ≥ 1, let f m (t) = L(x, U (t, 2 −m )). From the proof of Theorem 3.4, we can see that almost surely the functions f m (t) are continuous and bounded. Hence, we have, almost surely, for all integers m, n ≥ 1,\\nIt follows from (4.9), (4.8) and the proof of Pitt [34], Proposition 3.1 that for every positive integer n ≥ 1,\\nx, u j (4.10)\\n× E exp i Let γ > 0 be a constant, the value of which will be determined later. We consider the random set\\nDenote by µ ω the restriction of the random measure L(x, ·) to I, that is, µ ω (E) = L(x, E ∩ I) for all Borel sets E ∈ R N + . We now take n = ⌊log m⌋, where ⌊y⌋ denotes the integer part of y. Then by applying (4.11) and by Stirling's formula, we have\\nprovided γ > 0 is chosen large enough, say, γ ≥ c 4,2 . This implies that\\nTherefore, with probability 1 for µ ω -almost all t ∈ I, we have (4.14)\\nFinally, for any r > 0 small enough, there exists an integer m such that 2 −m ≤ r < 2 −m+1 and (4.13) is applicable. Since ϕ t is increasing near 0, (4.7) follows from (4.13) and a monotonicity argument.\\nRecall that the pointwise Hölder exponent of a random field {X(t), t ∈ R N } at a point t ∈ R N is defined by We remark that the corresponding problem of finding the Hausdorff dimensions of the level sets of multifractional Brownian motion has been investigated by Boufoussi et al. [13] and the Hausdorff dimensions of the level sets of fractional Brownian sheets were studied in Ayache and Xiao [6]. As shown by the following theorem, the fractal structure of Γ x is much richer than the level sets of multifractional Brownian motion and is locally reminiscent of the level sets of a fractional Brownian sheet.\\nBrownian sheet with Hurst functionals H ℓ (t) (ℓ = 1, . . . , N ). For any interval I ∈ A, let t * ∈ I be a point satisfying\\nH ℓ (t * ) > d, then for any x ∈ R d , with positive probability,\\nRemark 5.2. It can be verified that if (4.5) holds for t = t * , then β τ (t * ) is the same as in (4.6) with t replaced by t * . In the special case where H(t) = H is a constant, β τ (t * ) reduces to the form derived in Ayache and Xiao [6] for the Hausdorff dimension of the level sets of a fractional Brownian sheet. When\\nH ℓ (t * ) = d, we believe that for every x ∈ R d , Γ x ∩ I = ∅ a.s. However, the method of this paper is not sufficient for proving this statement.\\nProof of Theorem 5.1. We will follow the proof of Ayache and Xiao [6] Theorem 5.\\nFirst, we prove\\nand Γ x ∩ I = ∅ a.s. when the right-hand side of (5.2) is negative. Without loss of any generality, we may assume that I = [a, a + h ] and h is small so that Lemma 2.2 is applicable. For an integer n ≥ 1, divide the interval I into m n ≤ n N ℓ=1 (H ℓ (t * )) −1 subrectangles R n,j of side lengths n −1/H ℓ (t * ) h (ℓ = 1, . . . , N ). Let 0 < ̺ < 1 be fixed and let κ n,j be the lower-left vertex of R n,j . Define ρ(s, t) = (E[B H(t) (t) − B H(s) (s)] 2 ) 1/2 , denote by N ρ (R n,j , ε) the smallest number of ρ-balls of radius ε needed to cover R n,j and denote by D the diameter of R n,j , that is, D := sup{ρ(s, t) : s, t ∈ R n,j }.\\nCombining the above inequality with Talagrand [36], Lemma 2.1 gives\\nThen for n sufficiently large, we have, for any fixed x ∈ I,\\nIn the above, we have applied the fact that Var(B H(t) 0 (t)) ≥ c for all t ∈ I to derive the last inequality. If\\n. Let N n be the number of rectangles R n,j such that x ∈ {B H(u) (u), u ∈ R n,j }. It follows from (5.4) that\\nSince the random variables N n are integer-valued, (5.5) and Fatou's lemma imply that a.s. N n = 0 for infinitely many n. Therefore, Γ x ∩ I = ∅ a.s. We now assume\\nBy Ayache and Xiao [6], Lemma 7, we have η k > 0. Define a covering {R ′ n,j } of Γ x ∩ I by R ′ n,j = R n,j if x ∈ {B H(u) (u), u ∈ R n,j } and R ′ n,j = ∅ otherwise. R ′ n,j can be covered by n N ℓ=k+1 (H k (t * ) −1 −H ℓ (t * ) −1 ) cubes of side length n −H k (t * ) −1 h. Thus, we can cover the level set Γ x ∩ I by a sequence of cubes of side length n −H k (t * ) −1 h. Let ̺ ∈ (0, 1) be small and let\\nClearly, η k (̺) > η k > 0. In the following, we prove that the Hausdorff dimension of Γ x ∩ I is bounded above by η k (̺) almost surely. To this end, by (5.4), we have\\nFatou's lemma implies that the η k (̺)-dimensional Hausdorff measure of Γ x ∩I is finite a.s. and thus dim H (Γ x ∩ I) ≤ η k (̺) almost surely. Letting ̺ ↓ 0 along the rational numbers, we obtain dim H (Γ x ∩ I) ≤ η k and therefore (5.2).\\nTo prove the lower bound in (5.1), we assume τ (t * ) = k, that is,\\n.\\nBy Condition A, there exists a positive number ς such that for all t ∈ I ς := [t * − ς , t * + ς ] ∩ I, we have\\nwhere H ℓ (1 ≤ ℓ ≤ N ) are defined as in (3.1) with I ς in place of I. Note that (5.7) and Ayache and Xiao [6], Lemma 3.3, imply that\\nThus, we can choose ̺ > 0 such that\\nIt is sufficient to prove that there is a constant c 5,3 > 0 such that\\nOur proof of (5.9) is based on the capacity argument due to Kahane [23]. Similar methods have been used by Adler [1], Testard [37] and Xiao [40], to mention just a few. Let M + γ be the space of all non-negative measures on R N with finite γ-energy. It is known (cf. Adler [1]) that M + γ is a complete metric space under the metric\\nWe define a sequence of random positive measures µ n on the Borel sets C ⊆ I ς by\\nIt follows from Kahane [23] or Testard [37] that if there are positive and finite constants c 5,4 and c 5,5 , independent of ̺ and such that E( µ n ) ≥ c 5,4 , E( µ n 2 ) ≤ c 5,5 , (5.12) E( µ n γ ) < +∞, (5.13) where µ n = µ n (I ς ), then there is a subsequence of {µ n }, say {µ n k }, such that µ n k → µ in M + γ and µ is strictly positive with probability ≥ c 2 5,4 /(2c 5,5 ). It follows from (4.8) that µ has its support in Γ x ∩ I ς almost surely. Hence, Frostman's theorem yields (5.9).\\nIt remains to verify (5.12) and (5.13). Let σ 2 (t) = Var(B H(t) (t)). By Fubini's theorem, we have\\nwhere the two inequalities follow from (5.16) and a change of variables. By using Lemma 2.9 in the same way, we see that E( µ n γ ) < +∞ for any γ defined in (5.8). This proves (5.13). Finally, by letting ̺ ↓ 0, the lower bound for the Hausdorff dimension in (5.1) follows from (5.9) and we have therefore proven Theorem 5.1.\\nThe proof of Theorem 5.1 suggests that we can consider the Hausdorff dimension of the level set in any neighborhood of a point t ∈ (0, ∞), provided N ℓ=1 1 H ℓ (t) > d. However, in order to obtain an almost sure result, we have to consider Γ x at a random level x = B H(t) (t). The following corollary can be considered as a local Hausdorff dimension result for the level sets of mfBs. where β τ (t 0 ) is defined by (4.6) with t 0 in place of t.\\nProof. For any t 0 ∈ (0, ∞) N such that On the other hand, by using an argument similar to the one used in the proof of Berman [9] Theorem 2.1 (see also the proof of Xiao [42] Theorem 1.1), we can show that for every ε > 0 and r ∈ (0, r 0 ) small enough, P dim H (Γ B H(t) (t) ∩ U (t 0 , r)) ≥ max s∈U(t 0 ,r) β τ (s) − ε a.e. t ∈ U (t 0 , r 0 ) = 1.\\n(5.23)\\nBy letting r ↓ 0 and ε ↓ 0 along the rational numbers, we see that (5.19) follows from (5.22) and (5.23).\\nRemark 5.4. Corollary 5.3 shows the explicit way in which the fractal properties of the multifractional Brownian sheet vary in space. In short, the local Hausdorff dimension is derived from the constant parameter formula. Fractional Brownian sheets are essentially fractional integrals of Brownian sheets (cf. Benson et al. [8] and Biermé, Meerschaert and Scheffler [12]). The term \\\"multifractional\\\" indicates that the order of fractional integration varies in space. It would be interesting to explore the connection between multifractional Brownian sheets and multifractals. For example, are the level sets of the mfBs multifractals and, if so, how do their structure functions depend on the Hurst index function H(t)?\"\n },\n {\n \"title\": \"Definition 1 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 2. 8 ..\",\n \"content\": \"\"\n },\n {\n \"title\": \"E\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 3. 1 .mfBs\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 3. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Remark 3. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \". 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 3. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 3. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 3 . 9 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"H ℓ (t 0 )\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 4 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 4. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4 . 3 ..\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 4. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 5. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"H\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Let H(t) = (H 1 (t), . . . , H N (t)) be a function in t ∈ R N + with values in (0, 1) N satisfying Condition A. A real-valued multifractional Brownian sheet {B ), t ∈ R N + } with functional Hurst index H(t)is defined as the following moving average Wiener integral:","content":"Definition 1 . 1 .\n11Let H(t) = (H 1 (t), . . . , H N (t)) be a function in t ∈ R N + with values in (0, 1) N satisfying Condition A. A real-valued multifractional Brownian sheet {B ), t ∈ R N + } with functional Hurst index H(t)is defined as the following moving average Wiener integral:"},{"id":"","caption":"Let 0 < h < 1 be a constant. Then for any δ > 2h, M > 0 and β > 0, there exists a positive and finite constant c 2,14 , depending only on δ, ε, β and M , such that for all 0 < a ≤ M , |s − r| 2h ] −β ds ≤ c 2,14 (a −(Let α, β and η be positive constants. For A > 0 and B > 0, let","content":"Lemma 2. 8 ..\n8Let 0 < h < 1 be a constant. Then for any δ > 2h, M > 0 and β > 0, there exists a positive and finite constant c 2,14 , depending only on δ, ε, β and M , such that for all 0 < a ≤ M , |s − r| 2h ] −β ds ≤ c 2,14 (a −(Let α, β and η be positive constants. For A > 0 and B > 0, let"},{"id":"","caption":"[(L(x, I) − L(y, I)) n ] = (2π) −nd I n R nd n j=1 e −i u j ,x − e −i u j ,y j , X(t j ) du dt,","content":"E\n[(L(x, I) − L(y, I)) n ] = (2π) −nd I n R nd n j=1 e −i u j ,x − e −i u j ,y j , X(t j ) du dt,"},{"id":"","caption":"Let I ∈ A and H be the vector defined in (3.1). If d < {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.","content":"Theorem 3. 1 .mfBs\n1Let I ∈ A and H be the vector defined in (3.1). If d < {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely."},{"id":"","caption":"Let I ∈ A be fixed. If N ℓ=1 1H ℓ (t) > d for all t ∈ I, then mfBs {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.Proof. Since the functions H 1 (t), . . . , H N (t) are uniformly continuous on I, we can divide I into subintervals {I p } such that for each I p , we haveN ℓ=1 1 H ℓ (p) > d, where H ℓ (p) = max t∈Ip H ℓ (t). It follows from Theorem 3.1 that on every I p , mfBs {B H(t) (t)} has an L 2 -integrable local time L(·, I p ) almost surely. This implies that {B H(t) (t)} has an L 2 -integrable local time on I, which concludes the proof of Corollary 3.2.","content":"Corollary 3. 2 .\n2Let I ∈ A be fixed. If N ℓ=1 1H ℓ (t) > d for all t ∈ I, then mfBs {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.Proof. Since the functions H 1 (t), . . . , H N (t) are uniformly continuous on I, we can divide I into subintervals {I p } such that for each I p , we haveN ℓ=1 1 H ℓ (p) > d, where H ℓ (p) = max t∈Ip H ℓ (t). It follows from Theorem 3.1 that on every I p , mfBs {B H(t) (t)} has an L 2 -integrable local time L(·, I p ) almost surely. This implies that {B H(t) (t)} has an L 2 -integrable local time on I, which concludes the proof of Corollary 3.2."},{"id":"","caption":"The condition in Corollary 3.2 is almost the best possible, in the sense that, if d > N ℓ=1 1 H ℓ (t) for some t ∈ I, then it can be proven using Geman and Horowitz [20] Theorem 21.9 that {B H(t) (t)} has no L 2 (R d × Ω)-integrable local times on I. In the case when d ≤ N ℓ=1 1","content":"Remark 3. 3 .\n3The condition in Corollary 3.2 is almost the best possible, in the sense that, if d > N ℓ=1 1 H ℓ (t) for some t ∈ I, then it can be proven using Geman and Horowitz [20] Theorem 21.9 that {B H(t) (t)} has no L 2 (R d × Ω)-integrable local times on I. In the case when d ≤ N ℓ=1 1"},{"id":"","caption":"Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying There then exists a positive constant c 3,2 , depending only on N , d, H and I, such that for all x ∈ R d , all subintervals T = [a, a + h ] ⊆ I with h > 0 small and all integers n ≥ 1,","content":". 5 .\n5Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying There then exists a positive constant c 3,2 , depending only on N , d, H and I, such that for all x ∈ R d , all subintervals T = [a, a + h ] ⊆ I with h > 0 small and all integers n ≥ 1,"},{"id":"","caption":"Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying (3.5). There then exists a positive and finite constant c n , depending on N, d, H, I and n, such that for all subintervals T = [a, a + h ] ⊆ I with h > 0 small, x, y ∈ R d with |x − y| ≤ 1, all even integers n ≥ 1 and all γ ∈ (0, 1 ∧ ατ 2τ ), E[(L(x, T ) − L(y, T )) n ] ≤ c n |x − y| nγ h n(βτ −Hτ γ) , (3.18)","content":"Lemma 3. 7 .\n7Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying (3.5). There then exists a positive and finite constant c n , depending on N, d, H, I and n, such that for all subintervals T = [a, a + h ] ⊆ I with h > 0 small, x, y ∈ R d with |x − y| ≤ 1, all even integers n ≥ 1 and all γ ∈ (0, 1 ∧ ατ 2τ ), E[(L(x, T ) − L(y, T )) n ] ≤ c n |x − y| nγ h n(βτ −Hτ γ) , (3.18)"},{"id":"","caption":"Let I ∈ A be fixed. If N ℓ=1 1 H ℓ (t) > d for all t ∈ I,then mfBs {B H(t) (t)} has a jointly continuous local time on I almost surely.","content":"Corollary 3. 8 .\n8Let I ∈ A be fixed. If N ℓ=1 1 H ℓ (t) > d for all t ∈ I,then mfBs {B H(t) (t)} has a jointly continuous local time on I almost surely."},{"id":"","caption":"Let t 0 ∈ [ε, Λ] N be fixed. If","content":"Corollary 3 . 9 .\n39Let t 0 ∈ [ε, Λ] N be fixed. If"},{"id":"","caption":"> d, then mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ) for some r 0 > 0 almost surely, where U (t 0 , r 0 ) is the open ball centered at t 0 with radius r 0 .","content":"H ℓ (t 0 )\n0> d, then mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ) for some r 0 > 0 almost surely, where U (t 0 , r 0 ) is the open ball centered at t 0 with radius r 0 ."},{"id":"","caption":"we can prove a similar result as in Lemma 2.2 (cf. Boufoussi, Dozzi and Guerbaz[13], Wu[38]). Hence, the existence of local times of the (N, d)-multifractional Brownian sheet { B H(t) is (even one-sided) sectorial locally non-deterministic (since the Fourier analytic technique employed in Wu and Xiao[39] for proving that fractional Brownian sheets are sectorial local non-deterministic fails when H(t) varies in t).It is an open problem to establish the joint continuity of the local times of { B H(t) (t)}.","content":"we can prove a similar result as in Lemma 2.2 (cf. Boufoussi, Dozzi and Guerbaz[13], Wu[38]). Hence, the existence of local times of the (N, d)-multifractional Brownian sheet { B H(t) is (even one-sided) sectorial locally non-deterministic (since the Fourier analytic technique employed in Wu and Xiao[39] for proving that fractional Brownian sheets are sectorial local non-deterministic fails when H(t) varies in t).It is an open problem to establish the joint continuity of the local times of { B H(t) (t)}."},{"id":"","caption":"Under the conditions of Lemma 3.5, there exists a positive and finite constant c 4,1 , depending only on N, d, H and I, such that for all a ∈ I and hypercubes T = [a, a + r ] ⊆ I with r < δ, where δ > 0 is the same as in Lemma 2.2, x ∈ R d and all integers n ≥ 1,","content":"Lemma 4 . 1 .\n41Under the conditions of Lemma 3.5, there exists a positive and finite constant c 4,1 , depending only on N, d, H and I, such that for all a ∈ I and hypercubes T = [a, a + r ] ⊆ I with r < δ, where δ > 0 is the same as in Lemma 2.2, x ∈ R d and all integers n ≥ 1,"},{"id":"","caption":"Under the conditions of Lemma 3.5, there exist positive constants c 4,2 , b (depending only on","content":"Lemma 4. 2 .\n2Under the conditions of Lemma 3.5, there exist positive constants c 4,2 , b (depending only on"},{"id":"","caption":"Let I ∈ A be a fixed interval and assume that d < For any fixed x ∈ R d , let L(x, ·) be the local time of {B H(t) (t)} at x which is a random measure supported on the level set. There then exists a positive and finite constant c 4,4 , independent of x, such that with probability 1, the following holds for L(x, ·)-almost all t ∈ I: lim sup r→0 L(x, U (t, r)) ϕ t (r) ≤ c 4,4 , (4.7)","content":"Theorem 4 . 3 ..\n43Let I ∈ A be a fixed interval and assume that d < For any fixed x ∈ R d , let L(x, ·) be the local time of {B H(t) (t)} at x which is a random measure supported on the level set. There then exists a positive and finite constant c 4,4 , independent of x, such that with probability 1, the following holds for L(x, ·)-almost all t ∈ I: lim sup r→0 L(x, U (t, r)) ϕ t (r) ≤ c 4,4 , (4.7)"},{"id":"","caption":", B H(s j ) (s j ) du ds, where u = (u 1 , . . . , u n+1 ) ∈ R (n+1)d and s = (t, s 1 , . . . , s n ). Similarly to the proof of (3.6), for sufficiently large m, we have that the right-hand side of equation (4.10) is at most n!) N −β τ (U ) 2 −mnβ τ (U ) , where c 4,4 is a positive finite constant depending only on N, d, H, and I.","content":", B H(s j ) (s j ) du ds, where u = (u 1 , . . . , u n+1 ) ∈ R (n+1)d and s = (t, s 1 , . . . , s n ). Similarly to the proof of (3.6), for sufficiently large m, we have that the right-hand side of equation (4.10) is at most n!) N −β τ (U ) 2 −mnβ τ (U ) , where c 4,4 is a positive finite constant depending only on N, d, H, and I."},{"id":"","caption":", we can see that for m sufficiently large, there exists a constant c 4,5 > 0 such that ϕ U (2 −m ) ≤ c 4,5 ϕ t (2 −m ). Therefore, we have lim sup m→∞ L(x, U (t, 2 −m )) ϕ t (2 −m ) ≤ c 4,6 .","content":", we can see that for m sufficiently large, there exists a constant c 4,5 > 0 such that ϕ U (2 −m ) ≤ c 4,5 ϕ t (2 −m ). Therefore, we have lim sup m→∞ L(x, U (t, 2 −m )) ϕ t (2 −m ) ≤ c 4,6 ."},{"id":"","caption":"4.3, and noting that L(·, t) vanishes outside some compact set U (depending on ω), we have the following corollary.","content":"4.3, and noting that L(·, t) vanishes outside some compact set U (depending on ω), we have the following corollary."},{"id":"","caption":"For every x ∈ R d , the pointwise Hölder exponent α L of L(x, t) at t satisfiesα L (t) ≥ β τ (t)a.s.(4.16) 5. Hausdorff dimensions of the level sets For x ∈ R d , let Γ x = {t ∈ (0, ∞) N : B H(t) (t) = x} be the level set of the multifractional Brownian sheet {B H(t) (t)}. In this section, we determine the Hausdorff dimension of Γ x .","content":"Corollary 4. 4 .\n4For every x ∈ R d , the pointwise Hölder exponent α L of L(x, t) at t satisfiesα L (t) ≥ β τ (t)a.s.(4.16) 5. Hausdorff dimensions of the level sets For x ∈ R d , let Γ x = {t ∈ (0, ∞) N : B H(t) (t) = x} be the level set of the multifractional Brownian sheet {B H(t) (t)}. In this section, we determine the Hausdorff dimension of Γ x ."},{"id":"","caption":"Denote by I 2d the identity matrix of order 2d, by Cov(B H(s) (s), B H(t) (t)) the covariance matrix of (B H(s) (s), B H(t) (t)), let Σ = n −1 I 2d + Cov(B H(s) (s), B H(t) (t)) and denote by (ξ, η) ′ the transpose of the row vector (ξ, η). Lemma 2.4 and the regularity of the H(·), it can be proven that fors, t ∈ I ς , ℓ − t ℓ | 2H ℓ (t) ≥ c 5,6 N ℓ=1 |s ℓ − t ℓ | 2H ℓ . ℓ − t ℓ | 2H ℓ ] d/2 ds dt := c 5,5 < ∞. |s ℓ − t ℓ |) γ ( N ℓ=1 |s ℓ − t ℓ | 2H ℓ )","content":"Denote by I 2d the identity matrix of order 2d, by Cov(B H(s) (s), B H(t) (t)) the covariance matrix of (B H(s) (s), B H(t) (t)), let Σ = n −1 I 2d + Cov(B H(s) (s), B H(t) (t)) and denote by (ξ, η) ′ the transpose of the row vector (ξ, η). Lemma 2.4 and the regularity of the H(·), it can be proven that fors, t ∈ I ς , ℓ − t ℓ | 2H ℓ (t) ≥ c 5,6 N ℓ=1 |s ℓ − t ℓ | 2H ℓ . ℓ − t ℓ | 2H ℓ ] d/2 ds dt := c 5,5 < ∞. |s ℓ − t ℓ |) γ ( N ℓ=1 |s ℓ − t ℓ | 2H ℓ )"},{"id":"","caption":"Let {B H(t) (t)} = {B H(t) (t), t ∈ R N + } be an (N, d)-multifractional Brownian sheet with Hurst functionals H ℓ (t) (ℓ = 1, . . . , N ). If t 0 ∈ (0, ∞) N satisfies N ℓ=1 1 H ℓ (t 0 ) > d, then there exists r 0 > 0 such that P lim r→0 dim H (Γ B H(t) (t) ∩ U (t 0 , r)) = β τ (t 0 )for a.e. t ∈ U (t 0 , r 0 ) = 1,(5.19)","content":"Corollary 5. 3 .\n3Let {B H(t) (t)} = {B H(t) (t), t ∈ R N + } be an (N, d)-multifractional Brownian sheet with Hurst functionals H ℓ (t) (ℓ = 1, . . . , N ). If t 0 ∈ (0, ∞) N satisfies N ℓ=1 1 H ℓ (t 0 ) > d, then there exists r 0 > 0 such that P lim r→0 dim H (Γ B H(t) (t) ∩ U (t 0 , r)) = β τ (t 0 )for a.e. t ∈ U (t 0 , r 0 ) = 1,(5.19)"},{"id":"","caption":"ℓ (t 0 ) > d, there exists a positive number r 0 such that for all s ∈ U (t 0 , r 0 ), we haveN ℓ=1 1H ℓ (s) > d. By Corollary 3.9, mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ). By (5.2), we have that for every 0 < r < r 0 and x ∈ R d ,dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) 5.20)and Fubini's theorem, we see thatP dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) β τ (s) , a.e. x ∈ R d = 1. (5.21)Since {B H(t) (t)} has a local time on U (t 0 , r 0 ), the occupation density formula (2.41) and (5.21) together imply thatP dim H (Γ B H(t) (t) ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r)β τ (s) a.e. t ∈ U (t 0 , r 0 ) = 1.(5.22)","content":"H\nℓ (t 0 ) > d, there exists a positive number r 0 such that for all s ∈ U (t 0 , r 0 ), we haveN ℓ=1 1H ℓ (s) > d. By Corollary 3.9, mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ). By (5.2), we have that for every 0 < r < r 0 and x ∈ R d ,dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) 5.20)and Fubini's theorem, we see thatP dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) β τ (s) , a.e. x ∈ R d = 1. (5.21)Since {B H(t) (t)} has a local time on U (t 0 , r 0 ), the occupation density formula (2.41) and (5.21) together imply thatP dim H (Γ B H(t) (t) ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r)β τ (s) a.e. t ∈ U (t 0 , r 0 ) = 1.(5.22)"},{"id":"","caption":"28) where det Cov(Z 1 , . . . , Z n ) denotes the determinant of the covariance matrix of the Gaussian random vector (Z 1 , . . . , Z n ).Remark 2.7. By (2.27) and Lemma 2.4, we derive that {B","content":"28) where det Cov(Z 1 , . . . , Z n ) denotes the determinant of the covariance matrix of the Gaussian random vector (Z 1 , . . . , Z n ).Remark 2.7. By (2.27) and Lemma 2.4, we derive that {B"},{"id":"","caption":"a multifractional Brownian sheet with values in R d . Let I ∈ A and let H be the vector defined in (3.1). If d <","content":"a multifractional Brownian sheet with values in R d . Let I ∈ A and let H be the vector defined in (3.1). If d <N \nℓ=1 \n1 \nH ℓ \n\n, \n\nthen {B H(t) (t)} has a jointly continuous local time on I."},{"id":"","caption":"The proof of Theorem 3.4 is similar to the proofs of Xiao and Zhang[45] Theorem 4.1 and Ayache, Wu and Xiao [5] Theorem 3.1; we include it here for completeness.","content":"The proof of Theorem 3.4 is similar to the proofs of Xiao and Zhang[45] Theorem 4.1 and Ayache, Wu and Xiao [5] Theorem 3.1; we include it here for completeness."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Let H(t) = (H 1 (t), . . . , H N (t)) be a function in t ∈ R N + with values in (0, 1) N satisfying Condition A. A real-valued multifractional Brownian sheet {B ), t ∈ R N + } with functional Hurst index H(t)is defined as the following moving average Wiener integral:\",\n \"content\": \"Definition 1 . 1 .\\n11Let H(t) = (H 1 (t), . . . , H N (t)) be a function in t ∈ R N + with values in (0, 1) N satisfying Condition A. A real-valued multifractional Brownian sheet {B ), t ∈ R N + } with functional Hurst index H(t)is defined as the following moving average Wiener integral:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let 0 < h < 1 be a constant. Then for any δ > 2h, M > 0 and β > 0, there exists a positive and finite constant c 2,14 , depending only on δ, ε, β and M , such that for all 0 < a ≤ M , |s − r| 2h ] −β ds ≤ c 2,14 (a −(Let α, β and η be positive constants. For A > 0 and B > 0, let\",\n \"content\": \"Lemma 2. 8 ..\\n8Let 0 < h < 1 be a constant. Then for any δ > 2h, M > 0 and β > 0, there exists a positive and finite constant c 2,14 , depending only on δ, ε, β and M , such that for all 0 < a ≤ M , |s − r| 2h ] −β ds ≤ c 2,14 (a −(Let α, β and η be positive constants. For A > 0 and B > 0, let\"\n },\n {\n \"id\": \"\",\n \"caption\": \"[(L(x, I) − L(y, I)) n ] = (2π) −nd I n R nd n j=1 e −i u j ,x − e −i u j ,y j , X(t j ) du dt,\",\n \"content\": \"E\\n[(L(x, I) − L(y, I)) n ] = (2π) −nd I n R nd n j=1 e −i u j ,x − e −i u j ,y j , X(t j ) du dt,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let I ∈ A and H be the vector defined in (3.1). If d < {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.\",\n \"content\": \"Theorem 3. 1 .mfBs\\n1Let I ∈ A and H be the vector defined in (3.1). If d < {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let I ∈ A be fixed. If N ℓ=1 1H ℓ (t) > d for all t ∈ I, then mfBs {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.Proof. Since the functions H 1 (t), . . . , H N (t) are uniformly continuous on I, we can divide I into subintervals {I p } such that for each I p , we haveN ℓ=1 1 H ℓ (p) > d, where H ℓ (p) = max t∈Ip H ℓ (t). It follows from Theorem 3.1 that on every I p , mfBs {B H(t) (t)} has an L 2 -integrable local time L(·, I p ) almost surely. This implies that {B H(t) (t)} has an L 2 -integrable local time on I, which concludes the proof of Corollary 3.2.\",\n \"content\": \"Corollary 3. 2 .\\n2Let I ∈ A be fixed. If N ℓ=1 1H ℓ (t) > d for all t ∈ I, then mfBs {B H(t) (t)} admits an L 2 -integrable local time L(·, I) almost surely.Proof. Since the functions H 1 (t), . . . , H N (t) are uniformly continuous on I, we can divide I into subintervals {I p } such that for each I p , we haveN ℓ=1 1 H ℓ (p) > d, where H ℓ (p) = max t∈Ip H ℓ (t). It follows from Theorem 3.1 that on every I p , mfBs {B H(t) (t)} has an L 2 -integrable local time L(·, I p ) almost surely. This implies that {B H(t) (t)} has an L 2 -integrable local time on I, which concludes the proof of Corollary 3.2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The condition in Corollary 3.2 is almost the best possible, in the sense that, if d > N ℓ=1 1 H ℓ (t) for some t ∈ I, then it can be proven using Geman and Horowitz [20] Theorem 21.9 that {B H(t) (t)} has no L 2 (R d × Ω)-integrable local times on I. In the case when d ≤ N ℓ=1 1\",\n \"content\": \"Remark 3. 3 .\\n3The condition in Corollary 3.2 is almost the best possible, in the sense that, if d > N ℓ=1 1 H ℓ (t) for some t ∈ I, then it can be proven using Geman and Horowitz [20] Theorem 21.9 that {B H(t) (t)} has no L 2 (R d × Ω)-integrable local times on I. In the case when d ≤ N ℓ=1 1\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying There then exists a positive constant c 3,2 , depending only on N , d, H and I, such that for all x ∈ R d , all subintervals T = [a, a + h ] ⊆ I with h > 0 small and all integers n ≥ 1,\",\n \"content\": \". 5 .\\n5Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying There then exists a positive constant c 3,2 , depending only on N , d, H and I, such that for all x ∈ R d , all subintervals T = [a, a + h ] ⊆ I with h > 0 small and all integers n ≥ 1,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying (3.5). There then exists a positive and finite constant c n , depending on N, d, H, I and n, such that for all subintervals T = [a, a + h ] ⊆ I with h > 0 small, x, y ∈ R d with |x − y| ≤ 1, all even integers n ≥ 1 and all γ ∈ (0, 1 ∧ ατ 2τ ), E[(L(x, T ) − L(y, T )) n ] ≤ c n |x − y| nγ h n(βτ −Hτ γ) , (3.18)\",\n \"content\": \"Lemma 3. 7 .\\n7Assume the conditions of Theorem 3.4 hold. Let τ be the unique integer in {1, . . . , N } satisfying (3.5). There then exists a positive and finite constant c n , depending on N, d, H, I and n, such that for all subintervals T = [a, a + h ] ⊆ I with h > 0 small, x, y ∈ R d with |x − y| ≤ 1, all even integers n ≥ 1 and all γ ∈ (0, 1 ∧ ατ 2τ ), E[(L(x, T ) − L(y, T )) n ] ≤ c n |x − y| nγ h n(βτ −Hτ γ) , (3.18)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let I ∈ A be fixed. If N ℓ=1 1 H ℓ (t) > d for all t ∈ I,then mfBs {B H(t) (t)} has a jointly continuous local time on I almost surely.\",\n \"content\": \"Corollary 3. 8 .\\n8Let I ∈ A be fixed. If N ℓ=1 1 H ℓ (t) > d for all t ∈ I,then mfBs {B H(t) (t)} has a jointly continuous local time on I almost surely.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let t 0 ∈ [ε, Λ] N be fixed. If\",\n \"content\": \"Corollary 3 . 9 .\\n39Let t 0 ∈ [ε, Λ] N be fixed. If\"\n },\n {\n \"id\": \"\",\n \"caption\": \"> d, then mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ) for some r 0 > 0 almost surely, where U (t 0 , r 0 ) is the open ball centered at t 0 with radius r 0 .\",\n \"content\": \"H ℓ (t 0 )\\n0> d, then mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ) for some r 0 > 0 almost surely, where U (t 0 , r 0 ) is the open ball centered at t 0 with radius r 0 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"we can prove a similar result as in Lemma 2.2 (cf. Boufoussi, Dozzi and Guerbaz[13], Wu[38]). Hence, the existence of local times of the (N, d)-multifractional Brownian sheet { B H(t) is (even one-sided) sectorial locally non-deterministic (since the Fourier analytic technique employed in Wu and Xiao[39] for proving that fractional Brownian sheets are sectorial local non-deterministic fails when H(t) varies in t).It is an open problem to establish the joint continuity of the local times of { B H(t) (t)}.\",\n \"content\": \"we can prove a similar result as in Lemma 2.2 (cf. Boufoussi, Dozzi and Guerbaz[13], Wu[38]). Hence, the existence of local times of the (N, d)-multifractional Brownian sheet { B H(t) is (even one-sided) sectorial locally non-deterministic (since the Fourier analytic technique employed in Wu and Xiao[39] for proving that fractional Brownian sheets are sectorial local non-deterministic fails when H(t) varies in t).It is an open problem to establish the joint continuity of the local times of { B H(t) (t)}.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Under the conditions of Lemma 3.5, there exists a positive and finite constant c 4,1 , depending only on N, d, H and I, such that for all a ∈ I and hypercubes T = [a, a + r ] ⊆ I with r < δ, where δ > 0 is the same as in Lemma 2.2, x ∈ R d and all integers n ≥ 1,\",\n \"content\": \"Lemma 4 . 1 .\\n41Under the conditions of Lemma 3.5, there exists a positive and finite constant c 4,1 , depending only on N, d, H and I, such that for all a ∈ I and hypercubes T = [a, a + r ] ⊆ I with r < δ, where δ > 0 is the same as in Lemma 2.2, x ∈ R d and all integers n ≥ 1,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Under the conditions of Lemma 3.5, there exist positive constants c 4,2 , b (depending only on\",\n \"content\": \"Lemma 4. 2 .\\n2Under the conditions of Lemma 3.5, there exist positive constants c 4,2 , b (depending only on\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let I ∈ A be a fixed interval and assume that d < For any fixed x ∈ R d , let L(x, ·) be the local time of {B H(t) (t)} at x which is a random measure supported on the level set. There then exists a positive and finite constant c 4,4 , independent of x, such that with probability 1, the following holds for L(x, ·)-almost all t ∈ I: lim sup r→0 L(x, U (t, r)) ϕ t (r) ≤ c 4,4 , (4.7)\",\n \"content\": \"Theorem 4 . 3 ..\\n43Let I ∈ A be a fixed interval and assume that d < For any fixed x ∈ R d , let L(x, ·) be the local time of {B H(t) (t)} at x which is a random measure supported on the level set. There then exists a positive and finite constant c 4,4 , independent of x, such that with probability 1, the following holds for L(x, ·)-almost all t ∈ I: lim sup r→0 L(x, U (t, r)) ϕ t (r) ≤ c 4,4 , (4.7)\"\n },\n {\n \"id\": \"\",\n \"caption\": \", B H(s j ) (s j ) du ds, where u = (u 1 , . . . , u n+1 ) ∈ R (n+1)d and s = (t, s 1 , . . . , s n ). Similarly to the proof of (3.6), for sufficiently large m, we have that the right-hand side of equation (4.10) is at most n!) N −β τ (U ) 2 −mnβ τ (U ) , where c 4,4 is a positive finite constant depending only on N, d, H, and I.\",\n \"content\": \", B H(s j ) (s j ) du ds, where u = (u 1 , . . . , u n+1 ) ∈ R (n+1)d and s = (t, s 1 , . . . , s n ). Similarly to the proof of (3.6), for sufficiently large m, we have that the right-hand side of equation (4.10) is at most n!) N −β τ (U ) 2 −mnβ τ (U ) , where c 4,4 is a positive finite constant depending only on N, d, H, and I.\"\n },\n {\n \"id\": \"\",\n \"caption\": \", we can see that for m sufficiently large, there exists a constant c 4,5 > 0 such that ϕ U (2 −m ) ≤ c 4,5 ϕ t (2 −m ). Therefore, we have lim sup m→∞ L(x, U (t, 2 −m )) ϕ t (2 −m ) ≤ c 4,6 .\",\n \"content\": \", we can see that for m sufficiently large, there exists a constant c 4,5 > 0 such that ϕ U (2 −m ) ≤ c 4,5 ϕ t (2 −m ). Therefore, we have lim sup m→∞ L(x, U (t, 2 −m )) ϕ t (2 −m ) ≤ c 4,6 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"4.3, and noting that L(·, t) vanishes outside some compact set U (depending on ω), we have the following corollary.\",\n \"content\": \"4.3, and noting that L(·, t) vanishes outside some compact set U (depending on ω), we have the following corollary.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"For every x ∈ R d , the pointwise Hölder exponent α L of L(x, t) at t satisfiesα L (t) ≥ β τ (t)a.s.(4.16) 5. Hausdorff dimensions of the level sets For x ∈ R d , let Γ x = {t ∈ (0, ∞) N : B H(t) (t) = x} be the level set of the multifractional Brownian sheet {B H(t) (t)}. In this section, we determine the Hausdorff dimension of Γ x .\",\n \"content\": \"Corollary 4. 4 .\\n4For every x ∈ R d , the pointwise Hölder exponent α L of L(x, t) at t satisfiesα L (t) ≥ β τ (t)a.s.(4.16) 5. Hausdorff dimensions of the level sets For x ∈ R d , let Γ x = {t ∈ (0, ∞) N : B H(t) (t) = x} be the level set of the multifractional Brownian sheet {B H(t) (t)}. In this section, we determine the Hausdorff dimension of Γ x .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Denote by I 2d the identity matrix of order 2d, by Cov(B H(s) (s), B H(t) (t)) the covariance matrix of (B H(s) (s), B H(t) (t)), let Σ = n −1 I 2d + Cov(B H(s) (s), B H(t) (t)) and denote by (ξ, η) ′ the transpose of the row vector (ξ, η). Lemma 2.4 and the regularity of the H(·), it can be proven that fors, t ∈ I ς , ℓ − t ℓ | 2H ℓ (t) ≥ c 5,6 N ℓ=1 |s ℓ − t ℓ | 2H ℓ . ℓ − t ℓ | 2H ℓ ] d/2 ds dt := c 5,5 < ∞. |s ℓ − t ℓ |) γ ( N ℓ=1 |s ℓ − t ℓ | 2H ℓ )\",\n \"content\": \"Denote by I 2d the identity matrix of order 2d, by Cov(B H(s) (s), B H(t) (t)) the covariance matrix of (B H(s) (s), B H(t) (t)), let Σ = n −1 I 2d + Cov(B H(s) (s), B H(t) (t)) and denote by (ξ, η) ′ the transpose of the row vector (ξ, η). Lemma 2.4 and the regularity of the H(·), it can be proven that fors, t ∈ I ς , ℓ − t ℓ | 2H ℓ (t) ≥ c 5,6 N ℓ=1 |s ℓ − t ℓ | 2H ℓ . ℓ − t ℓ | 2H ℓ ] d/2 ds dt := c 5,5 < ∞. |s ℓ − t ℓ |) γ ( N ℓ=1 |s ℓ − t ℓ | 2H ℓ )\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let {B H(t) (t)} = {B H(t) (t), t ∈ R N + } be an (N, d)-multifractional Brownian sheet with Hurst functionals H ℓ (t) (ℓ = 1, . . . , N ). If t 0 ∈ (0, ∞) N satisfies N ℓ=1 1 H ℓ (t 0 ) > d, then there exists r 0 > 0 such that P lim r→0 dim H (Γ B H(t) (t) ∩ U (t 0 , r)) = β τ (t 0 )for a.e. t ∈ U (t 0 , r 0 ) = 1,(5.19)\",\n \"content\": \"Corollary 5. 3 .\\n3Let {B H(t) (t)} = {B H(t) (t), t ∈ R N + } be an (N, d)-multifractional Brownian sheet with Hurst functionals H ℓ (t) (ℓ = 1, . . . , N ). If t 0 ∈ (0, ∞) N satisfies N ℓ=1 1 H ℓ (t 0 ) > d, then there exists r 0 > 0 such that P lim r→0 dim H (Γ B H(t) (t) ∩ U (t 0 , r)) = β τ (t 0 )for a.e. t ∈ U (t 0 , r 0 ) = 1,(5.19)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"ℓ (t 0 ) > d, there exists a positive number r 0 such that for all s ∈ U (t 0 , r 0 ), we haveN ℓ=1 1H ℓ (s) > d. By Corollary 3.9, mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ). By (5.2), we have that for every 0 < r < r 0 and x ∈ R d ,dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) 5.20)and Fubini's theorem, we see thatP dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) β τ (s) , a.e. x ∈ R d = 1. (5.21)Since {B H(t) (t)} has a local time on U (t 0 , r 0 ), the occupation density formula (2.41) and (5.21) together imply thatP dim H (Γ B H(t) (t) ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r)β τ (s) a.e. t ∈ U (t 0 , r 0 ) = 1.(5.22)\",\n \"content\": \"H\\nℓ (t 0 ) > d, there exists a positive number r 0 such that for all s ∈ U (t 0 , r 0 ), we haveN ℓ=1 1H ℓ (s) > d. By Corollary 3.9, mfBs {B H(t) (t)} has a jointly continuous local time on U (t 0 , r 0 ). By (5.2), we have that for every 0 < r < r 0 and x ∈ R d ,dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) 5.20)and Fubini's theorem, we see thatP dim H (Γ x ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r) β τ (s) , a.e. x ∈ R d = 1. (5.21)Since {B H(t) (t)} has a local time on U (t 0 , r 0 ), the occupation density formula (2.41) and (5.21) together imply thatP dim H (Γ B H(t) (t) ∩ U (t 0 , r)) ≤ max s∈U(t 0 ,r)β τ (s) a.e. t ∈ U (t 0 , r 0 ) = 1.(5.22)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"28) where det Cov(Z 1 , . . . , Z n ) denotes the determinant of the covariance matrix of the Gaussian random vector (Z 1 , . . . , Z n ).Remark 2.7. By (2.27) and Lemma 2.4, we derive that {B\",\n \"content\": \"28) where det Cov(Z 1 , . . . , Z n ) denotes the determinant of the covariance matrix of the Gaussian random vector (Z 1 , . . . , Z n ).Remark 2.7. By (2.27) and Lemma 2.4, we derive that {B\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a multifractional Brownian sheet with values in R d . Let I ∈ A and let H be the vector defined in (3.1). If d <\",\n \"content\": \"a multifractional Brownian sheet with values in R d . Let I ∈ A and let H be the vector defined in (3.1). If d <N \\nℓ=1 \\n1 \\nH ℓ \\n\\n, \\n\\nthen {B H(t) (t)} has a jointly continuous local time on I.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The proof of Theorem 3.4 is similar to the proofs of Xiao and Zhang[45] Theorem 4.1 and Ayache, Wu and Xiao [5] Theorem 3.1; we include it here for completeness.\",\n \"content\": \"The proof of Theorem 3.4 is similar to the proofs of Xiao and Zhang[45] Theorem 4.1 and Ayache, Wu and Xiao [5] Theorem 3.1; we include it here for completeness.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The Geometry of Random Fields. R J Adler, Wiley611857New YorkAdler, R.J. (1981). The Geometry of Random Fields. New York: Wiley. MR0611857","doi":""},{"text":"The covariance structure of multifractional Brownian motion, with application to long range dependence. Proceeding of ICASSP. A Ayache, S Cohen, J Lévy-Véhel, IstambulAyache, A., Cohen, S. and Lévy-Véhel, J. (2002). The covariance structure of multifractional Brownian motion, with application to long range dependence. Proceeding of ICASSP. Istambul.","doi":""},{"text":"Fractional and multifractional Brownian sheet. A Ayache, S Léger, PreprintAyache, A. and Léger, S. (2000). Fractional and multifractional Brownian sheet. Preprint.","doi":""},{"text":"A Ayache, S Léger, M Pontier, Drap Brownien fractionnaire. Potential Theory. 171906407Ayache, A., Léger, S. and Pontier, M. (2002). Drap Brownien fractionnaire. Potential The- ory 17 31-43. MR1906407","doi":""},{"text":"Joint continuity of the local times of fractional Brownian sheets. A Ayache, D Wu, Y Xiao, Ann. Inst. H. Poincaré Probab. Statist. To appearAyache, A., Wu, D. and Xiao, Y. (2008). Joint continuity of the local times of fractional Brownian sheets. Ann. Inst. H. Poincaré Probab. Statist. To appear.","doi":""},{"text":"Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. A Ayache, Y Xiao, J. Fourier Anal. Appl. 11Ayache, A. and Xiao, Y. (2005). Asymptotic properties and Hausdorff dimensions of frac- tional Brownian sheets. J. Fourier Anal. Appl. 11 407-439. MR2169474","doi":""},{"text":"Elliptic Gaussian random processes. A Benassi, S Jaffard, D Roux, Rev. Mat. Iberoamericana. 131462329Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 19-90. MR1462329","doi":""},{"text":"Aquifer Operator-Scaling and the effect on solute mixing and dispersion. D Benson, M M M Meerschaert, B Bäumer, H P Scheffler, Water Resour. Res. 42Benson, D. Meerschaert, M.M.M., Bäumer, B. and Scheffler, H.P. (2006). Aquifer Operator- Scaling and the effect on solute mixing and dispersion Water Resour. Res. 42 1-18.","doi":""},{"text":"Gaussian processes with stationary increments: Local times and sample function properties. S M Berman, Ann. Math. Statist. 41272035Berman, S.M. (1970). Gaussian processes with stationary increments: Local times and sample function properties. Ann. Math. Statist. 41 1260-1272. MR0272035","doi":""},{"text":"Gaussian sample function: Uniform dimension and Hölder conditions nowhere. S M Berman, Nagoya Math. J. 46Berman, S.M. (1972). Gaussian sample function: Uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46 63-86. MR0307320","doi":""},{"text":"Local non-determinism and local times of Gaussian processes. S M Berman, MR0317397Indiana Univ. Math. J. 23Berman, S.M. (1973). Local non-determinism and local times of Gaussian processes. Indiana Univ. Math. J. 23 69-94. MR0317397","doi":"MR0317397"},{"text":"Operator scaling stable random fields. H Biermé, M M Meerschaert, H.-P Scheffler, Stochastic. Process. Appl. 1172290879Biermé, H., Meerschaert, M.M. and Scheffler, H.-P. (2007). Operator scaling stable random fields. Stochastic. Process. Appl. 117 312-332. MR2290879","doi":""},{"text":"On the local time of the multifractional Brownian motion. B Boufoussi, M Dozzi, R Guerbaz, Stochastics. 78Boufoussi, B., Dozzi, M. and Guerbaz, R. (2006). On the local time of the multifractional Brownian motion. Stochastics 78 33-49. MR2219711","doi":""},{"text":"Sample path properties of the local time of multifractional Brownian motion. B Boufoussi, M Dozzi, R Guerbaz, Bernoulli. 13Boufoussi, B., Dozzi, M. and Guerbaz, R. (2007). Sample path properties of the local time of multifractional Brownian motion. Bernoulli 13 849-867. MR2348754","doi":""},{"text":"Joint continuity of Gaussian local times. J Cuzick, J Dupreez, Ann. Probab. 10Cuzick, J. and DuPreez, J. (1982). Joint continuity of Gaussian local times. Ann. Probab. 10 810-817. MR0659550","doi":""},{"text":"V Dobrić, F M Ojeda, Fractional Brownian fields, duality, and martingales. IMS Lecture Notes: High Dimensional Probability. Beachwood, OHInstitute of Mathematical Statistics51Dobrić, V. and Ojeda, F.M. (2006). Fractional Brownian fields, duality, and martingales. IMS Lecture Notes: High Dimensional Probability 51 77-95. Beachwood, OH: Institute of Mathematical Statistics.","doi":""},{"text":"Occupation density and sample path properties of N -parameter processes. M Dozzi, Topics in Spatial Stochastic Processes. Martina Franca; BerlinSpringer18021975519Dozzi, M. (2002). Occupation density and sample path properties of N -parameter processes. In Topics in Spatial Stochastic Processes (Martina Franca, 2001) 127-166. Lecture Notes in Math. 1802. Berlin: Springer. MR1975519","doi":""},{"text":"Estimates for the small ball probabilities of the fractional Brownian sheet. T Dunker, J. Theoret. Probab. 131777539Dunker, T. (2000). Estimates for the small ball probabilities of the fractional Brownian sheet. J. Theoret. Probab. 13 357-382. MR1777539","doi":""},{"text":"Sample function properties of multi-parameter stable processes. W Ehm, Z. Wahrsch. Verw. Gebiete. 56618272Ehm, W. (1981). Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw. Gebiete 56 195-228. MR0618272","doi":""},{"text":"Occupation densities. D Geman, J Horowitz, Ann. Probab. 8Geman, D. and Horowitz, J. (1980). Occupation densities. Ann. Probab. 8 1-67. MR0556414","doi":""},{"text":"Inequalities. G H Hardy, Cambridge Univ. PressCambridgeHardy, G.H. (1934). Inequalities. Cambridge: Cambridge Univ. Press.","doi":""},{"text":"From N parameter fractional Brownian motions to N parameter multifractional Brownian motions. Rocky Mount. E Herbin, J. Math. 362274895Herbin, E. (2006). From N parameter fractional Brownian motions to N parameter multi- fractional Brownian motions. Rocky Mount. J. Math. 36 1249-1284. MR2274895","doi":""},{"text":"Some Random Series of Functions. J.-P Kahane, Cambridge Univ. Press8330732nd ed. CambridgeKahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge: Cambridge Univ. Press. MR0833073","doi":""},{"text":"On the fractional anisotropic Wiener field. A Kamont, Probab. Math. Statist. 161407935Kamont, A. (1996). On the fractional anisotropic Wiener field. Probab. Math. Statist. 16 85-98. MR1407935","doi":""},{"text":"Multiparameter Processes: An Introduction to Random Fields. D Khoshnevisan, Springer1914748New YorkKhoshnevisan, D. (2002). Multiparameter Processes: An Introduction to Random Fields. New York: Springer. MR1914748","doi":""},{"text":"Sectorial local non-determinism and geometric properties of the Brownian sheet. D Khoshnevisan, D Wu, Y Xiao, Electronic J. Probab. 11Khoshnevisan, D., Wu, D. and Xiao, Y. (2006). Sectorial local non-determinism and geo- metric properties of the Brownian sheet, Electronic J. Probab. 11 817-843. MR2261054","doi":""},{"text":"Images of the Brownian sheet. D Khoshnevisan, Y Xiao, Trans. Amer. Math. Soc. 3592299449Khoshnevisan, D. and Xiao, Y. (2007). Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 3125-3151. MR2299449","doi":""},{"text":"Multifractional Brownian motion: definition and preliminary results. J Lévy-Véhel, R F Peltier, RR-2645Technical ReportINRIALévy-Véhel, J. and Peltier, R.F. (1996). Multifractional Brownian motion: definition and preliminary results. Technical Report RR-2645, INRIA.","doi":"RR-2645"},{"text":"Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type. S C Lim, J. Phys. A: Math. Gen. 341819933Lim, S.C. (2001). Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type. J. Phys. A: Math. Gen. 34 1301-1310. MR1819933","doi":""},{"text":"On some possible generalizations of fractional Brownian motion. S C Lim, S V Muniandy, Phys. Lett. A. 266Lim, S.C. and Muniandy, S.V. (2000). On some possible generalizations of fractional Brow- nian motion. Phys. Lett. A 266 140-145. MR1741314","doi":""},{"text":"Fractional Brownian motion fractional noises and applications. B Mandelbrot, J Van Ness, SIAM Rev. 10242239Mandelbrot, B. and Van Ness, J. (1968). Fractional Brownian motion fractional noises and applications. SIAM Rev. 10 422-437. MR0242239","doi":""},{"text":"Small deviations for some multi-parameter Gaussian processes. D M Mason, Z Shi, J. Theoret. Probab. 141822902Mason, D.M. and Shi, Z. (2001). Small deviations for some multi-parameter Gaussian pro- cesses. J. Theoret. Probab. 14 213-239. MR1822902","doi":""},{"text":"Multiparameter fractional Brownian motion and quasilinear stochastic partial differential equations. B Øksendal, T Zhang, MR1922562Stochastics Stochastics Rep. 71Øksendal, B. and Zhang, T. (2000). Multiparameter fractional Brownian motion and quasi- linear stochastic partial differential equations. Stochastics Stochastics Rep. 71 141-163. MR1922562","doi":"MR1922562"},{"text":"Local times for Gaussian vector fields. L D Pitt, 309- 330. MR0471055Indiana Univ. Math. J. 27Pitt, L.D. (1978). Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 309- 330. MR0471055","doi":"309- 330. MR0471055"},{"text":"How rich is the class of multifractional Brownian motions? Stochastic Process. S Stoev, M S Taqqu, Appl. 116Stoev, S. and Taqqu, M.S. (2006). How rich is the class of multifractional Brownian motions? Stochastic Process. Appl. 116 200-221. MR2197974","doi":""},{"text":"Hausdorff measure of trajectories of multiparameter fractional Brownian motion. M Talagrand, Ann. Probab. 231334170Talagrand, M. (1995). Hausdorff measure of trajectories of multiparameter fractional Brow- nian motion. Ann. Probab. 23 767-775. MR1334170","doi":""},{"text":"Polarité, points multiples et géométrie de certain processus gaussiens. F Testard, U.P.S. ToulousePubl. du Laboratoire de Statistique et Probabilités de lTestard, F. (1986). Polarité, points multiples et géométrie de certain processus gaussiens. Publ. du Laboratoire de Statistique et Probabilités de l'U.P.S. Toulouse 01-86.","doi":""},{"text":"Dimension results for multifractional Brownian sheets. D Wu, J. Math. Phys. 482337692Paper 073511Wu, D. (2007). Dimension results for multifractional Brownian sheets. J. Math. Phys. 48 Paper 073511. MR2337692","doi":""},{"text":"Geometric properties of fractional Brownian sheets. D Wu, Y Xiao, J. Fourier Anal. Appl. 13Wu, D. and Xiao, Y. (2007). Geometric properties of fractional Brownian sheets. J. Fourier Anal. Appl. 13 1-37. MR2296726","doi":""},{"text":"Dimension results for Gaussian vector fields and index-α stable fields. Y Xiao, Ann. Probab. 231330771Xiao, Y. (1995). Dimension results for Gaussian vector fields and index-α stable fields. Ann. Probab. 23 273-291. MR1330771","doi":""},{"text":"Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Y Xiao, Probab. Theory Related Fields. 1091469923Xiao, Y. (1997). Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields 109 129-157. MR1469923","doi":""},{"text":"The Hausdorff dimension of the level sets of stable processes in random scenery. Y Xiao, Acta. Sci. Math. (Szeged). 651702175Xiao, Y. (1999). The Hausdorff dimension of the level sets of stable processes in random scenery. Acta. Sci. Math. (Szeged) 65 373-383. MR1702175","doi":""},{"text":"Properties of local non-determinism of Gaussian and stable random fields and their applications. Y Xiao, 157-193. MR2225751Ann. Fac. Sci. Toulouse Math. Xiao, Y. (2006). Properties of local non-determinism of Gaussian and stable random fields and their applications. Ann. Fac. Sci. Toulouse Math. XV 157-193. MR2225751","doi":"157-193. MR2225751"},{"text":"Sample path properties of anisotropic Gaussian random fields. Y Xiao, PreprintXiao, Y. (2007). Sample path properties of anisotropic Gaussian random fields. Preprint.","doi":""},{"text":"Local times of fractional Brownian sheets. Y Xiao, T Zhang, Probab. Theory Related Fields. 124Xiao, Y. and Zhang, T. (2002). Local times of fractional Brownian sheets. Probab. Theory Related Fields 124 204-226. MR1936017","doi":""}],"string":"[\n {\n \"text\": \"The Geometry of Random Fields. R J Adler, Wiley611857New YorkAdler, R.J. (1981). The Geometry of Random Fields. New York: Wiley. MR0611857\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The covariance structure of multifractional Brownian motion, with application to long range dependence. Proceeding of ICASSP. A Ayache, S Cohen, J Lévy-Véhel, IstambulAyache, A., Cohen, S. and Lévy-Véhel, J. (2002). The covariance structure of multifractional Brownian motion, with application to long range dependence. Proceeding of ICASSP. Istambul.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fractional and multifractional Brownian sheet. A Ayache, S Léger, PreprintAyache, A. and Léger, S. (2000). Fractional and multifractional Brownian sheet. Preprint.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Ayache, S Léger, M Pontier, Drap Brownien fractionnaire. Potential Theory. 171906407Ayache, A., Léger, S. and Pontier, M. (2002). Drap Brownien fractionnaire. Potential The- ory 17 31-43. MR1906407\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Joint continuity of the local times of fractional Brownian sheets. A Ayache, D Wu, Y Xiao, Ann. Inst. H. Poincaré Probab. Statist. To appearAyache, A., Wu, D. and Xiao, Y. (2008). Joint continuity of the local times of fractional Brownian sheets. Ann. Inst. H. Poincaré Probab. Statist. To appear.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. A Ayache, Y Xiao, J. Fourier Anal. Appl. 11Ayache, A. and Xiao, Y. (2005). Asymptotic properties and Hausdorff dimensions of frac- tional Brownian sheets. J. Fourier Anal. Appl. 11 407-439. MR2169474\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Elliptic Gaussian random processes. A Benassi, S Jaffard, D Roux, Rev. Mat. Iberoamericana. 131462329Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13 19-90. MR1462329\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Aquifer Operator-Scaling and the effect on solute mixing and dispersion. D Benson, M M M Meerschaert, B Bäumer, H P Scheffler, Water Resour. Res. 42Benson, D. Meerschaert, M.M.M., Bäumer, B. and Scheffler, H.P. (2006). Aquifer Operator- Scaling and the effect on solute mixing and dispersion Water Resour. Res. 42 1-18.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gaussian processes with stationary increments: Local times and sample function properties. S M Berman, Ann. Math. Statist. 41272035Berman, S.M. (1970). Gaussian processes with stationary increments: Local times and sample function properties. Ann. Math. Statist. 41 1260-1272. MR0272035\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gaussian sample function: Uniform dimension and Hölder conditions nowhere. S M Berman, Nagoya Math. J. 46Berman, S.M. (1972). Gaussian sample function: Uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46 63-86. MR0307320\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Local non-determinism and local times of Gaussian processes. S M Berman, MR0317397Indiana Univ. Math. J. 23Berman, S.M. (1973). Local non-determinism and local times of Gaussian processes. Indiana Univ. Math. J. 23 69-94. MR0317397\",\n \"doi\": \"MR0317397\"\n },\n {\n \"text\": \"Operator scaling stable random fields. H Biermé, M M Meerschaert, H.-P Scheffler, Stochastic. Process. Appl. 1172290879Biermé, H., Meerschaert, M.M. and Scheffler, H.-P. (2007). Operator scaling stable random fields. Stochastic. Process. Appl. 117 312-332. MR2290879\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the local time of the multifractional Brownian motion. B Boufoussi, M Dozzi, R Guerbaz, Stochastics. 78Boufoussi, B., Dozzi, M. and Guerbaz, R. (2006). On the local time of the multifractional Brownian motion. Stochastics 78 33-49. MR2219711\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sample path properties of the local time of multifractional Brownian motion. B Boufoussi, M Dozzi, R Guerbaz, Bernoulli. 13Boufoussi, B., Dozzi, M. and Guerbaz, R. (2007). Sample path properties of the local time of multifractional Brownian motion. Bernoulli 13 849-867. MR2348754\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Joint continuity of Gaussian local times. J Cuzick, J Dupreez, Ann. Probab. 10Cuzick, J. and DuPreez, J. (1982). Joint continuity of Gaussian local times. Ann. Probab. 10 810-817. MR0659550\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V Dobrić, F M Ojeda, Fractional Brownian fields, duality, and martingales. IMS Lecture Notes: High Dimensional Probability. Beachwood, OHInstitute of Mathematical Statistics51Dobrić, V. and Ojeda, F.M. (2006). Fractional Brownian fields, duality, and martingales. IMS Lecture Notes: High Dimensional Probability 51 77-95. Beachwood, OH: Institute of Mathematical Statistics.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Occupation density and sample path properties of N -parameter processes. M Dozzi, Topics in Spatial Stochastic Processes. Martina Franca; BerlinSpringer18021975519Dozzi, M. (2002). Occupation density and sample path properties of N -parameter processes. In Topics in Spatial Stochastic Processes (Martina Franca, 2001) 127-166. Lecture Notes in Math. 1802. Berlin: Springer. MR1975519\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Estimates for the small ball probabilities of the fractional Brownian sheet. T Dunker, J. Theoret. Probab. 131777539Dunker, T. (2000). Estimates for the small ball probabilities of the fractional Brownian sheet. J. Theoret. Probab. 13 357-382. MR1777539\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sample function properties of multi-parameter stable processes. W Ehm, Z. Wahrsch. Verw. Gebiete. 56618272Ehm, W. (1981). Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw. Gebiete 56 195-228. MR0618272\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Occupation densities. D Geman, J Horowitz, Ann. Probab. 8Geman, D. and Horowitz, J. (1980). Occupation densities. Ann. Probab. 8 1-67. MR0556414\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Inequalities. G H Hardy, Cambridge Univ. PressCambridgeHardy, G.H. (1934). Inequalities. Cambridge: Cambridge Univ. Press.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"From N parameter fractional Brownian motions to N parameter multifractional Brownian motions. Rocky Mount. E Herbin, J. Math. 362274895Herbin, E. (2006). From N parameter fractional Brownian motions to N parameter multi- fractional Brownian motions. Rocky Mount. J. Math. 36 1249-1284. MR2274895\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Some Random Series of Functions. J.-P Kahane, Cambridge Univ. Press8330732nd ed. CambridgeKahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge: Cambridge Univ. Press. MR0833073\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the fractional anisotropic Wiener field. A Kamont, Probab. Math. Statist. 161407935Kamont, A. (1996). On the fractional anisotropic Wiener field. Probab. Math. Statist. 16 85-98. MR1407935\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multiparameter Processes: An Introduction to Random Fields. D Khoshnevisan, Springer1914748New YorkKhoshnevisan, D. (2002). Multiparameter Processes: An Introduction to Random Fields. New York: Springer. MR1914748\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sectorial local non-determinism and geometric properties of the Brownian sheet. D Khoshnevisan, D Wu, Y Xiao, Electronic J. Probab. 11Khoshnevisan, D., Wu, D. and Xiao, Y. (2006). Sectorial local non-determinism and geo- metric properties of the Brownian sheet, Electronic J. Probab. 11 817-843. MR2261054\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Images of the Brownian sheet. D Khoshnevisan, Y Xiao, Trans. Amer. Math. Soc. 3592299449Khoshnevisan, D. and Xiao, Y. (2007). Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 3125-3151. MR2299449\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multifractional Brownian motion: definition and preliminary results. J Lévy-Véhel, R F Peltier, RR-2645Technical ReportINRIALévy-Véhel, J. and Peltier, R.F. (1996). Multifractional Brownian motion: definition and preliminary results. Technical Report RR-2645, INRIA.\",\n \"doi\": \"RR-2645\"\n },\n {\n \"text\": \"Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type. S C Lim, J. Phys. A: Math. Gen. 341819933Lim, S.C. (2001). Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type. J. Phys. A: Math. Gen. 34 1301-1310. MR1819933\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On some possible generalizations of fractional Brownian motion. S C Lim, S V Muniandy, Phys. Lett. A. 266Lim, S.C. and Muniandy, S.V. (2000). On some possible generalizations of fractional Brow- nian motion. Phys. Lett. A 266 140-145. MR1741314\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fractional Brownian motion fractional noises and applications. B Mandelbrot, J Van Ness, SIAM Rev. 10242239Mandelbrot, B. and Van Ness, J. (1968). Fractional Brownian motion fractional noises and applications. SIAM Rev. 10 422-437. MR0242239\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Small deviations for some multi-parameter Gaussian processes. D M Mason, Z Shi, J. Theoret. Probab. 141822902Mason, D.M. and Shi, Z. (2001). Small deviations for some multi-parameter Gaussian pro- cesses. J. Theoret. Probab. 14 213-239. MR1822902\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multiparameter fractional Brownian motion and quasilinear stochastic partial differential equations. B Øksendal, T Zhang, MR1922562Stochastics Stochastics Rep. 71Øksendal, B. and Zhang, T. (2000). Multiparameter fractional Brownian motion and quasi- linear stochastic partial differential equations. Stochastics Stochastics Rep. 71 141-163. MR1922562\",\n \"doi\": \"MR1922562\"\n },\n {\n \"text\": \"Local times for Gaussian vector fields. L D Pitt, 309- 330. MR0471055Indiana Univ. Math. J. 27Pitt, L.D. (1978). Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 309- 330. MR0471055\",\n \"doi\": \"309- 330. MR0471055\"\n },\n {\n \"text\": \"How rich is the class of multifractional Brownian motions? Stochastic Process. S Stoev, M S Taqqu, Appl. 116Stoev, S. and Taqqu, M.S. (2006). How rich is the class of multifractional Brownian motions? Stochastic Process. Appl. 116 200-221. MR2197974\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hausdorff measure of trajectories of multiparameter fractional Brownian motion. M Talagrand, Ann. Probab. 231334170Talagrand, M. (1995). Hausdorff measure of trajectories of multiparameter fractional Brow- nian motion. Ann. Probab. 23 767-775. MR1334170\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Polarité, points multiples et géométrie de certain processus gaussiens. F Testard, U.P.S. ToulousePubl. du Laboratoire de Statistique et Probabilités de lTestard, F. (1986). Polarité, points multiples et géométrie de certain processus gaussiens. Publ. du Laboratoire de Statistique et Probabilités de l'U.P.S. Toulouse 01-86.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dimension results for multifractional Brownian sheets. D Wu, J. Math. Phys. 482337692Paper 073511Wu, D. (2007). Dimension results for multifractional Brownian sheets. J. Math. Phys. 48 Paper 073511. MR2337692\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geometric properties of fractional Brownian sheets. D Wu, Y Xiao, J. Fourier Anal. Appl. 13Wu, D. and Xiao, Y. (2007). Geometric properties of fractional Brownian sheets. J. Fourier Anal. Appl. 13 1-37. MR2296726\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dimension results for Gaussian vector fields and index-α stable fields. Y Xiao, Ann. Probab. 231330771Xiao, Y. (1995). Dimension results for Gaussian vector fields and index-α stable fields. Ann. Probab. 23 273-291. MR1330771\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Y Xiao, Probab. Theory Related Fields. 1091469923Xiao, Y. (1997). Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields 109 129-157. MR1469923\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Hausdorff dimension of the level sets of stable processes in random scenery. Y Xiao, Acta. Sci. Math. (Szeged). 651702175Xiao, Y. (1999). The Hausdorff dimension of the level sets of stable processes in random scenery. Acta. Sci. Math. (Szeged) 65 373-383. MR1702175\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Properties of local non-determinism of Gaussian and stable random fields and their applications. Y Xiao, 157-193. MR2225751Ann. Fac. Sci. Toulouse Math. Xiao, Y. (2006). Properties of local non-determinism of Gaussian and stable random fields and their applications. Ann. Fac. Sci. Toulouse Math. XV 157-193. MR2225751\",\n \"doi\": \"157-193. MR2225751\"\n },\n {\n \"text\": \"Sample path properties of anisotropic Gaussian random fields. Y Xiao, PreprintXiao, Y. (2007). Sample path properties of anisotropic Gaussian random fields. Preprint.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Local times of fractional Brownian sheets. Y Xiao, T Zhang, Probab. Theory Related Fields. 124Xiao, Y. and Zhang, T. (2002). Local times of fractional Brownian sheets. Probab. Theory Related Fields 124 204-226. MR1936017\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":35,"cells":{"text":{"kind":"string","value":"\nPlasma surrogate modelling using Fourier neural operators\n9 April 2024\n\nVignesh Gopakumar vignesh.gopakumar@ukaea.uk 0000-0003-0904-3448\nStanislas Pamela 0000-0003-0904-3448\nLorenzo Zanisi \nZongyi Li \nAnder Gray \nDaniel Brennand \nNitesh Bhatia \nGregory Stathopoulos \nMatt Kusner \nMarc Peter Deisenroth \nAnima Anandkumar \n\n1 UK Atomic Energy Authority,\nOxford OX14 3EB, United Kingdom of Great Britain and Northern Ireland London WC1E 6BT, United Kingdom of Great Britain and Northern Ireland Liverpool L69 3BX, United Kingdom of Great Britain and Northern Ireland Pasadena, CA 91125, United States of America\n\n\n2 University College London,\n\n\n\n3 University of Liverpool,\n\n\n\n4 Caltech,\n\n\nPlasma surrogate modelling using Fourier neural operators\n9 April 202419D1778EF9CBA88838BA5E5240AEC9DA10.1088/1741-4326/ad313aReceived 2 November 2023, revised 27 February 2024 Accepted for publication 7 March 2024machine learningMHDplasma confinementsurrogate modelneural operatorsdigital twinmagnetohydrodynamics\nPredicting plasma evolution within a Tokamak reactor is crucial to realizing the goal of sustainable fusion.Capabilities in forecasting the spatio-temporal evolution of plasma rapidly and accurately allow us to quickly iterate over design and control strategies on current Tokamak devices and future reactors.Modelling plasma evolution using numerical solvers is often expensive, consuming many hours on supercomputers, and hence, we need alternative inexpensive surrogate models.We demonstrate accurate predictions of plasma evolution both in simulation and experimental domains using deep learning-based surrogate modelling tools, viz., Fourier neural operators (FNO).We show that FNO has a speedup of six orders of magnitude over traditional solvers in predicting the plasma dynamics simulated from magnetohydrodynamic models, while maintaining a high accuracy (Mean Squared Error in the normalised domain ≈10 −5 ).Our modified version of the FNO is capable of solving multi-variable Partial Differential Equations, and can capture the dependence among the different variables in a single model.FNOs can also predict plasma evolution on real-world experimental data observed by the cameras positioned within the MAST Tokamak, i.e. cameras looking across the central solenoid and the divertor in the Tokamak.We show that FNOs are able to accurately forecast the evolution of plasma and have the potential to be deployed for real-time monitoring.We also illustrate their capability in forecasting the plasma shape, the locations of interactions of the plasma with the central solenoid and the divertor for the full (available) duration of the plasma shot within MAST.The FNO offers a viable alternative for surrogate modelling as it is quick to train and infer, and requires fewer data points, while being able to do zero-shot super-resolution and getting high-fidelity solutions.\n\nIntroduction\n\nPredicting the evolution of plasma dynamics is crucial to building sustainable fusion devices as it allows us to anticipate plasma behaviour and make real-time decisions over the input control parameters and alter the trajectory of plasma evolution.Such predictions also allow us to improve our understanding of the physics governing plasma interactions.In particular, Magnetohydrodynamic (MHD) modelling is of critical importance as it improves our understanding of the dynamical evolution of filaments that burst out of the plasma edge and reach the Tokamak's first wall, as shown in figure 1(a) [1].These plasma filaments, called edge-localised modes (ELMs), are described by non-linear MHD, and can be modelled numerically, as shown in figure 1(b).\n\nIn fusion research, prediction of future plasma states is mostly undertaken using physics-based numerical modelling codes, such as BOUT++ [2], JOREK [3], JINTRAC [4] and SOLPS [5].However, these simulations are computationally intensive, taking weeks for effective simulations to be run on high-performance computers [2].They are also often complex in terms of software infrastructure and advanced solver library dependencies, making them difficult to deploy for rapid iterative modelling.With numerical solvers, we also encounter cases where the simulations do not converge due to model mismatch or other issues.\n\nMachine learning (ML) offers an alternative data-driven path for obtaining quick, inexpensive approximations to numerical simulations, allowing for rapid iterative modelling [6].Within plasma physics, ML-based surrogate models have been used to provide quick approximations for a range of solutions, from emulating turbulent transport [7][8][9], estimating the Tritium breeding ratio across Tokamak designs [10] to modelling edge plasma [11,12].\n\nRecently, a new class of data-driven ML models, known as neural operators (NO), has recently been at the forefront of surrogate modelling.Unlike other surrogate models, they directly learn the solution operator of the partial differential equation (PDE) in a discretization-convergent manner [13,14].Specifically, NO can predict at any resolution, unlike standard neural networks like convolutional neural networks.In other words, they learn the underlying continuum, making them suitable for learning solutions of PDEs.The output of NO converges to a unique function upon mesh refinement [15].NO approaches are more data-efficient compared to other surrogate models [16], converge faster than traditional surrogate models, and can generalise better to various PDE settings.Considering these advantages, NO-based surrogate models have found application in emulating the plasma evolution across the domain of a Tokamak [17,18].\n\n\nOur Contributions\n\n• We demonstrate trained Fourier neural operators (FNOs) that can be deployed as an emulator to perform real-time forecasting of the evolution of plasma as characterised by fast cameras on the MAST Tokamak.Our studies show that the FNO is capable of modelling the plasma across the full (available) shot duration ranging from the ramp-up to the L and H modes of operation.• We devise a modified version of the FNO, capable of handling multiple variables that govern a family of PDEs.The multi-variable FNO allows us to pass field information of Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020.\n\nall the interested variables to the model, allowing for learning and encoding the mutual dependence of variables, which is essential for capturing the non-linear effects.• We demonstrate the utility of the multi-variable FNO in (surrogate) modelling the evolution of plasma across a range of reduced MHD cases with increasing complexity.We conduct an extensive analysis of the various features of the FNO, performing ablation studies that evaluate the impact of various choices of the Hyperparameters across the model and training.\n\nIn this paper, we undertake an extensive study of the FNO due to its previously established superior cost-accuracy trade-off in both training and testing [16].We use NO to model plasma within fusion devices, in both the simulation space and within the experimental domain.For the simulation domain, we consider the reduced MHD models to provide data to train the neural operator [3].We study the solution of MHD equations in two spatial dimensions with multiple variables of interest.\n\nThe simulations that we are interested in modelling represent the diffusion of a blob of plasma within a specified domain.\n\nThe simulations presented here describe a simplified scenario to that of a full ELM simulation as given in figure 1(b).We iterate over increasingly complex cases of fluid models that govern the diffusion of a plasma blob in toroidal geometry.\n\nIn addition to numerical modelling, we demonstrate that FNOs can be trained to predict the evolution of experimental plasma as observed by high-speed cameras on the MAST spherical Tokamak.The ability to predict future plasma states based on camera imaging, as well as other plasma diagnostics, could allow us to design efficient control feedback loops for future fusion reactors.We build emulators of the fast camera that views across the central solenoid and at the divertor.We find these models to be capable of accurately predicting plasma evolution in real-time.\n\nIn this study, we use FNOs for both MHD simulations and experimental camera data due to the structural similarities of the plasma observed in both.The visual similarity of the MHD modelled in the simulation, and that captured by camera images in a fusion device is elucidated in figure 1(a).The fast camera images capture the MHD behaviour that is observed experimentally.Through this work, the FNO's utility is demonstrated in predicting both experimental and simulation dynamics.Although this study considers a simplified simulation setup, with filament cross-sections, in a 2D poloidal plane, instead of the full 3D representation, this paper lays the groundwork for future studies where surrogates could be trained from both simulation and experimental data simultaneously.However, in section 3.1.10,arguments of the challenges and potential methods of extending to a 3D modelling case are discussed.\n\nA preliminary version of this work was presented at the AI for Science workshop during NeurIPS 2022, where we demonstrated, as a proof-of-concept, the utility of the FNO in modelling MHD scenarios [18].In the previous work, we conducted a benchmark of the performance of the FNO against more traditional surrogate model architectures such as Convolutional-Long-short term memory [20] and U-Net [21].Our findings established that the FNO takes significantly less time to train, requires fewer parameters, and outperforms the Convolutional-LSTM and the U-Net in emulating the diffusion of a blob of plasma described by reduced MHD [18].\n\nThe paper is organised as follows: section 2 produces a full description of the datasets and methods involved in the paper, ranging from the physics description of the simulations, the diagnostic setup of the experimental camera data and a formal definition of the FNO, its training configuration and the extension to the multi-variable FNO.Section 3 outlines the results of our experiments with using the FNO to learn and predict results from various simulation settings and camera configurations.The section comprises an in-depth analysis of the multivariable FNO trained to solve different PDEs, exploring the impact of hyperparameters, training regime, and the training data.We also include a performance comparison of the individual FNO with that of the multi-variable FNO.Section 3 is concluded with the experimental results from the FNO trained to model the plasma evolution as captured by the visible camera looking at the central solenoid and the divertor.We summarise our key results in section 4 and conclude with a discussion on future extensions of the work.\n\n\nDatasets and methods\n\n\nMHD simulations of plasma Blobs\n\nThe FNO as a Neural-PDE solver is tested on three datasets of simplified MHD models, simulating the radial convection of plasma blobs in toroidal geometry, with coordinates system (R, Z, ϕ).These three cases are designed with increasing complexity to test the utility, versatility, and capabilities of the FNO to emulate MHD models.The simplified MHD model is similar to the so-called Reduced-MHD model from [3], but with all magnetic fields and currents kept to zero, which is similar to the Euler system, and can be summarised by the continuity, momentum, and pressure equations:\n∂ρ ∂t = −∇ • (ρ⃗ v) + D∇ 2 ρ (1) ρ ∂⃗ v ∂t = −ρ⃗ v • ∇⃗ v − ∇p + µ∇ 2 ⃗ v (2) ∂p ∂t = −⃗ v • ∇p − γp∇ •⃗ v + κ∇ 2 T (3)\nwhere ρ is the density, p the pressure, T the temperature, and ⃗ v the velocity.D is the diffusion coefficient, µ the viscosity, and κ the thermal conductivity.The ratio of specific heats γ is taken to be that of a monatomic gas, 5 3 .To obtain the Reduced-MHD model, the momentum equation ( 2) is projected in the poloidal plane using the operation ∇ϕ • ∇ × (R 2 . ..), while assuming the poloidal velocity takes the form ⃗ v = R 2 ∇ϕ × ∇Φ, where the electric potential Φ is equivalent to the velocity stream function in toroidal coordinates.This projection results in a system of three equations for the three variables ρ, Φ, T. A pseudo-variable is also used for the vorticity to reduce the level of derivatives in the system: w = ∇ 2 Φ.We refer the reader to [3] and references therein for the full system of Reduced-MHD equations.\n\nThe simulations are all run in a square slab geometry of 1 m in width and height, centred at major radius R = 10 m.The runs are performed with toroidally axisymmetric plasma blobs initialised on top of a low background density and temperature.In the absence of a plasma current equilibrium to hold the density blobs in place, the pressure gradient term in the momentum equation generates a buoyancy effect, causing the blobs to move outwards.The simulations are run with the JOREK code, which is routinely used to model non-linear MHD instabilities in realistic geometries [3].In this paper, we restrict ourselves to two simplified MHD models.\n\nA further reduction of the model can be obtained by using constant temperature and removing equation (3) from the system, therefore evolving only ρ and Φ.With this isothermal model, the pressure gradient term in the momentum equation (2), which is proportional to ρ, acts as a buoyancy force on the blob, affecting its momentum ρ⃗ v, which is also proportional to ρ.In other words, the velocity of the blob will be the same no matter what its density is.To have blobs moving at different speeds, the temperature inside the blob is needed: the higher the temperature, the faster the blob will move.The values of the diffusive parameters are kept constant inside the entire domain, for all simulations, at\nD = 3.4 m 2 s −1 , µ = 2.25 × 10 −6 kg.m −1 s −1 , and κ = 2.25 × 10 −7 kg.m −1 s −1 .\nThe background density level is set to 10 18 m −3 , and the background temperature is set to 1 eV (for simulations that evolve temperature).\n\nThree datasets are considered: with a single isothermal blob, with a single blob with non-uniform temperature, and with multiple blobs with non-uniform temperature.To create the datasets, the blob's initial conditions are varied.The sampled parameters are the number of blobs, their positions, their width, and their density and temperature amplitudes.This is summarised in table 1.For datasets with a single blob, the initial position is fixed at the centre of the domain, (R, Z) = (10.0,0.0).The smaller single-blob datasets are composed of 120 simulations (100 for training, 20 for testing), while the main dataset with multiple blobs contains 2000 simulations.\n\nAll cases are run with the same time-step size of 0.15 µs, for 2000 time-steps in total.This is long enough for the blobs to evolve radially until they reach the wall (where Dirichlet boundary conditions are applied) and disappear through both convective mixing and diffusion.Since a small time-step size is necessary in the simulations to ensure numerical stability, the datasets are down-sampled to 200 slices for the training set.Likewise, the 2D poloidal grid resolution, which is 200 × 200 bi-cubic finite-elements in the simulations, is down-sampled to a regular grid of 100 × 100.The smaller single-blob datasets were run on the EOSC-hub cloud using 16 cores per run, and the larger dataset of 2000 multi-blob simulations was run on CINECA's Marconi HPC cluster using 48 cores per run.Each simulation requires approximately 350 core hours.Though each simulation is down-sampled to 200 time instances, we keep the focus of our emulator on short-term evolution up to the first 50 time steps.\n\nSimulations of blobs in slab geometry are related to turbulence and MHD instabilities at the plasma edge of a Tokamak by assuming that they represent a poloidal cross-section of a filament.As seen in figure 1, filament structures are aligned to the magnetic field, they are elongated in the toroidal direction, but have a nearly circular cross section in the poloidal plane.Such slab simulations represent the dynamic evolution of these filaments in the plane perpendicular to the magnetic field.Extensive modelling of blobs in slab geometry can be found in [22,23] and references therein.Figure 2 shows an example of a numerical simulation of multiple blobs evolving for the first 500 steps (out of 2000), with the density, temperature, and vorticity.Filaments with higher temperatures can be seen to move at a faster velocity.\n\n\nFast-camera images on MAST Tokamak\n\nMAST was equipped with fast camera imaging diagnostics at several points across the Tokamak for capturing images within the visible spectrum, capturing the plasma evolution in real-time 5 .\n\nThese Photron cameras 6 are essential diagnostics that contributed to fundamental understandings of key plasma phenomena in Tokamaks in recent years [24].While most of the camera data has been used for qualitative analysis in the past, it can be exploited using more advanced methods to provide statistical insight into plasma turbulence [25], as well as global instabilities (disruptions) [26].Within the scope of this work, we focus on two camera views.The first looks at the central solenoid and the plasma confined around it, extending to the walls and providing a holistic view of the plasma (rbb camera).\n\nThe second camera configuration under study is that providing the divertor-view, looking at the bottom of the Tokamak, where energy travels from the plasma to the exhaust region (rba camera).The views of the fast cameras on MAST in the absence of the plasma is visualised in figure 3. Figure 1 shows a view of the main plasma as seen by the camera, together with a synthetic rendering (right) of the camera, simulating the MHD physics.The cameras have a temporal resolution of 1.2 ms, with spatial resolution varying according to the desired calibration setup for each experiment.They are tuned to visible wavelength and, therefore, capture mostly the Balmer D α light emitted mostly from the plasma edge.As seen in figure 1, the fast-visible cameras are able to capture complex dynamics of plasma filamentary eruptions at the plasma edge, the ELMs [24], hence being able to predict real-time evolution allows us to anticipate such plasma instabilities and other confinement related characteristics.Within the scope of this project, our focus relies on modelling the plasma confinement and its interactions with wall accurately, leaving the modelling of ELMs as the next potential research avenue.\n\n\nFNO\n\nThe FNO [27] is a neural network-based operator learning method [13] that learns a mapping between two function spaces from a finite collection of observed input-output pairs.In the Neural Operator (NO) setting, we learn the mapping\nNO : G θ : A → U (4)\nG θ is a neural network configuration parameterised by θ and is composed of three elements:\n\n(i) Lifting: A fully local, point-wise operation that projects the input domain to a higher dimensional latent representation a ∈ R da → ν 0 ∈ R dν 0 .(ii) Iterative Kernel Integration: Expressed as a sum of a local, linear operator and a non-local integral kernel operator, that maps from ν t → ν t+1 , iterated from ν 0 to ν T .The kernel integration is followed by a point-wise nonlinearity.\nν t+1 = σ (Wν t (x) + κ (a; ϕ) ν t (x))(5)\n(iii) Projection: Similar to lifting, characterised by a fully local point-wise operation that maps the latent representation to that of the output domain\nν T ∈ R dν T → u ∈ R du .\nIn equation (5), σ denotes the nonlinear activation, W represents the local, linear operator, and κ represents the chosen kernel which is parameterised by ϕ. x represents the input that is fed into that layer.\n\nConventionally, the lifting and projection within an NO are characterised by linear layers acting on the respective feature spaces.The kernel integration operator is characterised as the neural layer that evaluates a chosen kernel over the higher dimensional latent representation fed into it:\n\n(κ (a; ϕ) ν t ) (x) = ˆD κ (x, y, a (x) , a (y) ; ϕ ) ν t (y) dy.(6) As demonstrated in [27], the FNO is devised by characterising the kernel integral operator as an operation within the Fourier space.Replacing κ with the Fourier integral operator F(κ ϕ ) we obtain\n(κ (a; ϕ) ν t ) (x) = F −1 (F (κ ϕ ) .F (ν t )) (x) . (7)\nHere, F and F −1 represent the Fourier and inverse Fourier transform, respectively.κ, the kernel is parameterised by learnable weights ϕ ∈ θ κ .Within the FNO, the kernel integrator reduces to the inverse Fourier transform of the inner product between the Fourier integral operator F(κ ϕ ) and the Fourier transform of the input to that Fourier layer F(ν t ).\n\nIn practice, the iterative kernel integral transform with Fourier transform is characterised by two learnable weight matrices R and W as shown in figure 4. R learns the mapping behaviour within the Fourier space, while W learns the mapping required within the input Euclidean space.R characterises the non-local integral kernel operator, while W represents the local linear operators.The weight matrix R within the Fourier layer enables convolution within the Fourier space, allowing the network to be parameterised directly in it.The grid discretisation is also fed in at each layer to help the network learn the discretised Fourier representation of the data and undergoes point-wise linear transformations, characterised by C, its operation is similar to that of W. The output of a Fourier layer can be expressed as where x is the input to that layer, y the output, g the grid discretisation, b the bias and σ is the nonlinear activation function.Equation ( 8) resembles a single Fourier layer within the FNO.Multiple Fourier layers are stacked on top of each other to complete the iterative kernel integral operation.The stacked Fourier layers are sandwiched between the lifting and projection operations to complete the FNO architecture.\ny = σ ( F −1 (RF (x)) + Wx + Cg + b ) ,(8)\n\nComparison to convolutions.\n\nThe spectral convolutions characterised by κ in equation ( 7) have an analogous relationship to the convolutional operation.Convolution, being a product of functions across a certain range, is an integral transform.Expressed across an infinite range, it takes the mathematical form\n(f * g) (t) = ˆ+∞ −∞ f (τ ) .g (t − τ ) dτ,(9)\nwhere f and g are functions over which the convolution operation is performed.By replacing the convolution operation with the Fourier transform in equation ( 9), we obtain\n(Ff) (k) = ˆ+∞ −∞ f (x) e −i2π τ k dτ. (10)\nComparing equations ( 9) and ( 10), it becomes evident how equation ( 7) is a special form of the convolutional operation, one that takes place in the Fourier space.In practice, both operations are applied across a discretisation, characterised by the data that is fed to the model, but since we decompose the data into Fourier modes within the FNO and apply learning across the decomposed components, we can learn continuous representations of the required mapping as opposed to learning discrete representations with conventional convolutions.Named spectral convolutions in [27], allow for several advantages over a standard convolutional neural network, where filters are directly learned without a spectral transform.Firstly, by learning within the Fourier space, we allow the network to identify global features that are embedded within the data, highlighting the dominant modes in the process.By performing the Fourier decomposition over the data as it parses through the network, the weights are tuned to identify and learn the modes that possess a majority of the information, allowing the network to generalise better.Secondly, the spectral convolutions being in the function space allows for learning the mapping in a discretisation convergent manner, allowing for learning mesh-free solutions.Lastly, the Fourier transform in equation ( 10) is implemented using a Fast Fourier Transform (FFT) [28] which has a computational complexity of O(n log n), whereas the standard convolution operation without spectral transform has a complexity of O(kn 2 ), where k is the size of the kernel and n is the representation dimension.However, within the FNO, we only deal with a complexity of O(nk max ) since we truncate the Fourier modes up to a desired hyperparameter k max .\n\n\nTraining configuration.\n\nFor emulating the simulations, we use the FNO-2d configuration [27], where a 2-d FNO which only convolves in space, is deployed in an autoregressive manner [29].The FNO takes in as its input the field values across a 2D grid for an initial set of time instances (T_in) along with the grid discretisation in both dimensions.Similar to spectral numerical solvers, the grid discretisation, fed in with the field data, provides better-structured information for discrete Fourier transform utilised within the FFT.The FNO takes the initial fields as the input and outputs a set of later time instances (step) of the field value across the same grid.The output of the FNO, coupled with the later time steps from the input, are then fed back in an autoregressive manner to estimate further field evolution in time, until the desired time instance (T_out) is reached.Within the autoregressive setting, we roll out the evolution of the fields up until the desired time step and then perform the backpropagation through time to enable learning of the time evolution of the PDE.For further information regarding the theory behind operator learning as ascribed by the FNO and its implementation in various case settings, we invite the reader to refer to the original work in [27].Unlike the FNO 3D configuration, we only perform Fourier transformations across the spatial dimensions and do not include the temporal dimension.We have chosen the 2D configuration of the FNO as it is lighter, quicker, and gives us the flexibility in deciding the length of the temporal evolution.\n\nFor the case of the FNO for camera, instead of performing an autoregressive time roll out, we perform a fixed time window mapping, where the FNO takes in the camera information for fixed time steps (T_in) and outputs the next set of frames characterising the future plasma state (step).The next set of inputs is obtained by sliding the interested time window by one, as shown in figure 5.\n\n\nMulti-variable FNO.\n\nAs depicted in equations ( 1)-( 3), the physics model represents a highly correlated and interdependent system where density, temperature, and electric potential factor into the time evolution equations of each other.Taking this family of PDEs into account, it is vital to construct a surrogate model that is capable of emulating all variables together, allowing for information exchange within the model.The original FNO [27] was designed to model for a single variable and needed to be modified to handle multiple variables.The work done in [18] demonstrates the impact of the FNO on predicting MHD evolution but modelled [18], it was permissible to model the variables independently.However, for the case under consideration, it was imperative that for physical consistency, we model the variables together.As we move into more complex MHD cases, we introduce more correlated variables such as density and temperature, which are both required to evolve together to fully grasp the propagation of the plasma blob.\n\nTo account for the multi-variable setting, we designed the FNO to take in the variables as a separate channel.Initial construction of the FNO would consider data in the format [batch size, temporal projection, spatial along x-axis, spatial along y-axis].We modified that model to consider data in the format [batch size, variables, temporal projection, spatial along x-axis, spatial along y-axis].This modification allows us to account for all the variables within a single model, which facilitates information exchange across them.By having the multiple variables of the physics model as a channel, we allow for the Fourier transform to be applied across the spatial domain for all variables of interest.The R matrix that allows for learning within the Fourier space in equation ( 8) is modified to account for the new variables, and the local, linear operator W acts across the variable and spatial domain.\n\nBy providing the variables as a separate channel across the FNO, we introduce the possibility of performing the operations of lifting and projection defined in section 2.3 not just across the temporal features but across the fields as well.Thus, within this configuration of a multi-variable FNO, the feature space becomes two-fold, accounting for the time as well as the variables.We also modify the FNO with skip connections to reduce the information loss as the data progresses through the network [30].The network architecture is outlined under Appendix B in table B1.\n\n\nResults\n\nPrior to elaborating on the finer details of the results from our experiments, we provide an overview of the performance obtained by the FNO surrogates across all the interested modelling tasks in table 2. Across our range of experiments, emulating the plasma, we find that the FNO achieves considerable accuracy (Mean Squared Error (MSE) in the normalised domain ∼10 −4 ).Despite being heavily parameterised, the FFT within the network helps the FNO achieve quick training and execution times for surrogate models [18].The Fourier features extracted help provide the model with implicit bias found in the data, allowing it to converge quicker on the spatial evolution of the plasma.By transforming the data to the Fourier space, we allow the network to easily extract the dominant features found in the data as they would be highlighted across the various Fourier modes.As mentioned in section 1, we focus our experiments solely on the FNO as the chosen surrogate.For a detailed benchmark of MHD emulators across various models, we refer to our prior work [18].\n\n\nMulti-variable FNO over MHD Simulations\n\nWe design and train multi-variable FNOs to model the three MHD cases under study: Single isothermal blob, single blob in a non-uniform temperature field, and multiple blobs in a nonuniform temperature field, as described in section 2.1.Each multi-variable FNO is constructed as described in section 2.3.3 and trained using the autoregressive framework outlined in section 2.3.2.\n\nFor each multi-variable FNO, we chose an input feed of 10 time steps of the field information (T_in = 10), and an output size of 5 (step = 5).Each time step within the FNO corresponds to 1.5 µs of simulation time.Each FNO was autoregressively rolled out until the 50th time instance, allowing the neural operator to learn the time evolution of the function till then (T_out = 50).Each Fourier layer within the multivariable FNO has 16 modes and a temporal channel width of 32 (unless mentioned otherwise).The temporal channel width in the FNO is akin to the channel width in a Convolutional Neural Network, where the temporal data is scaled in the FNO channels instead of the RGB color space in the CNN channels [31].The channel variable width was maintained at the number of variables that the FNO learned.The grid discretisation associated with the field data is added to the training data within each model's forward call as an addition to the temporal channel, providing further field information to the model.We utilise the discretisation used within JOREK to generate the training data as the grid features within the FNO.We deploy a two-fold normalisation strategy considering the nature of the dataset.The physical field information represented within the MHD cases is in different scales, with densities ranging from 0 to 10 20 m −3 and temperature ranging up to 100 eV.Since we consider the gradual diffusion of an inhomogeneous density blob(s), the data distribution within the spatial domain is severely imbalanced.Taking these aspects of the training data into consideration, initially the field values are scaled down by a factor, indicative of the range of operation found within the simulation itself.The physics scaling would divide the data by the order of magnitude around which the plasma operates for each variable.We perform this by scaling the field value with the order of magnitude of its maximum found in the training data.This is followed up by a linear range scaling, allowing the field values to lie between −1 and 1.The normalisation pipeline is implemented separately for each of the variables that the multi-variable FNO accounts for.Each model is trained for up to 500 epochs using the Adam optimiser [32] with a step-decaying learning rate.The learning rate is initially set to 0.005 and scheduled to decrease by half after every 100 epochs.The input, output dimensions and the model architecture used for each case is given in B.\n\nThe loss function is expressed as an equally weighted linear combination of two elements: the reconstruction error and the difference error, expressed as (11) where ỹt represents the FNO output at the time step t, y t represents the labelled (actual) data at time frame t, step represents the step size of the FNO output, T represents the full length of the autoregressive time roll-out and N, the batch size or the number of simulations we are learning at a time over which we sum over.\n\nThe reconstruction error helps the FNO to learn to minimise the prediction error for a particular instance in time, whereas the difference error helps learn the time evolution of the plasma better as it minimises the temporal differences that arise in the autoregressive roll-out of the FNO.The loss curves associated with each experiment can be found in figure 6.\n\nThe focus of this research relies on the short-term evolution of up to 50 time steps from a total of 200 time steps, seeing how well the multi-variable FNO can model the evolution of all the physical variables in an integrated manner.We restrict ourselves to the short-term due to the errors arising from the autoregressive time rollouts, which are further explored in sections 3.1.4-3.1.6.To see the performance of the FNO deployed in an autoregressive setting for the full time range of the simulation, see appendix C.\n\n\nSingle blob-isothermal.\n\nFor the first case, that of a single blob in an isothermal setting (hereby referred to as an isothermal blob), the multi-variable FNO models the evolution of the two variables: density ρ and electric potential Φ. Considering the simple structure of the physics, the FNO performs well in (surrogate) modelling the evolution of both the variables arriving at an MSE (in the normalised domain) of 7.58 × 10 −5 in 48 min on a single Nvidia A100 chip.The multi-variable FNO is capable of emulating the evolution of the plasma up to the 50th time step in 0.025 s, offering a speedup of 6 orders of magnitude as shown in figure 7. Further plots showing the contrast of the FNO output with that of JOREK by way of discrepancy plots can be found under appendix A in figure A2.\n\n\nSingle blob with non-uniform temperature.\n\nMoving up in complexity, in the case of a single density and temperature blob, the FNO is adapted to model density, electric potential, and temperature, three highly correlated variables as described in section 2.1.Even considering the additional physics, the multi-variable FNO demonstrates the ability to learn the global features of blob diffusion.It characterises the movements of the blob, its interaction with the walls, and its eventual dissipation.We notice that with the added field variables, the FNO performs well in predicting the evolution within the initial timesteps, but loses rigour of structure as we roll further out in time.Despite the challenge of preserving minor features further into time, the multivariable FNO achieves an MSE (in the normalised domain) of 2.56 × 10 −5 as shown in figure 8.The model trains in 1 h 37 min on a single Nvidia A100 chip.The multi-variable FNO is capable of (surrogate) modelling the evolution of the plasma up to the 50th time step in 0.03 seconds.Further plots showing the con-trast of the FNO output with that of JOREK by way of discrep-ancy plots can be found under appendix A in figure A1.\n\n\nMulti-Blobs with\n\nNon-uniform temperature.Figure 9 shows the capability of our newly designed multi-variable FNO in emulating the evolution of density, electric potential, and temperature for the case with multiple density and temperature blobs.This presents a more challenging case to model as the distribution of the training data is wide due to it being generated from a much larger parameter space as described in table 1.With a training dataset of 1500 simulations, the model trains in 11 h and 22 min on a single Nvidia A100 chip and achieves MSE(in the normalised domain) ∼0.0001, capturing the larger, more global, features associated with the evolution of plasma characteristics.However, due to the truncated nature of the implementation of the linear operator over the Fourier Transform, the modelling gets limited to the more dominant modes, smoothing out the less dominant, minor, finer features.The multi-variable FNO is capable of (surrogate) modelling the evolution of the plasma up to the 50th time step in 0.03 s.Further plots showing the contrast of the FNO output with that of JOREK by way of discrepancy plots can be found under appendix A from figures A3 to A5.\n\nThe FNO deployed in the autoregressive manner is well equipped to model the evolution of the plasma for short time roll-outs, but as we extend the prediction further in the temporal domain, it moves further away from the ground truth.This arises mainly due to the training regime which we deploy the FNO.Being deployed in an autoregressive structure, the prediction errors accumulate over longer time roll-outs.This is illustrated in figure 10.Though we demonstrate the results obtained with 1500 simulation data points, for the rest of the ablation studies (except for sections 3.1.7and 3.1.8)demonstrated in this work, we restrict ourselves to the initial 250 as it allows us to compares against the prior 2 cases where we operate in a relatively data-scarce setting.\n\n\nIndividual FNO vs multi-variable FNO.\n\nMHDs governing the evolution of a plasma state is characterised by the amalgamation of Navier-Stokes equation of fluid dynamics and Maxwell's equations of electromagnetism [33].The evolution of MHD fields involves modelling multi-physics environments with coupled variables that are integrated together.\n\nConsidering this interplay of different field variables, it was imperative that we devise the multi-variable FNO, where all the interested and dependent variables can be modelled together.\n\nOur multi-variable FNO, as described in section 2.3.3,achieves the required multi-physics modelling, with the same computational costs as training individual FNOs for each variable.The multi-variable FNO has approximately the same number of parameters as the different individual FNOs combined, and thus takes the same time as training two or three different FNO models.As shown in figure 11, the multi-variable FNO achieves better performance with much less error than the individual FNOs. Figure 11 demonstrates that by integrating the different field variables within the model, we can reduce the error associated with the autoregressive time rollout, dampening the error growth.It must be noted that since each variable has a different normalisation applied to it (the same scheme, but different normalisation), the comparison within the normalised domain is consistent within the same variable.\n\nThus, we conclude that for highly correlated multi-physics systems, such as those described by the MHD equations, multi-variable FNO, which captures the interdependence of the variables within a single model, yields a much more physically consistent result than the variables modelled individually by separate FNOs with no methods of exploiting the relationship across them.\n\n3.1.5.Impact of step size.The FNO, devised to be utilised in an autoregressive configuration, is only exposed to the first T_in time steps as inputs to the model.The rest of the inputs for emulating the time evolution are obtained by feeding back the output into the model in a recursive manner.The model follows the layout of a Markov decision process [34] as it relies on the previously generated time steps to provide all the information pertaining to the state of the system, without having a memory component attached to it.As the model trains, the autoregressive output generated from the input is compared against that of the simulation data to learn the mapping in a supervising manner.When deployed at inference, the FNO relies purely on the initial time steps to perform the time rollouts of the plasma evolution.\n\nThus, the choice of the output size (step size), i.e, how much further in time the FNO predicts the time evolution with each forward pass, essentially provides disconnects across time for the model, which is evident by looking at figure 12.At each time step that is a multiple of the step size, we notice a spike in the cumulative error of the model.This spike denotes a disconnection that occurs as the model performs the autoregressive time evolution.In figure 12, the orange line indicates the error growth for a step size of 1, where there is no spike characterising the disconnection.For the red line with a Figure 13.Ablation study of multi-variable FNOs with different truncated mode numbers.Error growth of FNOs with different mode numbers as they are autoregressively evolved in time for the Multiblob case.Errors decrease with a higher number of modes, and performance nearly plateaus at higher modes, indicating the existence of dominant features for the problem up to a certain number of modes.For each model, the width of the temporal channel was fixed at 32, and that of the variable channel was fixed at 3. step size of 5, we notice that there is a spike at every 5th time step starting form 10 and similarly for the green line indicative of models with a step size of 10, the spike is observed at every 10th time step.Throughout the scope of this study for surrogate modelling plasma simulations, we restrict ourselves to step size = 5 as we find it offers the best training efficiency, optimum performance with less computational time.\n\nRegardless of this error spike, the FNO performs well in (surrogate) modelling the various MHD cases under study.We notice that the multi-variable FNO with a step size of 1 performs the best with a smooth and dampened error growth.This is expected as the network parameters are being used to perform mapping for one time step rather than several steps, effectively providing more parameters per step.We notice that the error decreases initially before it increases and believe it is because of the autoregressive structure of the training regime.In forward propagation, the model uses the time step data first from the data and then from the model itself.This results in a distribution shift problem because the inputs are no longer solely from ground truth data, the distribution learned during training will always be an approximation of the true data distribution [35].The pushforward trick demonstrated in [36], offers an alternative training regime that allows us to account for this behaviour.\n\n3.1.6.Mode ablations.The Fourier integral operator within the Fourier layer of the FNO truncates the Fourier series up to a maximal number of modes k max [27].This presents an additional hyperparameter to each Fourier layer of the FNO that determines the extent to which Fourier modes are required to be learned to map the spatio-temporal evolution.Within our experiments, we ensure that for each multi-variable FNO, the modes at each Fourier layer are fixed.Figure 13 shows the impact the number of modes has on the FNO in emulating the MHD case with multiple density and temperature blobs (Case 3).For a smaller number of modes, the FNO demonstrates a higher error value at each time step, whereas for higher mode numbers, we notice that the error is relatively lower.We also notice that the performance of the FNO plateaus at 8 modes and does not increase further as the mode numbers are increased.This arises due to the fact that the dominant features that are characteristic of the system we are learning saturates at the 8 th mode, and no more dominant features are found at higher modes for the multi-blob case.This is consistent with the mode selection that one has to perform for each problem that a neural operator is applied to.The effect of this is studied in detail [37].\n\nWithin the mode ablation study, we notice that at lower modes, though the error of the model is higher, it produces smoother field profiles than those at higher modes.In figure 14, we see that the FNO with 2 modes, generates fields that are smoother and less noisy, especially at the initial time steps.However, they soon lose the general structure as they progress further in time.In the FNO with higher modes (k max = 8, 32), the FNO output is much noisier than those at lower modes, but preserves the general structure of the temporal evolution better.\n\n3.1.7.Impact of training dataset size.Figure 15 visualises the impact the size of the training dataset has on the performance of the multi-variable FNO.For this purpose, we generate a larger dataset using the same scheme as given in table 1.We generate a total of 1835 JOREK simulation data points, out of which we set aside 85 as the test dataset.We train multiple multi-variable FNOs with training sizes ranging from 250 to 1750 simulations and compare their performance on the test dataset.All the models are constructed and trained under identical conditions, with the exception of the training data.We find that the performance of the FNO increases as we increase the training size, with model error decreasing by nearly half.However, we notice the gains made by added simulation data do not scale well at all.As shown in figure 15, improvement in performance is only marginal in comparison to the added computational costs associated with the training is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better.as well as the simulation run times.This is further illustrated in the loss curves of each associated experiment shown in figure 16.\n\n\nZero-shot super-resolution.\n\nAs described in section 2.3, the NO framework allows learning within the function space as opposed to the Euclidean space as with traditional neural networks [13].By learning the mapping between infinite-dimensional function spaces, the framework performs operator learning, allowing for characterising discretisation within our model.Within the FNO, discretisation invariance is built in by learning within the Fourier space rather than the Euclidean space on which the solutions exist, as demonstrated in equation (8).Being discretisation-convergent, the FNO learns information up to a certain refinement of the mesh resolution, higher than what we had provided in the initial We also notice the quick over-fitting tendency of the FNO as observed in [38].\n\ntraining dataset [15].The truncated nature of the FNO characterises the model as a band-limited function.Coupled with the discrete modal representation in the Fourier space, the FNO converges to a fixed performance at higher resolution [15].This discretisation-convergence allows us to perform zeroshot super-resolution, trained on a lower resolution, and directly evaluated on a higher resolution [27].In figure 17, we demonstrate this capability for zero-shot super-resolution, where we take the model trained on a spatial discretisation of 100 × 100 used for the prior experiments and then deploy it to emulate the evolution of multiple blobs at a spatial discretisation of 500 × 500.Even for grids at 5 times larger resolution, the FNO is capable of emulating the plasma with considerable accuracy, deviating from the true solution by an MSE (in the normalised domain)of only 7.90 × 10 −5 in the normalised domain.They share the same model parameters among different discretisations of the underlying function space that we are attempting to learn.Plots showing the contrast of the super-resolved FNO output with that of JOREK by way of discrepancy plots can be found in figure A6 in appendix A.\n\n\nExtrapolating outside training domain.\n\n\nNeural\n\nNetworks trained for modelling a particular task have been found to suffer from a general lack of extrapolation capabilities [39].The FNO relying on a continuous gradientdescent based optimisation of the deep learning parameters exhibit the same behaviour.The models perform well within the domain of the training data, as can be explained by the associated Neural Tangent Kernels (NTK) [40] and fail to perform well outside the training regime.\n\nFor our multi-variable FNO that models the evolution of multiple blobs in reduced MHD, we can empirically validate the above-mentioned statements.When inferred on a case where there are 13 blobs with amplitude and density twice more and width 1.5 times more than the maximum used within the training data, we observe that the FNO can model the plasma evolution at the earliest prediction time instance and fails to capture the evolution of the plasma behaviour further in time as evidenced in the figure 18.\n\n\nScaling to 3D.\n\nThere exist several challenges in the extending the FNO to model the evolution of multi-variable PDEs to 3D settings.The primary concern is of the memory constraints needed for a 3D multi-variable FNO when being evaluated autoregressively in time, leading to GPU bottlenecks.Recent work in performing Tensor factorisation over the FNO weights, allows representing the network in a lower rank approximation [41], offers potential to those GPU training bottlenecks.\n\nAnother challenge associated with the scaling is that of the computational costs associated with data generation.In 3D, the numerical solvers that generate the training datasets are exponentially more intensive than in the 2D settings, limiting the amount of training data that could be generated.Deploying Active Learning (AL) methods, which allow for more informed sampling of the parameter domain space offers a solution to this.Within AL, the model is initially trained on a coarsely sampled, limited dataset and then evaluated across the interested domain.Performance (or uncertainty) of the model within the interested domain informs the next data generation points, which are added to the training data, improving the model accuracy.AL allows the model to maximise performance, minimise uncertainty within data scarce scenarios [42].Within the scope of plasma physics, AL has been found to reduce the data requirement of surrogate models by a factor of 20 in modelling Gyrokinetic Turbulence [43].The issue of data-scarcity arising from the profound computational intensiveness of 3D simulations could also be combat by using physics-informed models, where the information regarding the PDE is hard-baked into the model architecture or the training regime.Recent works on physics informed neural operators has shown promise in fine-tuning coarsely trained models with PDE constraints within MHD scenarios [44].\n\nWhile scaling to 3D settings it might be the case that the choice of operator within the Neural Operator might be limiting and not representative of the true solution, thus there might be a requirement of a combination of different operators implemented over different families of basis functions within the model to account for the varied physics that is found within the PDEs.One last challenge to consider is that proposed by the autoregressive layout of the FNO.The addition of another dimension will contribute further to the error growth in time as noted in figure 10.Though this remains a standing problem within the domain of Neural-PDEs, recent works have shown promise in solving this challenge by providing further conditioning of the longer time roll-outs [45].\n\n\nFNO over MAST camera\n\nAs described in section 2.2, we conduct exploratory research in looking at using the FNO as an emulator for performing predictive inference in real-time of the plasma evolution within the MAST fusion device.For this purpose, we deploy FNOs with the same architecture as that used for emulating the Individual FNOs (given under appendix D in tables D1 and D2).However, considering the nature of the camera dataset, we implement a different training and inference regime from that of the autoregressive structure used to solve PDEs.\n\nIn the case of the MHD simulations, the evolution of plasma is a Markovian Process, with no element of noise found within the dataset.However, within the case of the experimental data, the images are still time-series data, but they are nonmarkovian, noisy, and represent only partial information about the state of the system.In addition, the dataset does not offer any information about the causal contributors, such as the various control inputs that influence the state of the system, and is assumed to be implicitly defined within the time evolution of the data.Taking this into account, implementing an autoregressive structure as demonstrated in [18], provided limited capabilities as the error growth across the time roll-out would be too large, not allowing us to perform longer time roll-outs.Within the scale of the experimental data, it might be the case that there might be new information fed into the system midway through it.By utilising an autoregressive rollout that relies solely on the initial time steps, the model will not be able to account for this new information added later to the system.Unlike in the case of modelling simulations, there is no actual need within the experimental space to rely just on the initial conditions and state of the system to perform the prediction.Data characterising the temporal evolution of the plasma is generated and captured by the diagnostic as the experiment is run and does not need any additional intervention to obtain it.Our training and inference setup takes advantage of this continuously generated data to build our predictive model.Instead of performing an autoregressive time rollout, we perform a fixed time window mapping (as outlined in figure figure 5), where the FNO takes in the camera information for fixed time steps (T_in) and outputs the next set of frames characterising the future plasma state (step).Being trained in this sequential pipeline, the model initially takes in the first 10 time instances (1-10 frames) to output the next 10 time instances (11-20 frames).With the next forward pass, the model takes in the other 10 time instances, but this time, forward by one frame (2-11 frames) to predict the next 10 frames to that input (12-21 frames).This input, characterised by a 10-frame window slid by one each forward pass, is run until we output the last required frame (T_out).Our trained FNO models the plasma at ramp-up, entering the flat-top regime, and in most cases, across the L-Mode and H-Mode operational regimes, covering the full (available) plasma duration.Since we are designing this to be run in real-time with the experiment, future time windows, as they become available, will be fed into the model to predict the future plasma states.The time resolution of the fast camera diagnostic is 1.2 ms, where the FNO only requires 6 ms to predict the next 12 ms of the plasma evolution, allowing us to run the model efficiently in real time.\n\nEven though we keep our focus on a limited number of shots curated according to the criteria given in section 3.2.1, as we structure them into input-output pairs by way of the sliding windows as described above, we essentially turn each shot characterising of approximately 200 frames into 180 inputoutput pairs.The FNO is exposed to dynamics of the plasma evolution across the interested time window of MAST operation, learning to predict from any intermediary states of the system to the next states of the system.The FNO is trained to learn how to predict the evolution of the plasma as witnessed by the camera for the full (available) range of operation of the Tokamak pulse.\n\nFor each camera FNO, we chose the input feed to consist of 10 time steps of the field information (T_in = 10), with an output size of 10 (step = 10).Each Fourier layer within the FNO has 8 modes and a temporal channel width of 16.The grid discretisation associated with the field data (obtained from the reactor CAD model) is added to the training data within each model's forward call as an addition to the temporal channel, providing further field information to the model.The data is normalised by following a linear range scaling, allowing the field values to lie between −1 and 1.Each model is trained for 500 epochs using the Adam optimiser [32] with a stepdecaying learning rate.The learning rate is initially set to 0.005 and scheduled to decrease by half after every 100 epochs.The model was trained using a Relative L2 loss function over the reconstruction error alone (see equation ( 11)).\n\n\nData selection.\n\nThroughout this study, we work on a selected number of plasma shots for each cameras, restricting ourselves to the M9 campaign on MAST.For the rbb camera, we restrict ourselves to 55 shots lying within the range: 30 250-30 431 from the MAST experimental database.For the rba camera we look at 48 shots lying within the same range 30 250-30 431.The shots were carefully curated from the above shot range by considering several factors.The first criterion was to ensure that the selected shot actually contained a confined plasma and was not a dummy shot like commissioning pulses.The second criterion was whether the shot had the desired duration of more than 100 time steps-looking at forecasting longer-term plasma evolution.The third one was selecting shots within the above-mentioned ranges that had the same camera calibration.Considering the FNO relied on a grid discretisation to leverage the Fourier modes, it was crucial to ensure that the pixels from the camera belonged to the same line of sight vectors.The camera calibrated for specific experimental runs captures the information of the plasma onto a uniform pixel grid, allowing us to perform Fourier transformations over them.By using calcam, a calibration tool developed by the European Atomic Energy Community [46] combined with 3D CAD models of MAST, we determine the domain range covered by each camera calibration.The upper and lower limits of the domain range of these images are then fixed onto uniform grids in both the R and Z axis, allowing us to create uniform grid discretisations underlying each camera image.The last criterion was to ensure that the camera images obtained from the diagnostic had the same spatial resolution across the dataset that could be fed into the FNO.For the rbb camera, we chose the shots with a resolution of (448 × 640), and for the rbb camera, we chose those with the resolution (400 × 512).However, the FNO offers discretisation convergence, the training setup benefits greatly from ensuring coherence along the spatial resolution.Figure 19 shows the plasma signals allows to demonstrate the diversity of the scenarios used to train and test the FNO modelling the cameras.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data.For a full list of the selected shots, refer appendix G.\n\n\nCamera viewing the central solenoid (rbb).\n\nThe evolution of the plasma within the Tokamak, as seen by the main camera, can be shown in figure 20.The plot shows the comparisons of the evolution as observed by the FNO with that of the camera spanning 10 frames predicted in one feed-forward iteration.The central solenoid serves as the backbone of the MAST magnet system, generating the majority of the magnetic fluxes as well as the plasma current.In figure 23, we portray the predictive capability of the FNO by comparing the last frame from multiple feed-forward instances taken from data across the interested length of the pulse.This allows us to compare that even for the longest time prediction, we can get reasonable predictions.In figure 23, we demonstrate that the FNO is capable of predicting the plasma in L-mode as well as in H-mode, characterising the plasma evolution across the full (available) range of the plasma shot.The main camera view, as shown in figure 1 The FNO deployed with the time-window pipeline (see figure 5), is able to capture the long-term dependencies that arise within the plasma confinement.The model is able to predict the evolution of the plasma from L-mode to H-mode as seen in figure 23.The prediction around the transition regime can be seen in Though the camera captures the confined plasma shape, it fails to capture the localised filaments.In figure 20, the trained FNO is deployed to model the evolution of plasma within the L-Mode of operation, whereas in figure 21 the FNO is modelling the region of H-mode within the same plasma shot.\n\nThe capability of the FNO in predicting across the L-H transition is further emphasised through figure 22, where we show the prediction around the transition region for shot number 30 428.We provide the camera image, FNO prediction and the absolute error at L-mode (162 ms), around the transition (172 ms) and with the plasma in H-mode (186 ms).The FNO is capable of modelling the plasma in both L-mode, Hmode and across the transition, but as the FNO is a regression model and not a classification model, without further analysis over the predicted solution, the model is unable to explicitly mention that it is capable of predicting the exact moment of transition.This will be a key area of focus for the next research avenue.\n\n\nCamera viewing the divertor (rba).\n\nThe evolution of the plasma within the Tokamak as seen by the divertor camera, along with predicted images from the FNO can be seen in figure 24.The plot shows the comparisons of the evolution as observed by the FNO with that of the camera spanning 10 frames predicted in one feed-forward iteration.The divertor of a Tokamak serves as the exhaust of the device, where the topology of the magnetic flux surfaces is designed to isolate the confined plasma from the first-wall of the device.In figure 25, we portray the predictive capability of the FNO by comparing the last frame from multiple feed-forward instances taken from data across the interested length of the pulse, demonstrating the predictive power of the FNO.The divertor camera looks at the lower divertor of MAST with a spatial resolution of 400 × 512.We sampled a total of 48 plasma shots from the MAST database, used 45 for training and 3 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 5815, and the test points grow to 701 data points.The model takes 9 hours 10 minutes on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.70 × 10 −4 .For further plots showcasing the forecasting capabilities of the FNO at the divertor plasma, refer to appendix F, specifically figures F1 and F2.\n\nThe FNO deployed for emulating the camera data accurately preserves the global features of the plasma evolution.It is able to clearly characterise the locations and intensities of the heat fluxes onto the solenoid and the divertors, allowing us to anticipate the heat flux in advance and initiate the appropriate actuator mechanism to minimise its impact on the various plasma-facing components.The FNO also captures the reflections generated within the device due to the interactions across the plasma and the plasma-facing components.The FNO, in its current form, smooths out the filaments but captures the general confined structure of the plasma as it evolves in time.As expected, the FNO learns characteristics extremely well, while failing to capture the localised phenomena with matching pixel-pixel accuracy.\n\n3.2.4.Limitations.We decided to look at a narrow shot corridor, since the data variability will be low, allowing us to demonstrate a functioning proof-of-concept.Our predictive model only takes in the camera diagnostic as the input and does not take into account any of the control signals, such as coil currents and heating parameters that influence the plasma evolution.However, we argue that within the past window of plasma observed from the camera data that is taken as the input, these control parameters and their influence are implicitly defined (with a latency), and thus, the model provides a suitable predictive tool for short-term predictions.The inclusion of various control signals, time matched with the camera inputs, will potentially help improve the performance of the model.Our future research trajectory will be heading in this direction.\n\n\nConclusion and discussion\n\nThrough this work, we demonstrate the utility of using the FNO as both a surrogate model for a plasma simulator and as a predictive emulator for a plasma experiment.We design a modified version of the FNO, that performs integrated emulation of multiple variables associated with a family of PDEs.Our analysis across a range of reduced MHD cases demonstrates that even in data-scarce scenarios, the FNO can be trained to act as a surrogate.Through this paper, we highlight the various features that make the FNO a natural choice for (surrogate) modelling MHD cases, all the while performing ablation studies on the various hyperparameters and the training regimes.We find that the FNO fits the requirements of a quick, cheap surrogate, but due to its auto-regressive nature, fails to capture the physics when deployed for long time rollouts.Being utilised for emulating the experiment, the FNO is capable of forecasting the evolution of the plasma as seen through the cameras on a fusion device.The FNO camera models are capable of identifying the regions of plasma interaction with the central solenoid and the divertor, while effectively predicting the plasma shape evolution.The FNO has shown versatile performance in being able to model the plasma confinement across both L-mode and H-mode of plasma operation across a diverse range of shots.The FNO for the camera can predict the plasma evolution in real-time and offers us the potential to be deployed within the planning stage for various control scenarios.Our work also demonstrates how we can perform zero-shot super-resolution across the pre-trained multi-variable FNO and obtain the plasma emulation at higher resolutions than it was trained on, without requiring any finetuning.\n\nIn continuing with this line of work, we will be exploring the utility of building more versatile and robust FNO models that can be deployed not just in a particular case, but across a range of simulation cases.While we build more general MHD models, we will also be looking at methods to better physics-inform the models [14], allowing us to finetune the general models to specific MHD instances.We will also be looking into methods of quantifying the uncertainty associated with the surrogate models and into AL pipelines that can facilitate the entire surrogate ecosystem.With the FNO for camera data, we are currently working on extending the utility across more plasma shots with a more diverse line of sights, across multiple campaigns while exploring the utility of state space models for longer predictions [47].The work will be looking at including various control signals, to help improve the performance of the real-time predictive emulator for the MAST Tokamak, performing longer forecasting.\n\n\nA.2. Single blob with non-uniform temperature\n\nFigure 1 .\n1\nFigure 1.(a): Fast camera image of a MAST plasma during an Edge-Localised-Mode, where plasma filaments are ejected outwards due to MHD instabilities at the plasma edge.Reprinted from [19], with the permission of AIP Publishing.(b): The evolution of the filamentary structures during the multi-mode ELM simulation imaged with a synthetic fast camera diagnostic (time given in ms).Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020.\n\n\nTable 1 .\n1\nDomain range and sampling strategies across the initial condition parameters.The case numbers represent the simulation settings in which that parameter was sampled.1-single isothermal blob, 2-single blob with non-uniform temperature, 3-multiple blobs with non-uniform temperature.U [a, b] refers to a uniform distribution bounded by a and b.\n\n\nFigure 2 .\n2\nFigure 2. Evolution of multiple blobs inside a slab for the first 500 time-steps of the numerical simulation (the full run is 2000 steps), taken at intervals of 100 steps (about 15 µs).The top row shows the density, the middle row the temperature, and the bottom row the vorticity.\n\n\nFigure 3 .\n3\nFigure 3. Views of the fast cameras on MAST in the absence of plasma, obtained using illuminations on the 3D CAD model of MAST.Camera shown in figure (a), looks across the central solenoid, while the camera in figure (b) shows the view at the divertor.\n\n\nFigure 4 .\n4\nFigure 4. Modified architecture of a single Fourier Layer.FNO is constructed by stacking multiple Fourier layers on top of each other, sandwiched by a lifting and projection operation.\n\n\nFigure 5 .\n5\nFigure 5. Schematic representation of the two kinds of training configurations deployed within this study.For the simulations, we use the autoregressive configuration (left).We only use the initial set of field values to propagate towards the desired time step, where the intermediary model output is fed back into the model as inputs to further the time evolution.For the experimental data, we use a sequential time window-based configuration.Here, the model output is not factored back into the model, but we perform a fixed window mapping across time.As we go across the time duration of the data, the window is slid further by one-time instance (right).In the example above, we have T_in = 3, step = 1, T_out = 8, the time focus of each forward propagation of the model is given along the x-axis while the evolution across the full-time domain is given along the y-axis.Each cell block represents the 2D field value at a certain instance of time.The input and output sizes shown in this diagram are representative only and vary across each experiment.\n\n\nFigure 6 .\n6\nFigure 6.Learning curves associated with training the multi-variable FNO across the various the various Reduced MHD Cases.Isothermal blob refers to the single blob diffusion under uniform temperature, Anisothermal blob refers to the single blob under a non-uniform temperature, and Anisothermal multiple Blobs extend that to the case with several density blobs under a non-uniform temperature distribution.\n\n\nFigure 7 .\n7\nFigure 7. Isothermal Blob: Temporal evolution of (a) density and (b) electric potential of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\n\n\nFigure 8 .\n8\nFigure 8. Single Blob : Temporal evolution of (a) density and (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\n\n\nFigure 9 .\n9\nFigure 9. Multiple Blobs with non-uniform temperature: Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\n\n\nFigure 10 .\n10\nFigure 10.Error growth across time roll-out of the FNO in the physical space.The error in emulating each variable of the reduced MHD model increases as we roll further out in time.\n\n\nFigure 11 .\n11\nFigure 11.Comparing the error growth (mean absolute error in the normalised domain) of different variables across time roll-out of the Individual FNO (Single) and the Multi-variable FNO (Multi) for the various MHD cases under study.(a) Single Isothermal Blob (b) Single Blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The multi-variable FNO with the same parameters as the individual FNOs demonstrates much better performance over time.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.\n\n\nFigure 12 .\n12\nFigure 12.Comparing the cumulative error growth of different variables across time roll-out for multi-variable FNOs with different step sizes.The autoregressive nature of the FNO brings about a spike in error, characterising a disconnection at the time step that is a multiple of the output size.(a) Isothermal blob (b) Single blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.\n\n\nFigure 14 .\n14\nFigure 14.Evolution of the electric potential (Multiblobs) in time as modelled by multi-variable FNOs at different modes.Top row of (a) shows the actual solution modelled using JOREK.The bottom row of (a), plots the time evolution of the FNO with 2 modes, top row of (b)is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better.\n\n\nFigure 15 .\n15\nFigure15.Impact of the size of the training dataset on the performance of the multi-variable FNO for simulating the multiple plasma blobs in a non-uniform temperature field.As we increase the availability of the training data for the multi-variable FNO, the model performance increases across the same test data.However, the gains obtained by increasing the training data are marginal compared to the additional compute required.The number of simulations used for training is given on the x-axis while the mean squared error is given on the y-axis.All training is performed on a single Nvidia A100.\n\n\nFigure 16 .\n16\nFigure 16.Learning curves associated with the models trained with data of different sizes.Training loss is represented by full lines, while the test loss is represented by dashed lines.With just 500 simulation instances, the model learns the physics of the model really well, which is only marginally improved by more training data.We also notice the quick over-fitting tendency of the FNO as observed in[38].\n\n\nFigure 17 .\n17\nFigure 17.Demonstrating Zero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE(in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the bottom series of plots demonstrate the time evolution as emulated by the multi-variable FNO.\n\n\nFigure 18 .\n18\nFigure 18.Outside the training domain: Inferring on a case with 13 blobs with amplitude and density twice more and the blob width 1.5 times more than the maximum used within the training data (see table 1).Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\n\n\nFigure 19 .\n19\nFigure 19.Temporal plots of (a) Plasma Current and (b) Total Heating Power of the shots selected for training and testing the FNO for modelling the camera data.The shots depicting the training data is given in black, whereas the test data is given in red.The plots help convey the diversity of plasma scenarios the FNO is trained to model over.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data.\n\n\n20 .\n20\nPredicting in L-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera with the plasma in L-mode.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\n\n\nFigure 21 .\n21\nFigure 21.Predicting in H-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera wiht the plasma in H-mod.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The domain is selected to showcase the performance of the FNO in mapping the plasma evolution in H-mode.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\n\n\n\n\nprovides a panoramic view of the reactor, showing the poloidal cross-sectional layout on both sides of the central solenoid.The main camera has a spatial resolution of 448 × 640 for the experiments that we sampled.We sampled a total of 55 plasma shots from the MAST database, used 50 for training and 5 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 7661, and the test points grow to 828 data points.The model takes 27 h 45 min on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.7 × 10 −3 .For further plots showcasing the forecasting capabilities of the FNO at the plasma around the central solenoid, refer to appendix E in figures E1 and E2.\n\n\nFigure 22 .\n22\nFigure 22.Predicting around the L-H transition: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.\n\n\nFigure 23 .\n23\nFigure 23.Long-term prediction: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.\n\n\nFigure 24 .\n24\nFigure 24.Single forward pass: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\n\n\nFigure 25 .\n25\nFigure 25.Long-term prediction: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\n\n\nFigure\n\nFigure Single Blob with non-uniform Temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\n\n\nFigure A4 .\nA4\nFigure A4.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\n\n\nFigure A5 .\nA5\nFigure A5.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\n\n\nFigure\n\nFigure DemonstratingZero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE (in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the middle series of plots demonstrate the time evolution as emulated by the multi-variable FNO and the bottom series indicates the absolute error across space.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nTable 2 .\n2\nOverview of the performance obtained by various FNO-based surrogate models built to emulate the plasma evolution in both simulations and experiments.The case represents the various simulation settings of JOREK or the camera diagnostic on MAST, for which we are training an emulator.The variables mention the number of variables from the PDE/diagnostic that we model together.Parameters refer to the number of parameters in the trained FNO, and MSE represents the mean squared error in the normalised domain.The Training Size represented the simulation/experimental input-output pairs.All models are trained on a single Nvidia A100 chip.\nCaseVariablesParametersTrain Time (m)Training SizeMSESingle blob-isothermal26321 000481007.58 × 10 −5Single blob-non-uniform temperature39467 000971602.56 × 10 −5Multiple blobs-non-uniform temperature39467 00068215001.33 × 10 −4Camera-Solenoid (rbb)1201 900156876611.97 × 10 −3Camera-Divertor (rba)1201 90093558513.80 × 10 −3each variable individually. Considering the simplicity of thereduced MHD case used in\nhttps://ccfe.ukaea.uk/research/mast-upgrade/.\nhttps://photron.com/.\nAppendix A. Error plotsAppendix B. Multi-variable FNO architectureThe architecture of the multi-variable FNO deployed in an auto-regressive manner for emulating the Reduced MHD equations.The model initially lifts the input data into a higher dimensional space using a linear layer.This is followed by 6 Fourier layers that learn the necessary modal mappings within the spatial domain, before being projected onto the lower dimensional space by a series of linear layers.Fourier 2d represents the 2D Fourier operator, features only extracted within the spatial domain, Conv3d represents the 3D convolutional layer across the variable channel and the spatial domain, Add operation adds the outputs from the Fourier layer and the convolutional layer together, GELU denotes the activation function: Gaussian Error Linear Unit.Appendix D. Camera FNO architectureThe architecture of the Individual FNO deployed in an auto-regressive manner for emulating the Camera data.The model initially lifts the input data into a higher dimensional space using a linear layer.This is followed by 6 Fourier layers that learn the necessary modal mappings within the spatial domain, before being projected onto the lower dimensional space by a series of linear layers.Fourier 2d represents the 2D Fourier operator, features only extracted within the spatial domain, Conv2d represents the 3D convolutional layer across the spatial domain, Add operation adds the outputs from the Fourier layer and the convolutional layer together, GELU denotes the activation function: Gaussian Error Linear Unit.Appendix G. MAST Shots used for camera dataShots used for the FNO modelling the plasma evolution as captured by the rbb camera looking at the central solenoid:Training shots:ORCID iDVignesh Gopakumar https://orcid.org/0000-0003-0904-\nSimulations of edge localised mode instabilities in MAST-U Super-X tokamak plasmas. S Smith, S Pamela, A Fil, M Hölzl, G Huijsmans, A Kirk, D Moulton, O Myatra, A Thornton, H Wilson, 10.1088/1741-4326/ab826aNucl. Fusion. 60660212020\n\nThree-dimensional plasma edge turbulence simulations of the Mega Ampere Spherical Tokamak and comparison with experimental measurements Plasma Phys. Control. F Riva, F Militello, S Elmore, J T Omotani, B Dudson, N R Walkden, 10.1088/1361-6587/ab3561Fusion. 61950132019the MAST team\n\nThe JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas. M Hoelzl, 10.1088/1741-4326/abf99fNucl. Fusion. 61650012021\n\nJINTRAC: a system of codes for integrated simulation of tokamak scenarios Plasma Fusion Res. M Romanelli, 10.1585/pfr.9.3403023201493403023\n\nThe new SOLPS-ITER code package. S Wiesen, 10.1016/j.jnucmat.2014.10.012J. Nucl. Mater. 4632015\n\nA Lavin, arXiv:2112.03235Simulation intelligence: towards a new generation of scientific methods CoRR. 2021\n\nFast modeling of turbulent transport in fusion plasmas using neural networks Phys. K L Van De Plassche, J Citrin, C Bourdelle, Y Camenen, F J Casson, V I Dagnelie, F Felici, A Ho, S Van Mulders, 10.1063/1.5134126Plasmas. 27223102020\n\n2021 Neural network surrogate of QuaLiKiz using JET experimental data to populate training space. A Ho, J Citrin, C Bourdelle, Y Camenen, F J Casson, K L Van De Plassche, H Weisen, 10.1063/5.0038290Phys. Plasmas. 2832305\n\nSelf-consistent core-pedestal transport simulations with neural network accelerated models. O Meneghini, 10.1088/1741-4326/aa7776Nucl. Fusion. 57860342017\n\n2023 Fast regression of the tritium breeding ratio in fusion reactors. G V G Mánek, V Gopakumar, N Nikolaou, J Shimwell, I Waldmann, 10.1088/2632-2153/acb2b3Mach. Learn.: Sci. Technol. 415008\n\nImage mapping the temporal evolution of edge characteristics in tokamaks using neural networks. V Gopakumar, D Samaddar, 10.1088/2632-2153/ab5639Mach. Learn.: Sci. Technol. 1150062020\n\nTowards fast surrogate models for interpolation of tokamak edge plasmas. S Dasbach, S Wiesen, 10.1016/j.nme.2023.101396Nucl. Mater. Energy. 341013962023\n\n2023 Neural operator: learning maps between function spaces. N Kovachki, Z Li, B Liu, K Azizzadenesheli, K Bhattacharya, A Stuart, A Anandkumar, J. Mach. Learn. Res. \n\nPhysics-informed neural operator for learning partial differential equations ACM. Z Li, H Zheng, N B Kovachki, D Jin, H Chen, B Liu, A Stuart, K Azizzadenesheli, A Anandkumar, 10.1145/3648506/ IMS J. Data Sci. 2022\n\nF Bartolucci, E De Bézenac, B Raonić, R Molinaro, S Mishra, R Alaifari, Representation equivalent neural operators: a framework for alias-free operator learning. 2023\n\nThe cost-accuracy trade-off in operator learning with neural networks. De Hoop, M Huang, D Z Qian, E Stuart, A M , 10.4208/jml.220509J. Mach. Learn. Res. 12992022\n\nFast dynamic 1D simulation of divertor plasmas with neural PDE surrogates. Y Poels, G Derks, E Westerhof, K Minartz, S Wiesen, V Menkovski, 10.1088/1741-4326/acf70dNucl. Fusion. 631260122023\n\nV Gopakumar, MAST TeamS Pamela, MAST TeamL Zanisi, MAST TeamZ Li, MAST TeamA Anandkumar, MAST TeamarXiv:2302.06542) 2023 Fourier neural operator for plasma modelling. \n\nOptical boundary reconstruction of tokamak plasmas for feedback control of plasma position and shape. G Hommen, M De Baar, G M Nuij, R Akers, M Steinbuch, 10.1063/1.3499219Rev. Sci. Instrum. 811135042010\n\nX Shi, Z Chen, H Wang, D.-Y Yeung, W.-K Wong, W.-C Woo, arXiv:1506.04214Convolutional LSTM network: a machine learning approach for precipitation nowcasting. 2015\n\nAssociation of genomic subtypes of lower-grade gliomas with shape features automatically extracted by a deep learning algorithm Comput. M Buda, A Saha, M A Mazurowski, 10.1016/j.compbiomed.2019.05.002Biol. Med. 1092019\n\nInvestigation of the effect of resistivity on scrape off layer filaments using three-dimensional simulations Phys. L Easy, F Militello, J Omotani, N R Walkden, B Dudson, 10.1063/1.4940330Plasmas. 23125122016\n\nOn the interaction of scrape off layer filaments Phys. F Militello, B Dudson, L Easy, A Kirk, P Naylor, 10.1088/1361-6587/aa9252Plasmas. 591250132017\n\net al (the MAST Team) 2006 Filament structures at the plasma edge on mast Plasma Phys. Control. A Kirk, 10.1088/0741-3335/48/12B/S41Fusion. 48B433\n\nThe physics of turbulence localised to the tokamak divertor volume. N Walkden, F Riva, J Harrison, F Militello, T Farley, J Omotani, B Lipschultz, 10.1038/s42005-022-00906-2Commun. Phys. 51392022\n\nC Ham, MAST using high speed visible camera data IAEA 2nd Technical Meeting on Plasma Disruptions and Their Mitigation (ITER Headquarters. 2022. 19-22 July 2022\n\nZ Li, N Kovachki, K Azizzadenesheli, B Liu, K Bhattacharya, A Stuart, A Anandkumar, arXiv:2010.08895Fourier neural operator for parametric partial differential equations. 2020\n\nThe fast Fourier transform IEEE Spectr. E O Brigham, R E Morrow, 10.1109/MSPEC.1967.521722019674\n\nM Dalal, A C Li, R Taori, arXiv:1910.07737Autoregressive models: what are they good for?. 2019\n\n. R K Srivastava, K Greff, J Schmidhuber, arXiv:1505.003872015Highway networks\n\nGradient-based learning applied to document recognition Proc. Y Lecun, L Bottou, Y Bengio, Haffner, 10.1109/5.726791IEEE. 861998\n\nD Kingma, J Ba, arXiv:1412.6980Adam: a method for stochastic optimization. 2014\n\nM Bellan, Fundamentals of Plasma Physics. Cambridge University Press2006\n\nA Markovian decision process. R Bellman, J. Math. Mech. 61957\n\nY Lee, Autoregressive renaissance in neural PDE solvers ICLR Blogposts 2023. Kigali Rwanda2023. 1-5 May\n\nJ Brandstetter, D Worrall, M Welling, arXiv:2202.03376Message passing neural PDE solvers. 2023\n\nJ Zhao, R J George, Z Li, A Anandkumar, arXiv:2211.15188Incremental spectral learning in Fourier neural operator. 2023\n\nS Lanthaler, Z Li, A M Stuart, arXiv:2304.13221The nonlocal neural operator: universal approximation. 2023\n\nCan neural networks extrapolate? Discussion of a theorem by. A Courtois, J.-M Morel, P Arias, 10.1007/s13398-023-01411-zPedro Domingos Rev. Real Acad. Cienc. Exactas Fis. Nat. A. 117792023\n\nA Jacot, F Gabriel, C Hongler, arXiv:1806.07572Neural tangent kernel: convergence and generalization in neural networks. 2020\n\nJ Kossaifi, N Kovachki, K Azizzadenesheli, A Anandkumar, arXiv:2310.00120Multi-grid tensorized Fourier neural operator for high-resolution PDEs. 2023\n\nC Aggarwal, X Kong, Q Gu, J Han, P Yu, Active Learning: A Survey. CRC Press2015\n\nJET Contributors 2024 Efficient training sets for surrogate models of tokamak turbulence with active deep ensembles. L Zanisi, A Ho, J Barr, T Madula, J Citrin, S Pamela, J Buchanan, F Casson, V Gopakumar, 10.1088/1741-4326/ad240dNucl. Fusion. 6436022\n\nApplications of physics informed neural operators. S G Rosofsky, H A Majed, E A Huerta, 10.1088/2632-2153/acd168DownloadArticlePDFMach. Learn.: Sci. Technol. 4250222023\n\nG Kohl, L.-W Chen, N Thuerey, arXiv:2309.01745Turbulent flow simulation using autoregressive conditional diffusion models. 2023\n\n. S Silburn, 10.5281/zenodo.68915042022\n\nFourier-RNNs for modelling noisy physics data IEEE-ICMLA Bahamas. V Gopakumar, S Pamela, L Zanisi, arXiv:2302.065342023. Dec\n"},"corpus_id":{"kind":"number","value":265129030,"string":"265,129,030"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2311.05967\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/abs-2311-05967\",\n \"doi\": \"10.1088/1741-4326/ad313a\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2311.05967v1.pdf\"\n ],\n \"pdf_sha\": \"ebe9e56ef598d8885f5eddccaf422da0e1a243cb\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://iopscience.iop.org/article/10.1088/1741-4326/ad313a/pdf\",\n \"status\": \"HYBRID\"\n }\n}"},"title":{"kind":"string","value":"Plasma surrogate modelling using Fourier neural operators"},"authors":{"kind":"list like","value":["Vignesh Gopakumar vignesh.gopakumar@ukaea.uk 0000-0003-0904-3448","Stanislas Pamela 0000-0003-0904-3448","Lorenzo Zanisi","Zongyi Li","Ander Gray","Daniel Brennand","Nitesh Bhatia","Gregory Stathopoulos","Matt Kusner","Marc Peter Deisenroth","Anima Anandkumar","1 UK Atomic Energy Authority,\nOxford OX14 3EB, United Kingdom of Great Britain and Northern Ireland London WC1E 6BT, United Kingdom of Great Britain and Northern Ireland Liverpool L69 3BX, United Kingdom of Great Britain and Northern Ireland Pasadena, CA 91125, United States of America","2 University College London,","3 University of Liverpool,","4 Caltech,"],"string":"[\n \"Vignesh Gopakumar vignesh.gopakumar@ukaea.uk 0000-0003-0904-3448\",\n \"Stanislas Pamela 0000-0003-0904-3448\",\n \"Lorenzo Zanisi\",\n \"Zongyi Li\",\n \"Ander Gray\",\n \"Daniel Brennand\",\n \"Nitesh Bhatia\",\n \"Gregory Stathopoulos\",\n \"Matt Kusner\",\n \"Marc Peter Deisenroth\",\n \"Anima Anandkumar\",\n \"1 UK Atomic Energy Authority,\\nOxford OX14 3EB, United Kingdom of Great Britain and Northern Ireland London WC1E 6BT, United Kingdom of Great Britain and Northern Ireland Liverpool L69 3BX, United Kingdom of Great Britain and Northern Ireland Pasadena, CA 91125, United States of America\",\n \"2 University College London,\",\n \"3 University of Liverpool,\",\n \"4 Caltech,\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Predicting plasma evolution within a Tokamak reactor is crucial to realizing the goal of sustainable fusion.Capabilities in forecasting the spatio-temporal evolution of plasma rapidly and accurately allow us to quickly iterate over design and control strategies on current Tokamak devices and future reactors.Modelling plasma evolution using numerical solvers is often expensive, consuming many hours on supercomputers, and hence, we need alternative inexpensive surrogate models.We demonstrate accurate predictions of plasma evolution both in simulation and experimental domains using deep learning-based surrogate modelling tools, viz., Fourier neural operators (FNO).We show that FNO has a speedup of six orders of magnitude over traditional solvers in predicting the plasma dynamics simulated from magnetohydrodynamic models, while maintaining a high accuracy (Mean Squared Error in the normalised domain ≈10 −5 ).Our modified version of the FNO is capable of solving multi-variable Partial Differential Equations, and can capture the dependence among the different variables in a single model.FNOs can also predict plasma evolution on real-world experimental data observed by the cameras positioned within the MAST Tokamak, i.e. cameras looking across the central solenoid and the divertor in the Tokamak.We show that FNOs are able to accurately forecast the evolution of plasma and have the potential to be deployed for real-time monitoring.We also illustrate their capability in forecasting the plasma shape, the locations of interactions of the plasma with the central solenoid and the divertor for the full (available) duration of the plasma shot within MAST.The FNO offers a viable alternative for surrogate modelling as it is quick to train and infer, and requires fewer data points, while being able to do zero-shot super-resolution and getting high-fidelity solutions."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Predicting the evolution of plasma dynamics is crucial to building sustainable fusion devices as it allows us to anticipate plasma behaviour and make real-time decisions over the input control parameters and alter the trajectory of plasma evolution.Such predictions also allow us to improve our understanding of the physics governing plasma interactions.In particular, Magnetohydrodynamic (MHD) modelling is of critical importance as it improves our understanding of the dynamical evolution of filaments that burst out of the plasma edge and reach the Tokamak's first wall, as shown in figure 1(a) [1].These plasma filaments, called edge-localised modes (ELMs), are described by non-linear MHD, and can be modelled numerically, as shown in figure 1(b).\nIn fusion research, prediction of future plasma states is mostly undertaken using physics-based numerical modelling codes, such as BOUT++ [2], JOREK [3], JINTRAC [4] and SOLPS [5].However, these simulations are computationally intensive, taking weeks for effective simulations to be run on high-performance computers [2].They are also often complex in terms of software infrastructure and advanced solver library dependencies, making them difficult to deploy for rapid iterative modelling.With numerical solvers, we also encounter cases where the simulations do not converge due to model mismatch or other issues.\nMachine learning (ML) offers an alternative data-driven path for obtaining quick, inexpensive approximations to numerical simulations, allowing for rapid iterative modelling [6].Within plasma physics, ML-based surrogate models have been used to provide quick approximations for a range of solutions, from emulating turbulent transport [7][8][9], estimating the Tritium breeding ratio across Tokamak designs [10] to modelling edge plasma [11,12].\nRecently, a new class of data-driven ML models, known as neural operators (NO), has recently been at the forefront of surrogate modelling.Unlike other surrogate models, they directly learn the solution operator of the partial differential equation (PDE) in a discretization-convergent manner [13,14].Specifically, NO can predict at any resolution, unlike standard neural networks like convolutional neural networks.In other words, they learn the underlying continuum, making them suitable for learning solutions of PDEs.The output of NO converges to a unique function upon mesh refinement [15].NO approaches are more data-efficient compared to other surrogate models [16], converge faster than traditional surrogate models, and can generalise better to various PDE settings.Considering these advantages, NO-based surrogate models have found application in emulating the plasma evolution across the domain of a Tokamak [17,18]."},{"title":"Our Contributions","content":"• We demonstrate trained Fourier neural operators (FNOs) that can be deployed as an emulator to perform real-time forecasting of the evolution of plasma as characterised by fast cameras on the MAST Tokamak.Our studies show that the FNO is capable of modelling the plasma across the full (available) shot duration ranging from the ramp-up to the L and H modes of operation.• We devise a modified version of the FNO, capable of handling multiple variables that govern a family of PDEs.The multi-variable FNO allows us to pass field information of Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020.\nall the interested variables to the model, allowing for learning and encoding the mutual dependence of variables, which is essential for capturing the non-linear effects.• We demonstrate the utility of the multi-variable FNO in (surrogate) modelling the evolution of plasma across a range of reduced MHD cases with increasing complexity.We conduct an extensive analysis of the various features of the FNO, performing ablation studies that evaluate the impact of various choices of the Hyperparameters across the model and training.\nIn this paper, we undertake an extensive study of the FNO due to its previously established superior cost-accuracy trade-off in both training and testing [16].We use NO to model plasma within fusion devices, in both the simulation space and within the experimental domain.For the simulation domain, we consider the reduced MHD models to provide data to train the neural operator [3].We study the solution of MHD equations in two spatial dimensions with multiple variables of interest.\nThe simulations that we are interested in modelling represent the diffusion of a blob of plasma within a specified domain.\nThe simulations presented here describe a simplified scenario to that of a full ELM simulation as given in figure 1(b).We iterate over increasingly complex cases of fluid models that govern the diffusion of a plasma blob in toroidal geometry.\nIn addition to numerical modelling, we demonstrate that FNOs can be trained to predict the evolution of experimental plasma as observed by high-speed cameras on the MAST spherical Tokamak.The ability to predict future plasma states based on camera imaging, as well as other plasma diagnostics, could allow us to design efficient control feedback loops for future fusion reactors.We build emulators of the fast camera that views across the central solenoid and at the divertor.We find these models to be capable of accurately predicting plasma evolution in real-time.\nIn this study, we use FNOs for both MHD simulations and experimental camera data due to the structural similarities of the plasma observed in both.The visual similarity of the MHD modelled in the simulation, and that captured by camera images in a fusion device is elucidated in figure 1(a).The fast camera images capture the MHD behaviour that is observed experimentally.Through this work, the FNO's utility is demonstrated in predicting both experimental and simulation dynamics.Although this study considers a simplified simulation setup, with filament cross-sections, in a 2D poloidal plane, instead of the full 3D representation, this paper lays the groundwork for future studies where surrogates could be trained from both simulation and experimental data simultaneously.However, in section 3.1.10,arguments of the challenges and potential methods of extending to a 3D modelling case are discussed.\nA preliminary version of this work was presented at the AI for Science workshop during NeurIPS 2022, where we demonstrated, as a proof-of-concept, the utility of the FNO in modelling MHD scenarios [18].In the previous work, we conducted a benchmark of the performance of the FNO against more traditional surrogate model architectures such as Convolutional-Long-short term memory [20] and U-Net [21].Our findings established that the FNO takes significantly less time to train, requires fewer parameters, and outperforms the Convolutional-LSTM and the U-Net in emulating the diffusion of a blob of plasma described by reduced MHD [18].\nThe paper is organised as follows: section 2 produces a full description of the datasets and methods involved in the paper, ranging from the physics description of the simulations, the diagnostic setup of the experimental camera data and a formal definition of the FNO, its training configuration and the extension to the multi-variable FNO.Section 3 outlines the results of our experiments with using the FNO to learn and predict results from various simulation settings and camera configurations.The section comprises an in-depth analysis of the multivariable FNO trained to solve different PDEs, exploring the impact of hyperparameters, training regime, and the training data.We also include a performance comparison of the individual FNO with that of the multi-variable FNO.Section 3 is concluded with the experimental results from the FNO trained to model the plasma evolution as captured by the visible camera looking at the central solenoid and the divertor.We summarise our key results in section 4 and conclude with a discussion on future extensions of the work."},{"title":"Datasets and methods","content":""},{"title":"MHD simulations of plasma Blobs","content":"The FNO as a Neural-PDE solver is tested on three datasets of simplified MHD models, simulating the radial convection of plasma blobs in toroidal geometry, with coordinates system (R, Z, ϕ).These three cases are designed with increasing complexity to test the utility, versatility, and capabilities of the FNO to emulate MHD models.The simplified MHD model is similar to the so-called Reduced-MHD model from [3], but with all magnetic fields and currents kept to zero, which is similar to the Euler system, and can be summarised by the continuity, momentum, and pressure equations:\nwhere ρ is the density, p the pressure, T the temperature, and ⃗ v the velocity.D is the diffusion coefficient, µ the viscosity, and κ the thermal conductivity.The ratio of specific heats γ is taken to be that of a monatomic gas, 5 3 .To obtain the Reduced-MHD model, the momentum equation ( 2) is projected in the poloidal plane using the operation ∇ϕ • ∇ × (R 2 . ..), while assuming the poloidal velocity takes the form ⃗ v = R 2 ∇ϕ × ∇Φ, where the electric potential Φ is equivalent to the velocity stream function in toroidal coordinates.This projection results in a system of three equations for the three variables ρ, Φ, T. A pseudo-variable is also used for the vorticity to reduce the level of derivatives in the system: w = ∇ 2 Φ.We refer the reader to [3] and references therein for the full system of Reduced-MHD equations.\nThe simulations are all run in a square slab geometry of 1 m in width and height, centred at major radius R = 10 m.The runs are performed with toroidally axisymmetric plasma blobs initialised on top of a low background density and temperature.In the absence of a plasma current equilibrium to hold the density blobs in place, the pressure gradient term in the momentum equation generates a buoyancy effect, causing the blobs to move outwards.The simulations are run with the JOREK code, which is routinely used to model non-linear MHD instabilities in realistic geometries [3].In this paper, we restrict ourselves to two simplified MHD models.\nA further reduction of the model can be obtained by using constant temperature and removing equation (3) from the system, therefore evolving only ρ and Φ.With this isothermal model, the pressure gradient term in the momentum equation (2), which is proportional to ρ, acts as a buoyancy force on the blob, affecting its momentum ρ⃗ v, which is also proportional to ρ.In other words, the velocity of the blob will be the same no matter what its density is.To have blobs moving at different speeds, the temperature inside the blob is needed: the higher the temperature, the faster the blob will move.The values of the diffusive parameters are kept constant inside the entire domain, for all simulations, at\nThe background density level is set to 10 18 m −3 , and the background temperature is set to 1 eV (for simulations that evolve temperature).\nThree datasets are considered: with a single isothermal blob, with a single blob with non-uniform temperature, and with multiple blobs with non-uniform temperature.To create the datasets, the blob's initial conditions are varied.The sampled parameters are the number of blobs, their positions, their width, and their density and temperature amplitudes.This is summarised in table 1.For datasets with a single blob, the initial position is fixed at the centre of the domain, (R, Z) = (10.0,0.0).The smaller single-blob datasets are composed of 120 simulations (100 for training, 20 for testing), while the main dataset with multiple blobs contains 2000 simulations.\nAll cases are run with the same time-step size of 0.15 µs, for 2000 time-steps in total.This is long enough for the blobs to evolve radially until they reach the wall (where Dirichlet boundary conditions are applied) and disappear through both convective mixing and diffusion.Since a small time-step size is necessary in the simulations to ensure numerical stability, the datasets are down-sampled to 200 slices for the training set.Likewise, the 2D poloidal grid resolution, which is 200 × 200 bi-cubic finite-elements in the simulations, is down-sampled to a regular grid of 100 × 100.The smaller single-blob datasets were run on the EOSC-hub cloud using 16 cores per run, and the larger dataset of 2000 multi-blob simulations was run on CINECA's Marconi HPC cluster using 48 cores per run.Each simulation requires approximately 350 core hours.Though each simulation is down-sampled to 200 time instances, we keep the focus of our emulator on short-term evolution up to the first 50 time steps.\nSimulations of blobs in slab geometry are related to turbulence and MHD instabilities at the plasma edge of a Tokamak by assuming that they represent a poloidal cross-section of a filament.As seen in figure 1, filament structures are aligned to the magnetic field, they are elongated in the toroidal direction, but have a nearly circular cross section in the poloidal plane.Such slab simulations represent the dynamic evolution of these filaments in the plane perpendicular to the magnetic field.Extensive modelling of blobs in slab geometry can be found in [22,23] and references therein.Figure 2 shows an example of a numerical simulation of multiple blobs evolving for the first 500 steps (out of 2000), with the density, temperature, and vorticity.Filaments with higher temperatures can be seen to move at a faster velocity."},{"title":"Fast-camera images on MAST Tokamak","content":"MAST was equipped with fast camera imaging diagnostics at several points across the Tokamak for capturing images within the visible spectrum, capturing the plasma evolution in real-time 5 .\nThese Photron cameras 6 are essential diagnostics that contributed to fundamental understandings of key plasma phenomena in Tokamaks in recent years [24].While most of the camera data has been used for qualitative analysis in the past, it can be exploited using more advanced methods to provide statistical insight into plasma turbulence [25], as well as global instabilities (disruptions) [26].Within the scope of this work, we focus on two camera views.The first looks at the central solenoid and the plasma confined around it, extending to the walls and providing a holistic view of the plasma (rbb camera).\nThe second camera configuration under study is that providing the divertor-view, looking at the bottom of the Tokamak, where energy travels from the plasma to the exhaust region (rba camera).The views of the fast cameras on MAST in the absence of the plasma is visualised in figure 3. Figure 1 shows a view of the main plasma as seen by the camera, together with a synthetic rendering (right) of the camera, simulating the MHD physics.The cameras have a temporal resolution of 1.2 ms, with spatial resolution varying according to the desired calibration setup for each experiment.They are tuned to visible wavelength and, therefore, capture mostly the Balmer D α light emitted mostly from the plasma edge.As seen in figure 1, the fast-visible cameras are able to capture complex dynamics of plasma filamentary eruptions at the plasma edge, the ELMs [24], hence being able to predict real-time evolution allows us to anticipate such plasma instabilities and other confinement related characteristics.Within the scope of this project, our focus relies on modelling the plasma confinement and its interactions with wall accurately, leaving the modelling of ELMs as the next potential research avenue."},{"title":"FNO","content":"The FNO [27] is a neural network-based operator learning method [13] that learns a mapping between two function spaces from a finite collection of observed input-output pairs.In the Neural Operator (NO) setting, we learn the mapping\nG θ is a neural network configuration parameterised by θ and is composed of three elements:\n(i) Lifting: A fully local, point-wise operation that projects the input domain to a higher dimensional latent representation a ∈ R da → ν 0 ∈ R dν 0 .(ii) Iterative Kernel Integration: Expressed as a sum of a local, linear operator and a non-local integral kernel operator, that maps from ν t → ν t+1 , iterated from ν 0 to ν T .The kernel integration is followed by a point-wise nonlinearity.\n(iii) Projection: Similar to lifting, characterised by a fully local point-wise operation that maps the latent representation to that of the output domain\nIn equation (5), σ denotes the nonlinear activation, W represents the local, linear operator, and κ represents the chosen kernel which is parameterised by ϕ. x represents the input that is fed into that layer.\nConventionally, the lifting and projection within an NO are characterised by linear layers acting on the respective feature spaces.The kernel integration operator is characterised as the neural layer that evaluates a chosen kernel over the higher dimensional latent representation fed into it:\n(κ (a; ϕ) ν t ) (x) = ˆD κ (x, y, a (x) , a (y) ; ϕ ) ν t (y) dy.(6) As demonstrated in [27], the FNO is devised by characterising the kernel integral operator as an operation within the Fourier space.Replacing κ with the Fourier integral operator F(κ ϕ ) we obtain\nHere, F and F −1 represent the Fourier and inverse Fourier transform, respectively.κ, the kernel is parameterised by learnable weights ϕ ∈ θ κ .Within the FNO, the kernel integrator reduces to the inverse Fourier transform of the inner product between the Fourier integral operator F(κ ϕ ) and the Fourier transform of the input to that Fourier layer F(ν t ).\nIn practice, the iterative kernel integral transform with Fourier transform is characterised by two learnable weight matrices R and W as shown in figure 4. R learns the mapping behaviour within the Fourier space, while W learns the mapping required within the input Euclidean space.R characterises the non-local integral kernel operator, while W represents the local linear operators.The weight matrix R within the Fourier layer enables convolution within the Fourier space, allowing the network to be parameterised directly in it.The grid discretisation is also fed in at each layer to help the network learn the discretised Fourier representation of the data and undergoes point-wise linear transformations, characterised by C, its operation is similar to that of W. The output of a Fourier layer can be expressed as where x is the input to that layer, y the output, g the grid discretisation, b the bias and σ is the nonlinear activation function.Equation ( 8) resembles a single Fourier layer within the FNO.Multiple Fourier layers are stacked on top of each other to complete the iterative kernel integral operation.The stacked Fourier layers are sandwiched between the lifting and projection operations to complete the FNO architecture."},{"title":"Comparison to convolutions.","content":"The spectral convolutions characterised by κ in equation ( 7) have an analogous relationship to the convolutional operation.Convolution, being a product of functions across a certain range, is an integral transform.Expressed across an infinite range, it takes the mathematical form\nwhere f and g are functions over which the convolution operation is performed.By replacing the convolution operation with the Fourier transform in equation ( 9), we obtain\nComparing equations ( 9) and ( 10), it becomes evident how equation ( 7) is a special form of the convolutional operation, one that takes place in the Fourier space.In practice, both operations are applied across a discretisation, characterised by the data that is fed to the model, but since we decompose the data into Fourier modes within the FNO and apply learning across the decomposed components, we can learn continuous representations of the required mapping as opposed to learning discrete representations with conventional convolutions.Named spectral convolutions in [27], allow for several advantages over a standard convolutional neural network, where filters are directly learned without a spectral transform.Firstly, by learning within the Fourier space, we allow the network to identify global features that are embedded within the data, highlighting the dominant modes in the process.By performing the Fourier decomposition over the data as it parses through the network, the weights are tuned to identify and learn the modes that possess a majority of the information, allowing the network to generalise better.Secondly, the spectral convolutions being in the function space allows for learning the mapping in a discretisation convergent manner, allowing for learning mesh-free solutions.Lastly, the Fourier transform in equation ( 10) is implemented using a Fast Fourier Transform (FFT) [28] which has a computational complexity of O(n log n), whereas the standard convolution operation without spectral transform has a complexity of O(kn 2 ), where k is the size of the kernel and n is the representation dimension.However, within the FNO, we only deal with a complexity of O(nk max ) since we truncate the Fourier modes up to a desired hyperparameter k max ."},{"title":"Training configuration.","content":"For emulating the simulations, we use the FNO-2d configuration [27], where a 2-d FNO which only convolves in space, is deployed in an autoregressive manner [29].The FNO takes in as its input the field values across a 2D grid for an initial set of time instances (T_in) along with the grid discretisation in both dimensions.Similar to spectral numerical solvers, the grid discretisation, fed in with the field data, provides better-structured information for discrete Fourier transform utilised within the FFT.The FNO takes the initial fields as the input and outputs a set of later time instances (step) of the field value across the same grid.The output of the FNO, coupled with the later time steps from the input, are then fed back in an autoregressive manner to estimate further field evolution in time, until the desired time instance (T_out) is reached.Within the autoregressive setting, we roll out the evolution of the fields up until the desired time step and then perform the backpropagation through time to enable learning of the time evolution of the PDE.For further information regarding the theory behind operator learning as ascribed by the FNO and its implementation in various case settings, we invite the reader to refer to the original work in [27].Unlike the FNO 3D configuration, we only perform Fourier transformations across the spatial dimensions and do not include the temporal dimension.We have chosen the 2D configuration of the FNO as it is lighter, quicker, and gives us the flexibility in deciding the length of the temporal evolution.\nFor the case of the FNO for camera, instead of performing an autoregressive time roll out, we perform a fixed time window mapping, where the FNO takes in the camera information for fixed time steps (T_in) and outputs the next set of frames characterising the future plasma state (step).The next set of inputs is obtained by sliding the interested time window by one, as shown in figure 5."},{"title":"Multi-variable FNO.","content":"As depicted in equations ( 1)-( 3), the physics model represents a highly correlated and interdependent system where density, temperature, and electric potential factor into the time evolution equations of each other.Taking this family of PDEs into account, it is vital to construct a surrogate model that is capable of emulating all variables together, allowing for information exchange within the model.The original FNO [27] was designed to model for a single variable and needed to be modified to handle multiple variables.The work done in [18] demonstrates the impact of the FNO on predicting MHD evolution but modelled [18], it was permissible to model the variables independently.However, for the case under consideration, it was imperative that for physical consistency, we model the variables together.As we move into more complex MHD cases, we introduce more correlated variables such as density and temperature, which are both required to evolve together to fully grasp the propagation of the plasma blob.\nTo account for the multi-variable setting, we designed the FNO to take in the variables as a separate channel.Initial construction of the FNO would consider data in the format [batch size, temporal projection, spatial along x-axis, spatial along y-axis].We modified that model to consider data in the format [batch size, variables, temporal projection, spatial along x-axis, spatial along y-axis].This modification allows us to account for all the variables within a single model, which facilitates information exchange across them.By having the multiple variables of the physics model as a channel, we allow for the Fourier transform to be applied across the spatial domain for all variables of interest.The R matrix that allows for learning within the Fourier space in equation ( 8) is modified to account for the new variables, and the local, linear operator W acts across the variable and spatial domain.\nBy providing the variables as a separate channel across the FNO, we introduce the possibility of performing the operations of lifting and projection defined in section 2.3 not just across the temporal features but across the fields as well.Thus, within this configuration of a multi-variable FNO, the feature space becomes two-fold, accounting for the time as well as the variables.We also modify the FNO with skip connections to reduce the information loss as the data progresses through the network [30].The network architecture is outlined under Appendix B in table B1."},{"title":"Results","content":"Prior to elaborating on the finer details of the results from our experiments, we provide an overview of the performance obtained by the FNO surrogates across all the interested modelling tasks in table 2. Across our range of experiments, emulating the plasma, we find that the FNO achieves considerable accuracy (Mean Squared Error (MSE) in the normalised domain ∼10 −4 ).Despite being heavily parameterised, the FFT within the network helps the FNO achieve quick training and execution times for surrogate models [18].The Fourier features extracted help provide the model with implicit bias found in the data, allowing it to converge quicker on the spatial evolution of the plasma.By transforming the data to the Fourier space, we allow the network to easily extract the dominant features found in the data as they would be highlighted across the various Fourier modes.As mentioned in section 1, we focus our experiments solely on the FNO as the chosen surrogate.For a detailed benchmark of MHD emulators across various models, we refer to our prior work [18]."},{"title":"Multi-variable FNO over MHD Simulations","content":"We design and train multi-variable FNOs to model the three MHD cases under study: Single isothermal blob, single blob in a non-uniform temperature field, and multiple blobs in a nonuniform temperature field, as described in section 2.1.Each multi-variable FNO is constructed as described in section 2.3.3 and trained using the autoregressive framework outlined in section 2.3.2.\nFor each multi-variable FNO, we chose an input feed of 10 time steps of the field information (T_in = 10), and an output size of 5 (step = 5).Each time step within the FNO corresponds to 1.5 µs of simulation time.Each FNO was autoregressively rolled out until the 50th time instance, allowing the neural operator to learn the time evolution of the function till then (T_out = 50).Each Fourier layer within the multivariable FNO has 16 modes and a temporal channel width of 32 (unless mentioned otherwise).The temporal channel width in the FNO is akin to the channel width in a Convolutional Neural Network, where the temporal data is scaled in the FNO channels instead of the RGB color space in the CNN channels [31].The channel variable width was maintained at the number of variables that the FNO learned.The grid discretisation associated with the field data is added to the training data within each model's forward call as an addition to the temporal channel, providing further field information to the model.We utilise the discretisation used within JOREK to generate the training data as the grid features within the FNO.We deploy a two-fold normalisation strategy considering the nature of the dataset.The physical field information represented within the MHD cases is in different scales, with densities ranging from 0 to 10 20 m −3 and temperature ranging up to 100 eV.Since we consider the gradual diffusion of an inhomogeneous density blob(s), the data distribution within the spatial domain is severely imbalanced.Taking these aspects of the training data into consideration, initially the field values are scaled down by a factor, indicative of the range of operation found within the simulation itself.The physics scaling would divide the data by the order of magnitude around which the plasma operates for each variable.We perform this by scaling the field value with the order of magnitude of its maximum found in the training data.This is followed up by a linear range scaling, allowing the field values to lie between −1 and 1.The normalisation pipeline is implemented separately for each of the variables that the multi-variable FNO accounts for.Each model is trained for up to 500 epochs using the Adam optimiser [32] with a step-decaying learning rate.The learning rate is initially set to 0.005 and scheduled to decrease by half after every 100 epochs.The input, output dimensions and the model architecture used for each case is given in B.\nThe loss function is expressed as an equally weighted linear combination of two elements: the reconstruction error and the difference error, expressed as (11) where ỹt represents the FNO output at the time step t, y t represents the labelled (actual) data at time frame t, step represents the step size of the FNO output, T represents the full length of the autoregressive time roll-out and N, the batch size or the number of simulations we are learning at a time over which we sum over.\nThe reconstruction error helps the FNO to learn to minimise the prediction error for a particular instance in time, whereas the difference error helps learn the time evolution of the plasma better as it minimises the temporal differences that arise in the autoregressive roll-out of the FNO.The loss curves associated with each experiment can be found in figure 6.\nThe focus of this research relies on the short-term evolution of up to 50 time steps from a total of 200 time steps, seeing how well the multi-variable FNO can model the evolution of all the physical variables in an integrated manner.We restrict ourselves to the short-term due to the errors arising from the autoregressive time rollouts, which are further explored in sections 3.1.4-3.1.6.To see the performance of the FNO deployed in an autoregressive setting for the full time range of the simulation, see appendix C."},{"title":"Single blob-isothermal.","content":"For the first case, that of a single blob in an isothermal setting (hereby referred to as an isothermal blob), the multi-variable FNO models the evolution of the two variables: density ρ and electric potential Φ. Considering the simple structure of the physics, the FNO performs well in (surrogate) modelling the evolution of both the variables arriving at an MSE (in the normalised domain) of 7.58 × 10 −5 in 48 min on a single Nvidia A100 chip.The multi-variable FNO is capable of emulating the evolution of the plasma up to the 50th time step in 0.025 s, offering a speedup of 6 orders of magnitude as shown in figure 7. Further plots showing the contrast of the FNO output with that of JOREK by way of discrepancy plots can be found under appendix A in figure A2."},{"title":"Single blob with non-uniform temperature.","content":"Moving up in complexity, in the case of a single density and temperature blob, the FNO is adapted to model density, electric potential, and temperature, three highly correlated variables as described in section 2.1.Even considering the additional physics, the multi-variable FNO demonstrates the ability to learn the global features of blob diffusion.It characterises the movements of the blob, its interaction with the walls, and its eventual dissipation.We notice that with the added field variables, the FNO performs well in predicting the evolution within the initial timesteps, but loses rigour of structure as we roll further out in time.Despite the challenge of preserving minor features further into time, the multivariable FNO achieves an MSE (in the normalised domain) of 2.56 × 10 −5 as shown in figure 8.The model trains in 1 h 37 min on a single Nvidia A100 chip.The multi-variable FNO is capable of (surrogate) modelling the evolution of the plasma up to the 50th time step in 0.03 seconds.Further plots showing the con-trast of the FNO output with that of JOREK by way of discrep-ancy plots can be found under appendix A in figure A1."},{"title":"Multi-Blobs with","content":"Non-uniform temperature.Figure 9 shows the capability of our newly designed multi-variable FNO in emulating the evolution of density, electric potential, and temperature for the case with multiple density and temperature blobs.This presents a more challenging case to model as the distribution of the training data is wide due to it being generated from a much larger parameter space as described in table 1.With a training dataset of 1500 simulations, the model trains in 11 h and 22 min on a single Nvidia A100 chip and achieves MSE(in the normalised domain) ∼0.0001, capturing the larger, more global, features associated with the evolution of plasma characteristics.However, due to the truncated nature of the implementation of the linear operator over the Fourier Transform, the modelling gets limited to the more dominant modes, smoothing out the less dominant, minor, finer features.The multi-variable FNO is capable of (surrogate) modelling the evolution of the plasma up to the 50th time step in 0.03 s.Further plots showing the contrast of the FNO output with that of JOREK by way of discrepancy plots can be found under appendix A from figures A3 to A5.\nThe FNO deployed in the autoregressive manner is well equipped to model the evolution of the plasma for short time roll-outs, but as we extend the prediction further in the temporal domain, it moves further away from the ground truth.This arises mainly due to the training regime which we deploy the FNO.Being deployed in an autoregressive structure, the prediction errors accumulate over longer time roll-outs.This is illustrated in figure 10.Though we demonstrate the results obtained with 1500 simulation data points, for the rest of the ablation studies (except for sections 3.1.7and 3.1.8)demonstrated in this work, we restrict ourselves to the initial 250 as it allows us to compares against the prior 2 cases where we operate in a relatively data-scarce setting."},{"title":"Individual FNO vs multi-variable FNO.","content":"MHDs governing the evolution of a plasma state is characterised by the amalgamation of Navier-Stokes equation of fluid dynamics and Maxwell's equations of electromagnetism [33].The evolution of MHD fields involves modelling multi-physics environments with coupled variables that are integrated together.\nConsidering this interplay of different field variables, it was imperative that we devise the multi-variable FNO, where all the interested and dependent variables can be modelled together.\nOur multi-variable FNO, as described in section 2.3.3,achieves the required multi-physics modelling, with the same computational costs as training individual FNOs for each variable.The multi-variable FNO has approximately the same number of parameters as the different individual FNOs combined, and thus takes the same time as training two or three different FNO models.As shown in figure 11, the multi-variable FNO achieves better performance with much less error than the individual FNOs. Figure 11 demonstrates that by integrating the different field variables within the model, we can reduce the error associated with the autoregressive time rollout, dampening the error growth.It must be noted that since each variable has a different normalisation applied to it (the same scheme, but different normalisation), the comparison within the normalised domain is consistent within the same variable.\nThus, we conclude that for highly correlated multi-physics systems, such as those described by the MHD equations, multi-variable FNO, which captures the interdependence of the variables within a single model, yields a much more physically consistent result than the variables modelled individually by separate FNOs with no methods of exploiting the relationship across them.\n3.1.5.Impact of step size.The FNO, devised to be utilised in an autoregressive configuration, is only exposed to the first T_in time steps as inputs to the model.The rest of the inputs for emulating the time evolution are obtained by feeding back the output into the model in a recursive manner.The model follows the layout of a Markov decision process [34] as it relies on the previously generated time steps to provide all the information pertaining to the state of the system, without having a memory component attached to it.As the model trains, the autoregressive output generated from the input is compared against that of the simulation data to learn the mapping in a supervising manner.When deployed at inference, the FNO relies purely on the initial time steps to perform the time rollouts of the plasma evolution.\nThus, the choice of the output size (step size), i.e, how much further in time the FNO predicts the time evolution with each forward pass, essentially provides disconnects across time for the model, which is evident by looking at figure 12.At each time step that is a multiple of the step size, we notice a spike in the cumulative error of the model.This spike denotes a disconnection that occurs as the model performs the autoregressive time evolution.In figure 12, the orange line indicates the error growth for a step size of 1, where there is no spike characterising the disconnection.For the red line with a Figure 13.Ablation study of multi-variable FNOs with different truncated mode numbers.Error growth of FNOs with different mode numbers as they are autoregressively evolved in time for the Multiblob case.Errors decrease with a higher number of modes, and performance nearly plateaus at higher modes, indicating the existence of dominant features for the problem up to a certain number of modes.For each model, the width of the temporal channel was fixed at 32, and that of the variable channel was fixed at 3. step size of 5, we notice that there is a spike at every 5th time step starting form 10 and similarly for the green line indicative of models with a step size of 10, the spike is observed at every 10th time step.Throughout the scope of this study for surrogate modelling plasma simulations, we restrict ourselves to step size = 5 as we find it offers the best training efficiency, optimum performance with less computational time.\nRegardless of this error spike, the FNO performs well in (surrogate) modelling the various MHD cases under study.We notice that the multi-variable FNO with a step size of 1 performs the best with a smooth and dampened error growth.This is expected as the network parameters are being used to perform mapping for one time step rather than several steps, effectively providing more parameters per step.We notice that the error decreases initially before it increases and believe it is because of the autoregressive structure of the training regime.In forward propagation, the model uses the time step data first from the data and then from the model itself.This results in a distribution shift problem because the inputs are no longer solely from ground truth data, the distribution learned during training will always be an approximation of the true data distribution [35].The pushforward trick demonstrated in [36], offers an alternative training regime that allows us to account for this behaviour.\n3.1.6.Mode ablations.The Fourier integral operator within the Fourier layer of the FNO truncates the Fourier series up to a maximal number of modes k max [27].This presents an additional hyperparameter to each Fourier layer of the FNO that determines the extent to which Fourier modes are required to be learned to map the spatio-temporal evolution.Within our experiments, we ensure that for each multi-variable FNO, the modes at each Fourier layer are fixed.Figure 13 shows the impact the number of modes has on the FNO in emulating the MHD case with multiple density and temperature blobs (Case 3).For a smaller number of modes, the FNO demonstrates a higher error value at each time step, whereas for higher mode numbers, we notice that the error is relatively lower.We also notice that the performance of the FNO plateaus at 8 modes and does not increase further as the mode numbers are increased.This arises due to the fact that the dominant features that are characteristic of the system we are learning saturates at the 8 th mode, and no more dominant features are found at higher modes for the multi-blob case.This is consistent with the mode selection that one has to perform for each problem that a neural operator is applied to.The effect of this is studied in detail [37].\nWithin the mode ablation study, we notice that at lower modes, though the error of the model is higher, it produces smoother field profiles than those at higher modes.In figure 14, we see that the FNO with 2 modes, generates fields that are smoother and less noisy, especially at the initial time steps.However, they soon lose the general structure as they progress further in time.In the FNO with higher modes (k max = 8, 32), the FNO output is much noisier than those at lower modes, but preserves the general structure of the temporal evolution better.\n3.1.7.Impact of training dataset size.Figure 15 visualises the impact the size of the training dataset has on the performance of the multi-variable FNO.For this purpose, we generate a larger dataset using the same scheme as given in table 1.We generate a total of 1835 JOREK simulation data points, out of which we set aside 85 as the test dataset.We train multiple multi-variable FNOs with training sizes ranging from 250 to 1750 simulations and compare their performance on the test dataset.All the models are constructed and trained under identical conditions, with the exception of the training data.We find that the performance of the FNO increases as we increase the training size, with model error decreasing by nearly half.However, we notice the gains made by added simulation data do not scale well at all.As shown in figure 15, improvement in performance is only marginal in comparison to the added computational costs associated with the training is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better.as well as the simulation run times.This is further illustrated in the loss curves of each associated experiment shown in figure 16."},{"title":"Zero-shot super-resolution.","content":"As described in section 2.3, the NO framework allows learning within the function space as opposed to the Euclidean space as with traditional neural networks [13].By learning the mapping between infinite-dimensional function spaces, the framework performs operator learning, allowing for characterising discretisation within our model.Within the FNO, discretisation invariance is built in by learning within the Fourier space rather than the Euclidean space on which the solutions exist, as demonstrated in equation (8).Being discretisation-convergent, the FNO learns information up to a certain refinement of the mesh resolution, higher than what we had provided in the initial We also notice the quick over-fitting tendency of the FNO as observed in [38].\ntraining dataset [15].The truncated nature of the FNO characterises the model as a band-limited function.Coupled with the discrete modal representation in the Fourier space, the FNO converges to a fixed performance at higher resolution [15].This discretisation-convergence allows us to perform zeroshot super-resolution, trained on a lower resolution, and directly evaluated on a higher resolution [27].In figure 17, we demonstrate this capability for zero-shot super-resolution, where we take the model trained on a spatial discretisation of 100 × 100 used for the prior experiments and then deploy it to emulate the evolution of multiple blobs at a spatial discretisation of 500 × 500.Even for grids at 5 times larger resolution, the FNO is capable of emulating the plasma with considerable accuracy, deviating from the true solution by an MSE (in the normalised domain)of only 7.90 × 10 −5 in the normalised domain.They share the same model parameters among different discretisations of the underlying function space that we are attempting to learn.Plots showing the contrast of the super-resolved FNO output with that of JOREK by way of discrepancy plots can be found in figure A6 in appendix A."},{"title":"Extrapolating outside training domain.","content":""},{"title":"Neural","content":"Networks trained for modelling a particular task have been found to suffer from a general lack of extrapolation capabilities [39].The FNO relying on a continuous gradientdescent based optimisation of the deep learning parameters exhibit the same behaviour.The models perform well within the domain of the training data, as can be explained by the associated Neural Tangent Kernels (NTK) [40] and fail to perform well outside the training regime.\nFor our multi-variable FNO that models the evolution of multiple blobs in reduced MHD, we can empirically validate the above-mentioned statements.When inferred on a case where there are 13 blobs with amplitude and density twice more and width 1.5 times more than the maximum used within the training data, we observe that the FNO can model the plasma evolution at the earliest prediction time instance and fails to capture the evolution of the plasma behaviour further in time as evidenced in the figure 18."},{"title":"Scaling to 3D.","content":"There exist several challenges in the extending the FNO to model the evolution of multi-variable PDEs to 3D settings.The primary concern is of the memory constraints needed for a 3D multi-variable FNO when being evaluated autoregressively in time, leading to GPU bottlenecks.Recent work in performing Tensor factorisation over the FNO weights, allows representing the network in a lower rank approximation [41], offers potential to those GPU training bottlenecks.\nAnother challenge associated with the scaling is that of the computational costs associated with data generation.In 3D, the numerical solvers that generate the training datasets are exponentially more intensive than in the 2D settings, limiting the amount of training data that could be generated.Deploying Active Learning (AL) methods, which allow for more informed sampling of the parameter domain space offers a solution to this.Within AL, the model is initially trained on a coarsely sampled, limited dataset and then evaluated across the interested domain.Performance (or uncertainty) of the model within the interested domain informs the next data generation points, which are added to the training data, improving the model accuracy.AL allows the model to maximise performance, minimise uncertainty within data scarce scenarios [42].Within the scope of plasma physics, AL has been found to reduce the data requirement of surrogate models by a factor of 20 in modelling Gyrokinetic Turbulence [43].The issue of data-scarcity arising from the profound computational intensiveness of 3D simulations could also be combat by using physics-informed models, where the information regarding the PDE is hard-baked into the model architecture or the training regime.Recent works on physics informed neural operators has shown promise in fine-tuning coarsely trained models with PDE constraints within MHD scenarios [44].\nWhile scaling to 3D settings it might be the case that the choice of operator within the Neural Operator might be limiting and not representative of the true solution, thus there might be a requirement of a combination of different operators implemented over different families of basis functions within the model to account for the varied physics that is found within the PDEs.One last challenge to consider is that proposed by the autoregressive layout of the FNO.The addition of another dimension will contribute further to the error growth in time as noted in figure 10.Though this remains a standing problem within the domain of Neural-PDEs, recent works have shown promise in solving this challenge by providing further conditioning of the longer time roll-outs [45]."},{"title":"FNO over MAST camera","content":"As described in section 2.2, we conduct exploratory research in looking at using the FNO as an emulator for performing predictive inference in real-time of the plasma evolution within the MAST fusion device.For this purpose, we deploy FNOs with the same architecture as that used for emulating the Individual FNOs (given under appendix D in tables D1 and D2).However, considering the nature of the camera dataset, we implement a different training and inference regime from that of the autoregressive structure used to solve PDEs.\nIn the case of the MHD simulations, the evolution of plasma is a Markovian Process, with no element of noise found within the dataset.However, within the case of the experimental data, the images are still time-series data, but they are nonmarkovian, noisy, and represent only partial information about the state of the system.In addition, the dataset does not offer any information about the causal contributors, such as the various control inputs that influence the state of the system, and is assumed to be implicitly defined within the time evolution of the data.Taking this into account, implementing an autoregressive structure as demonstrated in [18], provided limited capabilities as the error growth across the time roll-out would be too large, not allowing us to perform longer time roll-outs.Within the scale of the experimental data, it might be the case that there might be new information fed into the system midway through it.By utilising an autoregressive rollout that relies solely on the initial time steps, the model will not be able to account for this new information added later to the system.Unlike in the case of modelling simulations, there is no actual need within the experimental space to rely just on the initial conditions and state of the system to perform the prediction.Data characterising the temporal evolution of the plasma is generated and captured by the diagnostic as the experiment is run and does not need any additional intervention to obtain it.Our training and inference setup takes advantage of this continuously generated data to build our predictive model.Instead of performing an autoregressive time rollout, we perform a fixed time window mapping (as outlined in figure figure 5), where the FNO takes in the camera information for fixed time steps (T_in) and outputs the next set of frames characterising the future plasma state (step).Being trained in this sequential pipeline, the model initially takes in the first 10 time instances (1-10 frames) to output the next 10 time instances (11-20 frames).With the next forward pass, the model takes in the other 10 time instances, but this time, forward by one frame (2-11 frames) to predict the next 10 frames to that input (12-21 frames).This input, characterised by a 10-frame window slid by one each forward pass, is run until we output the last required frame (T_out).Our trained FNO models the plasma at ramp-up, entering the flat-top regime, and in most cases, across the L-Mode and H-Mode operational regimes, covering the full (available) plasma duration.Since we are designing this to be run in real-time with the experiment, future time windows, as they become available, will be fed into the model to predict the future plasma states.The time resolution of the fast camera diagnostic is 1.2 ms, where the FNO only requires 6 ms to predict the next 12 ms of the plasma evolution, allowing us to run the model efficiently in real time.\nEven though we keep our focus on a limited number of shots curated according to the criteria given in section 3.2.1, as we structure them into input-output pairs by way of the sliding windows as described above, we essentially turn each shot characterising of approximately 200 frames into 180 inputoutput pairs.The FNO is exposed to dynamics of the plasma evolution across the interested time window of MAST operation, learning to predict from any intermediary states of the system to the next states of the system.The FNO is trained to learn how to predict the evolution of the plasma as witnessed by the camera for the full (available) range of operation of the Tokamak pulse.\nFor each camera FNO, we chose the input feed to consist of 10 time steps of the field information (T_in = 10), with an output size of 10 (step = 10).Each Fourier layer within the FNO has 8 modes and a temporal channel width of 16.The grid discretisation associated with the field data (obtained from the reactor CAD model) is added to the training data within each model's forward call as an addition to the temporal channel, providing further field information to the model.The data is normalised by following a linear range scaling, allowing the field values to lie between −1 and 1.Each model is trained for 500 epochs using the Adam optimiser [32] with a stepdecaying learning rate.The learning rate is initially set to 0.005 and scheduled to decrease by half after every 100 epochs.The model was trained using a Relative L2 loss function over the reconstruction error alone (see equation ( 11))."},{"title":"Data selection.","content":"Throughout this study, we work on a selected number of plasma shots for each cameras, restricting ourselves to the M9 campaign on MAST.For the rbb camera, we restrict ourselves to 55 shots lying within the range: 30 250-30 431 from the MAST experimental database.For the rba camera we look at 48 shots lying within the same range 30 250-30 431.The shots were carefully curated from the above shot range by considering several factors.The first criterion was to ensure that the selected shot actually contained a confined plasma and was not a dummy shot like commissioning pulses.The second criterion was whether the shot had the desired duration of more than 100 time steps-looking at forecasting longer-term plasma evolution.The third one was selecting shots within the above-mentioned ranges that had the same camera calibration.Considering the FNO relied on a grid discretisation to leverage the Fourier modes, it was crucial to ensure that the pixels from the camera belonged to the same line of sight vectors.The camera calibrated for specific experimental runs captures the information of the plasma onto a uniform pixel grid, allowing us to perform Fourier transformations over them.By using calcam, a calibration tool developed by the European Atomic Energy Community [46] combined with 3D CAD models of MAST, we determine the domain range covered by each camera calibration.The upper and lower limits of the domain range of these images are then fixed onto uniform grids in both the R and Z axis, allowing us to create uniform grid discretisations underlying each camera image.The last criterion was to ensure that the camera images obtained from the diagnostic had the same spatial resolution across the dataset that could be fed into the FNO.For the rbb camera, we chose the shots with a resolution of (448 × 640), and for the rbb camera, we chose those with the resolution (400 × 512).However, the FNO offers discretisation convergence, the training setup benefits greatly from ensuring coherence along the spatial resolution.Figure 19 shows the plasma signals allows to demonstrate the diversity of the scenarios used to train and test the FNO modelling the cameras.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data.For a full list of the selected shots, refer appendix G."},{"title":"Camera viewing the central solenoid (rbb).","content":"The evolution of the plasma within the Tokamak, as seen by the main camera, can be shown in figure 20.The plot shows the comparisons of the evolution as observed by the FNO with that of the camera spanning 10 frames predicted in one feed-forward iteration.The central solenoid serves as the backbone of the MAST magnet system, generating the majority of the magnetic fluxes as well as the plasma current.In figure 23, we portray the predictive capability of the FNO by comparing the last frame from multiple feed-forward instances taken from data across the interested length of the pulse.This allows us to compare that even for the longest time prediction, we can get reasonable predictions.In figure 23, we demonstrate that the FNO is capable of predicting the plasma in L-mode as well as in H-mode, characterising the plasma evolution across the full (available) range of the plasma shot.The main camera view, as shown in figure 1 The FNO deployed with the time-window pipeline (see figure 5), is able to capture the long-term dependencies that arise within the plasma confinement.The model is able to predict the evolution of the plasma from L-mode to H-mode as seen in figure 23.The prediction around the transition regime can be seen in Though the camera captures the confined plasma shape, it fails to capture the localised filaments.In figure 20, the trained FNO is deployed to model the evolution of plasma within the L-Mode of operation, whereas in figure 21 the FNO is modelling the region of H-mode within the same plasma shot.\nThe capability of the FNO in predicting across the L-H transition is further emphasised through figure 22, where we show the prediction around the transition region for shot number 30 428.We provide the camera image, FNO prediction and the absolute error at L-mode (162 ms), around the transition (172 ms) and with the plasma in H-mode (186 ms).The FNO is capable of modelling the plasma in both L-mode, Hmode and across the transition, but as the FNO is a regression model and not a classification model, without further analysis over the predicted solution, the model is unable to explicitly mention that it is capable of predicting the exact moment of transition.This will be a key area of focus for the next research avenue."},{"title":"Camera viewing the divertor (rba).","content":"The evolution of the plasma within the Tokamak as seen by the divertor camera, along with predicted images from the FNO can be seen in figure 24.The plot shows the comparisons of the evolution as observed by the FNO with that of the camera spanning 10 frames predicted in one feed-forward iteration.The divertor of a Tokamak serves as the exhaust of the device, where the topology of the magnetic flux surfaces is designed to isolate the confined plasma from the first-wall of the device.In figure 25, we portray the predictive capability of the FNO by comparing the last frame from multiple feed-forward instances taken from data across the interested length of the pulse, demonstrating the predictive power of the FNO.The divertor camera looks at the lower divertor of MAST with a spatial resolution of 400 × 512.We sampled a total of 48 plasma shots from the MAST database, used 45 for training and 3 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 5815, and the test points grow to 701 data points.The model takes 9 hours 10 minutes on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.70 × 10 −4 .For further plots showcasing the forecasting capabilities of the FNO at the divertor plasma, refer to appendix F, specifically figures F1 and F2.\nThe FNO deployed for emulating the camera data accurately preserves the global features of the plasma evolution.It is able to clearly characterise the locations and intensities of the heat fluxes onto the solenoid and the divertors, allowing us to anticipate the heat flux in advance and initiate the appropriate actuator mechanism to minimise its impact on the various plasma-facing components.The FNO also captures the reflections generated within the device due to the interactions across the plasma and the plasma-facing components.The FNO, in its current form, smooths out the filaments but captures the general confined structure of the plasma as it evolves in time.As expected, the FNO learns characteristics extremely well, while failing to capture the localised phenomena with matching pixel-pixel accuracy.\n3.2.4.Limitations.We decided to look at a narrow shot corridor, since the data variability will be low, allowing us to demonstrate a functioning proof-of-concept.Our predictive model only takes in the camera diagnostic as the input and does not take into account any of the control signals, such as coil currents and heating parameters that influence the plasma evolution.However, we argue that within the past window of plasma observed from the camera data that is taken as the input, these control parameters and their influence are implicitly defined (with a latency), and thus, the model provides a suitable predictive tool for short-term predictions.The inclusion of various control signals, time matched with the camera inputs, will potentially help improve the performance of the model.Our future research trajectory will be heading in this direction."},{"title":"Conclusion and discussion","content":"Through this work, we demonstrate the utility of using the FNO as both a surrogate model for a plasma simulator and as a predictive emulator for a plasma experiment.We design a modified version of the FNO, that performs integrated emulation of multiple variables associated with a family of PDEs.Our analysis across a range of reduced MHD cases demonstrates that even in data-scarce scenarios, the FNO can be trained to act as a surrogate.Through this paper, we highlight the various features that make the FNO a natural choice for (surrogate) modelling MHD cases, all the while performing ablation studies on the various hyperparameters and the training regimes.We find that the FNO fits the requirements of a quick, cheap surrogate, but due to its auto-regressive nature, fails to capture the physics when deployed for long time rollouts.Being utilised for emulating the experiment, the FNO is capable of forecasting the evolution of the plasma as seen through the cameras on a fusion device.The FNO camera models are capable of identifying the regions of plasma interaction with the central solenoid and the divertor, while effectively predicting the plasma shape evolution.The FNO has shown versatile performance in being able to model the plasma confinement across both L-mode and H-mode of plasma operation across a diverse range of shots.The FNO for the camera can predict the plasma evolution in real-time and offers us the potential to be deployed within the planning stage for various control scenarios.Our work also demonstrates how we can perform zero-shot super-resolution across the pre-trained multi-variable FNO and obtain the plasma emulation at higher resolutions than it was trained on, without requiring any finetuning.\nIn continuing with this line of work, we will be exploring the utility of building more versatile and robust FNO models that can be deployed not just in a particular case, but across a range of simulation cases.While we build more general MHD models, we will also be looking at methods to better physics-inform the models [14], allowing us to finetune the general models to specific MHD instances.We will also be looking into methods of quantifying the uncertainty associated with the surrogate models and into AL pipelines that can facilitate the entire surrogate ecosystem.With the FNO for camera data, we are currently working on extending the utility across more plasma shots with a more diverse line of sights, across multiple campaigns while exploring the utility of state space models for longer predictions [47].The work will be looking at including various control signals, to help improve the performance of the real-time predictive emulator for the MAST Tokamak, performing longer forecasting."},{"title":"A.2. Single blob with non-uniform temperature","content":""},{"title":"Figure 1 .","content":"Figure 1.(a): Fast camera image of a MAST plasma during an Edge-Localised-Mode, where plasma filaments are ejected outwards due to MHD instabilities at the plasma edge.Reprinted from [19], with the permission of AIP Publishing.(b): The evolution of the filamentary structures during the multi-mode ELM simulation imaged with a synthetic fast camera diagnostic (time given in ms).Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020."},{"title":"Table 1 .","content":"Domain range and sampling strategies across the initial condition parameters.The case numbers represent the simulation settings in which that parameter was sampled.1-single isothermal blob, 2-single blob with non-uniform temperature, 3-multiple blobs with non-uniform temperature.U [a, b] refers to a uniform distribution bounded by a and b."},{"title":"Figure 2 .","content":"Figure 2. Evolution of multiple blobs inside a slab for the first 500 time-steps of the numerical simulation (the full run is 2000 steps), taken at intervals of 100 steps (about 15 µs).The top row shows the density, the middle row the temperature, and the bottom row the vorticity."},{"title":"Figure 3 .","content":"Figure 3. Views of the fast cameras on MAST in the absence of plasma, obtained using illuminations on the 3D CAD model of MAST.Camera shown in figure (a), looks across the central solenoid, while the camera in figure (b) shows the view at the divertor."},{"title":"Figure 4 .","content":"Figure 4. Modified architecture of a single Fourier Layer.FNO is constructed by stacking multiple Fourier layers on top of each other, sandwiched by a lifting and projection operation."},{"title":"Figure 5 .","content":"Figure 5. Schematic representation of the two kinds of training configurations deployed within this study.For the simulations, we use the autoregressive configuration (left).We only use the initial set of field values to propagate towards the desired time step, where the intermediary model output is fed back into the model as inputs to further the time evolution.For the experimental data, we use a sequential time window-based configuration.Here, the model output is not factored back into the model, but we perform a fixed window mapping across time.As we go across the time duration of the data, the window is slid further by one-time instance (right).In the example above, we have T_in = 3, step = 1, T_out = 8, the time focus of each forward propagation of the model is given along the x-axis while the evolution across the full-time domain is given along the y-axis.Each cell block represents the 2D field value at a certain instance of time.The input and output sizes shown in this diagram are representative only and vary across each experiment."},{"title":"Figure 6 .","content":"Figure 6.Learning curves associated with training the multi-variable FNO across the various the various Reduced MHD Cases.Isothermal blob refers to the single blob diffusion under uniform temperature, Anisothermal blob refers to the single blob under a non-uniform temperature, and Anisothermal multiple Blobs extend that to the case with several density blobs under a non-uniform temperature distribution."},{"title":"Figure 7 .","content":"Figure 7. Isothermal Blob: Temporal evolution of (a) density and (b) electric potential of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"title":"Figure 8 .","content":"Figure 8. Single Blob : Temporal evolution of (a) density and (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"title":"Figure 9 .","content":"Figure 9. Multiple Blobs with non-uniform temperature: Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"title":"Figure 10 .","content":"Figure 10.Error growth across time roll-out of the FNO in the physical space.The error in emulating each variable of the reduced MHD model increases as we roll further out in time."},{"title":"Figure 11 .","content":"Figure 11.Comparing the error growth (mean absolute error in the normalised domain) of different variables across time roll-out of the Individual FNO (Single) and the Multi-variable FNO (Multi) for the various MHD cases under study.(a) Single Isothermal Blob (b) Single Blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The multi-variable FNO with the same parameters as the individual FNOs demonstrates much better performance over time.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout."},{"title":"Figure 12 .","content":"Figure 12.Comparing the cumulative error growth of different variables across time roll-out for multi-variable FNOs with different step sizes.The autoregressive nature of the FNO brings about a spike in error, characterising a disconnection at the time step that is a multiple of the output size.(a) Isothermal blob (b) Single blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout."},{"title":"Figure 14 .","content":"Figure 14.Evolution of the electric potential (Multiblobs) in time as modelled by multi-variable FNOs at different modes.Top row of (a) shows the actual solution modelled using JOREK.The bottom row of (a), plots the time evolution of the FNO with 2 modes, top row of (b)is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better."},{"title":"Figure 15 .","content":"Figure15.Impact of the size of the training dataset on the performance of the multi-variable FNO for simulating the multiple plasma blobs in a non-uniform temperature field.As we increase the availability of the training data for the multi-variable FNO, the model performance increases across the same test data.However, the gains obtained by increasing the training data are marginal compared to the additional compute required.The number of simulations used for training is given on the x-axis while the mean squared error is given on the y-axis.All training is performed on a single Nvidia A100."},{"title":"Figure 16 .","content":"Figure 16.Learning curves associated with the models trained with data of different sizes.Training loss is represented by full lines, while the test loss is represented by dashed lines.With just 500 simulation instances, the model learns the physics of the model really well, which is only marginally improved by more training data.We also notice the quick over-fitting tendency of the FNO as observed in[38]."},{"title":"Figure 17 .","content":"Figure 17.Demonstrating Zero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE(in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the bottom series of plots demonstrate the time evolution as emulated by the multi-variable FNO."},{"title":"Figure 18 .","content":"Figure 18.Outside the training domain: Inferring on a case with 13 blobs with amplitude and density twice more and the blob width 1.5 times more than the maximum used within the training data (see table 1).Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"title":"Figure 19 .","content":"Figure 19.Temporal plots of (a) Plasma Current and (b) Total Heating Power of the shots selected for training and testing the FNO for modelling the camera data.The shots depicting the training data is given in black, whereas the test data is given in red.The plots help convey the diversity of plasma scenarios the FNO is trained to model over.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data."},{"title":"20 .","content":"Predicting in L-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera with the plasma in L-mode.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera."},{"title":"Figure 21 .","content":"Figure 21.Predicting in H-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera wiht the plasma in H-mod.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The domain is selected to showcase the performance of the FNO in mapping the plasma evolution in H-mode.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\nprovides a panoramic view of the reactor, showing the poloidal cross-sectional layout on both sides of the central solenoid.The main camera has a spatial resolution of 448 × 640 for the experiments that we sampled.We sampled a total of 55 plasma shots from the MAST database, used 50 for training and 5 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 7661, and the test points grow to 828 data points.The model takes 27 h 45 min on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.7 × 10 −3 .For further plots showcasing the forecasting capabilities of the FNO at the plasma around the central solenoid, refer to appendix E in figures E1 and E2."},{"title":"Figure 22 .","content":"Figure 22.Predicting around the L-H transition: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments."},{"title":"Figure 23 .","content":"Figure 23.Long-term prediction: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments."},{"title":"Figure 24 .","content":"Figure 24.Single forward pass: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera."},{"title":"Figure 25 .","content":"Figure 25.Long-term prediction: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera."},{"title":"Figure","content":"Figure Single Blob with non-uniform Temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"title":"Figure A4 .","content":"Figure A4.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"title":"Figure A5 .","content":"Figure A5.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"title":"Figure","content":"Figure DemonstratingZero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE (in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the middle series of plots demonstrate the time evolution as emulated by the multi-variable FNO and the bottom series indicates the absolute error across space."},{"title":"Table 2 .","content":"Overview of the performance obtained by various FNO-based surrogate models built to emulate the plasma evolution in both simulations and experiments.The case represents the various simulation settings of JOREK or the camera diagnostic on MAST, for which we are training an emulator.The variables mention the number of variables from the PDE/diagnostic that we model together.Parameters refer to the number of parameters in the trained FNO, and MSE represents the mean squared error in the normalised domain.The Training Size represented the simulation/experimental input-output pairs.All models are trained on a single Nvidia A100 chip."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Predicting the evolution of plasma dynamics is crucial to building sustainable fusion devices as it allows us to anticipate plasma behaviour and make real-time decisions over the input control parameters and alter the trajectory of plasma evolution.Such predictions also allow us to improve our understanding of the physics governing plasma interactions.In particular, Magnetohydrodynamic (MHD) modelling is of critical importance as it improves our understanding of the dynamical evolution of filaments that burst out of the plasma edge and reach the Tokamak's first wall, as shown in figure 1(a) [1].These plasma filaments, called edge-localised modes (ELMs), are described by non-linear MHD, and can be modelled numerically, as shown in figure 1(b).\\nIn fusion research, prediction of future plasma states is mostly undertaken using physics-based numerical modelling codes, such as BOUT++ [2], JOREK [3], JINTRAC [4] and SOLPS [5].However, these simulations are computationally intensive, taking weeks for effective simulations to be run on high-performance computers [2].They are also often complex in terms of software infrastructure and advanced solver library dependencies, making them difficult to deploy for rapid iterative modelling.With numerical solvers, we also encounter cases where the simulations do not converge due to model mismatch or other issues.\\nMachine learning (ML) offers an alternative data-driven path for obtaining quick, inexpensive approximations to numerical simulations, allowing for rapid iterative modelling [6].Within plasma physics, ML-based surrogate models have been used to provide quick approximations for a range of solutions, from emulating turbulent transport [7][8][9], estimating the Tritium breeding ratio across Tokamak designs [10] to modelling edge plasma [11,12].\\nRecently, a new class of data-driven ML models, known as neural operators (NO), has recently been at the forefront of surrogate modelling.Unlike other surrogate models, they directly learn the solution operator of the partial differential equation (PDE) in a discretization-convergent manner [13,14].Specifically, NO can predict at any resolution, unlike standard neural networks like convolutional neural networks.In other words, they learn the underlying continuum, making them suitable for learning solutions of PDEs.The output of NO converges to a unique function upon mesh refinement [15].NO approaches are more data-efficient compared to other surrogate models [16], converge faster than traditional surrogate models, and can generalise better to various PDE settings.Considering these advantages, NO-based surrogate models have found application in emulating the plasma evolution across the domain of a Tokamak [17,18].\"\n },\n {\n \"title\": \"Our Contributions\",\n \"content\": \"• We demonstrate trained Fourier neural operators (FNOs) that can be deployed as an emulator to perform real-time forecasting of the evolution of plasma as characterised by fast cameras on the MAST Tokamak.Our studies show that the FNO is capable of modelling the plasma across the full (available) shot duration ranging from the ramp-up to the L and H modes of operation.• We devise a modified version of the FNO, capable of handling multiple variables that govern a family of PDEs.The multi-variable FNO allows us to pass field information of Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020.\\nall the interested variables to the model, allowing for learning and encoding the mutual dependence of variables, which is essential for capturing the non-linear effects.• We demonstrate the utility of the multi-variable FNO in (surrogate) modelling the evolution of plasma across a range of reduced MHD cases with increasing complexity.We conduct an extensive analysis of the various features of the FNO, performing ablation studies that evaluate the impact of various choices of the Hyperparameters across the model and training.\\nIn this paper, we undertake an extensive study of the FNO due to its previously established superior cost-accuracy trade-off in both training and testing [16].We use NO to model plasma within fusion devices, in both the simulation space and within the experimental domain.For the simulation domain, we consider the reduced MHD models to provide data to train the neural operator [3].We study the solution of MHD equations in two spatial dimensions with multiple variables of interest.\\nThe simulations that we are interested in modelling represent the diffusion of a blob of plasma within a specified domain.\\nThe simulations presented here describe a simplified scenario to that of a full ELM simulation as given in figure 1(b).We iterate over increasingly complex cases of fluid models that govern the diffusion of a plasma blob in toroidal geometry.\\nIn addition to numerical modelling, we demonstrate that FNOs can be trained to predict the evolution of experimental plasma as observed by high-speed cameras on the MAST spherical Tokamak.The ability to predict future plasma states based on camera imaging, as well as other plasma diagnostics, could allow us to design efficient control feedback loops for future fusion reactors.We build emulators of the fast camera that views across the central solenoid and at the divertor.We find these models to be capable of accurately predicting plasma evolution in real-time.\\nIn this study, we use FNOs for both MHD simulations and experimental camera data due to the structural similarities of the plasma observed in both.The visual similarity of the MHD modelled in the simulation, and that captured by camera images in a fusion device is elucidated in figure 1(a).The fast camera images capture the MHD behaviour that is observed experimentally.Through this work, the FNO's utility is demonstrated in predicting both experimental and simulation dynamics.Although this study considers a simplified simulation setup, with filament cross-sections, in a 2D poloidal plane, instead of the full 3D representation, this paper lays the groundwork for future studies where surrogates could be trained from both simulation and experimental data simultaneously.However, in section 3.1.10,arguments of the challenges and potential methods of extending to a 3D modelling case are discussed.\\nA preliminary version of this work was presented at the AI for Science workshop during NeurIPS 2022, where we demonstrated, as a proof-of-concept, the utility of the FNO in modelling MHD scenarios [18].In the previous work, we conducted a benchmark of the performance of the FNO against more traditional surrogate model architectures such as Convolutional-Long-short term memory [20] and U-Net [21].Our findings established that the FNO takes significantly less time to train, requires fewer parameters, and outperforms the Convolutional-LSTM and the U-Net in emulating the diffusion of a blob of plasma described by reduced MHD [18].\\nThe paper is organised as follows: section 2 produces a full description of the datasets and methods involved in the paper, ranging from the physics description of the simulations, the diagnostic setup of the experimental camera data and a formal definition of the FNO, its training configuration and the extension to the multi-variable FNO.Section 3 outlines the results of our experiments with using the FNO to learn and predict results from various simulation settings and camera configurations.The section comprises an in-depth analysis of the multivariable FNO trained to solve different PDEs, exploring the impact of hyperparameters, training regime, and the training data.We also include a performance comparison of the individual FNO with that of the multi-variable FNO.Section 3 is concluded with the experimental results from the FNO trained to model the plasma evolution as captured by the visible camera looking at the central solenoid and the divertor.We summarise our key results in section 4 and conclude with a discussion on future extensions of the work.\"\n },\n {\n \"title\": \"Datasets and methods\",\n \"content\": \"\"\n },\n {\n \"title\": \"MHD simulations of plasma Blobs\",\n \"content\": \"The FNO as a Neural-PDE solver is tested on three datasets of simplified MHD models, simulating the radial convection of plasma blobs in toroidal geometry, with coordinates system (R, Z, ϕ).These three cases are designed with increasing complexity to test the utility, versatility, and capabilities of the FNO to emulate MHD models.The simplified MHD model is similar to the so-called Reduced-MHD model from [3], but with all magnetic fields and currents kept to zero, which is similar to the Euler system, and can be summarised by the continuity, momentum, and pressure equations:\\nwhere ρ is the density, p the pressure, T the temperature, and ⃗ v the velocity.D is the diffusion coefficient, µ the viscosity, and κ the thermal conductivity.The ratio of specific heats γ is taken to be that of a monatomic gas, 5 3 .To obtain the Reduced-MHD model, the momentum equation ( 2) is projected in the poloidal plane using the operation ∇ϕ • ∇ × (R 2 . ..), while assuming the poloidal velocity takes the form ⃗ v = R 2 ∇ϕ × ∇Φ, where the electric potential Φ is equivalent to the velocity stream function in toroidal coordinates.This projection results in a system of three equations for the three variables ρ, Φ, T. A pseudo-variable is also used for the vorticity to reduce the level of derivatives in the system: w = ∇ 2 Φ.We refer the reader to [3] and references therein for the full system of Reduced-MHD equations.\\nThe simulations are all run in a square slab geometry of 1 m in width and height, centred at major radius R = 10 m.The runs are performed with toroidally axisymmetric plasma blobs initialised on top of a low background density and temperature.In the absence of a plasma current equilibrium to hold the density blobs in place, the pressure gradient term in the momentum equation generates a buoyancy effect, causing the blobs to move outwards.The simulations are run with the JOREK code, which is routinely used to model non-linear MHD instabilities in realistic geometries [3].In this paper, we restrict ourselves to two simplified MHD models.\\nA further reduction of the model can be obtained by using constant temperature and removing equation (3) from the system, therefore evolving only ρ and Φ.With this isothermal model, the pressure gradient term in the momentum equation (2), which is proportional to ρ, acts as a buoyancy force on the blob, affecting its momentum ρ⃗ v, which is also proportional to ρ.In other words, the velocity of the blob will be the same no matter what its density is.To have blobs moving at different speeds, the temperature inside the blob is needed: the higher the temperature, the faster the blob will move.The values of the diffusive parameters are kept constant inside the entire domain, for all simulations, at\\nThe background density level is set to 10 18 m −3 , and the background temperature is set to 1 eV (for simulations that evolve temperature).\\nThree datasets are considered: with a single isothermal blob, with a single blob with non-uniform temperature, and with multiple blobs with non-uniform temperature.To create the datasets, the blob's initial conditions are varied.The sampled parameters are the number of blobs, their positions, their width, and their density and temperature amplitudes.This is summarised in table 1.For datasets with a single blob, the initial position is fixed at the centre of the domain, (R, Z) = (10.0,0.0).The smaller single-blob datasets are composed of 120 simulations (100 for training, 20 for testing), while the main dataset with multiple blobs contains 2000 simulations.\\nAll cases are run with the same time-step size of 0.15 µs, for 2000 time-steps in total.This is long enough for the blobs to evolve radially until they reach the wall (where Dirichlet boundary conditions are applied) and disappear through both convective mixing and diffusion.Since a small time-step size is necessary in the simulations to ensure numerical stability, the datasets are down-sampled to 200 slices for the training set.Likewise, the 2D poloidal grid resolution, which is 200 × 200 bi-cubic finite-elements in the simulations, is down-sampled to a regular grid of 100 × 100.The smaller single-blob datasets were run on the EOSC-hub cloud using 16 cores per run, and the larger dataset of 2000 multi-blob simulations was run on CINECA's Marconi HPC cluster using 48 cores per run.Each simulation requires approximately 350 core hours.Though each simulation is down-sampled to 200 time instances, we keep the focus of our emulator on short-term evolution up to the first 50 time steps.\\nSimulations of blobs in slab geometry are related to turbulence and MHD instabilities at the plasma edge of a Tokamak by assuming that they represent a poloidal cross-section of a filament.As seen in figure 1, filament structures are aligned to the magnetic field, they are elongated in the toroidal direction, but have a nearly circular cross section in the poloidal plane.Such slab simulations represent the dynamic evolution of these filaments in the plane perpendicular to the magnetic field.Extensive modelling of blobs in slab geometry can be found in [22,23] and references therein.Figure 2 shows an example of a numerical simulation of multiple blobs evolving for the first 500 steps (out of 2000), with the density, temperature, and vorticity.Filaments with higher temperatures can be seen to move at a faster velocity.\"\n },\n {\n \"title\": \"Fast-camera images on MAST Tokamak\",\n \"content\": \"MAST was equipped with fast camera imaging diagnostics at several points across the Tokamak for capturing images within the visible spectrum, capturing the plasma evolution in real-time 5 .\\nThese Photron cameras 6 are essential diagnostics that contributed to fundamental understandings of key plasma phenomena in Tokamaks in recent years [24].While most of the camera data has been used for qualitative analysis in the past, it can be exploited using more advanced methods to provide statistical insight into plasma turbulence [25], as well as global instabilities (disruptions) [26].Within the scope of this work, we focus on two camera views.The first looks at the central solenoid and the plasma confined around it, extending to the walls and providing a holistic view of the plasma (rbb camera).\\nThe second camera configuration under study is that providing the divertor-view, looking at the bottom of the Tokamak, where energy travels from the plasma to the exhaust region (rba camera).The views of the fast cameras on MAST in the absence of the plasma is visualised in figure 3. Figure 1 shows a view of the main plasma as seen by the camera, together with a synthetic rendering (right) of the camera, simulating the MHD physics.The cameras have a temporal resolution of 1.2 ms, with spatial resolution varying according to the desired calibration setup for each experiment.They are tuned to visible wavelength and, therefore, capture mostly the Balmer D α light emitted mostly from the plasma edge.As seen in figure 1, the fast-visible cameras are able to capture complex dynamics of plasma filamentary eruptions at the plasma edge, the ELMs [24], hence being able to predict real-time evolution allows us to anticipate such plasma instabilities and other confinement related characteristics.Within the scope of this project, our focus relies on modelling the plasma confinement and its interactions with wall accurately, leaving the modelling of ELMs as the next potential research avenue.\"\n },\n {\n \"title\": \"FNO\",\n \"content\": \"The FNO [27] is a neural network-based operator learning method [13] that learns a mapping between two function spaces from a finite collection of observed input-output pairs.In the Neural Operator (NO) setting, we learn the mapping\\nG θ is a neural network configuration parameterised by θ and is composed of three elements:\\n(i) Lifting: A fully local, point-wise operation that projects the input domain to a higher dimensional latent representation a ∈ R da → ν 0 ∈ R dν 0 .(ii) Iterative Kernel Integration: Expressed as a sum of a local, linear operator and a non-local integral kernel operator, that maps from ν t → ν t+1 , iterated from ν 0 to ν T .The kernel integration is followed by a point-wise nonlinearity.\\n(iii) Projection: Similar to lifting, characterised by a fully local point-wise operation that maps the latent representation to that of the output domain\\nIn equation (5), σ denotes the nonlinear activation, W represents the local, linear operator, and κ represents the chosen kernel which is parameterised by ϕ. x represents the input that is fed into that layer.\\nConventionally, the lifting and projection within an NO are characterised by linear layers acting on the respective feature spaces.The kernel integration operator is characterised as the neural layer that evaluates a chosen kernel over the higher dimensional latent representation fed into it:\\n(κ (a; ϕ) ν t ) (x) = ˆD κ (x, y, a (x) , a (y) ; ϕ ) ν t (y) dy.(6) As demonstrated in [27], the FNO is devised by characterising the kernel integral operator as an operation within the Fourier space.Replacing κ with the Fourier integral operator F(κ ϕ ) we obtain\\nHere, F and F −1 represent the Fourier and inverse Fourier transform, respectively.κ, the kernel is parameterised by learnable weights ϕ ∈ θ κ .Within the FNO, the kernel integrator reduces to the inverse Fourier transform of the inner product between the Fourier integral operator F(κ ϕ ) and the Fourier transform of the input to that Fourier layer F(ν t ).\\nIn practice, the iterative kernel integral transform with Fourier transform is characterised by two learnable weight matrices R and W as shown in figure 4. R learns the mapping behaviour within the Fourier space, while W learns the mapping required within the input Euclidean space.R characterises the non-local integral kernel operator, while W represents the local linear operators.The weight matrix R within the Fourier layer enables convolution within the Fourier space, allowing the network to be parameterised directly in it.The grid discretisation is also fed in at each layer to help the network learn the discretised Fourier representation of the data and undergoes point-wise linear transformations, characterised by C, its operation is similar to that of W. The output of a Fourier layer can be expressed as where x is the input to that layer, y the output, g the grid discretisation, b the bias and σ is the nonlinear activation function.Equation ( 8) resembles a single Fourier layer within the FNO.Multiple Fourier layers are stacked on top of each other to complete the iterative kernel integral operation.The stacked Fourier layers are sandwiched between the lifting and projection operations to complete the FNO architecture.\"\n },\n {\n \"title\": \"Comparison to convolutions.\",\n \"content\": \"The spectral convolutions characterised by κ in equation ( 7) have an analogous relationship to the convolutional operation.Convolution, being a product of functions across a certain range, is an integral transform.Expressed across an infinite range, it takes the mathematical form\\nwhere f and g are functions over which the convolution operation is performed.By replacing the convolution operation with the Fourier transform in equation ( 9), we obtain\\nComparing equations ( 9) and ( 10), it becomes evident how equation ( 7) is a special form of the convolutional operation, one that takes place in the Fourier space.In practice, both operations are applied across a discretisation, characterised by the data that is fed to the model, but since we decompose the data into Fourier modes within the FNO and apply learning across the decomposed components, we can learn continuous representations of the required mapping as opposed to learning discrete representations with conventional convolutions.Named spectral convolutions in [27], allow for several advantages over a standard convolutional neural network, where filters are directly learned without a spectral transform.Firstly, by learning within the Fourier space, we allow the network to identify global features that are embedded within the data, highlighting the dominant modes in the process.By performing the Fourier decomposition over the data as it parses through the network, the weights are tuned to identify and learn the modes that possess a majority of the information, allowing the network to generalise better.Secondly, the spectral convolutions being in the function space allows for learning the mapping in a discretisation convergent manner, allowing for learning mesh-free solutions.Lastly, the Fourier transform in equation ( 10) is implemented using a Fast Fourier Transform (FFT) [28] which has a computational complexity of O(n log n), whereas the standard convolution operation without spectral transform has a complexity of O(kn 2 ), where k is the size of the kernel and n is the representation dimension.However, within the FNO, we only deal with a complexity of O(nk max ) since we truncate the Fourier modes up to a desired hyperparameter k max .\"\n },\n {\n \"title\": \"Training configuration.\",\n \"content\": \"For emulating the simulations, we use the FNO-2d configuration [27], where a 2-d FNO which only convolves in space, is deployed in an autoregressive manner [29].The FNO takes in as its input the field values across a 2D grid for an initial set of time instances (T_in) along with the grid discretisation in both dimensions.Similar to spectral numerical solvers, the grid discretisation, fed in with the field data, provides better-structured information for discrete Fourier transform utilised within the FFT.The FNO takes the initial fields as the input and outputs a set of later time instances (step) of the field value across the same grid.The output of the FNO, coupled with the later time steps from the input, are then fed back in an autoregressive manner to estimate further field evolution in time, until the desired time instance (T_out) is reached.Within the autoregressive setting, we roll out the evolution of the fields up until the desired time step and then perform the backpropagation through time to enable learning of the time evolution of the PDE.For further information regarding the theory behind operator learning as ascribed by the FNO and its implementation in various case settings, we invite the reader to refer to the original work in [27].Unlike the FNO 3D configuration, we only perform Fourier transformations across the spatial dimensions and do not include the temporal dimension.We have chosen the 2D configuration of the FNO as it is lighter, quicker, and gives us the flexibility in deciding the length of the temporal evolution.\\nFor the case of the FNO for camera, instead of performing an autoregressive time roll out, we perform a fixed time window mapping, where the FNO takes in the camera information for fixed time steps (T_in) and outputs the next set of frames characterising the future plasma state (step).The next set of inputs is obtained by sliding the interested time window by one, as shown in figure 5.\"\n },\n {\n \"title\": \"Multi-variable FNO.\",\n \"content\": \"As depicted in equations ( 1)-( 3), the physics model represents a highly correlated and interdependent system where density, temperature, and electric potential factor into the time evolution equations of each other.Taking this family of PDEs into account, it is vital to construct a surrogate model that is capable of emulating all variables together, allowing for information exchange within the model.The original FNO [27] was designed to model for a single variable and needed to be modified to handle multiple variables.The work done in [18] demonstrates the impact of the FNO on predicting MHD evolution but modelled [18], it was permissible to model the variables independently.However, for the case under consideration, it was imperative that for physical consistency, we model the variables together.As we move into more complex MHD cases, we introduce more correlated variables such as density and temperature, which are both required to evolve together to fully grasp the propagation of the plasma blob.\\nTo account for the multi-variable setting, we designed the FNO to take in the variables as a separate channel.Initial construction of the FNO would consider data in the format [batch size, temporal projection, spatial along x-axis, spatial along y-axis].We modified that model to consider data in the format [batch size, variables, temporal projection, spatial along x-axis, spatial along y-axis].This modification allows us to account for all the variables within a single model, which facilitates information exchange across them.By having the multiple variables of the physics model as a channel, we allow for the Fourier transform to be applied across the spatial domain for all variables of interest.The R matrix that allows for learning within the Fourier space in equation ( 8) is modified to account for the new variables, and the local, linear operator W acts across the variable and spatial domain.\\nBy providing the variables as a separate channel across the FNO, we introduce the possibility of performing the operations of lifting and projection defined in section 2.3 not just across the temporal features but across the fields as well.Thus, within this configuration of a multi-variable FNO, the feature space becomes two-fold, accounting for the time as well as the variables.We also modify the FNO with skip connections to reduce the information loss as the data progresses through the network [30].The network architecture is outlined under Appendix B in table B1.\"\n },\n {\n \"title\": \"Results\",\n \"content\": \"Prior to elaborating on the finer details of the results from our experiments, we provide an overview of the performance obtained by the FNO surrogates across all the interested modelling tasks in table 2. Across our range of experiments, emulating the plasma, we find that the FNO achieves considerable accuracy (Mean Squared Error (MSE) in the normalised domain ∼10 −4 ).Despite being heavily parameterised, the FFT within the network helps the FNO achieve quick training and execution times for surrogate models [18].The Fourier features extracted help provide the model with implicit bias found in the data, allowing it to converge quicker on the spatial evolution of the plasma.By transforming the data to the Fourier space, we allow the network to easily extract the dominant features found in the data as they would be highlighted across the various Fourier modes.As mentioned in section 1, we focus our experiments solely on the FNO as the chosen surrogate.For a detailed benchmark of MHD emulators across various models, we refer to our prior work [18].\"\n },\n {\n \"title\": \"Multi-variable FNO over MHD Simulations\",\n \"content\": \"We design and train multi-variable FNOs to model the three MHD cases under study: Single isothermal blob, single blob in a non-uniform temperature field, and multiple blobs in a nonuniform temperature field, as described in section 2.1.Each multi-variable FNO is constructed as described in section 2.3.3 and trained using the autoregressive framework outlined in section 2.3.2.\\nFor each multi-variable FNO, we chose an input feed of 10 time steps of the field information (T_in = 10), and an output size of 5 (step = 5).Each time step within the FNO corresponds to 1.5 µs of simulation time.Each FNO was autoregressively rolled out until the 50th time instance, allowing the neural operator to learn the time evolution of the function till then (T_out = 50).Each Fourier layer within the multivariable FNO has 16 modes and a temporal channel width of 32 (unless mentioned otherwise).The temporal channel width in the FNO is akin to the channel width in a Convolutional Neural Network, where the temporal data is scaled in the FNO channels instead of the RGB color space in the CNN channels [31].The channel variable width was maintained at the number of variables that the FNO learned.The grid discretisation associated with the field data is added to the training data within each model's forward call as an addition to the temporal channel, providing further field information to the model.We utilise the discretisation used within JOREK to generate the training data as the grid features within the FNO.We deploy a two-fold normalisation strategy considering the nature of the dataset.The physical field information represented within the MHD cases is in different scales, with densities ranging from 0 to 10 20 m −3 and temperature ranging up to 100 eV.Since we consider the gradual diffusion of an inhomogeneous density blob(s), the data distribution within the spatial domain is severely imbalanced.Taking these aspects of the training data into consideration, initially the field values are scaled down by a factor, indicative of the range of operation found within the simulation itself.The physics scaling would divide the data by the order of magnitude around which the plasma operates for each variable.We perform this by scaling the field value with the order of magnitude of its maximum found in the training data.This is followed up by a linear range scaling, allowing the field values to lie between −1 and 1.The normalisation pipeline is implemented separately for each of the variables that the multi-variable FNO accounts for.Each model is trained for up to 500 epochs using the Adam optimiser [32] with a step-decaying learning rate.The learning rate is initially set to 0.005 and scheduled to decrease by half after every 100 epochs.The input, output dimensions and the model architecture used for each case is given in B.\\nThe loss function is expressed as an equally weighted linear combination of two elements: the reconstruction error and the difference error, expressed as (11) where ỹt represents the FNO output at the time step t, y t represents the labelled (actual) data at time frame t, step represents the step size of the FNO output, T represents the full length of the autoregressive time roll-out and N, the batch size or the number of simulations we are learning at a time over which we sum over.\\nThe reconstruction error helps the FNO to learn to minimise the prediction error for a particular instance in time, whereas the difference error helps learn the time evolution of the plasma better as it minimises the temporal differences that arise in the autoregressive roll-out of the FNO.The loss curves associated with each experiment can be found in figure 6.\\nThe focus of this research relies on the short-term evolution of up to 50 time steps from a total of 200 time steps, seeing how well the multi-variable FNO can model the evolution of all the physical variables in an integrated manner.We restrict ourselves to the short-term due to the errors arising from the autoregressive time rollouts, which are further explored in sections 3.1.4-3.1.6.To see the performance of the FNO deployed in an autoregressive setting for the full time range of the simulation, see appendix C.\"\n },\n {\n \"title\": \"Single blob-isothermal.\",\n \"content\": \"For the first case, that of a single blob in an isothermal setting (hereby referred to as an isothermal blob), the multi-variable FNO models the evolution of the two variables: density ρ and electric potential Φ. Considering the simple structure of the physics, the FNO performs well in (surrogate) modelling the evolution of both the variables arriving at an MSE (in the normalised domain) of 7.58 × 10 −5 in 48 min on a single Nvidia A100 chip.The multi-variable FNO is capable of emulating the evolution of the plasma up to the 50th time step in 0.025 s, offering a speedup of 6 orders of magnitude as shown in figure 7. Further plots showing the contrast of the FNO output with that of JOREK by way of discrepancy plots can be found under appendix A in figure A2.\"\n },\n {\n \"title\": \"Single blob with non-uniform temperature.\",\n \"content\": \"Moving up in complexity, in the case of a single density and temperature blob, the FNO is adapted to model density, electric potential, and temperature, three highly correlated variables as described in section 2.1.Even considering the additional physics, the multi-variable FNO demonstrates the ability to learn the global features of blob diffusion.It characterises the movements of the blob, its interaction with the walls, and its eventual dissipation.We notice that with the added field variables, the FNO performs well in predicting the evolution within the initial timesteps, but loses rigour of structure as we roll further out in time.Despite the challenge of preserving minor features further into time, the multivariable FNO achieves an MSE (in the normalised domain) of 2.56 × 10 −5 as shown in figure 8.The model trains in 1 h 37 min on a single Nvidia A100 chip.The multi-variable FNO is capable of (surrogate) modelling the evolution of the plasma up to the 50th time step in 0.03 seconds.Further plots showing the con-trast of the FNO output with that of JOREK by way of discrep-ancy plots can be found under appendix A in figure A1.\"\n },\n {\n \"title\": \"Multi-Blobs with\",\n \"content\": \"Non-uniform temperature.Figure 9 shows the capability of our newly designed multi-variable FNO in emulating the evolution of density, electric potential, and temperature for the case with multiple density and temperature blobs.This presents a more challenging case to model as the distribution of the training data is wide due to it being generated from a much larger parameter space as described in table 1.With a training dataset of 1500 simulations, the model trains in 11 h and 22 min on a single Nvidia A100 chip and achieves MSE(in the normalised domain) ∼0.0001, capturing the larger, more global, features associated with the evolution of plasma characteristics.However, due to the truncated nature of the implementation of the linear operator over the Fourier Transform, the modelling gets limited to the more dominant modes, smoothing out the less dominant, minor, finer features.The multi-variable FNO is capable of (surrogate) modelling the evolution of the plasma up to the 50th time step in 0.03 s.Further plots showing the contrast of the FNO output with that of JOREK by way of discrepancy plots can be found under appendix A from figures A3 to A5.\\nThe FNO deployed in the autoregressive manner is well equipped to model the evolution of the plasma for short time roll-outs, but as we extend the prediction further in the temporal domain, it moves further away from the ground truth.This arises mainly due to the training regime which we deploy the FNO.Being deployed in an autoregressive structure, the prediction errors accumulate over longer time roll-outs.This is illustrated in figure 10.Though we demonstrate the results obtained with 1500 simulation data points, for the rest of the ablation studies (except for sections 3.1.7and 3.1.8)demonstrated in this work, we restrict ourselves to the initial 250 as it allows us to compares against the prior 2 cases where we operate in a relatively data-scarce setting.\"\n },\n {\n \"title\": \"Individual FNO vs multi-variable FNO.\",\n \"content\": \"MHDs governing the evolution of a plasma state is characterised by the amalgamation of Navier-Stokes equation of fluid dynamics and Maxwell's equations of electromagnetism [33].The evolution of MHD fields involves modelling multi-physics environments with coupled variables that are integrated together.\\nConsidering this interplay of different field variables, it was imperative that we devise the multi-variable FNO, where all the interested and dependent variables can be modelled together.\\nOur multi-variable FNO, as described in section 2.3.3,achieves the required multi-physics modelling, with the same computational costs as training individual FNOs for each variable.The multi-variable FNO has approximately the same number of parameters as the different individual FNOs combined, and thus takes the same time as training two or three different FNO models.As shown in figure 11, the multi-variable FNO achieves better performance with much less error than the individual FNOs. Figure 11 demonstrates that by integrating the different field variables within the model, we can reduce the error associated with the autoregressive time rollout, dampening the error growth.It must be noted that since each variable has a different normalisation applied to it (the same scheme, but different normalisation), the comparison within the normalised domain is consistent within the same variable.\\nThus, we conclude that for highly correlated multi-physics systems, such as those described by the MHD equations, multi-variable FNO, which captures the interdependence of the variables within a single model, yields a much more physically consistent result than the variables modelled individually by separate FNOs with no methods of exploiting the relationship across them.\\n3.1.5.Impact of step size.The FNO, devised to be utilised in an autoregressive configuration, is only exposed to the first T_in time steps as inputs to the model.The rest of the inputs for emulating the time evolution are obtained by feeding back the output into the model in a recursive manner.The model follows the layout of a Markov decision process [34] as it relies on the previously generated time steps to provide all the information pertaining to the state of the system, without having a memory component attached to it.As the model trains, the autoregressive output generated from the input is compared against that of the simulation data to learn the mapping in a supervising manner.When deployed at inference, the FNO relies purely on the initial time steps to perform the time rollouts of the plasma evolution.\\nThus, the choice of the output size (step size), i.e, how much further in time the FNO predicts the time evolution with each forward pass, essentially provides disconnects across time for the model, which is evident by looking at figure 12.At each time step that is a multiple of the step size, we notice a spike in the cumulative error of the model.This spike denotes a disconnection that occurs as the model performs the autoregressive time evolution.In figure 12, the orange line indicates the error growth for a step size of 1, where there is no spike characterising the disconnection.For the red line with a Figure 13.Ablation study of multi-variable FNOs with different truncated mode numbers.Error growth of FNOs with different mode numbers as they are autoregressively evolved in time for the Multiblob case.Errors decrease with a higher number of modes, and performance nearly plateaus at higher modes, indicating the existence of dominant features for the problem up to a certain number of modes.For each model, the width of the temporal channel was fixed at 32, and that of the variable channel was fixed at 3. step size of 5, we notice that there is a spike at every 5th time step starting form 10 and similarly for the green line indicative of models with a step size of 10, the spike is observed at every 10th time step.Throughout the scope of this study for surrogate modelling plasma simulations, we restrict ourselves to step size = 5 as we find it offers the best training efficiency, optimum performance with less computational time.\\nRegardless of this error spike, the FNO performs well in (surrogate) modelling the various MHD cases under study.We notice that the multi-variable FNO with a step size of 1 performs the best with a smooth and dampened error growth.This is expected as the network parameters are being used to perform mapping for one time step rather than several steps, effectively providing more parameters per step.We notice that the error decreases initially before it increases and believe it is because of the autoregressive structure of the training regime.In forward propagation, the model uses the time step data first from the data and then from the model itself.This results in a distribution shift problem because the inputs are no longer solely from ground truth data, the distribution learned during training will always be an approximation of the true data distribution [35].The pushforward trick demonstrated in [36], offers an alternative training regime that allows us to account for this behaviour.\\n3.1.6.Mode ablations.The Fourier integral operator within the Fourier layer of the FNO truncates the Fourier series up to a maximal number of modes k max [27].This presents an additional hyperparameter to each Fourier layer of the FNO that determines the extent to which Fourier modes are required to be learned to map the spatio-temporal evolution.Within our experiments, we ensure that for each multi-variable FNO, the modes at each Fourier layer are fixed.Figure 13 shows the impact the number of modes has on the FNO in emulating the MHD case with multiple density and temperature blobs (Case 3).For a smaller number of modes, the FNO demonstrates a higher error value at each time step, whereas for higher mode numbers, we notice that the error is relatively lower.We also notice that the performance of the FNO plateaus at 8 modes and does not increase further as the mode numbers are increased.This arises due to the fact that the dominant features that are characteristic of the system we are learning saturates at the 8 th mode, and no more dominant features are found at higher modes for the multi-blob case.This is consistent with the mode selection that one has to perform for each problem that a neural operator is applied to.The effect of this is studied in detail [37].\\nWithin the mode ablation study, we notice that at lower modes, though the error of the model is higher, it produces smoother field profiles than those at higher modes.In figure 14, we see that the FNO with 2 modes, generates fields that are smoother and less noisy, especially at the initial time steps.However, they soon lose the general structure as they progress further in time.In the FNO with higher modes (k max = 8, 32), the FNO output is much noisier than those at lower modes, but preserves the general structure of the temporal evolution better.\\n3.1.7.Impact of training dataset size.Figure 15 visualises the impact the size of the training dataset has on the performance of the multi-variable FNO.For this purpose, we generate a larger dataset using the same scheme as given in table 1.We generate a total of 1835 JOREK simulation data points, out of which we set aside 85 as the test dataset.We train multiple multi-variable FNOs with training sizes ranging from 250 to 1750 simulations and compare their performance on the test dataset.All the models are constructed and trained under identical conditions, with the exception of the training data.We find that the performance of the FNO increases as we increase the training size, with model error decreasing by nearly half.However, we notice the gains made by added simulation data do not scale well at all.As shown in figure 15, improvement in performance is only marginal in comparison to the added computational costs associated with the training is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better.as well as the simulation run times.This is further illustrated in the loss curves of each associated experiment shown in figure 16.\"\n },\n {\n \"title\": \"Zero-shot super-resolution.\",\n \"content\": \"As described in section 2.3, the NO framework allows learning within the function space as opposed to the Euclidean space as with traditional neural networks [13].By learning the mapping between infinite-dimensional function spaces, the framework performs operator learning, allowing for characterising discretisation within our model.Within the FNO, discretisation invariance is built in by learning within the Fourier space rather than the Euclidean space on which the solutions exist, as demonstrated in equation (8).Being discretisation-convergent, the FNO learns information up to a certain refinement of the mesh resolution, higher than what we had provided in the initial We also notice the quick over-fitting tendency of the FNO as observed in [38].\\ntraining dataset [15].The truncated nature of the FNO characterises the model as a band-limited function.Coupled with the discrete modal representation in the Fourier space, the FNO converges to a fixed performance at higher resolution [15].This discretisation-convergence allows us to perform zeroshot super-resolution, trained on a lower resolution, and directly evaluated on a higher resolution [27].In figure 17, we demonstrate this capability for zero-shot super-resolution, where we take the model trained on a spatial discretisation of 100 × 100 used for the prior experiments and then deploy it to emulate the evolution of multiple blobs at a spatial discretisation of 500 × 500.Even for grids at 5 times larger resolution, the FNO is capable of emulating the plasma with considerable accuracy, deviating from the true solution by an MSE (in the normalised domain)of only 7.90 × 10 −5 in the normalised domain.They share the same model parameters among different discretisations of the underlying function space that we are attempting to learn.Plots showing the contrast of the super-resolved FNO output with that of JOREK by way of discrepancy plots can be found in figure A6 in appendix A.\"\n },\n {\n \"title\": \"Extrapolating outside training domain.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Neural\",\n \"content\": \"Networks trained for modelling a particular task have been found to suffer from a general lack of extrapolation capabilities [39].The FNO relying on a continuous gradientdescent based optimisation of the deep learning parameters exhibit the same behaviour.The models perform well within the domain of the training data, as can be explained by the associated Neural Tangent Kernels (NTK) [40] and fail to perform well outside the training regime.\\nFor our multi-variable FNO that models the evolution of multiple blobs in reduced MHD, we can empirically validate the above-mentioned statements.When inferred on a case where there are 13 blobs with amplitude and density twice more and width 1.5 times more than the maximum used within the training data, we observe that the FNO can model the plasma evolution at the earliest prediction time instance and fails to capture the evolution of the plasma behaviour further in time as evidenced in the figure 18.\"\n },\n {\n \"title\": \"Scaling to 3D.\",\n \"content\": \"There exist several challenges in the extending the FNO to model the evolution of multi-variable PDEs to 3D settings.The primary concern is of the memory constraints needed for a 3D multi-variable FNO when being evaluated autoregressively in time, leading to GPU bottlenecks.Recent work in performing Tensor factorisation over the FNO weights, allows representing the network in a lower rank approximation [41], offers potential to those GPU training bottlenecks.\\nAnother challenge associated with the scaling is that of the computational costs associated with data generation.In 3D, the numerical solvers that generate the training datasets are exponentially more intensive than in the 2D settings, limiting the amount of training data that could be generated.Deploying Active Learning (AL) methods, which allow for more informed sampling of the parameter domain space offers a solution to this.Within AL, the model is initially trained on a coarsely sampled, limited dataset and then evaluated across the interested domain.Performance (or uncertainty) of the model within the interested domain informs the next data generation points, which are added to the training data, improving the model accuracy.AL allows the model to maximise performance, minimise uncertainty within data scarce scenarios [42].Within the scope of plasma physics, AL has been found to reduce the data requirement of surrogate models by a factor of 20 in modelling Gyrokinetic Turbulence [43].The issue of data-scarcity arising from the profound computational intensiveness of 3D simulations could also be combat by using physics-informed models, where the information regarding the PDE is hard-baked into the model architecture or the training regime.Recent works on physics informed neural operators has shown promise in fine-tuning coarsely trained models with PDE constraints within MHD scenarios [44].\\nWhile scaling to 3D settings it might be the case that the choice of operator within the Neural Operator might be limiting and not representative of the true solution, thus there might be a requirement of a combination of different operators implemented over different families of basis functions within the model to account for the varied physics that is found within the PDEs.One last challenge to consider is that proposed by the autoregressive layout of the FNO.The addition of another dimension will contribute further to the error growth in time as noted in figure 10.Though this remains a standing problem within the domain of Neural-PDEs, recent works have shown promise in solving this challenge by providing further conditioning of the longer time roll-outs [45].\"\n },\n {\n \"title\": \"FNO over MAST camera\",\n \"content\": \"As described in section 2.2, we conduct exploratory research in looking at using the FNO as an emulator for performing predictive inference in real-time of the plasma evolution within the MAST fusion device.For this purpose, we deploy FNOs with the same architecture as that used for emulating the Individual FNOs (given under appendix D in tables D1 and D2).However, considering the nature of the camera dataset, we implement a different training and inference regime from that of the autoregressive structure used to solve PDEs.\\nIn the case of the MHD simulations, the evolution of plasma is a Markovian Process, with no element of noise found within the dataset.However, within the case of the experimental data, the images are still time-series data, but they are nonmarkovian, noisy, and represent only partial information about the state of the system.In addition, the dataset does not offer any information about the causal contributors, such as the various control inputs that influence the state of the system, and is assumed to be implicitly defined within the time evolution of the data.Taking this into account, implementing an autoregressive structure as demonstrated in [18], provided limited capabilities as the error growth across the time roll-out would be too large, not allowing us to perform longer time roll-outs.Within the scale of the experimental data, it might be the case that there might be new information fed into the system midway through it.By utilising an autoregressive rollout that relies solely on the initial time steps, the model will not be able to account for this new information added later to the system.Unlike in the case of modelling simulations, there is no actual need within the experimental space to rely just on the initial conditions and state of the system to perform the prediction.Data characterising the temporal evolution of the plasma is generated and captured by the diagnostic as the experiment is run and does not need any additional intervention to obtain it.Our training and inference setup takes advantage of this continuously generated data to build our predictive model.Instead of performing an autoregressive time rollout, we perform a fixed time window mapping (as outlined in figure figure 5), where the FNO takes in the camera information for fixed time steps (T_in) and outputs the next set of frames characterising the future plasma state (step).Being trained in this sequential pipeline, the model initially takes in the first 10 time instances (1-10 frames) to output the next 10 time instances (11-20 frames).With the next forward pass, the model takes in the other 10 time instances, but this time, forward by one frame (2-11 frames) to predict the next 10 frames to that input (12-21 frames).This input, characterised by a 10-frame window slid by one each forward pass, is run until we output the last required frame (T_out).Our trained FNO models the plasma at ramp-up, entering the flat-top regime, and in most cases, across the L-Mode and H-Mode operational regimes, covering the full (available) plasma duration.Since we are designing this to be run in real-time with the experiment, future time windows, as they become available, will be fed into the model to predict the future plasma states.The time resolution of the fast camera diagnostic is 1.2 ms, where the FNO only requires 6 ms to predict the next 12 ms of the plasma evolution, allowing us to run the model efficiently in real time.\\nEven though we keep our focus on a limited number of shots curated according to the criteria given in section 3.2.1, as we structure them into input-output pairs by way of the sliding windows as described above, we essentially turn each shot characterising of approximately 200 frames into 180 inputoutput pairs.The FNO is exposed to dynamics of the plasma evolution across the interested time window of MAST operation, learning to predict from any intermediary states of the system to the next states of the system.The FNO is trained to learn how to predict the evolution of the plasma as witnessed by the camera for the full (available) range of operation of the Tokamak pulse.\\nFor each camera FNO, we chose the input feed to consist of 10 time steps of the field information (T_in = 10), with an output size of 10 (step = 10).Each Fourier layer within the FNO has 8 modes and a temporal channel width of 16.The grid discretisation associated with the field data (obtained from the reactor CAD model) is added to the training data within each model's forward call as an addition to the temporal channel, providing further field information to the model.The data is normalised by following a linear range scaling, allowing the field values to lie between −1 and 1.Each model is trained for 500 epochs using the Adam optimiser [32] with a stepdecaying learning rate.The learning rate is initially set to 0.005 and scheduled to decrease by half after every 100 epochs.The model was trained using a Relative L2 loss function over the reconstruction error alone (see equation ( 11)).\"\n },\n {\n \"title\": \"Data selection.\",\n \"content\": \"Throughout this study, we work on a selected number of plasma shots for each cameras, restricting ourselves to the M9 campaign on MAST.For the rbb camera, we restrict ourselves to 55 shots lying within the range: 30 250-30 431 from the MAST experimental database.For the rba camera we look at 48 shots lying within the same range 30 250-30 431.The shots were carefully curated from the above shot range by considering several factors.The first criterion was to ensure that the selected shot actually contained a confined plasma and was not a dummy shot like commissioning pulses.The second criterion was whether the shot had the desired duration of more than 100 time steps-looking at forecasting longer-term plasma evolution.The third one was selecting shots within the above-mentioned ranges that had the same camera calibration.Considering the FNO relied on a grid discretisation to leverage the Fourier modes, it was crucial to ensure that the pixels from the camera belonged to the same line of sight vectors.The camera calibrated for specific experimental runs captures the information of the plasma onto a uniform pixel grid, allowing us to perform Fourier transformations over them.By using calcam, a calibration tool developed by the European Atomic Energy Community [46] combined with 3D CAD models of MAST, we determine the domain range covered by each camera calibration.The upper and lower limits of the domain range of these images are then fixed onto uniform grids in both the R and Z axis, allowing us to create uniform grid discretisations underlying each camera image.The last criterion was to ensure that the camera images obtained from the diagnostic had the same spatial resolution across the dataset that could be fed into the FNO.For the rbb camera, we chose the shots with a resolution of (448 × 640), and for the rbb camera, we chose those with the resolution (400 × 512).However, the FNO offers discretisation convergence, the training setup benefits greatly from ensuring coherence along the spatial resolution.Figure 19 shows the plasma signals allows to demonstrate the diversity of the scenarios used to train and test the FNO modelling the cameras.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data.For a full list of the selected shots, refer appendix G.\"\n },\n {\n \"title\": \"Camera viewing the central solenoid (rbb).\",\n \"content\": \"The evolution of the plasma within the Tokamak, as seen by the main camera, can be shown in figure 20.The plot shows the comparisons of the evolution as observed by the FNO with that of the camera spanning 10 frames predicted in one feed-forward iteration.The central solenoid serves as the backbone of the MAST magnet system, generating the majority of the magnetic fluxes as well as the plasma current.In figure 23, we portray the predictive capability of the FNO by comparing the last frame from multiple feed-forward instances taken from data across the interested length of the pulse.This allows us to compare that even for the longest time prediction, we can get reasonable predictions.In figure 23, we demonstrate that the FNO is capable of predicting the plasma in L-mode as well as in H-mode, characterising the plasma evolution across the full (available) range of the plasma shot.The main camera view, as shown in figure 1 The FNO deployed with the time-window pipeline (see figure 5), is able to capture the long-term dependencies that arise within the plasma confinement.The model is able to predict the evolution of the plasma from L-mode to H-mode as seen in figure 23.The prediction around the transition regime can be seen in Though the camera captures the confined plasma shape, it fails to capture the localised filaments.In figure 20, the trained FNO is deployed to model the evolution of plasma within the L-Mode of operation, whereas in figure 21 the FNO is modelling the region of H-mode within the same plasma shot.\\nThe capability of the FNO in predicting across the L-H transition is further emphasised through figure 22, where we show the prediction around the transition region for shot number 30 428.We provide the camera image, FNO prediction and the absolute error at L-mode (162 ms), around the transition (172 ms) and with the plasma in H-mode (186 ms).The FNO is capable of modelling the plasma in both L-mode, Hmode and across the transition, but as the FNO is a regression model and not a classification model, without further analysis over the predicted solution, the model is unable to explicitly mention that it is capable of predicting the exact moment of transition.This will be a key area of focus for the next research avenue.\"\n },\n {\n \"title\": \"Camera viewing the divertor (rba).\",\n \"content\": \"The evolution of the plasma within the Tokamak as seen by the divertor camera, along with predicted images from the FNO can be seen in figure 24.The plot shows the comparisons of the evolution as observed by the FNO with that of the camera spanning 10 frames predicted in one feed-forward iteration.The divertor of a Tokamak serves as the exhaust of the device, where the topology of the magnetic flux surfaces is designed to isolate the confined plasma from the first-wall of the device.In figure 25, we portray the predictive capability of the FNO by comparing the last frame from multiple feed-forward instances taken from data across the interested length of the pulse, demonstrating the predictive power of the FNO.The divertor camera looks at the lower divertor of MAST with a spatial resolution of 400 × 512.We sampled a total of 48 plasma shots from the MAST database, used 45 for training and 3 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 5815, and the test points grow to 701 data points.The model takes 9 hours 10 minutes on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.70 × 10 −4 .For further plots showcasing the forecasting capabilities of the FNO at the divertor plasma, refer to appendix F, specifically figures F1 and F2.\\nThe FNO deployed for emulating the camera data accurately preserves the global features of the plasma evolution.It is able to clearly characterise the locations and intensities of the heat fluxes onto the solenoid and the divertors, allowing us to anticipate the heat flux in advance and initiate the appropriate actuator mechanism to minimise its impact on the various plasma-facing components.The FNO also captures the reflections generated within the device due to the interactions across the plasma and the plasma-facing components.The FNO, in its current form, smooths out the filaments but captures the general confined structure of the plasma as it evolves in time.As expected, the FNO learns characteristics extremely well, while failing to capture the localised phenomena with matching pixel-pixel accuracy.\\n3.2.4.Limitations.We decided to look at a narrow shot corridor, since the data variability will be low, allowing us to demonstrate a functioning proof-of-concept.Our predictive model only takes in the camera diagnostic as the input and does not take into account any of the control signals, such as coil currents and heating parameters that influence the plasma evolution.However, we argue that within the past window of plasma observed from the camera data that is taken as the input, these control parameters and their influence are implicitly defined (with a latency), and thus, the model provides a suitable predictive tool for short-term predictions.The inclusion of various control signals, time matched with the camera inputs, will potentially help improve the performance of the model.Our future research trajectory will be heading in this direction.\"\n },\n {\n \"title\": \"Conclusion and discussion\",\n \"content\": \"Through this work, we demonstrate the utility of using the FNO as both a surrogate model for a plasma simulator and as a predictive emulator for a plasma experiment.We design a modified version of the FNO, that performs integrated emulation of multiple variables associated with a family of PDEs.Our analysis across a range of reduced MHD cases demonstrates that even in data-scarce scenarios, the FNO can be trained to act as a surrogate.Through this paper, we highlight the various features that make the FNO a natural choice for (surrogate) modelling MHD cases, all the while performing ablation studies on the various hyperparameters and the training regimes.We find that the FNO fits the requirements of a quick, cheap surrogate, but due to its auto-regressive nature, fails to capture the physics when deployed for long time rollouts.Being utilised for emulating the experiment, the FNO is capable of forecasting the evolution of the plasma as seen through the cameras on a fusion device.The FNO camera models are capable of identifying the regions of plasma interaction with the central solenoid and the divertor, while effectively predicting the plasma shape evolution.The FNO has shown versatile performance in being able to model the plasma confinement across both L-mode and H-mode of plasma operation across a diverse range of shots.The FNO for the camera can predict the plasma evolution in real-time and offers us the potential to be deployed within the planning stage for various control scenarios.Our work also demonstrates how we can perform zero-shot super-resolution across the pre-trained multi-variable FNO and obtain the plasma emulation at higher resolutions than it was trained on, without requiring any finetuning.\\nIn continuing with this line of work, we will be exploring the utility of building more versatile and robust FNO models that can be deployed not just in a particular case, but across a range of simulation cases.While we build more general MHD models, we will also be looking at methods to better physics-inform the models [14], allowing us to finetune the general models to specific MHD instances.We will also be looking into methods of quantifying the uncertainty associated with the surrogate models and into AL pipelines that can facilitate the entire surrogate ecosystem.With the FNO for camera data, we are currently working on extending the utility across more plasma shots with a more diverse line of sights, across multiple campaigns while exploring the utility of state space models for longer predictions [47].The work will be looking at including various control signals, to help improve the performance of the real-time predictive emulator for the MAST Tokamak, performing longer forecasting.\"\n },\n {\n \"title\": \"A.2. Single blob with non-uniform temperature\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"Figure 1.(a): Fast camera image of a MAST plasma during an Edge-Localised-Mode, where plasma filaments are ejected outwards due to MHD instabilities at the plasma edge.Reprinted from [19], with the permission of AIP Publishing.(b): The evolution of the filamentary structures during the multi-mode ELM simulation imaged with a synthetic fast camera diagnostic (time given in ms).Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020.\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"Domain range and sampling strategies across the initial condition parameters.The case numbers represent the simulation settings in which that parameter was sampled.1-single isothermal blob, 2-single blob with non-uniform temperature, 3-multiple blobs with non-uniform temperature.U [a, b] refers to a uniform distribution bounded by a and b.\"\n },\n {\n \"title\": \"Figure 2 .\",\n \"content\": \"Figure 2. Evolution of multiple blobs inside a slab for the first 500 time-steps of the numerical simulation (the full run is 2000 steps), taken at intervals of 100 steps (about 15 µs).The top row shows the density, the middle row the temperature, and the bottom row the vorticity.\"\n },\n {\n \"title\": \"Figure 3 .\",\n \"content\": \"Figure 3. Views of the fast cameras on MAST in the absence of plasma, obtained using illuminations on the 3D CAD model of MAST.Camera shown in figure (a), looks across the central solenoid, while the camera in figure (b) shows the view at the divertor.\"\n },\n {\n \"title\": \"Figure 4 .\",\n \"content\": \"Figure 4. Modified architecture of a single Fourier Layer.FNO is constructed by stacking multiple Fourier layers on top of each other, sandwiched by a lifting and projection operation.\"\n },\n {\n \"title\": \"Figure 5 .\",\n \"content\": \"Figure 5. Schematic representation of the two kinds of training configurations deployed within this study.For the simulations, we use the autoregressive configuration (left).We only use the initial set of field values to propagate towards the desired time step, where the intermediary model output is fed back into the model as inputs to further the time evolution.For the experimental data, we use a sequential time window-based configuration.Here, the model output is not factored back into the model, but we perform a fixed window mapping across time.As we go across the time duration of the data, the window is slid further by one-time instance (right).In the example above, we have T_in = 3, step = 1, T_out = 8, the time focus of each forward propagation of the model is given along the x-axis while the evolution across the full-time domain is given along the y-axis.Each cell block represents the 2D field value at a certain instance of time.The input and output sizes shown in this diagram are representative only and vary across each experiment.\"\n },\n {\n \"title\": \"Figure 6 .\",\n \"content\": \"Figure 6.Learning curves associated with training the multi-variable FNO across the various the various Reduced MHD Cases.Isothermal blob refers to the single blob diffusion under uniform temperature, Anisothermal blob refers to the single blob under a non-uniform temperature, and Anisothermal multiple Blobs extend that to the case with several density blobs under a non-uniform temperature distribution.\"\n },\n {\n \"title\": \"Figure 7 .\",\n \"content\": \"Figure 7. Isothermal Blob: Temporal evolution of (a) density and (b) electric potential of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"title\": \"Figure 8 .\",\n \"content\": \"Figure 8. Single Blob : Temporal evolution of (a) density and (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"title\": \"Figure 9 .\",\n \"content\": \"Figure 9. Multiple Blobs with non-uniform temperature: Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"title\": \"Figure 10 .\",\n \"content\": \"Figure 10.Error growth across time roll-out of the FNO in the physical space.The error in emulating each variable of the reduced MHD model increases as we roll further out in time.\"\n },\n {\n \"title\": \"Figure 11 .\",\n \"content\": \"Figure 11.Comparing the error growth (mean absolute error in the normalised domain) of different variables across time roll-out of the Individual FNO (Single) and the Multi-variable FNO (Multi) for the various MHD cases under study.(a) Single Isothermal Blob (b) Single Blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The multi-variable FNO with the same parameters as the individual FNOs demonstrates much better performance over time.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.\"\n },\n {\n \"title\": \"Figure 12 .\",\n \"content\": \"Figure 12.Comparing the cumulative error growth of different variables across time roll-out for multi-variable FNOs with different step sizes.The autoregressive nature of the FNO brings about a spike in error, characterising a disconnection at the time step that is a multiple of the output size.(a) Isothermal blob (b) Single blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.\"\n },\n {\n \"title\": \"Figure 14 .\",\n \"content\": \"Figure 14.Evolution of the electric potential (Multiblobs) in time as modelled by multi-variable FNOs at different modes.Top row of (a) shows the actual solution modelled using JOREK.The bottom row of (a), plots the time evolution of the FNO with 2 modes, top row of (b)is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better.\"\n },\n {\n \"title\": \"Figure 15 .\",\n \"content\": \"Figure15.Impact of the size of the training dataset on the performance of the multi-variable FNO for simulating the multiple plasma blobs in a non-uniform temperature field.As we increase the availability of the training data for the multi-variable FNO, the model performance increases across the same test data.However, the gains obtained by increasing the training data are marginal compared to the additional compute required.The number of simulations used for training is given on the x-axis while the mean squared error is given on the y-axis.All training is performed on a single Nvidia A100.\"\n },\n {\n \"title\": \"Figure 16 .\",\n \"content\": \"Figure 16.Learning curves associated with the models trained with data of different sizes.Training loss is represented by full lines, while the test loss is represented by dashed lines.With just 500 simulation instances, the model learns the physics of the model really well, which is only marginally improved by more training data.We also notice the quick over-fitting tendency of the FNO as observed in[38].\"\n },\n {\n \"title\": \"Figure 17 .\",\n \"content\": \"Figure 17.Demonstrating Zero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE(in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the bottom series of plots demonstrate the time evolution as emulated by the multi-variable FNO.\"\n },\n {\n \"title\": \"Figure 18 .\",\n \"content\": \"Figure 18.Outside the training domain: Inferring on a case with 13 blobs with amplitude and density twice more and the blob width 1.5 times more than the maximum used within the training data (see table 1).Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"title\": \"Figure 19 .\",\n \"content\": \"Figure 19.Temporal plots of (a) Plasma Current and (b) Total Heating Power of the shots selected for training and testing the FNO for modelling the camera data.The shots depicting the training data is given in black, whereas the test data is given in red.The plots help convey the diversity of plasma scenarios the FNO is trained to model over.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data.\"\n },\n {\n \"title\": \"20 .\",\n \"content\": \"Predicting in L-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera with the plasma in L-mode.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\"\n },\n {\n \"title\": \"Figure 21 .\",\n \"content\": \"Figure 21.Predicting in H-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera wiht the plasma in H-mod.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The domain is selected to showcase the performance of the FNO in mapping the plasma evolution in H-mode.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\\nprovides a panoramic view of the reactor, showing the poloidal cross-sectional layout on both sides of the central solenoid.The main camera has a spatial resolution of 448 × 640 for the experiments that we sampled.We sampled a total of 55 plasma shots from the MAST database, used 50 for training and 5 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 7661, and the test points grow to 828 data points.The model takes 27 h 45 min on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.7 × 10 −3 .For further plots showcasing the forecasting capabilities of the FNO at the plasma around the central solenoid, refer to appendix E in figures E1 and E2.\"\n },\n {\n \"title\": \"Figure 22 .\",\n \"content\": \"Figure 22.Predicting around the L-H transition: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.\"\n },\n {\n \"title\": \"Figure 23 .\",\n \"content\": \"Figure 23.Long-term prediction: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.\"\n },\n {\n \"title\": \"Figure 24 .\",\n \"content\": \"Figure 24.Single forward pass: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\"\n },\n {\n \"title\": \"Figure 25 .\",\n \"content\": \"Figure 25.Long-term prediction: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\"\n },\n {\n \"title\": \"Figure\",\n \"content\": \"Figure Single Blob with non-uniform Temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"title\": \"Figure A4 .\",\n \"content\": \"Figure A4.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"title\": \"Figure A5 .\",\n \"content\": \"Figure A5.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"title\": \"Figure\",\n \"content\": \"Figure DemonstratingZero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE (in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the middle series of plots demonstrate the time evolution as emulated by the multi-variable FNO and the bottom series indicates the absolute error across space.\"\n },\n {\n \"title\": \"Table 2 .\",\n \"content\": \"Overview of the performance obtained by various FNO-based surrogate models built to emulate the plasma evolution in both simulations and experiments.The case represents the various simulation settings of JOREK or the camera diagnostic on MAST, for which we are training an emulator.The variables mention the number of variables from the PDE/diagnostic that we model together.Parameters refer to the number of parameters in the trained FNO, and MSE represents the mean squared error in the normalised domain.The Training Size represented the simulation/experimental input-output pairs.All models are trained on a single Nvidia A100 chip.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Figure 1.(a): Fast camera image of a MAST plasma during an Edge-Localised-Mode, where plasma filaments are ejected outwards due to MHD instabilities at the plasma edge.Reprinted from [19], with the permission of AIP Publishing.(b): The evolution of the filamentary structures during the multi-mode ELM simulation imaged with a synthetic fast camera diagnostic (time given in ms).Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020.","content":"Figure 1 .\n1\nFigure 1.(a): Fast camera image of a MAST plasma during an Edge-Localised-Mode, where plasma filaments are ejected outwards due to MHD instabilities at the plasma edge.Reprinted from [19], with the permission of AIP Publishing.(b): The evolution of the filamentary structures during the multi-mode ELM simulation imaged with a synthetic fast camera diagnostic (time given in ms).Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020."},{"id":"","caption":"Domain range and sampling strategies across the initial condition parameters.The case numbers represent the simulation settings in which that parameter was sampled.1-single isothermal blob, 2-single blob with non-uniform temperature, 3-multiple blobs with non-uniform temperature.U [a, b] refers to a uniform distribution bounded by a and b.","content":"Table 1 .\n1\nDomain range and sampling strategies across the initial condition parameters.The case numbers represent the simulation settings in which that parameter was sampled.1-single isothermal blob, 2-single blob with non-uniform temperature, 3-multiple blobs with non-uniform temperature.U [a, b] refers to a uniform distribution bounded by a and b."},{"id":"","caption":"Figure 2. Evolution of multiple blobs inside a slab for the first 500 time-steps of the numerical simulation (the full run is 2000 steps), taken at intervals of 100 steps (about 15 µs).The top row shows the density, the middle row the temperature, and the bottom row the vorticity.","content":"Figure 2 .\n2\nFigure 2. Evolution of multiple blobs inside a slab for the first 500 time-steps of the numerical simulation (the full run is 2000 steps), taken at intervals of 100 steps (about 15 µs).The top row shows the density, the middle row the temperature, and the bottom row the vorticity."},{"id":"","caption":"Figure 3. Views of the fast cameras on MAST in the absence of plasma, obtained using illuminations on the 3D CAD model of MAST.Camera shown in figure (a), looks across the central solenoid, while the camera in figure (b) shows the view at the divertor.","content":"Figure 3 .\n3\nFigure 3. Views of the fast cameras on MAST in the absence of plasma, obtained using illuminations on the 3D CAD model of MAST.Camera shown in figure (a), looks across the central solenoid, while the camera in figure (b) shows the view at the divertor."},{"id":"","caption":"Figure 4. Modified architecture of a single Fourier Layer.FNO is constructed by stacking multiple Fourier layers on top of each other, sandwiched by a lifting and projection operation.","content":"Figure 4 .\n4\nFigure 4. Modified architecture of a single Fourier Layer.FNO is constructed by stacking multiple Fourier layers on top of each other, sandwiched by a lifting and projection operation."},{"id":"","caption":"Figure 5. Schematic representation of the two kinds of training configurations deployed within this study.For the simulations, we use the autoregressive configuration (left).We only use the initial set of field values to propagate towards the desired time step, where the intermediary model output is fed back into the model as inputs to further the time evolution.For the experimental data, we use a sequential time window-based configuration.Here, the model output is not factored back into the model, but we perform a fixed window mapping across time.As we go across the time duration of the data, the window is slid further by one-time instance (right).In the example above, we have T_in = 3, step = 1, T_out = 8, the time focus of each forward propagation of the model is given along the x-axis while the evolution across the full-time domain is given along the y-axis.Each cell block represents the 2D field value at a certain instance of time.The input and output sizes shown in this diagram are representative only and vary across each experiment.","content":"Figure 5 .\n5\nFigure 5. Schematic representation of the two kinds of training configurations deployed within this study.For the simulations, we use the autoregressive configuration (left).We only use the initial set of field values to propagate towards the desired time step, where the intermediary model output is fed back into the model as inputs to further the time evolution.For the experimental data, we use a sequential time window-based configuration.Here, the model output is not factored back into the model, but we perform a fixed window mapping across time.As we go across the time duration of the data, the window is slid further by one-time instance (right).In the example above, we have T_in = 3, step = 1, T_out = 8, the time focus of each forward propagation of the model is given along the x-axis while the evolution across the full-time domain is given along the y-axis.Each cell block represents the 2D field value at a certain instance of time.The input and output sizes shown in this diagram are representative only and vary across each experiment."},{"id":"","caption":"Figure 6.Learning curves associated with training the multi-variable FNO across the various the various Reduced MHD Cases.Isothermal blob refers to the single blob diffusion under uniform temperature, Anisothermal blob refers to the single blob under a non-uniform temperature, and Anisothermal multiple Blobs extend that to the case with several density blobs under a non-uniform temperature distribution.","content":"Figure 6 .\n6\nFigure 6.Learning curves associated with training the multi-variable FNO across the various the various Reduced MHD Cases.Isothermal blob refers to the single blob diffusion under uniform temperature, Anisothermal blob refers to the single blob under a non-uniform temperature, and Anisothermal multiple Blobs extend that to the case with several density blobs under a non-uniform temperature distribution."},{"id":"","caption":"Figure 7. Isothermal Blob: Temporal evolution of (a) density and (b) electric potential of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.","content":"Figure 7 .\n7\nFigure 7. Isothermal Blob: Temporal evolution of (a) density and (b) electric potential of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"id":"","caption":"Figure 8. Single Blob : Temporal evolution of (a) density and (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.","content":"Figure 8 .\n8\nFigure 8. Single Blob : Temporal evolution of (a) density and (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"id":"","caption":"Figure 9. Multiple Blobs with non-uniform temperature: Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.","content":"Figure 9 .\n9\nFigure 9. Multiple Blobs with non-uniform temperature: Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"id":"","caption":"Figure 10.Error growth across time roll-out of the FNO in the physical space.The error in emulating each variable of the reduced MHD model increases as we roll further out in time.","content":"Figure 10 .\n10\nFigure 10.Error growth across time roll-out of the FNO in the physical space.The error in emulating each variable of the reduced MHD model increases as we roll further out in time."},{"id":"","caption":"Figure 11.Comparing the error growth (mean absolute error in the normalised domain) of different variables across time roll-out of the Individual FNO (Single) and the Multi-variable FNO (Multi) for the various MHD cases under study.(a) Single Isothermal Blob (b) Single Blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The multi-variable FNO with the same parameters as the individual FNOs demonstrates much better performance over time.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.","content":"Figure 11 .\n11\nFigure 11.Comparing the error growth (mean absolute error in the normalised domain) of different variables across time roll-out of the Individual FNO (Single) and the Multi-variable FNO (Multi) for the various MHD cases under study.(a) Single Isothermal Blob (b) Single Blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The multi-variable FNO with the same parameters as the individual FNOs demonstrates much better performance over time.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout."},{"id":"","caption":"Figure 12.Comparing the cumulative error growth of different variables across time roll-out for multi-variable FNOs with different step sizes.The autoregressive nature of the FNO brings about a spike in error, characterising a disconnection at the time step that is a multiple of the output size.(a) Isothermal blob (b) Single blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.","content":"Figure 12 .\n12\nFigure 12.Comparing the cumulative error growth of different variables across time roll-out for multi-variable FNOs with different step sizes.The autoregressive nature of the FNO brings about a spike in error, characterising a disconnection at the time step that is a multiple of the output size.(a) Isothermal blob (b) Single blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout."},{"id":"","caption":"Figure 14.Evolution of the electric potential (Multiblobs) in time as modelled by multi-variable FNOs at different modes.Top row of (a) shows the actual solution modelled using JOREK.The bottom row of (a), plots the time evolution of the FNO with 2 modes, top row of (b)is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better.","content":"Figure 14 .\n14\nFigure 14.Evolution of the electric potential (Multiblobs) in time as modelled by multi-variable FNOs at different modes.Top row of (a) shows the actual solution modelled using JOREK.The bottom row of (a), plots the time evolution of the FNO with 2 modes, top row of (b)is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better."},{"id":"","caption":"Figure15.Impact of the size of the training dataset on the performance of the multi-variable FNO for simulating the multiple plasma blobs in a non-uniform temperature field.As we increase the availability of the training data for the multi-variable FNO, the model performance increases across the same test data.However, the gains obtained by increasing the training data are marginal compared to the additional compute required.The number of simulations used for training is given on the x-axis while the mean squared error is given on the y-axis.All training is performed on a single Nvidia A100.","content":"Figure 15 .\n15\nFigure15.Impact of the size of the training dataset on the performance of the multi-variable FNO for simulating the multiple plasma blobs in a non-uniform temperature field.As we increase the availability of the training data for the multi-variable FNO, the model performance increases across the same test data.However, the gains obtained by increasing the training data are marginal compared to the additional compute required.The number of simulations used for training is given on the x-axis while the mean squared error is given on the y-axis.All training is performed on a single Nvidia A100."},{"id":"","caption":"Figure 16.Learning curves associated with the models trained with data of different sizes.Training loss is represented by full lines, while the test loss is represented by dashed lines.With just 500 simulation instances, the model learns the physics of the model really well, which is only marginally improved by more training data.We also notice the quick over-fitting tendency of the FNO as observed in[38].","content":"Figure 16 .\n16\nFigure 16.Learning curves associated with the models trained with data of different sizes.Training loss is represented by full lines, while the test loss is represented by dashed lines.With just 500 simulation instances, the model learns the physics of the model really well, which is only marginally improved by more training data.We also notice the quick over-fitting tendency of the FNO as observed in[38]."},{"id":"","caption":"Figure 17.Demonstrating Zero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE(in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the bottom series of plots demonstrate the time evolution as emulated by the multi-variable FNO.","content":"Figure 17 .\n17\nFigure 17.Demonstrating Zero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE(in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the bottom series of plots demonstrate the time evolution as emulated by the multi-variable FNO."},{"id":"","caption":"Figure 18.Outside the training domain: Inferring on a case with 13 blobs with amplitude and density twice more and the blob width 1.5 times more than the maximum used within the training data (see table 1).Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.","content":"Figure 18 .\n18\nFigure 18.Outside the training domain: Inferring on a case with 13 blobs with amplitude and density twice more and the blob width 1.5 times more than the maximum used within the training data (see table 1).Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"id":"","caption":"Figure 19.Temporal plots of (a) Plasma Current and (b) Total Heating Power of the shots selected for training and testing the FNO for modelling the camera data.The shots depicting the training data is given in black, whereas the test data is given in red.The plots help convey the diversity of plasma scenarios the FNO is trained to model over.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data.","content":"Figure 19 .\n19\nFigure 19.Temporal plots of (a) Plasma Current and (b) Total Heating Power of the shots selected for training and testing the FNO for modelling the camera data.The shots depicting the training data is given in black, whereas the test data is given in red.The plots help convey the diversity of plasma scenarios the FNO is trained to model over.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data."},{"id":"","caption":"Predicting in L-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera with the plasma in L-mode.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.","content":"20 .\n20\nPredicting in L-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera with the plasma in L-mode.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera."},{"id":"","caption":"Figure 21.Predicting in H-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera wiht the plasma in H-mod.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The domain is selected to showcase the performance of the FNO in mapping the plasma evolution in H-mode.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.","content":"Figure 21 .\n21\nFigure 21.Predicting in H-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera wiht the plasma in H-mod.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The domain is selected to showcase the performance of the FNO in mapping the plasma evolution in H-mode.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera."},{"id":"","caption":"provides a panoramic view of the reactor, showing the poloidal cross-sectional layout on both sides of the central solenoid.The main camera has a spatial resolution of 448 × 640 for the experiments that we sampled.We sampled a total of 55 plasma shots from the MAST database, used 50 for training and 5 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 7661, and the test points grow to 828 data points.The model takes 27 h 45 min on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.7 × 10 −3 .For further plots showcasing the forecasting capabilities of the FNO at the plasma around the central solenoid, refer to appendix E in figures E1 and E2.","content":"provides a panoramic view of the reactor, showing the poloidal cross-sectional layout on both sides of the central solenoid.The main camera has a spatial resolution of 448 × 640 for the experiments that we sampled.We sampled a total of 55 plasma shots from the MAST database, used 50 for training and 5 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 7661, and the test points grow to 828 data points.The model takes 27 h 45 min on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.7 × 10 −3 .For further plots showcasing the forecasting capabilities of the FNO at the plasma around the central solenoid, refer to appendix E in figures E1 and E2."},{"id":"","caption":"Figure 22.Predicting around the L-H transition: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.","content":"Figure 22 .\n22\nFigure 22.Predicting around the L-H transition: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments."},{"id":"","caption":"Figure 23.Long-term prediction: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.","content":"Figure 23 .\n23\nFigure 23.Long-term prediction: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments."},{"id":"","caption":"Figure 24.Single forward pass: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.","content":"Figure 24 .\n24\nFigure 24.Single forward pass: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera."},{"id":"","caption":"Figure 25.Long-term prediction: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.","content":"Figure 25 .\n25\nFigure 25.Long-term prediction: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera."},{"id":"","caption":"Figure Single Blob with non-uniform Temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.","content":"Figure\n\nFigure Single Blob with non-uniform Temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"id":"","caption":"Figure A4.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.","content":"Figure A4 .\nA4\nFigure A4.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"id":"","caption":"Figure A5.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.","content":"Figure A5 .\nA5\nFigure A5.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis."},{"id":"","caption":"Figure DemonstratingZero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE (in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the middle series of plots demonstrate the time evolution as emulated by the multi-variable FNO and the bottom series indicates the absolute error across space.","content":"Figure\n\nFigure DemonstratingZero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE (in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the middle series of plots demonstrate the time evolution as emulated by the multi-variable FNO and the bottom series indicates the absolute error across space."},{"id":"","caption":"","content":""},{"id":"","caption":"","content":""},{"id":"","caption":"","content":""},{"id":"","caption":"","content":""},{"id":"","caption":"","content":""},{"id":"","caption":"","content":""},{"id":"","caption":"Overview of the performance obtained by various FNO-based surrogate models built to emulate the plasma evolution in both simulations and experiments.The case represents the various simulation settings of JOREK or the camera diagnostic on MAST, for which we are training an emulator.The variables mention the number of variables from the PDE/diagnostic that we model together.Parameters refer to the number of parameters in the trained FNO, and MSE represents the mean squared error in the normalised domain.The Training Size represented the simulation/experimental input-output pairs.All models are trained on a single Nvidia A100 chip.","content":"Table 2 .\n2\nOverview of the performance obtained by various FNO-based surrogate models built to emulate the plasma evolution in both simulations and experiments.The case represents the various simulation settings of JOREK or the camera diagnostic on MAST, for which we are training an emulator.The variables mention the number of variables from the PDE/diagnostic that we model together.Parameters refer to the number of parameters in the trained FNO, and MSE represents the mean squared error in the normalised domain.The Training Size represented the simulation/experimental input-output pairs.All models are trained on a single Nvidia A100 chip.\nCaseVariablesParametersTrain Time (m)Training SizeMSESingle blob-isothermal26321 000481007.58 × 10 −5Single blob-non-uniform temperature39467 000971602.56 × 10 −5Multiple blobs-non-uniform temperature39467 00068215001.33 × 10 −4Camera-Solenoid (rbb)1201 900156876611.97 × 10 −3Camera-Divertor (rba)1201 90093558513.80 × 10 −3each variable individually. Considering the simplicity of thereduced MHD case used in"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Figure 1.(a): Fast camera image of a MAST plasma during an Edge-Localised-Mode, where plasma filaments are ejected outwards due to MHD instabilities at the plasma edge.Reprinted from [19], with the permission of AIP Publishing.(b): The evolution of the filamentary structures during the multi-mode ELM simulation imaged with a synthetic fast camera diagnostic (time given in ms).Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020.\",\n \"content\": \"Figure 1 .\\n1\\nFigure 1.(a): Fast camera image of a MAST plasma during an Edge-Localised-Mode, where plasma filaments are ejected outwards due to MHD instabilities at the plasma edge.Reprinted from [19], with the permission of AIP Publishing.(b): The evolution of the filamentary structures during the multi-mode ELM simulation imaged with a synthetic fast camera diagnostic (time given in ms).Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.Reproduced courtesy of IAEA. Figure from [1].© EURATOM 2020.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Domain range and sampling strategies across the initial condition parameters.The case numbers represent the simulation settings in which that parameter was sampled.1-single isothermal blob, 2-single blob with non-uniform temperature, 3-multiple blobs with non-uniform temperature.U [a, b] refers to a uniform distribution bounded by a and b.\",\n \"content\": \"Table 1 .\\n1\\nDomain range and sampling strategies across the initial condition parameters.The case numbers represent the simulation settings in which that parameter was sampled.1-single isothermal blob, 2-single blob with non-uniform temperature, 3-multiple blobs with non-uniform temperature.U [a, b] refers to a uniform distribution bounded by a and b.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 2. Evolution of multiple blobs inside a slab for the first 500 time-steps of the numerical simulation (the full run is 2000 steps), taken at intervals of 100 steps (about 15 µs).The top row shows the density, the middle row the temperature, and the bottom row the vorticity.\",\n \"content\": \"Figure 2 .\\n2\\nFigure 2. Evolution of multiple blobs inside a slab for the first 500 time-steps of the numerical simulation (the full run is 2000 steps), taken at intervals of 100 steps (about 15 µs).The top row shows the density, the middle row the temperature, and the bottom row the vorticity.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 3. Views of the fast cameras on MAST in the absence of plasma, obtained using illuminations on the 3D CAD model of MAST.Camera shown in figure (a), looks across the central solenoid, while the camera in figure (b) shows the view at the divertor.\",\n \"content\": \"Figure 3 .\\n3\\nFigure 3. Views of the fast cameras on MAST in the absence of plasma, obtained using illuminations on the 3D CAD model of MAST.Camera shown in figure (a), looks across the central solenoid, while the camera in figure (b) shows the view at the divertor.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 4. Modified architecture of a single Fourier Layer.FNO is constructed by stacking multiple Fourier layers on top of each other, sandwiched by a lifting and projection operation.\",\n \"content\": \"Figure 4 .\\n4\\nFigure 4. Modified architecture of a single Fourier Layer.FNO is constructed by stacking multiple Fourier layers on top of each other, sandwiched by a lifting and projection operation.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 5. Schematic representation of the two kinds of training configurations deployed within this study.For the simulations, we use the autoregressive configuration (left).We only use the initial set of field values to propagate towards the desired time step, where the intermediary model output is fed back into the model as inputs to further the time evolution.For the experimental data, we use a sequential time window-based configuration.Here, the model output is not factored back into the model, but we perform a fixed window mapping across time.As we go across the time duration of the data, the window is slid further by one-time instance (right).In the example above, we have T_in = 3, step = 1, T_out = 8, the time focus of each forward propagation of the model is given along the x-axis while the evolution across the full-time domain is given along the y-axis.Each cell block represents the 2D field value at a certain instance of time.The input and output sizes shown in this diagram are representative only and vary across each experiment.\",\n \"content\": \"Figure 5 .\\n5\\nFigure 5. Schematic representation of the two kinds of training configurations deployed within this study.For the simulations, we use the autoregressive configuration (left).We only use the initial set of field values to propagate towards the desired time step, where the intermediary model output is fed back into the model as inputs to further the time evolution.For the experimental data, we use a sequential time window-based configuration.Here, the model output is not factored back into the model, but we perform a fixed window mapping across time.As we go across the time duration of the data, the window is slid further by one-time instance (right).In the example above, we have T_in = 3, step = 1, T_out = 8, the time focus of each forward propagation of the model is given along the x-axis while the evolution across the full-time domain is given along the y-axis.Each cell block represents the 2D field value at a certain instance of time.The input and output sizes shown in this diagram are representative only and vary across each experiment.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 6.Learning curves associated with training the multi-variable FNO across the various the various Reduced MHD Cases.Isothermal blob refers to the single blob diffusion under uniform temperature, Anisothermal blob refers to the single blob under a non-uniform temperature, and Anisothermal multiple Blobs extend that to the case with several density blobs under a non-uniform temperature distribution.\",\n \"content\": \"Figure 6 .\\n6\\nFigure 6.Learning curves associated with training the multi-variable FNO across the various the various Reduced MHD Cases.Isothermal blob refers to the single blob diffusion under uniform temperature, Anisothermal blob refers to the single blob under a non-uniform temperature, and Anisothermal multiple Blobs extend that to the case with several density blobs under a non-uniform temperature distribution.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 7. Isothermal Blob: Temporal evolution of (a) density and (b) electric potential of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\",\n \"content\": \"Figure 7 .\\n7\\nFigure 7. Isothermal Blob: Temporal evolution of (a) density and (b) electric potential of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 8. Single Blob : Temporal evolution of (a) density and (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\",\n \"content\": \"Figure 8 .\\n8\\nFigure 8. Single Blob : Temporal evolution of (a) density and (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 9. Multiple Blobs with non-uniform temperature: Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\",\n \"content\": \"Figure 9 .\\n9\\nFigure 9. Multiple Blobs with non-uniform temperature: Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 10.Error growth across time roll-out of the FNO in the physical space.The error in emulating each variable of the reduced MHD model increases as we roll further out in time.\",\n \"content\": \"Figure 10 .\\n10\\nFigure 10.Error growth across time roll-out of the FNO in the physical space.The error in emulating each variable of the reduced MHD model increases as we roll further out in time.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 11.Comparing the error growth (mean absolute error in the normalised domain) of different variables across time roll-out of the Individual FNO (Single) and the Multi-variable FNO (Multi) for the various MHD cases under study.(a) Single Isothermal Blob (b) Single Blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The multi-variable FNO with the same parameters as the individual FNOs demonstrates much better performance over time.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.\",\n \"content\": \"Figure 11 .\\n11\\nFigure 11.Comparing the error growth (mean absolute error in the normalised domain) of different variables across time roll-out of the Individual FNO (Single) and the Multi-variable FNO (Multi) for the various MHD cases under study.(a) Single Isothermal Blob (b) Single Blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The multi-variable FNO with the same parameters as the individual FNOs demonstrates much better performance over time.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 12.Comparing the cumulative error growth of different variables across time roll-out for multi-variable FNOs with different step sizes.The autoregressive nature of the FNO brings about a spike in error, characterising a disconnection at the time step that is a multiple of the output size.(a) Isothermal blob (b) Single blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.\",\n \"content\": \"Figure 12 .\\n12\\nFigure 12.Comparing the cumulative error growth of different variables across time roll-out for multi-variable FNOs with different step sizes.The autoregressive nature of the FNO brings about a spike in error, characterising a disconnection at the time step that is a multiple of the output size.(a) Isothermal blob (b) Single blob with non-uniform temperature (c) Multi-blobs with non-uniform temperature.The y-axis represents the mean absolute error for each case, and the x-axis represents the time iteration of the autoregressive rollout.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 14.Evolution of the electric potential (Multiblobs) in time as modelled by multi-variable FNOs at different modes.Top row of (a) shows the actual solution modelled using JOREK.The bottom row of (a), plots the time evolution of the FNO with 2 modes, top row of (b)is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better.\",\n \"content\": \"Figure 14 .\\n14\\nFigure 14.Evolution of the electric potential (Multiblobs) in time as modelled by multi-variable FNOs at different modes.Top row of (a) shows the actual solution modelled using JOREK.The bottom row of (a), plots the time evolution of the FNO with 2 modes, top row of (b)is that of a model with 8 modes, and the bottom row of (b) is the FNO with 32 modes.FNO with lower modes produce less noise, but fail to predict the longer evolution structures, while those with higher modes are inherently noisier and capture the long-term predictions better.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure15.Impact of the size of the training dataset on the performance of the multi-variable FNO for simulating the multiple plasma blobs in a non-uniform temperature field.As we increase the availability of the training data for the multi-variable FNO, the model performance increases across the same test data.However, the gains obtained by increasing the training data are marginal compared to the additional compute required.The number of simulations used for training is given on the x-axis while the mean squared error is given on the y-axis.All training is performed on a single Nvidia A100.\",\n \"content\": \"Figure 15 .\\n15\\nFigure15.Impact of the size of the training dataset on the performance of the multi-variable FNO for simulating the multiple plasma blobs in a non-uniform temperature field.As we increase the availability of the training data for the multi-variable FNO, the model performance increases across the same test data.However, the gains obtained by increasing the training data are marginal compared to the additional compute required.The number of simulations used for training is given on the x-axis while the mean squared error is given on the y-axis.All training is performed on a single Nvidia A100.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 16.Learning curves associated with the models trained with data of different sizes.Training loss is represented by full lines, while the test loss is represented by dashed lines.With just 500 simulation instances, the model learns the physics of the model really well, which is only marginally improved by more training data.We also notice the quick over-fitting tendency of the FNO as observed in[38].\",\n \"content\": \"Figure 16 .\\n16\\nFigure 16.Learning curves associated with the models trained with data of different sizes.Training loss is represented by full lines, while the test loss is represented by dashed lines.With just 500 simulation instances, the model learns the physics of the model really well, which is only marginally improved by more training data.We also notice the quick over-fitting tendency of the FNO as observed in[38].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 17.Demonstrating Zero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE(in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the bottom series of plots demonstrate the time evolution as emulated by the multi-variable FNO.\",\n \"content\": \"Figure 17 .\\n17\\nFigure 17.Demonstrating Zero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE(in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the bottom series of plots demonstrate the time evolution as emulated by the multi-variable FNO.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 18.Outside the training domain: Inferring on a case with 13 blobs with amplitude and density twice more and the blob width 1.5 times more than the maximum used within the training data (see table 1).Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\",\n \"content\": \"Figure 18 .\\n18\\nFigure 18.Outside the training domain: Inferring on a case with 13 blobs with amplitude and density twice more and the blob width 1.5 times more than the maximum used within the training data (see table 1).Temporal evolution of (a) density (b) electric potential and (c) temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 19.Temporal plots of (a) Plasma Current and (b) Total Heating Power of the shots selected for training and testing the FNO for modelling the camera data.The shots depicting the training data is given in black, whereas the test data is given in red.The plots help convey the diversity of plasma scenarios the FNO is trained to model over.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data.\",\n \"content\": \"Figure 19 .\\n19\\nFigure 19.Temporal plots of (a) Plasma Current and (b) Total Heating Power of the shots selected for training and testing the FNO for modelling the camera data.The shots depicting the training data is given in black, whereas the test data is given in red.The plots help convey the diversity of plasma scenarios the FNO is trained to model over.The training data consists of multiple different scenarios, allowing for building a general model.However, the test data is selected to be largely representative of the scenarios found within the training data.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Predicting in L-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera with the plasma in L-mode.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\",\n \"content\": \"20 .\\n20\\nPredicting in L-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera with the plasma in L-mode.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 21.Predicting in H-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera wiht the plasma in H-mod.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The domain is selected to showcase the performance of the FNO in mapping the plasma evolution in H-mode.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\",\n \"content\": \"Figure 21 .\\n21\\nFigure 21.Predicting in H-mode: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera wiht the plasma in H-mod.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The domain is selected to showcase the performance of the FNO in mapping the plasma evolution in H-mode.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"provides a panoramic view of the reactor, showing the poloidal cross-sectional layout on both sides of the central solenoid.The main camera has a spatial resolution of 448 × 640 for the experiments that we sampled.We sampled a total of 55 plasma shots from the MAST database, used 50 for training and 5 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 7661, and the test points grow to 828 data points.The model takes 27 h 45 min on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.7 × 10 −3 .For further plots showcasing the forecasting capabilities of the FNO at the plasma around the central solenoid, refer to appendix E in figures E1 and E2.\",\n \"content\": \"provides a panoramic view of the reactor, showing the poloidal cross-sectional layout on both sides of the central solenoid.The main camera has a spatial resolution of 448 × 640 for the experiments that we sampled.We sampled a total of 55 plasma shots from the MAST database, used 50 for training and 5 for testing.Once we structure the data into windowed input-output pairs, the training size of the dataset grows to 7661, and the test points grow to 828 data points.The model takes 27 h 45 min on a Nvidia A100 chip.With the above-mentioned training configuration, the model achieves an MSE (in the normalised domain) of 1.7 × 10 −3 .For further plots showcasing the forecasting capabilities of the FNO at the plasma around the central solenoid, refer to appendix E in figures E1 and E2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 22.Predicting around the L-H transition: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.\",\n \"content\": \"Figure 22 .\\n22\\nFigure 22.Predicting around the L-H transition: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 23.Long-term prediction: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.\",\n \"content\": \"Figure 23 .\\n23\\nFigure 23.Long-term prediction: Temporal evolution of the plasma around the central solenoid within MAST as perceived by the main camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment, the middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.Though the camera captures the confined plasma shape, it fails to capture the localised filaments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 24.Single forward pass: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\",\n \"content\": \"Figure 24 .\\n24\\nFigure 24.Single forward pass: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the plasma evolution across one feed-forward operation of the FNO spanning 10 frames ∼12 ms.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 25.Long-term prediction: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\",\n \"content\": \"Figure 25 .\\n25\\nFigure 25.Long-term prediction: Temporal evolution of the plasma at the divertor within MAST as perceived by the divertor camera.The image compares the last frame of the predicted plasma evolution across multiple feed-forward instances across the duration of the pulse.The top layer shows the actual data obtained from the experiment.The middle layer displays the emulation performed by the FNO.The bottom layer showcases the absolute error across the prediction and the camera.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure Single Blob with non-uniform Temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\",\n \"content\": \"Figure\\n\\nFigure Single Blob with non-uniform Temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure A4.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\",\n \"content\": \"Figure A4 .\\nA4\\nFigure A4.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure A5.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\",\n \"content\": \"Figure A5 .\\nA5\\nFigure A5.Multiple Blobs with non-uniform temperature: Temporal evolution of (a) the density and (b) the potential, (c) the Temperature of the plasma evolution as obtained using the JOREK code (top of each image), that of the trained multi-variable FNO (middle of each figure) and the absolute error across both (bottom of each figure).The spatial domain is given in toroidal geometry characterised by R in the x-axis and Z in the y-axis.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure DemonstratingZero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE (in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the middle series of plots demonstrate the time evolution as emulated by the multi-variable FNO and the bottom series indicates the absolute error across space.\",\n \"content\": \"Figure\\n\\nFigure DemonstratingZero-Shot Super-Resolution on the multi-variable FNO.The FNO being trained on a grid resolution of 100 × 100 is further deployed to model the evolution the the multiple blobs at a resolution of 500 × 500.The model is capable of performing zero-shot super-resolution, achieving an MSE (in the normalised domain) of 7.90 × 10 −5 at a grid resolution 5 × more resolved than the one it was being trained on without the requirement of any model engineering or fine-tuning.The plots demonstrate the evolution of density, electric potential, and temperature as emulated by the FNO.For plots (a), (b), and (c), the top series of plots demonstrate the time evolution of the ground truth while the middle series of plots demonstrate the time evolution as emulated by the multi-variable FNO and the bottom series indicates the absolute error across space.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Overview of the performance obtained by various FNO-based surrogate models built to emulate the plasma evolution in both simulations and experiments.The case represents the various simulation settings of JOREK or the camera diagnostic on MAST, for which we are training an emulator.The variables mention the number of variables from the PDE/diagnostic that we model together.Parameters refer to the number of parameters in the trained FNO, and MSE represents the mean squared error in the normalised domain.The Training Size represented the simulation/experimental input-output pairs.All models are trained on a single Nvidia A100 chip.\",\n \"content\": \"Table 2 .\\n2\\nOverview of the performance obtained by various FNO-based surrogate models built to emulate the plasma evolution in both simulations and experiments.The case represents the various simulation settings of JOREK or the camera diagnostic on MAST, for which we are training an emulator.The variables mention the number of variables from the PDE/diagnostic that we model together.Parameters refer to the number of parameters in the trained FNO, and MSE represents the mean squared error in the normalised domain.The Training Size represented the simulation/experimental input-output pairs.All models are trained on a single Nvidia A100 chip.\\nCaseVariablesParametersTrain Time (m)Training SizeMSESingle blob-isothermal26321 000481007.58 × 10 −5Single blob-non-uniform temperature39467 000971602.56 × 10 −5Multiple blobs-non-uniform temperature39467 00068215001.33 × 10 −4Camera-Solenoid (rbb)1201 900156876611.97 × 10 −3Camera-Divertor (rba)1201 90093558513.80 × 10 −3each variable individually. Considering the simplicity of thereduced MHD case used in\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Simulations of edge localised mode instabilities in MAST-U Super-X tokamak plasmas. S Smith, S Pamela, A Fil, M Hölzl, G Huijsmans, A Kirk, D Moulton, O Myatra, A Thornton, H Wilson, 10.1088/1741-4326/ab826aNucl. Fusion. 60660212020","doi":""},{"text":"Three-dimensional plasma edge turbulence simulations of the Mega Ampere Spherical Tokamak and comparison with experimental measurements Plasma Phys. Control. F Riva, F Militello, S Elmore, J T Omotani, B Dudson, N R Walkden, 10.1088/1361-6587/ab3561Fusion. 61950132019the MAST team","doi":""},{"text":"The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas. M Hoelzl, 10.1088/1741-4326/abf99fNucl. Fusion. 61650012021","doi":""},{"text":"JINTRAC: a system of codes for integrated simulation of tokamak scenarios Plasma Fusion Res. M Romanelli, 10.1585/pfr.9.3403023201493403023","doi":""},{"text":"The new SOLPS-ITER code package. S Wiesen, 10.1016/j.jnucmat.2014.10.012J. Nucl. Mater. 4632015","doi":""},{"text":"A Lavin, arXiv:2112.03235Simulation intelligence: towards a new generation of scientific methods CoRR. 2021","doi":""},{"text":"Fast modeling of turbulent transport in fusion plasmas using neural networks Phys. K L Van De Plassche, J Citrin, C Bourdelle, Y Camenen, F J Casson, V I Dagnelie, F Felici, A Ho, S Van Mulders, 10.1063/1.5134126Plasmas. 27223102020","doi":""},{"text":"2021 Neural network surrogate of QuaLiKiz using JET experimental data to populate training space. A Ho, J Citrin, C Bourdelle, Y Camenen, F J Casson, K L Van De Plassche, H Weisen, 10.1063/5.0038290Phys. Plasmas. 2832305","doi":""},{"text":"Self-consistent core-pedestal transport simulations with neural network accelerated models. O Meneghini, 10.1088/1741-4326/aa7776Nucl. Fusion. 57860342017","doi":""},{"text":"2023 Fast regression of the tritium breeding ratio in fusion reactors. G V G Mánek, V Gopakumar, N Nikolaou, J Shimwell, I Waldmann, 10.1088/2632-2153/acb2b3Mach. Learn.: Sci. Technol. 415008","doi":""},{"text":"Image mapping the temporal evolution of edge characteristics in tokamaks using neural networks. V Gopakumar, D Samaddar, 10.1088/2632-2153/ab5639Mach. Learn.: Sci. Technol. 1150062020","doi":""},{"text":"Towards fast surrogate models for interpolation of tokamak edge plasmas. S Dasbach, S Wiesen, 10.1016/j.nme.2023.101396Nucl. Mater. Energy. 341013962023","doi":""},{"text":"2023 Neural operator: learning maps between function spaces. N Kovachki, Z Li, B Liu, K Azizzadenesheli, K Bhattacharya, A Stuart, A Anandkumar, J. Mach. Learn. Res.","doi":""},{"text":"Physics-informed neural operator for learning partial differential equations ACM. Z Li, H Zheng, N B Kovachki, D Jin, H Chen, B Liu, A Stuart, K Azizzadenesheli, A Anandkumar, 10.1145/3648506/ IMS J. Data Sci. 2022","doi":""},{"text":"F Bartolucci, E De Bézenac, B Raonić, R Molinaro, S Mishra, R Alaifari, Representation equivalent neural operators: a framework for alias-free operator learning. 2023","doi":""},{"text":"The cost-accuracy trade-off in operator learning with neural networks. De Hoop, M Huang, D Z Qian, E Stuart, A M , 10.4208/jml.220509J. Mach. Learn. Res. 12992022","doi":""},{"text":"Fast dynamic 1D simulation of divertor plasmas with neural PDE surrogates. Y Poels, G Derks, E Westerhof, K Minartz, S Wiesen, V Menkovski, 10.1088/1741-4326/acf70dNucl. Fusion. 631260122023","doi":""},{"text":"V Gopakumar, MAST TeamS Pamela, MAST TeamL Zanisi, MAST TeamZ Li, MAST TeamA Anandkumar, MAST TeamarXiv:2302.06542) 2023 Fourier neural operator for plasma modelling.","doi":""},{"text":"Optical boundary reconstruction of tokamak plasmas for feedback control of plasma position and shape. G Hommen, M De Baar, G M Nuij, R Akers, M Steinbuch, 10.1063/1.3499219Rev. Sci. Instrum. 811135042010","doi":""},{"text":"X Shi, Z Chen, H Wang, D.-Y Yeung, W.-K Wong, W.-C Woo, arXiv:1506.04214Convolutional LSTM network: a machine learning approach for precipitation nowcasting. 2015","doi":""},{"text":"Association of genomic subtypes of lower-grade gliomas with shape features automatically extracted by a deep learning algorithm Comput. M Buda, A Saha, M A Mazurowski, 10.1016/j.compbiomed.2019.05.002Biol. Med. 1092019","doi":""},{"text":"Investigation of the effect of resistivity on scrape off layer filaments using three-dimensional simulations Phys. L Easy, F Militello, J Omotani, N R Walkden, B Dudson, 10.1063/1.4940330Plasmas. 23125122016","doi":""},{"text":"On the interaction of scrape off layer filaments Phys. F Militello, B Dudson, L Easy, A Kirk, P Naylor, 10.1088/1361-6587/aa9252Plasmas. 591250132017","doi":""},{"text":"et al (the MAST Team) 2006 Filament structures at the plasma edge on mast Plasma Phys. Control. A Kirk, 10.1088/0741-3335/48/12B/S41Fusion. 48B433","doi":""},{"text":"The physics of turbulence localised to the tokamak divertor volume. N Walkden, F Riva, J Harrison, F Militello, T Farley, J Omotani, B Lipschultz, 10.1038/s42005-022-00906-2Commun. Phys. 51392022","doi":""},{"text":"C Ham, MAST using high speed visible camera data IAEA 2nd Technical Meeting on Plasma Disruptions and Their Mitigation (ITER Headquarters. 2022. 19-22 July 2022","doi":""},{"text":"Z Li, N Kovachki, K Azizzadenesheli, B Liu, K Bhattacharya, A Stuart, A Anandkumar, arXiv:2010.08895Fourier neural operator for parametric partial differential equations. 2020","doi":""},{"text":"The fast Fourier transform IEEE Spectr. E O Brigham, R E Morrow, 10.1109/MSPEC.1967.521722019674","doi":""},{"text":"M Dalal, A C Li, R Taori, arXiv:1910.07737Autoregressive models: what are they good for?. 2019","doi":""},{"text":". R K Srivastava, K Greff, J Schmidhuber, arXiv:1505.003872015Highway networks","doi":""},{"text":"Gradient-based learning applied to document recognition Proc. Y Lecun, L Bottou, Y Bengio, Haffner, 10.1109/5.726791IEEE. 861998","doi":""},{"text":"D Kingma, J Ba, arXiv:1412.6980Adam: a method for stochastic optimization. 2014","doi":""},{"text":"M Bellan, Fundamentals of Plasma Physics. Cambridge University Press2006","doi":""},{"text":"A Markovian decision process. R Bellman, J. Math. Mech. 61957","doi":""},{"text":"Y Lee, Autoregressive renaissance in neural PDE solvers ICLR Blogposts 2023. Kigali Rwanda2023. 1-5 May","doi":""},{"text":"J Brandstetter, D Worrall, M Welling, arXiv:2202.03376Message passing neural PDE solvers. 2023","doi":""},{"text":"J Zhao, R J George, Z Li, A Anandkumar, arXiv:2211.15188Incremental spectral learning in Fourier neural operator. 2023","doi":""},{"text":"S Lanthaler, Z Li, A M Stuart, arXiv:2304.13221The nonlocal neural operator: universal approximation. 2023","doi":""},{"text":"Can neural networks extrapolate? Discussion of a theorem by. A Courtois, J.-M Morel, P Arias, 10.1007/s13398-023-01411-zPedro Domingos Rev. Real Acad. Cienc. Exactas Fis. Nat. A. 117792023","doi":""},{"text":"A Jacot, F Gabriel, C Hongler, arXiv:1806.07572Neural tangent kernel: convergence and generalization in neural networks. 2020","doi":""},{"text":"J Kossaifi, N Kovachki, K Azizzadenesheli, A Anandkumar, arXiv:2310.00120Multi-grid tensorized Fourier neural operator for high-resolution PDEs. 2023","doi":""},{"text":"C Aggarwal, X Kong, Q Gu, J Han, P Yu, Active Learning: A Survey. CRC Press2015","doi":""},{"text":"JET Contributors 2024 Efficient training sets for surrogate models of tokamak turbulence with active deep ensembles. L Zanisi, A Ho, J Barr, T Madula, J Citrin, S Pamela, J Buchanan, F Casson, V Gopakumar, 10.1088/1741-4326/ad240dNucl. Fusion. 6436022","doi":""},{"text":"Applications of physics informed neural operators. S G Rosofsky, H A Majed, E A Huerta, 10.1088/2632-2153/acd168DownloadArticlePDFMach. Learn.: Sci. Technol. 4250222023","doi":""},{"text":"G Kohl, L.-W Chen, N Thuerey, arXiv:2309.01745Turbulent flow simulation using autoregressive conditional diffusion models. 2023","doi":""},{"text":". S Silburn, 10.5281/zenodo.68915042022","doi":""},{"text":"Fourier-RNNs for modelling noisy physics data IEEE-ICMLA Bahamas. V Gopakumar, S Pamela, L Zanisi, arXiv:2302.065342023. Dec","doi":""}],"string":"[\n {\n \"text\": \"Simulations of edge localised mode instabilities in MAST-U Super-X tokamak plasmas. S Smith, S Pamela, A Fil, M Hölzl, G Huijsmans, A Kirk, D Moulton, O Myatra, A Thornton, H Wilson, 10.1088/1741-4326/ab826aNucl. Fusion. 60660212020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Three-dimensional plasma edge turbulence simulations of the Mega Ampere Spherical Tokamak and comparison with experimental measurements Plasma Phys. Control. F Riva, F Militello, S Elmore, J T Omotani, B Dudson, N R Walkden, 10.1088/1361-6587/ab3561Fusion. 61950132019the MAST team\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas. M Hoelzl, 10.1088/1741-4326/abf99fNucl. Fusion. 61650012021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"JINTRAC: a system of codes for integrated simulation of tokamak scenarios Plasma Fusion Res. M Romanelli, 10.1585/pfr.9.3403023201493403023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The new SOLPS-ITER code package. S Wiesen, 10.1016/j.jnucmat.2014.10.012J. Nucl. Mater. 4632015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Lavin, arXiv:2112.03235Simulation intelligence: towards a new generation of scientific methods CoRR. 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fast modeling of turbulent transport in fusion plasmas using neural networks Phys. K L Van De Plassche, J Citrin, C Bourdelle, Y Camenen, F J Casson, V I Dagnelie, F Felici, A Ho, S Van Mulders, 10.1063/1.5134126Plasmas. 27223102020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"2021 Neural network surrogate of QuaLiKiz using JET experimental data to populate training space. A Ho, J Citrin, C Bourdelle, Y Camenen, F J Casson, K L Van De Plassche, H Weisen, 10.1063/5.0038290Phys. Plasmas. 2832305\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Self-consistent core-pedestal transport simulations with neural network accelerated models. O Meneghini, 10.1088/1741-4326/aa7776Nucl. Fusion. 57860342017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"2023 Fast regression of the tritium breeding ratio in fusion reactors. G V G Mánek, V Gopakumar, N Nikolaou, J Shimwell, I Waldmann, 10.1088/2632-2153/acb2b3Mach. Learn.: Sci. Technol. 415008\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Image mapping the temporal evolution of edge characteristics in tokamaks using neural networks. V Gopakumar, D Samaddar, 10.1088/2632-2153/ab5639Mach. Learn.: Sci. Technol. 1150062020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Towards fast surrogate models for interpolation of tokamak edge plasmas. S Dasbach, S Wiesen, 10.1016/j.nme.2023.101396Nucl. Mater. Energy. 341013962023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"2023 Neural operator: learning maps between function spaces. N Kovachki, Z Li, B Liu, K Azizzadenesheli, K Bhattacharya, A Stuart, A Anandkumar, J. Mach. Learn. Res.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Physics-informed neural operator for learning partial differential equations ACM. Z Li, H Zheng, N B Kovachki, D Jin, H Chen, B Liu, A Stuart, K Azizzadenesheli, A Anandkumar, 10.1145/3648506/ IMS J. Data Sci. 2022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"F Bartolucci, E De Bézenac, B Raonić, R Molinaro, S Mishra, R Alaifari, Representation equivalent neural operators: a framework for alias-free operator learning. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The cost-accuracy trade-off in operator learning with neural networks. De Hoop, M Huang, D Z Qian, E Stuart, A M , 10.4208/jml.220509J. Mach. Learn. Res. 12992022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fast dynamic 1D simulation of divertor plasmas with neural PDE surrogates. Y Poels, G Derks, E Westerhof, K Minartz, S Wiesen, V Menkovski, 10.1088/1741-4326/acf70dNucl. Fusion. 631260122023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V Gopakumar, MAST TeamS Pamela, MAST TeamL Zanisi, MAST TeamZ Li, MAST TeamA Anandkumar, MAST TeamarXiv:2302.06542) 2023 Fourier neural operator for plasma modelling.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical boundary reconstruction of tokamak plasmas for feedback control of plasma position and shape. G Hommen, M De Baar, G M Nuij, R Akers, M Steinbuch, 10.1063/1.3499219Rev. Sci. Instrum. 811135042010\",\n \"doi\": \"\"\n },\n {\n \"text\": \"X Shi, Z Chen, H Wang, D.-Y Yeung, W.-K Wong, W.-C Woo, arXiv:1506.04214Convolutional LSTM network: a machine learning approach for precipitation nowcasting. 2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Association of genomic subtypes of lower-grade gliomas with shape features automatically extracted by a deep learning algorithm Comput. M Buda, A Saha, M A Mazurowski, 10.1016/j.compbiomed.2019.05.002Biol. Med. 1092019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Investigation of the effect of resistivity on scrape off layer filaments using three-dimensional simulations Phys. L Easy, F Militello, J Omotani, N R Walkden, B Dudson, 10.1063/1.4940330Plasmas. 23125122016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the interaction of scrape off layer filaments Phys. F Militello, B Dudson, L Easy, A Kirk, P Naylor, 10.1088/1361-6587/aa9252Plasmas. 591250132017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"et al (the MAST Team) 2006 Filament structures at the plasma edge on mast Plasma Phys. Control. A Kirk, 10.1088/0741-3335/48/12B/S41Fusion. 48B433\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The physics of turbulence localised to the tokamak divertor volume. N Walkden, F Riva, J Harrison, F Militello, T Farley, J Omotani, B Lipschultz, 10.1038/s42005-022-00906-2Commun. Phys. 51392022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Ham, MAST using high speed visible camera data IAEA 2nd Technical Meeting on Plasma Disruptions and Their Mitigation (ITER Headquarters. 2022. 19-22 July 2022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Z Li, N Kovachki, K Azizzadenesheli, B Liu, K Bhattacharya, A Stuart, A Anandkumar, arXiv:2010.08895Fourier neural operator for parametric partial differential equations. 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The fast Fourier transform IEEE Spectr. E O Brigham, R E Morrow, 10.1109/MSPEC.1967.521722019674\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Dalal, A C Li, R Taori, arXiv:1910.07737Autoregressive models: what are they good for?. 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R K Srivastava, K Greff, J Schmidhuber, arXiv:1505.003872015Highway networks\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gradient-based learning applied to document recognition Proc. Y Lecun, L Bottou, Y Bengio, Haffner, 10.1109/5.726791IEEE. 861998\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Kingma, J Ba, arXiv:1412.6980Adam: a method for stochastic optimization. 2014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Bellan, Fundamentals of Plasma Physics. Cambridge University Press2006\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Markovian decision process. R Bellman, J. Math. Mech. 61957\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Y Lee, Autoregressive renaissance in neural PDE solvers ICLR Blogposts 2023. Kigali Rwanda2023. 1-5 May\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Brandstetter, D Worrall, M Welling, arXiv:2202.03376Message passing neural PDE solvers. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Zhao, R J George, Z Li, A Anandkumar, arXiv:2211.15188Incremental spectral learning in Fourier neural operator. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Lanthaler, Z Li, A M Stuart, arXiv:2304.13221The nonlocal neural operator: universal approximation. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Can neural networks extrapolate? Discussion of a theorem by. A Courtois, J.-M Morel, P Arias, 10.1007/s13398-023-01411-zPedro Domingos Rev. Real Acad. Cienc. Exactas Fis. Nat. A. 117792023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Jacot, F Gabriel, C Hongler, arXiv:1806.07572Neural tangent kernel: convergence and generalization in neural networks. 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Kossaifi, N Kovachki, K Azizzadenesheli, A Anandkumar, arXiv:2310.00120Multi-grid tensorized Fourier neural operator for high-resolution PDEs. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C Aggarwal, X Kong, Q Gu, J Han, P Yu, Active Learning: A Survey. CRC Press2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"JET Contributors 2024 Efficient training sets for surrogate models of tokamak turbulence with active deep ensembles. L Zanisi, A Ho, J Barr, T Madula, J Citrin, S Pamela, J Buchanan, F Casson, V Gopakumar, 10.1088/1741-4326/ad240dNucl. Fusion. 6436022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Applications of physics informed neural operators. S G Rosofsky, H A Majed, E A Huerta, 10.1088/2632-2153/acd168DownloadArticlePDFMach. Learn.: Sci. Technol. 4250222023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G Kohl, L.-W Chen, N Thuerey, arXiv:2309.01745Turbulent flow simulation using autoregressive conditional diffusion models. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Silburn, 10.5281/zenodo.68915042022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fourier-RNNs for modelling noisy physics data IEEE-ICMLA Bahamas. V Gopakumar, S Pamela, L Zanisi, arXiv:2302.065342023. Dec\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":36,"cells":{"text":{"kind":"string","value":"\nAnalysis of the C 2 (d 3 Π g -a 3 Π u ) Swan bands as a thermometric probe in CO 2 microwave plasmas\n\n\nEmile Carbone \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany\n\nFederico D'isa \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany\n\nAnte Hecimovic \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany\n\nUrsel Fantz \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany\n\nAnalysis of the C 2 (d 3 Π g -a 3 Π u ) Swan bands as a thermometric probe in CO 2 microwave plasmas\nPublished as: Plasma Sources Sciences and Technology (2020)29055003\nThe optical emission spectra of high pressure CO 2 microwave plasmas are usually dominated by the C 2 Swan bands. In this paper, the use of the C 2 Swan bands for estimating the gas temperature in CO 2 microwave plasmas is assessed. State by state fitting is employed to check the correctness of assuming a Boltzmann distribution for the rotational and vibrational distribution functions and, within statistical and systematic uncertainties, the C 2 Swan band can be fitted accurately with a single temperature for rotational and vibrational levels. The processes leading to the production of the C 2 molecule and particularly its d 3 Π g state are briefly reviewed as well as collisional relaxation times of the latter. It is concluded that its rotational temperature can be associated to the gas temperature of the CO 2 microwave plasma and the results are moreover cross-checked by adding a small amount of N 2 in the discharge and measuring the CN violet band system. The 2.45 GHz plasma source is analyzed in the pressure range 180-925 mbar, for input microwave powers ranging from 0.9 -3 kW and with gas flow rates of 5-100 L/min. An intense C 2 Swan bands emission spectrum can be measured only when the plasma is operated in contracted regime. A unique temperature of about 6000 ± 500 K is obtained for all investigated conditions. A spectroscopic database is constructed using the recent compilation and calculations by Brooke et al.[1] of the line strengths and molecular constants for the C 2 (d 3 Π g -a 3 Π u ) Swan bands system and made available as 1 arXiv:1911.13121v2 [physics.plasm-ph] 29 Apr 2020 Supplementary Material in a format compatible with the open source MassiveOES software.\n\nIntroduction\n\nMany strategies are being currently studied for the recycling and valorization of CO 2 , a potent greenhouse gas in the atmosphere [2]. Production of liquid fuels from renewable energies is one of the pathways for re-utilization of CO 2 . The first step for converting CO 2 into added value chemicals is its conversion into carbon monoxide CO [3,4]. For practical applications, it is desirable to operate with high gas flows near atmospheric pressure (or above) in order to avoid bulky components, reduce costs and save energy for the compression steps [5]. Previous studies have indicated that non-equilibrium microwave plasmas are particularly suitable for high energy efficiencies η conversion of CO 2 into CO with η ≤ 80% and can operate at high pressure and with high flow rates [6]. CO 2 based plasmas have applications in fields such as lasers [7], plasma polymers treatment [8] and also have potential in plasma medicine [9]. The proposed mechanism for explaining high energy efficiencies for CO 2 conversion into CO is via the so-called vibrational ladder climbing mechanism. Relatively low energy electrons excite preferentially the assymetric vibrational modes of the CO 2 ground state CO 2 (00i) molecule and vibrational energy exchange between CO 2 molecules leads ultimately to their decomposition [10]. To favor this mechanism the key ingredient is to keep the plasma out of equilibrium with the gas temperature T g much smaller than the effective electron temperature T e [11,12,13]. Low gas temperatures (i.e. T g 3000 K) are required in order to avoid too many losses of CO 2 (00i) states by collisions with the background gas leading to gas heating by vibrational to translation relaxation processes [14,15]. For high energy efficiencies, calculations also indicate that formed O atoms need to react with highly vibrationally excited CO 2 molecules and to produce another CO molecule and a O 2 molecule [16]. On the other hand, at high gas temperatures (i.e. T gas ≥ 3000 K), thermal dissociation is the dominant mechanism for CO 2 dissociation with high conversion rates and is limited to energy efficiencies up to ∼ 50 % [17,18].\n\nFor studying the mechanisms limiting the plasma energy efficiency of the plasma, it is critical to have tools for assessing the degree of non-equilibrium in the plasma phase and notably its gas temperature. In high pressure (i.e. sub-up to atmospheric pressure) microwave CO 2 plasmas, one of the most dominant emission feature is the C 2 (d 3 Π g -a 3 Π u ) Swan bands system. In figure 1 an overview spectrum of a pure CO 2 microwave plasma (see section 2 for details on the plasma setup) is shown. In addition to a weak continuum emission 1 and typical atomic oxygen and carbon lines 2 , the C 2 Swan bands are the dominant molecular emission structure in the UV and visible range. Such spectrum is typical and has been measured by several authors [16,21,22,23,24,25,26,27,28]. To obtain reliable and accurate information about the plasma, it is then critical to have tools for evaluating the emission spectrum of the C 2 molecule and assess the accuracy of the results. Usually an equivalence between the rotational temperature of emitting species and the gas temperature is assumed but this hypothesis needs to be assessed. Also, in certain cases non-Boltzmann distributed rotational distribution function are measured experimentally [29]. In combustion studies, it has been observed that the C 2 Swan band can exhibit overpopulation of the high vibrational levels. The latter emission spectrum is known as the \"high pressure bands\"of the C 2 system [30]. This makes the analysis of C 2 spectra more complicated and can induce systematic errors in the determination of the rotational temperature (T rot ) of the C 2 (d 3 Π g ) state if the vibrational levels are assumed to take a Boltzmann distribution function. For CO 2 plasmas, an evaluation of potential deviations of C 2 Swan bands spectra from Boltzmann equilibrium and the assessment of T rot as a measure of the gas temperature have not been reported in the literature yet.\n\nIn this paper we present the spectral fitting analysis of the C 2 Swan band using a newly implemented spectroscopic database in the open source MassiveOES software [31,32]. The database is constructed using the recent compilation and calculations by Brooke et al. [1] of the line strengths and molecular constants for the C 2 (d 3 Π g -a 3 Π u ) Swan band system. C 2 Swan bands spectra measured on a 2.45 GHz CO 2 microwave plasma source are reported and carefully analyzed. By adding small amount of nitrogen in the plasma, the obtained temperatures from the Swan band fitting are compared with the one obtained by fitting the CN violet band system. Uncertainties related to the method used for fitting the C 2 spectrum, potential deviations from a Boltzmann distribution for the rotational and vibrational distribution functions and the assumption of fitting the spectra with a single or two temperatures (i.e. rotational and vibrational temperatures) are discussed and compared with a state-by-state fitting of the experimental spectra. Finally, currrent knowledge about rotational and vibrational relaxation processes of the C 2 (d 3 Π g ) state is presented in section 6 and the use of T rot for measuring the gas temperature of CO 2 microwave plasmas is discussed.\n\n\nExperimental setup\n\nA microwave device which can ignite an atmospheric pressure plasma torch using only microwave power without the use of additional igniters [33] is studied. A modified version of this device has been built so that it can be operated from low (∼10 mbar) up to atmospheric pressure in molecular gases such as N 2 and CO 2 . Stable operation of the plasma is obtained both in continuous and pulsed regime in a large parameters range. The microwaves are coupled via a WR340 waveguide (TE 01 mode) into a cylindrical cavity with low Q-factor where the plasma is sustained inside a quartz tube (see figure 2 for more details). The quartz tube has inner/outer diameters of 26/30 mm. Figure 1: Overview spectra using a low resolution spectrometer (instrumental profile of about 1.2 nm) of the plasma emission at low and high pressures. The high pressure spectrum is characteristic for all the conditions of the present study where the plasma is in contracted mode which occurs for power ≥ 900 W at pressures above 120 mbar. Note that the spectra are not calibrated for the sensitivity response of the spectrometer and optics.\n\nFor the ignition of the plasma, a coaxial cavity made of a waveguide to cylindrical resonator with high Q-factor and an inner conductor with conical shape is integrated at the bottom of the quartz tube. The coaxial cavity serves also for the injection of the gases using a 4-inlets tangential gas injection system. The coaxial conical pin vertical position can be adjusted for selecting different modes and enhance the electric field at its tip to ignite the plasma. The plasma can be operated in the power range of 600 W-3 kW, with gas flow rates between 5-100 L/min and pressures from 10 up to 960 mbar.\n\nThe high resolution measurements were acquired using a Czerny-Turner 1 m focal length SPEX1000 spectrometer with an entrance slit of about 30 µm and a dispersion grating of 1800 g/mm. An Andor iDus 420 CCD camera was used to record the spectra. The spectral resolution in first order is 20 pm at 550 nm. The wavelength calibration of the spectrometer was done using a mercury lamp. The light is collected using a system of two irises and a lens. The lens focuses the light into an optical fiber system that is used for coupling the light to the entrance slit of the spectrometer. The irises system acts as a collimator and a spatial resolution of 1 mm is obtained. The collecting optics, filters and the spectrometer have been absolute calibrated between 400 nm to 850 nm with an Ulbricht sphere to obtain the absolute photon flux from the plasma. A longpass filter at 400 nm (FEL0400 from Thorlabs) is used to suppress any contribution from second order emission overlapping with the Swan band emission. For the CN violet system measurements, a 280 nm longpass filter (FEL280 from Thorlabs) is used. Additional low-resolution survey spectra are acquired using an Ocean Optics S2000 spectrometer with 1.2 nm resolution.\n\n3 Sources of C 2 and of the C 2 (d 3 Π g ) state\n\nThe primary building blocks of C 2 are C atoms and it is therefore of interest to look first at sources of C atoms. No direct production channel from the CO 2 molecule is known by electron impact dissociation [34,35]. Electron impact dissociation processes of CO 2 lead to O and CO fragments either as neutral (ground or excited states). Electron dissociative attachment to CO 2 peaks at low electron energies < 10 eV and only produces CO [36] and O − . Electron impact ionization of CO 2 leading to formation of C + has a threshold cross section of 27.8 eV (compared to 13.8 eV for CO + 2 ) according to Itikawa [37] and can therefore be considered as negligible while considering typical electron energy distribution function in CO 2 plasmas [38]. Dissociative recombination of CO + 2 lead mostly to CO and O products and a small fraction of C atoms [39]. However, CO + 2 is lost by charge exchange and clustering reactions and is not a dominant ionic species in the plasma [40]. CO 2 species cannot then be considered as a primary source of C atoms and CO can be regarded as its primary source.\n\nElectron impact dissociation and recombination processes of CO and CO + have been extensively reviewed (see for instance [42,39]) and we will focus here on neutral processes leading to the formation of C atoms or even directly to C 2 molecules. A reason for focusing on these processes is also the knowledge that, in high pressure discharges with high degree of vibrational excitation, neutral processes are expected to play an important role for molecules dissociation [6]. Moreover, in section 7 are reported high gas temperatures for which thermal (i.e. neutral) processes dominates dissociation dynamics [18].\n\nTo understand the formation of the C 2 d 3 Π g state, as it will be discussed in the following, not only source terms of C atoms but also of the C 2 O should be considered. Vibrationally excited CO molecules lead to the formation of carbon atoms via thermal decomposition [43] or via the so-called Boudouard mechanism [44] \nCO(X 1 Σ g , v) + CO(X 1 Σ g , w) → CO 2 + C(1)\nwhere v and w are vibrational quantum numbers. 3 This process is very 3 Alternatively of having two CO molecules in their ground state X 1 Σ g in the reaction 1, Figure 3: Schematic representation of the potential energy curves of the lower triplet electronic states of the C 2 molecule as well as of the 1 5 Π g state and their dissociation limit. For more information see ref. [41]. efficient and researchers have reported that discharges into carbon monoxide even at pressures of a few Torr lead to deposition of solid carbon on the walls and a decrease of the pressure in the closed cell [46] . A competing process to reaction 1 is the formation of a C 2 O molecule [45,47] CO(\nX 1 Σ g , v) + CO(X 1 Σ g , w) → C 2 O + O.(2)\nSuch processes are efficient only when significant vibrational excitation of CO molecules is present and/or high gas temperatures. This is the case in the present study with reported temperatures of 6000 K (see section 7) where CO 2 is thermally dissociated, and production of C atoms happens in the gas phase [6].\n\nThe formation of the C 2 molecule from the three lowest electronic configurations of the carbon atom which are the 2s 2 2p 2 ( 3 P J , 1 D 2 and 1 S 0 ) states, where the ground state is the 3 P 0 state, was discussed by Ballik and Ramsay [48] while considering the potential energy curves of C 2 . In figure 3 are shown the lower triplet electronic levels of the C 2 molecule. For a recent overview of the potential energy curves of the C 2 molecule we refer to Martin [41]. The one of the species reacting can be a CO(a 3 Π,v) excited state formed from energy exchange between two CO (X 1 Σ g , v) molecules [45].\nd 3 Π g state 4 is formed via C( 3 P J ) + C( 3 P J ) + M → C 2 (d 3 Π g ) + M(3)\nwhere M stands for a third body that stabilizes the transition state. They noted that there is an avoided crossing with the C 2 (e 3 Π g ) state that dissociate into C ( 3 P) + C( 1 D) and so that the C( 1 D) does not recombine into the C 2 (d 3 Π g ) state. The C 2 Swan band can then potentially be used as marker for the recombination of ground state C atoms.\n\nIn high pressure discharges, a preferential emission of the C 2 (d 3 Π g , v = 6) state can be observed which is known as the \"C 2 high pressure band\". C 2 O molecule was proposed by Kunz et al. [49] for explaining the preferential emission of the C 2 (d 3 Π g , v = 6) state. The C 2 O molecule 5 reacts with a C atom and forms an electronically excited C 2 molecule [50]\nC 2 O + C → C * 2 + CO.(4)\nNaegeli and Palmer proposed that the initial electronic state in reaction 4 is the C 2 (b 3 Σ − g ) state that crosses to the C 2 (d 3 Π g ) state at v=6 and leads to the high pressure C 2 band [51]. Through a kinetic study of a CO plasma afterglow, Gosse et al. [52] supported the hypothesis that C 2 (d 3 Π g ) state is formed via reaction 4. Brewer et al [53] stated that formation of the high pressure band from a predissociation mechanism is unlikely and suggested that a resonant transfer from another molecule happens. However, Caubet and Dorthe [54], C 2 O may also be a source of the 5 Π g state via\nC 2 O(X 3 Σ + ) + C( 3 P ) → C 2 ( 5 Π g , v = 0) + CO(X 1 Σ + ).(5)\nThey argued that based on the correlation diagram for the reactants and products of the reaction C 2 O + C that the 5 Π g state of C 2 is correlated to the ground state of the reactants while the d 3 Π g state is not. We note that previously, because of the absence of observation of C 2 O by absorption, Kini 4 In Ballik and Ramsay this state is denoted as the A 3 Π g state which is a common notation found for the Swan band in the literature with the lower state of the transition defined as X 3 Π u . 5 Note that in addition to reaction 2, the C 2 O molecule can be formed from the recombination of a C atom with CO via\nC + CO + M → C 2 O + M.\nThe formation of C 2 O in pulsed radiolysis of CO was shown first by Willis and Devillers [47] and they showed that adding CO 2 did not affect the kinetics of C 2 O. Kinetics of this molecule discussed from kinetic analysis of pure CO discharges can then probably be extended to mixtures of CO/CO 2 . and Savadatti [46] proposed the following mechanism for the formation of the high pressure C 2 (d 3 Π g , v = 6) Swan band\nC 2 (a 3 Π u ) + C 3 (m) → C 2 (d 3 Π g , v = 6) + C 3(6)\nwhere the C 3 (m) molecule is a postulated metastable species with an energy of 3.7 eV. Little and Browne [55] argued against the possibility of the C 3 as intermediate species for the formation of the high pressure Swan band. Indeed, the C 2 high pressure band is observed in a large range of experimental conditions where C 3 formation is not favored and they also advanced a lack of evidence for a metastable state that should have an energy above thẽ A 1 Π u state which has 3.1 eV internal energy and is responsible for the C 3 comet band (Ã 1 Π u -X 1 Σ + g transition). Recent spectroscopic investigation and calculations of the energy levels of the C 3 molecule indicate that the lowest triplet state C 3 (ã 3 Π u ) has an energy of about 2.1 eV and theb 3 Π g an energy of 2.9 eV [56,57] which are well below the 3.7 eV energy required for the process 6. The C 3 molecule can therefore be ruled out as source of the C 2 high pressure band.\n\nLittle and Browne [55] proposed a general mechanism for the formation of the high pressure Swan band via the following three steps\nC( 3 P ) + C( 3 P ) + M → C 2 ( 5 Π g , v) + M C 2 ( 5 Π g , v) + M → C 2 ( 5 Π g , v = 0) + M C 2 ( 5 Π g , v = 0) + M → C 2 (d 3 Π g , v = 6) + M\nwhere the quintet 5 Π g is a metastable state for which a crossing exists between the v=6 level of the d 3 Π g state and the v=0 level of the quintet state. Kirby and Liu calculated its energy to lies 3.85 ± 0.34 eV above the ground state of C 2 [58] while Little and Browne estimated a value of 3.79 eV. Such mechanism essentially allow to explain previous observation that the high pressure band is rotationally colder than the \"normal\" Swan band emission that can be observed in the same discharge [59]. Also, high resolution analysis of the C 2 (d 3 Π g , v = 6) low rotation energy levels shows that both odd and even rotational numbers are perturbed with increase of intensity [60] which indicates that a Σ state is not involved in the perturbation but a state with Λ = 0 fills the lower rotational levels of the C 2 (d 3 Π g , v = 6) state [54]. A detailed analysis shows in fact that the b 3 Σ − g state perturb the J=19 and 21 of the v=6 upper state of the Swan band but lead to a decrease of the intensity of these lines [55]. It can be concluded that the b 3 Σ − g state is not at the origin of the high pressure band and that the quintet 5 Π g state is likely its source.\n\nThe proposal that C atoms recombine via the quintet 5 Π g state which then populate the C 2 (d 3 Π g , v = 6) state (i.e. to generate the C 2 high pressure band) does not exclude the direct recombination of C atoms into the d 3 Π g state. One should see them as processes that happen simultaneously with different probabilities. Looking at the energy potential curves of the C 2 molecules, one can see that the\nX 1 Σ + g , A 1 Π u , B 1 ∆ g , B' 1 Σ + g , C 1 Π g , a 3 Π u , b 3 Σ − g , c 3 Σ + u\n, d 3 Π g and 5 Π g all have as dissociation limit two C( 3 P) atoms [41]. The branching ratios for the recombination into these different electronic states are however not known although it appears from experiments discussed before that the d 3 Π g and 5 Π g are the ones first (and mainly) populated.\n\nIn conclusion, we can say that there are several concurring production mechanisms of the C 2 (d 3 Π g ) state either from recombination of C atoms or, in the case of CO containing plasmas, via the C 2 O molecule. The presence or absence of some of these mecanisms for which branching can unfortunately hardly be estimated will affect particularly the nascent population distribution of vibrational states of the C 2 (d 3 Π g ) state.\n\n\nMolecular spectra calculations\n\nIn order to reproduce experimental spectra, it is necessary to first model the population distribution of states and, using molecular constants and transitions probabilities, one can simulate a spectrum knowing the instrumental profile of the apparatus. The partition functions of rovibrational distributions under the assumption of a Maxwell-Boltzmann distribution used to generate so-called Boltzmann plots are given in section 4.1. The molecular constants used for calculating both the C 2 Swan band and the CN violet system and implemented into MassiveOES and available as Supplementary Data are described in sections 4.2 and 4.3.\n\n\nThermal population distributions\n\nThe distribution of energy between the internal degree of freedom of a diatomic molecule is described by the following partition function [61] Q = e νmax ν=0 Jmax J=Λ g Λ,hf s (2S + 1) (2J + 1)\n× exp − hc k B T (T e + G(ν) + F ν (J) − E 0 ) (7)\nwhere e denotes electronic states with projection of orbital angular momentum quantum number Λ and total spin quantum number S, and ν and J are the vibrational and rotational quantum numbers, respectively. g Λ,hf s is the statistical weight factor related to Λ-doubling and the nuclear hyperfine structure. T e , G(ν) and F ν (J) represent the electronic, vibrational and rotational energies in wavenumber units. E 0 is the energy of the lowest state of the molecule (i.e. ν = 0, J = Λ for the ground electronic state). T is the temperature of the heat's bath in the case where all internal degrees of freedom of the molecular gas are thermalized.\n\nAssuming that the internal degrees of freedom are independent, which is already implicitly assumed in the formalism used to derive equation 7, 6 the partition function for rotational and vibrational states can be rewritten as\nQ(ν, J) ∼ = Q rot (J, T rot )Q vib (ν, T vib )(8)\nwhere T rot and T vib are the temperature for the rotational and vibrational distribution functions in the case that they do not equilibrate with each other but obey Boltzmann distribution functions.\n\nUsing the Maxwell-Boltzmann distribution law, stating that the number of molecules dN E having an energy between E and E + dE is proportional to e (−E/k B T ) dE and using equations 7 and 8, the population density for rotational and vibrational levels can be defined as [63,62] \nN (ν, J) = N (ν) Q rot (2J + 1) exp − F ν (J) k B T rot (9) N (ν) = N e Q vib exp − G(ν)hc k B T vib(10)\nwhere N e is the total density of the electronic state e and N (ν) is the density of the ν level summed up over its rotational distribution.\n\nAssuming a Boltzmann distribution for the rotational and vibrational levels, the intensity of a line for a transition (ν → ν , J → J , Ω → Ω ) is given by\nI(λ) = N e Q e − E(ν ) k b T vib (2J + 1)e − E(J ,Ω ) k b T rot A ν ν J J Ω Ω · ψ(λ − λ 0 ) (11)\nwhere ψ(λ − λ 0 ) is the convoluted line profile and instrumental profile of the spectrometer and λ 0 the central wavelength position of the line. E(ν) = G(ν)hc and E(J , Ω ) ≈ F ν (J ) where the effect on energy level splitting due to Λ-doubling is expressed as a function of Ω (cf. equation 11). We use here the convention following Herzberg [63] that the rotational and vibrational quantum numbers of the upper level are defined by the index J', v and the lower level is described by J\" and v .\n\n\nSwan band database: description, implementation, benchmark\n\nC 2 is a homonuclear diatomic molecule and the Swan band is an electronically allowed dipole transition between two 3 Π states due to change of symmetry of the wavefunction between the upper and lower state g → u. These are triplet states and for each rotational quantum number J, there are sub-bands\n3 Π 0 − 3 Π 0 , 3 Π 1 − 3 Π 1 and 3 Π 2 − 3 Π 2 .\nThe sub-bands are dominated by P and R-branches (corresponding to selection rule ∆J = J − J = −1 and +1 respectively). Only weak Q-branches (∆J = 0) exist and none for the 3 Π 0 − 3 Π 0 sub-band as only ∆J = ±1 transitions occur for Ω = 0 where Ω is the total electronic angular momentum along the internuclear axis is designated with Ω = Λ + Σ. Σ is the projection of the spin quantum number S onto the internuclear axis. Λ is the quantum number for the component of the total angular momentum along the internuclear axis. It is the absolute value of the projected component of the angular momentum L on the internuclear axis M L with Λ =| M L | [64]. Π states have by definition Λ = 1 and for the Swan band we have triplet states d 3 Π g -a 3 Π u for which S=1 and so Σ = M S = −1, 0, +1. The coupling between the rotation of the nuclei and L additionally leads to a splitting into two components for each J value when Λ = 0. This is called Λ-doubling (see below). As the C 2 molecule is symmetric and the carbon atom has a zero nuclear spin, transition between symmetric and asymmetric wavefunctions are forbidden. The statistical weight due to nuclear spin I of the anti-symmetric rotational levels is given by I(I+1) and as for C 2 we have I=0, they are absent. However, for Λ = 1, we have symmetric and antisymmetric combination of wave functions due to Λ-doubling and therefore both even and odd rotational numbers are present. In the P, Q and R branches, only one of the Λ-doubled levels are present alternatively which leads to the appearance that the even levels are displaced to one side and the odd levels to the other side of a mean position for a given branch progression.\n\nWith each alternate line that is simply absent because of I=0, this is often referred to as the \"staggering\"effect. We note that the statistic weight (g Λ,hf s in equation 7) due to the nuclear spin for individual rotational levels induce an alternance of intensities for sub-bands of electronic transitions with Λ = 0.\n\nFor more details on the rovibrational structure and lines positions of the C 2 Swan band system we refer to Herzberg [63] and Pellerin [65].\n\nIn figure 4, a high resolution spectrum from 450 nm up to 565 nm is presented where the main transition groups ∆ν and the position of the vibrational transition band heads are identified. Transitions between two electronic states can be ordered following the difference in vibrational quantum numbers during the radiative transition between the upper and lower state ∆ν = v − v . For instance, in figure 4 the transition group ∆ν = +1 corresponds to the rovibrational serie\nd 3 Π g (v )-a 3 Π u (v − 1). Note that the d 3 Π g (v = 0)\nlevel is therefore not part of this transition group. The ∆ν = −2, +2 transition groups can also be seen experimentally but are not shown here. 1.0x10 19 1.5x10 19 2.0x10 19 2.5x10 19 1-0 The spectroscopic database that we implemented into the fitting program of molecular emission spectra MassiveOES [31] is derived from the work of Brooke et al. [1]. As initial input, MassiveOES requires rovibrationally resolved Einstein coefficients for the transitions between the upper and lower state that can be generally calculated using the following equation [66]:\n5-4 3-2 2-1 (nm) Photon flux (ph m -2 s -1 nm -1 ) '-v'' 4-3 6-5 2-2 1-1 0-0 2-3 1-2 0-1A (v ,J )→(v ,J ) = 16πν 3 S ∆J J 3 0 hc 3 (2J + 1) < Ψ v J | R e (r) | Ψ v J > 2(12)\nwhere S ∆J J is the Höln-London factor and\n< Ψ v J | R e (r) | Ψ v J > is the transition dipole moment (TDM).\nBrooke et al. [1] used PGOPHER, a program developed by Colin Western, for calculating the Einstein A (v ,J )→(v ,J ) coefficients [67]. PGOPHER requires as input a set of molecular constants and the band strengths for each vibrational band v → v . The purely vibrational transition dipole moments (TDMs) were calculated using the program LEVEL [68], a program that can solve the 1D Schrödinger equation for diatomic molecules when provided with a potential energy curve and the electronic TDM. The set of molecular constants for fitting RKR potential curves of the upper and lower electronic levels involved into the Swan band transition are listed in [1]. With the output of LEVEL, PGOPHER is then able to calculate the rotational TDMs and the Höln-London factors required to compute the numerical values of equation 12. The original simulations by Brooke et al were extended to a maximum rotational quantum number J of 300 by running again PGOPHER with the same molecular constants. This allows to properly calculate the rotational partition function in MassiveOES for the high rotational temperatures encountered in the present study. It should however be noted that, while doing this, the calculated energy levels have increasing error for increasing J in energy and line positions. The molecular constants used for the C 2 Swan band are given in Appendix A and the output calculation of PGOPHER as Supplementary Material to this paper.\n\nExtensive comparison of calculated line positions versus experimental ones were already done in the literature (see Brooke et al. [1] and references therein). We additionally performed a cross-check with the high resolution degenerate four-wave mixing experiments of Lloyd and Stewart [69]. The difference between their measured line positions for the ∆ν = 0 transition group and our calculated ones are shown in figure 5. One can see that, excepted for two outliers, the line positions are calculated with an accuracy much better than ± 0.05 cm −1 . Experimental data for higher J numbers is however not available for performing further benchmark. Experimentally we have a line profile of 22 pm corresponding to about 0.8 cm −1 which is much broader than the line position uncertainty λ err < 1 pm. Consequently, using calculated line positions for the fitting of the experimental spectra induces no additional error related to uncertainties in line position assignment. We note at this point that high resolution measurements of line positions were done up to J = 80 and only for some vibrational levels by Tanabashi et al. [70].\n\nThe database of lines and their transition probabilities is compiled in a SQL database, that is then read by MassiveOES. The database consists of three tables for listing the lines, the identity of rovibrational \"upper\" and \"lower\" states respectively. Each element of the table corresponds to a single line in the spectrum for which energy of the transition, air wavelength, vacuum wavelength, Einstein coefficient, upper state identity and lower state identity are given. The \"upper state\" tag (as well as the \"lower state\" tag) links the line with the energy level from which is originated, that will be used to calculate the intensity of the line. The effect of molecular perturbations ) and 1 5 Π g states [1] are combined in the table as additional transitions from the upper state, such that, the rovibrational population distribution is calculated without extra energy levels.\n\n\nCN violet system emission\n\nThe CN molecular violet system corresponds to the B 2 Σ + → X 2 Σ + transition of CN which is an heteronuclear diatomic molecule. The required molecular constants have been taken from S. Ram et al. [71]. The transition probabilities between two vibrational states calculated by solving the integral of the vibrational wavefunctions of the upper and lower states and TDM Ψ ν (r)R e (r)Ψ ν (r) were extracted from LIFBASE [72]. The molecular constants for calculating the Rydberg-Klein-Rees (RKR) potential curves of the X 2 Σ + and B 2 Σ + states were taken from [73,74]). For the TDMs R e (r), the values of Ito et al. [75] were taken. These molecular constants were then used as input for PGOPHER to calculate the Höln-London factors of the CN violet system with J number up to 180. For the CN violet system some perturbations also exist but those are not included in the calculations as they do not affect significantly the spectra. In Appendix B are given all the constants used in the calculation. As sanity check, a one to one comparison between our PGOPHER calculations and LIFBASE was made and they yield the same results both for line positions and intensities.\n\n5 Spectral analysis of the C 2 Swan band 5\n\n\n.1 Non-Boltzmann fitting of the Swan band\n\nIn non-equilibrium plasmas, rotational and vibrational distributions do not always thermalize because of competition between production and destruction processes and a molecular spectrum can then not be fitted using single rotational T rot and vibrational T vib temperatures [29]. As discussed in section 3, the C 2 ( 3 Π g ) state is well known for overpopulations in the density of its high vibrational levels and particularly the v = 6 level. In this section we investigate high resolution spectra of the C 2 Swan band in order to assess possible deviations from thermal equilibrium of the rovibrational populations.\n\nIn figure 6, individual P and R lines from the v = 0 level are identified (the Q lines are too weak to be distinguished, see section 4.2) from the ∆ν = 0 transition group and their area fitted with a Voigt profile (with gaussian and lorentzian half-width of respectively σ G =9.5 pm and σ L = 4 pm) to take into account the apparatus profile determined from Hg lines using a low pressure Hg lamp. They were selected only when they can be separated from neighboring lines and clearly identified. Note that the fine structure of the rotational levels could not be resolved and so that the lines are weighted over the rotational fine structure (i.e. rotational sub-bands). The logarithm of the density of each individual rotational level is plotted as function of the rotational energy (i.e. Boltzmann plot). All levels fit on a single line which can be fitted with a rotational temperature of 6200±160 K. Here and in the following, we will quote for each fitting method, its related statistical errors. No deviation is observed neither for low or high rotational levels indicating that the levels are thermalized. We note here that levels with approximately J>120 do not contribute noticably to the emission spectra and are not visible.\n\nThe thermalization of the rotational levels of the d 3 Π g state is further assessed using the option in MassiveOES of state-by-state fitting [32]. State by state fitting here means that the fitting routine evaluates the density of each upper rotational state while minimizing the residue for all transitions from the same upper state (i.e. different transition groups, P, Q and R branches and Λ-doubling). Allowing to fit independently the density of all rotational states helps to reduce the overall residue of the fit but only marginally. In figure 7, the population density of the rotational levels versus their energy is plotted. All rotational population densities fit quite well on a single line indicating no deviation from a thermal rotational distribution function. The lowest rotational levels are not shown because they are experimentally too much overlapped and the state by state fitting routine cannot converge (see below for a discussion of those). The highest rotational levels are widely scattered due to the very low signal to noise ratio. A rotational temperature T rot =5500± 500 K is obtained from the fitting of figure 7. From this first fit, some systematic deviations are apparent. A closer inspection indicates the presence of some outliers points which have densities off by an order of magnitude or more without any systematic trend. Removing these outliers which have a non-neglible weight for the fitting of a linear slope in logarithmic plot, the statistical uncertainty of the Boltzmann plot is significantly reduced. The inverse slope of the Boltzmann plot gives a temperature of 6200±150 K. This result agrees with the line by line Boltzmann plot analysis made in figure 6. Fitting the complete ∆ν = 0 transition group while assuming a Boltzmann distribution (i.e envelope fitting) for rotational and vibrational distribution gives similarly a temperature of 6000±200 K in excellent agreement with the state to state fitting analysis. The result of the fit using a Boltzmann distribution is also shown in figure 7 (note the logscale) as well and one can see that the residues following a Boltzmann fitting or a state by state fitting are similar.\n\nIn figure 8a, the state by state fitting analysis is extended to the ∆ν = −1 and ∆ν = +1 transition groups for the rotational and vibrational states of the d 3 Π g state. One can see that the rotational levels of the v=1 and v=2 levels also follow a Boltzmann distribution with the same slope as for the ν = 0 level although the signal is more noisy. Also population densities of the lower rotational states can be analysis and no systematic deviations from a single slope can be found.\n\nAfter integration over the rotational distribution the density of the vibrational states can be obtained (cf. equation 9). The resulting Boltzmann plot for the vibrational states is shown in figure 8b. Their population distribution can be described by a single temperature of 7200± 1200 K. For comparison, we also performed a state by state fitting of the ∆ν = +1 transition group that is the most sensitive to the vibrational temperature (see section 5.2). The rotational temperature was fixed to 6200 K following the analysis of the ∆ν = 0 transition group. For the ∆ν = +1 transition group, the density of each ν level is adjusted independently and the results are shown in figure 9. One can see that all vibrational levels fit perfectly on a single line with T vib =6000± 400 K. The larger uncertainties in the case of the state by state fitting when all three transition groups ∆ν = +1, 0, −1 are taken into account stems from the lower signal to noise ratio for the ∆ν = −1 transition group. The result of the fitting of the 3 transition groups for the vibrational distribution of the C 2 d 3 Π g state shown in figure 8b may let someone thing that the C 2 Swan band system follows a two-temperature vibrational distribution function. The vibrational state to state analysis of the most sensitive ∆ν = +1 transition group done in figure 9 (which is more sensitive to T vib as discussed in section 5.2) show that there is no evidence of a twotemperatures vibrational states distribution function. Fitting the ∆ν = +1 transition group alone, which is the one usually measured in the literature for studying the so-called C 2 high pressure bands as it is the most sensitive for determining vibrational states densities, show that the vibrational distribution function of the d 3 Π g state is very well described by a single temperature. Considering the noise in the state by state ro-vibrational fitting shown in figure 8a for ν = 1 and 2 levels, it is in fact quite remarkable to see that the fitted vibrational temperature still agrees with the more precise fitting given in figure 9 within their own error bars. Note that for the vibrational state to state fitting of the ∆ν = +1 transition group, the rotational temperature was set using the T rot obtained from the more sensitive ∆ν = 0 transition group.\n\nFrom this analysis, it appears that for the present plasma conditions, the vibrational levels are completely thermalized and no overpopulation of the higher vibrational levels and more particularly v=6, occurs. Analysis of other spectra as the one discussed above yield the same results and no deviations from Boltzmann equilibrium either for the rotational or vibrational levels could be found for the conditions of the present study (cf. section 7). Either the processes discussion in section 3 leading to the so-called C 2 high pressure bands are absent in the present discharge or the levels thermalize within the effective (radiative) lifetime of the state (see section 6 for a discussion). Different methods of fitting give similar results and no evidence of deviations from thermal distributions can be found within error margins. For the following, Boltzmann distributions are then assumed and an envelope fitting of the spectra is done. For measuring the rotational temperature, the ∆ν = 0 transition group is chosen while all three transition groups are used for the determination of the vibrational temperature. For more discussion on this, we refer to section 5.2.\n\n\nT rot and T vib from the C 2 Swan band: sensitivity and uncertainty analysis\n\nThe convergence of a fit is usually assessed by looking at the value of the least mean square error and minimizing the root mean squared error (RM SE = i (y i − y i ) 2 ). It is possible to use the computed residue for assessing how sensitive is the minimization routine for given external parameters that are being fitted. Information can be extracted from a dataset (here an experimental spectrum) only when the variation of signal is statistically significant. It usually means that the correlation between a signal and the event we want to measure must be significantly larger that the statistical noise. Theoretical spectra can be computed for given fixed rotational and vibrational temperatures and compared with another theoretical spectrum in terms of their respective RMSEs [76]. In the case where the (thermal) white noise is the dominant source of uncertainty in the fitting of an experimental spectrum, the amplitude of the noise can be directly correlated to an iso-line inside which the \"true value\" stands. Nassar [77] evaluated the errors induced by a white noise on T rot determination while fitting the band head of the ∆ν = 0 transition group and obtained errors in the order of the signal to noise ratio or lower. This is however without considering systematic errors that are induced by performing mathematical operations on the data such a baseline correction or intensity calibration. These errors are additive and amplify the uncertainties on the values obtained through a fit. Also, one needs to take additionally into account the uncertainties in the molecular constants (Franck-Condon and Höln-London factors).\n\nIn figure 10, such analysis is first made for the ∆ν = 0 alone and for the ∆ν = −1, 0, +1 transition groups of the C 2 Swan band by comparing simulated spectra with theoretical spectra with T rot = T vib taking values of 1000, 3000, 5000, 6000 and 7500 K with 5% surimposed noise. The steepness of the RMSE around the equilibrium point (i.e. true value) obtained by the minimization routine is a good indication of accuracy with which a parameter can be fitted (here T rot and T vib ). It appears clearly that the ∆ν = 0 is rather imprecise for the determination of the vibrational temperature of the d 3 Π g state and that typical errors are in the order of several 1000 K. The minimum for the rotational temperature is on the contrary quite steep indicating that the band has a good sensitivity for T rot determination. The ∆ν = −1, +1 transition groups have on the other hand a much higher sensitivity to T vib . A comparison between the ∆ν = 0 fit and the fit of all three transition groups seems to indicate at first sight that both allow to obtain the rotational temperature with rather good sensitivity. However, in the case of ∆ν = −1, 0, +1 transition groups fit, one can see that the isolines are stretched on a diagonal (i.e. not perpendicular to the axes) which means that T rot and T vib are correlated during the minimization (i.e. if an error is made in the parameters minimization used for the determination of one variable, the error will propagate to the other). The parametrization of the spectrum in terms of independent variables (T rot , T vib , instrumental profile,...) in a feature space (i.e. a space made of independent variables that are combinations of the original variables) is not entirely successful and some of the new variables are not orthogonal. This is a well known issue in the field of Principal Components Analysis (PCA) [78]. In the present case, when the isolines are equidistant in the vertical and horizontal directions, one can say that the determination of T rot and T vib are statistically independent. Note that a fitting routine gives for any given fit a RMSE but its absolute value carries no intrinsic information. It has a meaning only in correlation with other RMSEs calculated around the (local) equilibrium point of the fit. The RMS error is then directly normalized to the equilibrium value.\n\nWhile fitting an experimental spectrum using the assumption of Maxwell-Boltzmann distribution for the rotational and vibrational states of the radiating molecule, there are additional parameters that need to be fixed. They are listed into Table 1. These parameters, excepted for the first three in the table are all experimental and depends of the spectrometer. A linear baseline is fitted with an offset value B base and a slope α base for substracting contributions related to noise and continuum emission from the plasma. The instrumental profile of the spectrometer is taken into account via σ L and σ G while the wavelength calibration of the spectrum is adjusted with a quadratic fit (λ start , ∆λ and ∆λ 2 parameters) and the wavelength λ is defined as\nλ = λ start + ∆λ · i + ∆λ 2 · i(13)\nwhere i is the pixel number. The density N of the upper excited state has no influence on the evaluation of the temperatures as it is related only to the integrated band intensity.\n\nThe parameters that have an influence on the determination of T rot and T vib are the line profile and the fitting of the baseline. By varying systematically these values and considering uncertainties on the Franck-Condon (FC) and Höln-London factors, we estimated conservative uncertainty margins of about ±500 K for the spectra fitted in the present study. This evaluation is confirmed by the analysis made in section 5.1 where statistical methods have been applied while using different analysis methods to evaluate the rovibrational distribution functions. The errors for the rotational temperature are in fact probably lower. Fitting of theoretical low resolution spectra generated using SPECAIR [79,80] Wavelength of the first point in the spectra ∆λ Distance between two points in the spectra ∆λ 2 Quadratic correction of the position of a point in the spectra difference in the order of 1% for the rotational temperature. For the vibrational temperature, we also performed a comparison using the transition dipole moments of Kokkin et al [82] and FC-factors obtained by Fantz and Wünderlich [81]. The main difference with the present values and those calculations are only in the absolute values of the FC-factors with a difference of about 20%. The latter difference is systematic and consequently only affects the determination of the density N of the upper electronic state (i.e. after summation over rotational and vibrational population densities). No significant difference for the determination of T vib while using the two datasets were found. In figure 11, the same analysis is made for an experimental spectrum. The minimization routine of MassiveOES is here used only for adjusting the area of the spectra (i.e. population density of the electronic state), baseline and linewidth. The spectra are calibrated in wavelength beforehand using the position of known lines and its contribution to fitting uncertainties can be in the present case (i.e high resolution) neglected. The results are similar to figure 10 and it can be concluded that the C 2 Swan ∆ν = 0 transition group is overall more sensitive and \"precise\"for the determination of T rot .\n\nFinally, in figure 12, the sensitivity of the ∆ν = 0 transition group for determining T rot on assumed/fitted values of T vib is assessed. An experimental spectrum for the ∆ν = 0 transition group is fitted while taking different values for T vib . One can see that the fitted rotational temperature is almost independent of the vibrational temperature. This is due to the low transition probabilities for the ∆ν = 0 transitions for the v=1,2 levels while higher levels are even not visible at all. We note that the almost independence of the fitted rotational temperature to T vib is true for the relatively high resolution of the present spectrum. Interdependence of T rot and T vib becomes larger for lower resolution spectra also for the ∆ν = 0 transition group.\n\nThe ∆ν = −1, ±2 transition groups usually suffer from low intensities because of their low transition probabilities. The resulting lower signal to noise ratios make these groups not prime candidates for temperatures determination. From figure 10 and 11 and the discussion before, it can be concluded that the C 2 Swan ∆ν = 0 transition group is the best candidate for the determination of T rot while ∆ν = ±1 should be used for the determination of T vib . We should however stress that the present \"error\"calculations made here cannot be generalized to any C 2 Swan band fitted spectrum but the trends will remain similar.\n\n6 Is C 2 Swan band emission a good thermometer?\n\nBefore presenting a detailed analysis of the gas temperature of the plasma torch, a comparative study of the temperatures of the C 2 Swan band and CN violet system is presented in section 6.1. The good agreement that is reported between these two species gives a hint that the temperatures measured are with their local environment and so correspond to the gas temperature T gas with a value of about 6000 K. At atmospheric pressure, we also measured the first negative and second positive system of N 2 while operating the present plasma torch in pure N 2 and obtained rotational and vibrational temperatures in the order of 6500 K [85]. The consistency in the obtained values (for high pressure, high power microwave plasmas) using different measurements methods are a first indication that the C 2 Swan band may also be used for gas temperature determination. As discussed in section 3, there are several pathways involving neutral species for the formation of the C 2 (d 3 Π g ) state. Depending of its formation mechanism, one can expect the nascent rovibrational distribution of an excited state formed by chemical reactions to follow a thermal distribution or to be strongly non-thermal (i.e. formation of the excited into specific rotational and or vibrational state(s). In the latter case, for being able to predict its emission spectrum, it is necessary to take into account collisional relaxation processes [86]. The rotational and vibrational temperatures do not then correspond to the gas temperature. In sections 6.2 and 6.3 are discussed the collisional relaxation kinetics of the C 2 (d 3 Π g ) state. These processes that lead to the thermalization of the C 2 (d 3 Π g ) state rotational and vibrational distribution functions define the cases when the C 2 Swan band emission spectrum can still be used to measure the gas temperature independently of the formation mechanism of the C 2 (d 3 Π g ) state.\n\n\nComparison with the CN violet system\n\nBy adding a small amount of nitrogen, one can observe a strong decrease of the Swan band emission intensity while the CN violet system increase drastically. Vacher, Lino da Silva et al. [87,88] studied also the effect of adding N 2 to a CO 2 inductively coupled atmospheric pressure plasma and observed similar trends. Such behavior was also seen by other authors in microwave plasmas and low and atmospheric pressure [89,90]. This tendency can be explained by competing mechanisms for the formation of C 2 and CN that goes in favor of the formation of CN with increasing N 2 addition and for which it was found that C atoms are the main precursor [91]. CN (B-X) violet system is often used as a molecular pyrometer for determination of gas temperature in N 2 plasma sources with hydrocarbons [29, 92,93,94]. In order to generate CN violet system emission, 0.3 L/min of N 2 was added to the plasma corresponding to 3% admixture. This is a small quantity that, in the case of a pure CO 2 plasma, does not perturb the electron properties and neutral particles energy balance of the plasma.\n\nIn an argon atmospheric pressure microwave plasma with nitrogen addition and CO 2 advection from ambient air, Ridenti and Amorim discussed the weak presence of the CN violet system that is strongly overlapping with nitrogen first negative and second positive systems [95]. The band head of the first negative system is well separated from the one of the CN violet system while N 2 second positive system has distinctive features that do not overlap either with the CN violet system. These two systems can therefore, using high resolution spectroscopy, easily be identified. Their presence was not detected in the present CO 2 microwave plasma with 3% N 2 addition. They are however observed in the present reactor while operated at atmospheric pressure with pure N 2 flow [85]. In figure 13, an example of CN spectrum and its resulting fitting are shown. A value of T rot =T vib = 6000 ± 500 K is obtained which are the same as for the Swan band system. Also the residue of the fit shows that no deviation from rotational or vibrational Boltzmann distribution can be observed experimentally. A comparison for several pressures between the measured rotational and vibrational temperatures obtained by fitting the two molecular bands systems is presented in figure 14. One can see that, within error bars, there are no significant differences for the two molecular systems and that the mean values agree between each other very well. Note that in the case of CN, also the rotational and vibrational temperatures were fitted independently (cf. method discussed in section 5.1). This confirms that the gas inside the plasma region where the light comes from is very hot with a temperature of about 6000 K both for rotational and vibrational internal degrees of freedom. The rovibrational distributions for those two systems can be essentially described with a single temperature.\n\n\nCollisional relaxation of vibrational states\n\nC 2 Swan bands emission during pulsed laser irradiation of soot or graphite is often used for determination of gas temperature in (sub-)microsecond and even nanosecond time scales (see for instance [96,97]). The spectral resolution of such spectra is however usually low (because of the relative low intensity of such discharges) and a detailed evalution of the thermalization of rotational and vibrational states is difficult. UV pulsed photolysis experiments by Faust et al. [59] showed that the formation of the high pressure band is occuring only on long time scales (t>> 1µs) while for time up to 1µs the Swan band emission presented a normal vibrational distribution structure. They concluded that two different processes contribute to the formation of the C 2 Swan bands and that vibrational relaxation of the v=6 level is inefficient for populating the v=0 level of the d 3 Π g state.\n\nBondybey [98] analyzed the C 2 Swan band fluorescence following two photon excitation in rare gas solids. He observed an erratic progression of lifetimes for different vibrational levels and concluded that the fast vibrational relaxation of the d 3 Π g v=3,4 states may occur through the b 3 Σ − g state, following a suggestion by Frosch [99]. Xuechu and Nanquan [100] studied the vibrational relaxation of the C 2 (d 3 Π g ) state by observing the chemiluminescence spectrum of the reaction Na + CCl 4 in a thermal bath of argon where the C 2 (d 3 Π g , v = 6) state is formed preferentially. From their pressure dependence study, they concluded that the thermal relaxation of the C 2 (d 3 Π g ) state by argon takes only a few collisions with a rate of 2.2 · 10 6 Torr −1 s −1 while they deduce even higher quenching rate by the sodium atoms.\n\nSeveral experiments have shown that vibrational thermalization of the C 2 (d 3 Π g ) state is relatively slow and potentially as long if not longer than the lifetime of the state itself excepted for the rate reported above by Xuechu and Nanquan [100]. It may therefore be surprising to see that the vibrational distribution function follows closely a Maxwell-Boltzmann distribution in the experiments reported in this paper. However, the present measurements are done at high temperatures (i.e. T > 1000 K) and one may expect that the nascent population distribution follows already closely a thermal distribution, partly because of the large translational motion of the reactants. A temperature of 6000 K corresponds to a kinetic energy of 0.5 eV which is more than the energy separation between individual vibrational levels.\n\nRich and Bergman [101] performed vibrational excitation of CO at room temperature using a CO laser and observed formation of the C 1 Π g and d 3 Π g states of C 2 . They claimed that the observation of the C 1 Π g state (at the origin of the C 1 Π g -A 1 Π u Deslandres-D'Azambuja system) was the indication that CO (v) states can electronically excite the C 2 molecule. Indeed, the C 1 Π g state can be formed only via the association of two C ( 1 D) states and in their experiment only C ( 3 P) ground state atoms were produced. However, an analysis of the potential energy curves of C 2 shows that the C 1 Π g has an avoided crossing with another 1 Π g state and consequently dissociates into two ground state C( 3 P) atoms [48]. Grigorian and Cenian [102] recently performed a similar experiment and reported also the formation of the e 3 Π g state (e 3 Π g -a 3 Π u , Fox-Herzberg system) which have for dissociation limit the C ( 3 P) and C( 1 D) states [41]. Moreover, Wallaart et al. [50,103] reported during a CO vibrational excitation experiment the formation of the D 1 Σ + u state (D 1 Σ + u -X 1 Σ + g , Mulliken system) in addition to the aforementioned ones. The D 1 Σ + u state has for dissociation limit the C ( 1 S)+C( 1 D) states [41]. Although the initial experiments of Rich and Bergman did not allow to support their hypothesis of direct electronic excitation of C 2 electronic states by CO(v), subsequent measurements confirm their proposal. These observations go in favor to the hypothesis that vibrationally excited CO molecules can directly populate C 2 electronically excited states and that multiple vibrational excitation quanta may occur simultaneously. We note that Wallaart et al. reported also the observation of radiative transitions from electronically excited C atoms which could also be excited by CO(v). The presence of C 2 (D 1 Σ + u ) and C 2 (e 3 Π g ) states in the aforementioned studies could therefore also be explained by the recombination of C atoms which are electronically excited by CO(v) after being formed by the Boudouard mechanism (cf. reaction 1). The relatively low density of C* atoms compared to CO(v) states however makes it difficult to assess how important that channel is in reality. Hack and Langel [104] analysed the recombination of C( 1 D) atoms and observed the production of C 2 (e 3 Π g , C 1 Π g ) electronic states but no recombination rate coefficient was reported in the literature yet. The usual observation of C 2 electronic states formation in C/CO mixtures without detection of any electronically excited states of C atoms (at the exception of the work of Wallart et al) tends to support that efficient energy exchange mechanisms between CO(v) and C 2 exist. Such processes will favor rapid thermalization of the vibrational states of the C 2 molecule. Similarly, one can expect that the CO 2 molecule also interact with C 2 molecules in similar fashion due to its rich vibrational structure. Based on the present discussion, it can be concluded that either the formation mechanism of the C 2 (d 3 Π g ) state has already a nascent (nearly)-thermalized vibrational distribution (which seems unlikely based on its formation mechanisms discussed in section 3) or that, not only collisional quenching processes, but also reactive energy exchanges leads to its (faster) thermalization. This would explain the invariant observation of thermalized vibrational distributions of the C 2 (d 3 Π g ) state in CO 2 microwave plasmas.\n\n\nCollisional relaxation of rotational states and gas temperature\n\nDirect measurements of thermalization rates are usually difficult to obtain and indirect estimations are usually done. Using the assumption that the C 2 molecule would be formed rotationally hot and that the pressure dependence of the C 2 Swan band temperature can be used as estimate of the rotational relaxation/equilibration with the thermal bath of molecules, Bleekrode determined a rotational relaxation time of 3µs in an oxyacetylene flame at low pressure (2-120 Torr) [105]. Using the same hypotheses, Kini and Savadatti [46] estimated a rotational relaxation time for the Swan band of 30 µs in a CO discharge in a pressure range of 2-24 Torr. The underlying assumptions were particularly strong as no elementary process for the formation of rotationally hot C 2 were identified. However, Brockhinke et al [106] recently performed picosecond laser induced fluorescence measurements on the ∆ν = 0 transition group of the C 2 (d 3 Π g -a 3 Π u ) Swan. They managed to evaluate rotational energy transfer (RET) and vibrational energy transfer (VET) by monitoring the collisional transfer of single optically pumped rovibrational state of the d 3 Π g state to adjacent states by fluorescence. The measurements were performed in an oxy-acetylene flame with a gas temperature of 2200 K. Such measurements allow to monitor individual state to state RET and VET collisional transfers and are therefore very precise. They obtained state-to-state collisional re-distribution rate for RET in the order of k RET ≈ 4 · 10 9 s −1 .\n\nDue to the high electronic quenching rate of the C 2 d 3 Π g states they could not quantify the VET rate but assessed that it was slower than k RET in agreement with aforementioned studies. The temperatures obtained by fitting the C 2 Swan band and the CN (B 2 Σ + -X 2 Σ + ) violet system have been compared and match very well as seen in figure 14. Contrary to the C 2 (d 3 Π g ) state, the CN (B 2 Σ + ) state does not have, to the best of our knowledge, known mechanisms leading to non-equilibrium rovibrational emission spectra [107]. Rotational relaxation times of the CN (B 2 Σ + ) state are also considerably shorter than its radiative lifetime [108,109]. For CN, the rotational and vibrational states distributions have time to thermalize before being observed by optical emission spectroscopy in continuous plasmas. The cross check between the rotational temperatures of the Swan band and the CN violet system gives additional confidence that their rotational temperature may be associated to the gas temperature. The reactive formation of molecules can only lead to an overpopulation of the high rotational levels of a molecule (because of energy exchange during the formation process) or to a thermal distribution. In the case of non-Boltzmann distribution of rotational levels, the temperature of the lower rotational levels is then usually taken as a measure of the gas (i.e. translational) temperature of the plasma [29]. In the present study (see figure 8a), two-temperature rotational distribution function are not found. Observing thermal distributions of rotational states of the C 2 (d 3 Π g ) state and knowing that its RET relaxation time is much shorter than its radiative lifetime gives confidence that T rot is a local measure of the translational temperature of the plasma. This is additionally supported by the equivalence between the rotational and vibrational temperatures both for the C 2 Swan bands and for the CN violet system. For high pressure CO 2 plasmas (i.e. typically 100 mbar and above), it can therefore be proposed to use the C 2 Swan bands rotational temperature as an evaluation tool for the local gas temperature of the plasma. We note however, that for a CO 2 plasma with a much lower gas temperature, similar analysis should be performed as in section 5.1.\n\n\nParametric analysis of the CO 2 microwave plasma torch\n\nHaving assessed that C 2 Swan band can be used for measuring the gas temperature in a CO 2 plasma, we report in this section a parametric study of C 2 T rot and T vib as a function of plasma parameters. Although both T rot and T vib can be associated to the gas temperature, they are reported here independently. The measurements are taken in the pressure regime above 120 mbar where the C 2 Swan band emission is the dominant feature in the plasma emission spectrum. Concomitantly, the plasma is contracted with a diameter of about 3-8 mm which depends mainly of the applied power. The emission is spatially discriminated using a system of two pin-holes and a lens coupled to a fiber and collected over a area of about 1 mm 2 . The latter corresponds to the spatial resolution attainable in the present setup. Each data point corresponds to the light collected along the light of sight (radial direction) and over a plasma slab of 1 mm in the axial direction. Interestingly, in the contracted regime of the plasma torch (i.e. above a pressure of about 120 mbar), axial and radial measurements of the C 2 rotational temperature show no variation for all studied plasma conditions. After substraction of the continuous baseline, only a spatial variation of the overall intensity of the C 2 Swan band is observed within signal to noise sensitivity. In the following, all data are consequently reported for measurements in the center of the resonator; i.e 20 mm above the bottom of the waveguide while looking across the axis of the quartz tube. In figure 15, the effect of the input microwave power on the rotational and vibration temperatures of the C 2 Swan bands are presented. The measurements are done near atmospheric pressure plasma (925 mbar) for a gas flow of 10 L/min. One can see that both the rotational and vibrational temperatures of C 2 remain constant and equal within error bars. The increase in power has only the consequence of increasing the intensity of the radiation from the C 2 (d 3 Π g ) state. This increase of intensity is correlated with an increase in plasma radius. Also, by imaging it is observed that the plasma becomes elongated in the effluent region with increasing power.\n\nIn order to cool down a high pressure plasma, one possible way is by increasing the gas flow. In figure 16, the variation of the C 2 rotational temperature versus CO 2 gas flow at a pressure of about 850 mbar is shown for an input power of 900 W. One can see that the temperature remains constant around 6000± 500 K. Such trend was also observed by Sun et al. [27] and Spencer and Gallimore [24] but for lower flow rates variation. Similar lack of influence of the gas flow was also observed in the case of an atmospheric pressure microwave plasma torch in nitrogen [110]. Increasing the gas flow by more than 10-fold appears in the present setup (and also for other high pressure microwave sources) to be an inefficient way of tuning the gas temperature despite reducing the specific energy input per molecule by the same factor. Such result is rather surprising but may be due to an inefficient mixing between the hot and cold gas region.\n\nA typical method for enhancing the non-equilibrium of a plasma discharge (e.g. reducing the gas temperature while the electron temperature remains in the order of 1 eV or more) is to reduce the gas pressure. In figure 17, the pressure is varied from near atmospheric pressure down to 180 mbar where the C 2 Swan band is still intense enough for getting a good signal to noise ratio. One can see that T vib and T rot are equal to each other within absolute uncertainties. One should note that in pure CO 2 microwave plasmas, lower temperatures have been measured but only at (much) lower pressures than in the current study and where C 2 Swan band is not observed [89,111]. On the other hand, den Harder et al. [17] and Groen et al [83] measured gas temperature in the order of 5000 to 6000 K in the same pressure range as for the present study.\n\n\nConclusions and outlook\n\nIn this paper, the potential sources of C 2 molecules in CO 2 plasmas and the known pathways leading to non-equilibrium vibrational distributions of the (d 3 Π g ) state are reviewed. A detailed analysis of high resolution spectra is performed in order to detect potential deviations of the rovibrational distribution functions from thermal equilibrium in a high pressure microwave plasma discharge. State of the art molecular constants were used in order to fit the spectra. It is found that Boltzmann distributions can be used both for rotational and vibrational states for all conditions in the present study. Consequently, potential source terms of the C 2 (d 3 Π g ) state in a CO 2 plasma do not lead to any observed deviation from a Boltzmann vibrational distribution function. The sensitivity of different parts of the spectra for determining T rot and T vib is evaluated. For a fast and accurate determination of the rotational temperature of the C 2 (d 3 Π g ) state, it is shown that the ∆ν = 0 transition group is the most accurate part of the spectrum to be taken into account.\n\nDue to fast thermalization of the rotational states of the C 2 (d 3 Π g ) state with the background gas, the measured T rot can be used for assessing the gas temperature of the plasma. Furthermore, the vibrational states of the C 2 (d 3 Π g ) level are found to be thermalized indicating the possible presence of fast thermalization processes.\n\nA parametric study is performed on a pure CO 2 microwave plasma torch and the both the vibrational and rotational temperatures are found to be insensitive to external plasma parameters such as gas flow and input power at pressures of 180 mbar and above and a value of 6000 ± 500 K is found. The latter can be associated to the gas temperature of the plasma. These results are in agreement with the hypothesis that chemical equilibrium of the species can be described by the gas temperature at which thermal dissociation play an important role. The only observable differences in the extended parameter range of the present study are in the radially integrated density of the C 2 (d 3 Π g ) state. We note that similar values as the one presented in this study were reported before in a certain number of studies but no assessment of non-equilibrium distributions was done. Also, the rotational temperature of C 2 was implicitely assumed previously to be the one of the gas temperature and it is now shown that such approximation in high pressure CO 2 microwave discharges appears to hold.\n\nThe C 2 (d 3 Π g ) state emits only in a narrow region in the center of the plasma and only the gas temperature in the core of the plasma can be measured. The region inside the plasma reactor that can be probed is only where C atoms are present and react to form C 2 (or CN when some N 2 is present) molecules. It was observed that C atoms are not observed by optical emission spectroscopy at pressures lower than the ones where the C 2 (d 3 Π g ) state is measured. To form carbon atoms, large power densities (i.e. high T vib and/or T gas ) are required as CO is the only species that can act as a primary source of carbon atoms. Interestingly, at the observed temperature of 6000 K, CO can already partially thermally dissociate [112]. This implies that observation of C and C 2 species is linked to regions with high gas temperatures where CO 2 molecules are readily dissociated.\n\nThe measurements reported in this paper show the presence of a hot core of gas but no insight in the temperature gradient at the edge of the plasma could be obtained as not radiative emitting molecular species can be observed at the edges of this hot core. High gas temperatures T≥ 3500K allow already to dissociate fully the CO 2 and partially the CO molecules through thermal processes [18,6]. In situ diagnostics techniques like Raman scattering and CARS [113,114] would bring additional understanding in the gas dynamics at the boundary between the hot plasma core and the colder edge.\n\n[21] Yacine Babou, Philippe Rivire, Marie-Yvonne Perrin, and Anouar\n\nSoufiani. Spectroscopic study of microwave plasmas of CO 2 and CO 2 -N 2 mixtures at atmospheric pressure. Plasma Sources Science and Technology, 17 (4) Figure 15: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of power input. The measurements are performed in the center of the resonator at 925 mbar with 10 L/min CO 2 gas flow. n is the laterally integrated density of the C 2 (d 3 Π g ) state. Figure 17: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of pressure. The measurements are performed in the center of the resonator using a 10 L/min CO 2 gas flow and 900 W input power.\nB 2 Σ + ν\nOrigin B ν 10 6 · D ν 10 −2 · γ 0 25797.86825 (43) \n\nFigure 2 :\n2Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented.\n\nFigure 4 :\n4Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min.\n\nFigure 5 :\n5Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19\n\nFigure 6 :\n6Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm.\n\nFigure 7 :Figure 10 :\n710Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K.\n\nFigure 11 :\n11Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm.\n\nFigure 12 :\n12Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6.\n\nFigure 13 :Figure 14 :\n1314CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow\n\nFigure 16 :\n16Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W.\n\n\n−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3\n\n\nor from Fantz and Wünderlich[81] leads toParameter \n\nmeaning \nN \nUpper state excited specie density \nT rot \nRotational temperature \nT vib \nVibrational temperature \nB base \nConstant baseline offset value versus λ \nα base \nbaseline slope \nσ G \nGaussian width of the Voigt profile \nσ L \nLorentzian width of the Voigt profile \nλ start \n\n\nTable 1 :\n1Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program.\n\n\nGatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system:045010, 2008. \n\n[22] Yong C Hong, Dong H Shin, Sang J Lee, Ye J Kim, Bong Ju Lee, and \nHan S Uhm. Generation of high-power torch plasma by a 915-MHz mi-\ncrowave system. IEEE Transactions on Plasma Science, 39(10):1958-\n1962, 2011. \n\n[23] D Le Quang, Y Babou, and P Andre. Investigations of a microwave \nplasma source operating with air, N 2 , CO 2 and argon gases. IOP \nConference Series: Materials Science and Engineering, 29(1):012009, \n2012. \n\n[24] L F Spencer and A D Gallimore. CO 2 dissociation in an atmospheric \npressure plasma/catalyst system: a study of efficiency. Plasma Sources \nScience and Technology, 22(1):015019, 2013. \n\n[25] Han S. Uhm, Hyoung S. Kwak, and Yong C. Hong. Carbon dioxide \nelimination and regeneration of resources in a microwave plasma torch. \nEnvironmental Pollution, 211:191 -197, 2016. \n\n[26] Constandinos M Mitsingas, Rajavasanth Rajasegar, Stephen Ham-\nmack, Hyungrok Do, and Tonghun Lee. High energy efficiency plasma \nconversion of CO 2 at atmospheric pressure using a direct coupled \nmicrowave plasma system. IEEE Transactions on Plasma Science, \n44(4):651-656, 2016. \n\n[27] Hojoong Sun, Jungwun Lee, Hyungrok Do, Seong-kyun Im, and Moon \nSoo Bak. Experimental and numerical studies on carbon dioxide de-\ncomposition in atmospheric electrodeless microwave plasmas. Journal \nof Applied Physics, 122(3):033303, 2017. \n\n[28] Se Min Chun, Yong Cheol Hong, and Dae Hyun Choi. Reforming \nof methane to syngas in a microwave plasma torch at atmospheric \npressure. Journal of CO2 Utilization, 19:221-229, 2017. \n\n[29] PJ Bruggeman, N Sadeghi, DC Schram, and V Linss. Gas tempera-\nture determination from rotational lines in non-equilibrium plasmas: a \nreview. Plasma Sources Science and Technology, 23(2):023001, 2014. \n\n[30] Reginald William Blake Pearse and Alfred Gordon Gaydon. Identifi-\ncation of molecular spectra. Chapman and Hall, 1976. \n[31] Jan Voráč, Petr Synek, Lucia Potočňáková, Jaroslav Hnilica, and Vít \nKudrle. Batch processing of overlapping molecular spectra as a tool \nfor spatio-temporal diagnostics of power modulated microwave plasma \njet. Plasma Sources Science and Technology, 26(2):025010, 2017. \n\n[32] Jan Voráč, Petr Synek, Vojtěch Procházka, and Tomáš Hoder. State-\nby-state emission spectra fitting for non-equilibrium plasmas: OH spec-\ntra of surface barrier discharge at argon/water interface. Journal of \nPhysics D: Applied Physics, 50(29):294002, 2017. \n\n[33] Martina Leins, Sandra Gaiser, Andreas Schulz, Matthias Walker, Uwe \nSchumacher, and Thomas Hirth. How to ignite an atmospheric pressure \nmicrowave plasma torch without any additional igniters. Journal of \nvisualized experiments: JoVE, (98), 2015. \n\n[34] JW McConkey, CP Malone, PV Johnson, C Winstead, V McKoy, and \nI Kanik. Electron impact dissociation of oxygen-containing molecules-a \ncritical review. Physics Reports, 466(1-3):1-103, 2008. \n\n[35] Marija Grofulović, Luís L Alves, and Vasco Guerra. Electron-neutral \nscattering cross sections for CO2: a complete and consistent set and \nan assessment of dissociation. Journal of Physics D: Applied Physics, \n49(39):395207, sep 2016. \n\n[36] Pamir Nag and Dhananjay Nandi. Dissociation dynamics in the disso-\nciative electron attachment to carbon dioxide. Phys. Rev. A, 91:052705, \nMay 2015. \n\n[37] Yukikazu Itikawa. Cross sections for electron collisions with carbon \ndioxide. Journal of Physical and Chemical Reference Data, 31(3):749-\n767, 2002. \n\n[38] L.D. Pietanza, G. Colonna, V. Laporta, R. Celiberto, G. DAmmando, \nA. Laricchiuta, and M. Capitelli. Influence of electron molecule \nresonant vibrational collisions over the symmetric mode and direct \nexcitation-dissociation cross sections of co2 on the electron energy dis-\ntribution function and dissociation mechanisms in cold pure co2 plas-\nmas. The Journal of Physical Chemistry A, 120(17):2614-2628, 2016. \nPMID: 27064438. \n\n[39] A.I. Florescu-Mitchell and J.B.A. Mitchell. Dissociative recombination. \nPhysics Reports, 430(5):277 -374, 2006. \n[83] P W C Groen, A J Wolf, T W H Righart, M C M van de Sanden, \nF J J Peeters, and W A Bongers. Numerical model for the determina-\ntion of the reduced electric field in a CO 2 microwave plasma derived \nby the principle of impedance matching. Plasma Sources Science and \nTechnology, 28(7):075016, jul 2019. \n\n[84] N ν \nOrigin \nA \nAD \nBν \n10 6 · Dν \nλ \no \np \nq \n0 \n19 378.46749(51) -14.00139(63) 0.0005068(83) 1.7455663(43) 6.8205(16) 0.03301(47) 0.61076(52) 0.003973(43) -0.0007752(43) \n1 \n21 132.14977(25) -13.87513(49) 0.0005740(83) 1.7254062(53) 7.0194(77) 0.02972(38) 0.61713(36) 0.004133(44) -0.0008171(43) \n2 \n22 848.3877(21) \n-13.8205(23) \n0.000600(43) \n1.704516(21) \n7.308(22) \n0.0253(41) \n0.6208(32) \n0.00624(38) \n-0.000835(14) \n3 \n24 524.2201(19) \n-13.5361(28) \n0.000775(17) \n1.681437(16) \n7.438(24) \n0.0470(26) \n0.5827(26) \n0.00579(17) \n-0.0008568(85) \n4 \n26 155.0448(29) \n-13.3892(50) \n0.001451(14) \n1.656859(26) \n7.684(43) \n0.0219(38) \n0.6313(32) \n0.00954(29) \n-0.000923(21) \n5 \n27 735.6720(43) \n-13.0324(66) \n0.000723(37) \n1.630205(23) \n8.573(32) \n0.0601(28) \n0.6161(23) \n0.00685(32) \n-0.000912(15) \n6 \n29 259.3548(36) \n-12.820(10) \n0.001203(56) \n1.599876(31) \n8.998(44) \n0.0529(71) \n0.5773(71) \n0.00874(47) \n-0.000986(22) \n7 \n30 717.9011(46) \n-12.3458(71) \n0.000814(41) \n1.566047(32) \n10.044(66) 0.0960(34) \n0.5532(31) \n0.00936(35) \n-0.001175(17) \n8 \n32 102.655(22) \n-12.107(22) \n0.00076(fixed) 1.52675(31) \n9.60(97) \n0.095(fixed) 0.546(22) \n0.0055(21) \n-0.00088(21) \n9 \n33 406.230(22) \n-11.698(39) \n0.00076(fixed) 1.485755(96) \n11.85(10) \n0.172(26) \n0.498(28) \n0.0097(14) \n-0.002062(44) \n10 34 626.7860(94) \n-11.297(15) \n0.00076(fixed) 1.441138(72) \n12.837(73) 0.115(16) \n0.399(12) \n0.00745(90) \n-0.000977(30) \n\n\n\nTable 2 :\n2Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)ν Origin \nA \nAD \nBν \n10 6 · Dν \nλ \no \np \nq \n0 0 \n-15.26986(43) 0.0002634(71) 1.6240452(44) 6.4506(19) -0.15450(36) 0.67525(35) 0.002537(42) -0.0005281(44) \n1 0.0001996(94) 1.5907513(61) 6.4527(44) -0.1526(12) \n0.6649(14) \n0.003132(77) -0.0006457(48) \n3 4784.0688(31) \n-15.1972(39) \n0.000186(42) \n1.574088(24) \n6.455(24) \n-0.1333(61) \n0.6815(51) \n0.00488(42) \n-0.000618(17) \n4 6332.1364(51) \n-15.2043(65) \n0.000318(36) \n1.557117(31) \n6.338(39) \n-0.1551(72) \n0.6674(67) \n0.00632(36) \n-0.000894(16) \n5 7856.8175(32) \n-15.2096(35) \n0.00025(fixed) 1.540139(24) \n6.312(35) \n-0.1492(36) \n0.6546(37) \n0.00734(25) \n-0.001246(12) \n6 9358.1565(40) \n-15.1646(60) \n0.000355(33) \n1.523439(26) \n6.034(38) \n-0.1551(46) \n0.6886(38) \n0.00504(32) \n-0.000676(16) \n7 10 836.1430(92) -15.085(11) \n0.00025(fixed) 1.50869(25) \n3.51(92) \n-0.1641(90) \n0.704(14) \n-0.0244(23) \n0.00632(49) \n8 12 290.7997(29) -15.1702(46) \n0.00025(fixed) 1.488684(28) \n5.329(52) \n-0.1665(36) \n0.6742(30) \n0.01449(29) \n-0.002053(23) \n9 13 722.0897(43) -15.0980(61) \n0.000419(31) \n1.472818(24) \n6.066(35) \n-0.1584(27) \n0.6926(23) \n0.00303(33) \n-0.000081(16) \n\n\n\nTable 3 :\n3Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6B Molecular constants for the calculations of \nthe CN violet system \n\n\nTable 4 :\n4Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.X 2 Σ + \n\n\nTable 5 :\n5Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.(480 \n485 \n490 \n495 \n500 \n505 \n510 \n515 \n\n0.0 \n\n5.0x10 \n\n18 \n\n1.0x10 \n\n19 \n\n1.5x10 \n\n19 \n\n2.0x10 \n\n19 \n\n2.5x10 \n\n19 \n\n3.0x10 \n\n19 \n\nR(91) \nR(84) \nR(88) \n\nR(95) \n\nR(98) \n\nR(77) \n\nR(65) \n\nR(58) \nR(62) \nR(48) P(72) \n\nP(71) \n\nP(68) \n\nP(57) \n\nP(43) \n\nP(44) \n\nP(45) \n\nP(29) \n\nP(26) \n\nl (nm) \n\nIntensity (ph m \n\n-2 \n\ns \n\n-1 \n\nnm \n\n-1 \n\n) \n\nP(25) \n\n\n\nTable 6 :\n6Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).ν \n− ν \nB 2 Σ + \n0 \n1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n\nX 2 \nΣ + \n\n\n\nTable 7 :\n7The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition.\nThis continuum emission which is the dominant emission feature in the effluent of the plasma and at its edges corresponds to chemiluminescence of CO 2 * formed by recombination of O+CO species[19,20].2 Carbon atom lines are only observed below 300 nm and above 900 nm and are not shown in figure 1.\nWhile defining the energy contributions of all internal degrees of freedom to a state defined by the quantum number ν, S, J and Λ as additive, we make the assumption that those degree or freedom are independent[62].\nAcknowledgmentsThe authors would like to acknowledge prof. Christophe Laux (Ecole Centrale, Paris) for giving access to his program Specair. Fruitful discussion and collaboration with our colleagues from Uni. Stuttgart (Irina Kistner, dr. Andreas Schulz, dr. Sandra Gaiser, dr. Matthias Walker), DIFFER (dr. Tiny Verreycken, dr. Floran Peeters, dr. Waldo Bongers) and KIT (Lucas Silberer, dr. Sergey Soldatov) are also gratefully acknowledged.Figure 9: State by state fitting of the vibrational distribution function of the C 2 (d 3 Π g ) state using the ∆ν = +1 band. The rotational temperature was fixed to 6200 K and the experimental spectrum constructed by adjusting the density of the v levels. The experimental conditions are the same as infigure 6.\nLine strengths and updated molecular constants for the C 2 Swan system. S A James, Peter F Brooke, Timothy W Bernath, George B Schmidt, Bacskay, Journal of Quantitative Spectroscopy and Radiative Transfer. 124James S.A. Brooke, Peter F. Bernath, Timothy W. Schmidt, and George B. Bacskay. Line strengths and updated molecular constants for the C 2 Swan system. Journal of Quantitative Spectroscopy and Radiative Transfer, 124:11 -20, 2013.\n\nCO 2 utilization: a process systems engineering vision. José F Araújo Ofélia De Queiroz, Rita Luiz De Medeiros, B Maria, Alves, CO 2 Sequestration and Valorization. InTechF Araújo Ofélia de Queiroz, José Luiz de Medeiros, and Rita Maria B Alves. CO 2 utilization: a process systems engineering vision. In CO 2 Sequestration and Valorization. InTech, 2014.\n\nAssessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Alexandre Lebouvier, Samuel A Iwarere, Philippe Dargenlieu, Deresh Ramjugernath, Laurent Fulcheri, Energy & Fuels. 275Alexandre Lebouvier, Samuel A. Iwarere, Philippe dArgenlieu, Deresh Ramjugernath, and Laurent Fulcheri. Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Energy & Fuels, 27(5):2712-2722, 2013.\n\nPlasma technology a novel solution for CO 2 conversion?. Ramses Snoeckx, Annemie Bogaerts, Chem. Soc. Rev. 46Ramses Snoeckx and Annemie Bogaerts. Plasma technology a novel solution for CO 2 conversion? Chem. Soc. Rev., 46:5805-5863, 2017.\n\nPlasma for electrification of chemical industry: a case study on CO 2 reduction. H G J Van Rooij, W A N Akse, M C M Bongers, Van De Sanden, Plasma Physics and Controlled Fusion. 60114019G J van Rooij, H N Akse, W A Bongers, and M C M van de Sanden. Plasma for electrification of chemical industry: a case study on CO 2 reduction. Plasma Physics and Controlled Fusion, 60(1):014019, oct 2017.\n\nPlasma Chemistry. Alexander Fridman, Cambridge Books OnlineAlexander Fridman. Plasma Chemistry. Cambridge University Press, 2008. Cambridge Books Online.\n\n. N N Sobolev, V Vasilii, Sokovikov, LASERS. Soviet Physics Uspekhi. 102153N N Sobolev and Vasilii V Sokovikov. CO 2 LASERS. Soviet Physics Uspekhi, 10(2):153, 1967.\n\nPlasma treatments and plasma deposition of polymers for biomedical applications. Pietro Favia, Riccardo Dagostino, Papers presented at the Fifth International Conference on Plasma Surface Engineering. 98Pietro Favia and Riccardo dAgostino. Plasma treatments and plasma deposition of polymers for biomedical applications. Surface and Coat- ings Technology, 98(1):1102 -1106, 1998. Papers presented at the Fifth International Conference on Plasma Surface Engineering.\n\nCarbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine. Emile Carbone, Claire Douat, 8Emile Carbone and Claire Douat. Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine, 8(1):93-120, 2018.\n\nThe physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. V D Rusanov, G V Fridman, Sholin, Soviet Physics Uspekhi. 246447V D Rusanov, A A Fridman, and G V Sholin. The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. Soviet Physics Uspekhi, 24(6):447, 1981.\n\nEvaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. T Kozák, A Bogaerts, Plasma Sources Science and Technology. 24115024T. Kozák and A. Bogaerts. Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. Plasma Sources Science and Technology, 24(1):015024, 2015.\n\nTaming microwave plasma to beat thermodynamics in CO 2 dissociation. Gj Van Rooij, Van Den, Bekerom, Harder, G Minea, Berden, R Bongers, Engeln, Mf Graswinckel, Mcm Zoethout, Van De Sanden, Faraday discussions. 183GJ van Rooij, DCM van den Bekerom, N den Harder, T Minea, G Berden, WA Bongers, R Engeln, MF Graswinckel, E Zoethout, and MCM van de Sanden. Taming microwave plasma to beat thermody- namics in CO 2 dissociation. Faraday discussions, 183:233-248, 2015.\n\nSelfconsistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. M Capitelli, G Colonna, L D Dammando, Pietanza, Plasma Sources Science and Technology. 26555009M Capitelli, G Colonna, G DAmmando, and L D Pietanza. Self- consistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. Plasma Sources Sci- ence and Technology, 26(5):055009, 2017.\n\nVibrational relaxation of anharmonic oscillators with exchange-dominated collisions. C E Treanor, J W Rich, R G Rehm, The Journal of Chemical Physics. 484C. E. Treanor, J W. Rich, and R.G. Rehm. Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. The Jour- nal of Chemical Physics, 48(4):1798-1807, 1968.\n\nThe nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. A R I Azizov, V K Vakar, M F Zhivotov, O Krotov, Zinov'ev, Bv Potapkin, Rusanov, A A V D Rusanov, Fridman, 28567R I Azizov, A K Vakar, V K Zhivotov, M F Krotov, O A Zinov'ev, BV Potapkin, A A Rusanov, V D Rusanov, and A A Fridman. The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. 28:567, 1983.\n\nStefan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dissociation of CO 2 for fuel synthesis. Plasma processes and polymers. Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, 141600126Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, Ste- fan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dis- sociation of CO 2 for fuel synthesis. Plasma processes and polymers, 14(6):1600126, 2017.\n\nHomogeneous co 2 conversion by microwave plasma: Wave propagation and diagnostics. Dirk C M Niek Den Harder, Van Den, Richard S Bekerom, Martijn F Al, Jose M Graswinckel, Palomares, J J Floran, Srinath Peeters, Teofil Ponduri, Waldo A Minea, Bongers, C M Mauritius, Gerard J Van De Sanden, Van Rooij, Plasma Processes and Polymers. 1461600120Niek den Harder, Dirk C. M. van den Bekerom, Richard S. Al, Mar- tijn F. Graswinckel, Jose M. Palomares, Floran J. J. Peeters, Srinath Ponduri, Teofil Minea, Waldo A. Bongers, Mauritius C. M. van de Sanden, and Gerard J. van Rooij. Homogeneous co 2 conversion by mi- crowave plasma: Wave propagation and diagnostics. Plasma Processes and Polymers, 14(6):1600120, 2017.\n\nThe importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Dirk Van Den, Jose Maria Palomares Bekerom, Tiny Linares, Eddie M Verreycken, Sander Van Veldhuizen, Waldo Nijdam, Richard Bongers, Gerard J Van Van De Sanden, Rooij, Plasma Sources Science and Technology. Dirk van den Bekerom, Jose Maria Palomares Linares, Tiny Verr- eycken, Eddie M Van Veldhuizen, Sander Nijdam, Waldo Bongers, Richard Van de Sanden, and Gerard J Van Rooij. The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Plasma Sources Sci- ence and Technology, 2018.\n\nOn use of CO 2 chemiluminescence for combustion metrics in natural gas fired reciprocating engines. B Sreenath, Gupta, P Bipin, Munidhar S Bihari, Raj R Biruduganti, James Sekar, Zigan, Proceedings of the Combustion Institute. the Combustion Institute33Sreenath B. Gupta, Bipin P. Bihari, Munidhar S. Biruduganti, Raj R. Sekar, and James Zigan. On use of CO 2 chemiluminescence for com- bustion metrics in natural gas fired reciprocating engines. Proceedings of the Combustion Institute, 33(2):3131 -3139, 2011.\n\nCO 2 * Chemiluminescence in Premixed Flames. J.-M Samaniego, F N Egolfopoulos, C T Bowman, Combustion Science and Technology. 1091-6J.-M. Samaniego, F. N. Egolfopoulos, and C. T. Bowman. CO 2 * Chemi- luminescence in Premixed Flames. Combustion Science and Technol- ogy, 109(1-6):183-203, 1995.\n\nInvestigation of the dynamics of co 2 plasmas using kinetic modelling. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, Ursel Hecimovic, Fantz, Van Dijk, 23rd International Symposium on Plasma Chemistry. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, A Heci- movic, Ursel Fantz, and J van Dijk. Investigation of the dynamics of co 2 plasmas using kinetic modelling. In 23rd International Symposium on Plasma Chemistry (ISPC 2017), 2017.\n\nC 2 spectroscopy and kinetics. Margarita Martin, Journal of Photochemistry and Photobiology A: Chemistry. 663Margarita Martin. C 2 spectroscopy and kinetics. Journal of Photo- chemistry and Photobiology A: Chemistry, 66(3):263-289, 1992.\n\nNon-equilibrium plasma kinetics of reacting CO: an improved state to state approach. G L D Pietanza, M Colonna, Capitelli, Plasma Sources Science and Technology. 2612125007L D Pietanza, G Colonna, and M Capitelli. Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. Plasma Sources Science and Technology, 26(12):125007, 2017.\n\nVibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. E V Kustova, A Aliat, A Chikhaoui, Chemical Physics Letters. 3445E.V. Kustova, A. Aliat, and A. Chikhaoui. Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. Chemical Physics Letters, 344(5):638 -646, 2001.\n\nIsotope effect in Boudouard disproportionation reaction in optically pumped CO. Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, Lempert, Rich, Chemical Physics. 3631-3Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, WR Lem- pert, JW Rich, and IV Adamovich. Isotope effect in Boudouard dis- proportionation reaction in optically pumped CO. Chemical Physics, 363(1-3):24-32, 2009.\n\nGas phase Boudouard reactions involving singlet-singlet and singlet-triplet CO vibrationally excited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. R P Patricia, Henrique Barreto, Alessandra F Euclides, Vincenzo Albernaz, Mario Aquilanti, Gaia Capitelli, Andrea Grossi, Sergey Lombardi, Federico Macheret, Palazzetti, The European Physical Journal D. 7110259Patricia RP Barreto, Henrique de O Euclides, Alessandra F Albernaz, Vincenzo Aquilanti, Mario Capitelli, Gaia Grossi, Andrea Lombardi, Sergey Macheret, and Federico Palazzetti. Gas phase Boudouard reac- tions involving singlet-singlet and singlet-triplet CO vibrationally ex- cited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. The European Physical Journal D, 71(10):259, 2017.\n\nInvestigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. M I Ks Kini, Savadatti, Journal of Physics B: Atomic and Molecular Physics. 1061139KS Kini and MI Savadatti. Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. Journal of Physics B: Atomic and Molecular Physics, 10(6):1139, 1977.\n\nThe pulse radiolysis of carbon monoxide. C Willis, C Devillers, Chemical Physics Letters. 21C. Willis and C. Devillers. The pulse radiolysis of carbon monoxide. Chemical Physics Letters, 2(1):51 -53, 1968.\n\nAn Extension of the Phillips System of C 2 and a Survey of C 2 States. E A Ballik, D A Ramsay, The Astrophysical Journal. 13784E. A. Ballik and D. A. Ramsay. An Extension of the Phillips System of C 2 and a Survey of C 2 States. The Astrophysical Journal, 137:84, 1963.\n\nMechanism for Excitation of the C 2 High-Pressure Bands. C Kunz, S Harteck, Dondes, The Journal of Chemical Physics. 4610C+ C 2 O → C 2 (A 3 Π)+ COC Kunz, P Harteck, and S Dondes. Mechanism for Excitation of the C 2 High-Pressure Bands [C+ C 2 O → C 2 (A 3 Π)+ CO]. The Journal of Chemical Physics, 46(10):4157-4158, 1967.\n\nC 2 formation in vibrationally excited CO. Chemical physics letters. Hl Wallaart, M-Y Piar, J-P Perrin, Martin, 246HL Wallaart, B Piar, M-Y Perrin, and J-P Martin. C 2 formation in vibrationally excited CO. Chemical physics letters, 246(6):587-593, 1995.\n\nOn the excitation mechanism of the C 2 high-pressure bands. D W Naegeli, H B Palmer, Journal of Molecular Spectroscopy. 261D.W. Naegeli and H.B. Palmer. On the excitation mechanism of the C 2 high-pressure bands. Journal of Molecular Spectroscopy, 26(1):152 -154, 1968.\n\nStudy of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. F Gosse, J C Sadeghi, Pebay-Peyroula, Chemical Physics Letters. 136F Gosse, N Sadeghi, and JC Pebay-Peyroula. Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. Chemical Physics Letters, 13(6):557-560, 1972.\n\nHeat of Sublimation and Dissociation Energy of Gaseous C 2. Leo Brewer, W T Hicks, O H Krikorian, The Journal of Chemical Physics. 361Leo Brewer, W. T. Hicks, and O. H. Krikorian. Heat of Sublimation and Dissociation Energy of Gaseous C 2 . The Journal of Chemical Physics, 36(1):182-188, 1962.\n\nOrigin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters. P Caubet, G Dorthe, 218P Caubet and G Dorthe. Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters, 218(5-6):529-536, 1994.\n\nOrigin of the high-pressure bands of C 2 . Chemical physics letters. Ce Little, Browne, 134CE Little and PG Browne. Origin of the high-pressure bands of C 2 . Chemical physics letters, 134(6):560-564, 1987.\n\nA new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. H Sasada, T Amano, C Jarman, P F Bernath, The Journal of Chemical Physics. 944H. Sasada, T. Amano, C. Jarman, and P. F. Bernath. A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. The Journal of Chem- ical Physics, 94(4):2401-2407, 1991.\n\nTheoretical investigation of excited states of C 3. A Terentyev, R Scholz, M Schreiber, G Seifert, The Journal of Chemical Physics. 12112A. Terentyev, R. Scholz, M. Schreiber, and G. Seifert. Theoretical investigation of excited states of C 3 . The Journal of Chemical Physics, 121(12):5767-5776, 2004.\n\nThe valence states of c 2 : A configuration interaction study. K Kirby, B Liu, The Journal of Chemical Physics. 702K. Kirby and B. Liu. The valence states of c 2 : A configuration inter- action study. The Journal of Chemical Physics, 70(2):893-900, 1979.\n\nTime-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Wl Faust, Goldberg, Craig Bb, R G Weiss, Chemical Physics Letters. 832WL Faust, LS Goldberg, BB Craig, and RG Weiss. Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Chemical Physics Letters, 83(2):265- 269, 1981.\n\nInvestigation of the Fine Structure of the C 2 High-Pressure Bands. H Meinel, Gerd Messerle, The Astrophysical Journal. 154381H Meinel and Gerd Messerle. Investigation of the Fine Structure of the C 2 High-Pressure Bands. The Astrophysical Journal, 154:381, 1968.\n\nPartition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. S Paul, Remo Barklem, Collet, Astronomy & Astrophysics. 58896Paul S Barklem and Remo Collet. Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. Astronomy & Astrophysics, 588:A96, 2016.\n\nStatistical thermodynamics. Donald Allan Mcquarrie, Harper Collins PublishersDonald Allan McQuarrie. Statistical thermodynamics. Harper Collins Publishers, 1973.\n\nMolecular spectra and molecular structure. Gerhard Herzberg, Spectra of diatomic molecules. New YorkVan Nostrand Reinhold12nd ed.Gerhard Herzberg. Molecular spectra and molecular structure. Vol. 1: Spectra of diatomic molecules. New York: Van Nostrand Reinhold, 1950, 2nd ed., 1950.\n\nAtomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas. S Svanberg, S Svanberg. Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas, 2012.\n\nApplication of the (0,0) Swan band spectrum for temperature measurements. S Pellerin, Musiol, Motret, J Pokrzywka, Chapelle, Journal of Physics D: Applied Physics. 29112850S Pellerin, K Musiol, O Motret, B Pokrzywka, and J Chapelle. Appli- cation of the (0,0) Swan band spectrum for temperature measurements. Journal of Physics D: Applied Physics, 29(11):2850, 1996.\n\nSpectra of atoms and molecules. F Peter, Bernath, Oxford university pressPeter F Bernath. Spectra of atoms and molecules. Oxford university press, 2015.\n\nPGOPHER: A program for simulating rotational, vibrational and electronic spectra. Colin M Western, Journal of Quantitative Spectroscopy and Radiative Transfer. 186Colin M Western. PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:221-242, 2017.\n\nLEVEL: A computer program for solving the radial Schrdinger equation for bound and quasibound levels. J Le Robert, Roy, Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting. 186Robert J. Le Roy. LEVEL: A computer program for solving the ra- dial Schrdinger equation for bound and quasibound levels. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:167 -178, 2017. Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting, October 2015.\n\nHigh resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. G M Lloyd, P Ewart, The Journal of Chemical Physics. 1101G. M. Lloyd and P. Ewart. High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. The Journal of Chemical Physics, 110(1):385-392, 1999.\n\nThe Swan System of C 2 : A Global Analysis of Fourier Transform Emission Spectra. A Tanabashi, T Hirao, T Amano, P F Bernath, The Astrophysical Journal Supplement Series. 1692472A. Tanabashi, T. Hirao, T. Amano, and P. F. Bernath. The Swan Sys- tem of C 2 : A Global Analysis of Fourier Transform Emission Spectra. The Astrophysical Journal Supplement Series, 169(2):472, 2007.\n\nFourier transform emission spectroscopy of the B 2 Σ + -X 2 Σ + system of CN. R S Ram, S P Davis, L Wallace, R Engleman, D R T Appadoo, P F Bernath, Journal of Molecular Spectroscopy. 2372R.S. Ram, S.P. Davis, L. Wallace, R. Engleman, D.R.T. Appadoo, and P.F. Bernath. Fourier transform emission spectroscopy of the B 2 Σ + - X 2 Σ + system of CN. Journal of Molecular Spectroscopy, 237(2):225 - 231, 2006.\n\nLifbase, database and spectral simulation for diatomic molecules. J Luque, 99SRI International ReportJ Luque. Lifbase, database and spectral simulation for diatomic molecules. SRI International Report, 99, 1999.\n\nExtensive analysis of the red system of the CN molecule with a high resolution Fourier Spectrometer. D Cerny, R Bacis, G Guelachvili, F Roux, Journal of Molecular Spectroscopy. 731D. Cerny, R. Bacis, G. Guelachvili, and F. Roux. Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spec- trometer. Journal of Molecular Spectroscopy, 73(1):154 -167, 1978.\n\nAnalysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i. Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, Journal of Molecular Spectroscopy. 1272Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i . Journal of Molecular Spectroscopy, 127(2):283 -303, 1988.\n\nElectronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, The Journal of Chemical Physics. 948Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. The Journal of Chemical Physics, 94(8):5353-5359, 1991.\n\nSpectroscopic studies and rotational and vibrational temperature measurements of atmospheric pressure normal glow plasma discharges in air. David Staack, Bakhtier Farouk, Alexander F Gutsol, Alexander A Fridman, Plasma Sources Science and Technology. 154818David Staack, Bakhtier Farouk, Alexander F Gutsol, and Alexander A Fridman. Spectroscopic studies and rotational and vibrational tem- perature measurements of atmospheric pressure normal glow plasma discharges in air. Plasma Sources Science and Technology, 15(4):818, 2006.\n\nC 2 Swan spectrum used as a molecular pyrometer in transferred arc and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 511112066H Nassar. C 2 Swan spectrum used as a molecular pyrometer in trans- ferred arc and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 511(1):012066, 2014.\n\narXiv:1404.1100Jonathon Shlens. A tutorial on principal component analysis. arXiv preprintJonathon Shlens. A tutorial on principal component analysis. arXiv preprint arXiv:1404.1100, 2014.\n\n. Christophe Laux, Specair, Christophe Laux. SPECAIR, 2012. Accessed: March 2018.\n\nModeling the spectral radiation of a methane/nitrogen plasma: Test case 4. L Caillaut, Andreasson, C O Risberg, Laux, Radiation of High Temperature Gases in Atmospheric Entry. 629L Caillaut, J Andreasson, J Risberg, and CO Laux. Modeling the spec- tral radiation of a methane/nitrogen plasma: Test case 4. In Radiation of High Temperature Gases in Atmospheric Entry, volume 629, 2006.\n\nFundamental data of diatomic molecules relevant for fusion. U Fantz, Wünderlich, Atomic and Plasma-Material Interaction Data for Fusion. 16U Fantz and D Wünderlich. Fundamental data of diatomic molecules relevant for fusion. In Atomic and Plasma-Material Interaction Data for Fusion, volume 16. IAEA, 2011.\n\nOscillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. Damian L Kokkin, George B Bacskay, Timothy W Schmidt, The Journal of Chemical Physics. 126884302Damian L. Kokkin, George B. Bacskay, and Timothy W. Schmidt. Os- cillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. The Journal of Chemical Physics, 126(8):084302, 2007.\n\nMelanie Heusing, and Michael Letzgus. Measurement and simulation of rotationally-resolved chemiluminescence spectra in flames. Andreas Brockhinke, Julia Krüger, Applied Physics B. 1073Andreas Brockhinke, Julia Krüger, Melanie Heusing, and Michael Let- zgus. Measurement and simulation of rotationally-resolved chemilumi- nescence spectra in flames. Applied Physics B, 107(3):539-549, 2012.\n\nRadiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: I. Experiment. D Vacher, Lino Da, P Silva, G André, M Faure, Dudeck, Plasma Sources Science and Technology. 17335012D Vacher, M Lino da Silva, P André, G Faure, and M Dudeck. Ra- diation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spec- tral region: I. Experiment. Plasma Sources Science and Technology, 17(3):035012, 2008.\n\nRadiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. M Lino Da Silva, D Vacher, M Dudeck, P André, G Faure, Plasma Sources Science and Technology. 17335013M Lino da Silva, D Vacher, M Dudeck, P André, and G Faure. Radi- ation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. Plasma Sources Science and Technology, 17(3):035013, 2008.\n\nThe behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. E A Timmermans, J Jonkers, M Rodero, Quintero, Sola, D Gamero, J A Schram, Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 547E.A.H Timmermans, J Jonkers, A Rodero, M.C Quintero, A Sola, A Gamero, D.C Schram, and J.A.M van der Mullen. The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 54(7):1085 -1098, 1999.\n\nThe behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 1. plasmas at atmospheric pressure. E A H Timmermans, J Jonkers, I A J Thomas, A Rodero, M C Quintero, A Sola, A Gamero, J A M Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 5311E.A.H. Timmermans, J. Jonkers, I.A.J. Thomas, A. Rodero, M.C. Quintero, A. Sola, A. Gamero, and J.A.M. van der Mullen. The behav- ior of molecules in microwave-induced plasmas studied by optical emis- sion spectroscopy. 1. plasmas at atmospheric pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 53(11):1553 -1566, 1998.\n\nElucidation of C 2 and CN formation mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Meirong Dong, C.-Y. George, Xianglei Chan, Jhanis J Mao, Jidong Gonzalez, Richard E Lu, Russo, Spectrochimica Acta Part B: Atomic Spectroscopy. 100Dedicated To Nicolo Omenetto On the Occasion of his 75th BirthdayMeirong Dong, George C.-Y. Chan, Xianglei Mao, Jhanis J. Gonzalez, Jidong Lu, and Richard E. Russo. Elucidation of C 2 and CN forma- tion mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Spectrochimica Acta Part B: Atomic Spec- troscopy, 100:62 -69, 2014. Dedicated To Nicolo Omenetto On the Occasion of his 75th Birthday.\n\nCN violet system spectrum used as a molecular pyrometer and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 37012050H Nassar. CN violet system spectrum used as a molecular pyrome- ter and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 370:012050, jun 2012.\n\nTemperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. D B Se Youn Moon, B Kim, W Gweon, Choe, Journal of Applied Physics. 105553307Se Youn Moon, D. B. Kim, B. Gweon, and W. Choe. Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. Journal of Applied Physics, 105(5):053307, 2009.\n\nTorcheà plasmaà excitation micro-onde: deux configurations complémentaires. A Ricard, H St-Onge, Malvos, Gicquel, M Hubert, Moisan, Journal de Physique III. 58A Ricard, L St-Onge, H Malvos, A Gicquel, J Hubert, and M Moisan. Torcheà plasmaà excitation micro-onde: deux config- urations complémentaires. Journal de Physique III, 5(8):1269-1285, 1995.\n\nCn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Ma Ridenti, Amorim, Plasma Chemistry and Plasma Processing. 38MA Ridenti and J Amorim. Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Plasma Chemistry and Plasma Processing, 38(2):311-329, 2018.\n\nRotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Kotaro Saito, Tetsuo Sakka, Yukio H Ogata, Journal of Applied Physics. 949Kotaro Saito, Tetsuo Sakka, and Yukio H. Ogata. Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Journal of Applied Physics, 94(9):5530-5536, 2003.\n\nSpontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Fabien Goulay, László Nemes, Paul E Schrader, Hope A Michelsen, Molecular Physics. 1087-9Fabien Goulay, László Nemes, Paul E Schrader, and Hope A Michelsen. Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Molecular Physics, 108(7-9):1013-1025, 2010.\n\nSequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. V E Bondybey, The Journal of Chemical Physics. 656V. E. Bondybey. Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. The Journal of Chemical Physics, 65(6):2296-2304, 1976.\n\nC 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. R P Frosch, The Journal of Chemical Physics. 546R. P. Frosch. C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. The Journal of Chemical Physics, 54(6):2660-2666, 1971.\n\nThe vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Li Xuechu, Lou Nanquan, Chemical Physics Letters. 751Li Xuechu and Lou Nanquan. The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Chemical Physics Letters, 75(1):110-114, 1980.\n\nC 2 and CN formation by optical pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. William Rich, Richard C Bergman, Chemical Physics. 441J William Rich and Richard C Bergman. C 2 and CN formation by opti- cal pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. Chemical Physics, 44(1):53-64, 1979.\n\nVibrational to electronic energy transfer from CO to C 2 molecules. G Grigorian, Cenian, Chemical Physics Letters. 4694-6G Grigorian and A Cenian. Vibrational to electronic energy transfer from CO to C 2 molecules. Chemical Physics Letters, 469(4-6):247-249, 2009.\n\nTransfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. L Hidde, Bruno Wallaart, Marie-Yvonne Piar, Jean-Pierre Perrin, Martin, Chemical Physics. 1961Hidde L. Wallaart, Bruno Piar, Marie-Yvonne Perrin, and Jean-Pierre Martin. Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. Chemical Physics, 196(1):149 -169, 1995.\n\nFormation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. W Hack, Chemical Physics Letters. 813W Hack and W Langel. Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. Chemical Physics Letters, 81(3):387 -390, 1981.\n\nPressure Dependence of Rotational Intensity Distribution in C 2 Swan Emission from LowPressure Oxyacetylene Flames. R Bleekrode, The Journal of Chemical Physics. 458R. Bleekrode. Pressure Dependence of Rotational Intensity Distribu- tion in C 2 Swan Emission from LowPressure Oxyacetylene Flames. The Journal of Chemical Physics, 45(8):3153-3154, 1966.\n\nEnergy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, Katharina Kohse-Höinghaus, The Journal of Physical Chemistry A. 1109Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, and Katharina Kohse-Höinghaus. Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. The Journal of Physical Chemistry A, 110(9):3028-3035, 2006.\n\nG Grigorian, A Cenian, Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing. 31G. Grigorian and A. Cenian. Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing, 31(2):337-352, Apr 2011.\n\nState-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2. Jingzhong Guo, C M Sadowski, Q Gao, F J Morgan, The Journal of Chemical Physics. 11317Jingzhong Guo, C. M. Sadowski, Q. Gao, and F. J. Morgan. State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2 . The Journal of Chemical Physics, 113(17):7276-7285, 2000.\n\nState-to-state and total rotational energy transfer rate constants for CN. M K Sophie, Jingzhong Brunet, Tucker Guo, S V Carrington, C M Filseth, Sadowski, N)+NO. The Journal of Chemical Physics. 1169Sophie M. K. Brunet, Jingzhong Guo, Tucker Carrington, S. V. Filseth, and C. M. Sadowski. State-to-state and total rotational energy trans- fer rate constants for CN(B 2 Σ + ,v=0,N)+H 2 , CN(X 2 Σ + ,v=2,N)+H 2 , D 2 , and CN(X 2 Σ + ,v=3,N)+NO. The Journal of Chemical Physics, 116(9):3617-3625, 2002.\n\nSpectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave induced nitrogen plasma torch. Chuan-Jie Chen, Shou-Zhe Li, Plasma Sources Science and Technology. 24335017Chuan-Jie Chen and Shou-Zhe Li. Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave in- duced nitrogen plasma torch. Plasma Sources Science and Technology, 24(3):035017, 2015.\n\nOptical characterization of a microwave pulsed discharge used for dissociation of CO 2. Tiago Silva, Nikolay Britun, Thomas Godfroid, Rony Snyders, Plasma Sources Science and Technology. 23225009Tiago Silva, Nikolay Britun, Thomas Godfroid, and Rony Snyders. Op- tical characterization of a microwave pulsed discharge used for dissoci- ation of CO 2 . Plasma Sources Science and Technology, 23(2):025009, 2014.\n\nThe dissociation of carbon monoxide. A R Fairbairn, Alfred Gordon Gaydon, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 312A. R. Fairbairn and Alfred Gordon Gaydon. The dissociation of carbon monoxide. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 312(1509):207-227, 1969.\n\nCoherent anti-Stokes Raman scattering and spontaneous Raman scattering diagnostics of nonequilibrium plasmas and flows. R Walter, Igor V Lempert, Adamovich, 2014-10-29T00:00:00Journal of Physics D: Applied Physics. 4743Walter R. Lempert and Igor V. Adamovich. Coherent anti-Stokes Ra- man scattering and spontaneous Raman scattering diagnostics of non- equilibrium plasmas and flows. Journal of Physics D: Applied Physics, 47(43):433001-433026, 2014-10-29T00:00:00.\n\nHow the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. D C M Van Den, J M Bekerom, E M Linares, S Van Veldhuizen, M C M Nijdam, G J Van De Sanden, Van Rooij, Appl. Opt. 5720D. C. M. van den Bekerom, J. M. Palomares Linares, E. M. van Veld- huizen, S. Nijdam, M. C. M. van de Sanden, and G. J. van Rooij. How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. Appl. Opt., 57(20):5694- 5702, Jul 2018.\n"},"corpus_id":{"kind":"number","value":208513924,"string":"208,513,924"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1911.13121\",\n \"mag\": 3041034824,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/1361-6595/ab74b4\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1911.13121v2.pdf\"\n ],\n \"pdf_sha\": \"40e745d72e45e6c04d9881618772a61fbbbc9a83\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://doi.org/10.1088/1361-6595/ab74b4\",\n \"status\": \"HYBRID\"\n }\n}"},"title":{"kind":"string","value":"Analysis of the C 2 (d 3 Π g -a 3 Π u ) Swan bands as a thermometric probe in CO 2 microwave plasmas"},"authors":{"kind":"list like","value":["Emile Carbone \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany","Federico D'isa \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany","Ante Hecimovic \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany","Ursel Fantz \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany","Emile Carbone \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany","Federico D'isa \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany","Ante Hecimovic \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany","Ursel Fantz \nMax Planck Institute for Plasma Physics\nBoltzmannstr. 2D-85748GarchingGermany"],"string":"[\n \"Emile Carbone \\nMax Planck Institute for Plasma Physics\\nBoltzmannstr. 2D-85748GarchingGermany\",\n \"Federico D'isa \\nMax Planck Institute for Plasma Physics\\nBoltzmannstr. 2D-85748GarchingGermany\",\n \"Ante Hecimovic \\nMax Planck Institute for Plasma Physics\\nBoltzmannstr. 2D-85748GarchingGermany\",\n \"Ursel Fantz \\nMax Planck Institute for Plasma Physics\\nBoltzmannstr. 2D-85748GarchingGermany\",\n \"Emile Carbone \\nMax Planck Institute for Plasma Physics\\nBoltzmannstr. 2D-85748GarchingGermany\",\n \"Federico D'isa \\nMax Planck Institute for Plasma Physics\\nBoltzmannstr. 2D-85748GarchingGermany\",\n \"Ante Hecimovic \\nMax Planck Institute for Plasma Physics\\nBoltzmannstr. 2D-85748GarchingGermany\",\n \"Ursel Fantz \\nMax Planck Institute for Plasma Physics\\nBoltzmannstr. 2D-85748GarchingGermany\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The optical emission spectra of high pressure CO 2 microwave plasmas are usually dominated by the C 2 Swan bands. In this paper, the use of the C 2 Swan bands for estimating the gas temperature in CO 2 microwave plasmas is assessed. State by state fitting is employed to check the correctness of assuming a Boltzmann distribution for the rotational and vibrational distribution functions and, within statistical and systematic uncertainties, the C 2 Swan band can be fitted accurately with a single temperature for rotational and vibrational levels. The processes leading to the production of the C 2 molecule and particularly its d 3 Π g state are briefly reviewed as well as collisional relaxation times of the latter. It is concluded that its rotational temperature can be associated to the gas temperature of the CO 2 microwave plasma and the results are moreover cross-checked by adding a small amount of N 2 in the discharge and measuring the CN violet band system. The 2.45 GHz plasma source is analyzed in the pressure range 180-925 mbar, for input microwave powers ranging from 0.9 -3 kW and with gas flow rates of 5-100 L/min. An intense C 2 Swan bands emission spectrum can be measured only when the plasma is operated in contracted regime. A unique temperature of about 6000 ± 500 K is obtained for all investigated conditions. A spectroscopic database is constructed using the recent compilation and calculations by Brooke et al.[1] of the line strengths and molecular constants for the C 2 (d 3 Π g -a 3 Π u ) Swan bands system and made available as 1 arXiv:1911.13121v2 [physics.plasm-ph] 29 Apr 2020 Supplementary Material in a format compatible with the open source MassiveOES software."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":""},{"title":"Introduction","content":"Many strategies are being currently studied for the recycling and valorization of CO 2 , a potent greenhouse gas in the atmosphere [2]. Production of liquid fuels from renewable energies is one of the pathways for re-utilization of CO 2 . The first step for converting CO 2 into added value chemicals is its conversion into carbon monoxide CO [3,4]. For practical applications, it is desirable to operate with high gas flows near atmospheric pressure (or above) in order to avoid bulky components, reduce costs and save energy for the compression steps [5]. Previous studies have indicated that non-equilibrium microwave plasmas are particularly suitable for high energy efficiencies η conversion of CO 2 into CO with η ≤ 80% and can operate at high pressure and with high flow rates [6]. CO 2 based plasmas have applications in fields such as lasers [7], plasma polymers treatment [8] and also have potential in plasma medicine [9]. The proposed mechanism for explaining high energy efficiencies for CO 2 conversion into CO is via the so-called vibrational ladder climbing mechanism. Relatively low energy electrons excite preferentially the assymetric vibrational modes of the CO 2 ground state CO 2 (00i) molecule and vibrational energy exchange between CO 2 molecules leads ultimately to their decomposition [10]. To favor this mechanism the key ingredient is to keep the plasma out of equilibrium with the gas temperature T g much smaller than the effective electron temperature T e [11,12,13]. Low gas temperatures (i.e. T g 3000 K) are required in order to avoid too many losses of CO 2 (00i) states by collisions with the background gas leading to gas heating by vibrational to translation relaxation processes [14,15]. For high energy efficiencies, calculations also indicate that formed O atoms need to react with highly vibrationally excited CO 2 molecules and to produce another CO molecule and a O 2 molecule [16]. On the other hand, at high gas temperatures (i.e. T gas ≥ 3000 K), thermal dissociation is the dominant mechanism for CO 2 dissociation with high conversion rates and is limited to energy efficiencies up to ∼ 50 % [17,18].\nMany strategies are being currently studied for the recycling and valorization of CO 2 , a potent greenhouse gas in the atmosphere [2]. Production of liquid fuels from renewable energies is one of the pathways for re-utilization of CO 2 . The first step for converting CO 2 into added value chemicals is its conversion into carbon monoxide CO [3,4]. For practical applications, it is desirable to operate with high gas flows near atmospheric pressure (or above) in order to avoid bulky components, reduce costs and save energy for the compression steps [5]. Previous studies have indicated that non-equilibrium microwave plasmas are particularly suitable for high energy efficiencies η conversion of CO 2 into CO with η ≤ 80% and can operate at high pressure and with high flow rates [6]. CO 2 based plasmas have applications in fields such as lasers [7], plasma polymers treatment [8] and also have potential in plasma medicine [9]. The proposed mechanism for explaining high energy efficiencies for CO 2 conversion into CO is via the so-called vibrational ladder climbing mechanism. Relatively low energy electrons excite preferentially the assymetric vibrational modes of the CO 2 ground state CO 2 (00i) molecule and vibrational energy exchange between CO 2 molecules leads ultimately to their decomposition [10]. To favor this mechanism the key ingredient is to keep the plasma out of equilibrium with the gas temperature T g much smaller than the effective electron temperature T e [11,12,13]. Low gas temperatures (i.e. T g 3000 K) are required in order to avoid too many losses of CO 2 (00i) states by collisions with the background gas leading to gas heating by vibrational to translation relaxation processes [14,15]. For high energy efficiencies, calculations also indicate that formed O atoms need to react with highly vibrationally excited CO 2 molecules and to produce another CO molecule and a O 2 molecule [16]. On the other hand, at high gas temperatures (i.e. T gas ≥ 3000 K), thermal dissociation is the dominant mechanism for CO 2 dissociation with high conversion rates and is limited to energy efficiencies up to ∼ 50 % [17,18].\nFor studying the mechanisms limiting the plasma energy efficiency of the plasma, it is critical to have tools for assessing the degree of non-equilibrium in the plasma phase and notably its gas temperature. In high pressure (i.e. sub-up to atmospheric pressure) microwave CO 2 plasmas, one of the most dominant emission feature is the C 2 (d 3 Π g -a 3 Π u ) Swan bands system. In figure 1 an overview spectrum of a pure CO 2 microwave plasma (see section 2 for details on the plasma setup) is shown. In addition to a weak continuum emission 1 and typical atomic oxygen and carbon lines 2 , the C 2 Swan bands are the dominant molecular emission structure in the UV and visible range. Such spectrum is typical and has been measured by several authors [16,21,22,23,24,25,26,27,28]. To obtain reliable and accurate information about the plasma, it is then critical to have tools for evaluating the emission spectrum of the C 2 molecule and assess the accuracy of the results. Usually an equivalence between the rotational temperature of emitting species and the gas temperature is assumed but this hypothesis needs to be assessed. Also, in certain cases non-Boltzmann distributed rotational distribution function are measured experimentally [29]. In combustion studies, it has been observed that the C 2 Swan band can exhibit overpopulation of the high vibrational levels. The latter emission spectrum is known as the \"high pressure bands\"of the C 2 system [30]. This makes the analysis of C 2 spectra more complicated and can induce systematic errors in the determination of the rotational temperature (T rot ) of the C 2 (d 3 Π g ) state if the vibrational levels are assumed to take a Boltzmann distribution function. For CO 2 plasmas, an evaluation of potential deviations of C 2 Swan bands spectra from Boltzmann equilibrium and the assessment of T rot as a measure of the gas temperature have not been reported in the literature yet.\nFor studying the mechanisms limiting the plasma energy efficiency of the plasma, it is critical to have tools for assessing the degree of non-equilibrium in the plasma phase and notably its gas temperature. In high pressure (i.e. sub-up to atmospheric pressure) microwave CO 2 plasmas, one of the most dominant emission feature is the C 2 (d 3 Π g -a 3 Π u ) Swan bands system. In figure 1 an overview spectrum of a pure CO 2 microwave plasma (see section 2 for details on the plasma setup) is shown. In addition to a weak continuum emission 1 and typical atomic oxygen and carbon lines 2 , the C 2 Swan bands are the dominant molecular emission structure in the UV and visible range. Such spectrum is typical and has been measured by several authors [16,21,22,23,24,25,26,27,28]. To obtain reliable and accurate information about the plasma, it is then critical to have tools for evaluating the emission spectrum of the C 2 molecule and assess the accuracy of the results. Usually an equivalence between the rotational temperature of emitting species and the gas temperature is assumed but this hypothesis needs to be assessed. Also, in certain cases non-Boltzmann distributed rotational distribution function are measured experimentally [29]. In combustion studies, it has been observed that the C 2 Swan band can exhibit overpopulation of the high vibrational levels. The latter emission spectrum is known as the \"high pressure bands\"of the C 2 system [30]. This makes the analysis of C 2 spectra more complicated and can induce systematic errors in the determination of the rotational temperature (T rot ) of the C 2 (d 3 Π g ) state if the vibrational levels are assumed to take a Boltzmann distribution function. For CO 2 plasmas, an evaluation of potential deviations of C 2 Swan bands spectra from Boltzmann equilibrium and the assessment of T rot as a measure of the gas temperature have not been reported in the literature yet.\nIn this paper we present the spectral fitting analysis of the C 2 Swan band using a newly implemented spectroscopic database in the open source MassiveOES software [31,32]. The database is constructed using the recent compilation and calculations by Brooke et al. [1] of the line strengths and molecular constants for the C 2 (d 3 Π g -a 3 Π u ) Swan band system. C 2 Swan bands spectra measured on a 2.45 GHz CO 2 microwave plasma source are reported and carefully analyzed. By adding small amount of nitrogen in the plasma, the obtained temperatures from the Swan band fitting are compared with the one obtained by fitting the CN violet band system. Uncertainties related to the method used for fitting the C 2 spectrum, potential deviations from a Boltzmann distribution for the rotational and vibrational distribution functions and the assumption of fitting the spectra with a single or two temperatures (i.e. rotational and vibrational temperatures) are discussed and compared with a state-by-state fitting of the experimental spectra. Finally, currrent knowledge about rotational and vibrational relaxation processes of the C 2 (d 3 Π g ) state is presented in section 6 and the use of T rot for measuring the gas temperature of CO 2 microwave plasmas is discussed.\nIn this paper we present the spectral fitting analysis of the C 2 Swan band using a newly implemented spectroscopic database in the open source MassiveOES software [31,32]. The database is constructed using the recent compilation and calculations by Brooke et al. [1] of the line strengths and molecular constants for the C 2 (d 3 Π g -a 3 Π u ) Swan band system. C 2 Swan bands spectra measured on a 2.45 GHz CO 2 microwave plasma source are reported and carefully analyzed. By adding small amount of nitrogen in the plasma, the obtained temperatures from the Swan band fitting are compared with the one obtained by fitting the CN violet band system. Uncertainties related to the method used for fitting the C 2 spectrum, potential deviations from a Boltzmann distribution for the rotational and vibrational distribution functions and the assumption of fitting the spectra with a single or two temperatures (i.e. rotational and vibrational temperatures) are discussed and compared with a state-by-state fitting of the experimental spectra. Finally, currrent knowledge about rotational and vibrational relaxation processes of the C 2 (d 3 Π g ) state is presented in section 6 and the use of T rot for measuring the gas temperature of CO 2 microwave plasmas is discussed."},{"title":"Experimental setup","content":""},{"title":"Experimental setup","content":"A microwave device which can ignite an atmospheric pressure plasma torch using only microwave power without the use of additional igniters [33] is studied. A modified version of this device has been built so that it can be operated from low (∼10 mbar) up to atmospheric pressure in molecular gases such as N 2 and CO 2 . Stable operation of the plasma is obtained both in continuous and pulsed regime in a large parameters range. The microwaves are coupled via a WR340 waveguide (TE 01 mode) into a cylindrical cavity with low Q-factor where the plasma is sustained inside a quartz tube (see figure 2 for more details). The quartz tube has inner/outer diameters of 26/30 mm. Figure 1: Overview spectra using a low resolution spectrometer (instrumental profile of about 1.2 nm) of the plasma emission at low and high pressures. The high pressure spectrum is characteristic for all the conditions of the present study where the plasma is in contracted mode which occurs for power ≥ 900 W at pressures above 120 mbar. Note that the spectra are not calibrated for the sensitivity response of the spectrometer and optics.\nA microwave device which can ignite an atmospheric pressure plasma torch using only microwave power without the use of additional igniters [33] is studied. A modified version of this device has been built so that it can be operated from low (∼10 mbar) up to atmospheric pressure in molecular gases such as N 2 and CO 2 . Stable operation of the plasma is obtained both in continuous and pulsed regime in a large parameters range. The microwaves are coupled via a WR340 waveguide (TE 01 mode) into a cylindrical cavity with low Q-factor where the plasma is sustained inside a quartz tube (see figure 2 for more details). The quartz tube has inner/outer diameters of 26/30 mm. Figure 1: Overview spectra using a low resolution spectrometer (instrumental profile of about 1.2 nm) of the plasma emission at low and high pressures. The high pressure spectrum is characteristic for all the conditions of the present study where the plasma is in contracted mode which occurs for power ≥ 900 W at pressures above 120 mbar. Note that the spectra are not calibrated for the sensitivity response of the spectrometer and optics.\nFor the ignition of the plasma, a coaxial cavity made of a waveguide to cylindrical resonator with high Q-factor and an inner conductor with conical shape is integrated at the bottom of the quartz tube. The coaxial cavity serves also for the injection of the gases using a 4-inlets tangential gas injection system. The coaxial conical pin vertical position can be adjusted for selecting different modes and enhance the electric field at its tip to ignite the plasma. The plasma can be operated in the power range of 600 W-3 kW, with gas flow rates between 5-100 L/min and pressures from 10 up to 960 mbar.\nFor the ignition of the plasma, a coaxial cavity made of a waveguide to cylindrical resonator with high Q-factor and an inner conductor with conical shape is integrated at the bottom of the quartz tube. The coaxial cavity serves also for the injection of the gases using a 4-inlets tangential gas injection system. The coaxial conical pin vertical position can be adjusted for selecting different modes and enhance the electric field at its tip to ignite the plasma. The plasma can be operated in the power range of 600 W-3 kW, with gas flow rates between 5-100 L/min and pressures from 10 up to 960 mbar.\nThe high resolution measurements were acquired using a Czerny-Turner 1 m focal length SPEX1000 spectrometer with an entrance slit of about 30 µm and a dispersion grating of 1800 g/mm. An Andor iDus 420 CCD camera was used to record the spectra. The spectral resolution in first order is 20 pm at 550 nm. The wavelength calibration of the spectrometer was done using a mercury lamp. The light is collected using a system of two irises and a lens. The lens focuses the light into an optical fiber system that is used for coupling the light to the entrance slit of the spectrometer. The irises system acts as a collimator and a spatial resolution of 1 mm is obtained. The collecting optics, filters and the spectrometer have been absolute calibrated between 400 nm to 850 nm with an Ulbricht sphere to obtain the absolute photon flux from the plasma. A longpass filter at 400 nm (FEL0400 from Thorlabs) is used to suppress any contribution from second order emission overlapping with the Swan band emission. For the CN violet system measurements, a 280 nm longpass filter (FEL280 from Thorlabs) is used. Additional low-resolution survey spectra are acquired using an Ocean Optics S2000 spectrometer with 1.2 nm resolution.\nThe high resolution measurements were acquired using a Czerny-Turner 1 m focal length SPEX1000 spectrometer with an entrance slit of about 30 µm and a dispersion grating of 1800 g/mm. An Andor iDus 420 CCD camera was used to record the spectra. The spectral resolution in first order is 20 pm at 550 nm. The wavelength calibration of the spectrometer was done using a mercury lamp. The light is collected using a system of two irises and a lens. The lens focuses the light into an optical fiber system that is used for coupling the light to the entrance slit of the spectrometer. The irises system acts as a collimator and a spatial resolution of 1 mm is obtained. The collecting optics, filters and the spectrometer have been absolute calibrated between 400 nm to 850 nm with an Ulbricht sphere to obtain the absolute photon flux from the plasma. A longpass filter at 400 nm (FEL0400 from Thorlabs) is used to suppress any contribution from second order emission overlapping with the Swan band emission. For the CN violet system measurements, a 280 nm longpass filter (FEL280 from Thorlabs) is used. Additional low-resolution survey spectra are acquired using an Ocean Optics S2000 spectrometer with 1.2 nm resolution.\n3 Sources of C 2 and of the C 2 (d 3 Π g ) state\n3 Sources of C 2 and of the C 2 (d 3 Π g ) state\nThe primary building blocks of C 2 are C atoms and it is therefore of interest to look first at sources of C atoms. No direct production channel from the CO 2 molecule is known by electron impact dissociation [34,35]. Electron impact dissociation processes of CO 2 lead to O and CO fragments either as neutral (ground or excited states). Electron dissociative attachment to CO 2 peaks at low electron energies < 10 eV and only produces CO [36] and O − . Electron impact ionization of CO 2 leading to formation of C + has a threshold cross section of 27.8 eV (compared to 13.8 eV for CO + 2 ) according to Itikawa [37] and can therefore be considered as negligible while considering typical electron energy distribution function in CO 2 plasmas [38]. Dissociative recombination of CO + 2 lead mostly to CO and O products and a small fraction of C atoms [39]. However, CO + 2 is lost by charge exchange and clustering reactions and is not a dominant ionic species in the plasma [40]. CO 2 species cannot then be considered as a primary source of C atoms and CO can be regarded as its primary source.\nThe primary building blocks of C 2 are C atoms and it is therefore of interest to look first at sources of C atoms. No direct production channel from the CO 2 molecule is known by electron impact dissociation [34,35]. Electron impact dissociation processes of CO 2 lead to O and CO fragments either as neutral (ground or excited states). Electron dissociative attachment to CO 2 peaks at low electron energies < 10 eV and only produces CO [36] and O − . Electron impact ionization of CO 2 leading to formation of C + has a threshold cross section of 27.8 eV (compared to 13.8 eV for CO + 2 ) according to Itikawa [37] and can therefore be considered as negligible while considering typical electron energy distribution function in CO 2 plasmas [38]. Dissociative recombination of CO + 2 lead mostly to CO and O products and a small fraction of C atoms [39]. However, CO + 2 is lost by charge exchange and clustering reactions and is not a dominant ionic species in the plasma [40]. CO 2 species cannot then be considered as a primary source of C atoms and CO can be regarded as its primary source.\nElectron impact dissociation and recombination processes of CO and CO + have been extensively reviewed (see for instance [42,39]) and we will focus here on neutral processes leading to the formation of C atoms or even directly to C 2 molecules. A reason for focusing on these processes is also the knowledge that, in high pressure discharges with high degree of vibrational excitation, neutral processes are expected to play an important role for molecules dissociation [6]. Moreover, in section 7 are reported high gas temperatures for which thermal (i.e. neutral) processes dominates dissociation dynamics [18].\nElectron impact dissociation and recombination processes of CO and CO + have been extensively reviewed (see for instance [42,39]) and we will focus here on neutral processes leading to the formation of C atoms or even directly to C 2 molecules. A reason for focusing on these processes is also the knowledge that, in high pressure discharges with high degree of vibrational excitation, neutral processes are expected to play an important role for molecules dissociation [6]. Moreover, in section 7 are reported high gas temperatures for which thermal (i.e. neutral) processes dominates dissociation dynamics [18].\nTo understand the formation of the C 2 d 3 Π g state, as it will be discussed in the following, not only source terms of C atoms but also of the C 2 O should be considered. Vibrationally excited CO molecules lead to the formation of carbon atoms via thermal decomposition [43] or via the so-called Boudouard mechanism [44]\nTo understand the formation of the C 2 d 3 Π g state, as it will be discussed in the following, not only source terms of C atoms but also of the C 2 O should be considered. Vibrationally excited CO molecules lead to the formation of carbon atoms via thermal decomposition [43] or via the so-called Boudouard mechanism [44]\nwhere v and w are vibrational quantum numbers. 3 This process is very 3 Alternatively of having two CO molecules in their ground state X 1 Σ g in the reaction 1, Figure 3: Schematic representation of the potential energy curves of the lower triplet electronic states of the C 2 molecule as well as of the 1 5 Π g state and their dissociation limit. For more information see ref. [41]. efficient and researchers have reported that discharges into carbon monoxide even at pressures of a few Torr lead to deposition of solid carbon on the walls and a decrease of the pressure in the closed cell [46] . A competing process to reaction 1 is the formation of a C 2 O molecule [45,47] CO(\nwhere v and w are vibrational quantum numbers. 3 This process is very 3 Alternatively of having two CO molecules in their ground state X 1 Σ g in the reaction 1, Figure 3: Schematic representation of the potential energy curves of the lower triplet electronic states of the C 2 molecule as well as of the 1 5 Π g state and their dissociation limit. For more information see ref. [41]. efficient and researchers have reported that discharges into carbon monoxide even at pressures of a few Torr lead to deposition of solid carbon on the walls and a decrease of the pressure in the closed cell [46] . A competing process to reaction 1 is the formation of a C 2 O molecule [45,47] CO(\nSuch processes are efficient only when significant vibrational excitation of CO molecules is present and/or high gas temperatures. This is the case in the present study with reported temperatures of 6000 K (see section 7) where CO 2 is thermally dissociated, and production of C atoms happens in the gas phase [6].\nSuch processes are efficient only when significant vibrational excitation of CO molecules is present and/or high gas temperatures. This is the case in the present study with reported temperatures of 6000 K (see section 7) where CO 2 is thermally dissociated, and production of C atoms happens in the gas phase [6].\nThe formation of the C 2 molecule from the three lowest electronic configurations of the carbon atom which are the 2s 2 2p 2 ( 3 P J , 1 D 2 and 1 S 0 ) states, where the ground state is the 3 P 0 state, was discussed by Ballik and Ramsay [48] while considering the potential energy curves of C 2 . In figure 3 are shown the lower triplet electronic levels of the C 2 molecule. For a recent overview of the potential energy curves of the C 2 molecule we refer to Martin [41]. The one of the species reacting can be a CO(a 3 Π,v) excited state formed from energy exchange between two CO (X 1 Σ g , v) molecules [45].\nThe formation of the C 2 molecule from the three lowest electronic configurations of the carbon atom which are the 2s 2 2p 2 ( 3 P J , 1 D 2 and 1 S 0 ) states, where the ground state is the 3 P 0 state, was discussed by Ballik and Ramsay [48] while considering the potential energy curves of C 2 . In figure 3 are shown the lower triplet electronic levels of the C 2 molecule. For a recent overview of the potential energy curves of the C 2 molecule we refer to Martin [41]. The one of the species reacting can be a CO(a 3 Π,v) excited state formed from energy exchange between two CO (X 1 Σ g , v) molecules [45].\nwhere M stands for a third body that stabilizes the transition state. They noted that there is an avoided crossing with the C 2 (e 3 Π g ) state that dissociate into C ( 3 P) + C( 1 D) and so that the C( 1 D) does not recombine into the C 2 (d 3 Π g ) state. The C 2 Swan band can then potentially be used as marker for the recombination of ground state C atoms.\nwhere M stands for a third body that stabilizes the transition state. They noted that there is an avoided crossing with the C 2 (e 3 Π g ) state that dissociate into C ( 3 P) + C( 1 D) and so that the C( 1 D) does not recombine into the C 2 (d 3 Π g ) state. The C 2 Swan band can then potentially be used as marker for the recombination of ground state C atoms.\nIn high pressure discharges, a preferential emission of the C 2 (d 3 Π g , v = 6) state can be observed which is known as the \"C 2 high pressure band\". C 2 O molecule was proposed by Kunz et al. [49] for explaining the preferential emission of the C 2 (d 3 Π g , v = 6) state. The C 2 O molecule 5 reacts with a C atom and forms an electronically excited C 2 molecule [50]\nIn high pressure discharges, a preferential emission of the C 2 (d 3 Π g , v = 6) state can be observed which is known as the \"C 2 high pressure band\". C 2 O molecule was proposed by Kunz et al. [49] for explaining the preferential emission of the C 2 (d 3 Π g , v = 6) state. The C 2 O molecule 5 reacts with a C atom and forms an electronically excited C 2 molecule [50]\nNaegeli and Palmer proposed that the initial electronic state in reaction 4 is the C 2 (b 3 Σ − g ) state that crosses to the C 2 (d 3 Π g ) state at v=6 and leads to the high pressure C 2 band [51]. Through a kinetic study of a CO plasma afterglow, Gosse et al. [52] supported the hypothesis that C 2 (d 3 Π g ) state is formed via reaction 4. Brewer et al [53] stated that formation of the high pressure band from a predissociation mechanism is unlikely and suggested that a resonant transfer from another molecule happens. However, Caubet and Dorthe [54], C 2 O may also be a source of the 5 Π g state via\nNaegeli and Palmer proposed that the initial electronic state in reaction 4 is the C 2 (b 3 Σ − g ) state that crosses to the C 2 (d 3 Π g ) state at v=6 and leads to the high pressure C 2 band [51]. Through a kinetic study of a CO plasma afterglow, Gosse et al. [52] supported the hypothesis that C 2 (d 3 Π g ) state is formed via reaction 4. Brewer et al [53] stated that formation of the high pressure band from a predissociation mechanism is unlikely and suggested that a resonant transfer from another molecule happens. However, Caubet and Dorthe [54], C 2 O may also be a source of the 5 Π g state via\nThey argued that based on the correlation diagram for the reactants and products of the reaction C 2 O + C that the 5 Π g state of C 2 is correlated to the ground state of the reactants while the d 3 Π g state is not. We note that previously, because of the absence of observation of C 2 O by absorption, Kini 4 In Ballik and Ramsay this state is denoted as the A 3 Π g state which is a common notation found for the Swan band in the literature with the lower state of the transition defined as X 3 Π u . 5 Note that in addition to reaction 2, the C 2 O molecule can be formed from the recombination of a C atom with CO via\nThey argued that based on the correlation diagram for the reactants and products of the reaction C 2 O + C that the 5 Π g state of C 2 is correlated to the ground state of the reactants while the d 3 Π g state is not. We note that previously, because of the absence of observation of C 2 O by absorption, Kini 4 In Ballik and Ramsay this state is denoted as the A 3 Π g state which is a common notation found for the Swan band in the literature with the lower state of the transition defined as X 3 Π u . 5 Note that in addition to reaction 2, the C 2 O molecule can be formed from the recombination of a C atom with CO via\nThe formation of C 2 O in pulsed radiolysis of CO was shown first by Willis and Devillers [47] and they showed that adding CO 2 did not affect the kinetics of C 2 O. Kinetics of this molecule discussed from kinetic analysis of pure CO discharges can then probably be extended to mixtures of CO/CO 2 . and Savadatti [46] proposed the following mechanism for the formation of the high pressure C 2 (d 3 Π g , v = 6) Swan band\nThe formation of C 2 O in pulsed radiolysis of CO was shown first by Willis and Devillers [47] and they showed that adding CO 2 did not affect the kinetics of C 2 O. Kinetics of this molecule discussed from kinetic analysis of pure CO discharges can then probably be extended to mixtures of CO/CO 2 . and Savadatti [46] proposed the following mechanism for the formation of the high pressure C 2 (d 3 Π g , v = 6) Swan band\nwhere the C 3 (m) molecule is a postulated metastable species with an energy of 3.7 eV. Little and Browne [55] argued against the possibility of the C 3 as intermediate species for the formation of the high pressure Swan band. Indeed, the C 2 high pressure band is observed in a large range of experimental conditions where C 3 formation is not favored and they also advanced a lack of evidence for a metastable state that should have an energy above thẽ A 1 Π u state which has 3.1 eV internal energy and is responsible for the C 3 comet band (Ã 1 Π u -X 1 Σ + g transition). Recent spectroscopic investigation and calculations of the energy levels of the C 3 molecule indicate that the lowest triplet state C 3 (ã 3 Π u ) has an energy of about 2.1 eV and theb 3 Π g an energy of 2.9 eV [56,57] which are well below the 3.7 eV energy required for the process 6. The C 3 molecule can therefore be ruled out as source of the C 2 high pressure band.\nwhere the C 3 (m) molecule is a postulated metastable species with an energy of 3.7 eV. Little and Browne [55] argued against the possibility of the C 3 as intermediate species for the formation of the high pressure Swan band. Indeed, the C 2 high pressure band is observed in a large range of experimental conditions where C 3 formation is not favored and they also advanced a lack of evidence for a metastable state that should have an energy above thẽ A 1 Π u state which has 3.1 eV internal energy and is responsible for the C 3 comet band (Ã 1 Π u -X 1 Σ + g transition). Recent spectroscopic investigation and calculations of the energy levels of the C 3 molecule indicate that the lowest triplet state C 3 (ã 3 Π u ) has an energy of about 2.1 eV and theb 3 Π g an energy of 2.9 eV [56,57] which are well below the 3.7 eV energy required for the process 6. The C 3 molecule can therefore be ruled out as source of the C 2 high pressure band.\nLittle and Browne [55] proposed a general mechanism for the formation of the high pressure Swan band via the following three steps\nLittle and Browne [55] proposed a general mechanism for the formation of the high pressure Swan band via the following three steps\nwhere the quintet 5 Π g is a metastable state for which a crossing exists between the v=6 level of the d 3 Π g state and the v=0 level of the quintet state. Kirby and Liu calculated its energy to lies 3.85 ± 0.34 eV above the ground state of C 2 [58] while Little and Browne estimated a value of 3.79 eV. Such mechanism essentially allow to explain previous observation that the high pressure band is rotationally colder than the \"normal\" Swan band emission that can be observed in the same discharge [59]. Also, high resolution analysis of the C 2 (d 3 Π g , v = 6) low rotation energy levels shows that both odd and even rotational numbers are perturbed with increase of intensity [60] which indicates that a Σ state is not involved in the perturbation but a state with Λ = 0 fills the lower rotational levels of the C 2 (d 3 Π g , v = 6) state [54]. A detailed analysis shows in fact that the b 3 Σ − g state perturb the J=19 and 21 of the v=6 upper state of the Swan band but lead to a decrease of the intensity of these lines [55]. It can be concluded that the b 3 Σ − g state is not at the origin of the high pressure band and that the quintet 5 Π g state is likely its source.\nwhere the quintet 5 Π g is a metastable state for which a crossing exists between the v=6 level of the d 3 Π g state and the v=0 level of the quintet state. Kirby and Liu calculated its energy to lies 3.85 ± 0.34 eV above the ground state of C 2 [58] while Little and Browne estimated a value of 3.79 eV. Such mechanism essentially allow to explain previous observation that the high pressure band is rotationally colder than the \"normal\" Swan band emission that can be observed in the same discharge [59]. Also, high resolution analysis of the C 2 (d 3 Π g , v = 6) low rotation energy levels shows that both odd and even rotational numbers are perturbed with increase of intensity [60] which indicates that a Σ state is not involved in the perturbation but a state with Λ = 0 fills the lower rotational levels of the C 2 (d 3 Π g , v = 6) state [54]. A detailed analysis shows in fact that the b 3 Σ − g state perturb the J=19 and 21 of the v=6 upper state of the Swan band but lead to a decrease of the intensity of these lines [55]. It can be concluded that the b 3 Σ − g state is not at the origin of the high pressure band and that the quintet 5 Π g state is likely its source.\nThe proposal that C atoms recombine via the quintet 5 Π g state which then populate the C 2 (d 3 Π g , v = 6) state (i.e. to generate the C 2 high pressure band) does not exclude the direct recombination of C atoms into the d 3 Π g state. One should see them as processes that happen simultaneously with different probabilities. Looking at the energy potential curves of the C 2 molecules, one can see that the\nThe proposal that C atoms recombine via the quintet 5 Π g state which then populate the C 2 (d 3 Π g , v = 6) state (i.e. to generate the C 2 high pressure band) does not exclude the direct recombination of C atoms into the d 3 Π g state. One should see them as processes that happen simultaneously with different probabilities. Looking at the energy potential curves of the C 2 molecules, one can see that the\n, d 3 Π g and 5 Π g all have as dissociation limit two C( 3 P) atoms [41]. The branching ratios for the recombination into these different electronic states are however not known although it appears from experiments discussed before that the d 3 Π g and 5 Π g are the ones first (and mainly) populated.\n, d 3 Π g and 5 Π g all have as dissociation limit two C( 3 P) atoms [41]. The branching ratios for the recombination into these different electronic states are however not known although it appears from experiments discussed before that the d 3 Π g and 5 Π g are the ones first (and mainly) populated.\nIn conclusion, we can say that there are several concurring production mechanisms of the C 2 (d 3 Π g ) state either from recombination of C atoms or, in the case of CO containing plasmas, via the C 2 O molecule. The presence or absence of some of these mecanisms for which branching can unfortunately hardly be estimated will affect particularly the nascent population distribution of vibrational states of the C 2 (d 3 Π g ) state.\nIn conclusion, we can say that there are several concurring production mechanisms of the C 2 (d 3 Π g ) state either from recombination of C atoms or, in the case of CO containing plasmas, via the C 2 O molecule. The presence or absence of some of these mecanisms for which branching can unfortunately hardly be estimated will affect particularly the nascent population distribution of vibrational states of the C 2 (d 3 Π g ) state."},{"title":"Molecular spectra calculations","content":""},{"title":"Molecular spectra calculations","content":"In order to reproduce experimental spectra, it is necessary to first model the population distribution of states and, using molecular constants and transitions probabilities, one can simulate a spectrum knowing the instrumental profile of the apparatus. The partition functions of rovibrational distributions under the assumption of a Maxwell-Boltzmann distribution used to generate so-called Boltzmann plots are given in section 4.1. The molecular constants used for calculating both the C 2 Swan band and the CN violet system and implemented into MassiveOES and available as Supplementary Data are described in sections 4.2 and 4.3.\nIn order to reproduce experimental spectra, it is necessary to first model the population distribution of states and, using molecular constants and transitions probabilities, one can simulate a spectrum knowing the instrumental profile of the apparatus. The partition functions of rovibrational distributions under the assumption of a Maxwell-Boltzmann distribution used to generate so-called Boltzmann plots are given in section 4.1. The molecular constants used for calculating both the C 2 Swan band and the CN violet system and implemented into MassiveOES and available as Supplementary Data are described in sections 4.2 and 4.3."},{"title":"Thermal population distributions","content":""},{"title":"Thermal population distributions","content":"The distribution of energy between the internal degree of freedom of a diatomic molecule is described by the following partition function [61] Q = e νmax ν=0 Jmax J=Λ g Λ,hf s (2S + 1) (2J + 1)\nThe distribution of energy between the internal degree of freedom of a diatomic molecule is described by the following partition function [61] Q = e νmax ν=0 Jmax J=Λ g Λ,hf s (2S + 1) (2J + 1)\nwhere e denotes electronic states with projection of orbital angular momentum quantum number Λ and total spin quantum number S, and ν and J are the vibrational and rotational quantum numbers, respectively. g Λ,hf s is the statistical weight factor related to Λ-doubling and the nuclear hyperfine structure. T e , G(ν) and F ν (J) represent the electronic, vibrational and rotational energies in wavenumber units. E 0 is the energy of the lowest state of the molecule (i.e. ν = 0, J = Λ for the ground electronic state). T is the temperature of the heat's bath in the case where all internal degrees of freedom of the molecular gas are thermalized.\nwhere e denotes electronic states with projection of orbital angular momentum quantum number Λ and total spin quantum number S, and ν and J are the vibrational and rotational quantum numbers, respectively. g Λ,hf s is the statistical weight factor related to Λ-doubling and the nuclear hyperfine structure. T e , G(ν) and F ν (J) represent the electronic, vibrational and rotational energies in wavenumber units. E 0 is the energy of the lowest state of the molecule (i.e. ν = 0, J = Λ for the ground electronic state). T is the temperature of the heat's bath in the case where all internal degrees of freedom of the molecular gas are thermalized.\nAssuming that the internal degrees of freedom are independent, which is already implicitly assumed in the formalism used to derive equation 7, 6 the partition function for rotational and vibrational states can be rewritten as\nAssuming that the internal degrees of freedom are independent, which is already implicitly assumed in the formalism used to derive equation 7, 6 the partition function for rotational and vibrational states can be rewritten as\nwhere T rot and T vib are the temperature for the rotational and vibrational distribution functions in the case that they do not equilibrate with each other but obey Boltzmann distribution functions.\nwhere T rot and T vib are the temperature for the rotational and vibrational distribution functions in the case that they do not equilibrate with each other but obey Boltzmann distribution functions.\nUsing the Maxwell-Boltzmann distribution law, stating that the number of molecules dN E having an energy between E and E + dE is proportional to e (−E/k B T ) dE and using equations 7 and 8, the population density for rotational and vibrational levels can be defined as [63,62]\nUsing the Maxwell-Boltzmann distribution law, stating that the number of molecules dN E having an energy between E and E + dE is proportional to e (−E/k B T ) dE and using equations 7 and 8, the population density for rotational and vibrational levels can be defined as [63,62]\nwhere N e is the total density of the electronic state e and N (ν) is the density of the ν level summed up over its rotational distribution.\nwhere N e is the total density of the electronic state e and N (ν) is the density of the ν level summed up over its rotational distribution.\nAssuming a Boltzmann distribution for the rotational and vibrational levels, the intensity of a line for a transition (ν → ν , J → J , Ω → Ω ) is given by\nAssuming a Boltzmann distribution for the rotational and vibrational levels, the intensity of a line for a transition (ν → ν , J → J , Ω → Ω ) is given by\nwhere ψ(λ − λ 0 ) is the convoluted line profile and instrumental profile of the spectrometer and λ 0 the central wavelength position of the line. E(ν) = G(ν)hc and E(J , Ω ) ≈ F ν (J ) where the effect on energy level splitting due to Λ-doubling is expressed as a function of Ω (cf. equation 11). We use here the convention following Herzberg [63] that the rotational and vibrational quantum numbers of the upper level are defined by the index J', v and the lower level is described by J\" and v .\nwhere ψ(λ − λ 0 ) is the convoluted line profile and instrumental profile of the spectrometer and λ 0 the central wavelength position of the line. E(ν) = G(ν)hc and E(J , Ω ) ≈ F ν (J ) where the effect on energy level splitting due to Λ-doubling is expressed as a function of Ω (cf. equation 11). We use here the convention following Herzberg [63] that the rotational and vibrational quantum numbers of the upper level are defined by the index J', v and the lower level is described by J\" and v ."},{"title":"Swan band database: description, implementation, benchmark","content":""},{"title":"Swan band database: description, implementation, benchmark","content":"C 2 is a homonuclear diatomic molecule and the Swan band is an electronically allowed dipole transition between two 3 Π states due to change of symmetry of the wavefunction between the upper and lower state g → u. These are triplet states and for each rotational quantum number J, there are sub-bands\nC 2 is a homonuclear diatomic molecule and the Swan band is an electronically allowed dipole transition between two 3 Π states due to change of symmetry of the wavefunction between the upper and lower state g → u. These are triplet states and for each rotational quantum number J, there are sub-bands\nThe sub-bands are dominated by P and R-branches (corresponding to selection rule ∆J = J − J = −1 and +1 respectively). Only weak Q-branches (∆J = 0) exist and none for the 3 Π 0 − 3 Π 0 sub-band as only ∆J = ±1 transitions occur for Ω = 0 where Ω is the total electronic angular momentum along the internuclear axis is designated with Ω = Λ + Σ. Σ is the projection of the spin quantum number S onto the internuclear axis. Λ is the quantum number for the component of the total angular momentum along the internuclear axis. It is the absolute value of the projected component of the angular momentum L on the internuclear axis M L with Λ =| M L | [64]. Π states have by definition Λ = 1 and for the Swan band we have triplet states d 3 Π g -a 3 Π u for which S=1 and so Σ = M S = −1, 0, +1. The coupling between the rotation of the nuclei and L additionally leads to a splitting into two components for each J value when Λ = 0. This is called Λ-doubling (see below). As the C 2 molecule is symmetric and the carbon atom has a zero nuclear spin, transition between symmetric and asymmetric wavefunctions are forbidden. The statistical weight due to nuclear spin I of the anti-symmetric rotational levels is given by I(I+1) and as for C 2 we have I=0, they are absent. However, for Λ = 1, we have symmetric and antisymmetric combination of wave functions due to Λ-doubling and therefore both even and odd rotational numbers are present. In the P, Q and R branches, only one of the Λ-doubled levels are present alternatively which leads to the appearance that the even levels are displaced to one side and the odd levels to the other side of a mean position for a given branch progression.\nThe sub-bands are dominated by P and R-branches (corresponding to selection rule ∆J = J − J = −1 and +1 respectively). Only weak Q-branches (∆J = 0) exist and none for the 3 Π 0 − 3 Π 0 sub-band as only ∆J = ±1 transitions occur for Ω = 0 where Ω is the total electronic angular momentum along the internuclear axis is designated with Ω = Λ + Σ. Σ is the projection of the spin quantum number S onto the internuclear axis. Λ is the quantum number for the component of the total angular momentum along the internuclear axis. It is the absolute value of the projected component of the angular momentum L on the internuclear axis M L with Λ =| M L | [64]. Π states have by definition Λ = 1 and for the Swan band we have triplet states d 3 Π g -a 3 Π u for which S=1 and so Σ = M S = −1, 0, +1. The coupling between the rotation of the nuclei and L additionally leads to a splitting into two components for each J value when Λ = 0. This is called Λ-doubling (see below). As the C 2 molecule is symmetric and the carbon atom has a zero nuclear spin, transition between symmetric and asymmetric wavefunctions are forbidden. The statistical weight due to nuclear spin I of the anti-symmetric rotational levels is given by I(I+1) and as for C 2 we have I=0, they are absent. However, for Λ = 1, we have symmetric and antisymmetric combination of wave functions due to Λ-doubling and therefore both even and odd rotational numbers are present. In the P, Q and R branches, only one of the Λ-doubled levels are present alternatively which leads to the appearance that the even levels are displaced to one side and the odd levels to the other side of a mean position for a given branch progression.\nWith each alternate line that is simply absent because of I=0, this is often referred to as the \"staggering\"effect. We note that the statistic weight (g Λ,hf s in equation 7) due to the nuclear spin for individual rotational levels induce an alternance of intensities for sub-bands of electronic transitions with Λ = 0.\nWith each alternate line that is simply absent because of I=0, this is often referred to as the \"staggering\"effect. We note that the statistic weight (g Λ,hf s in equation 7) due to the nuclear spin for individual rotational levels induce an alternance of intensities for sub-bands of electronic transitions with Λ = 0.\nFor more details on the rovibrational structure and lines positions of the C 2 Swan band system we refer to Herzberg [63] and Pellerin [65].\nFor more details on the rovibrational structure and lines positions of the C 2 Swan band system we refer to Herzberg [63] and Pellerin [65].\nIn figure 4, a high resolution spectrum from 450 nm up to 565 nm is presented where the main transition groups ∆ν and the position of the vibrational transition band heads are identified. Transitions between two electronic states can be ordered following the difference in vibrational quantum numbers during the radiative transition between the upper and lower state ∆ν = v − v . For instance, in figure 4 the transition group ∆ν = +1 corresponds to the rovibrational serie\nIn figure 4, a high resolution spectrum from 450 nm up to 565 nm is presented where the main transition groups ∆ν and the position of the vibrational transition band heads are identified. Transitions between two electronic states can be ordered following the difference in vibrational quantum numbers during the radiative transition between the upper and lower state ∆ν = v − v . For instance, in figure 4 the transition group ∆ν = +1 corresponds to the rovibrational serie\nlevel is therefore not part of this transition group. The ∆ν = −2, +2 transition groups can also be seen experimentally but are not shown here. 1.0x10 19 1.5x10 19 2.0x10 19 2.5x10 19 1-0 The spectroscopic database that we implemented into the fitting program of molecular emission spectra MassiveOES [31] is derived from the work of Brooke et al. [1]. As initial input, MassiveOES requires rovibrationally resolved Einstein coefficients for the transitions between the upper and lower state that can be generally calculated using the following equation [66]:\nlevel is therefore not part of this transition group. The ∆ν = −2, +2 transition groups can also be seen experimentally but are not shown here. 1.0x10 19 1.5x10 19 2.0x10 19 2.5x10 19 1-0 The spectroscopic database that we implemented into the fitting program of molecular emission spectra MassiveOES [31] is derived from the work of Brooke et al. [1]. As initial input, MassiveOES requires rovibrationally resolved Einstein coefficients for the transitions between the upper and lower state that can be generally calculated using the following equation [66]:\nwhere S ∆J J is the Höln-London factor and\nwhere S ∆J J is the Höln-London factor and\nBrooke et al. [1] used PGOPHER, a program developed by Colin Western, for calculating the Einstein A (v ,J )→(v ,J ) coefficients [67]. PGOPHER requires as input a set of molecular constants and the band strengths for each vibrational band v → v . The purely vibrational transition dipole moments (TDMs) were calculated using the program LEVEL [68], a program that can solve the 1D Schrödinger equation for diatomic molecules when provided with a potential energy curve and the electronic TDM. The set of molecular constants for fitting RKR potential curves of the upper and lower electronic levels involved into the Swan band transition are listed in [1]. With the output of LEVEL, PGOPHER is then able to calculate the rotational TDMs and the Höln-London factors required to compute the numerical values of equation 12. The original simulations by Brooke et al were extended to a maximum rotational quantum number J of 300 by running again PGOPHER with the same molecular constants. This allows to properly calculate the rotational partition function in MassiveOES for the high rotational temperatures encountered in the present study. It should however be noted that, while doing this, the calculated energy levels have increasing error for increasing J in energy and line positions. The molecular constants used for the C 2 Swan band are given in Appendix A and the output calculation of PGOPHER as Supplementary Material to this paper.\nBrooke et al. [1] used PGOPHER, a program developed by Colin Western, for calculating the Einstein A (v ,J )→(v ,J ) coefficients [67]. PGOPHER requires as input a set of molecular constants and the band strengths for each vibrational band v → v . The purely vibrational transition dipole moments (TDMs) were calculated using the program LEVEL [68], a program that can solve the 1D Schrödinger equation for diatomic molecules when provided with a potential energy curve and the electronic TDM. The set of molecular constants for fitting RKR potential curves of the upper and lower electronic levels involved into the Swan band transition are listed in [1]. With the output of LEVEL, PGOPHER is then able to calculate the rotational TDMs and the Höln-London factors required to compute the numerical values of equation 12. The original simulations by Brooke et al were extended to a maximum rotational quantum number J of 300 by running again PGOPHER with the same molecular constants. This allows to properly calculate the rotational partition function in MassiveOES for the high rotational temperatures encountered in the present study. It should however be noted that, while doing this, the calculated energy levels have increasing error for increasing J in energy and line positions. The molecular constants used for the C 2 Swan band are given in Appendix A and the output calculation of PGOPHER as Supplementary Material to this paper.\nExtensive comparison of calculated line positions versus experimental ones were already done in the literature (see Brooke et al. [1] and references therein). We additionally performed a cross-check with the high resolution degenerate four-wave mixing experiments of Lloyd and Stewart [69]. The difference between their measured line positions for the ∆ν = 0 transition group and our calculated ones are shown in figure 5. One can see that, excepted for two outliers, the line positions are calculated with an accuracy much better than ± 0.05 cm −1 . Experimental data for higher J numbers is however not available for performing further benchmark. Experimentally we have a line profile of 22 pm corresponding to about 0.8 cm −1 which is much broader than the line position uncertainty λ err < 1 pm. Consequently, using calculated line positions for the fitting of the experimental spectra induces no additional error related to uncertainties in line position assignment. We note at this point that high resolution measurements of line positions were done up to J = 80 and only for some vibrational levels by Tanabashi et al. [70].\nExtensive comparison of calculated line positions versus experimental ones were already done in the literature (see Brooke et al. [1] and references therein). We additionally performed a cross-check with the high resolution degenerate four-wave mixing experiments of Lloyd and Stewart [69]. The difference between their measured line positions for the ∆ν = 0 transition group and our calculated ones are shown in figure 5. One can see that, excepted for two outliers, the line positions are calculated with an accuracy much better than ± 0.05 cm −1 . Experimental data for higher J numbers is however not available for performing further benchmark. Experimentally we have a line profile of 22 pm corresponding to about 0.8 cm −1 which is much broader than the line position uncertainty λ err < 1 pm. Consequently, using calculated line positions for the fitting of the experimental spectra induces no additional error related to uncertainties in line position assignment. We note at this point that high resolution measurements of line positions were done up to J = 80 and only for some vibrational levels by Tanabashi et al. [70].\nThe database of lines and their transition probabilities is compiled in a SQL database, that is then read by MassiveOES. The database consists of three tables for listing the lines, the identity of rovibrational \"upper\" and \"lower\" states respectively. Each element of the table corresponds to a single line in the spectrum for which energy of the transition, air wavelength, vacuum wavelength, Einstein coefficient, upper state identity and lower state identity are given. The \"upper state\" tag (as well as the \"lower state\" tag) links the line with the energy level from which is originated, that will be used to calculate the intensity of the line. The effect of molecular perturbations ) and 1 5 Π g states [1] are combined in the table as additional transitions from the upper state, such that, the rovibrational population distribution is calculated without extra energy levels.\nThe database of lines and their transition probabilities is compiled in a SQL database, that is then read by MassiveOES. The database consists of three tables for listing the lines, the identity of rovibrational \"upper\" and \"lower\" states respectively. Each element of the table corresponds to a single line in the spectrum for which energy of the transition, air wavelength, vacuum wavelength, Einstein coefficient, upper state identity and lower state identity are given. The \"upper state\" tag (as well as the \"lower state\" tag) links the line with the energy level from which is originated, that will be used to calculate the intensity of the line. The effect of molecular perturbations ) and 1 5 Π g states [1] are combined in the table as additional transitions from the upper state, such that, the rovibrational population distribution is calculated without extra energy levels."},{"title":"CN violet system emission","content":""},{"title":"CN violet system emission","content":"The CN molecular violet system corresponds to the B 2 Σ + → X 2 Σ + transition of CN which is an heteronuclear diatomic molecule. The required molecular constants have been taken from S. Ram et al. [71]. The transition probabilities between two vibrational states calculated by solving the integral of the vibrational wavefunctions of the upper and lower states and TDM Ψ ν (r)R e (r)Ψ ν (r) were extracted from LIFBASE [72]. The molecular constants for calculating the Rydberg-Klein-Rees (RKR) potential curves of the X 2 Σ + and B 2 Σ + states were taken from [73,74]). For the TDMs R e (r), the values of Ito et al. [75] were taken. These molecular constants were then used as input for PGOPHER to calculate the Höln-London factors of the CN violet system with J number up to 180. For the CN violet system some perturbations also exist but those are not included in the calculations as they do not affect significantly the spectra. In Appendix B are given all the constants used in the calculation. As sanity check, a one to one comparison between our PGOPHER calculations and LIFBASE was made and they yield the same results both for line positions and intensities.\nThe CN molecular violet system corresponds to the B 2 Σ + → X 2 Σ + transition of CN which is an heteronuclear diatomic molecule. The required molecular constants have been taken from S. Ram et al. [71]. The transition probabilities between two vibrational states calculated by solving the integral of the vibrational wavefunctions of the upper and lower states and TDM Ψ ν (r)R e (r)Ψ ν (r) were extracted from LIFBASE [72]. The molecular constants for calculating the Rydberg-Klein-Rees (RKR) potential curves of the X 2 Σ + and B 2 Σ + states were taken from [73,74]). For the TDMs R e (r), the values of Ito et al. [75] were taken. These molecular constants were then used as input for PGOPHER to calculate the Höln-London factors of the CN violet system with J number up to 180. For the CN violet system some perturbations also exist but those are not included in the calculations as they do not affect significantly the spectra. In Appendix B are given all the constants used in the calculation. As sanity check, a one to one comparison between our PGOPHER calculations and LIFBASE was made and they yield the same results both for line positions and intensities.\n5 Spectral analysis of the C 2 Swan band 5\n5 Spectral analysis of the C 2 Swan band 5"},{"title":".1 Non-Boltzmann fitting of the Swan band","content":""},{"title":".1 Non-Boltzmann fitting of the Swan band","content":"In non-equilibrium plasmas, rotational and vibrational distributions do not always thermalize because of competition between production and destruction processes and a molecular spectrum can then not be fitted using single rotational T rot and vibrational T vib temperatures [29]. As discussed in section 3, the C 2 ( 3 Π g ) state is well known for overpopulations in the density of its high vibrational levels and particularly the v = 6 level. In this section we investigate high resolution spectra of the C 2 Swan band in order to assess possible deviations from thermal equilibrium of the rovibrational populations.\nIn non-equilibrium plasmas, rotational and vibrational distributions do not always thermalize because of competition between production and destruction processes and a molecular spectrum can then not be fitted using single rotational T rot and vibrational T vib temperatures [29]. As discussed in section 3, the C 2 ( 3 Π g ) state is well known for overpopulations in the density of its high vibrational levels and particularly the v = 6 level. In this section we investigate high resolution spectra of the C 2 Swan band in order to assess possible deviations from thermal equilibrium of the rovibrational populations.\nIn figure 6, individual P and R lines from the v = 0 level are identified (the Q lines are too weak to be distinguished, see section 4.2) from the ∆ν = 0 transition group and their area fitted with a Voigt profile (with gaussian and lorentzian half-width of respectively σ G =9.5 pm and σ L = 4 pm) to take into account the apparatus profile determined from Hg lines using a low pressure Hg lamp. They were selected only when they can be separated from neighboring lines and clearly identified. Note that the fine structure of the rotational levels could not be resolved and so that the lines are weighted over the rotational fine structure (i.e. rotational sub-bands). The logarithm of the density of each individual rotational level is plotted as function of the rotational energy (i.e. Boltzmann plot). All levels fit on a single line which can be fitted with a rotational temperature of 6200±160 K. Here and in the following, we will quote for each fitting method, its related statistical errors. No deviation is observed neither for low or high rotational levels indicating that the levels are thermalized. We note here that levels with approximately J>120 do not contribute noticably to the emission spectra and are not visible.\nIn figure 6, individual P and R lines from the v = 0 level are identified (the Q lines are too weak to be distinguished, see section 4.2) from the ∆ν = 0 transition group and their area fitted with a Voigt profile (with gaussian and lorentzian half-width of respectively σ G =9.5 pm and σ L = 4 pm) to take into account the apparatus profile determined from Hg lines using a low pressure Hg lamp. They were selected only when they can be separated from neighboring lines and clearly identified. Note that the fine structure of the rotational levels could not be resolved and so that the lines are weighted over the rotational fine structure (i.e. rotational sub-bands). The logarithm of the density of each individual rotational level is plotted as function of the rotational energy (i.e. Boltzmann plot). All levels fit on a single line which can be fitted with a rotational temperature of 6200±160 K. Here and in the following, we will quote for each fitting method, its related statistical errors. No deviation is observed neither for low or high rotational levels indicating that the levels are thermalized. We note here that levels with approximately J>120 do not contribute noticably to the emission spectra and are not visible.\nThe thermalization of the rotational levels of the d 3 Π g state is further assessed using the option in MassiveOES of state-by-state fitting [32]. State by state fitting here means that the fitting routine evaluates the density of each upper rotational state while minimizing the residue for all transitions from the same upper state (i.e. different transition groups, P, Q and R branches and Λ-doubling). Allowing to fit independently the density of all rotational states helps to reduce the overall residue of the fit but only marginally. In figure 7, the population density of the rotational levels versus their energy is plotted. All rotational population densities fit quite well on a single line indicating no deviation from a thermal rotational distribution function. The lowest rotational levels are not shown because they are experimentally too much overlapped and the state by state fitting routine cannot converge (see below for a discussion of those). The highest rotational levels are widely scattered due to the very low signal to noise ratio. A rotational temperature T rot =5500± 500 K is obtained from the fitting of figure 7. From this first fit, some systematic deviations are apparent. A closer inspection indicates the presence of some outliers points which have densities off by an order of magnitude or more without any systematic trend. Removing these outliers which have a non-neglible weight for the fitting of a linear slope in logarithmic plot, the statistical uncertainty of the Boltzmann plot is significantly reduced. The inverse slope of the Boltzmann plot gives a temperature of 6200±150 K. This result agrees with the line by line Boltzmann plot analysis made in figure 6. Fitting the complete ∆ν = 0 transition group while assuming a Boltzmann distribution (i.e envelope fitting) for rotational and vibrational distribution gives similarly a temperature of 6000±200 K in excellent agreement with the state to state fitting analysis. The result of the fit using a Boltzmann distribution is also shown in figure 7 (note the logscale) as well and one can see that the residues following a Boltzmann fitting or a state by state fitting are similar.\nThe thermalization of the rotational levels of the d 3 Π g state is further assessed using the option in MassiveOES of state-by-state fitting [32]. State by state fitting here means that the fitting routine evaluates the density of each upper rotational state while minimizing the residue for all transitions from the same upper state (i.e. different transition groups, P, Q and R branches and Λ-doubling). Allowing to fit independently the density of all rotational states helps to reduce the overall residue of the fit but only marginally. In figure 7, the population density of the rotational levels versus their energy is plotted. All rotational population densities fit quite well on a single line indicating no deviation from a thermal rotational distribution function. The lowest rotational levels are not shown because they are experimentally too much overlapped and the state by state fitting routine cannot converge (see below for a discussion of those). The highest rotational levels are widely scattered due to the very low signal to noise ratio. A rotational temperature T rot =5500± 500 K is obtained from the fitting of figure 7. From this first fit, some systematic deviations are apparent. A closer inspection indicates the presence of some outliers points which have densities off by an order of magnitude or more without any systematic trend. Removing these outliers which have a non-neglible weight for the fitting of a linear slope in logarithmic plot, the statistical uncertainty of the Boltzmann plot is significantly reduced. The inverse slope of the Boltzmann plot gives a temperature of 6200±150 K. This result agrees with the line by line Boltzmann plot analysis made in figure 6. Fitting the complete ∆ν = 0 transition group while assuming a Boltzmann distribution (i.e envelope fitting) for rotational and vibrational distribution gives similarly a temperature of 6000±200 K in excellent agreement with the state to state fitting analysis. The result of the fit using a Boltzmann distribution is also shown in figure 7 (note the logscale) as well and one can see that the residues following a Boltzmann fitting or a state by state fitting are similar.\nIn figure 8a, the state by state fitting analysis is extended to the ∆ν = −1 and ∆ν = +1 transition groups for the rotational and vibrational states of the d 3 Π g state. One can see that the rotational levels of the v=1 and v=2 levels also follow a Boltzmann distribution with the same slope as for the ν = 0 level although the signal is more noisy. Also population densities of the lower rotational states can be analysis and no systematic deviations from a single slope can be found.\nIn figure 8a, the state by state fitting analysis is extended to the ∆ν = −1 and ∆ν = +1 transition groups for the rotational and vibrational states of the d 3 Π g state. One can see that the rotational levels of the v=1 and v=2 levels also follow a Boltzmann distribution with the same slope as for the ν = 0 level although the signal is more noisy. Also population densities of the lower rotational states can be analysis and no systematic deviations from a single slope can be found.\nAfter integration over the rotational distribution the density of the vibrational states can be obtained (cf. equation 9). The resulting Boltzmann plot for the vibrational states is shown in figure 8b. Their population distribution can be described by a single temperature of 7200± 1200 K. For comparison, we also performed a state by state fitting of the ∆ν = +1 transition group that is the most sensitive to the vibrational temperature (see section 5.2). The rotational temperature was fixed to 6200 K following the analysis of the ∆ν = 0 transition group. For the ∆ν = +1 transition group, the density of each ν level is adjusted independently and the results are shown in figure 9. One can see that all vibrational levels fit perfectly on a single line with T vib =6000± 400 K. The larger uncertainties in the case of the state by state fitting when all three transition groups ∆ν = +1, 0, −1 are taken into account stems from the lower signal to noise ratio for the ∆ν = −1 transition group. The result of the fitting of the 3 transition groups for the vibrational distribution of the C 2 d 3 Π g state shown in figure 8b may let someone thing that the C 2 Swan band system follows a two-temperature vibrational distribution function. The vibrational state to state analysis of the most sensitive ∆ν = +1 transition group done in figure 9 (which is more sensitive to T vib as discussed in section 5.2) show that there is no evidence of a twotemperatures vibrational states distribution function. Fitting the ∆ν = +1 transition group alone, which is the one usually measured in the literature for studying the so-called C 2 high pressure bands as it is the most sensitive for determining vibrational states densities, show that the vibrational distribution function of the d 3 Π g state is very well described by a single temperature. Considering the noise in the state by state ro-vibrational fitting shown in figure 8a for ν = 1 and 2 levels, it is in fact quite remarkable to see that the fitted vibrational temperature still agrees with the more precise fitting given in figure 9 within their own error bars. Note that for the vibrational state to state fitting of the ∆ν = +1 transition group, the rotational temperature was set using the T rot obtained from the more sensitive ∆ν = 0 transition group.\nAfter integration over the rotational distribution the density of the vibrational states can be obtained (cf. equation 9). The resulting Boltzmann plot for the vibrational states is shown in figure 8b. Their population distribution can be described by a single temperature of 7200± 1200 K. For comparison, we also performed a state by state fitting of the ∆ν = +1 transition group that is the most sensitive to the vibrational temperature (see section 5.2). The rotational temperature was fixed to 6200 K following the analysis of the ∆ν = 0 transition group. For the ∆ν = +1 transition group, the density of each ν level is adjusted independently and the results are shown in figure 9. One can see that all vibrational levels fit perfectly on a single line with T vib =6000± 400 K. The larger uncertainties in the case of the state by state fitting when all three transition groups ∆ν = +1, 0, −1 are taken into account stems from the lower signal to noise ratio for the ∆ν = −1 transition group. The result of the fitting of the 3 transition groups for the vibrational distribution of the C 2 d 3 Π g state shown in figure 8b may let someone thing that the C 2 Swan band system follows a two-temperature vibrational distribution function. The vibrational state to state analysis of the most sensitive ∆ν = +1 transition group done in figure 9 (which is more sensitive to T vib as discussed in section 5.2) show that there is no evidence of a twotemperatures vibrational states distribution function. Fitting the ∆ν = +1 transition group alone, which is the one usually measured in the literature for studying the so-called C 2 high pressure bands as it is the most sensitive for determining vibrational states densities, show that the vibrational distribution function of the d 3 Π g state is very well described by a single temperature. Considering the noise in the state by state ro-vibrational fitting shown in figure 8a for ν = 1 and 2 levels, it is in fact quite remarkable to see that the fitted vibrational temperature still agrees with the more precise fitting given in figure 9 within their own error bars. Note that for the vibrational state to state fitting of the ∆ν = +1 transition group, the rotational temperature was set using the T rot obtained from the more sensitive ∆ν = 0 transition group.\nFrom this analysis, it appears that for the present plasma conditions, the vibrational levels are completely thermalized and no overpopulation of the higher vibrational levels and more particularly v=6, occurs. Analysis of other spectra as the one discussed above yield the same results and no deviations from Boltzmann equilibrium either for the rotational or vibrational levels could be found for the conditions of the present study (cf. section 7). Either the processes discussion in section 3 leading to the so-called C 2 high pressure bands are absent in the present discharge or the levels thermalize within the effective (radiative) lifetime of the state (see section 6 for a discussion). Different methods of fitting give similar results and no evidence of deviations from thermal distributions can be found within error margins. For the following, Boltzmann distributions are then assumed and an envelope fitting of the spectra is done. For measuring the rotational temperature, the ∆ν = 0 transition group is chosen while all three transition groups are used for the determination of the vibrational temperature. For more discussion on this, we refer to section 5.2.\nFrom this analysis, it appears that for the present plasma conditions, the vibrational levels are completely thermalized and no overpopulation of the higher vibrational levels and more particularly v=6, occurs. Analysis of other spectra as the one discussed above yield the same results and no deviations from Boltzmann equilibrium either for the rotational or vibrational levels could be found for the conditions of the present study (cf. section 7). Either the processes discussion in section 3 leading to the so-called C 2 high pressure bands are absent in the present discharge or the levels thermalize within the effective (radiative) lifetime of the state (see section 6 for a discussion). Different methods of fitting give similar results and no evidence of deviations from thermal distributions can be found within error margins. For the following, Boltzmann distributions are then assumed and an envelope fitting of the spectra is done. For measuring the rotational temperature, the ∆ν = 0 transition group is chosen while all three transition groups are used for the determination of the vibrational temperature. For more discussion on this, we refer to section 5.2."},{"title":"T rot and T vib from the C 2 Swan band: sensitivity and uncertainty analysis","content":""},{"title":"T rot and T vib from the C 2 Swan band: sensitivity and uncertainty analysis","content":"The convergence of a fit is usually assessed by looking at the value of the least mean square error and minimizing the root mean squared error (RM SE = i (y i − y i ) 2 ). It is possible to use the computed residue for assessing how sensitive is the minimization routine for given external parameters that are being fitted. Information can be extracted from a dataset (here an experimental spectrum) only when the variation of signal is statistically significant. It usually means that the correlation between a signal and the event we want to measure must be significantly larger that the statistical noise. Theoretical spectra can be computed for given fixed rotational and vibrational temperatures and compared with another theoretical spectrum in terms of their respective RMSEs [76]. In the case where the (thermal) white noise is the dominant source of uncertainty in the fitting of an experimental spectrum, the amplitude of the noise can be directly correlated to an iso-line inside which the \"true value\" stands. Nassar [77] evaluated the errors induced by a white noise on T rot determination while fitting the band head of the ∆ν = 0 transition group and obtained errors in the order of the signal to noise ratio or lower. This is however without considering systematic errors that are induced by performing mathematical operations on the data such a baseline correction or intensity calibration. These errors are additive and amplify the uncertainties on the values obtained through a fit. Also, one needs to take additionally into account the uncertainties in the molecular constants (Franck-Condon and Höln-London factors).\nThe convergence of a fit is usually assessed by looking at the value of the least mean square error and minimizing the root mean squared error (RM SE = i (y i − y i ) 2 ). It is possible to use the computed residue for assessing how sensitive is the minimization routine for given external parameters that are being fitted. Information can be extracted from a dataset (here an experimental spectrum) only when the variation of signal is statistically significant. It usually means that the correlation between a signal and the event we want to measure must be significantly larger that the statistical noise. Theoretical spectra can be computed for given fixed rotational and vibrational temperatures and compared with another theoretical spectrum in terms of their respective RMSEs [76]. In the case where the (thermal) white noise is the dominant source of uncertainty in the fitting of an experimental spectrum, the amplitude of the noise can be directly correlated to an iso-line inside which the \"true value\" stands. Nassar [77] evaluated the errors induced by a white noise on T rot determination while fitting the band head of the ∆ν = 0 transition group and obtained errors in the order of the signal to noise ratio or lower. This is however without considering systematic errors that are induced by performing mathematical operations on the data such a baseline correction or intensity calibration. These errors are additive and amplify the uncertainties on the values obtained through a fit. Also, one needs to take additionally into account the uncertainties in the molecular constants (Franck-Condon and Höln-London factors).\nIn figure 10, such analysis is first made for the ∆ν = 0 alone and for the ∆ν = −1, 0, +1 transition groups of the C 2 Swan band by comparing simulated spectra with theoretical spectra with T rot = T vib taking values of 1000, 3000, 5000, 6000 and 7500 K with 5% surimposed noise. The steepness of the RMSE around the equilibrium point (i.e. true value) obtained by the minimization routine is a good indication of accuracy with which a parameter can be fitted (here T rot and T vib ). It appears clearly that the ∆ν = 0 is rather imprecise for the determination of the vibrational temperature of the d 3 Π g state and that typical errors are in the order of several 1000 K. The minimum for the rotational temperature is on the contrary quite steep indicating that the band has a good sensitivity for T rot determination. The ∆ν = −1, +1 transition groups have on the other hand a much higher sensitivity to T vib . A comparison between the ∆ν = 0 fit and the fit of all three transition groups seems to indicate at first sight that both allow to obtain the rotational temperature with rather good sensitivity. However, in the case of ∆ν = −1, 0, +1 transition groups fit, one can see that the isolines are stretched on a diagonal (i.e. not perpendicular to the axes) which means that T rot and T vib are correlated during the minimization (i.e. if an error is made in the parameters minimization used for the determination of one variable, the error will propagate to the other). The parametrization of the spectrum in terms of independent variables (T rot , T vib , instrumental profile,...) in a feature space (i.e. a space made of independent variables that are combinations of the original variables) is not entirely successful and some of the new variables are not orthogonal. This is a well known issue in the field of Principal Components Analysis (PCA) [78]. In the present case, when the isolines are equidistant in the vertical and horizontal directions, one can say that the determination of T rot and T vib are statistically independent. Note that a fitting routine gives for any given fit a RMSE but its absolute value carries no intrinsic information. It has a meaning only in correlation with other RMSEs calculated around the (local) equilibrium point of the fit. The RMS error is then directly normalized to the equilibrium value.\nIn figure 10, such analysis is first made for the ∆ν = 0 alone and for the ∆ν = −1, 0, +1 transition groups of the C 2 Swan band by comparing simulated spectra with theoretical spectra with T rot = T vib taking values of 1000, 3000, 5000, 6000 and 7500 K with 5% surimposed noise. The steepness of the RMSE around the equilibrium point (i.e. true value) obtained by the minimization routine is a good indication of accuracy with which a parameter can be fitted (here T rot and T vib ). It appears clearly that the ∆ν = 0 is rather imprecise for the determination of the vibrational temperature of the d 3 Π g state and that typical errors are in the order of several 1000 K. The minimum for the rotational temperature is on the contrary quite steep indicating that the band has a good sensitivity for T rot determination. The ∆ν = −1, +1 transition groups have on the other hand a much higher sensitivity to T vib . A comparison between the ∆ν = 0 fit and the fit of all three transition groups seems to indicate at first sight that both allow to obtain the rotational temperature with rather good sensitivity. However, in the case of ∆ν = −1, 0, +1 transition groups fit, one can see that the isolines are stretched on a diagonal (i.e. not perpendicular to the axes) which means that T rot and T vib are correlated during the minimization (i.e. if an error is made in the parameters minimization used for the determination of one variable, the error will propagate to the other). The parametrization of the spectrum in terms of independent variables (T rot , T vib , instrumental profile,...) in a feature space (i.e. a space made of independent variables that are combinations of the original variables) is not entirely successful and some of the new variables are not orthogonal. This is a well known issue in the field of Principal Components Analysis (PCA) [78]. In the present case, when the isolines are equidistant in the vertical and horizontal directions, one can say that the determination of T rot and T vib are statistically independent. Note that a fitting routine gives for any given fit a RMSE but its absolute value carries no intrinsic information. It has a meaning only in correlation with other RMSEs calculated around the (local) equilibrium point of the fit. The RMS error is then directly normalized to the equilibrium value.\nWhile fitting an experimental spectrum using the assumption of Maxwell-Boltzmann distribution for the rotational and vibrational states of the radiating molecule, there are additional parameters that need to be fixed. They are listed into Table 1. These parameters, excepted for the first three in the table are all experimental and depends of the spectrometer. A linear baseline is fitted with an offset value B base and a slope α base for substracting contributions related to noise and continuum emission from the plasma. The instrumental profile of the spectrometer is taken into account via σ L and σ G while the wavelength calibration of the spectrum is adjusted with a quadratic fit (λ start , ∆λ and ∆λ 2 parameters) and the wavelength λ is defined as\nWhile fitting an experimental spectrum using the assumption of Maxwell-Boltzmann distribution for the rotational and vibrational states of the radiating molecule, there are additional parameters that need to be fixed. They are listed into Table 1. These parameters, excepted for the first three in the table are all experimental and depends of the spectrometer. A linear baseline is fitted with an offset value B base and a slope α base for substracting contributions related to noise and continuum emission from the plasma. The instrumental profile of the spectrometer is taken into account via σ L and σ G while the wavelength calibration of the spectrum is adjusted with a quadratic fit (λ start , ∆λ and ∆λ 2 parameters) and the wavelength λ is defined as\nwhere i is the pixel number. The density N of the upper excited state has no influence on the evaluation of the temperatures as it is related only to the integrated band intensity.\nwhere i is the pixel number. The density N of the upper excited state has no influence on the evaluation of the temperatures as it is related only to the integrated band intensity.\nThe parameters that have an influence on the determination of T rot and T vib are the line profile and the fitting of the baseline. By varying systematically these values and considering uncertainties on the Franck-Condon (FC) and Höln-London factors, we estimated conservative uncertainty margins of about ±500 K for the spectra fitted in the present study. This evaluation is confirmed by the analysis made in section 5.1 where statistical methods have been applied while using different analysis methods to evaluate the rovibrational distribution functions. The errors for the rotational temperature are in fact probably lower. Fitting of theoretical low resolution spectra generated using SPECAIR [79,80] Wavelength of the first point in the spectra ∆λ Distance between two points in the spectra ∆λ 2 Quadratic correction of the position of a point in the spectra difference in the order of 1% for the rotational temperature. For the vibrational temperature, we also performed a comparison using the transition dipole moments of Kokkin et al [82] and FC-factors obtained by Fantz and Wünderlich [81]. The main difference with the present values and those calculations are only in the absolute values of the FC-factors with a difference of about 20%. The latter difference is systematic and consequently only affects the determination of the density N of the upper electronic state (i.e. after summation over rotational and vibrational population densities). No significant difference for the determination of T vib while using the two datasets were found. In figure 11, the same analysis is made for an experimental spectrum. The minimization routine of MassiveOES is here used only for adjusting the area of the spectra (i.e. population density of the electronic state), baseline and linewidth. The spectra are calibrated in wavelength beforehand using the position of known lines and its contribution to fitting uncertainties can be in the present case (i.e high resolution) neglected. The results are similar to figure 10 and it can be concluded that the C 2 Swan ∆ν = 0 transition group is overall more sensitive and \"precise\"for the determination of T rot .\nThe parameters that have an influence on the determination of T rot and T vib are the line profile and the fitting of the baseline. By varying systematically these values and considering uncertainties on the Franck-Condon (FC) and Höln-London factors, we estimated conservative uncertainty margins of about ±500 K for the spectra fitted in the present study. This evaluation is confirmed by the analysis made in section 5.1 where statistical methods have been applied while using different analysis methods to evaluate the rovibrational distribution functions. The errors for the rotational temperature are in fact probably lower. Fitting of theoretical low resolution spectra generated using SPECAIR [79,80] Wavelength of the first point in the spectra ∆λ Distance between two points in the spectra ∆λ 2 Quadratic correction of the position of a point in the spectra difference in the order of 1% for the rotational temperature. For the vibrational temperature, we also performed a comparison using the transition dipole moments of Kokkin et al [82] and FC-factors obtained by Fantz and Wünderlich [81]. The main difference with the present values and those calculations are only in the absolute values of the FC-factors with a difference of about 20%. The latter difference is systematic and consequently only affects the determination of the density N of the upper electronic state (i.e. after summation over rotational and vibrational population densities). No significant difference for the determination of T vib while using the two datasets were found. In figure 11, the same analysis is made for an experimental spectrum. The minimization routine of MassiveOES is here used only for adjusting the area of the spectra (i.e. population density of the electronic state), baseline and linewidth. The spectra are calibrated in wavelength beforehand using the position of known lines and its contribution to fitting uncertainties can be in the present case (i.e high resolution) neglected. The results are similar to figure 10 and it can be concluded that the C 2 Swan ∆ν = 0 transition group is overall more sensitive and \"precise\"for the determination of T rot .\nFinally, in figure 12, the sensitivity of the ∆ν = 0 transition group for determining T rot on assumed/fitted values of T vib is assessed. An experimental spectrum for the ∆ν = 0 transition group is fitted while taking different values for T vib . One can see that the fitted rotational temperature is almost independent of the vibrational temperature. This is due to the low transition probabilities for the ∆ν = 0 transitions for the v=1,2 levels while higher levels are even not visible at all. We note that the almost independence of the fitted rotational temperature to T vib is true for the relatively high resolution of the present spectrum. Interdependence of T rot and T vib becomes larger for lower resolution spectra also for the ∆ν = 0 transition group.\nFinally, in figure 12, the sensitivity of the ∆ν = 0 transition group for determining T rot on assumed/fitted values of T vib is assessed. An experimental spectrum for the ∆ν = 0 transition group is fitted while taking different values for T vib . One can see that the fitted rotational temperature is almost independent of the vibrational temperature. This is due to the low transition probabilities for the ∆ν = 0 transitions for the v=1,2 levels while higher levels are even not visible at all. We note that the almost independence of the fitted rotational temperature to T vib is true for the relatively high resolution of the present spectrum. Interdependence of T rot and T vib becomes larger for lower resolution spectra also for the ∆ν = 0 transition group.\nThe ∆ν = −1, ±2 transition groups usually suffer from low intensities because of their low transition probabilities. The resulting lower signal to noise ratios make these groups not prime candidates for temperatures determination. From figure 10 and 11 and the discussion before, it can be concluded that the C 2 Swan ∆ν = 0 transition group is the best candidate for the determination of T rot while ∆ν = ±1 should be used for the determination of T vib . We should however stress that the present \"error\"calculations made here cannot be generalized to any C 2 Swan band fitted spectrum but the trends will remain similar.\nThe ∆ν = −1, ±2 transition groups usually suffer from low intensities because of their low transition probabilities. The resulting lower signal to noise ratios make these groups not prime candidates for temperatures determination. From figure 10 and 11 and the discussion before, it can be concluded that the C 2 Swan ∆ν = 0 transition group is the best candidate for the determination of T rot while ∆ν = ±1 should be used for the determination of T vib . We should however stress that the present \"error\"calculations made here cannot be generalized to any C 2 Swan band fitted spectrum but the trends will remain similar.\n6 Is C 2 Swan band emission a good thermometer?\n6 Is C 2 Swan band emission a good thermometer?\nBefore presenting a detailed analysis of the gas temperature of the plasma torch, a comparative study of the temperatures of the C 2 Swan band and CN violet system is presented in section 6.1. The good agreement that is reported between these two species gives a hint that the temperatures measured are with their local environment and so correspond to the gas temperature T gas with a value of about 6000 K. At atmospheric pressure, we also measured the first negative and second positive system of N 2 while operating the present plasma torch in pure N 2 and obtained rotational and vibrational temperatures in the order of 6500 K [85]. The consistency in the obtained values (for high pressure, high power microwave plasmas) using different measurements methods are a first indication that the C 2 Swan band may also be used for gas temperature determination. As discussed in section 3, there are several pathways involving neutral species for the formation of the C 2 (d 3 Π g ) state. Depending of its formation mechanism, one can expect the nascent rovibrational distribution of an excited state formed by chemical reactions to follow a thermal distribution or to be strongly non-thermal (i.e. formation of the excited into specific rotational and or vibrational state(s). In the latter case, for being able to predict its emission spectrum, it is necessary to take into account collisional relaxation processes [86]. The rotational and vibrational temperatures do not then correspond to the gas temperature. In sections 6.2 and 6.3 are discussed the collisional relaxation kinetics of the C 2 (d 3 Π g ) state. These processes that lead to the thermalization of the C 2 (d 3 Π g ) state rotational and vibrational distribution functions define the cases when the C 2 Swan band emission spectrum can still be used to measure the gas temperature independently of the formation mechanism of the C 2 (d 3 Π g ) state.\nBefore presenting a detailed analysis of the gas temperature of the plasma torch, a comparative study of the temperatures of the C 2 Swan band and CN violet system is presented in section 6.1. The good agreement that is reported between these two species gives a hint that the temperatures measured are with their local environment and so correspond to the gas temperature T gas with a value of about 6000 K. At atmospheric pressure, we also measured the first negative and second positive system of N 2 while operating the present plasma torch in pure N 2 and obtained rotational and vibrational temperatures in the order of 6500 K [85]. The consistency in the obtained values (for high pressure, high power microwave plasmas) using different measurements methods are a first indication that the C 2 Swan band may also be used for gas temperature determination. As discussed in section 3, there are several pathways involving neutral species for the formation of the C 2 (d 3 Π g ) state. Depending of its formation mechanism, one can expect the nascent rovibrational distribution of an excited state formed by chemical reactions to follow a thermal distribution or to be strongly non-thermal (i.e. formation of the excited into specific rotational and or vibrational state(s). In the latter case, for being able to predict its emission spectrum, it is necessary to take into account collisional relaxation processes [86]. The rotational and vibrational temperatures do not then correspond to the gas temperature. In sections 6.2 and 6.3 are discussed the collisional relaxation kinetics of the C 2 (d 3 Π g ) state. These processes that lead to the thermalization of the C 2 (d 3 Π g ) state rotational and vibrational distribution functions define the cases when the C 2 Swan band emission spectrum can still be used to measure the gas temperature independently of the formation mechanism of the C 2 (d 3 Π g ) state."},{"title":"Comparison with the CN violet system","content":""},{"title":"Comparison with the CN violet system","content":"By adding a small amount of nitrogen, one can observe a strong decrease of the Swan band emission intensity while the CN violet system increase drastically. Vacher, Lino da Silva et al. [87,88] studied also the effect of adding N 2 to a CO 2 inductively coupled atmospheric pressure plasma and observed similar trends. Such behavior was also seen by other authors in microwave plasmas and low and atmospheric pressure [89,90]. This tendency can be explained by competing mechanisms for the formation of C 2 and CN that goes in favor of the formation of CN with increasing N 2 addition and for which it was found that C atoms are the main precursor [91]. CN (B-X) violet system is often used as a molecular pyrometer for determination of gas temperature in N 2 plasma sources with hydrocarbons [29, 92,93,94]. In order to generate CN violet system emission, 0.3 L/min of N 2 was added to the plasma corresponding to 3% admixture. This is a small quantity that, in the case of a pure CO 2 plasma, does not perturb the electron properties and neutral particles energy balance of the plasma.\nBy adding a small amount of nitrogen, one can observe a strong decrease of the Swan band emission intensity while the CN violet system increase drastically. Vacher, Lino da Silva et al. [87,88] studied also the effect of adding N 2 to a CO 2 inductively coupled atmospheric pressure plasma and observed similar trends. Such behavior was also seen by other authors in microwave plasmas and low and atmospheric pressure [89,90]. This tendency can be explained by competing mechanisms for the formation of C 2 and CN that goes in favor of the formation of CN with increasing N 2 addition and for which it was found that C atoms are the main precursor [91]. CN (B-X) violet system is often used as a molecular pyrometer for determination of gas temperature in N 2 plasma sources with hydrocarbons [29, 92,93,94]. In order to generate CN violet system emission, 0.3 L/min of N 2 was added to the plasma corresponding to 3% admixture. This is a small quantity that, in the case of a pure CO 2 plasma, does not perturb the electron properties and neutral particles energy balance of the plasma.\nIn an argon atmospheric pressure microwave plasma with nitrogen addition and CO 2 advection from ambient air, Ridenti and Amorim discussed the weak presence of the CN violet system that is strongly overlapping with nitrogen first negative and second positive systems [95]. The band head of the first negative system is well separated from the one of the CN violet system while N 2 second positive system has distinctive features that do not overlap either with the CN violet system. These two systems can therefore, using high resolution spectroscopy, easily be identified. Their presence was not detected in the present CO 2 microwave plasma with 3% N 2 addition. They are however observed in the present reactor while operated at atmospheric pressure with pure N 2 flow [85]. In figure 13, an example of CN spectrum and its resulting fitting are shown. A value of T rot =T vib = 6000 ± 500 K is obtained which are the same as for the Swan band system. Also the residue of the fit shows that no deviation from rotational or vibrational Boltzmann distribution can be observed experimentally. A comparison for several pressures between the measured rotational and vibrational temperatures obtained by fitting the two molecular bands systems is presented in figure 14. One can see that, within error bars, there are no significant differences for the two molecular systems and that the mean values agree between each other very well. Note that in the case of CN, also the rotational and vibrational temperatures were fitted independently (cf. method discussed in section 5.1). This confirms that the gas inside the plasma region where the light comes from is very hot with a temperature of about 6000 K both for rotational and vibrational internal degrees of freedom. The rovibrational distributions for those two systems can be essentially described with a single temperature.\nIn an argon atmospheric pressure microwave plasma with nitrogen addition and CO 2 advection from ambient air, Ridenti and Amorim discussed the weak presence of the CN violet system that is strongly overlapping with nitrogen first negative and second positive systems [95]. The band head of the first negative system is well separated from the one of the CN violet system while N 2 second positive system has distinctive features that do not overlap either with the CN violet system. These two systems can therefore, using high resolution spectroscopy, easily be identified. Their presence was not detected in the present CO 2 microwave plasma with 3% N 2 addition. They are however observed in the present reactor while operated at atmospheric pressure with pure N 2 flow [85]. In figure 13, an example of CN spectrum and its resulting fitting are shown. A value of T rot =T vib = 6000 ± 500 K is obtained which are the same as for the Swan band system. Also the residue of the fit shows that no deviation from rotational or vibrational Boltzmann distribution can be observed experimentally. A comparison for several pressures between the measured rotational and vibrational temperatures obtained by fitting the two molecular bands systems is presented in figure 14. One can see that, within error bars, there are no significant differences for the two molecular systems and that the mean values agree between each other very well. Note that in the case of CN, also the rotational and vibrational temperatures were fitted independently (cf. method discussed in section 5.1). This confirms that the gas inside the plasma region where the light comes from is very hot with a temperature of about 6000 K both for rotational and vibrational internal degrees of freedom. The rovibrational distributions for those two systems can be essentially described with a single temperature."},{"title":"Collisional relaxation of vibrational states","content":""},{"title":"Collisional relaxation of vibrational states","content":"C 2 Swan bands emission during pulsed laser irradiation of soot or graphite is often used for determination of gas temperature in (sub-)microsecond and even nanosecond time scales (see for instance [96,97]). The spectral resolution of such spectra is however usually low (because of the relative low intensity of such discharges) and a detailed evalution of the thermalization of rotational and vibrational states is difficult. UV pulsed photolysis experiments by Faust et al. [59] showed that the formation of the high pressure band is occuring only on long time scales (t>> 1µs) while for time up to 1µs the Swan band emission presented a normal vibrational distribution structure. They concluded that two different processes contribute to the formation of the C 2 Swan bands and that vibrational relaxation of the v=6 level is inefficient for populating the v=0 level of the d 3 Π g state.\nC 2 Swan bands emission during pulsed laser irradiation of soot or graphite is often used for determination of gas temperature in (sub-)microsecond and even nanosecond time scales (see for instance [96,97]). The spectral resolution of such spectra is however usually low (because of the relative low intensity of such discharges) and a detailed evalution of the thermalization of rotational and vibrational states is difficult. UV pulsed photolysis experiments by Faust et al. [59] showed that the formation of the high pressure band is occuring only on long time scales (t>> 1µs) while for time up to 1µs the Swan band emission presented a normal vibrational distribution structure. They concluded that two different processes contribute to the formation of the C 2 Swan bands and that vibrational relaxation of the v=6 level is inefficient for populating the v=0 level of the d 3 Π g state.\nBondybey [98] analyzed the C 2 Swan band fluorescence following two photon excitation in rare gas solids. He observed an erratic progression of lifetimes for different vibrational levels and concluded that the fast vibrational relaxation of the d 3 Π g v=3,4 states may occur through the b 3 Σ − g state, following a suggestion by Frosch [99]. Xuechu and Nanquan [100] studied the vibrational relaxation of the C 2 (d 3 Π g ) state by observing the chemiluminescence spectrum of the reaction Na + CCl 4 in a thermal bath of argon where the C 2 (d 3 Π g , v = 6) state is formed preferentially. From their pressure dependence study, they concluded that the thermal relaxation of the C 2 (d 3 Π g ) state by argon takes only a few collisions with a rate of 2.2 · 10 6 Torr −1 s −1 while they deduce even higher quenching rate by the sodium atoms.\nBondybey [98] analyzed the C 2 Swan band fluorescence following two photon excitation in rare gas solids. He observed an erratic progression of lifetimes for different vibrational levels and concluded that the fast vibrational relaxation of the d 3 Π g v=3,4 states may occur through the b 3 Σ − g state, following a suggestion by Frosch [99]. Xuechu and Nanquan [100] studied the vibrational relaxation of the C 2 (d 3 Π g ) state by observing the chemiluminescence spectrum of the reaction Na + CCl 4 in a thermal bath of argon where the C 2 (d 3 Π g , v = 6) state is formed preferentially. From their pressure dependence study, they concluded that the thermal relaxation of the C 2 (d 3 Π g ) state by argon takes only a few collisions with a rate of 2.2 · 10 6 Torr −1 s −1 while they deduce even higher quenching rate by the sodium atoms.\nSeveral experiments have shown that vibrational thermalization of the C 2 (d 3 Π g ) state is relatively slow and potentially as long if not longer than the lifetime of the state itself excepted for the rate reported above by Xuechu and Nanquan [100]. It may therefore be surprising to see that the vibrational distribution function follows closely a Maxwell-Boltzmann distribution in the experiments reported in this paper. However, the present measurements are done at high temperatures (i.e. T > 1000 K) and one may expect that the nascent population distribution follows already closely a thermal distribution, partly because of the large translational motion of the reactants. A temperature of 6000 K corresponds to a kinetic energy of 0.5 eV which is more than the energy separation between individual vibrational levels.\nSeveral experiments have shown that vibrational thermalization of the C 2 (d 3 Π g ) state is relatively slow and potentially as long if not longer than the lifetime of the state itself excepted for the rate reported above by Xuechu and Nanquan [100]. It may therefore be surprising to see that the vibrational distribution function follows closely a Maxwell-Boltzmann distribution in the experiments reported in this paper. However, the present measurements are done at high temperatures (i.e. T > 1000 K) and one may expect that the nascent population distribution follows already closely a thermal distribution, partly because of the large translational motion of the reactants. A temperature of 6000 K corresponds to a kinetic energy of 0.5 eV which is more than the energy separation between individual vibrational levels.\nRich and Bergman [101] performed vibrational excitation of CO at room temperature using a CO laser and observed formation of the C 1 Π g and d 3 Π g states of C 2 . They claimed that the observation of the C 1 Π g state (at the origin of the C 1 Π g -A 1 Π u Deslandres-D'Azambuja system) was the indication that CO (v) states can electronically excite the C 2 molecule. Indeed, the C 1 Π g state can be formed only via the association of two C ( 1 D) states and in their experiment only C ( 3 P) ground state atoms were produced. However, an analysis of the potential energy curves of C 2 shows that the C 1 Π g has an avoided crossing with another 1 Π g state and consequently dissociates into two ground state C( 3 P) atoms [48]. Grigorian and Cenian [102] recently performed a similar experiment and reported also the formation of the e 3 Π g state (e 3 Π g -a 3 Π u , Fox-Herzberg system) which have for dissociation limit the C ( 3 P) and C( 1 D) states [41]. Moreover, Wallaart et al. [50,103] reported during a CO vibrational excitation experiment the formation of the D 1 Σ + u state (D 1 Σ + u -X 1 Σ + g , Mulliken system) in addition to the aforementioned ones. The D 1 Σ + u state has for dissociation limit the C ( 1 S)+C( 1 D) states [41]. Although the initial experiments of Rich and Bergman did not allow to support their hypothesis of direct electronic excitation of C 2 electronic states by CO(v), subsequent measurements confirm their proposal. These observations go in favor to the hypothesis that vibrationally excited CO molecules can directly populate C 2 electronically excited states and that multiple vibrational excitation quanta may occur simultaneously. We note that Wallaart et al. reported also the observation of radiative transitions from electronically excited C atoms which could also be excited by CO(v). The presence of C 2 (D 1 Σ + u ) and C 2 (e 3 Π g ) states in the aforementioned studies could therefore also be explained by the recombination of C atoms which are electronically excited by CO(v) after being formed by the Boudouard mechanism (cf. reaction 1). The relatively low density of C* atoms compared to CO(v) states however makes it difficult to assess how important that channel is in reality. Hack and Langel [104] analysed the recombination of C( 1 D) atoms and observed the production of C 2 (e 3 Π g , C 1 Π g ) electronic states but no recombination rate coefficient was reported in the literature yet. The usual observation of C 2 electronic states formation in C/CO mixtures without detection of any electronically excited states of C atoms (at the exception of the work of Wallart et al) tends to support that efficient energy exchange mechanisms between CO(v) and C 2 exist. Such processes will favor rapid thermalization of the vibrational states of the C 2 molecule. Similarly, one can expect that the CO 2 molecule also interact with C 2 molecules in similar fashion due to its rich vibrational structure. Based on the present discussion, it can be concluded that either the formation mechanism of the C 2 (d 3 Π g ) state has already a nascent (nearly)-thermalized vibrational distribution (which seems unlikely based on its formation mechanisms discussed in section 3) or that, not only collisional quenching processes, but also reactive energy exchanges leads to its (faster) thermalization. This would explain the invariant observation of thermalized vibrational distributions of the C 2 (d 3 Π g ) state in CO 2 microwave plasmas.\nRich and Bergman [101] performed vibrational excitation of CO at room temperature using a CO laser and observed formation of the C 1 Π g and d 3 Π g states of C 2 . They claimed that the observation of the C 1 Π g state (at the origin of the C 1 Π g -A 1 Π u Deslandres-D'Azambuja system) was the indication that CO (v) states can electronically excite the C 2 molecule. Indeed, the C 1 Π g state can be formed only via the association of two C ( 1 D) states and in their experiment only C ( 3 P) ground state atoms were produced. However, an analysis of the potential energy curves of C 2 shows that the C 1 Π g has an avoided crossing with another 1 Π g state and consequently dissociates into two ground state C( 3 P) atoms [48]. Grigorian and Cenian [102] recently performed a similar experiment and reported also the formation of the e 3 Π g state (e 3 Π g -a 3 Π u , Fox-Herzberg system) which have for dissociation limit the C ( 3 P) and C( 1 D) states [41]. Moreover, Wallaart et al. [50,103] reported during a CO vibrational excitation experiment the formation of the D 1 Σ + u state (D 1 Σ + u -X 1 Σ + g , Mulliken system) in addition to the aforementioned ones. The D 1 Σ + u state has for dissociation limit the C ( 1 S)+C( 1 D) states [41]. Although the initial experiments of Rich and Bergman did not allow to support their hypothesis of direct electronic excitation of C 2 electronic states by CO(v), subsequent measurements confirm their proposal. These observations go in favor to the hypothesis that vibrationally excited CO molecules can directly populate C 2 electronically excited states and that multiple vibrational excitation quanta may occur simultaneously. We note that Wallaart et al. reported also the observation of radiative transitions from electronically excited C atoms which could also be excited by CO(v). The presence of C 2 (D 1 Σ + u ) and C 2 (e 3 Π g ) states in the aforementioned studies could therefore also be explained by the recombination of C atoms which are electronically excited by CO(v) after being formed by the Boudouard mechanism (cf. reaction 1). The relatively low density of C* atoms compared to CO(v) states however makes it difficult to assess how important that channel is in reality. Hack and Langel [104] analysed the recombination of C( 1 D) atoms and observed the production of C 2 (e 3 Π g , C 1 Π g ) electronic states but no recombination rate coefficient was reported in the literature yet. The usual observation of C 2 electronic states formation in C/CO mixtures without detection of any electronically excited states of C atoms (at the exception of the work of Wallart et al) tends to support that efficient energy exchange mechanisms between CO(v) and C 2 exist. Such processes will favor rapid thermalization of the vibrational states of the C 2 molecule. Similarly, one can expect that the CO 2 molecule also interact with C 2 molecules in similar fashion due to its rich vibrational structure. Based on the present discussion, it can be concluded that either the formation mechanism of the C 2 (d 3 Π g ) state has already a nascent (nearly)-thermalized vibrational distribution (which seems unlikely based on its formation mechanisms discussed in section 3) or that, not only collisional quenching processes, but also reactive energy exchanges leads to its (faster) thermalization. This would explain the invariant observation of thermalized vibrational distributions of the C 2 (d 3 Π g ) state in CO 2 microwave plasmas."},{"title":"Collisional relaxation of rotational states and gas temperature","content":""},{"title":"Collisional relaxation of rotational states and gas temperature","content":"Direct measurements of thermalization rates are usually difficult to obtain and indirect estimations are usually done. Using the assumption that the C 2 molecule would be formed rotationally hot and that the pressure dependence of the C 2 Swan band temperature can be used as estimate of the rotational relaxation/equilibration with the thermal bath of molecules, Bleekrode determined a rotational relaxation time of 3µs in an oxyacetylene flame at low pressure (2-120 Torr) [105]. Using the same hypotheses, Kini and Savadatti [46] estimated a rotational relaxation time for the Swan band of 30 µs in a CO discharge in a pressure range of 2-24 Torr. The underlying assumptions were particularly strong as no elementary process for the formation of rotationally hot C 2 were identified. However, Brockhinke et al [106] recently performed picosecond laser induced fluorescence measurements on the ∆ν = 0 transition group of the C 2 (d 3 Π g -a 3 Π u ) Swan. They managed to evaluate rotational energy transfer (RET) and vibrational energy transfer (VET) by monitoring the collisional transfer of single optically pumped rovibrational state of the d 3 Π g state to adjacent states by fluorescence. The measurements were performed in an oxy-acetylene flame with a gas temperature of 2200 K. Such measurements allow to monitor individual state to state RET and VET collisional transfers and are therefore very precise. They obtained state-to-state collisional re-distribution rate for RET in the order of k RET ≈ 4 · 10 9 s −1 .\nDirect measurements of thermalization rates are usually difficult to obtain and indirect estimations are usually done. Using the assumption that the C 2 molecule would be formed rotationally hot and that the pressure dependence of the C 2 Swan band temperature can be used as estimate of the rotational relaxation/equilibration with the thermal bath of molecules, Bleekrode determined a rotational relaxation time of 3µs in an oxyacetylene flame at low pressure (2-120 Torr) [105]. Using the same hypotheses, Kini and Savadatti [46] estimated a rotational relaxation time for the Swan band of 30 µs in a CO discharge in a pressure range of 2-24 Torr. The underlying assumptions were particularly strong as no elementary process for the formation of rotationally hot C 2 were identified. However, Brockhinke et al [106] recently performed picosecond laser induced fluorescence measurements on the ∆ν = 0 transition group of the C 2 (d 3 Π g -a 3 Π u ) Swan. They managed to evaluate rotational energy transfer (RET) and vibrational energy transfer (VET) by monitoring the collisional transfer of single optically pumped rovibrational state of the d 3 Π g state to adjacent states by fluorescence. The measurements were performed in an oxy-acetylene flame with a gas temperature of 2200 K. Such measurements allow to monitor individual state to state RET and VET collisional transfers and are therefore very precise. They obtained state-to-state collisional re-distribution rate for RET in the order of k RET ≈ 4 · 10 9 s −1 .\nDue to the high electronic quenching rate of the C 2 d 3 Π g states they could not quantify the VET rate but assessed that it was slower than k RET in agreement with aforementioned studies. The temperatures obtained by fitting the C 2 Swan band and the CN (B 2 Σ + -X 2 Σ + ) violet system have been compared and match very well as seen in figure 14. Contrary to the C 2 (d 3 Π g ) state, the CN (B 2 Σ + ) state does not have, to the best of our knowledge, known mechanisms leading to non-equilibrium rovibrational emission spectra [107]. Rotational relaxation times of the CN (B 2 Σ + ) state are also considerably shorter than its radiative lifetime [108,109]. For CN, the rotational and vibrational states distributions have time to thermalize before being observed by optical emission spectroscopy in continuous plasmas. The cross check between the rotational temperatures of the Swan band and the CN violet system gives additional confidence that their rotational temperature may be associated to the gas temperature. The reactive formation of molecules can only lead to an overpopulation of the high rotational levels of a molecule (because of energy exchange during the formation process) or to a thermal distribution. In the case of non-Boltzmann distribution of rotational levels, the temperature of the lower rotational levels is then usually taken as a measure of the gas (i.e. translational) temperature of the plasma [29]. In the present study (see figure 8a), two-temperature rotational distribution function are not found. Observing thermal distributions of rotational states of the C 2 (d 3 Π g ) state and knowing that its RET relaxation time is much shorter than its radiative lifetime gives confidence that T rot is a local measure of the translational temperature of the plasma. This is additionally supported by the equivalence between the rotational and vibrational temperatures both for the C 2 Swan bands and for the CN violet system. For high pressure CO 2 plasmas (i.e. typically 100 mbar and above), it can therefore be proposed to use the C 2 Swan bands rotational temperature as an evaluation tool for the local gas temperature of the plasma. We note however, that for a CO 2 plasma with a much lower gas temperature, similar analysis should be performed as in section 5.1.\nDue to the high electronic quenching rate of the C 2 d 3 Π g states they could not quantify the VET rate but assessed that it was slower than k RET in agreement with aforementioned studies. The temperatures obtained by fitting the C 2 Swan band and the CN (B 2 Σ + -X 2 Σ + ) violet system have been compared and match very well as seen in figure 14. Contrary to the C 2 (d 3 Π g ) state, the CN (B 2 Σ + ) state does not have, to the best of our knowledge, known mechanisms leading to non-equilibrium rovibrational emission spectra [107]. Rotational relaxation times of the CN (B 2 Σ + ) state are also considerably shorter than its radiative lifetime [108,109]. For CN, the rotational and vibrational states distributions have time to thermalize before being observed by optical emission spectroscopy in continuous plasmas. The cross check between the rotational temperatures of the Swan band and the CN violet system gives additional confidence that their rotational temperature may be associated to the gas temperature. The reactive formation of molecules can only lead to an overpopulation of the high rotational levels of a molecule (because of energy exchange during the formation process) or to a thermal distribution. In the case of non-Boltzmann distribution of rotational levels, the temperature of the lower rotational levels is then usually taken as a measure of the gas (i.e. translational) temperature of the plasma [29]. In the present study (see figure 8a), two-temperature rotational distribution function are not found. Observing thermal distributions of rotational states of the C 2 (d 3 Π g ) state and knowing that its RET relaxation time is much shorter than its radiative lifetime gives confidence that T rot is a local measure of the translational temperature of the plasma. This is additionally supported by the equivalence between the rotational and vibrational temperatures both for the C 2 Swan bands and for the CN violet system. For high pressure CO 2 plasmas (i.e. typically 100 mbar and above), it can therefore be proposed to use the C 2 Swan bands rotational temperature as an evaluation tool for the local gas temperature of the plasma. We note however, that for a CO 2 plasma with a much lower gas temperature, similar analysis should be performed as in section 5.1."},{"title":"Parametric analysis of the CO 2 microwave plasma torch","content":""},{"title":"Parametric analysis of the CO 2 microwave plasma torch","content":"Having assessed that C 2 Swan band can be used for measuring the gas temperature in a CO 2 plasma, we report in this section a parametric study of C 2 T rot and T vib as a function of plasma parameters. Although both T rot and T vib can be associated to the gas temperature, they are reported here independently. The measurements are taken in the pressure regime above 120 mbar where the C 2 Swan band emission is the dominant feature in the plasma emission spectrum. Concomitantly, the plasma is contracted with a diameter of about 3-8 mm which depends mainly of the applied power. The emission is spatially discriminated using a system of two pin-holes and a lens coupled to a fiber and collected over a area of about 1 mm 2 . The latter corresponds to the spatial resolution attainable in the present setup. Each data point corresponds to the light collected along the light of sight (radial direction) and over a plasma slab of 1 mm in the axial direction. Interestingly, in the contracted regime of the plasma torch (i.e. above a pressure of about 120 mbar), axial and radial measurements of the C 2 rotational temperature show no variation for all studied plasma conditions. After substraction of the continuous baseline, only a spatial variation of the overall intensity of the C 2 Swan band is observed within signal to noise sensitivity. In the following, all data are consequently reported for measurements in the center of the resonator; i.e 20 mm above the bottom of the waveguide while looking across the axis of the quartz tube. In figure 15, the effect of the input microwave power on the rotational and vibration temperatures of the C 2 Swan bands are presented. The measurements are done near atmospheric pressure plasma (925 mbar) for a gas flow of 10 L/min. One can see that both the rotational and vibrational temperatures of C 2 remain constant and equal within error bars. The increase in power has only the consequence of increasing the intensity of the radiation from the C 2 (d 3 Π g ) state. This increase of intensity is correlated with an increase in plasma radius. Also, by imaging it is observed that the plasma becomes elongated in the effluent region with increasing power.\nHaving assessed that C 2 Swan band can be used for measuring the gas temperature in a CO 2 plasma, we report in this section a parametric study of C 2 T rot and T vib as a function of plasma parameters. Although both T rot and T vib can be associated to the gas temperature, they are reported here independently. The measurements are taken in the pressure regime above 120 mbar where the C 2 Swan band emission is the dominant feature in the plasma emission spectrum. Concomitantly, the plasma is contracted with a diameter of about 3-8 mm which depends mainly of the applied power. The emission is spatially discriminated using a system of two pin-holes and a lens coupled to a fiber and collected over a area of about 1 mm 2 . The latter corresponds to the spatial resolution attainable in the present setup. Each data point corresponds to the light collected along the light of sight (radial direction) and over a plasma slab of 1 mm in the axial direction. Interestingly, in the contracted regime of the plasma torch (i.e. above a pressure of about 120 mbar), axial and radial measurements of the C 2 rotational temperature show no variation for all studied plasma conditions. After substraction of the continuous baseline, only a spatial variation of the overall intensity of the C 2 Swan band is observed within signal to noise sensitivity. In the following, all data are consequently reported for measurements in the center of the resonator; i.e 20 mm above the bottom of the waveguide while looking across the axis of the quartz tube. In figure 15, the effect of the input microwave power on the rotational and vibration temperatures of the C 2 Swan bands are presented. The measurements are done near atmospheric pressure plasma (925 mbar) for a gas flow of 10 L/min. One can see that both the rotational and vibrational temperatures of C 2 remain constant and equal within error bars. The increase in power has only the consequence of increasing the intensity of the radiation from the C 2 (d 3 Π g ) state. This increase of intensity is correlated with an increase in plasma radius. Also, by imaging it is observed that the plasma becomes elongated in the effluent region with increasing power.\nIn order to cool down a high pressure plasma, one possible way is by increasing the gas flow. In figure 16, the variation of the C 2 rotational temperature versus CO 2 gas flow at a pressure of about 850 mbar is shown for an input power of 900 W. One can see that the temperature remains constant around 6000± 500 K. Such trend was also observed by Sun et al. [27] and Spencer and Gallimore [24] but for lower flow rates variation. Similar lack of influence of the gas flow was also observed in the case of an atmospheric pressure microwave plasma torch in nitrogen [110]. Increasing the gas flow by more than 10-fold appears in the present setup (and also for other high pressure microwave sources) to be an inefficient way of tuning the gas temperature despite reducing the specific energy input per molecule by the same factor. Such result is rather surprising but may be due to an inefficient mixing between the hot and cold gas region.\nIn order to cool down a high pressure plasma, one possible way is by increasing the gas flow. In figure 16, the variation of the C 2 rotational temperature versus CO 2 gas flow at a pressure of about 850 mbar is shown for an input power of 900 W. One can see that the temperature remains constant around 6000± 500 K. Such trend was also observed by Sun et al. [27] and Spencer and Gallimore [24] but for lower flow rates variation. Similar lack of influence of the gas flow was also observed in the case of an atmospheric pressure microwave plasma torch in nitrogen [110]. Increasing the gas flow by more than 10-fold appears in the present setup (and also for other high pressure microwave sources) to be an inefficient way of tuning the gas temperature despite reducing the specific energy input per molecule by the same factor. Such result is rather surprising but may be due to an inefficient mixing between the hot and cold gas region.\nA typical method for enhancing the non-equilibrium of a plasma discharge (e.g. reducing the gas temperature while the electron temperature remains in the order of 1 eV or more) is to reduce the gas pressure. In figure 17, the pressure is varied from near atmospheric pressure down to 180 mbar where the C 2 Swan band is still intense enough for getting a good signal to noise ratio. One can see that T vib and T rot are equal to each other within absolute uncertainties. One should note that in pure CO 2 microwave plasmas, lower temperatures have been measured but only at (much) lower pressures than in the current study and where C 2 Swan band is not observed [89,111]. On the other hand, den Harder et al. [17] and Groen et al [83] measured gas temperature in the order of 5000 to 6000 K in the same pressure range as for the present study.\nA typical method for enhancing the non-equilibrium of a plasma discharge (e.g. reducing the gas temperature while the electron temperature remains in the order of 1 eV or more) is to reduce the gas pressure. In figure 17, the pressure is varied from near atmospheric pressure down to 180 mbar where the C 2 Swan band is still intense enough for getting a good signal to noise ratio. One can see that T vib and T rot are equal to each other within absolute uncertainties. One should note that in pure CO 2 microwave plasmas, lower temperatures have been measured but only at (much) lower pressures than in the current study and where C 2 Swan band is not observed [89,111]. On the other hand, den Harder et al. [17] and Groen et al [83] measured gas temperature in the order of 5000 to 6000 K in the same pressure range as for the present study."},{"title":"Conclusions and outlook","content":""},{"title":"Conclusions and outlook","content":"In this paper, the potential sources of C 2 molecules in CO 2 plasmas and the known pathways leading to non-equilibrium vibrational distributions of the (d 3 Π g ) state are reviewed. A detailed analysis of high resolution spectra is performed in order to detect potential deviations of the rovibrational distribution functions from thermal equilibrium in a high pressure microwave plasma discharge. State of the art molecular constants were used in order to fit the spectra. It is found that Boltzmann distributions can be used both for rotational and vibrational states for all conditions in the present study. Consequently, potential source terms of the C 2 (d 3 Π g ) state in a CO 2 plasma do not lead to any observed deviation from a Boltzmann vibrational distribution function. The sensitivity of different parts of the spectra for determining T rot and T vib is evaluated. For a fast and accurate determination of the rotational temperature of the C 2 (d 3 Π g ) state, it is shown that the ∆ν = 0 transition group is the most accurate part of the spectrum to be taken into account.\nIn this paper, the potential sources of C 2 molecules in CO 2 plasmas and the known pathways leading to non-equilibrium vibrational distributions of the (d 3 Π g ) state are reviewed. A detailed analysis of high resolution spectra is performed in order to detect potential deviations of the rovibrational distribution functions from thermal equilibrium in a high pressure microwave plasma discharge. State of the art molecular constants were used in order to fit the spectra. It is found that Boltzmann distributions can be used both for rotational and vibrational states for all conditions in the present study. Consequently, potential source terms of the C 2 (d 3 Π g ) state in a CO 2 plasma do not lead to any observed deviation from a Boltzmann vibrational distribution function. The sensitivity of different parts of the spectra for determining T rot and T vib is evaluated. For a fast and accurate determination of the rotational temperature of the C 2 (d 3 Π g ) state, it is shown that the ∆ν = 0 transition group is the most accurate part of the spectrum to be taken into account.\nDue to fast thermalization of the rotational states of the C 2 (d 3 Π g ) state with the background gas, the measured T rot can be used for assessing the gas temperature of the plasma. Furthermore, the vibrational states of the C 2 (d 3 Π g ) level are found to be thermalized indicating the possible presence of fast thermalization processes.\nDue to fast thermalization of the rotational states of the C 2 (d 3 Π g ) state with the background gas, the measured T rot can be used for assessing the gas temperature of the plasma. Furthermore, the vibrational states of the C 2 (d 3 Π g ) level are found to be thermalized indicating the possible presence of fast thermalization processes.\nA parametric study is performed on a pure CO 2 microwave plasma torch and the both the vibrational and rotational temperatures are found to be insensitive to external plasma parameters such as gas flow and input power at pressures of 180 mbar and above and a value of 6000 ± 500 K is found. The latter can be associated to the gas temperature of the plasma. These results are in agreement with the hypothesis that chemical equilibrium of the species can be described by the gas temperature at which thermal dissociation play an important role. The only observable differences in the extended parameter range of the present study are in the radially integrated density of the C 2 (d 3 Π g ) state. We note that similar values as the one presented in this study were reported before in a certain number of studies but no assessment of non-equilibrium distributions was done. Also, the rotational temperature of C 2 was implicitely assumed previously to be the one of the gas temperature and it is now shown that such approximation in high pressure CO 2 microwave discharges appears to hold.\nA parametric study is performed on a pure CO 2 microwave plasma torch and the both the vibrational and rotational temperatures are found to be insensitive to external plasma parameters such as gas flow and input power at pressures of 180 mbar and above and a value of 6000 ± 500 K is found. The latter can be associated to the gas temperature of the plasma. These results are in agreement with the hypothesis that chemical equilibrium of the species can be described by the gas temperature at which thermal dissociation play an important role. The only observable differences in the extended parameter range of the present study are in the radially integrated density of the C 2 (d 3 Π g ) state. We note that similar values as the one presented in this study were reported before in a certain number of studies but no assessment of non-equilibrium distributions was done. Also, the rotational temperature of C 2 was implicitely assumed previously to be the one of the gas temperature and it is now shown that such approximation in high pressure CO 2 microwave discharges appears to hold.\nThe C 2 (d 3 Π g ) state emits only in a narrow region in the center of the plasma and only the gas temperature in the core of the plasma can be measured. The region inside the plasma reactor that can be probed is only where C atoms are present and react to form C 2 (or CN when some N 2 is present) molecules. It was observed that C atoms are not observed by optical emission spectroscopy at pressures lower than the ones where the C 2 (d 3 Π g ) state is measured. To form carbon atoms, large power densities (i.e. high T vib and/or T gas ) are required as CO is the only species that can act as a primary source of carbon atoms. Interestingly, at the observed temperature of 6000 K, CO can already partially thermally dissociate [112]. This implies that observation of C and C 2 species is linked to regions with high gas temperatures where CO 2 molecules are readily dissociated.\nThe C 2 (d 3 Π g ) state emits only in a narrow region in the center of the plasma and only the gas temperature in the core of the plasma can be measured. The region inside the plasma reactor that can be probed is only where C atoms are present and react to form C 2 (or CN when some N 2 is present) molecules. It was observed that C atoms are not observed by optical emission spectroscopy at pressures lower than the ones where the C 2 (d 3 Π g ) state is measured. To form carbon atoms, large power densities (i.e. high T vib and/or T gas ) are required as CO is the only species that can act as a primary source of carbon atoms. Interestingly, at the observed temperature of 6000 K, CO can already partially thermally dissociate [112]. This implies that observation of C and C 2 species is linked to regions with high gas temperatures where CO 2 molecules are readily dissociated.\nThe measurements reported in this paper show the presence of a hot core of gas but no insight in the temperature gradient at the edge of the plasma could be obtained as not radiative emitting molecular species can be observed at the edges of this hot core. High gas temperatures T≥ 3500K allow already to dissociate fully the CO 2 and partially the CO molecules through thermal processes [18,6]. In situ diagnostics techniques like Raman scattering and CARS [113,114] would bring additional understanding in the gas dynamics at the boundary between the hot plasma core and the colder edge.\nThe measurements reported in this paper show the presence of a hot core of gas but no insight in the temperature gradient at the edge of the plasma could be obtained as not radiative emitting molecular species can be observed at the edges of this hot core. High gas temperatures T≥ 3500K allow already to dissociate fully the CO 2 and partially the CO molecules through thermal processes [18,6]. In situ diagnostics techniques like Raman scattering and CARS [113,114] would bring additional understanding in the gas dynamics at the boundary between the hot plasma core and the colder edge.\n[21] Yacine Babou, Philippe Rivire, Marie-Yvonne Perrin, and Anouar\n[21] Yacine Babou, Philippe Rivire, Marie-Yvonne Perrin, and Anouar\nSoufiani. Spectroscopic study of microwave plasmas of CO 2 and CO 2 -N 2 mixtures at atmospheric pressure. Plasma Sources Science and Technology, 17 (4) Figure 15: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of power input. The measurements are performed in the center of the resonator at 925 mbar with 10 L/min CO 2 gas flow. n is the laterally integrated density of the C 2 (d 3 Π g ) state. Figure 17: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of pressure. The measurements are performed in the center of the resonator using a 10 L/min CO 2 gas flow and 900 W input power.\nSoufiani. Spectroscopic study of microwave plasmas of CO 2 and CO 2 -N 2 mixtures at atmospheric pressure. Plasma Sources Science and Technology, 17 (4) Figure 15: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of power input. The measurements are performed in the center of the resonator at 925 mbar with 10 L/min CO 2 gas flow. n is the laterally integrated density of the C 2 (d 3 Π g ) state. Figure 17: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of pressure. The measurements are performed in the center of the resonator using a 10 L/min CO 2 gas flow and 900 W input power.\nOrigin B ν 10 6 · D ν 10 −2 · γ 0 25797.86825 (43)\nOrigin B ν 10 6 · D ν 10 −2 · γ 0 25797.86825 (43)"},{"title":"Figure 2 :","content":""},{"title":"Figure 2 :","content":""},{"title":"Figure 4 :","content":""},{"title":"Figure 4 :","content":""},{"title":"Figure 5 :","content":""},{"title":"Figure 5 :","content":""},{"title":"Figure 6 :","content":""},{"title":"Figure 6 :","content":""},{"title":"Figure 7 :Figure 10 :","content":""},{"title":"Figure 7 :Figure 10 :","content":""},{"title":"Figure 11 :","content":""},{"title":"Figure 11 :","content":""},{"title":"Figure 12 :","content":""},{"title":"Figure 12 :","content":""},{"title":"Figure 13 :Figure 14 :","content":""},{"title":"Figure 13 :Figure 14 :","content":""},{"title":"Figure 16 :","content":""},{"title":"Figure 16 :","content":""},{"title":"Table 1 :","content":""},{"title":"Table 1 :","content":""},{"title":"Table 2 :","content":""},{"title":"Table 2 :","content":""},{"title":"Table 3 :","content":""},{"title":"Table 3 :","content":""},{"title":"Table 4 :","content":""},{"title":"Table 4 :","content":""},{"title":"Table 5 :","content":""},{"title":"Table 5 :","content":""},{"title":"Table 6 :","content":""},{"title":"Table 6 :","content":""},{"title":"Table 7 :","content":""},{"title":"Table 7 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"\"\n },\n {\n \"title\": \"Introduction\",\n \"content\": \"Many strategies are being currently studied for the recycling and valorization of CO 2 , a potent greenhouse gas in the atmosphere [2]. Production of liquid fuels from renewable energies is one of the pathways for re-utilization of CO 2 . The first step for converting CO 2 into added value chemicals is its conversion into carbon monoxide CO [3,4]. For practical applications, it is desirable to operate with high gas flows near atmospheric pressure (or above) in order to avoid bulky components, reduce costs and save energy for the compression steps [5]. Previous studies have indicated that non-equilibrium microwave plasmas are particularly suitable for high energy efficiencies η conversion of CO 2 into CO with η ≤ 80% and can operate at high pressure and with high flow rates [6]. CO 2 based plasmas have applications in fields such as lasers [7], plasma polymers treatment [8] and also have potential in plasma medicine [9]. The proposed mechanism for explaining high energy efficiencies for CO 2 conversion into CO is via the so-called vibrational ladder climbing mechanism. Relatively low energy electrons excite preferentially the assymetric vibrational modes of the CO 2 ground state CO 2 (00i) molecule and vibrational energy exchange between CO 2 molecules leads ultimately to their decomposition [10]. To favor this mechanism the key ingredient is to keep the plasma out of equilibrium with the gas temperature T g much smaller than the effective electron temperature T e [11,12,13]. Low gas temperatures (i.e. T g 3000 K) are required in order to avoid too many losses of CO 2 (00i) states by collisions with the background gas leading to gas heating by vibrational to translation relaxation processes [14,15]. For high energy efficiencies, calculations also indicate that formed O atoms need to react with highly vibrationally excited CO 2 molecules and to produce another CO molecule and a O 2 molecule [16]. On the other hand, at high gas temperatures (i.e. T gas ≥ 3000 K), thermal dissociation is the dominant mechanism for CO 2 dissociation with high conversion rates and is limited to energy efficiencies up to ∼ 50 % [17,18].\\nMany strategies are being currently studied for the recycling and valorization of CO 2 , a potent greenhouse gas in the atmosphere [2]. Production of liquid fuels from renewable energies is one of the pathways for re-utilization of CO 2 . The first step for converting CO 2 into added value chemicals is its conversion into carbon monoxide CO [3,4]. For practical applications, it is desirable to operate with high gas flows near atmospheric pressure (or above) in order to avoid bulky components, reduce costs and save energy for the compression steps [5]. Previous studies have indicated that non-equilibrium microwave plasmas are particularly suitable for high energy efficiencies η conversion of CO 2 into CO with η ≤ 80% and can operate at high pressure and with high flow rates [6]. CO 2 based plasmas have applications in fields such as lasers [7], plasma polymers treatment [8] and also have potential in plasma medicine [9]. The proposed mechanism for explaining high energy efficiencies for CO 2 conversion into CO is via the so-called vibrational ladder climbing mechanism. Relatively low energy electrons excite preferentially the assymetric vibrational modes of the CO 2 ground state CO 2 (00i) molecule and vibrational energy exchange between CO 2 molecules leads ultimately to their decomposition [10]. To favor this mechanism the key ingredient is to keep the plasma out of equilibrium with the gas temperature T g much smaller than the effective electron temperature T e [11,12,13]. Low gas temperatures (i.e. T g 3000 K) are required in order to avoid too many losses of CO 2 (00i) states by collisions with the background gas leading to gas heating by vibrational to translation relaxation processes [14,15]. For high energy efficiencies, calculations also indicate that formed O atoms need to react with highly vibrationally excited CO 2 molecules and to produce another CO molecule and a O 2 molecule [16]. On the other hand, at high gas temperatures (i.e. T gas ≥ 3000 K), thermal dissociation is the dominant mechanism for CO 2 dissociation with high conversion rates and is limited to energy efficiencies up to ∼ 50 % [17,18].\\nFor studying the mechanisms limiting the plasma energy efficiency of the plasma, it is critical to have tools for assessing the degree of non-equilibrium in the plasma phase and notably its gas temperature. In high pressure (i.e. sub-up to atmospheric pressure) microwave CO 2 plasmas, one of the most dominant emission feature is the C 2 (d 3 Π g -a 3 Π u ) Swan bands system. In figure 1 an overview spectrum of a pure CO 2 microwave plasma (see section 2 for details on the plasma setup) is shown. In addition to a weak continuum emission 1 and typical atomic oxygen and carbon lines 2 , the C 2 Swan bands are the dominant molecular emission structure in the UV and visible range. Such spectrum is typical and has been measured by several authors [16,21,22,23,24,25,26,27,28]. To obtain reliable and accurate information about the plasma, it is then critical to have tools for evaluating the emission spectrum of the C 2 molecule and assess the accuracy of the results. Usually an equivalence between the rotational temperature of emitting species and the gas temperature is assumed but this hypothesis needs to be assessed. Also, in certain cases non-Boltzmann distributed rotational distribution function are measured experimentally [29]. In combustion studies, it has been observed that the C 2 Swan band can exhibit overpopulation of the high vibrational levels. The latter emission spectrum is known as the \\\"high pressure bands\\\"of the C 2 system [30]. This makes the analysis of C 2 spectra more complicated and can induce systematic errors in the determination of the rotational temperature (T rot ) of the C 2 (d 3 Π g ) state if the vibrational levels are assumed to take a Boltzmann distribution function. For CO 2 plasmas, an evaluation of potential deviations of C 2 Swan bands spectra from Boltzmann equilibrium and the assessment of T rot as a measure of the gas temperature have not been reported in the literature yet.\\nFor studying the mechanisms limiting the plasma energy efficiency of the plasma, it is critical to have tools for assessing the degree of non-equilibrium in the plasma phase and notably its gas temperature. In high pressure (i.e. sub-up to atmospheric pressure) microwave CO 2 plasmas, one of the most dominant emission feature is the C 2 (d 3 Π g -a 3 Π u ) Swan bands system. In figure 1 an overview spectrum of a pure CO 2 microwave plasma (see section 2 for details on the plasma setup) is shown. In addition to a weak continuum emission 1 and typical atomic oxygen and carbon lines 2 , the C 2 Swan bands are the dominant molecular emission structure in the UV and visible range. Such spectrum is typical and has been measured by several authors [16,21,22,23,24,25,26,27,28]. To obtain reliable and accurate information about the plasma, it is then critical to have tools for evaluating the emission spectrum of the C 2 molecule and assess the accuracy of the results. Usually an equivalence between the rotational temperature of emitting species and the gas temperature is assumed but this hypothesis needs to be assessed. Also, in certain cases non-Boltzmann distributed rotational distribution function are measured experimentally [29]. In combustion studies, it has been observed that the C 2 Swan band can exhibit overpopulation of the high vibrational levels. The latter emission spectrum is known as the \\\"high pressure bands\\\"of the C 2 system [30]. This makes the analysis of C 2 spectra more complicated and can induce systematic errors in the determination of the rotational temperature (T rot ) of the C 2 (d 3 Π g ) state if the vibrational levels are assumed to take a Boltzmann distribution function. For CO 2 plasmas, an evaluation of potential deviations of C 2 Swan bands spectra from Boltzmann equilibrium and the assessment of T rot as a measure of the gas temperature have not been reported in the literature yet.\\nIn this paper we present the spectral fitting analysis of the C 2 Swan band using a newly implemented spectroscopic database in the open source MassiveOES software [31,32]. The database is constructed using the recent compilation and calculations by Brooke et al. [1] of the line strengths and molecular constants for the C 2 (d 3 Π g -a 3 Π u ) Swan band system. C 2 Swan bands spectra measured on a 2.45 GHz CO 2 microwave plasma source are reported and carefully analyzed. By adding small amount of nitrogen in the plasma, the obtained temperatures from the Swan band fitting are compared with the one obtained by fitting the CN violet band system. Uncertainties related to the method used for fitting the C 2 spectrum, potential deviations from a Boltzmann distribution for the rotational and vibrational distribution functions and the assumption of fitting the spectra with a single or two temperatures (i.e. rotational and vibrational temperatures) are discussed and compared with a state-by-state fitting of the experimental spectra. Finally, currrent knowledge about rotational and vibrational relaxation processes of the C 2 (d 3 Π g ) state is presented in section 6 and the use of T rot for measuring the gas temperature of CO 2 microwave plasmas is discussed.\\nIn this paper we present the spectral fitting analysis of the C 2 Swan band using a newly implemented spectroscopic database in the open source MassiveOES software [31,32]. The database is constructed using the recent compilation and calculations by Brooke et al. [1] of the line strengths and molecular constants for the C 2 (d 3 Π g -a 3 Π u ) Swan band system. C 2 Swan bands spectra measured on a 2.45 GHz CO 2 microwave plasma source are reported and carefully analyzed. By adding small amount of nitrogen in the plasma, the obtained temperatures from the Swan band fitting are compared with the one obtained by fitting the CN violet band system. Uncertainties related to the method used for fitting the C 2 spectrum, potential deviations from a Boltzmann distribution for the rotational and vibrational distribution functions and the assumption of fitting the spectra with a single or two temperatures (i.e. rotational and vibrational temperatures) are discussed and compared with a state-by-state fitting of the experimental spectra. Finally, currrent knowledge about rotational and vibrational relaxation processes of the C 2 (d 3 Π g ) state is presented in section 6 and the use of T rot for measuring the gas temperature of CO 2 microwave plasmas is discussed.\"\n },\n {\n \"title\": \"Experimental setup\",\n \"content\": \"\"\n },\n {\n \"title\": \"Experimental setup\",\n \"content\": \"A microwave device which can ignite an atmospheric pressure plasma torch using only microwave power without the use of additional igniters [33] is studied. A modified version of this device has been built so that it can be operated from low (∼10 mbar) up to atmospheric pressure in molecular gases such as N 2 and CO 2 . Stable operation of the plasma is obtained both in continuous and pulsed regime in a large parameters range. The microwaves are coupled via a WR340 waveguide (TE 01 mode) into a cylindrical cavity with low Q-factor where the plasma is sustained inside a quartz tube (see figure 2 for more details). The quartz tube has inner/outer diameters of 26/30 mm. Figure 1: Overview spectra using a low resolution spectrometer (instrumental profile of about 1.2 nm) of the plasma emission at low and high pressures. The high pressure spectrum is characteristic for all the conditions of the present study where the plasma is in contracted mode which occurs for power ≥ 900 W at pressures above 120 mbar. Note that the spectra are not calibrated for the sensitivity response of the spectrometer and optics.\\nA microwave device which can ignite an atmospheric pressure plasma torch using only microwave power without the use of additional igniters [33] is studied. A modified version of this device has been built so that it can be operated from low (∼10 mbar) up to atmospheric pressure in molecular gases such as N 2 and CO 2 . Stable operation of the plasma is obtained both in continuous and pulsed regime in a large parameters range. The microwaves are coupled via a WR340 waveguide (TE 01 mode) into a cylindrical cavity with low Q-factor where the plasma is sustained inside a quartz tube (see figure 2 for more details). The quartz tube has inner/outer diameters of 26/30 mm. Figure 1: Overview spectra using a low resolution spectrometer (instrumental profile of about 1.2 nm) of the plasma emission at low and high pressures. The high pressure spectrum is characteristic for all the conditions of the present study where the plasma is in contracted mode which occurs for power ≥ 900 W at pressures above 120 mbar. Note that the spectra are not calibrated for the sensitivity response of the spectrometer and optics.\\nFor the ignition of the plasma, a coaxial cavity made of a waveguide to cylindrical resonator with high Q-factor and an inner conductor with conical shape is integrated at the bottom of the quartz tube. The coaxial cavity serves also for the injection of the gases using a 4-inlets tangential gas injection system. The coaxial conical pin vertical position can be adjusted for selecting different modes and enhance the electric field at its tip to ignite the plasma. The plasma can be operated in the power range of 600 W-3 kW, with gas flow rates between 5-100 L/min and pressures from 10 up to 960 mbar.\\nFor the ignition of the plasma, a coaxial cavity made of a waveguide to cylindrical resonator with high Q-factor and an inner conductor with conical shape is integrated at the bottom of the quartz tube. The coaxial cavity serves also for the injection of the gases using a 4-inlets tangential gas injection system. The coaxial conical pin vertical position can be adjusted for selecting different modes and enhance the electric field at its tip to ignite the plasma. The plasma can be operated in the power range of 600 W-3 kW, with gas flow rates between 5-100 L/min and pressures from 10 up to 960 mbar.\\nThe high resolution measurements were acquired using a Czerny-Turner 1 m focal length SPEX1000 spectrometer with an entrance slit of about 30 µm and a dispersion grating of 1800 g/mm. An Andor iDus 420 CCD camera was used to record the spectra. The spectral resolution in first order is 20 pm at 550 nm. The wavelength calibration of the spectrometer was done using a mercury lamp. The light is collected using a system of two irises and a lens. The lens focuses the light into an optical fiber system that is used for coupling the light to the entrance slit of the spectrometer. The irises system acts as a collimator and a spatial resolution of 1 mm is obtained. The collecting optics, filters and the spectrometer have been absolute calibrated between 400 nm to 850 nm with an Ulbricht sphere to obtain the absolute photon flux from the plasma. A longpass filter at 400 nm (FEL0400 from Thorlabs) is used to suppress any contribution from second order emission overlapping with the Swan band emission. For the CN violet system measurements, a 280 nm longpass filter (FEL280 from Thorlabs) is used. Additional low-resolution survey spectra are acquired using an Ocean Optics S2000 spectrometer with 1.2 nm resolution.\\nThe high resolution measurements were acquired using a Czerny-Turner 1 m focal length SPEX1000 spectrometer with an entrance slit of about 30 µm and a dispersion grating of 1800 g/mm. An Andor iDus 420 CCD camera was used to record the spectra. The spectral resolution in first order is 20 pm at 550 nm. The wavelength calibration of the spectrometer was done using a mercury lamp. The light is collected using a system of two irises and a lens. The lens focuses the light into an optical fiber system that is used for coupling the light to the entrance slit of the spectrometer. The irises system acts as a collimator and a spatial resolution of 1 mm is obtained. The collecting optics, filters and the spectrometer have been absolute calibrated between 400 nm to 850 nm with an Ulbricht sphere to obtain the absolute photon flux from the plasma. A longpass filter at 400 nm (FEL0400 from Thorlabs) is used to suppress any contribution from second order emission overlapping with the Swan band emission. For the CN violet system measurements, a 280 nm longpass filter (FEL280 from Thorlabs) is used. Additional low-resolution survey spectra are acquired using an Ocean Optics S2000 spectrometer with 1.2 nm resolution.\\n3 Sources of C 2 and of the C 2 (d 3 Π g ) state\\n3 Sources of C 2 and of the C 2 (d 3 Π g ) state\\nThe primary building blocks of C 2 are C atoms and it is therefore of interest to look first at sources of C atoms. No direct production channel from the CO 2 molecule is known by electron impact dissociation [34,35]. Electron impact dissociation processes of CO 2 lead to O and CO fragments either as neutral (ground or excited states). Electron dissociative attachment to CO 2 peaks at low electron energies < 10 eV and only produces CO [36] and O − . Electron impact ionization of CO 2 leading to formation of C + has a threshold cross section of 27.8 eV (compared to 13.8 eV for CO + 2 ) according to Itikawa [37] and can therefore be considered as negligible while considering typical electron energy distribution function in CO 2 plasmas [38]. Dissociative recombination of CO + 2 lead mostly to CO and O products and a small fraction of C atoms [39]. However, CO + 2 is lost by charge exchange and clustering reactions and is not a dominant ionic species in the plasma [40]. CO 2 species cannot then be considered as a primary source of C atoms and CO can be regarded as its primary source.\\nThe primary building blocks of C 2 are C atoms and it is therefore of interest to look first at sources of C atoms. No direct production channel from the CO 2 molecule is known by electron impact dissociation [34,35]. Electron impact dissociation processes of CO 2 lead to O and CO fragments either as neutral (ground or excited states). Electron dissociative attachment to CO 2 peaks at low electron energies < 10 eV and only produces CO [36] and O − . Electron impact ionization of CO 2 leading to formation of C + has a threshold cross section of 27.8 eV (compared to 13.8 eV for CO + 2 ) according to Itikawa [37] and can therefore be considered as negligible while considering typical electron energy distribution function in CO 2 plasmas [38]. Dissociative recombination of CO + 2 lead mostly to CO and O products and a small fraction of C atoms [39]. However, CO + 2 is lost by charge exchange and clustering reactions and is not a dominant ionic species in the plasma [40]. CO 2 species cannot then be considered as a primary source of C atoms and CO can be regarded as its primary source.\\nElectron impact dissociation and recombination processes of CO and CO + have been extensively reviewed (see for instance [42,39]) and we will focus here on neutral processes leading to the formation of C atoms or even directly to C 2 molecules. A reason for focusing on these processes is also the knowledge that, in high pressure discharges with high degree of vibrational excitation, neutral processes are expected to play an important role for molecules dissociation [6]. Moreover, in section 7 are reported high gas temperatures for which thermal (i.e. neutral) processes dominates dissociation dynamics [18].\\nElectron impact dissociation and recombination processes of CO and CO + have been extensively reviewed (see for instance [42,39]) and we will focus here on neutral processes leading to the formation of C atoms or even directly to C 2 molecules. A reason for focusing on these processes is also the knowledge that, in high pressure discharges with high degree of vibrational excitation, neutral processes are expected to play an important role for molecules dissociation [6]. Moreover, in section 7 are reported high gas temperatures for which thermal (i.e. neutral) processes dominates dissociation dynamics [18].\\nTo understand the formation of the C 2 d 3 Π g state, as it will be discussed in the following, not only source terms of C atoms but also of the C 2 O should be considered. Vibrationally excited CO molecules lead to the formation of carbon atoms via thermal decomposition [43] or via the so-called Boudouard mechanism [44]\\nTo understand the formation of the C 2 d 3 Π g state, as it will be discussed in the following, not only source terms of C atoms but also of the C 2 O should be considered. Vibrationally excited CO molecules lead to the formation of carbon atoms via thermal decomposition [43] or via the so-called Boudouard mechanism [44]\\nwhere v and w are vibrational quantum numbers. 3 This process is very 3 Alternatively of having two CO molecules in their ground state X 1 Σ g in the reaction 1, Figure 3: Schematic representation of the potential energy curves of the lower triplet electronic states of the C 2 molecule as well as of the 1 5 Π g state and their dissociation limit. For more information see ref. [41]. efficient and researchers have reported that discharges into carbon monoxide even at pressures of a few Torr lead to deposition of solid carbon on the walls and a decrease of the pressure in the closed cell [46] . A competing process to reaction 1 is the formation of a C 2 O molecule [45,47] CO(\\nwhere v and w are vibrational quantum numbers. 3 This process is very 3 Alternatively of having two CO molecules in their ground state X 1 Σ g in the reaction 1, Figure 3: Schematic representation of the potential energy curves of the lower triplet electronic states of the C 2 molecule as well as of the 1 5 Π g state and their dissociation limit. For more information see ref. [41]. efficient and researchers have reported that discharges into carbon monoxide even at pressures of a few Torr lead to deposition of solid carbon on the walls and a decrease of the pressure in the closed cell [46] . A competing process to reaction 1 is the formation of a C 2 O molecule [45,47] CO(\\nSuch processes are efficient only when significant vibrational excitation of CO molecules is present and/or high gas temperatures. This is the case in the present study with reported temperatures of 6000 K (see section 7) where CO 2 is thermally dissociated, and production of C atoms happens in the gas phase [6].\\nSuch processes are efficient only when significant vibrational excitation of CO molecules is present and/or high gas temperatures. This is the case in the present study with reported temperatures of 6000 K (see section 7) where CO 2 is thermally dissociated, and production of C atoms happens in the gas phase [6].\\nThe formation of the C 2 molecule from the three lowest electronic configurations of the carbon atom which are the 2s 2 2p 2 ( 3 P J , 1 D 2 and 1 S 0 ) states, where the ground state is the 3 P 0 state, was discussed by Ballik and Ramsay [48] while considering the potential energy curves of C 2 . In figure 3 are shown the lower triplet electronic levels of the C 2 molecule. For a recent overview of the potential energy curves of the C 2 molecule we refer to Martin [41]. The one of the species reacting can be a CO(a 3 Π,v) excited state formed from energy exchange between two CO (X 1 Σ g , v) molecules [45].\\nThe formation of the C 2 molecule from the three lowest electronic configurations of the carbon atom which are the 2s 2 2p 2 ( 3 P J , 1 D 2 and 1 S 0 ) states, where the ground state is the 3 P 0 state, was discussed by Ballik and Ramsay [48] while considering the potential energy curves of C 2 . In figure 3 are shown the lower triplet electronic levels of the C 2 molecule. For a recent overview of the potential energy curves of the C 2 molecule we refer to Martin [41]. The one of the species reacting can be a CO(a 3 Π,v) excited state formed from energy exchange between two CO (X 1 Σ g , v) molecules [45].\\nwhere M stands for a third body that stabilizes the transition state. They noted that there is an avoided crossing with the C 2 (e 3 Π g ) state that dissociate into C ( 3 P) + C( 1 D) and so that the C( 1 D) does not recombine into the C 2 (d 3 Π g ) state. The C 2 Swan band can then potentially be used as marker for the recombination of ground state C atoms.\\nwhere M stands for a third body that stabilizes the transition state. They noted that there is an avoided crossing with the C 2 (e 3 Π g ) state that dissociate into C ( 3 P) + C( 1 D) and so that the C( 1 D) does not recombine into the C 2 (d 3 Π g ) state. The C 2 Swan band can then potentially be used as marker for the recombination of ground state C atoms.\\nIn high pressure discharges, a preferential emission of the C 2 (d 3 Π g , v = 6) state can be observed which is known as the \\\"C 2 high pressure band\\\". C 2 O molecule was proposed by Kunz et al. [49] for explaining the preferential emission of the C 2 (d 3 Π g , v = 6) state. The C 2 O molecule 5 reacts with a C atom and forms an electronically excited C 2 molecule [50]\\nIn high pressure discharges, a preferential emission of the C 2 (d 3 Π g , v = 6) state can be observed which is known as the \\\"C 2 high pressure band\\\". C 2 O molecule was proposed by Kunz et al. [49] for explaining the preferential emission of the C 2 (d 3 Π g , v = 6) state. The C 2 O molecule 5 reacts with a C atom and forms an electronically excited C 2 molecule [50]\\nNaegeli and Palmer proposed that the initial electronic state in reaction 4 is the C 2 (b 3 Σ − g ) state that crosses to the C 2 (d 3 Π g ) state at v=6 and leads to the high pressure C 2 band [51]. Through a kinetic study of a CO plasma afterglow, Gosse et al. [52] supported the hypothesis that C 2 (d 3 Π g ) state is formed via reaction 4. Brewer et al [53] stated that formation of the high pressure band from a predissociation mechanism is unlikely and suggested that a resonant transfer from another molecule happens. However, Caubet and Dorthe [54], C 2 O may also be a source of the 5 Π g state via\\nNaegeli and Palmer proposed that the initial electronic state in reaction 4 is the C 2 (b 3 Σ − g ) state that crosses to the C 2 (d 3 Π g ) state at v=6 and leads to the high pressure C 2 band [51]. Through a kinetic study of a CO plasma afterglow, Gosse et al. [52] supported the hypothesis that C 2 (d 3 Π g ) state is formed via reaction 4. Brewer et al [53] stated that formation of the high pressure band from a predissociation mechanism is unlikely and suggested that a resonant transfer from another molecule happens. However, Caubet and Dorthe [54], C 2 O may also be a source of the 5 Π g state via\\nThey argued that based on the correlation diagram for the reactants and products of the reaction C 2 O + C that the 5 Π g state of C 2 is correlated to the ground state of the reactants while the d 3 Π g state is not. We note that previously, because of the absence of observation of C 2 O by absorption, Kini 4 In Ballik and Ramsay this state is denoted as the A 3 Π g state which is a common notation found for the Swan band in the literature with the lower state of the transition defined as X 3 Π u . 5 Note that in addition to reaction 2, the C 2 O molecule can be formed from the recombination of a C atom with CO via\\nThey argued that based on the correlation diagram for the reactants and products of the reaction C 2 O + C that the 5 Π g state of C 2 is correlated to the ground state of the reactants while the d 3 Π g state is not. We note that previously, because of the absence of observation of C 2 O by absorption, Kini 4 In Ballik and Ramsay this state is denoted as the A 3 Π g state which is a common notation found for the Swan band in the literature with the lower state of the transition defined as X 3 Π u . 5 Note that in addition to reaction 2, the C 2 O molecule can be formed from the recombination of a C atom with CO via\\nThe formation of C 2 O in pulsed radiolysis of CO was shown first by Willis and Devillers [47] and they showed that adding CO 2 did not affect the kinetics of C 2 O. Kinetics of this molecule discussed from kinetic analysis of pure CO discharges can then probably be extended to mixtures of CO/CO 2 . and Savadatti [46] proposed the following mechanism for the formation of the high pressure C 2 (d 3 Π g , v = 6) Swan band\\nThe formation of C 2 O in pulsed radiolysis of CO was shown first by Willis and Devillers [47] and they showed that adding CO 2 did not affect the kinetics of C 2 O. Kinetics of this molecule discussed from kinetic analysis of pure CO discharges can then probably be extended to mixtures of CO/CO 2 . and Savadatti [46] proposed the following mechanism for the formation of the high pressure C 2 (d 3 Π g , v = 6) Swan band\\nwhere the C 3 (m) molecule is a postulated metastable species with an energy of 3.7 eV. Little and Browne [55] argued against the possibility of the C 3 as intermediate species for the formation of the high pressure Swan band. Indeed, the C 2 high pressure band is observed in a large range of experimental conditions where C 3 formation is not favored and they also advanced a lack of evidence for a metastable state that should have an energy above thẽ A 1 Π u state which has 3.1 eV internal energy and is responsible for the C 3 comet band (Ã 1 Π u -X 1 Σ + g transition). Recent spectroscopic investigation and calculations of the energy levels of the C 3 molecule indicate that the lowest triplet state C 3 (ã 3 Π u ) has an energy of about 2.1 eV and theb 3 Π g an energy of 2.9 eV [56,57] which are well below the 3.7 eV energy required for the process 6. The C 3 molecule can therefore be ruled out as source of the C 2 high pressure band.\\nwhere the C 3 (m) molecule is a postulated metastable species with an energy of 3.7 eV. Little and Browne [55] argued against the possibility of the C 3 as intermediate species for the formation of the high pressure Swan band. Indeed, the C 2 high pressure band is observed in a large range of experimental conditions where C 3 formation is not favored and they also advanced a lack of evidence for a metastable state that should have an energy above thẽ A 1 Π u state which has 3.1 eV internal energy and is responsible for the C 3 comet band (Ã 1 Π u -X 1 Σ + g transition). Recent spectroscopic investigation and calculations of the energy levels of the C 3 molecule indicate that the lowest triplet state C 3 (ã 3 Π u ) has an energy of about 2.1 eV and theb 3 Π g an energy of 2.9 eV [56,57] which are well below the 3.7 eV energy required for the process 6. The C 3 molecule can therefore be ruled out as source of the C 2 high pressure band.\\nLittle and Browne [55] proposed a general mechanism for the formation of the high pressure Swan band via the following three steps\\nLittle and Browne [55] proposed a general mechanism for the formation of the high pressure Swan band via the following three steps\\nwhere the quintet 5 Π g is a metastable state for which a crossing exists between the v=6 level of the d 3 Π g state and the v=0 level of the quintet state. Kirby and Liu calculated its energy to lies 3.85 ± 0.34 eV above the ground state of C 2 [58] while Little and Browne estimated a value of 3.79 eV. Such mechanism essentially allow to explain previous observation that the high pressure band is rotationally colder than the \\\"normal\\\" Swan band emission that can be observed in the same discharge [59]. Also, high resolution analysis of the C 2 (d 3 Π g , v = 6) low rotation energy levels shows that both odd and even rotational numbers are perturbed with increase of intensity [60] which indicates that a Σ state is not involved in the perturbation but a state with Λ = 0 fills the lower rotational levels of the C 2 (d 3 Π g , v = 6) state [54]. A detailed analysis shows in fact that the b 3 Σ − g state perturb the J=19 and 21 of the v=6 upper state of the Swan band but lead to a decrease of the intensity of these lines [55]. It can be concluded that the b 3 Σ − g state is not at the origin of the high pressure band and that the quintet 5 Π g state is likely its source.\\nwhere the quintet 5 Π g is a metastable state for which a crossing exists between the v=6 level of the d 3 Π g state and the v=0 level of the quintet state. Kirby and Liu calculated its energy to lies 3.85 ± 0.34 eV above the ground state of C 2 [58] while Little and Browne estimated a value of 3.79 eV. Such mechanism essentially allow to explain previous observation that the high pressure band is rotationally colder than the \\\"normal\\\" Swan band emission that can be observed in the same discharge [59]. Also, high resolution analysis of the C 2 (d 3 Π g , v = 6) low rotation energy levels shows that both odd and even rotational numbers are perturbed with increase of intensity [60] which indicates that a Σ state is not involved in the perturbation but a state with Λ = 0 fills the lower rotational levels of the C 2 (d 3 Π g , v = 6) state [54]. A detailed analysis shows in fact that the b 3 Σ − g state perturb the J=19 and 21 of the v=6 upper state of the Swan band but lead to a decrease of the intensity of these lines [55]. It can be concluded that the b 3 Σ − g state is not at the origin of the high pressure band and that the quintet 5 Π g state is likely its source.\\nThe proposal that C atoms recombine via the quintet 5 Π g state which then populate the C 2 (d 3 Π g , v = 6) state (i.e. to generate the C 2 high pressure band) does not exclude the direct recombination of C atoms into the d 3 Π g state. One should see them as processes that happen simultaneously with different probabilities. Looking at the energy potential curves of the C 2 molecules, one can see that the\\nThe proposal that C atoms recombine via the quintet 5 Π g state which then populate the C 2 (d 3 Π g , v = 6) state (i.e. to generate the C 2 high pressure band) does not exclude the direct recombination of C atoms into the d 3 Π g state. One should see them as processes that happen simultaneously with different probabilities. Looking at the energy potential curves of the C 2 molecules, one can see that the\\n, d 3 Π g and 5 Π g all have as dissociation limit two C( 3 P) atoms [41]. The branching ratios for the recombination into these different electronic states are however not known although it appears from experiments discussed before that the d 3 Π g and 5 Π g are the ones first (and mainly) populated.\\n, d 3 Π g and 5 Π g all have as dissociation limit two C( 3 P) atoms [41]. The branching ratios for the recombination into these different electronic states are however not known although it appears from experiments discussed before that the d 3 Π g and 5 Π g are the ones first (and mainly) populated.\\nIn conclusion, we can say that there are several concurring production mechanisms of the C 2 (d 3 Π g ) state either from recombination of C atoms or, in the case of CO containing plasmas, via the C 2 O molecule. The presence or absence of some of these mecanisms for which branching can unfortunately hardly be estimated will affect particularly the nascent population distribution of vibrational states of the C 2 (d 3 Π g ) state.\\nIn conclusion, we can say that there are several concurring production mechanisms of the C 2 (d 3 Π g ) state either from recombination of C atoms or, in the case of CO containing plasmas, via the C 2 O molecule. The presence or absence of some of these mecanisms for which branching can unfortunately hardly be estimated will affect particularly the nascent population distribution of vibrational states of the C 2 (d 3 Π g ) state.\"\n },\n {\n \"title\": \"Molecular spectra calculations\",\n \"content\": \"\"\n },\n {\n \"title\": \"Molecular spectra calculations\",\n \"content\": \"In order to reproduce experimental spectra, it is necessary to first model the population distribution of states and, using molecular constants and transitions probabilities, one can simulate a spectrum knowing the instrumental profile of the apparatus. The partition functions of rovibrational distributions under the assumption of a Maxwell-Boltzmann distribution used to generate so-called Boltzmann plots are given in section 4.1. The molecular constants used for calculating both the C 2 Swan band and the CN violet system and implemented into MassiveOES and available as Supplementary Data are described in sections 4.2 and 4.3.\\nIn order to reproduce experimental spectra, it is necessary to first model the population distribution of states and, using molecular constants and transitions probabilities, one can simulate a spectrum knowing the instrumental profile of the apparatus. The partition functions of rovibrational distributions under the assumption of a Maxwell-Boltzmann distribution used to generate so-called Boltzmann plots are given in section 4.1. The molecular constants used for calculating both the C 2 Swan band and the CN violet system and implemented into MassiveOES and available as Supplementary Data are described in sections 4.2 and 4.3.\"\n },\n {\n \"title\": \"Thermal population distributions\",\n \"content\": \"\"\n },\n {\n \"title\": \"Thermal population distributions\",\n \"content\": \"The distribution of energy between the internal degree of freedom of a diatomic molecule is described by the following partition function [61] Q = e νmax ν=0 Jmax J=Λ g Λ,hf s (2S + 1) (2J + 1)\\nThe distribution of energy between the internal degree of freedom of a diatomic molecule is described by the following partition function [61] Q = e νmax ν=0 Jmax J=Λ g Λ,hf s (2S + 1) (2J + 1)\\nwhere e denotes electronic states with projection of orbital angular momentum quantum number Λ and total spin quantum number S, and ν and J are the vibrational and rotational quantum numbers, respectively. g Λ,hf s is the statistical weight factor related to Λ-doubling and the nuclear hyperfine structure. T e , G(ν) and F ν (J) represent the electronic, vibrational and rotational energies in wavenumber units. E 0 is the energy of the lowest state of the molecule (i.e. ν = 0, J = Λ for the ground electronic state). T is the temperature of the heat's bath in the case where all internal degrees of freedom of the molecular gas are thermalized.\\nwhere e denotes electronic states with projection of orbital angular momentum quantum number Λ and total spin quantum number S, and ν and J are the vibrational and rotational quantum numbers, respectively. g Λ,hf s is the statistical weight factor related to Λ-doubling and the nuclear hyperfine structure. T e , G(ν) and F ν (J) represent the electronic, vibrational and rotational energies in wavenumber units. E 0 is the energy of the lowest state of the molecule (i.e. ν = 0, J = Λ for the ground electronic state). T is the temperature of the heat's bath in the case where all internal degrees of freedom of the molecular gas are thermalized.\\nAssuming that the internal degrees of freedom are independent, which is already implicitly assumed in the formalism used to derive equation 7, 6 the partition function for rotational and vibrational states can be rewritten as\\nAssuming that the internal degrees of freedom are independent, which is already implicitly assumed in the formalism used to derive equation 7, 6 the partition function for rotational and vibrational states can be rewritten as\\nwhere T rot and T vib are the temperature for the rotational and vibrational distribution functions in the case that they do not equilibrate with each other but obey Boltzmann distribution functions.\\nwhere T rot and T vib are the temperature for the rotational and vibrational distribution functions in the case that they do not equilibrate with each other but obey Boltzmann distribution functions.\\nUsing the Maxwell-Boltzmann distribution law, stating that the number of molecules dN E having an energy between E and E + dE is proportional to e (−E/k B T ) dE and using equations 7 and 8, the population density for rotational and vibrational levels can be defined as [63,62]\\nUsing the Maxwell-Boltzmann distribution law, stating that the number of molecules dN E having an energy between E and E + dE is proportional to e (−E/k B T ) dE and using equations 7 and 8, the population density for rotational and vibrational levels can be defined as [63,62]\\nwhere N e is the total density of the electronic state e and N (ν) is the density of the ν level summed up over its rotational distribution.\\nwhere N e is the total density of the electronic state e and N (ν) is the density of the ν level summed up over its rotational distribution.\\nAssuming a Boltzmann distribution for the rotational and vibrational levels, the intensity of a line for a transition (ν → ν , J → J , Ω → Ω ) is given by\\nAssuming a Boltzmann distribution for the rotational and vibrational levels, the intensity of a line for a transition (ν → ν , J → J , Ω → Ω ) is given by\\nwhere ψ(λ − λ 0 ) is the convoluted line profile and instrumental profile of the spectrometer and λ 0 the central wavelength position of the line. E(ν) = G(ν)hc and E(J , Ω ) ≈ F ν (J ) where the effect on energy level splitting due to Λ-doubling is expressed as a function of Ω (cf. equation 11). We use here the convention following Herzberg [63] that the rotational and vibrational quantum numbers of the upper level are defined by the index J', v and the lower level is described by J\\\" and v .\\nwhere ψ(λ − λ 0 ) is the convoluted line profile and instrumental profile of the spectrometer and λ 0 the central wavelength position of the line. E(ν) = G(ν)hc and E(J , Ω ) ≈ F ν (J ) where the effect on energy level splitting due to Λ-doubling is expressed as a function of Ω (cf. equation 11). We use here the convention following Herzberg [63] that the rotational and vibrational quantum numbers of the upper level are defined by the index J', v and the lower level is described by J\\\" and v .\"\n },\n {\n \"title\": \"Swan band database: description, implementation, benchmark\",\n \"content\": \"\"\n },\n {\n \"title\": \"Swan band database: description, implementation, benchmark\",\n \"content\": \"C 2 is a homonuclear diatomic molecule and the Swan band is an electronically allowed dipole transition between two 3 Π states due to change of symmetry of the wavefunction between the upper and lower state g → u. These are triplet states and for each rotational quantum number J, there are sub-bands\\nC 2 is a homonuclear diatomic molecule and the Swan band is an electronically allowed dipole transition between two 3 Π states due to change of symmetry of the wavefunction between the upper and lower state g → u. These are triplet states and for each rotational quantum number J, there are sub-bands\\nThe sub-bands are dominated by P and R-branches (corresponding to selection rule ∆J = J − J = −1 and +1 respectively). Only weak Q-branches (∆J = 0) exist and none for the 3 Π 0 − 3 Π 0 sub-band as only ∆J = ±1 transitions occur for Ω = 0 where Ω is the total electronic angular momentum along the internuclear axis is designated with Ω = Λ + Σ. Σ is the projection of the spin quantum number S onto the internuclear axis. Λ is the quantum number for the component of the total angular momentum along the internuclear axis. It is the absolute value of the projected component of the angular momentum L on the internuclear axis M L with Λ =| M L | [64]. Π states have by definition Λ = 1 and for the Swan band we have triplet states d 3 Π g -a 3 Π u for which S=1 and so Σ = M S = −1, 0, +1. The coupling between the rotation of the nuclei and L additionally leads to a splitting into two components for each J value when Λ = 0. This is called Λ-doubling (see below). As the C 2 molecule is symmetric and the carbon atom has a zero nuclear spin, transition between symmetric and asymmetric wavefunctions are forbidden. The statistical weight due to nuclear spin I of the anti-symmetric rotational levels is given by I(I+1) and as for C 2 we have I=0, they are absent. However, for Λ = 1, we have symmetric and antisymmetric combination of wave functions due to Λ-doubling and therefore both even and odd rotational numbers are present. In the P, Q and R branches, only one of the Λ-doubled levels are present alternatively which leads to the appearance that the even levels are displaced to one side and the odd levels to the other side of a mean position for a given branch progression.\\nThe sub-bands are dominated by P and R-branches (corresponding to selection rule ∆J = J − J = −1 and +1 respectively). Only weak Q-branches (∆J = 0) exist and none for the 3 Π 0 − 3 Π 0 sub-band as only ∆J = ±1 transitions occur for Ω = 0 where Ω is the total electronic angular momentum along the internuclear axis is designated with Ω = Λ + Σ. Σ is the projection of the spin quantum number S onto the internuclear axis. Λ is the quantum number for the component of the total angular momentum along the internuclear axis. It is the absolute value of the projected component of the angular momentum L on the internuclear axis M L with Λ =| M L | [64]. Π states have by definition Λ = 1 and for the Swan band we have triplet states d 3 Π g -a 3 Π u for which S=1 and so Σ = M S = −1, 0, +1. The coupling between the rotation of the nuclei and L additionally leads to a splitting into two components for each J value when Λ = 0. This is called Λ-doubling (see below). As the C 2 molecule is symmetric and the carbon atom has a zero nuclear spin, transition between symmetric and asymmetric wavefunctions are forbidden. The statistical weight due to nuclear spin I of the anti-symmetric rotational levels is given by I(I+1) and as for C 2 we have I=0, they are absent. However, for Λ = 1, we have symmetric and antisymmetric combination of wave functions due to Λ-doubling and therefore both even and odd rotational numbers are present. In the P, Q and R branches, only one of the Λ-doubled levels are present alternatively which leads to the appearance that the even levels are displaced to one side and the odd levels to the other side of a mean position for a given branch progression.\\nWith each alternate line that is simply absent because of I=0, this is often referred to as the \\\"staggering\\\"effect. We note that the statistic weight (g Λ,hf s in equation 7) due to the nuclear spin for individual rotational levels induce an alternance of intensities for sub-bands of electronic transitions with Λ = 0.\\nWith each alternate line that is simply absent because of I=0, this is often referred to as the \\\"staggering\\\"effect. We note that the statistic weight (g Λ,hf s in equation 7) due to the nuclear spin for individual rotational levels induce an alternance of intensities for sub-bands of electronic transitions with Λ = 0.\\nFor more details on the rovibrational structure and lines positions of the C 2 Swan band system we refer to Herzberg [63] and Pellerin [65].\\nFor more details on the rovibrational structure and lines positions of the C 2 Swan band system we refer to Herzberg [63] and Pellerin [65].\\nIn figure 4, a high resolution spectrum from 450 nm up to 565 nm is presented where the main transition groups ∆ν and the position of the vibrational transition band heads are identified. Transitions between two electronic states can be ordered following the difference in vibrational quantum numbers during the radiative transition between the upper and lower state ∆ν = v − v . For instance, in figure 4 the transition group ∆ν = +1 corresponds to the rovibrational serie\\nIn figure 4, a high resolution spectrum from 450 nm up to 565 nm is presented where the main transition groups ∆ν and the position of the vibrational transition band heads are identified. Transitions between two electronic states can be ordered following the difference in vibrational quantum numbers during the radiative transition between the upper and lower state ∆ν = v − v . For instance, in figure 4 the transition group ∆ν = +1 corresponds to the rovibrational serie\\nlevel is therefore not part of this transition group. The ∆ν = −2, +2 transition groups can also be seen experimentally but are not shown here. 1.0x10 19 1.5x10 19 2.0x10 19 2.5x10 19 1-0 The spectroscopic database that we implemented into the fitting program of molecular emission spectra MassiveOES [31] is derived from the work of Brooke et al. [1]. As initial input, MassiveOES requires rovibrationally resolved Einstein coefficients for the transitions between the upper and lower state that can be generally calculated using the following equation [66]:\\nlevel is therefore not part of this transition group. The ∆ν = −2, +2 transition groups can also be seen experimentally but are not shown here. 1.0x10 19 1.5x10 19 2.0x10 19 2.5x10 19 1-0 The spectroscopic database that we implemented into the fitting program of molecular emission spectra MassiveOES [31] is derived from the work of Brooke et al. [1]. As initial input, MassiveOES requires rovibrationally resolved Einstein coefficients for the transitions between the upper and lower state that can be generally calculated using the following equation [66]:\\nwhere S ∆J J is the Höln-London factor and\\nwhere S ∆J J is the Höln-London factor and\\nBrooke et al. [1] used PGOPHER, a program developed by Colin Western, for calculating the Einstein A (v ,J )→(v ,J ) coefficients [67]. PGOPHER requires as input a set of molecular constants and the band strengths for each vibrational band v → v . The purely vibrational transition dipole moments (TDMs) were calculated using the program LEVEL [68], a program that can solve the 1D Schrödinger equation for diatomic molecules when provided with a potential energy curve and the electronic TDM. The set of molecular constants for fitting RKR potential curves of the upper and lower electronic levels involved into the Swan band transition are listed in [1]. With the output of LEVEL, PGOPHER is then able to calculate the rotational TDMs and the Höln-London factors required to compute the numerical values of equation 12. The original simulations by Brooke et al were extended to a maximum rotational quantum number J of 300 by running again PGOPHER with the same molecular constants. This allows to properly calculate the rotational partition function in MassiveOES for the high rotational temperatures encountered in the present study. It should however be noted that, while doing this, the calculated energy levels have increasing error for increasing J in energy and line positions. The molecular constants used for the C 2 Swan band are given in Appendix A and the output calculation of PGOPHER as Supplementary Material to this paper.\\nBrooke et al. [1] used PGOPHER, a program developed by Colin Western, for calculating the Einstein A (v ,J )→(v ,J ) coefficients [67]. PGOPHER requires as input a set of molecular constants and the band strengths for each vibrational band v → v . The purely vibrational transition dipole moments (TDMs) were calculated using the program LEVEL [68], a program that can solve the 1D Schrödinger equation for diatomic molecules when provided with a potential energy curve and the electronic TDM. The set of molecular constants for fitting RKR potential curves of the upper and lower electronic levels involved into the Swan band transition are listed in [1]. With the output of LEVEL, PGOPHER is then able to calculate the rotational TDMs and the Höln-London factors required to compute the numerical values of equation 12. The original simulations by Brooke et al were extended to a maximum rotational quantum number J of 300 by running again PGOPHER with the same molecular constants. This allows to properly calculate the rotational partition function in MassiveOES for the high rotational temperatures encountered in the present study. It should however be noted that, while doing this, the calculated energy levels have increasing error for increasing J in energy and line positions. The molecular constants used for the C 2 Swan band are given in Appendix A and the output calculation of PGOPHER as Supplementary Material to this paper.\\nExtensive comparison of calculated line positions versus experimental ones were already done in the literature (see Brooke et al. [1] and references therein). We additionally performed a cross-check with the high resolution degenerate four-wave mixing experiments of Lloyd and Stewart [69]. The difference between their measured line positions for the ∆ν = 0 transition group and our calculated ones are shown in figure 5. One can see that, excepted for two outliers, the line positions are calculated with an accuracy much better than ± 0.05 cm −1 . Experimental data for higher J numbers is however not available for performing further benchmark. Experimentally we have a line profile of 22 pm corresponding to about 0.8 cm −1 which is much broader than the line position uncertainty λ err < 1 pm. Consequently, using calculated line positions for the fitting of the experimental spectra induces no additional error related to uncertainties in line position assignment. We note at this point that high resolution measurements of line positions were done up to J = 80 and only for some vibrational levels by Tanabashi et al. [70].\\nExtensive comparison of calculated line positions versus experimental ones were already done in the literature (see Brooke et al. [1] and references therein). We additionally performed a cross-check with the high resolution degenerate four-wave mixing experiments of Lloyd and Stewart [69]. The difference between their measured line positions for the ∆ν = 0 transition group and our calculated ones are shown in figure 5. One can see that, excepted for two outliers, the line positions are calculated with an accuracy much better than ± 0.05 cm −1 . Experimental data for higher J numbers is however not available for performing further benchmark. Experimentally we have a line profile of 22 pm corresponding to about 0.8 cm −1 which is much broader than the line position uncertainty λ err < 1 pm. Consequently, using calculated line positions for the fitting of the experimental spectra induces no additional error related to uncertainties in line position assignment. We note at this point that high resolution measurements of line positions were done up to J = 80 and only for some vibrational levels by Tanabashi et al. [70].\\nThe database of lines and their transition probabilities is compiled in a SQL database, that is then read by MassiveOES. The database consists of three tables for listing the lines, the identity of rovibrational \\\"upper\\\" and \\\"lower\\\" states respectively. Each element of the table corresponds to a single line in the spectrum for which energy of the transition, air wavelength, vacuum wavelength, Einstein coefficient, upper state identity and lower state identity are given. The \\\"upper state\\\" tag (as well as the \\\"lower state\\\" tag) links the line with the energy level from which is originated, that will be used to calculate the intensity of the line. The effect of molecular perturbations ) and 1 5 Π g states [1] are combined in the table as additional transitions from the upper state, such that, the rovibrational population distribution is calculated without extra energy levels.\\nThe database of lines and their transition probabilities is compiled in a SQL database, that is then read by MassiveOES. The database consists of three tables for listing the lines, the identity of rovibrational \\\"upper\\\" and \\\"lower\\\" states respectively. Each element of the table corresponds to a single line in the spectrum for which energy of the transition, air wavelength, vacuum wavelength, Einstein coefficient, upper state identity and lower state identity are given. The \\\"upper state\\\" tag (as well as the \\\"lower state\\\" tag) links the line with the energy level from which is originated, that will be used to calculate the intensity of the line. The effect of molecular perturbations ) and 1 5 Π g states [1] are combined in the table as additional transitions from the upper state, such that, the rovibrational population distribution is calculated without extra energy levels.\"\n },\n {\n \"title\": \"CN violet system emission\",\n \"content\": \"\"\n },\n {\n \"title\": \"CN violet system emission\",\n \"content\": \"The CN molecular violet system corresponds to the B 2 Σ + → X 2 Σ + transition of CN which is an heteronuclear diatomic molecule. The required molecular constants have been taken from S. Ram et al. [71]. The transition probabilities between two vibrational states calculated by solving the integral of the vibrational wavefunctions of the upper and lower states and TDM Ψ ν (r)R e (r)Ψ ν (r) were extracted from LIFBASE [72]. The molecular constants for calculating the Rydberg-Klein-Rees (RKR) potential curves of the X 2 Σ + and B 2 Σ + states were taken from [73,74]). For the TDMs R e (r), the values of Ito et al. [75] were taken. These molecular constants were then used as input for PGOPHER to calculate the Höln-London factors of the CN violet system with J number up to 180. For the CN violet system some perturbations also exist but those are not included in the calculations as they do not affect significantly the spectra. In Appendix B are given all the constants used in the calculation. As sanity check, a one to one comparison between our PGOPHER calculations and LIFBASE was made and they yield the same results both for line positions and intensities.\\nThe CN molecular violet system corresponds to the B 2 Σ + → X 2 Σ + transition of CN which is an heteronuclear diatomic molecule. The required molecular constants have been taken from S. Ram et al. [71]. The transition probabilities between two vibrational states calculated by solving the integral of the vibrational wavefunctions of the upper and lower states and TDM Ψ ν (r)R e (r)Ψ ν (r) were extracted from LIFBASE [72]. The molecular constants for calculating the Rydberg-Klein-Rees (RKR) potential curves of the X 2 Σ + and B 2 Σ + states were taken from [73,74]). For the TDMs R e (r), the values of Ito et al. [75] were taken. These molecular constants were then used as input for PGOPHER to calculate the Höln-London factors of the CN violet system with J number up to 180. For the CN violet system some perturbations also exist but those are not included in the calculations as they do not affect significantly the spectra. In Appendix B are given all the constants used in the calculation. As sanity check, a one to one comparison between our PGOPHER calculations and LIFBASE was made and they yield the same results both for line positions and intensities.\\n5 Spectral analysis of the C 2 Swan band 5\\n5 Spectral analysis of the C 2 Swan band 5\"\n },\n {\n \"title\": \".1 Non-Boltzmann fitting of the Swan band\",\n \"content\": \"\"\n },\n {\n \"title\": \".1 Non-Boltzmann fitting of the Swan band\",\n \"content\": \"In non-equilibrium plasmas, rotational and vibrational distributions do not always thermalize because of competition between production and destruction processes and a molecular spectrum can then not be fitted using single rotational T rot and vibrational T vib temperatures [29]. As discussed in section 3, the C 2 ( 3 Π g ) state is well known for overpopulations in the density of its high vibrational levels and particularly the v = 6 level. In this section we investigate high resolution spectra of the C 2 Swan band in order to assess possible deviations from thermal equilibrium of the rovibrational populations.\\nIn non-equilibrium plasmas, rotational and vibrational distributions do not always thermalize because of competition between production and destruction processes and a molecular spectrum can then not be fitted using single rotational T rot and vibrational T vib temperatures [29]. As discussed in section 3, the C 2 ( 3 Π g ) state is well known for overpopulations in the density of its high vibrational levels and particularly the v = 6 level. In this section we investigate high resolution spectra of the C 2 Swan band in order to assess possible deviations from thermal equilibrium of the rovibrational populations.\\nIn figure 6, individual P and R lines from the v = 0 level are identified (the Q lines are too weak to be distinguished, see section 4.2) from the ∆ν = 0 transition group and their area fitted with a Voigt profile (with gaussian and lorentzian half-width of respectively σ G =9.5 pm and σ L = 4 pm) to take into account the apparatus profile determined from Hg lines using a low pressure Hg lamp. They were selected only when they can be separated from neighboring lines and clearly identified. Note that the fine structure of the rotational levels could not be resolved and so that the lines are weighted over the rotational fine structure (i.e. rotational sub-bands). The logarithm of the density of each individual rotational level is plotted as function of the rotational energy (i.e. Boltzmann plot). All levels fit on a single line which can be fitted with a rotational temperature of 6200±160 K. Here and in the following, we will quote for each fitting method, its related statistical errors. No deviation is observed neither for low or high rotational levels indicating that the levels are thermalized. We note here that levels with approximately J>120 do not contribute noticably to the emission spectra and are not visible.\\nIn figure 6, individual P and R lines from the v = 0 level are identified (the Q lines are too weak to be distinguished, see section 4.2) from the ∆ν = 0 transition group and their area fitted with a Voigt profile (with gaussian and lorentzian half-width of respectively σ G =9.5 pm and σ L = 4 pm) to take into account the apparatus profile determined from Hg lines using a low pressure Hg lamp. They were selected only when they can be separated from neighboring lines and clearly identified. Note that the fine structure of the rotational levels could not be resolved and so that the lines are weighted over the rotational fine structure (i.e. rotational sub-bands). The logarithm of the density of each individual rotational level is plotted as function of the rotational energy (i.e. Boltzmann plot). All levels fit on a single line which can be fitted with a rotational temperature of 6200±160 K. Here and in the following, we will quote for each fitting method, its related statistical errors. No deviation is observed neither for low or high rotational levels indicating that the levels are thermalized. We note here that levels with approximately J>120 do not contribute noticably to the emission spectra and are not visible.\\nThe thermalization of the rotational levels of the d 3 Π g state is further assessed using the option in MassiveOES of state-by-state fitting [32]. State by state fitting here means that the fitting routine evaluates the density of each upper rotational state while minimizing the residue for all transitions from the same upper state (i.e. different transition groups, P, Q and R branches and Λ-doubling). Allowing to fit independently the density of all rotational states helps to reduce the overall residue of the fit but only marginally. In figure 7, the population density of the rotational levels versus their energy is plotted. All rotational population densities fit quite well on a single line indicating no deviation from a thermal rotational distribution function. The lowest rotational levels are not shown because they are experimentally too much overlapped and the state by state fitting routine cannot converge (see below for a discussion of those). The highest rotational levels are widely scattered due to the very low signal to noise ratio. A rotational temperature T rot =5500± 500 K is obtained from the fitting of figure 7. From this first fit, some systematic deviations are apparent. A closer inspection indicates the presence of some outliers points which have densities off by an order of magnitude or more without any systematic trend. Removing these outliers which have a non-neglible weight for the fitting of a linear slope in logarithmic plot, the statistical uncertainty of the Boltzmann plot is significantly reduced. The inverse slope of the Boltzmann plot gives a temperature of 6200±150 K. This result agrees with the line by line Boltzmann plot analysis made in figure 6. Fitting the complete ∆ν = 0 transition group while assuming a Boltzmann distribution (i.e envelope fitting) for rotational and vibrational distribution gives similarly a temperature of 6000±200 K in excellent agreement with the state to state fitting analysis. The result of the fit using a Boltzmann distribution is also shown in figure 7 (note the logscale) as well and one can see that the residues following a Boltzmann fitting or a state by state fitting are similar.\\nThe thermalization of the rotational levels of the d 3 Π g state is further assessed using the option in MassiveOES of state-by-state fitting [32]. State by state fitting here means that the fitting routine evaluates the density of each upper rotational state while minimizing the residue for all transitions from the same upper state (i.e. different transition groups, P, Q and R branches and Λ-doubling). Allowing to fit independently the density of all rotational states helps to reduce the overall residue of the fit but only marginally. In figure 7, the population density of the rotational levels versus their energy is plotted. All rotational population densities fit quite well on a single line indicating no deviation from a thermal rotational distribution function. The lowest rotational levels are not shown because they are experimentally too much overlapped and the state by state fitting routine cannot converge (see below for a discussion of those). The highest rotational levels are widely scattered due to the very low signal to noise ratio. A rotational temperature T rot =5500± 500 K is obtained from the fitting of figure 7. From this first fit, some systematic deviations are apparent. A closer inspection indicates the presence of some outliers points which have densities off by an order of magnitude or more without any systematic trend. Removing these outliers which have a non-neglible weight for the fitting of a linear slope in logarithmic plot, the statistical uncertainty of the Boltzmann plot is significantly reduced. The inverse slope of the Boltzmann plot gives a temperature of 6200±150 K. This result agrees with the line by line Boltzmann plot analysis made in figure 6. Fitting the complete ∆ν = 0 transition group while assuming a Boltzmann distribution (i.e envelope fitting) for rotational and vibrational distribution gives similarly a temperature of 6000±200 K in excellent agreement with the state to state fitting analysis. The result of the fit using a Boltzmann distribution is also shown in figure 7 (note the logscale) as well and one can see that the residues following a Boltzmann fitting or a state by state fitting are similar.\\nIn figure 8a, the state by state fitting analysis is extended to the ∆ν = −1 and ∆ν = +1 transition groups for the rotational and vibrational states of the d 3 Π g state. One can see that the rotational levels of the v=1 and v=2 levels also follow a Boltzmann distribution with the same slope as for the ν = 0 level although the signal is more noisy. Also population densities of the lower rotational states can be analysis and no systematic deviations from a single slope can be found.\\nIn figure 8a, the state by state fitting analysis is extended to the ∆ν = −1 and ∆ν = +1 transition groups for the rotational and vibrational states of the d 3 Π g state. One can see that the rotational levels of the v=1 and v=2 levels also follow a Boltzmann distribution with the same slope as for the ν = 0 level although the signal is more noisy. Also population densities of the lower rotational states can be analysis and no systematic deviations from a single slope can be found.\\nAfter integration over the rotational distribution the density of the vibrational states can be obtained (cf. equation 9). The resulting Boltzmann plot for the vibrational states is shown in figure 8b. Their population distribution can be described by a single temperature of 7200± 1200 K. For comparison, we also performed a state by state fitting of the ∆ν = +1 transition group that is the most sensitive to the vibrational temperature (see section 5.2). The rotational temperature was fixed to 6200 K following the analysis of the ∆ν = 0 transition group. For the ∆ν = +1 transition group, the density of each ν level is adjusted independently and the results are shown in figure 9. One can see that all vibrational levels fit perfectly on a single line with T vib =6000± 400 K. The larger uncertainties in the case of the state by state fitting when all three transition groups ∆ν = +1, 0, −1 are taken into account stems from the lower signal to noise ratio for the ∆ν = −1 transition group. The result of the fitting of the 3 transition groups for the vibrational distribution of the C 2 d 3 Π g state shown in figure 8b may let someone thing that the C 2 Swan band system follows a two-temperature vibrational distribution function. The vibrational state to state analysis of the most sensitive ∆ν = +1 transition group done in figure 9 (which is more sensitive to T vib as discussed in section 5.2) show that there is no evidence of a twotemperatures vibrational states distribution function. Fitting the ∆ν = +1 transition group alone, which is the one usually measured in the literature for studying the so-called C 2 high pressure bands as it is the most sensitive for determining vibrational states densities, show that the vibrational distribution function of the d 3 Π g state is very well described by a single temperature. Considering the noise in the state by state ro-vibrational fitting shown in figure 8a for ν = 1 and 2 levels, it is in fact quite remarkable to see that the fitted vibrational temperature still agrees with the more precise fitting given in figure 9 within their own error bars. Note that for the vibrational state to state fitting of the ∆ν = +1 transition group, the rotational temperature was set using the T rot obtained from the more sensitive ∆ν = 0 transition group.\\nAfter integration over the rotational distribution the density of the vibrational states can be obtained (cf. equation 9). The resulting Boltzmann plot for the vibrational states is shown in figure 8b. Their population distribution can be described by a single temperature of 7200± 1200 K. For comparison, we also performed a state by state fitting of the ∆ν = +1 transition group that is the most sensitive to the vibrational temperature (see section 5.2). The rotational temperature was fixed to 6200 K following the analysis of the ∆ν = 0 transition group. For the ∆ν = +1 transition group, the density of each ν level is adjusted independently and the results are shown in figure 9. One can see that all vibrational levels fit perfectly on a single line with T vib =6000± 400 K. The larger uncertainties in the case of the state by state fitting when all three transition groups ∆ν = +1, 0, −1 are taken into account stems from the lower signal to noise ratio for the ∆ν = −1 transition group. The result of the fitting of the 3 transition groups for the vibrational distribution of the C 2 d 3 Π g state shown in figure 8b may let someone thing that the C 2 Swan band system follows a two-temperature vibrational distribution function. The vibrational state to state analysis of the most sensitive ∆ν = +1 transition group done in figure 9 (which is more sensitive to T vib as discussed in section 5.2) show that there is no evidence of a twotemperatures vibrational states distribution function. Fitting the ∆ν = +1 transition group alone, which is the one usually measured in the literature for studying the so-called C 2 high pressure bands as it is the most sensitive for determining vibrational states densities, show that the vibrational distribution function of the d 3 Π g state is very well described by a single temperature. Considering the noise in the state by state ro-vibrational fitting shown in figure 8a for ν = 1 and 2 levels, it is in fact quite remarkable to see that the fitted vibrational temperature still agrees with the more precise fitting given in figure 9 within their own error bars. Note that for the vibrational state to state fitting of the ∆ν = +1 transition group, the rotational temperature was set using the T rot obtained from the more sensitive ∆ν = 0 transition group.\\nFrom this analysis, it appears that for the present plasma conditions, the vibrational levels are completely thermalized and no overpopulation of the higher vibrational levels and more particularly v=6, occurs. Analysis of other spectra as the one discussed above yield the same results and no deviations from Boltzmann equilibrium either for the rotational or vibrational levels could be found for the conditions of the present study (cf. section 7). Either the processes discussion in section 3 leading to the so-called C 2 high pressure bands are absent in the present discharge or the levels thermalize within the effective (radiative) lifetime of the state (see section 6 for a discussion). Different methods of fitting give similar results and no evidence of deviations from thermal distributions can be found within error margins. For the following, Boltzmann distributions are then assumed and an envelope fitting of the spectra is done. For measuring the rotational temperature, the ∆ν = 0 transition group is chosen while all three transition groups are used for the determination of the vibrational temperature. For more discussion on this, we refer to section 5.2.\\nFrom this analysis, it appears that for the present plasma conditions, the vibrational levels are completely thermalized and no overpopulation of the higher vibrational levels and more particularly v=6, occurs. Analysis of other spectra as the one discussed above yield the same results and no deviations from Boltzmann equilibrium either for the rotational or vibrational levels could be found for the conditions of the present study (cf. section 7). Either the processes discussion in section 3 leading to the so-called C 2 high pressure bands are absent in the present discharge or the levels thermalize within the effective (radiative) lifetime of the state (see section 6 for a discussion). Different methods of fitting give similar results and no evidence of deviations from thermal distributions can be found within error margins. For the following, Boltzmann distributions are then assumed and an envelope fitting of the spectra is done. For measuring the rotational temperature, the ∆ν = 0 transition group is chosen while all three transition groups are used for the determination of the vibrational temperature. For more discussion on this, we refer to section 5.2.\"\n },\n {\n \"title\": \"T rot and T vib from the C 2 Swan band: sensitivity and uncertainty analysis\",\n \"content\": \"\"\n },\n {\n \"title\": \"T rot and T vib from the C 2 Swan band: sensitivity and uncertainty analysis\",\n \"content\": \"The convergence of a fit is usually assessed by looking at the value of the least mean square error and minimizing the root mean squared error (RM SE = i (y i − y i ) 2 ). It is possible to use the computed residue for assessing how sensitive is the minimization routine for given external parameters that are being fitted. Information can be extracted from a dataset (here an experimental spectrum) only when the variation of signal is statistically significant. It usually means that the correlation between a signal and the event we want to measure must be significantly larger that the statistical noise. Theoretical spectra can be computed for given fixed rotational and vibrational temperatures and compared with another theoretical spectrum in terms of their respective RMSEs [76]. In the case where the (thermal) white noise is the dominant source of uncertainty in the fitting of an experimental spectrum, the amplitude of the noise can be directly correlated to an iso-line inside which the \\\"true value\\\" stands. Nassar [77] evaluated the errors induced by a white noise on T rot determination while fitting the band head of the ∆ν = 0 transition group and obtained errors in the order of the signal to noise ratio or lower. This is however without considering systematic errors that are induced by performing mathematical operations on the data such a baseline correction or intensity calibration. These errors are additive and amplify the uncertainties on the values obtained through a fit. Also, one needs to take additionally into account the uncertainties in the molecular constants (Franck-Condon and Höln-London factors).\\nThe convergence of a fit is usually assessed by looking at the value of the least mean square error and minimizing the root mean squared error (RM SE = i (y i − y i ) 2 ). It is possible to use the computed residue for assessing how sensitive is the minimization routine for given external parameters that are being fitted. Information can be extracted from a dataset (here an experimental spectrum) only when the variation of signal is statistically significant. It usually means that the correlation between a signal and the event we want to measure must be significantly larger that the statistical noise. Theoretical spectra can be computed for given fixed rotational and vibrational temperatures and compared with another theoretical spectrum in terms of their respective RMSEs [76]. In the case where the (thermal) white noise is the dominant source of uncertainty in the fitting of an experimental spectrum, the amplitude of the noise can be directly correlated to an iso-line inside which the \\\"true value\\\" stands. Nassar [77] evaluated the errors induced by a white noise on T rot determination while fitting the band head of the ∆ν = 0 transition group and obtained errors in the order of the signal to noise ratio or lower. This is however without considering systematic errors that are induced by performing mathematical operations on the data such a baseline correction or intensity calibration. These errors are additive and amplify the uncertainties on the values obtained through a fit. Also, one needs to take additionally into account the uncertainties in the molecular constants (Franck-Condon and Höln-London factors).\\nIn figure 10, such analysis is first made for the ∆ν = 0 alone and for the ∆ν = −1, 0, +1 transition groups of the C 2 Swan band by comparing simulated spectra with theoretical spectra with T rot = T vib taking values of 1000, 3000, 5000, 6000 and 7500 K with 5% surimposed noise. The steepness of the RMSE around the equilibrium point (i.e. true value) obtained by the minimization routine is a good indication of accuracy with which a parameter can be fitted (here T rot and T vib ). It appears clearly that the ∆ν = 0 is rather imprecise for the determination of the vibrational temperature of the d 3 Π g state and that typical errors are in the order of several 1000 K. The minimum for the rotational temperature is on the contrary quite steep indicating that the band has a good sensitivity for T rot determination. The ∆ν = −1, +1 transition groups have on the other hand a much higher sensitivity to T vib . A comparison between the ∆ν = 0 fit and the fit of all three transition groups seems to indicate at first sight that both allow to obtain the rotational temperature with rather good sensitivity. However, in the case of ∆ν = −1, 0, +1 transition groups fit, one can see that the isolines are stretched on a diagonal (i.e. not perpendicular to the axes) which means that T rot and T vib are correlated during the minimization (i.e. if an error is made in the parameters minimization used for the determination of one variable, the error will propagate to the other). The parametrization of the spectrum in terms of independent variables (T rot , T vib , instrumental profile,...) in a feature space (i.e. a space made of independent variables that are combinations of the original variables) is not entirely successful and some of the new variables are not orthogonal. This is a well known issue in the field of Principal Components Analysis (PCA) [78]. In the present case, when the isolines are equidistant in the vertical and horizontal directions, one can say that the determination of T rot and T vib are statistically independent. Note that a fitting routine gives for any given fit a RMSE but its absolute value carries no intrinsic information. It has a meaning only in correlation with other RMSEs calculated around the (local) equilibrium point of the fit. The RMS error is then directly normalized to the equilibrium value.\\nIn figure 10, such analysis is first made for the ∆ν = 0 alone and for the ∆ν = −1, 0, +1 transition groups of the C 2 Swan band by comparing simulated spectra with theoretical spectra with T rot = T vib taking values of 1000, 3000, 5000, 6000 and 7500 K with 5% surimposed noise. The steepness of the RMSE around the equilibrium point (i.e. true value) obtained by the minimization routine is a good indication of accuracy with which a parameter can be fitted (here T rot and T vib ). It appears clearly that the ∆ν = 0 is rather imprecise for the determination of the vibrational temperature of the d 3 Π g state and that typical errors are in the order of several 1000 K. The minimum for the rotational temperature is on the contrary quite steep indicating that the band has a good sensitivity for T rot determination. The ∆ν = −1, +1 transition groups have on the other hand a much higher sensitivity to T vib . A comparison between the ∆ν = 0 fit and the fit of all three transition groups seems to indicate at first sight that both allow to obtain the rotational temperature with rather good sensitivity. However, in the case of ∆ν = −1, 0, +1 transition groups fit, one can see that the isolines are stretched on a diagonal (i.e. not perpendicular to the axes) which means that T rot and T vib are correlated during the minimization (i.e. if an error is made in the parameters minimization used for the determination of one variable, the error will propagate to the other). The parametrization of the spectrum in terms of independent variables (T rot , T vib , instrumental profile,...) in a feature space (i.e. a space made of independent variables that are combinations of the original variables) is not entirely successful and some of the new variables are not orthogonal. This is a well known issue in the field of Principal Components Analysis (PCA) [78]. In the present case, when the isolines are equidistant in the vertical and horizontal directions, one can say that the determination of T rot and T vib are statistically independent. Note that a fitting routine gives for any given fit a RMSE but its absolute value carries no intrinsic information. It has a meaning only in correlation with other RMSEs calculated around the (local) equilibrium point of the fit. The RMS error is then directly normalized to the equilibrium value.\\nWhile fitting an experimental spectrum using the assumption of Maxwell-Boltzmann distribution for the rotational and vibrational states of the radiating molecule, there are additional parameters that need to be fixed. They are listed into Table 1. These parameters, excepted for the first three in the table are all experimental and depends of the spectrometer. A linear baseline is fitted with an offset value B base and a slope α base for substracting contributions related to noise and continuum emission from the plasma. The instrumental profile of the spectrometer is taken into account via σ L and σ G while the wavelength calibration of the spectrum is adjusted with a quadratic fit (λ start , ∆λ and ∆λ 2 parameters) and the wavelength λ is defined as\\nWhile fitting an experimental spectrum using the assumption of Maxwell-Boltzmann distribution for the rotational and vibrational states of the radiating molecule, there are additional parameters that need to be fixed. They are listed into Table 1. These parameters, excepted for the first three in the table are all experimental and depends of the spectrometer. A linear baseline is fitted with an offset value B base and a slope α base for substracting contributions related to noise and continuum emission from the plasma. The instrumental profile of the spectrometer is taken into account via σ L and σ G while the wavelength calibration of the spectrum is adjusted with a quadratic fit (λ start , ∆λ and ∆λ 2 parameters) and the wavelength λ is defined as\\nwhere i is the pixel number. The density N of the upper excited state has no influence on the evaluation of the temperatures as it is related only to the integrated band intensity.\\nwhere i is the pixel number. The density N of the upper excited state has no influence on the evaluation of the temperatures as it is related only to the integrated band intensity.\\nThe parameters that have an influence on the determination of T rot and T vib are the line profile and the fitting of the baseline. By varying systematically these values and considering uncertainties on the Franck-Condon (FC) and Höln-London factors, we estimated conservative uncertainty margins of about ±500 K for the spectra fitted in the present study. This evaluation is confirmed by the analysis made in section 5.1 where statistical methods have been applied while using different analysis methods to evaluate the rovibrational distribution functions. The errors for the rotational temperature are in fact probably lower. Fitting of theoretical low resolution spectra generated using SPECAIR [79,80] Wavelength of the first point in the spectra ∆λ Distance between two points in the spectra ∆λ 2 Quadratic correction of the position of a point in the spectra difference in the order of 1% for the rotational temperature. For the vibrational temperature, we also performed a comparison using the transition dipole moments of Kokkin et al [82] and FC-factors obtained by Fantz and Wünderlich [81]. The main difference with the present values and those calculations are only in the absolute values of the FC-factors with a difference of about 20%. The latter difference is systematic and consequently only affects the determination of the density N of the upper electronic state (i.e. after summation over rotational and vibrational population densities). No significant difference for the determination of T vib while using the two datasets were found. In figure 11, the same analysis is made for an experimental spectrum. The minimization routine of MassiveOES is here used only for adjusting the area of the spectra (i.e. population density of the electronic state), baseline and linewidth. The spectra are calibrated in wavelength beforehand using the position of known lines and its contribution to fitting uncertainties can be in the present case (i.e high resolution) neglected. The results are similar to figure 10 and it can be concluded that the C 2 Swan ∆ν = 0 transition group is overall more sensitive and \\\"precise\\\"for the determination of T rot .\\nThe parameters that have an influence on the determination of T rot and T vib are the line profile and the fitting of the baseline. By varying systematically these values and considering uncertainties on the Franck-Condon (FC) and Höln-London factors, we estimated conservative uncertainty margins of about ±500 K for the spectra fitted in the present study. This evaluation is confirmed by the analysis made in section 5.1 where statistical methods have been applied while using different analysis methods to evaluate the rovibrational distribution functions. The errors for the rotational temperature are in fact probably lower. Fitting of theoretical low resolution spectra generated using SPECAIR [79,80] Wavelength of the first point in the spectra ∆λ Distance between two points in the spectra ∆λ 2 Quadratic correction of the position of a point in the spectra difference in the order of 1% for the rotational temperature. For the vibrational temperature, we also performed a comparison using the transition dipole moments of Kokkin et al [82] and FC-factors obtained by Fantz and Wünderlich [81]. The main difference with the present values and those calculations are only in the absolute values of the FC-factors with a difference of about 20%. The latter difference is systematic and consequently only affects the determination of the density N of the upper electronic state (i.e. after summation over rotational and vibrational population densities). No significant difference for the determination of T vib while using the two datasets were found. In figure 11, the same analysis is made for an experimental spectrum. The minimization routine of MassiveOES is here used only for adjusting the area of the spectra (i.e. population density of the electronic state), baseline and linewidth. The spectra are calibrated in wavelength beforehand using the position of known lines and its contribution to fitting uncertainties can be in the present case (i.e high resolution) neglected. The results are similar to figure 10 and it can be concluded that the C 2 Swan ∆ν = 0 transition group is overall more sensitive and \\\"precise\\\"for the determination of T rot .\\nFinally, in figure 12, the sensitivity of the ∆ν = 0 transition group for determining T rot on assumed/fitted values of T vib is assessed. An experimental spectrum for the ∆ν = 0 transition group is fitted while taking different values for T vib . One can see that the fitted rotational temperature is almost independent of the vibrational temperature. This is due to the low transition probabilities for the ∆ν = 0 transitions for the v=1,2 levels while higher levels are even not visible at all. We note that the almost independence of the fitted rotational temperature to T vib is true for the relatively high resolution of the present spectrum. Interdependence of T rot and T vib becomes larger for lower resolution spectra also for the ∆ν = 0 transition group.\\nFinally, in figure 12, the sensitivity of the ∆ν = 0 transition group for determining T rot on assumed/fitted values of T vib is assessed. An experimental spectrum for the ∆ν = 0 transition group is fitted while taking different values for T vib . One can see that the fitted rotational temperature is almost independent of the vibrational temperature. This is due to the low transition probabilities for the ∆ν = 0 transitions for the v=1,2 levels while higher levels are even not visible at all. We note that the almost independence of the fitted rotational temperature to T vib is true for the relatively high resolution of the present spectrum. Interdependence of T rot and T vib becomes larger for lower resolution spectra also for the ∆ν = 0 transition group.\\nThe ∆ν = −1, ±2 transition groups usually suffer from low intensities because of their low transition probabilities. The resulting lower signal to noise ratios make these groups not prime candidates for temperatures determination. From figure 10 and 11 and the discussion before, it can be concluded that the C 2 Swan ∆ν = 0 transition group is the best candidate for the determination of T rot while ∆ν = ±1 should be used for the determination of T vib . We should however stress that the present \\\"error\\\"calculations made here cannot be generalized to any C 2 Swan band fitted spectrum but the trends will remain similar.\\nThe ∆ν = −1, ±2 transition groups usually suffer from low intensities because of their low transition probabilities. The resulting lower signal to noise ratios make these groups not prime candidates for temperatures determination. From figure 10 and 11 and the discussion before, it can be concluded that the C 2 Swan ∆ν = 0 transition group is the best candidate for the determination of T rot while ∆ν = ±1 should be used for the determination of T vib . We should however stress that the present \\\"error\\\"calculations made here cannot be generalized to any C 2 Swan band fitted spectrum but the trends will remain similar.\\n6 Is C 2 Swan band emission a good thermometer?\\n6 Is C 2 Swan band emission a good thermometer?\\nBefore presenting a detailed analysis of the gas temperature of the plasma torch, a comparative study of the temperatures of the C 2 Swan band and CN violet system is presented in section 6.1. The good agreement that is reported between these two species gives a hint that the temperatures measured are with their local environment and so correspond to the gas temperature T gas with a value of about 6000 K. At atmospheric pressure, we also measured the first negative and second positive system of N 2 while operating the present plasma torch in pure N 2 and obtained rotational and vibrational temperatures in the order of 6500 K [85]. The consistency in the obtained values (for high pressure, high power microwave plasmas) using different measurements methods are a first indication that the C 2 Swan band may also be used for gas temperature determination. As discussed in section 3, there are several pathways involving neutral species for the formation of the C 2 (d 3 Π g ) state. Depending of its formation mechanism, one can expect the nascent rovibrational distribution of an excited state formed by chemical reactions to follow a thermal distribution or to be strongly non-thermal (i.e. formation of the excited into specific rotational and or vibrational state(s). In the latter case, for being able to predict its emission spectrum, it is necessary to take into account collisional relaxation processes [86]. The rotational and vibrational temperatures do not then correspond to the gas temperature. In sections 6.2 and 6.3 are discussed the collisional relaxation kinetics of the C 2 (d 3 Π g ) state. These processes that lead to the thermalization of the C 2 (d 3 Π g ) state rotational and vibrational distribution functions define the cases when the C 2 Swan band emission spectrum can still be used to measure the gas temperature independently of the formation mechanism of the C 2 (d 3 Π g ) state.\\nBefore presenting a detailed analysis of the gas temperature of the plasma torch, a comparative study of the temperatures of the C 2 Swan band and CN violet system is presented in section 6.1. The good agreement that is reported between these two species gives a hint that the temperatures measured are with their local environment and so correspond to the gas temperature T gas with a value of about 6000 K. At atmospheric pressure, we also measured the first negative and second positive system of N 2 while operating the present plasma torch in pure N 2 and obtained rotational and vibrational temperatures in the order of 6500 K [85]. The consistency in the obtained values (for high pressure, high power microwave plasmas) using different measurements methods are a first indication that the C 2 Swan band may also be used for gas temperature determination. As discussed in section 3, there are several pathways involving neutral species for the formation of the C 2 (d 3 Π g ) state. Depending of its formation mechanism, one can expect the nascent rovibrational distribution of an excited state formed by chemical reactions to follow a thermal distribution or to be strongly non-thermal (i.e. formation of the excited into specific rotational and or vibrational state(s). In the latter case, for being able to predict its emission spectrum, it is necessary to take into account collisional relaxation processes [86]. The rotational and vibrational temperatures do not then correspond to the gas temperature. In sections 6.2 and 6.3 are discussed the collisional relaxation kinetics of the C 2 (d 3 Π g ) state. These processes that lead to the thermalization of the C 2 (d 3 Π g ) state rotational and vibrational distribution functions define the cases when the C 2 Swan band emission spectrum can still be used to measure the gas temperature independently of the formation mechanism of the C 2 (d 3 Π g ) state.\"\n },\n {\n \"title\": \"Comparison with the CN violet system\",\n \"content\": \"\"\n },\n {\n \"title\": \"Comparison with the CN violet system\",\n \"content\": \"By adding a small amount of nitrogen, one can observe a strong decrease of the Swan band emission intensity while the CN violet system increase drastically. Vacher, Lino da Silva et al. [87,88] studied also the effect of adding N 2 to a CO 2 inductively coupled atmospheric pressure plasma and observed similar trends. Such behavior was also seen by other authors in microwave plasmas and low and atmospheric pressure [89,90]. This tendency can be explained by competing mechanisms for the formation of C 2 and CN that goes in favor of the formation of CN with increasing N 2 addition and for which it was found that C atoms are the main precursor [91]. CN (B-X) violet system is often used as a molecular pyrometer for determination of gas temperature in N 2 plasma sources with hydrocarbons [29, 92,93,94]. In order to generate CN violet system emission, 0.3 L/min of N 2 was added to the plasma corresponding to 3% admixture. This is a small quantity that, in the case of a pure CO 2 plasma, does not perturb the electron properties and neutral particles energy balance of the plasma.\\nBy adding a small amount of nitrogen, one can observe a strong decrease of the Swan band emission intensity while the CN violet system increase drastically. Vacher, Lino da Silva et al. [87,88] studied also the effect of adding N 2 to a CO 2 inductively coupled atmospheric pressure plasma and observed similar trends. Such behavior was also seen by other authors in microwave plasmas and low and atmospheric pressure [89,90]. This tendency can be explained by competing mechanisms for the formation of C 2 and CN that goes in favor of the formation of CN with increasing N 2 addition and for which it was found that C atoms are the main precursor [91]. CN (B-X) violet system is often used as a molecular pyrometer for determination of gas temperature in N 2 plasma sources with hydrocarbons [29, 92,93,94]. In order to generate CN violet system emission, 0.3 L/min of N 2 was added to the plasma corresponding to 3% admixture. This is a small quantity that, in the case of a pure CO 2 plasma, does not perturb the electron properties and neutral particles energy balance of the plasma.\\nIn an argon atmospheric pressure microwave plasma with nitrogen addition and CO 2 advection from ambient air, Ridenti and Amorim discussed the weak presence of the CN violet system that is strongly overlapping with nitrogen first negative and second positive systems [95]. The band head of the first negative system is well separated from the one of the CN violet system while N 2 second positive system has distinctive features that do not overlap either with the CN violet system. These two systems can therefore, using high resolution spectroscopy, easily be identified. Their presence was not detected in the present CO 2 microwave plasma with 3% N 2 addition. They are however observed in the present reactor while operated at atmospheric pressure with pure N 2 flow [85]. In figure 13, an example of CN spectrum and its resulting fitting are shown. A value of T rot =T vib = 6000 ± 500 K is obtained which are the same as for the Swan band system. Also the residue of the fit shows that no deviation from rotational or vibrational Boltzmann distribution can be observed experimentally. A comparison for several pressures between the measured rotational and vibrational temperatures obtained by fitting the two molecular bands systems is presented in figure 14. One can see that, within error bars, there are no significant differences for the two molecular systems and that the mean values agree between each other very well. Note that in the case of CN, also the rotational and vibrational temperatures were fitted independently (cf. method discussed in section 5.1). This confirms that the gas inside the plasma region where the light comes from is very hot with a temperature of about 6000 K both for rotational and vibrational internal degrees of freedom. The rovibrational distributions for those two systems can be essentially described with a single temperature.\\nIn an argon atmospheric pressure microwave plasma with nitrogen addition and CO 2 advection from ambient air, Ridenti and Amorim discussed the weak presence of the CN violet system that is strongly overlapping with nitrogen first negative and second positive systems [95]. The band head of the first negative system is well separated from the one of the CN violet system while N 2 second positive system has distinctive features that do not overlap either with the CN violet system. These two systems can therefore, using high resolution spectroscopy, easily be identified. Their presence was not detected in the present CO 2 microwave plasma with 3% N 2 addition. They are however observed in the present reactor while operated at atmospheric pressure with pure N 2 flow [85]. In figure 13, an example of CN spectrum and its resulting fitting are shown. A value of T rot =T vib = 6000 ± 500 K is obtained which are the same as for the Swan band system. Also the residue of the fit shows that no deviation from rotational or vibrational Boltzmann distribution can be observed experimentally. A comparison for several pressures between the measured rotational and vibrational temperatures obtained by fitting the two molecular bands systems is presented in figure 14. One can see that, within error bars, there are no significant differences for the two molecular systems and that the mean values agree between each other very well. Note that in the case of CN, also the rotational and vibrational temperatures were fitted independently (cf. method discussed in section 5.1). This confirms that the gas inside the plasma region where the light comes from is very hot with a temperature of about 6000 K both for rotational and vibrational internal degrees of freedom. The rovibrational distributions for those two systems can be essentially described with a single temperature.\"\n },\n {\n \"title\": \"Collisional relaxation of vibrational states\",\n \"content\": \"\"\n },\n {\n \"title\": \"Collisional relaxation of vibrational states\",\n \"content\": \"C 2 Swan bands emission during pulsed laser irradiation of soot or graphite is often used for determination of gas temperature in (sub-)microsecond and even nanosecond time scales (see for instance [96,97]). The spectral resolution of such spectra is however usually low (because of the relative low intensity of such discharges) and a detailed evalution of the thermalization of rotational and vibrational states is difficult. UV pulsed photolysis experiments by Faust et al. [59] showed that the formation of the high pressure band is occuring only on long time scales (t>> 1µs) while for time up to 1µs the Swan band emission presented a normal vibrational distribution structure. They concluded that two different processes contribute to the formation of the C 2 Swan bands and that vibrational relaxation of the v=6 level is inefficient for populating the v=0 level of the d 3 Π g state.\\nC 2 Swan bands emission during pulsed laser irradiation of soot or graphite is often used for determination of gas temperature in (sub-)microsecond and even nanosecond time scales (see for instance [96,97]). The spectral resolution of such spectra is however usually low (because of the relative low intensity of such discharges) and a detailed evalution of the thermalization of rotational and vibrational states is difficult. UV pulsed photolysis experiments by Faust et al. [59] showed that the formation of the high pressure band is occuring only on long time scales (t>> 1µs) while for time up to 1µs the Swan band emission presented a normal vibrational distribution structure. They concluded that two different processes contribute to the formation of the C 2 Swan bands and that vibrational relaxation of the v=6 level is inefficient for populating the v=0 level of the d 3 Π g state.\\nBondybey [98] analyzed the C 2 Swan band fluorescence following two photon excitation in rare gas solids. He observed an erratic progression of lifetimes for different vibrational levels and concluded that the fast vibrational relaxation of the d 3 Π g v=3,4 states may occur through the b 3 Σ − g state, following a suggestion by Frosch [99]. Xuechu and Nanquan [100] studied the vibrational relaxation of the C 2 (d 3 Π g ) state by observing the chemiluminescence spectrum of the reaction Na + CCl 4 in a thermal bath of argon where the C 2 (d 3 Π g , v = 6) state is formed preferentially. From their pressure dependence study, they concluded that the thermal relaxation of the C 2 (d 3 Π g ) state by argon takes only a few collisions with a rate of 2.2 · 10 6 Torr −1 s −1 while they deduce even higher quenching rate by the sodium atoms.\\nBondybey [98] analyzed the C 2 Swan band fluorescence following two photon excitation in rare gas solids. He observed an erratic progression of lifetimes for different vibrational levels and concluded that the fast vibrational relaxation of the d 3 Π g v=3,4 states may occur through the b 3 Σ − g state, following a suggestion by Frosch [99]. Xuechu and Nanquan [100] studied the vibrational relaxation of the C 2 (d 3 Π g ) state by observing the chemiluminescence spectrum of the reaction Na + CCl 4 in a thermal bath of argon where the C 2 (d 3 Π g , v = 6) state is formed preferentially. From their pressure dependence study, they concluded that the thermal relaxation of the C 2 (d 3 Π g ) state by argon takes only a few collisions with a rate of 2.2 · 10 6 Torr −1 s −1 while they deduce even higher quenching rate by the sodium atoms.\\nSeveral experiments have shown that vibrational thermalization of the C 2 (d 3 Π g ) state is relatively slow and potentially as long if not longer than the lifetime of the state itself excepted for the rate reported above by Xuechu and Nanquan [100]. It may therefore be surprising to see that the vibrational distribution function follows closely a Maxwell-Boltzmann distribution in the experiments reported in this paper. However, the present measurements are done at high temperatures (i.e. T > 1000 K) and one may expect that the nascent population distribution follows already closely a thermal distribution, partly because of the large translational motion of the reactants. A temperature of 6000 K corresponds to a kinetic energy of 0.5 eV which is more than the energy separation between individual vibrational levels.\\nSeveral experiments have shown that vibrational thermalization of the C 2 (d 3 Π g ) state is relatively slow and potentially as long if not longer than the lifetime of the state itself excepted for the rate reported above by Xuechu and Nanquan [100]. It may therefore be surprising to see that the vibrational distribution function follows closely a Maxwell-Boltzmann distribution in the experiments reported in this paper. However, the present measurements are done at high temperatures (i.e. T > 1000 K) and one may expect that the nascent population distribution follows already closely a thermal distribution, partly because of the large translational motion of the reactants. A temperature of 6000 K corresponds to a kinetic energy of 0.5 eV which is more than the energy separation between individual vibrational levels.\\nRich and Bergman [101] performed vibrational excitation of CO at room temperature using a CO laser and observed formation of the C 1 Π g and d 3 Π g states of C 2 . They claimed that the observation of the C 1 Π g state (at the origin of the C 1 Π g -A 1 Π u Deslandres-D'Azambuja system) was the indication that CO (v) states can electronically excite the C 2 molecule. Indeed, the C 1 Π g state can be formed only via the association of two C ( 1 D) states and in their experiment only C ( 3 P) ground state atoms were produced. However, an analysis of the potential energy curves of C 2 shows that the C 1 Π g has an avoided crossing with another 1 Π g state and consequently dissociates into two ground state C( 3 P) atoms [48]. Grigorian and Cenian [102] recently performed a similar experiment and reported also the formation of the e 3 Π g state (e 3 Π g -a 3 Π u , Fox-Herzberg system) which have for dissociation limit the C ( 3 P) and C( 1 D) states [41]. Moreover, Wallaart et al. [50,103] reported during a CO vibrational excitation experiment the formation of the D 1 Σ + u state (D 1 Σ + u -X 1 Σ + g , Mulliken system) in addition to the aforementioned ones. The D 1 Σ + u state has for dissociation limit the C ( 1 S)+C( 1 D) states [41]. Although the initial experiments of Rich and Bergman did not allow to support their hypothesis of direct electronic excitation of C 2 electronic states by CO(v), subsequent measurements confirm their proposal. These observations go in favor to the hypothesis that vibrationally excited CO molecules can directly populate C 2 electronically excited states and that multiple vibrational excitation quanta may occur simultaneously. We note that Wallaart et al. reported also the observation of radiative transitions from electronically excited C atoms which could also be excited by CO(v). The presence of C 2 (D 1 Σ + u ) and C 2 (e 3 Π g ) states in the aforementioned studies could therefore also be explained by the recombination of C atoms which are electronically excited by CO(v) after being formed by the Boudouard mechanism (cf. reaction 1). The relatively low density of C* atoms compared to CO(v) states however makes it difficult to assess how important that channel is in reality. Hack and Langel [104] analysed the recombination of C( 1 D) atoms and observed the production of C 2 (e 3 Π g , C 1 Π g ) electronic states but no recombination rate coefficient was reported in the literature yet. The usual observation of C 2 electronic states formation in C/CO mixtures without detection of any electronically excited states of C atoms (at the exception of the work of Wallart et al) tends to support that efficient energy exchange mechanisms between CO(v) and C 2 exist. Such processes will favor rapid thermalization of the vibrational states of the C 2 molecule. Similarly, one can expect that the CO 2 molecule also interact with C 2 molecules in similar fashion due to its rich vibrational structure. Based on the present discussion, it can be concluded that either the formation mechanism of the C 2 (d 3 Π g ) state has already a nascent (nearly)-thermalized vibrational distribution (which seems unlikely based on its formation mechanisms discussed in section 3) or that, not only collisional quenching processes, but also reactive energy exchanges leads to its (faster) thermalization. This would explain the invariant observation of thermalized vibrational distributions of the C 2 (d 3 Π g ) state in CO 2 microwave plasmas.\\nRich and Bergman [101] performed vibrational excitation of CO at room temperature using a CO laser and observed formation of the C 1 Π g and d 3 Π g states of C 2 . They claimed that the observation of the C 1 Π g state (at the origin of the C 1 Π g -A 1 Π u Deslandres-D'Azambuja system) was the indication that CO (v) states can electronically excite the C 2 molecule. Indeed, the C 1 Π g state can be formed only via the association of two C ( 1 D) states and in their experiment only C ( 3 P) ground state atoms were produced. However, an analysis of the potential energy curves of C 2 shows that the C 1 Π g has an avoided crossing with another 1 Π g state and consequently dissociates into two ground state C( 3 P) atoms [48]. Grigorian and Cenian [102] recently performed a similar experiment and reported also the formation of the e 3 Π g state (e 3 Π g -a 3 Π u , Fox-Herzberg system) which have for dissociation limit the C ( 3 P) and C( 1 D) states [41]. Moreover, Wallaart et al. [50,103] reported during a CO vibrational excitation experiment the formation of the D 1 Σ + u state (D 1 Σ + u -X 1 Σ + g , Mulliken system) in addition to the aforementioned ones. The D 1 Σ + u state has for dissociation limit the C ( 1 S)+C( 1 D) states [41]. Although the initial experiments of Rich and Bergman did not allow to support their hypothesis of direct electronic excitation of C 2 electronic states by CO(v), subsequent measurements confirm their proposal. These observations go in favor to the hypothesis that vibrationally excited CO molecules can directly populate C 2 electronically excited states and that multiple vibrational excitation quanta may occur simultaneously. We note that Wallaart et al. reported also the observation of radiative transitions from electronically excited C atoms which could also be excited by CO(v). The presence of C 2 (D 1 Σ + u ) and C 2 (e 3 Π g ) states in the aforementioned studies could therefore also be explained by the recombination of C atoms which are electronically excited by CO(v) after being formed by the Boudouard mechanism (cf. reaction 1). The relatively low density of C* atoms compared to CO(v) states however makes it difficult to assess how important that channel is in reality. Hack and Langel [104] analysed the recombination of C( 1 D) atoms and observed the production of C 2 (e 3 Π g , C 1 Π g ) electronic states but no recombination rate coefficient was reported in the literature yet. The usual observation of C 2 electronic states formation in C/CO mixtures without detection of any electronically excited states of C atoms (at the exception of the work of Wallart et al) tends to support that efficient energy exchange mechanisms between CO(v) and C 2 exist. Such processes will favor rapid thermalization of the vibrational states of the C 2 molecule. Similarly, one can expect that the CO 2 molecule also interact with C 2 molecules in similar fashion due to its rich vibrational structure. Based on the present discussion, it can be concluded that either the formation mechanism of the C 2 (d 3 Π g ) state has already a nascent (nearly)-thermalized vibrational distribution (which seems unlikely based on its formation mechanisms discussed in section 3) or that, not only collisional quenching processes, but also reactive energy exchanges leads to its (faster) thermalization. This would explain the invariant observation of thermalized vibrational distributions of the C 2 (d 3 Π g ) state in CO 2 microwave plasmas.\"\n },\n {\n \"title\": \"Collisional relaxation of rotational states and gas temperature\",\n \"content\": \"\"\n },\n {\n \"title\": \"Collisional relaxation of rotational states and gas temperature\",\n \"content\": \"Direct measurements of thermalization rates are usually difficult to obtain and indirect estimations are usually done. Using the assumption that the C 2 molecule would be formed rotationally hot and that the pressure dependence of the C 2 Swan band temperature can be used as estimate of the rotational relaxation/equilibration with the thermal bath of molecules, Bleekrode determined a rotational relaxation time of 3µs in an oxyacetylene flame at low pressure (2-120 Torr) [105]. Using the same hypotheses, Kini and Savadatti [46] estimated a rotational relaxation time for the Swan band of 30 µs in a CO discharge in a pressure range of 2-24 Torr. The underlying assumptions were particularly strong as no elementary process for the formation of rotationally hot C 2 were identified. However, Brockhinke et al [106] recently performed picosecond laser induced fluorescence measurements on the ∆ν = 0 transition group of the C 2 (d 3 Π g -a 3 Π u ) Swan. They managed to evaluate rotational energy transfer (RET) and vibrational energy transfer (VET) by monitoring the collisional transfer of single optically pumped rovibrational state of the d 3 Π g state to adjacent states by fluorescence. The measurements were performed in an oxy-acetylene flame with a gas temperature of 2200 K. Such measurements allow to monitor individual state to state RET and VET collisional transfers and are therefore very precise. They obtained state-to-state collisional re-distribution rate for RET in the order of k RET ≈ 4 · 10 9 s −1 .\\nDirect measurements of thermalization rates are usually difficult to obtain and indirect estimations are usually done. Using the assumption that the C 2 molecule would be formed rotationally hot and that the pressure dependence of the C 2 Swan band temperature can be used as estimate of the rotational relaxation/equilibration with the thermal bath of molecules, Bleekrode determined a rotational relaxation time of 3µs in an oxyacetylene flame at low pressure (2-120 Torr) [105]. Using the same hypotheses, Kini and Savadatti [46] estimated a rotational relaxation time for the Swan band of 30 µs in a CO discharge in a pressure range of 2-24 Torr. The underlying assumptions were particularly strong as no elementary process for the formation of rotationally hot C 2 were identified. However, Brockhinke et al [106] recently performed picosecond laser induced fluorescence measurements on the ∆ν = 0 transition group of the C 2 (d 3 Π g -a 3 Π u ) Swan. They managed to evaluate rotational energy transfer (RET) and vibrational energy transfer (VET) by monitoring the collisional transfer of single optically pumped rovibrational state of the d 3 Π g state to adjacent states by fluorescence. The measurements were performed in an oxy-acetylene flame with a gas temperature of 2200 K. Such measurements allow to monitor individual state to state RET and VET collisional transfers and are therefore very precise. They obtained state-to-state collisional re-distribution rate for RET in the order of k RET ≈ 4 · 10 9 s −1 .\\nDue to the high electronic quenching rate of the C 2 d 3 Π g states they could not quantify the VET rate but assessed that it was slower than k RET in agreement with aforementioned studies. The temperatures obtained by fitting the C 2 Swan band and the CN (B 2 Σ + -X 2 Σ + ) violet system have been compared and match very well as seen in figure 14. Contrary to the C 2 (d 3 Π g ) state, the CN (B 2 Σ + ) state does not have, to the best of our knowledge, known mechanisms leading to non-equilibrium rovibrational emission spectra [107]. Rotational relaxation times of the CN (B 2 Σ + ) state are also considerably shorter than its radiative lifetime [108,109]. For CN, the rotational and vibrational states distributions have time to thermalize before being observed by optical emission spectroscopy in continuous plasmas. The cross check between the rotational temperatures of the Swan band and the CN violet system gives additional confidence that their rotational temperature may be associated to the gas temperature. The reactive formation of molecules can only lead to an overpopulation of the high rotational levels of a molecule (because of energy exchange during the formation process) or to a thermal distribution. In the case of non-Boltzmann distribution of rotational levels, the temperature of the lower rotational levels is then usually taken as a measure of the gas (i.e. translational) temperature of the plasma [29]. In the present study (see figure 8a), two-temperature rotational distribution function are not found. Observing thermal distributions of rotational states of the C 2 (d 3 Π g ) state and knowing that its RET relaxation time is much shorter than its radiative lifetime gives confidence that T rot is a local measure of the translational temperature of the plasma. This is additionally supported by the equivalence between the rotational and vibrational temperatures both for the C 2 Swan bands and for the CN violet system. For high pressure CO 2 plasmas (i.e. typically 100 mbar and above), it can therefore be proposed to use the C 2 Swan bands rotational temperature as an evaluation tool for the local gas temperature of the plasma. We note however, that for a CO 2 plasma with a much lower gas temperature, similar analysis should be performed as in section 5.1.\\nDue to the high electronic quenching rate of the C 2 d 3 Π g states they could not quantify the VET rate but assessed that it was slower than k RET in agreement with aforementioned studies. The temperatures obtained by fitting the C 2 Swan band and the CN (B 2 Σ + -X 2 Σ + ) violet system have been compared and match very well as seen in figure 14. Contrary to the C 2 (d 3 Π g ) state, the CN (B 2 Σ + ) state does not have, to the best of our knowledge, known mechanisms leading to non-equilibrium rovibrational emission spectra [107]. Rotational relaxation times of the CN (B 2 Σ + ) state are also considerably shorter than its radiative lifetime [108,109]. For CN, the rotational and vibrational states distributions have time to thermalize before being observed by optical emission spectroscopy in continuous plasmas. The cross check between the rotational temperatures of the Swan band and the CN violet system gives additional confidence that their rotational temperature may be associated to the gas temperature. The reactive formation of molecules can only lead to an overpopulation of the high rotational levels of a molecule (because of energy exchange during the formation process) or to a thermal distribution. In the case of non-Boltzmann distribution of rotational levels, the temperature of the lower rotational levels is then usually taken as a measure of the gas (i.e. translational) temperature of the plasma [29]. In the present study (see figure 8a), two-temperature rotational distribution function are not found. Observing thermal distributions of rotational states of the C 2 (d 3 Π g ) state and knowing that its RET relaxation time is much shorter than its radiative lifetime gives confidence that T rot is a local measure of the translational temperature of the plasma. This is additionally supported by the equivalence between the rotational and vibrational temperatures both for the C 2 Swan bands and for the CN violet system. For high pressure CO 2 plasmas (i.e. typically 100 mbar and above), it can therefore be proposed to use the C 2 Swan bands rotational temperature as an evaluation tool for the local gas temperature of the plasma. We note however, that for a CO 2 plasma with a much lower gas temperature, similar analysis should be performed as in section 5.1.\"\n },\n {\n \"title\": \"Parametric analysis of the CO 2 microwave plasma torch\",\n \"content\": \"\"\n },\n {\n \"title\": \"Parametric analysis of the CO 2 microwave plasma torch\",\n \"content\": \"Having assessed that C 2 Swan band can be used for measuring the gas temperature in a CO 2 plasma, we report in this section a parametric study of C 2 T rot and T vib as a function of plasma parameters. Although both T rot and T vib can be associated to the gas temperature, they are reported here independently. The measurements are taken in the pressure regime above 120 mbar where the C 2 Swan band emission is the dominant feature in the plasma emission spectrum. Concomitantly, the plasma is contracted with a diameter of about 3-8 mm which depends mainly of the applied power. The emission is spatially discriminated using a system of two pin-holes and a lens coupled to a fiber and collected over a area of about 1 mm 2 . The latter corresponds to the spatial resolution attainable in the present setup. Each data point corresponds to the light collected along the light of sight (radial direction) and over a plasma slab of 1 mm in the axial direction. Interestingly, in the contracted regime of the plasma torch (i.e. above a pressure of about 120 mbar), axial and radial measurements of the C 2 rotational temperature show no variation for all studied plasma conditions. After substraction of the continuous baseline, only a spatial variation of the overall intensity of the C 2 Swan band is observed within signal to noise sensitivity. In the following, all data are consequently reported for measurements in the center of the resonator; i.e 20 mm above the bottom of the waveguide while looking across the axis of the quartz tube. In figure 15, the effect of the input microwave power on the rotational and vibration temperatures of the C 2 Swan bands are presented. The measurements are done near atmospheric pressure plasma (925 mbar) for a gas flow of 10 L/min. One can see that both the rotational and vibrational temperatures of C 2 remain constant and equal within error bars. The increase in power has only the consequence of increasing the intensity of the radiation from the C 2 (d 3 Π g ) state. This increase of intensity is correlated with an increase in plasma radius. Also, by imaging it is observed that the plasma becomes elongated in the effluent region with increasing power.\\nHaving assessed that C 2 Swan band can be used for measuring the gas temperature in a CO 2 plasma, we report in this section a parametric study of C 2 T rot and T vib as a function of plasma parameters. Although both T rot and T vib can be associated to the gas temperature, they are reported here independently. The measurements are taken in the pressure regime above 120 mbar where the C 2 Swan band emission is the dominant feature in the plasma emission spectrum. Concomitantly, the plasma is contracted with a diameter of about 3-8 mm which depends mainly of the applied power. The emission is spatially discriminated using a system of two pin-holes and a lens coupled to a fiber and collected over a area of about 1 mm 2 . The latter corresponds to the spatial resolution attainable in the present setup. Each data point corresponds to the light collected along the light of sight (radial direction) and over a plasma slab of 1 mm in the axial direction. Interestingly, in the contracted regime of the plasma torch (i.e. above a pressure of about 120 mbar), axial and radial measurements of the C 2 rotational temperature show no variation for all studied plasma conditions. After substraction of the continuous baseline, only a spatial variation of the overall intensity of the C 2 Swan band is observed within signal to noise sensitivity. In the following, all data are consequently reported for measurements in the center of the resonator; i.e 20 mm above the bottom of the waveguide while looking across the axis of the quartz tube. In figure 15, the effect of the input microwave power on the rotational and vibration temperatures of the C 2 Swan bands are presented. The measurements are done near atmospheric pressure plasma (925 mbar) for a gas flow of 10 L/min. One can see that both the rotational and vibrational temperatures of C 2 remain constant and equal within error bars. The increase in power has only the consequence of increasing the intensity of the radiation from the C 2 (d 3 Π g ) state. This increase of intensity is correlated with an increase in plasma radius. Also, by imaging it is observed that the plasma becomes elongated in the effluent region with increasing power.\\nIn order to cool down a high pressure plasma, one possible way is by increasing the gas flow. In figure 16, the variation of the C 2 rotational temperature versus CO 2 gas flow at a pressure of about 850 mbar is shown for an input power of 900 W. One can see that the temperature remains constant around 6000± 500 K. Such trend was also observed by Sun et al. [27] and Spencer and Gallimore [24] but for lower flow rates variation. Similar lack of influence of the gas flow was also observed in the case of an atmospheric pressure microwave plasma torch in nitrogen [110]. Increasing the gas flow by more than 10-fold appears in the present setup (and also for other high pressure microwave sources) to be an inefficient way of tuning the gas temperature despite reducing the specific energy input per molecule by the same factor. Such result is rather surprising but may be due to an inefficient mixing between the hot and cold gas region.\\nIn order to cool down a high pressure plasma, one possible way is by increasing the gas flow. In figure 16, the variation of the C 2 rotational temperature versus CO 2 gas flow at a pressure of about 850 mbar is shown for an input power of 900 W. One can see that the temperature remains constant around 6000± 500 K. Such trend was also observed by Sun et al. [27] and Spencer and Gallimore [24] but for lower flow rates variation. Similar lack of influence of the gas flow was also observed in the case of an atmospheric pressure microwave plasma torch in nitrogen [110]. Increasing the gas flow by more than 10-fold appears in the present setup (and also for other high pressure microwave sources) to be an inefficient way of tuning the gas temperature despite reducing the specific energy input per molecule by the same factor. Such result is rather surprising but may be due to an inefficient mixing between the hot and cold gas region.\\nA typical method for enhancing the non-equilibrium of a plasma discharge (e.g. reducing the gas temperature while the electron temperature remains in the order of 1 eV or more) is to reduce the gas pressure. In figure 17, the pressure is varied from near atmospheric pressure down to 180 mbar where the C 2 Swan band is still intense enough for getting a good signal to noise ratio. One can see that T vib and T rot are equal to each other within absolute uncertainties. One should note that in pure CO 2 microwave plasmas, lower temperatures have been measured but only at (much) lower pressures than in the current study and where C 2 Swan band is not observed [89,111]. On the other hand, den Harder et al. [17] and Groen et al [83] measured gas temperature in the order of 5000 to 6000 K in the same pressure range as for the present study.\\nA typical method for enhancing the non-equilibrium of a plasma discharge (e.g. reducing the gas temperature while the electron temperature remains in the order of 1 eV or more) is to reduce the gas pressure. In figure 17, the pressure is varied from near atmospheric pressure down to 180 mbar where the C 2 Swan band is still intense enough for getting a good signal to noise ratio. One can see that T vib and T rot are equal to each other within absolute uncertainties. One should note that in pure CO 2 microwave plasmas, lower temperatures have been measured but only at (much) lower pressures than in the current study and where C 2 Swan band is not observed [89,111]. On the other hand, den Harder et al. [17] and Groen et al [83] measured gas temperature in the order of 5000 to 6000 K in the same pressure range as for the present study.\"\n },\n {\n \"title\": \"Conclusions and outlook\",\n \"content\": \"\"\n },\n {\n \"title\": \"Conclusions and outlook\",\n \"content\": \"In this paper, the potential sources of C 2 molecules in CO 2 plasmas and the known pathways leading to non-equilibrium vibrational distributions of the (d 3 Π g ) state are reviewed. A detailed analysis of high resolution spectra is performed in order to detect potential deviations of the rovibrational distribution functions from thermal equilibrium in a high pressure microwave plasma discharge. State of the art molecular constants were used in order to fit the spectra. It is found that Boltzmann distributions can be used both for rotational and vibrational states for all conditions in the present study. Consequently, potential source terms of the C 2 (d 3 Π g ) state in a CO 2 plasma do not lead to any observed deviation from a Boltzmann vibrational distribution function. The sensitivity of different parts of the spectra for determining T rot and T vib is evaluated. For a fast and accurate determination of the rotational temperature of the C 2 (d 3 Π g ) state, it is shown that the ∆ν = 0 transition group is the most accurate part of the spectrum to be taken into account.\\nIn this paper, the potential sources of C 2 molecules in CO 2 plasmas and the known pathways leading to non-equilibrium vibrational distributions of the (d 3 Π g ) state are reviewed. A detailed analysis of high resolution spectra is performed in order to detect potential deviations of the rovibrational distribution functions from thermal equilibrium in a high pressure microwave plasma discharge. State of the art molecular constants were used in order to fit the spectra. It is found that Boltzmann distributions can be used both for rotational and vibrational states for all conditions in the present study. Consequently, potential source terms of the C 2 (d 3 Π g ) state in a CO 2 plasma do not lead to any observed deviation from a Boltzmann vibrational distribution function. The sensitivity of different parts of the spectra for determining T rot and T vib is evaluated. For a fast and accurate determination of the rotational temperature of the C 2 (d 3 Π g ) state, it is shown that the ∆ν = 0 transition group is the most accurate part of the spectrum to be taken into account.\\nDue to fast thermalization of the rotational states of the C 2 (d 3 Π g ) state with the background gas, the measured T rot can be used for assessing the gas temperature of the plasma. Furthermore, the vibrational states of the C 2 (d 3 Π g ) level are found to be thermalized indicating the possible presence of fast thermalization processes.\\nDue to fast thermalization of the rotational states of the C 2 (d 3 Π g ) state with the background gas, the measured T rot can be used for assessing the gas temperature of the plasma. Furthermore, the vibrational states of the C 2 (d 3 Π g ) level are found to be thermalized indicating the possible presence of fast thermalization processes.\\nA parametric study is performed on a pure CO 2 microwave plasma torch and the both the vibrational and rotational temperatures are found to be insensitive to external plasma parameters such as gas flow and input power at pressures of 180 mbar and above and a value of 6000 ± 500 K is found. The latter can be associated to the gas temperature of the plasma. These results are in agreement with the hypothesis that chemical equilibrium of the species can be described by the gas temperature at which thermal dissociation play an important role. The only observable differences in the extended parameter range of the present study are in the radially integrated density of the C 2 (d 3 Π g ) state. We note that similar values as the one presented in this study were reported before in a certain number of studies but no assessment of non-equilibrium distributions was done. Also, the rotational temperature of C 2 was implicitely assumed previously to be the one of the gas temperature and it is now shown that such approximation in high pressure CO 2 microwave discharges appears to hold.\\nA parametric study is performed on a pure CO 2 microwave plasma torch and the both the vibrational and rotational temperatures are found to be insensitive to external plasma parameters such as gas flow and input power at pressures of 180 mbar and above and a value of 6000 ± 500 K is found. The latter can be associated to the gas temperature of the plasma. These results are in agreement with the hypothesis that chemical equilibrium of the species can be described by the gas temperature at which thermal dissociation play an important role. The only observable differences in the extended parameter range of the present study are in the radially integrated density of the C 2 (d 3 Π g ) state. We note that similar values as the one presented in this study were reported before in a certain number of studies but no assessment of non-equilibrium distributions was done. Also, the rotational temperature of C 2 was implicitely assumed previously to be the one of the gas temperature and it is now shown that such approximation in high pressure CO 2 microwave discharges appears to hold.\\nThe C 2 (d 3 Π g ) state emits only in a narrow region in the center of the plasma and only the gas temperature in the core of the plasma can be measured. The region inside the plasma reactor that can be probed is only where C atoms are present and react to form C 2 (or CN when some N 2 is present) molecules. It was observed that C atoms are not observed by optical emission spectroscopy at pressures lower than the ones where the C 2 (d 3 Π g ) state is measured. To form carbon atoms, large power densities (i.e. high T vib and/or T gas ) are required as CO is the only species that can act as a primary source of carbon atoms. Interestingly, at the observed temperature of 6000 K, CO can already partially thermally dissociate [112]. This implies that observation of C and C 2 species is linked to regions with high gas temperatures where CO 2 molecules are readily dissociated.\\nThe C 2 (d 3 Π g ) state emits only in a narrow region in the center of the plasma and only the gas temperature in the core of the plasma can be measured. The region inside the plasma reactor that can be probed is only where C atoms are present and react to form C 2 (or CN when some N 2 is present) molecules. It was observed that C atoms are not observed by optical emission spectroscopy at pressures lower than the ones where the C 2 (d 3 Π g ) state is measured. To form carbon atoms, large power densities (i.e. high T vib and/or T gas ) are required as CO is the only species that can act as a primary source of carbon atoms. Interestingly, at the observed temperature of 6000 K, CO can already partially thermally dissociate [112]. This implies that observation of C and C 2 species is linked to regions with high gas temperatures where CO 2 molecules are readily dissociated.\\nThe measurements reported in this paper show the presence of a hot core of gas but no insight in the temperature gradient at the edge of the plasma could be obtained as not radiative emitting molecular species can be observed at the edges of this hot core. High gas temperatures T≥ 3500K allow already to dissociate fully the CO 2 and partially the CO molecules through thermal processes [18,6]. In situ diagnostics techniques like Raman scattering and CARS [113,114] would bring additional understanding in the gas dynamics at the boundary between the hot plasma core and the colder edge.\\nThe measurements reported in this paper show the presence of a hot core of gas but no insight in the temperature gradient at the edge of the plasma could be obtained as not radiative emitting molecular species can be observed at the edges of this hot core. High gas temperatures T≥ 3500K allow already to dissociate fully the CO 2 and partially the CO molecules through thermal processes [18,6]. In situ diagnostics techniques like Raman scattering and CARS [113,114] would bring additional understanding in the gas dynamics at the boundary between the hot plasma core and the colder edge.\\n[21] Yacine Babou, Philippe Rivire, Marie-Yvonne Perrin, and Anouar\\n[21] Yacine Babou, Philippe Rivire, Marie-Yvonne Perrin, and Anouar\\nSoufiani. Spectroscopic study of microwave plasmas of CO 2 and CO 2 -N 2 mixtures at atmospheric pressure. Plasma Sources Science and Technology, 17 (4) Figure 15: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of power input. The measurements are performed in the center of the resonator at 925 mbar with 10 L/min CO 2 gas flow. n is the laterally integrated density of the C 2 (d 3 Π g ) state. Figure 17: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of pressure. The measurements are performed in the center of the resonator using a 10 L/min CO 2 gas flow and 900 W input power.\\nSoufiani. Spectroscopic study of microwave plasmas of CO 2 and CO 2 -N 2 mixtures at atmospheric pressure. Plasma Sources Science and Technology, 17 (4) Figure 15: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of power input. The measurements are performed in the center of the resonator at 925 mbar with 10 L/min CO 2 gas flow. n is the laterally integrated density of the C 2 (d 3 Π g ) state. Figure 17: Rotational and vibrational temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 and ∆ν = 0, ±1 transition groups respectively as a function of pressure. The measurements are performed in the center of the resonator using a 10 L/min CO 2 gas flow and 900 W input power.\\nOrigin B ν 10 6 · D ν 10 −2 · γ 0 25797.86825 (43)\\nOrigin B ν 10 6 · D ν 10 −2 · γ 0 25797.86825 (43)\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7 :Figure 10 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7 :Figure 10 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 11 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 11 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 12 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 12 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 13 :Figure 14 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 13 :Figure 14 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 16 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 16 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 2 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 5 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 5 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 6 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 6 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 7 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 7 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented.","content":"Figure 2 :\n2Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented."},{"id":"","caption":"Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented.","content":"Figure 2 :\n2Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented."},{"id":"","caption":"Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min.","content":"Figure 4 :\n4Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min."},{"id":"","caption":"Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min.","content":"Figure 4 :\n4Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min."},{"id":"","caption":"Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19","content":"Figure 5 :\n5Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19"},{"id":"","caption":"Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19","content":"Figure 5 :\n5Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19"},{"id":"","caption":"Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm.","content":"Figure 6 :\n6Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm."},{"id":"","caption":"Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm.","content":"Figure 6 :\n6Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm."},{"id":"","caption":"Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K.","content":"Figure 7 :Figure 10 :\n710Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K."},{"id":"","caption":"Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K.","content":"Figure 7 :Figure 10 :\n710Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K."},{"id":"","caption":"Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm.","content":"Figure 11 :\n11Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm."},{"id":"","caption":"Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm.","content":"Figure 11 :\n11Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm."},{"id":"","caption":"Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6.","content":"Figure 12 :\n12Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6."},{"id":"","caption":"Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6.","content":"Figure 12 :\n12Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6."},{"id":"","caption":"CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow","content":"Figure 13 :Figure 14 :\n1314CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow"},{"id":"","caption":"CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow","content":"Figure 13 :Figure 14 :\n1314CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow"},{"id":"","caption":"Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W.","content":"Figure 16 :\n16Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W."},{"id":"","caption":"Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W.","content":"Figure 16 :\n16Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W."},{"id":"","caption":"−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3","content":"−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3"},{"id":"","caption":"−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3","content":"−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3"},{"id":"","caption":"or from Fantz and Wünderlich[81] leads to","content":"or from Fantz and Wünderlich[81] leads toParameter \n\nmeaning \nN \nUpper state excited specie density \nT rot \nRotational temperature \nT vib \nVibrational temperature \nB base \nConstant baseline offset value versus λ \nα base \nbaseline slope \nσ G \nGaussian width of the Voigt profile \nσ L \nLorentzian width of the Voigt profile \nλ start"},{"id":"","caption":"or from Fantz and Wünderlich[81] leads to","content":"or from Fantz and Wünderlich[81] leads toParameter \n\nmeaning \nN \nUpper state excited specie density \nT rot \nRotational temperature \nT vib \nVibrational temperature \nB base \nConstant baseline offset value versus λ \nα base \nbaseline slope \nσ G \nGaussian width of the Voigt profile \nσ L \nLorentzian width of the Voigt profile \nλ start"},{"id":"","caption":"Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program.","content":"Table 1 :\n1Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program."},{"id":"","caption":"Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program.","content":"Table 1 :\n1Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program."},{"id":"","caption":"Gatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system","content":"Gatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system:045010, 2008. \n\n[22] Yong C Hong, Dong H Shin, Sang J Lee, Ye J Kim, Bong Ju Lee, and \nHan S Uhm. Generation of high-power torch plasma by a 915-MHz mi-\ncrowave system. IEEE Transactions on Plasma Science, 39(10):1958-\n1962, 2011. \n\n[23] D Le Quang, Y Babou, and P Andre. Investigations of a microwave \nplasma source operating with air, N 2 , CO 2 and argon gases. IOP \nConference Series: Materials Science and Engineering, 29(1):012009, \n2012. \n\n[24] L F Spencer and A D Gallimore. CO 2 dissociation in an atmospheric \npressure plasma/catalyst system: a study of efficiency. Plasma Sources \nScience and Technology, 22(1):015019, 2013. \n\n[25] Han S. Uhm, Hyoung S. Kwak, and Yong C. Hong. Carbon dioxide \nelimination and regeneration of resources in a microwave plasma torch. \nEnvironmental Pollution, 211:191 -197, 2016. \n\n[26] Constandinos M Mitsingas, Rajavasanth Rajasegar, Stephen Ham-\nmack, Hyungrok Do, and Tonghun Lee. High energy efficiency plasma \nconversion of CO 2 at atmospheric pressure using a direct coupled \nmicrowave plasma system. IEEE Transactions on Plasma Science, \n44(4):651-656, 2016. \n\n[27] Hojoong Sun, Jungwun Lee, Hyungrok Do, Seong-kyun Im, and Moon \nSoo Bak. Experimental and numerical studies on carbon dioxide de-\ncomposition in atmospheric electrodeless microwave plasmas. Journal \nof Applied Physics, 122(3):033303, 2017. \n\n[28] Se Min Chun, Yong Cheol Hong, and Dae Hyun Choi. Reforming \nof methane to syngas in a microwave plasma torch at atmospheric \npressure. Journal of CO2 Utilization, 19:221-229, 2017. \n\n[29] PJ Bruggeman, N Sadeghi, DC Schram, and V Linss. Gas tempera-\nture determination from rotational lines in non-equilibrium plasmas: a \nreview. Plasma Sources Science and Technology, 23(2):023001, 2014. \n\n[30] Reginald William Blake Pearse and Alfred Gordon Gaydon. Identifi-\ncation of molecular spectra. Chapman and Hall, 1976. \n[31] Jan Voráč, Petr Synek, Lucia Potočňáková, Jaroslav Hnilica, and Vít \nKudrle. Batch processing of overlapping molecular spectra as a tool \nfor spatio-temporal diagnostics of power modulated microwave plasma \njet. Plasma Sources Science and Technology, 26(2):025010, 2017. \n\n[32] Jan Voráč, Petr Synek, Vojtěch Procházka, and Tomáš Hoder. State-\nby-state emission spectra fitting for non-equilibrium plasmas: OH spec-\ntra of surface barrier discharge at argon/water interface. Journal of \nPhysics D: Applied Physics, 50(29):294002, 2017. \n\n[33] Martina Leins, Sandra Gaiser, Andreas Schulz, Matthias Walker, Uwe \nSchumacher, and Thomas Hirth. How to ignite an atmospheric pressure \nmicrowave plasma torch without any additional igniters. Journal of \nvisualized experiments: JoVE, (98), 2015. \n\n[34] JW McConkey, CP Malone, PV Johnson, C Winstead, V McKoy, and \nI Kanik. Electron impact dissociation of oxygen-containing molecules-a \ncritical review. Physics Reports, 466(1-3):1-103, 2008. \n\n[35] Marija Grofulović, Luís L Alves, and Vasco Guerra. Electron-neutral \nscattering cross sections for CO2: a complete and consistent set and \nan assessment of dissociation. Journal of Physics D: Applied Physics, \n49(39):395207, sep 2016. \n\n[36] Pamir Nag and Dhananjay Nandi. Dissociation dynamics in the disso-\nciative electron attachment to carbon dioxide. Phys. Rev. A, 91:052705, \nMay 2015. \n\n[37] Yukikazu Itikawa. Cross sections for electron collisions with carbon \ndioxide. Journal of Physical and Chemical Reference Data, 31(3):749-\n767, 2002. \n\n[38] L.D. Pietanza, G. Colonna, V. Laporta, R. Celiberto, G. DAmmando, \nA. Laricchiuta, and M. Capitelli. Influence of electron molecule \nresonant vibrational collisions over the symmetric mode and direct \nexcitation-dissociation cross sections of co2 on the electron energy dis-\ntribution function and dissociation mechanisms in cold pure co2 plas-\nmas. The Journal of Physical Chemistry A, 120(17):2614-2628, 2016. \nPMID: 27064438. \n\n[39] A.I. Florescu-Mitchell and J.B.A. Mitchell. Dissociative recombination. \nPhysics Reports, 430(5):277 -374, 2006. \n[83] P W C Groen, A J Wolf, T W H Righart, M C M van de Sanden, \nF J J Peeters, and W A Bongers. Numerical model for the determina-\ntion of the reduced electric field in a CO 2 microwave plasma derived \nby the principle of impedance matching. Plasma Sources Science and \nTechnology, 28(7):075016, jul 2019. \n\n[84] N ν \nOrigin \nA \nAD \nBν \n10 6 · Dν \nλ \no \np \nq \n0 \n19 378.46749(51) -14.00139(63) 0.0005068(83) 1.7455663(43) 6.8205(16) 0.03301(47) 0.61076(52) 0.003973(43) -0.0007752(43) \n1 \n21 132.14977(25) -13.87513(49) 0.0005740(83) 1.7254062(53) 7.0194(77) 0.02972(38) 0.61713(36) 0.004133(44) -0.0008171(43) \n2 \n22 848.3877(21) \n-13.8205(23) \n0.000600(43) \n1.704516(21) \n7.308(22) \n0.0253(41) \n0.6208(32) \n0.00624(38) \n-0.000835(14) \n3 \n24 524.2201(19) \n-13.5361(28) \n0.000775(17) \n1.681437(16) \n7.438(24) \n0.0470(26) \n0.5827(26) \n0.00579(17) \n-0.0008568(85) \n4 \n26 155.0448(29) \n-13.3892(50) \n0.001451(14) \n1.656859(26) \n7.684(43) \n0.0219(38) \n0.6313(32) \n0.00954(29) \n-0.000923(21) \n5 \n27 735.6720(43) \n-13.0324(66) \n0.000723(37) \n1.630205(23) \n8.573(32) \n0.0601(28) \n0.6161(23) \n0.00685(32) \n-0.000912(15) \n6 \n29 259.3548(36) \n-12.820(10) \n0.001203(56) \n1.599876(31) \n8.998(44) \n0.0529(71) \n0.5773(71) \n0.00874(47) \n-0.000986(22) \n7 \n30 717.9011(46) \n-12.3458(71) \n0.000814(41) \n1.566047(32) \n10.044(66) 0.0960(34) \n0.5532(31) \n0.00936(35) \n-0.001175(17) \n8 \n32 102.655(22) \n-12.107(22) \n0.00076(fixed) 1.52675(31) \n9.60(97) \n0.095(fixed) 0.546(22) \n0.0055(21) \n-0.00088(21) \n9 \n33 406.230(22) \n-11.698(39) \n0.00076(fixed) 1.485755(96) \n11.85(10) \n0.172(26) \n0.498(28) \n0.0097(14) \n-0.002062(44) \n10 34 626.7860(94) \n-11.297(15) \n0.00076(fixed) 1.441138(72) \n12.837(73) 0.115(16) \n0.399(12) \n0.00745(90) \n-0.000977(30)"},{"id":"","caption":"Gatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system","content":"Gatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system:045010, 2008. \n\n[22] Yong C Hong, Dong H Shin, Sang J Lee, Ye J Kim, Bong Ju Lee, and \nHan S Uhm. Generation of high-power torch plasma by a 915-MHz mi-\ncrowave system. IEEE Transactions on Plasma Science, 39(10):1958-\n1962, 2011. \n\n[23] D Le Quang, Y Babou, and P Andre. Investigations of a microwave \nplasma source operating with air, N 2 , CO 2 and argon gases. IOP \nConference Series: Materials Science and Engineering, 29(1):012009, \n2012. \n\n[24] L F Spencer and A D Gallimore. CO 2 dissociation in an atmospheric \npressure plasma/catalyst system: a study of efficiency. Plasma Sources \nScience and Technology, 22(1):015019, 2013. \n\n[25] Han S. Uhm, Hyoung S. Kwak, and Yong C. Hong. Carbon dioxide \nelimination and regeneration of resources in a microwave plasma torch. \nEnvironmental Pollution, 211:191 -197, 2016. \n\n[26] Constandinos M Mitsingas, Rajavasanth Rajasegar, Stephen Ham-\nmack, Hyungrok Do, and Tonghun Lee. High energy efficiency plasma \nconversion of CO 2 at atmospheric pressure using a direct coupled \nmicrowave plasma system. IEEE Transactions on Plasma Science, \n44(4):651-656, 2016. \n\n[27] Hojoong Sun, Jungwun Lee, Hyungrok Do, Seong-kyun Im, and Moon \nSoo Bak. Experimental and numerical studies on carbon dioxide de-\ncomposition in atmospheric electrodeless microwave plasmas. Journal \nof Applied Physics, 122(3):033303, 2017. \n\n[28] Se Min Chun, Yong Cheol Hong, and Dae Hyun Choi. Reforming \nof methane to syngas in a microwave plasma torch at atmospheric \npressure. Journal of CO2 Utilization, 19:221-229, 2017. \n\n[29] PJ Bruggeman, N Sadeghi, DC Schram, and V Linss. Gas tempera-\nture determination from rotational lines in non-equilibrium plasmas: a \nreview. Plasma Sources Science and Technology, 23(2):023001, 2014. \n\n[30] Reginald William Blake Pearse and Alfred Gordon Gaydon. Identifi-\ncation of molecular spectra. Chapman and Hall, 1976. \n[31] Jan Voráč, Petr Synek, Lucia Potočňáková, Jaroslav Hnilica, and Vít \nKudrle. Batch processing of overlapping molecular spectra as a tool \nfor spatio-temporal diagnostics of power modulated microwave plasma \njet. Plasma Sources Science and Technology, 26(2):025010, 2017. \n\n[32] Jan Voráč, Petr Synek, Vojtěch Procházka, and Tomáš Hoder. State-\nby-state emission spectra fitting for non-equilibrium plasmas: OH spec-\ntra of surface barrier discharge at argon/water interface. Journal of \nPhysics D: Applied Physics, 50(29):294002, 2017. \n\n[33] Martina Leins, Sandra Gaiser, Andreas Schulz, Matthias Walker, Uwe \nSchumacher, and Thomas Hirth. How to ignite an atmospheric pressure \nmicrowave plasma torch without any additional igniters. Journal of \nvisualized experiments: JoVE, (98), 2015. \n\n[34] JW McConkey, CP Malone, PV Johnson, C Winstead, V McKoy, and \nI Kanik. Electron impact dissociation of oxygen-containing molecules-a \ncritical review. Physics Reports, 466(1-3):1-103, 2008. \n\n[35] Marija Grofulović, Luís L Alves, and Vasco Guerra. Electron-neutral \nscattering cross sections for CO2: a complete and consistent set and \nan assessment of dissociation. Journal of Physics D: Applied Physics, \n49(39):395207, sep 2016. \n\n[36] Pamir Nag and Dhananjay Nandi. Dissociation dynamics in the disso-\nciative electron attachment to carbon dioxide. Phys. Rev. A, 91:052705, \nMay 2015. \n\n[37] Yukikazu Itikawa. Cross sections for electron collisions with carbon \ndioxide. Journal of Physical and Chemical Reference Data, 31(3):749-\n767, 2002. \n\n[38] L.D. Pietanza, G. Colonna, V. Laporta, R. Celiberto, G. DAmmando, \nA. Laricchiuta, and M. Capitelli. Influence of electron molecule \nresonant vibrational collisions over the symmetric mode and direct \nexcitation-dissociation cross sections of co2 on the electron energy dis-\ntribution function and dissociation mechanisms in cold pure co2 plas-\nmas. The Journal of Physical Chemistry A, 120(17):2614-2628, 2016. \nPMID: 27064438. \n\n[39] A.I. Florescu-Mitchell and J.B.A. Mitchell. Dissociative recombination. \nPhysics Reports, 430(5):277 -374, 2006. \n[83] P W C Groen, A J Wolf, T W H Righart, M C M van de Sanden, \nF J J Peeters, and W A Bongers. Numerical model for the determina-\ntion of the reduced electric field in a CO 2 microwave plasma derived \nby the principle of impedance matching. Plasma Sources Science and \nTechnology, 28(7):075016, jul 2019. \n\n[84] N ν \nOrigin \nA \nAD \nBν \n10 6 · Dν \nλ \no \np \nq \n0 \n19 378.46749(51) -14.00139(63) 0.0005068(83) 1.7455663(43) 6.8205(16) 0.03301(47) 0.61076(52) 0.003973(43) -0.0007752(43) \n1 \n21 132.14977(25) -13.87513(49) 0.0005740(83) 1.7254062(53) 7.0194(77) 0.02972(38) 0.61713(36) 0.004133(44) -0.0008171(43) \n2 \n22 848.3877(21) \n-13.8205(23) \n0.000600(43) \n1.704516(21) \n7.308(22) \n0.0253(41) \n0.6208(32) \n0.00624(38) \n-0.000835(14) \n3 \n24 524.2201(19) \n-13.5361(28) \n0.000775(17) \n1.681437(16) \n7.438(24) \n0.0470(26) \n0.5827(26) \n0.00579(17) \n-0.0008568(85) \n4 \n26 155.0448(29) \n-13.3892(50) \n0.001451(14) \n1.656859(26) \n7.684(43) \n0.0219(38) \n0.6313(32) \n0.00954(29) \n-0.000923(21) \n5 \n27 735.6720(43) \n-13.0324(66) \n0.000723(37) \n1.630205(23) \n8.573(32) \n0.0601(28) \n0.6161(23) \n0.00685(32) \n-0.000912(15) \n6 \n29 259.3548(36) \n-12.820(10) \n0.001203(56) \n1.599876(31) \n8.998(44) \n0.0529(71) \n0.5773(71) \n0.00874(47) \n-0.000986(22) \n7 \n30 717.9011(46) \n-12.3458(71) \n0.000814(41) \n1.566047(32) \n10.044(66) 0.0960(34) \n0.5532(31) \n0.00936(35) \n-0.001175(17) \n8 \n32 102.655(22) \n-12.107(22) \n0.00076(fixed) 1.52675(31) \n9.60(97) \n0.095(fixed) 0.546(22) \n0.0055(21) \n-0.00088(21) \n9 \n33 406.230(22) \n-11.698(39) \n0.00076(fixed) 1.485755(96) \n11.85(10) \n0.172(26) \n0.498(28) \n0.0097(14) \n-0.002062(44) \n10 34 626.7860(94) \n-11.297(15) \n0.00076(fixed) 1.441138(72) \n12.837(73) 0.115(16) \n0.399(12) \n0.00745(90) \n-0.000977(30)"},{"id":"","caption":"Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)","content":"Table 2 :\n2Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)ν Origin \nA \nAD \nBν \n10 6 · Dν \nλ \no \np \nq \n0 0 \n-15.26986(43) 0.0002634(71) 1.6240452(44) 6.4506(19) -0.15450(36) 0.67525(35) 0.002537(42) -0.0005281(44) \n1 0.0001996(94) 1.5907513(61) 6.4527(44) -0.1526(12) \n0.6649(14) \n0.003132(77) -0.0006457(48) \n3 4784.0688(31) \n-15.1972(39) \n0.000186(42) \n1.574088(24) \n6.455(24) \n-0.1333(61) \n0.6815(51) \n0.00488(42) \n-0.000618(17) \n4 6332.1364(51) \n-15.2043(65) \n0.000318(36) \n1.557117(31) \n6.338(39) \n-0.1551(72) \n0.6674(67) \n0.00632(36) \n-0.000894(16) \n5 7856.8175(32) \n-15.2096(35) \n0.00025(fixed) 1.540139(24) \n6.312(35) \n-0.1492(36) \n0.6546(37) \n0.00734(25) \n-0.001246(12) \n6 9358.1565(40) \n-15.1646(60) \n0.000355(33) \n1.523439(26) \n6.034(38) \n-0.1551(46) \n0.6886(38) \n0.00504(32) \n-0.000676(16) \n7 10 836.1430(92) -15.085(11) \n0.00025(fixed) 1.50869(25) \n3.51(92) \n-0.1641(90) \n0.704(14) \n-0.0244(23) \n0.00632(49) \n8 12 290.7997(29) -15.1702(46) \n0.00025(fixed) 1.488684(28) \n5.329(52) \n-0.1665(36) \n0.6742(30) \n0.01449(29) \n-0.002053(23) \n9 13 722.0897(43) -15.0980(61) \n0.000419(31) \n1.472818(24) \n6.066(35) \n-0.1584(27) \n0.6926(23) \n0.00303(33) \n-0.000081(16)"},{"id":"","caption":"Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)","content":"Table 2 :\n2Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)ν Origin \nA \nAD \nBν \n10 6 · Dν \nλ \no \np \nq \n0 0 \n-15.26986(43) 0.0002634(71) 1.6240452(44) 6.4506(19) -0.15450(36) 0.67525(35) 0.002537(42) -0.0005281(44) \n1 0.0001996(94) 1.5907513(61) 6.4527(44) -0.1526(12) \n0.6649(14) \n0.003132(77) -0.0006457(48) \n3 4784.0688(31) \n-15.1972(39) \n0.000186(42) \n1.574088(24) \n6.455(24) \n-0.1333(61) \n0.6815(51) \n0.00488(42) \n-0.000618(17) \n4 6332.1364(51) \n-15.2043(65) \n0.000318(36) \n1.557117(31) \n6.338(39) \n-0.1551(72) \n0.6674(67) \n0.00632(36) \n-0.000894(16) \n5 7856.8175(32) \n-15.2096(35) \n0.00025(fixed) 1.540139(24) \n6.312(35) \n-0.1492(36) \n0.6546(37) \n0.00734(25) \n-0.001246(12) \n6 9358.1565(40) \n-15.1646(60) \n0.000355(33) \n1.523439(26) \n6.034(38) \n-0.1551(46) \n0.6886(38) \n0.00504(32) \n-0.000676(16) \n7 10 836.1430(92) -15.085(11) \n0.00025(fixed) 1.50869(25) \n3.51(92) \n-0.1641(90) \n0.704(14) \n-0.0244(23) \n0.00632(49) \n8 12 290.7997(29) -15.1702(46) \n0.00025(fixed) 1.488684(28) \n5.329(52) \n-0.1665(36) \n0.6742(30) \n0.01449(29) \n-0.002053(23) \n9 13 722.0897(43) -15.0980(61) \n0.000419(31) \n1.472818(24) \n6.066(35) \n-0.1584(27) \n0.6926(23) \n0.00303(33) \n-0.000081(16)"},{"id":"","caption":"Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6","content":"Table 3 :\n3Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6B Molecular constants for the calculations of \nthe CN violet system"},{"id":"","caption":"Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6","content":"Table 3 :\n3Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6B Molecular constants for the calculations of \nthe CN violet system"},{"id":"","caption":"Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.","content":"Table 4 :\n4Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.X 2 Σ +"},{"id":"","caption":"Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.","content":"Table 4 :\n4Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.X 2 Σ +"},{"id":"","caption":"Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.","content":"Table 5 :\n5Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.(480 \n485 \n490 \n495 \n500 \n505 \n510 \n515 \n\n0.0 \n\n5.0x10 \n\n18 \n\n1.0x10 \n\n19 \n\n1.5x10 \n\n19 \n\n2.0x10 \n\n19 \n\n2.5x10 \n\n19 \n\n3.0x10 \n\n19 \n\nR(91) \nR(84) \nR(88) \n\nR(95) \n\nR(98) \n\nR(77) \n\nR(65) \n\nR(58) \nR(62) \nR(48) P(72) \n\nP(71) \n\nP(68) \n\nP(57) \n\nP(43) \n\nP(44) \n\nP(45) \n\nP(29) \n\nP(26) \n\nl (nm) \n\nIntensity (ph m \n\n-2 \n\ns \n\n-1 \n\nnm \n\n-1 \n\n) \n\nP(25)"},{"id":"","caption":"Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.","content":"Table 5 :\n5Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.(480 \n485 \n490 \n495 \n500 \n505 \n510 \n515 \n\n0.0 \n\n5.0x10 \n\n18 \n\n1.0x10 \n\n19 \n\n1.5x10 \n\n19 \n\n2.0x10 \n\n19 \n\n2.5x10 \n\n19 \n\n3.0x10 \n\n19 \n\nR(91) \nR(84) \nR(88) \n\nR(95) \n\nR(98) \n\nR(77) \n\nR(65) \n\nR(58) \nR(62) \nR(48) P(72) \n\nP(71) \n\nP(68) \n\nP(57) \n\nP(43) \n\nP(44) \n\nP(45) \n\nP(29) \n\nP(26) \n\nl (nm) \n\nIntensity (ph m \n\n-2 \n\ns \n\n-1 \n\nnm \n\n-1 \n\n) \n\nP(25)"},{"id":"","caption":"Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).","content":"Table 6 :\n6Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).ν \n− ν \nB 2 Σ + \n0 \n1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n\nX 2 \nΣ +"},{"id":"","caption":"Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).","content":"Table 6 :\n6Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).ν \n− ν \nB 2 Σ + \n0 \n1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n\nX 2 \nΣ +"},{"id":"","caption":"The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition.","content":"Table 7 :\n7The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition."},{"id":"","caption":"The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition.","content":"Table 7 :\n7The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented.\",\n \"content\": \"Figure 2 :\\n2Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented.\",\n \"content\": \"Figure 2 :\\n2Schematic representation of the 2.45 GHz plasma source. The gases are tangentially injected at the bottom of the coaxial cavity (only two of the 4-inlets are represented.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min.\",\n \"content\": \"Figure 4 :\\n4Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min.\",\n \"content\": \"Figure 4 :\\n4Experimental C 2 Swan bands emission between 450 nm and 567 nm. The spectrum was acquired in the center of the resonator for a power of 2.7 kW and a flow of 10 L/min.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19\",\n \"content\": \"Figure 5 :\\n5Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19\",\n \"content\": \"Figure 5 :\\n5Difference between the calculated line position ν cal and the experimental one ν exp as function of the rotational quantum number of the lower state. Red and blue dots indicates the P and R branches respectively. Square, circle and diamond indicate the different fine structure sub-bands. All lines comes from the ν = 0 level and only the main transitions are considered which corresponds to ∆Ω = 0. on the energy level and transition probability of the d 3 Π g ν= 4 and 6 is included in the Einstein coefficient and energy of the transitions. The extra lines due to the perturbations for the d 3 Π g state by the b 3 Σ − g (ν = 16, 19\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm.\",\n \"content\": \"Figure 6 :\\n6Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm.\",\n \"content\": \"Figure 6 :\\n6Selection of isolated lines of the ∆ν = 0 C 2 Swan band and Boltzmann plot fitting to obtain the rotational temperature of the d 3 Π g state. Experimental conditions are: 2.7 kW input power, 10 L/min of CO 2 , an operating pressure of 920 mbar. The measurements are performed at a height of 20 mm (i.e. middle) inside the resonator and for the radial position r=0 mm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K.\",\n \"content\": \"Figure 7 :Figure 10 :\\n710Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K.\",\n \"content\": \"Figure 7 :Figure 10 :\\n710Analysis of the ∆ν = 0 transition group of the C 2 Swan band: a) Comparison between Boltzmann and state by state fitting. Note the logarithmic and linear scales used for presenting the fitted spectra. b) Boltzmann plot of the rotational population densities obtained from the state by state fitting. The experimental conditions are the same as infigure 6. I m and I s stand respectively for the intensity of the measured and simulated spectra. Sensitivity of the ∆ν = 0 transition group (475-517 nm) and of the ∆ν = −1, 0, +1 transition groups (450-567 nm) of the C 2 Swan band while fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done against a theoretical spectrum using a Voigt profile of 20 pm (equal gaussian and lorentzian contributions) for T rot = T vib fixed at values of 1000, 3000, 5000, 6000 and 7500 K.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm.\",\n \"content\": \"Figure 11 :\\n11Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm.\",\n \"content\": \"Figure 11 :\\n11Sensitivity of the ∆ν = 0 (on the left) and ∆ν = 0, ±1 (on the right) transition groups of the C 2 Swan band for fitting simultaneously the rotational and vibrational temperature of the d 3 Π g state. The calculations are done for a spectrum acquired with a resolution of about 22 pm.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6.\",\n \"content\": \"Figure 12 :\\n12Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6.\",\n \"content\": \"Figure 12 :\\n12Fitting of an experimental C 2 ∆ν = 0 Swan band transition group with fixed vibrational temperatures in the spectral range 480-516 nm. On top are superposed spectra normalized to the band head onto the experimental spectrum. On the bottom, are the fitted rotational temperatures while fixing the vibrational temperature. The experimental conditions are the same as in figure 6.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow\",\n \"content\": \"Figure 13 :Figure 14 :\\n1314CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow\"\n },\n {\n \"id\": \"\",\n \"caption\": \"CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow\",\n \"content\": \"Figure 13 :Figure 14 :\\n1314CN violet system emission between 345 nm and 425 nm corresponding to a typical temperature of T rot ≈ T vib 6000 K. The spectrum was acquired in the center of the resonator for a power of 900 W and a flow of 10 L/min of CO 2 and 0.3 L/min of N 2 . Comparison of the rotational and vibrational temperatures of the C 2 Swan band obtained by fitting the ∆ν = −1, 0, +1 transition groups and the CN violet band (fitting of ∆ν = −1, 0, +1 transition groups) as a function of pressure. The measurements are performed in the center of the resonator using 10 L/min CO 2 gas flow and 900 W input power. N 2 added gas flow\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W.\",\n \"content\": \"Figure 16 :\\n16Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W.\",\n \"content\": \"Figure 16 :\\n16Rotational/gas temperature of the C 2 Swan band obtained by fitting the ∆ν = 0 transition group as a function of the gas flow. The measurements are performed in the center of the resonator for a pressure of 850 mbar and an input power of 900 W.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3\",\n \"content\": \"−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3\"\n },\n {\n \"id\": \"\",\n \"caption\": \"−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3\",\n \"content\": \"−2 3.8010 −3 4.0210 −5 1.0610 −6 4.3910 −8 2.2710 −9 1.1610 −11 1.2210 −11 2.9410 −15 1 3.6710 −3 3.3610 −2 6.5010 −3 7.8310 −5 5.1710 −6 1.3910 −7 1.5810 −8 2.1110 −12 5.3510 −11 2 2.4710 −4 6.0410 −3 2.8010 −2 8.3810 −3 9.4310 −5 1.4910 −5 2.3410 −7 5.9810 −8 2.3110 −10 3 1.2110 −5 6.1710 −4 7.5010 −3 2.3710 −2 9.6410 −3 8.2810 −5 3.2710 −5 2.3210 −7 1.7410 −7 4 3.2110 −7 4.1410 −5 1.0310 −3 8.3210 −3 2.0410 −2 1.0410 −2 5.0810 −5 6.0210 −5 7.8910 −8 5 1.9110 −10 1.5010 −6 8.8110 −5 1.4310 −3 8.6810 −3 1.8010 −2 1.0810 −2 1.5410 −5 9.6710 −5 6 1.6610 −9 4.0010 −9 4.1510 −6 1.5010 −4 1.7910 −3 8.7210 −3 1.6210 −2 1.0810 −2 1.8010 −7 7 5.4010 −10 7.2110 −9 2.7210 −8 8.7910 −6 2.2210 −4 2.0910 −3 8.5710 −3 1.4810 −2 1.0510 −2 8 7.2310 −11 2.8910 −9 1.7210 −8 1.1210 −7 1.5710 −5 3.0110 −4 2.3210 −3 8.2610 −3 1.3810 −2 9 2.7610 −12 4.3910 −10 8.6510 −9 2.7510 −8 3.2310 −7 2.5310 −5 3.8210 −4 2.5010 −3 7.8510 −3\"\n },\n {\n \"id\": \"\",\n \"caption\": \"or from Fantz and Wünderlich[81] leads to\",\n \"content\": \"or from Fantz and Wünderlich[81] leads toParameter \\n\\nmeaning \\nN \\nUpper state excited specie density \\nT rot \\nRotational temperature \\nT vib \\nVibrational temperature \\nB base \\nConstant baseline offset value versus λ \\nα base \\nbaseline slope \\nσ G \\nGaussian width of the Voigt profile \\nσ L \\nLorentzian width of the Voigt profile \\nλ start\"\n },\n {\n \"id\": \"\",\n \"caption\": \"or from Fantz and Wünderlich[81] leads to\",\n \"content\": \"or from Fantz and Wünderlich[81] leads toParameter \\n\\nmeaning \\nN \\nUpper state excited specie density \\nT rot \\nRotational temperature \\nT vib \\nVibrational temperature \\nB base \\nConstant baseline offset value versus λ \\nα base \\nbaseline slope \\nσ G \\nGaussian width of the Voigt profile \\nσ L \\nLorentzian width of the Voigt profile \\nλ start\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program.\",\n \"content\": \"Table 1 :\\n1Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program.\",\n \"content\": \"Table 1 :\\n1Total amount of fitting parameters while making a Boltzmann population fit with the MassiveOES fitting program.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Gatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system\",\n \"content\": \"Gatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system:045010, 2008. \\n\\n[22] Yong C Hong, Dong H Shin, Sang J Lee, Ye J Kim, Bong Ju Lee, and \\nHan S Uhm. Generation of high-power torch plasma by a 915-MHz mi-\\ncrowave system. IEEE Transactions on Plasma Science, 39(10):1958-\\n1962, 2011. \\n\\n[23] D Le Quang, Y Babou, and P Andre. Investigations of a microwave \\nplasma source operating with air, N 2 , CO 2 and argon gases. IOP \\nConference Series: Materials Science and Engineering, 29(1):012009, \\n2012. \\n\\n[24] L F Spencer and A D Gallimore. CO 2 dissociation in an atmospheric \\npressure plasma/catalyst system: a study of efficiency. Plasma Sources \\nScience and Technology, 22(1):015019, 2013. \\n\\n[25] Han S. Uhm, Hyoung S. Kwak, and Yong C. Hong. Carbon dioxide \\nelimination and regeneration of resources in a microwave plasma torch. \\nEnvironmental Pollution, 211:191 -197, 2016. \\n\\n[26] Constandinos M Mitsingas, Rajavasanth Rajasegar, Stephen Ham-\\nmack, Hyungrok Do, and Tonghun Lee. High energy efficiency plasma \\nconversion of CO 2 at atmospheric pressure using a direct coupled \\nmicrowave plasma system. IEEE Transactions on Plasma Science, \\n44(4):651-656, 2016. \\n\\n[27] Hojoong Sun, Jungwun Lee, Hyungrok Do, Seong-kyun Im, and Moon \\nSoo Bak. Experimental and numerical studies on carbon dioxide de-\\ncomposition in atmospheric electrodeless microwave plasmas. Journal \\nof Applied Physics, 122(3):033303, 2017. \\n\\n[28] Se Min Chun, Yong Cheol Hong, and Dae Hyun Choi. Reforming \\nof methane to syngas in a microwave plasma torch at atmospheric \\npressure. Journal of CO2 Utilization, 19:221-229, 2017. \\n\\n[29] PJ Bruggeman, N Sadeghi, DC Schram, and V Linss. Gas tempera-\\nture determination from rotational lines in non-equilibrium plasmas: a \\nreview. Plasma Sources Science and Technology, 23(2):023001, 2014. \\n\\n[30] Reginald William Blake Pearse and Alfred Gordon Gaydon. Identifi-\\ncation of molecular spectra. Chapman and Hall, 1976. \\n[31] Jan Voráč, Petr Synek, Lucia Potočňáková, Jaroslav Hnilica, and Vít \\nKudrle. Batch processing of overlapping molecular spectra as a tool \\nfor spatio-temporal diagnostics of power modulated microwave plasma \\njet. Plasma Sources Science and Technology, 26(2):025010, 2017. \\n\\n[32] Jan Voráč, Petr Synek, Vojtěch Procházka, and Tomáš Hoder. State-\\nby-state emission spectra fitting for non-equilibrium plasmas: OH spec-\\ntra of surface barrier discharge at argon/water interface. Journal of \\nPhysics D: Applied Physics, 50(29):294002, 2017. \\n\\n[33] Martina Leins, Sandra Gaiser, Andreas Schulz, Matthias Walker, Uwe \\nSchumacher, and Thomas Hirth. How to ignite an atmospheric pressure \\nmicrowave plasma torch without any additional igniters. Journal of \\nvisualized experiments: JoVE, (98), 2015. \\n\\n[34] JW McConkey, CP Malone, PV Johnson, C Winstead, V McKoy, and \\nI Kanik. Electron impact dissociation of oxygen-containing molecules-a \\ncritical review. Physics Reports, 466(1-3):1-103, 2008. \\n\\n[35] Marija Grofulović, Luís L Alves, and Vasco Guerra. Electron-neutral \\nscattering cross sections for CO2: a complete and consistent set and \\nan assessment of dissociation. Journal of Physics D: Applied Physics, \\n49(39):395207, sep 2016. \\n\\n[36] Pamir Nag and Dhananjay Nandi. Dissociation dynamics in the disso-\\nciative electron attachment to carbon dioxide. Phys. Rev. A, 91:052705, \\nMay 2015. \\n\\n[37] Yukikazu Itikawa. Cross sections for electron collisions with carbon \\ndioxide. Journal of Physical and Chemical Reference Data, 31(3):749-\\n767, 2002. \\n\\n[38] L.D. Pietanza, G. Colonna, V. Laporta, R. Celiberto, G. DAmmando, \\nA. Laricchiuta, and M. Capitelli. Influence of electron molecule \\nresonant vibrational collisions over the symmetric mode and direct \\nexcitation-dissociation cross sections of co2 on the electron energy dis-\\ntribution function and dissociation mechanisms in cold pure co2 plas-\\nmas. The Journal of Physical Chemistry A, 120(17):2614-2628, 2016. \\nPMID: 27064438. \\n\\n[39] A.I. Florescu-Mitchell and J.B.A. Mitchell. Dissociative recombination. \\nPhysics Reports, 430(5):277 -374, 2006. \\n[83] P W C Groen, A J Wolf, T W H Righart, M C M van de Sanden, \\nF J J Peeters, and W A Bongers. Numerical model for the determina-\\ntion of the reduced electric field in a CO 2 microwave plasma derived \\nby the principle of impedance matching. Plasma Sources Science and \\nTechnology, 28(7):075016, jul 2019. \\n\\n[84] N ν \\nOrigin \\nA \\nAD \\nBν \\n10 6 · Dν \\nλ \\no \\np \\nq \\n0 \\n19 378.46749(51) -14.00139(63) 0.0005068(83) 1.7455663(43) 6.8205(16) 0.03301(47) 0.61076(52) 0.003973(43) -0.0007752(43) \\n1 \\n21 132.14977(25) -13.87513(49) 0.0005740(83) 1.7254062(53) 7.0194(77) 0.02972(38) 0.61713(36) 0.004133(44) -0.0008171(43) \\n2 \\n22 848.3877(21) \\n-13.8205(23) \\n0.000600(43) \\n1.704516(21) \\n7.308(22) \\n0.0253(41) \\n0.6208(32) \\n0.00624(38) \\n-0.000835(14) \\n3 \\n24 524.2201(19) \\n-13.5361(28) \\n0.000775(17) \\n1.681437(16) \\n7.438(24) \\n0.0470(26) \\n0.5827(26) \\n0.00579(17) \\n-0.0008568(85) \\n4 \\n26 155.0448(29) \\n-13.3892(50) \\n0.001451(14) \\n1.656859(26) \\n7.684(43) \\n0.0219(38) \\n0.6313(32) \\n0.00954(29) \\n-0.000923(21) \\n5 \\n27 735.6720(43) \\n-13.0324(66) \\n0.000723(37) \\n1.630205(23) \\n8.573(32) \\n0.0601(28) \\n0.6161(23) \\n0.00685(32) \\n-0.000912(15) \\n6 \\n29 259.3548(36) \\n-12.820(10) \\n0.001203(56) \\n1.599876(31) \\n8.998(44) \\n0.0529(71) \\n0.5773(71) \\n0.00874(47) \\n-0.000986(22) \\n7 \\n30 717.9011(46) \\n-12.3458(71) \\n0.000814(41) \\n1.566047(32) \\n10.044(66) 0.0960(34) \\n0.5532(31) \\n0.00936(35) \\n-0.001175(17) \\n8 \\n32 102.655(22) \\n-12.107(22) \\n0.00076(fixed) 1.52675(31) \\n9.60(97) \\n0.095(fixed) 0.546(22) \\n0.0055(21) \\n-0.00088(21) \\n9 \\n33 406.230(22) \\n-11.698(39) \\n0.00076(fixed) 1.485755(96) \\n11.85(10) \\n0.172(26) \\n0.498(28) \\n0.0097(14) \\n-0.002062(44) \\n10 34 626.7860(94) \\n-11.297(15) \\n0.00076(fixed) 1.441138(72) \\n12.837(73) 0.115(16) \\n0.399(12) \\n0.00745(90) \\n-0.000977(30)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Gatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system\",\n \"content\": \"Gatti, S Ponduri, F J J Peeters, D C M van den Bekerom, T Minea, P Tosi, M C M van de Sanden, and G J van Rooij. Preferential vibrational excitation in microwave nitrogen plasma assessed by raman scattering. Plasma Sources Science and Technology, 27(5):055006, may 2018. [85] Emile Carbone, Federico D'Isa, Ante Hecimovic, and Ursel Fantz. Optical emission spectroscopy characterization of a kHz pulsed atmospheric pressure N 2 microwave plasma. In APS Meeting Abstracts, 2018.A Molecular constants for the calculations of the C 2 Swan band system:045010, 2008. \\n\\n[22] Yong C Hong, Dong H Shin, Sang J Lee, Ye J Kim, Bong Ju Lee, and \\nHan S Uhm. Generation of high-power torch plasma by a 915-MHz mi-\\ncrowave system. IEEE Transactions on Plasma Science, 39(10):1958-\\n1962, 2011. \\n\\n[23] D Le Quang, Y Babou, and P Andre. Investigations of a microwave \\nplasma source operating with air, N 2 , CO 2 and argon gases. IOP \\nConference Series: Materials Science and Engineering, 29(1):012009, \\n2012. \\n\\n[24] L F Spencer and A D Gallimore. CO 2 dissociation in an atmospheric \\npressure plasma/catalyst system: a study of efficiency. Plasma Sources \\nScience and Technology, 22(1):015019, 2013. \\n\\n[25] Han S. Uhm, Hyoung S. Kwak, and Yong C. Hong. Carbon dioxide \\nelimination and regeneration of resources in a microwave plasma torch. \\nEnvironmental Pollution, 211:191 -197, 2016. \\n\\n[26] Constandinos M Mitsingas, Rajavasanth Rajasegar, Stephen Ham-\\nmack, Hyungrok Do, and Tonghun Lee. High energy efficiency plasma \\nconversion of CO 2 at atmospheric pressure using a direct coupled \\nmicrowave plasma system. IEEE Transactions on Plasma Science, \\n44(4):651-656, 2016. \\n\\n[27] Hojoong Sun, Jungwun Lee, Hyungrok Do, Seong-kyun Im, and Moon \\nSoo Bak. Experimental and numerical studies on carbon dioxide de-\\ncomposition in atmospheric electrodeless microwave plasmas. Journal \\nof Applied Physics, 122(3):033303, 2017. \\n\\n[28] Se Min Chun, Yong Cheol Hong, and Dae Hyun Choi. Reforming \\nof methane to syngas in a microwave plasma torch at atmospheric \\npressure. Journal of CO2 Utilization, 19:221-229, 2017. \\n\\n[29] PJ Bruggeman, N Sadeghi, DC Schram, and V Linss. Gas tempera-\\nture determination from rotational lines in non-equilibrium plasmas: a \\nreview. Plasma Sources Science and Technology, 23(2):023001, 2014. \\n\\n[30] Reginald William Blake Pearse and Alfred Gordon Gaydon. Identifi-\\ncation of molecular spectra. Chapman and Hall, 1976. \\n[31] Jan Voráč, Petr Synek, Lucia Potočňáková, Jaroslav Hnilica, and Vít \\nKudrle. Batch processing of overlapping molecular spectra as a tool \\nfor spatio-temporal diagnostics of power modulated microwave plasma \\njet. Plasma Sources Science and Technology, 26(2):025010, 2017. \\n\\n[32] Jan Voráč, Petr Synek, Vojtěch Procházka, and Tomáš Hoder. State-\\nby-state emission spectra fitting for non-equilibrium plasmas: OH spec-\\ntra of surface barrier discharge at argon/water interface. Journal of \\nPhysics D: Applied Physics, 50(29):294002, 2017. \\n\\n[33] Martina Leins, Sandra Gaiser, Andreas Schulz, Matthias Walker, Uwe \\nSchumacher, and Thomas Hirth. How to ignite an atmospheric pressure \\nmicrowave plasma torch without any additional igniters. Journal of \\nvisualized experiments: JoVE, (98), 2015. \\n\\n[34] JW McConkey, CP Malone, PV Johnson, C Winstead, V McKoy, and \\nI Kanik. Electron impact dissociation of oxygen-containing molecules-a \\ncritical review. Physics Reports, 466(1-3):1-103, 2008. \\n\\n[35] Marija Grofulović, Luís L Alves, and Vasco Guerra. Electron-neutral \\nscattering cross sections for CO2: a complete and consistent set and \\nan assessment of dissociation. Journal of Physics D: Applied Physics, \\n49(39):395207, sep 2016. \\n\\n[36] Pamir Nag and Dhananjay Nandi. Dissociation dynamics in the disso-\\nciative electron attachment to carbon dioxide. Phys. Rev. A, 91:052705, \\nMay 2015. \\n\\n[37] Yukikazu Itikawa. Cross sections for electron collisions with carbon \\ndioxide. Journal of Physical and Chemical Reference Data, 31(3):749-\\n767, 2002. \\n\\n[38] L.D. Pietanza, G. Colonna, V. Laporta, R. Celiberto, G. DAmmando, \\nA. Laricchiuta, and M. Capitelli. Influence of electron molecule \\nresonant vibrational collisions over the symmetric mode and direct \\nexcitation-dissociation cross sections of co2 on the electron energy dis-\\ntribution function and dissociation mechanisms in cold pure co2 plas-\\nmas. The Journal of Physical Chemistry A, 120(17):2614-2628, 2016. \\nPMID: 27064438. \\n\\n[39] A.I. Florescu-Mitchell and J.B.A. Mitchell. Dissociative recombination. \\nPhysics Reports, 430(5):277 -374, 2006. \\n[83] P W C Groen, A J Wolf, T W H Righart, M C M van de Sanden, \\nF J J Peeters, and W A Bongers. Numerical model for the determina-\\ntion of the reduced electric field in a CO 2 microwave plasma derived \\nby the principle of impedance matching. Plasma Sources Science and \\nTechnology, 28(7):075016, jul 2019. \\n\\n[84] N ν \\nOrigin \\nA \\nAD \\nBν \\n10 6 · Dν \\nλ \\no \\np \\nq \\n0 \\n19 378.46749(51) -14.00139(63) 0.0005068(83) 1.7455663(43) 6.8205(16) 0.03301(47) 0.61076(52) 0.003973(43) -0.0007752(43) \\n1 \\n21 132.14977(25) -13.87513(49) 0.0005740(83) 1.7254062(53) 7.0194(77) 0.02972(38) 0.61713(36) 0.004133(44) -0.0008171(43) \\n2 \\n22 848.3877(21) \\n-13.8205(23) \\n0.000600(43) \\n1.704516(21) \\n7.308(22) \\n0.0253(41) \\n0.6208(32) \\n0.00624(38) \\n-0.000835(14) \\n3 \\n24 524.2201(19) \\n-13.5361(28) \\n0.000775(17) \\n1.681437(16) \\n7.438(24) \\n0.0470(26) \\n0.5827(26) \\n0.00579(17) \\n-0.0008568(85) \\n4 \\n26 155.0448(29) \\n-13.3892(50) \\n0.001451(14) \\n1.656859(26) \\n7.684(43) \\n0.0219(38) \\n0.6313(32) \\n0.00954(29) \\n-0.000923(21) \\n5 \\n27 735.6720(43) \\n-13.0324(66) \\n0.000723(37) \\n1.630205(23) \\n8.573(32) \\n0.0601(28) \\n0.6161(23) \\n0.00685(32) \\n-0.000912(15) \\n6 \\n29 259.3548(36) \\n-12.820(10) \\n0.001203(56) \\n1.599876(31) \\n8.998(44) \\n0.0529(71) \\n0.5773(71) \\n0.00874(47) \\n-0.000986(22) \\n7 \\n30 717.9011(46) \\n-12.3458(71) \\n0.000814(41) \\n1.566047(32) \\n10.044(66) 0.0960(34) \\n0.5532(31) \\n0.00936(35) \\n-0.001175(17) \\n8 \\n32 102.655(22) \\n-12.107(22) \\n0.00076(fixed) 1.52675(31) \\n9.60(97) \\n0.095(fixed) 0.546(22) \\n0.0055(21) \\n-0.00088(21) \\n9 \\n33 406.230(22) \\n-11.698(39) \\n0.00076(fixed) 1.485755(96) \\n11.85(10) \\n0.172(26) \\n0.498(28) \\n0.0097(14) \\n-0.002062(44) \\n10 34 626.7860(94) \\n-11.297(15) \\n0.00076(fixed) 1.441138(72) \\n12.837(73) 0.115(16) \\n0.399(12) \\n0.00745(90) \\n-0.000977(30)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)\",\n \"content\": \"Table 2 :\\n2Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)ν Origin \\nA \\nAD \\nBν \\n10 6 · Dν \\nλ \\no \\np \\nq \\n0 0 \\n-15.26986(43) 0.0002634(71) 1.6240452(44) 6.4506(19) -0.15450(36) 0.67525(35) 0.002537(42) -0.0005281(44) \\n1 0.0001996(94) 1.5907513(61) 6.4527(44) -0.1526(12) \\n0.6649(14) \\n0.003132(77) -0.0006457(48) \\n3 4784.0688(31) \\n-15.1972(39) \\n0.000186(42) \\n1.574088(24) \\n6.455(24) \\n-0.1333(61) \\n0.6815(51) \\n0.00488(42) \\n-0.000618(17) \\n4 6332.1364(51) \\n-15.2043(65) \\n0.000318(36) \\n1.557117(31) \\n6.338(39) \\n-0.1551(72) \\n0.6674(67) \\n0.00632(36) \\n-0.000894(16) \\n5 7856.8175(32) \\n-15.2096(35) \\n0.00025(fixed) 1.540139(24) \\n6.312(35) \\n-0.1492(36) \\n0.6546(37) \\n0.00734(25) \\n-0.001246(12) \\n6 9358.1565(40) \\n-15.1646(60) \\n0.000355(33) \\n1.523439(26) \\n6.034(38) \\n-0.1551(46) \\n0.6886(38) \\n0.00504(32) \\n-0.000676(16) \\n7 10 836.1430(92) -15.085(11) \\n0.00025(fixed) 1.50869(25) \\n3.51(92) \\n-0.1641(90) \\n0.704(14) \\n-0.0244(23) \\n0.00632(49) \\n8 12 290.7997(29) -15.1702(46) \\n0.00025(fixed) 1.488684(28) \\n5.329(52) \\n-0.1665(36) \\n0.6742(30) \\n0.01449(29) \\n-0.002053(23) \\n9 13 722.0897(43) -15.0980(61) \\n0.000419(31) \\n1.472818(24) \\n6.066(35) \\n-0.1584(27) \\n0.6926(23) \\n0.00303(33) \\n-0.000081(16)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)\",\n \"content\": \"Table 2 :\\n2Molecular constants (in cm −1 ) for the d 3 Π g electronic state used in the calculations performed by Brooke et al. [1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=1 the additional constant H= 2.14(30) · 10 −11 is used. 1618.02244(53) -15.25197(61) 0.0002266(73) 1.6074266(44) 6.4439(21) -0.15373(51) 0.67017(51) 0.002705(44) -0.0005772(42) 2 3212.72793(96) -15.2328(15)ν Origin \\nA \\nAD \\nBν \\n10 6 · Dν \\nλ \\no \\np \\nq \\n0 0 \\n-15.26986(43) 0.0002634(71) 1.6240452(44) 6.4506(19) -0.15450(36) 0.67525(35) 0.002537(42) -0.0005281(44) \\n1 0.0001996(94) 1.5907513(61) 6.4527(44) -0.1526(12) \\n0.6649(14) \\n0.003132(77) -0.0006457(48) \\n3 4784.0688(31) \\n-15.1972(39) \\n0.000186(42) \\n1.574088(24) \\n6.455(24) \\n-0.1333(61) \\n0.6815(51) \\n0.00488(42) \\n-0.000618(17) \\n4 6332.1364(51) \\n-15.2043(65) \\n0.000318(36) \\n1.557117(31) \\n6.338(39) \\n-0.1551(72) \\n0.6674(67) \\n0.00632(36) \\n-0.000894(16) \\n5 7856.8175(32) \\n-15.2096(35) \\n0.00025(fixed) 1.540139(24) \\n6.312(35) \\n-0.1492(36) \\n0.6546(37) \\n0.00734(25) \\n-0.001246(12) \\n6 9358.1565(40) \\n-15.1646(60) \\n0.000355(33) \\n1.523439(26) \\n6.034(38) \\n-0.1551(46) \\n0.6886(38) \\n0.00504(32) \\n-0.000676(16) \\n7 10 836.1430(92) -15.085(11) \\n0.00025(fixed) 1.50869(25) \\n3.51(92) \\n-0.1641(90) \\n0.704(14) \\n-0.0244(23) \\n0.00632(49) \\n8 12 290.7997(29) -15.1702(46) \\n0.00025(fixed) 1.488684(28) \\n5.329(52) \\n-0.1665(36) \\n0.6742(30) \\n0.01449(29) \\n-0.002053(23) \\n9 13 722.0897(43) -15.0980(61) \\n0.000419(31) \\n1.472818(24) \\n6.066(35) \\n-0.1584(27) \\n0.6926(23) \\n0.00303(33) \\n-0.000081(16)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6\",\n \"content\": \"Table 3 :\\n3Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6B Molecular constants for the calculations of \\nthe CN violet system\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6\",\n \"content\": \"Table 3 :\\n3Molecular constants (in cm −1 ) for the a 3 Π u electronic state used in the calculations performed by Brooke et al.[1]. The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). For the vibrational level ν=0 the additional constants H= 6.73(16) · 10 −11 , o D = −6.86(114) · 10 −6 and q D = −9.61(41) · 10 −9 are used. +6 2.814 10 +6 2.809 10 +5 4.333 10 +3 2.033 10 +2 3.642 10 +1 2.470 10 −2 2.140 10 −1 9.989 10 −4 3.827 10 −3 3.140 10 −9 1 2.135 10 +6 3.427 10 +6 4.072 10 +6 6.429 10 +5 8.720 10 +3 1.608 10 +3 1.591 10 +2 4.744 10 +0 1.822 10 +0 4.947 10 −2 3.540 10 −2 2 3.832 10 +5 2.746 10 +6 1.270 10 +6 4.422 10 +6 9.615 10 +5 7.432 10 +3 6.154 10 +3 3.223 10 +2 6.379 10 +1 4.108 10 +0 1.567 10 +0 3 5.590 10 +4 8.273 10 +5 2.568 10 +6 3.236 10 +5 4.301 10 +6 1.168 10 +6 1.085 10 +3 1.805 10 +4 1.707 10 +2 3.472 10 +2 1.346 10 −2 4 7.224 10 +3 1.710 10 +5 1.169 10 +6 2.066 10 +6 2.505 10 +4 4.005 10 +6 1.149 10 +6 4.993 10 +3 3.581 10 +4 1.535 10 +2 9.452 10 +2 5 8.592 10 +2 2.886 10 +4 3.215 10 +5 1.352 10 +6 1.510 10 +6 1.513 10 +4 3.459 10 +6 1.120 10 +6 5.272 10 +4 4.975 10 +4 4.445 10 +3 6 9.574 10 +1 4.280 10 +3 6.793 10 +4 4.745 10 +5 1.381 10 +6 1.038 10 +6 8.028 10 +4 3.438 10 +6 8.558 10 +5 1.863 10 +5 4.075 10 +4 7 1.006 10 +1 5.775 10 +2 1.218 10 +4 1.218 10 +5 6.009 10 +5 1.302 10 +6 6.443 10 +5 1.343 10 +5 3.253 10 +6 4.834 10 +5 4.111 10 +5 8 9.931 10 −1 7.206 10 +1 1.945 10 +3 2.582 10 +4 1.830 10 +5 6.840 10 +5 1.087 10 +6 4.546 10 +5 1.285 10 +5 3.081 10 +6 1.342 10 +5 9 9.005 10 −2 8.335 10 +0 2.828 10 +2 4.785 10 +3 4.495 10 +4 2.421 10 +5 6.721 10 +5 9.993 10 +5 3.120 10 +5 8.080 10 +4 2.824 10 +6B Molecular constants for the calculations of \\nthe CN violet system\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.\",\n \"content\": \"Table 4 :\\n4Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.X 2 Σ +\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.\",\n \"content\": \"Table 4 :\\n4Vibrational transition probabilities (in s −1 ) calculated by Brooke at al.[1] for the d 3 Π g -a 3 Π u transition. The transition probabilities are calculated as the sum of all the possible transition from a J'=1,Ω =0 energy level.X 2 Σ +\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.\",\n \"content\": \"Table 5 :\\n5Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.(480 \\n485 \\n490 \\n495 \\n500 \\n505 \\n510 \\n515 \\n\\n0.0 \\n\\n5.0x10 \\n\\n18 \\n\\n1.0x10 \\n\\n19 \\n\\n1.5x10 \\n\\n19 \\n\\n2.0x10 \\n\\n19 \\n\\n2.5x10 \\n\\n19 \\n\\n3.0x10 \\n\\n19 \\n\\nR(91) \\nR(84) \\nR(88) \\n\\nR(95) \\n\\nR(98) \\n\\nR(77) \\n\\nR(65) \\n\\nR(58) \\nR(62) \\nR(48) P(72) \\n\\nP(71) \\n\\nP(68) \\n\\nP(57) \\n\\nP(43) \\n\\nP(44) \\n\\nP(45) \\n\\nP(29) \\n\\nP(26) \\n\\nl (nm) \\n\\nIntensity (ph m \\n\\n-2 \\n\\ns \\n\\n-1 \\n\\nnm \\n\\n-1 \\n\\n) \\n\\nP(25)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.\",\n \"content\": \"Table 5 :\\n5Molecular constants (in cm −1 ) for the X 2 Σ + used in the calculations (in cm −1 ).The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants). a) Identification of individual lines from the P and R branches of the v=0 level.(b) Boltzmann plot constructed using selected lines of the ∆ν = 0 transition group.(480 \\n485 \\n490 \\n495 \\n500 \\n505 \\n510 \\n515 \\n\\n0.0 \\n\\n5.0x10 \\n\\n18 \\n\\n1.0x10 \\n\\n19 \\n\\n1.5x10 \\n\\n19 \\n\\n2.0x10 \\n\\n19 \\n\\n2.5x10 \\n\\n19 \\n\\n3.0x10 \\n\\n19 \\n\\nR(91) \\nR(84) \\nR(88) \\n\\nR(95) \\n\\nR(98) \\n\\nR(77) \\n\\nR(65) \\n\\nR(58) \\nR(62) \\nR(48) P(72) \\n\\nP(71) \\n\\nP(68) \\n\\nP(57) \\n\\nP(43) \\n\\nP(44) \\n\\nP(45) \\n\\nP(29) \\n\\nP(26) \\n\\nl (nm) \\n\\nIntensity (ph m \\n\\n-2 \\n\\ns \\n\\n-1 \\n\\nnm \\n\\n-1 \\n\\n) \\n\\nP(25)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).\",\n \"content\": \"Table 6 :\\n6Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).ν \\n− ν \\nB 2 Σ + \\n0 \\n1 \\n2 \\n3 \\n4 \\n5 \\n6 \\n7 \\n8 \\n\\nX 2 \\nΣ +\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).\",\n \"content\": \"Table 6 :\\n6Molecular constants (in cm −1 ) for the B 2 Σ + used in the calculations (in cm −1 ). The uncertainties are given in parenthesis (i.e. one standard deviation to the last signicant digits of the constants).ν \\n− ν \\nB 2 Σ + \\n0 \\n1 \\n2 \\n3 \\n4 \\n5 \\n6 \\n7 \\n8 \\n\\nX 2 \\nΣ +\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition.\",\n \"content\": \"Table 7 :\\n7The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition.\",\n \"content\": \"Table 7 :\\n7The table shows the transition probabilities extracted from LIF-BASE for the CN (B 2 Σ + -X 2 Σ + ) transition.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Line strengths and updated molecular constants for the C 2 Swan system. S A James, Peter F Brooke, Timothy W Bernath, George B Schmidt, Bacskay, Journal of Quantitative Spectroscopy and Radiative Transfer. 124James S.A. Brooke, Peter F. Bernath, Timothy W. Schmidt, and George B. Bacskay. Line strengths and updated molecular constants for the C 2 Swan system. Journal of Quantitative Spectroscopy and Radiative Transfer, 124:11 -20, 2013.","doi":""},{"text":"CO 2 utilization: a process systems engineering vision. José F Araújo Ofélia De Queiroz, Rita Luiz De Medeiros, B Maria, Alves, CO 2 Sequestration and Valorization. InTechF Araújo Ofélia de Queiroz, José Luiz de Medeiros, and Rita Maria B Alves. CO 2 utilization: a process systems engineering vision. In CO 2 Sequestration and Valorization. InTech, 2014.","doi":""},{"text":"Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Alexandre Lebouvier, Samuel A Iwarere, Philippe Dargenlieu, Deresh Ramjugernath, Laurent Fulcheri, Energy & Fuels. 275Alexandre Lebouvier, Samuel A. Iwarere, Philippe dArgenlieu, Deresh Ramjugernath, and Laurent Fulcheri. Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Energy & Fuels, 27(5):2712-2722, 2013.","doi":""},{"text":"Plasma technology a novel solution for CO 2 conversion?. Ramses Snoeckx, Annemie Bogaerts, Chem. Soc. Rev. 46Ramses Snoeckx and Annemie Bogaerts. Plasma technology a novel solution for CO 2 conversion? Chem. Soc. Rev., 46:5805-5863, 2017.","doi":""},{"text":"Plasma for electrification of chemical industry: a case study on CO 2 reduction. H G J Van Rooij, W A N Akse, M C M Bongers, Van De Sanden, Plasma Physics and Controlled Fusion. 60114019G J van Rooij, H N Akse, W A Bongers, and M C M van de Sanden. Plasma for electrification of chemical industry: a case study on CO 2 reduction. Plasma Physics and Controlled Fusion, 60(1):014019, oct 2017.","doi":""},{"text":"Plasma Chemistry. Alexander Fridman, Cambridge Books OnlineAlexander Fridman. Plasma Chemistry. Cambridge University Press, 2008. Cambridge Books Online.","doi":""},{"text":". N N Sobolev, V Vasilii, Sokovikov, LASERS. Soviet Physics Uspekhi. 102153N N Sobolev and Vasilii V Sokovikov. CO 2 LASERS. Soviet Physics Uspekhi, 10(2):153, 1967.","doi":""},{"text":"Plasma treatments and plasma deposition of polymers for biomedical applications. Pietro Favia, Riccardo Dagostino, Papers presented at the Fifth International Conference on Plasma Surface Engineering. 98Pietro Favia and Riccardo dAgostino. Plasma treatments and plasma deposition of polymers for biomedical applications. Surface and Coat- ings Technology, 98(1):1102 -1106, 1998. Papers presented at the Fifth International Conference on Plasma Surface Engineering.","doi":""},{"text":"Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine. Emile Carbone, Claire Douat, 8Emile Carbone and Claire Douat. Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine, 8(1):93-120, 2018.","doi":""},{"text":"The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. V D Rusanov, G V Fridman, Sholin, Soviet Physics Uspekhi. 246447V D Rusanov, A A Fridman, and G V Sholin. The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. Soviet Physics Uspekhi, 24(6):447, 1981.","doi":""},{"text":"Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. T Kozák, A Bogaerts, Plasma Sources Science and Technology. 24115024T. Kozák and A. Bogaerts. Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. Plasma Sources Science and Technology, 24(1):015024, 2015.","doi":""},{"text":"Taming microwave plasma to beat thermodynamics in CO 2 dissociation. Gj Van Rooij, Van Den, Bekerom, Harder, G Minea, Berden, R Bongers, Engeln, Mf Graswinckel, Mcm Zoethout, Van De Sanden, Faraday discussions. 183GJ van Rooij, DCM van den Bekerom, N den Harder, T Minea, G Berden, WA Bongers, R Engeln, MF Graswinckel, E Zoethout, and MCM van de Sanden. Taming microwave plasma to beat thermody- namics in CO 2 dissociation. Faraday discussions, 183:233-248, 2015.","doi":""},{"text":"Selfconsistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. M Capitelli, G Colonna, L D Dammando, Pietanza, Plasma Sources Science and Technology. 26555009M Capitelli, G Colonna, G DAmmando, and L D Pietanza. Self- consistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. Plasma Sources Sci- ence and Technology, 26(5):055009, 2017.","doi":""},{"text":"Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. C E Treanor, J W Rich, R G Rehm, The Journal of Chemical Physics. 484C. E. Treanor, J W. Rich, and R.G. Rehm. Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. The Jour- nal of Chemical Physics, 48(4):1798-1807, 1968.","doi":""},{"text":"The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. A R I Azizov, V K Vakar, M F Zhivotov, O Krotov, Zinov'ev, Bv Potapkin, Rusanov, A A V D Rusanov, Fridman, 28567R I Azizov, A K Vakar, V K Zhivotov, M F Krotov, O A Zinov'ev, BV Potapkin, A A Rusanov, V D Rusanov, and A A Fridman. The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. 28:567, 1983.","doi":""},{"text":"Stefan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dissociation of CO 2 for fuel synthesis. Plasma processes and polymers. Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, 141600126Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, Ste- fan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dis- sociation of CO 2 for fuel synthesis. Plasma processes and polymers, 14(6):1600126, 2017.","doi":""},{"text":"Homogeneous co 2 conversion by microwave plasma: Wave propagation and diagnostics. Dirk C M Niek Den Harder, Van Den, Richard S Bekerom, Martijn F Al, Jose M Graswinckel, Palomares, J J Floran, Srinath Peeters, Teofil Ponduri, Waldo A Minea, Bongers, C M Mauritius, Gerard J Van De Sanden, Van Rooij, Plasma Processes and Polymers. 1461600120Niek den Harder, Dirk C. M. van den Bekerom, Richard S. Al, Mar- tijn F. Graswinckel, Jose M. Palomares, Floran J. J. Peeters, Srinath Ponduri, Teofil Minea, Waldo A. Bongers, Mauritius C. M. van de Sanden, and Gerard J. van Rooij. Homogeneous co 2 conversion by mi- crowave plasma: Wave propagation and diagnostics. Plasma Processes and Polymers, 14(6):1600120, 2017.","doi":""},{"text":"The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Dirk Van Den, Jose Maria Palomares Bekerom, Tiny Linares, Eddie M Verreycken, Sander Van Veldhuizen, Waldo Nijdam, Richard Bongers, Gerard J Van Van De Sanden, Rooij, Plasma Sources Science and Technology. Dirk van den Bekerom, Jose Maria Palomares Linares, Tiny Verr- eycken, Eddie M Van Veldhuizen, Sander Nijdam, Waldo Bongers, Richard Van de Sanden, and Gerard J Van Rooij. The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Plasma Sources Sci- ence and Technology, 2018.","doi":""},{"text":"On use of CO 2 chemiluminescence for combustion metrics in natural gas fired reciprocating engines. B Sreenath, Gupta, P Bipin, Munidhar S Bihari, Raj R Biruduganti, James Sekar, Zigan, Proceedings of the Combustion Institute. the Combustion Institute33Sreenath B. Gupta, Bipin P. Bihari, Munidhar S. Biruduganti, Raj R. Sekar, and James Zigan. On use of CO 2 chemiluminescence for com- bustion metrics in natural gas fired reciprocating engines. Proceedings of the Combustion Institute, 33(2):3131 -3139, 2011.","doi":""},{"text":"CO 2 * Chemiluminescence in Premixed Flames. J.-M Samaniego, F N Egolfopoulos, C T Bowman, Combustion Science and Technology. 1091-6J.-M. Samaniego, F. N. Egolfopoulos, and C. T. Bowman. CO 2 * Chemi- luminescence in Premixed Flames. Combustion Science and Technol- ogy, 109(1-6):183-203, 1995.","doi":""},{"text":"Investigation of the dynamics of co 2 plasmas using kinetic modelling. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, Ursel Hecimovic, Fantz, Van Dijk, 23rd International Symposium on Plasma Chemistry. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, A Heci- movic, Ursel Fantz, and J van Dijk. Investigation of the dynamics of co 2 plasmas using kinetic modelling. In 23rd International Symposium on Plasma Chemistry (ISPC 2017), 2017.","doi":""},{"text":"C 2 spectroscopy and kinetics. Margarita Martin, Journal of Photochemistry and Photobiology A: Chemistry. 663Margarita Martin. C 2 spectroscopy and kinetics. Journal of Photo- chemistry and Photobiology A: Chemistry, 66(3):263-289, 1992.","doi":""},{"text":"Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. G L D Pietanza, M Colonna, Capitelli, Plasma Sources Science and Technology. 2612125007L D Pietanza, G Colonna, and M Capitelli. Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. Plasma Sources Science and Technology, 26(12):125007, 2017.","doi":""},{"text":"Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. E V Kustova, A Aliat, A Chikhaoui, Chemical Physics Letters. 3445E.V. Kustova, A. Aliat, and A. Chikhaoui. Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. Chemical Physics Letters, 344(5):638 -646, 2001.","doi":""},{"text":"Isotope effect in Boudouard disproportionation reaction in optically pumped CO. Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, Lempert, Rich, Chemical Physics. 3631-3Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, WR Lem- pert, JW Rich, and IV Adamovich. Isotope effect in Boudouard dis- proportionation reaction in optically pumped CO. Chemical Physics, 363(1-3):24-32, 2009.","doi":""},{"text":"Gas phase Boudouard reactions involving singlet-singlet and singlet-triplet CO vibrationally excited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. R P Patricia, Henrique Barreto, Alessandra F Euclides, Vincenzo Albernaz, Mario Aquilanti, Gaia Capitelli, Andrea Grossi, Sergey Lombardi, Federico Macheret, Palazzetti, The European Physical Journal D. 7110259Patricia RP Barreto, Henrique de O Euclides, Alessandra F Albernaz, Vincenzo Aquilanti, Mario Capitelli, Gaia Grossi, Andrea Lombardi, Sergey Macheret, and Federico Palazzetti. Gas phase Boudouard reac- tions involving singlet-singlet and singlet-triplet CO vibrationally ex- cited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. The European Physical Journal D, 71(10):259, 2017.","doi":""},{"text":"Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. M I Ks Kini, Savadatti, Journal of Physics B: Atomic and Molecular Physics. 1061139KS Kini and MI Savadatti. Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. Journal of Physics B: Atomic and Molecular Physics, 10(6):1139, 1977.","doi":""},{"text":"The pulse radiolysis of carbon monoxide. C Willis, C Devillers, Chemical Physics Letters. 21C. Willis and C. Devillers. The pulse radiolysis of carbon monoxide. Chemical Physics Letters, 2(1):51 -53, 1968.","doi":""},{"text":"An Extension of the Phillips System of C 2 and a Survey of C 2 States. E A Ballik, D A Ramsay, The Astrophysical Journal. 13784E. A. Ballik and D. A. Ramsay. An Extension of the Phillips System of C 2 and a Survey of C 2 States. The Astrophysical Journal, 137:84, 1963.","doi":""},{"text":"Mechanism for Excitation of the C 2 High-Pressure Bands. C Kunz, S Harteck, Dondes, The Journal of Chemical Physics. 4610C+ C 2 O → C 2 (A 3 Π)+ COC Kunz, P Harteck, and S Dondes. Mechanism for Excitation of the C 2 High-Pressure Bands [C+ C 2 O → C 2 (A 3 Π)+ CO]. The Journal of Chemical Physics, 46(10):4157-4158, 1967.","doi":""},{"text":"C 2 formation in vibrationally excited CO. Chemical physics letters. Hl Wallaart, M-Y Piar, J-P Perrin, Martin, 246HL Wallaart, B Piar, M-Y Perrin, and J-P Martin. C 2 formation in vibrationally excited CO. Chemical physics letters, 246(6):587-593, 1995.","doi":""},{"text":"On the excitation mechanism of the C 2 high-pressure bands. D W Naegeli, H B Palmer, Journal of Molecular Spectroscopy. 261D.W. Naegeli and H.B. Palmer. On the excitation mechanism of the C 2 high-pressure bands. Journal of Molecular Spectroscopy, 26(1):152 -154, 1968.","doi":""},{"text":"Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. F Gosse, J C Sadeghi, Pebay-Peyroula, Chemical Physics Letters. 136F Gosse, N Sadeghi, and JC Pebay-Peyroula. Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. Chemical Physics Letters, 13(6):557-560, 1972.","doi":""},{"text":"Heat of Sublimation and Dissociation Energy of Gaseous C 2. Leo Brewer, W T Hicks, O H Krikorian, The Journal of Chemical Physics. 361Leo Brewer, W. T. Hicks, and O. H. Krikorian. Heat of Sublimation and Dissociation Energy of Gaseous C 2 . The Journal of Chemical Physics, 36(1):182-188, 1962.","doi":""},{"text":"Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters. P Caubet, G Dorthe, 218P Caubet and G Dorthe. Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters, 218(5-6):529-536, 1994.","doi":""},{"text":"Origin of the high-pressure bands of C 2 . Chemical physics letters. Ce Little, Browne, 134CE Little and PG Browne. Origin of the high-pressure bands of C 2 . Chemical physics letters, 134(6):560-564, 1987.","doi":""},{"text":"A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. H Sasada, T Amano, C Jarman, P F Bernath, The Journal of Chemical Physics. 944H. Sasada, T. Amano, C. Jarman, and P. F. Bernath. A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. The Journal of Chem- ical Physics, 94(4):2401-2407, 1991.","doi":""},{"text":"Theoretical investigation of excited states of C 3. A Terentyev, R Scholz, M Schreiber, G Seifert, The Journal of Chemical Physics. 12112A. Terentyev, R. Scholz, M. Schreiber, and G. Seifert. Theoretical investigation of excited states of C 3 . The Journal of Chemical Physics, 121(12):5767-5776, 2004.","doi":""},{"text":"The valence states of c 2 : A configuration interaction study. K Kirby, B Liu, The Journal of Chemical Physics. 702K. Kirby and B. Liu. The valence states of c 2 : A configuration inter- action study. The Journal of Chemical Physics, 70(2):893-900, 1979.","doi":""},{"text":"Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Wl Faust, Goldberg, Craig Bb, R G Weiss, Chemical Physics Letters. 832WL Faust, LS Goldberg, BB Craig, and RG Weiss. Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Chemical Physics Letters, 83(2):265- 269, 1981.","doi":""},{"text":"Investigation of the Fine Structure of the C 2 High-Pressure Bands. H Meinel, Gerd Messerle, The Astrophysical Journal. 154381H Meinel and Gerd Messerle. Investigation of the Fine Structure of the C 2 High-Pressure Bands. The Astrophysical Journal, 154:381, 1968.","doi":""},{"text":"Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. S Paul, Remo Barklem, Collet, Astronomy & Astrophysics. 58896Paul S Barklem and Remo Collet. Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. Astronomy & Astrophysics, 588:A96, 2016.","doi":""},{"text":"Statistical thermodynamics. Donald Allan Mcquarrie, Harper Collins PublishersDonald Allan McQuarrie. Statistical thermodynamics. Harper Collins Publishers, 1973.","doi":""},{"text":"Molecular spectra and molecular structure. Gerhard Herzberg, Spectra of diatomic molecules. New YorkVan Nostrand Reinhold12nd ed.Gerhard Herzberg. Molecular spectra and molecular structure. Vol. 1: Spectra of diatomic molecules. New York: Van Nostrand Reinhold, 1950, 2nd ed., 1950.","doi":""},{"text":"Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas. S Svanberg, S Svanberg. Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas, 2012.","doi":""},{"text":"Application of the (0,0) Swan band spectrum for temperature measurements. S Pellerin, Musiol, Motret, J Pokrzywka, Chapelle, Journal of Physics D: Applied Physics. 29112850S Pellerin, K Musiol, O Motret, B Pokrzywka, and J Chapelle. Appli- cation of the (0,0) Swan band spectrum for temperature measurements. Journal of Physics D: Applied Physics, 29(11):2850, 1996.","doi":""},{"text":"Spectra of atoms and molecules. F Peter, Bernath, Oxford university pressPeter F Bernath. Spectra of atoms and molecules. Oxford university press, 2015.","doi":""},{"text":"PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Colin M Western, Journal of Quantitative Spectroscopy and Radiative Transfer. 186Colin M Western. PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:221-242, 2017.","doi":""},{"text":"LEVEL: A computer program for solving the radial Schrdinger equation for bound and quasibound levels. J Le Robert, Roy, Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting. 186Robert J. Le Roy. LEVEL: A computer program for solving the ra- dial Schrdinger equation for bound and quasibound levels. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:167 -178, 2017. Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting, October 2015.","doi":""},{"text":"High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. G M Lloyd, P Ewart, The Journal of Chemical Physics. 1101G. M. Lloyd and P. Ewart. High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. The Journal of Chemical Physics, 110(1):385-392, 1999.","doi":""},{"text":"The Swan System of C 2 : A Global Analysis of Fourier Transform Emission Spectra. A Tanabashi, T Hirao, T Amano, P F Bernath, The Astrophysical Journal Supplement Series. 1692472A. Tanabashi, T. Hirao, T. Amano, and P. F. Bernath. The Swan Sys- tem of C 2 : A Global Analysis of Fourier Transform Emission Spectra. The Astrophysical Journal Supplement Series, 169(2):472, 2007.","doi":""},{"text":"Fourier transform emission spectroscopy of the B 2 Σ + -X 2 Σ + system of CN. R S Ram, S P Davis, L Wallace, R Engleman, D R T Appadoo, P F Bernath, Journal of Molecular Spectroscopy. 2372R.S. Ram, S.P. Davis, L. Wallace, R. Engleman, D.R.T. Appadoo, and P.F. Bernath. Fourier transform emission spectroscopy of the B 2 Σ + - X 2 Σ + system of CN. Journal of Molecular Spectroscopy, 237(2):225 - 231, 2006.","doi":""},{"text":"Lifbase, database and spectral simulation for diatomic molecules. J Luque, 99SRI International ReportJ Luque. Lifbase, database and spectral simulation for diatomic molecules. SRI International Report, 99, 1999.","doi":""},{"text":"Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spectrometer. D Cerny, R Bacis, G Guelachvili, F Roux, Journal of Molecular Spectroscopy. 731D. Cerny, R. Bacis, G. Guelachvili, and F. Roux. Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spec- trometer. Journal of Molecular Spectroscopy, 73(1):154 -167, 1978.","doi":""},{"text":"Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i. Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, Journal of Molecular Spectroscopy. 1272Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i . Journal of Molecular Spectroscopy, 127(2):283 -303, 1988.","doi":""},{"text":"Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, The Journal of Chemical Physics. 948Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. The Journal of Chemical Physics, 94(8):5353-5359, 1991.","doi":""},{"text":"Spectroscopic studies and rotational and vibrational temperature measurements of atmospheric pressure normal glow plasma discharges in air. David Staack, Bakhtier Farouk, Alexander F Gutsol, Alexander A Fridman, Plasma Sources Science and Technology. 154818David Staack, Bakhtier Farouk, Alexander F Gutsol, and Alexander A Fridman. Spectroscopic studies and rotational and vibrational tem- perature measurements of atmospheric pressure normal glow plasma discharges in air. Plasma Sources Science and Technology, 15(4):818, 2006.","doi":""},{"text":"C 2 Swan spectrum used as a molecular pyrometer in transferred arc and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 511112066H Nassar. C 2 Swan spectrum used as a molecular pyrometer in trans- ferred arc and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 511(1):012066, 2014.","doi":""},{"text":"arXiv:1404.1100Jonathon Shlens. A tutorial on principal component analysis. arXiv preprintJonathon Shlens. A tutorial on principal component analysis. arXiv preprint arXiv:1404.1100, 2014.","doi":""},{"text":". Christophe Laux, Specair, Christophe Laux. SPECAIR, 2012. Accessed: March 2018.","doi":""},{"text":"Modeling the spectral radiation of a methane/nitrogen plasma: Test case 4. L Caillaut, Andreasson, C O Risberg, Laux, Radiation of High Temperature Gases in Atmospheric Entry. 629L Caillaut, J Andreasson, J Risberg, and CO Laux. Modeling the spec- tral radiation of a methane/nitrogen plasma: Test case 4. In Radiation of High Temperature Gases in Atmospheric Entry, volume 629, 2006.","doi":""},{"text":"Fundamental data of diatomic molecules relevant for fusion. U Fantz, Wünderlich, Atomic and Plasma-Material Interaction Data for Fusion. 16U Fantz and D Wünderlich. Fundamental data of diatomic molecules relevant for fusion. In Atomic and Plasma-Material Interaction Data for Fusion, volume 16. IAEA, 2011.","doi":""},{"text":"Oscillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. Damian L Kokkin, George B Bacskay, Timothy W Schmidt, The Journal of Chemical Physics. 126884302Damian L. Kokkin, George B. Bacskay, and Timothy W. Schmidt. Os- cillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. The Journal of Chemical Physics, 126(8):084302, 2007.","doi":""},{"text":"Melanie Heusing, and Michael Letzgus. Measurement and simulation of rotationally-resolved chemiluminescence spectra in flames. Andreas Brockhinke, Julia Krüger, Applied Physics B. 1073Andreas Brockhinke, Julia Krüger, Melanie Heusing, and Michael Let- zgus. Measurement and simulation of rotationally-resolved chemilumi- nescence spectra in flames. Applied Physics B, 107(3):539-549, 2012.","doi":""},{"text":"Radiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: I. Experiment. D Vacher, Lino Da, P Silva, G André, M Faure, Dudeck, Plasma Sources Science and Technology. 17335012D Vacher, M Lino da Silva, P André, G Faure, and M Dudeck. Ra- diation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spec- tral region: I. Experiment. Plasma Sources Science and Technology, 17(3):035012, 2008.","doi":""},{"text":"Radiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. M Lino Da Silva, D Vacher, M Dudeck, P André, G Faure, Plasma Sources Science and Technology. 17335013M Lino da Silva, D Vacher, M Dudeck, P André, and G Faure. Radi- ation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. Plasma Sources Science and Technology, 17(3):035013, 2008.","doi":""},{"text":"The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. E A Timmermans, J Jonkers, M Rodero, Quintero, Sola, D Gamero, J A Schram, Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 547E.A.H Timmermans, J Jonkers, A Rodero, M.C Quintero, A Sola, A Gamero, D.C Schram, and J.A.M van der Mullen. The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 54(7):1085 -1098, 1999.","doi":""},{"text":"The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 1. plasmas at atmospheric pressure. E A H Timmermans, J Jonkers, I A J Thomas, A Rodero, M C Quintero, A Sola, A Gamero, J A M Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 5311E.A.H. Timmermans, J. Jonkers, I.A.J. Thomas, A. Rodero, M.C. Quintero, A. Sola, A. Gamero, and J.A.M. van der Mullen. The behav- ior of molecules in microwave-induced plasmas studied by optical emis- sion spectroscopy. 1. plasmas at atmospheric pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 53(11):1553 -1566, 1998.","doi":""},{"text":"Elucidation of C 2 and CN formation mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Meirong Dong, C.-Y. George, Xianglei Chan, Jhanis J Mao, Jidong Gonzalez, Richard E Lu, Russo, Spectrochimica Acta Part B: Atomic Spectroscopy. 100Dedicated To Nicolo Omenetto On the Occasion of his 75th BirthdayMeirong Dong, George C.-Y. Chan, Xianglei Mao, Jhanis J. Gonzalez, Jidong Lu, and Richard E. Russo. Elucidation of C 2 and CN forma- tion mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Spectrochimica Acta Part B: Atomic Spec- troscopy, 100:62 -69, 2014. Dedicated To Nicolo Omenetto On the Occasion of his 75th Birthday.","doi":""},{"text":"CN violet system spectrum used as a molecular pyrometer and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 37012050H Nassar. CN violet system spectrum used as a molecular pyrome- ter and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 370:012050, jun 2012.","doi":""},{"text":"Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. D B Se Youn Moon, B Kim, W Gweon, Choe, Journal of Applied Physics. 105553307Se Youn Moon, D. B. Kim, B. Gweon, and W. Choe. Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. Journal of Applied Physics, 105(5):053307, 2009.","doi":""},{"text":"Torcheà plasmaà excitation micro-onde: deux configurations complémentaires. A Ricard, H St-Onge, Malvos, Gicquel, M Hubert, Moisan, Journal de Physique III. 58A Ricard, L St-Onge, H Malvos, A Gicquel, J Hubert, and M Moisan. Torcheà plasmaà excitation micro-onde: deux config- urations complémentaires. Journal de Physique III, 5(8):1269-1285, 1995.","doi":""},{"text":"Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Ma Ridenti, Amorim, Plasma Chemistry and Plasma Processing. 38MA Ridenti and J Amorim. Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Plasma Chemistry and Plasma Processing, 38(2):311-329, 2018.","doi":""},{"text":"Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Kotaro Saito, Tetsuo Sakka, Yukio H Ogata, Journal of Applied Physics. 949Kotaro Saito, Tetsuo Sakka, and Yukio H. Ogata. Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Journal of Applied Physics, 94(9):5530-5536, 2003.","doi":""},{"text":"Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Fabien Goulay, László Nemes, Paul E Schrader, Hope A Michelsen, Molecular Physics. 1087-9Fabien Goulay, László Nemes, Paul E Schrader, and Hope A Michelsen. Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Molecular Physics, 108(7-9):1013-1025, 2010.","doi":""},{"text":"Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. V E Bondybey, The Journal of Chemical Physics. 656V. E. Bondybey. Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. The Journal of Chemical Physics, 65(6):2296-2304, 1976.","doi":""},{"text":"C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. R P Frosch, The Journal of Chemical Physics. 546R. P. Frosch. C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. The Journal of Chemical Physics, 54(6):2660-2666, 1971.","doi":""},{"text":"The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Li Xuechu, Lou Nanquan, Chemical Physics Letters. 751Li Xuechu and Lou Nanquan. The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Chemical Physics Letters, 75(1):110-114, 1980.","doi":""},{"text":"C 2 and CN formation by optical pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. William Rich, Richard C Bergman, Chemical Physics. 441J William Rich and Richard C Bergman. C 2 and CN formation by opti- cal pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. Chemical Physics, 44(1):53-64, 1979.","doi":""},{"text":"Vibrational to electronic energy transfer from CO to C 2 molecules. G Grigorian, Cenian, Chemical Physics Letters. 4694-6G Grigorian and A Cenian. Vibrational to electronic energy transfer from CO to C 2 molecules. Chemical Physics Letters, 469(4-6):247-249, 2009.","doi":""},{"text":"Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. L Hidde, Bruno Wallaart, Marie-Yvonne Piar, Jean-Pierre Perrin, Martin, Chemical Physics. 1961Hidde L. Wallaart, Bruno Piar, Marie-Yvonne Perrin, and Jean-Pierre Martin. Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. Chemical Physics, 196(1):149 -169, 1995.","doi":""},{"text":"Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. W Hack, Chemical Physics Letters. 813W Hack and W Langel. Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. Chemical Physics Letters, 81(3):387 -390, 1981.","doi":""},{"text":"Pressure Dependence of Rotational Intensity Distribution in C 2 Swan Emission from LowPressure Oxyacetylene Flames. R Bleekrode, The Journal of Chemical Physics. 458R. Bleekrode. Pressure Dependence of Rotational Intensity Distribu- tion in C 2 Swan Emission from LowPressure Oxyacetylene Flames. The Journal of Chemical Physics, 45(8):3153-3154, 1966.","doi":""},{"text":"Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, Katharina Kohse-Höinghaus, The Journal of Physical Chemistry A. 1109Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, and Katharina Kohse-Höinghaus. Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. The Journal of Physical Chemistry A, 110(9):3028-3035, 2006.","doi":""},{"text":"G Grigorian, A Cenian, Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing. 31G. Grigorian and A. Cenian. Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing, 31(2):337-352, Apr 2011.","doi":""},{"text":"State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2. Jingzhong Guo, C M Sadowski, Q Gao, F J Morgan, The Journal of Chemical Physics. 11317Jingzhong Guo, C. M. Sadowski, Q. Gao, and F. J. Morgan. State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2 . The Journal of Chemical Physics, 113(17):7276-7285, 2000.","doi":""},{"text":"State-to-state and total rotational energy transfer rate constants for CN. M K Sophie, Jingzhong Brunet, Tucker Guo, S V Carrington, C M Filseth, Sadowski, N)+NO. The Journal of Chemical Physics. 1169Sophie M. K. Brunet, Jingzhong Guo, Tucker Carrington, S. V. Filseth, and C. M. Sadowski. State-to-state and total rotational energy trans- fer rate constants for CN(B 2 Σ + ,v=0,N)+H 2 , CN(X 2 Σ + ,v=2,N)+H 2 , D 2 , and CN(X 2 Σ + ,v=3,N)+NO. The Journal of Chemical Physics, 116(9):3617-3625, 2002.","doi":""},{"text":"Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave induced nitrogen plasma torch. Chuan-Jie Chen, Shou-Zhe Li, Plasma Sources Science and Technology. 24335017Chuan-Jie Chen and Shou-Zhe Li. Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave in- duced nitrogen plasma torch. Plasma Sources Science and Technology, 24(3):035017, 2015.","doi":""},{"text":"Optical characterization of a microwave pulsed discharge used for dissociation of CO 2. Tiago Silva, Nikolay Britun, Thomas Godfroid, Rony Snyders, Plasma Sources Science and Technology. 23225009Tiago Silva, Nikolay Britun, Thomas Godfroid, and Rony Snyders. Op- tical characterization of a microwave pulsed discharge used for dissoci- ation of CO 2 . Plasma Sources Science and Technology, 23(2):025009, 2014.","doi":""},{"text":"The dissociation of carbon monoxide. A R Fairbairn, Alfred Gordon Gaydon, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 312A. R. Fairbairn and Alfred Gordon Gaydon. The dissociation of carbon monoxide. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 312(1509):207-227, 1969.","doi":""},{"text":"Coherent anti-Stokes Raman scattering and spontaneous Raman scattering diagnostics of nonequilibrium plasmas and flows. R Walter, Igor V Lempert, Adamovich, 2014-10-29T00:00:00Journal of Physics D: Applied Physics. 4743Walter R. Lempert and Igor V. Adamovich. Coherent anti-Stokes Ra- man scattering and spontaneous Raman scattering diagnostics of non- equilibrium plasmas and flows. Journal of Physics D: Applied Physics, 47(43):433001-433026, 2014-10-29T00:00:00.","doi":""},{"text":"How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. D C M Van Den, J M Bekerom, E M Linares, S Van Veldhuizen, M C M Nijdam, G J Van De Sanden, Van Rooij, Appl. Opt. 5720D. C. M. van den Bekerom, J. M. Palomares Linares, E. M. van Veld- huizen, S. Nijdam, M. C. M. van de Sanden, and G. J. van Rooij. How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. Appl. Opt., 57(20):5694- 5702, Jul 2018.","doi":""},{"text":"Line strengths and updated molecular constants for the C 2 Swan system. S A James, Peter F Brooke, Timothy W Bernath, George B Schmidt, Bacskay, Journal of Quantitative Spectroscopy and Radiative Transfer. 124James S.A. Brooke, Peter F. Bernath, Timothy W. Schmidt, and George B. Bacskay. Line strengths and updated molecular constants for the C 2 Swan system. Journal of Quantitative Spectroscopy and Radiative Transfer, 124:11 -20, 2013.","doi":""},{"text":"CO 2 utilization: a process systems engineering vision. José F Araújo Ofélia De Queiroz, Rita Luiz De Medeiros, B Maria, Alves, CO 2 Sequestration and Valorization. InTechF Araújo Ofélia de Queiroz, José Luiz de Medeiros, and Rita Maria B Alves. CO 2 utilization: a process systems engineering vision. In CO 2 Sequestration and Valorization. InTech, 2014.","doi":""},{"text":"Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Alexandre Lebouvier, Samuel A Iwarere, Philippe Dargenlieu, Deresh Ramjugernath, Laurent Fulcheri, Energy & Fuels. 275Alexandre Lebouvier, Samuel A. Iwarere, Philippe dArgenlieu, Deresh Ramjugernath, and Laurent Fulcheri. Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Energy & Fuels, 27(5):2712-2722, 2013.","doi":""},{"text":"Plasma technology a novel solution for CO 2 conversion?. Ramses Snoeckx, Annemie Bogaerts, Chem. Soc. Rev. 46Ramses Snoeckx and Annemie Bogaerts. Plasma technology a novel solution for CO 2 conversion? Chem. Soc. Rev., 46:5805-5863, 2017.","doi":""},{"text":"Plasma for electrification of chemical industry: a case study on CO 2 reduction. H G J Van Rooij, W A N Akse, M C M Bongers, Van De Sanden, Plasma Physics and Controlled Fusion. 60114019G J van Rooij, H N Akse, W A Bongers, and M C M van de Sanden. Plasma for electrification of chemical industry: a case study on CO 2 reduction. Plasma Physics and Controlled Fusion, 60(1):014019, oct 2017.","doi":""},{"text":"Plasma Chemistry. Alexander Fridman, Cambridge Books OnlineAlexander Fridman. Plasma Chemistry. Cambridge University Press, 2008. Cambridge Books Online.","doi":""},{"text":". N N Sobolev, V Vasilii, Sokovikov, LASERS. Soviet Physics Uspekhi. 102153N N Sobolev and Vasilii V Sokovikov. CO 2 LASERS. Soviet Physics Uspekhi, 10(2):153, 1967.","doi":""},{"text":"Plasma treatments and plasma deposition of polymers for biomedical applications. Pietro Favia, Riccardo Dagostino, Papers presented at the Fifth International Conference on Plasma Surface Engineering. 98Pietro Favia and Riccardo dAgostino. Plasma treatments and plasma deposition of polymers for biomedical applications. Surface and Coat- ings Technology, 98(1):1102 -1106, 1998. Papers presented at the Fifth International Conference on Plasma Surface Engineering.","doi":""},{"text":"Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine. Emile Carbone, Claire Douat, 8Emile Carbone and Claire Douat. Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine, 8(1):93-120, 2018.","doi":""},{"text":"The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. V D Rusanov, G V Fridman, Sholin, Soviet Physics Uspekhi. 246447V D Rusanov, A A Fridman, and G V Sholin. The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. Soviet Physics Uspekhi, 24(6):447, 1981.","doi":""},{"text":"Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. T Kozák, A Bogaerts, Plasma Sources Science and Technology. 24115024T. Kozák and A. Bogaerts. Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. Plasma Sources Science and Technology, 24(1):015024, 2015.","doi":""},{"text":"Taming microwave plasma to beat thermodynamics in CO 2 dissociation. Gj Van Rooij, Van Den, Bekerom, Harder, G Minea, Berden, R Bongers, Engeln, Mf Graswinckel, Mcm Zoethout, Van De Sanden, Faraday discussions. 183GJ van Rooij, DCM van den Bekerom, N den Harder, T Minea, G Berden, WA Bongers, R Engeln, MF Graswinckel, E Zoethout, and MCM van de Sanden. Taming microwave plasma to beat thermody- namics in CO 2 dissociation. Faraday discussions, 183:233-248, 2015.","doi":""},{"text":"Selfconsistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. M Capitelli, G Colonna, L D Dammando, Pietanza, Plasma Sources Science and Technology. 26555009M Capitelli, G Colonna, G DAmmando, and L D Pietanza. Self- consistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. Plasma Sources Sci- ence and Technology, 26(5):055009, 2017.","doi":""},{"text":"Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. C E Treanor, J W Rich, R G Rehm, The Journal of Chemical Physics. 484C. E. Treanor, J W. Rich, and R.G. Rehm. Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. The Jour- nal of Chemical Physics, 48(4):1798-1807, 1968.","doi":""},{"text":"The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. A R I Azizov, V K Vakar, M F Zhivotov, O Krotov, Zinov'ev, Bv Potapkin, Rusanov, A A V D Rusanov, Fridman, 28567R I Azizov, A K Vakar, V K Zhivotov, M F Krotov, O A Zinov'ev, BV Potapkin, A A Rusanov, V D Rusanov, and A A Fridman. The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. 28:567, 1983.","doi":""},{"text":"Stefan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dissociation of CO 2 for fuel synthesis. Plasma processes and polymers. Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, 141600126Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, Ste- fan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dis- sociation of CO 2 for fuel synthesis. Plasma processes and polymers, 14(6):1600126, 2017.","doi":""},{"text":"Homogeneous co 2 conversion by microwave plasma: Wave propagation and diagnostics. Dirk C M Niek Den Harder, Van Den, Richard S Bekerom, Martijn F Al, Jose M Graswinckel, Palomares, J J Floran, Srinath Peeters, Teofil Ponduri, Waldo A Minea, Bongers, C M Mauritius, Gerard J Van De Sanden, Van Rooij, Plasma Processes and Polymers. 1461600120Niek den Harder, Dirk C. M. van den Bekerom, Richard S. Al, Mar- tijn F. Graswinckel, Jose M. Palomares, Floran J. J. Peeters, Srinath Ponduri, Teofil Minea, Waldo A. Bongers, Mauritius C. M. van de Sanden, and Gerard J. van Rooij. Homogeneous co 2 conversion by mi- crowave plasma: Wave propagation and diagnostics. Plasma Processes and Polymers, 14(6):1600120, 2017.","doi":""},{"text":"The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Dirk Van Den, Jose Maria Palomares Bekerom, Tiny Linares, Eddie M Verreycken, Sander Van Veldhuizen, Waldo Nijdam, Richard Bongers, Gerard J Van Van De Sanden, Rooij, Plasma Sources Science and Technology. Dirk van den Bekerom, Jose Maria Palomares Linares, Tiny Verr- eycken, Eddie M Van Veldhuizen, Sander Nijdam, Waldo Bongers, Richard Van de Sanden, and Gerard J Van Rooij. The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Plasma Sources Sci- ence and Technology, 2018.","doi":""},{"text":"On use of CO 2 chemiluminescence for combustion metrics in natural gas fired reciprocating engines. B Sreenath, Gupta, P Bipin, Munidhar S Bihari, Raj R Biruduganti, James Sekar, Zigan, Proceedings of the Combustion Institute. the Combustion Institute33Sreenath B. Gupta, Bipin P. Bihari, Munidhar S. Biruduganti, Raj R. Sekar, and James Zigan. On use of CO 2 chemiluminescence for com- bustion metrics in natural gas fired reciprocating engines. Proceedings of the Combustion Institute, 33(2):3131 -3139, 2011.","doi":""},{"text":"CO 2 * Chemiluminescence in Premixed Flames. J.-M Samaniego, F N Egolfopoulos, C T Bowman, Combustion Science and Technology. 1091-6J.-M. Samaniego, F. N. Egolfopoulos, and C. T. Bowman. CO 2 * Chemi- luminescence in Premixed Flames. Combustion Science and Technol- ogy, 109(1-6):183-203, 1995.","doi":""},{"text":"Investigation of the dynamics of co 2 plasmas using kinetic modelling. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, Ursel Hecimovic, Fantz, Van Dijk, 23rd International Symposium on Plasma Chemistry. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, A Heci- movic, Ursel Fantz, and J van Dijk. Investigation of the dynamics of co 2 plasmas using kinetic modelling. In 23rd International Symposium on Plasma Chemistry (ISPC 2017), 2017.","doi":""},{"text":"C 2 spectroscopy and kinetics. Margarita Martin, Journal of Photochemistry and Photobiology A: Chemistry. 663Margarita Martin. C 2 spectroscopy and kinetics. Journal of Photo- chemistry and Photobiology A: Chemistry, 66(3):263-289, 1992.","doi":""},{"text":"Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. G L D Pietanza, M Colonna, Capitelli, Plasma Sources Science and Technology. 2612125007L D Pietanza, G Colonna, and M Capitelli. Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. Plasma Sources Science and Technology, 26(12):125007, 2017.","doi":""},{"text":"Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. E V Kustova, A Aliat, A Chikhaoui, Chemical Physics Letters. 3445E.V. Kustova, A. Aliat, and A. Chikhaoui. Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. Chemical Physics Letters, 344(5):638 -646, 2001.","doi":""},{"text":"Isotope effect in Boudouard disproportionation reaction in optically pumped CO. Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, Lempert, Rich, Chemical Physics. 3631-3Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, WR Lem- pert, JW Rich, and IV Adamovich. Isotope effect in Boudouard dis- proportionation reaction in optically pumped CO. Chemical Physics, 363(1-3):24-32, 2009.","doi":""},{"text":"Gas phase Boudouard reactions involving singlet-singlet and singlet-triplet CO vibrationally excited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. R P Patricia, Henrique Barreto, Alessandra F Euclides, Vincenzo Albernaz, Mario Aquilanti, Gaia Capitelli, Andrea Grossi, Sergey Lombardi, Federico Macheret, Palazzetti, The European Physical Journal D. 7110259Patricia RP Barreto, Henrique de O Euclides, Alessandra F Albernaz, Vincenzo Aquilanti, Mario Capitelli, Gaia Grossi, Andrea Lombardi, Sergey Macheret, and Federico Palazzetti. Gas phase Boudouard reac- tions involving singlet-singlet and singlet-triplet CO vibrationally ex- cited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. The European Physical Journal D, 71(10):259, 2017.","doi":""},{"text":"Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. M I Ks Kini, Savadatti, Journal of Physics B: Atomic and Molecular Physics. 1061139KS Kini and MI Savadatti. Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. Journal of Physics B: Atomic and Molecular Physics, 10(6):1139, 1977.","doi":""},{"text":"The pulse radiolysis of carbon monoxide. C Willis, C Devillers, Chemical Physics Letters. 21C. Willis and C. Devillers. The pulse radiolysis of carbon monoxide. Chemical Physics Letters, 2(1):51 -53, 1968.","doi":""},{"text":"An Extension of the Phillips System of C 2 and a Survey of C 2 States. E A Ballik, D A Ramsay, The Astrophysical Journal. 13784E. A. Ballik and D. A. Ramsay. An Extension of the Phillips System of C 2 and a Survey of C 2 States. The Astrophysical Journal, 137:84, 1963.","doi":""},{"text":"Mechanism for Excitation of the C 2 High-Pressure Bands. C Kunz, S Harteck, Dondes, The Journal of Chemical Physics. 4610C+ C 2 O → C 2 (A 3 Π)+ COC Kunz, P Harteck, and S Dondes. Mechanism for Excitation of the C 2 High-Pressure Bands [C+ C 2 O → C 2 (A 3 Π)+ CO]. The Journal of Chemical Physics, 46(10):4157-4158, 1967.","doi":""},{"text":"C 2 formation in vibrationally excited CO. Chemical physics letters. Hl Wallaart, M-Y Piar, J-P Perrin, Martin, 246HL Wallaart, B Piar, M-Y Perrin, and J-P Martin. C 2 formation in vibrationally excited CO. Chemical physics letters, 246(6):587-593, 1995.","doi":""},{"text":"On the excitation mechanism of the C 2 high-pressure bands. D W Naegeli, H B Palmer, Journal of Molecular Spectroscopy. 261D.W. Naegeli and H.B. Palmer. On the excitation mechanism of the C 2 high-pressure bands. Journal of Molecular Spectroscopy, 26(1):152 -154, 1968.","doi":""},{"text":"Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. F Gosse, J C Sadeghi, Pebay-Peyroula, Chemical Physics Letters. 136F Gosse, N Sadeghi, and JC Pebay-Peyroula. Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. Chemical Physics Letters, 13(6):557-560, 1972.","doi":""},{"text":"Heat of Sublimation and Dissociation Energy of Gaseous C 2. Leo Brewer, W T Hicks, O H Krikorian, The Journal of Chemical Physics. 361Leo Brewer, W. T. Hicks, and O. H. Krikorian. Heat of Sublimation and Dissociation Energy of Gaseous C 2 . The Journal of Chemical Physics, 36(1):182-188, 1962.","doi":""},{"text":"Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters. P Caubet, G Dorthe, 218P Caubet and G Dorthe. Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters, 218(5-6):529-536, 1994.","doi":""},{"text":"Origin of the high-pressure bands of C 2 . Chemical physics letters. Ce Little, Browne, 134CE Little and PG Browne. Origin of the high-pressure bands of C 2 . Chemical physics letters, 134(6):560-564, 1987.","doi":""},{"text":"A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. H Sasada, T Amano, C Jarman, P F Bernath, The Journal of Chemical Physics. 944H. Sasada, T. Amano, C. Jarman, and P. F. Bernath. A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. The Journal of Chem- ical Physics, 94(4):2401-2407, 1991.","doi":""},{"text":"Theoretical investigation of excited states of C 3. A Terentyev, R Scholz, M Schreiber, G Seifert, The Journal of Chemical Physics. 12112A. Terentyev, R. Scholz, M. Schreiber, and G. Seifert. Theoretical investigation of excited states of C 3 . The Journal of Chemical Physics, 121(12):5767-5776, 2004.","doi":""},{"text":"The valence states of c 2 : A configuration interaction study. K Kirby, B Liu, The Journal of Chemical Physics. 702K. Kirby and B. Liu. The valence states of c 2 : A configuration inter- action study. The Journal of Chemical Physics, 70(2):893-900, 1979.","doi":""},{"text":"Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Wl Faust, Goldberg, Craig Bb, R G Weiss, Chemical Physics Letters. 832WL Faust, LS Goldberg, BB Craig, and RG Weiss. Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Chemical Physics Letters, 83(2):265- 269, 1981.","doi":""},{"text":"Investigation of the Fine Structure of the C 2 High-Pressure Bands. H Meinel, Gerd Messerle, The Astrophysical Journal. 154381H Meinel and Gerd Messerle. Investigation of the Fine Structure of the C 2 High-Pressure Bands. The Astrophysical Journal, 154:381, 1968.","doi":""},{"text":"Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. S Paul, Remo Barklem, Collet, Astronomy & Astrophysics. 58896Paul S Barklem and Remo Collet. Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. Astronomy & Astrophysics, 588:A96, 2016.","doi":""},{"text":"Statistical thermodynamics. Donald Allan Mcquarrie, Harper Collins PublishersDonald Allan McQuarrie. Statistical thermodynamics. Harper Collins Publishers, 1973.","doi":""},{"text":"Molecular spectra and molecular structure. Gerhard Herzberg, Spectra of diatomic molecules. New YorkVan Nostrand Reinhold12nd ed.Gerhard Herzberg. Molecular spectra and molecular structure. Vol. 1: Spectra of diatomic molecules. New York: Van Nostrand Reinhold, 1950, 2nd ed., 1950.","doi":""},{"text":"Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas. S Svanberg, S Svanberg. Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas, 2012.","doi":""},{"text":"Application of the (0,0) Swan band spectrum for temperature measurements. S Pellerin, Musiol, Motret, J Pokrzywka, Chapelle, Journal of Physics D: Applied Physics. 29112850S Pellerin, K Musiol, O Motret, B Pokrzywka, and J Chapelle. Appli- cation of the (0,0) Swan band spectrum for temperature measurements. Journal of Physics D: Applied Physics, 29(11):2850, 1996.","doi":""},{"text":"Spectra of atoms and molecules. F Peter, Bernath, Oxford university pressPeter F Bernath. Spectra of atoms and molecules. Oxford university press, 2015.","doi":""},{"text":"PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Colin M Western, Journal of Quantitative Spectroscopy and Radiative Transfer. 186Colin M Western. PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:221-242, 2017.","doi":""},{"text":"LEVEL: A computer program for solving the radial Schrdinger equation for bound and quasibound levels. J Le Robert, Roy, Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting. 186Robert J. Le Roy. LEVEL: A computer program for solving the ra- dial Schrdinger equation for bound and quasibound levels. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:167 -178, 2017. Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting, October 2015.","doi":""},{"text":"High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. G M Lloyd, P Ewart, The Journal of Chemical Physics. 1101G. M. Lloyd and P. Ewart. High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. The Journal of Chemical Physics, 110(1):385-392, 1999.","doi":""},{"text":"The Swan System of C 2 : A Global Analysis of Fourier Transform Emission Spectra. A Tanabashi, T Hirao, T Amano, P F Bernath, The Astrophysical Journal Supplement Series. 1692472A. Tanabashi, T. Hirao, T. Amano, and P. F. Bernath. The Swan Sys- tem of C 2 : A Global Analysis of Fourier Transform Emission Spectra. The Astrophysical Journal Supplement Series, 169(2):472, 2007.","doi":""},{"text":"Fourier transform emission spectroscopy of the B 2 Σ + -X 2 Σ + system of CN. R S Ram, S P Davis, L Wallace, R Engleman, D R T Appadoo, P F Bernath, Journal of Molecular Spectroscopy. 2372R.S. Ram, S.P. Davis, L. Wallace, R. Engleman, D.R.T. Appadoo, and P.F. Bernath. Fourier transform emission spectroscopy of the B 2 Σ + - X 2 Σ + system of CN. Journal of Molecular Spectroscopy, 237(2):225 - 231, 2006.","doi":""},{"text":"Lifbase, database and spectral simulation for diatomic molecules. J Luque, 99SRI International ReportJ Luque. Lifbase, database and spectral simulation for diatomic molecules. SRI International Report, 99, 1999.","doi":""},{"text":"Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spectrometer. D Cerny, R Bacis, G Guelachvili, F Roux, Journal of Molecular Spectroscopy. 731D. Cerny, R. Bacis, G. Guelachvili, and F. Roux. Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spec- trometer. Journal of Molecular Spectroscopy, 73(1):154 -167, 1978.","doi":""},{"text":"Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i. Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, Journal of Molecular Spectroscopy. 1272Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i . Journal of Molecular Spectroscopy, 127(2):283 -303, 1988.","doi":""},{"text":"Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, The Journal of Chemical Physics. 948Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. The Journal of Chemical Physics, 94(8):5353-5359, 1991.","doi":""},{"text":"Spectroscopic studies and rotational and vibrational temperature measurements of atmospheric pressure normal glow plasma discharges in air. David Staack, Bakhtier Farouk, Alexander F Gutsol, Alexander A Fridman, Plasma Sources Science and Technology. 154818David Staack, Bakhtier Farouk, Alexander F Gutsol, and Alexander A Fridman. Spectroscopic studies and rotational and vibrational tem- perature measurements of atmospheric pressure normal glow plasma discharges in air. Plasma Sources Science and Technology, 15(4):818, 2006.","doi":""},{"text":"C 2 Swan spectrum used as a molecular pyrometer in transferred arc and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 511112066H Nassar. C 2 Swan spectrum used as a molecular pyrometer in trans- ferred arc and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 511(1):012066, 2014.","doi":""},{"text":"arXiv:1404.1100Jonathon Shlens. A tutorial on principal component analysis. arXiv preprintJonathon Shlens. A tutorial on principal component analysis. arXiv preprint arXiv:1404.1100, 2014.","doi":""},{"text":". Christophe Laux, Specair, Christophe Laux. SPECAIR, 2012. Accessed: March 2018.","doi":""},{"text":"Modeling the spectral radiation of a methane/nitrogen plasma: Test case 4. L Caillaut, Andreasson, C O Risberg, Laux, Radiation of High Temperature Gases in Atmospheric Entry. 629L Caillaut, J Andreasson, J Risberg, and CO Laux. Modeling the spec- tral radiation of a methane/nitrogen plasma: Test case 4. In Radiation of High Temperature Gases in Atmospheric Entry, volume 629, 2006.","doi":""},{"text":"Fundamental data of diatomic molecules relevant for fusion. U Fantz, Wünderlich, Atomic and Plasma-Material Interaction Data for Fusion. 16U Fantz and D Wünderlich. Fundamental data of diatomic molecules relevant for fusion. In Atomic and Plasma-Material Interaction Data for Fusion, volume 16. IAEA, 2011.","doi":""},{"text":"Oscillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. Damian L Kokkin, George B Bacskay, Timothy W Schmidt, The Journal of Chemical Physics. 126884302Damian L. Kokkin, George B. Bacskay, and Timothy W. Schmidt. Os- cillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. The Journal of Chemical Physics, 126(8):084302, 2007.","doi":""},{"text":"Melanie Heusing, and Michael Letzgus. Measurement and simulation of rotationally-resolved chemiluminescence spectra in flames. Andreas Brockhinke, Julia Krüger, Applied Physics B. 1073Andreas Brockhinke, Julia Krüger, Melanie Heusing, and Michael Let- zgus. Measurement and simulation of rotationally-resolved chemilumi- nescence spectra in flames. Applied Physics B, 107(3):539-549, 2012.","doi":""},{"text":"Radiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: I. Experiment. D Vacher, Lino Da, P Silva, G André, M Faure, Dudeck, Plasma Sources Science and Technology. 17335012D Vacher, M Lino da Silva, P André, G Faure, and M Dudeck. Ra- diation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spec- tral region: I. Experiment. Plasma Sources Science and Technology, 17(3):035012, 2008.","doi":""},{"text":"Radiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. M Lino Da Silva, D Vacher, M Dudeck, P André, G Faure, Plasma Sources Science and Technology. 17335013M Lino da Silva, D Vacher, M Dudeck, P André, and G Faure. Radi- ation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. Plasma Sources Science and Technology, 17(3):035013, 2008.","doi":""},{"text":"The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. E A Timmermans, J Jonkers, M Rodero, Quintero, Sola, D Gamero, J A Schram, Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 547E.A.H Timmermans, J Jonkers, A Rodero, M.C Quintero, A Sola, A Gamero, D.C Schram, and J.A.M van der Mullen. The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 54(7):1085 -1098, 1999.","doi":""},{"text":"The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 1. plasmas at atmospheric pressure. E A H Timmermans, J Jonkers, I A J Thomas, A Rodero, M C Quintero, A Sola, A Gamero, J A M Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 5311E.A.H. Timmermans, J. Jonkers, I.A.J. Thomas, A. Rodero, M.C. Quintero, A. Sola, A. Gamero, and J.A.M. van der Mullen. The behav- ior of molecules in microwave-induced plasmas studied by optical emis- sion spectroscopy. 1. plasmas at atmospheric pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 53(11):1553 -1566, 1998.","doi":""},{"text":"Elucidation of C 2 and CN formation mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Meirong Dong, C.-Y. George, Xianglei Chan, Jhanis J Mao, Jidong Gonzalez, Richard E Lu, Russo, Spectrochimica Acta Part B: Atomic Spectroscopy. 100Dedicated To Nicolo Omenetto On the Occasion of his 75th BirthdayMeirong Dong, George C.-Y. Chan, Xianglei Mao, Jhanis J. Gonzalez, Jidong Lu, and Richard E. Russo. Elucidation of C 2 and CN forma- tion mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Spectrochimica Acta Part B: Atomic Spec- troscopy, 100:62 -69, 2014. Dedicated To Nicolo Omenetto On the Occasion of his 75th Birthday.","doi":""},{"text":"CN violet system spectrum used as a molecular pyrometer and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 37012050H Nassar. CN violet system spectrum used as a molecular pyrome- ter and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 370:012050, jun 2012.","doi":""},{"text":"Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. D B Se Youn Moon, B Kim, W Gweon, Choe, Journal of Applied Physics. 105553307Se Youn Moon, D. B. Kim, B. Gweon, and W. Choe. Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. Journal of Applied Physics, 105(5):053307, 2009.","doi":""},{"text":"Torcheà plasmaà excitation micro-onde: deux configurations complémentaires. A Ricard, H St-Onge, Malvos, Gicquel, M Hubert, Moisan, Journal de Physique III. 58A Ricard, L St-Onge, H Malvos, A Gicquel, J Hubert, and M Moisan. Torcheà plasmaà excitation micro-onde: deux config- urations complémentaires. Journal de Physique III, 5(8):1269-1285, 1995.","doi":""},{"text":"Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Ma Ridenti, Amorim, Plasma Chemistry and Plasma Processing. 38MA Ridenti and J Amorim. Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Plasma Chemistry and Plasma Processing, 38(2):311-329, 2018.","doi":""},{"text":"Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Kotaro Saito, Tetsuo Sakka, Yukio H Ogata, Journal of Applied Physics. 949Kotaro Saito, Tetsuo Sakka, and Yukio H. Ogata. Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Journal of Applied Physics, 94(9):5530-5536, 2003.","doi":""},{"text":"Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Fabien Goulay, László Nemes, Paul E Schrader, Hope A Michelsen, Molecular Physics. 1087-9Fabien Goulay, László Nemes, Paul E Schrader, and Hope A Michelsen. Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Molecular Physics, 108(7-9):1013-1025, 2010.","doi":""},{"text":"Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. V E Bondybey, The Journal of Chemical Physics. 656V. E. Bondybey. Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. The Journal of Chemical Physics, 65(6):2296-2304, 1976.","doi":""},{"text":"C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. R P Frosch, The Journal of Chemical Physics. 546R. P. Frosch. C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. The Journal of Chemical Physics, 54(6):2660-2666, 1971.","doi":""},{"text":"The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Li Xuechu, Lou Nanquan, Chemical Physics Letters. 751Li Xuechu and Lou Nanquan. The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Chemical Physics Letters, 75(1):110-114, 1980.","doi":""},{"text":"C 2 and CN formation by optical pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. William Rich, Richard C Bergman, Chemical Physics. 441J William Rich and Richard C Bergman. C 2 and CN formation by opti- cal pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. Chemical Physics, 44(1):53-64, 1979.","doi":""},{"text":"Vibrational to electronic energy transfer from CO to C 2 molecules. G Grigorian, Cenian, Chemical Physics Letters. 4694-6G Grigorian and A Cenian. Vibrational to electronic energy transfer from CO to C 2 molecules. Chemical Physics Letters, 469(4-6):247-249, 2009.","doi":""},{"text":"Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. L Hidde, Bruno Wallaart, Marie-Yvonne Piar, Jean-Pierre Perrin, Martin, Chemical Physics. 1961Hidde L. Wallaart, Bruno Piar, Marie-Yvonne Perrin, and Jean-Pierre Martin. Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. Chemical Physics, 196(1):149 -169, 1995.","doi":""},{"text":"Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. W Hack, Chemical Physics Letters. 813W Hack and W Langel. Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. Chemical Physics Letters, 81(3):387 -390, 1981.","doi":""},{"text":"Pressure Dependence of Rotational Intensity Distribution in C 2 Swan Emission from LowPressure Oxyacetylene Flames. R Bleekrode, The Journal of Chemical Physics. 458R. Bleekrode. Pressure Dependence of Rotational Intensity Distribu- tion in C 2 Swan Emission from LowPressure Oxyacetylene Flames. The Journal of Chemical Physics, 45(8):3153-3154, 1966.","doi":""},{"text":"Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, Katharina Kohse-Höinghaus, The Journal of Physical Chemistry A. 1109Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, and Katharina Kohse-Höinghaus. Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. The Journal of Physical Chemistry A, 110(9):3028-3035, 2006.","doi":""},{"text":"G Grigorian, A Cenian, Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing. 31G. Grigorian and A. Cenian. Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing, 31(2):337-352, Apr 2011.","doi":""},{"text":"State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2. Jingzhong Guo, C M Sadowski, Q Gao, F J Morgan, The Journal of Chemical Physics. 11317Jingzhong Guo, C. M. Sadowski, Q. Gao, and F. J. Morgan. State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2 . The Journal of Chemical Physics, 113(17):7276-7285, 2000.","doi":""},{"text":"State-to-state and total rotational energy transfer rate constants for CN. M K Sophie, Jingzhong Brunet, Tucker Guo, S V Carrington, C M Filseth, Sadowski, N)+NO. The Journal of Chemical Physics. 1169Sophie M. K. Brunet, Jingzhong Guo, Tucker Carrington, S. V. Filseth, and C. M. Sadowski. State-to-state and total rotational energy trans- fer rate constants for CN(B 2 Σ + ,v=0,N)+H 2 , CN(X 2 Σ + ,v=2,N)+H 2 , D 2 , and CN(X 2 Σ + ,v=3,N)+NO. The Journal of Chemical Physics, 116(9):3617-3625, 2002.","doi":""},{"text":"Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave induced nitrogen plasma torch. Chuan-Jie Chen, Shou-Zhe Li, Plasma Sources Science and Technology. 24335017Chuan-Jie Chen and Shou-Zhe Li. Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave in- duced nitrogen plasma torch. Plasma Sources Science and Technology, 24(3):035017, 2015.","doi":""},{"text":"Optical characterization of a microwave pulsed discharge used for dissociation of CO 2. Tiago Silva, Nikolay Britun, Thomas Godfroid, Rony Snyders, Plasma Sources Science and Technology. 23225009Tiago Silva, Nikolay Britun, Thomas Godfroid, and Rony Snyders. Op- tical characterization of a microwave pulsed discharge used for dissoci- ation of CO 2 . Plasma Sources Science and Technology, 23(2):025009, 2014.","doi":""},{"text":"The dissociation of carbon monoxide. A R Fairbairn, Alfred Gordon Gaydon, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 312A. R. Fairbairn and Alfred Gordon Gaydon. The dissociation of carbon monoxide. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 312(1509):207-227, 1969.","doi":""},{"text":"Coherent anti-Stokes Raman scattering and spontaneous Raman scattering diagnostics of nonequilibrium plasmas and flows. R Walter, Igor V Lempert, Adamovich, 2014-10-29T00:00:00Journal of Physics D: Applied Physics. 4743Walter R. Lempert and Igor V. Adamovich. Coherent anti-Stokes Ra- man scattering and spontaneous Raman scattering diagnostics of non- equilibrium plasmas and flows. Journal of Physics D: Applied Physics, 47(43):433001-433026, 2014-10-29T00:00:00.","doi":""},{"text":"How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. D C M Van Den, J M Bekerom, E M Linares, S Van Veldhuizen, M C M Nijdam, G J Van De Sanden, Van Rooij, Appl. Opt. 5720D. C. M. van den Bekerom, J. M. Palomares Linares, E. M. van Veld- huizen, S. Nijdam, M. C. M. van de Sanden, and G. J. van Rooij. How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. Appl. Opt., 57(20):5694- 5702, Jul 2018.","doi":""}],"string":"[\n {\n \"text\": \"Line strengths and updated molecular constants for the C 2 Swan system. S A James, Peter F Brooke, Timothy W Bernath, George B Schmidt, Bacskay, Journal of Quantitative Spectroscopy and Radiative Transfer. 124James S.A. Brooke, Peter F. Bernath, Timothy W. Schmidt, and George B. Bacskay. Line strengths and updated molecular constants for the C 2 Swan system. Journal of Quantitative Spectroscopy and Radiative Transfer, 124:11 -20, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"CO 2 utilization: a process systems engineering vision. José F Araújo Ofélia De Queiroz, Rita Luiz De Medeiros, B Maria, Alves, CO 2 Sequestration and Valorization. InTechF Araújo Ofélia de Queiroz, José Luiz de Medeiros, and Rita Maria B Alves. CO 2 utilization: a process systems engineering vision. In CO 2 Sequestration and Valorization. InTech, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Alexandre Lebouvier, Samuel A Iwarere, Philippe Dargenlieu, Deresh Ramjugernath, Laurent Fulcheri, Energy & Fuels. 275Alexandre Lebouvier, Samuel A. Iwarere, Philippe dArgenlieu, Deresh Ramjugernath, and Laurent Fulcheri. Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Energy & Fuels, 27(5):2712-2722, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasma technology a novel solution for CO 2 conversion?. Ramses Snoeckx, Annemie Bogaerts, Chem. Soc. Rev. 46Ramses Snoeckx and Annemie Bogaerts. Plasma technology a novel solution for CO 2 conversion? Chem. Soc. Rev., 46:5805-5863, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasma for electrification of chemical industry: a case study on CO 2 reduction. H G J Van Rooij, W A N Akse, M C M Bongers, Van De Sanden, Plasma Physics and Controlled Fusion. 60114019G J van Rooij, H N Akse, W A Bongers, and M C M van de Sanden. Plasma for electrification of chemical industry: a case study on CO 2 reduction. Plasma Physics and Controlled Fusion, 60(1):014019, oct 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasma Chemistry. Alexander Fridman, Cambridge Books OnlineAlexander Fridman. Plasma Chemistry. Cambridge University Press, 2008. Cambridge Books Online.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N N Sobolev, V Vasilii, Sokovikov, LASERS. Soviet Physics Uspekhi. 102153N N Sobolev and Vasilii V Sokovikov. CO 2 LASERS. Soviet Physics Uspekhi, 10(2):153, 1967.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasma treatments and plasma deposition of polymers for biomedical applications. Pietro Favia, Riccardo Dagostino, Papers presented at the Fifth International Conference on Plasma Surface Engineering. 98Pietro Favia and Riccardo dAgostino. Plasma treatments and plasma deposition of polymers for biomedical applications. Surface and Coat- ings Technology, 98(1):1102 -1106, 1998. Papers presented at the Fifth International Conference on Plasma Surface Engineering.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine. Emile Carbone, Claire Douat, 8Emile Carbone and Claire Douat. Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine, 8(1):93-120, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. V D Rusanov, G V Fridman, Sholin, Soviet Physics Uspekhi. 246447V D Rusanov, A A Fridman, and G V Sholin. The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. Soviet Physics Uspekhi, 24(6):447, 1981.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. T Kozák, A Bogaerts, Plasma Sources Science and Technology. 24115024T. Kozák and A. Bogaerts. Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. Plasma Sources Science and Technology, 24(1):015024, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Taming microwave plasma to beat thermodynamics in CO 2 dissociation. Gj Van Rooij, Van Den, Bekerom, Harder, G Minea, Berden, R Bongers, Engeln, Mf Graswinckel, Mcm Zoethout, Van De Sanden, Faraday discussions. 183GJ van Rooij, DCM van den Bekerom, N den Harder, T Minea, G Berden, WA Bongers, R Engeln, MF Graswinckel, E Zoethout, and MCM van de Sanden. Taming microwave plasma to beat thermody- namics in CO 2 dissociation. Faraday discussions, 183:233-248, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Selfconsistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. M Capitelli, G Colonna, L D Dammando, Pietanza, Plasma Sources Science and Technology. 26555009M Capitelli, G Colonna, G DAmmando, and L D Pietanza. Self- consistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. Plasma Sources Sci- ence and Technology, 26(5):055009, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. C E Treanor, J W Rich, R G Rehm, The Journal of Chemical Physics. 484C. E. Treanor, J W. Rich, and R.G. Rehm. Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. The Jour- nal of Chemical Physics, 48(4):1798-1807, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. A R I Azizov, V K Vakar, M F Zhivotov, O Krotov, Zinov'ev, Bv Potapkin, Rusanov, A A V D Rusanov, Fridman, 28567R I Azizov, A K Vakar, V K Zhivotov, M F Krotov, O A Zinov'ev, BV Potapkin, A A Rusanov, V D Rusanov, and A A Fridman. The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. 28:567, 1983.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stefan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dissociation of CO 2 for fuel synthesis. Plasma processes and polymers. Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, 141600126Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, Ste- fan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dis- sociation of CO 2 for fuel synthesis. Plasma processes and polymers, 14(6):1600126, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Homogeneous co 2 conversion by microwave plasma: Wave propagation and diagnostics. Dirk C M Niek Den Harder, Van Den, Richard S Bekerom, Martijn F Al, Jose M Graswinckel, Palomares, J J Floran, Srinath Peeters, Teofil Ponduri, Waldo A Minea, Bongers, C M Mauritius, Gerard J Van De Sanden, Van Rooij, Plasma Processes and Polymers. 1461600120Niek den Harder, Dirk C. M. van den Bekerom, Richard S. Al, Mar- tijn F. Graswinckel, Jose M. Palomares, Floran J. J. Peeters, Srinath Ponduri, Teofil Minea, Waldo A. Bongers, Mauritius C. M. van de Sanden, and Gerard J. van Rooij. Homogeneous co 2 conversion by mi- crowave plasma: Wave propagation and diagnostics. Plasma Processes and Polymers, 14(6):1600120, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Dirk Van Den, Jose Maria Palomares Bekerom, Tiny Linares, Eddie M Verreycken, Sander Van Veldhuizen, Waldo Nijdam, Richard Bongers, Gerard J Van Van De Sanden, Rooij, Plasma Sources Science and Technology. Dirk van den Bekerom, Jose Maria Palomares Linares, Tiny Verr- eycken, Eddie M Van Veldhuizen, Sander Nijdam, Waldo Bongers, Richard Van de Sanden, and Gerard J Van Rooij. The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Plasma Sources Sci- ence and Technology, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On use of CO 2 chemiluminescence for combustion metrics in natural gas fired reciprocating engines. B Sreenath, Gupta, P Bipin, Munidhar S Bihari, Raj R Biruduganti, James Sekar, Zigan, Proceedings of the Combustion Institute. the Combustion Institute33Sreenath B. Gupta, Bipin P. Bihari, Munidhar S. Biruduganti, Raj R. Sekar, and James Zigan. On use of CO 2 chemiluminescence for com- bustion metrics in natural gas fired reciprocating engines. Proceedings of the Combustion Institute, 33(2):3131 -3139, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"CO 2 * Chemiluminescence in Premixed Flames. J.-M Samaniego, F N Egolfopoulos, C T Bowman, Combustion Science and Technology. 1091-6J.-M. Samaniego, F. N. Egolfopoulos, and C. T. Bowman. CO 2 * Chemi- luminescence in Premixed Flames. Combustion Science and Technol- ogy, 109(1-6):183-203, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Investigation of the dynamics of co 2 plasmas using kinetic modelling. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, Ursel Hecimovic, Fantz, Van Dijk, 23rd International Symposium on Plasma Chemistry. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, A Heci- movic, Ursel Fantz, and J van Dijk. Investigation of the dynamics of co 2 plasmas using kinetic modelling. In 23rd International Symposium on Plasma Chemistry (ISPC 2017), 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 spectroscopy and kinetics. Margarita Martin, Journal of Photochemistry and Photobiology A: Chemistry. 663Margarita Martin. C 2 spectroscopy and kinetics. Journal of Photo- chemistry and Photobiology A: Chemistry, 66(3):263-289, 1992.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. G L D Pietanza, M Colonna, Capitelli, Plasma Sources Science and Technology. 2612125007L D Pietanza, G Colonna, and M Capitelli. Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. Plasma Sources Science and Technology, 26(12):125007, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. E V Kustova, A Aliat, A Chikhaoui, Chemical Physics Letters. 3445E.V. Kustova, A. Aliat, and A. Chikhaoui. Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. Chemical Physics Letters, 344(5):638 -646, 2001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Isotope effect in Boudouard disproportionation reaction in optically pumped CO. Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, Lempert, Rich, Chemical Physics. 3631-3Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, WR Lem- pert, JW Rich, and IV Adamovich. Isotope effect in Boudouard dis- proportionation reaction in optically pumped CO. Chemical Physics, 363(1-3):24-32, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gas phase Boudouard reactions involving singlet-singlet and singlet-triplet CO vibrationally excited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. R P Patricia, Henrique Barreto, Alessandra F Euclides, Vincenzo Albernaz, Mario Aquilanti, Gaia Capitelli, Andrea Grossi, Sergey Lombardi, Federico Macheret, Palazzetti, The European Physical Journal D. 7110259Patricia RP Barreto, Henrique de O Euclides, Alessandra F Albernaz, Vincenzo Aquilanti, Mario Capitelli, Gaia Grossi, Andrea Lombardi, Sergey Macheret, and Federico Palazzetti. Gas phase Boudouard reac- tions involving singlet-singlet and singlet-triplet CO vibrationally ex- cited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. The European Physical Journal D, 71(10):259, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. M I Ks Kini, Savadatti, Journal of Physics B: Atomic and Molecular Physics. 1061139KS Kini and MI Savadatti. Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. Journal of Physics B: Atomic and Molecular Physics, 10(6):1139, 1977.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The pulse radiolysis of carbon monoxide. C Willis, C Devillers, Chemical Physics Letters. 21C. Willis and C. Devillers. The pulse radiolysis of carbon monoxide. Chemical Physics Letters, 2(1):51 -53, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An Extension of the Phillips System of C 2 and a Survey of C 2 States. E A Ballik, D A Ramsay, The Astrophysical Journal. 13784E. A. Ballik and D. A. Ramsay. An Extension of the Phillips System of C 2 and a Survey of C 2 States. The Astrophysical Journal, 137:84, 1963.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mechanism for Excitation of the C 2 High-Pressure Bands. C Kunz, S Harteck, Dondes, The Journal of Chemical Physics. 4610C+ C 2 O → C 2 (A 3 Π)+ COC Kunz, P Harteck, and S Dondes. Mechanism for Excitation of the C 2 High-Pressure Bands [C+ C 2 O → C 2 (A 3 Π)+ CO]. The Journal of Chemical Physics, 46(10):4157-4158, 1967.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 formation in vibrationally excited CO. Chemical physics letters. Hl Wallaart, M-Y Piar, J-P Perrin, Martin, 246HL Wallaart, B Piar, M-Y Perrin, and J-P Martin. C 2 formation in vibrationally excited CO. Chemical physics letters, 246(6):587-593, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the excitation mechanism of the C 2 high-pressure bands. D W Naegeli, H B Palmer, Journal of Molecular Spectroscopy. 261D.W. Naegeli and H.B. Palmer. On the excitation mechanism of the C 2 high-pressure bands. Journal of Molecular Spectroscopy, 26(1):152 -154, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. F Gosse, J C Sadeghi, Pebay-Peyroula, Chemical Physics Letters. 136F Gosse, N Sadeghi, and JC Pebay-Peyroula. Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. Chemical Physics Letters, 13(6):557-560, 1972.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Heat of Sublimation and Dissociation Energy of Gaseous C 2. Leo Brewer, W T Hicks, O H Krikorian, The Journal of Chemical Physics. 361Leo Brewer, W. T. Hicks, and O. H. Krikorian. Heat of Sublimation and Dissociation Energy of Gaseous C 2 . The Journal of Chemical Physics, 36(1):182-188, 1962.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters. P Caubet, G Dorthe, 218P Caubet and G Dorthe. Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters, 218(5-6):529-536, 1994.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Origin of the high-pressure bands of C 2 . Chemical physics letters. Ce Little, Browne, 134CE Little and PG Browne. Origin of the high-pressure bands of C 2 . Chemical physics letters, 134(6):560-564, 1987.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. H Sasada, T Amano, C Jarman, P F Bernath, The Journal of Chemical Physics. 944H. Sasada, T. Amano, C. Jarman, and P. F. Bernath. A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. The Journal of Chem- ical Physics, 94(4):2401-2407, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Theoretical investigation of excited states of C 3. A Terentyev, R Scholz, M Schreiber, G Seifert, The Journal of Chemical Physics. 12112A. Terentyev, R. Scholz, M. Schreiber, and G. Seifert. Theoretical investigation of excited states of C 3 . The Journal of Chemical Physics, 121(12):5767-5776, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The valence states of c 2 : A configuration interaction study. K Kirby, B Liu, The Journal of Chemical Physics. 702K. Kirby and B. Liu. The valence states of c 2 : A configuration inter- action study. The Journal of Chemical Physics, 70(2):893-900, 1979.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Wl Faust, Goldberg, Craig Bb, R G Weiss, Chemical Physics Letters. 832WL Faust, LS Goldberg, BB Craig, and RG Weiss. Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Chemical Physics Letters, 83(2):265- 269, 1981.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Investigation of the Fine Structure of the C 2 High-Pressure Bands. H Meinel, Gerd Messerle, The Astrophysical Journal. 154381H Meinel and Gerd Messerle. Investigation of the Fine Structure of the C 2 High-Pressure Bands. The Astrophysical Journal, 154:381, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. S Paul, Remo Barklem, Collet, Astronomy & Astrophysics. 58896Paul S Barklem and Remo Collet. Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. Astronomy & Astrophysics, 588:A96, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statistical thermodynamics. Donald Allan Mcquarrie, Harper Collins PublishersDonald Allan McQuarrie. Statistical thermodynamics. Harper Collins Publishers, 1973.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Molecular spectra and molecular structure. Gerhard Herzberg, Spectra of diatomic molecules. New YorkVan Nostrand Reinhold12nd ed.Gerhard Herzberg. Molecular spectra and molecular structure. Vol. 1: Spectra of diatomic molecules. New York: Van Nostrand Reinhold, 1950, 2nd ed., 1950.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas. S Svanberg, S Svanberg. Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Application of the (0,0) Swan band spectrum for temperature measurements. S Pellerin, Musiol, Motret, J Pokrzywka, Chapelle, Journal of Physics D: Applied Physics. 29112850S Pellerin, K Musiol, O Motret, B Pokrzywka, and J Chapelle. Appli- cation of the (0,0) Swan band spectrum for temperature measurements. Journal of Physics D: Applied Physics, 29(11):2850, 1996.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spectra of atoms and molecules. F Peter, Bernath, Oxford university pressPeter F Bernath. Spectra of atoms and molecules. Oxford university press, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Colin M Western, Journal of Quantitative Spectroscopy and Radiative Transfer. 186Colin M Western. PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:221-242, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"LEVEL: A computer program for solving the radial Schrdinger equation for bound and quasibound levels. J Le Robert, Roy, Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting. 186Robert J. Le Roy. LEVEL: A computer program for solving the ra- dial Schrdinger equation for bound and quasibound levels. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:167 -178, 2017. Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting, October 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. G M Lloyd, P Ewart, The Journal of Chemical Physics. 1101G. M. Lloyd and P. Ewart. High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. The Journal of Chemical Physics, 110(1):385-392, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Swan System of C 2 : A Global Analysis of Fourier Transform Emission Spectra. A Tanabashi, T Hirao, T Amano, P F Bernath, The Astrophysical Journal Supplement Series. 1692472A. Tanabashi, T. Hirao, T. Amano, and P. F. Bernath. The Swan Sys- tem of C 2 : A Global Analysis of Fourier Transform Emission Spectra. The Astrophysical Journal Supplement Series, 169(2):472, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fourier transform emission spectroscopy of the B 2 Σ + -X 2 Σ + system of CN. R S Ram, S P Davis, L Wallace, R Engleman, D R T Appadoo, P F Bernath, Journal of Molecular Spectroscopy. 2372R.S. Ram, S.P. Davis, L. Wallace, R. Engleman, D.R.T. Appadoo, and P.F. Bernath. Fourier transform emission spectroscopy of the B 2 Σ + - X 2 Σ + system of CN. Journal of Molecular Spectroscopy, 237(2):225 - 231, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lifbase, database and spectral simulation for diatomic molecules. J Luque, 99SRI International ReportJ Luque. Lifbase, database and spectral simulation for diatomic molecules. SRI International Report, 99, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spectrometer. D Cerny, R Bacis, G Guelachvili, F Roux, Journal of Molecular Spectroscopy. 731D. Cerny, R. Bacis, G. Guelachvili, and F. Roux. Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spec- trometer. Journal of Molecular Spectroscopy, 73(1):154 -167, 1978.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i. Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, Journal of Molecular Spectroscopy. 1272Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i . Journal of Molecular Spectroscopy, 127(2):283 -303, 1988.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, The Journal of Chemical Physics. 948Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. The Journal of Chemical Physics, 94(8):5353-5359, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spectroscopic studies and rotational and vibrational temperature measurements of atmospheric pressure normal glow plasma discharges in air. David Staack, Bakhtier Farouk, Alexander F Gutsol, Alexander A Fridman, Plasma Sources Science and Technology. 154818David Staack, Bakhtier Farouk, Alexander F Gutsol, and Alexander A Fridman. Spectroscopic studies and rotational and vibrational tem- perature measurements of atmospheric pressure normal glow plasma discharges in air. Plasma Sources Science and Technology, 15(4):818, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 Swan spectrum used as a molecular pyrometer in transferred arc and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 511112066H Nassar. C 2 Swan spectrum used as a molecular pyrometer in trans- ferred arc and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 511(1):012066, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"arXiv:1404.1100Jonathon Shlens. A tutorial on principal component analysis. arXiv preprintJonathon Shlens. A tutorial on principal component analysis. arXiv preprint arXiv:1404.1100, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Christophe Laux, Specair, Christophe Laux. SPECAIR, 2012. Accessed: March 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Modeling the spectral radiation of a methane/nitrogen plasma: Test case 4. L Caillaut, Andreasson, C O Risberg, Laux, Radiation of High Temperature Gases in Atmospheric Entry. 629L Caillaut, J Andreasson, J Risberg, and CO Laux. Modeling the spec- tral radiation of a methane/nitrogen plasma: Test case 4. In Radiation of High Temperature Gases in Atmospheric Entry, volume 629, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fundamental data of diatomic molecules relevant for fusion. U Fantz, Wünderlich, Atomic and Plasma-Material Interaction Data for Fusion. 16U Fantz and D Wünderlich. Fundamental data of diatomic molecules relevant for fusion. In Atomic and Plasma-Material Interaction Data for Fusion, volume 16. IAEA, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Oscillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. Damian L Kokkin, George B Bacskay, Timothy W Schmidt, The Journal of Chemical Physics. 126884302Damian L. Kokkin, George B. Bacskay, and Timothy W. Schmidt. Os- cillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. The Journal of Chemical Physics, 126(8):084302, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Melanie Heusing, and Michael Letzgus. Measurement and simulation of rotationally-resolved chemiluminescence spectra in flames. Andreas Brockhinke, Julia Krüger, Applied Physics B. 1073Andreas Brockhinke, Julia Krüger, Melanie Heusing, and Michael Let- zgus. Measurement and simulation of rotationally-resolved chemilumi- nescence spectra in flames. Applied Physics B, 107(3):539-549, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Radiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: I. Experiment. D Vacher, Lino Da, P Silva, G André, M Faure, Dudeck, Plasma Sources Science and Technology. 17335012D Vacher, M Lino da Silva, P André, G Faure, and M Dudeck. Ra- diation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spec- tral region: I. Experiment. Plasma Sources Science and Technology, 17(3):035012, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Radiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. M Lino Da Silva, D Vacher, M Dudeck, P André, G Faure, Plasma Sources Science and Technology. 17335013M Lino da Silva, D Vacher, M Dudeck, P André, and G Faure. Radi- ation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. Plasma Sources Science and Technology, 17(3):035013, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. E A Timmermans, J Jonkers, M Rodero, Quintero, Sola, D Gamero, J A Schram, Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 547E.A.H Timmermans, J Jonkers, A Rodero, M.C Quintero, A Sola, A Gamero, D.C Schram, and J.A.M van der Mullen. The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 54(7):1085 -1098, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 1. plasmas at atmospheric pressure. E A H Timmermans, J Jonkers, I A J Thomas, A Rodero, M C Quintero, A Sola, A Gamero, J A M Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 5311E.A.H. Timmermans, J. Jonkers, I.A.J. Thomas, A. Rodero, M.C. Quintero, A. Sola, A. Gamero, and J.A.M. van der Mullen. The behav- ior of molecules in microwave-induced plasmas studied by optical emis- sion spectroscopy. 1. plasmas at atmospheric pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 53(11):1553 -1566, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Elucidation of C 2 and CN formation mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Meirong Dong, C.-Y. George, Xianglei Chan, Jhanis J Mao, Jidong Gonzalez, Richard E Lu, Russo, Spectrochimica Acta Part B: Atomic Spectroscopy. 100Dedicated To Nicolo Omenetto On the Occasion of his 75th BirthdayMeirong Dong, George C.-Y. Chan, Xianglei Mao, Jhanis J. Gonzalez, Jidong Lu, and Richard E. Russo. Elucidation of C 2 and CN forma- tion mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Spectrochimica Acta Part B: Atomic Spec- troscopy, 100:62 -69, 2014. Dedicated To Nicolo Omenetto On the Occasion of his 75th Birthday.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"CN violet system spectrum used as a molecular pyrometer and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 37012050H Nassar. CN violet system spectrum used as a molecular pyrome- ter and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 370:012050, jun 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. D B Se Youn Moon, B Kim, W Gweon, Choe, Journal of Applied Physics. 105553307Se Youn Moon, D. B. Kim, B. Gweon, and W. Choe. Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. Journal of Applied Physics, 105(5):053307, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Torcheà plasmaà excitation micro-onde: deux configurations complémentaires. A Ricard, H St-Onge, Malvos, Gicquel, M Hubert, Moisan, Journal de Physique III. 58A Ricard, L St-Onge, H Malvos, A Gicquel, J Hubert, and M Moisan. Torcheà plasmaà excitation micro-onde: deux config- urations complémentaires. Journal de Physique III, 5(8):1269-1285, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Ma Ridenti, Amorim, Plasma Chemistry and Plasma Processing. 38MA Ridenti and J Amorim. Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Plasma Chemistry and Plasma Processing, 38(2):311-329, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Kotaro Saito, Tetsuo Sakka, Yukio H Ogata, Journal of Applied Physics. 949Kotaro Saito, Tetsuo Sakka, and Yukio H. Ogata. Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Journal of Applied Physics, 94(9):5530-5536, 2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Fabien Goulay, László Nemes, Paul E Schrader, Hope A Michelsen, Molecular Physics. 1087-9Fabien Goulay, László Nemes, Paul E Schrader, and Hope A Michelsen. Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Molecular Physics, 108(7-9):1013-1025, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. V E Bondybey, The Journal of Chemical Physics. 656V. E. Bondybey. Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. The Journal of Chemical Physics, 65(6):2296-2304, 1976.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. R P Frosch, The Journal of Chemical Physics. 546R. P. Frosch. C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. The Journal of Chemical Physics, 54(6):2660-2666, 1971.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Li Xuechu, Lou Nanquan, Chemical Physics Letters. 751Li Xuechu and Lou Nanquan. The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Chemical Physics Letters, 75(1):110-114, 1980.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 and CN formation by optical pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. William Rich, Richard C Bergman, Chemical Physics. 441J William Rich and Richard C Bergman. C 2 and CN formation by opti- cal pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. Chemical Physics, 44(1):53-64, 1979.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vibrational to electronic energy transfer from CO to C 2 molecules. G Grigorian, Cenian, Chemical Physics Letters. 4694-6G Grigorian and A Cenian. Vibrational to electronic energy transfer from CO to C 2 molecules. Chemical Physics Letters, 469(4-6):247-249, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. L Hidde, Bruno Wallaart, Marie-Yvonne Piar, Jean-Pierre Perrin, Martin, Chemical Physics. 1961Hidde L. Wallaart, Bruno Piar, Marie-Yvonne Perrin, and Jean-Pierre Martin. Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. Chemical Physics, 196(1):149 -169, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. W Hack, Chemical Physics Letters. 813W Hack and W Langel. Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. Chemical Physics Letters, 81(3):387 -390, 1981.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pressure Dependence of Rotational Intensity Distribution in C 2 Swan Emission from LowPressure Oxyacetylene Flames. R Bleekrode, The Journal of Chemical Physics. 458R. Bleekrode. Pressure Dependence of Rotational Intensity Distribu- tion in C 2 Swan Emission from LowPressure Oxyacetylene Flames. The Journal of Chemical Physics, 45(8):3153-3154, 1966.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, Katharina Kohse-Höinghaus, The Journal of Physical Chemistry A. 1109Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, and Katharina Kohse-Höinghaus. Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. The Journal of Physical Chemistry A, 110(9):3028-3035, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G Grigorian, A Cenian, Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing. 31G. Grigorian and A. Cenian. Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing, 31(2):337-352, Apr 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2. Jingzhong Guo, C M Sadowski, Q Gao, F J Morgan, The Journal of Chemical Physics. 11317Jingzhong Guo, C. M. Sadowski, Q. Gao, and F. J. Morgan. State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2 . The Journal of Chemical Physics, 113(17):7276-7285, 2000.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"State-to-state and total rotational energy transfer rate constants for CN. M K Sophie, Jingzhong Brunet, Tucker Guo, S V Carrington, C M Filseth, Sadowski, N)+NO. The Journal of Chemical Physics. 1169Sophie M. K. Brunet, Jingzhong Guo, Tucker Carrington, S. V. Filseth, and C. M. Sadowski. State-to-state and total rotational energy trans- fer rate constants for CN(B 2 Σ + ,v=0,N)+H 2 , CN(X 2 Σ + ,v=2,N)+H 2 , D 2 , and CN(X 2 Σ + ,v=3,N)+NO. The Journal of Chemical Physics, 116(9):3617-3625, 2002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave induced nitrogen plasma torch. Chuan-Jie Chen, Shou-Zhe Li, Plasma Sources Science and Technology. 24335017Chuan-Jie Chen and Shou-Zhe Li. Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave in- duced nitrogen plasma torch. Plasma Sources Science and Technology, 24(3):035017, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical characterization of a microwave pulsed discharge used for dissociation of CO 2. Tiago Silva, Nikolay Britun, Thomas Godfroid, Rony Snyders, Plasma Sources Science and Technology. 23225009Tiago Silva, Nikolay Britun, Thomas Godfroid, and Rony Snyders. Op- tical characterization of a microwave pulsed discharge used for dissoci- ation of CO 2 . Plasma Sources Science and Technology, 23(2):025009, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The dissociation of carbon monoxide. A R Fairbairn, Alfred Gordon Gaydon, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 312A. R. Fairbairn and Alfred Gordon Gaydon. The dissociation of carbon monoxide. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 312(1509):207-227, 1969.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Coherent anti-Stokes Raman scattering and spontaneous Raman scattering diagnostics of nonequilibrium plasmas and flows. R Walter, Igor V Lempert, Adamovich, 2014-10-29T00:00:00Journal of Physics D: Applied Physics. 4743Walter R. Lempert and Igor V. Adamovich. Coherent anti-Stokes Ra- man scattering and spontaneous Raman scattering diagnostics of non- equilibrium plasmas and flows. Journal of Physics D: Applied Physics, 47(43):433001-433026, 2014-10-29T00:00:00.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. D C M Van Den, J M Bekerom, E M Linares, S Van Veldhuizen, M C M Nijdam, G J Van De Sanden, Van Rooij, Appl. Opt. 5720D. C. M. van den Bekerom, J. M. Palomares Linares, E. M. van Veld- huizen, S. Nijdam, M. C. M. van de Sanden, and G. J. van Rooij. How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. Appl. Opt., 57(20):5694- 5702, Jul 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Line strengths and updated molecular constants for the C 2 Swan system. S A James, Peter F Brooke, Timothy W Bernath, George B Schmidt, Bacskay, Journal of Quantitative Spectroscopy and Radiative Transfer. 124James S.A. Brooke, Peter F. Bernath, Timothy W. Schmidt, and George B. Bacskay. Line strengths and updated molecular constants for the C 2 Swan system. Journal of Quantitative Spectroscopy and Radiative Transfer, 124:11 -20, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"CO 2 utilization: a process systems engineering vision. José F Araújo Ofélia De Queiroz, Rita Luiz De Medeiros, B Maria, Alves, CO 2 Sequestration and Valorization. InTechF Araújo Ofélia de Queiroz, José Luiz de Medeiros, and Rita Maria B Alves. CO 2 utilization: a process systems engineering vision. In CO 2 Sequestration and Valorization. InTech, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Alexandre Lebouvier, Samuel A Iwarere, Philippe Dargenlieu, Deresh Ramjugernath, Laurent Fulcheri, Energy & Fuels. 275Alexandre Lebouvier, Samuel A. Iwarere, Philippe dArgenlieu, Deresh Ramjugernath, and Laurent Fulcheri. Assessment of Carbon Dioxide Dissociation as a New Route for Syngas Production: A Comparative Review and Potential of Plasma-Based Technologies. Energy & Fuels, 27(5):2712-2722, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasma technology a novel solution for CO 2 conversion?. Ramses Snoeckx, Annemie Bogaerts, Chem. Soc. Rev. 46Ramses Snoeckx and Annemie Bogaerts. Plasma technology a novel solution for CO 2 conversion? Chem. Soc. Rev., 46:5805-5863, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasma for electrification of chemical industry: a case study on CO 2 reduction. H G J Van Rooij, W A N Akse, M C M Bongers, Van De Sanden, Plasma Physics and Controlled Fusion. 60114019G J van Rooij, H N Akse, W A Bongers, and M C M van de Sanden. Plasma for electrification of chemical industry: a case study on CO 2 reduction. Plasma Physics and Controlled Fusion, 60(1):014019, oct 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasma Chemistry. Alexander Fridman, Cambridge Books OnlineAlexander Fridman. Plasma Chemistry. Cambridge University Press, 2008. Cambridge Books Online.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N N Sobolev, V Vasilii, Sokovikov, LASERS. Soviet Physics Uspekhi. 102153N N Sobolev and Vasilii V Sokovikov. CO 2 LASERS. Soviet Physics Uspekhi, 10(2):153, 1967.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Plasma treatments and plasma deposition of polymers for biomedical applications. Pietro Favia, Riccardo Dagostino, Papers presented at the Fifth International Conference on Plasma Surface Engineering. 98Pietro Favia and Riccardo dAgostino. Plasma treatments and plasma deposition of polymers for biomedical applications. Surface and Coat- ings Technology, 98(1):1102 -1106, 1998. Papers presented at the Fifth International Conference on Plasma Surface Engineering.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine. Emile Carbone, Claire Douat, 8Emile Carbone and Claire Douat. Carbon monoxide in plasma medicine and agriculture: Just a foe or a potential friend? Plasma Medicine, 8(1):93-120, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. V D Rusanov, G V Fridman, Sholin, Soviet Physics Uspekhi. 246447V D Rusanov, A A Fridman, and G V Sholin. The physics of a chemically active plasma with non-equilibrium vibrational excitation of molecules. Soviet Physics Uspekhi, 24(6):447, 1981.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. T Kozák, A Bogaerts, Plasma Sources Science and Technology. 24115024T. Kozák and A. Bogaerts. Evaluation of the energy efficiency of CO 2 conversion in microwave discharges using a reaction kinetics model. Plasma Sources Science and Technology, 24(1):015024, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Taming microwave plasma to beat thermodynamics in CO 2 dissociation. Gj Van Rooij, Van Den, Bekerom, Harder, G Minea, Berden, R Bongers, Engeln, Mf Graswinckel, Mcm Zoethout, Van De Sanden, Faraday discussions. 183GJ van Rooij, DCM van den Bekerom, N den Harder, T Minea, G Berden, WA Bongers, R Engeln, MF Graswinckel, E Zoethout, and MCM van de Sanden. Taming microwave plasma to beat thermody- namics in CO 2 dissociation. Faraday discussions, 183:233-248, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Selfconsistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. M Capitelli, G Colonna, L D Dammando, Pietanza, Plasma Sources Science and Technology. 26555009M Capitelli, G Colonna, G DAmmando, and L D Pietanza. Self- consistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas. Plasma Sources Sci- ence and Technology, 26(5):055009, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. C E Treanor, J W Rich, R G Rehm, The Journal of Chemical Physics. 484C. E. Treanor, J W. Rich, and R.G. Rehm. Vibrational relaxation of anharmonic oscillators with exchange-dominated collisions. The Jour- nal of Chemical Physics, 48(4):1798-1807, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. A R I Azizov, V K Vakar, M F Zhivotov, O Krotov, Zinov'ev, Bv Potapkin, Rusanov, A A V D Rusanov, Fridman, 28567R I Azizov, A K Vakar, V K Zhivotov, M F Krotov, O A Zinov'ev, BV Potapkin, A A Rusanov, V D Rusanov, and A A Fridman. The nonequilibrium plasmachemical process of decomposition of CO 2 in a supersonic SHF discharge. 28:567, 1983.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stefan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dissociation of CO 2 for fuel synthesis. Plasma processes and polymers. Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, 141600126Waldo Bongers, Henny Bouwmeester, Bram Wolf, Floran Peeters, Ste- fan Welzel, Dirk van den Bekerom, Niek den Harder, Adelbert Goede, Martijn Graswinckel, Pieter Willem Groen, et al. Plasma-driven dis- sociation of CO 2 for fuel synthesis. Plasma processes and polymers, 14(6):1600126, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Homogeneous co 2 conversion by microwave plasma: Wave propagation and diagnostics. Dirk C M Niek Den Harder, Van Den, Richard S Bekerom, Martijn F Al, Jose M Graswinckel, Palomares, J J Floran, Srinath Peeters, Teofil Ponduri, Waldo A Minea, Bongers, C M Mauritius, Gerard J Van De Sanden, Van Rooij, Plasma Processes and Polymers. 1461600120Niek den Harder, Dirk C. M. van den Bekerom, Richard S. Al, Mar- tijn F. Graswinckel, Jose M. Palomares, Floran J. J. Peeters, Srinath Ponduri, Teofil Minea, Waldo A. Bongers, Mauritius C. M. van de Sanden, and Gerard J. van Rooij. Homogeneous co 2 conversion by mi- crowave plasma: Wave propagation and diagnostics. Plasma Processes and Polymers, 14(6):1600120, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Dirk Van Den, Jose Maria Palomares Bekerom, Tiny Linares, Eddie M Verreycken, Sander Van Veldhuizen, Waldo Nijdam, Richard Bongers, Gerard J Van Van De Sanden, Rooij, Plasma Sources Science and Technology. Dirk van den Bekerom, Jose Maria Palomares Linares, Tiny Verr- eycken, Eddie M Van Veldhuizen, Sander Nijdam, Waldo Bongers, Richard Van de Sanden, and Gerard J Van Rooij. The importance of thermal dissociation in CO 2 microwave discharges investigated by power pulsing and rotational Raman scattering. Plasma Sources Sci- ence and Technology, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On use of CO 2 chemiluminescence for combustion metrics in natural gas fired reciprocating engines. B Sreenath, Gupta, P Bipin, Munidhar S Bihari, Raj R Biruduganti, James Sekar, Zigan, Proceedings of the Combustion Institute. the Combustion Institute33Sreenath B. Gupta, Bipin P. Bihari, Munidhar S. Biruduganti, Raj R. Sekar, and James Zigan. On use of CO 2 chemiluminescence for com- bustion metrics in natural gas fired reciprocating engines. Proceedings of the Combustion Institute, 33(2):3131 -3139, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"CO 2 * Chemiluminescence in Premixed Flames. J.-M Samaniego, F N Egolfopoulos, C T Bowman, Combustion Science and Technology. 1091-6J.-M. Samaniego, F. N. Egolfopoulos, and C. T. Bowman. CO 2 * Chemi- luminescence in Premixed Flames. Combustion Science and Technol- ogy, 109(1-6):183-203, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Investigation of the dynamics of co 2 plasmas using kinetic modelling. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, Ursel Hecimovic, Fantz, Van Dijk, 23rd International Symposium on Plasma Chemistry. Emile Carbone, Luca Vialetto, Stef Bardoel, Peter Koelman, A Heci- movic, Ursel Fantz, and J van Dijk. Investigation of the dynamics of co 2 plasmas using kinetic modelling. In 23rd International Symposium on Plasma Chemistry (ISPC 2017), 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 spectroscopy and kinetics. Margarita Martin, Journal of Photochemistry and Photobiology A: Chemistry. 663Margarita Martin. C 2 spectroscopy and kinetics. Journal of Photo- chemistry and Photobiology A: Chemistry, 66(3):263-289, 1992.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. G L D Pietanza, M Colonna, Capitelli, Plasma Sources Science and Technology. 2612125007L D Pietanza, G Colonna, and M Capitelli. Non-equilibrium plasma kinetics of reacting CO: an improved state to state approach. Plasma Sources Science and Technology, 26(12):125007, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. E V Kustova, A Aliat, A Chikhaoui, Chemical Physics Letters. 3445E.V. Kustova, A. Aliat, and A. Chikhaoui. Vibrationelectronic and chemical kinetics of non-equilibrium radiative diatomic gas flows. Chemical Physics Letters, 344(5):638 -646, 2001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Isotope effect in Boudouard disproportionation reaction in optically pumped CO. Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, Lempert, Rich, Chemical Physics. 3631-3Anna Serdyuchenko, Eugene Mintusov, Kraig Frederickson, WR Lem- pert, JW Rich, and IV Adamovich. Isotope effect in Boudouard dis- proportionation reaction in optically pumped CO. Chemical Physics, 363(1-3):24-32, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gas phase Boudouard reactions involving singlet-singlet and singlet-triplet CO vibrationally excited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. R P Patricia, Henrique Barreto, Alessandra F Euclides, Vincenzo Albernaz, Mario Aquilanti, Gaia Capitelli, Andrea Grossi, Sergey Lombardi, Federico Macheret, Palazzetti, The European Physical Journal D. 7110259Patricia RP Barreto, Henrique de O Euclides, Alessandra F Albernaz, Vincenzo Aquilanti, Mario Capitelli, Gaia Grossi, Andrea Lombardi, Sergey Macheret, and Federico Palazzetti. Gas phase Boudouard reac- tions involving singlet-singlet and singlet-triplet CO vibrationally ex- cited states: implications for the non-equilibrium vibrational kinetics of CO/CO 2 plasmas. The European Physical Journal D, 71(10):259, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. M I Ks Kini, Savadatti, Journal of Physics B: Atomic and Molecular Physics. 1061139KS Kini and MI Savadatti. Investigation of the C 2 (A 3 Π g X' 3 Π u ) band system excited in a discharge through carbon monoxide. Journal of Physics B: Atomic and Molecular Physics, 10(6):1139, 1977.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The pulse radiolysis of carbon monoxide. C Willis, C Devillers, Chemical Physics Letters. 21C. Willis and C. Devillers. The pulse radiolysis of carbon monoxide. Chemical Physics Letters, 2(1):51 -53, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An Extension of the Phillips System of C 2 and a Survey of C 2 States. E A Ballik, D A Ramsay, The Astrophysical Journal. 13784E. A. Ballik and D. A. Ramsay. An Extension of the Phillips System of C 2 and a Survey of C 2 States. The Astrophysical Journal, 137:84, 1963.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mechanism for Excitation of the C 2 High-Pressure Bands. C Kunz, S Harteck, Dondes, The Journal of Chemical Physics. 4610C+ C 2 O → C 2 (A 3 Π)+ COC Kunz, P Harteck, and S Dondes. Mechanism for Excitation of the C 2 High-Pressure Bands [C+ C 2 O → C 2 (A 3 Π)+ CO]. The Journal of Chemical Physics, 46(10):4157-4158, 1967.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 formation in vibrationally excited CO. Chemical physics letters. Hl Wallaart, M-Y Piar, J-P Perrin, Martin, 246HL Wallaart, B Piar, M-Y Perrin, and J-P Martin. C 2 formation in vibrationally excited CO. Chemical physics letters, 246(6):587-593, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the excitation mechanism of the C 2 high-pressure bands. D W Naegeli, H B Palmer, Journal of Molecular Spectroscopy. 261D.W. Naegeli and H.B. Palmer. On the excitation mechanism of the C 2 high-pressure bands. Journal of Molecular Spectroscopy, 26(1):152 -154, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. F Gosse, J C Sadeghi, Pebay-Peyroula, Chemical Physics Letters. 136F Gosse, N Sadeghi, and JC Pebay-Peyroula. Study of mechanisms producing C 2 *(A 3 Π g ) molecules in a CO afterglow. Chemical Physics Letters, 13(6):557-560, 1972.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Heat of Sublimation and Dissociation Energy of Gaseous C 2. Leo Brewer, W T Hicks, O H Krikorian, The Journal of Chemical Physics. 361Leo Brewer, W. T. Hicks, and O. H. Krikorian. Heat of Sublimation and Dissociation Energy of Gaseous C 2 . The Journal of Chemical Physics, 36(1):182-188, 1962.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters. P Caubet, G Dorthe, 218P Caubet and G Dorthe. Origin of C 2 high-pressure bands observed in the products of a microwave discharge through CO. Chemical physics letters, 218(5-6):529-536, 1994.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Origin of the high-pressure bands of C 2 . Chemical physics letters. Ce Little, Browne, 134CE Little and PG Browne. Origin of the high-pressure bands of C 2 . Chemical physics letters, 134(6):560-564, 1987.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. H Sasada, T Amano, C Jarman, P F Bernath, The Journal of Chemical Physics. 944H. Sasada, T. Amano, C. Jarman, and P. F. Bernath. A new triplet band system of C 3 : Theb 3 Π g −ã 3 Π u transition. The Journal of Chem- ical Physics, 94(4):2401-2407, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Theoretical investigation of excited states of C 3. A Terentyev, R Scholz, M Schreiber, G Seifert, The Journal of Chemical Physics. 12112A. Terentyev, R. Scholz, M. Schreiber, and G. Seifert. Theoretical investigation of excited states of C 3 . The Journal of Chemical Physics, 121(12):5767-5776, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The valence states of c 2 : A configuration interaction study. K Kirby, B Liu, The Journal of Chemical Physics. 702K. Kirby and B. Liu. The valence states of c 2 : A configuration inter- action study. The Journal of Chemical Physics, 70(2):893-900, 1979.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Wl Faust, Goldberg, Craig Bb, R G Weiss, Chemical Physics Letters. 832WL Faust, LS Goldberg, BB Craig, and RG Weiss. Time-resolved C 2 Swan emission from short-pulse UV fragmentation of CO: evidence for two C 2 formation mechanisms. Chemical Physics Letters, 83(2):265- 269, 1981.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Investigation of the Fine Structure of the C 2 High-Pressure Bands. H Meinel, Gerd Messerle, The Astrophysical Journal. 154381H Meinel and Gerd Messerle. Investigation of the Fine Structure of the C 2 High-Pressure Bands. The Astrophysical Journal, 154:381, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. S Paul, Remo Barklem, Collet, Astronomy & Astrophysics. 58896Paul S Barklem and Remo Collet. Partition functions and equilibrium constants for diatomic molecules and atoms of astrophysical interest. Astronomy & Astrophysics, 588:A96, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statistical thermodynamics. Donald Allan Mcquarrie, Harper Collins PublishersDonald Allan McQuarrie. Statistical thermodynamics. Harper Collins Publishers, 1973.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Molecular spectra and molecular structure. Gerhard Herzberg, Spectra of diatomic molecules. New YorkVan Nostrand Reinhold12nd ed.Gerhard Herzberg. Molecular spectra and molecular structure. Vol. 1: Spectra of diatomic molecules. New York: Van Nostrand Reinhold, 1950, 2nd ed., 1950.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas. S Svanberg, S Svanberg. Atomic and Molecular Spectroscopy: Basic Aspects and Practical Applications. Springer series on atoms and plasmas, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Application of the (0,0) Swan band spectrum for temperature measurements. S Pellerin, Musiol, Motret, J Pokrzywka, Chapelle, Journal of Physics D: Applied Physics. 29112850S Pellerin, K Musiol, O Motret, B Pokrzywka, and J Chapelle. Appli- cation of the (0,0) Swan band spectrum for temperature measurements. Journal of Physics D: Applied Physics, 29(11):2850, 1996.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spectra of atoms and molecules. F Peter, Bernath, Oxford university pressPeter F Bernath. Spectra of atoms and molecules. Oxford university press, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Colin M Western, Journal of Quantitative Spectroscopy and Radiative Transfer. 186Colin M Western. PGOPHER: A program for simulating rotational, vibrational and electronic spectra. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:221-242, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"LEVEL: A computer program for solving the radial Schrdinger equation for bound and quasibound levels. J Le Robert, Roy, Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting. 186Robert J. Le Roy. LEVEL: A computer program for solving the ra- dial Schrdinger equation for bound and quasibound levels. Journal of Quantitative Spectroscopy and Radiative Transfer, 186:167 -178, 2017. Satellite Remote Sensing and Spectroscopy: Joint ACE-Odin Meeting, October 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. G M Lloyd, P Ewart, The Journal of Chemical Physics. 1101G. M. Lloyd and P. Ewart. High resolution spectroscopy and spectral simulation of C 2 using degenerate four-wave mixing. The Journal of Chemical Physics, 110(1):385-392, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Swan System of C 2 : A Global Analysis of Fourier Transform Emission Spectra. A Tanabashi, T Hirao, T Amano, P F Bernath, The Astrophysical Journal Supplement Series. 1692472A. Tanabashi, T. Hirao, T. Amano, and P. F. Bernath. The Swan Sys- tem of C 2 : A Global Analysis of Fourier Transform Emission Spectra. The Astrophysical Journal Supplement Series, 169(2):472, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fourier transform emission spectroscopy of the B 2 Σ + -X 2 Σ + system of CN. R S Ram, S P Davis, L Wallace, R Engleman, D R T Appadoo, P F Bernath, Journal of Molecular Spectroscopy. 2372R.S. Ram, S.P. Davis, L. Wallace, R. Engleman, D.R.T. Appadoo, and P.F. Bernath. Fourier transform emission spectroscopy of the B 2 Σ + - X 2 Σ + system of CN. Journal of Molecular Spectroscopy, 237(2):225 - 231, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lifbase, database and spectral simulation for diatomic molecules. J Luque, 99SRI International ReportJ Luque. Lifbase, database and spectral simulation for diatomic molecules. SRI International Report, 99, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spectrometer. D Cerny, R Bacis, G Guelachvili, F Roux, Journal of Molecular Spectroscopy. 731D. Cerny, R. Bacis, G. Guelachvili, and F. Roux. Extensive analysis of the red system of the CN molecule with a high resolution Fourier Spec- trometer. Journal of Molecular Spectroscopy, 73(1):154 -167, 1978.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i. Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, Journal of Molecular Spectroscopy. 1272Haruhiko Ito, Yasushi Ozaki, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Analysis of the B 2 Σ + -A 2 Π i perturbations in the CN(B 2 Σ + -X 2 Σ + ) main band system: I. Molecular constants for B 2 Σ + and A 2 Π i . Journal of Molecular Spectroscopy, 127(2):283 -303, 1988.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, Kozo Kuchitsu, The Journal of Chemical Physics. 948Haruhiko Ito, Kaoru Suzuki, Tamotsu Kondow, and Kozo Kuchitsu. Electronic transition moment for the B 2 Σ + -X 2 Σ + emission of CN. Analysis of dependence on the internuclear distance. The Journal of Chemical Physics, 94(8):5353-5359, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spectroscopic studies and rotational and vibrational temperature measurements of atmospheric pressure normal glow plasma discharges in air. David Staack, Bakhtier Farouk, Alexander F Gutsol, Alexander A Fridman, Plasma Sources Science and Technology. 154818David Staack, Bakhtier Farouk, Alexander F Gutsol, and Alexander A Fridman. Spectroscopic studies and rotational and vibrational tem- perature measurements of atmospheric pressure normal glow plasma discharges in air. Plasma Sources Science and Technology, 15(4):818, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 Swan spectrum used as a molecular pyrometer in transferred arc and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 511112066H Nassar. C 2 Swan spectrum used as a molecular pyrometer in trans- ferred arc and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 511(1):012066, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"arXiv:1404.1100Jonathon Shlens. A tutorial on principal component analysis. arXiv preprintJonathon Shlens. A tutorial on principal component analysis. arXiv preprint arXiv:1404.1100, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Christophe Laux, Specair, Christophe Laux. SPECAIR, 2012. Accessed: March 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Modeling the spectral radiation of a methane/nitrogen plasma: Test case 4. L Caillaut, Andreasson, C O Risberg, Laux, Radiation of High Temperature Gases in Atmospheric Entry. 629L Caillaut, J Andreasson, J Risberg, and CO Laux. Modeling the spec- tral radiation of a methane/nitrogen plasma: Test case 4. In Radiation of High Temperature Gases in Atmospheric Entry, volume 629, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fundamental data of diatomic molecules relevant for fusion. U Fantz, Wünderlich, Atomic and Plasma-Material Interaction Data for Fusion. 16U Fantz and D Wünderlich. Fundamental data of diatomic molecules relevant for fusion. In Atomic and Plasma-Material Interaction Data for Fusion, volume 16. IAEA, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Oscillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. Damian L Kokkin, George B Bacskay, Timothy W Schmidt, The Journal of Chemical Physics. 126884302Damian L. Kokkin, George B. Bacskay, and Timothy W. Schmidt. Os- cillator strengths and radiative lifetimes for C 2 : Swan, Ballik-Ramsay, Phillips, and d 3 Π g ← c 3 Σ + u systems. The Journal of Chemical Physics, 126(8):084302, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Melanie Heusing, and Michael Letzgus. Measurement and simulation of rotationally-resolved chemiluminescence spectra in flames. Andreas Brockhinke, Julia Krüger, Applied Physics B. 1073Andreas Brockhinke, Julia Krüger, Melanie Heusing, and Michael Let- zgus. Measurement and simulation of rotationally-resolved chemilumi- nescence spectra in flames. Applied Physics B, 107(3):539-549, 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Radiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: I. Experiment. D Vacher, Lino Da, P Silva, G André, M Faure, Dudeck, Plasma Sources Science and Technology. 17335012D Vacher, M Lino da Silva, P André, G Faure, and M Dudeck. Ra- diation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spec- tral region: I. Experiment. Plasma Sources Science and Technology, 17(3):035012, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Radiation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. M Lino Da Silva, D Vacher, M Dudeck, P André, G Faure, Plasma Sources Science and Technology. 17335013M Lino da Silva, D Vacher, M Dudeck, P André, and G Faure. Radi- ation from an equilibrium CO 2 -N 2 plasma in the 250-850nm spectral region: II. Spectral modelling. Plasma Sources Science and Technology, 17(3):035013, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. E A Timmermans, J Jonkers, M Rodero, Quintero, Sola, D Gamero, J A Schram, Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 547E.A.H Timmermans, J Jonkers, A Rodero, M.C Quintero, A Sola, A Gamero, D.C Schram, and J.A.M van der Mullen. The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 2: Plasmas at reduced pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 54(7):1085 -1098, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The behavior of molecules in microwave-induced plasmas studied by optical emission spectroscopy. 1. plasmas at atmospheric pressure. E A H Timmermans, J Jonkers, I A J Thomas, A Rodero, M C Quintero, A Sola, A Gamero, J A M Van Der Mullen, Spectrochimica Acta Part B: Atomic Spectroscopy. 5311E.A.H. Timmermans, J. Jonkers, I.A.J. Thomas, A. Rodero, M.C. Quintero, A. Sola, A. Gamero, and J.A.M. van der Mullen. The behav- ior of molecules in microwave-induced plasmas studied by optical emis- sion spectroscopy. 1. plasmas at atmospheric pressure. Spectrochimica Acta Part B: Atomic Spectroscopy, 53(11):1553 -1566, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Elucidation of C 2 and CN formation mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Meirong Dong, C.-Y. George, Xianglei Chan, Jhanis J Mao, Jidong Gonzalez, Richard E Lu, Russo, Spectrochimica Acta Part B: Atomic Spectroscopy. 100Dedicated To Nicolo Omenetto On the Occasion of his 75th BirthdayMeirong Dong, George C.-Y. Chan, Xianglei Mao, Jhanis J. Gonzalez, Jidong Lu, and Richard E. Russo. Elucidation of C 2 and CN forma- tion mechanisms in laser-induced plasmas through correlation analysis of carbon isotopic ratio. Spectrochimica Acta Part B: Atomic Spec- troscopy, 100:62 -69, 2014. Dedicated To Nicolo Omenetto On the Occasion of his 75th Birthday.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"CN violet system spectrum used as a molecular pyrometer and the influence noise to signal ratio on the temperature values. H Nassar, Journal of Physics: Conference Series. 37012050H Nassar. CN violet system spectrum used as a molecular pyrome- ter and the influence noise to signal ratio on the temperature values. Journal of Physics: Conference Series, 370:012050, jun 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. D B Se Youn Moon, B Kim, W Gweon, Choe, Journal of Applied Physics. 105553307Se Youn Moon, D. B. Kim, B. Gweon, and W. Choe. Temperature measurement of an atmospheric pressure arc discharge plasma jet using the diatomic CN (B 2 Σ + -X 2 Σ + , violet system) molecular spectra. Journal of Applied Physics, 105(5):053307, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Torcheà plasmaà excitation micro-onde: deux configurations complémentaires. A Ricard, H St-Onge, Malvos, Gicquel, M Hubert, Moisan, Journal de Physique III. 58A Ricard, L St-Onge, H Malvos, A Gicquel, J Hubert, and M Moisan. Torcheà plasmaà excitation micro-onde: deux config- urations complémentaires. Journal de Physique III, 5(8):1269-1285, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Ma Ridenti, Amorim, Plasma Chemistry and Plasma Processing. 38MA Ridenti and J Amorim. Cn (b 2 σ + → x 2 σ + ) violet system in a cold atmospheric-pressure argon plasma jet. Plasma Chemistry and Plasma Processing, 38(2):311-329, 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Kotaro Saito, Tetsuo Sakka, Yukio H Ogata, Journal of Applied Physics. 949Kotaro Saito, Tetsuo Sakka, and Yukio H. Ogata. Rotational spectra and temperature evaluation of C 2 molecules produced by pulsed laser irradiation to a graphite-water interface. Journal of Applied Physics, 94(9):5530-5536, 2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Fabien Goulay, László Nemes, Paul E Schrader, Hope A Michelsen, Molecular Physics. 1087-9Fabien Goulay, László Nemes, Paul E Schrader, and Hope A Michelsen. Spontaneous emission from C 2 (d 3 Π g ) and C 3 (A 1 Π u ) during laser irradiation of soot particles. Molecular Physics, 108(7-9):1013-1025, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. V E Bondybey, The Journal of Chemical Physics. 656V. E. Bondybey. Sequential two photon excitation of the C 2 Swan transitions and C 2 relaxation dynamics in rare gas solids. The Journal of Chemical Physics, 65(6):2296-2304, 1976.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. R P Frosch, The Journal of Chemical Physics. 546R. P. Frosch. C 2 and C 2 -Spectra Produced by the X Irradiation of Acetylene in RareGas Matrices. The Journal of Chemical Physics, 54(6):2660-2666, 1971.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Li Xuechu, Lou Nanquan, Chemical Physics Letters. 751Li Xuechu and Lou Nanquan. The vibrational population inversion and relaxation kinetics of the C * 2 , d 3 Π g state. Chemical Physics Letters, 75(1):110-114, 1980.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"C 2 and CN formation by optical pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. William Rich, Richard C Bergman, Chemical Physics. 441J William Rich and Richard C Bergman. C 2 and CN formation by opti- cal pumping of CO/Ar and CO/N 2 /Ar mixtures at room temperature. Chemical Physics, 44(1):53-64, 1979.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vibrational to electronic energy transfer from CO to C 2 molecules. G Grigorian, Cenian, Chemical Physics Letters. 4694-6G Grigorian and A Cenian. Vibrational to electronic energy transfer from CO to C 2 molecules. Chemical Physics Letters, 469(4-6):247-249, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. L Hidde, Bruno Wallaart, Marie-Yvonne Piar, Jean-Pierre Perrin, Martin, Chemical Physics. 1961Hidde L. Wallaart, Bruno Piar, Marie-Yvonne Perrin, and Jean-Pierre Martin. Transfer of vibrational energy to electronic excited states and vibration enhanced carbon production in optically excited VV pumped CO. Chemical Physics, 196(1):149 -169, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. W Hack, Chemical Physics Letters. 813W Hack and W Langel. Formation of C 2 (e 3 Π g , C 1 Π g ) in the photolysis of CO(X 1 Σ + ) at 193 nm with an ArF laser. Chemical Physics Letters, 81(3):387 -390, 1981.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pressure Dependence of Rotational Intensity Distribution in C 2 Swan Emission from LowPressure Oxyacetylene Flames. R Bleekrode, The Journal of Chemical Physics. 458R. Bleekrode. Pressure Dependence of Rotational Intensity Distribu- tion in C 2 Swan Emission from LowPressure Oxyacetylene Flames. The Journal of Chemical Physics, 45(8):3153-3154, 1966.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, Katharina Kohse-Höinghaus, The Journal of Physical Chemistry A. 1109Andreas Brockhinke, Michael Letzgus, Stefanie Rinne, and Katharina Kohse-Höinghaus. Energy Transfer in the d 3 Π g -a 3 Π u (0-0) Swan Bands of C 2 : Implications for Quantitative Measurements. The Journal of Physical Chemistry A, 110(9):3028-3035, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G Grigorian, A Cenian, Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing. 31G. Grigorian and A. Cenian. Formation and Excitation of CN Molecules in He-CO-N 2 -O 2 Discharge Plasmas. Plasma Chemistry and Plasma Processing, 31(2):337-352, Apr 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2. Jingzhong Guo, C M Sadowski, Q Gao, F J Morgan, The Journal of Chemical Physics. 11317Jingzhong Guo, C. M. Sadowski, Q. Gao, and F. J. Morgan. State-to-state and total rotational energy transfer rate constants for CN(B 2 Σ + ,v=0,N)+He and C 2 N 2 . The Journal of Chemical Physics, 113(17):7276-7285, 2000.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"State-to-state and total rotational energy transfer rate constants for CN. M K Sophie, Jingzhong Brunet, Tucker Guo, S V Carrington, C M Filseth, Sadowski, N)+NO. The Journal of Chemical Physics. 1169Sophie M. K. Brunet, Jingzhong Guo, Tucker Carrington, S. V. Filseth, and C. M. Sadowski. State-to-state and total rotational energy trans- fer rate constants for CN(B 2 Σ + ,v=0,N)+H 2 , CN(X 2 Σ + ,v=2,N)+H 2 , D 2 , and CN(X 2 Σ + ,v=3,N)+NO. The Journal of Chemical Physics, 116(9):3617-3625, 2002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave induced nitrogen plasma torch. Chuan-Jie Chen, Shou-Zhe Li, Plasma Sources Science and Technology. 24335017Chuan-Jie Chen and Shou-Zhe Li. Spectroscopic measurement of plasma gas temperature of the atmospheric-pressure microwave in- duced nitrogen plasma torch. Plasma Sources Science and Technology, 24(3):035017, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optical characterization of a microwave pulsed discharge used for dissociation of CO 2. Tiago Silva, Nikolay Britun, Thomas Godfroid, Rony Snyders, Plasma Sources Science and Technology. 23225009Tiago Silva, Nikolay Britun, Thomas Godfroid, and Rony Snyders. Op- tical characterization of a microwave pulsed discharge used for dissoci- ation of CO 2 . Plasma Sources Science and Technology, 23(2):025009, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The dissociation of carbon monoxide. A R Fairbairn, Alfred Gordon Gaydon, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences. 312A. R. Fairbairn and Alfred Gordon Gaydon. The dissociation of carbon monoxide. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 312(1509):207-227, 1969.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Coherent anti-Stokes Raman scattering and spontaneous Raman scattering diagnostics of nonequilibrium plasmas and flows. R Walter, Igor V Lempert, Adamovich, 2014-10-29T00:00:00Journal of Physics D: Applied Physics. 4743Walter R. Lempert and Igor V. Adamovich. Coherent anti-Stokes Ra- man scattering and spontaneous Raman scattering diagnostics of non- equilibrium plasmas and flows. Journal of Physics D: Applied Physics, 47(43):433001-433026, 2014-10-29T00:00:00.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. D C M Van Den, J M Bekerom, E M Linares, S Van Veldhuizen, M C M Nijdam, G J Van De Sanden, Van Rooij, Appl. Opt. 5720D. C. M. van den Bekerom, J. M. Palomares Linares, E. M. van Veld- huizen, S. Nijdam, M. C. M. van de Sanden, and G. J. van Rooij. How the alternating degeneracy in rotational Raman spectra of CO 2 and C 2 H 2 reveals the vibrational temperature. Appl. Opt., 57(20):5694- 5702, Jul 2018.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":37,"cells":{"text":{"kind":"string","value":"\nHamiltonian reduction of the U EM (1) gauged three flavour WZW model\narXiv:hep-th/9506025v1 5 Jun 1995 June 1995\n\nJ E Paschalis paschalis@olymp.ccf.auth.grporfyriadis@olymp.ccf.auth.gr \nP I Porfyriadis \nTbilisi State University\nTbilisiGeorgia\n\n\nDepartment of Theoretical Physics\nUniversity of Thessaloniki\n54006ThessalonikiGRGreece\n\nHamiltonian reduction of the U EM (1) gauged three flavour WZW model\narXiv:hep-th/9506025v1 5 Jun 1995 June 1995\nThe three-flavour Wess-Zumino model coupled to electromagnetism is treated as a constraint system using the Faddeev-Jackiw method. Expanding into series of powers of the Goldstone boson fields and keeping terms up to second and third order we obtain Coulomb-gauge hamiltonian densities.\n\nIntroduction\n\nThe Faddeev-Jackiv method [1] provides a simple and straightforward way to deal with constraint systems without having to distinguish between primary and secondary, first class and second class constraints. A lot of work has been done in this field by several authors. The equivalence of the Faddeev-Jackiw approach to the Dirac's method is discussed in [2,3]. Its extension in supersymmetry is given in [4,5]. Application of the approach to the light cone quantum field theory is given in [6,7], and to hidden symmetries in [8]. Further elaboration and applications of the method in different cases is given in [9] . A formulation, intimately related to the FJ approach, for constructing unconstraint hamiltonians starting from general first order lagrangians is given in [10].\n\nIn a previous paper [11] we used the Faddeev-Jackiw method [1] to treat the two-flavour WZW model coupled to electromagnetism [12,13]. In this work we extend this analysis to the SU(3) case. The gauge invariant action under the electromagnetic group gauge transformations is given by\nΓ ef f (U, A µ ) = Γ EM (A µ ) + Γ σ (U, A µ ) + Γ W ZW (U) + Γ W ZW (U, A µ ) ,(1)\nwhere the first term is the action of the free electromagnetic field and the sum of the rest three terms constitute the action of the gauged Wess-Zumino-Witten model\nΓ EM (A µ ) = − 1 4 d 4 xF µν F µν , Γ σ (U, A µ ) = − f 2 π 16 d 4 xtr (R µ R µ ) = − f 2 π 16 d 4 xtr (r µ r µ ) + if 2 π e 8 d 4 xA µ tr [Q(r µ − l µ )] + f 2 π e 2 8 d 4 xA µ A µ tr (Q 2 − U † QUQ) ,(2)Γ W ZW (U) = − iN c 240π 2 d 5 xǫ ijklm tr (l i l j l k l l l m ) , Γ W ZW (U, A µ ) = − N c e 48π 2 d 4 xǫ µναβ A µ tr [Q(r ν r α r β + l ν l α l β )] + iN c e 2 24π 2 d 4 xǫ µναβ A µ (∂ ν A α )tr [Q 2 (r β + l β ) + 1 2 QU † QUr β + 1 2 QUQU † l β ] ,\nwhere U = exp (2iθ a λ a /f π ) , a = 1, ..., 8 is an element of SU(3)\nr µ = U † ∂ µ U , R µ = U † D µ U , l µ = (∂ µ U)U † , L µ = (D µ U)U † .\nSee Appendix for notation.\n\nThe model describes succesfully the low energy dynamics of photon and the eight Goldstone bosons including their interactions related to the axial anomaly.\n\nAs in the two-flavour case we expand the U fields into series of powers of the Goldstone boson fields θ a . Then using the expression for Γ ef f (U, A µ ) given in (1) we obtain lagrangian densities containing terms initially up to two and afterwards up to three Goldstone boson fields. These lagrangian densities are constrained and we treat them using the Faddeev-Jackiw approach. The general case which involves any number of Goldstone bosons is still out of hand.\n\n2 Expanding up to second order in the Goldstone boson fields\n\nWe expand the U field into series of powers of the Goldstone boson fields\nU = 1 + 2i f π θ a λ a + ... , a = 1, ..., 8\nand we substitute back into the expression (1) for the effective action. The resulting lagrangian density with terms up to second order in θ a is given by\nL ef f = L EM + L (2) σ + L (2) W ZW + O(θ 3 ) ,(3)L EM = − 1 4 F µν F µν , L (2) σ = 1 2 (∂ µ θ a )(∂ µ θ a ) + eA µ (θ 2 ∂ µ θ 1 − θ 1 ∂ µ θ 2 + θ 5 ∂ µ θ 4 − θ 4 ∂ µ θ 5 ) + e 2 2 A µ A µ (θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 ) , L(2)W ZW = − N c e 2 12π 2 f π ǫ µναβ A µ (∂ ν A α )(∂ β θ 3 + 1 √ 3 ∂ β θ 8 ) .\nWe can easily check that the U EM (1) gauge invariance is not lost. In the two flavour case [11] the expression for the corresponding lagrangian density can be deduced from (3) by omitting those terms which include powers or derivatives of θ 4 up to θ 8 . We should also notice that the term Γ W ZW (U) of the effective action given in (2) which is zero in the two flavour case, contributes, in the three flavour case, terms involving five or more Goldstone boson fields. In the non-covariant notation, (3) can be written as follows\nL ef f = −E ·Ȧ + 1 2θ a 2 + eA 0 (θ 2θ1 − θ 1θ2 + θ 5θ4 − θ 4θ5 ) + A 0 ∇ · E + e 2 2 (A 2 0 − A 2 )(θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 ) − 1 2 (E 2 + B 2 + (∇θ a ) 2 ) + eA · (θ 2 ∇θ 1 − θ 1 ∇θ 2 + θ 5 ∇θ 4 − θ 4 ∇θ 5 ) + N c e 2 6π 2 f π (E · B)(θ 3 + 1 √ 3 θ 8 ) ,(4)\nwhere E = −Ȧ − ∇A 0 , B = ∇ × A are the electric and magnetic fields. We see that the dynamical variables in (4) which span the configuration space of the lagrangian density are the three components of the vector potential and the eight Goldstone boson fields. The scalar potential A 0 will be treated not as a dynamical variable but as a Lagrange multiplier since there is no time derivative of it in (4). Now according to Faddeev and Jackiw [1] we must construct out of (4), which is second order in time derivatives of the θ a fields, a new lagrangian density which is first order in time derivatives of all the dynamical variables. This can be accomplished by enlarging the configuration space so that it includes the whole phase space of the model. The enlarged configuration space coordinates will include the three vector potential components, the eight θ a fields and the corresponding canonical momenta. The canonical momenta conjugate to A are the three component of −E, and those conjugate to θ a are given by\np 1 = ∂L ef f ∂θ 1 =θ 1 + eA 0 θ 2 , p 2 = ∂L ef f ∂θ 2 =θ 2 − eA 0 θ 1 , p 3 = ∂L ef f ∂θ 3 =θ 3 , p 4 = ∂L ef f ∂θ 4 =θ 4 + eA 0 θ 5 , p 5 = ∂L ef f ∂θ 5 =θ 5 − eA 0 θ 4 , p 6 = ∂L ef f ∂θ 6 =θ 6 , p 7 = ∂L ef f ∂θ 7 =θ 7 , p 8 = ∂L ef f ∂θ 8 =θ 8 .\nSo the resulting, first order in time derivatives, lagrangian density is given by\nL ef f = −E ·Ȧ + p aθa − H (2) (E, A, p a , θ a ) − A 0 (ρ (2) − ∇ · E) + O(θ 3 ) ,(5)H (2) = 1 2 [E 2 + B 2 + (∇θ a ) 2 + p 2 a ] + e 2 2 A 2 (θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 ) + eA · (θ 1 ∇θ 2 − θ 2 ∇θ 1 + θ 4 ∇θ 5 − θ 5 ∇θ 4 ) − N c e 2 6π 2 f π (E · B)(θ 3 + 1 √ 3 θ 8 ) , ρ (2) = e(p 2 θ 1 − p 1 θ 2 + p 5 θ 4 − p 4 θ 5 ) .\nThe expression (p aθa − E ·Ȧ)dt, called the canonical one-form, can be written up to a total time derivative as\n1 2 ξ i ω ij dξ j ,(6)\nwhere\nξ i = A i , i = 1, ..., 3 ξ i = θ i , i = 4, ..., 11 ξ i = −E i , i = 12, ..., 14 ξ i = p i , i = 15, ..., 22\nand ω ij is the symplectic 22 × 22 matrix\nω ij = 0 I −I 0 ij\nAs we mentioned before A 0 is the Lagrange multiplier. The expression ρ (2) − ∇ · E which is multiplied by A 0 is the only constraint in (5). The next step is to solve the equation of the constraint\n∇ · E − ρ (2) = 0 ,(7)\nand incorporate the solution into the expression (5) for the lagrangian density. In order to do that we decompose the electric field E and the vector potential A into transverse and longitudinal components\nE = E T + E L , A = A T + A L , ∇ · E T = 0 , ∇ × E L = 0 , ∇ · A T = 0 , ∇ × A L = 0 .\nThen (7) implies that\nE L = ∇ ∇ 2 ρ (2) ≡ − 1 4π d 3 y 1 | y − x | ∇ρ (2) (y) .(8)\nSubstituting into (5) the expression of E L given in (8) we get (apart from a total divergence)\nL ef f = −E T ·Ȧ T + ρ (2) ∇ ∇ 2 ·Ȧ L + p aθa − 1 2 [E 2 T + B 2 − ρ (2) 1 ∇ 2 ρ (2) + (∇θ a ) 2 + p 2 a ] − eA · (θ 1 ∇θ 2 − θ 2 ∇θ 1 + θ 4 ∇θ 5 − θ 5 ∇θ 4 ) − e 2 2 A 2 (θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 ) + N c e 2 6π 2 f π [E T · B + ( ∇ ∇ 2 ρ (2) ) · B](θ 3 + 1 √ 3 θ 8 ) .(9)\nWe see that the canonical one-form in (9) has lost the standard form as given in (6) because of the term ρ (2) ∇ ∇ 2 ·Ȧ L dt. According to Darboux's theorem [1] we can perform coordinate transformations so that the canonical one-form gets back the standard form. This coordinate transformations called Darboux's transformations can be written as follows p 1 → p 1 cos α + p 2 sin α , θ 1 → θ 1 cos α + θ 2 sin α ,\np 2 → p 2 cos α − p 1 sin α , θ 2 → θ 2 cos α − θ 1 sin α , p 4 → p 4 cos α + p 5 sin α , θ 4 → θ 4 cos α + θ 5 sin α ,(10)p 5 → p 5 cos α − p 4 sin α , θ 5 → θ 5 cos α − θ 4 sin α , p 3 → p 3 , θ 3 → θ 3 , p 6 → p 6 , θ 6 → θ 6 , p 7 → p 7 , θ 7 → θ 7 , p 8 → p 8 , θ 8 → θ 8 , E T → E T , A T → A T ,\nwhere α = e ∇ ∇ 2 ·A L . We substitute (10) into the expression (9) for the lagrangian density and we find out that all the terms which include powers or derivatives of A L cancel out exactly. Only terms with the physical transverse components of the vector potential survive, and the canonical one-form in (9) acquires the standard form. The resulting lagrangian density is given by\nL ef f = −E T ·Ȧ T + p aθa − H (2) C ,(11)\nwhere\nH (2) C = 1 2 [E 2 T + B 2 − ρ (2) 1 ∇ 2 ρ (2) + (∇θ a ) 2 + p 2 a ] + eA T · (θ 1 ∇θ 2 − θ 2 ∇θ 1 + θ 4 ∇θ 5 − θ 5 ∇θ 4 ) + e 2 2 A 2 T (θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 ) − N c e 2 6π 2 f π [E T · B + ( ∇ ∇ 2 ρ (2) ) · B](θ 3 + 1 √ 3 θ 8 ) ,(12)\nis the expression for the hamiltonian density of the model. We see that without mentioning gauge fixing the solution of the constraint lead to a Coulomb gauge hamiltonian density .\n\n( Note that B = ∇ × A T ).\n\n\nKeeping third order terms\n\nWe now proceed with the expansion and keep terms up to the third order in the Goldstone boson fields. We obtain the following expression for the resulting U EM (1) gauge invariant effective lagrangian density\nL ef f = L EM + L (2) σ + L (2) W ZW + L (3) W ZW + O(θ 4 ) ,(13)\nwhere the expression for L (2) σ and L\n\nW ZW are given in (3) and\nL (3) W ZW = − N c e 3π 2 f 3 π ǫ µναβ (∂ µ A ν )(θ 1 ∂ α θ 2 − θ 2 ∂ α θ 1 + θ 4 ∂ α θ 5 − θ 5 ∂ α θ 4 )(∂ β θ 3 + 1 √ 3 ∂ β θ 8 ) − N c e √ 3π 2 f 3 π ǫ µναβ (∂ µ A ν )(θ 7 ∂ α θ 6 − θ 6 ∂ α θ 7 )∂ β θ 8 + N c e 2 18π 2 f 3 π ǫ µναβ (∂ µ A ν )(∂ α A β ){[4(θ 2 1 + θ 2 2 ) + 5(θ 2 4 + θ 2 5 )]θ 3 + √ 3[2(θ 2 1 + θ 2 2 ) + θ 2 4 + θ 2 5 ]θ 8 + 2[(θ 1 θ 5 − θ 2 θ 4 )θ 7 + (θ 1 θ 4 + θ 2 θ 5 )θ 6 ]} − N c e 2 3π 2 f 3 π ǫ µναβ A µ (∂ ν A α )(θ 3 + 1 √ 3 θ 8 )∂ β (θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 ) .\nWe see that only Γ W ZW (U, A µ ) ( the part of the Wess-Zumino-Witten action which describes the anomalous interaction of the Goldstone bosons with the photons ) gives extra contribution at this order. Next we derive the expressions for the canonical momenta p a conjugate to the Goldstone boson fields θ a . We substitute into (13) and we get the following expression for the effective lagrangian density in the enlarged configuration space\nL ef f = −E ·Ȧ + p aθa − H (2) − H (3) − A 0 (ρ (2) + ρ (3) − ∇ · E) + O(θ 4 ) ,(14)\nwhere the expression for H (2) is given in (12) and\nH (3) = − N c e 3π 2 f 3 π [E × ∇(θ 3 + 1 √ 3 θ 8 ) − (p 3 + 1 √ 3 p 8 )B)] · (θ 1 ∇θ 2 − θ 2 ∇θ 1 + θ 4 ∇θ 5 − θ 5 ∇θ 4 ) − N c e √ 3π 2 f 3 π (E × ∇θ 8 − p 8 B) · (θ 7 ∇θ 6 − θ 6 ∇θ 7 ) + N c e 2 9π 2 f 3 π (E · B){[4(θ 2 1 + θ 2 2 ) + 5(θ 2 4 + θ 2 5 )]θ 3 + √ 3[2(θ 2 1 + θ 2 2 ) + θ 2 4 + θ 2 5 ]θ 8 + 2[(θ 1 θ 5 − θ 2 θ 4 )θ 7 + (θ 1 θ 4 + θ 2 θ 5 )θ 6 ]} − N c e 2 3π 2 f 3 π (E × A) · [∇(θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 )](θ 3 + 1 √ 3 θ 8 ) − 2N c e 2 3π 2 f 3 π (A · B)(p 1 θ 1 + p 2 θ 2 + p 4 θ 4 + p 5 θ 5 )(θ 3 + 1 √ 3 θ 8 ) − N c e 3π 2 f 3 π [B · ∇(θ 3 + 1 √ 3 θ 8 )](p 2 θ 1 − p 1 θ 2 + p 5 θ 4 − p 4 θ 5 ) − N c e √ 3π 2 f 3 π (B · ∇θ 8 )(p 6 θ 7 − p 7 θ 6 ) ,(15)ρ (2) = e(p 2 θ 1 − p 1 θ 2 + p 5 θ 4 − p 4 θ 5 ) , ρ (3) = − N c e 2 3π 2 f 3 π ∇ · [B(θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 )(θ 3 + 1 √ 3 θ 8 )] .\nThe scalar potential A 0 is again the Lagrange multiplier and the equation of the constraint is given by\n∇ · E − (ρ (2) + ρ (3) ) = 0 .(16)\nThe equation (16) has similar structure as in the previous case (7) so we proceed similarly. We decompose the electric field E and the vector potential A into transverse and longitudinal components. Then (16) implies\nE L = ∇ ∇ 2 (ρ (2) + ρ (3) ) ≡ − 1 4π d 3 y 1 | y − x | ∇(ρ (2) (y) + ρ (3) (y)) .(17)\nNext we substitute the expression for E L given in (17) into (14) and we obtain a lagrangian density whose canonical one-form is given (apart from a total divergence) by\n− E T ·Ȧ T + ρ (2) ∇ ∇ 2 ·Ȧ L + ρ (3) ∇ ∇ 2 ·Ȧ L + p aθa ,(18)\nThis expression must be diagonalized so that it acquires the standard form (6). In order to do this we proceed in two steps as in the SU(2) case [11]. First we perform the Darboux's transformations given in (10) which lead to partial diagonalization of (18). The new lagrangian density has a canonical one-form given by\n−E T ·Ȧ T + ρ (3) ∇ ∇ 2 ·Ȧ L + p aθa ,\nThe term ρ (3) ∇ ∇ 2 ·Ȧ L can be written as follows apart from a total time derivative\nρ (3) ∇ ∇ 2 ·Ȧ L = − N c e 2 3π 2 f 3 π [(∇φ × A L ) ·Ȧ T + (B · A L )φ] , where φ = (θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 )(θ 3 + 1 √ 3 θ 8 ) .\nNow we proceed with the second step. We perform the following Darboux's transformations\nE T → E T − Nce 2 3π 2 f 3 π ∇[(θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 )(θ 3 + 1 √ 3 θ 8 )] × A L , p i → p i + 2Nce 2 3π 2 f 3 π (B · A L )(θ 3 + 1 √ 3 θ 8 )θ i , i = 1, 2, 4, 5 p 3 → p 3 + Nce 2 3π 2 f 3 π (B · A L )(θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 ) , p 8 → p 8 + Nce 2 3 √ 3π 2 f 3 π (B · A L )(θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 ) , A T → A T , p 6 → p 6 , p 7 → p 7 , θ a → θ a , a = 1, ..., 8 .\nThese transformations complete the diagonalization of the canonical one-form and we end up with a lagrangian density where the longitudinal part of the vector potential A L cancels out exactly\nL = −E T ·Ȧ T + p aθa − H (2) C − H (3) C + O(θ 4 ) .(19)\nThe expression for H (2) C is given in (12) and\nH (3) C = − N c e 3π 2 f 3 π [E T × ∇(θ 3 + 1 √ 3 θ 8 ) − (p 3 + 1 √ 3 p 8 )B)] · (θ 1 ∇θ 2 − θ 2 ∇θ 1 + θ 4 ∇θ 5 − θ 5 ∇θ 4 ) − N c e √ 3π 2 f 3 π (E T × ∇θ 8 − p 8 B) · (θ 7 ∇θ 6 − θ 6 ∇θ 7 ) + N c e 2 9π 2 f 3 π (E T · B){[4(θ 2 1 + θ 2 2 ) + 5(θ 2 4 + θ 2 5 )]θ 3 + √ 3[2(θ 2 1 + θ 2 2 ) + θ 2 4 + θ 2 5 ]θ 8 + 2[(θ 1 θ 5 − θ 2 θ 4 )θ 7 + (θ 1 θ 4 + θ 2 θ 5 )θ 6 ]} − N c e 2 3π 2 f 3 π (E T × A T ) · [∇(θ 2 1 + θ 2 2 + θ 2 4 + θ 2 5 )](θ 3 + 1 √ 3 θ 8 ) − 2N c e 2 3π 2 f 3 π (A T · B)(p 1 θ 1 + p 2 θ 2 + p 4 θ 4 + p 5 θ 5 )(θ 3 + 1 √ 3 θ 8 ) − N c e 3π 2 f 3 π [B · ∇(θ 3 + 1 √ 3 θ 8 )](p 2 θ 1 − p 1 θ 2 + p 5 θ 4 − p 4 θ 5 ) − N c e √ 3π 2 f 3 π (B · ∇θ 8 )(p 6 θ 7 − p 7 θ 6 ) .(20)H (2) C + H(3)\nC is the expression for the Coulomb-gauge hamiltonian density. In terms of the usual pseudoscalar fields π ± = 1\n√ 2 (θ 1 ∓ iθ 2 ) , K ± = 1 √ 2 (θ 4 ∓ iθ 5 ) , K • = 1 √ 2 (θ 6 − iθ 7 ) , K • = 1 √ 2 (θ 6 + iθ 7 )\n, π • = θ 3 , η 8 = θ 8 and the corresponding canonical momenta p i (i = π ± , π • , K ± , K • , K • , η 8 ) the expressions for H (2) C and H\n\nC are given by\nH (2) C = 1 2 [E 2 T + B 2 − ρ (2) 1 ∇ 2 ρ (2) + p 2 π • + p 2 η 8 + 2p π + p π − + 2p K + p K − + 2p K • p K • + (∇π • ) 2 + (∇η 8 ) 2 + 2(∇π + ) · (∇π − ) + 2(∇K + ) · (∇K − ) + 2(∇K • ) · (∇K • )] − ieA T · (π + ∇π − − π − ∇π + + K + ∇K − − K − ∇K + ) + e 2 A 2 T (π + π − + K + K − ) − N c e 2 6π 2 f π [E T · B + ( ∇ ∇ 2 ρ (2) ) · B](π • + 1 √ 3 η 8 ) , H(3)C = iN c e 3π 2 f 3 π [E T × ∇(π • + 1 √ 3 η 8 ) − (p π • + 1 √ 3 p η 8 )B)] · (π + ∇π − + K + ∇K − ) + iN c e √ 3π 2 f 3 π (E T × ∇η 8 − p η 8 B) · (K • ∇K • ) + N c e 2 9π 2 f 3 π (E T · B)[(4π + π − + 5K + K − )π • + √ 3(2π + π − + K + K − )η 8 + 2 √ 2π + K − K • ] − N c e 2 3π 2 f 3 π (E T × A T ) · [∇(π + π − + K + K − )](π • + 1 √ 3 η 8 ) − 2N c e 2 3π 2 f 3 π (A T · B)(p π + π + + p K + K + )(π • + 1 √ 3 η 8 ) + iN c e 3π 2 f 3 π [B · ∇(π • + 1 √ 3 η 8 )](p π + π + + p K + K + ) + iN c e √ 3π 2 f 3 π (B · ∇η 8 )p K • K • + h.c. ,\nρ (2) = −ie(p π + π + − p π − π − + p K + K + − p K − K − ) .\n\n\nConclusion\n\nIn this work we applied the Faddeev and Jackiw method for constrained systems to the U EM (1) gauged three flavour WZW model. We expanded the U-field into series of powers of the Goldstone boson fields and we limited ourselves to lagrangian densites with terms up to the third power in θ a . After treating these constrained langrangian densities using the FJ approach we ended up with unconstrained Coulomb-gauge hamiltonians, as in the case of QED [1]. The general case which involves any number of Goldstone boson fields is currently under investigation.\n\n\nAppendix\n\nOur metric is g µν = diag(1, −1, −1, −1) , Q = diag(2/3, −1/3, −1/3) is the charge matrix, D µ = ∂ µ + ieA µ [Q, ] denote the covariant derivative. λ a , a = 1, ..., 8 are the SU (3) generators tr (λ a λ b ) = 2δ ab . We choose e > 0 so that the electric charge of the electron is −e. We define ǫ 0123 = 1 .\n \n. L Faddeev, R Jackiw, Phys.Rev.Lett. 601692L.Faddeev, R.Jackiw, Phys.Rev.Lett. 60 (1988) 1692;\n\nR Jackiw, hep-th/9306075Constraint) Quantization Without Tears. Report No. CTP-2215R.Jackiw, (Constraint) Quantiza- tion Without Tears, Report No. CTP-2215, hep-th/9306075.\n\n. J Govaerts, Int. J. Mod. Phys. 53625J.Govaerts, Int. J. Mod. Phys. A5 (1990) 3625.\n\n. H Montani, Int.Mod.Phys. 84319H.Montani, Int.Mod.Phys. A8 (1993) 4319\n\n. D S Kulshreshtha, H J W Muller-Kirsten, Phys.Rev. 433376D.S.Kulshreshtha, H.J.W. Muller-Kirsten, Phys.Rev. D43 (1991) 3376.\n\n. J Barcelos-Neto, E S Cheb-Terrab, Int.J.Mod.Phys. 95563J.Barcelos-Neto, E.S.Cheb-Terrab, Int.J.Mod.Phys. A9 (1994) 5563;\n\n. Z.Phys. 54133Z.Phys. C54 (1992) 133.\n\n. J W Jun, C Jue, Phys. Rev. 502939J.W.Jun, C.Jue, Phys. Rev. D50 (1994) 2939.\n\n. O V Jacob, Phys. Rev. 505289O.V.Jacob, Phys. Rev. D50 (1994) 5289.\n\nFaddeev-Jackiw approach to hidden symmetries. C Wotzasek, hep-th/9502042C.Wotzasek, Faddeev-Jackiw approach to hidden symmetries, UR-1401, hep-th/9502042.\n\n. J Barcelos-Neto, C Wotzasek, Mod.Phys.Lett. 71737J.Barcelos-Neto and C.Wotzasek, Mod.Phys.Lett. A7 (1992) 1737;\n\nD45 (1992) 1410; C.Wotzasek. M M Barreira, C Wotzasek, Mod.Phys.Lett. 82509Phys.Rev.M.M.Horta Barreira and C.Wotzasek, Phys.Rev. D45 (1992) 1410; C.Wotzasek,Mod.Phys.Lett. A8 (1993) 2509;\n\n. H Montani, C Wotzasek, Mod.Phys.Lett. 83387H.Montani and C.Wotzasek, Mod.Phys.Lett. A8 (1993) 3387;\n\n. J Barcelos-Neto, E Vasquez, Z.Phys. 61695J.Barcelos-Neto, E.Vasquez, Z.Phys. C61 (1994) 695;\n\n. J Barcelos-Neto, Phys.Rev. 491012J.Barcelos-Neto, Phys.Rev. D49 (1994) 1012.\n\n. C Cronstrom, M Noga, Nucl. Phys. 428449C.Cronstrom, M. Noga, Nucl. Phys. B428 (1994) 449;\n\nHamiltonian formulation of dynamical systems. C Cronstrom, NORDITA-94/51 PReportC.Cronstrom, Hamiltonian for- mulation of dynamical systems, Report No. NORDITA-94/51 P.\n\nApplication of the Faddeev-Jackiw formalism to the gauged WZW model. J E Paschalis, P I Porfyriadis, No. THES-TP 95/039503009ReportJ.E.Paschalis, P.I.Porfyriadis,Application of the Faddeev-Jackiw formalism to the gauged WZW model,Report No. THES-TP 95/03, hep-th 9503009.\n\n. E Witten, Nucl.Phys. 223422E.Witten, Nucl.Phys. B223 (1983) 422;\n\n. O Kaymakcalan, S Rajeev, J Schechter, Phys.Rev. 30594O.Kaymakcalan, S.Rajeev and J.Schechter, Phys.Rev. D30 (1984) 594;\n\n. N K Pak, P Rossi, Nucl.Phys. 250279N.K.Pak and P.Rossi, Nucl.Phys. B250 (1985) 279.\n\nJ F Donoghue, E Golowich, B R Holstein, Dynamics of the Standard Model, p.201. (Cambridge Monographs on Particle Physics. J.F.Donoghue, E.Golowich, B.R.Holstein, Dynamics of the Standard Model, p.201. (Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology).\n"},"corpus_id":{"kind":"number","value":118934432,"string":"118,934,432"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"hep-th/9506025\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/hep-th/9506025v1.pdf\"\n ],\n \"pdf_sha\": \"0165c04bb1e1c139ab553371559487d1ad3620a7\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Hamiltonian reduction of the U EM (1) gauged three flavour WZW model"},"authors":{"kind":"list like","value":["J E Paschalis paschalis@olymp.ccf.auth.grporfyriadis@olymp.ccf.auth.gr","P I Porfyriadis \nTbilisi State University\nTbilisiGeorgia","Department of Theoretical Physics\nUniversity of Thessaloniki\n54006ThessalonikiGRGreece"],"string":"[\n \"J E Paschalis paschalis@olymp.ccf.auth.grporfyriadis@olymp.ccf.auth.gr\",\n \"P I Porfyriadis \\nTbilisi State University\\nTbilisiGeorgia\",\n \"Department of Theoretical Physics\\nUniversity of Thessaloniki\\n54006ThessalonikiGRGreece\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The three-flavour Wess-Zumino model coupled to electromagnetism is treated as a constraint system using the Faddeev-Jackiw method. Expanding into series of powers of the Goldstone boson fields and keeping terms up to second and third order we obtain Coulomb-gauge hamiltonian densities."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The Faddeev-Jackiv method [1] provides a simple and straightforward way to deal with constraint systems without having to distinguish between primary and secondary, first class and second class constraints. A lot of work has been done in this field by several authors. The equivalence of the Faddeev-Jackiw approach to the Dirac's method is discussed in [2,3]. Its extension in supersymmetry is given in [4,5]. Application of the approach to the light cone quantum field theory is given in [6,7], and to hidden symmetries in [8]. Further elaboration and applications of the method in different cases is given in [9] . A formulation, intimately related to the FJ approach, for constructing unconstraint hamiltonians starting from general first order lagrangians is given in [10].\nIn a previous paper [11] we used the Faddeev-Jackiw method [1] to treat the two-flavour WZW model coupled to electromagnetism [12,13]. In this work we extend this analysis to the SU(3) case. The gauge invariant action under the electromagnetic group gauge transformations is given by\nwhere the first term is the action of the free electromagnetic field and the sum of the rest three terms constitute the action of the gauged Wess-Zumino-Witten model\nwhere U = exp (2iθ a λ a /f π ) , a = 1, ..., 8 is an element of SU(3)\nSee Appendix for notation.\nThe model describes succesfully the low energy dynamics of photon and the eight Goldstone bosons including their interactions related to the axial anomaly.\nAs in the two-flavour case we expand the U fields into series of powers of the Goldstone boson fields θ a . Then using the expression for Γ ef f (U, A µ ) given in (1) we obtain lagrangian densities containing terms initially up to two and afterwards up to three Goldstone boson fields. These lagrangian densities are constrained and we treat them using the Faddeev-Jackiw approach. The general case which involves any number of Goldstone bosons is still out of hand.\n2 Expanding up to second order in the Goldstone boson fields\nWe expand the U field into series of powers of the Goldstone boson fields\nand we substitute back into the expression (1) for the effective action. The resulting lagrangian density with terms up to second order in θ a is given by\nWe can easily check that the U EM (1) gauge invariance is not lost. In the two flavour case [11] the expression for the corresponding lagrangian density can be deduced from (3) by omitting those terms which include powers or derivatives of θ 4 up to θ 8 . We should also notice that the term Γ W ZW (U) of the effective action given in (2) which is zero in the two flavour case, contributes, in the three flavour case, terms involving five or more Goldstone boson fields. In the non-covariant notation, (3) can be written as follows\nwhere E = −Ȧ − ∇A 0 , B = ∇ × A are the electric and magnetic fields. We see that the dynamical variables in (4) which span the configuration space of the lagrangian density are the three components of the vector potential and the eight Goldstone boson fields. The scalar potential A 0 will be treated not as a dynamical variable but as a Lagrange multiplier since there is no time derivative of it in (4). Now according to Faddeev and Jackiw [1] we must construct out of (4), which is second order in time derivatives of the θ a fields, a new lagrangian density which is first order in time derivatives of all the dynamical variables. This can be accomplished by enlarging the configuration space so that it includes the whole phase space of the model. The enlarged configuration space coordinates will include the three vector potential components, the eight θ a fields and the corresponding canonical momenta. The canonical momenta conjugate to A are the three component of −E, and those conjugate to θ a are given by\nSo the resulting, first order in time derivatives, lagrangian density is given by\nThe expression (p aθa − E ·Ȧ)dt, called the canonical one-form, can be written up to a total time derivative as\nwhere\nand ω ij is the symplectic 22 × 22 matrix\nAs we mentioned before A 0 is the Lagrange multiplier. The expression ρ (2) − ∇ · E which is multiplied by A 0 is the only constraint in (5). The next step is to solve the equation of the constraint\nand incorporate the solution into the expression (5) for the lagrangian density. In order to do that we decompose the electric field E and the vector potential A into transverse and longitudinal components\nThen (7) implies that\nSubstituting into (5) the expression of E L given in (8) we get (apart from a total divergence)\nWe see that the canonical one-form in (9) has lost the standard form as given in (6) because of the term ρ (2) ∇ ∇ 2 ·Ȧ L dt. According to Darboux's theorem [1] we can perform coordinate transformations so that the canonical one-form gets back the standard form. This coordinate transformations called Darboux's transformations can be written as follows p 1 → p 1 cos α + p 2 sin α , θ 1 → θ 1 cos α + θ 2 sin α ,\nwhere α = e ∇ ∇ 2 ·A L . We substitute (10) into the expression (9) for the lagrangian density and we find out that all the terms which include powers or derivatives of A L cancel out exactly. Only terms with the physical transverse components of the vector potential survive, and the canonical one-form in (9) acquires the standard form. The resulting lagrangian density is given by\nwhere\nis the expression for the hamiltonian density of the model. We see that without mentioning gauge fixing the solution of the constraint lead to a Coulomb gauge hamiltonian density .\n( Note that B = ∇ × A T )."},{"title":"Keeping third order terms","content":"We now proceed with the expansion and keep terms up to the third order in the Goldstone boson fields. We obtain the following expression for the resulting U EM (1) gauge invariant effective lagrangian density\nwhere the expression for L (2) σ and L\nW ZW are given in (3) and\nWe see that only Γ W ZW (U, A µ ) ( the part of the Wess-Zumino-Witten action which describes the anomalous interaction of the Goldstone bosons with the photons ) gives extra contribution at this order. Next we derive the expressions for the canonical momenta p a conjugate to the Goldstone boson fields θ a . We substitute into (13) and we get the following expression for the effective lagrangian density in the enlarged configuration space\nwhere the expression for H (2) is given in (12) and\nThe scalar potential A 0 is again the Lagrange multiplier and the equation of the constraint is given by\nThe equation (16) has similar structure as in the previous case (7) so we proceed similarly. We decompose the electric field E and the vector potential A into transverse and longitudinal components. Then (16) implies\nNext we substitute the expression for E L given in (17) into (14) and we obtain a lagrangian density whose canonical one-form is given (apart from a total divergence) by\nThis expression must be diagonalized so that it acquires the standard form (6). In order to do this we proceed in two steps as in the SU(2) case [11]. First we perform the Darboux's transformations given in (10) which lead to partial diagonalization of (18). The new lagrangian density has a canonical one-form given by\nThe term ρ (3) ∇ ∇ 2 ·Ȧ L can be written as follows apart from a total time derivative\nNow we proceed with the second step. We perform the following Darboux's transformations\nThese transformations complete the diagonalization of the canonical one-form and we end up with a lagrangian density where the longitudinal part of the vector potential A L cancels out exactly\nThe expression for H (2) C is given in (12) and\nC is the expression for the Coulomb-gauge hamiltonian density. In terms of the usual pseudoscalar fields π ± = 1\n, π • = θ 3 , η 8 = θ 8 and the corresponding canonical momenta p i (i = π ± , π • , K ± , K • , K • , η 8 ) the expressions for H (2) C and H\nC are given by\nρ (2) = −ie(p π + π + − p π − π − + p K + K + − p K − K − ) ."},{"title":"Conclusion","content":"In this work we applied the Faddeev and Jackiw method for constrained systems to the U EM (1) gauged three flavour WZW model. We expanded the U-field into series of powers of the Goldstone boson fields and we limited ourselves to lagrangian densites with terms up to the third power in θ a . After treating these constrained langrangian densities using the FJ approach we ended up with unconstrained Coulomb-gauge hamiltonians, as in the case of QED [1]. The general case which involves any number of Goldstone boson fields is currently under investigation."},{"title":"Appendix","content":"Our metric is g µν = diag(1, −1, −1, −1) , Q = diag(2/3, −1/3, −1/3) is the charge matrix, D µ = ∂ µ + ieA µ [Q, ] denote the covariant derivative. λ a , a = 1, ..., 8 are the SU (3) generators tr (λ a λ b ) = 2δ ab . We choose e > 0 so that the electric charge of the electron is −e. We define ǫ 0123 = 1 ."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The Faddeev-Jackiv method [1] provides a simple and straightforward way to deal with constraint systems without having to distinguish between primary and secondary, first class and second class constraints. A lot of work has been done in this field by several authors. The equivalence of the Faddeev-Jackiw approach to the Dirac's method is discussed in [2,3]. Its extension in supersymmetry is given in [4,5]. Application of the approach to the light cone quantum field theory is given in [6,7], and to hidden symmetries in [8]. Further elaboration and applications of the method in different cases is given in [9] . A formulation, intimately related to the FJ approach, for constructing unconstraint hamiltonians starting from general first order lagrangians is given in [10].\\nIn a previous paper [11] we used the Faddeev-Jackiw method [1] to treat the two-flavour WZW model coupled to electromagnetism [12,13]. In this work we extend this analysis to the SU(3) case. The gauge invariant action under the electromagnetic group gauge transformations is given by\\nwhere the first term is the action of the free electromagnetic field and the sum of the rest three terms constitute the action of the gauged Wess-Zumino-Witten model\\nwhere U = exp (2iθ a λ a /f π ) , a = 1, ..., 8 is an element of SU(3)\\nSee Appendix for notation.\\nThe model describes succesfully the low energy dynamics of photon and the eight Goldstone bosons including their interactions related to the axial anomaly.\\nAs in the two-flavour case we expand the U fields into series of powers of the Goldstone boson fields θ a . Then using the expression for Γ ef f (U, A µ ) given in (1) we obtain lagrangian densities containing terms initially up to two and afterwards up to three Goldstone boson fields. These lagrangian densities are constrained and we treat them using the Faddeev-Jackiw approach. The general case which involves any number of Goldstone bosons is still out of hand.\\n2 Expanding up to second order in the Goldstone boson fields\\nWe expand the U field into series of powers of the Goldstone boson fields\\nand we substitute back into the expression (1) for the effective action. The resulting lagrangian density with terms up to second order in θ a is given by\\nWe can easily check that the U EM (1) gauge invariance is not lost. In the two flavour case [11] the expression for the corresponding lagrangian density can be deduced from (3) by omitting those terms which include powers or derivatives of θ 4 up to θ 8 . We should also notice that the term Γ W ZW (U) of the effective action given in (2) which is zero in the two flavour case, contributes, in the three flavour case, terms involving five or more Goldstone boson fields. In the non-covariant notation, (3) can be written as follows\\nwhere E = −Ȧ − ∇A 0 , B = ∇ × A are the electric and magnetic fields. We see that the dynamical variables in (4) which span the configuration space of the lagrangian density are the three components of the vector potential and the eight Goldstone boson fields. The scalar potential A 0 will be treated not as a dynamical variable but as a Lagrange multiplier since there is no time derivative of it in (4). Now according to Faddeev and Jackiw [1] we must construct out of (4), which is second order in time derivatives of the θ a fields, a new lagrangian density which is first order in time derivatives of all the dynamical variables. This can be accomplished by enlarging the configuration space so that it includes the whole phase space of the model. The enlarged configuration space coordinates will include the three vector potential components, the eight θ a fields and the corresponding canonical momenta. The canonical momenta conjugate to A are the three component of −E, and those conjugate to θ a are given by\\nSo the resulting, first order in time derivatives, lagrangian density is given by\\nThe expression (p aθa − E ·Ȧ)dt, called the canonical one-form, can be written up to a total time derivative as\\nwhere\\nand ω ij is the symplectic 22 × 22 matrix\\nAs we mentioned before A 0 is the Lagrange multiplier. The expression ρ (2) − ∇ · E which is multiplied by A 0 is the only constraint in (5). The next step is to solve the equation of the constraint\\nand incorporate the solution into the expression (5) for the lagrangian density. In order to do that we decompose the electric field E and the vector potential A into transverse and longitudinal components\\nThen (7) implies that\\nSubstituting into (5) the expression of E L given in (8) we get (apart from a total divergence)\\nWe see that the canonical one-form in (9) has lost the standard form as given in (6) because of the term ρ (2) ∇ ∇ 2 ·Ȧ L dt. According to Darboux's theorem [1] we can perform coordinate transformations so that the canonical one-form gets back the standard form. This coordinate transformations called Darboux's transformations can be written as follows p 1 → p 1 cos α + p 2 sin α , θ 1 → θ 1 cos α + θ 2 sin α ,\\nwhere α = e ∇ ∇ 2 ·A L . We substitute (10) into the expression (9) for the lagrangian density and we find out that all the terms which include powers or derivatives of A L cancel out exactly. Only terms with the physical transverse components of the vector potential survive, and the canonical one-form in (9) acquires the standard form. The resulting lagrangian density is given by\\nwhere\\nis the expression for the hamiltonian density of the model. We see that without mentioning gauge fixing the solution of the constraint lead to a Coulomb gauge hamiltonian density .\\n( Note that B = ∇ × A T ).\"\n },\n {\n \"title\": \"Keeping third order terms\",\n \"content\": \"We now proceed with the expansion and keep terms up to the third order in the Goldstone boson fields. We obtain the following expression for the resulting U EM (1) gauge invariant effective lagrangian density\\nwhere the expression for L (2) σ and L\\nW ZW are given in (3) and\\nWe see that only Γ W ZW (U, A µ ) ( the part of the Wess-Zumino-Witten action which describes the anomalous interaction of the Goldstone bosons with the photons ) gives extra contribution at this order. Next we derive the expressions for the canonical momenta p a conjugate to the Goldstone boson fields θ a . We substitute into (13) and we get the following expression for the effective lagrangian density in the enlarged configuration space\\nwhere the expression for H (2) is given in (12) and\\nThe scalar potential A 0 is again the Lagrange multiplier and the equation of the constraint is given by\\nThe equation (16) has similar structure as in the previous case (7) so we proceed similarly. We decompose the electric field E and the vector potential A into transverse and longitudinal components. Then (16) implies\\nNext we substitute the expression for E L given in (17) into (14) and we obtain a lagrangian density whose canonical one-form is given (apart from a total divergence) by\\nThis expression must be diagonalized so that it acquires the standard form (6). In order to do this we proceed in two steps as in the SU(2) case [11]. First we perform the Darboux's transformations given in (10) which lead to partial diagonalization of (18). The new lagrangian density has a canonical one-form given by\\nThe term ρ (3) ∇ ∇ 2 ·Ȧ L can be written as follows apart from a total time derivative\\nNow we proceed with the second step. We perform the following Darboux's transformations\\nThese transformations complete the diagonalization of the canonical one-form and we end up with a lagrangian density where the longitudinal part of the vector potential A L cancels out exactly\\nThe expression for H (2) C is given in (12) and\\nC is the expression for the Coulomb-gauge hamiltonian density. In terms of the usual pseudoscalar fields π ± = 1\\n, π • = θ 3 , η 8 = θ 8 and the corresponding canonical momenta p i (i = π ± , π • , K ± , K • , K • , η 8 ) the expressions for H (2) C and H\\nC are given by\\nρ (2) = −ie(p π + π + − p π − π − + p K + K + − p K − K − ) .\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"In this work we applied the Faddeev and Jackiw method for constrained systems to the U EM (1) gauged three flavour WZW model. We expanded the U-field into series of powers of the Goldstone boson fields and we limited ourselves to lagrangian densites with terms up to the third power in θ a . After treating these constrained langrangian densities using the FJ approach we ended up with unconstrained Coulomb-gauge hamiltonians, as in the case of QED [1]. The general case which involves any number of Goldstone boson fields is currently under investigation.\"\n },\n {\n \"title\": \"Appendix\",\n \"content\": \"Our metric is g µν = diag(1, −1, −1, −1) , Q = diag(2/3, −1/3, −1/3) is the charge matrix, D µ = ∂ µ + ieA µ [Q, ] denote the covariant derivative. λ a , a = 1, ..., 8 are the SU (3) generators tr (λ a λ b ) = 2δ ab . We choose e > 0 so that the electric charge of the electron is −e. We define ǫ 0123 = 1 .\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". L Faddeev, R Jackiw, Phys.Rev.Lett. 601692L.Faddeev, R.Jackiw, Phys.Rev.Lett. 60 (1988) 1692;","doi":""},{"text":"R Jackiw, hep-th/9306075Constraint) Quantization Without Tears. Report No. CTP-2215R.Jackiw, (Constraint) Quantiza- tion Without Tears, Report No. CTP-2215, hep-th/9306075.","doi":"hep-th/9306075"},{"text":". J Govaerts, Int. J. Mod. Phys. 53625J.Govaerts, Int. J. Mod. Phys. A5 (1990) 3625.","doi":""},{"text":". H Montani, Int.Mod.Phys. 84319H.Montani, Int.Mod.Phys. A8 (1993) 4319","doi":""},{"text":". D S Kulshreshtha, H J W Muller-Kirsten, Phys.Rev. 433376D.S.Kulshreshtha, H.J.W. Muller-Kirsten, Phys.Rev. D43 (1991) 3376.","doi":""},{"text":". J Barcelos-Neto, E S Cheb-Terrab, Int.J.Mod.Phys. 95563J.Barcelos-Neto, E.S.Cheb-Terrab, Int.J.Mod.Phys. A9 (1994) 5563;","doi":""},{"text":". Z.Phys. 54133Z.Phys. C54 (1992) 133.","doi":""},{"text":". J W Jun, C Jue, Phys. Rev. 502939J.W.Jun, C.Jue, Phys. Rev. D50 (1994) 2939.","doi":""},{"text":". O V Jacob, Phys. Rev. 505289O.V.Jacob, Phys. Rev. D50 (1994) 5289.","doi":""},{"text":"Faddeev-Jackiw approach to hidden symmetries. C Wotzasek, hep-th/9502042C.Wotzasek, Faddeev-Jackiw approach to hidden symmetries, UR-1401, hep-th/9502042.","doi":"hep-th/9502042"},{"text":". J Barcelos-Neto, C Wotzasek, Mod.Phys.Lett. 71737J.Barcelos-Neto and C.Wotzasek, Mod.Phys.Lett. A7 (1992) 1737;","doi":""},{"text":"D45 (1992) 1410; C.Wotzasek. M M Barreira, C Wotzasek, Mod.Phys.Lett. 82509Phys.Rev.M.M.Horta Barreira and C.Wotzasek, Phys.Rev. D45 (1992) 1410; C.Wotzasek,Mod.Phys.Lett. A8 (1993) 2509;","doi":""},{"text":". H Montani, C Wotzasek, Mod.Phys.Lett. 83387H.Montani and C.Wotzasek, Mod.Phys.Lett. A8 (1993) 3387;","doi":""},{"text":". J Barcelos-Neto, E Vasquez, Z.Phys. 61695J.Barcelos-Neto, E.Vasquez, Z.Phys. C61 (1994) 695;","doi":""},{"text":". J Barcelos-Neto, Phys.Rev. 491012J.Barcelos-Neto, Phys.Rev. D49 (1994) 1012.","doi":""},{"text":". C Cronstrom, M Noga, Nucl. Phys. 428449C.Cronstrom, M. Noga, Nucl. Phys. B428 (1994) 449;","doi":""},{"text":"Hamiltonian formulation of dynamical systems. C Cronstrom, NORDITA-94/51 PReportC.Cronstrom, Hamiltonian for- mulation of dynamical systems, Report No. NORDITA-94/51 P.","doi":"NORDITA-94/51 P"},{"text":"Application of the Faddeev-Jackiw formalism to the gauged WZW model. J E Paschalis, P I Porfyriadis, No. THES-TP 95/039503009ReportJ.E.Paschalis, P.I.Porfyriadis,Application of the Faddeev-Jackiw formalism to the gauged WZW model,Report No. THES-TP 95/03, hep-th 9503009.","doi":"No. THES-TP 95/03"},{"text":". E Witten, Nucl.Phys. 223422E.Witten, Nucl.Phys. B223 (1983) 422;","doi":""},{"text":". O Kaymakcalan, S Rajeev, J Schechter, Phys.Rev. 30594O.Kaymakcalan, S.Rajeev and J.Schechter, Phys.Rev. D30 (1984) 594;","doi":""},{"text":". N K Pak, P Rossi, Nucl.Phys. 250279N.K.Pak and P.Rossi, Nucl.Phys. B250 (1985) 279.","doi":""},{"text":"J F Donoghue, E Golowich, B R Holstein, Dynamics of the Standard Model, p.201. (Cambridge Monographs on Particle Physics. J.F.Donoghue, E.Golowich, B.R.Holstein, Dynamics of the Standard Model, p.201. (Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology).","doi":""}],"string":"[\n {\n \"text\": \". L Faddeev, R Jackiw, Phys.Rev.Lett. 601692L.Faddeev, R.Jackiw, Phys.Rev.Lett. 60 (1988) 1692;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R Jackiw, hep-th/9306075Constraint) Quantization Without Tears. Report No. CTP-2215R.Jackiw, (Constraint) Quantiza- tion Without Tears, Report No. CTP-2215, hep-th/9306075.\",\n \"doi\": \"hep-th/9306075\"\n },\n {\n \"text\": \". J Govaerts, Int. J. Mod. Phys. 53625J.Govaerts, Int. J. Mod. Phys. A5 (1990) 3625.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Montani, Int.Mod.Phys. 84319H.Montani, Int.Mod.Phys. A8 (1993) 4319\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D S Kulshreshtha, H J W Muller-Kirsten, Phys.Rev. 433376D.S.Kulshreshtha, H.J.W. Muller-Kirsten, Phys.Rev. D43 (1991) 3376.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Barcelos-Neto, E S Cheb-Terrab, Int.J.Mod.Phys. 95563J.Barcelos-Neto, E.S.Cheb-Terrab, Int.J.Mod.Phys. A9 (1994) 5563;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z.Phys. 54133Z.Phys. C54 (1992) 133.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J W Jun, C Jue, Phys. Rev. 502939J.W.Jun, C.Jue, Phys. Rev. D50 (1994) 2939.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O V Jacob, Phys. Rev. 505289O.V.Jacob, Phys. Rev. D50 (1994) 5289.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Faddeev-Jackiw approach to hidden symmetries. C Wotzasek, hep-th/9502042C.Wotzasek, Faddeev-Jackiw approach to hidden symmetries, UR-1401, hep-th/9502042.\",\n \"doi\": \"hep-th/9502042\"\n },\n {\n \"text\": \". J Barcelos-Neto, C Wotzasek, Mod.Phys.Lett. 71737J.Barcelos-Neto and C.Wotzasek, Mod.Phys.Lett. A7 (1992) 1737;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D45 (1992) 1410; C.Wotzasek. M M Barreira, C Wotzasek, Mod.Phys.Lett. 82509Phys.Rev.M.M.Horta Barreira and C.Wotzasek, Phys.Rev. D45 (1992) 1410; C.Wotzasek,Mod.Phys.Lett. A8 (1993) 2509;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Montani, C Wotzasek, Mod.Phys.Lett. 83387H.Montani and C.Wotzasek, Mod.Phys.Lett. A8 (1993) 3387;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Barcelos-Neto, E Vasquez, Z.Phys. 61695J.Barcelos-Neto, E.Vasquez, Z.Phys. C61 (1994) 695;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Barcelos-Neto, Phys.Rev. 491012J.Barcelos-Neto, Phys.Rev. D49 (1994) 1012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Cronstrom, M Noga, Nucl. Phys. 428449C.Cronstrom, M. Noga, Nucl. Phys. B428 (1994) 449;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hamiltonian formulation of dynamical systems. C Cronstrom, NORDITA-94/51 PReportC.Cronstrom, Hamiltonian for- mulation of dynamical systems, Report No. NORDITA-94/51 P.\",\n \"doi\": \"NORDITA-94/51 P\"\n },\n {\n \"text\": \"Application of the Faddeev-Jackiw formalism to the gauged WZW model. J E Paschalis, P I Porfyriadis, No. THES-TP 95/039503009ReportJ.E.Paschalis, P.I.Porfyriadis,Application of the Faddeev-Jackiw formalism to the gauged WZW model,Report No. THES-TP 95/03, hep-th 9503009.\",\n \"doi\": \"No. THES-TP 95/03\"\n },\n {\n \"text\": \". E Witten, Nucl.Phys. 223422E.Witten, Nucl.Phys. B223 (1983) 422;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Kaymakcalan, S Rajeev, J Schechter, Phys.Rev. 30594O.Kaymakcalan, S.Rajeev and J.Schechter, Phys.Rev. D30 (1984) 594;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N K Pak, P Rossi, Nucl.Phys. 250279N.K.Pak and P.Rossi, Nucl.Phys. B250 (1985) 279.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J F Donoghue, E Golowich, B R Holstein, Dynamics of the Standard Model, p.201. (Cambridge Monographs on Particle Physics. J.F.Donoghue, E.Golowich, B.R.Holstein, Dynamics of the Standard Model, p.201. (Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":38,"cells":{"text":{"kind":"string","value":"\n\nOctober 29, 2018\n\nMahendra Sinha \nDepartment of Physics\nPresidency College\n700 073CalcuttaIndia\n\nRoy \nDepartment of Physics\nPresidency College\n700 073CalcuttaIndia\nOctober 29, 2018arXiv:quant-ph/0210048v1 8 Oct 2002 Green's Function Approach to B. E. Condensation in the alkali atomsnumbers: 0375Fi0260Cb3280Pj Keywords : B E CondensateEf- fective InteractionGreen'Function\nA new scheme has been proposed to solve the B. E. condenstates in terms of Green's function approach. It has been shown that the radial wave function of two interacting atoms, moving in a common harmonic oscillator potential modified by an effctive interaction, satisfies an intregral equation whose kernel is separable. The solution of the integral equation can be written in terms of the harmonic oscillator wave functions. The ground state wave function of the system can be written in terms of these solutions.\n\nThe recent experimental realization of Bose-Einstein condensation [BEC] in ultra cold atomic gases has generated intensive theoretical and practical interest regarding the properties of Bose gas. Non-linear Schrodinger equation in this context known as Gross-Pitaevskii [ G-P ] equation [1],has been used to study the properties of the interacting spinless Bosons , trapped in harmonic potential . Altough the G-P equation is widely accepeted as a workable model for description of the B E condenstates at low temperature, the actual dynamics of B E condenstates is not fully captured by the G-P equation because of the non-integrability of the G-P equation and explicit solution is not known [2]. It is to be noted that for the positive scattering length, the G-P equation is generally solved numerically only under cylindrically symmetric systems. The Thomas Fermi approximation [3] has been introduced to get some analytical solution for the description of B E condenstates. On the other hand for the negative scattering length the G-P equation is yet to be properly investigated [3]properly. Another alternative approach to study the dynamics of the B.E.condenstates is the time dependent variation technique [4] in which one assumes fixed profile of the condenstate and computes the evolution of some parameters such as the width by variational techniques. It should also be mentioned that the negative scattering length provides stable solution of the G-P equation only under certain conditions for the number of particles and the size of the trap [5]. When those conditions are not fulfiled the condenstates become unstable. Therefore realization of large condenstates will provide reliable experimental data for the precise understanding of the condensation process [6]. In this paper we shall present a new solution for the condenstate on the basis of Green's function approach of scattering theory. Similar method was used in the treatment of the nuclear many body problems by Bethe and Goldstone; Brueckner and Gammel [7] for the understanding of the various aspects nuclear many body problem. We have derived an integral equation for the description of two interacting bosons in the presence of many bosons. We shall consider a system of N spinless interacting atoms all having the same hyperfine species, in equilibrium and at very low temperature and densities characteristic of alkali atoms under Bose-Einstein condensation (BEC) condition. We assume that each of the atoms is moving in the influence of a common harmonic oscillator potential which is called a trapping potential(V ext ) and an effective two-body interaction acting between them which modifies their behaviour . The essential questions we wish to investigate are : How does the effective two body interaction modify the two-body wave function?. How does the energies of the two interacting bosons system change when the two-body effective interaction is switched on ? It is welknown that the scattering length (a s ) is very small compared to the interatomic distant(r int ) [8] under BEC in the dilute atomic alkali gases. Therefore the behaviour of the atoms is extremely sensitive to the details of the trap and the effective two-body interaction. Since the effective two-body interaction is very weak the quantitative calculation based on perturbation theory in the interatomic interaction is hoped to be highly reliable for the alkali gases. Furthermore the de Broglie wave length associated with individual atom is comparable to interatomic r int distance therefore the correlation between atoms becomes important [7]. Hence the construction of the many -body condenstate state wave function of the system in terms of two-body wave function will be interesting. The standard many-body wave function in terms of Hartree-Fock ansatz using the single particle solution of the G-P equation does not take into accoutn the effect of correlation between atoms which is very important in the case of BEC alkali gases.In this letter we have presented an analytical formalism to investigate the above aspects of BEC problem without any numerical estimate for a particular choice of the effective interaction.That will be communicated shortly.\n\nLet us consider a system of interacting N spinless atoms and now arbitrarily pick out any two interacting spinless atoms, say 1 and 2, denoting their relative and center-of-mass co-ordinates, by r and R respectively . We shall investigate the state (φ) of these two interacting atoms trapped in a common harmonic potential in the presence (N − 2) interacting atoms. The most general states of the many-body system will then be described by a wave function Ψ( r, R, χ, t) where χ represents schemitically the co-ordinates of the remaining (N − 2) particles. It is the dependence on r that is crucial but the parameter dependence of Ψ on R, χ and t will be ignored . In the present context we assume that simple BEC is realised in the two-particle state φ( r) whose equatiom of motion will be determined using Schrodinger equation. From the knowledge of the wave-function of two-atom system φ( r), the many-body ground state of B E condenstate system of alkali atoms can be written. The hamiltonian for the two-atom system in the common harmonic oscillator potential of frequency ω is given by\nH 0 = p 2 1 2m + p 2 2 2m + mω 2 r 2 1 2 + mω 2 r 2 2 2 .(1)\nIntroducing the cannonical co-ordinate transform\nr = r 2 − r 1 , p = ( p 2 − p 1 ) 2 ,(2)\nand\nR = ( r 2 + r 1 ) 2 , P = ( p 2 + p 1 ).(3)\nThe hamiltonian is transformed to\nH 0 = (P 2 + M 2 ω 2 R 2 ) 2M + (p 2 + µ 2 ω 2 r 2 ) 2µ ,(4)\nwhere M=2m,µ = m 2 . The Schrodinger equation for the system withh = 1 is\n[ (−∇ 2 R + M 2 ω 2 R 2 ) 2M + (−∇ 2 r + µ 2 ω 2 r 2 ) 2µ ]φ( r)φ( R) = E 0 φ( r)φ( R). (5)\nThe Schrodinger equation for the relative motion is then\nH 0 φ( r) = (− ∇ 2 2µ + kr 2 )φ(r) = E 0 φ( r),(6)\nwhere\nk = µω 2 2 = mω 2 4 .(7)\nThe solutions are given by\nφ m nl ( r) = R nl (r) r Y lm (r),(8)\nwhere Y lm (r) are the usual spherical harmonics. The radial wave function R nl (r) is given by\nR nl (r) = N nl exp(− νr 2 2 )r l+1 v nl (r),(9)\nwhere ν = ωμ h and v nl (r) is the associated Laguerre polynomial:\n\nv nl (r) = L \n\nN nl is determind by the normalization condition and E 0 nl =hω(2n + l + 3 2 ). We shall now treat the effective spin independent two-body interaction v(r) which is very weak between two atoms , as a peturbation and solve the Schrodinger equation using the Green's function method to determine the perturbed wave function for the two-atom system. The unperturbed wave function is given by\nφ nl ( r) = R nl (r) r Y lm (r).(11)\nSimilarly the perturbed wave function can be written as\nψ n 1 l 1 (r) = u n 1 l 1 r Y l 1 m 1 (r).(12)\nThe Schrodinger equation for the perturbed system is\n[H 0 + v(r)] u n 1 l 1 r Y m 1 l 1 (r) = E n 1 l 1 u n 1 l 1 r Y m 1 l 1 (r),(13)\nwhere E n 1 l 1 is the perturbed energy eigen value. We now write\nH 0 R nl (r) r Y lm (r) = 1 r [− 1 m ∂ 2 ∂r 2 + V l (r)]R nl (r)Y nl (r) = E 0 nl R nl (r) r Y nl (r). (14)\nMultiplying the perturbed equation(13) by Y * n 1 l 1 (r) on the left and integrating over the solid angle dr we obtain\n[E n 1 l 1 − (− 1 m d 2 dr 2 + V l 1 (r)]u n 1 l 1 (r) = v l 1 (r)u n 1 l 1 (r),(15)\nwhere\nv l 1 (r) = d rY * l 1 m 1 (r)v(r)Y l 1 m 1 (r)(16)\nIn order to solve this equation we introduce Green's G n 1 l ′ 1 ( r, r ′ ) corresponding to equ.(15) satisfies the equation\n(E n 1 l 1 − H 0 )G n 1 l 1 ( r, r ′ ) = δ( r − r').(17)\nMultiplying through on the left by Y * l 1 m 1 (r) and integrate over the solid angle dr we obtain\n[E n l l 1 − (− 1 m d 2 dr 2 ) + V l 1 (r)]g n 1 l 1 (r, r ′ ) = δ(r − r ′ ),(18)\nwhere\ng n 1 l 1 (r, r ′ ) = n 2 R n 2 l 1 (r)R n 2 l 1 (r ′ ) E n 1 l 1 ,n 2 l 1 ,(19)\nwhere\nE n 1 l 1 ,n 2 l 2 = (E n 1 l 1 − E 0 n 2 l 1 ).(20)\nHence the perturbed wave function u n 1 l 1 (r) satisfies an integral equation with a separable kernel. Therefore we write\nu n 1 l 1 (r) = ∞ 0 dr ′ g n 1 l 1 (r, r ′ )v l 1 (r ′ )u n 1 l 1 (r ′ ).(21)\nFinally we write\nu n 1 l 1 (r) = ∞ n 2 =0 R n 2 l 1 (r) E n 1 l 1 , n 2 l 1 ∞ 0 dr ′ R n 2 l 1 (r ′ )v l 1 (r ′ )u n 1 l 1 (r ′ ).(22)\nThe equation for R nl 1 (r) is\n[E 0 nl 1 − (− 1 m d 2 dr 2 + V l1 (r))]R nl 1 (r) = o.(23)\nMultiplying equ.(15)and equ.(23) on the left by R nl 1 (r) and u n 1 l 1 respectively subtracting and integrating over r ′ we obtain\nE n 1 l 1 − E 0 nl 1 = ∞ 0 drR nl 1 (r)v l (r)u n 1 l 1 (r) ∞ 0 drR nl 1 (r)u n 1 l 1 (r)(24)\nand the perturbed wavefunction u n 1 l 1 (r) in a closed form can be written as\nu n 1 l 1 (r) = ∞ n 2 =0 K n 1 l 1 ,n 2 l 1 R n 2 l 1 (r),(25)\nwhere K n 1 l 1 ,n 2 l 1 = 1 E n 1 l 1 ,n 2 ,l 1 ∞ 0 drR n 2 l 1 (r)v l 1 (r)u n 1 l 1 (r).\n\n. To this expression of u n 1 l 1 we should add the solution of (E n 1 l 1 −H 0 )χ n 1 l 1 = 0. Since E n 1 l 1 = E 0 nl 1 the equation (E n 1 ,l 1 −H 0 )χ n 1 l 1 = 0 has no solution.Thus it is not necessary to add a particular solution to the u n 1 l 1 . From equations(25) and (26)it follows that\nK n 1 l 1 ,n 2 l 1 = 1 E n 1 l 1 ,n 2 l 1 ∞ n 3 =0 K n 1 l 1 ,n 3 l 1 ∞ 0 drR n 2 l 1 (r)v l 1 (r)R n 3 l 1 (r).(27)\n.\n\n\nResults and Discussion\n\nThe main conclusions of the paper are the following: It should be noted that n 3 can in principle run from o to ∞, the above equ. (27) reprsents an infinite set of simultaneous homogeneous algebraic equations in K n 1 l 1 ,n 2 l 1 . The consistency of the these equations imposes the condition of vanishing of the determinant formed from the coefficients of K n 1 l 1 ,n 2 l 1 . This condition determines the energy shift E n 1 l 1 ,nl 1 of the two interacting atoms. The corresponding eigenvectors when substituted in eq.(26) determine the perturbed radial function u n 1 l 1 (r). Since B E condensation takes place strictly in the ground state of the system therefore a few values of n 3 in the sum will provide a very reliable estimate of the two-particle wave function. Because of the separable structure of eq.(21) we can also include various types of central and non central of two-body effective interaction and the variation of the scattering length can also be incorporated to investigate sensitivity of the two-body condenstate wave function and the stability of the Bose-Einstein condensation in alkali atoms on the effective two-body interaction which can not be so easily taken care in the formalism of G-P equation. When the effective two-body interaction becomes non-central in nature then the radial equation for u n 1 l 1 (r) i.e eq.(21) will satisfy a coupled integral equation. When the effective two-body interaction is a pseudo-potential type, as commonly used in the calculation of G-P equation, the determination of the two-body wave function u n 1 l 1 (r) becomes extermely simple in this formalism. We can also investigate the importance two-particle correlation in BEC. Finally the construction of the condenstate wave function describing the BEC of the in terms of the calculated two-particle wave function to study various aspects of BEC becomes possible. These are the main conclusions of the paper.\n\n\nReferences\n \n. L P Pitaevskii, Sov,Phys. 13451L.P.Pitaevskii, Sov,Phys. JEPT13, 451(1961).\n\n. P Gross, Nuovo Cimento. 20454P.Gross, Nuovo Cimento 20, 454(1961).\n\n. F Dalfovo, L Pitaevskii, S Stringari, Rev.Mod.Phys. 71463F.Dalfovo, L.P Pitaevskii, and S.Stringari, Rev.Mod.Phys.71, 463(1999).\n\n. F Dalfovo, S Stringari, Phys. Rev.A53. 2477F.Dalfovo and S.Stringari, Phys. Rev.A53, 2477(1996).\n\n. P A Rupecht, M Edwards, K Burnett, C W Clark, Phys.Rev. 544178P.A. Rupecht, M.Edwards, K.Burnett,and C.W.Clark, Phys.Rev.A54, 4178(1996).\n\n. M Holland, J Cooper, Phys.Rev. 531954M.Holland and J.Cooper, Phys.Rev.A53, R1954(1996).\n\n. M Edwards, R J Dodd, C W Clark, P A Rupecht, K Burnett, Phys.Rev. 531950M.Edwards, R.J.Dodd, C.W.Clark ,P.A.Rupecht and K.Burnett, Phys.Rev.A53 R1950(1996).\n\n. V M Perez-Garcia, H Michinel, J I Cirac, M Lewenstein, P Zoller, Phys.Rev. V.M.Perez-Garcia, H.Michinel, J.I.Cirac, M.Lewenstein and P.Zoller, Phys.Rev.A56 1424(1997).\n\n. G Baym, C J Pethick, Phys.Rev.Lett. 766G.Baym and C.J.Pethick, Phys.Rev.Lett.76 6,1996).\n\n. E V Shuryak, Phys.Rev.A54. 3151E.V.Shuryak, Phys.Rev.A54 3151(1996).\n\n. H A Bethe, J Goldstone, Proc.Roy.Soc. 19856H.A. Bethe and J Goldstone, Proc.Roy.Soc.(London)A198, 56(1958).\n\n. K A Brueckner, J L Gammel, Phys. Rev. 1091023K.A. Brueckner and J.L.Gammel, Phys. Rev.109, 1023(1958).\n\n. A J Leggett, Rev.Mod.Phys. 731A.J. Leggett, Rev.Mod.Phys.73, 1(2001).\n"},"corpus_id":{"kind":"number","value":14831625,"string":"14,831,625"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"quant-ph/0210048\",\n \"mag\": 1664908431,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/quant-ph/0210048v1.pdf\"\n ],\n \"pdf_sha\": \"00d04d1ff7c4a8cce1c17c8fa70549e66984a422\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":""},"authors":{"kind":"list like","value":["Mahendra Sinha \nDepartment of Physics\nPresidency College\n700 073CalcuttaIndia","Roy \nDepartment of Physics\nPresidency College\n700 073CalcuttaIndia"],"string":"[\n \"Mahendra Sinha \\nDepartment of Physics\\nPresidency College\\n700 073CalcuttaIndia\",\n \"Roy \\nDepartment of Physics\\nPresidency College\\n700 073CalcuttaIndia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"A new scheme has been proposed to solve the B. E. condenstates in terms of Green's function approach. It has been shown that the radial wave function of two interacting atoms, moving in a common harmonic oscillator potential modified by an effctive interaction, satisfies an intregral equation whose kernel is separable. The solution of the integral equation can be written in terms of the harmonic oscillator wave functions. The ground state wave function of the system can be written in terms of these solutions."},"sections":{"kind":"list like","value":[{"title":"Results and Discussion","content":"The main conclusions of the paper are the following: It should be noted that n 3 can in principle run from o to ∞, the above equ. (27) reprsents an infinite set of simultaneous homogeneous algebraic equations in K n 1 l 1 ,n 2 l 1 . The consistency of the these equations imposes the condition of vanishing of the determinant formed from the coefficients of K n 1 l 1 ,n 2 l 1 . This condition determines the energy shift E n 1 l 1 ,nl 1 of the two interacting atoms. The corresponding eigenvectors when substituted in eq.(26) determine the perturbed radial function u n 1 l 1 (r). Since B E condensation takes place strictly in the ground state of the system therefore a few values of n 3 in the sum will provide a very reliable estimate of the two-particle wave function. Because of the separable structure of eq.(21) we can also include various types of central and non central of two-body effective interaction and the variation of the scattering length can also be incorporated to investigate sensitivity of the two-body condenstate wave function and the stability of the Bose-Einstein condensation in alkali atoms on the effective two-body interaction which can not be so easily taken care in the formalism of G-P equation. When the effective two-body interaction becomes non-central in nature then the radial equation for u n 1 l 1 (r) i.e eq.(21) will satisfy a coupled integral equation. When the effective two-body interaction is a pseudo-potential type, as commonly used in the calculation of G-P equation, the determination of the two-body wave function u n 1 l 1 (r) becomes extermely simple in this formalism. We can also investigate the importance two-particle correlation in BEC. Finally the construction of the condenstate wave function describing the BEC of the in terms of the calculated two-particle wave function to study various aspects of BEC becomes possible. These are the main conclusions of the paper."},{"title":"References","content":""}],"string":"[\n {\n \"title\": \"Results and Discussion\",\n \"content\": \"The main conclusions of the paper are the following: It should be noted that n 3 can in principle run from o to ∞, the above equ. (27) reprsents an infinite set of simultaneous homogeneous algebraic equations in K n 1 l 1 ,n 2 l 1 . The consistency of the these equations imposes the condition of vanishing of the determinant formed from the coefficients of K n 1 l 1 ,n 2 l 1 . This condition determines the energy shift E n 1 l 1 ,nl 1 of the two interacting atoms. The corresponding eigenvectors when substituted in eq.(26) determine the perturbed radial function u n 1 l 1 (r). Since B E condensation takes place strictly in the ground state of the system therefore a few values of n 3 in the sum will provide a very reliable estimate of the two-particle wave function. Because of the separable structure of eq.(21) we can also include various types of central and non central of two-body effective interaction and the variation of the scattering length can also be incorporated to investigate sensitivity of the two-body condenstate wave function and the stability of the Bose-Einstein condensation in alkali atoms on the effective two-body interaction which can not be so easily taken care in the formalism of G-P equation. When the effective two-body interaction becomes non-central in nature then the radial equation for u n 1 l 1 (r) i.e eq.(21) will satisfy a coupled integral equation. When the effective two-body interaction is a pseudo-potential type, as commonly used in the calculation of G-P equation, the determination of the two-body wave function u n 1 l 1 (r) becomes extermely simple in this formalism. We can also investigate the importance two-particle correlation in BEC. Finally the construction of the condenstate wave function describing the BEC of the in terms of the calculated two-particle wave function to study various aspects of BEC becomes possible. These are the main conclusions of the paper.\"\n },\n {\n \"title\": \"References\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". L P Pitaevskii, Sov,Phys. 13451L.P.Pitaevskii, Sov,Phys. JEPT13, 451(1961).","doi":""},{"text":". P Gross, Nuovo Cimento. 20454P.Gross, Nuovo Cimento 20, 454(1961).","doi":""},{"text":". F Dalfovo, L Pitaevskii, S Stringari, Rev.Mod.Phys. 71463F.Dalfovo, L.P Pitaevskii, and S.Stringari, Rev.Mod.Phys.71, 463(1999).","doi":""},{"text":". F Dalfovo, S Stringari, Phys. Rev.A53. 2477F.Dalfovo and S.Stringari, Phys. Rev.A53, 2477(1996).","doi":""},{"text":". P A Rupecht, M Edwards, K Burnett, C W Clark, Phys.Rev. 544178P.A. Rupecht, M.Edwards, K.Burnett,and C.W.Clark, Phys.Rev.A54, 4178(1996).","doi":""},{"text":". M Holland, J Cooper, Phys.Rev. 531954M.Holland and J.Cooper, Phys.Rev.A53, R1954(1996).","doi":""},{"text":". M Edwards, R J Dodd, C W Clark, P A Rupecht, K Burnett, Phys.Rev. 531950M.Edwards, R.J.Dodd, C.W.Clark ,P.A.Rupecht and K.Burnett, Phys.Rev.A53 R1950(1996).","doi":""},{"text":". V M Perez-Garcia, H Michinel, J I Cirac, M Lewenstein, P Zoller, Phys.Rev. V.M.Perez-Garcia, H.Michinel, J.I.Cirac, M.Lewenstein and P.Zoller, Phys.Rev.A56 1424(1997).","doi":""},{"text":". G Baym, C J Pethick, Phys.Rev.Lett. 766G.Baym and C.J.Pethick, Phys.Rev.Lett.76 6,1996).","doi":""},{"text":". E V Shuryak, Phys.Rev.A54. 3151E.V.Shuryak, Phys.Rev.A54 3151(1996).","doi":""},{"text":". H A Bethe, J Goldstone, Proc.Roy.Soc. 19856H.A. Bethe and J Goldstone, Proc.Roy.Soc.(London)A198, 56(1958).","doi":""},{"text":". K A Brueckner, J L Gammel, Phys. Rev. 1091023K.A. Brueckner and J.L.Gammel, Phys. Rev.109, 1023(1958).","doi":""},{"text":". A J Leggett, Rev.Mod.Phys. 731A.J. Leggett, Rev.Mod.Phys.73, 1(2001).","doi":""}],"string":"[\n {\n \"text\": \". L P Pitaevskii, Sov,Phys. 13451L.P.Pitaevskii, Sov,Phys. JEPT13, 451(1961).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Gross, Nuovo Cimento. 20454P.Gross, Nuovo Cimento 20, 454(1961).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Dalfovo, L Pitaevskii, S Stringari, Rev.Mod.Phys. 71463F.Dalfovo, L.P Pitaevskii, and S.Stringari, Rev.Mod.Phys.71, 463(1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Dalfovo, S Stringari, Phys. Rev.A53. 2477F.Dalfovo and S.Stringari, Phys. Rev.A53, 2477(1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P A Rupecht, M Edwards, K Burnett, C W Clark, Phys.Rev. 544178P.A. Rupecht, M.Edwards, K.Burnett,and C.W.Clark, Phys.Rev.A54, 4178(1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Holland, J Cooper, Phys.Rev. 531954M.Holland and J.Cooper, Phys.Rev.A53, R1954(1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Edwards, R J Dodd, C W Clark, P A Rupecht, K Burnett, Phys.Rev. 531950M.Edwards, R.J.Dodd, C.W.Clark ,P.A.Rupecht and K.Burnett, Phys.Rev.A53 R1950(1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V M Perez-Garcia, H Michinel, J I Cirac, M Lewenstein, P Zoller, Phys.Rev. V.M.Perez-Garcia, H.Michinel, J.I.Cirac, M.Lewenstein and P.Zoller, Phys.Rev.A56 1424(1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Baym, C J Pethick, Phys.Rev.Lett. 766G.Baym and C.J.Pethick, Phys.Rev.Lett.76 6,1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E V Shuryak, Phys.Rev.A54. 3151E.V.Shuryak, Phys.Rev.A54 3151(1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H A Bethe, J Goldstone, Proc.Roy.Soc. 19856H.A. Bethe and J Goldstone, Proc.Roy.Soc.(London)A198, 56(1958).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K A Brueckner, J L Gammel, Phys. Rev. 1091023K.A. Brueckner and J.L.Gammel, Phys. Rev.109, 1023(1958).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A J Leggett, Rev.Mod.Phys. 731A.J. Leggett, Rev.Mod.Phys.73, 1(2001).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":39,"cells":{"text":{"kind":"string","value":"\nFaster Fuzzing: Reinitialization with Deep Neural Models\n8 Nov 2017\n\nNicole Nichols nicole.nichols@pnnl.govmark.raugas@pnnl.govrobert.jasper@pnnl.govnathan.hilliard@pnnl.gov \nPacific Northwest National Laboratory\n1100 Dexter Avenue, Suite 50098109SeattleWA\n\nMark Raugas \nPacific Northwest National Laboratory\n1100 Dexter Avenue, Suite 50098109SeattleWA\n\nRobert Jasper \nPacific Northwest National Laboratory\n1100 Dexter Avenue, Suite 50098109SeattleWA\n\nNathan Hilliard \nPacific Northwest National Laboratory\n1100 Dexter Avenue, Suite 50098109SeattleWA\n\nFaster Fuzzing: Reinitialization with Deep Neural Models\n8 Nov 2017\nWe improve the performance of the American Fuzzy Lop (AFL) fuzz testing framework by using Generative Adversarial Network (GAN) models to reinitialize he system with novel seed files. We assess performance based on the temporal rate at which we produce novel and unseen code paths. We compare this approach to seed file generation from a random draw of bytes observed in the training seed files. The code path lengths and variations were not sufficiently diverse to fully replace AFL input generation. However, augmenting native AFL with these additional code paths demonstrated improvements over AFL alone. Specifically, experiments showed the GAN was faster and more effective than the LSTM and outperformed a random augmentation strategy, as measured by the number of unique code paths discovered. GAN helps AFL discover 14.23% more code paths than the random strategy in the same amount of CPU time, finds 6.16% more unique code paths, and finds paths that are on average 13.84% longer. Using GAN shows promise as a reinitialization strategy for AFL to help the fuzzer exercise deep paths in software.\n\nIntroduction\n\nIdentifying software defects through testing is a challenging problem. Over the years, a number of approaches have been developed to test software, including random mutation testing (black box fuzzing) Doupé u. a. (2012) ;Woo u. a. (2013), abstract interpretation (of either source or machine code) Cousot und Cousot (1977) ;Cadar u. a. (2008) ;Ma u. a. (2011), and property based testing Arts u. a. (2006); Claessen und Hughes (2011).\n\nMethods such as symbolic and concolic execution have increased the fidelity of analyses run over programs Schwartz u. a. (2010). The development of Satisfiability Modulo Theories (SMT) solvers such as Z3, Boolector, and others have allowed for powerful programmatic analysis of reasoning about software De Moura und Bjørner (2008); Brummayer und Biere (2009). Separation logic has allowed for analyses to be applied to complicated data structures Reynolds (2002) ;Dongol u. a. (2015).\n\nAmerican Fuzzy Lop (AFL) is an advanced fuzzing framework that has been used to discover a number of novel software vulnerabilities (https://github.com/mrash/afl-cve). AFL uses random mutations of byte strings to identify unique code paths and discover defects in target programs. The inputs that successfully generated unique code paths are then documented as \"seed files\". We propose to use these native seed files as training data for deep generative models to create augmented seed files. Our proposed reinitialization methods are a scalable process that can improve the time to discovery of software defects.\n\nOther researchers have used machine learning to augment fuzzing frameworks including: Godefroid u. a. (2017), Wang u. a. (2017). To identify deeper bugs and code paths, Steelix Li u. a. (2017) uses a program-state based binary fuzzing approach and Driller Stephens u. a. (2016) demonstrates a hybrid approach using fuzzing and selective concolic execution. AFLFAST Böhme u. a. (2016) extends AFL using Markov chain models. Deep Neural Networks (DNNs) Bengio u. a. (2015), have had great success in the fields of Natural Language Processing (NLP) Jones (2014); Wu u. a. (2016), Computer Vision Krizhevsky u. a. (2012, and the playing of bounded games such as Go Mnih u. a. (2013) or video games like ATARI Silver u. a. (2016). Can these DNN's help existing program analysis tools perform better? In our work we investigate that question. We augment AFL Zalewski (2015), an advanced fuzzing framework, using Generative Adversarial Networks (GAN) Goodfellow u. a. (2014) and Long Short Term Memory (LSTM) Sak u. a. (2014) to increase its rate of unique code path discovery.\n\nOur work quantifies the benefits that augmentation strategies such as generative models can provide, even when limited by small quantities of training data. By periodically perturbing the state of AFL as it explores the input space, we are able to improve its performance, as measured by unique code paths. Specifically, we test our approach on the software ecosystem surrounding the Ethereum Wood (2014) project. As a financial system, correctness of the Ethereum code base is important for guaranteeing that transactions or calculations run without fault. We choose ethkey as an initial fuzzing target. Ethkey is a small C++ program provided as part of the cpp-ethereum project used to load, parse, and perform maintenance on Ethereum wallets, and importantly, takes a simple input file, making it easy to test with AFL.\n\n\nExperimental Design\n\nFirst, we describe the basic functionality of AFL, highlighting the key features that connect with the proposed augmentation framework. Next, we describe the methodology used to create the LSTM and GAN generated seed files. As a baseline, we also consider random generation of seed files from the training data used to construct the LSTM and GAN models. AFL has extensions to the GCC compiler which in conjunction with Genetic Algorithms, it uses to create seed files. Each seed file documents the input that yielded a unique code path, the time of discovery, and is used as the basis for mutation (or fuzzing) to generate future seed files. Our augmentation strategy takes advantage of the fact that if an external tool places additional seed files in the AFL working directory, AFL will use those files as inputs in subsequent fuzzing runs.\n\nTo produce the training data for our methods, we run AFL on a target program P for a fixed amount of time. AFL generates an initial set of seed files S = {S 0 , ..., S K } for each unique execution trace τ taken through P . We use S as training examples for the LSTM and GAN models, which are both trained using Keras Chollet (2017).\n\nOur LSTM is trained from the concatenation of AFL-generated seed file corpus S into a single file and generates new seed files with a maximum length of 40 characters. The LSTM model has a 128 wide initial layer, an internal dense layer, and a final softmax activation layer. To train the LSTM model, we use RMS propagation as our optimizer and a categorical cross-entropy loss function. The model takes in a seed sequence sampled from the training corpus and predicts the next character in the sequence. We additionally tune a separate temperature parameter to diversify the output seed files from the network. The generated seed files are noted as S L .\n\nIn our GAN architecture, two models are built, a generator G, which is pitted against a discriminator D. G is optimized to generate realistic output, and the discriminator D has the task of predicting if the data is true or fake. This training strategy is unsupervised and particularly expressive. The generative model G, is a fully connected 2 layer DNN with a ReLU non-linearity as the inner activation and a tanh output activation. It is trained with a binary cross-entropy loss function via stochastic gradient descent. The discriminative model D is a 3 layer DNN, but the first layer has 25% dropout followed by two fully connected layers. It uses an Adam optimizer for stochastic gradient descent and the seed files resulting from the GAN process are noted as S G . Table 1: Initial Run: |C| is the number of seeds generated after reinitializing AFL. We observe GAN or LSTM allows discovery of novel code paths at a quicker rate than restarting AFL using a random sampling of bytes. L(C) is the number of unique code path lengths l(c i ) associated to input files c i in the set C of LSTM, GAN, or uniform Random reinitialization. Table 2: Synthetic Seed Files: Random sampling, LSTM, and GAN can be used to produce synthetic seed files for AFL. We compute statistics on the synthetic files from the 3 strategies. The synthetic files are not themselves deep or varied code paths, but can be used to reinitilize AFL.\nClass C |C| L(C) % Unique µ(L(C)) σ(L(C)) AFL\nAdditionally, given the native AFL seed files, S, we randomly draw bytes from this training set and produce new, random seed files S R of the same length as S G . This serves as a baseline to determine if the added time and complexity of GAN and LSTM based seed generation are truly providing an advantage over a simple strategy of randomly perturbing AFL's state.\n\nSmall Experiment: The seed files (S R , S G , and S L ) alone are not an end goal. However, we are interested in characterizing their variability and other properties as they will provide a set of initial conditions when AFL is restarted. In a fuzzing run on a single CPU core, we produce 936 unique code paths used to train initial GAN and LSTM models. Random seed generation is performed by drawing random bytes from /dev/urandom. For each method, we generate 200 samples, reinitialize AFL on a single CPU with only the seed files of one method and run for an additional 72 hours to measure the impact on code path discovery. Both LSTM and GAN models sightly out-perform random sampling for AFL reinitialization. We summarize the resulting mean time to discover new code paths in Table 1.\n\nEach seed file produces a program trace when supplied as an input to ethkey. Code paths that have different lengths will differ in at least one basic block or branch instruction. The unique code path length is fast to compute but only provides a lower bound on the number of unique code paths exercised by the testing framework, across fuzzing runs using AFL. Two code paths with the same length can result from unique traces, thus detailed evaluation is needed to determine the true uniqueness of a code path from seed file execution.\n\nLarge Experiment: To demonstrate the scalability of this augmentation strategy, we performed an extended run of AFL on 200 CPU cores for 72 hours. Each core in the AFL run stopped finding seed files after the first 10 to 12 hours of fuzzing and accumulated a total of 39,185 seed files across 49 workers. All seed files produced within a given node are known to be unique, due to AFL's internal book keeping mechanism. However, seed files whose content are different across nodes, could in principle exercise the same code path. By measuring the length of each program trace (code path), we can compute a lower bound on the number of unique paths discovered by only counting paths that have a unique length. After removing identical seed files from across the nodes, and seed files that resulted in the same code path length, we estimate 802 of the initial files were duplicates from the independent worker nodes. Removing those duplicates resulted in a total of 38,384 unique files.\n\nWe then trained GAN and LSTM networks on the total corpus of unique seed files and generated approximately 20,000 samples from each method, respectively, to use as synthetic seed files in order to reinitialize AFL. GAN took approximately 30 minutes to train and generate synthetic seed files, while LSTM took 14 hours to do so.\n\nClass C |C| L(C) Novel L(C) Rate Novel Rate µ(L(C)) σ(L(C)) Rand Table 3: Sustained Run: With 38,000 training seed file samples, we compare the seed files generated after reinitialization from random, GAN, and LSTM generation methods. L(C) Rate is the speedup over the random strategy of discovery of code paths with unique length per second. Novel Rate is the speedup over random for unique lengths not found in the training set.\n\nIn Table 2 we summarize the mean and variance of the length of program traces (i.e., code paths) associated with the seed files from native AFL and from the synthetic generation methods for this larger experiment. The synthetic seed files, when provided as inputs to the program under test, do not cause deep paths to be explored, compared to AFL. So, we cannot simply replace AFL with a generative model. Instead, we seek to combine generative models with AFL to boost its performance. We see from this data that, in fact, the seed files generated by LSTM and GAN are not representative of the distribution S in terms of the mean and variance of code paths generated. This reinforces the need to use S G and S L as an augmentation strategy rather than a direct replacement of AFL seeds.\n\nNext, we performed 24 hours of fuzzing with GAN, LSTM, and a random reinitilialzation strategy using a random sampling of bytes from the initial seed files (i.e., performing no learning on the seed files). Table 3 summarizes our results. All three strategies allowed for additional seed files to be generated. The GAN-based approach produced seed files 14.23% quicker than the random approach and 60.72% faster than using LSTM. We do lose 30 minutes of training time for GAN that could otherwise be used for fuzzing using the random sampling method; discounting by this amount of time reduces the code path rate to an 11.85% improvement. However, we are most interested in unique code paths. GAN found the greatest number of seed files whose associated code paths had lengths not found in the initial fuzzing run, outperforming the random control approach by 6.16%. The average code path length discovered by GAN was 13.84% longer than the random control, so GAN is capable of exercising deeper paths in the program. The LSTM model underperformed both GAN and random sampling and took a substantially longer time (14 hours) to train.\n\n\nConclusions\n\nIn this work, we explored the utility of augmenting random mutation testing with deep neural models. Natively AFL, combines file mutation strategies from Genetic Algorithms with program instrumentation via the use of compiler plugins. We observed the most improvement in AFL's performance when we restart a fuzzing run mid-course, using novel seed files built from a GAN model. Though the synthetic seed file statistics on average had similar path length, the GAN out-performed reinitialization from a random or LSTM strategy when restarting the fuzzing system. The LSTM model was deficient in both training time and code path discovery time. Both approaches used no manual analysis or information about file formats for the program under test. The GAN and random strategies both improve the performance of AFL, even though the internal state of the program is never directly exposed.\n\nFuture work of interest includes experimentation on additional targets, including the DARPA Cyber Grand Challenge problems, open source OS network services, bytecode interpreters, and other system applications and programs where input data is easily generated. We also plan to explore exposing the internal state of the program under test in order to define a reward function for reinforcement learning. We envision this internal state could be exposed by: 1) the instrumentation AFL adds to programs via its GCC compiler plugins, 2) using Intel's PIN tool to output the length of each code path or summary information about a given trace 3) recording program traces using a replay framework such as Mozilla's rr tool in order to collect additional descriptive statistics.\n AcknowledgementsThe authors would like to thank Court Corley, Nathan Hodas, and Sam Winters for useful discussions. The research described in this paper is part of the Deep Science Initiative at Pacific Northwest National Laboratory. It was conducted under the Laboratory Directed Research and Development Program at PNNL, a multi-program national laboratory operated by Battelle for the U.S. Department of Energy.\nTesting telecoms software with quviq quickcheck. Thomas ; Arts, Hughes, ; John, Johansson, ; Joakim, Ulf Wiger, Proceedings of the. the. 2006] ARTS, Thomas ; HUGHES, John ; JOHANSSON, Joakim ; WIGER, Ulf: Testing telecoms software with quviq quickcheck. In: Proceedings of the 2006 ACM SIGPLAN workshop on Erlang ACM (Veranst.), 2006, S. 2-10\n\n. Yoshua ; Bengio, Ian J ; Goodfellow, Aaron Courville, Nature. 521Bengio u. a. 2015[Bengio u. a. 2015] BENGIO, Yoshua ; GOODFELLOW, Ian J. ; COURVILLE, Aaron: Deep learn- ing. In: Nature 521 (2015), S. 436-444\n\n. Marcel ; Böhme, Pham, Van-Thuan, Böhme u. a. 2016[Böhme u. a. 2016] BÖHME, Marcel ; PHAM, Van-Thuan ;\n\nAbhik Roychoudhury, http:/doi.acm.org/10.1145/2976749.2978428978-1-4503-4139-4Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. the 2016 ACM SIGSAC Conference on Computer and Communications SecurityNew York, NY, USAACM16Coveragebased Greybox Fuzzing As Markov ChainROYCHOUDHURY, Abhik: Coverage- based Greybox Fuzzing As Markov Chain. In: Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. New York, NY, USA : ACM, 2016 (CCS '16), S. 1032-1043. -URL http://doi.acm.org/10.1145/2976749.2978428. -ISBN 978-1-4503-4139-4\n\nSurvey of Automated Vulnerability Detection and Exploit Generation Techniques in Cyber Reasoning Systems. Teresa N Brooks, CoRR abs/1702.06162[Brooks 2017] BROOKS, Teresa N.: Survey of Automated Vulnerability Detection and Exploit Generation Techniques in Cyber Reasoning Systems. In: CoRR abs/1702.06162 (2017). -URL http://arxiv.org/abs/1702.06162\n\nBoolector: An efficient SMT solver for bit-vectors and arrays. Robert ; Brummayer, Armin Biere, Tools and Algorithms for the Construction and Analysis of Systems. Brummayer und Biere[Brummayer und Biere 2009] BRUMMAYER, Robert ; BIERE, Armin: Boolector: An efficient SMT solver for bit-vectors and arrays. In: Tools and Algorithms for the Construction and Analysis of Systems (2009), S. 174-177\n\nKLEE: Unassisted and Automatic Generation of High-Coverage Tests for Complex Systems Programs. Cristian ; Cadar, Dunbar, ; Daniel, Engler, R U Dawson, OSDI Bd. 8Cadar u. a.[Cadar u. a. 2008] CADAR, Cristian ; DUNBAR, Daniel ; ENGLER, Dawson R. u. a.: KLEE: Unassisted and Automatic Generation of High-Coverage Tests for Complex Systems Programs. In: OSDI Bd. 8, 2008, S. 209-224\n\nQuickCheck: a lightweight tool for random testing of Haskell programs. Claessen, ; Koen, John Hughes, CHOLLET, François: Keras. 46Acm sigplan notices. Nr. 4, S[Chollet 2017] CHOLLET, François: Keras (2015). In: URL http://keras. io (2017) [Claessen und Hughes 2011] CLAESSEN, Koen ; HUGHES, John: QuickCheck: a lightweight tool for random testing of Haskell programs. In: Acm sigplan notices 46 (2011), Nr. 4, S. 53-64\n\nAbstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. Patrick ; Cousot, Radhia Cousot, Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACM. the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACMCousot und Cousot[Cousot und Cousot 1977] COUSOT, Patrick ; COUSOT, Radhia: Abstract interpretation: a uni- fied lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACM (Veranst.), 1977, S. 238-252\n\nZ3: An efficient SMT solver. De Moura, ; Leonardo, Nikolaj Bjørner, Tools and Algorithms for the Construction and Analysis of Systems. De Moura und Bjørner[De Moura und Bjørner 2008] DE MOURA, Leonardo ; BJØRNER, Nikolaj: Z3: An efficient SMT solver. In: Tools and Algorithms for the Construction and Analysis of Systems (2008), S. 337-340\n\nA program construction and verification tool for separation logic. Dongol, ; Brijesh, Gomes, B ; Victor, Georg Struth, International Conference on Mathematics of Program Construction Springer. Dongol u. a. 2015[Dongol u. a. 2015] DONGOL, Brijesh ; GOMES, Victor B. ; STRUTH, Georg: A program con- struction and verification tool for separation logic. In: International Conference on Mathematics of Program Construction Springer (Veranst.), 2015, S. 137-158\n\nEnemy of the State: A State-Aware Black-Box Web Vulnerability Scanner. Adam ; Doupé, Cavedon, ; Ludovico, Kruegel, ; Christopher, Giovanni Vigna, USENIX Security Symposium Bd. 14Doupé u. a. 2012[Doupé u. a. 2012] DOUPÉ, Adam ; CAVEDON, Ludovico ; KRUEGEL, Christopher ; VIGNA, Giovanni: Enemy of the State: A State-Aware Black-Box Web Vulnerability Scanner. In: USENIX Security Symposium Bd. 14, 2012\n\nLearn&fuzz: Machine learning for input fuzzing. Patrice ; Godefroid, Peleg, ; Hila, Rishabh Singh, arXiv:1701.07232arXiv preprintGodefroid u. a. 2017[Godefroid u. a. 2017] GODEFROID, Patrice ; PELEG, Hila ; SINGH, Rishabh: Learn&fuzz: Machine learning for input fuzzing. In: arXiv preprint arXiv:1701.07232 (2017)\n\nIan ; Goodfellow, Pouget-Abadie, ; Jean, Mirza, ; Mehdi, Xu, ; Warde-Farley Bing, ; David, Ozair, ; Sherjil, Courville, ; Aaron, Yoshua Bengio, Advances in neural information processing systems. Generative adversarial nets[Goodfellow u. a. 2014] GOODFELLOW, Ian ; POUGET-ABADIE, Jean ; MIRZA, Mehdi ; XU, Bing ; WARDE-FARLEY, David ; OZAIR, Sherjil ; COURVILLE, Aaron ; BENGIO, Yoshua: Gen- erative adversarial nets. In: Advances in neural information processing systems, 2014, S. 2672- 2680\n\nThe learning machines. Nicola ; Jones, Alex ; Krizhevsky, Sutskever, ; Ilya, Geoffrey E Hinton, Nr. 7482Imagenet classification with deep convolutional neural networks. 505Advances in neural information processing systems[Jones 2014] JONES, Nicola: The learning machines. In: Nature 505 (2014), Nr. 7482, S. 146 [Krizhevsky u. a. 2012] KRIZHEVSKY, Alex ; SUTSKEVER, Ilya ; HINTON, Geoffrey E.: Ima- genet classification with deep convolutional neural networks. In: Advances in neural information processing systems, 2012, S. 1097-1105\n\n. Yuekang ; Li, Chen, ; Bihuan, Chandramohan, ; Mahinthan, Lin, Shang-Wei, LI, Yuekang ; CHEN, Bihuan ; CHANDRAMOHAN, Mahinthan ; LIN, Shang-Wei ;\n\nSteelix: Program-state Based Binary Fuzzing. Yang ; Liu, Alwen Tiu, http:/doi.acm.org/10.1145/3106237.3106295978-1-4503-5105-8Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering. the 2017 11th Joint Meeting on Foundations of Software EngineeringNew York, NY, USAACMLIU, Yang ; TIU, Alwen: Steelix: Program-state Based Binary Fuzzing. In: Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineer- ing. New York, NY, USA : ACM, 2017 (ESEC/FSE 2017), S. 627-637. -URL http://doi.acm.org/10.1145/3106237.3106295. -ISBN 978-1-4503-5105-8\n\n. Kin - Ma, Keung, Ma u. a. 2011[Ma u. a. 2011] MA, Kin-Keung ;\n\nDirected symbolic execution. Yit Phang, ; Khoo, Foster, ; Jeffrey, Michael Hicks, Static Analysis. YIT PHANG, Khoo ; FOSTER, Jeffrey ; HICKS, Michael: Directed symbolic execution. In: Static Analysis (2011), S. 95-111\n\n. Mnih, ; Volodymyr, Kavukcuoglu, ; Koray, Silver, ; David, Alex Graves, Mnih u. a. 2013[Mnih u. a. 2013] MNIH, Volodymyr ; KAVUKCUOGLU, Koray ; SILVER, David ; GRAVES, Alex ;\n\nPlaying atari with deep reinforcement learning. Antonoglou, ; Ioannis, Wierstra, ; Daan, Martin Riedmiller, arXiv:1312.5602arXiv preprintANTONOGLOU, Ioannis ; WIERSTRA, Daan ; RIEDMILLER, Martin: Playing atari with deep reinforcement learning. In: arXiv preprint arXiv:1312.5602 (2013)\n\nSeparation logic: A logic for shared mutable data structures. John C Reynolds, Logic in Computer Science, 2002. Proceedings. 17th Annual IEEE Symposium on IEEE. [Reynolds 2002] REYNOLDS, John C.: Separation logic: A logic for shared mutable data struc- tures. In: Logic in Computer Science, 2002. Proceedings. 17th Annual IEEE Symposium on IEEE (Veranst.), 2002, S. 55-74\n\nLong short-term memory recurrent neural network architectures for large scale acoustic modeling. Sak, ; Haşim, Senior, ; Andrew, Françoise Beaufays, Fifteenth Annual Conference of the International Speech Communication Association. Sak u. a. 2014[Sak u. a. 2014] SAK, Haşim ; SENIOR, Andrew ; BEAUFAYS, Françoise: Long short-term memory recurrent neural network architectures for large scale acoustic modeling. In: Fifteenth Annual Conference of the International Speech Communication Association, 2014\n\nAll you ever wanted to know about dynamic taint analysis and forward symbolic execution (but might have been afraid to ask). Edward J ; Schwartz, Avgerinos, ; Thanassis, David Brumley, Security and privacy (SP), 2010 IEEE symposium on IEEE. Schwartz u. a. 2010[Schwartz u. a. 2010] SCHWARTZ, Edward J. ; AVGERINOS, Thanassis ; BRUMLEY, David: All you ever wanted to know about dynamic taint analysis and forward symbolic execution (but might have been afraid to ask). In: Security and privacy (SP), 2010 IEEE symposium on IEEE (Veranst.), 2010, S. 317-331\n\n. Yan ; Shoshitaishvili, Wang, ; Ruoyu, Christopher Salls, Shoshitaishvili u. a. 2016[Shoshitaishvili u. a. 2016] SHOSHITAISHVILI, Yan ; WANG, Ruoyu ; SALLS, Christopher ;\n\nSok:(state of) the art of war: Offensive techniques in binary analysis. Nick ; Stephens, Polino, ; Mario, Dutcher, ; Andrew, Grosen, ; John, Feng, ; Siji, Christophe ; Hauser, Christopher U A Kruegel, Security and Privacy (SP), 2016 IEEE Symposium on IEEE. STEPHENS, Nick ; POLINO, Mario ; DUTCHER, Andrew ; GROSEN, John ; FENG, Siji ; HAUSER, Christophe ; KRUEGEL, Christopher u. a.: Sok:(state of) the art of war: Offensive techniques in binary analysis. In: Security and Privacy (SP), 2016 IEEE Symposium on IEEE (Veranst.), 2016, S. 138-157\n\n. David ; Silver, Aja ; Huang, Chris J ; Maddison, Guez, ; Arthur, Sifre, ; Laurent, Van Den, Driessche, ; George, Schrittwieser, ; Julian, Ioannis Antonoglou, Silver u. a. 2016[Silver u. a. 2016] SILVER, David ; HUANG, Aja ; MADDISON, Chris J. ; GUEZ, Arthur ; SIFRE, Laurent ; VAN DEN DRIESSCHE, George ; SCHRITTWIESER, Julian ; ANTONOGLOU, Ioannis ;\n\nMastering the game of Go with deep neural networks and tree search. Veda ; Panneershelvam, Marc U A Lanctot, Nr. 7587Nature. 529PANNEERSHELVAM, Veda ; LANCTOT, Marc u. a.: Mastering the game of Go with deep neural networks and tree search. In: Nature 529 (2016), Nr. 7587, S. 484-489\n\n. Nick ; Stephens, Grosen, ; John, Salls, ; Christopher, Andrew Dutcher, Stephens u. a. 2016[Stephens u. a. 2016] STEPHENS, Nick ; GROSEN, John ; SALLS, Christopher ; DUTCHER, An- drew ;\n\n. Ruoyu ; Wang, Corbetta, ; Jacopo, Shoshitaishvili, ; Yan, Christopher Kruegel, WANG, Ruoyu ; CORBETTA, Jacopo ; SHOSHITAISHVILI, Yan ; KRUEGEL, Christopher ;\n\nGiovanni Vigna, Driller: Augmenting Fuzzing Through Selective Symbolic Execution. 16NDSS BdVIGNA, Giovanni: Driller: Augmenting Fuzzing Through Selective Symbolic Execution. In: NDSS Bd. 16, 2016, S. 1-16\n\nSkyfire: Data-driven seed generation for fuzzing. Junjie ; Wang, Chen, ; Bihuan, Wei, ; Lei, Yang Liu, [Wang u. a. 2017] WANG, Junjie ; CHEN, Bihuan ; WEI, Lei ; LIU, Yang: Skyfire: Data-driven seed generation for fuzzing. 2017\n\nScheduling black-box mutational fuzzing. Woo, ; Maverick, Sang K ; Cha, Gottlieb, ; Samantha, David Brumley, Proceedings of the. theWoo u. a. 2013[Woo u. a. 2013] WOO, Maverick ; CHA, Sang K. ; GOTTLIEB, Samantha ; BRUMLEY, David: Scheduling black-box mutational fuzzing. In: Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security ACM (Veranst.), 2013, S. 511-522\n\nEthereum: A secure decentralised generalised transaction ledger. ] Wood, Gavin, Ethereum Project Yellow Paper. 151[Wood 2014] WOOD, Gavin: Ethereum: A secure decentralised generalised transaction ledger. In: Ethereum Project Yellow Paper 151 (2014)\n\nGoogle's neural machine translation system: Bridging the gap between human and machine translation. Yonghui ; Wu, Schuster, ; Mike, Chen, ; Zhifeng, Le, V ; Quoc, Norouzi, ; Mohammad, Wolfgang ; Macherey, Krikun, ; Maxim, Cao, ; Yuan, Gao, ; Qin, Klaus U Macherey, arXiv:1609.08144arXiv preprintWu u. a. 2016[Wu u. a. 2016] WU, Yonghui ; SCHUSTER, Mike ; CHEN, Zhifeng ; LE, Quoc V. ; NOROUZI, Mohammad ; MACHEREY, Wolfgang ; KRIKUN, Maxim ; CAO, Yuan ; GAO, Qin ; MACHEREY, Klaus u. a.: Google's neural machine translation system: Bridging the gap between human and machine translation. In: arXiv preprint arXiv:1609.08144 (2016)\n\nZALEWSKI, Michał: American Fuzzy Lop (AFL) fuzzer. [Zalewski 2015] ZALEWSKI, Michał: American Fuzzy Lop (AFL) fuzzer (2015). 2015\n"},"corpus_id":{"kind":"number","value":6521088,"string":"6,521,088"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1711.02807\",\n \"mag\": 2767634461,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/abs-1711-02807\",\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1711.02807v1.pdf\"\n ],\n \"pdf_sha\": \"b6ade209fba99c0bb17b979252bdbaf354ca743b\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Faster Fuzzing: Reinitialization with Deep Neural Models"},"authors":{"kind":"list like","value":["Nicole Nichols nicole.nichols@pnnl.govmark.raugas@pnnl.govrobert.jasper@pnnl.govnathan.hilliard@pnnl.gov \nPacific Northwest National Laboratory\n1100 Dexter Avenue, Suite 50098109SeattleWA","Mark Raugas \nPacific Northwest National Laboratory\n1100 Dexter Avenue, Suite 50098109SeattleWA","Robert Jasper \nPacific Northwest National Laboratory\n1100 Dexter Avenue, Suite 50098109SeattleWA","Nathan Hilliard \nPacific Northwest National Laboratory\n1100 Dexter Avenue, Suite 50098109SeattleWA"],"string":"[\n \"Nicole Nichols nicole.nichols@pnnl.govmark.raugas@pnnl.govrobert.jasper@pnnl.govnathan.hilliard@pnnl.gov \\nPacific Northwest National Laboratory\\n1100 Dexter Avenue, Suite 50098109SeattleWA\",\n \"Mark Raugas \\nPacific Northwest National Laboratory\\n1100 Dexter Avenue, Suite 50098109SeattleWA\",\n \"Robert Jasper \\nPacific Northwest National Laboratory\\n1100 Dexter Avenue, Suite 50098109SeattleWA\",\n \"Nathan Hilliard \\nPacific Northwest National Laboratory\\n1100 Dexter Avenue, Suite 50098109SeattleWA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We improve the performance of the American Fuzzy Lop (AFL) fuzz testing framework by using Generative Adversarial Network (GAN) models to reinitialize he system with novel seed files. We assess performance based on the temporal rate at which we produce novel and unseen code paths. We compare this approach to seed file generation from a random draw of bytes observed in the training seed files. The code path lengths and variations were not sufficiently diverse to fully replace AFL input generation. However, augmenting native AFL with these additional code paths demonstrated improvements over AFL alone. Specifically, experiments showed the GAN was faster and more effective than the LSTM and outperformed a random augmentation strategy, as measured by the number of unique code paths discovered. GAN helps AFL discover 14.23% more code paths than the random strategy in the same amount of CPU time, finds 6.16% more unique code paths, and finds paths that are on average 13.84% longer. Using GAN shows promise as a reinitialization strategy for AFL to help the fuzzer exercise deep paths in software."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Identifying software defects through testing is a challenging problem. Over the years, a number of approaches have been developed to test software, including random mutation testing (black box fuzzing) Doupé u. a. (2012) ;Woo u. a. (2013), abstract interpretation (of either source or machine code) Cousot und Cousot (1977) ;Cadar u. a. (2008) ;Ma u. a. (2011), and property based testing Arts u. a. (2006); Claessen und Hughes (2011).\nMethods such as symbolic and concolic execution have increased the fidelity of analyses run over programs Schwartz u. a. (2010). The development of Satisfiability Modulo Theories (SMT) solvers such as Z3, Boolector, and others have allowed for powerful programmatic analysis of reasoning about software De Moura und Bjørner (2008); Brummayer und Biere (2009). Separation logic has allowed for analyses to be applied to complicated data structures Reynolds (2002) ;Dongol u. a. (2015).\nAmerican Fuzzy Lop (AFL) is an advanced fuzzing framework that has been used to discover a number of novel software vulnerabilities (https://github.com/mrash/afl-cve). AFL uses random mutations of byte strings to identify unique code paths and discover defects in target programs. The inputs that successfully generated unique code paths are then documented as \"seed files\". We propose to use these native seed files as training data for deep generative models to create augmented seed files. Our proposed reinitialization methods are a scalable process that can improve the time to discovery of software defects.\nOther researchers have used machine learning to augment fuzzing frameworks including: Godefroid u. a. (2017), Wang u. a. (2017). To identify deeper bugs and code paths, Steelix Li u. a. (2017) uses a program-state based binary fuzzing approach and Driller Stephens u. a. (2016) demonstrates a hybrid approach using fuzzing and selective concolic execution. AFLFAST Böhme u. a. (2016) extends AFL using Markov chain models. Deep Neural Networks (DNNs) Bengio u. a. (2015), have had great success in the fields of Natural Language Processing (NLP) Jones (2014); Wu u. a. (2016), Computer Vision Krizhevsky u. a. (2012, and the playing of bounded games such as Go Mnih u. a. (2013) or video games like ATARI Silver u. a. (2016). Can these DNN's help existing program analysis tools perform better? In our work we investigate that question. We augment AFL Zalewski (2015), an advanced fuzzing framework, using Generative Adversarial Networks (GAN) Goodfellow u. a. (2014) and Long Short Term Memory (LSTM) Sak u. a. (2014) to increase its rate of unique code path discovery.\nOur work quantifies the benefits that augmentation strategies such as generative models can provide, even when limited by small quantities of training data. By periodically perturbing the state of AFL as it explores the input space, we are able to improve its performance, as measured by unique code paths. Specifically, we test our approach on the software ecosystem surrounding the Ethereum Wood (2014) project. As a financial system, correctness of the Ethereum code base is important for guaranteeing that transactions or calculations run without fault. We choose ethkey as an initial fuzzing target. Ethkey is a small C++ program provided as part of the cpp-ethereum project used to load, parse, and perform maintenance on Ethereum wallets, and importantly, takes a simple input file, making it easy to test with AFL."},{"title":"Experimental Design","content":"First, we describe the basic functionality of AFL, highlighting the key features that connect with the proposed augmentation framework. Next, we describe the methodology used to create the LSTM and GAN generated seed files. As a baseline, we also consider random generation of seed files from the training data used to construct the LSTM and GAN models. AFL has extensions to the GCC compiler which in conjunction with Genetic Algorithms, it uses to create seed files. Each seed file documents the input that yielded a unique code path, the time of discovery, and is used as the basis for mutation (or fuzzing) to generate future seed files. Our augmentation strategy takes advantage of the fact that if an external tool places additional seed files in the AFL working directory, AFL will use those files as inputs in subsequent fuzzing runs.\nTo produce the training data for our methods, we run AFL on a target program P for a fixed amount of time. AFL generates an initial set of seed files S = {S 0 , ..., S K } for each unique execution trace τ taken through P . We use S as training examples for the LSTM and GAN models, which are both trained using Keras Chollet (2017).\nOur LSTM is trained from the concatenation of AFL-generated seed file corpus S into a single file and generates new seed files with a maximum length of 40 characters. The LSTM model has a 128 wide initial layer, an internal dense layer, and a final softmax activation layer. To train the LSTM model, we use RMS propagation as our optimizer and a categorical cross-entropy loss function. The model takes in a seed sequence sampled from the training corpus and predicts the next character in the sequence. We additionally tune a separate temperature parameter to diversify the output seed files from the network. The generated seed files are noted as S L .\nIn our GAN architecture, two models are built, a generator G, which is pitted against a discriminator D. G is optimized to generate realistic output, and the discriminator D has the task of predicting if the data is true or fake. This training strategy is unsupervised and particularly expressive. The generative model G, is a fully connected 2 layer DNN with a ReLU non-linearity as the inner activation and a tanh output activation. It is trained with a binary cross-entropy loss function via stochastic gradient descent. The discriminative model D is a 3 layer DNN, but the first layer has 25% dropout followed by two fully connected layers. It uses an Adam optimizer for stochastic gradient descent and the seed files resulting from the GAN process are noted as S G . Table 1: Initial Run: |C| is the number of seeds generated after reinitializing AFL. We observe GAN or LSTM allows discovery of novel code paths at a quicker rate than restarting AFL using a random sampling of bytes. L(C) is the number of unique code path lengths l(c i ) associated to input files c i in the set C of LSTM, GAN, or uniform Random reinitialization. Table 2: Synthetic Seed Files: Random sampling, LSTM, and GAN can be used to produce synthetic seed files for AFL. We compute statistics on the synthetic files from the 3 strategies. The synthetic files are not themselves deep or varied code paths, but can be used to reinitilize AFL.\nAdditionally, given the native AFL seed files, S, we randomly draw bytes from this training set and produce new, random seed files S R of the same length as S G . This serves as a baseline to determine if the added time and complexity of GAN and LSTM based seed generation are truly providing an advantage over a simple strategy of randomly perturbing AFL's state.\nSmall Experiment: The seed files (S R , S G , and S L ) alone are not an end goal. However, we are interested in characterizing their variability and other properties as they will provide a set of initial conditions when AFL is restarted. In a fuzzing run on a single CPU core, we produce 936 unique code paths used to train initial GAN and LSTM models. Random seed generation is performed by drawing random bytes from /dev/urandom. For each method, we generate 200 samples, reinitialize AFL on a single CPU with only the seed files of one method and run for an additional 72 hours to measure the impact on code path discovery. Both LSTM and GAN models sightly out-perform random sampling for AFL reinitialization. We summarize the resulting mean time to discover new code paths in Table 1.\nEach seed file produces a program trace when supplied as an input to ethkey. Code paths that have different lengths will differ in at least one basic block or branch instruction. The unique code path length is fast to compute but only provides a lower bound on the number of unique code paths exercised by the testing framework, across fuzzing runs using AFL. Two code paths with the same length can result from unique traces, thus detailed evaluation is needed to determine the true uniqueness of a code path from seed file execution.\nLarge Experiment: To demonstrate the scalability of this augmentation strategy, we performed an extended run of AFL on 200 CPU cores for 72 hours. Each core in the AFL run stopped finding seed files after the first 10 to 12 hours of fuzzing and accumulated a total of 39,185 seed files across 49 workers. All seed files produced within a given node are known to be unique, due to AFL's internal book keeping mechanism. However, seed files whose content are different across nodes, could in principle exercise the same code path. By measuring the length of each program trace (code path), we can compute a lower bound on the number of unique paths discovered by only counting paths that have a unique length. After removing identical seed files from across the nodes, and seed files that resulted in the same code path length, we estimate 802 of the initial files were duplicates from the independent worker nodes. Removing those duplicates resulted in a total of 38,384 unique files.\nWe then trained GAN and LSTM networks on the total corpus of unique seed files and generated approximately 20,000 samples from each method, respectively, to use as synthetic seed files in order to reinitialize AFL. GAN took approximately 30 minutes to train and generate synthetic seed files, while LSTM took 14 hours to do so.\nClass C |C| L(C) Novel L(C) Rate Novel Rate µ(L(C)) σ(L(C)) Rand Table 3: Sustained Run: With 38,000 training seed file samples, we compare the seed files generated after reinitialization from random, GAN, and LSTM generation methods. L(C) Rate is the speedup over the random strategy of discovery of code paths with unique length per second. Novel Rate is the speedup over random for unique lengths not found in the training set.\nIn Table 2 we summarize the mean and variance of the length of program traces (i.e., code paths) associated with the seed files from native AFL and from the synthetic generation methods for this larger experiment. The synthetic seed files, when provided as inputs to the program under test, do not cause deep paths to be explored, compared to AFL. So, we cannot simply replace AFL with a generative model. Instead, we seek to combine generative models with AFL to boost its performance. We see from this data that, in fact, the seed files generated by LSTM and GAN are not representative of the distribution S in terms of the mean and variance of code paths generated. This reinforces the need to use S G and S L as an augmentation strategy rather than a direct replacement of AFL seeds.\nNext, we performed 24 hours of fuzzing with GAN, LSTM, and a random reinitilialzation strategy using a random sampling of bytes from the initial seed files (i.e., performing no learning on the seed files). Table 3 summarizes our results. All three strategies allowed for additional seed files to be generated. The GAN-based approach produced seed files 14.23% quicker than the random approach and 60.72% faster than using LSTM. We do lose 30 minutes of training time for GAN that could otherwise be used for fuzzing using the random sampling method; discounting by this amount of time reduces the code path rate to an 11.85% improvement. However, we are most interested in unique code paths. GAN found the greatest number of seed files whose associated code paths had lengths not found in the initial fuzzing run, outperforming the random control approach by 6.16%. The average code path length discovered by GAN was 13.84% longer than the random control, so GAN is capable of exercising deeper paths in the program. The LSTM model underperformed both GAN and random sampling and took a substantially longer time (14 hours) to train."},{"title":"Conclusions","content":"In this work, we explored the utility of augmenting random mutation testing with deep neural models. Natively AFL, combines file mutation strategies from Genetic Algorithms with program instrumentation via the use of compiler plugins. We observed the most improvement in AFL's performance when we restart a fuzzing run mid-course, using novel seed files built from a GAN model. Though the synthetic seed file statistics on average had similar path length, the GAN out-performed reinitialization from a random or LSTM strategy when restarting the fuzzing system. The LSTM model was deficient in both training time and code path discovery time. Both approaches used no manual analysis or information about file formats for the program under test. The GAN and random strategies both improve the performance of AFL, even though the internal state of the program is never directly exposed.\nFuture work of interest includes experimentation on additional targets, including the DARPA Cyber Grand Challenge problems, open source OS network services, bytecode interpreters, and other system applications and programs where input data is easily generated. We also plan to explore exposing the internal state of the program under test in order to define a reward function for reinforcement learning. We envision this internal state could be exposed by: 1) the instrumentation AFL adds to programs via its GCC compiler plugins, 2) using Intel's PIN tool to output the length of each code path or summary information about a given trace 3) recording program traces using a replay framework such as Mozilla's rr tool in order to collect additional descriptive statistics."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Identifying software defects through testing is a challenging problem. Over the years, a number of approaches have been developed to test software, including random mutation testing (black box fuzzing) Doupé u. a. (2012) ;Woo u. a. (2013), abstract interpretation (of either source or machine code) Cousot und Cousot (1977) ;Cadar u. a. (2008) ;Ma u. a. (2011), and property based testing Arts u. a. (2006); Claessen und Hughes (2011).\\nMethods such as symbolic and concolic execution have increased the fidelity of analyses run over programs Schwartz u. a. (2010). The development of Satisfiability Modulo Theories (SMT) solvers such as Z3, Boolector, and others have allowed for powerful programmatic analysis of reasoning about software De Moura und Bjørner (2008); Brummayer und Biere (2009). Separation logic has allowed for analyses to be applied to complicated data structures Reynolds (2002) ;Dongol u. a. (2015).\\nAmerican Fuzzy Lop (AFL) is an advanced fuzzing framework that has been used to discover a number of novel software vulnerabilities (https://github.com/mrash/afl-cve). AFL uses random mutations of byte strings to identify unique code paths and discover defects in target programs. The inputs that successfully generated unique code paths are then documented as \\\"seed files\\\". We propose to use these native seed files as training data for deep generative models to create augmented seed files. Our proposed reinitialization methods are a scalable process that can improve the time to discovery of software defects.\\nOther researchers have used machine learning to augment fuzzing frameworks including: Godefroid u. a. (2017), Wang u. a. (2017). To identify deeper bugs and code paths, Steelix Li u. a. (2017) uses a program-state based binary fuzzing approach and Driller Stephens u. a. (2016) demonstrates a hybrid approach using fuzzing and selective concolic execution. AFLFAST Böhme u. a. (2016) extends AFL using Markov chain models. Deep Neural Networks (DNNs) Bengio u. a. (2015), have had great success in the fields of Natural Language Processing (NLP) Jones (2014); Wu u. a. (2016), Computer Vision Krizhevsky u. a. (2012, and the playing of bounded games such as Go Mnih u. a. (2013) or video games like ATARI Silver u. a. (2016). Can these DNN's help existing program analysis tools perform better? In our work we investigate that question. We augment AFL Zalewski (2015), an advanced fuzzing framework, using Generative Adversarial Networks (GAN) Goodfellow u. a. (2014) and Long Short Term Memory (LSTM) Sak u. a. (2014) to increase its rate of unique code path discovery.\\nOur work quantifies the benefits that augmentation strategies such as generative models can provide, even when limited by small quantities of training data. By periodically perturbing the state of AFL as it explores the input space, we are able to improve its performance, as measured by unique code paths. Specifically, we test our approach on the software ecosystem surrounding the Ethereum Wood (2014) project. As a financial system, correctness of the Ethereum code base is important for guaranteeing that transactions or calculations run without fault. We choose ethkey as an initial fuzzing target. Ethkey is a small C++ program provided as part of the cpp-ethereum project used to load, parse, and perform maintenance on Ethereum wallets, and importantly, takes a simple input file, making it easy to test with AFL.\"\n },\n {\n \"title\": \"Experimental Design\",\n \"content\": \"First, we describe the basic functionality of AFL, highlighting the key features that connect with the proposed augmentation framework. Next, we describe the methodology used to create the LSTM and GAN generated seed files. As a baseline, we also consider random generation of seed files from the training data used to construct the LSTM and GAN models. AFL has extensions to the GCC compiler which in conjunction with Genetic Algorithms, it uses to create seed files. Each seed file documents the input that yielded a unique code path, the time of discovery, and is used as the basis for mutation (or fuzzing) to generate future seed files. Our augmentation strategy takes advantage of the fact that if an external tool places additional seed files in the AFL working directory, AFL will use those files as inputs in subsequent fuzzing runs.\\nTo produce the training data for our methods, we run AFL on a target program P for a fixed amount of time. AFL generates an initial set of seed files S = {S 0 , ..., S K } for each unique execution trace τ taken through P . We use S as training examples for the LSTM and GAN models, which are both trained using Keras Chollet (2017).\\nOur LSTM is trained from the concatenation of AFL-generated seed file corpus S into a single file and generates new seed files with a maximum length of 40 characters. The LSTM model has a 128 wide initial layer, an internal dense layer, and a final softmax activation layer. To train the LSTM model, we use RMS propagation as our optimizer and a categorical cross-entropy loss function. The model takes in a seed sequence sampled from the training corpus and predicts the next character in the sequence. We additionally tune a separate temperature parameter to diversify the output seed files from the network. The generated seed files are noted as S L .\\nIn our GAN architecture, two models are built, a generator G, which is pitted against a discriminator D. G is optimized to generate realistic output, and the discriminator D has the task of predicting if the data is true or fake. This training strategy is unsupervised and particularly expressive. The generative model G, is a fully connected 2 layer DNN with a ReLU non-linearity as the inner activation and a tanh output activation. It is trained with a binary cross-entropy loss function via stochastic gradient descent. The discriminative model D is a 3 layer DNN, but the first layer has 25% dropout followed by two fully connected layers. It uses an Adam optimizer for stochastic gradient descent and the seed files resulting from the GAN process are noted as S G . Table 1: Initial Run: |C| is the number of seeds generated after reinitializing AFL. We observe GAN or LSTM allows discovery of novel code paths at a quicker rate than restarting AFL using a random sampling of bytes. L(C) is the number of unique code path lengths l(c i ) associated to input files c i in the set C of LSTM, GAN, or uniform Random reinitialization. Table 2: Synthetic Seed Files: Random sampling, LSTM, and GAN can be used to produce synthetic seed files for AFL. We compute statistics on the synthetic files from the 3 strategies. The synthetic files are not themselves deep or varied code paths, but can be used to reinitilize AFL.\\nAdditionally, given the native AFL seed files, S, we randomly draw bytes from this training set and produce new, random seed files S R of the same length as S G . This serves as a baseline to determine if the added time and complexity of GAN and LSTM based seed generation are truly providing an advantage over a simple strategy of randomly perturbing AFL's state.\\nSmall Experiment: The seed files (S R , S G , and S L ) alone are not an end goal. However, we are interested in characterizing their variability and other properties as they will provide a set of initial conditions when AFL is restarted. In a fuzzing run on a single CPU core, we produce 936 unique code paths used to train initial GAN and LSTM models. Random seed generation is performed by drawing random bytes from /dev/urandom. For each method, we generate 200 samples, reinitialize AFL on a single CPU with only the seed files of one method and run for an additional 72 hours to measure the impact on code path discovery. Both LSTM and GAN models sightly out-perform random sampling for AFL reinitialization. We summarize the resulting mean time to discover new code paths in Table 1.\\nEach seed file produces a program trace when supplied as an input to ethkey. Code paths that have different lengths will differ in at least one basic block or branch instruction. The unique code path length is fast to compute but only provides a lower bound on the number of unique code paths exercised by the testing framework, across fuzzing runs using AFL. Two code paths with the same length can result from unique traces, thus detailed evaluation is needed to determine the true uniqueness of a code path from seed file execution.\\nLarge Experiment: To demonstrate the scalability of this augmentation strategy, we performed an extended run of AFL on 200 CPU cores for 72 hours. Each core in the AFL run stopped finding seed files after the first 10 to 12 hours of fuzzing and accumulated a total of 39,185 seed files across 49 workers. All seed files produced within a given node are known to be unique, due to AFL's internal book keeping mechanism. However, seed files whose content are different across nodes, could in principle exercise the same code path. By measuring the length of each program trace (code path), we can compute a lower bound on the number of unique paths discovered by only counting paths that have a unique length. After removing identical seed files from across the nodes, and seed files that resulted in the same code path length, we estimate 802 of the initial files were duplicates from the independent worker nodes. Removing those duplicates resulted in a total of 38,384 unique files.\\nWe then trained GAN and LSTM networks on the total corpus of unique seed files and generated approximately 20,000 samples from each method, respectively, to use as synthetic seed files in order to reinitialize AFL. GAN took approximately 30 minutes to train and generate synthetic seed files, while LSTM took 14 hours to do so.\\nClass C |C| L(C) Novel L(C) Rate Novel Rate µ(L(C)) σ(L(C)) Rand Table 3: Sustained Run: With 38,000 training seed file samples, we compare the seed files generated after reinitialization from random, GAN, and LSTM generation methods. L(C) Rate is the speedup over the random strategy of discovery of code paths with unique length per second. Novel Rate is the speedup over random for unique lengths not found in the training set.\\nIn Table 2 we summarize the mean and variance of the length of program traces (i.e., code paths) associated with the seed files from native AFL and from the synthetic generation methods for this larger experiment. The synthetic seed files, when provided as inputs to the program under test, do not cause deep paths to be explored, compared to AFL. So, we cannot simply replace AFL with a generative model. Instead, we seek to combine generative models with AFL to boost its performance. We see from this data that, in fact, the seed files generated by LSTM and GAN are not representative of the distribution S in terms of the mean and variance of code paths generated. This reinforces the need to use S G and S L as an augmentation strategy rather than a direct replacement of AFL seeds.\\nNext, we performed 24 hours of fuzzing with GAN, LSTM, and a random reinitilialzation strategy using a random sampling of bytes from the initial seed files (i.e., performing no learning on the seed files). Table 3 summarizes our results. All three strategies allowed for additional seed files to be generated. The GAN-based approach produced seed files 14.23% quicker than the random approach and 60.72% faster than using LSTM. We do lose 30 minutes of training time for GAN that could otherwise be used for fuzzing using the random sampling method; discounting by this amount of time reduces the code path rate to an 11.85% improvement. However, we are most interested in unique code paths. GAN found the greatest number of seed files whose associated code paths had lengths not found in the initial fuzzing run, outperforming the random control approach by 6.16%. The average code path length discovered by GAN was 13.84% longer than the random control, so GAN is capable of exercising deeper paths in the program. The LSTM model underperformed both GAN and random sampling and took a substantially longer time (14 hours) to train.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"In this work, we explored the utility of augmenting random mutation testing with deep neural models. Natively AFL, combines file mutation strategies from Genetic Algorithms with program instrumentation via the use of compiler plugins. We observed the most improvement in AFL's performance when we restart a fuzzing run mid-course, using novel seed files built from a GAN model. Though the synthetic seed file statistics on average had similar path length, the GAN out-performed reinitialization from a random or LSTM strategy when restarting the fuzzing system. The LSTM model was deficient in both training time and code path discovery time. Both approaches used no manual analysis or information about file formats for the program under test. The GAN and random strategies both improve the performance of AFL, even though the internal state of the program is never directly exposed.\\nFuture work of interest includes experimentation on additional targets, including the DARPA Cyber Grand Challenge problems, open source OS network services, bytecode interpreters, and other system applications and programs where input data is easily generated. We also plan to explore exposing the internal state of the program under test in order to define a reward function for reinforcement learning. We envision this internal state could be exposed by: 1) the instrumentation AFL adds to programs via its GCC compiler plugins, 2) using Intel's PIN tool to output the length of each code path or summary information about a given trace 3) recording program traces using a replay framework such as Mozilla's rr tool in order to collect additional descriptive statistics.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Testing telecoms software with quviq quickcheck. Thomas ; Arts, Hughes, ; John, Johansson, ; Joakim, Ulf Wiger, Proceedings of the. the. 2006] ARTS, Thomas ; HUGHES, John ; JOHANSSON, Joakim ; WIGER, Ulf: Testing telecoms software with quviq quickcheck. In: Proceedings of the 2006 ACM SIGPLAN workshop on Erlang ACM (Veranst.), 2006, S. 2-10","doi":""},{"text":". Yoshua ; Bengio, Ian J ; Goodfellow, Aaron Courville, Nature. 521Bengio u. a. 2015[Bengio u. a. 2015] BENGIO, Yoshua ; GOODFELLOW, Ian J. ; COURVILLE, Aaron: Deep learn- ing. In: Nature 521 (2015), S. 436-444","doi":""},{"text":". Marcel ; Böhme, Pham, Van-Thuan, Böhme u. a. 2016[Böhme u. a. 2016] BÖHME, Marcel ; PHAM, Van-Thuan ;","doi":""},{"text":"Abhik Roychoudhury, http:/doi.acm.org/10.1145/2976749.2978428978-1-4503-4139-4Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. the 2016 ACM SIGSAC Conference on Computer and Communications SecurityNew York, NY, USAACM16Coveragebased Greybox Fuzzing As Markov ChainROYCHOUDHURY, Abhik: Coverage- based Greybox Fuzzing As Markov Chain. In: Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. New York, NY, USA : ACM, 2016 (CCS '16), S. 1032-1043. -URL http://doi.acm.org/10.1145/2976749.2978428. -ISBN 978-1-4503-4139-4","doi":"http:/doi.acm.org/10.1145/2976749.2978428"},{"text":"Survey of Automated Vulnerability Detection and Exploit Generation Techniques in Cyber Reasoning Systems. Teresa N Brooks, CoRR abs/1702.06162[Brooks 2017] BROOKS, Teresa N.: Survey of Automated Vulnerability Detection and Exploit Generation Techniques in Cyber Reasoning Systems. In: CoRR abs/1702.06162 (2017). -URL http://arxiv.org/abs/1702.06162","doi":""},{"text":"Boolector: An efficient SMT solver for bit-vectors and arrays. Robert ; Brummayer, Armin Biere, Tools and Algorithms for the Construction and Analysis of Systems. Brummayer und Biere[Brummayer und Biere 2009] BRUMMAYER, Robert ; BIERE, Armin: Boolector: An efficient SMT solver for bit-vectors and arrays. In: Tools and Algorithms for the Construction and Analysis of Systems (2009), S. 174-177","doi":""},{"text":"KLEE: Unassisted and Automatic Generation of High-Coverage Tests for Complex Systems Programs. Cristian ; Cadar, Dunbar, ; Daniel, Engler, R U Dawson, OSDI Bd. 8Cadar u. a.[Cadar u. a. 2008] CADAR, Cristian ; DUNBAR, Daniel ; ENGLER, Dawson R. u. a.: KLEE: Unassisted and Automatic Generation of High-Coverage Tests for Complex Systems Programs. In: OSDI Bd. 8, 2008, S. 209-224","doi":""},{"text":"QuickCheck: a lightweight tool for random testing of Haskell programs. Claessen, ; Koen, John Hughes, CHOLLET, François: Keras. 46Acm sigplan notices. Nr. 4, S[Chollet 2017] CHOLLET, François: Keras (2015). In: URL http://keras. io (2017) [Claessen und Hughes 2011] CLAESSEN, Koen ; HUGHES, John: QuickCheck: a lightweight tool for random testing of Haskell programs. In: Acm sigplan notices 46 (2011), Nr. 4, S. 53-64","doi":""},{"text":"Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. Patrick ; Cousot, Radhia Cousot, Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACM. the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACMCousot und Cousot[Cousot und Cousot 1977] COUSOT, Patrick ; COUSOT, Radhia: Abstract interpretation: a uni- fied lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACM (Veranst.), 1977, S. 238-252","doi":""},{"text":"Z3: An efficient SMT solver. De Moura, ; Leonardo, Nikolaj Bjørner, Tools and Algorithms for the Construction and Analysis of Systems. De Moura und Bjørner[De Moura und Bjørner 2008] DE MOURA, Leonardo ; BJØRNER, Nikolaj: Z3: An efficient SMT solver. In: Tools and Algorithms for the Construction and Analysis of Systems (2008), S. 337-340","doi":""},{"text":"A program construction and verification tool for separation logic. Dongol, ; Brijesh, Gomes, B ; Victor, Georg Struth, International Conference on Mathematics of Program Construction Springer. Dongol u. a. 2015[Dongol u. a. 2015] DONGOL, Brijesh ; GOMES, Victor B. ; STRUTH, Georg: A program con- struction and verification tool for separation logic. In: International Conference on Mathematics of Program Construction Springer (Veranst.), 2015, S. 137-158","doi":""},{"text":"Enemy of the State: A State-Aware Black-Box Web Vulnerability Scanner. Adam ; Doupé, Cavedon, ; Ludovico, Kruegel, ; Christopher, Giovanni Vigna, USENIX Security Symposium Bd. 14Doupé u. a. 2012[Doupé u. a. 2012] DOUPÉ, Adam ; CAVEDON, Ludovico ; KRUEGEL, Christopher ; VIGNA, Giovanni: Enemy of the State: A State-Aware Black-Box Web Vulnerability Scanner. In: USENIX Security Symposium Bd. 14, 2012","doi":""},{"text":"Learn&fuzz: Machine learning for input fuzzing. Patrice ; Godefroid, Peleg, ; Hila, Rishabh Singh, arXiv:1701.07232arXiv preprintGodefroid u. a. 2017[Godefroid u. a. 2017] GODEFROID, Patrice ; PELEG, Hila ; SINGH, Rishabh: Learn&fuzz: Machine learning for input fuzzing. In: arXiv preprint arXiv:1701.07232 (2017)","doi":""},{"text":"Ian ; Goodfellow, Pouget-Abadie, ; Jean, Mirza, ; Mehdi, Xu, ; Warde-Farley Bing, ; David, Ozair, ; Sherjil, Courville, ; Aaron, Yoshua Bengio, Advances in neural information processing systems. Generative adversarial nets[Goodfellow u. a. 2014] GOODFELLOW, Ian ; POUGET-ABADIE, Jean ; MIRZA, Mehdi ; XU, Bing ; WARDE-FARLEY, David ; OZAIR, Sherjil ; COURVILLE, Aaron ; BENGIO, Yoshua: Gen- erative adversarial nets. In: Advances in neural information processing systems, 2014, S. 2672- 2680","doi":""},{"text":"The learning machines. Nicola ; Jones, Alex ; Krizhevsky, Sutskever, ; Ilya, Geoffrey E Hinton, Nr. 7482Imagenet classification with deep convolutional neural networks. 505Advances in neural information processing systems[Jones 2014] JONES, Nicola: The learning machines. In: Nature 505 (2014), Nr. 7482, S. 146 [Krizhevsky u. a. 2012] KRIZHEVSKY, Alex ; SUTSKEVER, Ilya ; HINTON, Geoffrey E.: Ima- genet classification with deep convolutional neural networks. In: Advances in neural information processing systems, 2012, S. 1097-1105","doi":""},{"text":". Yuekang ; Li, Chen, ; Bihuan, Chandramohan, ; Mahinthan, Lin, Shang-Wei, LI, Yuekang ; CHEN, Bihuan ; CHANDRAMOHAN, Mahinthan ; LIN, Shang-Wei ;","doi":""},{"text":"Steelix: Program-state Based Binary Fuzzing. Yang ; Liu, Alwen Tiu, http:/doi.acm.org/10.1145/3106237.3106295978-1-4503-5105-8Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering. the 2017 11th Joint Meeting on Foundations of Software EngineeringNew York, NY, USAACMLIU, Yang ; TIU, Alwen: Steelix: Program-state Based Binary Fuzzing. In: Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineer- ing. New York, NY, USA : ACM, 2017 (ESEC/FSE 2017), S. 627-637. -URL http://doi.acm.org/10.1145/3106237.3106295. -ISBN 978-1-4503-5105-8","doi":""},{"text":". Kin - Ma, Keung, Ma u. a. 2011[Ma u. a. 2011] MA, Kin-Keung ;","doi":""},{"text":"Directed symbolic execution. Yit Phang, ; Khoo, Foster, ; Jeffrey, Michael Hicks, Static Analysis. YIT PHANG, Khoo ; FOSTER, Jeffrey ; HICKS, Michael: Directed symbolic execution. In: Static Analysis (2011), S. 95-111","doi":""},{"text":". Mnih, ; Volodymyr, Kavukcuoglu, ; Koray, Silver, ; David, Alex Graves, Mnih u. a. 2013[Mnih u. a. 2013] MNIH, Volodymyr ; KAVUKCUOGLU, Koray ; SILVER, David ; GRAVES, Alex ;","doi":""},{"text":"Playing atari with deep reinforcement learning. Antonoglou, ; Ioannis, Wierstra, ; Daan, Martin Riedmiller, arXiv:1312.5602arXiv preprintANTONOGLOU, Ioannis ; WIERSTRA, Daan ; RIEDMILLER, Martin: Playing atari with deep reinforcement learning. In: arXiv preprint arXiv:1312.5602 (2013)","doi":""},{"text":"Separation logic: A logic for shared mutable data structures. John C Reynolds, Logic in Computer Science, 2002. Proceedings. 17th Annual IEEE Symposium on IEEE. [Reynolds 2002] REYNOLDS, John C.: Separation logic: A logic for shared mutable data struc- tures. In: Logic in Computer Science, 2002. Proceedings. 17th Annual IEEE Symposium on IEEE (Veranst.), 2002, S. 55-74","doi":""},{"text":"Long short-term memory recurrent neural network architectures for large scale acoustic modeling. Sak, ; Haşim, Senior, ; Andrew, Françoise Beaufays, Fifteenth Annual Conference of the International Speech Communication Association. Sak u. a. 2014[Sak u. a. 2014] SAK, Haşim ; SENIOR, Andrew ; BEAUFAYS, Françoise: Long short-term memory recurrent neural network architectures for large scale acoustic modeling. In: Fifteenth Annual Conference of the International Speech Communication Association, 2014","doi":""},{"text":"All you ever wanted to know about dynamic taint analysis and forward symbolic execution (but might have been afraid to ask). Edward J ; Schwartz, Avgerinos, ; Thanassis, David Brumley, Security and privacy (SP), 2010 IEEE symposium on IEEE. Schwartz u. a. 2010[Schwartz u. a. 2010] SCHWARTZ, Edward J. ; AVGERINOS, Thanassis ; BRUMLEY, David: All you ever wanted to know about dynamic taint analysis and forward symbolic execution (but might have been afraid to ask). In: Security and privacy (SP), 2010 IEEE symposium on IEEE (Veranst.), 2010, S. 317-331","doi":""},{"text":". Yan ; Shoshitaishvili, Wang, ; Ruoyu, Christopher Salls, Shoshitaishvili u. a. 2016[Shoshitaishvili u. a. 2016] SHOSHITAISHVILI, Yan ; WANG, Ruoyu ; SALLS, Christopher ;","doi":""},{"text":"Sok:(state of) the art of war: Offensive techniques in binary analysis. Nick ; Stephens, Polino, ; Mario, Dutcher, ; Andrew, Grosen, ; John, Feng, ; Siji, Christophe ; Hauser, Christopher U A Kruegel, Security and Privacy (SP), 2016 IEEE Symposium on IEEE. STEPHENS, Nick ; POLINO, Mario ; DUTCHER, Andrew ; GROSEN, John ; FENG, Siji ; HAUSER, Christophe ; KRUEGEL, Christopher u. a.: Sok:(state of) the art of war: Offensive techniques in binary analysis. In: Security and Privacy (SP), 2016 IEEE Symposium on IEEE (Veranst.), 2016, S. 138-157","doi":""},{"text":". David ; Silver, Aja ; Huang, Chris J ; Maddison, Guez, ; Arthur, Sifre, ; Laurent, Van Den, Driessche, ; George, Schrittwieser, ; Julian, Ioannis Antonoglou, Silver u. a. 2016[Silver u. a. 2016] SILVER, David ; HUANG, Aja ; MADDISON, Chris J. ; GUEZ, Arthur ; SIFRE, Laurent ; VAN DEN DRIESSCHE, George ; SCHRITTWIESER, Julian ; ANTONOGLOU, Ioannis ;","doi":""},{"text":"Mastering the game of Go with deep neural networks and tree search. Veda ; Panneershelvam, Marc U A Lanctot, Nr. 7587Nature. 529PANNEERSHELVAM, Veda ; LANCTOT, Marc u. a.: Mastering the game of Go with deep neural networks and tree search. In: Nature 529 (2016), Nr. 7587, S. 484-489","doi":""},{"text":". Nick ; Stephens, Grosen, ; John, Salls, ; Christopher, Andrew Dutcher, Stephens u. a. 2016[Stephens u. a. 2016] STEPHENS, Nick ; GROSEN, John ; SALLS, Christopher ; DUTCHER, An- drew ;","doi":""},{"text":". Ruoyu ; Wang, Corbetta, ; Jacopo, Shoshitaishvili, ; Yan, Christopher Kruegel, WANG, Ruoyu ; CORBETTA, Jacopo ; SHOSHITAISHVILI, Yan ; KRUEGEL, Christopher ;","doi":""},{"text":"Giovanni Vigna, Driller: Augmenting Fuzzing Through Selective Symbolic Execution. 16NDSS BdVIGNA, Giovanni: Driller: Augmenting Fuzzing Through Selective Symbolic Execution. In: NDSS Bd. 16, 2016, S. 1-16","doi":""},{"text":"Skyfire: Data-driven seed generation for fuzzing. Junjie ; Wang, Chen, ; Bihuan, Wei, ; Lei, Yang Liu, [Wang u. a. 2017] WANG, Junjie ; CHEN, Bihuan ; WEI, Lei ; LIU, Yang: Skyfire: Data-driven seed generation for fuzzing. 2017","doi":""},{"text":"Scheduling black-box mutational fuzzing. Woo, ; Maverick, Sang K ; Cha, Gottlieb, ; Samantha, David Brumley, Proceedings of the. theWoo u. a. 2013[Woo u. a. 2013] WOO, Maverick ; CHA, Sang K. ; GOTTLIEB, Samantha ; BRUMLEY, David: Scheduling black-box mutational fuzzing. In: Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security ACM (Veranst.), 2013, S. 511-522","doi":""},{"text":"Ethereum: A secure decentralised generalised transaction ledger. ] Wood, Gavin, Ethereum Project Yellow Paper. 151[Wood 2014] WOOD, Gavin: Ethereum: A secure decentralised generalised transaction ledger. In: Ethereum Project Yellow Paper 151 (2014)","doi":""},{"text":"Google's neural machine translation system: Bridging the gap between human and machine translation. Yonghui ; Wu, Schuster, ; Mike, Chen, ; Zhifeng, Le, V ; Quoc, Norouzi, ; Mohammad, Wolfgang ; Macherey, Krikun, ; Maxim, Cao, ; Yuan, Gao, ; Qin, Klaus U Macherey, arXiv:1609.08144arXiv preprintWu u. a. 2016[Wu u. a. 2016] WU, Yonghui ; SCHUSTER, Mike ; CHEN, Zhifeng ; LE, Quoc V. ; NOROUZI, Mohammad ; MACHEREY, Wolfgang ; KRIKUN, Maxim ; CAO, Yuan ; GAO, Qin ; MACHEREY, Klaus u. a.: Google's neural machine translation system: Bridging the gap between human and machine translation. In: arXiv preprint arXiv:1609.08144 (2016)","doi":""},{"text":"ZALEWSKI, Michał: American Fuzzy Lop (AFL) fuzzer. [Zalewski 2015] ZALEWSKI, Michał: American Fuzzy Lop (AFL) fuzzer (2015). 2015","doi":""}],"string":"[\n {\n \"text\": \"Testing telecoms software with quviq quickcheck. Thomas ; Arts, Hughes, ; John, Johansson, ; Joakim, Ulf Wiger, Proceedings of the. the. 2006] ARTS, Thomas ; HUGHES, John ; JOHANSSON, Joakim ; WIGER, Ulf: Testing telecoms software with quviq quickcheck. In: Proceedings of the 2006 ACM SIGPLAN workshop on Erlang ACM (Veranst.), 2006, S. 2-10\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Yoshua ; Bengio, Ian J ; Goodfellow, Aaron Courville, Nature. 521Bengio u. a. 2015[Bengio u. a. 2015] BENGIO, Yoshua ; GOODFELLOW, Ian J. ; COURVILLE, Aaron: Deep learn- ing. In: Nature 521 (2015), S. 436-444\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Marcel ; Böhme, Pham, Van-Thuan, Böhme u. a. 2016[Böhme u. a. 2016] BÖHME, Marcel ; PHAM, Van-Thuan ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Abhik Roychoudhury, http:/doi.acm.org/10.1145/2976749.2978428978-1-4503-4139-4Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. the 2016 ACM SIGSAC Conference on Computer and Communications SecurityNew York, NY, USAACM16Coveragebased Greybox Fuzzing As Markov ChainROYCHOUDHURY, Abhik: Coverage- based Greybox Fuzzing As Markov Chain. In: Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security. New York, NY, USA : ACM, 2016 (CCS '16), S. 1032-1043. -URL http://doi.acm.org/10.1145/2976749.2978428. -ISBN 978-1-4503-4139-4\",\n \"doi\": \"http:/doi.acm.org/10.1145/2976749.2978428\"\n },\n {\n \"text\": \"Survey of Automated Vulnerability Detection and Exploit Generation Techniques in Cyber Reasoning Systems. Teresa N Brooks, CoRR abs/1702.06162[Brooks 2017] BROOKS, Teresa N.: Survey of Automated Vulnerability Detection and Exploit Generation Techniques in Cyber Reasoning Systems. In: CoRR abs/1702.06162 (2017). -URL http://arxiv.org/abs/1702.06162\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Boolector: An efficient SMT solver for bit-vectors and arrays. Robert ; Brummayer, Armin Biere, Tools and Algorithms for the Construction and Analysis of Systems. Brummayer und Biere[Brummayer und Biere 2009] BRUMMAYER, Robert ; BIERE, Armin: Boolector: An efficient SMT solver for bit-vectors and arrays. In: Tools and Algorithms for the Construction and Analysis of Systems (2009), S. 174-177\",\n \"doi\": \"\"\n },\n {\n \"text\": \"KLEE: Unassisted and Automatic Generation of High-Coverage Tests for Complex Systems Programs. Cristian ; Cadar, Dunbar, ; Daniel, Engler, R U Dawson, OSDI Bd. 8Cadar u. a.[Cadar u. a. 2008] CADAR, Cristian ; DUNBAR, Daniel ; ENGLER, Dawson R. u. a.: KLEE: Unassisted and Automatic Generation of High-Coverage Tests for Complex Systems Programs. In: OSDI Bd. 8, 2008, S. 209-224\",\n \"doi\": \"\"\n },\n {\n \"text\": \"QuickCheck: a lightweight tool for random testing of Haskell programs. Claessen, ; Koen, John Hughes, CHOLLET, François: Keras. 46Acm sigplan notices. Nr. 4, S[Chollet 2017] CHOLLET, François: Keras (2015). In: URL http://keras. io (2017) [Claessen und Hughes 2011] CLAESSEN, Koen ; HUGHES, John: QuickCheck: a lightweight tool for random testing of Haskell programs. In: Acm sigplan notices 46 (2011), Nr. 4, S. 53-64\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. Patrick ; Cousot, Radhia Cousot, Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACM. the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACMCousot und Cousot[Cousot und Cousot 1977] COUSOT, Patrick ; COUSOT, Radhia: Abstract interpretation: a uni- fied lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proceedings of the 4th ACM SIGACT-SIGPLAN symposium on Principles of programming languages ACM (Veranst.), 1977, S. 238-252\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Z3: An efficient SMT solver. De Moura, ; Leonardo, Nikolaj Bjørner, Tools and Algorithms for the Construction and Analysis of Systems. De Moura und Bjørner[De Moura und Bjørner 2008] DE MOURA, Leonardo ; BJØRNER, Nikolaj: Z3: An efficient SMT solver. In: Tools and Algorithms for the Construction and Analysis of Systems (2008), S. 337-340\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A program construction and verification tool for separation logic. Dongol, ; Brijesh, Gomes, B ; Victor, Georg Struth, International Conference on Mathematics of Program Construction Springer. Dongol u. a. 2015[Dongol u. a. 2015] DONGOL, Brijesh ; GOMES, Victor B. ; STRUTH, Georg: A program con- struction and verification tool for separation logic. In: International Conference on Mathematics of Program Construction Springer (Veranst.), 2015, S. 137-158\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Enemy of the State: A State-Aware Black-Box Web Vulnerability Scanner. Adam ; Doupé, Cavedon, ; Ludovico, Kruegel, ; Christopher, Giovanni Vigna, USENIX Security Symposium Bd. 14Doupé u. a. 2012[Doupé u. a. 2012] DOUPÉ, Adam ; CAVEDON, Ludovico ; KRUEGEL, Christopher ; VIGNA, Giovanni: Enemy of the State: A State-Aware Black-Box Web Vulnerability Scanner. In: USENIX Security Symposium Bd. 14, 2012\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learn&fuzz: Machine learning for input fuzzing. Patrice ; Godefroid, Peleg, ; Hila, Rishabh Singh, arXiv:1701.07232arXiv preprintGodefroid u. a. 2017[Godefroid u. a. 2017] GODEFROID, Patrice ; PELEG, Hila ; SINGH, Rishabh: Learn&fuzz: Machine learning for input fuzzing. In: arXiv preprint arXiv:1701.07232 (2017)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ian ; Goodfellow, Pouget-Abadie, ; Jean, Mirza, ; Mehdi, Xu, ; Warde-Farley Bing, ; David, Ozair, ; Sherjil, Courville, ; Aaron, Yoshua Bengio, Advances in neural information processing systems. Generative adversarial nets[Goodfellow u. a. 2014] GOODFELLOW, Ian ; POUGET-ABADIE, Jean ; MIRZA, Mehdi ; XU, Bing ; WARDE-FARLEY, David ; OZAIR, Sherjil ; COURVILLE, Aaron ; BENGIO, Yoshua: Gen- erative adversarial nets. In: Advances in neural information processing systems, 2014, S. 2672- 2680\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The learning machines. Nicola ; Jones, Alex ; Krizhevsky, Sutskever, ; Ilya, Geoffrey E Hinton, Nr. 7482Imagenet classification with deep convolutional neural networks. 505Advances in neural information processing systems[Jones 2014] JONES, Nicola: The learning machines. In: Nature 505 (2014), Nr. 7482, S. 146 [Krizhevsky u. a. 2012] KRIZHEVSKY, Alex ; SUTSKEVER, Ilya ; HINTON, Geoffrey E.: Ima- genet classification with deep convolutional neural networks. In: Advances in neural information processing systems, 2012, S. 1097-1105\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Yuekang ; Li, Chen, ; Bihuan, Chandramohan, ; Mahinthan, Lin, Shang-Wei, LI, Yuekang ; CHEN, Bihuan ; CHANDRAMOHAN, Mahinthan ; LIN, Shang-Wei ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Steelix: Program-state Based Binary Fuzzing. Yang ; Liu, Alwen Tiu, http:/doi.acm.org/10.1145/3106237.3106295978-1-4503-5105-8Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering. the 2017 11th Joint Meeting on Foundations of Software EngineeringNew York, NY, USAACMLIU, Yang ; TIU, Alwen: Steelix: Program-state Based Binary Fuzzing. In: Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineer- ing. New York, NY, USA : ACM, 2017 (ESEC/FSE 2017), S. 627-637. -URL http://doi.acm.org/10.1145/3106237.3106295. -ISBN 978-1-4503-5105-8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Kin - Ma, Keung, Ma u. a. 2011[Ma u. a. 2011] MA, Kin-Keung ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Directed symbolic execution. Yit Phang, ; Khoo, Foster, ; Jeffrey, Michael Hicks, Static Analysis. YIT PHANG, Khoo ; FOSTER, Jeffrey ; HICKS, Michael: Directed symbolic execution. In: Static Analysis (2011), S. 95-111\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Mnih, ; Volodymyr, Kavukcuoglu, ; Koray, Silver, ; David, Alex Graves, Mnih u. a. 2013[Mnih u. a. 2013] MNIH, Volodymyr ; KAVUKCUOGLU, Koray ; SILVER, David ; GRAVES, Alex ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Playing atari with deep reinforcement learning. Antonoglou, ; Ioannis, Wierstra, ; Daan, Martin Riedmiller, arXiv:1312.5602arXiv preprintANTONOGLOU, Ioannis ; WIERSTRA, Daan ; RIEDMILLER, Martin: Playing atari with deep reinforcement learning. In: arXiv preprint arXiv:1312.5602 (2013)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Separation logic: A logic for shared mutable data structures. John C Reynolds, Logic in Computer Science, 2002. Proceedings. 17th Annual IEEE Symposium on IEEE. [Reynolds 2002] REYNOLDS, John C.: Separation logic: A logic for shared mutable data struc- tures. In: Logic in Computer Science, 2002. Proceedings. 17th Annual IEEE Symposium on IEEE (Veranst.), 2002, S. 55-74\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Long short-term memory recurrent neural network architectures for large scale acoustic modeling. Sak, ; Haşim, Senior, ; Andrew, Françoise Beaufays, Fifteenth Annual Conference of the International Speech Communication Association. Sak u. a. 2014[Sak u. a. 2014] SAK, Haşim ; SENIOR, Andrew ; BEAUFAYS, Françoise: Long short-term memory recurrent neural network architectures for large scale acoustic modeling. In: Fifteenth Annual Conference of the International Speech Communication Association, 2014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"All you ever wanted to know about dynamic taint analysis and forward symbolic execution (but might have been afraid to ask). Edward J ; Schwartz, Avgerinos, ; Thanassis, David Brumley, Security and privacy (SP), 2010 IEEE symposium on IEEE. Schwartz u. a. 2010[Schwartz u. a. 2010] SCHWARTZ, Edward J. ; AVGERINOS, Thanassis ; BRUMLEY, David: All you ever wanted to know about dynamic taint analysis and forward symbolic execution (but might have been afraid to ask). In: Security and privacy (SP), 2010 IEEE symposium on IEEE (Veranst.), 2010, S. 317-331\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Yan ; Shoshitaishvili, Wang, ; Ruoyu, Christopher Salls, Shoshitaishvili u. a. 2016[Shoshitaishvili u. a. 2016] SHOSHITAISHVILI, Yan ; WANG, Ruoyu ; SALLS, Christopher ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sok:(state of) the art of war: Offensive techniques in binary analysis. Nick ; Stephens, Polino, ; Mario, Dutcher, ; Andrew, Grosen, ; John, Feng, ; Siji, Christophe ; Hauser, Christopher U A Kruegel, Security and Privacy (SP), 2016 IEEE Symposium on IEEE. STEPHENS, Nick ; POLINO, Mario ; DUTCHER, Andrew ; GROSEN, John ; FENG, Siji ; HAUSER, Christophe ; KRUEGEL, Christopher u. a.: Sok:(state of) the art of war: Offensive techniques in binary analysis. In: Security and Privacy (SP), 2016 IEEE Symposium on IEEE (Veranst.), 2016, S. 138-157\",\n \"doi\": \"\"\n },\n {\n \"text\": \". David ; Silver, Aja ; Huang, Chris J ; Maddison, Guez, ; Arthur, Sifre, ; Laurent, Van Den, Driessche, ; George, Schrittwieser, ; Julian, Ioannis Antonoglou, Silver u. a. 2016[Silver u. a. 2016] SILVER, David ; HUANG, Aja ; MADDISON, Chris J. ; GUEZ, Arthur ; SIFRE, Laurent ; VAN DEN DRIESSCHE, George ; SCHRITTWIESER, Julian ; ANTONOGLOU, Ioannis ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mastering the game of Go with deep neural networks and tree search. Veda ; Panneershelvam, Marc U A Lanctot, Nr. 7587Nature. 529PANNEERSHELVAM, Veda ; LANCTOT, Marc u. a.: Mastering the game of Go with deep neural networks and tree search. In: Nature 529 (2016), Nr. 7587, S. 484-489\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Nick ; Stephens, Grosen, ; John, Salls, ; Christopher, Andrew Dutcher, Stephens u. a. 2016[Stephens u. a. 2016] STEPHENS, Nick ; GROSEN, John ; SALLS, Christopher ; DUTCHER, An- drew ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Ruoyu ; Wang, Corbetta, ; Jacopo, Shoshitaishvili, ; Yan, Christopher Kruegel, WANG, Ruoyu ; CORBETTA, Jacopo ; SHOSHITAISHVILI, Yan ; KRUEGEL, Christopher ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Giovanni Vigna, Driller: Augmenting Fuzzing Through Selective Symbolic Execution. 16NDSS BdVIGNA, Giovanni: Driller: Augmenting Fuzzing Through Selective Symbolic Execution. In: NDSS Bd. 16, 2016, S. 1-16\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Skyfire: Data-driven seed generation for fuzzing. Junjie ; Wang, Chen, ; Bihuan, Wei, ; Lei, Yang Liu, [Wang u. a. 2017] WANG, Junjie ; CHEN, Bihuan ; WEI, Lei ; LIU, Yang: Skyfire: Data-driven seed generation for fuzzing. 2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scheduling black-box mutational fuzzing. Woo, ; Maverick, Sang K ; Cha, Gottlieb, ; Samantha, David Brumley, Proceedings of the. theWoo u. a. 2013[Woo u. a. 2013] WOO, Maverick ; CHA, Sang K. ; GOTTLIEB, Samantha ; BRUMLEY, David: Scheduling black-box mutational fuzzing. In: Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security ACM (Veranst.), 2013, S. 511-522\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ethereum: A secure decentralised generalised transaction ledger. ] Wood, Gavin, Ethereum Project Yellow Paper. 151[Wood 2014] WOOD, Gavin: Ethereum: A secure decentralised generalised transaction ledger. In: Ethereum Project Yellow Paper 151 (2014)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Google's neural machine translation system: Bridging the gap between human and machine translation. Yonghui ; Wu, Schuster, ; Mike, Chen, ; Zhifeng, Le, V ; Quoc, Norouzi, ; Mohammad, Wolfgang ; Macherey, Krikun, ; Maxim, Cao, ; Yuan, Gao, ; Qin, Klaus U Macherey, arXiv:1609.08144arXiv preprintWu u. a. 2016[Wu u. a. 2016] WU, Yonghui ; SCHUSTER, Mike ; CHEN, Zhifeng ; LE, Quoc V. ; NOROUZI, Mohammad ; MACHEREY, Wolfgang ; KRIKUN, Maxim ; CAO, Yuan ; GAO, Qin ; MACHEREY, Klaus u. a.: Google's neural machine translation system: Bridging the gap between human and machine translation. In: arXiv preprint arXiv:1609.08144 (2016)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ZALEWSKI, Michał: American Fuzzy Lop (AFL) fuzzer. [Zalewski 2015] ZALEWSKI, Michał: American Fuzzy Lop (AFL) fuzzer (2015). 2015\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":40,"cells":{"text":{"kind":"string","value":"\nOn the theory of ideal Bose-gas at a finite particle number\n9 Nov 2020\n\nA I Bugrij \nBogolyubov Institute for Theoretical Physics\nNAS of Ukraine\n14-b, Metrolohichna Str03143KyivUkraine\n\nV M Loktev \nBogolyubov Institute for Theoretical Physics\nNAS of Ukraine\n14-b, Metrolohichna Str03143KyivUkraine\n\nNational Technical University of Ukraine \"Igor Sikorsky Kyiv Politechnical Institute\" 37\nPeremogy Ave03056KyivUkraine\n\nOn the theory of ideal Bose-gas at a finite particle number\n9 Nov 20200530Jp0530-d0530Ch Keywords: ideal Bose-gasBose-distributioncanonical ensembles\nThe ideal Bose-gas with finite number N of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymptotic expressions (in the case N ≫ 1) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential µ of the ideal Bose-gas can lie in the range −∞ < µ < ∞ oppositely to the widely adopted opinion that the value of this potential is negative.\n\nIntroduction\n\nThe theory of ideal Bose-gas is referred to the old and rather well studied field of statistical physics of macroscopic systems [1][2][3][4]. However, the real experiments aimed at its verification are performed with systems consisting of a finite number N of the particles. For example, in the experiments with atomic Bose-gases [5][6][7][8] whose results are usually interpreted as the experimental confirmation of the phenomenon of Bose-condensation, the number of particles is at most ∼ 10 4 , or does not attain the Avogadro number. At the same time, no corresponding calculations for the ideal Bose-gas consisting of a finite number of particles are available in the literature. Therefore, the goal of the our paper is to fill this gap. Moreover, we will calculate the observable quantities for all basic statistical ensembles, namely, the grand canonical (GCE), canonical (CE), and microcanonical (MCE) ones.\n\n2 Ideal Bose-gas A quantum particle located in the vessel with impermeable walls with volume V has the discrete energy spectrum ε k , where the index k runs the values 0, 1, . . . , ∞, as the energy ε k of the state increases.\n\nConsider the system consisting of N bosons non-interacting with one another that are placed into the vessel. The system is characterized by a configuration [n], i.e., by a set of the occupation numbers n k . Each of them indicates the number of particles in the k-th state with the energy ε k . It is obvious that n k cannot exceed the number N of particles in the system. In this case, their total number and total energy of the system are given by the equalities:\nN = ∞ k=0 n k , E = ∞ k=0 n k ε k .(1)\nLet us place the vessel into a thermostat with the temperature T . Under the action of thermal fluctuations on the vessel walls, the configuration [n] will vary (stochastically) with the time. As is known, from experiments one obtain the time-averaged values of the observable quantities. In statistical physics for their calculations the ergodic hypothesis is as a rule accepted. According to it, the mean over the time coincides with the mean over the ensemble.\n\nBy definition, the ensemble is a collection of systems with all possible configurations or, in other words, with different distributions of particles over states. The ensembles can be very different. Each specific ensemble is determined by the own distribution function f [n], and the most popular among them are, as mentioned above, GCE, CE, and MCE.\n\n\nGrand canonical ensemble\n\nThe distribution function for GCE takes the form\nf [n] = e −β(E−µN ) = k e −n k β(ε k −µ) ,(2)\nwhere β = (k B T ) −1 stands for the reciprocal temperature, and µ is as above the chemical potential.\n\nSince f [n] (2) is factorized, the partition function of such system can be easily calculated [9]:\nZ = [n] f [n] = N n0=0 e −n0β(ε0−µ) N n1=0 e −n1β(ε1−µ) . . . N n k =0 e −n k β(ε k −µ) . . . = k 1 − e −β(ε k −µ)(N +1) 1 − e −β(ε k −µ) .(3)\nThe mean value of the number of occupation can be also determined and takes the form\nn k = Z −1 [n] n k f [n] = n k − (N + 1)m k ,(4)\nwhere n k and m k are defined as follows:\nn k = 1 e β(ε k −µ) − 1 , m k = 1 e β(ε k −µ)(N +1) − 1 .\nIt is seen that the first term on the right-hand side of (4) is the ordinary Bose-distribution, and the second one introduces the dependence on the number of particles of the system into the average over the ensemble. This point is worth to be noted, since the mean number of occupation following from relations (3) and (4) coincides formally with that for the parastatistics [3] under the condition p = N , though it is quite obvious that the order p of the parastatistics and the number N are different physical quantities without any connection with each other. In addition, the derivation of the corresponding formula in [3] seems not to be quite proper, because the Stirling factorial formula is used in it for the quantities less than 1. Nevertheless, the final result turns out proper and coincides with formula (4) which is obtained with the help of exact calculations.\n\nIt is worth to note that the number n k has a finite value for −∞ < µ < ∞, rather than only for µ < 0. So, the latter is the assertion which is not absolutely true. Indeed, the number of occupation for the lowest state\nn 0 = 1 e β(ε0−µ) − 1 − N + 1 e β(ε0−µ)(N +1) − 1 .(5)\nFrom whence, we get n 0 = N for µ → ∞, n 0 = N/2 for µ → ε 0 , and n 0 = 0 for µ → −∞. In this case, the parameters β and µ can be presented in terms of N and E by means of the system of equations (cf. (1))\nN = ∞ k=0 n k , E = ∞ k=0 n k ε k .(6)\n\nCanonical ensemble\n\nUnlike the previous case, the distribution function for CE takes the form\nf CE [n] = f [n]δ N − k n k .(7)\nThe calculation of the partition function becomes harder due to the presence of the δ-function in (7), but the exact analytic calculation can be carried out.\n\nWe represent δ(N − k n k ) as the integral:\nδ(N − k n k ) = 1 2π π −π dx e ixN k e −ixn k .(8)\nThen the partition function of CE takes the form\nZ CE = [n] f CE [n] = 1 2π π −π dx e −w(x)(9)\nwhere\nw(x) = −ixN − k ln 1 − e −[β(ε k −µ)+ix](N +1) 1 − e −[β(ε k −µ)+ix] ,(10)n CE k = Z −1 CE 2π π −π dx e −w(x) 1 e β(ε k −µ)+ix − 1 − N + 1 e [β(ε k −µ)+ix](N +1) − 1 .(11)\nFor N ≫ 1, integrals (9) and (11) can be calculated with the help of the saddle-point method. Corresponding saddle point is determined from the equation\n∂w(x) ∂x = 0 = −i N − k 1 e β(ε k −µ)+ix − 1 − N + 1 e [β(ε k −µ)+ix](N +1) − 1\nwhich holds, due to equalities (6), for x = 0. The second derivative at the saddle point takes the form\nw xx = ∂ 2 w(x) ∂x 2 x=0 = k e β(ε k −µ) (e β (ε k − µ) − 1) 2 − (N + 1) 2 e β(ε k −µ)(N +1) (e β(ε k −µ)(N +1) − 1) 2 .\n(12) As a result, if the condition N ≫ 1 holds, we obtain\nZ CE = Z √ 2πw xx .(13)\nThe mean value of the number of occupation in CE for N ≫ 1 is\nn CE k = −β −1 ∂ ln Z CE ∂ε k = n k − m CE k ,(14)\nwhere\nm CE k = 1 2w xx n k (n k + 1)(2n k + 1) − (N + 1) 3 m k (m k + 1)(2m k + 1) ,(15)\nand the quantities n k and m k are defined in (4). The direct verification gives that the numbers of occupation in the ground state and in the excited ones in CE are, respectively, larger and less, than in GCE.\n\n\nMicrocanonical ensemble\n\nThe distribution function for MCE is easily set and takes the form\nf MCE [n] = f [n]δ(N − k n k )δ(E − k n k ε k ) .(16)\nFor the second δ-function, we also use the integral representation analogous to (8), namely:\nδ E − k n k ε k = 1 2π π −π dye iyE k e −iyn k ε k ,(17)\nand can easily sum over the configurations. Then, for the partition function in MCE, we get (cf. (9))\nZ MCE = [n] f MCE [n] = 1 (2π) 2 π −π dxdy e −w(x,y) ,(18)\nwhere\nw(x, y) = −iN x − iEy − k ln 1 − e −[β(ε k −µ)+ix+iyε k ](N +1) 1 − e −[β(ε k −µ)+ix+iyε k ] .(19)\nIn this case, the mean value of the number of occupation is presented in terms of the integral\nn MCE k = Z −1 MCE (2π) 2 π −π dxdy e −w(x,y) × × 1 e β(ε k −µ)+ix+iyε k − 1 − N + 1 e [β(ε k −µ)+ix+iyε k ](N +1) − 1 ,(20)\nwhich can be considered as a generalization of integral (11). In the calculation of the partition function (18), we take N ≫ 1 and use the saddle-point method whose point has the coordinates x = y = 0 . As a result, we obtain\nZ MCE = 1 2π Z √ d ,(21)\nwhere\nd = w xx w yy − w 2 xy ,(22)w xx = ∂ 2 w(x, y) ∂x 2 x=y=0 , w xy = ∂ 2 w(x, y) ∂x∂y x=y=0 , w yy = ∂ 2 w(x, y) ∂y 2 x=y=0 , n MCE k = n k − m MCE k ,(23)\nand\nm MCE k = 1 2d w yy + ε 2 k w xx b k − 2ε k w xx − w xy a k β ,(24)\na k = n k (n k + 1) − (N + 1) 2 m k (m k + 1), b k = n k (n k + 1)(2n k + 1) − (N + 1) 3 m k (m k + 1)(2m k + 1).\n\nAs is seen from formulas (22) and (23), the mean values of the numbers of occupation of the ground state and of the excited ones in MCE are, respectively, larger and less, than in the above-considered GCE and CE.\n\n\nConclusions\n\nThe above results are different from the commonly known ones (see, e.g., [1][2][3][4]) and supplement them. In our opinion, the cause lies in that the previous studies (at least, the studies of systems in which the number of particles (bosons, in this case) is given) neglected some obvious fact, namely, the physical limitation imposed on the numbers of occupation n k ≤ N , which requires a special consideration.\n\nNevertheless, even for a sufficiently large number of bosons that usually corresponds to the experiments on the Bose-Einstein condensation of cold atomic gases, their thermodynamic quantities (in particular, the condensate density) depend, as is shown above, on the number N . Therefore, we believe that the presented results can be useful for the thermodynamics and statistical physics of Bose-systems with finite numbers of particles.\n\nWe are grateful to A.S. Kovalev for his interest in the problem and the useful discussions.\n\nThe work was performed in the frame of the programs 0117U00236 and 0116U003191 (State CPCEC 6541210 and 6541230) and the Special scientific program 0117U00240 of the Department of Physics and Astronomy of the NAS of Ukraine.\n\n\nСписок литературы\n \nStatistical Physics. L D Landau, E M Lifshitz, Pergamon PressNew YorkL.D. Landau and E.M. Lifshitz. Statistical Physics. Pergamon Press, New York (1980).\n\nK Huang, Statistical Mechanics. New YorkWileyK. Huang. Statistical Mechanics. Wiley, New York (1987).\n\nStatistical Physics. A Isihara, Acad. PressNew YorkA. Isihara. Statistical Physics. Acad. Press, New York (1971).\n\n. N N Bogolyubov, Selected Works. 2351N.N.Bogolyubov. Selected Works, Vol. 2, Naukova Dumka, Kiev, 1970, p. 351.\n\n. M H Anderson, J R Ensher, M R Vatthews, C E Wieman, E A Cornell, Science. 269M.H. Anderson, J.R.Ensher, M.R.Vatthews, C.E.Wieman, and E.A.Cornell. Science 269, 198-201 (1995).\n\n. K Burnett, M Edwards, C W Clark, Physics Today. 52K.Burnett, M.Edwards, and C.W.Clark. Physics Today 52, 37-42 (1999).\n\n. Y Shin, M Saba, A Schirotzek, T A Pasquini, A E Leanhardt, D E Prithard, W Kttterle, Phys. Rev. Lett. 92150401Y.Shin, M.Saba, A.Schirotzek, T.A.Pasquini, A.E.Leanhardt, D.E.Prithard, and W.Kttterle. Phys. Rev. Lett. 92, 150401 (2004).\n\n. G K Chaudhary, A Chattopadhyay, R Ramakumar, Int. J. Mod. Phys. B. 25G.K.Chaudhary, A.Chattopadhyay, and R.Ramakumar, Int. J. Mod. Phys. B 25, 3927-3940 (2012).\n\n. A I Bugrij, V M Loktev, LTP. 472A.I.Bugrij and V.M.Loktev. LTP 47, No. 2 (2021).\n"},"corpus_id":{"kind":"number","value":226281656,"string":"226,281,656"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2011.04625\",\n \"mag\": 3098998719,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2011.04625v1.pdf\"\n ],\n \"pdf_sha\": \"c5d76f70a82ef95a4285dc4b411737dd39d54a0b\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"On the theory of ideal Bose-gas at a finite particle number"},"authors":{"kind":"list like","value":["A I Bugrij \nBogolyubov Institute for Theoretical Physics\nNAS of Ukraine\n14-b, Metrolohichna Str03143KyivUkraine","V M Loktev \nBogolyubov Institute for Theoretical Physics\nNAS of Ukraine\n14-b, Metrolohichna Str03143KyivUkraine\n\nNational Technical University of Ukraine \"Igor Sikorsky Kyiv Politechnical Institute\" 37\nPeremogy Ave03056KyivUkraine"],"string":"[\n \"A I Bugrij \\nBogolyubov Institute for Theoretical Physics\\nNAS of Ukraine\\n14-b, Metrolohichna Str03143KyivUkraine\",\n \"V M Loktev \\nBogolyubov Institute for Theoretical Physics\\nNAS of Ukraine\\n14-b, Metrolohichna Str03143KyivUkraine\\n\\nNational Technical University of Ukraine \\\"Igor Sikorsky Kyiv Politechnical Institute\\\" 37\\nPeremogy Ave03056KyivUkraine\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The ideal Bose-gas with finite number N of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymptotic expressions (in the case N ≫ 1) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential µ of the ideal Bose-gas can lie in the range −∞ < µ < ∞ oppositely to the widely adopted opinion that the value of this potential is negative."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The theory of ideal Bose-gas is referred to the old and rather well studied field of statistical physics of macroscopic systems [1][2][3][4]. However, the real experiments aimed at its verification are performed with systems consisting of a finite number N of the particles. For example, in the experiments with atomic Bose-gases [5][6][7][8] whose results are usually interpreted as the experimental confirmation of the phenomenon of Bose-condensation, the number of particles is at most ∼ 10 4 , or does not attain the Avogadro number. At the same time, no corresponding calculations for the ideal Bose-gas consisting of a finite number of particles are available in the literature. Therefore, the goal of the our paper is to fill this gap. Moreover, we will calculate the observable quantities for all basic statistical ensembles, namely, the grand canonical (GCE), canonical (CE), and microcanonical (MCE) ones.\n2 Ideal Bose-gas A quantum particle located in the vessel with impermeable walls with volume V has the discrete energy spectrum ε k , where the index k runs the values 0, 1, . . . , ∞, as the energy ε k of the state increases.\nConsider the system consisting of N bosons non-interacting with one another that are placed into the vessel. The system is characterized by a configuration [n], i.e., by a set of the occupation numbers n k . Each of them indicates the number of particles in the k-th state with the energy ε k . It is obvious that n k cannot exceed the number N of particles in the system. In this case, their total number and total energy of the system are given by the equalities:\nLet us place the vessel into a thermostat with the temperature T . Under the action of thermal fluctuations on the vessel walls, the configuration [n] will vary (stochastically) with the time. As is known, from experiments one obtain the time-averaged values of the observable quantities. In statistical physics for their calculations the ergodic hypothesis is as a rule accepted. According to it, the mean over the time coincides with the mean over the ensemble.\nBy definition, the ensemble is a collection of systems with all possible configurations or, in other words, with different distributions of particles over states. The ensembles can be very different. Each specific ensemble is determined by the own distribution function f [n], and the most popular among them are, as mentioned above, GCE, CE, and MCE."},{"title":"Grand canonical ensemble","content":"The distribution function for GCE takes the form\nwhere β = (k B T ) −1 stands for the reciprocal temperature, and µ is as above the chemical potential.\nSince f [n] (2) is factorized, the partition function of such system can be easily calculated [9]:\nThe mean value of the number of occupation can be also determined and takes the form\nwhere n k and m k are defined as follows:\nIt is seen that the first term on the right-hand side of (4) is the ordinary Bose-distribution, and the second one introduces the dependence on the number of particles of the system into the average over the ensemble. This point is worth to be noted, since the mean number of occupation following from relations (3) and (4) coincides formally with that for the parastatistics [3] under the condition p = N , though it is quite obvious that the order p of the parastatistics and the number N are different physical quantities without any connection with each other. In addition, the derivation of the corresponding formula in [3] seems not to be quite proper, because the Stirling factorial formula is used in it for the quantities less than 1. Nevertheless, the final result turns out proper and coincides with formula (4) which is obtained with the help of exact calculations.\nIt is worth to note that the number n k has a finite value for −∞ < µ < ∞, rather than only for µ < 0. So, the latter is the assertion which is not absolutely true. Indeed, the number of occupation for the lowest state\nFrom whence, we get n 0 = N for µ → ∞, n 0 = N/2 for µ → ε 0 , and n 0 = 0 for µ → −∞. In this case, the parameters β and µ can be presented in terms of N and E by means of the system of equations (cf. (1))"},{"title":"Canonical ensemble","content":"Unlike the previous case, the distribution function for CE takes the form\nThe calculation of the partition function becomes harder due to the presence of the δ-function in (7), but the exact analytic calculation can be carried out.\nWe represent δ(N − k n k ) as the integral:\nThen the partition function of CE takes the form\nwhere\nFor N ≫ 1, integrals (9) and (11) can be calculated with the help of the saddle-point method. Corresponding saddle point is determined from the equation\nwhich holds, due to equalities (6), for x = 0. The second derivative at the saddle point takes the form\n(12) As a result, if the condition N ≫ 1 holds, we obtain\nThe mean value of the number of occupation in CE for N ≫ 1 is\nwhere\nand the quantities n k and m k are defined in (4). The direct verification gives that the numbers of occupation in the ground state and in the excited ones in CE are, respectively, larger and less, than in GCE."},{"title":"Microcanonical ensemble","content":"The distribution function for MCE is easily set and takes the form\nFor the second δ-function, we also use the integral representation analogous to (8), namely:\nand can easily sum over the configurations. Then, for the partition function in MCE, we get (cf. (9))\nwhere\nIn this case, the mean value of the number of occupation is presented in terms of the integral\nwhich can be considered as a generalization of integral (11). In the calculation of the partition function (18), we take N ≫ 1 and use the saddle-point method whose point has the coordinates x = y = 0 . As a result, we obtain\nwhere\nand\na k = n k (n k + 1) − (N + 1) 2 m k (m k + 1), b k = n k (n k + 1)(2n k + 1) − (N + 1) 3 m k (m k + 1)(2m k + 1).\nAs is seen from formulas (22) and (23), the mean values of the numbers of occupation of the ground state and of the excited ones in MCE are, respectively, larger and less, than in the above-considered GCE and CE."},{"title":"Conclusions","content":"The above results are different from the commonly known ones (see, e.g., [1][2][3][4]) and supplement them. In our opinion, the cause lies in that the previous studies (at least, the studies of systems in which the number of particles (bosons, in this case) is given) neglected some obvious fact, namely, the physical limitation imposed on the numbers of occupation n k ≤ N , which requires a special consideration.\nNevertheless, even for a sufficiently large number of bosons that usually corresponds to the experiments on the Bose-Einstein condensation of cold atomic gases, their thermodynamic quantities (in particular, the condensate density) depend, as is shown above, on the number N . Therefore, we believe that the presented results can be useful for the thermodynamics and statistical physics of Bose-systems with finite numbers of particles.\nWe are grateful to A.S. Kovalev for his interest in the problem and the useful discussions.\nThe work was performed in the frame of the programs 0117U00236 and 0116U003191 (State CPCEC 6541210 and 6541230) and the Special scientific program 0117U00240 of the Department of Physics and Astronomy of the NAS of Ukraine."},{"title":"Список литературы","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The theory of ideal Bose-gas is referred to the old and rather well studied field of statistical physics of macroscopic systems [1][2][3][4]. However, the real experiments aimed at its verification are performed with systems consisting of a finite number N of the particles. For example, in the experiments with atomic Bose-gases [5][6][7][8] whose results are usually interpreted as the experimental confirmation of the phenomenon of Bose-condensation, the number of particles is at most ∼ 10 4 , or does not attain the Avogadro number. At the same time, no corresponding calculations for the ideal Bose-gas consisting of a finite number of particles are available in the literature. Therefore, the goal of the our paper is to fill this gap. Moreover, we will calculate the observable quantities for all basic statistical ensembles, namely, the grand canonical (GCE), canonical (CE), and microcanonical (MCE) ones.\\n2 Ideal Bose-gas A quantum particle located in the vessel with impermeable walls with volume V has the discrete energy spectrum ε k , where the index k runs the values 0, 1, . . . , ∞, as the energy ε k of the state increases.\\nConsider the system consisting of N bosons non-interacting with one another that are placed into the vessel. The system is characterized by a configuration [n], i.e., by a set of the occupation numbers n k . Each of them indicates the number of particles in the k-th state with the energy ε k . It is obvious that n k cannot exceed the number N of particles in the system. In this case, their total number and total energy of the system are given by the equalities:\\nLet us place the vessel into a thermostat with the temperature T . Under the action of thermal fluctuations on the vessel walls, the configuration [n] will vary (stochastically) with the time. As is known, from experiments one obtain the time-averaged values of the observable quantities. In statistical physics for their calculations the ergodic hypothesis is as a rule accepted. According to it, the mean over the time coincides with the mean over the ensemble.\\nBy definition, the ensemble is a collection of systems with all possible configurations or, in other words, with different distributions of particles over states. The ensembles can be very different. Each specific ensemble is determined by the own distribution function f [n], and the most popular among them are, as mentioned above, GCE, CE, and MCE.\"\n },\n {\n \"title\": \"Grand canonical ensemble\",\n \"content\": \"The distribution function for GCE takes the form\\nwhere β = (k B T ) −1 stands for the reciprocal temperature, and µ is as above the chemical potential.\\nSince f [n] (2) is factorized, the partition function of such system can be easily calculated [9]:\\nThe mean value of the number of occupation can be also determined and takes the form\\nwhere n k and m k are defined as follows:\\nIt is seen that the first term on the right-hand side of (4) is the ordinary Bose-distribution, and the second one introduces the dependence on the number of particles of the system into the average over the ensemble. This point is worth to be noted, since the mean number of occupation following from relations (3) and (4) coincides formally with that for the parastatistics [3] under the condition p = N , though it is quite obvious that the order p of the parastatistics and the number N are different physical quantities without any connection with each other. In addition, the derivation of the corresponding formula in [3] seems not to be quite proper, because the Stirling factorial formula is used in it for the quantities less than 1. Nevertheless, the final result turns out proper and coincides with formula (4) which is obtained with the help of exact calculations.\\nIt is worth to note that the number n k has a finite value for −∞ < µ < ∞, rather than only for µ < 0. So, the latter is the assertion which is not absolutely true. Indeed, the number of occupation for the lowest state\\nFrom whence, we get n 0 = N for µ → ∞, n 0 = N/2 for µ → ε 0 , and n 0 = 0 for µ → −∞. In this case, the parameters β and µ can be presented in terms of N and E by means of the system of equations (cf. (1))\"\n },\n {\n \"title\": \"Canonical ensemble\",\n \"content\": \"Unlike the previous case, the distribution function for CE takes the form\\nThe calculation of the partition function becomes harder due to the presence of the δ-function in (7), but the exact analytic calculation can be carried out.\\nWe represent δ(N − k n k ) as the integral:\\nThen the partition function of CE takes the form\\nwhere\\nFor N ≫ 1, integrals (9) and (11) can be calculated with the help of the saddle-point method. Corresponding saddle point is determined from the equation\\nwhich holds, due to equalities (6), for x = 0. The second derivative at the saddle point takes the form\\n(12) As a result, if the condition N ≫ 1 holds, we obtain\\nThe mean value of the number of occupation in CE for N ≫ 1 is\\nwhere\\nand the quantities n k and m k are defined in (4). The direct verification gives that the numbers of occupation in the ground state and in the excited ones in CE are, respectively, larger and less, than in GCE.\"\n },\n {\n \"title\": \"Microcanonical ensemble\",\n \"content\": \"The distribution function for MCE is easily set and takes the form\\nFor the second δ-function, we also use the integral representation analogous to (8), namely:\\nand can easily sum over the configurations. Then, for the partition function in MCE, we get (cf. (9))\\nwhere\\nIn this case, the mean value of the number of occupation is presented in terms of the integral\\nwhich can be considered as a generalization of integral (11). In the calculation of the partition function (18), we take N ≫ 1 and use the saddle-point method whose point has the coordinates x = y = 0 . As a result, we obtain\\nwhere\\nand\\na k = n k (n k + 1) − (N + 1) 2 m k (m k + 1), b k = n k (n k + 1)(2n k + 1) − (N + 1) 3 m k (m k + 1)(2m k + 1).\\nAs is seen from formulas (22) and (23), the mean values of the numbers of occupation of the ground state and of the excited ones in MCE are, respectively, larger and less, than in the above-considered GCE and CE.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"The above results are different from the commonly known ones (see, e.g., [1][2][3][4]) and supplement them. In our opinion, the cause lies in that the previous studies (at least, the studies of systems in which the number of particles (bosons, in this case) is given) neglected some obvious fact, namely, the physical limitation imposed on the numbers of occupation n k ≤ N , which requires a special consideration.\\nNevertheless, even for a sufficiently large number of bosons that usually corresponds to the experiments on the Bose-Einstein condensation of cold atomic gases, their thermodynamic quantities (in particular, the condensate density) depend, as is shown above, on the number N . Therefore, we believe that the presented results can be useful for the thermodynamics and statistical physics of Bose-systems with finite numbers of particles.\\nWe are grateful to A.S. Kovalev for his interest in the problem and the useful discussions.\\nThe work was performed in the frame of the programs 0117U00236 and 0116U003191 (State CPCEC 6541210 and 6541230) and the Special scientific program 0117U00240 of the Department of Physics and Astronomy of the NAS of Ukraine.\"\n },\n {\n \"title\": \"Список литературы\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Statistical Physics. L D Landau, E M Lifshitz, Pergamon PressNew YorkL.D. Landau and E.M. Lifshitz. Statistical Physics. Pergamon Press, New York (1980).","doi":""},{"text":"K Huang, Statistical Mechanics. New YorkWileyK. Huang. Statistical Mechanics. Wiley, New York (1987).","doi":""},{"text":"Statistical Physics. A Isihara, Acad. PressNew YorkA. Isihara. Statistical Physics. Acad. Press, New York (1971).","doi":""},{"text":". N N Bogolyubov, Selected Works. 2351N.N.Bogolyubov. Selected Works, Vol. 2, Naukova Dumka, Kiev, 1970, p. 351.","doi":""},{"text":". M H Anderson, J R Ensher, M R Vatthews, C E Wieman, E A Cornell, Science. 269M.H. Anderson, J.R.Ensher, M.R.Vatthews, C.E.Wieman, and E.A.Cornell. Science 269, 198-201 (1995).","doi":""},{"text":". K Burnett, M Edwards, C W Clark, Physics Today. 52K.Burnett, M.Edwards, and C.W.Clark. Physics Today 52, 37-42 (1999).","doi":""},{"text":". Y Shin, M Saba, A Schirotzek, T A Pasquini, A E Leanhardt, D E Prithard, W Kttterle, Phys. Rev. Lett. 92150401Y.Shin, M.Saba, A.Schirotzek, T.A.Pasquini, A.E.Leanhardt, D.E.Prithard, and W.Kttterle. Phys. Rev. Lett. 92, 150401 (2004).","doi":""},{"text":". G K Chaudhary, A Chattopadhyay, R Ramakumar, Int. J. Mod. Phys. B. 25G.K.Chaudhary, A.Chattopadhyay, and R.Ramakumar, Int. J. Mod. Phys. B 25, 3927-3940 (2012).","doi":""},{"text":". A I Bugrij, V M Loktev, LTP. 472A.I.Bugrij and V.M.Loktev. LTP 47, No. 2 (2021).","doi":""}],"string":"[\n {\n \"text\": \"Statistical Physics. L D Landau, E M Lifshitz, Pergamon PressNew YorkL.D. Landau and E.M. Lifshitz. Statistical Physics. Pergamon Press, New York (1980).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"K Huang, Statistical Mechanics. New YorkWileyK. Huang. Statistical Mechanics. Wiley, New York (1987).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statistical Physics. A Isihara, Acad. PressNew YorkA. Isihara. Statistical Physics. Acad. Press, New York (1971).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N N Bogolyubov, Selected Works. 2351N.N.Bogolyubov. Selected Works, Vol. 2, Naukova Dumka, Kiev, 1970, p. 351.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M H Anderson, J R Ensher, M R Vatthews, C E Wieman, E A Cornell, Science. 269M.H. Anderson, J.R.Ensher, M.R.Vatthews, C.E.Wieman, and E.A.Cornell. Science 269, 198-201 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Burnett, M Edwards, C W Clark, Physics Today. 52K.Burnett, M.Edwards, and C.W.Clark. Physics Today 52, 37-42 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Shin, M Saba, A Schirotzek, T A Pasquini, A E Leanhardt, D E Prithard, W Kttterle, Phys. Rev. Lett. 92150401Y.Shin, M.Saba, A.Schirotzek, T.A.Pasquini, A.E.Leanhardt, D.E.Prithard, and W.Kttterle. Phys. Rev. Lett. 92, 150401 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G K Chaudhary, A Chattopadhyay, R Ramakumar, Int. J. Mod. Phys. B. 25G.K.Chaudhary, A.Chattopadhyay, and R.Ramakumar, Int. J. Mod. Phys. B 25, 3927-3940 (2012).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A I Bugrij, V M Loktev, LTP. 472A.I.Bugrij and V.M.Loktev. LTP 47, No. 2 (2021).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":41,"cells":{"text":{"kind":"string","value":"\nEntanglement entropy for even spheres\n27 Sep 2010\n\nJ S Dowker \nSchool of Physics and Astronomy\nTheory Group\nThe University of Manchester\nManchesterEngland\n\nEntanglement entropy for even spheres\n27 Sep 2010\nThe coefficient of the logarithmic term in the entropy on even spheres is re-computed by the local technique of integrating the finite temperature energy density up to the horizon on static d-dimensional de Sitter space and thence finding the entropy by thermodynamics. Numeric evaluation yields the known answer i.e. (minus) the conformal anomaly on the d-sphere. The de Sitter quantities are obtained by conformal transformation of the Rindler ones, themselves obtained, for convenience, from those around a cosmic string. The expressions are given in terms of generalised Bernoulli polynomials for which an identity is derived. The arising spherical conformal anomaly is discussed and a formula is given for it for Branson's higher GJMS Laplacian, P 2k , as an oscillating polynomial in the level, k.\n\nIntroduction.\n\nIn a previous work, [1], I have computed the universal logarithmic term that occurs in the expansion of the entanglement entropy associated with even spheres in the special case when the conically deformed Euclidean space-time used in the entropy recipe is the cyclically factored sphere, S d /Z q . Then the spatial surface is a (d − 2)-sphere (with vanishing extrinsic curvature).\n\nIn this note I return to the same problem but the method this time uses the thermodynamic way of introducing a conical singularity, [2], and harks back, in detail, to a computation of the geometric entropy on de Sitter space, [3], by straightforward integration of the energy density off shell and then picking out the log term in the entropy before going back on shell.\n\nThis off shell procedure is well known and is used mostly in connection with the black hole geometry. Fursaev and Miele, [4], discuss the specific application to de Sitter space, relevant to the present situation. See also De Nardo et al, [5].\n\nA recent paper by Marolf et al also sets up thermodynamics on static de Sitter by integrating the energy density holographically generated.\n\nCalculations of spherical entanglement entropy have been performed by Casini and Huerta, [6], Solodukhin [7,8] and Lohmeyer et al., [9].\n\n\nThe log coefficient in four dimensions\n\nThe geometry I treat is de Sitter space and the method, and formulae, are to be found in [3]. The novelty here is that the expansion away from the horizon, which there stopped at the most divergent term, is here taken far enough to give the logarithm. This is somewhat elementary but the different approach makes contact with the more geometric and holographic methods. Similar, but not identical, calculations have been done over the past fifteen years.\n\nI start by writing the de Sitter metric in the static form,\nds 2 = 4a 2 (1 + Z 2 ) 2 Z 2 d(t/a) 2 − dZ 2 − a 2 1 − Z 2 1 + Z 2 dθ 2 + sin θ 2 dφ 2 ,(1)\nwhere Z is related to the usual static radius r by Z 2 = (a − r)/(a + r). For convenience I set a, the radius of the de Sitter sphere, to unity. There is a slight computational advantage in using Z rather than r, and the metric exhibits a Rindlerlike part. The part of the manifold I am interested in corresponds to the range, 0 ≤ Z ≤ 1, the lower limit being the location of the horizon where the field theory averages diverge. In this section I use a lower cut-off, Z ≥ ǫ/2, to define the global quantities. The basic idea is to compute the energy density, T 0 0 β , at a finite temperature, T = 1/β. Then integrate this over the region indicated to give the internal energy, E(β), and next find the entropy from,\nS(β) = βE ′ (β) − β E ′ (β) dβ = β β dE ′ (β) ,(2)\nwhere E ′ (β) is the finite temperature correction to the energy. The last act is to go on shell by setting β = 2π, the Gibbons-Hawking temperature associated with the horizon. At this temperature, static de Sitter turns into zero temperature de Sitter, [10], the state being the de Sitter invariant 'Euclidean' vacuum because the corresponding Green functions are related by the thermalisation, [11] 2 . See also Laflamme, [12].\n\nExpressions for T 0 0 β in four dimensions were given in [3]. There are various means of finding them. They can be calculated directly in de Sitter space from the re-periodised Green function, [11]. They can be found by conformally transforming the expressions in Rindler space, which can themselves be obtained from those around a cosmic string by transposition of coordinates. This last is the path I will follow here because the cosmic string results have already been calculated in d dimensions, [13]. I give a few subsidiary details. It will be noticed that B in the present situation plays the role that q did in [1].\n\nFor convenience I repeat the formulae on Rindler space with metric,\nds 2 = Z 2 d t 2 − dZ 2 − dx 2 − dy 2 ,(3)T 0 0 β = 1 480π 2 Z 4 (B 2 − 1)(B 2 + 1) , j = 0 = 1 1920π 2 Z 4 (B 2 − 1)(7B 2 + 17) , j = 1/2 = 1 240π 2 Z 4 (B 2 − 1)(B 2 + 11) , j = 1 ,(4)\nwhere B = 2π/β. These quantities vanish on shell, i.e. when B = 1, which is a consequence of the subtraction of the Minkowski values. In fact the last terms in (4), the zero temperature values, do not contribute to the entropy, (2), as is easily seen, and we could simply concentrate on the remaining finite temperature corrections.\n\nThe transition to de Sitter space uses the Bunch-Davies-Brown-Cassidy (BD-BC) relation between the averaged energy momentum tensor densities on conformally flat manifolds developed in [10], where the original references are given. It is not necessary to give this relation in detail. It connects tensor densities under conformal transformations in the expected manner, but there is an additional piece resulting from the integration of the conformal anomaly. For spaces conformal to Rindler, as here, with static conformal factors, the relation also holds for finite temperature quantities. The addition, however, is independent of temperature and can be ignored when calculating the entropy. The finite temperature corrections are thus homogeneously related by 3 ,\n√ g T 0 0 ′ β = √ g R T R 0 0 ′ β ,(5)\nwhere R refers to Rindler (the expressions (4)) and the dash signifies the finite temperature corrections. The relation of the coordinates implied in (5) is explained in [10]. The upshot is that the finite temperature corrections in static de Sitter can be obtained from (4) by the replacement, 1\nZ 4 → (1 + Z 2 ) 4 16Z 4 .(6)\nThe total de Sitter thermal energy is the integral,\nE ′ j (β) = 1 ǫ/2 √ g dZ dθ dφ T 0 0 ′ β = P j (B) − P j (∞) 16π 1 ǫ/2 (1 − Z 2 ) 2 Z 3 dZ ,\nwhere P j (B) is the polynomial in (4), j indicating the spin i.e.\nT R,0 0 = P j (B) 16π 2 Z 4 .\nThe constant, zero de Sitter temperature, term, P j (∞), will not contribute to the entropy when (2) is constructed and hence can be dropped. It then is easy to show that,\nS j (B) = B 1 B dP j (B) 1 ǫ/2 (1 − Z 2 ) 2 8Z 3 dZ ,(7)\nwhich, on shell, gives the simple expression, (using P j (1) = 0),\nS j (B) β=2π = 1 dB P j (B) B 2 1 ǫ/2 (1 − Z 2 ) 2 8Z 3 dZ = 1 ǫ 2 + log(ǫ/2) + ǫ 2 16 1 4 1 dB B 2 P j (B) .(8)\nI concentrate on the log ǫ term as the others are not universal. Its coefficient is the number,\nC j = 1 4 1 dB B 2 P j (B) ,\nwhich is easily computed. I find\nC 0 = 1 90 , C 1/2 = 11 180 , C 1 = 16 45 ,\nand I should comment on these values. The first two agree with (minus) the accepted values of the conformal anomaly on the four sphere if I note that I am using a Weyl neutrino, as in [10]. However the spin one value differs from the standard Maxwell conformal anomaly, −62/90, e.g. [15]. In [16], a relevant heat-kernel coefficient (that for transverse vectors) is given as −2/45.\n\n\nExtension to higher dimensions\n\nI now extend the procedure to d dimensions, but only for conformal scalars. The two factors in (8) extend separately. This is similar to the analysis of Solodukhin, [8], and Casini and Huerta, [6], where the formulae, following the geometry, likewise split into two.\n\nThe second integral in (8) is geometric and the extension to d dimensions consists of replacing the two-sphere metric in (1) by that for a (d − 2)-sphere. This trivially alters the |g| measure.\n\nThe first integral is field-theoretic. I will again obtain it from the expressions for T 0 0 around a cosmic string in d dimensions, [17], via the Rindler values which are related by,\nT 0 0 Rindler = −(d − 1) T 0 0 string ,\nusing tracelessness of the local T ν µ .\n\nThe general structure is now,\nT 0 0 Rindler = P (d, B) 2 d π d/2 Z d .\nThe BD-BC relation, (5), is still valid for the finite temperature corrections, which is all I need. It is fortunate the conformal anomaly contribution is not required since its d-dimensional form is unknown.\n\nAfter the Rindler to de Sitter replacement, cf (6),\n1 Z d → (1 + Z 2 ) d 2 d Z d ,(9)\nthe expression for the entropy on shell becomes, as before,\nS j (β) β=2π = |S d−2 | 2 2d−3 π d/2−1 1 dB P (d, B) B 2 1 ǫ/2 (1 − Z 2 ) d−2 Z d−1 dZ .(10)\nPicking out the log term in (10) I find its coefficient to be,\nC(d) = −(−1) d/2 |S d−2 | 2 2d−3 π d/2−1 d − 2 d/2 − 1 1 dB P (d, B) B 2 = (−1) d/2 1 2 d−2 Γ(d/2) 1 dB P (d, B) B 2 .(11)\nPerhaps a more fundamental quantity is the free energy and I just write down its off shell finite temperature correction,\nF ′ j (B) = |S d−2 | B 2 2d−4 π d/2−2 B dB P (d, B) B 2 1 ǫ/2 (1 − Z 2 ) d−2 Z d−1 dZ .(12)\nThe polynomial, P (d, B), is actually the combination of two polynomials (see below). However, for the moment, I just present some explicit examples which can be read off from the formulae in [13], especially equns. (10) and (11). 4 I list a few of the P (d, B),\nP (6, B) = 1 945 (B 2 − 1)(10B 4 + 31B 2 + 31) P (8, B) = 1 4725 (B 2 − 1)(7B 2 + 17)(3B 4 + 10B 2 + 17) P (10, B) = 4 51975 (B 2 − 1)(30B 8 + 261B 6 + 1031B 4 + 2219B 2 + 2219) .(13)\n4 The expressions in [13] are for complex scalars. Hence there is a factor of two difference.\n\nThe polynomials do not contain a B 2 term, which is generally true and a consequence of conformal invariance (for scalars).\n\nCompact expressions can be given in terms of generalised Bernoulli polynomials, as briefly mentioned in [13]. Algebra produces the explicit formula in d dimensions as the combination of two polynomials,\nP (d, B) = − 2 d−1 (−1) d/2 Γ(d/2) d (d − 2)! B (d+1) d (d/2|B, 1) + d(d −2) 4 B (d−1) d−2 (d/2 − 1|B, 1) ,(14)\nand it is this combination that accounts for the cancellation of the B 2 terms. Each term in (14) vanishes at B = 1 by virtue of a standard property of Bernoulli polynomials, [18], [19].\n\nSubstitution into (11) then produces, on shell, the log coefficient,\nC(d) = 2 d (d − 2)! 1 dB B 2 B (d+1) d (d/2|B, 1) + d(d − 2) 4 B (d−1) d−2 (d/2 − 1|B, 1) ,(15)\nwhose numerical evaluation yields (minus) the scalar conformal anomalies on the dsphere, as do the calculations mentioned earlier. The bare evaluation itself gives no clue as to why this should be so. In our previous approach, [1], the analytical reason was that the derivative of the conformal anomaly on the orbifolded d-sphere was zero for the non-singular sphere, i.e. on shell, and I make some relevant remarks in the next section.\n\nThe conformal anomaly values are available in the old literature and, for convenience, and interest, I give a few remarks and specifics in the next section.\n\n\nConformal anomaly on spheres\n\nAs has been used above, the conformal anomaly is the constant term in the heat-kernel expansion. 5 There are a number of ways of finding this, based on the standard spectral information for the (conformal) Laplacian on d-spheres (which can be traced back to George Green in the 1830s).\n\nThere is no need to elaborate, as the work of Copeland and Toms, [20], covers the necessary ground. One technique uses the values of the corresponding ζfunction, the required analytic continuation of which was done by expressing it as a series of Hurwitz ζ-functions, which have known properties. Birmingham, [21], employs a similar approach. A list of the numerical results can be found in [20] for scalars and spinors.\n\nAmong other works, I mention the interesting one by Cappelli and D'Appolonio, [22], which employs basically the same method, only slightly more streamlined. It also lists the values.\n\nA different organisation of the spectral information was given in [23,24] and resulted in the compact expression for the heat-kernel coefficients (the conformal anomaly) on a d-sphere,\nC d/2 = 1 d! B (d) d (d/2 − 1 1) + B (d) d (d/2 1) ,(16)\nwhich rapidly evaluates to the known numbers. Spelling out some calculational details, this evaluation can be done using the integral relation, [18], p.171,\nB (n−1) ν (x |1) = 1 0 dt B (n) ν (x + t |1) ,\nwhich produces, from (16),\nC d/2 = 1 d! 1 0 dt B (d+1) d (d/2 + t |1) + B (d+1) d (d/2 − t |1) ,(17)\nwith the product form, (e.g. [18] p.186, [25], §8),\nB (d+1) d (x 1) = (x − 1)(x − 2) . . . (x − d) .\nThis yields an expression identical to that given by Diaz, [26],\nC d/2 = 2 (−1) d d! 1 0 dt d/2−1 i=0 (i 2 − t 2 ) ,(18)\nwhich he obtains on the basis of a holographic technique as the simplest example of a more general construction of the conformal anomaly associated with Branson's GJMS higher Laplacian operators, P 2k , on the sphere, e.g. Branson, [27], Graham, [28], Gover [29], Juhl, [30]. It is possible to show 6 , using the approach of [24], that the general result at level k is,\nC d/2 (k) = 1 k d! k−1 j=0 B (d) d (d/2 + j |1) + B (d) d (d/2 + j + 1 |1) = 2(−1) d/2 k d! k 0 dt d/2−1 i=1 i 2 − t 2 ,(19)\n6 The details will be presented at another time.\n\nwhich evaluates to a polynomial in k, for a given d; for example,\nC 2 (k) = 2 3. 5! k 2 (3k 2 − 5) C 3 (k) = 2 3. 7! k 2 (3k 4 − 21k 2 + 28) C 4 (k) = 2 5. 9! k 2 (5k 6 − 90k 4 + 441k 2 − 540) C 5 (k) = 2 3. 11! k 2 (3k 8 − 110k 6 + 1287k 4 − 5412k 2 + 6336) C 6 (k) = 2 105. 13! k 2 (105k 10 − 6825k 8 + 155155k 6 − 1490775k 4 + 5753748k 2 − 6552000) .(20)\nEvaluation gives agreement with the numbers found by Diaz using holography, [26], eqn.(7.1) and Table 3.\n\nThe extrema of these polynomials lie in the region |k| < d/2 and are located, for positive k, closely below the integers which fact corresponds to the alternating signs at the 'physical' values, k = 1, . . . , d/2.\n\nIn this approach, the operator existence condition on the level, k ≤ d/2 (for d even), is equivalent to the absence of a zero mode in the spectrum of P 2k . (Actually for the Neumann modes on the hemisphere.) It may be possible to relax this condition.\n\nIt is worth noting that (17), (18) and (19) do not involve the Bernoulli numbers, unlike some previously mentioned computations.\n\nReturning to the simplest case, k = 1, equating (16) and (15) gives an identity between generalised Bernoulli polynomials. In fact there is the off shell identity\n1 B B (d) d (d/2 − 1 | B, 1) + B (d) d (d/2 | B, 1) = 2(d − 1) B dB B 2 B (d+1) d (d/2|B, 1) + d(d − 2) 4 B (d−1) d−2 (d/2 − 1|B, 1) .(21)\nThe left-hand side is d! times the conformal anomaly on the orbifolded dsphere (or periodic lune), S d /Z B , discussed in [1] where it was proved that it had an extremum when B = 1, i.e. on shell. This can be confirmed here immediately since (each term in) the integrand vanishes on shell. Reference to (12) shows that this corresponds to an extremum for β F , in thermodynamic language.\n\nBy applying Barnes' derivative formula,\n∂ ∂B 1 B B (d) k (a| B, 1) = − 1 B 2 B (d+1) k (a + B| B, B, 1) , = −(−1) k 1 B 2 B (d+1) k (d − 1 − a + B| B, B, 1) ,(22)\nto (21), I obtain the off shell identity (d is even),\nB (d+1) d (d/2 | B, 1)+ d(d − 2) 4 B (d−1) d−2 (d/2 − 1 | B, 1) = − 1 d(d − 1) B (d+1) d (d/2 + B B, B, 1) ,(23)\nwhich is not obvious without further manipulation. It furnishes a simpler formula for the scalar conformal polynomials, P (d, B), (14),\nP (d, B) = − 2 d−1 (−1) d/2 Γ(d/2) d 2 (d − 1)! B (d+1) d (d/2 + B B, B, 1) .(24)\n\nComments\n\nThe calculation of the entropy in section 4 has similarities with others in this topic, e.g. [8,6]. However, in these, the entropy is usually computed from the global effective action while I have employed a local method which involves integrating the thermal average of the canonical energy obtained, e.g., from the coincidence limit of a bilinear operator acting on the thermal Green function. This gives the total internal energy from which the free energy can be determined and thence the entropy.\n\nThe numerical evaluation is algorithmic and offers no obvious reason why the log coefficient is (minus) the conformal anomaly, although, being conformally invariant, there is not much else it could be. In (15) one has a rather complicated formula for calculating the conformal anomaly on spheres.\n\nI have taken a simple minded approach to the conical singularity and have integrated the local energy density using a basic cut-off, resorting neither to smoothing the geometry nor to integrating against a test function. These procedures might appear more rigorous, but have their own ambiguities. I believe the present method is sufficient to achieve its aim. I also note that the formulae encountered are closely related to expressions occurring in the holographic calculations of Ryu and Takayanagi of the entanglement entropy for a disk, [31] §7.1.2, and also to those of Graham, [32], on volume renormalisation.\n\nThe finite temperature averaged stress tensor on static de Sitter that I have used does not agree with that derived by Marolf et al, [33], on the basis of a dS/CFT duality.\n However see[4].\nThis is already evident for global quantities from the considerations in[14] \nThe trace anomaly is a constant times the conformal anomaly.\nAcknowledgments I wish to thank Danilo Diaz for drawing my attention to the utility of the AdS/CFT correspondence for a computation of the conformal anomaly.\n. J S Dowker, ArXiv:1007.3865Hyperspherical entanglement entropy. Dowker,J.S. Hyperspherical entanglement entropy, ArXiv:1007.3865.\n\n. C G Callan, F Wilczek, Phys. Letts. 33355Callan,C.G. and Wilczek,F. Phys. Letts. B333 (1994) 55.\n\n. J S Dowker, Class. Quant. Grav. 1155Dowker,J.S. Class. Quant. Grav. 11 (1994) L55.\n\n. D V Fursaev, G Miele, Phys. Rev. 49987Fursaev,D.V. and Miele,G. Phys. Rev. D49 (1994) 987.\n\n. L De Nardo, D V Fursaev, G Miele, Class. Quant. Grav. 143269De Nardo,L., Fursaev,D.V. and Miele,G. Class. Quant. Grav. 14 (1987) 3269.\n\n. H Casini, M Huerta, arXiv:1007.1813Entanglement entropy for the n-sphereCasini,H. and Huerta,M. Entanglement entropy for the n-sphere, arXiv:1007.1813.\n\n. S N Solodukhin, Phys. Letts. 665305Solodukhin,S.N. Phys. Letts. B665 (2008) 305.\n\nEntanglement entropy on round spheres. S N Solodukhin, ArXiv: 1008.4314Solodukhin,S.N. Entanglement entropy on round spheres ArXiv: 1008.4314.\n\n. R Lohmayer, H Neuberger, A Schwimmer, S Theisen, Phys. Letts. 685222Lohmayer,R., Neuberger,H, Schwimmer,A. and Theisen,S. Phys. Letts. B685 (2010) 222.\n\n. P Candelas, J S Dowker, Phys. Rev. 192902Candelas,P. and Dowker,J.S. Phys. Rev. D19 (1979) 2902.\n\n. J S Dowker, Phys. Rev. 181856Dowker,J.S. Phys. Rev. D18 (1978) 1856.\n\n. R Laflamme, Nucl. Phys. 324233Laflamme,R. Nucl. Phys. B324 (1989) 233.\n\n. J S Dowker, Phys. Rev. 363742Dowker,J.S. Phys. Rev. D36 (1987) 3742.\n\n. J S Dowker, G Kennedy, J. Phys. 11895Dowker,J.S. and Kennedy,G. J. Phys. A11 (1978) 895.\n\nQuantum fields in curved space. N D Birrell, P C W Davies, Cambridge Univ. PressCambridgeBirrell,N.D, and Davies,P.C.W. Quantum fields in curved space (Cambridge Univ. Press, Cambridge, 1982).\n\n. E Elizalde, M Lygren, D V Vassilevich, J. Math. Phys. 373105Elizalde,E., Lygren, M. and Vassilevich, D.V. J. Math. Phys. 37 (1996) 3105.\n\n. J S Dowker, Phys. Rev. 363095Dowker,J.S. Phys. Rev. D36 (1987) 3095.\n\n. N E Nörlund, Acta Mathematica. 43121Nörlund,N.E. Acta Mathematica 43 (1922) 121.\n\nL M Milne-Thomson, The Calculus of Finite Differences. LondonMacMillanMilne-Thomson,L.M. The Calculus of Finite Differences (MacMillan,London, 1933).\n\n. E Copeland, D J Toms, Class. Quant. Grav. 3431Copeland,E. and Toms,D.J. Class. Quant. Grav. 3 (1986) 431.\n\n. D Birmingham, Phys. Rev. 363037Birmingham,D. Phys. Rev. D36 (1987) 3037.\n\n. A Cappelli, Phys. Lett. 48787Cappelli,A. and D'Appollonio, Phys. Lett. 487B (2000) 87.\n\n. J S Dowker, Comm, Math. Phys. 162633Dowker,J.S. Comm. Math. Phys. 162 (1994) 633.\n\n. J S Dowker, J. Math. Phys. 354989Dowker,J.S. J. Math. Phys. 35 (1994) 4989.\n\n. E W Barnes, Trans. Camb. Phil. Soc. 19426Barnes,E.W. Trans. Camb. Phil. Soc. 19 (1903) 426.\n\n. D E Diaz, JHEP. 7103Diaz,D.E. JHEP 7 (2008)103.\n\n. T Branson, Trans.Am.Math.Soc. 3473671Branson,T. Trans.Am.Math.Soc. 347 (1995) 3671.\n\n. C R Graham, Sigma, 3121Graham,C.R. SIGMA 3 (2007) 121.\n\nLaplacian operators and Q-curvature on conformally Einstein manifolds ArXiv. A R Gover, math.DG/0506037Gover,A.R. Laplacian operators and Q-curvature on conformally Einstein man- ifolds ArXiv:math.DG/0506037.\n\nOn conformally covariant powers of the Laplacian ArXiv:math. A Juhl, /0905.3993Juhl,A. On conformally covariant powers of the Laplacian ArXiv:math.DG/0905.3993.\n\n. S Ryu, T Takayanagi, JHEP. 060845Ryu,S. and Takayanagi,T. JHEP 0608(2006)045.\n\n. C R Graham, Rend, Circ.Mat.Palermo Suppl. No. 6331Graham,C.R. Rend.Circ.Mat.Palermo Suppl. No.63 (2000) 31.\n\nD Marolf, M Rangamani, M Van Raamsdonk, Holographic Models of de Sitter QFTs. Marolf,D., Rangamani,M. and Van Raamsdonk,M. Holographic Models of de Sitter QFTs, ArXiv: 1007.3996.\n"},"corpus_id":{"kind":"number","value":119605808,"string":"119,605,808"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1009.3854\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1009.3854v2.pdf\"\n ],\n \"pdf_sha\": \"9c2404de0a281864d1adec6bb7a8d744be69c94c\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Entanglement entropy for even spheres"},"authors":{"kind":"list like","value":["J S Dowker \nSchool of Physics and Astronomy\nTheory Group\nThe University of Manchester\nManchesterEngland","J S Dowker \nSchool of Physics and Astronomy\nTheory Group\nThe University of Manchester\nManchesterEngland"],"string":"[\n \"J S Dowker \\nSchool of Physics and Astronomy\\nTheory Group\\nThe University of Manchester\\nManchesterEngland\",\n \"J S Dowker \\nSchool of Physics and Astronomy\\nTheory Group\\nThe University of Manchester\\nManchesterEngland\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The coefficient of the logarithmic term in the entropy on even spheres is re-computed by the local technique of integrating the finite temperature energy density up to the horizon on static d-dimensional de Sitter space and thence finding the entropy by thermodynamics. Numeric evaluation yields the known answer i.e. (minus) the conformal anomaly on the d-sphere. The de Sitter quantities are obtained by conformal transformation of the Rindler ones, themselves obtained, for convenience, from those around a cosmic string. The expressions are given in terms of generalised Bernoulli polynomials for which an identity is derived. The arising spherical conformal anomaly is discussed and a formula is given for it for Branson's higher GJMS Laplacian, P 2k , as an oscillating polynomial in the level, k."},"sections":{"kind":"list like","value":[{"title":"Introduction.","content":""},{"title":"Introduction.","content":"In a previous work, [1], I have computed the universal logarithmic term that occurs in the expansion of the entanglement entropy associated with even spheres in the special case when the conically deformed Euclidean space-time used in the entropy recipe is the cyclically factored sphere, S d /Z q . Then the spatial surface is a (d − 2)-sphere (with vanishing extrinsic curvature).\nIn a previous work, [1], I have computed the universal logarithmic term that occurs in the expansion of the entanglement entropy associated with even spheres in the special case when the conically deformed Euclidean space-time used in the entropy recipe is the cyclically factored sphere, S d /Z q . Then the spatial surface is a (d − 2)-sphere (with vanishing extrinsic curvature).\nIn this note I return to the same problem but the method this time uses the thermodynamic way of introducing a conical singularity, [2], and harks back, in detail, to a computation of the geometric entropy on de Sitter space, [3], by straightforward integration of the energy density off shell and then picking out the log term in the entropy before going back on shell.\nIn this note I return to the same problem but the method this time uses the thermodynamic way of introducing a conical singularity, [2], and harks back, in detail, to a computation of the geometric entropy on de Sitter space, [3], by straightforward integration of the energy density off shell and then picking out the log term in the entropy before going back on shell.\nThis off shell procedure is well known and is used mostly in connection with the black hole geometry. Fursaev and Miele, [4], discuss the specific application to de Sitter space, relevant to the present situation. See also De Nardo et al, [5].\nThis off shell procedure is well known and is used mostly in connection with the black hole geometry. Fursaev and Miele, [4], discuss the specific application to de Sitter space, relevant to the present situation. See also De Nardo et al, [5].\nA recent paper by Marolf et al also sets up thermodynamics on static de Sitter by integrating the energy density holographically generated.\nA recent paper by Marolf et al also sets up thermodynamics on static de Sitter by integrating the energy density holographically generated.\nCalculations of spherical entanglement entropy have been performed by Casini and Huerta, [6], Solodukhin [7,8] and Lohmeyer et al., [9].\nCalculations of spherical entanglement entropy have been performed by Casini and Huerta, [6], Solodukhin [7,8] and Lohmeyer et al., [9]."},{"title":"The log coefficient in four dimensions","content":""},{"title":"The log coefficient in four dimensions","content":"The geometry I treat is de Sitter space and the method, and formulae, are to be found in [3]. The novelty here is that the expansion away from the horizon, which there stopped at the most divergent term, is here taken far enough to give the logarithm. This is somewhat elementary but the different approach makes contact with the more geometric and holographic methods. Similar, but not identical, calculations have been done over the past fifteen years.\nThe geometry I treat is de Sitter space and the method, and formulae, are to be found in [3]. The novelty here is that the expansion away from the horizon, which there stopped at the most divergent term, is here taken far enough to give the logarithm. This is somewhat elementary but the different approach makes contact with the more geometric and holographic methods. Similar, but not identical, calculations have been done over the past fifteen years.\nI start by writing the de Sitter metric in the static form,\nI start by writing the de Sitter metric in the static form,\nwhere Z is related to the usual static radius r by Z 2 = (a − r)/(a + r). For convenience I set a, the radius of the de Sitter sphere, to unity. There is a slight computational advantage in using Z rather than r, and the metric exhibits a Rindlerlike part. The part of the manifold I am interested in corresponds to the range, 0 ≤ Z ≤ 1, the lower limit being the location of the horizon where the field theory averages diverge. In this section I use a lower cut-off, Z ≥ ǫ/2, to define the global quantities. The basic idea is to compute the energy density, T 0 0 β , at a finite temperature, T = 1/β. Then integrate this over the region indicated to give the internal energy, E(β), and next find the entropy from,\nwhere Z is related to the usual static radius r by Z 2 = (a − r)/(a + r). For convenience I set a, the radius of the de Sitter sphere, to unity. There is a slight computational advantage in using Z rather than r, and the metric exhibits a Rindlerlike part. The part of the manifold I am interested in corresponds to the range, 0 ≤ Z ≤ 1, the lower limit being the location of the horizon where the field theory averages diverge. In this section I use a lower cut-off, Z ≥ ǫ/2, to define the global quantities. The basic idea is to compute the energy density, T 0 0 β , at a finite temperature, T = 1/β. Then integrate this over the region indicated to give the internal energy, E(β), and next find the entropy from,\nwhere E ′ (β) is the finite temperature correction to the energy. The last act is to go on shell by setting β = 2π, the Gibbons-Hawking temperature associated with the horizon. At this temperature, static de Sitter turns into zero temperature de Sitter, [10], the state being the de Sitter invariant 'Euclidean' vacuum because the corresponding Green functions are related by the thermalisation, [11] 2 . See also Laflamme, [12].\nwhere E ′ (β) is the finite temperature correction to the energy. The last act is to go on shell by setting β = 2π, the Gibbons-Hawking temperature associated with the horizon. At this temperature, static de Sitter turns into zero temperature de Sitter, [10], the state being the de Sitter invariant 'Euclidean' vacuum because the corresponding Green functions are related by the thermalisation, [11] 2 . See also Laflamme, [12].\nExpressions for T 0 0 β in four dimensions were given in [3]. There are various means of finding them. They can be calculated directly in de Sitter space from the re-periodised Green function, [11]. They can be found by conformally transforming the expressions in Rindler space, which can themselves be obtained from those around a cosmic string by transposition of coordinates. This last is the path I will follow here because the cosmic string results have already been calculated in d dimensions, [13]. I give a few subsidiary details. It will be noticed that B in the present situation plays the role that q did in [1].\nExpressions for T 0 0 β in four dimensions were given in [3]. There are various means of finding them. They can be calculated directly in de Sitter space from the re-periodised Green function, [11]. They can be found by conformally transforming the expressions in Rindler space, which can themselves be obtained from those around a cosmic string by transposition of coordinates. This last is the path I will follow here because the cosmic string results have already been calculated in d dimensions, [13]. I give a few subsidiary details. It will be noticed that B in the present situation plays the role that q did in [1].\nFor convenience I repeat the formulae on Rindler space with metric,\nFor convenience I repeat the formulae on Rindler space with metric,\nwhere B = 2π/β. These quantities vanish on shell, i.e. when B = 1, which is a consequence of the subtraction of the Minkowski values. In fact the last terms in (4), the zero temperature values, do not contribute to the entropy, (2), as is easily seen, and we could simply concentrate on the remaining finite temperature corrections.\nwhere B = 2π/β. These quantities vanish on shell, i.e. when B = 1, which is a consequence of the subtraction of the Minkowski values. In fact the last terms in (4), the zero temperature values, do not contribute to the entropy, (2), as is easily seen, and we could simply concentrate on the remaining finite temperature corrections.\nThe transition to de Sitter space uses the Bunch-Davies-Brown-Cassidy (BD-BC) relation between the averaged energy momentum tensor densities on conformally flat manifolds developed in [10], where the original references are given. It is not necessary to give this relation in detail. It connects tensor densities under conformal transformations in the expected manner, but there is an additional piece resulting from the integration of the conformal anomaly. For spaces conformal to Rindler, as here, with static conformal factors, the relation also holds for finite temperature quantities. The addition, however, is independent of temperature and can be ignored when calculating the entropy. The finite temperature corrections are thus homogeneously related by 3 ,\nThe transition to de Sitter space uses the Bunch-Davies-Brown-Cassidy (BD-BC) relation between the averaged energy momentum tensor densities on conformally flat manifolds developed in [10], where the original references are given. It is not necessary to give this relation in detail. It connects tensor densities under conformal transformations in the expected manner, but there is an additional piece resulting from the integration of the conformal anomaly. For spaces conformal to Rindler, as here, with static conformal factors, the relation also holds for finite temperature quantities. The addition, however, is independent of temperature and can be ignored when calculating the entropy. The finite temperature corrections are thus homogeneously related by 3 ,\nwhere R refers to Rindler (the expressions (4)) and the dash signifies the finite temperature corrections. The relation of the coordinates implied in (5) is explained in [10]. The upshot is that the finite temperature corrections in static de Sitter can be obtained from (4) by the replacement, 1\nwhere R refers to Rindler (the expressions (4)) and the dash signifies the finite temperature corrections. The relation of the coordinates implied in (5) is explained in [10]. The upshot is that the finite temperature corrections in static de Sitter can be obtained from (4) by the replacement, 1\nThe total de Sitter thermal energy is the integral,\nThe total de Sitter thermal energy is the integral,\nwhere P j (B) is the polynomial in (4), j indicating the spin i.e.\nwhere P j (B) is the polynomial in (4), j indicating the spin i.e.\nThe constant, zero de Sitter temperature, term, P j (∞), will not contribute to the entropy when (2) is constructed and hence can be dropped. It then is easy to show that,\nThe constant, zero de Sitter temperature, term, P j (∞), will not contribute to the entropy when (2) is constructed and hence can be dropped. It then is easy to show that,\nwhich, on shell, gives the simple expression, (using P j (1) = 0),\nwhich, on shell, gives the simple expression, (using P j (1) = 0),\nI concentrate on the log ǫ term as the others are not universal. Its coefficient is the number,\nI concentrate on the log ǫ term as the others are not universal. Its coefficient is the number,\nwhich is easily computed. I find\nwhich is easily computed. I find\nand I should comment on these values. The first two agree with (minus) the accepted values of the conformal anomaly on the four sphere if I note that I am using a Weyl neutrino, as in [10]. However the spin one value differs from the standard Maxwell conformal anomaly, −62/90, e.g. [15]. In [16], a relevant heat-kernel coefficient (that for transverse vectors) is given as −2/45.\nand I should comment on these values. The first two agree with (minus) the accepted values of the conformal anomaly on the four sphere if I note that I am using a Weyl neutrino, as in [10]. However the spin one value differs from the standard Maxwell conformal anomaly, −62/90, e.g. [15]. In [16], a relevant heat-kernel coefficient (that for transverse vectors) is given as −2/45."},{"title":"Extension to higher dimensions","content":""},{"title":"Extension to higher dimensions","content":"I now extend the procedure to d dimensions, but only for conformal scalars. The two factors in (8) extend separately. This is similar to the analysis of Solodukhin, [8], and Casini and Huerta, [6], where the formulae, following the geometry, likewise split into two.\nI now extend the procedure to d dimensions, but only for conformal scalars. The two factors in (8) extend separately. This is similar to the analysis of Solodukhin, [8], and Casini and Huerta, [6], where the formulae, following the geometry, likewise split into two.\nThe second integral in (8) is geometric and the extension to d dimensions consists of replacing the two-sphere metric in (1) by that for a (d − 2)-sphere. This trivially alters the |g| measure.\nThe second integral in (8) is geometric and the extension to d dimensions consists of replacing the two-sphere metric in (1) by that for a (d − 2)-sphere. This trivially alters the |g| measure.\nThe first integral is field-theoretic. I will again obtain it from the expressions for T 0 0 around a cosmic string in d dimensions, [17], via the Rindler values which are related by,\nThe first integral is field-theoretic. I will again obtain it from the expressions for T 0 0 around a cosmic string in d dimensions, [17], via the Rindler values which are related by,\nusing tracelessness of the local T ν µ .\nusing tracelessness of the local T ν µ .\nThe general structure is now,\nThe general structure is now,\nThe BD-BC relation, (5), is still valid for the finite temperature corrections, which is all I need. It is fortunate the conformal anomaly contribution is not required since its d-dimensional form is unknown.\nThe BD-BC relation, (5), is still valid for the finite temperature corrections, which is all I need. It is fortunate the conformal anomaly contribution is not required since its d-dimensional form is unknown.\nAfter the Rindler to de Sitter replacement, cf (6),\nAfter the Rindler to de Sitter replacement, cf (6),\nthe expression for the entropy on shell becomes, as before,\nthe expression for the entropy on shell becomes, as before,\nPicking out the log term in (10) I find its coefficient to be,\nPicking out the log term in (10) I find its coefficient to be,\nPerhaps a more fundamental quantity is the free energy and I just write down its off shell finite temperature correction,\nPerhaps a more fundamental quantity is the free energy and I just write down its off shell finite temperature correction,\nThe polynomial, P (d, B), is actually the combination of two polynomials (see below). However, for the moment, I just present some explicit examples which can be read off from the formulae in [13], especially equns. (10) and (11). 4 I list a few of the P (d, B),\nThe polynomial, P (d, B), is actually the combination of two polynomials (see below). However, for the moment, I just present some explicit examples which can be read off from the formulae in [13], especially equns. (10) and (11). 4 I list a few of the P (d, B),\n4 The expressions in [13] are for complex scalars. Hence there is a factor of two difference.\n4 The expressions in [13] are for complex scalars. Hence there is a factor of two difference.\nThe polynomials do not contain a B 2 term, which is generally true and a consequence of conformal invariance (for scalars).\nThe polynomials do not contain a B 2 term, which is generally true and a consequence of conformal invariance (for scalars).\nCompact expressions can be given in terms of generalised Bernoulli polynomials, as briefly mentioned in [13]. Algebra produces the explicit formula in d dimensions as the combination of two polynomials,\nCompact expressions can be given in terms of generalised Bernoulli polynomials, as briefly mentioned in [13]. Algebra produces the explicit formula in d dimensions as the combination of two polynomials,\nand it is this combination that accounts for the cancellation of the B 2 terms. Each term in (14) vanishes at B = 1 by virtue of a standard property of Bernoulli polynomials, [18], [19].\nand it is this combination that accounts for the cancellation of the B 2 terms. Each term in (14) vanishes at B = 1 by virtue of a standard property of Bernoulli polynomials, [18], [19].\nSubstitution into (11) then produces, on shell, the log coefficient,\nSubstitution into (11) then produces, on shell, the log coefficient,\nwhose numerical evaluation yields (minus) the scalar conformal anomalies on the dsphere, as do the calculations mentioned earlier. The bare evaluation itself gives no clue as to why this should be so. In our previous approach, [1], the analytical reason was that the derivative of the conformal anomaly on the orbifolded d-sphere was zero for the non-singular sphere, i.e. on shell, and I make some relevant remarks in the next section.\nwhose numerical evaluation yields (minus) the scalar conformal anomalies on the dsphere, as do the calculations mentioned earlier. The bare evaluation itself gives no clue as to why this should be so. In our previous approach, [1], the analytical reason was that the derivative of the conformal anomaly on the orbifolded d-sphere was zero for the non-singular sphere, i.e. on shell, and I make some relevant remarks in the next section.\nThe conformal anomaly values are available in the old literature and, for convenience, and interest, I give a few remarks and specifics in the next section.\nThe conformal anomaly values are available in the old literature and, for convenience, and interest, I give a few remarks and specifics in the next section."},{"title":"Conformal anomaly on spheres","content":""},{"title":"Conformal anomaly on spheres","content":"As has been used above, the conformal anomaly is the constant term in the heat-kernel expansion. 5 There are a number of ways of finding this, based on the standard spectral information for the (conformal) Laplacian on d-spheres (which can be traced back to George Green in the 1830s).\nAs has been used above, the conformal anomaly is the constant term in the heat-kernel expansion. 5 There are a number of ways of finding this, based on the standard spectral information for the (conformal) Laplacian on d-spheres (which can be traced back to George Green in the 1830s).\nThere is no need to elaborate, as the work of Copeland and Toms, [20], covers the necessary ground. One technique uses the values of the corresponding ζfunction, the required analytic continuation of which was done by expressing it as a series of Hurwitz ζ-functions, which have known properties. Birmingham, [21], employs a similar approach. A list of the numerical results can be found in [20] for scalars and spinors.\nThere is no need to elaborate, as the work of Copeland and Toms, [20], covers the necessary ground. One technique uses the values of the corresponding ζfunction, the required analytic continuation of which was done by expressing it as a series of Hurwitz ζ-functions, which have known properties. Birmingham, [21], employs a similar approach. A list of the numerical results can be found in [20] for scalars and spinors.\nAmong other works, I mention the interesting one by Cappelli and D'Appolonio, [22], which employs basically the same method, only slightly more streamlined. It also lists the values.\nAmong other works, I mention the interesting one by Cappelli and D'Appolonio, [22], which employs basically the same method, only slightly more streamlined. It also lists the values.\nA different organisation of the spectral information was given in [23,24] and resulted in the compact expression for the heat-kernel coefficients (the conformal anomaly) on a d-sphere,\nA different organisation of the spectral information was given in [23,24] and resulted in the compact expression for the heat-kernel coefficients (the conformal anomaly) on a d-sphere,\nwhich rapidly evaluates to the known numbers. Spelling out some calculational details, this evaluation can be done using the integral relation, [18], p.171,\nwhich rapidly evaluates to the known numbers. Spelling out some calculational details, this evaluation can be done using the integral relation, [18], p.171,\nwhich produces, from (16),\nwhich produces, from (16),\nwith the product form, (e.g. [18] p.186, [25], §8),\nwith the product form, (e.g. [18] p.186, [25], §8),\nThis yields an expression identical to that given by Diaz, [26],\nThis yields an expression identical to that given by Diaz, [26],\nwhich he obtains on the basis of a holographic technique as the simplest example of a more general construction of the conformal anomaly associated with Branson's GJMS higher Laplacian operators, P 2k , on the sphere, e.g. Branson, [27], Graham, [28], Gover [29], Juhl, [30]. It is possible to show 6 , using the approach of [24], that the general result at level k is,\nwhich he obtains on the basis of a holographic technique as the simplest example of a more general construction of the conformal anomaly associated with Branson's GJMS higher Laplacian operators, P 2k , on the sphere, e.g. Branson, [27], Graham, [28], Gover [29], Juhl, [30]. It is possible to show 6 , using the approach of [24], that the general result at level k is,\n6 The details will be presented at another time.\n6 The details will be presented at another time.\nwhich evaluates to a polynomial in k, for a given d; for example,\nwhich evaluates to a polynomial in k, for a given d; for example,\nEvaluation gives agreement with the numbers found by Diaz using holography, [26], eqn.(7.1) and Table 3.\nEvaluation gives agreement with the numbers found by Diaz using holography, [26], eqn.(7.1) and Table 3.\nThe extrema of these polynomials lie in the region |k| < d/2 and are located, for positive k, closely below the integers which fact corresponds to the alternating signs at the 'physical' values, k = 1, . . . , d/2.\nThe extrema of these polynomials lie in the region |k| < d/2 and are located, for positive k, closely below the integers which fact corresponds to the alternating signs at the 'physical' values, k = 1, . . . , d/2.\nIn this approach, the operator existence condition on the level, k ≤ d/2 (for d even), is equivalent to the absence of a zero mode in the spectrum of P 2k . (Actually for the Neumann modes on the hemisphere.) It may be possible to relax this condition.\nIn this approach, the operator existence condition on the level, k ≤ d/2 (for d even), is equivalent to the absence of a zero mode in the spectrum of P 2k . (Actually for the Neumann modes on the hemisphere.) It may be possible to relax this condition.\nIt is worth noting that (17), (18) and (19) do not involve the Bernoulli numbers, unlike some previously mentioned computations.\nIt is worth noting that (17), (18) and (19) do not involve the Bernoulli numbers, unlike some previously mentioned computations.\nReturning to the simplest case, k = 1, equating (16) and (15) gives an identity between generalised Bernoulli polynomials. In fact there is the off shell identity\nReturning to the simplest case, k = 1, equating (16) and (15) gives an identity between generalised Bernoulli polynomials. In fact there is the off shell identity\nThe left-hand side is d! times the conformal anomaly on the orbifolded dsphere (or periodic lune), S d /Z B , discussed in [1] where it was proved that it had an extremum when B = 1, i.e. on shell. This can be confirmed here immediately since (each term in) the integrand vanishes on shell. Reference to (12) shows that this corresponds to an extremum for β F , in thermodynamic language.\nThe left-hand side is d! times the conformal anomaly on the orbifolded dsphere (or periodic lune), S d /Z B , discussed in [1] where it was proved that it had an extremum when B = 1, i.e. on shell. This can be confirmed here immediately since (each term in) the integrand vanishes on shell. Reference to (12) shows that this corresponds to an extremum for β F , in thermodynamic language.\nBy applying Barnes' derivative formula,\nBy applying Barnes' derivative formula,\nto (21), I obtain the off shell identity (d is even),\nto (21), I obtain the off shell identity (d is even),\nwhich is not obvious without further manipulation. It furnishes a simpler formula for the scalar conformal polynomials, P (d, B), (14),\nwhich is not obvious without further manipulation. It furnishes a simpler formula for the scalar conformal polynomials, P (d, B), (14),"},{"title":"Comments","content":""},{"title":"Comments","content":"The calculation of the entropy in section 4 has similarities with others in this topic, e.g. [8,6]. However, in these, the entropy is usually computed from the global effective action while I have employed a local method which involves integrating the thermal average of the canonical energy obtained, e.g., from the coincidence limit of a bilinear operator acting on the thermal Green function. This gives the total internal energy from which the free energy can be determined and thence the entropy.\nThe calculation of the entropy in section 4 has similarities with others in this topic, e.g. [8,6]. However, in these, the entropy is usually computed from the global effective action while I have employed a local method which involves integrating the thermal average of the canonical energy obtained, e.g., from the coincidence limit of a bilinear operator acting on the thermal Green function. This gives the total internal energy from which the free energy can be determined and thence the entropy.\nThe numerical evaluation is algorithmic and offers no obvious reason why the log coefficient is (minus) the conformal anomaly, although, being conformally invariant, there is not much else it could be. In (15) one has a rather complicated formula for calculating the conformal anomaly on spheres.\nThe numerical evaluation is algorithmic and offers no obvious reason why the log coefficient is (minus) the conformal anomaly, although, being conformally invariant, there is not much else it could be. In (15) one has a rather complicated formula for calculating the conformal anomaly on spheres.\nI have taken a simple minded approach to the conical singularity and have integrated the local energy density using a basic cut-off, resorting neither to smoothing the geometry nor to integrating against a test function. These procedures might appear more rigorous, but have their own ambiguities. I believe the present method is sufficient to achieve its aim. I also note that the formulae encountered are closely related to expressions occurring in the holographic calculations of Ryu and Takayanagi of the entanglement entropy for a disk, [31] §7.1.2, and also to those of Graham, [32], on volume renormalisation.\nI have taken a simple minded approach to the conical singularity and have integrated the local energy density using a basic cut-off, resorting neither to smoothing the geometry nor to integrating against a test function. These procedures might appear more rigorous, but have their own ambiguities. I believe the present method is sufficient to achieve its aim. I also note that the formulae encountered are closely related to expressions occurring in the holographic calculations of Ryu and Takayanagi of the entanglement entropy for a disk, [31] §7.1.2, and also to those of Graham, [32], on volume renormalisation.\nThe finite temperature averaged stress tensor on static de Sitter that I have used does not agree with that derived by Marolf et al, [33], on the basis of a dS/CFT duality.\nThe finite temperature averaged stress tensor on static de Sitter that I have used does not agree with that derived by Marolf et al, [33], on the basis of a dS/CFT duality."}],"string":"[\n {\n \"title\": \"Introduction.\",\n \"content\": \"\"\n },\n {\n \"title\": \"Introduction.\",\n \"content\": \"In a previous work, [1], I have computed the universal logarithmic term that occurs in the expansion of the entanglement entropy associated with even spheres in the special case when the conically deformed Euclidean space-time used in the entropy recipe is the cyclically factored sphere, S d /Z q . Then the spatial surface is a (d − 2)-sphere (with vanishing extrinsic curvature).\\nIn a previous work, [1], I have computed the universal logarithmic term that occurs in the expansion of the entanglement entropy associated with even spheres in the special case when the conically deformed Euclidean space-time used in the entropy recipe is the cyclically factored sphere, S d /Z q . Then the spatial surface is a (d − 2)-sphere (with vanishing extrinsic curvature).\\nIn this note I return to the same problem but the method this time uses the thermodynamic way of introducing a conical singularity, [2], and harks back, in detail, to a computation of the geometric entropy on de Sitter space, [3], by straightforward integration of the energy density off shell and then picking out the log term in the entropy before going back on shell.\\nIn this note I return to the same problem but the method this time uses the thermodynamic way of introducing a conical singularity, [2], and harks back, in detail, to a computation of the geometric entropy on de Sitter space, [3], by straightforward integration of the energy density off shell and then picking out the log term in the entropy before going back on shell.\\nThis off shell procedure is well known and is used mostly in connection with the black hole geometry. Fursaev and Miele, [4], discuss the specific application to de Sitter space, relevant to the present situation. See also De Nardo et al, [5].\\nThis off shell procedure is well known and is used mostly in connection with the black hole geometry. Fursaev and Miele, [4], discuss the specific application to de Sitter space, relevant to the present situation. See also De Nardo et al, [5].\\nA recent paper by Marolf et al also sets up thermodynamics on static de Sitter by integrating the energy density holographically generated.\\nA recent paper by Marolf et al also sets up thermodynamics on static de Sitter by integrating the energy density holographically generated.\\nCalculations of spherical entanglement entropy have been performed by Casini and Huerta, [6], Solodukhin [7,8] and Lohmeyer et al., [9].\\nCalculations of spherical entanglement entropy have been performed by Casini and Huerta, [6], Solodukhin [7,8] and Lohmeyer et al., [9].\"\n },\n {\n \"title\": \"The log coefficient in four dimensions\",\n \"content\": \"\"\n },\n {\n \"title\": \"The log coefficient in four dimensions\",\n \"content\": \"The geometry I treat is de Sitter space and the method, and formulae, are to be found in [3]. The novelty here is that the expansion away from the horizon, which there stopped at the most divergent term, is here taken far enough to give the logarithm. This is somewhat elementary but the different approach makes contact with the more geometric and holographic methods. Similar, but not identical, calculations have been done over the past fifteen years.\\nThe geometry I treat is de Sitter space and the method, and formulae, are to be found in [3]. The novelty here is that the expansion away from the horizon, which there stopped at the most divergent term, is here taken far enough to give the logarithm. This is somewhat elementary but the different approach makes contact with the more geometric and holographic methods. Similar, but not identical, calculations have been done over the past fifteen years.\\nI start by writing the de Sitter metric in the static form,\\nI start by writing the de Sitter metric in the static form,\\nwhere Z is related to the usual static radius r by Z 2 = (a − r)/(a + r). For convenience I set a, the radius of the de Sitter sphere, to unity. There is a slight computational advantage in using Z rather than r, and the metric exhibits a Rindlerlike part. The part of the manifold I am interested in corresponds to the range, 0 ≤ Z ≤ 1, the lower limit being the location of the horizon where the field theory averages diverge. In this section I use a lower cut-off, Z ≥ ǫ/2, to define the global quantities. The basic idea is to compute the energy density, T 0 0 β , at a finite temperature, T = 1/β. Then integrate this over the region indicated to give the internal energy, E(β), and next find the entropy from,\\nwhere Z is related to the usual static radius r by Z 2 = (a − r)/(a + r). For convenience I set a, the radius of the de Sitter sphere, to unity. There is a slight computational advantage in using Z rather than r, and the metric exhibits a Rindlerlike part. The part of the manifold I am interested in corresponds to the range, 0 ≤ Z ≤ 1, the lower limit being the location of the horizon where the field theory averages diverge. In this section I use a lower cut-off, Z ≥ ǫ/2, to define the global quantities. The basic idea is to compute the energy density, T 0 0 β , at a finite temperature, T = 1/β. Then integrate this over the region indicated to give the internal energy, E(β), and next find the entropy from,\\nwhere E ′ (β) is the finite temperature correction to the energy. The last act is to go on shell by setting β = 2π, the Gibbons-Hawking temperature associated with the horizon. At this temperature, static de Sitter turns into zero temperature de Sitter, [10], the state being the de Sitter invariant 'Euclidean' vacuum because the corresponding Green functions are related by the thermalisation, [11] 2 . See also Laflamme, [12].\\nwhere E ′ (β) is the finite temperature correction to the energy. The last act is to go on shell by setting β = 2π, the Gibbons-Hawking temperature associated with the horizon. At this temperature, static de Sitter turns into zero temperature de Sitter, [10], the state being the de Sitter invariant 'Euclidean' vacuum because the corresponding Green functions are related by the thermalisation, [11] 2 . See also Laflamme, [12].\\nExpressions for T 0 0 β in four dimensions were given in [3]. There are various means of finding them. They can be calculated directly in de Sitter space from the re-periodised Green function, [11]. They can be found by conformally transforming the expressions in Rindler space, which can themselves be obtained from those around a cosmic string by transposition of coordinates. This last is the path I will follow here because the cosmic string results have already been calculated in d dimensions, [13]. I give a few subsidiary details. It will be noticed that B in the present situation plays the role that q did in [1].\\nExpressions for T 0 0 β in four dimensions were given in [3]. There are various means of finding them. They can be calculated directly in de Sitter space from the re-periodised Green function, [11]. They can be found by conformally transforming the expressions in Rindler space, which can themselves be obtained from those around a cosmic string by transposition of coordinates. This last is the path I will follow here because the cosmic string results have already been calculated in d dimensions, [13]. I give a few subsidiary details. It will be noticed that B in the present situation plays the role that q did in [1].\\nFor convenience I repeat the formulae on Rindler space with metric,\\nFor convenience I repeat the formulae on Rindler space with metric,\\nwhere B = 2π/β. These quantities vanish on shell, i.e. when B = 1, which is a consequence of the subtraction of the Minkowski values. In fact the last terms in (4), the zero temperature values, do not contribute to the entropy, (2), as is easily seen, and we could simply concentrate on the remaining finite temperature corrections.\\nwhere B = 2π/β. These quantities vanish on shell, i.e. when B = 1, which is a consequence of the subtraction of the Minkowski values. In fact the last terms in (4), the zero temperature values, do not contribute to the entropy, (2), as is easily seen, and we could simply concentrate on the remaining finite temperature corrections.\\nThe transition to de Sitter space uses the Bunch-Davies-Brown-Cassidy (BD-BC) relation between the averaged energy momentum tensor densities on conformally flat manifolds developed in [10], where the original references are given. It is not necessary to give this relation in detail. It connects tensor densities under conformal transformations in the expected manner, but there is an additional piece resulting from the integration of the conformal anomaly. For spaces conformal to Rindler, as here, with static conformal factors, the relation also holds for finite temperature quantities. The addition, however, is independent of temperature and can be ignored when calculating the entropy. The finite temperature corrections are thus homogeneously related by 3 ,\\nThe transition to de Sitter space uses the Bunch-Davies-Brown-Cassidy (BD-BC) relation between the averaged energy momentum tensor densities on conformally flat manifolds developed in [10], where the original references are given. It is not necessary to give this relation in detail. It connects tensor densities under conformal transformations in the expected manner, but there is an additional piece resulting from the integration of the conformal anomaly. For spaces conformal to Rindler, as here, with static conformal factors, the relation also holds for finite temperature quantities. The addition, however, is independent of temperature and can be ignored when calculating the entropy. The finite temperature corrections are thus homogeneously related by 3 ,\\nwhere R refers to Rindler (the expressions (4)) and the dash signifies the finite temperature corrections. The relation of the coordinates implied in (5) is explained in [10]. The upshot is that the finite temperature corrections in static de Sitter can be obtained from (4) by the replacement, 1\\nwhere R refers to Rindler (the expressions (4)) and the dash signifies the finite temperature corrections. The relation of the coordinates implied in (5) is explained in [10]. The upshot is that the finite temperature corrections in static de Sitter can be obtained from (4) by the replacement, 1\\nThe total de Sitter thermal energy is the integral,\\nThe total de Sitter thermal energy is the integral,\\nwhere P j (B) is the polynomial in (4), j indicating the spin i.e.\\nwhere P j (B) is the polynomial in (4), j indicating the spin i.e.\\nThe constant, zero de Sitter temperature, term, P j (∞), will not contribute to the entropy when (2) is constructed and hence can be dropped. It then is easy to show that,\\nThe constant, zero de Sitter temperature, term, P j (∞), will not contribute to the entropy when (2) is constructed and hence can be dropped. It then is easy to show that,\\nwhich, on shell, gives the simple expression, (using P j (1) = 0),\\nwhich, on shell, gives the simple expression, (using P j (1) = 0),\\nI concentrate on the log ǫ term as the others are not universal. Its coefficient is the number,\\nI concentrate on the log ǫ term as the others are not universal. Its coefficient is the number,\\nwhich is easily computed. I find\\nwhich is easily computed. I find\\nand I should comment on these values. The first two agree with (minus) the accepted values of the conformal anomaly on the four sphere if I note that I am using a Weyl neutrino, as in [10]. However the spin one value differs from the standard Maxwell conformal anomaly, −62/90, e.g. [15]. In [16], a relevant heat-kernel coefficient (that for transverse vectors) is given as −2/45.\\nand I should comment on these values. The first two agree with (minus) the accepted values of the conformal anomaly on the four sphere if I note that I am using a Weyl neutrino, as in [10]. However the spin one value differs from the standard Maxwell conformal anomaly, −62/90, e.g. [15]. In [16], a relevant heat-kernel coefficient (that for transverse vectors) is given as −2/45.\"\n },\n {\n \"title\": \"Extension to higher dimensions\",\n \"content\": \"\"\n },\n {\n \"title\": \"Extension to higher dimensions\",\n \"content\": \"I now extend the procedure to d dimensions, but only for conformal scalars. The two factors in (8) extend separately. This is similar to the analysis of Solodukhin, [8], and Casini and Huerta, [6], where the formulae, following the geometry, likewise split into two.\\nI now extend the procedure to d dimensions, but only for conformal scalars. The two factors in (8) extend separately. This is similar to the analysis of Solodukhin, [8], and Casini and Huerta, [6], where the formulae, following the geometry, likewise split into two.\\nThe second integral in (8) is geometric and the extension to d dimensions consists of replacing the two-sphere metric in (1) by that for a (d − 2)-sphere. This trivially alters the |g| measure.\\nThe second integral in (8) is geometric and the extension to d dimensions consists of replacing the two-sphere metric in (1) by that for a (d − 2)-sphere. This trivially alters the |g| measure.\\nThe first integral is field-theoretic. I will again obtain it from the expressions for T 0 0 around a cosmic string in d dimensions, [17], via the Rindler values which are related by,\\nThe first integral is field-theoretic. I will again obtain it from the expressions for T 0 0 around a cosmic string in d dimensions, [17], via the Rindler values which are related by,\\nusing tracelessness of the local T ν µ .\\nusing tracelessness of the local T ν µ .\\nThe general structure is now,\\nThe general structure is now,\\nThe BD-BC relation, (5), is still valid for the finite temperature corrections, which is all I need. It is fortunate the conformal anomaly contribution is not required since its d-dimensional form is unknown.\\nThe BD-BC relation, (5), is still valid for the finite temperature corrections, which is all I need. It is fortunate the conformal anomaly contribution is not required since its d-dimensional form is unknown.\\nAfter the Rindler to de Sitter replacement, cf (6),\\nAfter the Rindler to de Sitter replacement, cf (6),\\nthe expression for the entropy on shell becomes, as before,\\nthe expression for the entropy on shell becomes, as before,\\nPicking out the log term in (10) I find its coefficient to be,\\nPicking out the log term in (10) I find its coefficient to be,\\nPerhaps a more fundamental quantity is the free energy and I just write down its off shell finite temperature correction,\\nPerhaps a more fundamental quantity is the free energy and I just write down its off shell finite temperature correction,\\nThe polynomial, P (d, B), is actually the combination of two polynomials (see below). However, for the moment, I just present some explicit examples which can be read off from the formulae in [13], especially equns. (10) and (11). 4 I list a few of the P (d, B),\\nThe polynomial, P (d, B), is actually the combination of two polynomials (see below). However, for the moment, I just present some explicit examples which can be read off from the formulae in [13], especially equns. (10) and (11). 4 I list a few of the P (d, B),\\n4 The expressions in [13] are for complex scalars. Hence there is a factor of two difference.\\n4 The expressions in [13] are for complex scalars. Hence there is a factor of two difference.\\nThe polynomials do not contain a B 2 term, which is generally true and a consequence of conformal invariance (for scalars).\\nThe polynomials do not contain a B 2 term, which is generally true and a consequence of conformal invariance (for scalars).\\nCompact expressions can be given in terms of generalised Bernoulli polynomials, as briefly mentioned in [13]. Algebra produces the explicit formula in d dimensions as the combination of two polynomials,\\nCompact expressions can be given in terms of generalised Bernoulli polynomials, as briefly mentioned in [13]. Algebra produces the explicit formula in d dimensions as the combination of two polynomials,\\nand it is this combination that accounts for the cancellation of the B 2 terms. Each term in (14) vanishes at B = 1 by virtue of a standard property of Bernoulli polynomials, [18], [19].\\nand it is this combination that accounts for the cancellation of the B 2 terms. Each term in (14) vanishes at B = 1 by virtue of a standard property of Bernoulli polynomials, [18], [19].\\nSubstitution into (11) then produces, on shell, the log coefficient,\\nSubstitution into (11) then produces, on shell, the log coefficient,\\nwhose numerical evaluation yields (minus) the scalar conformal anomalies on the dsphere, as do the calculations mentioned earlier. The bare evaluation itself gives no clue as to why this should be so. In our previous approach, [1], the analytical reason was that the derivative of the conformal anomaly on the orbifolded d-sphere was zero for the non-singular sphere, i.e. on shell, and I make some relevant remarks in the next section.\\nwhose numerical evaluation yields (minus) the scalar conformal anomalies on the dsphere, as do the calculations mentioned earlier. The bare evaluation itself gives no clue as to why this should be so. In our previous approach, [1], the analytical reason was that the derivative of the conformal anomaly on the orbifolded d-sphere was zero for the non-singular sphere, i.e. on shell, and I make some relevant remarks in the next section.\\nThe conformal anomaly values are available in the old literature and, for convenience, and interest, I give a few remarks and specifics in the next section.\\nThe conformal anomaly values are available in the old literature and, for convenience, and interest, I give a few remarks and specifics in the next section.\"\n },\n {\n \"title\": \"Conformal anomaly on spheres\",\n \"content\": \"\"\n },\n {\n \"title\": \"Conformal anomaly on spheres\",\n \"content\": \"As has been used above, the conformal anomaly is the constant term in the heat-kernel expansion. 5 There are a number of ways of finding this, based on the standard spectral information for the (conformal) Laplacian on d-spheres (which can be traced back to George Green in the 1830s).\\nAs has been used above, the conformal anomaly is the constant term in the heat-kernel expansion. 5 There are a number of ways of finding this, based on the standard spectral information for the (conformal) Laplacian on d-spheres (which can be traced back to George Green in the 1830s).\\nThere is no need to elaborate, as the work of Copeland and Toms, [20], covers the necessary ground. One technique uses the values of the corresponding ζfunction, the required analytic continuation of which was done by expressing it as a series of Hurwitz ζ-functions, which have known properties. Birmingham, [21], employs a similar approach. A list of the numerical results can be found in [20] for scalars and spinors.\\nThere is no need to elaborate, as the work of Copeland and Toms, [20], covers the necessary ground. One technique uses the values of the corresponding ζfunction, the required analytic continuation of which was done by expressing it as a series of Hurwitz ζ-functions, which have known properties. Birmingham, [21], employs a similar approach. A list of the numerical results can be found in [20] for scalars and spinors.\\nAmong other works, I mention the interesting one by Cappelli and D'Appolonio, [22], which employs basically the same method, only slightly more streamlined. It also lists the values.\\nAmong other works, I mention the interesting one by Cappelli and D'Appolonio, [22], which employs basically the same method, only slightly more streamlined. It also lists the values.\\nA different organisation of the spectral information was given in [23,24] and resulted in the compact expression for the heat-kernel coefficients (the conformal anomaly) on a d-sphere,\\nA different organisation of the spectral information was given in [23,24] and resulted in the compact expression for the heat-kernel coefficients (the conformal anomaly) on a d-sphere,\\nwhich rapidly evaluates to the known numbers. Spelling out some calculational details, this evaluation can be done using the integral relation, [18], p.171,\\nwhich rapidly evaluates to the known numbers. Spelling out some calculational details, this evaluation can be done using the integral relation, [18], p.171,\\nwhich produces, from (16),\\nwhich produces, from (16),\\nwith the product form, (e.g. [18] p.186, [25], §8),\\nwith the product form, (e.g. [18] p.186, [25], §8),\\nThis yields an expression identical to that given by Diaz, [26],\\nThis yields an expression identical to that given by Diaz, [26],\\nwhich he obtains on the basis of a holographic technique as the simplest example of a more general construction of the conformal anomaly associated with Branson's GJMS higher Laplacian operators, P 2k , on the sphere, e.g. Branson, [27], Graham, [28], Gover [29], Juhl, [30]. It is possible to show 6 , using the approach of [24], that the general result at level k is,\\nwhich he obtains on the basis of a holographic technique as the simplest example of a more general construction of the conformal anomaly associated with Branson's GJMS higher Laplacian operators, P 2k , on the sphere, e.g. Branson, [27], Graham, [28], Gover [29], Juhl, [30]. It is possible to show 6 , using the approach of [24], that the general result at level k is,\\n6 The details will be presented at another time.\\n6 The details will be presented at another time.\\nwhich evaluates to a polynomial in k, for a given d; for example,\\nwhich evaluates to a polynomial in k, for a given d; for example,\\nEvaluation gives agreement with the numbers found by Diaz using holography, [26], eqn.(7.1) and Table 3.\\nEvaluation gives agreement with the numbers found by Diaz using holography, [26], eqn.(7.1) and Table 3.\\nThe extrema of these polynomials lie in the region |k| < d/2 and are located, for positive k, closely below the integers which fact corresponds to the alternating signs at the 'physical' values, k = 1, . . . , d/2.\\nThe extrema of these polynomials lie in the region |k| < d/2 and are located, for positive k, closely below the integers which fact corresponds to the alternating signs at the 'physical' values, k = 1, . . . , d/2.\\nIn this approach, the operator existence condition on the level, k ≤ d/2 (for d even), is equivalent to the absence of a zero mode in the spectrum of P 2k . (Actually for the Neumann modes on the hemisphere.) It may be possible to relax this condition.\\nIn this approach, the operator existence condition on the level, k ≤ d/2 (for d even), is equivalent to the absence of a zero mode in the spectrum of P 2k . (Actually for the Neumann modes on the hemisphere.) It may be possible to relax this condition.\\nIt is worth noting that (17), (18) and (19) do not involve the Bernoulli numbers, unlike some previously mentioned computations.\\nIt is worth noting that (17), (18) and (19) do not involve the Bernoulli numbers, unlike some previously mentioned computations.\\nReturning to the simplest case, k = 1, equating (16) and (15) gives an identity between generalised Bernoulli polynomials. In fact there is the off shell identity\\nReturning to the simplest case, k = 1, equating (16) and (15) gives an identity between generalised Bernoulli polynomials. In fact there is the off shell identity\\nThe left-hand side is d! times the conformal anomaly on the orbifolded dsphere (or periodic lune), S d /Z B , discussed in [1] where it was proved that it had an extremum when B = 1, i.e. on shell. This can be confirmed here immediately since (each term in) the integrand vanishes on shell. Reference to (12) shows that this corresponds to an extremum for β F , in thermodynamic language.\\nThe left-hand side is d! times the conformal anomaly on the orbifolded dsphere (or periodic lune), S d /Z B , discussed in [1] where it was proved that it had an extremum when B = 1, i.e. on shell. This can be confirmed here immediately since (each term in) the integrand vanishes on shell. Reference to (12) shows that this corresponds to an extremum for β F , in thermodynamic language.\\nBy applying Barnes' derivative formula,\\nBy applying Barnes' derivative formula,\\nto (21), I obtain the off shell identity (d is even),\\nto (21), I obtain the off shell identity (d is even),\\nwhich is not obvious without further manipulation. It furnishes a simpler formula for the scalar conformal polynomials, P (d, B), (14),\\nwhich is not obvious without further manipulation. It furnishes a simpler formula for the scalar conformal polynomials, P (d, B), (14),\"\n },\n {\n \"title\": \"Comments\",\n \"content\": \"\"\n },\n {\n \"title\": \"Comments\",\n \"content\": \"The calculation of the entropy in section 4 has similarities with others in this topic, e.g. [8,6]. However, in these, the entropy is usually computed from the global effective action while I have employed a local method which involves integrating the thermal average of the canonical energy obtained, e.g., from the coincidence limit of a bilinear operator acting on the thermal Green function. This gives the total internal energy from which the free energy can be determined and thence the entropy.\\nThe calculation of the entropy in section 4 has similarities with others in this topic, e.g. [8,6]. However, in these, the entropy is usually computed from the global effective action while I have employed a local method which involves integrating the thermal average of the canonical energy obtained, e.g., from the coincidence limit of a bilinear operator acting on the thermal Green function. This gives the total internal energy from which the free energy can be determined and thence the entropy.\\nThe numerical evaluation is algorithmic and offers no obvious reason why the log coefficient is (minus) the conformal anomaly, although, being conformally invariant, there is not much else it could be. In (15) one has a rather complicated formula for calculating the conformal anomaly on spheres.\\nThe numerical evaluation is algorithmic and offers no obvious reason why the log coefficient is (minus) the conformal anomaly, although, being conformally invariant, there is not much else it could be. In (15) one has a rather complicated formula for calculating the conformal anomaly on spheres.\\nI have taken a simple minded approach to the conical singularity and have integrated the local energy density using a basic cut-off, resorting neither to smoothing the geometry nor to integrating against a test function. These procedures might appear more rigorous, but have their own ambiguities. I believe the present method is sufficient to achieve its aim. I also note that the formulae encountered are closely related to expressions occurring in the holographic calculations of Ryu and Takayanagi of the entanglement entropy for a disk, [31] §7.1.2, and also to those of Graham, [32], on volume renormalisation.\\nI have taken a simple minded approach to the conical singularity and have integrated the local energy density using a basic cut-off, resorting neither to smoothing the geometry nor to integrating against a test function. These procedures might appear more rigorous, but have their own ambiguities. I believe the present method is sufficient to achieve its aim. I also note that the formulae encountered are closely related to expressions occurring in the holographic calculations of Ryu and Takayanagi of the entanglement entropy for a disk, [31] §7.1.2, and also to those of Graham, [32], on volume renormalisation.\\nThe finite temperature averaged stress tensor on static de Sitter that I have used does not agree with that derived by Marolf et al, [33], on the basis of a dS/CFT duality.\\nThe finite temperature averaged stress tensor on static de Sitter that I have used does not agree with that derived by Marolf et al, [33], on the basis of a dS/CFT duality.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". J S Dowker, ArXiv:1007.3865Hyperspherical entanglement entropy. Dowker,J.S. Hyperspherical entanglement entropy, ArXiv:1007.3865.","doi":""},{"text":". C G Callan, F Wilczek, Phys. Letts. 33355Callan,C.G. and Wilczek,F. Phys. Letts. B333 (1994) 55.","doi":""},{"text":". J S Dowker, Class. Quant. Grav. 1155Dowker,J.S. Class. Quant. Grav. 11 (1994) L55.","doi":""},{"text":". D V Fursaev, G Miele, Phys. Rev. 49987Fursaev,D.V. and Miele,G. Phys. Rev. D49 (1994) 987.","doi":""},{"text":". L De Nardo, D V Fursaev, G Miele, Class. Quant. Grav. 143269De Nardo,L., Fursaev,D.V. and Miele,G. Class. Quant. Grav. 14 (1987) 3269.","doi":""},{"text":". H Casini, M Huerta, arXiv:1007.1813Entanglement entropy for the n-sphereCasini,H. and Huerta,M. Entanglement entropy for the n-sphere, arXiv:1007.1813.","doi":""},{"text":". S N Solodukhin, Phys. Letts. 665305Solodukhin,S.N. Phys. Letts. B665 (2008) 305.","doi":""},{"text":"Entanglement entropy on round spheres. S N Solodukhin, ArXiv: 1008.4314Solodukhin,S.N. Entanglement entropy on round spheres ArXiv: 1008.4314.","doi":""},{"text":". R Lohmayer, H Neuberger, A Schwimmer, S Theisen, Phys. Letts. 685222Lohmayer,R., Neuberger,H, Schwimmer,A. and Theisen,S. Phys. Letts. B685 (2010) 222.","doi":""},{"text":". P Candelas, J S Dowker, Phys. Rev. 192902Candelas,P. and Dowker,J.S. Phys. Rev. D19 (1979) 2902.","doi":""},{"text":". J S Dowker, Phys. Rev. 181856Dowker,J.S. Phys. Rev. D18 (1978) 1856.","doi":""},{"text":". R Laflamme, Nucl. Phys. 324233Laflamme,R. Nucl. Phys. B324 (1989) 233.","doi":""},{"text":". J S Dowker, Phys. Rev. 363742Dowker,J.S. Phys. Rev. D36 (1987) 3742.","doi":""},{"text":". J S Dowker, G Kennedy, J. Phys. 11895Dowker,J.S. and Kennedy,G. J. Phys. A11 (1978) 895.","doi":""},{"text":"Quantum fields in curved space. N D Birrell, P C W Davies, Cambridge Univ. PressCambridgeBirrell,N.D, and Davies,P.C.W. Quantum fields in curved space (Cambridge Univ. Press, Cambridge, 1982).","doi":""},{"text":". E Elizalde, M Lygren, D V Vassilevich, J. Math. Phys. 373105Elizalde,E., Lygren, M. and Vassilevich, D.V. J. Math. Phys. 37 (1996) 3105.","doi":""},{"text":". J S Dowker, Phys. Rev. 363095Dowker,J.S. Phys. Rev. D36 (1987) 3095.","doi":""},{"text":". N E Nörlund, Acta Mathematica. 43121Nörlund,N.E. Acta Mathematica 43 (1922) 121.","doi":""},{"text":"L M Milne-Thomson, The Calculus of Finite Differences. LondonMacMillanMilne-Thomson,L.M. The Calculus of Finite Differences (MacMillan,London, 1933).","doi":""},{"text":". E Copeland, D J Toms, Class. Quant. Grav. 3431Copeland,E. and Toms,D.J. Class. Quant. Grav. 3 (1986) 431.","doi":""},{"text":". D Birmingham, Phys. Rev. 363037Birmingham,D. Phys. Rev. D36 (1987) 3037.","doi":""},{"text":". A Cappelli, Phys. Lett. 48787Cappelli,A. and D'Appollonio, Phys. Lett. 487B (2000) 87.","doi":""},{"text":". J S Dowker, Comm, Math. Phys. 162633Dowker,J.S. Comm. Math. Phys. 162 (1994) 633.","doi":""},{"text":". J S Dowker, J. Math. Phys. 354989Dowker,J.S. J. Math. Phys. 35 (1994) 4989.","doi":""},{"text":". E W Barnes, Trans. Camb. Phil. Soc. 19426Barnes,E.W. Trans. Camb. Phil. Soc. 19 (1903) 426.","doi":""},{"text":". D E Diaz, JHEP. 7103Diaz,D.E. JHEP 7 (2008)103.","doi":""},{"text":". T Branson, Trans.Am.Math.Soc. 3473671Branson,T. Trans.Am.Math.Soc. 347 (1995) 3671.","doi":""},{"text":". C R Graham, Sigma, 3121Graham,C.R. SIGMA 3 (2007) 121.","doi":""},{"text":"Laplacian operators and Q-curvature on conformally Einstein manifolds ArXiv. A R Gover, math.DG/0506037Gover,A.R. Laplacian operators and Q-curvature on conformally Einstein man- ifolds ArXiv:math.DG/0506037.","doi":""},{"text":"On conformally covariant powers of the Laplacian ArXiv:math. A Juhl, /0905.3993Juhl,A. On conformally covariant powers of the Laplacian ArXiv:math.DG/0905.3993.","doi":""},{"text":". S Ryu, T Takayanagi, JHEP. 060845Ryu,S. and Takayanagi,T. JHEP 0608(2006)045.","doi":""},{"text":". C R Graham, Rend, Circ.Mat.Palermo Suppl. No. 6331Graham,C.R. Rend.Circ.Mat.Palermo Suppl. No.63 (2000) 31.","doi":""},{"text":"D Marolf, M Rangamani, M Van Raamsdonk, Holographic Models of de Sitter QFTs. Marolf,D., Rangamani,M. and Van Raamsdonk,M. Holographic Models of de Sitter QFTs, ArXiv: 1007.3996.","doi":""},{"text":". J S Dowker, ArXiv:1007.3865Hyperspherical entanglement entropy. Dowker,J.S. Hyperspherical entanglement entropy, ArXiv:1007.3865.","doi":""},{"text":". C G Callan, F Wilczek, Phys. Letts. 33355Callan,C.G. and Wilczek,F. Phys. Letts. B333 (1994) 55.","doi":""},{"text":". J S Dowker, Class. Quant. Grav. 1155Dowker,J.S. Class. Quant. Grav. 11 (1994) L55.","doi":""},{"text":". D V Fursaev, G Miele, Phys. Rev. 49987Fursaev,D.V. and Miele,G. Phys. Rev. D49 (1994) 987.","doi":""},{"text":". L De Nardo, D V Fursaev, G Miele, Class. Quant. Grav. 143269De Nardo,L., Fursaev,D.V. and Miele,G. Class. Quant. Grav. 14 (1987) 3269.","doi":""},{"text":". H Casini, M Huerta, arXiv:1007.1813Entanglement entropy for the n-sphereCasini,H. and Huerta,M. Entanglement entropy for the n-sphere, arXiv:1007.1813.","doi":""},{"text":". S N Solodukhin, Phys. Letts. 665305Solodukhin,S.N. Phys. Letts. B665 (2008) 305.","doi":""},{"text":"Entanglement entropy on round spheres. S N Solodukhin, ArXiv: 1008.4314Solodukhin,S.N. Entanglement entropy on round spheres ArXiv: 1008.4314.","doi":""},{"text":". R Lohmayer, H Neuberger, A Schwimmer, S Theisen, Phys. Letts. 685222Lohmayer,R., Neuberger,H, Schwimmer,A. and Theisen,S. Phys. Letts. B685 (2010) 222.","doi":""},{"text":". P Candelas, J S Dowker, Phys. Rev. 192902Candelas,P. and Dowker,J.S. Phys. Rev. D19 (1979) 2902.","doi":""},{"text":". J S Dowker, Phys. Rev. 181856Dowker,J.S. Phys. Rev. D18 (1978) 1856.","doi":""},{"text":". R Laflamme, Nucl. Phys. 324233Laflamme,R. Nucl. Phys. B324 (1989) 233.","doi":""},{"text":". J S Dowker, Phys. Rev. 363742Dowker,J.S. Phys. Rev. D36 (1987) 3742.","doi":""},{"text":". J S Dowker, G Kennedy, J. Phys. 11895Dowker,J.S. and Kennedy,G. J. Phys. A11 (1978) 895.","doi":""},{"text":"Quantum fields in curved space. N D Birrell, P C W Davies, Cambridge Univ. PressCambridgeBirrell,N.D, and Davies,P.C.W. Quantum fields in curved space (Cambridge Univ. Press, Cambridge, 1982).","doi":""},{"text":". E Elizalde, M Lygren, D V Vassilevich, J. Math. Phys. 373105Elizalde,E., Lygren, M. and Vassilevich, D.V. J. Math. Phys. 37 (1996) 3105.","doi":""},{"text":". J S Dowker, Phys. Rev. 363095Dowker,J.S. Phys. Rev. D36 (1987) 3095.","doi":""},{"text":". N E Nörlund, Acta Mathematica. 43121Nörlund,N.E. Acta Mathematica 43 (1922) 121.","doi":""},{"text":"L M Milne-Thomson, The Calculus of Finite Differences. LondonMacMillanMilne-Thomson,L.M. The Calculus of Finite Differences (MacMillan,London, 1933).","doi":""},{"text":". E Copeland, D J Toms, Class. Quant. Grav. 3431Copeland,E. and Toms,D.J. Class. Quant. Grav. 3 (1986) 431.","doi":""},{"text":". D Birmingham, Phys. Rev. 363037Birmingham,D. Phys. Rev. D36 (1987) 3037.","doi":""},{"text":". A Cappelli, Phys. Lett. 48787Cappelli,A. and D'Appollonio, Phys. Lett. 487B (2000) 87.","doi":""},{"text":". J S Dowker, Comm, Math. Phys. 162633Dowker,J.S. Comm. Math. Phys. 162 (1994) 633.","doi":""},{"text":". J S Dowker, J. Math. Phys. 354989Dowker,J.S. J. Math. Phys. 35 (1994) 4989.","doi":""},{"text":". E W Barnes, Trans. Camb. Phil. Soc. 19426Barnes,E.W. Trans. Camb. Phil. Soc. 19 (1903) 426.","doi":""},{"text":". D E Diaz, JHEP. 7103Diaz,D.E. JHEP 7 (2008)103.","doi":""},{"text":". T Branson, Trans.Am.Math.Soc. 3473671Branson,T. Trans.Am.Math.Soc. 347 (1995) 3671.","doi":""},{"text":". C R Graham, Sigma, 3121Graham,C.R. SIGMA 3 (2007) 121.","doi":""},{"text":"Laplacian operators and Q-curvature on conformally Einstein manifolds ArXiv. A R Gover, math.DG/0506037Gover,A.R. Laplacian operators and Q-curvature on conformally Einstein man- ifolds ArXiv:math.DG/0506037.","doi":""},{"text":"On conformally covariant powers of the Laplacian ArXiv:math. A Juhl, /0905.3993Juhl,A. On conformally covariant powers of the Laplacian ArXiv:math.DG/0905.3993.","doi":""},{"text":". S Ryu, T Takayanagi, JHEP. 060845Ryu,S. and Takayanagi,T. JHEP 0608(2006)045.","doi":""},{"text":". C R Graham, Rend, Circ.Mat.Palermo Suppl. No. 6331Graham,C.R. Rend.Circ.Mat.Palermo Suppl. No.63 (2000) 31.","doi":""},{"text":"D Marolf, M Rangamani, M Van Raamsdonk, Holographic Models of de Sitter QFTs. Marolf,D., Rangamani,M. and Van Raamsdonk,M. Holographic Models of de Sitter QFTs, ArXiv: 1007.3996.","doi":""}],"string":"[\n {\n \"text\": \". J S Dowker, ArXiv:1007.3865Hyperspherical entanglement entropy. Dowker,J.S. Hyperspherical entanglement entropy, ArXiv:1007.3865.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C G Callan, F Wilczek, Phys. Letts. 33355Callan,C.G. and Wilczek,F. Phys. Letts. B333 (1994) 55.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Class. Quant. Grav. 1155Dowker,J.S. Class. Quant. Grav. 11 (1994) L55.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D V Fursaev, G Miele, Phys. Rev. 49987Fursaev,D.V. and Miele,G. Phys. Rev. D49 (1994) 987.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L De Nardo, D V Fursaev, G Miele, Class. Quant. Grav. 143269De Nardo,L., Fursaev,D.V. and Miele,G. Class. Quant. Grav. 14 (1987) 3269.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Casini, M Huerta, arXiv:1007.1813Entanglement entropy for the n-sphereCasini,H. and Huerta,M. Entanglement entropy for the n-sphere, arXiv:1007.1813.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S N Solodukhin, Phys. Letts. 665305Solodukhin,S.N. Phys. Letts. B665 (2008) 305.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Entanglement entropy on round spheres. S N Solodukhin, ArXiv: 1008.4314Solodukhin,S.N. Entanglement entropy on round spheres ArXiv: 1008.4314.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Lohmayer, H Neuberger, A Schwimmer, S Theisen, Phys. Letts. 685222Lohmayer,R., Neuberger,H, Schwimmer,A. and Theisen,S. Phys. Letts. B685 (2010) 222.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Candelas, J S Dowker, Phys. Rev. 192902Candelas,P. and Dowker,J.S. Phys. Rev. D19 (1979) 2902.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Phys. Rev. 181856Dowker,J.S. Phys. Rev. D18 (1978) 1856.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Laflamme, Nucl. Phys. 324233Laflamme,R. Nucl. Phys. B324 (1989) 233.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Phys. Rev. 363742Dowker,J.S. Phys. Rev. D36 (1987) 3742.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, G Kennedy, J. Phys. 11895Dowker,J.S. and Kennedy,G. J. Phys. A11 (1978) 895.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum fields in curved space. N D Birrell, P C W Davies, Cambridge Univ. PressCambridgeBirrell,N.D, and Davies,P.C.W. Quantum fields in curved space (Cambridge Univ. Press, Cambridge, 1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Elizalde, M Lygren, D V Vassilevich, J. Math. Phys. 373105Elizalde,E., Lygren, M. and Vassilevich, D.V. J. Math. Phys. 37 (1996) 3105.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Phys. Rev. 363095Dowker,J.S. Phys. Rev. D36 (1987) 3095.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E Nörlund, Acta Mathematica. 43121Nörlund,N.E. Acta Mathematica 43 (1922) 121.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L M Milne-Thomson, The Calculus of Finite Differences. LondonMacMillanMilne-Thomson,L.M. The Calculus of Finite Differences (MacMillan,London, 1933).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Copeland, D J Toms, Class. Quant. Grav. 3431Copeland,E. and Toms,D.J. Class. Quant. Grav. 3 (1986) 431.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Birmingham, Phys. Rev. 363037Birmingham,D. Phys. Rev. D36 (1987) 3037.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Cappelli, Phys. Lett. 48787Cappelli,A. and D'Appollonio, Phys. Lett. 487B (2000) 87.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Comm, Math. Phys. 162633Dowker,J.S. Comm. Math. Phys. 162 (1994) 633.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, J. Math. Phys. 354989Dowker,J.S. J. Math. Phys. 35 (1994) 4989.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E W Barnes, Trans. Camb. Phil. Soc. 19426Barnes,E.W. Trans. Camb. Phil. Soc. 19 (1903) 426.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D E Diaz, JHEP. 7103Diaz,D.E. JHEP 7 (2008)103.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Branson, Trans.Am.Math.Soc. 3473671Branson,T. Trans.Am.Math.Soc. 347 (1995) 3671.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C R Graham, Sigma, 3121Graham,C.R. SIGMA 3 (2007) 121.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Laplacian operators and Q-curvature on conformally Einstein manifolds ArXiv. A R Gover, math.DG/0506037Gover,A.R. Laplacian operators and Q-curvature on conformally Einstein man- ifolds ArXiv:math.DG/0506037.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On conformally covariant powers of the Laplacian ArXiv:math. A Juhl, /0905.3993Juhl,A. On conformally covariant powers of the Laplacian ArXiv:math.DG/0905.3993.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Ryu, T Takayanagi, JHEP. 060845Ryu,S. and Takayanagi,T. JHEP 0608(2006)045.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C R Graham, Rend, Circ.Mat.Palermo Suppl. No. 6331Graham,C.R. Rend.Circ.Mat.Palermo Suppl. No.63 (2000) 31.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Marolf, M Rangamani, M Van Raamsdonk, Holographic Models of de Sitter QFTs. Marolf,D., Rangamani,M. and Van Raamsdonk,M. Holographic Models of de Sitter QFTs, ArXiv: 1007.3996.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, ArXiv:1007.3865Hyperspherical entanglement entropy. Dowker,J.S. Hyperspherical entanglement entropy, ArXiv:1007.3865.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C G Callan, F Wilczek, Phys. Letts. 33355Callan,C.G. and Wilczek,F. Phys. Letts. B333 (1994) 55.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Class. Quant. Grav. 1155Dowker,J.S. Class. Quant. Grav. 11 (1994) L55.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D V Fursaev, G Miele, Phys. Rev. 49987Fursaev,D.V. and Miele,G. Phys. Rev. D49 (1994) 987.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L De Nardo, D V Fursaev, G Miele, Class. Quant. Grav. 143269De Nardo,L., Fursaev,D.V. and Miele,G. Class. Quant. Grav. 14 (1987) 3269.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Casini, M Huerta, arXiv:1007.1813Entanglement entropy for the n-sphereCasini,H. and Huerta,M. Entanglement entropy for the n-sphere, arXiv:1007.1813.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S N Solodukhin, Phys. Letts. 665305Solodukhin,S.N. Phys. Letts. B665 (2008) 305.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Entanglement entropy on round spheres. S N Solodukhin, ArXiv: 1008.4314Solodukhin,S.N. Entanglement entropy on round spheres ArXiv: 1008.4314.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Lohmayer, H Neuberger, A Schwimmer, S Theisen, Phys. Letts. 685222Lohmayer,R., Neuberger,H, Schwimmer,A. and Theisen,S. Phys. Letts. B685 (2010) 222.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Candelas, J S Dowker, Phys. Rev. 192902Candelas,P. and Dowker,J.S. Phys. Rev. D19 (1979) 2902.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Phys. Rev. 181856Dowker,J.S. Phys. Rev. D18 (1978) 1856.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Laflamme, Nucl. Phys. 324233Laflamme,R. Nucl. Phys. B324 (1989) 233.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Phys. Rev. 363742Dowker,J.S. Phys. Rev. D36 (1987) 3742.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, G Kennedy, J. Phys. 11895Dowker,J.S. and Kennedy,G. J. Phys. A11 (1978) 895.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum fields in curved space. N D Birrell, P C W Davies, Cambridge Univ. PressCambridgeBirrell,N.D, and Davies,P.C.W. Quantum fields in curved space (Cambridge Univ. Press, Cambridge, 1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Elizalde, M Lygren, D V Vassilevich, J. Math. Phys. 373105Elizalde,E., Lygren, M. and Vassilevich, D.V. J. Math. Phys. 37 (1996) 3105.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Phys. Rev. 363095Dowker,J.S. Phys. Rev. D36 (1987) 3095.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E Nörlund, Acta Mathematica. 43121Nörlund,N.E. Acta Mathematica 43 (1922) 121.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L M Milne-Thomson, The Calculus of Finite Differences. LondonMacMillanMilne-Thomson,L.M. The Calculus of Finite Differences (MacMillan,London, 1933).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Copeland, D J Toms, Class. Quant. Grav. 3431Copeland,E. and Toms,D.J. Class. Quant. Grav. 3 (1986) 431.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Birmingham, Phys. Rev. 363037Birmingham,D. Phys. Rev. D36 (1987) 3037.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Cappelli, Phys. Lett. 48787Cappelli,A. and D'Appollonio, Phys. Lett. 487B (2000) 87.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, Comm, Math. Phys. 162633Dowker,J.S. Comm. Math. Phys. 162 (1994) 633.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J S Dowker, J. Math. Phys. 354989Dowker,J.S. J. Math. Phys. 35 (1994) 4989.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E W Barnes, Trans. Camb. Phil. Soc. 19426Barnes,E.W. Trans. Camb. Phil. Soc. 19 (1903) 426.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D E Diaz, JHEP. 7103Diaz,D.E. JHEP 7 (2008)103.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Branson, Trans.Am.Math.Soc. 3473671Branson,T. Trans.Am.Math.Soc. 347 (1995) 3671.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C R Graham, Sigma, 3121Graham,C.R. SIGMA 3 (2007) 121.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Laplacian operators and Q-curvature on conformally Einstein manifolds ArXiv. A R Gover, math.DG/0506037Gover,A.R. Laplacian operators and Q-curvature on conformally Einstein man- ifolds ArXiv:math.DG/0506037.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On conformally covariant powers of the Laplacian ArXiv:math. A Juhl, /0905.3993Juhl,A. On conformally covariant powers of the Laplacian ArXiv:math.DG/0905.3993.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Ryu, T Takayanagi, JHEP. 060845Ryu,S. and Takayanagi,T. JHEP 0608(2006)045.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C R Graham, Rend, Circ.Mat.Palermo Suppl. No. 6331Graham,C.R. Rend.Circ.Mat.Palermo Suppl. No.63 (2000) 31.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Marolf, M Rangamani, M Van Raamsdonk, Holographic Models of de Sitter QFTs. Marolf,D., Rangamani,M. and Van Raamsdonk,M. Holographic Models of de Sitter QFTs, ArXiv: 1007.3996.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":42,"cells":{"text":{"kind":"string","value":"\nStatistical Modeling of Primary Ewing Tumours of the Bone\nJune-July 2005\n\nSreepurna Malakar \nDepartment of Chemistry and Biochemistry\nUniversity of Alaska\nU.S.A\n\nF Smarandache \nDepartment of Mathematics\nUniversity of New Mexico\nU.S.A\n\nS Bhattacharya sbhattacharya@alaskapacific.edu \nDepartment of Business Administration\nAlaska Pacific University\nU.S.A\n\nStatistical Modeling of Primary Ewing Tumours of the Bone\n\nInternational Journal of Tomography & Statistics (IJTS)\n3JJ05June-July 2005\nThis short technical paper advocates a bootstrapping algorithm from which we can form a statistically reliable opinion based on limited clinically observed data, regarding whether an osteo-hyperplasia could actually be a case of Ewing's osteosarcoma. The basic premise underlying our methodology is that a primary bone tumour, if it is indeed Ewing's osteosarcoma, cannot increase in volume beyond some critical limit without showing metastasis. We propose a statistical method to extrapolate such critical limit to primary tumour volume. Our model does not involve any physiological variables but rather is entirely based on time series observations of increase in primary tumour volume from the point of initial detection to the actual detection of metastases.KEY WORDSEwing's bone tumour, multi-cellular spheroids, linear difference equations\n\nINTRODUCTION\n\nTo date, oncogenetic studies of EWS/FLI-11 induced malignant transformation have largely relied upon experimental manipulation of Ewing's bone tumour cell lines and fibroblasts that have been induced to express the oncogene. It has been shown that the biology of Ewing's tumour cells in vitro is dramatically different between cells grown as mono-layers and cells grown as anchorage-independent, multi-cellular spheroids (MCS). The latter is more representative of primary Ewing's tumour in vivo (Lawlor et. al, 2002). MCS are clusters of cancer cells, used in the laboratory to study the early stages of avascular tumour growth. Mature MCS possess a well-defined structure, comprising a central core of necrotic i.e. dead cells, surrounded by a layer of non-proliferating, quiescent cells, with proliferating cells restricted to the outer, nutrient-rich layer of the tumour. As such, they are often used to assess the efficacy of new anti-cancer drugs and treatment therapies. The majority of mathematical models focus on the growth of MCS or avascular tumour growth. Most recent works have focused on the evolution of MCS growing in response to a single, externally-supplied nutrient, such as oxygen or glucose, and usually two growth inhibitors (Marusic et. al., 1994).\n\nMathematical models of MCS growth typically consist of an ordinary differential equation (ODE) coupled to one or more reaction-diffusion equations (RDEs). The ODE is derived from mass conservation and describes the evolution of the outer tumour boundary, whereas the RDEs describe the distribution within the tumour of vital nutrients such as oxygen and glucose and growth inhibitors (Dorman and Deutsch, 2002). However studies of this type, no matter how mathematically refined, often fall short of direct clinical applicability because of rather rigorous restrictions imposed on the boundary conditions. Moreover, these models focus more on the structural evolution of a tumour that is already positively classified as cancerous rather than on the clinically pertinent question of whether an initially benign growth can at a subsequent stage become invasive and show metastases (De Vita et. al., 2001).\n\nWhat we therefore aim to devise in our present paper is a bootstrapping algorithm from which we can form an educated opinion based on clinically observed data, regarding whether a bone growth initially diagnosed as benign can subsequently prove to be malignant (i.e. specifically, a case of Ewing's osteosarcoma) . The strength of our proposed algorithm lies mainly in its computational simplicity -our model does not involve any physiological variables but is entirely based on time series observations of progression in tumour volume from the first observation point till detection of metastases.\n\n\nLITERATURE SUPPORT\n\nIn a clinical study conducted by Hense et. al. (1999), restricted to patients with suspected Ewing's sarcoma, tumour volumes of more than 100 ml and the presence of primary metastases were identified as determinants of poor prognosis in patients with such tumours. Diagnoses of primary tumours were ascertained exclusively by biopsies. The diagnosis of primary metastases was based on thoracic computed tomography or on whole body bone scans. It was observed that of 559 of the patients (approx. 68% in a total sample size of 821) had a volume above 100 ml with smaller tumours being more common in childhood than in late adolescence and early adulthood. Extensive volumes were observed in almost 90% of the tumours located in femur and pelvis while they were less common in other sites (p < 0.001). On average, 26% of all patients were detected with clinically apparent primary metastases.\n\nThe detection rate of metastases was markedly higher in patients diagnosed after 1991 (p < 0.001). Primary metastases were also significantly more common for tumours originating in the pelvis and for other tumours in the Ewing's family of tumours (EFT); mainly the peripheral neuro-ectodermal tumours (PNET); (p < 0.01). Tumours greater than 100 ml were positively associated with metastatic disease (p < 0.001). Multivariate analyses, which included simultaneously all univariate predictors in a logistic regression model, indicated the observed associations were mostly unconfounded. Further it has been found that the metastatic potential of human tumours is encoded in the bulk of a primary tumour, thus challenging the notion that metastases arise from sparse cells within a primary tumour that have the ability to metastasize (Sridhar Ramaswamy et. al., 2003). These studies lend credence to our fundamental premise about a critical primary tumour volume being used as a classification factor to distinguish between benign and potentially malignant bone growth.\n\n\nSTATISTICAL MODELLING METHODOLOGY\n\nAssuming that the temporal drift process governing the progression in size of a primary Ewing tumour of the bone to be linear (the computationally simplest process), we suggest a straightforward computational technique to generate a large family of possible tumour propagation paths based on clinically observed growth patterns under laboratory conditions. In case the governing process is decidedly non-linear, then our proposed scheme would not be applicable and in such a case one will have to rely on a completely non-parametric classification technique like e.g. an Artificial Neural Network (ANN).\n\nOur proposed approach is a bootstrapping one, whereby a linear autoregression model is fitted through the origin to the observation data in the first stage. If one or more beta coefficients are found to be significant at least at a 95% level for the fitted model then, in the second stage, the autoregression equation is formulated and solved as a linear difference equation to extract the governing equation.\n\nIn the final stage, the governing equation obtained as above is plotted, for different values of the constant coefficients, as a family of possible temporal progression curves generated to explain the propagation property of that particular strain of tumour. The critical volume of the primary growth can thereafter be visually extrapolated from the observed cluster of points where the generated family of primary tumour progression curves shows a definite uptrend vis-à-vis the actual progression curve.\n\nIf no beta coefficient is found to be significant in the first stage, a non-linear temporal progression process is strongly suspected and the algorithm terminates without proceeding onto the subsequent stages, thereby implicitly recommending the problem to a non-parametric classification model.\n\nThe mathematical structure of our proposed model may be given as follows:\n\nProgression in primary Ewing tumour size over time expressed as an n-step general autoregressive process through the origin:\n\nFormulated as a linear, difference equation we can write:\n\nTaking S t common and applying the negative shift operator throughout, we get:\n\nNow applying the positive shift operator throughout we get:\n\nThe characteristic equation of the above form is then obtained as follows:\n\nHere r is the root of the characteristic equation. After solving for r, the governing equation can be derived in accordance with the well-known analytical solution techniques for ordinary linear difference equations (Kelly and Peterson, 2000).\n\n\nSIMULATED CLINICAL STUDY\n\nWe set up a simulated clinical study applying our modeling methodology with the following hypothetical primary Ewing tumour progression data adapted from the clinical study of Hense et. al. (1999) as given in Table I below: 84 IJTS, Vol. 3, No. JJ05, June-July 2005 The temporal progression path of the primary growth from the point of first detection to the onset of metastasis is plotted above in Figure I.\n\nWe have fitted an AR (2) model to the primary tumour growth data as follows:\n\n\nE (S t ) = -1.01081081S t-1 + 5.32365561S t-2 (VI)\n\nThe R 2 of the fitted model is approximately 0.8311 and the F-statistic is 9.83832\n\n\nInternational Journal of Tomography & Statistics\n\nwith an associated p-value of approximately 0.04812. Therefore the fitted model definitely has an overall predictive utility at the 5% level of significance. The residuals of the above AR (2) fitted model are given in Table II as follows:\n\nThe average of the residuals comes to 3.044841. Therefore the linear difference equation to be solved in this case is as follows:\n\nX t = -1.01081081X t-1 + 5.32365561X t-2 + 3.\n\n\n(VII)\n\nApplying usual solution techniques, the general solution to equation (VII) is obtained as follows:\n\n\nX t = c 1 (2.43124756) t + c 2 (-3.44205837) t (VIII)\n\nHere c 1 and c 2 are the constant coefficients which may now be suitably varied to generate family of possible primary tumour progression curves as in Figure II below: 86 IJTS, Vol. 3, No. JJ05, June-July 2005 In the above plot, we have varied c 2 in the range 0.01 to 0.10 and imposed the condition c 1 = 1 -c 2 . The other obvious condition is that choice of c 1 and c 2 would be such as to rule out any absurd case of negative volume. Of course the choice of the governing equation parameters would also depend on specific clinical considerations (King, 2000).\n\n\nCONCLUSION\n\nFrom Figure II, it becomes visually apparent that continuing increase in the observed size of the primary growth beyond approximately 52 ml. in volume would be potentially malignant as this would imply that the tumour would possibly keep exhibiting uncontrolled progression till it shows metastasis. This could also be obtained arithmetically as the average volume for t = 5. Therefore the critical volume could be fixed around 2 ml. as per the computational results obtained in our illustrative example.\n\nThough our computational study is intended to be purely illustrative as we have worked with hypothetical figures and hence cannot yield any clinical conclusion, we believe we have hereby aptly demonstrated the essential algorithm of our statistical approach and justified its practical usability under laboratory settings. This is not intended to be a clinical paper and in defense, it may be stated that given the immense potential of computer modeling and simulation in present times, a number of mathematical tumor growth studies are being conducted now that are essentially theoretical rather than clinical in nature (Swanson et. al. 2003;Tabatabai, Williams and Bursac, 2005).\n\nWe have used a difference equation model rather than a differential equation one because under practical laboratory settings, observations cannot be made continuously but only at discrete time intervals. However, as already stated previously, the numerical example we have provided is purely demonstrative -its purpose is illustrating the computational modeling technique rather than drawing clinically viable conclusions based on the model coefficients. Having said that, we definitely feel that there is an immediate scope of taking our line of research further forward by actually implementing an autoregressive process to model in vitro growth of MCS with clinically observed data.\n ACKNOWLEDGEMENTWe sincerely thank all the referees for their valued comments and observations.\nCancer -Principles & Practice of Oncology. V Devita, T , S Hellman, S A Rosenberg, Lippincott Williams & WilkinsPhiladelphia, USA6th EdDeVita, V., T., S. Hellman, and S.A. Rosenberg, 2001, Cancer -Principles & Practice of Oncology, 6th Ed., Lippincott Williams & Wilkins, Philadelphia, USA\n\nModeling of self-organized, avascular tumour growth with a hybrid cellular automaton. S Dormann, A Deutsch, last accessed on 09/09/2003Dormann, S. and A. Deutsch, 2003, Modeling of self-organized, avascular tumour growth with a hybrid cellular automaton, www.bioinfo.de/isb/2002/02/0035/main. html, (last accessed on 09/09/2003)\n\nFactors associated with tumor volume and primary metastases in Ewing tumors: results from the (EI) CESS studies. H W Hense, S Ahrens, M Paulussen, M Lehnert, H Jurgens, Annals of Oncology. 109Hense, H.W., S. Ahrens, M. Paulussen, M. Lehnert and H. Jurgens, 1999, Factors associated with tumor volume and primary metastases in Ewing tumors: results from the (EI) CESS studies, Annals of Oncology, 10(9), 1073-1077\n\nDifference Equations -An Introduction with Applications. Walter G Kelly, Allan C Peterson, Academic PressMassachusetts, Burlington, USA2nd EdKelly, Walter G. and Allan C. Peterson, 2000,. Difference Equations -An Introduction with Applications, 2nd Ed., Academic Press, Massachusetts, Burlington, USA\n\nCancer Biology. Roger J B King, Pearson Education LimitedEssex, UK2nd EdKing, Roger J.B., 2000, Cancer Biology, 2nd Ed., Pearson Education Limited, Essex, UK\n\nAnchorage-Independent multicellular spheroids as an in vitro model of growth signaling in Ewing tumors. E R Lawlor, C Scheel, J Irving, P H B Sorensen, Oncogene. 21Lawlor E.R., C. Scheel, J. Irving, and P.H.B. Sorensen, 2002, Anchorage- Independent multicellular spheroids as an in vitro model of growth signaling in Ewing tumors, Oncogene, 21, 307-318\n\nTumor growth in vivo and as multicellular spheroids compared by mathematical models. M Marusic, Z Bajzer, S Vuk-Pavlovic, J P Freyer, Bulletin of Mathematical Biology. 564Marusic, M., Z. Bajzer, S. Vuk-Pavlovic, and J.P. Freyer, 1994, Tumor growth in vivo and as multicellular spheroids compared by mathematical models, Bulletin of Mathematical Biology, 56(4), 617-31\n\nEvidence for a Molecular Signature of Metastasis in Primary Solid Tumours. S Ramaswamy, K N Ross, E S Lander, T R Golub, last accessed on 09/09/2003Ramaswamy, S., K.N. Ross, E.S. Lander and T.R. Golub, Evidence for a Molecular Signature of Metastasis in Primary Solid Tumours, www.genome.wi.mit.edu/cancer/solid_tumor_metastasis, (last accessed on 09/09/2003)\n\nVirtual and real brain tumors: using mathematical model to quantify glioma growth and invasion. K R Swanson, C Bridge, J D Murray, E C AlvordJr, Journal of the Neurological Sciences. 216Swanson, K.R., C. Bridge, J.D. Murray, and E.C. Alvord Jr., 2003, Virtual and real brain tumors: using mathematical model to quantify glioma growth and invasion, Journal of the Neurological Sciences, 216, 1 -10\n\nHyperbolastic growth models: theory and application. M Tabatabai, D K Williams, Z Bursac, Theoretical Biology and Medical Modelling. 21488in printTabatabai, M., D.K. Williams, and Z. Bursac, 2005, Hyperbolastic growth models: theory and application, Theoretical Biology and Medical Modelling, 2(14), (in print) 88\n\n. IJTS. 3JJ05IJTS, Vol. 3, No. JJ05, June-July 2005\n"},"corpus_id":{"kind":"number","value":15625075,"string":"15,625,075"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"q-bio/0504010\",\n \"mag\": 2963837295,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [],\n \"pdf_sha\": \"3749178d2d690db4a37f35cb8b55f3ccc0b9eac5\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Statistical Modeling of Primary Ewing Tumours of the Bone"},"authors":{"kind":"list like","value":["Sreepurna Malakar \nDepartment of Chemistry and Biochemistry\nUniversity of Alaska\nU.S.A","F Smarandache \nDepartment of Mathematics\nUniversity of New Mexico\nU.S.A","S Bhattacharya sbhattacharya@alaskapacific.edu \nDepartment of Business Administration\nAlaska Pacific University\nU.S.A"],"string":"[\n \"Sreepurna Malakar \\nDepartment of Chemistry and Biochemistry\\nUniversity of Alaska\\nU.S.A\",\n \"F Smarandache \\nDepartment of Mathematics\\nUniversity of New Mexico\\nU.S.A\",\n \"S Bhattacharya sbhattacharya@alaskapacific.edu \\nDepartment of Business Administration\\nAlaska Pacific University\\nU.S.A\"\n]","isConversational":false},"abstract":{"kind":"string","value":"This short technical paper advocates a bootstrapping algorithm from which we can form a statistically reliable opinion based on limited clinically observed data, regarding whether an osteo-hyperplasia could actually be a case of Ewing's osteosarcoma. The basic premise underlying our methodology is that a primary bone tumour, if it is indeed Ewing's osteosarcoma, cannot increase in volume beyond some critical limit without showing metastasis. We propose a statistical method to extrapolate such critical limit to primary tumour volume. Our model does not involve any physiological variables but rather is entirely based on time series observations of increase in primary tumour volume from the point of initial detection to the actual detection of metastases.KEY WORDSEwing's bone tumour, multi-cellular spheroids, linear difference equations"},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"To date, oncogenetic studies of EWS/FLI-11 induced malignant transformation have largely relied upon experimental manipulation of Ewing's bone tumour cell lines and fibroblasts that have been induced to express the oncogene. It has been shown that the biology of Ewing's tumour cells in vitro is dramatically different between cells grown as mono-layers and cells grown as anchorage-independent, multi-cellular spheroids (MCS). The latter is more representative of primary Ewing's tumour in vivo (Lawlor et. al, 2002). MCS are clusters of cancer cells, used in the laboratory to study the early stages of avascular tumour growth. Mature MCS possess a well-defined structure, comprising a central core of necrotic i.e. dead cells, surrounded by a layer of non-proliferating, quiescent cells, with proliferating cells restricted to the outer, nutrient-rich layer of the tumour. As such, they are often used to assess the efficacy of new anti-cancer drugs and treatment therapies. The majority of mathematical models focus on the growth of MCS or avascular tumour growth. Most recent works have focused on the evolution of MCS growing in response to a single, externally-supplied nutrient, such as oxygen or glucose, and usually two growth inhibitors (Marusic et. al., 1994).\nMathematical models of MCS growth typically consist of an ordinary differential equation (ODE) coupled to one or more reaction-diffusion equations (RDEs). The ODE is derived from mass conservation and describes the evolution of the outer tumour boundary, whereas the RDEs describe the distribution within the tumour of vital nutrients such as oxygen and glucose and growth inhibitors (Dorman and Deutsch, 2002). However studies of this type, no matter how mathematically refined, often fall short of direct clinical applicability because of rather rigorous restrictions imposed on the boundary conditions. Moreover, these models focus more on the structural evolution of a tumour that is already positively classified as cancerous rather than on the clinically pertinent question of whether an initially benign growth can at a subsequent stage become invasive and show metastases (De Vita et. al., 2001).\nWhat we therefore aim to devise in our present paper is a bootstrapping algorithm from which we can form an educated opinion based on clinically observed data, regarding whether a bone growth initially diagnosed as benign can subsequently prove to be malignant (i.e. specifically, a case of Ewing's osteosarcoma) . The strength of our proposed algorithm lies mainly in its computational simplicity -our model does not involve any physiological variables but is entirely based on time series observations of progression in tumour volume from the first observation point till detection of metastases."},{"title":"LITERATURE SUPPORT","content":"In a clinical study conducted by Hense et. al. (1999), restricted to patients with suspected Ewing's sarcoma, tumour volumes of more than 100 ml and the presence of primary metastases were identified as determinants of poor prognosis in patients with such tumours. Diagnoses of primary tumours were ascertained exclusively by biopsies. The diagnosis of primary metastases was based on thoracic computed tomography or on whole body bone scans. It was observed that of 559 of the patients (approx. 68% in a total sample size of 821) had a volume above 100 ml with smaller tumours being more common in childhood than in late adolescence and early adulthood. Extensive volumes were observed in almost 90% of the tumours located in femur and pelvis while they were less common in other sites (p < 0.001). On average, 26% of all patients were detected with clinically apparent primary metastases.\nThe detection rate of metastases was markedly higher in patients diagnosed after 1991 (p < 0.001). Primary metastases were also significantly more common for tumours originating in the pelvis and for other tumours in the Ewing's family of tumours (EFT); mainly the peripheral neuro-ectodermal tumours (PNET); (p < 0.01). Tumours greater than 100 ml were positively associated with metastatic disease (p < 0.001). Multivariate analyses, which included simultaneously all univariate predictors in a logistic regression model, indicated the observed associations were mostly unconfounded. Further it has been found that the metastatic potential of human tumours is encoded in the bulk of a primary tumour, thus challenging the notion that metastases arise from sparse cells within a primary tumour that have the ability to metastasize (Sridhar Ramaswamy et. al., 2003). These studies lend credence to our fundamental premise about a critical primary tumour volume being used as a classification factor to distinguish between benign and potentially malignant bone growth."},{"title":"STATISTICAL MODELLING METHODOLOGY","content":"Assuming that the temporal drift process governing the progression in size of a primary Ewing tumour of the bone to be linear (the computationally simplest process), we suggest a straightforward computational technique to generate a large family of possible tumour propagation paths based on clinically observed growth patterns under laboratory conditions. In case the governing process is decidedly non-linear, then our proposed scheme would not be applicable and in such a case one will have to rely on a completely non-parametric classification technique like e.g. an Artificial Neural Network (ANN).\nOur proposed approach is a bootstrapping one, whereby a linear autoregression model is fitted through the origin to the observation data in the first stage. If one or more beta coefficients are found to be significant at least at a 95% level for the fitted model then, in the second stage, the autoregression equation is formulated and solved as a linear difference equation to extract the governing equation.\nIn the final stage, the governing equation obtained as above is plotted, for different values of the constant coefficients, as a family of possible temporal progression curves generated to explain the propagation property of that particular strain of tumour. The critical volume of the primary growth can thereafter be visually extrapolated from the observed cluster of points where the generated family of primary tumour progression curves shows a definite uptrend vis-à-vis the actual progression curve.\nIf no beta coefficient is found to be significant in the first stage, a non-linear temporal progression process is strongly suspected and the algorithm terminates without proceeding onto the subsequent stages, thereby implicitly recommending the problem to a non-parametric classification model.\nThe mathematical structure of our proposed model may be given as follows:\nProgression in primary Ewing tumour size over time expressed as an n-step general autoregressive process through the origin:\nFormulated as a linear, difference equation we can write:\nTaking S t common and applying the negative shift operator throughout, we get:\nNow applying the positive shift operator throughout we get:\nThe characteristic equation of the above form is then obtained as follows:\nHere r is the root of the characteristic equation. After solving for r, the governing equation can be derived in accordance with the well-known analytical solution techniques for ordinary linear difference equations (Kelly and Peterson, 2000)."},{"title":"SIMULATED CLINICAL STUDY","content":"We set up a simulated clinical study applying our modeling methodology with the following hypothetical primary Ewing tumour progression data adapted from the clinical study of Hense et. al. (1999) as given in Table I below: 84 IJTS, Vol. 3, No. JJ05, June-July 2005 The temporal progression path of the primary growth from the point of first detection to the onset of metastasis is plotted above in Figure I.\nWe have fitted an AR (2) model to the primary tumour growth data as follows:"},{"title":"E (S t ) = -1.01081081S t-1 + 5.32365561S t-2 (VI)","content":"The R 2 of the fitted model is approximately 0.8311 and the F-statistic is 9.83832"},{"title":"International Journal of Tomography & Statistics","content":"with an associated p-value of approximately 0.04812. Therefore the fitted model definitely has an overall predictive utility at the 5% level of significance. The residuals of the above AR (2) fitted model are given in Table II as follows:\nThe average of the residuals comes to 3.044841. Therefore the linear difference equation to be solved in this case is as follows:\nX t = -1.01081081X t-1 + 5.32365561X t-2 + 3."},{"title":"(VII)","content":"Applying usual solution techniques, the general solution to equation (VII) is obtained as follows:"},{"title":"X t = c 1 (2.43124756) t + c 2 (-3.44205837) t (VIII)","content":"Here c 1 and c 2 are the constant coefficients which may now be suitably varied to generate family of possible primary tumour progression curves as in Figure II below: 86 IJTS, Vol. 3, No. JJ05, June-July 2005 In the above plot, we have varied c 2 in the range 0.01 to 0.10 and imposed the condition c 1 = 1 -c 2 . The other obvious condition is that choice of c 1 and c 2 would be such as to rule out any absurd case of negative volume. Of course the choice of the governing equation parameters would also depend on specific clinical considerations (King, 2000)."},{"title":"CONCLUSION","content":"From Figure II, it becomes visually apparent that continuing increase in the observed size of the primary growth beyond approximately 52 ml. in volume would be potentially malignant as this would imply that the tumour would possibly keep exhibiting uncontrolled progression till it shows metastasis. This could also be obtained arithmetically as the average volume for t = 5. Therefore the critical volume could be fixed around 2 ml. as per the computational results obtained in our illustrative example.\nThough our computational study is intended to be purely illustrative as we have worked with hypothetical figures and hence cannot yield any clinical conclusion, we believe we have hereby aptly demonstrated the essential algorithm of our statistical approach and justified its practical usability under laboratory settings. This is not intended to be a clinical paper and in defense, it may be stated that given the immense potential of computer modeling and simulation in present times, a number of mathematical tumor growth studies are being conducted now that are essentially theoretical rather than clinical in nature (Swanson et. al. 2003;Tabatabai, Williams and Bursac, 2005).\nWe have used a difference equation model rather than a differential equation one because under practical laboratory settings, observations cannot be made continuously but only at discrete time intervals. However, as already stated previously, the numerical example we have provided is purely demonstrative -its purpose is illustrating the computational modeling technique rather than drawing clinically viable conclusions based on the model coefficients. Having said that, we definitely feel that there is an immediate scope of taking our line of research further forward by actually implementing an autoregressive process to model in vitro growth of MCS with clinically observed data."}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"To date, oncogenetic studies of EWS/FLI-11 induced malignant transformation have largely relied upon experimental manipulation of Ewing's bone tumour cell lines and fibroblasts that have been induced to express the oncogene. It has been shown that the biology of Ewing's tumour cells in vitro is dramatically different between cells grown as mono-layers and cells grown as anchorage-independent, multi-cellular spheroids (MCS). The latter is more representative of primary Ewing's tumour in vivo (Lawlor et. al, 2002). MCS are clusters of cancer cells, used in the laboratory to study the early stages of avascular tumour growth. Mature MCS possess a well-defined structure, comprising a central core of necrotic i.e. dead cells, surrounded by a layer of non-proliferating, quiescent cells, with proliferating cells restricted to the outer, nutrient-rich layer of the tumour. As such, they are often used to assess the efficacy of new anti-cancer drugs and treatment therapies. The majority of mathematical models focus on the growth of MCS or avascular tumour growth. Most recent works have focused on the evolution of MCS growing in response to a single, externally-supplied nutrient, such as oxygen or glucose, and usually two growth inhibitors (Marusic et. al., 1994).\\nMathematical models of MCS growth typically consist of an ordinary differential equation (ODE) coupled to one or more reaction-diffusion equations (RDEs). The ODE is derived from mass conservation and describes the evolution of the outer tumour boundary, whereas the RDEs describe the distribution within the tumour of vital nutrients such as oxygen and glucose and growth inhibitors (Dorman and Deutsch, 2002). However studies of this type, no matter how mathematically refined, often fall short of direct clinical applicability because of rather rigorous restrictions imposed on the boundary conditions. Moreover, these models focus more on the structural evolution of a tumour that is already positively classified as cancerous rather than on the clinically pertinent question of whether an initially benign growth can at a subsequent stage become invasive and show metastases (De Vita et. al., 2001).\\nWhat we therefore aim to devise in our present paper is a bootstrapping algorithm from which we can form an educated opinion based on clinically observed data, regarding whether a bone growth initially diagnosed as benign can subsequently prove to be malignant (i.e. specifically, a case of Ewing's osteosarcoma) . The strength of our proposed algorithm lies mainly in its computational simplicity -our model does not involve any physiological variables but is entirely based on time series observations of progression in tumour volume from the first observation point till detection of metastases.\"\n },\n {\n \"title\": \"LITERATURE SUPPORT\",\n \"content\": \"In a clinical study conducted by Hense et. al. (1999), restricted to patients with suspected Ewing's sarcoma, tumour volumes of more than 100 ml and the presence of primary metastases were identified as determinants of poor prognosis in patients with such tumours. Diagnoses of primary tumours were ascertained exclusively by biopsies. The diagnosis of primary metastases was based on thoracic computed tomography or on whole body bone scans. It was observed that of 559 of the patients (approx. 68% in a total sample size of 821) had a volume above 100 ml with smaller tumours being more common in childhood than in late adolescence and early adulthood. Extensive volumes were observed in almost 90% of the tumours located in femur and pelvis while they were less common in other sites (p < 0.001). On average, 26% of all patients were detected with clinically apparent primary metastases.\\nThe detection rate of metastases was markedly higher in patients diagnosed after 1991 (p < 0.001). Primary metastases were also significantly more common for tumours originating in the pelvis and for other tumours in the Ewing's family of tumours (EFT); mainly the peripheral neuro-ectodermal tumours (PNET); (p < 0.01). Tumours greater than 100 ml were positively associated with metastatic disease (p < 0.001). Multivariate analyses, which included simultaneously all univariate predictors in a logistic regression model, indicated the observed associations were mostly unconfounded. Further it has been found that the metastatic potential of human tumours is encoded in the bulk of a primary tumour, thus challenging the notion that metastases arise from sparse cells within a primary tumour that have the ability to metastasize (Sridhar Ramaswamy et. al., 2003). These studies lend credence to our fundamental premise about a critical primary tumour volume being used as a classification factor to distinguish between benign and potentially malignant bone growth.\"\n },\n {\n \"title\": \"STATISTICAL MODELLING METHODOLOGY\",\n \"content\": \"Assuming that the temporal drift process governing the progression in size of a primary Ewing tumour of the bone to be linear (the computationally simplest process), we suggest a straightforward computational technique to generate a large family of possible tumour propagation paths based on clinically observed growth patterns under laboratory conditions. In case the governing process is decidedly non-linear, then our proposed scheme would not be applicable and in such a case one will have to rely on a completely non-parametric classification technique like e.g. an Artificial Neural Network (ANN).\\nOur proposed approach is a bootstrapping one, whereby a linear autoregression model is fitted through the origin to the observation data in the first stage. If one or more beta coefficients are found to be significant at least at a 95% level for the fitted model then, in the second stage, the autoregression equation is formulated and solved as a linear difference equation to extract the governing equation.\\nIn the final stage, the governing equation obtained as above is plotted, for different values of the constant coefficients, as a family of possible temporal progression curves generated to explain the propagation property of that particular strain of tumour. The critical volume of the primary growth can thereafter be visually extrapolated from the observed cluster of points where the generated family of primary tumour progression curves shows a definite uptrend vis-à-vis the actual progression curve.\\nIf no beta coefficient is found to be significant in the first stage, a non-linear temporal progression process is strongly suspected and the algorithm terminates without proceeding onto the subsequent stages, thereby implicitly recommending the problem to a non-parametric classification model.\\nThe mathematical structure of our proposed model may be given as follows:\\nProgression in primary Ewing tumour size over time expressed as an n-step general autoregressive process through the origin:\\nFormulated as a linear, difference equation we can write:\\nTaking S t common and applying the negative shift operator throughout, we get:\\nNow applying the positive shift operator throughout we get:\\nThe characteristic equation of the above form is then obtained as follows:\\nHere r is the root of the characteristic equation. After solving for r, the governing equation can be derived in accordance with the well-known analytical solution techniques for ordinary linear difference equations (Kelly and Peterson, 2000).\"\n },\n {\n \"title\": \"SIMULATED CLINICAL STUDY\",\n \"content\": \"We set up a simulated clinical study applying our modeling methodology with the following hypothetical primary Ewing tumour progression data adapted from the clinical study of Hense et. al. (1999) as given in Table I below: 84 IJTS, Vol. 3, No. JJ05, June-July 2005 The temporal progression path of the primary growth from the point of first detection to the onset of metastasis is plotted above in Figure I.\\nWe have fitted an AR (2) model to the primary tumour growth data as follows:\"\n },\n {\n \"title\": \"E (S t ) = -1.01081081S t-1 + 5.32365561S t-2 (VI)\",\n \"content\": \"The R 2 of the fitted model is approximately 0.8311 and the F-statistic is 9.83832\"\n },\n {\n \"title\": \"International Journal of Tomography & Statistics\",\n \"content\": \"with an associated p-value of approximately 0.04812. Therefore the fitted model definitely has an overall predictive utility at the 5% level of significance. The residuals of the above AR (2) fitted model are given in Table II as follows:\\nThe average of the residuals comes to 3.044841. Therefore the linear difference equation to be solved in this case is as follows:\\nX t = -1.01081081X t-1 + 5.32365561X t-2 + 3.\"\n },\n {\n \"title\": \"(VII)\",\n \"content\": \"Applying usual solution techniques, the general solution to equation (VII) is obtained as follows:\"\n },\n {\n \"title\": \"X t = c 1 (2.43124756) t + c 2 (-3.44205837) t (VIII)\",\n \"content\": \"Here c 1 and c 2 are the constant coefficients which may now be suitably varied to generate family of possible primary tumour progression curves as in Figure II below: 86 IJTS, Vol. 3, No. JJ05, June-July 2005 In the above plot, we have varied c 2 in the range 0.01 to 0.10 and imposed the condition c 1 = 1 -c 2 . The other obvious condition is that choice of c 1 and c 2 would be such as to rule out any absurd case of negative volume. Of course the choice of the governing equation parameters would also depend on specific clinical considerations (King, 2000).\"\n },\n {\n \"title\": \"CONCLUSION\",\n \"content\": \"From Figure II, it becomes visually apparent that continuing increase in the observed size of the primary growth beyond approximately 52 ml. in volume would be potentially malignant as this would imply that the tumour would possibly keep exhibiting uncontrolled progression till it shows metastasis. This could also be obtained arithmetically as the average volume for t = 5. Therefore the critical volume could be fixed around 2 ml. as per the computational results obtained in our illustrative example.\\nThough our computational study is intended to be purely illustrative as we have worked with hypothetical figures and hence cannot yield any clinical conclusion, we believe we have hereby aptly demonstrated the essential algorithm of our statistical approach and justified its practical usability under laboratory settings. This is not intended to be a clinical paper and in defense, it may be stated that given the immense potential of computer modeling and simulation in present times, a number of mathematical tumor growth studies are being conducted now that are essentially theoretical rather than clinical in nature (Swanson et. al. 2003;Tabatabai, Williams and Bursac, 2005).\\nWe have used a difference equation model rather than a differential equation one because under practical laboratory settings, observations cannot be made continuously but only at discrete time intervals. However, as already stated previously, the numerical example we have provided is purely demonstrative -its purpose is illustrating the computational modeling technique rather than drawing clinically viable conclusions based on the model coefficients. Having said that, we definitely feel that there is an immediate scope of taking our line of research further forward by actually implementing an autoregressive process to model in vitro growth of MCS with clinically observed data.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Cancer -Principles & Practice of Oncology. V Devita, T , S Hellman, S A Rosenberg, Lippincott Williams & WilkinsPhiladelphia, USA6th EdDeVita, V., T., S. Hellman, and S.A. Rosenberg, 2001, Cancer -Principles & Practice of Oncology, 6th Ed., Lippincott Williams & Wilkins, Philadelphia, USA","doi":""},{"text":"Modeling of self-organized, avascular tumour growth with a hybrid cellular automaton. S Dormann, A Deutsch, last accessed on 09/09/2003Dormann, S. and A. Deutsch, 2003, Modeling of self-organized, avascular tumour growth with a hybrid cellular automaton, www.bioinfo.de/isb/2002/02/0035/main. html, (last accessed on 09/09/2003)","doi":""},{"text":"Factors associated with tumor volume and primary metastases in Ewing tumors: results from the (EI) CESS studies. H W Hense, S Ahrens, M Paulussen, M Lehnert, H Jurgens, Annals of Oncology. 109Hense, H.W., S. Ahrens, M. Paulussen, M. Lehnert and H. Jurgens, 1999, Factors associated with tumor volume and primary metastases in Ewing tumors: results from the (EI) CESS studies, Annals of Oncology, 10(9), 1073-1077","doi":""},{"text":"Difference Equations -An Introduction with Applications. Walter G Kelly, Allan C Peterson, Academic PressMassachusetts, Burlington, USA2nd EdKelly, Walter G. and Allan C. Peterson, 2000,. Difference Equations -An Introduction with Applications, 2nd Ed., Academic Press, Massachusetts, Burlington, USA","doi":""},{"text":"Cancer Biology. Roger J B King, Pearson Education LimitedEssex, UK2nd EdKing, Roger J.B., 2000, Cancer Biology, 2nd Ed., Pearson Education Limited, Essex, UK","doi":""},{"text":"Anchorage-Independent multicellular spheroids as an in vitro model of growth signaling in Ewing tumors. E R Lawlor, C Scheel, J Irving, P H B Sorensen, Oncogene. 21Lawlor E.R., C. Scheel, J. Irving, and P.H.B. Sorensen, 2002, Anchorage- Independent multicellular spheroids as an in vitro model of growth signaling in Ewing tumors, Oncogene, 21, 307-318","doi":""},{"text":"Tumor growth in vivo and as multicellular spheroids compared by mathematical models. M Marusic, Z Bajzer, S Vuk-Pavlovic, J P Freyer, Bulletin of Mathematical Biology. 564Marusic, M., Z. Bajzer, S. Vuk-Pavlovic, and J.P. Freyer, 1994, Tumor growth in vivo and as multicellular spheroids compared by mathematical models, Bulletin of Mathematical Biology, 56(4), 617-31","doi":""},{"text":"Evidence for a Molecular Signature of Metastasis in Primary Solid Tumours. S Ramaswamy, K N Ross, E S Lander, T R Golub, last accessed on 09/09/2003Ramaswamy, S., K.N. Ross, E.S. Lander and T.R. Golub, Evidence for a Molecular Signature of Metastasis in Primary Solid Tumours, www.genome.wi.mit.edu/cancer/solid_tumor_metastasis, (last accessed on 09/09/2003)","doi":""},{"text":"Virtual and real brain tumors: using mathematical model to quantify glioma growth and invasion. K R Swanson, C Bridge, J D Murray, E C AlvordJr, Journal of the Neurological Sciences. 216Swanson, K.R., C. Bridge, J.D. Murray, and E.C. Alvord Jr., 2003, Virtual and real brain tumors: using mathematical model to quantify glioma growth and invasion, Journal of the Neurological Sciences, 216, 1 -10","doi":""},{"text":"Hyperbolastic growth models: theory and application. M Tabatabai, D K Williams, Z Bursac, Theoretical Biology and Medical Modelling. 21488in printTabatabai, M., D.K. Williams, and Z. Bursac, 2005, Hyperbolastic growth models: theory and application, Theoretical Biology and Medical Modelling, 2(14), (in print) 88","doi":""},{"text":". IJTS. 3JJ05IJTS, Vol. 3, No. JJ05, June-July 2005","doi":""}],"string":"[\n {\n \"text\": \"Cancer -Principles & Practice of Oncology. V Devita, T , S Hellman, S A Rosenberg, Lippincott Williams & WilkinsPhiladelphia, USA6th EdDeVita, V., T., S. Hellman, and S.A. Rosenberg, 2001, Cancer -Principles & Practice of Oncology, 6th Ed., Lippincott Williams & Wilkins, Philadelphia, USA\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Modeling of self-organized, avascular tumour growth with a hybrid cellular automaton. S Dormann, A Deutsch, last accessed on 09/09/2003Dormann, S. and A. Deutsch, 2003, Modeling of self-organized, avascular tumour growth with a hybrid cellular automaton, www.bioinfo.de/isb/2002/02/0035/main. html, (last accessed on 09/09/2003)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Factors associated with tumor volume and primary metastases in Ewing tumors: results from the (EI) CESS studies. H W Hense, S Ahrens, M Paulussen, M Lehnert, H Jurgens, Annals of Oncology. 109Hense, H.W., S. Ahrens, M. Paulussen, M. Lehnert and H. Jurgens, 1999, Factors associated with tumor volume and primary metastases in Ewing tumors: results from the (EI) CESS studies, Annals of Oncology, 10(9), 1073-1077\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Difference Equations -An Introduction with Applications. Walter G Kelly, Allan C Peterson, Academic PressMassachusetts, Burlington, USA2nd EdKelly, Walter G. and Allan C. Peterson, 2000,. Difference Equations -An Introduction with Applications, 2nd Ed., Academic Press, Massachusetts, Burlington, USA\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cancer Biology. Roger J B King, Pearson Education LimitedEssex, UK2nd EdKing, Roger J.B., 2000, Cancer Biology, 2nd Ed., Pearson Education Limited, Essex, UK\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Anchorage-Independent multicellular spheroids as an in vitro model of growth signaling in Ewing tumors. E R Lawlor, C Scheel, J Irving, P H B Sorensen, Oncogene. 21Lawlor E.R., C. Scheel, J. Irving, and P.H.B. Sorensen, 2002, Anchorage- Independent multicellular spheroids as an in vitro model of growth signaling in Ewing tumors, Oncogene, 21, 307-318\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tumor growth in vivo and as multicellular spheroids compared by mathematical models. M Marusic, Z Bajzer, S Vuk-Pavlovic, J P Freyer, Bulletin of Mathematical Biology. 564Marusic, M., Z. Bajzer, S. Vuk-Pavlovic, and J.P. Freyer, 1994, Tumor growth in vivo and as multicellular spheroids compared by mathematical models, Bulletin of Mathematical Biology, 56(4), 617-31\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Evidence for a Molecular Signature of Metastasis in Primary Solid Tumours. S Ramaswamy, K N Ross, E S Lander, T R Golub, last accessed on 09/09/2003Ramaswamy, S., K.N. Ross, E.S. Lander and T.R. Golub, Evidence for a Molecular Signature of Metastasis in Primary Solid Tumours, www.genome.wi.mit.edu/cancer/solid_tumor_metastasis, (last accessed on 09/09/2003)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Virtual and real brain tumors: using mathematical model to quantify glioma growth and invasion. K R Swanson, C Bridge, J D Murray, E C AlvordJr, Journal of the Neurological Sciences. 216Swanson, K.R., C. Bridge, J.D. Murray, and E.C. Alvord Jr., 2003, Virtual and real brain tumors: using mathematical model to quantify glioma growth and invasion, Journal of the Neurological Sciences, 216, 1 -10\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hyperbolastic growth models: theory and application. M Tabatabai, D K Williams, Z Bursac, Theoretical Biology and Medical Modelling. 21488in printTabatabai, M., D.K. Williams, and Z. Bursac, 2005, Hyperbolastic growth models: theory and application, Theoretical Biology and Medical Modelling, 2(14), (in print) 88\",\n \"doi\": \"\"\n },\n {\n \"text\": \". IJTS. 3JJ05IJTS, Vol. 3, No. JJ05, June-July 2005\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":43,"cells":{"text":{"kind":"string","value":"\nBlow up criterion for incompressible nematic liquid crystal flows\n27 Oct 2012\n\nYinghui Zhang \nDepartment of Mathematics\nHunan Institute of Science and Technology\n414006YueyangChina\n\nZhong Tan \nGuochun Wu \nSchool of Mathematical Sciences\nXiamen University\n361005FujianChina\n\nBlow up criterion for incompressible nematic liquid crystal flows\n27 Oct 2012Incompressible nematic liquid crystal flowBlow up criterion MSC2010 35Q3576D03\nIn this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if (u, d) is smooth up to time T provided that T 0 ∇ × u(t, ·) BM O(R 3 ) + ∇d(t, ·) 8 L 4 (R 3 ) dt < ∞.\n\nIntroduction\n\nWe consider the following hydrodynamic system modeling the flow of liquid crystal materials in dimension three (see [2,3,8,10]) and references therein):\nu t + u · ∇u − ν△u + ∇P = −∇d · △d,\n(1.1a)\n∂ t d + u · ∇d = △d + |∇d| 2 d,(1.\n\n1b)\n\n∇ · u = 0 |d| = 1, (1.1c)\n\nfor (t, x) ∈ [0, +∞) × R 3 . Here u : R 3 → R 3 represents the velocity field of the incompressible viscous fluid, ν > 0 is the kinematic viscosity, P : R 3 → R represents the pressure function, d : R 3 → S 2 represents the macroscopic average of the nematic liquid crystal orientation field, ∇· and △ denote the divergence operator and the laplace operator respectively. We are interested in the Cauchy problem (1.1) with the initial value (u(x, 0), d(x, 0)) = (u 0 (x), d 0 (x)) (1.2) satisfying the following compatibility condition:\n\n∇ · u 0 (x) = 0, |d 0 (x)| = 1, lim\n|x|→∞ d 0 (x) = a ∈ S 2 ,(1.3)\nwhere a is a given unit vector.\n\nThe above system is a simplified version of the Ericksen-Leslie model, which reduces to the Ossen-Frank model in the static case, for the hydrodynamics of nematic liquid crystals developed during the period of 1958 through 1968 [2,3,8]. It is a macroscopic continuum description of the time evolution of the materials under the influence of both the flow field u(x, t), the macroscopic description of the microscopic orientation configurations d(x, t) of rod-like liquid crystals. Roughly speaking, the system (1.1) is a coupling between the non-homogeneous Navier-Stokes equation and the transported flow harmonic maps. Due to the physical importance and mathematical challenges, the study on nematic liquid crystals has attracted many physicists and mathematicians. The mathematical analysis of the liquid crystal flows was initiated by Lin [9], Lin and Liu in [10,11]. For any bounded smooth domain in R 2 , Lin , Lin and Wang [12] have proved the global existence of Leray-Hopf type weak solutions to system (1.1) which are smooth everywhere except on finitely many time slices (see [4] for the whole space). The uniqueness of weak solutions in two dimension was studied by [13,15]. Recently, Hong and Xin [5] studied the global existence for general Ericksen-Leslie system in dimension two. However, the global existence of weak solutions to the incompressible nematic liquid crystal flow equation (1.1) in three dimension with large initial data is still an outstanding open question.\n\nThe local well-posedness of the Cauchy problem of system (1.1) is rather standard (see [4,6,12]). At present, there is no global-in-time existence theory for classical solutions to system (1.1). In this paper, we will consider the short time classical solution to (1.1)-(1.2) and some criterion that characterizes the first finite singular time. Motivated by the famous work [1], Huang and Wang [6] have obtained a BKM type blow-up criterion. However, the techniques involved in this paper are much different from [6], which we believe that the result may have its own interest. Our main results are formulated as the following theorem:\nTheorem 1.1. For u 0 ∈ H s (R 3 ) with ∇ · u 0 = 0 and d 0 − a ∈ H s+1 (R 3 ) with |d 0 | = 1 for s ≥ 3. Suppose that (u, d) is a smooth solution to the system (1.1)-(1.2), then for given T > 0, (u, d) is smooth up to time T provided that T 0 ∇ × u(t, ·) BM O(R 3 ) + ∇d(t, ·) 8 L 4 (R 3 ) dt < ∞. (1.4)\nNotations. We denote by L p , W m,p the usual Lebesgue and Sobolev spaces on R 3 and H m = W m,2 , with norms · L p , · W m,p and · H m respectively. For the sake of conciseness, we do not distinguish functional space when scalar-valued or vector-valued functions are involved. We denote\n∇ = ∂ x = (∂ 1 , ∂ 2 , ∂ 3 ), where ∂ i = ∂ x i , ∇ i = ∂ i and put ∂ l x f = ∇ l f = ∇(∇ l−1 f ).\nWe assume C be a positive generic constant throughout this paper that may vary at different places and the integration domain R 3 will be always omitted without any ambiguity. Finally, ·, · denotes the inner-product in L 2 (R 3 ).\n\nRemark 1.1. It is standard that the condition (1.3) is preserved by the flow. In fact, first notice that the divergence free of the velocity field u can be justified by the initial assumption that ∇ · u = 0. Indeed, this can be easily and formally observed by take ∇· to the momentum equation. Moreover, applying the maximum principle to the equation for |d| 2 , one also can easily see that |d| = 1 under the initial assumption that |d 0 | = 1.\n\n\nProof of Theorem 1.1\n\nWe prove our theorem in this section. Without loss of generality, we assume ν = 1. The first bright idea to reduce many complicated computations lies in that we just need to do the lowest order and highest order energy estimates for the solutions. This is motivated by the following observation:\nf 2 H k ≤ C (f, ∇ k f ) 2 L 2 , ∀f ∈ H k . (2.1)\nThis inequality (2.1) can be easily proved by combing Young's inequality and Gagliardo-Nirenberg's inequality.\n∇ i f L p ≤ C(p) f α L q ∇ k f 1−α L r , ∀f ∈ H k (2.2) where 1 p − i 3 = 1 q α + ( 1 r − k 3 )(1 − α) with i ≤ k.\nIn order to prove Theorem 1.1, we also need the following logarithmic Sobolev's which is proved in [7] and is an improved version of that in [1].\n\nLemma 2.1. For p > n, the following logarithmic Sobolev's embedding theorem holds for all divergence free vector fields:\n∇f L ∞ ≤ C[1 + f L 2 + ∇ × f BM O ln(1 + f W 2,p )]. (2.3)\nNow we are in a position to prove our Theorem 1.1.\n\nProof of Theorem 1.1 First of all, for classical solutions to (1.1)-(1.2), one has the following basic energy law:\nu(t, ·) 2 L 2 + ∇d(t, ·) 2 L 2 + t 0 ( ∇u(s, ·) 2 L 2 + △d(s, ·) + |∇d| 2 d(s, ·) 2 L 2 )ds = u 0 2 L 2 + ∇d 0 2 L 2 ∀t > 0. (2.4)\nLet's concentrate on the case s = 3. For each multi-index α with |α| ≤ 3, by applying ∂ α x to (1.1a) and ∂ α+1\n\nx to (1.1b), multiplying them by ∂ α x u, ∂ α+1\n\nx d respectively and then integrating them over R 3 , we have\n1 2 d dt ∂ α x (u, ∇d) 2 L 2 + ∂ α x (∇u, △d) 2 L 2 = − ∂ α x (u · ∇u), ∂ α x u − ∂ α x (△d · ∇d), ∂ α x u − ∂ α+1 x (u · ∇d), ∂ α+1 x d + ∂ α+1 x (|∇d| 2 d), ∂ α+1 x d = 4 i=1 I |α|,i . (2.5)\nwhere I |α|,i are the corresponding terms in the above equation which will be estimated as follows. Now for |α| = 1 in (2.5), integrating by parts and using the divergence free condition ∇ · u = 0, we have\n|I 1,1 | = | ∂ 1 x (u · ∇u), ∂ 1 x u | = | ∂ 1 x u · ∇u, ∂ 1 x u | ≤ C ∇u L ∞ ∇u 2 L 2 .\nCombining Cauchy's inequality, Sobolev's inequality and the fact |∇d| 2 = −d · △d (since |d| = 1) gives\n|I 1,2 | = | ∂ 1 x (△d · ∇d), ∂ 1 x u | = | △d · ∇d, ∂ 2 x u | ≤ C △d 3 L 3 + 1 8 ∇ 2 u 2 L 2 ≤ C ∇d 6 5 L 4 ∇△d 9 5 L 2 + 1 8 ∇ 2 u 2 L 2 ≤ C ∇d 12 L 4 + 1 8 ( ∇△d 2 L 2 + ∇ 2 u 2 L 2 ).\nSimilarly,\n|I 1,3 | = | ∂ 2 x u · ∇d + ∂ x u · ∇∂ x d, ∂ 2 x d | ≤ C( ∇u L ∞ ∇ 2 d 2 L 2 + ∇d 12 L 4 ) + 1 8 ( ∇△d 2 L 2 + ∇ 2 u 2 L 2 ), |I 1,4 | = | ∂ 1 x (|∇d| 2 d), ∂ x △d | ≤ C ∇d 12 L 4 + 1 8 ∇△d 2 L 2 .\nTaking the above estimates in (2.5) for |α| = 1, we arrive at\nd dt (∇u, ∇ 2 d) 2 L 2 + (∇ 2 u, ∇ 3 d) 2 L 2 ≤ C[ ∇u L ∞ ( ∇u 2 L 2 + ∇ 2 d 2 L 2 ) + ∇d 12 L 4 ].\n(2.6)\n\nNext we derive an estimate for ∇d 4 L 4 . Differentiating (1.1b) with respect to x, we have\n∂ x d t − △∂ x d = ∂ x (|∇d| 2 d − u · ∇d). (2.7)\nWe multiply (2.7) by |∇d| 2 ∂ x d and integrate over R 3 to obtain that\n1 4 d dt ∇d 4 L 4 + |∇d||∇ 2 d| 2 L 2 + 1 2 ∂ x (|∇d| 2 ) 2 L 2 = |∇d| 2 ∂ x d, ∂ x (|∇d| 2 d − u · ∇d) = ∇d 6 L 6 + |∇d| 2 ∂ x d, 2∇d∇∂ x dd − ∂ x u · ∇d) ≤ C( ∇d 3 L 3 + ∇u L ∞ ∇d 4 L 4 ) ≤ C( ∇u L ∞ ∇d 4 L 4 + ∇d 12 L 4 ) + 1 8 ∇ 3 d 2 L 2 .\n(2.8)\n\nWe add (2.8) to (2.6) to obtain\nd dt ( (∇u, ∇ 2 d) 2 L 2 + ∇d 4 L 4 ) + (∇ 2 u, ∇ 3 d, |∇d||∇ 2 d|) 2 L 2 ≤ C( ∇u L ∞ + ∇d 8 L 4 )( (∇u, ∇ 2 d) 2 L 2 + ∇d 4 L 4 ), which gives that (∇u, ∇ 2 d)(t, ·) 2 L 2 + ∇d(t, ·) 4 L 4 + t t 0 (∇ 2 u, ∇ 3 d, |∇d||∇ 2 d|) 2 L 2 ds ≤ Ce C t t 0 ( ∇u L ∞ + ∇d 8 L 4 )ds ( (∇u, ∇ 2 d)(t 0 , ·) 2 L 2 + ∇d(t 0 , ·) 4 L 4 ).\n(2.9)\n\nNoting the condition (1.4) in Theorem 1.1, thus for any fixed small constant ε > 0, there exists T * (ε) < T such that\nT T * ( ∇ × u BM O + ∇d 8 L 4 )ds ≤ ε. (2.10)\nDefine the temporal energy functional:\nH(t) = sup T * ≤s≤t ( ∇ 3 u(s, ·) 2 L 2 + ∇ 4 d(s, ·) 2 L 2 )\nBy (2.1), (2.3), (2.4), (2.9) and (2.10), we arrive at\n(∇u, ∇ 2 d)(t, ·) 2 L 2 + ∇d(t, ·) 4 L 4 + t T * (∇ 2 u, ∇ 3 d, |∇d||∇ 2 d|) 2 L 2 ds ≤ C * e t T * C 0 ∇×u BM O ln(1+ (u,∇d) 2 H 3 )ds ≤ C * e C 0 ε ln(1+H(t)) = C * (1 + H(t)) C 0 ε , ∀ T * ≤ t < T,(2.11)\nwhere C 0 depends on (u 0 , ∇d 0 ) 2 L 2 and C * depends on (∇u, ∇ 2 d) 2 L 2 + ∇d 4 L 4 . Next for |α| = 3. For I 3,1 , we need to use the following Moser-type inequality (see [14, p. 43\n]): D s (f g) L 2 ≤ C( g L ∞ ∇ s f L 2 + f L ∞ ∇ s g L 2 ). (2.12)\nThus we have\n|I 3,1 | = | ∂ 2 x div(u ⊗ u), ∂ 4 x u | ≤ C ∇ 3 (u ⊗ u) 2 L 2 + 1 16 ∇ 4 u 2 L 2 ≤ C u 2 L ∞ ∇ 3 u 2 L 2 + 1 16 ∇ 4 u 2 L 2 ≤ C{ ∇u 5 6 L 2 ∇ 4 u 1 6 L 2 ∇u 1 3 L 2 ∇ 4 u 2 3 L 2 } 2 + 1 16 ∇ 4 u 2 L 2 ≤ C ∇u 14 L 2 + 1 8 ∇ 4 u 2 L 2 .\n(2.13)\n\nFor I 3,2 , we apply (2.12) and the fact △d · ∇d = ∇ · (∇d ⊗ ∇d − 1 2 |∇d| 2 I) to obtain that\n|I 3,2 | = | ∂ 2 x ∇ · (∇d ⊗ ∇d − 1 2 |∇d| 2 I), ∂ 4 x u | ≤ C ∇ 3 (∇d ⊗ ∇d − 1 2 |∇d| 2 I) 2 L 2 + 1 8 ∇ 4 2 L 2 ≤ C ∇d 2 L ∞ ∇ 4 d 2 L 2 + 1 8 ∇ 4 u 2 L 2 ≤ C{ ∇ 2 d 5 6 L 2 ∇ 5 d 1 6 L 2 ∇ 2 d 1 3 L 2 ∇ 5 d 2 3 L 2 } 2 + 1 8 ∇ 4 u 2 L 2 ≤ C ∇ 2 d 14 L 2 + 1 8 ( ∇ 4 u 2 L 2 + ∇ 5 d 2 L 2 ).\n(2.14)\n\nSimilar as the proof of (2.14), I 3,3 , I 3,4 can be bounded as follow:\n|I 3,3 | = | ∂ 3 x (u · ∇d), ∂ 5 x d | ≤ C ∂ 3 x (u · ∇d) 2 L 2 + 1 16 ∇ 5 d 2 L 2 ≤ C( ∇d 2 L ∞ ∇ 3 u 2 L 2 + u 2 L ∞ ∇ 4 u 2 L 2 ) + 1 16 ∇ 4 u 2 L 2 ≤ C{ ∇ 2 d 5 6 L 2 ∇ 5 d 1 6 L 2 ∇u 1 3 L 2 ∇ 4 u 2 3 L 2 + ∇u 5 6 L 2 ∇ 4 u 1 6 L 2 ∇ 2 d 1 3 L 2 ∇ 5 d 2 3 L 2 } 2 + 1 16 ∇ 4 u 2 L 2 ≤ C( ∇ 2 d 14 L 2 + ∇u 14 L 2 ) + 1 8 ( ∇ 4 u 2 L 2 + ∇ 5 d 2 L 2 ),(2.15)\nand\n|I 3,4 | = | ∂ 3 x (|∇d| 2 d), ∂ 5 x d | ≤ C ∂ 3 x (|∇d| 2 d) 2 L 2 + 1 32 ∇ 5 d 2 L 2 ≤ C( ∇ 3 |∇d| 2 2 L 2 + |∇d| 2 2 L ∞ ∇ 3 d 2 L 2 ) + 1 32 ∇ 5 d 2 L 2 ≤ C( ∇d 2 L ∞ ∇ 4 d 2 L 2 + △d 2 L ∞ ∇ 3 d 2 L 2 ) + 1 32 ∇ 5 d 2 L 2 ≤ C( ∇ 2 d 14 L 2 + { ∇ 2 d 1 2 L 2 ∇ 5 d 1 2 L 2 ∇ 2 d 2 3 L 2 ∇ 5 d 1 3 L 2 } 2 ) + 1 16 ∇ 5 d 2 L 2 ≤ C ∇ 2 d 14 L 2 + 1 8 ∇ 5 d 2 L 2 .\n(2.16)\n\nPutting (2.13)-(2.16) into (2.5) for |α| = 3 and by (2.11), we arrive at d dt ( ∇ 3 u 2 L 2 + ∇ 4 d 2 L 2 ) + ∇ 4 u 2 L 2 + ∇ 5 d 2 L 2 ≤ C( ∇ 2 d 14 L 2 + ∇u 14 L 2 ) ≤ C * (1 + H(t)) C 0 ε for all T * ≤ t < T . Integrating the above inequality with respect to time from T * to t ∈ [T * , T ), we obtain 1 + ∇ 3 u(t, ·) 2 L 2 + ∇ 4 d(t, ·) 2 L 2 ≤ 1 + ∇ 3 u(T * , ·) 2 L 2 + ∇ 4 d(T * , ·) 2 L 2 + C * (1 + H(t)) C 0 ε .\n\n(2.17)\n\nIf we choose ε small such that C 0 ε < 1 2 , then (2.17) implies\n1 + H(t) ≤ C(1 + ∇ 3 u(T * , ·) 2 L 2 + ∇ 4 d(T * , ·) 2 L 2 ),\nfor all T * ≤ t < T . Since the right hand side of above inequality is independent of t for T * ≤ t < T , we conclude that the above inequality is valid for t = T which means that u(T, ·) ∈ H 3 (R 3 ) and d(T, ·)−a ∈ H 4 (R 3 ). Thus the proof of Theorem 1.1 is completed.\n \nRemarks on the breakdown of smooth solutions for the 3-D Euler equation. J T Beale, T Kato, A Majda, Commun. Math. Phys. 94J. T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equation. Commun. Math. Phys. 94: 61-66.\n\nThe Physics of Liquid Crystals. P G De Gennes, OxfordP. G. DE Gennes, The Physics of Liquid Crystals. Oxford, 1974.\n\nHydrostatic theory of liquid crystal. J L Ericksen, Arch. Ration. Mech. Anal. 9J. L. Ericksen, Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9 (1962), 371-378.\n\nGlobal existence of solutions of the simplified Ericksen-Leslie system in R 2. M C Hong, Calc. Var. Partial Differential Equations. 40M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in R 2 . Calc. Var. Partial Differential Equations 40 (2011), 15-36.\n\nGlobal existence of solutions of the Liquid Crystal flow for the Oseen-Frank model in R 2. M C Hong, Z P Xin, Adv. Math. 231M. C. Hong, Z. P. Xin, Global existence of solutions of the Liquid Crystal flow for the Oseen- Frank model in R 2 . Adv. Math., 231 (2012), 1364-1400.\n\nBlow up criterion for nematic liquid crystal flows. T Huang, C Y Wang, Comm. Partial Differ. Equ. 37T. Huang, C. Y. Wang, Blow up criterion for nematic liquid crystal flows. Comm. Partial Differ. Equ. 37 (2012), 875-884.\n\nBilinear estimates in BMO and the Navier-Stokes equations. H Kozono, Y Taniuchi, Math. Z. 235H. Kozono, Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235 (2000), 173-194.\n\nSome constitutive equations for liquid crystals. F M Leslie, Arch. Ration. Mech. Anal. 28F. M. Leslie, Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28 (1962), 265-283.\n\nNonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. F H Lin, Commun. Pure Appl. Math. 42F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Math. 42 (1989), 789-814.\n\nNonparabolic dissipative systems modeling the flow of liquid crystals. F H Lin, C Liu, Commun. Pure Appl. Math. 48F. H. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48 (1995), 501-537.\n\nPartial regularity of the dynamic system modeling the flow of liquid crystals. F H Lin, C Liu, Discrete Contin. Dyn. Syst. 2F. H. Lin, C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 2 (1996), 1-22.\n\nLiquid crystal flows in two dimensions. F H Lin, J Lin, C Y Wang, Arch. Rational Mech. Anal. 197F. H. Lin, J. Lin, C. Y. Wang, Liquid crystal flows in two dimensions. Arch. Rational Mech. Anal. 197 (2010), 297-336.\n\nOn the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. F H Lin, J Lin, C Y Wang, Chinese Ann. Math. 31F. H. Lin, J. Lin, C. Y. Wang, On the uniqueness of heat flow of harmonic maps and hydrody- namic flow of nematic liquid crystals. Chinese Ann. Math. 31B: 921-928.\n\nCompressible fluid flow and system of conservation laws in several space variables. A Majda, Applied Mathematical Sciences. 53Springer-VerlagA. Majda, Compressible fluid flow and system of conservation laws in several space variables. Applied Mathematical Sciences, 53, Springer-Verlag, NewYork, 1984.\n\nGlobal regularity and uniqueness of weak solution for the 2-D liquid crystal flows. X Xu, Z Zhang, J. Differential Equations. 252X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. J. Differential Equations 252 (2012), 1169-1181.\n"},"corpus_id":{"kind":"number","value":119264097,"string":"119,264,097"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1210.4140\",\n \"mag\": 2057178348,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1210.4140v5.pdf\"\n ],\n \"pdf_sha\": \"b25e5bdf346006a872169aa53120c3aa50a09dfd\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Blow up criterion for incompressible nematic liquid crystal flows"},"authors":{"kind":"list like","value":["Yinghui Zhang \nDepartment of Mathematics\nHunan Institute of Science and Technology\n414006YueyangChina","Zhong Tan","Guochun Wu \nSchool of Mathematical Sciences\nXiamen University\n361005FujianChina"],"string":"[\n \"Yinghui Zhang \\nDepartment of Mathematics\\nHunan Institute of Science and Technology\\n414006YueyangChina\",\n \"Zhong Tan\",\n \"Guochun Wu \\nSchool of Mathematical Sciences\\nXiamen University\\n361005FujianChina\"\n]","isConversational":false},"abstract":{"kind":"string","value":"In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if (u, d) is smooth up to time T provided that T 0 ∇ × u(t, ·) BM O(R 3 ) + ∇d(t, ·) 8 L 4 (R 3 ) dt < ∞."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"We consider the following hydrodynamic system modeling the flow of liquid crystal materials in dimension three (see [2,3,8,10]) and references therein):\n(1.1a)"},{"title":"1b)","content":"∇ · u = 0 |d| = 1, (1.1c)\nfor (t, x) ∈ [0, +∞) × R 3 . Here u : R 3 → R 3 represents the velocity field of the incompressible viscous fluid, ν > 0 is the kinematic viscosity, P : R 3 → R represents the pressure function, d : R 3 → S 2 represents the macroscopic average of the nematic liquid crystal orientation field, ∇· and △ denote the divergence operator and the laplace operator respectively. We are interested in the Cauchy problem (1.1) with the initial value (u(x, 0), d(x, 0)) = (u 0 (x), d 0 (x)) (1.2) satisfying the following compatibility condition:\n∇ · u 0 (x) = 0, |d 0 (x)| = 1, lim\nwhere a is a given unit vector.\nThe above system is a simplified version of the Ericksen-Leslie model, which reduces to the Ossen-Frank model in the static case, for the hydrodynamics of nematic liquid crystals developed during the period of 1958 through 1968 [2,3,8]. It is a macroscopic continuum description of the time evolution of the materials under the influence of both the flow field u(x, t), the macroscopic description of the microscopic orientation configurations d(x, t) of rod-like liquid crystals. Roughly speaking, the system (1.1) is a coupling between the non-homogeneous Navier-Stokes equation and the transported flow harmonic maps. Due to the physical importance and mathematical challenges, the study on nematic liquid crystals has attracted many physicists and mathematicians. The mathematical analysis of the liquid crystal flows was initiated by Lin [9], Lin and Liu in [10,11]. For any bounded smooth domain in R 2 , Lin , Lin and Wang [12] have proved the global existence of Leray-Hopf type weak solutions to system (1.1) which are smooth everywhere except on finitely many time slices (see [4] for the whole space). The uniqueness of weak solutions in two dimension was studied by [13,15]. Recently, Hong and Xin [5] studied the global existence for general Ericksen-Leslie system in dimension two. However, the global existence of weak solutions to the incompressible nematic liquid crystal flow equation (1.1) in three dimension with large initial data is still an outstanding open question.\nThe local well-posedness of the Cauchy problem of system (1.1) is rather standard (see [4,6,12]). At present, there is no global-in-time existence theory for classical solutions to system (1.1). In this paper, we will consider the short time classical solution to (1.1)-(1.2) and some criterion that characterizes the first finite singular time. Motivated by the famous work [1], Huang and Wang [6] have obtained a BKM type blow-up criterion. However, the techniques involved in this paper are much different from [6], which we believe that the result may have its own interest. Our main results are formulated as the following theorem:\nNotations. We denote by L p , W m,p the usual Lebesgue and Sobolev spaces on R 3 and H m = W m,2 , with norms · L p , · W m,p and · H m respectively. For the sake of conciseness, we do not distinguish functional space when scalar-valued or vector-valued functions are involved. We denote\nWe assume C be a positive generic constant throughout this paper that may vary at different places and the integration domain R 3 will be always omitted without any ambiguity. Finally, ·, · denotes the inner-product in L 2 (R 3 ).\nRemark 1.1. It is standard that the condition (1.3) is preserved by the flow. In fact, first notice that the divergence free of the velocity field u can be justified by the initial assumption that ∇ · u = 0. Indeed, this can be easily and formally observed by take ∇· to the momentum equation. Moreover, applying the maximum principle to the equation for |d| 2 , one also can easily see that |d| = 1 under the initial assumption that |d 0 | = 1."},{"title":"Proof of Theorem 1.1","content":"We prove our theorem in this section. Without loss of generality, we assume ν = 1. The first bright idea to reduce many complicated computations lies in that we just need to do the lowest order and highest order energy estimates for the solutions. This is motivated by the following observation:\nThis inequality (2.1) can be easily proved by combing Young's inequality and Gagliardo-Nirenberg's inequality.\nIn order to prove Theorem 1.1, we also need the following logarithmic Sobolev's which is proved in [7] and is an improved version of that in [1].\nLemma 2.1. For p > n, the following logarithmic Sobolev's embedding theorem holds for all divergence free vector fields:\nNow we are in a position to prove our Theorem 1.1.\nProof of Theorem 1.1 First of all, for classical solutions to (1.1)-(1.2), one has the following basic energy law:\nLet's concentrate on the case s = 3. For each multi-index α with |α| ≤ 3, by applying ∂ α x to (1.1a) and ∂ α+1\nx to (1.1b), multiplying them by ∂ α x u, ∂ α+1\nx d respectively and then integrating them over R 3 , we have\nwhere I |α|,i are the corresponding terms in the above equation which will be estimated as follows. Now for |α| = 1 in (2.5), integrating by parts and using the divergence free condition ∇ · u = 0, we have\nCombining Cauchy's inequality, Sobolev's inequality and the fact |∇d| 2 = −d · △d (since |d| = 1) gives\nSimilarly,\nTaking the above estimates in (2.5) for |α| = 1, we arrive at\n(2.6)\nNext we derive an estimate for ∇d 4 L 4 . Differentiating (1.1b) with respect to x, we have\nWe multiply (2.7) by |∇d| 2 ∂ x d and integrate over R 3 to obtain that\n(2.8)\nWe add (2.8) to (2.6) to obtain\n(2.9)\nNoting the condition (1.4) in Theorem 1.1, thus for any fixed small constant ε > 0, there exists T * (ε) < T such that\nDefine the temporal energy functional:\nBy (2.1), (2.3), (2.4), (2.9) and (2.10), we arrive at\nwhere C 0 depends on (u 0 , ∇d 0 ) 2 L 2 and C * depends on (∇u, ∇ 2 d) 2 L 2 + ∇d 4 L 4 . Next for |α| = 3. For I 3,1 , we need to use the following Moser-type inequality (see [14, p. 43\nThus we have\n(2.13)\nFor I 3,2 , we apply (2.12) and the fact △d · ∇d = ∇ · (∇d ⊗ ∇d − 1 2 |∇d| 2 I) to obtain that\n(2.14)\nSimilar as the proof of (2.14), I 3,3 , I 3,4 can be bounded as follow:\nand\n(2.16)\nPutting (2.13)-(2.16) into (2.5) for |α| = 3 and by (2.11), we arrive at d dt ( ∇ 3 u 2 L 2 + ∇ 4 d 2 L 2 ) + ∇ 4 u 2 L 2 + ∇ 5 d 2 L 2 ≤ C( ∇ 2 d 14 L 2 + ∇u 14 L 2 ) ≤ C * (1 + H(t)) C 0 ε for all T * ≤ t < T . Integrating the above inequality with respect to time from T * to t ∈ [T * , T ), we obtain 1 + ∇ 3 u(t, ·) 2 L 2 + ∇ 4 d(t, ·) 2 L 2 ≤ 1 + ∇ 3 u(T * , ·) 2 L 2 + ∇ 4 d(T * , ·) 2 L 2 + C * (1 + H(t)) C 0 ε .\n(2.17)\nIf we choose ε small such that C 0 ε < 1 2 , then (2.17) implies\nfor all T * ≤ t < T . Since the right hand side of above inequality is independent of t for T * ≤ t < T , we conclude that the above inequality is valid for t = T which means that u(T, ·) ∈ H 3 (R 3 ) and d(T, ·)−a ∈ H 4 (R 3 ). Thus the proof of Theorem 1.1 is completed."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"We consider the following hydrodynamic system modeling the flow of liquid crystal materials in dimension three (see [2,3,8,10]) and references therein):\\n(1.1a)\"\n },\n {\n \"title\": \"1b)\",\n \"content\": \"∇ · u = 0 |d| = 1, (1.1c)\\nfor (t, x) ∈ [0, +∞) × R 3 . Here u : R 3 → R 3 represents the velocity field of the incompressible viscous fluid, ν > 0 is the kinematic viscosity, P : R 3 → R represents the pressure function, d : R 3 → S 2 represents the macroscopic average of the nematic liquid crystal orientation field, ∇· and △ denote the divergence operator and the laplace operator respectively. We are interested in the Cauchy problem (1.1) with the initial value (u(x, 0), d(x, 0)) = (u 0 (x), d 0 (x)) (1.2) satisfying the following compatibility condition:\\n∇ · u 0 (x) = 0, |d 0 (x)| = 1, lim\\nwhere a is a given unit vector.\\nThe above system is a simplified version of the Ericksen-Leslie model, which reduces to the Ossen-Frank model in the static case, for the hydrodynamics of nematic liquid crystals developed during the period of 1958 through 1968 [2,3,8]. It is a macroscopic continuum description of the time evolution of the materials under the influence of both the flow field u(x, t), the macroscopic description of the microscopic orientation configurations d(x, t) of rod-like liquid crystals. Roughly speaking, the system (1.1) is a coupling between the non-homogeneous Navier-Stokes equation and the transported flow harmonic maps. Due to the physical importance and mathematical challenges, the study on nematic liquid crystals has attracted many physicists and mathematicians. The mathematical analysis of the liquid crystal flows was initiated by Lin [9], Lin and Liu in [10,11]. For any bounded smooth domain in R 2 , Lin , Lin and Wang [12] have proved the global existence of Leray-Hopf type weak solutions to system (1.1) which are smooth everywhere except on finitely many time slices (see [4] for the whole space). The uniqueness of weak solutions in two dimension was studied by [13,15]. Recently, Hong and Xin [5] studied the global existence for general Ericksen-Leslie system in dimension two. However, the global existence of weak solutions to the incompressible nematic liquid crystal flow equation (1.1) in three dimension with large initial data is still an outstanding open question.\\nThe local well-posedness of the Cauchy problem of system (1.1) is rather standard (see [4,6,12]). At present, there is no global-in-time existence theory for classical solutions to system (1.1). In this paper, we will consider the short time classical solution to (1.1)-(1.2) and some criterion that characterizes the first finite singular time. Motivated by the famous work [1], Huang and Wang [6] have obtained a BKM type blow-up criterion. However, the techniques involved in this paper are much different from [6], which we believe that the result may have its own interest. Our main results are formulated as the following theorem:\\nNotations. We denote by L p , W m,p the usual Lebesgue and Sobolev spaces on R 3 and H m = W m,2 , with norms · L p , · W m,p and · H m respectively. For the sake of conciseness, we do not distinguish functional space when scalar-valued or vector-valued functions are involved. We denote\\nWe assume C be a positive generic constant throughout this paper that may vary at different places and the integration domain R 3 will be always omitted without any ambiguity. Finally, ·, · denotes the inner-product in L 2 (R 3 ).\\nRemark 1.1. It is standard that the condition (1.3) is preserved by the flow. In fact, first notice that the divergence free of the velocity field u can be justified by the initial assumption that ∇ · u = 0. Indeed, this can be easily and formally observed by take ∇· to the momentum equation. Moreover, applying the maximum principle to the equation for |d| 2 , one also can easily see that |d| = 1 under the initial assumption that |d 0 | = 1.\"\n },\n {\n \"title\": \"Proof of Theorem 1.1\",\n \"content\": \"We prove our theorem in this section. Without loss of generality, we assume ν = 1. The first bright idea to reduce many complicated computations lies in that we just need to do the lowest order and highest order energy estimates for the solutions. This is motivated by the following observation:\\nThis inequality (2.1) can be easily proved by combing Young's inequality and Gagliardo-Nirenberg's inequality.\\nIn order to prove Theorem 1.1, we also need the following logarithmic Sobolev's which is proved in [7] and is an improved version of that in [1].\\nLemma 2.1. For p > n, the following logarithmic Sobolev's embedding theorem holds for all divergence free vector fields:\\nNow we are in a position to prove our Theorem 1.1.\\nProof of Theorem 1.1 First of all, for classical solutions to (1.1)-(1.2), one has the following basic energy law:\\nLet's concentrate on the case s = 3. For each multi-index α with |α| ≤ 3, by applying ∂ α x to (1.1a) and ∂ α+1\\nx to (1.1b), multiplying them by ∂ α x u, ∂ α+1\\nx d respectively and then integrating them over R 3 , we have\\nwhere I |α|,i are the corresponding terms in the above equation which will be estimated as follows. Now for |α| = 1 in (2.5), integrating by parts and using the divergence free condition ∇ · u = 0, we have\\nCombining Cauchy's inequality, Sobolev's inequality and the fact |∇d| 2 = −d · △d (since |d| = 1) gives\\nSimilarly,\\nTaking the above estimates in (2.5) for |α| = 1, we arrive at\\n(2.6)\\nNext we derive an estimate for ∇d 4 L 4 . Differentiating (1.1b) with respect to x, we have\\nWe multiply (2.7) by |∇d| 2 ∂ x d and integrate over R 3 to obtain that\\n(2.8)\\nWe add (2.8) to (2.6) to obtain\\n(2.9)\\nNoting the condition (1.4) in Theorem 1.1, thus for any fixed small constant ε > 0, there exists T * (ε) < T such that\\nDefine the temporal energy functional:\\nBy (2.1), (2.3), (2.4), (2.9) and (2.10), we arrive at\\nwhere C 0 depends on (u 0 , ∇d 0 ) 2 L 2 and C * depends on (∇u, ∇ 2 d) 2 L 2 + ∇d 4 L 4 . Next for |α| = 3. For I 3,1 , we need to use the following Moser-type inequality (see [14, p. 43\\nThus we have\\n(2.13)\\nFor I 3,2 , we apply (2.12) and the fact △d · ∇d = ∇ · (∇d ⊗ ∇d − 1 2 |∇d| 2 I) to obtain that\\n(2.14)\\nSimilar as the proof of (2.14), I 3,3 , I 3,4 can be bounded as follow:\\nand\\n(2.16)\\nPutting (2.13)-(2.16) into (2.5) for |α| = 3 and by (2.11), we arrive at d dt ( ∇ 3 u 2 L 2 + ∇ 4 d 2 L 2 ) + ∇ 4 u 2 L 2 + ∇ 5 d 2 L 2 ≤ C( ∇ 2 d 14 L 2 + ∇u 14 L 2 ) ≤ C * (1 + H(t)) C 0 ε for all T * ≤ t < T . Integrating the above inequality with respect to time from T * to t ∈ [T * , T ), we obtain 1 + ∇ 3 u(t, ·) 2 L 2 + ∇ 4 d(t, ·) 2 L 2 ≤ 1 + ∇ 3 u(T * , ·) 2 L 2 + ∇ 4 d(T * , ·) 2 L 2 + C * (1 + H(t)) C 0 ε .\\n(2.17)\\nIf we choose ε small such that C 0 ε < 1 2 , then (2.17) implies\\nfor all T * ≤ t < T . Since the right hand side of above inequality is independent of t for T * ≤ t < T , we conclude that the above inequality is valid for t = T which means that u(T, ·) ∈ H 3 (R 3 ) and d(T, ·)−a ∈ H 4 (R 3 ). Thus the proof of Theorem 1.1 is completed.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Remarks on the breakdown of smooth solutions for the 3-D Euler equation. J T Beale, T Kato, A Majda, Commun. Math. Phys. 94J. T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equation. Commun. Math. Phys. 94: 61-66.","doi":""},{"text":"The Physics of Liquid Crystals. P G De Gennes, OxfordP. G. DE Gennes, The Physics of Liquid Crystals. Oxford, 1974.","doi":""},{"text":"Hydrostatic theory of liquid crystal. J L Ericksen, Arch. Ration. Mech. Anal. 9J. L. Ericksen, Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9 (1962), 371-378.","doi":""},{"text":"Global existence of solutions of the simplified Ericksen-Leslie system in R 2. M C Hong, Calc. Var. Partial Differential Equations. 40M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in R 2 . Calc. Var. Partial Differential Equations 40 (2011), 15-36.","doi":""},{"text":"Global existence of solutions of the Liquid Crystal flow for the Oseen-Frank model in R 2. M C Hong, Z P Xin, Adv. Math. 231M. C. Hong, Z. P. Xin, Global existence of solutions of the Liquid Crystal flow for the Oseen- Frank model in R 2 . Adv. Math., 231 (2012), 1364-1400.","doi":""},{"text":"Blow up criterion for nematic liquid crystal flows. T Huang, C Y Wang, Comm. Partial Differ. Equ. 37T. Huang, C. Y. Wang, Blow up criterion for nematic liquid crystal flows. Comm. Partial Differ. Equ. 37 (2012), 875-884.","doi":""},{"text":"Bilinear estimates in BMO and the Navier-Stokes equations. H Kozono, Y Taniuchi, Math. Z. 235H. Kozono, Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235 (2000), 173-194.","doi":""},{"text":"Some constitutive equations for liquid crystals. F M Leslie, Arch. Ration. Mech. Anal. 28F. M. Leslie, Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28 (1962), 265-283.","doi":""},{"text":"Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. F H Lin, Commun. Pure Appl. Math. 42F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Math. 42 (1989), 789-814.","doi":""},{"text":"Nonparabolic dissipative systems modeling the flow of liquid crystals. F H Lin, C Liu, Commun. Pure Appl. Math. 48F. H. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48 (1995), 501-537.","doi":""},{"text":"Partial regularity of the dynamic system modeling the flow of liquid crystals. F H Lin, C Liu, Discrete Contin. Dyn. Syst. 2F. H. Lin, C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 2 (1996), 1-22.","doi":""},{"text":"Liquid crystal flows in two dimensions. F H Lin, J Lin, C Y Wang, Arch. Rational Mech. Anal. 197F. H. Lin, J. Lin, C. Y. Wang, Liquid crystal flows in two dimensions. Arch. Rational Mech. Anal. 197 (2010), 297-336.","doi":""},{"text":"On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. F H Lin, J Lin, C Y Wang, Chinese Ann. Math. 31F. H. Lin, J. Lin, C. Y. Wang, On the uniqueness of heat flow of harmonic maps and hydrody- namic flow of nematic liquid crystals. Chinese Ann. Math. 31B: 921-928.","doi":""},{"text":"Compressible fluid flow and system of conservation laws in several space variables. A Majda, Applied Mathematical Sciences. 53Springer-VerlagA. Majda, Compressible fluid flow and system of conservation laws in several space variables. Applied Mathematical Sciences, 53, Springer-Verlag, NewYork, 1984.","doi":""},{"text":"Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. X Xu, Z Zhang, J. Differential Equations. 252X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. J. Differential Equations 252 (2012), 1169-1181.","doi":""}],"string":"[\n {\n \"text\": \"Remarks on the breakdown of smooth solutions for the 3-D Euler equation. J T Beale, T Kato, A Majda, Commun. Math. Phys. 94J. T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equation. Commun. Math. Phys. 94: 61-66.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Physics of Liquid Crystals. P G De Gennes, OxfordP. G. DE Gennes, The Physics of Liquid Crystals. Oxford, 1974.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hydrostatic theory of liquid crystal. J L Ericksen, Arch. Ration. Mech. Anal. 9J. L. Ericksen, Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9 (1962), 371-378.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Global existence of solutions of the simplified Ericksen-Leslie system in R 2. M C Hong, Calc. Var. Partial Differential Equations. 40M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in R 2 . Calc. Var. Partial Differential Equations 40 (2011), 15-36.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Global existence of solutions of the Liquid Crystal flow for the Oseen-Frank model in R 2. M C Hong, Z P Xin, Adv. Math. 231M. C. Hong, Z. P. Xin, Global existence of solutions of the Liquid Crystal flow for the Oseen- Frank model in R 2 . Adv. Math., 231 (2012), 1364-1400.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Blow up criterion for nematic liquid crystal flows. T Huang, C Y Wang, Comm. Partial Differ. Equ. 37T. Huang, C. Y. Wang, Blow up criterion for nematic liquid crystal flows. Comm. Partial Differ. Equ. 37 (2012), 875-884.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bilinear estimates in BMO and the Navier-Stokes equations. H Kozono, Y Taniuchi, Math. Z. 235H. Kozono, Y. Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235 (2000), 173-194.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Some constitutive equations for liquid crystals. F M Leslie, Arch. Ration. Mech. Anal. 28F. M. Leslie, Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28 (1962), 265-283.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. F H Lin, Commun. Pure Appl. Math. 42F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Math. 42 (1989), 789-814.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonparabolic dissipative systems modeling the flow of liquid crystals. F H Lin, C Liu, Commun. Pure Appl. Math. 48F. H. Lin, C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48 (1995), 501-537.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Partial regularity of the dynamic system modeling the flow of liquid crystals. F H Lin, C Liu, Discrete Contin. Dyn. Syst. 2F. H. Lin, C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 2 (1996), 1-22.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Liquid crystal flows in two dimensions. F H Lin, J Lin, C Y Wang, Arch. Rational Mech. Anal. 197F. H. Lin, J. Lin, C. Y. Wang, Liquid crystal flows in two dimensions. Arch. Rational Mech. Anal. 197 (2010), 297-336.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. F H Lin, J Lin, C Y Wang, Chinese Ann. Math. 31F. H. Lin, J. Lin, C. Y. Wang, On the uniqueness of heat flow of harmonic maps and hydrody- namic flow of nematic liquid crystals. Chinese Ann. Math. 31B: 921-928.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Compressible fluid flow and system of conservation laws in several space variables. A Majda, Applied Mathematical Sciences. 53Springer-VerlagA. Majda, Compressible fluid flow and system of conservation laws in several space variables. Applied Mathematical Sciences, 53, Springer-Verlag, NewYork, 1984.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. X Xu, Z Zhang, J. Differential Equations. 252X. Xu and Z. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows. J. Differential Equations 252 (2012), 1169-1181.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":44,"cells":{"text":{"kind":"string","value":"\nLOCALLY POTENTIALLY EQUIVALENT GALOIS REPRESENTATIONS II\n12 Jan 2019\n\nVijay M Patankar \nC S Rajan \nLOCALLY POTENTIALLY EQUIVALENT GALOIS REPRESENTATIONS II\n12 Jan 2019\nSuppose ρ 1 , ρ 2 are two ℓ-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of the first representation is connected, and the representations are locally potentially equivalent at a set of places of positive upper density. We classify such pairs of representations and show that up to twisting by an arbitrary representation, it is given by a pair of representations one of which is trivial and the other abelian.Consequently, we obtain that the representations are (globally) potentially equivalent, provided one of the following conditions hold: (a) the ranks of the algebraic monodromy groups are equal; (b) the first representation is absolutely irreducible and (c) the algebraic monodromy group of the second representation is also connected.\n\nIntroduction\n\nTwo n-dimensional linear representations ρ 1 , ρ 2 of a group Γ are said to be potentially equivalent if they become isomorphic upon restriction to a subgroup of finite index in Γ. In this article, we are interested in proving potential equivalence of representations of the absolute Galois group of a global field, which are locally potentially equivalent at a sufficiently large number of places of the global field.\n\nFor a field K, denote by G K := Gal(K/K) an absolute Galois group of K, wherē K denote a seperable algebraic closure of K. Let K be a global field. For a place v of K let K v denote the completion of K at v. Choosing a place w ofK lying above v, allows us to identity G Kv with the decomposition subgroup D w of G K . As w varies this gives a conjugacy class of subgroups of G K . Given a representation ρ of G K as above, define the localization (or the local component) ρ v of ρ at v, to be the representation of G Kv obtained by restricting ρ to a decomposition subgroup. This is well defined upto isomorphism.\n\nLet F be a local field of characteristic zero and residue characteristic ℓ relatively prime to the characteristic of K. Suppose ρ i : G K → GL n (F ), i = 1, 2 are two continuous semisimple ℓ-adic representations of G K , unramified outside a finite set Σ ram of places of K containing the archimedean places of K. Let T be a set of place of K. Define ρ 1 and ρ 2 to be locally potentially equivalent at T , if for each v ∈ T , the restrictions ρ 1,v and ρ 2,v to G Kv are potentially equivalent.\n\nGiven a continuous representation ρ : G K → GL n (F ) and a place v of K where ρ is unramified, let ρ(σ v ) denote the Frobenius conjugacy class in the image group G K /Ker(ρ) ≃ ρ(G K ) ⊂ GL n (F ). By an abuse of notation, we will also continue to denote by ρ(σ v ) an element in the associated conjugacy class. The representations ρ 1 and ρ 2 are locally potentially equivalent at a prime v not in Σ ram , precisely when the eigenvalues of the Frobenius conjugacy classes ρ 1 (σ v ) and ρ 2 (σ v ) differ by roots of unity.\n\nFor a representation ρ : G K → GL n (F ), let G ρ be the algebraic monodromy group attached to ρ over F , i.e., the smallest algebraic subgroup G ρ of GL n defined over F such that ρ(G K ) ⊂ G ρ (F ). Denote by G i the algebraic monodromy groups associated to the representations ρ i for i = 1, 2. We recall the upper density of a set S of finite places of K is defined as:\nud(S) := lim sup x→∞ #{v ∈ S| Nv ≤ x}/π(x),\nwhere π(x) is the number of finite places v of K with Nv ≤ x. Here Nv denotes the cardinality of the residue field k v of K v .\n\nOur aim in this paper is to establish a theorem of the following type: assume that the algebraic monodromy group of either ρ 1 or ρ 2 is connected and the set of places v of K where the localizations ρ 1,v and ρ 2,v are potentially equivalent has positive upper density. Then ρ 1 and ρ 2 are (globally) potentially equivalent.\n\nIn [Ra1], it is proved that if the characters of ρ 1 and ρ 2 agree at a set of unramified places having positive upper density and G 1 is connected, then ρ 1 and ρ 2 are potentially equivalent.\n\nIn ( [PR]), we had erroneously claimed that for representations as above, if G 1 is connected and the upper density of the set T of places of K at which ρ 1 and ρ 2 are locally potentially equivalent is positive, then the representations ρ 1 and ρ 2 are potentially equivalent. It was pointed out by J.-P. Serre that this claim is wrong (the proof given in ( [PR]) however goes through if we assume that the density of T is one).\n\nThe problem was that in the course of the proof (see Remark 2.2), the following assertion was made: let G be a reductive algebraic group, and let m be a number divisible by the exponent of the group G/G 0 . Then for any connected component\nG φ of G, the m-th power map G φ → G 0 is surjective, where G 0 is the connected component of identity in G.\nExample 1.1. Serre gave the following counter-example: take G to be the normalizer of the diagonal torus in SL(2). The square map sends the non-identity component to scalar matrices (±Id) and is not surjective. In terms of representations, let p 1 denote the trivial two dimensional representation and p 2 the natural two dimensional representation of G. The characteristic polynomials of both these representations are equal evaluated at the fourth power of elements of the non identity connected component of G.\n\nExample 1.2. Based on this, an arithmetical counterexample can be given to the claim made in [PR]. Let L = Q(i) be the field of Gaussian numbers, and fix an embedding of L into C. Associated to an elliptic curve E with complex multiplication by Z[i], there exists, by the theory of complex multiplication, a continuous character ψ w : G L → L * w for any finite place w of L with the following property: at a finite place v of L with residue field coprime to the residue characteristic of L w and unramified for E, the Frobenius element π v := ψ(σ v ) generates a prime ideal in Z[i]. Further ψ(σ v )ψ τ (σ v ) is equal to p (resp. to p 2 ), when Nv = p (resp. Nv = p 2 ), where v divides the rational prime p of Q and τ denotes the non-trivial element of Gal(L/Q).\n\nLet χ : G Q → Q * ℓ denote the ℓ-adic cyclotomic character. We have for p coprime to ℓ, χ(σ p ) = p. Consider the two representations,\nρ 1 = χ ⊕ χ and ρ 2 = Ind G Q G K (ψ 2 ).\nLet T be the set of rational primes p inert in k and unramified for ρ 2 . The characteristic polynomials of ρ 1 (σ p ) and ρ 2 (σ p ) for p ∈ T , are respectively of the form (X − p) 2 and X 2 + p 2 . Hence for i = 1, 2, ρ i (σ p ) 4 = p 4 I, where I is the identity matrix. Thus these representations are potentially equivalent at T .\n\nThe algebraic monodromy group G 1 of ρ 1 is isomorphic to G m . For any natural number m, the characters ψ m and (ψ τ ) m are not equal up to multiplication by a root of unity, by considering the values of the character at places of degree one over Q. It follows that the algebraic monodromy group G 2 is isomorphic to the normalizer of the Weil restriction of scalars R L/Q (G m ) ⊂ GL 2 . Thus G 1 is connected, the representations ρ 1 and ρ 2 are potentially equivalent at a set of places of density 1 2 , but are not potentially equivalent yielding a contradiction to Theorem 2.1 proved in [PR]. Example 1.3. Once there is a pair (ρ 1 , ρ 2 ) of representations of G K which fails to be potentially equivalent, then any twist (ρ 1 ⊗ η, ρ 2 ⊗ η), where η is a finite dimensional linear representation of G K , satisfies the hypothesis of being locally potentially equivalent at T . Since the characters are non-vanishing in some neighbourhood of identity, the twists are not potentially equivalent either.\n\nFurther, if K ⊃ K 0 , and the representations ρ 1 ⊗ η and ρ 2 ⊗ η are respectively restrictions to G K of representations η 1 and η 2 of G K 0 , then the pair of representations (η 1 , η 2 ) will also provide a counterexample. These representations will be locally potentially equivalent at the unramified set of places T 0 of K 0 that lie below T , but will not be potentially equivalent.\n\n1.1. Theorems. Our main theorem says that the failue of ρ 1 and ρ 2 being potentially isomorphic stems from the presence of a summand generalizing the above examples, where upto a twist the first representation is trivial and the second representation is abelian.\n\nTheorem 1.1. Let K be a global field and S be a set of places of K containing the archimedean places and of density zero. Let F be a non-archimedean local field of characteristic zero, and ρ 1 , ρ 2 : G K → GL n (F ) be semisimple continuous representations unramified outside S, satisfying the following hypothesis: H1: The algebraic monodromy group G 1 of ρ 1 is connected. H2: There exists a set of places T of K disjoint from S having positive upper density such that the representations ρ 1 and ρ 2 are locally potentially equivalent at T , i.e., for each v ∈ T , there exists a natural number k v ≥ 1 such that the ρ 1 (σ v ) kv and ρ 2 (σ v ) kv are conjugate in GL n (F ), where ρ 1 (σ v ) and ρ 2 (σ v ) are respectively the Frobenius conjugacy classes at v of ρ 1 and ρ 2 .\n\nThen there exists a finite extension L of K, a finite extension E of F , a set of places T L of positive density lying above the set of places T of K, and decompositions\nρ 1 | G L ≃ ⊕ t i=1 ρ 1,i and ρ 2 | G L ≃ ⊕ t i=1 ρ 2,i ,\nwhere ρ 1,i and ρ 2,i are representations of G L to GL n i (E) for some n i ≥ 1. Further, for each i, there exists an absolutely irreducible representation r i of G L defined over E, and representations ρ ′ 1,i , ρ ′ 2,i of G L into GL n i /dim(r i ) (E) satisfying the following:\n\n(1) ρ ′ 1,i is potentially trivial and ρ ′ 2,i is potentially abelian.\n(2) ρ 1,i = r i ⊗ ρ ′ 1,i and ρ 2,i = r i ⊗ ρ ′ 2,i . (3)\nThe representations ρ ′ 1,i and ρ ′ 2,i are locally potentially equivalent at T L . Remark 1.1. In [KPR], we had given a proof of the above theorem in the context of non-CM elliptic curves. One of the motivations was to conclude that an elliptic curve has CM by F , if the imaginary quadratic field F occurs as the Frobenius field at a set of primes of positive upper density.\n\nWe now present a different version of the foregoing theorem in terms of character values evaluated at a given power of Frobenius classes: Theorem 1.2. With notation as in Theorem 1.1, assume that the following hypothesis H2 ′ is satisfied instead of H2: H2 ′ : There exists a set of places T of K disjoint from S having positive upper density, and a natural number m such that for v ∈ T ,\n(1.1) Tr(ρ 1 (σ v ) m ) = Tr(ρ 2 (σ v ) m ).\nThen the conclusion of Theorem 1.2 remain valid, except that Condition (3) of the conclusion should be replaced by the condition,\nTr(ρ ′ 1,i (σ w ) m ) = Tr(ρ ′ 2,i (σ w ) m ), w ∈ T L .\nRemark 1.2. Theorem 1.2 differs from Theorem 1.1, in that apart from bounding k v uniformly independent of v ∈ T , we are able to work with character values of the Frobenius classes, rather than the m-th powers of the Frobenius classes being conjugate. We don't know whether we can replace H2 ′ with the following condition:\nTr(ρ 1 (σ v ) mv ) = Tr(ρ 2 (σ v ) mv ),\nwhere m v are natural numbers depending on v.\n\nThe proof of these theorems follow along the lines of ( [Se]) and ( [Ra1]), by considering the algebraic monodromy groups of the representations involved and converting the problem to one on algebraic groups by means of an algebraic Chebotarev density theorem (see Theorem 2.1). The algebraic formulation allows us to base change to complex numbers and to employ an unitary trick used in [Ra1]. The crucial observation out here is to interpret the consequence of the unitary trick and convert the problem to one involving the first representation and twist of the second representation by an automorphism of finite order. This allows us to get at the algebraic structure of the representations satisfying the hypothesis of Theorem 1.2.\n\n1.2. Potential equivalence. The examples given above suggest the possibility of proving potential equivalence of ρ 1 and ρ 2 (assuming G 1 is connected and upper density of T is positive), under some additional natural hypothesis on the nature of the representations ρ 1 and ρ 2 . As corollaries of the theorems stated above, we obtain the following:\n\nCorollary 1.1. With hypothesis as in Theorem 1.2 (or Theorem 1.1), assume further that either of the following conditions hold:\n\n(1) The algebraic ranks of G 1 and G 0 2 are equal, where G 0 2 is the connected component of identity of G 2 .\n\n(2) ρ 1 is absolutely irreducible.\n\nThen ρ 1 and ρ 2 are potentially equivalent.\n\nWe also deduce potential isomorphy invoking the hypothesis that the algebraic monodromy groups of both the representations are connected, by a direct use of the algebraic machinery: Theorem 1.3. With hypothesis as in Theorem 1.2, assume further that G 2 is connected. Then ρ 1 and ρ 2 are potentially equivalent.\n\n2. Proof of Theorem 1.2 2.1. A uniform bound for the exponents in Theorem 1.1. We first show that the exponents k v , v ∈ T appearing in Theorem 1.1 can be uniformly bounded, using the fact that there are only finitely many roots of unity in any non-archimedean local field:\n\nLemma 2.1. Let σ 1 and σ 2 be two semisimple elements in GL n (F ) where F is local field (finite extension of Q ℓ ). Suppose there exists a non-zero integer k such that σ k 1 and σ k 1 are conjugate in GL n (F ). Then, there exists a positive integer m depending only on n and F such that σ m 1 and σ m 2 are conjugate in GL n (F ). Remark 2.1. Since we are working in GL n , two elements are conjugate in GL n (F ) if and only if they are conjugate in GL n (F ).\n\nProof. Choose an algebraic closureF of F . Let F ′ be the extension of F inF generated by the eigenvalues of σ 1 and σ 2 inF . It is easy to see that\n[F ′ : F ] ≤ (n!) 2 .\nThe number of roots of unity contained in such a field F ′ is bounded above by some positive integer m 0 depending only on [F ′ : Q ℓ ], thus depending only on n and [F :\nQ ℓ ].\nLet {α 1 , · · · , α n } (respectively {β 1 , · · · , β n }) be the eigenvalues of σ 1 (respectively σ 2 ). Since by our hypothesis σ k 1 is conjugate of σ k 2 we have up to a permutation,\nα k i = β k i , ∀ 1 ≤ i ≤ n.\nHence α i and β i differ by a root of unity, which lies in F ′ . Thus from the above comment, for m = m 0 ! we have:\nα m i = β m i , ∀ 1 ≤ i ≤ n.\nBut since both σ 1 and σ 2 are semisimple elements in GL n (F ), σ m 1 and σ m 2 are conjugate in GL n (F ).\n\nIt follows from this lemma, upon assuming the hypothesis of Theorem 1.1, there exists a positive integer m independent of v ∈ T , and such that for all v ∈ T , ρ 1 (σ v ) m and ρ 2 (σ v ) m are conjugate in GL n (F ).\n\n\n2.2.\n\nAn application of an algebraic Chebotarev density theorem. We recall Theorem 3 of [Ra1], an algebraic interpretation of results proved in Section 6 (especially Proposition 15) of [Se], giving an algebraic formulation of the Chebotarev density theorem for the density of places satisfying an algebraic conjugacy condition:\nTheorem 2.1. [Ra1,\nTheorem 3] Let M be an algebraic group defined over a l-adic local field F of characteristic zero. Suppose\nρ : G K → M(F )\nis a continuous representation unramified outside a finite set of places of K.\n\nSuppose X is a closed subscheme of M defined over F and stable under the adjoint action of M on itself. Let C := X(F ) ∩ ρ(G K ). Let Σ u denote the set of finite places of K at which ρ is unramified. Then the set\nS := {v ∈ Σ u | ρ(σ v ) ⊂ C}.\nhas a density given by\nd(S) = |Ψ| |Φ| ,\nwhere Ψ is the set of those φ ∈ Φ such that the corresponding connected component G φ of G is contained in X.\n\nCorollary 2.1. Let ρ be a semisimple continuous ℓ-adic representation of G K to GL n (F ) unramified outside a finite set of places of K. Then there is a density one set of places of K at which ρ is unramified and the corresponding Frobenius conjugacy class is semisimple.\n\nProof. Since the representations are assumed to be semi-simple, the algebraic monodromy groups are reductive algebraic group defined over F . The corollary follows from the fact that the semisimple elements in a reductive group G contain a Zariski dense open subset of G.\n\nLet m be a natural number. Consider the following Zariski closed, invariant subsets of GL n × GL n : 2 )), j = 1, · · · , n}, where Λ j denotes the j-th exterior power representation of GL n . For semisimple elements, the condition that (g 1 , g 2 ) ∈ Y m , is equialent to saying that g m 1 and g m 2 are conjugate.\nX m := {(g 1 , g 2 ) ∈ GL n × GL n | Trace(g m 1 ) = Trace(g m 2 )} Y m := {(g 1 , g 2 ) ∈ GL n × GL n | Trace(Λ j (g m 1 )) = Trace(Λ j (g m\nGiven the hypothesis of Theorem 1.2 (resp. Theorem 1.1), we have\nρ(σ v ) ∈ X m (F ) (resp. ρ(σ v ) ∈ Y m (F )) ∀v ∈ T ′ , where ρ = ρ 1 ⊕ ρ 2 : G K → GL n (F ) × GL n (F ) is the direct sum of the representations ρ 1 and ρ 2 .\nIn what follows we present the proof for Theorem 1.2, and make remarks only as required for the proof of Theorem 1.1.\n\nLet G denote the algebraic monodromy group of ρ. Since T is of positive upper density, by Theorem 2.1 above, there exists a connected component G φ of G such that G φ is contained in X m .\n\nWe are led to consider the following problem in the context of algebraic groups: let G ⊂ GL n × GL n be a reductive algebraic group and p 1 , p 2 denote the two projections. Assume that the image p 1 (G) = G 1 is connected. Suppose that there is a connected component G φ of G contained inside X m . What can we conclude about the representations p 1 and p 2 of G? For the rest of the paper, we will be following this notation.\n\nWe first observe the equivalence of the representations when G is connected:\n\nProposition 2.2. With the above notation, suppose G is connected and G ⊂ X m . Then the representations p 1 and p 2 are equivalent.\n\nProof. Since G is connected, the m-th power map x → x m from G to G is dominant. Hence G ⊂ X 1 , and the representations p 1 and p 2 are equivalent.\n\nRemark 2.2. The surjectivity of the m-th power map fails when we consider it between connected components. In [PR] the argument continued as follows: it can be assumed that m is chosen such that the m-th power map sends G φ to G 0 , the connected component of identity in G. The image will be contained inside the subvariety X 1 . Upon the erroneous assumption that the m-th power map is surjective, one concludes that G 0 ⊂ X 1 and this implies potential equivalence of ρ 1 and ρ 2 . But this assumption is wrong, as was pointed out by J.-P. Serre.\n\n\nA unitary trick.\n\nOne of the advantages with the algebraic formulation is that it allows base change to the field of complex numbers, which makes it amenable to transcendental methods. Choosing an isomorphism of the algebraic closure of F with C, we consider the analogous problem over complex numbers.\n\nLet U be a maximal compact subgroup of G(C) which we will assume is contained inside U(n) × U(n), where U(n) ⊂ GL n (C) is the (standard) group of unitary n × n matrices. The group U is Zariski dense in G. Hence the intersection U φ := U ∩ G φ is Zariski dense in G φ . In particular, it is non-empty.\n\nThe image p 1 (U) of the projection of U to the first factor is a maximal compact subgroup of G 1 (C). By hypothesis G 1 is connected. Hence p 1 (U) is connected and equal to p 1 (U 0 ), where U 0 := U ∩ G 0 .\n\nSince G φ ⊂ X m , there exists an element of the form (I n , j) ∈ U φ ⊂ X m (C) where I n denotes the identity matrix in GL n (C). We have, n = Tr(I m n ) = Tr(j m ). Since the only unitary matrix in with trace equal to n is the identity matrix, we conclude that j m = I n . Thus, G φ = G 0 (I n , j) with j m = I n . Let J = (I n , j). The condition G φ ⊂ X m , translates to the condition (xJ) m ∈ X m for any x ∈ G 0 . Remark 2.3. In [Ra1], we considered the case m = 1. In this case, we have that (I n , I n ) ∈ G φ (C), and this implies that G φ = G 0 , and the representations p 1 and p 2 are isomorphic restricted to G 0 . Hence we conclude that ρ 1 and ρ 2 are potentially isomorphic.\n\nFor m ≥ 2, the significance of the unitary trick lies in the following crucial observation:\n(2.1) (xJ) m = xJ · · · xJ = xJxJ −1 J 2 xJ −2 · · · J m−1 xJ −(m−1) J m−1 J, x ∈ G 0 .\nLet θ(x) = JxJ −1 denote the automorphism of finite order (dividing m) of G 0 induced by the conjugation action of J. Since by the unitary trick J m = (I n , I n ), the above equation becomes,\n(2.2) (xJ) m = xθ(x)θ 2 (x) · · · θ m−1 (x), x ∈ G 0 .\nThe condition G φ ⊂ X m now translates to the following condition on the representations p 1 and p 2 of G 0 :\n(2.3) Tr(p 1 (x) m ) = Tr p 2 (xθ(x)θ 2 (x) · · · θ m−1 (x)) , x ∈ G 0 .\nThis calculation can be reversed. Given the validity of Equation (2.3), we have an equality for y = xJ ∈ G φ , x ∈ G 0 , (2.4) Tr(p 1 (y) m ) = Tr(p 2 (y) m ).\n\nWe observe that since J has finite order, it is semisimple with eigenvalues roots of unity. Hence J and the automorphism θ are defined over a finite extension F ′ of F .\n\n\n2.4.\n\nAlgebraic analogue of Theorem 1.2. In this section, we consider the subgroup of G ⊂ GL(n) × GL(n) generated by G 0 and the the connected component G φ , as considered above. By an abuse of notation, we continue to denote this subgroup by G. The group G is defined over F , and over a finite extension F ′ of F , is isomorphic to the group generated by G 0 and the element J = (1, j) ∈ G(F ′ ). The conjugation action by the element (1, j) induces the automorphism θ on G 0 defined over F ′ . We decompose G with respect to the action of θ as follows:\n\n(1) The derived subgroup G ′ of G 0 is a connected semisimple group defined over F . (2) The connected component of the center Z of G 0 is invariant under θ.\n\nThe automorphism θ leaves stable the groups G ′ and Z (considered as subgroups over F ′ ), and there is a decomposition\nG 0 = G ′ Z and G ′ ∩ Z is finite.\nWe further decompose Z with respect to the action of θ.\n\nConsider the lattice X * (Z) of characters of Z. Define endomorphisms of X * (Z), I θ = 1 − θ and N θ = 1 + θ · · · θ m−1 . Let X θ be the subgroup of X * (Z) on which θ acts trivially and Z θ be the corresponding subtorus of Z defined over F ′ . This is the maximial subtorus of Z on which θ acts trivially.\n(3) Decompose the space X * (Z) ⊗ Q = X θ ⊗ Q ⊕ Y , where Y is the kernel of N θ . Choose a lattice L 0 ⊂ Y and let L θ = m−1 i=1 θ i (L)\n. The lattice L θ is θ-invariant and is the character group of a θ-stable subtorus Z θ of Z. The invariants of θ acting on Z θ and G ′ Z θ ∩ Z θ are finite groups.\n\n(4) Let Z θ <J> be the subgroup of G generated by Z θ and the element J. The groups G ′ Z θ and Z θ <J> are normal in G, and there is a decomposition\nG = G ′ Z θ (Z θ <J>) and G ′ Z θ ∩ Z θ <J> is finite.\nWe have the following algebraic analogue of Theorem 1.2:\n\nTheorem 2.3. Let G be as above. Suppose that the following condition is satisfied for x ∈ G 0 :\n\n(2.5) Tr(p 1 (x m )) = Tr(p 2 (xθ(x) · · · θ m−1 (x))).\n\nWith respect to the decomposition G = G ′ Z θ Z θ <J> given above, the following hold:\n\n(1) θ acts trivially on G ′ Z θ , and\np 1 | G ′ Z θ ≃ p 2 | G ′ Z θ\n(2) p 1 restricted to Z θ <J> is trivial.\n\n(3) There exists a finite extension E of F ′ and decompositions defined over E, p 1 ≃ ⊕ t i=1 p 1,i and p 2 ≃ ⊕ t i=1 p 2,i , where p 1,i and p 2,i are representations of G to GL n i for some n i ≥ 1. For 1 ≤ i ≤ t, there exists an absolutely irreducible representations R i of G and representations p ′ 1,i , p ′ 2,i of Z θ <J> into GL n i /dim(R i ) (E), satisfying:\n\n(a) As representations of the group G ′ Z θ × Z θ <J>,\np 1,i = R i ⊗ p ′ 1,i and p 2,i = R i ⊗ p ′ 2,i . (b) The representation p ′ 1,i is trivial. (c) The representations p ′ 1,i , p ′ 2,i satisfy, (2.6) n i = Tr(p ′ 1,i (x m )) = Tr(p ′ 2,i (xθ(x) · · · θ m−1 (x))), x ∈ Z θ .\nEquivalently, for y = xJ belonging to the coset Z θ J, n i = Tr(p ′ 1,i (y m )) = Tr(p ′ 2,i (y m )). Remark 2.4. A similar analogous statement can be made for Theorem 1.1, where the hypothesis is modified by considering the analogue of Equation (2.5) for all exterior powers. Similarly the conclusion can be strengthened to say that the equality of Equation (2.6) holds for all exterior powers. Since Z θ is a torus, the elements in the image of Z θ <J> will be semisimple for any linear representation of Z θ <J>. From this and the equality of traces for all exterior powers, one concludes that the elements p ′ 1,i (y m ) and p ′ 2,i (y m ) are conjugate for y = xJ ∈ Z θ J, x ∈ Z θ . Example 2.1. The calculation given by Equation (2.2) can be reversed to give examples generalizing those given by Serre. Let T be a torus equipped with an automorphism θ of finite order, say d. Assume that T = T θ , i.e., the θ-invariants of T is a finite group. Form the semidirect product T ⋊ Z/dZ, where 1 ∈ Z/dZ acts via θ on T . For an element xθ, x ∈ T belonging to the connected component containing θ,\n(2.7) (xθ) d = xθ(x) · · · θ d−1 (x) = N θ (x).\nAs T is commutative, the element N θ (x) is θ-invariant. Hence there exists a natural number m, such that (xθ) m = 1. For any linear representation of T ⋊ Z/dZ of dimension d, R((xθ) m ) = 1. Thus the trivial representation of T of dimension d and the representation R are 'potentially equivalent' on the coset containing θ. (1) and (2) of Theorem 2.3. Let H denote the connected component of identity of Ker(p 1 | G ′ ). The group H is semisimple, and since p 1 = p 1 •θ, H is θ-stable. Let H θ denote the identity connected component of the fixed points of θ acting on H.\n\n\nProof of Parts\n\nRestricted to H θ , Equation (2.5) gives the identity:\nn = Tr(p 1 (x m )) = Tr(p 2 (x m )), x ∈ H θ ,\nwhere we have used the fact that p 1 restricted to H is trivial. By Proposition 2.2, the representation p 2 restricted to H θ is trivial. Since G ′ ⊂ GL n ×GL n , the representation p 2 restricted to H, and in particular to H θ is injective. Hence H θ is trivial.\n\nThe following theorem is classical (see for instance [K,Chapter 8]) or [BM]), and we give a proof in our context in the following section):\n\nTheorem 2.4. Let H be a (non-trivial) connected semisimple group over F , and let θ be a finite order automorphism of H. Then the connected component H θ of the fixed points of θ is a non-trivial reductive group.\n\nFrom this theorem and the triviality of H θ , it follows that H is trivial, i.e., the kernel of p 1 | G ′ is finite. From the equality p 1 • θ = p 1 , we get for g ∈ G ′ that θ(g) = zg for some z in kernel of p 1 , i.e., θ(g)g −1 = z. The finiteness of the kernel and the connectedness of G ′ implies that θ acts trivially on G ′ . Thus θ acts trivially on G ′ Z θ . Proposition 2.2 applied to Equation (2.5) restricted to G ′ Z θ yields Part (i) of Theorem 2.3.\n\nThe proof of Part (ii) of Theorem 2.3, follows from the fact that θ acts trivially on the first co-ordinate. Since Z θ has finite θ-invariants, being connected, it cannot have a non-trivial quotient with trivial action of θ. The element J projects trivially to the first component by our choice of J.\n\n2.6. Part (3) of Theorem 2.3. Let E be a finite extension of F ′ , over which the representations p 1 and p 2 breaks up as direct sums of absolutely irreducible representations of G. By Part (1), after a change of basis of one of the representations we can assume that\np 1 | G ′ Z θ = p 2 | G ′ Z θ Let E n = ⊕ t i=1\nV i be the decomposition into isotypical components of absolutely irreducible representations of the representation p 1 | G ′ Z θ . These components are stabilized by G as G ′ Z θ is normal in G. Since p 1 restricted to Z θ <J> is trivial, the representation p 1 factors via G/Z θ which is a quotient of G ′ Z θ . Let R i denote the isomorphism class of an absolutely irreducible representation of G (acting via p 1 ) on V i . Considered as representations of the group G ′ Z θ × Z θ <J>, the representation p 2,i can be written as, p 2,i ≃ ⊕ n j j=1 R i ⊗ η ij , for some collection of absolutely irreducible representations η ij of Z θ <J>. Define p ′ 2,i = ⊕ n j j=1 η ij . Since p 1 is trivial restricted to Z θ <J >, this gives the decomposition in Part (3.a), and also proves Part (3.b) of Theorem 2.3.\n\nTo prove Part (3.c), in terms of the decompositions, Equation (2.5) can be written as,\nt i=1 Tr(R i (g m ))Tr(p ′ 1,i (x m )) = t i=1\nTr(R i (g m ))Tr(p ′ 2,i (xθ(x) · · · θ m−1 (x))), for g ∈ G ′ Z θ and x ∈ Z θ . Since G ′ Z θ is connected, the map g → g m is surjective. The representations R i are distinct irreducible representations of G ′ Z θ . Hence by the linear independence of characters, we obtain Part (3.c) of Theorem 2.3.\n\n\n2.7.\n\nProof of Theorem 1.2. We deduce now Theorem 1.2 from Theorem 2.3. From the arguments of Sections 2.2 and 2.3, consider the group G 1 generated by G 0 and the connected component G φ (as is done in Section 2.4). The group ρ −1 (G 1 (F )) is of finite index in G K and is of the form G K 1 for some finite extension K 1 of K. We apply Theorem 2.3 to G 1 and let E be the field given in Part (3) of Theorem 2.3.\n\nThe natural map π from the product of groups\nπ : G ′ Z θ (E) × Z θ <J> (E) → G 1 (E),\nhas finite kernel A. Choose a θ-stable open subgroup M ′ (resp. M θ ) of G ′ Z θ (E) (resp. Z θ <J > (E)), such that M ′ × M θ intersects A only at the identity element. Let M θ <J> denote the open subgroup of Z θ <J> (E) generated by M θ and J. This group is Zariski dense in Z θ <J>.\n\nLet M ′ × M θ <J > maps isomorphically via π to the its image group denoted by M in G 1 (E). This is an open subgroup of G 1 (E). The intersection ρ(G K 1 ) ∩ M is of finite index in ρ(G K 1 ), and is of the form ρ(G L ) for some finite extension L of K. Thus we can consider the map ρ :\nG L → G 1 (E) to factor via G ′ Z θ (E) × Z θ <J > (E).\nComposing the maps R i , p ′ 1,i and p ′ 2,i with the map ρ defines respectively linear representations r i , ρ ′ 1,i and ρ ′ 2,i of G L .\n\nFrom the construction, the algebraic monodromy of the representations ρ ′ 2,i is Z θ < J >. The set of places T L is defined to be those places of L such that the image of the Frobenius conjugacy class by the representations ρ ′ 2,i lands inside the component Z θ J of Z θ <J>, and hence it is of positive density by Chebotarev density theorem. Theorem 1.2 follows now from Theorem 2.3.\n\n2.8. Proof of Corollary 1.1. (1) If the ranks are equal, then for each i, the representations ρ ′ 2,i are finite. This means that the representations are potentially equivalent.\n\n(2) If ρ 1 is absolutely irreducible, then t = 1 and the dimension of the representations ρ ′ 1,1 and ρ ′ 2,1 is one. Since they are also potentially equivalent, the values of ρ ′ 2,1 lie in roots of unity of F , which is a finite set. Hence the representations are potentially equivalent.\n\n2.9. Proof of Theorem 2.4. For the sake of completeness of exposition, we give a proof of Theorem 2.4 in the following context which is sufficient for our purpose (see [K,Chapter 8] for a more detailed and complete exposition): we take F = C, G ⊂ GL(n, C) and the automorphism θ is induced by conjugation by an unitary matrix. In this case, θ commutes with the Cartan involution θ c : A → tĀ−1 . Hence the Cartain involution fixes H θ , and it follows that H θ is a reductive group.\n\nTo see that the connected component H θ of the fixed points of θ is non-trivial, it suffices to work with a compact form of H. Let H denote the complexification of the Lie algebra of the compact form. We continue to denote by θ the inducted action on the Lie algebra. Decompose H = ⊕ m−1 i=0 H i as eigenspaces for the action of θ on H,\nwhere H j = {X ∈ H | θ(X) = ζ j m X}.\nHere ζ m is a fixed choice of a primitive m-th root of unity.\n\nLet I be the set of indices for which H i is non-zero, and let k ∈ I be such that ζ k m is a generator for the subgroup of the roots of unity generated by ζ j m , j ∈ I. For X ∈ H k and Y ∈ H l ,\nθ(ad(X)(Y )) = θ([X, Y ]) = [θ(X), θ(Y )] = ζ k+l m [X, Y ].\nSuppose 0 is not in I. For any n, ad(X) n (Y ) belongs to ζ nk+l m -eigenspace of θ. The assumption on k implies that for some n, ad(X) n will annihilate the space H l . Choosing n appropriately, this implies that ad(X) is nilpotent on H. But then there are no such elements in H. Hence 0 ∈ I and this implies that H θ is non-trivial.\n\n2.10. Images of connected components. The observation given by Equation (2.1) allows us to conclude that the images of connected components of semisimple algebraic groups with respect to the power maps P n will not collapse.\n\nTheorem 2.5. Let G be a semisimple algebraic group over an algebraically closed field F of characteristic zero. Let G φ denote a connected component of G. Suppose that the order of G φ is n. For a sufficiently large multiple m of n, the image of G φ by P m contains a (Zariski open) neighbourhood of identity of a non-trivial connected reductive group.\n\nProof. We show that there exists a torsion element j in G φ of order m. Write\nG 0 = r i=1 H i ,\nwhere H i are the connected normal almost simple subgroups of G 0 . Elements x of G φ induce automorphisms θ x of G 0 by conjugation, and two such automorphisms are related by an inner automorphism. Hence there exists some k, such that the automorphisms θ k x will send H i to H i for each i. Consequently, this gives an automorphism ofH i whereH i is the simply connected cover of H i , and thus defines an element of r i=1 Out(H i ). It is known that for any connected, simply connected, almost simple group H, the sequence 1 → Inn(H) → Aut(H) → Out(H) → 1, splits. Since Out(H) is finite, this gives an j ∈ G φ such that the automorphism θ j has finite order.\n\nLet θ j (x) = jxj −1 be the automorphism induced on G 0 by conjugation by j. From Equation (2.2), the image of P m will contain elements of the form xθ(x)θ 2 (x) · · · θ m−1 (x), where x ranges over the elements of G 0 . In particular, the image will contain elements of the form x m for x ∈ (G 0 ) θ , where (G 0 ) θ denotes the connected component of identity of the fixed points of θ acting on G 0 . By Theorem 2.4, this is a non-trivial connected reductive group. The image of the map x → x m from (G 0 ) θ to itself is dominant and contains a neighbourhood of identity of (G 0 ) θ , and this proves the theorem.\n\n\nProof of Theorem 1.3\n\nFor the proof of Theorem 1.3, we first argue as in Section 2.2, and conclude by Theorem 2.1 that there exists a connected component G φ of G such that G φ is contained in X m . Consider the map P m : x → x m from G to itself.\n\nClaim: The m-th power map from G φ to G 0 is dominant.\n\nGranting the validity of the claim, this implies that G 0 ⊂ X 1 . This implies that the representations p 1 and p 2 are equivalent, hence the representations ρ 1 and ρ 2 are potentially equivalent.\n\nTo prove the claim, we first observe that since G 2 is connected and G 0 projects onto G 2 , the projection to the second factor of G φ is equal to G 2 .\n\nWe now use a fact from algebraic geometry (Exercise 3.7, Hartshorne): given X, Y are quasi-projective varieties of the same dimension and a dominant morphism π : X → Y , there exists open subsets U ⊂ Y and V ⊂ π −1 (U), such that π | V : V → U is finite.\n\nApplying this to m-th power maps G 2 → G 2 and G 1 → G 1 , and making using of the fact that for any connected algebraic group the m-th power map P m is dominant, we obtain open subsets U 2 ⊂ G 2 , U 1 ⊂ G 1 and V 2 ⊂ G 2 , V 1 ⊂ G 1 such that P m : V i → U i is a finite map for i = 1, 2.\n\nWe have G φ ⊂ G 1 ×G 2 , and V 1 ×V 2 ⊂ G 1 ×G 2 . Since the projection map G φ → G 2 is dominant, the sets pr −1 2 (V 2 ) ∩ G φ and pr −1 1 (V 1 ) ∩ G φ are non-empty open subsets. Since G φ is an integral scheme, the intersection of these two open subsets which is V = G φ ∩ V 1 × V 2 is a non-empty open subset of G φ .\n\nWe have a commutative diagram,\nV − −− → V 1 × V 2 G 0 − −− → U 1 × U 2\nSince the map V 1 × V 2 → U 1 × U 2 is finite, its restriction to V is finite onto its image. Hence the image of P m contains a subvariety of dimension equal to that of G 0 . This implies that X 1 ∩ G 0 contains a subscheme of dimension equal to dimension of G 0 . As G 0 is connected and X 1 ∩ G 0 is a closed subvariety of G 0 , we get G 0 ⊂ X 1 .\n\nThis gives a proof of Theorem 1.3.\n Acknowledgement. Our sincere thanks to J.-P. Serre for pointing out an error in our earlier paper ( [PR]), and for his continued interest which motivated us to persist with this question.\nA Borel, G D Mostow, On Semi-Simple Automorphisms of Lie Algebras. 61A. Borel and G. D. Mostow, On Semi-Simple Automorphisms of Lie Algebras, Annals of Mathematics Second Series, Vol. 61, No. 3 (May, 1955), pp. 389-405.\n\nInfinite dimensional Lie algebras. V G Kac, Cambridge University PressCambridgeThird edition. xxii+400 ppV. G. Kac, Infinite dimensional Lie algebras, Third edition. Cambridge University Press, Cambridge, 1990. xxii+400 pp.\n\nLocally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves. M Kulkarni, V M Patankar, C S Rajan, J. Number Theory. 164M. Kulkarni, V. M. Patankar and C. S. Rajan, Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves, J. Number Theory, 164 (2016) 87-102.\n\nLocally potentially equivalent Galois representations. M Vijay, C S Patankar, Rajan, J. of Ramanujan Math. Soc. 271Vijay M. Patankar and C. S. Rajan, Locally potentially equivalent Galois representations, J. of Ramanujan Math. Soc., Vol. 27, No. 1, (2012), 77-90.\n\nOn strong multiplicity one for l-adic representations. C S Rajan, International Math. Res. Notices. 3C. S. Rajan, On strong multiplicity one for l-adic representations, International Math. Res. Notices, 1998, No.3, pp. 161-172.\n\nJ.-P Serre, Quelques applications du théorème de densité de Chebotarev. 54J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. HautesÉtudes Sc. Publ. Math. 54 (1981) 323-4-1.\n\nemail: vijaypatankar@gmail.com School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road. New Delhi, INDIA; BombaySchool of Physical Sciences, Jawaharlal Nehru UniversityINDIA 400005., email: rajan@math.tifr.res.inSchool of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067., email: vijaypatankar@gmail.com School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Colaba, Bombay, INDIA 400005., email: rajan@math.tifr.res.in\n"},"corpus_id":{"kind":"number","value":118217588,"string":"118,217,588"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1901.03820\",\n \"mag\": 1656227185,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1901.03820v1.pdf\"\n ],\n \"pdf_sha\": \"87a3ffe1e291e6027bfacc0c67bc7b04bf8b60a8\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"LOCALLY POTENTIALLY EQUIVALENT GALOIS REPRESENTATIONS II"},"authors":{"kind":"list like","value":["Vijay M Patankar","C S Rajan"],"string":"[\n \"Vijay M Patankar\",\n \"C S Rajan\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Suppose ρ 1 , ρ 2 are two ℓ-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of the first representation is connected, and the representations are locally potentially equivalent at a set of places of positive upper density. We classify such pairs of representations and show that up to twisting by an arbitrary representation, it is given by a pair of representations one of which is trivial and the other abelian.Consequently, we obtain that the representations are (globally) potentially equivalent, provided one of the following conditions hold: (a) the ranks of the algebraic monodromy groups are equal; (b) the first representation is absolutely irreducible and (c) the algebraic monodromy group of the second representation is also connected."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Two n-dimensional linear representations ρ 1 , ρ 2 of a group Γ are said to be potentially equivalent if they become isomorphic upon restriction to a subgroup of finite index in Γ. In this article, we are interested in proving potential equivalence of representations of the absolute Galois group of a global field, which are locally potentially equivalent at a sufficiently large number of places of the global field.\nFor a field K, denote by G K := Gal(K/K) an absolute Galois group of K, wherē K denote a seperable algebraic closure of K. Let K be a global field. For a place v of K let K v denote the completion of K at v. Choosing a place w ofK lying above v, allows us to identity G Kv with the decomposition subgroup D w of G K . As w varies this gives a conjugacy class of subgroups of G K . Given a representation ρ of G K as above, define the localization (or the local component) ρ v of ρ at v, to be the representation of G Kv obtained by restricting ρ to a decomposition subgroup. This is well defined upto isomorphism.\nLet F be a local field of characteristic zero and residue characteristic ℓ relatively prime to the characteristic of K. Suppose ρ i : G K → GL n (F ), i = 1, 2 are two continuous semisimple ℓ-adic representations of G K , unramified outside a finite set Σ ram of places of K containing the archimedean places of K. Let T be a set of place of K. Define ρ 1 and ρ 2 to be locally potentially equivalent at T , if for each v ∈ T , the restrictions ρ 1,v and ρ 2,v to G Kv are potentially equivalent.\nGiven a continuous representation ρ : G K → GL n (F ) and a place v of K where ρ is unramified, let ρ(σ v ) denote the Frobenius conjugacy class in the image group G K /Ker(ρ) ≃ ρ(G K ) ⊂ GL n (F ). By an abuse of notation, we will also continue to denote by ρ(σ v ) an element in the associated conjugacy class. The representations ρ 1 and ρ 2 are locally potentially equivalent at a prime v not in Σ ram , precisely when the eigenvalues of the Frobenius conjugacy classes ρ 1 (σ v ) and ρ 2 (σ v ) differ by roots of unity.\nFor a representation ρ : G K → GL n (F ), let G ρ be the algebraic monodromy group attached to ρ over F , i.e., the smallest algebraic subgroup G ρ of GL n defined over F such that ρ(G K ) ⊂ G ρ (F ). Denote by G i the algebraic monodromy groups associated to the representations ρ i for i = 1, 2. We recall the upper density of a set S of finite places of K is defined as:\nwhere π(x) is the number of finite places v of K with Nv ≤ x. Here Nv denotes the cardinality of the residue field k v of K v .\nOur aim in this paper is to establish a theorem of the following type: assume that the algebraic monodromy group of either ρ 1 or ρ 2 is connected and the set of places v of K where the localizations ρ 1,v and ρ 2,v are potentially equivalent has positive upper density. Then ρ 1 and ρ 2 are (globally) potentially equivalent.\nIn [Ra1], it is proved that if the characters of ρ 1 and ρ 2 agree at a set of unramified places having positive upper density and G 1 is connected, then ρ 1 and ρ 2 are potentially equivalent.\nIn ( [PR]), we had erroneously claimed that for representations as above, if G 1 is connected and the upper density of the set T of places of K at which ρ 1 and ρ 2 are locally potentially equivalent is positive, then the representations ρ 1 and ρ 2 are potentially equivalent. It was pointed out by J.-P. Serre that this claim is wrong (the proof given in ( [PR]) however goes through if we assume that the density of T is one).\nThe problem was that in the course of the proof (see Remark 2.2), the following assertion was made: let G be a reductive algebraic group, and let m be a number divisible by the exponent of the group G/G 0 . Then for any connected component\nExample 1.1. Serre gave the following counter-example: take G to be the normalizer of the diagonal torus in SL(2). The square map sends the non-identity component to scalar matrices (±Id) and is not surjective. In terms of representations, let p 1 denote the trivial two dimensional representation and p 2 the natural two dimensional representation of G. The characteristic polynomials of both these representations are equal evaluated at the fourth power of elements of the non identity connected component of G.\nExample 1.2. Based on this, an arithmetical counterexample can be given to the claim made in [PR]. Let L = Q(i) be the field of Gaussian numbers, and fix an embedding of L into C. Associated to an elliptic curve E with complex multiplication by Z[i], there exists, by the theory of complex multiplication, a continuous character ψ w : G L → L * w for any finite place w of L with the following property: at a finite place v of L with residue field coprime to the residue characteristic of L w and unramified for E, the Frobenius element π v := ψ(σ v ) generates a prime ideal in Z[i]. Further ψ(σ v )ψ τ (σ v ) is equal to p (resp. to p 2 ), when Nv = p (resp. Nv = p 2 ), where v divides the rational prime p of Q and τ denotes the non-trivial element of Gal(L/Q).\nLet χ : G Q → Q * ℓ denote the ℓ-adic cyclotomic character. We have for p coprime to ℓ, χ(σ p ) = p. Consider the two representations,\nLet T be the set of rational primes p inert in k and unramified for ρ 2 . The characteristic polynomials of ρ 1 (σ p ) and ρ 2 (σ p ) for p ∈ T , are respectively of the form (X − p) 2 and X 2 + p 2 . Hence for i = 1, 2, ρ i (σ p ) 4 = p 4 I, where I is the identity matrix. Thus these representations are potentially equivalent at T .\nThe algebraic monodromy group G 1 of ρ 1 is isomorphic to G m . For any natural number m, the characters ψ m and (ψ τ ) m are not equal up to multiplication by a root of unity, by considering the values of the character at places of degree one over Q. It follows that the algebraic monodromy group G 2 is isomorphic to the normalizer of the Weil restriction of scalars R L/Q (G m ) ⊂ GL 2 . Thus G 1 is connected, the representations ρ 1 and ρ 2 are potentially equivalent at a set of places of density 1 2 , but are not potentially equivalent yielding a contradiction to Theorem 2.1 proved in [PR]. Example 1.3. Once there is a pair (ρ 1 , ρ 2 ) of representations of G K which fails to be potentially equivalent, then any twist (ρ 1 ⊗ η, ρ 2 ⊗ η), where η is a finite dimensional linear representation of G K , satisfies the hypothesis of being locally potentially equivalent at T . Since the characters are non-vanishing in some neighbourhood of identity, the twists are not potentially equivalent either.\nFurther, if K ⊃ K 0 , and the representations ρ 1 ⊗ η and ρ 2 ⊗ η are respectively restrictions to G K of representations η 1 and η 2 of G K 0 , then the pair of representations (η 1 , η 2 ) will also provide a counterexample. These representations will be locally potentially equivalent at the unramified set of places T 0 of K 0 that lie below T , but will not be potentially equivalent.\n1.1. Theorems. Our main theorem says that the failue of ρ 1 and ρ 2 being potentially isomorphic stems from the presence of a summand generalizing the above examples, where upto a twist the first representation is trivial and the second representation is abelian.\nTheorem 1.1. Let K be a global field and S be a set of places of K containing the archimedean places and of density zero. Let F be a non-archimedean local field of characteristic zero, and ρ 1 , ρ 2 : G K → GL n (F ) be semisimple continuous representations unramified outside S, satisfying the following hypothesis: H1: The algebraic monodromy group G 1 of ρ 1 is connected. H2: There exists a set of places T of K disjoint from S having positive upper density such that the representations ρ 1 and ρ 2 are locally potentially equivalent at T , i.e., for each v ∈ T , there exists a natural number k v ≥ 1 such that the ρ 1 (σ v ) kv and ρ 2 (σ v ) kv are conjugate in GL n (F ), where ρ 1 (σ v ) and ρ 2 (σ v ) are respectively the Frobenius conjugacy classes at v of ρ 1 and ρ 2 .\nThen there exists a finite extension L of K, a finite extension E of F , a set of places T L of positive density lying above the set of places T of K, and decompositions\nwhere ρ 1,i and ρ 2,i are representations of G L to GL n i (E) for some n i ≥ 1. Further, for each i, there exists an absolutely irreducible representation r i of G L defined over E, and representations ρ ′ 1,i , ρ ′ 2,i of G L into GL n i /dim(r i ) (E) satisfying the following:\n(1) ρ ′ 1,i is potentially trivial and ρ ′ 2,i is potentially abelian.\nThe representations ρ ′ 1,i and ρ ′ 2,i are locally potentially equivalent at T L . Remark 1.1. In [KPR], we had given a proof of the above theorem in the context of non-CM elliptic curves. One of the motivations was to conclude that an elliptic curve has CM by F , if the imaginary quadratic field F occurs as the Frobenius field at a set of primes of positive upper density.\nWe now present a different version of the foregoing theorem in terms of character values evaluated at a given power of Frobenius classes: Theorem 1.2. With notation as in Theorem 1.1, assume that the following hypothesis H2 ′ is satisfied instead of H2: H2 ′ : There exists a set of places T of K disjoint from S having positive upper density, and a natural number m such that for v ∈ T ,\nThen the conclusion of Theorem 1.2 remain valid, except that Condition (3) of the conclusion should be replaced by the condition,\nRemark 1.2. Theorem 1.2 differs from Theorem 1.1, in that apart from bounding k v uniformly independent of v ∈ T , we are able to work with character values of the Frobenius classes, rather than the m-th powers of the Frobenius classes being conjugate. We don't know whether we can replace H2 ′ with the following condition:\nwhere m v are natural numbers depending on v.\nThe proof of these theorems follow along the lines of ( [Se]) and ( [Ra1]), by considering the algebraic monodromy groups of the representations involved and converting the problem to one on algebraic groups by means of an algebraic Chebotarev density theorem (see Theorem 2.1). The algebraic formulation allows us to base change to complex numbers and to employ an unitary trick used in [Ra1]. The crucial observation out here is to interpret the consequence of the unitary trick and convert the problem to one involving the first representation and twist of the second representation by an automorphism of finite order. This allows us to get at the algebraic structure of the representations satisfying the hypothesis of Theorem 1.2.\n1.2. Potential equivalence. The examples given above suggest the possibility of proving potential equivalence of ρ 1 and ρ 2 (assuming G 1 is connected and upper density of T is positive), under some additional natural hypothesis on the nature of the representations ρ 1 and ρ 2 . As corollaries of the theorems stated above, we obtain the following:\nCorollary 1.1. With hypothesis as in Theorem 1.2 (or Theorem 1.1), assume further that either of the following conditions hold:\n(1) The algebraic ranks of G 1 and G 0 2 are equal, where G 0 2 is the connected component of identity of G 2 .\n(2) ρ 1 is absolutely irreducible.\nThen ρ 1 and ρ 2 are potentially equivalent.\nWe also deduce potential isomorphy invoking the hypothesis that the algebraic monodromy groups of both the representations are connected, by a direct use of the algebraic machinery: Theorem 1.3. With hypothesis as in Theorem 1.2, assume further that G 2 is connected. Then ρ 1 and ρ 2 are potentially equivalent.\n2. Proof of Theorem 1.2 2.1. A uniform bound for the exponents in Theorem 1.1. We first show that the exponents k v , v ∈ T appearing in Theorem 1.1 can be uniformly bounded, using the fact that there are only finitely many roots of unity in any non-archimedean local field:\nLemma 2.1. Let σ 1 and σ 2 be two semisimple elements in GL n (F ) where F is local field (finite extension of Q ℓ ). Suppose there exists a non-zero integer k such that σ k 1 and σ k 1 are conjugate in GL n (F ). Then, there exists a positive integer m depending only on n and F such that σ m 1 and σ m 2 are conjugate in GL n (F ). Remark 2.1. Since we are working in GL n , two elements are conjugate in GL n (F ) if and only if they are conjugate in GL n (F ).\nProof. Choose an algebraic closureF of F . Let F ′ be the extension of F inF generated by the eigenvalues of σ 1 and σ 2 inF . It is easy to see that\nThe number of roots of unity contained in such a field F ′ is bounded above by some positive integer m 0 depending only on [F ′ : Q ℓ ], thus depending only on n and [F :\nLet {α 1 , · · · , α n } (respectively {β 1 , · · · , β n }) be the eigenvalues of σ 1 (respectively σ 2 ). Since by our hypothesis σ k 1 is conjugate of σ k 2 we have up to a permutation,\nHence α i and β i differ by a root of unity, which lies in F ′ . Thus from the above comment, for m = m 0 ! we have:\nBut since both σ 1 and σ 2 are semisimple elements in GL n (F ), σ m 1 and σ m 2 are conjugate in GL n (F ).\nIt follows from this lemma, upon assuming the hypothesis of Theorem 1.1, there exists a positive integer m independent of v ∈ T , and such that for all v ∈ T , ρ 1 (σ v ) m and ρ 2 (σ v ) m are conjugate in GL n (F )."},{"title":"2.2.","content":"An application of an algebraic Chebotarev density theorem. We recall Theorem 3 of [Ra1], an algebraic interpretation of results proved in Section 6 (especially Proposition 15) of [Se], giving an algebraic formulation of the Chebotarev density theorem for the density of places satisfying an algebraic conjugacy condition:\nTheorem 3] Let M be an algebraic group defined over a l-adic local field F of characteristic zero. Suppose\nis a continuous representation unramified outside a finite set of places of K.\nSuppose X is a closed subscheme of M defined over F and stable under the adjoint action of M on itself. Let C := X(F ) ∩ ρ(G K ). Let Σ u denote the set of finite places of K at which ρ is unramified. Then the set\nhas a density given by\nwhere Ψ is the set of those φ ∈ Φ such that the corresponding connected component G φ of G is contained in X.\nCorollary 2.1. Let ρ be a semisimple continuous ℓ-adic representation of G K to GL n (F ) unramified outside a finite set of places of K. Then there is a density one set of places of K at which ρ is unramified and the corresponding Frobenius conjugacy class is semisimple.\nProof. Since the representations are assumed to be semi-simple, the algebraic monodromy groups are reductive algebraic group defined over F . The corollary follows from the fact that the semisimple elements in a reductive group G contain a Zariski dense open subset of G.\nLet m be a natural number. Consider the following Zariski closed, invariant subsets of GL n × GL n : 2 )), j = 1, · · · , n}, where Λ j denotes the j-th exterior power representation of GL n . For semisimple elements, the condition that (g 1 , g 2 ) ∈ Y m , is equialent to saying that g m 1 and g m 2 are conjugate.\nGiven the hypothesis of Theorem 1.2 (resp. Theorem 1.1), we have\nIn what follows we present the proof for Theorem 1.2, and make remarks only as required for the proof of Theorem 1.1.\nLet G denote the algebraic monodromy group of ρ. Since T is of positive upper density, by Theorem 2.1 above, there exists a connected component G φ of G such that G φ is contained in X m .\nWe are led to consider the following problem in the context of algebraic groups: let G ⊂ GL n × GL n be a reductive algebraic group and p 1 , p 2 denote the two projections. Assume that the image p 1 (G) = G 1 is connected. Suppose that there is a connected component G φ of G contained inside X m . What can we conclude about the representations p 1 and p 2 of G? For the rest of the paper, we will be following this notation.\nWe first observe the equivalence of the representations when G is connected:\nProposition 2.2. With the above notation, suppose G is connected and G ⊂ X m . Then the representations p 1 and p 2 are equivalent.\nProof. Since G is connected, the m-th power map x → x m from G to G is dominant. Hence G ⊂ X 1 , and the representations p 1 and p 2 are equivalent.\nRemark 2.2. The surjectivity of the m-th power map fails when we consider it between connected components. In [PR] the argument continued as follows: it can be assumed that m is chosen such that the m-th power map sends G φ to G 0 , the connected component of identity in G. The image will be contained inside the subvariety X 1 . Upon the erroneous assumption that the m-th power map is surjective, one concludes that G 0 ⊂ X 1 and this implies potential equivalence of ρ 1 and ρ 2 . But this assumption is wrong, as was pointed out by J.-P. Serre."},{"title":"A unitary trick.","content":"One of the advantages with the algebraic formulation is that it allows base change to the field of complex numbers, which makes it amenable to transcendental methods. Choosing an isomorphism of the algebraic closure of F with C, we consider the analogous problem over complex numbers.\nLet U be a maximal compact subgroup of G(C) which we will assume is contained inside U(n) × U(n), where U(n) ⊂ GL n (C) is the (standard) group of unitary n × n matrices. The group U is Zariski dense in G. Hence the intersection U φ := U ∩ G φ is Zariski dense in G φ . In particular, it is non-empty.\nThe image p 1 (U) of the projection of U to the first factor is a maximal compact subgroup of G 1 (C). By hypothesis G 1 is connected. Hence p 1 (U) is connected and equal to p 1 (U 0 ), where U 0 := U ∩ G 0 .\nSince G φ ⊂ X m , there exists an element of the form (I n , j) ∈ U φ ⊂ X m (C) where I n denotes the identity matrix in GL n (C). We have, n = Tr(I m n ) = Tr(j m ). Since the only unitary matrix in with trace equal to n is the identity matrix, we conclude that j m = I n . Thus, G φ = G 0 (I n , j) with j m = I n . Let J = (I n , j). The condition G φ ⊂ X m , translates to the condition (xJ) m ∈ X m for any x ∈ G 0 . Remark 2.3. In [Ra1], we considered the case m = 1. In this case, we have that (I n , I n ) ∈ G φ (C), and this implies that G φ = G 0 , and the representations p 1 and p 2 are isomorphic restricted to G 0 . Hence we conclude that ρ 1 and ρ 2 are potentially isomorphic.\nFor m ≥ 2, the significance of the unitary trick lies in the following crucial observation:\nLet θ(x) = JxJ −1 denote the automorphism of finite order (dividing m) of G 0 induced by the conjugation action of J. Since by the unitary trick J m = (I n , I n ), the above equation becomes,\nThe condition G φ ⊂ X m now translates to the following condition on the representations p 1 and p 2 of G 0 :\nThis calculation can be reversed. Given the validity of Equation (2.3), we have an equality for y = xJ ∈ G φ , x ∈ G 0 , (2.4) Tr(p 1 (y) m ) = Tr(p 2 (y) m ).\nWe observe that since J has finite order, it is semisimple with eigenvalues roots of unity. Hence J and the automorphism θ are defined over a finite extension F ′ of F ."},{"title":"2.4.","content":"Algebraic analogue of Theorem 1.2. In this section, we consider the subgroup of G ⊂ GL(n) × GL(n) generated by G 0 and the the connected component G φ , as considered above. By an abuse of notation, we continue to denote this subgroup by G. The group G is defined over F , and over a finite extension F ′ of F , is isomorphic to the group generated by G 0 and the element J = (1, j) ∈ G(F ′ ). The conjugation action by the element (1, j) induces the automorphism θ on G 0 defined over F ′ . We decompose G with respect to the action of θ as follows:\n(1) The derived subgroup G ′ of G 0 is a connected semisimple group defined over F . (2) The connected component of the center Z of G 0 is invariant under θ.\nThe automorphism θ leaves stable the groups G ′ and Z (considered as subgroups over F ′ ), and there is a decomposition\nWe further decompose Z with respect to the action of θ.\nConsider the lattice X * (Z) of characters of Z. Define endomorphisms of X * (Z), I θ = 1 − θ and N θ = 1 + θ · · · θ m−1 . Let X θ be the subgroup of X * (Z) on which θ acts trivially and Z θ be the corresponding subtorus of Z defined over F ′ . This is the maximial subtorus of Z on which θ acts trivially.\n. The lattice L θ is θ-invariant and is the character group of a θ-stable subtorus Z θ of Z. The invariants of θ acting on Z θ and G ′ Z θ ∩ Z θ are finite groups.\n(4) Let Z θ <J> be the subgroup of G generated by Z θ and the element J. The groups G ′ Z θ and Z θ <J> are normal in G, and there is a decomposition\nWe have the following algebraic analogue of Theorem 1.2:\nTheorem 2.3. Let G be as above. Suppose that the following condition is satisfied for x ∈ G 0 :\n(2.5) Tr(p 1 (x m )) = Tr(p 2 (xθ(x) · · · θ m−1 (x))).\nWith respect to the decomposition G = G ′ Z θ Z θ <J> given above, the following hold:\n(1) θ acts trivially on G ′ Z θ , and\n(2) p 1 restricted to Z θ <J> is trivial.\n(3) There exists a finite extension E of F ′ and decompositions defined over E, p 1 ≃ ⊕ t i=1 p 1,i and p 2 ≃ ⊕ t i=1 p 2,i , where p 1,i and p 2,i are representations of G to GL n i for some n i ≥ 1. For 1 ≤ i ≤ t, there exists an absolutely irreducible representations R i of G and representations p ′ 1,i , p ′ 2,i of Z θ <J> into GL n i /dim(R i ) (E), satisfying:\n(a) As representations of the group G ′ Z θ × Z θ <J>,\nEquivalently, for y = xJ belonging to the coset Z θ J, n i = Tr(p ′ 1,i (y m )) = Tr(p ′ 2,i (y m )). Remark 2.4. A similar analogous statement can be made for Theorem 1.1, where the hypothesis is modified by considering the analogue of Equation (2.5) for all exterior powers. Similarly the conclusion can be strengthened to say that the equality of Equation (2.6) holds for all exterior powers. Since Z θ is a torus, the elements in the image of Z θ <J> will be semisimple for any linear representation of Z θ <J>. From this and the equality of traces for all exterior powers, one concludes that the elements p ′ 1,i (y m ) and p ′ 2,i (y m ) are conjugate for y = xJ ∈ Z θ J, x ∈ Z θ . Example 2.1. The calculation given by Equation (2.2) can be reversed to give examples generalizing those given by Serre. Let T be a torus equipped with an automorphism θ of finite order, say d. Assume that T = T θ , i.e., the θ-invariants of T is a finite group. Form the semidirect product T ⋊ Z/dZ, where 1 ∈ Z/dZ acts via θ on T . For an element xθ, x ∈ T belonging to the connected component containing θ,\nAs T is commutative, the element N θ (x) is θ-invariant. Hence there exists a natural number m, such that (xθ) m = 1. For any linear representation of T ⋊ Z/dZ of dimension d, R((xθ) m ) = 1. Thus the trivial representation of T of dimension d and the representation R are 'potentially equivalent' on the coset containing θ. (1) and (2) of Theorem 2.3. Let H denote the connected component of identity of Ker(p 1 | G ′ ). The group H is semisimple, and since p 1 = p 1 •θ, H is θ-stable. Let H θ denote the identity connected component of the fixed points of θ acting on H."},{"title":"Proof of Parts","content":"Restricted to H θ , Equation (2.5) gives the identity:\nwhere we have used the fact that p 1 restricted to H is trivial. By Proposition 2.2, the representation p 2 restricted to H θ is trivial. Since G ′ ⊂ GL n ×GL n , the representation p 2 restricted to H, and in particular to H θ is injective. Hence H θ is trivial.\nThe following theorem is classical (see for instance [K,Chapter 8]) or [BM]), and we give a proof in our context in the following section):\nTheorem 2.4. Let H be a (non-trivial) connected semisimple group over F , and let θ be a finite order automorphism of H. Then the connected component H θ of the fixed points of θ is a non-trivial reductive group.\nFrom this theorem and the triviality of H θ , it follows that H is trivial, i.e., the kernel of p 1 | G ′ is finite. From the equality p 1 • θ = p 1 , we get for g ∈ G ′ that θ(g) = zg for some z in kernel of p 1 , i.e., θ(g)g −1 = z. The finiteness of the kernel and the connectedness of G ′ implies that θ acts trivially on G ′ . Thus θ acts trivially on G ′ Z θ . Proposition 2.2 applied to Equation (2.5) restricted to G ′ Z θ yields Part (i) of Theorem 2.3.\nThe proof of Part (ii) of Theorem 2.3, follows from the fact that θ acts trivially on the first co-ordinate. Since Z θ has finite θ-invariants, being connected, it cannot have a non-trivial quotient with trivial action of θ. The element J projects trivially to the first component by our choice of J.\n2.6. Part (3) of Theorem 2.3. Let E be a finite extension of F ′ , over which the representations p 1 and p 2 breaks up as direct sums of absolutely irreducible representations of G. By Part (1), after a change of basis of one of the representations we can assume that\nV i be the decomposition into isotypical components of absolutely irreducible representations of the representation p 1 | G ′ Z θ . These components are stabilized by G as G ′ Z θ is normal in G. Since p 1 restricted to Z θ <J> is trivial, the representation p 1 factors via G/Z θ which is a quotient of G ′ Z θ . Let R i denote the isomorphism class of an absolutely irreducible representation of G (acting via p 1 ) on V i . Considered as representations of the group G ′ Z θ × Z θ <J>, the representation p 2,i can be written as, p 2,i ≃ ⊕ n j j=1 R i ⊗ η ij , for some collection of absolutely irreducible representations η ij of Z θ <J>. Define p ′ 2,i = ⊕ n j j=1 η ij . Since p 1 is trivial restricted to Z θ <J >, this gives the decomposition in Part (3.a), and also proves Part (3.b) of Theorem 2.3.\nTo prove Part (3.c), in terms of the decompositions, Equation (2.5) can be written as,\nTr(R i (g m ))Tr(p ′ 2,i (xθ(x) · · · θ m−1 (x))), for g ∈ G ′ Z θ and x ∈ Z θ . Since G ′ Z θ is connected, the map g → g m is surjective. The representations R i are distinct irreducible representations of G ′ Z θ . Hence by the linear independence of characters, we obtain Part (3.c) of Theorem 2.3."},{"title":"2.7.","content":"Proof of Theorem 1.2. We deduce now Theorem 1.2 from Theorem 2.3. From the arguments of Sections 2.2 and 2.3, consider the group G 1 generated by G 0 and the connected component G φ (as is done in Section 2.4). The group ρ −1 (G 1 (F )) is of finite index in G K and is of the form G K 1 for some finite extension K 1 of K. We apply Theorem 2.3 to G 1 and let E be the field given in Part (3) of Theorem 2.3.\nThe natural map π from the product of groups\nhas finite kernel A. Choose a θ-stable open subgroup M ′ (resp. M θ ) of G ′ Z θ (E) (resp. Z θ <J > (E)), such that M ′ × M θ intersects A only at the identity element. Let M θ <J> denote the open subgroup of Z θ <J> (E) generated by M θ and J. This group is Zariski dense in Z θ <J>.\nLet M ′ × M θ <J > maps isomorphically via π to the its image group denoted by M in G 1 (E). This is an open subgroup of G 1 (E). The intersection ρ(G K 1 ) ∩ M is of finite index in ρ(G K 1 ), and is of the form ρ(G L ) for some finite extension L of K. Thus we can consider the map ρ :\nComposing the maps R i , p ′ 1,i and p ′ 2,i with the map ρ defines respectively linear representations r i , ρ ′ 1,i and ρ ′ 2,i of G L .\nFrom the construction, the algebraic monodromy of the representations ρ ′ 2,i is Z θ < J >. The set of places T L is defined to be those places of L such that the image of the Frobenius conjugacy class by the representations ρ ′ 2,i lands inside the component Z θ J of Z θ <J>, and hence it is of positive density by Chebotarev density theorem. Theorem 1.2 follows now from Theorem 2.3.\n2.8. Proof of Corollary 1.1. (1) If the ranks are equal, then for each i, the representations ρ ′ 2,i are finite. This means that the representations are potentially equivalent.\n(2) If ρ 1 is absolutely irreducible, then t = 1 and the dimension of the representations ρ ′ 1,1 and ρ ′ 2,1 is one. Since they are also potentially equivalent, the values of ρ ′ 2,1 lie in roots of unity of F , which is a finite set. Hence the representations are potentially equivalent.\n2.9. Proof of Theorem 2.4. For the sake of completeness of exposition, we give a proof of Theorem 2.4 in the following context which is sufficient for our purpose (see [K,Chapter 8] for a more detailed and complete exposition): we take F = C, G ⊂ GL(n, C) and the automorphism θ is induced by conjugation by an unitary matrix. In this case, θ commutes with the Cartan involution θ c : A → tĀ−1 . Hence the Cartain involution fixes H θ , and it follows that H θ is a reductive group.\nTo see that the connected component H θ of the fixed points of θ is non-trivial, it suffices to work with a compact form of H. Let H denote the complexification of the Lie algebra of the compact form. We continue to denote by θ the inducted action on the Lie algebra. Decompose H = ⊕ m−1 i=0 H i as eigenspaces for the action of θ on H,\nHere ζ m is a fixed choice of a primitive m-th root of unity.\nLet I be the set of indices for which H i is non-zero, and let k ∈ I be such that ζ k m is a generator for the subgroup of the roots of unity generated by ζ j m , j ∈ I. For X ∈ H k and Y ∈ H l ,\nSuppose 0 is not in I. For any n, ad(X) n (Y ) belongs to ζ nk+l m -eigenspace of θ. The assumption on k implies that for some n, ad(X) n will annihilate the space H l . Choosing n appropriately, this implies that ad(X) is nilpotent on H. But then there are no such elements in H. Hence 0 ∈ I and this implies that H θ is non-trivial.\n2.10. Images of connected components. The observation given by Equation (2.1) allows us to conclude that the images of connected components of semisimple algebraic groups with respect to the power maps P n will not collapse.\nTheorem 2.5. Let G be a semisimple algebraic group over an algebraically closed field F of characteristic zero. Let G φ denote a connected component of G. Suppose that the order of G φ is n. For a sufficiently large multiple m of n, the image of G φ by P m contains a (Zariski open) neighbourhood of identity of a non-trivial connected reductive group.\nProof. We show that there exists a torsion element j in G φ of order m. Write\nwhere H i are the connected normal almost simple subgroups of G 0 . Elements x of G φ induce automorphisms θ x of G 0 by conjugation, and two such automorphisms are related by an inner automorphism. Hence there exists some k, such that the automorphisms θ k x will send H i to H i for each i. Consequently, this gives an automorphism ofH i whereH i is the simply connected cover of H i , and thus defines an element of r i=1 Out(H i ). It is known that for any connected, simply connected, almost simple group H, the sequence 1 → Inn(H) → Aut(H) → Out(H) → 1, splits. Since Out(H) is finite, this gives an j ∈ G φ such that the automorphism θ j has finite order.\nLet θ j (x) = jxj −1 be the automorphism induced on G 0 by conjugation by j. From Equation (2.2), the image of P m will contain elements of the form xθ(x)θ 2 (x) · · · θ m−1 (x), where x ranges over the elements of G 0 . In particular, the image will contain elements of the form x m for x ∈ (G 0 ) θ , where (G 0 ) θ denotes the connected component of identity of the fixed points of θ acting on G 0 . By Theorem 2.4, this is a non-trivial connected reductive group. The image of the map x → x m from (G 0 ) θ to itself is dominant and contains a neighbourhood of identity of (G 0 ) θ , and this proves the theorem."},{"title":"Proof of Theorem 1.3","content":"For the proof of Theorem 1.3, we first argue as in Section 2.2, and conclude by Theorem 2.1 that there exists a connected component G φ of G such that G φ is contained in X m . Consider the map P m : x → x m from G to itself.\nClaim: The m-th power map from G φ to G 0 is dominant.\nGranting the validity of the claim, this implies that G 0 ⊂ X 1 . This implies that the representations p 1 and p 2 are equivalent, hence the representations ρ 1 and ρ 2 are potentially equivalent.\nTo prove the claim, we first observe that since G 2 is connected and G 0 projects onto G 2 , the projection to the second factor of G φ is equal to G 2 .\nWe now use a fact from algebraic geometry (Exercise 3.7, Hartshorne): given X, Y are quasi-projective varieties of the same dimension and a dominant morphism π : X → Y , there exists open subsets U ⊂ Y and V ⊂ π −1 (U), such that π | V : V → U is finite.\nApplying this to m-th power maps G 2 → G 2 and G 1 → G 1 , and making using of the fact that for any connected algebraic group the m-th power map P m is dominant, we obtain open subsets U 2 ⊂ G 2 , U 1 ⊂ G 1 and V 2 ⊂ G 2 , V 1 ⊂ G 1 such that P m : V i → U i is a finite map for i = 1, 2.\nWe have G φ ⊂ G 1 ×G 2 , and V 1 ×V 2 ⊂ G 1 ×G 2 . Since the projection map G φ → G 2 is dominant, the sets pr −1 2 (V 2 ) ∩ G φ and pr −1 1 (V 1 ) ∩ G φ are non-empty open subsets. Since G φ is an integral scheme, the intersection of these two open subsets which is V = G φ ∩ V 1 × V 2 is a non-empty open subset of G φ .\nWe have a commutative diagram,\nSince the map V 1 × V 2 → U 1 × U 2 is finite, its restriction to V is finite onto its image. Hence the image of P m contains a subvariety of dimension equal to that of G 0 . This implies that X 1 ∩ G 0 contains a subscheme of dimension equal to dimension of G 0 . As G 0 is connected and X 1 ∩ G 0 is a closed subvariety of G 0 , we get G 0 ⊂ X 1 .\nThis gives a proof of Theorem 1.3."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Two n-dimensional linear representations ρ 1 , ρ 2 of a group Γ are said to be potentially equivalent if they become isomorphic upon restriction to a subgroup of finite index in Γ. In this article, we are interested in proving potential equivalence of representations of the absolute Galois group of a global field, which are locally potentially equivalent at a sufficiently large number of places of the global field.\\nFor a field K, denote by G K := Gal(K/K) an absolute Galois group of K, wherē K denote a seperable algebraic closure of K. Let K be a global field. For a place v of K let K v denote the completion of K at v. Choosing a place w ofK lying above v, allows us to identity G Kv with the decomposition subgroup D w of G K . As w varies this gives a conjugacy class of subgroups of G K . Given a representation ρ of G K as above, define the localization (or the local component) ρ v of ρ at v, to be the representation of G Kv obtained by restricting ρ to a decomposition subgroup. This is well defined upto isomorphism.\\nLet F be a local field of characteristic zero and residue characteristic ℓ relatively prime to the characteristic of K. Suppose ρ i : G K → GL n (F ), i = 1, 2 are two continuous semisimple ℓ-adic representations of G K , unramified outside a finite set Σ ram of places of K containing the archimedean places of K. Let T be a set of place of K. Define ρ 1 and ρ 2 to be locally potentially equivalent at T , if for each v ∈ T , the restrictions ρ 1,v and ρ 2,v to G Kv are potentially equivalent.\\nGiven a continuous representation ρ : G K → GL n (F ) and a place v of K where ρ is unramified, let ρ(σ v ) denote the Frobenius conjugacy class in the image group G K /Ker(ρ) ≃ ρ(G K ) ⊂ GL n (F ). By an abuse of notation, we will also continue to denote by ρ(σ v ) an element in the associated conjugacy class. The representations ρ 1 and ρ 2 are locally potentially equivalent at a prime v not in Σ ram , precisely when the eigenvalues of the Frobenius conjugacy classes ρ 1 (σ v ) and ρ 2 (σ v ) differ by roots of unity.\\nFor a representation ρ : G K → GL n (F ), let G ρ be the algebraic monodromy group attached to ρ over F , i.e., the smallest algebraic subgroup G ρ of GL n defined over F such that ρ(G K ) ⊂ G ρ (F ). Denote by G i the algebraic monodromy groups associated to the representations ρ i for i = 1, 2. We recall the upper density of a set S of finite places of K is defined as:\\nwhere π(x) is the number of finite places v of K with Nv ≤ x. Here Nv denotes the cardinality of the residue field k v of K v .\\nOur aim in this paper is to establish a theorem of the following type: assume that the algebraic monodromy group of either ρ 1 or ρ 2 is connected and the set of places v of K where the localizations ρ 1,v and ρ 2,v are potentially equivalent has positive upper density. Then ρ 1 and ρ 2 are (globally) potentially equivalent.\\nIn [Ra1], it is proved that if the characters of ρ 1 and ρ 2 agree at a set of unramified places having positive upper density and G 1 is connected, then ρ 1 and ρ 2 are potentially equivalent.\\nIn ( [PR]), we had erroneously claimed that for representations as above, if G 1 is connected and the upper density of the set T of places of K at which ρ 1 and ρ 2 are locally potentially equivalent is positive, then the representations ρ 1 and ρ 2 are potentially equivalent. It was pointed out by J.-P. Serre that this claim is wrong (the proof given in ( [PR]) however goes through if we assume that the density of T is one).\\nThe problem was that in the course of the proof (see Remark 2.2), the following assertion was made: let G be a reductive algebraic group, and let m be a number divisible by the exponent of the group G/G 0 . Then for any connected component\\nExample 1.1. Serre gave the following counter-example: take G to be the normalizer of the diagonal torus in SL(2). The square map sends the non-identity component to scalar matrices (±Id) and is not surjective. In terms of representations, let p 1 denote the trivial two dimensional representation and p 2 the natural two dimensional representation of G. The characteristic polynomials of both these representations are equal evaluated at the fourth power of elements of the non identity connected component of G.\\nExample 1.2. Based on this, an arithmetical counterexample can be given to the claim made in [PR]. Let L = Q(i) be the field of Gaussian numbers, and fix an embedding of L into C. Associated to an elliptic curve E with complex multiplication by Z[i], there exists, by the theory of complex multiplication, a continuous character ψ w : G L → L * w for any finite place w of L with the following property: at a finite place v of L with residue field coprime to the residue characteristic of L w and unramified for E, the Frobenius element π v := ψ(σ v ) generates a prime ideal in Z[i]. Further ψ(σ v )ψ τ (σ v ) is equal to p (resp. to p 2 ), when Nv = p (resp. Nv = p 2 ), where v divides the rational prime p of Q and τ denotes the non-trivial element of Gal(L/Q).\\nLet χ : G Q → Q * ℓ denote the ℓ-adic cyclotomic character. We have for p coprime to ℓ, χ(σ p ) = p. Consider the two representations,\\nLet T be the set of rational primes p inert in k and unramified for ρ 2 . The characteristic polynomials of ρ 1 (σ p ) and ρ 2 (σ p ) for p ∈ T , are respectively of the form (X − p) 2 and X 2 + p 2 . Hence for i = 1, 2, ρ i (σ p ) 4 = p 4 I, where I is the identity matrix. Thus these representations are potentially equivalent at T .\\nThe algebraic monodromy group G 1 of ρ 1 is isomorphic to G m . For any natural number m, the characters ψ m and (ψ τ ) m are not equal up to multiplication by a root of unity, by considering the values of the character at places of degree one over Q. It follows that the algebraic monodromy group G 2 is isomorphic to the normalizer of the Weil restriction of scalars R L/Q (G m ) ⊂ GL 2 . Thus G 1 is connected, the representations ρ 1 and ρ 2 are potentially equivalent at a set of places of density 1 2 , but are not potentially equivalent yielding a contradiction to Theorem 2.1 proved in [PR]. Example 1.3. Once there is a pair (ρ 1 , ρ 2 ) of representations of G K which fails to be potentially equivalent, then any twist (ρ 1 ⊗ η, ρ 2 ⊗ η), where η is a finite dimensional linear representation of G K , satisfies the hypothesis of being locally potentially equivalent at T . Since the characters are non-vanishing in some neighbourhood of identity, the twists are not potentially equivalent either.\\nFurther, if K ⊃ K 0 , and the representations ρ 1 ⊗ η and ρ 2 ⊗ η are respectively restrictions to G K of representations η 1 and η 2 of G K 0 , then the pair of representations (η 1 , η 2 ) will also provide a counterexample. These representations will be locally potentially equivalent at the unramified set of places T 0 of K 0 that lie below T , but will not be potentially equivalent.\\n1.1. Theorems. Our main theorem says that the failue of ρ 1 and ρ 2 being potentially isomorphic stems from the presence of a summand generalizing the above examples, where upto a twist the first representation is trivial and the second representation is abelian.\\nTheorem 1.1. Let K be a global field and S be a set of places of K containing the archimedean places and of density zero. Let F be a non-archimedean local field of characteristic zero, and ρ 1 , ρ 2 : G K → GL n (F ) be semisimple continuous representations unramified outside S, satisfying the following hypothesis: H1: The algebraic monodromy group G 1 of ρ 1 is connected. H2: There exists a set of places T of K disjoint from S having positive upper density such that the representations ρ 1 and ρ 2 are locally potentially equivalent at T , i.e., for each v ∈ T , there exists a natural number k v ≥ 1 such that the ρ 1 (σ v ) kv and ρ 2 (σ v ) kv are conjugate in GL n (F ), where ρ 1 (σ v ) and ρ 2 (σ v ) are respectively the Frobenius conjugacy classes at v of ρ 1 and ρ 2 .\\nThen there exists a finite extension L of K, a finite extension E of F , a set of places T L of positive density lying above the set of places T of K, and decompositions\\nwhere ρ 1,i and ρ 2,i are representations of G L to GL n i (E) for some n i ≥ 1. Further, for each i, there exists an absolutely irreducible representation r i of G L defined over E, and representations ρ ′ 1,i , ρ ′ 2,i of G L into GL n i /dim(r i ) (E) satisfying the following:\\n(1) ρ ′ 1,i is potentially trivial and ρ ′ 2,i is potentially abelian.\\nThe representations ρ ′ 1,i and ρ ′ 2,i are locally potentially equivalent at T L . Remark 1.1. In [KPR], we had given a proof of the above theorem in the context of non-CM elliptic curves. One of the motivations was to conclude that an elliptic curve has CM by F , if the imaginary quadratic field F occurs as the Frobenius field at a set of primes of positive upper density.\\nWe now present a different version of the foregoing theorem in terms of character values evaluated at a given power of Frobenius classes: Theorem 1.2. With notation as in Theorem 1.1, assume that the following hypothesis H2 ′ is satisfied instead of H2: H2 ′ : There exists a set of places T of K disjoint from S having positive upper density, and a natural number m such that for v ∈ T ,\\nThen the conclusion of Theorem 1.2 remain valid, except that Condition (3) of the conclusion should be replaced by the condition,\\nRemark 1.2. Theorem 1.2 differs from Theorem 1.1, in that apart from bounding k v uniformly independent of v ∈ T , we are able to work with character values of the Frobenius classes, rather than the m-th powers of the Frobenius classes being conjugate. We don't know whether we can replace H2 ′ with the following condition:\\nwhere m v are natural numbers depending on v.\\nThe proof of these theorems follow along the lines of ( [Se]) and ( [Ra1]), by considering the algebraic monodromy groups of the representations involved and converting the problem to one on algebraic groups by means of an algebraic Chebotarev density theorem (see Theorem 2.1). The algebraic formulation allows us to base change to complex numbers and to employ an unitary trick used in [Ra1]. The crucial observation out here is to interpret the consequence of the unitary trick and convert the problem to one involving the first representation and twist of the second representation by an automorphism of finite order. This allows us to get at the algebraic structure of the representations satisfying the hypothesis of Theorem 1.2.\\n1.2. Potential equivalence. The examples given above suggest the possibility of proving potential equivalence of ρ 1 and ρ 2 (assuming G 1 is connected and upper density of T is positive), under some additional natural hypothesis on the nature of the representations ρ 1 and ρ 2 . As corollaries of the theorems stated above, we obtain the following:\\nCorollary 1.1. With hypothesis as in Theorem 1.2 (or Theorem 1.1), assume further that either of the following conditions hold:\\n(1) The algebraic ranks of G 1 and G 0 2 are equal, where G 0 2 is the connected component of identity of G 2 .\\n(2) ρ 1 is absolutely irreducible.\\nThen ρ 1 and ρ 2 are potentially equivalent.\\nWe also deduce potential isomorphy invoking the hypothesis that the algebraic monodromy groups of both the representations are connected, by a direct use of the algebraic machinery: Theorem 1.3. With hypothesis as in Theorem 1.2, assume further that G 2 is connected. Then ρ 1 and ρ 2 are potentially equivalent.\\n2. Proof of Theorem 1.2 2.1. A uniform bound for the exponents in Theorem 1.1. We first show that the exponents k v , v ∈ T appearing in Theorem 1.1 can be uniformly bounded, using the fact that there are only finitely many roots of unity in any non-archimedean local field:\\nLemma 2.1. Let σ 1 and σ 2 be two semisimple elements in GL n (F ) where F is local field (finite extension of Q ℓ ). Suppose there exists a non-zero integer k such that σ k 1 and σ k 1 are conjugate in GL n (F ). Then, there exists a positive integer m depending only on n and F such that σ m 1 and σ m 2 are conjugate in GL n (F ). Remark 2.1. Since we are working in GL n , two elements are conjugate in GL n (F ) if and only if they are conjugate in GL n (F ).\\nProof. Choose an algebraic closureF of F . Let F ′ be the extension of F inF generated by the eigenvalues of σ 1 and σ 2 inF . It is easy to see that\\nThe number of roots of unity contained in such a field F ′ is bounded above by some positive integer m 0 depending only on [F ′ : Q ℓ ], thus depending only on n and [F :\\nLet {α 1 , · · · , α n } (respectively {β 1 , · · · , β n }) be the eigenvalues of σ 1 (respectively σ 2 ). Since by our hypothesis σ k 1 is conjugate of σ k 2 we have up to a permutation,\\nHence α i and β i differ by a root of unity, which lies in F ′ . Thus from the above comment, for m = m 0 ! we have:\\nBut since both σ 1 and σ 2 are semisimple elements in GL n (F ), σ m 1 and σ m 2 are conjugate in GL n (F ).\\nIt follows from this lemma, upon assuming the hypothesis of Theorem 1.1, there exists a positive integer m independent of v ∈ T , and such that for all v ∈ T , ρ 1 (σ v ) m and ρ 2 (σ v ) m are conjugate in GL n (F ).\"\n },\n {\n \"title\": \"2.2.\",\n \"content\": \"An application of an algebraic Chebotarev density theorem. We recall Theorem 3 of [Ra1], an algebraic interpretation of results proved in Section 6 (especially Proposition 15) of [Se], giving an algebraic formulation of the Chebotarev density theorem for the density of places satisfying an algebraic conjugacy condition:\\nTheorem 3] Let M be an algebraic group defined over a l-adic local field F of characteristic zero. Suppose\\nis a continuous representation unramified outside a finite set of places of K.\\nSuppose X is a closed subscheme of M defined over F and stable under the adjoint action of M on itself. Let C := X(F ) ∩ ρ(G K ). Let Σ u denote the set of finite places of K at which ρ is unramified. Then the set\\nhas a density given by\\nwhere Ψ is the set of those φ ∈ Φ such that the corresponding connected component G φ of G is contained in X.\\nCorollary 2.1. Let ρ be a semisimple continuous ℓ-adic representation of G K to GL n (F ) unramified outside a finite set of places of K. Then there is a density one set of places of K at which ρ is unramified and the corresponding Frobenius conjugacy class is semisimple.\\nProof. Since the representations are assumed to be semi-simple, the algebraic monodromy groups are reductive algebraic group defined over F . The corollary follows from the fact that the semisimple elements in a reductive group G contain a Zariski dense open subset of G.\\nLet m be a natural number. Consider the following Zariski closed, invariant subsets of GL n × GL n : 2 )), j = 1, · · · , n}, where Λ j denotes the j-th exterior power representation of GL n . For semisimple elements, the condition that (g 1 , g 2 ) ∈ Y m , is equialent to saying that g m 1 and g m 2 are conjugate.\\nGiven the hypothesis of Theorem 1.2 (resp. Theorem 1.1), we have\\nIn what follows we present the proof for Theorem 1.2, and make remarks only as required for the proof of Theorem 1.1.\\nLet G denote the algebraic monodromy group of ρ. Since T is of positive upper density, by Theorem 2.1 above, there exists a connected component G φ of G such that G φ is contained in X m .\\nWe are led to consider the following problem in the context of algebraic groups: let G ⊂ GL n × GL n be a reductive algebraic group and p 1 , p 2 denote the two projections. Assume that the image p 1 (G) = G 1 is connected. Suppose that there is a connected component G φ of G contained inside X m . What can we conclude about the representations p 1 and p 2 of G? For the rest of the paper, we will be following this notation.\\nWe first observe the equivalence of the representations when G is connected:\\nProposition 2.2. With the above notation, suppose G is connected and G ⊂ X m . Then the representations p 1 and p 2 are equivalent.\\nProof. Since G is connected, the m-th power map x → x m from G to G is dominant. Hence G ⊂ X 1 , and the representations p 1 and p 2 are equivalent.\\nRemark 2.2. The surjectivity of the m-th power map fails when we consider it between connected components. In [PR] the argument continued as follows: it can be assumed that m is chosen such that the m-th power map sends G φ to G 0 , the connected component of identity in G. The image will be contained inside the subvariety X 1 . Upon the erroneous assumption that the m-th power map is surjective, one concludes that G 0 ⊂ X 1 and this implies potential equivalence of ρ 1 and ρ 2 . But this assumption is wrong, as was pointed out by J.-P. Serre.\"\n },\n {\n \"title\": \"A unitary trick.\",\n \"content\": \"One of the advantages with the algebraic formulation is that it allows base change to the field of complex numbers, which makes it amenable to transcendental methods. Choosing an isomorphism of the algebraic closure of F with C, we consider the analogous problem over complex numbers.\\nLet U be a maximal compact subgroup of G(C) which we will assume is contained inside U(n) × U(n), where U(n) ⊂ GL n (C) is the (standard) group of unitary n × n matrices. The group U is Zariski dense in G. Hence the intersection U φ := U ∩ G φ is Zariski dense in G φ . In particular, it is non-empty.\\nThe image p 1 (U) of the projection of U to the first factor is a maximal compact subgroup of G 1 (C). By hypothesis G 1 is connected. Hence p 1 (U) is connected and equal to p 1 (U 0 ), where U 0 := U ∩ G 0 .\\nSince G φ ⊂ X m , there exists an element of the form (I n , j) ∈ U φ ⊂ X m (C) where I n denotes the identity matrix in GL n (C). We have, n = Tr(I m n ) = Tr(j m ). Since the only unitary matrix in with trace equal to n is the identity matrix, we conclude that j m = I n . Thus, G φ = G 0 (I n , j) with j m = I n . Let J = (I n , j). The condition G φ ⊂ X m , translates to the condition (xJ) m ∈ X m for any x ∈ G 0 . Remark 2.3. In [Ra1], we considered the case m = 1. In this case, we have that (I n , I n ) ∈ G φ (C), and this implies that G φ = G 0 , and the representations p 1 and p 2 are isomorphic restricted to G 0 . Hence we conclude that ρ 1 and ρ 2 are potentially isomorphic.\\nFor m ≥ 2, the significance of the unitary trick lies in the following crucial observation:\\nLet θ(x) = JxJ −1 denote the automorphism of finite order (dividing m) of G 0 induced by the conjugation action of J. Since by the unitary trick J m = (I n , I n ), the above equation becomes,\\nThe condition G φ ⊂ X m now translates to the following condition on the representations p 1 and p 2 of G 0 :\\nThis calculation can be reversed. Given the validity of Equation (2.3), we have an equality for y = xJ ∈ G φ , x ∈ G 0 , (2.4) Tr(p 1 (y) m ) = Tr(p 2 (y) m ).\\nWe observe that since J has finite order, it is semisimple with eigenvalues roots of unity. Hence J and the automorphism θ are defined over a finite extension F ′ of F .\"\n },\n {\n \"title\": \"2.4.\",\n \"content\": \"Algebraic analogue of Theorem 1.2. In this section, we consider the subgroup of G ⊂ GL(n) × GL(n) generated by G 0 and the the connected component G φ , as considered above. By an abuse of notation, we continue to denote this subgroup by G. The group G is defined over F , and over a finite extension F ′ of F , is isomorphic to the group generated by G 0 and the element J = (1, j) ∈ G(F ′ ). The conjugation action by the element (1, j) induces the automorphism θ on G 0 defined over F ′ . We decompose G with respect to the action of θ as follows:\\n(1) The derived subgroup G ′ of G 0 is a connected semisimple group defined over F . (2) The connected component of the center Z of G 0 is invariant under θ.\\nThe automorphism θ leaves stable the groups G ′ and Z (considered as subgroups over F ′ ), and there is a decomposition\\nWe further decompose Z with respect to the action of θ.\\nConsider the lattice X * (Z) of characters of Z. Define endomorphisms of X * (Z), I θ = 1 − θ and N θ = 1 + θ · · · θ m−1 . Let X θ be the subgroup of X * (Z) on which θ acts trivially and Z θ be the corresponding subtorus of Z defined over F ′ . This is the maximial subtorus of Z on which θ acts trivially.\\n. The lattice L θ is θ-invariant and is the character group of a θ-stable subtorus Z θ of Z. The invariants of θ acting on Z θ and G ′ Z θ ∩ Z θ are finite groups.\\n(4) Let Z θ <J> be the subgroup of G generated by Z θ and the element J. The groups G ′ Z θ and Z θ <J> are normal in G, and there is a decomposition\\nWe have the following algebraic analogue of Theorem 1.2:\\nTheorem 2.3. Let G be as above. Suppose that the following condition is satisfied for x ∈ G 0 :\\n(2.5) Tr(p 1 (x m )) = Tr(p 2 (xθ(x) · · · θ m−1 (x))).\\nWith respect to the decomposition G = G ′ Z θ Z θ <J> given above, the following hold:\\n(1) θ acts trivially on G ′ Z θ , and\\n(2) p 1 restricted to Z θ <J> is trivial.\\n(3) There exists a finite extension E of F ′ and decompositions defined over E, p 1 ≃ ⊕ t i=1 p 1,i and p 2 ≃ ⊕ t i=1 p 2,i , where p 1,i and p 2,i are representations of G to GL n i for some n i ≥ 1. For 1 ≤ i ≤ t, there exists an absolutely irreducible representations R i of G and representations p ′ 1,i , p ′ 2,i of Z θ <J> into GL n i /dim(R i ) (E), satisfying:\\n(a) As representations of the group G ′ Z θ × Z θ <J>,\\nEquivalently, for y = xJ belonging to the coset Z θ J, n i = Tr(p ′ 1,i (y m )) = Tr(p ′ 2,i (y m )). Remark 2.4. A similar analogous statement can be made for Theorem 1.1, where the hypothesis is modified by considering the analogue of Equation (2.5) for all exterior powers. Similarly the conclusion can be strengthened to say that the equality of Equation (2.6) holds for all exterior powers. Since Z θ is a torus, the elements in the image of Z θ <J> will be semisimple for any linear representation of Z θ <J>. From this and the equality of traces for all exterior powers, one concludes that the elements p ′ 1,i (y m ) and p ′ 2,i (y m ) are conjugate for y = xJ ∈ Z θ J, x ∈ Z θ . Example 2.1. The calculation given by Equation (2.2) can be reversed to give examples generalizing those given by Serre. Let T be a torus equipped with an automorphism θ of finite order, say d. Assume that T = T θ , i.e., the θ-invariants of T is a finite group. Form the semidirect product T ⋊ Z/dZ, where 1 ∈ Z/dZ acts via θ on T . For an element xθ, x ∈ T belonging to the connected component containing θ,\\nAs T is commutative, the element N θ (x) is θ-invariant. Hence there exists a natural number m, such that (xθ) m = 1. For any linear representation of T ⋊ Z/dZ of dimension d, R((xθ) m ) = 1. Thus the trivial representation of T of dimension d and the representation R are 'potentially equivalent' on the coset containing θ. (1) and (2) of Theorem 2.3. Let H denote the connected component of identity of Ker(p 1 | G ′ ). The group H is semisimple, and since p 1 = p 1 •θ, H is θ-stable. Let H θ denote the identity connected component of the fixed points of θ acting on H.\"\n },\n {\n \"title\": \"Proof of Parts\",\n \"content\": \"Restricted to H θ , Equation (2.5) gives the identity:\\nwhere we have used the fact that p 1 restricted to H is trivial. By Proposition 2.2, the representation p 2 restricted to H θ is trivial. Since G ′ ⊂ GL n ×GL n , the representation p 2 restricted to H, and in particular to H θ is injective. Hence H θ is trivial.\\nThe following theorem is classical (see for instance [K,Chapter 8]) or [BM]), and we give a proof in our context in the following section):\\nTheorem 2.4. Let H be a (non-trivial) connected semisimple group over F , and let θ be a finite order automorphism of H. Then the connected component H θ of the fixed points of θ is a non-trivial reductive group.\\nFrom this theorem and the triviality of H θ , it follows that H is trivial, i.e., the kernel of p 1 | G ′ is finite. From the equality p 1 • θ = p 1 , we get for g ∈ G ′ that θ(g) = zg for some z in kernel of p 1 , i.e., θ(g)g −1 = z. The finiteness of the kernel and the connectedness of G ′ implies that θ acts trivially on G ′ . Thus θ acts trivially on G ′ Z θ . Proposition 2.2 applied to Equation (2.5) restricted to G ′ Z θ yields Part (i) of Theorem 2.3.\\nThe proof of Part (ii) of Theorem 2.3, follows from the fact that θ acts trivially on the first co-ordinate. Since Z θ has finite θ-invariants, being connected, it cannot have a non-trivial quotient with trivial action of θ. The element J projects trivially to the first component by our choice of J.\\n2.6. Part (3) of Theorem 2.3. Let E be a finite extension of F ′ , over which the representations p 1 and p 2 breaks up as direct sums of absolutely irreducible representations of G. By Part (1), after a change of basis of one of the representations we can assume that\\nV i be the decomposition into isotypical components of absolutely irreducible representations of the representation p 1 | G ′ Z θ . These components are stabilized by G as G ′ Z θ is normal in G. Since p 1 restricted to Z θ <J> is trivial, the representation p 1 factors via G/Z θ which is a quotient of G ′ Z θ . Let R i denote the isomorphism class of an absolutely irreducible representation of G (acting via p 1 ) on V i . Considered as representations of the group G ′ Z θ × Z θ <J>, the representation p 2,i can be written as, p 2,i ≃ ⊕ n j j=1 R i ⊗ η ij , for some collection of absolutely irreducible representations η ij of Z θ <J>. Define p ′ 2,i = ⊕ n j j=1 η ij . Since p 1 is trivial restricted to Z θ <J >, this gives the decomposition in Part (3.a), and also proves Part (3.b) of Theorem 2.3.\\nTo prove Part (3.c), in terms of the decompositions, Equation (2.5) can be written as,\\nTr(R i (g m ))Tr(p ′ 2,i (xθ(x) · · · θ m−1 (x))), for g ∈ G ′ Z θ and x ∈ Z θ . Since G ′ Z θ is connected, the map g → g m is surjective. The representations R i are distinct irreducible representations of G ′ Z θ . Hence by the linear independence of characters, we obtain Part (3.c) of Theorem 2.3.\"\n },\n {\n \"title\": \"2.7.\",\n \"content\": \"Proof of Theorem 1.2. We deduce now Theorem 1.2 from Theorem 2.3. From the arguments of Sections 2.2 and 2.3, consider the group G 1 generated by G 0 and the connected component G φ (as is done in Section 2.4). The group ρ −1 (G 1 (F )) is of finite index in G K and is of the form G K 1 for some finite extension K 1 of K. We apply Theorem 2.3 to G 1 and let E be the field given in Part (3) of Theorem 2.3.\\nThe natural map π from the product of groups\\nhas finite kernel A. Choose a θ-stable open subgroup M ′ (resp. M θ ) of G ′ Z θ (E) (resp. Z θ <J > (E)), such that M ′ × M θ intersects A only at the identity element. Let M θ <J> denote the open subgroup of Z θ <J> (E) generated by M θ and J. This group is Zariski dense in Z θ <J>.\\nLet M ′ × M θ <J > maps isomorphically via π to the its image group denoted by M in G 1 (E). This is an open subgroup of G 1 (E). The intersection ρ(G K 1 ) ∩ M is of finite index in ρ(G K 1 ), and is of the form ρ(G L ) for some finite extension L of K. Thus we can consider the map ρ :\\nComposing the maps R i , p ′ 1,i and p ′ 2,i with the map ρ defines respectively linear representations r i , ρ ′ 1,i and ρ ′ 2,i of G L .\\nFrom the construction, the algebraic monodromy of the representations ρ ′ 2,i is Z θ < J >. The set of places T L is defined to be those places of L such that the image of the Frobenius conjugacy class by the representations ρ ′ 2,i lands inside the component Z θ J of Z θ <J>, and hence it is of positive density by Chebotarev density theorem. Theorem 1.2 follows now from Theorem 2.3.\\n2.8. Proof of Corollary 1.1. (1) If the ranks are equal, then for each i, the representations ρ ′ 2,i are finite. This means that the representations are potentially equivalent.\\n(2) If ρ 1 is absolutely irreducible, then t = 1 and the dimension of the representations ρ ′ 1,1 and ρ ′ 2,1 is one. Since they are also potentially equivalent, the values of ρ ′ 2,1 lie in roots of unity of F , which is a finite set. Hence the representations are potentially equivalent.\\n2.9. Proof of Theorem 2.4. For the sake of completeness of exposition, we give a proof of Theorem 2.4 in the following context which is sufficient for our purpose (see [K,Chapter 8] for a more detailed and complete exposition): we take F = C, G ⊂ GL(n, C) and the automorphism θ is induced by conjugation by an unitary matrix. In this case, θ commutes with the Cartan involution θ c : A → tĀ−1 . Hence the Cartain involution fixes H θ , and it follows that H θ is a reductive group.\\nTo see that the connected component H θ of the fixed points of θ is non-trivial, it suffices to work with a compact form of H. Let H denote the complexification of the Lie algebra of the compact form. We continue to denote by θ the inducted action on the Lie algebra. Decompose H = ⊕ m−1 i=0 H i as eigenspaces for the action of θ on H,\\nHere ζ m is a fixed choice of a primitive m-th root of unity.\\nLet I be the set of indices for which H i is non-zero, and let k ∈ I be such that ζ k m is a generator for the subgroup of the roots of unity generated by ζ j m , j ∈ I. For X ∈ H k and Y ∈ H l ,\\nSuppose 0 is not in I. For any n, ad(X) n (Y ) belongs to ζ nk+l m -eigenspace of θ. The assumption on k implies that for some n, ad(X) n will annihilate the space H l . Choosing n appropriately, this implies that ad(X) is nilpotent on H. But then there are no such elements in H. Hence 0 ∈ I and this implies that H θ is non-trivial.\\n2.10. Images of connected components. The observation given by Equation (2.1) allows us to conclude that the images of connected components of semisimple algebraic groups with respect to the power maps P n will not collapse.\\nTheorem 2.5. Let G be a semisimple algebraic group over an algebraically closed field F of characteristic zero. Let G φ denote a connected component of G. Suppose that the order of G φ is n. For a sufficiently large multiple m of n, the image of G φ by P m contains a (Zariski open) neighbourhood of identity of a non-trivial connected reductive group.\\nProof. We show that there exists a torsion element j in G φ of order m. Write\\nwhere H i are the connected normal almost simple subgroups of G 0 . Elements x of G φ induce automorphisms θ x of G 0 by conjugation, and two such automorphisms are related by an inner automorphism. Hence there exists some k, such that the automorphisms θ k x will send H i to H i for each i. Consequently, this gives an automorphism ofH i whereH i is the simply connected cover of H i , and thus defines an element of r i=1 Out(H i ). It is known that for any connected, simply connected, almost simple group H, the sequence 1 → Inn(H) → Aut(H) → Out(H) → 1, splits. Since Out(H) is finite, this gives an j ∈ G φ such that the automorphism θ j has finite order.\\nLet θ j (x) = jxj −1 be the automorphism induced on G 0 by conjugation by j. From Equation (2.2), the image of P m will contain elements of the form xθ(x)θ 2 (x) · · · θ m−1 (x), where x ranges over the elements of G 0 . In particular, the image will contain elements of the form x m for x ∈ (G 0 ) θ , where (G 0 ) θ denotes the connected component of identity of the fixed points of θ acting on G 0 . By Theorem 2.4, this is a non-trivial connected reductive group. The image of the map x → x m from (G 0 ) θ to itself is dominant and contains a neighbourhood of identity of (G 0 ) θ , and this proves the theorem.\"\n },\n {\n \"title\": \"Proof of Theorem 1.3\",\n \"content\": \"For the proof of Theorem 1.3, we first argue as in Section 2.2, and conclude by Theorem 2.1 that there exists a connected component G φ of G such that G φ is contained in X m . Consider the map P m : x → x m from G to itself.\\nClaim: The m-th power map from G φ to G 0 is dominant.\\nGranting the validity of the claim, this implies that G 0 ⊂ X 1 . This implies that the representations p 1 and p 2 are equivalent, hence the representations ρ 1 and ρ 2 are potentially equivalent.\\nTo prove the claim, we first observe that since G 2 is connected and G 0 projects onto G 2 , the projection to the second factor of G φ is equal to G 2 .\\nWe now use a fact from algebraic geometry (Exercise 3.7, Hartshorne): given X, Y are quasi-projective varieties of the same dimension and a dominant morphism π : X → Y , there exists open subsets U ⊂ Y and V ⊂ π −1 (U), such that π | V : V → U is finite.\\nApplying this to m-th power maps G 2 → G 2 and G 1 → G 1 , and making using of the fact that for any connected algebraic group the m-th power map P m is dominant, we obtain open subsets U 2 ⊂ G 2 , U 1 ⊂ G 1 and V 2 ⊂ G 2 , V 1 ⊂ G 1 such that P m : V i → U i is a finite map for i = 1, 2.\\nWe have G φ ⊂ G 1 ×G 2 , and V 1 ×V 2 ⊂ G 1 ×G 2 . Since the projection map G φ → G 2 is dominant, the sets pr −1 2 (V 2 ) ∩ G φ and pr −1 1 (V 1 ) ∩ G φ are non-empty open subsets. Since G φ is an integral scheme, the intersection of these two open subsets which is V = G φ ∩ V 1 × V 2 is a non-empty open subset of G φ .\\nWe have a commutative diagram,\\nSince the map V 1 × V 2 → U 1 × U 2 is finite, its restriction to V is finite onto its image. Hence the image of P m contains a subvariety of dimension equal to that of G 0 . This implies that X 1 ∩ G 0 contains a subscheme of dimension equal to dimension of G 0 . As G 0 is connected and X 1 ∩ G 0 is a closed subvariety of G 0 , we get G 0 ⊂ X 1 .\\nThis gives a proof of Theorem 1.3.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"A Borel, G D Mostow, On Semi-Simple Automorphisms of Lie Algebras. 61A. Borel and G. D. Mostow, On Semi-Simple Automorphisms of Lie Algebras, Annals of Mathematics Second Series, Vol. 61, No. 3 (May, 1955), pp. 389-405.","doi":""},{"text":"Infinite dimensional Lie algebras. V G Kac, Cambridge University PressCambridgeThird edition. xxii+400 ppV. G. Kac, Infinite dimensional Lie algebras, Third edition. Cambridge University Press, Cambridge, 1990. xxii+400 pp.","doi":""},{"text":"Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves. M Kulkarni, V M Patankar, C S Rajan, J. Number Theory. 164M. Kulkarni, V. M. Patankar and C. S. Rajan, Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves, J. Number Theory, 164 (2016) 87-102.","doi":""},{"text":"Locally potentially equivalent Galois representations. M Vijay, C S Patankar, Rajan, J. of Ramanujan Math. Soc. 271Vijay M. Patankar and C. S. Rajan, Locally potentially equivalent Galois representations, J. of Ramanujan Math. Soc., Vol. 27, No. 1, (2012), 77-90.","doi":""},{"text":"On strong multiplicity one for l-adic representations. C S Rajan, International Math. Res. Notices. 3C. S. Rajan, On strong multiplicity one for l-adic representations, International Math. Res. Notices, 1998, No.3, pp. 161-172.","doi":""},{"text":"J.-P Serre, Quelques applications du théorème de densité de Chebotarev. 54J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. HautesÉtudes Sc. Publ. Math. 54 (1981) 323-4-1.","doi":""},{"text":"email: vijaypatankar@gmail.com School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road. New Delhi, INDIA; BombaySchool of Physical Sciences, Jawaharlal Nehru UniversityINDIA 400005., email: rajan@math.tifr.res.inSchool of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067., email: vijaypatankar@gmail.com School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Colaba, Bombay, INDIA 400005., email: rajan@math.tifr.res.in","doi":""}],"string":"[\n {\n \"text\": \"A Borel, G D Mostow, On Semi-Simple Automorphisms of Lie Algebras. 61A. Borel and G. D. Mostow, On Semi-Simple Automorphisms of Lie Algebras, Annals of Mathematics Second Series, Vol. 61, No. 3 (May, 1955), pp. 389-405.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Infinite dimensional Lie algebras. V G Kac, Cambridge University PressCambridgeThird edition. xxii+400 ppV. G. Kac, Infinite dimensional Lie algebras, Third edition. Cambridge University Press, Cambridge, 1990. xxii+400 pp.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves. M Kulkarni, V M Patankar, C S Rajan, J. Number Theory. 164M. Kulkarni, V. M. Patankar and C. S. Rajan, Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves, J. Number Theory, 164 (2016) 87-102.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Locally potentially equivalent Galois representations. M Vijay, C S Patankar, Rajan, J. of Ramanujan Math. Soc. 271Vijay M. Patankar and C. S. Rajan, Locally potentially equivalent Galois representations, J. of Ramanujan Math. Soc., Vol. 27, No. 1, (2012), 77-90.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On strong multiplicity one for l-adic representations. C S Rajan, International Math. Res. Notices. 3C. S. Rajan, On strong multiplicity one for l-adic representations, International Math. Res. Notices, 1998, No.3, pp. 161-172.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J.-P Serre, Quelques applications du théorème de densité de Chebotarev. 54J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. HautesÉtudes Sc. Publ. Math. 54 (1981) 323-4-1.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"email: vijaypatankar@gmail.com School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road. New Delhi, INDIA; BombaySchool of Physical Sciences, Jawaharlal Nehru UniversityINDIA 400005., email: rajan@math.tifr.res.inSchool of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067., email: vijaypatankar@gmail.com School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Colaba, Bombay, INDIA 400005., email: rajan@math.tifr.res.in\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":45,"cells":{"text":{"kind":"string","value":"\nClassical programmability is enough for quantum circuits universality in approximate sense\n20 Mar 2001\n\nAlexander Yu Vlasov \nFRC/IRH\n197101 Mira Street 8St.-PetersburgRussia\n\nClassical programmability is enough for quantum circuits universality in approximate sense\n20 Mar 2001arXiv:quant-ph/0103119v1 Extended Abstract\nIt was shown by M. A. Nielsen and I. L Chuang[1], that it is impossible to build strictly universal programmable quantum gate array, that could perform any unitary operation precisely and it was suggested to use probabilistic gate arrays instead. In present work is shown, that if to use more physical and weak condition of universality (suggested already in earliest work by D. Deutsch[3]) and to talk about simulation with arbitrary, but finite precision, then it is possible to build universal programmable gate array. But now the same no-go theorem by Nielsen and Chuang [1] will have new interesting consequence -controlling programs for the gate arrays can be considered as pure classical. More detailed design of such deterministic quantum gate arrays universal \"in approximate sense\" is considered in the paper.\n\nIntroduction\n\nIn the paper [1] was discussed conception of programmable quantum gate arrays, i.e., some quantum circuits are acting on a system in form |d; P ≡ |d ⊗ |P considered as data register |d and program register (or simply program) |P . Similar with conception of usual classical computer, it was considered, that circuit acts as some fixed unitary transformation U on whole system and different transformations U P of data related only with content P of program register, i.e:\nU |d ⊗ |P = (U P |d ) ⊗ |P ′ .(1)\nIt should be emphasized, here U P is same for any state of data register |d i.e., depends only on program |P , and states of these two registers are not entangled before and after application of U.\n\nIn relation with such a definition in [1] was noticed, that if to write the Eq. (1) for two different programs |P and |Q and corresponding unitary data transformations |U P and |U Q , it is simple to find by considering scalar products of both parts, that if U P = φ U Q for some complex number φ then the states of program register must be orthogonal for different programs:\nP | Q = 0.(2)\nThe property Eq. (2) treated in [1] as demonstration that \"... no universal quantum gate array (of finite extent) can be realized. More specifically, we show that every implementable unitary operation requires an extra Hilbert space dimension in the program register.\" Due to such a problem in the article [1] was suggested idea of (exactly) universal stochastic programmable gate array developed further by other authors [10].\n\nIt was no-go theorem for \"universality in exact sense\" [2], but fortunately already in initial definition of universal quantum computer [3] was shown possibility of using some discrete everywhere dense subset in the whole continuous space of unitary operators, it is called sometime \"universality in approximate sense\" [2].\n\nHere is shown, that in the approximate sense such universal programmable quantum gate arrays with possibility of approximation of any unitary transformation of data register with given precision are really exist and constructions are quite simple and may be described directly. But the property Eq. (2) is again very important, because if all different possible states of program register are orthogonal, it is possible without lost of generality to find implementation of same array with all different programs are elements of computational basis and superpositions of the states for the program register are never used and so any such array can be designed with possibility of using only classical programs. It is shown and discussed with more extent in relation with particular design used below.\n\n\nUniversal programmable quantum gate arrays\n\nIt is more convenient here to use notation with program register first, i.e., |P ; d instead of |d; P . The reason is more simple form of operations like\nCONTROL-U : 1 0 0 0 0 1 0 0 0 0 u 00 u 01 0 0 u 10 u 11 .(3)\nSuch operator acts on second qubit as U only if first qubit is |1 , but |0 does not change anything. A straightforward generalization for arbitrary N × N matrix U and one program qubit is 2N × 2N matrix:\n1 0 0 U ,(4)\nwhere 1 and 0 are N × N unit and zero matrices.\n\nLet us now consider simple programmable gate array with dimension of program is M and data register is N . For quantum computation with qubits M = 2 m and N = 2 n . Let us introduce a matrix (cf. [5]) S = S {U1,...,UM } :\nS = U 1 0 U 2 . . . 0 U M .(5)\nwhere\nU 1 , . . . , U M are M matrices N × N . It is clear, that such N M × N M matrix S = S {U1,.\n..,UM } acts as:\nS |P ⊗ |d = |P ⊗ (U P |d ).(6)\nand it corresponds to Eq. (1) with P = P ′ is simply number of matrix U P . Such matrix was suggested for description of conditional quantum dynamics in [5], but Eq. (2) introduced in [1] and discussed here shows also, that we do not need more general control with superpositions of basis states of program (control register).\n\nLet us consider a case with {U 1 , . . . , U M } are universal set of gates (in approximate sense) for some quantum circuits with n qubits together with unit matrix U 0 ≡ 1. Here N = 2 n and for design with matrix S size of program is small enough -number of qubits in the register is m = ⌈log 2 M ⌉. But if due to technical problem with implementation we should use set of CONTROL-U i gates Eq. (4) instead of S Eq. (5), then number of program qubits are m = M .\n\nWe may use small universal sets with only one and two-qubit gates [2,7,8,9]. For example, if to choose a set with n + 2 one-qubit gates and n − 1 two-gates acting on pairs of neighboring qubits from [9] together with unit matrix, then number of qubit in control register is m = ⌈log 2 (2n + 2)⌉ in best case and even in worst case discussed earlier it is 2n + 2.\n\nLet us now extend program register up to km qubits, i.e. state of arrays described now as |P k ; . . . ; P 2 ; P 1 ; d and together with matrix S acting only on last m + n qubits |P 1 ; d as described above let us consider unitary matrix R acting on program register as right cyclic shift:\nR |P k ; . . . ; P 2 ; P 1 ; d = |P 1 ; P k ; . . . ; P 2 ; d(7)\nIt is clear, that if U = RS, then operator U k can perform with data arbitrary sequence of up to k operators {U 1 , . . . , U M }: U k |0; . . . ; 0; P l ; . . . ; P 1 ⊗ |d = |0; . . . ; 0; P l ; . . . ; P 1 ⊗ (U P l · · · U P1 |d ) (8) here U 0 is unit operator used to fill out k−l positions for l < k and U P l · · · U P1 ∼ = U is arbitrary product of gates from the finite universal set {U 1 , . . . , U M } for approximation of any data gate U with necessary precision. Usually number of terms in product l ≫ M . If minimal error of simulation is given, it is possible to find necessary number k and then U k is universal circuit with bounded chosen accuracy.\n\nThe idle k − l steps make more clear advantage of a design well known in classical theory of computations -instead of consideration of whole circuit U k it is possible to introduce notion of one computational step U. In such approach action of quantum circuit similar with usual CPU \"timing\" with repeating of same U till halting.\n\nIt is also useful, then due to algorithm of approximation we need to apply same loop many times, for example instead of some operator U may be used \"j-th root\" U 1/j [6,9] and the circuit should be applied j times and can be expressed as U k j .\n\nHere were examples of some general architecture for programmable quantum gate arrays -we have n data qubits and program with p + m qubits of two different kinds. Here are m control qubits of quantum controller together with n data qubits form input for \"quantum step operator\" S described by Eq. (5).\n\nGeneralization of R operator is not necessary simple rotation, because such model may use too many space. The R is equivalent of reversible classical 1 circuit acting on p + m (qu)bit program register with purpose to generate necessary index of universal quantum gate for quantum controller during simulation.\n\nIt should be mentioned also yet another advantage and special property of \"pseudo-classical\", i.e., orthogonal states of program register. Only for such kind of programs it is possible to make measurements of state without destruction and so use tools like halt bit. It is also further justification of notion of variable lenght algorithms already used above.\n\nThe consideration shows possible model of programmable \"quantum chip\". It has three different kinds of \"wiring\": quantum, intermediate and classical \"buses\" with d, m and p (qu)bits respectively and two different kinds of circuits: quantum controller acting on quantum bus and controlled by intermediate bus and reversible classical (or Qu-ERCC) circuit with input via classical bus for control of intermediate bus.\n\nDue to result about orthogonality of programs [1] the design is general for deterministic circuits with pure states, and does not need some special quantum control, i.e, superpositions in program register, only data is necessary quantum.\n\nOn the other hand, for circuits with mixed states, stochastic programmable quantum arrays [1,10] control already is not limited by the (pseudo) classical case discussed here.\n \"Qu-ERCC\" -Quantum Equivalent of Reversible Classical Circuit, or simply \"classical gate\" in terminology used by[6] \n\nProgrammable quantum gate arrays. M A Nielsen, I Chuang, Phys. Rev. Lett. 79M. A. Nielsen and I. L Chuang, \"Programmable quantum gate arrays,\" Phys. Rev. Lett. 79 321-324, (1997).\n\nAn introduction to quantum complexity theory. R Cleve, Preprint quant-ph/9906111R. Cleve, \"An introduction to quantum complexity theory,\" Preprint quant-ph/9906111, (1999).\n\nQuantum Theory, the Church-Turing Principle and the Universal Quantum Computer. D Deutsch, Proc. R. Soc. London A. 400D. Deutsch, \"Quantum Theory, the Church-Turing Principle and the Uni- versal Quantum Computer,\" Proc. R. Soc. London A 400, 97-117 (1985).\n\nQuantum Computational Networks. D Deutsch, Proc. R. Soc. London A. 425D. Deutsch, \"Quantum Computational Networks,\" Proc. R. Soc. London A 425, 73-90 (1989).\n\nConditional quantum dynamics and logic gates. A Barenco, D Deutsch, A K Ekert, R Jozsa, Phys. Rev. Lett. 74A. Barenco, D. Deutsch, A. K. Ekert, and R. Jozsa, \"Conditional quantum dynamics and logic gates,\" Phys. Rev. Lett. 74, 20, 4083-4086 (1995).\n\nUniversality in quantum computation. D Deutsch, A Barenco, A Ekert, Proc. R. Soc. London A. 449669D. Deutsch, A. Barenco, and A. Ekert, \"Universality in quantum computa- tion,\" Proc. R. Soc. London A 449, 669 (1995).\n\nElementary gates for quantum computation. A Barenco, C H Bennett, R Cleve, D P Divincenzo, N Margolus, P W Shor, T Sleator, J A Smolin, H Weinfurter, Phys. Rev. A. 52A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. W. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, \"Elementary gates for quantum computation,\" Phys. Rev. A 52, 5, 3457-3467 (1995);\n\nTwo-bit gates are universal for quantum computation. D P Divincenzo, Phys. Rev. A. 51D. P. DiVincenzo, \"Two-bit gates are universal for quantum computation,\" Phys. Rev. A 51, 1015 -1022 (1995).\n\nClifford algebras and universal sets of quantum gates. A Yu, Vlasov, quant-ph/0010071Phys. Rev. A. Eprintaccepted for publication inA. Yu. Vlasov, \"Clifford algebras and universal sets of quantum gates,\" E- print: quant-ph/0010071 (2000) [accepted for publication in Phys. Rev. A (2001)].\n\nStoring quantum dynamics in quantum states: stochastic programmable gate for U (1) operations. G Vidal, L Masanes, J I Cirac, quant-ph/0102037PreprintG. Vidal, L. Masanes, and J. I. Cirac, \"Storing quantum dynamics in quan- tum states: stochastic programmable gate for U (1) operations,\" Preprint quant-ph/0102037, (2001).\n"},"corpus_id":{"kind":"number","value":16220335,"string":"16,220,335"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"quant-ph/0103119\",\n \"mag\": 1558144723,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/quant-ph/0103119v1.pdf\"\n ],\n \"pdf_sha\": \"c084b819a64a39d31016abed81d2cc12bbd483d7\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Classical programmability is enough for quantum circuits universality in approximate sense"},"authors":{"kind":"list like","value":["Alexander Yu Vlasov \nFRC/IRH\n197101 Mira Street 8St.-PetersburgRussia"],"string":"[\n \"Alexander Yu Vlasov \\nFRC/IRH\\n197101 Mira Street 8St.-PetersburgRussia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"It was shown by M. A. Nielsen and I. L Chuang[1], that it is impossible to build strictly universal programmable quantum gate array, that could perform any unitary operation precisely and it was suggested to use probabilistic gate arrays instead. In present work is shown, that if to use more physical and weak condition of universality (suggested already in earliest work by D. Deutsch[3]) and to talk about simulation with arbitrary, but finite precision, then it is possible to build universal programmable gate array. But now the same no-go theorem by Nielsen and Chuang [1] will have new interesting consequence -controlling programs for the gate arrays can be considered as pure classical. More detailed design of such deterministic quantum gate arrays universal \"in approximate sense\" is considered in the paper."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"In the paper [1] was discussed conception of programmable quantum gate arrays, i.e., some quantum circuits are acting on a system in form |d; P ≡ |d ⊗ |P considered as data register |d and program register (or simply program) |P . Similar with conception of usual classical computer, it was considered, that circuit acts as some fixed unitary transformation U on whole system and different transformations U P of data related only with content P of program register, i.e:\nIt should be emphasized, here U P is same for any state of data register |d i.e., depends only on program |P , and states of these two registers are not entangled before and after application of U.\nIn relation with such a definition in [1] was noticed, that if to write the Eq. (1) for two different programs |P and |Q and corresponding unitary data transformations |U P and |U Q , it is simple to find by considering scalar products of both parts, that if U P = φ U Q for some complex number φ then the states of program register must be orthogonal for different programs:\nThe property Eq. (2) treated in [1] as demonstration that \"... no universal quantum gate array (of finite extent) can be realized. More specifically, we show that every implementable unitary operation requires an extra Hilbert space dimension in the program register.\" Due to such a problem in the article [1] was suggested idea of (exactly) universal stochastic programmable gate array developed further by other authors [10].\nIt was no-go theorem for \"universality in exact sense\" [2], but fortunately already in initial definition of universal quantum computer [3] was shown possibility of using some discrete everywhere dense subset in the whole continuous space of unitary operators, it is called sometime \"universality in approximate sense\" [2].\nHere is shown, that in the approximate sense such universal programmable quantum gate arrays with possibility of approximation of any unitary transformation of data register with given precision are really exist and constructions are quite simple and may be described directly. But the property Eq. (2) is again very important, because if all different possible states of program register are orthogonal, it is possible without lost of generality to find implementation of same array with all different programs are elements of computational basis and superpositions of the states for the program register are never used and so any such array can be designed with possibility of using only classical programs. It is shown and discussed with more extent in relation with particular design used below."},{"title":"Universal programmable quantum gate arrays","content":"It is more convenient here to use notation with program register first, i.e., |P ; d instead of |d; P . The reason is more simple form of operations like\nSuch operator acts on second qubit as U only if first qubit is |1 , but |0 does not change anything. A straightforward generalization for arbitrary N × N matrix U and one program qubit is 2N × 2N matrix:\nwhere 1 and 0 are N × N unit and zero matrices.\nLet us now consider simple programmable gate array with dimension of program is M and data register is N . For quantum computation with qubits M = 2 m and N = 2 n . Let us introduce a matrix (cf. [5]) S = S {U1,...,UM } :\nwhere\n..,UM } acts as:\nand it corresponds to Eq. (1) with P = P ′ is simply number of matrix U P . Such matrix was suggested for description of conditional quantum dynamics in [5], but Eq. (2) introduced in [1] and discussed here shows also, that we do not need more general control with superpositions of basis states of program (control register).\nLet us consider a case with {U 1 , . . . , U M } are universal set of gates (in approximate sense) for some quantum circuits with n qubits together with unit matrix U 0 ≡ 1. Here N = 2 n and for design with matrix S size of program is small enough -number of qubits in the register is m = ⌈log 2 M ⌉. But if due to technical problem with implementation we should use set of CONTROL-U i gates Eq. (4) instead of S Eq. (5), then number of program qubits are m = M .\nWe may use small universal sets with only one and two-qubit gates [2,7,8,9]. For example, if to choose a set with n + 2 one-qubit gates and n − 1 two-gates acting on pairs of neighboring qubits from [9] together with unit matrix, then number of qubit in control register is m = ⌈log 2 (2n + 2)⌉ in best case and even in worst case discussed earlier it is 2n + 2.\nLet us now extend program register up to km qubits, i.e. state of arrays described now as |P k ; . . . ; P 2 ; P 1 ; d and together with matrix S acting only on last m + n qubits |P 1 ; d as described above let us consider unitary matrix R acting on program register as right cyclic shift:\nIt is clear, that if U = RS, then operator U k can perform with data arbitrary sequence of up to k operators {U 1 , . . . , U M }: U k |0; . . . ; 0; P l ; . . . ; P 1 ⊗ |d = |0; . . . ; 0; P l ; . . . ; P 1 ⊗ (U P l · · · U P1 |d ) (8) here U 0 is unit operator used to fill out k−l positions for l < k and U P l · · · U P1 ∼ = U is arbitrary product of gates from the finite universal set {U 1 , . . . , U M } for approximation of any data gate U with necessary precision. Usually number of terms in product l ≫ M . If minimal error of simulation is given, it is possible to find necessary number k and then U k is universal circuit with bounded chosen accuracy.\nThe idle k − l steps make more clear advantage of a design well known in classical theory of computations -instead of consideration of whole circuit U k it is possible to introduce notion of one computational step U. In such approach action of quantum circuit similar with usual CPU \"timing\" with repeating of same U till halting.\nIt is also useful, then due to algorithm of approximation we need to apply same loop many times, for example instead of some operator U may be used \"j-th root\" U 1/j [6,9] and the circuit should be applied j times and can be expressed as U k j .\nHere were examples of some general architecture for programmable quantum gate arrays -we have n data qubits and program with p + m qubits of two different kinds. Here are m control qubits of quantum controller together with n data qubits form input for \"quantum step operator\" S described by Eq. (5).\nGeneralization of R operator is not necessary simple rotation, because such model may use too many space. The R is equivalent of reversible classical 1 circuit acting on p + m (qu)bit program register with purpose to generate necessary index of universal quantum gate for quantum controller during simulation.\nIt should be mentioned also yet another advantage and special property of \"pseudo-classical\", i.e., orthogonal states of program register. Only for such kind of programs it is possible to make measurements of state without destruction and so use tools like halt bit. It is also further justification of notion of variable lenght algorithms already used above.\nThe consideration shows possible model of programmable \"quantum chip\". It has three different kinds of \"wiring\": quantum, intermediate and classical \"buses\" with d, m and p (qu)bits respectively and two different kinds of circuits: quantum controller acting on quantum bus and controlled by intermediate bus and reversible classical (or Qu-ERCC) circuit with input via classical bus for control of intermediate bus.\nDue to result about orthogonality of programs [1] the design is general for deterministic circuits with pure states, and does not need some special quantum control, i.e, superpositions in program register, only data is necessary quantum.\nOn the other hand, for circuits with mixed states, stochastic programmable quantum arrays [1,10] control already is not limited by the (pseudo) classical case discussed here."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"In the paper [1] was discussed conception of programmable quantum gate arrays, i.e., some quantum circuits are acting on a system in form |d; P ≡ |d ⊗ |P considered as data register |d and program register (or simply program) |P . Similar with conception of usual classical computer, it was considered, that circuit acts as some fixed unitary transformation U on whole system and different transformations U P of data related only with content P of program register, i.e:\\nIt should be emphasized, here U P is same for any state of data register |d i.e., depends only on program |P , and states of these two registers are not entangled before and after application of U.\\nIn relation with such a definition in [1] was noticed, that if to write the Eq. (1) for two different programs |P and |Q and corresponding unitary data transformations |U P and |U Q , it is simple to find by considering scalar products of both parts, that if U P = φ U Q for some complex number φ then the states of program register must be orthogonal for different programs:\\nThe property Eq. (2) treated in [1] as demonstration that \\\"... no universal quantum gate array (of finite extent) can be realized. More specifically, we show that every implementable unitary operation requires an extra Hilbert space dimension in the program register.\\\" Due to such a problem in the article [1] was suggested idea of (exactly) universal stochastic programmable gate array developed further by other authors [10].\\nIt was no-go theorem for \\\"universality in exact sense\\\" [2], but fortunately already in initial definition of universal quantum computer [3] was shown possibility of using some discrete everywhere dense subset in the whole continuous space of unitary operators, it is called sometime \\\"universality in approximate sense\\\" [2].\\nHere is shown, that in the approximate sense such universal programmable quantum gate arrays with possibility of approximation of any unitary transformation of data register with given precision are really exist and constructions are quite simple and may be described directly. But the property Eq. (2) is again very important, because if all different possible states of program register are orthogonal, it is possible without lost of generality to find implementation of same array with all different programs are elements of computational basis and superpositions of the states for the program register are never used and so any such array can be designed with possibility of using only classical programs. It is shown and discussed with more extent in relation with particular design used below.\"\n },\n {\n \"title\": \"Universal programmable quantum gate arrays\",\n \"content\": \"It is more convenient here to use notation with program register first, i.e., |P ; d instead of |d; P . The reason is more simple form of operations like\\nSuch operator acts on second qubit as U only if first qubit is |1 , but |0 does not change anything. A straightforward generalization for arbitrary N × N matrix U and one program qubit is 2N × 2N matrix:\\nwhere 1 and 0 are N × N unit and zero matrices.\\nLet us now consider simple programmable gate array with dimension of program is M and data register is N . For quantum computation with qubits M = 2 m and N = 2 n . Let us introduce a matrix (cf. [5]) S = S {U1,...,UM } :\\nwhere\\n..,UM } acts as:\\nand it corresponds to Eq. (1) with P = P ′ is simply number of matrix U P . Such matrix was suggested for description of conditional quantum dynamics in [5], but Eq. (2) introduced in [1] and discussed here shows also, that we do not need more general control with superpositions of basis states of program (control register).\\nLet us consider a case with {U 1 , . . . , U M } are universal set of gates (in approximate sense) for some quantum circuits with n qubits together with unit matrix U 0 ≡ 1. Here N = 2 n and for design with matrix S size of program is small enough -number of qubits in the register is m = ⌈log 2 M ⌉. But if due to technical problem with implementation we should use set of CONTROL-U i gates Eq. (4) instead of S Eq. (5), then number of program qubits are m = M .\\nWe may use small universal sets with only one and two-qubit gates [2,7,8,9]. For example, if to choose a set with n + 2 one-qubit gates and n − 1 two-gates acting on pairs of neighboring qubits from [9] together with unit matrix, then number of qubit in control register is m = ⌈log 2 (2n + 2)⌉ in best case and even in worst case discussed earlier it is 2n + 2.\\nLet us now extend program register up to km qubits, i.e. state of arrays described now as |P k ; . . . ; P 2 ; P 1 ; d and together with matrix S acting only on last m + n qubits |P 1 ; d as described above let us consider unitary matrix R acting on program register as right cyclic shift:\\nIt is clear, that if U = RS, then operator U k can perform with data arbitrary sequence of up to k operators {U 1 , . . . , U M }: U k |0; . . . ; 0; P l ; . . . ; P 1 ⊗ |d = |0; . . . ; 0; P l ; . . . ; P 1 ⊗ (U P l · · · U P1 |d ) (8) here U 0 is unit operator used to fill out k−l positions for l < k and U P l · · · U P1 ∼ = U is arbitrary product of gates from the finite universal set {U 1 , . . . , U M } for approximation of any data gate U with necessary precision. Usually number of terms in product l ≫ M . If minimal error of simulation is given, it is possible to find necessary number k and then U k is universal circuit with bounded chosen accuracy.\\nThe idle k − l steps make more clear advantage of a design well known in classical theory of computations -instead of consideration of whole circuit U k it is possible to introduce notion of one computational step U. In such approach action of quantum circuit similar with usual CPU \\\"timing\\\" with repeating of same U till halting.\\nIt is also useful, then due to algorithm of approximation we need to apply same loop many times, for example instead of some operator U may be used \\\"j-th root\\\" U 1/j [6,9] and the circuit should be applied j times and can be expressed as U k j .\\nHere were examples of some general architecture for programmable quantum gate arrays -we have n data qubits and program with p + m qubits of two different kinds. Here are m control qubits of quantum controller together with n data qubits form input for \\\"quantum step operator\\\" S described by Eq. (5).\\nGeneralization of R operator is not necessary simple rotation, because such model may use too many space. The R is equivalent of reversible classical 1 circuit acting on p + m (qu)bit program register with purpose to generate necessary index of universal quantum gate for quantum controller during simulation.\\nIt should be mentioned also yet another advantage and special property of \\\"pseudo-classical\\\", i.e., orthogonal states of program register. Only for such kind of programs it is possible to make measurements of state without destruction and so use tools like halt bit. It is also further justification of notion of variable lenght algorithms already used above.\\nThe consideration shows possible model of programmable \\\"quantum chip\\\". It has three different kinds of \\\"wiring\\\": quantum, intermediate and classical \\\"buses\\\" with d, m and p (qu)bits respectively and two different kinds of circuits: quantum controller acting on quantum bus and controlled by intermediate bus and reversible classical (or Qu-ERCC) circuit with input via classical bus for control of intermediate bus.\\nDue to result about orthogonality of programs [1] the design is general for deterministic circuits with pure states, and does not need some special quantum control, i.e, superpositions in program register, only data is necessary quantum.\\nOn the other hand, for circuits with mixed states, stochastic programmable quantum arrays [1,10] control already is not limited by the (pseudo) classical case discussed here.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Programmable quantum gate arrays. M A Nielsen, I Chuang, Phys. Rev. Lett. 79M. A. Nielsen and I. L Chuang, \"Programmable quantum gate arrays,\" Phys. Rev. Lett. 79 321-324, (1997).","doi":""},{"text":"An introduction to quantum complexity theory. R Cleve, Preprint quant-ph/9906111R. Cleve, \"An introduction to quantum complexity theory,\" Preprint quant-ph/9906111, (1999).","doi":""},{"text":"Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. D Deutsch, Proc. R. Soc. London A. 400D. Deutsch, \"Quantum Theory, the Church-Turing Principle and the Uni- versal Quantum Computer,\" Proc. R. Soc. London A 400, 97-117 (1985).","doi":""},{"text":"Quantum Computational Networks. D Deutsch, Proc. R. Soc. London A. 425D. Deutsch, \"Quantum Computational Networks,\" Proc. R. Soc. London A 425, 73-90 (1989).","doi":""},{"text":"Conditional quantum dynamics and logic gates. A Barenco, D Deutsch, A K Ekert, R Jozsa, Phys. Rev. Lett. 74A. Barenco, D. Deutsch, A. K. Ekert, and R. Jozsa, \"Conditional quantum dynamics and logic gates,\" Phys. Rev. Lett. 74, 20, 4083-4086 (1995).","doi":""},{"text":"Universality in quantum computation. D Deutsch, A Barenco, A Ekert, Proc. R. Soc. London A. 449669D. Deutsch, A. Barenco, and A. Ekert, \"Universality in quantum computa- tion,\" Proc. R. Soc. London A 449, 669 (1995).","doi":""},{"text":"Elementary gates for quantum computation. A Barenco, C H Bennett, R Cleve, D P Divincenzo, N Margolus, P W Shor, T Sleator, J A Smolin, H Weinfurter, Phys. Rev. A. 52A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. W. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, \"Elementary gates for quantum computation,\" Phys. Rev. A 52, 5, 3457-3467 (1995);","doi":""},{"text":"Two-bit gates are universal for quantum computation. D P Divincenzo, Phys. Rev. A. 51D. P. DiVincenzo, \"Two-bit gates are universal for quantum computation,\" Phys. Rev. A 51, 1015 -1022 (1995).","doi":""},{"text":"Clifford algebras and universal sets of quantum gates. A Yu, Vlasov, quant-ph/0010071Phys. Rev. A. Eprintaccepted for publication inA. Yu. Vlasov, \"Clifford algebras and universal sets of quantum gates,\" E- print: quant-ph/0010071 (2000) [accepted for publication in Phys. Rev. A (2001)].","doi":"quant-ph/0010071"},{"text":"Storing quantum dynamics in quantum states: stochastic programmable gate for U (1) operations. G Vidal, L Masanes, J I Cirac, quant-ph/0102037PreprintG. Vidal, L. Masanes, and J. I. Cirac, \"Storing quantum dynamics in quan- tum states: stochastic programmable gate for U (1) operations,\" Preprint quant-ph/0102037, (2001).","doi":"quant-ph/0102037"}],"string":"[\n {\n \"text\": \"Programmable quantum gate arrays. M A Nielsen, I Chuang, Phys. Rev. Lett. 79M. A. Nielsen and I. L Chuang, \\\"Programmable quantum gate arrays,\\\" Phys. Rev. Lett. 79 321-324, (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An introduction to quantum complexity theory. R Cleve, Preprint quant-ph/9906111R. Cleve, \\\"An introduction to quantum complexity theory,\\\" Preprint quant-ph/9906111, (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. D Deutsch, Proc. R. Soc. London A. 400D. Deutsch, \\\"Quantum Theory, the Church-Turing Principle and the Uni- versal Quantum Computer,\\\" Proc. R. Soc. London A 400, 97-117 (1985).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum Computational Networks. D Deutsch, Proc. R. Soc. London A. 425D. Deutsch, \\\"Quantum Computational Networks,\\\" Proc. R. Soc. London A 425, 73-90 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conditional quantum dynamics and logic gates. A Barenco, D Deutsch, A K Ekert, R Jozsa, Phys. Rev. Lett. 74A. Barenco, D. Deutsch, A. K. Ekert, and R. Jozsa, \\\"Conditional quantum dynamics and logic gates,\\\" Phys. Rev. Lett. 74, 20, 4083-4086 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universality in quantum computation. D Deutsch, A Barenco, A Ekert, Proc. R. Soc. London A. 449669D. Deutsch, A. Barenco, and A. Ekert, \\\"Universality in quantum computa- tion,\\\" Proc. R. Soc. London A 449, 669 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Elementary gates for quantum computation. A Barenco, C H Bennett, R Cleve, D P Divincenzo, N Margolus, P W Shor, T Sleator, J A Smolin, H Weinfurter, Phys. Rev. A. 52A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. W. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, \\\"Elementary gates for quantum computation,\\\" Phys. Rev. A 52, 5, 3457-3467 (1995);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Two-bit gates are universal for quantum computation. D P Divincenzo, Phys. Rev. A. 51D. P. DiVincenzo, \\\"Two-bit gates are universal for quantum computation,\\\" Phys. Rev. A 51, 1015 -1022 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Clifford algebras and universal sets of quantum gates. A Yu, Vlasov, quant-ph/0010071Phys. Rev. A. Eprintaccepted for publication inA. Yu. Vlasov, \\\"Clifford algebras and universal sets of quantum gates,\\\" E- print: quant-ph/0010071 (2000) [accepted for publication in Phys. Rev. A (2001)].\",\n \"doi\": \"quant-ph/0010071\"\n },\n {\n \"text\": \"Storing quantum dynamics in quantum states: stochastic programmable gate for U (1) operations. G Vidal, L Masanes, J I Cirac, quant-ph/0102037PreprintG. Vidal, L. Masanes, and J. I. Cirac, \\\"Storing quantum dynamics in quan- tum states: stochastic programmable gate for U (1) operations,\\\" Preprint quant-ph/0102037, (2001).\",\n \"doi\": \"quant-ph/0102037\"\n }\n]","isConversational":false}}},{"rowIdx":46,"cells":{"text":{"kind":"string","value":"\nModel Photospheres with Accelerated Lambda Iteration\n2003\n\nI Hubeny \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nD Mihalas \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nK Werner \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nKlaus Werner \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nJochen L Deetjen \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nStefan Dreizler \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nThorsten Nagel \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nThomas Rauch \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nSonja L Schuh \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany\n\nModel Photospheres with Accelerated Lambda Iteration\n\nStellar Atmosphere Modeling ASP Conference Series\nxxx2003\nWe address the classical stellar-atmosphere problem and describe our method of numerical solution in detail. The problem consists of the solution of the radiation transfer equation under the constraints of hydrostatic, radiative and statistical equilibrium (non-LTE). We employ the Accelerated Lambda Iteration (ALI) technique, and use statistical methods to construct non-LTE metal-line-blanketed model-atmospheres.2 Werner, Deetjen, Dreizler, Nagel, Rauch, & Schuh neously with the equations for hydrostatic and radiative equilibrium, together with the statistical equilibrium (rate equations).A stellar atmosphere radiates into circumstellar space and thus is an open thermodynamic system; hence it cannot be in thermodynamic equilibrium (TE), and cannot be characterized by a single temperature. The idea of \"Local Thermodynamic Equilibrium\" (LTE) is a hypothesis which assumes that while TE does not hold for the atmosphere as a whole, it can be applied in small volume elements. In this case the atomic occupation numbers depend only on the local electron temperature and electron density via the Saha-Boltzmann equations. But this approximation can be valid only in the limit that collision rates dominate radiative rates, and photon-mean-free-paths are small. Models in which the Saha-Boltzmann equations are replaced by the physically more accurate rate equations are called non-LTE (or NLTE) models. This designation is somewhat imprecise because the velocity distribution of particles is still assumed to be Maxwellian, i.e. we can still define a local temperature. NLTE calculations are more costly than LTE calculations, however, and it is hard to predict a priori whether NLTE effects will be important in a specific problem. Generally, NLTE effects are large at high temperatures and low densities, which imply intense radiation fields, hence frequent radiative processes (which have a non-local nature and, in particular, respond to the presence of the open boundary), and less-frequent particle collisions, which tend to enforce LTE conditions. Abandonment of the LTE assumption leads to a much more difficult modelatmosphere problem, because of subtle couplings between the radiation transfer and statistical equilibrium equations. The pioneering work by Auer & Mihalas (1969) provided a basic tool for making such models. But the numerical problem going from LTE to realistic NLTE models has been solved only recently, and is the topic of this paper. We now have much more powerful tools to compute non-classical models, which account for very complex opacities, and solve the radiation transfer equations in more general environments, for example in expanding stellar atmospheres. These are the topics of other papers in this volume.Stellar-atmosphere modeling has made huge progress in recent years. This advance was achieved by the development of new numerical techniques for modelconstruction, and by the availability of atomic data for many species. And these achievements became possible only with an enormous increase of computing power. Model atmospheres assuming LTE have been highly refined by the inclusion of many more atomic and molecular opacity sources, and effective numerical techniques for LTE model-computation have been available for years. The progress is most remarkable in the field of NLTE model atmospheres. Replacement of the Saha-Boltzmann equations by atomic rate-equations requires radically different numerical-solution techniques, otherwise metal-opacities cannot be taken into account at all. Such techniques have been developed with great success during the last decade, inspired by important papers byCannon (1973) andScharmer (1981). The Accelerated Lambda-Iteration Method (ALI) is at the heart of this development. Combined with statistical representations of line-opacities, we are finally able to compute metal-line-blanketed NLTE models, including many millions of spectral lines, with a very high level of sophistication.\n\nIntroduction\n\nEssentially all our knowledge about the structure and evolution of stars, hence about galactic evolution in general, rests on the interpretation of their electromagnetic spectrum. Therefore quantitative analysis of stellar spectra is one of the most important tools of modern astrophysics. The formation of the observed spectrum is confined to the atmosphere, a very thin layer on top of the stellar core. Spectral analysis is performed by modeling the temperature-and pressure-stratification of the atmosphere and computing synthetic spectra which are then compared to observation. Fitting synthetic spectra to the data yields basic photospheric parameters: effective temperature, surface gravity, and chemical composition. Comparison with theoretical evolutionary calculations allows one to derive stellar parameters: mass, radius, and total luminosity.\n\nThe classical stellar-atmosphere problem considers radiation transfer through the outermost layers of a star into free space under three assumptions. First it is assumed that the atmosphere is in hydrostatic equilibrium, thus, the matter which interacts with photons is at rest. Second, the transfer of energy through the atmosphere is assumed to be done entirely by photons, i.e. heatconduction and convection are regarded as negligible (radiative equilibrium). But the effectiveness of photon-transfer depends on the opacity and emissivity of the matter, which are strongly state-and frequency-dependent quantities. They depend in detail on the occupation numbers of atomic/ionic levels, which, in turn, are determined by the local temperature and electron density, and the radiation field, whose nature is non-local. The occupation number of any atomic level is determined by a balance among radiative and collisional population and de-population processes (statistical equilibrium; our third assumption), i.e. the interaction of atoms with other particles and photons. Mathematically, the whole problem requires the solution of the radiation transfer equations simulta-In this paper we discuss the basic ideas behind the new numerical methods for NLTE modeling. We begin by presenting the classical model-atmosphere problem and introduce the basic equations. Then we focus on the ALI solutionmethod, and our numerical implementation of it. We then briefly describe the solution of the NLTE metal-line-blanketing problem.\n\n\nStatement of the problem and overview of the solution method\n\nWe assume plane-parallel geometry, which is well justified for most stars because the atmospheres are thin compared to the stellar radius. The only parameters needed to characterize uniquely such an atmosphere are the effective temperature (T eff ), a measure for the amount of energy transported through the atmosphere per unit area and time (see Eq. 15), the surface gravity (g), and the chemical composition. Generalization to spherical symmetry to account for extended (static) atmospheres mainly affects the radiation transfer equation and is straightforward (Mihalas & Hummer 1974). A spherical-symmetry variant of our code has been written by Nagel et al. (2001). To construct model atmospheres we must solve simultaneously a set of equations that is highly coupled and non-linear. Because of the coupling, no equation determines uniquely a single quantityall equations jointly determine each of the state parameters. Neverlethess, each of them is usually thought of as determining, more or less, a particular quantity. These equations are:\n\n• The radiation transfer equations which are solved for the angle-integrated mean intensities J i , i = 1, . . . , N F , on a pre-chosen frequency grid comprising N F points. The formal solution is given by J = ΛS, where S is the source function as defined later (Eq. 11), and Λ is the transport operator. Although Λ is written as an operator, one may think of it as a process of obtaining the mean intensity from the source function.\n\n• The hydrostatic-equilibrium equation which determines the total particle density N .\n\n• The radiative-equilibrium equation from which the temperature T follows.\n\n• The particle-conservation equation, determining the electron density n e .\n\n• The statistical-equilibrium equations, which are solved for the population densities n i , i = 1, . . . , N L of the atomic levels that are allowed to depart from LTE (NLTE levels).\n\n• The equation that defines a fictitious massive-particle density n H , which is introduced for convenience in the solution procedure.\n\nThis set of equations has to be solved at each point d of a grid comprising N D depth points. Thus we are looking for solution vectors ψ ′ d = (n 1 , . . . , n N L , n e , T, n H , N, J 1 , . . . , J N F ), d = 1, . . . , N D.\n\nThe Complete Linearization (CL) method (Auer & Mihalas 1969) solves this set by linearizing the equations with respect to all variables. The basic advantage of the ALI (or \"operator splitting\") method over CL is that it allows one to eliminate, at the outset, the explicit appearance of the mean intensities J i from the solution scheme by expressing these variables by the current, yet-tobe determined, occupation numbers and temperature. This is accomplished by an iteration procedure which may be written as (suppressing indices indicating depth and frequency dependence of variables):\nJ n = Λ ⋆ S n + (Λ − Λ ⋆ )S n−1 .(2)\nThis means that the actual mean intensity at any iteration step n is computed by applying an approximate lambda operator (ALO) Λ ⋆ on the actual source function S n plus a correction term that is computed from quantities known from the previous iteration step. This correction term includes the exact lambda operator Λ, which guarantees convergence to the exact solution of the radiation transfer problem J = ΛS. The use of Λ in Eq. 2 only indicates that a formal solution of the transfer equation is performed, but in our application the operator is not constructed explicitly. Instead we employ the Feautrier (1964) solution scheme (Mihalas 1978) or a short-characteristic method (Olson & Kunasz 1987) to solve the transfer equation as a differential equation. The resulting set of equations for the reduced solution vectors\nψ d = (n 1 , . . . , n N L , n e , T, n H , N ), d = 1, . . . , N D(3)\nis of course still non-linear. The solution is obtained by linearization and iteration which is performed either with a usual Newton-Raphson iteration or by other, much faster methods like the quasi-Newton or Kantorovich variants (Dreizler & Werner 1991, Hubeny & Lanz 1992) (see Sect. 5.4.). The first modelatmosphere calculations with the ALI method were performed by Werner (1986). Another advantage of the ALI method is that explicit depth coupling of the solution vectors Eq. 1 through the transfer equation can be avoided if one uses diagonal (i.e. local) ALOs. Then the solution vectors Eq. 3 are independent from each other and the solution procedure within one iteration step of Eq. 2 is much more straightforward. Depth coupling is provided by the correction which contains the exact solution of the transfer equation. The hydrostatic equation, which also gives explicit depth coupling, may be taken out of the set of equations and can -as experience shows -be solved in between two iteration steps of Eq. 2. Then full advantage of a local ALO can be taken. The linearized system may be written as:\nψ d = ψ 0 d + δψ d ,(4)\nwhere ψ 0 d is the current estimate for the solution vector at depth d and δψ d is the correction vector to be computed. Using a tri-diagonal Λ ⋆ operator the resulting system for δψ d is -like in the classical CL scheme -of block tri-diagonal form coupling each depth point d to its nearest neighbors d ± 1:\nγ d δψ d−1 + β d δψ d + α d δψ d+1 = c d .(5)\nThe quantities α, β, γ are (N N × N N ) matrices where N N is the total number of physical variables, i.e., N N = N L+4, and c d is the residual error in the equations. The solution is obtained by the standard Feautrier scheme. With starting values D 1 = β −1 1 (-α 1 ) and v 1 = β −1 1 c 1 we sweep from the outer boundary of the atmosphere inside and calculate at each depth:\nD d = (β d + γ d D d−1 ) −1 (−α d ) v d = (β d + γ d D d−1 ) −1 (c d − γ d v d−1 ).(6)\nAt the inner boundary we have D N D = 0 and by sweeping backwards to the surface we calculate the correction vectors, first δψ N D = v N D , and then successively δψ d = D d δψ d+1 + v d . As already mentioned, the system Eq. 5 breaks into N D independent equations δψ d = β −1 d c d (d = 1, . . . , N D) when a local Λ ⋆ operator is used. The additional numerical effort to set up the subdiagonal matrices and matrix multiplications in the tri-diagonal case is outweighed by the faster global convergence of the ALI cycle, accomplished by the explicit depth coupling in the linearization procedure (Werner 1989).\n\nThe principal advantage of the ALI over the CL method becomes clear at this point. Each matrix inversion in Eq. 6 requires (N L + 4) 3 operations whereas in the CL method (N L + N F + 4) 3 operations are needed. Since the number of frequency points N F is much larger than the number of levels N L, the matrix inversion in the CL approach is dominated by N F .\n\nRecent developments deal with the problem that the total number of atomic levels tractable in NLTE with the ALI method described so far is restricted to the order of 250, from experience with our model atmosphere code PRO2. This limit is a consequence of the non-linearity of the equations, and in order to overcome it, measures must be taken in order to achieve a linear system whose numerical solution is much more stable. Such a pre-conditioning procedure was first applied in the ALI context by Werner & Husfeld (1985) using the coresaturation method (Rybicki 1972). More advanced work achieves linearity by replacing the Λ operator with the Ψ operator (and by judiciously considering some populations as \"old\" and some as \"new\" ones within an ALI step) which is formally defined by writing:\nJ ν = Ψ ν η ν , i.e. Ψ ν ≡ Λ ν /χ ν ,(7)\nwhere the total opacity χ ν (as defined in Sect. 3.6.) is calculated from the previous ALI cycle. The advantage is that the emissivity η ν (Sect. 3.6.) is linear in the populations, whereas the source function S ν is not. Hence the new operator Ψ gives the solution of the transfer problem by acting on a linear function. This idea was proposed by Rybicki & Hummer (1991) who applied it to the line-formation problem, i.e. regarding the atmospheric structure as fixed. Hauschildt (1993) and Hauschildt & Baron (1999) generalized it to solve the full model-atmosphere problem. In addition, splitting the set of statistical equations, and solving it separately for each chemical element, means that now many hundreds of levels per species are tractable in NLTE (see Dreizler, this volume, for our experience). A very robust method and fast variant of the ALI method, the ALI/CL hybrid scheme, permits the linearization of the radiation field for selected frequencies (Hubeny & Lanz 1995), but it is not implemented in PRO2.\n\n\nBasic equations\n\n\nRadiation transfer\n\nAny numerical method requires a formal solution (i.e. atmospheric structure already given) of the radiation transfer problem. We disregard here polarization effects. (An implementation of an ALI technique to treat polarized radiation is discussed by Deetjen et al. in this volume.) Radiation transfer at any depth point is described by the transfer equation:\n±µ ∂I ν (±µ) ∂τ ν = S ν − I ν (±µ), µ ∈ [0, 1],(8)\nwritten separately for positive and negative µ (the angle-cosine between the direction of propagation and the outward-directed normal to the surface), i.e. for the inward and outward directional intensities I(µ) at frequency ν. Here τ ν is the optical depth (defined in terms of the column mass m used in the other structural equations by dτ ν = dmχ ν /ρ, where ρ is the mass density), and S ν is the local source function. Introducing the Feautrier intensity\nu νµ ≡ (I ν (µ) + I ν (−µ)) /2(9)\nwe obtain the second-order form (Mihalas 1978, p. 151):\nµ 2 ∂ 2 u νµ ∂τ 2 ν = u νµ − S ν , µ ∈ [0, 1].(10)\nWe may separate the Thomson emissivity term (scattering from free electrons, assumed coherent, with cross-section σ e ) from the source function so that\nS ν = S ′ ν + n e σ e J ν /χ ν ,(11)\nwhere S ′ ν is the ratio of thermal emissivity to total opacity as described in detail below (Sect. 3.6.): S ′ ν = η ν /χ ν . The mean intensity is the angular integral over the Feautrier intensity, hence the transfer equation becomes\nµ 2 ∂ 2 u νµ ∂τ 2 ν = u νµ − S ′ ν − n e σ e χ ν 1 0 u νµ dµ.(12)\nThomson scattering complicates the situation by its explicit angle coupling but the solution can be obtained with the standard Feautrier scheme. Assuming complete frequency-redistribution in spectral lines (Mihalas 1978, p. 29), no explicit frequency coupling occurs so that a parallel solution for all frequencies enables very efficient vectorization on the computer. The following boundary conditions are used for the transfer equation. At the inner boundary where the optical depth is at maximum, τ = τ max , we have\nµ ∂u νµ ∂τ ν τmax = I + νµ − u νµ (τ max ),(13)\nwhere we specify I + νµ = I ν (+µ, τ max ) from the diffusion approximation:\nI + νµ = B ν + 3µ χ ν ∂B ν ∂T H ∞ 0 1 χν ∂Bν ∂T dν .(14)\nHere B ν is the Planck function and H the nominal (frequency-integrated) Eddington flux:\nH = σ R T 4 eff /4π,(15)\nwith the Stefan-Boltzmann constant σ R . At the outer boundary we take τ ν = τ min = m 1 χ ν /2ρ, assuming that χ is a linear function of m for m < m 1 . Since τ min = 0, it is not exactly valid to assume no incident radiation at the stellar surface. Instead we specify I − νµ = I ν (−µ, τ min ) after Scharmer & Nordlund (1982):\nI − νµ = S ν (τ min )[1 − exp(−τ min /µ)](16)\nwhich follows from Eq. 8 assuming S(τ ) = S(τ min ) for τ < τ min . Then we get:\nµ ∂u νµ ∂τ ν τ min = u νµ (τ min ) − I − νµ .(17)\nThe boundary conditions are discretized performing Taylor expansions which yield second-order accuracy (Mihalas 1978, p. 155).\n\n\nStatistical equilibrium\n\nThe statistical equilibrium equations are set up according to Mihalas (1978, p. 127). The number of atomic levels, ionization stages, and chemical species, as well as all radiative and collisional transitions are taken from the input model atom supplied by the user. Ionization into excited states of the next ionization stage is taken into account. Dielectronic recombination and autoionization processes can also be included in the model atom.\n\nRate equations As usual, the atomic energy levels are ordered sequentially by increasing excitation energy, starting with the lowest ionization stage. Then for each atomic level i of any ionization stage of any element, the rate equation describes the equilibrium of rates into and rates out of this level:\nn i i =j P ij − j =i n j P ji = 0.(18)\nThe rate coefficients P ij have radiative and collisional components: P ij = R ij + C ij . Radiative upward and downward rates are respectively given by:\nR ij = 4π ∞ 0 σ ij (ν) hν J ν dν,(19)R ji = n i n j ⋆ 4π ∞ 0 σ ij (ν) hν 2hν 3 c 2 + J ν e −hν/kT dν.(20)\n8\n\n\nWerner, Deetjen, Dreizler, Nagel, Rauch, & Schuh\n\nPhoton cross-sections are denoted by σ ij (ν). (n i /n j ) ⋆ is the Boltzmann LTE population ratio in the case of line transitions: g i /g j exp(−hν ij /kT ), where the g i,j are the statistical weights. The LTE population number of a particular level is defined relative to the ground state of the next ion, so that in the case of recombination from a ground state n + 1 we have by definition (n i /n j ) ⋆ = n e φ i (T ) with the Saha-Boltzmann factor\nφ i (T ) = 2.07 · 10 −16 g i g + 1 T −3/2 e hν i /kT ,(21)\nwhere hν i is the ionization potential of the level i. Care must be taken in the case of recombination from an excited level into the next lower ion. Then (n i /n j ) ⋆ = n e φ i · φ + 1 /φ j . Dielectronic recombination is included following Mihalas & Hummer (1973). Assuming now that j is a ground state of ion k, then the recombination rate into level i of ion k − 1 via an autoionization level c (with ionization potential hν c , having a negative value when lying above the ionization limit) is:\nR ji = 8π 2 e 2 mc 3 n e φ i f ic e h(νc−ν i )/kT ν 2 c 1 + c 2 2hν 3 cJ .(22)\nThe reverse process, the autoionization rate, is given by:\nR ij = 4π 2 e 2 hmc 1 ν c f icJ .(23)\nThe oscillator strength for the stabilizing transition (i.e. transition i → c) is denoted by f ic , andJ is the mean intensity averaged over the line profile. Our program simply takes J ν from the continuum frequency point closest to the transition frequency, which is reasonable because the autoionization line profiles are extremely broad. The population of autoionization levels is assumed to be in LTE and therefore such levels do not appear explicitly in the rate equations.\n\nThe computation of collisional rates is generally dependent on the specific ion or even transition. Several options, covering the most important cases, may be chosen by the user.\n\nAbundance-definition equation The rate equation for the highest level of a given chemical species is redundant. It is replaced by the abundance definition equation. Summation over all levels usually includes not only NLTE levels but also levels which are treated in LTE, according to the specification in the model atom. Denoting the number of ionization stages of species k with N ION (k), the number of NLTE and LTE levels per ion with N L(l) and LT E(l), respectively, we can write:\nN ION (k) l=1 N L(l) i=1 n kli + LT E(l) i=1 n ⋆ kli = y k (N − n e ).(24)\ny k is the number fraction of element k.\n\n\nModel Photospheres with Accelerated Lambda Iteration\n\n\n9\n\nCharge conservation We close the system of statistical equilibrium equations by demanding charge conservation. We denote the total number of chemical species with NATOM, the charge of ion l with q(l) (in units of the electron charge) and write:\nNATOM k=1 N ION (k) l=1 q(l) N L(l) i=1 n kli + LT E(l) i=1 n ⋆ kli = n e .(25)\nComplete statistical equilibrium equations We introduce a vector comprising the occupation numbers of all NLTE levels, n = (n 1 , . . . , n N L ). Then the statistical equilibrium equation is written as:\nAn = b.(26)\nThe gross structure of the rate matrix A is of block matrix form, because transitions between levels occur within one ionization stage or to the ground state of the next ion. The structure is complicated by ionizations into excited levels and by the abundance definition and charge conservation equations which give additional non-zero elements in the corresponding lines of A.\n\n\nRadiative equilibrium\n\nRadiative equilibrium can be enforced by adjusting the temperature stratification either during the linearization procedure or in between ALI iterations. In the former case a linear combination of two different formulations is used and in the latter case the classical Unsöld-Lucy temperature correction procedure (Lucy 1964) is utilized. The latter is particularly interesting, because it allows one to exploit the block form of the rate-coefficient matrix. This fact allows an economic block-by-block solution followed by a subsequent Unsöld-Lucy temperature correction step. On the other hand, however, this correction procedure may decelerate the global convergence behavior of the ALI iteration. The two forms of expressing the radiative equilibrium condition follow from the requirement that the energy emitted by a volume element per unit time is equal to the absorbed energy per unit time (integral form):\n∞ 0 χ ν (S ν − J ν ) dν = 0,(27)\nwhere scattering terms in χ ν and S ν cancel out. In principle, this formulation is equivalent to demanding flux-constancy throughout the atmosphere (differential form)\n∞ 0 ∂ ∂τ ν (f ν J ν ) dν − H = 0,(28)\nwhere H (Eq. 15) is the nominal flux. Here f ν is the variable Eddington factor, defined as\nf ν = 1 0 µ 2 u νµ dµ 1 0 u νµ dµ,(29)\nwhich is computed from the Feautrier intensity u νµ (Eq. 9) after the formal solution. As discussed e.g. in Hubeny (1988) the differential form is more accurate at large depths, while the integral form behaves numerically superior at small depths. Instead of arbitrarily selecting that depth in the atmosphere where we switch from one formulation to the other, we use a linear combination of both constraint equations, which guarantees a smooth transition with depth, based on physical grounds (Hubeny & Lanz 1992, Hamann 1994:\n1 κ J ∞ 0 χ ν (S ν − J ν ) dν +Λ ⋆ J ∞ 0 ∂ ∂τ ν (f ν J ν ) dν −Λ ⋆ J H − F 0 = 0.(30)\nFor details on this equation and on our implementation of the Unsöld-Lucy procedure see Dreizler (this volume). We note that explicit depth coupling is introduced by the differential form Eq. 28 through the derivative ∂/∂τ ν even if a purely local Λ ⋆ operator is used. Therefore the linearization procedure can no longer be performed independently at each depth point and the question at which boundary to start becomes relevant. Numerical experience shows that it is essential to start at the outer boundary and proceed inwards. If a tri-diagonal operator is used, nearest neighbor depth coupling is introduced anyhow.\n\n\nHydrostatic equilibrium\n\nWe write the equation for hydrostatic equilibrium as Mihalas (1978, p. 170):\ndP dm = g.(31)\nP is the total pressure comprising gas, radiation and turbulent pressures, so that:\nd dm N kT + 4π c ∞ 0 f ν J ν dν + 1 2 ρv 2 turb = g(32)\nwith Boltzmann's constant k and the turbulent velocity v turb . The hydrostatic equation may either be solved simultaneously with all other equations or separately between iterations. The overall convergence behavior is usually the same in both cases. If taken into the linearization scheme, and a local Λ ⋆ operator is used, then, as in the case of the radiative equilibrium equation, explicit depthcoupling enters via the depth derivative d/dm. Again, solution of the linearized equations must proceed inwards starting at the outer boundary. The starting value at the first depth point (subscript d = 1) is:\nN 1 kT 1 + 1 2 ρ 1 v 2 turb (m 1 ) = m 1 g − 4π c ∞ 0 χ 1,ν ρ 1 h ν J ν,k dν ,(33)\nwhere h ν is the variable Eddington factor denoting the ratio of H ν /J ν at the surface, kept fixed during linearization. We are also able to account for element separation resulting from pressure diffusion, which is the process that governs the spectroscopic appearance e.g. of white dwarfs. The numerical method and some results are described by Dreizler & Wolff (1999) and Schuh et al. (this volume).\n\n\nParticle-conservation and fictitious massive-particle density\n\nThe total particle density N is the sum of electron density plus the populations of all atomic/ionic states, in all LTE and NLTE levels:\nN = n e + NATOM k=1 N ION (k) l=1 N L(l) i=1 n kli + LT E(l) i=1 n ⋆ kli .(34)\nThe fictitious massive-particle density n H is introduced for notational convenience. It is defined by:\nn H = NATOM k=1 m k N ION (k) l=1 N L(l) i=1 n kli + LT E(l) i=1 n ⋆ kli .(35)\nThe mass of a chemical species in AMU is denoted by m k . Introducing the mass of a hydrogen atom m H , we may write the material density simply as\nρ = n H m H .(36)\n\nOpacity and emissivity\n\nThermal opacity and emissivity are made up by radiative bound-bound, boundfree and free-free transitions. For each species we compute and sum:\nκ ν = N ION l=1 N L(l) i=1 N L(l) j>i σ li→lj (ν) n li − n lj g li g lj e −h(ν−ν ij )/kT(37)+ N L(l) i=1 N L(l+1) j>i σ li→l+1,k (ν) n li − n ⋆ li e −hν/kT + n e σ kk (l, ν) 1 − e −hν/kT N L(l+1) i=1 n l+1,i + LT E(l+1) i=1 n ⋆ l+1,i \nwhere the total opacity includes Thomson scattering, i.e. χ ν = κ ν + n e σ e , and\nη ν 2hν 3 /c 2 = N ION l=1 N L(l) i=1 N L(l) j>i σ li→lj (ν)n lj g li g lj e −h(ν−ν ij )/kT(38)+ N L(l) i=1 N L(l+1) j>i σ li→l+1,k (ν)n ⋆ li e −hν/kT + n e σ kk (l, ν)e −hν/kT N L(l+1) i=1 n l+1,i + LT E(l+1) i=1 n ⋆ l+1,i .\nσ li→l+1,k (ν) denotes the cross-section for photoionization from level i of ion l into level k of ion l + 1. The double summation over the bound-free continua takes into account the possibility that a particular level may be ionized into more than one level of the next high ion. Again, note the definition of the LTE population number n ⋆ li in this case, which depends on the level (l + 1, k) of the parent ion:\nn ⋆ li = n l+1,k n e φ li φ l+1,1 φ l+1,k .(39)\nNote also that the concept of LTE levels (whose occupation numbers enter, e.g. the number-and charge-conservation equations) in the atomic models of complex ions is therefore not unambiguous. Our code always assumes that LTE levels in the model atoms are populated in LTE with respect to the ground state of the upper ion. The source function used for the approximate radiation transfer is η ν /κ ν , thus it excludes Thomson scattering. For the exact formal solution of course, the total opacity χ ν in the expression Eq. 11 includes the Thomson term (n e σ e ).\n\n\nAtomic level dissolution by plasma perturbations\n\nAs high-lying atomic levels are strongly perturbed by other charged particles in the plasma they are broadened and finally dissolved. This effect is observable by line merging at series limits and must be accounted for in line profile analyses. Moreover, line-overlap couples the radiation field in many lines and flux-blocking can strongly affect the global atmospheric structure. Numerically, we treat the level dissolution in terms of occupation probabilities, which for LTE plasmas can be defined as the ratio of the level populations to those in absence of perturbations. A phenomenological theory for these quantities was given in Hummer & Mihalas (1988). The non-trivial generalization to NLTE plasmas was made by Hubeny et al. (1994). In practice an individual occupation-probability factor (depending on T, n e , and principal quantum number), is applied to each atomic level, which describes the probability that the level is dissolved. Furthermore, the rate equations Eq. 18 must be generalized in a unique and unambiguous manner. For details see Hubeny et al. (1994).\n\n\nThe Accelerated Lambda Iteration (ALI)\n\nIn all constraint equations described above the mean intensities J ν are replaced by the approximate radiation field Eq. 2 in order to eliminate these variables from the solution vector Eq. 1. In principle the ALO may be of arbitrary form as long as the iteration procedure converges. In practice, however, an optimum choice is desired in order to achieve convergence with a minimum amount of iteration steps. The history of the ALOs is interesting, and was summarized in detail by Hubeny (1992). Of utmost importance were two papers by Olson and collaborators (Olson et al. 1986, Olson & Kunasz 1987 who overcame the major drawback of early ALOs, namely the occurrence of free parameters controlling the convergence process, and thus found the optimum choice of ALOs. Our model atmosphere program permits the use of either a diagonal or a tri-diagonal ALO, both are set up following Olson & Kunasz (1987).\n\n\nDiagonal (local) lambda operators\n\nIn this case the mean intensity J d at a particular depth d in the current iteration step is computed solely from the local source function S d and a correction term ∆J d , the latter depending on the source functions (at all depths) from the previous iteration. Dropping the iteration count and introducing indices denoting depth points we can rewrite Eq. 2 as:\nJ d = Λ ⋆ d,d S d + ∆J d .(40)\nIn discrete form we can think of Λ ⋆ as a matrix acting on a vector whose elements comprise the source functions of all depths. Then Λ ⋆ d,d is the diagonal element of the Λ ⋆ matrix corresponding to depth point d. Writing Λ ⋆ d,d ≡ B d (for numerical computation see Eq. 44 below) we have a purely local expression for the mean intensity:\nJ d = B d S d + ∆J d .(41)\n\nTridiagonal (non-local) lambda operators\n\nMuch better convergence is obtained if the mean intensity is computed not only from the local source function, but also from the source function of the neighboring depths points. Then the matrix representation of Λ ⋆ is of tri-diagonal form and we may write:\nJ d = C d−1 S d−1 + B d S d + A d+1 S d+1 + ∆J d ,(42)\nwhere C d−1 and A d+1 represent the upper and lower subdiagonal elements of Λ ⋆ , and S d±1 the source functions at the adjacent depths. By analogy the correction term becomes:\n∆J d = Λ d,d ′ S d ′ − (C d−1 S d−1 + B d S d + A d+1 S d+1 ).(43)\nAgain all quantities for the computation of ∆J d are known from the previous iteration, and the first term gives the exact formal solution of the transfer equation. We emphasize again that the source functions in Eq. 42 are to be computed from the correct occupation numbers and temperature, all of which are still unknown. We therefore have a non-linear set of equations for these quantities, which must be solved by either Newton-Raphson iteration or other techniques, resulting in the solution of a tri-diagonal linear equation of the form Eq. 5. As was shown in Olson et al. (1986) the elements of the optimum Λ ⋆ matrix are given by the corresponding elements of the exact Λ matrix. The diagonal and subdiagonal elements are computed following Olson & Kunasz (1987):\nA d+1 = 1 0 e −∆τ d e −∆τ d−1 − 1 ∆τ d−1 − e −∆τ d − 1 ∆τ d dµ 2 , 1 − B d = 1 0 1 − e −∆τ d−1 ∆τ d−1 + 1 − e −∆τ d ∆τ d dµ 2 , C d−1 = 1 0 e −∆τ d−1 e −∆τ d − 1 ∆τ d − e −∆τ d−1 − 1 ∆τ d−1 dµ 2 ,(44)\nwith ∆τ d−1 ≡ (τ d − τ d−1 )/µ. At large optical depths with increasing ∆τ steps (the depth grid is roughly equidistant in log τ ) the subdiagonals A d+1 and C d−1 vanish, and the diagonal B d approaches unity, reflecting the fact that the radiation field is more and more determined by local properties of the matter. At very small optical depths all elements of Λ ⋆ vanish, reflecting the non-localness of the radiation field at these depths.\n\n\nAcceleration of convergence\n\nOur code takes advantage of an acceleration scheme to speed up convergence of the iteration cycle Eq. 2. We implemented the scheme originally proposed by Ng (1974) and later by Auer (1987). It extrapolates the correction vector δψ d from the previous three iterations. From our experience the extrapolation often yields over-corrections, resulting in alternating convergence or even divergence. In contrast, use of a tri-diagonal ALO usually results in satisfactorily fast convergence, so that the acceleration scheme is rarely used.\n\n\nSolution of the non-linear equations by iteration\n\nAt each depth the complete set of non-linear equations Eq. 2 for a single iteration step comprises the equations for statistical, radiative, and hydrostatic equilibrium, and the particle-conservation equation. For the numerical solution we introduce discrete depth and frequency grids. The equations are then linearized and solved by a suitable iterative scheme. Explicit angle-dependence of the radiation field is not required here and is consequently eliminated by the use of variable Eddington factors. Angle dependence is considered only in the formal solution of the transfer equation. Our program requires an input modelatmosphere structure as a starting approximation, together with an atomic data file, and a frequency grid. Depth and frequency grids are therefore set up in advance by separate programs.\n\n\nDiscretization\n\nAfter a depth-grid is chosen, we start by computing a gray LTE continuum model using the Unsöld-Lucy temperature-correction procedure. Depth points (typical number is 90) are chosen in equidistant steps on a logarithmic (Rosseland) optical depth scale. The converged LTE model (temperature and density structure, given on a column mass depth scale) is written to a file that is read by PRO2. The NLTE code uses the column mass as an independent depth variable. The frequency grid is determined by the atomic-data input file. Frequency points are set blue-and red-ward of each absorption edge, and for each spectral line. Gaps are filled up by setting continuum points. Finally, the quadrature weights are computed. Frequency integrals appearing e.g. in Eq. 30 are replaced by quadrature sums, and depth derivatives by difference quotients.\n\n\nLinearization\n\nAll variables x are replaced by x → x+δx where δx denotes a small perturbation of x. Terms not linear in these perturbations are neglected. The perturbations are expressed by perturbations of the basic variables:\nδx = ∂x ∂T δT + ∂x ∂n e δn e + ∂x ∂N δN + ∂x ∂n H δn H + N L l=1 ∂x ∂n l δn l .(45)\nAs an illustrative example we linearize the equation for radiative equilibrium.\n\nMost other linearized equations may be found in Werner (1986). Assigning two indices (d for depth and i for frequency of a grid with NF points) to the variables and denoting the quadrature weights with w i Eq. 30 becomes:\nN F i=1 w i ( χ dī κ J [δS di − δJ di ] + δχ di [S di − J di ]) +Λ ⋆ J N F i=1 w i ∆τ i (δJ di f di − δJ d−1,i f d−1,i ) = F 0 +Λ ⋆ J H − N F i=1 w i χ dī κ J (S di − J di ) −Λ ⋆ J N F i=1 w i ∆τ i (f di J di − f d−1,i J d−1,i ). (46)\nNote that we do not linearize ∆τ i . Because of this, convergence properties may deteriorate significantly in some cases. Perturbations δS di , δχ di are expressed by Eq. 45, and the perturbation of the mean intensity J di is, according to Eq. 42, given through the perturbations of the source function at the current and the two adjacent depths:\nδJ di = C d−1,i δS d−1,i + B di δS di + A d+1,i δS d+1,i ,(47)\nwhere A, B, C are the Λ matrix elements from Eq. 44. The δJ d−1,i contain the term C d−2,i δS d−2,i which is neglected because we want to account only for nearest neighbor coupling. We write δS di with the help of Eq. 45 and observe that for any variable z:\n∂S di ∂z = 1 χ di ∂η di ∂z − S di ∂χ di ∂z .(48)\nDerivatives of opacity and emissivity with respect to temperature, electron and population densities are computed from analytical expressions (see e.g. Mihalas et al. 1975, Werner 1987. We finally get from Eq. 46:\nδT d−1,i N F i − w ī κ J ∂S d−1,i ∂T χ di C d−1,i +Λ ⋆ J N F i w i ∆τ i (f di C d−1,i − f d−1,i B d−1,i ) ∂S d−1,i ∂T + δT d N F i w ī κ J ∂S di ∂T χ di (1 − B di ) + ∂χ di ∂T (S di − J di ) +Λ ⋆ J N F i w i ∆τ i (f di B di − f d−1,i A di ) ∂S di ∂T + δT d+1,i N F i − w ī κ J ∂S d+1,i ∂T χ di A d+1,i +Λ ⋆ J N F i w i ∆τ i (f di A d+1,i − f d−1,i B d+1,i ) ∂S d+1,i ∂T + δn e d−1,i {· · ·} + δn e d,i {· · ·} + δn e d+1,i {· · ·} + N L l=1 δn l d−1,i {· · ·} + N L l=1 δn l d,i {· · ·} + N L l=1 δn l d+1,i {· · ·} = r.h.s.(49)\nThe curly brackets {· · ·} denote terms that are similar to those multiplied by perturbations of the temperature. Instead of partial derivatives with respect to T , they contain derivatives with respect to n e and the populations n l . They all represent coefficients of the matrices α, β, γ in Eq. 5.\n\n\nNewton-Raphson iteration\n\nAs described in Sect. 2. the linearized equations have a tri-diagonal block-matrix form, see Eq. 5. Inversion of the grand matrix (≡ T sized (N N · N D) × (N N · N D), i.e. about 10 4 × 10 4 in typical applications) is performed with a block-Gaussian elimination scheme, which means that our iteration of the non-linear equations is a multi-dimensional Newton-Raphson method. The problem is structurally simplified when explicit-depth coupling is avoided by the use of a local ALO, however, the numerical effort is not reduced much, because in both cases the main effort lies with the inversion of matrices sized N N × N N . The Newton-Raphson iteration involves two numerically expensive steps, first setting up the Jacobian (comprising α, β, γ) and then inverting it. Additionally, the matrix inversions in Eq. 6 limit their size to about N N = 250 because otherwise numerical accuracy is lost. Two variants recently introduced in stellar atmosphere calculations are able to improve both numerical accuracy and, best of all, computational speed.\n\n\nAlternative fast solution techniques for non-linear equations: Broyden-and Kantorovich-variants\n\nBroyden's (1965) variant belongs to the family of quasi-Newton methods; it was first used in model-atmosphere calculations in Dreizler & Werner (1991), Hamann et al. 1991, Koesterke et al. (1992. It avoids the repeated set-up of the Jacobian by the use of an update formula. In addition, it also gives an update formula for the inverse Jacobian. In the case of a local ALO the solution of the linearized system at any depth is:\nδψ k = β −1 k c k .(50)\nLet β −1 k be the k-th iterate of the inverse Jacobian, then an update can be found from:\nβ −1 k+1 = β −1 k + (s k − β −1 k y k )⊗(s T k β −1 k ) s T k β −1 k y k ,(51)\nwhere ⊗ denotes the dyadic product, and where we have defined:\n\ns k ≡ δψ k solution vector of preceding linearization, y k ≡ c k+1 − c k difference of actual and preceding residuum.\n\nThe convergence rate is super-linear, i.e. slower than the quadratic rate of the Newton-Raphson method; but this is more than compensated by the tremendous speed-up for a single iteration step. It is not always necessary to begin the iteration with the calculation of an exact Jacobian and its inversion. Experience shows that in an advanced stage of the overall (ALI-) iteration Eq. 2 (i.e. when corrections become small, of the order 1%) we can start the linearization cycle Eq. 51 by using the inverse Jacobian from the previous overall iteration. Computational speed-up is extreme in this case, however, it requires storage of the Jacobians of all depths. More difficult is its application to the tri-diagonal ALO case. Here we have to update the grand matrix T which, as already mentioned, is of block tri-diagonal form. We cannot update its inverse, because it is never computed explicitly. Furthermore we need an update formula that preserves the block tridiagonal form which is a prerequisite for its inversion by the Feautrier scheme Eq. 6. Such a formula was found by Schubert (1970):\nT k+1 = T k + (y k − T k s k ) ⊗s T k s T ks k ,(52)\nwheres k ≡ Zs k with the structure matrix Z as defined by:\nZ ij = 1 if T ij = 0 0 if T ij = 0.\nThe vectors s k and y k are defined as above but now they span quantities over all depth instead of a single depth point. With this formula we obtain new submatrices α, β, γ and c with which the Feautrier scheme Eq. 6 is solved again. This procedure saves the computation of derivatives. Another feature realized in our program also saves the repeated inversion of q ≡ (β d + γ d D d−1 ) by updating its inverse with the Broyden formula Eq. 51. Similar to the diagonal ALO case it is also possible to pass starting matrices from one overall iteration Eq. 2 to the next for the update of T and the matrix q −1 . In both cases the user specifies two threshold values for the maximum relative correction in δψ which cause the program to switch from Newton-Raphson to Broyden stages 1 and 2. During stage 1, each new overall cycle Eq. 2 is started with an exact calculation and inversion of all matrices involved, and in stage 2 these matrices are passed through each iteration. Another variant, the Kantorovich (1949) method has been introduced into model-atmosphere calculations (Hubeny & Lanz 1992). It is more simple and straightforward to implement. This method simply keeps the Jacobian fixed during the linearization cycle; it is surprisingly stable. In fact it turns out to be even more stable (i.e. it can be utilized in an earlier stage of iteration) than the Broyden method in the tri-diagonal ALO case. The user of PRO2 may choose this variant in two stages in analogy to the Broyden variant. It was found that in stage 2 it is necessary to update the Jacobian every 5 to 10 overall iterations in order to prevent divergence.\n\n\nNLTE metal-line blanketing\n\nDespite the increase in capacity for solving NLTE model-atmosphere problems given by the ALI method combined with pre-conditioning techniques, the blanketing by millions of lines from the iron-group elements arising from transitions among some 10 5 levels can be attacked only with statistical methods. These were introduced into NLTE model-atmosphere work by Anderson (1989Anderson ( , 1991. At the outset, model atoms are constructed by combining many thousands of levels into a relatively small number of superlevels which can be treated by ALI (or other) methods. Then, in order to reduce the computational effort, two approaches were developed which vastly decrease the number of frequency points (hence the number of transfer equations to be solved) needed to describe properly the complex frequency dependence of the opacity. These two approaches have their roots in LTE modeling techniques, where statistical methods are applied for the same reason in the treatment of opacity: The Opacity Distribution Function (ODF) and Opacity Sampling (OS) approaches. Both are based on the fact that the opacity (in the LTE approximation) is a function of two only local thermodynamic quantities. Roughly speaking, each opacity source can be written in terms of a population density and a photon cross-section for the respective radiative transition: κ ν ∼ n l σ lu (ν). In LTE the population follows from the Saha-Boltzmann equations, hence n l = n l (n e , T ). The OS and ODF methods use such pre-tabulated (on a very fine frequency mesh) κ ν (n e , T ) during the model atmosphere calculations. The NLTE situation is more complicated, because pre-tabulation of opacities is not useful. The occupation numbers at any depth now also depend explicitly on the radiation field (via the rate equations which replace the TE Saha-Boltzmann statistics) and thus on the populations in every other depth of the atmosphere. As a consequence, the OS and ODF methods are not applied to opacity tabulations, but to tabulations of the photon cross-sections σ(ν). These do depend on local quantities only, e.g. line broadening by Stark-and Doppler-effects is calculated from T and n e . In the NLTE case cross-sections take over the role which the opacity played in the LTE case. So, strictly speaking, the designation OS and ODF is not quite correct in the NLTE context.\n\nThe strategy in our code is the following. Before any model atmosphere calculation is started, the atomic data are prepared by constructing superlevels, and the cross-sections for superlines. Then these cross-sections are either sampled on a coarse frequency grid or ODFs are constructed. These data are put into the model atom which is read by the code. The code does not know if OS or ODFs are used, i.e. it is written to be independent of these approaches.\n\nFurther details about our implementation are described in Dreizler & Werner (1993), Haas et al. (1996), Werner & Dreizler (1999), and by Rauch & Deetjen in this volume.\n\n\nSummary\n\nWe have presented in detail our technique for numerical solution of the classical model-atmosphere problem. The construction of metal-line-blanketed models in hydrostatic and radiative equilibrium under NLTE conditions was the last and\n longest-standing problem of classical model atmosphere theory. It has finally been solved with a high degree of sophistication. The essential milestones for this development, starting from the pioneering work ofAuer & Mihalas (1969)are:• Introduction of Accelerated Lambda Iteration (ALI, or \"operator splitting\") methods, based upon early work byCannon (1973)andScharmer (1981). The first ALI model atmospheres were constructed byWerner (1986).• Introduction of statistical approaches to treat the iron-group elements in NLTE byAnderson (1989Anderson ( , 1991.• Linear formulation of the statistical-equilibrium equations(Rybicki & Hummer 1991, Hauschildt 1993).• Computation of atomic data byKurucz (1991), by the Opacity Project(Seaton et al. 1994) and subsequent improvements, and by the Iron Project(Hummer et al. 1993).\n. L S Anderson, ApJ. 339558Anderson, L. S. 1989, ApJ, 339, 558\n\nL S Anderson, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D. G. HummerDordrechtKluwer34129Anderson, L. S. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D. G. Hummer, (Dordrecht: Kluwer), 29\n\n. L H Auer, Numerical Radiative Transfer, ed. W. KalkofenCambridge University Press101Auer, L. H. 1987, in Numerical Radiative Transfer, ed. W. Kalkofen, Cambridge University Press, 101\n\n. L H Auer, D Mihalas, ApJ. 158641Auer, L. H., & Mihalas, D. 1969, ApJ, 158, 641\n\n. C G Broyden, Math. Comp. 19577Broyden, C. G. 1965, Math. Comp., 19, 577\n\n. C J Cannon, ApJ. 185621Cannon, C. J. 1973, ApJ, 185, 621\n\nS Dreizler, K Werner, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer341155Dreizler, S., & Werner, K. 1991, in Stellar Atmospheres: Beyond Classical Mod- els, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 155\n\n. S Dreizler, K Werner, A&A. 278199Dreizler, S., & Werner, K. 1993, A&A, 278, 199\n\n. S Dreizler, B Wolff, A&A. 348189Dreizler, S., & Wolff, B. 1999, A&A, 348, 189\n\n. P Feautrier, C. R. Acad. Sci. 2583189Feautrier, P. 1964, C. R. Acad. Sci. Paris, 258, 3189\n\n. S Haas, S Dreizler, U Heber, S Jeffery, K Werner, A&A. 311669Haas, S., Dreizler, S., Heber, U., Jeffery, S., & Werner, K. 1996, A&A, 311, 669\n\nW.-R Hamann, W.-R Hamann, L Koesterke, U Wessolowski, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer34169Hamann, W.-R. 1994, private communication Hamann, W.-R., Koesterke, L., & Wessolowski, U. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 69\n\n. P H Hauschildt, JQSRT. 50301Hauschildt, P. H. 1993, JQSRT, 50, 301\n\nP H Hauschildt, E Baron, Computational Astrophysics. H. Riffert & K. Werner10941Hauschildt, P. H., & Baron, E. 1999, in Computational Astrophysics, ed. H. Rif- fert & K. Werner, Journal of Computational and Applied Mathematics, 109, 41\n\n. Deetjen Werner, Dreizler, Nagel, Rauch, I Schuh Hubeny, Comput. Phys. Commun. 52103Werner, Deetjen, Dreizler, Nagel, Rauch, & Schuh Hubeny, I. 1988, Comput. Phys. Commun., 52, 103\n\nI Hubeny, Atmospheres of Early-Type Stars. U. Heber & C.S. JefferyBerlinSpringer401377Hubeny, I. 1992, in Atmospheres of Early-Type Stars, ed. U. Heber & C.S. Jeffery, Lecture Notes in Physics Vol. 401, (Berlin: Springer), 377\n\n. I Hubeny, T Lanz, A&A. 262501Hubeny, I., & Lanz, T. 1992, A&A, 262, 501\n\n. I Hubeny, T Lanz, ApJ. 439875Hubeny, I., & Lanz, T. 1995, ApJ, 439, 875\n\n. I Hubeny, D G Hummer, T Lanz, A&A. 282157Hubeny, I., Hummer, D. G., & Lanz, T. 1994, A&A, 282, 157\n\n. D G Hummer, D Mihalas, ApJ. 331794Hummer, D. G., & Mihalas, D. 1988, ApJ, 331, 794\n\n. D G Hummer, K A Berrington, W Eissner, A K Pradhan, H E Saraph, J A Tully, A&A. 279298Hummer, D. G., Berrington, K. A., Eissner, W., Pradhan, A. K., Saraph, H. E., & Tully, J. A. 1993, A&A, 279, 298\n\n. L V Kantorovich, Trudy Mat. Inst. Steklova, XXVIII. 104Kantorovich L. V. 1949, Trudy Mat. Inst. Steklova, XXVIII, 104\n\n. L Koesterke, W.-R Hamann, P Kosmol, A&A. 255490Koesterke, L., Hamann, W.-R., & Kosmol, P. 1992, A&A, 255, 490\n\nR L Kurucz, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer341441Kurucz, R. L. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 441\n\nL B Lucy, First Harvard Smithsonian Conference on Stellar Atmospheres. Cambridge, Massachusetts, 93Lucy, L. B. 1964, in First Harvard Smithsonian Conference on Stellar Atmo- spheres, Smithsonian Astrophysical Observatory Special Report No. 167, Cambridge, Massachusetts, 93\n\nD Mihalas, D Mihalas, D G Hummer, Stellar Atmospheres. San FranciscoFreeman1798272nd editionMihalas, D. 1978, Stellar Atmospheres, 2nd edition, (San Francisco: Freeman) Mihalas, D., & Hummer, D. G. 1973, ApJ, 179, 827\n\n. D Mihalas, D G Hummer, ApJS. 265343Mihalas, D., & Hummer, D. G. 1974, ApJS, 265, 343\n\nD Mihalas, L H Auer, J N Heasley, NCAR Technical Note STR-104. BoulderNational Center for Atmospheric ResearchMihalas, D., Auer, L. H., & Heasley, J. N. 1975, NCAR Technical Note STR-104, (Boulder: National Center for Atmospheric Research)\n\n. T Nagel, S Dreizler, K Werner, ASP Conf. Ser. White Dwarfs, ed. J. L. Provencal, H. L. Shipman, J. MacDonald & S. Goodchild22686ASPNagel, T., Dreizler, S., & Werner, K. 2001, in White Dwarfs, ed. J. L. Provencal, H. L. Shipman, J. MacDonald & S. Goodchild, ASP Conf. Ser. Vol. 226, (San Francisco: ASP), 86\n\n. K C Ng, J. Chem. Phys. 612680Ng, K. C. 1974, J. Chem. Phys., 61, 2680\n\n. G L Olson, P B Kunasz, JQSRT. 38325Olson, G. L., & Kunasz, P. B. 1987, JQSRT, 38, 325\n\n. G L Olson, L H Auer, J R Buchler, JQSRT. 35431Olson, G. L., Auer, L. H., & Buchler, J. R. 1986, JQSRT, 35, 431\n\n1972 in Line Formation in the Presence of Magnetic Fields. G B Rybicki, G ; R, L Athay, & G House, Jr Newkirk, Boulder: High Altitude Observatory. 145Rybicki, G. B. 1972 in Line Formation in the Presence of Magnetic Fields, ed. R. G. Athay, L. House & G. Newkirk, Jr., (Boulder: High Altitude Observatory), 145\n\n. G B Rybicki, D G Hummer, A&A. 245171Rybicki, G. B., & Hummer, D. G. 1991, A&A, 245, 171\n\n. G B Scharmer, ApJ. 249720Scharmer, G. B. 1981, ApJ, 249, 720\n\n. G B Scharmer, A Nordlund, StockholmScharmer, G. B., & Nordlund, A. 1982, Stockholm Observatory Report 19\n\n. L K Schubert, Math. Comp. 2427Schubert, L. K. 1970, Math. Comp., 24, 27\n\n. M J Seaton, Y Yan, D Mihalas, A K Pradhan, MNRAS. 266805Seaton, M. J., Yan, Y., Mihalas, D.,& Pradhan, A. K. 1994, MNRAS, 266, 805\n\n. K Werner, A&A. 161177Werner, K. 1986, A&A, 161, 177\n\n. K Werner, A&A. 226265Universität Kiel WernerPhD ThesisWerner, K. 1987, PhD Thesis, Universität Kiel Werner, K. 1989, A&A, 226, 265\n\n. K Werner, D Husfeld, A&A. 148417Werner, K., & Husfeld, D. 1985, A&A, 148, 417\n\nK Werner, S Dreizler, Computational Astrophysics. H. Riffert & K. Werner10965Werner, K., & Dreizler, S. 1999, in Computational Astrophysics, ed. H. Riffert & K. Werner, Journal of Computational and Applied Mathematics, 109, 65\n"},"corpus_id":{"kind":"number","value":119031851,"string":"119,031,851"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"astro-ph/0209535\",\n \"mag\": 3095813580,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/astro-ph/0209535v1.pdf\"\n ],\n \"pdf_sha\": \"7774ef1a5e9aa2ad8ac1a9bf4d8622aa52b05178\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Model Photospheres with Accelerated Lambda Iteration"},"authors":{"kind":"list like","value":["I Hubeny \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany","D Mihalas \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany","K Werner \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany","Klaus Werner \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany","Jochen L Deetjen \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany","Stefan Dreizler \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany","Thorsten Nagel \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany","Thomas Rauch \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany","Sonja L Schuh \nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 1D-72076TübingenGermany"],"string":"[\n \"I Hubeny \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\",\n \"D Mihalas \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\",\n \"K Werner \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\",\n \"Klaus Werner \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\",\n \"Jochen L Deetjen \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\",\n \"Stefan Dreizler \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\",\n \"Thorsten Nagel \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\",\n \"Thomas Rauch \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\",\n \"Sonja L Schuh \\nInstitut für Astronomie und Astrophysik\\nUniversität Tübingen\\nSand 1D-72076TübingenGermany\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We address the classical stellar-atmosphere problem and describe our method of numerical solution in detail. The problem consists of the solution of the radiation transfer equation under the constraints of hydrostatic, radiative and statistical equilibrium (non-LTE). We employ the Accelerated Lambda Iteration (ALI) technique, and use statistical methods to construct non-LTE metal-line-blanketed model-atmospheres.2 Werner, Deetjen, Dreizler, Nagel, Rauch, & Schuh neously with the equations for hydrostatic and radiative equilibrium, together with the statistical equilibrium (rate equations).A stellar atmosphere radiates into circumstellar space and thus is an open thermodynamic system; hence it cannot be in thermodynamic equilibrium (TE), and cannot be characterized by a single temperature. The idea of \"Local Thermodynamic Equilibrium\" (LTE) is a hypothesis which assumes that while TE does not hold for the atmosphere as a whole, it can be applied in small volume elements. In this case the atomic occupation numbers depend only on the local electron temperature and electron density via the Saha-Boltzmann equations. But this approximation can be valid only in the limit that collision rates dominate radiative rates, and photon-mean-free-paths are small. Models in which the Saha-Boltzmann equations are replaced by the physically more accurate rate equations are called non-LTE (or NLTE) models. This designation is somewhat imprecise because the velocity distribution of particles is still assumed to be Maxwellian, i.e. we can still define a local temperature. NLTE calculations are more costly than LTE calculations, however, and it is hard to predict a priori whether NLTE effects will be important in a specific problem. Generally, NLTE effects are large at high temperatures and low densities, which imply intense radiation fields, hence frequent radiative processes (which have a non-local nature and, in particular, respond to the presence of the open boundary), and less-frequent particle collisions, which tend to enforce LTE conditions. Abandonment of the LTE assumption leads to a much more difficult modelatmosphere problem, because of subtle couplings between the radiation transfer and statistical equilibrium equations. The pioneering work by Auer & Mihalas (1969) provided a basic tool for making such models. But the numerical problem going from LTE to realistic NLTE models has been solved only recently, and is the topic of this paper. We now have much more powerful tools to compute non-classical models, which account for very complex opacities, and solve the radiation transfer equations in more general environments, for example in expanding stellar atmospheres. These are the topics of other papers in this volume.Stellar-atmosphere modeling has made huge progress in recent years. This advance was achieved by the development of new numerical techniques for modelconstruction, and by the availability of atomic data for many species. And these achievements became possible only with an enormous increase of computing power. Model atmospheres assuming LTE have been highly refined by the inclusion of many more atomic and molecular opacity sources, and effective numerical techniques for LTE model-computation have been available for years. The progress is most remarkable in the field of NLTE model atmospheres. Replacement of the Saha-Boltzmann equations by atomic rate-equations requires radically different numerical-solution techniques, otherwise metal-opacities cannot be taken into account at all. Such techniques have been developed with great success during the last decade, inspired by important papers byCannon (1973) andScharmer (1981). The Accelerated Lambda-Iteration Method (ALI) is at the heart of this development. Combined with statistical representations of line-opacities, we are finally able to compute metal-line-blanketed NLTE models, including many millions of spectral lines, with a very high level of sophistication."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Essentially all our knowledge about the structure and evolution of stars, hence about galactic evolution in general, rests on the interpretation of their electromagnetic spectrum. Therefore quantitative analysis of stellar spectra is one of the most important tools of modern astrophysics. The formation of the observed spectrum is confined to the atmosphere, a very thin layer on top of the stellar core. Spectral analysis is performed by modeling the temperature-and pressure-stratification of the atmosphere and computing synthetic spectra which are then compared to observation. Fitting synthetic spectra to the data yields basic photospheric parameters: effective temperature, surface gravity, and chemical composition. Comparison with theoretical evolutionary calculations allows one to derive stellar parameters: mass, radius, and total luminosity.\nThe classical stellar-atmosphere problem considers radiation transfer through the outermost layers of a star into free space under three assumptions. First it is assumed that the atmosphere is in hydrostatic equilibrium, thus, the matter which interacts with photons is at rest. Second, the transfer of energy through the atmosphere is assumed to be done entirely by photons, i.e. heatconduction and convection are regarded as negligible (radiative equilibrium). But the effectiveness of photon-transfer depends on the opacity and emissivity of the matter, which are strongly state-and frequency-dependent quantities. They depend in detail on the occupation numbers of atomic/ionic levels, which, in turn, are determined by the local temperature and electron density, and the radiation field, whose nature is non-local. The occupation number of any atomic level is determined by a balance among radiative and collisional population and de-population processes (statistical equilibrium; our third assumption), i.e. the interaction of atoms with other particles and photons. Mathematically, the whole problem requires the solution of the radiation transfer equations simulta-In this paper we discuss the basic ideas behind the new numerical methods for NLTE modeling. We begin by presenting the classical model-atmosphere problem and introduce the basic equations. Then we focus on the ALI solutionmethod, and our numerical implementation of it. We then briefly describe the solution of the NLTE metal-line-blanketing problem."},{"title":"Statement of the problem and overview of the solution method","content":"We assume plane-parallel geometry, which is well justified for most stars because the atmospheres are thin compared to the stellar radius. The only parameters needed to characterize uniquely such an atmosphere are the effective temperature (T eff ), a measure for the amount of energy transported through the atmosphere per unit area and time (see Eq. 15), the surface gravity (g), and the chemical composition. Generalization to spherical symmetry to account for extended (static) atmospheres mainly affects the radiation transfer equation and is straightforward (Mihalas & Hummer 1974). A spherical-symmetry variant of our code has been written by Nagel et al. (2001). To construct model atmospheres we must solve simultaneously a set of equations that is highly coupled and non-linear. Because of the coupling, no equation determines uniquely a single quantityall equations jointly determine each of the state parameters. Neverlethess, each of them is usually thought of as determining, more or less, a particular quantity. These equations are:\n• The radiation transfer equations which are solved for the angle-integrated mean intensities J i , i = 1, . . . , N F , on a pre-chosen frequency grid comprising N F points. The formal solution is given by J = ΛS, where S is the source function as defined later (Eq. 11), and Λ is the transport operator. Although Λ is written as an operator, one may think of it as a process of obtaining the mean intensity from the source function.\n• The hydrostatic-equilibrium equation which determines the total particle density N .\n• The radiative-equilibrium equation from which the temperature T follows.\n• The particle-conservation equation, determining the electron density n e .\n• The statistical-equilibrium equations, which are solved for the population densities n i , i = 1, . . . , N L of the atomic levels that are allowed to depart from LTE (NLTE levels).\n• The equation that defines a fictitious massive-particle density n H , which is introduced for convenience in the solution procedure.\nThis set of equations has to be solved at each point d of a grid comprising N D depth points. Thus we are looking for solution vectors ψ ′ d = (n 1 , . . . , n N L , n e , T, n H , N, J 1 , . . . , J N F ), d = 1, . . . , N D.\nThe Complete Linearization (CL) method (Auer & Mihalas 1969) solves this set by linearizing the equations with respect to all variables. The basic advantage of the ALI (or \"operator splitting\") method over CL is that it allows one to eliminate, at the outset, the explicit appearance of the mean intensities J i from the solution scheme by expressing these variables by the current, yet-tobe determined, occupation numbers and temperature. This is accomplished by an iteration procedure which may be written as (suppressing indices indicating depth and frequency dependence of variables):\nThis means that the actual mean intensity at any iteration step n is computed by applying an approximate lambda operator (ALO) Λ ⋆ on the actual source function S n plus a correction term that is computed from quantities known from the previous iteration step. This correction term includes the exact lambda operator Λ, which guarantees convergence to the exact solution of the radiation transfer problem J = ΛS. The use of Λ in Eq. 2 only indicates that a formal solution of the transfer equation is performed, but in our application the operator is not constructed explicitly. Instead we employ the Feautrier (1964) solution scheme (Mihalas 1978) or a short-characteristic method (Olson & Kunasz 1987) to solve the transfer equation as a differential equation. The resulting set of equations for the reduced solution vectors\nis of course still non-linear. The solution is obtained by linearization and iteration which is performed either with a usual Newton-Raphson iteration or by other, much faster methods like the quasi-Newton or Kantorovich variants (Dreizler & Werner 1991, Hubeny & Lanz 1992) (see Sect. 5.4.). The first modelatmosphere calculations with the ALI method were performed by Werner (1986). Another advantage of the ALI method is that explicit depth coupling of the solution vectors Eq. 1 through the transfer equation can be avoided if one uses diagonal (i.e. local) ALOs. Then the solution vectors Eq. 3 are independent from each other and the solution procedure within one iteration step of Eq. 2 is much more straightforward. Depth coupling is provided by the correction which contains the exact solution of the transfer equation. The hydrostatic equation, which also gives explicit depth coupling, may be taken out of the set of equations and can -as experience shows -be solved in between two iteration steps of Eq. 2. Then full advantage of a local ALO can be taken. The linearized system may be written as:\nwhere ψ 0 d is the current estimate for the solution vector at depth d and δψ d is the correction vector to be computed. Using a tri-diagonal Λ ⋆ operator the resulting system for δψ d is -like in the classical CL scheme -of block tri-diagonal form coupling each depth point d to its nearest neighbors d ± 1:\nThe quantities α, β, γ are (N N × N N ) matrices where N N is the total number of physical variables, i.e., N N = N L+4, and c d is the residual error in the equations. The solution is obtained by the standard Feautrier scheme. With starting values D 1 = β −1 1 (-α 1 ) and v 1 = β −1 1 c 1 we sweep from the outer boundary of the atmosphere inside and calculate at each depth:\nAt the inner boundary we have D N D = 0 and by sweeping backwards to the surface we calculate the correction vectors, first δψ N D = v N D , and then successively δψ d = D d δψ d+1 + v d . As already mentioned, the system Eq. 5 breaks into N D independent equations δψ d = β −1 d c d (d = 1, . . . , N D) when a local Λ ⋆ operator is used. The additional numerical effort to set up the subdiagonal matrices and matrix multiplications in the tri-diagonal case is outweighed by the faster global convergence of the ALI cycle, accomplished by the explicit depth coupling in the linearization procedure (Werner 1989).\nThe principal advantage of the ALI over the CL method becomes clear at this point. Each matrix inversion in Eq. 6 requires (N L + 4) 3 operations whereas in the CL method (N L + N F + 4) 3 operations are needed. Since the number of frequency points N F is much larger than the number of levels N L, the matrix inversion in the CL approach is dominated by N F .\nRecent developments deal with the problem that the total number of atomic levels tractable in NLTE with the ALI method described so far is restricted to the order of 250, from experience with our model atmosphere code PRO2. This limit is a consequence of the non-linearity of the equations, and in order to overcome it, measures must be taken in order to achieve a linear system whose numerical solution is much more stable. Such a pre-conditioning procedure was first applied in the ALI context by Werner & Husfeld (1985) using the coresaturation method (Rybicki 1972). More advanced work achieves linearity by replacing the Λ operator with the Ψ operator (and by judiciously considering some populations as \"old\" and some as \"new\" ones within an ALI step) which is formally defined by writing:\nwhere the total opacity χ ν (as defined in Sect. 3.6.) is calculated from the previous ALI cycle. The advantage is that the emissivity η ν (Sect. 3.6.) is linear in the populations, whereas the source function S ν is not. Hence the new operator Ψ gives the solution of the transfer problem by acting on a linear function. This idea was proposed by Rybicki & Hummer (1991) who applied it to the line-formation problem, i.e. regarding the atmospheric structure as fixed. Hauschildt (1993) and Hauschildt & Baron (1999) generalized it to solve the full model-atmosphere problem. In addition, splitting the set of statistical equations, and solving it separately for each chemical element, means that now many hundreds of levels per species are tractable in NLTE (see Dreizler, this volume, for our experience). A very robust method and fast variant of the ALI method, the ALI/CL hybrid scheme, permits the linearization of the radiation field for selected frequencies (Hubeny & Lanz 1995), but it is not implemented in PRO2."},{"title":"Basic equations","content":""},{"title":"Radiation transfer","content":"Any numerical method requires a formal solution (i.e. atmospheric structure already given) of the radiation transfer problem. We disregard here polarization effects. (An implementation of an ALI technique to treat polarized radiation is discussed by Deetjen et al. in this volume.) Radiation transfer at any depth point is described by the transfer equation:\nwritten separately for positive and negative µ (the angle-cosine between the direction of propagation and the outward-directed normal to the surface), i.e. for the inward and outward directional intensities I(µ) at frequency ν. Here τ ν is the optical depth (defined in terms of the column mass m used in the other structural equations by dτ ν = dmχ ν /ρ, where ρ is the mass density), and S ν is the local source function. Introducing the Feautrier intensity\nwe obtain the second-order form (Mihalas 1978, p. 151):\nWe may separate the Thomson emissivity term (scattering from free electrons, assumed coherent, with cross-section σ e ) from the source function so that\nwhere S ′ ν is the ratio of thermal emissivity to total opacity as described in detail below (Sect. 3.6.): S ′ ν = η ν /χ ν . The mean intensity is the angular integral over the Feautrier intensity, hence the transfer equation becomes\nThomson scattering complicates the situation by its explicit angle coupling but the solution can be obtained with the standard Feautrier scheme. Assuming complete frequency-redistribution in spectral lines (Mihalas 1978, p. 29), no explicit frequency coupling occurs so that a parallel solution for all frequencies enables very efficient vectorization on the computer. The following boundary conditions are used for the transfer equation. At the inner boundary where the optical depth is at maximum, τ = τ max , we have\nwhere we specify I + νµ = I ν (+µ, τ max ) from the diffusion approximation:\nHere B ν is the Planck function and H the nominal (frequency-integrated) Eddington flux:\nwith the Stefan-Boltzmann constant σ R . At the outer boundary we take τ ν = τ min = m 1 χ ν /2ρ, assuming that χ is a linear function of m for m < m 1 . Since τ min = 0, it is not exactly valid to assume no incident radiation at the stellar surface. Instead we specify I − νµ = I ν (−µ, τ min ) after Scharmer & Nordlund (1982):\nwhich follows from Eq. 8 assuming S(τ ) = S(τ min ) for τ < τ min . Then we get:\nThe boundary conditions are discretized performing Taylor expansions which yield second-order accuracy (Mihalas 1978, p. 155)."},{"title":"Statistical equilibrium","content":"The statistical equilibrium equations are set up according to Mihalas (1978, p. 127). The number of atomic levels, ionization stages, and chemical species, as well as all radiative and collisional transitions are taken from the input model atom supplied by the user. Ionization into excited states of the next ionization stage is taken into account. Dielectronic recombination and autoionization processes can also be included in the model atom.\nRate equations As usual, the atomic energy levels are ordered sequentially by increasing excitation energy, starting with the lowest ionization stage. Then for each atomic level i of any ionization stage of any element, the rate equation describes the equilibrium of rates into and rates out of this level:\nThe rate coefficients P ij have radiative and collisional components: P ij = R ij + C ij . Radiative upward and downward rates are respectively given by:\n8"},{"title":"Werner, Deetjen, Dreizler, Nagel, Rauch, & Schuh","content":"Photon cross-sections are denoted by σ ij (ν). (n i /n j ) ⋆ is the Boltzmann LTE population ratio in the case of line transitions: g i /g j exp(−hν ij /kT ), where the g i,j are the statistical weights. The LTE population number of a particular level is defined relative to the ground state of the next ion, so that in the case of recombination from a ground state n + 1 we have by definition (n i /n j ) ⋆ = n e φ i (T ) with the Saha-Boltzmann factor\nwhere hν i is the ionization potential of the level i. Care must be taken in the case of recombination from an excited level into the next lower ion. Then (n i /n j ) ⋆ = n e φ i · φ + 1 /φ j . Dielectronic recombination is included following Mihalas & Hummer (1973). Assuming now that j is a ground state of ion k, then the recombination rate into level i of ion k − 1 via an autoionization level c (with ionization potential hν c , having a negative value when lying above the ionization limit) is:\nThe reverse process, the autoionization rate, is given by:\nThe oscillator strength for the stabilizing transition (i.e. transition i → c) is denoted by f ic , andJ is the mean intensity averaged over the line profile. Our program simply takes J ν from the continuum frequency point closest to the transition frequency, which is reasonable because the autoionization line profiles are extremely broad. The population of autoionization levels is assumed to be in LTE and therefore such levels do not appear explicitly in the rate equations.\nThe computation of collisional rates is generally dependent on the specific ion or even transition. Several options, covering the most important cases, may be chosen by the user.\nAbundance-definition equation The rate equation for the highest level of a given chemical species is redundant. It is replaced by the abundance definition equation. Summation over all levels usually includes not only NLTE levels but also levels which are treated in LTE, according to the specification in the model atom. Denoting the number of ionization stages of species k with N ION (k), the number of NLTE and LTE levels per ion with N L(l) and LT E(l), respectively, we can write:\ny k is the number fraction of element k."},{"title":"Model Photospheres with Accelerated Lambda Iteration","content":""},{"title":"9","content":"Charge conservation We close the system of statistical equilibrium equations by demanding charge conservation. We denote the total number of chemical species with NATOM, the charge of ion l with q(l) (in units of the electron charge) and write:\nComplete statistical equilibrium equations We introduce a vector comprising the occupation numbers of all NLTE levels, n = (n 1 , . . . , n N L ). Then the statistical equilibrium equation is written as:\nThe gross structure of the rate matrix A is of block matrix form, because transitions between levels occur within one ionization stage or to the ground state of the next ion. The structure is complicated by ionizations into excited levels and by the abundance definition and charge conservation equations which give additional non-zero elements in the corresponding lines of A."},{"title":"Radiative equilibrium","content":"Radiative equilibrium can be enforced by adjusting the temperature stratification either during the linearization procedure or in between ALI iterations. In the former case a linear combination of two different formulations is used and in the latter case the classical Unsöld-Lucy temperature correction procedure (Lucy 1964) is utilized. The latter is particularly interesting, because it allows one to exploit the block form of the rate-coefficient matrix. This fact allows an economic block-by-block solution followed by a subsequent Unsöld-Lucy temperature correction step. On the other hand, however, this correction procedure may decelerate the global convergence behavior of the ALI iteration. The two forms of expressing the radiative equilibrium condition follow from the requirement that the energy emitted by a volume element per unit time is equal to the absorbed energy per unit time (integral form):\nwhere scattering terms in χ ν and S ν cancel out. In principle, this formulation is equivalent to demanding flux-constancy throughout the atmosphere (differential form)\nwhere H (Eq. 15) is the nominal flux. Here f ν is the variable Eddington factor, defined as\nwhich is computed from the Feautrier intensity u νµ (Eq. 9) after the formal solution. As discussed e.g. in Hubeny (1988) the differential form is more accurate at large depths, while the integral form behaves numerically superior at small depths. Instead of arbitrarily selecting that depth in the atmosphere where we switch from one formulation to the other, we use a linear combination of both constraint equations, which guarantees a smooth transition with depth, based on physical grounds (Hubeny & Lanz 1992, Hamann 1994:\nFor details on this equation and on our implementation of the Unsöld-Lucy procedure see Dreizler (this volume). We note that explicit depth coupling is introduced by the differential form Eq. 28 through the derivative ∂/∂τ ν even if a purely local Λ ⋆ operator is used. Therefore the linearization procedure can no longer be performed independently at each depth point and the question at which boundary to start becomes relevant. Numerical experience shows that it is essential to start at the outer boundary and proceed inwards. If a tri-diagonal operator is used, nearest neighbor depth coupling is introduced anyhow."},{"title":"Hydrostatic equilibrium","content":"We write the equation for hydrostatic equilibrium as Mihalas (1978, p. 170):\nP is the total pressure comprising gas, radiation and turbulent pressures, so that:\nwith Boltzmann's constant k and the turbulent velocity v turb . The hydrostatic equation may either be solved simultaneously with all other equations or separately between iterations. The overall convergence behavior is usually the same in both cases. If taken into the linearization scheme, and a local Λ ⋆ operator is used, then, as in the case of the radiative equilibrium equation, explicit depthcoupling enters via the depth derivative d/dm. Again, solution of the linearized equations must proceed inwards starting at the outer boundary. The starting value at the first depth point (subscript d = 1) is:\nwhere h ν is the variable Eddington factor denoting the ratio of H ν /J ν at the surface, kept fixed during linearization. We are also able to account for element separation resulting from pressure diffusion, which is the process that governs the spectroscopic appearance e.g. of white dwarfs. The numerical method and some results are described by Dreizler & Wolff (1999) and Schuh et al. (this volume)."},{"title":"Particle-conservation and fictitious massive-particle density","content":"The total particle density N is the sum of electron density plus the populations of all atomic/ionic states, in all LTE and NLTE levels:\nThe fictitious massive-particle density n H is introduced for notational convenience. It is defined by:\nThe mass of a chemical species in AMU is denoted by m k . Introducing the mass of a hydrogen atom m H , we may write the material density simply as"},{"title":"Opacity and emissivity","content":"Thermal opacity and emissivity are made up by radiative bound-bound, boundfree and free-free transitions. For each species we compute and sum:\nwhere the total opacity includes Thomson scattering, i.e. χ ν = κ ν + n e σ e , and\nσ li→l+1,k (ν) denotes the cross-section for photoionization from level i of ion l into level k of ion l + 1. The double summation over the bound-free continua takes into account the possibility that a particular level may be ionized into more than one level of the next high ion. Again, note the definition of the LTE population number n ⋆ li in this case, which depends on the level (l + 1, k) of the parent ion:\nNote also that the concept of LTE levels (whose occupation numbers enter, e.g. the number-and charge-conservation equations) in the atomic models of complex ions is therefore not unambiguous. Our code always assumes that LTE levels in the model atoms are populated in LTE with respect to the ground state of the upper ion. The source function used for the approximate radiation transfer is η ν /κ ν , thus it excludes Thomson scattering. For the exact formal solution of course, the total opacity χ ν in the expression Eq. 11 includes the Thomson term (n e σ e )."},{"title":"Atomic level dissolution by plasma perturbations","content":"As high-lying atomic levels are strongly perturbed by other charged particles in the plasma they are broadened and finally dissolved. This effect is observable by line merging at series limits and must be accounted for in line profile analyses. Moreover, line-overlap couples the radiation field in many lines and flux-blocking can strongly affect the global atmospheric structure. Numerically, we treat the level dissolution in terms of occupation probabilities, which for LTE plasmas can be defined as the ratio of the level populations to those in absence of perturbations. A phenomenological theory for these quantities was given in Hummer & Mihalas (1988). The non-trivial generalization to NLTE plasmas was made by Hubeny et al. (1994). In practice an individual occupation-probability factor (depending on T, n e , and principal quantum number), is applied to each atomic level, which describes the probability that the level is dissolved. Furthermore, the rate equations Eq. 18 must be generalized in a unique and unambiguous manner. For details see Hubeny et al. (1994)."},{"title":"The Accelerated Lambda Iteration (ALI)","content":"In all constraint equations described above the mean intensities J ν are replaced by the approximate radiation field Eq. 2 in order to eliminate these variables from the solution vector Eq. 1. In principle the ALO may be of arbitrary form as long as the iteration procedure converges. In practice, however, an optimum choice is desired in order to achieve convergence with a minimum amount of iteration steps. The history of the ALOs is interesting, and was summarized in detail by Hubeny (1992). Of utmost importance were two papers by Olson and collaborators (Olson et al. 1986, Olson & Kunasz 1987 who overcame the major drawback of early ALOs, namely the occurrence of free parameters controlling the convergence process, and thus found the optimum choice of ALOs. Our model atmosphere program permits the use of either a diagonal or a tri-diagonal ALO, both are set up following Olson & Kunasz (1987)."},{"title":"Diagonal (local) lambda operators","content":"In this case the mean intensity J d at a particular depth d in the current iteration step is computed solely from the local source function S d and a correction term ∆J d , the latter depending on the source functions (at all depths) from the previous iteration. Dropping the iteration count and introducing indices denoting depth points we can rewrite Eq. 2 as:\nIn discrete form we can think of Λ ⋆ as a matrix acting on a vector whose elements comprise the source functions of all depths. Then Λ ⋆ d,d is the diagonal element of the Λ ⋆ matrix corresponding to depth point d. Writing Λ ⋆ d,d ≡ B d (for numerical computation see Eq. 44 below) we have a purely local expression for the mean intensity:"},{"title":"Tridiagonal (non-local) lambda operators","content":"Much better convergence is obtained if the mean intensity is computed not only from the local source function, but also from the source function of the neighboring depths points. Then the matrix representation of Λ ⋆ is of tri-diagonal form and we may write:\nwhere C d−1 and A d+1 represent the upper and lower subdiagonal elements of Λ ⋆ , and S d±1 the source functions at the adjacent depths. By analogy the correction term becomes:\nAgain all quantities for the computation of ∆J d are known from the previous iteration, and the first term gives the exact formal solution of the transfer equation. We emphasize again that the source functions in Eq. 42 are to be computed from the correct occupation numbers and temperature, all of which are still unknown. We therefore have a non-linear set of equations for these quantities, which must be solved by either Newton-Raphson iteration or other techniques, resulting in the solution of a tri-diagonal linear equation of the form Eq. 5. As was shown in Olson et al. (1986) the elements of the optimum Λ ⋆ matrix are given by the corresponding elements of the exact Λ matrix. The diagonal and subdiagonal elements are computed following Olson & Kunasz (1987):\nwith ∆τ d−1 ≡ (τ d − τ d−1 )/µ. At large optical depths with increasing ∆τ steps (the depth grid is roughly equidistant in log τ ) the subdiagonals A d+1 and C d−1 vanish, and the diagonal B d approaches unity, reflecting the fact that the radiation field is more and more determined by local properties of the matter. At very small optical depths all elements of Λ ⋆ vanish, reflecting the non-localness of the radiation field at these depths."},{"title":"Acceleration of convergence","content":"Our code takes advantage of an acceleration scheme to speed up convergence of the iteration cycle Eq. 2. We implemented the scheme originally proposed by Ng (1974) and later by Auer (1987). It extrapolates the correction vector δψ d from the previous three iterations. From our experience the extrapolation often yields over-corrections, resulting in alternating convergence or even divergence. In contrast, use of a tri-diagonal ALO usually results in satisfactorily fast convergence, so that the acceleration scheme is rarely used."},{"title":"Solution of the non-linear equations by iteration","content":"At each depth the complete set of non-linear equations Eq. 2 for a single iteration step comprises the equations for statistical, radiative, and hydrostatic equilibrium, and the particle-conservation equation. For the numerical solution we introduce discrete depth and frequency grids. The equations are then linearized and solved by a suitable iterative scheme. Explicit angle-dependence of the radiation field is not required here and is consequently eliminated by the use of variable Eddington factors. Angle dependence is considered only in the formal solution of the transfer equation. Our program requires an input modelatmosphere structure as a starting approximation, together with an atomic data file, and a frequency grid. Depth and frequency grids are therefore set up in advance by separate programs."},{"title":"Discretization","content":"After a depth-grid is chosen, we start by computing a gray LTE continuum model using the Unsöld-Lucy temperature-correction procedure. Depth points (typical number is 90) are chosen in equidistant steps on a logarithmic (Rosseland) optical depth scale. The converged LTE model (temperature and density structure, given on a column mass depth scale) is written to a file that is read by PRO2. The NLTE code uses the column mass as an independent depth variable. The frequency grid is determined by the atomic-data input file. Frequency points are set blue-and red-ward of each absorption edge, and for each spectral line. Gaps are filled up by setting continuum points. Finally, the quadrature weights are computed. Frequency integrals appearing e.g. in Eq. 30 are replaced by quadrature sums, and depth derivatives by difference quotients."},{"title":"Linearization","content":"All variables x are replaced by x → x+δx where δx denotes a small perturbation of x. Terms not linear in these perturbations are neglected. The perturbations are expressed by perturbations of the basic variables:\nAs an illustrative example we linearize the equation for radiative equilibrium.\nMost other linearized equations may be found in Werner (1986). Assigning two indices (d for depth and i for frequency of a grid with NF points) to the variables and denoting the quadrature weights with w i Eq. 30 becomes:\nNote that we do not linearize ∆τ i . Because of this, convergence properties may deteriorate significantly in some cases. Perturbations δS di , δχ di are expressed by Eq. 45, and the perturbation of the mean intensity J di is, according to Eq. 42, given through the perturbations of the source function at the current and the two adjacent depths:\nwhere A, B, C are the Λ matrix elements from Eq. 44. The δJ d−1,i contain the term C d−2,i δS d−2,i which is neglected because we want to account only for nearest neighbor coupling. We write δS di with the help of Eq. 45 and observe that for any variable z:\nDerivatives of opacity and emissivity with respect to temperature, electron and population densities are computed from analytical expressions (see e.g. Mihalas et al. 1975, Werner 1987. We finally get from Eq. 46:\nThe curly brackets {· · ·} denote terms that are similar to those multiplied by perturbations of the temperature. Instead of partial derivatives with respect to T , they contain derivatives with respect to n e and the populations n l . They all represent coefficients of the matrices α, β, γ in Eq. 5."},{"title":"Newton-Raphson iteration","content":"As described in Sect. 2. the linearized equations have a tri-diagonal block-matrix form, see Eq. 5. Inversion of the grand matrix (≡ T sized (N N · N D) × (N N · N D), i.e. about 10 4 × 10 4 in typical applications) is performed with a block-Gaussian elimination scheme, which means that our iteration of the non-linear equations is a multi-dimensional Newton-Raphson method. The problem is structurally simplified when explicit-depth coupling is avoided by the use of a local ALO, however, the numerical effort is not reduced much, because in both cases the main effort lies with the inversion of matrices sized N N × N N . The Newton-Raphson iteration involves two numerically expensive steps, first setting up the Jacobian (comprising α, β, γ) and then inverting it. Additionally, the matrix inversions in Eq. 6 limit their size to about N N = 250 because otherwise numerical accuracy is lost. Two variants recently introduced in stellar atmosphere calculations are able to improve both numerical accuracy and, best of all, computational speed."},{"title":"Alternative fast solution techniques for non-linear equations: Broyden-and Kantorovich-variants","content":"Broyden's (1965) variant belongs to the family of quasi-Newton methods; it was first used in model-atmosphere calculations in Dreizler & Werner (1991), Hamann et al. 1991, Koesterke et al. (1992. It avoids the repeated set-up of the Jacobian by the use of an update formula. In addition, it also gives an update formula for the inverse Jacobian. In the case of a local ALO the solution of the linearized system at any depth is:\nLet β −1 k be the k-th iterate of the inverse Jacobian, then an update can be found from:\nwhere ⊗ denotes the dyadic product, and where we have defined:\ns k ≡ δψ k solution vector of preceding linearization, y k ≡ c k+1 − c k difference of actual and preceding residuum.\nThe convergence rate is super-linear, i.e. slower than the quadratic rate of the Newton-Raphson method; but this is more than compensated by the tremendous speed-up for a single iteration step. It is not always necessary to begin the iteration with the calculation of an exact Jacobian and its inversion. Experience shows that in an advanced stage of the overall (ALI-) iteration Eq. 2 (i.e. when corrections become small, of the order 1%) we can start the linearization cycle Eq. 51 by using the inverse Jacobian from the previous overall iteration. Computational speed-up is extreme in this case, however, it requires storage of the Jacobians of all depths. More difficult is its application to the tri-diagonal ALO case. Here we have to update the grand matrix T which, as already mentioned, is of block tri-diagonal form. We cannot update its inverse, because it is never computed explicitly. Furthermore we need an update formula that preserves the block tridiagonal form which is a prerequisite for its inversion by the Feautrier scheme Eq. 6. Such a formula was found by Schubert (1970):\nwheres k ≡ Zs k with the structure matrix Z as defined by:\nThe vectors s k and y k are defined as above but now they span quantities over all depth instead of a single depth point. With this formula we obtain new submatrices α, β, γ and c with which the Feautrier scheme Eq. 6 is solved again. This procedure saves the computation of derivatives. Another feature realized in our program also saves the repeated inversion of q ≡ (β d + γ d D d−1 ) by updating its inverse with the Broyden formula Eq. 51. Similar to the diagonal ALO case it is also possible to pass starting matrices from one overall iteration Eq. 2 to the next for the update of T and the matrix q −1 . In both cases the user specifies two threshold values for the maximum relative correction in δψ which cause the program to switch from Newton-Raphson to Broyden stages 1 and 2. During stage 1, each new overall cycle Eq. 2 is started with an exact calculation and inversion of all matrices involved, and in stage 2 these matrices are passed through each iteration. Another variant, the Kantorovich (1949) method has been introduced into model-atmosphere calculations (Hubeny & Lanz 1992). It is more simple and straightforward to implement. This method simply keeps the Jacobian fixed during the linearization cycle; it is surprisingly stable. In fact it turns out to be even more stable (i.e. it can be utilized in an earlier stage of iteration) than the Broyden method in the tri-diagonal ALO case. The user of PRO2 may choose this variant in two stages in analogy to the Broyden variant. It was found that in stage 2 it is necessary to update the Jacobian every 5 to 10 overall iterations in order to prevent divergence."},{"title":"NLTE metal-line blanketing","content":"Despite the increase in capacity for solving NLTE model-atmosphere problems given by the ALI method combined with pre-conditioning techniques, the blanketing by millions of lines from the iron-group elements arising from transitions among some 10 5 levels can be attacked only with statistical methods. These were introduced into NLTE model-atmosphere work by Anderson (1989Anderson ( , 1991. At the outset, model atoms are constructed by combining many thousands of levels into a relatively small number of superlevels which can be treated by ALI (or other) methods. Then, in order to reduce the computational effort, two approaches were developed which vastly decrease the number of frequency points (hence the number of transfer equations to be solved) needed to describe properly the complex frequency dependence of the opacity. These two approaches have their roots in LTE modeling techniques, where statistical methods are applied for the same reason in the treatment of opacity: The Opacity Distribution Function (ODF) and Opacity Sampling (OS) approaches. Both are based on the fact that the opacity (in the LTE approximation) is a function of two only local thermodynamic quantities. Roughly speaking, each opacity source can be written in terms of a population density and a photon cross-section for the respective radiative transition: κ ν ∼ n l σ lu (ν). In LTE the population follows from the Saha-Boltzmann equations, hence n l = n l (n e , T ). The OS and ODF methods use such pre-tabulated (on a very fine frequency mesh) κ ν (n e , T ) during the model atmosphere calculations. The NLTE situation is more complicated, because pre-tabulation of opacities is not useful. The occupation numbers at any depth now also depend explicitly on the radiation field (via the rate equations which replace the TE Saha-Boltzmann statistics) and thus on the populations in every other depth of the atmosphere. As a consequence, the OS and ODF methods are not applied to opacity tabulations, but to tabulations of the photon cross-sections σ(ν). These do depend on local quantities only, e.g. line broadening by Stark-and Doppler-effects is calculated from T and n e . In the NLTE case cross-sections take over the role which the opacity played in the LTE case. So, strictly speaking, the designation OS and ODF is not quite correct in the NLTE context.\nThe strategy in our code is the following. Before any model atmosphere calculation is started, the atomic data are prepared by constructing superlevels, and the cross-sections for superlines. Then these cross-sections are either sampled on a coarse frequency grid or ODFs are constructed. These data are put into the model atom which is read by the code. The code does not know if OS or ODFs are used, i.e. it is written to be independent of these approaches.\nFurther details about our implementation are described in Dreizler & Werner (1993), Haas et al. (1996), Werner & Dreizler (1999), and by Rauch & Deetjen in this volume."},{"title":"Summary","content":"We have presented in detail our technique for numerical solution of the classical model-atmosphere problem. The construction of metal-line-blanketed models in hydrostatic and radiative equilibrium under NLTE conditions was the last and"}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Essentially all our knowledge about the structure and evolution of stars, hence about galactic evolution in general, rests on the interpretation of their electromagnetic spectrum. Therefore quantitative analysis of stellar spectra is one of the most important tools of modern astrophysics. The formation of the observed spectrum is confined to the atmosphere, a very thin layer on top of the stellar core. Spectral analysis is performed by modeling the temperature-and pressure-stratification of the atmosphere and computing synthetic spectra which are then compared to observation. Fitting synthetic spectra to the data yields basic photospheric parameters: effective temperature, surface gravity, and chemical composition. Comparison with theoretical evolutionary calculations allows one to derive stellar parameters: mass, radius, and total luminosity.\\nThe classical stellar-atmosphere problem considers radiation transfer through the outermost layers of a star into free space under three assumptions. First it is assumed that the atmosphere is in hydrostatic equilibrium, thus, the matter which interacts with photons is at rest. Second, the transfer of energy through the atmosphere is assumed to be done entirely by photons, i.e. heatconduction and convection are regarded as negligible (radiative equilibrium). But the effectiveness of photon-transfer depends on the opacity and emissivity of the matter, which are strongly state-and frequency-dependent quantities. They depend in detail on the occupation numbers of atomic/ionic levels, which, in turn, are determined by the local temperature and electron density, and the radiation field, whose nature is non-local. The occupation number of any atomic level is determined by a balance among radiative and collisional population and de-population processes (statistical equilibrium; our third assumption), i.e. the interaction of atoms with other particles and photons. Mathematically, the whole problem requires the solution of the radiation transfer equations simulta-In this paper we discuss the basic ideas behind the new numerical methods for NLTE modeling. We begin by presenting the classical model-atmosphere problem and introduce the basic equations. Then we focus on the ALI solutionmethod, and our numerical implementation of it. We then briefly describe the solution of the NLTE metal-line-blanketing problem.\"\n },\n {\n \"title\": \"Statement of the problem and overview of the solution method\",\n \"content\": \"We assume plane-parallel geometry, which is well justified for most stars because the atmospheres are thin compared to the stellar radius. The only parameters needed to characterize uniquely such an atmosphere are the effective temperature (T eff ), a measure for the amount of energy transported through the atmosphere per unit area and time (see Eq. 15), the surface gravity (g), and the chemical composition. Generalization to spherical symmetry to account for extended (static) atmospheres mainly affects the radiation transfer equation and is straightforward (Mihalas & Hummer 1974). A spherical-symmetry variant of our code has been written by Nagel et al. (2001). To construct model atmospheres we must solve simultaneously a set of equations that is highly coupled and non-linear. Because of the coupling, no equation determines uniquely a single quantityall equations jointly determine each of the state parameters. Neverlethess, each of them is usually thought of as determining, more or less, a particular quantity. These equations are:\\n• The radiation transfer equations which are solved for the angle-integrated mean intensities J i , i = 1, . . . , N F , on a pre-chosen frequency grid comprising N F points. The formal solution is given by J = ΛS, where S is the source function as defined later (Eq. 11), and Λ is the transport operator. Although Λ is written as an operator, one may think of it as a process of obtaining the mean intensity from the source function.\\n• The hydrostatic-equilibrium equation which determines the total particle density N .\\n• The radiative-equilibrium equation from which the temperature T follows.\\n• The particle-conservation equation, determining the electron density n e .\\n• The statistical-equilibrium equations, which are solved for the population densities n i , i = 1, . . . , N L of the atomic levels that are allowed to depart from LTE (NLTE levels).\\n• The equation that defines a fictitious massive-particle density n H , which is introduced for convenience in the solution procedure.\\nThis set of equations has to be solved at each point d of a grid comprising N D depth points. Thus we are looking for solution vectors ψ ′ d = (n 1 , . . . , n N L , n e , T, n H , N, J 1 , . . . , J N F ), d = 1, . . . , N D.\\nThe Complete Linearization (CL) method (Auer & Mihalas 1969) solves this set by linearizing the equations with respect to all variables. The basic advantage of the ALI (or \\\"operator splitting\\\") method over CL is that it allows one to eliminate, at the outset, the explicit appearance of the mean intensities J i from the solution scheme by expressing these variables by the current, yet-tobe determined, occupation numbers and temperature. This is accomplished by an iteration procedure which may be written as (suppressing indices indicating depth and frequency dependence of variables):\\nThis means that the actual mean intensity at any iteration step n is computed by applying an approximate lambda operator (ALO) Λ ⋆ on the actual source function S n plus a correction term that is computed from quantities known from the previous iteration step. This correction term includes the exact lambda operator Λ, which guarantees convergence to the exact solution of the radiation transfer problem J = ΛS. The use of Λ in Eq. 2 only indicates that a formal solution of the transfer equation is performed, but in our application the operator is not constructed explicitly. Instead we employ the Feautrier (1964) solution scheme (Mihalas 1978) or a short-characteristic method (Olson & Kunasz 1987) to solve the transfer equation as a differential equation. The resulting set of equations for the reduced solution vectors\\nis of course still non-linear. The solution is obtained by linearization and iteration which is performed either with a usual Newton-Raphson iteration or by other, much faster methods like the quasi-Newton or Kantorovich variants (Dreizler & Werner 1991, Hubeny & Lanz 1992) (see Sect. 5.4.). The first modelatmosphere calculations with the ALI method were performed by Werner (1986). Another advantage of the ALI method is that explicit depth coupling of the solution vectors Eq. 1 through the transfer equation can be avoided if one uses diagonal (i.e. local) ALOs. Then the solution vectors Eq. 3 are independent from each other and the solution procedure within one iteration step of Eq. 2 is much more straightforward. Depth coupling is provided by the correction which contains the exact solution of the transfer equation. The hydrostatic equation, which also gives explicit depth coupling, may be taken out of the set of equations and can -as experience shows -be solved in between two iteration steps of Eq. 2. Then full advantage of a local ALO can be taken. The linearized system may be written as:\\nwhere ψ 0 d is the current estimate for the solution vector at depth d and δψ d is the correction vector to be computed. Using a tri-diagonal Λ ⋆ operator the resulting system for δψ d is -like in the classical CL scheme -of block tri-diagonal form coupling each depth point d to its nearest neighbors d ± 1:\\nThe quantities α, β, γ are (N N × N N ) matrices where N N is the total number of physical variables, i.e., N N = N L+4, and c d is the residual error in the equations. The solution is obtained by the standard Feautrier scheme. With starting values D 1 = β −1 1 (-α 1 ) and v 1 = β −1 1 c 1 we sweep from the outer boundary of the atmosphere inside and calculate at each depth:\\nAt the inner boundary we have D N D = 0 and by sweeping backwards to the surface we calculate the correction vectors, first δψ N D = v N D , and then successively δψ d = D d δψ d+1 + v d . As already mentioned, the system Eq. 5 breaks into N D independent equations δψ d = β −1 d c d (d = 1, . . . , N D) when a local Λ ⋆ operator is used. The additional numerical effort to set up the subdiagonal matrices and matrix multiplications in the tri-diagonal case is outweighed by the faster global convergence of the ALI cycle, accomplished by the explicit depth coupling in the linearization procedure (Werner 1989).\\nThe principal advantage of the ALI over the CL method becomes clear at this point. Each matrix inversion in Eq. 6 requires (N L + 4) 3 operations whereas in the CL method (N L + N F + 4) 3 operations are needed. Since the number of frequency points N F is much larger than the number of levels N L, the matrix inversion in the CL approach is dominated by N F .\\nRecent developments deal with the problem that the total number of atomic levels tractable in NLTE with the ALI method described so far is restricted to the order of 250, from experience with our model atmosphere code PRO2. This limit is a consequence of the non-linearity of the equations, and in order to overcome it, measures must be taken in order to achieve a linear system whose numerical solution is much more stable. Such a pre-conditioning procedure was first applied in the ALI context by Werner & Husfeld (1985) using the coresaturation method (Rybicki 1972). More advanced work achieves linearity by replacing the Λ operator with the Ψ operator (and by judiciously considering some populations as \\\"old\\\" and some as \\\"new\\\" ones within an ALI step) which is formally defined by writing:\\nwhere the total opacity χ ν (as defined in Sect. 3.6.) is calculated from the previous ALI cycle. The advantage is that the emissivity η ν (Sect. 3.6.) is linear in the populations, whereas the source function S ν is not. Hence the new operator Ψ gives the solution of the transfer problem by acting on a linear function. This idea was proposed by Rybicki & Hummer (1991) who applied it to the line-formation problem, i.e. regarding the atmospheric structure as fixed. Hauschildt (1993) and Hauschildt & Baron (1999) generalized it to solve the full model-atmosphere problem. In addition, splitting the set of statistical equations, and solving it separately for each chemical element, means that now many hundreds of levels per species are tractable in NLTE (see Dreizler, this volume, for our experience). A very robust method and fast variant of the ALI method, the ALI/CL hybrid scheme, permits the linearization of the radiation field for selected frequencies (Hubeny & Lanz 1995), but it is not implemented in PRO2.\"\n },\n {\n \"title\": \"Basic equations\",\n \"content\": \"\"\n },\n {\n \"title\": \"Radiation transfer\",\n \"content\": \"Any numerical method requires a formal solution (i.e. atmospheric structure already given) of the radiation transfer problem. We disregard here polarization effects. (An implementation of an ALI technique to treat polarized radiation is discussed by Deetjen et al. in this volume.) Radiation transfer at any depth point is described by the transfer equation:\\nwritten separately for positive and negative µ (the angle-cosine between the direction of propagation and the outward-directed normal to the surface), i.e. for the inward and outward directional intensities I(µ) at frequency ν. Here τ ν is the optical depth (defined in terms of the column mass m used in the other structural equations by dτ ν = dmχ ν /ρ, where ρ is the mass density), and S ν is the local source function. Introducing the Feautrier intensity\\nwe obtain the second-order form (Mihalas 1978, p. 151):\\nWe may separate the Thomson emissivity term (scattering from free electrons, assumed coherent, with cross-section σ e ) from the source function so that\\nwhere S ′ ν is the ratio of thermal emissivity to total opacity as described in detail below (Sect. 3.6.): S ′ ν = η ν /χ ν . The mean intensity is the angular integral over the Feautrier intensity, hence the transfer equation becomes\\nThomson scattering complicates the situation by its explicit angle coupling but the solution can be obtained with the standard Feautrier scheme. Assuming complete frequency-redistribution in spectral lines (Mihalas 1978, p. 29), no explicit frequency coupling occurs so that a parallel solution for all frequencies enables very efficient vectorization on the computer. The following boundary conditions are used for the transfer equation. At the inner boundary where the optical depth is at maximum, τ = τ max , we have\\nwhere we specify I + νµ = I ν (+µ, τ max ) from the diffusion approximation:\\nHere B ν is the Planck function and H the nominal (frequency-integrated) Eddington flux:\\nwith the Stefan-Boltzmann constant σ R . At the outer boundary we take τ ν = τ min = m 1 χ ν /2ρ, assuming that χ is a linear function of m for m < m 1 . Since τ min = 0, it is not exactly valid to assume no incident radiation at the stellar surface. Instead we specify I − νµ = I ν (−µ, τ min ) after Scharmer & Nordlund (1982):\\nwhich follows from Eq. 8 assuming S(τ ) = S(τ min ) for τ < τ min . Then we get:\\nThe boundary conditions are discretized performing Taylor expansions which yield second-order accuracy (Mihalas 1978, p. 155).\"\n },\n {\n \"title\": \"Statistical equilibrium\",\n \"content\": \"The statistical equilibrium equations are set up according to Mihalas (1978, p. 127). The number of atomic levels, ionization stages, and chemical species, as well as all radiative and collisional transitions are taken from the input model atom supplied by the user. Ionization into excited states of the next ionization stage is taken into account. Dielectronic recombination and autoionization processes can also be included in the model atom.\\nRate equations As usual, the atomic energy levels are ordered sequentially by increasing excitation energy, starting with the lowest ionization stage. Then for each atomic level i of any ionization stage of any element, the rate equation describes the equilibrium of rates into and rates out of this level:\\nThe rate coefficients P ij have radiative and collisional components: P ij = R ij + C ij . Radiative upward and downward rates are respectively given by:\\n8\"\n },\n {\n \"title\": \"Werner, Deetjen, Dreizler, Nagel, Rauch, & Schuh\",\n \"content\": \"Photon cross-sections are denoted by σ ij (ν). (n i /n j ) ⋆ is the Boltzmann LTE population ratio in the case of line transitions: g i /g j exp(−hν ij /kT ), where the g i,j are the statistical weights. The LTE population number of a particular level is defined relative to the ground state of the next ion, so that in the case of recombination from a ground state n + 1 we have by definition (n i /n j ) ⋆ = n e φ i (T ) with the Saha-Boltzmann factor\\nwhere hν i is the ionization potential of the level i. Care must be taken in the case of recombination from an excited level into the next lower ion. Then (n i /n j ) ⋆ = n e φ i · φ + 1 /φ j . Dielectronic recombination is included following Mihalas & Hummer (1973). Assuming now that j is a ground state of ion k, then the recombination rate into level i of ion k − 1 via an autoionization level c (with ionization potential hν c , having a negative value when lying above the ionization limit) is:\\nThe reverse process, the autoionization rate, is given by:\\nThe oscillator strength for the stabilizing transition (i.e. transition i → c) is denoted by f ic , andJ is the mean intensity averaged over the line profile. Our program simply takes J ν from the continuum frequency point closest to the transition frequency, which is reasonable because the autoionization line profiles are extremely broad. The population of autoionization levels is assumed to be in LTE and therefore such levels do not appear explicitly in the rate equations.\\nThe computation of collisional rates is generally dependent on the specific ion or even transition. Several options, covering the most important cases, may be chosen by the user.\\nAbundance-definition equation The rate equation for the highest level of a given chemical species is redundant. It is replaced by the abundance definition equation. Summation over all levels usually includes not only NLTE levels but also levels which are treated in LTE, according to the specification in the model atom. Denoting the number of ionization stages of species k with N ION (k), the number of NLTE and LTE levels per ion with N L(l) and LT E(l), respectively, we can write:\\ny k is the number fraction of element k.\"\n },\n {\n \"title\": \"Model Photospheres with Accelerated Lambda Iteration\",\n \"content\": \"\"\n },\n {\n \"title\": \"9\",\n \"content\": \"Charge conservation We close the system of statistical equilibrium equations by demanding charge conservation. We denote the total number of chemical species with NATOM, the charge of ion l with q(l) (in units of the electron charge) and write:\\nComplete statistical equilibrium equations We introduce a vector comprising the occupation numbers of all NLTE levels, n = (n 1 , . . . , n N L ). Then the statistical equilibrium equation is written as:\\nThe gross structure of the rate matrix A is of block matrix form, because transitions between levels occur within one ionization stage or to the ground state of the next ion. The structure is complicated by ionizations into excited levels and by the abundance definition and charge conservation equations which give additional non-zero elements in the corresponding lines of A.\"\n },\n {\n \"title\": \"Radiative equilibrium\",\n \"content\": \"Radiative equilibrium can be enforced by adjusting the temperature stratification either during the linearization procedure or in between ALI iterations. In the former case a linear combination of two different formulations is used and in the latter case the classical Unsöld-Lucy temperature correction procedure (Lucy 1964) is utilized. The latter is particularly interesting, because it allows one to exploit the block form of the rate-coefficient matrix. This fact allows an economic block-by-block solution followed by a subsequent Unsöld-Lucy temperature correction step. On the other hand, however, this correction procedure may decelerate the global convergence behavior of the ALI iteration. The two forms of expressing the radiative equilibrium condition follow from the requirement that the energy emitted by a volume element per unit time is equal to the absorbed energy per unit time (integral form):\\nwhere scattering terms in χ ν and S ν cancel out. In principle, this formulation is equivalent to demanding flux-constancy throughout the atmosphere (differential form)\\nwhere H (Eq. 15) is the nominal flux. Here f ν is the variable Eddington factor, defined as\\nwhich is computed from the Feautrier intensity u νµ (Eq. 9) after the formal solution. As discussed e.g. in Hubeny (1988) the differential form is more accurate at large depths, while the integral form behaves numerically superior at small depths. Instead of arbitrarily selecting that depth in the atmosphere where we switch from one formulation to the other, we use a linear combination of both constraint equations, which guarantees a smooth transition with depth, based on physical grounds (Hubeny & Lanz 1992, Hamann 1994:\\nFor details on this equation and on our implementation of the Unsöld-Lucy procedure see Dreizler (this volume). We note that explicit depth coupling is introduced by the differential form Eq. 28 through the derivative ∂/∂τ ν even if a purely local Λ ⋆ operator is used. Therefore the linearization procedure can no longer be performed independently at each depth point and the question at which boundary to start becomes relevant. Numerical experience shows that it is essential to start at the outer boundary and proceed inwards. If a tri-diagonal operator is used, nearest neighbor depth coupling is introduced anyhow.\"\n },\n {\n \"title\": \"Hydrostatic equilibrium\",\n \"content\": \"We write the equation for hydrostatic equilibrium as Mihalas (1978, p. 170):\\nP is the total pressure comprising gas, radiation and turbulent pressures, so that:\\nwith Boltzmann's constant k and the turbulent velocity v turb . The hydrostatic equation may either be solved simultaneously with all other equations or separately between iterations. The overall convergence behavior is usually the same in both cases. If taken into the linearization scheme, and a local Λ ⋆ operator is used, then, as in the case of the radiative equilibrium equation, explicit depthcoupling enters via the depth derivative d/dm. Again, solution of the linearized equations must proceed inwards starting at the outer boundary. The starting value at the first depth point (subscript d = 1) is:\\nwhere h ν is the variable Eddington factor denoting the ratio of H ν /J ν at the surface, kept fixed during linearization. We are also able to account for element separation resulting from pressure diffusion, which is the process that governs the spectroscopic appearance e.g. of white dwarfs. The numerical method and some results are described by Dreizler & Wolff (1999) and Schuh et al. (this volume).\"\n },\n {\n \"title\": \"Particle-conservation and fictitious massive-particle density\",\n \"content\": \"The total particle density N is the sum of electron density plus the populations of all atomic/ionic states, in all LTE and NLTE levels:\\nThe fictitious massive-particle density n H is introduced for notational convenience. It is defined by:\\nThe mass of a chemical species in AMU is denoted by m k . Introducing the mass of a hydrogen atom m H , we may write the material density simply as\"\n },\n {\n \"title\": \"Opacity and emissivity\",\n \"content\": \"Thermal opacity and emissivity are made up by radiative bound-bound, boundfree and free-free transitions. For each species we compute and sum:\\nwhere the total opacity includes Thomson scattering, i.e. χ ν = κ ν + n e σ e , and\\nσ li→l+1,k (ν) denotes the cross-section for photoionization from level i of ion l into level k of ion l + 1. The double summation over the bound-free continua takes into account the possibility that a particular level may be ionized into more than one level of the next high ion. Again, note the definition of the LTE population number n ⋆ li in this case, which depends on the level (l + 1, k) of the parent ion:\\nNote also that the concept of LTE levels (whose occupation numbers enter, e.g. the number-and charge-conservation equations) in the atomic models of complex ions is therefore not unambiguous. Our code always assumes that LTE levels in the model atoms are populated in LTE with respect to the ground state of the upper ion. The source function used for the approximate radiation transfer is η ν /κ ν , thus it excludes Thomson scattering. For the exact formal solution of course, the total opacity χ ν in the expression Eq. 11 includes the Thomson term (n e σ e ).\"\n },\n {\n \"title\": \"Atomic level dissolution by plasma perturbations\",\n \"content\": \"As high-lying atomic levels are strongly perturbed by other charged particles in the plasma they are broadened and finally dissolved. This effect is observable by line merging at series limits and must be accounted for in line profile analyses. Moreover, line-overlap couples the radiation field in many lines and flux-blocking can strongly affect the global atmospheric structure. Numerically, we treat the level dissolution in terms of occupation probabilities, which for LTE plasmas can be defined as the ratio of the level populations to those in absence of perturbations. A phenomenological theory for these quantities was given in Hummer & Mihalas (1988). The non-trivial generalization to NLTE plasmas was made by Hubeny et al. (1994). In practice an individual occupation-probability factor (depending on T, n e , and principal quantum number), is applied to each atomic level, which describes the probability that the level is dissolved. Furthermore, the rate equations Eq. 18 must be generalized in a unique and unambiguous manner. For details see Hubeny et al. (1994).\"\n },\n {\n \"title\": \"The Accelerated Lambda Iteration (ALI)\",\n \"content\": \"In all constraint equations described above the mean intensities J ν are replaced by the approximate radiation field Eq. 2 in order to eliminate these variables from the solution vector Eq. 1. In principle the ALO may be of arbitrary form as long as the iteration procedure converges. In practice, however, an optimum choice is desired in order to achieve convergence with a minimum amount of iteration steps. The history of the ALOs is interesting, and was summarized in detail by Hubeny (1992). Of utmost importance were two papers by Olson and collaborators (Olson et al. 1986, Olson & Kunasz 1987 who overcame the major drawback of early ALOs, namely the occurrence of free parameters controlling the convergence process, and thus found the optimum choice of ALOs. Our model atmosphere program permits the use of either a diagonal or a tri-diagonal ALO, both are set up following Olson & Kunasz (1987).\"\n },\n {\n \"title\": \"Diagonal (local) lambda operators\",\n \"content\": \"In this case the mean intensity J d at a particular depth d in the current iteration step is computed solely from the local source function S d and a correction term ∆J d , the latter depending on the source functions (at all depths) from the previous iteration. Dropping the iteration count and introducing indices denoting depth points we can rewrite Eq. 2 as:\\nIn discrete form we can think of Λ ⋆ as a matrix acting on a vector whose elements comprise the source functions of all depths. Then Λ ⋆ d,d is the diagonal element of the Λ ⋆ matrix corresponding to depth point d. Writing Λ ⋆ d,d ≡ B d (for numerical computation see Eq. 44 below) we have a purely local expression for the mean intensity:\"\n },\n {\n \"title\": \"Tridiagonal (non-local) lambda operators\",\n \"content\": \"Much better convergence is obtained if the mean intensity is computed not only from the local source function, but also from the source function of the neighboring depths points. Then the matrix representation of Λ ⋆ is of tri-diagonal form and we may write:\\nwhere C d−1 and A d+1 represent the upper and lower subdiagonal elements of Λ ⋆ , and S d±1 the source functions at the adjacent depths. By analogy the correction term becomes:\\nAgain all quantities for the computation of ∆J d are known from the previous iteration, and the first term gives the exact formal solution of the transfer equation. We emphasize again that the source functions in Eq. 42 are to be computed from the correct occupation numbers and temperature, all of which are still unknown. We therefore have a non-linear set of equations for these quantities, which must be solved by either Newton-Raphson iteration or other techniques, resulting in the solution of a tri-diagonal linear equation of the form Eq. 5. As was shown in Olson et al. (1986) the elements of the optimum Λ ⋆ matrix are given by the corresponding elements of the exact Λ matrix. The diagonal and subdiagonal elements are computed following Olson & Kunasz (1987):\\nwith ∆τ d−1 ≡ (τ d − τ d−1 )/µ. At large optical depths with increasing ∆τ steps (the depth grid is roughly equidistant in log τ ) the subdiagonals A d+1 and C d−1 vanish, and the diagonal B d approaches unity, reflecting the fact that the radiation field is more and more determined by local properties of the matter. At very small optical depths all elements of Λ ⋆ vanish, reflecting the non-localness of the radiation field at these depths.\"\n },\n {\n \"title\": \"Acceleration of convergence\",\n \"content\": \"Our code takes advantage of an acceleration scheme to speed up convergence of the iteration cycle Eq. 2. We implemented the scheme originally proposed by Ng (1974) and later by Auer (1987). It extrapolates the correction vector δψ d from the previous three iterations. From our experience the extrapolation often yields over-corrections, resulting in alternating convergence or even divergence. In contrast, use of a tri-diagonal ALO usually results in satisfactorily fast convergence, so that the acceleration scheme is rarely used.\"\n },\n {\n \"title\": \"Solution of the non-linear equations by iteration\",\n \"content\": \"At each depth the complete set of non-linear equations Eq. 2 for a single iteration step comprises the equations for statistical, radiative, and hydrostatic equilibrium, and the particle-conservation equation. For the numerical solution we introduce discrete depth and frequency grids. The equations are then linearized and solved by a suitable iterative scheme. Explicit angle-dependence of the radiation field is not required here and is consequently eliminated by the use of variable Eddington factors. Angle dependence is considered only in the formal solution of the transfer equation. Our program requires an input modelatmosphere structure as a starting approximation, together with an atomic data file, and a frequency grid. Depth and frequency grids are therefore set up in advance by separate programs.\"\n },\n {\n \"title\": \"Discretization\",\n \"content\": \"After a depth-grid is chosen, we start by computing a gray LTE continuum model using the Unsöld-Lucy temperature-correction procedure. Depth points (typical number is 90) are chosen in equidistant steps on a logarithmic (Rosseland) optical depth scale. The converged LTE model (temperature and density structure, given on a column mass depth scale) is written to a file that is read by PRO2. The NLTE code uses the column mass as an independent depth variable. The frequency grid is determined by the atomic-data input file. Frequency points are set blue-and red-ward of each absorption edge, and for each spectral line. Gaps are filled up by setting continuum points. Finally, the quadrature weights are computed. Frequency integrals appearing e.g. in Eq. 30 are replaced by quadrature sums, and depth derivatives by difference quotients.\"\n },\n {\n \"title\": \"Linearization\",\n \"content\": \"All variables x are replaced by x → x+δx where δx denotes a small perturbation of x. Terms not linear in these perturbations are neglected. The perturbations are expressed by perturbations of the basic variables:\\nAs an illustrative example we linearize the equation for radiative equilibrium.\\nMost other linearized equations may be found in Werner (1986). Assigning two indices (d for depth and i for frequency of a grid with NF points) to the variables and denoting the quadrature weights with w i Eq. 30 becomes:\\nNote that we do not linearize ∆τ i . Because of this, convergence properties may deteriorate significantly in some cases. Perturbations δS di , δχ di are expressed by Eq. 45, and the perturbation of the mean intensity J di is, according to Eq. 42, given through the perturbations of the source function at the current and the two adjacent depths:\\nwhere A, B, C are the Λ matrix elements from Eq. 44. The δJ d−1,i contain the term C d−2,i δS d−2,i which is neglected because we want to account only for nearest neighbor coupling. We write δS di with the help of Eq. 45 and observe that for any variable z:\\nDerivatives of opacity and emissivity with respect to temperature, electron and population densities are computed from analytical expressions (see e.g. Mihalas et al. 1975, Werner 1987. We finally get from Eq. 46:\\nThe curly brackets {· · ·} denote terms that are similar to those multiplied by perturbations of the temperature. Instead of partial derivatives with respect to T , they contain derivatives with respect to n e and the populations n l . They all represent coefficients of the matrices α, β, γ in Eq. 5.\"\n },\n {\n \"title\": \"Newton-Raphson iteration\",\n \"content\": \"As described in Sect. 2. the linearized equations have a tri-diagonal block-matrix form, see Eq. 5. Inversion of the grand matrix (≡ T sized (N N · N D) × (N N · N D), i.e. about 10 4 × 10 4 in typical applications) is performed with a block-Gaussian elimination scheme, which means that our iteration of the non-linear equations is a multi-dimensional Newton-Raphson method. The problem is structurally simplified when explicit-depth coupling is avoided by the use of a local ALO, however, the numerical effort is not reduced much, because in both cases the main effort lies with the inversion of matrices sized N N × N N . The Newton-Raphson iteration involves two numerically expensive steps, first setting up the Jacobian (comprising α, β, γ) and then inverting it. Additionally, the matrix inversions in Eq. 6 limit their size to about N N = 250 because otherwise numerical accuracy is lost. Two variants recently introduced in stellar atmosphere calculations are able to improve both numerical accuracy and, best of all, computational speed.\"\n },\n {\n \"title\": \"Alternative fast solution techniques for non-linear equations: Broyden-and Kantorovich-variants\",\n \"content\": \"Broyden's (1965) variant belongs to the family of quasi-Newton methods; it was first used in model-atmosphere calculations in Dreizler & Werner (1991), Hamann et al. 1991, Koesterke et al. (1992. It avoids the repeated set-up of the Jacobian by the use of an update formula. In addition, it also gives an update formula for the inverse Jacobian. In the case of a local ALO the solution of the linearized system at any depth is:\\nLet β −1 k be the k-th iterate of the inverse Jacobian, then an update can be found from:\\nwhere ⊗ denotes the dyadic product, and where we have defined:\\ns k ≡ δψ k solution vector of preceding linearization, y k ≡ c k+1 − c k difference of actual and preceding residuum.\\nThe convergence rate is super-linear, i.e. slower than the quadratic rate of the Newton-Raphson method; but this is more than compensated by the tremendous speed-up for a single iteration step. It is not always necessary to begin the iteration with the calculation of an exact Jacobian and its inversion. Experience shows that in an advanced stage of the overall (ALI-) iteration Eq. 2 (i.e. when corrections become small, of the order 1%) we can start the linearization cycle Eq. 51 by using the inverse Jacobian from the previous overall iteration. Computational speed-up is extreme in this case, however, it requires storage of the Jacobians of all depths. More difficult is its application to the tri-diagonal ALO case. Here we have to update the grand matrix T which, as already mentioned, is of block tri-diagonal form. We cannot update its inverse, because it is never computed explicitly. Furthermore we need an update formula that preserves the block tridiagonal form which is a prerequisite for its inversion by the Feautrier scheme Eq. 6. Such a formula was found by Schubert (1970):\\nwheres k ≡ Zs k with the structure matrix Z as defined by:\\nThe vectors s k and y k are defined as above but now they span quantities over all depth instead of a single depth point. With this formula we obtain new submatrices α, β, γ and c with which the Feautrier scheme Eq. 6 is solved again. This procedure saves the computation of derivatives. Another feature realized in our program also saves the repeated inversion of q ≡ (β d + γ d D d−1 ) by updating its inverse with the Broyden formula Eq. 51. Similar to the diagonal ALO case it is also possible to pass starting matrices from one overall iteration Eq. 2 to the next for the update of T and the matrix q −1 . In both cases the user specifies two threshold values for the maximum relative correction in δψ which cause the program to switch from Newton-Raphson to Broyden stages 1 and 2. During stage 1, each new overall cycle Eq. 2 is started with an exact calculation and inversion of all matrices involved, and in stage 2 these matrices are passed through each iteration. Another variant, the Kantorovich (1949) method has been introduced into model-atmosphere calculations (Hubeny & Lanz 1992). It is more simple and straightforward to implement. This method simply keeps the Jacobian fixed during the linearization cycle; it is surprisingly stable. In fact it turns out to be even more stable (i.e. it can be utilized in an earlier stage of iteration) than the Broyden method in the tri-diagonal ALO case. The user of PRO2 may choose this variant in two stages in analogy to the Broyden variant. It was found that in stage 2 it is necessary to update the Jacobian every 5 to 10 overall iterations in order to prevent divergence.\"\n },\n {\n \"title\": \"NLTE metal-line blanketing\",\n \"content\": \"Despite the increase in capacity for solving NLTE model-atmosphere problems given by the ALI method combined with pre-conditioning techniques, the blanketing by millions of lines from the iron-group elements arising from transitions among some 10 5 levels can be attacked only with statistical methods. These were introduced into NLTE model-atmosphere work by Anderson (1989Anderson ( , 1991. At the outset, model atoms are constructed by combining many thousands of levels into a relatively small number of superlevels which can be treated by ALI (or other) methods. Then, in order to reduce the computational effort, two approaches were developed which vastly decrease the number of frequency points (hence the number of transfer equations to be solved) needed to describe properly the complex frequency dependence of the opacity. These two approaches have their roots in LTE modeling techniques, where statistical methods are applied for the same reason in the treatment of opacity: The Opacity Distribution Function (ODF) and Opacity Sampling (OS) approaches. Both are based on the fact that the opacity (in the LTE approximation) is a function of two only local thermodynamic quantities. Roughly speaking, each opacity source can be written in terms of a population density and a photon cross-section for the respective radiative transition: κ ν ∼ n l σ lu (ν). In LTE the population follows from the Saha-Boltzmann equations, hence n l = n l (n e , T ). The OS and ODF methods use such pre-tabulated (on a very fine frequency mesh) κ ν (n e , T ) during the model atmosphere calculations. The NLTE situation is more complicated, because pre-tabulation of opacities is not useful. The occupation numbers at any depth now also depend explicitly on the radiation field (via the rate equations which replace the TE Saha-Boltzmann statistics) and thus on the populations in every other depth of the atmosphere. As a consequence, the OS and ODF methods are not applied to opacity tabulations, but to tabulations of the photon cross-sections σ(ν). These do depend on local quantities only, e.g. line broadening by Stark-and Doppler-effects is calculated from T and n e . In the NLTE case cross-sections take over the role which the opacity played in the LTE case. So, strictly speaking, the designation OS and ODF is not quite correct in the NLTE context.\\nThe strategy in our code is the following. Before any model atmosphere calculation is started, the atomic data are prepared by constructing superlevels, and the cross-sections for superlines. Then these cross-sections are either sampled on a coarse frequency grid or ODFs are constructed. These data are put into the model atom which is read by the code. The code does not know if OS or ODFs are used, i.e. it is written to be independent of these approaches.\\nFurther details about our implementation are described in Dreizler & Werner (1993), Haas et al. (1996), Werner & Dreizler (1999), and by Rauch & Deetjen in this volume.\"\n },\n {\n \"title\": \"Summary\",\n \"content\": \"We have presented in detail our technique for numerical solution of the classical model-atmosphere problem. The construction of metal-line-blanketed models in hydrostatic and radiative equilibrium under NLTE conditions was the last and\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". L S Anderson, ApJ. 339558Anderson, L. S. 1989, ApJ, 339, 558","doi":""},{"text":"L S Anderson, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D. G. HummerDordrechtKluwer34129Anderson, L. S. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D. G. Hummer, (Dordrecht: Kluwer), 29","doi":""},{"text":". L H Auer, Numerical Radiative Transfer, ed. W. KalkofenCambridge University Press101Auer, L. H. 1987, in Numerical Radiative Transfer, ed. W. Kalkofen, Cambridge University Press, 101","doi":""},{"text":". L H Auer, D Mihalas, ApJ. 158641Auer, L. H., & Mihalas, D. 1969, ApJ, 158, 641","doi":""},{"text":". C G Broyden, Math. Comp. 19577Broyden, C. G. 1965, Math. Comp., 19, 577","doi":""},{"text":". C J Cannon, ApJ. 185621Cannon, C. J. 1973, ApJ, 185, 621","doi":""},{"text":"S Dreizler, K Werner, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer341155Dreizler, S., & Werner, K. 1991, in Stellar Atmospheres: Beyond Classical Mod- els, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 155","doi":""},{"text":". S Dreizler, K Werner, A&A. 278199Dreizler, S., & Werner, K. 1993, A&A, 278, 199","doi":""},{"text":". S Dreizler, B Wolff, A&A. 348189Dreizler, S., & Wolff, B. 1999, A&A, 348, 189","doi":""},{"text":". P Feautrier, C. R. Acad. Sci. 2583189Feautrier, P. 1964, C. R. Acad. Sci. Paris, 258, 3189","doi":""},{"text":". S Haas, S Dreizler, U Heber, S Jeffery, K Werner, A&A. 311669Haas, S., Dreizler, S., Heber, U., Jeffery, S., & Werner, K. 1996, A&A, 311, 669","doi":""},{"text":"W.-R Hamann, W.-R Hamann, L Koesterke, U Wessolowski, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer34169Hamann, W.-R. 1994, private communication Hamann, W.-R., Koesterke, L., & Wessolowski, U. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 69","doi":""},{"text":". P H Hauschildt, JQSRT. 50301Hauschildt, P. H. 1993, JQSRT, 50, 301","doi":""},{"text":"P H Hauschildt, E Baron, Computational Astrophysics. H. Riffert & K. Werner10941Hauschildt, P. H., & Baron, E. 1999, in Computational Astrophysics, ed. H. Rif- fert & K. Werner, Journal of Computational and Applied Mathematics, 109, 41","doi":""},{"text":". Deetjen Werner, Dreizler, Nagel, Rauch, I Schuh Hubeny, Comput. Phys. Commun. 52103Werner, Deetjen, Dreizler, Nagel, Rauch, & Schuh Hubeny, I. 1988, Comput. Phys. Commun., 52, 103","doi":""},{"text":"I Hubeny, Atmospheres of Early-Type Stars. U. Heber & C.S. JefferyBerlinSpringer401377Hubeny, I. 1992, in Atmospheres of Early-Type Stars, ed. U. Heber & C.S. Jeffery, Lecture Notes in Physics Vol. 401, (Berlin: Springer), 377","doi":""},{"text":". I Hubeny, T Lanz, A&A. 262501Hubeny, I., & Lanz, T. 1992, A&A, 262, 501","doi":""},{"text":". I Hubeny, T Lanz, ApJ. 439875Hubeny, I., & Lanz, T. 1995, ApJ, 439, 875","doi":""},{"text":". I Hubeny, D G Hummer, T Lanz, A&A. 282157Hubeny, I., Hummer, D. G., & Lanz, T. 1994, A&A, 282, 157","doi":""},{"text":". D G Hummer, D Mihalas, ApJ. 331794Hummer, D. G., & Mihalas, D. 1988, ApJ, 331, 794","doi":""},{"text":". D G Hummer, K A Berrington, W Eissner, A K Pradhan, H E Saraph, J A Tully, A&A. 279298Hummer, D. G., Berrington, K. A., Eissner, W., Pradhan, A. K., Saraph, H. E., & Tully, J. A. 1993, A&A, 279, 298","doi":""},{"text":". L V Kantorovich, Trudy Mat. Inst. Steklova, XXVIII. 104Kantorovich L. V. 1949, Trudy Mat. Inst. Steklova, XXVIII, 104","doi":""},{"text":". L Koesterke, W.-R Hamann, P Kosmol, A&A. 255490Koesterke, L., Hamann, W.-R., & Kosmol, P. 1992, A&A, 255, 490","doi":""},{"text":"R L Kurucz, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer341441Kurucz, R. L. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 441","doi":""},{"text":"L B Lucy, First Harvard Smithsonian Conference on Stellar Atmospheres. Cambridge, Massachusetts, 93Lucy, L. B. 1964, in First Harvard Smithsonian Conference on Stellar Atmo- spheres, Smithsonian Astrophysical Observatory Special Report No. 167, Cambridge, Massachusetts, 93","doi":""},{"text":"D Mihalas, D Mihalas, D G Hummer, Stellar Atmospheres. San FranciscoFreeman1798272nd editionMihalas, D. 1978, Stellar Atmospheres, 2nd edition, (San Francisco: Freeman) Mihalas, D., & Hummer, D. G. 1973, ApJ, 179, 827","doi":""},{"text":". D Mihalas, D G Hummer, ApJS. 265343Mihalas, D., & Hummer, D. G. 1974, ApJS, 265, 343","doi":""},{"text":"D Mihalas, L H Auer, J N Heasley, NCAR Technical Note STR-104. BoulderNational Center for Atmospheric ResearchMihalas, D., Auer, L. H., & Heasley, J. N. 1975, NCAR Technical Note STR-104, (Boulder: National Center for Atmospheric Research)","doi":""},{"text":". T Nagel, S Dreizler, K Werner, ASP Conf. Ser. White Dwarfs, ed. J. L. Provencal, H. L. Shipman, J. MacDonald & S. Goodchild22686ASPNagel, T., Dreizler, S., & Werner, K. 2001, in White Dwarfs, ed. J. L. Provencal, H. L. Shipman, J. MacDonald & S. Goodchild, ASP Conf. Ser. Vol. 226, (San Francisco: ASP), 86","doi":""},{"text":". K C Ng, J. Chem. Phys. 612680Ng, K. C. 1974, J. Chem. Phys., 61, 2680","doi":""},{"text":". G L Olson, P B Kunasz, JQSRT. 38325Olson, G. L., & Kunasz, P. B. 1987, JQSRT, 38, 325","doi":""},{"text":". G L Olson, L H Auer, J R Buchler, JQSRT. 35431Olson, G. L., Auer, L. H., & Buchler, J. R. 1986, JQSRT, 35, 431","doi":""},{"text":"1972 in Line Formation in the Presence of Magnetic Fields. G B Rybicki, G ; R, L Athay, & G House, Jr Newkirk, Boulder: High Altitude Observatory. 145Rybicki, G. B. 1972 in Line Formation in the Presence of Magnetic Fields, ed. R. G. Athay, L. House & G. Newkirk, Jr., (Boulder: High Altitude Observatory), 145","doi":""},{"text":". G B Rybicki, D G Hummer, A&A. 245171Rybicki, G. B., & Hummer, D. G. 1991, A&A, 245, 171","doi":""},{"text":". G B Scharmer, ApJ. 249720Scharmer, G. B. 1981, ApJ, 249, 720","doi":""},{"text":". G B Scharmer, A Nordlund, StockholmScharmer, G. B., & Nordlund, A. 1982, Stockholm Observatory Report 19","doi":""},{"text":". L K Schubert, Math. Comp. 2427Schubert, L. K. 1970, Math. Comp., 24, 27","doi":""},{"text":". M J Seaton, Y Yan, D Mihalas, A K Pradhan, MNRAS. 266805Seaton, M. J., Yan, Y., Mihalas, D.,& Pradhan, A. K. 1994, MNRAS, 266, 805","doi":""},{"text":". K Werner, A&A. 161177Werner, K. 1986, A&A, 161, 177","doi":""},{"text":". K Werner, A&A. 226265Universität Kiel WernerPhD ThesisWerner, K. 1987, PhD Thesis, Universität Kiel Werner, K. 1989, A&A, 226, 265","doi":""},{"text":". K Werner, D Husfeld, A&A. 148417Werner, K., & Husfeld, D. 1985, A&A, 148, 417","doi":""},{"text":"K Werner, S Dreizler, Computational Astrophysics. H. Riffert & K. Werner10965Werner, K., & Dreizler, S. 1999, in Computational Astrophysics, ed. H. Riffert & K. Werner, Journal of Computational and Applied Mathematics, 109, 65","doi":""}],"string":"[\n {\n \"text\": \". L S Anderson, ApJ. 339558Anderson, L. S. 1989, ApJ, 339, 558\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L S Anderson, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D. G. HummerDordrechtKluwer34129Anderson, L. S. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D. G. Hummer, (Dordrecht: Kluwer), 29\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L H Auer, Numerical Radiative Transfer, ed. W. KalkofenCambridge University Press101Auer, L. H. 1987, in Numerical Radiative Transfer, ed. W. Kalkofen, Cambridge University Press, 101\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L H Auer, D Mihalas, ApJ. 158641Auer, L. H., & Mihalas, D. 1969, ApJ, 158, 641\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C G Broyden, Math. Comp. 19577Broyden, C. G. 1965, Math. Comp., 19, 577\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C J Cannon, ApJ. 185621Cannon, C. J. 1973, ApJ, 185, 621\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Dreizler, K Werner, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer341155Dreizler, S., & Werner, K. 1991, in Stellar Atmospheres: Beyond Classical Mod- els, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 155\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Dreizler, K Werner, A&A. 278199Dreizler, S., & Werner, K. 1993, A&A, 278, 199\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Dreizler, B Wolff, A&A. 348189Dreizler, S., & Wolff, B. 1999, A&A, 348, 189\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Feautrier, C. R. Acad. Sci. 2583189Feautrier, P. 1964, C. R. Acad. Sci. Paris, 258, 3189\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Haas, S Dreizler, U Heber, S Jeffery, K Werner, A&A. 311669Haas, S., Dreizler, S., Heber, U., Jeffery, S., & Werner, K. 1996, A&A, 311, 669\",\n \"doi\": \"\"\n },\n {\n \"text\": \"W.-R Hamann, W.-R Hamann, L Koesterke, U Wessolowski, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer34169Hamann, W.-R. 1994, private communication Hamann, W.-R., Koesterke, L., & Wessolowski, U. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 69\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P H Hauschildt, JQSRT. 50301Hauschildt, P. H. 1993, JQSRT, 50, 301\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P H Hauschildt, E Baron, Computational Astrophysics. H. Riffert & K. Werner10941Hauschildt, P. H., & Baron, E. 1999, in Computational Astrophysics, ed. H. Rif- fert & K. Werner, Journal of Computational and Applied Mathematics, 109, 41\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Deetjen Werner, Dreizler, Nagel, Rauch, I Schuh Hubeny, Comput. Phys. Commun. 52103Werner, Deetjen, Dreizler, Nagel, Rauch, & Schuh Hubeny, I. 1988, Comput. Phys. Commun., 52, 103\",\n \"doi\": \"\"\n },\n {\n \"text\": \"I Hubeny, Atmospheres of Early-Type Stars. U. Heber & C.S. JefferyBerlinSpringer401377Hubeny, I. 1992, in Atmospheres of Early-Type Stars, ed. U. Heber & C.S. Jeffery, Lecture Notes in Physics Vol. 401, (Berlin: Springer), 377\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Hubeny, T Lanz, A&A. 262501Hubeny, I., & Lanz, T. 1992, A&A, 262, 501\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Hubeny, T Lanz, ApJ. 439875Hubeny, I., & Lanz, T. 1995, ApJ, 439, 875\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Hubeny, D G Hummer, T Lanz, A&A. 282157Hubeny, I., Hummer, D. G., & Lanz, T. 1994, A&A, 282, 157\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D G Hummer, D Mihalas, ApJ. 331794Hummer, D. G., & Mihalas, D. 1988, ApJ, 331, 794\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D G Hummer, K A Berrington, W Eissner, A K Pradhan, H E Saraph, J A Tully, A&A. 279298Hummer, D. G., Berrington, K. A., Eissner, W., Pradhan, A. K., Saraph, H. E., & Tully, J. A. 1993, A&A, 279, 298\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L V Kantorovich, Trudy Mat. Inst. Steklova, XXVIII. 104Kantorovich L. V. 1949, Trudy Mat. Inst. Steklova, XXVIII, 104\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Koesterke, W.-R Hamann, P Kosmol, A&A. 255490Koesterke, L., Hamann, W.-R., & Kosmol, P. 1992, A&A, 255, 490\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R L Kurucz, Stellar Atmospheres: Beyond Classical Models. L. Crivellari, I. Hubeny & D.G. HummerDordrechtKluwer341441Kurucz, R. L. 1991, in Stellar Atmospheres: Beyond Classical Models, NATO ASI Series C, Vol. 341, ed. L. Crivellari, I. Hubeny & D.G. Hummer, (Dordrecht: Kluwer), 441\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L B Lucy, First Harvard Smithsonian Conference on Stellar Atmospheres. Cambridge, Massachusetts, 93Lucy, L. B. 1964, in First Harvard Smithsonian Conference on Stellar Atmo- spheres, Smithsonian Astrophysical Observatory Special Report No. 167, Cambridge, Massachusetts, 93\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Mihalas, D Mihalas, D G Hummer, Stellar Atmospheres. San FranciscoFreeman1798272nd editionMihalas, D. 1978, Stellar Atmospheres, 2nd edition, (San Francisco: Freeman) Mihalas, D., & Hummer, D. G. 1973, ApJ, 179, 827\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Mihalas, D G Hummer, ApJS. 265343Mihalas, D., & Hummer, D. G. 1974, ApJS, 265, 343\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Mihalas, L H Auer, J N Heasley, NCAR Technical Note STR-104. BoulderNational Center for Atmospheric ResearchMihalas, D., Auer, L. H., & Heasley, J. N. 1975, NCAR Technical Note STR-104, (Boulder: National Center for Atmospheric Research)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Nagel, S Dreizler, K Werner, ASP Conf. Ser. White Dwarfs, ed. J. L. Provencal, H. L. Shipman, J. MacDonald & S. Goodchild22686ASPNagel, T., Dreizler, S., & Werner, K. 2001, in White Dwarfs, ed. J. L. Provencal, H. L. Shipman, J. MacDonald & S. Goodchild, ASP Conf. Ser. Vol. 226, (San Francisco: ASP), 86\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K C Ng, J. Chem. Phys. 612680Ng, K. C. 1974, J. Chem. Phys., 61, 2680\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G L Olson, P B Kunasz, JQSRT. 38325Olson, G. L., & Kunasz, P. B. 1987, JQSRT, 38, 325\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G L Olson, L H Auer, J R Buchler, JQSRT. 35431Olson, G. L., Auer, L. H., & Buchler, J. R. 1986, JQSRT, 35, 431\",\n \"doi\": \"\"\n },\n {\n \"text\": \"1972 in Line Formation in the Presence of Magnetic Fields. G B Rybicki, G ; R, L Athay, & G House, Jr Newkirk, Boulder: High Altitude Observatory. 145Rybicki, G. B. 1972 in Line Formation in the Presence of Magnetic Fields, ed. R. G. Athay, L. House & G. Newkirk, Jr., (Boulder: High Altitude Observatory), 145\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G B Rybicki, D G Hummer, A&A. 245171Rybicki, G. B., & Hummer, D. G. 1991, A&A, 245, 171\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G B Scharmer, ApJ. 249720Scharmer, G. B. 1981, ApJ, 249, 720\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G B Scharmer, A Nordlund, StockholmScharmer, G. B., & Nordlund, A. 1982, Stockholm Observatory Report 19\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L K Schubert, Math. Comp. 2427Schubert, L. K. 1970, Math. Comp., 24, 27\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M J Seaton, Y Yan, D Mihalas, A K Pradhan, MNRAS. 266805Seaton, M. J., Yan, Y., Mihalas, D.,& Pradhan, A. K. 1994, MNRAS, 266, 805\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Werner, A&A. 161177Werner, K. 1986, A&A, 161, 177\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Werner, A&A. 226265Universität Kiel WernerPhD ThesisWerner, K. 1987, PhD Thesis, Universität Kiel Werner, K. 1989, A&A, 226, 265\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Werner, D Husfeld, A&A. 148417Werner, K., & Husfeld, D. 1985, A&A, 148, 417\",\n \"doi\": \"\"\n },\n {\n \"text\": \"K Werner, S Dreizler, Computational Astrophysics. H. Riffert & K. Werner10965Werner, K., & Dreizler, S. 1999, in Computational Astrophysics, ed. H. Riffert & K. Werner, Journal of Computational and Applied Mathematics, 109, 65\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":47,"cells":{"text":{"kind":"string","value":"\nTravelling waves in the Fisher-KPP equation with nonlinear diffusion and a non-Lipschitzian reaction term\n\n\nPavel Drábek \nTravelling waves in the Fisher-KPP equation with nonlinear diffusion and a non-Lipschitzian reaction term\n\nWe consider a one-dimensional reaction-diffusion equation of Fisher-Kolmogoroff-Petrovsky-Piscounoff type. We investigate the effect of the interaction between the nonlinear diffusion coefficient and the reaction term on the existence and nonexistence of travelling waves. Our diffusion coefficient is allowed to be degenerate or singular at both equilibrium points, 0 and 1, while the reaction term need not be differentiable. These facts influence the existence and qualitative properties of travelling waves in a substantial way. This is a joint work with Peter Takac."},"corpus_id":{"kind":"number","value":119639399,"string":"119,639,399"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1803.10306\",\n \"mag\": 2794596604,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [],\n \"pdf_sha\": \"e7432e1ba10486080bbb87ac9a7bb859eae3bd4d\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Travelling waves in the Fisher-KPP equation with nonlinear diffusion and a non-Lipschitzian reaction term"},"authors":{"kind":"list like","value":["Pavel Drábek"],"string":"[\n \"Pavel Drábek\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We consider a one-dimensional reaction-diffusion equation of Fisher-Kolmogoroff-Petrovsky-Piscounoff type. We investigate the effect of the interaction between the nonlinear diffusion coefficient and the reaction term on the existence and nonexistence of travelling waves. Our diffusion coefficient is allowed to be degenerate or singular at both equilibrium points, 0 and 1, while the reaction term need not be differentiable. These facts influence the existence and qualitative properties of travelling waves in a substantial way. This is a joint work with Peter Takac."},"sections":{"kind":"list like","value":[],"string":"[]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[],"string":"[]","isConversational":false}}},{"rowIdx":48,"cells":{"text":{"kind":"string","value":"\nÒÓØ ÓÒ Ò Ö ØÝÔ × Å ÖØ Ò Ð Ö Ù Ù×Ø ½ ¸¾¼¼\n17 Aug 2006\n\nÒÓØ ÓÒ Ò Ö ØÝÔ × Å ÖØ Ò Ð Ö Ù Ù×Ø ½ ¸¾¼¼\n17 Aug 2006\n\n×ØÖ Ø ÁÒ ×Ø Ð Ð Ò ÖÓÙÔ¸Û Ö Ø Ö Þ Ò Ö ØÝÔ × Ó Ó× Ø× Ó ØÝÔ ¹ Ò Ð ×Ù ÖÓÙÔ׺ Ì ÓÐÐÓÛ Ò Ö Ñ Ö × Ô ÖØ Ó Ø ÓÐ ÐÓÖ º ÁØ Û × ×Ø Ø Ý Ø ÙØ ÓÖ ½ ¼ Ò Ð ØØ Ö ØÓ Ò Ð Ä × Öº Ì ÓÖ Ñ ½º Ä Ø G ×Ø Ð Ð Ò ÖÓÙÔº a¸b Ò c Ô ÖÛ × Ò Ô Ò ÒØ ÓÚ Ö 0 Ò a + b + c = 0º Ì Ò ½º Ì ×ØÖÓÒ ØÝÔ × Ó a¸b Ò c ÐÐ Ú Ø × Ñ ÓÒÒ Ø ×Ø Ð Þ Ö U º ¾º a¸b¸ Ò c Ö Ò Ö Ð Ñ ÒØ× Ó acl eq (0) Ò Ð Ó× Ø× Ó U º Á G × ØÓØ ÐÐÝ ØÖ Ò× Ò Òظ Ø ÓÐÐÓÛ× Ø Ø a¸b Ò c Ú Ø × Ñ ÅÓÖÐ Ý Ö Ò ÓÚ Ö 0¸Ò Ñ ÐÝ Ò Ö Ò Ó U º ÅÓÖ ÓÚ Ö U × Ò Ð º Ä Ø p Ò q ×ØÖÓÒ ØÝÔ × ÓÚ Ö 0º Ì Ò Hom(p, q) = {g ∈ G | ∀a a |= p |g ⇒ a + g |= q |g } × Ò acl eq (0) ØÝÔ Ò Ð ´ G × ØÓØ ÐÐÝ ØÖ Ò× Ò ÒØ Ò Ð µ ×Ù × Ø Ó Gº Ç ÓÙÖ× Hom(p, p) = Stab(p)º Ä ÑÑ ¾º Ä Ø p¸q Ò r ×ØÖÓÒ ØÝÔ ×º Ì Ò • ÓÖ ÐÐ A ⊂ G Ò g ∈ Hom(p, q) Û Ú a |= p |g,A ⇒ a + g |= q |g,A .\n• Hom(p, q) + Hom(q, r) ⊂ Hom(p, r)\n• Hom(p, q) = −Hom(q, p) = Hom(−q, −p) • 0 ∈ Hom(p, p) • Á Hom(p, q) × ÒÓÒ ÑÔØÝ Ò G ØÓØ ÐÐÝ ØÖ Ò× Ò ÒØM R(Hom(p, q)) ≤ MR(p) = MR(q). ½ ÈÖÓÓ º Ä Ø a Ö Ð Þ Ø ÓÒ Ó p¸Û × Ò Ô Ò ÒØ Ó g, Aº Ì Ò a | ⌣g A ÑÔÐ × Ø Ø a + g | ⌣g Aº Ë Ò a + g | ⌣ g¸Û Ú a + g | ⌣ g, Aº Ä Ø g ∈ Hom(p, q)¸h ∈ Hom(q, r) Ò a Ö Ð Þ Ø ÓÒ Ó p Û × Ò Ô Ò ÒØ Ó g + hº Ï Ñ Ý ××ÙÑ Ø Ø a × Ò Ô Ò ÒØ Ó g, hº Ì Ò a + g × Ö Ð Þ Ø ÓÒ Ó q¸Û × Ð×Ó Ò Ô Ò ÒØ Ó g, hº Ì Ö ÓÖ a + g + h × Ö Ð Þ Ø ÓÒ Ó r¸Û × Ò Ô Ò ÒØ Ó g, h Ò Ø Ö ÓÖ Ó g + hº Ä Ø g ∈ Hom(p, q) Ò a Ö Ð Þ Ø ÓÒ Ó p¸Û × Ò Ô Ò ÒØ Ó gº Ì Ò ×\nMR(a) = MR(a/g) = MR(a + g/g) = MR(a + g).\n\nÌ × ÑÔÐ × MR(p) = MR(q)º ÖÓÑ MR(g) = MR(g/a) = MR(a + g/a) ≤ MR(a + g)\nÓÐÐÓÛ× MR(Hom(p, q)) ≤ MR(q)º ××ÙÑ ÒÓÛ Ø Ø a¸b Ò c Ö × Ò Ø Ø ÓÖ Ñ¸p¸q Ò r Ø ×ØÖÓÒ ØÝÔ × Ó a¸b Ò cº Ì Ò¸ØÖ Ú ÐÐÝ• p(G) ⊂ Hom(q, −r) = Hom(r, −q) • q(G) ⊂ Hom(r, −p)) = Hom(p, −r) • r(G) ⊂ Hom(p, −q) = Hom(q, −p)º ÁØ ÓÐÐÓÛ× • p(G) − p(G) ⊂ Stab(q), Stab(r) • q(G) − q(G) ⊂ Stab(r), Stab(p) • r(G) − r(G) ⊂ Stab(p), Stab(q)º Ë Ò ÓÒ Ø ÓØ Ö Ò • Stab(p) ⊂ p(G) − p(G) • Stab(q) ⊂ q(G) − q(G) • Stab(r) ⊂ r(G) − r(G)¸ ÐÐ ×Ø Ð Þ Ö× Ö ÕÙ Ð ØÓ U = p(G) − p(G) = q(G) − q(G) = r(G) − r(G).\nÌ Ö ÓÖ p(G)¸q(G)¸r(G) ÐÐ Ð Ò acl eq (0) Ò Ð Ó× Ø× Ó U º Ë Ò U × Ø ×Ø Ð Þ Ö Ó Ó Ø × ØÝÔ ×¸p¸q¸r Ö Ò Ö ØÝÔ × Ò U × ÓÒÒ Ø º Ì × ÔÖÓÚ × Ø Ø ÓÖ Ñº\n "},"corpus_id":{"kind":"number","value":119651624,"string":"119,651,624"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"math/0608433\",\n \"mag\": 1678121057,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/math/0608433v1.pdf\"\n ],\n \"pdf_sha\": \"dfdb550004a0c2188ebcd0780be26a4dbf72be32\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"ÒÓØ ÓÒ Ò Ö ØÝÔ × Å ÖØ Ò Ð Ö Ù Ù×Ø ½ ¸¾¼¼"},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"abstract":{"kind":"string","value":""},"sections":{"kind":"list like","value":[],"string":"[]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[],"string":"[]","isConversational":false}}},{"rowIdx":49,"cells":{"text":{"kind":"string","value":"\nObserving the Effects of Chemistry on Exoplanets and Planet Formation\n15 Oct 2018\n\nBrett A Mcguire bmcguire@nrao.edu \nNational Radio Astronomy Observatory\n22903CharlottesvilleVA\n\nEdwin Bergin \nDepartment of Astronomy\nUniversity of Michigan\n48109Ann ArborMI\n\nGeoffrey A Blake \nDivision of Geological and Planetary Sciences\nCalifornia Institute of Technology\n91125PasadenaCA\n\nDivision of Chemistry and Chemical Engineering\nCalifornia Institute of Technology\n91125PasadenaCA\n\nAndrew M Burkhardt \nHarvard-Smithsonian Center for Astrophysics\n02138CambridgeMA\n\nL Ilsedore Cleeves \nDepartment of Astronomy\nUniversity of Virginia\n22904CharlottesvilleVA\n\nRyan A Loomis \nNational Radio Astronomy Observatory\n22903CharlottesvilleVA\n\nAnthony J Remijan \nNational Radio Astronomy Observatory\n22903CharlottesvilleVA\n\nChristopher N Shingledecker \nMax Planck Institute for Extraterrestrial Physics\nGarchingGermany\n\nInstitute for Theoretical Chemistry\nUniversity of Stuttgart\nGermany\n\nEric R Willis \nDepartment of Chemistry\nUniversity of Virginia\n22904CharlottesvilleVA\n\nObserving the Effects of Chemistry on Exoplanets and Planet Formation\n15 Oct 2018\n\n\nBackground\n\nThe location of condensation fronts (snow lines) in protoplanetary disks (PPDs) can have substantial effects on the physical and chemical evolution of planets in the system. Indeed, many models predict that these condensation fronts are the locations where planetesimal formation begins (Ros & Johansen 2013;Schoonenberg & Ormel 2017;Drazkowska & Alibert 2017). The location of these condensation fronts is determined both by the incident radiation from the protostar, but also, in large part, by the chemical composition of the gas and ice. The desorption temperature of the bulk ice varies significantly with the chemical composition and mixing ratio (Collings et al. 2004), even of minor constituents. This can be further complicated by a variance in the sticking efficiency of the icy grains with composition (Wang et al. 2005). It is clear that the underlying chemical composition in these ices, even at low mixing ratios, can have a substantial effect on the efficiency of planet formation.\n\nFurther, the relative gas vs solid-phase inventories at snow lines govern the composition of the planets and their atmospheres. These compositions (e.g., the C/O ratio in planetary atmospheres; Öberg et al. 2011) may therefore be useful probes of the history of exoplanets. Effectively using observed exoplanet elemental ratios to test those models therefore relies heavily on our ability to accurately interpret chemical abun-dances measured in disks to infer the inventory of undetectable species, and how these measurements affect observed elemental ratios.\n\nConstraining the chemical compositions in PPDs is challenging. The most complex gas-phase species yet detected in PPDs are methanol (CH 3 OH; Walsh et al. 2016) and methyl cyanide (CH 3 CN; Öberg et al. 2015), and more complex molecules are unlikley to be observed (Walsh et al. 2014). Further, in cold or shielded regions within disks, ices may be the major repository of C and O (Bergin et al. 2016). Direct observation of ices typically require space-based infrared facilities and a target source with a sufficiently bright background continuum and favorable viewing geometry. Even in massive young stellar objects with substantial ice columns, the list of firmly-detected ice species numbers less than a dozen (Boogert et al. 2015). Our knowledge of gas and ice compositions in disks must therefore rely on astrochemical models, using the detectable simple gas species as the primary constraint.\n\nThese models are not yet robust, despite substantial efforts (Garrod 2013;Herbst 2017;Wakelam et al. 2015). To reliably infer chemical compositions in PPDs, in which chemistry is closely coupled to physical conditions (temperature, density, UV flux, evolving dust populations, etc.) that vary with radius and scale height, necessitates that these models be well-constrained by observations in earlier stages of star formation, where more robust inventories can be observed. Here, we briefly discuss the state-of-the-art in progressively earlier stages of chemical evolution, and outline future directions and the technology and community support needed to enable them.\n\n\nChemical evolution in PPDs ...\n\nA number of larger scale studies of disk physical properties and chemistry have been enabled by the sensitivity and resolution of ALMA. Efforts to measure fundamental properties such as bulk mass (Ansdell et al. 2016), and larger compositional studies examining nitrogen (Guzmán et al. 2017) and deuterium ratios, are now possible for larger samples sizes. With robust chemical modeling, ALMA observations of simple hydrocarbons (e.g., C 2 H) can perhaps be used to infer C/O ratios indirectly (Bergin et al. 2016). As well, the JWST and, if commissioned, Origins Space Telescope, have significant potential to observe major ice components (e.g. H 2 O, CO, CO 2 , CH 3 OH) toward a wide variety of targets. Yet, these infrared facilities will still largely be blind to the complex chemistry occurring within the ices, and to the combined ice/gas coupling in warmer gas. That insight is only achievable through radio observations of gas-phase species, that are sensitive to populations orders of magnitude lower in abundance, coupled with chemical modeling.\n\n\n... and in complex, star-forming regions\n\nSgr B2(N), Orion-KL, and NGC 6334I are the most chemically and physically complex stages of star formation observed and modeled (Garrod et al. 2008;Crockett et al. 2014;McGuire et al. 2017). The extreme physical environments, interplay between the gas and ice, and complex structures often result in models which can reproduce observed abundances in, at best, a qualitative fashion (Laas et al. 2011;Garrod et al. 2008;Garrod 2013). These types of physical and chemical interactions are the same that are at play in PPDs, albeit on a much larger scale. It is therefore these environments that provide a unique test environment for the similar PDR style models applied to PPDs. Improvements in these models are most likely to be achieved by testing them against a wide range of environments from quiescent clouds to massive star forming regions.\n\nExisting facilities in the (sub-)millimeter regime, particularly ALMA, are capable of reaching the line-confusion limit in broadband line surveys of complex, star forming regions in a matter of hours. That is, the density of molecular lines in the resulting spectra is such that there is at least one spectral line every FWHM; there are no linefree channels. When line-confusion is reached, no additional information can be gained from longer integration times at the same spatial resolution. We are rapidly approaching the point at which deep drill observations with ALMA and the GBT will no longer produce new spectral lines. The end result of this limit is that no further detections of new species can be made, and thus the refinement of chemical networks will be limited to those species detectable at the confusion limit.\n\nIn the cm-wave regime, however, the most sensitive survey, the Prebiotic Interstellar Molecular Survey (PRIMOS; Neill et al. 2012) with the most sensitive facility (the GBT) toward the most line-rich source (Sgr B2) has lines in only ∼10-20% of its channels at cm-wavelengths. Diminishing returns on sensitivity for this survey have already set in, and to increase the sensitivity substantially would require hundreds of hours of dedicated observations for every ∼2 GHz window. Further, ALMA observations at higher frequencies are revealing that the region displays widely variable physical conditions within a relatively small environment, variation that is lost to single-dish GBT observations (Belloche et al. 2016). This offers a valuable opportunity to probe environment-induced chemical differentiation, but only with spatially-resolved observations which are not possible with the GBT.\n\nWhile the brightest lines in Sgr B2 are readily detectable with extant facilities such as the VLA and ATCA (Corby et al. 2015), the complex species are not efficiently observable, especially at high angular resolution. The only viable solution to this problem is the construction of a next-generation interferometer like the next-generation Very Large Array (ngVLA). Under the current design concept (Selina & Murphy 2017), the ngVLA is likely to not only be capable of spatially resolving chemistry in Sgr B2(N), but to reach the line-confusion limit even at cm-wavelengths, providing substantial new, complex molecular detections and their correlation with the physical conditions at work.\n\n\nApplications to PPDs\n\nOnce refined, these models can be used to accurately interpret observations from ALMA targeting the outermost condensation fronts within disks, primarily CO snow lines (Qi et al. 2015). Application in the terrestrial planet forming region, near the H 2 O snow line, will require AU-scale resolution with high sensitivity. A high-frequency ngVLA is the only facility currently in design with the required specifications. Further, while these models can then interpret results for C and O in disks, observations of the major N-bearing species, NH 3 will still be required at similar scales. An ngVLA operating at least through the 24 GHz NH 3 transition is therefore necessary.\n\nAn ngVLA design concept is needed that supports broadband, high surface-brightness sensitivity capabilities at the small to modest (0.1-3 ′′ ) scales required to probe the effects of the widely variable physical conditions on the chemical evolution in sources like Sgr B2 and unlock chemical inventories obscured by line confusion suffered at extant facilities like ALMA. The same instrument should be capable of observing NH 3 and imaging the planet forming zone inside the water snow line in disks, requiring AUscale (mas) resolution capabilities.\n\n\nChemical evolution in protostellar environments\n\nThe protostellar stage of star formation spans a wide range of sources, from 'hot corinos' such as IRAS 16293 (Jørgensen et al. 2016) to chemically rich outflow sources like L1157 (Mendoza et al. 2014). The molecular outflows from active young protostars like L1157 provide a unique environment to test the immediate impact of rapid changes in temperature and density on 'pristine' molecular clouds. In these sources, distance from the protostar along the outflow corresponds directly to time, allowing modeling of temporal evolution. Hot corinos are the next stage after L1157 and before hot cores like Sgr B2(N).\n\nThe best method for constraining the chemistry in these sources is broadband spectral line surveys. These sources are relatively compact, are beam-diluted in single-dish observations, and their linewidths are typically such that high-sensitivity data are often at or approaching line confusion limits, necessitating interferometric observation (Jørgensen et al. 2016). Because they are typically moderately warm (50-150 K; Jørgensen et al. 2016;McGuire et al. 2015), complex molecules display bright features in the 1-3 mm window well-covered by ALMA, NOEMA, and the SMA.\n\nThe SOLIS project at NOEMA and IRAM, which examines a sample of solar-type protostellar sources (Ceccarelli et al. 2017), and the PILS project at ALMA, which has focused particularly on an exquisitely detailed examination of the IRAS 16293 system (Jørgensen et al. 2016), are the primary active dedicated investigations into this class of objects. Extant observational facilities are likely sufficient to provide the required insight. Ongoing and future efforts on ALMA, NOEMA, and the SMA examining a large sample of protostellar environments are critical to constraining the effects of physical environment on chemical evolution and incorporating these effects into models.\n\n\nChemical evolution in quiescent environments\n\nCold (T ex ∼10 K), dark, starless clouds are an ideal environment in which to test the time evolution of chemical reaction networks under 'constant' physical conditions. The typical sources for exploring this stage of star and planet formation are highly extended (θ s > 30 ′′ ). At these excitation temperatures, most molecules, especially those more complex than CH 3 OH, will display bright emission at cm wavelengths. The best strategy for obtaining the required abundance and excitation information is therefore deep, broadband spectral line surveys at cm-wavelengths using single-dish facilities combined with chemical modeling (Cordiner et al. 2017;Kaifu et al. 2004;McGuire et al. 2018). The best extant instrument for this work is the Green Bank Telescope (GBT).\n\nEven in a source as extended as TMC-1, there exists evidence for large chemical variation over small spatial scales (0.02-0.03 pc; Peng et al. 1998;Dickens et al. 2001), and observations of such would probe the effects of the very earliest stages of collapse on the chemistry. These observations will require high spatial resolution, high surfacebrightness sensitivity, and broad spectral bandwidth to be efficient. Because of the cold temperatures, this further requires a cm-wavelength facility. While the VLA satisfies this need, it lacks the combination of surface brightness sensitivity and bandwidth necessary to conduct a sensitive line survey with any efficiency. The proposed ngVLA, in its current conceptual design, would prove more than optimal for this task.\n\nThe chemical evolution of all ranges of environments from planetary atmospheres (Arsenovic et al. 2016) to disk midplanes (Cleeves et al. 2013) to molecular clouds (Hudson et al. 2008) is impacted by radiation. The last few years have seen significant advances in modeling efforts to accurately include these effects (Shingledecker & Herbst 2018), but these efforts are young, and require substantial training on 'simple' environments like quiescent clouds where the radiation effects can be easily decoupled from dynamical physical events such as outflows and shocks. In short, efforts at the GBT and other large single-dish facilities are needed to survey quiescent chemical clouds. These efforts would be greatly bolstered by the development of broadband capabilities on the ngVLA to enable investigations of the chemical evolution upon initial gravitational collapse.\n\n\nSummary\n\nA combination of observational and theoretical efforts are needed that are focused on placing rigorous constraints on models of chemical evolution at each stage of the star and planet formation process. The observational requirements of these investigations necessitate a commitment by the community to support extant facilities and the development of technological advancements for the same, as well as the development of a next-generation VLA with high sensitivity and exquisite spatial resolution. The ultimate goal should be producing models that can be used to better calibrate our estimates of gas and ice compositions in PPDs, given the observations of simple gas-phase species achievable in these sources, and the construction of a radio facility capable of observationally exploiting these models in situ during planet formation.\n \n. M Ansdell, J P Williams, N Van Der Marel, The Astrophysical Journal. 8281Ansdell, M., Williams, J. P., van der Marel, N., et al. 2016, The Astrophysical Journal, 828, 1\n\n. P Arsenovic, E Rozanov, A Stenke, Journal of Atmospheric and Solar-Terrestrial Physics. 149180Arsenovic, P., Rozanov, E., Stenke, A., et al. 2016, Journal of Atmospheric and Solar-Terrestrial Physics, 149, 180\n\n. A Belloche, H S P Müller, R T Garrod, K M Menten, Astronomy & Astrophysics. 58791Belloche, A., Müller, H. S. P., Garrod, R. T., & Menten, K. M. 2016, Astronomy & Astro- physics, 587, A91\n\n. E A Bergin, F Du, L I Cleeves, The Astrophysical Journal. 8311Bergin, E. A., Du, F., Cleeves, L. I., et al. 2016, The Astrophysical Journal, 831, 1\n\n. A C A Boogert, P A Gerakines, D C B Whittet, Annual Review of Astronomy and Astrophysics. 53541Boogert, A. C. A., Gerakines, P. A., & Whittet, D. C. B. 2015, Annual Review of Astronomy and Astrophysics, 53, 541\n\n. C Ceccarelli, P Caselli, F Fontani, The Astrophysical Journal. 850176Ceccarelli, C., Caselli, P., Fontani, F., et al. 2017, The Astrophysical Journal, 850, 176\n\n. L I Cleeves, F C Adams, E A Bergin, R Visser, The Astrophysical Journal. 77728Cleeves, L. I., Adams, F. C., Bergin, E. A., & Visser, R. 2013, The Astrophysical Journal, 777, 28\n\n. M P Collings, M A Anderson, R Chen, Monthly Notices of the Royal Astronomical Society. 3541133Collings, M. P., Anderson, M. A., Chen, R., et al. 2004, Monthly Notices of the Royal Astro- nomical Society, 354, 1133\n\n. J F Corby, P A Jones, M R Cunningham, Monthly Notices of the Royal Astronomical Society. 4523969Corby, J. F., Jones, P. A., Cunningham, M. R., et al. 2015, Monthly Notices of the Royal Astronomical Society, 452, 3969\n\n. M A Cordiner, S B Charnley, Z Kisiel, B A Mcguire, Y.-J Kuan, The Astrophysical Journal. 850187Cordiner, M. A., Charnley, S. B., Kisiel, Z., McGuire, B. A., & Kuan, Y.-J. 2017, The Astro- physical Journal, 850, 187\n\n. N R Crockett, E A Bergin, J L Neill, The Astrophysical Journal. 787112Crockett, N. R., Bergin, E. A., Neill, J. L., et al. 2014, The Astrophysical Journal, 787, 112\n\n. J E Dickens, W D Langer, T Velusamy, The Astrophysical Journal. 558693Dickens, J. E., Langer, W. D., & Velusamy, T. 2001, The Astrophysical Journal, 558, 693\n\n. J Drazkowska, Y Alibert, Astronomy & Astrophysics. 60892Drazkowska, J., & Alibert, Y. 2017, Astronomy & Astrophysics, 608, A92\n\n. R T Garrod, The Astrophysical Journal. 76560Garrod, R. T. 2013, The Astrophysical Journal, 765, 60\n\n. R T Garrod, S L W Weaver, E Herbst, Astrophys. J. 682283Garrod, R. T., Weaver, S. L. W., & Herbst, E. 2008, Astrophys. J., 682, 283\n\n. V V Guzmán, K I Öberg, J Huang, R Loomis, C Qi, The Astrophysical Journal. 8361Guzmán, V. V., Öberg, K. I., Huang, J., Loomis, R., & Qi, C. 2017, The Astrophysical Journal, 836, 1\n\n. E Herbst, International Reviews in Physical Chemistry. 36287Herbst, E. 2017, International Reviews in Physical Chemistry, 36, 287\n\n. J Huang, K I Öberg, C Qi, The Astrophysical Journal. 8351Huang, J., Öberg, K. I., Qi, C., et al. 2017, The Astrophysical Journal, 835, 1\n\n. R L Hudson, M H Moore, J P Dworkin, M P Martin, Z D Pozun, Astrobiology. 8771Hudson, R. L., Moore, M. H., Dworkin, J. P., Martin, M. P., & Pozun, Z. D. 2008, Astrobiology, 8, 771\n\n. J K Jørgensen, M Van Der Wiel, A Coutens, Astronomy & Astrophysics. 595117Jørgensen, J. K., van der Wiel, M., Coutens, A., et al. 2016, Astronomy & Astrophysics, 595, A117\n\n. N Kaifu, M Ohishi, K Kawaguchi, Publications of the Astronomical Society of Japan. 5669Kaifu, N., Ohishi, M., Kawaguchi, K., et al. 2004, Publications of the Astronomical Society of Japan, 56, 69\n\n. J C Laas, R T Garrod, E Herbst, S L Weaver, The Astrophysical Journal. 72871Laas, J. C., Garrod, R. T., Herbst, E., & Widicus Weaver, S. L. 2011, The Astrophysical Journal, 728, 71\n\n. B A Mcguire, A M Burkhardt, S V Kalenskii, Science. 359202McGuire, B. A., Burkhardt, A. M., Kalenskii, S. V., et al. 2018, Science, 359, 202\n\n. B A Mcguire, P B Carroll, N M Dollhopf, The Astrophysical Journal. 8121McGuire, B. A., Carroll, P. B., Dollhopf, N. M., et al. 2015, The Astrophysical Journal, 812, 1\n\n. B A Mcguire, C N Shingledecker, E R Willis, The Astrophysical Journal Letters. 85146McGuire, B. A., Shingledecker, C. N., Willis, E. R., et al. 2017, The Astrophysical Journal Letters, 851, L46\n\n. E Mendoza, B Lefloch, A Lopez-Sepulcre, Monthly Notices of the Royal Astronomical Society. 445151Mendoza, E., Lefloch, B., Lopez-Sepulcre, A., et al. 2014, Monthly Notices of the Royal As- tronomical Society, 445, 151\n\n. J L Neill, M T Muckle, D P Zaleski, The Astrophysical Journal. 755153Neill, J. L., Muckle, M. T., Zaleski, D. P., et al. 2012, The Astrophysical Journal, 755, 153\n\n. K I Öberg, V V Guzmán, K Furuya, Nature. 520Öberg, K. I., Guzmán, V. V., Furuya, K., et al. 2015, Nature, 520, 198\n\n. K I Öberg, R Murray-Clay, E A Bergin, The Astrophysical Journal. 74316Öberg, K. I., Murray-Clay, R., & Bergin, E. A. 2011, The Astrophysical Journal, 743, L16\n\n. R Peng, W D Langer, T Velusamy, T B H Kuiper, S Levin, The Astrophysical Journal. 497842Peng, R., Langer, W. D., Velusamy, T., Kuiper, T. B. H., & Levin, S. 1998, The Astrophysical Journal, 497, 842\n\n. C Qi, K I Öberg, S M Andrews, The Astrophysical Journal. 8131Qi, C., Öberg, K. I., Andrews, S. M., et al. 2015, The Astrophysical Journal, 813, 1\n\n. K Ros, A Johansen, Astronomy & Astrophysics. 552137Ros, K., & Johansen, A. 2013, Astronomy & Astrophysics, 552, A137\n\n. D Schoonenberg, C W Ormel, Astronomy & Astrophysics. 60221Schoonenberg, D., & Ormel, C. W. 2017, Astronomy & Astrophysics, 602, A21\n\nR Selina, E J Murphy, ngVLA Memo Series. 17Selina, R., & Murphy, E. J. 2017, in ngVLA Memo Series, 17\n\n. C N Shingledecker, E Herbst, Physical Chemistry Chemical Physics. 205359Shingledecker, C. N., & Herbst, E. 2018, Physical Chemistry Chemical Physics, 20, 5359\n\n. V Wakelam, J C Loison, E Herbst, The Astrophysical Journal Supplement Series. 2171Wakelam, V., Loison, J. C., Herbst, E., et al. 2015, The Astrophysical Journal Supplement Series, 217, 1\n\n. C Walsh, T J Millar, H Nomura, Astronomy & Astrophysics. 56333Walsh, C., Millar, T. J., Nomura, H., et al. 2014, Astronomy & Astrophysics, 563, A33\n\n. C Walsh, R A Loomis, K I Öberg, The Astrophysical Journal Letters. 82310Walsh, C., Loomis, R. A., Öberg, K. I., et al. 2016, The Astrophysical Journal Letters, 823, L10\n\n. H Wang, R C Bell, M J Iedema, A A Tsekouras, J P Cowin, The Astrophysical Journal. 6201027Wang, H., Bell, R. C., Iedema, M. J., Tsekouras, A. A., & Cowin, J. P. 2005, The Astrophysical Journal, 620, 1027\n"},"corpus_id":{"kind":"number","value":118947830,"string":"118,947,830"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1810.09550\",\n \"mag\": 2944267837,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1810.09550v1.pdf\"\n ],\n \"pdf_sha\": \"7d8f275f7445cb06ce5ac565042f05a70daa35d6\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Observing the Effects of Chemistry on Exoplanets and Planet Formation"},"authors":{"kind":"list like","value":["Brett A Mcguire bmcguire@nrao.edu \nNational Radio Astronomy Observatory\n22903CharlottesvilleVA","Edwin Bergin \nDepartment of Astronomy\nUniversity of Michigan\n48109Ann ArborMI","Geoffrey A Blake \nDivision of Geological and Planetary Sciences\nCalifornia Institute of Technology\n91125PasadenaCA\n\nDivision of Chemistry and Chemical Engineering\nCalifornia Institute of Technology\n91125PasadenaCA","Andrew M Burkhardt \nHarvard-Smithsonian Center for Astrophysics\n02138CambridgeMA","L Ilsedore Cleeves \nDepartment of Astronomy\nUniversity of Virginia\n22904CharlottesvilleVA","Ryan A Loomis \nNational Radio Astronomy Observatory\n22903CharlottesvilleVA","Anthony J Remijan \nNational Radio Astronomy Observatory\n22903CharlottesvilleVA","Christopher N Shingledecker \nMax Planck Institute for Extraterrestrial Physics\nGarchingGermany\n\nInstitute for Theoretical Chemistry\nUniversity of Stuttgart\nGermany","Eric R Willis \nDepartment of Chemistry\nUniversity of Virginia\n22904CharlottesvilleVA"],"string":"[\n \"Brett A Mcguire bmcguire@nrao.edu \\nNational Radio Astronomy Observatory\\n22903CharlottesvilleVA\",\n \"Edwin Bergin \\nDepartment of Astronomy\\nUniversity of Michigan\\n48109Ann ArborMI\",\n \"Geoffrey A Blake \\nDivision of Geological and Planetary Sciences\\nCalifornia Institute of Technology\\n91125PasadenaCA\\n\\nDivision of Chemistry and Chemical Engineering\\nCalifornia Institute of Technology\\n91125PasadenaCA\",\n \"Andrew M Burkhardt \\nHarvard-Smithsonian Center for Astrophysics\\n02138CambridgeMA\",\n \"L Ilsedore Cleeves \\nDepartment of Astronomy\\nUniversity of Virginia\\n22904CharlottesvilleVA\",\n \"Ryan A Loomis \\nNational Radio Astronomy Observatory\\n22903CharlottesvilleVA\",\n \"Anthony J Remijan \\nNational Radio Astronomy Observatory\\n22903CharlottesvilleVA\",\n \"Christopher N Shingledecker \\nMax Planck Institute for Extraterrestrial Physics\\nGarchingGermany\\n\\nInstitute for Theoretical Chemistry\\nUniversity of Stuttgart\\nGermany\",\n \"Eric R Willis \\nDepartment of Chemistry\\nUniversity of Virginia\\n22904CharlottesvilleVA\"\n]","isConversational":false},"abstract":{"kind":"string","value":""},"sections":{"kind":"list like","value":[{"title":"Background","content":"The location of condensation fronts (snow lines) in protoplanetary disks (PPDs) can have substantial effects on the physical and chemical evolution of planets in the system. Indeed, many models predict that these condensation fronts are the locations where planetesimal formation begins (Ros & Johansen 2013;Schoonenberg & Ormel 2017;Drazkowska & Alibert 2017). The location of these condensation fronts is determined both by the incident radiation from the protostar, but also, in large part, by the chemical composition of the gas and ice. The desorption temperature of the bulk ice varies significantly with the chemical composition and mixing ratio (Collings et al. 2004), even of minor constituents. This can be further complicated by a variance in the sticking efficiency of the icy grains with composition (Wang et al. 2005). It is clear that the underlying chemical composition in these ices, even at low mixing ratios, can have a substantial effect on the efficiency of planet formation.\nFurther, the relative gas vs solid-phase inventories at snow lines govern the composition of the planets and their atmospheres. These compositions (e.g., the C/O ratio in planetary atmospheres; Öberg et al. 2011) may therefore be useful probes of the history of exoplanets. Effectively using observed exoplanet elemental ratios to test those models therefore relies heavily on our ability to accurately interpret chemical abun-dances measured in disks to infer the inventory of undetectable species, and how these measurements affect observed elemental ratios.\nConstraining the chemical compositions in PPDs is challenging. The most complex gas-phase species yet detected in PPDs are methanol (CH 3 OH; Walsh et al. 2016) and methyl cyanide (CH 3 CN; Öberg et al. 2015), and more complex molecules are unlikley to be observed (Walsh et al. 2014). Further, in cold or shielded regions within disks, ices may be the major repository of C and O (Bergin et al. 2016). Direct observation of ices typically require space-based infrared facilities and a target source with a sufficiently bright background continuum and favorable viewing geometry. Even in massive young stellar objects with substantial ice columns, the list of firmly-detected ice species numbers less than a dozen (Boogert et al. 2015). Our knowledge of gas and ice compositions in disks must therefore rely on astrochemical models, using the detectable simple gas species as the primary constraint.\nThese models are not yet robust, despite substantial efforts (Garrod 2013;Herbst 2017;Wakelam et al. 2015). To reliably infer chemical compositions in PPDs, in which chemistry is closely coupled to physical conditions (temperature, density, UV flux, evolving dust populations, etc.) that vary with radius and scale height, necessitates that these models be well-constrained by observations in earlier stages of star formation, where more robust inventories can be observed. Here, we briefly discuss the state-of-the-art in progressively earlier stages of chemical evolution, and outline future directions and the technology and community support needed to enable them."},{"title":"Chemical evolution in PPDs ...","content":"A number of larger scale studies of disk physical properties and chemistry have been enabled by the sensitivity and resolution of ALMA. Efforts to measure fundamental properties such as bulk mass (Ansdell et al. 2016), and larger compositional studies examining nitrogen (Guzmán et al. 2017) and deuterium ratios, are now possible for larger samples sizes. With robust chemical modeling, ALMA observations of simple hydrocarbons (e.g., C 2 H) can perhaps be used to infer C/O ratios indirectly (Bergin et al. 2016). As well, the JWST and, if commissioned, Origins Space Telescope, have significant potential to observe major ice components (e.g. H 2 O, CO, CO 2 , CH 3 OH) toward a wide variety of targets. Yet, these infrared facilities will still largely be blind to the complex chemistry occurring within the ices, and to the combined ice/gas coupling in warmer gas. That insight is only achievable through radio observations of gas-phase species, that are sensitive to populations orders of magnitude lower in abundance, coupled with chemical modeling."},{"title":"... and in complex, star-forming regions","content":"Sgr B2(N), Orion-KL, and NGC 6334I are the most chemically and physically complex stages of star formation observed and modeled (Garrod et al. 2008;Crockett et al. 2014;McGuire et al. 2017). The extreme physical environments, interplay between the gas and ice, and complex structures often result in models which can reproduce observed abundances in, at best, a qualitative fashion (Laas et al. 2011;Garrod et al. 2008;Garrod 2013). These types of physical and chemical interactions are the same that are at play in PPDs, albeit on a much larger scale. It is therefore these environments that provide a unique test environment for the similar PDR style models applied to PPDs. Improvements in these models are most likely to be achieved by testing them against a wide range of environments from quiescent clouds to massive star forming regions.\nExisting facilities in the (sub-)millimeter regime, particularly ALMA, are capable of reaching the line-confusion limit in broadband line surveys of complex, star forming regions in a matter of hours. That is, the density of molecular lines in the resulting spectra is such that there is at least one spectral line every FWHM; there are no linefree channels. When line-confusion is reached, no additional information can be gained from longer integration times at the same spatial resolution. We are rapidly approaching the point at which deep drill observations with ALMA and the GBT will no longer produce new spectral lines. The end result of this limit is that no further detections of new species can be made, and thus the refinement of chemical networks will be limited to those species detectable at the confusion limit.\nIn the cm-wave regime, however, the most sensitive survey, the Prebiotic Interstellar Molecular Survey (PRIMOS; Neill et al. 2012) with the most sensitive facility (the GBT) toward the most line-rich source (Sgr B2) has lines in only ∼10-20% of its channels at cm-wavelengths. Diminishing returns on sensitivity for this survey have already set in, and to increase the sensitivity substantially would require hundreds of hours of dedicated observations for every ∼2 GHz window. Further, ALMA observations at higher frequencies are revealing that the region displays widely variable physical conditions within a relatively small environment, variation that is lost to single-dish GBT observations (Belloche et al. 2016). This offers a valuable opportunity to probe environment-induced chemical differentiation, but only with spatially-resolved observations which are not possible with the GBT.\nWhile the brightest lines in Sgr B2 are readily detectable with extant facilities such as the VLA and ATCA (Corby et al. 2015), the complex species are not efficiently observable, especially at high angular resolution. The only viable solution to this problem is the construction of a next-generation interferometer like the next-generation Very Large Array (ngVLA). Under the current design concept (Selina & Murphy 2017), the ngVLA is likely to not only be capable of spatially resolving chemistry in Sgr B2(N), but to reach the line-confusion limit even at cm-wavelengths, providing substantial new, complex molecular detections and their correlation with the physical conditions at work."},{"title":"Applications to PPDs","content":"Once refined, these models can be used to accurately interpret observations from ALMA targeting the outermost condensation fronts within disks, primarily CO snow lines (Qi et al. 2015). Application in the terrestrial planet forming region, near the H 2 O snow line, will require AU-scale resolution with high sensitivity. A high-frequency ngVLA is the only facility currently in design with the required specifications. Further, while these models can then interpret results for C and O in disks, observations of the major N-bearing species, NH 3 will still be required at similar scales. An ngVLA operating at least through the 24 GHz NH 3 transition is therefore necessary.\nAn ngVLA design concept is needed that supports broadband, high surface-brightness sensitivity capabilities at the small to modest (0.1-3 ′′ ) scales required to probe the effects of the widely variable physical conditions on the chemical evolution in sources like Sgr B2 and unlock chemical inventories obscured by line confusion suffered at extant facilities like ALMA. The same instrument should be capable of observing NH 3 and imaging the planet forming zone inside the water snow line in disks, requiring AUscale (mas) resolution capabilities."},{"title":"Chemical evolution in protostellar environments","content":"The protostellar stage of star formation spans a wide range of sources, from 'hot corinos' such as IRAS 16293 (Jørgensen et al. 2016) to chemically rich outflow sources like L1157 (Mendoza et al. 2014). The molecular outflows from active young protostars like L1157 provide a unique environment to test the immediate impact of rapid changes in temperature and density on 'pristine' molecular clouds. In these sources, distance from the protostar along the outflow corresponds directly to time, allowing modeling of temporal evolution. Hot corinos are the next stage after L1157 and before hot cores like Sgr B2(N).\nThe best method for constraining the chemistry in these sources is broadband spectral line surveys. These sources are relatively compact, are beam-diluted in single-dish observations, and their linewidths are typically such that high-sensitivity data are often at or approaching line confusion limits, necessitating interferometric observation (Jørgensen et al. 2016). Because they are typically moderately warm (50-150 K; Jørgensen et al. 2016;McGuire et al. 2015), complex molecules display bright features in the 1-3 mm window well-covered by ALMA, NOEMA, and the SMA.\nThe SOLIS project at NOEMA and IRAM, which examines a sample of solar-type protostellar sources (Ceccarelli et al. 2017), and the PILS project at ALMA, which has focused particularly on an exquisitely detailed examination of the IRAS 16293 system (Jørgensen et al. 2016), are the primary active dedicated investigations into this class of objects. Extant observational facilities are likely sufficient to provide the required insight. Ongoing and future efforts on ALMA, NOEMA, and the SMA examining a large sample of protostellar environments are critical to constraining the effects of physical environment on chemical evolution and incorporating these effects into models."},{"title":"Chemical evolution in quiescent environments","content":"Cold (T ex ∼10 K), dark, starless clouds are an ideal environment in which to test the time evolution of chemical reaction networks under 'constant' physical conditions. The typical sources for exploring this stage of star and planet formation are highly extended (θ s > 30 ′′ ). At these excitation temperatures, most molecules, especially those more complex than CH 3 OH, will display bright emission at cm wavelengths. The best strategy for obtaining the required abundance and excitation information is therefore deep, broadband spectral line surveys at cm-wavelengths using single-dish facilities combined with chemical modeling (Cordiner et al. 2017;Kaifu et al. 2004;McGuire et al. 2018). The best extant instrument for this work is the Green Bank Telescope (GBT).\nEven in a source as extended as TMC-1, there exists evidence for large chemical variation over small spatial scales (0.02-0.03 pc; Peng et al. 1998;Dickens et al. 2001), and observations of such would probe the effects of the very earliest stages of collapse on the chemistry. These observations will require high spatial resolution, high surfacebrightness sensitivity, and broad spectral bandwidth to be efficient. Because of the cold temperatures, this further requires a cm-wavelength facility. While the VLA satisfies this need, it lacks the combination of surface brightness sensitivity and bandwidth necessary to conduct a sensitive line survey with any efficiency. The proposed ngVLA, in its current conceptual design, would prove more than optimal for this task.\nThe chemical evolution of all ranges of environments from planetary atmospheres (Arsenovic et al. 2016) to disk midplanes (Cleeves et al. 2013) to molecular clouds (Hudson et al. 2008) is impacted by radiation. The last few years have seen significant advances in modeling efforts to accurately include these effects (Shingledecker & Herbst 2018), but these efforts are young, and require substantial training on 'simple' environments like quiescent clouds where the radiation effects can be easily decoupled from dynamical physical events such as outflows and shocks. In short, efforts at the GBT and other large single-dish facilities are needed to survey quiescent chemical clouds. These efforts would be greatly bolstered by the development of broadband capabilities on the ngVLA to enable investigations of the chemical evolution upon initial gravitational collapse."},{"title":"Summary","content":"A combination of observational and theoretical efforts are needed that are focused on placing rigorous constraints on models of chemical evolution at each stage of the star and planet formation process. The observational requirements of these investigations necessitate a commitment by the community to support extant facilities and the development of technological advancements for the same, as well as the development of a next-generation VLA with high sensitivity and exquisite spatial resolution. The ultimate goal should be producing models that can be used to better calibrate our estimates of gas and ice compositions in PPDs, given the observations of simple gas-phase species achievable in these sources, and the construction of a radio facility capable of observationally exploiting these models in situ during planet formation."}],"string":"[\n {\n \"title\": \"Background\",\n \"content\": \"The location of condensation fronts (snow lines) in protoplanetary disks (PPDs) can have substantial effects on the physical and chemical evolution of planets in the system. Indeed, many models predict that these condensation fronts are the locations where planetesimal formation begins (Ros & Johansen 2013;Schoonenberg & Ormel 2017;Drazkowska & Alibert 2017). The location of these condensation fronts is determined both by the incident radiation from the protostar, but also, in large part, by the chemical composition of the gas and ice. The desorption temperature of the bulk ice varies significantly with the chemical composition and mixing ratio (Collings et al. 2004), even of minor constituents. This can be further complicated by a variance in the sticking efficiency of the icy grains with composition (Wang et al. 2005). It is clear that the underlying chemical composition in these ices, even at low mixing ratios, can have a substantial effect on the efficiency of planet formation.\\nFurther, the relative gas vs solid-phase inventories at snow lines govern the composition of the planets and their atmospheres. These compositions (e.g., the C/O ratio in planetary atmospheres; Öberg et al. 2011) may therefore be useful probes of the history of exoplanets. Effectively using observed exoplanet elemental ratios to test those models therefore relies heavily on our ability to accurately interpret chemical abun-dances measured in disks to infer the inventory of undetectable species, and how these measurements affect observed elemental ratios.\\nConstraining the chemical compositions in PPDs is challenging. The most complex gas-phase species yet detected in PPDs are methanol (CH 3 OH; Walsh et al. 2016) and methyl cyanide (CH 3 CN; Öberg et al. 2015), and more complex molecules are unlikley to be observed (Walsh et al. 2014). Further, in cold or shielded regions within disks, ices may be the major repository of C and O (Bergin et al. 2016). Direct observation of ices typically require space-based infrared facilities and a target source with a sufficiently bright background continuum and favorable viewing geometry. Even in massive young stellar objects with substantial ice columns, the list of firmly-detected ice species numbers less than a dozen (Boogert et al. 2015). Our knowledge of gas and ice compositions in disks must therefore rely on astrochemical models, using the detectable simple gas species as the primary constraint.\\nThese models are not yet robust, despite substantial efforts (Garrod 2013;Herbst 2017;Wakelam et al. 2015). To reliably infer chemical compositions in PPDs, in which chemistry is closely coupled to physical conditions (temperature, density, UV flux, evolving dust populations, etc.) that vary with radius and scale height, necessitates that these models be well-constrained by observations in earlier stages of star formation, where more robust inventories can be observed. Here, we briefly discuss the state-of-the-art in progressively earlier stages of chemical evolution, and outline future directions and the technology and community support needed to enable them.\"\n },\n {\n \"title\": \"Chemical evolution in PPDs ...\",\n \"content\": \"A number of larger scale studies of disk physical properties and chemistry have been enabled by the sensitivity and resolution of ALMA. Efforts to measure fundamental properties such as bulk mass (Ansdell et al. 2016), and larger compositional studies examining nitrogen (Guzmán et al. 2017) and deuterium ratios, are now possible for larger samples sizes. With robust chemical modeling, ALMA observations of simple hydrocarbons (e.g., C 2 H) can perhaps be used to infer C/O ratios indirectly (Bergin et al. 2016). As well, the JWST and, if commissioned, Origins Space Telescope, have significant potential to observe major ice components (e.g. H 2 O, CO, CO 2 , CH 3 OH) toward a wide variety of targets. Yet, these infrared facilities will still largely be blind to the complex chemistry occurring within the ices, and to the combined ice/gas coupling in warmer gas. That insight is only achievable through radio observations of gas-phase species, that are sensitive to populations orders of magnitude lower in abundance, coupled with chemical modeling.\"\n },\n {\n \"title\": \"... and in complex, star-forming regions\",\n \"content\": \"Sgr B2(N), Orion-KL, and NGC 6334I are the most chemically and physically complex stages of star formation observed and modeled (Garrod et al. 2008;Crockett et al. 2014;McGuire et al. 2017). The extreme physical environments, interplay between the gas and ice, and complex structures often result in models which can reproduce observed abundances in, at best, a qualitative fashion (Laas et al. 2011;Garrod et al. 2008;Garrod 2013). These types of physical and chemical interactions are the same that are at play in PPDs, albeit on a much larger scale. It is therefore these environments that provide a unique test environment for the similar PDR style models applied to PPDs. Improvements in these models are most likely to be achieved by testing them against a wide range of environments from quiescent clouds to massive star forming regions.\\nExisting facilities in the (sub-)millimeter regime, particularly ALMA, are capable of reaching the line-confusion limit in broadband line surveys of complex, star forming regions in a matter of hours. That is, the density of molecular lines in the resulting spectra is such that there is at least one spectral line every FWHM; there are no linefree channels. When line-confusion is reached, no additional information can be gained from longer integration times at the same spatial resolution. We are rapidly approaching the point at which deep drill observations with ALMA and the GBT will no longer produce new spectral lines. The end result of this limit is that no further detections of new species can be made, and thus the refinement of chemical networks will be limited to those species detectable at the confusion limit.\\nIn the cm-wave regime, however, the most sensitive survey, the Prebiotic Interstellar Molecular Survey (PRIMOS; Neill et al. 2012) with the most sensitive facility (the GBT) toward the most line-rich source (Sgr B2) has lines in only ∼10-20% of its channels at cm-wavelengths. Diminishing returns on sensitivity for this survey have already set in, and to increase the sensitivity substantially would require hundreds of hours of dedicated observations for every ∼2 GHz window. Further, ALMA observations at higher frequencies are revealing that the region displays widely variable physical conditions within a relatively small environment, variation that is lost to single-dish GBT observations (Belloche et al. 2016). This offers a valuable opportunity to probe environment-induced chemical differentiation, but only with spatially-resolved observations which are not possible with the GBT.\\nWhile the brightest lines in Sgr B2 are readily detectable with extant facilities such as the VLA and ATCA (Corby et al. 2015), the complex species are not efficiently observable, especially at high angular resolution. The only viable solution to this problem is the construction of a next-generation interferometer like the next-generation Very Large Array (ngVLA). Under the current design concept (Selina & Murphy 2017), the ngVLA is likely to not only be capable of spatially resolving chemistry in Sgr B2(N), but to reach the line-confusion limit even at cm-wavelengths, providing substantial new, complex molecular detections and their correlation with the physical conditions at work.\"\n },\n {\n \"title\": \"Applications to PPDs\",\n \"content\": \"Once refined, these models can be used to accurately interpret observations from ALMA targeting the outermost condensation fronts within disks, primarily CO snow lines (Qi et al. 2015). Application in the terrestrial planet forming region, near the H 2 O snow line, will require AU-scale resolution with high sensitivity. A high-frequency ngVLA is the only facility currently in design with the required specifications. Further, while these models can then interpret results for C and O in disks, observations of the major N-bearing species, NH 3 will still be required at similar scales. An ngVLA operating at least through the 24 GHz NH 3 transition is therefore necessary.\\nAn ngVLA design concept is needed that supports broadband, high surface-brightness sensitivity capabilities at the small to modest (0.1-3 ′′ ) scales required to probe the effects of the widely variable physical conditions on the chemical evolution in sources like Sgr B2 and unlock chemical inventories obscured by line confusion suffered at extant facilities like ALMA. The same instrument should be capable of observing NH 3 and imaging the planet forming zone inside the water snow line in disks, requiring AUscale (mas) resolution capabilities.\"\n },\n {\n \"title\": \"Chemical evolution in protostellar environments\",\n \"content\": \"The protostellar stage of star formation spans a wide range of sources, from 'hot corinos' such as IRAS 16293 (Jørgensen et al. 2016) to chemically rich outflow sources like L1157 (Mendoza et al. 2014). The molecular outflows from active young protostars like L1157 provide a unique environment to test the immediate impact of rapid changes in temperature and density on 'pristine' molecular clouds. In these sources, distance from the protostar along the outflow corresponds directly to time, allowing modeling of temporal evolution. Hot corinos are the next stage after L1157 and before hot cores like Sgr B2(N).\\nThe best method for constraining the chemistry in these sources is broadband spectral line surveys. These sources are relatively compact, are beam-diluted in single-dish observations, and their linewidths are typically such that high-sensitivity data are often at or approaching line confusion limits, necessitating interferometric observation (Jørgensen et al. 2016). Because they are typically moderately warm (50-150 K; Jørgensen et al. 2016;McGuire et al. 2015), complex molecules display bright features in the 1-3 mm window well-covered by ALMA, NOEMA, and the SMA.\\nThe SOLIS project at NOEMA and IRAM, which examines a sample of solar-type protostellar sources (Ceccarelli et al. 2017), and the PILS project at ALMA, which has focused particularly on an exquisitely detailed examination of the IRAS 16293 system (Jørgensen et al. 2016), are the primary active dedicated investigations into this class of objects. Extant observational facilities are likely sufficient to provide the required insight. Ongoing and future efforts on ALMA, NOEMA, and the SMA examining a large sample of protostellar environments are critical to constraining the effects of physical environment on chemical evolution and incorporating these effects into models.\"\n },\n {\n \"title\": \"Chemical evolution in quiescent environments\",\n \"content\": \"Cold (T ex ∼10 K), dark, starless clouds are an ideal environment in which to test the time evolution of chemical reaction networks under 'constant' physical conditions. The typical sources for exploring this stage of star and planet formation are highly extended (θ s > 30 ′′ ). At these excitation temperatures, most molecules, especially those more complex than CH 3 OH, will display bright emission at cm wavelengths. The best strategy for obtaining the required abundance and excitation information is therefore deep, broadband spectral line surveys at cm-wavelengths using single-dish facilities combined with chemical modeling (Cordiner et al. 2017;Kaifu et al. 2004;McGuire et al. 2018). The best extant instrument for this work is the Green Bank Telescope (GBT).\\nEven in a source as extended as TMC-1, there exists evidence for large chemical variation over small spatial scales (0.02-0.03 pc; Peng et al. 1998;Dickens et al. 2001), and observations of such would probe the effects of the very earliest stages of collapse on the chemistry. These observations will require high spatial resolution, high surfacebrightness sensitivity, and broad spectral bandwidth to be efficient. Because of the cold temperatures, this further requires a cm-wavelength facility. While the VLA satisfies this need, it lacks the combination of surface brightness sensitivity and bandwidth necessary to conduct a sensitive line survey with any efficiency. The proposed ngVLA, in its current conceptual design, would prove more than optimal for this task.\\nThe chemical evolution of all ranges of environments from planetary atmospheres (Arsenovic et al. 2016) to disk midplanes (Cleeves et al. 2013) to molecular clouds (Hudson et al. 2008) is impacted by radiation. The last few years have seen significant advances in modeling efforts to accurately include these effects (Shingledecker & Herbst 2018), but these efforts are young, and require substantial training on 'simple' environments like quiescent clouds where the radiation effects can be easily decoupled from dynamical physical events such as outflows and shocks. In short, efforts at the GBT and other large single-dish facilities are needed to survey quiescent chemical clouds. These efforts would be greatly bolstered by the development of broadband capabilities on the ngVLA to enable investigations of the chemical evolution upon initial gravitational collapse.\"\n },\n {\n \"title\": \"Summary\",\n \"content\": \"A combination of observational and theoretical efforts are needed that are focused on placing rigorous constraints on models of chemical evolution at each stage of the star and planet formation process. The observational requirements of these investigations necessitate a commitment by the community to support extant facilities and the development of technological advancements for the same, as well as the development of a next-generation VLA with high sensitivity and exquisite spatial resolution. The ultimate goal should be producing models that can be used to better calibrate our estimates of gas and ice compositions in PPDs, given the observations of simple gas-phase species achievable in these sources, and the construction of a radio facility capable of observationally exploiting these models in situ during planet formation.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". M Ansdell, J P Williams, N Van Der Marel, The Astrophysical Journal. 8281Ansdell, M., Williams, J. P., van der Marel, N., et al. 2016, The Astrophysical Journal, 828, 1","doi":""},{"text":". P Arsenovic, E Rozanov, A Stenke, Journal of Atmospheric and Solar-Terrestrial Physics. 149180Arsenovic, P., Rozanov, E., Stenke, A., et al. 2016, Journal of Atmospheric and Solar-Terrestrial Physics, 149, 180","doi":""},{"text":". A Belloche, H S P Müller, R T Garrod, K M Menten, Astronomy & Astrophysics. 58791Belloche, A., Müller, H. S. P., Garrod, R. T., & Menten, K. M. 2016, Astronomy & Astro- physics, 587, A91","doi":""},{"text":". E A Bergin, F Du, L I Cleeves, The Astrophysical Journal. 8311Bergin, E. A., Du, F., Cleeves, L. I., et al. 2016, The Astrophysical Journal, 831, 1","doi":""},{"text":". A C A Boogert, P A Gerakines, D C B Whittet, Annual Review of Astronomy and Astrophysics. 53541Boogert, A. C. A., Gerakines, P. A., & Whittet, D. C. B. 2015, Annual Review of Astronomy and Astrophysics, 53, 541","doi":""},{"text":". C Ceccarelli, P Caselli, F Fontani, The Astrophysical Journal. 850176Ceccarelli, C., Caselli, P., Fontani, F., et al. 2017, The Astrophysical Journal, 850, 176","doi":""},{"text":". L I Cleeves, F C Adams, E A Bergin, R Visser, The Astrophysical Journal. 77728Cleeves, L. I., Adams, F. C., Bergin, E. A., & Visser, R. 2013, The Astrophysical Journal, 777, 28","doi":""},{"text":". M P Collings, M A Anderson, R Chen, Monthly Notices of the Royal Astronomical Society. 3541133Collings, M. P., Anderson, M. A., Chen, R., et al. 2004, Monthly Notices of the Royal Astro- nomical Society, 354, 1133","doi":""},{"text":". J F Corby, P A Jones, M R Cunningham, Monthly Notices of the Royal Astronomical Society. 4523969Corby, J. F., Jones, P. A., Cunningham, M. R., et al. 2015, Monthly Notices of the Royal Astronomical Society, 452, 3969","doi":""},{"text":". M A Cordiner, S B Charnley, Z Kisiel, B A Mcguire, Y.-J Kuan, The Astrophysical Journal. 850187Cordiner, M. A., Charnley, S. B., Kisiel, Z., McGuire, B. A., & Kuan, Y.-J. 2017, The Astro- physical Journal, 850, 187","doi":""},{"text":". N R Crockett, E A Bergin, J L Neill, The Astrophysical Journal. 787112Crockett, N. R., Bergin, E. A., Neill, J. L., et al. 2014, The Astrophysical Journal, 787, 112","doi":""},{"text":". J E Dickens, W D Langer, T Velusamy, The Astrophysical Journal. 558693Dickens, J. E., Langer, W. D., & Velusamy, T. 2001, The Astrophysical Journal, 558, 693","doi":""},{"text":". J Drazkowska, Y Alibert, Astronomy & Astrophysics. 60892Drazkowska, J., & Alibert, Y. 2017, Astronomy & Astrophysics, 608, A92","doi":""},{"text":". R T Garrod, The Astrophysical Journal. 76560Garrod, R. T. 2013, The Astrophysical Journal, 765, 60","doi":""},{"text":". R T Garrod, S L W Weaver, E Herbst, Astrophys. J. 682283Garrod, R. T., Weaver, S. L. W., & Herbst, E. 2008, Astrophys. J., 682, 283","doi":""},{"text":". V V Guzmán, K I Öberg, J Huang, R Loomis, C Qi, The Astrophysical Journal. 8361Guzmán, V. V., Öberg, K. I., Huang, J., Loomis, R., & Qi, C. 2017, The Astrophysical Journal, 836, 1","doi":""},{"text":". E Herbst, International Reviews in Physical Chemistry. 36287Herbst, E. 2017, International Reviews in Physical Chemistry, 36, 287","doi":""},{"text":". J Huang, K I Öberg, C Qi, The Astrophysical Journal. 8351Huang, J., Öberg, K. I., Qi, C., et al. 2017, The Astrophysical Journal, 835, 1","doi":""},{"text":". R L Hudson, M H Moore, J P Dworkin, M P Martin, Z D Pozun, Astrobiology. 8771Hudson, R. L., Moore, M. H., Dworkin, J. P., Martin, M. P., & Pozun, Z. D. 2008, Astrobiology, 8, 771","doi":""},{"text":". J K Jørgensen, M Van Der Wiel, A Coutens, Astronomy & Astrophysics. 595117Jørgensen, J. K., van der Wiel, M., Coutens, A., et al. 2016, Astronomy & Astrophysics, 595, A117","doi":""},{"text":". N Kaifu, M Ohishi, K Kawaguchi, Publications of the Astronomical Society of Japan. 5669Kaifu, N., Ohishi, M., Kawaguchi, K., et al. 2004, Publications of the Astronomical Society of Japan, 56, 69","doi":""},{"text":". J C Laas, R T Garrod, E Herbst, S L Weaver, The Astrophysical Journal. 72871Laas, J. C., Garrod, R. T., Herbst, E., & Widicus Weaver, S. L. 2011, The Astrophysical Journal, 728, 71","doi":""},{"text":". B A Mcguire, A M Burkhardt, S V Kalenskii, Science. 359202McGuire, B. A., Burkhardt, A. M., Kalenskii, S. V., et al. 2018, Science, 359, 202","doi":""},{"text":". B A Mcguire, P B Carroll, N M Dollhopf, The Astrophysical Journal. 8121McGuire, B. A., Carroll, P. B., Dollhopf, N. M., et al. 2015, The Astrophysical Journal, 812, 1","doi":""},{"text":". B A Mcguire, C N Shingledecker, E R Willis, The Astrophysical Journal Letters. 85146McGuire, B. A., Shingledecker, C. N., Willis, E. R., et al. 2017, The Astrophysical Journal Letters, 851, L46","doi":""},{"text":". E Mendoza, B Lefloch, A Lopez-Sepulcre, Monthly Notices of the Royal Astronomical Society. 445151Mendoza, E., Lefloch, B., Lopez-Sepulcre, A., et al. 2014, Monthly Notices of the Royal As- tronomical Society, 445, 151","doi":""},{"text":". J L Neill, M T Muckle, D P Zaleski, The Astrophysical Journal. 755153Neill, J. L., Muckle, M. T., Zaleski, D. P., et al. 2012, The Astrophysical Journal, 755, 153","doi":""},{"text":". K I Öberg, V V Guzmán, K Furuya, Nature. 520Öberg, K. I., Guzmán, V. V., Furuya, K., et al. 2015, Nature, 520, 198","doi":""},{"text":". K I Öberg, R Murray-Clay, E A Bergin, The Astrophysical Journal. 74316Öberg, K. I., Murray-Clay, R., & Bergin, E. A. 2011, The Astrophysical Journal, 743, L16","doi":""},{"text":". R Peng, W D Langer, T Velusamy, T B H Kuiper, S Levin, The Astrophysical Journal. 497842Peng, R., Langer, W. D., Velusamy, T., Kuiper, T. B. H., & Levin, S. 1998, The Astrophysical Journal, 497, 842","doi":""},{"text":". C Qi, K I Öberg, S M Andrews, The Astrophysical Journal. 8131Qi, C., Öberg, K. I., Andrews, S. M., et al. 2015, The Astrophysical Journal, 813, 1","doi":""},{"text":". K Ros, A Johansen, Astronomy & Astrophysics. 552137Ros, K., & Johansen, A. 2013, Astronomy & Astrophysics, 552, A137","doi":""},{"text":". D Schoonenberg, C W Ormel, Astronomy & Astrophysics. 60221Schoonenberg, D., & Ormel, C. W. 2017, Astronomy & Astrophysics, 602, A21","doi":""},{"text":"R Selina, E J Murphy, ngVLA Memo Series. 17Selina, R., & Murphy, E. J. 2017, in ngVLA Memo Series, 17","doi":""},{"text":". C N Shingledecker, E Herbst, Physical Chemistry Chemical Physics. 205359Shingledecker, C. N., & Herbst, E. 2018, Physical Chemistry Chemical Physics, 20, 5359","doi":""},{"text":". V Wakelam, J C Loison, E Herbst, The Astrophysical Journal Supplement Series. 2171Wakelam, V., Loison, J. C., Herbst, E., et al. 2015, The Astrophysical Journal Supplement Series, 217, 1","doi":""},{"text":". C Walsh, T J Millar, H Nomura, Astronomy & Astrophysics. 56333Walsh, C., Millar, T. J., Nomura, H., et al. 2014, Astronomy & Astrophysics, 563, A33","doi":""},{"text":". C Walsh, R A Loomis, K I Öberg, The Astrophysical Journal Letters. 82310Walsh, C., Loomis, R. A., Öberg, K. I., et al. 2016, The Astrophysical Journal Letters, 823, L10","doi":""},{"text":". H Wang, R C Bell, M J Iedema, A A Tsekouras, J P Cowin, The Astrophysical Journal. 6201027Wang, H., Bell, R. C., Iedema, M. J., Tsekouras, A. A., & Cowin, J. P. 2005, The Astrophysical Journal, 620, 1027","doi":""}],"string":"[\n {\n \"text\": \". M Ansdell, J P Williams, N Van Der Marel, The Astrophysical Journal. 8281Ansdell, M., Williams, J. P., van der Marel, N., et al. 2016, The Astrophysical Journal, 828, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Arsenovic, E Rozanov, A Stenke, Journal of Atmospheric and Solar-Terrestrial Physics. 149180Arsenovic, P., Rozanov, E., Stenke, A., et al. 2016, Journal of Atmospheric and Solar-Terrestrial Physics, 149, 180\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Belloche, H S P Müller, R T Garrod, K M Menten, Astronomy & Astrophysics. 58791Belloche, A., Müller, H. S. P., Garrod, R. T., & Menten, K. M. 2016, Astronomy & Astro- physics, 587, A91\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E A Bergin, F Du, L I Cleeves, The Astrophysical Journal. 8311Bergin, E. A., Du, F., Cleeves, L. I., et al. 2016, The Astrophysical Journal, 831, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A C A Boogert, P A Gerakines, D C B Whittet, Annual Review of Astronomy and Astrophysics. 53541Boogert, A. C. A., Gerakines, P. A., & Whittet, D. C. B. 2015, Annual Review of Astronomy and Astrophysics, 53, 541\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Ceccarelli, P Caselli, F Fontani, The Astrophysical Journal. 850176Ceccarelli, C., Caselli, P., Fontani, F., et al. 2017, The Astrophysical Journal, 850, 176\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L I Cleeves, F C Adams, E A Bergin, R Visser, The Astrophysical Journal. 77728Cleeves, L. I., Adams, F. C., Bergin, E. A., & Visser, R. 2013, The Astrophysical Journal, 777, 28\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M P Collings, M A Anderson, R Chen, Monthly Notices of the Royal Astronomical Society. 3541133Collings, M. P., Anderson, M. A., Chen, R., et al. 2004, Monthly Notices of the Royal Astro- nomical Society, 354, 1133\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J F Corby, P A Jones, M R Cunningham, Monthly Notices of the Royal Astronomical Society. 4523969Corby, J. F., Jones, P. A., Cunningham, M. R., et al. 2015, Monthly Notices of the Royal Astronomical Society, 452, 3969\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M A Cordiner, S B Charnley, Z Kisiel, B A Mcguire, Y.-J Kuan, The Astrophysical Journal. 850187Cordiner, M. A., Charnley, S. B., Kisiel, Z., McGuire, B. A., & Kuan, Y.-J. 2017, The Astro- physical Journal, 850, 187\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N R Crockett, E A Bergin, J L Neill, The Astrophysical Journal. 787112Crockett, N. R., Bergin, E. A., Neill, J. L., et al. 2014, The Astrophysical Journal, 787, 112\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J E Dickens, W D Langer, T Velusamy, The Astrophysical Journal. 558693Dickens, J. E., Langer, W. D., & Velusamy, T. 2001, The Astrophysical Journal, 558, 693\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Drazkowska, Y Alibert, Astronomy & Astrophysics. 60892Drazkowska, J., & Alibert, Y. 2017, Astronomy & Astrophysics, 608, A92\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R T Garrod, The Astrophysical Journal. 76560Garrod, R. T. 2013, The Astrophysical Journal, 765, 60\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R T Garrod, S L W Weaver, E Herbst, Astrophys. J. 682283Garrod, R. T., Weaver, S. L. W., & Herbst, E. 2008, Astrophys. J., 682, 283\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V V Guzmán, K I Öberg, J Huang, R Loomis, C Qi, The Astrophysical Journal. 8361Guzmán, V. V., Öberg, K. I., Huang, J., Loomis, R., & Qi, C. 2017, The Astrophysical Journal, 836, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Herbst, International Reviews in Physical Chemistry. 36287Herbst, E. 2017, International Reviews in Physical Chemistry, 36, 287\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Huang, K I Öberg, C Qi, The Astrophysical Journal. 8351Huang, J., Öberg, K. I., Qi, C., et al. 2017, The Astrophysical Journal, 835, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R L Hudson, M H Moore, J P Dworkin, M P Martin, Z D Pozun, Astrobiology. 8771Hudson, R. L., Moore, M. H., Dworkin, J. P., Martin, M. P., & Pozun, Z. D. 2008, Astrobiology, 8, 771\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J K Jørgensen, M Van Der Wiel, A Coutens, Astronomy & Astrophysics. 595117Jørgensen, J. K., van der Wiel, M., Coutens, A., et al. 2016, Astronomy & Astrophysics, 595, A117\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Kaifu, M Ohishi, K Kawaguchi, Publications of the Astronomical Society of Japan. 5669Kaifu, N., Ohishi, M., Kawaguchi, K., et al. 2004, Publications of the Astronomical Society of Japan, 56, 69\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J C Laas, R T Garrod, E Herbst, S L Weaver, The Astrophysical Journal. 72871Laas, J. C., Garrod, R. T., Herbst, E., & Widicus Weaver, S. L. 2011, The Astrophysical Journal, 728, 71\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B A Mcguire, A M Burkhardt, S V Kalenskii, Science. 359202McGuire, B. A., Burkhardt, A. M., Kalenskii, S. V., et al. 2018, Science, 359, 202\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B A Mcguire, P B Carroll, N M Dollhopf, The Astrophysical Journal. 8121McGuire, B. A., Carroll, P. B., Dollhopf, N. M., et al. 2015, The Astrophysical Journal, 812, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B A Mcguire, C N Shingledecker, E R Willis, The Astrophysical Journal Letters. 85146McGuire, B. A., Shingledecker, C. N., Willis, E. R., et al. 2017, The Astrophysical Journal Letters, 851, L46\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Mendoza, B Lefloch, A Lopez-Sepulcre, Monthly Notices of the Royal Astronomical Society. 445151Mendoza, E., Lefloch, B., Lopez-Sepulcre, A., et al. 2014, Monthly Notices of the Royal As- tronomical Society, 445, 151\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J L Neill, M T Muckle, D P Zaleski, The Astrophysical Journal. 755153Neill, J. L., Muckle, M. T., Zaleski, D. P., et al. 2012, The Astrophysical Journal, 755, 153\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K I Öberg, V V Guzmán, K Furuya, Nature. 520Öberg, K. I., Guzmán, V. V., Furuya, K., et al. 2015, Nature, 520, 198\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K I Öberg, R Murray-Clay, E A Bergin, The Astrophysical Journal. 74316Öberg, K. I., Murray-Clay, R., & Bergin, E. A. 2011, The Astrophysical Journal, 743, L16\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Peng, W D Langer, T Velusamy, T B H Kuiper, S Levin, The Astrophysical Journal. 497842Peng, R., Langer, W. D., Velusamy, T., Kuiper, T. B. H., & Levin, S. 1998, The Astrophysical Journal, 497, 842\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Qi, K I Öberg, S M Andrews, The Astrophysical Journal. 8131Qi, C., Öberg, K. I., Andrews, S. M., et al. 2015, The Astrophysical Journal, 813, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Ros, A Johansen, Astronomy & Astrophysics. 552137Ros, K., & Johansen, A. 2013, Astronomy & Astrophysics, 552, A137\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Schoonenberg, C W Ormel, Astronomy & Astrophysics. 60221Schoonenberg, D., & Ormel, C. W. 2017, Astronomy & Astrophysics, 602, A21\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R Selina, E J Murphy, ngVLA Memo Series. 17Selina, R., & Murphy, E. J. 2017, in ngVLA Memo Series, 17\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C N Shingledecker, E Herbst, Physical Chemistry Chemical Physics. 205359Shingledecker, C. N., & Herbst, E. 2018, Physical Chemistry Chemical Physics, 20, 5359\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Wakelam, J C Loison, E Herbst, The Astrophysical Journal Supplement Series. 2171Wakelam, V., Loison, J. C., Herbst, E., et al. 2015, The Astrophysical Journal Supplement Series, 217, 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Walsh, T J Millar, H Nomura, Astronomy & Astrophysics. 56333Walsh, C., Millar, T. J., Nomura, H., et al. 2014, Astronomy & Astrophysics, 563, A33\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Walsh, R A Loomis, K I Öberg, The Astrophysical Journal Letters. 82310Walsh, C., Loomis, R. A., Öberg, K. I., et al. 2016, The Astrophysical Journal Letters, 823, L10\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Wang, R C Bell, M J Iedema, A A Tsekouras, J P Cowin, The Astrophysical Journal. 6201027Wang, H., Bell, R. C., Iedema, M. J., Tsekouras, A. A., & Cowin, J. P. 2005, The Astrophysical Journal, 620, 1027\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":50,"cells":{"text":{"kind":"string","value":"\nArguments Against a Finite N = 8 Supergravity\n25 May 2012\n\nTom Banks \nNHETC and Department of Physics and Astronomy\nRutgers University\n08854-8019PiscatawayNJUSA\n\nSCIPP and Department of Physics\nUniversity of California\n95064-1077Santa CruzCAUSA\n\nArguments Against a Finite N = 8 Supergravity\n25 May 2012\nI present arguments based on dispersion theory and the existence of charged BPS states of small charge, that maximally supersymmetric SUGRA is not a finite theory. The essence of the argument is that the perturbative construction of the theory implies that scattering amplitudes are invariant under a continuous SU(8) global symmetry, which is broken by contributions from virtual BPS states. Conventional estimates of the contributions of those states above production threshold, shows that they should be order one. If the amplitude satisfies a dispersion relation with a finite number of subtractions, as has recently been argued by Giddings and Porto [1], then SU(8) violating amplitudes appear at finite orders in the low energy expansion. We also make a brief remark about the inconsistency of the idea of classicalization in non-gravitational theories.\n\nIntroduction\n\nThere has been a lot of interest in recent years in the question of whether N = 8 supergravity (SUGRA) has a finite S-matrix in perturbation theory [2]. Most of the debate has centered around detailed perturbative calculations and the question of whether the finite answers that have been obtained could be explained in terms of symmetries, or were surprises that might point to new principles at work, which could guarantee finiteness to all orders. The work of [9] shows that all existing calculations can be explained by symmetries, and that the first real test of finiteness will come at 7 loops. I believe that this is now the consensus in the field.\n\nI've always been skeptical about the finiteness arguments, for reasons that I will outline below. At the Penn State Conference in the Fall of 2009 I gave a talk claiming that the putative finite theory could not be a real theory of gravitation, because it could not contain the BPS black hole states that can be found as solutions of the classical equations of N = 8 SUGRA. In that talk, in order to make a comparison with the way in which \"nonperturbative\" states affect string perturbation theory, I emphasized the black hole states with E(7, 7) invariant J 4 = 0, which go to zero mass at the boundaries of moduli space. In the string theoretic realizations of maximal SUGRA, these are elementary particles in the appropriate limits, and have very small entropy. The entropy is of course an invariant on the moduli space, and, as a consequence, the corresponding classical solutions have zero area horizons and are singular.\n\nIn [10] it was suggested that a version of the theory omitting only these singular black holes might be consistent. While I disagree with this claim, it seems easier to avoid the J 4 = 0 black holes, because the argument does not depend on them. Instead, it depends only on the following basic points • Theories of Quantum gravity do not have global continuous symmetries. This is basically a consequence of the existence of black holes and Hawking radiation. A global continuous symmetry implies that we can make black holes with size of order the Planck scale, and masses of order the Planck mass, with arbitrarily large numbers of quantum states, corresponding to high dimensional representations of the group. This violates the Covariant Entropy Bound [12], and it also leads to a variety of paradoxes [6]. In particular, the SU(8) global symmetry of N = 8 SUGRA must be broken. The methods that are used to compute finite N = 8 loop amplitudes would guarantee that to all orders in the loop expansion, SU (8) is a valid symmetry.\n\n• As a consequence of the Dirac quantization condition, the spectrum of BPS black holes is not SU (8) invariant. Using unitarity, we can then prove that amplitudes above the threshold for black hole production for scattering of particles in the SUGRA multiplet, also violate SU (8). SU (8) invariance follows from SUSY for four point amplitudes, so the simplest such amplitude is a five point function. An example is the 2 → 3 process, in which two gravitons scatter to produce two gravitons and a scalar. The gravitons are singlets, and the scalar is in the 70 of SU (8). This amplitude is known to be non-vanishing in perturbative string theory [11], where perturbative winding and Kaluza Klein modes are known to be dual to BPS black holes. [11] shows that even tree level string theory breaks the SU(8) symmetry down to SU(4) × SU (4). Loops of BPS states break it to a discrete group. Green, Ooguri and Schwarz [5] have shown that there are no limits of string theory in which all the BPS states decouple.\n\n• One often hears the phrase, \"black holes are non-perturbative states and their production will not appear in perturbation theory\". This is a mistake. The phrase is only correct for black holes, like those of large charge, whose horizon radius is much larger than the Planck scale. We argue that states of small charge are produced with amplitudes of order 1 above threshold. The thresholds are of order the Planck scale and so is the physical size of the state (by extrapolation from the classical solutions). We recall that the production cross sections for solitons are suppressed exponentially in the ratio of their physical size to their Compton wavelength. There is no such suppression for black holes of small charge, whether or not their invariant entropy vanishes. We also argue that the existence of the small charge black holes follows from the existence of classical solutions plus cluster decomposition 1 .\n\n• The rest of the argument follows if the SU(8) violating 5 point amplitude satisfies a dispersion relation in the center of mass energy, with a finite number of subtractions, when the other kinematic invariants are fixed. Such a representation shows that the amplitude has a finite perturbation series in powers of the energy over the Planck mass, which contradicts the claim that there is an SU(8) invariant finite power series, constructed by the methods of [2] 2 , which coincides with the amplitudes that include contributions from BPS states.\n\nThere are two reasons to worry about the existence of such a dispersion relation. At fixed impact parameter, we expect graviton scattering at sufficiently high energy to be 1 The last argument is due to Andy Strominger. 2 Although there are many alternative methods of calculation in the papers of reference [2], all but the string theory based method begin with manifestly SU (8) invariant tree level amplitudes and use unitarity and analyticity to construct the loops. Interestingly, the methods of Bern, Dixon and collaborators use dimensional regularization to deal with infrared divergences. Zvi Bern has informed me that this breaks the manifest SU (8) symmetry, though the final results are SU (8) invariant in four dimensions. One would imagine that, since IR divergences come only from soft graviton emission, and gravitons are SU (8) singlets, one is guaranteed to get invariant results unless genuine UV divergences are encountered. dominated by black hole production (these are neutral black holes). As a consequence of the thermal nature of black hole evaporation, any exclusive channel with a small number of high energy particles, is exponentially suppressed. This means that, at fixed impact parameter, the amplitude cannot be power law bounded in all directions in the complex energy plane, because it violates the Cerulus-Martin bound [7]. However, Giddings and Porto have recently argued that at fixed kinematic invariants, the shadow scattering of the black hole absorption process, produces an amplitude that does satisfy the CM bound.\n\nThe other reason one might worry about the dispersion relation is that IR divergences would cause a singularity at zero energy, which prevent us from pulling the contour around the origin to complete the argument. Since the amplitude vanishes in perturbation theory, there is no perturbative evidence for such an IR singularity. Furthermore, in perturbative string theory, where such power law corrections to the SU(8) violating amplitude certainly exist, and which has the same low energy spectrum and interactions, there is again no evidence than the amplitude is non-analytic at zero.\n\nTo summarize: The SU(8) violating five point amplitude, which follows from the existence of black holes of small charge above the production threshold for such black holes, has an analytic continuation which has a power law tale, and contradicts the existence of an SU (8) invariant method for constructing a finite unitary S-matrix to all orders in a power series in energy divided by the Planck mass.\n\nThere is a logical possibility that some of the methods in [2] might produce a finite result to all orders in perturbation theory, and provide evidence for a model of quantum gravity, which did not contain the BPS states, that are the focus of this article. While I cannot provide a string free refutation of this claim, it is worth recalling how the \"UV\" divergence of the pure SUGRA amplitudes would be interpreted in the string based computations of [4]. Their computations try to define SUGRA as a certain limit of Type II string amplitudes. All such limits are equivalent to weakly coupled string theory, or 11D SUGRA, on a large torus, so in this method of calculation, the UV divergence must be interpreted as an IR divergence coming from non-perturbative states whose mass goes to zero in this limit.\n\nThere is one other bit of information that emerged from thinking about the consequences of a finite N = 8 SUGRA, which applies to any putatively finite but perturbatively nonrenormalizable theory [8]. In such a theory the expansion parameter is energy divided by some scale. On the other hand, simple diagram counting shows that field theoretic perturbation series are divergent and lead to essential singularities in the coupling 3 In such a theory, this would mean essential singularities at zero energy. In the theories discussed in [8], nothing prevents us from giving mass to all the fields in the Lagrangian. Rigorous theorems of S-matrix theory then preclude the existence of essential singularities, either at the origin or at a finite mass. The point is that in a theory with a mass gap, the number of states at finite energy is bounded and all singularities are multi-particle cuts, for a finite number of particles. Thus, we can refute the claims that any such theory is finite, unless some principle prevents us from giving mass to the particles. Of course, in N = 8 SUGRA, there is such a principle, so this argument is not applicable. That is why we went through the more elaborate discussion outlined above.\n\nOne might want to preserve the claim to a finite N = 8 SUGRA for a bit longer by insisting that the finite theory does not have any charged states in it. The classical Lagrangian has solutions corresponding to a pair of oppositely charged BPS black holes moving away from each other, with arbitrary initial velocity, from an arbitrary initial separation. For some values of these parameters the pair will separate and reach infinity, while for others they will collapse and form a neutral black hole. The usual rules of crossing symmetry suggest that charged particles pairs will be produced in the Hawking radiation of neutral black holes, so it is likely that one would have to forbid the existence of ordinary Schwarzschild black holes as well. Indeed, we've already argued that the covariant entropy bound implies that processes involving the formation and decay of neutral black holes cannot preserve SU (8) symmetry.\n\nThe other option is would be to declare that the theory contained no neutral black holes either, but then its claim to be a theory of gravitation would be empty. The rest of this paper will be devoted to a more detailed discussion of the SU (8) violating five point amplitude.\n\n\nProduction of Soliton States\n\nThe Lagrangian of N = 8 SUGRA contains 28 U(1) gauge bosons. Among the solutions of the theory are BPS dyonic black holes with arbitrary continuous values of the electric and magnetic charges. The theory has a classical E(7, 7) symmetry, which is partly realized by duality rotations in the 56 dimensional charge space. Dirac quantization restricts the space of charges to an integer lattice, which is invariant only under an integer subgroup E(7, 7; Z). The 70 scalar fields of the theory lie in the coset E(7, 7)/SU (8), so only the SU(8) symmetry acts linearly on scattering amplitudes for excitations around a particular point in moduli space. The coset transformations instead give rise to soft scalar low energy theorems for these amplitudes. Recently, these symmetry constraints have been shown to guarantee the finiteness of all amplitudes that have so far been computed in this theory, but fail to rule out a divergence at 7 loops.\n\nThe entropy of these BPS black holes is computed using the Bekenstein-Hawking law, which determines the large entropy asymptotics. It is given by\nS = π J 4 ,\nwhere J 4 is an E(7, 7) invariant, quartic in the charges. When the charges are quantized it is invariant only under the integer subgroup. These are extremal black holes, so the charge also determines the mass, which is not invariant under SU (8).\n\nThe charges live on a Lorentzian lattice and the invariant J 4 can be positive, negative or zero. The states with zero J 4 go to zero mass at certain boundaries of moduli space. The corresponding classical solutions are singular, because they have horizons of zero area. The authors of reference [10] argued that these singular solutions might not be real quantum states of the theory in question. While I do not agree with those arguments, the main discussion of the present paper will be restricted to states with non-zero J 4 .\n\nThe realization of N = 8 SUGRA as the low energy effective field theory of Type II string theory compactified on a 6− torus of stringy size, sheds light on these states. Some of the BPS sectors are realized in terms of Kaluza-Klein momenta, and wrapping numbers of elementary strings and D-branes on cycles of the torus [13]. As shown by Witten [14], all the boundaries of moduli space are dual (via E(7, 7; Z) rotations) to regimes in which weakly coupled Type II string theory, or 11 dimensional SUGRA, are compactified on a torus of growing size. The J 4 = 0 states are precisely those which become massless particles at one of these boundaries. This explains the vanishing of the area term in their entropy formula (the entropy is constant on moduli space).\n\nWhen the string coupling is weak, and all geometrical moduli are of order string scale or larger, we can do reliable perturbative calculations of amplitudes involving virtual charged BPS states. These lead to SU(8) violation. In string perturbation theory, only states with J 4 = 0 contribute to the amplitudes. We can also do reliable weak coupling calculations of amplitudes involving loops of wrapped D-branes with J 4 = 0. These are T-dual to D-instanton calculations and are of order e − c g S . We will see below that this exponential suppression should be interpreted as an exponential of the linear size of the wrapped D-brane state, divided by its Compton wavelength. Here we note only that this is a mathematical fact. The size of a D-brane can be measured by scattering gravitons from it, and this leads to a size of order string scale [15]. The Compton wavelength is smaller than this by a factor g S . 4 Note further, that, if we imagine extrapolating to g s ∼ 1, there is every indication that the contribution of all of these BPS states to SU(8) violating amplitudes will become of order one in eleven dimensional Planck units. Elvang and Kiermaier [11] have computed the lowest dimension SU (8) violating operator in the effective field theory for the SUGRA multiplet, to lowest order in Type II string perturbation theory. It has the form\ne −6φ R 4 ,\nwhere φ is the dimensionless dilaton field. This term comes from explicit violation of SU (8) in tree level string diagrams involving massive non-BPS string states, but it is of the same form as one would expect for the leading low energy contribution from loops of BPS states whose masses violate SU (8). In different regions of the moduli space E(7, 7)/SU (8), the field φ would be replaced by other members of the 70 multiplet of Goldstone bosons. Naively extrapolating to the regime g S ∼ 1 one finds that this term is scaled by the 11 D Planck mass, with a coefficient of order 1. Of course, in this regime, the calculation of [11] is not valid, but since there is no small dimensionless parameter, it is hard to see how the term could be significantly smaller than the extrapolation of the perturbative calculation.\n\nSome of the advocates of a finite N = 8 theory emphasize that their theory may have nothing to do with string theory and may be a new paradigm for the approach to quantum gravity as a quantum field theory. Therefore, I must emphasize that the above remarks about string theory are meant to be suggestive only. Type II string theory with g S ∼ 1 and a torus of 11 dimensional Planck size is a finite theory with a single parameter, with dimensions of mass. Like the would be finite N = 8 theory, it has BPS states, whose mass formula violates the continuous symmetry of the leading term in the low energy effective Lagrangian. Unlike that hypothetical theory, the violation shows up in the low energy power series expansion of the scattering amplitudes of the particles in the SUGRA multiplet. What I will try to demonstrate below is that rather general principles imply that any sensible theory containing these BPS states will behave like string theory does and not like the hypothetical finite theory is supposed to.\n\n\nBrief outline of finite calculations\n\nThis tiny section is not meant to be even a cursory introduction to the remarkable set of methods that have been developed to calculate the S-matrix in N = 8 SUGRA and similar theories. These include spinor helicity methods, tree level recursion relations based on analyticity in the complex momentum plane, and standard as well as novel unitarity cut techniques for calculating loop amplitudes. Using dimensional regularization, some authors have even found a way to get around the traditional dispersion techniques for calculating the real parts of amplitudes. The point is that in 4 − ǫ dimensions, even \"polynomial\" contributions to amplitudes have an imaginary part, as a consequence of the irrational scaling exponents. Although the dimensionally continued amplitudes have no Hilbert space interpretation, the do satisfy the cutting rules, which are the perturbative expression of unitarity.\n\nThe important observation is that, as long as these methods give finite results, those results will be SU(8) covariant 5 . The tree amplitudes are all SU(8) covariant. The loops are computed from the trees by expressing their discontinuities in terms of lower order amplitudes, and so SU(8) covariance follows by induction, as long as the procedure gives finite and unambiguous answers at each step. In particular, the five point amplitude that will be the focus of our attention, which transforms as a 70, will be identically zero to all orders in the momentum expansion. We will show that if we accept the existence of BPS states, with charges of order 1, which are produced above threshold with amplitudes of order 1, that the all orders vanishing of this amplitude contradicts some very general principles of S-matrix theory.\n\n\nDispersion relations in models of quantum gravity\n\nThe S-matrix elements for the scattering amplitude of two gravitons to two gravitons plus a scalar are conveniently classified according to the helicities of the incoming and outgoing gravitons. Each helicity amplitude has a fixed dependence on components of the momenta, multiplied by unknown scalar functions. The scalar amplitudes in the 2 → 3 S-matrix element depend on the square of the center of mass energy s 12 ≡ s, and the other invariants s ij ≡ p i · p j , (with i = j), subject to the constraints j s ij = 0. There are 5 independent invariants. We will write dispersion relations for these scalar functions in the variable s, with the other independent invariants held fixed.\n\nFeynman diagram calculations in low energy field theory, as well as perturbative string theory calculations indicate that the (somewhat vague) holomorphy properties of scattering amplitudes, which are postulated in analytic S-matrix theory, remain true in models of quantum gravity in asymptotically flat space. Amplitudes are holomorphic functions of the invariants, with isolated singularities corresponding to the production of real on-shell physical particle states.\n\nDispersion relations employ this assumption of holomorphy with an assumption that amplitudes are polynomially bounded in the entire complex s plane. This assumption is valid in low energy effective field theory, but when the perturbation parameter has dimensions of length to a positive power, the order of the polynomial increases with the order of perturbation theory.\n\nIn fact, at fixed impact parameter, simple considerations of black hole physics suggest that the amplitudes fall off exponentially at large s. The argument is simple. As s increases, the colliding particles in the center of mass frame will have a gravitational field, as seen by a distant observer\nds 2 = −dt 2 (1 − 2 √ sG N r ) + dr 2 (1 − 2 √ sG N r ) + r 2 dΩ 2 .\nWhen the impact parameter is less than 2 √ sG N , the colliding particles are inside the horizon of the black hole. It is extremely plausible (in general relativity, this is called The Hoop Conjecture) that in such a kinematic regime, particle collisions lead to black hole formation, with probability that rapidly approaches 1 as s goes to infinity with fixed impact parameter. When s is far above the threshold for black hole production at some fixed impact parameter, we can treat the black hole by semi-classical techniques. In particular, Hawking's calculation of black hole decay tells us that the probability of producing a particular final state, with a small number of particles carrying all of the outgoing energy, is exponentially suppressed. The argument is simple counting. The black hole decay is thermal, and the black hole entropy increases linearly with s.\n\nGiddings and Porto [1] have argued that for fixed s ij this behavior implies that the standard assumption of polynomial boundedness in the complex s plane is still valid. The amplitudes at fixed kinematic invariants are integrals over all impact parameters. Since we have argued that for small enough impact parameter, in a range growing like √ s , the amplitudes fall off exponentially, the high energy behavior at fixed s ij is dominated by large impact parameters, where it may be estimated by the eikonal approximation. The eikonal behavior obeys the Cerulus-Martin lower bound on the real axis, and there is every indication that the amplitude is power law bounded in the entire complex s plane, for fixed values of the other s ij , and should satisfy a dispersion relation with a finite number of subtractions. For the 2 → 2 amplitude in N = 8 SUGRA, we expect the right and left hand cuts, which contribute to the dispersion integrand, to meet at the origin. However, for the SU(8) violating 5-point amplitude at hand, one would imagine that the lowest contributing threshold would correspond to the production of pairs of the lightest charged BPS states, whose masses are of order the Planck scale 6 . The amplitudes will have IR divergences, corresponding to emission of soft gravitons from incoming and outgoing particles, as they accelerate, but these will not lead to a lowering of the threshold for real production of particles whose mass violates SU (8). Both the spectrum and perturbative interactions of the SUGRA multiplet preserve SU (8).\n\nWe have thus argued that SU (8) violating helicity amplitudes may be written as\nA h (s) = K h (s, s ij )[P (s) + ∞ s 0 dx (x − s) N ρ(x)],\nwhere N is an integer we have not determined, and K h is a known kinematic helicity factor. P (s) is a polynomial, and ρ is the sum of the discontinuities across the right and left hand cuts. s 0 is the squared mass of the lightest BPS state contributing to the discontinuity. It is of order m 2 P . P and ρ depend implicitly on the other s ij , which we take to be fixed at non-zero values ≪ m 2 P . The low energy expansion of this amplitude involves the moments\nM k = ∞ 1 ρ(ys 0 )y −(N +k) = 1 0 dzσ(z)z N +k−2 ,\nwhere σ(z) ≡ ρ( s 0 z ), and the integrals are all power law convergent at z = 0. The assumption of a convergent perturbation expansion, constructed in terms of recursion relations and perturbative unitarity, implies that A h (s) vanishes to all orders in s m 2 P . This in turn implies that all moments above some k max vanish. The function f (y) = ρ(ys 0 )y −N , is square integrable with respect to the measure dµ = dy y kmax . Indeed, it is finite at every y, because it is a sum of discontinuities across multiparticle cuts, and the integral ∞ 1 dy y 2N +kmax |ρ(ys 0 )| 2 , is convergent at infinity because of the convergence of the original dispersion relation.\n\nThe vanishing of the higher moments is equivalent to the statement that f (y) is orthogonal to all of the functions y −k for non-negative k. These functions are square integrable and linearly independent. By Gram-Schmidt orthogonalization, we can use them to construct an orthonormal basis. Consequently f (y) = 0. This contradicts the explicit finite contribution to f (y) slightly above the production threshold for the lightest BPS dyon-anti-dyon pair.\n\n\nProduction cross sections for BPS monopoles\n\nIt is sometimes argued that the BPS states are \"non-perturbative\", and should not show up in the perturbation expansion. This response is too superficial, and a proper understanding with exchange of massive string modes at tree level. Loops of J 4 = 0 states break SU (8) to a discrete subgroup, while the states with J 4 = 0 are non-perturbative in g S because their size is much larger than their Compton wavelength. For string theory with g S and radii in string units all of order one, or any model whose only parameter is the 4D Planck mass, the threshold for SU (8) violating intermediate states will be given by this scale. of why soliton production is suppressed in conventional quantum field theory shows that there is no similar suppression of production of BPS states of small charge.\n\nIndeed, in conventional field theory, solitons have masses of order m g 2 , where m is the mass of perturbative excitations, and g the dimensionless coupling. Energy denominator suppression would indicate that they contribute to scattering of perturbative excitations at finite orders in perturbation theory, but this is not true. The example of magnetic monopoles in the SO(3) Georgi-Glashow model, is typical. Consider the contribution of monopole production to the total cross section for annihilation of two perturbative charged excitations, at lowest order in the electro-magnetic coupling. It is determined by the vacuum to monopole-anti-monopole matrix element of the electromagnetic current:\n0|J µ (0)|MM .\nBy crossing symmetry, this is the analytic continuation of the monopole form factor\nM(0)|J 0 |M(q) ,\nto time-like q, ≥ twice the monopole mass. The usual rules of semi-classical physics, tell us that the monopole form factor in the monopole rest frame is simply the classical electromagnetic current of the monopole field configuration. The current is a smooth function j(xm), where m is the mass of the charged vector boson. The Riemann-Lebesgue lemma, tells us that at large q the function falls off faster than any power. It is typically of order\ne −a |ql m ,\nwhere the constant a is of order one and represents the closest singularity of j(xm) to the real axis in the complex xm plane. This leads to the estimate\ne − 2a g 2 ,\nfor the monopole production amplitude. It is clear from this calculation that the relevant parameter is the ratio between the physical size of the soliton, and its Compton wave-length. Non-perturbative suppression of soliton production means that the amplitudes are exponentially small in this ratio. Although I've given this argument in the context of the Georgi-Glashow model, this explanation of the non-perturbative suppression of soliton production amplitudes above threshold is universal in all quantum field theories.\n\nThe lesson for production of charged BPS states in N = 8 SUGRA is obvious. The scale of variation of the fields is the Schwarzschild radius, which goes like the square root of the E 7,7 invariant, J 4 , that determines the entropy. Indeed, this is the only dimensionless parameter, which characterizes these solutions. We conclude that when J 4 is of order one, there is no suppression of BPS production amplitudes.\n\nOne might attempt to argue that we only have reliable evidence for the existence of these BPS states when J 4 is large and the semi-classical description of the states is valid. However, this is enough to prove the existence of states of large charge of opposite sign and almost equal magnitude. Cluster decomposition and charge conservation assure us that these can annihilate to states of small charge, which should then Hawking radiate down to the BPS state of small charge. Previously this argument was made only for states of J 4 = 0, and 7 arguments were given that those states might not exist. We've now seen that the argument holds just as well for states with small non-zero J 4 , and that the existence of such states follows from the existence of non-singular BPS black holes.\n\nWe thus have arguments based on very general principles, that SU (8) violation should occur at finite orders in the expansion in energy over the Planck scale. This manifestly contradicts the claim that the techniques for constructing finite amplitudes will construct the correct amplitudes to all orders, since those techniques preserve SU (8). With more work, one could perhaps give an estimate of the order in the loop expansion at which divergences occur. Elvang and Kiermaier [11], have found the leading low energy contribution to SU (8) violating amplitudes in perturbative string theory. These come from non-BPS massive string modes, at tree level in the g S expansion 8 . By unitarity, these give higher order contributions to the 2 → 2 super-graviton amplitude, which cannot be captured by the methods that give finite perturbative amplitudes. Perturbative finiteness should fail before these contributions become manifest. Henriette Elvang (private communication) has pointed out that if we assume the SU(8) violating 5 point function appears at the order in the energy expansion indicated by the lowest dimension operator that appears in string theory, then the induced unitary correction to 2 → 2 graviton scattering first shows up at 8 loops in the SUGRA loop expansion. This is consistent with the conjecture that finiteness will fail at 7 loops. Unfortunately, our general argument does not rule out the possibility that some finite order of the perturbation expansion of SU(8) violating amplitudes vanishes. String theory predicts definite orders, but if one takes the point of view that the hypothetical finite perturbation series defines a separate theory of quantum gravity, disjoint from string theory, one cannot counter the argument by citing string theory results.\n\nFinally I'd like to note that, although I've concentrated on BPS states, where SU(8) violation is manifest in the mass spectrum, ordinary Schwarzschild black holes are expected to lead to SU(8) violation as well. A continuous global symmetry like SU(8) violates the Covariant Entropy Bound for black holes. By dropping neutral particles with SU(8) quantum numbers into a black hole, we can create an infinite number of distinct black holes with the same Schwarzschild field, but transforming in different SU (8) representations. Graviton-graviton scattering, at fixed impact parameter less than the Schwarzschild radius corresponding to the center of mass energy, is expected to be dominated by black hole production. Once √ s > M P , this occurs at impact parameters larger than the Planck scale. Black hole production and decay must violate SU (8) to avoid the violation of the covariant entropy bound. As a consequence, it appears that there are only three logical responses to the arguments presented in this paper.\n\n• The SU(8) invariant perturbation series fails to be finite at some order. This would indicate that N = 8 SUGRA is just an effective field theory, which requires UV completion. As occurs in string theory, the UV completion could violate SU (8) and be consistent with the existence of charged BPS states.\n\n• The SU(8) invariant perturbation expansion is finite to all orders. If the considerations of this paper are correct, it cannot be the perturbation series of a theory which contains charged BPS states, and probably not of one which contains any kind of black hole. Despite the fact that it contains a graviton, one would feel uncomfortable calling such a model a quantum theory of gravity.\n\n• One of the assumptions of this paper is incorrect, most likely the assumption that amplitudes are polynomially bounded in the cut complex energy plane. The results of Giddings and Porto make it plausible that gravitational amplitudes obey such a bound, but by their nature they cannot constitute a proof of that fact.\n Modern on shell methods for computing the S-matrix automatically combine together many diagrams and exhibit cancelations due to gauge invariance explicitly. However, the n ways of combining the on shell legs in an n-particle unitarity cut, mirrors the source of the factorial growth of diagrams. The path integral derivation of the degree of divergence of the perturbation series applies equally well to on shell amplitudes.\nNote that this in no way contradicts the observation of[16] that D-branes in e.g. Type IIA, have a core of order the eleven dimensional Planck length, which is revealed in D-brane-D-brane scattering at low velocity.\nBut see the remarks in Footnote 2.\nHere I'm assuming an hypothetical theory in which the leading source of SU (8) violation comes from BPS states with J 4 = 0. In string perturbation theory, breaking to a continuous subgroup occurs already\nIn my opinion, specious,8 Which is not the same as the N = 8 loop expansion.\nAcknowledgementsI would like to thank Dan Freedman, Andy Strominger, Greg Moore, Renata Kallosh, Lance Dixon, Stanley Deser, Kelly Stelle, Michael Green, Michael Kiermaier, Dan Harlow, Rafael Porto, Zvi Bern, and particularly Henriette Elvang, for numerous conversations which led me to an understanding of the issues discussed in this paper. This work was supported in part by the Department of Energy.\nThe Gravitational S-matrix. S B Giddings, R A Porto, arXiv:0908.0004Phys. Rev. D. 8125002hep-thS. B. Giddings and R. A. Porto, \"The Gravitational S-matrix,\" Phys. Rev. D 81, 025002 (2010) [arXiv:0908.0004 [hep-th]].\n\nThree Loop Counterterms For Extended Supergravity. S Deser, J H Kay, K S Stelle, ; S Deser, J H B Kay ; M, J G Green, P Russo, Vanhove, arXiv:hep-th/0610299Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity. 3899JHEPS. Deser, J. H. Kay and K. S. Stelle, \"Renormalizability Properties of Supergravity,\" Phys. Rev. Lett. 38, 527 (1977). ; S. Deser and J. H. Kay, \"Three Loop Counterterms For Extended Supergravity,\" Phys. Lett. B 76, 400 (1978). ; M. B. Green, J. G. Russo and P. Vanhove, \"Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity,\" JHEP 0702 (2007) 099 [arXiv:hep-th/0610299];\n\nUltraviolet Properties of Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0611273Phys. Rev. Lett. 98131602M. B. Green, J. G. Russo and P. Vanhove, \"Ultraviolet Properties of Maximal Supergravity,\" Phys. Rev. Lett. 98 (2007) 131602 [arXiv:hep-th/0611273];\n\nM B Green, J G Russo, P Vanhove, arXiv:1002.3805String Theory Dualities and Supergravity Divergences. 100675hep-thM. B. Green, J. G. Russo and P. Van- hove, \"String Theory Dualities and Supergravity Divergences,\" JHEP 1006 (2010) 075 [arXiv:1002.3805 [hep-th]];\n\nThe Critical Ultraviolet Behaviour of N = 8. P Vanhove, P. Vanhove, \"The Critical Ultraviolet Behaviour of N = 8\n\n. Supergravity Amplitudes, arXiv:1004.1392hep-thSupergravity Amplitudes,\" arXiv:1004.1392 [hep-th];\n\n5 Loops in 24/5 Dimensions. J Bjornsson, M B Green, arXiv:1004.2692JHEP. 1008132hep-thJ. Bjornsson and M. B. Green, \"5 Loops in 24/5 Dimensions,\" JHEP 1008 (2010) 132 [arXiv:1004.2692 [hep-th]];\n\nAmplitudes and Ultraviolet Behavior of N = 8 Supergravity. Z Bern, J J Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:1103.1848hep-th. and references thereinZ. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \"Amplitudes and Ultravio- let Behavior of N = 8 Supergravity,\" arXiv:1103.1848 [hep-th], and references therein;\n\nSimplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes. Z Bern, J J M Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:1201.5366hep-thZ. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \"Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Am- plitudes,\" arXiv:1201.5366 [hep-th];\n\nThree-Loop Superfiniteness of N=8 Supergravity. Z Bern, J J Carrasco, L J Dixon, H Johansson, D A Kosower, R Roiban, hep-th/0702112Phys. Rev. Lett. 98161303Z. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson, D. A. Kosower and R. Roiban, \"Three-Loop Superfiniteness of N=8 Supergravity,\" Phys. Rev. Lett. 98, 161303 (2007) [hep-th/0702112];\n\nThe Ultraviolet Behavior of N=8 Supergravity at Four Loops. Z Bern, J J Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:0905.2326Phys. Rev. Lett. 10381301hep-thZ. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \"The Ultraviolet Behavior of N=8 Supergravity at Four Loops,\" Phys. Rev. Lett. 103, 081301 (2009) [arXiv:0905.2326 [hep-th]];\n\nStringy KLT relations, global symmetries, and E 7(7) violation. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 1010108hep-thH. Elvang and M. Kiermaier, \"Stringy KLT relations, global symmetries, and E 7(7) violation,\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]];\n\nCounterterms vs. Dualities. Guillaume Bossard, Hermann Nicolai, arXiv:1105.1273JHEP. 110874hep-thGuillaume Bossard, Hermann Nicolai (Potsdam, Max Planck Inst.). \"Counterterms vs. Dualities\", JHEP 1108 (2011) 074 arXiv:1105.1273 [hep-th];\n\nE7(7) symmetry in perturbatively quantised N=8 supergravity. Guillaume Bossard, Christian Hillmann, Hermann Nicolai, arXiv:1007.5472JHEP. 101252hep-thGuillaume Bossard, Christian Hillmann, Hermann Nicolai \"E7(7) symmetry in pertur- batively quantised N=8 supergravity\" JHEP 1012 (2010) 052, arXiv:1007.5472 [hep-th] ;\n\nAbsence of Three-Loop Four-Point Divergences in N=4 Supergravity. Z Bern, S Davies, T Dennen, Y. -T Huang, arXiv:1202.3423hep-thZ. Bern, S. Davies, T. Dennen and Y. -t. Huang, \"Absence of Three-Loop Four- Point Divergences in N=4 Supergravity,\" arXiv:1202.3423 [hep-th];\n\nN=8 Supergravity on the Light Cone. R Kallosh, arXiv:0903.4630Phys. Rev. D. 80105022hep-thR. Kallosh, \"N=8 Supergravity on the Light Cone,\" Phys. Rev. D 80, 105022 (2009) [arXiv:0903.4630 [hep-th]];\n\nSymmetry and Finiteness of N=8 Supergravity. R Kallosh, arXiv:1103.4115JHEP. 1203783hep-thR. Kallosh, \"E 7(7) Symmetry and Finiteness of N=8 Supergravity,\" JHEP 1203, 083 (2012) [arXiv:1103.4115 [hep-th]];\n\n) Current Conservation. R Kallosh, arXiv:1104.5480N=8 Counterterms and E. 7773JHEP. hep-thR. Kallosh, \"N=8 Counterterms and E 7(7) Current Conservation,\" JHEP 1106, 073 (2011) [arXiv:1104.5480 [hep-th]];\n\nOn Absence of 3-loop Divergence in N=4 Supergravity. R Kallosh, arXiv:1202.4690hep-thR. Kallosh, \"On Absence of 3-loop Divergence in N=4 Supergravity,\" arXiv:1202.4690 [hep-th];\n\n5 loops in 24/5 dimensions. J Bjornsson, M B Green, arXiv:1004.2692JHEP. 1008132hep-thJ. Bjornsson and M. B. Green, \"5 loops in 24/5 dimensions,\" JHEP 1008, 132 (2010) [arXiv:1004.2692 [hep-th]].\n\nThe N=8 Supergravity In Superspace. L Brink, P S Howe, Phys. Lett. B. 88268L. Brink and P. S. Howe, \"The N=8 Supergravity In Su- perspace,\" Phys. Lett. B 88, 268 (1979);\n\nSupergravity In Superspace. P S Howe, Nucl. Phys. B. 199309P. S. Howe, \"Supergravity In Superspace,\" Nucl. Phys. B 199 (1982) 309;\n\nCounterterms in extended supergravities. R E Kallosh, Phys. Lett. B. 99122R. E. Kallosh, \"Counterterms in extended supergrav- ities,\" Phys. Lett. B 99, 122 (1981);\n\nHigher Order Invariants In Extended Supergravity. P S Howe, U Lindstrom, Nucl. Phys. B. 181487P. S. Howe and U. Lindstrom, \"Higher Order Invariants In Extended Supergravity,\" Nucl. Phys. B 181, 487 (1981);\n\nSuperactions. P S Howe, K S Stelle, P K Townsend, Nucl. Phys. B. 191445P. S. Howe, K. S. Stelle and P. K. Townsend, \"Superactions,\" Nucl. Phys. B 191, 445 (1981);\n\nIntegral invariants in N=4 SYM and the effective action for coincident D-branes. J M Drummond, P J Heslop, P S Howe, S F Kerstan, hep-th/0305202JHEP. 030816J. M. Drummond, P. J. Heslop, P. S. Howe and S. F. Kerstan, \"Integral invariants in N=4 SYM and the effective action for coincident D-branes,\" JHEP 0308, 016 (2003) [hep-th/0305202];\n\nA Note on N=8 counterterms. J M Drummond, P J Heslop, P S Howe, arXiv:1008.4939hep-thJ. M. Drummond, P. J. Heslop and P. S. Howe, \"A Note on N=8 coun- terterms,\" arXiv:1008.4939 [hep-th];\n\n) symmetry in maximal supergravity. J Broedel, L J Dixon, arXiv:0911.5704JHEP. 100573R**4 counterterm and. hep-thJ. Broedel and L. J. Dixon, \"R**4 counterterm and E(7)(7) symmetry in maximal supergravity,\" JHEP 1005, 003 (2010) [arXiv:0911.5704 [hep-th]];\n\nOn duality symmetries of supergravity invariants. G Bossard, P S Howe, K S Stelle, arXiv:1009.0743JHEP. 110120hep-thG. Bossard, P. S. Howe and K. S. Stelle, \"On duality symmetries of super- gravity invariants,\" JHEP 1101, 020 (2011) [arXiv:1009.0743 [hep-th]];\n\nThe Ultra-violet question in maximally supersymmetric field theories. G Bossard, P S Howe, K S Stelle, arXiv:0901.4661Gen. Rel. Grav. 41hep-thG. Bossard, P. S. Howe and K. S. Stelle, \"The Ultra-violet question in maximally supersymmetric field theories,\" Gen. Rel. Grav. 41, 919 (2009) [arXiv:0901.4661 [hep-th]];\n\nThe Ultraviolet Behavior Of N=4 Yang-mills And The Power Counting Of Extended Superspace. N Marcus, A Sagnotti, Nucl. Phys. B. 25677N. Marcus and A. Sagnotti, \"The Ultraviolet Behavior Of N=4 Yang-mills And The Power Counting Of Extended Superspace,\" Nucl. Phys. B 256, 77 (1985);\n\nP S Howe, K S Stelle, The Ultraviolet Properties Of Supersymmetric Field Theories. 41871P. S. Howe and K. S. Stelle, \"The Ultraviolet Properties Of Supersymmetric Field Theories,\" Int. J. Mod. Phys. A 4, 1871 (1989);\n\nThe vanishing volume of D=4 superspace. G Bossard, P S Howe, K S Stelle, P Vanhove, arXiv:1105.6087Class. Quant. Grav. 28215005hep-thG. Bossard, P. S. Howe, K. S. Stelle and P. Vanhove, \"The vanishing volume of D=4 superspace,\" Class. Quant. Grav. 28, 215005 (2011) [arXiv:1105.6087 [hep-th]];\n\nAn R 4 non-renormalisation theorem in N=4 supergravity. P Tourkine, P Vanhove, arXiv:1202.3692hep-thP. Tourkine and P. Vanhove, \"An R 4 non-renormalisation theorem in N=4 supergravity,\" arXiv:1202.3692 [hep-th].\n\nNon-Renormalisation Conditions in Type II String Theory and Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0610299JHEP. 070299M. B. Green, J. G. Russo and P. Vanhove, \"Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity,\" JHEP 0702 (2007) 099 [arXiv:hep- th/0610299];\n\nUltraviolet Properties of Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0611273Phys. Rev. Lett. 98131602M. B. Green, J. G. Russo and P. Vanhove, \"Ultraviolet Properties of Maximal Supergravity,\" Phys. Rev. Lett. 98 (2007) 131602 [arXiv:hep-th/0611273];\n\nString Theory Dualities and Supergravity Divergences. M B Green, J G Russo, P Vanhove, arXiv:1002.3805JHEP. 100675hep-thM. B. Green, J. G. Russo and P. Vanhove, \"String Theory Dualities and Supergravity Divergences,\" JHEP 1006 (2010) 075 [arXiv:1002.3805 [hep-th]];\n\nNondecoupling of Maximal Supergravity from the Superstring. M B Green, H Ooguri, J H Schwarz, arXiv:0704.0777Phys. Rev. Lett. 9941601hep-thM. B. Green, H. Ooguri and J. H. Schwarz, \"Nondecoupling of Maximal Supergravity from the Superstring,\" Phys. Rev. Lett. 99, 041601 (2007) [arXiv:0704.0777 [hep-th]].\n\nTrouble for remnants. L Susskind, hep-th/9501106L. Susskind, \"Trouble for remnants,\" hep-th/9501106.\n\n. F Cerulus, A Martin, Phys. Letters. 880F. Cerulus and A. Martin, Phys. Letters 8, 80 (1964).\n\nUV-Completion by Classicalization. G Dvali, G F Giudice, C Gomez, A Kehagias, arXiv:1010.1415JHEP. 1108hep-phG. Dvali, G. F. Giudice, C. Gomez and A. Kehagias, \"UV-Completion by Classicaliza- tion,\" JHEP 1108, 108 (2011) [arXiv:1010.1415 [hep-ph]].\n\nStringy KLT relations, global symmetries. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 10107108) violation. hep-thH. Elvang and M. Kiermaier, \"Stringy KLT relations, global symmetries, and E 7(7) vio- lation,\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]].\n\nA simple approach to counterterms in N=8 supergravity. H Elvang, D Z Freedman, M Kiermaier, arXiv:1003.5018JHEP. 101116hep-thH. Elvang, D. Z. Freedman and M. Kiermaier, \"A simple approach to counterterms in N=8 supergravity,\" JHEP 1011, 016 (2010) [arXiv:1003.5018 [hep-th]].\n\nE7(7) constraints on counterterms in N=8 supergravity. N Beisert, H Elvang, D Z Freedman, M Kiermaier, A Morales, S Stieberger, arXiv:1009.1643Phys. Lett. B. 694265hep-thN. Beisert, H. Elvang, D. Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, \"E7(7) constraints on counterterms in N=8 supergravity,\" Phys. Lett. B 694, 265 (2010) [arXiv:1009.1643 [hep-th]];\n\nSUSY Ward identities, Superamplitudes, and Counterterms. H Elvang, D Z Freedman, M Kiermaier, arXiv:1012.3401J. Phys. A A. 44454009hep-thH. El- vang, D. Z. Freedman and M. Kiermaier, \"SUSY Ward identities, Superamplitudes, and Counterterms,\" J. Phys. A A 44, 454009 (2011) [arXiv:1012.3401 [hep-th]].\n\nPerturbative and Non-perturbative N =8 Supergravity. M Bianchi, S Ferrara, R Kallosh, arXiv:0910.3674Phys. Lett. B. 690328hep-thM. Bianchi, S. Ferrara and R. Kallosh, \"Perturbative and Non-perturbative N =8 Su- pergravity,\" Phys. Lett. B 690, 328 (2010) [arXiv:0910.3674 [hep-th]].\n\nStringy KLT relations, global symmetries. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 10107108) violation. hep-thH. Elvang and M. Kiermaier, \"Stringy KLT relations, global symmetries, and E 7(7) violation,\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]].\n\nHolography and cosmology. W Fischler, L Susskind, hep-th/9806039W. Fischler and L. Susskind, \"Holography and cosmology,\" hep-th/9806039;\n\nThe Holographic principle. R Bousso, hep-th/0203101Rev. Mod. Phys. 74and references thereinR. Bousso, \"The Holographic principle,\" Rev. Mod. Phys. 74, 825 (2002) [hep-th/0203101], and references therein.\n\nHow to count the states of extremal black holes in N=8 supergravity. V Balasubramanian, hep-th/9712215*Cargese 1997, Strings, branes and dualities* 399-410. V. Balasubramanian, \"How to count the states of extremal black holes in N=8 super- gravity,\" In *Cargese 1997, Strings, branes and dualities* 399-410 [hep-th/9712215].\n\nString theory dynamics in various dimensions. E Witten, hep-th/9503124Nucl. Phys. B. 44385E. Witten, \"String theory dynamics in various dimensions,\" Nucl. Phys. B 443, 85 (1995) [hep-th/9503124].\n\nScattering of strings from D-branes. A Hashimoto, I R Klebanov, hep-th/9611214Nucl. Phys. Proc. Suppl. 55A. Hashimoto and I. R. Klebanov, \"Scattering of strings from D-branes,\" Nucl. Phys. Proc. Suppl. 55B, 118 (1997) [hep-th/9611214].\n\nD-branes and short distances in string theory. M R Douglas, D N Kabat, P Pouliot, S H Shenker, hep-th/9608024Nucl. Phys. B. 48585M. R. Douglas, D. N. Kabat, P. Pouliot and S. H. Shenker, \"D-branes and short distances in string theory,\" Nucl. Phys. B 485, 85 (1997) [hep-th/9608024].\n"},"corpus_id":{"kind":"number","value":119266117,"string":"119,266,117"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1205.5768\",\n \"mag\": 2131922329,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1205.5768v1.pdf\"\n ],\n \"pdf_sha\": \"c412f9204d95b45660da2d3d563b2f2862c04e18\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Arguments Against a Finite N = 8 Supergravity"},"authors":{"kind":"list like","value":["Tom Banks \nNHETC and Department of Physics and Astronomy\nRutgers University\n08854-8019PiscatawayNJUSA\n\nSCIPP and Department of Physics\nUniversity of California\n95064-1077Santa CruzCAUSA"],"string":"[\n \"Tom Banks \\nNHETC and Department of Physics and Astronomy\\nRutgers University\\n08854-8019PiscatawayNJUSA\\n\\nSCIPP and Department of Physics\\nUniversity of California\\n95064-1077Santa CruzCAUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"I present arguments based on dispersion theory and the existence of charged BPS states of small charge, that maximally supersymmetric SUGRA is not a finite theory. The essence of the argument is that the perturbative construction of the theory implies that scattering amplitudes are invariant under a continuous SU(8) global symmetry, which is broken by contributions from virtual BPS states. Conventional estimates of the contributions of those states above production threshold, shows that they should be order one. If the amplitude satisfies a dispersion relation with a finite number of subtractions, as has recently been argued by Giddings and Porto [1], then SU(8) violating amplitudes appear at finite orders in the low energy expansion. We also make a brief remark about the inconsistency of the idea of classicalization in non-gravitational theories."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"There has been a lot of interest in recent years in the question of whether N = 8 supergravity (SUGRA) has a finite S-matrix in perturbation theory [2]. Most of the debate has centered around detailed perturbative calculations and the question of whether the finite answers that have been obtained could be explained in terms of symmetries, or were surprises that might point to new principles at work, which could guarantee finiteness to all orders. The work of [9] shows that all existing calculations can be explained by symmetries, and that the first real test of finiteness will come at 7 loops. I believe that this is now the consensus in the field.\nI've always been skeptical about the finiteness arguments, for reasons that I will outline below. At the Penn State Conference in the Fall of 2009 I gave a talk claiming that the putative finite theory could not be a real theory of gravitation, because it could not contain the BPS black hole states that can be found as solutions of the classical equations of N = 8 SUGRA. In that talk, in order to make a comparison with the way in which \"nonperturbative\" states affect string perturbation theory, I emphasized the black hole states with E(7, 7) invariant J 4 = 0, which go to zero mass at the boundaries of moduli space. In the string theoretic realizations of maximal SUGRA, these are elementary particles in the appropriate limits, and have very small entropy. The entropy is of course an invariant on the moduli space, and, as a consequence, the corresponding classical solutions have zero area horizons and are singular.\nIn [10] it was suggested that a version of the theory omitting only these singular black holes might be consistent. While I disagree with this claim, it seems easier to avoid the J 4 = 0 black holes, because the argument does not depend on them. Instead, it depends only on the following basic points • Theories of Quantum gravity do not have global continuous symmetries. This is basically a consequence of the existence of black holes and Hawking radiation. A global continuous symmetry implies that we can make black holes with size of order the Planck scale, and masses of order the Planck mass, with arbitrarily large numbers of quantum states, corresponding to high dimensional representations of the group. This violates the Covariant Entropy Bound [12], and it also leads to a variety of paradoxes [6]. In particular, the SU(8) global symmetry of N = 8 SUGRA must be broken. The methods that are used to compute finite N = 8 loop amplitudes would guarantee that to all orders in the loop expansion, SU (8) is a valid symmetry.\n• As a consequence of the Dirac quantization condition, the spectrum of BPS black holes is not SU (8) invariant. Using unitarity, we can then prove that amplitudes above the threshold for black hole production for scattering of particles in the SUGRA multiplet, also violate SU (8). SU (8) invariance follows from SUSY for four point amplitudes, so the simplest such amplitude is a five point function. An example is the 2 → 3 process, in which two gravitons scatter to produce two gravitons and a scalar. The gravitons are singlets, and the scalar is in the 70 of SU (8). This amplitude is known to be non-vanishing in perturbative string theory [11], where perturbative winding and Kaluza Klein modes are known to be dual to BPS black holes. [11] shows that even tree level string theory breaks the SU(8) symmetry down to SU(4) × SU (4). Loops of BPS states break it to a discrete group. Green, Ooguri and Schwarz [5] have shown that there are no limits of string theory in which all the BPS states decouple.\n• One often hears the phrase, \"black holes are non-perturbative states and their production will not appear in perturbation theory\". This is a mistake. The phrase is only correct for black holes, like those of large charge, whose horizon radius is much larger than the Planck scale. We argue that states of small charge are produced with amplitudes of order 1 above threshold. The thresholds are of order the Planck scale and so is the physical size of the state (by extrapolation from the classical solutions). We recall that the production cross sections for solitons are suppressed exponentially in the ratio of their physical size to their Compton wavelength. There is no such suppression for black holes of small charge, whether or not their invariant entropy vanishes. We also argue that the existence of the small charge black holes follows from the existence of classical solutions plus cluster decomposition 1 .\n• The rest of the argument follows if the SU(8) violating 5 point amplitude satisfies a dispersion relation in the center of mass energy, with a finite number of subtractions, when the other kinematic invariants are fixed. Such a representation shows that the amplitude has a finite perturbation series in powers of the energy over the Planck mass, which contradicts the claim that there is an SU(8) invariant finite power series, constructed by the methods of [2] 2 , which coincides with the amplitudes that include contributions from BPS states.\nThere are two reasons to worry about the existence of such a dispersion relation. At fixed impact parameter, we expect graviton scattering at sufficiently high energy to be 1 The last argument is due to Andy Strominger. 2 Although there are many alternative methods of calculation in the papers of reference [2], all but the string theory based method begin with manifestly SU (8) invariant tree level amplitudes and use unitarity and analyticity to construct the loops. Interestingly, the methods of Bern, Dixon and collaborators use dimensional regularization to deal with infrared divergences. Zvi Bern has informed me that this breaks the manifest SU (8) symmetry, though the final results are SU (8) invariant in four dimensions. One would imagine that, since IR divergences come only from soft graviton emission, and gravitons are SU (8) singlets, one is guaranteed to get invariant results unless genuine UV divergences are encountered. dominated by black hole production (these are neutral black holes). As a consequence of the thermal nature of black hole evaporation, any exclusive channel with a small number of high energy particles, is exponentially suppressed. This means that, at fixed impact parameter, the amplitude cannot be power law bounded in all directions in the complex energy plane, because it violates the Cerulus-Martin bound [7]. However, Giddings and Porto have recently argued that at fixed kinematic invariants, the shadow scattering of the black hole absorption process, produces an amplitude that does satisfy the CM bound.\nThe other reason one might worry about the dispersion relation is that IR divergences would cause a singularity at zero energy, which prevent us from pulling the contour around the origin to complete the argument. Since the amplitude vanishes in perturbation theory, there is no perturbative evidence for such an IR singularity. Furthermore, in perturbative string theory, where such power law corrections to the SU(8) violating amplitude certainly exist, and which has the same low energy spectrum and interactions, there is again no evidence than the amplitude is non-analytic at zero.\nTo summarize: The SU(8) violating five point amplitude, which follows from the existence of black holes of small charge above the production threshold for such black holes, has an analytic continuation which has a power law tale, and contradicts the existence of an SU (8) invariant method for constructing a finite unitary S-matrix to all orders in a power series in energy divided by the Planck mass.\nThere is a logical possibility that some of the methods in [2] might produce a finite result to all orders in perturbation theory, and provide evidence for a model of quantum gravity, which did not contain the BPS states, that are the focus of this article. While I cannot provide a string free refutation of this claim, it is worth recalling how the \"UV\" divergence of the pure SUGRA amplitudes would be interpreted in the string based computations of [4]. Their computations try to define SUGRA as a certain limit of Type II string amplitudes. All such limits are equivalent to weakly coupled string theory, or 11D SUGRA, on a large torus, so in this method of calculation, the UV divergence must be interpreted as an IR divergence coming from non-perturbative states whose mass goes to zero in this limit.\nThere is one other bit of information that emerged from thinking about the consequences of a finite N = 8 SUGRA, which applies to any putatively finite but perturbatively nonrenormalizable theory [8]. In such a theory the expansion parameter is energy divided by some scale. On the other hand, simple diagram counting shows that field theoretic perturbation series are divergent and lead to essential singularities in the coupling 3 In such a theory, this would mean essential singularities at zero energy. In the theories discussed in [8], nothing prevents us from giving mass to all the fields in the Lagrangian. Rigorous theorems of S-matrix theory then preclude the existence of essential singularities, either at the origin or at a finite mass. The point is that in a theory with a mass gap, the number of states at finite energy is bounded and all singularities are multi-particle cuts, for a finite number of particles. Thus, we can refute the claims that any such theory is finite, unless some principle prevents us from giving mass to the particles. Of course, in N = 8 SUGRA, there is such a principle, so this argument is not applicable. That is why we went through the more elaborate discussion outlined above.\nOne might want to preserve the claim to a finite N = 8 SUGRA for a bit longer by insisting that the finite theory does not have any charged states in it. The classical Lagrangian has solutions corresponding to a pair of oppositely charged BPS black holes moving away from each other, with arbitrary initial velocity, from an arbitrary initial separation. For some values of these parameters the pair will separate and reach infinity, while for others they will collapse and form a neutral black hole. The usual rules of crossing symmetry suggest that charged particles pairs will be produced in the Hawking radiation of neutral black holes, so it is likely that one would have to forbid the existence of ordinary Schwarzschild black holes as well. Indeed, we've already argued that the covariant entropy bound implies that processes involving the formation and decay of neutral black holes cannot preserve SU (8) symmetry.\nThe other option is would be to declare that the theory contained no neutral black holes either, but then its claim to be a theory of gravitation would be empty. The rest of this paper will be devoted to a more detailed discussion of the SU (8) violating five point amplitude."},{"title":"Production of Soliton States","content":"The Lagrangian of N = 8 SUGRA contains 28 U(1) gauge bosons. Among the solutions of the theory are BPS dyonic black holes with arbitrary continuous values of the electric and magnetic charges. The theory has a classical E(7, 7) symmetry, which is partly realized by duality rotations in the 56 dimensional charge space. Dirac quantization restricts the space of charges to an integer lattice, which is invariant only under an integer subgroup E(7, 7; Z). The 70 scalar fields of the theory lie in the coset E(7, 7)/SU (8), so only the SU(8) symmetry acts linearly on scattering amplitudes for excitations around a particular point in moduli space. The coset transformations instead give rise to soft scalar low energy theorems for these amplitudes. Recently, these symmetry constraints have been shown to guarantee the finiteness of all amplitudes that have so far been computed in this theory, but fail to rule out a divergence at 7 loops.\nThe entropy of these BPS black holes is computed using the Bekenstein-Hawking law, which determines the large entropy asymptotics. It is given by\nwhere J 4 is an E(7, 7) invariant, quartic in the charges. When the charges are quantized it is invariant only under the integer subgroup. These are extremal black holes, so the charge also determines the mass, which is not invariant under SU (8).\nThe charges live on a Lorentzian lattice and the invariant J 4 can be positive, negative or zero. The states with zero J 4 go to zero mass at certain boundaries of moduli space. The corresponding classical solutions are singular, because they have horizons of zero area. The authors of reference [10] argued that these singular solutions might not be real quantum states of the theory in question. While I do not agree with those arguments, the main discussion of the present paper will be restricted to states with non-zero J 4 .\nThe realization of N = 8 SUGRA as the low energy effective field theory of Type II string theory compactified on a 6− torus of stringy size, sheds light on these states. Some of the BPS sectors are realized in terms of Kaluza-Klein momenta, and wrapping numbers of elementary strings and D-branes on cycles of the torus [13]. As shown by Witten [14], all the boundaries of moduli space are dual (via E(7, 7; Z) rotations) to regimes in which weakly coupled Type II string theory, or 11 dimensional SUGRA, are compactified on a torus of growing size. The J 4 = 0 states are precisely those which become massless particles at one of these boundaries. This explains the vanishing of the area term in their entropy formula (the entropy is constant on moduli space).\nWhen the string coupling is weak, and all geometrical moduli are of order string scale or larger, we can do reliable perturbative calculations of amplitudes involving virtual charged BPS states. These lead to SU(8) violation. In string perturbation theory, only states with J 4 = 0 contribute to the amplitudes. We can also do reliable weak coupling calculations of amplitudes involving loops of wrapped D-branes with J 4 = 0. These are T-dual to D-instanton calculations and are of order e − c g S . We will see below that this exponential suppression should be interpreted as an exponential of the linear size of the wrapped D-brane state, divided by its Compton wavelength. Here we note only that this is a mathematical fact. The size of a D-brane can be measured by scattering gravitons from it, and this leads to a size of order string scale [15]. The Compton wavelength is smaller than this by a factor g S . 4 Note further, that, if we imagine extrapolating to g s ∼ 1, there is every indication that the contribution of all of these BPS states to SU(8) violating amplitudes will become of order one in eleven dimensional Planck units. Elvang and Kiermaier [11] have computed the lowest dimension SU (8) violating operator in the effective field theory for the SUGRA multiplet, to lowest order in Type II string perturbation theory. It has the form\nwhere φ is the dimensionless dilaton field. This term comes from explicit violation of SU (8) in tree level string diagrams involving massive non-BPS string states, but it is of the same form as one would expect for the leading low energy contribution from loops of BPS states whose masses violate SU (8). In different regions of the moduli space E(7, 7)/SU (8), the field φ would be replaced by other members of the 70 multiplet of Goldstone bosons. Naively extrapolating to the regime g S ∼ 1 one finds that this term is scaled by the 11 D Planck mass, with a coefficient of order 1. Of course, in this regime, the calculation of [11] is not valid, but since there is no small dimensionless parameter, it is hard to see how the term could be significantly smaller than the extrapolation of the perturbative calculation.\nSome of the advocates of a finite N = 8 theory emphasize that their theory may have nothing to do with string theory and may be a new paradigm for the approach to quantum gravity as a quantum field theory. Therefore, I must emphasize that the above remarks about string theory are meant to be suggestive only. Type II string theory with g S ∼ 1 and a torus of 11 dimensional Planck size is a finite theory with a single parameter, with dimensions of mass. Like the would be finite N = 8 theory, it has BPS states, whose mass formula violates the continuous symmetry of the leading term in the low energy effective Lagrangian. Unlike that hypothetical theory, the violation shows up in the low energy power series expansion of the scattering amplitudes of the particles in the SUGRA multiplet. What I will try to demonstrate below is that rather general principles imply that any sensible theory containing these BPS states will behave like string theory does and not like the hypothetical finite theory is supposed to."},{"title":"Brief outline of finite calculations","content":"This tiny section is not meant to be even a cursory introduction to the remarkable set of methods that have been developed to calculate the S-matrix in N = 8 SUGRA and similar theories. These include spinor helicity methods, tree level recursion relations based on analyticity in the complex momentum plane, and standard as well as novel unitarity cut techniques for calculating loop amplitudes. Using dimensional regularization, some authors have even found a way to get around the traditional dispersion techniques for calculating the real parts of amplitudes. The point is that in 4 − ǫ dimensions, even \"polynomial\" contributions to amplitudes have an imaginary part, as a consequence of the irrational scaling exponents. Although the dimensionally continued amplitudes have no Hilbert space interpretation, the do satisfy the cutting rules, which are the perturbative expression of unitarity.\nThe important observation is that, as long as these methods give finite results, those results will be SU(8) covariant 5 . The tree amplitudes are all SU(8) covariant. The loops are computed from the trees by expressing their discontinuities in terms of lower order amplitudes, and so SU(8) covariance follows by induction, as long as the procedure gives finite and unambiguous answers at each step. In particular, the five point amplitude that will be the focus of our attention, which transforms as a 70, will be identically zero to all orders in the momentum expansion. We will show that if we accept the existence of BPS states, with charges of order 1, which are produced above threshold with amplitudes of order 1, that the all orders vanishing of this amplitude contradicts some very general principles of S-matrix theory."},{"title":"Dispersion relations in models of quantum gravity","content":"The S-matrix elements for the scattering amplitude of two gravitons to two gravitons plus a scalar are conveniently classified according to the helicities of the incoming and outgoing gravitons. Each helicity amplitude has a fixed dependence on components of the momenta, multiplied by unknown scalar functions. The scalar amplitudes in the 2 → 3 S-matrix element depend on the square of the center of mass energy s 12 ≡ s, and the other invariants s ij ≡ p i · p j , (with i = j), subject to the constraints j s ij = 0. There are 5 independent invariants. We will write dispersion relations for these scalar functions in the variable s, with the other independent invariants held fixed.\nFeynman diagram calculations in low energy field theory, as well as perturbative string theory calculations indicate that the (somewhat vague) holomorphy properties of scattering amplitudes, which are postulated in analytic S-matrix theory, remain true in models of quantum gravity in asymptotically flat space. Amplitudes are holomorphic functions of the invariants, with isolated singularities corresponding to the production of real on-shell physical particle states.\nDispersion relations employ this assumption of holomorphy with an assumption that amplitudes are polynomially bounded in the entire complex s plane. This assumption is valid in low energy effective field theory, but when the perturbation parameter has dimensions of length to a positive power, the order of the polynomial increases with the order of perturbation theory.\nIn fact, at fixed impact parameter, simple considerations of black hole physics suggest that the amplitudes fall off exponentially at large s. The argument is simple. As s increases, the colliding particles in the center of mass frame will have a gravitational field, as seen by a distant observer\nWhen the impact parameter is less than 2 √ sG N , the colliding particles are inside the horizon of the black hole. It is extremely plausible (in general relativity, this is called The Hoop Conjecture) that in such a kinematic regime, particle collisions lead to black hole formation, with probability that rapidly approaches 1 as s goes to infinity with fixed impact parameter. When s is far above the threshold for black hole production at some fixed impact parameter, we can treat the black hole by semi-classical techniques. In particular, Hawking's calculation of black hole decay tells us that the probability of producing a particular final state, with a small number of particles carrying all of the outgoing energy, is exponentially suppressed. The argument is simple counting. The black hole decay is thermal, and the black hole entropy increases linearly with s.\nGiddings and Porto [1] have argued that for fixed s ij this behavior implies that the standard assumption of polynomial boundedness in the complex s plane is still valid. The amplitudes at fixed kinematic invariants are integrals over all impact parameters. Since we have argued that for small enough impact parameter, in a range growing like √ s , the amplitudes fall off exponentially, the high energy behavior at fixed s ij is dominated by large impact parameters, where it may be estimated by the eikonal approximation. The eikonal behavior obeys the Cerulus-Martin lower bound on the real axis, and there is every indication that the amplitude is power law bounded in the entire complex s plane, for fixed values of the other s ij , and should satisfy a dispersion relation with a finite number of subtractions. For the 2 → 2 amplitude in N = 8 SUGRA, we expect the right and left hand cuts, which contribute to the dispersion integrand, to meet at the origin. However, for the SU(8) violating 5-point amplitude at hand, one would imagine that the lowest contributing threshold would correspond to the production of pairs of the lightest charged BPS states, whose masses are of order the Planck scale 6 . The amplitudes will have IR divergences, corresponding to emission of soft gravitons from incoming and outgoing particles, as they accelerate, but these will not lead to a lowering of the threshold for real production of particles whose mass violates SU (8). Both the spectrum and perturbative interactions of the SUGRA multiplet preserve SU (8).\nWe have thus argued that SU (8) violating helicity amplitudes may be written as\nwhere N is an integer we have not determined, and K h is a known kinematic helicity factor. P (s) is a polynomial, and ρ is the sum of the discontinuities across the right and left hand cuts. s 0 is the squared mass of the lightest BPS state contributing to the discontinuity. It is of order m 2 P . P and ρ depend implicitly on the other s ij , which we take to be fixed at non-zero values ≪ m 2 P . The low energy expansion of this amplitude involves the moments\nwhere σ(z) ≡ ρ( s 0 z ), and the integrals are all power law convergent at z = 0. The assumption of a convergent perturbation expansion, constructed in terms of recursion relations and perturbative unitarity, implies that A h (s) vanishes to all orders in s m 2 P . This in turn implies that all moments above some k max vanish. The function f (y) = ρ(ys 0 )y −N , is square integrable with respect to the measure dµ = dy y kmax . Indeed, it is finite at every y, because it is a sum of discontinuities across multiparticle cuts, and the integral ∞ 1 dy y 2N +kmax |ρ(ys 0 )| 2 , is convergent at infinity because of the convergence of the original dispersion relation.\nThe vanishing of the higher moments is equivalent to the statement that f (y) is orthogonal to all of the functions y −k for non-negative k. These functions are square integrable and linearly independent. By Gram-Schmidt orthogonalization, we can use them to construct an orthonormal basis. Consequently f (y) = 0. This contradicts the explicit finite contribution to f (y) slightly above the production threshold for the lightest BPS dyon-anti-dyon pair."},{"title":"Production cross sections for BPS monopoles","content":"It is sometimes argued that the BPS states are \"non-perturbative\", and should not show up in the perturbation expansion. This response is too superficial, and a proper understanding with exchange of massive string modes at tree level. Loops of J 4 = 0 states break SU (8) to a discrete subgroup, while the states with J 4 = 0 are non-perturbative in g S because their size is much larger than their Compton wavelength. For string theory with g S and radii in string units all of order one, or any model whose only parameter is the 4D Planck mass, the threshold for SU (8) violating intermediate states will be given by this scale. of why soliton production is suppressed in conventional quantum field theory shows that there is no similar suppression of production of BPS states of small charge.\nIndeed, in conventional field theory, solitons have masses of order m g 2 , where m is the mass of perturbative excitations, and g the dimensionless coupling. Energy denominator suppression would indicate that they contribute to scattering of perturbative excitations at finite orders in perturbation theory, but this is not true. The example of magnetic monopoles in the SO(3) Georgi-Glashow model, is typical. Consider the contribution of monopole production to the total cross section for annihilation of two perturbative charged excitations, at lowest order in the electro-magnetic coupling. It is determined by the vacuum to monopole-anti-monopole matrix element of the electromagnetic current:\nBy crossing symmetry, this is the analytic continuation of the monopole form factor\nto time-like q, ≥ twice the monopole mass. The usual rules of semi-classical physics, tell us that the monopole form factor in the monopole rest frame is simply the classical electromagnetic current of the monopole field configuration. The current is a smooth function j(xm), where m is the mass of the charged vector boson. The Riemann-Lebesgue lemma, tells us that at large q the function falls off faster than any power. It is typically of order\nwhere the constant a is of order one and represents the closest singularity of j(xm) to the real axis in the complex xm plane. This leads to the estimate\nfor the monopole production amplitude. It is clear from this calculation that the relevant parameter is the ratio between the physical size of the soliton, and its Compton wave-length. Non-perturbative suppression of soliton production means that the amplitudes are exponentially small in this ratio. Although I've given this argument in the context of the Georgi-Glashow model, this explanation of the non-perturbative suppression of soliton production amplitudes above threshold is universal in all quantum field theories.\nThe lesson for production of charged BPS states in N = 8 SUGRA is obvious. The scale of variation of the fields is the Schwarzschild radius, which goes like the square root of the E 7,7 invariant, J 4 , that determines the entropy. Indeed, this is the only dimensionless parameter, which characterizes these solutions. We conclude that when J 4 is of order one, there is no suppression of BPS production amplitudes.\nOne might attempt to argue that we only have reliable evidence for the existence of these BPS states when J 4 is large and the semi-classical description of the states is valid. However, this is enough to prove the existence of states of large charge of opposite sign and almost equal magnitude. Cluster decomposition and charge conservation assure us that these can annihilate to states of small charge, which should then Hawking radiate down to the BPS state of small charge. Previously this argument was made only for states of J 4 = 0, and 7 arguments were given that those states might not exist. We've now seen that the argument holds just as well for states with small non-zero J 4 , and that the existence of such states follows from the existence of non-singular BPS black holes.\nWe thus have arguments based on very general principles, that SU (8) violation should occur at finite orders in the expansion in energy over the Planck scale. This manifestly contradicts the claim that the techniques for constructing finite amplitudes will construct the correct amplitudes to all orders, since those techniques preserve SU (8). With more work, one could perhaps give an estimate of the order in the loop expansion at which divergences occur. Elvang and Kiermaier [11], have found the leading low energy contribution to SU (8) violating amplitudes in perturbative string theory. These come from non-BPS massive string modes, at tree level in the g S expansion 8 . By unitarity, these give higher order contributions to the 2 → 2 super-graviton amplitude, which cannot be captured by the methods that give finite perturbative amplitudes. Perturbative finiteness should fail before these contributions become manifest. Henriette Elvang (private communication) has pointed out that if we assume the SU(8) violating 5 point function appears at the order in the energy expansion indicated by the lowest dimension operator that appears in string theory, then the induced unitary correction to 2 → 2 graviton scattering first shows up at 8 loops in the SUGRA loop expansion. This is consistent with the conjecture that finiteness will fail at 7 loops. Unfortunately, our general argument does not rule out the possibility that some finite order of the perturbation expansion of SU(8) violating amplitudes vanishes. String theory predicts definite orders, but if one takes the point of view that the hypothetical finite perturbation series defines a separate theory of quantum gravity, disjoint from string theory, one cannot counter the argument by citing string theory results.\nFinally I'd like to note that, although I've concentrated on BPS states, where SU(8) violation is manifest in the mass spectrum, ordinary Schwarzschild black holes are expected to lead to SU(8) violation as well. A continuous global symmetry like SU(8) violates the Covariant Entropy Bound for black holes. By dropping neutral particles with SU(8) quantum numbers into a black hole, we can create an infinite number of distinct black holes with the same Schwarzschild field, but transforming in different SU (8) representations. Graviton-graviton scattering, at fixed impact parameter less than the Schwarzschild radius corresponding to the center of mass energy, is expected to be dominated by black hole production. Once √ s > M P , this occurs at impact parameters larger than the Planck scale. Black hole production and decay must violate SU (8) to avoid the violation of the covariant entropy bound. As a consequence, it appears that there are only three logical responses to the arguments presented in this paper.\n• The SU(8) invariant perturbation series fails to be finite at some order. This would indicate that N = 8 SUGRA is just an effective field theory, which requires UV completion. As occurs in string theory, the UV completion could violate SU (8) and be consistent with the existence of charged BPS states.\n• The SU(8) invariant perturbation expansion is finite to all orders. If the considerations of this paper are correct, it cannot be the perturbation series of a theory which contains charged BPS states, and probably not of one which contains any kind of black hole. Despite the fact that it contains a graviton, one would feel uncomfortable calling such a model a quantum theory of gravity.\n• One of the assumptions of this paper is incorrect, most likely the assumption that amplitudes are polynomially bounded in the cut complex energy plane. The results of Giddings and Porto make it plausible that gravitational amplitudes obey such a bound, but by their nature they cannot constitute a proof of that fact."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"There has been a lot of interest in recent years in the question of whether N = 8 supergravity (SUGRA) has a finite S-matrix in perturbation theory [2]. Most of the debate has centered around detailed perturbative calculations and the question of whether the finite answers that have been obtained could be explained in terms of symmetries, or were surprises that might point to new principles at work, which could guarantee finiteness to all orders. The work of [9] shows that all existing calculations can be explained by symmetries, and that the first real test of finiteness will come at 7 loops. I believe that this is now the consensus in the field.\\nI've always been skeptical about the finiteness arguments, for reasons that I will outline below. At the Penn State Conference in the Fall of 2009 I gave a talk claiming that the putative finite theory could not be a real theory of gravitation, because it could not contain the BPS black hole states that can be found as solutions of the classical equations of N = 8 SUGRA. In that talk, in order to make a comparison with the way in which \\\"nonperturbative\\\" states affect string perturbation theory, I emphasized the black hole states with E(7, 7) invariant J 4 = 0, which go to zero mass at the boundaries of moduli space. In the string theoretic realizations of maximal SUGRA, these are elementary particles in the appropriate limits, and have very small entropy. The entropy is of course an invariant on the moduli space, and, as a consequence, the corresponding classical solutions have zero area horizons and are singular.\\nIn [10] it was suggested that a version of the theory omitting only these singular black holes might be consistent. While I disagree with this claim, it seems easier to avoid the J 4 = 0 black holes, because the argument does not depend on them. Instead, it depends only on the following basic points • Theories of Quantum gravity do not have global continuous symmetries. This is basically a consequence of the existence of black holes and Hawking radiation. A global continuous symmetry implies that we can make black holes with size of order the Planck scale, and masses of order the Planck mass, with arbitrarily large numbers of quantum states, corresponding to high dimensional representations of the group. This violates the Covariant Entropy Bound [12], and it also leads to a variety of paradoxes [6]. In particular, the SU(8) global symmetry of N = 8 SUGRA must be broken. The methods that are used to compute finite N = 8 loop amplitudes would guarantee that to all orders in the loop expansion, SU (8) is a valid symmetry.\\n• As a consequence of the Dirac quantization condition, the spectrum of BPS black holes is not SU (8) invariant. Using unitarity, we can then prove that amplitudes above the threshold for black hole production for scattering of particles in the SUGRA multiplet, also violate SU (8). SU (8) invariance follows from SUSY for four point amplitudes, so the simplest such amplitude is a five point function. An example is the 2 → 3 process, in which two gravitons scatter to produce two gravitons and a scalar. The gravitons are singlets, and the scalar is in the 70 of SU (8). This amplitude is known to be non-vanishing in perturbative string theory [11], where perturbative winding and Kaluza Klein modes are known to be dual to BPS black holes. [11] shows that even tree level string theory breaks the SU(8) symmetry down to SU(4) × SU (4). Loops of BPS states break it to a discrete group. Green, Ooguri and Schwarz [5] have shown that there are no limits of string theory in which all the BPS states decouple.\\n• One often hears the phrase, \\\"black holes are non-perturbative states and their production will not appear in perturbation theory\\\". This is a mistake. The phrase is only correct for black holes, like those of large charge, whose horizon radius is much larger than the Planck scale. We argue that states of small charge are produced with amplitudes of order 1 above threshold. The thresholds are of order the Planck scale and so is the physical size of the state (by extrapolation from the classical solutions). We recall that the production cross sections for solitons are suppressed exponentially in the ratio of their physical size to their Compton wavelength. There is no such suppression for black holes of small charge, whether or not their invariant entropy vanishes. We also argue that the existence of the small charge black holes follows from the existence of classical solutions plus cluster decomposition 1 .\\n• The rest of the argument follows if the SU(8) violating 5 point amplitude satisfies a dispersion relation in the center of mass energy, with a finite number of subtractions, when the other kinematic invariants are fixed. Such a representation shows that the amplitude has a finite perturbation series in powers of the energy over the Planck mass, which contradicts the claim that there is an SU(8) invariant finite power series, constructed by the methods of [2] 2 , which coincides with the amplitudes that include contributions from BPS states.\\nThere are two reasons to worry about the existence of such a dispersion relation. At fixed impact parameter, we expect graviton scattering at sufficiently high energy to be 1 The last argument is due to Andy Strominger. 2 Although there are many alternative methods of calculation in the papers of reference [2], all but the string theory based method begin with manifestly SU (8) invariant tree level amplitudes and use unitarity and analyticity to construct the loops. Interestingly, the methods of Bern, Dixon and collaborators use dimensional regularization to deal with infrared divergences. Zvi Bern has informed me that this breaks the manifest SU (8) symmetry, though the final results are SU (8) invariant in four dimensions. One would imagine that, since IR divergences come only from soft graviton emission, and gravitons are SU (8) singlets, one is guaranteed to get invariant results unless genuine UV divergences are encountered. dominated by black hole production (these are neutral black holes). As a consequence of the thermal nature of black hole evaporation, any exclusive channel with a small number of high energy particles, is exponentially suppressed. This means that, at fixed impact parameter, the amplitude cannot be power law bounded in all directions in the complex energy plane, because it violates the Cerulus-Martin bound [7]. However, Giddings and Porto have recently argued that at fixed kinematic invariants, the shadow scattering of the black hole absorption process, produces an amplitude that does satisfy the CM bound.\\nThe other reason one might worry about the dispersion relation is that IR divergences would cause a singularity at zero energy, which prevent us from pulling the contour around the origin to complete the argument. Since the amplitude vanishes in perturbation theory, there is no perturbative evidence for such an IR singularity. Furthermore, in perturbative string theory, where such power law corrections to the SU(8) violating amplitude certainly exist, and which has the same low energy spectrum and interactions, there is again no evidence than the amplitude is non-analytic at zero.\\nTo summarize: The SU(8) violating five point amplitude, which follows from the existence of black holes of small charge above the production threshold for such black holes, has an analytic continuation which has a power law tale, and contradicts the existence of an SU (8) invariant method for constructing a finite unitary S-matrix to all orders in a power series in energy divided by the Planck mass.\\nThere is a logical possibility that some of the methods in [2] might produce a finite result to all orders in perturbation theory, and provide evidence for a model of quantum gravity, which did not contain the BPS states, that are the focus of this article. While I cannot provide a string free refutation of this claim, it is worth recalling how the \\\"UV\\\" divergence of the pure SUGRA amplitudes would be interpreted in the string based computations of [4]. Their computations try to define SUGRA as a certain limit of Type II string amplitudes. All such limits are equivalent to weakly coupled string theory, or 11D SUGRA, on a large torus, so in this method of calculation, the UV divergence must be interpreted as an IR divergence coming from non-perturbative states whose mass goes to zero in this limit.\\nThere is one other bit of information that emerged from thinking about the consequences of a finite N = 8 SUGRA, which applies to any putatively finite but perturbatively nonrenormalizable theory [8]. In such a theory the expansion parameter is energy divided by some scale. On the other hand, simple diagram counting shows that field theoretic perturbation series are divergent and lead to essential singularities in the coupling 3 In such a theory, this would mean essential singularities at zero energy. In the theories discussed in [8], nothing prevents us from giving mass to all the fields in the Lagrangian. Rigorous theorems of S-matrix theory then preclude the existence of essential singularities, either at the origin or at a finite mass. The point is that in a theory with a mass gap, the number of states at finite energy is bounded and all singularities are multi-particle cuts, for a finite number of particles. Thus, we can refute the claims that any such theory is finite, unless some principle prevents us from giving mass to the particles. Of course, in N = 8 SUGRA, there is such a principle, so this argument is not applicable. That is why we went through the more elaborate discussion outlined above.\\nOne might want to preserve the claim to a finite N = 8 SUGRA for a bit longer by insisting that the finite theory does not have any charged states in it. The classical Lagrangian has solutions corresponding to a pair of oppositely charged BPS black holes moving away from each other, with arbitrary initial velocity, from an arbitrary initial separation. For some values of these parameters the pair will separate and reach infinity, while for others they will collapse and form a neutral black hole. The usual rules of crossing symmetry suggest that charged particles pairs will be produced in the Hawking radiation of neutral black holes, so it is likely that one would have to forbid the existence of ordinary Schwarzschild black holes as well. Indeed, we've already argued that the covariant entropy bound implies that processes involving the formation and decay of neutral black holes cannot preserve SU (8) symmetry.\\nThe other option is would be to declare that the theory contained no neutral black holes either, but then its claim to be a theory of gravitation would be empty. The rest of this paper will be devoted to a more detailed discussion of the SU (8) violating five point amplitude.\"\n },\n {\n \"title\": \"Production of Soliton States\",\n \"content\": \"The Lagrangian of N = 8 SUGRA contains 28 U(1) gauge bosons. Among the solutions of the theory are BPS dyonic black holes with arbitrary continuous values of the electric and magnetic charges. The theory has a classical E(7, 7) symmetry, which is partly realized by duality rotations in the 56 dimensional charge space. Dirac quantization restricts the space of charges to an integer lattice, which is invariant only under an integer subgroup E(7, 7; Z). The 70 scalar fields of the theory lie in the coset E(7, 7)/SU (8), so only the SU(8) symmetry acts linearly on scattering amplitudes for excitations around a particular point in moduli space. The coset transformations instead give rise to soft scalar low energy theorems for these amplitudes. Recently, these symmetry constraints have been shown to guarantee the finiteness of all amplitudes that have so far been computed in this theory, but fail to rule out a divergence at 7 loops.\\nThe entropy of these BPS black holes is computed using the Bekenstein-Hawking law, which determines the large entropy asymptotics. It is given by\\nwhere J 4 is an E(7, 7) invariant, quartic in the charges. When the charges are quantized it is invariant only under the integer subgroup. These are extremal black holes, so the charge also determines the mass, which is not invariant under SU (8).\\nThe charges live on a Lorentzian lattice and the invariant J 4 can be positive, negative or zero. The states with zero J 4 go to zero mass at certain boundaries of moduli space. The corresponding classical solutions are singular, because they have horizons of zero area. The authors of reference [10] argued that these singular solutions might not be real quantum states of the theory in question. While I do not agree with those arguments, the main discussion of the present paper will be restricted to states with non-zero J 4 .\\nThe realization of N = 8 SUGRA as the low energy effective field theory of Type II string theory compactified on a 6− torus of stringy size, sheds light on these states. Some of the BPS sectors are realized in terms of Kaluza-Klein momenta, and wrapping numbers of elementary strings and D-branes on cycles of the torus [13]. As shown by Witten [14], all the boundaries of moduli space are dual (via E(7, 7; Z) rotations) to regimes in which weakly coupled Type II string theory, or 11 dimensional SUGRA, are compactified on a torus of growing size. The J 4 = 0 states are precisely those which become massless particles at one of these boundaries. This explains the vanishing of the area term in their entropy formula (the entropy is constant on moduli space).\\nWhen the string coupling is weak, and all geometrical moduli are of order string scale or larger, we can do reliable perturbative calculations of amplitudes involving virtual charged BPS states. These lead to SU(8) violation. In string perturbation theory, only states with J 4 = 0 contribute to the amplitudes. We can also do reliable weak coupling calculations of amplitudes involving loops of wrapped D-branes with J 4 = 0. These are T-dual to D-instanton calculations and are of order e − c g S . We will see below that this exponential suppression should be interpreted as an exponential of the linear size of the wrapped D-brane state, divided by its Compton wavelength. Here we note only that this is a mathematical fact. The size of a D-brane can be measured by scattering gravitons from it, and this leads to a size of order string scale [15]. The Compton wavelength is smaller than this by a factor g S . 4 Note further, that, if we imagine extrapolating to g s ∼ 1, there is every indication that the contribution of all of these BPS states to SU(8) violating amplitudes will become of order one in eleven dimensional Planck units. Elvang and Kiermaier [11] have computed the lowest dimension SU (8) violating operator in the effective field theory for the SUGRA multiplet, to lowest order in Type II string perturbation theory. It has the form\\nwhere φ is the dimensionless dilaton field. This term comes from explicit violation of SU (8) in tree level string diagrams involving massive non-BPS string states, but it is of the same form as one would expect for the leading low energy contribution from loops of BPS states whose masses violate SU (8). In different regions of the moduli space E(7, 7)/SU (8), the field φ would be replaced by other members of the 70 multiplet of Goldstone bosons. Naively extrapolating to the regime g S ∼ 1 one finds that this term is scaled by the 11 D Planck mass, with a coefficient of order 1. Of course, in this regime, the calculation of [11] is not valid, but since there is no small dimensionless parameter, it is hard to see how the term could be significantly smaller than the extrapolation of the perturbative calculation.\\nSome of the advocates of a finite N = 8 theory emphasize that their theory may have nothing to do with string theory and may be a new paradigm for the approach to quantum gravity as a quantum field theory. Therefore, I must emphasize that the above remarks about string theory are meant to be suggestive only. Type II string theory with g S ∼ 1 and a torus of 11 dimensional Planck size is a finite theory with a single parameter, with dimensions of mass. Like the would be finite N = 8 theory, it has BPS states, whose mass formula violates the continuous symmetry of the leading term in the low energy effective Lagrangian. Unlike that hypothetical theory, the violation shows up in the low energy power series expansion of the scattering amplitudes of the particles in the SUGRA multiplet. What I will try to demonstrate below is that rather general principles imply that any sensible theory containing these BPS states will behave like string theory does and not like the hypothetical finite theory is supposed to.\"\n },\n {\n \"title\": \"Brief outline of finite calculations\",\n \"content\": \"This tiny section is not meant to be even a cursory introduction to the remarkable set of methods that have been developed to calculate the S-matrix in N = 8 SUGRA and similar theories. These include spinor helicity methods, tree level recursion relations based on analyticity in the complex momentum plane, and standard as well as novel unitarity cut techniques for calculating loop amplitudes. Using dimensional regularization, some authors have even found a way to get around the traditional dispersion techniques for calculating the real parts of amplitudes. The point is that in 4 − ǫ dimensions, even \\\"polynomial\\\" contributions to amplitudes have an imaginary part, as a consequence of the irrational scaling exponents. Although the dimensionally continued amplitudes have no Hilbert space interpretation, the do satisfy the cutting rules, which are the perturbative expression of unitarity.\\nThe important observation is that, as long as these methods give finite results, those results will be SU(8) covariant 5 . The tree amplitudes are all SU(8) covariant. The loops are computed from the trees by expressing their discontinuities in terms of lower order amplitudes, and so SU(8) covariance follows by induction, as long as the procedure gives finite and unambiguous answers at each step. In particular, the five point amplitude that will be the focus of our attention, which transforms as a 70, will be identically zero to all orders in the momentum expansion. We will show that if we accept the existence of BPS states, with charges of order 1, which are produced above threshold with amplitudes of order 1, that the all orders vanishing of this amplitude contradicts some very general principles of S-matrix theory.\"\n },\n {\n \"title\": \"Dispersion relations in models of quantum gravity\",\n \"content\": \"The S-matrix elements for the scattering amplitude of two gravitons to two gravitons plus a scalar are conveniently classified according to the helicities of the incoming and outgoing gravitons. Each helicity amplitude has a fixed dependence on components of the momenta, multiplied by unknown scalar functions. The scalar amplitudes in the 2 → 3 S-matrix element depend on the square of the center of mass energy s 12 ≡ s, and the other invariants s ij ≡ p i · p j , (with i = j), subject to the constraints j s ij = 0. There are 5 independent invariants. We will write dispersion relations for these scalar functions in the variable s, with the other independent invariants held fixed.\\nFeynman diagram calculations in low energy field theory, as well as perturbative string theory calculations indicate that the (somewhat vague) holomorphy properties of scattering amplitudes, which are postulated in analytic S-matrix theory, remain true in models of quantum gravity in asymptotically flat space. Amplitudes are holomorphic functions of the invariants, with isolated singularities corresponding to the production of real on-shell physical particle states.\\nDispersion relations employ this assumption of holomorphy with an assumption that amplitudes are polynomially bounded in the entire complex s plane. This assumption is valid in low energy effective field theory, but when the perturbation parameter has dimensions of length to a positive power, the order of the polynomial increases with the order of perturbation theory.\\nIn fact, at fixed impact parameter, simple considerations of black hole physics suggest that the amplitudes fall off exponentially at large s. The argument is simple. As s increases, the colliding particles in the center of mass frame will have a gravitational field, as seen by a distant observer\\nWhen the impact parameter is less than 2 √ sG N , the colliding particles are inside the horizon of the black hole. It is extremely plausible (in general relativity, this is called The Hoop Conjecture) that in such a kinematic regime, particle collisions lead to black hole formation, with probability that rapidly approaches 1 as s goes to infinity with fixed impact parameter. When s is far above the threshold for black hole production at some fixed impact parameter, we can treat the black hole by semi-classical techniques. In particular, Hawking's calculation of black hole decay tells us that the probability of producing a particular final state, with a small number of particles carrying all of the outgoing energy, is exponentially suppressed. The argument is simple counting. The black hole decay is thermal, and the black hole entropy increases linearly with s.\\nGiddings and Porto [1] have argued that for fixed s ij this behavior implies that the standard assumption of polynomial boundedness in the complex s plane is still valid. The amplitudes at fixed kinematic invariants are integrals over all impact parameters. Since we have argued that for small enough impact parameter, in a range growing like √ s , the amplitudes fall off exponentially, the high energy behavior at fixed s ij is dominated by large impact parameters, where it may be estimated by the eikonal approximation. The eikonal behavior obeys the Cerulus-Martin lower bound on the real axis, and there is every indication that the amplitude is power law bounded in the entire complex s plane, for fixed values of the other s ij , and should satisfy a dispersion relation with a finite number of subtractions. For the 2 → 2 amplitude in N = 8 SUGRA, we expect the right and left hand cuts, which contribute to the dispersion integrand, to meet at the origin. However, for the SU(8) violating 5-point amplitude at hand, one would imagine that the lowest contributing threshold would correspond to the production of pairs of the lightest charged BPS states, whose masses are of order the Planck scale 6 . The amplitudes will have IR divergences, corresponding to emission of soft gravitons from incoming and outgoing particles, as they accelerate, but these will not lead to a lowering of the threshold for real production of particles whose mass violates SU (8). Both the spectrum and perturbative interactions of the SUGRA multiplet preserve SU (8).\\nWe have thus argued that SU (8) violating helicity amplitudes may be written as\\nwhere N is an integer we have not determined, and K h is a known kinematic helicity factor. P (s) is a polynomial, and ρ is the sum of the discontinuities across the right and left hand cuts. s 0 is the squared mass of the lightest BPS state contributing to the discontinuity. It is of order m 2 P . P and ρ depend implicitly on the other s ij , which we take to be fixed at non-zero values ≪ m 2 P . The low energy expansion of this amplitude involves the moments\\nwhere σ(z) ≡ ρ( s 0 z ), and the integrals are all power law convergent at z = 0. The assumption of a convergent perturbation expansion, constructed in terms of recursion relations and perturbative unitarity, implies that A h (s) vanishes to all orders in s m 2 P . This in turn implies that all moments above some k max vanish. The function f (y) = ρ(ys 0 )y −N , is square integrable with respect to the measure dµ = dy y kmax . Indeed, it is finite at every y, because it is a sum of discontinuities across multiparticle cuts, and the integral ∞ 1 dy y 2N +kmax |ρ(ys 0 )| 2 , is convergent at infinity because of the convergence of the original dispersion relation.\\nThe vanishing of the higher moments is equivalent to the statement that f (y) is orthogonal to all of the functions y −k for non-negative k. These functions are square integrable and linearly independent. By Gram-Schmidt orthogonalization, we can use them to construct an orthonormal basis. Consequently f (y) = 0. This contradicts the explicit finite contribution to f (y) slightly above the production threshold for the lightest BPS dyon-anti-dyon pair.\"\n },\n {\n \"title\": \"Production cross sections for BPS monopoles\",\n \"content\": \"It is sometimes argued that the BPS states are \\\"non-perturbative\\\", and should not show up in the perturbation expansion. This response is too superficial, and a proper understanding with exchange of massive string modes at tree level. Loops of J 4 = 0 states break SU (8) to a discrete subgroup, while the states with J 4 = 0 are non-perturbative in g S because their size is much larger than their Compton wavelength. For string theory with g S and radii in string units all of order one, or any model whose only parameter is the 4D Planck mass, the threshold for SU (8) violating intermediate states will be given by this scale. of why soliton production is suppressed in conventional quantum field theory shows that there is no similar suppression of production of BPS states of small charge.\\nIndeed, in conventional field theory, solitons have masses of order m g 2 , where m is the mass of perturbative excitations, and g the dimensionless coupling. Energy denominator suppression would indicate that they contribute to scattering of perturbative excitations at finite orders in perturbation theory, but this is not true. The example of magnetic monopoles in the SO(3) Georgi-Glashow model, is typical. Consider the contribution of monopole production to the total cross section for annihilation of two perturbative charged excitations, at lowest order in the electro-magnetic coupling. It is determined by the vacuum to monopole-anti-monopole matrix element of the electromagnetic current:\\nBy crossing symmetry, this is the analytic continuation of the monopole form factor\\nto time-like q, ≥ twice the monopole mass. The usual rules of semi-classical physics, tell us that the monopole form factor in the monopole rest frame is simply the classical electromagnetic current of the monopole field configuration. The current is a smooth function j(xm), where m is the mass of the charged vector boson. The Riemann-Lebesgue lemma, tells us that at large q the function falls off faster than any power. It is typically of order\\nwhere the constant a is of order one and represents the closest singularity of j(xm) to the real axis in the complex xm plane. This leads to the estimate\\nfor the monopole production amplitude. It is clear from this calculation that the relevant parameter is the ratio between the physical size of the soliton, and its Compton wave-length. Non-perturbative suppression of soliton production means that the amplitudes are exponentially small in this ratio. Although I've given this argument in the context of the Georgi-Glashow model, this explanation of the non-perturbative suppression of soliton production amplitudes above threshold is universal in all quantum field theories.\\nThe lesson for production of charged BPS states in N = 8 SUGRA is obvious. The scale of variation of the fields is the Schwarzschild radius, which goes like the square root of the E 7,7 invariant, J 4 , that determines the entropy. Indeed, this is the only dimensionless parameter, which characterizes these solutions. We conclude that when J 4 is of order one, there is no suppression of BPS production amplitudes.\\nOne might attempt to argue that we only have reliable evidence for the existence of these BPS states when J 4 is large and the semi-classical description of the states is valid. However, this is enough to prove the existence of states of large charge of opposite sign and almost equal magnitude. Cluster decomposition and charge conservation assure us that these can annihilate to states of small charge, which should then Hawking radiate down to the BPS state of small charge. Previously this argument was made only for states of J 4 = 0, and 7 arguments were given that those states might not exist. We've now seen that the argument holds just as well for states with small non-zero J 4 , and that the existence of such states follows from the existence of non-singular BPS black holes.\\nWe thus have arguments based on very general principles, that SU (8) violation should occur at finite orders in the expansion in energy over the Planck scale. This manifestly contradicts the claim that the techniques for constructing finite amplitudes will construct the correct amplitudes to all orders, since those techniques preserve SU (8). With more work, one could perhaps give an estimate of the order in the loop expansion at which divergences occur. Elvang and Kiermaier [11], have found the leading low energy contribution to SU (8) violating amplitudes in perturbative string theory. These come from non-BPS massive string modes, at tree level in the g S expansion 8 . By unitarity, these give higher order contributions to the 2 → 2 super-graviton amplitude, which cannot be captured by the methods that give finite perturbative amplitudes. Perturbative finiteness should fail before these contributions become manifest. Henriette Elvang (private communication) has pointed out that if we assume the SU(8) violating 5 point function appears at the order in the energy expansion indicated by the lowest dimension operator that appears in string theory, then the induced unitary correction to 2 → 2 graviton scattering first shows up at 8 loops in the SUGRA loop expansion. This is consistent with the conjecture that finiteness will fail at 7 loops. Unfortunately, our general argument does not rule out the possibility that some finite order of the perturbation expansion of SU(8) violating amplitudes vanishes. String theory predicts definite orders, but if one takes the point of view that the hypothetical finite perturbation series defines a separate theory of quantum gravity, disjoint from string theory, one cannot counter the argument by citing string theory results.\\nFinally I'd like to note that, although I've concentrated on BPS states, where SU(8) violation is manifest in the mass spectrum, ordinary Schwarzschild black holes are expected to lead to SU(8) violation as well. A continuous global symmetry like SU(8) violates the Covariant Entropy Bound for black holes. By dropping neutral particles with SU(8) quantum numbers into a black hole, we can create an infinite number of distinct black holes with the same Schwarzschild field, but transforming in different SU (8) representations. Graviton-graviton scattering, at fixed impact parameter less than the Schwarzschild radius corresponding to the center of mass energy, is expected to be dominated by black hole production. Once √ s > M P , this occurs at impact parameters larger than the Planck scale. Black hole production and decay must violate SU (8) to avoid the violation of the covariant entropy bound. As a consequence, it appears that there are only three logical responses to the arguments presented in this paper.\\n• The SU(8) invariant perturbation series fails to be finite at some order. This would indicate that N = 8 SUGRA is just an effective field theory, which requires UV completion. As occurs in string theory, the UV completion could violate SU (8) and be consistent with the existence of charged BPS states.\\n• The SU(8) invariant perturbation expansion is finite to all orders. If the considerations of this paper are correct, it cannot be the perturbation series of a theory which contains charged BPS states, and probably not of one which contains any kind of black hole. Despite the fact that it contains a graviton, one would feel uncomfortable calling such a model a quantum theory of gravity.\\n• One of the assumptions of this paper is incorrect, most likely the assumption that amplitudes are polynomially bounded in the cut complex energy plane. The results of Giddings and Porto make it plausible that gravitational amplitudes obey such a bound, but by their nature they cannot constitute a proof of that fact.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The Gravitational S-matrix. S B Giddings, R A Porto, arXiv:0908.0004Phys. Rev. D. 8125002hep-thS. B. Giddings and R. A. Porto, \"The Gravitational S-matrix,\" Phys. Rev. D 81, 025002 (2010) [arXiv:0908.0004 [hep-th]].","doi":"arXiv:0908.0004"},{"text":"Three Loop Counterterms For Extended Supergravity. S Deser, J H Kay, K S Stelle, ; S Deser, J H B Kay ; M, J G Green, P Russo, Vanhove, arXiv:hep-th/0610299Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity. 3899JHEPS. Deser, J. H. Kay and K. S. Stelle, \"Renormalizability Properties of Supergravity,\" Phys. Rev. Lett. 38, 527 (1977). ; S. Deser and J. H. Kay, \"Three Loop Counterterms For Extended Supergravity,\" Phys. Lett. B 76, 400 (1978). ; M. B. Green, J. G. Russo and P. Vanhove, \"Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity,\" JHEP 0702 (2007) 099 [arXiv:hep-th/0610299];","doi":"arXiv:hep-th/0610299"},{"text":"Ultraviolet Properties of Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0611273Phys. Rev. Lett. 98131602M. B. Green, J. G. Russo and P. Vanhove, \"Ultraviolet Properties of Maximal Supergravity,\" Phys. Rev. Lett. 98 (2007) 131602 [arXiv:hep-th/0611273];","doi":"arXiv:hep-th/0611273"},{"text":"M B Green, J G Russo, P Vanhove, arXiv:1002.3805String Theory Dualities and Supergravity Divergences. 100675hep-thM. B. Green, J. G. Russo and P. Van- hove, \"String Theory Dualities and Supergravity Divergences,\" JHEP 1006 (2010) 075 [arXiv:1002.3805 [hep-th]];","doi":"arXiv:1002.3805"},{"text":"The Critical Ultraviolet Behaviour of N = 8. P Vanhove, P. Vanhove, \"The Critical Ultraviolet Behaviour of N = 8","doi":""},{"text":". Supergravity Amplitudes, arXiv:1004.1392hep-thSupergravity Amplitudes,\" arXiv:1004.1392 [hep-th];","doi":"arXiv:1004.1392"},{"text":"5 Loops in 24/5 Dimensions. J Bjornsson, M B Green, arXiv:1004.2692JHEP. 1008132hep-thJ. Bjornsson and M. B. Green, \"5 Loops in 24/5 Dimensions,\" JHEP 1008 (2010) 132 [arXiv:1004.2692 [hep-th]];","doi":"arXiv:1004.2692"},{"text":"Amplitudes and Ultraviolet Behavior of N = 8 Supergravity. Z Bern, J J Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:1103.1848hep-th. and references thereinZ. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \"Amplitudes and Ultravio- let Behavior of N = 8 Supergravity,\" arXiv:1103.1848 [hep-th], and references therein;","doi":"arXiv:1103.1848"},{"text":"Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes. Z Bern, J J M Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:1201.5366hep-thZ. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \"Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Am- plitudes,\" arXiv:1201.5366 [hep-th];","doi":"arXiv:1201.5366"},{"text":"Three-Loop Superfiniteness of N=8 Supergravity. Z Bern, J J Carrasco, L J Dixon, H Johansson, D A Kosower, R Roiban, hep-th/0702112Phys. Rev. Lett. 98161303Z. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson, D. A. Kosower and R. Roiban, \"Three-Loop Superfiniteness of N=8 Supergravity,\" Phys. Rev. Lett. 98, 161303 (2007) [hep-th/0702112];","doi":"hep-th/0702112"},{"text":"The Ultraviolet Behavior of N=8 Supergravity at Four Loops. Z Bern, J J Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:0905.2326Phys. Rev. Lett. 10381301hep-thZ. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \"The Ultraviolet Behavior of N=8 Supergravity at Four Loops,\" Phys. Rev. Lett. 103, 081301 (2009) [arXiv:0905.2326 [hep-th]];","doi":"arXiv:0905.2326"},{"text":"Stringy KLT relations, global symmetries, and E 7(7) violation. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 1010108hep-thH. Elvang and M. Kiermaier, \"Stringy KLT relations, global symmetries, and E 7(7) violation,\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]];","doi":"arXiv:1007.4813"},{"text":"Counterterms vs. Dualities. Guillaume Bossard, Hermann Nicolai, arXiv:1105.1273JHEP. 110874hep-thGuillaume Bossard, Hermann Nicolai (Potsdam, Max Planck Inst.). \"Counterterms vs. Dualities\", JHEP 1108 (2011) 074 arXiv:1105.1273 [hep-th];","doi":"arXiv:1105.1273"},{"text":"E7(7) symmetry in perturbatively quantised N=8 supergravity. Guillaume Bossard, Christian Hillmann, Hermann Nicolai, arXiv:1007.5472JHEP. 101252hep-thGuillaume Bossard, Christian Hillmann, Hermann Nicolai \"E7(7) symmetry in pertur- batively quantised N=8 supergravity\" JHEP 1012 (2010) 052, arXiv:1007.5472 [hep-th] ;","doi":"arXiv:1007.5472"},{"text":"Absence of Three-Loop Four-Point Divergences in N=4 Supergravity. Z Bern, S Davies, T Dennen, Y. -T Huang, arXiv:1202.3423hep-thZ. Bern, S. Davies, T. Dennen and Y. -t. Huang, \"Absence of Three-Loop Four- Point Divergences in N=4 Supergravity,\" arXiv:1202.3423 [hep-th];","doi":"arXiv:1202.3423"},{"text":"N=8 Supergravity on the Light Cone. R Kallosh, arXiv:0903.4630Phys. Rev. D. 80105022hep-thR. Kallosh, \"N=8 Supergravity on the Light Cone,\" Phys. Rev. D 80, 105022 (2009) [arXiv:0903.4630 [hep-th]];","doi":"arXiv:0903.4630"},{"text":"Symmetry and Finiteness of N=8 Supergravity. R Kallosh, arXiv:1103.4115JHEP. 1203783hep-thR. Kallosh, \"E 7(7) Symmetry and Finiteness of N=8 Supergravity,\" JHEP 1203, 083 (2012) [arXiv:1103.4115 [hep-th]];","doi":"arXiv:1103.4115"},{"text":") Current Conservation. R Kallosh, arXiv:1104.5480N=8 Counterterms and E. 7773JHEP. hep-thR. Kallosh, \"N=8 Counterterms and E 7(7) Current Conservation,\" JHEP 1106, 073 (2011) [arXiv:1104.5480 [hep-th]];","doi":"arXiv:1104.5480"},{"text":"On Absence of 3-loop Divergence in N=4 Supergravity. R Kallosh, arXiv:1202.4690hep-thR. Kallosh, \"On Absence of 3-loop Divergence in N=4 Supergravity,\" arXiv:1202.4690 [hep-th];","doi":"arXiv:1202.4690"},{"text":"5 loops in 24/5 dimensions. J Bjornsson, M B Green, arXiv:1004.2692JHEP. 1008132hep-thJ. Bjornsson and M. B. Green, \"5 loops in 24/5 dimensions,\" JHEP 1008, 132 (2010) [arXiv:1004.2692 [hep-th]].","doi":"arXiv:1004.2692"},{"text":"The N=8 Supergravity In Superspace. L Brink, P S Howe, Phys. Lett. B. 88268L. Brink and P. S. Howe, \"The N=8 Supergravity In Su- perspace,\" Phys. Lett. B 88, 268 (1979);","doi":""},{"text":"Supergravity In Superspace. P S Howe, Nucl. Phys. B. 199309P. S. Howe, \"Supergravity In Superspace,\" Nucl. Phys. B 199 (1982) 309;","doi":""},{"text":"Counterterms in extended supergravities. R E Kallosh, Phys. Lett. B. 99122R. E. Kallosh, \"Counterterms in extended supergrav- ities,\" Phys. Lett. B 99, 122 (1981);","doi":""},{"text":"Higher Order Invariants In Extended Supergravity. P S Howe, U Lindstrom, Nucl. Phys. B. 181487P. S. Howe and U. Lindstrom, \"Higher Order Invariants In Extended Supergravity,\" Nucl. Phys. B 181, 487 (1981);","doi":""},{"text":"Superactions. P S Howe, K S Stelle, P K Townsend, Nucl. Phys. B. 191445P. S. Howe, K. S. Stelle and P. K. Townsend, \"Superactions,\" Nucl. Phys. B 191, 445 (1981);","doi":""},{"text":"Integral invariants in N=4 SYM and the effective action for coincident D-branes. J M Drummond, P J Heslop, P S Howe, S F Kerstan, hep-th/0305202JHEP. 030816J. M. Drummond, P. J. Heslop, P. S. Howe and S. F. Kerstan, \"Integral invariants in N=4 SYM and the effective action for coincident D-branes,\" JHEP 0308, 016 (2003) [hep-th/0305202];","doi":"hep-th/0305202"},{"text":"A Note on N=8 counterterms. J M Drummond, P J Heslop, P S Howe, arXiv:1008.4939hep-thJ. M. Drummond, P. J. Heslop and P. S. Howe, \"A Note on N=8 coun- terterms,\" arXiv:1008.4939 [hep-th];","doi":"arXiv:1008.4939"},{"text":") symmetry in maximal supergravity. J Broedel, L J Dixon, arXiv:0911.5704JHEP. 100573R**4 counterterm and. hep-thJ. Broedel and L. J. Dixon, \"R**4 counterterm and E(7)(7) symmetry in maximal supergravity,\" JHEP 1005, 003 (2010) [arXiv:0911.5704 [hep-th]];","doi":"arXiv:0911.5704"},{"text":"On duality symmetries of supergravity invariants. G Bossard, P S Howe, K S Stelle, arXiv:1009.0743JHEP. 110120hep-thG. Bossard, P. S. Howe and K. S. Stelle, \"On duality symmetries of super- gravity invariants,\" JHEP 1101, 020 (2011) [arXiv:1009.0743 [hep-th]];","doi":"arXiv:1009.0743"},{"text":"The Ultra-violet question in maximally supersymmetric field theories. G Bossard, P S Howe, K S Stelle, arXiv:0901.4661Gen. Rel. Grav. 41hep-thG. Bossard, P. S. Howe and K. S. Stelle, \"The Ultra-violet question in maximally supersymmetric field theories,\" Gen. Rel. Grav. 41, 919 (2009) [arXiv:0901.4661 [hep-th]];","doi":"arXiv:0901.4661"},{"text":"The Ultraviolet Behavior Of N=4 Yang-mills And The Power Counting Of Extended Superspace. N Marcus, A Sagnotti, Nucl. Phys. B. 25677N. Marcus and A. Sagnotti, \"The Ultraviolet Behavior Of N=4 Yang-mills And The Power Counting Of Extended Superspace,\" Nucl. Phys. B 256, 77 (1985);","doi":""},{"text":"P S Howe, K S Stelle, The Ultraviolet Properties Of Supersymmetric Field Theories. 41871P. S. Howe and K. S. Stelle, \"The Ultraviolet Properties Of Supersymmetric Field Theories,\" Int. J. Mod. Phys. A 4, 1871 (1989);","doi":""},{"text":"The vanishing volume of D=4 superspace. G Bossard, P S Howe, K S Stelle, P Vanhove, arXiv:1105.6087Class. Quant. Grav. 28215005hep-thG. Bossard, P. S. Howe, K. S. Stelle and P. Vanhove, \"The vanishing volume of D=4 superspace,\" Class. Quant. Grav. 28, 215005 (2011) [arXiv:1105.6087 [hep-th]];","doi":"arXiv:1105.6087"},{"text":"An R 4 non-renormalisation theorem in N=4 supergravity. P Tourkine, P Vanhove, arXiv:1202.3692hep-thP. Tourkine and P. Vanhove, \"An R 4 non-renormalisation theorem in N=4 supergravity,\" arXiv:1202.3692 [hep-th].","doi":"arXiv:1202.3692"},{"text":"Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0610299JHEP. 070299M. B. Green, J. G. Russo and P. Vanhove, \"Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity,\" JHEP 0702 (2007) 099 [arXiv:hep- th/0610299];","doi":"arXiv:hep-th/0610299"},{"text":"Ultraviolet Properties of Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0611273Phys. Rev. Lett. 98131602M. B. Green, J. G. Russo and P. Vanhove, \"Ultraviolet Properties of Maximal Supergravity,\" Phys. Rev. Lett. 98 (2007) 131602 [arXiv:hep-th/0611273];","doi":"arXiv:hep-th/0611273"},{"text":"String Theory Dualities and Supergravity Divergences. M B Green, J G Russo, P Vanhove, arXiv:1002.3805JHEP. 100675hep-thM. B. Green, J. G. Russo and P. Vanhove, \"String Theory Dualities and Supergravity Divergences,\" JHEP 1006 (2010) 075 [arXiv:1002.3805 [hep-th]];","doi":"arXiv:1002.3805"},{"text":"Nondecoupling of Maximal Supergravity from the Superstring. M B Green, H Ooguri, J H Schwarz, arXiv:0704.0777Phys. Rev. Lett. 9941601hep-thM. B. Green, H. Ooguri and J. H. Schwarz, \"Nondecoupling of Maximal Supergravity from the Superstring,\" Phys. Rev. Lett. 99, 041601 (2007) [arXiv:0704.0777 [hep-th]].","doi":"arXiv:0704.0777"},{"text":"Trouble for remnants. L Susskind, hep-th/9501106L. Susskind, \"Trouble for remnants,\" hep-th/9501106.","doi":"hep-th/9501106"},{"text":". F Cerulus, A Martin, Phys. Letters. 880F. Cerulus and A. Martin, Phys. Letters 8, 80 (1964).","doi":""},{"text":"UV-Completion by Classicalization. G Dvali, G F Giudice, C Gomez, A Kehagias, arXiv:1010.1415JHEP. 1108hep-phG. Dvali, G. F. Giudice, C. Gomez and A. Kehagias, \"UV-Completion by Classicaliza- tion,\" JHEP 1108, 108 (2011) [arXiv:1010.1415 [hep-ph]].","doi":"arXiv:1010.1415"},{"text":"Stringy KLT relations, global symmetries. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 10107108) violation. hep-thH. Elvang and M. Kiermaier, \"Stringy KLT relations, global symmetries, and E 7(7) vio- lation,\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]].","doi":"arXiv:1007.4813"},{"text":"A simple approach to counterterms in N=8 supergravity. H Elvang, D Z Freedman, M Kiermaier, arXiv:1003.5018JHEP. 101116hep-thH. Elvang, D. Z. Freedman and M. Kiermaier, \"A simple approach to counterterms in N=8 supergravity,\" JHEP 1011, 016 (2010) [arXiv:1003.5018 [hep-th]].","doi":"arXiv:1003.5018"},{"text":"E7(7) constraints on counterterms in N=8 supergravity. N Beisert, H Elvang, D Z Freedman, M Kiermaier, A Morales, S Stieberger, arXiv:1009.1643Phys. Lett. B. 694265hep-thN. Beisert, H. Elvang, D. Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, \"E7(7) constraints on counterterms in N=8 supergravity,\" Phys. Lett. B 694, 265 (2010) [arXiv:1009.1643 [hep-th]];","doi":"arXiv:1009.1643"},{"text":"SUSY Ward identities, Superamplitudes, and Counterterms. H Elvang, D Z Freedman, M Kiermaier, arXiv:1012.3401J. Phys. A A. 44454009hep-thH. El- vang, D. Z. Freedman and M. Kiermaier, \"SUSY Ward identities, Superamplitudes, and Counterterms,\" J. Phys. A A 44, 454009 (2011) [arXiv:1012.3401 [hep-th]].","doi":"arXiv:1012.3401"},{"text":"Perturbative and Non-perturbative N =8 Supergravity. M Bianchi, S Ferrara, R Kallosh, arXiv:0910.3674Phys. Lett. B. 690328hep-thM. Bianchi, S. Ferrara and R. Kallosh, \"Perturbative and Non-perturbative N =8 Su- pergravity,\" Phys. Lett. B 690, 328 (2010) [arXiv:0910.3674 [hep-th]].","doi":"arXiv:0910.3674"},{"text":"Stringy KLT relations, global symmetries. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 10107108) violation. hep-thH. Elvang and M. Kiermaier, \"Stringy KLT relations, global symmetries, and E 7(7) violation,\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]].","doi":"arXiv:1007.4813"},{"text":"Holography and cosmology. W Fischler, L Susskind, hep-th/9806039W. Fischler and L. Susskind, \"Holography and cosmology,\" hep-th/9806039;","doi":"hep-th/9806039"},{"text":"The Holographic principle. R Bousso, hep-th/0203101Rev. Mod. Phys. 74and references thereinR. Bousso, \"The Holographic principle,\" Rev. Mod. Phys. 74, 825 (2002) [hep-th/0203101], and references therein.","doi":"hep-th/0203101"},{"text":"How to count the states of extremal black holes in N=8 supergravity. V Balasubramanian, hep-th/9712215*Cargese 1997, Strings, branes and dualities* 399-410. V. Balasubramanian, \"How to count the states of extremal black holes in N=8 super- gravity,\" In *Cargese 1997, Strings, branes and dualities* 399-410 [hep-th/9712215].","doi":"hep-th/9712215"},{"text":"String theory dynamics in various dimensions. E Witten, hep-th/9503124Nucl. Phys. B. 44385E. Witten, \"String theory dynamics in various dimensions,\" Nucl. Phys. B 443, 85 (1995) [hep-th/9503124].","doi":"hep-th/9503124"},{"text":"Scattering of strings from D-branes. A Hashimoto, I R Klebanov, hep-th/9611214Nucl. Phys. Proc. Suppl. 55A. Hashimoto and I. R. Klebanov, \"Scattering of strings from D-branes,\" Nucl. Phys. Proc. Suppl. 55B, 118 (1997) [hep-th/9611214].","doi":"hep-th/9611214"},{"text":"D-branes and short distances in string theory. M R Douglas, D N Kabat, P Pouliot, S H Shenker, hep-th/9608024Nucl. Phys. B. 48585M. R. Douglas, D. N. Kabat, P. Pouliot and S. H. Shenker, \"D-branes and short distances in string theory,\" Nucl. Phys. B 485, 85 (1997) [hep-th/9608024].","doi":"hep-th/9608024"}],"string":"[\n {\n \"text\": \"The Gravitational S-matrix. S B Giddings, R A Porto, arXiv:0908.0004Phys. Rev. D. 8125002hep-thS. B. Giddings and R. A. Porto, \\\"The Gravitational S-matrix,\\\" Phys. Rev. D 81, 025002 (2010) [arXiv:0908.0004 [hep-th]].\",\n \"doi\": \"arXiv:0908.0004\"\n },\n {\n \"text\": \"Three Loop Counterterms For Extended Supergravity. S Deser, J H Kay, K S Stelle, ; S Deser, J H B Kay ; M, J G Green, P Russo, Vanhove, arXiv:hep-th/0610299Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity. 3899JHEPS. Deser, J. H. Kay and K. S. Stelle, \\\"Renormalizability Properties of Supergravity,\\\" Phys. Rev. Lett. 38, 527 (1977). ; S. Deser and J. H. Kay, \\\"Three Loop Counterterms For Extended Supergravity,\\\" Phys. Lett. B 76, 400 (1978). ; M. B. Green, J. G. Russo and P. Vanhove, \\\"Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity,\\\" JHEP 0702 (2007) 099 [arXiv:hep-th/0610299];\",\n \"doi\": \"arXiv:hep-th/0610299\"\n },\n {\n \"text\": \"Ultraviolet Properties of Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0611273Phys. Rev. Lett. 98131602M. B. Green, J. G. Russo and P. Vanhove, \\\"Ultraviolet Properties of Maximal Supergravity,\\\" Phys. Rev. Lett. 98 (2007) 131602 [arXiv:hep-th/0611273];\",\n \"doi\": \"arXiv:hep-th/0611273\"\n },\n {\n \"text\": \"M B Green, J G Russo, P Vanhove, arXiv:1002.3805String Theory Dualities and Supergravity Divergences. 100675hep-thM. B. Green, J. G. Russo and P. Van- hove, \\\"String Theory Dualities and Supergravity Divergences,\\\" JHEP 1006 (2010) 075 [arXiv:1002.3805 [hep-th]];\",\n \"doi\": \"arXiv:1002.3805\"\n },\n {\n \"text\": \"The Critical Ultraviolet Behaviour of N = 8. P Vanhove, P. Vanhove, \\\"The Critical Ultraviolet Behaviour of N = 8\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Supergravity Amplitudes, arXiv:1004.1392hep-thSupergravity Amplitudes,\\\" arXiv:1004.1392 [hep-th];\",\n \"doi\": \"arXiv:1004.1392\"\n },\n {\n \"text\": \"5 Loops in 24/5 Dimensions. J Bjornsson, M B Green, arXiv:1004.2692JHEP. 1008132hep-thJ. Bjornsson and M. B. Green, \\\"5 Loops in 24/5 Dimensions,\\\" JHEP 1008 (2010) 132 [arXiv:1004.2692 [hep-th]];\",\n \"doi\": \"arXiv:1004.2692\"\n },\n {\n \"text\": \"Amplitudes and Ultraviolet Behavior of N = 8 Supergravity. Z Bern, J J Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:1103.1848hep-th. and references thereinZ. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \\\"Amplitudes and Ultravio- let Behavior of N = 8 Supergravity,\\\" arXiv:1103.1848 [hep-th], and references therein;\",\n \"doi\": \"arXiv:1103.1848\"\n },\n {\n \"text\": \"Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes. Z Bern, J J M Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:1201.5366hep-thZ. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \\\"Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Am- plitudes,\\\" arXiv:1201.5366 [hep-th];\",\n \"doi\": \"arXiv:1201.5366\"\n },\n {\n \"text\": \"Three-Loop Superfiniteness of N=8 Supergravity. Z Bern, J J Carrasco, L J Dixon, H Johansson, D A Kosower, R Roiban, hep-th/0702112Phys. Rev. Lett. 98161303Z. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson, D. A. Kosower and R. Roiban, \\\"Three-Loop Superfiniteness of N=8 Supergravity,\\\" Phys. Rev. Lett. 98, 161303 (2007) [hep-th/0702112];\",\n \"doi\": \"hep-th/0702112\"\n },\n {\n \"text\": \"The Ultraviolet Behavior of N=8 Supergravity at Four Loops. Z Bern, J J Carrasco, L J Dixon, H Johansson, R Roiban, arXiv:0905.2326Phys. Rev. Lett. 10381301hep-thZ. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, \\\"The Ultraviolet Behavior of N=8 Supergravity at Four Loops,\\\" Phys. Rev. Lett. 103, 081301 (2009) [arXiv:0905.2326 [hep-th]];\",\n \"doi\": \"arXiv:0905.2326\"\n },\n {\n \"text\": \"Stringy KLT relations, global symmetries, and E 7(7) violation. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 1010108hep-thH. Elvang and M. Kiermaier, \\\"Stringy KLT relations, global symmetries, and E 7(7) violation,\\\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]];\",\n \"doi\": \"arXiv:1007.4813\"\n },\n {\n \"text\": \"Counterterms vs. Dualities. Guillaume Bossard, Hermann Nicolai, arXiv:1105.1273JHEP. 110874hep-thGuillaume Bossard, Hermann Nicolai (Potsdam, Max Planck Inst.). \\\"Counterterms vs. Dualities\\\", JHEP 1108 (2011) 074 arXiv:1105.1273 [hep-th];\",\n \"doi\": \"arXiv:1105.1273\"\n },\n {\n \"text\": \"E7(7) symmetry in perturbatively quantised N=8 supergravity. Guillaume Bossard, Christian Hillmann, Hermann Nicolai, arXiv:1007.5472JHEP. 101252hep-thGuillaume Bossard, Christian Hillmann, Hermann Nicolai \\\"E7(7) symmetry in pertur- batively quantised N=8 supergravity\\\" JHEP 1012 (2010) 052, arXiv:1007.5472 [hep-th] ;\",\n \"doi\": \"arXiv:1007.5472\"\n },\n {\n \"text\": \"Absence of Three-Loop Four-Point Divergences in N=4 Supergravity. Z Bern, S Davies, T Dennen, Y. -T Huang, arXiv:1202.3423hep-thZ. Bern, S. Davies, T. Dennen and Y. -t. Huang, \\\"Absence of Three-Loop Four- Point Divergences in N=4 Supergravity,\\\" arXiv:1202.3423 [hep-th];\",\n \"doi\": \"arXiv:1202.3423\"\n },\n {\n \"text\": \"N=8 Supergravity on the Light Cone. R Kallosh, arXiv:0903.4630Phys. Rev. D. 80105022hep-thR. Kallosh, \\\"N=8 Supergravity on the Light Cone,\\\" Phys. Rev. D 80, 105022 (2009) [arXiv:0903.4630 [hep-th]];\",\n \"doi\": \"arXiv:0903.4630\"\n },\n {\n \"text\": \"Symmetry and Finiteness of N=8 Supergravity. R Kallosh, arXiv:1103.4115JHEP. 1203783hep-thR. Kallosh, \\\"E 7(7) Symmetry and Finiteness of N=8 Supergravity,\\\" JHEP 1203, 083 (2012) [arXiv:1103.4115 [hep-th]];\",\n \"doi\": \"arXiv:1103.4115\"\n },\n {\n \"text\": \") Current Conservation. R Kallosh, arXiv:1104.5480N=8 Counterterms and E. 7773JHEP. hep-thR. Kallosh, \\\"N=8 Counterterms and E 7(7) Current Conservation,\\\" JHEP 1106, 073 (2011) [arXiv:1104.5480 [hep-th]];\",\n \"doi\": \"arXiv:1104.5480\"\n },\n {\n \"text\": \"On Absence of 3-loop Divergence in N=4 Supergravity. R Kallosh, arXiv:1202.4690hep-thR. Kallosh, \\\"On Absence of 3-loop Divergence in N=4 Supergravity,\\\" arXiv:1202.4690 [hep-th];\",\n \"doi\": \"arXiv:1202.4690\"\n },\n {\n \"text\": \"5 loops in 24/5 dimensions. J Bjornsson, M B Green, arXiv:1004.2692JHEP. 1008132hep-thJ. Bjornsson and M. B. Green, \\\"5 loops in 24/5 dimensions,\\\" JHEP 1008, 132 (2010) [arXiv:1004.2692 [hep-th]].\",\n \"doi\": \"arXiv:1004.2692\"\n },\n {\n \"text\": \"The N=8 Supergravity In Superspace. L Brink, P S Howe, Phys. Lett. B. 88268L. Brink and P. S. Howe, \\\"The N=8 Supergravity In Su- perspace,\\\" Phys. Lett. B 88, 268 (1979);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Supergravity In Superspace. P S Howe, Nucl. Phys. B. 199309P. S. Howe, \\\"Supergravity In Superspace,\\\" Nucl. Phys. B 199 (1982) 309;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Counterterms in extended supergravities. R E Kallosh, Phys. Lett. B. 99122R. E. Kallosh, \\\"Counterterms in extended supergrav- ities,\\\" Phys. Lett. B 99, 122 (1981);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Higher Order Invariants In Extended Supergravity. P S Howe, U Lindstrom, Nucl. Phys. B. 181487P. S. Howe and U. Lindstrom, \\\"Higher Order Invariants In Extended Supergravity,\\\" Nucl. Phys. B 181, 487 (1981);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Superactions. P S Howe, K S Stelle, P K Townsend, Nucl. Phys. B. 191445P. S. Howe, K. S. Stelle and P. K. Townsend, \\\"Superactions,\\\" Nucl. Phys. B 191, 445 (1981);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Integral invariants in N=4 SYM and the effective action for coincident D-branes. J M Drummond, P J Heslop, P S Howe, S F Kerstan, hep-th/0305202JHEP. 030816J. M. Drummond, P. J. Heslop, P. S. Howe and S. F. Kerstan, \\\"Integral invariants in N=4 SYM and the effective action for coincident D-branes,\\\" JHEP 0308, 016 (2003) [hep-th/0305202];\",\n \"doi\": \"hep-th/0305202\"\n },\n {\n \"text\": \"A Note on N=8 counterterms. J M Drummond, P J Heslop, P S Howe, arXiv:1008.4939hep-thJ. M. Drummond, P. J. Heslop and P. S. Howe, \\\"A Note on N=8 coun- terterms,\\\" arXiv:1008.4939 [hep-th];\",\n \"doi\": \"arXiv:1008.4939\"\n },\n {\n \"text\": \") symmetry in maximal supergravity. J Broedel, L J Dixon, arXiv:0911.5704JHEP. 100573R**4 counterterm and. hep-thJ. Broedel and L. J. Dixon, \\\"R**4 counterterm and E(7)(7) symmetry in maximal supergravity,\\\" JHEP 1005, 003 (2010) [arXiv:0911.5704 [hep-th]];\",\n \"doi\": \"arXiv:0911.5704\"\n },\n {\n \"text\": \"On duality symmetries of supergravity invariants. G Bossard, P S Howe, K S Stelle, arXiv:1009.0743JHEP. 110120hep-thG. Bossard, P. S. Howe and K. S. Stelle, \\\"On duality symmetries of super- gravity invariants,\\\" JHEP 1101, 020 (2011) [arXiv:1009.0743 [hep-th]];\",\n \"doi\": \"arXiv:1009.0743\"\n },\n {\n \"text\": \"The Ultra-violet question in maximally supersymmetric field theories. G Bossard, P S Howe, K S Stelle, arXiv:0901.4661Gen. Rel. Grav. 41hep-thG. Bossard, P. S. Howe and K. S. Stelle, \\\"The Ultra-violet question in maximally supersymmetric field theories,\\\" Gen. Rel. Grav. 41, 919 (2009) [arXiv:0901.4661 [hep-th]];\",\n \"doi\": \"arXiv:0901.4661\"\n },\n {\n \"text\": \"The Ultraviolet Behavior Of N=4 Yang-mills And The Power Counting Of Extended Superspace. N Marcus, A Sagnotti, Nucl. Phys. B. 25677N. Marcus and A. Sagnotti, \\\"The Ultraviolet Behavior Of N=4 Yang-mills And The Power Counting Of Extended Superspace,\\\" Nucl. Phys. B 256, 77 (1985);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P S Howe, K S Stelle, The Ultraviolet Properties Of Supersymmetric Field Theories. 41871P. S. Howe and K. S. Stelle, \\\"The Ultraviolet Properties Of Supersymmetric Field Theories,\\\" Int. J. Mod. Phys. A 4, 1871 (1989);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The vanishing volume of D=4 superspace. G Bossard, P S Howe, K S Stelle, P Vanhove, arXiv:1105.6087Class. Quant. Grav. 28215005hep-thG. Bossard, P. S. Howe, K. S. Stelle and P. Vanhove, \\\"The vanishing volume of D=4 superspace,\\\" Class. Quant. Grav. 28, 215005 (2011) [arXiv:1105.6087 [hep-th]];\",\n \"doi\": \"arXiv:1105.6087\"\n },\n {\n \"text\": \"An R 4 non-renormalisation theorem in N=4 supergravity. P Tourkine, P Vanhove, arXiv:1202.3692hep-thP. Tourkine and P. Vanhove, \\\"An R 4 non-renormalisation theorem in N=4 supergravity,\\\" arXiv:1202.3692 [hep-th].\",\n \"doi\": \"arXiv:1202.3692\"\n },\n {\n \"text\": \"Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0610299JHEP. 070299M. B. Green, J. G. Russo and P. Vanhove, \\\"Non-Renormalisation Conditions in Type II String Theory and Maximal Supergravity,\\\" JHEP 0702 (2007) 099 [arXiv:hep- th/0610299];\",\n \"doi\": \"arXiv:hep-th/0610299\"\n },\n {\n \"text\": \"Ultraviolet Properties of Maximal Supergravity. M B Green, J G Russo, P Vanhove, arXiv:hep-th/0611273Phys. Rev. Lett. 98131602M. B. Green, J. G. Russo and P. Vanhove, \\\"Ultraviolet Properties of Maximal Supergravity,\\\" Phys. Rev. Lett. 98 (2007) 131602 [arXiv:hep-th/0611273];\",\n \"doi\": \"arXiv:hep-th/0611273\"\n },\n {\n \"text\": \"String Theory Dualities and Supergravity Divergences. M B Green, J G Russo, P Vanhove, arXiv:1002.3805JHEP. 100675hep-thM. B. Green, J. G. Russo and P. Vanhove, \\\"String Theory Dualities and Supergravity Divergences,\\\" JHEP 1006 (2010) 075 [arXiv:1002.3805 [hep-th]];\",\n \"doi\": \"arXiv:1002.3805\"\n },\n {\n \"text\": \"Nondecoupling of Maximal Supergravity from the Superstring. M B Green, H Ooguri, J H Schwarz, arXiv:0704.0777Phys. Rev. Lett. 9941601hep-thM. B. Green, H. Ooguri and J. H. Schwarz, \\\"Nondecoupling of Maximal Supergravity from the Superstring,\\\" Phys. Rev. Lett. 99, 041601 (2007) [arXiv:0704.0777 [hep-th]].\",\n \"doi\": \"arXiv:0704.0777\"\n },\n {\n \"text\": \"Trouble for remnants. L Susskind, hep-th/9501106L. Susskind, \\\"Trouble for remnants,\\\" hep-th/9501106.\",\n \"doi\": \"hep-th/9501106\"\n },\n {\n \"text\": \". F Cerulus, A Martin, Phys. Letters. 880F. Cerulus and A. Martin, Phys. Letters 8, 80 (1964).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"UV-Completion by Classicalization. G Dvali, G F Giudice, C Gomez, A Kehagias, arXiv:1010.1415JHEP. 1108hep-phG. Dvali, G. F. Giudice, C. Gomez and A. Kehagias, \\\"UV-Completion by Classicaliza- tion,\\\" JHEP 1108, 108 (2011) [arXiv:1010.1415 [hep-ph]].\",\n \"doi\": \"arXiv:1010.1415\"\n },\n {\n \"text\": \"Stringy KLT relations, global symmetries. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 10107108) violation. hep-thH. Elvang and M. Kiermaier, \\\"Stringy KLT relations, global symmetries, and E 7(7) vio- lation,\\\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]].\",\n \"doi\": \"arXiv:1007.4813\"\n },\n {\n \"text\": \"A simple approach to counterterms in N=8 supergravity. H Elvang, D Z Freedman, M Kiermaier, arXiv:1003.5018JHEP. 101116hep-thH. Elvang, D. Z. Freedman and M. Kiermaier, \\\"A simple approach to counterterms in N=8 supergravity,\\\" JHEP 1011, 016 (2010) [arXiv:1003.5018 [hep-th]].\",\n \"doi\": \"arXiv:1003.5018\"\n },\n {\n \"text\": \"E7(7) constraints on counterterms in N=8 supergravity. N Beisert, H Elvang, D Z Freedman, M Kiermaier, A Morales, S Stieberger, arXiv:1009.1643Phys. Lett. B. 694265hep-thN. Beisert, H. Elvang, D. Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, \\\"E7(7) constraints on counterterms in N=8 supergravity,\\\" Phys. Lett. B 694, 265 (2010) [arXiv:1009.1643 [hep-th]];\",\n \"doi\": \"arXiv:1009.1643\"\n },\n {\n \"text\": \"SUSY Ward identities, Superamplitudes, and Counterterms. H Elvang, D Z Freedman, M Kiermaier, arXiv:1012.3401J. Phys. A A. 44454009hep-thH. El- vang, D. Z. Freedman and M. Kiermaier, \\\"SUSY Ward identities, Superamplitudes, and Counterterms,\\\" J. Phys. A A 44, 454009 (2011) [arXiv:1012.3401 [hep-th]].\",\n \"doi\": \"arXiv:1012.3401\"\n },\n {\n \"text\": \"Perturbative and Non-perturbative N =8 Supergravity. M Bianchi, S Ferrara, R Kallosh, arXiv:0910.3674Phys. Lett. B. 690328hep-thM. Bianchi, S. Ferrara and R. Kallosh, \\\"Perturbative and Non-perturbative N =8 Su- pergravity,\\\" Phys. Lett. B 690, 328 (2010) [arXiv:0910.3674 [hep-th]].\",\n \"doi\": \"arXiv:0910.3674\"\n },\n {\n \"text\": \"Stringy KLT relations, global symmetries. H Elvang, M Kiermaier, arXiv:1007.4813JHEP. 10107108) violation. hep-thH. Elvang and M. Kiermaier, \\\"Stringy KLT relations, global symmetries, and E 7(7) violation,\\\" JHEP 1010, 108 (2010) [arXiv:1007.4813 [hep-th]].\",\n \"doi\": \"arXiv:1007.4813\"\n },\n {\n \"text\": \"Holography and cosmology. W Fischler, L Susskind, hep-th/9806039W. Fischler and L. Susskind, \\\"Holography and cosmology,\\\" hep-th/9806039;\",\n \"doi\": \"hep-th/9806039\"\n },\n {\n \"text\": \"The Holographic principle. R Bousso, hep-th/0203101Rev. Mod. Phys. 74and references thereinR. Bousso, \\\"The Holographic principle,\\\" Rev. Mod. Phys. 74, 825 (2002) [hep-th/0203101], and references therein.\",\n \"doi\": \"hep-th/0203101\"\n },\n {\n \"text\": \"How to count the states of extremal black holes in N=8 supergravity. V Balasubramanian, hep-th/9712215*Cargese 1997, Strings, branes and dualities* 399-410. V. Balasubramanian, \\\"How to count the states of extremal black holes in N=8 super- gravity,\\\" In *Cargese 1997, Strings, branes and dualities* 399-410 [hep-th/9712215].\",\n \"doi\": \"hep-th/9712215\"\n },\n {\n \"text\": \"String theory dynamics in various dimensions. E Witten, hep-th/9503124Nucl. Phys. B. 44385E. Witten, \\\"String theory dynamics in various dimensions,\\\" Nucl. Phys. B 443, 85 (1995) [hep-th/9503124].\",\n \"doi\": \"hep-th/9503124\"\n },\n {\n \"text\": \"Scattering of strings from D-branes. A Hashimoto, I R Klebanov, hep-th/9611214Nucl. Phys. Proc. Suppl. 55A. Hashimoto and I. R. Klebanov, \\\"Scattering of strings from D-branes,\\\" Nucl. Phys. Proc. Suppl. 55B, 118 (1997) [hep-th/9611214].\",\n \"doi\": \"hep-th/9611214\"\n },\n {\n \"text\": \"D-branes and short distances in string theory. M R Douglas, D N Kabat, P Pouliot, S H Shenker, hep-th/9608024Nucl. Phys. B. 48585M. R. Douglas, D. N. Kabat, P. Pouliot and S. H. Shenker, \\\"D-branes and short distances in string theory,\\\" Nucl. Phys. B 485, 85 (1997) [hep-th/9608024].\",\n \"doi\": \"hep-th/9608024\"\n }\n]","isConversational":false}}},{"rowIdx":51,"cells":{"text":{"kind":"string","value":"\nWorld Wide Student Laboratory Description of Project\n24 Jul 1998 February 20, 1995\n\nAnatoli Arodzero \nDepartment of Physics\nUniversity of Oregon Eugene\n97403OregonUSA\n\nWorld Wide Student Laboratory Description of Project\n24 Jul 1998 February 20, 1995arXiv:physics/9806044v2 [physics.ed-ph] Revised June 19, 1998 *\n\n\nIntroduction\n\nThis paper describes a new framework in educational technology -the World Wide Student Laboratory Project. Its main goals are to improve the efficiency of educating students in the principles of advanced experimental research, to stimulate student's interest in science and to provide professional resources to educators.\n\nThe World Wide Student Laboratory ( WWSL ) Project expands upon traditional laboratory study. It allows students to study natural phenomena while learning an Internet-based telecommunication and computer technology and studying the methods of transmission, processing and analyzing experimental data.\n\nIn this publication, the concepts of the World Wide Student Laboratory are outlined, a possible idea for structure is discussed and examples of various laboratory studies (projects) are shown.\n\n\nConcept of the World Wide Student Laboratory\n\nBased on the Internet, the World Wide Web (WWW) has been rapidly growing over the last several years. The WWW is a combination of a large number of computer servers which can send documents or \"pages\" to Internet clients using a Web-browser software for navigation between the different \"sites.\"\n\nFrom the beginning the WWW has primarily been used for educational purposes in three ways [1]:\n\n-to provide students with a wider access to information; -as a communications tool supplementing traditional forms of education, raising the efficiency of interaction between the educators and students;\n\n-as \"a virtual classroom\", \" a virtual laboratory\", as an integrated interface for distance learning.\n\nThe student's laboratory practice work is the main form of experimental research study. The efficiency of traditional labs have certain limits: -often students do not have enough opportunities to deeply explore a number of important sections of laboratory courses . Not all universities can support a wide spectrum of laboratory research setups because of their high price, service cost, the lack of necessary space, etc.; -more often laboratory classes are too short to obtain persuasive research results of the given process or phenomenon; -in many cases, the laboratory instruction and student research projects do not fully reflect the current level of professional experimental research.\n\nIn the framework of the WWSL project, we offer new methods of educational technology which complement the traditional methods and upgrade the standard of experimental research study even with the use of existing lab equipment. These methods may also significantly reduce the above mentioned limits of laboratory practice work.\n\nThe concept of WWSL may be expressed in the following way:\n\nWorld Wide Student Laboratory (WWSL) is a dynamic international Internet-based collaboration of resources from the educational and research labs of universities, institutions, government research centers and companies. The main element of WWSL is the educational or research setup. This setup has an interactive connection through the Internet with a particular thematic WWSL-Site (Research Group Site). Through the Research Group Sites, students, under the instruction of an educator, are able to control experiments, take, process and analyse data, regardless of their location. The results obtained by each student are available to all other students who are participants of the particular research group and can be used in their own projects. 1 The main features of the WWSL project involve the following:\n\n-the possibility of studying and demonstrating certain effects and phenomena unobservable in traditional lab studies. For example, the study of the effects which require simultaneous tests in different geographical points of the world, in different environments, during long time intervals, etc.;\n\n-as a result of unification of the experimental data obtained from a network of setups, it is possible to study \"subtle\" processes and/or processes which require a large volume of data. This allows exploration of the origins of instrumental or methodological errors and demonstrates methods of extracting information from noisy data;\n\n-it is possible to run certain experimental projects in parallel with mathematical modelling. Integrating the two allows more global understanding of the phenomena;\n\n-any student who has access to the Internet, regardless of their location, may become a member of the WWSL. This also makes the WWSL a perfect tool for distance education;\n\n-students have around the clock access to experimental setups, and the ability to work on their own schedule;\n\n-educators have the possibility to use the data of experiments on-line through the WWSL for lectures; -by using the WWSL, students can have access to the real data from \"professional\" scientific experiments, and educators may use this \"live\" data for educational purposes;\n\n-the use by WWSL of well known elements of the Internet such as chat rooms, white-boards, etc. can allow students and educators to have \"global seminars \" for each experiment;\n\n-individual student projects can be integrated in the form of a final report of the research work. The participation in making such a report will teach the student the ethics of joint research, increase motivation and give significance to their work; -setting up new WWSL software modules may well be an attractive and challenging project for students in computer science;\n\n-members of WWSL may not only be students from universities but also from colleges and high schools.\n\nWWSL should enrich the personal research results of each student and help lab studies approach the top-level scientific research. It is important to emphasise that WWSL is not a \"virtual\" laboratory. It is a real lab where actual experiments (with all real-world \"noisy\" effects) can be made.\n\nThe traditional form of conducting some experiments is still preferable for laboratory studies. However, from our point of view, the following projects would be better served by WWSL:\n\n-projects that require a long time to complete; -projects which require a certain preliminary theoretical and/or practical preparation; -projects which require repeated change of experimental conditions;\n\n-projects which require special conditions for their fulfilment.\n\n\nThe Structure of WWSL\n\nPresently, the World Wide Student Laboratory is developing using the following structure:\n\nThe general coordination of WWSL operations is conducted by the WWSL Coordination Center which fulfils the following functions:\n\n-supports the official Web-site of the WWSL Project; -registers organizations and individuals which will be participants in the WWSL;\n\n-monitors status of WWSL activity; -manages the development of methodology of WWSL studies and its data;\n\n-maintains the WWSL-FAQ, chat rooms, etc.\n\nA Research Group Site would be created to conduct a particular series of student lab studies. It combines the sites of Experimental Setups of participating organizations. If it is required, the activity of each Research Group can be defined by a Research Program which is prepared by Educators from participating Universities. The Research Group Site has the following functions:\n\n-registers the participants in given lab studies; -integrates Student's Sites who are participating in a given project;\n\n-provides access to experimental data from all Experimental Setups;\n\n-maintains Research Program if required, and/or a schedule for taking data;\n\n-maintains the pages with information necessary to complete these studies;\n\n-maintains the FAQ and chat room page; -maintains a page with test questions; -maintains a page for a combined student report.\n\nThe Site of the Experimental Setup may be located at a computer which is a part of the setup and is used for data storage, data processing and analysis of the results from that setup. It is maintained by the Educator in charge of the setup.\n\nStudents use sites they must create (analogous to an electronic lab book) for access to WWSL resources, to control experimental setups and for processing and analysing experimental data. Their site would also be used to prepare the final report of a completed experiment.\n\nThe structure of WWSL is not complicated and uses existing technology. Although it can take advanage of then, it does not rely on new technical developments at student or research laboratories or on the Internet.\n\nPresently, the first stage of the WWSL development has been completed [2]. An Official Site of the WWSL Project has been created on the University of Oregon Physics Department's web server, in Eugene Oregon, USA. On that server, a Site for the \"Cosmic Rays\" Research Group has been organised. It combines two student research setups, one of which is located at the University of Oregon, the other is located at the Department of Physics at N. E. Bauman Moscow State Technical University, Moscow, Russia.\n\nThe next stage is going to involve a number of projects for students who are studying General Physics, Radiophysics, Statistical Physics and Biophysics. The following projects are under discussion and/or development:\n\n-lifetime of cosmic ray muons; -Mossbauer effect; -Coriolis acceleration; -nuclear magnetic resonance; -study of the accuracy of global positioning system; -study of the correlation between the processes of different bi-ological systems.\n\n\nExamples of possible WWSL projects\n\nLet us now consider some possible student projects which can be completed in laboratory setups for studying the properties of cosmic rays. These studies are part of the curriculum of physics and engineering majors in many universities [3], [4].\n\nThe setup consists of two detectors, usually scintillating. An absorber of variable thickness (a set of lead or tungsten plates) is sandwiched between them. The signals from the detectors are read out to a coincidence unit. The unit selects events due to cosmic ray particles passing through both detectors. The output signals from coincidence units are collected by a data acquisition system and a computer.\n\nStudent projects which can be pursued using these setups can be divided into two groups:\n\n-projects for studying certain properties of cosmic rays, for studying of experimental methods and apparatus of cosmic ray physics and high energy physics;\n\n-projects for studying the properties of random processes (in this case, the flux of particles of cosmic rays is considered as a generator of random processes).\n\nThe first group may contain the following projects:\n\n-study of the composition of cosmic rays near the earth surface;\n\n-study of the angular distribution of the cosmic ray flux; -study of the fluctuation of energy loss of charged particles; -study of the effect of the detector's dead time on the character of the statistical distribution of the signals.\n\nThe second group may contain the following projects:\n\n-properties of Poisson data of random events; -study of the relationship between binomial, Poisson and normal distributions;\n\n-test of the hypotheses presented by different statistical distributions;\n\n-the \"waiting time paradox\" for Poisson processes; -study of correlations in stochastic processes.\n The idea of such a organisation of lab studies came to me in 1987, when I was project manager of development of the new Department of Fundamental Science at the Moscow State Technical University. But the Internet at that time was weakly developed.\nThe experimental data collected with the setups for the detection of cosmic rays allows students to fulfil the studies of all the topics listed above simultaneously, and WWSL provides the tools to study them fully.\nThe World Wide Web as an Instructional Tool. J M Barrie, D E Presti, Science. 274J. M. Barrie, D. E. Presti. The World Wide Web as an Instructional Tool. Science, Vol. 274, 1996, pp. 371-372.\n\nStudy of composition of cosmic rays. Guide for students. A Arodzero, P Gramenitsky, V Rodionov, 10Moscow, MSTUin RussianA. Arodzero, P. Gramenitsky, V. Rodionov. Study of composition of cos- mic rays. Guide for students. Moscow, MSTU, 1980. 10p. (in Russian).\n\n. Physics Laboratory Studies. Edited by L. Goldin. Moscow, Nauka. 704in RussianPhysics Laboratory Studies. Edited by L. Goldin. Moscow, Nauka, 1983. 704 p. (in Russian).\n"},"corpus_id":{"kind":"number","value":117494903,"string":"117,494,903"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"physics/9806044\",\n \"mag\": 1562268838,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/physics/9806044v2.pdf\"\n ],\n \"pdf_sha\": \"95af9c0118e3884755df61723372d843b206e4c5\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"World Wide Student Laboratory Description of Project"},"authors":{"kind":"list like","value":["Anatoli Arodzero \nDepartment of Physics\nUniversity of Oregon Eugene\n97403OregonUSA"],"string":"[\n \"Anatoli Arodzero \\nDepartment of Physics\\nUniversity of Oregon Eugene\\n97403OregonUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":""},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"This paper describes a new framework in educational technology -the World Wide Student Laboratory Project. Its main goals are to improve the efficiency of educating students in the principles of advanced experimental research, to stimulate student's interest in science and to provide professional resources to educators.\nThe World Wide Student Laboratory ( WWSL ) Project expands upon traditional laboratory study. It allows students to study natural phenomena while learning an Internet-based telecommunication and computer technology and studying the methods of transmission, processing and analyzing experimental data.\nIn this publication, the concepts of the World Wide Student Laboratory are outlined, a possible idea for structure is discussed and examples of various laboratory studies (projects) are shown."},{"title":"Concept of the World Wide Student Laboratory","content":"Based on the Internet, the World Wide Web (WWW) has been rapidly growing over the last several years. The WWW is a combination of a large number of computer servers which can send documents or \"pages\" to Internet clients using a Web-browser software for navigation between the different \"sites.\"\nFrom the beginning the WWW has primarily been used for educational purposes in three ways [1]:\n-to provide students with a wider access to information; -as a communications tool supplementing traditional forms of education, raising the efficiency of interaction between the educators and students;\n-as \"a virtual classroom\", \" a virtual laboratory\", as an integrated interface for distance learning.\nThe student's laboratory practice work is the main form of experimental research study. The efficiency of traditional labs have certain limits: -often students do not have enough opportunities to deeply explore a number of important sections of laboratory courses . Not all universities can support a wide spectrum of laboratory research setups because of their high price, service cost, the lack of necessary space, etc.; -more often laboratory classes are too short to obtain persuasive research results of the given process or phenomenon; -in many cases, the laboratory instruction and student research projects do not fully reflect the current level of professional experimental research.\nIn the framework of the WWSL project, we offer new methods of educational technology which complement the traditional methods and upgrade the standard of experimental research study even with the use of existing lab equipment. These methods may also significantly reduce the above mentioned limits of laboratory practice work.\nThe concept of WWSL may be expressed in the following way:\nWorld Wide Student Laboratory (WWSL) is a dynamic international Internet-based collaboration of resources from the educational and research labs of universities, institutions, government research centers and companies. The main element of WWSL is the educational or research setup. This setup has an interactive connection through the Internet with a particular thematic WWSL-Site (Research Group Site). Through the Research Group Sites, students, under the instruction of an educator, are able to control experiments, take, process and analyse data, regardless of their location. The results obtained by each student are available to all other students who are participants of the particular research group and can be used in their own projects. 1 The main features of the WWSL project involve the following:\n-the possibility of studying and demonstrating certain effects and phenomena unobservable in traditional lab studies. For example, the study of the effects which require simultaneous tests in different geographical points of the world, in different environments, during long time intervals, etc.;\n-as a result of unification of the experimental data obtained from a network of setups, it is possible to study \"subtle\" processes and/or processes which require a large volume of data. This allows exploration of the origins of instrumental or methodological errors and demonstrates methods of extracting information from noisy data;\n-it is possible to run certain experimental projects in parallel with mathematical modelling. Integrating the two allows more global understanding of the phenomena;\n-any student who has access to the Internet, regardless of their location, may become a member of the WWSL. This also makes the WWSL a perfect tool for distance education;\n-students have around the clock access to experimental setups, and the ability to work on their own schedule;\n-educators have the possibility to use the data of experiments on-line through the WWSL for lectures; -by using the WWSL, students can have access to the real data from \"professional\" scientific experiments, and educators may use this \"live\" data for educational purposes;\n-the use by WWSL of well known elements of the Internet such as chat rooms, white-boards, etc. can allow students and educators to have \"global seminars \" for each experiment;\n-individual student projects can be integrated in the form of a final report of the research work. The participation in making such a report will teach the student the ethics of joint research, increase motivation and give significance to their work; -setting up new WWSL software modules may well be an attractive and challenging project for students in computer science;\n-members of WWSL may not only be students from universities but also from colleges and high schools.\nWWSL should enrich the personal research results of each student and help lab studies approach the top-level scientific research. It is important to emphasise that WWSL is not a \"virtual\" laboratory. It is a real lab where actual experiments (with all real-world \"noisy\" effects) can be made.\nThe traditional form of conducting some experiments is still preferable for laboratory studies. However, from our point of view, the following projects would be better served by WWSL:\n-projects that require a long time to complete; -projects which require a certain preliminary theoretical and/or practical preparation; -projects which require repeated change of experimental conditions;\n-projects which require special conditions for their fulfilment."},{"title":"The Structure of WWSL","content":"Presently, the World Wide Student Laboratory is developing using the following structure:\nThe general coordination of WWSL operations is conducted by the WWSL Coordination Center which fulfils the following functions:\n-supports the official Web-site of the WWSL Project; -registers organizations and individuals which will be participants in the WWSL;\n-monitors status of WWSL activity; -manages the development of methodology of WWSL studies and its data;\n-maintains the WWSL-FAQ, chat rooms, etc.\nA Research Group Site would be created to conduct a particular series of student lab studies. It combines the sites of Experimental Setups of participating organizations. If it is required, the activity of each Research Group can be defined by a Research Program which is prepared by Educators from participating Universities. The Research Group Site has the following functions:\n-registers the participants in given lab studies; -integrates Student's Sites who are participating in a given project;\n-provides access to experimental data from all Experimental Setups;\n-maintains Research Program if required, and/or a schedule for taking data;\n-maintains the pages with information necessary to complete these studies;\n-maintains the FAQ and chat room page; -maintains a page with test questions; -maintains a page for a combined student report.\nThe Site of the Experimental Setup may be located at a computer which is a part of the setup and is used for data storage, data processing and analysis of the results from that setup. It is maintained by the Educator in charge of the setup.\nStudents use sites they must create (analogous to an electronic lab book) for access to WWSL resources, to control experimental setups and for processing and analysing experimental data. Their site would also be used to prepare the final report of a completed experiment.\nThe structure of WWSL is not complicated and uses existing technology. Although it can take advanage of then, it does not rely on new technical developments at student or research laboratories or on the Internet.\nPresently, the first stage of the WWSL development has been completed [2]. An Official Site of the WWSL Project has been created on the University of Oregon Physics Department's web server, in Eugene Oregon, USA. On that server, a Site for the \"Cosmic Rays\" Research Group has been organised. It combines two student research setups, one of which is located at the University of Oregon, the other is located at the Department of Physics at N. E. Bauman Moscow State Technical University, Moscow, Russia.\nThe next stage is going to involve a number of projects for students who are studying General Physics, Radiophysics, Statistical Physics and Biophysics. The following projects are under discussion and/or development:\n-lifetime of cosmic ray muons; -Mossbauer effect; -Coriolis acceleration; -nuclear magnetic resonance; -study of the accuracy of global positioning system; -study of the correlation between the processes of different bi-ological systems."},{"title":"Examples of possible WWSL projects","content":"Let us now consider some possible student projects which can be completed in laboratory setups for studying the properties of cosmic rays. These studies are part of the curriculum of physics and engineering majors in many universities [3], [4].\nThe setup consists of two detectors, usually scintillating. An absorber of variable thickness (a set of lead or tungsten plates) is sandwiched between them. The signals from the detectors are read out to a coincidence unit. The unit selects events due to cosmic ray particles passing through both detectors. The output signals from coincidence units are collected by a data acquisition system and a computer.\nStudent projects which can be pursued using these setups can be divided into two groups:\n-projects for studying certain properties of cosmic rays, for studying of experimental methods and apparatus of cosmic ray physics and high energy physics;\n-projects for studying the properties of random processes (in this case, the flux of particles of cosmic rays is considered as a generator of random processes).\nThe first group may contain the following projects:\n-study of the composition of cosmic rays near the earth surface;\n-study of the angular distribution of the cosmic ray flux; -study of the fluctuation of energy loss of charged particles; -study of the effect of the detector's dead time on the character of the statistical distribution of the signals.\nThe second group may contain the following projects:\n-properties of Poisson data of random events; -study of the relationship between binomial, Poisson and normal distributions;\n-test of the hypotheses presented by different statistical distributions;\n-the \"waiting time paradox\" for Poisson processes; -study of correlations in stochastic processes."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"This paper describes a new framework in educational technology -the World Wide Student Laboratory Project. Its main goals are to improve the efficiency of educating students in the principles of advanced experimental research, to stimulate student's interest in science and to provide professional resources to educators.\\nThe World Wide Student Laboratory ( WWSL ) Project expands upon traditional laboratory study. It allows students to study natural phenomena while learning an Internet-based telecommunication and computer technology and studying the methods of transmission, processing and analyzing experimental data.\\nIn this publication, the concepts of the World Wide Student Laboratory are outlined, a possible idea for structure is discussed and examples of various laboratory studies (projects) are shown.\"\n },\n {\n \"title\": \"Concept of the World Wide Student Laboratory\",\n \"content\": \"Based on the Internet, the World Wide Web (WWW) has been rapidly growing over the last several years. The WWW is a combination of a large number of computer servers which can send documents or \\\"pages\\\" to Internet clients using a Web-browser software for navigation between the different \\\"sites.\\\"\\nFrom the beginning the WWW has primarily been used for educational purposes in three ways [1]:\\n-to provide students with a wider access to information; -as a communications tool supplementing traditional forms of education, raising the efficiency of interaction between the educators and students;\\n-as \\\"a virtual classroom\\\", \\\" a virtual laboratory\\\", as an integrated interface for distance learning.\\nThe student's laboratory practice work is the main form of experimental research study. The efficiency of traditional labs have certain limits: -often students do not have enough opportunities to deeply explore a number of important sections of laboratory courses . Not all universities can support a wide spectrum of laboratory research setups because of their high price, service cost, the lack of necessary space, etc.; -more often laboratory classes are too short to obtain persuasive research results of the given process or phenomenon; -in many cases, the laboratory instruction and student research projects do not fully reflect the current level of professional experimental research.\\nIn the framework of the WWSL project, we offer new methods of educational technology which complement the traditional methods and upgrade the standard of experimental research study even with the use of existing lab equipment. These methods may also significantly reduce the above mentioned limits of laboratory practice work.\\nThe concept of WWSL may be expressed in the following way:\\nWorld Wide Student Laboratory (WWSL) is a dynamic international Internet-based collaboration of resources from the educational and research labs of universities, institutions, government research centers and companies. The main element of WWSL is the educational or research setup. This setup has an interactive connection through the Internet with a particular thematic WWSL-Site (Research Group Site). Through the Research Group Sites, students, under the instruction of an educator, are able to control experiments, take, process and analyse data, regardless of their location. The results obtained by each student are available to all other students who are participants of the particular research group and can be used in their own projects. 1 The main features of the WWSL project involve the following:\\n-the possibility of studying and demonstrating certain effects and phenomena unobservable in traditional lab studies. For example, the study of the effects which require simultaneous tests in different geographical points of the world, in different environments, during long time intervals, etc.;\\n-as a result of unification of the experimental data obtained from a network of setups, it is possible to study \\\"subtle\\\" processes and/or processes which require a large volume of data. This allows exploration of the origins of instrumental or methodological errors and demonstrates methods of extracting information from noisy data;\\n-it is possible to run certain experimental projects in parallel with mathematical modelling. Integrating the two allows more global understanding of the phenomena;\\n-any student who has access to the Internet, regardless of their location, may become a member of the WWSL. This also makes the WWSL a perfect tool for distance education;\\n-students have around the clock access to experimental setups, and the ability to work on their own schedule;\\n-educators have the possibility to use the data of experiments on-line through the WWSL for lectures; -by using the WWSL, students can have access to the real data from \\\"professional\\\" scientific experiments, and educators may use this \\\"live\\\" data for educational purposes;\\n-the use by WWSL of well known elements of the Internet such as chat rooms, white-boards, etc. can allow students and educators to have \\\"global seminars \\\" for each experiment;\\n-individual student projects can be integrated in the form of a final report of the research work. The participation in making such a report will teach the student the ethics of joint research, increase motivation and give significance to their work; -setting up new WWSL software modules may well be an attractive and challenging project for students in computer science;\\n-members of WWSL may not only be students from universities but also from colleges and high schools.\\nWWSL should enrich the personal research results of each student and help lab studies approach the top-level scientific research. It is important to emphasise that WWSL is not a \\\"virtual\\\" laboratory. It is a real lab where actual experiments (with all real-world \\\"noisy\\\" effects) can be made.\\nThe traditional form of conducting some experiments is still preferable for laboratory studies. However, from our point of view, the following projects would be better served by WWSL:\\n-projects that require a long time to complete; -projects which require a certain preliminary theoretical and/or practical preparation; -projects which require repeated change of experimental conditions;\\n-projects which require special conditions for their fulfilment.\"\n },\n {\n \"title\": \"The Structure of WWSL\",\n \"content\": \"Presently, the World Wide Student Laboratory is developing using the following structure:\\nThe general coordination of WWSL operations is conducted by the WWSL Coordination Center which fulfils the following functions:\\n-supports the official Web-site of the WWSL Project; -registers organizations and individuals which will be participants in the WWSL;\\n-monitors status of WWSL activity; -manages the development of methodology of WWSL studies and its data;\\n-maintains the WWSL-FAQ, chat rooms, etc.\\nA Research Group Site would be created to conduct a particular series of student lab studies. It combines the sites of Experimental Setups of participating organizations. If it is required, the activity of each Research Group can be defined by a Research Program which is prepared by Educators from participating Universities. The Research Group Site has the following functions:\\n-registers the participants in given lab studies; -integrates Student's Sites who are participating in a given project;\\n-provides access to experimental data from all Experimental Setups;\\n-maintains Research Program if required, and/or a schedule for taking data;\\n-maintains the pages with information necessary to complete these studies;\\n-maintains the FAQ and chat room page; -maintains a page with test questions; -maintains a page for a combined student report.\\nThe Site of the Experimental Setup may be located at a computer which is a part of the setup and is used for data storage, data processing and analysis of the results from that setup. It is maintained by the Educator in charge of the setup.\\nStudents use sites they must create (analogous to an electronic lab book) for access to WWSL resources, to control experimental setups and for processing and analysing experimental data. Their site would also be used to prepare the final report of a completed experiment.\\nThe structure of WWSL is not complicated and uses existing technology. Although it can take advanage of then, it does not rely on new technical developments at student or research laboratories or on the Internet.\\nPresently, the first stage of the WWSL development has been completed [2]. An Official Site of the WWSL Project has been created on the University of Oregon Physics Department's web server, in Eugene Oregon, USA. On that server, a Site for the \\\"Cosmic Rays\\\" Research Group has been organised. It combines two student research setups, one of which is located at the University of Oregon, the other is located at the Department of Physics at N. E. Bauman Moscow State Technical University, Moscow, Russia.\\nThe next stage is going to involve a number of projects for students who are studying General Physics, Radiophysics, Statistical Physics and Biophysics. The following projects are under discussion and/or development:\\n-lifetime of cosmic ray muons; -Mossbauer effect; -Coriolis acceleration; -nuclear magnetic resonance; -study of the accuracy of global positioning system; -study of the correlation between the processes of different bi-ological systems.\"\n },\n {\n \"title\": \"Examples of possible WWSL projects\",\n \"content\": \"Let us now consider some possible student projects which can be completed in laboratory setups for studying the properties of cosmic rays. These studies are part of the curriculum of physics and engineering majors in many universities [3], [4].\\nThe setup consists of two detectors, usually scintillating. An absorber of variable thickness (a set of lead or tungsten plates) is sandwiched between them. The signals from the detectors are read out to a coincidence unit. The unit selects events due to cosmic ray particles passing through both detectors. The output signals from coincidence units are collected by a data acquisition system and a computer.\\nStudent projects which can be pursued using these setups can be divided into two groups:\\n-projects for studying certain properties of cosmic rays, for studying of experimental methods and apparatus of cosmic ray physics and high energy physics;\\n-projects for studying the properties of random processes (in this case, the flux of particles of cosmic rays is considered as a generator of random processes).\\nThe first group may contain the following projects:\\n-study of the composition of cosmic rays near the earth surface;\\n-study of the angular distribution of the cosmic ray flux; -study of the fluctuation of energy loss of charged particles; -study of the effect of the detector's dead time on the character of the statistical distribution of the signals.\\nThe second group may contain the following projects:\\n-properties of Poisson data of random events; -study of the relationship between binomial, Poisson and normal distributions;\\n-test of the hypotheses presented by different statistical distributions;\\n-the \\\"waiting time paradox\\\" for Poisson processes; -study of correlations in stochastic processes.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The World Wide Web as an Instructional Tool. J M Barrie, D E Presti, Science. 274J. M. Barrie, D. E. Presti. The World Wide Web as an Instructional Tool. Science, Vol. 274, 1996, pp. 371-372.","doi":""},{"text":"Study of composition of cosmic rays. Guide for students. A Arodzero, P Gramenitsky, V Rodionov, 10Moscow, MSTUin RussianA. Arodzero, P. Gramenitsky, V. Rodionov. Study of composition of cos- mic rays. Guide for students. Moscow, MSTU, 1980. 10p. (in Russian).","doi":""},{"text":". Physics Laboratory Studies. Edited by L. Goldin. Moscow, Nauka. 704in RussianPhysics Laboratory Studies. Edited by L. Goldin. Moscow, Nauka, 1983. 704 p. (in Russian).","doi":""}],"string":"[\n {\n \"text\": \"The World Wide Web as an Instructional Tool. J M Barrie, D E Presti, Science. 274J. M. Barrie, D. E. Presti. The World Wide Web as an Instructional Tool. Science, Vol. 274, 1996, pp. 371-372.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Study of composition of cosmic rays. Guide for students. A Arodzero, P Gramenitsky, V Rodionov, 10Moscow, MSTUin RussianA. Arodzero, P. Gramenitsky, V. Rodionov. Study of composition of cos- mic rays. Guide for students. Moscow, MSTU, 1980. 10p. (in Russian).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Physics Laboratory Studies. Edited by L. Goldin. Moscow, Nauka. 704in RussianPhysics Laboratory Studies. Edited by L. Goldin. Moscow, Nauka, 1983. 704 p. (in Russian).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":52,"cells":{"text":{"kind":"string","value":"\nCan 2HDM support fermion-stabilized bubbles of false vacuum?\n1 Jul 2007\n\nI P Ivanov \nInteractions Fondamentales en Physique et en Astrophysique\nUniversité de Liège\nAllée du 6 Août, 17, bâtiment B5aB-4000LiègeBelgium\n\nCan 2HDM support fermion-stabilized bubbles of false vacuum?\n1 Jul 2007arXiv:0706.4332v2 [hep-ph] Sobolev Institute of Mathematics, ac. Koptyug av. 4, 630090, Novosibirsk, Russia\nThe Higgs potential of the two-Higgs-doublet model can have several minima with different properties. We discuss a possibility that a heavy fermion, if trapped in a microscopic false vacuum bubble, might become light enough to prevent the bubble from the collapse.\n\nThe Standard Model relies on the Higgs mechanism of the electroweak symmetry breaking, whose details are not yet known. Its simplest realization is based on a single weak isodoublet of scalar fields, while extended versions deal with more elaborate scalar sectors, see reviews [1,2]. The two-Higgs-doublet model [3], where one introduces two Higgs doublets φ 1 and φ 2 , is one of the most economic extensions of the Higgs sector beyond the Standard Model. The minimal supersymmetric extension of the Standard Model (MSSM) uses precisely a specific version of the 2HDM to break the electroweak symmetry.\n\nThe Higgs potential of the most general 2HDM V H = V 2 +V 4 is conventionally parametrized as\nV 2 = − 1 2 m 2 11 (φ † 1 φ 1 ) + m 2 22 (φ † 2 φ 2 ) + m 2 12 (φ † 1 φ 2 ) + m 2 * 12 (φ † 2 φ 1 ) ; V 4 = λ 1 2 (φ † 1 φ 1 ) 2 + λ 2 2 (φ † 2 φ 2 ) 2 + λ 3 (φ † 1 φ 1 )(φ † 2 φ 2 ) + λ 4 (φ † 1 φ 2 )(φ † 2 φ 1 )(1)+ 1 2 λ 5 (φ † 1 φ 2 ) 2 + λ * 5 (φ † 2 φ 1 ) 2 + λ 6 (φ † 1 φ 1 ) + λ 7 (φ † 2 φ 2 ) (φ † 1 φ 2 ) + h.c. .\nIt contains 14 free parameters: real m 2 11 , m 2 22 , λ 1 , λ 2 , λ 3 , λ 4 and complex m 2 12 , λ 5 , λ 6 , λ 7 . The large number of free parameters makes the analysis of the most general 2HDM and its phenomenological consequences rather complicated. Even the first step, straightforward minimization of the potential is prohibitively difficult for the most general 2HDM. This is why the most activity so far has been focused on some particular version of 2HDM with a strongly reduced number of free parameters.\n\nOn the other hand, the most general 2HDM has a remarkably rich spectrum of phases, which is lost in many specific version of 2HDM. The number of extrema, their nature, and various phase transitions in the most general 2HDM can be studied without explicit algebraic minimization of the potential, within the geometric approach developed in [4]. This approach helps easily see various possibilities offered by the 2HDM.\n\nIn this Letter we discuss a particular possibility that might be realized in 2HDM: a microscopic bubble of the false vacuum stabilized by a heavy fermion trapped in it.\n\nThe idea goes as follows. 2HDM can have several minima with different properties [4,5]. One assumes that our world is in the global minimum (true vacuum). If a false vacuum bubbles is produced, it quickly collapses due to the large surface tension [6]. However, one can construct a 2HDM, in which a fermion that is heavy in our world can become much lighter inside the false vacuum bubble. This mass difference might be sufficient to compensate the extra energy associated with the bubble surface tension, thus stabilizing the bubble against the collapse.\n\nHere, we present quasiclassical estimates indicating that such a possibility should not be neglected. It is of course difficult to say how reliable these quasiclassical estimates of the properties of microscopic bubbles are. A more accurate analysis is needed to establish whether it can be indeed realized in 2HDM without violation of the experimental bounds. If 2HDM indeed supports such a possibility, it will have remarkable phenomenological consequences.\n\nLet us first describe the construction of a 2HDM that contains two nearly degenerate minima with almost equal v 2 1 + v 2 2 but very different fermion masses. We start with the compact representation of the Higgs potential of 2HDM introduced in [4,7]:\nV = −M µ r µ + 1 2 Λ µν r µ r ν .(2)\nHere the Higgs fields appear as components of a four-vector r µ = (r 0 , r i ) with\nr 0 = (Φ † Φ) = (φ † 1 φ 1 ) + (φ † 2 φ 2 ) , r i = (Φ † σ i Φ) = (φ † 2 φ 1 ) + (φ † 1 φ 2 ) −i[(φ † 1 φ 2 ) − (φ † 2 φ 1 )] (φ † 1 φ 1 ) − (φ † 2 φ 2 ) ,(3)\nand the neutral vacuum we consider corresponds to vacuum points r µ lying on the forward lightcone, r µ r µ = 0. Thus, the orbit space of the 2HDM Higgs potential is equipped with the Minkowski space structure. The group of special linear transformations of doublets φ 1 and φ 2 induces the Lorentz group SO(1, 3) of transformations of r µ . The ten quartic couplings λ i of the most general 2HDM define the real symmetric tensor Λ µν , while the four free parameters of the quadratic term form the real four-vector M µ . The explicit expressions for Λ µν and M µ can be found in [4,7]. Here we mention only the fact that Λ µν can be always diagonalized by an SO(1, 3) transformation. To simplify the discussion we assume that Λ µν is already diagonal.\n\nOne of the features of 2HDM is the possibility to spontaneously break discrete symmetries. The most studied case is the spontaneous CP -violation in the Higgs sector, [8]. It corresponds to the situation when M 2 = 0 but r µ has a non-zero component r 2 , which according to (3) corresponds to a nonzero relative phase between v 1 and v 2 .\n\nAnalogously, one can consider a situation when M 1 = 0, but r 1 = 0 (see Fig. 1, left). It corresponds to a situation when the Higgs potential respects Z 2 symmetry (i.e. symmetry Figure 1: Location of the minima in the orbit space of a specific 2HDM. One starts from a nearly degenerate situation with, right, and then rotates the construction to obtain model discussed in the text, left.\n\nunder sign change of one of the doublets), but it is spontaneously violated in each of the two degenerate minima (i.e. minima correspond to non-zero v.e.v.'s of both doublets). The requirements for such a situation to take place is that M µ must lie inside the corresponding caustic cone in the Minkowski space (see details in [4]). For simplicity, we also take M 2 = 0 and r 2 = 0. Note that v 2 0 = v 2 1 + v 2 2 = (246GeV) 2 is the same for both minima. In addition, one can always choose parameters λ i such that there is no other minimum of the potential. All these criteria are realized, for instance, in the 2HDM with the following parameters \nλ 1 = λ 2 > 0 , λ 5 < 0 , |λ 5 | > λ 4 , λ 3 = λ 6 = λ 7 = 0 ,\nOne can remove degeneracy by slightly shifting M µ away from the Λ µν eigenaxis (by taking small but nonzero Re m 2 12 ), which makes one of the minima (the true vacuum) slightly deeper than the other. This difference can be made small enough so that it does not significantly affect the values of v i and the energetic considerations below. Now we rotate this construction to obtain the one shown in Fig. 1, right. This can be done by an appropriate change of parameters of the Higgs potential, for example, by introducing some real λ 6 = −λ 7 together with matching Re m 2 12 . The two nearly degenerate minima now correspond to\ntrue vacuum: v 1 = v 0 cos β t , v 2 = v 0 sin β t ≡ v t ; false vacuum: v 1 = v 0 cos β f , v 2 = v 0 sin β f ≡ v f .(5)\nAngles β t and β f can be chosen at will. The choice we make is β t being noticably larger than β f . Angles α i shown in Fig. 1 are twice β i . Let us now suppose that the heavy fermion gets its mass via coupling only to the second doublet. In the true and false vacuum we have, respectively,\nm t = g v t √ 2 , m f = g v f √ 2 .(6)\nLet us now focus on the Higgs potential along the least energy path in the Higgs space that connects the two minima. If the minima are close to each other, we can approximate the path by a straight line. Let h be the neutral scalar field along this direction, whose zero is taken just between the minima. Then the Higgs lagrangian density is approximately\nL = 1 2 (∂ µ h)(∂ µ h) − λ 2 1 2 h 2 − 1 2 v t − v f 2 2 2 + L 0 ,(7)\nwhere L 0 includes all other terms and does not significantly change between the two minima.\n\nHere λ is the coefficient of the quartic potential along the chosen direction. The Higgs boson mass along this direction in each of the two minima is\nm h = √ λ v t − v f 2 .(8)\nWe need to estimate the energy associated with the corresponding barrier. Consider now a spherical bubble of the false vacuum of radius ξ. We take this radius large enough to allow the fermion (with its false vacuum mass!) to fit into the bubble: ξ = 1/m f . A typical radial field profile of the bubble is the following: false vacuum at r ∼ < ξ, barrier around r ∼ ξ and the true vacuum at r ∼ > ξ. For estimate, we assume that the barrier wall spans from r = ξ/2 to r = 3ξ/2 and that the field changes by about (v t − v f )/2 over this distance. Then, the energy associated with the static bubble of such a field profile is approximately\n∆E H ∼ 4π 3 3 2 3 − 1 2 3 ξ 3 · 1 2 v t − v f 2 2 1 ξ 2 + λ 4 v t − v f 2 4 = 13π 6 ξ v t − v f 2 2 1 + λ 4 v t − v f 2 2 ξ 2 .(9)\nThe stability condition is that the fermion mass difference between the true and false vacua be larger than this barrier energy\n∆E f = g v t − v f √ 2 > ∆E H .(10)\nSubstituting ξ = √ 2/(gv f ), we get\ng 2 > 13π 12 x 1 + λ 8g 2 x 2 , where x ≡ v t − v f v f .(11)\nWe now consider two options for the heavy fermion: the top-quark and a new very heavy fermion.\n\nTop quark. The natural candidate for the heavy fermion would be the top-quark. Satisfying this inequality without generating too light Higgs boson implies not only large g, but also large λ, as can be seen from (6) together with (8). The exact experimental limit on the lightest Higgs mass is not known in the most general 2HDM, so we take the PDG bound on the lightest supersymmetric Higgs, m H > 89.8 GeV, [10]. Then the above inequality can be satisfied, for example, with λ = 12, g = 2.2, and x = 1 so that v t = 2v f = 0.447v 0 and m H = 95 GeV. This implies, however, that the bubble itself is rather light, ∼ 80 GeV with this particular choice of parameters. It is even lighter than the Higgs boson itself, which seems to be a very unlikely possibility.\n\nAccording to [9], where the tree-level unitarity constraints were presented for the most general 2HDM, values of λ larger than ∼ 10 lead to intense coupling regime in the Higgs sector, where the conclusions based on the tree-level analysis are hardly reliable. However, the above estimate is very sensitive to numerical factors. It remains to be seen if a more accurate calculation of ∆E H can make λ smaller.\n\nLet us briefly discuss the phenomenological properties of such a bubble if it can be indeed realized in 2HDM. This bubble should manifest itself as a particle with a mass below the known t-quark mass. Its lifetime will depend on the stability of the fermion in the false vacuum (the bubble cannot collapse before the fermion decays). Since its mass in the false vacuum is several times smaller than in the true vacuum, its width should be below 1 GeV scale. Not only the smaller mass reduces the phase space, but it also makes its decay to bW + subthreshold (the W mass does not change significantly in the false vacuum considered, since v 0 is kept the same). Its production at colliders should be a rare process since an energetic collision must produce simultaneously a top-quark and a specific Higgs field configuration around it.\n\nNew very heavy fermions. If new very heavy fermions exist and if their masses are also generated by the Higgs mechanism, they will readily provide a large enough Yukawa constant g to satisfy (11). A very heavy fermion also allows for a sufficiently heavy bubble. However, it will also lead to large loop corrections to the Higgs potential, so it remains to be seen whether how strongly the above estimates get modified.\n Acknowledgements. I am thankful to Ilya Ginzburg for stimulating discussions of various aspects of 2HDM. The work was supported by FNRS and partly by grants RFBR 05-02-16211 and NSh-5362.2006.2.\nJ F Gunion, H E Haber, G Kane, S Dawson, The Higgs Hunter's Guide. ReadingAddison-WesleyJ.F. Gunion, H.E. Haber, G. Kane, S. Dawson, The Higgs Hunter's Guide (Addison- Wesley, Reading, 1990).\n\nE Accomando, arXiv:hep-ph/0608079Workshop on CP studies and non-standard Higgs physics. E. Accomando et al., \"Workshop on CP studies and non-standard Higgs physics,\" arXiv:hep-ph/0608079.\n\n. T D Lee, Phys. Rev. D. 81226T. D. Lee, Phys. Rev. D 8 (1973) 1226.\n\n. I P Ivanov, Phys. Rev. D. 7535001I. P. Ivanov, Phys. Rev. D 75 (2007) 035001.\n\n. A Barroso, P M Ferreira, R F Santos ; I, K A Ginzburg, Kanishev, arXiv:hep-ph/0702098arXiv:0704.3664hep-phA. Barroso, P. M. Ferreira and R. Santos, arXiv:hep-ph/0702098; I. F. Ginzburg and K. A. Kanishev, arXiv:0704.3664 [hep-ph].\n\n. Y B Zeldovich, I Y Kobzarev, L B Okun, Zh. Eksp. Teor. Fiz. 673Sov. Phys. JETPY. B. Zeldovich, I. Y. Kobzarev and L. B. Okun, Zh. Eksp. Teor. Fiz. 67 (1974) 3 [Sov. Phys. JETP 40 (1974) 1];\n\n. I Y Kobzarev, L B Okun, M B Voloshin, Sov. J. Nucl. Phys. 201229Yad. Fiz.I. Y. Kobzarev, L. B. Okun and M. B. Voloshin, Sov. J. Nucl. Phys. 20 (1975) 644 [Yad. Fiz. 20 (1974) 1229];\n\n. S R Coleman, Phys. Rev. D. 152929Erratum-ibid. D 16 (1977) 1248S. R. Coleman, Phys. Rev. D 15 (1977) 2929 [Erratum-ibid. D 16 (1977) 1248].\n\n. C C Nishi, Phys. Rev. D. 7436003C. C. Nishi, Phys. Rev. D 74 (2006) 036003.\n\n. I F Ginzburg, M Krawczyk, Phys. Rev. D. 72115013I. F. Ginzburg and M. Krawczyk, Phys. Rev. D 72 (2005) 115013;\n\n. J F Gunion, H E Haber, Phys. Rev. D. 7295002J. F. Gunion and H. E. Haber, Phys. Rev. D 72 (2005) 095002.\n\n. I F Ginzburg, I P Ivanov, Phys. Rev. D. 72115010I. F. Ginzburg and I. P. Ivanov, Phys. Rev. D 72 (2005) 115010.\n\n. W.-M Yao, J. Phys. G. 331W.-M. Yao et al., J. Phys. G 33 (2006) 1.\n"},"corpus_id":{"kind":"number","value":2854202,"string":"2,854,202"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0706.4332\",\n \"mag\": 1532339105,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0706.4332v2.pdf\"\n ],\n \"pdf_sha\": \"6dd8cc13ebff32cb9456607e3e5f6295a01cc1b3\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Can 2HDM support fermion-stabilized bubbles of false vacuum?"},"authors":{"kind":"list like","value":["I P Ivanov \nInteractions Fondamentales en Physique et en Astrophysique\nUniversité de Liège\nAllée du 6 Août, 17, bâtiment B5aB-4000LiègeBelgium"],"string":"[\n \"I P Ivanov \\nInteractions Fondamentales en Physique et en Astrophysique\\nUniversité de Liège\\nAllée du 6 Août, 17, bâtiment B5aB-4000LiègeBelgium\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The Higgs potential of the two-Higgs-doublet model can have several minima with different properties. We discuss a possibility that a heavy fermion, if trapped in a microscopic false vacuum bubble, might become light enough to prevent the bubble from the collapse."},"sections":{"kind":"list like","value":[],"string":"[]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"J F Gunion, H E Haber, G Kane, S Dawson, The Higgs Hunter's Guide. ReadingAddison-WesleyJ.F. Gunion, H.E. Haber, G. Kane, S. Dawson, The Higgs Hunter's Guide (Addison- Wesley, Reading, 1990).","doi":""},{"text":"E Accomando, arXiv:hep-ph/0608079Workshop on CP studies and non-standard Higgs physics. E. Accomando et al., \"Workshop on CP studies and non-standard Higgs physics,\" arXiv:hep-ph/0608079.","doi":""},{"text":". T D Lee, Phys. Rev. D. 81226T. D. Lee, Phys. Rev. D 8 (1973) 1226.","doi":""},{"text":". I P Ivanov, Phys. Rev. D. 7535001I. P. Ivanov, Phys. Rev. D 75 (2007) 035001.","doi":""},{"text":". A Barroso, P M Ferreira, R F Santos ; I, K A Ginzburg, Kanishev, arXiv:hep-ph/0702098arXiv:0704.3664hep-phA. Barroso, P. M. Ferreira and R. Santos, arXiv:hep-ph/0702098; I. F. Ginzburg and K. A. Kanishev, arXiv:0704.3664 [hep-ph].","doi":""},{"text":". Y B Zeldovich, I Y Kobzarev, L B Okun, Zh. Eksp. Teor. Fiz. 673Sov. Phys. JETPY. B. Zeldovich, I. Y. Kobzarev and L. B. Okun, Zh. Eksp. Teor. Fiz. 67 (1974) 3 [Sov. Phys. JETP 40 (1974) 1];","doi":""},{"text":". I Y Kobzarev, L B Okun, M B Voloshin, Sov. J. Nucl. Phys. 201229Yad. Fiz.I. Y. Kobzarev, L. B. Okun and M. B. Voloshin, Sov. J. Nucl. Phys. 20 (1975) 644 [Yad. Fiz. 20 (1974) 1229];","doi":""},{"text":". S R Coleman, Phys. Rev. D. 152929Erratum-ibid. D 16 (1977) 1248S. R. Coleman, Phys. Rev. D 15 (1977) 2929 [Erratum-ibid. D 16 (1977) 1248].","doi":""},{"text":". C C Nishi, Phys. Rev. D. 7436003C. C. Nishi, Phys. Rev. D 74 (2006) 036003.","doi":""},{"text":". I F Ginzburg, M Krawczyk, Phys. Rev. D. 72115013I. F. Ginzburg and M. Krawczyk, Phys. Rev. D 72 (2005) 115013;","doi":""},{"text":". J F Gunion, H E Haber, Phys. Rev. D. 7295002J. F. Gunion and H. E. Haber, Phys. Rev. D 72 (2005) 095002.","doi":""},{"text":". I F Ginzburg, I P Ivanov, Phys. Rev. D. 72115010I. F. Ginzburg and I. P. Ivanov, Phys. Rev. D 72 (2005) 115010.","doi":""},{"text":". W.-M Yao, J. Phys. G. 331W.-M. Yao et al., J. Phys. G 33 (2006) 1.","doi":""}],"string":"[\n {\n \"text\": \"J F Gunion, H E Haber, G Kane, S Dawson, The Higgs Hunter's Guide. ReadingAddison-WesleyJ.F. Gunion, H.E. Haber, G. Kane, S. Dawson, The Higgs Hunter's Guide (Addison- Wesley, Reading, 1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"E Accomando, arXiv:hep-ph/0608079Workshop on CP studies and non-standard Higgs physics. E. Accomando et al., \\\"Workshop on CP studies and non-standard Higgs physics,\\\" arXiv:hep-ph/0608079.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T D Lee, Phys. Rev. D. 81226T. D. Lee, Phys. Rev. D 8 (1973) 1226.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I P Ivanov, Phys. Rev. D. 7535001I. P. Ivanov, Phys. Rev. D 75 (2007) 035001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Barroso, P M Ferreira, R F Santos ; I, K A Ginzburg, Kanishev, arXiv:hep-ph/0702098arXiv:0704.3664hep-phA. Barroso, P. M. Ferreira and R. Santos, arXiv:hep-ph/0702098; I. F. Ginzburg and K. A. Kanishev, arXiv:0704.3664 [hep-ph].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y B Zeldovich, I Y Kobzarev, L B Okun, Zh. Eksp. Teor. Fiz. 673Sov. Phys. JETPY. B. Zeldovich, I. Y. Kobzarev and L. B. Okun, Zh. Eksp. Teor. Fiz. 67 (1974) 3 [Sov. Phys. JETP 40 (1974) 1];\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Y Kobzarev, L B Okun, M B Voloshin, Sov. J. Nucl. Phys. 201229Yad. Fiz.I. Y. Kobzarev, L. B. Okun and M. B. Voloshin, Sov. J. Nucl. Phys. 20 (1975) 644 [Yad. Fiz. 20 (1974) 1229];\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S R Coleman, Phys. Rev. D. 152929Erratum-ibid. D 16 (1977) 1248S. R. Coleman, Phys. Rev. D 15 (1977) 2929 [Erratum-ibid. D 16 (1977) 1248].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C C Nishi, Phys. Rev. D. 7436003C. C. Nishi, Phys. Rev. D 74 (2006) 036003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I F Ginzburg, M Krawczyk, Phys. Rev. D. 72115013I. F. Ginzburg and M. Krawczyk, Phys. Rev. D 72 (2005) 115013;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J F Gunion, H E Haber, Phys. Rev. D. 7295002J. F. Gunion and H. E. Haber, Phys. Rev. D 72 (2005) 095002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I F Ginzburg, I P Ivanov, Phys. Rev. D. 72115010I. F. Ginzburg and I. P. Ivanov, Phys. Rev. D 72 (2005) 115010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W.-M Yao, J. Phys. G. 331W.-M. Yao et al., J. Phys. G 33 (2006) 1.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":53,"cells":{"text":{"kind":"string","value":"\nCHERN-SIMONS QUANTUM FIELD REPRESENTATION OF PLANAR FERROMAGNETS\nNov 1994\n\nL Martina \nO K Pashaev \nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nG Soliani \n\nDipartimento di Fisica dell'Universita' and Sezione INFN\n73100LecceItaly\n\nCHERN-SIMONS QUANTUM FIELD REPRESENTATION OF PLANAR FERROMAGNETS\nNov 1994arXiv:hep-th/9411120v1 16\nWe represent the two -dimensional planar classical continuous Heisenberg spin model as a constrained Chern-Simons gauged nonlinear Schrödinger system. The hamiltonian structure of the model is studied, allowing the quantization of the theory by the gauge invariant approach. A preliminary study of the quantum states is displayed and several physical consequences in terms of anyons are discussed.\n\n1.\n\nChern-Simons (CS) gauged models have recently attracted a great attention in the description of several phenomena in condensed matter physics as the Fractional Quantum Hall Effect (FQHE) and the High Temperature Superconductivity [1,2]. In this context, models of the Landau-Ginzburg type have been proposed as effective field-theories [3,4]. In the stationary limit, soliton solutions to the self-dual reductions of these models appear. These can be exploited to represent Laughlin's quasiparticles and anyons [5]. On the other hand, the present authors showed that the self-dual CS system can be obtained, in the tangent space representation, from the stationary two-dimensional Landau -Lifshitz equation (LLE) [6]. This is the classical continuous isotropic version of the Heisenberg model, obtained by the discrete quantum version in the framework of the coherent state representation [7]. In particular, soliton solutions of the former system turn out to be related to magnetic vortices of the LLE. More generally, the LLE in 2+1 dimensions takes the form of a constrained nonlinear Schrödinger system (cNLS) for two charged matter fields, coupled to a CS gauge potential [8]. This correspondence suggests an explanation to the phenemenological analogy between the behaviour of magnetic vortices [9] and Hall particles in an external magnetic field [10].\n\nSec. 2 is devoted to a brief review of the Landau -Lifshitz model in 2+1\n\ndimensions and its tangent space formulation, that is the CS gauged cNLS system. Furthermore, the conservation of some associated physical quantities is discussed. In Sec. 3 we study the lagrangian formulation and the hamiltonian structure of the CS gauged cNLS system. Finally, in Sec. 4 we apply the method of gauge invariant quantization to the above mentioned system.\n\n\n2.\n\nIn 2+1 dimensions the LLE for the local magnetization S reads\nS t = S × ∇ 2 S,(1)\nwhere S belongs to the 2-dimensional sphere S 2 . In terms of the stereographic projection\nS + = S 1 + i S 2 = 2 ζ 1 + |ζ | 2 , S 3 = 1 − |ζ | 2 1 + |ζ | 2(2)\nEq. (1) becomes\n1 4 i ζ t + ζ z z = 2 ζ z ζ z 1 + |ζ | 2 ζ (z = x 1 + ix 2 , z = x 1 − ix 2 ).(3)\nWith ferromagnetic boundary conditions, that is S(r, t) → (0, 0, −1) for |r| → ∞, the (x 1 , x 2 ) plane is compactified. Thus S defines maps of S 2 → S 2 , which are partitioned into homotopy classes, labeled by the integer-valued topological charge\nQ = 1 4π S · (∂ 1 S × ∂ 2 S) d 2 x = 1 π |ζ z | 2 − |ζ z | 2 1 + |ζ | 2 2 d 2 x.(4)\nThe energy of the system takes the form\nW = 1 2 ∇ S 2 d 2 x = 4 |ζ z | 2 + |ζ z | 2 1 + |ζ | 2 2 d 2 x,(5)\nwhich compared with Eq. (4) leads to the Bogomol'nyi inequality W ≥ 4π | Q |. When the equality holds, the field ζ has to be an (anti)holomorphic function of z, leading to the Belavin-Polyakov instantons [11]. In our approach the spin S is represented by the matrix S = S α σ α (α = 1, 2, 3), where σ α are the Pauli matrices. Then, we diagonalize S by resorting to a right-invariant local U (1) transformation g, such that S = gσ 3 g −1 . The matrix g belongs to U (2) and has the form\ng = ± [2 (1 + S 3 )] −1/2 1 + S 3 −S + S + 1 + S 3 exp iσ 3 η 2 ,(6)\nwhere η is an arbitrary real function. The corresponding chiral current J µ (µ = 0, 1, 2) is decomposable into a diagonal and an off-diagonal part:\nJ µ = g −1 ∂ µ g = i 4 σ 3 V µ + 0 − q µ q µ 0 .(7)\nBy definition, the chiral current J µ satifies the zero curvature condition\n∂ µ J ν − ∂ ν J µ + [J µ , J ν ] = 0.(8)\nIts off-diagonal and diagonal parts are\nD µ q ν = D ν q µ ,(9)[D µ , D ν ] = − i 2 (∂ µ V ν − ∂ ν V µ ) = −2 q µ q ν − q µ q ν ,(10)\nwhere we have introduced the covariant derivative D µ = ∂ µ − i 2 V µ . In the variables q µ , the equation of motion (1) becomes\nq 0 = iD m q m ,(11)\nwhich allows us to eliminate the function q 0 from Eqs. (10)(11). Now, let us introduce the new fields\nψ ± = (q 1 ± i q 2 ) 2 √ π , A 0 = V 0 − 8(| ψ + | 2 + | ψ − | 2 ), A i = V i (i = 1, 2)(12)\nand, correspondingly, the new covariant derivatives D µ = ∂ µ − (i/2) A µ . Then we notice that the expressions of ψ ± and A j in terms of ζ are\nψ ± = 1 2 √ π (∂ 1 ± i∂ 2 ) ζ (1 + |ζ| 2 ) e iη , A j = 2i (ζ∂ jζ −ζ∂ j ζ) (1 + |ζ| 2 ) + 2∂ j η.(13)\nFor µ = 1 and ν = 2, we see that Eq. (9) takes the form\nγ ≡ (D 1 + iD 2 ) ψ − − (D 1 − iD 2 ) ψ + = 0.(14)\nThis equation can be interpreted as a geometrical constraint for the CS gauged NLS system, arising from the remaining equations (9) and Eq. (10), i.e.\n\niD 0 ψ ± + D 1 2 + D 2 2 ψ ± + 8π |ψ ± | 2 ψ ± = 0, (15.a)\n∂ 2 A 1 − ∂ 1 A 2 = 8π J + 0 − J − 0 , (15.b) ∂ 0 A j − ∂ j A 0 = −8π ε jℓ J + l − J − l ,(15.c)\nwhere\nJ ± 0 = |ψ ± | 2 , J ± l = i (ψ ±Dlψ ± −ψ ± D l ψ ± ) (l = 1, 2) .(16)\nHere we introduce the charge density currents satisfying the continuity equation ∂ µ J ± µ = 0. Then, the matter fields ψ ± carry a conserved electric charge and we can interprete the quantity\nQ = J + 0 − J − 0 d 2 x(17)\nas the (conserved) total electric charge. In terms of ψ ± it is easy to see that Q and Q coincide, thus we are led to the quantization of the total electric charge. Furthermore, Eq. (15.b) implies the quantization of the magnetic flux: Φ =\nBd 2 x = (∂ 1 A 2 − ∂ 2 A 1 ) d 2 x = −8πQ.\nMoreover, the continuity equations for the currents J ± µ predict that the total number of particles (anyons)\nN = J + 0 + J − 0 d 2 x(18)\nis also a conserved quantity. Now, comparing Eq. (5) with (18) , we obtain W = 4πN . Then, in the tangent space formulation, the Bogomol'nyi inequality takes the form |Q | ≤ N . Of course, the equality holds when ψ + , or ψ − , vanishes. In such a case, the system (14 -15) reduces to the self-dual CS model [2]. This model is completely determined in terms of the solutions of the Liouville equation. This simple observation enables us to identify the magnetic vortices, or equivalently the instantons of Belavin-Polyakov type, with the solitons of the self-dual CS model, that is with the anyons. In particular, from (17) we have an explicit correspondence between the magnetic topological charge and the anyon electric charge, whose quantization finds here a geometrical interpretation. In ref. [8] the more general static case (D 1 + iD 2 ) ψ − = (D 1 − iD 2 ) ψ + = 0 is treated. This leads to a special reduction of the euclidean conformal affine Toda system, which contains the sinh-Gordon, or the Poisson-Boltzmann equation, in a particular gauge. Finally, we notice that the constraint (14) is identically satisfied using the expressions (13). Indeed, we find the identity\n(D 1 + iD 2 ) ψ − = (D 1 − iD 2 ) ψ + = 1 2 ζ z z − 2 ζ z ζ z 1 + |ζ | 2 ζ e iη 1 + |ζ | 2 . (19)\nEquating to zero the quantity in the square brackets, we recognize the stationary LLE in the stereographic variable (see Eq. (3)). Thus, all static solutions to Eq.\n\n(1) are mapped, modulo gauge transformations, into solutions to the sinh-Gordon reduction studied in ref. [8].\n\n\n3.\n\nSystem (15) can be derived from the Lagrangian\nL = i 2 (ψD 0 ψ −D 0ψ ψ) −D aψ D a ψ + 4π (ψσ 3 ψ)(ψψ) + 1 32π ǫ µνλ A µ ∂ ν A λ ,(20)\nwhere we have used the spinor notation ψ T = (ψ + , ψ − ), withψ = ψ + σ 3 , and the summation over a = 1, 2 is understood. Let us observe that the strength of the selfinteraction and the CS coupling are fixed by the original spin model. The specific ratio between these constants allows us to perform the self-dual reduction, which is a very special subcase in more general and the phenomenological theories of CS type [2]. Moreover, in the effective theories for FQHE the CS coupling is related to the so-called filling factor ν, which in the present case is 1/2, where the units h = c = 1 and e = 1/2 have been used. This result is consistent with the so-called chiral spin fluid model [1 -12]. Moreover, recent experimental observations in double-layer samples indicate the compatibility of ν, with values which are inverse of even numbers [13]. A final remark concerns the parity invariance of the Lagrangian (20), which changes the total sign under the transformation\nx 1 → x 1 , x 2 → −x 2 ψ + ↔ ψ − , D 1 + iD 2 ↔ D 1 − iD 2 .(21)\nSo that equations of motion and the constraint (14) are both left invariant. This fact is an obvious consequence of the parity invariance of the spin model (1) and it is another example, against the common belief, that the presence of the CS interaction breaks the P and T invariance. Now let us enter into the detail of the hamiltonian structure of the model (14 -15) by resorting to the first order Lagrangian method proposed in ref. [14], which implements the classical Dirac method for constrained Hamiltonian systems [15].\n\nFirst, we consider as dynamical variables ξ = ψ + ,ψ + , ψ − ,ψ − , A 1 , A 2 and,\ndefining a = ( i 2ψ + , − i 2 ψ + , − i 2ψ − , i 2 ψ − , 1 32 π A 2 , − 1 32 π A 1 ), we rearrange (20) in the form L ′ = a mξ m − H(ξ) + 1 2 A 0 Γ 1 ,(22)H =D aψ D a ψ − 4π (ψσ 3 ψ)(ψψ)(23)Γ 1 ≡ δL δA 0 =ψψ + 1 8 π ǫ ij ∂ i A j = 0.(24)\nThus L ′ describes the system (15 a,c) constrained by Eq. (24), that is the Chern-Simons version of the Gauss law (15.b). In (22) A 0 plays the role of Lagrange multiplier. Furthermore, H would be the density of the Hamiltonian H. In fact, the corresponding Euler-Lagrange equations take the hamiltonian forṁ\nξ i = f ij −1 ∂H ∂ξ j = {ξ i , H },(25)\nwhere f ml = ∂a l /∂ξ m − ∂a m /∂ξ l is a nonsingular symplectic matrix and we have defined the Poisson brackets by {ξ m , ξ l } = f −1 ml δ 2 (x − y). Specifically, in our case the nonvanishing Poisson brackets are\n{ψ ± (x),ψ ± (y)} = ∓ i δ 2 (x − y), {A i (x), A j (y)} = 16π ǫ ij δ 2 (x − y).(26)\nBy using Eq. (33), one easily shows thatΓ 1 = {Γ 1 , H} = 0; consequently, in phase space the Gauss' law defines an invariant submanifold under the dynamics. On the other hand, Eq. (14) provides the further complex primary constraint γ = 0, whose real and immaginary parts will be denoted by Γ 2 and Γ 3 . By evaluating the Poisson brackets, we find that {γ, H} ≈ 0, i.e.γ is vanishing only on the submanifold defined by (14) or, in other words, Γ 1 and Γ 2 are weakly invariant under time evolution. Furthermore, they do not produce secondary constraints. However, since {Γ 1 , Γ 2 , Γ 3 } is a set of first class constraints (i.e., {Γ i , H} ≈ 0 and det ({Γ i , Γ j }) ≈ 0 ), we need a gauge-fixing condition, for example the Coulomb gauge condition Γ 4 = ∂ i A i ≈ 0, which is the most used in the litterature [16,17,18]. So, we are able to compute the Dirac brackets, defined by\n{ξ m , ξ l } D ≡ {ξ m , ξ l } − {ξ m , Γ s }(d −1 ) ss ′ {Γ s ′ , ξ l }, where d ss ′ = {Γ s , Γ s ′ }.\nIn our case, the nonvanishing matrix elements d ss ′ are distributions in terms of δ 2 (x − y) and its derivatives. Then, inversion can be performed only in implicit form. In order to avoid this type of problem, one can look for a different gauge-fixing condition. For instance, a more general non-local gauge fixing can be choosen [19], or the so-called \"axial gauge\" [20]. Nevertheless, in all generality the corresponding Dirac brackets are rational in the field variables. Then a quantization of the model seems to be rather difficult, because of the problem arising in the ordering and the inversion of the operators involved. This is one of the main reasons which address us to look for an alternative approach, which is invariant under the choice of the gauge.\n\n\n4.\n\nIn this Section, we will adopt a slightly different point of view, in order to tackle the above mentioned problems. Moreover, we would like to understand the quantum meaning of the geometrical constraints γ, as seen separately from the basic CS-NLS field theory. Then, a possible route consists in quantizing system (14 -15) in two steps: 1. first, we apply the procedure of the gauge-invariant quantization to system (15) following the approach suggested in ref. [21]; 2. then, we use the constraint γ to select the physical states among all the gauge invariant quantum states.\n\nFirst of all, let us change the hamiltonian description by taking\nH ′ = H (ξ) + f 0 Γ 0 + 1 2 A 0 Γ 1 ,(27)\nwhere we have as primary constraint Γ 0 ≡ π 0 = 0, π 0 being the momentum conjugated to the canonical variable A 0 , such that {A 0 , π 0 } = δ 2 (x − y). This constraint is implicit in the previous formulation, since ∂L/∂Ȧ 0 = 0. Γ 1 is now a secondary constraint, arising from the Poisson bracket {π 0 , H}. Finally, in the expression (27) f 0 is an arbitrary function, which defines the evolution of A 0 . Furthermore, noticing that the special canonical pair (A 1 , A 2 ) breaks the rotational covariance of the theory, it is more convenient to consider the decomposition of the vector potential into a longitudinal and a transverse part\nA i (x) = A L i + A T i = 2∂ i η(x) + ǫ ij ∂ j χ(x).\nRecalling that the magnetic field is B = ǫ ij ∂ i A j = −∂ 2 χ, we derive formally the useful relation\nA T i (x) = −ǫ ij (∂ −1 j B)(x),(28)\nwhere we have defined\n∂ −1 j f (x) = 1 2π ∂ (x) j ln |x − y|f (y)d 2 y.\nAnalogously, it is convenient to express ψ ± in terms of the canonical variables (P ± , Q ± ) by\nψ ± = 1 √ 2 (P ± ∓ iQ ± ) .(29)\nHence, the Poisson structure associated with the hamiltonian system given by (27) is\n{Q ± (x), P ± (y)} = δ 2 (x − y), {A 0 (x), π 0 (y)} = δ 2 (x − y), {B(y), η(x)} = 8 π δ 2 (x − y).(30)\nFurthermore, the equations of motion are supplemented by the first class constraints Γ 0 ≈ Γ 1 ≈ 0. Now, we are look for a suitable canonical transformation, such that some of the new momenta are equal to the constraints. In such a way, the coordinates canonically conjugated to these momenta have an arbitrary time evolution and are remnants of the gauge invariance of the theory. The remaining canonical variables define the prominent dynamical degrees of freedom of the theory.\n\nFirst, let us denote by (Q * ± , P * ± ), (A * 0 , π * 0 ), (η * , π * 1 ) new canonically conjugated coordinates, where the momenta π * i have to be set equal to the constraints Γ i . This can be done through a suitable generating function F = F Q ± , A 0 , η; P * ± , π * 0 , π * 1 , satisfying the set of generalized Hamilton-Jacobi equations\nπ * i = Γ i Q ± , A 0 , η; ∂F ∂Q ± , ∂F ∂A 0 , ∂F ∂η .(31)\nThe integrability of this system is assured by {Γ 0 , Γ 1 } = 0, moreover its general solution is given by\nF = π * 0 A 0 + π * 1 + 1 2 P * 2 + − 1 2 P * 2 − η + P * 2 + − Q 2 + dQ + + P * 2 − − Q 2 dQ − .(32)\nThe corresponding canonical transformation reads\n π 0 = π * 0 , A 0 = A * 0 , − 1 8 π B = π * 1 + 1 2 P * 2 + − 1 2 P * 2 − , η = η * , Q ± = P * ± sin Q * ± P * ± ∓ η * , P ± = P * ± cos Q * ± P * ± ∓ η * .(33)\nFinally, the new gauge-invariant degrees of freedom are\nΦ ± = 1 √ 2 P * ± exp ∓ i Q * ± P * ± = ψ ± exp (−iη) , π * ± = Φ ± ,(34)\nwhich fulfill the canonical brackets\n{Φ ± (x), Φ ± (y)} = ∓ i δ 2 (x − y).(35)\nIn terms of these new variables, the Hamiltonian density becomes\nH = (∂ a + i 2 A T a )Φ + (∂ a − i 2 A T a )Φ + − (∂ a + i 2 A T a )Φ − (∂ a − i 2 A T a )Φ − −4 π (|Φ + | 4 − |Φ − | 4 )] + f 0 π * 0 + 1 2 A * 0 π * 1 ,(36)where A T a (x) = 8 π ǫ ab ∂ −1 b (|Φ + | 2 − |Φ − | 2 )(x)\n. Furthermore, the constraint γ takes the form\nγ = (∂ + − i 2 A T + )Φ − − (∂ − − i 2 A T − )Φ + ,(37)\nwhere ∂ ± = ∂ 1 ± i ∂ 2 and A T ± = A T 1 ± i A T 2 . Now, let us observe that the relations (13) are algebraically invertible in terms of gauge invariant quantities when the topological density is non vanishing, namely\nζ = i 4 √ π Φ − A T + − Φ + A T − |Φ + | 2 − |Φ − | 2 .(38)\nThen, we can directly reconstruct the spin field in terms of Φ ± , although it turns out to be a non local field in view of the expressions for A T ± . In this regard, let us observe that, in the one-dimensional completely integrable case, the relation between ζ and the NLS matter field is also analytic in nature, but we need to solve the associated linear spectral problem.\n\nFurthermore, keeping in mind that the current densities (16) are manifestly gauge invariant, we can give the expressions of the linear and angular momentum for the model (14 -15) in a gauge invariant form.\n\nFirstly, in opposition to what happens for the electric charge densities, the linear momenta P ± m = J ± m d 2 x, associated with the Φ ± fields separately, are no longer conserved, but they evolve according to Newton's law\nd dt P ± m = 8 ǫ mn (J − 0 J + n − J + 0 J − n )d 2 x.(39)\nWe notice that both Φ ± are subjected to the same force. Therefore, the total kinetic momentum\nP m = P + m − P − m = Φ + ∂ m Φ + − Φ − ∂ m Φ − d 2 x + c.c.(40)\nis conserved, the \" − \" comes from the \"negative mass\" associated to Φ − . The total angular momentum M = M + − M − is also conserved , with\nM ± = x 1 J ± 2 − x 2 J ± 1 d 2 x = ℓ ± + s ±(41)\nand\nl ± = −i Φ ± (x 1 ∂ 2 − x 2 ∂ 1 ) Φ ± d 2 x + c.c., s ± = |Φ ± | 2 A T 1 x 2 − A T 2 x 1 d 2 x.(42)\nThe quantities l = l + − l − and s = s + − s − are interpreted as orbital and spin angular momentum, respectively. In particular, the general result s = 2 Q 2 suggests that vortices have non zero momentum also in the static case. Finally, since the magnetization M = S d 2 x is a conserved non-local quantity for the LLE system, via Eqs. (38) and (2) we find a further gauge invariant conserved quantity for the system (14 -15) , related to the original global SU (2) symmetry.\n\nWe are now ready to pass on the quantum theory in a straightforward manner, just by replacing the canonical brackets by i times the commutators (or the anticommutators, if the matter fields are supposed to be fermionic). At the quantum level, the first class constraints Γ i become the operatorsΓ i , which must annihilate the physical states. This fact implies that the quantum states are independent of A 0 and invariant under time-independent gauge-transformations. Therefore, the gauge -invariant operators of the theory must commute withΓ i . Then, recalling the Poisson brackets for the Hamiltonian system described by (36), we can establish the correspondence Φ ± →Φ ± and Φ ± →Φ † ± , with the equal time (anti-) commutation relations\n[Φ ± (x),Φ † ± (y)] ± = ± δ 2 (x − y).(43)\nIn the physical gauge-invariant subspace of the full Hilbert space, the Hamiltonian is given byĤ\n= {−Φ † + (∂ a − i 2 A T a ) 2Φ + +Φ † − (∂ a − i 2 A T a ) 2Φ − −4 π : (Φ † +Φ + ) 2 − (Φ † −Φ − ) 2 :}d 2 x,(44)\nwhere the normal product operator : : and A T a = 8 π ǫ ab ∂ −1 b (Φ † +Φ + −Φ † −Φ − )(x) have been used. The fieldsΦ ± are local and invariant under local gauge transformations generated by the first-class constraints. Therefore, they may be exploited to study the breaking of global simmetries.\n\nThe quantum version of relations (30) leads to [B(x),η(y)] = 8πiδ 2 (x − y), which implies the relation exp (iη (y))B (x) exp (−iη (y)) =B (x) + 8 π δ 2 (x − y) .\n\nThis result is used to prove thatΦ ± are the gauge invariant operators which create a charge-solenoid composite, having magnetic flux equal to ±8 π . The Hamiltonian (44) commutes with the particle number operatorŝ\nN ± = Φ † ±Φ ± d 2 x.(46)\nHence, we can diagonalize them simultaneously via the Fock representation for a quantum state with N = N + + N − particles, which is given by\n|N + , N − > = N + i=1 d 2 x + i N − j=1 d 2 x − j Ψ x + 1 , . . . , x + N + , x − 1 , . . . , x − N − Φ † + x + 1 . . .Φ † + x + N + Φ † − x − 1 . . .Φ † − x − N − |0 >,(47)\nwhere the vacuum state is defined byΦ ± |0 >= 0. The function Ψ is an arbitrary element of the Hilbert space L 2 [R 2(N + +N − ) ] obeying the Schrödinger equation for (N + + N − )-bodies. Furthermore, the state (47) is gauge invariant by construction. Thus, we have selected in the full Hilbert space only those states which contain the whole physical information.\n\nNevertheless, up to now we have dealt with a quite general \"anyonic\" field theory, without using the constraint γ, which in some sense takes into account the properties of the original spin field model. As a consequence, \"quantizing\" Eq. (37) in the formγ\n=D +Φ− (x) −D −Φ+ (x) ,(48)\nwe should obtain the physical states by requiring thatγ|N + , N − >= 0. However, for arbitrary occupation numbers (N + , N − ) this equation leads to complicated relations for the wave function Ψ. Here we display the result only for the finite boson case |N + , 0 , namely Ψ x + 1 , · · · , x + N + = F z + 1 , · · · , z + N + i < j\nx + i − x + j − 2 ,(49)\nwhere F is a holomorphic function of its arguments. By using a singular gauge transformation [23][24], the expression (49) takes the form of the Laughlin wave function [25] , describing a condensate state of bosonic solitons, which may be related with quantum disorderd state of the original ferromagnets [12]. Furthermore, it should be proved that the quantum state corresponding to the wave function (49) is actually an eigenvector of the Hamiltonian (44). This work is in progress, and we have strong indications in the positive sense, because at the classical level the Hamiltonian (23) can be written as a linear combination of the constraint γ and Γ 1 . This work was supported in part by MURST of Italy and by INFN -Sezione di Lecce. One of the authors (O. K. P.) thanks the Department of Physics of Lecce University for the warm hospitality.\n \n. F A Zee, Prog. Theor. Phys. Suppl. 10777F. A. Zee, Prog. Theor. Phys. Suppl. 107, 77 (1992).\n\nR E Prange, S M Girvin, The Quantum Hall Effect. New YorkSpringer-VerlagR. E. Prange and S. M. Girvin, The Quantum Hall Effect (Springer-Verlag, New York, 1990).\n\n. S C Zhang, T H Hansson, S Kivelson, Phys. Rev. Lett. 6282S. C. Zhang, T. H. Hansson and S. Kivelson, Phys. Rev. Lett. 62, 82 (1989).\n\n. Z F Ezawa, M Hotta, A Iwazaki, Progr. Theor. Phys. Suppl. 107185Z. F. Ezawa, M. Hotta and A. Iwazaki,Progr. Theor. Phys. Suppl. 107, 185 (1992).\n\n. R Jackiw, S-Y Pi, Phys. Rev. Lett. 642682R. Jackiw and S-Y. Pi, Phys. Rev. Lett. 64, 2969 (1990); 66, 2682 (1990);\n\n. Phys. Rev. D. 423500Phys. Rev. D 42, 3500 (1990).\n\n. L Martina, O K Pashaev, G Soliani, Mod. Phys. Lett. A. 83241L. Martina, O.K. Pashaev, G. Soliani, Mod. Phys. Lett. A 8, 3241 (1993).\n\nIntegrable Pseudospin Models in Condensed Matter. V G Makhankov, O K Pashaev, Sov. Sci. Rev., Math. Phys. 91V. G. Makhankov and O.K. Pashaev, Integrable Pseudospin Models in Con- densed Matter, in Sov. Sci. Rev., Math. Phys. 9, 1 (1992).\n\n. L Martina, O K Pashaev, G Soliani, Phys. Rev. B. 4815787L. Martina, O.K. Pashaev, G. Soliani, Phys. Rev. B 48, 15787 (1993).\n\nFerromagnetodynamics, the dynamics of magnetic bubbles, domains and domain wall. T H O'dell, WileyNew YorkT. H. O'Dell, Ferromagnetodynamics, the dynamics of magnetic bubbles, do- mains and domain wall, New York, Wiley (1981).\n\n. N Papanicolau, T N Tomaras, Nucl. Phys B. 360425N. Papanicolau and T. N. Tomaras, Nucl. Phys B 360, 425 (1991).\n\n. A A Belavin, A M Polyakov, JETP Lett. 22245A. A. Belavin and A. M. Polyakov,JETP Lett. 22, 245 (1975).\n\nA Zee, Physics in (2+1)-dimensions. Y.M. Cho ed.SingaporeWorld ScientificA. Zee, Physics in (2+1)-dimensions Y.M. Cho ed. (World Scientific, Singa- pore, 1992).\n\nPaired Hall states in double layer electron system. M Greiter, X G Wen, F Wilczek, IASSNS-HEP-92/1Nucl. Phys. 374M. Greiter, X.G. Wen and F. Wilczek, Nucl. Phys. B374, 567 (1992), \"Paired Hall states in double layer electron system\", IASSNS-HEP-92/1\n\n. L Faddeev, R Jackiw, Phys. Rev. Lett. 60692L. Faddeev, R. Jackiw, Phys. Rev. Lett. 60,692 (1988).\n\nP A M Dirac, Lectures on Quantum Mechanics. New YorkYeshiva UniversityP. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964).\n\n. P De Sousa Gerbert, Phys. Rev. D. 42543P. de Sousa Gerbert, Phys. Rev. D 42, 543 (1990).\n\n. T Matsuyama, Phys. Lett. B. 22899T. Matsuyama, Phys. Lett. B 228, 99 (1989).\n\n. C G Han, Phys. Rev. D. 475521C. G. Han, Phys. Rev. D 47, 5521 (1993).\n\n. R Banerjee, A Chatterjee, V V Sreedhar, Ann. Phys. 222254R. Banerjee, A. Chatterjee and V. V. Sreedhar, Ann. Phys.222,254 (1993).\n\nM Chaichian, N F Nelipa, Introduction to gauge field theory. BerlinSpringer-VerlagM. Chaichian and N. F. Nelipa, Introduction to gauge field theory ( Springer- Verlag, Berlin 1984).\n\n. D Boyanovsky, E T Newman, C Rovelli, Phys. Rev. D. 451210D. Boyanovsky, E. T. Newman and C. Rovelli, Phys. Rev. D 45, 1210 (1992);\n\n. D Boyanovsky, Int. J. Mod. Phys. A. 75917D. Boyanovsky, Int. J. Mod. Phys. A 7, 5917, (1992).\n\n. N K Pak, R Percacci, J. Math. Phys. 302951N. K. Pak and R. Percacci, J. Math. Phys. 30, 2951 (1989);\n\n. N K Pak, Progr. Theor. Phys. 88585N. K.L Pak, Progr. Theor. Phys. 88, 585 (1992).\n\n. F Wilczek, Phys. Rev. Lett. 481144F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982).\n\n. S Girvin, Phys. Rev. Lett. 651489S. Girvin et al., Phys. Rev. Lett. 65, 1489, (1992).\n\n. R B Laughlin, Phys. Rev. Lett. 501395R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).\n"},"corpus_id":{"kind":"number","value":15406411,"string":"15,406,411"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"hep-th/9411120\",\n \"mag\": 2963398774,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/hep-th/9411120v1.pdf\"\n ],\n \"pdf_sha\": \"1dd0a9f165861ed3e24750928aebf25d4c917572\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"CHERN-SIMONS QUANTUM FIELD REPRESENTATION OF PLANAR FERROMAGNETS"},"authors":{"kind":"list like","value":["L Martina","O K Pashaev \nJoint Institute for Nuclear Research\n141980DubnaRussia","G Soliani","Dipartimento di Fisica dell'Universita' and Sezione INFN\n73100LecceItaly"],"string":"[\n \"L Martina\",\n \"O K Pashaev \\nJoint Institute for Nuclear Research\\n141980DubnaRussia\",\n \"G Soliani\",\n \"Dipartimento di Fisica dell'Universita' and Sezione INFN\\n73100LecceItaly\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We represent the two -dimensional planar classical continuous Heisenberg spin model as a constrained Chern-Simons gauged nonlinear Schrödinger system. The hamiltonian structure of the model is studied, allowing the quantization of the theory by the gauge invariant approach. A preliminary study of the quantum states is displayed and several physical consequences in terms of anyons are discussed."},"sections":{"kind":"list like","value":[{"title":"1.","content":"Chern-Simons (CS) gauged models have recently attracted a great attention in the description of several phenomena in condensed matter physics as the Fractional Quantum Hall Effect (FQHE) and the High Temperature Superconductivity [1,2]. In this context, models of the Landau-Ginzburg type have been proposed as effective field-theories [3,4]. In the stationary limit, soliton solutions to the self-dual reductions of these models appear. These can be exploited to represent Laughlin's quasiparticles and anyons [5]. On the other hand, the present authors showed that the self-dual CS system can be obtained, in the tangent space representation, from the stationary two-dimensional Landau -Lifshitz equation (LLE) [6]. This is the classical continuous isotropic version of the Heisenberg model, obtained by the discrete quantum version in the framework of the coherent state representation [7]. In particular, soliton solutions of the former system turn out to be related to magnetic vortices of the LLE. More generally, the LLE in 2+1 dimensions takes the form of a constrained nonlinear Schrödinger system (cNLS) for two charged matter fields, coupled to a CS gauge potential [8]. This correspondence suggests an explanation to the phenemenological analogy between the behaviour of magnetic vortices [9] and Hall particles in an external magnetic field [10].\nSec. 2 is devoted to a brief review of the Landau -Lifshitz model in 2+1\ndimensions and its tangent space formulation, that is the CS gauged cNLS system. Furthermore, the conservation of some associated physical quantities is discussed. In Sec. 3 we study the lagrangian formulation and the hamiltonian structure of the CS gauged cNLS system. Finally, in Sec. 4 we apply the method of gauge invariant quantization to the above mentioned system."},{"title":"2.","content":"In 2+1 dimensions the LLE for the local magnetization S reads\nwhere S belongs to the 2-dimensional sphere S 2 . In terms of the stereographic projection\nEq. (1) becomes\nWith ferromagnetic boundary conditions, that is S(r, t) → (0, 0, −1) for |r| → ∞, the (x 1 , x 2 ) plane is compactified. Thus S defines maps of S 2 → S 2 , which are partitioned into homotopy classes, labeled by the integer-valued topological charge\nThe energy of the system takes the form\nwhich compared with Eq. (4) leads to the Bogomol'nyi inequality W ≥ 4π | Q |. When the equality holds, the field ζ has to be an (anti)holomorphic function of z, leading to the Belavin-Polyakov instantons [11]. In our approach the spin S is represented by the matrix S = S α σ α (α = 1, 2, 3), where σ α are the Pauli matrices. Then, we diagonalize S by resorting to a right-invariant local U (1) transformation g, such that S = gσ 3 g −1 . The matrix g belongs to U (2) and has the form\nwhere η is an arbitrary real function. The corresponding chiral current J µ (µ = 0, 1, 2) is decomposable into a diagonal and an off-diagonal part:\nBy definition, the chiral current J µ satifies the zero curvature condition\nIts off-diagonal and diagonal parts are\nwhere we have introduced the covariant derivative D µ = ∂ µ − i 2 V µ . In the variables q µ , the equation of motion (1) becomes\nwhich allows us to eliminate the function q 0 from Eqs. (10)(11). Now, let us introduce the new fields\nand, correspondingly, the new covariant derivatives D µ = ∂ µ − (i/2) A µ . Then we notice that the expressions of ψ ± and A j in terms of ζ are\nFor µ = 1 and ν = 2, we see that Eq. (9) takes the form\nThis equation can be interpreted as a geometrical constraint for the CS gauged NLS system, arising from the remaining equations (9) and Eq. (10), i.e.\niD 0 ψ ± + D 1 2 + D 2 2 ψ ± + 8π |ψ ± | 2 ψ ± = 0, (15.a)\nwhere\nHere we introduce the charge density currents satisfying the continuity equation ∂ µ J ± µ = 0. Then, the matter fields ψ ± carry a conserved electric charge and we can interprete the quantity\nas the (conserved) total electric charge. In terms of ψ ± it is easy to see that Q and Q coincide, thus we are led to the quantization of the total electric charge. Furthermore, Eq. (15.b) implies the quantization of the magnetic flux: Φ =\nMoreover, the continuity equations for the currents J ± µ predict that the total number of particles (anyons)\nis also a conserved quantity. Now, comparing Eq. (5) with (18) , we obtain W = 4πN . Then, in the tangent space formulation, the Bogomol'nyi inequality takes the form |Q | ≤ N . Of course, the equality holds when ψ + , or ψ − , vanishes. In such a case, the system (14 -15) reduces to the self-dual CS model [2]. This model is completely determined in terms of the solutions of the Liouville equation. This simple observation enables us to identify the magnetic vortices, or equivalently the instantons of Belavin-Polyakov type, with the solitons of the self-dual CS model, that is with the anyons. In particular, from (17) we have an explicit correspondence between the magnetic topological charge and the anyon electric charge, whose quantization finds here a geometrical interpretation. In ref. [8] the more general static case (D 1 + iD 2 ) ψ − = (D 1 − iD 2 ) ψ + = 0 is treated. This leads to a special reduction of the euclidean conformal affine Toda system, which contains the sinh-Gordon, or the Poisson-Boltzmann equation, in a particular gauge. Finally, we notice that the constraint (14) is identically satisfied using the expressions (13). Indeed, we find the identity\nEquating to zero the quantity in the square brackets, we recognize the stationary LLE in the stereographic variable (see Eq. (3)). Thus, all static solutions to Eq.\n(1) are mapped, modulo gauge transformations, into solutions to the sinh-Gordon reduction studied in ref. [8]."},{"title":"3.","content":"System (15) can be derived from the Lagrangian\nwhere we have used the spinor notation ψ T = (ψ + , ψ − ), withψ = ψ + σ 3 , and the summation over a = 1, 2 is understood. Let us observe that the strength of the selfinteraction and the CS coupling are fixed by the original spin model. The specific ratio between these constants allows us to perform the self-dual reduction, which is a very special subcase in more general and the phenomenological theories of CS type [2]. Moreover, in the effective theories for FQHE the CS coupling is related to the so-called filling factor ν, which in the present case is 1/2, where the units h = c = 1 and e = 1/2 have been used. This result is consistent with the so-called chiral spin fluid model [1 -12]. Moreover, recent experimental observations in double-layer samples indicate the compatibility of ν, with values which are inverse of even numbers [13]. A final remark concerns the parity invariance of the Lagrangian (20), which changes the total sign under the transformation\nSo that equations of motion and the constraint (14) are both left invariant. This fact is an obvious consequence of the parity invariance of the spin model (1) and it is another example, against the common belief, that the presence of the CS interaction breaks the P and T invariance. Now let us enter into the detail of the hamiltonian structure of the model (14 -15) by resorting to the first order Lagrangian method proposed in ref. [14], which implements the classical Dirac method for constrained Hamiltonian systems [15].\nFirst, we consider as dynamical variables ξ = ψ + ,ψ + , ψ − ,ψ − , A 1 , A 2 and,\nThus L ′ describes the system (15 a,c) constrained by Eq. (24), that is the Chern-Simons version of the Gauss law (15.b). In (22) A 0 plays the role of Lagrange multiplier. Furthermore, H would be the density of the Hamiltonian H. In fact, the corresponding Euler-Lagrange equations take the hamiltonian forṁ\nwhere f ml = ∂a l /∂ξ m − ∂a m /∂ξ l is a nonsingular symplectic matrix and we have defined the Poisson brackets by {ξ m , ξ l } = f −1 ml δ 2 (x − y). Specifically, in our case the nonvanishing Poisson brackets are\nBy using Eq. (33), one easily shows thatΓ 1 = {Γ 1 , H} = 0; consequently, in phase space the Gauss' law defines an invariant submanifold under the dynamics. On the other hand, Eq. (14) provides the further complex primary constraint γ = 0, whose real and immaginary parts will be denoted by Γ 2 and Γ 3 . By evaluating the Poisson brackets, we find that {γ, H} ≈ 0, i.e.γ is vanishing only on the submanifold defined by (14) or, in other words, Γ 1 and Γ 2 are weakly invariant under time evolution. Furthermore, they do not produce secondary constraints. However, since {Γ 1 , Γ 2 , Γ 3 } is a set of first class constraints (i.e., {Γ i , H} ≈ 0 and det ({Γ i , Γ j }) ≈ 0 ), we need a gauge-fixing condition, for example the Coulomb gauge condition Γ 4 = ∂ i A i ≈ 0, which is the most used in the litterature [16,17,18]. So, we are able to compute the Dirac brackets, defined by\nIn our case, the nonvanishing matrix elements d ss ′ are distributions in terms of δ 2 (x − y) and its derivatives. Then, inversion can be performed only in implicit form. In order to avoid this type of problem, one can look for a different gauge-fixing condition. For instance, a more general non-local gauge fixing can be choosen [19], or the so-called \"axial gauge\" [20]. Nevertheless, in all generality the corresponding Dirac brackets are rational in the field variables. Then a quantization of the model seems to be rather difficult, because of the problem arising in the ordering and the inversion of the operators involved. This is one of the main reasons which address us to look for an alternative approach, which is invariant under the choice of the gauge."},{"title":"4.","content":"In this Section, we will adopt a slightly different point of view, in order to tackle the above mentioned problems. Moreover, we would like to understand the quantum meaning of the geometrical constraints γ, as seen separately from the basic CS-NLS field theory. Then, a possible route consists in quantizing system (14 -15) in two steps: 1. first, we apply the procedure of the gauge-invariant quantization to system (15) following the approach suggested in ref. [21]; 2. then, we use the constraint γ to select the physical states among all the gauge invariant quantum states.\nFirst of all, let us change the hamiltonian description by taking\nwhere we have as primary constraint Γ 0 ≡ π 0 = 0, π 0 being the momentum conjugated to the canonical variable A 0 , such that {A 0 , π 0 } = δ 2 (x − y). This constraint is implicit in the previous formulation, since ∂L/∂Ȧ 0 = 0. Γ 1 is now a secondary constraint, arising from the Poisson bracket {π 0 , H}. Finally, in the expression (27) f 0 is an arbitrary function, which defines the evolution of A 0 . Furthermore, noticing that the special canonical pair (A 1 , A 2 ) breaks the rotational covariance of the theory, it is more convenient to consider the decomposition of the vector potential into a longitudinal and a transverse part\nRecalling that the magnetic field is B = ǫ ij ∂ i A j = −∂ 2 χ, we derive formally the useful relation\nwhere we have defined\nAnalogously, it is convenient to express ψ ± in terms of the canonical variables (P ± , Q ± ) by\nHence, the Poisson structure associated with the hamiltonian system given by (27) is\nFurthermore, the equations of motion are supplemented by the first class constraints Γ 0 ≈ Γ 1 ≈ 0. Now, we are look for a suitable canonical transformation, such that some of the new momenta are equal to the constraints. In such a way, the coordinates canonically conjugated to these momenta have an arbitrary time evolution and are remnants of the gauge invariance of the theory. The remaining canonical variables define the prominent dynamical degrees of freedom of the theory.\nFirst, let us denote by (Q * ± , P * ± ), (A * 0 , π * 0 ), (η * , π * 1 ) new canonically conjugated coordinates, where the momenta π * i have to be set equal to the constraints Γ i . This can be done through a suitable generating function F = F Q ± , A 0 , η; P * ± , π * 0 , π * 1 , satisfying the set of generalized Hamilton-Jacobi equations\nThe integrability of this system is assured by {Γ 0 , Γ 1 } = 0, moreover its general solution is given by\nThe corresponding canonical transformation reads\nFinally, the new gauge-invariant degrees of freedom are\nwhich fulfill the canonical brackets\nIn terms of these new variables, the Hamiltonian density becomes\n. Furthermore, the constraint γ takes the form\nwhere ∂ ± = ∂ 1 ± i ∂ 2 and A T ± = A T 1 ± i A T 2 . Now, let us observe that the relations (13) are algebraically invertible in terms of gauge invariant quantities when the topological density is non vanishing, namely\nThen, we can directly reconstruct the spin field in terms of Φ ± , although it turns out to be a non local field in view of the expressions for A T ± . In this regard, let us observe that, in the one-dimensional completely integrable case, the relation between ζ and the NLS matter field is also analytic in nature, but we need to solve the associated linear spectral problem.\nFurthermore, keeping in mind that the current densities (16) are manifestly gauge invariant, we can give the expressions of the linear and angular momentum for the model (14 -15) in a gauge invariant form.\nFirstly, in opposition to what happens for the electric charge densities, the linear momenta P ± m = J ± m d 2 x, associated with the Φ ± fields separately, are no longer conserved, but they evolve according to Newton's law\nWe notice that both Φ ± are subjected to the same force. Therefore, the total kinetic momentum\nis conserved, the \" − \" comes from the \"negative mass\" associated to Φ − . The total angular momentum M = M + − M − is also conserved , with\nand\nThe quantities l = l + − l − and s = s + − s − are interpreted as orbital and spin angular momentum, respectively. In particular, the general result s = 2 Q 2 suggests that vortices have non zero momentum also in the static case. Finally, since the magnetization M = S d 2 x is a conserved non-local quantity for the LLE system, via Eqs. (38) and (2) we find a further gauge invariant conserved quantity for the system (14 -15) , related to the original global SU (2) symmetry.\nWe are now ready to pass on the quantum theory in a straightforward manner, just by replacing the canonical brackets by i times the commutators (or the anticommutators, if the matter fields are supposed to be fermionic). At the quantum level, the first class constraints Γ i become the operatorsΓ i , which must annihilate the physical states. This fact implies that the quantum states are independent of A 0 and invariant under time-independent gauge-transformations. Therefore, the gauge -invariant operators of the theory must commute withΓ i . Then, recalling the Poisson brackets for the Hamiltonian system described by (36), we can establish the correspondence Φ ± →Φ ± and Φ ± →Φ † ± , with the equal time (anti-) commutation relations\nIn the physical gauge-invariant subspace of the full Hilbert space, the Hamiltonian is given byĤ\nwhere the normal product operator : : and A T a = 8 π ǫ ab ∂ −1 b (Φ † +Φ + −Φ † −Φ − )(x) have been used. The fieldsΦ ± are local and invariant under local gauge transformations generated by the first-class constraints. Therefore, they may be exploited to study the breaking of global simmetries.\nThe quantum version of relations (30) leads to [B(x),η(y)] = 8πiδ 2 (x − y), which implies the relation exp (iη (y))B (x) exp (−iη (y)) =B (x) + 8 π δ 2 (x − y) .\nThis result is used to prove thatΦ ± are the gauge invariant operators which create a charge-solenoid composite, having magnetic flux equal to ±8 π . The Hamiltonian (44) commutes with the particle number operatorŝ\nHence, we can diagonalize them simultaneously via the Fock representation for a quantum state with N = N + + N − particles, which is given by\nwhere the vacuum state is defined byΦ ± |0 >= 0. The function Ψ is an arbitrary element of the Hilbert space L 2 [R 2(N + +N − ) ] obeying the Schrödinger equation for (N + + N − )-bodies. Furthermore, the state (47) is gauge invariant by construction. Thus, we have selected in the full Hilbert space only those states which contain the whole physical information.\nNevertheless, up to now we have dealt with a quite general \"anyonic\" field theory, without using the constraint γ, which in some sense takes into account the properties of the original spin field model. As a consequence, \"quantizing\" Eq. (37) in the formγ\nwe should obtain the physical states by requiring thatγ|N + , N − >= 0. However, for arbitrary occupation numbers (N + , N − ) this equation leads to complicated relations for the wave function Ψ. Here we display the result only for the finite boson case |N + , 0 , namely Ψ x + 1 , · · · , x + N + = F z + 1 , · · · , z + N + i < j\nwhere F is a holomorphic function of its arguments. By using a singular gauge transformation [23][24], the expression (49) takes the form of the Laughlin wave function [25] , describing a condensate state of bosonic solitons, which may be related with quantum disorderd state of the original ferromagnets [12]. Furthermore, it should be proved that the quantum state corresponding to the wave function (49) is actually an eigenvector of the Hamiltonian (44). This work is in progress, and we have strong indications in the positive sense, because at the classical level the Hamiltonian (23) can be written as a linear combination of the constraint γ and Γ 1 . This work was supported in part by MURST of Italy and by INFN -Sezione di Lecce. One of the authors (O. K. P.) thanks the Department of Physics of Lecce University for the warm hospitality."}],"string":"[\n {\n \"title\": \"1.\",\n \"content\": \"Chern-Simons (CS) gauged models have recently attracted a great attention in the description of several phenomena in condensed matter physics as the Fractional Quantum Hall Effect (FQHE) and the High Temperature Superconductivity [1,2]. In this context, models of the Landau-Ginzburg type have been proposed as effective field-theories [3,4]. In the stationary limit, soliton solutions to the self-dual reductions of these models appear. These can be exploited to represent Laughlin's quasiparticles and anyons [5]. On the other hand, the present authors showed that the self-dual CS system can be obtained, in the tangent space representation, from the stationary two-dimensional Landau -Lifshitz equation (LLE) [6]. This is the classical continuous isotropic version of the Heisenberg model, obtained by the discrete quantum version in the framework of the coherent state representation [7]. In particular, soliton solutions of the former system turn out to be related to magnetic vortices of the LLE. More generally, the LLE in 2+1 dimensions takes the form of a constrained nonlinear Schrödinger system (cNLS) for two charged matter fields, coupled to a CS gauge potential [8]. This correspondence suggests an explanation to the phenemenological analogy between the behaviour of magnetic vortices [9] and Hall particles in an external magnetic field [10].\\nSec. 2 is devoted to a brief review of the Landau -Lifshitz model in 2+1\\ndimensions and its tangent space formulation, that is the CS gauged cNLS system. Furthermore, the conservation of some associated physical quantities is discussed. In Sec. 3 we study the lagrangian formulation and the hamiltonian structure of the CS gauged cNLS system. Finally, in Sec. 4 we apply the method of gauge invariant quantization to the above mentioned system.\"\n },\n {\n \"title\": \"2.\",\n \"content\": \"In 2+1 dimensions the LLE for the local magnetization S reads\\nwhere S belongs to the 2-dimensional sphere S 2 . In terms of the stereographic projection\\nEq. (1) becomes\\nWith ferromagnetic boundary conditions, that is S(r, t) → (0, 0, −1) for |r| → ∞, the (x 1 , x 2 ) plane is compactified. Thus S defines maps of S 2 → S 2 , which are partitioned into homotopy classes, labeled by the integer-valued topological charge\\nThe energy of the system takes the form\\nwhich compared with Eq. (4) leads to the Bogomol'nyi inequality W ≥ 4π | Q |. When the equality holds, the field ζ has to be an (anti)holomorphic function of z, leading to the Belavin-Polyakov instantons [11]. In our approach the spin S is represented by the matrix S = S α σ α (α = 1, 2, 3), where σ α are the Pauli matrices. Then, we diagonalize S by resorting to a right-invariant local U (1) transformation g, such that S = gσ 3 g −1 . The matrix g belongs to U (2) and has the form\\nwhere η is an arbitrary real function. The corresponding chiral current J µ (µ = 0, 1, 2) is decomposable into a diagonal and an off-diagonal part:\\nBy definition, the chiral current J µ satifies the zero curvature condition\\nIts off-diagonal and diagonal parts are\\nwhere we have introduced the covariant derivative D µ = ∂ µ − i 2 V µ . In the variables q µ , the equation of motion (1) becomes\\nwhich allows us to eliminate the function q 0 from Eqs. (10)(11). Now, let us introduce the new fields\\nand, correspondingly, the new covariant derivatives D µ = ∂ µ − (i/2) A µ . Then we notice that the expressions of ψ ± and A j in terms of ζ are\\nFor µ = 1 and ν = 2, we see that Eq. (9) takes the form\\nThis equation can be interpreted as a geometrical constraint for the CS gauged NLS system, arising from the remaining equations (9) and Eq. (10), i.e.\\niD 0 ψ ± + D 1 2 + D 2 2 ψ ± + 8π |ψ ± | 2 ψ ± = 0, (15.a)\\nwhere\\nHere we introduce the charge density currents satisfying the continuity equation ∂ µ J ± µ = 0. Then, the matter fields ψ ± carry a conserved electric charge and we can interprete the quantity\\nas the (conserved) total electric charge. In terms of ψ ± it is easy to see that Q and Q coincide, thus we are led to the quantization of the total electric charge. Furthermore, Eq. (15.b) implies the quantization of the magnetic flux: Φ =\\nMoreover, the continuity equations for the currents J ± µ predict that the total number of particles (anyons)\\nis also a conserved quantity. Now, comparing Eq. (5) with (18) , we obtain W = 4πN . Then, in the tangent space formulation, the Bogomol'nyi inequality takes the form |Q | ≤ N . Of course, the equality holds when ψ + , or ψ − , vanishes. In such a case, the system (14 -15) reduces to the self-dual CS model [2]. This model is completely determined in terms of the solutions of the Liouville equation. This simple observation enables us to identify the magnetic vortices, or equivalently the instantons of Belavin-Polyakov type, with the solitons of the self-dual CS model, that is with the anyons. In particular, from (17) we have an explicit correspondence between the magnetic topological charge and the anyon electric charge, whose quantization finds here a geometrical interpretation. In ref. [8] the more general static case (D 1 + iD 2 ) ψ − = (D 1 − iD 2 ) ψ + = 0 is treated. This leads to a special reduction of the euclidean conformal affine Toda system, which contains the sinh-Gordon, or the Poisson-Boltzmann equation, in a particular gauge. Finally, we notice that the constraint (14) is identically satisfied using the expressions (13). Indeed, we find the identity\\nEquating to zero the quantity in the square brackets, we recognize the stationary LLE in the stereographic variable (see Eq. (3)). Thus, all static solutions to Eq.\\n(1) are mapped, modulo gauge transformations, into solutions to the sinh-Gordon reduction studied in ref. [8].\"\n },\n {\n \"title\": \"3.\",\n \"content\": \"System (15) can be derived from the Lagrangian\\nwhere we have used the spinor notation ψ T = (ψ + , ψ − ), withψ = ψ + σ 3 , and the summation over a = 1, 2 is understood. Let us observe that the strength of the selfinteraction and the CS coupling are fixed by the original spin model. The specific ratio between these constants allows us to perform the self-dual reduction, which is a very special subcase in more general and the phenomenological theories of CS type [2]. Moreover, in the effective theories for FQHE the CS coupling is related to the so-called filling factor ν, which in the present case is 1/2, where the units h = c = 1 and e = 1/2 have been used. This result is consistent with the so-called chiral spin fluid model [1 -12]. Moreover, recent experimental observations in double-layer samples indicate the compatibility of ν, with values which are inverse of even numbers [13]. A final remark concerns the parity invariance of the Lagrangian (20), which changes the total sign under the transformation\\nSo that equations of motion and the constraint (14) are both left invariant. This fact is an obvious consequence of the parity invariance of the spin model (1) and it is another example, against the common belief, that the presence of the CS interaction breaks the P and T invariance. Now let us enter into the detail of the hamiltonian structure of the model (14 -15) by resorting to the first order Lagrangian method proposed in ref. [14], which implements the classical Dirac method for constrained Hamiltonian systems [15].\\nFirst, we consider as dynamical variables ξ = ψ + ,ψ + , ψ − ,ψ − , A 1 , A 2 and,\\nThus L ′ describes the system (15 a,c) constrained by Eq. (24), that is the Chern-Simons version of the Gauss law (15.b). In (22) A 0 plays the role of Lagrange multiplier. Furthermore, H would be the density of the Hamiltonian H. In fact, the corresponding Euler-Lagrange equations take the hamiltonian forṁ\\nwhere f ml = ∂a l /∂ξ m − ∂a m /∂ξ l is a nonsingular symplectic matrix and we have defined the Poisson brackets by {ξ m , ξ l } = f −1 ml δ 2 (x − y). Specifically, in our case the nonvanishing Poisson brackets are\\nBy using Eq. (33), one easily shows thatΓ 1 = {Γ 1 , H} = 0; consequently, in phase space the Gauss' law defines an invariant submanifold under the dynamics. On the other hand, Eq. (14) provides the further complex primary constraint γ = 0, whose real and immaginary parts will be denoted by Γ 2 and Γ 3 . By evaluating the Poisson brackets, we find that {γ, H} ≈ 0, i.e.γ is vanishing only on the submanifold defined by (14) or, in other words, Γ 1 and Γ 2 are weakly invariant under time evolution. Furthermore, they do not produce secondary constraints. However, since {Γ 1 , Γ 2 , Γ 3 } is a set of first class constraints (i.e., {Γ i , H} ≈ 0 and det ({Γ i , Γ j }) ≈ 0 ), we need a gauge-fixing condition, for example the Coulomb gauge condition Γ 4 = ∂ i A i ≈ 0, which is the most used in the litterature [16,17,18]. So, we are able to compute the Dirac brackets, defined by\\nIn our case, the nonvanishing matrix elements d ss ′ are distributions in terms of δ 2 (x − y) and its derivatives. Then, inversion can be performed only in implicit form. In order to avoid this type of problem, one can look for a different gauge-fixing condition. For instance, a more general non-local gauge fixing can be choosen [19], or the so-called \\\"axial gauge\\\" [20]. Nevertheless, in all generality the corresponding Dirac brackets are rational in the field variables. Then a quantization of the model seems to be rather difficult, because of the problem arising in the ordering and the inversion of the operators involved. This is one of the main reasons which address us to look for an alternative approach, which is invariant under the choice of the gauge.\"\n },\n {\n \"title\": \"4.\",\n \"content\": \"In this Section, we will adopt a slightly different point of view, in order to tackle the above mentioned problems. Moreover, we would like to understand the quantum meaning of the geometrical constraints γ, as seen separately from the basic CS-NLS field theory. Then, a possible route consists in quantizing system (14 -15) in two steps: 1. first, we apply the procedure of the gauge-invariant quantization to system (15) following the approach suggested in ref. [21]; 2. then, we use the constraint γ to select the physical states among all the gauge invariant quantum states.\\nFirst of all, let us change the hamiltonian description by taking\\nwhere we have as primary constraint Γ 0 ≡ π 0 = 0, π 0 being the momentum conjugated to the canonical variable A 0 , such that {A 0 , π 0 } = δ 2 (x − y). This constraint is implicit in the previous formulation, since ∂L/∂Ȧ 0 = 0. Γ 1 is now a secondary constraint, arising from the Poisson bracket {π 0 , H}. Finally, in the expression (27) f 0 is an arbitrary function, which defines the evolution of A 0 . Furthermore, noticing that the special canonical pair (A 1 , A 2 ) breaks the rotational covariance of the theory, it is more convenient to consider the decomposition of the vector potential into a longitudinal and a transverse part\\nRecalling that the magnetic field is B = ǫ ij ∂ i A j = −∂ 2 χ, we derive formally the useful relation\\nwhere we have defined\\nAnalogously, it is convenient to express ψ ± in terms of the canonical variables (P ± , Q ± ) by\\nHence, the Poisson structure associated with the hamiltonian system given by (27) is\\nFurthermore, the equations of motion are supplemented by the first class constraints Γ 0 ≈ Γ 1 ≈ 0. Now, we are look for a suitable canonical transformation, such that some of the new momenta are equal to the constraints. In such a way, the coordinates canonically conjugated to these momenta have an arbitrary time evolution and are remnants of the gauge invariance of the theory. The remaining canonical variables define the prominent dynamical degrees of freedom of the theory.\\nFirst, let us denote by (Q * ± , P * ± ), (A * 0 , π * 0 ), (η * , π * 1 ) new canonically conjugated coordinates, where the momenta π * i have to be set equal to the constraints Γ i . This can be done through a suitable generating function F = F Q ± , A 0 , η; P * ± , π * 0 , π * 1 , satisfying the set of generalized Hamilton-Jacobi equations\\nThe integrability of this system is assured by {Γ 0 , Γ 1 } = 0, moreover its general solution is given by\\nThe corresponding canonical transformation reads\\nFinally, the new gauge-invariant degrees of freedom are\\nwhich fulfill the canonical brackets\\nIn terms of these new variables, the Hamiltonian density becomes\\n. Furthermore, the constraint γ takes the form\\nwhere ∂ ± = ∂ 1 ± i ∂ 2 and A T ± = A T 1 ± i A T 2 . Now, let us observe that the relations (13) are algebraically invertible in terms of gauge invariant quantities when the topological density is non vanishing, namely\\nThen, we can directly reconstruct the spin field in terms of Φ ± , although it turns out to be a non local field in view of the expressions for A T ± . In this regard, let us observe that, in the one-dimensional completely integrable case, the relation between ζ and the NLS matter field is also analytic in nature, but we need to solve the associated linear spectral problem.\\nFurthermore, keeping in mind that the current densities (16) are manifestly gauge invariant, we can give the expressions of the linear and angular momentum for the model (14 -15) in a gauge invariant form.\\nFirstly, in opposition to what happens for the electric charge densities, the linear momenta P ± m = J ± m d 2 x, associated with the Φ ± fields separately, are no longer conserved, but they evolve according to Newton's law\\nWe notice that both Φ ± are subjected to the same force. Therefore, the total kinetic momentum\\nis conserved, the \\\" − \\\" comes from the \\\"negative mass\\\" associated to Φ − . The total angular momentum M = M + − M − is also conserved , with\\nand\\nThe quantities l = l + − l − and s = s + − s − are interpreted as orbital and spin angular momentum, respectively. In particular, the general result s = 2 Q 2 suggests that vortices have non zero momentum also in the static case. Finally, since the magnetization M = S d 2 x is a conserved non-local quantity for the LLE system, via Eqs. (38) and (2) we find a further gauge invariant conserved quantity for the system (14 -15) , related to the original global SU (2) symmetry.\\nWe are now ready to pass on the quantum theory in a straightforward manner, just by replacing the canonical brackets by i times the commutators (or the anticommutators, if the matter fields are supposed to be fermionic). At the quantum level, the first class constraints Γ i become the operatorsΓ i , which must annihilate the physical states. This fact implies that the quantum states are independent of A 0 and invariant under time-independent gauge-transformations. Therefore, the gauge -invariant operators of the theory must commute withΓ i . Then, recalling the Poisson brackets for the Hamiltonian system described by (36), we can establish the correspondence Φ ± →Φ ± and Φ ± →Φ † ± , with the equal time (anti-) commutation relations\\nIn the physical gauge-invariant subspace of the full Hilbert space, the Hamiltonian is given byĤ\\nwhere the normal product operator : : and A T a = 8 π ǫ ab ∂ −1 b (Φ † +Φ + −Φ † −Φ − )(x) have been used. The fieldsΦ ± are local and invariant under local gauge transformations generated by the first-class constraints. Therefore, they may be exploited to study the breaking of global simmetries.\\nThe quantum version of relations (30) leads to [B(x),η(y)] = 8πiδ 2 (x − y), which implies the relation exp (iη (y))B (x) exp (−iη (y)) =B (x) + 8 π δ 2 (x − y) .\\nThis result is used to prove thatΦ ± are the gauge invariant operators which create a charge-solenoid composite, having magnetic flux equal to ±8 π . The Hamiltonian (44) commutes with the particle number operatorŝ\\nHence, we can diagonalize them simultaneously via the Fock representation for a quantum state with N = N + + N − particles, which is given by\\nwhere the vacuum state is defined byΦ ± |0 >= 0. The function Ψ is an arbitrary element of the Hilbert space L 2 [R 2(N + +N − ) ] obeying the Schrödinger equation for (N + + N − )-bodies. Furthermore, the state (47) is gauge invariant by construction. Thus, we have selected in the full Hilbert space only those states which contain the whole physical information.\\nNevertheless, up to now we have dealt with a quite general \\\"anyonic\\\" field theory, without using the constraint γ, which in some sense takes into account the properties of the original spin field model. As a consequence, \\\"quantizing\\\" Eq. (37) in the formγ\\nwe should obtain the physical states by requiring thatγ|N + , N − >= 0. However, for arbitrary occupation numbers (N + , N − ) this equation leads to complicated relations for the wave function Ψ. Here we display the result only for the finite boson case |N + , 0 , namely Ψ x + 1 , · · · , x + N + = F z + 1 , · · · , z + N + i < j\\nwhere F is a holomorphic function of its arguments. By using a singular gauge transformation [23][24], the expression (49) takes the form of the Laughlin wave function [25] , describing a condensate state of bosonic solitons, which may be related with quantum disorderd state of the original ferromagnets [12]. Furthermore, it should be proved that the quantum state corresponding to the wave function (49) is actually an eigenvector of the Hamiltonian (44). This work is in progress, and we have strong indications in the positive sense, because at the classical level the Hamiltonian (23) can be written as a linear combination of the constraint γ and Γ 1 . This work was supported in part by MURST of Italy and by INFN -Sezione di Lecce. One of the authors (O. K. P.) thanks the Department of Physics of Lecce University for the warm hospitality.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". F A Zee, Prog. Theor. Phys. Suppl. 10777F. A. Zee, Prog. Theor. Phys. Suppl. 107, 77 (1992).","doi":""},{"text":"R E Prange, S M Girvin, The Quantum Hall Effect. New YorkSpringer-VerlagR. E. Prange and S. M. Girvin, The Quantum Hall Effect (Springer-Verlag, New York, 1990).","doi":""},{"text":". S C Zhang, T H Hansson, S Kivelson, Phys. Rev. Lett. 6282S. C. Zhang, T. H. Hansson and S. Kivelson, Phys. Rev. Lett. 62, 82 (1989).","doi":""},{"text":". Z F Ezawa, M Hotta, A Iwazaki, Progr. Theor. Phys. Suppl. 107185Z. F. Ezawa, M. Hotta and A. Iwazaki,Progr. Theor. Phys. Suppl. 107, 185 (1992).","doi":""},{"text":". R Jackiw, S-Y Pi, Phys. Rev. Lett. 642682R. Jackiw and S-Y. Pi, Phys. Rev. Lett. 64, 2969 (1990); 66, 2682 (1990);","doi":""},{"text":". Phys. Rev. D. 423500Phys. Rev. D 42, 3500 (1990).","doi":""},{"text":". L Martina, O K Pashaev, G Soliani, Mod. Phys. Lett. A. 83241L. Martina, O.K. Pashaev, G. Soliani, Mod. Phys. Lett. A 8, 3241 (1993).","doi":""},{"text":"Integrable Pseudospin Models in Condensed Matter. V G Makhankov, O K Pashaev, Sov. Sci. Rev., Math. Phys. 91V. G. Makhankov and O.K. Pashaev, Integrable Pseudospin Models in Con- densed Matter, in Sov. Sci. Rev., Math. Phys. 9, 1 (1992).","doi":""},{"text":". L Martina, O K Pashaev, G Soliani, Phys. Rev. B. 4815787L. Martina, O.K. Pashaev, G. Soliani, Phys. Rev. B 48, 15787 (1993).","doi":""},{"text":"Ferromagnetodynamics, the dynamics of magnetic bubbles, domains and domain wall. T H O'dell, WileyNew YorkT. H. O'Dell, Ferromagnetodynamics, the dynamics of magnetic bubbles, do- mains and domain wall, New York, Wiley (1981).","doi":""},{"text":". N Papanicolau, T N Tomaras, Nucl. Phys B. 360425N. Papanicolau and T. N. Tomaras, Nucl. Phys B 360, 425 (1991).","doi":""},{"text":". A A Belavin, A M Polyakov, JETP Lett. 22245A. A. Belavin and A. M. Polyakov,JETP Lett. 22, 245 (1975).","doi":""},{"text":"A Zee, Physics in (2+1)-dimensions. Y.M. Cho ed.SingaporeWorld ScientificA. Zee, Physics in (2+1)-dimensions Y.M. Cho ed. (World Scientific, Singa- pore, 1992).","doi":""},{"text":"Paired Hall states in double layer electron system. M Greiter, X G Wen, F Wilczek, IASSNS-HEP-92/1Nucl. Phys. 374M. Greiter, X.G. Wen and F. Wilczek, Nucl. Phys. B374, 567 (1992), \"Paired Hall states in double layer electron system\", IASSNS-HEP-92/1","doi":"IASSNS-HEP-92/1"},{"text":". L Faddeev, R Jackiw, Phys. Rev. Lett. 60692L. Faddeev, R. Jackiw, Phys. Rev. Lett. 60,692 (1988).","doi":""},{"text":"P A M Dirac, Lectures on Quantum Mechanics. New YorkYeshiva UniversityP. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964).","doi":""},{"text":". P De Sousa Gerbert, Phys. Rev. D. 42543P. de Sousa Gerbert, Phys. Rev. D 42, 543 (1990).","doi":""},{"text":". T Matsuyama, Phys. Lett. B. 22899T. Matsuyama, Phys. Lett. B 228, 99 (1989).","doi":""},{"text":". C G Han, Phys. Rev. D. 475521C. G. Han, Phys. Rev. D 47, 5521 (1993).","doi":""},{"text":". R Banerjee, A Chatterjee, V V Sreedhar, Ann. Phys. 222254R. Banerjee, A. Chatterjee and V. V. Sreedhar, Ann. Phys.222,254 (1993).","doi":""},{"text":"M Chaichian, N F Nelipa, Introduction to gauge field theory. BerlinSpringer-VerlagM. Chaichian and N. F. Nelipa, Introduction to gauge field theory ( Springer- Verlag, Berlin 1984).","doi":""},{"text":". D Boyanovsky, E T Newman, C Rovelli, Phys. Rev. D. 451210D. Boyanovsky, E. T. Newman and C. Rovelli, Phys. Rev. D 45, 1210 (1992);","doi":""},{"text":". D Boyanovsky, Int. J. Mod. Phys. A. 75917D. Boyanovsky, Int. J. Mod. Phys. A 7, 5917, (1992).","doi":""},{"text":". N K Pak, R Percacci, J. Math. Phys. 302951N. K. Pak and R. Percacci, J. Math. Phys. 30, 2951 (1989);","doi":""},{"text":". N K Pak, Progr. Theor. Phys. 88585N. K.L Pak, Progr. Theor. Phys. 88, 585 (1992).","doi":""},{"text":". F Wilczek, Phys. Rev. Lett. 481144F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982).","doi":""},{"text":". S Girvin, Phys. Rev. Lett. 651489S. Girvin et al., Phys. Rev. Lett. 65, 1489, (1992).","doi":""},{"text":". R B Laughlin, Phys. Rev. Lett. 501395R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).","doi":""}],"string":"[\n {\n \"text\": \". F A Zee, Prog. Theor. Phys. Suppl. 10777F. A. Zee, Prog. Theor. Phys. Suppl. 107, 77 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R E Prange, S M Girvin, The Quantum Hall Effect. New YorkSpringer-VerlagR. E. Prange and S. M. Girvin, The Quantum Hall Effect (Springer-Verlag, New York, 1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S C Zhang, T H Hansson, S Kivelson, Phys. Rev. Lett. 6282S. C. Zhang, T. H. Hansson and S. Kivelson, Phys. Rev. Lett. 62, 82 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Z F Ezawa, M Hotta, A Iwazaki, Progr. Theor. Phys. Suppl. 107185Z. F. Ezawa, M. Hotta and A. Iwazaki,Progr. Theor. Phys. Suppl. 107, 185 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Jackiw, S-Y Pi, Phys. Rev. Lett. 642682R. Jackiw and S-Y. Pi, Phys. Rev. Lett. 64, 2969 (1990); 66, 2682 (1990);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. D. 423500Phys. Rev. D 42, 3500 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Martina, O K Pashaev, G Soliani, Mod. Phys. Lett. A. 83241L. Martina, O.K. Pashaev, G. Soliani, Mod. Phys. Lett. A 8, 3241 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Integrable Pseudospin Models in Condensed Matter. V G Makhankov, O K Pashaev, Sov. Sci. Rev., Math. Phys. 91V. G. Makhankov and O.K. Pashaev, Integrable Pseudospin Models in Con- densed Matter, in Sov. Sci. Rev., Math. Phys. 9, 1 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Martina, O K Pashaev, G Soliani, Phys. Rev. B. 4815787L. Martina, O.K. Pashaev, G. Soliani, Phys. Rev. B 48, 15787 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ferromagnetodynamics, the dynamics of magnetic bubbles, domains and domain wall. T H O'dell, WileyNew YorkT. H. O'Dell, Ferromagnetodynamics, the dynamics of magnetic bubbles, do- mains and domain wall, New York, Wiley (1981).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Papanicolau, T N Tomaras, Nucl. Phys B. 360425N. Papanicolau and T. N. Tomaras, Nucl. Phys B 360, 425 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A A Belavin, A M Polyakov, JETP Lett. 22245A. A. Belavin and A. M. Polyakov,JETP Lett. 22, 245 (1975).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Zee, Physics in (2+1)-dimensions. Y.M. Cho ed.SingaporeWorld ScientificA. Zee, Physics in (2+1)-dimensions Y.M. Cho ed. (World Scientific, Singa- pore, 1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Paired Hall states in double layer electron system. M Greiter, X G Wen, F Wilczek, IASSNS-HEP-92/1Nucl. Phys. 374M. Greiter, X.G. Wen and F. Wilczek, Nucl. Phys. B374, 567 (1992), \\\"Paired Hall states in double layer electron system\\\", IASSNS-HEP-92/1\",\n \"doi\": \"IASSNS-HEP-92/1\"\n },\n {\n \"text\": \". L Faddeev, R Jackiw, Phys. Rev. Lett. 60692L. Faddeev, R. Jackiw, Phys. Rev. Lett. 60,692 (1988).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P A M Dirac, Lectures on Quantum Mechanics. New YorkYeshiva UniversityP. A. M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P De Sousa Gerbert, Phys. Rev. D. 42543P. de Sousa Gerbert, Phys. Rev. D 42, 543 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Matsuyama, Phys. Lett. B. 22899T. Matsuyama, Phys. Lett. B 228, 99 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C G Han, Phys. Rev. D. 475521C. G. Han, Phys. Rev. D 47, 5521 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Banerjee, A Chatterjee, V V Sreedhar, Ann. Phys. 222254R. Banerjee, A. Chatterjee and V. V. Sreedhar, Ann. Phys.222,254 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Chaichian, N F Nelipa, Introduction to gauge field theory. BerlinSpringer-VerlagM. Chaichian and N. F. Nelipa, Introduction to gauge field theory ( Springer- Verlag, Berlin 1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Boyanovsky, E T Newman, C Rovelli, Phys. Rev. D. 451210D. Boyanovsky, E. T. Newman and C. Rovelli, Phys. Rev. D 45, 1210 (1992);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Boyanovsky, Int. J. Mod. Phys. A. 75917D. Boyanovsky, Int. J. Mod. Phys. A 7, 5917, (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N K Pak, R Percacci, J. Math. Phys. 302951N. K. Pak and R. Percacci, J. Math. Phys. 30, 2951 (1989);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N K Pak, Progr. Theor. Phys. 88585N. K.L Pak, Progr. Theor. Phys. 88, 585 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Wilczek, Phys. Rev. Lett. 481144F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Girvin, Phys. Rev. Lett. 651489S. Girvin et al., Phys. Rev. Lett. 65, 1489, (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R B Laughlin, Phys. Rev. Lett. 501395R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":54,"cells":{"text":{"kind":"string","value":"\nGravity from strings : personal reminiscence on early developments *\n31 Dec 2008 January, 2008\n\nTamiaki Yoneya \nInstitute of Physics\nUniversity of Tokyo Komaba\nMeguro-ku153-8902TokyoJapan\n\nGravity from strings : personal reminiscence on early developments *\n31 Dec 2008 January, 2008@\nI discuss the early developments of string theory with respect to its connection with gauge theory and general relativity from my own perspective. The period covered is mainly from 1969 to 1974, during which I became involved in research on dual string models as a graduate student. My thinking towards the recognition of string theory as an extended quantum theory of gravity is described. Some retrospective remarks on my later works related to this subject are also given. * Invited contribution to the book \"The Birth of String Theory\".\n\n1. Prologue : an encounter with the dual string model I entered the graduate school of Hokkaido University, Sapporo, in April, 1969. My advisor, Akira Kanazawa who was an expert in dispersion-theoretic approach to strong interactions, proposed having a seminar on Regge pole theory regularly every week together with a few other students. Comparing with various marvelous theories which I had learnt during undergraduate studies, however, the Regge pole theory was somewhat disappointing for me. I felt that it was too formal and phenomenological in its nature without much physical content, except for interesting mathematical physics related to the notion of complex angular momentum. Looking for some more favorable topics, I became interested in studying the quantum field theory of composite particles, which, I thought, might be useful to explain the Regge-pole behavior from the dynamics of fundamental particles. I read many papers related to this problem such as those on compositeness criteria, on the definition of asymptotic field for a composite particle, and especially on the Bethe-Salpeter equation. Although I felt that this subject was not yet what I really would want to pursue, I learnt much about quantum field theory by studying these papers.\n\nWhile still seeking subjects for my reseach, some senior students told me that there was actually a spectacular new development trigged off by a proposal made a year ago by Veneziano [Ven68]. After reading the paper of Veneziano and some others which extended the Veneziano amplitude to various directions, I gradually became convinced that this had to be the subject I should choose. In particular, when I was exposed to a short but remarkable preprint by Susskind [Sus69] giving a physical interpretation of the Veneziano formula in terms of vibrating strings (or 'rubber band' in Susskind's terminology), I was struck by the simplicity of the idea. My interest on dual models was further strengthened by reading the paper by Nambu [Nam70], containing similar discussions. In particular, I was intrigued by the concept of 'master field' (in today's language, string field), which had been emphasized by Nambu. A little later I also came to be fascinated by the very attractive world-sheet picture of Fairlie and Nielsen [FN70].\n\nIn spite of encountering such stunning ideas, my interests in dual models remained mostly within the realm of more formal aspects of dual models during my years in graduate school, as was perhaps common among many other young students who became interested in this field in the same period. By the beginning of the autumn of 1969, I started to study various available preprints, which were very rapidly growing. They included works on factorization, operator formalism, and unitrarization program.\n\nOne of the papers which I was most interested in during this period was the one by Fubini and Veneziano [FV70]. They introduced an extremely elegant formulation of factorization and duality for general n-point amplitudes in terms of a vertex operator : exp ik µ Q µ (z): for tachon as the ground state of the dual string. I thought that this formalism not only was suggestive from the physical viewpoint of vibrating strings, but also was quite important in making manifest the key symmetry structure, Möbius symmetry or SL(2, R) invariance, underlying the basic duality property of the amplitudes as had been exhibited by the Koba-Nielsen representation [KN69]. On the other hand, from the viewpoint of the unitarization program which had been initiated in a remarkable work by Kikkawa, Sakita and Virasoro (KSV) [KSV69], it was a crucial step to construct amplitudes or vertices with the most general external lines corresponding to general excited states of strings. Attempts towards such 'Reggeon' vertices were started by Sciuto [Sci69]. During the years from 1969 to 1971, there appeared a large number of papers discussing this problem.\n\nIn the Japanese system of graduate school, we usually have to present a thesis in order to finish the first two years, 'master course', of graduate study before entering the next stage, 'doctor course', of three years. As the subject of my master thesis, I decided to try, by utilizing the symmetry structure uncovered by the Fubini-Veneziano formalism, to establish a formalism of the Reggeon vertices such that the duality symmetry and factorization property were exhibited as manifestly as possible. I realized that a similar direction had been pursued in some papers that I became aware of, such as those by Lovelace [Lov70] and Olive [Oli71]. I was able to achieve such a goal by adopting a general method proposed earlier by Shapiro [Sha70b] for factorizing the generalized Veneziano amplitudes. I applied this method to a multiple factorization of the Fubini-Veneziano's opertorial form of the n-point amplitudes. The resulting general n-point Reggeon vertex enjoyed desired symmetry properties with respect to gauge, twisting, factorization and duality transformations. It also contained the previous results due to Lovelace and Olive as special cases. This small work [Yon71] became my first full paper. Through this work, I acquired a certain personal confidence in engaging myself further in this research field, although I was working alone in a way almost completely isolated from the centers of dual model research. After many years, I met C. Montonen at the Strings 2002 conference in Cambridge. According to him, my work had some influence on his work [CM72] with E. Corrigan in which they discussed a group theoretical formulation of the Reggeon vertices. In fact, I remember that my preprint had been cited in their paper. It was an unexpected surprise for me to have an enjoyable conversation related to such old works more than three decades later.\n\n\nConnection of dual models to field theories\n\nFrom these early days of my studies on dual models, there had been one basic question which was increasingly occupying my mind. That was on the relationship between dual models and ordinary field theory. In characterizing the Veneziano amplitude, it had been emphasized that the amplitude could be expanded into a sum over an infinite set of either s-channel poles or t-channel poles, but not of both. Adding both would amount to doublecounting. This is of course the statement of channel duality, which had motivated the original proposal of the Veneziano amplitude. However, the unitarization program started by the KSV paper seemed to be an extension of the usual Feynman-graph expansion in ordinary field theory, in which we have to sum both s-and t-channel diagrams. One of my questions was whether it would in principle be possible to find a field theory from which such an extended dual Feynman-graph expansion would be derived, perhaps by introducing some appropriate interactions for Nambu's master field.\n\nAnother related question was on the interpretation of massless spinning excitations. The existence of such states in dual string models is an inevitable consequence of conformal symmetry associated with the Virasoro operators. In contrast to this, the existence of massless spinning fields in the framework of ordinary field theory is related to local gauge symmetries. During my master-course years, I became strongly interested in the local gauge principle, through reading the original papers by Yang and Mills and by Utiyama, in addition to those related to composite particles as mentioned above. Because my main interest had been in strong interactions, I was naturally seeking papers aiming applications of the local gauge principle to strong interactions. In regard to later works following the above classical works, the paper most intriguing to me was the one by J. J. Sakurai [Sak60]. I was deeply impressed by a strong persuasive power of this work. I thought that if the local gauge symmetry could be such a fundamental principle for strong interation, it should also play some role in dual models, and then the existence of massless spinning states in the latter had to be a clue.\n\n\nA gauge theory interpretation of massless spinning states\n\nAfter submitting my master thesis in the beginning of 1971, I decided to pursue the second question above as my next theme. I attempted first to interpret the interaction vertices of the massless vector states of open strings in terms of a local gauge transformation for string wave functions. I soon realized that all those vertices in the Veneziano, Neveu-Schwarz [NS71] and Ramond [Ram71] models were precisely obtained by a minimal substitution p µ (σ) → p µ (σ) − gA µ (x(0))δ(σ) with A µ = ∂ µ A µ = 0 corresponding to a particular class of local-gauge transformations on the master field |Ψ , such as\n|Ψ → e igλ(x µ (0)) |Ψ (2.1)\nunder an on-shell condition λ(x) = 0 for the gauge function λ(x µ ), where x µ (0) is the string coordinate at an end point σ = 0 of open strings. Namely, I showed that the vertices obtained from the Virasoro or super-Virasoro operators in this way coincided with those obtained by factorizing the pole residues at corresponding massless poles. I interpreted this simple fact as clear evidence for the fact that these massless vector states themselves should be regarded as gauge bosons and that the world-sheet conformal symmetry was the mechanism which made the local gauge symmetry intrinsic to dual strings. In modern language, this formulation is of course equivalent to the vertex operator introduced as a deformation of the world-sheet action by background gauge field at the boundary of open-string world sheets. When I was pursuing this idea, I became aware of some prior works which were trying to apply local gauge principles to dual models, such as those by Kikkawa-Sato [KS70] and by Manassah-Matsuda [MM71]. However, the standpoint common to these two works was entirely different from my viewpoint, in the sense that they were aiming toward constructing off-shell currents which would make possible the coupling of electromagnetic and weak interactions to dual strings. Hence, in their works, the gauge bosons coupled to the Veneziano amplitudes were nothing to do with the massless spinning states of dual strings themselves. I wrote up a paper on my result, emphasizing that the conformally invariant dual models of open strings should be regarded as embodying local gauge principle intrinsically, rather than interpreting the latter as some external structure to be artificially adjoined to the dual models. I submitted it to Nuovo Cim. Unfortunately this manuscript was rejected for the reason, as far as I remember, that the vertices discussed in it were not new. This rejection disappointed me. However, to my knowledge at that time, no one had expressed the same viewpoint as mine on the meaning of local gauge principle in string theory.\n\nA few months after this experience, I received a letter commenting on my preprint on the gauge principle in dual models from M. Minami, who was at the Research Institute of Mathematical Sciences (RIMS) in Kyoto and was actively engaged in dual models. To his letter, there was an attached copy of a preprint by Neveu-Scherk [NS72] on the application of Scherk's work [Sch71] on the zero-slope limit to the dual pion model of Neveu and Schwarz, making a connection to gauge theory. Minami also mentioned some works by Nakanishi [Nak71] on a crossing-symmetric decomposition of the integral representation of the Veneziano-type amplitudes in such a way that each decomposed piece corresponds to a usual Feynman graph only with pole singularities in a single definite set of channels. All these works which I had not been aware of until that time were a big surprise to me. As for the rejected manuscript which I was almost trying to forget, I decided to rewrite it in a more compact form. I sent it to Prog. Theory. Phys. [Yon72a] in which I added an acknowledgement to Minami.\n\nMinami's letter encouraged me to pursue my interpretation further. I immediately hit upon an idea of extending it to the case of closed strings, namely to the Virasoro-Shapiro amplitudes [Vir69] [Sha70a]. However, making a connection to gravity seemed to be a rather bizarre thing to do, since at that time everyone thought that the dual string model was aiming at the construction of a definitive theory for hadrons and their strong interactions. Actually, during my undergraduate years, I had been quite serious about Einstein's general relativity theory and had studied even some of the original papers. In fact, Einstein had been basic impetus to me, since I had started out on trying to comprehend his book 'The meaning of relativity' during my high school day. However, after entering the graduate school and being influenced by the atmosphere of the particle physics community, I became quite strongly prejudiced against general relativity, regarding it as a sort of a relic of the past. In the case of the vector gauge principle, I was able to free myself from similar prejudice by reading the Sakurai paper and others. I hoped that the connection to gauge theory would be useful in trying to give masses to the unwanted massless states, in the way ρ mesons were related to the gauge principle as Sakurai had advocated. But in the case of gravity, to my knowledge at that time, there had been no such attempt. It took another year for me to come back to this idea again.\n\nExtension of the Nakanishi decomposition and a desperate attempt toward string field theory Studying Nakanishi's beautiful papers on the decomposition of the integral representation for the n-point dual amplitudes, I gradually returned to my first question concerning the relationship between dual models and field theory. From the viewpoint of my question whether the dual amplitudes could be derived from some field theory, I thought that Nakanishi's way of decomposing the amplitudes had to be one of the necessary steps. Then, I realized that what I had done in my master thesis would fit quite nicely in oder to reformulate Nakanishi's result in such a way that the rule of decomposition is expressed as a generalized Feynman-like rule. That would make it possible to extend the decomposition to arbitrary loop diagrams. It had been well known that a naive application of the operator formalism, in the case of non-planar diagrams, led to a divergence difficulty [KT70], known as the 'periodicity' problem, of integrating over one and the same contribution infinitely many times. I was able to confirm that the rules I had derived on the basis of my previous Reggeon-vertex paper indeed resolved this difficulty as expected, because my new Feynman-like rule automatically gave a definite prescription for the integration region for arbitrary diagrams. Also this rule involving only a generalized 3-point vertex naturally reduced to that of the φ 3 theory in the zero-slope limit discussed by Scherk. I was certainly excited by this result, since this strongly suggested that the rule I had derived might correspond precisely to the Feynman rule of a desired string field theory. When I was finishing the paper about this result in May 1972, I was staying about a month at the Yukawa Institute in Kyoto, having discussions on this and related problems with Nakanishi at RIMS. As soon as I completed the paper [Yon72b] on this result in Kyoto, I turned to attempt constructing such a formalism. Seeking some hint for a new field theory underlying the dual string models, I studied some old works on infinite-component field theories and especially non-local field theories, including papers by Yukawa [KY68]. Unfortunately, however, I did not get really useful insight, since most of those works had been essentially restricted to free theories. In addition to this, the rule I had presented was using a very cumbersome language, which had inherited from Nakanishi's original method and my own formulation of the general Reggeon vertex, and hence did not allow simple interpretation in terms of more intuitive world-sheet picture. What I was trying to do is equivalent to obtaining a covariant triangulation of the moduli space of Riemann surfaces with boundaries and punctures such that it corresponds to a second-quantized field theory of interacting open strings. More than a decade later in 1985, Witten achieved this goal [Wit86] in his proposal of a covariant open string field theory, on the basis of the BRST operator formalism.\n\nA version of fully interacting string field theories first appeared on the scene in 1974, when Kaku and Kikkawa published their work [KK74]. Their formulation was based on Mandelstam's light-cone representation [Man73] of the dual amplitudes which had appeared about a year later after my failed attempt towards string field theory. The lightcone string field theory provided another possible, albeit non-covariant, triangulation of the moduli space of Riemann surfaces.\n\n\nConnection of dual models to general relativity\n\nBy the failure of my desperate attempt toward string field theory, I was quite exhausted. To recover from a depressed mood which I fell into, I had to spend almost half a year from the summer of 1972, wandering about various other topics. From about this period onward, many people were departing from the dual string models. But I had some feeling that the dual models themselves contained something more to be uncovered, despite their immediate inapplicability to the physics of hadronic interactions. I was also influenced by a rapid resurgence of gauge field theories, which had been initiated by 't Hooft's famous work on the renormalizability of non-abelian gauge field theories. I was now thinking seriously about my previous idea on a possible connection between dual string models in the case of closed strings and general relativity. In studying the progress in gauge field theories, I was gradually freeing myself from the prejudice on general relativity. I thought that as the Yang-Mills theory had turned out to be useful in this way, general relativity might also become important for particle physics someday in some unexpected way. I also felt that it was good for me to return, now in the last stage of my graduate study, to general relativity which after all had been the prime impetus toward theoretical physics for me.\n\nMy thinking was something like this: One of the useful viewpoints on dual string models which had been made clear by Scherk's work was that the slope parameter α ′ should be treated as a fundamental constant as the Planck constant h had been for quantum theory. It would be very exciting if we had some kind of concrete correspondence principle between dual string models and general relativity, in an analogous way as in the case of the relation between, say, the canonical commutation relation in quantum mechancis and the Poisson bracket in classical mechanics. In any case, I had first to study a nontrivial example of the zero-slope limit for the Virasoro-Shapiro amplitudes. I decided to compute a 4-point amplitude involving two massive scalars and two gravitons, since the coupling of graviton to scalars was certainly the simplest case, as had been treated in most of old literatures which I had come across in this subject. All such papers of course only discussed the case of 4 dimensions, and I was not attempting to generalize them to critical dimensions, since for discussing only the properties of massless modes of strings without loop corrections we could restrict ourselves to lower dimensions. Consequently, the Newton constant was identified to be proportional to g 2 α ′ with g being the 3-point coupling constant of closed strings. Staying in 4 dimensions made possible for me to assign an arbitrary (real) mass to the scalar (tachyonic in critical dimensions) ground state treated as a matter field coupled with gravity, using additional components of momentum along the extra dimensions. In the early summer of 1973, I decided to publish the result [Yon73] of such a computation first in a letter form.\n\nIn parallel to this computation, I was pursuing a question on how to establish the correspondence to all orders with respect to graviton interactions in the tree approximation, or in other words to the tree amplitudes with arbitrary number (= n) of gravitons and with a fixed number (=2) of massive scalars. Since general relativity is a non-polynomial field theory, I needed a certain general theorem for the purpose of establishing the connection in a recursive way on the basis of my explicit computation of the (2+2)-point amplitude. I found a paper which was quite suitable for this aim. That was the one by W. Wyss [Wys65] who had given a very clear argument on how the non-polynomial Einstein Lagrangian is obtained, strarting from the lowest order coupling between a massive scalar field and a massless spin 2 field.\n\nThe second and more ambitious question was, as alluded to already, to find an appropriate correspondence principle which would lead from general relativity to the dual strings in a more or less unique way. An ideal thing which I was dreaming of would be to find something analogous to what had been happening in the discovery of quantum mechancis. I was also recalling P. Ramond's argument [Ram71] in his proposal of the dual fermion model as a generalization of the Dirac equation. Unfortunately, however, I was not able to find such a correspondence principle in any satisfactory form. I could give only a modest discussion, which was related to a generalization of the equivalence principle, about the generating functional for the S-matrix for (n + 2)-point scatterings: The S-matrix on the string side can be obtained from that of field theory by making a simple reinterpretation for the space-time energy momentum tensor of a massive scalar field.\n\nThe third point I was thinking about was a question whether there could exist an extension of the Nakanishi decompostion to the general Virasoro-Shapiro amplitudes. That would make it possible to connect both sides in a more direct way by comparing the Feynman rules on both sides. I failed to arrive at really useful conclusion on this question, except for an expectation that in this case the decomposition would require most probably an infinite number of higher-point vertices as we go to the higher-point graviton amplitudes. † Although these failures somewhat disappointed me, I decided to write a full paper on the connection between the dual closed strings and general relativity. The title of the paper that I chose was 'Connection of Dual Models to Electrodynamics and Gravidynamics', which was sent to Prog. Theor. Phys. [Yon74a] in the early autumn of 1973. The reason why I put also the word 'Electrodynamics' was that I included in the paper an argument for the similar case of the open-string scattering of n massless vectors and 2 massive scalars. That was intended for the purpose of emphasizing the similarity and, simultaneously, the contrast of the arguments between electrodynamics and gravidynamics, especially with respect to the second question above.\n\nI remember that, while writing this paper, my mind was in a somewhat perplexed state. One of the reasons was that I was not completely sure about my standpoint in interpreting the connection of the dual models to gravity. After my first seminar on my paper in our high-energy laboratory, a senior student (S. Araki) privately asked me something like this ‡ ; \"So, are you claiming that the dual model is more fundamental than the Einstein theory, replacing the latter by the dual model?\" I was slightly upset by this pressing question. As far as I remember, my answer was \"Well, as I have said, I could not find any definite principle, relating the dual model to general relativity. If I could have provided such a deeper explanation, I might have a right of claiming that. All what I can say is simply a fact that we can extract general relativity from the dual model in this way.\" Rather than thinking toward the application of my results to the real world, I was mostly occupied by a desire to understand a deeper reason for why the theory of relativistic strings happens to automatically encompass general relativity, the latter being based on such deep principles as the equivalence principle and general covariance. I confess that I did not have sufficient courage to make such a bold statement, since I had been too strongly worried by lack of fundamental principles for explaining the emergence of gravity from strings.\n\nFurthermore, my mind was also entangled with another competing viewpoint on the relationship between the dual string models and field theory. Already in 1970, there had been suggestive discussions on planar 'fishnet' diagrams by Sakita-Virasoro [SV70] and Nielson-Olesen [NO70]. § Their elegant observation had also been quite attractive to me as another possible attitude towards an understanding of the relation between dual models and local field theories. Moreover, in 1973, Nielsen and Olesen had published an interesting pair of papers [NO73a] [NO73b] in which they discussed the possibility of embedding relativistic strings in local field theories, either as a gauge theory with a nonlinear field equation corresponding to the lagrangian − √ F 2 /2α ′ or as a vortex solution to a gauge-Higgs system, respectively. I became aware of these preprints just about the time when I was working on the connection of dual models to gravity on the basis of the zeroslope limit. These works were suggesting that the dual string might well be a consequence from non-linear gauge-field theories, resulting from their nontrivial dynamics. This is a view which is completely opposite to the derivation of field theories from the dual models in the zero-slope limit. I was asking myself, \"Does this then imply that general relativity can be contained in gauge theory?\".\n\nI have not been able to resolve this puzzle for many years. When I was invited in 1975 from the journal 'Butsuri', a periodical (in Japanese) of the Physical Society of Japan, to write a brief report [Yon76a] on the recent development of dual models with respect to the relationship between the dual models and field theory, I began the article as: \"Since the dual resonance model has started out without any explicit connection to field theory, it has been one of the basic questions how the relationship between the dual model and field theory should be understood. There are two possible answers. One is that the dual ‡ I cannot confirm this to Araki, since unfortunately he passed away in 1994.\n\n§ These works were precursor to the seminal work [THo74] of 't Hooft on planar diagrams in the large N limit.\n\nstring is something which is beyond the framework of ordinary local field theory, and the latter should be interpreted as a particular limiting case of the dual string model. Another possibility would be that although the dual model cannot be understood using the traditional perturbative methods of field theory, they may be properly derived from field theory by means of some non-perturbative treatments of its complicated non-linear dynamics. In the present report, I will discuss the recent developments from the first viewpoint, relegating the second to other opportunities.\"\n\nIn the next year, several months after the submission of my paper, I received a preprint by Scherk and Schwarz [SS74a] (see also [SS74b]) in which they had studied the zero-slope limit of closed strings. In contrast to my approach of studying the coupling of gravitons to an artificial massive scalar state mainly for a methodological reason, they treated the amplitudes with only massless states. In 1975, furthermore, they made a bold proposal [SS75] that the dual string models should be interpreted as a unified theory of quarks and gluons. I was quite impressed by their works, especially by their clear attitude with respect to the interpretation of the connection of the dual models to gravity.\n\nAt any rate, there were several reactions to my work. In the beginning of 1974, I was invited to give a talk in a workshop organized by the GRG (general relativity and gravitation) group in Japan, and I met some of the leading Japanese physicists working in this field, including R. Utiyama. In the early summer of the same year, I received a letter from Y. Hara who was then staying in Europe, telling me that the connection of dual models to gravity was discussed by D. Olive in an international conference in London. Hara encouraged me to go to the United states for my postdoctoral studies. Subsequently, I received a letter from B. Sakita asking me whether I was interested in joining his group at the City College, New York. I also met K. Kikkawa who came back to Japan from there. Very stimulating conversations with him were most encouraging. Since I was actually about to be offered a faculty position at Hokkaido University, I joined Sakita's group, from 1976 to 1978 during which I met A. Jevicki and S. Wadia, after fulfilling my teaching duty at Hokkaido for one and a half years. In 1980, I moved to the Komaba campus of the University of Tokyo.\n\nIn these periods, my interest in dual strings was almost completely diminishing, because I was sparked by the fascinating development of gauge theories, especially by various new discoveries in non-perturbative aspects of gauge theories, such as magnetic monoples, instantons and lattice gauge theories. I became interested more and more in the dynamics of non-Abelian gauge theories, especially in the question of quark confinement, on which I was mainly working for a few years from the late 70s to the early 80s, though my papers at that time included actually an attempt [Yon81] to derive a version of string field theory as an effective theory of hadronic strings directly from lattice Yang-Mills theory: In fact, I considered a generalized gauge transformation, which was not so dissimilar to (2.1), of string field associated with Wilson loops to apply it for a reformulation of topological structure of the algebra of Wilson and 't Hooft loops. Then from the late summer in 1983, I stayed at CERN for one year. During my stay at CERN, M. Green made a visit and gave a seminar. Green's talk was quite stimulating to me, especially through his enthusiasm for string theory. As far as I remember, he gave a review of his works with Schwarz, but not yet on anomaly cancellation. From about this time, I became gradually inclined again to string theory, and the impact of their epoch-making work [GM84] on the anomaly cancellation brought me back to this field in the mid 80s. In fact, the works which I published during my stay at CERN were on the Liouville field theory and 2 dimensional gravity, close cousins of string theory.\n\n\nFurther related works\n\nBefore concluding this section, let me describe some of my further works, which are directly related to my earlier papers. After submitting the first full paper on gravitystring connection in 1973, I continued for sometime to think again about the possibility of establishing a Nakanishi-type decomposition for closed strings, since I thought that my discussion in connecting the closed string to general relativity should be improved to the level of a comparison between Feynman on both sides. In particular, I wanted to understand more clearly the mechanism of generating higher order interaction terms directly from the effect of massive excited states of the closed string. But convincing myself that this task was too difficult, I soon turned myself to more modest problems. First I went back to the connection of open strings to non-Abelian gauge theory, and decided to establish a detailed connection at the level of Feynman rules [Yon74b] on the basis of my previous work on the general Reggeon vertex and the Nakanishi decomposition.\n\nThen, after a mathematical work [Yon75a] on an extension of the Fubini-Veneziano operator formalism using superfields which natually arose by rewriting the Fairlie-Martin representation for the Neveu-Schwarz model, I also studied the coupling of graviton and other massless closed-string states to fermionic open strings. That was an extention of the formalism which I had developed earlier in the case of massless vectors. I showed [Yon75b][Yon76c] that the vertex operators for these states were obtained by means of an extended local gauge principle similar to the ordinary tetrad formalism of local Lorentz gauge symmetry for the Dirac field, and discussed the associated geometric theory in the zero-slope limit. In particular in [Yon76c], I made a conjecture on how these properties would be realized in full-fledged string field theories. Ten years later, I showed the validity of this conjecture, when I discussed the symmetry structure associated with massless closed string states in the light-cone string field theory [Yon85] [Yon87a]. In fact, this result could have been obtained in the mid 70s if I had been devoting myself more seriously to the problem.\n\nAfter sending the paper on this subject to Phys. Rev. Lett., I received a preliminary draft of the astonishing paper by Witten on a covariant open-string field theory [Wit86], in which he gave an impressive answer to the problem against which I had struggled in vain more than 10 years ago. But I was pleased to see that he had cited my latest work on the light-cone string field theory. The connection of Witten's open string field theory to the Nakanishi decomposition was noticed in a later paper by Giddings and Martinec [GM86].\n\nThe above idea behind the paper [Yon85] is also related to my conjecture of a purely cubic action which I have proposed [Yon87b] in the spring of 1986 at the ICOBAN '86 international workshop on Grand Unification, held at the city of Toyama located near the site of the Kamiokande neutrino detector. In this workshop, there were also other talks on string theory, including those by D. Gross and A. Strominger. Actually I had discussed this conjecture earlier in a workshop in Kyoto in the beginning of the same year or perhaps in the end of 1985. To my surprise, after only one or two months from the Toyama workshop, I received preprints on possible realizations of my conjecture by Santa Barbara group [HLTS86] and by Kyoto group [HIKK86], both mentioning my previous talk in Toyama.\n\nFinally, let me also mention that the expectation of a hidden correspondence principle, which I had been seeking without success in the 70s for characterizing the departure of string theory from general relativity, was basically the motivation for my later proposal [Yon87c] [Yon89] in 1987 of an uncertainty relation for space-time. A decade later, this idea turned out to be of some relevance for a qualitative understanding [LY97] of the characteristic scales of D-brane dynamics. I have written several papers [Yon00] related to this subject during years around the mid 1990s to the early 2000s.\n\n\nEpilogue\n\nRecalling what I was thinking during my student years from 1969 to 1974, I find somewhat surprisingly how often my later works have been rooted in this period: I am still in quest for better formulation of strings and D-branes along various lines which germinated, at least partially, from these early ideas. I was working almost alone, but with occasional kind help from other workers and, especially, in an environment of great progress which was occuring in both string and gauge theories. I was fortunate in being able to start working in such a dramatic period of particle physics.\n\nAfter more than three decades passed, the developments in string theory in the recent 10 years through the conception of the gravity-gauge correspondence (or AdS/CFT correspondence) are now throwing new light on my old perplexing question, the question of whether 'from general relativity to gauge theory' or 'from gauge theory to general relativity'. With respect to this puzzle, personally, a work [OY98] which I did with Y. Okawa in 1998 on a derivation of the gravitational 3-body force for D0 branes from the D0 gauge theory (or M(atrix) theory [BFSS97]) was a small revelation to me. This work enabled me to convince myself that general relativity may indeed be contained in the quantum dynamics of gauge field theories, when interpreted as the theory for D branes.\n\nIn a sense, as if we are watching a revolving stage, the string theory came back to a scene, with which it had started in the early 70s, and is providing an entirely new perspective on gauge theory and even on hadron physics. However, we had to admit that we are still far from grasping fundamental principles behind string theory. Hopefully, we will see further dramatic turns in the next 30 years.\n\nI would like to thank the organizers for inviting me to participate in the meeting \"The birth of string theory\" and in the present book. I also would like to thank S. Wadia for useful comments on the manuscript.\n † Many years later, I have presented some considerations which I had made at that time along this direction in a review talk[Yon86] on string field theory in a workshop held at Komaba in 1986. In the late 80s, essentially the same problem was discussed by several groups in attempting to extend Witten's covariant open string field theory to closed strings. A systematic discussion along this line was given by Zwiebach[Zwi92].\n\nM theory as a matrix model: A Conjecture. T Banks, W Fischler, S Shenker, L Susskind, Phys. Rev. 555112Banks T., Fischler W., Shenker S. and Susskind L. (1997), \"M theory as a matrix model: A Conjecture\", Phys. Rev. D55, 5112.\n\nGeneral dual operatorial vertices. E Corrigan, C Montonen, Nucl. Phys. 3658Corrigan E. and Montonen C. (1972), \"General dual operatorial vertices.\", Nucl. Phys. 36, 58.\n\nHarmonic-oscillator analogy for the veneziano model. S Fubini, G Veneziano, Nuovo Cimento 67A. 29Fubini S. and Veneziano G.(1970), \"Harmonic-oscillator analogy for the veneziano model\", Nuovo Cimento 67A , 29.\n\nS B Giddings, E Martinec, Conformal Geometry and String Field Theory. 27891Giddings S. B. and Martinec E. (1986), \"Conformal Geometry and String Field Theory\", Nucl. Phys. B278, 91.\n\nAnomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory. M Green, J H Schwarz, Phys. Lett. 149117Green M. and Schwarz J. H. (1984), \"Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory\", Phys. Lett. 149B , 117.\n\nPregeometrical string field theory: creation of space-time and motion. H Hata, K Itoh, T Kugo, H Kunitomo, K Ogawa, Phys. Lett. 175138Hata H., Itoh K., Kugo T., Kunitomo H. and Ogawa K.(1986), \"Pregeometrical string field theory: creation of space-time and motion\", Phys. Lett. 175, 138.\n\nA Purely Cubic Action for String Field Theory. G T Horowitz, J Lykken, R Rohm, A Strominger, Phys. Rev. Lett. 57283Horowitz G. T., Lykken J., Rohm R. and Strominger A. (1986), \"A Purely Cubic Action for String Field Theory\", Phys. Rev. Lett. 57 , 283.\n\nM Kaku, K Kikkawa, The Field Theory of Relativistic Strings. 101110Pt. 1. TreesKaku M. and Kikkawa K. (1974), \"The Field Theory of Relativistic Strings, Pt. 1. Trees\", Phys. Rev. D10, 1110.\n\nUnitary nonplanar closed loops. M Kaku, C B Thorn, Phys. Rev. 12860Kaku M. and Thorn C. B. (1970), \"Unitary nonplanar closed loops\", Phys. Rev. D1, 2860.\n\nManifestly crossing invariant parametrization of n meson amplitude. Z Koba, H B Nielsen, Nucl. Phys. 12517Koba Z. and Nielsen H. B. (1969), \"Manifestly crossing invariant parametrization of n meson amplitude\", Nucl. Phys. B12, 517.\n\nFeynman-like diagrams compatible with duality. I: Planar diagrams. K Kikkawa, B Sakita, M A Virasoro, Phys. Rev. 1841701Kikkawa K., Sakita B. and Virasoro M. A.(1969), \"Feynman-like diagrams compatible with duality. I: Planar diagrams\", Phys. Rev. 184, 1701.\n\nNonhadronic interactions in the dual resonance model. K Kikkawa, H Sato, Phys. Lett. 32280Kikkawa K. and Sato H.(1970), \"Nonhadronic interactions in the dual resonance model\", Phys. Lett. 32B, 280.\n\nD particle dynamics and the space-time uncertainty relation. M Li, T Yoneya, Phys. Rev. Lett. 781219Li M. and Yoneya T. (1997), \"D particle dynamics and the space-time uncertainty relation\", Phys. Rev. Lett. 78, 1219.\n\nSimple n-reggeon vertex. C Lovelace, Phys. Lett. 32490Lovelace C.(1970), \"Simple n-reggeon vertex\", Phys. Lett. 32B, 490.\n\nInteracting string picture of dual resonance models. S Mandelstam, Nucl. Phys. 64205Mandelstam S. (1973), \"Interacting string picture of dual resonance models\", Nucl. Phys. B64, 205.\n\nJ T Mannasah, S Matsuda, Hadron electrodynamics in the resonance model -deep-inelastic e p scattering. 4882e e colliding-beam processes, and other applicationsMannasah J. T. and Matsuda S.(1971), \"Hadron electrodynamics in the resonance model -deep-inelastic e p scattering, e e colliding-beam processes, and other applications\", Phys. Rev. D4, 882.\n\nCrossing-symmetric decomposition of the n-point Veneziano formula into tree-graph integrals. 2. koba-nielsen representation. N Nakanishi, Prog. Theor. Phys. 45919and reference thereinNakanishi N. (1971), \"Crossing-symmetric decomposition of the n-point Veneziano for- mula into tree-graph integrals. 2. koba-nielsen representation\", Prog. Theor. Phys. 45, 919 and reference therein.\n\nQuark model and the factorization of the Veneziano model. Y Nambu, Proc. int. conf. on symmetries and quark models, p269, Gordon and Breach. int. conf. on symmetries and quark models, p269, Gordon and BreachNambu Y. (1970),\"Quark model and the factorization of the Veneziano model\", in Proc. int. conf. on symmetries and quark models, p269, Gordon and Breach, 1970.\n\nAn analog model for ksv theory. D B Fairlie, H B Nielsen, Nucl.Phys. 20637Fairlie D. B.Nielsen H. B. . (1970), \"An analog model for ksv theory\", Nucl.Phys. B20, 637.\n\nA Parton view on dual amplitudes. H B Nielsen, P Olesen, Phys. Lett. 32203Nielsen H. B. and Olesen P. (1970),\"A Parton view on dual amplitudes\", Phys. Lett. 32B, 203.\n\nLocal field theory of the dual string. H B Nielsen, P Olesen, Nucl. Phys. 57367Nielsen H. B. and Olesen P.(1973),\"Local field theory of the dual string\", Nucl. Phys. B57, 367.\n\nVortex Line Models for Dual Strings. H B Nielsen, P Olesen, Nucl. Phys. 6145Nielsen H. B. and Olesen P.(1973), \"Vortex Line Models for Dual Strings\", Nucl. Phys.61(1973) 45.\n\nQuark Model of Dual Pions. A Neveu, J H Schwarz, Phys. Rev. 4529Neveu A. and Schwarz J. H. (1971), \"Quark Model of Dual Pions\", Phys. Rev.D4, 529.\n\nConnection between Yang-Mills fields and dual models. A Neveu, J Scherk, Nucl. Phys. 36155Neveu A. and Scherk J. (1972), \"Connection between Yang-Mills fields and dual mod- els.\", Nucl. Phys.B36, 155.\n\nMultibody interactions of D particles in supergravity and matrix theory. Y Okawa, T Yoneya, Nucl. Phys. 53867Okawa Y. and Yoneya T. (1998), \"Multibody interactions of D particles in supergravity and matrix theory\", Nucl. Phys. B538, 67.\n\nOperator vertices and propagators in dual theories. D I Olive, Nuovo Cimento 3A. 399Olive D.I. (1971),\"Operator vertices and propagators in dual theories\", Nuovo Cimento 3A, 399.\n\nDual Theory for Free Fermions. P Ramond, Phys. Rev. 12415Ramond P. (1971), \"Dual Theory for Free Fermions\", Phys. Rev. D1, 2415.\n\nDynamical model of dual amplitudes. B Sakita, M A Virasoro, Phys. Rev. Lett. 241146Sakita B. and Virasoro M. A. (1970), \"Dynamical model of dual amplitudes\", Phys. Rev. Lett.24, 1146.\n\nTheory of strong interactions. J J Sakurai, Ann. Phys. (N. Y. ). 111Sakurai J. J., \"Theory of strong interactions\", Ann. Phys. (N. Y. ) 11, 1.\n\nZero-slope limit of the dual resonance model. J Scherk, Nucl. Phys. 31222Scherk J. (1971), \"Zero-slope limit of the dual resonance model\", Nucl. Phys. B31, 222.\n\nDual models for nonhadrons. J Scherk, J H Schwarz, Nucl. Phys. 81118Scherk J. and Schwarz J. H. (1974), \"Dual models for nonhadrons\", Nucl. Phys. B81,118;\n\nDual Models and the Geometry of Space-Time. J Scherk, J H Schwarz, Phys. Lett. 52347Scherk J. and Schwarz J. H.(1974), \"Dual Models and the Geometry of Space-Time\", Phys. Lett. 52B, 347.\n\nDual field theory of quarks and gluons. J Scherk, J H Schwarz, P hys. Lett. 57463Scherk J. and Schwarz J. H.(1975), \"Dual field theory of quarks and gluons.\", P hys. Lett. 57B, 463.\n\nThe general vertex function in dual resonance models. S Sciuto, Lett. Nuovo Cim. 3411Sciuto S. (1969),\"The general vertex function in dual resonance models\", Lett. Nuovo Cim. 3, 411.\n\nGeneral factorization of the n-point beta function into two parts in a moebius-invariant manner. J A Shapiro, Phys. Rev. 13371Shapiro J. A. (1970), \"General factorization of the n-point beta function into two parts in a moebius-invariant manner\", Phys. Rev. D1, 3371.\n\nElectrostatic analog for the virasoro model. J A Shapiro, Phys. Lett. 33361Shapiro J. A. (1970), \"Electrostatic analog for the virasoro model\", Phys. Lett. 33B, 361.\n\nHarmonic-oscillator analogy for the veneziano model. L Susskind, Phys. Rev. Lett. 23545Susskind L. (1969),\"Harmonic-oscillator analogy for the veneziano model\", Phys. Rev. Lett. 23, 545.\n\nA planar diagram theory for strong interactions. G 't Hooft, Nucl. Phys. 72461't Hooft G. (1974),\"A planar diagram theory for strong interactions\", Nucl. Phys. B72, 461.\n\nConstruction of a crossing -symmetric, Regge behaved amplitude for linearly rising trajectories. G Veneziano, Nuovo Cimento A57. Veneziano G. (1968),\"Construction of a crossing -symmetric, Regge behaved amplitude for linearly rising trajectories\",Nuovo Cimento A57, 190.\n\nAlternative constructions of crossing-symmetric amplitudes with regge behavior. M A Virasoro, Phys. Rev. 1772309Virasoro M. A. (1969), \"Alternative constructions of crossing-symmetric amplitudes with regge behavior\", Phys. Rev. 177, 2309\n\nNoncommutative geometry and string field theory. E Witten, Nucl. Phys. 268253Witten E. (1986), \"Noncommutative geometry and string field theory\", Nucl. Phys. B268, 253.\n\n. W Wyss, Zur Unizität der Gravitationstheorie. 38469Helv. Phys. ActaWyss W. (1965), \"Zur Unizität der Gravitationstheorie\", Helv. Phys. Acta 38 (1965), 469.\n\nGeneral reggeon vertex in dual theory. T Yoneya, Prog. Theor. Phys. 461192Yoneya T. (1971), \"General reggeon vertex in dual theory\", Prog. Theor. Phys. 46, 1192.\n\nNote on the local gauge principle in Conformal Dual Models. T Yoneya, Prog. Theor. Phys. 48616Yoneya T. (1972), \"Note on the local gauge principle in Conformal Dual Models\", Prog. Theor. Phys. 48, 616.\n\nFeynman-like rules for the dual-resonance model on the basis of the Nakanishi decomposition. T Yoneya, Prog. Theor. Phys. 482044Yoneya T. (1972), \"Feynman-like rules for the dual-resonance model on the basis of the Nakanishi decomposition\", Prog. Theor. Phys. 48, 2044.\n\nQuantum gravity and the zero slope limit of the generalized Virasoro model. T Yoneya, Lett. Nuovo Cim. 8951Yoneya T. (1973), \"Quantum gravity and the zero slope limit of the generalized Virasoro model\", Lett. Nuovo Cim. 8, 951.\n\nConnection of dual models to electrodynamics and gravidynamics. T Yoneya, Prog. Theor. Phys. 51Yoneya T. (1974), \"Connection of dual models to electrodynamics and gravidynamics\", Prog. Theor. Phys. 51, 1907.\n\nGauge freedom in dual Feynman like rules and its relation to the massless Yang-Mills field theory. T Yoneya, Prog. Theor. Phys. 521355Yoneya T. (1974), \"Gauge freedom in dual Feynman like rules and its relation to the massless Yang-Mills field theory\", Prog. Theor. Phys. 52, 1355.\n\nGrassmann algebraic approach to the Neveu-Schwarz model and representation of supermobius algebra. T Yoneya, Prog. Theor. Phys. 54526Yoneya T. (1975), \"Grassmann algebraic approach to the Neveu-Schwarz model and representation of supermobius algebra\", Prog. Theor. Phys. 54, 526.\n\nInteracting fermionic and Pomeronic strings: Gravitational interaction of the Ramond fermion. T Yoneya, Nuovo Cimento 27A. 440Yoneya T. (1975), \"Interacting fermionic and Pomeronic strings: Gravitational inter- action of the Ramond fermion\", Nuovo Cimento 27A, 440.\n\nDual model and field theory. T Yoneya, 3146Yoneya T. (1976),\"Dual model and field theory\", Butsuri 31, 46.\n\nGeometry, gravity and dual strings. T Yoneya, Prog. Theor. Phys. 561310Yoneya T. (1976), \"Geometry, gravity and dual strings\", Prog. Theor. Phys. 56, 1310.\n\nA path-functional field theory of lattice gauge models and the large-N limit. T Yoneya, Nucl. Phys. 183471Yoneya T. (1981),\"A path-functional field theory of lattice gauge models and the large- N limit\", Nucl. Phys. 183, 471.\n\nSpace-time local symmetry of string field theory. T Yoneya, Phys. Rev. Lett. 551828Yoneya T. (1985), \"Space-time local symmetry of string field theory\", Phys. Rev. Lett. 55, 1828.\n\nString field theory. T Yoneya, D59 in the proceedings of the workshop published in Soryuushiron-Kenkyuu. 74Yoneya T. (1986),\"String field theory\", p. D59 in the proceedings of the workshop published in Soryuushiron-Kenkyuu 74 (1986-1987), No. 4.\n\nString coupling constant and dilaton vacuum expectation value in string field theory. T Yoneya, Phys. Lett. 19776Yoneya, T. (1987), \"String coupling constant and dilaton vacuum expectation value in string field theory\", Phys. Lett. 197B, 76.\n\nApproaches to string field theory. T Yoneya, Seventh Workshop on Grand Unification/ICOBAN'86. World ScientificYoneya T. (1987),\"Approaches to string field theory\", p. 508 in Seventh Workshop on Grand Unification/ICOBAN'86, World Scientific.\n\nDuality and indeterminacy principle in string theory. T Yoneya, in Wandering in the field, Festschrift for Prof. Kazuhiko Nishijima on the occation of his sixtieth birthday. World Scientific419Yoneya T. (1987),\"Duality and indeterminacy principle in string theory\", p. 419 in Wandering in the field, Festschrift for Prof. Kazuhiko Nishijima on the occation of his sixtieth birthday, World Scientific, 1987.\n\nOn the interpretation of minimal length in string theories. T Yoneya, Mod.Phys.Lett. 41587Yoneya T.(1989),\"On the interpretation of minimal length in string theories\", Mod.Phys.Lett.A4, 1587.\n\nString theory and space-time uncertainty principle. Prog. Theor. Phys. 1031081For a review on later development, see Yoneya T., \"String theory and space-time un- certainty principle\", Prog. Theor. Phys. 103, 1081.\n\nField theory of elementary domains and particles. 1. Y Katayama, H Yukawa, Prog. Theor. Phys. Suppl. 41, 1Katayama Y. and Yukawa H. (1968), \"Field theory of elementary domains and particles. 1.\", Prog. Theor. Phys. Suppl. 41, 1.\n\nClosed string field theory: An Introduction. B Zwiebach, hep-th/9305026647Les Houches Summer SchoolFor a review on this subject, see e. g. Zwiebach B.,\"Closed string field theory: An Introduction\", p. 647 in Les Houches Summer School 1992, hep-th/9305026.\n"},"corpus_id":{"kind":"number","value":16174980,"string":"16,174,980"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0901.0079\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1017/cbo9780511977725.041\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0901.0079v1.pdf\"\n ],\n \"pdf_sha\": \"998c902cc1e9ce9089a6c8937c3209b239f26d8e\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Gravity from strings : personal reminiscence on early developments *"},"authors":{"kind":"list like","value":["Tamiaki Yoneya \nInstitute of Physics\nUniversity of Tokyo Komaba\nMeguro-ku153-8902TokyoJapan"],"string":"[\n \"Tamiaki Yoneya \\nInstitute of Physics\\nUniversity of Tokyo Komaba\\nMeguro-ku153-8902TokyoJapan\"\n]","isConversational":false},"abstract":{"kind":"string","value":"I discuss the early developments of string theory with respect to its connection with gauge theory and general relativity from my own perspective. The period covered is mainly from 1969 to 1974, during which I became involved in research on dual string models as a graduate student. My thinking towards the recognition of string theory as an extended quantum theory of gravity is described. Some retrospective remarks on my later works related to this subject are also given. * Invited contribution to the book \"The Birth of String Theory\"."},"sections":{"kind":"list like","value":[{"title":"Connection of dual models to field theories","content":"From these early days of my studies on dual models, there had been one basic question which was increasingly occupying my mind. That was on the relationship between dual models and ordinary field theory. In characterizing the Veneziano amplitude, it had been emphasized that the amplitude could be expanded into a sum over an infinite set of either s-channel poles or t-channel poles, but not of both. Adding both would amount to doublecounting. This is of course the statement of channel duality, which had motivated the original proposal of the Veneziano amplitude. However, the unitarization program started by the KSV paper seemed to be an extension of the usual Feynman-graph expansion in ordinary field theory, in which we have to sum both s-and t-channel diagrams. One of my questions was whether it would in principle be possible to find a field theory from which such an extended dual Feynman-graph expansion would be derived, perhaps by introducing some appropriate interactions for Nambu's master field.\nAnother related question was on the interpretation of massless spinning excitations. The existence of such states in dual string models is an inevitable consequence of conformal symmetry associated with the Virasoro operators. In contrast to this, the existence of massless spinning fields in the framework of ordinary field theory is related to local gauge symmetries. During my master-course years, I became strongly interested in the local gauge principle, through reading the original papers by Yang and Mills and by Utiyama, in addition to those related to composite particles as mentioned above. Because my main interest had been in strong interactions, I was naturally seeking papers aiming applications of the local gauge principle to strong interactions. In regard to later works following the above classical works, the paper most intriguing to me was the one by J. J. Sakurai [Sak60]. I was deeply impressed by a strong persuasive power of this work. I thought that if the local gauge symmetry could be such a fundamental principle for strong interation, it should also play some role in dual models, and then the existence of massless spinning states in the latter had to be a clue."},{"title":"A gauge theory interpretation of massless spinning states","content":"After submitting my master thesis in the beginning of 1971, I decided to pursue the second question above as my next theme. I attempted first to interpret the interaction vertices of the massless vector states of open strings in terms of a local gauge transformation for string wave functions. I soon realized that all those vertices in the Veneziano, Neveu-Schwarz [NS71] and Ramond [Ram71] models were precisely obtained by a minimal substitution p µ (σ) → p µ (σ) − gA µ (x(0))δ(σ) with A µ = ∂ µ A µ = 0 corresponding to a particular class of local-gauge transformations on the master field |Ψ , such as\nunder an on-shell condition λ(x) = 0 for the gauge function λ(x µ ), where x µ (0) is the string coordinate at an end point σ = 0 of open strings. Namely, I showed that the vertices obtained from the Virasoro or super-Virasoro operators in this way coincided with those obtained by factorizing the pole residues at corresponding massless poles. I interpreted this simple fact as clear evidence for the fact that these massless vector states themselves should be regarded as gauge bosons and that the world-sheet conformal symmetry was the mechanism which made the local gauge symmetry intrinsic to dual strings. In modern language, this formulation is of course equivalent to the vertex operator introduced as a deformation of the world-sheet action by background gauge field at the boundary of open-string world sheets. When I was pursuing this idea, I became aware of some prior works which were trying to apply local gauge principles to dual models, such as those by Kikkawa-Sato [KS70] and by Manassah-Matsuda [MM71]. However, the standpoint common to these two works was entirely different from my viewpoint, in the sense that they were aiming toward constructing off-shell currents which would make possible the coupling of electromagnetic and weak interactions to dual strings. Hence, in their works, the gauge bosons coupled to the Veneziano amplitudes were nothing to do with the massless spinning states of dual strings themselves. I wrote up a paper on my result, emphasizing that the conformally invariant dual models of open strings should be regarded as embodying local gauge principle intrinsically, rather than interpreting the latter as some external structure to be artificially adjoined to the dual models. I submitted it to Nuovo Cim. Unfortunately this manuscript was rejected for the reason, as far as I remember, that the vertices discussed in it were not new. This rejection disappointed me. However, to my knowledge at that time, no one had expressed the same viewpoint as mine on the meaning of local gauge principle in string theory.\nA few months after this experience, I received a letter commenting on my preprint on the gauge principle in dual models from M. Minami, who was at the Research Institute of Mathematical Sciences (RIMS) in Kyoto and was actively engaged in dual models. To his letter, there was an attached copy of a preprint by Neveu-Scherk [NS72] on the application of Scherk's work [Sch71] on the zero-slope limit to the dual pion model of Neveu and Schwarz, making a connection to gauge theory. Minami also mentioned some works by Nakanishi [Nak71] on a crossing-symmetric decomposition of the integral representation of the Veneziano-type amplitudes in such a way that each decomposed piece corresponds to a usual Feynman graph only with pole singularities in a single definite set of channels. All these works which I had not been aware of until that time were a big surprise to me. As for the rejected manuscript which I was almost trying to forget, I decided to rewrite it in a more compact form. I sent it to Prog. Theory. Phys. [Yon72a] in which I added an acknowledgement to Minami.\nMinami's letter encouraged me to pursue my interpretation further. I immediately hit upon an idea of extending it to the case of closed strings, namely to the Virasoro-Shapiro amplitudes [Vir69] [Sha70a]. However, making a connection to gravity seemed to be a rather bizarre thing to do, since at that time everyone thought that the dual string model was aiming at the construction of a definitive theory for hadrons and their strong interactions. Actually, during my undergraduate years, I had been quite serious about Einstein's general relativity theory and had studied even some of the original papers. In fact, Einstein had been basic impetus to me, since I had started out on trying to comprehend his book 'The meaning of relativity' during my high school day. However, after entering the graduate school and being influenced by the atmosphere of the particle physics community, I became quite strongly prejudiced against general relativity, regarding it as a sort of a relic of the past. In the case of the vector gauge principle, I was able to free myself from similar prejudice by reading the Sakurai paper and others. I hoped that the connection to gauge theory would be useful in trying to give masses to the unwanted massless states, in the way ρ mesons were related to the gauge principle as Sakurai had advocated. But in the case of gravity, to my knowledge at that time, there had been no such attempt. It took another year for me to come back to this idea again.\nExtension of the Nakanishi decomposition and a desperate attempt toward string field theory Studying Nakanishi's beautiful papers on the decomposition of the integral representation for the n-point dual amplitudes, I gradually returned to my first question concerning the relationship between dual models and field theory. From the viewpoint of my question whether the dual amplitudes could be derived from some field theory, I thought that Nakanishi's way of decomposing the amplitudes had to be one of the necessary steps. Then, I realized that what I had done in my master thesis would fit quite nicely in oder to reformulate Nakanishi's result in such a way that the rule of decomposition is expressed as a generalized Feynman-like rule. That would make it possible to extend the decomposition to arbitrary loop diagrams. It had been well known that a naive application of the operator formalism, in the case of non-planar diagrams, led to a divergence difficulty [KT70], known as the 'periodicity' problem, of integrating over one and the same contribution infinitely many times. I was able to confirm that the rules I had derived on the basis of my previous Reggeon-vertex paper indeed resolved this difficulty as expected, because my new Feynman-like rule automatically gave a definite prescription for the integration region for arbitrary diagrams. Also this rule involving only a generalized 3-point vertex naturally reduced to that of the φ 3 theory in the zero-slope limit discussed by Scherk. I was certainly excited by this result, since this strongly suggested that the rule I had derived might correspond precisely to the Feynman rule of a desired string field theory. When I was finishing the paper about this result in May 1972, I was staying about a month at the Yukawa Institute in Kyoto, having discussions on this and related problems with Nakanishi at RIMS. As soon as I completed the paper [Yon72b] on this result in Kyoto, I turned to attempt constructing such a formalism. Seeking some hint for a new field theory underlying the dual string models, I studied some old works on infinite-component field theories and especially non-local field theories, including papers by Yukawa [KY68]. Unfortunately, however, I did not get really useful insight, since most of those works had been essentially restricted to free theories. In addition to this, the rule I had presented was using a very cumbersome language, which had inherited from Nakanishi's original method and my own formulation of the general Reggeon vertex, and hence did not allow simple interpretation in terms of more intuitive world-sheet picture. What I was trying to do is equivalent to obtaining a covariant triangulation of the moduli space of Riemann surfaces with boundaries and punctures such that it corresponds to a second-quantized field theory of interacting open strings. More than a decade later in 1985, Witten achieved this goal [Wit86] in his proposal of a covariant open string field theory, on the basis of the BRST operator formalism.\nA version of fully interacting string field theories first appeared on the scene in 1974, when Kaku and Kikkawa published their work [KK74]. Their formulation was based on Mandelstam's light-cone representation [Man73] of the dual amplitudes which had appeared about a year later after my failed attempt towards string field theory. The lightcone string field theory provided another possible, albeit non-covariant, triangulation of the moduli space of Riemann surfaces."},{"title":"Connection of dual models to general relativity","content":"By the failure of my desperate attempt toward string field theory, I was quite exhausted. To recover from a depressed mood which I fell into, I had to spend almost half a year from the summer of 1972, wandering about various other topics. From about this period onward, many people were departing from the dual string models. But I had some feeling that the dual models themselves contained something more to be uncovered, despite their immediate inapplicability to the physics of hadronic interactions. I was also influenced by a rapid resurgence of gauge field theories, which had been initiated by 't Hooft's famous work on the renormalizability of non-abelian gauge field theories. I was now thinking seriously about my previous idea on a possible connection between dual string models in the case of closed strings and general relativity. In studying the progress in gauge field theories, I was gradually freeing myself from the prejudice on general relativity. I thought that as the Yang-Mills theory had turned out to be useful in this way, general relativity might also become important for particle physics someday in some unexpected way. I also felt that it was good for me to return, now in the last stage of my graduate study, to general relativity which after all had been the prime impetus toward theoretical physics for me.\nMy thinking was something like this: One of the useful viewpoints on dual string models which had been made clear by Scherk's work was that the slope parameter α ′ should be treated as a fundamental constant as the Planck constant h had been for quantum theory. It would be very exciting if we had some kind of concrete correspondence principle between dual string models and general relativity, in an analogous way as in the case of the relation between, say, the canonical commutation relation in quantum mechancis and the Poisson bracket in classical mechanics. In any case, I had first to study a nontrivial example of the zero-slope limit for the Virasoro-Shapiro amplitudes. I decided to compute a 4-point amplitude involving two massive scalars and two gravitons, since the coupling of graviton to scalars was certainly the simplest case, as had been treated in most of old literatures which I had come across in this subject. All such papers of course only discussed the case of 4 dimensions, and I was not attempting to generalize them to critical dimensions, since for discussing only the properties of massless modes of strings without loop corrections we could restrict ourselves to lower dimensions. Consequently, the Newton constant was identified to be proportional to g 2 α ′ with g being the 3-point coupling constant of closed strings. Staying in 4 dimensions made possible for me to assign an arbitrary (real) mass to the scalar (tachyonic in critical dimensions) ground state treated as a matter field coupled with gravity, using additional components of momentum along the extra dimensions. In the early summer of 1973, I decided to publish the result [Yon73] of such a computation first in a letter form.\nIn parallel to this computation, I was pursuing a question on how to establish the correspondence to all orders with respect to graviton interactions in the tree approximation, or in other words to the tree amplitudes with arbitrary number (= n) of gravitons and with a fixed number (=2) of massive scalars. Since general relativity is a non-polynomial field theory, I needed a certain general theorem for the purpose of establishing the connection in a recursive way on the basis of my explicit computation of the (2+2)-point amplitude. I found a paper which was quite suitable for this aim. That was the one by W. Wyss [Wys65] who had given a very clear argument on how the non-polynomial Einstein Lagrangian is obtained, strarting from the lowest order coupling between a massive scalar field and a massless spin 2 field.\nThe second and more ambitious question was, as alluded to already, to find an appropriate correspondence principle which would lead from general relativity to the dual strings in a more or less unique way. An ideal thing which I was dreaming of would be to find something analogous to what had been happening in the discovery of quantum mechancis. I was also recalling P. Ramond's argument [Ram71] in his proposal of the dual fermion model as a generalization of the Dirac equation. Unfortunately, however, I was not able to find such a correspondence principle in any satisfactory form. I could give only a modest discussion, which was related to a generalization of the equivalence principle, about the generating functional for the S-matrix for (n + 2)-point scatterings: The S-matrix on the string side can be obtained from that of field theory by making a simple reinterpretation for the space-time energy momentum tensor of a massive scalar field.\nThe third point I was thinking about was a question whether there could exist an extension of the Nakanishi decompostion to the general Virasoro-Shapiro amplitudes. That would make it possible to connect both sides in a more direct way by comparing the Feynman rules on both sides. I failed to arrive at really useful conclusion on this question, except for an expectation that in this case the decomposition would require most probably an infinite number of higher-point vertices as we go to the higher-point graviton amplitudes. † Although these failures somewhat disappointed me, I decided to write a full paper on the connection between the dual closed strings and general relativity. The title of the paper that I chose was 'Connection of Dual Models to Electrodynamics and Gravidynamics', which was sent to Prog. Theor. Phys. [Yon74a] in the early autumn of 1973. The reason why I put also the word 'Electrodynamics' was that I included in the paper an argument for the similar case of the open-string scattering of n massless vectors and 2 massive scalars. That was intended for the purpose of emphasizing the similarity and, simultaneously, the contrast of the arguments between electrodynamics and gravidynamics, especially with respect to the second question above.\nI remember that, while writing this paper, my mind was in a somewhat perplexed state. One of the reasons was that I was not completely sure about my standpoint in interpreting the connection of the dual models to gravity. After my first seminar on my paper in our high-energy laboratory, a senior student (S. Araki) privately asked me something like this ‡ ; \"So, are you claiming that the dual model is more fundamental than the Einstein theory, replacing the latter by the dual model?\" I was slightly upset by this pressing question. As far as I remember, my answer was \"Well, as I have said, I could not find any definite principle, relating the dual model to general relativity. If I could have provided such a deeper explanation, I might have a right of claiming that. All what I can say is simply a fact that we can extract general relativity from the dual model in this way.\" Rather than thinking toward the application of my results to the real world, I was mostly occupied by a desire to understand a deeper reason for why the theory of relativistic strings happens to automatically encompass general relativity, the latter being based on such deep principles as the equivalence principle and general covariance. I confess that I did not have sufficient courage to make such a bold statement, since I had been too strongly worried by lack of fundamental principles for explaining the emergence of gravity from strings.\nFurthermore, my mind was also entangled with another competing viewpoint on the relationship between the dual string models and field theory. Already in 1970, there had been suggestive discussions on planar 'fishnet' diagrams by Sakita-Virasoro [SV70] and Nielson-Olesen [NO70]. § Their elegant observation had also been quite attractive to me as another possible attitude towards an understanding of the relation between dual models and local field theories. Moreover, in 1973, Nielsen and Olesen had published an interesting pair of papers [NO73a] [NO73b] in which they discussed the possibility of embedding relativistic strings in local field theories, either as a gauge theory with a nonlinear field equation corresponding to the lagrangian − √ F 2 /2α ′ or as a vortex solution to a gauge-Higgs system, respectively. I became aware of these preprints just about the time when I was working on the connection of dual models to gravity on the basis of the zeroslope limit. These works were suggesting that the dual string might well be a consequence from non-linear gauge-field theories, resulting from their nontrivial dynamics. This is a view which is completely opposite to the derivation of field theories from the dual models in the zero-slope limit. I was asking myself, \"Does this then imply that general relativity can be contained in gauge theory?\".\nI have not been able to resolve this puzzle for many years. When I was invited in 1975 from the journal 'Butsuri', a periodical (in Japanese) of the Physical Society of Japan, to write a brief report [Yon76a] on the recent development of dual models with respect to the relationship between the dual models and field theory, I began the article as: \"Since the dual resonance model has started out without any explicit connection to field theory, it has been one of the basic questions how the relationship between the dual model and field theory should be understood. There are two possible answers. One is that the dual ‡ I cannot confirm this to Araki, since unfortunately he passed away in 1994.\n§ These works were precursor to the seminal work [THo74] of 't Hooft on planar diagrams in the large N limit.\nstring is something which is beyond the framework of ordinary local field theory, and the latter should be interpreted as a particular limiting case of the dual string model. Another possibility would be that although the dual model cannot be understood using the traditional perturbative methods of field theory, they may be properly derived from field theory by means of some non-perturbative treatments of its complicated non-linear dynamics. In the present report, I will discuss the recent developments from the first viewpoint, relegating the second to other opportunities.\"\nIn the next year, several months after the submission of my paper, I received a preprint by Scherk and Schwarz [SS74a] (see also [SS74b]) in which they had studied the zero-slope limit of closed strings. In contrast to my approach of studying the coupling of gravitons to an artificial massive scalar state mainly for a methodological reason, they treated the amplitudes with only massless states. In 1975, furthermore, they made a bold proposal [SS75] that the dual string models should be interpreted as a unified theory of quarks and gluons. I was quite impressed by their works, especially by their clear attitude with respect to the interpretation of the connection of the dual models to gravity.\nAt any rate, there were several reactions to my work. In the beginning of 1974, I was invited to give a talk in a workshop organized by the GRG (general relativity and gravitation) group in Japan, and I met some of the leading Japanese physicists working in this field, including R. Utiyama. In the early summer of the same year, I received a letter from Y. Hara who was then staying in Europe, telling me that the connection of dual models to gravity was discussed by D. Olive in an international conference in London. Hara encouraged me to go to the United states for my postdoctoral studies. Subsequently, I received a letter from B. Sakita asking me whether I was interested in joining his group at the City College, New York. I also met K. Kikkawa who came back to Japan from there. Very stimulating conversations with him were most encouraging. Since I was actually about to be offered a faculty position at Hokkaido University, I joined Sakita's group, from 1976 to 1978 during which I met A. Jevicki and S. Wadia, after fulfilling my teaching duty at Hokkaido for one and a half years. In 1980, I moved to the Komaba campus of the University of Tokyo.\nIn these periods, my interest in dual strings was almost completely diminishing, because I was sparked by the fascinating development of gauge theories, especially by various new discoveries in non-perturbative aspects of gauge theories, such as magnetic monoples, instantons and lattice gauge theories. I became interested more and more in the dynamics of non-Abelian gauge theories, especially in the question of quark confinement, on which I was mainly working for a few years from the late 70s to the early 80s, though my papers at that time included actually an attempt [Yon81] to derive a version of string field theory as an effective theory of hadronic strings directly from lattice Yang-Mills theory: In fact, I considered a generalized gauge transformation, which was not so dissimilar to (2.1), of string field associated with Wilson loops to apply it for a reformulation of topological structure of the algebra of Wilson and 't Hooft loops. Then from the late summer in 1983, I stayed at CERN for one year. During my stay at CERN, M. Green made a visit and gave a seminar. Green's talk was quite stimulating to me, especially through his enthusiasm for string theory. As far as I remember, he gave a review of his works with Schwarz, but not yet on anomaly cancellation. From about this time, I became gradually inclined again to string theory, and the impact of their epoch-making work [GM84] on the anomaly cancellation brought me back to this field in the mid 80s. In fact, the works which I published during my stay at CERN were on the Liouville field theory and 2 dimensional gravity, close cousins of string theory."},{"title":"Further related works","content":"Before concluding this section, let me describe some of my further works, which are directly related to my earlier papers. After submitting the first full paper on gravitystring connection in 1973, I continued for sometime to think again about the possibility of establishing a Nakanishi-type decomposition for closed strings, since I thought that my discussion in connecting the closed string to general relativity should be improved to the level of a comparison between Feynman on both sides. In particular, I wanted to understand more clearly the mechanism of generating higher order interaction terms directly from the effect of massive excited states of the closed string. But convincing myself that this task was too difficult, I soon turned myself to more modest problems. First I went back to the connection of open strings to non-Abelian gauge theory, and decided to establish a detailed connection at the level of Feynman rules [Yon74b] on the basis of my previous work on the general Reggeon vertex and the Nakanishi decomposition.\nThen, after a mathematical work [Yon75a] on an extension of the Fubini-Veneziano operator formalism using superfields which natually arose by rewriting the Fairlie-Martin representation for the Neveu-Schwarz model, I also studied the coupling of graviton and other massless closed-string states to fermionic open strings. That was an extention of the formalism which I had developed earlier in the case of massless vectors. I showed [Yon75b][Yon76c] that the vertex operators for these states were obtained by means of an extended local gauge principle similar to the ordinary tetrad formalism of local Lorentz gauge symmetry for the Dirac field, and discussed the associated geometric theory in the zero-slope limit. In particular in [Yon76c], I made a conjecture on how these properties would be realized in full-fledged string field theories. Ten years later, I showed the validity of this conjecture, when I discussed the symmetry structure associated with massless closed string states in the light-cone string field theory [Yon85] [Yon87a]. In fact, this result could have been obtained in the mid 70s if I had been devoting myself more seriously to the problem.\nAfter sending the paper on this subject to Phys. Rev. Lett., I received a preliminary draft of the astonishing paper by Witten on a covariant open-string field theory [Wit86], in which he gave an impressive answer to the problem against which I had struggled in vain more than 10 years ago. But I was pleased to see that he had cited my latest work on the light-cone string field theory. The connection of Witten's open string field theory to the Nakanishi decomposition was noticed in a later paper by Giddings and Martinec [GM86].\nThe above idea behind the paper [Yon85] is also related to my conjecture of a purely cubic action which I have proposed [Yon87b] in the spring of 1986 at the ICOBAN '86 international workshop on Grand Unification, held at the city of Toyama located near the site of the Kamiokande neutrino detector. In this workshop, there were also other talks on string theory, including those by D. Gross and A. Strominger. Actually I had discussed this conjecture earlier in a workshop in Kyoto in the beginning of the same year or perhaps in the end of 1985. To my surprise, after only one or two months from the Toyama workshop, I received preprints on possible realizations of my conjecture by Santa Barbara group [HLTS86] and by Kyoto group [HIKK86], both mentioning my previous talk in Toyama.\nFinally, let me also mention that the expectation of a hidden correspondence principle, which I had been seeking without success in the 70s for characterizing the departure of string theory from general relativity, was basically the motivation for my later proposal [Yon87c] [Yon89] in 1987 of an uncertainty relation for space-time. A decade later, this idea turned out to be of some relevance for a qualitative understanding [LY97] of the characteristic scales of D-brane dynamics. I have written several papers [Yon00] related to this subject during years around the mid 1990s to the early 2000s."},{"title":"Epilogue","content":"Recalling what I was thinking during my student years from 1969 to 1974, I find somewhat surprisingly how often my later works have been rooted in this period: I am still in quest for better formulation of strings and D-branes along various lines which germinated, at least partially, from these early ideas. I was working almost alone, but with occasional kind help from other workers and, especially, in an environment of great progress which was occuring in both string and gauge theories. I was fortunate in being able to start working in such a dramatic period of particle physics.\nAfter more than three decades passed, the developments in string theory in the recent 10 years through the conception of the gravity-gauge correspondence (or AdS/CFT correspondence) are now throwing new light on my old perplexing question, the question of whether 'from general relativity to gauge theory' or 'from gauge theory to general relativity'. With respect to this puzzle, personally, a work [OY98] which I did with Y. Okawa in 1998 on a derivation of the gravitational 3-body force for D0 branes from the D0 gauge theory (or M(atrix) theory [BFSS97]) was a small revelation to me. This work enabled me to convince myself that general relativity may indeed be contained in the quantum dynamics of gauge field theories, when interpreted as the theory for D branes.\nIn a sense, as if we are watching a revolving stage, the string theory came back to a scene, with which it had started in the early 70s, and is providing an entirely new perspective on gauge theory and even on hadron physics. However, we had to admit that we are still far from grasping fundamental principles behind string theory. Hopefully, we will see further dramatic turns in the next 30 years.\nI would like to thank the organizers for inviting me to participate in the meeting \"The birth of string theory\" and in the present book. I also would like to thank S. Wadia for useful comments on the manuscript."}],"string":"[\n {\n \"title\": \"Connection of dual models to field theories\",\n \"content\": \"From these early days of my studies on dual models, there had been one basic question which was increasingly occupying my mind. That was on the relationship between dual models and ordinary field theory. In characterizing the Veneziano amplitude, it had been emphasized that the amplitude could be expanded into a sum over an infinite set of either s-channel poles or t-channel poles, but not of both. Adding both would amount to doublecounting. This is of course the statement of channel duality, which had motivated the original proposal of the Veneziano amplitude. However, the unitarization program started by the KSV paper seemed to be an extension of the usual Feynman-graph expansion in ordinary field theory, in which we have to sum both s-and t-channel diagrams. One of my questions was whether it would in principle be possible to find a field theory from which such an extended dual Feynman-graph expansion would be derived, perhaps by introducing some appropriate interactions for Nambu's master field.\\nAnother related question was on the interpretation of massless spinning excitations. The existence of such states in dual string models is an inevitable consequence of conformal symmetry associated with the Virasoro operators. In contrast to this, the existence of massless spinning fields in the framework of ordinary field theory is related to local gauge symmetries. During my master-course years, I became strongly interested in the local gauge principle, through reading the original papers by Yang and Mills and by Utiyama, in addition to those related to composite particles as mentioned above. Because my main interest had been in strong interactions, I was naturally seeking papers aiming applications of the local gauge principle to strong interactions. In regard to later works following the above classical works, the paper most intriguing to me was the one by J. J. Sakurai [Sak60]. I was deeply impressed by a strong persuasive power of this work. I thought that if the local gauge symmetry could be such a fundamental principle for strong interation, it should also play some role in dual models, and then the existence of massless spinning states in the latter had to be a clue.\"\n },\n {\n \"title\": \"A gauge theory interpretation of massless spinning states\",\n \"content\": \"After submitting my master thesis in the beginning of 1971, I decided to pursue the second question above as my next theme. I attempted first to interpret the interaction vertices of the massless vector states of open strings in terms of a local gauge transformation for string wave functions. I soon realized that all those vertices in the Veneziano, Neveu-Schwarz [NS71] and Ramond [Ram71] models were precisely obtained by a minimal substitution p µ (σ) → p µ (σ) − gA µ (x(0))δ(σ) with A µ = ∂ µ A µ = 0 corresponding to a particular class of local-gauge transformations on the master field |Ψ , such as\\nunder an on-shell condition λ(x) = 0 for the gauge function λ(x µ ), where x µ (0) is the string coordinate at an end point σ = 0 of open strings. Namely, I showed that the vertices obtained from the Virasoro or super-Virasoro operators in this way coincided with those obtained by factorizing the pole residues at corresponding massless poles. I interpreted this simple fact as clear evidence for the fact that these massless vector states themselves should be regarded as gauge bosons and that the world-sheet conformal symmetry was the mechanism which made the local gauge symmetry intrinsic to dual strings. In modern language, this formulation is of course equivalent to the vertex operator introduced as a deformation of the world-sheet action by background gauge field at the boundary of open-string world sheets. When I was pursuing this idea, I became aware of some prior works which were trying to apply local gauge principles to dual models, such as those by Kikkawa-Sato [KS70] and by Manassah-Matsuda [MM71]. However, the standpoint common to these two works was entirely different from my viewpoint, in the sense that they were aiming toward constructing off-shell currents which would make possible the coupling of electromagnetic and weak interactions to dual strings. Hence, in their works, the gauge bosons coupled to the Veneziano amplitudes were nothing to do with the massless spinning states of dual strings themselves. I wrote up a paper on my result, emphasizing that the conformally invariant dual models of open strings should be regarded as embodying local gauge principle intrinsically, rather than interpreting the latter as some external structure to be artificially adjoined to the dual models. I submitted it to Nuovo Cim. Unfortunately this manuscript was rejected for the reason, as far as I remember, that the vertices discussed in it were not new. This rejection disappointed me. However, to my knowledge at that time, no one had expressed the same viewpoint as mine on the meaning of local gauge principle in string theory.\\nA few months after this experience, I received a letter commenting on my preprint on the gauge principle in dual models from M. Minami, who was at the Research Institute of Mathematical Sciences (RIMS) in Kyoto and was actively engaged in dual models. To his letter, there was an attached copy of a preprint by Neveu-Scherk [NS72] on the application of Scherk's work [Sch71] on the zero-slope limit to the dual pion model of Neveu and Schwarz, making a connection to gauge theory. Minami also mentioned some works by Nakanishi [Nak71] on a crossing-symmetric decomposition of the integral representation of the Veneziano-type amplitudes in such a way that each decomposed piece corresponds to a usual Feynman graph only with pole singularities in a single definite set of channels. All these works which I had not been aware of until that time were a big surprise to me. As for the rejected manuscript which I was almost trying to forget, I decided to rewrite it in a more compact form. I sent it to Prog. Theory. Phys. [Yon72a] in which I added an acknowledgement to Minami.\\nMinami's letter encouraged me to pursue my interpretation further. I immediately hit upon an idea of extending it to the case of closed strings, namely to the Virasoro-Shapiro amplitudes [Vir69] [Sha70a]. However, making a connection to gravity seemed to be a rather bizarre thing to do, since at that time everyone thought that the dual string model was aiming at the construction of a definitive theory for hadrons and their strong interactions. Actually, during my undergraduate years, I had been quite serious about Einstein's general relativity theory and had studied even some of the original papers. In fact, Einstein had been basic impetus to me, since I had started out on trying to comprehend his book 'The meaning of relativity' during my high school day. However, after entering the graduate school and being influenced by the atmosphere of the particle physics community, I became quite strongly prejudiced against general relativity, regarding it as a sort of a relic of the past. In the case of the vector gauge principle, I was able to free myself from similar prejudice by reading the Sakurai paper and others. I hoped that the connection to gauge theory would be useful in trying to give masses to the unwanted massless states, in the way ρ mesons were related to the gauge principle as Sakurai had advocated. But in the case of gravity, to my knowledge at that time, there had been no such attempt. It took another year for me to come back to this idea again.\\nExtension of the Nakanishi decomposition and a desperate attempt toward string field theory Studying Nakanishi's beautiful papers on the decomposition of the integral representation for the n-point dual amplitudes, I gradually returned to my first question concerning the relationship between dual models and field theory. From the viewpoint of my question whether the dual amplitudes could be derived from some field theory, I thought that Nakanishi's way of decomposing the amplitudes had to be one of the necessary steps. Then, I realized that what I had done in my master thesis would fit quite nicely in oder to reformulate Nakanishi's result in such a way that the rule of decomposition is expressed as a generalized Feynman-like rule. That would make it possible to extend the decomposition to arbitrary loop diagrams. It had been well known that a naive application of the operator formalism, in the case of non-planar diagrams, led to a divergence difficulty [KT70], known as the 'periodicity' problem, of integrating over one and the same contribution infinitely many times. I was able to confirm that the rules I had derived on the basis of my previous Reggeon-vertex paper indeed resolved this difficulty as expected, because my new Feynman-like rule automatically gave a definite prescription for the integration region for arbitrary diagrams. Also this rule involving only a generalized 3-point vertex naturally reduced to that of the φ 3 theory in the zero-slope limit discussed by Scherk. I was certainly excited by this result, since this strongly suggested that the rule I had derived might correspond precisely to the Feynman rule of a desired string field theory. When I was finishing the paper about this result in May 1972, I was staying about a month at the Yukawa Institute in Kyoto, having discussions on this and related problems with Nakanishi at RIMS. As soon as I completed the paper [Yon72b] on this result in Kyoto, I turned to attempt constructing such a formalism. Seeking some hint for a new field theory underlying the dual string models, I studied some old works on infinite-component field theories and especially non-local field theories, including papers by Yukawa [KY68]. Unfortunately, however, I did not get really useful insight, since most of those works had been essentially restricted to free theories. In addition to this, the rule I had presented was using a very cumbersome language, which had inherited from Nakanishi's original method and my own formulation of the general Reggeon vertex, and hence did not allow simple interpretation in terms of more intuitive world-sheet picture. What I was trying to do is equivalent to obtaining a covariant triangulation of the moduli space of Riemann surfaces with boundaries and punctures such that it corresponds to a second-quantized field theory of interacting open strings. More than a decade later in 1985, Witten achieved this goal [Wit86] in his proposal of a covariant open string field theory, on the basis of the BRST operator formalism.\\nA version of fully interacting string field theories first appeared on the scene in 1974, when Kaku and Kikkawa published their work [KK74]. Their formulation was based on Mandelstam's light-cone representation [Man73] of the dual amplitudes which had appeared about a year later after my failed attempt towards string field theory. The lightcone string field theory provided another possible, albeit non-covariant, triangulation of the moduli space of Riemann surfaces.\"\n },\n {\n \"title\": \"Connection of dual models to general relativity\",\n \"content\": \"By the failure of my desperate attempt toward string field theory, I was quite exhausted. To recover from a depressed mood which I fell into, I had to spend almost half a year from the summer of 1972, wandering about various other topics. From about this period onward, many people were departing from the dual string models. But I had some feeling that the dual models themselves contained something more to be uncovered, despite their immediate inapplicability to the physics of hadronic interactions. I was also influenced by a rapid resurgence of gauge field theories, which had been initiated by 't Hooft's famous work on the renormalizability of non-abelian gauge field theories. I was now thinking seriously about my previous idea on a possible connection between dual string models in the case of closed strings and general relativity. In studying the progress in gauge field theories, I was gradually freeing myself from the prejudice on general relativity. I thought that as the Yang-Mills theory had turned out to be useful in this way, general relativity might also become important for particle physics someday in some unexpected way. I also felt that it was good for me to return, now in the last stage of my graduate study, to general relativity which after all had been the prime impetus toward theoretical physics for me.\\nMy thinking was something like this: One of the useful viewpoints on dual string models which had been made clear by Scherk's work was that the slope parameter α ′ should be treated as a fundamental constant as the Planck constant h had been for quantum theory. It would be very exciting if we had some kind of concrete correspondence principle between dual string models and general relativity, in an analogous way as in the case of the relation between, say, the canonical commutation relation in quantum mechancis and the Poisson bracket in classical mechanics. In any case, I had first to study a nontrivial example of the zero-slope limit for the Virasoro-Shapiro amplitudes. I decided to compute a 4-point amplitude involving two massive scalars and two gravitons, since the coupling of graviton to scalars was certainly the simplest case, as had been treated in most of old literatures which I had come across in this subject. All such papers of course only discussed the case of 4 dimensions, and I was not attempting to generalize them to critical dimensions, since for discussing only the properties of massless modes of strings without loop corrections we could restrict ourselves to lower dimensions. Consequently, the Newton constant was identified to be proportional to g 2 α ′ with g being the 3-point coupling constant of closed strings. Staying in 4 dimensions made possible for me to assign an arbitrary (real) mass to the scalar (tachyonic in critical dimensions) ground state treated as a matter field coupled with gravity, using additional components of momentum along the extra dimensions. In the early summer of 1973, I decided to publish the result [Yon73] of such a computation first in a letter form.\\nIn parallel to this computation, I was pursuing a question on how to establish the correspondence to all orders with respect to graviton interactions in the tree approximation, or in other words to the tree amplitudes with arbitrary number (= n) of gravitons and with a fixed number (=2) of massive scalars. Since general relativity is a non-polynomial field theory, I needed a certain general theorem for the purpose of establishing the connection in a recursive way on the basis of my explicit computation of the (2+2)-point amplitude. I found a paper which was quite suitable for this aim. That was the one by W. Wyss [Wys65] who had given a very clear argument on how the non-polynomial Einstein Lagrangian is obtained, strarting from the lowest order coupling between a massive scalar field and a massless spin 2 field.\\nThe second and more ambitious question was, as alluded to already, to find an appropriate correspondence principle which would lead from general relativity to the dual strings in a more or less unique way. An ideal thing which I was dreaming of would be to find something analogous to what had been happening in the discovery of quantum mechancis. I was also recalling P. Ramond's argument [Ram71] in his proposal of the dual fermion model as a generalization of the Dirac equation. Unfortunately, however, I was not able to find such a correspondence principle in any satisfactory form. I could give only a modest discussion, which was related to a generalization of the equivalence principle, about the generating functional for the S-matrix for (n + 2)-point scatterings: The S-matrix on the string side can be obtained from that of field theory by making a simple reinterpretation for the space-time energy momentum tensor of a massive scalar field.\\nThe third point I was thinking about was a question whether there could exist an extension of the Nakanishi decompostion to the general Virasoro-Shapiro amplitudes. That would make it possible to connect both sides in a more direct way by comparing the Feynman rules on both sides. I failed to arrive at really useful conclusion on this question, except for an expectation that in this case the decomposition would require most probably an infinite number of higher-point vertices as we go to the higher-point graviton amplitudes. † Although these failures somewhat disappointed me, I decided to write a full paper on the connection between the dual closed strings and general relativity. The title of the paper that I chose was 'Connection of Dual Models to Electrodynamics and Gravidynamics', which was sent to Prog. Theor. Phys. [Yon74a] in the early autumn of 1973. The reason why I put also the word 'Electrodynamics' was that I included in the paper an argument for the similar case of the open-string scattering of n massless vectors and 2 massive scalars. That was intended for the purpose of emphasizing the similarity and, simultaneously, the contrast of the arguments between electrodynamics and gravidynamics, especially with respect to the second question above.\\nI remember that, while writing this paper, my mind was in a somewhat perplexed state. One of the reasons was that I was not completely sure about my standpoint in interpreting the connection of the dual models to gravity. After my first seminar on my paper in our high-energy laboratory, a senior student (S. Araki) privately asked me something like this ‡ ; \\\"So, are you claiming that the dual model is more fundamental than the Einstein theory, replacing the latter by the dual model?\\\" I was slightly upset by this pressing question. As far as I remember, my answer was \\\"Well, as I have said, I could not find any definite principle, relating the dual model to general relativity. If I could have provided such a deeper explanation, I might have a right of claiming that. All what I can say is simply a fact that we can extract general relativity from the dual model in this way.\\\" Rather than thinking toward the application of my results to the real world, I was mostly occupied by a desire to understand a deeper reason for why the theory of relativistic strings happens to automatically encompass general relativity, the latter being based on such deep principles as the equivalence principle and general covariance. I confess that I did not have sufficient courage to make such a bold statement, since I had been too strongly worried by lack of fundamental principles for explaining the emergence of gravity from strings.\\nFurthermore, my mind was also entangled with another competing viewpoint on the relationship between the dual string models and field theory. Already in 1970, there had been suggestive discussions on planar 'fishnet' diagrams by Sakita-Virasoro [SV70] and Nielson-Olesen [NO70]. § Their elegant observation had also been quite attractive to me as another possible attitude towards an understanding of the relation between dual models and local field theories. Moreover, in 1973, Nielsen and Olesen had published an interesting pair of papers [NO73a] [NO73b] in which they discussed the possibility of embedding relativistic strings in local field theories, either as a gauge theory with a nonlinear field equation corresponding to the lagrangian − √ F 2 /2α ′ or as a vortex solution to a gauge-Higgs system, respectively. I became aware of these preprints just about the time when I was working on the connection of dual models to gravity on the basis of the zeroslope limit. These works were suggesting that the dual string might well be a consequence from non-linear gauge-field theories, resulting from their nontrivial dynamics. This is a view which is completely opposite to the derivation of field theories from the dual models in the zero-slope limit. I was asking myself, \\\"Does this then imply that general relativity can be contained in gauge theory?\\\".\\nI have not been able to resolve this puzzle for many years. When I was invited in 1975 from the journal 'Butsuri', a periodical (in Japanese) of the Physical Society of Japan, to write a brief report [Yon76a] on the recent development of dual models with respect to the relationship between the dual models and field theory, I began the article as: \\\"Since the dual resonance model has started out without any explicit connection to field theory, it has been one of the basic questions how the relationship between the dual model and field theory should be understood. There are two possible answers. One is that the dual ‡ I cannot confirm this to Araki, since unfortunately he passed away in 1994.\\n§ These works were precursor to the seminal work [THo74] of 't Hooft on planar diagrams in the large N limit.\\nstring is something which is beyond the framework of ordinary local field theory, and the latter should be interpreted as a particular limiting case of the dual string model. Another possibility would be that although the dual model cannot be understood using the traditional perturbative methods of field theory, they may be properly derived from field theory by means of some non-perturbative treatments of its complicated non-linear dynamics. In the present report, I will discuss the recent developments from the first viewpoint, relegating the second to other opportunities.\\\"\\nIn the next year, several months after the submission of my paper, I received a preprint by Scherk and Schwarz [SS74a] (see also [SS74b]) in which they had studied the zero-slope limit of closed strings. In contrast to my approach of studying the coupling of gravitons to an artificial massive scalar state mainly for a methodological reason, they treated the amplitudes with only massless states. In 1975, furthermore, they made a bold proposal [SS75] that the dual string models should be interpreted as a unified theory of quarks and gluons. I was quite impressed by their works, especially by their clear attitude with respect to the interpretation of the connection of the dual models to gravity.\\nAt any rate, there were several reactions to my work. In the beginning of 1974, I was invited to give a talk in a workshop organized by the GRG (general relativity and gravitation) group in Japan, and I met some of the leading Japanese physicists working in this field, including R. Utiyama. In the early summer of the same year, I received a letter from Y. Hara who was then staying in Europe, telling me that the connection of dual models to gravity was discussed by D. Olive in an international conference in London. Hara encouraged me to go to the United states for my postdoctoral studies. Subsequently, I received a letter from B. Sakita asking me whether I was interested in joining his group at the City College, New York. I also met K. Kikkawa who came back to Japan from there. Very stimulating conversations with him were most encouraging. Since I was actually about to be offered a faculty position at Hokkaido University, I joined Sakita's group, from 1976 to 1978 during which I met A. Jevicki and S. Wadia, after fulfilling my teaching duty at Hokkaido for one and a half years. In 1980, I moved to the Komaba campus of the University of Tokyo.\\nIn these periods, my interest in dual strings was almost completely diminishing, because I was sparked by the fascinating development of gauge theories, especially by various new discoveries in non-perturbative aspects of gauge theories, such as magnetic monoples, instantons and lattice gauge theories. I became interested more and more in the dynamics of non-Abelian gauge theories, especially in the question of quark confinement, on which I was mainly working for a few years from the late 70s to the early 80s, though my papers at that time included actually an attempt [Yon81] to derive a version of string field theory as an effective theory of hadronic strings directly from lattice Yang-Mills theory: In fact, I considered a generalized gauge transformation, which was not so dissimilar to (2.1), of string field associated with Wilson loops to apply it for a reformulation of topological structure of the algebra of Wilson and 't Hooft loops. Then from the late summer in 1983, I stayed at CERN for one year. During my stay at CERN, M. Green made a visit and gave a seminar. Green's talk was quite stimulating to me, especially through his enthusiasm for string theory. As far as I remember, he gave a review of his works with Schwarz, but not yet on anomaly cancellation. From about this time, I became gradually inclined again to string theory, and the impact of their epoch-making work [GM84] on the anomaly cancellation brought me back to this field in the mid 80s. In fact, the works which I published during my stay at CERN were on the Liouville field theory and 2 dimensional gravity, close cousins of string theory.\"\n },\n {\n \"title\": \"Further related works\",\n \"content\": \"Before concluding this section, let me describe some of my further works, which are directly related to my earlier papers. After submitting the first full paper on gravitystring connection in 1973, I continued for sometime to think again about the possibility of establishing a Nakanishi-type decomposition for closed strings, since I thought that my discussion in connecting the closed string to general relativity should be improved to the level of a comparison between Feynman on both sides. In particular, I wanted to understand more clearly the mechanism of generating higher order interaction terms directly from the effect of massive excited states of the closed string. But convincing myself that this task was too difficult, I soon turned myself to more modest problems. First I went back to the connection of open strings to non-Abelian gauge theory, and decided to establish a detailed connection at the level of Feynman rules [Yon74b] on the basis of my previous work on the general Reggeon vertex and the Nakanishi decomposition.\\nThen, after a mathematical work [Yon75a] on an extension of the Fubini-Veneziano operator formalism using superfields which natually arose by rewriting the Fairlie-Martin representation for the Neveu-Schwarz model, I also studied the coupling of graviton and other massless closed-string states to fermionic open strings. That was an extention of the formalism which I had developed earlier in the case of massless vectors. I showed [Yon75b][Yon76c] that the vertex operators for these states were obtained by means of an extended local gauge principle similar to the ordinary tetrad formalism of local Lorentz gauge symmetry for the Dirac field, and discussed the associated geometric theory in the zero-slope limit. In particular in [Yon76c], I made a conjecture on how these properties would be realized in full-fledged string field theories. Ten years later, I showed the validity of this conjecture, when I discussed the symmetry structure associated with massless closed string states in the light-cone string field theory [Yon85] [Yon87a]. In fact, this result could have been obtained in the mid 70s if I had been devoting myself more seriously to the problem.\\nAfter sending the paper on this subject to Phys. Rev. Lett., I received a preliminary draft of the astonishing paper by Witten on a covariant open-string field theory [Wit86], in which he gave an impressive answer to the problem against which I had struggled in vain more than 10 years ago. But I was pleased to see that he had cited my latest work on the light-cone string field theory. The connection of Witten's open string field theory to the Nakanishi decomposition was noticed in a later paper by Giddings and Martinec [GM86].\\nThe above idea behind the paper [Yon85] is also related to my conjecture of a purely cubic action which I have proposed [Yon87b] in the spring of 1986 at the ICOBAN '86 international workshop on Grand Unification, held at the city of Toyama located near the site of the Kamiokande neutrino detector. In this workshop, there were also other talks on string theory, including those by D. Gross and A. Strominger. Actually I had discussed this conjecture earlier in a workshop in Kyoto in the beginning of the same year or perhaps in the end of 1985. To my surprise, after only one or two months from the Toyama workshop, I received preprints on possible realizations of my conjecture by Santa Barbara group [HLTS86] and by Kyoto group [HIKK86], both mentioning my previous talk in Toyama.\\nFinally, let me also mention that the expectation of a hidden correspondence principle, which I had been seeking without success in the 70s for characterizing the departure of string theory from general relativity, was basically the motivation for my later proposal [Yon87c] [Yon89] in 1987 of an uncertainty relation for space-time. A decade later, this idea turned out to be of some relevance for a qualitative understanding [LY97] of the characteristic scales of D-brane dynamics. I have written several papers [Yon00] related to this subject during years around the mid 1990s to the early 2000s.\"\n },\n {\n \"title\": \"Epilogue\",\n \"content\": \"Recalling what I was thinking during my student years from 1969 to 1974, I find somewhat surprisingly how often my later works have been rooted in this period: I am still in quest for better formulation of strings and D-branes along various lines which germinated, at least partially, from these early ideas. I was working almost alone, but with occasional kind help from other workers and, especially, in an environment of great progress which was occuring in both string and gauge theories. I was fortunate in being able to start working in such a dramatic period of particle physics.\\nAfter more than three decades passed, the developments in string theory in the recent 10 years through the conception of the gravity-gauge correspondence (or AdS/CFT correspondence) are now throwing new light on my old perplexing question, the question of whether 'from general relativity to gauge theory' or 'from gauge theory to general relativity'. With respect to this puzzle, personally, a work [OY98] which I did with Y. Okawa in 1998 on a derivation of the gravitational 3-body force for D0 branes from the D0 gauge theory (or M(atrix) theory [BFSS97]) was a small revelation to me. This work enabled me to convince myself that general relativity may indeed be contained in the quantum dynamics of gauge field theories, when interpreted as the theory for D branes.\\nIn a sense, as if we are watching a revolving stage, the string theory came back to a scene, with which it had started in the early 70s, and is providing an entirely new perspective on gauge theory and even on hadron physics. However, we had to admit that we are still far from grasping fundamental principles behind string theory. Hopefully, we will see further dramatic turns in the next 30 years.\\nI would like to thank the organizers for inviting me to participate in the meeting \\\"The birth of string theory\\\" and in the present book. I also would like to thank S. Wadia for useful comments on the manuscript.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"M theory as a matrix model: A Conjecture. T Banks, W Fischler, S Shenker, L Susskind, Phys. Rev. 555112Banks T., Fischler W., Shenker S. and Susskind L. (1997), \"M theory as a matrix model: A Conjecture\", Phys. Rev. D55, 5112.","doi":""},{"text":"General dual operatorial vertices. E Corrigan, C Montonen, Nucl. Phys. 3658Corrigan E. and Montonen C. (1972), \"General dual operatorial vertices.\", Nucl. Phys. 36, 58.","doi":""},{"text":"Harmonic-oscillator analogy for the veneziano model. S Fubini, G Veneziano, Nuovo Cimento 67A. 29Fubini S. and Veneziano G.(1970), \"Harmonic-oscillator analogy for the veneziano model\", Nuovo Cimento 67A , 29.","doi":""},{"text":"S B Giddings, E Martinec, Conformal Geometry and String Field Theory. 27891Giddings S. B. and Martinec E. (1986), \"Conformal Geometry and String Field Theory\", Nucl. Phys. B278, 91.","doi":""},{"text":"Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory. M Green, J H Schwarz, Phys. Lett. 149117Green M. and Schwarz J. H. (1984), \"Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory\", Phys. Lett. 149B , 117.","doi":""},{"text":"Pregeometrical string field theory: creation of space-time and motion. H Hata, K Itoh, T Kugo, H Kunitomo, K Ogawa, Phys. Lett. 175138Hata H., Itoh K., Kugo T., Kunitomo H. and Ogawa K.(1986), \"Pregeometrical string field theory: creation of space-time and motion\", Phys. Lett. 175, 138.","doi":""},{"text":"A Purely Cubic Action for String Field Theory. G T Horowitz, J Lykken, R Rohm, A Strominger, Phys. Rev. Lett. 57283Horowitz G. T., Lykken J., Rohm R. and Strominger A. (1986), \"A Purely Cubic Action for String Field Theory\", Phys. Rev. Lett. 57 , 283.","doi":""},{"text":"M Kaku, K Kikkawa, The Field Theory of Relativistic Strings. 101110Pt. 1. TreesKaku M. and Kikkawa K. (1974), \"The Field Theory of Relativistic Strings, Pt. 1. Trees\", Phys. Rev. D10, 1110.","doi":""},{"text":"Unitary nonplanar closed loops. M Kaku, C B Thorn, Phys. Rev. 12860Kaku M. and Thorn C. B. (1970), \"Unitary nonplanar closed loops\", Phys. Rev. D1, 2860.","doi":""},{"text":"Manifestly crossing invariant parametrization of n meson amplitude. Z Koba, H B Nielsen, Nucl. Phys. 12517Koba Z. and Nielsen H. B. (1969), \"Manifestly crossing invariant parametrization of n meson amplitude\", Nucl. Phys. B12, 517.","doi":""},{"text":"Feynman-like diagrams compatible with duality. I: Planar diagrams. K Kikkawa, B Sakita, M A Virasoro, Phys. Rev. 1841701Kikkawa K., Sakita B. and Virasoro M. A.(1969), \"Feynman-like diagrams compatible with duality. I: Planar diagrams\", Phys. Rev. 184, 1701.","doi":""},{"text":"Nonhadronic interactions in the dual resonance model. K Kikkawa, H Sato, Phys. Lett. 32280Kikkawa K. and Sato H.(1970), \"Nonhadronic interactions in the dual resonance model\", Phys. Lett. 32B, 280.","doi":""},{"text":"D particle dynamics and the space-time uncertainty relation. M Li, T Yoneya, Phys. Rev. Lett. 781219Li M. and Yoneya T. (1997), \"D particle dynamics and the space-time uncertainty relation\", Phys. Rev. Lett. 78, 1219.","doi":""},{"text":"Simple n-reggeon vertex. C Lovelace, Phys. Lett. 32490Lovelace C.(1970), \"Simple n-reggeon vertex\", Phys. Lett. 32B, 490.","doi":""},{"text":"Interacting string picture of dual resonance models. S Mandelstam, Nucl. Phys. 64205Mandelstam S. (1973), \"Interacting string picture of dual resonance models\", Nucl. Phys. B64, 205.","doi":""},{"text":"J T Mannasah, S Matsuda, Hadron electrodynamics in the resonance model -deep-inelastic e p scattering. 4882e e colliding-beam processes, and other applicationsMannasah J. T. and Matsuda S.(1971), \"Hadron electrodynamics in the resonance model -deep-inelastic e p scattering, e e colliding-beam processes, and other applications\", Phys. Rev. D4, 882.","doi":""},{"text":"Crossing-symmetric decomposition of the n-point Veneziano formula into tree-graph integrals. 2. koba-nielsen representation. N Nakanishi, Prog. Theor. Phys. 45919and reference thereinNakanishi N. (1971), \"Crossing-symmetric decomposition of the n-point Veneziano for- mula into tree-graph integrals. 2. koba-nielsen representation\", Prog. Theor. Phys. 45, 919 and reference therein.","doi":""},{"text":"Quark model and the factorization of the Veneziano model. Y Nambu, Proc. int. conf. on symmetries and quark models, p269, Gordon and Breach. int. conf. on symmetries and quark models, p269, Gordon and BreachNambu Y. (1970),\"Quark model and the factorization of the Veneziano model\", in Proc. int. conf. on symmetries and quark models, p269, Gordon and Breach, 1970.","doi":""},{"text":"An analog model for ksv theory. D B Fairlie, H B Nielsen, Nucl.Phys. 20637Fairlie D. B.Nielsen H. B. . (1970), \"An analog model for ksv theory\", Nucl.Phys. B20, 637.","doi":""},{"text":"A Parton view on dual amplitudes. H B Nielsen, P Olesen, Phys. Lett. 32203Nielsen H. B. and Olesen P. (1970),\"A Parton view on dual amplitudes\", Phys. Lett. 32B, 203.","doi":""},{"text":"Local field theory of the dual string. H B Nielsen, P Olesen, Nucl. Phys. 57367Nielsen H. B. and Olesen P.(1973),\"Local field theory of the dual string\", Nucl. Phys. B57, 367.","doi":""},{"text":"Vortex Line Models for Dual Strings. H B Nielsen, P Olesen, Nucl. Phys. 6145Nielsen H. B. and Olesen P.(1973), \"Vortex Line Models for Dual Strings\", Nucl. Phys.61(1973) 45.","doi":""},{"text":"Quark Model of Dual Pions. A Neveu, J H Schwarz, Phys. Rev. 4529Neveu A. and Schwarz J. H. (1971), \"Quark Model of Dual Pions\", Phys. Rev.D4, 529.","doi":""},{"text":"Connection between Yang-Mills fields and dual models. A Neveu, J Scherk, Nucl. Phys. 36155Neveu A. and Scherk J. (1972), \"Connection between Yang-Mills fields and dual mod- els.\", Nucl. Phys.B36, 155.","doi":""},{"text":"Multibody interactions of D particles in supergravity and matrix theory. Y Okawa, T Yoneya, Nucl. Phys. 53867Okawa Y. and Yoneya T. (1998), \"Multibody interactions of D particles in supergravity and matrix theory\", Nucl. Phys. B538, 67.","doi":""},{"text":"Operator vertices and propagators in dual theories. D I Olive, Nuovo Cimento 3A. 399Olive D.I. (1971),\"Operator vertices and propagators in dual theories\", Nuovo Cimento 3A, 399.","doi":""},{"text":"Dual Theory for Free Fermions. P Ramond, Phys. Rev. 12415Ramond P. (1971), \"Dual Theory for Free Fermions\", Phys. Rev. D1, 2415.","doi":""},{"text":"Dynamical model of dual amplitudes. B Sakita, M A Virasoro, Phys. Rev. Lett. 241146Sakita B. and Virasoro M. A. (1970), \"Dynamical model of dual amplitudes\", Phys. Rev. Lett.24, 1146.","doi":""},{"text":"Theory of strong interactions. J J Sakurai, Ann. Phys. (N. Y. ). 111Sakurai J. J., \"Theory of strong interactions\", Ann. Phys. (N. Y. ) 11, 1.","doi":""},{"text":"Zero-slope limit of the dual resonance model. J Scherk, Nucl. Phys. 31222Scherk J. (1971), \"Zero-slope limit of the dual resonance model\", Nucl. Phys. B31, 222.","doi":""},{"text":"Dual models for nonhadrons. J Scherk, J H Schwarz, Nucl. Phys. 81118Scherk J. and Schwarz J. H. (1974), \"Dual models for nonhadrons\", Nucl. Phys. B81,118;","doi":""},{"text":"Dual Models and the Geometry of Space-Time. J Scherk, J H Schwarz, Phys. Lett. 52347Scherk J. and Schwarz J. H.(1974), \"Dual Models and the Geometry of Space-Time\", Phys. Lett. 52B, 347.","doi":""},{"text":"Dual field theory of quarks and gluons. J Scherk, J H Schwarz, P hys. Lett. 57463Scherk J. and Schwarz J. H.(1975), \"Dual field theory of quarks and gluons.\", P hys. Lett. 57B, 463.","doi":""},{"text":"The general vertex function in dual resonance models. S Sciuto, Lett. Nuovo Cim. 3411Sciuto S. (1969),\"The general vertex function in dual resonance models\", Lett. Nuovo Cim. 3, 411.","doi":""},{"text":"General factorization of the n-point beta function into two parts in a moebius-invariant manner. J A Shapiro, Phys. Rev. 13371Shapiro J. A. (1970), \"General factorization of the n-point beta function into two parts in a moebius-invariant manner\", Phys. Rev. D1, 3371.","doi":""},{"text":"Electrostatic analog for the virasoro model. J A Shapiro, Phys. Lett. 33361Shapiro J. A. (1970), \"Electrostatic analog for the virasoro model\", Phys. Lett. 33B, 361.","doi":""},{"text":"Harmonic-oscillator analogy for the veneziano model. L Susskind, Phys. Rev. Lett. 23545Susskind L. (1969),\"Harmonic-oscillator analogy for the veneziano model\", Phys. Rev. Lett. 23, 545.","doi":""},{"text":"A planar diagram theory for strong interactions. G 't Hooft, Nucl. Phys. 72461't Hooft G. (1974),\"A planar diagram theory for strong interactions\", Nucl. Phys. B72, 461.","doi":""},{"text":"Construction of a crossing -symmetric, Regge behaved amplitude for linearly rising trajectories. G Veneziano, Nuovo Cimento A57. Veneziano G. (1968),\"Construction of a crossing -symmetric, Regge behaved amplitude for linearly rising trajectories\",Nuovo Cimento A57, 190.","doi":""},{"text":"Alternative constructions of crossing-symmetric amplitudes with regge behavior. M A Virasoro, Phys. Rev. 1772309Virasoro M. A. (1969), \"Alternative constructions of crossing-symmetric amplitudes with regge behavior\", Phys. Rev. 177, 2309","doi":""},{"text":"Noncommutative geometry and string field theory. E Witten, Nucl. Phys. 268253Witten E. (1986), \"Noncommutative geometry and string field theory\", Nucl. Phys. B268, 253.","doi":""},{"text":". W Wyss, Zur Unizität der Gravitationstheorie. 38469Helv. Phys. ActaWyss W. (1965), \"Zur Unizität der Gravitationstheorie\", Helv. Phys. Acta 38 (1965), 469.","doi":""},{"text":"General reggeon vertex in dual theory. T Yoneya, Prog. Theor. Phys. 461192Yoneya T. (1971), \"General reggeon vertex in dual theory\", Prog. Theor. Phys. 46, 1192.","doi":""},{"text":"Note on the local gauge principle in Conformal Dual Models. T Yoneya, Prog. Theor. Phys. 48616Yoneya T. (1972), \"Note on the local gauge principle in Conformal Dual Models\", Prog. Theor. Phys. 48, 616.","doi":""},{"text":"Feynman-like rules for the dual-resonance model on the basis of the Nakanishi decomposition. T Yoneya, Prog. Theor. Phys. 482044Yoneya T. (1972), \"Feynman-like rules for the dual-resonance model on the basis of the Nakanishi decomposition\", Prog. Theor. Phys. 48, 2044.","doi":""},{"text":"Quantum gravity and the zero slope limit of the generalized Virasoro model. T Yoneya, Lett. Nuovo Cim. 8951Yoneya T. (1973), \"Quantum gravity and the zero slope limit of the generalized Virasoro model\", Lett. Nuovo Cim. 8, 951.","doi":""},{"text":"Connection of dual models to electrodynamics and gravidynamics. T Yoneya, Prog. Theor. Phys. 51Yoneya T. (1974), \"Connection of dual models to electrodynamics and gravidynamics\", Prog. Theor. Phys. 51, 1907.","doi":""},{"text":"Gauge freedom in dual Feynman like rules and its relation to the massless Yang-Mills field theory. T Yoneya, Prog. Theor. Phys. 521355Yoneya T. (1974), \"Gauge freedom in dual Feynman like rules and its relation to the massless Yang-Mills field theory\", Prog. Theor. Phys. 52, 1355.","doi":""},{"text":"Grassmann algebraic approach to the Neveu-Schwarz model and representation of supermobius algebra. T Yoneya, Prog. Theor. Phys. 54526Yoneya T. (1975), \"Grassmann algebraic approach to the Neveu-Schwarz model and representation of supermobius algebra\", Prog. Theor. Phys. 54, 526.","doi":""},{"text":"Interacting fermionic and Pomeronic strings: Gravitational interaction of the Ramond fermion. T Yoneya, Nuovo Cimento 27A. 440Yoneya T. (1975), \"Interacting fermionic and Pomeronic strings: Gravitational inter- action of the Ramond fermion\", Nuovo Cimento 27A, 440.","doi":""},{"text":"Dual model and field theory. T Yoneya, 3146Yoneya T. (1976),\"Dual model and field theory\", Butsuri 31, 46.","doi":""},{"text":"Geometry, gravity and dual strings. T Yoneya, Prog. Theor. Phys. 561310Yoneya T. (1976), \"Geometry, gravity and dual strings\", Prog. Theor. Phys. 56, 1310.","doi":""},{"text":"A path-functional field theory of lattice gauge models and the large-N limit. T Yoneya, Nucl. Phys. 183471Yoneya T. (1981),\"A path-functional field theory of lattice gauge models and the large- N limit\", Nucl. Phys. 183, 471.","doi":""},{"text":"Space-time local symmetry of string field theory. T Yoneya, Phys. Rev. Lett. 551828Yoneya T. (1985), \"Space-time local symmetry of string field theory\", Phys. Rev. Lett. 55, 1828.","doi":""},{"text":"String field theory. T Yoneya, D59 in the proceedings of the workshop published in Soryuushiron-Kenkyuu. 74Yoneya T. (1986),\"String field theory\", p. D59 in the proceedings of the workshop published in Soryuushiron-Kenkyuu 74 (1986-1987), No. 4.","doi":""},{"text":"String coupling constant and dilaton vacuum expectation value in string field theory. T Yoneya, Phys. Lett. 19776Yoneya, T. (1987), \"String coupling constant and dilaton vacuum expectation value in string field theory\", Phys. Lett. 197B, 76.","doi":""},{"text":"Approaches to string field theory. T Yoneya, Seventh Workshop on Grand Unification/ICOBAN'86. World ScientificYoneya T. (1987),\"Approaches to string field theory\", p. 508 in Seventh Workshop on Grand Unification/ICOBAN'86, World Scientific.","doi":""},{"text":"Duality and indeterminacy principle in string theory. T Yoneya, in Wandering in the field, Festschrift for Prof. Kazuhiko Nishijima on the occation of his sixtieth birthday. World Scientific419Yoneya T. (1987),\"Duality and indeterminacy principle in string theory\", p. 419 in Wandering in the field, Festschrift for Prof. Kazuhiko Nishijima on the occation of his sixtieth birthday, World Scientific, 1987.","doi":""},{"text":"On the interpretation of minimal length in string theories. T Yoneya, Mod.Phys.Lett. 41587Yoneya T.(1989),\"On the interpretation of minimal length in string theories\", Mod.Phys.Lett.A4, 1587.","doi":""},{"text":"String theory and space-time uncertainty principle. Prog. Theor. Phys. 1031081For a review on later development, see Yoneya T., \"String theory and space-time un- certainty principle\", Prog. Theor. Phys. 103, 1081.","doi":""},{"text":"Field theory of elementary domains and particles. 1. Y Katayama, H Yukawa, Prog. Theor. Phys. Suppl. 41, 1Katayama Y. and Yukawa H. (1968), \"Field theory of elementary domains and particles. 1.\", Prog. Theor. Phys. Suppl. 41, 1.","doi":""},{"text":"Closed string field theory: An Introduction. B Zwiebach, hep-th/9305026647Les Houches Summer SchoolFor a review on this subject, see e. g. Zwiebach B.,\"Closed string field theory: An Introduction\", p. 647 in Les Houches Summer School 1992, hep-th/9305026.","doi":"hep-th/9305026"}],"string":"[\n {\n \"text\": \"M theory as a matrix model: A Conjecture. T Banks, W Fischler, S Shenker, L Susskind, Phys. Rev. 555112Banks T., Fischler W., Shenker S. and Susskind L. (1997), \\\"M theory as a matrix model: A Conjecture\\\", Phys. Rev. D55, 5112.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"General dual operatorial vertices. E Corrigan, C Montonen, Nucl. Phys. 3658Corrigan E. and Montonen C. (1972), \\\"General dual operatorial vertices.\\\", Nucl. Phys. 36, 58.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harmonic-oscillator analogy for the veneziano model. S Fubini, G Veneziano, Nuovo Cimento 67A. 29Fubini S. and Veneziano G.(1970), \\\"Harmonic-oscillator analogy for the veneziano model\\\", Nuovo Cimento 67A , 29.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S B Giddings, E Martinec, Conformal Geometry and String Field Theory. 27891Giddings S. B. and Martinec E. (1986), \\\"Conformal Geometry and String Field Theory\\\", Nucl. Phys. B278, 91.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory. M Green, J H Schwarz, Phys. Lett. 149117Green M. and Schwarz J. H. (1984), \\\"Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory\\\", Phys. Lett. 149B , 117.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pregeometrical string field theory: creation of space-time and motion. H Hata, K Itoh, T Kugo, H Kunitomo, K Ogawa, Phys. Lett. 175138Hata H., Itoh K., Kugo T., Kunitomo H. and Ogawa K.(1986), \\\"Pregeometrical string field theory: creation of space-time and motion\\\", Phys. Lett. 175, 138.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Purely Cubic Action for String Field Theory. G T Horowitz, J Lykken, R Rohm, A Strominger, Phys. Rev. Lett. 57283Horowitz G. T., Lykken J., Rohm R. and Strominger A. (1986), \\\"A Purely Cubic Action for String Field Theory\\\", Phys. Rev. Lett. 57 , 283.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Kaku, K Kikkawa, The Field Theory of Relativistic Strings. 101110Pt. 1. TreesKaku M. and Kikkawa K. (1974), \\\"The Field Theory of Relativistic Strings, Pt. 1. Trees\\\", Phys. Rev. D10, 1110.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unitary nonplanar closed loops. M Kaku, C B Thorn, Phys. Rev. 12860Kaku M. and Thorn C. B. (1970), \\\"Unitary nonplanar closed loops\\\", Phys. Rev. D1, 2860.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Manifestly crossing invariant parametrization of n meson amplitude. Z Koba, H B Nielsen, Nucl. Phys. 12517Koba Z. and Nielsen H. B. (1969), \\\"Manifestly crossing invariant parametrization of n meson amplitude\\\", Nucl. Phys. B12, 517.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Feynman-like diagrams compatible with duality. I: Planar diagrams. K Kikkawa, B Sakita, M A Virasoro, Phys. Rev. 1841701Kikkawa K., Sakita B. and Virasoro M. A.(1969), \\\"Feynman-like diagrams compatible with duality. I: Planar diagrams\\\", Phys. Rev. 184, 1701.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonhadronic interactions in the dual resonance model. K Kikkawa, H Sato, Phys. Lett. 32280Kikkawa K. and Sato H.(1970), \\\"Nonhadronic interactions in the dual resonance model\\\", Phys. Lett. 32B, 280.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D particle dynamics and the space-time uncertainty relation. M Li, T Yoneya, Phys. Rev. Lett. 781219Li M. and Yoneya T. (1997), \\\"D particle dynamics and the space-time uncertainty relation\\\", Phys. Rev. Lett. 78, 1219.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Simple n-reggeon vertex. C Lovelace, Phys. Lett. 32490Lovelace C.(1970), \\\"Simple n-reggeon vertex\\\", Phys. Lett. 32B, 490.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interacting string picture of dual resonance models. S Mandelstam, Nucl. Phys. 64205Mandelstam S. (1973), \\\"Interacting string picture of dual resonance models\\\", Nucl. Phys. B64, 205.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J T Mannasah, S Matsuda, Hadron electrodynamics in the resonance model -deep-inelastic e p scattering. 4882e e colliding-beam processes, and other applicationsMannasah J. T. and Matsuda S.(1971), \\\"Hadron electrodynamics in the resonance model -deep-inelastic e p scattering, e e colliding-beam processes, and other applications\\\", Phys. Rev. D4, 882.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Crossing-symmetric decomposition of the n-point Veneziano formula into tree-graph integrals. 2. koba-nielsen representation. N Nakanishi, Prog. Theor. Phys. 45919and reference thereinNakanishi N. (1971), \\\"Crossing-symmetric decomposition of the n-point Veneziano for- mula into tree-graph integrals. 2. koba-nielsen representation\\\", Prog. Theor. Phys. 45, 919 and reference therein.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quark model and the factorization of the Veneziano model. Y Nambu, Proc. int. conf. on symmetries and quark models, p269, Gordon and Breach. int. conf. on symmetries and quark models, p269, Gordon and BreachNambu Y. (1970),\\\"Quark model and the factorization of the Veneziano model\\\", in Proc. int. conf. on symmetries and quark models, p269, Gordon and Breach, 1970.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An analog model for ksv theory. D B Fairlie, H B Nielsen, Nucl.Phys. 20637Fairlie D. B.Nielsen H. B. . (1970), \\\"An analog model for ksv theory\\\", Nucl.Phys. B20, 637.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Parton view on dual amplitudes. H B Nielsen, P Olesen, Phys. Lett. 32203Nielsen H. B. and Olesen P. (1970),\\\"A Parton view on dual amplitudes\\\", Phys. Lett. 32B, 203.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Local field theory of the dual string. H B Nielsen, P Olesen, Nucl. Phys. 57367Nielsen H. B. and Olesen P.(1973),\\\"Local field theory of the dual string\\\", Nucl. Phys. B57, 367.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Vortex Line Models for Dual Strings. H B Nielsen, P Olesen, Nucl. Phys. 6145Nielsen H. B. and Olesen P.(1973), \\\"Vortex Line Models for Dual Strings\\\", Nucl. Phys.61(1973) 45.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quark Model of Dual Pions. A Neveu, J H Schwarz, Phys. Rev. 4529Neveu A. and Schwarz J. H. (1971), \\\"Quark Model of Dual Pions\\\", Phys. Rev.D4, 529.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Connection between Yang-Mills fields and dual models. A Neveu, J Scherk, Nucl. Phys. 36155Neveu A. and Scherk J. (1972), \\\"Connection between Yang-Mills fields and dual mod- els.\\\", Nucl. Phys.B36, 155.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multibody interactions of D particles in supergravity and matrix theory. Y Okawa, T Yoneya, Nucl. Phys. 53867Okawa Y. and Yoneya T. (1998), \\\"Multibody interactions of D particles in supergravity and matrix theory\\\", Nucl. Phys. B538, 67.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Operator vertices and propagators in dual theories. D I Olive, Nuovo Cimento 3A. 399Olive D.I. (1971),\\\"Operator vertices and propagators in dual theories\\\", Nuovo Cimento 3A, 399.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dual Theory for Free Fermions. P Ramond, Phys. Rev. 12415Ramond P. (1971), \\\"Dual Theory for Free Fermions\\\", Phys. Rev. D1, 2415.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dynamical model of dual amplitudes. B Sakita, M A Virasoro, Phys. Rev. Lett. 241146Sakita B. and Virasoro M. A. (1970), \\\"Dynamical model of dual amplitudes\\\", Phys. Rev. Lett.24, 1146.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Theory of strong interactions. J J Sakurai, Ann. Phys. (N. Y. ). 111Sakurai J. J., \\\"Theory of strong interactions\\\", Ann. Phys. (N. Y. ) 11, 1.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Zero-slope limit of the dual resonance model. J Scherk, Nucl. Phys. 31222Scherk J. (1971), \\\"Zero-slope limit of the dual resonance model\\\", Nucl. Phys. B31, 222.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dual models for nonhadrons. J Scherk, J H Schwarz, Nucl. Phys. 81118Scherk J. and Schwarz J. H. (1974), \\\"Dual models for nonhadrons\\\", Nucl. Phys. B81,118;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dual Models and the Geometry of Space-Time. J Scherk, J H Schwarz, Phys. Lett. 52347Scherk J. and Schwarz J. H.(1974), \\\"Dual Models and the Geometry of Space-Time\\\", Phys. Lett. 52B, 347.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dual field theory of quarks and gluons. J Scherk, J H Schwarz, P hys. Lett. 57463Scherk J. and Schwarz J. H.(1975), \\\"Dual field theory of quarks and gluons.\\\", P hys. Lett. 57B, 463.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The general vertex function in dual resonance models. S Sciuto, Lett. Nuovo Cim. 3411Sciuto S. (1969),\\\"The general vertex function in dual resonance models\\\", Lett. Nuovo Cim. 3, 411.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"General factorization of the n-point beta function into two parts in a moebius-invariant manner. J A Shapiro, Phys. Rev. 13371Shapiro J. A. (1970), \\\"General factorization of the n-point beta function into two parts in a moebius-invariant manner\\\", Phys. Rev. D1, 3371.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Electrostatic analog for the virasoro model. J A Shapiro, Phys. Lett. 33361Shapiro J. A. (1970), \\\"Electrostatic analog for the virasoro model\\\", Phys. Lett. 33B, 361.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harmonic-oscillator analogy for the veneziano model. L Susskind, Phys. Rev. Lett. 23545Susskind L. (1969),\\\"Harmonic-oscillator analogy for the veneziano model\\\", Phys. Rev. Lett. 23, 545.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A planar diagram theory for strong interactions. G 't Hooft, Nucl. Phys. 72461't Hooft G. (1974),\\\"A planar diagram theory for strong interactions\\\", Nucl. Phys. B72, 461.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Construction of a crossing -symmetric, Regge behaved amplitude for linearly rising trajectories. G Veneziano, Nuovo Cimento A57. Veneziano G. (1968),\\\"Construction of a crossing -symmetric, Regge behaved amplitude for linearly rising trajectories\\\",Nuovo Cimento A57, 190.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Alternative constructions of crossing-symmetric amplitudes with regge behavior. M A Virasoro, Phys. Rev. 1772309Virasoro M. A. (1969), \\\"Alternative constructions of crossing-symmetric amplitudes with regge behavior\\\", Phys. Rev. 177, 2309\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Noncommutative geometry and string field theory. E Witten, Nucl. Phys. 268253Witten E. (1986), \\\"Noncommutative geometry and string field theory\\\", Nucl. Phys. B268, 253.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W Wyss, Zur Unizität der Gravitationstheorie. 38469Helv. Phys. ActaWyss W. (1965), \\\"Zur Unizität der Gravitationstheorie\\\", Helv. Phys. Acta 38 (1965), 469.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"General reggeon vertex in dual theory. T Yoneya, Prog. Theor. Phys. 461192Yoneya T. (1971), \\\"General reggeon vertex in dual theory\\\", Prog. Theor. Phys. 46, 1192.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Note on the local gauge principle in Conformal Dual Models. T Yoneya, Prog. Theor. Phys. 48616Yoneya T. (1972), \\\"Note on the local gauge principle in Conformal Dual Models\\\", Prog. Theor. Phys. 48, 616.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Feynman-like rules for the dual-resonance model on the basis of the Nakanishi decomposition. T Yoneya, Prog. Theor. Phys. 482044Yoneya T. (1972), \\\"Feynman-like rules for the dual-resonance model on the basis of the Nakanishi decomposition\\\", Prog. Theor. Phys. 48, 2044.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum gravity and the zero slope limit of the generalized Virasoro model. T Yoneya, Lett. Nuovo Cim. 8951Yoneya T. (1973), \\\"Quantum gravity and the zero slope limit of the generalized Virasoro model\\\", Lett. Nuovo Cim. 8, 951.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Connection of dual models to electrodynamics and gravidynamics. T Yoneya, Prog. Theor. Phys. 51Yoneya T. (1974), \\\"Connection of dual models to electrodynamics and gravidynamics\\\", Prog. Theor. Phys. 51, 1907.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gauge freedom in dual Feynman like rules and its relation to the massless Yang-Mills field theory. T Yoneya, Prog. Theor. Phys. 521355Yoneya T. (1974), \\\"Gauge freedom in dual Feynman like rules and its relation to the massless Yang-Mills field theory\\\", Prog. Theor. Phys. 52, 1355.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Grassmann algebraic approach to the Neveu-Schwarz model and representation of supermobius algebra. T Yoneya, Prog. Theor. Phys. 54526Yoneya T. (1975), \\\"Grassmann algebraic approach to the Neveu-Schwarz model and representation of supermobius algebra\\\", Prog. Theor. Phys. 54, 526.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interacting fermionic and Pomeronic strings: Gravitational interaction of the Ramond fermion. T Yoneya, Nuovo Cimento 27A. 440Yoneya T. (1975), \\\"Interacting fermionic and Pomeronic strings: Gravitational inter- action of the Ramond fermion\\\", Nuovo Cimento 27A, 440.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dual model and field theory. T Yoneya, 3146Yoneya T. (1976),\\\"Dual model and field theory\\\", Butsuri 31, 46.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Geometry, gravity and dual strings. T Yoneya, Prog. Theor. Phys. 561310Yoneya T. (1976), \\\"Geometry, gravity and dual strings\\\", Prog. Theor. Phys. 56, 1310.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A path-functional field theory of lattice gauge models and the large-N limit. T Yoneya, Nucl. Phys. 183471Yoneya T. (1981),\\\"A path-functional field theory of lattice gauge models and the large- N limit\\\", Nucl. Phys. 183, 471.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Space-time local symmetry of string field theory. T Yoneya, Phys. Rev. Lett. 551828Yoneya T. (1985), \\\"Space-time local symmetry of string field theory\\\", Phys. Rev. Lett. 55, 1828.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"String field theory. T Yoneya, D59 in the proceedings of the workshop published in Soryuushiron-Kenkyuu. 74Yoneya T. (1986),\\\"String field theory\\\", p. D59 in the proceedings of the workshop published in Soryuushiron-Kenkyuu 74 (1986-1987), No. 4.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"String coupling constant and dilaton vacuum expectation value in string field theory. T Yoneya, Phys. Lett. 19776Yoneya, T. (1987), \\\"String coupling constant and dilaton vacuum expectation value in string field theory\\\", Phys. Lett. 197B, 76.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Approaches to string field theory. T Yoneya, Seventh Workshop on Grand Unification/ICOBAN'86. World ScientificYoneya T. (1987),\\\"Approaches to string field theory\\\", p. 508 in Seventh Workshop on Grand Unification/ICOBAN'86, World Scientific.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Duality and indeterminacy principle in string theory. T Yoneya, in Wandering in the field, Festschrift for Prof. Kazuhiko Nishijima on the occation of his sixtieth birthday. World Scientific419Yoneya T. (1987),\\\"Duality and indeterminacy principle in string theory\\\", p. 419 in Wandering in the field, Festschrift for Prof. Kazuhiko Nishijima on the occation of his sixtieth birthday, World Scientific, 1987.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the interpretation of minimal length in string theories. T Yoneya, Mod.Phys.Lett. 41587Yoneya T.(1989),\\\"On the interpretation of minimal length in string theories\\\", Mod.Phys.Lett.A4, 1587.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"String theory and space-time uncertainty principle. Prog. Theor. Phys. 1031081For a review on later development, see Yoneya T., \\\"String theory and space-time un- certainty principle\\\", Prog. Theor. Phys. 103, 1081.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Field theory of elementary domains and particles. 1. Y Katayama, H Yukawa, Prog. Theor. Phys. Suppl. 41, 1Katayama Y. and Yukawa H. (1968), \\\"Field theory of elementary domains and particles. 1.\\\", Prog. Theor. Phys. Suppl. 41, 1.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Closed string field theory: An Introduction. B Zwiebach, hep-th/9305026647Les Houches Summer SchoolFor a review on this subject, see e. g. Zwiebach B.,\\\"Closed string field theory: An Introduction\\\", p. 647 in Les Houches Summer School 1992, hep-th/9305026.\",\n \"doi\": \"hep-th/9305026\"\n }\n]","isConversational":false}}},{"rowIdx":55,"cells":{"text":{"kind":"string","value":"\nVisDA-2021 Competition: Universal Domain Adaptation to Improve Performance on Out-of-Distribution Data\n23 Jul 2021\n\nDina Bashkirova \nBoston University\n\n\nEqual Contribution\n\n\nDan Hendrycks \nBerkeley\n\nEqual Contribution\n\n\nDonghyun Kim \nBoston University\n\n\nEqual Contribution\n\n\nSamarth Mishra \nBoston University\n\n\nEqual Contribution\n\n\nKate Saenko \nBoston University\n\n\nMIT-IBM Watson AI\n\n\nEqual Contribution\n\n\nKuniaki Saito \nBoston University\n\n\nEqual Contribution\n\n\nPiotr Teterwak \nBoston University\n\n\nEqual Contribution\n\n\nBen Usman \nBoston University\n\n\nEqual Contribution\n\n\nVisDA-2021 Competition: Universal Domain Adaptation to Improve Performance on Out-of-Distribution Data\n23 Jul 202182070F1B27B9079B3A79B4C52D3E377CarXiv:2107.11011v1[cs.LG]\nProgress in machine learning is typically measured by training and testing a model on the same distribution of data, i.e., the same domain.This over-estimates future accuracy on out-of-distribution data.The Visual Domain Adaptation (VisDA) 2021 competition tests models' ability to adapt to novel test distributions and handle distributional shift.We set up unsupervised domain adaptation challenges for image classifiers and will evaluate adaptation to novel viewpoints, backgrounds, modalities and degradation in quality.Our challenge draws on large-scale publically available datasets but constructs the evaluation across domains, rather that the traditional in-domain benchmarking.Furthermore, we focus on the difficult \"universal\" setting where, in addition to input distribution drift, methods may encounter missing and/or novel classes in the target dataset.Performance will be measured using a rigorous protocol, comparing to state-of-the-art domain adaptation methods with the help of established metrics.We believe that the competition will encourage further improvement in machine learning methods' ability to handle realistic data in many deployment scenarios.See\n\nIntroduction\n\nIn machine learning, \"dataset bias\" happens when the training data is not representative of future test data.Finite datasets cannot include all variations possible in the real world, so every machine learning dataset is biased in some way.Yet, machine learning progress Figure 1: An illustration of the Universal Domain Adaptation challenge posed by our competition.Given a labeled dataset (source) and an unlabeled dataset (target), the task is to achieve the best possible performance on the target dataset.\"Universal\" refers to fact that, in addition to the input distribution shift, there will be a category shift between the source and target domains.Specifically, the exact category overlap between source and target will be unknown a priori and may include: all classes being shared, missing classes in source, missing classes in target, or both [21].\n\nis traditionally measured by testing on in-distribution data: almost every new approach is trained on an i.i.d.subset and evaluated on another i.i.d.subset from the same original dataset.\n\nThis traditional evaluation obscures the real danger that models will fail on new data distributions.For example, a pedestrian detector trained on pictures of people in the sidewalk could fail on jaywalkers, if the original data collection omitted them.A medical classifier could fail on data from a new hospital or a slightly different patient population.While deep neural networks have significantly improved performance on recognition tasks [4,22,14,19,7], they still suffer from poor generalization to out-of-domain data [25].\n\nDomain adaptation techniques aim to evaluate and improve performance on outof-domain data.The Unsupervised Domain Adaptation (UDA) setting transfers models from a label-rich source domain to an unlabeled target domain without additional supervision.Recent UDA methods achieve this through unsupervised learning on the target domain, e.g., by minimizing the feature distribution shift between source and target domains [5,15,23], classifier confusion [13], clustering [21], and pseudo-label based methods [28].Promising UDA results have been demonstrated on image classification [16,27,2,26,24], semantic segmentation [12] and object detection [3] tasks.\n\nOur competition asks teams to solve the Universal Domain Adaptation (UniDA) task on visual data.The task is as follows: given a labeled dataset (source) and an unlabeled dataset (target), achieve the best possible performance on the target dataset.\n\n\"Universal\" refers to fact that, in addition to the input distribution shift, there is an unspecified category overlap between the source and target domains.Specifically, the exact category overlap between source and target will be unknown a priori and may include: full overlap, missing classes in source, missing classes in target, or both [21], as illustrated in Fig. 1.Robustness or Domain Generalization is the task of performing well on a target domain without training on it.Common methods include clever augmentations of the training data [8,10] and image stylization [6].\n\n\nMotivation\n\nThe domain adaptation problem is highly relevant because generalization to out-of-distribution data is a crucial problem at the heart of machine learning.Humans can generalize across a wide variety of domains, but ML algorithms tend to fail on anything that does not look like their training data.Yet the vast majority of algorithms are evaluated in-distribution, hiding this Achilles' heel of supervised learning.While the domain adaptation sub-field does evaluate the performance of algorithms on out-of-sample data, and has a long history as a research community, the task is still somewhat niche.\n\nTraditional unsupervised domain adaptation (UDA) methods assume that all source categories are present in the target domain, however, in practice, little might be known about the category overlap between the two domains.Our competition is the first to address this more universally applicable UniDA setting.\n\n\nApplications\n\nDistribution shift occurs in many, if not all, real-life application scenarios.When models are trained on offline datasets and deployed in the real world (e.g., a self-driving car, a hospital, etc.) the kinds of data they see inevitably shifts and changes relative to the static training distribution.Collecting more training data in each new domain is not always feasible and interferes with the operation of the system (e.g., imagine having to constantly take it offline to gather more data.)Furthermore, even very large labeled datasets like ImageNet have been found to have a distribution shift from similar datasets collected in the same way [18], so dataset bias is always going to be a problem with finite datasets.\n\nConsidering the ubiquity of the problem, there is a strong need for machine learning algorithms that generalize beyond their training data, or at least have the ability to adapt to novel distributions without requiring human supervision.Such algorithms would significantly impact many applications of machine learning.Since our competition focuses on visual data (although the problem also occurs in other modalities), we can imagine several real-world applications where our task is useful: autonomous vehicles navigating in a new environment, a robot encountering objects in the real world and dealing with changing pose, lighting and other factors, or a medical imaging application receiving data from a novel facility or sensor.\n\n\nRelated Competitions\n\nMembers of our team have organized a similar Visual Domain Adaptation (VisDA) challenge at several computer vision conferences in recent years:\n\n• The 1st VISDA (2017) challenge aimed to test domain adaptation methods' ability to transfer source knowledge and adapt it to novel target domains, focusing on Sim2Real transfer from a synthetic source domain to a real target domain.It featured an object classification and a semantic image segmentation track.\n\n• The 2nd VISDA (2018) also tackled the Sim2Real problem, but featured object detection and open-set classification tracks.\n\n• The 3rd VISDA (2019) promoted the multi-source and the semi-supervised domain adaptation settings on a newly collected 6-domain DomainNet dataset [17] (real, clipart, painting, drawing, infograph and sketch domains).\n\n• The 4th VISDA (2020) focused on domain adaptive instance retrieval, where the source and target domains have completely different classes (instance IDs), for example, pedestrian IDs.\n\nThese challenges were very well received by the computer vision community, with many teams participating and advancing the state-of-the-art in the process.However, the proposed UniDA competition would be the first domain adaptation competition at NeurIPS (or any other machine learning conference) as far as we know.Also, this time we will use different datasets that put emphasis on different types of realistic distribution shifts.The task will also be different: universal domain adaptation (UniDA) where we have unknown overlap between the source and target classes.This more challenging and realistic setting will advance the state-of-the-art in domain transfer even further towards being able to run off the shelf.\n\nThere have been several related competitions at NeurIPS.AutoML for Lifelong Machine Learning (NeurIPS'18 competition) addressed concept drift in lifelong learning, which is different from unsupervised domain adaptation.Inclusive Images evaluated classification on images drawn from geographic regions underrepresented in the training data.However they provided a labeled validation set from the target distribution, while our competition is focused on truly 'novel' distributions for which no labels were provided.Also, their distributional shift was more narrowly defined as a change in the geographic location where the image was collected by a user (via a crowdsourcing app).\n\nPredicting Generalization in Deep Learning (NeurIPS'20 competition) invited competitors to design metrics that accurately predict the generalization performance of deep neural networks without using a test set.The competitors were asked to implement a function that takes a trained model and its training data, and returns a single scalar that correlates with the generalization ability of the model.Our competition is somewhat related in its desire to measure generalization, but addresses a very different task.We do not predict a given models' performance, but evaluate it's actual performance.We also consider generalization to novel domains rather than the same distribution of data.\n\n3 VisDA-21 Competition\n\n\nDatasets\n\nIn the competition, we use ImageNet [20] as our source data.This is a large-scale annotated dataset.contains 1.4M images from 1,000 categories collected from Web. ImageNet While deep learning models works well on the test set in ImageNet, these can learn biased representation to wrong textures ques [6] and often do not perform well on data which contains domain-shift including the changes in artistic visual styles, viewpoints, illumination, corruptions, and so on.\n\nThe target domains in the competition use images from the following publicly available datasets.\n\n• ObjectNet [1] contains 50,000 images containing 313 object classes.Only 113 classes out of the 313 classes are overlapped with ImageNet.The dataset is both easier than ImageNet -objects are largely centered and unoccluded -and harder, due to controlled variations in pose, background and viewpoints.\n\n• ImageNet-R [8] contains 30,000 images of the 200 classes in ImageNet (partial DA).The images contains different visual styles and textures.\n\n• ImageNet-C [9] contains the same validation images with 1,000 classes in ImageNet (closed set DA) but consists of 15 diverse corruption types, such as blur or noise, with different level of severity.\n\n• ImageNet-O [11] is a dataset built for out-of-distribution detection using imagenet models.It contains a set of 2000 images from Imagenet-22K which do not appear in Imagenet-1K, and are typically classified with high confidence by Imagenet-1K classifiers.\n\nUnder the setting of unsupervised domain adaptation, we allow the access to each unlabeled target domain during training.As we tackle the problem of universal domain adaptation, the goal is to provide a single unified method which performs well on all cross dataset scenarios including full class overlap (closed set), some missing classes in the target (partial set), some novel classes in the target (open set), and a mix of these (open-partial domain adaptation).\n\nWe expect these domain shifts to be difficult and will adjust the number of classes accordingly to make the tasks feasible.For example, when tested on ObjectNet, detectors trained on ImageNet lose 40-45% in performance [1].\n\nIn addition to the above datasets, we provide development datasets so that teams can tune model architectures and hyperparameters without touching the test datasets.The development datasets are similar to the test datasets in that they are mix of publicly available datasets; however the mixing ratios are different to prevent overfitting as a viable strategy.\n\n\nTask\n\nThe objective of the competition is to leverage labeled data in the source domain and unlabeled data from the target domain to classify these target domain examples.Keeping with the universal domain adaptation problem, both the source and target domains will have classes that are not found in the other domain.For the images from classes only in the target domain, the task is to detect them as a \"novel\" class not found in the source domain (see Sec. 3.3 for evaluation metrics).\n\nFor development/validation, teams are provided a mix of images with labels from Ob-jectNet, ImageNet-R, ImageNet-C and ImageNet-O.A disjoint set of images from these datasets is used as the test set.In addition, the test set also includes images corrupted via methods different from the ones used in ImageNet-C.A description of these corruptions would be disclosed after the competition concludes.\n\nTeams are asked to submit their predictions on the validation set to find their position on a competition leaderboard based on the metrics described in Sec.3.3.Finally towards the end of the competition, teams will be asked to submit their predictions on the test set, for which labels would not be available to the teams.The final leaderboard will be decided by the metrics on this set.\n\n\nMetrics\n\nWe will use two metrics to evaluate performance during the challenge.\n\n\nAUC\n\nOne desirable characteristic of the model is to be able identify which samples in the evaluation data belong to novel classes, in other words those which were not present in the source data.To do this, we ask the the competitiors to give a scalar novelty score and compute Area Under the ROC Curve (AUC).The AUC measures unknown (outlier) category detection and is the area under the ROC curve, which is computed by thresholding a score that represents how likely the input is to be an unknown class.\n\n\nInstance-wise Accuracy\n\nWe not only want to identify novel classes as OOD data, we also want to have high accuracy on classes present in the source data.Thus, we also employ the instance-wise accuracy over all samples to better understand the behavior of models.\n\n\nDevelopment Kit\n\nThe competition provides a development kit (see http://ai.bu.edu/visda-2021/.)The kit contains links to the training, development and test data (test data will be released in the test phase).It also provides baseline code and instructions on how to submit results to the evaluation server, as well as the implementation of metrics.\n\n\nProtocol\n\nThe competition will have two phases.During phase I, the training data (ImageNet as a source domain) and validation domains (a mixture of ObjectNet, ImageNet-R, ImageNet-C and ImageNet-O as target domains) will be available to participants to aid in model development and hyperparameter validation.While we allow participants to use labels for validation domains, high performance on validation domains does not guarantee high performance on test data.During phase II, the test data will be made available for a brief period (two weeks) so that participants can train their models on test data (no labels available for test data) and submit label predictions to the evaluation server.\n\nTuning on Test Data.An important part of the protocol is that tuning hyperparameters or stopping criteria on the target test labels will NOT be allowed.This is because the target data is unsupervised, so supervised tuning is not possible.Therefore successful approaches will have come up with ways to tune without target-domain labels.\n\n\nCompetition Results\n\nThis section will be updated with the winning methods and results after the competition's completion.\n Appendix: Past Related Workshops and Tutorials\nObjectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition models. A Barbu, D Mayo, J Alverio, W Luo, C Wang, D Gutfreund, J Tenenbaum, B Katz, Advances in neural information processing systems. 201932\n\nTransferability vs. discriminability: Batch spectral penalization for adversarial domain adaptation. X Chen, S Wang, M Long, J Wang, 2019\n\nDomain adaptive faster r-cnn for object detection in the wild. Y Chen, W Li, C Sakaridis, D Dai, L Van Gool, 2018\n\nImagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, 2009\n\nUnsupervised domain adaptation by backpropagation. Y Ganin, V Lempitsky, 2014\n\nR Geirhos, P Rubisch, C Michaelis, M Bethge, F A Wichmann, W Brendel, arXiv:1811.12231Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. 2018arXiv preprint\n\n. K He, G Gkioxari, P Dollár, R Girshick, Mask r-cnn. 2017\n\nThe many faces of robustness: A critical analysis of outof-distribution generalization. D Hendrycks, S Basart, N Mu, S Kadavath, F Wang, E Dorundo, R Desai, T Zhu, S Parajuli, M Guo, arXiv:2006.162412020arXiv preprint\n\nD Hendrycks, T Dietterich, arXiv:1903.12261Benchmarking neural network robustness to common corruptions and perturbations. 2019arXiv preprint\n\nAugmix: A simple method to improve robustness and uncertainty under data shift. D Hendrycks, * , N Mu, * , E D Cubuk, B Zoph, J Gilmer, B Lakshminarayanan, International Conference on Learning Representations. 2020\n\nNatural adversarial examples. D Hendrycks, K Zhao, S Basart, J Steinhardt, D Song, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern Recognition2021\n\nJ Hoffman, D Wang, F Yu, T Darrell, arXiv:1612.02649Fcns in the wild: Pixel-level adversarial and constraint-based adaptation. 2016arXiv preprint\n\nMinimum class confusion for versatile domain adaptation. Y Jin, X Wang, M Long, J Wang, 2020\n\nImagenet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, 2012\n\nLearning transferable features with deep adaptation networks. M Long, Y Cao, J Wang, M I Jordan, 2015\n\nConditional adversarial domain adaptation. M Long, Z Cao, J Wang, M I Jordan, 2018\n\nMoment matching for multi-source domain adaptation. X Peng, Q Bai, X Xia, Z Huang, K Saenko, B Wang, 2019\n\nDo imagenet classifiers generalize to imagenet?. B Recht, R Roelofs, L Schmidt, V Shankar, CoRR, abs/1902.108112019\n\nFaster r-cnn: Towards real-time object detection with region proposal networks. S Ren, K He, R Girshick, J Sun, 2015\n\nImagenet large scale visual recognition challenge. O Russakovsky, J Deng, H Su, J Krause, S Satheesh, S Ma, Z Huang, A Karpathy, A Khosla, M Bernstein, International journal of computer vision. 11532015\n\nUniversal domain adaptation through self supervision. K Saito, D Kim, S Sclaroff, K Saenko, arXiv:2002.079532020arXiv preprint\n\nVery deep convolutional networks for large-scale image recognition. K Simonyan, A Zisserman, arXiv2014\n\nReturn of frustratingly easy domain adaptation. B Sun, J Feng, K Saenko, AAAI. 2016\n\nUnsupervised domain adaptation via structurally regularized deep clustering. H Tang, K Chen, K Jia, 2020\n\nDeep domain confusion: Maximizing for domain invariance. E Tzeng, J Hoffman, N Zhang, K Saenko, T Darrell, 2014arXiv\n\nLarger norm more transferable: An adaptive feature norm approach for unsupervised domain adaptation. R Xu, G Li, J Yang, L Lin, 2019\n\nConfidence regularized self-training. Y Zou, Z Yu, X Liu, B Kumar, J Wang, 2019\n\nUnsupervised domain adaptation for semantic segmentation via class-balanced self-training. Y Zou, Z Yu, B Vijaya, J Kumar, Wang, 2018\n\nDomain Adaptation: Theory and Application. Workshop Nips2011, On, \n\nNIPS2011, Workshop on Challenges in Learning Hierarchical Models: Transfer Learning and Optimization. \n\nCVPR2012, half day Tutorial on Domain Transfer Learning for Vision Applications 4. IJCV2013, Special Issue on Domain Adaptation for Vision Applications 5. ICCV2013, Workshop on Visual Domain Adaptation and Dataset Bias (VisDA). \n\nNIPS2013, Workshop on New Directions in Transfer and Multi-Task: Learning Across Domains and Tasks 7. ECCV2014, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV). \n\nECCV2014, Tutorial onDomain Adaptation and Transfer Learning 9. ECML2014, Workshop on Learning over Multiple Contexts. \n\nNIPS2014, Workshop on Transfer and Multi-Task Learning: Theory Meets Practice. \n\nSpecial Issue on Multi-Task Learning, Domain Adaptation and Transfer Learning 12. ICDM2015. Workshop on Practical Transfer Learning 13. ICCV2015, Workshop on Transferringand Adapting Source Knowledge (TASK) in Computer Vision (CV). \n\nNIPS2015, Workshop on Transfer and Multi-Task Learning: Trends and New Perspectives. \n\nECCV2016, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV). \n\nICCV2017, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV). \n\nBigdata2017, Workshop on Big Data Transfer Learning (BDTL) 18. AAMAS2017, Workshop on Transfer in Reinforcement Learning (TiRL). \n\nWorkshop on Domain Adaptation and Few-Shot Learning (Open MIC) 20. ECCV2018, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV). ACCV 2018\n\nTutorial on Transferring Knowledge Across Domains: an Introduction to Deep Domain Adaptation. ICIAP2019\n\nICCV2019, tutorial on Visual Learning with Limited Labeled Data. \n\nICCV2019, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV). \n\nECCV2020, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV). \n"},"corpus_id":{"kind":"number","value":236318217,"string":"236,318,217"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2107.11011\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/nips/BashkirovaHKLMR21\",\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2107.11011v1.pdf\"\n ],\n \"pdf_sha\": \"5ec04ea2f106f8b0455d938ead0fb4f6aa86ed65\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"VisDA-2021 Competition: Universal Domain Adaptation to Improve Performance on Out-of-Distribution Data"},"authors":{"kind":"list like","value":["Dina Bashkirova \nBoston University\n\n\nEqual Contribution","Dan Hendrycks \nBerkeley\n\nEqual Contribution","Donghyun Kim \nBoston University\n\n\nEqual Contribution","Samarth Mishra \nBoston University\n\n\nEqual Contribution","Kate Saenko \nBoston University\n\n\nMIT-IBM Watson AI\n\n\nEqual Contribution","Kuniaki Saito \nBoston University\n\n\nEqual Contribution","Piotr Teterwak \nBoston University\n\n\nEqual Contribution","Ben Usman \nBoston University\n\n\nEqual Contribution"],"string":"[\n \"Dina Bashkirova \\nBoston University\\n\\n\\nEqual Contribution\",\n \"Dan Hendrycks \\nBerkeley\\n\\nEqual Contribution\",\n \"Donghyun Kim \\nBoston University\\n\\n\\nEqual Contribution\",\n \"Samarth Mishra \\nBoston University\\n\\n\\nEqual Contribution\",\n \"Kate Saenko \\nBoston University\\n\\n\\nMIT-IBM Watson AI\\n\\n\\nEqual Contribution\",\n \"Kuniaki Saito \\nBoston University\\n\\n\\nEqual Contribution\",\n \"Piotr Teterwak \\nBoston University\\n\\n\\nEqual Contribution\",\n \"Ben Usman \\nBoston University\\n\\n\\nEqual Contribution\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Progress in machine learning is typically measured by training and testing a model on the same distribution of data, i.e., the same domain.This over-estimates future accuracy on out-of-distribution data.The Visual Domain Adaptation (VisDA) 2021 competition tests models' ability to adapt to novel test distributions and handle distributional shift.We set up unsupervised domain adaptation challenges for image classifiers and will evaluate adaptation to novel viewpoints, backgrounds, modalities and degradation in quality.Our challenge draws on large-scale publically available datasets but constructs the evaluation across domains, rather that the traditional in-domain benchmarking.Furthermore, we focus on the difficult \"universal\" setting where, in addition to input distribution drift, methods may encounter missing and/or novel classes in the target dataset.Performance will be measured using a rigorous protocol, comparing to state-of-the-art domain adaptation methods with the help of established metrics.We believe that the competition will encourage further improvement in machine learning methods' ability to handle realistic data in many deployment scenarios.See"},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"In machine learning, \"dataset bias\" happens when the training data is not representative of future test data.Finite datasets cannot include all variations possible in the real world, so every machine learning dataset is biased in some way.Yet, machine learning progress Figure 1: An illustration of the Universal Domain Adaptation challenge posed by our competition.Given a labeled dataset (source) and an unlabeled dataset (target), the task is to achieve the best possible performance on the target dataset.\"Universal\" refers to fact that, in addition to the input distribution shift, there will be a category shift between the source and target domains.Specifically, the exact category overlap between source and target will be unknown a priori and may include: all classes being shared, missing classes in source, missing classes in target, or both [21].\nis traditionally measured by testing on in-distribution data: almost every new approach is trained on an i.i.d.subset and evaluated on another i.i.d.subset from the same original dataset.\nThis traditional evaluation obscures the real danger that models will fail on new data distributions.For example, a pedestrian detector trained on pictures of people in the sidewalk could fail on jaywalkers, if the original data collection omitted them.A medical classifier could fail on data from a new hospital or a slightly different patient population.While deep neural networks have significantly improved performance on recognition tasks [4,22,14,19,7], they still suffer from poor generalization to out-of-domain data [25].\nDomain adaptation techniques aim to evaluate and improve performance on outof-domain data.The Unsupervised Domain Adaptation (UDA) setting transfers models from a label-rich source domain to an unlabeled target domain without additional supervision.Recent UDA methods achieve this through unsupervised learning on the target domain, e.g., by minimizing the feature distribution shift between source and target domains [5,15,23], classifier confusion [13], clustering [21], and pseudo-label based methods [28].Promising UDA results have been demonstrated on image classification [16,27,2,26,24], semantic segmentation [12] and object detection [3] tasks.\nOur competition asks teams to solve the Universal Domain Adaptation (UniDA) task on visual data.The task is as follows: given a labeled dataset (source) and an unlabeled dataset (target), achieve the best possible performance on the target dataset.\n\"Universal\" refers to fact that, in addition to the input distribution shift, there is an unspecified category overlap between the source and target domains.Specifically, the exact category overlap between source and target will be unknown a priori and may include: full overlap, missing classes in source, missing classes in target, or both [21], as illustrated in Fig. 1.Robustness or Domain Generalization is the task of performing well on a target domain without training on it.Common methods include clever augmentations of the training data [8,10] and image stylization [6]."},{"title":"Motivation","content":"The domain adaptation problem is highly relevant because generalization to out-of-distribution data is a crucial problem at the heart of machine learning.Humans can generalize across a wide variety of domains, but ML algorithms tend to fail on anything that does not look like their training data.Yet the vast majority of algorithms are evaluated in-distribution, hiding this Achilles' heel of supervised learning.While the domain adaptation sub-field does evaluate the performance of algorithms on out-of-sample data, and has a long history as a research community, the task is still somewhat niche.\nTraditional unsupervised domain adaptation (UDA) methods assume that all source categories are present in the target domain, however, in practice, little might be known about the category overlap between the two domains.Our competition is the first to address this more universally applicable UniDA setting."},{"title":"Applications","content":"Distribution shift occurs in many, if not all, real-life application scenarios.When models are trained on offline datasets and deployed in the real world (e.g., a self-driving car, a hospital, etc.) the kinds of data they see inevitably shifts and changes relative to the static training distribution.Collecting more training data in each new domain is not always feasible and interferes with the operation of the system (e.g., imagine having to constantly take it offline to gather more data.)Furthermore, even very large labeled datasets like ImageNet have been found to have a distribution shift from similar datasets collected in the same way [18], so dataset bias is always going to be a problem with finite datasets.\nConsidering the ubiquity of the problem, there is a strong need for machine learning algorithms that generalize beyond their training data, or at least have the ability to adapt to novel distributions without requiring human supervision.Such algorithms would significantly impact many applications of machine learning.Since our competition focuses on visual data (although the problem also occurs in other modalities), we can imagine several real-world applications where our task is useful: autonomous vehicles navigating in a new environment, a robot encountering objects in the real world and dealing with changing pose, lighting and other factors, or a medical imaging application receiving data from a novel facility or sensor."},{"title":"Related Competitions","content":"Members of our team have organized a similar Visual Domain Adaptation (VisDA) challenge at several computer vision conferences in recent years:\n• The 1st VISDA (2017) challenge aimed to test domain adaptation methods' ability to transfer source knowledge and adapt it to novel target domains, focusing on Sim2Real transfer from a synthetic source domain to a real target domain.It featured an object classification and a semantic image segmentation track.\n• The 2nd VISDA (2018) also tackled the Sim2Real problem, but featured object detection and open-set classification tracks.\n• The 3rd VISDA (2019) promoted the multi-source and the semi-supervised domain adaptation settings on a newly collected 6-domain DomainNet dataset [17] (real, clipart, painting, drawing, infograph and sketch domains).\n• The 4th VISDA (2020) focused on domain adaptive instance retrieval, where the source and target domains have completely different classes (instance IDs), for example, pedestrian IDs.\nThese challenges were very well received by the computer vision community, with many teams participating and advancing the state-of-the-art in the process.However, the proposed UniDA competition would be the first domain adaptation competition at NeurIPS (or any other machine learning conference) as far as we know.Also, this time we will use different datasets that put emphasis on different types of realistic distribution shifts.The task will also be different: universal domain adaptation (UniDA) where we have unknown overlap between the source and target classes.This more challenging and realistic setting will advance the state-of-the-art in domain transfer even further towards being able to run off the shelf.\nThere have been several related competitions at NeurIPS.AutoML for Lifelong Machine Learning (NeurIPS'18 competition) addressed concept drift in lifelong learning, which is different from unsupervised domain adaptation.Inclusive Images evaluated classification on images drawn from geographic regions underrepresented in the training data.However they provided a labeled validation set from the target distribution, while our competition is focused on truly 'novel' distributions for which no labels were provided.Also, their distributional shift was more narrowly defined as a change in the geographic location where the image was collected by a user (via a crowdsourcing app).\nPredicting Generalization in Deep Learning (NeurIPS'20 competition) invited competitors to design metrics that accurately predict the generalization performance of deep neural networks without using a test set.The competitors were asked to implement a function that takes a trained model and its training data, and returns a single scalar that correlates with the generalization ability of the model.Our competition is somewhat related in its desire to measure generalization, but addresses a very different task.We do not predict a given models' performance, but evaluate it's actual performance.We also consider generalization to novel domains rather than the same distribution of data.\n3 VisDA-21 Competition"},{"title":"Datasets","content":"In the competition, we use ImageNet [20] as our source data.This is a large-scale annotated dataset.contains 1.4M images from 1,000 categories collected from Web. ImageNet While deep learning models works well on the test set in ImageNet, these can learn biased representation to wrong textures ques [6] and often do not perform well on data which contains domain-shift including the changes in artistic visual styles, viewpoints, illumination, corruptions, and so on.\nThe target domains in the competition use images from the following publicly available datasets.\n• ObjectNet [1] contains 50,000 images containing 313 object classes.Only 113 classes out of the 313 classes are overlapped with ImageNet.The dataset is both easier than ImageNet -objects are largely centered and unoccluded -and harder, due to controlled variations in pose, background and viewpoints.\n• ImageNet-R [8] contains 30,000 images of the 200 classes in ImageNet (partial DA).The images contains different visual styles and textures.\n• ImageNet-C [9] contains the same validation images with 1,000 classes in ImageNet (closed set DA) but consists of 15 diverse corruption types, such as blur or noise, with different level of severity.\n• ImageNet-O [11] is a dataset built for out-of-distribution detection using imagenet models.It contains a set of 2000 images from Imagenet-22K which do not appear in Imagenet-1K, and are typically classified with high confidence by Imagenet-1K classifiers.\nUnder the setting of unsupervised domain adaptation, we allow the access to each unlabeled target domain during training.As we tackle the problem of universal domain adaptation, the goal is to provide a single unified method which performs well on all cross dataset scenarios including full class overlap (closed set), some missing classes in the target (partial set), some novel classes in the target (open set), and a mix of these (open-partial domain adaptation).\nWe expect these domain shifts to be difficult and will adjust the number of classes accordingly to make the tasks feasible.For example, when tested on ObjectNet, detectors trained on ImageNet lose 40-45% in performance [1].\nIn addition to the above datasets, we provide development datasets so that teams can tune model architectures and hyperparameters without touching the test datasets.The development datasets are similar to the test datasets in that they are mix of publicly available datasets; however the mixing ratios are different to prevent overfitting as a viable strategy."},{"title":"Task","content":"The objective of the competition is to leverage labeled data in the source domain and unlabeled data from the target domain to classify these target domain examples.Keeping with the universal domain adaptation problem, both the source and target domains will have classes that are not found in the other domain.For the images from classes only in the target domain, the task is to detect them as a \"novel\" class not found in the source domain (see Sec. 3.3 for evaluation metrics).\nFor development/validation, teams are provided a mix of images with labels from Ob-jectNet, ImageNet-R, ImageNet-C and ImageNet-O.A disjoint set of images from these datasets is used as the test set.In addition, the test set also includes images corrupted via methods different from the ones used in ImageNet-C.A description of these corruptions would be disclosed after the competition concludes.\nTeams are asked to submit their predictions on the validation set to find their position on a competition leaderboard based on the metrics described in Sec.3.3.Finally towards the end of the competition, teams will be asked to submit their predictions on the test set, for which labels would not be available to the teams.The final leaderboard will be decided by the metrics on this set."},{"title":"Metrics","content":"We will use two metrics to evaluate performance during the challenge."},{"title":"AUC","content":"One desirable characteristic of the model is to be able identify which samples in the evaluation data belong to novel classes, in other words those which were not present in the source data.To do this, we ask the the competitiors to give a scalar novelty score and compute Area Under the ROC Curve (AUC).The AUC measures unknown (outlier) category detection and is the area under the ROC curve, which is computed by thresholding a score that represents how likely the input is to be an unknown class."},{"title":"Instance-wise Accuracy","content":"We not only want to identify novel classes as OOD data, we also want to have high accuracy on classes present in the source data.Thus, we also employ the instance-wise accuracy over all samples to better understand the behavior of models."},{"title":"Development Kit","content":"The competition provides a development kit (see http://ai.bu.edu/visda-2021/.)The kit contains links to the training, development and test data (test data will be released in the test phase).It also provides baseline code and instructions on how to submit results to the evaluation server, as well as the implementation of metrics."},{"title":"Protocol","content":"The competition will have two phases.During phase I, the training data (ImageNet as a source domain) and validation domains (a mixture of ObjectNet, ImageNet-R, ImageNet-C and ImageNet-O as target domains) will be available to participants to aid in model development and hyperparameter validation.While we allow participants to use labels for validation domains, high performance on validation domains does not guarantee high performance on test data.During phase II, the test data will be made available for a brief period (two weeks) so that participants can train their models on test data (no labels available for test data) and submit label predictions to the evaluation server.\nTuning on Test Data.An important part of the protocol is that tuning hyperparameters or stopping criteria on the target test labels will NOT be allowed.This is because the target data is unsupervised, so supervised tuning is not possible.Therefore successful approaches will have come up with ways to tune without target-domain labels."},{"title":"Competition Results","content":"This section will be updated with the winning methods and results after the competition's completion."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"In machine learning, \\\"dataset bias\\\" happens when the training data is not representative of future test data.Finite datasets cannot include all variations possible in the real world, so every machine learning dataset is biased in some way.Yet, machine learning progress Figure 1: An illustration of the Universal Domain Adaptation challenge posed by our competition.Given a labeled dataset (source) and an unlabeled dataset (target), the task is to achieve the best possible performance on the target dataset.\\\"Universal\\\" refers to fact that, in addition to the input distribution shift, there will be a category shift between the source and target domains.Specifically, the exact category overlap between source and target will be unknown a priori and may include: all classes being shared, missing classes in source, missing classes in target, or both [21].\\nis traditionally measured by testing on in-distribution data: almost every new approach is trained on an i.i.d.subset and evaluated on another i.i.d.subset from the same original dataset.\\nThis traditional evaluation obscures the real danger that models will fail on new data distributions.For example, a pedestrian detector trained on pictures of people in the sidewalk could fail on jaywalkers, if the original data collection omitted them.A medical classifier could fail on data from a new hospital or a slightly different patient population.While deep neural networks have significantly improved performance on recognition tasks [4,22,14,19,7], they still suffer from poor generalization to out-of-domain data [25].\\nDomain adaptation techniques aim to evaluate and improve performance on outof-domain data.The Unsupervised Domain Adaptation (UDA) setting transfers models from a label-rich source domain to an unlabeled target domain without additional supervision.Recent UDA methods achieve this through unsupervised learning on the target domain, e.g., by minimizing the feature distribution shift between source and target domains [5,15,23], classifier confusion [13], clustering [21], and pseudo-label based methods [28].Promising UDA results have been demonstrated on image classification [16,27,2,26,24], semantic segmentation [12] and object detection [3] tasks.\\nOur competition asks teams to solve the Universal Domain Adaptation (UniDA) task on visual data.The task is as follows: given a labeled dataset (source) and an unlabeled dataset (target), achieve the best possible performance on the target dataset.\\n\\\"Universal\\\" refers to fact that, in addition to the input distribution shift, there is an unspecified category overlap between the source and target domains.Specifically, the exact category overlap between source and target will be unknown a priori and may include: full overlap, missing classes in source, missing classes in target, or both [21], as illustrated in Fig. 1.Robustness or Domain Generalization is the task of performing well on a target domain without training on it.Common methods include clever augmentations of the training data [8,10] and image stylization [6].\"\n },\n {\n \"title\": \"Motivation\",\n \"content\": \"The domain adaptation problem is highly relevant because generalization to out-of-distribution data is a crucial problem at the heart of machine learning.Humans can generalize across a wide variety of domains, but ML algorithms tend to fail on anything that does not look like their training data.Yet the vast majority of algorithms are evaluated in-distribution, hiding this Achilles' heel of supervised learning.While the domain adaptation sub-field does evaluate the performance of algorithms on out-of-sample data, and has a long history as a research community, the task is still somewhat niche.\\nTraditional unsupervised domain adaptation (UDA) methods assume that all source categories are present in the target domain, however, in practice, little might be known about the category overlap between the two domains.Our competition is the first to address this more universally applicable UniDA setting.\"\n },\n {\n \"title\": \"Applications\",\n \"content\": \"Distribution shift occurs in many, if not all, real-life application scenarios.When models are trained on offline datasets and deployed in the real world (e.g., a self-driving car, a hospital, etc.) the kinds of data they see inevitably shifts and changes relative to the static training distribution.Collecting more training data in each new domain is not always feasible and interferes with the operation of the system (e.g., imagine having to constantly take it offline to gather more data.)Furthermore, even very large labeled datasets like ImageNet have been found to have a distribution shift from similar datasets collected in the same way [18], so dataset bias is always going to be a problem with finite datasets.\\nConsidering the ubiquity of the problem, there is a strong need for machine learning algorithms that generalize beyond their training data, or at least have the ability to adapt to novel distributions without requiring human supervision.Such algorithms would significantly impact many applications of machine learning.Since our competition focuses on visual data (although the problem also occurs in other modalities), we can imagine several real-world applications where our task is useful: autonomous vehicles navigating in a new environment, a robot encountering objects in the real world and dealing with changing pose, lighting and other factors, or a medical imaging application receiving data from a novel facility or sensor.\"\n },\n {\n \"title\": \"Related Competitions\",\n \"content\": \"Members of our team have organized a similar Visual Domain Adaptation (VisDA) challenge at several computer vision conferences in recent years:\\n• The 1st VISDA (2017) challenge aimed to test domain adaptation methods' ability to transfer source knowledge and adapt it to novel target domains, focusing on Sim2Real transfer from a synthetic source domain to a real target domain.It featured an object classification and a semantic image segmentation track.\\n• The 2nd VISDA (2018) also tackled the Sim2Real problem, but featured object detection and open-set classification tracks.\\n• The 3rd VISDA (2019) promoted the multi-source and the semi-supervised domain adaptation settings on a newly collected 6-domain DomainNet dataset [17] (real, clipart, painting, drawing, infograph and sketch domains).\\n• The 4th VISDA (2020) focused on domain adaptive instance retrieval, where the source and target domains have completely different classes (instance IDs), for example, pedestrian IDs.\\nThese challenges were very well received by the computer vision community, with many teams participating and advancing the state-of-the-art in the process.However, the proposed UniDA competition would be the first domain adaptation competition at NeurIPS (or any other machine learning conference) as far as we know.Also, this time we will use different datasets that put emphasis on different types of realistic distribution shifts.The task will also be different: universal domain adaptation (UniDA) where we have unknown overlap between the source and target classes.This more challenging and realistic setting will advance the state-of-the-art in domain transfer even further towards being able to run off the shelf.\\nThere have been several related competitions at NeurIPS.AutoML for Lifelong Machine Learning (NeurIPS'18 competition) addressed concept drift in lifelong learning, which is different from unsupervised domain adaptation.Inclusive Images evaluated classification on images drawn from geographic regions underrepresented in the training data.However they provided a labeled validation set from the target distribution, while our competition is focused on truly 'novel' distributions for which no labels were provided.Also, their distributional shift was more narrowly defined as a change in the geographic location where the image was collected by a user (via a crowdsourcing app).\\nPredicting Generalization in Deep Learning (NeurIPS'20 competition) invited competitors to design metrics that accurately predict the generalization performance of deep neural networks without using a test set.The competitors were asked to implement a function that takes a trained model and its training data, and returns a single scalar that correlates with the generalization ability of the model.Our competition is somewhat related in its desire to measure generalization, but addresses a very different task.We do not predict a given models' performance, but evaluate it's actual performance.We also consider generalization to novel domains rather than the same distribution of data.\\n3 VisDA-21 Competition\"\n },\n {\n \"title\": \"Datasets\",\n \"content\": \"In the competition, we use ImageNet [20] as our source data.This is a large-scale annotated dataset.contains 1.4M images from 1,000 categories collected from Web. ImageNet While deep learning models works well on the test set in ImageNet, these can learn biased representation to wrong textures ques [6] and often do not perform well on data which contains domain-shift including the changes in artistic visual styles, viewpoints, illumination, corruptions, and so on.\\nThe target domains in the competition use images from the following publicly available datasets.\\n• ObjectNet [1] contains 50,000 images containing 313 object classes.Only 113 classes out of the 313 classes are overlapped with ImageNet.The dataset is both easier than ImageNet -objects are largely centered and unoccluded -and harder, due to controlled variations in pose, background and viewpoints.\\n• ImageNet-R [8] contains 30,000 images of the 200 classes in ImageNet (partial DA).The images contains different visual styles and textures.\\n• ImageNet-C [9] contains the same validation images with 1,000 classes in ImageNet (closed set DA) but consists of 15 diverse corruption types, such as blur or noise, with different level of severity.\\n• ImageNet-O [11] is a dataset built for out-of-distribution detection using imagenet models.It contains a set of 2000 images from Imagenet-22K which do not appear in Imagenet-1K, and are typically classified with high confidence by Imagenet-1K classifiers.\\nUnder the setting of unsupervised domain adaptation, we allow the access to each unlabeled target domain during training.As we tackle the problem of universal domain adaptation, the goal is to provide a single unified method which performs well on all cross dataset scenarios including full class overlap (closed set), some missing classes in the target (partial set), some novel classes in the target (open set), and a mix of these (open-partial domain adaptation).\\nWe expect these domain shifts to be difficult and will adjust the number of classes accordingly to make the tasks feasible.For example, when tested on ObjectNet, detectors trained on ImageNet lose 40-45% in performance [1].\\nIn addition to the above datasets, we provide development datasets so that teams can tune model architectures and hyperparameters without touching the test datasets.The development datasets are similar to the test datasets in that they are mix of publicly available datasets; however the mixing ratios are different to prevent overfitting as a viable strategy.\"\n },\n {\n \"title\": \"Task\",\n \"content\": \"The objective of the competition is to leverage labeled data in the source domain and unlabeled data from the target domain to classify these target domain examples.Keeping with the universal domain adaptation problem, both the source and target domains will have classes that are not found in the other domain.For the images from classes only in the target domain, the task is to detect them as a \\\"novel\\\" class not found in the source domain (see Sec. 3.3 for evaluation metrics).\\nFor development/validation, teams are provided a mix of images with labels from Ob-jectNet, ImageNet-R, ImageNet-C and ImageNet-O.A disjoint set of images from these datasets is used as the test set.In addition, the test set also includes images corrupted via methods different from the ones used in ImageNet-C.A description of these corruptions would be disclosed after the competition concludes.\\nTeams are asked to submit their predictions on the validation set to find their position on a competition leaderboard based on the metrics described in Sec.3.3.Finally towards the end of the competition, teams will be asked to submit their predictions on the test set, for which labels would not be available to the teams.The final leaderboard will be decided by the metrics on this set.\"\n },\n {\n \"title\": \"Metrics\",\n \"content\": \"We will use two metrics to evaluate performance during the challenge.\"\n },\n {\n \"title\": \"AUC\",\n \"content\": \"One desirable characteristic of the model is to be able identify which samples in the evaluation data belong to novel classes, in other words those which were not present in the source data.To do this, we ask the the competitiors to give a scalar novelty score and compute Area Under the ROC Curve (AUC).The AUC measures unknown (outlier) category detection and is the area under the ROC curve, which is computed by thresholding a score that represents how likely the input is to be an unknown class.\"\n },\n {\n \"title\": \"Instance-wise Accuracy\",\n \"content\": \"We not only want to identify novel classes as OOD data, we also want to have high accuracy on classes present in the source data.Thus, we also employ the instance-wise accuracy over all samples to better understand the behavior of models.\"\n },\n {\n \"title\": \"Development Kit\",\n \"content\": \"The competition provides a development kit (see http://ai.bu.edu/visda-2021/.)The kit contains links to the training, development and test data (test data will be released in the test phase).It also provides baseline code and instructions on how to submit results to the evaluation server, as well as the implementation of metrics.\"\n },\n {\n \"title\": \"Protocol\",\n \"content\": \"The competition will have two phases.During phase I, the training data (ImageNet as a source domain) and validation domains (a mixture of ObjectNet, ImageNet-R, ImageNet-C and ImageNet-O as target domains) will be available to participants to aid in model development and hyperparameter validation.While we allow participants to use labels for validation domains, high performance on validation domains does not guarantee high performance on test data.During phase II, the test data will be made available for a brief period (two weeks) so that participants can train their models on test data (no labels available for test data) and submit label predictions to the evaluation server.\\nTuning on Test Data.An important part of the protocol is that tuning hyperparameters or stopping criteria on the target test labels will NOT be allowed.This is because the target data is unsupervised, so supervised tuning is not possible.Therefore successful approaches will have come up with ways to tune without target-domain labels.\"\n },\n {\n \"title\": \"Competition Results\",\n \"content\": \"This section will be updated with the winning methods and results after the competition's completion.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Objectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition models. A Barbu, D Mayo, J Alverio, W Luo, C Wang, D Gutfreund, J Tenenbaum, B Katz, Advances in neural information processing systems. 201932","doi":""},{"text":"Transferability vs. discriminability: Batch spectral penalization for adversarial domain adaptation. X Chen, S Wang, M Long, J Wang, 2019","doi":""},{"text":"Domain adaptive faster r-cnn for object detection in the wild. Y Chen, W Li, C Sakaridis, D Dai, L Van Gool, 2018","doi":""},{"text":"Imagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, 2009","doi":""},{"text":"Unsupervised domain adaptation by backpropagation. Y Ganin, V Lempitsky, 2014","doi":""},{"text":"R Geirhos, P Rubisch, C Michaelis, M Bethge, F A Wichmann, W Brendel, arXiv:1811.12231Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. 2018arXiv preprint","doi":""},{"text":". K He, G Gkioxari, P Dollár, R Girshick, Mask r-cnn. 2017","doi":""},{"text":"The many faces of robustness: A critical analysis of outof-distribution generalization. D Hendrycks, S Basart, N Mu, S Kadavath, F Wang, E Dorundo, R Desai, T Zhu, S Parajuli, M Guo, arXiv:2006.162412020arXiv preprint","doi":""},{"text":"D Hendrycks, T Dietterich, arXiv:1903.12261Benchmarking neural network robustness to common corruptions and perturbations. 2019arXiv preprint","doi":""},{"text":"Augmix: A simple method to improve robustness and uncertainty under data shift. D Hendrycks, * , N Mu, * , E D Cubuk, B Zoph, J Gilmer, B Lakshminarayanan, International Conference on Learning Representations. 2020","doi":""},{"text":"Natural adversarial examples. D Hendrycks, K Zhao, S Basart, J Steinhardt, D Song, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern Recognition2021","doi":""},{"text":"J Hoffman, D Wang, F Yu, T Darrell, arXiv:1612.02649Fcns in the wild: Pixel-level adversarial and constraint-based adaptation. 2016arXiv preprint","doi":""},{"text":"Minimum class confusion for versatile domain adaptation. Y Jin, X Wang, M Long, J Wang, 2020","doi":""},{"text":"Imagenet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, 2012","doi":""},{"text":"Learning transferable features with deep adaptation networks. M Long, Y Cao, J Wang, M I Jordan, 2015","doi":""},{"text":"Conditional adversarial domain adaptation. M Long, Z Cao, J Wang, M I Jordan, 2018","doi":""},{"text":"Moment matching for multi-source domain adaptation. X Peng, Q Bai, X Xia, Z Huang, K Saenko, B Wang, 2019","doi":""},{"text":"Do imagenet classifiers generalize to imagenet?. B Recht, R Roelofs, L Schmidt, V Shankar, CoRR, abs/1902.108112019","doi":""},{"text":"Faster r-cnn: Towards real-time object detection with region proposal networks. S Ren, K He, R Girshick, J Sun, 2015","doi":""},{"text":"Imagenet large scale visual recognition challenge. O Russakovsky, J Deng, H Su, J Krause, S Satheesh, S Ma, Z Huang, A Karpathy, A Khosla, M Bernstein, International journal of computer vision. 11532015","doi":""},{"text":"Universal domain adaptation through self supervision. K Saito, D Kim, S Sclaroff, K Saenko, arXiv:2002.079532020arXiv preprint","doi":""},{"text":"Very deep convolutional networks for large-scale image recognition. K Simonyan, A Zisserman, arXiv2014","doi":""},{"text":"Return of frustratingly easy domain adaptation. B Sun, J Feng, K Saenko, AAAI. 2016","doi":""},{"text":"Unsupervised domain adaptation via structurally regularized deep clustering. H Tang, K Chen, K Jia, 2020","doi":""},{"text":"Deep domain confusion: Maximizing for domain invariance. E Tzeng, J Hoffman, N Zhang, K Saenko, T Darrell, 2014arXiv","doi":""},{"text":"Larger norm more transferable: An adaptive feature norm approach for unsupervised domain adaptation. R Xu, G Li, J Yang, L Lin, 2019","doi":""},{"text":"Confidence regularized self-training. Y Zou, Z Yu, X Liu, B Kumar, J Wang, 2019","doi":""},{"text":"Unsupervised domain adaptation for semantic segmentation via class-balanced self-training. Y Zou, Z Yu, B Vijaya, J Kumar, Wang, 2018","doi":""},{"text":"Domain Adaptation: Theory and Application. Workshop Nips2011, On,","doi":""},{"text":"NIPS2011, Workshop on Challenges in Learning Hierarchical Models: Transfer Learning and Optimization.","doi":""},{"text":"CVPR2012, half day Tutorial on Domain Transfer Learning for Vision Applications 4. IJCV2013, Special Issue on Domain Adaptation for Vision Applications 5. ICCV2013, Workshop on Visual Domain Adaptation and Dataset Bias (VisDA).","doi":""},{"text":"NIPS2013, Workshop on New Directions in Transfer and Multi-Task: Learning Across Domains and Tasks 7. ECCV2014, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).","doi":""},{"text":"ECCV2014, Tutorial onDomain Adaptation and Transfer Learning 9. ECML2014, Workshop on Learning over Multiple Contexts.","doi":""},{"text":"NIPS2014, Workshop on Transfer and Multi-Task Learning: Theory Meets Practice.","doi":""},{"text":"Special Issue on Multi-Task Learning, Domain Adaptation and Transfer Learning 12. ICDM2015. Workshop on Practical Transfer Learning 13. ICCV2015, Workshop on Transferringand Adapting Source Knowledge (TASK) in Computer Vision (CV).","doi":""},{"text":"NIPS2015, Workshop on Transfer and Multi-Task Learning: Trends and New Perspectives.","doi":""},{"text":"ECCV2016, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).","doi":""},{"text":"ICCV2017, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).","doi":""},{"text":"Bigdata2017, Workshop on Big Data Transfer Learning (BDTL) 18. AAMAS2017, Workshop on Transfer in Reinforcement Learning (TiRL).","doi":""},{"text":"Workshop on Domain Adaptation and Few-Shot Learning (Open MIC) 20. ECCV2018, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV). ACCV 2018","doi":""},{"text":"Tutorial on Transferring Knowledge Across Domains: an Introduction to Deep Domain Adaptation. ICIAP2019","doi":""},{"text":"ICCV2019, tutorial on Visual Learning with Limited Labeled Data.","doi":""},{"text":"ICCV2019, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).","doi":""},{"text":"ECCV2020, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).","doi":""}],"string":"[\n {\n \"text\": \"Objectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition models. A Barbu, D Mayo, J Alverio, W Luo, C Wang, D Gutfreund, J Tenenbaum, B Katz, Advances in neural information processing systems. 201932\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transferability vs. discriminability: Batch spectral penalization for adversarial domain adaptation. X Chen, S Wang, M Long, J Wang, 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Domain adaptive faster r-cnn for object detection in the wild. Y Chen, W Li, C Sakaridis, D Dai, L Van Gool, 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imagenet: A large-scale hierarchical image database. J Deng, W Dong, R Socher, L.-J Li, K Li, L Fei-Fei, 2009\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unsupervised domain adaptation by backpropagation. Y Ganin, V Lempitsky, 2014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R Geirhos, P Rubisch, C Michaelis, M Bethge, F A Wichmann, W Brendel, arXiv:1811.12231Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. 2018arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K He, G Gkioxari, P Dollár, R Girshick, Mask r-cnn. 2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The many faces of robustness: A critical analysis of outof-distribution generalization. D Hendrycks, S Basart, N Mu, S Kadavath, F Wang, E Dorundo, R Desai, T Zhu, S Parajuli, M Guo, arXiv:2006.162412020arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"D Hendrycks, T Dietterich, arXiv:1903.12261Benchmarking neural network robustness to common corruptions and perturbations. 2019arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Augmix: A simple method to improve robustness and uncertainty under data shift. D Hendrycks, * , N Mu, * , E D Cubuk, B Zoph, J Gilmer, B Lakshminarayanan, International Conference on Learning Representations. 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Natural adversarial examples. D Hendrycks, K Zhao, S Basart, J Steinhardt, D Song, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. the IEEE/CVF Conference on Computer Vision and Pattern Recognition2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Hoffman, D Wang, F Yu, T Darrell, arXiv:1612.02649Fcns in the wild: Pixel-level adversarial and constraint-based adaptation. 2016arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Minimum class confusion for versatile domain adaptation. Y Jin, X Wang, M Long, J Wang, 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imagenet classification with deep convolutional neural networks. A Krizhevsky, I Sutskever, G E Hinton, 2012\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning transferable features with deep adaptation networks. M Long, Y Cao, J Wang, M I Jordan, 2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Conditional adversarial domain adaptation. M Long, Z Cao, J Wang, M I Jordan, 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Moment matching for multi-source domain adaptation. X Peng, Q Bai, X Xia, Z Huang, K Saenko, B Wang, 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Do imagenet classifiers generalize to imagenet?. B Recht, R Roelofs, L Schmidt, V Shankar, CoRR, abs/1902.108112019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Faster r-cnn: Towards real-time object detection with region proposal networks. S Ren, K He, R Girshick, J Sun, 2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imagenet large scale visual recognition challenge. O Russakovsky, J Deng, H Su, J Krause, S Satheesh, S Ma, Z Huang, A Karpathy, A Khosla, M Bernstein, International journal of computer vision. 11532015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Universal domain adaptation through self supervision. K Saito, D Kim, S Sclaroff, K Saenko, arXiv:2002.079532020arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Very deep convolutional networks for large-scale image recognition. K Simonyan, A Zisserman, arXiv2014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Return of frustratingly easy domain adaptation. B Sun, J Feng, K Saenko, AAAI. 2016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unsupervised domain adaptation via structurally regularized deep clustering. H Tang, K Chen, K Jia, 2020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deep domain confusion: Maximizing for domain invariance. E Tzeng, J Hoffman, N Zhang, K Saenko, T Darrell, 2014arXiv\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Larger norm more transferable: An adaptive feature norm approach for unsupervised domain adaptation. R Xu, G Li, J Yang, L Lin, 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Confidence regularized self-training. Y Zou, Z Yu, X Liu, B Kumar, J Wang, 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unsupervised domain adaptation for semantic segmentation via class-balanced self-training. Y Zou, Z Yu, B Vijaya, J Kumar, Wang, 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Domain Adaptation: Theory and Application. Workshop Nips2011, On,\",\n \"doi\": \"\"\n },\n {\n \"text\": \"NIPS2011, Workshop on Challenges in Learning Hierarchical Models: Transfer Learning and Optimization.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"CVPR2012, half day Tutorial on Domain Transfer Learning for Vision Applications 4. IJCV2013, Special Issue on Domain Adaptation for Vision Applications 5. ICCV2013, Workshop on Visual Domain Adaptation and Dataset Bias (VisDA).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"NIPS2013, Workshop on New Directions in Transfer and Multi-Task: Learning Across Domains and Tasks 7. ECCV2014, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ECCV2014, Tutorial onDomain Adaptation and Transfer Learning 9. ECML2014, Workshop on Learning over Multiple Contexts.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"NIPS2014, Workshop on Transfer and Multi-Task Learning: Theory Meets Practice.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Special Issue on Multi-Task Learning, Domain Adaptation and Transfer Learning 12. ICDM2015. Workshop on Practical Transfer Learning 13. ICCV2015, Workshop on Transferringand Adapting Source Knowledge (TASK) in Computer Vision (CV).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"NIPS2015, Workshop on Transfer and Multi-Task Learning: Trends and New Perspectives.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ECCV2016, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ICCV2017, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bigdata2017, Workshop on Big Data Transfer Learning (BDTL) 18. AAMAS2017, Workshop on Transfer in Reinforcement Learning (TiRL).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Workshop on Domain Adaptation and Few-Shot Learning (Open MIC) 20. ECCV2018, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV). ACCV 2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tutorial on Transferring Knowledge Across Domains: an Introduction to Deep Domain Adaptation. ICIAP2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ICCV2019, tutorial on Visual Learning with Limited Labeled Data.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ICCV2019, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ECCV2020, Workshop on Transferring and Adapting Source Knowledge (TASK) in Computer Vision (CV).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":56,"cells":{"text":{"kind":"string","value":"\nSupplementary Material: Closed-loop robots driven by short-term synaptic plasticity: Emergent explorative vs. limit-cycle locomotion\n\n\nLaura Martin \nCorrespondence: Bulcsú Sándor\n\n\nBulcsú Sándor \nCorrespondence: Bulcsú Sándor\n\n\nClaudius Gros \nCorrespondence: Bulcsú Sándor\n\n\nSupplementary Material: Closed-loop robots driven by short-term synaptic plasticity: Emergent explorative vs. limit-cycle locomotion\n\nSUPPLEMENTARY VIDEOSHere we provide supplementary videos to the different dynamical behaviors discussed in the main paper. The simulations can be reproduced by using the LpzRobots simulation software, available on http: //robot.informatik.uni-leipzig.de/software https://github.com/georgmartius/lpzrobots.Regular motion patternsThe different locomotion modes found for U max = 1 (seeFig. 3) are compared in Suppl. Video 1. Six robots are created with the same settings, except for the (w 0 , z 0 ) parameter pairs, corresponding to the examples presented in Fig. 4: T1: (280, 650), T2: (230, 415), C1: (190, 600), S1: (250, 530), S2: (240, 380), S3: (220, 470). As result of locomotion the robots interact. Following the individual trajectories, among other interesting phenomena, autonomous mode switching is observed after collisions:• The magenta robot displays the T1 mode, which coexists with the S1 and C1 motion patterns. Bumping into the red robot, first the center of the star-shaped trajectory is shifted. After the second collision it switches to the C1 mode, precessing transiently to the left, while as a result of a third collision it sets into the rightwards circulating mode.• The light-blue (cyan) colored robot stays in T2 mode even after the collision, since for these parameters (seeFig. 4) other stable motion pattern does not exist.• The same holds for the C1 mode as well, demonstrated by the orange robot. Note that the radii of the circular patterns generated by the orange and the magenta robots differ significantly, an effect of the significantly different w 0 parameters.• The S1 mode is visualized by the dark-blue spherical robot. The collision induces an autonomous switch to C1.• S2 mode is essential for our results, being a singleton locomotive mode in its parameter domain. Hence interactions with other robots or walls can only change the direction of translation. In Suppl. Video 1 one can observe seven such collisions.• The green robot, originally started in S3 mode, changes to S1 on an early impact.\n\n• The magenta robot displays the T1 mode, which coexists with the S1 and C1 motion patterns. Bumping into the red robot, first the center of the star-shaped trajectory is shifted. After the second collision it switches to the C1 mode, precessing transiently to the left, while as a result of a third collision it sets into the rightwards circulating mode.\n\n• The light-blue (cyan) colored robot stays in T2 mode even after the collision, since for these parameters (see Fig. 4) other stable motion pattern does not exist.\n\n• The same holds for the C1 mode as well, demonstrated by the orange robot. Note that the radii of the circular patterns generated by the orange and the magenta robots differ significantly, an effect of the significantly different w 0 parameters.\n\n• The S1 mode is visualized by the dark-blue spherical robot. The collision induces an autonomous switch to C1.\n\n• S2 mode is essential for our results, being a singleton locomotive mode in its parameter domain. Hence interactions with other robots or walls can only change the direction of translation. In Suppl. Video 1 one can observe seven such collisions.\n\nSupplementary Video 1. Examples of regular motions patterns T1, T2, C1, S1, S2, S3, generated by identical robots, using U max = 1. The (w 0 , z 0 ) parameter pairs are set for each robots separately, corresponding to the values used for the close-up of the trajectories in Fig. 4 (having initially the same order and color coding). When several modes coexist for the same parameters, robots interacting with each other can change their behavior, choosing an other working mode. See for instance the green robot, changing from S3 to S1 after colliding with the yellow one. See: Video 1.avi.\n\nSupplementary Video 2. Close-up showing the dynamics of the robot in the chaotic mode, using U max = 1 and (w 0 , z 0 ) = (210, 400). The smaller balls fixed to the end of the rods are guides to the eyes, indicating the negative direction along the axes, without any physical influence on the dynamics. The obstacles, having similar masses to the mass of the robot, are pushed apart in different directions. See: Video 2.avi.\n\n\nPlayful behavior\n\nThe playful behavior of the spherical robot is shown in Suppl. Video 2 and 3, for the parameter settings used for Fig. 6, U max = 4 and (w 0 , z 0 ) = (210, 400). Autonomous mode switching can be observed between the coexisting chaotic meandering and C1 modes, pushing objects, of masses similar to its own body mass, in a playful manner.\n\n\nExplorative behavior\n\nIn Suppl. Video 4 the two types of chaotic explorations are compared in a maze, also shown in Fig. 10, using U max = 1 and U max = 4. Additional stable modes also arise in environments different from the bare plane ground (see the locomotion along the walls).\n\nSupplementary Video 3. The playful behavior of the robot in the closed playground, viewed from the top. The parameters are set as for Suppl. Video 2, corresponding to the chaotic mode. The blue trajectory, plotted on the ground, traces the C1 locomotion pattern occasionally out, which is unstable for these settings (as discussed in the main text of the paper). See: Video 3.avi.\n\nSupplementary Video 4. Two robots showing explorative chaotic behavior, placed in a maze with fixed walls, using U max = 1, (w 0 , z 0 ) = (210, 400) for the cyan, and U max = 4, (w 0 , z 0 ) = (180, 80) for the dark-blue colored robot. Note that the robots also interact occasionally. See: Video 4.avi.\n\n\nCHAOTIC MODES\n\nTo examine the different modes from a dynamical systems point of view, one can consider the corresponding attractors in the subspace of internal variables (see also (Sándor et al., 2015)). The chaotic attractor corresponding to the explorative behavior with abrupt change of directions, found for U max = 1, has a densely filling structure, as shown in Suppl. Fig. 5. For U max = 4, with the angle of propagation only smoothly diffusing, however, the topology resembles closed braids, a sign of partial predictability, typically found close to period doubling bifurcations (Wernecke et al., 2016). As a result of the attractor's topology the discrete mode switching can never occur for U max = 4 (see Fig. 12).\n\nNote that for U max = 4 there are totally three chaotic attractors in the phase space of internal variables, which are ultimately generating the same type of explorative behavior, being related by symmetry operations (corresponding to the permutation of colors in Fig. 12). Hence, in this case discrete mode switching (between these attractors), similar to the S2 case, can be achieved by adding external noise.\n Supplementary Material3 ), using (w 0 , z 0 ) = (210, 400) with U max = 1 on the left, and (w 0 , z 0 ) = (180, 80) with U max = 4 on the right (only one shown of the three).\nThe sensorimotor loop as a dynamical system: How regular motion primitives may emerge from self-organized limit cycles. B Sándor, T Jahn, L Martin, C Gros, Frontiers in Robotics and AI. 231Sándor, B., Jahn, T., Martin, L., and Gros, C. (2015). The sensorimotor loop as a dynamical system: How regular motion primitives may emerge from self-organized limit cycles. Frontiers in Robotics and AI 2, 31\n\n. H Wernecke, B Sándor, C Gros, arXiv:1605.05616arXiv preprintPartially predictable chaosWernecke, H., Sándor, B., and Gros, C. (2016). Partially predictable chaos. arXiv preprint arXiv:1605.05616\n"},"corpus_id":{"kind":"number","value":3859018,"string":"3,859,018"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1608.02838\",\n \"mag\": 3098302626,\n \"acl\": null,\n \"pubmed\": \"27803661\",\n \"pubmedcentral\": 5067527,\n \"dblp\": \"journals/corr/abs-1608-02838\",\n \"doi\": \"10.3389/fnbot.2016.00012\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [],\n \"pdf_sha\": \"96bbdf370fee496beefb2233a83fa8185bb5de7b\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://www.frontiersin.org/articles/10.3389/fnbot.2016.00012/pdf\",\n \"status\": \"GOLD\"\n }\n}"},"title":{"kind":"string","value":"Supplementary Material: Closed-loop robots driven by short-term synaptic plasticity: Emergent explorative vs. limit-cycle locomotion"},"authors":{"kind":"list like","value":["Laura Martin \nCorrespondence: Bulcsú Sándor","Bulcsú Sándor \nCorrespondence: Bulcsú Sándor","Claudius Gros \nCorrespondence: Bulcsú Sándor"],"string":"[\n \"Laura Martin \\nCorrespondence: Bulcsú Sándor\",\n \"Bulcsú Sándor \\nCorrespondence: Bulcsú Sándor\",\n \"Claudius Gros \\nCorrespondence: Bulcsú Sándor\"\n]","isConversational":false},"abstract":{"kind":"string","value":"SUPPLEMENTARY VIDEOSHere we provide supplementary videos to the different dynamical behaviors discussed in the main paper. The simulations can be reproduced by using the LpzRobots simulation software, available on http: //robot.informatik.uni-leipzig.de/software https://github.com/georgmartius/lpzrobots.Regular motion patternsThe different locomotion modes found for U max = 1 (seeFig. 3) are compared in Suppl. Video 1. Six robots are created with the same settings, except for the (w 0 , z 0 ) parameter pairs, corresponding to the examples presented in Fig. 4: T1: (280, 650), T2: (230, 415), C1: (190, 600), S1: (250, 530), S2: (240, 380), S3: (220, 470). As result of locomotion the robots interact. Following the individual trajectories, among other interesting phenomena, autonomous mode switching is observed after collisions:• The magenta robot displays the T1 mode, which coexists with the S1 and C1 motion patterns. Bumping into the red robot, first the center of the star-shaped trajectory is shifted. After the second collision it switches to the C1 mode, precessing transiently to the left, while as a result of a third collision it sets into the rightwards circulating mode.• The light-blue (cyan) colored robot stays in T2 mode even after the collision, since for these parameters (seeFig. 4) other stable motion pattern does not exist.• The same holds for the C1 mode as well, demonstrated by the orange robot. Note that the radii of the circular patterns generated by the orange and the magenta robots differ significantly, an effect of the significantly different w 0 parameters.• The S1 mode is visualized by the dark-blue spherical robot. The collision induces an autonomous switch to C1.• S2 mode is essential for our results, being a singleton locomotive mode in its parameter domain. Hence interactions with other robots or walls can only change the direction of translation. In Suppl. Video 1 one can observe seven such collisions.• The green robot, originally started in S3 mode, changes to S1 on an early impact."},"sections":{"kind":"list like","value":[{"title":"Playful behavior","content":"The playful behavior of the spherical robot is shown in Suppl. Video 2 and 3, for the parameter settings used for Fig. 6, U max = 4 and (w 0 , z 0 ) = (210, 400). Autonomous mode switching can be observed between the coexisting chaotic meandering and C1 modes, pushing objects, of masses similar to its own body mass, in a playful manner."},{"title":"Explorative behavior","content":"In Suppl. Video 4 the two types of chaotic explorations are compared in a maze, also shown in Fig. 10, using U max = 1 and U max = 4. Additional stable modes also arise in environments different from the bare plane ground (see the locomotion along the walls).\nSupplementary Video 3. The playful behavior of the robot in the closed playground, viewed from the top. The parameters are set as for Suppl. Video 2, corresponding to the chaotic mode. The blue trajectory, plotted on the ground, traces the C1 locomotion pattern occasionally out, which is unstable for these settings (as discussed in the main text of the paper). See: Video 3.avi.\nSupplementary Video 4. Two robots showing explorative chaotic behavior, placed in a maze with fixed walls, using U max = 1, (w 0 , z 0 ) = (210, 400) for the cyan, and U max = 4, (w 0 , z 0 ) = (180, 80) for the dark-blue colored robot. Note that the robots also interact occasionally. See: Video 4.avi."},{"title":"CHAOTIC MODES","content":"To examine the different modes from a dynamical systems point of view, one can consider the corresponding attractors in the subspace of internal variables (see also (Sándor et al., 2015)). The chaotic attractor corresponding to the explorative behavior with abrupt change of directions, found for U max = 1, has a densely filling structure, as shown in Suppl. Fig. 5. For U max = 4, with the angle of propagation only smoothly diffusing, however, the topology resembles closed braids, a sign of partial predictability, typically found close to period doubling bifurcations (Wernecke et al., 2016). As a result of the attractor's topology the discrete mode switching can never occur for U max = 4 (see Fig. 12).\nNote that for U max = 4 there are totally three chaotic attractors in the phase space of internal variables, which are ultimately generating the same type of explorative behavior, being related by symmetry operations (corresponding to the permutation of colors in Fig. 12). Hence, in this case discrete mode switching (between these attractors), similar to the S2 case, can be achieved by adding external noise."}],"string":"[\n {\n \"title\": \"Playful behavior\",\n \"content\": \"The playful behavior of the spherical robot is shown in Suppl. Video 2 and 3, for the parameter settings used for Fig. 6, U max = 4 and (w 0 , z 0 ) = (210, 400). Autonomous mode switching can be observed between the coexisting chaotic meandering and C1 modes, pushing objects, of masses similar to its own body mass, in a playful manner.\"\n },\n {\n \"title\": \"Explorative behavior\",\n \"content\": \"In Suppl. Video 4 the two types of chaotic explorations are compared in a maze, also shown in Fig. 10, using U max = 1 and U max = 4. Additional stable modes also arise in environments different from the bare plane ground (see the locomotion along the walls).\\nSupplementary Video 3. The playful behavior of the robot in the closed playground, viewed from the top. The parameters are set as for Suppl. Video 2, corresponding to the chaotic mode. The blue trajectory, plotted on the ground, traces the C1 locomotion pattern occasionally out, which is unstable for these settings (as discussed in the main text of the paper). See: Video 3.avi.\\nSupplementary Video 4. Two robots showing explorative chaotic behavior, placed in a maze with fixed walls, using U max = 1, (w 0 , z 0 ) = (210, 400) for the cyan, and U max = 4, (w 0 , z 0 ) = (180, 80) for the dark-blue colored robot. Note that the robots also interact occasionally. See: Video 4.avi.\"\n },\n {\n \"title\": \"CHAOTIC MODES\",\n \"content\": \"To examine the different modes from a dynamical systems point of view, one can consider the corresponding attractors in the subspace of internal variables (see also (Sándor et al., 2015)). The chaotic attractor corresponding to the explorative behavior with abrupt change of directions, found for U max = 1, has a densely filling structure, as shown in Suppl. Fig. 5. For U max = 4, with the angle of propagation only smoothly diffusing, however, the topology resembles closed braids, a sign of partial predictability, typically found close to period doubling bifurcations (Wernecke et al., 2016). As a result of the attractor's topology the discrete mode switching can never occur for U max = 4 (see Fig. 12).\\nNote that for U max = 4 there are totally three chaotic attractors in the phase space of internal variables, which are ultimately generating the same type of explorative behavior, being related by symmetry operations (corresponding to the permutation of colors in Fig. 12). Hence, in this case discrete mode switching (between these attractors), similar to the S2 case, can be achieved by adding external noise.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The sensorimotor loop as a dynamical system: How regular motion primitives may emerge from self-organized limit cycles. B Sándor, T Jahn, L Martin, C Gros, Frontiers in Robotics and AI. 231Sándor, B., Jahn, T., Martin, L., and Gros, C. (2015). The sensorimotor loop as a dynamical system: How regular motion primitives may emerge from self-organized limit cycles. Frontiers in Robotics and AI 2, 31","doi":""},{"text":". H Wernecke, B Sándor, C Gros, arXiv:1605.05616arXiv preprintPartially predictable chaosWernecke, H., Sándor, B., and Gros, C. (2016). Partially predictable chaos. arXiv preprint arXiv:1605.05616","doi":"arXiv:1605.05616"}],"string":"[\n {\n \"text\": \"The sensorimotor loop as a dynamical system: How regular motion primitives may emerge from self-organized limit cycles. B Sándor, T Jahn, L Martin, C Gros, Frontiers in Robotics and AI. 231Sándor, B., Jahn, T., Martin, L., and Gros, C. (2015). The sensorimotor loop as a dynamical system: How regular motion primitives may emerge from self-organized limit cycles. Frontiers in Robotics and AI 2, 31\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Wernecke, B Sándor, C Gros, arXiv:1605.05616arXiv preprintPartially predictable chaosWernecke, H., Sándor, B., and Gros, C. (2016). Partially predictable chaos. arXiv preprint arXiv:1605.05616\",\n \"doi\": \"arXiv:1605.05616\"\n }\n]","isConversational":false}}},{"rowIdx":57,"cells":{"text":{"kind":"string","value":"\nIs gravitational entropy quantized ?\n9 Jul 2008\n\nDawood Kothawala \nIUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia\n\nT Padmanabhan \nIUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia\n\nSudipta Sarkar \nIUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia\n\nIs gravitational entropy quantized ?\n9 Jul 2008(Dated: July 9, 2008)\nIn Einstein's gravity, the entropy of horizons is proportional to their area. Several arguments given in the literature suggest that, in this context, both area and entropy should be quantized with an equally spaced spectrum for large quantum numbers. But in more general theories (like, for e.g, in the black hole solutions of Gauss-Bonnet or Lanczos-Lovelock gravity) the horizon entropy is not proportional to area and the question arises as to which of the two (if at all) will have this property. We give a general argument that in all Lanczos-Lovelock theories of gravity, it is the entropy that has equally spaced spectrum. In the case of Gauss-Bonnet gravity, we use the asymptotic form of quasi normal mode frequencies to explicitly demonstrate this result. Hence, the concept of a quantum of area in Einstein Hilbert (EH) gravity needs to be replaced by a concept of quantum of entropy in a more general context. PACS numbers: 04.62.+v, 04.60.-mIt was conjectured by Bekenstein [1] long back that, in a quantum theory, the black hole area would be represented by a quantum operator with a discrete spectrum of eigenvalues. Bekenstein showed that the area of a classical black hole behaves like an adiabatic invariant, and so, according to Ehrenfest's theorem, the corresponding quantum operator must have a discrete spectrum. It was also known that, when a quantum particle is captured by a (non extremal) black hole its area increases by a minimum non-zero value [1, 2, 3] which is independent of the black hole parameters. This argument also suggests an equidistant spacing of area levels, with a well-defined notion of a quantum of area. The fundamental constants G, c and combine to give a quantity with the dimensions of area A P = (G /c 3 ) = 10 −66 cm 2 , which is quite suggestive[4]and sets the scale in area quantization.In Einstein's gravity, entropy of the horizon is proportional to its area. Hence one could equivalently claim that it is the gravitational entropy which has an equidistant spectrum with a well-defined notion of quantum of entropy. But, when one considers the natural generalization of Einstein gravity by including higher derivative correction terms to the original Einstein-Hilbert action, no such trivial relationship remains valid between horizon area and associated entropy. One such higher derivative theory which has attracted a fair amount of attention is Lanczos-Lovelock (LL) gravity [5], of which the lowest order correction appears as a Gauss-Bonnet (GB) term in D(> 4) dimensions. These lagrangians have the unique feature that the field equations obtained from them are quasi linear, as a result of which, the initial value problem remains well defined. More importantly, several features related to horizon thermodynamics, which were first discovered in the context of Einstein's theory [6], continues * Electronic address: dawood@iucaa.ernet.in † Electronic address: paddy@iucaa.ernet.in ‡ Electronic address: sudipta@iucaa.ernet.in to be valid in LL gravity models[7,8].Black hole solutions in the LL gravity are well studied in the literature. For these spacetimes, the notion of entropy can be defined using Wald's formalism[9], where entropy is associated with the Noether charge of the diffeomorphism invariance symmetry of the theory. The entropy calculated from this approach turns out to be no longer proportional to horizon area. The question then arises as to whether it is the quantum of area or quantum of entropy (if at all either) which arises in a natural manner in these models. We attempt to answer this question in this paper.We will first provide a very general argument which suggests that it is the entropy which is quantized with equidistant spectrum in the case of LL gravity and then provide an explicit proof for the result in the context of GB gravity.In any geometrical description of gravity that obeys the principle of equivalence and is based on a nontrivial metric, the propagation of light rays will be affected by gravity. This, in turn, leads to regions of spacetime which are causally inaccessible to classes of observers. (These two features are reasonably independent of the precise field equations which determine the metric.). The fact that any observer has a right to formulate physical theories in a given coordinate system entirely in terms of the variables that an observer using that coordinate system can access, imposes strong constraints on the nature of the action functional A grav which can be used to describe gravity[10]. Suppose we divide the space-time manifold into two regions separated by a null hypersurface H and choose a coordinate system such that H acts as a horizon for the observer on one side (say side 1). The effective theory for the observer on side 1 (with the degrees of freedom formally denoted by g 1 ) is obtained by integrating out the variables on the inaccessible side (side 2): In the semiclassical limit, saddle-point integration leads to the exponential of the classical action A grav (class) evaluated on-shell. The effective theory on side 1 is thus described\n\nIn Einstein's gravity, the entropy of horizons is proportional to their area. Several arguments given in the literature suggest that, in this context, both area and entropy should be quantized with an equally spaced spectrum for large quantum numbers. But in more general theories (like, for e.g, in the black hole solutions of Gauss-Bonnet or Lanczos-Lovelock gravity) the horizon entropy is not proportional to area and the question arises as to which of the two (if at all) will have this property. We give a general argument that in all Lanczos-Lovelock theories of gravity, it is the entropy that has equally spaced spectrum. In the case of Gauss-Bonnet gravity, we use the asymptotic form of quasi normal mode frequencies to explicitly demonstrate this result. Hence, the concept of a quantum of area in Einstein Hilbert (EH) gravity needs to be replaced by a concept of quantum of entropy in a more general context. PACS numbers: 04.62.+v, It was conjectured by Bekenstein [1] long back that, in a quantum theory, the black hole area would be represented by a quantum operator with a discrete spectrum of eigenvalues. Bekenstein showed that the area of a classical black hole behaves like an adiabatic invariant, and so, according to Ehrenfest's theorem, the corresponding quantum operator must have a discrete spectrum. It was also known that, when a quantum particle is captured by a (non extremal) black hole its area increases by a minimum non-zero value [1,2,3] which is independent of the black hole parameters. This argument also suggests an equidistant spacing of area levels, with a well-defined notion of a quantum of area. The fundamental constants G, c and combine to give a quantity with the dimensions of area A P = (G /c 3 ) = 10 −66 cm 2 , which is quite suggestive [4] and sets the scale in area quantization.\n\nIn Einstein's gravity, entropy of the horizon is proportional to its area. Hence one could equivalently claim that it is the gravitational entropy which has an equidistant spectrum with a well-defined notion of quantum of entropy. But, when one considers the natural generalization of Einstein gravity by including higher derivative correction terms to the original Einstein-Hilbert action, no such trivial relationship remains valid between horizon area and associated entropy. One such higher derivative theory which has attracted a fair amount of attention is Lanczos-Lovelock (LL) gravity [5], of which the lowest order correction appears as a Gauss-Bonnet (GB) term in D(> 4) dimensions. These lagrangians have the unique feature that the field equations obtained from them are quasi linear, as a result of which, the initial value problem remains well defined. More importantly, several features related to horizon thermodynamics, which were first discovered in the context of Einstein's theory [6], continues * Electronic address: dawood@iucaa.ernet.in † Electronic address: paddy@iucaa.ernet.in ‡ Electronic address: sudipta@iucaa.ernet.in to be valid in LL gravity models [7,8].\n\nBlack hole solutions in the LL gravity are well studied in the literature. For these spacetimes, the notion of entropy can be defined using Wald's formalism [9], where entropy is associated with the Noether charge of the diffeomorphism invariance symmetry of the theory. The entropy calculated from this approach turns out to be no longer proportional to horizon area. The question then arises as to whether it is the quantum of area or quantum of entropy (if at all either) which arises in a natural manner in these models. We attempt to answer this question in this paper.\n\nWe will first provide a very general argument which suggests that it is the entropy which is quantized with equidistant spectrum in the case of LL gravity and then provide an explicit proof for the result in the context of GB gravity.\n\nIn any geometrical description of gravity that obeys the principle of equivalence and is based on a nontrivial metric, the propagation of light rays will be affected by gravity. This, in turn, leads to regions of spacetime which are causally inaccessible to classes of observers. (These two features are reasonably independent of the precise field equations which determine the metric.). The fact that any observer has a right to formulate physical theories in a given coordinate system entirely in terms of the variables that an observer using that coordinate system can access, imposes strong constraints on the nature of the action functional A grav which can be used to describe gravity [10]. Suppose we divide the space-time manifold into two regions separated by a null hypersurface H and choose a coordinate system such that H acts as a horizon for the observer on one side (say side 1). The effective theory for the observer on side 1 (with the degrees of freedom formally denoted by g 1 ) is obtained by integrating out the variables on the inaccessible side (side 2): In the semiclassical limit, saddle-point integration leads to the exponential of the classical action A grav (class) evaluated on-shell. The effective theory on side 1 is thus described (1) Since the effects of the unobserved degrees of freedom g 2 can only be encoded in the geometry of the (shared) boundary between regions 1 and 2, we get the constraint that the on-shell value of the action A grav (g class 2 ) must be expressible in terms of the boundary geometry which could be expressible in terms of g 1 itself. That is, (2) This is a non-trivial requirement on any geometrical theory of gravity. Further since the boundary term depends on the choice of the coordinate system (or foliation), in which H acts as a one-way membrane, A sur (g 1 ) will in general depend on the coordinate choice for the observer.\nby the action A WKB eff (g 1 ), with exp[iA WKB eff (g 1 )] ≃ exp[i(A grav (g 1 ) + A grav (class))].A grav (g class 2 ) = A sur (g 1 ), and exp[iA WKB eff (g 1 )] = exp[i(A grav (g 1 ) + A sur (g 1 ))].\nClassically, with the boundary variables held fixed, the equations of motion remain unaffected by the existence of a (total divergence) boundary term; hence the fact that the boundary term is not generally covariant is unimportant for classical theory. This is, of course, not true in semiclassical/quantum theory. But since the quantum theory is governed by exp[iA eff ] rather than by A eff , the boundary term will have no effect in the quantum theory, if the quantum processes keep exp[iA sur ] single-valued. This is equivalent to demanding that the boundary term satisfies the quantization condition A sur = 2πn. (More precisely, the change in the surface term ∆A sur = 2π; this is irrelevant for our purpose when we work in the semiclassical limit of large n).\n\nIt is now worth noting that the lagrangian in all the LL models (of which Einstein-Hilbert action is just a special case) can be expressed [8] as a sum of a bulk and total divergence terms, L = L bulk + L sur with L sur integrating to give a surface term in the action. There is a peculiar 'holographic' relationship between L bulk and L sur in all these models with the same information being coded in both the bulk and surface terms [see eq.(41) of ref. [8]]. The on-shell value of the surface term in all these action functionals is proportional to the Wald entropy of the horizon [8,12]. We can now see how a condition like A sur = 2πn can lead to quantization of Wald entropy.\n\nIn the case of Einstein-Hilbert action, A sur is well defined and is given by the standard Gibbons-Hawking-York term. As pointed out in ref. [10] the surface term will give the entropy -equal to one quarter of horizon area -and both will have an equally-spaced spectrum. When we proceed to general LL gravity, the correct (surface) counterterm which should be added to the higher derivative action is unknown and the action principle, using metric as dynamical variable, is actually ill-defined. There is, however, an alternative approach we can follow to obtain meaningful results for the LL gravity.\n\nSo far we did not have to specify the exact nature of the degrees of freedom g 1 , g 2 in the above discussion. Interestingly enough these arguments go through unhindered, when one formulates gravity as an emergent phenomenon without treating the metric as dynamical variables in the theory [11]. In this approach one proceeds along the following lines:\n\nAround any event in spacetime one can introduce a local inertial frame and -by boosting with a uniform acceleration -a local Rindler frame. The effective long range variables in emergent gravity approach are the normals n a to the null surfaces which act as local Rindler horizons. (One can think of n a as the 'fluid velocity' of a virtual null fluid in the spacetime.). The total action is now [11,12] taken to be A tot = A grav + A matt where\nA grav = −4 V d D x √ −gP cd ab ∇ c n a ∇ d n b(3)\nis determined by a fourth rank tensor P abcd which can be expressed as a derivative of the LL lagrangian and\nA matt = V d D x √ −gT ab n a n b(4)\nMaximizing A tot with respect to all n a leads to the field equations of the LL theory. (All these aspects are described in detail in ref. [12] and hence not repeated here.)\n\nThe key result we need here is that the on-shell value of the total action is given by\nA tot | on−shell = 4 ∂V d D−1 Σ a P abcd n c ∇ b n d(5)\nwhich can be shown to be identically equal to the Wald entropy of the horizon in the LL theory [8,12]. (In the case of lowest order LL theory -which is just Einstein gravity -the expression for A| on−shell will be one quarter of the transverse area of the horizon.) The emergent gravity approach is strongly motivated by thermodynamic considerations andclassically -the maximization of the action can be thought of as maximization of the entropy. In this context, Eq. (5) will also give the entropy of the local Rindler horizon for each Rindler observer, which can be interpreted as due to integrating out the inaccessible degrees of freedom behind the local Rindler horizon. In the semiclassical limit, the on-shell value of the action will be related to the phase of the semiclassical wave function Ψ ∝ exp (iA| on−shell ) This expression, of course, should be generally covariant but as it stands it explicitly depends on the Rindler observer chosen to define the horizon. Hence we can ensure observer independence of semiclassical gravity only if we assume\nA Wald = A| on−shell = 2πn(6)\nNote that we are again obtaining the quantization condition from the phase of the semiclassical wavefunction which is completely in accord with previous approaches to this problem. The holographic relation between surface and bulk terms underscores how the surface term captures the dynamical information contained in the bulk.\n\nWhile this gives a general result that in LL theories it is the entropy of the horizon that is quantized, it would be nice if the result could be reinforced by an explicit calculation within the standard context. Fortunately, this can be done for GB theory using the arguments suggested by Hod [13] based on quasinormal modes of black hole oscillations.\n\nHod started from Bekenstein's arguments regarding quantum area spectrum of a non extremal Kerr-Newman black hole, and showed that the spacing of area eigenvalues can be fixed by associating the classical limit of the quasinormal mode frequencies, ω c , with the large n limit of the quantum area spectrum, in the spirit of Bohr's correspondence principle (n being the quantum number). Specifically, for a Schwarzschild black hole of mass M in (3 + 1) dimensions, the absorption of a quantum of energy ω c , (in units with = 1) would lead to change in the black hole area eigenvalues as, ∆A ≡ A n+1 − A n = (∂A/∂M )ω c and for entropy ∆S = (∂S/∂M )ω c . In the case of a Schwarzschild black hole the level spacing of both area and entropy eigenvalues were indeed found to be equidistant, allowing one to associate the notion of a minimum unit, a quantum, of area and entropy.\n\nWe will use these ideas in the context of 5D GB black holes, using the numerically known form of the quasinormal mode frequencies. We show that, the form of the highly damped quasi normal modes of these black holes suggest that it is the entropy which has a equally spaced spectrum. The Gauss-Bonnet (GB) Lagrangian L in D dimensions is given by [18],\n(16πG)L = R + α GB (R 2 − 4R ab R ab + R abcd R abcd )\nStatic, spherically symmetric black hole solutions in this theory is of the form,\nds 2 = −f (r)dt 2 + f (r) −1 dr 2 + r 2 dΩ D−2 where, f (r) = 1 + r 2 2α 1 − 1 + 4 α ̟ r D−1 1/2\nHere, α = (D − 3)(D − 4)α GB and ̟ is related to the ADM mass M by the relationship,\n̟ = 16πG (D − 2)Σ D−2 M\nwhere Σ D−2 is the volume of unit (D − 2) sphere. The Hawking temperature T , and entropy S for this spacetime are,\nT = D − 3 4πr + r 2 + r 2 + + 2α + α D − 5 D − 3 1 r 2 + + 2α S = A 4G 1 + 2α D − 2 D − 4 A Σ D−2 −2/(D−2) where A = Σ D−2 r D−2 +\nis the horizon area. The location of the horizon is found as roots of q(r + ) = 0, where q(r) = r D−3 + α r D−5 − ̟, and for the horizon to exist at all, one must also have r 2 + + 2α ≥ 0. The highly damped quasi normal modes for the GB black holes (when ω I ≫ ω R ) has been worked out for D = 5. These QNM frequencies are given by [16] ω(n) −→ n→∞ T ln Q + i(2πT )n (The imaginary part can be understood in terms of a scattering matrix formalism; see e.g., [17]). We now use the Hod conjecture to obtain the entropy spacing for this spacetime. Accordingly, we identify as the relevant frequency ω c the real part of ω, i.e., we take ω c = T ln Q. The entropy spacing is then given by\nS n+1 − S n = ∂S ∂M ω c = ln Q\nClearly the spacing ∆S ≡ S n+1 − S n is a constant. This result depends essentially only on the fact that Re ω c ∝ T leading to (∂S/∂M )ω c ∝ T (∂S/∂M ) which is a constant. For GB black holes the area is a function of entropy, A = F (S) which is not linear. Hence, for the area spectrum for this class of black holes we get\nA n+1 − A n = ∂A ∂M ω c = g(A n ) ln Q(7)\nwhere, g(A n ) = dF/dS is given by,\ng(A n ) = 4 1 − 2α A n Σ 3 −2/3(8)\nwhich is correct to O(α). We therefore find that the entropy eigenvalues are discrete and equally spaced but the area spacing is not equidistant. Hence, for GB gravity, the notion of quantum of entropy is more natural than the quantum of area.\n\nWe shall now comment on the value of ln Q which determines the actual value of quantum of entropy. Originally, in the case of Einstein gravity, the results of Bekenstein and Hod lead to the picture of a quantum black hole with the horizon built out of patches of area ǫ A P , the most natural choice for ǫ being the (constant) spacing of eigenvalues of the area quantum operator, A. Hod further argued that one should take, for ω c , the real part of the quasi normal mode frequencies after the imaginary part has been sent to infinity. This leads to ǫ = 4 ln k, k being some integer. The numerical, as well as later analytical results for the quasi normal mode frequencies of spherically symmetric black holes in 3+1 dimensions, give k = 3.\n\nRecently, Maggiore [14] has put forth another argument which leads to identifying the transition frequency between large n levels with the classical limit (rather than the real part of QNM frequencies, as was done by Hod). This gives ǫ = 8π, consistent with earlier arguments of Bekenstein. While the specific value of area spacing is important for a statistical definition of entropy, it does not seem to be absolutely essential since, in a semiclassical description, the number of microstates need not exactly come out to be an integer, as was argued by Maggiore. Recently, all these arguments have been applied for the case of more general black holes in the context of Einstein gravity and it has been argued that in all such cases, when properly analyzed, one finds an equally spaced area spectrum [15]. The suggestion by Maggiore [14] to associate this classical limit with transition frequencies (n+ 1) → n as n → ∞ leads to the replacement: ln Q → 2π, which gives ∆S = 2π (in units of ). Thus, we obtain, for the quantum of entropy, a value of 2π in agreement with the general arguments given earlier.\n\nThe broader picture which emerges from this analysis can be summarized along these lines: (a) In any theory which obeys the principle of equivalence, the gravitational field will be described at long wavelengths by a spacetime metric. (b) Around any event in spacetime, one can introduce a local inertial frame and -by boosting with an acceleration -a local Rindler frame. The observers using this coordinate system will have a local Rindler horizon with a temperature and entropy associated with the virtual deformations of the horizon. (c) Classically, we interpret A tot in Eq. (3) as the total entropy which is maximized for all the Rindler observers to give the field equations of the theory (which are the same as the equations of LL gravity). The on-shell value of the action giving the Wald entropy of the horizon which is interpreted as due to modes which are inaccessible to the given observer. (d) In the semiclassical limit the A tot is interpreted as an action and its value will affect the phase of the semiclassical wave function. (e) The observer independence of the semiclassical gravity requires this phase -i.e., the Wald entropy of the horizon -to be quantized in units of 2π. (Of course, mathematically, one could have treated A tot as the action functional even in classical theory.)\n\nWhen we take the lowest order LL theory, we reproduce Einstein gravity and the quantization condition becomes equivalent to area quantization of the horizon as discussed several times in the literature. At the next order, we have the GB theory for which we have explicitly demonstrated the quantization of entropy. We believe that, once we have the structure of QNM in the case of LL theory -which, as far as we know, has not yet been explicitly worked out -the analysis given above can be repeated to give an explicit demonstration of this result. Since entropy is directly related with information content, the quantum of gravitational entropy points out a new and intriguing relationship between gravity, quantum theory and thermodynamics.\n\nDK and SS are supported by the Council of Scientific & Industrial Research, India.\n \n. J D Bekenstein, Lett, Nuovo Cimento. 11J.D. Bekenstein, Lett. Nuovo Cimento 11, 467, (1974);\n\n. J D Bekenstein, gr-qc/9710076J.D. Bekenstein [gr-qc/9710076].\n\n. D Christodoulou, Phys. Rev. Lett. 25Christodoulou, D, Phys. Rev. Lett. 25, 1596-1597, (1970).\n\nThis result only depends on spherical symmetry and the first law of black hole thermodynamics, and can be easily extended to black holes in Gauss-Bonnet gravity, leading to the same result for minimum increase of entropy. This result only depends on spherical symmetry and the first law of black hole thermodynamics, and can be easily extended to black holes in Gauss-Bonnet gravity, leading to the same result for minimum increase of entropy.\n\n. B S Dewitt, Phys. Rev. Lett. 13114B. S. DeWitt, Phys. Rev. Lett., 13, 114 (1964);\n\n. T Padmanabhan, Ann. Phys. (N.Y.). 16538T. Pad- manabhan Ann. Phys. (N.Y.), 165, 38 (1985);\n\n. Class. Quantum Grav. 4107Class. Quantum Grav. 4, L107 (1987);\n\n. A Ashtekar, Phys. Rev. Lett. 69237A. Ashtekar et al., Phys. Rev. Lett., 69, 237 (1992);\n\n. T Padmanabhan, Phys. Rev. T. Padmanabhan Phys. Rev.\n\n. Lett, hep-th/9608182781854Lett. 78, 1854 (1997) [hep-th/9608182];\n\n. Phys. Rev. D. 576206Phys. Rev. D 57, 6206 (1998);\n\n. L J Garay, Int. J. Mod. Phys. 10145L.J. Garay, Int. J. Mod. Phys. A10, 145 (1995).\n\n. C Lanczos, Z.Phys. 73147C. Lanczos, Z.Phys. 73, 147 (1932);\n\n. Annals Math. 39842Annals Math. 39, 842 (1938);\n\nFor a recent review, see Nathalie Deruelle and John Madore. D Lovelock, gr-qc/0305004Jour. Math. Phys. 12498D. Lovelock, Jour. Math. Phys., 12, 498 (1971). For a recent review, see Nathalie Deruelle and John Madore, [gr-qc/0305004], (2003).\n\n. T Padmanabhan, gr-qc/0204019Class. Quan. Grav. 19T Padmanabhan,Class. Quan. Grav., 19, 5387, (2002) [gr-qc/0204019];\n\n. T Padmanabhan, gr-qc/0205090Gen. Rel. Grav. 342029T. Padmanabhan, Gen. Rel. Grav., 34, 2029, (2002) [gr-qc/0205090];\n\n. R-G Cai, L-M Cao, gr-qc/0611071R-G Cai, L-M Cao, [gr-qc/0611071];\n\n. M Akbar, R-G Cai, hep-th/0609128M Akbar, R-G Cai, [hep-th/0609128];\n\n. D Kothawala, S Sarkar, T Padmanabhan, gr-qc/0701002Phys. Letts, B. 652D. Kothawala, S. Sarkar, T. Padmanabhan, Phys. Letts, B 652, 338-342 (2007) [gr-qc/0701002]\n\n. A Paranjape, S Sarkar, T Padmanabhan, hep-th/0607240Phys. Rev. D74. 104015A Paranjape, S Sarkar and T Padmanabhan, Phys. Rev. D74, 104015, (2006) [hep-th/0607240].\n\n. A Mukhopadhyay, Padmanabhan, Phys. Rev. D74. 124023A Mukhopadhyay, T Padmanabhan, Phys. Rev. D74,124023, (2006).\n\n. R M Wald, Phys. Rev. D. 483427R. M. Wald, Phys. Rev. D 48, 3427 (1993);\n\n. V Iyer, R M Wald, Phys. Rev. D. 52V. Iyer, R. M. Wald, Phys. Rev. D 52, 4430-4439 (1995).\n\nThe arguments were given for the case of Einstein's theory in these references but they continue to be valid for any gravitational action. T Padmanabhan, Apoorva Patel, gr-qc/0311036T. Padmanabhan, Phys. Reports. 406T. Padmanabhan, Apoorva Patel [gr-qc/0309053], T. Padmanabhan, Phys. Reports, 406, 49, (2005) [gr-qc/0311036]. The arguments were given for the case of Einstein's theory in these references but they continue to be valid for any gravitational action.\n\n. T Padmanabhan, arXiv:0706.1654AIP Conf. Proc. 939T. Padmanabhan, AIP Conf. Proc. 939, 114-123, (2007) [arXiv:0706.1654]\n\n. T Padmanabhan, arXiv:0705.2533Gen.Rel.Grav. 40T. Padmanabhan, Gen.Rel.Grav., 40, 529-564 (2008) [arXiv:0705.2533];\n\n. T Padmanabhan, A Paranjape, Phys.Rev. D75. 064004T. Padmanabhan, A. Paranjape, Phys.Rev. D75 064004, (2007).\n\n. Shahar Hod, Phys. Rev. Lett. 814293Shahar Hod, Phys. Rev. Lett. 81, 4293, (1998).\n\n. Michele Maggiore, Phys. Rev. Lett. 100141301Maggiore, Michele, Phys. Rev. Lett. 100, 141301, (2008).\n\n. C Elias, Vagenas, arXiv:0804.3264v2Elias C. Vagenas [arXiv:0804.3264v2];\n\n. A J M Medved, arXiv:0804.4346v2A.J.M. Medved [arXiv:0804.4346v2].\n\n. G Ramin, Daghigh, Class. Quantum Grav. 24Int.J.Mod.Phys. ARamin G Daghigh et. al. .Class. Quantum Grav. 24, 1981-1991, (2007). Sayan K. Chakrabarti, Kumar S. Gupta. Int.J.Mod.Phys. A, 21, 3565-3574 , (2006).\n\n. T Padmanabhan, gr-qc/0310027Class. Quan. Grav. 211T. Padmanabhan, Class. Quan. Grav., 21, L1 (2004) [gr-qc/0310027];\n\n. T Roy Choudhury, T Padmanabhan, gr-qc/0311064Phys. Rev.D, D. 6964033T.Roy Choudhury, T. Padmanabhan, Phys. Rev.D, D 69 064033 (2004) [gr-qc/0311064]\n\n. T Jacobson, R C Myers, Phys. Rev. Lett. 70Jacobson T and Myers R C , Phys. Rev. Lett. 70, 3684, (1993);\n\n. R Myers, J Z Simon, Phys. Rev. D. 382434Myers R C and Simon J Z , Phys. Rev. D, 38, 2434, (1998)\n"},"corpus_id":{"kind":"number","value":36431992,"string":"36,431,992"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0807.1481\",\n \"mag\": 2112232996,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevd.78.104018\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0807.1481v1.pdf\"\n ],\n \"pdf_sha\": \"d006c95b7ea7457f3b3d270a7cbf1aa3ae17beef\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/0807.1481\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Is gravitational entropy quantized ?"},"authors":{"kind":"list like","value":["Dawood Kothawala \nIUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia","T Padmanabhan \nIUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia","Sudipta Sarkar \nIUCAA\nPost Bag 4, Pune -411 007GaneshkhindIndia"],"string":"[\n \"Dawood Kothawala \\nIUCAA\\nPost Bag 4, Pune -411 007GaneshkhindIndia\",\n \"T Padmanabhan \\nIUCAA\\nPost Bag 4, Pune -411 007GaneshkhindIndia\",\n \"Sudipta Sarkar \\nIUCAA\\nPost Bag 4, Pune -411 007GaneshkhindIndia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"In Einstein's gravity, the entropy of horizons is proportional to their area. Several arguments given in the literature suggest that, in this context, both area and entropy should be quantized with an equally spaced spectrum for large quantum numbers. But in more general theories (like, for e.g, in the black hole solutions of Gauss-Bonnet or Lanczos-Lovelock gravity) the horizon entropy is not proportional to area and the question arises as to which of the two (if at all) will have this property. We give a general argument that in all Lanczos-Lovelock theories of gravity, it is the entropy that has equally spaced spectrum. In the case of Gauss-Bonnet gravity, we use the asymptotic form of quasi normal mode frequencies to explicitly demonstrate this result. Hence, the concept of a quantum of area in Einstein Hilbert (EH) gravity needs to be replaced by a concept of quantum of entropy in a more general context. PACS numbers: 04.62.+v, 04.60.-mIt was conjectured by Bekenstein [1] long back that, in a quantum theory, the black hole area would be represented by a quantum operator with a discrete spectrum of eigenvalues. Bekenstein showed that the area of a classical black hole behaves like an adiabatic invariant, and so, according to Ehrenfest's theorem, the corresponding quantum operator must have a discrete spectrum. It was also known that, when a quantum particle is captured by a (non extremal) black hole its area increases by a minimum non-zero value [1, 2, 3] which is independent of the black hole parameters. This argument also suggests an equidistant spacing of area levels, with a well-defined notion of a quantum of area. The fundamental constants G, c and combine to give a quantity with the dimensions of area A P = (G /c 3 ) = 10 −66 cm 2 , which is quite suggestive[4]and sets the scale in area quantization.In Einstein's gravity, entropy of the horizon is proportional to its area. Hence one could equivalently claim that it is the gravitational entropy which has an equidistant spectrum with a well-defined notion of quantum of entropy. But, when one considers the natural generalization of Einstein gravity by including higher derivative correction terms to the original Einstein-Hilbert action, no such trivial relationship remains valid between horizon area and associated entropy. One such higher derivative theory which has attracted a fair amount of attention is Lanczos-Lovelock (LL) gravity [5], of which the lowest order correction appears as a Gauss-Bonnet (GB) term in D(> 4) dimensions. These lagrangians have the unique feature that the field equations obtained from them are quasi linear, as a result of which, the initial value problem remains well defined. More importantly, several features related to horizon thermodynamics, which were first discovered in the context of Einstein's theory [6], continues * Electronic address: dawood@iucaa.ernet.in † Electronic address: paddy@iucaa.ernet.in ‡ Electronic address: sudipta@iucaa.ernet.in to be valid in LL gravity models[7,8].Black hole solutions in the LL gravity are well studied in the literature. For these spacetimes, the notion of entropy can be defined using Wald's formalism[9], where entropy is associated with the Noether charge of the diffeomorphism invariance symmetry of the theory. The entropy calculated from this approach turns out to be no longer proportional to horizon area. The question then arises as to whether it is the quantum of area or quantum of entropy (if at all either) which arises in a natural manner in these models. We attempt to answer this question in this paper.We will first provide a very general argument which suggests that it is the entropy which is quantized with equidistant spectrum in the case of LL gravity and then provide an explicit proof for the result in the context of GB gravity.In any geometrical description of gravity that obeys the principle of equivalence and is based on a nontrivial metric, the propagation of light rays will be affected by gravity. This, in turn, leads to regions of spacetime which are causally inaccessible to classes of observers. (These two features are reasonably independent of the precise field equations which determine the metric.). The fact that any observer has a right to formulate physical theories in a given coordinate system entirely in terms of the variables that an observer using that coordinate system can access, imposes strong constraints on the nature of the action functional A grav which can be used to describe gravity[10]. Suppose we divide the space-time manifold into two regions separated by a null hypersurface H and choose a coordinate system such that H acts as a horizon for the observer on one side (say side 1). The effective theory for the observer on side 1 (with the degrees of freedom formally denoted by g 1 ) is obtained by integrating out the variables on the inaccessible side (side 2): In the semiclassical limit, saddle-point integration leads to the exponential of the classical action A grav (class) evaluated on-shell. The effective theory on side 1 is thus described"},"sections":{"kind":"list like","value":[],"string":"[]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". J D Bekenstein, Lett, Nuovo Cimento. 11J.D. Bekenstein, Lett. Nuovo Cimento 11, 467, (1974);","doi":""},{"text":". J D Bekenstein, gr-qc/9710076J.D. Bekenstein [gr-qc/9710076].","doi":"gr-qc/9710076"},{"text":". D Christodoulou, Phys. Rev. Lett. 25Christodoulou, D, Phys. Rev. Lett. 25, 1596-1597, (1970).","doi":""},{"text":"This result only depends on spherical symmetry and the first law of black hole thermodynamics, and can be easily extended to black holes in Gauss-Bonnet gravity, leading to the same result for minimum increase of entropy. This result only depends on spherical symmetry and the first law of black hole thermodynamics, and can be easily extended to black holes in Gauss-Bonnet gravity, leading to the same result for minimum increase of entropy.","doi":""},{"text":". B S Dewitt, Phys. Rev. Lett. 13114B. S. DeWitt, Phys. Rev. Lett., 13, 114 (1964);","doi":""},{"text":". T Padmanabhan, Ann. Phys. (N.Y.). 16538T. Pad- manabhan Ann. Phys. (N.Y.), 165, 38 (1985);","doi":""},{"text":". Class. Quantum Grav. 4107Class. Quantum Grav. 4, L107 (1987);","doi":""},{"text":". A Ashtekar, Phys. Rev. Lett. 69237A. Ashtekar et al., Phys. Rev. Lett., 69, 237 (1992);","doi":""},{"text":". T Padmanabhan, Phys. Rev. T. Padmanabhan Phys. Rev.","doi":""},{"text":". Lett, hep-th/9608182781854Lett. 78, 1854 (1997) [hep-th/9608182];","doi":"hep-th/9608182"},{"text":". Phys. Rev. D. 576206Phys. Rev. D 57, 6206 (1998);","doi":""},{"text":". L J Garay, Int. J. Mod. Phys. 10145L.J. Garay, Int. J. Mod. Phys. A10, 145 (1995).","doi":""},{"text":". C Lanczos, Z.Phys. 73147C. Lanczos, Z.Phys. 73, 147 (1932);","doi":""},{"text":". Annals Math. 39842Annals Math. 39, 842 (1938);","doi":""},{"text":"For a recent review, see Nathalie Deruelle and John Madore. D Lovelock, gr-qc/0305004Jour. Math. Phys. 12498D. Lovelock, Jour. Math. Phys., 12, 498 (1971). For a recent review, see Nathalie Deruelle and John Madore, [gr-qc/0305004], (2003).","doi":"gr-qc/0305004"},{"text":". T Padmanabhan, gr-qc/0204019Class. Quan. Grav. 19T Padmanabhan,Class. Quan. Grav., 19, 5387, (2002) [gr-qc/0204019];","doi":"gr-qc/0204019"},{"text":". T Padmanabhan, gr-qc/0205090Gen. Rel. Grav. 342029T. Padmanabhan, Gen. Rel. Grav., 34, 2029, (2002) [gr-qc/0205090];","doi":"gr-qc/0205090"},{"text":". R-G Cai, L-M Cao, gr-qc/0611071R-G Cai, L-M Cao, [gr-qc/0611071];","doi":"gr-qc/0611071"},{"text":". M Akbar, R-G Cai, hep-th/0609128M Akbar, R-G Cai, [hep-th/0609128];","doi":"hep-th/0609128"},{"text":". D Kothawala, S Sarkar, T Padmanabhan, gr-qc/0701002Phys. Letts, B. 652D. Kothawala, S. Sarkar, T. Padmanabhan, Phys. Letts, B 652, 338-342 (2007) [gr-qc/0701002]","doi":"gr-qc/0701002"},{"text":". A Paranjape, S Sarkar, T Padmanabhan, hep-th/0607240Phys. Rev. D74. 104015A Paranjape, S Sarkar and T Padmanabhan, Phys. Rev. D74, 104015, (2006) [hep-th/0607240].","doi":"hep-th/0607240"},{"text":". A Mukhopadhyay, Padmanabhan, Phys. Rev. D74. 124023A Mukhopadhyay, T Padmanabhan, Phys. Rev. D74,124023, (2006).","doi":""},{"text":". R M Wald, Phys. Rev. D. 483427R. M. Wald, Phys. Rev. D 48, 3427 (1993);","doi":""},{"text":". V Iyer, R M Wald, Phys. Rev. D. 52V. Iyer, R. M. Wald, Phys. Rev. D 52, 4430-4439 (1995).","doi":""},{"text":"The arguments were given for the case of Einstein's theory in these references but they continue to be valid for any gravitational action. T Padmanabhan, Apoorva Patel, gr-qc/0311036T. Padmanabhan, Phys. Reports. 406T. Padmanabhan, Apoorva Patel [gr-qc/0309053], T. Padmanabhan, Phys. Reports, 406, 49, (2005) [gr-qc/0311036]. The arguments were given for the case of Einstein's theory in these references but they continue to be valid for any gravitational action.","doi":"gr-qc/0311036"},{"text":". T Padmanabhan, arXiv:0706.1654AIP Conf. Proc. 939T. Padmanabhan, AIP Conf. Proc. 939, 114-123, (2007) [arXiv:0706.1654]","doi":"arXiv:0706.1654"},{"text":". T Padmanabhan, arXiv:0705.2533Gen.Rel.Grav. 40T. Padmanabhan, Gen.Rel.Grav., 40, 529-564 (2008) [arXiv:0705.2533];","doi":"arXiv:0705.2533"},{"text":". T Padmanabhan, A Paranjape, Phys.Rev. D75. 064004T. Padmanabhan, A. Paranjape, Phys.Rev. D75 064004, (2007).","doi":""},{"text":". Shahar Hod, Phys. Rev. Lett. 814293Shahar Hod, Phys. Rev. Lett. 81, 4293, (1998).","doi":""},{"text":". Michele Maggiore, Phys. Rev. Lett. 100141301Maggiore, Michele, Phys. Rev. Lett. 100, 141301, (2008).","doi":""},{"text":". C Elias, Vagenas, arXiv:0804.3264v2Elias C. Vagenas [arXiv:0804.3264v2];","doi":"arXiv:0804.3264v2"},{"text":". A J M Medved, arXiv:0804.4346v2A.J.M. Medved [arXiv:0804.4346v2].","doi":"arXiv:0804.4346v2"},{"text":". G Ramin, Daghigh, Class. Quantum Grav. 24Int.J.Mod.Phys. ARamin G Daghigh et. al. .Class. Quantum Grav. 24, 1981-1991, (2007). Sayan K. Chakrabarti, Kumar S. Gupta. Int.J.Mod.Phys. A, 21, 3565-3574 , (2006).","doi":""},{"text":". T Padmanabhan, gr-qc/0310027Class. Quan. Grav. 211T. Padmanabhan, Class. Quan. Grav., 21, L1 (2004) [gr-qc/0310027];","doi":"gr-qc/0310027"},{"text":". T Roy Choudhury, T Padmanabhan, gr-qc/0311064Phys. Rev.D, D. 6964033T.Roy Choudhury, T. Padmanabhan, Phys. Rev.D, D 69 064033 (2004) [gr-qc/0311064]","doi":"gr-qc/0311064"},{"text":". T Jacobson, R C Myers, Phys. Rev. Lett. 70Jacobson T and Myers R C , Phys. Rev. Lett. 70, 3684, (1993);","doi":""},{"text":". R Myers, J Z Simon, Phys. Rev. D. 382434Myers R C and Simon J Z , Phys. Rev. D, 38, 2434, (1998)","doi":""}],"string":"[\n {\n \"text\": \". J D Bekenstein, Lett, Nuovo Cimento. 11J.D. Bekenstein, Lett. Nuovo Cimento 11, 467, (1974);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Bekenstein, gr-qc/9710076J.D. Bekenstein [gr-qc/9710076].\",\n \"doi\": \"gr-qc/9710076\"\n },\n {\n \"text\": \". D Christodoulou, Phys. Rev. Lett. 25Christodoulou, D, Phys. Rev. Lett. 25, 1596-1597, (1970).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"This result only depends on spherical symmetry and the first law of black hole thermodynamics, and can be easily extended to black holes in Gauss-Bonnet gravity, leading to the same result for minimum increase of entropy. This result only depends on spherical symmetry and the first law of black hole thermodynamics, and can be easily extended to black holes in Gauss-Bonnet gravity, leading to the same result for minimum increase of entropy.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B S Dewitt, Phys. Rev. Lett. 13114B. S. DeWitt, Phys. Rev. Lett., 13, 114 (1964);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Padmanabhan, Ann. Phys. (N.Y.). 16538T. Pad- manabhan Ann. Phys. (N.Y.), 165, 38 (1985);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Class. Quantum Grav. 4107Class. Quantum Grav. 4, L107 (1987);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Ashtekar, Phys. Rev. Lett. 69237A. Ashtekar et al., Phys. Rev. Lett., 69, 237 (1992);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Padmanabhan, Phys. Rev. T. Padmanabhan Phys. Rev.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Lett, hep-th/9608182781854Lett. 78, 1854 (1997) [hep-th/9608182];\",\n \"doi\": \"hep-th/9608182\"\n },\n {\n \"text\": \". Phys. Rev. D. 576206Phys. Rev. D 57, 6206 (1998);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L J Garay, Int. J. Mod. Phys. 10145L.J. Garay, Int. J. Mod. Phys. A10, 145 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Lanczos, Z.Phys. 73147C. Lanczos, Z.Phys. 73, 147 (1932);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Annals Math. 39842Annals Math. 39, 842 (1938);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"For a recent review, see Nathalie Deruelle and John Madore. D Lovelock, gr-qc/0305004Jour. Math. Phys. 12498D. Lovelock, Jour. Math. Phys., 12, 498 (1971). For a recent review, see Nathalie Deruelle and John Madore, [gr-qc/0305004], (2003).\",\n \"doi\": \"gr-qc/0305004\"\n },\n {\n \"text\": \". T Padmanabhan, gr-qc/0204019Class. Quan. Grav. 19T Padmanabhan,Class. Quan. Grav., 19, 5387, (2002) [gr-qc/0204019];\",\n \"doi\": \"gr-qc/0204019\"\n },\n {\n \"text\": \". T Padmanabhan, gr-qc/0205090Gen. Rel. Grav. 342029T. Padmanabhan, Gen. Rel. Grav., 34, 2029, (2002) [gr-qc/0205090];\",\n \"doi\": \"gr-qc/0205090\"\n },\n {\n \"text\": \". R-G Cai, L-M Cao, gr-qc/0611071R-G Cai, L-M Cao, [gr-qc/0611071];\",\n \"doi\": \"gr-qc/0611071\"\n },\n {\n \"text\": \". M Akbar, R-G Cai, hep-th/0609128M Akbar, R-G Cai, [hep-th/0609128];\",\n \"doi\": \"hep-th/0609128\"\n },\n {\n \"text\": \". D Kothawala, S Sarkar, T Padmanabhan, gr-qc/0701002Phys. Letts, B. 652D. Kothawala, S. Sarkar, T. Padmanabhan, Phys. Letts, B 652, 338-342 (2007) [gr-qc/0701002]\",\n \"doi\": \"gr-qc/0701002\"\n },\n {\n \"text\": \". A Paranjape, S Sarkar, T Padmanabhan, hep-th/0607240Phys. Rev. D74. 104015A Paranjape, S Sarkar and T Padmanabhan, Phys. Rev. D74, 104015, (2006) [hep-th/0607240].\",\n \"doi\": \"hep-th/0607240\"\n },\n {\n \"text\": \". A Mukhopadhyay, Padmanabhan, Phys. Rev. D74. 124023A Mukhopadhyay, T Padmanabhan, Phys. Rev. D74,124023, (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R M Wald, Phys. Rev. D. 483427R. M. Wald, Phys. Rev. D 48, 3427 (1993);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Iyer, R M Wald, Phys. Rev. D. 52V. Iyer, R. M. Wald, Phys. Rev. D 52, 4430-4439 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The arguments were given for the case of Einstein's theory in these references but they continue to be valid for any gravitational action. T Padmanabhan, Apoorva Patel, gr-qc/0311036T. Padmanabhan, Phys. Reports. 406T. Padmanabhan, Apoorva Patel [gr-qc/0309053], T. Padmanabhan, Phys. Reports, 406, 49, (2005) [gr-qc/0311036]. The arguments were given for the case of Einstein's theory in these references but they continue to be valid for any gravitational action.\",\n \"doi\": \"gr-qc/0311036\"\n },\n {\n \"text\": \". T Padmanabhan, arXiv:0706.1654AIP Conf. Proc. 939T. Padmanabhan, AIP Conf. Proc. 939, 114-123, (2007) [arXiv:0706.1654]\",\n \"doi\": \"arXiv:0706.1654\"\n },\n {\n \"text\": \". T Padmanabhan, arXiv:0705.2533Gen.Rel.Grav. 40T. Padmanabhan, Gen.Rel.Grav., 40, 529-564 (2008) [arXiv:0705.2533];\",\n \"doi\": \"arXiv:0705.2533\"\n },\n {\n \"text\": \". T Padmanabhan, A Paranjape, Phys.Rev. D75. 064004T. Padmanabhan, A. Paranjape, Phys.Rev. D75 064004, (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Shahar Hod, Phys. Rev. Lett. 814293Shahar Hod, Phys. Rev. Lett. 81, 4293, (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Michele Maggiore, Phys. Rev. Lett. 100141301Maggiore, Michele, Phys. Rev. Lett. 100, 141301, (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Elias, Vagenas, arXiv:0804.3264v2Elias C. Vagenas [arXiv:0804.3264v2];\",\n \"doi\": \"arXiv:0804.3264v2\"\n },\n {\n \"text\": \". A J M Medved, arXiv:0804.4346v2A.J.M. Medved [arXiv:0804.4346v2].\",\n \"doi\": \"arXiv:0804.4346v2\"\n },\n {\n \"text\": \". G Ramin, Daghigh, Class. Quantum Grav. 24Int.J.Mod.Phys. ARamin G Daghigh et. al. .Class. Quantum Grav. 24, 1981-1991, (2007). Sayan K. Chakrabarti, Kumar S. Gupta. Int.J.Mod.Phys. A, 21, 3565-3574 , (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Padmanabhan, gr-qc/0310027Class. Quan. Grav. 211T. Padmanabhan, Class. Quan. Grav., 21, L1 (2004) [gr-qc/0310027];\",\n \"doi\": \"gr-qc/0310027\"\n },\n {\n \"text\": \". T Roy Choudhury, T Padmanabhan, gr-qc/0311064Phys. Rev.D, D. 6964033T.Roy Choudhury, T. Padmanabhan, Phys. Rev.D, D 69 064033 (2004) [gr-qc/0311064]\",\n \"doi\": \"gr-qc/0311064\"\n },\n {\n \"text\": \". T Jacobson, R C Myers, Phys. Rev. Lett. 70Jacobson T and Myers R C , Phys. Rev. Lett. 70, 3684, (1993);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Myers, J Z Simon, Phys. Rev. D. 382434Myers R C and Simon J Z , Phys. Rev. D, 38, 2434, (1998)\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":58,"cells":{"text":{"kind":"string","value":"\nDerivation of Generalized Thomas-Bargmann-Michel-Telegdi Equation for a Particle with Electric Dipole Moment\n9 Nov 2013 May 11, 2014\n\nTakeshi Fukuyama \nResearch Center for Nuclear Physics (RCNP)\nOsaka University\n567-0047IbarakiOsakaJapan\n\nAlexander J Silenko \nInstitute of Nuclear Problems\nBelarusian State University\n220030MinskBelarus\n\nBogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research\n141980DubnaRussia\n\nDerivation of Generalized Thomas-Bargmann-Michel-Telegdi Equation for a Particle with Electric Dipole Moment\n9 Nov 2013 May 11, 2014electric dipole moment, equation of spin motion PACS numbers: 1340Em; 2110Ky\nGeneral classical equation of spin motion is explicitly derived for a particle with magnetic and electric dipole moments in electromagnetic fields. Equation describing the spin motion relative to the momentum direction in storage rings is also obtained.\n\nElectric dipole moments (EDMs) of particles, atoms and molecules are smoking guns of new physics beyond the Standard Model (New Physics BSM). There are many ongoing and near future experiments measuring EDMs and anomalous magnetic dipole moments (MDMs) of charged particles [1]. One of the most important equations is the following equation of spin motion,\ndζ dt = Ω s × ζ,(1)\nwhere\nΩ s = − e m G + 1 γ B − G + 1 γ + 1 v × E + η 2 (v × B + E) .(2)\nIn this equation, ζ is the rest frame spin pseudovector, G = (g −2)/2, g = 2µm/(es), η = 2dm/(es), γ = ǫ/m is the Lorentz factor, ǫ is the kinetic energy including the rest one, and s is the spin. We consider the case of s = 1/2 and use the system of units = 1, c = 1. However, to be curious enough, the explicit derivation of this equation has not been published [2]. There are several confusions on interpretations of this equation. In the present letter, we derive it explicitly. It is necessary to take into account that the EDM and MDM terms are not fully symmetrical. The difference is caused by the Dirac (\"normal\") magnetic moment, the Thomas precession and the evolution of particle momentum. The last two effects depend on the electric charge of the particle.\n\nIn the relativistic theory, the spin pseudovector is not conserved. We should obtain an equation of motion for the spin of the moving particle. For that purpose, it is convenient to introduce the spin 4-pseudovector and the momentum 4-vector, a µ and p µ , whose definition in the particle rest frame is given by [3] a µ = (0, ζ), p µ = (m, 0).\n\nSo, in any frame\na µ p µ = 0, a µ a µ = −ζ 2 .(4)\nIn a frame moving with velocity v = p/ǫ, the a µ = (a 0 , a) 4-pseudovector is defined by\na µ = (a 0 , a), a = ζ + p(ζ · p) m(ǫ + m) , a 0 = a · p ǫ = p · ζ m , a 2 = ζ 2 + (p · ζ) 2 m 2 .(5)\nThe relativistic equation of spin motion in an electromagnetic field using this 4-pseudovector has the form\nda µ dτ = αF µν a ν + βu µ F νλ u ν a λ + γF * µν a ν + δu µ F * νλ u ν a λ(6)\nwith F * µν = 1 2 ǫ µνρσ F ρσ . Here α, β, γ, δ are coefficients whose meanings are determined as follows. In the rest frame, Eq. (6) becomes\nda i dt = dζ i dt = αF ij ζ j + γF * ij ζ j = α(ζ × B) i + γ(E × ζ) i .(7)\nIn this frame, the equation of spin motion is\ndζ dt = 2µζ × B + 2dζ × E.(8)\nComparing this equation with Eq. (7), we obtain\nα = 2µ, γ = −2d.(9)\nThe value of β results from the equation of motion\nm du µ dτ = eF µν u ν(10)\nand from a µ u µ = 0 that\nu µ da µ dτ = −a µ du µ dτ = e m F µν u µ a ν .(11)\nOn the other hand, multiplying Eq. (6) by u µ and taking u µ u µ = 1 into account, we obtain\nu µ da µ dτ = (2µ + β)F µν u µ a ν + (−2d + δ)F * µν u µ a ν .(12)\nThen\nβ = −2 µ − e 2m ≡ −2µ ′ , δ = 2d.(13)\nAs a result, the equation of spin motion takes the form\nda µ dτ = 2µF µν a ν − 2µ ′ u µ F νλ u ν a λ − 2d(F * µν a ν − u µ F * νλ u ν a λ ).(14)\nThis is the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation [4,5,6] added by the EDM terms.\n\nIn the original paper of Bargmann, Michel, and Telegdi [6], an extension of the equation of spin motion due to an electric dipole moment has already been discussed. However, such an extension has been based on a dual transformation of F µν into F * µν and an explicit derivation of Eq. (14) has not been presented.\n\nThe spatial part of this equation is presented by\nda dt = da dt M DM + da dt EDM , da dt M DM = 2 γ µ {a × B + (a · v)E} + µ ′ γ 2 v {−a · E + v · (a × B) + (a · v)(v · E)} , da dt EDM = − 2d γ (a · v)B − a × E + γ 2 v {−a · B − v · (a × E) + (a · v)(v · B)} ,(15)\nwhere γ = ǫ/m. We consider the evolution of ζ. Since\nζ = a − v(a · v)ǫ ǫ + m , it is defined by dζ dt = dζ dt M DM + dζ dt EDM , dζ dt M DM = da dt M DM − vǫ ǫ + m da dt M DM · v − (a · v)ǫ ǫ + m · dv dt − vǫ ǫ + m a · dv dt − v(a · v)m (ǫ + m) 2 · dǫ dt , dζ dt EDM = da dt EDM − vǫ ǫ + m da dt EDM · v .(16)\nEvidently, the quantity dζ dt M DM is expressed by the T-BMT equation. When we use the decomposition of the equation of motion (10) into spatial and temporal components,\ndv dt = e mγ [E + v × B − v(v · E)] , dǫ dt = ev · E,(17)\ntedious but simple calculations result in\ndζ dt M DM = 2µm + 2µ ′ (ǫ − m) ǫ ζ × B + 2µ ′ ǫ ǫ + m (v · B)(v × ζ) + 2µm + 2µ ′ ǫ ǫ + m ζ × (E × v) = e m G + 1 γ ζ × B + Gγ γ + 1 (v · B)v × ζ + G − 1 γ + 1 ζ × (E × v) .(18)\nWe present a more detailed derivation of contribution of the EDM. Substituting the relation a = ζ + v(ζ · v)γ 2 /(γ + 1) into Eq. (15) results in\nda dt EDM = 2d 1 γ (ζ × E) + γ γ + 1 (v × E)(ζ · v) − (ζ · v)B + γv(ζ · B) − γ 2 γ + 1 v(v · B)(ζ · v) + γv(v · (ζ × E)) .(19)\nSince\n− vǫ ǫ + m da dt EDM · v = −2d γv γ + 1 γ(v · (ζ × E)) + γ 2 − 1 γ (ζ · B) − γ(ζ · v)(v · B) ,\n(20) the result is given by\ndζ dt EDM = 2d ζ × E + γ γ + 1 (v · E)(v × ζ) − ζ × (B × v) = eη 2m ζ × E + γ γ + 1 (v · E)(v × ζ) − ζ × (B × v) .(21)\nThe resulting angular velocity of spin precession is\nΩ s = − e m G + 1 γ B − γG γ + 1 (v · B)v − G + 1 γ + 1 v × E + η 2 E − γ γ + 1 (v · E)v + v × B .(22)\nEquation (22) coincides with that obtained by Nelson et al. [2] with the use of dual transformation of terms proportional to G. The same dual transformation G → η, B → E, E → −B has been applied by Khriplovich [7] for the derivation of this equation. However, the problem is not too simple. The three-component spin ζ is defined in the rest frame, whereas E and B are the fields in the laboratory frame. In addition, one also needs to take into account the nontrivial effect of the Thomas precession [4,5]. We suppose that the explicit derivation of Eq. (22) which is basic for all EDM experiments is necessary.\n\nThis equation also coincides with classical limits of quantum-mechanical equations obtained for spin-1/2 and spin-1 particles in Refs. [8,9] and [10], respectively.\n\n\nThe experimental situation usual for EDM experiments in storage rings consists in\nB · v = 0, E · v = 0.(23)\nIn this case, Eq. (22) takes the form (2). In relation to the EDM experiments in storage rings, taking into account a longitudinal magnetic field may be important for calculations of systematical errors. A longitudinal electric field is usually associated with a beam acceleration. This field can be applied in the EDM experiments performed by the resonance method [11].\n\nOne usually considers the spin motion relative to the beam direction and rewrites Eq. (17) in terms of the unit vector in direction of the velocity (momentum), N = v/v = p/p:\ndN dt =v v − v v 3 (v ·v) = Ω p × N, Ω p = e mγ N × E v − B ,\nwhere Ω p is the angular velocity of rotation of the velocity, momentum, and beam directions. Thus, the angular velocity of the spin rotation relative to the beam direction is given by\nΩ = Ω s − Ω p = − e m GB − G − 1 γ 2 − 1 v × E + η 2 (E + v × B) .(24)\nFor more detailed derivations, especially for calculations of systematical errors, it is helpful to use the cylindrical coordinates [12].\n\nThus, the general classical equation of spin motion has been explicitly derived for the particle with magnetic and electric dipole moments in electromagnetic fields. This equation coincides with the classical limit of corresponding quantum-mechanical equations previously obtained for spin-1/2 and spin-1 particles in Refs. [8,9] and [10], respectively. We also present the equation of spin motion relative to the momentum direction in storage rings.\n AcknowledgementsWe are grateful to J. Pretz and E.D. Commins for valuable arguments. The work of T.F. is supported in part by the Grant-in-Aid for Science Research from the Ministry of Education, Science and Culture of Japan (No. 21104004). The work of A.S. is supported by the Belarusian Republican Foundation for Fundamental Research (Grant No. Φ12D-002).\n. T Fukuyama, arXiv:1201.4252Int. J. Mod. Phys. A. 271230015e-PrintT. Fukuyama, Int. J. Mod. Phys. A 27, 1230015 (2012); e-Print: arXiv:1201.4252.\n\n. D F Nelson, A A Schupp, R W Pidd, H R Crane, Phys. Rev. Lett. 2492D.F. Nelson, A.A. Schupp, R.W. Pidd, and H.R. Crane, Phys. Rev. Lett. 2, 492 (1959).\n\nV B Berestetskii, E M Lifshitz, L P Pitaevskii, Relativistic Quantum Theory. Pergamon PressV.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Relativistic Quantum Theory (Pergamon Press 1973).\n\n. L H Thomas, Nature. 117514L. H. Thomas, Nature (London) 117, 514 (1926).\n\n. L H Thomas, Philos. Mag. 31L. H. Thomas, Philos. Mag. 3, 1 (1927).\n\n. V Bargmann, L Michel, V L Telegdi, Phys. Rev. Lett. 2435V. Bargmann, L. Michel and V.L. Telegdi, Phys. Rev. Lett. 2, 435 (1959).\n\n. I B Khriplovich, Phys. Lett. B. 44498I. B. Khriplovich, Phys. Lett. B 444, 98 (1998).\n\n. M Nowakowski, E A Paschos, J M Rodriguez, Eur. J. Phys. 26545M. Nowakowski, E. A. Paschos, and J. M. Rodriguez, Eur. J. Phys. 26, 545 (2005).\n\n. A J Silenko, Izv. Vyssh. Uchebn. Zaved., Fiz. 48Russ. Phys. J.A. J. Silenko, Izv. Vyssh. Uchebn. Zaved., Fiz. 48, 9 (2005) [Russ. Phys. J. 48, 788 (2005)].\n\n. A J Silenko, Phys. Rev. D. 8773015A. J. Silenko, Phys. Rev. D 87, 073015 (2013).\n\n. Y F Orlov, W M Morse, Y K Semertzidis, Phys. Rev. Lett. 96214802Y.F. Orlov, W.M. Morse, and Y.K. Semertzidis, Phys. Rev. Lett. 96, 214802 (2006).\n\n. A J Silenko, Phys. Rev. ST Accel. Beams. 934003A. J. Silenko, Phys. Rev. ST Accel. Beams 9, 034003 (2006).\n"},"corpus_id":{"kind":"number","value":10910105,"string":"10,910,105"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1308.1580\",\n \"mag\": 3124137577,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1142/s0217751x13501479\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1308.1580v3.pdf\"\n ],\n \"pdf_sha\": \"4fb623ad27f0170855f43f7a6c21bbe5ac3b518e\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1308.1580\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Derivation of Generalized Thomas-Bargmann-Michel-Telegdi Equation for a Particle with Electric Dipole Moment"},"authors":{"kind":"list like","value":["Takeshi Fukuyama \nResearch Center for Nuclear Physics (RCNP)\nOsaka University\n567-0047IbarakiOsakaJapan","Alexander J Silenko \nInstitute of Nuclear Problems\nBelarusian State University\n220030MinskBelarus\n\nBogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research\n141980DubnaRussia"],"string":"[\n \"Takeshi Fukuyama \\nResearch Center for Nuclear Physics (RCNP)\\nOsaka University\\n567-0047IbarakiOsakaJapan\",\n \"Alexander J Silenko \\nInstitute of Nuclear Problems\\nBelarusian State University\\n220030MinskBelarus\\n\\nBogoliubov Laboratory of Theoretical Physics\\nJoint Institute for Nuclear Research\\n141980DubnaRussia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"General classical equation of spin motion is explicitly derived for a particle with magnetic and electric dipole moments in electromagnetic fields. Equation describing the spin motion relative to the momentum direction in storage rings is also obtained."},"sections":{"kind":"list like","value":[{"title":"The experimental situation usual for EDM experiments in storage rings consists in","content":"In this case, Eq. (22) takes the form (2). In relation to the EDM experiments in storage rings, taking into account a longitudinal magnetic field may be important for calculations of systematical errors. A longitudinal electric field is usually associated with a beam acceleration. This field can be applied in the EDM experiments performed by the resonance method [11].\nOne usually considers the spin motion relative to the beam direction and rewrites Eq. (17) in terms of the unit vector in direction of the velocity (momentum), N = v/v = p/p:\nwhere Ω p is the angular velocity of rotation of the velocity, momentum, and beam directions. Thus, the angular velocity of the spin rotation relative to the beam direction is given by\nFor more detailed derivations, especially for calculations of systematical errors, it is helpful to use the cylindrical coordinates [12].\nThus, the general classical equation of spin motion has been explicitly derived for the particle with magnetic and electric dipole moments in electromagnetic fields. This equation coincides with the classical limit of corresponding quantum-mechanical equations previously obtained for spin-1/2 and spin-1 particles in Refs. [8,9] and [10], respectively. We also present the equation of spin motion relative to the momentum direction in storage rings."}],"string":"[\n {\n \"title\": \"The experimental situation usual for EDM experiments in storage rings consists in\",\n \"content\": \"In this case, Eq. (22) takes the form (2). In relation to the EDM experiments in storage rings, taking into account a longitudinal magnetic field may be important for calculations of systematical errors. A longitudinal electric field is usually associated with a beam acceleration. This field can be applied in the EDM experiments performed by the resonance method [11].\\nOne usually considers the spin motion relative to the beam direction and rewrites Eq. (17) in terms of the unit vector in direction of the velocity (momentum), N = v/v = p/p:\\nwhere Ω p is the angular velocity of rotation of the velocity, momentum, and beam directions. Thus, the angular velocity of the spin rotation relative to the beam direction is given by\\nFor more detailed derivations, especially for calculations of systematical errors, it is helpful to use the cylindrical coordinates [12].\\nThus, the general classical equation of spin motion has been explicitly derived for the particle with magnetic and electric dipole moments in electromagnetic fields. This equation coincides with the classical limit of corresponding quantum-mechanical equations previously obtained for spin-1/2 and spin-1 particles in Refs. [8,9] and [10], respectively. We also present the equation of spin motion relative to the momentum direction in storage rings.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". T Fukuyama, arXiv:1201.4252Int. J. Mod. Phys. A. 271230015e-PrintT. Fukuyama, Int. J. Mod. Phys. A 27, 1230015 (2012); e-Print: arXiv:1201.4252.","doi":""},{"text":". D F Nelson, A A Schupp, R W Pidd, H R Crane, Phys. Rev. Lett. 2492D.F. Nelson, A.A. Schupp, R.W. Pidd, and H.R. Crane, Phys. Rev. Lett. 2, 492 (1959).","doi":""},{"text":"V B Berestetskii, E M Lifshitz, L P Pitaevskii, Relativistic Quantum Theory. Pergamon PressV.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Relativistic Quantum Theory (Pergamon Press 1973).","doi":""},{"text":". L H Thomas, Nature. 117514L. H. Thomas, Nature (London) 117, 514 (1926).","doi":""},{"text":". L H Thomas, Philos. Mag. 31L. H. Thomas, Philos. Mag. 3, 1 (1927).","doi":""},{"text":". V Bargmann, L Michel, V L Telegdi, Phys. Rev. Lett. 2435V. Bargmann, L. Michel and V.L. Telegdi, Phys. Rev. Lett. 2, 435 (1959).","doi":""},{"text":". I B Khriplovich, Phys. Lett. B. 44498I. B. Khriplovich, Phys. Lett. B 444, 98 (1998).","doi":""},{"text":". M Nowakowski, E A Paschos, J M Rodriguez, Eur. J. Phys. 26545M. Nowakowski, E. A. Paschos, and J. M. Rodriguez, Eur. J. Phys. 26, 545 (2005).","doi":""},{"text":". A J Silenko, Izv. Vyssh. Uchebn. Zaved., Fiz. 48Russ. Phys. J.A. J. Silenko, Izv. Vyssh. Uchebn. Zaved., Fiz. 48, 9 (2005) [Russ. Phys. J. 48, 788 (2005)].","doi":""},{"text":". A J Silenko, Phys. Rev. D. 8773015A. J. Silenko, Phys. Rev. D 87, 073015 (2013).","doi":""},{"text":". Y F Orlov, W M Morse, Y K Semertzidis, Phys. Rev. Lett. 96214802Y.F. Orlov, W.M. Morse, and Y.K. Semertzidis, Phys. Rev. Lett. 96, 214802 (2006).","doi":""},{"text":". A J Silenko, Phys. Rev. ST Accel. Beams. 934003A. J. Silenko, Phys. Rev. ST Accel. Beams 9, 034003 (2006).","doi":""}],"string":"[\n {\n \"text\": \". T Fukuyama, arXiv:1201.4252Int. J. Mod. Phys. A. 271230015e-PrintT. Fukuyama, Int. J. Mod. Phys. A 27, 1230015 (2012); e-Print: arXiv:1201.4252.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D F Nelson, A A Schupp, R W Pidd, H R Crane, Phys. Rev. Lett. 2492D.F. Nelson, A.A. Schupp, R.W. Pidd, and H.R. Crane, Phys. Rev. Lett. 2, 492 (1959).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V B Berestetskii, E M Lifshitz, L P Pitaevskii, Relativistic Quantum Theory. Pergamon PressV.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Relativistic Quantum Theory (Pergamon Press 1973).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L H Thomas, Nature. 117514L. H. Thomas, Nature (London) 117, 514 (1926).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L H Thomas, Philos. Mag. 31L. H. Thomas, Philos. Mag. 3, 1 (1927).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Bargmann, L Michel, V L Telegdi, Phys. Rev. Lett. 2435V. Bargmann, L. Michel and V.L. Telegdi, Phys. Rev. Lett. 2, 435 (1959).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I B Khriplovich, Phys. Lett. B. 44498I. B. Khriplovich, Phys. Lett. B 444, 98 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Nowakowski, E A Paschos, J M Rodriguez, Eur. J. Phys. 26545M. Nowakowski, E. A. Paschos, and J. M. Rodriguez, Eur. J. Phys. 26, 545 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A J Silenko, Izv. Vyssh. Uchebn. Zaved., Fiz. 48Russ. Phys. J.A. J. Silenko, Izv. Vyssh. Uchebn. Zaved., Fiz. 48, 9 (2005) [Russ. Phys. J. 48, 788 (2005)].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A J Silenko, Phys. Rev. D. 8773015A. J. Silenko, Phys. Rev. D 87, 073015 (2013).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y F Orlov, W M Morse, Y K Semertzidis, Phys. Rev. Lett. 96214802Y.F. Orlov, W.M. Morse, and Y.K. Semertzidis, Phys. Rev. Lett. 96, 214802 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A J Silenko, Phys. Rev. ST Accel. Beams. 934003A. J. Silenko, Phys. Rev. ST Accel. Beams 9, 034003 (2006).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":59,"cells":{"text":{"kind":"string","value":"\nOn the BCH formula of Rezek and Kosloff\n6 Dec 2010\n\nJan Naudts \nDepartement Fysica\nUniversiteit Antwerpen\nUniversiteitsplein 12610AntwerpenBelgium\n\nWinny O' \nDepartement Fysica\nUniversiteit Antwerpen\nUniversiteitsplein 12610AntwerpenBelgium\n\nKelly De Galway \nDepartement Fysica\nUniversiteit Antwerpen\nUniversiteitsplein 12610AntwerpenBelgium\n\nOn the BCH formula of Rezek and Kosloff\n6 Dec 2010\nThe BCH formula of Rezek and Kosloff is a convenient tool to handle a family of density matrices, which occurs in the study of quantum heat engines. We prove the formula using a known argument from Lie theory.\n\nIntroduction\n\nThe Hamiltonian\nH = 1 2 ω(a † a + aa † )(1)\nof the quantum harmonic oscillator belongs to the Lie algebra su(1,1) with generators S 1 = 1 4 ((a † ) 2 + a 2 ),\nS 2 = i 4 ((a † ) 2 − a 2 ), S 3 = 1 4 (a † a + aa † ).(2)\nAs a consequence, it is possible to write down simplified Baker-Campbell-Haussdorf (BCH) relations [1,2]. These have been used to study the quantum harmonic oscillator in a time-dependent external field [3,4,5,6,7,8,9]. The topic of the present paper is a new BCH relation, introduced recently by Rezek and Kosloff [10]. They consider the family of density matrices\nρ = 1 Z(β, γ) e γa 2 e −βH e γ(a † ) 2(3)\nwith real β and complex γ, and with Z(β, γ) = Tr e γa 2 e −βH e γ(a † ) 2 .\n\nBecause all operators appearing in (3) belong to the Lie algebra it is clear from the general Baker-Campbell-Haussdorf relation that it must be possible to write\ne γa 2 e −βH e γ(a † ) 2 = e χa 2 −ξH+χ(a † ) 2(5)\nThe explicit expression of the coefficients χ and ξ as a function of β and γ is found in the Appendix of [10]. The functions were derived [11] using the algebraic manipulation software Mathematica. Note that the special case of (5) with ξ = 0 appeared in the physics literature before (see Example I of Section II of [12]; see also [13,14]). In the present paper the relation (5) is derived using the argument of [12].\n\nThe relation (5) is of interest in its own. But it is also very useful in the study of quasi-stationary processes [10,15]. Indeed, from (5) it is clear that the density matrix ρ describes a system in thermal equilibrium at inverse temperature ξ. On the other hand, the expression (3) is more convenient for practical calculations.\n\nWe derive the BCH formula in the next section. In Section 3 follows a similar BCH formula valid for su (2). In the final section follows a short discussion.\n\n\nThe identity\n\nThe the l.h.s. of (5) can be written in terms of the generators of the Lie algebra su(1,1) as\ne 2γ(S 1 +iS 2 ) e −2β ωS 3 e 2γ(S 1 −iS 2 ) .(6)\nThese generators satisfy the commutation relations\n[S 1 , S 2 ] = iS 3 , [S 2 , S 3 ] = −iS 1 , [S 3 , S 1 ] = −iS 2 .(7)\nIntroduce su(2) generators T 1 = −iS 1 , T 2 = iS 2 , and T 3 = S 3 . Then (6) becomes\nX ≡ e 2γ(iT 1 +T 2 ) e −2β ωT 3 e 2γ(iT 1 −T 2 ) .(8)\nThe relation (5) does not depend on the choice of the representation of the su(2) algebra. Therefore, we may change it. A favourable choice is that of the Pauli spin matrices σ α = 2T α . Using that (σ 1 ± iσ 2 ) 2 = 0 and σ 2 α = I the calculation becomes very easy. One obtains\nX = e iγ(σ 1 −iσ 2 ) e −β ωσ 3 e iγ(σ 1 +iσ 2 ) = (I + iγ(σ 1 − iσ 2 ))(cosh( ω) − σ 3 sinh( ω))(I + iγ(σ 1 + iσ 2 )) = e −β ωσ 3 − 2κ|γ| 2 + iκ(γ + γ)σ 1 + κ(γ − γ)σ 2 + 2κ|γ| 2 σ 3 ,(9)\nwith κ = e −β ω as before. On the other hand is\nexp χa 2 − ξH + χ(a † ) 2 = exp (2χ(S 1 + iS 2 ) − 2ξ ωS 3 + 2χ(S 1 − iS 2 )) = exp (2χ(iT 1 + T 2 ) − 2ξ ωT 3 + 2χ(iT 1 − T 2 )) .(10)\nIn the Pauli spin representation this becomes e Y with\nY = i(χ + χ)σ 1 + (χ − χ)σ 2 − ξ ωσ 3 .(11)\nBecause the Pauli matrices anti-commute and their squares equal I there follows that\nY 2 = λ 2 I with λ = ξ 2 ( ω) 2 − 4|χ| 2 .(12)\nHence one obtains\ne Y = cosh(λ) + 1 λ sinh(λ)Y.(13)\nComparison with (9) gives the 4 conditions cosh(λ) = α + κ,\n1 λ sinh(λ)(χ + χ) = κ(γ + γ),(14)1 λ sinh(λ)(χ − χ) = κ(γ − γ),(15)1 λ sinh(λ)ξ ω = α,(16)\nwith\nα = sinh(β ω) − 2κ|γ| 2 = 1 2κ 1 − κ 2 − 4κ|γ| 2 .(18)\nThe solution of these equations is\nξ = α ω λ sinh(λ) ,(19)χ = κ λ sinh(λ) γ.(20)\nwith sinh(λ) = α 2 − 4κ 2 |γ| 2 (21)\n\nThese results coincide with those found in the Appendix of [10]. Note that the expressions for ξ and χ can be inverted easily. Given ξ and χ one obtains λ from (12). Then α follows by inverting (19). This gives\nα = ωξ sinh(λ) λ .(22)\nNext β is obtained from (14)\nκ = cosh(λ) − α.(23)\nFinally, γ follows from (20)\nγ = sinh(λ) κλ χ.(24)\n3 An example with SU(2) symmetry Formulas similar to (5) can be derived for other symmetry groups than SU(1,1). For instance, in the case of SU(2) one has e γσ + e −βσz e γσ − = exp (χσ + − ξσ z + χσ − )\n\nwith σ ± = 1 2 (σ x ± iσ y ). Using σ 2 ± = 0, σ 2 z = I, σ ± σ z = ∓σ ± , and σ + σ − =\n1 2 (1 + σ z ) the l.h.s. becomes l.h.s. = (1 + γσ + ) (cosh(β) − sinh(β)σ z ) (1 + γσ − ) = cosh(β) + 1 2 |γ| 2 e β + e β (γσ + + γσ − ) − (sinh(β) − 1 2 e β |γ| 2 )σ z .(26)\nThe r.h.s. of (25) is evaluated using\n(χσ + − ξσ z + χσ − ) 2 = λ 2 I,(27)\nwith λ = ξ 2 + |χ| 2 . One finds\nr.h.s. = cosh(λ) + 1 λ sinh(λ) (χσ + − ξσ z + χσ − ) .(28)\nEquating both expressions yields the set of equations\ncosh(β) + 1 2 e β |γ| 2 = cosh(λ),(29)− sinh(β) + 1 2 e β |γ| 2 = − 1 λ sinh(λ)ξ,(30)γe β = 1 λ sinh(λ)χ.(31)\nGiven ξ and χ, the value of λ can be obtained from its definition. The solution then reads\ne β = cosh(λ) + 1 λ sinh(λ)ξ γ = 1 λ sinh(λ) cosh(λ) + 1 λ sinh(λ)ξ χ.(32)\nConversely, given β and γ one obtains λ from (29). Then ξ and χ follow from (30) and (31), respectively.\n\n\nDiscussion\n\nThe BCH relation of Rezek and Kosloff is somewhat special because it is written in a form suited for application to density matrices. Similar results found in the literature [1,2,3,4,5,6,7,8,9] aim at the calculation of time evolution operators and refer to similarity transformations, this is, to expressions of the form e A Be −A . But the l.h.s. of (5) is not a similarity transformation. This is precisely the reason why this BCH relation is of interest! The change of the spectrum implies that the average energy H = Tr ρH will depend on the value of the parameter γ. This dependence is essential in the context of heat engines.\n \nLie algebraic solution of linear differential equations. J Wei, E Norman, J. Math. Phys. 4J. Wei and E. Norman, Lie algebraic solution of linear differential equations, J. Math. Phys. 4, 575 -582 (1963).\n\nOn global representations of the solutions of linear differential equations as a product of exponentials. J Wei, E Norman, Proc. Am. Math. Soc. 15J. Wei and E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Am. Math. Soc. 15, 327 -334 (1964).\n\nAn exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field. H R LewisJr, W B Riesenfeld, J. Math. Phys. 10H.R. Lewis, Jr., W.B. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10, 1458 -1473 (1969).\n\nHarmonic oscillator with time-dependent mass and frequency and a perturbative potential. C M A Dantas, I A Pedrosa, B Baseia, Phys. Rev. A. 45C.M.A. Dantas, I.A. Pedrosa, and B. Baseia, Harmonic oscillator with time-dependent mass and frequency and a perturbative potential, Phys. Rev. A 45, 1320 -1324 (1992).\n\nWave function in the invariant representation and squeezed-state function of the time-dependent harmonic oscillator. Kyu Hwang Yeon, Hyon Ju Kim, Chung In Um, Thomas F George, Lakshmi N Pandey, Phys. Rev. A. 50Kyu Hwang Yeon, Hyon Ju Kim, Chung In Um, Thomas F. George and Lakshmi N. Pandey, Wave function in the invariant representation and squeezed-state function of the time-dependent harmonic oscillator, Phys. Rev. A 50, 1035 -1039 (1994).\n\nUnitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential. D.-Y. Song, Phys. Rev. A. 6214103D.-Y. Song, Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential, Phys. Rev. A 62, 014103 (2000).\n\nNonequilibrium work distribution of a quantum harmonic oscillator. S Deffner, E Lutz, Phys. Rev. E. 7721128S. Deffner and E. Lutz, Nonequilibrium work distribution of a quantum harmonic oscillator, Phys. Rev. E 77, 021128 (2008).\n\nGeneral solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach. M Kuna, J Naudts, Rep. Math. Phys. 65M. Kuna and J. Naudts, General solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach, Rep. Math. Phys. 65, 77-107 (2010).\n\nAnalytic solutions for a three-level system in a time-dependent field. J Naudts, W O'kelly De Galway, to appear in Physica DJ. Naudts and W. O'Kelly de Galway, Analytic solutions for a three-level system in a time-dependent field, to appear in Physica D.\n\nIrreversible performance of a quantum harmonic heat engine. Y Rezek, R Kosloff, New J. Phys. 883Y. Rezek and R. Kosloff, Irreversible performance of a quantum harmonic heat engine, New J. Phys. 8, 83 (2006).\n\n. Y Rezek, private communicationY. Rezek, private communication, 2010.\n\nBaker-Campbell-Hausdorff formulas. R Gilmore, J. Math. Phys. 15R. Gilmore, Baker-Campbell-Hausdorff formulas, J. Math. Phys. 15, 2090 - 2092 (1974).\n\nImpossibility of naively generalizing squeezed coherent states. R A Fisher, M M Nieto, V D Sandberg, Phys. Rev. 29R.A. Fisher, M.M. Nieto, V.D. Sandberg, Impossibility of naively generalizing squeezed coherent states, Phys. Rev. D29, 1107 -1110 (1984).\n\nBaker-Campbell-Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators. D R Truax, Phys. Rev. 31D.R. Truax, Baker-Campbell-Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators, Phys. Rev. D31, 1988 -1991 (1985).\n\nMaximum work in minimum time from a conservative quantum system. P Salamon, K H Hoffmann, Y Rezek, R Kosloff, Phys. Chem. Chem. Phys. 11P. Salamon, K. H. Hoffmann, Y. Rezek and R. Kosloff, Maximum work in minimum time from a conservative quantum system, Phys. Chem. Chem. Phys. 11, 1027-1032 (2009).\n"},"corpus_id":{"kind":"number","value":119649989,"string":"119,649,989"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1012.1111\",\n \"mag\": 2000793599,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1016/j.physa.2011.04.035\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1012.1111v1.pdf\"\n ],\n \"pdf_sha\": \"2c744c9cf198988213ee78761ffeb36f102bf80e\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://lirias.kuleuven.be/bitstream/123456789/331285/1/science.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"On the BCH formula of Rezek and Kosloff"},"authors":{"kind":"list like","value":["Jan Naudts \nDepartement Fysica\nUniversiteit Antwerpen\nUniversiteitsplein 12610AntwerpenBelgium","Winny O' \nDepartement Fysica\nUniversiteit Antwerpen\nUniversiteitsplein 12610AntwerpenBelgium","Kelly De Galway \nDepartement Fysica\nUniversiteit Antwerpen\nUniversiteitsplein 12610AntwerpenBelgium"],"string":"[\n \"Jan Naudts \\nDepartement Fysica\\nUniversiteit Antwerpen\\nUniversiteitsplein 12610AntwerpenBelgium\",\n \"Winny O' \\nDepartement Fysica\\nUniversiteit Antwerpen\\nUniversiteitsplein 12610AntwerpenBelgium\",\n \"Kelly De Galway \\nDepartement Fysica\\nUniversiteit Antwerpen\\nUniversiteitsplein 12610AntwerpenBelgium\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The BCH formula of Rezek and Kosloff is a convenient tool to handle a family of density matrices, which occurs in the study of quantum heat engines. We prove the formula using a known argument from Lie theory."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The Hamiltonian\nof the quantum harmonic oscillator belongs to the Lie algebra su(1,1) with generators S 1 = 1 4 ((a † ) 2 + a 2 ),\nAs a consequence, it is possible to write down simplified Baker-Campbell-Haussdorf (BCH) relations [1,2]. These have been used to study the quantum harmonic oscillator in a time-dependent external field [3,4,5,6,7,8,9]. The topic of the present paper is a new BCH relation, introduced recently by Rezek and Kosloff [10]. They consider the family of density matrices\nwith real β and complex γ, and with Z(β, γ) = Tr e γa 2 e −βH e γ(a † ) 2 .\nBecause all operators appearing in (3) belong to the Lie algebra it is clear from the general Baker-Campbell-Haussdorf relation that it must be possible to write\nThe explicit expression of the coefficients χ and ξ as a function of β and γ is found in the Appendix of [10]. The functions were derived [11] using the algebraic manipulation software Mathematica. Note that the special case of (5) with ξ = 0 appeared in the physics literature before (see Example I of Section II of [12]; see also [13,14]). In the present paper the relation (5) is derived using the argument of [12].\nThe relation (5) is of interest in its own. But it is also very useful in the study of quasi-stationary processes [10,15]. Indeed, from (5) it is clear that the density matrix ρ describes a system in thermal equilibrium at inverse temperature ξ. On the other hand, the expression (3) is more convenient for practical calculations.\nWe derive the BCH formula in the next section. In Section 3 follows a similar BCH formula valid for su (2). In the final section follows a short discussion."},{"title":"The identity","content":"The the l.h.s. of (5) can be written in terms of the generators of the Lie algebra su(1,1) as\nThese generators satisfy the commutation relations\nIntroduce su(2) generators T 1 = −iS 1 , T 2 = iS 2 , and T 3 = S 3 . Then (6) becomes\nThe relation (5) does not depend on the choice of the representation of the su(2) algebra. Therefore, we may change it. A favourable choice is that of the Pauli spin matrices σ α = 2T α . Using that (σ 1 ± iσ 2 ) 2 = 0 and σ 2 α = I the calculation becomes very easy. One obtains\nwith κ = e −β ω as before. On the other hand is\nIn the Pauli spin representation this becomes e Y with\nBecause the Pauli matrices anti-commute and their squares equal I there follows that\nHence one obtains\nComparison with (9) gives the 4 conditions cosh(λ) = α + κ,\nwith\nThe solution of these equations is\nwith sinh(λ) = α 2 − 4κ 2 |γ| 2 (21)\nThese results coincide with those found in the Appendix of [10]. Note that the expressions for ξ and χ can be inverted easily. Given ξ and χ one obtains λ from (12). Then α follows by inverting (19). This gives\nNext β is obtained from (14)\nFinally, γ follows from (20)\n3 An example with SU(2) symmetry Formulas similar to (5) can be derived for other symmetry groups than SU(1,1). For instance, in the case of SU(2) one has e γσ + e −βσz e γσ − = exp (χσ + − ξσ z + χσ − )\nwith σ ± = 1 2 (σ x ± iσ y ). Using σ 2 ± = 0, σ 2 z = I, σ ± σ z = ∓σ ± , and σ + σ − =\nThe r.h.s. of (25) is evaluated using\nwith λ = ξ 2 + |χ| 2 . One finds\nEquating both expressions yields the set of equations\nGiven ξ and χ, the value of λ can be obtained from its definition. The solution then reads\nConversely, given β and γ one obtains λ from (29). Then ξ and χ follow from (30) and (31), respectively."},{"title":"Discussion","content":"The BCH relation of Rezek and Kosloff is somewhat special because it is written in a form suited for application to density matrices. Similar results found in the literature [1,2,3,4,5,6,7,8,9] aim at the calculation of time evolution operators and refer to similarity transformations, this is, to expressions of the form e A Be −A . But the l.h.s. of (5) is not a similarity transformation. This is precisely the reason why this BCH relation is of interest! The change of the spectrum implies that the average energy H = Tr ρH will depend on the value of the parameter γ. This dependence is essential in the context of heat engines."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The Hamiltonian\\nof the quantum harmonic oscillator belongs to the Lie algebra su(1,1) with generators S 1 = 1 4 ((a † ) 2 + a 2 ),\\nAs a consequence, it is possible to write down simplified Baker-Campbell-Haussdorf (BCH) relations [1,2]. These have been used to study the quantum harmonic oscillator in a time-dependent external field [3,4,5,6,7,8,9]. The topic of the present paper is a new BCH relation, introduced recently by Rezek and Kosloff [10]. They consider the family of density matrices\\nwith real β and complex γ, and with Z(β, γ) = Tr e γa 2 e −βH e γ(a † ) 2 .\\nBecause all operators appearing in (3) belong to the Lie algebra it is clear from the general Baker-Campbell-Haussdorf relation that it must be possible to write\\nThe explicit expression of the coefficients χ and ξ as a function of β and γ is found in the Appendix of [10]. The functions were derived [11] using the algebraic manipulation software Mathematica. Note that the special case of (5) with ξ = 0 appeared in the physics literature before (see Example I of Section II of [12]; see also [13,14]). In the present paper the relation (5) is derived using the argument of [12].\\nThe relation (5) is of interest in its own. But it is also very useful in the study of quasi-stationary processes [10,15]. Indeed, from (5) it is clear that the density matrix ρ describes a system in thermal equilibrium at inverse temperature ξ. On the other hand, the expression (3) is more convenient for practical calculations.\\nWe derive the BCH formula in the next section. In Section 3 follows a similar BCH formula valid for su (2). In the final section follows a short discussion.\"\n },\n {\n \"title\": \"The identity\",\n \"content\": \"The the l.h.s. of (5) can be written in terms of the generators of the Lie algebra su(1,1) as\\nThese generators satisfy the commutation relations\\nIntroduce su(2) generators T 1 = −iS 1 , T 2 = iS 2 , and T 3 = S 3 . Then (6) becomes\\nThe relation (5) does not depend on the choice of the representation of the su(2) algebra. Therefore, we may change it. A favourable choice is that of the Pauli spin matrices σ α = 2T α . Using that (σ 1 ± iσ 2 ) 2 = 0 and σ 2 α = I the calculation becomes very easy. One obtains\\nwith κ = e −β ω as before. On the other hand is\\nIn the Pauli spin representation this becomes e Y with\\nBecause the Pauli matrices anti-commute and their squares equal I there follows that\\nHence one obtains\\nComparison with (9) gives the 4 conditions cosh(λ) = α + κ,\\nwith\\nThe solution of these equations is\\nwith sinh(λ) = α 2 − 4κ 2 |γ| 2 (21)\\nThese results coincide with those found in the Appendix of [10]. Note that the expressions for ξ and χ can be inverted easily. Given ξ and χ one obtains λ from (12). Then α follows by inverting (19). This gives\\nNext β is obtained from (14)\\nFinally, γ follows from (20)\\n3 An example with SU(2) symmetry Formulas similar to (5) can be derived for other symmetry groups than SU(1,1). For instance, in the case of SU(2) one has e γσ + e −βσz e γσ − = exp (χσ + − ξσ z + χσ − )\\nwith σ ± = 1 2 (σ x ± iσ y ). Using σ 2 ± = 0, σ 2 z = I, σ ± σ z = ∓σ ± , and σ + σ − =\\nThe r.h.s. of (25) is evaluated using\\nwith λ = ξ 2 + |χ| 2 . One finds\\nEquating both expressions yields the set of equations\\nGiven ξ and χ, the value of λ can be obtained from its definition. The solution then reads\\nConversely, given β and γ one obtains λ from (29). Then ξ and χ follow from (30) and (31), respectively.\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"The BCH relation of Rezek and Kosloff is somewhat special because it is written in a form suited for application to density matrices. Similar results found in the literature [1,2,3,4,5,6,7,8,9] aim at the calculation of time evolution operators and refer to similarity transformations, this is, to expressions of the form e A Be −A . But the l.h.s. of (5) is not a similarity transformation. This is precisely the reason why this BCH relation is of interest! The change of the spectrum implies that the average energy H = Tr ρH will depend on the value of the parameter γ. This dependence is essential in the context of heat engines.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Lie algebraic solution of linear differential equations. J Wei, E Norman, J. Math. Phys. 4J. Wei and E. Norman, Lie algebraic solution of linear differential equations, J. Math. Phys. 4, 575 -582 (1963).","doi":""},{"text":"On global representations of the solutions of linear differential equations as a product of exponentials. J Wei, E Norman, Proc. Am. Math. Soc. 15J. Wei and E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Am. Math. Soc. 15, 327 -334 (1964).","doi":""},{"text":"An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field. H R LewisJr, W B Riesenfeld, J. Math. Phys. 10H.R. Lewis, Jr., W.B. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10, 1458 -1473 (1969).","doi":""},{"text":"Harmonic oscillator with time-dependent mass and frequency and a perturbative potential. C M A Dantas, I A Pedrosa, B Baseia, Phys. Rev. A. 45C.M.A. Dantas, I.A. Pedrosa, and B. Baseia, Harmonic oscillator with time-dependent mass and frequency and a perturbative potential, Phys. Rev. A 45, 1320 -1324 (1992).","doi":""},{"text":"Wave function in the invariant representation and squeezed-state function of the time-dependent harmonic oscillator. Kyu Hwang Yeon, Hyon Ju Kim, Chung In Um, Thomas F George, Lakshmi N Pandey, Phys. Rev. A. 50Kyu Hwang Yeon, Hyon Ju Kim, Chung In Um, Thomas F. George and Lakshmi N. Pandey, Wave function in the invariant representation and squeezed-state function of the time-dependent harmonic oscillator, Phys. Rev. A 50, 1035 -1039 (1994).","doi":""},{"text":"Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential. D.-Y. Song, Phys. Rev. A. 6214103D.-Y. Song, Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential, Phys. Rev. A 62, 014103 (2000).","doi":""},{"text":"Nonequilibrium work distribution of a quantum harmonic oscillator. S Deffner, E Lutz, Phys. Rev. E. 7721128S. Deffner and E. Lutz, Nonequilibrium work distribution of a quantum harmonic oscillator, Phys. Rev. E 77, 021128 (2008).","doi":""},{"text":"General solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach. M Kuna, J Naudts, Rep. Math. Phys. 65M. Kuna and J. Naudts, General solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach, Rep. Math. Phys. 65, 77-107 (2010).","doi":""},{"text":"Analytic solutions for a three-level system in a time-dependent field. J Naudts, W O'kelly De Galway, to appear in Physica DJ. Naudts and W. O'Kelly de Galway, Analytic solutions for a three-level system in a time-dependent field, to appear in Physica D.","doi":""},{"text":"Irreversible performance of a quantum harmonic heat engine. Y Rezek, R Kosloff, New J. Phys. 883Y. Rezek and R. Kosloff, Irreversible performance of a quantum harmonic heat engine, New J. Phys. 8, 83 (2006).","doi":""},{"text":". Y Rezek, private communicationY. Rezek, private communication, 2010.","doi":""},{"text":"Baker-Campbell-Hausdorff formulas. R Gilmore, J. Math. Phys. 15R. Gilmore, Baker-Campbell-Hausdorff formulas, J. Math. Phys. 15, 2090 - 2092 (1974).","doi":""},{"text":"Impossibility of naively generalizing squeezed coherent states. R A Fisher, M M Nieto, V D Sandberg, Phys. Rev. 29R.A. Fisher, M.M. Nieto, V.D. Sandberg, Impossibility of naively generalizing squeezed coherent states, Phys. Rev. D29, 1107 -1110 (1984).","doi":""},{"text":"Baker-Campbell-Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators. D R Truax, Phys. Rev. 31D.R. Truax, Baker-Campbell-Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators, Phys. Rev. D31, 1988 -1991 (1985).","doi":""},{"text":"Maximum work in minimum time from a conservative quantum system. P Salamon, K H Hoffmann, Y Rezek, R Kosloff, Phys. Chem. Chem. Phys. 11P. Salamon, K. H. Hoffmann, Y. Rezek and R. Kosloff, Maximum work in minimum time from a conservative quantum system, Phys. Chem. Chem. Phys. 11, 1027-1032 (2009).","doi":""}],"string":"[\n {\n \"text\": \"Lie algebraic solution of linear differential equations. J Wei, E Norman, J. Math. Phys. 4J. Wei and E. Norman, Lie algebraic solution of linear differential equations, J. Math. Phys. 4, 575 -582 (1963).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On global representations of the solutions of linear differential equations as a product of exponentials. J Wei, E Norman, Proc. Am. Math. Soc. 15J. Wei and E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Am. Math. Soc. 15, 327 -334 (1964).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field. H R LewisJr, W B Riesenfeld, J. Math. Phys. 10H.R. Lewis, Jr., W.B. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10, 1458 -1473 (1969).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harmonic oscillator with time-dependent mass and frequency and a perturbative potential. C M A Dantas, I A Pedrosa, B Baseia, Phys. Rev. A. 45C.M.A. Dantas, I.A. Pedrosa, and B. Baseia, Harmonic oscillator with time-dependent mass and frequency and a perturbative potential, Phys. Rev. A 45, 1320 -1324 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Wave function in the invariant representation and squeezed-state function of the time-dependent harmonic oscillator. Kyu Hwang Yeon, Hyon Ju Kim, Chung In Um, Thomas F George, Lakshmi N Pandey, Phys. Rev. A. 50Kyu Hwang Yeon, Hyon Ju Kim, Chung In Um, Thomas F. George and Lakshmi N. Pandey, Wave function in the invariant representation and squeezed-state function of the time-dependent harmonic oscillator, Phys. Rev. A 50, 1035 -1039 (1994).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential. D.-Y. Song, Phys. Rev. A. 6214103D.-Y. Song, Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential, Phys. Rev. A 62, 014103 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonequilibrium work distribution of a quantum harmonic oscillator. S Deffner, E Lutz, Phys. Rev. E. 7721128S. Deffner and E. Lutz, Nonequilibrium work distribution of a quantum harmonic oscillator, Phys. Rev. E 77, 021128 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"General solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach. M Kuna, J Naudts, Rep. Math. Phys. 65M. Kuna and J. Naudts, General solutions of quantum mechanical equations of motion with time-dependent Hamiltonians: a Lie algebraic approach, Rep. Math. Phys. 65, 77-107 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Analytic solutions for a three-level system in a time-dependent field. J Naudts, W O'kelly De Galway, to appear in Physica DJ. Naudts and W. O'Kelly de Galway, Analytic solutions for a three-level system in a time-dependent field, to appear in Physica D.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Irreversible performance of a quantum harmonic heat engine. Y Rezek, R Kosloff, New J. Phys. 883Y. Rezek and R. Kosloff, Irreversible performance of a quantum harmonic heat engine, New J. Phys. 8, 83 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Rezek, private communicationY. Rezek, private communication, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Baker-Campbell-Hausdorff formulas. R Gilmore, J. Math. Phys. 15R. Gilmore, Baker-Campbell-Hausdorff formulas, J. Math. Phys. 15, 2090 - 2092 (1974).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Impossibility of naively generalizing squeezed coherent states. R A Fisher, M M Nieto, V D Sandberg, Phys. Rev. 29R.A. Fisher, M.M. Nieto, V.D. Sandberg, Impossibility of naively generalizing squeezed coherent states, Phys. Rev. D29, 1107 -1110 (1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Baker-Campbell-Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators. D R Truax, Phys. Rev. 31D.R. Truax, Baker-Campbell-Hausdorff relations and unitarity of SU(2) and SU(1,1) squeeze operators, Phys. Rev. D31, 1988 -1991 (1985).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Maximum work in minimum time from a conservative quantum system. P Salamon, K H Hoffmann, Y Rezek, R Kosloff, Phys. Chem. Chem. Phys. 11P. Salamon, K. H. Hoffmann, Y. Rezek and R. Kosloff, Maximum work in minimum time from a conservative quantum system, Phys. Chem. Chem. Phys. 11, 1027-1032 (2009).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":60,"cells":{"text":{"kind":"string","value":"\nEntropy operator and associated Wigner function\n5 Mar 2013\n\nH Moya-Cessa \nCoordinación de Optica\nINAOE\nPueApdo. Postal 51 y 21672000PueblaMexico\n\nEntropy operator and associated Wigner function\n5 Mar 2013numbers: 0210Ud4250Dv0365Ud4250Lc\nWe show how the entropy operators for two subsystems may be calculated. In the case of the atom-field interaction we obtain the associated Wigner function for the entropy operator for the quantized field.\n\nI. INTRODUCTION\n\nIt is well known that when two subsystems A and B interact the entropy of the total system and the subsystem entropies obey a triangle inequality [1] \n|S A − S B | ≤ S AB ≤ S A + S B .(1)\nThis result implies that if both subsystems are initially in pure states, the total entropy is zero and both subsystems entropies will be equal after they interact. Here we would like to arrive to the result that, if initially the two subsystems are in pure states any function the density matrix of subsystem A is equal to the function of the density matrix of subsystem B, and in particular obtain that both subsystems entropies are equal, without using the Araki-Lieb theorem. We would also like to find the entropy operator\nS A(B) = − lnρ A(B) ,(2)\nfor any of the subsystems. In particular we will consider later a two-level atom-field interaction, but the results may be generalized to other kind of subsystems, for instance atom-atom interaction, atom-many atoms interaction, N-level atom-field interaction, etc.\n\nThe density matrix for a two-level system interacting with another subsystem B is given byρ = |c c| |c s|\n|s c| |s s| (3)\nwhere |c (|s ) is the unnormalized wave function of the second system corresponding to the excited (ground) state of the two level system. From the total density matrix we may obtain the subsystem density matrices aŝ\nρ A = c|c s|c c|s s|s ≡ ρ 11 ρ 12 ρ 21 ρ 22 ,(4)\nandρ B = |c c| + |s s|.\n\nFrom the density matrix for subsystem B one can not see a clear way to calculate the entropy operator as powers of ρ B get complicated to be obtained. To make the calculation easier we state that if two subsystems are initially in pure states, after interaction, the trace of any function of one of the subsystems density matrix is equal to the trace of the function of the other subsystem's density matrix.\n\nWe can prove this, by using the following relation valid for two interacting systems that before interaction were in pure stateŝ\nρ n+1 B = T r A {ρ(t)ρ n A (t)},(6)\nwith this relation it is easy to show that\nT r B {ρ n+1 B (t)} = T r A {ρ n+1 A (t)}.(7)\nIn particular with the expression (6) we demonstrate that S B = S A . Of course it is also true\nthatρ n+1 A = T r B {ρ(t)ρ n B (t)}. (8) II. PROVINGρ n+1 B = T r A {ρ(t)ρ n A (t)} BY INDUCTION\nWe can prove relation (6) by induction. For this we need to findρ n A , to this end we writê\nρ A = 1 2 + δ 2 ρ 12 ρ 21 − δ 2 ≡ 1 2 +R,(9)\nwhere δ = ρ 11 − ρ 22 and 1 is the 2 × 2 unit density matrix. We can then find simplŷ\nρ n A = 1 2 +R n = n m=0 n m 1 2 n−mR m .(10)\nWe split the above sum into two sums, one with odd powers ofR and one with even powers. We also use thatR\n2m = ǫ 2m ,R 2m+1 =R ǫ ǫ 2m+1 (11) with ǫ = δ 2 4 + |ρ 12 | 2 1/2\n. Therefore we can writê\nρ n A = 1 2 1 2 + ǫ n + 1 2 − ǫ n 1 +R 2ǫ 1 2 + ǫ n − 1 2 − ǫ n(12)\nIn terms of ρ A the above equation is written aŝ\nρ n A = G(n)ρ A − ||ρ A ||G(n − 1)1 (13) where G(n) = 1 2ǫ 1 2 + ǫ n − 1 2 − ǫ n (14)\nwith the determinant ||ρ A (t)|| = 1 4 − ǫ 2 . Note that we have written ρ n A in terms of ρ A and the unity matrix. We could have done it via Cayley-Hamilton theorem [2], which states that any (square) matrix obeys its characteristic equation, i.e. a 2 × 2 matrix may be written as we did, as the square of the matrix (and any other power of it) may be related to the matrix and the unity matrix.\n\nThen relation (6) my be proved by induction: for n = 1, we havê\nρ 2 B = T r A {ρ(t)ρ A (t)}.(15)\nWe now assume it to be correct for n = k + 1\nρ k+1 B = T r A {ρ(t)ρ k A (t)}(16)\nand prove (6) for n = k + 1. By using (13) we can writê\nρ k+1 B =ρ 2 B G(k) −ρ B ||ρ A (t)||G(k − 1)(17)\nNote that any power ofρ B may be written in terms ofρ B andρ 2 B (k ≥ 2). Multiplying the above equation byρ B we obtain\nρ k+2 B =ρ 3 B G(k) −ρ 2 B ||ρ A (t)||G(k − 1)(18)\nWe obtainρ 3 B from (6) aŝ\nρ 3 B = T r A {ρ(t)ρ 2 A (t)} = T r A {ρ(t)[ρ A (t) + (ǫ 2 − 1 4 )]}(19)\nwhere for the second equality we have used the Cayley-Hamilton theorem for the atomic density matrix:ρ 2 (18) and after some algebra we findρ\nA (t) =ρ A (t) − ||ρ A (t)||1. Inserting (19) ink+2 B =ρ 2 B G(k + 1) −ρ B ||ρ A (t)||G(k) (20) orρ k+2 B = T r A {ρ(t)ρ k+1 A (t)}(21)\nthat ends the prove of relation (6) by induction.\n\n\nIII. ATOMIC ENTROPY OPERATOR\n\nWith the tools we have developed up to here we can study the two-level atom-field interaction [3] or equivalently the ion-laser interaction [4,5] and construct atom and field entropy operators. In the off-resonant atom-field interaction, i.e. the dispersive interaction, the (unnormalized) states |c and |s read [6] |c\n= 1 √ 2 |βe −iχt , |s = 1 √ 2 |βe iχt (22) where |β = e −|β| 2 /2 ∞ k=0 β k √\nk! |k is the initial coherent state for the field, χ is the interaction constant. The atom was considered initially in a superposition of excited and ground |ψ A = 1 √ 2 (|e + |g ). We write the entropy operator aŝ\nS A = lnρ −1 A = ln(1 −ρ A ) − ln ||ρ A ||,(23)\nsuch that the expectation value ofŜ A is the entropy. In the above expression we have\nused thatρ −1 A (t) =ξ A /||ρ A (t)|| with the purity operatorξ A = 1 −ρ A (t). We can develop ln(1 − x) = − ∞ n=1\nx n n and use (13) to find\nS A = F 1ρA + F 2 1(24)\nwith\nF 1 = 1 2ǫ ln 1 − 2ǫ 1 + 2ǫ F 2 = − 1 2 ln ||ρ A (t)|| + 1 2ǫ ln 1 − 2ǫ 1 + 2ǫ(25)\nNote thatŜ A is linear in the atomic density matrix as expected from Cayley-Hamilton's theorem (for 2 × 2 matrices). From (24) we can calculate the atomic (field) entropy and the atomic (field) entropy fluctuations\nŜ A = F 2 + F 1 (1 − 2||ρ A (t)||),(26)\nand\n∆Ŝ A = Ŝ 2 A − Ŝ A 2 = ln 1 − 2ǫ 1 + 2ǫ ||ρ A (t)|| 1/2 .(27)\n\nIV. FIELD ENTROPY OPERATOR\n\nWe use the expression forρ n B in terms ofρ A to write the field entropy operator in terms of the atomic density operatorŜ\nB = T r A {ρ(t)Ŝ A (t)ρ −1 A (t)}(28)\nNote that we can write the atomic entropy operator in terms of the operator used to define concurrence [7] becauseρ −1\nA (t) = 1 ||ρ A (t)||σ yρ * A (t)σ y =ρ A (t) ||ρ A (t)|| i.e. S B = − 1 ||ρ A (t)|| T r A {ρ(t)Ŝ AρA (t)}(29)\nBy inserting (24) into (28) we obtain\nS B = T r A {ρ(t)(F 1 + F 2ρ −1 A (t))}(30)\nusing the expression of the inverse of the atomic density operator in terms of the purity operator,the entropy may be written aŝ\nS B = T r A {ρ(t)(F 1 + F 2 ||ρ A (t)|| [1 −ρ A (t)])}(31)\nIn terms of the field density matrix the field entropy operator iŝ\nS B = F 1 + F 2 ||ρ A (t)|| ρ B (t) − F 2 ||ρ A (t)||ρ 2 B (t)(32)\nfrom (5) we can writeρ 2 B (t) aŝ ρ 2 B = |c c| c|c + |s s| s|s + |c s| c|s + |s c| s|c .\n\nFrom (22) we have that c|c = s|s = 1/2 and that c|s = s|c * = exp(−|β| 2 [1 − e 2iχt ])/2.\n\nThe entropy operator for the field is written aŝ\nS B = F 1 + F 2 2||ρ A (t)|| (|c c| + |s s|) − F 2 ||ρ A (t)||\n( s|c |s c| + c|s |c s|)\n\nWe have chosen the dispersive interaction so that all the terms forming the field density matrix and its square are coherent states, such that we can calculate the Wigner function associated to (32) in an easy way. We do it by means of the formula [8] W S (α) = ∞ n=0 (−1) n α, n|Ŝ B |α, n\n\nwhere |α, n are the so-called displaced number states [9]. The explicit expression for the above equation is W S (α) = e −2β 2 −2|α| 2 +4βαx cos χt F 1 + F 2 2||ρ A (t)|| cosh(4βα y sin χt)\n\n− e −2β 2 −2|α| 2 +4βαx cos χt F 2 2||ρ A (t)|| cos(2β[β sin 2χt − 2α x sin χt]).\n\n\nV. CONCLUSIONS\n\nWe have shown that the entropy operator for the subsystems may be obtained and in the case one of those subsystems is the quantized field. The conection with quasiprobability distribution functions [10,11], in particular the association to the Wigner function for the entropy has been obtained.\n \n. H Araki, E Lieb, Commun. Math. Phys. 18160H. Araki and E. Lieb, Commun. Math. Phys. 18, 160 (1970).\n\nLinear Algebra. R B J T Allenby ; Edward, Arnold , Modular Mathematics. R.B.J.T. Allenby (1995) \"Linear Algebra\", Modular Mathematics, Edward Arnold.\n\nE T Jaynes, F W Cummings, Proc. IEEE. IEEE5181E.T. Jaynes and F.W. Cummings, Proc. IEEE 51, 81 (1963).\n\n. H Moya-Cessa, P Tombesi, Phys. Rev. A. 6125401H. Moya-Cessa and P. Tombesi, Phys. Rev. A 61, 025401 (2000).\n\n. H Moya-Cessa, D Jonathan, P L Knight, J. of Mod. Optics. 50265H. Moya-Cessa, D. Jonathan and P.L. Knight, J. of Mod. Optics 50, 265 (2003).\n\n. L G Lutterbach, L Davidovich, Phys. Rev. Lett. 782547L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett 78, 2547 (1997).\n\n. W K Wootters, Phys. Rev. Lett. 802245W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).\n\nQuantum Optics in Phase Space. W P Schleich, Wiley-VCHW.P. Schleich, \"Quantum Optics in Phase Space\", (Wiley-VCH, 2001).\n\n. F A M De Oliveira, M S Kim, P L Knight, V Buzek, Phys. Rev. A. 412645F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek Phys. Rev. A 41, 2645 (1990).\n\n. H Moya-Cessa, S M Dutra, J A Roversi, A Vidiella-Barranco, J. of Mod. Optics. 46555H. Moya-Cessa, S.M. Dutra, J.A. Roversi, and A. Vidiella-Barranco, J. of Mod. Optics 46, 555 (1999).\n\n. H Moya-Cessa, J A Roversi, S M Dutra, A Vidiella-Barranco, Phys. Rev. A. 604029H. Moya-Cessa, J.A. Roversi, S.M. Dutra, and A. Vidiella-Barranco, Phys. Rev. A 60, 4029 (1999).\n"},"corpus_id":{"kind":"number","value":119246375,"string":"119,246,375"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1303.1042\",\n \"mag\": 2158987970,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1142/s021974990700261x\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1303.1042v1.pdf\"\n ],\n \"pdf_sha\": \"7973fc89ea4e0b88dd4df54025f1c14b9553cdba\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1303.1042\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Entropy operator and associated Wigner function"},"authors":{"kind":"list like","value":["H Moya-Cessa \nCoordinación de Optica\nINAOE\nPueApdo. Postal 51 y 21672000PueblaMexico"],"string":"[\n \"H Moya-Cessa \\nCoordinación de Optica\\nINAOE\\nPueApdo. Postal 51 y 21672000PueblaMexico\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We show how the entropy operators for two subsystems may be calculated. In the case of the atom-field interaction we obtain the associated Wigner function for the entropy operator for the quantized field."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"It is well known that when two subsystems A and B interact the entropy of the total system and the subsystem entropies obey a triangle inequality [1]\nThis result implies that if both subsystems are initially in pure states, the total entropy is zero and both subsystems entropies will be equal after they interact. Here we would like to arrive to the result that, if initially the two subsystems are in pure states any function the density matrix of subsystem A is equal to the function of the density matrix of subsystem B, and in particular obtain that both subsystems entropies are equal, without using the Araki-Lieb theorem. We would also like to find the entropy operator\nfor any of the subsystems. In particular we will consider later a two-level atom-field interaction, but the results may be generalized to other kind of subsystems, for instance atom-atom interaction, atom-many atoms interaction, N-level atom-field interaction, etc.\nThe density matrix for a two-level system interacting with another subsystem B is given byρ = |c c| |c s|\nwhere |c (|s ) is the unnormalized wave function of the second system corresponding to the excited (ground) state of the two level system. From the total density matrix we may obtain the subsystem density matrices aŝ\nandρ B = |c c| + |s s|.\nFrom the density matrix for subsystem B one can not see a clear way to calculate the entropy operator as powers of ρ B get complicated to be obtained. To make the calculation easier we state that if two subsystems are initially in pure states, after interaction, the trace of any function of one of the subsystems density matrix is equal to the trace of the function of the other subsystem's density matrix.\nWe can prove this, by using the following relation valid for two interacting systems that before interaction were in pure stateŝ\nwith this relation it is easy to show that\nIn particular with the expression (6) we demonstrate that S B = S A . Of course it is also true\nWe can prove relation (6) by induction. For this we need to findρ n A , to this end we writê\nwhere δ = ρ 11 − ρ 22 and 1 is the 2 × 2 unit density matrix. We can then find simplŷ\nWe split the above sum into two sums, one with odd powers ofR and one with even powers. We also use thatR\n. Therefore we can writê\nIn terms of ρ A the above equation is written aŝ\nwith the determinant ||ρ A (t)|| = 1 4 − ǫ 2 . Note that we have written ρ n A in terms of ρ A and the unity matrix. We could have done it via Cayley-Hamilton theorem [2], which states that any (square) matrix obeys its characteristic equation, i.e. a 2 × 2 matrix may be written as we did, as the square of the matrix (and any other power of it) may be related to the matrix and the unity matrix.\nThen relation (6) my be proved by induction: for n = 1, we havê\nWe now assume it to be correct for n = k + 1\nand prove (6) for n = k + 1. By using (13) we can writê\nNote that any power ofρ B may be written in terms ofρ B andρ 2 B (k ≥ 2). Multiplying the above equation byρ B we obtain\nWe obtainρ 3 B from (6) aŝ\nwhere for the second equality we have used the Cayley-Hamilton theorem for the atomic density matrix:ρ 2 (18) and after some algebra we findρ\nthat ends the prove of relation (6) by induction."},{"title":"III. ATOMIC ENTROPY OPERATOR","content":"With the tools we have developed up to here we can study the two-level atom-field interaction [3] or equivalently the ion-laser interaction [4,5] and construct atom and field entropy operators. In the off-resonant atom-field interaction, i.e. the dispersive interaction, the (unnormalized) states |c and |s read [6] |c\nk! |k is the initial coherent state for the field, χ is the interaction constant. The atom was considered initially in a superposition of excited and ground |ψ A = 1 √ 2 (|e + |g ). We write the entropy operator aŝ\nsuch that the expectation value ofŜ A is the entropy. In the above expression we have\nx n n and use (13) to find\nwith\nNote thatŜ A is linear in the atomic density matrix as expected from Cayley-Hamilton's theorem (for 2 × 2 matrices). From (24) we can calculate the atomic (field) entropy and the atomic (field) entropy fluctuations\nand"},{"title":"IV. FIELD ENTROPY OPERATOR","content":"We use the expression forρ n B in terms ofρ A to write the field entropy operator in terms of the atomic density operatorŜ\nNote that we can write the atomic entropy operator in terms of the operator used to define concurrence [7] becauseρ −1\nBy inserting (24) into (28) we obtain\nusing the expression of the inverse of the atomic density operator in terms of the purity operator,the entropy may be written aŝ\nIn terms of the field density matrix the field entropy operator iŝ\nfrom (5) we can writeρ 2 B (t) aŝ ρ 2 B = |c c| c|c + |s s| s|s + |c s| c|s + |s c| s|c .\nFrom (22) we have that c|c = s|s = 1/2 and that c|s = s|c * = exp(−|β| 2 [1 − e 2iχt ])/2.\nThe entropy operator for the field is written aŝ\n( s|c |s c| + c|s |c s|)\nWe have chosen the dispersive interaction so that all the terms forming the field density matrix and its square are coherent states, such that we can calculate the Wigner function associated to (32) in an easy way. We do it by means of the formula [8] W S (α) = ∞ n=0 (−1) n α, n|Ŝ B |α, n\nwhere |α, n are the so-called displaced number states [9]. The explicit expression for the above equation is W S (α) = e −2β 2 −2|α| 2 +4βαx cos χt F 1 + F 2 2||ρ A (t)|| cosh(4βα y sin χt)\n− e −2β 2 −2|α| 2 +4βαx cos χt F 2 2||ρ A (t)|| cos(2β[β sin 2χt − 2α x sin χt])."},{"title":"V. CONCLUSIONS","content":"We have shown that the entropy operator for the subsystems may be obtained and in the case one of those subsystems is the quantized field. The conection with quasiprobability distribution functions [10,11], in particular the association to the Wigner function for the entropy has been obtained."}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"It is well known that when two subsystems A and B interact the entropy of the total system and the subsystem entropies obey a triangle inequality [1]\\nThis result implies that if both subsystems are initially in pure states, the total entropy is zero and both subsystems entropies will be equal after they interact. Here we would like to arrive to the result that, if initially the two subsystems are in pure states any function the density matrix of subsystem A is equal to the function of the density matrix of subsystem B, and in particular obtain that both subsystems entropies are equal, without using the Araki-Lieb theorem. We would also like to find the entropy operator\\nfor any of the subsystems. In particular we will consider later a two-level atom-field interaction, but the results may be generalized to other kind of subsystems, for instance atom-atom interaction, atom-many atoms interaction, N-level atom-field interaction, etc.\\nThe density matrix for a two-level system interacting with another subsystem B is given byρ = |c c| |c s|\\nwhere |c (|s ) is the unnormalized wave function of the second system corresponding to the excited (ground) state of the two level system. From the total density matrix we may obtain the subsystem density matrices aŝ\\nandρ B = |c c| + |s s|.\\nFrom the density matrix for subsystem B one can not see a clear way to calculate the entropy operator as powers of ρ B get complicated to be obtained. To make the calculation easier we state that if two subsystems are initially in pure states, after interaction, the trace of any function of one of the subsystems density matrix is equal to the trace of the function of the other subsystem's density matrix.\\nWe can prove this, by using the following relation valid for two interacting systems that before interaction were in pure stateŝ\\nwith this relation it is easy to show that\\nIn particular with the expression (6) we demonstrate that S B = S A . Of course it is also true\\nWe can prove relation (6) by induction. For this we need to findρ n A , to this end we writê\\nwhere δ = ρ 11 − ρ 22 and 1 is the 2 × 2 unit density matrix. We can then find simplŷ\\nWe split the above sum into two sums, one with odd powers ofR and one with even powers. We also use thatR\\n. Therefore we can writê\\nIn terms of ρ A the above equation is written aŝ\\nwith the determinant ||ρ A (t)|| = 1 4 − ǫ 2 . Note that we have written ρ n A in terms of ρ A and the unity matrix. We could have done it via Cayley-Hamilton theorem [2], which states that any (square) matrix obeys its characteristic equation, i.e. a 2 × 2 matrix may be written as we did, as the square of the matrix (and any other power of it) may be related to the matrix and the unity matrix.\\nThen relation (6) my be proved by induction: for n = 1, we havê\\nWe now assume it to be correct for n = k + 1\\nand prove (6) for n = k + 1. By using (13) we can writê\\nNote that any power ofρ B may be written in terms ofρ B andρ 2 B (k ≥ 2). Multiplying the above equation byρ B we obtain\\nWe obtainρ 3 B from (6) aŝ\\nwhere for the second equality we have used the Cayley-Hamilton theorem for the atomic density matrix:ρ 2 (18) and after some algebra we findρ\\nthat ends the prove of relation (6) by induction.\"\n },\n {\n \"title\": \"III. ATOMIC ENTROPY OPERATOR\",\n \"content\": \"With the tools we have developed up to here we can study the two-level atom-field interaction [3] or equivalently the ion-laser interaction [4,5] and construct atom and field entropy operators. In the off-resonant atom-field interaction, i.e. the dispersive interaction, the (unnormalized) states |c and |s read [6] |c\\nk! |k is the initial coherent state for the field, χ is the interaction constant. The atom was considered initially in a superposition of excited and ground |ψ A = 1 √ 2 (|e + |g ). We write the entropy operator aŝ\\nsuch that the expectation value ofŜ A is the entropy. In the above expression we have\\nx n n and use (13) to find\\nwith\\nNote thatŜ A is linear in the atomic density matrix as expected from Cayley-Hamilton's theorem (for 2 × 2 matrices). From (24) we can calculate the atomic (field) entropy and the atomic (field) entropy fluctuations\\nand\"\n },\n {\n \"title\": \"IV. FIELD ENTROPY OPERATOR\",\n \"content\": \"We use the expression forρ n B in terms ofρ A to write the field entropy operator in terms of the atomic density operatorŜ\\nNote that we can write the atomic entropy operator in terms of the operator used to define concurrence [7] becauseρ −1\\nBy inserting (24) into (28) we obtain\\nusing the expression of the inverse of the atomic density operator in terms of the purity operator,the entropy may be written aŝ\\nIn terms of the field density matrix the field entropy operator iŝ\\nfrom (5) we can writeρ 2 B (t) aŝ ρ 2 B = |c c| c|c + |s s| s|s + |c s| c|s + |s c| s|c .\\nFrom (22) we have that c|c = s|s = 1/2 and that c|s = s|c * = exp(−|β| 2 [1 − e 2iχt ])/2.\\nThe entropy operator for the field is written aŝ\\n( s|c |s c| + c|s |c s|)\\nWe have chosen the dispersive interaction so that all the terms forming the field density matrix and its square are coherent states, such that we can calculate the Wigner function associated to (32) in an easy way. We do it by means of the formula [8] W S (α) = ∞ n=0 (−1) n α, n|Ŝ B |α, n\\nwhere |α, n are the so-called displaced number states [9]. The explicit expression for the above equation is W S (α) = e −2β 2 −2|α| 2 +4βαx cos χt F 1 + F 2 2||ρ A (t)|| cosh(4βα y sin χt)\\n− e −2β 2 −2|α| 2 +4βαx cos χt F 2 2||ρ A (t)|| cos(2β[β sin 2χt − 2α x sin χt]).\"\n },\n {\n \"title\": \"V. CONCLUSIONS\",\n \"content\": \"We have shown that the entropy operator for the subsystems may be obtained and in the case one of those subsystems is the quantized field. The conection with quasiprobability distribution functions [10,11], in particular the association to the Wigner function for the entropy has been obtained.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". H Araki, E Lieb, Commun. Math. Phys. 18160H. Araki and E. Lieb, Commun. Math. Phys. 18, 160 (1970).","doi":""},{"text":"Linear Algebra. R B J T Allenby ; Edward, Arnold , Modular Mathematics. R.B.J.T. Allenby (1995) \"Linear Algebra\", Modular Mathematics, Edward Arnold.","doi":""},{"text":"E T Jaynes, F W Cummings, Proc. IEEE. IEEE5181E.T. Jaynes and F.W. Cummings, Proc. IEEE 51, 81 (1963).","doi":""},{"text":". H Moya-Cessa, P Tombesi, Phys. Rev. A. 6125401H. Moya-Cessa and P. Tombesi, Phys. Rev. A 61, 025401 (2000).","doi":""},{"text":". H Moya-Cessa, D Jonathan, P L Knight, J. of Mod. Optics. 50265H. Moya-Cessa, D. Jonathan and P.L. Knight, J. of Mod. Optics 50, 265 (2003).","doi":""},{"text":". L G Lutterbach, L Davidovich, Phys. Rev. Lett. 782547L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett 78, 2547 (1997).","doi":""},{"text":". W K Wootters, Phys. Rev. Lett. 802245W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).","doi":""},{"text":"Quantum Optics in Phase Space. W P Schleich, Wiley-VCHW.P. Schleich, \"Quantum Optics in Phase Space\", (Wiley-VCH, 2001).","doi":""},{"text":". F A M De Oliveira, M S Kim, P L Knight, V Buzek, Phys. Rev. A. 412645F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek Phys. Rev. A 41, 2645 (1990).","doi":""},{"text":". H Moya-Cessa, S M Dutra, J A Roversi, A Vidiella-Barranco, J. of Mod. Optics. 46555H. Moya-Cessa, S.M. Dutra, J.A. Roversi, and A. Vidiella-Barranco, J. of Mod. Optics 46, 555 (1999).","doi":""},{"text":". H Moya-Cessa, J A Roversi, S M Dutra, A Vidiella-Barranco, Phys. Rev. A. 604029H. Moya-Cessa, J.A. Roversi, S.M. Dutra, and A. Vidiella-Barranco, Phys. Rev. A 60, 4029 (1999).","doi":""}],"string":"[\n {\n \"text\": \". H Araki, E Lieb, Commun. Math. Phys. 18160H. Araki and E. Lieb, Commun. Math. Phys. 18, 160 (1970).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Linear Algebra. R B J T Allenby ; Edward, Arnold , Modular Mathematics. R.B.J.T. Allenby (1995) \\\"Linear Algebra\\\", Modular Mathematics, Edward Arnold.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"E T Jaynes, F W Cummings, Proc. IEEE. IEEE5181E.T. Jaynes and F.W. Cummings, Proc. IEEE 51, 81 (1963).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Moya-Cessa, P Tombesi, Phys. Rev. A. 6125401H. Moya-Cessa and P. Tombesi, Phys. Rev. A 61, 025401 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Moya-Cessa, D Jonathan, P L Knight, J. of Mod. Optics. 50265H. Moya-Cessa, D. Jonathan and P.L. Knight, J. of Mod. Optics 50, 265 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L G Lutterbach, L Davidovich, Phys. Rev. Lett. 782547L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett 78, 2547 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". W K Wootters, Phys. Rev. Lett. 802245W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum Optics in Phase Space. W P Schleich, Wiley-VCHW.P. Schleich, \\\"Quantum Optics in Phase Space\\\", (Wiley-VCH, 2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F A M De Oliveira, M S Kim, P L Knight, V Buzek, Phys. Rev. A. 412645F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V. Buzek Phys. Rev. A 41, 2645 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Moya-Cessa, S M Dutra, J A Roversi, A Vidiella-Barranco, J. of Mod. Optics. 46555H. Moya-Cessa, S.M. Dutra, J.A. Roversi, and A. Vidiella-Barranco, J. of Mod. Optics 46, 555 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Moya-Cessa, J A Roversi, S M Dutra, A Vidiella-Barranco, Phys. Rev. A. 604029H. Moya-Cessa, J.A. Roversi, S.M. Dutra, and A. Vidiella-Barranco, Phys. Rev. A 60, 4029 (1999).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":61,"cells":{"text":{"kind":"string","value":"\nComment on the new AdS universë\n25 Nov 2008\n\nOzgür Sarıoglu *sarioglu@metu.edu.tr†btekin@metu.edu.tr \nDepartment of Physics\nFaculty of Arts and Sciences\nMiddle East Technical University\n06531AnkaraTurkey\n\nBayram Tekin \nDepartment of Physics\nFaculty of Arts and Sciences\nMiddle East Technical University\n06531AnkaraTurkey\n\nComment on the new AdS universë\n25 Nov 2008(Dated: November 25, 2008)PACS numbers: 04.20.-q, 04.20.Jb, 04.50-h\nWe show that Bonnor's new Anti-de Sitter (AdS) universe and its D-dimensional generalization is the previously studied AdS soliton.Recently, Bonnor [1] noted a second non-singular solution to cosmological Einstein equations R µν = Λg µν in four dimensions, with Λ < 0; the first obviously being the maximally symmetric AdS spacetime. In this comment, we show that Bonnor's \"non-uniform AdS universe\" is nothing but the AdS soliton, whose higher dimensional generalizations also exist[2].The metric introduced in [1] belongs to a family of cylindrically symmetric, static spacetimes that are solutions to the four dimensional Einstein equations with a negative cosmological constant, and that were independently found a long time ago by Linet [3] and Tian[4]. Various properties of these metrics were studied in[5], where non-singular sheet sources of them were also found and their relation to the black-string solutions were examined. Roughly speaking, these can be given the interpretation of infinite rods whose mass per unit length is defined through a parameter σ with a specific range. The spacetime that Bonnor considered is the one obtained by setting this parameter σ = 0 (see[5]and [1] for details).The generalization of this solution to D = n + 1 dimensions follows as\nds 2 = dρ 2 + cosh 4/n nρ 2ℓ − dt 2 + 4ℓ 2 n 2 tanh 2 nρ 2ℓ dφ 2 + n−2 i=1 dx 2 i ,(1)\nwhich solves\nR µν = − n ℓ 2 g µν ,\nwith the ranges t, x i ∈ R (where i = 1, . . . , n − 2), ρ ∈ R + and φ ∈ [0, 2π). When n = 3, this reduces to the metric in [1]. For n = 2, one simply gets the AdS solution in 2 + 1 dimensions. (1) has no singularities or horizons. Any invariant that involves only the curvature scalar or the Ricci tensor is identical to its counterpart belonging to the usual maximally symmetric AdS. However, the regularity and the non-uniformity of the solution is apparent from the Kretschmann scalar\nR µνλσ R µνλσ = 1 ℓ 4 cosh 4 nρ 2ℓ n(n − 2)(n − 1) 2 + n(n + 1) 2 1 + cosh nρ ℓ 2 .(2)\nLet us show that (1) is a special case of the AdS soliton [2], which reads\nds 2 = r 2 ℓ 2 1 − r n 0 r n dτ 2 + n−2 i=1 (dx i ) 2 − dt 2 + 1 − r n 0 r n −1 ℓ 2 r 2 dr 2 .\n(3) * Electronic address: sarioglu@metu.edu.tr † Electronic address: btekin@metu.edu.tr (3) was obtained by the double analytic continuation of a near extremal (n − 1)-brane solution.\n\nx i and the t denote the coordinates on the \"brane\" and r ≥ r 0 , where r 0 is a free parameter. A conical singularity is avoided if τ has a period β = 4πℓ 2 /(nr 0 ) [2].\n\nConsider the coordinate transformation r = r 0 cosh 2/n (nρ/2ℓ), which respects the proper ranges of r and ρ [7]. Then (3) turns into\nds 2 = dρ 2 + r 2 0 ℓ 2 cosh 4/n nρ 2ℓ − dt 2 + tanh 2 nρ 2ℓ dτ 2 + n−2 i=1 dx 2 i .(4)\nChoosing r 0 = ℓ and τ = 2ℓφ/n, one obtains (1). Finally, it is obvious that (1) does not make sense in D = 1 + 1 dimensions. However, one can separately consider the D = 2 Euclidean space. There, the cigar soliton of Witten [6] \nds 2 = dρ 2 + tanh 2 ρ dφ 2(5)\ncan be thought of as the D = 2 analog of (1). Note that, the cigar is also singularity free with a curvature scalar R = 4/ cosh 2 ρ but, of course, it is asymptotically flat unlike (1). To conclude, we have shown that Bonnor's solution [1] and its D-dimensional generalization is the AdS soliton [2].\n\n\nI. ACKNOWLEDGMENTS\n\nIn the initial version of this note, we weren't aware that Bonnor's solution is a special case of the AdS soliton. We kindly thank Roberto Emparan who brought this to our attention. That led to a complete revision of our work. This work is partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK). B.T. is also partially supported by the TÜBİTAK Kariyer Grant 104T177.\n \n. W B Bonnor, Class. Quantum Grav. 25225005W.B. Bonnor, Class. Quantum Grav. 25, 225005 (2008).\n\n. G T Horowitz, R C Myers, Phys. Rev. D. 5926005G.T. Horowitz and R.C. Myers, Phys. Rev. D 59, 026005 (1999).\n\n. B Linet, J. Math. Phys. 271817B. Linet, J. Math. Phys. 27, 1817 (1986).\n\n. Q Tian, Phys. Rev. D. 333549Q. Tian, Phys. Rev. D 33, 3549 (1986).\n\n. M Zofka, J Bicak, arXiv:0712.2144Class. Quantum Grav. 2515011gr-qcM. Zofka and J. Bicak, Class. Quantum Grav. 25, 015011 (2008) [arXiv:0712.2144 [gr-qc]].\n\n. E Witten, Phys. Rev. D. 44314E. Witten, Phys. Rev. D 44, 314 (1991).\n\nThis transformation was suggested by Roberto Emparan. This transformation was suggested by Roberto Emparan.\n"},"corpus_id":{"kind":"number","value":119117439,"string":"119,117,439"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0811.3683\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/0264-9381/26/4/048001\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0811.3683v2.pdf\"\n ],\n \"pdf_sha\": \"c9d8555a71fd6e6a253c59cf5d19774fee35ba9c\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/0811.3683\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Comment on the new AdS universë"},"authors":{"kind":"list like","value":["Ozgür Sarıoglu *sarioglu@metu.edu.tr†btekin@metu.edu.tr \nDepartment of Physics\nFaculty of Arts and Sciences\nMiddle East Technical University\n06531AnkaraTurkey","Bayram Tekin \nDepartment of Physics\nFaculty of Arts and Sciences\nMiddle East Technical University\n06531AnkaraTurkey"],"string":"[\n \"Ozgür Sarıoglu *sarioglu@metu.edu.tr†btekin@metu.edu.tr \\nDepartment of Physics\\nFaculty of Arts and Sciences\\nMiddle East Technical University\\n06531AnkaraTurkey\",\n \"Bayram Tekin \\nDepartment of Physics\\nFaculty of Arts and Sciences\\nMiddle East Technical University\\n06531AnkaraTurkey\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We show that Bonnor's new Anti-de Sitter (AdS) universe and its D-dimensional generalization is the previously studied AdS soliton.Recently, Bonnor [1] noted a second non-singular solution to cosmological Einstein equations R µν = Λg µν in four dimensions, with Λ < 0; the first obviously being the maximally symmetric AdS spacetime. In this comment, we show that Bonnor's \"non-uniform AdS universe\" is nothing but the AdS soliton, whose higher dimensional generalizations also exist[2].The metric introduced in [1] belongs to a family of cylindrically symmetric, static spacetimes that are solutions to the four dimensional Einstein equations with a negative cosmological constant, and that were independently found a long time ago by Linet [3] and Tian[4]. Various properties of these metrics were studied in[5], where non-singular sheet sources of them were also found and their relation to the black-string solutions were examined. Roughly speaking, these can be given the interpretation of infinite rods whose mass per unit length is defined through a parameter σ with a specific range. The spacetime that Bonnor considered is the one obtained by setting this parameter σ = 0 (see[5]and [1] for details).The generalization of this solution to D = n + 1 dimensions follows as"},"sections":{"kind":"list like","value":[{"title":"I. ACKNOWLEDGMENTS","content":"In the initial version of this note, we weren't aware that Bonnor's solution is a special case of the AdS soliton. We kindly thank Roberto Emparan who brought this to our attention. That led to a complete revision of our work. This work is partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK). B.T. is also partially supported by the TÜBİTAK Kariyer Grant 104T177."}],"string":"[\n {\n \"title\": \"I. ACKNOWLEDGMENTS\",\n \"content\": \"In the initial version of this note, we weren't aware that Bonnor's solution is a special case of the AdS soliton. We kindly thank Roberto Emparan who brought this to our attention. That led to a complete revision of our work. This work is partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK). B.T. is also partially supported by the TÜBİTAK Kariyer Grant 104T177.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". W B Bonnor, Class. Quantum Grav. 25225005W.B. Bonnor, Class. Quantum Grav. 25, 225005 (2008).","doi":""},{"text":". G T Horowitz, R C Myers, Phys. Rev. D. 5926005G.T. Horowitz and R.C. Myers, Phys. Rev. D 59, 026005 (1999).","doi":""},{"text":". B Linet, J. Math. Phys. 271817B. Linet, J. Math. Phys. 27, 1817 (1986).","doi":""},{"text":". Q Tian, Phys. Rev. D. 333549Q. Tian, Phys. Rev. D 33, 3549 (1986).","doi":""},{"text":". M Zofka, J Bicak, arXiv:0712.2144Class. Quantum Grav. 2515011gr-qcM. Zofka and J. Bicak, Class. Quantum Grav. 25, 015011 (2008) [arXiv:0712.2144 [gr-qc]].","doi":""},{"text":". E Witten, Phys. Rev. D. 44314E. Witten, Phys. Rev. D 44, 314 (1991).","doi":""},{"text":"This transformation was suggested by Roberto Emparan. This transformation was suggested by Roberto Emparan.","doi":""}],"string":"[\n {\n \"text\": \". W B Bonnor, Class. Quantum Grav. 25225005W.B. Bonnor, Class. Quantum Grav. 25, 225005 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G T Horowitz, R C Myers, Phys. Rev. D. 5926005G.T. Horowitz and R.C. Myers, Phys. Rev. D 59, 026005 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Linet, J. Math. Phys. 271817B. Linet, J. Math. Phys. 27, 1817 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Q Tian, Phys. Rev. D. 333549Q. Tian, Phys. Rev. D 33, 3549 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Zofka, J Bicak, arXiv:0712.2144Class. Quantum Grav. 2515011gr-qcM. Zofka and J. Bicak, Class. Quantum Grav. 25, 015011 (2008) [arXiv:0712.2144 [gr-qc]].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Witten, Phys. Rev. D. 44314E. Witten, Phys. Rev. D 44, 314 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"This transformation was suggested by Roberto Emparan. This transformation was suggested by Roberto Emparan.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":62,"cells":{"text":{"kind":"string","value":"\nEFFECTS OF BBN ON POPULATION III STARS\n18 Dec 2003 February 7, 2020\n\nF Iocco iocco@na.infn.it \nComplesso Universitario di M.te S.Angelo\nvia Cinthia80126NapoliItaly\n\nEFFECTS OF BBN ON POPULATION III STARS\n18 Dec 2003 February 7, 202017:36 WSPC/Trim Size: 9.75in x 6.5in for Proceedings proc\nThe presence (or absence) of CNO elements in the primordial gas determines different behaviours in population III stars formation and evolution: we therefore present an analysis of the main channels for the synthesis of these elements in BBN in order to understand, within a reliable interval, their abundance in the primordial material.\n\nIntroduction\n\nRecent studies on population III (popIII) showed how interesting this pregalactic generation of stars is for our overall understanding of cosmology: their contribution to the first stucture formation and to reionization makes them an extremely promising topic, (see for example 1 ). In light of the huge number of works published in the last years, (see 2 and references therein), it's likely to picture the population III as a self-killing generation of very massive stars, which evolve fastly and eventually explode as Pair Instability Supernovae, ejecting in the outer space the metals they have produced during their life, (see for example 3 ). The initial mass function (IMF) of these stars is determined from the absence of metals in the material from which they form out: there is a threshold in the metallicity of the gas above which the IMF shows the usual Salpeter form, and below which the IMF is the top-heavy one shown from 4 . The paucity of metals affects also the evolution of popIII: as widely known stars more massive than 3-4 M ⊙ burn hydrogen via CNO cycle, so that the scarcity of CNO catalyzers in popIII forces the main sequence through an \"unusual\" pp burning of hydrogen, resulting in an expansion and extreme heating of the core 5 . This phenomenon shows a threshold in CNO elements abundance, above which the star experience a \"normal\" main sequence. In light of all this it would be extremely important to find out with reliable precision the amount of metals in the primordial gas.\n\n\nMetal nucleosynthesis in BBN\n\nThe successes of BBN predictions, obtained comparing the theoretical estimates of light elements with the observed abudances, show the deep knowledge of nucleosynthesis phenomenon we have today -at least for what deals with light elements 6 . In particular the updating of the cross-section data and the knowledge of the main synthesis channels is essential in order to estimate the primordial elemental abundances and their uncertainties. Such a precision in the light nuclide predictions is not yet available for the heavier CNO nuuclides, nor in the connections between light and heavy elements. Historycally this is due to the very low final abundance of these elements which does not justify an effort to determine them, as long as it is impossible to compare the predictions with observational data. In light of the importance of metals (and more specifically of CNO elements) in POPIII stars theory we have performed an analysis of the BBN network toward carbon, nytrogen and oxygen, identifying the main reaction channels and updating, where possible, the cross-section data with newest ones. Our study showed that the main channel for carbon production in stars, namely the 2 4 He(α, γ) 12 C which provides a direct connection between light and heavy elements, is suppressed in BBN as a consequence of the very low density of the plasma; we have identified as main channels for carbon synthesis the 7 Li(α, γ) 11 B(p, γ) 12 C, 7 Li(α, γ) 11 B(d, n) 12 C and 7 Be(α, γ) 11 C, together with the back reaction 11 C(n, α)2 4 He and 11 B(p, α)2 4 He; heavier elements build up through progressive proton, deuteron and neutron captures upon carbon nuclei. These results, obtained by computing the contributions of the different reactions to the Boltzmann equation which describes the elemental abundance evolution, show that the heavy elements synthesis in BBN is strongly related to the intermediate element (such as Li, Be and B) one, thus emphasizing the necesssity of a complete revision of the \"intermediate\" and \"upper\" part of the network. We have also noted that most of the reactions involving isotopes of hydrogen and helium but H and 4 He are neglected in the existing code -at least for what deals with the \"metal network\"; we have added a first set of new reactions to study how they affect the heavy-nuclide synthesis, and results will be presented in a future paper.\n \n. R Cen, ApJ. 591R. Cen; ApJ.591:L5-L8,2003.\n\n. V Bromm, R B Larson, astro-ph/0311019V. Bromm and R.B. Larson; astro-ph/0311019.\n\n. R Salvaterra, A Ferrara, ; Mnrot, 34017R. Salvaterra, A. Ferrara; MNROT.340:L17,2003.\n\n. Nakamura E Umemura, astro-ph/0106009Nakamura e Umemura; astro-ph/0106009.\n\n. I Dominguez, O Straniero, M Limongi, A Chieffi, astro-ph/9912425I. Dominguez, O. Straniero, M. Limongi and A. Chieffi; astro-ph/9912425.\n\n. A Cuoco, F Iocco, G Mangano, G Miele, O Pisanti, P D Serpico, astro-ph/0307213A. Cuoco, F. Iocco, G. Mangano, G. Miele, O. Pisanti and P.D. Serpico; astro-ph/0307213.\n"},"corpus_id":{"kind":"number","value":119032389,"string":"119,032,389"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"astro-ph/0312473\",\n \"mag\": 3102449031,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1142/9789812702999_0043\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/astro-ph/0312473v1.pdf\"\n ],\n \"pdf_sha\": \"7d194f46c6fb425b37faf679ea9560b0f208767a\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/astro-ph/0312473\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"EFFECTS OF BBN ON POPULATION III STARS"},"authors":{"kind":"list like","value":["F Iocco iocco@na.infn.it \nComplesso Universitario di M.te S.Angelo\nvia Cinthia80126NapoliItaly"],"string":"[\n \"F Iocco iocco@na.infn.it \\nComplesso Universitario di M.te S.Angelo\\nvia Cinthia80126NapoliItaly\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The presence (or absence) of CNO elements in the primordial gas determines different behaviours in population III stars formation and evolution: we therefore present an analysis of the main channels for the synthesis of these elements in BBN in order to understand, within a reliable interval, their abundance in the primordial material."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Recent studies on population III (popIII) showed how interesting this pregalactic generation of stars is for our overall understanding of cosmology: their contribution to the first stucture formation and to reionization makes them an extremely promising topic, (see for example 1 ). In light of the huge number of works published in the last years, (see 2 and references therein), it's likely to picture the population III as a self-killing generation of very massive stars, which evolve fastly and eventually explode as Pair Instability Supernovae, ejecting in the outer space the metals they have produced during their life, (see for example 3 ). The initial mass function (IMF) of these stars is determined from the absence of metals in the material from which they form out: there is a threshold in the metallicity of the gas above which the IMF shows the usual Salpeter form, and below which the IMF is the top-heavy one shown from 4 . The paucity of metals affects also the evolution of popIII: as widely known stars more massive than 3-4 M ⊙ burn hydrogen via CNO cycle, so that the scarcity of CNO catalyzers in popIII forces the main sequence through an \"unusual\" pp burning of hydrogen, resulting in an expansion and extreme heating of the core 5 . This phenomenon shows a threshold in CNO elements abundance, above which the star experience a \"normal\" main sequence. In light of all this it would be extremely important to find out with reliable precision the amount of metals in the primordial gas."},{"title":"Metal nucleosynthesis in BBN","content":"The successes of BBN predictions, obtained comparing the theoretical estimates of light elements with the observed abudances, show the deep knowledge of nucleosynthesis phenomenon we have today -at least for what deals with light elements 6 . In particular the updating of the cross-section data and the knowledge of the main synthesis channels is essential in order to estimate the primordial elemental abundances and their uncertainties. Such a precision in the light nuclide predictions is not yet available for the heavier CNO nuuclides, nor in the connections between light and heavy elements. Historycally this is due to the very low final abundance of these elements which does not justify an effort to determine them, as long as it is impossible to compare the predictions with observational data. In light of the importance of metals (and more specifically of CNO elements) in POPIII stars theory we have performed an analysis of the BBN network toward carbon, nytrogen and oxygen, identifying the main reaction channels and updating, where possible, the cross-section data with newest ones. Our study showed that the main channel for carbon production in stars, namely the 2 4 He(α, γ) 12 C which provides a direct connection between light and heavy elements, is suppressed in BBN as a consequence of the very low density of the plasma; we have identified as main channels for carbon synthesis the 7 Li(α, γ) 11 B(p, γ) 12 C, 7 Li(α, γ) 11 B(d, n) 12 C and 7 Be(α, γ) 11 C, together with the back reaction 11 C(n, α)2 4 He and 11 B(p, α)2 4 He; heavier elements build up through progressive proton, deuteron and neutron captures upon carbon nuclei. These results, obtained by computing the contributions of the different reactions to the Boltzmann equation which describes the elemental abundance evolution, show that the heavy elements synthesis in BBN is strongly related to the intermediate element (such as Li, Be and B) one, thus emphasizing the necesssity of a complete revision of the \"intermediate\" and \"upper\" part of the network. We have also noted that most of the reactions involving isotopes of hydrogen and helium but H and 4 He are neglected in the existing code -at least for what deals with the \"metal network\"; we have added a first set of new reactions to study how they affect the heavy-nuclide synthesis, and results will be presented in a future paper."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Recent studies on population III (popIII) showed how interesting this pregalactic generation of stars is for our overall understanding of cosmology: their contribution to the first stucture formation and to reionization makes them an extremely promising topic, (see for example 1 ). In light of the huge number of works published in the last years, (see 2 and references therein), it's likely to picture the population III as a self-killing generation of very massive stars, which evolve fastly and eventually explode as Pair Instability Supernovae, ejecting in the outer space the metals they have produced during their life, (see for example 3 ). The initial mass function (IMF) of these stars is determined from the absence of metals in the material from which they form out: there is a threshold in the metallicity of the gas above which the IMF shows the usual Salpeter form, and below which the IMF is the top-heavy one shown from 4 . The paucity of metals affects also the evolution of popIII: as widely known stars more massive than 3-4 M ⊙ burn hydrogen via CNO cycle, so that the scarcity of CNO catalyzers in popIII forces the main sequence through an \\\"unusual\\\" pp burning of hydrogen, resulting in an expansion and extreme heating of the core 5 . This phenomenon shows a threshold in CNO elements abundance, above which the star experience a \\\"normal\\\" main sequence. In light of all this it would be extremely important to find out with reliable precision the amount of metals in the primordial gas.\"\n },\n {\n \"title\": \"Metal nucleosynthesis in BBN\",\n \"content\": \"The successes of BBN predictions, obtained comparing the theoretical estimates of light elements with the observed abudances, show the deep knowledge of nucleosynthesis phenomenon we have today -at least for what deals with light elements 6 . In particular the updating of the cross-section data and the knowledge of the main synthesis channels is essential in order to estimate the primordial elemental abundances and their uncertainties. Such a precision in the light nuclide predictions is not yet available for the heavier CNO nuuclides, nor in the connections between light and heavy elements. Historycally this is due to the very low final abundance of these elements which does not justify an effort to determine them, as long as it is impossible to compare the predictions with observational data. In light of the importance of metals (and more specifically of CNO elements) in POPIII stars theory we have performed an analysis of the BBN network toward carbon, nytrogen and oxygen, identifying the main reaction channels and updating, where possible, the cross-section data with newest ones. Our study showed that the main channel for carbon production in stars, namely the 2 4 He(α, γ) 12 C which provides a direct connection between light and heavy elements, is suppressed in BBN as a consequence of the very low density of the plasma; we have identified as main channels for carbon synthesis the 7 Li(α, γ) 11 B(p, γ) 12 C, 7 Li(α, γ) 11 B(d, n) 12 C and 7 Be(α, γ) 11 C, together with the back reaction 11 C(n, α)2 4 He and 11 B(p, α)2 4 He; heavier elements build up through progressive proton, deuteron and neutron captures upon carbon nuclei. These results, obtained by computing the contributions of the different reactions to the Boltzmann equation which describes the elemental abundance evolution, show that the heavy elements synthesis in BBN is strongly related to the intermediate element (such as Li, Be and B) one, thus emphasizing the necesssity of a complete revision of the \\\"intermediate\\\" and \\\"upper\\\" part of the network. We have also noted that most of the reactions involving isotopes of hydrogen and helium but H and 4 He are neglected in the existing code -at least for what deals with the \\\"metal network\\\"; we have added a first set of new reactions to study how they affect the heavy-nuclide synthesis, and results will be presented in a future paper.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". R Cen, ApJ. 591R. Cen; ApJ.591:L5-L8,2003.","doi":""},{"text":". V Bromm, R B Larson, astro-ph/0311019V. Bromm and R.B. Larson; astro-ph/0311019.","doi":""},{"text":". R Salvaterra, A Ferrara, ; Mnrot, 34017R. Salvaterra, A. Ferrara; MNROT.340:L17,2003.","doi":""},{"text":". Nakamura E Umemura, astro-ph/0106009Nakamura e Umemura; astro-ph/0106009.","doi":""},{"text":". I Dominguez, O Straniero, M Limongi, A Chieffi, astro-ph/9912425I. Dominguez, O. Straniero, M. Limongi and A. Chieffi; astro-ph/9912425.","doi":""},{"text":". A Cuoco, F Iocco, G Mangano, G Miele, O Pisanti, P D Serpico, astro-ph/0307213A. Cuoco, F. Iocco, G. Mangano, G. Miele, O. Pisanti and P.D. Serpico; astro-ph/0307213.","doi":""}],"string":"[\n {\n \"text\": \". R Cen, ApJ. 591R. Cen; ApJ.591:L5-L8,2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Bromm, R B Larson, astro-ph/0311019V. Bromm and R.B. Larson; astro-ph/0311019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Salvaterra, A Ferrara, ; Mnrot, 34017R. Salvaterra, A. Ferrara; MNROT.340:L17,2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Nakamura E Umemura, astro-ph/0106009Nakamura e Umemura; astro-ph/0106009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Dominguez, O Straniero, M Limongi, A Chieffi, astro-ph/9912425I. Dominguez, O. Straniero, M. Limongi and A. Chieffi; astro-ph/9912425.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Cuoco, F Iocco, G Mangano, G Miele, O Pisanti, P D Serpico, astro-ph/0307213A. Cuoco, F. Iocco, G. Mangano, G. Miele, O. Pisanti and P.D. Serpico; astro-ph/0307213.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":63,"cells":{"text":{"kind":"string","value":"\nPERSPECTIVES ON LORENTZ AND CPT VIOLATION\n5 Feb 2008 IUHET 513, September 2007\n\nV Alan Kostelecký \nPhysics Department\nIndiana University Bloomington\n47405INU.S.A\n\nPERSPECTIVES ON LORENTZ AND CPT VIOLATION\n5 Feb 2008 IUHET 513, September 2007This talk offers some comments and perspectives on Lorentz and CPT violation.\n\n\nIntroduction\n\nLorentz symmetry is the invariance of physical laws under rotations and boosts. As a global symmetry over Minkowski spacetime, it underlies the theory of Special Relativity and the Standard Model of particle physics, where it is intimately tied to CPT invariance. As a local symmetry of freely falling frames, it is an essential component of General Relativity. Nonetheless, the possibility exists that nature may exhibit tiny violations of Lorentz symmetry. This talk presents some perspectives on the basic motivations and reasoning in this subject.\n\nSince Lorentz symmetry has been verified in many experiments, as has CPT invariance, it is reasonable to ask why relativity violations are worth considering. A sufficient theoretical motivation is the need for a consistent description of Lorentz and CPT violation to offer guidance for experimental tests. However, a stronger motivation is the prospect that Lorentz and CPT violation can serve as a sensitive potential signal for physics at the Planck scale. In fact, the present interest in the subject was triggered by the realization that natural mechanisms for Lorentz and CPT violation exist in unified theories at the Planck scale. 1 The large range of existing phenomenological and experimental activities stems from the application of effective field theory 2 and the construction of the Standard-Model Extension (SME) 3,4 to catalogue and predict observable effects.\n\n\nApproaches and origins\n\nThe study of Lorentz and CPT violation can be approached on three distinct levels. First, at the level of fundamental theory at the unification scale, one can investigate possible mechanisms and determine their features and implications. Second, at the level of theory at accessible scales, one can seek a description of the resulting effects that is quantitative, general, and compatible with the established physics of the Standard Model and General Relativity. Finally, at the level of observation and experiment, one can study and perform both high sensitivity tests and broad searches, preferably ones that are feasible with existing or near-future technology.\n\nThe first point to establish is whether Lorentz violation can indeed occur in a fundamental theory. Without at least one viable mechanism, the interest of the idea would be much reduced. A plausible origin for Lorentz violation has been identified in string field theory, 1 which has interactions with a generic structure that could in principle trigger spontaneous violation of Lorentz symmetry and generate vacuum expectation values for Lorentz tensor fields. More recently, numerous other mechanisms for Lorentz violation at the fundamental level have been proposed including, for example, ones involving noncommutative field theories, 5 spacetime-varying fields, 6 quantum gravity, 7 random-dynamics models, 8 multiverses, 9 brane-world scenarios, 10 supersymmetry, 11 and massive gravity. 12 Given that mechanisms exist for Lorentz and CPT violation in an underlying theory, it is natural to ask about the consequences for physics at observable scales. In particular, the question arises as to the best method to develop a description of the possible effects.\n\n\nDescribing Lorentz violation\n\nSome key criteria offer valuable guidance in the search for a suitable theoretical framework for describing Lorentz violation at attainable scales. The first is coordinate independence. It has been accepted since long before 1905 that the physics of a system should be independent of a change of observer coordinates. This holds whether a coordinate change is implemented via a Lorentz transformation or in any other way.\n\nThe second is realism. Since 1905, when virtually no fundamental particles were known and quantum physics was at its dawn, thousands of people have invested millions of person-hours and billions of dollars in establishing the Standard Model of particle physics and General Relativity as an accurate description of nature. To be of real interest nowadays, any proposed theoretical framework for Lorentz violation must incorporate this well-established physics.\n\nThe third is generality. No compelling evidence for Lorentz violation exists at present. Physics is therefore currently in the position of searching for a violation, as opposed to attempting to understand an observed effect.\n\nIn the searching phase, it is desirable to have the most general possible formulation so that no region is left unexplored. This is in strong contrast to the modeling phase, where considerations such as simplicity are important in attempts to understand a known effect. Armed with these criteria, we can follow the basic reasoning that leads to the application of effective field theory and the construction of the SME.\n\n\nModified Lorentz transformations\n\nSince the essential content of Special Relativity is the idea that physics is invariant under Lorentz transformations, the most obvious approach to describing relativity violations is to investigate modifications of the Lorentz transformations. In fact, the literature since 1905 abounds with various ad hoc proposals of this type. However, independent of any specific proposal, this approach has some serious disadvantages.\n\nOne is that a textbook Lorentz transformation acts on the observer and therefore corresponds merely to a change of coordinates, i.e., a change of reference frame. However, according to the above criterion of coordinate independence, a frame change cannot have physical implications by itself.\n\nThe key feature of Special Relativity is really the requirement that the equations for the system being observed must be covariant under a Lorentz transformation, which intrinsically assumes that Lorentz symmetry is exact. This approach is therefore problematic for investigating violations.\n\nIt is of course possible to construct special models imposing form covariance of the system under some ad hoc alternative transformation. However, any specific such proposal runs counter to the criterion of generality. Moreover, some kinds of violations are difficult and perhaps even impossible to countenance via this approach. For example, Lorentz violation in nature might well be particle-species dependent, but it is very challenging to formulate a description of this flavor dependence based on modified Lorentz transformations of the observer. The criterion of realism presents a further substantial obstacle, since it is awkward at best to implement such models in the context of the Standard Model and General Relativity.\n\n\nModified dispersion laws\n\nThe above discussion suggests that a general and realistic investigation of Lorentz violation is most naturally performed directly in terms of the properties of a system rather than via modifications of the Lorentz transformations. A simple implementation of this is to study modifications of particle dispersion laws. However, this also suffers from serious drawbacks.\n\nOne issue involves the criterion of generality. Modifications of dispersion laws can only describe changes in the free propagation of particles and perhaps also partially account for interaction kinematics. However, physics is far more than free propagation, and this approach therefore disregards a large range of interesting Lorentz-violating effects involving interactions.\n\nThere are also various issues associated with the choice of modifications to the dispersion law. Not all choices are compatible with desirable features such as originating from an action. Also, meaningful physical measurements must necessarily compare two quantities, so some choices may be unphysical. In particular, calculations with a modified dispersion law that yield apparent changes of properties are insufficient by themselves to demonstrate physical Lorentz violation. A simple example of a modified dispersion law with no observable consequences in Minkowski spacetime is 3 p µ p µ = m 2 + a µ p µ , where a µ is a prescribed set of four numbers in a given frame. Direct calculations with this dispersion law appear to give Lorentz-violating properties that depend on the preferred vector a µ , but in fact they are unobservable because a µ can be eliminated via a physically irrelevant redefinition of the energy and momentum. The observability of modifications to a dispersion law can be challenging to demonstrate.\n\n\nEffective field theory and the SME\n\nWe see that the desiderata for a satisfactory description of Lorentz violation include a comprehensive treatment of free and interacting effects in all particle species. Remarkably, a model-independent and general approach of this type exists.\n\nThe key is to take advantage of the idea that Lorentz violation at attainable energies can be described using effective field theory, independent of the underlying mechanism. 2,13 Starting from the Standard Model coupled to General Relativity, we can add to the action all possible scalar terms formed by contracting operators for Lorentz violation with coefficients that control the size of the effects. The operators are naturally ordered according to their mass dimension. The resulting realistic effective field theory is the SME. 3,4 Since CPT violation in realistic field theories comes with Lorentz violation, 14 the SME also incorporates general CPT violation.\n\nBy virtue of its construction, the SME satisfies the three guiding crite-ria of coordinate invariance, realism, and generality. Moreover, it handles simultaneously all particle species, including both propagation and interaction properties, so its equations of motion contain all action-compatible modifications of realistic dispersion laws. The coordinate invariance implies that physics is unaffected by observer frame changes, including Lorentz and other transformations, while particle transformations can produce observable effects of Lorentz violation. The primary disadvantage of the SME approach is its comparative complexity and the investment required to become proficient with its use. However, this is outweighed by its advantages as a realistic, general, and calculable framework for describing Lorentz violation. Judicious choices establishing relations among the SME coefficients for Lorentz violation yield elegant and simple models that can serve as a theorist's playground, while the general case offers guidance for broad-based experimental searches.\n\n\nGravity and Lorentz violation\n\nThe SME allows for both global 3 and local 4 Lorentz violation, and interesting effects arise from local Lorentz violation in the gravitational context. In general, local Lorentz violation can be understood as arising when a nonzero coefficient t abc... for Lorentz violation exists in local freely falling frames. 4 The cofficient t abc... can be converted to a coefficient t λµν... on the spacetime manifold using the vierbein e a µ . One result is that spontaneous violation of local Lorentz symmetry is always accompanied by spontaneous diffeomorphism violation, and vice versa. 15 A nonzero coefficient t abc... is the vacuum value of a local Lorentz tensor field, and it implies Lorentz violation because it is invariant instead of transforming like a tensor under particle transformations. The vierbein ensures that there is a corresponding spacetime tensor field with vacuum value t λµν... on the spacetime manifold, which in turn implies spontaneous diffeomorphism breaking because it is invariant instead of transforming like a tensor under particle diffeomorphisms.\n\nA more surprising result is that explicit Lorentz violation is generically incompatible with Riemann geometry. 4 Explicit violation occurs when the SME coefficients are externally prescribed, but the ensuing equations of motion turn out to be inconsistent with the Bianchi identities. This result also holds in Riemann-Cartan spacetime. However, spontaneous violation evades the difficulty because it generates the SME coefficients dynamically, thereby ensuring compatibility with the underlying spacetime geometry.\n\nSpontaneous local Lorentz violation is accompanied by up to 10 Nambu-Goldstone (NG) modes. 15 With a suitable choice of gauge, these modes can be identified with components of the vierbein normally associated with local Lorentz and diffeomorphism gauge freedoms. The physical role of the NG modes varies, but in general they represent long-range forces that can be problematic for phenomenology. However, in certain models the NG modes can be interpreted as photons, thus offering the intriguing prospect that the existence of light could be a consequence of Lorentz violation instead of local U(1) gauge invariance, with concomitant observable signals. 15,16 A similar interpretation is possible for the graviton. 17 Other potential experimental signals arise from NG modes in the gravity 18 and matter 19 sectors, and from torsion. 20 The spectrum of vacuum excitations typically also includes massive modes that may lead to additional observable effects. 21 \n\n\nThe search for signals\n\nThe SME predicts some unique signals, such as rotational, sidereal, and annual variations. The effects are likely to be heavily suppressed, perhaps as some power of the ratio of an accessible scale to the underlying scale, but they could be detected using sensitive tools such as interferometry. For example, meson interferometry offers the potential to identify flavorand direction-dependent energy shifts of mesons relative to antimesons, 22 while exquisite interferometric sensitivity to polarization-dependent effects of photons is attained using cosmological birefringence. 23 Conceivably, SME effects might even be reflected in existing data, such as those for flavor oscillations of neutrinos. 24 Overall, an impressive range of sensitivities in the matter, gauge, and gravitational sectors of the SME has been achieved. 25 Despite a decade of intense activity, most of the SME coefficient space is still unexplored by experiments, and many basic theoretical issues are unaddressed. The study of relativity violations remains fascinating, with the enticing prospect of identifying a signal from the Planck scale.\n AcknowledgmentsThis work was supported in part by DoE grant DE-FG02-91ER40661 and NASA grant NAG3-2914.\n. V A Kostelecký, S Samuel, Phys. Rev. D. 39683V.A. Kostelecký and S. Samuel, Phys. Rev. D 39, 683 (1989);\n\n. V A Kostelecký, R Potting, Nucl. Phys. B. 359545V.A. Kost- elecký and R. Potting, Nucl. Phys. B 359, 545 (1991).\n\n. V A Kostelecký, R Potting, Phys. Rev. D. 513923V.A. Kostelecký and R. Potting, Phys. Rev. D 51, 3923 (1995).\n\n. D Colladay, V A Kostelecký, Phys. Rev. D. 556760D. Colladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997);\n\n. Phys. Rev. D. 58116002Phys. Rev. D 58, 116002 (1998).\n\n. V A Kostelecký, Phys. Rev. D. 69105009V.A. Kostelecký, Phys. Rev. D 69, 105009 (2004).\n\n. See, I Example, Mocioiu, Phys. Lett. B. 489390See, for example, I. Mocioiu et al., Phys. Lett. B 489, 390 (2000);\n\n. S M Carroll, Phys. Rev. Lett. 87141601S.M. Carroll et al., Phys. Rev. Lett. 87, 141601 (2001).\n\n. V A Kostelecký, Phys. Rev. D. 68123511V.A. Kostelecký et al., Phys. Rev. D 68, 123511 (2003).\n\n. See, G Example, Amelino-Camelia, AIP Conf. Proc. 75830See, for example, G. Amelino-Camelia et al., AIP Conf. Proc. 758, 30 (2005);\n\n. N E Mavromatos, Lect. Notes Phys. 669245N.E. Mavromatos, Lect. Notes Phys. 669, 245 (2005);\n\n. Y Bonder, D Sudarsky, arXiv:0709.0551Y. Bonder and D. Sudarsky, arXiv:0709.0551.\n\n. C D Froggatt, H B Nielsen, hep-ph/0211106C.D. Froggatt and H.B. Nielsen, hep-ph/0211106.\n\n. J D Bjorken, Phys. Rev. D. 6743508J.D. Bjorken, Phys. Rev. D 67, 043508 (2003).\n\n. See, C P Example, Burgess, JHEP. 020343See, for example, C.P. Burgess et al., JHEP 0203, 043 (2002).\n\n. M Berger, V A Kostelecký, Phys. Rev. D. 6591701M. Berger and V.A. Kostelecký, Phys. Rev. D 65, 091701(R) (2002);\n\n. P A Bolokhov, Phys. Rev. D. 7215013P.A. Bolokhov et al., Phys. Rev. D 72, 015013 (2005).\n\n. See, G Example, Dvali, Phys. Rev. D. 7644028See, for example, G. Dvali et al., Phys. Rev. D 76, 044028 (2007);\n\n. D S Gorbunov, S M Sibiryakov, JHEP. 050982D.S. Gorbunov and S.M. Sibiryakov, JHEP 0509, 082 (2005);\n\n. M V Libanov, V A Rubakov, JHEP. 05081M.V. Libanov and V.A. Rubakov, JHEP 0508, 001 (2005);\n\n. N Arkani-Hamed, JHEP. 050729N. Arkani-Hamed et al., JHEP 0507, 029 (2005);\n\n. V A Kostelecký, S Samuel, Phys. Rev. D. 421289V.A. Kostelecký and S. Samuel, Phys. Rev. D 42, 1289 (1990);\n\n. Phys. Rev. Lett. 661811Phys. Rev. Lett. 66, 1811 (1991).\n\n. V A Kostelecký, R Lehnert, Phys. Rev. D. 6365008V.A. Kostelecký and R. Lehnert, Phys. Rev. D 63, 065008 (2001).\n\n. O W Greenberg, Phys. Rev. Lett. 89231602O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002).\n\n. R Bluhm, V A Kostelecký, Phys. Rev. D. 7165008R. Bluhm and V.A. Kostelecký, Phys. Rev. D 71, 065008 (2005).\n\n. B , V A Kostelecký, Phys. Lett. B. 628106B. Altschul and V.A. Kostelecký, Phys. Lett. B 628, 106 (2005).\n\n. V A Kostelecký, R Potting, Gen. Rel. Grav. 371675V.A. Kostelecký and R. Potting, Gen. Rel. Grav. 37, 1675 (2005).\n\n. Q G Bailey, V A Kostelecký, Phys. Rev. D. 7445001Q.G. Bailey and V.A. Kostelecký, Phys. Rev. D 74, 045001 (2006).\n\n. V A Kostelecký, J D Tasson, in preparationV.A. Kostelecký and J.D. Tasson, in preparation.\n\n. V A Kostelecký, N Russell, J D Tasson, arXiv:0712.4393V.A. Kostelecký, N. Russell, and J.D. Tasson, arXiv:0712.4393.\n\n. R Bluhm, arxiv:0712.4119Phys. Rev. D. 401886R. Bluhm et al., arxiv:0712.4119; V.A. Kostelecký and S. Samuel, Phys. Rev. D 40, 1886 (1989);\n\n. Phys. Rev. Lett. 63224Phys. Rev. Lett. 63, 224 (1989).\n\nKTeV), hep-ex/0112046. H Nguyen, KLOEA. Di Domenico et al.these proceedingsH. Nguyen (KTeV), hep-ex/0112046; A. Di Domenico et al. (KLOE), these proceedings;\n\n. B Aubert, BaBararXiv:0711.2713B. Aubert et al. (BaBar), hep-ex/0607103; arXiv:0711.2713;\n\n. D P Stoker, BaBarthese proceedingsD.P. Stoker (BaBar), these proceedings;\n\n. J M Link, Phys. Lett. B. 5567FOCUS)J.M. Link et al. (FOCUS), Phys. Lett. B 556, 7 (2003);\n\n. V A Kostelecký, Phys. Rev. Lett. 801818V.A. Kostelecký, Phys. Rev. Lett. 80, 1818 (1998);\n\n. Phys. Rev. D. 6116002Phys. Rev. D 61, 016002 (2000);\n\n. Phys. Rev. D. 6476001Phys. Rev. D 64, 076001 (2001).\n\n. V A Kostelecký, M Mewes, Phys. Rev. Lett. 87251304V.A. Kostelecký and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001);\n\n. Phys. Rev. D. 6656005Phys. Rev. D 66, 056005 (2002);\n\n. Phys. Rev. Lett. 97140401Phys. Rev. Lett. 97, 140401 (2006);\n\n. Phys. Rev. Lett. 9911601Phys. Rev. Lett. 99, 011601 (2007).\n\n. L B Auerbach, Phys. Rev. D. 7276004L.B. Auerbach et al., Phys. Rev. D 72, 076004 (2005);\n\n. B J Rebel, S F Mufson, these proceedingsB.J. Rebel and S.F. Mufson, these proceedings;\n\n. V A Kostelecký, M Mewes, Phys. Rev. D. 6916005V.A. Kostelecký and M. Mewes, Phys. Rev. D 69, 016005 (2004);\n\n. Phys. Rev. D. 7076002Phys. Rev. DPhys. Rev. D 70, 031902(R) (2004); Phys. Rev. D 70, 076002 (2004);\n\n. T Katori, Phys. Rev. D. 74105009T. Katori et al., Phys. Rev. D 74, 105009 (2006);\n\n. V Barger, Phys. Lett. B. 653267V. Barger et al., Phys. Lett. B 653, 267 (2007);\n\n. K Whisnant, these proceedingsK. Whisnant, these proceedings.\n"},"corpus_id":{"kind":"number","value":17539445,"string":"17,539,445"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0802.0581\",\n \"mag\": 2950720952,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1142/9789812779519_0008\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0802.0581v1.pdf\"\n ],\n \"pdf_sha\": \"06fa6e82519e5ec51bfd513b504137521bcc45c0\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/0802.0581v1.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"PERSPECTIVES ON LORENTZ AND CPT VIOLATION"},"authors":{"kind":"list like","value":["V Alan Kostelecký \nPhysics Department\nIndiana University Bloomington\n47405INU.S.A"],"string":"[\n \"V Alan Kostelecký \\nPhysics Department\\nIndiana University Bloomington\\n47405INU.S.A\"\n]","isConversational":false},"abstract":{"kind":"string","value":""},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Lorentz symmetry is the invariance of physical laws under rotations and boosts. As a global symmetry over Minkowski spacetime, it underlies the theory of Special Relativity and the Standard Model of particle physics, where it is intimately tied to CPT invariance. As a local symmetry of freely falling frames, it is an essential component of General Relativity. Nonetheless, the possibility exists that nature may exhibit tiny violations of Lorentz symmetry. This talk presents some perspectives on the basic motivations and reasoning in this subject.\nSince Lorentz symmetry has been verified in many experiments, as has CPT invariance, it is reasonable to ask why relativity violations are worth considering. A sufficient theoretical motivation is the need for a consistent description of Lorentz and CPT violation to offer guidance for experimental tests. However, a stronger motivation is the prospect that Lorentz and CPT violation can serve as a sensitive potential signal for physics at the Planck scale. In fact, the present interest in the subject was triggered by the realization that natural mechanisms for Lorentz and CPT violation exist in unified theories at the Planck scale. 1 The large range of existing phenomenological and experimental activities stems from the application of effective field theory 2 and the construction of the Standard-Model Extension (SME) 3,4 to catalogue and predict observable effects."},{"title":"Approaches and origins","content":"The study of Lorentz and CPT violation can be approached on three distinct levels. First, at the level of fundamental theory at the unification scale, one can investigate possible mechanisms and determine their features and implications. Second, at the level of theory at accessible scales, one can seek a description of the resulting effects that is quantitative, general, and compatible with the established physics of the Standard Model and General Relativity. Finally, at the level of observation and experiment, one can study and perform both high sensitivity tests and broad searches, preferably ones that are feasible with existing or near-future technology.\nThe first point to establish is whether Lorentz violation can indeed occur in a fundamental theory. Without at least one viable mechanism, the interest of the idea would be much reduced. A plausible origin for Lorentz violation has been identified in string field theory, 1 which has interactions with a generic structure that could in principle trigger spontaneous violation of Lorentz symmetry and generate vacuum expectation values for Lorentz tensor fields. More recently, numerous other mechanisms for Lorentz violation at the fundamental level have been proposed including, for example, ones involving noncommutative field theories, 5 spacetime-varying fields, 6 quantum gravity, 7 random-dynamics models, 8 multiverses, 9 brane-world scenarios, 10 supersymmetry, 11 and massive gravity. 12 Given that mechanisms exist for Lorentz and CPT violation in an underlying theory, it is natural to ask about the consequences for physics at observable scales. In particular, the question arises as to the best method to develop a description of the possible effects."},{"title":"Describing Lorentz violation","content":"Some key criteria offer valuable guidance in the search for a suitable theoretical framework for describing Lorentz violation at attainable scales. The first is coordinate independence. It has been accepted since long before 1905 that the physics of a system should be independent of a change of observer coordinates. This holds whether a coordinate change is implemented via a Lorentz transformation or in any other way.\nThe second is realism. Since 1905, when virtually no fundamental particles were known and quantum physics was at its dawn, thousands of people have invested millions of person-hours and billions of dollars in establishing the Standard Model of particle physics and General Relativity as an accurate description of nature. To be of real interest nowadays, any proposed theoretical framework for Lorentz violation must incorporate this well-established physics.\nThe third is generality. No compelling evidence for Lorentz violation exists at present. Physics is therefore currently in the position of searching for a violation, as opposed to attempting to understand an observed effect.\nIn the searching phase, it is desirable to have the most general possible formulation so that no region is left unexplored. This is in strong contrast to the modeling phase, where considerations such as simplicity are important in attempts to understand a known effect. Armed with these criteria, we can follow the basic reasoning that leads to the application of effective field theory and the construction of the SME."},{"title":"Modified Lorentz transformations","content":"Since the essential content of Special Relativity is the idea that physics is invariant under Lorentz transformations, the most obvious approach to describing relativity violations is to investigate modifications of the Lorentz transformations. In fact, the literature since 1905 abounds with various ad hoc proposals of this type. However, independent of any specific proposal, this approach has some serious disadvantages.\nOne is that a textbook Lorentz transformation acts on the observer and therefore corresponds merely to a change of coordinates, i.e., a change of reference frame. However, according to the above criterion of coordinate independence, a frame change cannot have physical implications by itself.\nThe key feature of Special Relativity is really the requirement that the equations for the system being observed must be covariant under a Lorentz transformation, which intrinsically assumes that Lorentz symmetry is exact. This approach is therefore problematic for investigating violations.\nIt is of course possible to construct special models imposing form covariance of the system under some ad hoc alternative transformation. However, any specific such proposal runs counter to the criterion of generality. Moreover, some kinds of violations are difficult and perhaps even impossible to countenance via this approach. For example, Lorentz violation in nature might well be particle-species dependent, but it is very challenging to formulate a description of this flavor dependence based on modified Lorentz transformations of the observer. The criterion of realism presents a further substantial obstacle, since it is awkward at best to implement such models in the context of the Standard Model and General Relativity."},{"title":"Modified dispersion laws","content":"The above discussion suggests that a general and realistic investigation of Lorentz violation is most naturally performed directly in terms of the properties of a system rather than via modifications of the Lorentz transformations. A simple implementation of this is to study modifications of particle dispersion laws. However, this also suffers from serious drawbacks.\nOne issue involves the criterion of generality. Modifications of dispersion laws can only describe changes in the free propagation of particles and perhaps also partially account for interaction kinematics. However, physics is far more than free propagation, and this approach therefore disregards a large range of interesting Lorentz-violating effects involving interactions.\nThere are also various issues associated with the choice of modifications to the dispersion law. Not all choices are compatible with desirable features such as originating from an action. Also, meaningful physical measurements must necessarily compare two quantities, so some choices may be unphysical. In particular, calculations with a modified dispersion law that yield apparent changes of properties are insufficient by themselves to demonstrate physical Lorentz violation. A simple example of a modified dispersion law with no observable consequences in Minkowski spacetime is 3 p µ p µ = m 2 + a µ p µ , where a µ is a prescribed set of four numbers in a given frame. Direct calculations with this dispersion law appear to give Lorentz-violating properties that depend on the preferred vector a µ , but in fact they are unobservable because a µ can be eliminated via a physically irrelevant redefinition of the energy and momentum. The observability of modifications to a dispersion law can be challenging to demonstrate."},{"title":"Effective field theory and the SME","content":"We see that the desiderata for a satisfactory description of Lorentz violation include a comprehensive treatment of free and interacting effects in all particle species. Remarkably, a model-independent and general approach of this type exists.\nThe key is to take advantage of the idea that Lorentz violation at attainable energies can be described using effective field theory, independent of the underlying mechanism. 2,13 Starting from the Standard Model coupled to General Relativity, we can add to the action all possible scalar terms formed by contracting operators for Lorentz violation with coefficients that control the size of the effects. The operators are naturally ordered according to their mass dimension. The resulting realistic effective field theory is the SME. 3,4 Since CPT violation in realistic field theories comes with Lorentz violation, 14 the SME also incorporates general CPT violation.\nBy virtue of its construction, the SME satisfies the three guiding crite-ria of coordinate invariance, realism, and generality. Moreover, it handles simultaneously all particle species, including both propagation and interaction properties, so its equations of motion contain all action-compatible modifications of realistic dispersion laws. The coordinate invariance implies that physics is unaffected by observer frame changes, including Lorentz and other transformations, while particle transformations can produce observable effects of Lorentz violation. The primary disadvantage of the SME approach is its comparative complexity and the investment required to become proficient with its use. However, this is outweighed by its advantages as a realistic, general, and calculable framework for describing Lorentz violation. Judicious choices establishing relations among the SME coefficients for Lorentz violation yield elegant and simple models that can serve as a theorist's playground, while the general case offers guidance for broad-based experimental searches."},{"title":"Gravity and Lorentz violation","content":"The SME allows for both global 3 and local 4 Lorentz violation, and interesting effects arise from local Lorentz violation in the gravitational context. In general, local Lorentz violation can be understood as arising when a nonzero coefficient t abc... for Lorentz violation exists in local freely falling frames. 4 The cofficient t abc... can be converted to a coefficient t λµν... on the spacetime manifold using the vierbein e a µ . One result is that spontaneous violation of local Lorentz symmetry is always accompanied by spontaneous diffeomorphism violation, and vice versa. 15 A nonzero coefficient t abc... is the vacuum value of a local Lorentz tensor field, and it implies Lorentz violation because it is invariant instead of transforming like a tensor under particle transformations. The vierbein ensures that there is a corresponding spacetime tensor field with vacuum value t λµν... on the spacetime manifold, which in turn implies spontaneous diffeomorphism breaking because it is invariant instead of transforming like a tensor under particle diffeomorphisms.\nA more surprising result is that explicit Lorentz violation is generically incompatible with Riemann geometry. 4 Explicit violation occurs when the SME coefficients are externally prescribed, but the ensuing equations of motion turn out to be inconsistent with the Bianchi identities. This result also holds in Riemann-Cartan spacetime. However, spontaneous violation evades the difficulty because it generates the SME coefficients dynamically, thereby ensuring compatibility with the underlying spacetime geometry.\nSpontaneous local Lorentz violation is accompanied by up to 10 Nambu-Goldstone (NG) modes. 15 With a suitable choice of gauge, these modes can be identified with components of the vierbein normally associated with local Lorentz and diffeomorphism gauge freedoms. The physical role of the NG modes varies, but in general they represent long-range forces that can be problematic for phenomenology. However, in certain models the NG modes can be interpreted as photons, thus offering the intriguing prospect that the existence of light could be a consequence of Lorentz violation instead of local U(1) gauge invariance, with concomitant observable signals. 15,16 A similar interpretation is possible for the graviton. 17 Other potential experimental signals arise from NG modes in the gravity 18 and matter 19 sectors, and from torsion. 20 The spectrum of vacuum excitations typically also includes massive modes that may lead to additional observable effects. 21"},{"title":"The search for signals","content":"The SME predicts some unique signals, such as rotational, sidereal, and annual variations. The effects are likely to be heavily suppressed, perhaps as some power of the ratio of an accessible scale to the underlying scale, but they could be detected using sensitive tools such as interferometry. For example, meson interferometry offers the potential to identify flavorand direction-dependent energy shifts of mesons relative to antimesons, 22 while exquisite interferometric sensitivity to polarization-dependent effects of photons is attained using cosmological birefringence. 23 Conceivably, SME effects might even be reflected in existing data, such as those for flavor oscillations of neutrinos. 24 Overall, an impressive range of sensitivities in the matter, gauge, and gravitational sectors of the SME has been achieved. 25 Despite a decade of intense activity, most of the SME coefficient space is still unexplored by experiments, and many basic theoretical issues are unaddressed. The study of relativity violations remains fascinating, with the enticing prospect of identifying a signal from the Planck scale."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Lorentz symmetry is the invariance of physical laws under rotations and boosts. As a global symmetry over Minkowski spacetime, it underlies the theory of Special Relativity and the Standard Model of particle physics, where it is intimately tied to CPT invariance. As a local symmetry of freely falling frames, it is an essential component of General Relativity. Nonetheless, the possibility exists that nature may exhibit tiny violations of Lorentz symmetry. This talk presents some perspectives on the basic motivations and reasoning in this subject.\\nSince Lorentz symmetry has been verified in many experiments, as has CPT invariance, it is reasonable to ask why relativity violations are worth considering. A sufficient theoretical motivation is the need for a consistent description of Lorentz and CPT violation to offer guidance for experimental tests. However, a stronger motivation is the prospect that Lorentz and CPT violation can serve as a sensitive potential signal for physics at the Planck scale. In fact, the present interest in the subject was triggered by the realization that natural mechanisms for Lorentz and CPT violation exist in unified theories at the Planck scale. 1 The large range of existing phenomenological and experimental activities stems from the application of effective field theory 2 and the construction of the Standard-Model Extension (SME) 3,4 to catalogue and predict observable effects.\"\n },\n {\n \"title\": \"Approaches and origins\",\n \"content\": \"The study of Lorentz and CPT violation can be approached on three distinct levels. First, at the level of fundamental theory at the unification scale, one can investigate possible mechanisms and determine their features and implications. Second, at the level of theory at accessible scales, one can seek a description of the resulting effects that is quantitative, general, and compatible with the established physics of the Standard Model and General Relativity. Finally, at the level of observation and experiment, one can study and perform both high sensitivity tests and broad searches, preferably ones that are feasible with existing or near-future technology.\\nThe first point to establish is whether Lorentz violation can indeed occur in a fundamental theory. Without at least one viable mechanism, the interest of the idea would be much reduced. A plausible origin for Lorentz violation has been identified in string field theory, 1 which has interactions with a generic structure that could in principle trigger spontaneous violation of Lorentz symmetry and generate vacuum expectation values for Lorentz tensor fields. More recently, numerous other mechanisms for Lorentz violation at the fundamental level have been proposed including, for example, ones involving noncommutative field theories, 5 spacetime-varying fields, 6 quantum gravity, 7 random-dynamics models, 8 multiverses, 9 brane-world scenarios, 10 supersymmetry, 11 and massive gravity. 12 Given that mechanisms exist for Lorentz and CPT violation in an underlying theory, it is natural to ask about the consequences for physics at observable scales. In particular, the question arises as to the best method to develop a description of the possible effects.\"\n },\n {\n \"title\": \"Describing Lorentz violation\",\n \"content\": \"Some key criteria offer valuable guidance in the search for a suitable theoretical framework for describing Lorentz violation at attainable scales. The first is coordinate independence. It has been accepted since long before 1905 that the physics of a system should be independent of a change of observer coordinates. This holds whether a coordinate change is implemented via a Lorentz transformation or in any other way.\\nThe second is realism. Since 1905, when virtually no fundamental particles were known and quantum physics was at its dawn, thousands of people have invested millions of person-hours and billions of dollars in establishing the Standard Model of particle physics and General Relativity as an accurate description of nature. To be of real interest nowadays, any proposed theoretical framework for Lorentz violation must incorporate this well-established physics.\\nThe third is generality. No compelling evidence for Lorentz violation exists at present. Physics is therefore currently in the position of searching for a violation, as opposed to attempting to understand an observed effect.\\nIn the searching phase, it is desirable to have the most general possible formulation so that no region is left unexplored. This is in strong contrast to the modeling phase, where considerations such as simplicity are important in attempts to understand a known effect. Armed with these criteria, we can follow the basic reasoning that leads to the application of effective field theory and the construction of the SME.\"\n },\n {\n \"title\": \"Modified Lorentz transformations\",\n \"content\": \"Since the essential content of Special Relativity is the idea that physics is invariant under Lorentz transformations, the most obvious approach to describing relativity violations is to investigate modifications of the Lorentz transformations. In fact, the literature since 1905 abounds with various ad hoc proposals of this type. However, independent of any specific proposal, this approach has some serious disadvantages.\\nOne is that a textbook Lorentz transformation acts on the observer and therefore corresponds merely to a change of coordinates, i.e., a change of reference frame. However, according to the above criterion of coordinate independence, a frame change cannot have physical implications by itself.\\nThe key feature of Special Relativity is really the requirement that the equations for the system being observed must be covariant under a Lorentz transformation, which intrinsically assumes that Lorentz symmetry is exact. This approach is therefore problematic for investigating violations.\\nIt is of course possible to construct special models imposing form covariance of the system under some ad hoc alternative transformation. However, any specific such proposal runs counter to the criterion of generality. Moreover, some kinds of violations are difficult and perhaps even impossible to countenance via this approach. For example, Lorentz violation in nature might well be particle-species dependent, but it is very challenging to formulate a description of this flavor dependence based on modified Lorentz transformations of the observer. The criterion of realism presents a further substantial obstacle, since it is awkward at best to implement such models in the context of the Standard Model and General Relativity.\"\n },\n {\n \"title\": \"Modified dispersion laws\",\n \"content\": \"The above discussion suggests that a general and realistic investigation of Lorentz violation is most naturally performed directly in terms of the properties of a system rather than via modifications of the Lorentz transformations. A simple implementation of this is to study modifications of particle dispersion laws. However, this also suffers from serious drawbacks.\\nOne issue involves the criterion of generality. Modifications of dispersion laws can only describe changes in the free propagation of particles and perhaps also partially account for interaction kinematics. However, physics is far more than free propagation, and this approach therefore disregards a large range of interesting Lorentz-violating effects involving interactions.\\nThere are also various issues associated with the choice of modifications to the dispersion law. Not all choices are compatible with desirable features such as originating from an action. Also, meaningful physical measurements must necessarily compare two quantities, so some choices may be unphysical. In particular, calculations with a modified dispersion law that yield apparent changes of properties are insufficient by themselves to demonstrate physical Lorentz violation. A simple example of a modified dispersion law with no observable consequences in Minkowski spacetime is 3 p µ p µ = m 2 + a µ p µ , where a µ is a prescribed set of four numbers in a given frame. Direct calculations with this dispersion law appear to give Lorentz-violating properties that depend on the preferred vector a µ , but in fact they are unobservable because a µ can be eliminated via a physically irrelevant redefinition of the energy and momentum. The observability of modifications to a dispersion law can be challenging to demonstrate.\"\n },\n {\n \"title\": \"Effective field theory and the SME\",\n \"content\": \"We see that the desiderata for a satisfactory description of Lorentz violation include a comprehensive treatment of free and interacting effects in all particle species. Remarkably, a model-independent and general approach of this type exists.\\nThe key is to take advantage of the idea that Lorentz violation at attainable energies can be described using effective field theory, independent of the underlying mechanism. 2,13 Starting from the Standard Model coupled to General Relativity, we can add to the action all possible scalar terms formed by contracting operators for Lorentz violation with coefficients that control the size of the effects. The operators are naturally ordered according to their mass dimension. The resulting realistic effective field theory is the SME. 3,4 Since CPT violation in realistic field theories comes with Lorentz violation, 14 the SME also incorporates general CPT violation.\\nBy virtue of its construction, the SME satisfies the three guiding crite-ria of coordinate invariance, realism, and generality. Moreover, it handles simultaneously all particle species, including both propagation and interaction properties, so its equations of motion contain all action-compatible modifications of realistic dispersion laws. The coordinate invariance implies that physics is unaffected by observer frame changes, including Lorentz and other transformations, while particle transformations can produce observable effects of Lorentz violation. The primary disadvantage of the SME approach is its comparative complexity and the investment required to become proficient with its use. However, this is outweighed by its advantages as a realistic, general, and calculable framework for describing Lorentz violation. Judicious choices establishing relations among the SME coefficients for Lorentz violation yield elegant and simple models that can serve as a theorist's playground, while the general case offers guidance for broad-based experimental searches.\"\n },\n {\n \"title\": \"Gravity and Lorentz violation\",\n \"content\": \"The SME allows for both global 3 and local 4 Lorentz violation, and interesting effects arise from local Lorentz violation in the gravitational context. In general, local Lorentz violation can be understood as arising when a nonzero coefficient t abc... for Lorentz violation exists in local freely falling frames. 4 The cofficient t abc... can be converted to a coefficient t λµν... on the spacetime manifold using the vierbein e a µ . One result is that spontaneous violation of local Lorentz symmetry is always accompanied by spontaneous diffeomorphism violation, and vice versa. 15 A nonzero coefficient t abc... is the vacuum value of a local Lorentz tensor field, and it implies Lorentz violation because it is invariant instead of transforming like a tensor under particle transformations. The vierbein ensures that there is a corresponding spacetime tensor field with vacuum value t λµν... on the spacetime manifold, which in turn implies spontaneous diffeomorphism breaking because it is invariant instead of transforming like a tensor under particle diffeomorphisms.\\nA more surprising result is that explicit Lorentz violation is generically incompatible with Riemann geometry. 4 Explicit violation occurs when the SME coefficients are externally prescribed, but the ensuing equations of motion turn out to be inconsistent with the Bianchi identities. This result also holds in Riemann-Cartan spacetime. However, spontaneous violation evades the difficulty because it generates the SME coefficients dynamically, thereby ensuring compatibility with the underlying spacetime geometry.\\nSpontaneous local Lorentz violation is accompanied by up to 10 Nambu-Goldstone (NG) modes. 15 With a suitable choice of gauge, these modes can be identified with components of the vierbein normally associated with local Lorentz and diffeomorphism gauge freedoms. The physical role of the NG modes varies, but in general they represent long-range forces that can be problematic for phenomenology. However, in certain models the NG modes can be interpreted as photons, thus offering the intriguing prospect that the existence of light could be a consequence of Lorentz violation instead of local U(1) gauge invariance, with concomitant observable signals. 15,16 A similar interpretation is possible for the graviton. 17 Other potential experimental signals arise from NG modes in the gravity 18 and matter 19 sectors, and from torsion. 20 The spectrum of vacuum excitations typically also includes massive modes that may lead to additional observable effects. 21\"\n },\n {\n \"title\": \"The search for signals\",\n \"content\": \"The SME predicts some unique signals, such as rotational, sidereal, and annual variations. The effects are likely to be heavily suppressed, perhaps as some power of the ratio of an accessible scale to the underlying scale, but they could be detected using sensitive tools such as interferometry. For example, meson interferometry offers the potential to identify flavorand direction-dependent energy shifts of mesons relative to antimesons, 22 while exquisite interferometric sensitivity to polarization-dependent effects of photons is attained using cosmological birefringence. 23 Conceivably, SME effects might even be reflected in existing data, such as those for flavor oscillations of neutrinos. 24 Overall, an impressive range of sensitivities in the matter, gauge, and gravitational sectors of the SME has been achieved. 25 Despite a decade of intense activity, most of the SME coefficient space is still unexplored by experiments, and many basic theoretical issues are unaddressed. The study of relativity violations remains fascinating, with the enticing prospect of identifying a signal from the Planck scale.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". V A Kostelecký, S Samuel, Phys. Rev. D. 39683V.A. Kostelecký and S. Samuel, Phys. Rev. D 39, 683 (1989);","doi":""},{"text":". V A Kostelecký, R Potting, Nucl. Phys. B. 359545V.A. Kost- elecký and R. Potting, Nucl. Phys. B 359, 545 (1991).","doi":""},{"text":". V A Kostelecký, R Potting, Phys. Rev. D. 513923V.A. Kostelecký and R. Potting, Phys. Rev. D 51, 3923 (1995).","doi":""},{"text":". D Colladay, V A Kostelecký, Phys. Rev. D. 556760D. Colladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997);","doi":""},{"text":". Phys. Rev. D. 58116002Phys. Rev. D 58, 116002 (1998).","doi":""},{"text":". V A Kostelecký, Phys. Rev. D. 69105009V.A. Kostelecký, Phys. Rev. D 69, 105009 (2004).","doi":""},{"text":". See, I Example, Mocioiu, Phys. Lett. B. 489390See, for example, I. Mocioiu et al., Phys. Lett. B 489, 390 (2000);","doi":""},{"text":". S M Carroll, Phys. Rev. Lett. 87141601S.M. Carroll et al., Phys. Rev. Lett. 87, 141601 (2001).","doi":""},{"text":". V A Kostelecký, Phys. Rev. D. 68123511V.A. Kostelecký et al., Phys. Rev. D 68, 123511 (2003).","doi":""},{"text":". See, G Example, Amelino-Camelia, AIP Conf. Proc. 75830See, for example, G. Amelino-Camelia et al., AIP Conf. Proc. 758, 30 (2005);","doi":""},{"text":". N E Mavromatos, Lect. Notes Phys. 669245N.E. Mavromatos, Lect. Notes Phys. 669, 245 (2005);","doi":""},{"text":". Y Bonder, D Sudarsky, arXiv:0709.0551Y. Bonder and D. Sudarsky, arXiv:0709.0551.","doi":""},{"text":". C D Froggatt, H B Nielsen, hep-ph/0211106C.D. Froggatt and H.B. Nielsen, hep-ph/0211106.","doi":""},{"text":". J D Bjorken, Phys. Rev. D. 6743508J.D. Bjorken, Phys. Rev. D 67, 043508 (2003).","doi":""},{"text":". See, C P Example, Burgess, JHEP. 020343See, for example, C.P. Burgess et al., JHEP 0203, 043 (2002).","doi":""},{"text":". M Berger, V A Kostelecký, Phys. Rev. D. 6591701M. Berger and V.A. Kostelecký, Phys. Rev. D 65, 091701(R) (2002);","doi":""},{"text":". P A Bolokhov, Phys. Rev. D. 7215013P.A. Bolokhov et al., Phys. Rev. D 72, 015013 (2005).","doi":""},{"text":". See, G Example, Dvali, Phys. Rev. D. 7644028See, for example, G. Dvali et al., Phys. Rev. D 76, 044028 (2007);","doi":""},{"text":". D S Gorbunov, S M Sibiryakov, JHEP. 050982D.S. Gorbunov and S.M. Sibiryakov, JHEP 0509, 082 (2005);","doi":""},{"text":". M V Libanov, V A Rubakov, JHEP. 05081M.V. Libanov and V.A. Rubakov, JHEP 0508, 001 (2005);","doi":""},{"text":". N Arkani-Hamed, JHEP. 050729N. Arkani-Hamed et al., JHEP 0507, 029 (2005);","doi":""},{"text":". V A Kostelecký, S Samuel, Phys. Rev. D. 421289V.A. Kostelecký and S. Samuel, Phys. Rev. D 42, 1289 (1990);","doi":""},{"text":". Phys. Rev. Lett. 661811Phys. Rev. Lett. 66, 1811 (1991).","doi":""},{"text":". V A Kostelecký, R Lehnert, Phys. Rev. D. 6365008V.A. Kostelecký and R. Lehnert, Phys. Rev. D 63, 065008 (2001).","doi":""},{"text":". O W Greenberg, Phys. Rev. Lett. 89231602O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002).","doi":""},{"text":". R Bluhm, V A Kostelecký, Phys. Rev. D. 7165008R. Bluhm and V.A. Kostelecký, Phys. Rev. D 71, 065008 (2005).","doi":""},{"text":". B , V A Kostelecký, Phys. Lett. B. 628106B. Altschul and V.A. Kostelecký, Phys. Lett. B 628, 106 (2005).","doi":""},{"text":". V A Kostelecký, R Potting, Gen. Rel. Grav. 371675V.A. Kostelecký and R. Potting, Gen. Rel. Grav. 37, 1675 (2005).","doi":""},{"text":". Q G Bailey, V A Kostelecký, Phys. Rev. D. 7445001Q.G. Bailey and V.A. Kostelecký, Phys. Rev. D 74, 045001 (2006).","doi":""},{"text":". V A Kostelecký, J D Tasson, in preparationV.A. Kostelecký and J.D. Tasson, in preparation.","doi":""},{"text":". V A Kostelecký, N Russell, J D Tasson, arXiv:0712.4393V.A. Kostelecký, N. Russell, and J.D. Tasson, arXiv:0712.4393.","doi":""},{"text":". R Bluhm, arxiv:0712.4119Phys. Rev. D. 401886R. Bluhm et al., arxiv:0712.4119; V.A. Kostelecký and S. Samuel, Phys. Rev. D 40, 1886 (1989);","doi":""},{"text":". Phys. Rev. Lett. 63224Phys. Rev. Lett. 63, 224 (1989).","doi":""},{"text":"KTeV), hep-ex/0112046. H Nguyen, KLOEA. Di Domenico et al.these proceedingsH. Nguyen (KTeV), hep-ex/0112046; A. Di Domenico et al. (KLOE), these proceedings;","doi":""},{"text":". B Aubert, BaBararXiv:0711.2713B. Aubert et al. (BaBar), hep-ex/0607103; arXiv:0711.2713;","doi":""},{"text":". D P Stoker, BaBarthese proceedingsD.P. Stoker (BaBar), these proceedings;","doi":""},{"text":". J M Link, Phys. Lett. B. 5567FOCUS)J.M. Link et al. (FOCUS), Phys. Lett. B 556, 7 (2003);","doi":""},{"text":". V A Kostelecký, Phys. Rev. Lett. 801818V.A. Kostelecký, Phys. Rev. Lett. 80, 1818 (1998);","doi":""},{"text":". Phys. Rev. D. 6116002Phys. Rev. D 61, 016002 (2000);","doi":""},{"text":". Phys. Rev. D. 6476001Phys. Rev. D 64, 076001 (2001).","doi":""},{"text":". V A Kostelecký, M Mewes, Phys. Rev. Lett. 87251304V.A. Kostelecký and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001);","doi":""},{"text":". Phys. Rev. D. 6656005Phys. Rev. D 66, 056005 (2002);","doi":""},{"text":". Phys. Rev. Lett. 97140401Phys. Rev. Lett. 97, 140401 (2006);","doi":""},{"text":". Phys. Rev. Lett. 9911601Phys. Rev. Lett. 99, 011601 (2007).","doi":""},{"text":". L B Auerbach, Phys. Rev. D. 7276004L.B. Auerbach et al., Phys. Rev. D 72, 076004 (2005);","doi":""},{"text":". B J Rebel, S F Mufson, these proceedingsB.J. Rebel and S.F. Mufson, these proceedings;","doi":""},{"text":". V A Kostelecký, M Mewes, Phys. Rev. D. 6916005V.A. Kostelecký and M. Mewes, Phys. Rev. D 69, 016005 (2004);","doi":""},{"text":". Phys. Rev. D. 7076002Phys. Rev. DPhys. Rev. D 70, 031902(R) (2004); Phys. Rev. D 70, 076002 (2004);","doi":""},{"text":". T Katori, Phys. Rev. D. 74105009T. Katori et al., Phys. Rev. D 74, 105009 (2006);","doi":""},{"text":". V Barger, Phys. Lett. B. 653267V. Barger et al., Phys. Lett. B 653, 267 (2007);","doi":""},{"text":". K Whisnant, these proceedingsK. Whisnant, these proceedings.","doi":""}],"string":"[\n {\n \"text\": \". V A Kostelecký, S Samuel, Phys. Rev. D. 39683V.A. Kostelecký and S. Samuel, Phys. Rev. D 39, 683 (1989);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, R Potting, Nucl. Phys. B. 359545V.A. Kost- elecký and R. Potting, Nucl. Phys. B 359, 545 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, R Potting, Phys. Rev. D. 513923V.A. Kostelecký and R. Potting, Phys. Rev. D 51, 3923 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Colladay, V A Kostelecký, Phys. Rev. D. 556760D. Colladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. D. 58116002Phys. Rev. D 58, 116002 (1998).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, Phys. Rev. D. 69105009V.A. Kostelecký, Phys. Rev. D 69, 105009 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". See, I Example, Mocioiu, Phys. Lett. B. 489390See, for example, I. Mocioiu et al., Phys. Lett. B 489, 390 (2000);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S M Carroll, Phys. Rev. Lett. 87141601S.M. Carroll et al., Phys. Rev. Lett. 87, 141601 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, Phys. Rev. D. 68123511V.A. Kostelecký et al., Phys. Rev. D 68, 123511 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". See, G Example, Amelino-Camelia, AIP Conf. Proc. 75830See, for example, G. Amelino-Camelia et al., AIP Conf. Proc. 758, 30 (2005);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N E Mavromatos, Lect. Notes Phys. 669245N.E. Mavromatos, Lect. Notes Phys. 669, 245 (2005);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Bonder, D Sudarsky, arXiv:0709.0551Y. Bonder and D. Sudarsky, arXiv:0709.0551.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C D Froggatt, H B Nielsen, hep-ph/0211106C.D. Froggatt and H.B. Nielsen, hep-ph/0211106.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Bjorken, Phys. Rev. D. 6743508J.D. Bjorken, Phys. Rev. D 67, 043508 (2003).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". See, C P Example, Burgess, JHEP. 020343See, for example, C.P. Burgess et al., JHEP 0203, 043 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Berger, V A Kostelecký, Phys. Rev. D. 6591701M. Berger and V.A. Kostelecký, Phys. Rev. D 65, 091701(R) (2002);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P A Bolokhov, Phys. Rev. D. 7215013P.A. Bolokhov et al., Phys. Rev. D 72, 015013 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". See, G Example, Dvali, Phys. Rev. D. 7644028See, for example, G. Dvali et al., Phys. Rev. D 76, 044028 (2007);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D S Gorbunov, S M Sibiryakov, JHEP. 050982D.S. Gorbunov and S.M. Sibiryakov, JHEP 0509, 082 (2005);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M V Libanov, V A Rubakov, JHEP. 05081M.V. Libanov and V.A. Rubakov, JHEP 0508, 001 (2005);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N Arkani-Hamed, JHEP. 050729N. Arkani-Hamed et al., JHEP 0507, 029 (2005);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, S Samuel, Phys. Rev. D. 421289V.A. Kostelecký and S. Samuel, Phys. Rev. D 42, 1289 (1990);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. Lett. 661811Phys. Rev. Lett. 66, 1811 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, R Lehnert, Phys. Rev. D. 6365008V.A. Kostelecký and R. Lehnert, Phys. Rev. D 63, 065008 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O W Greenberg, Phys. Rev. Lett. 89231602O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Bluhm, V A Kostelecký, Phys. Rev. D. 7165008R. Bluhm and V.A. Kostelecký, Phys. Rev. D 71, 065008 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B , V A Kostelecký, Phys. Lett. B. 628106B. Altschul and V.A. Kostelecký, Phys. Lett. B 628, 106 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, R Potting, Gen. Rel. Grav. 371675V.A. Kostelecký and R. Potting, Gen. Rel. Grav. 37, 1675 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Q G Bailey, V A Kostelecký, Phys. Rev. D. 7445001Q.G. Bailey and V.A. Kostelecký, Phys. Rev. D 74, 045001 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, J D Tasson, in preparationV.A. Kostelecký and J.D. Tasson, in preparation.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, N Russell, J D Tasson, arXiv:0712.4393V.A. Kostelecký, N. Russell, and J.D. Tasson, arXiv:0712.4393.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Bluhm, arxiv:0712.4119Phys. Rev. D. 401886R. Bluhm et al., arxiv:0712.4119; V.A. Kostelecký and S. Samuel, Phys. Rev. D 40, 1886 (1989);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. Lett. 63224Phys. Rev. Lett. 63, 224 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"KTeV), hep-ex/0112046. H Nguyen, KLOEA. Di Domenico et al.these proceedingsH. Nguyen (KTeV), hep-ex/0112046; A. Di Domenico et al. (KLOE), these proceedings;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Aubert, BaBararXiv:0711.2713B. Aubert et al. (BaBar), hep-ex/0607103; arXiv:0711.2713;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D P Stoker, BaBarthese proceedingsD.P. Stoker (BaBar), these proceedings;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J M Link, Phys. Lett. B. 5567FOCUS)J.M. Link et al. (FOCUS), Phys. Lett. B 556, 7 (2003);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, Phys. Rev. Lett. 801818V.A. Kostelecký, Phys. Rev. Lett. 80, 1818 (1998);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. D. 6116002Phys. Rev. D 61, 016002 (2000);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. D. 6476001Phys. Rev. D 64, 076001 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, M Mewes, Phys. Rev. Lett. 87251304V.A. Kostelecký and M. Mewes, Phys. Rev. Lett. 87, 251304 (2001);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. D. 6656005Phys. Rev. D 66, 056005 (2002);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. Lett. 97140401Phys. Rev. Lett. 97, 140401 (2006);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. Lett. 9911601Phys. Rev. Lett. 99, 011601 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L B Auerbach, Phys. Rev. D. 7276004L.B. Auerbach et al., Phys. Rev. D 72, 076004 (2005);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B J Rebel, S F Mufson, these proceedingsB.J. Rebel and S.F. Mufson, these proceedings;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V A Kostelecký, M Mewes, Phys. Rev. D. 6916005V.A. Kostelecký and M. Mewes, Phys. Rev. D 69, 016005 (2004);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. D. 7076002Phys. Rev. DPhys. Rev. D 70, 031902(R) (2004); Phys. Rev. D 70, 076002 (2004);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Katori, Phys. Rev. D. 74105009T. Katori et al., Phys. Rev. D 74, 105009 (2006);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Barger, Phys. Lett. B. 653267V. Barger et al., Phys. Lett. B 653, 267 (2007);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Whisnant, these proceedingsK. Whisnant, these proceedings.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":64,"cells":{"text":{"kind":"string","value":"\nMechanical First Law of Black Hole Spacetimes with Cosmological Constant and Its Application to Schwarzschild-de Sitter Spacetime First law of BH spacetime with Λ and its application to SdS 2\n25 Mar 2009\n\nMiho Urano 1urano@gravity.phys.nagoya-u.ac.jp \nAkira Tomimatsu \nHiromi Saida saida@daido-it.ac.jp \n\nDepartment of Physics\nGraduate School of Science\nDepartment of Physics\nNagoya University\n464-8602NagoyaJapan\n\n\nDaido Institute of Technology\n457-8530NagoyaJapan ‡\n\nMechanical First Law of Black Hole Spacetimes with Cosmological Constant and Its Application to Schwarzschild-de Sitter Spacetime First law of BH spacetime with Λ and its application to SdS 2\n25 Mar 2009Submitted to: Classical and Quantum Gravity (Accepted for Publication, 23 March 2009) ‡ Name of institution changes to Daido University at April 2009.numbers: 0470-s0470Bw0470Dy\nThe mechanical first law (MFL) of black hole spacetimes is a geometrical relation which relates variations of mass parameter and horizon area. While it is well known that the MFL of asymptotic flat black hole is equivalent to its thermodynamical first law, however we do not know the detail of MFL of black hole spacetimes with cosmological constant which possess black hole and cosmological event horizons. Then this paper aims to formulate an MFL of the two-horizon spacetimes. For this purpose, we try to include the effects of two horizons in the MFL. To do so, we make use of the Iyer-Wald formalism and extend it to regard the mass parameter and the cosmological constant as two independent variables which make it possible to treat the two horizons on the same footing. Our extended Iyer-Wald formalism preserves the existence of conserved Noether current and its associated Noether charge, and gives the abstract form of MFL of black hole spacetimes with cosmological constant. Then, as a representative application of that formalism, we derive the MFL of Schwarzschildde Sitter (SdS) spacetime. Our MFL of SdS spacetime relates the variations of three quantities; the mass parameter, the total area of two horizons and the volume enclosed by two horizons. If our MFL is regarded as a thermodynamical first law of SdS spacetime, it offers a thermodynamically consistent description of SdS black hole evaporation process: The mass decreases while the volume and the entropy increase. In our suggestion, the generalized second law is not needed to ensure the second law of SdS thermodynamics for its evaporation process.\n\nIntroduction\n\nBlack hole thermodynamics has already been established well for spacetimes with a single black hole event horizon [1]. However there is no consistent thermodynamical formulation for black hole spacetimes with cosmological constant Λ which possess a black hole event horizon (BEH) and a cosmological event horizon (CEH). For example, in Schwarzschild-de Sitter (SdS) spacetime, the Hawking temperature of BEH is higher than that of CEH [2]. This temperature difference causes difficulties in formulating SdS thermodynamics, since BEH (CEH) seems in some non-equilibrium state under the influence of Hawking radiation of different temperature by CEH (BEH). In general, the similar difficulty arises also for any two-horizon spacetime with Λ. § However we may be able to search for thermodynamical first law of two-horizon spacetimes with Λ through the mechanical first law (MFL). The MFL is a geometrical relation which relates the variations of mass parameter, horizon area and the other supplemental quantities. For example, the MFL of Schwarzschild black hole is equivalent to its thermodynamical first law, and formulated by using the mass parameter as only one variable which describes the effect of a single horizon on the MFL. This fact of Schwarzschild black hole leads us to expect that the MFL of two-horizon spacetimes with Λ can be a candidate of a thermodynamical first law of the spacetimes, and that the existence of two horizons requires two independent variables in the resultant MFL of the spacetimes to include the effects of two horizons. Then we adopt the following working hypothesis to search for the MFL of two-horizon spacetimes with Λ:\n\nWorking Hypothesis (Two Independent Variables): Generally for spacetimes with cosmological constant which possess BEH and CEH, the mass parameter M and the cosmological constant Λ are regarded as two independent variables in the MFL of the spacetimes.\n\nIndeed it will be shown in Sec.3 that the MFL of SdS spacetime becomes mathematically inconsistent if Λ is not an independent variable. When one considers a non-variable Λ as a physical situation, it is obtained by setting the variation of Λ zero (δΛ = 0) in the MFL after constructing it with regarding Λ as an independent variable. In such case, the variable Λ is interpreted as a \"working variable\" to obtain the MFL of two-horizon spacetimes with Λ. On the other hand for SdS spacetime, the search for an MFL with and without regarding Λ as an independent variable, has already been tried in some existing works [3,4]. Those works make use of some conserved quantities defined by some integrals (e.g. Hamiltonian, action integral and so on). One can obtain some candidates of MFL or thermodynamical first law of SdS spacetime with regarding the integral quantities as the variables in the first law (see [4] for example). However, when one uses § Some proposals for thermodynamics of BEH and CEH are already given for a case with some special matter fields and for an extreme case with magnetic/electric charge [11]. These examples are artificial to vanish the temperature difference of BEH and CEH. But in this paper we consider more general case which is not extremal and does not depend on artificial matter fields. the integral quantities, there arises a problem of the choice of integration constant (e.g. the so-called boundary counter-term or subtraction term to eliminate some divergent term in Hamiltonian or action integral).\n\nThen, in order to derive the MFL of two-horizon spacetimes with Λ, we make use of the Iyer-Wald formalism which is free from the problem of integration constant [5]. We extend the original Iyer-Wald formalism to make it so general to be applicable to any spacetime with Λ under the working hypothesis of two independent variables. The extended Iyer-Wald formalism preserves the existence of conserved Noether current and its associated Noether charge, and enables us to observe how the resultant MFL is related to the Noether charge which is locally constructed from the metric (free from the problem of integration constant).\n\nThe general extension of Iyer-Wald formalism to include the variable Λ gives simply an abstract form of MFL of two-horizon spacetimes with Λ, since the metric is not concretely specified. Then, as a representative application of our extended Iyer-Wald formalism, we derive the MFL of SdS spacetime by that formalism. Moreover, we examine what the MFL of SdS spacetime implies for SdS black hole evaporation if the MFL is regarded as a thermodynamical first law.\n\nHere let us note that Ref. [4] has already treated Λ as an independent variable for SdS spacetime. (In [3], Λ is not an independent variable but a complete constant.) In [4], the variable Λ is concluded from mathematical consistency of the MFL of SdS spacetime, although there remains an uncertainty of integration constant in the resultant MFL given in [4]. However we will show in this paper that the extended Iyer-Wald formalism gives the same MFL with Ref. [4] in a straightforward way without the problem of integration constant. Furthermore, as the advantage of our procedure, the extended Iyer-Wald formalism enables us to introduce naturally a state variable which represents the size of the two-horizon system. Then, consequently, we will show that the new state variable of system size leads natural definitions of effective temperature and pressure of two-horizon system. Those state variables are suitable to describe the SdS black hole evaporation process. Such good state variables are not given in [4] and also the evaporation process is not described in [4]. This paper is organized as follows. Sec.2 is devoted to the extension of Iyer-Wald formalism to include the variable Λ and obtain the abstract form of MFL of any twohorizon spacetimes with Λ. Sec.3 is for the application of extended Iyer-Wald formalism to SdS spacetime and obtain the MFL of SdS spacetime. In that section, the state variables of system size, effective temperature and pressure are given, and then the mathematical consistency of the MFL with variable Λ is also examined. In Sec.4, we discuss what is implied about SdS black hole evaporation if our MFL of SdS spacetime is regarded as a thermodynamical first law, and give a comment about the non-equilibrium nature due to the difference of Hawking temperatures of BEH and CEH.\n\nThroughout this paper, we use the Planck units, c = = G = k B = 1.\n\n\nIyer-Wald formalism with variable Λ\n\nFor simplicity we consider the empty spacetime in n dimensions without matter fields and start with the diffeomorphism invariant Lagrangian n-form L(g, Λ) = L(g, Λ) ε, where g denotes the metric g µν , ε is the volume n-form and L(g, Λ) is the Lagrangian scalar density. The inclusion of matter fields is very straightforward in following discussions. In the next section, L is specified to be the ordinary Einstein-Hilbert form in four dimensions,\nL = 1 16π (R − 2Λ) ε ,(2.1)\nwhere R is the Ricci scalar and Λ is the variable cosmological constant. But in this section L(g, Λ) is an arbitrary Lagrangian of the metric and the variable Λ. The first variation of this Lagrangian is expressed as\nδL = E g δg + dΘ(g, δg) + E Λ δΛ ,(2.2)\nwhere δg is the abbreviation of metric variation δg µν , Θ is the (n − 1)-form called the symplectic potential which corresponds to the boundary term in the variational principle of action integral, E g = E g ε gives the Einstein equation of the metric by E g = 0 and E Λ = E Λ ε is the variation of L with respect to the working variable Λ,\nE Λ = ∂L ∂Λ ε . (2.3)\nFor L in Eq.(2.1), we get E Λ = −(1/8π) ε which will be used in the next section.\n\nHere we have to emphasize the following two remarks: The first one is on the socalled on-shell condition, which restricts the metric (and the matter field if it exists) to be the solution of equation of motion. When the on-shell condition in the presence of metric variation is going to be required later in this section, the variation δg µν is to be understood as a solution of the \"extended\" linearized Einstein equation due to the variable Λ,\nδG µν + Λ δg µν + g µν δΛ = 0 ,(2.4)\nwhere G µν is the Einstein tensor and g µν is the \"unperturbed\" metric given by the Einstein equation E g = 0 . (The right-hand side would not be zero but be the variation δT µν of stress-energy tensor if matter fields exit.) The third term g µν δΛ does not appear for the ordinary linearized equation with completely constant Λ, but now it appears due to the variable Λ. Under the working hypothesis of two independent variables, we can regard Λ not as the kin of universal constants like Newton's constant but as the kin of constants of motion like mass parameter M which is a variable in asymptotic flat black hole thermodynamics. The second remark we should emphasize here is on the variational principle. Exactly speaking, the variational principle gives the equations of motion of dynamical variables via the vanishing variation of action integral I with respect to the dynamical variable This g µν can also be regarded as a \"background\" metric of the perturbation δg µν . φ(x) which depends generally on the spacetime points, δI/δφ(x) = 0, where x represents the spacetime dependence. However, the \"working\" variable Λ is not regarded as any dynamical variable and has no spacetime dependence. The Λ is simply a working variable to ensure the mathematical consistency of the resultant MFL of two-horizon spacetimes. Hence, even if E Λ is set zero, it can never be interpreted as any equation of motion of dynamical variable. This means that the on-shell condition requires only the Einstein equation E g = 0 (and the extended linearized equation (2.4) under the presence of variation δg µν ), and E Λ is non-vanishing (E Λ ≡ 0). Now it is recognized that the physical principle we rely on is the \"extended\" variational principle in which the variations are taken with respect to not only the dynamical variable g µν (and matter field if it exists) but also the working variable Λ, while the equations of motion are given by the variation of Lagrangian with respect to dynamical variables.\n\nAs a by-product of the above two remarks, it will be shown below that, in our extended Iyer-Wald formalism, the conserved Noether current and its associated Noether charge are defined in the same way as in the original Iyer-Wald formalism. Our \"extension\" of Iyer-Wald formalism has three meanings; (1) the extended Iyer-Wald formalism includes the variable Λ, (2) the Λ is not a dynamical variable but simply the working variable which means E Λ does not give any equation of motion, and (3) the conserved Noether current is obtained with the same definition as in the original Iyer-Wald formalism.\n\nThen let us proceed to the extension of Iyer-Wald formalism. Eq.(2.2) is the starting point. The extended Iyer-Wald formalism with variable Λ differs from the original formalism on the following two points: One of them is a manifest point expressed by the third term in the right-hand side of Eq.(2.2). That term does not arise in the original Iyer-Wald formalism, but arises in our extended formalism by the variation δΛ in, for example, the second term in the right-hand side of Eq.(2.1). Another point is a subtle point included in the metric variation δg µν . In our extended formalism, the variation δg µν also gives rise to the variation δΛ if the concrete form of the metric depends on Λ as for SdS, de Sitter and Anti-de Sitter spacetimes.\n\nTo formulate the abstract form of MFL with the variable Λ, let us follow the same procedure of the original Iyer-Wald formalism [5]. If we introduce an arbitrary vector field ξ µ , which is not a dynamical variable in L, and consider the variation given by the Lie derivative δ = L ξ along ξ µ , then we get from Eq.(2.2), 5) where ξ · L := ξ µ L µν 1 ···ν n−1 and a relation L ξ A = ξ · dA + d(ξ · A) of the Lie and exterior derivatives of a form A is used. The (n − 1)-from in the left-hand side, Θ(g, L ξ g) − ξ · L =: J ξ , is called the Noether current. Here note that, since Λ has no spacetime dependence, L ξ Λ ≡ 0 holds. Therefore, when the on-shell condition E g = 0 is required, the Noether current is closed dJ ξ = 0 which guarantees the local existence of the (n − 2)-form Q ξ called the Noether charge,\nd [Θ(g, L ξ g) − ξ · L] = −E g L ξ g − E Λ L ξ Λ ,(2.dQ ξ := J ξ = Θ(g, L ξ g) − ξ · L . (2.6)\nThis Q ξ is locally constructed from the on-shell metric which satisfies the Einstein equation. Note that the conservation of the Noether current dJ ξ = 0 under the onshell condition corresponds to the Noether's theorem, and the Noether charge Q ξ is the conserved charge of J ξ associated with the symmetry generator ξ. The definition of Q ξ in Eq.(2.6) is the same with that in the original Iyer-Wald formalism. Here it should also be emphasized that the existence condition of Q ξ is the on-shell condition E g = 0 for the metric g µν appearing in Eq. (2.6). This condition is the same with that in the original Iyer-Wald formalism. Hence we find that, in our extended Iyer-Wald formalism, the conserved Noether current and its associated Noether charge are defined in the same way as in the original Iyer-Wald formalism. Next, since the vector ξ µ (not the 1-form ξ µ ) is not a dynamical variable, its variation does not exist (δξ µ ≡ 0) under the variation of dynamical and working variables. Then, from the variation of J ξ , the following relation is obtained:\nδJ ξ = δΘ(g, L ξ g) − ξ · δL = ω(g, δg, L ξ g) + d [ξ · Θ(g, δg)] − ξ · E Λ δΛ , (2.7)\nwhere ω is defined as ω(g, δg, L ξ g) := δΘ(g, L ξ g) − L ξ Θ(g, L ξ g) and the on-shell condition E g = 0 is required. Here we must note that, as explained in the second paragraph of this section, the on-shell condition under the presence of metric variation δg µν denotes that the unperturbed matric g µν and the variation δg µν appearing in Eq.(2.7) satisfy, respectively, the Einstein equation E g = 0 and the extended linearized Einstein equation (2.4). Furthermore, as noted in [5], the (n − 1)-form ω vanishes (ω = 0), when ξ is the generator of a symmetry of all dynamical variables in L, i.e. L ξ g µν = 0. For stationary spacetimes, the timelike Killing vector can be regarded as the symmetry generator ξ. Hence, at least for the stationary case, we get\nξ · E Λ δΛ + δJ ξ − d [ξ · Θ(g, δg)] = 0 . (2.8)\nHere the Noether charge defined in Eq.(2.6) gives δJ ξ = d(δQ ξ ). Therefore, by integrating Eq.(2.8) on a hypersurface Σ and applying the Stokes' theorem, we obtain\nΣ ξ · E Λ δΛ + ∂Σ [ δQ ξ − ξ · Θ(g, δg) ] = 0 . (2.9)\nThe first term is due to the variable Λ. There are three conditions to hold Eq.(2.9);\n\n(1) the Einstein equation E g = 0 for the unperturbed metric g µν , (2) the extended linearized Einstein equation (2.4) of the metric variation δg µν , and (3) the existence of the symmetry generator ξ to give L ξ g = 0.\n\nIn the original Iyer-Wald formalism, the first term in Eq.(2.9) disappears. Ref. [5] shows that, when the original formalism is applied to asymptotic flat black holes, Eq.(2.9) without the first term reduces to the MFL of those black holes. Therefore we can regard Eq.(2.9) as the primitive MFL of any spacetime with Λ in any dimensions. ¶ ¶ The primitive MFL (2.9) seems applicable to spacetimes without CEH like asymptotic Anti-de Sitter black holes. When Eq.(2.9) is applied to such black holes, the two degrees of freedom expressed by M and Λ may describe two independent effects in the MFL; one of them is due to the BEH, and another is due to the \"wall\" given by the infinitely large gravitational potential barrier of Anti-de Sitter metric.\n\n\nMechanical first law of SdS spacetime\n\n\nPreparations\n\nBefore deriving the MFL of SdS spacetime from Eq.(2.9), we summarize the SdS metric and the MFL of asymptotic flat black holes derived from the original Iyer-Wald formalism. For the first, the line element of SdS spacetime is\nds 2 = −f (r) dt 2 + dr 2 f (r) + r 2 dθ 2 + sin 2 θdϕ 2 , (3.1) where f (r) := 1 − 2M r − Λ 3 r 2 .(3.\n2)\n\nThe equation f (r) = 0 has two positive roots and one negative root for the parameter range,\n0 < 9 M 2 Λ < 1 . (3.3)\nThroughout this paper, our discussion is restricted in this range. This parameter range guarantees the existence of BEH and CEH. The radius of BEH r b and that of CEH r c are given by, respectively, the smaller positive root and the larger positive root of f (r) = 0. Then the parameter range (3.3) gives the relations of these radii,\n2M < r b < 3M < 1 √ Λ < r c < 3 √ Λ . (3.4)\nThe equations f (r b ) = 0 and f (r c ) = 0 are rearranged to\nM = r b r c (r b + r c ) 2 (r 2 b + r b r c + r 2 c ) , Λ = 3 r 2 b + r b r c + r 2 c . (3.5)\nSince SdS spacetime is static, each event horizon possesses the bifurcation sphere, on which the timelike Killing vector ξ := ∂ t vanishes. The surface gravity of BEH κ b and that of CEH κ c are defined by the following equations evaluated at the horizons,\nξ ν ∇ ν ξ µ BEH = κ b ξ µ BEH , ξ ν ∇ ν ξ µ CEH = κ c ξ µ CEH .(3.6)\nIn general, the numerical value of surface gravity depends on the normalization of ξ.\n\nWith the normalization ξ := ∂ t , we get\nκ b = 1 2 df (r) dr r=r b = 1 − Λ r 2 b 2 r b , κ c = 1 2 df (r) dr r=rc = 1 − Λ r 2 c 2 r c . (3.7)\nThen, from Eq.(3.4), the following relations hold,\nκ b > 0 , κ c < 0 , κ b > |κ c | .(3.8)\nNext, we summarize the MFL of asymptotic flat black holes derived from the original Iyer-Wald formalism [5]: The first term in Eq.(2.9) does not exist in the original formalism. When the original formalism is applied to asymptotic flat black holes and Σ connects the bifurcation sphere B of the horizon and the spatial infinity ∞, then Eq.(2.9) without the first term is rearranged to\n∞ [ δQ ξ − ξ · Θ(g, δg) ] − B δQ ξ = 0 ,(3.9)\nwhere ξ := ∂ t with the usual static time t and ξ = 0 on B is used. As shown in [5], this reduces to the MFL of asymptotic flat black holes,\nδM = κ BH 8π δA BH + work terms ,(3.10)\nwhere M is the ADM mass of black hole, A BH is the area of B, κ BH is the surface gravity of the horizon with an appropriate normalization of Killing vector, the \"work terms\" are given by electromagnetic fields and angular momentum for Reissner-Nordström and Kerr black holes [6]. (This MFL in Eq.(3.10) is a differential form of the so-called Smarr's formula [7].) It should be emphasized that the first term (integration on ∞) in the lefthand side of Eq. (3.9) gives the variation of the ADM mass δM and the \"work terms\" in Eq. (3.10), and the second term (integration on B) in Eq.(3.9) gives the \"heat term\" (κ BH /8π) δA BH . In asymptotic flat black hole thermodynamics, the MFL in Eq.(3.10) is equivalent to the thermodynamical first law of asymptotic flat black holes in which M is regarded as the internal energy, and A BH /4 is regarded as the entropy (the so-called entropy-area law ) due to the Hawking temperature κ BH /2π [1].\n\n\nApplication of extended Iyer-Wald formalism to SdS spacetime\n\nNow we apply Eq.(2.9) to SdS spacetime to obtain the appropriately expressed MFL of SdS spacetime. Here the meaning of \"appropriate expression\" is that the mass parameter M of SdS spacetime is expressed as a function of two independent quantities which seem correspond to the entropy and the state variable of system size when the MFL is regarded as a thermodynamical first law. The quantity corresponding to entropy gives the \"heat term\" in MFL, and the quantity corresponding to system size gives the \"work term\" in MFL. In thermodynamical first law of ordinary laboratory systems, the heat term is the product of temperature and variation of entropy, and the work term is the product of pressure and variation of system size. Concerning the quantity S which seems correspond to the entropy of SdS spacetime, we refer simply to the entropy-area law of asymptotic flat black holes and define S by the total spatial area of BEH and CEH:\n\nS := π r 2 c + π r 2 b . (3.11) Here note that the horizon radii r b and r c are regarded as two independent variables due to the working hypothesis of two independent variables, and that the total horizon area includes manifestly the effects of BEH and CEH through these two variables r b and r c . When this definition (3.11) is adopted, the variation δS gives the \"heat term\" in the appropriate MFL of SdS spacetime. Next, we have to define the system size V whose variation δV composes the \"work term\" in the appropriate MFL of SdS spacetime. Here, for the system size of SdS spacetime, it seems reasonable to consider the three dimensional volume of a spacelike hypersurface connecting BEH and CEH as the object of thermodynamical interest, since the Hawking radiations of two horizons coexist there. As such hypersurface, let us consider Σ bif which connects the bifurcation spheres of BEH and CEH at t = constant. Furthermore let us refer to the first term in Eq. (2.9). In that term, the integral Σ bif ξ · E Λ appears as a natural quantity of three dimensional volume. Then concerning the quantity V which seems correspond to the system size of SdS spacetime, we adopt the following definition:\nV := Σ bif ξ · E Λ = 4 π 3 r 3 c − r 3 b ,(3.12)\nwhere ξ := ∂ t . The definitions (3.11) and (3.12) are equivalent to search for the MFL of SdS spacetime which is expressed as δM = T eff δS − p eff δV , (3.13) where the concrete forms of coefficients T eff and p eff will be obtained below. These coefficients T eff and p eff are simply the partial derivatives of M. If the MFL in Eq.(3.13) is regarded as a thermodynamical first law of SdS spacetime, then T eff and p eff are interpreted as, respectively, effective temperature and pressure. Therefore, because the temperature and pressure are positive definite for ordinary laboratory systems, it is natural to require that the appropriately expressed MFL should satisfy the following requirement:\n\nT eff > 0 , p eff > 0 . (3.14) If the MFL in Eq.(3.13) is regarded as a thermodynamical first law of SdS spacetime, then the positivity of T eff and p eff makes it plausible to call T eff and p eff , respectively, the effective temperature and pressure. A comment on T eff in relation to non-equilibrium nature of SdS spacetime will be given in Sec.4. The appropriately expressed MFL of SdS spacetime should be obtained by using the definitions (3.11) and (3.12) under the requirement (3.14) with regarding M and Λ as two independent variables. To obtain the appropriate MFL, we must be careful to carry out the integrations in Eq. (2.9). According to asymptotic flat case summarized in the previous subsection, we expect that, when the second term (surface term) in Eq.(2.9) is evaluated on the bifurcation sphere in SdS spacetime, it should give the \"heat term\" T eff δS in the MFL of SdS spacetime (3.13). But, when the second term in Eq.(2.9) is evaluated on some sphere which is not the bifurcation sphere, it should contribute to the mass variation δM in Eq. (3.13), as for the ADM mass in asymptotic flat case (3.10). And the first term (3-volume term) in Eq. (2.9) should contribute to the \"work term\" −p eff δV in Eq. (3.13). Then, since there seems no natural sphere except for the bifurcation spheres of horizons, one problem arises; how to prepare the sphere to get the mass variation in the MFL of SdS spacetime. Hence, in order to extract the mass variation from Eq.(2.9), we adopt the strategy composed of three steps as follows:\n\nStep 1: Consider two hypersurfaces, Σ b and Σ c , at t = constant. Here Σ b covers the region r b < r < r cut and Σ c covers r cut < r < r c , where r cut is a working variable which is introduced to extract the mass variation δM and must not appear in the resultant MFL of SdS spacetime.\n\nStep 2: Carry out the integrations in Eq.(2.9) for the hypersurface Σ b . Do the same for Σ c .\n\nStep 3: Combine those integrated equations to introduce the variations of M, S and V into the appropriately expressed MFL of SdS spacetime (3.13).\n\nNote that the working variable r cut is introduced simply to carry out the integrations in Eq.(2.9) as a \"piecewise\" integration. If the resultant MFL obtained in the step 3 depends on r cut , this strategy is not suitable for the construction of the appropriate MFL of SdS spacetime. However in fact, as will be shown below, the result does not depend on r cut and we will succeed to obtain the appropriately expressed MFL of SdS spacetime. Let us carry out the step 2. We calculate Eq.(2.9) with the hypersurfaces Σ b and Σ c which are introduced in the step 1. In the following calculations, B b and B c denote, respectively, the bifurcation spheres of BEH and CEH, and B cut is the sphere of radius r cut which is the outer (inner) boundary of Σ b (Σ c ). As mentioned in Sec.2, we use the Lagrangian L(g, Λ) ε in Eq.(2.1) with the variable Λ, which gives E Λ = −(1/8π) ε. The metric is given in Eq. (3.1), and the timelike Killing vector ∂ t is regarded as the symmetry generator ξ := ∂ t .\n\nEq.(2.9) with Σ b in the SdS spacetime reads\n− 1 8π Σ b ξ · ε δΛ − B b δQ ξ + Bcut [ δQ ξ − ξ · Θ (g, δg) ] = 0 ,(3.15)\nwhere ξ = 0 at B b is used in the second term, and the sign of the second and third terms are determined by the normal directions to Σ b at B b and B cut which are, respectively, in-ward and out-ward pointing along the \"r-axis\". The first term in the left-hand side of Eq.(3.15) is a simple three dimensional volume integral in Σ b ,\n− 1 8π Σ b ξ · ε δΛ = − 1 6 r 3 cut δΛ + 1 6 r 3 b δΛ. (3.16)\nFor the second term in the left-hand side of Eq.(3.15), note that the symplectic potential Θ includes the (n − 1)-volume form as its factor by definition (2.2), and consequently the Noether charge Q ξ includes the (n − 2)-volume form as its factor by definition (2.6). (n = 4 for SdS spacetime.) This implies that the integral and the variation δ of Noether charge is commutative, B b δQ ξ = δ B b Q ξ . On the other hand, we can find for the Lagrangian in Eq.(2.1) with the variable Λ that the Noether charge Q ξ takes the same form with the ordinary general relativity without Λ [5],\n(Q ξ ) µν = − 1 16π ε µναβ ∇ α ξ β .(3.17)\nThen, referring to the proof of the theorem 6.1 in the original Iyer-Wald formalism [5] which holds even with the variable Λ, we get\nδ B b (Q ξ ) ab = − κ b 16π δ B bǭ αβ ε αβ ab ,(3.18)\nwhere the indices a and b are the abstract indices [8],ǭ αβ is the bi-normal 2-form to B b which satisfies ∇ µ ξ ν = κ bǭµν on B b , and the vanishing variation of surface gravity at\nB b (δκ b | B b = 0) is used + .\nHence we obtain for the second term in Eq. (3.15),\nB b δQ ξ = κ b 16π δ 2 B b d 2 Ω b = κ b 2π δ πr 2 b ,(3.19)\nwhere d 2 Ω b := r 2 b sin θ dθ dϕ is the area element on B b . Let us proceed to the calculation of the third term in the left-hand side of Eq. (3.15). For the first we calculate the integral of Noether charge Br Q ξ , where B r is a two-sphere of an arbitrary radius r. Note that the θ-ϕ component of Q ξ in Eq. (3.17) contributes to the integral,\n(Q ξ ) θϕ = − 1 16π ε θϕ αβ ∇ α ξ β = − 1 16π ∂g tt ∂r ξ t − det(g µν ) dθ ∧ dϕ . (3.20)\nThis gives\nBr Q ξ = − 1 16π Br d 2 Ω r ∂g tt ∂r ξ t = 1 16π df (r) dr 4πr 2 ,(3.21)\nwhere d 2 Ω r := r 2 sin θ dθ dϕ is the area element on B r and the definitions ξ := ∂ t and g tt = −f (r) in Eq.(3.2) are used. Then, using the relation Br δQ ξ = δ Br Q ξ , the first integral in the third term in Eq.(3.15) becomes\nBcut δQ ξ = δ Bcut Q ξ = δ r 2 cut f ′ (r cut ) 4 = r 2 cut 4 δf ′ (r cut ) ,(3.22)\nwhere f ′ := df /dr and Eq.(3.21) is used in the second equality. Next, the explicit form of the symplectic potential Θ for the Lagrangian (2.1) with the variable Λ is\nΘ λµν = 1 16π ε λµνα g αβ g στ [ ∇ τ (δg βσ ) − ∇ β (δg στ ) ] . (3.23)\nThis is the same form with the ordinary general relativity without Λ [5]. Here, under the variations of δM and δΛ, the metric variation δg µν is given from Eq.(3.1),\nδ(ds 2 ) = δg µν dx µ dx ν = −δf (r) dt 2 + −δf (r) f 2 (r) dr 2 ,(3.24)\nwhere the spherical part vanishes since that part does not depend on M and Λ, and\nδf (r) = − 2 r δM − r 2 3 δΛ .\n(3.25) + By definition of κ b in Eq. (3.6) and δξ µ = 0, we get ξ ν (δΓ µ να ) ξ α = (δκ b ) ξ µ at the BEH, where Γ µ να is the Christoffel symbol. When this is evaluated by, for example, the Kruskal-Szekeres type coordinate covering the inside and outside of BEH, we can find the limiting behavior δκ b → 0 as approaches the bifurcation sphere\nB b in BEH, i.e. δκ b B b = 0.\nIt should be emphasized here that the metric variation (3.24) is a static solution of the extended linearized Einstein equation (2.4) and satisfies the condition to ensure Eq. (2.9). Furthermore note that, for the integrand ξ α Θ αµν in the integral Bcut ξ · Θ, the indices µ and ν denotes the tangential components to B cut (θ-ϕ component), ξ α Θ α θϕ . Then we obtain\nBcut ξ · Θ = − 1 16π Bcut d 2 Ω cut g rβ g στ [ ∇ τ (δg βσ ) − ∇ β (δg στ ) ] = − 1 16π Bcut d 2 Ω cut −δf ′ (r) − 2 r δf (r) = r 2 cut 4 δf ′ (r cut ) + 2 r cut δf (r cut ) ,(3.26)\nwhere d 2 Ω cut := r 2 cut sin θ dθ dϕ is the area element on B cut , and ξ := ∂ t and ε tµνr = −ε trµν are used in the first equality. Hence the third term in Eq. (3.15) becomes\nBcut [ δQ ξ − ξ · Θ (g, δg) ] = δM + 1 6 r 3 cut δΛ . (3.27)\nThen, collecting Eqs. (3.16), (3.19) and (3.27), we obtain from Eq.(3.15),\nδM = κ b 2π δ πr 2 b − 1 6 r 3 b δΛ . (3.28)\nIt should be noted that the working variable r cut does not appear in this equation. Next turn our calculation to Eq.(2.9) with the hypersurface Σ c , which reads\n− 1 8π Σc ξ · ε δΛ + Bc δQ ξ − Bcut [ δQ ξ − ξ · Θ (g, δg) ] = 0 ,(3.29)\nwhere ξ = 0 at B c is used in the second term, and the sign of the second and third terms are determined by the normal directions to Σ c at B c and B cut which are, respectively, out-ward and in-ward pointing along the \"r-axis\". Then, following the same calculations to obtain Eq.(3.28), we obtain from Eq. (3.29),\nδM = κ c 2π δ πr 2 c − 1 6 r 3 c δΛ . (3.30)\nIt should be noted that the working variable r cut does not appear in this equation. So far we have carried out the step 2 to obtain Eqs. (3.28) and (3.30). Before proceeding to the step 3 of our strategy, let us comment on Ref. [4]: As mentioned in Sec. 1,Ref.[4] has already discussed the MFL of SdS spacetime with variable Λ using some integral quantities as state variables, and obtained two MFLs separately for BEH and CEH. Those MFLs in [4] are the same with our Eqs. (3.28) and (3.30). Although the derivation of MFLs in [4] includes the problem of the choice of integration constant, but our derivation based on the extended Iyer-Wald formalism is free from such problem. Furthermore, the integral quantity used in [4] requires to place a boundary (corresponding to B cut in our step 2) at CEH in deriving Eq. (3.28) and at BEH in deriving Eq. (3.30). If we interpret the boundary as the position of an observer, it seems not to be physically acceptable to place the observer at the event horizon. However our Indeed, Ref. [4] has already obtained Eqs. (3.28) and (3.30) (whose combination gives Eq.(3.34)) by using some integral quantities as state variables. The advantages of using the extended Iyer-Wald formalism are the following three points: (1) it is free from the problem of integration constant as mentioned in Sec.1, (2) it offers naturally the definition of V in Eq. (3.12), and (3) it gives the positive definite effective temperature T eff and pressure p eff which, together with V , are suitable to describe the SdS black hole evaporation process as discussed in the next section.\n\nFinally in this section, for the completeness of our discussion, let us consider the case that Λ is not regarded as an independent variable. In this case, one obtains a relation, δS = 2π (1/κ b + 1/κ c ) δM, from the definition (3.11). By rearranging this relation appropriately and introducing V defined in Eq.(3.12), one can formally obtain the MFL in Eq.(3.34) with the same coefficients in Eqs. (3.35) and (3.36). However in this case, we find relations; ∂M(S, V )/∂S = (dS/dM) −1 ≡ T eff and −∂M/∂V = −(dV /dM) −1 ≡ −p eff . These contradict the definition of coefficients T eff and p eff in Eq. (3.34). By the reductive absurdity, this fact indicates the necessity of \"variable Λ\" for a mathematically consistent MFL of SdS spacetime of the form in Eq. (3.34). Here note that the same claim, the necessity of variable Λ, is already suggested in [4], although the MFL in [4] is given in a different expression from Eq. (3.34) and includes the uncertainty of integration constant of conserved quantities as mentioned in Sec.1. The necessity of variable Λ seems a universal fact for mathematically consistent MFL of SdS spacetime.\n\n\nDiscussions\n\nLet us discuss what is implied by our appropriate MFL in Eq. (3.34) if it is regarded as a thermodynamical first law of SdS spacetime. As an interesting process, we treat the SdS black hole evaporation process at constant Λ.\n\nBefore considering SdS black hole evaporation, let us recall the so-called generalized second law in the evaporation process of Schwarzschild black hole. In Schwarzschild thermodynamics, the thermodynamical first law is given by Eq.(3.10) with setting the \"work terms\" zero, δM = T BH δS BH , where T BH := κ BH /2π = 1/8πM is the Hawking temperature and S BH := A BH /4 = π(2M) 2 is the black hole entropy. When Schwarzschild black hole evaporates, the mass energy M decreases and consequently S BH decreases. Here note that, the evaporation of isolated black hole is an irreversible adiabatic process. Hence, if the total entropy of the whole system is given by S BH , the decrease of S BH contradicts the second law of thermodynamics which requires the increase of total entropy for irreversible adiabatic processes. Then the generalized second law claims that the total entropy of black hole and matter fields of Hawking radiation increases for the evaporation process of isolated black holes [9,10]. Therefore, at least for the evaporation of asymptotic flat black holes, the generalized second law is necessary to hold the validity of thermodynamical formulation of black holes.\n\nThen, proceed to the discussion of SdS black hole evaporation at constant Λ. When SdS black hole evaporates, it seems reasonable to consider the mass parameter decreases, δM < 0. Here, for any process at constant Λ, our MFL in Eq. (3.34) and δV in Eq. (3.33) are rearranged to δS = 2π (1/κ b + 1/κ c ) δM and δV = 4π (r c /κ c − r b /κ b ) δM. Then, we get δS > 0 and δV > 0 due to δM < 0 and Eq. (3.8). The expansion of volume δV > 0 is a reasonable result if p eff is interpreted as a pressure. The increase of entropy δS > 0 denotes that the generalized second law is not needed to ensure the second law of SdS thermodynamics for its evaporation process at constant Λ. Hence our appropriate MFL in Eq.(3.34) suggests a thermodynamically consistent description of SdS black hole evaporation at constant Λ. Finally let us comment on the coefficient T eff in Eq. (3.35). This T eff is regarded as a temperature if our MFL in Eq.(3.34) is regarded as a thermodynamical first law. Then one may wonder about what the physical meaning of the temperature is, since the SdS spacetime is essentially a non-equilibrium system due to the difference of Hawking temperatures of BEH and CEH [2]. Here let us point out that there is a long history of research on two-temperature non-equilibrium systems, and there are many proposals on the definition of non-equilibrium temperature of non-equilibrium systems (see, for example, references of works in [10]). No commonly accepted definition of nonequilibrium temperature exists as present. Therefore, from the point of view of nonequilibrium thermodynamics, the suggestion of defining a non-equilibrium temperature is meaningful at present. If the SdS thermodynamics is formulated with regarding our MFL in Eq.(3.34) as a thermodynamical first law, then our T eff in Eq. (3.35), which can be expressed as a linear combination of the surface gravities of BEH and CEH, may be understood as one suggestion of an effective temperature of a non-equilibrium system. At present, the physical meaning of T eff is an open issue which requires further researches on non-equilibrium nature of SdS spacetime. The research on SdS thermodynamics provides the stage for the intersection of gravitational physics and non-equilibrium physics.\n AcknowledgmentsOne of authors H.S. is supported by the Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Culture, Sports, Science and Technology, Japan (Young Scientists (B) 19740149).surface B cut may be appropriate as a candidate of the position of observer who measures thermodynamical quantities of two-horizon system. Although Eqs.(3.28)and(3.30)are regarded as MFLs of SdS spacetime in Ref.[4], our aim is to propose a single formula(3.13)as the appropriately expressed MFL of SdS spacetime. The procedure to obtain the appropriate MFL is the step 3 of our strategy. Then, let us carry out the step 3: Combining Eqs.(3.28)and(3.30)together with the definition (3.11), we getOn the other hand, we get from Eq.(3.5),(3.32)These variations of horizon radii together with the definition (3.12) give the volume variation,Then, substituting this δV into Eq.(3.31), we obtain the appropriate MFL of SdS spacetime(3.13),where the coefficients arewhere x := r b /r c and 0 < x < 1. We find that T eff and p eff are positive definite, T eff > 0 and p eff > 0, and the requirement (3.14) is satisfied. Hence the MFL in Eq.(3.34) with the coefficients given above is the appropriately expressed MFL of SdS spacetime.Here note that, although we refer to the entropy-area law of asymptotic flat black hole thermodynamics as a motivation to adopt the definition (3.11), it does not mean to assume S in Eq.(3.11)to be the physical entropy of SdS spacetime. It is a future task to resolve the issue whether this S is really a physical entropy in SdS thermodynamics. Even if S is not a physical entropy, our MFL in Eq.(3.34)suggests the mass formula which relates mass parameter M and total horizon area S. (The mass formula of asymptotic flat black holes is the Smarr's formula[7].)Our derivation of the appropriately expressed MFL (3.34) is based on the extended Iyer-Wald formalism. However, since the relation among M, S and V is definitely determined by the metric (3.1), the relation (3.34) must be obtained from S and V defined in Eqs.(3.11)and (3.12) without using the extended Iyer-Wald formalism.\nBlack Holes and Entropy. J D Bekenstein, Phys. Rev. 72333J.D. Bekenstein, Black Holes and Entropy, Phys. Rev. D7 (1973) 2333.\n\nGeneralized second law of thermodynamics in black-hole physics. J D Bekenstein, Phys. Rev. 93292J.D. Bekenstein, Generalized second law of thermodynamics in black-hole physics, Phys. Rev. D9 (1974) 3292.\n\nGravitational Radiation from Colliding Black Holes. S W Hawking, Phys. Rev. Lett. 261344S.W. Hawking, Gravitational Radiation from Colliding Black Holes, Phys. Rev. Lett. 26 (1971) 1344.\n\nParticle Creation by Black Holes. S W Hawking, Commun. Math. Phys. 43199S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199.\n\nS W Hawking, D N Page, Thermodynamics of Black Holes in Anti-de Sitter Space. 87577S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in Anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577.\n\nThird Law of Black-Hole Dynamics: A Formulation and Proof. W Israel, Phys. Rev. Lett. 57397W. Israel, Third Law of Black-Hole Dynamics: A Formulation and Proof, Phys. Rev. Lett. 57 (1986) 397.\n\nBlack-hole thermodynamics and the Euclidean Einstein action. J W York, Jr , Phys. Rev. 332092J.W. York,Jr., Black-hole thermodynamics and the Euclidean Einstein action Phys. Rev. D33 (1986) 2092.\n\nCharged black hole in a grand canonical ensemble. H W Braden, J D Brown, B F Whiting, J W York, Jr , Phys. Rev. 423376H.W. Braden, J.D. Brown, B.F. Whiting and J.W. York,Jr., Charged black hole in a grand canonical ensemble, Phys. Rev. D42 (1990) 3376.\n\nComplex Kerr-Newman Geometry and Black-Hole Thermodynamics. J David, E A Martinez, J W York, Jr , Phys. Rev. Lett. 662281J. David, E.A. Martinez and J.W. York,Jr., Complex Kerr-Newman Geometry and Black-Hole Thermodynamics, Phys. Rev. Lett.66 (1991) 2281.\n\nCosmological event horizons, thermodynamics, and particle creation. G W Gibbons, S W Hawking, Phys. Rev. 152738G.W. Gibbons and S.W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D15 (1977) 2738.\n\nStability of gravity with a cosmological constant. L F Abbot, S Deser, Nucl. Phys. 19576L.F. Abbot and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B195 (1982) 76.\n\nEntropy and Holography in Asymptotically de Sitter Spaces. V Balasubramanian, J Deboer, D Minic, Mass, Phys. Rev. 65123508V. Balasubramanian, J. deBoer and D. Minic, Mass, Entropy and Holography in Asymptotically de Sitter Spaces, Phys. Rev. D65 (2002) 123508.\n\nMass and Entropy of Schwarzschild-de Sitter black holes and the de Sitter/CFT Correspondence. A M Ghezelbash, R B Mann, J. High Energy Phys. 02015ActionA.M. Ghezelbash and R.B. Mann, Action, Mass and Entropy of Schwarzschild-de Sitter black holes and the de Sitter/CFT Correspondence, J. High Energy Phys. 0201 (2002) 005.\n\nde Sitter black holes with either of the two horizons as a boundary. A Gomberoff, C Teitelboim, Phys. Rev. 67104024A.Gomberoff and C.Teitelboim, de Sitter black holes with either of the two horizons as a boundary, Phys. Rev. D67 (2003) 104024.\n\nBlack Holes in de Sitter Space: Masses, Energies and Entropy Bounds. A Corichi, A Gomberoff, Phys. Rev. 6964016A. Corichi and A. Gomberoff, Black Holes in de Sitter Space: Masses, Energies and Entropy Bounds, Phys. Rev. D69 (2004) 064016.\n\nThermodynamics of de Sitter black holes: Thermal cosmological constant. Y Sekiwa, Phys. Rev. 7384009Y.Sekiwa, Thermodynamics of de Sitter black holes: Thermal cosmological constant, Phys. Rev. D73 (2006) 084009.\n\nBlack hole entropy is the Noether charge. R M Wald, Phys. Rev. 483427R.M.Wald, Black hole entropy is the Noether charge, Phys. Rev. D48 (1993) R3427.\n\nSome properties of the Noether charge and a proposal for dynamical black hole entropy. V Iyer, R M Wald, Phys. Rev. 50846V.Iyer and R.M.Wald, Some properties of the Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D50 (1994) 846.\n\nThe Four Laws of Black Hole Mechanics. J M Bardeen, B Carter, S W Hawking, Commun. Math. Phys. 31161J.M. Bardeen, B. Carter and S.W. Hawking, The Four Laws of Black Hole Mechanics, Commun. Math. Phys. 31 (1973) 161.\n\nMass Formula for Kerr Black Holes. L Smarr, Phys. Rev. Lett. 3071L. Smarr, Mass Formula for Kerr Black Holes, Phys. Rev. Lett. 30 (1973) 71.\n\nR M Wald, General Relativity. The Univ. of Chicago PressR.M.Wald, General Relativity, The Univ. of Chicago Press, 1984.\n\nAcceleration radiation and the generalized second law of thermodynamics. W G Unruh, R M Wald, Phys. Rev. 25942W.G. Unruh and R.M. Wald, Acceleration radiation and the generalized second law of thermodynamics, Phys. Rev. D25 (1982) 942.\n\nEntropy bounds, acceleration radiation, and the generalized second law. W G Unruh, R M Wald, Phys. Rev. 272271W.G. Unruh and R.M. Wald, Entropy bounds, acceleration radiation, and the generalized second law, Phys. Rev. D27 (1983) 2271.\n\nProof of Generalized Second Law for Quasistationary Semiclassical Black Holes. V P Frolov, D N Page, Phys. Rev. Lett. 713902V.P. Frolov and D.N. Page, Proof of Generalized Second Law for Quasistationary Semiclassical Black Holes, Phys. Rev. Lett. 71 (1993) 3902.\n\nProof of Classical Version of the Bousso Entropy Bound and of the Generalized Second Law. E E Flanagan, D Marolf, R M Wald, Phys. Rev. 6284035E.E. Flanagan, D. Marolf and R.M. Wald, Proof of Classical Version of the Bousso Entropy Bound and of the Generalized Second Law, Phys. Rev. D62 (2000) 084035.\n\nTwo-temperature Steady State Thermodynamics for a Radiation Field. H Saida, Physica. 356481H. Saida, Two-temperature Steady State Thermodynamics for a Radiation Field, Physica A356 (2005) 481.\n\nThe generalized second law and the black hole evaporation in an empty space as a nonequilibrium process. H Saida, Class. Quant. Grav. 236227H. Saida, The generalized second law and the black hole evaporation in an empty space as a nonequilibrium process, Class. Quant. Grav. 23 (2006) 6227.\n\nBlack hole evaporation as a nonequilibrium process. H Saida, arXiv:0811.1622Classical and Quantum Gravity Research. M.N. Christiansen and T.K. Rasmussen8Nova Science Publgr-qcH. Saida, Black hole evaporation as a nonequilibrium process, Chapter 8 (arXiv:0811.1622 [gr-qc]) in the edited book by M.N. Christiansen and T.K. Rasmussen, Classical and Quantum Gravity Research, Nova Science Publ., 2008.\n\nEuclidean action and the thermodynamics of manifolds without boundary. G Hayward, Phys. Rev. 413248G. Hayward, Euclidean action and the thermodynamics of manifolds without boundary, Phys. Rev. D41 (1990) 3248.\n\nDuality between Electric and Magnetic Black Holes. S W Hawking, S F Ross, Phys. Rev. 525865S.W. Hawking and S.F. Ross, Duality between Electric and Magnetic Black Holes, Phys. Rev. D52 (1995) 5865.\n"},"corpus_id":{"kind":"number","value":918655,"string":"918,655"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0903.4230\",\n \"mag\": 2012067565,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/0264-9381/26/10/105010\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0903.4230v1.pdf\"\n ],\n \"pdf_sha\": \"4ce514b0fb1e8d1442b2d0778e5ac3d9b148491f\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/0903.4230\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Mechanical First Law of Black Hole Spacetimes with Cosmological Constant and Its Application to Schwarzschild-de Sitter Spacetime First law of BH spacetime with Λ and its application to SdS 2"},"authors":{"kind":"list like","value":["Miho Urano 1urano@gravity.phys.nagoya-u.ac.jp","Akira Tomimatsu","Hiromi Saida saida@daido-it.ac.jp","Department of Physics\nGraduate School of Science\nDepartment of Physics\nNagoya University\n464-8602NagoyaJapan","Daido Institute of Technology\n457-8530NagoyaJapan ‡"],"string":"[\n \"Miho Urano 1urano@gravity.phys.nagoya-u.ac.jp\",\n \"Akira Tomimatsu\",\n \"Hiromi Saida saida@daido-it.ac.jp\",\n \"Department of Physics\\nGraduate School of Science\\nDepartment of Physics\\nNagoya University\\n464-8602NagoyaJapan\",\n \"Daido Institute of Technology\\n457-8530NagoyaJapan ‡\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The mechanical first law (MFL) of black hole spacetimes is a geometrical relation which relates variations of mass parameter and horizon area. While it is well known that the MFL of asymptotic flat black hole is equivalent to its thermodynamical first law, however we do not know the detail of MFL of black hole spacetimes with cosmological constant which possess black hole and cosmological event horizons. Then this paper aims to formulate an MFL of the two-horizon spacetimes. For this purpose, we try to include the effects of two horizons in the MFL. To do so, we make use of the Iyer-Wald formalism and extend it to regard the mass parameter and the cosmological constant as two independent variables which make it possible to treat the two horizons on the same footing. Our extended Iyer-Wald formalism preserves the existence of conserved Noether current and its associated Noether charge, and gives the abstract form of MFL of black hole spacetimes with cosmological constant. Then, as a representative application of that formalism, we derive the MFL of Schwarzschildde Sitter (SdS) spacetime. Our MFL of SdS spacetime relates the variations of three quantities; the mass parameter, the total area of two horizons and the volume enclosed by two horizons. If our MFL is regarded as a thermodynamical first law of SdS spacetime, it offers a thermodynamically consistent description of SdS black hole evaporation process: The mass decreases while the volume and the entropy increase. In our suggestion, the generalized second law is not needed to ensure the second law of SdS thermodynamics for its evaporation process."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Black hole thermodynamics has already been established well for spacetimes with a single black hole event horizon [1]. However there is no consistent thermodynamical formulation for black hole spacetimes with cosmological constant Λ which possess a black hole event horizon (BEH) and a cosmological event horizon (CEH). For example, in Schwarzschild-de Sitter (SdS) spacetime, the Hawking temperature of BEH is higher than that of CEH [2]. This temperature difference causes difficulties in formulating SdS thermodynamics, since BEH (CEH) seems in some non-equilibrium state under the influence of Hawking radiation of different temperature by CEH (BEH). In general, the similar difficulty arises also for any two-horizon spacetime with Λ. § However we may be able to search for thermodynamical first law of two-horizon spacetimes with Λ through the mechanical first law (MFL). The MFL is a geometrical relation which relates the variations of mass parameter, horizon area and the other supplemental quantities. For example, the MFL of Schwarzschild black hole is equivalent to its thermodynamical first law, and formulated by using the mass parameter as only one variable which describes the effect of a single horizon on the MFL. This fact of Schwarzschild black hole leads us to expect that the MFL of two-horizon spacetimes with Λ can be a candidate of a thermodynamical first law of the spacetimes, and that the existence of two horizons requires two independent variables in the resultant MFL of the spacetimes to include the effects of two horizons. Then we adopt the following working hypothesis to search for the MFL of two-horizon spacetimes with Λ:\nWorking Hypothesis (Two Independent Variables): Generally for spacetimes with cosmological constant which possess BEH and CEH, the mass parameter M and the cosmological constant Λ are regarded as two independent variables in the MFL of the spacetimes.\nIndeed it will be shown in Sec.3 that the MFL of SdS spacetime becomes mathematically inconsistent if Λ is not an independent variable. When one considers a non-variable Λ as a physical situation, it is obtained by setting the variation of Λ zero (δΛ = 0) in the MFL after constructing it with regarding Λ as an independent variable. In such case, the variable Λ is interpreted as a \"working variable\" to obtain the MFL of two-horizon spacetimes with Λ. On the other hand for SdS spacetime, the search for an MFL with and without regarding Λ as an independent variable, has already been tried in some existing works [3,4]. Those works make use of some conserved quantities defined by some integrals (e.g. Hamiltonian, action integral and so on). One can obtain some candidates of MFL or thermodynamical first law of SdS spacetime with regarding the integral quantities as the variables in the first law (see [4] for example). However, when one uses § Some proposals for thermodynamics of BEH and CEH are already given for a case with some special matter fields and for an extreme case with magnetic/electric charge [11]. These examples are artificial to vanish the temperature difference of BEH and CEH. But in this paper we consider more general case which is not extremal and does not depend on artificial matter fields. the integral quantities, there arises a problem of the choice of integration constant (e.g. the so-called boundary counter-term or subtraction term to eliminate some divergent term in Hamiltonian or action integral).\nThen, in order to derive the MFL of two-horizon spacetimes with Λ, we make use of the Iyer-Wald formalism which is free from the problem of integration constant [5]. We extend the original Iyer-Wald formalism to make it so general to be applicable to any spacetime with Λ under the working hypothesis of two independent variables. The extended Iyer-Wald formalism preserves the existence of conserved Noether current and its associated Noether charge, and enables us to observe how the resultant MFL is related to the Noether charge which is locally constructed from the metric (free from the problem of integration constant).\nThe general extension of Iyer-Wald formalism to include the variable Λ gives simply an abstract form of MFL of two-horizon spacetimes with Λ, since the metric is not concretely specified. Then, as a representative application of our extended Iyer-Wald formalism, we derive the MFL of SdS spacetime by that formalism. Moreover, we examine what the MFL of SdS spacetime implies for SdS black hole evaporation if the MFL is regarded as a thermodynamical first law.\nHere let us note that Ref. [4] has already treated Λ as an independent variable for SdS spacetime. (In [3], Λ is not an independent variable but a complete constant.) In [4], the variable Λ is concluded from mathematical consistency of the MFL of SdS spacetime, although there remains an uncertainty of integration constant in the resultant MFL given in [4]. However we will show in this paper that the extended Iyer-Wald formalism gives the same MFL with Ref. [4] in a straightforward way without the problem of integration constant. Furthermore, as the advantage of our procedure, the extended Iyer-Wald formalism enables us to introduce naturally a state variable which represents the size of the two-horizon system. Then, consequently, we will show that the new state variable of system size leads natural definitions of effective temperature and pressure of two-horizon system. Those state variables are suitable to describe the SdS black hole evaporation process. Such good state variables are not given in [4] and also the evaporation process is not described in [4]. This paper is organized as follows. Sec.2 is devoted to the extension of Iyer-Wald formalism to include the variable Λ and obtain the abstract form of MFL of any twohorizon spacetimes with Λ. Sec.3 is for the application of extended Iyer-Wald formalism to SdS spacetime and obtain the MFL of SdS spacetime. In that section, the state variables of system size, effective temperature and pressure are given, and then the mathematical consistency of the MFL with variable Λ is also examined. In Sec.4, we discuss what is implied about SdS black hole evaporation if our MFL of SdS spacetime is regarded as a thermodynamical first law, and give a comment about the non-equilibrium nature due to the difference of Hawking temperatures of BEH and CEH.\nThroughout this paper, we use the Planck units, c = = G = k B = 1."},{"title":"Iyer-Wald formalism with variable Λ","content":"For simplicity we consider the empty spacetime in n dimensions without matter fields and start with the diffeomorphism invariant Lagrangian n-form L(g, Λ) = L(g, Λ) ε, where g denotes the metric g µν , ε is the volume n-form and L(g, Λ) is the Lagrangian scalar density. The inclusion of matter fields is very straightforward in following discussions. In the next section, L is specified to be the ordinary Einstein-Hilbert form in four dimensions,\nwhere R is the Ricci scalar and Λ is the variable cosmological constant. But in this section L(g, Λ) is an arbitrary Lagrangian of the metric and the variable Λ. The first variation of this Lagrangian is expressed as\nwhere δg is the abbreviation of metric variation δg µν , Θ is the (n − 1)-form called the symplectic potential which corresponds to the boundary term in the variational principle of action integral, E g = E g ε gives the Einstein equation of the metric by E g = 0 and E Λ = E Λ ε is the variation of L with respect to the working variable Λ,\nFor L in Eq.(2.1), we get E Λ = −(1/8π) ε which will be used in the next section.\nHere we have to emphasize the following two remarks: The first one is on the socalled on-shell condition, which restricts the metric (and the matter field if it exists) to be the solution of equation of motion. When the on-shell condition in the presence of metric variation is going to be required later in this section, the variation δg µν is to be understood as a solution of the \"extended\" linearized Einstein equation due to the variable Λ,\nwhere G µν is the Einstein tensor and g µν is the \"unperturbed\" metric given by the Einstein equation E g = 0 . (The right-hand side would not be zero but be the variation δT µν of stress-energy tensor if matter fields exit.) The third term g µν δΛ does not appear for the ordinary linearized equation with completely constant Λ, but now it appears due to the variable Λ. Under the working hypothesis of two independent variables, we can regard Λ not as the kin of universal constants like Newton's constant but as the kin of constants of motion like mass parameter M which is a variable in asymptotic flat black hole thermodynamics. The second remark we should emphasize here is on the variational principle. Exactly speaking, the variational principle gives the equations of motion of dynamical variables via the vanishing variation of action integral I with respect to the dynamical variable This g µν can also be regarded as a \"background\" metric of the perturbation δg µν . φ(x) which depends generally on the spacetime points, δI/δφ(x) = 0, where x represents the spacetime dependence. However, the \"working\" variable Λ is not regarded as any dynamical variable and has no spacetime dependence. The Λ is simply a working variable to ensure the mathematical consistency of the resultant MFL of two-horizon spacetimes. Hence, even if E Λ is set zero, it can never be interpreted as any equation of motion of dynamical variable. This means that the on-shell condition requires only the Einstein equation E g = 0 (and the extended linearized equation (2.4) under the presence of variation δg µν ), and E Λ is non-vanishing (E Λ ≡ 0). Now it is recognized that the physical principle we rely on is the \"extended\" variational principle in which the variations are taken with respect to not only the dynamical variable g µν (and matter field if it exists) but also the working variable Λ, while the equations of motion are given by the variation of Lagrangian with respect to dynamical variables.\nAs a by-product of the above two remarks, it will be shown below that, in our extended Iyer-Wald formalism, the conserved Noether current and its associated Noether charge are defined in the same way as in the original Iyer-Wald formalism. Our \"extension\" of Iyer-Wald formalism has three meanings; (1) the extended Iyer-Wald formalism includes the variable Λ, (2) the Λ is not a dynamical variable but simply the working variable which means E Λ does not give any equation of motion, and (3) the conserved Noether current is obtained with the same definition as in the original Iyer-Wald formalism.\nThen let us proceed to the extension of Iyer-Wald formalism. Eq.(2.2) is the starting point. The extended Iyer-Wald formalism with variable Λ differs from the original formalism on the following two points: One of them is a manifest point expressed by the third term in the right-hand side of Eq.(2.2). That term does not arise in the original Iyer-Wald formalism, but arises in our extended formalism by the variation δΛ in, for example, the second term in the right-hand side of Eq.(2.1). Another point is a subtle point included in the metric variation δg µν . In our extended formalism, the variation δg µν also gives rise to the variation δΛ if the concrete form of the metric depends on Λ as for SdS, de Sitter and Anti-de Sitter spacetimes.\nTo formulate the abstract form of MFL with the variable Λ, let us follow the same procedure of the original Iyer-Wald formalism [5]. If we introduce an arbitrary vector field ξ µ , which is not a dynamical variable in L, and consider the variation given by the Lie derivative δ = L ξ along ξ µ , then we get from Eq.(2.2), 5) where ξ · L := ξ µ L µν 1 ···ν n−1 and a relation L ξ A = ξ · dA + d(ξ · A) of the Lie and exterior derivatives of a form A is used. The (n − 1)-from in the left-hand side, Θ(g, L ξ g) − ξ · L =: J ξ , is called the Noether current. Here note that, since Λ has no spacetime dependence, L ξ Λ ≡ 0 holds. Therefore, when the on-shell condition E g = 0 is required, the Noether current is closed dJ ξ = 0 which guarantees the local existence of the (n − 2)-form Q ξ called the Noether charge,\nThis Q ξ is locally constructed from the on-shell metric which satisfies the Einstein equation. Note that the conservation of the Noether current dJ ξ = 0 under the onshell condition corresponds to the Noether's theorem, and the Noether charge Q ξ is the conserved charge of J ξ associated with the symmetry generator ξ. The definition of Q ξ in Eq.(2.6) is the same with that in the original Iyer-Wald formalism. Here it should also be emphasized that the existence condition of Q ξ is the on-shell condition E g = 0 for the metric g µν appearing in Eq. (2.6). This condition is the same with that in the original Iyer-Wald formalism. Hence we find that, in our extended Iyer-Wald formalism, the conserved Noether current and its associated Noether charge are defined in the same way as in the original Iyer-Wald formalism. Next, since the vector ξ µ (not the 1-form ξ µ ) is not a dynamical variable, its variation does not exist (δξ µ ≡ 0) under the variation of dynamical and working variables. Then, from the variation of J ξ , the following relation is obtained:\nwhere ω is defined as ω(g, δg, L ξ g) := δΘ(g, L ξ g) − L ξ Θ(g, L ξ g) and the on-shell condition E g = 0 is required. Here we must note that, as explained in the second paragraph of this section, the on-shell condition under the presence of metric variation δg µν denotes that the unperturbed matric g µν and the variation δg µν appearing in Eq.(2.7) satisfy, respectively, the Einstein equation E g = 0 and the extended linearized Einstein equation (2.4). Furthermore, as noted in [5], the (n − 1)-form ω vanishes (ω = 0), when ξ is the generator of a symmetry of all dynamical variables in L, i.e. L ξ g µν = 0. For stationary spacetimes, the timelike Killing vector can be regarded as the symmetry generator ξ. Hence, at least for the stationary case, we get\nHere the Noether charge defined in Eq.(2.6) gives δJ ξ = d(δQ ξ ). Therefore, by integrating Eq.(2.8) on a hypersurface Σ and applying the Stokes' theorem, we obtain\nThe first term is due to the variable Λ. There are three conditions to hold Eq.(2.9);\n(1) the Einstein equation E g = 0 for the unperturbed metric g µν , (2) the extended linearized Einstein equation (2.4) of the metric variation δg µν , and (3) the existence of the symmetry generator ξ to give L ξ g = 0.\nIn the original Iyer-Wald formalism, the first term in Eq.(2.9) disappears. Ref. [5] shows that, when the original formalism is applied to asymptotic flat black holes, Eq.(2.9) without the first term reduces to the MFL of those black holes. Therefore we can regard Eq.(2.9) as the primitive MFL of any spacetime with Λ in any dimensions. ¶ ¶ The primitive MFL (2.9) seems applicable to spacetimes without CEH like asymptotic Anti-de Sitter black holes. When Eq.(2.9) is applied to such black holes, the two degrees of freedom expressed by M and Λ may describe two independent effects in the MFL; one of them is due to the BEH, and another is due to the \"wall\" given by the infinitely large gravitational potential barrier of Anti-de Sitter metric."},{"title":"Mechanical first law of SdS spacetime","content":""},{"title":"Preparations","content":"Before deriving the MFL of SdS spacetime from Eq.(2.9), we summarize the SdS metric and the MFL of asymptotic flat black holes derived from the original Iyer-Wald formalism. For the first, the line element of SdS spacetime is\n2)\nThe equation f (r) = 0 has two positive roots and one negative root for the parameter range,\nThroughout this paper, our discussion is restricted in this range. This parameter range guarantees the existence of BEH and CEH. The radius of BEH r b and that of CEH r c are given by, respectively, the smaller positive root and the larger positive root of f (r) = 0. Then the parameter range (3.3) gives the relations of these radii,\nThe equations f (r b ) = 0 and f (r c ) = 0 are rearranged to\nSince SdS spacetime is static, each event horizon possesses the bifurcation sphere, on which the timelike Killing vector ξ := ∂ t vanishes. The surface gravity of BEH κ b and that of CEH κ c are defined by the following equations evaluated at the horizons,\nIn general, the numerical value of surface gravity depends on the normalization of ξ.\nWith the normalization ξ := ∂ t , we get\nThen, from Eq.(3.4), the following relations hold,\nNext, we summarize the MFL of asymptotic flat black holes derived from the original Iyer-Wald formalism [5]: The first term in Eq.(2.9) does not exist in the original formalism. When the original formalism is applied to asymptotic flat black holes and Σ connects the bifurcation sphere B of the horizon and the spatial infinity ∞, then Eq.(2.9) without the first term is rearranged to\nwhere ξ := ∂ t with the usual static time t and ξ = 0 on B is used. As shown in [5], this reduces to the MFL of asymptotic flat black holes,\nwhere M is the ADM mass of black hole, A BH is the area of B, κ BH is the surface gravity of the horizon with an appropriate normalization of Killing vector, the \"work terms\" are given by electromagnetic fields and angular momentum for Reissner-Nordström and Kerr black holes [6]. (This MFL in Eq.(3.10) is a differential form of the so-called Smarr's formula [7].) It should be emphasized that the first term (integration on ∞) in the lefthand side of Eq. (3.9) gives the variation of the ADM mass δM and the \"work terms\" in Eq. (3.10), and the second term (integration on B) in Eq.(3.9) gives the \"heat term\" (κ BH /8π) δA BH . In asymptotic flat black hole thermodynamics, the MFL in Eq.(3.10) is equivalent to the thermodynamical first law of asymptotic flat black holes in which M is regarded as the internal energy, and A BH /4 is regarded as the entropy (the so-called entropy-area law ) due to the Hawking temperature κ BH /2π [1]."},{"title":"Application of extended Iyer-Wald formalism to SdS spacetime","content":"Now we apply Eq.(2.9) to SdS spacetime to obtain the appropriately expressed MFL of SdS spacetime. Here the meaning of \"appropriate expression\" is that the mass parameter M of SdS spacetime is expressed as a function of two independent quantities which seem correspond to the entropy and the state variable of system size when the MFL is regarded as a thermodynamical first law. The quantity corresponding to entropy gives the \"heat term\" in MFL, and the quantity corresponding to system size gives the \"work term\" in MFL. In thermodynamical first law of ordinary laboratory systems, the heat term is the product of temperature and variation of entropy, and the work term is the product of pressure and variation of system size. Concerning the quantity S which seems correspond to the entropy of SdS spacetime, we refer simply to the entropy-area law of asymptotic flat black holes and define S by the total spatial area of BEH and CEH:\nS := π r 2 c + π r 2 b . (3.11) Here note that the horizon radii r b and r c are regarded as two independent variables due to the working hypothesis of two independent variables, and that the total horizon area includes manifestly the effects of BEH and CEH through these two variables r b and r c . When this definition (3.11) is adopted, the variation δS gives the \"heat term\" in the appropriate MFL of SdS spacetime. Next, we have to define the system size V whose variation δV composes the \"work term\" in the appropriate MFL of SdS spacetime. Here, for the system size of SdS spacetime, it seems reasonable to consider the three dimensional volume of a spacelike hypersurface connecting BEH and CEH as the object of thermodynamical interest, since the Hawking radiations of two horizons coexist there. As such hypersurface, let us consider Σ bif which connects the bifurcation spheres of BEH and CEH at t = constant. Furthermore let us refer to the first term in Eq. (2.9). In that term, the integral Σ bif ξ · E Λ appears as a natural quantity of three dimensional volume. Then concerning the quantity V which seems correspond to the system size of SdS spacetime, we adopt the following definition:\nwhere ξ := ∂ t . The definitions (3.11) and (3.12) are equivalent to search for the MFL of SdS spacetime which is expressed as δM = T eff δS − p eff δV , (3.13) where the concrete forms of coefficients T eff and p eff will be obtained below. These coefficients T eff and p eff are simply the partial derivatives of M. If the MFL in Eq.(3.13) is regarded as a thermodynamical first law of SdS spacetime, then T eff and p eff are interpreted as, respectively, effective temperature and pressure. Therefore, because the temperature and pressure are positive definite for ordinary laboratory systems, it is natural to require that the appropriately expressed MFL should satisfy the following requirement:\nT eff > 0 , p eff > 0 . (3.14) If the MFL in Eq.(3.13) is regarded as a thermodynamical first law of SdS spacetime, then the positivity of T eff and p eff makes it plausible to call T eff and p eff , respectively, the effective temperature and pressure. A comment on T eff in relation to non-equilibrium nature of SdS spacetime will be given in Sec.4. The appropriately expressed MFL of SdS spacetime should be obtained by using the definitions (3.11) and (3.12) under the requirement (3.14) with regarding M and Λ as two independent variables. To obtain the appropriate MFL, we must be careful to carry out the integrations in Eq. (2.9). According to asymptotic flat case summarized in the previous subsection, we expect that, when the second term (surface term) in Eq.(2.9) is evaluated on the bifurcation sphere in SdS spacetime, it should give the \"heat term\" T eff δS in the MFL of SdS spacetime (3.13). But, when the second term in Eq.(2.9) is evaluated on some sphere which is not the bifurcation sphere, it should contribute to the mass variation δM in Eq. (3.13), as for the ADM mass in asymptotic flat case (3.10). And the first term (3-volume term) in Eq. (2.9) should contribute to the \"work term\" −p eff δV in Eq. (3.13). Then, since there seems no natural sphere except for the bifurcation spheres of horizons, one problem arises; how to prepare the sphere to get the mass variation in the MFL of SdS spacetime. Hence, in order to extract the mass variation from Eq.(2.9), we adopt the strategy composed of three steps as follows:\nStep 1: Consider two hypersurfaces, Σ b and Σ c , at t = constant. Here Σ b covers the region r b < r < r cut and Σ c covers r cut < r < r c , where r cut is a working variable which is introduced to extract the mass variation δM and must not appear in the resultant MFL of SdS spacetime.\nStep 2: Carry out the integrations in Eq.(2.9) for the hypersurface Σ b . Do the same for Σ c .\nStep 3: Combine those integrated equations to introduce the variations of M, S and V into the appropriately expressed MFL of SdS spacetime (3.13).\nNote that the working variable r cut is introduced simply to carry out the integrations in Eq.(2.9) as a \"piecewise\" integration. If the resultant MFL obtained in the step 3 depends on r cut , this strategy is not suitable for the construction of the appropriate MFL of SdS spacetime. However in fact, as will be shown below, the result does not depend on r cut and we will succeed to obtain the appropriately expressed MFL of SdS spacetime. Let us carry out the step 2. We calculate Eq.(2.9) with the hypersurfaces Σ b and Σ c which are introduced in the step 1. In the following calculations, B b and B c denote, respectively, the bifurcation spheres of BEH and CEH, and B cut is the sphere of radius r cut which is the outer (inner) boundary of Σ b (Σ c ). As mentioned in Sec.2, we use the Lagrangian L(g, Λ) ε in Eq.(2.1) with the variable Λ, which gives E Λ = −(1/8π) ε. The metric is given in Eq. (3.1), and the timelike Killing vector ∂ t is regarded as the symmetry generator ξ := ∂ t .\nEq.(2.9) with Σ b in the SdS spacetime reads\nwhere ξ = 0 at B b is used in the second term, and the sign of the second and third terms are determined by the normal directions to Σ b at B b and B cut which are, respectively, in-ward and out-ward pointing along the \"r-axis\". The first term in the left-hand side of Eq.(3.15) is a simple three dimensional volume integral in Σ b ,\nFor the second term in the left-hand side of Eq.(3.15), note that the symplectic potential Θ includes the (n − 1)-volume form as its factor by definition (2.2), and consequently the Noether charge Q ξ includes the (n − 2)-volume form as its factor by definition (2.6). (n = 4 for SdS spacetime.) This implies that the integral and the variation δ of Noether charge is commutative, B b δQ ξ = δ B b Q ξ . On the other hand, we can find for the Lagrangian in Eq.(2.1) with the variable Λ that the Noether charge Q ξ takes the same form with the ordinary general relativity without Λ [5],\nThen, referring to the proof of the theorem 6.1 in the original Iyer-Wald formalism [5] which holds even with the variable Λ, we get\nwhere the indices a and b are the abstract indices [8],ǭ αβ is the bi-normal 2-form to B b which satisfies ∇ µ ξ ν = κ bǭµν on B b , and the vanishing variation of surface gravity at\nHence we obtain for the second term in Eq. (3.15),\nwhere d 2 Ω b := r 2 b sin θ dθ dϕ is the area element on B b . Let us proceed to the calculation of the third term in the left-hand side of Eq. (3.15). For the first we calculate the integral of Noether charge Br Q ξ , where B r is a two-sphere of an arbitrary radius r. Note that the θ-ϕ component of Q ξ in Eq. (3.17) contributes to the integral,\nThis gives\nwhere d 2 Ω r := r 2 sin θ dθ dϕ is the area element on B r and the definitions ξ := ∂ t and g tt = −f (r) in Eq.(3.2) are used. Then, using the relation Br δQ ξ = δ Br Q ξ , the first integral in the third term in Eq.(3.15) becomes\nwhere f ′ := df /dr and Eq.(3.21) is used in the second equality. Next, the explicit form of the symplectic potential Θ for the Lagrangian (2.1) with the variable Λ is\nThis is the same form with the ordinary general relativity without Λ [5]. Here, under the variations of δM and δΛ, the metric variation δg µν is given from Eq.(3.1),\nwhere the spherical part vanishes since that part does not depend on M and Λ, and\n(3.25) + By definition of κ b in Eq. (3.6) and δξ µ = 0, we get ξ ν (δΓ µ να ) ξ α = (δκ b ) ξ µ at the BEH, where Γ µ να is the Christoffel symbol. When this is evaluated by, for example, the Kruskal-Szekeres type coordinate covering the inside and outside of BEH, we can find the limiting behavior δκ b → 0 as approaches the bifurcation sphere\nIt should be emphasized here that the metric variation (3.24) is a static solution of the extended linearized Einstein equation (2.4) and satisfies the condition to ensure Eq. (2.9). Furthermore note that, for the integrand ξ α Θ αµν in the integral Bcut ξ · Θ, the indices µ and ν denotes the tangential components to B cut (θ-ϕ component), ξ α Θ α θϕ . Then we obtain\nwhere d 2 Ω cut := r 2 cut sin θ dθ dϕ is the area element on B cut , and ξ := ∂ t and ε tµνr = −ε trµν are used in the first equality. Hence the third term in Eq. (3.15) becomes\nThen, collecting Eqs. (3.16), (3.19) and (3.27), we obtain from Eq.(3.15),\nIt should be noted that the working variable r cut does not appear in this equation. Next turn our calculation to Eq.(2.9) with the hypersurface Σ c , which reads\nwhere ξ = 0 at B c is used in the second term, and the sign of the second and third terms are determined by the normal directions to Σ c at B c and B cut which are, respectively, out-ward and in-ward pointing along the \"r-axis\". Then, following the same calculations to obtain Eq.(3.28), we obtain from Eq. (3.29),\nIt should be noted that the working variable r cut does not appear in this equation. So far we have carried out the step 2 to obtain Eqs. (3.28) and (3.30). Before proceeding to the step 3 of our strategy, let us comment on Ref. [4]: As mentioned in Sec. 1,Ref.[4] has already discussed the MFL of SdS spacetime with variable Λ using some integral quantities as state variables, and obtained two MFLs separately for BEH and CEH. Those MFLs in [4] are the same with our Eqs. (3.28) and (3.30). Although the derivation of MFLs in [4] includes the problem of the choice of integration constant, but our derivation based on the extended Iyer-Wald formalism is free from such problem. Furthermore, the integral quantity used in [4] requires to place a boundary (corresponding to B cut in our step 2) at CEH in deriving Eq. (3.28) and at BEH in deriving Eq. (3.30). If we interpret the boundary as the position of an observer, it seems not to be physically acceptable to place the observer at the event horizon. However our Indeed, Ref. [4] has already obtained Eqs. (3.28) and (3.30) (whose combination gives Eq.(3.34)) by using some integral quantities as state variables. The advantages of using the extended Iyer-Wald formalism are the following three points: (1) it is free from the problem of integration constant as mentioned in Sec.1, (2) it offers naturally the definition of V in Eq. (3.12), and (3) it gives the positive definite effective temperature T eff and pressure p eff which, together with V , are suitable to describe the SdS black hole evaporation process as discussed in the next section.\nFinally in this section, for the completeness of our discussion, let us consider the case that Λ is not regarded as an independent variable. In this case, one obtains a relation, δS = 2π (1/κ b + 1/κ c ) δM, from the definition (3.11). By rearranging this relation appropriately and introducing V defined in Eq.(3.12), one can formally obtain the MFL in Eq.(3.34) with the same coefficients in Eqs. (3.35) and (3.36). However in this case, we find relations; ∂M(S, V )/∂S = (dS/dM) −1 ≡ T eff and −∂M/∂V = −(dV /dM) −1 ≡ −p eff . These contradict the definition of coefficients T eff and p eff in Eq. (3.34). By the reductive absurdity, this fact indicates the necessity of \"variable Λ\" for a mathematically consistent MFL of SdS spacetime of the form in Eq. (3.34). Here note that the same claim, the necessity of variable Λ, is already suggested in [4], although the MFL in [4] is given in a different expression from Eq. (3.34) and includes the uncertainty of integration constant of conserved quantities as mentioned in Sec.1. The necessity of variable Λ seems a universal fact for mathematically consistent MFL of SdS spacetime."},{"title":"Discussions","content":"Let us discuss what is implied by our appropriate MFL in Eq. (3.34) if it is regarded as a thermodynamical first law of SdS spacetime. As an interesting process, we treat the SdS black hole evaporation process at constant Λ.\nBefore considering SdS black hole evaporation, let us recall the so-called generalized second law in the evaporation process of Schwarzschild black hole. In Schwarzschild thermodynamics, the thermodynamical first law is given by Eq.(3.10) with setting the \"work terms\" zero, δM = T BH δS BH , where T BH := κ BH /2π = 1/8πM is the Hawking temperature and S BH := A BH /4 = π(2M) 2 is the black hole entropy. When Schwarzschild black hole evaporates, the mass energy M decreases and consequently S BH decreases. Here note that, the evaporation of isolated black hole is an irreversible adiabatic process. Hence, if the total entropy of the whole system is given by S BH , the decrease of S BH contradicts the second law of thermodynamics which requires the increase of total entropy for irreversible adiabatic processes. Then the generalized second law claims that the total entropy of black hole and matter fields of Hawking radiation increases for the evaporation process of isolated black holes [9,10]. Therefore, at least for the evaporation of asymptotic flat black holes, the generalized second law is necessary to hold the validity of thermodynamical formulation of black holes.\nThen, proceed to the discussion of SdS black hole evaporation at constant Λ. When SdS black hole evaporates, it seems reasonable to consider the mass parameter decreases, δM < 0. Here, for any process at constant Λ, our MFL in Eq. (3.34) and δV in Eq. (3.33) are rearranged to δS = 2π (1/κ b + 1/κ c ) δM and δV = 4π (r c /κ c − r b /κ b ) δM. Then, we get δS > 0 and δV > 0 due to δM < 0 and Eq. (3.8). The expansion of volume δV > 0 is a reasonable result if p eff is interpreted as a pressure. The increase of entropy δS > 0 denotes that the generalized second law is not needed to ensure the second law of SdS thermodynamics for its evaporation process at constant Λ. Hence our appropriate MFL in Eq.(3.34) suggests a thermodynamically consistent description of SdS black hole evaporation at constant Λ. Finally let us comment on the coefficient T eff in Eq. (3.35). This T eff is regarded as a temperature if our MFL in Eq.(3.34) is regarded as a thermodynamical first law. Then one may wonder about what the physical meaning of the temperature is, since the SdS spacetime is essentially a non-equilibrium system due to the difference of Hawking temperatures of BEH and CEH [2]. Here let us point out that there is a long history of research on two-temperature non-equilibrium systems, and there are many proposals on the definition of non-equilibrium temperature of non-equilibrium systems (see, for example, references of works in [10]). No commonly accepted definition of nonequilibrium temperature exists as present. Therefore, from the point of view of nonequilibrium thermodynamics, the suggestion of defining a non-equilibrium temperature is meaningful at present. If the SdS thermodynamics is formulated with regarding our MFL in Eq.(3.34) as a thermodynamical first law, then our T eff in Eq. (3.35), which can be expressed as a linear combination of the surface gravities of BEH and CEH, may be understood as one suggestion of an effective temperature of a non-equilibrium system. At present, the physical meaning of T eff is an open issue which requires further researches on non-equilibrium nature of SdS spacetime. The research on SdS thermodynamics provides the stage for the intersection of gravitational physics and non-equilibrium physics."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Black hole thermodynamics has already been established well for spacetimes with a single black hole event horizon [1]. However there is no consistent thermodynamical formulation for black hole spacetimes with cosmological constant Λ which possess a black hole event horizon (BEH) and a cosmological event horizon (CEH). For example, in Schwarzschild-de Sitter (SdS) spacetime, the Hawking temperature of BEH is higher than that of CEH [2]. This temperature difference causes difficulties in formulating SdS thermodynamics, since BEH (CEH) seems in some non-equilibrium state under the influence of Hawking radiation of different temperature by CEH (BEH). In general, the similar difficulty arises also for any two-horizon spacetime with Λ. § However we may be able to search for thermodynamical first law of two-horizon spacetimes with Λ through the mechanical first law (MFL). The MFL is a geometrical relation which relates the variations of mass parameter, horizon area and the other supplemental quantities. For example, the MFL of Schwarzschild black hole is equivalent to its thermodynamical first law, and formulated by using the mass parameter as only one variable which describes the effect of a single horizon on the MFL. This fact of Schwarzschild black hole leads us to expect that the MFL of two-horizon spacetimes with Λ can be a candidate of a thermodynamical first law of the spacetimes, and that the existence of two horizons requires two independent variables in the resultant MFL of the spacetimes to include the effects of two horizons. Then we adopt the following working hypothesis to search for the MFL of two-horizon spacetimes with Λ:\\nWorking Hypothesis (Two Independent Variables): Generally for spacetimes with cosmological constant which possess BEH and CEH, the mass parameter M and the cosmological constant Λ are regarded as two independent variables in the MFL of the spacetimes.\\nIndeed it will be shown in Sec.3 that the MFL of SdS spacetime becomes mathematically inconsistent if Λ is not an independent variable. When one considers a non-variable Λ as a physical situation, it is obtained by setting the variation of Λ zero (δΛ = 0) in the MFL after constructing it with regarding Λ as an independent variable. In such case, the variable Λ is interpreted as a \\\"working variable\\\" to obtain the MFL of two-horizon spacetimes with Λ. On the other hand for SdS spacetime, the search for an MFL with and without regarding Λ as an independent variable, has already been tried in some existing works [3,4]. Those works make use of some conserved quantities defined by some integrals (e.g. Hamiltonian, action integral and so on). One can obtain some candidates of MFL or thermodynamical first law of SdS spacetime with regarding the integral quantities as the variables in the first law (see [4] for example). However, when one uses § Some proposals for thermodynamics of BEH and CEH are already given for a case with some special matter fields and for an extreme case with magnetic/electric charge [11]. These examples are artificial to vanish the temperature difference of BEH and CEH. But in this paper we consider more general case which is not extremal and does not depend on artificial matter fields. the integral quantities, there arises a problem of the choice of integration constant (e.g. the so-called boundary counter-term or subtraction term to eliminate some divergent term in Hamiltonian or action integral).\\nThen, in order to derive the MFL of two-horizon spacetimes with Λ, we make use of the Iyer-Wald formalism which is free from the problem of integration constant [5]. We extend the original Iyer-Wald formalism to make it so general to be applicable to any spacetime with Λ under the working hypothesis of two independent variables. The extended Iyer-Wald formalism preserves the existence of conserved Noether current and its associated Noether charge, and enables us to observe how the resultant MFL is related to the Noether charge which is locally constructed from the metric (free from the problem of integration constant).\\nThe general extension of Iyer-Wald formalism to include the variable Λ gives simply an abstract form of MFL of two-horizon spacetimes with Λ, since the metric is not concretely specified. Then, as a representative application of our extended Iyer-Wald formalism, we derive the MFL of SdS spacetime by that formalism. Moreover, we examine what the MFL of SdS spacetime implies for SdS black hole evaporation if the MFL is regarded as a thermodynamical first law.\\nHere let us note that Ref. [4] has already treated Λ as an independent variable for SdS spacetime. (In [3], Λ is not an independent variable but a complete constant.) In [4], the variable Λ is concluded from mathematical consistency of the MFL of SdS spacetime, although there remains an uncertainty of integration constant in the resultant MFL given in [4]. However we will show in this paper that the extended Iyer-Wald formalism gives the same MFL with Ref. [4] in a straightforward way without the problem of integration constant. Furthermore, as the advantage of our procedure, the extended Iyer-Wald formalism enables us to introduce naturally a state variable which represents the size of the two-horizon system. Then, consequently, we will show that the new state variable of system size leads natural definitions of effective temperature and pressure of two-horizon system. Those state variables are suitable to describe the SdS black hole evaporation process. Such good state variables are not given in [4] and also the evaporation process is not described in [4]. This paper is organized as follows. Sec.2 is devoted to the extension of Iyer-Wald formalism to include the variable Λ and obtain the abstract form of MFL of any twohorizon spacetimes with Λ. Sec.3 is for the application of extended Iyer-Wald formalism to SdS spacetime and obtain the MFL of SdS spacetime. In that section, the state variables of system size, effective temperature and pressure are given, and then the mathematical consistency of the MFL with variable Λ is also examined. In Sec.4, we discuss what is implied about SdS black hole evaporation if our MFL of SdS spacetime is regarded as a thermodynamical first law, and give a comment about the non-equilibrium nature due to the difference of Hawking temperatures of BEH and CEH.\\nThroughout this paper, we use the Planck units, c = = G = k B = 1.\"\n },\n {\n \"title\": \"Iyer-Wald formalism with variable Λ\",\n \"content\": \"For simplicity we consider the empty spacetime in n dimensions without matter fields and start with the diffeomorphism invariant Lagrangian n-form L(g, Λ) = L(g, Λ) ε, where g denotes the metric g µν , ε is the volume n-form and L(g, Λ) is the Lagrangian scalar density. The inclusion of matter fields is very straightforward in following discussions. In the next section, L is specified to be the ordinary Einstein-Hilbert form in four dimensions,\\nwhere R is the Ricci scalar and Λ is the variable cosmological constant. But in this section L(g, Λ) is an arbitrary Lagrangian of the metric and the variable Λ. The first variation of this Lagrangian is expressed as\\nwhere δg is the abbreviation of metric variation δg µν , Θ is the (n − 1)-form called the symplectic potential which corresponds to the boundary term in the variational principle of action integral, E g = E g ε gives the Einstein equation of the metric by E g = 0 and E Λ = E Λ ε is the variation of L with respect to the working variable Λ,\\nFor L in Eq.(2.1), we get E Λ = −(1/8π) ε which will be used in the next section.\\nHere we have to emphasize the following two remarks: The first one is on the socalled on-shell condition, which restricts the metric (and the matter field if it exists) to be the solution of equation of motion. When the on-shell condition in the presence of metric variation is going to be required later in this section, the variation δg µν is to be understood as a solution of the \\\"extended\\\" linearized Einstein equation due to the variable Λ,\\nwhere G µν is the Einstein tensor and g µν is the \\\"unperturbed\\\" metric given by the Einstein equation E g = 0 . (The right-hand side would not be zero but be the variation δT µν of stress-energy tensor if matter fields exit.) The third term g µν δΛ does not appear for the ordinary linearized equation with completely constant Λ, but now it appears due to the variable Λ. Under the working hypothesis of two independent variables, we can regard Λ not as the kin of universal constants like Newton's constant but as the kin of constants of motion like mass parameter M which is a variable in asymptotic flat black hole thermodynamics. The second remark we should emphasize here is on the variational principle. Exactly speaking, the variational principle gives the equations of motion of dynamical variables via the vanishing variation of action integral I with respect to the dynamical variable This g µν can also be regarded as a \\\"background\\\" metric of the perturbation δg µν . φ(x) which depends generally on the spacetime points, δI/δφ(x) = 0, where x represents the spacetime dependence. However, the \\\"working\\\" variable Λ is not regarded as any dynamical variable and has no spacetime dependence. The Λ is simply a working variable to ensure the mathematical consistency of the resultant MFL of two-horizon spacetimes. Hence, even if E Λ is set zero, it can never be interpreted as any equation of motion of dynamical variable. This means that the on-shell condition requires only the Einstein equation E g = 0 (and the extended linearized equation (2.4) under the presence of variation δg µν ), and E Λ is non-vanishing (E Λ ≡ 0). Now it is recognized that the physical principle we rely on is the \\\"extended\\\" variational principle in which the variations are taken with respect to not only the dynamical variable g µν (and matter field if it exists) but also the working variable Λ, while the equations of motion are given by the variation of Lagrangian with respect to dynamical variables.\\nAs a by-product of the above two remarks, it will be shown below that, in our extended Iyer-Wald formalism, the conserved Noether current and its associated Noether charge are defined in the same way as in the original Iyer-Wald formalism. Our \\\"extension\\\" of Iyer-Wald formalism has three meanings; (1) the extended Iyer-Wald formalism includes the variable Λ, (2) the Λ is not a dynamical variable but simply the working variable which means E Λ does not give any equation of motion, and (3) the conserved Noether current is obtained with the same definition as in the original Iyer-Wald formalism.\\nThen let us proceed to the extension of Iyer-Wald formalism. Eq.(2.2) is the starting point. The extended Iyer-Wald formalism with variable Λ differs from the original formalism on the following two points: One of them is a manifest point expressed by the third term in the right-hand side of Eq.(2.2). That term does not arise in the original Iyer-Wald formalism, but arises in our extended formalism by the variation δΛ in, for example, the second term in the right-hand side of Eq.(2.1). Another point is a subtle point included in the metric variation δg µν . In our extended formalism, the variation δg µν also gives rise to the variation δΛ if the concrete form of the metric depends on Λ as for SdS, de Sitter and Anti-de Sitter spacetimes.\\nTo formulate the abstract form of MFL with the variable Λ, let us follow the same procedure of the original Iyer-Wald formalism [5]. If we introduce an arbitrary vector field ξ µ , which is not a dynamical variable in L, and consider the variation given by the Lie derivative δ = L ξ along ξ µ , then we get from Eq.(2.2), 5) where ξ · L := ξ µ L µν 1 ···ν n−1 and a relation L ξ A = ξ · dA + d(ξ · A) of the Lie and exterior derivatives of a form A is used. The (n − 1)-from in the left-hand side, Θ(g, L ξ g) − ξ · L =: J ξ , is called the Noether current. Here note that, since Λ has no spacetime dependence, L ξ Λ ≡ 0 holds. Therefore, when the on-shell condition E g = 0 is required, the Noether current is closed dJ ξ = 0 which guarantees the local existence of the (n − 2)-form Q ξ called the Noether charge,\\nThis Q ξ is locally constructed from the on-shell metric which satisfies the Einstein equation. Note that the conservation of the Noether current dJ ξ = 0 under the onshell condition corresponds to the Noether's theorem, and the Noether charge Q ξ is the conserved charge of J ξ associated with the symmetry generator ξ. The definition of Q ξ in Eq.(2.6) is the same with that in the original Iyer-Wald formalism. Here it should also be emphasized that the existence condition of Q ξ is the on-shell condition E g = 0 for the metric g µν appearing in Eq. (2.6). This condition is the same with that in the original Iyer-Wald formalism. Hence we find that, in our extended Iyer-Wald formalism, the conserved Noether current and its associated Noether charge are defined in the same way as in the original Iyer-Wald formalism. Next, since the vector ξ µ (not the 1-form ξ µ ) is not a dynamical variable, its variation does not exist (δξ µ ≡ 0) under the variation of dynamical and working variables. Then, from the variation of J ξ , the following relation is obtained:\\nwhere ω is defined as ω(g, δg, L ξ g) := δΘ(g, L ξ g) − L ξ Θ(g, L ξ g) and the on-shell condition E g = 0 is required. Here we must note that, as explained in the second paragraph of this section, the on-shell condition under the presence of metric variation δg µν denotes that the unperturbed matric g µν and the variation δg µν appearing in Eq.(2.7) satisfy, respectively, the Einstein equation E g = 0 and the extended linearized Einstein equation (2.4). Furthermore, as noted in [5], the (n − 1)-form ω vanishes (ω = 0), when ξ is the generator of a symmetry of all dynamical variables in L, i.e. L ξ g µν = 0. For stationary spacetimes, the timelike Killing vector can be regarded as the symmetry generator ξ. Hence, at least for the stationary case, we get\\nHere the Noether charge defined in Eq.(2.6) gives δJ ξ = d(δQ ξ ). Therefore, by integrating Eq.(2.8) on a hypersurface Σ and applying the Stokes' theorem, we obtain\\nThe first term is due to the variable Λ. There are three conditions to hold Eq.(2.9);\\n(1) the Einstein equation E g = 0 for the unperturbed metric g µν , (2) the extended linearized Einstein equation (2.4) of the metric variation δg µν , and (3) the existence of the symmetry generator ξ to give L ξ g = 0.\\nIn the original Iyer-Wald formalism, the first term in Eq.(2.9) disappears. Ref. [5] shows that, when the original formalism is applied to asymptotic flat black holes, Eq.(2.9) without the first term reduces to the MFL of those black holes. Therefore we can regard Eq.(2.9) as the primitive MFL of any spacetime with Λ in any dimensions. ¶ ¶ The primitive MFL (2.9) seems applicable to spacetimes without CEH like asymptotic Anti-de Sitter black holes. When Eq.(2.9) is applied to such black holes, the two degrees of freedom expressed by M and Λ may describe two independent effects in the MFL; one of them is due to the BEH, and another is due to the \\\"wall\\\" given by the infinitely large gravitational potential barrier of Anti-de Sitter metric.\"\n },\n {\n \"title\": \"Mechanical first law of SdS spacetime\",\n \"content\": \"\"\n },\n {\n \"title\": \"Preparations\",\n \"content\": \"Before deriving the MFL of SdS spacetime from Eq.(2.9), we summarize the SdS metric and the MFL of asymptotic flat black holes derived from the original Iyer-Wald formalism. For the first, the line element of SdS spacetime is\\n2)\\nThe equation f (r) = 0 has two positive roots and one negative root for the parameter range,\\nThroughout this paper, our discussion is restricted in this range. This parameter range guarantees the existence of BEH and CEH. The radius of BEH r b and that of CEH r c are given by, respectively, the smaller positive root and the larger positive root of f (r) = 0. Then the parameter range (3.3) gives the relations of these radii,\\nThe equations f (r b ) = 0 and f (r c ) = 0 are rearranged to\\nSince SdS spacetime is static, each event horizon possesses the bifurcation sphere, on which the timelike Killing vector ξ := ∂ t vanishes. The surface gravity of BEH κ b and that of CEH κ c are defined by the following equations evaluated at the horizons,\\nIn general, the numerical value of surface gravity depends on the normalization of ξ.\\nWith the normalization ξ := ∂ t , we get\\nThen, from Eq.(3.4), the following relations hold,\\nNext, we summarize the MFL of asymptotic flat black holes derived from the original Iyer-Wald formalism [5]: The first term in Eq.(2.9) does not exist in the original formalism. When the original formalism is applied to asymptotic flat black holes and Σ connects the bifurcation sphere B of the horizon and the spatial infinity ∞, then Eq.(2.9) without the first term is rearranged to\\nwhere ξ := ∂ t with the usual static time t and ξ = 0 on B is used. As shown in [5], this reduces to the MFL of asymptotic flat black holes,\\nwhere M is the ADM mass of black hole, A BH is the area of B, κ BH is the surface gravity of the horizon with an appropriate normalization of Killing vector, the \\\"work terms\\\" are given by electromagnetic fields and angular momentum for Reissner-Nordström and Kerr black holes [6]. (This MFL in Eq.(3.10) is a differential form of the so-called Smarr's formula [7].) It should be emphasized that the first term (integration on ∞) in the lefthand side of Eq. (3.9) gives the variation of the ADM mass δM and the \\\"work terms\\\" in Eq. (3.10), and the second term (integration on B) in Eq.(3.9) gives the \\\"heat term\\\" (κ BH /8π) δA BH . In asymptotic flat black hole thermodynamics, the MFL in Eq.(3.10) is equivalent to the thermodynamical first law of asymptotic flat black holes in which M is regarded as the internal energy, and A BH /4 is regarded as the entropy (the so-called entropy-area law ) due to the Hawking temperature κ BH /2π [1].\"\n },\n {\n \"title\": \"Application of extended Iyer-Wald formalism to SdS spacetime\",\n \"content\": \"Now we apply Eq.(2.9) to SdS spacetime to obtain the appropriately expressed MFL of SdS spacetime. Here the meaning of \\\"appropriate expression\\\" is that the mass parameter M of SdS spacetime is expressed as a function of two independent quantities which seem correspond to the entropy and the state variable of system size when the MFL is regarded as a thermodynamical first law. The quantity corresponding to entropy gives the \\\"heat term\\\" in MFL, and the quantity corresponding to system size gives the \\\"work term\\\" in MFL. In thermodynamical first law of ordinary laboratory systems, the heat term is the product of temperature and variation of entropy, and the work term is the product of pressure and variation of system size. Concerning the quantity S which seems correspond to the entropy of SdS spacetime, we refer simply to the entropy-area law of asymptotic flat black holes and define S by the total spatial area of BEH and CEH:\\nS := π r 2 c + π r 2 b . (3.11) Here note that the horizon radii r b and r c are regarded as two independent variables due to the working hypothesis of two independent variables, and that the total horizon area includes manifestly the effects of BEH and CEH through these two variables r b and r c . When this definition (3.11) is adopted, the variation δS gives the \\\"heat term\\\" in the appropriate MFL of SdS spacetime. Next, we have to define the system size V whose variation δV composes the \\\"work term\\\" in the appropriate MFL of SdS spacetime. Here, for the system size of SdS spacetime, it seems reasonable to consider the three dimensional volume of a spacelike hypersurface connecting BEH and CEH as the object of thermodynamical interest, since the Hawking radiations of two horizons coexist there. As such hypersurface, let us consider Σ bif which connects the bifurcation spheres of BEH and CEH at t = constant. Furthermore let us refer to the first term in Eq. (2.9). In that term, the integral Σ bif ξ · E Λ appears as a natural quantity of three dimensional volume. Then concerning the quantity V which seems correspond to the system size of SdS spacetime, we adopt the following definition:\\nwhere ξ := ∂ t . The definitions (3.11) and (3.12) are equivalent to search for the MFL of SdS spacetime which is expressed as δM = T eff δS − p eff δV , (3.13) where the concrete forms of coefficients T eff and p eff will be obtained below. These coefficients T eff and p eff are simply the partial derivatives of M. If the MFL in Eq.(3.13) is regarded as a thermodynamical first law of SdS spacetime, then T eff and p eff are interpreted as, respectively, effective temperature and pressure. Therefore, because the temperature and pressure are positive definite for ordinary laboratory systems, it is natural to require that the appropriately expressed MFL should satisfy the following requirement:\\nT eff > 0 , p eff > 0 . (3.14) If the MFL in Eq.(3.13) is regarded as a thermodynamical first law of SdS spacetime, then the positivity of T eff and p eff makes it plausible to call T eff and p eff , respectively, the effective temperature and pressure. A comment on T eff in relation to non-equilibrium nature of SdS spacetime will be given in Sec.4. The appropriately expressed MFL of SdS spacetime should be obtained by using the definitions (3.11) and (3.12) under the requirement (3.14) with regarding M and Λ as two independent variables. To obtain the appropriate MFL, we must be careful to carry out the integrations in Eq. (2.9). According to asymptotic flat case summarized in the previous subsection, we expect that, when the second term (surface term) in Eq.(2.9) is evaluated on the bifurcation sphere in SdS spacetime, it should give the \\\"heat term\\\" T eff δS in the MFL of SdS spacetime (3.13). But, when the second term in Eq.(2.9) is evaluated on some sphere which is not the bifurcation sphere, it should contribute to the mass variation δM in Eq. (3.13), as for the ADM mass in asymptotic flat case (3.10). And the first term (3-volume term) in Eq. (2.9) should contribute to the \\\"work term\\\" −p eff δV in Eq. (3.13). Then, since there seems no natural sphere except for the bifurcation spheres of horizons, one problem arises; how to prepare the sphere to get the mass variation in the MFL of SdS spacetime. Hence, in order to extract the mass variation from Eq.(2.9), we adopt the strategy composed of three steps as follows:\\nStep 1: Consider two hypersurfaces, Σ b and Σ c , at t = constant. Here Σ b covers the region r b < r < r cut and Σ c covers r cut < r < r c , where r cut is a working variable which is introduced to extract the mass variation δM and must not appear in the resultant MFL of SdS spacetime.\\nStep 2: Carry out the integrations in Eq.(2.9) for the hypersurface Σ b . Do the same for Σ c .\\nStep 3: Combine those integrated equations to introduce the variations of M, S and V into the appropriately expressed MFL of SdS spacetime (3.13).\\nNote that the working variable r cut is introduced simply to carry out the integrations in Eq.(2.9) as a \\\"piecewise\\\" integration. If the resultant MFL obtained in the step 3 depends on r cut , this strategy is not suitable for the construction of the appropriate MFL of SdS spacetime. However in fact, as will be shown below, the result does not depend on r cut and we will succeed to obtain the appropriately expressed MFL of SdS spacetime. Let us carry out the step 2. We calculate Eq.(2.9) with the hypersurfaces Σ b and Σ c which are introduced in the step 1. In the following calculations, B b and B c denote, respectively, the bifurcation spheres of BEH and CEH, and B cut is the sphere of radius r cut which is the outer (inner) boundary of Σ b (Σ c ). As mentioned in Sec.2, we use the Lagrangian L(g, Λ) ε in Eq.(2.1) with the variable Λ, which gives E Λ = −(1/8π) ε. The metric is given in Eq. (3.1), and the timelike Killing vector ∂ t is regarded as the symmetry generator ξ := ∂ t .\\nEq.(2.9) with Σ b in the SdS spacetime reads\\nwhere ξ = 0 at B b is used in the second term, and the sign of the second and third terms are determined by the normal directions to Σ b at B b and B cut which are, respectively, in-ward and out-ward pointing along the \\\"r-axis\\\". The first term in the left-hand side of Eq.(3.15) is a simple three dimensional volume integral in Σ b ,\\nFor the second term in the left-hand side of Eq.(3.15), note that the symplectic potential Θ includes the (n − 1)-volume form as its factor by definition (2.2), and consequently the Noether charge Q ξ includes the (n − 2)-volume form as its factor by definition (2.6). (n = 4 for SdS spacetime.) This implies that the integral and the variation δ of Noether charge is commutative, B b δQ ξ = δ B b Q ξ . On the other hand, we can find for the Lagrangian in Eq.(2.1) with the variable Λ that the Noether charge Q ξ takes the same form with the ordinary general relativity without Λ [5],\\nThen, referring to the proof of the theorem 6.1 in the original Iyer-Wald formalism [5] which holds even with the variable Λ, we get\\nwhere the indices a and b are the abstract indices [8],ǭ αβ is the bi-normal 2-form to B b which satisfies ∇ µ ξ ν = κ bǭµν on B b , and the vanishing variation of surface gravity at\\nHence we obtain for the second term in Eq. (3.15),\\nwhere d 2 Ω b := r 2 b sin θ dθ dϕ is the area element on B b . Let us proceed to the calculation of the third term in the left-hand side of Eq. (3.15). For the first we calculate the integral of Noether charge Br Q ξ , where B r is a two-sphere of an arbitrary radius r. Note that the θ-ϕ component of Q ξ in Eq. (3.17) contributes to the integral,\\nThis gives\\nwhere d 2 Ω r := r 2 sin θ dθ dϕ is the area element on B r and the definitions ξ := ∂ t and g tt = −f (r) in Eq.(3.2) are used. Then, using the relation Br δQ ξ = δ Br Q ξ , the first integral in the third term in Eq.(3.15) becomes\\nwhere f ′ := df /dr and Eq.(3.21) is used in the second equality. Next, the explicit form of the symplectic potential Θ for the Lagrangian (2.1) with the variable Λ is\\nThis is the same form with the ordinary general relativity without Λ [5]. Here, under the variations of δM and δΛ, the metric variation δg µν is given from Eq.(3.1),\\nwhere the spherical part vanishes since that part does not depend on M and Λ, and\\n(3.25) + By definition of κ b in Eq. (3.6) and δξ µ = 0, we get ξ ν (δΓ µ να ) ξ α = (δκ b ) ξ µ at the BEH, where Γ µ να is the Christoffel symbol. When this is evaluated by, for example, the Kruskal-Szekeres type coordinate covering the inside and outside of BEH, we can find the limiting behavior δκ b → 0 as approaches the bifurcation sphere\\nIt should be emphasized here that the metric variation (3.24) is a static solution of the extended linearized Einstein equation (2.4) and satisfies the condition to ensure Eq. (2.9). Furthermore note that, for the integrand ξ α Θ αµν in the integral Bcut ξ · Θ, the indices µ and ν denotes the tangential components to B cut (θ-ϕ component), ξ α Θ α θϕ . Then we obtain\\nwhere d 2 Ω cut := r 2 cut sin θ dθ dϕ is the area element on B cut , and ξ := ∂ t and ε tµνr = −ε trµν are used in the first equality. Hence the third term in Eq. (3.15) becomes\\nThen, collecting Eqs. (3.16), (3.19) and (3.27), we obtain from Eq.(3.15),\\nIt should be noted that the working variable r cut does not appear in this equation. Next turn our calculation to Eq.(2.9) with the hypersurface Σ c , which reads\\nwhere ξ = 0 at B c is used in the second term, and the sign of the second and third terms are determined by the normal directions to Σ c at B c and B cut which are, respectively, out-ward and in-ward pointing along the \\\"r-axis\\\". Then, following the same calculations to obtain Eq.(3.28), we obtain from Eq. (3.29),\\nIt should be noted that the working variable r cut does not appear in this equation. So far we have carried out the step 2 to obtain Eqs. (3.28) and (3.30). Before proceeding to the step 3 of our strategy, let us comment on Ref. [4]: As mentioned in Sec. 1,Ref.[4] has already discussed the MFL of SdS spacetime with variable Λ using some integral quantities as state variables, and obtained two MFLs separately for BEH and CEH. Those MFLs in [4] are the same with our Eqs. (3.28) and (3.30). Although the derivation of MFLs in [4] includes the problem of the choice of integration constant, but our derivation based on the extended Iyer-Wald formalism is free from such problem. Furthermore, the integral quantity used in [4] requires to place a boundary (corresponding to B cut in our step 2) at CEH in deriving Eq. (3.28) and at BEH in deriving Eq. (3.30). If we interpret the boundary as the position of an observer, it seems not to be physically acceptable to place the observer at the event horizon. However our Indeed, Ref. [4] has already obtained Eqs. (3.28) and (3.30) (whose combination gives Eq.(3.34)) by using some integral quantities as state variables. The advantages of using the extended Iyer-Wald formalism are the following three points: (1) it is free from the problem of integration constant as mentioned in Sec.1, (2) it offers naturally the definition of V in Eq. (3.12), and (3) it gives the positive definite effective temperature T eff and pressure p eff which, together with V , are suitable to describe the SdS black hole evaporation process as discussed in the next section.\\nFinally in this section, for the completeness of our discussion, let us consider the case that Λ is not regarded as an independent variable. In this case, one obtains a relation, δS = 2π (1/κ b + 1/κ c ) δM, from the definition (3.11). By rearranging this relation appropriately and introducing V defined in Eq.(3.12), one can formally obtain the MFL in Eq.(3.34) with the same coefficients in Eqs. (3.35) and (3.36). However in this case, we find relations; ∂M(S, V )/∂S = (dS/dM) −1 ≡ T eff and −∂M/∂V = −(dV /dM) −1 ≡ −p eff . These contradict the definition of coefficients T eff and p eff in Eq. (3.34). By the reductive absurdity, this fact indicates the necessity of \\\"variable Λ\\\" for a mathematically consistent MFL of SdS spacetime of the form in Eq. (3.34). Here note that the same claim, the necessity of variable Λ, is already suggested in [4], although the MFL in [4] is given in a different expression from Eq. (3.34) and includes the uncertainty of integration constant of conserved quantities as mentioned in Sec.1. The necessity of variable Λ seems a universal fact for mathematically consistent MFL of SdS spacetime.\"\n },\n {\n \"title\": \"Discussions\",\n \"content\": \"Let us discuss what is implied by our appropriate MFL in Eq. (3.34) if it is regarded as a thermodynamical first law of SdS spacetime. As an interesting process, we treat the SdS black hole evaporation process at constant Λ.\\nBefore considering SdS black hole evaporation, let us recall the so-called generalized second law in the evaporation process of Schwarzschild black hole. In Schwarzschild thermodynamics, the thermodynamical first law is given by Eq.(3.10) with setting the \\\"work terms\\\" zero, δM = T BH δS BH , where T BH := κ BH /2π = 1/8πM is the Hawking temperature and S BH := A BH /4 = π(2M) 2 is the black hole entropy. When Schwarzschild black hole evaporates, the mass energy M decreases and consequently S BH decreases. Here note that, the evaporation of isolated black hole is an irreversible adiabatic process. Hence, if the total entropy of the whole system is given by S BH , the decrease of S BH contradicts the second law of thermodynamics which requires the increase of total entropy for irreversible adiabatic processes. Then the generalized second law claims that the total entropy of black hole and matter fields of Hawking radiation increases for the evaporation process of isolated black holes [9,10]. Therefore, at least for the evaporation of asymptotic flat black holes, the generalized second law is necessary to hold the validity of thermodynamical formulation of black holes.\\nThen, proceed to the discussion of SdS black hole evaporation at constant Λ. When SdS black hole evaporates, it seems reasonable to consider the mass parameter decreases, δM < 0. Here, for any process at constant Λ, our MFL in Eq. (3.34) and δV in Eq. (3.33) are rearranged to δS = 2π (1/κ b + 1/κ c ) δM and δV = 4π (r c /κ c − r b /κ b ) δM. Then, we get δS > 0 and δV > 0 due to δM < 0 and Eq. (3.8). The expansion of volume δV > 0 is a reasonable result if p eff is interpreted as a pressure. The increase of entropy δS > 0 denotes that the generalized second law is not needed to ensure the second law of SdS thermodynamics for its evaporation process at constant Λ. Hence our appropriate MFL in Eq.(3.34) suggests a thermodynamically consistent description of SdS black hole evaporation at constant Λ. Finally let us comment on the coefficient T eff in Eq. (3.35). This T eff is regarded as a temperature if our MFL in Eq.(3.34) is regarded as a thermodynamical first law. Then one may wonder about what the physical meaning of the temperature is, since the SdS spacetime is essentially a non-equilibrium system due to the difference of Hawking temperatures of BEH and CEH [2]. Here let us point out that there is a long history of research on two-temperature non-equilibrium systems, and there are many proposals on the definition of non-equilibrium temperature of non-equilibrium systems (see, for example, references of works in [10]). No commonly accepted definition of nonequilibrium temperature exists as present. Therefore, from the point of view of nonequilibrium thermodynamics, the suggestion of defining a non-equilibrium temperature is meaningful at present. If the SdS thermodynamics is formulated with regarding our MFL in Eq.(3.34) as a thermodynamical first law, then our T eff in Eq. (3.35), which can be expressed as a linear combination of the surface gravities of BEH and CEH, may be understood as one suggestion of an effective temperature of a non-equilibrium system. At present, the physical meaning of T eff is an open issue which requires further researches on non-equilibrium nature of SdS spacetime. The research on SdS thermodynamics provides the stage for the intersection of gravitational physics and non-equilibrium physics.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Black Holes and Entropy. J D Bekenstein, Phys. Rev. 72333J.D. Bekenstein, Black Holes and Entropy, Phys. Rev. D7 (1973) 2333.","doi":""},{"text":"Generalized second law of thermodynamics in black-hole physics. J D Bekenstein, Phys. Rev. 93292J.D. Bekenstein, Generalized second law of thermodynamics in black-hole physics, Phys. Rev. D9 (1974) 3292.","doi":""},{"text":"Gravitational Radiation from Colliding Black Holes. S W Hawking, Phys. Rev. Lett. 261344S.W. Hawking, Gravitational Radiation from Colliding Black Holes, Phys. Rev. Lett. 26 (1971) 1344.","doi":""},{"text":"Particle Creation by Black Holes. S W Hawking, Commun. Math. Phys. 43199S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199.","doi":""},{"text":"S W Hawking, D N Page, Thermodynamics of Black Holes in Anti-de Sitter Space. 87577S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in Anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577.","doi":""},{"text":"Third Law of Black-Hole Dynamics: A Formulation and Proof. W Israel, Phys. Rev. Lett. 57397W. Israel, Third Law of Black-Hole Dynamics: A Formulation and Proof, Phys. Rev. Lett. 57 (1986) 397.","doi":""},{"text":"Black-hole thermodynamics and the Euclidean Einstein action. J W York, Jr , Phys. Rev. 332092J.W. York,Jr., Black-hole thermodynamics and the Euclidean Einstein action Phys. Rev. D33 (1986) 2092.","doi":""},{"text":"Charged black hole in a grand canonical ensemble. H W Braden, J D Brown, B F Whiting, J W York, Jr , Phys. Rev. 423376H.W. Braden, J.D. Brown, B.F. Whiting and J.W. York,Jr., Charged black hole in a grand canonical ensemble, Phys. Rev. D42 (1990) 3376.","doi":""},{"text":"Complex Kerr-Newman Geometry and Black-Hole Thermodynamics. J David, E A Martinez, J W York, Jr , Phys. Rev. Lett. 662281J. David, E.A. Martinez and J.W. York,Jr., Complex Kerr-Newman Geometry and Black-Hole Thermodynamics, Phys. Rev. Lett.66 (1991) 2281.","doi":""},{"text":"Cosmological event horizons, thermodynamics, and particle creation. G W Gibbons, S W Hawking, Phys. Rev. 152738G.W. Gibbons and S.W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D15 (1977) 2738.","doi":""},{"text":"Stability of gravity with a cosmological constant. L F Abbot, S Deser, Nucl. Phys. 19576L.F. Abbot and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B195 (1982) 76.","doi":""},{"text":"Entropy and Holography in Asymptotically de Sitter Spaces. V Balasubramanian, J Deboer, D Minic, Mass, Phys. Rev. 65123508V. Balasubramanian, J. deBoer and D. Minic, Mass, Entropy and Holography in Asymptotically de Sitter Spaces, Phys. Rev. D65 (2002) 123508.","doi":""},{"text":"Mass and Entropy of Schwarzschild-de Sitter black holes and the de Sitter/CFT Correspondence. A M Ghezelbash, R B Mann, J. High Energy Phys. 02015ActionA.M. Ghezelbash and R.B. Mann, Action, Mass and Entropy of Schwarzschild-de Sitter black holes and the de Sitter/CFT Correspondence, J. High Energy Phys. 0201 (2002) 005.","doi":""},{"text":"de Sitter black holes with either of the two horizons as a boundary. A Gomberoff, C Teitelboim, Phys. Rev. 67104024A.Gomberoff and C.Teitelboim, de Sitter black holes with either of the two horizons as a boundary, Phys. Rev. D67 (2003) 104024.","doi":""},{"text":"Black Holes in de Sitter Space: Masses, Energies and Entropy Bounds. A Corichi, A Gomberoff, Phys. Rev. 6964016A. Corichi and A. Gomberoff, Black Holes in de Sitter Space: Masses, Energies and Entropy Bounds, Phys. Rev. D69 (2004) 064016.","doi":""},{"text":"Thermodynamics of de Sitter black holes: Thermal cosmological constant. Y Sekiwa, Phys. Rev. 7384009Y.Sekiwa, Thermodynamics of de Sitter black holes: Thermal cosmological constant, Phys. Rev. D73 (2006) 084009.","doi":""},{"text":"Black hole entropy is the Noether charge. R M Wald, Phys. Rev. 483427R.M.Wald, Black hole entropy is the Noether charge, Phys. Rev. D48 (1993) R3427.","doi":""},{"text":"Some properties of the Noether charge and a proposal for dynamical black hole entropy. V Iyer, R M Wald, Phys. Rev. 50846V.Iyer and R.M.Wald, Some properties of the Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D50 (1994) 846.","doi":""},{"text":"The Four Laws of Black Hole Mechanics. J M Bardeen, B Carter, S W Hawking, Commun. Math. Phys. 31161J.M. Bardeen, B. Carter and S.W. Hawking, The Four Laws of Black Hole Mechanics, Commun. Math. Phys. 31 (1973) 161.","doi":""},{"text":"Mass Formula for Kerr Black Holes. L Smarr, Phys. Rev. Lett. 3071L. Smarr, Mass Formula for Kerr Black Holes, Phys. Rev. Lett. 30 (1973) 71.","doi":""},{"text":"R M Wald, General Relativity. The Univ. of Chicago PressR.M.Wald, General Relativity, The Univ. of Chicago Press, 1984.","doi":""},{"text":"Acceleration radiation and the generalized second law of thermodynamics. W G Unruh, R M Wald, Phys. Rev. 25942W.G. Unruh and R.M. Wald, Acceleration radiation and the generalized second law of thermodynamics, Phys. Rev. D25 (1982) 942.","doi":""},{"text":"Entropy bounds, acceleration radiation, and the generalized second law. W G Unruh, R M Wald, Phys. Rev. 272271W.G. Unruh and R.M. Wald, Entropy bounds, acceleration radiation, and the generalized second law, Phys. Rev. D27 (1983) 2271.","doi":""},{"text":"Proof of Generalized Second Law for Quasistationary Semiclassical Black Holes. V P Frolov, D N Page, Phys. Rev. Lett. 713902V.P. Frolov and D.N. Page, Proof of Generalized Second Law for Quasistationary Semiclassical Black Holes, Phys. Rev. Lett. 71 (1993) 3902.","doi":""},{"text":"Proof of Classical Version of the Bousso Entropy Bound and of the Generalized Second Law. E E Flanagan, D Marolf, R M Wald, Phys. Rev. 6284035E.E. Flanagan, D. Marolf and R.M. Wald, Proof of Classical Version of the Bousso Entropy Bound and of the Generalized Second Law, Phys. Rev. D62 (2000) 084035.","doi":""},{"text":"Two-temperature Steady State Thermodynamics for a Radiation Field. H Saida, Physica. 356481H. Saida, Two-temperature Steady State Thermodynamics for a Radiation Field, Physica A356 (2005) 481.","doi":""},{"text":"The generalized second law and the black hole evaporation in an empty space as a nonequilibrium process. H Saida, Class. Quant. Grav. 236227H. Saida, The generalized second law and the black hole evaporation in an empty space as a nonequilibrium process, Class. Quant. Grav. 23 (2006) 6227.","doi":""},{"text":"Black hole evaporation as a nonequilibrium process. H Saida, arXiv:0811.1622Classical and Quantum Gravity Research. M.N. Christiansen and T.K. Rasmussen8Nova Science Publgr-qcH. Saida, Black hole evaporation as a nonequilibrium process, Chapter 8 (arXiv:0811.1622 [gr-qc]) in the edited book by M.N. Christiansen and T.K. Rasmussen, Classical and Quantum Gravity Research, Nova Science Publ., 2008.","doi":"arXiv:0811.1622"},{"text":"Euclidean action and the thermodynamics of manifolds without boundary. G Hayward, Phys. Rev. 413248G. Hayward, Euclidean action and the thermodynamics of manifolds without boundary, Phys. Rev. D41 (1990) 3248.","doi":""},{"text":"Duality between Electric and Magnetic Black Holes. S W Hawking, S F Ross, Phys. Rev. 525865S.W. Hawking and S.F. Ross, Duality between Electric and Magnetic Black Holes, Phys. Rev. D52 (1995) 5865.","doi":""}],"string":"[\n {\n \"text\": \"Black Holes and Entropy. J D Bekenstein, Phys. Rev. 72333J.D. Bekenstein, Black Holes and Entropy, Phys. Rev. D7 (1973) 2333.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Generalized second law of thermodynamics in black-hole physics. J D Bekenstein, Phys. Rev. 93292J.D. Bekenstein, Generalized second law of thermodynamics in black-hole physics, Phys. Rev. D9 (1974) 3292.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Gravitational Radiation from Colliding Black Holes. S W Hawking, Phys. Rev. Lett. 261344S.W. Hawking, Gravitational Radiation from Colliding Black Holes, Phys. Rev. Lett. 26 (1971) 1344.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Particle Creation by Black Holes. S W Hawking, Commun. Math. Phys. 43199S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S W Hawking, D N Page, Thermodynamics of Black Holes in Anti-de Sitter Space. 87577S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in Anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Third Law of Black-Hole Dynamics: A Formulation and Proof. W Israel, Phys. Rev. Lett. 57397W. Israel, Third Law of Black-Hole Dynamics: A Formulation and Proof, Phys. Rev. Lett. 57 (1986) 397.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Black-hole thermodynamics and the Euclidean Einstein action. J W York, Jr , Phys. Rev. 332092J.W. York,Jr., Black-hole thermodynamics and the Euclidean Einstein action Phys. Rev. D33 (1986) 2092.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Charged black hole in a grand canonical ensemble. H W Braden, J D Brown, B F Whiting, J W York, Jr , Phys. Rev. 423376H.W. Braden, J.D. Brown, B.F. Whiting and J.W. York,Jr., Charged black hole in a grand canonical ensemble, Phys. Rev. D42 (1990) 3376.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Complex Kerr-Newman Geometry and Black-Hole Thermodynamics. J David, E A Martinez, J W York, Jr , Phys. Rev. Lett. 662281J. David, E.A. Martinez and J.W. York,Jr., Complex Kerr-Newman Geometry and Black-Hole Thermodynamics, Phys. Rev. Lett.66 (1991) 2281.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cosmological event horizons, thermodynamics, and particle creation. G W Gibbons, S W Hawking, Phys. Rev. 152738G.W. Gibbons and S.W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D15 (1977) 2738.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Stability of gravity with a cosmological constant. L F Abbot, S Deser, Nucl. Phys. 19576L.F. Abbot and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B195 (1982) 76.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Entropy and Holography in Asymptotically de Sitter Spaces. V Balasubramanian, J Deboer, D Minic, Mass, Phys. Rev. 65123508V. Balasubramanian, J. deBoer and D. Minic, Mass, Entropy and Holography in Asymptotically de Sitter Spaces, Phys. Rev. D65 (2002) 123508.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mass and Entropy of Schwarzschild-de Sitter black holes and the de Sitter/CFT Correspondence. A M Ghezelbash, R B Mann, J. High Energy Phys. 02015ActionA.M. Ghezelbash and R.B. Mann, Action, Mass and Entropy of Schwarzschild-de Sitter black holes and the de Sitter/CFT Correspondence, J. High Energy Phys. 0201 (2002) 005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"de Sitter black holes with either of the two horizons as a boundary. A Gomberoff, C Teitelboim, Phys. Rev. 67104024A.Gomberoff and C.Teitelboim, de Sitter black holes with either of the two horizons as a boundary, Phys. Rev. D67 (2003) 104024.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Black Holes in de Sitter Space: Masses, Energies and Entropy Bounds. A Corichi, A Gomberoff, Phys. Rev. 6964016A. Corichi and A. Gomberoff, Black Holes in de Sitter Space: Masses, Energies and Entropy Bounds, Phys. Rev. D69 (2004) 064016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Thermodynamics of de Sitter black holes: Thermal cosmological constant. Y Sekiwa, Phys. Rev. 7384009Y.Sekiwa, Thermodynamics of de Sitter black holes: Thermal cosmological constant, Phys. Rev. D73 (2006) 084009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Black hole entropy is the Noether charge. R M Wald, Phys. Rev. 483427R.M.Wald, Black hole entropy is the Noether charge, Phys. Rev. D48 (1993) R3427.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Some properties of the Noether charge and a proposal for dynamical black hole entropy. V Iyer, R M Wald, Phys. Rev. 50846V.Iyer and R.M.Wald, Some properties of the Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D50 (1994) 846.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Four Laws of Black Hole Mechanics. J M Bardeen, B Carter, S W Hawking, Commun. Math. Phys. 31161J.M. Bardeen, B. Carter and S.W. Hawking, The Four Laws of Black Hole Mechanics, Commun. Math. Phys. 31 (1973) 161.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mass Formula for Kerr Black Holes. L Smarr, Phys. Rev. Lett. 3071L. Smarr, Mass Formula for Kerr Black Holes, Phys. Rev. Lett. 30 (1973) 71.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R M Wald, General Relativity. The Univ. of Chicago PressR.M.Wald, General Relativity, The Univ. of Chicago Press, 1984.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Acceleration radiation and the generalized second law of thermodynamics. W G Unruh, R M Wald, Phys. Rev. 25942W.G. Unruh and R.M. Wald, Acceleration radiation and the generalized second law of thermodynamics, Phys. Rev. D25 (1982) 942.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Entropy bounds, acceleration radiation, and the generalized second law. W G Unruh, R M Wald, Phys. Rev. 272271W.G. Unruh and R.M. Wald, Entropy bounds, acceleration radiation, and the generalized second law, Phys. Rev. D27 (1983) 2271.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Proof of Generalized Second Law for Quasistationary Semiclassical Black Holes. V P Frolov, D N Page, Phys. Rev. Lett. 713902V.P. Frolov and D.N. Page, Proof of Generalized Second Law for Quasistationary Semiclassical Black Holes, Phys. Rev. Lett. 71 (1993) 3902.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Proof of Classical Version of the Bousso Entropy Bound and of the Generalized Second Law. E E Flanagan, D Marolf, R M Wald, Phys. Rev. 6284035E.E. Flanagan, D. Marolf and R.M. Wald, Proof of Classical Version of the Bousso Entropy Bound and of the Generalized Second Law, Phys. Rev. D62 (2000) 084035.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Two-temperature Steady State Thermodynamics for a Radiation Field. H Saida, Physica. 356481H. Saida, Two-temperature Steady State Thermodynamics for a Radiation Field, Physica A356 (2005) 481.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The generalized second law and the black hole evaporation in an empty space as a nonequilibrium process. H Saida, Class. Quant. Grav. 236227H. Saida, The generalized second law and the black hole evaporation in an empty space as a nonequilibrium process, Class. Quant. Grav. 23 (2006) 6227.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Black hole evaporation as a nonequilibrium process. H Saida, arXiv:0811.1622Classical and Quantum Gravity Research. M.N. Christiansen and T.K. Rasmussen8Nova Science Publgr-qcH. Saida, Black hole evaporation as a nonequilibrium process, Chapter 8 (arXiv:0811.1622 [gr-qc]) in the edited book by M.N. Christiansen and T.K. Rasmussen, Classical and Quantum Gravity Research, Nova Science Publ., 2008.\",\n \"doi\": \"arXiv:0811.1622\"\n },\n {\n \"text\": \"Euclidean action and the thermodynamics of manifolds without boundary. G Hayward, Phys. Rev. 413248G. Hayward, Euclidean action and the thermodynamics of manifolds without boundary, Phys. Rev. D41 (1990) 3248.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Duality between Electric and Magnetic Black Holes. S W Hawking, S F Ross, Phys. Rev. 525865S.W. Hawking and S.F. Ross, Duality between Electric and Magnetic Black Holes, Phys. Rev. D52 (1995) 5865.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":65,"cells":{"text":{"kind":"string","value":"\nUniversality in Effective Strings\n4 Nov 1998 (November 17, 2021)\n\nS Jaimungal \nDepartment of Physics and Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia\n\nG W Semenoff \nDepartment of Physics and Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia\n\nK Zarembo \nDepartment of Physics and Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia\n\nUniversality in Effective Strings\n4 Nov 1998 (November 17, 2021)1127+d1125Pm1110Kk\nWe demonstrate that, due to the finite thickness of domain walls, and the consequent ambiguity in defining their locations, the effective string description obtained by integrating out bulk degrees of freedom contains ambiguities in the coefficients of the various geometric terms. The only term with unambiguous coefficient is the zeroth order Nambu-Goto term. We argue that fermionic ghost fields which implement gauge-fixing act to balance these ambiguities. The renormalized string tension, obtained after integrating out both bulk and world-sheet degrees of freedom, can be defined in a scheme independent manner; and we compute the explicit finite expressions, to one-loop, for the case of compact quantum electrodynamics and ϕ 4 theory.\n\nPACS: 11.27.+d, 11.25.Pm, 11.10.Kk A long standing problem in the physics of interfaces in three dimensional systems is to describe the interface dynamics as a theory of fluctuating surfaces analogous to an effective Euclidean string theory [1][2][3]. The interface surface can be interpreted as the world-sheet of an effective string in three-dimensions, while the two phases on either side of the interface represent the vacuum expectation values of a fundamental field. For example in the Ising model, the field is the spin operator, and the interface is the set of links about which the spin changes sign. An effective string action of this surface has a typical geometric expansion which begins with the Nambu-Goto term (the area), next the extrinsic curvature and then higher order curvature corrections. It is usually argued that the higher order corrections are irrelevant as far as the low energy dynamics is concerned, and perturbation theory consisting in keeping only the area term and extrinsic curvature is valid. However, we will demonstrate that such an ansatz is in fact ambiguous from the outset. In particular, we will demonstrate that coefficients of the higher order curvatures are completely arbitrary, and depend upon the precise prescription implemented in defining the position of the interface. Such an ambiguity is a direct consequence of the finite thickness of the interface region. In spite of this ambiguity an effective string description is possible if one includes in the action fermionic degrees of freedom which reinstate the scheme independent nature of the fundamental action.\n\nIn this work we concentrate on interfaces which occur in three-dimensional compact quantum electrodynamics (QED) [4]. The analysis can be easily applied to any other field theory in three-dimensions which has a soliton solution to the classical equations of motion. In compact QED, monopole instantons cause the electric fields between two charged particles to form a flux tube, the potential between electric charges grows linearly with the distance between them, and the charges are confined [4]. This picture of confinement is, however, a purely classi-cal one. In the full quantum theory the string of electric flux, along with the magnetic fields in the bulk, are not rigid but rather fluctuate.\n\nCompact QED can be regarded as the low-energy effective theory for the Georgi-Glashow model where the SU (2) symmetry is spontaneously broken to U (1). The monopoles appearing in the broken gauge theory are the classical solutions of the original Higgs model which have finite Euclidean action [5]. The collection of monopoles behave as a gas of charged particles interacting through a Coulomb force. Since the charges are of order √ 4π/g, where g is the gauge coupling, the monopole configurations can be treated using a semi-classical approximation in the limit of weak gauge coupling. The system then reduces to the classical thermodynamics of a Coulomb gas and the partition function is a Sine-Gordon (SG) theory [4]:\nZ = [dϕ] exp − g 2 32π 2 d 3 x (∂ϕ) 2 −2m 2 : cos ϕ :(1)\nThe monopole density is the coefficient of the cosine interaction, ζ = g 2 m 2 /32π 2 . The Grand canonical partition function for the monopoles is recovered by expanding in powers of ζ and performing the functional integral over ϕ. In the presence of the monopole solution the photon becomes massive and Wilson loop correlators obey an area law [4]. The relevant order parameter in the original gauge theory (compact QED) is the vacuum expectation value of the Wilson loop. There is a natural mapping between this correlator and the following correlation function in the SG model [4,6]:\nW (C) ≡ exp i 2 C dA QED = exp g 2 16π S ⋆dϕ SG(2)\nHere S is an arbitrary surface bounded by the contour C.\n\nThe result can be shown to be independent of the choice of this surface. We are interested in the behaviour of (2) for large loops. The contour C will be assumed to lie at infinity in the x 3 = 0 plane. In this case, it is possible to reformulate the problem. The operator on the right hand side of (2) introduces a source for the SG field or, equivalently, one may assume that ϕ experiences a jump of magnitude 2π across the surface S. Since the potential is periodic in ϕ, this jump can be eliminated by shifting ϕ on one side of the surface by 2π. This shift renders the field continuous, however, it changes the boundary conditions as x 3 → +∞. Consequently, performing such a shift on ϕ reduces (2) to an evaluation of the path integral 1 with the boundary conditions ϕ → 0 as x 3 → −∞ and ϕ → 2π as x 3 → +∞. These boundary conditions are precisely what is required for field configurations in the presence of a domain wall. The domain wall in this case describes a world sheet of a string of electric flux created by charges at infinity. The first step in obtaining an effective string action is to obtain the classical solution satisfying the appropriate boundary conditions. The solution is the SG soliton:\nϕ cl (x) = 4 arctan e mx3 ,(3)\nwhich corresponds to a domain wall in the x 3 = 0 plane. The position of this domain wall is in fact ambiguous. Conventionally, its position is given by the surface on which ϕ = π [6,7]. However, other definitions are also possible, for example, the surface on which the energy density is maximal [8,9]. These definitions, although agreeing at the classical level, do not agree once quantum corrections are included. Nevertheless, they will yield the same result within an order of m −1 . This uncertainty is due to the finite thickness of the domain wall and we will argue that this ambiguity translates into the nonuniversality of the world-sheet action. Upon inserting (3) as a classical background field in the functional integral over ϕ in (1), the collective coordinate method can be used to separate the integral over fluctuations into an integral over the domain wall position, which we describe by a height function f (x 1 , x 2 ), and over the field fluctuations in the bulk:\nZ = [dϕ][df ]∆ FP [ϕ]δ [K[f, φ]] e −SSG[ϕ](4)\nwhere\nK[f, φ] = dx 3 K( x, x 3 −f )ϕ(x)−π and ∆ FP [ϕ] = δK[f, φ]/δf is the Faddeev-Popov determinant, x is (x 1 , x 2 )\nand S SG is the same as in (1). A particular definition of the domain wall position corresponds to choosing a kernel K. There is no unique choice of this function. The definition of [6,7] corresponds to K = δ(x 3 − f ), while the standard collective coordinate method [8,9], in which the fluctuations of the domain wall are associated with quasi-zero modes in the background of the classical solution (3),\ncorresponds to K = ϕ ′ cl (x 3 − f ).\nIntegrating over ϕ in (4) yields an effective action for the coordinate f ( x) of the domain wall. In the semiclassical approximation ϕ is replaced by its classical solution. The effective action is then given by,\nS eff. = σ 0 1 + 1 2 ( ∇f ) 2 + O ( ∇f ) 2 (5)\nwhere the string tension is determined by the mass of the SG soliton [4,10]:\nσ 0 = g 2 32π 2 (∂ 3 ϕ cl ) 2 + 2m 2 cos(ϕ cl ) dx 3 = g 2 m 2π 2\nIn S eff. all higher order corrections, coming from tree level diagrams, were ignored. However, if terms which would contribute to higher curvature corrections are ignored, it is possible to re-sum an infinite subset of the terms which were previously ignored. This can be achieved by considering a domain wall solution which is curved, unfortunately, such configurations are not solutions of the SG equations of motion. They can, on the other-hand, be considered as a constraint solution [11] -the solution of the equations of motion with a source term proportional to the argument of the delta function in (4). This can be included in the action by introducing a Lagrange multiplier field. For slowly varying f this equation can be solved by perturbation theory in the derivatives of f . If only the first derivatives of f are taken into account (i.e. ignoring higher curvature corrections), the solution can be constructed by the following simple arguments [2]. When one neglects higher derivatives this implies that f is a linear function of its arguments, which corresponds to the plane domain wall rotated through an angle θ with tan θ = √ G where G ≡ 1 + ( ∇f ) 2 . The classical solution in this case is obtained from (3) by rotation:\nϕ(x) = ϕ cl ((x 3 − f )/ √ G)\n. This solution is exact in the approximation of constant ∇f , since no source term is required to produce it. Consequently, the Lagrange multiplier appears only in the next order of the derivative expansion and is proportional to ∇ 2 f . The effective action with the above solution is equal to the area of the domain wall. The rotated wall area element contains the factor 1/ cos θ = √ G, and the re-summed effective action is therefore,\nS eff = σ 0 d 2 x G ( x).(6)\nSince G is the determinant of the induced metric on the string world sheet we have obtained the Nambu-Goto action. The next term in the derivative expansion will be of order of the Lagrange multiplier squared, that is ( ∇ 2 f ) 2 , and will depend on the choice of the constraint. This arbitrariness leads to an ambiguity in the coefficient of this term, which in the covariant description corresponds to the extrinsic curvature squared. It appears that the area term (6) is the only universal part of the effective action. This is not surprising, since the parameter of the derivative expansion is k/m, where k is a momentum of the excitation on the string world sheet. Higher derivative terms become important when k ∼ m, i.e., when the wave-length of the excitation is of the same order of the thickness of the domain wall. Such excitations are of course indistinguishable from the fluctuations of the SG field in the bulk. As such, including such fluctuations in the effective string action leads to ambiguities. Of course, the computation of any physical quantity must be invariant under any choice of the constraint. It is the Faddeev-Popov determinant which cancels the ambiguities arising in the bosonic sector of the theory -the full effective action contains the fermionic ghosts coming from the determinant.\n\nThe universality of the Nambu-Goto term stems from the rotational invariance of the original model. This holds even when quantum corrections are included as long as the constraint respects rotational symmetry. Constraints which do not respect this symmetry would produce actions that are not reparametrization invariant. We have explicitly checked that when the kernel in (4) is chosen to be K( x,\nx 3 − f ) = ϕ ′ cl (x 3 − f )\n, which is not rotationally invariant, the one-loop correction to the constant and to the ( ∇f ) 2 terms in (6) disagree.\n\nLet us now consider the quantum corrections. The string tension gains quantum corrections from fluctuations of the domain wall and of the field ϕ in the bulk. To study how these two corrections are correlated we calculate the one-loop corrections to the string tension. The parameter of the loop expansion, m/g 2 , is small since m 2 is proportional to the monopole density. These corrections will be computed in the background field method starting from the classical solution (3). The oneloop corrections in 3D SG theory are linearly divergent, which leads to ambiguities in the definition of dimensional quantities like the string tension. However, the monopole gas representation implies the particular UV regularization based on the normal ordering prescription. To implement this prescription in the background field method, it is instructive to first study the one-loop corrections to the general classical solution. Expanding the quantum field as ϕ = ϕ cl + η and integrating out η, we obtain:\nS bare 1 = 1 2 Tr ln −∂ 2 + m 2 cos ϕ cl −∂ 2 + m 2 .\nThe expansion of S 1 in the powers of ϕ cl yields the usual Feynman diagrams while the normal ordering prescription consists in throwing out bubble diagrams (Fig. 1). Thus, to one-loop, the normal ordering is implemented by adding the following counter-term to the effective action: 1. Tadpole diagrams to be subtracted in the normal ordering prescription.\nS 1 = 1 2 Tr ln −∂ 2 + m 2 cos ϕ cl −∂ 2 + m 2 ✫✪ ✬✩ ❅ ❅ ♣ ♣ ♣ FIG.− m 2 2 d 3 x(cos ϕ cl − 1) d 3 p (2π) 3 1 p 2 + m 2(7)\nThis expression is free of all UV divergences. Notice that since the expansion of the renormalized effective action S 1 begins with ϕ 4 cl terms, the mass of the photon does not acquire quantum corrections at one loop.\n\nThe spectrum of the operator describing the excitations of the soliton are well known. It consists of plane waves in the x 1 and x 2 directions; while the x 3 spectrum is gapped possessing a zero mode and then a set of continuum states corresponding to the scattering of plane waves off the potential given by the solitonic solution.\n\nInserting the analytic form of the spectrum into (7) we find that the one-loop correction to the string tension is,\n∆σ 1 = 1 2 d 3 p (2π) 3 ln(p 2 + m 2 )ν(p 3 ) + 1 2 d 2 p (2π) 2 ln( p 2 ) + 2m d 3 p (2π) 3 1 p 2 + m 2 = − m 2 4π .(8)\nHere ν(p 3 ) is the difference of the density of scattering states in the presence and absence of the kink, and can be obtained using the exact eigenmodes of the linearized SG equation [12]:\nν(p 3 ) = dx 3 ψ * p3 (x 3 )ψ p3 (x 3 ) − 1 = − 2m p 2\n3 + m 2 The first term in (8) corresponds to the trace over continuum states; the second term corresponds to the quasizero mode contribution; and the last term is the counterterm prescribed by normal ordering. The sum of these three terms is UV finite, although each term diverges individually. This leaves room for some ambiguity in the final finite result depending on what regularization scheme is implemented. However, if the sum of the terms is placed under one single integral, so that the function being integrated over has finite UV behaviour, the answer must be regularization independent. The difficulty in implementing such a scheme is that the quasi-zero mode is a two dimensional integral, while all others are threedimensional. It is possible to integrate out the p 3 component of the three-dimensional integrals first, which then leaves a single two dimensional integral to perform. This is the scheme that was used in (8).\n\nThus far, we have shown that after including one-loop effects the string tension in the SG model is,\nσ = g 2 m 2π 2 1 − π 2 m g 2 .\nWe performed analogous calculation for ϕ 4 theory with the potential λ(ϕ 2 − m 2 /λ) 2 /2, the result is,\nσ = 4 3 m 3 λ 1 − 9 32π (4 − ln 3) λ m .\nAs mentioned earlier, different regularization schemes will lead to different final answers. The authors of [13] performed similar calculations on the ϕ 4 theory using the zeta-function regularization. That computation, however, did not include the integral over the quasi-zero branch of the theory, and therefore corresponds to obtaining the one-loop correction to the effective string tension rather than the one-loop renormalized string tension.\n\nExcluding that branch still leads to finite results in their regularization scheme since it is insensitive to power like divergences. Unfortunately, it yields an answer incompatible with the ansatz of first integrating out p 3 and then performing the finite integrals. This is to be expected, since the quasi-zero branch should and must contribute to the renormalized string tension.\n\nIn the preceding, all modes were including in computing the determinant. However, it is possible to integrate out only the scattering states, i.e. to omit the second term in (8), and obtain an effective action for the quasizero branch (as in [13]). This branch contains the modes responsible for shifting the surface in the x 3 direction. Omitting the second term in 8 is equivalent to keeping f fixed and integrating out only bulk modes. The one-loop correction with only bulk modes included is badly UV divergent and is not regularization independent. In the naive cut-off regularization the effective string tension for the f fields will be ∼ Λ 2 ln(Λ 2 /m 2 ) as can be easily checked from (8). Of course, hard modes (with k ≫ m) of the field f cancel this divergence, rendering the physical string tension finite. The implication is that the one-loop renormalization of the string tension is known precisely, even though all the interaction terms are not known.\n\nTo summarize we have argued that the accuracy of the macroscopic description of the confining string in compact QED is limited by the fact that the string is not infinitely thin. As a consequence the higher-derivative terms in the world-sheet action are not universal. Formally, it is possible to obtain an effective string theory action by integrating out ϕ in (4) exactly. However, the world-sheet action will be scheme-dependent and must be supplemented by the fermionic ghosts coming from the constraints. In addition, the action contains rather peculiar divergences and the finite physical quantities obtained from such an effective action appear only after delicate cancellations between these divergences and the contribution of hard modes of the string coordinates.\n\nIt is worth mentioning that the derivation of the effective domain wall action by collective coordinate method in one dimension lower would lead to essentially the same conclusions. In 2D theory, which can be thought of as a high-temperature reduction of 3D one [14], domain walls correspond to soliton paths. Solitons in 2D SG theory are known to be described by a local field theory, the Thirring model [15] and soliton operators [16] look very much like the dimensional reduction of the Wilson loops (2) [14]. Where the only difference is a local factor rendering the solitons fermionic. Fermion propogators have a well defined sum-over-path representation where the action is the supersymmetrized length of the world-line. Nevertheless, at weak SG coupling, solitons cannot be described by the world-line theory, since the four-fermion interaction in the Thirring model, which corresponds to a contact interaction in the sum-over-path picture, becomes infinitely strong.\n\nWe are grateful to K. Selivanov and A. Zhitnitsky for discussions. This work is supported in part by NSERC of Canada, a NATO Science Fellowship and, in part, by RFFI grant 98-01-00327.\n \n. D Jasnow, Rep. Prog. Phys. 471059D. Jasnow, Rep. Prog. Phys 47 (1984) 1059.\n\n. R K P Zia, Nucl. Phys. 676B251[FS13R.K.P. Zia, Nucl. Phys. B251[FS13] (1985) 676.\n\n. G Münster, Nucl. Phys. 340559G. Münster, Nucl. Phys. B340 (1990) 559;\n\nG Münster, hep- th/9802006Critical Phenomena, Strings, and Interfaces. G. Münster, Critical Phenomena, Strings, and Interfaces, hep- th/9802006.\n\n. A M Polyakov, Nucl. Phys. 120429A.M. Polyakov, Nucl. Phys. B120 (1977) 429.\n\n. G Hooft, Nucl. Phys. 79276G. 't Hooft, Nucl. Phys. B79 (1974) 276;\n\n. A M Polyakov, JETP Lett. 20194Pis'ma ZhETF 20 (1974) 430A.M. Polyakov, JETP Lett. 20 (1974) 194 [Pis'ma ZhETF 20 (1974) 430].\n\n. A M Polyakov, Nucl. Phys. 48623A.M. Polyakov, Nucl. Phys. B486 (1997) 23.\n\n. T H R Skyrme, Proc. Roy. Soc. 262556Nucl. Phys.T.H.R. Skyrme, Proc. Roy. Soc. A262 (1961) 237; , Nucl. Phys. 31 (1962) 556;\n\n. J K Perring, T H R Skyrme, Nucl. Phys. 31550J.K. Perring and T.H.R. Skyrme, Nucl. Phys. 31 (1962) 550.\n\n. J L Gervais, B Sakita, Phys. Rev. 112943J.L. Gervais and B. Sakita, Phys. Rev. D11 (1975) 2943;\n\n. J L Gervais, A Jevicki, B Sakita, Phys. Rev. 121038J.L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038.\n\n. J L Gervais, A Jevicki, B Sakita, Phys. Rev. 121038J.L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038;\n\n. E Tomboulis, Phys. Rev. 121678E. Tomboulis, Phys. Rev. D12 (1975) 1678.\n\n. N J Snyderman, Nucl. Phys. 218381N.J. Snyderman, Nucl. Phys. B218 (1983) 381.\n\n. I Affleck, Nucl. Phys. 191429I. Affleck, Nucl. Phys. B191 (1981) 429.\n\n. J Goldstone, R Jackiw, Phys. Rev. 111486J. Goldstone and R. Jackiw, Phys. Rev. D11 (1975) 1486.\n\n. R V Konoplich, Nucl. Phys. 323660R.V. Konoplich, Nucl. Phys. B323 (1989) 660.\n\n. N O Agasyan, K Zarembo, Phys. Rev. 572475N.O. Agasyan and K.Zarembo, Phys. Rev. D57 (1998) 2475.\n\n. S Coleman, Phys. Rev. 112088S. Coleman, Phys. Rev. D11 (1975) 2088.\n\n. S Mandelstam, Phys. Rev. 113026S. Mandelstam, Phys. Rev. D11 (1975) 3026.\n"},"corpus_id":{"kind":"number","value":30130722,"string":"30,130,722"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"hep-ph/9811238\",\n \"mag\": 2098855373,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1134/1.568059\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/hep-ph/9811238v1.pdf\"\n ],\n \"pdf_sha\": \"4a5e82d51db9c2ba14dc081dcbfbd4c5b1246b6a\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/hep-ph/9811238\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Universality in Effective Strings"},"authors":{"kind":"list like","value":["S Jaimungal \nDepartment of Physics and Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia","G W Semenoff \nDepartment of Physics and Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia","K Zarembo \nDepartment of Physics and Astronomy\nUniversity of British Columbia\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia"],"string":"[\n \"S Jaimungal \\nDepartment of Physics and Astronomy\\nUniversity of British Columbia\\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia\",\n \"G W Semenoff \\nDepartment of Physics and Astronomy\\nUniversity of British Columbia\\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia\",\n \"K Zarembo \\nDepartment of Physics and Astronomy\\nUniversity of British Columbia\\n6224 Agricultural RoadV6T 1Z1VancouverBritish Columbia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We demonstrate that, due to the finite thickness of domain walls, and the consequent ambiguity in defining their locations, the effective string description obtained by integrating out bulk degrees of freedom contains ambiguities in the coefficients of the various geometric terms. The only term with unambiguous coefficient is the zeroth order Nambu-Goto term. We argue that fermionic ghost fields which implement gauge-fixing act to balance these ambiguities. The renormalized string tension, obtained after integrating out both bulk and world-sheet degrees of freedom, can be defined in a scheme independent manner; and we compute the explicit finite expressions, to one-loop, for the case of compact quantum electrodynamics and ϕ 4 theory."},"sections":{"kind":"list like","value":[],"string":"[]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". D Jasnow, Rep. Prog. Phys. 471059D. Jasnow, Rep. Prog. Phys 47 (1984) 1059.","doi":""},{"text":". R K P Zia, Nucl. Phys. 676B251[FS13R.K.P. Zia, Nucl. Phys. B251[FS13] (1985) 676.","doi":""},{"text":". G Münster, Nucl. Phys. 340559G. Münster, Nucl. Phys. B340 (1990) 559;","doi":""},{"text":"G Münster, hep- th/9802006Critical Phenomena, Strings, and Interfaces. G. Münster, Critical Phenomena, Strings, and Interfaces, hep- th/9802006.","doi":""},{"text":". A M Polyakov, Nucl. Phys. 120429A.M. Polyakov, Nucl. Phys. B120 (1977) 429.","doi":""},{"text":". G Hooft, Nucl. Phys. 79276G. 't Hooft, Nucl. Phys. B79 (1974) 276;","doi":""},{"text":". A M Polyakov, JETP Lett. 20194Pis'ma ZhETF 20 (1974) 430A.M. Polyakov, JETP Lett. 20 (1974) 194 [Pis'ma ZhETF 20 (1974) 430].","doi":""},{"text":". A M Polyakov, Nucl. Phys. 48623A.M. Polyakov, Nucl. Phys. B486 (1997) 23.","doi":""},{"text":". T H R Skyrme, Proc. Roy. Soc. 262556Nucl. Phys.T.H.R. Skyrme, Proc. Roy. Soc. A262 (1961) 237; , Nucl. Phys. 31 (1962) 556;","doi":""},{"text":". J K Perring, T H R Skyrme, Nucl. Phys. 31550J.K. Perring and T.H.R. Skyrme, Nucl. Phys. 31 (1962) 550.","doi":""},{"text":". J L Gervais, B Sakita, Phys. Rev. 112943J.L. Gervais and B. Sakita, Phys. Rev. D11 (1975) 2943;","doi":""},{"text":". J L Gervais, A Jevicki, B Sakita, Phys. Rev. 121038J.L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038.","doi":""},{"text":". J L Gervais, A Jevicki, B Sakita, Phys. Rev. 121038J.L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038;","doi":""},{"text":". E Tomboulis, Phys. Rev. 121678E. Tomboulis, Phys. Rev. D12 (1975) 1678.","doi":""},{"text":". N J Snyderman, Nucl. Phys. 218381N.J. Snyderman, Nucl. Phys. B218 (1983) 381.","doi":""},{"text":". I Affleck, Nucl. Phys. 191429I. Affleck, Nucl. Phys. B191 (1981) 429.","doi":""},{"text":". J Goldstone, R Jackiw, Phys. Rev. 111486J. Goldstone and R. Jackiw, Phys. Rev. D11 (1975) 1486.","doi":""},{"text":". R V Konoplich, Nucl. Phys. 323660R.V. Konoplich, Nucl. Phys. B323 (1989) 660.","doi":""},{"text":". N O Agasyan, K Zarembo, Phys. Rev. 572475N.O. Agasyan and K.Zarembo, Phys. Rev. D57 (1998) 2475.","doi":""},{"text":". S Coleman, Phys. Rev. 112088S. Coleman, Phys. Rev. D11 (1975) 2088.","doi":""},{"text":". S Mandelstam, Phys. Rev. 113026S. Mandelstam, Phys. Rev. D11 (1975) 3026.","doi":""}],"string":"[\n {\n \"text\": \". D Jasnow, Rep. Prog. Phys. 471059D. Jasnow, Rep. Prog. Phys 47 (1984) 1059.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R K P Zia, Nucl. Phys. 676B251[FS13R.K.P. Zia, Nucl. Phys. B251[FS13] (1985) 676.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Münster, Nucl. Phys. 340559G. Münster, Nucl. Phys. B340 (1990) 559;\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G Münster, hep- th/9802006Critical Phenomena, Strings, and Interfaces. G. Münster, Critical Phenomena, Strings, and Interfaces, hep- th/9802006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A M Polyakov, Nucl. Phys. 120429A.M. Polyakov, Nucl. Phys. B120 (1977) 429.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Hooft, Nucl. Phys. 79276G. 't Hooft, Nucl. Phys. B79 (1974) 276;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A M Polyakov, JETP Lett. 20194Pis'ma ZhETF 20 (1974) 430A.M. Polyakov, JETP Lett. 20 (1974) 194 [Pis'ma ZhETF 20 (1974) 430].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A M Polyakov, Nucl. Phys. 48623A.M. Polyakov, Nucl. Phys. B486 (1997) 23.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T H R Skyrme, Proc. Roy. Soc. 262556Nucl. Phys.T.H.R. Skyrme, Proc. Roy. Soc. A262 (1961) 237; , Nucl. Phys. 31 (1962) 556;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J K Perring, T H R Skyrme, Nucl. Phys. 31550J.K. Perring and T.H.R. Skyrme, Nucl. Phys. 31 (1962) 550.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J L Gervais, B Sakita, Phys. Rev. 112943J.L. Gervais and B. Sakita, Phys. Rev. D11 (1975) 2943;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J L Gervais, A Jevicki, B Sakita, Phys. Rev. 121038J.L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J L Gervais, A Jevicki, B Sakita, Phys. Rev. 121038J.L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D12 (1975) 1038;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Tomboulis, Phys. Rev. 121678E. Tomboulis, Phys. Rev. D12 (1975) 1678.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N J Snyderman, Nucl. Phys. 218381N.J. Snyderman, Nucl. Phys. B218 (1983) 381.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Affleck, Nucl. Phys. 191429I. Affleck, Nucl. Phys. B191 (1981) 429.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Goldstone, R Jackiw, Phys. Rev. 111486J. Goldstone and R. Jackiw, Phys. Rev. D11 (1975) 1486.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R V Konoplich, Nucl. Phys. 323660R.V. Konoplich, Nucl. Phys. B323 (1989) 660.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". N O Agasyan, K Zarembo, Phys. Rev. 572475N.O. Agasyan and K.Zarembo, Phys. Rev. D57 (1998) 2475.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Coleman, Phys. Rev. 112088S. Coleman, Phys. Rev. D11 (1975) 2088.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Mandelstam, Phys. Rev. 113026S. Mandelstam, Phys. Rev. D11 (1975) 3026.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":66,"cells":{"text":{"kind":"string","value":"\nA Generic Approach to Detect Design Patterns in Model Transformations Using a String-Matching Algorithm\n\n\nChihab Eddine Mokaddem \nHouari Sahraoui \nEugene Syriani \nA Generic Approach to Detect Design Patterns in Model Transformations Using a String-Matching Algorithm\nReceived: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)Design pattern · Model transformation · Pattern detection · String matching · Bit-vector · Model-driven engineering\nMaintaining software artifacts is among the hardest tasks an engineer faces. Like any other piece of code, model transformations developed by engineers are also subject to maintenance. To facilitate the comprehension of programs, software engineers rely on many techniques, such as design pattern detection. Therefore, detecting design patterns in model transformation implementations is of tremendous value for developers. In this paper, we propose a generic technique to detect design patterns and their variations in model transformation implementations automatically. It takes as input a set of model transformation rules and the participants of a model transformation design pattern to find occurrences of the latter in the former. The technique also detects certain kinds of degenerate forms of the pattern, thus indicating potential opportunities to improve the model transformation implementation.\n\nIntroduction\n\nModel transformation is now the mainstream paradigm to manipulate models in model-driven software engineering (MDE) [6]. Designing model transformations is a tedious task. Moreover, like any other code artifact, model transformations evolve and should be maintained. To assist developers in writing and maintaining model transformations, several design patterns have been proposed [11,22]. These presumably facilitate the comprehension and manipulation of transformation programs [1].\n\nDetecting instances of a pattern in a transformation provides valuable information to the developer, such as understanding high-level concepts used, and identifying refactoring and reuse opportunities. This helps, among others, improving the documentation. However, as for general programs, developers do not always implement perfectly a pattern in model transformations. Hence, design pattern detection should identify both complete and incomplete occurrences.\n\nDetecting design patterns in model transformations did not get much attention so far from the modeling community. To the best of our knowledge, only the work in [28] has attempted to automatically detect design patterns in model transformations. Preliminary results showed that this is an effective technique to find complete and approximate design pattern occurrences. However, this technique has performance limitations as it relies on a rule inference engine that is time and memory consuming. Another limitation of this technique is the need to specify a set of detection rules for each pattern.\n\nTo find inspiration on how to detect patterns in transformations, we looked at the active community of design pattern detection in object-oriented programs. As reported in [1], there are dozens of detection approaches for this family of programs. However, as mentioned in [13], these approaches also suffer from performance problems, because detecting complete and incomplete occurrences is generally costly in time, due to the large search-space that includes all possible combinations of classes. These approaches are also prone to return many false positives, impeding program comprehension, and cluttering the maintainers' cognitive capabilities. To address the performance issues, the work by Kaczor et al. [20] uses a string matching technique inspired by pattern matching algorithms in bioinformatics to identify pattern occurrences in object-oriented programs. These algorithms allow to efficiently process a large amount of data if the problem to solve can be encoded as a string matching one.\n\nIn this paper, we propose a generic technique to detect design pattern occurrences in model transformation implementations, without writing detection code for each design pattern, its variants, and approximations. Like in Kaczor et al., we rely on a bit-vector algorithm that has proven to be efficient for string matching problems [30]. The challenge we faced is how to encode model transformations, which are sets of rules linked by control schemes, as strings. The same challenge arises also in the encoding of the patterns as strings. We succeed to encode the participants of a patterns as strings, but had to complete our approach by a manual step to combine the identified participant instances to form pattern occurrences. Thus, the detection consists in an automated step that matches the participant strings of a pattern with rule strings of transformation, and a manual step to complete the occurrences. In addition to the performance, an advantage of using this approach is the fact both complete and incomplete occurrences can be detected.\n\nWe evaluated our approach on a set of 18 transformations. Our results show that patterns are actually used in transformations and that our approach is able to detect them. Moreover, we found that patterns are not always used independently, but in combination with other patterns.\n\nThe rest of the paper is structured as follows. In Section 2, we first introduce the basic notions used in our work, and then, discuss the related work. Section 3 details the different steps of our approach, whereas Section 4 describes the different forms of patterns that can be detected by our approach. We provide an evaluation of the detection approach in Section 5. Finally, we discuss the limits of our approach in Section 6, and conclude in Section 7.\n \nA survey on design pattern detection approaches. M Al-Obeidallah, M Petridis, S Kapetanakis, International Journal of Software Engineering. 7Al-Obeidallah, M., Petridis, M., Kapetanakis, S.: A survey on design pattern detection approaches. International Journal of Software Engineering 7, 41-59 (2016)\n\nRule-based detection of design patterns in program code. A Alnusair, T Zhao, G Yan, 10.1007/s10009-013-0292-zSTTT. 163Alnusair, A., Zhao, T., Yan, G.: Rule-based detection of design patterns in program code. STTT 16(3), 315-334 (2014). DOI 10.1007/s10009-013-0292-z. URL https://doi.org/ 10.1007/s10009-013-0292-z\n\nDesign pattern recovery in object-oriented software. G Antoniol, R Fiutem, L Cristoforetti, 10.1109/WPC.1998.6933426th International Workshop on Program Comprehension (IWPC '98). Ischia, ItalyAntoniol, G., Fiutem, R., Cristoforetti, L.: Design pattern recovery in object-oriented software. In: 6th International Workshop on Program Comprehension (IWPC '98), June 24-26, 1998, Ischia, Italy, pp. 153-160 (1998). DOI 10.1109/WPC.1998.693342. URL https://doi.org/10.1109/WPC.1998.693342\n\nDesign pattern detection in java systems: A dynamic analysis based approach. F Arcelli, F Perin, C Raibulet, S Ravani, 10.1007/978-3-642-14819-4_12Evaluation of Novel Approaches to Software Engineering -3rd and 4th International Conferences. Madeira, Portugal; Milan, ItalyRevised Selected PapersArcelli, F., Perin, F., Raibulet, C., Ravani, S.: Design pattern detection in java sys- tems: A dynamic analysis based approach. In: Evaluation of Novel Approaches to Soft- ware Engineering -3rd and 4th International Conferences, ENASE 2008/2009, Fun- chal, Madeira, Portugal, May 4-7, 2008 / Milan, Italy, May 9-10, 2009. Revised Se- lected Papers, pp. 163-179 (2009). DOI 10.1007/978-3-642-14819-4\\ 12. URL https: //doi.org/10.1007/978-3-642-14819-4_12\n\nHenshin: Advanced concepts and tools for in-place EMF model transformations. T Arendt, E Biermann, S Jurack, C Krause, G Taentzer, 10.1007/978-3-642-16145-2_9Model Driven Engineering Languages and Systems -13th International Conference, MODELS 2010. D.C. Petriu, N. Rouquette, Ø. HaugenOslo, NorwaySpringer6394Proceedings, Part IArendt, T., Biermann, E., Jurack, S., Krause, C., Taentzer, G.: Henshin: Advanced con- cepts and tools for in-place EMF model transformations. In: D.C. Petriu, N. Rouquette, Ø. Haugen (eds.) Model Driven Engineering Languages and Systems -13th Interna- tional Conference, MODELS 2010, Oslo, Norway, October 3-8, 2010, Proceedings, Part I, Lecture Notes in Computer Science, vol. 6394, pp. 121-135. Springer (2010). DOI 10.1007/978-3-642-16145-2\\ 9. URL https://doi.org/10.1007/978-3-642-16145-2_9\n\nSystematic mapping study of model transformations for concrete problems. E Batot, H A Sahraoui, E Syriani, P Molins, W Sboui, 10.5220/0005657301760183MODELSWARD 2016 -Proceedings of the 4rd International Conference on Model-Driven Engineering and Software Development. S. Hammoudi, L.F. Pires, B. Selic, P. DesfrayRome, ItalySciTePressBatot, E., Sahraoui, H.A., Syriani, E., Molins, P., Sboui, W.: Systematic mapping study of model transformations for concrete problems. In: S. Hammoudi, L.F. Pires, B. Selic, P. Desfray (eds.) MODELSWARD 2016 -Proceedings of the 4rd International Conference on Model-Driven Engineering and Software Development, Rome, Italy, 19-21 February, 2016, pp. 176-183. SciTePress (2016). DOI 10.5220/0005657301760183. URL https: //doi.org/10.5220/0005657301760183\n\nVector algorithms for approximate string matching. A Bergeron, S Hamel, 10.1142/S0129054102000947Int. J. Found. Comput. Sci. 131Bergeron, A., Hamel, S.: Vector algorithms for approximate string matching. Int. J. Found. Comput. Sci. 13(1), 53-66 (2002). DOI 10.1142/S0129054102000947. URL https://doi. org/10.1142/S0129054102000947\n\nLinear Programming and Extensions. G B Dantizig, Princeton University PressDantizig, G.B.: Linear Programming and Extensions. Princeton University Press (1991). URL http://www.jstor.org/stable/j.ctt1cx3tvg\n\nCombining clustering and pattern detection for the reengineering of component-based software systems. M Von Detten, S Becker, 10.1145/2000259.20002657th International Conference on the Quality of Software Architectures, QoSA 2011 and 2nd International Symposium on Architecting Critical Systems, ISARCS 2011. Boulder, CO, USAvon Detten, M., Becker, S.: Combining clustering and pattern detection for the reengineer- ing of component-based software systems. In: 7th International Conference on the Quality of Software Architectures, QoSA 2011 and 2nd International Symposium on Architecting Critical Systems, ISARCS 2011. Boulder, CO, USA, June 20-24, 2011, Proceedings, pp. 23-32 (2011). DOI 10.1145/2000259.2000265. URL https://doi.org/10.1145/2000259. 2000265\n\nDp-miner: Design pattern discovery using matrix. J Dong, D S Lad, Y Zhao, 14th Annual IEEE International Conference and Workshop on Engineering of Computer Based Systems. Tucson, Arizona, USADong, J., Lad, D.S., Zhao, Y.: Dp-miner: Design pattern discovery using matrix. In: 14th Annual IEEE International Conference and Workshop on Engineering of Computer Based Systems (ECBS 2007), 26-29 March 2007, Tucson, Arizona, USA, pp. 371-380 (2007).\n\n. 10.1109/ECBS.2007.33DOI 10.1109/ECBS.2007.33. URL https://doi.org/10.1109/ECBS.2007.33\n\nDesign pattern oriented development of model transformations. H Ergin, E Syriani, J Gray, 10.1016/j.cl.2016.07.004Computer Languages, Systems & Structures. 46Ergin, H., Syriani, E., Gray, J.: Design pattern oriented development of model transfor- mations. Computer Languages, Systems & Structures 46, 106-139 (2016). DOI http:// dx.doi.org/10.1016/j.cl.2016.07.004. URL https://doi.org/10.1016/j.cl.2016.07.004\n\nColumbus -reverse engineering tool and schema for C++. R Ferenc, Á Beszédes, M Tarkiainen, T Gyimóthy, 10.1109/ICSM.2002.116776418th International Conference on Software Maintenance (ICSM 2002), Maintaining Distributed Heterogeneous Systems. Montreal, Quebec, CanadaFerenc, R., Beszédes,Á., Tarkiainen, M., Gyimóthy, T.: Columbus -reverse engineer- ing tool and schema for C++. In: 18th International Conference on Software Mainte- nance (ICSM 2002), Maintaining Distributed Heterogeneous Systems, 3-6 October 2002, Montreal, Quebec, Canada, pp. 172-181 (2002). DOI 10.1109/ICSM.2002.1167764. URL https://doi.org/10.1109/ICSM.2002.1167764\n\nImproving design-pattern identification: a new approach and an exploratory study. Y Guéhéneuc, J Guyomarc'h, H A Sahraoui, 10.1007/s11219-009-9082-ySoftw. Qual. J. 181Guéhéneuc, Y., Guyomarc'h, J., Sahraoui, H.A.: Improving design-pattern identification: a new approach and an exploratory study. Softw. Qual. J. 18(1), 145-174 (2010). DOI 10.1007/s11219-009-9082-y. URL https://doi.org/10.1007/s11219-009-9082-y\n\nFingerprinting design patterns. Y Guéhéneuc, H A Sahraoui, F Zaidi, 10.1109/WCRE.2004.2111th Working Conference on Reverse Engineering. Delft, The NetherlandsGuéhéneuc, Y., Sahraoui, H.A., Zaidi, F.: Fingerprinting design patterns. In: 11th Working Conference on Reverse Engineering, WCRE 2004, Delft, The Netherlands, November 8-12, 2004, pp. 172-181 (2004). DOI 10.1109/WCRE.2004.21. URL https://doi.org/10.1109/ WCRE.2004.21\n\nSimulation of NFA in approximate string and sequence matching. J Holub, Proceedings of the Prague Stringology Club Workshop. the Prague Stringology Club WorkshopPrague, Czech RepublicDepartment of Computer Science and Engineering, Faculty of Electrical Engineering, Czech Technical UniversityHolub, J.: Simulation of NFA in approximate string and sequence matching. In: Proceed- ings of the Prague Stringology Club Workshop 1997, Prague, Czech Republic, July 7, 1997, pp. 39-46. Department of Computer Science and Engineering, Faculty of Electrical En- gineering, Czech Technical University (1997). URL http://www.stringology.org/event/ 1997/p5.html\n\nReusable model transformation patterns. M Iacob, M W A Steen, L Heerink, 10.1109/EDOCW.2008.51Workshops Proceedings of the 12th International IEEE Enterprise Distributed Object Computing Conference. Munich, GermanyIacob, M., Steen, M.W.A., Heerink, L.: Reusable model transformation patterns. In: Workshops Proceedings of the 12th International IEEE Enterprise Distributed Object Computing Conference, ECOCW 2008, 16 September 2008, Munich, Germany, pp. 1-10 (2008). DOI 10.1109/EDOCW.2008.51. URL https://doi.org/10.1109/EDOCW.2008.51\n\nAbstracting complex languages through transformation and composition. J Johannes, S Zschaler, M A Fernández, A Castillo, D S Kolovos, R F Paige, 10.1007/978-3-642-04425-0_41Proceedings, Lecture Notes in Computer Science. A. Schürr, B. SelicLecture Notes in Computer ScienceDenver, CO, USASpringer579512th International ConferenceJohannes, J., Zschaler, S., Fernández, M.A., Castillo, A., Kolovos, D.S., Paige, R.F.: Ab- stracting complex languages through transformation and composition. In: A. Schürr, B. Selic (eds.) Model Driven Engineering Languages and Systems, 12th International Conference, MODELS 2009, Denver, CO, USA, October 4-9, 2009. Proceedings, Lec- ture Notes in Computer Science, vol. 5795, pp. 546-550. Springer (2009). DOI 10.1007/978-3-642-04425-0\\ 41. URL https://doi.org/10.1007/978-3-642-04425-0_41\n\nATL: A model transformation tool. F Jouault, F Allilaire, J Bézivin, I Kurtev, 10.1016/j.scico.2007.08.002Sci. Comput. Program. 721-2Jouault, F., Allilaire, F., Bézivin, J., Kurtev, I.: ATL: A model transformation tool. Sci. Comput. Program. 72(1-2), 31-39 (2008). DOI 10.1016/j.scico.2007.08.002. URL https: //doi.org/10.1016/j.scico.2007.08.002\n\nSolving the TTC 2011 reengineering case with henshin. S Jurack, J Tietje, 10.4204/EPTCS.74.17Proceedings Fifth Transformation Tool Contest, TTC 2011. Fifth Transformation Tool Contest, TTC 2011Zürich, SwitzerlandJurack, S., Tietje, J.: Solving the TTC 2011 reengineering case with henshin. In: Pro- ceedings Fifth Transformation Tool Contest, TTC 2011, Zürich, Switzerland, June 29-30 2011., pp. 181-203 (2011). DOI 10.4204/EPTCS.74.17. URL https://doi.org/10.4204/ EPTCS.74.17\n\nIdentification of design motifs with pattern matching algorithms. O Kaczor, Y Guéhéneuc, S Hamel, 10.1016/j.infsof.2009.08.006Information & Software Technology. 522Kaczor, O., Guéhéneuc, Y., Hamel, S.: Identification of design motifs with pattern match- ing algorithms. Information & Software Technology 52(2), 152-168 (2010). DOI 10.1016/j.infsof.2009.08.006. URL https://doi.org/10.1016/j.infsof.2009.08.006\n\nDesign recovery by automated search for structural design patterns in object-oriented software. C Kramer, L Prechelt, 10.1109/WCRE.1996.558905Proceedings of WCRE '96: 4rd Working Conference on Reverse Engineering. WCRE '96: 4rd Working Conference on Reverse EngineeringKramer, C., Prechelt, L.: Design recovery by automated search for structural design pat- terns in object-oriented software. In: Proceedings of WCRE '96: 4rd Working Conference on Reverse Engineering, pp. 208-215 (1996). DOI 10.1109/WCRE.1996.558905\n\nModel-transformation design patterns. K Lano, S K Rahimi, 10.1109/TSE.2014.2354344IEEE Transactions on Software Engineering. 4012Lano, K., Rahimi, S.K.: Model-transformation design patterns. IEEE Transactions on Software Engineering 40(12), 1224-1259 (2014). DOI 10.1109/TSE.2014.2354344. URL https://doi.org/10.1109/TSE.2014.2354344\n\nA survey of model transformation design pattern usage. K Lano, S K Rahimi, S Y Tehrani, M Sharbaf, Theory and Practice of Model Transformation -10th Internat. Lano, K., Rahimi, S.K., Tehrani, S.Y., Sharbaf, M.: A survey of model transformation design pattern usage. In: Theory and Practice of Model Transformation -10th Internat.\n\n10.1007/978-3-319-61473-1_8Conf, ICMT 2017, Proceedings. Springer10374Conf, ICMT 2017, Proceedings, LNCS, vol. 10374, pp. 108-118. Springer (2017). DOI 10.1007/978-3-319-61473-1\\ 8. URL https://doi.org/10.1007/978-3-319-61473-1_8\n\nA survey of model transformation design patterns in practice. K Lano, S K Rahimi, S Y Tehrani, M Sharbaf, 10.1016/j.jss.2018.03.001Journal of Systems and Software. 140Lano, K., Rahimi, S.K., Tehrani, S.Y., Sharbaf, M.: A survey of model transformation design patterns in practice. Journal of Systems and Software 140, 48-73 (2018). DOI 10.1016/j.jss.2018.03.001. URL https://doi.org/10.1016/j.jss.2018.03.001\n\nDesign pattern recovery through visual language parsing and source code analysis. A D Lucia, V Deufemia, C Gravino, M Risi, 10.1016/j.jss.2009.02.012Journal of Systems and Software. 827Lucia, A.D., Deufemia, V., Gravino, C., Risi, M.: Design pattern recovery through visual language parsing and source code analysis. Journal of Systems and Software 82(7), 1177- 1193 (2009). DOI 10.1016/j.jss.2009.02.012. URL https://doi.org/10.1016/j.jss.2009. 02.012\n\nDesign pattern detection based on the graph theory. B B Mayvan, A Rasoolzadegan, 10.1016/j.knosys.2017.01.007120Knowledge-Based SystemsMayvan, B.B., Rasoolzadegan, A.: Design pattern detection based on the graph theory. Knowledge-Based Systems 120, 211-225 (2017). DOI 10.1016/j.knosys.2017.01.007. URL https://doi.org/10.1016/j.knosys.2017.01.007\n\nDesign pattern detection using ontology. M Y Mhawish, M Gupta, Mhawish, M.Y., Gupta, M.: Design pattern detection using ontology (2018)\n\nTowards rule-based detection of design patterns in model transformations. C Mokaddem, H Sahraoui, E Syriani, 10.1007/978-3-319-46613-2_14System Analysis and Modeling. Technology-Specific Aspects of Models -9th International Conference. SAM 2016, Saint-Melo, France; Saint-MaloSpringer9959LNCSMokaddem, C., Sahraoui, H., Syriani, E.: Towards rule-based detection of design patterns in model transformations. In: System Analysis and Modeling. Technology-Specific Aspects of Models -9th International Conference, SAM 2016, Saint-Melo, France, October 3-4, 2016, Proceedings, LNCS, vol. 9959, pp. 211-225. Springer, Saint-Malo (2016). DOI 10.1007/978-3-319-46613-2\\ 14. URL https://doi.org/10.1007/978-3-319-46613-2_14\n\nRecommending model refactoring rules from refactoring examples. C Mokaddem, H Sahraoui, E Syriani, 10.1145/3239372.3239406Proceedings of the 21th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems. A. Wasowski, R.F. Paige, Ø. Haugenthe 21th ACM/IEEE International Conference on Model Driven Engineering Languages and SystemsCopenhagen, DenmarkACMMokaddem, C., Sahraoui, H., Syriani, E.: Recommending model refactoring rules from refactoring examples. In: A. Wasowski, R.F. Paige, Ø. Haugen (eds.) Proceedings of the 21th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems, MODELS 2018, Copenhagen, Denmark, October 14-19, 2018, pp. 257-266. ACM (2018). DOI 10.1145/3239372.3239406. URL https://doi.org/10.1145/3239372.3239406\n\nA fast bit-vector algorithm for approximate string matching based on dynamic programming. G Myers, 10.1145/316542.316550Journal of the ACM. 463Myers, G.: A fast bit-vector algorithm for approximate string matching based on dynamic programming. Journal of the ACM 46(3), 395-415 (1999). DOI 10.1145/316542.316550. URL https://doi.org/10.1145/316542.316550\n\nA general method applicable to the search for similarities in the amino acid sequence of two proteins. S B Needleman, C D Wunsch, 10.1016/0022-2836(70)90057-4.URLhttp:/www.sciencedirect.com/science/article/pii/0022283670900574Journal of Molecular Biology. 483Needleman, S.B., Wunsch, C.D.: A general method applicable to the search for similarities in the amino acid sequence of two proteins. Journal of Molecular Biology 48(3), 443 -453 (1970). DOI https://doi.org/10.1016/0022-2836(70)90057-4. URL http://www. sciencedirect.com/science/article/pii/0022283670900574\n\nTowards pattern-based design recovery. J Niere, W Schäfer, J P Wadsack, L Wendehals, J Welsh, 10.1145/581339.581382Proceedings of the 24th International Conference on Software Engineering, ICSE. the 24th International Conference on Software Engineering, ICSEOrlando, Florida, USANiere, J., Schäfer, W., Wadsack, J.P., Wendehals, L., Welsh, J.: Towards pattern-based design recovery. In: Proceedings of the 24th International Conference on Software Engi- neering, ICSE 2002, 19-25 May 2002, Orlando, Florida, USA, pp. 338-348 (2002). DOI 10.1145/581339.581382. URL https://doi.org/10.1145/581339.581382\n\nFull contract verification for ATL using symbolic execution. B J Oakes, J Troya, L Lúcio, M Wimmer, 10.1007/s10270-016-0548-710.1007/ s10270-016-0548-7Software and Systems Modeling. 173Oakes, B.J., Troya, J., Lúcio, L., Wimmer, M.: Full contract verification for ATL using symbolic execution. Software and Systems Modeling 17(3), 815-849 (2018). DOI 10.1007/ s10270-016-0548-7. URL https://doi.org/10.1007/s10270-016-0548-7\n\nSoftware metrics by architectural pattern mining. J Paakki, A Karhinen, J Gustafsson, L Nenonen, A Verkamo, Proceedings of the International Conference on Software: Theory and Practice. the International Conference on Software: Theory and PracticePaakki, J., Karhinen, A., Gustafsson, J., Nenonen, L., Verkamo, A.: Software metrics by architectural pattern mining. Proceedings of the International Conference on Software: Theory and Practice (16th IFIP World Computer Congress) (2000)\n\nA comparative study on string matching algorithm of biological sequences. P Pandiselvam, T Marimuthu, R Lawrance, CoRR) abs/1401.7416Computing Research. Pandiselvam, P., Marimuthu, T., Lawrance, R.: A comparative study on string matching al- gorithm of biological sequences. Computing Research Repository (CoRR) abs/1401.7416 (2014). URL http://arxiv.org/abs/1401.7416\n\nAn approach for reverse engineering of design patterns. I Philippow, D Streitferdt, M Riebisch, S Naumann, 10.1007/s10270-004-0059-9Software and Systems Modeling. 41Philippow, I., Streitferdt, D., Riebisch, M., Naumann, S.: An approach for reverse engi- neering of design patterns. Software and Systems Modeling 4(1), 55-70 (2005). DOI 10.1007/s10270-004-0059-9. URL https://doi.org/10.1007/s10270-004-0059-9\n\nFrom secure business process modeling to design-level security verification. Q Ramadan, M Salnitri, D Strüber, J Jürjens, P Giorgini, 10.1109/MODELS.2017.1020th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems. Austin, TX, USAIEEE Computer SocietyRamadan, Q., Salnitri, M., Strüber, D., Jürjens, J., Giorgini, P.: From secure business process modeling to design-level security verification. In: 20th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems, MODELS 2017, Austin, TX, USA, September 17-22, 2017, pp. 123-133. IEEE Computer Society (2017). DOI 10.1109/MODELS.2017.10. URL https://doi.org/10.1109/MODELS.2017.10\n\nDesign pattern recovery based on annotations. G Rasool, I Philippow, P Mäder, 10.1016/j.advengsoft.2009.10.014Advances in Engineering Software. 414Rasool, G., Philippow, I., Mäder, P.: Design pattern recovery based on annotations. Ad- vances in Engineering Software 41(4), 519-526 (2010). DOI 10.1016/j.advengsoft.2009.10. 014. URL https://doi.org/10.1016/j.advengsoft.2009.10.014\n\nTranslation of ATL to AGT and application to a code generator for simulink. Software and Systems Modeling. E Richa, E Borde, L Pautet, 10.1007/s10270-017-0607-818Richa, E., Borde, E., Pautet, L.: Translation of ATL to AGT and application to a code generator for simulink. Software and Systems Modeling 18(1), 321-344 (2019). DOI 10.1007/s10270-017-0607-8. URL https://doi.org/10.1007/s10270-017-0607-8\n\nReverse engineering of design patterns from java source code. N Shi, R A Olsson, 10.1109/ASE.2006.5721st IEEE/ACM International Conference on Automated Software Engineering (ASE 2006). Tokyo, JapanShi, N., Olsson, R.A.: Reverse engineering of design patterns from java source code. In: 21st IEEE/ACM International Conference on Automated Software Engineering (ASE 2006), 18-22 September 2006, Tokyo, Japan, pp. 123-134 (2006). DOI 10.1109/ASE.2006.57. URL https://doi.org/10.1109/ASE.2006.57\n\nIdentification of common molecular subsequences. T Smith, M Waterman, 10.1016/0022-2836(81)90087-5.URLhttp:/www.sciencedirect.com/science/article/pii/0022283681900875Journal of Molecular Biology. 1471Smith, T., Waterman, M.: Identification of common molecular subsequences. Journal of Molecular Biology 147(1), 195 -197 (1981). DOI https://doi.org/10.1016/0022-2836(81) 90087-5. URL http://www.sciencedirect.com/science/article/pii/0022283681900875\n\nG Srinivasan, Operations Research : Principles and Applications. Prentice-Hall of IndiaSrinivasan, G.: Operations Research : Principles and Applications. Prentice-Hall of India (2010). URL https://books.google.ca/books?id=wYfxffB62NUC\n\nA Modular Timed Graph Transformation Language for Simulation-based Design. E Syriani, H Vangheluwe, 10.1007/s10270-011-0205-0Software & Systems Modeling. 122Syriani, E., Vangheluwe, H.: A Modular Timed Graph Transformation Language for Simulation-based Design. Software & Systems Modeling 12(2), 387-414 (2013). DOI 10.1007/s10270-011-0205-0. URL https://doi.org/10.1007/s10270-011-0205-0\n\nDesign pattern detection using similarity scoring. N Tsantalis, A Chatzigeorgiou, G Stephanides, S T Halkidis, 10.1109/TSE.2006.11210.1109/ TSE.2006.112IEEE Trans. Software Eng. 3211Tsantalis, N., Chatzigeorgiou, A., Stephanides, G., Halkidis, S.T.: Design pattern detection using similarity scoring. IEEE Trans. Software Eng. 32(11), 896-909 (2006). DOI 10.1109/ TSE.2006.112. URL https://doi.org/10.1109/TSE.2006.112\n\nDetecting design patterns in objectoriented program source code by using metrics and machine learning. S Uchiyama, A Kubo, H Washizaki, Y Fukazawa, 10.4236/ jsea.2014.712086Uchiyama, S., Kubo, A., Washizaki, H., Fukazawa, Y.: Detecting design patterns in object- oriented program source code by using metrics and machine learning (2014). DOI 10.4236/ jsea.2014.712086\n\nA comprehensive approach to the recovery of design pattern instances based on sub-patterns and method signatures. D Yu, Y Zhang, Z Chen, 10.1016/j.jss.2015.01.019Journal of Systems and Software. 103Yu, D., Zhang, Y., Chen, Z.: A comprehensive approach to the recovery of design pattern instances based on sub-patterns and method signatures. Journal of Systems and Software 103, 1-16 (2015). DOI 10.1016/j.jss.2015.01.019. URL https://doi.org/10.1016/j.jss. 2015.01.019\n\nOn applying machine learning techniques for design pattern detection. M Zanoni, F A Fontana, F Stella, 10.1016/j.jss.2015.01.03710.1016/ j.jss.2015.01.037Journal of Systems and Software. 103Zanoni, M., Fontana, F.A., Stella, F.: On applying machine learning techniques for design pattern detection. Journal of Systems and Software 103, 102-117 (2015). DOI 10.1016/ j.jss.2015.01.037. URL https://doi.org/10.1016/j.jss.2015.01.037\n"},"corpus_id":{"kind":"number","value":222291604,"string":"222,291,604"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2010.04759\",\n \"mag\": 3092605954,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/sosym/MokaddemSS22\",\n \"doi\": \"10.1007/s10270-021-00936-4\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2010.04759v1.pdf\"\n ],\n \"pdf_sha\": \"48610488cfcc158b7da7a8ddcd415c61f1b00c89\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/2010.04759\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"A Generic Approach to Detect Design Patterns in Model Transformations Using a String-Matching Algorithm"},"authors":{"kind":"list like","value":["Chihab Eddine Mokaddem","Houari Sahraoui","Eugene Syriani"],"string":"[\n \"Chihab Eddine Mokaddem\",\n \"Houari Sahraoui\",\n \"Eugene Syriani\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Maintaining software artifacts is among the hardest tasks an engineer faces. Like any other piece of code, model transformations developed by engineers are also subject to maintenance. To facilitate the comprehension of programs, software engineers rely on many techniques, such as design pattern detection. Therefore, detecting design patterns in model transformation implementations is of tremendous value for developers. In this paper, we propose a generic technique to detect design patterns and their variations in model transformation implementations automatically. It takes as input a set of model transformation rules and the participants of a model transformation design pattern to find occurrences of the latter in the former. The technique also detects certain kinds of degenerate forms of the pattern, thus indicating potential opportunities to improve the model transformation implementation."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Model transformation is now the mainstream paradigm to manipulate models in model-driven software engineering (MDE) [6]. Designing model transformations is a tedious task. Moreover, like any other code artifact, model transformations evolve and should be maintained. To assist developers in writing and maintaining model transformations, several design patterns have been proposed [11,22]. These presumably facilitate the comprehension and manipulation of transformation programs [1].\nDetecting instances of a pattern in a transformation provides valuable information to the developer, such as understanding high-level concepts used, and identifying refactoring and reuse opportunities. This helps, among others, improving the documentation. However, as for general programs, developers do not always implement perfectly a pattern in model transformations. Hence, design pattern detection should identify both complete and incomplete occurrences.\nDetecting design patterns in model transformations did not get much attention so far from the modeling community. To the best of our knowledge, only the work in [28] has attempted to automatically detect design patterns in model transformations. Preliminary results showed that this is an effective technique to find complete and approximate design pattern occurrences. However, this technique has performance limitations as it relies on a rule inference engine that is time and memory consuming. Another limitation of this technique is the need to specify a set of detection rules for each pattern.\nTo find inspiration on how to detect patterns in transformations, we looked at the active community of design pattern detection in object-oriented programs. As reported in [1], there are dozens of detection approaches for this family of programs. However, as mentioned in [13], these approaches also suffer from performance problems, because detecting complete and incomplete occurrences is generally costly in time, due to the large search-space that includes all possible combinations of classes. These approaches are also prone to return many false positives, impeding program comprehension, and cluttering the maintainers' cognitive capabilities. To address the performance issues, the work by Kaczor et al. [20] uses a string matching technique inspired by pattern matching algorithms in bioinformatics to identify pattern occurrences in object-oriented programs. These algorithms allow to efficiently process a large amount of data if the problem to solve can be encoded as a string matching one.\nIn this paper, we propose a generic technique to detect design pattern occurrences in model transformation implementations, without writing detection code for each design pattern, its variants, and approximations. Like in Kaczor et al., we rely on a bit-vector algorithm that has proven to be efficient for string matching problems [30]. The challenge we faced is how to encode model transformations, which are sets of rules linked by control schemes, as strings. The same challenge arises also in the encoding of the patterns as strings. We succeed to encode the participants of a patterns as strings, but had to complete our approach by a manual step to combine the identified participant instances to form pattern occurrences. Thus, the detection consists in an automated step that matches the participant strings of a pattern with rule strings of transformation, and a manual step to complete the occurrences. In addition to the performance, an advantage of using this approach is the fact both complete and incomplete occurrences can be detected.\nWe evaluated our approach on a set of 18 transformations. Our results show that patterns are actually used in transformations and that our approach is able to detect them. Moreover, we found that patterns are not always used independently, but in combination with other patterns.\nThe rest of the paper is structured as follows. In Section 2, we first introduce the basic notions used in our work, and then, discuss the related work. Section 3 details the different steps of our approach, whereas Section 4 describes the different forms of patterns that can be detected by our approach. We provide an evaluation of the detection approach in Section 5. Finally, we discuss the limits of our approach in Section 6, and conclude in Section 7."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Model transformation is now the mainstream paradigm to manipulate models in model-driven software engineering (MDE) [6]. Designing model transformations is a tedious task. Moreover, like any other code artifact, model transformations evolve and should be maintained. To assist developers in writing and maintaining model transformations, several design patterns have been proposed [11,22]. These presumably facilitate the comprehension and manipulation of transformation programs [1].\\nDetecting instances of a pattern in a transformation provides valuable information to the developer, such as understanding high-level concepts used, and identifying refactoring and reuse opportunities. This helps, among others, improving the documentation. However, as for general programs, developers do not always implement perfectly a pattern in model transformations. Hence, design pattern detection should identify both complete and incomplete occurrences.\\nDetecting design patterns in model transformations did not get much attention so far from the modeling community. To the best of our knowledge, only the work in [28] has attempted to automatically detect design patterns in model transformations. Preliminary results showed that this is an effective technique to find complete and approximate design pattern occurrences. However, this technique has performance limitations as it relies on a rule inference engine that is time and memory consuming. Another limitation of this technique is the need to specify a set of detection rules for each pattern.\\nTo find inspiration on how to detect patterns in transformations, we looked at the active community of design pattern detection in object-oriented programs. As reported in [1], there are dozens of detection approaches for this family of programs. However, as mentioned in [13], these approaches also suffer from performance problems, because detecting complete and incomplete occurrences is generally costly in time, due to the large search-space that includes all possible combinations of classes. These approaches are also prone to return many false positives, impeding program comprehension, and cluttering the maintainers' cognitive capabilities. To address the performance issues, the work by Kaczor et al. [20] uses a string matching technique inspired by pattern matching algorithms in bioinformatics to identify pattern occurrences in object-oriented programs. These algorithms allow to efficiently process a large amount of data if the problem to solve can be encoded as a string matching one.\\nIn this paper, we propose a generic technique to detect design pattern occurrences in model transformation implementations, without writing detection code for each design pattern, its variants, and approximations. Like in Kaczor et al., we rely on a bit-vector algorithm that has proven to be efficient for string matching problems [30]. The challenge we faced is how to encode model transformations, which are sets of rules linked by control schemes, as strings. The same challenge arises also in the encoding of the patterns as strings. We succeed to encode the participants of a patterns as strings, but had to complete our approach by a manual step to combine the identified participant instances to form pattern occurrences. Thus, the detection consists in an automated step that matches the participant strings of a pattern with rule strings of transformation, and a manual step to complete the occurrences. In addition to the performance, an advantage of using this approach is the fact both complete and incomplete occurrences can be detected.\\nWe evaluated our approach on a set of 18 transformations. Our results show that patterns are actually used in transformations and that our approach is able to detect them. Moreover, we found that patterns are not always used independently, but in combination with other patterns.\\nThe rest of the paper is structured as follows. In Section 2, we first introduce the basic notions used in our work, and then, discuss the related work. Section 3 details the different steps of our approach, whereas Section 4 describes the different forms of patterns that can be detected by our approach. We provide an evaluation of the detection approach in Section 5. Finally, we discuss the limits of our approach in Section 6, and conclude in Section 7.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"A survey on design pattern detection approaches. M Al-Obeidallah, M Petridis, S Kapetanakis, International Journal of Software Engineering. 7Al-Obeidallah, M., Petridis, M., Kapetanakis, S.: A survey on design pattern detection approaches. International Journal of Software Engineering 7, 41-59 (2016)","doi":""},{"text":"Rule-based detection of design patterns in program code. A Alnusair, T Zhao, G Yan, 10.1007/s10009-013-0292-zSTTT. 163Alnusair, A., Zhao, T., Yan, G.: Rule-based detection of design patterns in program code. STTT 16(3), 315-334 (2014). DOI 10.1007/s10009-013-0292-z. URL https://doi.org/ 10.1007/s10009-013-0292-z","doi":"10.1007/s10009-013-0292-z"},{"text":"Design pattern recovery in object-oriented software. G Antoniol, R Fiutem, L Cristoforetti, 10.1109/WPC.1998.6933426th International Workshop on Program Comprehension (IWPC '98). Ischia, ItalyAntoniol, G., Fiutem, R., Cristoforetti, L.: Design pattern recovery in object-oriented software. In: 6th International Workshop on Program Comprehension (IWPC '98), June 24-26, 1998, Ischia, Italy, pp. 153-160 (1998). DOI 10.1109/WPC.1998.693342. URL https://doi.org/10.1109/WPC.1998.693342","doi":"10.1109/WPC.1998.693342"},{"text":"Design pattern detection in java systems: A dynamic analysis based approach. F Arcelli, F Perin, C Raibulet, S Ravani, 10.1007/978-3-642-14819-4_12Evaluation of Novel Approaches to Software Engineering -3rd and 4th International Conferences. Madeira, Portugal; Milan, ItalyRevised Selected PapersArcelli, F., Perin, F., Raibulet, C., Ravani, S.: Design pattern detection in java sys- tems: A dynamic analysis based approach. In: Evaluation of Novel Approaches to Soft- ware Engineering -3rd and 4th International Conferences, ENASE 2008/2009, Fun- chal, Madeira, Portugal, May 4-7, 2008 / Milan, Italy, May 9-10, 2009. Revised Se- lected Papers, pp. 163-179 (2009). DOI 10.1007/978-3-642-14819-4\\ 12. URL https: //doi.org/10.1007/978-3-642-14819-4_12","doi":"10.1007/978-3-642-14819-4_12"},{"text":"Henshin: Advanced concepts and tools for in-place EMF model transformations. T Arendt, E Biermann, S Jurack, C Krause, G Taentzer, 10.1007/978-3-642-16145-2_9Model Driven Engineering Languages and Systems -13th International Conference, MODELS 2010. D.C. Petriu, N. Rouquette, Ø. HaugenOslo, NorwaySpringer6394Proceedings, Part IArendt, T., Biermann, E., Jurack, S., Krause, C., Taentzer, G.: Henshin: Advanced con- cepts and tools for in-place EMF model transformations. In: D.C. Petriu, N. Rouquette, Ø. Haugen (eds.) Model Driven Engineering Languages and Systems -13th Interna- tional Conference, MODELS 2010, Oslo, Norway, October 3-8, 2010, Proceedings, Part I, Lecture Notes in Computer Science, vol. 6394, pp. 121-135. Springer (2010). DOI 10.1007/978-3-642-16145-2\\ 9. URL https://doi.org/10.1007/978-3-642-16145-2_9","doi":"10.1007/978-3-642-16145-2_9"},{"text":"Systematic mapping study of model transformations for concrete problems. E Batot, H A Sahraoui, E Syriani, P Molins, W Sboui, 10.5220/0005657301760183MODELSWARD 2016 -Proceedings of the 4rd International Conference on Model-Driven Engineering and Software Development. S. Hammoudi, L.F. Pires, B. Selic, P. DesfrayRome, ItalySciTePressBatot, E., Sahraoui, H.A., Syriani, E., Molins, P., Sboui, W.: Systematic mapping study of model transformations for concrete problems. In: S. Hammoudi, L.F. Pires, B. Selic, P. Desfray (eds.) MODELSWARD 2016 -Proceedings of the 4rd International Conference on Model-Driven Engineering and Software Development, Rome, Italy, 19-21 February, 2016, pp. 176-183. SciTePress (2016). DOI 10.5220/0005657301760183. URL https: //doi.org/10.5220/0005657301760183","doi":"10.5220/0005657301760183"},{"text":"Vector algorithms for approximate string matching. A Bergeron, S Hamel, 10.1142/S0129054102000947Int. J. Found. Comput. Sci. 131Bergeron, A., Hamel, S.: Vector algorithms for approximate string matching. Int. J. Found. Comput. Sci. 13(1), 53-66 (2002). DOI 10.1142/S0129054102000947. URL https://doi. org/10.1142/S0129054102000947","doi":"10.1142/S0129054102000947"},{"text":"Linear Programming and Extensions. G B Dantizig, Princeton University PressDantizig, G.B.: Linear Programming and Extensions. Princeton University Press (1991). URL http://www.jstor.org/stable/j.ctt1cx3tvg","doi":""},{"text":"Combining clustering and pattern detection for the reengineering of component-based software systems. M Von Detten, S Becker, 10.1145/2000259.20002657th International Conference on the Quality of Software Architectures, QoSA 2011 and 2nd International Symposium on Architecting Critical Systems, ISARCS 2011. Boulder, CO, USAvon Detten, M., Becker, S.: Combining clustering and pattern detection for the reengineer- ing of component-based software systems. In: 7th International Conference on the Quality of Software Architectures, QoSA 2011 and 2nd International Symposium on Architecting Critical Systems, ISARCS 2011. Boulder, CO, USA, June 20-24, 2011, Proceedings, pp. 23-32 (2011). DOI 10.1145/2000259.2000265. URL https://doi.org/10.1145/2000259. 2000265","doi":"10.1145/2000259.2000265"},{"text":"Dp-miner: Design pattern discovery using matrix. J Dong, D S Lad, Y Zhao, 14th Annual IEEE International Conference and Workshop on Engineering of Computer Based Systems. Tucson, Arizona, USADong, J., Lad, D.S., Zhao, Y.: Dp-miner: Design pattern discovery using matrix. In: 14th Annual IEEE International Conference and Workshop on Engineering of Computer Based Systems (ECBS 2007), 26-29 March 2007, Tucson, Arizona, USA, pp. 371-380 (2007).","doi":""},{"text":". 10.1109/ECBS.2007.33DOI 10.1109/ECBS.2007.33. URL https://doi.org/10.1109/ECBS.2007.33","doi":"10.1109/ECBS.2007.33"},{"text":"Design pattern oriented development of model transformations. H Ergin, E Syriani, J Gray, 10.1016/j.cl.2016.07.004Computer Languages, Systems & Structures. 46Ergin, H., Syriani, E., Gray, J.: Design pattern oriented development of model transfor- mations. Computer Languages, Systems & Structures 46, 106-139 (2016). DOI http:// dx.doi.org/10.1016/j.cl.2016.07.004. URL https://doi.org/10.1016/j.cl.2016.07.004","doi":"10.1016/j.cl.2016.07.004"},{"text":"Columbus -reverse engineering tool and schema for C++. R Ferenc, Á Beszédes, M Tarkiainen, T Gyimóthy, 10.1109/ICSM.2002.116776418th International Conference on Software Maintenance (ICSM 2002), Maintaining Distributed Heterogeneous Systems. Montreal, Quebec, CanadaFerenc, R., Beszédes,Á., Tarkiainen, M., Gyimóthy, T.: Columbus -reverse engineer- ing tool and schema for C++. In: 18th International Conference on Software Mainte- nance (ICSM 2002), Maintaining Distributed Heterogeneous Systems, 3-6 October 2002, Montreal, Quebec, Canada, pp. 172-181 (2002). DOI 10.1109/ICSM.2002.1167764. URL https://doi.org/10.1109/ICSM.2002.1167764","doi":"10.1109/ICSM.2002.1167764"},{"text":"Improving design-pattern identification: a new approach and an exploratory study. Y Guéhéneuc, J Guyomarc'h, H A Sahraoui, 10.1007/s11219-009-9082-ySoftw. Qual. J. 181Guéhéneuc, Y., Guyomarc'h, J., Sahraoui, H.A.: Improving design-pattern identification: a new approach and an exploratory study. Softw. Qual. J. 18(1), 145-174 (2010). DOI 10.1007/s11219-009-9082-y. URL https://doi.org/10.1007/s11219-009-9082-y","doi":"10.1007/s11219-009-9082-y"},{"text":"Fingerprinting design patterns. Y Guéhéneuc, H A Sahraoui, F Zaidi, 10.1109/WCRE.2004.2111th Working Conference on Reverse Engineering. Delft, The NetherlandsGuéhéneuc, Y., Sahraoui, H.A., Zaidi, F.: Fingerprinting design patterns. In: 11th Working Conference on Reverse Engineering, WCRE 2004, Delft, The Netherlands, November 8-12, 2004, pp. 172-181 (2004). DOI 10.1109/WCRE.2004.21. URL https://doi.org/10.1109/ WCRE.2004.21","doi":"10.1109/WCRE.2004.21"},{"text":"Simulation of NFA in approximate string and sequence matching. J Holub, Proceedings of the Prague Stringology Club Workshop. the Prague Stringology Club WorkshopPrague, Czech RepublicDepartment of Computer Science and Engineering, Faculty of Electrical Engineering, Czech Technical UniversityHolub, J.: Simulation of NFA in approximate string and sequence matching. In: Proceed- ings of the Prague Stringology Club Workshop 1997, Prague, Czech Republic, July 7, 1997, pp. 39-46. Department of Computer Science and Engineering, Faculty of Electrical En- gineering, Czech Technical University (1997). URL http://www.stringology.org/event/ 1997/p5.html","doi":""},{"text":"Reusable model transformation patterns. M Iacob, M W A Steen, L Heerink, 10.1109/EDOCW.2008.51Workshops Proceedings of the 12th International IEEE Enterprise Distributed Object Computing Conference. Munich, GermanyIacob, M., Steen, M.W.A., Heerink, L.: Reusable model transformation patterns. In: Workshops Proceedings of the 12th International IEEE Enterprise Distributed Object Computing Conference, ECOCW 2008, 16 September 2008, Munich, Germany, pp. 1-10 (2008). DOI 10.1109/EDOCW.2008.51. URL https://doi.org/10.1109/EDOCW.2008.51","doi":"10.1109/EDOCW.2008.51"},{"text":"Abstracting complex languages through transformation and composition. J Johannes, S Zschaler, M A Fernández, A Castillo, D S Kolovos, R F Paige, 10.1007/978-3-642-04425-0_41Proceedings, Lecture Notes in Computer Science. A. Schürr, B. SelicLecture Notes in Computer ScienceDenver, CO, USASpringer579512th International ConferenceJohannes, J., Zschaler, S., Fernández, M.A., Castillo, A., Kolovos, D.S., Paige, R.F.: Ab- stracting complex languages through transformation and composition. In: A. Schürr, B. Selic (eds.) Model Driven Engineering Languages and Systems, 12th International Conference, MODELS 2009, Denver, CO, USA, October 4-9, 2009. Proceedings, Lec- ture Notes in Computer Science, vol. 5795, pp. 546-550. Springer (2009). DOI 10.1007/978-3-642-04425-0\\ 41. URL https://doi.org/10.1007/978-3-642-04425-0_41","doi":"10.1007/978-3-642-04425-0_41"},{"text":"ATL: A model transformation tool. F Jouault, F Allilaire, J Bézivin, I Kurtev, 10.1016/j.scico.2007.08.002Sci. Comput. Program. 721-2Jouault, F., Allilaire, F., Bézivin, J., Kurtev, I.: ATL: A model transformation tool. Sci. Comput. Program. 72(1-2), 31-39 (2008). DOI 10.1016/j.scico.2007.08.002. URL https: //doi.org/10.1016/j.scico.2007.08.002","doi":"10.1016/j.scico.2007.08.002"},{"text":"Solving the TTC 2011 reengineering case with henshin. S Jurack, J Tietje, 10.4204/EPTCS.74.17Proceedings Fifth Transformation Tool Contest, TTC 2011. Fifth Transformation Tool Contest, TTC 2011Zürich, SwitzerlandJurack, S., Tietje, J.: Solving the TTC 2011 reengineering case with henshin. In: Pro- ceedings Fifth Transformation Tool Contest, TTC 2011, Zürich, Switzerland, June 29-30 2011., pp. 181-203 (2011). DOI 10.4204/EPTCS.74.17. URL https://doi.org/10.4204/ EPTCS.74.17","doi":"10.4204/EPTCS.74.17"},{"text":"Identification of design motifs with pattern matching algorithms. O Kaczor, Y Guéhéneuc, S Hamel, 10.1016/j.infsof.2009.08.006Information & Software Technology. 522Kaczor, O., Guéhéneuc, Y., Hamel, S.: Identification of design motifs with pattern match- ing algorithms. Information & Software Technology 52(2), 152-168 (2010). DOI 10.1016/j.infsof.2009.08.006. URL https://doi.org/10.1016/j.infsof.2009.08.006","doi":"10.1016/j.infsof.2009.08.006"},{"text":"Design recovery by automated search for structural design patterns in object-oriented software. C Kramer, L Prechelt, 10.1109/WCRE.1996.558905Proceedings of WCRE '96: 4rd Working Conference on Reverse Engineering. WCRE '96: 4rd Working Conference on Reverse EngineeringKramer, C., Prechelt, L.: Design recovery by automated search for structural design pat- terns in object-oriented software. In: Proceedings of WCRE '96: 4rd Working Conference on Reverse Engineering, pp. 208-215 (1996). DOI 10.1109/WCRE.1996.558905","doi":"10.1109/WCRE.1996.558905"},{"text":"Model-transformation design patterns. K Lano, S K Rahimi, 10.1109/TSE.2014.2354344IEEE Transactions on Software Engineering. 4012Lano, K., Rahimi, S.K.: Model-transformation design patterns. IEEE Transactions on Software Engineering 40(12), 1224-1259 (2014). DOI 10.1109/TSE.2014.2354344. URL https://doi.org/10.1109/TSE.2014.2354344","doi":"10.1109/TSE.2014.2354344"},{"text":"A survey of model transformation design pattern usage. K Lano, S K Rahimi, S Y Tehrani, M Sharbaf, Theory and Practice of Model Transformation -10th Internat. Lano, K., Rahimi, S.K., Tehrani, S.Y., Sharbaf, M.: A survey of model transformation design pattern usage. In: Theory and Practice of Model Transformation -10th Internat.","doi":""},{"text":"10.1007/978-3-319-61473-1_8Conf, ICMT 2017, Proceedings. Springer10374Conf, ICMT 2017, Proceedings, LNCS, vol. 10374, pp. 108-118. Springer (2017). DOI 10.1007/978-3-319-61473-1\\ 8. URL https://doi.org/10.1007/978-3-319-61473-1_8","doi":"10.1007/978-3-319-61473-1_8"},{"text":"A survey of model transformation design patterns in practice. K Lano, S K Rahimi, S Y Tehrani, M Sharbaf, 10.1016/j.jss.2018.03.001Journal of Systems and Software. 140Lano, K., Rahimi, S.K., Tehrani, S.Y., Sharbaf, M.: A survey of model transformation design patterns in practice. Journal of Systems and Software 140, 48-73 (2018). DOI 10.1016/j.jss.2018.03.001. URL https://doi.org/10.1016/j.jss.2018.03.001","doi":"10.1016/j.jss.2018.03.001"},{"text":"Design pattern recovery through visual language parsing and source code analysis. A D Lucia, V Deufemia, C Gravino, M Risi, 10.1016/j.jss.2009.02.012Journal of Systems and Software. 827Lucia, A.D., Deufemia, V., Gravino, C., Risi, M.: Design pattern recovery through visual language parsing and source code analysis. Journal of Systems and Software 82(7), 1177- 1193 (2009). DOI 10.1016/j.jss.2009.02.012. URL https://doi.org/10.1016/j.jss.2009. 02.012","doi":"10.1016/j.jss.2009.02.012"},{"text":"Design pattern detection based on the graph theory. B B Mayvan, A Rasoolzadegan, 10.1016/j.knosys.2017.01.007120Knowledge-Based SystemsMayvan, B.B., Rasoolzadegan, A.: Design pattern detection based on the graph theory. Knowledge-Based Systems 120, 211-225 (2017). DOI 10.1016/j.knosys.2017.01.007. URL https://doi.org/10.1016/j.knosys.2017.01.007","doi":"10.1016/j.knosys.2017.01.007"},{"text":"Design pattern detection using ontology. M Y Mhawish, M Gupta, Mhawish, M.Y., Gupta, M.: Design pattern detection using ontology (2018)","doi":""},{"text":"Towards rule-based detection of design patterns in model transformations. C Mokaddem, H Sahraoui, E Syriani, 10.1007/978-3-319-46613-2_14System Analysis and Modeling. Technology-Specific Aspects of Models -9th International Conference. SAM 2016, Saint-Melo, France; Saint-MaloSpringer9959LNCSMokaddem, C., Sahraoui, H., Syriani, E.: Towards rule-based detection of design patterns in model transformations. In: System Analysis and Modeling. Technology-Specific Aspects of Models -9th International Conference, SAM 2016, Saint-Melo, France, October 3-4, 2016, Proceedings, LNCS, vol. 9959, pp. 211-225. Springer, Saint-Malo (2016). DOI 10.1007/978-3-319-46613-2\\ 14. URL https://doi.org/10.1007/978-3-319-46613-2_14","doi":"10.1007/978-3-319-46613-2_14"},{"text":"Recommending model refactoring rules from refactoring examples. C Mokaddem, H Sahraoui, E Syriani, 10.1145/3239372.3239406Proceedings of the 21th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems. A. Wasowski, R.F. Paige, Ø. Haugenthe 21th ACM/IEEE International Conference on Model Driven Engineering Languages and SystemsCopenhagen, DenmarkACMMokaddem, C., Sahraoui, H., Syriani, E.: Recommending model refactoring rules from refactoring examples. In: A. Wasowski, R.F. Paige, Ø. Haugen (eds.) Proceedings of the 21th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems, MODELS 2018, Copenhagen, Denmark, October 14-19, 2018, pp. 257-266. ACM (2018). DOI 10.1145/3239372.3239406. URL https://doi.org/10.1145/3239372.3239406","doi":"10.1145/3239372.3239406"},{"text":"A fast bit-vector algorithm for approximate string matching based on dynamic programming. G Myers, 10.1145/316542.316550Journal of the ACM. 463Myers, G.: A fast bit-vector algorithm for approximate string matching based on dynamic programming. Journal of the ACM 46(3), 395-415 (1999). DOI 10.1145/316542.316550. URL https://doi.org/10.1145/316542.316550","doi":"10.1145/316542.316550"},{"text":"A general method applicable to the search for similarities in the amino acid sequence of two proteins. S B Needleman, C D Wunsch, 10.1016/0022-2836(70)90057-4.URLhttp:/www.sciencedirect.com/science/article/pii/0022283670900574Journal of Molecular Biology. 483Needleman, S.B., Wunsch, C.D.: A general method applicable to the search for similarities in the amino acid sequence of two proteins. Journal of Molecular Biology 48(3), 443 -453 (1970). DOI https://doi.org/10.1016/0022-2836(70)90057-4. URL http://www. sciencedirect.com/science/article/pii/0022283670900574","doi":"10.1016/0022-2836(70)90057-4.URLhttp:/www.sciencedirect.com/science/article/pii/0022283670900574"},{"text":"Towards pattern-based design recovery. J Niere, W Schäfer, J P Wadsack, L Wendehals, J Welsh, 10.1145/581339.581382Proceedings of the 24th International Conference on Software Engineering, ICSE. the 24th International Conference on Software Engineering, ICSEOrlando, Florida, USANiere, J., Schäfer, W., Wadsack, J.P., Wendehals, L., Welsh, J.: Towards pattern-based design recovery. In: Proceedings of the 24th International Conference on Software Engi- neering, ICSE 2002, 19-25 May 2002, Orlando, Florida, USA, pp. 338-348 (2002). DOI 10.1145/581339.581382. URL https://doi.org/10.1145/581339.581382","doi":"10.1145/581339.581382"},{"text":"Full contract verification for ATL using symbolic execution. B J Oakes, J Troya, L Lúcio, M Wimmer, 10.1007/s10270-016-0548-710.1007/ s10270-016-0548-7Software and Systems Modeling. 173Oakes, B.J., Troya, J., Lúcio, L., Wimmer, M.: Full contract verification for ATL using symbolic execution. Software and Systems Modeling 17(3), 815-849 (2018). DOI 10.1007/ s10270-016-0548-7. URL https://doi.org/10.1007/s10270-016-0548-7","doi":"10.1007/s10270-016-0548-7"},{"text":"Software metrics by architectural pattern mining. J Paakki, A Karhinen, J Gustafsson, L Nenonen, A Verkamo, Proceedings of the International Conference on Software: Theory and Practice. the International Conference on Software: Theory and PracticePaakki, J., Karhinen, A., Gustafsson, J., Nenonen, L., Verkamo, A.: Software metrics by architectural pattern mining. Proceedings of the International Conference on Software: Theory and Practice (16th IFIP World Computer Congress) (2000)","doi":""},{"text":"A comparative study on string matching algorithm of biological sequences. P Pandiselvam, T Marimuthu, R Lawrance, CoRR) abs/1401.7416Computing Research. Pandiselvam, P., Marimuthu, T., Lawrance, R.: A comparative study on string matching al- gorithm of biological sequences. Computing Research Repository (CoRR) abs/1401.7416 (2014). URL http://arxiv.org/abs/1401.7416","doi":""},{"text":"An approach for reverse engineering of design patterns. I Philippow, D Streitferdt, M Riebisch, S Naumann, 10.1007/s10270-004-0059-9Software and Systems Modeling. 41Philippow, I., Streitferdt, D., Riebisch, M., Naumann, S.: An approach for reverse engi- neering of design patterns. Software and Systems Modeling 4(1), 55-70 (2005). DOI 10.1007/s10270-004-0059-9. URL https://doi.org/10.1007/s10270-004-0059-9","doi":""},{"text":"From secure business process modeling to design-level security verification. Q Ramadan, M Salnitri, D Strüber, J Jürjens, P Giorgini, 10.1109/MODELS.2017.1020th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems. Austin, TX, USAIEEE Computer SocietyRamadan, Q., Salnitri, M., Strüber, D., Jürjens, J., Giorgini, P.: From secure business process modeling to design-level security verification. In: 20th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems, MODELS 2017, Austin, TX, USA, September 17-22, 2017, pp. 123-133. IEEE Computer Society (2017). DOI 10.1109/MODELS.2017.10. URL https://doi.org/10.1109/MODELS.2017.10","doi":""},{"text":"Design pattern recovery based on annotations. G Rasool, I Philippow, P Mäder, 10.1016/j.advengsoft.2009.10.014Advances in Engineering Software. 414Rasool, G., Philippow, I., Mäder, P.: Design pattern recovery based on annotations. Ad- vances in Engineering Software 41(4), 519-526 (2010). DOI 10.1016/j.advengsoft.2009.10. 014. URL https://doi.org/10.1016/j.advengsoft.2009.10.014","doi":""},{"text":"Translation of ATL to AGT and application to a code generator for simulink. Software and Systems Modeling. E Richa, E Borde, L Pautet, 10.1007/s10270-017-0607-818Richa, E., Borde, E., Pautet, L.: Translation of ATL to AGT and application to a code generator for simulink. Software and Systems Modeling 18(1), 321-344 (2019). DOI 10.1007/s10270-017-0607-8. URL https://doi.org/10.1007/s10270-017-0607-8","doi":""},{"text":"Reverse engineering of design patterns from java source code. N Shi, R A Olsson, 10.1109/ASE.2006.5721st IEEE/ACM International Conference on Automated Software Engineering (ASE 2006). Tokyo, JapanShi, N., Olsson, R.A.: Reverse engineering of design patterns from java source code. In: 21st IEEE/ACM International Conference on Automated Software Engineering (ASE 2006), 18-22 September 2006, Tokyo, Japan, pp. 123-134 (2006). DOI 10.1109/ASE.2006.57. URL https://doi.org/10.1109/ASE.2006.57","doi":""},{"text":"Identification of common molecular subsequences. T Smith, M Waterman, 10.1016/0022-2836(81)90087-5.URLhttp:/www.sciencedirect.com/science/article/pii/0022283681900875Journal of Molecular Biology. 1471Smith, T., Waterman, M.: Identification of common molecular subsequences. Journal of Molecular Biology 147(1), 195 -197 (1981). DOI https://doi.org/10.1016/0022-2836(81) 90087-5. URL http://www.sciencedirect.com/science/article/pii/0022283681900875","doi":""},{"text":"G Srinivasan, Operations Research : Principles and Applications. Prentice-Hall of IndiaSrinivasan, G.: Operations Research : Principles and Applications. Prentice-Hall of India (2010). URL https://books.google.ca/books?id=wYfxffB62NUC","doi":""},{"text":"A Modular Timed Graph Transformation Language for Simulation-based Design. E Syriani, H Vangheluwe, 10.1007/s10270-011-0205-0Software & Systems Modeling. 122Syriani, E., Vangheluwe, H.: A Modular Timed Graph Transformation Language for Simulation-based Design. Software & Systems Modeling 12(2), 387-414 (2013). DOI 10.1007/s10270-011-0205-0. URL https://doi.org/10.1007/s10270-011-0205-0","doi":""},{"text":"Design pattern detection using similarity scoring. N Tsantalis, A Chatzigeorgiou, G Stephanides, S T Halkidis, 10.1109/TSE.2006.11210.1109/ TSE.2006.112IEEE Trans. Software Eng. 3211Tsantalis, N., Chatzigeorgiou, A., Stephanides, G., Halkidis, S.T.: Design pattern detection using similarity scoring. IEEE Trans. Software Eng. 32(11), 896-909 (2006). DOI 10.1109/ TSE.2006.112. URL https://doi.org/10.1109/TSE.2006.112","doi":""},{"text":"Detecting design patterns in objectoriented program source code by using metrics and machine learning. S Uchiyama, A Kubo, H Washizaki, Y Fukazawa, 10.4236/ jsea.2014.712086Uchiyama, S., Kubo, A., Washizaki, H., Fukazawa, Y.: Detecting design patterns in object- oriented program source code by using metrics and machine learning (2014). DOI 10.4236/ jsea.2014.712086","doi":""},{"text":"A comprehensive approach to the recovery of design pattern instances based on sub-patterns and method signatures. D Yu, Y Zhang, Z Chen, 10.1016/j.jss.2015.01.019Journal of Systems and Software. 103Yu, D., Zhang, Y., Chen, Z.: A comprehensive approach to the recovery of design pattern instances based on sub-patterns and method signatures. Journal of Systems and Software 103, 1-16 (2015). DOI 10.1016/j.jss.2015.01.019. URL https://doi.org/10.1016/j.jss. 2015.01.019","doi":""},{"text":"On applying machine learning techniques for design pattern detection. M Zanoni, F A Fontana, F Stella, 10.1016/j.jss.2015.01.03710.1016/ j.jss.2015.01.037Journal of Systems and Software. 103Zanoni, M., Fontana, F.A., Stella, F.: On applying machine learning techniques for design pattern detection. Journal of Systems and Software 103, 102-117 (2015). DOI 10.1016/ j.jss.2015.01.037. URL https://doi.org/10.1016/j.jss.2015.01.037","doi":""}],"string":"[\n {\n \"text\": \"A survey on design pattern detection approaches. M Al-Obeidallah, M Petridis, S Kapetanakis, International Journal of Software Engineering. 7Al-Obeidallah, M., Petridis, M., Kapetanakis, S.: A survey on design pattern detection approaches. International Journal of Software Engineering 7, 41-59 (2016)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rule-based detection of design patterns in program code. A Alnusair, T Zhao, G Yan, 10.1007/s10009-013-0292-zSTTT. 163Alnusair, A., Zhao, T., Yan, G.: Rule-based detection of design patterns in program code. STTT 16(3), 315-334 (2014). DOI 10.1007/s10009-013-0292-z. URL https://doi.org/ 10.1007/s10009-013-0292-z\",\n \"doi\": \"10.1007/s10009-013-0292-z\"\n },\n {\n \"text\": \"Design pattern recovery in object-oriented software. G Antoniol, R Fiutem, L Cristoforetti, 10.1109/WPC.1998.6933426th International Workshop on Program Comprehension (IWPC '98). Ischia, ItalyAntoniol, G., Fiutem, R., Cristoforetti, L.: Design pattern recovery in object-oriented software. In: 6th International Workshop on Program Comprehension (IWPC '98), June 24-26, 1998, Ischia, Italy, pp. 153-160 (1998). DOI 10.1109/WPC.1998.693342. URL https://doi.org/10.1109/WPC.1998.693342\",\n \"doi\": \"10.1109/WPC.1998.693342\"\n },\n {\n \"text\": \"Design pattern detection in java systems: A dynamic analysis based approach. F Arcelli, F Perin, C Raibulet, S Ravani, 10.1007/978-3-642-14819-4_12Evaluation of Novel Approaches to Software Engineering -3rd and 4th International Conferences. Madeira, Portugal; Milan, ItalyRevised Selected PapersArcelli, F., Perin, F., Raibulet, C., Ravani, S.: Design pattern detection in java sys- tems: A dynamic analysis based approach. In: Evaluation of Novel Approaches to Soft- ware Engineering -3rd and 4th International Conferences, ENASE 2008/2009, Fun- chal, Madeira, Portugal, May 4-7, 2008 / Milan, Italy, May 9-10, 2009. Revised Se- lected Papers, pp. 163-179 (2009). DOI 10.1007/978-3-642-14819-4\\\\ 12. URL https: //doi.org/10.1007/978-3-642-14819-4_12\",\n \"doi\": \"10.1007/978-3-642-14819-4_12\"\n },\n {\n \"text\": \"Henshin: Advanced concepts and tools for in-place EMF model transformations. T Arendt, E Biermann, S Jurack, C Krause, G Taentzer, 10.1007/978-3-642-16145-2_9Model Driven Engineering Languages and Systems -13th International Conference, MODELS 2010. D.C. Petriu, N. Rouquette, Ø. HaugenOslo, NorwaySpringer6394Proceedings, Part IArendt, T., Biermann, E., Jurack, S., Krause, C., Taentzer, G.: Henshin: Advanced con- cepts and tools for in-place EMF model transformations. In: D.C. Petriu, N. Rouquette, Ø. Haugen (eds.) Model Driven Engineering Languages and Systems -13th Interna- tional Conference, MODELS 2010, Oslo, Norway, October 3-8, 2010, Proceedings, Part I, Lecture Notes in Computer Science, vol. 6394, pp. 121-135. Springer (2010). DOI 10.1007/978-3-642-16145-2\\\\ 9. URL https://doi.org/10.1007/978-3-642-16145-2_9\",\n \"doi\": \"10.1007/978-3-642-16145-2_9\"\n },\n {\n \"text\": \"Systematic mapping study of model transformations for concrete problems. E Batot, H A Sahraoui, E Syriani, P Molins, W Sboui, 10.5220/0005657301760183MODELSWARD 2016 -Proceedings of the 4rd International Conference on Model-Driven Engineering and Software Development. S. Hammoudi, L.F. Pires, B. Selic, P. DesfrayRome, ItalySciTePressBatot, E., Sahraoui, H.A., Syriani, E., Molins, P., Sboui, W.: Systematic mapping study of model transformations for concrete problems. In: S. Hammoudi, L.F. Pires, B. Selic, P. Desfray (eds.) MODELSWARD 2016 -Proceedings of the 4rd International Conference on Model-Driven Engineering and Software Development, Rome, Italy, 19-21 February, 2016, pp. 176-183. SciTePress (2016). DOI 10.5220/0005657301760183. URL https: //doi.org/10.5220/0005657301760183\",\n \"doi\": \"10.5220/0005657301760183\"\n },\n {\n \"text\": \"Vector algorithms for approximate string matching. A Bergeron, S Hamel, 10.1142/S0129054102000947Int. J. Found. Comput. Sci. 131Bergeron, A., Hamel, S.: Vector algorithms for approximate string matching. Int. J. Found. Comput. Sci. 13(1), 53-66 (2002). DOI 10.1142/S0129054102000947. URL https://doi. org/10.1142/S0129054102000947\",\n \"doi\": \"10.1142/S0129054102000947\"\n },\n {\n \"text\": \"Linear Programming and Extensions. G B Dantizig, Princeton University PressDantizig, G.B.: Linear Programming and Extensions. Princeton University Press (1991). URL http://www.jstor.org/stable/j.ctt1cx3tvg\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Combining clustering and pattern detection for the reengineering of component-based software systems. M Von Detten, S Becker, 10.1145/2000259.20002657th International Conference on the Quality of Software Architectures, QoSA 2011 and 2nd International Symposium on Architecting Critical Systems, ISARCS 2011. Boulder, CO, USAvon Detten, M., Becker, S.: Combining clustering and pattern detection for the reengineer- ing of component-based software systems. In: 7th International Conference on the Quality of Software Architectures, QoSA 2011 and 2nd International Symposium on Architecting Critical Systems, ISARCS 2011. Boulder, CO, USA, June 20-24, 2011, Proceedings, pp. 23-32 (2011). DOI 10.1145/2000259.2000265. URL https://doi.org/10.1145/2000259. 2000265\",\n \"doi\": \"10.1145/2000259.2000265\"\n },\n {\n \"text\": \"Dp-miner: Design pattern discovery using matrix. J Dong, D S Lad, Y Zhao, 14th Annual IEEE International Conference and Workshop on Engineering of Computer Based Systems. Tucson, Arizona, USADong, J., Lad, D.S., Zhao, Y.: Dp-miner: Design pattern discovery using matrix. In: 14th Annual IEEE International Conference and Workshop on Engineering of Computer Based Systems (ECBS 2007), 26-29 March 2007, Tucson, Arizona, USA, pp. 371-380 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". 10.1109/ECBS.2007.33DOI 10.1109/ECBS.2007.33. URL https://doi.org/10.1109/ECBS.2007.33\",\n \"doi\": \"10.1109/ECBS.2007.33\"\n },\n {\n \"text\": \"Design pattern oriented development of model transformations. H Ergin, E Syriani, J Gray, 10.1016/j.cl.2016.07.004Computer Languages, Systems & Structures. 46Ergin, H., Syriani, E., Gray, J.: Design pattern oriented development of model transfor- mations. Computer Languages, Systems & Structures 46, 106-139 (2016). DOI http:// dx.doi.org/10.1016/j.cl.2016.07.004. URL https://doi.org/10.1016/j.cl.2016.07.004\",\n \"doi\": \"10.1016/j.cl.2016.07.004\"\n },\n {\n \"text\": \"Columbus -reverse engineering tool and schema for C++. R Ferenc, Á Beszédes, M Tarkiainen, T Gyimóthy, 10.1109/ICSM.2002.116776418th International Conference on Software Maintenance (ICSM 2002), Maintaining Distributed Heterogeneous Systems. Montreal, Quebec, CanadaFerenc, R., Beszédes,Á., Tarkiainen, M., Gyimóthy, T.: Columbus -reverse engineer- ing tool and schema for C++. In: 18th International Conference on Software Mainte- nance (ICSM 2002), Maintaining Distributed Heterogeneous Systems, 3-6 October 2002, Montreal, Quebec, Canada, pp. 172-181 (2002). DOI 10.1109/ICSM.2002.1167764. URL https://doi.org/10.1109/ICSM.2002.1167764\",\n \"doi\": \"10.1109/ICSM.2002.1167764\"\n },\n {\n \"text\": \"Improving design-pattern identification: a new approach and an exploratory study. Y Guéhéneuc, J Guyomarc'h, H A Sahraoui, 10.1007/s11219-009-9082-ySoftw. Qual. J. 181Guéhéneuc, Y., Guyomarc'h, J., Sahraoui, H.A.: Improving design-pattern identification: a new approach and an exploratory study. Softw. Qual. J. 18(1), 145-174 (2010). DOI 10.1007/s11219-009-9082-y. URL https://doi.org/10.1007/s11219-009-9082-y\",\n \"doi\": \"10.1007/s11219-009-9082-y\"\n },\n {\n \"text\": \"Fingerprinting design patterns. Y Guéhéneuc, H A Sahraoui, F Zaidi, 10.1109/WCRE.2004.2111th Working Conference on Reverse Engineering. Delft, The NetherlandsGuéhéneuc, Y., Sahraoui, H.A., Zaidi, F.: Fingerprinting design patterns. In: 11th Working Conference on Reverse Engineering, WCRE 2004, Delft, The Netherlands, November 8-12, 2004, pp. 172-181 (2004). DOI 10.1109/WCRE.2004.21. URL https://doi.org/10.1109/ WCRE.2004.21\",\n \"doi\": \"10.1109/WCRE.2004.21\"\n },\n {\n \"text\": \"Simulation of NFA in approximate string and sequence matching. J Holub, Proceedings of the Prague Stringology Club Workshop. the Prague Stringology Club WorkshopPrague, Czech RepublicDepartment of Computer Science and Engineering, Faculty of Electrical Engineering, Czech Technical UniversityHolub, J.: Simulation of NFA in approximate string and sequence matching. In: Proceed- ings of the Prague Stringology Club Workshop 1997, Prague, Czech Republic, July 7, 1997, pp. 39-46. Department of Computer Science and Engineering, Faculty of Electrical En- gineering, Czech Technical University (1997). URL http://www.stringology.org/event/ 1997/p5.html\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reusable model transformation patterns. M Iacob, M W A Steen, L Heerink, 10.1109/EDOCW.2008.51Workshops Proceedings of the 12th International IEEE Enterprise Distributed Object Computing Conference. Munich, GermanyIacob, M., Steen, M.W.A., Heerink, L.: Reusable model transformation patterns. In: Workshops Proceedings of the 12th International IEEE Enterprise Distributed Object Computing Conference, ECOCW 2008, 16 September 2008, Munich, Germany, pp. 1-10 (2008). DOI 10.1109/EDOCW.2008.51. URL https://doi.org/10.1109/EDOCW.2008.51\",\n \"doi\": \"10.1109/EDOCW.2008.51\"\n },\n {\n \"text\": \"Abstracting complex languages through transformation and composition. J Johannes, S Zschaler, M A Fernández, A Castillo, D S Kolovos, R F Paige, 10.1007/978-3-642-04425-0_41Proceedings, Lecture Notes in Computer Science. A. Schürr, B. SelicLecture Notes in Computer ScienceDenver, CO, USASpringer579512th International ConferenceJohannes, J., Zschaler, S., Fernández, M.A., Castillo, A., Kolovos, D.S., Paige, R.F.: Ab- stracting complex languages through transformation and composition. In: A. Schürr, B. Selic (eds.) Model Driven Engineering Languages and Systems, 12th International Conference, MODELS 2009, Denver, CO, USA, October 4-9, 2009. Proceedings, Lec- ture Notes in Computer Science, vol. 5795, pp. 546-550. Springer (2009). DOI 10.1007/978-3-642-04425-0\\\\ 41. URL https://doi.org/10.1007/978-3-642-04425-0_41\",\n \"doi\": \"10.1007/978-3-642-04425-0_41\"\n },\n {\n \"text\": \"ATL: A model transformation tool. F Jouault, F Allilaire, J Bézivin, I Kurtev, 10.1016/j.scico.2007.08.002Sci. Comput. Program. 721-2Jouault, F., Allilaire, F., Bézivin, J., Kurtev, I.: ATL: A model transformation tool. Sci. Comput. Program. 72(1-2), 31-39 (2008). DOI 10.1016/j.scico.2007.08.002. URL https: //doi.org/10.1016/j.scico.2007.08.002\",\n \"doi\": \"10.1016/j.scico.2007.08.002\"\n },\n {\n \"text\": \"Solving the TTC 2011 reengineering case with henshin. S Jurack, J Tietje, 10.4204/EPTCS.74.17Proceedings Fifth Transformation Tool Contest, TTC 2011. Fifth Transformation Tool Contest, TTC 2011Zürich, SwitzerlandJurack, S., Tietje, J.: Solving the TTC 2011 reengineering case with henshin. In: Pro- ceedings Fifth Transformation Tool Contest, TTC 2011, Zürich, Switzerland, June 29-30 2011., pp. 181-203 (2011). DOI 10.4204/EPTCS.74.17. URL https://doi.org/10.4204/ EPTCS.74.17\",\n \"doi\": \"10.4204/EPTCS.74.17\"\n },\n {\n \"text\": \"Identification of design motifs with pattern matching algorithms. O Kaczor, Y Guéhéneuc, S Hamel, 10.1016/j.infsof.2009.08.006Information & Software Technology. 522Kaczor, O., Guéhéneuc, Y., Hamel, S.: Identification of design motifs with pattern match- ing algorithms. Information & Software Technology 52(2), 152-168 (2010). DOI 10.1016/j.infsof.2009.08.006. URL https://doi.org/10.1016/j.infsof.2009.08.006\",\n \"doi\": \"10.1016/j.infsof.2009.08.006\"\n },\n {\n \"text\": \"Design recovery by automated search for structural design patterns in object-oriented software. C Kramer, L Prechelt, 10.1109/WCRE.1996.558905Proceedings of WCRE '96: 4rd Working Conference on Reverse Engineering. WCRE '96: 4rd Working Conference on Reverse EngineeringKramer, C., Prechelt, L.: Design recovery by automated search for structural design pat- terns in object-oriented software. In: Proceedings of WCRE '96: 4rd Working Conference on Reverse Engineering, pp. 208-215 (1996). DOI 10.1109/WCRE.1996.558905\",\n \"doi\": \"10.1109/WCRE.1996.558905\"\n },\n {\n \"text\": \"Model-transformation design patterns. K Lano, S K Rahimi, 10.1109/TSE.2014.2354344IEEE Transactions on Software Engineering. 4012Lano, K., Rahimi, S.K.: Model-transformation design patterns. IEEE Transactions on Software Engineering 40(12), 1224-1259 (2014). DOI 10.1109/TSE.2014.2354344. URL https://doi.org/10.1109/TSE.2014.2354344\",\n \"doi\": \"10.1109/TSE.2014.2354344\"\n },\n {\n \"text\": \"A survey of model transformation design pattern usage. K Lano, S K Rahimi, S Y Tehrani, M Sharbaf, Theory and Practice of Model Transformation -10th Internat. Lano, K., Rahimi, S.K., Tehrani, S.Y., Sharbaf, M.: A survey of model transformation design pattern usage. In: Theory and Practice of Model Transformation -10th Internat.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"10.1007/978-3-319-61473-1_8Conf, ICMT 2017, Proceedings. Springer10374Conf, ICMT 2017, Proceedings, LNCS, vol. 10374, pp. 108-118. Springer (2017). DOI 10.1007/978-3-319-61473-1\\\\ 8. URL https://doi.org/10.1007/978-3-319-61473-1_8\",\n \"doi\": \"10.1007/978-3-319-61473-1_8\"\n },\n {\n \"text\": \"A survey of model transformation design patterns in practice. K Lano, S K Rahimi, S Y Tehrani, M Sharbaf, 10.1016/j.jss.2018.03.001Journal of Systems and Software. 140Lano, K., Rahimi, S.K., Tehrani, S.Y., Sharbaf, M.: A survey of model transformation design patterns in practice. Journal of Systems and Software 140, 48-73 (2018). DOI 10.1016/j.jss.2018.03.001. URL https://doi.org/10.1016/j.jss.2018.03.001\",\n \"doi\": \"10.1016/j.jss.2018.03.001\"\n },\n {\n \"text\": \"Design pattern recovery through visual language parsing and source code analysis. A D Lucia, V Deufemia, C Gravino, M Risi, 10.1016/j.jss.2009.02.012Journal of Systems and Software. 827Lucia, A.D., Deufemia, V., Gravino, C., Risi, M.: Design pattern recovery through visual language parsing and source code analysis. Journal of Systems and Software 82(7), 1177- 1193 (2009). DOI 10.1016/j.jss.2009.02.012. URL https://doi.org/10.1016/j.jss.2009. 02.012\",\n \"doi\": \"10.1016/j.jss.2009.02.012\"\n },\n {\n \"text\": \"Design pattern detection based on the graph theory. B B Mayvan, A Rasoolzadegan, 10.1016/j.knosys.2017.01.007120Knowledge-Based SystemsMayvan, B.B., Rasoolzadegan, A.: Design pattern detection based on the graph theory. Knowledge-Based Systems 120, 211-225 (2017). DOI 10.1016/j.knosys.2017.01.007. URL https://doi.org/10.1016/j.knosys.2017.01.007\",\n \"doi\": \"10.1016/j.knosys.2017.01.007\"\n },\n {\n \"text\": \"Design pattern detection using ontology. M Y Mhawish, M Gupta, Mhawish, M.Y., Gupta, M.: Design pattern detection using ontology (2018)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Towards rule-based detection of design patterns in model transformations. C Mokaddem, H Sahraoui, E Syriani, 10.1007/978-3-319-46613-2_14System Analysis and Modeling. Technology-Specific Aspects of Models -9th International Conference. SAM 2016, Saint-Melo, France; Saint-MaloSpringer9959LNCSMokaddem, C., Sahraoui, H., Syriani, E.: Towards rule-based detection of design patterns in model transformations. In: System Analysis and Modeling. Technology-Specific Aspects of Models -9th International Conference, SAM 2016, Saint-Melo, France, October 3-4, 2016, Proceedings, LNCS, vol. 9959, pp. 211-225. Springer, Saint-Malo (2016). DOI 10.1007/978-3-319-46613-2\\\\ 14. URL https://doi.org/10.1007/978-3-319-46613-2_14\",\n \"doi\": \"10.1007/978-3-319-46613-2_14\"\n },\n {\n \"text\": \"Recommending model refactoring rules from refactoring examples. C Mokaddem, H Sahraoui, E Syriani, 10.1145/3239372.3239406Proceedings of the 21th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems. A. Wasowski, R.F. Paige, Ø. Haugenthe 21th ACM/IEEE International Conference on Model Driven Engineering Languages and SystemsCopenhagen, DenmarkACMMokaddem, C., Sahraoui, H., Syriani, E.: Recommending model refactoring rules from refactoring examples. In: A. Wasowski, R.F. Paige, Ø. Haugen (eds.) Proceedings of the 21th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems, MODELS 2018, Copenhagen, Denmark, October 14-19, 2018, pp. 257-266. ACM (2018). DOI 10.1145/3239372.3239406. URL https://doi.org/10.1145/3239372.3239406\",\n \"doi\": \"10.1145/3239372.3239406\"\n },\n {\n \"text\": \"A fast bit-vector algorithm for approximate string matching based on dynamic programming. G Myers, 10.1145/316542.316550Journal of the ACM. 463Myers, G.: A fast bit-vector algorithm for approximate string matching based on dynamic programming. Journal of the ACM 46(3), 395-415 (1999). DOI 10.1145/316542.316550. URL https://doi.org/10.1145/316542.316550\",\n \"doi\": \"10.1145/316542.316550\"\n },\n {\n \"text\": \"A general method applicable to the search for similarities in the amino acid sequence of two proteins. S B Needleman, C D Wunsch, 10.1016/0022-2836(70)90057-4.URLhttp:/www.sciencedirect.com/science/article/pii/0022283670900574Journal of Molecular Biology. 483Needleman, S.B., Wunsch, C.D.: A general method applicable to the search for similarities in the amino acid sequence of two proteins. Journal of Molecular Biology 48(3), 443 -453 (1970). DOI https://doi.org/10.1016/0022-2836(70)90057-4. URL http://www. sciencedirect.com/science/article/pii/0022283670900574\",\n \"doi\": \"10.1016/0022-2836(70)90057-4.URLhttp:/www.sciencedirect.com/science/article/pii/0022283670900574\"\n },\n {\n \"text\": \"Towards pattern-based design recovery. J Niere, W Schäfer, J P Wadsack, L Wendehals, J Welsh, 10.1145/581339.581382Proceedings of the 24th International Conference on Software Engineering, ICSE. the 24th International Conference on Software Engineering, ICSEOrlando, Florida, USANiere, J., Schäfer, W., Wadsack, J.P., Wendehals, L., Welsh, J.: Towards pattern-based design recovery. In: Proceedings of the 24th International Conference on Software Engi- neering, ICSE 2002, 19-25 May 2002, Orlando, Florida, USA, pp. 338-348 (2002). DOI 10.1145/581339.581382. URL https://doi.org/10.1145/581339.581382\",\n \"doi\": \"10.1145/581339.581382\"\n },\n {\n \"text\": \"Full contract verification for ATL using symbolic execution. B J Oakes, J Troya, L Lúcio, M Wimmer, 10.1007/s10270-016-0548-710.1007/ s10270-016-0548-7Software and Systems Modeling. 173Oakes, B.J., Troya, J., Lúcio, L., Wimmer, M.: Full contract verification for ATL using symbolic execution. Software and Systems Modeling 17(3), 815-849 (2018). DOI 10.1007/ s10270-016-0548-7. URL https://doi.org/10.1007/s10270-016-0548-7\",\n \"doi\": \"10.1007/s10270-016-0548-7\"\n },\n {\n \"text\": \"Software metrics by architectural pattern mining. J Paakki, A Karhinen, J Gustafsson, L Nenonen, A Verkamo, Proceedings of the International Conference on Software: Theory and Practice. the International Conference on Software: Theory and PracticePaakki, J., Karhinen, A., Gustafsson, J., Nenonen, L., Verkamo, A.: Software metrics by architectural pattern mining. Proceedings of the International Conference on Software: Theory and Practice (16th IFIP World Computer Congress) (2000)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A comparative study on string matching algorithm of biological sequences. P Pandiselvam, T Marimuthu, R Lawrance, CoRR) abs/1401.7416Computing Research. Pandiselvam, P., Marimuthu, T., Lawrance, R.: A comparative study on string matching al- gorithm of biological sequences. Computing Research Repository (CoRR) abs/1401.7416 (2014). URL http://arxiv.org/abs/1401.7416\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An approach for reverse engineering of design patterns. I Philippow, D Streitferdt, M Riebisch, S Naumann, 10.1007/s10270-004-0059-9Software and Systems Modeling. 41Philippow, I., Streitferdt, D., Riebisch, M., Naumann, S.: An approach for reverse engi- neering of design patterns. Software and Systems Modeling 4(1), 55-70 (2005). DOI 10.1007/s10270-004-0059-9. URL https://doi.org/10.1007/s10270-004-0059-9\",\n \"doi\": \"\"\n },\n {\n \"text\": \"From secure business process modeling to design-level security verification. Q Ramadan, M Salnitri, D Strüber, J Jürjens, P Giorgini, 10.1109/MODELS.2017.1020th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems. Austin, TX, USAIEEE Computer SocietyRamadan, Q., Salnitri, M., Strüber, D., Jürjens, J., Giorgini, P.: From secure business process modeling to design-level security verification. In: 20th ACM/IEEE International Conference on Model Driven Engineering Languages and Systems, MODELS 2017, Austin, TX, USA, September 17-22, 2017, pp. 123-133. IEEE Computer Society (2017). DOI 10.1109/MODELS.2017.10. URL https://doi.org/10.1109/MODELS.2017.10\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Design pattern recovery based on annotations. G Rasool, I Philippow, P Mäder, 10.1016/j.advengsoft.2009.10.014Advances in Engineering Software. 414Rasool, G., Philippow, I., Mäder, P.: Design pattern recovery based on annotations. Ad- vances in Engineering Software 41(4), 519-526 (2010). DOI 10.1016/j.advengsoft.2009.10. 014. URL https://doi.org/10.1016/j.advengsoft.2009.10.014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Translation of ATL to AGT and application to a code generator for simulink. Software and Systems Modeling. E Richa, E Borde, L Pautet, 10.1007/s10270-017-0607-818Richa, E., Borde, E., Pautet, L.: Translation of ATL to AGT and application to a code generator for simulink. Software and Systems Modeling 18(1), 321-344 (2019). DOI 10.1007/s10270-017-0607-8. URL https://doi.org/10.1007/s10270-017-0607-8\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reverse engineering of design patterns from java source code. N Shi, R A Olsson, 10.1109/ASE.2006.5721st IEEE/ACM International Conference on Automated Software Engineering (ASE 2006). Tokyo, JapanShi, N., Olsson, R.A.: Reverse engineering of design patterns from java source code. In: 21st IEEE/ACM International Conference on Automated Software Engineering (ASE 2006), 18-22 September 2006, Tokyo, Japan, pp. 123-134 (2006). DOI 10.1109/ASE.2006.57. URL https://doi.org/10.1109/ASE.2006.57\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Identification of common molecular subsequences. T Smith, M Waterman, 10.1016/0022-2836(81)90087-5.URLhttp:/www.sciencedirect.com/science/article/pii/0022283681900875Journal of Molecular Biology. 1471Smith, T., Waterman, M.: Identification of common molecular subsequences. Journal of Molecular Biology 147(1), 195 -197 (1981). DOI https://doi.org/10.1016/0022-2836(81) 90087-5. URL http://www.sciencedirect.com/science/article/pii/0022283681900875\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G Srinivasan, Operations Research : Principles and Applications. Prentice-Hall of IndiaSrinivasan, G.: Operations Research : Principles and Applications. Prentice-Hall of India (2010). URL https://books.google.ca/books?id=wYfxffB62NUC\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Modular Timed Graph Transformation Language for Simulation-based Design. E Syriani, H Vangheluwe, 10.1007/s10270-011-0205-0Software & Systems Modeling. 122Syriani, E., Vangheluwe, H.: A Modular Timed Graph Transformation Language for Simulation-based Design. Software & Systems Modeling 12(2), 387-414 (2013). DOI 10.1007/s10270-011-0205-0. URL https://doi.org/10.1007/s10270-011-0205-0\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Design pattern detection using similarity scoring. N Tsantalis, A Chatzigeorgiou, G Stephanides, S T Halkidis, 10.1109/TSE.2006.11210.1109/ TSE.2006.112IEEE Trans. Software Eng. 3211Tsantalis, N., Chatzigeorgiou, A., Stephanides, G., Halkidis, S.T.: Design pattern detection using similarity scoring. IEEE Trans. Software Eng. 32(11), 896-909 (2006). DOI 10.1109/ TSE.2006.112. URL https://doi.org/10.1109/TSE.2006.112\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Detecting design patterns in objectoriented program source code by using metrics and machine learning. S Uchiyama, A Kubo, H Washizaki, Y Fukazawa, 10.4236/ jsea.2014.712086Uchiyama, S., Kubo, A., Washizaki, H., Fukazawa, Y.: Detecting design patterns in object- oriented program source code by using metrics and machine learning (2014). DOI 10.4236/ jsea.2014.712086\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A comprehensive approach to the recovery of design pattern instances based on sub-patterns and method signatures. D Yu, Y Zhang, Z Chen, 10.1016/j.jss.2015.01.019Journal of Systems and Software. 103Yu, D., Zhang, Y., Chen, Z.: A comprehensive approach to the recovery of design pattern instances based on sub-patterns and method signatures. Journal of Systems and Software 103, 1-16 (2015). DOI 10.1016/j.jss.2015.01.019. URL https://doi.org/10.1016/j.jss. 2015.01.019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On applying machine learning techniques for design pattern detection. M Zanoni, F A Fontana, F Stella, 10.1016/j.jss.2015.01.03710.1016/ j.jss.2015.01.037Journal of Systems and Software. 103Zanoni, M., Fontana, F.A., Stella, F.: On applying machine learning techniques for design pattern detection. Journal of Systems and Software 103, 102-117 (2015). DOI 10.1016/ j.jss.2015.01.037. URL https://doi.org/10.1016/j.jss.2015.01.037\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":67,"cells":{"text":{"kind":"string","value":"\n\n29 Mar 2016 March 30, 2016\n\nAndrei Khrennikov \nInternational Center for Mathematical Modeling in Physics\nCognitive Science Linnaeus University\nEngineeringVäxjöEconomicsSweden\n29 Mar 2016 March 30, 2016After Bell\nWe analyze foundational consequences of recently announced loophole free tests of violation of Bell's inequality. We consider two interpretations of these remarkable experiments. By the conventional one \"Einstein was wrong and Bohr was right, there is spooky action at a distance, quantum realism is incompatible with locality.\" However, we show that it is still possible to treat quantum mechanics without appealing to nonlocality or denying realism. We hope that this note will attract attention of the experts in quantum foundations and convince them to come with their own comments on the final Bell's test.\n\nIntroduction\n\nIt finally happened! In 2015 three world's leading experimental groups working on foundational aspects of quantum mechanics (QM) announced (practically simultaneously) that they had performed the loophole free tests of violation of the Bell inequality [1], [2]: the groups of Ronald Hanson (Delf University of Technology) [3], Anton Zeilinger (University of Vienna) [4] and Linden Shalm (NIST, Boulder) [5]. This is definitely a great event in quantum foundations; the more so because it took so long to evolve from the pioneer experiment of Alain Aspect [6] to these final Bell's tests. (Besides, some experts in quantum foundations presented the arguments, in the spirit of Heisenberg uncertainty principle, that the locality and detection loopholes could not be closed in one experiment [7]).\n\nI was surprised to see that this event did not generate a new wave of the quantum foundational enthusiasm, neither in the quantum community nor in general mass-media (even oriented to popularization of science). 1 One of the reasons for this rather mild reaction is that, as was already mentioned, the result was commonly expected. 2 A similar reason is that the Bell inequality has already been widely \"sold\". In literature and talks the Bell argument is typically presented as if everything has already been experimentally proven; scientists working on closing loopholes were considered merely as polishing the famous Aspect's experiment [6]. Those who are closer to quantum foundations could additionally point that G. Weihs [8] contributed to complete Aspect's experiment by closing the locality loophole.\n\nTherefore, it would be great if the recent publications [3]- [5] ignited a serious discussion on the possible impact of this event of the realization of the totally loophole free Bell test. Such a discussion is especially important because conclusions presented in [3]- [5], see also comments on these tests in [9], [10], present only a part of the wide spectrum of views on Bell's argument. Although the presented \"conventional viewpoint\" dominates in the quantum community, it would be natural to represent other, so to say, singular, parts of this spectrum of viewpoints, see, e.g., the recent comment of M. Kupczynski [11].\n\nWe briefly remind the conventional viewpoint presented in [1], [2] and in hundreds of articles and monographs, e.g., [12]- [14], [3]- [10]. It was finally confirmed experimentally that 1. CV1: Einstein was wrong and Bohr was right; 2. CV2: there is spooky action at a distance;\n\n3. CV3: quantum realism is incompatible with locality. 1 The situation reminds me of what happened in mathematics after the famous Fermat theorem had been finally proven. Mathematicians were definitely not able to \"sell\" this great achievement, even in the scientific community; the test-question: who did prove the Fermat theorem? 2 One very famous experimenter whose group at that time worked hard to perform a loophole free Bell's test told me that he would be really excited if in this test the Bell equality would not be violated...\n\nThe views of those who disagree with the presented \"conventional position\" are characterized by the high degree of diversity [15]- [26]. Therefore I shall not try to elaborate on some common \"non-conventional viewpoints\", but present only my own position [25]:\n\n1. NCV1: both Einstein and Bohr were right; 2. NCV2: there is no need in spooky action at a distance;\n\n3. NCV3: quantum realism is compatible with locality.\n\nIn section 2 I shall question the CV1-CV3 viewpoint and try to justify the NCV1-NCV3 viewpoint; in particular, I confront the \"action at a distance interpretation\" with the Copenhagen interpretation of QM. (It is surprising that one may combine without cognitive dissonance these two interpretations.) Then to discuss the issue of realism in QM I appeal to its onticepistemic analysis in the spirit of Atmanspacher and Primas [27]. From this viewpoint, Bell's argument can be treated as the conjecture that ontic states can be identified with epistemic ones. We also discuss this conjecture by appealing to the old Bild conception (Hertz, Boltzman, Schrödinger) about the two descriptive levels of nature, theoretical and observational, see, e.g., [28] and chapter 1 of monograph [29]. Our conclusion is that the rejection of Bell's conjecture as the result of the recent experiments cannot be treated as the impossibility to keep the realist viewpoint. There is neither need in action at a distance. Section 3 is started with the presentation of Kolmogorov's interpretation [30] of classical probability (CP) as the observational theory (describing the epistemic states of nature). By Kolmogorov CP is a contextual theory assigning probability spaces to experimental contexts, complexes of experimental physical conditions. This position leads to the contextual representation of the probabilistic structure of Bell's experimental test [23], [25].\n\nIn section 4 I present my personal picture of future development of quantum foundations, in \"after Bell epoch\": from the total rejection of Bell's conjecture to novel studies on the two descriptive levels approach to QM. In contrast to rather common opinion (see, e.g., Aspect's paper [9] entitled \"Closing the door on Einstein and Bohr's quantum debate\" and Wiseman's paper [10] entitled \"Quantum physics: Death by experiment for local realism\"), for me the final Bell test did not imply the total impossibility to \"go beyond quantum\" [29]. The main message of this test is that the way to a proper subquantum model is more tricky than it was hypothesized by Bell. 3 2 Schrödinger, Einstein, Bohr, and Bell: philosophy meets quantums physics\n\nTypically the output of Bell's argument (nowadays experimentally confirmed) is formulated as QM is incompatible with local realism. Thus either one has to reject the possibility of the realist description of quantum phenomena or to imagine nonlocal reality of the Bohmian type which is even more exotic and mystical than quantum surreality. Since Einstein dreamed for the realist interpretation of quantum phenomena and he considered nonlocality as an \"absurd alternative\" to realism, nowadays it is clear (for everybody, besides a few outsiders, see, e.g., [29], [20], [16]- [26], [11]) that he was wrong and automatically (as Einstein's opponent) Bohr was right, see, e.g., [9]. In general in the modern quantum community it is fashionable to be a follower of Bohr.\n\n\nBohr versus Bell\n\nFirst of all, I want to point to one of the main interpretational problems of the conventional viewpoint on the output of Bell's test, see CV1-CV3. This is an attempt to present CV1-CV3 in a single package and consistently with the Copenhagen interpretation of QM. However, none of the fathers of QM, neither Bohr, Heisenberg, Fock, Landau, Pauli nor Einstein, could even imagine that spooky action at a distance would be taken seriously in the quantum foundations. Thus, one has to recognize that the modern nonlocal feature of QM motivated by Bell's argument has nothing to do with Copenhagen. One can agree or disagree with this statement, but in any event it has to be presented and discussed. Thus to be consistent one has to speak about the revolutionary changes in quantum foundations generated by Bell's argument and experimental tests confirming it. The possibility of these changes was not even imagined by earlier Copenhagenists.\n\nOne can still argue that \"Copenhagenists were against realism\" and, hence, Bell's test can be considered as supporting the Copenhagen position...\n\n\nRealism: philosophy meets physics\n\nRealism is a complex issue, both in physics and philosophy. One cannot debate it without taking into account the recent progress in philosophical studies. And I want to analyze Bell's argument in the light of such studies, namely, the ontic-epistemic viewpoint on physical reality, firmly established in quantum foundations by H. Atmanspacher and Primas [27].\n\nThere are ontic states, assigned to physical systems as \"they are\", and epistemic states representing knowledge that observers gain from measurements on physical systems. QM is about epistemic states. This is in complete agreement with the Copenhagen interpretation of QM, especially Bohr's views.\n\nIt often escapes notice that the EPR-argument [31] was about ontic states; it was directed to show that quantum states are epistemic states. For Einstein, QM was incomplete in the sense that there should exists a finer description of physical processes in the microworld than given by QM. The states of such a subquantum model are ontic states. Einstein and his coauthors did not have any intention to treat the wave function ψ as the ontic state; for them, it was the epistemic state representing knowledge which can be earn from measurements.\n\nThis separation between the two descriptive levels is very important for understanding the real tragedy of QM: misunderstanding about the issue of completeness of QM, between Einstein and Bohr, see [31], [32]. As was pointed out, Einstein's incompleteness is so to say ontic incompleteness of QM, but Bohr's completeness is epistemic completeness of QM. Bohr in his famous reply to Einstein [32] wanted to show that QM is complete at the epistemic level. However, Einstein did not have doubts in epistemic completeness of QM. On the other hand, it seems that Bohr was not so strongly against the possibility of creation of subquantum models; he simply considered such studies as meaningless, because for him physics was only about epistemic states. 4 (This picture of Bohr's views resulted from my long discus-sions with A. Plotnitsky, cf. [33]). However, it seems that Einstein and Bohr did not understand that their debate could be resolved in purely philosophic (ontic-epistemic) terms. Now, let us consider Bell's project from the philosophic viewpoint, by using the ontic-epistemic approach. In philosophic terms, Bell's conjecture was that O=E: Ontic states of quantum systems can be identified with epistemic states.\n\nSince the values of observables are assigned directly to the ontic states,\nλ → a(λ),(1)\nthese states also have to interpreted as the epistemic states. Here a is an observable and the result of a measurement is determined directly by the parameter λ. Hence, λ has to be considered as the epistemic state. At the same time Bell treated λ as representing the state of reality as it is (λ has no relation to context of measurement of a). Hence, it has to be considered as the ontic state. The final loophole free Bell tests demonstrated that this conjecture was wrong (if there is no action at a distance, see NCV2).\n\nHowever, this statement is not the same as to state that realism is incompatible with QM. The impossibility to identify two descriptive levels, ontic and epistemic, is not rejection of the possibility of the realist description.\n\n\nImpact of Bell's test for Copenhagenists\n\nFor Bohr, only epistemic states had physical meaning. Of course, he did not use this terminology, but his statements match it very well; for example [34]:\n\n\"This crucial point, which was to become a main theme of the discussions reported in the following, implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear. In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of empiricism as the foundational principle of science. This principle is the cornerstone of the Copenhagen interpretation of QM. At the same time it is not so common to refer to Mach as one of fathers of this interpretation; in particular, because his orthodox empiricism once had already led to aggressive denial of a basic and nowadays commonly recognized physical structure -the atomic structure of matter.\n\nsubdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle cannot be controlled. Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.\"\n\nTherefore it seems that the problem of identification of ontic and epistemic states was of no interest for Bohr and \"old Copenhagenists.\"\n\n\nBild concept: Schrödinger versus Bell\n\nWe remark that the ontic-epistemic approach can be considered as the modern version of the old Bild-conception tradition (Hertz, Boltzmann and their followers as well as Einstein) [28], [29], chapter 1. In QM this tradition was especially strongly presented in views of Schrödinger who inherited this tradition from his teacher Exner. By the Bild-conception one has to separate two descriptive levels, one is theoretical and another is observational. The latter can be identified with the modernly used epistemic level. The theoretical level, as preceding the observational level, can be coupled to the ontic level of description. However, it seems that these notions cannot be completely identified. The theoretical level was not about \"reality as it is\", but about its theoretical mathematical model. Schrödinger emphasized the impossibility to identify these two descriptive levels: theory and observations are not necessarily related in a term-to-term correspondence and a certain degree of independence exists between them. He also point out that already physicists of 19th century, e.g., Hertz, Boltzmann understood this well. 5 As was emphasized by D'Agostino [28], p. 351, Schrödinger called \"the classical ideal of uninterrupted continuous description\", at both observables' and theoretical levels, an \"old way\", meaning, of course, that this ideal is no longer attainable. He acknowledged that this problem was at the center of the scientific debate in the Nineteenth and Twentieth centuries as well, [35], p.24:\n\n\"Very similar declarations...(were) made again and again by competent physicists a long time ago, all through the Nineteenth Century and the early days of our century...they were aware that the desire for having a clear picture necessarily led one to encumber it with unwarranted details,.'\n\n\nDe Broglie: hidden probabilities\n\nThe excellent presentation of De Broglie's viewpoint on quantum measurement and its consequences for Bell's argument can be found in the paper of G. Lochak [17]. We now present this measurement theory briefly by trying to refine it from the presence of the physical guiding waves, De Broglie's waves.\n\nIn his measurement theory De Broglie assigned the special value to the position measurement: every physical quantity can be measured only via the final detection of the position, e.g., localization of a detector produced a click. The unitary equivalence of various representations in the Hilbert space formalism is just a mathematical feature of the theory. To measure the concrete physical quantity A, systems emitted by a source have to pass an analyzer, e.g., PBS, and then approach the corresponding detector. Probabilities determined by detectors after passing of the analyzer are called the present probabilities; probabilities before the analyzer, corresponding to the prepared pure state, are called the predicted probabilities. The main point of De Broglie's probabilistic considerations is that it is impossible to combine the predicted and present probabilities straightforwardly, i.e., by using the rules of the classical probability calculus, see section 3. In particular, it is impossible to combine the predicted and present probabilities corresponding to two arbitrary analyzers withing classical probability theory, by using the joint probability distribution.\n\nHowever, this fact (a simple consequence of the quantum formalism leading to \"interference of probabilities\", see [25] for details) does not contradict to a possibility to introduce hidden variables, because (according De Broglie) their probability distribution is hidden, i.e., we have to consider the third type of probabilities, hidden probabilities. The main point of De Broglie's analysis is that there is no reason to identify these hidden probabilities, probabilities for ontic states with probabilities produced by QM, namely, the predicted and present probabilities (which are epistemic probabilities). Thus similarly to Schrödinger, Einstein, Atmanspacher, Primas and the author of this paper, De Broglie and Lochak were against identification of ontic and epistemic states and more generally their probability distributions (the latter is the main point of a long series of publications of the author of this note, starting with [36]).\n\nFinally, we mention the guiding wave element of De Broglie's picture. The physical guiding wave appears as the element of the De Broglie's ontic description of micro-phenomena, i.e., as one of variables representing the ontic state. We emphasize that the above scheme with hidden (ontic) and observational (epistemic) probabilities can be explored in general, i.e., without claiming that a subquantum model has to be of the guiding wave type. Here we cannot discuss even briefly the difference in views of L. De Broglie and D. Bohm. We just remark that anybody who read De Broglie personally [37] would never say that De Broglie double solution model and so-called Bohmian mechanics coincide up to the degree of mathematical rigorousness. In particular, we stress that the double solution model is local (as was emphasized by De Broglie [37]) and Bohmian mechanics is nonlocal.\n\n\nEinstein and Bohr\n\nThe above discussion about the ontic-epistemic philosophy and the corresponding structuring the notion of realism demonstrated that the Bell argument has no direct relation neither to the position of Bohr nor Einstein and, hence, it cannot be used to justify the position of one of them in contrast to another. In particular, Einstein would say that he never tried to identify the ontic states with the epistemic states. As was emphasized, Bohr and other Copenhagenists were interested only in the epistemic states. Schrödinger would say that he is not surprised that the coupling between the (possible) theoretic and observational levels is not so straightforward, see (1), as in the Bell-argument.\n\nI hope that the statement NCV1 has been completely justified: both Einstein and Bohr were right, but with respect to the corresponding levels of description.\n\n\nBell versus von Neumann\n\nWe remind that Bell started his foundational project with hard critique [2] of von Neumann's theorem [38]. Nowadays the latter is practically forgotten, not least because Bell's critique. However, we can ask ourselves: Was von Neumann's theorem really so bad?\n\nVon Neumann did not identify the ontic and epistemic states. He established the special rules of correspondence between the two descriptive levels. In principle, he proceeded in the Bild concept tradition: there are theoretical and observational models and correspondence rules. Bell criticized von Neumann's image of a possible coupling between the theoretical and observational models and their critique was reasonable. In particular, by von Neumann the correspondence between ontic and epistemic quantities should be additive, a + b →â +b,\n\nwhich seems to be unacceptable for incompatible observables. However, from the philosophic viewpoint the sharp distinguishing between the two descriptive levels present in von Neumann theorem (personally he called this statement \"ansatz\") is preferable comparing with simple identification of these levels. A follower of Einstein would say [25] that von Neumann simply was not lucky to find the right correspondence rules. As was emphasized in [25], the \"no-go\" activity in establishing the correspondence between the two levels of description is totally meaningless, since there is an infinite variety of possibilities to establish the rules of such a correspondence, see, e.g., [29].\n\n\n\"Bell was wrong\" community\n\nIt is useful to analyze from the ontic-epistemic viewpoint not only the position of the conventional quantum (and more specifically quantum information) community, see CV1-CV3, but even of \"Bell was wrong\" community. It seems that the majority of opponents of the CV1-CV2 interpretation do not understand the philosophic meaning of the Bell conjecture, O=E. Typically one from the anti-Bell opposition tries to show that observational (epistemic) probabilities can violate Bell's inequality, or to show that the standard proofs of the Bell inequality do not work, or to construct an epistemic (local) model producing correlations violating Bell's inequality. In more tricky studies there is even a subquantum model and special rules coupling two descriptive levels, but, of course, such a correspondence is more complex than conjectured in O=E. Unfortunately, such studies are not treated from the double descriptive viewpoint; in particular, the rules of correspondence between subquantum and quantum models are not specified, cf., however, with [29].\n\nThen it is claimed that \"Bell's theorem\" (argument) was wrong. My impression is that this was precisely the way of treatement of the Bell argument by the main anti-Bell gurus: L. Accardi, K. Hess and W. Philipp, M. Kupczynski, H. De Raedt, K., Michielsen T. Nieuwenhuizen and partially myself (although later I proceeded in the De Broglie-Lochak manner by distinguishing hidden probabilities form epistemic). The claim \"Bell was wrong\" acts as a red muleta used by Spanish matadors to make bulls an-gry and generates the strong critical feedback from the pro-Bell community. This \"Bell was wrong\" position was not fruitful for quantum foundations, it induced brutal debates (in particular, during the Växjö series of annual conferences, 2001-2015). By understanding that Bell's argument is based on the O=E conjecture and that activity to test Bell's inequality is directed to check this conjecture we would escape these debates or least to make them less brutal.\n\nIn short, it is definitely possible to construct local probabilistic models violating the Bell inequality, but for such models the Bell conjecture O=E is clearly wrong; in the same way the standard proofs of the Bell inequality work only under the Bell conjecture.\n\n\nClassical probability meets quantum physics\n\nBell's argument is of the probabilistic nature and therefore it is natural to analyze it from the viewpoint of foundations of probability theory. The latter was formalized in the measure-theoretic framework by A. N. Kolmogorov [30] in 1933. As any scientific theory, it was endowed with the corresponding interpretation [30].\n\nBy Kolmogorov [30], see also [25] for discussions, any complex of physical conditions C, experimental context 6 , determines its own probability space P C -the system of events F ≡ F C and the probability P ≡ P C . 7 Classical probability (CP) theory was designed to serve the epistemic description. Here the positions of Kolmogorov and Bohr are very close. In CP complementarity is expressed in assigning different probability spaces to different contexts. 8 6 Here, see [25], the notion of \"context\" started to be in the use. It is important to point out that here context represents \"total experimental arrangement\". Thus contextuality is understood wider than just dependence on measurement of a compatible observable. Our viewpoint on contextuality matches better the views of Bohr than commonly used, e.g., in discussions on Bell's argument, \"restricted contextuality\". We follow Bohr's recommendation that all experimental arrangement has to be taken into account. 7 The system of events has the structure of Boolean σ-algebra and the probability is the σ-additive measure normalized by 1. Here σ express countability: F is closed with respect to countable unions and intersections and probability is countably additive. 8 One of the problems of modern CP is that this clear contextual message of Kolmogorov is not emphasized in presentation of CP in modern textbooks; thus the young generation of probabilists lost this original multi-context and multi-space picture of CP. Typically the context-indexing is omitted. At the beginning it was done only to simplify notations and However, often we have to work with probabilistic data collected for a family of contexts. Therefore CP has to provide some mechanism of embedding of multi-contextual data in a single probability space. It is important to emphasize that there are two main CP constructions of such embedding:\n\n1. Marginal embedding: contextual probabilities are represented as marginal probabilities for a single probability measure P.\n\n\n2.\n\nConditional embedding: contextual probabilities are represented as marginal probabilities for some probability for a single probability measure P.\n\nThe first (straightforward) embedding was excellently realized by A. N. Kolmogorov in his famous theorem [30] about existence of the probability measure P (defined on the space of trajectories) for a stochastic process determined by its probability distributions for all finite sequences of instances of time, P t 1 ...tn . However, this construction can work only in special cases, for special families of contexts. (Therefore Kolmogorov's theorem about the existence of a probability space for a stochastic process is so famous: he found such a special (and very important) case.) It is well known in CP, since the work of Boole, that probability distributions for random variables measured for different contexts can be incompatible, i.e., it is impossible reproduce them as marginal probabilities from the the single probability P [39]. 9 In fact, the inequality providing the necessary condition of the existence of such P derived by Boole coincides with Bell's inequality; therefore some authors even proposed to speak about the Boole-Bell inequality [23]. The complete description of conditions of existence of the straightforward embedding of contextual probabilities in a single probability space was presented in the paper of Vorobj'ev [40] which contains all Bell's type inequalities known in quantum physics (as well as still unknown).\n\nprobabilists (at least representatives of the Soviet school) remembered well Kolmogorov's slogan: \"no probability unrelated to experimental context\". When, say 15 years ago, I spoke with two former students of Kolmogorov, A. N. Shiryaev and A. V. Bulinski, and tried to teach them that CP is contextual, they immediately pointed to section 2 of Kolmogorov's book [30]. For them, it was totally evident that each experimental context generates its own probability space. It seems that this rule is forgotten in the modern probabilistic community. 9 In the quantum foundational community this historical fact was \"discovered\" by I. Pitowsky; in particular, he pointed to it in his talk at Växjö-2001 conference. For many years, I tried to advertise it for experts in quantum foundations, but without any success.\n\nWe remark that the marginal embedding is based on the identification of the ontic and epistemic descriptions. The epistemic probabilities for fixed contexts are identified with the marginal probabilities with respect to the probability distribution P which can be treated as the ontic probability. In particular, the Kolmogorov theorem on the probability space for a stochastic process can be applied only the assumption of the ontic-epistemic identification; without this assumption one has to use theory of quantum stochastic processes [41].\n\nHowever, in contrast to QM, the \"no-go\" activity, the impossibility theorems about striaghtforward marginal embedding was not so popular in CP. And it is clear why. There is widely explored the second embedding procedure based on the identification of concrete contextual probabilities not with marginal, but with conditional probabilities with respect to a single probability measure P. This is so-called randomization procedure. It is especially important in statistics. We shall present it for the concrete multi-contextual test -the Bell test.\n\n\nContextual probabilistic structure of Bell's test\n\nThe considerations of this section are presented in the form of a formal mathematical model in the article [42].\n\nHow would Kolmogorov handle Bell's argument? For Kolmogorov, it is based on consideration of a few experimental contexts C θ i θ ′ j corresponding to fixing the pairs of orientations of PBSs. There are a few probability spaces corresponding to these measurement contexts. Experimental statistical data collected for context C θ i θ ′ j has to be handled in its own probability space\nP C θ i θ ′ j .\n• Can Bell's type inequalities be derived in such multi-space probabilistic framework? Not!\n\n• Would be Kolmogorov surprised by violation of Bell's inequality for multi-space data? Not at all! Moreover, he would be surprised if it were not violated.\n\n• Can one construct a single probability space representing all contexts C θ i θ ′ j ? Yes! Construction is known as randomization. In short randomization can be described as follows:\n\n• A). Choose the probabilities of selections of contexts.\n\n• B). Determine probability distributions for fixed contexts, P C θ i θ ′ j .\n\n• C). Construct probability P serving for all experimental contexts: combine probabilities for fixed contexts P C θ i θ ′ j with probabilities of context-\n\n\nselections. 10\n\nThis single probability measure P represents the multi-context experiment. We stress that, for it, it is possible to derive Bell's inequality, since the latter is just a theorem of CP [25].\n\nIn this framework the probability distributions for fixed contexts P C θ i θ ′ j (which are obtained by experimenters as frequencies) appear as conditional probabilities with respect to fixing selections of the parameters θ i , θ ′ j (contexts C θ i θ ′ j ). However, there is no reason to hope that these conditional probabilities and corresponding conditional quantities, e.g., correlations, would satisfy Bell's inequality, see [42].\n\nIn CP the magnitude ∆ of violation of Bell's inequality can be treated as the numerical measure of multi-space probabilistic structure of the experiment. Thus ∆ represents the degree of contextuality (which is understood a la Bohr as taking into account all experimental arrangement). What does such contextuality mean from the QM foundational viewpoint? Contextuality exhibits itself only the existence of incompatible contexts. Therefore ∆ is also the measure of incompatibility-complementarity. From this viewpoint, the experimental tests violating Bell's inequality confirmed complementarity not only for local observables, such as position and momentum or two spin projections of a single electron, but even for nonlocal observables such as projections of spins measured for a pair of electrons.\n\n\nKolmogorov versus Bell\n\nAs we have seen, the CP description of Bell's test does not lead to revolutionary consequences in the form CV2, CV3. And this is completely clear why. The above CP model of the probabilistic structure of Bell's test is epistemic -it takes into account randomness of selection of experimental contexts, i.e., random generators used for this aim are also considered as measurement devices. In principle, one can say that this is the end of the CP-story about Bell's test.\n\nHowever, we can continue our discussion and represent its output in the hidden variable form. The latter will be not real ontic hidden variables, but so to say \"epistemic hidden variables\".\n\nCP teaches us that all random influences involved in the tests have to incorporated into the model. Probabilist would consider [42] not the\"Bell's map\" (1), but a map of the form:\nω → a(ω), ω = (λ, λ L , λ R ),(3)\nwhere λ L , λ R are random parameters determining with some probabilities, p L , p R outputs of random generators L and R which in turn determine settings. 11 Of course, not only random generators contribute to randomness generated by experimental equipment. There is a lot of randomness in PBSs, in detectors, in optical fibers. This randomness also has to be taken into account by modifying (3).\n\n\nClassical probability, operational approach and Bell's argument\n\nThe above CP-analysis of Bell's test motivates us to comment the representation of observables in the form (3) from the operational viewpoint which is widely used in QM, e.g., [19]. By it there is a preparation procedure and there is a measurement procedure. From the operational viewpoint, in Bell's considerations the preparation procedure is performed by the source of entangled systems and the detection procedure by detectors. However, there are, e.g., random generators determining the orientations of PBSs. They exist! One cannot simply ignore them. In the consistent operational approach they have to taken into account. In the model represented by (3) they are included in the preparation procedure. In fact, we follow De Broglie's who emphasized the role of analyzers in production of the present probabilities.\n\nIn the Bell test for photons PBSs are precisely De Broglie's analyzers.\n\n\nWith Ockham's razor against nonlocality\n\nWe emphasize that we have not appealed to spooky action at a distance. Contextuality 12 leads to violation of Bell's inequality. This motivates NCV2 -\"there is no need in spooky action at a distance\", although not rejection of CV2. In principle, one can still play under the assumption CV2 -if one likes spooky action and it helps her to create the consistent picture of the world. However, there is no need in it. This is the good case to use Ockham's razor, see [43] for discussion. This viewpoint, that by taking into account Kolmogorov probabilistic contextuality of the Bell test one need not mention nonlocality at all, is often criticized by piointing out that contexts C θ i θ ′ j are \"nonlocal\", since their are based on orientations of two spatially separated PBSs. However, this classical nonlocality of spatial location of experimental equipment has nothing to do with action at a distance.\n\nAnother point which was discussed in [7] is that in the Bell conjecture the notion of \"locality\" is represented in very specific form of action at a distance [2]. Where are there space-time variables? Action at a distance locality is not at all Einsteinian locality of relativity theory. In principle, one can try to couple \"Bellian and Einsteinian localities\" by taking into account the space-time dependence of correlations for entangled systems [44]. However, by proceeding this way we lose the simple prequantum-quantum (ontic-epistemic) correspondence given by (1). The real space-time correlations contain the at least contribution of media for signal propagation; then one has to take into account dispersion and losses in this media, finally the temporal structure of functioning of photo-detectors [45]. Thus the model becomes epistemic; violation of Bell's inequality by such correlations is not surprising.\n\n\nQuantum realism\n\nBell's argument strongly supported anti-realist tendency in the quantum community. It was very supporting for development of a variety of information interpretations of QM -with two bright examples, the Zeilinger-Brukner information interpretation [50] and Fuchs' QBism [51], [52]. Nowadays a randomly chosen representative of the quantum community would say with very high probability (I account it as 99%) that realism and QM are incompatible.\n\nI totally disagree with this position, see, e.g., my manifest [?]may be naive, since at that time I was not so well educated in quantum foundational issues). I repeat that the successful Bell's test only rejects the straightforward coupling (1) between the ontic and epistemic or more generally theoretic and observation levels. Other maps can be constructed and the fact that we still do not have a commonly acceptable map may just mean lack of imagination, cf. with the famous citation of Bell [46], p. 15: \"what is proved, by impossibility proofs, is lack of imagination.\"\n\nHowever, realism discussed in section 2 is merely subquantum realism. What is about, so to say, genuine quantum realism?\n\nAs was shown, CP modeling of physical experiments, including multicontextual experiments such as Bell's test, does not lead to rejection of realism. This is realism of experimental contexts and observations performed for these contexts. This is precisely \"physical realism\", defined as the statement that the goal of physics is to study entities of the natural world, existing independently from any particular observer's perception, and obeying universal and intelligible rules, see Auffèves and Grangier [47].\n\nIn my works this realism was treated as contextual realism of QM [25]. However, this terminology might be misleading, since in quantum foundations contextuality is treated restrictively, as dependence on outcomes of joint measurement of a compatible observable. We can also speak about epistemic realism.\n\nFinally, we remark that physical (contextual) realism matches well with Bohr's views.\n\nOne may even say that my contextual quantum realism (similarly quantum realism of Grangier) 13 is trivially and commonly acceptable. In fact, this is not the case. And even if it were the case, it is very important to emphasize realism of QM. The main problem is wide spread of identifying of ontic and epistemic realisms. As was discussed, the impossibility of this identification was interpreted as the impossibility to keep ontic realism (at least locally). However, what is the most surprising this rejection of (local) ontic realism is often represented as total rejection of realism in QM.\n\n\nVäxjö interpretation of quantum mechanics\n\nThe Växjö interpretation (VI) of QM is the realist statistical local and contextual interpretation of QM [53]- [55]. We shall briefly explain the meanings assigned to these terms.\n\n• The easiest for explanation is \"statistical\". By VI probabilities are interpreted statistically, i.e., they are related to ensembles of systems and not individual systems.\n\n• VI is local, simply no need even to rise the issue of nonlocality -in complete agreement with Einstein who mentioned nonlocality is an absurd alternative to realism.\n\n• VI considers both realisms: \"subquantum realism\" (cf. Schrödinger, Einstein, De Broglie, Lochak, Atmanspacher and Primas) and \"contextual realism\" (cf. Bohr, Auffèves and Grangier). The first one is more challenging. We all still suffer of the lack of imagination... The second one was clearly presented in section 3.5.\n\n• VI is contextual, as any interpretation referring to two descriptive levels, because any epistemic model is contextual as taking into account all experimental arrangement.\n\n\nFuture\n\nRecent successful tests for Bell's inequality can be interpreted in various ways, with two extremes, CV1-CV3 and NC1-NCV3, with numerous intermediate interpretations. We can say that experimenters did their job and excellently. However, the loophole free Bell tests should not be interpreted as finalizing the century long discussion about the possibility to construct a theoretical subquantum model coupled to the quantum observational model. My interpretation is that the successful realization of Bell's test confirmed that\n\n• We have to distinguish sharply two descriptive levels, theoretical and observational (ontic and epistemic).\n\n• The correspondence between them can be very tricky (so, not so straightforward as in the Bell conjecture O=E). 14 • Contextuality plays the crucial role in coupling of the two descriptive levels.\n\nThus people looking for new insights on QM have to work hardly to find the proper subquantum theoretical description by relaxing constraints on the rules of correspondence between the two descriptive levels.\n\nThe final loophole free test says goodbye to the last hopes to identify the observational and ontic descriptions. One has to look for more complex maps\nf : SUBQM → QM.(4)\nThis modeling has to be physically constructive, i.e., from the very beginning one should put efforts not to beat some no-go theorem(s), but to create a physically meaningful theoretical model 15 SUBQM endowed with a physically meaningful map f. A partial experimental verification is a part of this great project which successful realization would finally resolve (as I believe!!!) the puzzle of QM.\n\nPersonally I still belief [29] in the possibility to construct a subquantum model of the classical random field type, cf. Einstein and Infeld [56], with nonlinear subquantum dynamics. This is also the main message of De Broglie. Although, in contrast to Einstein and Infeld and the author of this note, he considered not the ordinary say classical electromagnetic field, but the so-called guiding field, this field is also physical. Thus one may hope to measure it, in some way. Coming back to the Einstein-Infeld hypothesis about classical subquantum field, we can say that, for a moment, experimentalists can try to proceed to new measurement technologies which would provide a possibility to measure characteristics of classical fields for quantum systems, e.g., the polarization vector.\n\n14 Cf., e.g., with prequantum classical statistical field theory [29]. From my viewpoint this model reproduce all probabilistic predictions of QM for one and bipartite systems; for tripartite systems the representation of entanglement becomes extremely technical and hardly interpretable; for quadripartite systems I was not able to resolve technical difficulties. 15 It may be surprising to hear this statement from a pure mathematician. However, in contrast to physicists who are so excited by the ability of mathematics to model everything in nature, I recognize that sometimes this ability looks like the ability of a prostitution to serve all men's desires. Thus a purely mathematical subquantum model is practically valueless.\n\n\nAcknowledgments\n\nI was lucky to discuss the Bell test and its possible interpretations with practically all world's leading experts in quantum foundations, those who firmly support the CV1-CV3 position as well as those who play with various variants of NCV1-NCV2. I would like thank all them for discussions and insights. I am especially thankful to A. Zeilinger who expressed in lovely Viennese-style discussions the firm anti-realist interpretation and K. Hess who expressed in the same lovely Viennese manner the firm realist position. It was very important for me that both are highly qualified physicists, that they do not simply play with mathematical formulas, but there is a lot of physical intuition behind their positions. And, of course, I was lucky to meet personally A. N. Kolmogorov and speak with him about foundations of CP (well, just 15 min conversation of the old and sick great man with the PhD-student who wrote a paper about a \"non-Komogorov probability model\").\n\nThe first test of this note was performed at March 18, 2016, at the Quantum Seminar of Stockholm University; I would like to thank all participants of this seminar and especially I. Bengtsson and M. A. M. Versteegh for constructive critique. For writing of this paper the email correspondence with H. Atmanspacher and M. Kupczynski was very stimulating.\n I also remark that even those who would not be convinced by my philosophic and probabilistic arguments should agree that we have no choice -we have to go beyond quantum, at least to solve the problem of merging of two great physical theories: QM (or more precisely QFT) and general relativity.\nN. Bohr as well as W. Heisenberg were strongly influenced by E. Mach who advocated\nBoltzmann asserted that only one half of our experience is ever experience.\nThis P is defined on the system of events F related to this randomized experiment; it contains not only events related to observations for fixed contexts C θiθ ′ j , but also events of selections of these contexts.\nIn Bell's test for the CHSH inequality each random generator selects a pair of angles with probabilities p L (i), P R (j), i, j = 1, 2.\nWe understand it as determination of probability space by experimental context.\nTo reconcile realism with QM, Grangier et al. also proceed in the contextual framework[47]-[48]. However, they understood contextuality differently from me and from the conventional viewpoint (as joint measurement with a compatible observable); for them, context is not simply a complex of experimental physical conditions, see[47]-[48] for details. Thus in this note we operate with three different types of contextuality.\nThis research is a part of the project \"Mathematical Modeling of Complex Hierarchic Systems\" of the Faculty of Technology of Linnaeus University; it is also supported by Quantum Bio-Informatic Center of Tokyo University of Science.\nOn the Einstein-Podolski-Rosen paradox. J S Bell, Physics. 1195J.S. Bell, \"On the Einstein-Podolski-Rosen paradox\", Physics 1, 195 (1964).\n\nJ Bell, Speakable and Unspeakable in Quantum Mechanics. CambridgeCambridge Univ. PressBell, J.: Speakable and Unspeakable in Quantum Mechanics. Cambridge Univ. Press, Cambridge (1987)\n\nExperimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km. B Hensen, arXiv:1508.05949Nature. 526682quant-phB. Hensen et al, Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km, Nature 526, 682 (2015); arXiv:1508.05949 [quant-ph] (2015)\n\nA significant-loophole-free test of Bell's theorem with entangled photons. M Giustina, arXiv: quant- ph1511.03190Phys. Rev. Lett. 115250401M. Giustina et al, A significant-loophole-free test of Bell's theorem with entangled photons, Phys. Rev. Lett. 115, 250401 (2015) [arXiv: quant- ph1511.03190].\n\nA strong loophole-free test of local realism. L K Shalm, arXiv:quant-ph1511.03189Phys. Rev. Lett. 115250402L. K. Shalm et al, A strong loophole-free test of local realism, Phys. Rev. Lett. 115, 250402 (2015) [arXiv:quant-ph1511.03189].\n\nThree Experimental Tests of Bell Inequalities by the Measurement of Polarization Correlations between Photons. A Aspect, Orsay PressOrsayAspect, A. (1983). Three Experimental Tests of Bell Inequalities by the Measurement of Polarization Correlations between Photons (Orsay Press, Orsay).\n\nLocal realistic representation for correlations in the original EPR model for position and momentum. A Khrennikov, I V Volovich, Soft Computing. 10521A. Khrennikov and I.V. Volovich, Local realistic representation for cor- relations in the original EPR model for position and momentum, Soft Computing 10, 521 (2005).\n\nEin Experiment zum Test der Bellschen Ungleichung unter Einsteinscher Lokalitat. G Weihs, University of Vienna PressViennaWeihs, G. (1999). Ein Experiment zum Test der Bellschen Ungleichung unter Einsteinscher Lokalitat (University of Vienna Press, Vienna).\n\nClosing the Door on Einstein and Bohr's Quantum Debate. A Aspect, Physics. 8123A. Aspect, Closing the Door on Einstein and Bohr's Quantum Debate, Physics 8, 123 (2015)\n\nQuantum physics: Death by experiment for local realism. H Wiseman, Nature. 526649650H. Wiseman, Quantum physics: Death by experiment for local realism. Nature 526, 649650 (2015).\n\nCan Einstein with Bohr Debate on Quantum Mechanics Be Closed?. Kupczynski, arXiv:1603.00266quant-phKupczynski, Can Einstein with Bohr Debate on Quantum Mechanics Be Closed? arXiv:1603.00266 [quant-ph].\n\n. J F Clauser, M A Horne, A Shimony, R A Holt, Phys. Rev. Lett. 2315Clauser J F, Horne M A, Shimony A and Holt R A 1969 Phys. Rev. Lett. 23 (15) 880-884\n\n. J F Clauser, M A Horne, Phys. Rev. D. 10Clauser J F and Horne M A 1974 Phys. Rev. D 10 526-535\n\n. J F Clauser, A Shimony, Rep. Prog. Phys. 41Clauser J F and Shimony A 1978 Rep. Prog. Phys. 41 1881-1927\n\nResolution of the Einstein-Podolsky-Rosen and Bell paradoxes. I Pitowsky, Erratum Phys. Rev. Lett. 481768Phys. Rev. Lett.Pitowsky, I. (1982). Resolution of the Einstein-Podolsky-Rosen and Bell paradoxes, Phys. Rev. Lett. 48, 1299; Erratum Phys. Rev. Lett. 48, 1768.\n\nBertrand's paradox and Bell's inequalities. M Kupczynski, Phys. Lett. A. 121Kupczynski, M. (1987). Bertrand's paradox and Bell's inequalities, Phys. Lett. A 121, pp. 205-207.\n\nDe Broglie's initial conception of De Broglie waves. G Lochak, The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birthday. Diner, S., Fargue, D. Lochak, G. and Selleri, F.DordrechtD. Reidel Publ. CompanyG. Lochak, De Broglie's initial conception of De Broglie waves. In: Diner, S., Fargue, D. Lochak, G. and Selleri, F. (eds) The Wave-Particle Du- alism. A Tribute to Louis de Broglie on his 90th Birthday, pp. 1-26. D. Reidel Publ. Company, Dordrecht (1984)\n\nThe probabilistic roots of the quantum mechanical paradoxes. L Accardi, The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birthday. Diner, S., Fargue, D. Lochak, G. and Selleri, F.DordrechtD. Reidel Publ. CompanyAccardi, L.: The probabilistic roots of the quantum mechanical para- doxes. In: Diner, S., Fargue, D. Lochak, G. and Selleri, F. (eds) The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birth- day, pp. 47-55. D. Reidel Publ. Company, Dordrecht (1984)\n\nFoundations of Quantum Mechanics, an Empiricists Approach. W M De Muynck, Kluwer, DordrechtDe Muynck, W. M. (2002). Foundations of Quantum Mechanics, an Empiricists Approach. (Kluwer, Dordrecht).\n\nK Hess, Einstein was right!. SingaporePan Stanford PublHess, K. (2014). Einstein was right! (Pan Stanford Publ., Singapore).\n\nEvent-based computer simulation model of Aspecttype experiments strictly satisfying Einstein's locality conditions. H De Raedt, K De Raedt, K Michielsen, K Keimpema, S Miyashita, Phys. Soc. Japan. 76104005De Raedt, H., De Raedt, K., Michielsen, K., Keimpema, K. and Miyashita, S. (2007). Event-based computer simulation model of Aspect- type experiments strictly satisfying Einstein's locality conditions, Phys. Soc. Japan 76, 104005.\n\nBrownian entanglement. A Allahverdyan, A Khrennikov, Yu, Th M Nieuwenhuizen, Phys. Rev. A. 71Allahverdyan, A., Khrennikov, A.Yu. and Nieuwenhuizen, Th.M.: Brow- nian entanglement. Phys. Rev. A 71, 032102-1 -032102-14 (2005).\n\nBell-Boole inequality: Nonlocality or probabilistic incompatibility of random variables?. A Khrennikov, Entropy. 102A. Khrennikov, Bell-Boole inequality: Nonlocality or probabilistic in- compatibility of random variables? Entropy, 10, N 2, 19-32 (2008).\n\nLocal Realist Approach and Numerical Simulation of Nonclassical Experiments in Quantum Mechanics. G Adenier, Vẍjö Univ. PressVẍjöAdenier, G.: Local Realist Approach and Numerical Simulation of Non- classical Experiments in Quantum Mechanics. Vẍjö Univ. Press, Vẍjö (2008)\n\nA Khrennikov, 978-1-4020-9592-4Contextual Approach to Quantum Formalism. SpringerA. Khrennikov, Contextual Approach to Quantum Formalism, Springer, ISBN 978-1-4020-9592-4 (2009).\n\nWhere Bell went wrong?. Th M Nieuwenhuizen, G Accardi, C A Adenier, G Fuchs, A Jaeger, Yu, Khrennikov, AIP Conf. Proc., Foundations of Probability and Physics -5 1101. J.-A. Larsson and S. StenholmMelville, NYNieuwenhuizen, Th. M. (2009). Where Bell went wrong?, in L. Accardi, G. Adenier, C. A. Fuchs, G. Jaeger, A. Yu. Khrennikov, J.-A. Larsson and S. Stenholm (eds.), AIP Conf. Proc., Foundations of Probability and Physics -5 1101 (Am. Inst. Phys., Melville, NY), pp. 127-133.\n\nEpistemic and ontic quantumrealities. H Atmanspacher, H Primas, Foundations of Probability and Physics-3. Adenier, G., Khrennikov, A. Yu.Melville, NYAmerican Institute of Physics750Ser. Conference ProceedingsAtmanspacher, H. and Primas, H.: Epistemic and ontic quantumreal- ities. In: Adenier, G., Khrennikov, A. Yu. (eds) Foundations of Prob- ability and Physics-3, pp. 49-62. American Institute of Physics, Ser. Conference Proceedings 750, Melville, NY (2005)\n\nContinuity and completeness in physical theory: Schrödinger's return to the wave interpretation of quantum mechanics in the 1950's. S D'agostino, E. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Bitbol, M. and O. DarrigolGif-sur-YvetteD'Agostino, S.: Continuity and completeness in physical theory: Schrödinger's return to the wave interpretation of quantum mechanics in the 1950's. In: Bitbol, M. and O. Darrigol (eds.) E. Schrödinger: Philos- ophy and the Birth of Quantum Mechanics, 339-360. Editions Frontieres, Gif-sur-Yvette (1992)\n\n. A Khrennikov Beyond, Quantum, Pan Stanford PublSingaporeA. Khrennikov Beyond Quantum, Pan Stanford Publ., Singapore, 2014.\n\nEnglish translation: Kolmogorov, A.N.: Foundations of Theory of Probability. A N Kolmogoroff, Chelsea Publishing CompanyBerlin; New YorkGrundbegriffe der WahrscheinlichkeitsrechnungKolmogoroff, A. N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer Verlag, Berlin (1933); English translation: Kolmogorov, A.N.: Foundations of Theory of Probability. Chelsea Publishing Company, New York (1956).\n\nCan Quantum-Mechanical Description of Physical Reality Be Considered Complete?. A Einstein, B Podolsky, N Rosen, Phys. Rev. 47A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical De- scription of Physical Reality Be Considered Complete? Phys. Rev. 47, 777-780 (1935).\n\nCan Quantum-Mechanical Description of Physical Reality be Considered Complete?. N Bohr, Phys. Rev. 48N. Bohr, Can Quantum-Mechanical Description of Physical Reality be Considered Complete?, Phys. Rev. 48, 696-702 (1935).\n\nReading Bohr: Physics and Philosophy. A Plotnitsky, SpringerDordrechtPlotnitsky, A.: Reading Bohr: Physics and Philosophy. Springer, Dor- drecht (2006)\n\nThe causality problem in atomic physics. N Bohr, The Philosophical Writings of Niels Bohr. J. Faye and H. J. FolseOx Bow Press4Supplementary PapersBohr, N. (1938). The causality problem in atomic physics, in J. Faye and H. J. Folse (eds.), (1987). The Philosophical Writings of Niels Bohr, Volume 4: Causality and Complementarity, Supplementary Papers (Ox Bow Press;\n\n. C T Woodbridge, Woodbridge, CT), pp. 94-121.\n\nE. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Editions Frontieres. Bitbol, M. and Darrigol, O.Gif-sur-YvetteBitbol, M. and Darrigol, O. (eds.): E. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Editions Frontieres, Gif-sur-Yvette (1992).\n\nStatistical measure of ensemble nonreproducibility and correction to Bell's inequality. A Khrennikov, Il Nuovo Cimento, B. 1152A. Khrennikov, Statistical measure of ensemble nonreproducibility and correction to Bell's inequality. Il Nuovo Cimento, B 115, N. 2, 179-184 (1999).\n\nThe Current Interpretation of Wave Mechanics. A Critical Study. L De Broglie, Andrade E Silva, J , ElsevierAmstredam-London-New Yorkde Broglie L and Andrade E Silva J (the author of a chapter) 1964 The Current Interpretation of Wave Mechanics. A Critical Study (Amstredam-London-New York: Elsevier)\n\n. J Von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton Univ. Pressvon Neumann J 1955 Mathematical Foundations of Quantum Mechanics (Princeton: Princeton Univ. Press)\n\nOn the theory of probabilities. G Boole, Phil. Trans. Royal Soc. London. 152Boole, G.: On the theory of probabilities. Phil. Trans. Royal Soc. Lon- don, 152, 225-242 (1862)\n\nConsistent families of measures and their extensions. N N Vorob'ev, Theory of Probability and its Applications. 7Vorob'ev, N. N. (1962). Consistent families of measures and their exten- sions, Theory of Probability and its Applications 7, pp. 147-162.\n\nOn the noncommutative Markov property. L Accardi, Func. Anal. Appl. 918Accardi L., On the noncommutative Markov property, Func. Anal. Appl. , 9 (1975) 18.\n\nCHSH inequality: quantum probabilities as classical conditional probabilities Found. A Khrennikov, Phys. 45A. Khrennikov, CHSH inequality: quantum probabilities as classical conditional probabilities Found. Phys. 45, N 7, 711-725 (2015).\n\nBell argument: Locality or Realism? Time to make the choice. A Khrennikov, arXiv:1108.0001v3quant-phA. Khrennikov, Bell argument: Locality or Realism? Time to make the choice. arXiv:1108.0001v3 [quant-ph]\n\nQuantization of propagating modes in optical fibers. A Khrennikov, B Nilsson, S Nordebo, I V Volovich, Phys. Scr. 8565404A. Khrennikov, B. Nilsson, S. Nordebo, and I. V. Volovich, Quantization of propagating modes in optical fibers, Phys. Scr., 85, 2012. 065404\n\nPhoton Flux and Distance from the Source: Consequences for Quantum Communication. Found. A Khrennikov, B Nilsson, S Nordebo, I Volovich, Physics. 44A. Khrennikov, B. Nilsson, S. Nordebo, I. Volovich, Photon Flux and Distance from the Source: Consequences for Quantum Communication. Found. Physics 44 389-405 (2014).\n\nJ Bell, On the impossible pilot wave, Ref.TH.3315-CERN. 15J. Bell, On the impossible pilot wave, Ref.TH.3315-CERN, 1982, p. 15\n\nViolation of Bell's inequalities in a quantum realistic framework. Special issue of Växjö-15 conference. A Auffèves, P Grangier, to be published in IJQIA. Auffèves and P. Grangier, Violation of Bell's inequalities in a quantum realistic framework. Special issue of Växjö-15 conference, to be published in IJQI.\n\nContextual objectivity: a realistic interpretation of quantum mechanics. P Grangier, arXiv:quant-ph/0012122European Journal of Physics. 23P. Grangier, Contextual objectivity: a realistic interpretation of quantum mechanics, European Journal of Physics 23, 331 (2002) [arXiv:quant-ph/0012122].\n\nContextual objectivity and the quantum formalism. P Grangier, arXiv:quant-ph/0407025IJQIP. Grangier, Contextual objectivity and the quantum formalism, IJQI, 3 (1): 17-22 (2005) [arXiv:quant-ph/0407025]\n\nInformation Invariance and Quantum Probabilities. C Brukner, A Zeilinger, Found. Phys. 39677C. Brukner, A. Zeilinger, Information Invariance and Quantum Proba- bilities, Found. Phys. 39, 677 (2009).\n\nQuantum mechanics as quantum information (and only a little more. C A Fuchs, Quantum Theory: Reconsideration of Foundations. A. KhrennikovVäxjöVäxjö University Press2Fuchs, C. A. (2002). Quantum mechanics as quantum information (and only a little more), in A. Khrennikov (ed.), Quantum Theory: Reconsid- eration of Foundations, Ser. Math. Modeling 2 (Växjö University Press, Växjö), pp. 463-543.\n\nThe anti-Växjö interpretation of quantum mechanics, Quantum Theory: Reconsideration of Foundations. C A Fuchs, Ser. Math. Model. 2Växjö University PressFuchs, C. A. (2002). The anti-Växjö interpretation of quantum me- chanics, Quantum Theory: Reconsideration of Foundations, Ser. Math. Model. 2 (Växjö University Press, Växjö), pp. 99-116.\n\nVäxjö interpretation of quantum mechanics. A Khrennikov, arXiv:quant-ph/0202107v2A. Khrennikov Quantum Theory:Reconsideration of Foundations. VäxjöVäxjö University Press2A. Khrennikov, Växjö interpretation of quantum mechanics. In: A. Khrennikov Quantum Theory:Reconsideration of Foundations\" Ser. Math. Model. 2, p.163-170, Växjö University Press, Växjö (2002). [arXiv:quant-ph/0202107v2]\n\n(ed) Quantum Theory: Reconsideration of Foundations-2. A Khrennikov, Ser. Math. Model. Khrennikov, A.Yu.10Växjö University PressVäxjö interpretation-2003: Realism of contextsA. Khrennikov, Växjö interpretation-2003: Realism of contexts. In: Khrennikov, A.Yu. (ed) Quantum Theory: Reconsideration of Foundations-2. Ser. Math. Model. 10, pp. 323-338. Växjö University Press, Växjö (2004).\n\nVäxjö interpretation of wave function. A Khrennikov, arXiv:quant-ph/1210.2390A. Khrennikov, Växjö interpretation of wave function: 2012. [arXiv:quant-ph/1210.2390]\n\nEvolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. A Einstein, L Infeld, Simon and SchusterNew-YorkEinstein, A. and Infeld, L.: Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. Simon and Schuster, New-York (1961)\n"},"corpus_id":{"kind":"number","value":99023948,"string":"99,023,948"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1603.08674\",\n \"mag\": 2512069842,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1002/prop.201600044\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1603.08674v1.pdf\"\n ],\n \"pdf_sha\": \"7b4ef0fdf5806b668ed34289f8f69e9801fc7922\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1603.08674\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":""},"authors":{"kind":"list like","value":["Andrei Khrennikov \nInternational Center for Mathematical Modeling in Physics\nCognitive Science Linnaeus University\nEngineeringVäxjöEconomicsSweden"],"string":"[\n \"Andrei Khrennikov \\nInternational Center for Mathematical Modeling in Physics\\nCognitive Science Linnaeus University\\nEngineeringVäxjöEconomicsSweden\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We analyze foundational consequences of recently announced loophole free tests of violation of Bell's inequality. We consider two interpretations of these remarkable experiments. By the conventional one \"Einstein was wrong and Bohr was right, there is spooky action at a distance, quantum realism is incompatible with locality.\" However, we show that it is still possible to treat quantum mechanics without appealing to nonlocality or denying realism. We hope that this note will attract attention of the experts in quantum foundations and convince them to come with their own comments on the final Bell's test."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"It finally happened! In 2015 three world's leading experimental groups working on foundational aspects of quantum mechanics (QM) announced (practically simultaneously) that they had performed the loophole free tests of violation of the Bell inequality [1], [2]: the groups of Ronald Hanson (Delf University of Technology) [3], Anton Zeilinger (University of Vienna) [4] and Linden Shalm (NIST, Boulder) [5]. This is definitely a great event in quantum foundations; the more so because it took so long to evolve from the pioneer experiment of Alain Aspect [6] to these final Bell's tests. (Besides, some experts in quantum foundations presented the arguments, in the spirit of Heisenberg uncertainty principle, that the locality and detection loopholes could not be closed in one experiment [7]).\nI was surprised to see that this event did not generate a new wave of the quantum foundational enthusiasm, neither in the quantum community nor in general mass-media (even oriented to popularization of science). 1 One of the reasons for this rather mild reaction is that, as was already mentioned, the result was commonly expected. 2 A similar reason is that the Bell inequality has already been widely \"sold\". In literature and talks the Bell argument is typically presented as if everything has already been experimentally proven; scientists working on closing loopholes were considered merely as polishing the famous Aspect's experiment [6]. Those who are closer to quantum foundations could additionally point that G. Weihs [8] contributed to complete Aspect's experiment by closing the locality loophole.\nTherefore, it would be great if the recent publications [3]- [5] ignited a serious discussion on the possible impact of this event of the realization of the totally loophole free Bell test. Such a discussion is especially important because conclusions presented in [3]- [5], see also comments on these tests in [9], [10], present only a part of the wide spectrum of views on Bell's argument. Although the presented \"conventional viewpoint\" dominates in the quantum community, it would be natural to represent other, so to say, singular, parts of this spectrum of viewpoints, see, e.g., the recent comment of M. Kupczynski [11].\nWe briefly remind the conventional viewpoint presented in [1], [2] and in hundreds of articles and monographs, e.g., [12]- [14], [3]- [10]. It was finally confirmed experimentally that 1. CV1: Einstein was wrong and Bohr was right; 2. CV2: there is spooky action at a distance;\n3. CV3: quantum realism is incompatible with locality. 1 The situation reminds me of what happened in mathematics after the famous Fermat theorem had been finally proven. Mathematicians were definitely not able to \"sell\" this great achievement, even in the scientific community; the test-question: who did prove the Fermat theorem? 2 One very famous experimenter whose group at that time worked hard to perform a loophole free Bell's test told me that he would be really excited if in this test the Bell equality would not be violated...\nThe views of those who disagree with the presented \"conventional position\" are characterized by the high degree of diversity [15]- [26]. Therefore I shall not try to elaborate on some common \"non-conventional viewpoints\", but present only my own position [25]:\n1. NCV1: both Einstein and Bohr were right; 2. NCV2: there is no need in spooky action at a distance;\n3. NCV3: quantum realism is compatible with locality.\nIn section 2 I shall question the CV1-CV3 viewpoint and try to justify the NCV1-NCV3 viewpoint; in particular, I confront the \"action at a distance interpretation\" with the Copenhagen interpretation of QM. (It is surprising that one may combine without cognitive dissonance these two interpretations.) Then to discuss the issue of realism in QM I appeal to its onticepistemic analysis in the spirit of Atmanspacher and Primas [27]. From this viewpoint, Bell's argument can be treated as the conjecture that ontic states can be identified with epistemic ones. We also discuss this conjecture by appealing to the old Bild conception (Hertz, Boltzman, Schrödinger) about the two descriptive levels of nature, theoretical and observational, see, e.g., [28] and chapter 1 of monograph [29]. Our conclusion is that the rejection of Bell's conjecture as the result of the recent experiments cannot be treated as the impossibility to keep the realist viewpoint. There is neither need in action at a distance. Section 3 is started with the presentation of Kolmogorov's interpretation [30] of classical probability (CP) as the observational theory (describing the epistemic states of nature). By Kolmogorov CP is a contextual theory assigning probability spaces to experimental contexts, complexes of experimental physical conditions. This position leads to the contextual representation of the probabilistic structure of Bell's experimental test [23], [25].\nIn section 4 I present my personal picture of future development of quantum foundations, in \"after Bell epoch\": from the total rejection of Bell's conjecture to novel studies on the two descriptive levels approach to QM. In contrast to rather common opinion (see, e.g., Aspect's paper [9] entitled \"Closing the door on Einstein and Bohr's quantum debate\" and Wiseman's paper [10] entitled \"Quantum physics: Death by experiment for local realism\"), for me the final Bell test did not imply the total impossibility to \"go beyond quantum\" [29]. The main message of this test is that the way to a proper subquantum model is more tricky than it was hypothesized by Bell. 3 2 Schrödinger, Einstein, Bohr, and Bell: philosophy meets quantums physics\nTypically the output of Bell's argument (nowadays experimentally confirmed) is formulated as QM is incompatible with local realism. Thus either one has to reject the possibility of the realist description of quantum phenomena or to imagine nonlocal reality of the Bohmian type which is even more exotic and mystical than quantum surreality. Since Einstein dreamed for the realist interpretation of quantum phenomena and he considered nonlocality as an \"absurd alternative\" to realism, nowadays it is clear (for everybody, besides a few outsiders, see, e.g., [29], [20], [16]- [26], [11]) that he was wrong and automatically (as Einstein's opponent) Bohr was right, see, e.g., [9]. In general in the modern quantum community it is fashionable to be a follower of Bohr."},{"title":"Bohr versus Bell","content":"First of all, I want to point to one of the main interpretational problems of the conventional viewpoint on the output of Bell's test, see CV1-CV3. This is an attempt to present CV1-CV3 in a single package and consistently with the Copenhagen interpretation of QM. However, none of the fathers of QM, neither Bohr, Heisenberg, Fock, Landau, Pauli nor Einstein, could even imagine that spooky action at a distance would be taken seriously in the quantum foundations. Thus, one has to recognize that the modern nonlocal feature of QM motivated by Bell's argument has nothing to do with Copenhagen. One can agree or disagree with this statement, but in any event it has to be presented and discussed. Thus to be consistent one has to speak about the revolutionary changes in quantum foundations generated by Bell's argument and experimental tests confirming it. The possibility of these changes was not even imagined by earlier Copenhagenists.\nOne can still argue that \"Copenhagenists were against realism\" and, hence, Bell's test can be considered as supporting the Copenhagen position..."},{"title":"Realism: philosophy meets physics","content":"Realism is a complex issue, both in physics and philosophy. One cannot debate it without taking into account the recent progress in philosophical studies. And I want to analyze Bell's argument in the light of such studies, namely, the ontic-epistemic viewpoint on physical reality, firmly established in quantum foundations by H. Atmanspacher and Primas [27].\nThere are ontic states, assigned to physical systems as \"they are\", and epistemic states representing knowledge that observers gain from measurements on physical systems. QM is about epistemic states. This is in complete agreement with the Copenhagen interpretation of QM, especially Bohr's views.\nIt often escapes notice that the EPR-argument [31] was about ontic states; it was directed to show that quantum states are epistemic states. For Einstein, QM was incomplete in the sense that there should exists a finer description of physical processes in the microworld than given by QM. The states of such a subquantum model are ontic states. Einstein and his coauthors did not have any intention to treat the wave function ψ as the ontic state; for them, it was the epistemic state representing knowledge which can be earn from measurements.\nThis separation between the two descriptive levels is very important for understanding the real tragedy of QM: misunderstanding about the issue of completeness of QM, between Einstein and Bohr, see [31], [32]. As was pointed out, Einstein's incompleteness is so to say ontic incompleteness of QM, but Bohr's completeness is epistemic completeness of QM. Bohr in his famous reply to Einstein [32] wanted to show that QM is complete at the epistemic level. However, Einstein did not have doubts in epistemic completeness of QM. On the other hand, it seems that Bohr was not so strongly against the possibility of creation of subquantum models; he simply considered such studies as meaningless, because for him physics was only about epistemic states. 4 (This picture of Bohr's views resulted from my long discus-sions with A. Plotnitsky, cf. [33]). However, it seems that Einstein and Bohr did not understand that their debate could be resolved in purely philosophic (ontic-epistemic) terms. Now, let us consider Bell's project from the philosophic viewpoint, by using the ontic-epistemic approach. In philosophic terms, Bell's conjecture was that O=E: Ontic states of quantum systems can be identified with epistemic states.\nSince the values of observables are assigned directly to the ontic states,\nthese states also have to interpreted as the epistemic states. Here a is an observable and the result of a measurement is determined directly by the parameter λ. Hence, λ has to be considered as the epistemic state. At the same time Bell treated λ as representing the state of reality as it is (λ has no relation to context of measurement of a). Hence, it has to be considered as the ontic state. The final loophole free Bell tests demonstrated that this conjecture was wrong (if there is no action at a distance, see NCV2).\nHowever, this statement is not the same as to state that realism is incompatible with QM. The impossibility to identify two descriptive levels, ontic and epistemic, is not rejection of the possibility of the realist description."},{"title":"Impact of Bell's test for Copenhagenists","content":"For Bohr, only epistemic states had physical meaning. Of course, he did not use this terminology, but his statements match it very well; for example [34]:\n\"This crucial point, which was to become a main theme of the discussions reported in the following, implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear. In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of empiricism as the foundational principle of science. This principle is the cornerstone of the Copenhagen interpretation of QM. At the same time it is not so common to refer to Mach as one of fathers of this interpretation; in particular, because his orthodox empiricism once had already led to aggressive denial of a basic and nowadays commonly recognized physical structure -the atomic structure of matter.\nsubdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle cannot be controlled. Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.\"\nTherefore it seems that the problem of identification of ontic and epistemic states was of no interest for Bohr and \"old Copenhagenists.\""},{"title":"Bild concept: Schrödinger versus Bell","content":"We remark that the ontic-epistemic approach can be considered as the modern version of the old Bild-conception tradition (Hertz, Boltzmann and their followers as well as Einstein) [28], [29], chapter 1. In QM this tradition was especially strongly presented in views of Schrödinger who inherited this tradition from his teacher Exner. By the Bild-conception one has to separate two descriptive levels, one is theoretical and another is observational. The latter can be identified with the modernly used epistemic level. The theoretical level, as preceding the observational level, can be coupled to the ontic level of description. However, it seems that these notions cannot be completely identified. The theoretical level was not about \"reality as it is\", but about its theoretical mathematical model. Schrödinger emphasized the impossibility to identify these two descriptive levels: theory and observations are not necessarily related in a term-to-term correspondence and a certain degree of independence exists between them. He also point out that already physicists of 19th century, e.g., Hertz, Boltzmann understood this well. 5 As was emphasized by D'Agostino [28], p. 351, Schrödinger called \"the classical ideal of uninterrupted continuous description\", at both observables' and theoretical levels, an \"old way\", meaning, of course, that this ideal is no longer attainable. He acknowledged that this problem was at the center of the scientific debate in the Nineteenth and Twentieth centuries as well, [35], p.24:\n\"Very similar declarations...(were) made again and again by competent physicists a long time ago, all through the Nineteenth Century and the early days of our century...they were aware that the desire for having a clear picture necessarily led one to encumber it with unwarranted details,.'"},{"title":"De Broglie: hidden probabilities","content":"The excellent presentation of De Broglie's viewpoint on quantum measurement and its consequences for Bell's argument can be found in the paper of G. Lochak [17]. We now present this measurement theory briefly by trying to refine it from the presence of the physical guiding waves, De Broglie's waves.\nIn his measurement theory De Broglie assigned the special value to the position measurement: every physical quantity can be measured only via the final detection of the position, e.g., localization of a detector produced a click. The unitary equivalence of various representations in the Hilbert space formalism is just a mathematical feature of the theory. To measure the concrete physical quantity A, systems emitted by a source have to pass an analyzer, e.g., PBS, and then approach the corresponding detector. Probabilities determined by detectors after passing of the analyzer are called the present probabilities; probabilities before the analyzer, corresponding to the prepared pure state, are called the predicted probabilities. The main point of De Broglie's probabilistic considerations is that it is impossible to combine the predicted and present probabilities straightforwardly, i.e., by using the rules of the classical probability calculus, see section 3. In particular, it is impossible to combine the predicted and present probabilities corresponding to two arbitrary analyzers withing classical probability theory, by using the joint probability distribution.\nHowever, this fact (a simple consequence of the quantum formalism leading to \"interference of probabilities\", see [25] for details) does not contradict to a possibility to introduce hidden variables, because (according De Broglie) their probability distribution is hidden, i.e., we have to consider the third type of probabilities, hidden probabilities. The main point of De Broglie's analysis is that there is no reason to identify these hidden probabilities, probabilities for ontic states with probabilities produced by QM, namely, the predicted and present probabilities (which are epistemic probabilities). Thus similarly to Schrödinger, Einstein, Atmanspacher, Primas and the author of this paper, De Broglie and Lochak were against identification of ontic and epistemic states and more generally their probability distributions (the latter is the main point of a long series of publications of the author of this note, starting with [36]).\nFinally, we mention the guiding wave element of De Broglie's picture. The physical guiding wave appears as the element of the De Broglie's ontic description of micro-phenomena, i.e., as one of variables representing the ontic state. We emphasize that the above scheme with hidden (ontic) and observational (epistemic) probabilities can be explored in general, i.e., without claiming that a subquantum model has to be of the guiding wave type. Here we cannot discuss even briefly the difference in views of L. De Broglie and D. Bohm. We just remark that anybody who read De Broglie personally [37] would never say that De Broglie double solution model and so-called Bohmian mechanics coincide up to the degree of mathematical rigorousness. In particular, we stress that the double solution model is local (as was emphasized by De Broglie [37]) and Bohmian mechanics is nonlocal."},{"title":"Einstein and Bohr","content":"The above discussion about the ontic-epistemic philosophy and the corresponding structuring the notion of realism demonstrated that the Bell argument has no direct relation neither to the position of Bohr nor Einstein and, hence, it cannot be used to justify the position of one of them in contrast to another. In particular, Einstein would say that he never tried to identify the ontic states with the epistemic states. As was emphasized, Bohr and other Copenhagenists were interested only in the epistemic states. Schrödinger would say that he is not surprised that the coupling between the (possible) theoretic and observational levels is not so straightforward, see (1), as in the Bell-argument.\nI hope that the statement NCV1 has been completely justified: both Einstein and Bohr were right, but with respect to the corresponding levels of description."},{"title":"Bell versus von Neumann","content":"We remind that Bell started his foundational project with hard critique [2] of von Neumann's theorem [38]. Nowadays the latter is practically forgotten, not least because Bell's critique. However, we can ask ourselves: Was von Neumann's theorem really so bad?\nVon Neumann did not identify the ontic and epistemic states. He established the special rules of correspondence between the two descriptive levels. In principle, he proceeded in the Bild concept tradition: there are theoretical and observational models and correspondence rules. Bell criticized von Neumann's image of a possible coupling between the theoretical and observational models and their critique was reasonable. In particular, by von Neumann the correspondence between ontic and epistemic quantities should be additive, a + b →â +b,\nwhich seems to be unacceptable for incompatible observables. However, from the philosophic viewpoint the sharp distinguishing between the two descriptive levels present in von Neumann theorem (personally he called this statement \"ansatz\") is preferable comparing with simple identification of these levels. A follower of Einstein would say [25] that von Neumann simply was not lucky to find the right correspondence rules. As was emphasized in [25], the \"no-go\" activity in establishing the correspondence between the two levels of description is totally meaningless, since there is an infinite variety of possibilities to establish the rules of such a correspondence, see, e.g., [29]."},{"title":"\"Bell was wrong\" community","content":"It is useful to analyze from the ontic-epistemic viewpoint not only the position of the conventional quantum (and more specifically quantum information) community, see CV1-CV3, but even of \"Bell was wrong\" community. It seems that the majority of opponents of the CV1-CV2 interpretation do not understand the philosophic meaning of the Bell conjecture, O=E. Typically one from the anti-Bell opposition tries to show that observational (epistemic) probabilities can violate Bell's inequality, or to show that the standard proofs of the Bell inequality do not work, or to construct an epistemic (local) model producing correlations violating Bell's inequality. In more tricky studies there is even a subquantum model and special rules coupling two descriptive levels, but, of course, such a correspondence is more complex than conjectured in O=E. Unfortunately, such studies are not treated from the double descriptive viewpoint; in particular, the rules of correspondence between subquantum and quantum models are not specified, cf., however, with [29].\nThen it is claimed that \"Bell's theorem\" (argument) was wrong. My impression is that this was precisely the way of treatement of the Bell argument by the main anti-Bell gurus: L. Accardi, K. Hess and W. Philipp, M. Kupczynski, H. De Raedt, K., Michielsen T. Nieuwenhuizen and partially myself (although later I proceeded in the De Broglie-Lochak manner by distinguishing hidden probabilities form epistemic). The claim \"Bell was wrong\" acts as a red muleta used by Spanish matadors to make bulls an-gry and generates the strong critical feedback from the pro-Bell community. This \"Bell was wrong\" position was not fruitful for quantum foundations, it induced brutal debates (in particular, during the Växjö series of annual conferences, 2001-2015). By understanding that Bell's argument is based on the O=E conjecture and that activity to test Bell's inequality is directed to check this conjecture we would escape these debates or least to make them less brutal.\nIn short, it is definitely possible to construct local probabilistic models violating the Bell inequality, but for such models the Bell conjecture O=E is clearly wrong; in the same way the standard proofs of the Bell inequality work only under the Bell conjecture."},{"title":"Classical probability meets quantum physics","content":"Bell's argument is of the probabilistic nature and therefore it is natural to analyze it from the viewpoint of foundations of probability theory. The latter was formalized in the measure-theoretic framework by A. N. Kolmogorov [30] in 1933. As any scientific theory, it was endowed with the corresponding interpretation [30].\nBy Kolmogorov [30], see also [25] for discussions, any complex of physical conditions C, experimental context 6 , determines its own probability space P C -the system of events F ≡ F C and the probability P ≡ P C . 7 Classical probability (CP) theory was designed to serve the epistemic description. Here the positions of Kolmogorov and Bohr are very close. In CP complementarity is expressed in assigning different probability spaces to different contexts. 8 6 Here, see [25], the notion of \"context\" started to be in the use. It is important to point out that here context represents \"total experimental arrangement\". Thus contextuality is understood wider than just dependence on measurement of a compatible observable. Our viewpoint on contextuality matches better the views of Bohr than commonly used, e.g., in discussions on Bell's argument, \"restricted contextuality\". We follow Bohr's recommendation that all experimental arrangement has to be taken into account. 7 The system of events has the structure of Boolean σ-algebra and the probability is the σ-additive measure normalized by 1. Here σ express countability: F is closed with respect to countable unions and intersections and probability is countably additive. 8 One of the problems of modern CP is that this clear contextual message of Kolmogorov is not emphasized in presentation of CP in modern textbooks; thus the young generation of probabilists lost this original multi-context and multi-space picture of CP. Typically the context-indexing is omitted. At the beginning it was done only to simplify notations and However, often we have to work with probabilistic data collected for a family of contexts. Therefore CP has to provide some mechanism of embedding of multi-contextual data in a single probability space. It is important to emphasize that there are two main CP constructions of such embedding:\n1. Marginal embedding: contextual probabilities are represented as marginal probabilities for a single probability measure P."},{"title":"2.","content":"Conditional embedding: contextual probabilities are represented as marginal probabilities for some probability for a single probability measure P.\nThe first (straightforward) embedding was excellently realized by A. N. Kolmogorov in his famous theorem [30] about existence of the probability measure P (defined on the space of trajectories) for a stochastic process determined by its probability distributions for all finite sequences of instances of time, P t 1 ...tn . However, this construction can work only in special cases, for special families of contexts. (Therefore Kolmogorov's theorem about the existence of a probability space for a stochastic process is so famous: he found such a special (and very important) case.) It is well known in CP, since the work of Boole, that probability distributions for random variables measured for different contexts can be incompatible, i.e., it is impossible reproduce them as marginal probabilities from the the single probability P [39]. 9 In fact, the inequality providing the necessary condition of the existence of such P derived by Boole coincides with Bell's inequality; therefore some authors even proposed to speak about the Boole-Bell inequality [23]. The complete description of conditions of existence of the straightforward embedding of contextual probabilities in a single probability space was presented in the paper of Vorobj'ev [40] which contains all Bell's type inequalities known in quantum physics (as well as still unknown).\nprobabilists (at least representatives of the Soviet school) remembered well Kolmogorov's slogan: \"no probability unrelated to experimental context\". When, say 15 years ago, I spoke with two former students of Kolmogorov, A. N. Shiryaev and A. V. Bulinski, and tried to teach them that CP is contextual, they immediately pointed to section 2 of Kolmogorov's book [30]. For them, it was totally evident that each experimental context generates its own probability space. It seems that this rule is forgotten in the modern probabilistic community. 9 In the quantum foundational community this historical fact was \"discovered\" by I. Pitowsky; in particular, he pointed to it in his talk at Växjö-2001 conference. For many years, I tried to advertise it for experts in quantum foundations, but without any success.\nWe remark that the marginal embedding is based on the identification of the ontic and epistemic descriptions. The epistemic probabilities for fixed contexts are identified with the marginal probabilities with respect to the probability distribution P which can be treated as the ontic probability. In particular, the Kolmogorov theorem on the probability space for a stochastic process can be applied only the assumption of the ontic-epistemic identification; without this assumption one has to use theory of quantum stochastic processes [41].\nHowever, in contrast to QM, the \"no-go\" activity, the impossibility theorems about striaghtforward marginal embedding was not so popular in CP. And it is clear why. There is widely explored the second embedding procedure based on the identification of concrete contextual probabilities not with marginal, but with conditional probabilities with respect to a single probability measure P. This is so-called randomization procedure. It is especially important in statistics. We shall present it for the concrete multi-contextual test -the Bell test."},{"title":"Contextual probabilistic structure of Bell's test","content":"The considerations of this section are presented in the form of a formal mathematical model in the article [42].\nHow would Kolmogorov handle Bell's argument? For Kolmogorov, it is based on consideration of a few experimental contexts C θ i θ ′ j corresponding to fixing the pairs of orientations of PBSs. There are a few probability spaces corresponding to these measurement contexts. Experimental statistical data collected for context C θ i θ ′ j has to be handled in its own probability space\n• Can Bell's type inequalities be derived in such multi-space probabilistic framework? Not!\n• Would be Kolmogorov surprised by violation of Bell's inequality for multi-space data? Not at all! Moreover, he would be surprised if it were not violated.\n• Can one construct a single probability space representing all contexts C θ i θ ′ j ? Yes! Construction is known as randomization. In short randomization can be described as follows:\n• A). Choose the probabilities of selections of contexts.\n• B). Determine probability distributions for fixed contexts, P C θ i θ ′ j .\n• C). Construct probability P serving for all experimental contexts: combine probabilities for fixed contexts P C θ i θ ′ j with probabilities of context-"},{"title":"selections. 10","content":"This single probability measure P represents the multi-context experiment. We stress that, for it, it is possible to derive Bell's inequality, since the latter is just a theorem of CP [25].\nIn this framework the probability distributions for fixed contexts P C θ i θ ′ j (which are obtained by experimenters as frequencies) appear as conditional probabilities with respect to fixing selections of the parameters θ i , θ ′ j (contexts C θ i θ ′ j ). However, there is no reason to hope that these conditional probabilities and corresponding conditional quantities, e.g., correlations, would satisfy Bell's inequality, see [42].\nIn CP the magnitude ∆ of violation of Bell's inequality can be treated as the numerical measure of multi-space probabilistic structure of the experiment. Thus ∆ represents the degree of contextuality (which is understood a la Bohr as taking into account all experimental arrangement). What does such contextuality mean from the QM foundational viewpoint? Contextuality exhibits itself only the existence of incompatible contexts. Therefore ∆ is also the measure of incompatibility-complementarity. From this viewpoint, the experimental tests violating Bell's inequality confirmed complementarity not only for local observables, such as position and momentum or two spin projections of a single electron, but even for nonlocal observables such as projections of spins measured for a pair of electrons."},{"title":"Kolmogorov versus Bell","content":"As we have seen, the CP description of Bell's test does not lead to revolutionary consequences in the form CV2, CV3. And this is completely clear why. The above CP model of the probabilistic structure of Bell's test is epistemic -it takes into account randomness of selection of experimental contexts, i.e., random generators used for this aim are also considered as measurement devices. In principle, one can say that this is the end of the CP-story about Bell's test.\nHowever, we can continue our discussion and represent its output in the hidden variable form. The latter will be not real ontic hidden variables, but so to say \"epistemic hidden variables\".\nCP teaches us that all random influences involved in the tests have to incorporated into the model. Probabilist would consider [42] not the\"Bell's map\" (1), but a map of the form:\nwhere λ L , λ R are random parameters determining with some probabilities, p L , p R outputs of random generators L and R which in turn determine settings. 11 Of course, not only random generators contribute to randomness generated by experimental equipment. There is a lot of randomness in PBSs, in detectors, in optical fibers. This randomness also has to be taken into account by modifying (3)."},{"title":"Classical probability, operational approach and Bell's argument","content":"The above CP-analysis of Bell's test motivates us to comment the representation of observables in the form (3) from the operational viewpoint which is widely used in QM, e.g., [19]. By it there is a preparation procedure and there is a measurement procedure. From the operational viewpoint, in Bell's considerations the preparation procedure is performed by the source of entangled systems and the detection procedure by detectors. However, there are, e.g., random generators determining the orientations of PBSs. They exist! One cannot simply ignore them. In the consistent operational approach they have to taken into account. In the model represented by (3) they are included in the preparation procedure. In fact, we follow De Broglie's who emphasized the role of analyzers in production of the present probabilities.\nIn the Bell test for photons PBSs are precisely De Broglie's analyzers."},{"title":"With Ockham's razor against nonlocality","content":"We emphasize that we have not appealed to spooky action at a distance. Contextuality 12 leads to violation of Bell's inequality. This motivates NCV2 -\"there is no need in spooky action at a distance\", although not rejection of CV2. In principle, one can still play under the assumption CV2 -if one likes spooky action and it helps her to create the consistent picture of the world. However, there is no need in it. This is the good case to use Ockham's razor, see [43] for discussion. This viewpoint, that by taking into account Kolmogorov probabilistic contextuality of the Bell test one need not mention nonlocality at all, is often criticized by piointing out that contexts C θ i θ ′ j are \"nonlocal\", since their are based on orientations of two spatially separated PBSs. However, this classical nonlocality of spatial location of experimental equipment has nothing to do with action at a distance.\nAnother point which was discussed in [7] is that in the Bell conjecture the notion of \"locality\" is represented in very specific form of action at a distance [2]. Where are there space-time variables? Action at a distance locality is not at all Einsteinian locality of relativity theory. In principle, one can try to couple \"Bellian and Einsteinian localities\" by taking into account the space-time dependence of correlations for entangled systems [44]. However, by proceeding this way we lose the simple prequantum-quantum (ontic-epistemic) correspondence given by (1). The real space-time correlations contain the at least contribution of media for signal propagation; then one has to take into account dispersion and losses in this media, finally the temporal structure of functioning of photo-detectors [45]. Thus the model becomes epistemic; violation of Bell's inequality by such correlations is not surprising."},{"title":"Quantum realism","content":"Bell's argument strongly supported anti-realist tendency in the quantum community. It was very supporting for development of a variety of information interpretations of QM -with two bright examples, the Zeilinger-Brukner information interpretation [50] and Fuchs' QBism [51], [52]. Nowadays a randomly chosen representative of the quantum community would say with very high probability (I account it as 99%) that realism and QM are incompatible.\nI totally disagree with this position, see, e.g., my manifest [?]may be naive, since at that time I was not so well educated in quantum foundational issues). I repeat that the successful Bell's test only rejects the straightforward coupling (1) between the ontic and epistemic or more generally theoretic and observation levels. Other maps can be constructed and the fact that we still do not have a commonly acceptable map may just mean lack of imagination, cf. with the famous citation of Bell [46], p. 15: \"what is proved, by impossibility proofs, is lack of imagination.\"\nHowever, realism discussed in section 2 is merely subquantum realism. What is about, so to say, genuine quantum realism?\nAs was shown, CP modeling of physical experiments, including multicontextual experiments such as Bell's test, does not lead to rejection of realism. This is realism of experimental contexts and observations performed for these contexts. This is precisely \"physical realism\", defined as the statement that the goal of physics is to study entities of the natural world, existing independently from any particular observer's perception, and obeying universal and intelligible rules, see Auffèves and Grangier [47].\nIn my works this realism was treated as contextual realism of QM [25]. However, this terminology might be misleading, since in quantum foundations contextuality is treated restrictively, as dependence on outcomes of joint measurement of a compatible observable. We can also speak about epistemic realism.\nFinally, we remark that physical (contextual) realism matches well with Bohr's views.\nOne may even say that my contextual quantum realism (similarly quantum realism of Grangier) 13 is trivially and commonly acceptable. In fact, this is not the case. And even if it were the case, it is very important to emphasize realism of QM. The main problem is wide spread of identifying of ontic and epistemic realisms. As was discussed, the impossibility of this identification was interpreted as the impossibility to keep ontic realism (at least locally). However, what is the most surprising this rejection of (local) ontic realism is often represented as total rejection of realism in QM."},{"title":"Växjö interpretation of quantum mechanics","content":"The Växjö interpretation (VI) of QM is the realist statistical local and contextual interpretation of QM [53]- [55]. We shall briefly explain the meanings assigned to these terms.\n• The easiest for explanation is \"statistical\". By VI probabilities are interpreted statistically, i.e., they are related to ensembles of systems and not individual systems.\n• VI is local, simply no need even to rise the issue of nonlocality -in complete agreement with Einstein who mentioned nonlocality is an absurd alternative to realism.\n• VI considers both realisms: \"subquantum realism\" (cf. Schrödinger, Einstein, De Broglie, Lochak, Atmanspacher and Primas) and \"contextual realism\" (cf. Bohr, Auffèves and Grangier). The first one is more challenging. We all still suffer of the lack of imagination... The second one was clearly presented in section 3.5.\n• VI is contextual, as any interpretation referring to two descriptive levels, because any epistemic model is contextual as taking into account all experimental arrangement."},{"title":"Future","content":"Recent successful tests for Bell's inequality can be interpreted in various ways, with two extremes, CV1-CV3 and NC1-NCV3, with numerous intermediate interpretations. We can say that experimenters did their job and excellently. However, the loophole free Bell tests should not be interpreted as finalizing the century long discussion about the possibility to construct a theoretical subquantum model coupled to the quantum observational model. My interpretation is that the successful realization of Bell's test confirmed that\n• We have to distinguish sharply two descriptive levels, theoretical and observational (ontic and epistemic).\n• The correspondence between them can be very tricky (so, not so straightforward as in the Bell conjecture O=E). 14 • Contextuality plays the crucial role in coupling of the two descriptive levels.\nThus people looking for new insights on QM have to work hardly to find the proper subquantum theoretical description by relaxing constraints on the rules of correspondence between the two descriptive levels.\nThe final loophole free test says goodbye to the last hopes to identify the observational and ontic descriptions. One has to look for more complex maps\nThis modeling has to be physically constructive, i.e., from the very beginning one should put efforts not to beat some no-go theorem(s), but to create a physically meaningful theoretical model 15 SUBQM endowed with a physically meaningful map f. A partial experimental verification is a part of this great project which successful realization would finally resolve (as I believe!!!) the puzzle of QM.\nPersonally I still belief [29] in the possibility to construct a subquantum model of the classical random field type, cf. Einstein and Infeld [56], with nonlinear subquantum dynamics. This is also the main message of De Broglie. Although, in contrast to Einstein and Infeld and the author of this note, he considered not the ordinary say classical electromagnetic field, but the so-called guiding field, this field is also physical. Thus one may hope to measure it, in some way. Coming back to the Einstein-Infeld hypothesis about classical subquantum field, we can say that, for a moment, experimentalists can try to proceed to new measurement technologies which would provide a possibility to measure characteristics of classical fields for quantum systems, e.g., the polarization vector.\n14 Cf., e.g., with prequantum classical statistical field theory [29]. From my viewpoint this model reproduce all probabilistic predictions of QM for one and bipartite systems; for tripartite systems the representation of entanglement becomes extremely technical and hardly interpretable; for quadripartite systems I was not able to resolve technical difficulties. 15 It may be surprising to hear this statement from a pure mathematician. However, in contrast to physicists who are so excited by the ability of mathematics to model everything in nature, I recognize that sometimes this ability looks like the ability of a prostitution to serve all men's desires. Thus a purely mathematical subquantum model is practically valueless."},{"title":"Acknowledgments","content":"I was lucky to discuss the Bell test and its possible interpretations with practically all world's leading experts in quantum foundations, those who firmly support the CV1-CV3 position as well as those who play with various variants of NCV1-NCV2. I would like thank all them for discussions and insights. I am especially thankful to A. Zeilinger who expressed in lovely Viennese-style discussions the firm anti-realist interpretation and K. Hess who expressed in the same lovely Viennese manner the firm realist position. It was very important for me that both are highly qualified physicists, that they do not simply play with mathematical formulas, but there is a lot of physical intuition behind their positions. And, of course, I was lucky to meet personally A. N. Kolmogorov and speak with him about foundations of CP (well, just 15 min conversation of the old and sick great man with the PhD-student who wrote a paper about a \"non-Komogorov probability model\").\nThe first test of this note was performed at March 18, 2016, at the Quantum Seminar of Stockholm University; I would like to thank all participants of this seminar and especially I. Bengtsson and M. A. M. Versteegh for constructive critique. For writing of this paper the email correspondence with H. Atmanspacher and M. Kupczynski was very stimulating."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"It finally happened! In 2015 three world's leading experimental groups working on foundational aspects of quantum mechanics (QM) announced (practically simultaneously) that they had performed the loophole free tests of violation of the Bell inequality [1], [2]: the groups of Ronald Hanson (Delf University of Technology) [3], Anton Zeilinger (University of Vienna) [4] and Linden Shalm (NIST, Boulder) [5]. This is definitely a great event in quantum foundations; the more so because it took so long to evolve from the pioneer experiment of Alain Aspect [6] to these final Bell's tests. (Besides, some experts in quantum foundations presented the arguments, in the spirit of Heisenberg uncertainty principle, that the locality and detection loopholes could not be closed in one experiment [7]).\\nI was surprised to see that this event did not generate a new wave of the quantum foundational enthusiasm, neither in the quantum community nor in general mass-media (even oriented to popularization of science). 1 One of the reasons for this rather mild reaction is that, as was already mentioned, the result was commonly expected. 2 A similar reason is that the Bell inequality has already been widely \\\"sold\\\". In literature and talks the Bell argument is typically presented as if everything has already been experimentally proven; scientists working on closing loopholes were considered merely as polishing the famous Aspect's experiment [6]. Those who are closer to quantum foundations could additionally point that G. Weihs [8] contributed to complete Aspect's experiment by closing the locality loophole.\\nTherefore, it would be great if the recent publications [3]- [5] ignited a serious discussion on the possible impact of this event of the realization of the totally loophole free Bell test. Such a discussion is especially important because conclusions presented in [3]- [5], see also comments on these tests in [9], [10], present only a part of the wide spectrum of views on Bell's argument. Although the presented \\\"conventional viewpoint\\\" dominates in the quantum community, it would be natural to represent other, so to say, singular, parts of this spectrum of viewpoints, see, e.g., the recent comment of M. Kupczynski [11].\\nWe briefly remind the conventional viewpoint presented in [1], [2] and in hundreds of articles and monographs, e.g., [12]- [14], [3]- [10]. It was finally confirmed experimentally that 1. CV1: Einstein was wrong and Bohr was right; 2. CV2: there is spooky action at a distance;\\n3. CV3: quantum realism is incompatible with locality. 1 The situation reminds me of what happened in mathematics after the famous Fermat theorem had been finally proven. Mathematicians were definitely not able to \\\"sell\\\" this great achievement, even in the scientific community; the test-question: who did prove the Fermat theorem? 2 One very famous experimenter whose group at that time worked hard to perform a loophole free Bell's test told me that he would be really excited if in this test the Bell equality would not be violated...\\nThe views of those who disagree with the presented \\\"conventional position\\\" are characterized by the high degree of diversity [15]- [26]. Therefore I shall not try to elaborate on some common \\\"non-conventional viewpoints\\\", but present only my own position [25]:\\n1. NCV1: both Einstein and Bohr were right; 2. NCV2: there is no need in spooky action at a distance;\\n3. NCV3: quantum realism is compatible with locality.\\nIn section 2 I shall question the CV1-CV3 viewpoint and try to justify the NCV1-NCV3 viewpoint; in particular, I confront the \\\"action at a distance interpretation\\\" with the Copenhagen interpretation of QM. (It is surprising that one may combine without cognitive dissonance these two interpretations.) Then to discuss the issue of realism in QM I appeal to its onticepistemic analysis in the spirit of Atmanspacher and Primas [27]. From this viewpoint, Bell's argument can be treated as the conjecture that ontic states can be identified with epistemic ones. We also discuss this conjecture by appealing to the old Bild conception (Hertz, Boltzman, Schrödinger) about the two descriptive levels of nature, theoretical and observational, see, e.g., [28] and chapter 1 of monograph [29]. Our conclusion is that the rejection of Bell's conjecture as the result of the recent experiments cannot be treated as the impossibility to keep the realist viewpoint. There is neither need in action at a distance. Section 3 is started with the presentation of Kolmogorov's interpretation [30] of classical probability (CP) as the observational theory (describing the epistemic states of nature). By Kolmogorov CP is a contextual theory assigning probability spaces to experimental contexts, complexes of experimental physical conditions. This position leads to the contextual representation of the probabilistic structure of Bell's experimental test [23], [25].\\nIn section 4 I present my personal picture of future development of quantum foundations, in \\\"after Bell epoch\\\": from the total rejection of Bell's conjecture to novel studies on the two descriptive levels approach to QM. In contrast to rather common opinion (see, e.g., Aspect's paper [9] entitled \\\"Closing the door on Einstein and Bohr's quantum debate\\\" and Wiseman's paper [10] entitled \\\"Quantum physics: Death by experiment for local realism\\\"), for me the final Bell test did not imply the total impossibility to \\\"go beyond quantum\\\" [29]. The main message of this test is that the way to a proper subquantum model is more tricky than it was hypothesized by Bell. 3 2 Schrödinger, Einstein, Bohr, and Bell: philosophy meets quantums physics\\nTypically the output of Bell's argument (nowadays experimentally confirmed) is formulated as QM is incompatible with local realism. Thus either one has to reject the possibility of the realist description of quantum phenomena or to imagine nonlocal reality of the Bohmian type which is even more exotic and mystical than quantum surreality. Since Einstein dreamed for the realist interpretation of quantum phenomena and he considered nonlocality as an \\\"absurd alternative\\\" to realism, nowadays it is clear (for everybody, besides a few outsiders, see, e.g., [29], [20], [16]- [26], [11]) that he was wrong and automatically (as Einstein's opponent) Bohr was right, see, e.g., [9]. In general in the modern quantum community it is fashionable to be a follower of Bohr.\"\n },\n {\n \"title\": \"Bohr versus Bell\",\n \"content\": \"First of all, I want to point to one of the main interpretational problems of the conventional viewpoint on the output of Bell's test, see CV1-CV3. This is an attempt to present CV1-CV3 in a single package and consistently with the Copenhagen interpretation of QM. However, none of the fathers of QM, neither Bohr, Heisenberg, Fock, Landau, Pauli nor Einstein, could even imagine that spooky action at a distance would be taken seriously in the quantum foundations. Thus, one has to recognize that the modern nonlocal feature of QM motivated by Bell's argument has nothing to do with Copenhagen. One can agree or disagree with this statement, but in any event it has to be presented and discussed. Thus to be consistent one has to speak about the revolutionary changes in quantum foundations generated by Bell's argument and experimental tests confirming it. The possibility of these changes was not even imagined by earlier Copenhagenists.\\nOne can still argue that \\\"Copenhagenists were against realism\\\" and, hence, Bell's test can be considered as supporting the Copenhagen position...\"\n },\n {\n \"title\": \"Realism: philosophy meets physics\",\n \"content\": \"Realism is a complex issue, both in physics and philosophy. One cannot debate it without taking into account the recent progress in philosophical studies. And I want to analyze Bell's argument in the light of such studies, namely, the ontic-epistemic viewpoint on physical reality, firmly established in quantum foundations by H. Atmanspacher and Primas [27].\\nThere are ontic states, assigned to physical systems as \\\"they are\\\", and epistemic states representing knowledge that observers gain from measurements on physical systems. QM is about epistemic states. This is in complete agreement with the Copenhagen interpretation of QM, especially Bohr's views.\\nIt often escapes notice that the EPR-argument [31] was about ontic states; it was directed to show that quantum states are epistemic states. For Einstein, QM was incomplete in the sense that there should exists a finer description of physical processes in the microworld than given by QM. The states of such a subquantum model are ontic states. Einstein and his coauthors did not have any intention to treat the wave function ψ as the ontic state; for them, it was the epistemic state representing knowledge which can be earn from measurements.\\nThis separation between the two descriptive levels is very important for understanding the real tragedy of QM: misunderstanding about the issue of completeness of QM, between Einstein and Bohr, see [31], [32]. As was pointed out, Einstein's incompleteness is so to say ontic incompleteness of QM, but Bohr's completeness is epistemic completeness of QM. Bohr in his famous reply to Einstein [32] wanted to show that QM is complete at the epistemic level. However, Einstein did not have doubts in epistemic completeness of QM. On the other hand, it seems that Bohr was not so strongly against the possibility of creation of subquantum models; he simply considered such studies as meaningless, because for him physics was only about epistemic states. 4 (This picture of Bohr's views resulted from my long discus-sions with A. Plotnitsky, cf. [33]). However, it seems that Einstein and Bohr did not understand that their debate could be resolved in purely philosophic (ontic-epistemic) terms. Now, let us consider Bell's project from the philosophic viewpoint, by using the ontic-epistemic approach. In philosophic terms, Bell's conjecture was that O=E: Ontic states of quantum systems can be identified with epistemic states.\\nSince the values of observables are assigned directly to the ontic states,\\nthese states also have to interpreted as the epistemic states. Here a is an observable and the result of a measurement is determined directly by the parameter λ. Hence, λ has to be considered as the epistemic state. At the same time Bell treated λ as representing the state of reality as it is (λ has no relation to context of measurement of a). Hence, it has to be considered as the ontic state. The final loophole free Bell tests demonstrated that this conjecture was wrong (if there is no action at a distance, see NCV2).\\nHowever, this statement is not the same as to state that realism is incompatible with QM. The impossibility to identify two descriptive levels, ontic and epistemic, is not rejection of the possibility of the realist description.\"\n },\n {\n \"title\": \"Impact of Bell's test for Copenhagenists\",\n \"content\": \"For Bohr, only epistemic states had physical meaning. Of course, he did not use this terminology, but his statements match it very well; for example [34]:\\n\\\"This crucial point, which was to become a main theme of the discussions reported in the following, implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear. In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of empiricism as the foundational principle of science. This principle is the cornerstone of the Copenhagen interpretation of QM. At the same time it is not so common to refer to Mach as one of fathers of this interpretation; in particular, because his orthodox empiricism once had already led to aggressive denial of a basic and nowadays commonly recognized physical structure -the atomic structure of matter.\\nsubdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle cannot be controlled. Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.\\\"\\nTherefore it seems that the problem of identification of ontic and epistemic states was of no interest for Bohr and \\\"old Copenhagenists.\\\"\"\n },\n {\n \"title\": \"Bild concept: Schrödinger versus Bell\",\n \"content\": \"We remark that the ontic-epistemic approach can be considered as the modern version of the old Bild-conception tradition (Hertz, Boltzmann and their followers as well as Einstein) [28], [29], chapter 1. In QM this tradition was especially strongly presented in views of Schrödinger who inherited this tradition from his teacher Exner. By the Bild-conception one has to separate two descriptive levels, one is theoretical and another is observational. The latter can be identified with the modernly used epistemic level. The theoretical level, as preceding the observational level, can be coupled to the ontic level of description. However, it seems that these notions cannot be completely identified. The theoretical level was not about \\\"reality as it is\\\", but about its theoretical mathematical model. Schrödinger emphasized the impossibility to identify these two descriptive levels: theory and observations are not necessarily related in a term-to-term correspondence and a certain degree of independence exists between them. He also point out that already physicists of 19th century, e.g., Hertz, Boltzmann understood this well. 5 As was emphasized by D'Agostino [28], p. 351, Schrödinger called \\\"the classical ideal of uninterrupted continuous description\\\", at both observables' and theoretical levels, an \\\"old way\\\", meaning, of course, that this ideal is no longer attainable. He acknowledged that this problem was at the center of the scientific debate in the Nineteenth and Twentieth centuries as well, [35], p.24:\\n\\\"Very similar declarations...(were) made again and again by competent physicists a long time ago, all through the Nineteenth Century and the early days of our century...they were aware that the desire for having a clear picture necessarily led one to encumber it with unwarranted details,.'\"\n },\n {\n \"title\": \"De Broglie: hidden probabilities\",\n \"content\": \"The excellent presentation of De Broglie's viewpoint on quantum measurement and its consequences for Bell's argument can be found in the paper of G. Lochak [17]. We now present this measurement theory briefly by trying to refine it from the presence of the physical guiding waves, De Broglie's waves.\\nIn his measurement theory De Broglie assigned the special value to the position measurement: every physical quantity can be measured only via the final detection of the position, e.g., localization of a detector produced a click. The unitary equivalence of various representations in the Hilbert space formalism is just a mathematical feature of the theory. To measure the concrete physical quantity A, systems emitted by a source have to pass an analyzer, e.g., PBS, and then approach the corresponding detector. Probabilities determined by detectors after passing of the analyzer are called the present probabilities; probabilities before the analyzer, corresponding to the prepared pure state, are called the predicted probabilities. The main point of De Broglie's probabilistic considerations is that it is impossible to combine the predicted and present probabilities straightforwardly, i.e., by using the rules of the classical probability calculus, see section 3. In particular, it is impossible to combine the predicted and present probabilities corresponding to two arbitrary analyzers withing classical probability theory, by using the joint probability distribution.\\nHowever, this fact (a simple consequence of the quantum formalism leading to \\\"interference of probabilities\\\", see [25] for details) does not contradict to a possibility to introduce hidden variables, because (according De Broglie) their probability distribution is hidden, i.e., we have to consider the third type of probabilities, hidden probabilities. The main point of De Broglie's analysis is that there is no reason to identify these hidden probabilities, probabilities for ontic states with probabilities produced by QM, namely, the predicted and present probabilities (which are epistemic probabilities). Thus similarly to Schrödinger, Einstein, Atmanspacher, Primas and the author of this paper, De Broglie and Lochak were against identification of ontic and epistemic states and more generally their probability distributions (the latter is the main point of a long series of publications of the author of this note, starting with [36]).\\nFinally, we mention the guiding wave element of De Broglie's picture. The physical guiding wave appears as the element of the De Broglie's ontic description of micro-phenomena, i.e., as one of variables representing the ontic state. We emphasize that the above scheme with hidden (ontic) and observational (epistemic) probabilities can be explored in general, i.e., without claiming that a subquantum model has to be of the guiding wave type. Here we cannot discuss even briefly the difference in views of L. De Broglie and D. Bohm. We just remark that anybody who read De Broglie personally [37] would never say that De Broglie double solution model and so-called Bohmian mechanics coincide up to the degree of mathematical rigorousness. In particular, we stress that the double solution model is local (as was emphasized by De Broglie [37]) and Bohmian mechanics is nonlocal.\"\n },\n {\n \"title\": \"Einstein and Bohr\",\n \"content\": \"The above discussion about the ontic-epistemic philosophy and the corresponding structuring the notion of realism demonstrated that the Bell argument has no direct relation neither to the position of Bohr nor Einstein and, hence, it cannot be used to justify the position of one of them in contrast to another. In particular, Einstein would say that he never tried to identify the ontic states with the epistemic states. As was emphasized, Bohr and other Copenhagenists were interested only in the epistemic states. Schrödinger would say that he is not surprised that the coupling between the (possible) theoretic and observational levels is not so straightforward, see (1), as in the Bell-argument.\\nI hope that the statement NCV1 has been completely justified: both Einstein and Bohr were right, but with respect to the corresponding levels of description.\"\n },\n {\n \"title\": \"Bell versus von Neumann\",\n \"content\": \"We remind that Bell started his foundational project with hard critique [2] of von Neumann's theorem [38]. Nowadays the latter is practically forgotten, not least because Bell's critique. However, we can ask ourselves: Was von Neumann's theorem really so bad?\\nVon Neumann did not identify the ontic and epistemic states. He established the special rules of correspondence between the two descriptive levels. In principle, he proceeded in the Bild concept tradition: there are theoretical and observational models and correspondence rules. Bell criticized von Neumann's image of a possible coupling between the theoretical and observational models and their critique was reasonable. In particular, by von Neumann the correspondence between ontic and epistemic quantities should be additive, a + b →â +b,\\nwhich seems to be unacceptable for incompatible observables. However, from the philosophic viewpoint the sharp distinguishing between the two descriptive levels present in von Neumann theorem (personally he called this statement \\\"ansatz\\\") is preferable comparing with simple identification of these levels. A follower of Einstein would say [25] that von Neumann simply was not lucky to find the right correspondence rules. As was emphasized in [25], the \\\"no-go\\\" activity in establishing the correspondence between the two levels of description is totally meaningless, since there is an infinite variety of possibilities to establish the rules of such a correspondence, see, e.g., [29].\"\n },\n {\n \"title\": \"\\\"Bell was wrong\\\" community\",\n \"content\": \"It is useful to analyze from the ontic-epistemic viewpoint not only the position of the conventional quantum (and more specifically quantum information) community, see CV1-CV3, but even of \\\"Bell was wrong\\\" community. It seems that the majority of opponents of the CV1-CV2 interpretation do not understand the philosophic meaning of the Bell conjecture, O=E. Typically one from the anti-Bell opposition tries to show that observational (epistemic) probabilities can violate Bell's inequality, or to show that the standard proofs of the Bell inequality do not work, or to construct an epistemic (local) model producing correlations violating Bell's inequality. In more tricky studies there is even a subquantum model and special rules coupling two descriptive levels, but, of course, such a correspondence is more complex than conjectured in O=E. Unfortunately, such studies are not treated from the double descriptive viewpoint; in particular, the rules of correspondence between subquantum and quantum models are not specified, cf., however, with [29].\\nThen it is claimed that \\\"Bell's theorem\\\" (argument) was wrong. My impression is that this was precisely the way of treatement of the Bell argument by the main anti-Bell gurus: L. Accardi, K. Hess and W. Philipp, M. Kupczynski, H. De Raedt, K., Michielsen T. Nieuwenhuizen and partially myself (although later I proceeded in the De Broglie-Lochak manner by distinguishing hidden probabilities form epistemic). The claim \\\"Bell was wrong\\\" acts as a red muleta used by Spanish matadors to make bulls an-gry and generates the strong critical feedback from the pro-Bell community. This \\\"Bell was wrong\\\" position was not fruitful for quantum foundations, it induced brutal debates (in particular, during the Växjö series of annual conferences, 2001-2015). By understanding that Bell's argument is based on the O=E conjecture and that activity to test Bell's inequality is directed to check this conjecture we would escape these debates or least to make them less brutal.\\nIn short, it is definitely possible to construct local probabilistic models violating the Bell inequality, but for such models the Bell conjecture O=E is clearly wrong; in the same way the standard proofs of the Bell inequality work only under the Bell conjecture.\"\n },\n {\n \"title\": \"Classical probability meets quantum physics\",\n \"content\": \"Bell's argument is of the probabilistic nature and therefore it is natural to analyze it from the viewpoint of foundations of probability theory. The latter was formalized in the measure-theoretic framework by A. N. Kolmogorov [30] in 1933. As any scientific theory, it was endowed with the corresponding interpretation [30].\\nBy Kolmogorov [30], see also [25] for discussions, any complex of physical conditions C, experimental context 6 , determines its own probability space P C -the system of events F ≡ F C and the probability P ≡ P C . 7 Classical probability (CP) theory was designed to serve the epistemic description. Here the positions of Kolmogorov and Bohr are very close. In CP complementarity is expressed in assigning different probability spaces to different contexts. 8 6 Here, see [25], the notion of \\\"context\\\" started to be in the use. It is important to point out that here context represents \\\"total experimental arrangement\\\". Thus contextuality is understood wider than just dependence on measurement of a compatible observable. Our viewpoint on contextuality matches better the views of Bohr than commonly used, e.g., in discussions on Bell's argument, \\\"restricted contextuality\\\". We follow Bohr's recommendation that all experimental arrangement has to be taken into account. 7 The system of events has the structure of Boolean σ-algebra and the probability is the σ-additive measure normalized by 1. Here σ express countability: F is closed with respect to countable unions and intersections and probability is countably additive. 8 One of the problems of modern CP is that this clear contextual message of Kolmogorov is not emphasized in presentation of CP in modern textbooks; thus the young generation of probabilists lost this original multi-context and multi-space picture of CP. Typically the context-indexing is omitted. At the beginning it was done only to simplify notations and However, often we have to work with probabilistic data collected for a family of contexts. Therefore CP has to provide some mechanism of embedding of multi-contextual data in a single probability space. It is important to emphasize that there are two main CP constructions of such embedding:\\n1. Marginal embedding: contextual probabilities are represented as marginal probabilities for a single probability measure P.\"\n },\n {\n \"title\": \"2.\",\n \"content\": \"Conditional embedding: contextual probabilities are represented as marginal probabilities for some probability for a single probability measure P.\\nThe first (straightforward) embedding was excellently realized by A. N. Kolmogorov in his famous theorem [30] about existence of the probability measure P (defined on the space of trajectories) for a stochastic process determined by its probability distributions for all finite sequences of instances of time, P t 1 ...tn . However, this construction can work only in special cases, for special families of contexts. (Therefore Kolmogorov's theorem about the existence of a probability space for a stochastic process is so famous: he found such a special (and very important) case.) It is well known in CP, since the work of Boole, that probability distributions for random variables measured for different contexts can be incompatible, i.e., it is impossible reproduce them as marginal probabilities from the the single probability P [39]. 9 In fact, the inequality providing the necessary condition of the existence of such P derived by Boole coincides with Bell's inequality; therefore some authors even proposed to speak about the Boole-Bell inequality [23]. The complete description of conditions of existence of the straightforward embedding of contextual probabilities in a single probability space was presented in the paper of Vorobj'ev [40] which contains all Bell's type inequalities known in quantum physics (as well as still unknown).\\nprobabilists (at least representatives of the Soviet school) remembered well Kolmogorov's slogan: \\\"no probability unrelated to experimental context\\\". When, say 15 years ago, I spoke with two former students of Kolmogorov, A. N. Shiryaev and A. V. Bulinski, and tried to teach them that CP is contextual, they immediately pointed to section 2 of Kolmogorov's book [30]. For them, it was totally evident that each experimental context generates its own probability space. It seems that this rule is forgotten in the modern probabilistic community. 9 In the quantum foundational community this historical fact was \\\"discovered\\\" by I. Pitowsky; in particular, he pointed to it in his talk at Växjö-2001 conference. For many years, I tried to advertise it for experts in quantum foundations, but without any success.\\nWe remark that the marginal embedding is based on the identification of the ontic and epistemic descriptions. The epistemic probabilities for fixed contexts are identified with the marginal probabilities with respect to the probability distribution P which can be treated as the ontic probability. In particular, the Kolmogorov theorem on the probability space for a stochastic process can be applied only the assumption of the ontic-epistemic identification; without this assumption one has to use theory of quantum stochastic processes [41].\\nHowever, in contrast to QM, the \\\"no-go\\\" activity, the impossibility theorems about striaghtforward marginal embedding was not so popular in CP. And it is clear why. There is widely explored the second embedding procedure based on the identification of concrete contextual probabilities not with marginal, but with conditional probabilities with respect to a single probability measure P. This is so-called randomization procedure. It is especially important in statistics. We shall present it for the concrete multi-contextual test -the Bell test.\"\n },\n {\n \"title\": \"Contextual probabilistic structure of Bell's test\",\n \"content\": \"The considerations of this section are presented in the form of a formal mathematical model in the article [42].\\nHow would Kolmogorov handle Bell's argument? For Kolmogorov, it is based on consideration of a few experimental contexts C θ i θ ′ j corresponding to fixing the pairs of orientations of PBSs. There are a few probability spaces corresponding to these measurement contexts. Experimental statistical data collected for context C θ i θ ′ j has to be handled in its own probability space\\n• Can Bell's type inequalities be derived in such multi-space probabilistic framework? Not!\\n• Would be Kolmogorov surprised by violation of Bell's inequality for multi-space data? Not at all! Moreover, he would be surprised if it were not violated.\\n• Can one construct a single probability space representing all contexts C θ i θ ′ j ? Yes! Construction is known as randomization. In short randomization can be described as follows:\\n• A). Choose the probabilities of selections of contexts.\\n• B). Determine probability distributions for fixed contexts, P C θ i θ ′ j .\\n• C). Construct probability P serving for all experimental contexts: combine probabilities for fixed contexts P C θ i θ ′ j with probabilities of context-\"\n },\n {\n \"title\": \"selections. 10\",\n \"content\": \"This single probability measure P represents the multi-context experiment. We stress that, for it, it is possible to derive Bell's inequality, since the latter is just a theorem of CP [25].\\nIn this framework the probability distributions for fixed contexts P C θ i θ ′ j (which are obtained by experimenters as frequencies) appear as conditional probabilities with respect to fixing selections of the parameters θ i , θ ′ j (contexts C θ i θ ′ j ). However, there is no reason to hope that these conditional probabilities and corresponding conditional quantities, e.g., correlations, would satisfy Bell's inequality, see [42].\\nIn CP the magnitude ∆ of violation of Bell's inequality can be treated as the numerical measure of multi-space probabilistic structure of the experiment. Thus ∆ represents the degree of contextuality (which is understood a la Bohr as taking into account all experimental arrangement). What does such contextuality mean from the QM foundational viewpoint? Contextuality exhibits itself only the existence of incompatible contexts. Therefore ∆ is also the measure of incompatibility-complementarity. From this viewpoint, the experimental tests violating Bell's inequality confirmed complementarity not only for local observables, such as position and momentum or two spin projections of a single electron, but even for nonlocal observables such as projections of spins measured for a pair of electrons.\"\n },\n {\n \"title\": \"Kolmogorov versus Bell\",\n \"content\": \"As we have seen, the CP description of Bell's test does not lead to revolutionary consequences in the form CV2, CV3. And this is completely clear why. The above CP model of the probabilistic structure of Bell's test is epistemic -it takes into account randomness of selection of experimental contexts, i.e., random generators used for this aim are also considered as measurement devices. In principle, one can say that this is the end of the CP-story about Bell's test.\\nHowever, we can continue our discussion and represent its output in the hidden variable form. The latter will be not real ontic hidden variables, but so to say \\\"epistemic hidden variables\\\".\\nCP teaches us that all random influences involved in the tests have to incorporated into the model. Probabilist would consider [42] not the\\\"Bell's map\\\" (1), but a map of the form:\\nwhere λ L , λ R are random parameters determining with some probabilities, p L , p R outputs of random generators L and R which in turn determine settings. 11 Of course, not only random generators contribute to randomness generated by experimental equipment. There is a lot of randomness in PBSs, in detectors, in optical fibers. This randomness also has to be taken into account by modifying (3).\"\n },\n {\n \"title\": \"Classical probability, operational approach and Bell's argument\",\n \"content\": \"The above CP-analysis of Bell's test motivates us to comment the representation of observables in the form (3) from the operational viewpoint which is widely used in QM, e.g., [19]. By it there is a preparation procedure and there is a measurement procedure. From the operational viewpoint, in Bell's considerations the preparation procedure is performed by the source of entangled systems and the detection procedure by detectors. However, there are, e.g., random generators determining the orientations of PBSs. They exist! One cannot simply ignore them. In the consistent operational approach they have to taken into account. In the model represented by (3) they are included in the preparation procedure. In fact, we follow De Broglie's who emphasized the role of analyzers in production of the present probabilities.\\nIn the Bell test for photons PBSs are precisely De Broglie's analyzers.\"\n },\n {\n \"title\": \"With Ockham's razor against nonlocality\",\n \"content\": \"We emphasize that we have not appealed to spooky action at a distance. Contextuality 12 leads to violation of Bell's inequality. This motivates NCV2 -\\\"there is no need in spooky action at a distance\\\", although not rejection of CV2. In principle, one can still play under the assumption CV2 -if one likes spooky action and it helps her to create the consistent picture of the world. However, there is no need in it. This is the good case to use Ockham's razor, see [43] for discussion. This viewpoint, that by taking into account Kolmogorov probabilistic contextuality of the Bell test one need not mention nonlocality at all, is often criticized by piointing out that contexts C θ i θ ′ j are \\\"nonlocal\\\", since their are based on orientations of two spatially separated PBSs. However, this classical nonlocality of spatial location of experimental equipment has nothing to do with action at a distance.\\nAnother point which was discussed in [7] is that in the Bell conjecture the notion of \\\"locality\\\" is represented in very specific form of action at a distance [2]. Where are there space-time variables? Action at a distance locality is not at all Einsteinian locality of relativity theory. In principle, one can try to couple \\\"Bellian and Einsteinian localities\\\" by taking into account the space-time dependence of correlations for entangled systems [44]. However, by proceeding this way we lose the simple prequantum-quantum (ontic-epistemic) correspondence given by (1). The real space-time correlations contain the at least contribution of media for signal propagation; then one has to take into account dispersion and losses in this media, finally the temporal structure of functioning of photo-detectors [45]. Thus the model becomes epistemic; violation of Bell's inequality by such correlations is not surprising.\"\n },\n {\n \"title\": \"Quantum realism\",\n \"content\": \"Bell's argument strongly supported anti-realist tendency in the quantum community. It was very supporting for development of a variety of information interpretations of QM -with two bright examples, the Zeilinger-Brukner information interpretation [50] and Fuchs' QBism [51], [52]. Nowadays a randomly chosen representative of the quantum community would say with very high probability (I account it as 99%) that realism and QM are incompatible.\\nI totally disagree with this position, see, e.g., my manifest [?]may be naive, since at that time I was not so well educated in quantum foundational issues). I repeat that the successful Bell's test only rejects the straightforward coupling (1) between the ontic and epistemic or more generally theoretic and observation levels. Other maps can be constructed and the fact that we still do not have a commonly acceptable map may just mean lack of imagination, cf. with the famous citation of Bell [46], p. 15: \\\"what is proved, by impossibility proofs, is lack of imagination.\\\"\\nHowever, realism discussed in section 2 is merely subquantum realism. What is about, so to say, genuine quantum realism?\\nAs was shown, CP modeling of physical experiments, including multicontextual experiments such as Bell's test, does not lead to rejection of realism. This is realism of experimental contexts and observations performed for these contexts. This is precisely \\\"physical realism\\\", defined as the statement that the goal of physics is to study entities of the natural world, existing independently from any particular observer's perception, and obeying universal and intelligible rules, see Auffèves and Grangier [47].\\nIn my works this realism was treated as contextual realism of QM [25]. However, this terminology might be misleading, since in quantum foundations contextuality is treated restrictively, as dependence on outcomes of joint measurement of a compatible observable. We can also speak about epistemic realism.\\nFinally, we remark that physical (contextual) realism matches well with Bohr's views.\\nOne may even say that my contextual quantum realism (similarly quantum realism of Grangier) 13 is trivially and commonly acceptable. In fact, this is not the case. And even if it were the case, it is very important to emphasize realism of QM. The main problem is wide spread of identifying of ontic and epistemic realisms. As was discussed, the impossibility of this identification was interpreted as the impossibility to keep ontic realism (at least locally). However, what is the most surprising this rejection of (local) ontic realism is often represented as total rejection of realism in QM.\"\n },\n {\n \"title\": \"Växjö interpretation of quantum mechanics\",\n \"content\": \"The Växjö interpretation (VI) of QM is the realist statistical local and contextual interpretation of QM [53]- [55]. We shall briefly explain the meanings assigned to these terms.\\n• The easiest for explanation is \\\"statistical\\\". By VI probabilities are interpreted statistically, i.e., they are related to ensembles of systems and not individual systems.\\n• VI is local, simply no need even to rise the issue of nonlocality -in complete agreement with Einstein who mentioned nonlocality is an absurd alternative to realism.\\n• VI considers both realisms: \\\"subquantum realism\\\" (cf. Schrödinger, Einstein, De Broglie, Lochak, Atmanspacher and Primas) and \\\"contextual realism\\\" (cf. Bohr, Auffèves and Grangier). The first one is more challenging. We all still suffer of the lack of imagination... The second one was clearly presented in section 3.5.\\n• VI is contextual, as any interpretation referring to two descriptive levels, because any epistemic model is contextual as taking into account all experimental arrangement.\"\n },\n {\n \"title\": \"Future\",\n \"content\": \"Recent successful tests for Bell's inequality can be interpreted in various ways, with two extremes, CV1-CV3 and NC1-NCV3, with numerous intermediate interpretations. We can say that experimenters did their job and excellently. However, the loophole free Bell tests should not be interpreted as finalizing the century long discussion about the possibility to construct a theoretical subquantum model coupled to the quantum observational model. My interpretation is that the successful realization of Bell's test confirmed that\\n• We have to distinguish sharply two descriptive levels, theoretical and observational (ontic and epistemic).\\n• The correspondence between them can be very tricky (so, not so straightforward as in the Bell conjecture O=E). 14 • Contextuality plays the crucial role in coupling of the two descriptive levels.\\nThus people looking for new insights on QM have to work hardly to find the proper subquantum theoretical description by relaxing constraints on the rules of correspondence between the two descriptive levels.\\nThe final loophole free test says goodbye to the last hopes to identify the observational and ontic descriptions. One has to look for more complex maps\\nThis modeling has to be physically constructive, i.e., from the very beginning one should put efforts not to beat some no-go theorem(s), but to create a physically meaningful theoretical model 15 SUBQM endowed with a physically meaningful map f. A partial experimental verification is a part of this great project which successful realization would finally resolve (as I believe!!!) the puzzle of QM.\\nPersonally I still belief [29] in the possibility to construct a subquantum model of the classical random field type, cf. Einstein and Infeld [56], with nonlinear subquantum dynamics. This is also the main message of De Broglie. Although, in contrast to Einstein and Infeld and the author of this note, he considered not the ordinary say classical electromagnetic field, but the so-called guiding field, this field is also physical. Thus one may hope to measure it, in some way. Coming back to the Einstein-Infeld hypothesis about classical subquantum field, we can say that, for a moment, experimentalists can try to proceed to new measurement technologies which would provide a possibility to measure characteristics of classical fields for quantum systems, e.g., the polarization vector.\\n14 Cf., e.g., with prequantum classical statistical field theory [29]. From my viewpoint this model reproduce all probabilistic predictions of QM for one and bipartite systems; for tripartite systems the representation of entanglement becomes extremely technical and hardly interpretable; for quadripartite systems I was not able to resolve technical difficulties. 15 It may be surprising to hear this statement from a pure mathematician. However, in contrast to physicists who are so excited by the ability of mathematics to model everything in nature, I recognize that sometimes this ability looks like the ability of a prostitution to serve all men's desires. Thus a purely mathematical subquantum model is practically valueless.\"\n },\n {\n \"title\": \"Acknowledgments\",\n \"content\": \"I was lucky to discuss the Bell test and its possible interpretations with practically all world's leading experts in quantum foundations, those who firmly support the CV1-CV3 position as well as those who play with various variants of NCV1-NCV2. I would like thank all them for discussions and insights. I am especially thankful to A. Zeilinger who expressed in lovely Viennese-style discussions the firm anti-realist interpretation and K. Hess who expressed in the same lovely Viennese manner the firm realist position. It was very important for me that both are highly qualified physicists, that they do not simply play with mathematical formulas, but there is a lot of physical intuition behind their positions. And, of course, I was lucky to meet personally A. N. Kolmogorov and speak with him about foundations of CP (well, just 15 min conversation of the old and sick great man with the PhD-student who wrote a paper about a \\\"non-Komogorov probability model\\\").\\nThe first test of this note was performed at March 18, 2016, at the Quantum Seminar of Stockholm University; I would like to thank all participants of this seminar and especially I. Bengtsson and M. A. M. Versteegh for constructive critique. For writing of this paper the email correspondence with H. Atmanspacher and M. Kupczynski was very stimulating.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":"On the Einstein-Podolski-Rosen paradox. J S Bell, Physics. 1195J.S. Bell, \"On the Einstein-Podolski-Rosen paradox\", Physics 1, 195 (1964).","doi":""},{"text":"J Bell, Speakable and Unspeakable in Quantum Mechanics. CambridgeCambridge Univ. PressBell, J.: Speakable and Unspeakable in Quantum Mechanics. Cambridge Univ. Press, Cambridge (1987)","doi":""},{"text":"Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km. B Hensen, arXiv:1508.05949Nature. 526682quant-phB. Hensen et al, Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km, Nature 526, 682 (2015); arXiv:1508.05949 [quant-ph] (2015)","doi":"arXiv:1508.05949"},{"text":"A significant-loophole-free test of Bell's theorem with entangled photons. M Giustina, arXiv: quant- ph1511.03190Phys. Rev. Lett. 115250401M. Giustina et al, A significant-loophole-free test of Bell's theorem with entangled photons, Phys. Rev. Lett. 115, 250401 (2015) [arXiv: quant- ph1511.03190].","doi":"arXiv: quant- ph1511.03190"},{"text":"A strong loophole-free test of local realism. L K Shalm, arXiv:quant-ph1511.03189Phys. Rev. Lett. 115250402L. K. Shalm et al, A strong loophole-free test of local realism, Phys. Rev. Lett. 115, 250402 (2015) [arXiv:quant-ph1511.03189].","doi":"arXiv:quant-ph1511.03189"},{"text":"Three Experimental Tests of Bell Inequalities by the Measurement of Polarization Correlations between Photons. A Aspect, Orsay PressOrsayAspect, A. (1983). Three Experimental Tests of Bell Inequalities by the Measurement of Polarization Correlations between Photons (Orsay Press, Orsay).","doi":""},{"text":"Local realistic representation for correlations in the original EPR model for position and momentum. A Khrennikov, I V Volovich, Soft Computing. 10521A. Khrennikov and I.V. Volovich, Local realistic representation for cor- relations in the original EPR model for position and momentum, Soft Computing 10, 521 (2005).","doi":""},{"text":"Ein Experiment zum Test der Bellschen Ungleichung unter Einsteinscher Lokalitat. G Weihs, University of Vienna PressViennaWeihs, G. (1999). Ein Experiment zum Test der Bellschen Ungleichung unter Einsteinscher Lokalitat (University of Vienna Press, Vienna).","doi":""},{"text":"Closing the Door on Einstein and Bohr's Quantum Debate. A Aspect, Physics. 8123A. Aspect, Closing the Door on Einstein and Bohr's Quantum Debate, Physics 8, 123 (2015)","doi":""},{"text":"Quantum physics: Death by experiment for local realism. H Wiseman, Nature. 526649650H. Wiseman, Quantum physics: Death by experiment for local realism. Nature 526, 649650 (2015).","doi":""},{"text":"Can Einstein with Bohr Debate on Quantum Mechanics Be Closed?. Kupczynski, arXiv:1603.00266quant-phKupczynski, Can Einstein with Bohr Debate on Quantum Mechanics Be Closed? arXiv:1603.00266 [quant-ph].","doi":"arXiv:1603.00266"},{"text":". J F Clauser, M A Horne, A Shimony, R A Holt, Phys. Rev. Lett. 2315Clauser J F, Horne M A, Shimony A and Holt R A 1969 Phys. Rev. Lett. 23 (15) 880-884","doi":""},{"text":". J F Clauser, M A Horne, Phys. Rev. D. 10Clauser J F and Horne M A 1974 Phys. Rev. D 10 526-535","doi":""},{"text":". J F Clauser, A Shimony, Rep. Prog. Phys. 41Clauser J F and Shimony A 1978 Rep. Prog. Phys. 41 1881-1927","doi":""},{"text":"Resolution of the Einstein-Podolsky-Rosen and Bell paradoxes. I Pitowsky, Erratum Phys. Rev. Lett. 481768Phys. Rev. Lett.Pitowsky, I. (1982). Resolution of the Einstein-Podolsky-Rosen and Bell paradoxes, Phys. Rev. Lett. 48, 1299; Erratum Phys. Rev. Lett. 48, 1768.","doi":""},{"text":"Bertrand's paradox and Bell's inequalities. M Kupczynski, Phys. Lett. A. 121Kupczynski, M. (1987). Bertrand's paradox and Bell's inequalities, Phys. Lett. A 121, pp. 205-207.","doi":""},{"text":"De Broglie's initial conception of De Broglie waves. G Lochak, The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birthday. Diner, S., Fargue, D. Lochak, G. and Selleri, F.DordrechtD. Reidel Publ. CompanyG. Lochak, De Broglie's initial conception of De Broglie waves. In: Diner, S., Fargue, D. Lochak, G. and Selleri, F. (eds) The Wave-Particle Du- alism. A Tribute to Louis de Broglie on his 90th Birthday, pp. 1-26. D. Reidel Publ. Company, Dordrecht (1984)","doi":""},{"text":"The probabilistic roots of the quantum mechanical paradoxes. L Accardi, The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birthday. Diner, S., Fargue, D. Lochak, G. and Selleri, F.DordrechtD. Reidel Publ. CompanyAccardi, L.: The probabilistic roots of the quantum mechanical para- doxes. In: Diner, S., Fargue, D. Lochak, G. and Selleri, F. (eds) The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birth- day, pp. 47-55. D. Reidel Publ. Company, Dordrecht (1984)","doi":""},{"text":"Foundations of Quantum Mechanics, an Empiricists Approach. W M De Muynck, Kluwer, DordrechtDe Muynck, W. M. (2002). Foundations of Quantum Mechanics, an Empiricists Approach. (Kluwer, Dordrecht).","doi":""},{"text":"K Hess, Einstein was right!. SingaporePan Stanford PublHess, K. (2014). Einstein was right! (Pan Stanford Publ., Singapore).","doi":""},{"text":"Event-based computer simulation model of Aspecttype experiments strictly satisfying Einstein's locality conditions. H De Raedt, K De Raedt, K Michielsen, K Keimpema, S Miyashita, Phys. Soc. Japan. 76104005De Raedt, H., De Raedt, K., Michielsen, K., Keimpema, K. and Miyashita, S. (2007). Event-based computer simulation model of Aspect- type experiments strictly satisfying Einstein's locality conditions, Phys. Soc. Japan 76, 104005.","doi":""},{"text":"Brownian entanglement. A Allahverdyan, A Khrennikov, Yu, Th M Nieuwenhuizen, Phys. Rev. A. 71Allahverdyan, A., Khrennikov, A.Yu. and Nieuwenhuizen, Th.M.: Brow- nian entanglement. Phys. Rev. A 71, 032102-1 -032102-14 (2005).","doi":""},{"text":"Bell-Boole inequality: Nonlocality or probabilistic incompatibility of random variables?. A Khrennikov, Entropy. 102A. Khrennikov, Bell-Boole inequality: Nonlocality or probabilistic in- compatibility of random variables? Entropy, 10, N 2, 19-32 (2008).","doi":""},{"text":"Local Realist Approach and Numerical Simulation of Nonclassical Experiments in Quantum Mechanics. G Adenier, Vẍjö Univ. PressVẍjöAdenier, G.: Local Realist Approach and Numerical Simulation of Non- classical Experiments in Quantum Mechanics. Vẍjö Univ. Press, Vẍjö (2008)","doi":""},{"text":"A Khrennikov, 978-1-4020-9592-4Contextual Approach to Quantum Formalism. SpringerA. Khrennikov, Contextual Approach to Quantum Formalism, Springer, ISBN 978-1-4020-9592-4 (2009).","doi":"978-1-4020-9592-4"},{"text":"Where Bell went wrong?. Th M Nieuwenhuizen, G Accardi, C A Adenier, G Fuchs, A Jaeger, Yu, Khrennikov, AIP Conf. Proc., Foundations of Probability and Physics -5 1101. J.-A. Larsson and S. StenholmMelville, NYNieuwenhuizen, Th. M. (2009). Where Bell went wrong?, in L. Accardi, G. Adenier, C. A. Fuchs, G. Jaeger, A. Yu. Khrennikov, J.-A. Larsson and S. Stenholm (eds.), AIP Conf. Proc., Foundations of Probability and Physics -5 1101 (Am. Inst. Phys., Melville, NY), pp. 127-133.","doi":""},{"text":"Epistemic and ontic quantumrealities. H Atmanspacher, H Primas, Foundations of Probability and Physics-3. Adenier, G., Khrennikov, A. Yu.Melville, NYAmerican Institute of Physics750Ser. Conference ProceedingsAtmanspacher, H. and Primas, H.: Epistemic and ontic quantumreal- ities. In: Adenier, G., Khrennikov, A. Yu. (eds) Foundations of Prob- ability and Physics-3, pp. 49-62. American Institute of Physics, Ser. Conference Proceedings 750, Melville, NY (2005)","doi":""},{"text":"Continuity and completeness in physical theory: Schrödinger's return to the wave interpretation of quantum mechanics in the 1950's. S D'agostino, E. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Bitbol, M. and O. DarrigolGif-sur-YvetteD'Agostino, S.: Continuity and completeness in physical theory: Schrödinger's return to the wave interpretation of quantum mechanics in the 1950's. In: Bitbol, M. and O. Darrigol (eds.) E. Schrödinger: Philos- ophy and the Birth of Quantum Mechanics, 339-360. Editions Frontieres, Gif-sur-Yvette (1992)","doi":""},{"text":". A Khrennikov Beyond, Quantum, Pan Stanford PublSingaporeA. Khrennikov Beyond Quantum, Pan Stanford Publ., Singapore, 2014.","doi":""},{"text":"English translation: Kolmogorov, A.N.: Foundations of Theory of Probability. A N Kolmogoroff, Chelsea Publishing CompanyBerlin; New YorkGrundbegriffe der WahrscheinlichkeitsrechnungKolmogoroff, A. N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer Verlag, Berlin (1933); English translation: Kolmogorov, A.N.: Foundations of Theory of Probability. Chelsea Publishing Company, New York (1956).","doi":""},{"text":"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?. A Einstein, B Podolsky, N Rosen, Phys. Rev. 47A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical De- scription of Physical Reality Be Considered Complete? Phys. Rev. 47, 777-780 (1935).","doi":""},{"text":"Can Quantum-Mechanical Description of Physical Reality be Considered Complete?. N Bohr, Phys. Rev. 48N. Bohr, Can Quantum-Mechanical Description of Physical Reality be Considered Complete?, Phys. Rev. 48, 696-702 (1935).","doi":""},{"text":"Reading Bohr: Physics and Philosophy. A Plotnitsky, SpringerDordrechtPlotnitsky, A.: Reading Bohr: Physics and Philosophy. Springer, Dor- drecht (2006)","doi":""},{"text":"The causality problem in atomic physics. N Bohr, The Philosophical Writings of Niels Bohr. J. Faye and H. J. FolseOx Bow Press4Supplementary PapersBohr, N. (1938). The causality problem in atomic physics, in J. Faye and H. J. Folse (eds.), (1987). The Philosophical Writings of Niels Bohr, Volume 4: Causality and Complementarity, Supplementary Papers (Ox Bow Press;","doi":""},{"text":". C T Woodbridge, Woodbridge, CT), pp. 94-121.","doi":""},{"text":"E. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Editions Frontieres. Bitbol, M. and Darrigol, O.Gif-sur-YvetteBitbol, M. and Darrigol, O. (eds.): E. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Editions Frontieres, Gif-sur-Yvette (1992).","doi":""},{"text":"Statistical measure of ensemble nonreproducibility and correction to Bell's inequality. A Khrennikov, Il Nuovo Cimento, B. 1152A. Khrennikov, Statistical measure of ensemble nonreproducibility and correction to Bell's inequality. Il Nuovo Cimento, B 115, N. 2, 179-184 (1999).","doi":""},{"text":"The Current Interpretation of Wave Mechanics. A Critical Study. L De Broglie, Andrade E Silva, J , ElsevierAmstredam-London-New Yorkde Broglie L and Andrade E Silva J (the author of a chapter) 1964 The Current Interpretation of Wave Mechanics. A Critical Study (Amstredam-London-New York: Elsevier)","doi":""},{"text":". J Von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton Univ. Pressvon Neumann J 1955 Mathematical Foundations of Quantum Mechanics (Princeton: Princeton Univ. Press)","doi":""},{"text":"On the theory of probabilities. G Boole, Phil. Trans. Royal Soc. London. 152Boole, G.: On the theory of probabilities. Phil. Trans. Royal Soc. Lon- don, 152, 225-242 (1862)","doi":""},{"text":"Consistent families of measures and their extensions. N N Vorob'ev, Theory of Probability and its Applications. 7Vorob'ev, N. N. (1962). Consistent families of measures and their exten- sions, Theory of Probability and its Applications 7, pp. 147-162.","doi":""},{"text":"On the noncommutative Markov property. L Accardi, Func. Anal. Appl. 918Accardi L., On the noncommutative Markov property, Func. Anal. Appl. , 9 (1975) 18.","doi":""},{"text":"CHSH inequality: quantum probabilities as classical conditional probabilities Found. A Khrennikov, Phys. 45A. Khrennikov, CHSH inequality: quantum probabilities as classical conditional probabilities Found. Phys. 45, N 7, 711-725 (2015).","doi":""},{"text":"Bell argument: Locality or Realism? Time to make the choice. A Khrennikov, arXiv:1108.0001v3quant-phA. Khrennikov, Bell argument: Locality or Realism? Time to make the choice. arXiv:1108.0001v3 [quant-ph]","doi":"arXiv:1108.0001v3"},{"text":"Quantization of propagating modes in optical fibers. A Khrennikov, B Nilsson, S Nordebo, I V Volovich, Phys. Scr. 8565404A. Khrennikov, B. Nilsson, S. Nordebo, and I. V. Volovich, Quantization of propagating modes in optical fibers, Phys. Scr., 85, 2012. 065404","doi":""},{"text":"Photon Flux and Distance from the Source: Consequences for Quantum Communication. Found. A Khrennikov, B Nilsson, S Nordebo, I Volovich, Physics. 44A. Khrennikov, B. Nilsson, S. Nordebo, I. Volovich, Photon Flux and Distance from the Source: Consequences for Quantum Communication. Found. Physics 44 389-405 (2014).","doi":""},{"text":"J Bell, On the impossible pilot wave, Ref.TH.3315-CERN. 15J. Bell, On the impossible pilot wave, Ref.TH.3315-CERN, 1982, p. 15","doi":""},{"text":"Violation of Bell's inequalities in a quantum realistic framework. Special issue of Växjö-15 conference. A Auffèves, P Grangier, to be published in IJQIA. Auffèves and P. Grangier, Violation of Bell's inequalities in a quantum realistic framework. Special issue of Växjö-15 conference, to be published in IJQI.","doi":""},{"text":"Contextual objectivity: a realistic interpretation of quantum mechanics. P Grangier, arXiv:quant-ph/0012122European Journal of Physics. 23P. Grangier, Contextual objectivity: a realistic interpretation of quantum mechanics, European Journal of Physics 23, 331 (2002) [arXiv:quant-ph/0012122].","doi":"arXiv:quant-ph/0012122"},{"text":"Contextual objectivity and the quantum formalism. P Grangier, arXiv:quant-ph/0407025IJQIP. Grangier, Contextual objectivity and the quantum formalism, IJQI, 3 (1): 17-22 (2005) [arXiv:quant-ph/0407025]","doi":"arXiv:quant-ph/0407025"},{"text":"Information Invariance and Quantum Probabilities. C Brukner, A Zeilinger, Found. Phys. 39677C. Brukner, A. Zeilinger, Information Invariance and Quantum Proba- bilities, Found. Phys. 39, 677 (2009).","doi":""},{"text":"Quantum mechanics as quantum information (and only a little more. C A Fuchs, Quantum Theory: Reconsideration of Foundations. A. KhrennikovVäxjöVäxjö University Press2Fuchs, C. A. (2002). Quantum mechanics as quantum information (and only a little more), in A. Khrennikov (ed.), Quantum Theory: Reconsid- eration of Foundations, Ser. Math. Modeling 2 (Växjö University Press, Växjö), pp. 463-543.","doi":""},{"text":"The anti-Växjö interpretation of quantum mechanics, Quantum Theory: Reconsideration of Foundations. C A Fuchs, Ser. Math. Model. 2Växjö University PressFuchs, C. A. (2002). The anti-Växjö interpretation of quantum me- chanics, Quantum Theory: Reconsideration of Foundations, Ser. Math. Model. 2 (Växjö University Press, Växjö), pp. 99-116.","doi":""},{"text":"Växjö interpretation of quantum mechanics. A Khrennikov, arXiv:quant-ph/0202107v2A. Khrennikov Quantum Theory:Reconsideration of Foundations. VäxjöVäxjö University Press2A. Khrennikov, Växjö interpretation of quantum mechanics. In: A. Khrennikov Quantum Theory:Reconsideration of Foundations\" Ser. Math. Model. 2, p.163-170, Växjö University Press, Växjö (2002). [arXiv:quant-ph/0202107v2]","doi":"arXiv:quant-ph/0202107v2"},{"text":"(ed) Quantum Theory: Reconsideration of Foundations-2. A Khrennikov, Ser. Math. Model. Khrennikov, A.Yu.10Växjö University PressVäxjö interpretation-2003: Realism of contextsA. Khrennikov, Växjö interpretation-2003: Realism of contexts. In: Khrennikov, A.Yu. (ed) Quantum Theory: Reconsideration of Foundations-2. Ser. Math. Model. 10, pp. 323-338. Växjö University Press, Växjö (2004).","doi":""},{"text":"Växjö interpretation of wave function. A Khrennikov, arXiv:quant-ph/1210.2390A. Khrennikov, Växjö interpretation of wave function: 2012. [arXiv:quant-ph/1210.2390]","doi":"arXiv:quant-ph/1210.2390"},{"text":"Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. A Einstein, L Infeld, Simon and SchusterNew-YorkEinstein, A. and Infeld, L.: Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. Simon and Schuster, New-York (1961)","doi":""}],"string":"[\n {\n \"text\": \"On the Einstein-Podolski-Rosen paradox. J S Bell, Physics. 1195J.S. Bell, \\\"On the Einstein-Podolski-Rosen paradox\\\", Physics 1, 195 (1964).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Bell, Speakable and Unspeakable in Quantum Mechanics. CambridgeCambridge Univ. PressBell, J.: Speakable and Unspeakable in Quantum Mechanics. Cambridge Univ. Press, Cambridge (1987)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km. B Hensen, arXiv:1508.05949Nature. 526682quant-phB. Hensen et al, Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km, Nature 526, 682 (2015); arXiv:1508.05949 [quant-ph] (2015)\",\n \"doi\": \"arXiv:1508.05949\"\n },\n {\n \"text\": \"A significant-loophole-free test of Bell's theorem with entangled photons. M Giustina, arXiv: quant- ph1511.03190Phys. Rev. Lett. 115250401M. Giustina et al, A significant-loophole-free test of Bell's theorem with entangled photons, Phys. Rev. Lett. 115, 250401 (2015) [arXiv: quant- ph1511.03190].\",\n \"doi\": \"arXiv: quant- ph1511.03190\"\n },\n {\n \"text\": \"A strong loophole-free test of local realism. L K Shalm, arXiv:quant-ph1511.03189Phys. Rev. Lett. 115250402L. K. Shalm et al, A strong loophole-free test of local realism, Phys. Rev. Lett. 115, 250402 (2015) [arXiv:quant-ph1511.03189].\",\n \"doi\": \"arXiv:quant-ph1511.03189\"\n },\n {\n \"text\": \"Three Experimental Tests of Bell Inequalities by the Measurement of Polarization Correlations between Photons. A Aspect, Orsay PressOrsayAspect, A. (1983). Three Experimental Tests of Bell Inequalities by the Measurement of Polarization Correlations between Photons (Orsay Press, Orsay).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Local realistic representation for correlations in the original EPR model for position and momentum. A Khrennikov, I V Volovich, Soft Computing. 10521A. Khrennikov and I.V. Volovich, Local realistic representation for cor- relations in the original EPR model for position and momentum, Soft Computing 10, 521 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ein Experiment zum Test der Bellschen Ungleichung unter Einsteinscher Lokalitat. G Weihs, University of Vienna PressViennaWeihs, G. (1999). Ein Experiment zum Test der Bellschen Ungleichung unter Einsteinscher Lokalitat (University of Vienna Press, Vienna).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Closing the Door on Einstein and Bohr's Quantum Debate. A Aspect, Physics. 8123A. Aspect, Closing the Door on Einstein and Bohr's Quantum Debate, Physics 8, 123 (2015)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum physics: Death by experiment for local realism. H Wiseman, Nature. 526649650H. Wiseman, Quantum physics: Death by experiment for local realism. Nature 526, 649650 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Can Einstein with Bohr Debate on Quantum Mechanics Be Closed?. Kupczynski, arXiv:1603.00266quant-phKupczynski, Can Einstein with Bohr Debate on Quantum Mechanics Be Closed? arXiv:1603.00266 [quant-ph].\",\n \"doi\": \"arXiv:1603.00266\"\n },\n {\n \"text\": \". J F Clauser, M A Horne, A Shimony, R A Holt, Phys. Rev. Lett. 2315Clauser J F, Horne M A, Shimony A and Holt R A 1969 Phys. Rev. Lett. 23 (15) 880-884\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J F Clauser, M A Horne, Phys. Rev. D. 10Clauser J F and Horne M A 1974 Phys. Rev. D 10 526-535\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J F Clauser, A Shimony, Rep. Prog. Phys. 41Clauser J F and Shimony A 1978 Rep. Prog. Phys. 41 1881-1927\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Resolution of the Einstein-Podolsky-Rosen and Bell paradoxes. I Pitowsky, Erratum Phys. Rev. Lett. 481768Phys. Rev. Lett.Pitowsky, I. (1982). Resolution of the Einstein-Podolsky-Rosen and Bell paradoxes, Phys. Rev. Lett. 48, 1299; Erratum Phys. Rev. Lett. 48, 1768.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bertrand's paradox and Bell's inequalities. M Kupczynski, Phys. Lett. A. 121Kupczynski, M. (1987). Bertrand's paradox and Bell's inequalities, Phys. Lett. A 121, pp. 205-207.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"De Broglie's initial conception of De Broglie waves. G Lochak, The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birthday. Diner, S., Fargue, D. Lochak, G. and Selleri, F.DordrechtD. Reidel Publ. CompanyG. Lochak, De Broglie's initial conception of De Broglie waves. In: Diner, S., Fargue, D. Lochak, G. and Selleri, F. (eds) The Wave-Particle Du- alism. A Tribute to Louis de Broglie on his 90th Birthday, pp. 1-26. D. Reidel Publ. Company, Dordrecht (1984)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The probabilistic roots of the quantum mechanical paradoxes. L Accardi, The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birthday. Diner, S., Fargue, D. Lochak, G. and Selleri, F.DordrechtD. Reidel Publ. CompanyAccardi, L.: The probabilistic roots of the quantum mechanical para- doxes. In: Diner, S., Fargue, D. Lochak, G. and Selleri, F. (eds) The Wave-Particle Dualism. A Tribute to Louis de Broglie on his 90th Birth- day, pp. 47-55. D. Reidel Publ. Company, Dordrecht (1984)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Foundations of Quantum Mechanics, an Empiricists Approach. W M De Muynck, Kluwer, DordrechtDe Muynck, W. M. (2002). Foundations of Quantum Mechanics, an Empiricists Approach. (Kluwer, Dordrecht).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"K Hess, Einstein was right!. SingaporePan Stanford PublHess, K. (2014). Einstein was right! (Pan Stanford Publ., Singapore).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Event-based computer simulation model of Aspecttype experiments strictly satisfying Einstein's locality conditions. H De Raedt, K De Raedt, K Michielsen, K Keimpema, S Miyashita, Phys. Soc. Japan. 76104005De Raedt, H., De Raedt, K., Michielsen, K., Keimpema, K. and Miyashita, S. (2007). Event-based computer simulation model of Aspect- type experiments strictly satisfying Einstein's locality conditions, Phys. Soc. Japan 76, 104005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brownian entanglement. A Allahverdyan, A Khrennikov, Yu, Th M Nieuwenhuizen, Phys. Rev. A. 71Allahverdyan, A., Khrennikov, A.Yu. and Nieuwenhuizen, Th.M.: Brow- nian entanglement. Phys. Rev. A 71, 032102-1 -032102-14 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bell-Boole inequality: Nonlocality or probabilistic incompatibility of random variables?. A Khrennikov, Entropy. 102A. Khrennikov, Bell-Boole inequality: Nonlocality or probabilistic in- compatibility of random variables? Entropy, 10, N 2, 19-32 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Local Realist Approach and Numerical Simulation of Nonclassical Experiments in Quantum Mechanics. G Adenier, Vẍjö Univ. PressVẍjöAdenier, G.: Local Realist Approach and Numerical Simulation of Non- classical Experiments in Quantum Mechanics. Vẍjö Univ. Press, Vẍjö (2008)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Khrennikov, 978-1-4020-9592-4Contextual Approach to Quantum Formalism. SpringerA. Khrennikov, Contextual Approach to Quantum Formalism, Springer, ISBN 978-1-4020-9592-4 (2009).\",\n \"doi\": \"978-1-4020-9592-4\"\n },\n {\n \"text\": \"Where Bell went wrong?. Th M Nieuwenhuizen, G Accardi, C A Adenier, G Fuchs, A Jaeger, Yu, Khrennikov, AIP Conf. Proc., Foundations of Probability and Physics -5 1101. J.-A. Larsson and S. StenholmMelville, NYNieuwenhuizen, Th. M. (2009). Where Bell went wrong?, in L. Accardi, G. Adenier, C. A. Fuchs, G. Jaeger, A. Yu. Khrennikov, J.-A. Larsson and S. Stenholm (eds.), AIP Conf. Proc., Foundations of Probability and Physics -5 1101 (Am. Inst. Phys., Melville, NY), pp. 127-133.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Epistemic and ontic quantumrealities. H Atmanspacher, H Primas, Foundations of Probability and Physics-3. Adenier, G., Khrennikov, A. Yu.Melville, NYAmerican Institute of Physics750Ser. Conference ProceedingsAtmanspacher, H. and Primas, H.: Epistemic and ontic quantumreal- ities. In: Adenier, G., Khrennikov, A. Yu. (eds) Foundations of Prob- ability and Physics-3, pp. 49-62. American Institute of Physics, Ser. Conference Proceedings 750, Melville, NY (2005)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Continuity and completeness in physical theory: Schrödinger's return to the wave interpretation of quantum mechanics in the 1950's. S D'agostino, E. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Bitbol, M. and O. DarrigolGif-sur-YvetteD'Agostino, S.: Continuity and completeness in physical theory: Schrödinger's return to the wave interpretation of quantum mechanics in the 1950's. In: Bitbol, M. and O. Darrigol (eds.) E. Schrödinger: Philos- ophy and the Birth of Quantum Mechanics, 339-360. Editions Frontieres, Gif-sur-Yvette (1992)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Khrennikov Beyond, Quantum, Pan Stanford PublSingaporeA. Khrennikov Beyond Quantum, Pan Stanford Publ., Singapore, 2014.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"English translation: Kolmogorov, A.N.: Foundations of Theory of Probability. A N Kolmogoroff, Chelsea Publishing CompanyBerlin; New YorkGrundbegriffe der WahrscheinlichkeitsrechnungKolmogoroff, A. N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer Verlag, Berlin (1933); English translation: Kolmogorov, A.N.: Foundations of Theory of Probability. Chelsea Publishing Company, New York (1956).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?. A Einstein, B Podolsky, N Rosen, Phys. Rev. 47A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical De- scription of Physical Reality Be Considered Complete? Phys. Rev. 47, 777-780 (1935).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Can Quantum-Mechanical Description of Physical Reality be Considered Complete?. N Bohr, Phys. Rev. 48N. Bohr, Can Quantum-Mechanical Description of Physical Reality be Considered Complete?, Phys. Rev. 48, 696-702 (1935).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reading Bohr: Physics and Philosophy. A Plotnitsky, SpringerDordrechtPlotnitsky, A.: Reading Bohr: Physics and Philosophy. Springer, Dor- drecht (2006)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The causality problem in atomic physics. N Bohr, The Philosophical Writings of Niels Bohr. J. Faye and H. J. FolseOx Bow Press4Supplementary PapersBohr, N. (1938). The causality problem in atomic physics, in J. Faye and H. J. Folse (eds.), (1987). The Philosophical Writings of Niels Bohr, Volume 4: Causality and Complementarity, Supplementary Papers (Ox Bow Press;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C T Woodbridge, Woodbridge, CT), pp. 94-121.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"E. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Editions Frontieres. Bitbol, M. and Darrigol, O.Gif-sur-YvetteBitbol, M. and Darrigol, O. (eds.): E. Schrödinger: Philosophy and the Birth of Quantum Mechanics. Editions Frontieres, Gif-sur-Yvette (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statistical measure of ensemble nonreproducibility and correction to Bell's inequality. A Khrennikov, Il Nuovo Cimento, B. 1152A. Khrennikov, Statistical measure of ensemble nonreproducibility and correction to Bell's inequality. Il Nuovo Cimento, B 115, N. 2, 179-184 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Current Interpretation of Wave Mechanics. A Critical Study. L De Broglie, Andrade E Silva, J , ElsevierAmstredam-London-New Yorkde Broglie L and Andrade E Silva J (the author of a chapter) 1964 The Current Interpretation of Wave Mechanics. A Critical Study (Amstredam-London-New York: Elsevier)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton Univ. Pressvon Neumann J 1955 Mathematical Foundations of Quantum Mechanics (Princeton: Princeton Univ. Press)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the theory of probabilities. G Boole, Phil. Trans. Royal Soc. London. 152Boole, G.: On the theory of probabilities. Phil. Trans. Royal Soc. Lon- don, 152, 225-242 (1862)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Consistent families of measures and their extensions. N N Vorob'ev, Theory of Probability and its Applications. 7Vorob'ev, N. N. (1962). Consistent families of measures and their exten- sions, Theory of Probability and its Applications 7, pp. 147-162.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the noncommutative Markov property. L Accardi, Func. Anal. Appl. 918Accardi L., On the noncommutative Markov property, Func. Anal. Appl. , 9 (1975) 18.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"CHSH inequality: quantum probabilities as classical conditional probabilities Found. A Khrennikov, Phys. 45A. Khrennikov, CHSH inequality: quantum probabilities as classical conditional probabilities Found. Phys. 45, N 7, 711-725 (2015).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bell argument: Locality or Realism? Time to make the choice. A Khrennikov, arXiv:1108.0001v3quant-phA. Khrennikov, Bell argument: Locality or Realism? Time to make the choice. arXiv:1108.0001v3 [quant-ph]\",\n \"doi\": \"arXiv:1108.0001v3\"\n },\n {\n \"text\": \"Quantization of propagating modes in optical fibers. A Khrennikov, B Nilsson, S Nordebo, I V Volovich, Phys. Scr. 8565404A. Khrennikov, B. Nilsson, S. Nordebo, and I. V. Volovich, Quantization of propagating modes in optical fibers, Phys. Scr., 85, 2012. 065404\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Photon Flux and Distance from the Source: Consequences for Quantum Communication. Found. A Khrennikov, B Nilsson, S Nordebo, I Volovich, Physics. 44A. Khrennikov, B. Nilsson, S. Nordebo, I. Volovich, Photon Flux and Distance from the Source: Consequences for Quantum Communication. Found. Physics 44 389-405 (2014).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Bell, On the impossible pilot wave, Ref.TH.3315-CERN. 15J. Bell, On the impossible pilot wave, Ref.TH.3315-CERN, 1982, p. 15\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Violation of Bell's inequalities in a quantum realistic framework. Special issue of Växjö-15 conference. A Auffèves, P Grangier, to be published in IJQIA. Auffèves and P. Grangier, Violation of Bell's inequalities in a quantum realistic framework. Special issue of Växjö-15 conference, to be published in IJQI.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Contextual objectivity: a realistic interpretation of quantum mechanics. P Grangier, arXiv:quant-ph/0012122European Journal of Physics. 23P. Grangier, Contextual objectivity: a realistic interpretation of quantum mechanics, European Journal of Physics 23, 331 (2002) [arXiv:quant-ph/0012122].\",\n \"doi\": \"arXiv:quant-ph/0012122\"\n },\n {\n \"text\": \"Contextual objectivity and the quantum formalism. P Grangier, arXiv:quant-ph/0407025IJQIP. Grangier, Contextual objectivity and the quantum formalism, IJQI, 3 (1): 17-22 (2005) [arXiv:quant-ph/0407025]\",\n \"doi\": \"arXiv:quant-ph/0407025\"\n },\n {\n \"text\": \"Information Invariance and Quantum Probabilities. C Brukner, A Zeilinger, Found. Phys. 39677C. Brukner, A. Zeilinger, Information Invariance and Quantum Proba- bilities, Found. Phys. 39, 677 (2009).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Quantum mechanics as quantum information (and only a little more. C A Fuchs, Quantum Theory: Reconsideration of Foundations. A. KhrennikovVäxjöVäxjö University Press2Fuchs, C. A. (2002). Quantum mechanics as quantum information (and only a little more), in A. Khrennikov (ed.), Quantum Theory: Reconsid- eration of Foundations, Ser. Math. Modeling 2 (Växjö University Press, Växjö), pp. 463-543.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The anti-Växjö interpretation of quantum mechanics, Quantum Theory: Reconsideration of Foundations. C A Fuchs, Ser. Math. Model. 2Växjö University PressFuchs, C. A. (2002). The anti-Växjö interpretation of quantum me- chanics, Quantum Theory: Reconsideration of Foundations, Ser. Math. Model. 2 (Växjö University Press, Växjö), pp. 99-116.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Växjö interpretation of quantum mechanics. A Khrennikov, arXiv:quant-ph/0202107v2A. Khrennikov Quantum Theory:Reconsideration of Foundations. VäxjöVäxjö University Press2A. Khrennikov, Växjö interpretation of quantum mechanics. In: A. Khrennikov Quantum Theory:Reconsideration of Foundations\\\" Ser. Math. Model. 2, p.163-170, Växjö University Press, Växjö (2002). [arXiv:quant-ph/0202107v2]\",\n \"doi\": \"arXiv:quant-ph/0202107v2\"\n },\n {\n \"text\": \"(ed) Quantum Theory: Reconsideration of Foundations-2. A Khrennikov, Ser. Math. Model. Khrennikov, A.Yu.10Växjö University PressVäxjö interpretation-2003: Realism of contextsA. Khrennikov, Växjö interpretation-2003: Realism of contexts. In: Khrennikov, A.Yu. (ed) Quantum Theory: Reconsideration of Foundations-2. Ser. Math. Model. 10, pp. 323-338. Växjö University Press, Växjö (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Växjö interpretation of wave function. A Khrennikov, arXiv:quant-ph/1210.2390A. Khrennikov, Växjö interpretation of wave function: 2012. [arXiv:quant-ph/1210.2390]\",\n \"doi\": \"arXiv:quant-ph/1210.2390\"\n },\n {\n \"text\": \"Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. A Einstein, L Infeld, Simon and SchusterNew-YorkEinstein, A. and Infeld, L.: Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. Simon and Schuster, New-York (1961)\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":68,"cells":{"text":{"kind":"string","value":"\nMagneto-elastic interaction in cubic helimagnets with B20 structure\n10 Apr 2008 (Dated: April 10, 2008)\n\nS V Maleyev \nPetersburg Nuclear Physics Institute\n188300Gatchina, PetersburgStRussia\n\nMagneto-elastic interaction in cubic helimagnets with B20 structure\n10 Apr 2008 (Dated: April 10, 2008)arXiv:0804.1678v1 [cond-mat.str-el]numbers: 0570F h7520E n7545+j\nThe magneto-elastic interaction in cubic helimagnets with B20 symmetry is considered. It is shown that this interaction is responsible for negative contribution to the square of the spin-wave gap ∆ which is alone has to disrupt assumed helical structure. It is suggested that competition between positive part of ∆ 2 I which stems from magnon-magnon interaction and its negative magneto-elastic part leads to the quantum phase transition observed at high pressure in M nSi and F eGe. This transition has to occur when ∆ 2 = 0. For M nSi from rough estimations at ambient pressure both parts ∆I and |∆ME| are comparable with the experimentally observed gap. The magneto-elastic interaction is responsible also for 2k modulation of the lattice where k is the helix wave-vector and contribution to the magnetic anisotropy.Experimental observation by x-ray and neutron scattering the lattice modulation allows determine the strength of anisotropic part of the magneto-elastic interaction responsible for above phenomena and the lattice helicity.\n\nI. INTRODUCTION\n\nHelical magnetic order stipulated by Dzyaloshinskii-Moriya interaction and its evolution in magnetic field attract a lot of attention in last years (see for example 1 and references therein). In this respect non-centrosymmetric cubic helimagnets such as M nSi, F eGe, F eCoSi with P 2 1 3 (B20) symmetry play special role. They are a subject of the intensive experimental and theoretical studies during several decades. Their single-handed helical structure was explained by Dzyaloshinskii 2 . The full set of interactions responsible for observed helical structure (Bak-Jensen model) was established later in 3 , 4 in agreement with existing experimental data (see for example 5 and references therein). The renascence in this field began with a discovery of the quantum phase transition to a disordered (partially ordered) state in M nSi at high pressure (magnetization and resistivity measurements 6 , 7 and neutron scattering 8 , 9 ). Recently similar transition was observed in F eGe 10 . The following properties of this state attract the main attention: i) non-Fermi-liquid conductivity, ii) spherical neutron scattering surface with the weak maxima along the 110 axes 8 , 11 , whereas at ambient pressure Bragg reflections and critical scattering were observed along 111 5 . These features and the structure of the partially ordered state were discussed in several theoretical papers (see 12,13,14,15 and references therein). It should be noted that the spherical scattering surface with maxima along 111 was observed at ambient pressure just above critical temperature T c ≃ 29K and explained using the Bak-Jensen model 16 . It was demonstrated also that this phase transition is of the first order one 17 Recently phase separation in M nSi near the quantum phase transition was observed by muon spin resonant method 18 .\n\nIn 1914 strong anisotropy of the spin-wave spectrum at low momenta was demonstrated: excitations with momentum along and perpendicular to the helix wave-vector k have linear and quadratic dispersion respectively. At the same time there is a contradiction between these papers. In 14 was claimed that the spin-waves are gapless Goldstone excitations due to translation invariance along the helix axis whereas in 19 the spin-wave gap was calculated in 1/S approximation. This contradiction was discussed in 20 . In brief its essence is following. In 14 the 1/S corrections to the spin-wave energy were not evaluated and the translation invariance was not proved. Meanwhile doing this the authors should meet a problem how to consider the Dzyaloshinskii-Moriya interaction. It contains two spin operators and if they belong to the single lattice point, the translation invariance holds and the gap is zero. However this interaction always acts between different spins, the translation changes the pair energy and the gap is not zero 21 .\n\nExistence of the gap ∆ is very important for correct description of the helix behavior in magnetic field H ⊥ perpendicular to the helix vector k. In the gapless case the spin-wave spectrum becomes unstable in infinitesimal H ⊥ in contradiction to the well-known experimental findings 19 and predictions of the phenomenological Landau-like theory 22 . In 19 was shown that the helix state remains stable if H ⊥ < ∆ √ 2 and then k begins rotate toward the field. Recently this prediction was confirmed using small angle polarized neutron scattering and was found that ∆ ≃ 13µeV for M nSi 23,24,25,26 .\n\nThe quantum phase transition to magnetically disordered state observed in M nSi and F eGe 7,8,9,10,11 is an another important problem. Recently it was considered on the base of phenomenological Landau theory 27 . However up to now we have not any attempt to understand microscopic origin of this transition. In this paper we consider the magneto-elastic (ME) interaction and evaluate its contribution to the square of the spin-wave gap ∆. We demonstrate that\n∆ 2 = ∆ 2 I + ∆ 2 ME ,(1)\nwhere ∆ 2 I was evaluated in 19 and ∆ 2 ME < 0 appears due to the ME considered as the second order perturbation. It is important to note that if ∆ 2 I is zero the ME has to disrupt the helical magnetic order. Rough estimations using existing experimental data at ambient pressure (see Sec.V) give |∆ ME | ∼ 7.6µeV and ∆ I = 4.0 ÷ 28µeV . Both contributions are comparable with experimental value determined in 23 . Hence at pressure two parts of ∆ 2 has to compete and the quantum phase transition to the partly disordered state occurs when ∆ = 0. Besides we estimated the ME contribution to the magnetic anisotropy and demonstrated that it is not small in comparison to the experimental value. We demonstrated also that the ME leads to the lattice deformation with the wave-vector 2k and evaluated intensities of corresponding super-lattice reflections. Their experimental study would allow determine the strength of anisotropic part of the ME interaction responsible for above mentioned phenomena and the lattice helicity. It has to be noted that the ME interaction in the Landau theory was investigated in 28,29 and the lattice deformation was predicted at the wave-vector k. This result does not contradict to ours as it was obtained for magnetized systems only.\n\nThe paper is organized as follows. In Sec.II we consider the ME in cubic helimagnets. Classical ground state energy and the lattice deformation is studied in Sec.III. The spin-wave-phonon interaction and the ME contribution to ∆ 2 is considered in Sec.IV. Obtained results, numerical estimations and experimental consequences are discussed in Sec.V and main results are summarized in Sec.IV In Appendixes A and B some mathematical details are considered. Appendix C is devoted to consideration super-lattice reflections near forbidden < n00 > Bragg peaks with odd n-s.\n\n\nII. MAGNETO-ELASTIC INTERACTION\n\nIn general form the magneto elastic energy is given by (see for example 30 )\nV ME = R S α R S β R B αβγµ U γµ (R)(2)\nwhere S α R is the spin components at the lattice point R , U γµ = (1/2)(∂u γ /∂R µ + ∂u µ /∂R γ ) is the deformation tensor and the lattice site displacement has well known form\nu(R) = e iq·R 1 2N M ω qj (e qj b qj + e −qj b + −qj ),(3)\nwhere e qj are vectors of the phonon polarization , b(b + ) their absorption (excitation) operators and e −qj = e * qj 31 . Tensor B is symmetric in (αβ) and (γµ) components. In cubic crystals we have following non-zero components 30\nB xxxx = B yyyy = B zzzz = B 1 , B xyxy = B yzyz = B zxzx = B 2 ,(4)\nand B xyxy = B yxxy = B xyyx etc. In isotropic medium we have\nB is αβµν = B is (δ αµ δ βν + δ αν δ βµ )/2,(5)\nand\nB 1 = B is , B 2 = B is /2.\nIn cubic helimagnets the lattice spin is given by 19\nS R = Ae ik·R (S ζ R cos α + iS η R − S ξ R sin α) + A * e −ik·R (S ζ R cos α − iS η R − S ξ R sin α) +ĉ(S ζ R sin α + S ξ R cos α) = AS A R + A * S A * R +ĉS c R ,(6)\nwhere < S ζ R > = 0 is an average value of the lattice spin, S η,ξ R describe its perpendicular motion, k is the helix wave-vector, A = (â − ib)/2, unit vectorsâ,b, andĉ form right-handed orthogonal frame, sin α = −H /H C where H is the magnetic field component along the helix vector k and H C is the critical field for transition to ferromagnetic state. According to 19 the vector k ĉ in arbitrary field.\n\nUsing standard definition S q = N −1/2 S R exp (−iq · R) in momentum space we obtain 19\nS q = S c qĉ + S A q A + S A * q A * S c q = S ζ q sin α + S ξ q cos α, S A q = S ζ q−k cos α − S ξ q−k sin α + iS η q−k , S A * q = S ζ q+k cos α − S ξ q+k sin α − iS η q+k ,(7)\nwhere S c q , S A q and S A * q are functions of q, q − k and q + k respectively. The spin components in Eq.(7) have well-known form\nS ζ q = N 1/2 Sδ q,0 − (a + a) q ; S η q = −i S/2[a q − a + −q − (a + a 2 ) q /2S]; S ξ = S/2[a q + a + −q − (a + a 2 )q/2S],(8)\nwhere a q and a + q are conventional spin-wave operators. In momentum space Eq.(2) is given by\nV ME = N −1/2 l,m=A,A * ,c S l q1 S m q2 BU −q1−q2 ,(9)\nThis expression is divided on three parts: direct (ĉĉ and A A * ) terms where the U tensor is k independent, first (cA), (cA * ) and second (A A and A * A * ) order umklapp terms where U operator depends on q 1 + q 2 ± k and q 1 + q 2 ± 2k respectively 32 . For their consideration we use following identity\nmnBU Q = iB A p=x,y,z m p n p Q p u p + iB 2 [(m · Q)(n · u) + (m · u)(n · Q)],(10)\nwhere B A = B 1 − 2B 2 is an anisotropic part of the tensor B and u = u Q .\n\nIn the case of uniform pressure P we have U αβ = −(P/3K)δ αβ where K is the bulk modulus and U q ∼ N 1/2 δ q,0 . As a result the umklapp terms are zero asĉ · A = A · A = 0 and V ME → −N B 1 S(S + 1)P/(3K). Hence the uniform pressure contributes to the classical part of the magneto-elastic ground state energy only. However it has to change basic parameters of the problem such as B 1,2 , sound velocities etc.\n\n\nIII. GROUND STATE ENERGY AND LATTICE DEFORMATION\n\nIn zero magnetic field sin α = 0 and we have a planar helix. In this case from Eqs. (7)(8)(9)(10) follows that in the classical part of the ME interaction the first-order umklapps are forbidden 33 and we obtain\nV ME = −2iN 1/2 S 2 B A k[(g · u) − c.c],(11)\nwhere g p = A 2 pĉ p and u p = u p −2k . For evaluation of the lattice deformation we must consider the elastic energy. Unlike Refs. 28,29 for simplicity we ignore in it the cubic symmetry as its principal symmetry breaking role has been taken into account above in Eq (11). In this case the unit cell energy is given by 34\nF (r) = Q[U αβ (r)U βα (r) + σU 2 αα (r)/(1 − 2σ)],(12)where Q = Ev 0 /[2(1 + σ)]\n, v 0 is the unit cell volume, and E and σ are Young modulus and Poisson coefficient respectively. As a result we have\nF 2k = 2Q[k 2 (u · u * ) + (k · u)(k · u * )/(1 − 2σ)].(13)\nFrom Eqs. (11)(12)(13) for the ME part of the ground state energy we obtain\nE ME = (2N S 4 B 2 A /Q){(w · w * ) + (w ·ĉ)(w * ·ĉ)/(1 − 2σ) − [(g · w) + c.c.]},(14)\nwhere\nu −2k = −i(B A S 2 N 1/2 /Qk)w.\nMinimum of this energy is evaluated in Appendix A and we have\nE ME = − N S 4 B 2 A (1 + σ) 4Ev 0 (G 1 − G 2 ) + (1 − 2σ) 2(1 − σ) G 2 ≃ − N Ev 0 g 2 A 4 (G 1 − G 2 + G 2 /2),(15)u −2k = −2iN 1/2 S 2 B A (1 + σ) Ev 0 k g * − (g * ·ĉ)ĉ 2(1 − σ) ≃ −(2ig A /k)[g * − (g * ·ĉ)ĉ/2],(16)\nwhere g A = S 2 B A /Ev 0 , cubic invariants G 1 = 16(g · g * ) and G 2 = 16(g ·ĉ)(g * ·ĉ) are considered in Appendix A and in the right hand side of both equations we neglect σ as it is usually small 34 .\n\n\nIV. MAGNON-PHONON INTERACTION\n\nWe consider now the magnon-phonon interaction. We are interested by terms which survive at q = 0 and contribute to the spin-wave gap as other terms are small corrections to the q-dependent part of the magnon dispersion considered in 19 and 14 . To single out them we have to replace in Eq.(9) one of S ζ operators by Sδ q±k,0 . As a result we obtain terms with phonon momenta q, q ± k, and q ± 2k. The former disappear at q = 0, the second are proportional to a q + a + −q and can not contribute to the gap (see below). Using identity (10) for the last AA end A * A * terms in the case of planar helix we have\nV 2k = −(2S) 3/2 ikB A [(a q − a + −q )g p u p −2k−q + (a −q − a + q )g * p u p 2k+q ],(17)\nFrom this equation for the magnon-magnon interaction we obtain\nV MM = 2(2kS 2 B A ) 2 /S (a q − a + −q ) g p D p r (∆, 2k)g * r (a −q − a + q ),(18)\nwhere we neglect q in comparison with ±2k . The phonon Green function can be represented as\nD p r (ω, Q) = D t (δ p r −Q pQr ) + D lQpQr ,(19)whereQ = Q/Q and D l(t) = [M (ω 2 − s 2 l(t) q 2 )] −1\nwhere l(t)labels longitudinal ( transverse) phonon mode and s l(t) is a corresponding sound velocity. We neglect optical branches as their contribution is of order (sk/θ D ) 2 ≪ 1 where θ D is Debye temperature.\n\nIn the linear spin-wave theory the Hamiltonian is given by 19\nH SW = [E q a + q a q + B q (a q a −q + a + −q a + q )/2](20)\nand the square of the spin-wave energy ǫ 2 q = E 2 q − B 2 q . As was shown in 19 E 0 = B 0 = Ak 2 /2 where A is the spin-wave stiffness at q ≫ k and we have gapless excitations. We assume that the ME interaction is weak and gives small corrections to A 0 and B 0 . In this case from Eqs. (18)(19)(20) for the magneto-elastic contribution to the square of the spin-wave gap we obtain\n∆ 2 ME = − Ak 2 (B A S 2 ) 2 2SM G 1 − G 2 s 2 t + G 2 s 2 l ≃ − Ak 2 Ev 0 g 2 A 4S (G 1 + G 2 )(21)\nwhere g A = S 2 B A /Ev 0 and neglecting σ we have M s 2 l = Ev 0 and s t = s l / √ 2. This expression is negative as it should be in the second order perturbation theory and ∆ 2 ME = 0 in < 100 > direction only (see Appendix A). Consideration of q ± k terms lead to expression similar to Eq.(17) with replacing a − a + → a + a + which do not contribute to the gap.\n\nThe spin-wave interaction considered in the 1/S approximation leads to positive contribution to ∆ 2 which is given by\n∆ 2 I = (Ak 2 ) 2 4SN D q D 0 ,(22)\nand D q is a form-factor of the Dzyaloshinskii interaction 19 , 35 . The helical structure can be stable if\n∆ 2 = ∆ 2 I + ∆ 2 ME > 0(23)\nand if ∆ I = 0 it can survive at k < 100 > only where ∆ ME = 0. Meanwhile it is well known that in M nSi and F eGe at low T the helix axis k < 111 > and ∆ I > |∆ ME |.\n\n\nV. EXPERIMENTAL CONSEQUENCES AND DISCUSSION\n\nFor discussion experimental consequences of the magneto-elastic interaction we have to know the Young modules E and anisotropic part of the magneto elastic interaction S 2 B A . For M nSi according to 17 , 36 the bulk modulus K = 1.37 × 10 6 bar and neglecting the Poisson coefficient σ we obtain E = 3K = 4, 11 × 10 6 bar and Ev 0 = 240eV (v 0 = 95 × 10 −24 cm 3 ).\n\nUnfortunately the value of B A S 2 is unknown. As we will see below it may be determined by x-ray and neutron scattering. Isotropic part of the ME interaction was studied by indirect method in 37 . Its contribution to the lattice constant ∆a/a ≃ −1.1 × 10 −4 at T = 0K was determined and the sum of the isotropic part of the ME and elastic energies can be represented as B is S 2 (∆a/a) + Kv 0 (3∆a/a) 2 /2.\n\nThis expression is minimal at g is = S 2 B is /Ev 0 = −3.3 × 10 −4 and from Eqs. (15) and (21) we obtain\nE ME = −6.5µeV {57mT }(g A /g is ) 2 (G 1 − G 2 /2),(25)∆ 2 ME = −(17µeV {0.15T }) 2 (g A /g is ) 2 (G 1 + G 2 ),(26)\nwhere we used S = 1.6, Ak 2 = H C = 0.6T and H C is a critical field for transition to ferromagnetic state 19 . We compare now above results with existing experimental data 23 . As was shown in 19 the ground state energy of the helical structure in magnetic field is given by\nE G = E A + E ME − SH 2 2H C − SH 2 ⊥ ∆ 2 4H C (∆ 2 − H 2 ⊥ /2) ,(27)\nwhere H (⊥) is a field component along (perpendicular) to the helix wave-vector k, E A = (S 2 F 0 k 2 − 3S 4 K)L/4, F 0 and K are constants of the anisotropic exchange and cubic anisotropy respectively 4,19 . The cubic invariant L = (g ·ĉ) is considered in Appendix A. Evolution of the helical structure in magnetic field was studied by small angle polarized neutron scattering in M nSi near T C 24 at low T 23 and in compound F eCoSi 25,26 . Two new characteristic fields were determined. In zero field the sample is in multidomain state with k along all < 111 > directions. Then for the field along one of < 111 > axes at H C1 the single domain state appears. With further field increasing the Bragg intensity demonstrated a cusp at H in . In 23 it was interpreted as instability of the k direction connected with the second term in Eq. (27). Indeed if H is slightly below ∆ √ 2 this term predominates and the vector k has to rotate perpendicular to the field but blocked by the anisotropy. Just below T C where the anisotropy is weak this rotation was observed in 24 . In M nSi we have H C1 = 80mT , H in 160mT and ∆ ≃ 110mT = 13µeV 23 .\n\nIt is obvious that the single domain state can be realized if SH 2\nCI /2H C ≃ 9mT is of order of E A +E ME . For < 111 > directions we have E ME ≃ 25(g A /g is ) 2 mT [See Eq.(A5)]\n. So this condition is fulfilled roughly. It is impossible to do more detailed analysis as we do not know E A and g A .\n\nThe invariant L has two extrema L = 2/3 and L = 0 at k along < 111 > and < 100 > directions respectively and a saddle points at < 110 > directions. Hence if one neglects the ME interaction the configuration with k < 110 > is forbidden 4 . The same holds for maxima of the critical fluctuations above T C ≃ 29K 16 . Meanwhile in M nSi at high pressure above the quantum critical point p c ≃ 14.6kbar maxima of the neutron scattering at < 110 > directions were observed 11 . In Appendix B we show that the ME interaction can not resolve this problem i.e. that < 111 > and < 100 > remain only possible k directions in zero magnetic field.\n\nLet us estimate now two contribution to the spin-wave gap given by Eqs. (22) and (26). We do not know real form of the ratio r = D q /D 0 in Eq. (22). For r = 1, Ak 2 = 0.6T and S = 1.6 we get maximal value ∆ 2 I max = (0.24T ) 2 . Minimal value of ∆ 2 I may be estimated assuming that in Eq.(22) q max = 0.024nm is a border of the Stoner continuum 38 . In this case we have ∆ 2 I min = (0.035T ) 2 . In Eq.(26) for < 111 > direction expression in the bracket is equal to 4/9 [see Eq.(A5)] and we get ∆ ME ≃ 0.067(g A /g is ) 2 T . So we see that at ambient pressure both contributions are comparable with the observed ∆ ≃ 0.11T . Hence we can do plausible assumption that the quantum phase transition observed at 14.6kbar 6,7,8,11 is a result of vanishing ∆ 2 . At higher pressure the helical structure becomes unstable.\n\nFor more precise estimations experimental measurement of magneto-elastic anisotropy S 2 (B 1 − 2B 2 ) would be important. We demonstrate now that it can be directly extracted from intensities of satellite peaks near nuclear Bragg reflections. Indeed using Eq.(16) we obtain 39\nδI ± (K) = (2g A /k) 2 |K ± · [g − (g ·ĉ)ĉ/2]| 2 |F (K ± )| 2 ,(28)\nwhere K ± = K± 2k, F (Q) is the nuclear structure factor, K is a reciprocal lattice point and g A = S 2 (B 1 − 2B 2 )/Ev 0 . Relative satellite intensities are given by\nδI ± (K)/I(K) ≃ (2g A /k) 2 |K · [g − (g ·ĉ)ĉ/2]| 2 .(29)\nIn zero magnetic field vectors k are along all < 111 > directions. If K k we have δI ± = 0. If however k (1, 1, 1) but K = (2πn/a)(1, 1, −1) we obtain\nδI ± I = 4πng A 9 √ 3ka 2 = 2.2 × 10 −5 ng A g is 2(30)\nwhere we have used ka = 0.17 19 . For K (1, ±1, 0) this expression has to be multiplied on 1/4. Similar results can be obtained for other K directions with one exclusion. We are interested by crystals with P 2 1 3 (B20) symmetry where < n, 0, 0 > Bragg reflections are forbidden if n is odd and observation of very weak super-lattice reflections would be more easier 40 . Hence this case has to be considered separately. We restrict ourselves by the case k (1, 1, 1) only. First of all for even n we have Eq.(30) with replacement 2.2 → 2.2/4 ≃ 0.55. For the odd n the Bragg intensities of the satellites are given by\nI ± (K) = 0.55 × 10 −5 ng A g is 2 |F Mn (Q) + F Si (Q)| 2 ,(31)\nwhere form-factors F j (Q) are given in Appendix C. They are not zero due 2k modulation. Unfortunately Eq. (31) has an additional small factor (2ka) 2 << 1 in comparison with the even-n case. Observation of these odd reflections provides a possibility to determine the lattice chirality [See Eq.(C6)] and its connections with the spin chirality studied by polarized neutrons 5,23,24,25,26 . It has to be noted that the lattice chirality in some cases was determined by anomalous x-ray scattering 5 and electron diffraction 41 .\n\nThere are eight domains in virgin sample corresponding k along < 111 > directions. In magnetic field H C1 ≃ 0.08T << H C the single domain state is realized with k along the field and satellite intensity increases. However further increasing of the field suppresses the helical structure and it disappears at H C ≃ 0.6T 23 . In intermediate region at H C1 < H < H C the lattice modulation with the wave-vector k has to appears also 28 . Along with discussed 2k lattice modulation at low field H < ∆ √ 2 the second order helix harmonic appears 19 . I was observed in 23,42 .\n\n\nVI. CONCLUSIONS\n\nWe considered the magneto-elastic interaction in cubic helimagnets with B20 structure and demonstrated that it deformed the lattice and gave a negative contribution to the square of the spin-wave gap ∆ 2 . Hence the helical structure is stabilized due to positive contribution to ∆ 2 which stems from the magnon-magnon interaction 19 . It was suggested that the quantum phase transition observed at pressure in M nSi and F eGe is a result o competition between these two parts of the gap and takes place when ∆ 2 = 0. This suggestion is supported by rough estimations at ambient pressure of both contributions to ∆ 2 for M nSi which have the same order and close to experimentally observed gap. It was discussed also how to measure directly anisotropic part of the ME interaction responsible for considered phenomena using x-ray and neutron scattering.\n\n\nACKNOWLEDGMENT\n\nThe author thanks S.V.Grigoriev and A.V.Syromyatnikov for interesting discussions. He is very grateful to S.M.Stishov and D.Yu.Chernyshov also for very important information. This work is supported in part by the RFBR (projects No 05-02-19889, 06-02-16702 and 07-02-01318) and the Russian State Programs \"Quantum Macrophysics\", \"Strongly correlated electrons in Semiconductors, Metals, Superconductors and magnetic Materials\" and \"Neutron research of solids\".\n\n\nAPPENDIX A\n\nIn this Appendix we calculate deformation of the lattice by the ME interactions, cubic invariants G 1,2 in Eqs. (15), (21), (25)(26) and analyse their properties.\n\nWe begin with the classical energy (14). It is minimal if\nw p + (w ·ĉ)ĉ p /(1 − 2σ) = g * p ,(A1)\nwhere p = x, y, z and g p = A 2 pĉ p and we have\nw p = g * p − (ĉ · g * )ĉ p /[2(1 − σ)], E CL = −(2N S 4 B 2 /Q)[(g · g * ) − (g ·ĉ)(ĉ · g * )/(1 − σ)](A2)\nwhere 16(g · g * ) = G 1 and 16(g ·ĉ)(ĉ · g * ) = G 2 .\n\nIn cubic xyz frame we can writê a = (cos ϑ cos ϕ, cosϑ sin ϕ, − sin ϑ);b = (sin ϕ, − cos ϕ, 0);ĉ = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ),\n\nand for three principal k-directions < 111 >, < 110 >, and < 100 > we have: {â = (1/ In this representation for G-functions we obtain G 1 = sin 2 ϑ[(cos 2 ϑ cos 2 ϕ + sin 2 ϕ) 2 cos 2 ϕ + (cos 2 ϑ sin 2 ϕ + cos 2 ϕ) 2 sin 2 ϕ + sin 2 ϑ cos 2 ϑ] G 2 = sin 4 ϑ{[cos 2 ϑ(1 + sin 4 ϕ + cos 4 ϕ) + 2 sin 2 ϕ cos 2 ϕ] 2 + 4(sin 2 ϕ − cos 2 ϕ) 2 cos 2 ϑ sin 2 ϕ cos 2 ϕ} (A4)\n\nFrom these equations follows that functions G 1 − G 2 and G 2 have extrema at < 111 >, < 110 > and < 100 > near which we have G 1 − G 2 = 4/9 − 20δϑ 2 /9 − 40δϕ 2 /27; G 2 = 8δϑ 2 /9 + 16δϕ 2 /27, < 111 >;\nG 1 − G 2 = 13δϑ 2 /4 + δϕ 2 ; G 2 = 1/4 − 2(δϑ 2 + δϕ 2 ), < 110 >; G 1 − G 2 = δϑ 2 + δϕ 2 ; G 2 = 4(δϑ 4 + δϕ 4 + δϑ 2 δϕ 2 ), < 100 >,(A5)\nwhere δϑ and δϕ are distances from corresponding extremal points. Hence in considered directions both functions G 1 − G 2 and G 2 have extrema and one can show that they have not other extrema. Contribution of the anisotropic exchange and cubic anisotropy to the classical energy is proportional to cubic invariant L given by 19\nL = 4\n|A p | 2ĉ2 p = sin 2 ϑ[(cos 2 ϑ cos 2 ϕ + sin 2 ϕ) cos 2 ϕ + (cos 2 ϑ sin 2 ϕ + cos 2 ϕ) sin 2 ϕ + cos 2 ϑ].\n\nAs above for three principal directions we have\nL = 2/3 − 4δϑ 2 /3 − 8δϕ 2 /9, < 111 >; 1/2 + δϑ 2 − 2δϕ 2 , < 110 >; 2(δϑ 2 + δϕ 2 ), < 100 >,(A7)\nand L has a saddle point at < 110 >.\n\n\nAPPENDIX B\n\nWe demonstrate now that in presence of the ME contribution to the ground state energy the < 111 > and < 100 > remain only possible stable directions for the vector k.\n\nFrom Eq. (27) at H = 0 follows\nE G = ΦL − Ψ(G 1 − G 2 + G 2 /2) = Ψf (y),(B1)\nwhere Ψ > 0 and y = Φ/Ψ. We have to study behavior of f (y) for three principal directions. For k < 111 > we obtain\nf (y) = (2/3)(y − 2/3) + (4/3)(−y + 4/3)(δϑ 2 + 2δϕ 2 ).(B2)\nand E G is stable if Φ < 4Ψ/3. In the < 110 > case we have\nf (y) = (1/2)(y − 1/4) + (y − 9/4)δϑ 2 − 2yδϕ 2 ,(B3)\nIn this configuration there is a saddle point as coefficients at deviations δϑ 2 and δϕ 2 can not be positive simultaneously. Finally if k < 100 > we have f (y) = 2(y − 1/2)(δϑ 2 + δϕ 2 ).\n\nThis configuration is stable if Φ > Ψ/2. However, comparing Eqs.(B2) and (B4) we see that configuration < 111 > in the region Ψ/2 < Φ < Ψ/3 has lower energy and < 100 > configuration is metastable. Hence we see that the magneto-elastic energy can not be responsible for the stability of < 110 > configuration.\n\n\nAPPENDIX C\n\nThere are two different ions in compounds with P 2 1 3 symmetry (M n and Si 4,5 ; F e and Ge 42 etc.) labeled below as 1 and 2 respectively. Each of them occupy in cubic unit cell four positions: ρ 1 = (x, x, x), ρ 2 = (1/2 + x, 1/2 − x, 1 − x), ρ 3 = (1 − x, 1/2 + x, 1/2 − x) and ρ 4 = (1/2 − x, 1 − x, 1/2 + x) (Right-handed structure) or ρ 1 = (x, x, x) ρ 2 = (1/2 − x, 1/2 + x, 1 − x), ρ 3 = (1/2 + x, 1 − x, 1/2 − x), and ρ 4 = (1 − x, 1/2 − x, 1/2 + x) (Lefthanded structure) 5 . For M nSi we have x 1 = 0.138 (M n) and x 2 = 0.846 (Si). It is interesting to note that these numbers very close to the ion positions in \"ideal\" B20 structure with x 1 = 1/4τ = 0.1545 and x 2 = 1 − x 1 = 0.8455 where τ = (1 + √ 5)/2 43 . We consider below the odd-n case only. As we have two different ions in the unit cell the total structure factor is a sum F (Q) = F 1 (Q) + F 2 (Q) where F j (Q) = f j (Q) e i(Q·ρ λ (xj )) (C1) and f j (Q) is a scattering amplitude for the j ion.\n\nThere are eight super-lattice reflections corresponding to Q 1 = (2πn + κ, κ, κ), Q 2 = (2πn − κ, κ, κ), Q 3 = (2πn + κ, −κ, κ), and Q 4 = (2π + κ, κ, −κ) where κ = ±2ka/ √ 3. Corresponding partial form-factors are given by\nF R (Q 1 ) = F L (Q 1 ) = f e 2πinx [e 3iκx − e iκ(2−x) ] ≃ if κ(4x − 2)e 2πinx ,(C2)F R (Q 2 ) = F L (Q 2 ) = f {e 2πinx [e iκx − e iκ(1−3x) ] + e −2iπnx [e iκx − e iκ(1+x) ]} ≃ if κ[(4x − 1)e 2πinx − e −2πnx ], (C3) F R (Q 3 ) = F L (Q 4 ) = f {e 2πinx [e iκx − e iκ(1+x) ] + e −2πinx [e iκ(1−3x) − e iκx ]} ≃ if κ[−e 2πinx + (1 − 4x)e −2πinx ], (C4) F R (Q 4 ) = F L (Q 3 ) = f e −2πinx [e iκ(1+x) − e iκ(1−3x) ] ≃ 4f iκxe −2πinx .(C5)\nFrom Eqs.(C4,5) follow the intensity ratio\nI R (Q 3 ) I R (Q 4 ) = I L (Q 4 ) I L (Q 3 ) ,(C6)\nand from measurement of the 2k lattice deformation one can determines the lattice chirality.\n \n. H Katsura, A V Balatsky, N Nagaosa, Phys.Rev.Lett. 9827203H.Katsura, A.V.Balatsky, N.Nagaosa, Phys.Rev.Lett. 98, 027203 (2007).\n\n. I E Dzyaloshinskii, Zh.Eksp. Teor.Fiz. 46960Sov.Phys.I.E.Dzyaloshinskii, Zh.Eksp. Teor.Fiz.46, 1420 (1964)[Sov.Phys.JETP19, 960 (1964)];\n\n. O Nakanishi, A Yanase, A Hasegava, M Kataoka, Solid State Commun. 35995O.Nakanishi, A.Yanase, A.Hasegava, M.Kataoka, Solid State Commun.35, 995 (1980).\n\n. P Bak, M Jensen, J.Phys. C. 13881P.Bak, M.Jensen, J.Phys. C 13, L881 (1980).\n\n. M Ishida, Y Endoh, S Mitsuda, Y Ishikawa, Tanaka, J.Phys.Soc.Jpn. 542975M.Ishida, Y.Endoh, S.Mitsuda, Y.Ishikawa, M Tanaka,J.Phys.Soc.Jpn. 54 2975 (1975).\n\n. C Pfleiderer, G J Macmillan, S R Julian, G G Lonzarich, Phys.Rev. B. 558330C.Pfleiderer, G.J.MacMillan, S.R.Julian, G.G.Lonzarich, Phys.Rev. B 55, 8330 (1997).\n\n. K Koyama, T Goto, T Kanomata, R Note, Phys.Rev. B. 62986K.Koyama, T.Goto, T.Kanomata, R.Note, Phys.Rev. B 62,986 (2000).\n\n. C Pfleiderer, S R Julian, G G Lonzarich, Nature. 414427C.Pfleiderer, S.R.Julian, G.G.Lonzarich, Nature(London) 414, 427 (2001).\n\n. C Pfleiderer, D Reznik, L Pintschovius, J Haug, Phys.Rev.Lett. 99156406C.Pfleiderer, D.Reznik, L.Pintschovius, and J.Haug, Phys.Rev.Lett. 99, 156406 (2007).\n\n. P Pedrazzini, H Wilhelm, D Jaccard, T Jalborg, M Schmidt, M Hanfland, H Q Akselrud, U Yuan, Yu Schwarz, F Grin, Steglich, Phys.Rev.Lett. 9847204P.Pedrazzini, H.Wilhelm, D.Jaccard,T.Jalborg,M.Schmidt, M. Hanfland, l. Akselrud, H.Q.Yuan, U.Schwarz, Yu.Grin, and F.Steglich, Phys.Rev.Lett., 98, 047204 (2007)\n\n. C Pfleiderer, D Reznik, L Pintschovius, H V Löhneysen, M Garst, A Rosh, Nature. 427227C.Pfleiderer, D.Reznik, L.Pintschovius, H.v.Löhneysen, M.Garst, A.Rosh, Nature (London)427, 227 (2004).\n\n. S Tewari, D Belitz, T R Kirpatrick, Phys.Rev.Lett. 9647207S.Tewari, D.Belitz, T.R.Kirpatrick, Phys.Rev.Lett.96, 047207 (2006).\n\n. U K Rössler, A N Bogdanov, C Pfleiderer, Nature. 442797U.K.Rössler, A.N.Bogdanov, C.Pfleiderer, Nature(London) 442, 797 (2006).\n\n. D Belitz, T R Kirpatrick, A Roch, Phys.Rev. B. 7354431D.Belitz, T.R.Kirpatrick, A.Roch, Phys.Rev. B 73, 054431 (2006).\n\n. B Binz, A Vishwanath, Phys.Rev. B. 74214408B.Binz, A.Vishwanath, Phys.Rev. B 74, 214408 (2006).\n\n. S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, R Georgii, P Böni, D Lamago, H Eckerlebe, K Pranzas, Phys.Rev. B. 72134420S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, R.Georgii, P.Böni, D.Lamago, H.Eckerlebe, K.Pranzas, Phys.Rev. B 72,134420 (2005).\n\n. S M Stishov, A B Petrova, S Khasanov, G Kh, A A Panova, J C Shikov, D Lashley, T A Wu, Logasso, Phys.Rev. B. 7652405S.M.Stishov, A.B.Petrova, S.Khasanov, G.Kh.Panova, A.A.Shikov, J.C.Lashley, D.Wu, and T.A.Logasso, Phys.Rev. B 76, 052405 (2007).\n\n. Y J T Uemura, I M Goko, Gat, Malureanu, Nature Physics. 329Y.J.Uemura. T.Goko, I.M.Gat.Malureanu et al., Nature Physics 3, 29 (2007).\n\n. S V Maleyev, Phys.Rev. B. 73174403S.V.Maleyev, Phys.Rev. B 73, 174403 (2006).\n\n. S V Maleyev, arXiv:0711.3547S.V.Maleyev, arXiv: 0711.3547.\n\nRotation two different spins on infinitesimal angle ϕ k does not change the exchange energy but add 2D P. 1∇1 −Rotation two different spins on infinitesimal angle ϕ k does not change the exchange energy but add 2D P 1,2 (∇1 −\n\nP = 2di, · Sq)( ϕ · S−q) − ( ϕ · Sq. q · S−q)] = 0 to the Dzyaloshinskii interaction= 2Di P [(q · Sq)( ϕ · S−q) − ( ϕ · Sq)(q · S−q)] = 0 to the Dzyaloshinskii interaction.\n\n. M L Plumer, M B Walker, J.Phys.C: Solid State Phys. 144689M.L.Plumer and M.B.Walker, J.Phys.C: Solid State Phys. 14, 4689 (1981).\n\n. S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, P Böni, R Georgii, D Lamago, H Eckerslebe, K Pranzas, Phys.Rev. B. 74214414S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, P.Böni, R.Georgii, D.Lamago, H.Eckerslebe and K.Pranzas, Phys.Rev. B 74,214414 (2006).\n\n. S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, H Eckerlebe, Phys.Rev. B. 73224440S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, and H.Eckerlebe, Phys.Rev. B 73,224440 (2006).\n\n. S V Grigoriev, S V Maleyev, V A Dyadkin, D Menzel, J Shoenes, H Eckerlebe, Phys.Pev. B. 7692407S.V.Grigoriev, S.V.Maleyev, V.A.Dyadkin, D.Menzel, J.Shoenes, and H.Eckerlebe, Phys.Pev. B 76,092407 (2007).\n\n. S V Grigoriev, V A Dyadkin, J Schoenes, Yu O Chetverikov, A I Okorokov, H Eckerlebe, S V Maleyev, Phys.Rev. B. 76224424S.V.Grigoriev, V.A.Dyadkin, J.Schoenes, Yu.O.Chetverikov, A.I.Okorokov, H.Eckerlebe, and S.V.Maleyev, Phys.Rev. B 76, 224424 (2007).\n\n. G A Gehring, arXiv:0711.2586G.A. Gehring, arXiv:0711.2586.\n\n. M L Plumer, M B Walker, J.Phys.C: Solid. State Phys. 157181M.L.Plumer and M.B.Walker, J.Phys.C: Solid. State Phys.15,7181 (1982).\n\n. M L Plumer, J.Phys.C: Solid.State.Phys. 174663M.L.Plumer, J.Phys.C: Solid.State.Phys., 17, 4663 (1984).\n\nL D Landau, E M Lifshitz, Electrodynamics of Continuous Media. OxfordPergamon PressL.D.Landau and E.M.Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1984).\n\n. L D Landau, E M Lifshitz, Statistical Physics, Pt. 1Pergamon PressL.D.Landau and E.M.Lifshitz, Statistical Physics, Pt 1 (Pergamon Press, Oxford,1969).\n\nAs in 19 we use term umklapp for processes mixing excitations with momenta k and ±k, ±2k etc. As in 19 we use term umklapp for processes mixing excitations with momenta k and ±k, ±2k etc..\n\n33 They are allowed in magnetized state only 28. 33 They are allowed in magnetized state only 28 .\n\nL D Landau, E M Lifshitz, Theory of Elasticity. OxfordPergamon PressL.D.Landau and E.M.Lifshitz, Theory of Elasticity (Pergamon Press, Oxford, 1986).\n\nAt H = 0 cubic anisotropy does not contribute to ∆ 2 and Eqs. in 19 are erroneousAt H = 0 cubic anisotropy does not contribute to ∆ 2 and Eqs.(52-53) in 19 are erroneous.\n\n. E L Gromnitskaya, Ref , 17E.L.Gromnitskaya, Ref.[23] in 17 .\n\n. M Matsunaga, Y Ishikawa, T Nakajima, J.Phys.Soc.Jpn. 511153M.Matsunaga, Y.Ishikawa, and T.Nakajima, J.Phys.Soc.Jpn. 51, 1153 (1982).\n\n. Y Ishikawa, G Shirane, J A Tarvin, M Kohgi, Phys.Rev. B. 164956Y.Ishikawa, G.Shirane, J.A.Tarvin, and M.Kohgi, Phys.Rev. B 16, 4956 (1977).\n\nEquations presented below are correct if the lattice and the helical structure have the same mosaic. For M nSi the magnetic mosaic is greater than the lattice one 23 and an additional small factor has to be. introduced in expressions for the relative intensitiesEquations presented below are correct if the lattice and the helical structure have the same mosaic. For M nSi the magnetic mosaic is greater than the lattice one 23 and an additional small factor has to be introduced in expressions for the relative intensities.\n\nCheryshov for corresponding explanation. Yu, I am thankful D.Yu.Cheryshov for corresponding explanation.\n\n. M Tanaka, H Takajoshi, M Ishida, Y Endoh, J.phys. Soc.Jpn. 542970M.Tanaka, H.Takajoshi, M.Ishida, and Y.Endoh, J.phys. Soc.Jpn., 54, 2970 (1985).\n\n. B Lebech, J Bernard, T Freltoft, J.Phys.: Condens.Matter. 16105B.Lebech, J.Bernard, and T.Freltoft, J.Phys.: Condens.Matter 1, 6105 (1989).\n\n. L Vocadlo, G D Price, I G Wood, Acta.Cryst., B. 55484L.Vocadlo, G.D.Price, and I.G.Wood, Acta.Cryst., B 55, 484 (1999).\n"},"corpus_id":{"kind":"number","value":41656615,"string":"41,656,615"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0804.1678\",\n \"mag\": 3103839033,\n \"acl\": null,\n \"pubmed\": \"21825348\",\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/0953-8984/21/14/146001\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0804.1678v1.pdf\"\n ],\n \"pdf_sha\": \"05fc6dd3d8664d454dbdb6af08c9dc5e16ce27eb\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/abs/0804.1678\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Magneto-elastic interaction in cubic helimagnets with B20 structure"},"authors":{"kind":"list like","value":["S V Maleyev \nPetersburg Nuclear Physics Institute\n188300Gatchina, PetersburgStRussia"],"string":"[\n \"S V Maleyev \\nPetersburg Nuclear Physics Institute\\n188300Gatchina, PetersburgStRussia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The magneto-elastic interaction in cubic helimagnets with B20 symmetry is considered. It is shown that this interaction is responsible for negative contribution to the square of the spin-wave gap ∆ which is alone has to disrupt assumed helical structure. It is suggested that competition between positive part of ∆ 2 I which stems from magnon-magnon interaction and its negative magneto-elastic part leads to the quantum phase transition observed at high pressure in M nSi and F eGe. This transition has to occur when ∆ 2 = 0. For M nSi from rough estimations at ambient pressure both parts ∆I and |∆ME| are comparable with the experimentally observed gap. The magneto-elastic interaction is responsible also for 2k modulation of the lattice where k is the helix wave-vector and contribution to the magnetic anisotropy.Experimental observation by x-ray and neutron scattering the lattice modulation allows determine the strength of anisotropic part of the magneto-elastic interaction responsible for above phenomena and the lattice helicity."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"Helical magnetic order stipulated by Dzyaloshinskii-Moriya interaction and its evolution in magnetic field attract a lot of attention in last years (see for example 1 and references therein). In this respect non-centrosymmetric cubic helimagnets such as M nSi, F eGe, F eCoSi with P 2 1 3 (B20) symmetry play special role. They are a subject of the intensive experimental and theoretical studies during several decades. Their single-handed helical structure was explained by Dzyaloshinskii 2 . The full set of interactions responsible for observed helical structure (Bak-Jensen model) was established later in 3 , 4 in agreement with existing experimental data (see for example 5 and references therein). The renascence in this field began with a discovery of the quantum phase transition to a disordered (partially ordered) state in M nSi at high pressure (magnetization and resistivity measurements 6 , 7 and neutron scattering 8 , 9 ). Recently similar transition was observed in F eGe 10 . The following properties of this state attract the main attention: i) non-Fermi-liquid conductivity, ii) spherical neutron scattering surface with the weak maxima along the 110 axes 8 , 11 , whereas at ambient pressure Bragg reflections and critical scattering were observed along 111 5 . These features and the structure of the partially ordered state were discussed in several theoretical papers (see 12,13,14,15 and references therein). It should be noted that the spherical scattering surface with maxima along 111 was observed at ambient pressure just above critical temperature T c ≃ 29K and explained using the Bak-Jensen model 16 . It was demonstrated also that this phase transition is of the first order one 17 Recently phase separation in M nSi near the quantum phase transition was observed by muon spin resonant method 18 .\nIn 1914 strong anisotropy of the spin-wave spectrum at low momenta was demonstrated: excitations with momentum along and perpendicular to the helix wave-vector k have linear and quadratic dispersion respectively. At the same time there is a contradiction between these papers. In 14 was claimed that the spin-waves are gapless Goldstone excitations due to translation invariance along the helix axis whereas in 19 the spin-wave gap was calculated in 1/S approximation. This contradiction was discussed in 20 . In brief its essence is following. In 14 the 1/S corrections to the spin-wave energy were not evaluated and the translation invariance was not proved. Meanwhile doing this the authors should meet a problem how to consider the Dzyaloshinskii-Moriya interaction. It contains two spin operators and if they belong to the single lattice point, the translation invariance holds and the gap is zero. However this interaction always acts between different spins, the translation changes the pair energy and the gap is not zero 21 .\nExistence of the gap ∆ is very important for correct description of the helix behavior in magnetic field H ⊥ perpendicular to the helix vector k. In the gapless case the spin-wave spectrum becomes unstable in infinitesimal H ⊥ in contradiction to the well-known experimental findings 19 and predictions of the phenomenological Landau-like theory 22 . In 19 was shown that the helix state remains stable if H ⊥ < ∆ √ 2 and then k begins rotate toward the field. Recently this prediction was confirmed using small angle polarized neutron scattering and was found that ∆ ≃ 13µeV for M nSi 23,24,25,26 .\nThe quantum phase transition to magnetically disordered state observed in M nSi and F eGe 7,8,9,10,11 is an another important problem. Recently it was considered on the base of phenomenological Landau theory 27 . However up to now we have not any attempt to understand microscopic origin of this transition. In this paper we consider the magneto-elastic (ME) interaction and evaluate its contribution to the square of the spin-wave gap ∆. We demonstrate that\nwhere ∆ 2 I was evaluated in 19 and ∆ 2 ME < 0 appears due to the ME considered as the second order perturbation. It is important to note that if ∆ 2 I is zero the ME has to disrupt the helical magnetic order. Rough estimations using existing experimental data at ambient pressure (see Sec.V) give |∆ ME | ∼ 7.6µeV and ∆ I = 4.0 ÷ 28µeV . Both contributions are comparable with experimental value determined in 23 . Hence at pressure two parts of ∆ 2 has to compete and the quantum phase transition to the partly disordered state occurs when ∆ = 0. Besides we estimated the ME contribution to the magnetic anisotropy and demonstrated that it is not small in comparison to the experimental value. We demonstrated also that the ME leads to the lattice deformation with the wave-vector 2k and evaluated intensities of corresponding super-lattice reflections. Their experimental study would allow determine the strength of anisotropic part of the ME interaction responsible for above mentioned phenomena and the lattice helicity. It has to be noted that the ME interaction in the Landau theory was investigated in 28,29 and the lattice deformation was predicted at the wave-vector k. This result does not contradict to ours as it was obtained for magnetized systems only.\nThe paper is organized as follows. In Sec.II we consider the ME in cubic helimagnets. Classical ground state energy and the lattice deformation is studied in Sec.III. The spin-wave-phonon interaction and the ME contribution to ∆ 2 is considered in Sec.IV. Obtained results, numerical estimations and experimental consequences are discussed in Sec.V and main results are summarized in Sec.IV In Appendixes A and B some mathematical details are considered. Appendix C is devoted to consideration super-lattice reflections near forbidden < n00 > Bragg peaks with odd n-s."},{"title":"II. MAGNETO-ELASTIC INTERACTION","content":"In general form the magneto elastic energy is given by (see for example 30 )\nwhere S α R is the spin components at the lattice point R , U γµ = (1/2)(∂u γ /∂R µ + ∂u µ /∂R γ ) is the deformation tensor and the lattice site displacement has well known form\nwhere e qj are vectors of the phonon polarization , b(b + ) their absorption (excitation) operators and e −qj = e * qj 31 . Tensor B is symmetric in (αβ) and (γµ) components. In cubic crystals we have following non-zero components 30\nand B xyxy = B yxxy = B xyyx etc. In isotropic medium we have\nand\nIn cubic helimagnets the lattice spin is given by 19\nwhere < S ζ R > = 0 is an average value of the lattice spin, S η,ξ R describe its perpendicular motion, k is the helix wave-vector, A = (â − ib)/2, unit vectorsâ,b, andĉ form right-handed orthogonal frame, sin α = −H /H C where H is the magnetic field component along the helix vector k and H C is the critical field for transition to ferromagnetic state. According to 19 the vector k ĉ in arbitrary field.\nUsing standard definition S q = N −1/2 S R exp (−iq · R) in momentum space we obtain 19\nwhere S c q , S A q and S A * q are functions of q, q − k and q + k respectively. The spin components in Eq.(7) have well-known form\nwhere a q and a + q are conventional spin-wave operators. In momentum space Eq.(2) is given by\nThis expression is divided on three parts: direct (ĉĉ and A A * ) terms where the U tensor is k independent, first (cA), (cA * ) and second (A A and A * A * ) order umklapp terms where U operator depends on q 1 + q 2 ± k and q 1 + q 2 ± 2k respectively 32 . For their consideration we use following identity\nwhere B A = B 1 − 2B 2 is an anisotropic part of the tensor B and u = u Q .\nIn the case of uniform pressure P we have U αβ = −(P/3K)δ αβ where K is the bulk modulus and U q ∼ N 1/2 δ q,0 . As a result the umklapp terms are zero asĉ · A = A · A = 0 and V ME → −N B 1 S(S + 1)P/(3K). Hence the uniform pressure contributes to the classical part of the magneto-elastic ground state energy only. However it has to change basic parameters of the problem such as B 1,2 , sound velocities etc."},{"title":"III. GROUND STATE ENERGY AND LATTICE DEFORMATION","content":"In zero magnetic field sin α = 0 and we have a planar helix. In this case from Eqs. (7)(8)(9)(10) follows that in the classical part of the ME interaction the first-order umklapps are forbidden 33 and we obtain\nwhere g p = A 2 pĉ p and u p = u p −2k . For evaluation of the lattice deformation we must consider the elastic energy. Unlike Refs. 28,29 for simplicity we ignore in it the cubic symmetry as its principal symmetry breaking role has been taken into account above in Eq (11). In this case the unit cell energy is given by 34\n, v 0 is the unit cell volume, and E and σ are Young modulus and Poisson coefficient respectively. As a result we have\nFrom Eqs. (11)(12)(13) for the ME part of the ground state energy we obtain\nwhere\nMinimum of this energy is evaluated in Appendix A and we have\nwhere g A = S 2 B A /Ev 0 , cubic invariants G 1 = 16(g · g * ) and G 2 = 16(g ·ĉ)(g * ·ĉ) are considered in Appendix A and in the right hand side of both equations we neglect σ as it is usually small 34 ."},{"title":"IV. MAGNON-PHONON INTERACTION","content":"We consider now the magnon-phonon interaction. We are interested by terms which survive at q = 0 and contribute to the spin-wave gap as other terms are small corrections to the q-dependent part of the magnon dispersion considered in 19 and 14 . To single out them we have to replace in Eq.(9) one of S ζ operators by Sδ q±k,0 . As a result we obtain terms with phonon momenta q, q ± k, and q ± 2k. The former disappear at q = 0, the second are proportional to a q + a + −q and can not contribute to the gap (see below). Using identity (10) for the last AA end A * A * terms in the case of planar helix we have\nFrom this equation for the magnon-magnon interaction we obtain\nwhere we neglect q in comparison with ±2k . The phonon Green function can be represented as\nwhere l(t)labels longitudinal ( transverse) phonon mode and s l(t) is a corresponding sound velocity. We neglect optical branches as their contribution is of order (sk/θ D ) 2 ≪ 1 where θ D is Debye temperature.\nIn the linear spin-wave theory the Hamiltonian is given by 19\nand the square of the spin-wave energy ǫ 2 q = E 2 q − B 2 q . As was shown in 19 E 0 = B 0 = Ak 2 /2 where A is the spin-wave stiffness at q ≫ k and we have gapless excitations. We assume that the ME interaction is weak and gives small corrections to A 0 and B 0 . In this case from Eqs. (18)(19)(20) for the magneto-elastic contribution to the square of the spin-wave gap we obtain\nwhere g A = S 2 B A /Ev 0 and neglecting σ we have M s 2 l = Ev 0 and s t = s l / √ 2. This expression is negative as it should be in the second order perturbation theory and ∆ 2 ME = 0 in < 100 > direction only (see Appendix A). Consideration of q ± k terms lead to expression similar to Eq.(17) with replacing a − a + → a + a + which do not contribute to the gap.\nThe spin-wave interaction considered in the 1/S approximation leads to positive contribution to ∆ 2 which is given by\nand D q is a form-factor of the Dzyaloshinskii interaction 19 , 35 . The helical structure can be stable if\nand if ∆ I = 0 it can survive at k < 100 > only where ∆ ME = 0. Meanwhile it is well known that in M nSi and F eGe at low T the helix axis k < 111 > and ∆ I > |∆ ME |."},{"title":"V. EXPERIMENTAL CONSEQUENCES AND DISCUSSION","content":"For discussion experimental consequences of the magneto-elastic interaction we have to know the Young modules E and anisotropic part of the magneto elastic interaction S 2 B A . For M nSi according to 17 , 36 the bulk modulus K = 1.37 × 10 6 bar and neglecting the Poisson coefficient σ we obtain E = 3K = 4, 11 × 10 6 bar and Ev 0 = 240eV (v 0 = 95 × 10 −24 cm 3 ).\nUnfortunately the value of B A S 2 is unknown. As we will see below it may be determined by x-ray and neutron scattering. Isotropic part of the ME interaction was studied by indirect method in 37 . Its contribution to the lattice constant ∆a/a ≃ −1.1 × 10 −4 at T = 0K was determined and the sum of the isotropic part of the ME and elastic energies can be represented as B is S 2 (∆a/a) + Kv 0 (3∆a/a) 2 /2.\nThis expression is minimal at g is = S 2 B is /Ev 0 = −3.3 × 10 −4 and from Eqs. (15) and (21) we obtain\nwhere we used S = 1.6, Ak 2 = H C = 0.6T and H C is a critical field for transition to ferromagnetic state 19 . We compare now above results with existing experimental data 23 . As was shown in 19 the ground state energy of the helical structure in magnetic field is given by\nwhere H (⊥) is a field component along (perpendicular) to the helix wave-vector k, E A = (S 2 F 0 k 2 − 3S 4 K)L/4, F 0 and K are constants of the anisotropic exchange and cubic anisotropy respectively 4,19 . The cubic invariant L = (g ·ĉ) is considered in Appendix A. Evolution of the helical structure in magnetic field was studied by small angle polarized neutron scattering in M nSi near T C 24 at low T 23 and in compound F eCoSi 25,26 . Two new characteristic fields were determined. In zero field the sample is in multidomain state with k along all < 111 > directions. Then for the field along one of < 111 > axes at H C1 the single domain state appears. With further field increasing the Bragg intensity demonstrated a cusp at H in . In 23 it was interpreted as instability of the k direction connected with the second term in Eq. (27). Indeed if H is slightly below ∆ √ 2 this term predominates and the vector k has to rotate perpendicular to the field but blocked by the anisotropy. Just below T C where the anisotropy is weak this rotation was observed in 24 . In M nSi we have H C1 = 80mT , H in 160mT and ∆ ≃ 110mT = 13µeV 23 .\nIt is obvious that the single domain state can be realized if SH 2\n. So this condition is fulfilled roughly. It is impossible to do more detailed analysis as we do not know E A and g A .\nThe invariant L has two extrema L = 2/3 and L = 0 at k along < 111 > and < 100 > directions respectively and a saddle points at < 110 > directions. Hence if one neglects the ME interaction the configuration with k < 110 > is forbidden 4 . The same holds for maxima of the critical fluctuations above T C ≃ 29K 16 . Meanwhile in M nSi at high pressure above the quantum critical point p c ≃ 14.6kbar maxima of the neutron scattering at < 110 > directions were observed 11 . In Appendix B we show that the ME interaction can not resolve this problem i.e. that < 111 > and < 100 > remain only possible k directions in zero magnetic field.\nLet us estimate now two contribution to the spin-wave gap given by Eqs. (22) and (26). We do not know real form of the ratio r = D q /D 0 in Eq. (22). For r = 1, Ak 2 = 0.6T and S = 1.6 we get maximal value ∆ 2 I max = (0.24T ) 2 . Minimal value of ∆ 2 I may be estimated assuming that in Eq.(22) q max = 0.024nm is a border of the Stoner continuum 38 . In this case we have ∆ 2 I min = (0.035T ) 2 . In Eq.(26) for < 111 > direction expression in the bracket is equal to 4/9 [see Eq.(A5)] and we get ∆ ME ≃ 0.067(g A /g is ) 2 T . So we see that at ambient pressure both contributions are comparable with the observed ∆ ≃ 0.11T . Hence we can do plausible assumption that the quantum phase transition observed at 14.6kbar 6,7,8,11 is a result of vanishing ∆ 2 . At higher pressure the helical structure becomes unstable.\nFor more precise estimations experimental measurement of magneto-elastic anisotropy S 2 (B 1 − 2B 2 ) would be important. We demonstrate now that it can be directly extracted from intensities of satellite peaks near nuclear Bragg reflections. Indeed using Eq.(16) we obtain 39\nwhere K ± = K± 2k, F (Q) is the nuclear structure factor, K is a reciprocal lattice point and g A = S 2 (B 1 − 2B 2 )/Ev 0 . Relative satellite intensities are given by\nIn zero magnetic field vectors k are along all < 111 > directions. If K k we have δI ± = 0. If however k (1, 1, 1) but K = (2πn/a)(1, 1, −1) we obtain\nwhere we have used ka = 0.17 19 . For K (1, ±1, 0) this expression has to be multiplied on 1/4. Similar results can be obtained for other K directions with one exclusion. We are interested by crystals with P 2 1 3 (B20) symmetry where < n, 0, 0 > Bragg reflections are forbidden if n is odd and observation of very weak super-lattice reflections would be more easier 40 . Hence this case has to be considered separately. We restrict ourselves by the case k (1, 1, 1) only. First of all for even n we have Eq.(30) with replacement 2.2 → 2.2/4 ≃ 0.55. For the odd n the Bragg intensities of the satellites are given by\nwhere form-factors F j (Q) are given in Appendix C. They are not zero due 2k modulation. Unfortunately Eq. (31) has an additional small factor (2ka) 2 << 1 in comparison with the even-n case. Observation of these odd reflections provides a possibility to determine the lattice chirality [See Eq.(C6)] and its connections with the spin chirality studied by polarized neutrons 5,23,24,25,26 . It has to be noted that the lattice chirality in some cases was determined by anomalous x-ray scattering 5 and electron diffraction 41 .\nThere are eight domains in virgin sample corresponding k along < 111 > directions. In magnetic field H C1 ≃ 0.08T << H C the single domain state is realized with k along the field and satellite intensity increases. However further increasing of the field suppresses the helical structure and it disappears at H C ≃ 0.6T 23 . In intermediate region at H C1 < H < H C the lattice modulation with the wave-vector k has to appears also 28 . Along with discussed 2k lattice modulation at low field H < ∆ √ 2 the second order helix harmonic appears 19 . I was observed in 23,42 ."},{"title":"VI. CONCLUSIONS","content":"We considered the magneto-elastic interaction in cubic helimagnets with B20 structure and demonstrated that it deformed the lattice and gave a negative contribution to the square of the spin-wave gap ∆ 2 . Hence the helical structure is stabilized due to positive contribution to ∆ 2 which stems from the magnon-magnon interaction 19 . It was suggested that the quantum phase transition observed at pressure in M nSi and F eGe is a result o competition between these two parts of the gap and takes place when ∆ 2 = 0. This suggestion is supported by rough estimations at ambient pressure of both contributions to ∆ 2 for M nSi which have the same order and close to experimentally observed gap. It was discussed also how to measure directly anisotropic part of the ME interaction responsible for considered phenomena using x-ray and neutron scattering."},{"title":"ACKNOWLEDGMENT","content":"The author thanks S.V.Grigoriev and A.V.Syromyatnikov for interesting discussions. He is very grateful to S.M.Stishov and D.Yu.Chernyshov also for very important information. This work is supported in part by the RFBR (projects No 05-02-19889, 06-02-16702 and 07-02-01318) and the Russian State Programs \"Quantum Macrophysics\", \"Strongly correlated electrons in Semiconductors, Metals, Superconductors and magnetic Materials\" and \"Neutron research of solids\"."},{"title":"APPENDIX A","content":"In this Appendix we calculate deformation of the lattice by the ME interactions, cubic invariants G 1,2 in Eqs. (15), (21), (25)(26) and analyse their properties.\nWe begin with the classical energy (14). It is minimal if\nwhere p = x, y, z and g p = A 2 pĉ p and we have\nwhere 16(g · g * ) = G 1 and 16(g ·ĉ)(ĉ · g * ) = G 2 .\nIn cubic xyz frame we can writê a = (cos ϑ cos ϕ, cosϑ sin ϕ, − sin ϑ);b = (sin ϕ, − cos ϕ, 0);ĉ = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ),\nand for three principal k-directions < 111 >, < 110 >, and < 100 > we have: {â = (1/ In this representation for G-functions we obtain G 1 = sin 2 ϑ[(cos 2 ϑ cos 2 ϕ + sin 2 ϕ) 2 cos 2 ϕ + (cos 2 ϑ sin 2 ϕ + cos 2 ϕ) 2 sin 2 ϕ + sin 2 ϑ cos 2 ϑ] G 2 = sin 4 ϑ{[cos 2 ϑ(1 + sin 4 ϕ + cos 4 ϕ) + 2 sin 2 ϕ cos 2 ϕ] 2 + 4(sin 2 ϕ − cos 2 ϕ) 2 cos 2 ϑ sin 2 ϕ cos 2 ϕ} (A4)\nFrom these equations follows that functions G 1 − G 2 and G 2 have extrema at < 111 >, < 110 > and < 100 > near which we have G 1 − G 2 = 4/9 − 20δϑ 2 /9 − 40δϕ 2 /27; G 2 = 8δϑ 2 /9 + 16δϕ 2 /27, < 111 >;\nwhere δϑ and δϕ are distances from corresponding extremal points. Hence in considered directions both functions G 1 − G 2 and G 2 have extrema and one can show that they have not other extrema. Contribution of the anisotropic exchange and cubic anisotropy to the classical energy is proportional to cubic invariant L given by 19\n|A p | 2ĉ2 p = sin 2 ϑ[(cos 2 ϑ cos 2 ϕ + sin 2 ϕ) cos 2 ϕ + (cos 2 ϑ sin 2 ϕ + cos 2 ϕ) sin 2 ϕ + cos 2 ϑ].\nAs above for three principal directions we have\nand L has a saddle point at < 110 >."},{"title":"APPENDIX B","content":"We demonstrate now that in presence of the ME contribution to the ground state energy the < 111 > and < 100 > remain only possible stable directions for the vector k.\nFrom Eq. (27) at H = 0 follows\nwhere Ψ > 0 and y = Φ/Ψ. We have to study behavior of f (y) for three principal directions. For k < 111 > we obtain\nand E G is stable if Φ < 4Ψ/3. In the < 110 > case we have\nIn this configuration there is a saddle point as coefficients at deviations δϑ 2 and δϕ 2 can not be positive simultaneously. Finally if k < 100 > we have f (y) = 2(y − 1/2)(δϑ 2 + δϕ 2 ).\nThis configuration is stable if Φ > Ψ/2. However, comparing Eqs.(B2) and (B4) we see that configuration < 111 > in the region Ψ/2 < Φ < Ψ/3 has lower energy and < 100 > configuration is metastable. Hence we see that the magneto-elastic energy can not be responsible for the stability of < 110 > configuration."},{"title":"APPENDIX C","content":"There are two different ions in compounds with P 2 1 3 symmetry (M n and Si 4,5 ; F e and Ge 42 etc.) labeled below as 1 and 2 respectively. Each of them occupy in cubic unit cell four positions: ρ 1 = (x, x, x), ρ 2 = (1/2 + x, 1/2 − x, 1 − x), ρ 3 = (1 − x, 1/2 + x, 1/2 − x) and ρ 4 = (1/2 − x, 1 − x, 1/2 + x) (Right-handed structure) or ρ 1 = (x, x, x) ρ 2 = (1/2 − x, 1/2 + x, 1 − x), ρ 3 = (1/2 + x, 1 − x, 1/2 − x), and ρ 4 = (1 − x, 1/2 − x, 1/2 + x) (Lefthanded structure) 5 . For M nSi we have x 1 = 0.138 (M n) and x 2 = 0.846 (Si). It is interesting to note that these numbers very close to the ion positions in \"ideal\" B20 structure with x 1 = 1/4τ = 0.1545 and x 2 = 1 − x 1 = 0.8455 where τ = (1 + √ 5)/2 43 . We consider below the odd-n case only. As we have two different ions in the unit cell the total structure factor is a sum F (Q) = F 1 (Q) + F 2 (Q) where F j (Q) = f j (Q) e i(Q·ρ λ (xj )) (C1) and f j (Q) is a scattering amplitude for the j ion.\nThere are eight super-lattice reflections corresponding to Q 1 = (2πn + κ, κ, κ), Q 2 = (2πn − κ, κ, κ), Q 3 = (2πn + κ, −κ, κ), and Q 4 = (2π + κ, κ, −κ) where κ = ±2ka/ √ 3. Corresponding partial form-factors are given by\nFrom Eqs.(C4,5) follow the intensity ratio\nand from measurement of the 2k lattice deformation one can determines the lattice chirality."}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Helical magnetic order stipulated by Dzyaloshinskii-Moriya interaction and its evolution in magnetic field attract a lot of attention in last years (see for example 1 and references therein). In this respect non-centrosymmetric cubic helimagnets such as M nSi, F eGe, F eCoSi with P 2 1 3 (B20) symmetry play special role. They are a subject of the intensive experimental and theoretical studies during several decades. Their single-handed helical structure was explained by Dzyaloshinskii 2 . The full set of interactions responsible for observed helical structure (Bak-Jensen model) was established later in 3 , 4 in agreement with existing experimental data (see for example 5 and references therein). The renascence in this field began with a discovery of the quantum phase transition to a disordered (partially ordered) state in M nSi at high pressure (magnetization and resistivity measurements 6 , 7 and neutron scattering 8 , 9 ). Recently similar transition was observed in F eGe 10 . The following properties of this state attract the main attention: i) non-Fermi-liquid conductivity, ii) spherical neutron scattering surface with the weak maxima along the 110 axes 8 , 11 , whereas at ambient pressure Bragg reflections and critical scattering were observed along 111 5 . These features and the structure of the partially ordered state were discussed in several theoretical papers (see 12,13,14,15 and references therein). It should be noted that the spherical scattering surface with maxima along 111 was observed at ambient pressure just above critical temperature T c ≃ 29K and explained using the Bak-Jensen model 16 . It was demonstrated also that this phase transition is of the first order one 17 Recently phase separation in M nSi near the quantum phase transition was observed by muon spin resonant method 18 .\\nIn 1914 strong anisotropy of the spin-wave spectrum at low momenta was demonstrated: excitations with momentum along and perpendicular to the helix wave-vector k have linear and quadratic dispersion respectively. At the same time there is a contradiction between these papers. In 14 was claimed that the spin-waves are gapless Goldstone excitations due to translation invariance along the helix axis whereas in 19 the spin-wave gap was calculated in 1/S approximation. This contradiction was discussed in 20 . In brief its essence is following. In 14 the 1/S corrections to the spin-wave energy were not evaluated and the translation invariance was not proved. Meanwhile doing this the authors should meet a problem how to consider the Dzyaloshinskii-Moriya interaction. It contains two spin operators and if they belong to the single lattice point, the translation invariance holds and the gap is zero. However this interaction always acts between different spins, the translation changes the pair energy and the gap is not zero 21 .\\nExistence of the gap ∆ is very important for correct description of the helix behavior in magnetic field H ⊥ perpendicular to the helix vector k. In the gapless case the spin-wave spectrum becomes unstable in infinitesimal H ⊥ in contradiction to the well-known experimental findings 19 and predictions of the phenomenological Landau-like theory 22 . In 19 was shown that the helix state remains stable if H ⊥ < ∆ √ 2 and then k begins rotate toward the field. Recently this prediction was confirmed using small angle polarized neutron scattering and was found that ∆ ≃ 13µeV for M nSi 23,24,25,26 .\\nThe quantum phase transition to magnetically disordered state observed in M nSi and F eGe 7,8,9,10,11 is an another important problem. Recently it was considered on the base of phenomenological Landau theory 27 . However up to now we have not any attempt to understand microscopic origin of this transition. In this paper we consider the magneto-elastic (ME) interaction and evaluate its contribution to the square of the spin-wave gap ∆. We demonstrate that\\nwhere ∆ 2 I was evaluated in 19 and ∆ 2 ME < 0 appears due to the ME considered as the second order perturbation. It is important to note that if ∆ 2 I is zero the ME has to disrupt the helical magnetic order. Rough estimations using existing experimental data at ambient pressure (see Sec.V) give |∆ ME | ∼ 7.6µeV and ∆ I = 4.0 ÷ 28µeV . Both contributions are comparable with experimental value determined in 23 . Hence at pressure two parts of ∆ 2 has to compete and the quantum phase transition to the partly disordered state occurs when ∆ = 0. Besides we estimated the ME contribution to the magnetic anisotropy and demonstrated that it is not small in comparison to the experimental value. We demonstrated also that the ME leads to the lattice deformation with the wave-vector 2k and evaluated intensities of corresponding super-lattice reflections. Their experimental study would allow determine the strength of anisotropic part of the ME interaction responsible for above mentioned phenomena and the lattice helicity. It has to be noted that the ME interaction in the Landau theory was investigated in 28,29 and the lattice deformation was predicted at the wave-vector k. This result does not contradict to ours as it was obtained for magnetized systems only.\\nThe paper is organized as follows. In Sec.II we consider the ME in cubic helimagnets. Classical ground state energy and the lattice deformation is studied in Sec.III. The spin-wave-phonon interaction and the ME contribution to ∆ 2 is considered in Sec.IV. Obtained results, numerical estimations and experimental consequences are discussed in Sec.V and main results are summarized in Sec.IV In Appendixes A and B some mathematical details are considered. Appendix C is devoted to consideration super-lattice reflections near forbidden < n00 > Bragg peaks with odd n-s.\"\n },\n {\n \"title\": \"II. MAGNETO-ELASTIC INTERACTION\",\n \"content\": \"In general form the magneto elastic energy is given by (see for example 30 )\\nwhere S α R is the spin components at the lattice point R , U γµ = (1/2)(∂u γ /∂R µ + ∂u µ /∂R γ ) is the deformation tensor and the lattice site displacement has well known form\\nwhere e qj are vectors of the phonon polarization , b(b + ) their absorption (excitation) operators and e −qj = e * qj 31 . Tensor B is symmetric in (αβ) and (γµ) components. In cubic crystals we have following non-zero components 30\\nand B xyxy = B yxxy = B xyyx etc. In isotropic medium we have\\nand\\nIn cubic helimagnets the lattice spin is given by 19\\nwhere < S ζ R > = 0 is an average value of the lattice spin, S η,ξ R describe its perpendicular motion, k is the helix wave-vector, A = (â − ib)/2, unit vectorsâ,b, andĉ form right-handed orthogonal frame, sin α = −H /H C where H is the magnetic field component along the helix vector k and H C is the critical field for transition to ferromagnetic state. According to 19 the vector k ĉ in arbitrary field.\\nUsing standard definition S q = N −1/2 S R exp (−iq · R) in momentum space we obtain 19\\nwhere S c q , S A q and S A * q are functions of q, q − k and q + k respectively. The spin components in Eq.(7) have well-known form\\nwhere a q and a + q are conventional spin-wave operators. In momentum space Eq.(2) is given by\\nThis expression is divided on three parts: direct (ĉĉ and A A * ) terms where the U tensor is k independent, first (cA), (cA * ) and second (A A and A * A * ) order umklapp terms where U operator depends on q 1 + q 2 ± k and q 1 + q 2 ± 2k respectively 32 . For their consideration we use following identity\\nwhere B A = B 1 − 2B 2 is an anisotropic part of the tensor B and u = u Q .\\nIn the case of uniform pressure P we have U αβ = −(P/3K)δ αβ where K is the bulk modulus and U q ∼ N 1/2 δ q,0 . As a result the umklapp terms are zero asĉ · A = A · A = 0 and V ME → −N B 1 S(S + 1)P/(3K). Hence the uniform pressure contributes to the classical part of the magneto-elastic ground state energy only. However it has to change basic parameters of the problem such as B 1,2 , sound velocities etc.\"\n },\n {\n \"title\": \"III. GROUND STATE ENERGY AND LATTICE DEFORMATION\",\n \"content\": \"In zero magnetic field sin α = 0 and we have a planar helix. In this case from Eqs. (7)(8)(9)(10) follows that in the classical part of the ME interaction the first-order umklapps are forbidden 33 and we obtain\\nwhere g p = A 2 pĉ p and u p = u p −2k . For evaluation of the lattice deformation we must consider the elastic energy. Unlike Refs. 28,29 for simplicity we ignore in it the cubic symmetry as its principal symmetry breaking role has been taken into account above in Eq (11). In this case the unit cell energy is given by 34\\n, v 0 is the unit cell volume, and E and σ are Young modulus and Poisson coefficient respectively. As a result we have\\nFrom Eqs. (11)(12)(13) for the ME part of the ground state energy we obtain\\nwhere\\nMinimum of this energy is evaluated in Appendix A and we have\\nwhere g A = S 2 B A /Ev 0 , cubic invariants G 1 = 16(g · g * ) and G 2 = 16(g ·ĉ)(g * ·ĉ) are considered in Appendix A and in the right hand side of both equations we neglect σ as it is usually small 34 .\"\n },\n {\n \"title\": \"IV. MAGNON-PHONON INTERACTION\",\n \"content\": \"We consider now the magnon-phonon interaction. We are interested by terms which survive at q = 0 and contribute to the spin-wave gap as other terms are small corrections to the q-dependent part of the magnon dispersion considered in 19 and 14 . To single out them we have to replace in Eq.(9) one of S ζ operators by Sδ q±k,0 . As a result we obtain terms with phonon momenta q, q ± k, and q ± 2k. The former disappear at q = 0, the second are proportional to a q + a + −q and can not contribute to the gap (see below). Using identity (10) for the last AA end A * A * terms in the case of planar helix we have\\nFrom this equation for the magnon-magnon interaction we obtain\\nwhere we neglect q in comparison with ±2k . The phonon Green function can be represented as\\nwhere l(t)labels longitudinal ( transverse) phonon mode and s l(t) is a corresponding sound velocity. We neglect optical branches as their contribution is of order (sk/θ D ) 2 ≪ 1 where θ D is Debye temperature.\\nIn the linear spin-wave theory the Hamiltonian is given by 19\\nand the square of the spin-wave energy ǫ 2 q = E 2 q − B 2 q . As was shown in 19 E 0 = B 0 = Ak 2 /2 where A is the spin-wave stiffness at q ≫ k and we have gapless excitations. We assume that the ME interaction is weak and gives small corrections to A 0 and B 0 . In this case from Eqs. (18)(19)(20) for the magneto-elastic contribution to the square of the spin-wave gap we obtain\\nwhere g A = S 2 B A /Ev 0 and neglecting σ we have M s 2 l = Ev 0 and s t = s l / √ 2. This expression is negative as it should be in the second order perturbation theory and ∆ 2 ME = 0 in < 100 > direction only (see Appendix A). Consideration of q ± k terms lead to expression similar to Eq.(17) with replacing a − a + → a + a + which do not contribute to the gap.\\nThe spin-wave interaction considered in the 1/S approximation leads to positive contribution to ∆ 2 which is given by\\nand D q is a form-factor of the Dzyaloshinskii interaction 19 , 35 . The helical structure can be stable if\\nand if ∆ I = 0 it can survive at k < 100 > only where ∆ ME = 0. Meanwhile it is well known that in M nSi and F eGe at low T the helix axis k < 111 > and ∆ I > |∆ ME |.\"\n },\n {\n \"title\": \"V. EXPERIMENTAL CONSEQUENCES AND DISCUSSION\",\n \"content\": \"For discussion experimental consequences of the magneto-elastic interaction we have to know the Young modules E and anisotropic part of the magneto elastic interaction S 2 B A . For M nSi according to 17 , 36 the bulk modulus K = 1.37 × 10 6 bar and neglecting the Poisson coefficient σ we obtain E = 3K = 4, 11 × 10 6 bar and Ev 0 = 240eV (v 0 = 95 × 10 −24 cm 3 ).\\nUnfortunately the value of B A S 2 is unknown. As we will see below it may be determined by x-ray and neutron scattering. Isotropic part of the ME interaction was studied by indirect method in 37 . Its contribution to the lattice constant ∆a/a ≃ −1.1 × 10 −4 at T = 0K was determined and the sum of the isotropic part of the ME and elastic energies can be represented as B is S 2 (∆a/a) + Kv 0 (3∆a/a) 2 /2.\\nThis expression is minimal at g is = S 2 B is /Ev 0 = −3.3 × 10 −4 and from Eqs. (15) and (21) we obtain\\nwhere we used S = 1.6, Ak 2 = H C = 0.6T and H C is a critical field for transition to ferromagnetic state 19 . We compare now above results with existing experimental data 23 . As was shown in 19 the ground state energy of the helical structure in magnetic field is given by\\nwhere H (⊥) is a field component along (perpendicular) to the helix wave-vector k, E A = (S 2 F 0 k 2 − 3S 4 K)L/4, F 0 and K are constants of the anisotropic exchange and cubic anisotropy respectively 4,19 . The cubic invariant L = (g ·ĉ) is considered in Appendix A. Evolution of the helical structure in magnetic field was studied by small angle polarized neutron scattering in M nSi near T C 24 at low T 23 and in compound F eCoSi 25,26 . Two new characteristic fields were determined. In zero field the sample is in multidomain state with k along all < 111 > directions. Then for the field along one of < 111 > axes at H C1 the single domain state appears. With further field increasing the Bragg intensity demonstrated a cusp at H in . In 23 it was interpreted as instability of the k direction connected with the second term in Eq. (27). Indeed if H is slightly below ∆ √ 2 this term predominates and the vector k has to rotate perpendicular to the field but blocked by the anisotropy. Just below T C where the anisotropy is weak this rotation was observed in 24 . In M nSi we have H C1 = 80mT , H in 160mT and ∆ ≃ 110mT = 13µeV 23 .\\nIt is obvious that the single domain state can be realized if SH 2\\n. So this condition is fulfilled roughly. It is impossible to do more detailed analysis as we do not know E A and g A .\\nThe invariant L has two extrema L = 2/3 and L = 0 at k along < 111 > and < 100 > directions respectively and a saddle points at < 110 > directions. Hence if one neglects the ME interaction the configuration with k < 110 > is forbidden 4 . The same holds for maxima of the critical fluctuations above T C ≃ 29K 16 . Meanwhile in M nSi at high pressure above the quantum critical point p c ≃ 14.6kbar maxima of the neutron scattering at < 110 > directions were observed 11 . In Appendix B we show that the ME interaction can not resolve this problem i.e. that < 111 > and < 100 > remain only possible k directions in zero magnetic field.\\nLet us estimate now two contribution to the spin-wave gap given by Eqs. (22) and (26). We do not know real form of the ratio r = D q /D 0 in Eq. (22). For r = 1, Ak 2 = 0.6T and S = 1.6 we get maximal value ∆ 2 I max = (0.24T ) 2 . Minimal value of ∆ 2 I may be estimated assuming that in Eq.(22) q max = 0.024nm is a border of the Stoner continuum 38 . In this case we have ∆ 2 I min = (0.035T ) 2 . In Eq.(26) for < 111 > direction expression in the bracket is equal to 4/9 [see Eq.(A5)] and we get ∆ ME ≃ 0.067(g A /g is ) 2 T . So we see that at ambient pressure both contributions are comparable with the observed ∆ ≃ 0.11T . Hence we can do plausible assumption that the quantum phase transition observed at 14.6kbar 6,7,8,11 is a result of vanishing ∆ 2 . At higher pressure the helical structure becomes unstable.\\nFor more precise estimations experimental measurement of magneto-elastic anisotropy S 2 (B 1 − 2B 2 ) would be important. We demonstrate now that it can be directly extracted from intensities of satellite peaks near nuclear Bragg reflections. Indeed using Eq.(16) we obtain 39\\nwhere K ± = K± 2k, F (Q) is the nuclear structure factor, K is a reciprocal lattice point and g A = S 2 (B 1 − 2B 2 )/Ev 0 . Relative satellite intensities are given by\\nIn zero magnetic field vectors k are along all < 111 > directions. If K k we have δI ± = 0. If however k (1, 1, 1) but K = (2πn/a)(1, 1, −1) we obtain\\nwhere we have used ka = 0.17 19 . For K (1, ±1, 0) this expression has to be multiplied on 1/4. Similar results can be obtained for other K directions with one exclusion. We are interested by crystals with P 2 1 3 (B20) symmetry where < n, 0, 0 > Bragg reflections are forbidden if n is odd and observation of very weak super-lattice reflections would be more easier 40 . Hence this case has to be considered separately. We restrict ourselves by the case k (1, 1, 1) only. First of all for even n we have Eq.(30) with replacement 2.2 → 2.2/4 ≃ 0.55. For the odd n the Bragg intensities of the satellites are given by\\nwhere form-factors F j (Q) are given in Appendix C. They are not zero due 2k modulation. Unfortunately Eq. (31) has an additional small factor (2ka) 2 << 1 in comparison with the even-n case. Observation of these odd reflections provides a possibility to determine the lattice chirality [See Eq.(C6)] and its connections with the spin chirality studied by polarized neutrons 5,23,24,25,26 . It has to be noted that the lattice chirality in some cases was determined by anomalous x-ray scattering 5 and electron diffraction 41 .\\nThere are eight domains in virgin sample corresponding k along < 111 > directions. In magnetic field H C1 ≃ 0.08T << H C the single domain state is realized with k along the field and satellite intensity increases. However further increasing of the field suppresses the helical structure and it disappears at H C ≃ 0.6T 23 . In intermediate region at H C1 < H < H C the lattice modulation with the wave-vector k has to appears also 28 . Along with discussed 2k lattice modulation at low field H < ∆ √ 2 the second order helix harmonic appears 19 . I was observed in 23,42 .\"\n },\n {\n \"title\": \"VI. CONCLUSIONS\",\n \"content\": \"We considered the magneto-elastic interaction in cubic helimagnets with B20 structure and demonstrated that it deformed the lattice and gave a negative contribution to the square of the spin-wave gap ∆ 2 . Hence the helical structure is stabilized due to positive contribution to ∆ 2 which stems from the magnon-magnon interaction 19 . It was suggested that the quantum phase transition observed at pressure in M nSi and F eGe is a result o competition between these two parts of the gap and takes place when ∆ 2 = 0. This suggestion is supported by rough estimations at ambient pressure of both contributions to ∆ 2 for M nSi which have the same order and close to experimentally observed gap. It was discussed also how to measure directly anisotropic part of the ME interaction responsible for considered phenomena using x-ray and neutron scattering.\"\n },\n {\n \"title\": \"ACKNOWLEDGMENT\",\n \"content\": \"The author thanks S.V.Grigoriev and A.V.Syromyatnikov for interesting discussions. He is very grateful to S.M.Stishov and D.Yu.Chernyshov also for very important information. This work is supported in part by the RFBR (projects No 05-02-19889, 06-02-16702 and 07-02-01318) and the Russian State Programs \\\"Quantum Macrophysics\\\", \\\"Strongly correlated electrons in Semiconductors, Metals, Superconductors and magnetic Materials\\\" and \\\"Neutron research of solids\\\".\"\n },\n {\n \"title\": \"APPENDIX A\",\n \"content\": \"In this Appendix we calculate deformation of the lattice by the ME interactions, cubic invariants G 1,2 in Eqs. (15), (21), (25)(26) and analyse their properties.\\nWe begin with the classical energy (14). It is minimal if\\nwhere p = x, y, z and g p = A 2 pĉ p and we have\\nwhere 16(g · g * ) = G 1 and 16(g ·ĉ)(ĉ · g * ) = G 2 .\\nIn cubic xyz frame we can writê a = (cos ϑ cos ϕ, cosϑ sin ϕ, − sin ϑ);b = (sin ϕ, − cos ϕ, 0);ĉ = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ),\\nand for three principal k-directions < 111 >, < 110 >, and < 100 > we have: {â = (1/ In this representation for G-functions we obtain G 1 = sin 2 ϑ[(cos 2 ϑ cos 2 ϕ + sin 2 ϕ) 2 cos 2 ϕ + (cos 2 ϑ sin 2 ϕ + cos 2 ϕ) 2 sin 2 ϕ + sin 2 ϑ cos 2 ϑ] G 2 = sin 4 ϑ{[cos 2 ϑ(1 + sin 4 ϕ + cos 4 ϕ) + 2 sin 2 ϕ cos 2 ϕ] 2 + 4(sin 2 ϕ − cos 2 ϕ) 2 cos 2 ϑ sin 2 ϕ cos 2 ϕ} (A4)\\nFrom these equations follows that functions G 1 − G 2 and G 2 have extrema at < 111 >, < 110 > and < 100 > near which we have G 1 − G 2 = 4/9 − 20δϑ 2 /9 − 40δϕ 2 /27; G 2 = 8δϑ 2 /9 + 16δϕ 2 /27, < 111 >;\\nwhere δϑ and δϕ are distances from corresponding extremal points. Hence in considered directions both functions G 1 − G 2 and G 2 have extrema and one can show that they have not other extrema. Contribution of the anisotropic exchange and cubic anisotropy to the classical energy is proportional to cubic invariant L given by 19\\n|A p | 2ĉ2 p = sin 2 ϑ[(cos 2 ϑ cos 2 ϕ + sin 2 ϕ) cos 2 ϕ + (cos 2 ϑ sin 2 ϕ + cos 2 ϕ) sin 2 ϕ + cos 2 ϑ].\\nAs above for three principal directions we have\\nand L has a saddle point at < 110 >.\"\n },\n {\n \"title\": \"APPENDIX B\",\n \"content\": \"We demonstrate now that in presence of the ME contribution to the ground state energy the < 111 > and < 100 > remain only possible stable directions for the vector k.\\nFrom Eq. (27) at H = 0 follows\\nwhere Ψ > 0 and y = Φ/Ψ. We have to study behavior of f (y) for three principal directions. For k < 111 > we obtain\\nand E G is stable if Φ < 4Ψ/3. In the < 110 > case we have\\nIn this configuration there is a saddle point as coefficients at deviations δϑ 2 and δϕ 2 can not be positive simultaneously. Finally if k < 100 > we have f (y) = 2(y − 1/2)(δϑ 2 + δϕ 2 ).\\nThis configuration is stable if Φ > Ψ/2. However, comparing Eqs.(B2) and (B4) we see that configuration < 111 > in the region Ψ/2 < Φ < Ψ/3 has lower energy and < 100 > configuration is metastable. Hence we see that the magneto-elastic energy can not be responsible for the stability of < 110 > configuration.\"\n },\n {\n \"title\": \"APPENDIX C\",\n \"content\": \"There are two different ions in compounds with P 2 1 3 symmetry (M n and Si 4,5 ; F e and Ge 42 etc.) labeled below as 1 and 2 respectively. Each of them occupy in cubic unit cell four positions: ρ 1 = (x, x, x), ρ 2 = (1/2 + x, 1/2 − x, 1 − x), ρ 3 = (1 − x, 1/2 + x, 1/2 − x) and ρ 4 = (1/2 − x, 1 − x, 1/2 + x) (Right-handed structure) or ρ 1 = (x, x, x) ρ 2 = (1/2 − x, 1/2 + x, 1 − x), ρ 3 = (1/2 + x, 1 − x, 1/2 − x), and ρ 4 = (1 − x, 1/2 − x, 1/2 + x) (Lefthanded structure) 5 . For M nSi we have x 1 = 0.138 (M n) and x 2 = 0.846 (Si). It is interesting to note that these numbers very close to the ion positions in \\\"ideal\\\" B20 structure with x 1 = 1/4τ = 0.1545 and x 2 = 1 − x 1 = 0.8455 where τ = (1 + √ 5)/2 43 . We consider below the odd-n case only. As we have two different ions in the unit cell the total structure factor is a sum F (Q) = F 1 (Q) + F 2 (Q) where F j (Q) = f j (Q) e i(Q·ρ λ (xj )) (C1) and f j (Q) is a scattering amplitude for the j ion.\\nThere are eight super-lattice reflections corresponding to Q 1 = (2πn + κ, κ, κ), Q 2 = (2πn − κ, κ, κ), Q 3 = (2πn + κ, −κ, κ), and Q 4 = (2π + κ, κ, −κ) where κ = ±2ka/ √ 3. Corresponding partial form-factors are given by\\nFrom Eqs.(C4,5) follow the intensity ratio\\nand from measurement of the 2k lattice deformation one can determines the lattice chirality.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". H Katsura, A V Balatsky, N Nagaosa, Phys.Rev.Lett. 9827203H.Katsura, A.V.Balatsky, N.Nagaosa, Phys.Rev.Lett. 98, 027203 (2007).","doi":""},{"text":". I E Dzyaloshinskii, Zh.Eksp. Teor.Fiz. 46960Sov.Phys.I.E.Dzyaloshinskii, Zh.Eksp. Teor.Fiz.46, 1420 (1964)[Sov.Phys.JETP19, 960 (1964)];","doi":""},{"text":". O Nakanishi, A Yanase, A Hasegava, M Kataoka, Solid State Commun. 35995O.Nakanishi, A.Yanase, A.Hasegava, M.Kataoka, Solid State Commun.35, 995 (1980).","doi":""},{"text":". P Bak, M Jensen, J.Phys. C. 13881P.Bak, M.Jensen, J.Phys. C 13, L881 (1980).","doi":""},{"text":". M Ishida, Y Endoh, S Mitsuda, Y Ishikawa, Tanaka, J.Phys.Soc.Jpn. 542975M.Ishida, Y.Endoh, S.Mitsuda, Y.Ishikawa, M Tanaka,J.Phys.Soc.Jpn. 54 2975 (1975).","doi":""},{"text":". C Pfleiderer, G J Macmillan, S R Julian, G G Lonzarich, Phys.Rev. B. 558330C.Pfleiderer, G.J.MacMillan, S.R.Julian, G.G.Lonzarich, Phys.Rev. B 55, 8330 (1997).","doi":""},{"text":". K Koyama, T Goto, T Kanomata, R Note, Phys.Rev. B. 62986K.Koyama, T.Goto, T.Kanomata, R.Note, Phys.Rev. B 62,986 (2000).","doi":""},{"text":". C Pfleiderer, S R Julian, G G Lonzarich, Nature. 414427C.Pfleiderer, S.R.Julian, G.G.Lonzarich, Nature(London) 414, 427 (2001).","doi":""},{"text":". C Pfleiderer, D Reznik, L Pintschovius, J Haug, Phys.Rev.Lett. 99156406C.Pfleiderer, D.Reznik, L.Pintschovius, and J.Haug, Phys.Rev.Lett. 99, 156406 (2007).","doi":""},{"text":". P Pedrazzini, H Wilhelm, D Jaccard, T Jalborg, M Schmidt, M Hanfland, H Q Akselrud, U Yuan, Yu Schwarz, F Grin, Steglich, Phys.Rev.Lett. 9847204P.Pedrazzini, H.Wilhelm, D.Jaccard,T.Jalborg,M.Schmidt, M. Hanfland, l. Akselrud, H.Q.Yuan, U.Schwarz, Yu.Grin, and F.Steglich, Phys.Rev.Lett., 98, 047204 (2007)","doi":""},{"text":". C Pfleiderer, D Reznik, L Pintschovius, H V Löhneysen, M Garst, A Rosh, Nature. 427227C.Pfleiderer, D.Reznik, L.Pintschovius, H.v.Löhneysen, M.Garst, A.Rosh, Nature (London)427, 227 (2004).","doi":""},{"text":". S Tewari, D Belitz, T R Kirpatrick, Phys.Rev.Lett. 9647207S.Tewari, D.Belitz, T.R.Kirpatrick, Phys.Rev.Lett.96, 047207 (2006).","doi":""},{"text":". U K Rössler, A N Bogdanov, C Pfleiderer, Nature. 442797U.K.Rössler, A.N.Bogdanov, C.Pfleiderer, Nature(London) 442, 797 (2006).","doi":""},{"text":". D Belitz, T R Kirpatrick, A Roch, Phys.Rev. B. 7354431D.Belitz, T.R.Kirpatrick, A.Roch, Phys.Rev. B 73, 054431 (2006).","doi":""},{"text":". B Binz, A Vishwanath, Phys.Rev. B. 74214408B.Binz, A.Vishwanath, Phys.Rev. B 74, 214408 (2006).","doi":""},{"text":". S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, R Georgii, P Böni, D Lamago, H Eckerlebe, K Pranzas, Phys.Rev. B. 72134420S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, R.Georgii, P.Böni, D.Lamago, H.Eckerlebe, K.Pranzas, Phys.Rev. B 72,134420 (2005).","doi":""},{"text":". S M Stishov, A B Petrova, S Khasanov, G Kh, A A Panova, J C Shikov, D Lashley, T A Wu, Logasso, Phys.Rev. B. 7652405S.M.Stishov, A.B.Petrova, S.Khasanov, G.Kh.Panova, A.A.Shikov, J.C.Lashley, D.Wu, and T.A.Logasso, Phys.Rev. B 76, 052405 (2007).","doi":""},{"text":". Y J T Uemura, I M Goko, Gat, Malureanu, Nature Physics. 329Y.J.Uemura. T.Goko, I.M.Gat.Malureanu et al., Nature Physics 3, 29 (2007).","doi":""},{"text":". S V Maleyev, Phys.Rev. B. 73174403S.V.Maleyev, Phys.Rev. B 73, 174403 (2006).","doi":""},{"text":". S V Maleyev, arXiv:0711.3547S.V.Maleyev, arXiv: 0711.3547.","doi":""},{"text":"Rotation two different spins on infinitesimal angle ϕ k does not change the exchange energy but add 2D P. 1∇1 −Rotation two different spins on infinitesimal angle ϕ k does not change the exchange energy but add 2D P 1,2 (∇1 −","doi":""},{"text":"P = 2di, · Sq)( ϕ · S−q) − ( ϕ · Sq. q · S−q)] = 0 to the Dzyaloshinskii interaction= 2Di P [(q · Sq)( ϕ · S−q) − ( ϕ · Sq)(q · S−q)] = 0 to the Dzyaloshinskii interaction.","doi":""},{"text":". M L Plumer, M B Walker, J.Phys.C: Solid State Phys. 144689M.L.Plumer and M.B.Walker, J.Phys.C: Solid State Phys. 14, 4689 (1981).","doi":""},{"text":". S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, P Böni, R Georgii, D Lamago, H Eckerslebe, K Pranzas, Phys.Rev. B. 74214414S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, P.Böni, R.Georgii, D.Lamago, H.Eckerslebe and K.Pranzas, Phys.Rev. B 74,214414 (2006).","doi":""},{"text":". S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, H Eckerlebe, Phys.Rev. B. 73224440S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, and H.Eckerlebe, Phys.Rev. B 73,224440 (2006).","doi":""},{"text":". S V Grigoriev, S V Maleyev, V A Dyadkin, D Menzel, J Shoenes, H Eckerlebe, Phys.Pev. B. 7692407S.V.Grigoriev, S.V.Maleyev, V.A.Dyadkin, D.Menzel, J.Shoenes, and H.Eckerlebe, Phys.Pev. B 76,092407 (2007).","doi":""},{"text":". S V Grigoriev, V A Dyadkin, J Schoenes, Yu O Chetverikov, A I Okorokov, H Eckerlebe, S V Maleyev, Phys.Rev. B. 76224424S.V.Grigoriev, V.A.Dyadkin, J.Schoenes, Yu.O.Chetverikov, A.I.Okorokov, H.Eckerlebe, and S.V.Maleyev, Phys.Rev. B 76, 224424 (2007).","doi":""},{"text":". G A Gehring, arXiv:0711.2586G.A. Gehring, arXiv:0711.2586.","doi":""},{"text":". M L Plumer, M B Walker, J.Phys.C: Solid. State Phys. 157181M.L.Plumer and M.B.Walker, J.Phys.C: Solid. State Phys.15,7181 (1982).","doi":""},{"text":". M L Plumer, J.Phys.C: Solid.State.Phys. 174663M.L.Plumer, J.Phys.C: Solid.State.Phys., 17, 4663 (1984).","doi":""},{"text":"L D Landau, E M Lifshitz, Electrodynamics of Continuous Media. OxfordPergamon PressL.D.Landau and E.M.Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1984).","doi":""},{"text":". L D Landau, E M Lifshitz, Statistical Physics, Pt. 1Pergamon PressL.D.Landau and E.M.Lifshitz, Statistical Physics, Pt 1 (Pergamon Press, Oxford,1969).","doi":""},{"text":"As in 19 we use term umklapp for processes mixing excitations with momenta k and ±k, ±2k etc. As in 19 we use term umklapp for processes mixing excitations with momenta k and ±k, ±2k etc..","doi":""},{"text":"33 They are allowed in magnetized state only 28. 33 They are allowed in magnetized state only 28 .","doi":""},{"text":"L D Landau, E M Lifshitz, Theory of Elasticity. OxfordPergamon PressL.D.Landau and E.M.Lifshitz, Theory of Elasticity (Pergamon Press, Oxford, 1986).","doi":""},{"text":"At H = 0 cubic anisotropy does not contribute to ∆ 2 and Eqs. in 19 are erroneousAt H = 0 cubic anisotropy does not contribute to ∆ 2 and Eqs.(52-53) in 19 are erroneous.","doi":""},{"text":". E L Gromnitskaya, Ref , 17E.L.Gromnitskaya, Ref.[23] in 17 .","doi":""},{"text":". M Matsunaga, Y Ishikawa, T Nakajima, J.Phys.Soc.Jpn. 511153M.Matsunaga, Y.Ishikawa, and T.Nakajima, J.Phys.Soc.Jpn. 51, 1153 (1982).","doi":""},{"text":". Y Ishikawa, G Shirane, J A Tarvin, M Kohgi, Phys.Rev. B. 164956Y.Ishikawa, G.Shirane, J.A.Tarvin, and M.Kohgi, Phys.Rev. B 16, 4956 (1977).","doi":""},{"text":"Equations presented below are correct if the lattice and the helical structure have the same mosaic. For M nSi the magnetic mosaic is greater than the lattice one 23 and an additional small factor has to be. introduced in expressions for the relative intensitiesEquations presented below are correct if the lattice and the helical structure have the same mosaic. For M nSi the magnetic mosaic is greater than the lattice one 23 and an additional small factor has to be introduced in expressions for the relative intensities.","doi":""},{"text":"Cheryshov for corresponding explanation. Yu, I am thankful D.Yu.Cheryshov for corresponding explanation.","doi":""},{"text":". M Tanaka, H Takajoshi, M Ishida, Y Endoh, J.phys. Soc.Jpn. 542970M.Tanaka, H.Takajoshi, M.Ishida, and Y.Endoh, J.phys. Soc.Jpn., 54, 2970 (1985).","doi":""},{"text":". B Lebech, J Bernard, T Freltoft, J.Phys.: Condens.Matter. 16105B.Lebech, J.Bernard, and T.Freltoft, J.Phys.: Condens.Matter 1, 6105 (1989).","doi":""},{"text":". L Vocadlo, G D Price, I G Wood, Acta.Cryst., B. 55484L.Vocadlo, G.D.Price, and I.G.Wood, Acta.Cryst., B 55, 484 (1999).","doi":""}],"string":"[\n {\n \"text\": \". H Katsura, A V Balatsky, N Nagaosa, Phys.Rev.Lett. 9827203H.Katsura, A.V.Balatsky, N.Nagaosa, Phys.Rev.Lett. 98, 027203 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I E Dzyaloshinskii, Zh.Eksp. Teor.Fiz. 46960Sov.Phys.I.E.Dzyaloshinskii, Zh.Eksp. Teor.Fiz.46, 1420 (1964)[Sov.Phys.JETP19, 960 (1964)];\",\n \"doi\": \"\"\n },\n {\n \"text\": \". O Nakanishi, A Yanase, A Hasegava, M Kataoka, Solid State Commun. 35995O.Nakanishi, A.Yanase, A.Hasegava, M.Kataoka, Solid State Commun.35, 995 (1980).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Bak, M Jensen, J.Phys. C. 13881P.Bak, M.Jensen, J.Phys. C 13, L881 (1980).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Ishida, Y Endoh, S Mitsuda, Y Ishikawa, Tanaka, J.Phys.Soc.Jpn. 542975M.Ishida, Y.Endoh, S.Mitsuda, Y.Ishikawa, M Tanaka,J.Phys.Soc.Jpn. 54 2975 (1975).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Pfleiderer, G J Macmillan, S R Julian, G G Lonzarich, Phys.Rev. B. 558330C.Pfleiderer, G.J.MacMillan, S.R.Julian, G.G.Lonzarich, Phys.Rev. B 55, 8330 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Koyama, T Goto, T Kanomata, R Note, Phys.Rev. B. 62986K.Koyama, T.Goto, T.Kanomata, R.Note, Phys.Rev. B 62,986 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Pfleiderer, S R Julian, G G Lonzarich, Nature. 414427C.Pfleiderer, S.R.Julian, G.G.Lonzarich, Nature(London) 414, 427 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Pfleiderer, D Reznik, L Pintschovius, J Haug, Phys.Rev.Lett. 99156406C.Pfleiderer, D.Reznik, L.Pintschovius, and J.Haug, Phys.Rev.Lett. 99, 156406 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Pedrazzini, H Wilhelm, D Jaccard, T Jalborg, M Schmidt, M Hanfland, H Q Akselrud, U Yuan, Yu Schwarz, F Grin, Steglich, Phys.Rev.Lett. 9847204P.Pedrazzini, H.Wilhelm, D.Jaccard,T.Jalborg,M.Schmidt, M. Hanfland, l. Akselrud, H.Q.Yuan, U.Schwarz, Yu.Grin, and F.Steglich, Phys.Rev.Lett., 98, 047204 (2007)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Pfleiderer, D Reznik, L Pintschovius, H V Löhneysen, M Garst, A Rosh, Nature. 427227C.Pfleiderer, D.Reznik, L.Pintschovius, H.v.Löhneysen, M.Garst, A.Rosh, Nature (London)427, 227 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Tewari, D Belitz, T R Kirpatrick, Phys.Rev.Lett. 9647207S.Tewari, D.Belitz, T.R.Kirpatrick, Phys.Rev.Lett.96, 047207 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". U K Rössler, A N Bogdanov, C Pfleiderer, Nature. 442797U.K.Rössler, A.N.Bogdanov, C.Pfleiderer, Nature(London) 442, 797 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Belitz, T R Kirpatrick, A Roch, Phys.Rev. B. 7354431D.Belitz, T.R.Kirpatrick, A.Roch, Phys.Rev. B 73, 054431 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Binz, A Vishwanath, Phys.Rev. B. 74214408B.Binz, A.Vishwanath, Phys.Rev. B 74, 214408 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, R Georgii, P Böni, D Lamago, H Eckerlebe, K Pranzas, Phys.Rev. B. 72134420S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, R.Georgii, P.Böni, D.Lamago, H.Eckerlebe, K.Pranzas, Phys.Rev. B 72,134420 (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S M Stishov, A B Petrova, S Khasanov, G Kh, A A Panova, J C Shikov, D Lashley, T A Wu, Logasso, Phys.Rev. B. 7652405S.M.Stishov, A.B.Petrova, S.Khasanov, G.Kh.Panova, A.A.Shikov, J.C.Lashley, D.Wu, and T.A.Logasso, Phys.Rev. B 76, 052405 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y J T Uemura, I M Goko, Gat, Malureanu, Nature Physics. 329Y.J.Uemura. T.Goko, I.M.Gat.Malureanu et al., Nature Physics 3, 29 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S V Maleyev, Phys.Rev. B. 73174403S.V.Maleyev, Phys.Rev. B 73, 174403 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S V Maleyev, arXiv:0711.3547S.V.Maleyev, arXiv: 0711.3547.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rotation two different spins on infinitesimal angle ϕ k does not change the exchange energy but add 2D P. 1∇1 −Rotation two different spins on infinitesimal angle ϕ k does not change the exchange energy but add 2D P 1,2 (∇1 −\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P = 2di, · Sq)( ϕ · S−q) − ( ϕ · Sq. q · S−q)] = 0 to the Dzyaloshinskii interaction= 2Di P [(q · Sq)( ϕ · S−q) − ( ϕ · Sq)(q · S−q)] = 0 to the Dzyaloshinskii interaction.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M L Plumer, M B Walker, J.Phys.C: Solid State Phys. 144689M.L.Plumer and M.B.Walker, J.Phys.C: Solid State Phys. 14, 4689 (1981).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, P Böni, R Georgii, D Lamago, H Eckerslebe, K Pranzas, Phys.Rev. B. 74214414S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, P.Böni, R.Georgii, D.Lamago, H.Eckerslebe and K.Pranzas, Phys.Rev. B 74,214414 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S V Grigoriev, S V Maleyev, A I Okorokov, Yu O Chetverikov, H Eckerlebe, Phys.Rev. B. 73224440S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, and H.Eckerlebe, Phys.Rev. B 73,224440 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S V Grigoriev, S V Maleyev, V A Dyadkin, D Menzel, J Shoenes, H Eckerlebe, Phys.Pev. B. 7692407S.V.Grigoriev, S.V.Maleyev, V.A.Dyadkin, D.Menzel, J.Shoenes, and H.Eckerlebe, Phys.Pev. B 76,092407 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S V Grigoriev, V A Dyadkin, J Schoenes, Yu O Chetverikov, A I Okorokov, H Eckerlebe, S V Maleyev, Phys.Rev. B. 76224424S.V.Grigoriev, V.A.Dyadkin, J.Schoenes, Yu.O.Chetverikov, A.I.Okorokov, H.Eckerlebe, and S.V.Maleyev, Phys.Rev. B 76, 224424 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G A Gehring, arXiv:0711.2586G.A. Gehring, arXiv:0711.2586.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M L Plumer, M B Walker, J.Phys.C: Solid. State Phys. 157181M.L.Plumer and M.B.Walker, J.Phys.C: Solid. State Phys.15,7181 (1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M L Plumer, J.Phys.C: Solid.State.Phys. 174663M.L.Plumer, J.Phys.C: Solid.State.Phys., 17, 4663 (1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L D Landau, E M Lifshitz, Electrodynamics of Continuous Media. OxfordPergamon PressL.D.Landau and E.M.Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L D Landau, E M Lifshitz, Statistical Physics, Pt. 1Pergamon PressL.D.Landau and E.M.Lifshitz, Statistical Physics, Pt 1 (Pergamon Press, Oxford,1969).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"As in 19 we use term umklapp for processes mixing excitations with momenta k and ±k, ±2k etc. As in 19 we use term umklapp for processes mixing excitations with momenta k and ±k, ±2k etc..\",\n \"doi\": \"\"\n },\n {\n \"text\": \"33 They are allowed in magnetized state only 28. 33 They are allowed in magnetized state only 28 .\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L D Landau, E M Lifshitz, Theory of Elasticity. OxfordPergamon PressL.D.Landau and E.M.Lifshitz, Theory of Elasticity (Pergamon Press, Oxford, 1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"At H = 0 cubic anisotropy does not contribute to ∆ 2 and Eqs. in 19 are erroneousAt H = 0 cubic anisotropy does not contribute to ∆ 2 and Eqs.(52-53) in 19 are erroneous.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E L Gromnitskaya, Ref , 17E.L.Gromnitskaya, Ref.[23] in 17 .\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Matsunaga, Y Ishikawa, T Nakajima, J.Phys.Soc.Jpn. 511153M.Matsunaga, Y.Ishikawa, and T.Nakajima, J.Phys.Soc.Jpn. 51, 1153 (1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Ishikawa, G Shirane, J A Tarvin, M Kohgi, Phys.Rev. B. 164956Y.Ishikawa, G.Shirane, J.A.Tarvin, and M.Kohgi, Phys.Rev. B 16, 4956 (1977).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Equations presented below are correct if the lattice and the helical structure have the same mosaic. For M nSi the magnetic mosaic is greater than the lattice one 23 and an additional small factor has to be. introduced in expressions for the relative intensitiesEquations presented below are correct if the lattice and the helical structure have the same mosaic. For M nSi the magnetic mosaic is greater than the lattice one 23 and an additional small factor has to be introduced in expressions for the relative intensities.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cheryshov for corresponding explanation. Yu, I am thankful D.Yu.Cheryshov for corresponding explanation.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Tanaka, H Takajoshi, M Ishida, Y Endoh, J.phys. Soc.Jpn. 542970M.Tanaka, H.Takajoshi, M.Ishida, and Y.Endoh, J.phys. Soc.Jpn., 54, 2970 (1985).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Lebech, J Bernard, T Freltoft, J.Phys.: Condens.Matter. 16105B.Lebech, J.Bernard, and T.Freltoft, J.Phys.: Condens.Matter 1, 6105 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Vocadlo, G D Price, I G Wood, Acta.Cryst., B. 55484L.Vocadlo, G.D.Price, and I.G.Wood, Acta.Cryst., B 55, 484 (1999).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":69,"cells":{"text":{"kind":"string","value":"\nDISTANCE MEASUREMENT AND κ-DEFORMED PROPAGATION OF LIGHT AND HEAVY PROBES\narXiv:gr-qc/9903066v1 17 Mar 1999\n\nGiovanni Amelino-Camelia \nInstitut de Physique\nUniversité de Neuchâtel\nrue Breguet 1CH-2000NeuchâtelSwitzerland\n\nJerzy Lukierski \nInstitute for Theoretical Physics\nUniversity of Wroc law\npl. M. Borna 9, 50-204 Wroc lawPoland\n\nAnatol Nowicki \nInstitute of Physics\nPedagogical University\npl. S lowiański 665-029Zielona GóraPoland\n\nDISTANCE MEASUREMENT AND κ-DEFORMED PROPAGATION OF LIGHT AND HEAVY PROBES\narXiv:gr-qc/9903066v1 17 Mar 1999NEIP-99-004 gr-qc/9903066 Int. J. Mod. Phys. A (in press)\nWe investigate the implications for the measurability of distances of a covariant dimensionful \"κ\" deformation of D = 4 relativistic symmetries, with quantum time coordinate and modified Heisenberg algebra. We show that the structure of the deformed mass-shell condition has significant implications for measurement procedures relying on light probes, whereas in the case of heavy probes the most sizeable effect is due to the nontrivial commutation relation between threemomenta and quantum time coordinate. We argue that these findings might indicate that κ-Poincaré symmetries capture some aspects of the physics of the Quantum-Gravity vacuum.\n\n. Introduction\n\nOne of the aspects of Quantum Gravity that has been most actively investigated is the possibility that there be a minimum uncertainty for the measurement of distances. The simplest and best understood proposal is the one of a \"minimum length\"\n∆x ≥ L min ,(1.1)\nwhich fits well the expectation of certain studies [2,3] of measurability in quantum gravity. Eq. (1.1) might also hold in critical sting theory as suggested by analyses of 1 Supported by a grant of the Swiss National Science Foundation 2 Partially supported by KBN grant 2P03B130.12 string collisions at Planckian energies, which were found [4] to be characterized by the following modified uncertainty relation ∆x ≥h ∆p + αG∆p , (1.2) where G = c 2 l 2 p /h is the gravitational coupling (Newton) constant, c and l p are the speed-of-light and Planck-length constants respectively, and α is a constant related to the string tension (Regge slope). Clearly Eq. (1.2) implies L min ∼ √h αG. While modified uncertainty relations are not necessarily associated to quantum groups [3], it is interesting that quantum-group descriptions are often available [5,6,7]; in particular, in Ref. [7] it was shown that the relations (1.1)-(1.2) can be associated to SU q (n) covariance.\n\nThe minimum-length scenario (1.1) would already require that the conceptual framework of Quantum Gravity be significantly different from the one of \"ordinary\" (non-gravitational) Quantum Mechanics 3 . However, as emphasized in Ref. [9], even more dramatic consequences for the measurability of distances appear to emerge from the analysis of additional contributions to the uncertainty relations which are associated to the fact that the limit of \"classical\" (infinitely massive) devices might not be accessible in Quantum Gravity. In particular, the analysis in Ref. [9] has led to the proposal of an alternative to the bound (1.1). By taking into account both the quantum nature of the agents involved in the measurement and the gravitational effects associated to the devices, one finds [9] that the measurability of distances is bound by a quantity that (as needed for the decoherence mechanism discussed in Ref. [10]) grows with the time required by the measurement procedure\nmin [∆L] ∼ l p cT s ∼ l p L s , (1.3)\nwhere L is the distance being measured, s is a length scale characterizing the spatial extension of the devices (e.g., clocks) used in the measurement, T is the time needed to complete the procedure of measuring L, and on the right-hand-side we used the fact that, assuming the measurement procedure uses massless probes, one has typically T ∼ L. Notice that for all acceptable values [9] of s (i.e. s < L) the bound (1.3) is more stringent than (1.1); this is a direct consequence of the fact that the analyses leading to (1.1) had implicitly relied on the availability of ideal classical devices in the measurement procedure. While, as mentioned above, critical string theories provide a framework for the bound (1.1), it appears that noncritical string theories might provide a framework for the bound (1.3). In particular, within \"Liouville\" noncritical string theories, in which the target time is identified with the Liouville mode [11], the nature of the dynamics of the light probes exchanged in a typical procedure of measurement of a distance was shown [12] to lead to a measurability bound of type (1.3).\n\nAn interesting problem is the one of finding a quantum-group (and quantum-Liealgebra) framework for (1.3), just like Ref. [7] has provided a quantum-group framework for (1.1)-(1.2). The notion of quantum group as a Hopf algebra permits to consider deformed symmetries; in fact, the Hopf algebra axioms provide simultaneously an algebraic generalization of the definition of Lie group as well as of Lie algebra. As exemplified by the formulae in the following section, the phase space containing the coordinate and momentum sectors can be described in the quantum-deformed case as a semidirect product of two dual Hopf algebras describing the coordinates and momentum sectors. Such a definition of quantum phase space has been first proposed by Majid [5], and it is endoved with the property that in the undeformed case (coordinates and momenta described by Abelian Hopf algebra with primitive coproducts) one obtains the standard quantum mechanical Heisenberg commutation relations. 4 The so-called κ-deformations [14,15,16,17] provide an example of this type of quantum deformations of relativistic symmetries, and one of us recently argued [18] that κ-deformed symmetries might provide an algebraic abstraction of the measurability bound (1.3). The analysis reported in [18] was somewhat preliminary since only the coordinate sector was considered, but the bound (1.3) emerged rather compellingly, as a direct consequence of the noncommuting space-time coordinates of κ-deformed Minkowski space. Encouraged by the findings of Ref. [18], in this paper we explore further the relation between κ-Poincaré and (1.3); specifically, we extend the analysis of Ref. [18] from the confines of the space-time coordinate sector to the full structure of the κ-deformed phase space. We primarily consider the κ-deformed Poincaré symmetries in the bicrossproduct basis [5,16], which appears to be a very natural framework for the quantum deformations of semidirect product algebras, and outside of the coordinate sector we identify two structures which could affect the analysis of Ref. [18]: the κ-deformed mass-shell condition, which is associated to the Casimir and suggests a modification of the propagation of the light probes exchanged during measurement, and the nontrivial commutation relation between three-momenta and quantum time coordinate, which we find to have important consequences for the analysis of the propagation of heavy probes exchanged during measurement. As discussed below, our analysis uncovers new nonnegligible contributions to the bound on the measurability of distances. These contributions are however comparable to the one identified in Ref. [18], and therefore the order of magnitude of the effect discussed in Ref. [18] is confirmed by our analysis. These findings provide additional evidence of a relation between κ-Poincaré and the bound (1.3), and thereby contribute to the development of a physical interpretation of this class of deformations. We hope that this will provide further motivation for experimentalists to investigate the theoretical framework here advocated, especially exploiting the recent remarkable discoveries [19] in the phenomenology of gamma-ray bursts that allow [20] a direct test 5 of some of the predictions of the κ deformations of Poincaré symmetries.\n\n2 κ-deformed quantum relativistic phase space\n\nThe standard form of the covariant fourdimensional Heisenberg commutation relations, describing quantum-mechanical covariant phase space looks as follows:\n[x µ , p ν ] = ihg µν , g µν = diag(−1, 1, 1, 1). (2.1)\nThe space-time coordinates x µ (µ = 0, 1, 2, 3) can be identified with the translation sector of the Poincaré group, and the fourmomenta p µ (µ = 0, 1, 2, 3) are given by the translation generators of the Poincaré algebra. In considering quantum deformations of relativistic symmetries as describing the modification of space-time structure one is lead to the study of the possible quantum Poincaré groups. 6 The classification of quantum deformations of D = 4 Poincaré groups in the framework of Hopf algebras was given by Podleś and Woronowicz ( [25]; see also [26]) and provides the most general class of noncommutative space-time coordinatesx ν allowed by the quantum-group formalism. If we assume that the quantum deformation does not affect the nonrelativistic kinematics, i.e. we preserve the nonrelativistic O(3) rotations classical and O(3) covariance, the only consistent class of noncommuting space-time coordinates is described by the relations of the κ-deformed Minkowski space with commuting classical space coordinates. In order to describe the relativistic phase space we start with the deformed Hopf algebra of fourmomentap µ written as follows\n[p 0 ,p k ] = 0 (2.2) ∆(p 0 ) =p 0 ⊗ 1 + 1 ⊗p 0 ∆(p k ) =p k ⊗ e αp 0 + e βp 0 ⊗p k (2.3)\nwith antipode and counit given by\nS(p k ) = −e −(α+β)p 0p k S(p 0 ) = −p 0 ǫ(p µ ) = 0 . (2.4)\nUsing the duality relations involving the fundamental constanth (Planck's constant)\n\nx µ ,p ν = −ihg µν g µν = (−1, 1, 1, 1) (2.5)\n\nwe obtain the noncommutative deformed configuration space X as a Hopf algebra with the following algebra and coalgebra structure 6 We take into consideration here only the genuine 10-generator quantum deformations of D = 4 Poincaré symmetries. In particular, the \"standard\" q-deformations are not considered. These qdeformations require adding an eleventh (dilatation) generator, i.e. one deals with the dilatation extended Poincaré algebra [23]. In such a case the corresponding quantum phase space is much more complicated (see, e.g., [24]), and the deformation parameter is dimensionless, rendering difficult the physical separation between the ordinary regime of commutative space-time coordinates and the shortdistance regime in which non-commutativity sets in.\n[x 0 ,x k ] = ih(β − α)x k , [x k ,x l ] = 0 , (2.6)∆(x µ ) =x µ ⊗ 1 + 1 ⊗x µ , (2.7a) S(x µ ) = −x µ , ǫ(x µ ) = 0 . (2.7b)\nThe deformed phase space can be considered as the vector space X ⊗ P with the product (see [5])\n(x ⊗ p)(x ⊗p) = x(p (1) ⊲x) ⊗ p (2)p (2.8)\nwhere left action is given by\np ⊲ x = p, x (2) x (1) (2.9)\nThe product (2.8) can be rewritten as the commutators between coordinates and momenta by using the obvious isomorphism x ∼ x ⊗ 1, p ∼ 1 ⊗ p. This procedure provides the following commutation relations (see also [27,28])\n[x k ,p l ] = ihδ kl e αp 0 , [x k ,p 0 ] = 0 , [x 0 ,p k ] = −ihβp k , [x 0 ,p 0 ] = −ih . (2.10)\nThe set of relations (2.2), (2.6) and (2.10) describes the deformed relativistic quantum phase space.\n\nIntroducing the dispersion of the observable a in quantum mechanical sense by\n∆(a) = a 2 − a 2 (2.11) we have ∆(a)∆(b) ≥ 1 2 | c | where c = [a, b] (2.12)\nWe obtain deformed uncertainty relations in the form\n∆x 0 ∆x k ≥h 2 |(β − α)|| x k | (2.13a) ∆p k ∆x l ≥h 2 δ kl e αp 0 (2.13b) ∆p 0 ∆x 0 ≥h 2 (2.13c) ∆p k ∆x 0 ≥h 2 | βp k | (2.13d)\nDepending on a choice of the parameters α and β we can distinguish the following cases (c-speed of light, κ-(mass like) deformation parameter):\ni) α = β = 0 ; standard form of nondeformed covariant phase space, ii) α = β ; trivially deformed phase space with commuting configuration space, iii) α = −β = 1 2κc\n; κ-deformed phase space in the standard basis (see [29]), iv) α = 0, β = − 1 κc ; κ-deformed phase space in the bicrossproduct basis (see [16]), v) α = 1 κc , β = 0 ; κ-deformed phase space in the bicrossproduct basis (see [17]), vi) α = 0, β = 1 κc ; κ-deformed phase space in the bicrossproduct basis (the case (v)) with transposed coproduct.\n\nThe Quantum-Gravity arguments advocated in the next sections make contact with the bicrossproduct basis, and therefore (also for definiteness) in the following we focus on the case (vi). The set of relations (2.2), (2.6) and (2.10) for our choice of parameters (vi) are the following\n[p 0 ,p k ] = 0 (2.14a) [x 0 ,x k ] = ih κcx k , [x k ,x l ] = 0 (2.14b) [x k ,p l ] = ihδ kl , [x k ,p 0 ] = 0 (2.14c) [x 0 ,p k ] = − ih κcp k , [x 0 ,p 0 ] = −ih (2.14d)\nand are κ-Poincaré covariant 7 . The modified covariant Heisenberg uncertainty relations follow from the relations (2.14), therefore we obtain κ-deformed uncertainty relations\n∆t∆x k ≥h 2κc 2 | x k | = 1 2 l κ c | x k | , (2.15a) ∆p k ∆x l ≥ 1 2h δ kl , (2.15b) ∆Ê∆t ≥ 1 2h , (2.15c) ∆p k ∆t ≥h 2κc 2 | p k | = 1 2 l κ c | p k | . (2.15d)\nwhere l κ =h κc describes the fundamental length at which the time variable should already be considered noncommutative. In comparison with the discussion in Ref. [18], which only considered the coordinate sector, the significant new element that emerged in our present analysis is the relation (2.15d). Interestingly, multiplying the three relations (2.15a), (2.15b) and (2.15d) one obtains\n(∆t) 2 (∆x l ∆p l ) 2 ≥h 8 l 2 κ c 2 | x l p l | (2.16)\n(where no sum over the index l is to be understood). This suggests that a wave packet with minimal standard (∆x∆p) uncertainty has the largest uncertainty in the localization of time. (In ordinary quantum mechanics l κ = 0 and there is no such correlation.) It is also interesting to consider the relation (2.15d) under the assumption that the three-momentap k can be expressed by a general formulap i = M(v 2 )v i , in which case\n∆p i = M ij ∆v k with M ij = M[δ ij + 2v i v j (ln M) ′ ]. Then (2.15d) implies ∆t∆v i ≥ l κ c M(v) M −1 ij (v) v j (2.17)\nBecause in part of our measurement analysis we shall consider light probes, we now discuss the modification of the kinematics of κ-deformed photons. We shall assume that the generators of the κ-deformed Poincaré algebra in bicrossproduct basis describes the \"physical\" generators of space-time symmetries. In the bicrossproduct basis the κdeformed mass Casimir takes the form\nC κ 2 = 1 c 2 P 2 e − P 0 κc − (2κ sinh P 0 2κc ) 2 = −M 2 ,(2.18)\nwhere P µ are the generators of space-time translations and M denotes the κ-invariant mass parameter. For M = 0 (κ-deformed photons) from (2.18) one obtains that (we identify P µ ≡p µ )p\n0 = κc ln(1 + | p| κc ) = | p| − | p| 2 2κc + O( 1 κ 2 ) (2.19)\nand in particular the velocity formula for massless κ-deformed quanta looks as follows 8\n(E = cp 0 ) v i = ∂E ∂p i = c 1 + | p| κcp i | p| (2.20a) or v = | v| = c 1 + | p| κc = c − | p| κ + O( 1 κ 2 ) (2.20b)\nThe inverse formula, which can be inserted in (2.17), looks as followŝ\np i = κ c v ( c v − 1)v i (2.21)\nand it is linear in the deformation parameter κ.\n\n\nMeasurement of distance and covariant κ-deformed phase space\n\nIn this section we analyze the measurement of the distance L between two bodies as it results from a plausible physical interpretation of the uncertainty relations (2.15a)-(2.15d). Like the related studies [9,10,12] we consider the procedure of measurement of distances set out by Wigner [31], which relies on the exchange of a probe/signal between the bodies. The distance is therefore measured as L = v T /2, where v is the velocity of the probe and T is the time (being measured by a clock) spent by the probe to go from one body to the other and return. In general the quantum mechanical nature of the agents intervening in the experiment introduces uncertainties in the measurement of L, and in particular one finds that 9\n∆L ≥ [∆L] clock + [∆L] probe ,(3.\n\n1)\n\ni.e. the uncertainty in the measurement of L receives of course contributions that originate from the quantum mechanical nature of the \"clock\" (the timing/triggering device employed in the measurement) and from the quantum mechanical nature of the probe exchanged between the bodies. A significant contribution to ∆L clock was uncovered in Ref. [9]; this results in the relation\n[∆L] clock ≥ l p cT s , (3.2)\nwhere s is a length scale characterizing the spatial extension of the clock (e.g., the radius of a spherically-symmetric clock) and T is the time needed to complete the procedure of measuring L (i.e. T is the time that the clock measures). Within ordinary quantum mechanics the quantum mechanical nature of the probe (while contributing in general to the uncertainty) does not contribute to the bound on the measurability of L (i.e. a suitable measurement set up can be found so that the quantum mechanical nature of the probe does not lead to a contribution to ∆L). It was shown in Ref. [18] that instead the kinematics of quantum κ-Minkowski space-time does lead to a nontrivial [∆L] probe , and interestingly this turns out to be of the same form of the [∆L] clock in (3.2). As announced in the Introduction we are interested in extending the analysis of Ref. [18] to include structure from the full κ-deformed phase space. We are also more general than Ref. [18] and other related work (see, e.g., Ref. [9,10,12]) in that we not only consider massless particles as the probes exchanged in the Wigner measurement, but we also consider the opposite limit in which the probes are ultraheavy.\n\n\nUsing a heavy probe\n\nIn general combining the contribution (3.2), which originates from the quantum mechanical nature of the clock, with uncertainties due to the quantum mechanical nature of the probe one finds that\n∆L ≥ l p cT s + ∆x + v ∆t + T ∆v (3.3)\nwhere ∆x and ∆t are the uncertainties on the space-time position 10 of the probe at the \"final time\" T , while ∆v is the uncertainty on the velocity of the probe. 9 Of course there are other contributions to ∆L (e.g., coming from the quantum mechanical nature of the other devices used in the experiment [9]); however, since they obviously contribute additively to the total uncertainty in the measurement of L, these uncertainties could only make stronger the bound derived in the following. 10 As implicit in the terminology here adopted, the Wigner measurement procedure is essentially one-dimensional, and the only relevant spatial coordinate is the one along the axis passing through the bodies whose distance is being measured.\n\nThe first contribution on the right-hand-side of (3.3) originates from the quantum mechanical nature of the clock, and it is interesting to notice that in the case of a heavy probe the proportionality to √ T of that term, which always signals decoherence effects (e.g., the more time goes by, the more the quantum clock decoheres according to the ideas in Refs. [9,10]), can be turned into a proportionality to L/v, i.e. the uncertainty actually diverges in the limit of vanishing velocity as expected in a context involving decoherence (small velocities imply large times).\n\nConcerning the contributions on the right-hand-side of (3.3) that originate from the quantum mechanical nature of the probe, it is interesting to observe that in ordinary quantum mechanics ∆x, ∆t and ∆v are not correlated and therefore they do not lead to a contribution to the bound on the measurability of L. However, the κ deformation induces correlations between ∆x, ∆t and ∆v. In particular, we observe that (2.15a)-(2.15d) imply (for an ideal heavy/nonrelativistic probe with p = Mv and interpreting the x on the right-hand-side of (2.15a) as the distance traveled by the probe) ∆v ≥ l κ v 2c ∆t (3.4) and ∆x ≥ l κ L 2c ∆t . (3.5) This relations together with the fact that v ∼ L/T allow to rewrite (3.3) as\n∆L ≥ l p cT s + l κ L 2c ∆t + L T ∆t + l κ L 2c ∆t . (3.6)\nThis uncertainty can be minimized by preparing the probe in a state with v ∼ cl p / √ sl κ ,\n\ni.e. T ∼ L √ sl κ /(cl p ), and ∆t ∼ l κ T /c, and this results in the measurability bound\nmin[∆L] ∼ Ll p l κ s . (3.7)\nThe fact that this bound emerging from our analysis of Wigner measurement using a heavy probe manifests the same √ L behavior encountered in the heuristic quantumgravity analysis of the clock involved in the measurement is a rather interesting aspect of the covariantly κ-deformed phase space. In fact, Eq.(3.4), which reflects the specific structure of the κ-deformed commutation relation between three-momenta and quantum time coordinate, plays a nontrivial role in establishing that the κ-deformed kinematics of the heavy probe leads to an uncertainty with this √ L behavior.\n\n\nUsing a massless probe\n\nOf course, also in the case of a Wigner measurement involving a massless probe one finds that ∆L ≥ l p cT s + ∆x + c ∆t + T ∆v , (3.8) and again the κ deformation induces correlations between ∆x, ∆t and ∆v. In particular, concerning the correlation between ∆x and ∆t using again (2.15a) one finds ∆t ≥h L 2κc 2 ∆x . (3.9) Moreover, if the probe is massless with modified velocity 11 (2.20b) one finds that\n∆v ∼ ∆P κ ∼h 2κ ∆x ,(3.10)\nwhere on the right-hand-side we used (2.15b). Using (3.9) and (3.10) one can rewrite (3.8) as 11) and therefore, also taking into account that L ∼ c T /2 and l κ ≡h/(κc), one finds that the minimal value of ∆L is obtained if (∆x) 2 ∼ Ll κ and this implies that the minimal uncertainty in the measurement of the distance L is\n∆L ≥ l p cT s + ∆x +h L 2κc ∆x +h T 2κ ∆x ,(3.min[∆L] ∼ Ll 2 p s + Ll κ (3.12)\nAgain we find the √ L behavior, and again the full structure of the covariantly κdeformed phase space advocated here plays a rather central role in obtaining this result; in fact, the relation (2.18) ensures that the fourth term on the right-hand side of Eq. (3.8) (which was not considered in Ref. [18]) is of the same order as the second term on the right-hand side of Eq. (3.8), which is the one considered in Ref. [18].\n\nWhile the √ L behavior is of course the most robust outcome of these analyses, it is interesting to notice the interplay between the scale l κ , which characterizes the κ deformation, and the scales s and l p , which characterize heuristic quantum-gravity arguments. Although the relative magnitude of these scales could only be determined once a full Quantum Gravity formalism became available, it is quite natural to guess that if κ deformations were to have physical applications it might be that l κ ∼ l p . Moreover, from the role of s in the measurement procedure it is clear [9,18] that s ≥ l p , and since the measurability bound should be a general property of the theory it is conceivable that also s ∼ l p . This, for example, appears to fit rather well the schemes, such as the one discussed in Ref. [32], in which \"fundamental clocks\" are intrinsic to the formulation of the quantum-gravity approach. For l κ ∼ l p ∼ s the heavy probe and the massless probe considered in this and in the previous subsection lead to exactly (up to an overall numerical factor of order 1) the same bound in the context of the Wigner measurement, and even the heuristic quantum-gravity measurement analysis of Ref. [9] reproduces this bound exactly (again up to an overall numerical factor of order 1). Nevertheless, especially in light of the fact that very little will be known about s until a fully consistent (and genuinely quantum) theory of gravity is available, it is interesting to observe that if s = l p (i.e. s > l p ) the Wigner measurement using a heavy probe is actually a \"better measurement\" (weaker bound) than its counterpart using a massless probe. Since most of the previous studies of quantum-gravity measurability bounds have relied on massless probes, our results suggest that a reanalysis of those studies might be necessary.\n\n\nClosing Remarks\n\nThe covariant κ-deformation of relativistic symmetries here considered, and the associated covariant κ-deformation of the Heisenberg algebra (2.14), has several appealing properties as a candidate for the high-energy modification of classical relativistic symmetries. It provides a rather moderate (at least in comparison with some of its alternatives) deformation of classical relativistic symmetries, which in particular reflects the reasonable expectation that, if any of the space-time coordinates is to be special, the special coordinate should be time. (Interestingly this intuition appears to be also realized in certain approaches to string theory, see e.g. Ref. [11].) As manifest in the relations (2.15a)-(2.15d), the κ-modifications of the covariant Heisenberg commutations relations are of quantum mechanical nature, i.e. proportional to the Planck constant h. This suggests that the κ-deformation can be related with the quantum corrections to the classical dynamics of the space-time geometry. In extending the analysis of Ref. [18] from the space-time coordinate sector to the full structure of the κ-deformed phase space, our analysis has provided additional evidence that the bounds on the measurability of distances associated with the uncertainty relations characterizing the κ-deformed covariant Heisenberg algebra (2.14) are consistent with independent heuristic quantum-gravity analyses of such measurability bounds. Based on this consistency between heuristic quantum-gravity measurability analysis and κ-Poincaré measurability analysis one is prompted to consider the possibility that [18,33] at length scales larger than the Planck length (but of course smaller than the length scales already probed experimentally) the κ-deformations of Poincaré symmetries might play a role in the effective description of Quantum Gravity. This is not completely surprising since some of the most popular Quantum-Gravity scenarios, such as the ones based on a spacetime lattice and the ones involving a foamy Quantum-Gravity vacuum, would plausibly lead [33] to deformations of Poincaré symmetries, and, in particular, the analysis of Ref. [12] appears to suggest that propagation in a foamy Quantum Gravity vacuum might be characterized by a κ-deformed dispersion relation.\n\nThe three-momentum-dependent (i.e. energy-dependent) \"speed of light\" (2.20b) is a novel phenomenon that arises in the framework here considered. As mentioned, it has the same functional form (upon appropriate identification between κ and the string scale) as the energy-dependent speed of light recently discussed [12] in the non-critical (\"Liouville\") string literature. Both in the κ-Poincaré and in the string theory contexts the deviation from ordinary physics, while very significant at the conceptual level, is rather marginal from the phenomenological viewpoint. For example, for photons of energies of order 1 GeV the Eq. (2.20b) entails a minuscule 10 −19 c correction with respect to the ordinary scenario with constant speed of light. At least when κ is identified with the Planck scale, the Eq. (2.20b) is completely consistent with presently available experimental data [12,21,22]. However (as already emphasized in Refs. [20,34] and references therein), some of the modern techniques of investigation of astrophysical phenomena could soon bring remarkable progress in the investigation of space-time symmetries. In our case the best laboratory appears to be provided by gamma-ray bursts, and, now that there is convincing evidence [19] that these bursts have cosmological origin, we can expect [20] that within a few years gamma-ray-burst data will test conclusively the κ deformations we considered.\n In particular, in Ref.[8] it was shown that when gravitational effects are taken into account in a (quantum) measurement process then the masses of the probes used in the measurement induce a change in the space-time metric and this is associated to the emergence of nonlocality. The nature of this gravitationally-induced nonlocality suggests[8] a modification of the fundamental commutators.\nIn the literature sometimes the semidirect product construction for two dual Hopf algebras describing respectively quantum Lie group and quantum Lie algebra is called \"Heisenberg double\" (see, e.g., Ref.[13]).5 Previous analyses attempting to bound the dimensionful parameter κ only probed values of κ that were several orderds of magnitude below the Planck scale (see, e.g., Ref.[12,21,22]), but now that we have definitive evidence that gamma-ray bursts are at cosmological distances we can expect[20] to probe values of κ all the way up to the Planck scale.\nThe κ-covariance of the relations (2.14b) has been shown firstly in Ref.[16]. The κ-covariance of the whole quantum κ-deformed Heisenberg algebra follows from the general properties of the semidirect product, defined by the relations (2.14b) and (2.14c-d). (see, e.g.,[30])\nThe relation (2.20a) is valid as a consequence of the Hamiltonian equation of motionẋ i = ∂H/∂p i − (x i /κ)∂H/∂x 0 . [See Ref. [17], Eq. (4.22).] For the κ-photon here considered, since H = H(p i ), the velocities are classical ([v i , v j ] = 0).\nIt is interesting to notice that κ-deformed mass-shell condition and κ-commutation relation between three-momenta and quantum time coordinate are somewhat related. In fact, for a minimumuncertainty state in the framework of κ-deformed kinematics one has ∆E ∆t ∼h/2 and ∆p ∆t ∼ l κ p/(2c), and this is consistent with a given dispersion relation E(p) only if E(p) ∼ (ch/l κ ) ln(p/p * ) (with p * a constant to be otherwise determined) which coincides with the asymptotic behavior of the κ-deformed dispersion relation (see(2.19)).\n\n. C A Mead, Phys. Rev. 135849C.A. Mead, Phys. Rev. 135B (1964) 849.\n\nL107, and the recent review. E G See, T Padmanabhan, Class. Quantum Grav. L.J. Garay4145Int. J. Mod. Phys.See, e.g., T. Padmanabhan, Class. Quantum Grav. 4 (1987) L107, and the recent review L.J. Garay, Int. J. Mod. Phys. A10 (1995) 145.\n\n. S Dopplicher, K Fredenhagen, J E Roberts, Phys. Lett. 33139S. Dopplicher, K. Fredenhagen, and J.E. Roberts, Phys. Lett. B331 (1994) 39;\n\n. Comm. Math. Phys. 172187Comm. Math. Phys. 172 (1995) 187.\n\n. G Veneziano, Europhys. Lett. 2199G. Veneziano, Europhys. Lett. 2 (1986) 199;\n\n. D J Gross, P F Mende, Nucl. Phys. 303407D.J. Gross and P.F. Mende, Nucl. Phys. B303 (1988) 407;\n\n. D Amati, M Ciafaloni, G Veneziano, Phys. Lett. 21641D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B216 (1989) 41;\n\n. K Konishi, G Paffuti, P Provero, Phys. Lett. 234276K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B234 (1990) 276;\n\n. T Yoneya, Mod. Phys. Lett. 41587T. Yoneya, Mod. Phys. Lett. A4, 1587 (1989);\n\n. Phys. Rev. Lett. 781219Phys. Rev. Lett. 78, 1219 (1997).\n\nS Majid, Foundations of Quantum Groups. Cambridge Univ. Press5S. Majid, Class. Quant. Grav. 5 (1987) 1587; \"Foundations of Quantum Groups\" (Cambridge Univ. Press., 1995).\n\n. M Maggiore, Phys. Lett. 30465M. Maggiore, Phys. Lett. B304 (1993) 65.\n\n. A Kempf, J. Math. Phys. 354483A. Kempf, J. Math. Phys. 35 (1994) 4483;\n\n. A Kempf, G Mangano, R B Mann, Phys. Rev. 521108A. Kempf, G. Mangano, and R.B. Mann, Phys. Rev. D52 (1995) 1108;\n\n. A Kempf, G Mangano, Phys. Rev. 557909A. Kempf and G. Mangano, Phys. Rev. D55 (1997) 7909.\n\n. D V Ahluwalia, Phys. Lett. 339301D.V. Ahluwalia, Phys. Lett. B339 (1994) 301.\n\n. G Amelino-Camelia, Mod. Phys. Lett. 91411G. Amelino-Camelia, Mod. Phys. Lett. A9 (1994) 3415; A11 (1996) 1411.\n\n. F Karolyhazy, Il Nuovo Cimento A42. 390F. Karolyhazy, Il Nuovo Cimento A42 (1966) 390;\n\n. L Diosi, B Lukacs, Phys. Lett. 142331L. Diosi, B. Lukacs, Phys. Lett. A142 (1989) 331;\n\n. Y J Ng, H Van Dam, Mod. Phys. Lett. 9335Y.J. Ng and H. Van Dam, Mod. Phys. Lett. A9 (1994) 335;\n\n. A10. 2801A10 (1995) 2801.\n\n. J Ellis, N E Mavromatos, D V Nanopoulos, Phys. Lett. 29337J. Ellis, N.E. Mavromatos, and D.V. Nanopoulos, Phys. Lett. B293 (1992) 37;\n\n. Mod. Phys. Lett. 10425Mod. Phys. Lett. A10 (1995) 425.\n\n. G Amelino-Camelia, J Ellis, N E Mavromatos, V D Nanopoulos, Int. Journ. Mod. Phys. 12607G. Amelino-Camelia, J. Ellis, N.E. Mavromatos and V.D. Nanopoulos, Int. Journ. Mod. Phys. A12 (1997) 607\n\n. A Yu, L D Alekseev, Faddeev, Comm. Math. Phys. 141413A.Yu. Alekseev, L.D. Faddeev, Comm. Math. Phys. 141 (1991) 413.\n\n. J Lukierski, A Nowicki, H Ruegg, V N Tolstoy, Phys. Lett. 264331J. Lukierski, A. Nowicki, H. Ruegg and V.N. Tolstoy, Phys. Lett. B264 (1991) 331.\n\n. J Lukierski, A Nowicki, H Ruegg, Phys. Lett. 293334J. Lukierski, A. Nowicki and H. Ruegg, Phys. Lett. B293 (1992) 334.\n\n. S Majid, H Ruegg, Phys. Lett. 334348S. Majid and H. Ruegg, Phys. Lett. B334 (1994) 348.\n\n. J Lukierski, H Ruegg, W J Zakrzewski, Ann. of Phys. 24390J. Lukierski, H. Ruegg and W.J. Zakrzewski, Ann. of Phys. 243 (1995) 90.\n\n. G Amelino-Camelia, Phys. Lett. 392283G. Amelino-Camelia, Phys. Lett. B392 (1997) 283.\n\n. See R Wijers, Nature. 393and references thereinSee R. Wijers, Nature 393 (1998) 13, and references therein.\n\n. G Amelino-Camelia, J Ellis, N E Mavromatos, D V Nanopoulos, S Sarkar, Nature. 393763G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, and S. Sarkar, Nature 393 (1998) 763.\n\n. G Domokos, S Kovesi-Domokos, Journ. Phys. 20G. Domokos and S. Kovesi-Domokos, Journ. Phys. G20 (1994) 1989.\n\n. J P Bowes, P D Jarvis, Class. Quant. Grav. 131405J.P. Bowes and P.D. Jarvis, Class. Quant. Grav. 13 (1996) 1405.\n\n. S Majid, Journ. Math. Phys. 342045S. Majid, Journ. Math. Phys. 34 (1993) 2045.\n\n. J Schwenk, J Wess, Phys. Lett. 291273J. Schwenk and J. Wess Phys. Lett. B291 (1992) 273;\n\n. M Fichtmüller, J Lorek, Wess, hep-th/9511106M. Fichtmüller, A Lorek, and J. Wess hep-th/9511106.\n\n. P Podleś, S L Woronowicz, Comm. Math. Phys. 17861P. Podleś, S.L. Woronowicz, Comm. Math. Phys. 178 (1996) 61.\n\n. P Podleś, Comm. Math. Phys. 181569P. Podleś, Comm. Math. Phys. 181 (1996) 569.\n\nJ Lukierski, A Nowicki, Quantum Group Symposium at GROUP21. H.D. Doebner, V.K. DobrevSofiaHeron Press186q-alg/9702003J. Lukierski, A. Nowicki, in \"Quantum Group Symposium at GROUP21\", ed. H.D. Doebner, V.K. Dobrev, Heron Press, Sofia 1997, p. 186; q-alg/9702003.\n\nA Nowicki, ; J Lukierski, M Mozrzymas, Warsaw, New Symmetries in the Theory of Fundamental Interactions. 43q-alg/ 9702004A. Nowicki, in \"New Symmetries in the Theory of Fundamental Interactions\", ed. J. Lukierski, M. Mozrzymas, PWN Warsaw 1997, p. 43; q-alg/ 9702004.\n\nKappa Deformed Phase Space and Uncertainty Relations. A Nowicki, math.QA/9803064A. Nowicki, \"Kappa Deformed Phase Space and Uncertainty Relations\", math.QA/9803064.\n\n. J Schupp, P Watts, B , Zumino Comm, Math. Phys. 157305J. Schupp, P. Watts and B. Zumino Comm. Math. Phys. 157 (1993) 305.\n\n. E P Wigner, Rev. Mod. Phys. 2941E.P. Wigner, Rev. Mod. Phys. 29 (1957) 41;\n\n. H Salecker, E P Wigner, Phys. Rev. 109571H. Salecker, E.P. Wigner, Phys. Rev. 109 (1958) 571.\n\n. C Rovelli, Class. Quantum Grav. 8317ibid. 8C. Rovelli, Class. Quantum Grav. 8 (1991) 297; ibid. 8 (1991) 317.\n\n. G Amelino-Camelia, Mod. Phys. Lett. 131319G. Amelino-Camelia, Mod. Phys. Lett. A13 (1998) 1319.\n\n. S L Glashow, Nucl. Phys. (Proc. Suppl.). 70180S.L. Glashow, Nucl. Phys. (Proc. Suppl.) B70 (1999) 180.\n"},"corpus_id":{"kind":"number","value":16943069,"string":"16,943,069"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"gr-qc/9903066\",\n \"mag\": 1996561603,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1142/s0217751x99002141\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/gr-qc/9903066v1.pdf\"\n ],\n \"pdf_sha\": \"48f2395789335229f69f156c4b37029efc7c8995\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/gr-qc/9903066\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"DISTANCE MEASUREMENT AND κ-DEFORMED PROPAGATION OF LIGHT AND HEAVY PROBES"},"authors":{"kind":"list like","value":["Giovanni Amelino-Camelia \nInstitut de Physique\nUniversité de Neuchâtel\nrue Breguet 1CH-2000NeuchâtelSwitzerland","Jerzy Lukierski \nInstitute for Theoretical Physics\nUniversity of Wroc law\npl. M. Borna 9, 50-204 Wroc lawPoland","Anatol Nowicki \nInstitute of Physics\nPedagogical University\npl. S lowiański 665-029Zielona GóraPoland"],"string":"[\n \"Giovanni Amelino-Camelia \\nInstitut de Physique\\nUniversité de Neuchâtel\\nrue Breguet 1CH-2000NeuchâtelSwitzerland\",\n \"Jerzy Lukierski \\nInstitute for Theoretical Physics\\nUniversity of Wroc law\\npl. M. Borna 9, 50-204 Wroc lawPoland\",\n \"Anatol Nowicki \\nInstitute of Physics\\nPedagogical University\\npl. S lowiański 665-029Zielona GóraPoland\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We investigate the implications for the measurability of distances of a covariant dimensionful \"κ\" deformation of D = 4 relativistic symmetries, with quantum time coordinate and modified Heisenberg algebra. We show that the structure of the deformed mass-shell condition has significant implications for measurement procedures relying on light probes, whereas in the case of heavy probes the most sizeable effect is due to the nontrivial commutation relation between threemomenta and quantum time coordinate. We argue that these findings might indicate that κ-Poincaré symmetries capture some aspects of the physics of the Quantum-Gravity vacuum."},"sections":{"kind":"list like","value":[{"title":". Introduction","content":"One of the aspects of Quantum Gravity that has been most actively investigated is the possibility that there be a minimum uncertainty for the measurement of distances. The simplest and best understood proposal is the one of a \"minimum length\"\nwhich fits well the expectation of certain studies [2,3] of measurability in quantum gravity. Eq. (1.1) might also hold in critical sting theory as suggested by analyses of 1 Supported by a grant of the Swiss National Science Foundation 2 Partially supported by KBN grant 2P03B130.12 string collisions at Planckian energies, which were found [4] to be characterized by the following modified uncertainty relation ∆x ≥h ∆p + αG∆p , (1.2) where G = c 2 l 2 p /h is the gravitational coupling (Newton) constant, c and l p are the speed-of-light and Planck-length constants respectively, and α is a constant related to the string tension (Regge slope). Clearly Eq. (1.2) implies L min ∼ √h αG. While modified uncertainty relations are not necessarily associated to quantum groups [3], it is interesting that quantum-group descriptions are often available [5,6,7]; in particular, in Ref. [7] it was shown that the relations (1.1)-(1.2) can be associated to SU q (n) covariance.\nThe minimum-length scenario (1.1) would already require that the conceptual framework of Quantum Gravity be significantly different from the one of \"ordinary\" (non-gravitational) Quantum Mechanics 3 . However, as emphasized in Ref. [9], even more dramatic consequences for the measurability of distances appear to emerge from the analysis of additional contributions to the uncertainty relations which are associated to the fact that the limit of \"classical\" (infinitely massive) devices might not be accessible in Quantum Gravity. In particular, the analysis in Ref. [9] has led to the proposal of an alternative to the bound (1.1). By taking into account both the quantum nature of the agents involved in the measurement and the gravitational effects associated to the devices, one finds [9] that the measurability of distances is bound by a quantity that (as needed for the decoherence mechanism discussed in Ref. [10]) grows with the time required by the measurement procedure\nwhere L is the distance being measured, s is a length scale characterizing the spatial extension of the devices (e.g., clocks) used in the measurement, T is the time needed to complete the procedure of measuring L, and on the right-hand-side we used the fact that, assuming the measurement procedure uses massless probes, one has typically T ∼ L. Notice that for all acceptable values [9] of s (i.e. s < L) the bound (1.3) is more stringent than (1.1); this is a direct consequence of the fact that the analyses leading to (1.1) had implicitly relied on the availability of ideal classical devices in the measurement procedure. While, as mentioned above, critical string theories provide a framework for the bound (1.1), it appears that noncritical string theories might provide a framework for the bound (1.3). In particular, within \"Liouville\" noncritical string theories, in which the target time is identified with the Liouville mode [11], the nature of the dynamics of the light probes exchanged in a typical procedure of measurement of a distance was shown [12] to lead to a measurability bound of type (1.3).\nAn interesting problem is the one of finding a quantum-group (and quantum-Liealgebra) framework for (1.3), just like Ref. [7] has provided a quantum-group framework for (1.1)-(1.2). The notion of quantum group as a Hopf algebra permits to consider deformed symmetries; in fact, the Hopf algebra axioms provide simultaneously an algebraic generalization of the definition of Lie group as well as of Lie algebra. As exemplified by the formulae in the following section, the phase space containing the coordinate and momentum sectors can be described in the quantum-deformed case as a semidirect product of two dual Hopf algebras describing the coordinates and momentum sectors. Such a definition of quantum phase space has been first proposed by Majid [5], and it is endoved with the property that in the undeformed case (coordinates and momenta described by Abelian Hopf algebra with primitive coproducts) one obtains the standard quantum mechanical Heisenberg commutation relations. 4 The so-called κ-deformations [14,15,16,17] provide an example of this type of quantum deformations of relativistic symmetries, and one of us recently argued [18] that κ-deformed symmetries might provide an algebraic abstraction of the measurability bound (1.3). The analysis reported in [18] was somewhat preliminary since only the coordinate sector was considered, but the bound (1.3) emerged rather compellingly, as a direct consequence of the noncommuting space-time coordinates of κ-deformed Minkowski space. Encouraged by the findings of Ref. [18], in this paper we explore further the relation between κ-Poincaré and (1.3); specifically, we extend the analysis of Ref. [18] from the confines of the space-time coordinate sector to the full structure of the κ-deformed phase space. We primarily consider the κ-deformed Poincaré symmetries in the bicrossproduct basis [5,16], which appears to be a very natural framework for the quantum deformations of semidirect product algebras, and outside of the coordinate sector we identify two structures which could affect the analysis of Ref. [18]: the κ-deformed mass-shell condition, which is associated to the Casimir and suggests a modification of the propagation of the light probes exchanged during measurement, and the nontrivial commutation relation between three-momenta and quantum time coordinate, which we find to have important consequences for the analysis of the propagation of heavy probes exchanged during measurement. As discussed below, our analysis uncovers new nonnegligible contributions to the bound on the measurability of distances. These contributions are however comparable to the one identified in Ref. [18], and therefore the order of magnitude of the effect discussed in Ref. [18] is confirmed by our analysis. These findings provide additional evidence of a relation between κ-Poincaré and the bound (1.3), and thereby contribute to the development of a physical interpretation of this class of deformations. We hope that this will provide further motivation for experimentalists to investigate the theoretical framework here advocated, especially exploiting the recent remarkable discoveries [19] in the phenomenology of gamma-ray bursts that allow [20] a direct test 5 of some of the predictions of the κ deformations of Poincaré symmetries.\n2 κ-deformed quantum relativistic phase space\nThe standard form of the covariant fourdimensional Heisenberg commutation relations, describing quantum-mechanical covariant phase space looks as follows:\nThe space-time coordinates x µ (µ = 0, 1, 2, 3) can be identified with the translation sector of the Poincaré group, and the fourmomenta p µ (µ = 0, 1, 2, 3) are given by the translation generators of the Poincaré algebra. In considering quantum deformations of relativistic symmetries as describing the modification of space-time structure one is lead to the study of the possible quantum Poincaré groups. 6 The classification of quantum deformations of D = 4 Poincaré groups in the framework of Hopf algebras was given by Podleś and Woronowicz ( [25]; see also [26]) and provides the most general class of noncommutative space-time coordinatesx ν allowed by the quantum-group formalism. If we assume that the quantum deformation does not affect the nonrelativistic kinematics, i.e. we preserve the nonrelativistic O(3) rotations classical and O(3) covariance, the only consistent class of noncommuting space-time coordinates is described by the relations of the κ-deformed Minkowski space with commuting classical space coordinates. In order to describe the relativistic phase space we start with the deformed Hopf algebra of fourmomentap µ written as follows\nwith antipode and counit given by\nUsing the duality relations involving the fundamental constanth (Planck's constant)\nx µ ,p ν = −ihg µν g µν = (−1, 1, 1, 1) (2.5)\nwe obtain the noncommutative deformed configuration space X as a Hopf algebra with the following algebra and coalgebra structure 6 We take into consideration here only the genuine 10-generator quantum deformations of D = 4 Poincaré symmetries. In particular, the \"standard\" q-deformations are not considered. These qdeformations require adding an eleventh (dilatation) generator, i.e. one deals with the dilatation extended Poincaré algebra [23]. In such a case the corresponding quantum phase space is much more complicated (see, e.g., [24]), and the deformation parameter is dimensionless, rendering difficult the physical separation between the ordinary regime of commutative space-time coordinates and the shortdistance regime in which non-commutativity sets in.\nThe deformed phase space can be considered as the vector space X ⊗ P with the product (see [5])\nwhere left action is given by\nThe product (2.8) can be rewritten as the commutators between coordinates and momenta by using the obvious isomorphism x ∼ x ⊗ 1, p ∼ 1 ⊗ p. This procedure provides the following commutation relations (see also [27,28])\nThe set of relations (2.2), (2.6) and (2.10) describes the deformed relativistic quantum phase space.\nIntroducing the dispersion of the observable a in quantum mechanical sense by\nWe obtain deformed uncertainty relations in the form\nDepending on a choice of the parameters α and β we can distinguish the following cases (c-speed of light, κ-(mass like) deformation parameter):\n; κ-deformed phase space in the standard basis (see [29]), iv) α = 0, β = − 1 κc ; κ-deformed phase space in the bicrossproduct basis (see [16]), v) α = 1 κc , β = 0 ; κ-deformed phase space in the bicrossproduct basis (see [17]), vi) α = 0, β = 1 κc ; κ-deformed phase space in the bicrossproduct basis (the case (v)) with transposed coproduct.\nThe Quantum-Gravity arguments advocated in the next sections make contact with the bicrossproduct basis, and therefore (also for definiteness) in the following we focus on the case (vi). The set of relations (2.2), (2.6) and (2.10) for our choice of parameters (vi) are the following\nand are κ-Poincaré covariant 7 . The modified covariant Heisenberg uncertainty relations follow from the relations (2.14), therefore we obtain κ-deformed uncertainty relations\nwhere l κ =h κc describes the fundamental length at which the time variable should already be considered noncommutative. In comparison with the discussion in Ref. [18], which only considered the coordinate sector, the significant new element that emerged in our present analysis is the relation (2.15d). Interestingly, multiplying the three relations (2.15a), (2.15b) and (2.15d) one obtains\n(where no sum over the index l is to be understood). This suggests that a wave packet with minimal standard (∆x∆p) uncertainty has the largest uncertainty in the localization of time. (In ordinary quantum mechanics l κ = 0 and there is no such correlation.) It is also interesting to consider the relation (2.15d) under the assumption that the three-momentap k can be expressed by a general formulap i = M(v 2 )v i , in which case\nBecause in part of our measurement analysis we shall consider light probes, we now discuss the modification of the kinematics of κ-deformed photons. We shall assume that the generators of the κ-deformed Poincaré algebra in bicrossproduct basis describes the \"physical\" generators of space-time symmetries. In the bicrossproduct basis the κdeformed mass Casimir takes the form\nwhere P µ are the generators of space-time translations and M denotes the κ-invariant mass parameter. For M = 0 (κ-deformed photons) from (2.18) one obtains that (we identify P µ ≡p µ )p\nand in particular the velocity formula for massless κ-deformed quanta looks as follows 8\nThe inverse formula, which can be inserted in (2.17), looks as followŝ\nand it is linear in the deformation parameter κ."},{"title":"Measurement of distance and covariant κ-deformed phase space","content":"In this section we analyze the measurement of the distance L between two bodies as it results from a plausible physical interpretation of the uncertainty relations (2.15a)-(2.15d). Like the related studies [9,10,12] we consider the procedure of measurement of distances set out by Wigner [31], which relies on the exchange of a probe/signal between the bodies. The distance is therefore measured as L = v T /2, where v is the velocity of the probe and T is the time (being measured by a clock) spent by the probe to go from one body to the other and return. In general the quantum mechanical nature of the agents intervening in the experiment introduces uncertainties in the measurement of L, and in particular one finds that 9"},{"title":"1)","content":"i.e. the uncertainty in the measurement of L receives of course contributions that originate from the quantum mechanical nature of the \"clock\" (the timing/triggering device employed in the measurement) and from the quantum mechanical nature of the probe exchanged between the bodies. A significant contribution to ∆L clock was uncovered in Ref. [9]; this results in the relation\nwhere s is a length scale characterizing the spatial extension of the clock (e.g., the radius of a spherically-symmetric clock) and T is the time needed to complete the procedure of measuring L (i.e. T is the time that the clock measures). Within ordinary quantum mechanics the quantum mechanical nature of the probe (while contributing in general to the uncertainty) does not contribute to the bound on the measurability of L (i.e. a suitable measurement set up can be found so that the quantum mechanical nature of the probe does not lead to a contribution to ∆L). It was shown in Ref. [18] that instead the kinematics of quantum κ-Minkowski space-time does lead to a nontrivial [∆L] probe , and interestingly this turns out to be of the same form of the [∆L] clock in (3.2). As announced in the Introduction we are interested in extending the analysis of Ref. [18] to include structure from the full κ-deformed phase space. We are also more general than Ref. [18] and other related work (see, e.g., Ref. [9,10,12]) in that we not only consider massless particles as the probes exchanged in the Wigner measurement, but we also consider the opposite limit in which the probes are ultraheavy."},{"title":"Using a heavy probe","content":"In general combining the contribution (3.2), which originates from the quantum mechanical nature of the clock, with uncertainties due to the quantum mechanical nature of the probe one finds that\nwhere ∆x and ∆t are the uncertainties on the space-time position 10 of the probe at the \"final time\" T , while ∆v is the uncertainty on the velocity of the probe. 9 Of course there are other contributions to ∆L (e.g., coming from the quantum mechanical nature of the other devices used in the experiment [9]); however, since they obviously contribute additively to the total uncertainty in the measurement of L, these uncertainties could only make stronger the bound derived in the following. 10 As implicit in the terminology here adopted, the Wigner measurement procedure is essentially one-dimensional, and the only relevant spatial coordinate is the one along the axis passing through the bodies whose distance is being measured.\nThe first contribution on the right-hand-side of (3.3) originates from the quantum mechanical nature of the clock, and it is interesting to notice that in the case of a heavy probe the proportionality to √ T of that term, which always signals decoherence effects (e.g., the more time goes by, the more the quantum clock decoheres according to the ideas in Refs. [9,10]), can be turned into a proportionality to L/v, i.e. the uncertainty actually diverges in the limit of vanishing velocity as expected in a context involving decoherence (small velocities imply large times).\nConcerning the contributions on the right-hand-side of (3.3) that originate from the quantum mechanical nature of the probe, it is interesting to observe that in ordinary quantum mechanics ∆x, ∆t and ∆v are not correlated and therefore they do not lead to a contribution to the bound on the measurability of L. However, the κ deformation induces correlations between ∆x, ∆t and ∆v. In particular, we observe that (2.15a)-(2.15d) imply (for an ideal heavy/nonrelativistic probe with p = Mv and interpreting the x on the right-hand-side of (2.15a) as the distance traveled by the probe) ∆v ≥ l κ v 2c ∆t (3.4) and ∆x ≥ l κ L 2c ∆t . (3.5) This relations together with the fact that v ∼ L/T allow to rewrite (3.3) as\nThis uncertainty can be minimized by preparing the probe in a state with v ∼ cl p / √ sl κ ,\ni.e. T ∼ L √ sl κ /(cl p ), and ∆t ∼ l κ T /c, and this results in the measurability bound\nThe fact that this bound emerging from our analysis of Wigner measurement using a heavy probe manifests the same √ L behavior encountered in the heuristic quantumgravity analysis of the clock involved in the measurement is a rather interesting aspect of the covariantly κ-deformed phase space. In fact, Eq.(3.4), which reflects the specific structure of the κ-deformed commutation relation between three-momenta and quantum time coordinate, plays a nontrivial role in establishing that the κ-deformed kinematics of the heavy probe leads to an uncertainty with this √ L behavior."},{"title":"Using a massless probe","content":"Of course, also in the case of a Wigner measurement involving a massless probe one finds that ∆L ≥ l p cT s + ∆x + c ∆t + T ∆v , (3.8) and again the κ deformation induces correlations between ∆x, ∆t and ∆v. In particular, concerning the correlation between ∆x and ∆t using again (2.15a) one finds ∆t ≥h L 2κc 2 ∆x . (3.9) Moreover, if the probe is massless with modified velocity 11 (2.20b) one finds that\nwhere on the right-hand-side we used (2.15b). Using (3.9) and (3.10) one can rewrite (3.8) as 11) and therefore, also taking into account that L ∼ c T /2 and l κ ≡h/(κc), one finds that the minimal value of ∆L is obtained if (∆x) 2 ∼ Ll κ and this implies that the minimal uncertainty in the measurement of the distance L is\nAgain we find the √ L behavior, and again the full structure of the covariantly κdeformed phase space advocated here plays a rather central role in obtaining this result; in fact, the relation (2.18) ensures that the fourth term on the right-hand side of Eq. (3.8) (which was not considered in Ref. [18]) is of the same order as the second term on the right-hand side of Eq. (3.8), which is the one considered in Ref. [18].\nWhile the √ L behavior is of course the most robust outcome of these analyses, it is interesting to notice the interplay between the scale l κ , which characterizes the κ deformation, and the scales s and l p , which characterize heuristic quantum-gravity arguments. Although the relative magnitude of these scales could only be determined once a full Quantum Gravity formalism became available, it is quite natural to guess that if κ deformations were to have physical applications it might be that l κ ∼ l p . Moreover, from the role of s in the measurement procedure it is clear [9,18] that s ≥ l p , and since the measurability bound should be a general property of the theory it is conceivable that also s ∼ l p . This, for example, appears to fit rather well the schemes, such as the one discussed in Ref. [32], in which \"fundamental clocks\" are intrinsic to the formulation of the quantum-gravity approach. For l κ ∼ l p ∼ s the heavy probe and the massless probe considered in this and in the previous subsection lead to exactly (up to an overall numerical factor of order 1) the same bound in the context of the Wigner measurement, and even the heuristic quantum-gravity measurement analysis of Ref. [9] reproduces this bound exactly (again up to an overall numerical factor of order 1). Nevertheless, especially in light of the fact that very little will be known about s until a fully consistent (and genuinely quantum) theory of gravity is available, it is interesting to observe that if s = l p (i.e. s > l p ) the Wigner measurement using a heavy probe is actually a \"better measurement\" (weaker bound) than its counterpart using a massless probe. Since most of the previous studies of quantum-gravity measurability bounds have relied on massless probes, our results suggest that a reanalysis of those studies might be necessary."},{"title":"Closing Remarks","content":"The covariant κ-deformation of relativistic symmetries here considered, and the associated covariant κ-deformation of the Heisenberg algebra (2.14), has several appealing properties as a candidate for the high-energy modification of classical relativistic symmetries. It provides a rather moderate (at least in comparison with some of its alternatives) deformation of classical relativistic symmetries, which in particular reflects the reasonable expectation that, if any of the space-time coordinates is to be special, the special coordinate should be time. (Interestingly this intuition appears to be also realized in certain approaches to string theory, see e.g. Ref. [11].) As manifest in the relations (2.15a)-(2.15d), the κ-modifications of the covariant Heisenberg commutations relations are of quantum mechanical nature, i.e. proportional to the Planck constant h. This suggests that the κ-deformation can be related with the quantum corrections to the classical dynamics of the space-time geometry. In extending the analysis of Ref. [18] from the space-time coordinate sector to the full structure of the κ-deformed phase space, our analysis has provided additional evidence that the bounds on the measurability of distances associated with the uncertainty relations characterizing the κ-deformed covariant Heisenberg algebra (2.14) are consistent with independent heuristic quantum-gravity analyses of such measurability bounds. Based on this consistency between heuristic quantum-gravity measurability analysis and κ-Poincaré measurability analysis one is prompted to consider the possibility that [18,33] at length scales larger than the Planck length (but of course smaller than the length scales already probed experimentally) the κ-deformations of Poincaré symmetries might play a role in the effective description of Quantum Gravity. This is not completely surprising since some of the most popular Quantum-Gravity scenarios, such as the ones based on a spacetime lattice and the ones involving a foamy Quantum-Gravity vacuum, would plausibly lead [33] to deformations of Poincaré symmetries, and, in particular, the analysis of Ref. [12] appears to suggest that propagation in a foamy Quantum Gravity vacuum might be characterized by a κ-deformed dispersion relation.\nThe three-momentum-dependent (i.e. energy-dependent) \"speed of light\" (2.20b) is a novel phenomenon that arises in the framework here considered. As mentioned, it has the same functional form (upon appropriate identification between κ and the string scale) as the energy-dependent speed of light recently discussed [12] in the non-critical (\"Liouville\") string literature. Both in the κ-Poincaré and in the string theory contexts the deviation from ordinary physics, while very significant at the conceptual level, is rather marginal from the phenomenological viewpoint. For example, for photons of energies of order 1 GeV the Eq. (2.20b) entails a minuscule 10 −19 c correction with respect to the ordinary scenario with constant speed of light. At least when κ is identified with the Planck scale, the Eq. (2.20b) is completely consistent with presently available experimental data [12,21,22]. However (as already emphasized in Refs. [20,34] and references therein), some of the modern techniques of investigation of astrophysical phenomena could soon bring remarkable progress in the investigation of space-time symmetries. In our case the best laboratory appears to be provided by gamma-ray bursts, and, now that there is convincing evidence [19] that these bursts have cosmological origin, we can expect [20] that within a few years gamma-ray-burst data will test conclusively the κ deformations we considered."}],"string":"[\n {\n \"title\": \". Introduction\",\n \"content\": \"One of the aspects of Quantum Gravity that has been most actively investigated is the possibility that there be a minimum uncertainty for the measurement of distances. The simplest and best understood proposal is the one of a \\\"minimum length\\\"\\nwhich fits well the expectation of certain studies [2,3] of measurability in quantum gravity. Eq. (1.1) might also hold in critical sting theory as suggested by analyses of 1 Supported by a grant of the Swiss National Science Foundation 2 Partially supported by KBN grant 2P03B130.12 string collisions at Planckian energies, which were found [4] to be characterized by the following modified uncertainty relation ∆x ≥h ∆p + αG∆p , (1.2) where G = c 2 l 2 p /h is the gravitational coupling (Newton) constant, c and l p are the speed-of-light and Planck-length constants respectively, and α is a constant related to the string tension (Regge slope). Clearly Eq. (1.2) implies L min ∼ √h αG. While modified uncertainty relations are not necessarily associated to quantum groups [3], it is interesting that quantum-group descriptions are often available [5,6,7]; in particular, in Ref. [7] it was shown that the relations (1.1)-(1.2) can be associated to SU q (n) covariance.\\nThe minimum-length scenario (1.1) would already require that the conceptual framework of Quantum Gravity be significantly different from the one of \\\"ordinary\\\" (non-gravitational) Quantum Mechanics 3 . However, as emphasized in Ref. [9], even more dramatic consequences for the measurability of distances appear to emerge from the analysis of additional contributions to the uncertainty relations which are associated to the fact that the limit of \\\"classical\\\" (infinitely massive) devices might not be accessible in Quantum Gravity. In particular, the analysis in Ref. [9] has led to the proposal of an alternative to the bound (1.1). By taking into account both the quantum nature of the agents involved in the measurement and the gravitational effects associated to the devices, one finds [9] that the measurability of distances is bound by a quantity that (as needed for the decoherence mechanism discussed in Ref. [10]) grows with the time required by the measurement procedure\\nwhere L is the distance being measured, s is a length scale characterizing the spatial extension of the devices (e.g., clocks) used in the measurement, T is the time needed to complete the procedure of measuring L, and on the right-hand-side we used the fact that, assuming the measurement procedure uses massless probes, one has typically T ∼ L. Notice that for all acceptable values [9] of s (i.e. s < L) the bound (1.3) is more stringent than (1.1); this is a direct consequence of the fact that the analyses leading to (1.1) had implicitly relied on the availability of ideal classical devices in the measurement procedure. While, as mentioned above, critical string theories provide a framework for the bound (1.1), it appears that noncritical string theories might provide a framework for the bound (1.3). In particular, within \\\"Liouville\\\" noncritical string theories, in which the target time is identified with the Liouville mode [11], the nature of the dynamics of the light probes exchanged in a typical procedure of measurement of a distance was shown [12] to lead to a measurability bound of type (1.3).\\nAn interesting problem is the one of finding a quantum-group (and quantum-Liealgebra) framework for (1.3), just like Ref. [7] has provided a quantum-group framework for (1.1)-(1.2). The notion of quantum group as a Hopf algebra permits to consider deformed symmetries; in fact, the Hopf algebra axioms provide simultaneously an algebraic generalization of the definition of Lie group as well as of Lie algebra. As exemplified by the formulae in the following section, the phase space containing the coordinate and momentum sectors can be described in the quantum-deformed case as a semidirect product of two dual Hopf algebras describing the coordinates and momentum sectors. Such a definition of quantum phase space has been first proposed by Majid [5], and it is endoved with the property that in the undeformed case (coordinates and momenta described by Abelian Hopf algebra with primitive coproducts) one obtains the standard quantum mechanical Heisenberg commutation relations. 4 The so-called κ-deformations [14,15,16,17] provide an example of this type of quantum deformations of relativistic symmetries, and one of us recently argued [18] that κ-deformed symmetries might provide an algebraic abstraction of the measurability bound (1.3). The analysis reported in [18] was somewhat preliminary since only the coordinate sector was considered, but the bound (1.3) emerged rather compellingly, as a direct consequence of the noncommuting space-time coordinates of κ-deformed Minkowski space. Encouraged by the findings of Ref. [18], in this paper we explore further the relation between κ-Poincaré and (1.3); specifically, we extend the analysis of Ref. [18] from the confines of the space-time coordinate sector to the full structure of the κ-deformed phase space. We primarily consider the κ-deformed Poincaré symmetries in the bicrossproduct basis [5,16], which appears to be a very natural framework for the quantum deformations of semidirect product algebras, and outside of the coordinate sector we identify two structures which could affect the analysis of Ref. [18]: the κ-deformed mass-shell condition, which is associated to the Casimir and suggests a modification of the propagation of the light probes exchanged during measurement, and the nontrivial commutation relation between three-momenta and quantum time coordinate, which we find to have important consequences for the analysis of the propagation of heavy probes exchanged during measurement. As discussed below, our analysis uncovers new nonnegligible contributions to the bound on the measurability of distances. These contributions are however comparable to the one identified in Ref. [18], and therefore the order of magnitude of the effect discussed in Ref. [18] is confirmed by our analysis. These findings provide additional evidence of a relation between κ-Poincaré and the bound (1.3), and thereby contribute to the development of a physical interpretation of this class of deformations. We hope that this will provide further motivation for experimentalists to investigate the theoretical framework here advocated, especially exploiting the recent remarkable discoveries [19] in the phenomenology of gamma-ray bursts that allow [20] a direct test 5 of some of the predictions of the κ deformations of Poincaré symmetries.\\n2 κ-deformed quantum relativistic phase space\\nThe standard form of the covariant fourdimensional Heisenberg commutation relations, describing quantum-mechanical covariant phase space looks as follows:\\nThe space-time coordinates x µ (µ = 0, 1, 2, 3) can be identified with the translation sector of the Poincaré group, and the fourmomenta p µ (µ = 0, 1, 2, 3) are given by the translation generators of the Poincaré algebra. In considering quantum deformations of relativistic symmetries as describing the modification of space-time structure one is lead to the study of the possible quantum Poincaré groups. 6 The classification of quantum deformations of D = 4 Poincaré groups in the framework of Hopf algebras was given by Podleś and Woronowicz ( [25]; see also [26]) and provides the most general class of noncommutative space-time coordinatesx ν allowed by the quantum-group formalism. If we assume that the quantum deformation does not affect the nonrelativistic kinematics, i.e. we preserve the nonrelativistic O(3) rotations classical and O(3) covariance, the only consistent class of noncommuting space-time coordinates is described by the relations of the κ-deformed Minkowski space with commuting classical space coordinates. In order to describe the relativistic phase space we start with the deformed Hopf algebra of fourmomentap µ written as follows\\nwith antipode and counit given by\\nUsing the duality relations involving the fundamental constanth (Planck's constant)\\nx µ ,p ν = −ihg µν g µν = (−1, 1, 1, 1) (2.5)\\nwe obtain the noncommutative deformed configuration space X as a Hopf algebra with the following algebra and coalgebra structure 6 We take into consideration here only the genuine 10-generator quantum deformations of D = 4 Poincaré symmetries. In particular, the \\\"standard\\\" q-deformations are not considered. These qdeformations require adding an eleventh (dilatation) generator, i.e. one deals with the dilatation extended Poincaré algebra [23]. In such a case the corresponding quantum phase space is much more complicated (see, e.g., [24]), and the deformation parameter is dimensionless, rendering difficult the physical separation between the ordinary regime of commutative space-time coordinates and the shortdistance regime in which non-commutativity sets in.\\nThe deformed phase space can be considered as the vector space X ⊗ P with the product (see [5])\\nwhere left action is given by\\nThe product (2.8) can be rewritten as the commutators between coordinates and momenta by using the obvious isomorphism x ∼ x ⊗ 1, p ∼ 1 ⊗ p. This procedure provides the following commutation relations (see also [27,28])\\nThe set of relations (2.2), (2.6) and (2.10) describes the deformed relativistic quantum phase space.\\nIntroducing the dispersion of the observable a in quantum mechanical sense by\\nWe obtain deformed uncertainty relations in the form\\nDepending on a choice of the parameters α and β we can distinguish the following cases (c-speed of light, κ-(mass like) deformation parameter):\\n; κ-deformed phase space in the standard basis (see [29]), iv) α = 0, β = − 1 κc ; κ-deformed phase space in the bicrossproduct basis (see [16]), v) α = 1 κc , β = 0 ; κ-deformed phase space in the bicrossproduct basis (see [17]), vi) α = 0, β = 1 κc ; κ-deformed phase space in the bicrossproduct basis (the case (v)) with transposed coproduct.\\nThe Quantum-Gravity arguments advocated in the next sections make contact with the bicrossproduct basis, and therefore (also for definiteness) in the following we focus on the case (vi). The set of relations (2.2), (2.6) and (2.10) for our choice of parameters (vi) are the following\\nand are κ-Poincaré covariant 7 . The modified covariant Heisenberg uncertainty relations follow from the relations (2.14), therefore we obtain κ-deformed uncertainty relations\\nwhere l κ =h κc describes the fundamental length at which the time variable should already be considered noncommutative. In comparison with the discussion in Ref. [18], which only considered the coordinate sector, the significant new element that emerged in our present analysis is the relation (2.15d). Interestingly, multiplying the three relations (2.15a), (2.15b) and (2.15d) one obtains\\n(where no sum over the index l is to be understood). This suggests that a wave packet with minimal standard (∆x∆p) uncertainty has the largest uncertainty in the localization of time. (In ordinary quantum mechanics l κ = 0 and there is no such correlation.) It is also interesting to consider the relation (2.15d) under the assumption that the three-momentap k can be expressed by a general formulap i = M(v 2 )v i , in which case\\nBecause in part of our measurement analysis we shall consider light probes, we now discuss the modification of the kinematics of κ-deformed photons. We shall assume that the generators of the κ-deformed Poincaré algebra in bicrossproduct basis describes the \\\"physical\\\" generators of space-time symmetries. In the bicrossproduct basis the κdeformed mass Casimir takes the form\\nwhere P µ are the generators of space-time translations and M denotes the κ-invariant mass parameter. For M = 0 (κ-deformed photons) from (2.18) one obtains that (we identify P µ ≡p µ )p\\nand in particular the velocity formula for massless κ-deformed quanta looks as follows 8\\nThe inverse formula, which can be inserted in (2.17), looks as followŝ\\nand it is linear in the deformation parameter κ.\"\n },\n {\n \"title\": \"Measurement of distance and covariant κ-deformed phase space\",\n \"content\": \"In this section we analyze the measurement of the distance L between two bodies as it results from a plausible physical interpretation of the uncertainty relations (2.15a)-(2.15d). Like the related studies [9,10,12] we consider the procedure of measurement of distances set out by Wigner [31], which relies on the exchange of a probe/signal between the bodies. The distance is therefore measured as L = v T /2, where v is the velocity of the probe and T is the time (being measured by a clock) spent by the probe to go from one body to the other and return. In general the quantum mechanical nature of the agents intervening in the experiment introduces uncertainties in the measurement of L, and in particular one finds that 9\"\n },\n {\n \"title\": \"1)\",\n \"content\": \"i.e. the uncertainty in the measurement of L receives of course contributions that originate from the quantum mechanical nature of the \\\"clock\\\" (the timing/triggering device employed in the measurement) and from the quantum mechanical nature of the probe exchanged between the bodies. A significant contribution to ∆L clock was uncovered in Ref. [9]; this results in the relation\\nwhere s is a length scale characterizing the spatial extension of the clock (e.g., the radius of a spherically-symmetric clock) and T is the time needed to complete the procedure of measuring L (i.e. T is the time that the clock measures). Within ordinary quantum mechanics the quantum mechanical nature of the probe (while contributing in general to the uncertainty) does not contribute to the bound on the measurability of L (i.e. a suitable measurement set up can be found so that the quantum mechanical nature of the probe does not lead to a contribution to ∆L). It was shown in Ref. [18] that instead the kinematics of quantum κ-Minkowski space-time does lead to a nontrivial [∆L] probe , and interestingly this turns out to be of the same form of the [∆L] clock in (3.2). As announced in the Introduction we are interested in extending the analysis of Ref. [18] to include structure from the full κ-deformed phase space. We are also more general than Ref. [18] and other related work (see, e.g., Ref. [9,10,12]) in that we not only consider massless particles as the probes exchanged in the Wigner measurement, but we also consider the opposite limit in which the probes are ultraheavy.\"\n },\n {\n \"title\": \"Using a heavy probe\",\n \"content\": \"In general combining the contribution (3.2), which originates from the quantum mechanical nature of the clock, with uncertainties due to the quantum mechanical nature of the probe one finds that\\nwhere ∆x and ∆t are the uncertainties on the space-time position 10 of the probe at the \\\"final time\\\" T , while ∆v is the uncertainty on the velocity of the probe. 9 Of course there are other contributions to ∆L (e.g., coming from the quantum mechanical nature of the other devices used in the experiment [9]); however, since they obviously contribute additively to the total uncertainty in the measurement of L, these uncertainties could only make stronger the bound derived in the following. 10 As implicit in the terminology here adopted, the Wigner measurement procedure is essentially one-dimensional, and the only relevant spatial coordinate is the one along the axis passing through the bodies whose distance is being measured.\\nThe first contribution on the right-hand-side of (3.3) originates from the quantum mechanical nature of the clock, and it is interesting to notice that in the case of a heavy probe the proportionality to √ T of that term, which always signals decoherence effects (e.g., the more time goes by, the more the quantum clock decoheres according to the ideas in Refs. [9,10]), can be turned into a proportionality to L/v, i.e. the uncertainty actually diverges in the limit of vanishing velocity as expected in a context involving decoherence (small velocities imply large times).\\nConcerning the contributions on the right-hand-side of (3.3) that originate from the quantum mechanical nature of the probe, it is interesting to observe that in ordinary quantum mechanics ∆x, ∆t and ∆v are not correlated and therefore they do not lead to a contribution to the bound on the measurability of L. However, the κ deformation induces correlations between ∆x, ∆t and ∆v. In particular, we observe that (2.15a)-(2.15d) imply (for an ideal heavy/nonrelativistic probe with p = Mv and interpreting the x on the right-hand-side of (2.15a) as the distance traveled by the probe) ∆v ≥ l κ v 2c ∆t (3.4) and ∆x ≥ l κ L 2c ∆t . (3.5) This relations together with the fact that v ∼ L/T allow to rewrite (3.3) as\\nThis uncertainty can be minimized by preparing the probe in a state with v ∼ cl p / √ sl κ ,\\ni.e. T ∼ L √ sl κ /(cl p ), and ∆t ∼ l κ T /c, and this results in the measurability bound\\nThe fact that this bound emerging from our analysis of Wigner measurement using a heavy probe manifests the same √ L behavior encountered in the heuristic quantumgravity analysis of the clock involved in the measurement is a rather interesting aspect of the covariantly κ-deformed phase space. In fact, Eq.(3.4), which reflects the specific structure of the κ-deformed commutation relation between three-momenta and quantum time coordinate, plays a nontrivial role in establishing that the κ-deformed kinematics of the heavy probe leads to an uncertainty with this √ L behavior.\"\n },\n {\n \"title\": \"Using a massless probe\",\n \"content\": \"Of course, also in the case of a Wigner measurement involving a massless probe one finds that ∆L ≥ l p cT s + ∆x + c ∆t + T ∆v , (3.8) and again the κ deformation induces correlations between ∆x, ∆t and ∆v. In particular, concerning the correlation between ∆x and ∆t using again (2.15a) one finds ∆t ≥h L 2κc 2 ∆x . (3.9) Moreover, if the probe is massless with modified velocity 11 (2.20b) one finds that\\nwhere on the right-hand-side we used (2.15b). Using (3.9) and (3.10) one can rewrite (3.8) as 11) and therefore, also taking into account that L ∼ c T /2 and l κ ≡h/(κc), one finds that the minimal value of ∆L is obtained if (∆x) 2 ∼ Ll κ and this implies that the minimal uncertainty in the measurement of the distance L is\\nAgain we find the √ L behavior, and again the full structure of the covariantly κdeformed phase space advocated here plays a rather central role in obtaining this result; in fact, the relation (2.18) ensures that the fourth term on the right-hand side of Eq. (3.8) (which was not considered in Ref. [18]) is of the same order as the second term on the right-hand side of Eq. (3.8), which is the one considered in Ref. [18].\\nWhile the √ L behavior is of course the most robust outcome of these analyses, it is interesting to notice the interplay between the scale l κ , which characterizes the κ deformation, and the scales s and l p , which characterize heuristic quantum-gravity arguments. Although the relative magnitude of these scales could only be determined once a full Quantum Gravity formalism became available, it is quite natural to guess that if κ deformations were to have physical applications it might be that l κ ∼ l p . Moreover, from the role of s in the measurement procedure it is clear [9,18] that s ≥ l p , and since the measurability bound should be a general property of the theory it is conceivable that also s ∼ l p . This, for example, appears to fit rather well the schemes, such as the one discussed in Ref. [32], in which \\\"fundamental clocks\\\" are intrinsic to the formulation of the quantum-gravity approach. For l κ ∼ l p ∼ s the heavy probe and the massless probe considered in this and in the previous subsection lead to exactly (up to an overall numerical factor of order 1) the same bound in the context of the Wigner measurement, and even the heuristic quantum-gravity measurement analysis of Ref. [9] reproduces this bound exactly (again up to an overall numerical factor of order 1). Nevertheless, especially in light of the fact that very little will be known about s until a fully consistent (and genuinely quantum) theory of gravity is available, it is interesting to observe that if s = l p (i.e. s > l p ) the Wigner measurement using a heavy probe is actually a \\\"better measurement\\\" (weaker bound) than its counterpart using a massless probe. Since most of the previous studies of quantum-gravity measurability bounds have relied on massless probes, our results suggest that a reanalysis of those studies might be necessary.\"\n },\n {\n \"title\": \"Closing Remarks\",\n \"content\": \"The covariant κ-deformation of relativistic symmetries here considered, and the associated covariant κ-deformation of the Heisenberg algebra (2.14), has several appealing properties as a candidate for the high-energy modification of classical relativistic symmetries. It provides a rather moderate (at least in comparison with some of its alternatives) deformation of classical relativistic symmetries, which in particular reflects the reasonable expectation that, if any of the space-time coordinates is to be special, the special coordinate should be time. (Interestingly this intuition appears to be also realized in certain approaches to string theory, see e.g. Ref. [11].) As manifest in the relations (2.15a)-(2.15d), the κ-modifications of the covariant Heisenberg commutations relations are of quantum mechanical nature, i.e. proportional to the Planck constant h. This suggests that the κ-deformation can be related with the quantum corrections to the classical dynamics of the space-time geometry. In extending the analysis of Ref. [18] from the space-time coordinate sector to the full structure of the κ-deformed phase space, our analysis has provided additional evidence that the bounds on the measurability of distances associated with the uncertainty relations characterizing the κ-deformed covariant Heisenberg algebra (2.14) are consistent with independent heuristic quantum-gravity analyses of such measurability bounds. Based on this consistency between heuristic quantum-gravity measurability analysis and κ-Poincaré measurability analysis one is prompted to consider the possibility that [18,33] at length scales larger than the Planck length (but of course smaller than the length scales already probed experimentally) the κ-deformations of Poincaré symmetries might play a role in the effective description of Quantum Gravity. This is not completely surprising since some of the most popular Quantum-Gravity scenarios, such as the ones based on a spacetime lattice and the ones involving a foamy Quantum-Gravity vacuum, would plausibly lead [33] to deformations of Poincaré symmetries, and, in particular, the analysis of Ref. [12] appears to suggest that propagation in a foamy Quantum Gravity vacuum might be characterized by a κ-deformed dispersion relation.\\nThe three-momentum-dependent (i.e. energy-dependent) \\\"speed of light\\\" (2.20b) is a novel phenomenon that arises in the framework here considered. As mentioned, it has the same functional form (upon appropriate identification between κ and the string scale) as the energy-dependent speed of light recently discussed [12] in the non-critical (\\\"Liouville\\\") string literature. Both in the κ-Poincaré and in the string theory contexts the deviation from ordinary physics, while very significant at the conceptual level, is rather marginal from the phenomenological viewpoint. For example, for photons of energies of order 1 GeV the Eq. (2.20b) entails a minuscule 10 −19 c correction with respect to the ordinary scenario with constant speed of light. At least when κ is identified with the Planck scale, the Eq. (2.20b) is completely consistent with presently available experimental data [12,21,22]. However (as already emphasized in Refs. [20,34] and references therein), some of the modern techniques of investigation of astrophysical phenomena could soon bring remarkable progress in the investigation of space-time symmetries. In our case the best laboratory appears to be provided by gamma-ray bursts, and, now that there is convincing evidence [19] that these bursts have cosmological origin, we can expect [20] that within a few years gamma-ray-burst data will test conclusively the κ deformations we considered.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". C A Mead, Phys. Rev. 135849C.A. Mead, Phys. Rev. 135B (1964) 849.","doi":""},{"text":"L107, and the recent review. E G See, T Padmanabhan, Class. Quantum Grav. L.J. Garay4145Int. J. Mod. Phys.See, e.g., T. Padmanabhan, Class. Quantum Grav. 4 (1987) L107, and the recent review L.J. Garay, Int. J. Mod. Phys. A10 (1995) 145.","doi":""},{"text":". S Dopplicher, K Fredenhagen, J E Roberts, Phys. Lett. 33139S. Dopplicher, K. Fredenhagen, and J.E. Roberts, Phys. Lett. B331 (1994) 39;","doi":""},{"text":". Comm. Math. Phys. 172187Comm. Math. Phys. 172 (1995) 187.","doi":""},{"text":". G Veneziano, Europhys. Lett. 2199G. Veneziano, Europhys. Lett. 2 (1986) 199;","doi":""},{"text":". D J Gross, P F Mende, Nucl. Phys. 303407D.J. Gross and P.F. Mende, Nucl. Phys. B303 (1988) 407;","doi":""},{"text":". D Amati, M Ciafaloni, G Veneziano, Phys. Lett. 21641D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B216 (1989) 41;","doi":""},{"text":". K Konishi, G Paffuti, P Provero, Phys. Lett. 234276K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B234 (1990) 276;","doi":""},{"text":". T Yoneya, Mod. Phys. Lett. 41587T. Yoneya, Mod. Phys. Lett. A4, 1587 (1989);","doi":""},{"text":". Phys. Rev. Lett. 781219Phys. Rev. Lett. 78, 1219 (1997).","doi":""},{"text":"S Majid, Foundations of Quantum Groups. Cambridge Univ. Press5S. Majid, Class. Quant. Grav. 5 (1987) 1587; \"Foundations of Quantum Groups\" (Cambridge Univ. Press., 1995).","doi":""},{"text":". M Maggiore, Phys. Lett. 30465M. Maggiore, Phys. Lett. B304 (1993) 65.","doi":""},{"text":". A Kempf, J. Math. Phys. 354483A. Kempf, J. Math. Phys. 35 (1994) 4483;","doi":""},{"text":". A Kempf, G Mangano, R B Mann, Phys. Rev. 521108A. Kempf, G. Mangano, and R.B. Mann, Phys. Rev. D52 (1995) 1108;","doi":""},{"text":". A Kempf, G Mangano, Phys. Rev. 557909A. Kempf and G. Mangano, Phys. Rev. D55 (1997) 7909.","doi":""},{"text":". D V Ahluwalia, Phys. Lett. 339301D.V. Ahluwalia, Phys. Lett. B339 (1994) 301.","doi":""},{"text":". G Amelino-Camelia, Mod. Phys. Lett. 91411G. Amelino-Camelia, Mod. Phys. Lett. A9 (1994) 3415; A11 (1996) 1411.","doi":""},{"text":". F Karolyhazy, Il Nuovo Cimento A42. 390F. Karolyhazy, Il Nuovo Cimento A42 (1966) 390;","doi":""},{"text":". L Diosi, B Lukacs, Phys. Lett. 142331L. Diosi, B. Lukacs, Phys. Lett. A142 (1989) 331;","doi":""},{"text":". Y J Ng, H Van Dam, Mod. Phys. Lett. 9335Y.J. Ng and H. Van Dam, Mod. Phys. Lett. A9 (1994) 335;","doi":""},{"text":". A10. 2801A10 (1995) 2801.","doi":""},{"text":". J Ellis, N E Mavromatos, D V Nanopoulos, Phys. Lett. 29337J. Ellis, N.E. Mavromatos, and D.V. Nanopoulos, Phys. Lett. B293 (1992) 37;","doi":""},{"text":". Mod. Phys. Lett. 10425Mod. Phys. Lett. A10 (1995) 425.","doi":""},{"text":". G Amelino-Camelia, J Ellis, N E Mavromatos, V D Nanopoulos, Int. Journ. Mod. Phys. 12607G. Amelino-Camelia, J. Ellis, N.E. Mavromatos and V.D. Nanopoulos, Int. Journ. Mod. Phys. A12 (1997) 607","doi":""},{"text":". A Yu, L D Alekseev, Faddeev, Comm. Math. Phys. 141413A.Yu. Alekseev, L.D. Faddeev, Comm. Math. Phys. 141 (1991) 413.","doi":""},{"text":". J Lukierski, A Nowicki, H Ruegg, V N Tolstoy, Phys. Lett. 264331J. Lukierski, A. Nowicki, H. Ruegg and V.N. Tolstoy, Phys. Lett. B264 (1991) 331.","doi":""},{"text":". J Lukierski, A Nowicki, H Ruegg, Phys. Lett. 293334J. Lukierski, A. Nowicki and H. Ruegg, Phys. Lett. B293 (1992) 334.","doi":""},{"text":". S Majid, H Ruegg, Phys. Lett. 334348S. Majid and H. Ruegg, Phys. Lett. B334 (1994) 348.","doi":""},{"text":". J Lukierski, H Ruegg, W J Zakrzewski, Ann. of Phys. 24390J. Lukierski, H. Ruegg and W.J. Zakrzewski, Ann. of Phys. 243 (1995) 90.","doi":""},{"text":". G Amelino-Camelia, Phys. Lett. 392283G. Amelino-Camelia, Phys. Lett. B392 (1997) 283.","doi":""},{"text":". See R Wijers, Nature. 393and references thereinSee R. Wijers, Nature 393 (1998) 13, and references therein.","doi":""},{"text":". G Amelino-Camelia, J Ellis, N E Mavromatos, D V Nanopoulos, S Sarkar, Nature. 393763G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, and S. Sarkar, Nature 393 (1998) 763.","doi":""},{"text":". G Domokos, S Kovesi-Domokos, Journ. Phys. 20G. Domokos and S. Kovesi-Domokos, Journ. Phys. G20 (1994) 1989.","doi":""},{"text":". J P Bowes, P D Jarvis, Class. Quant. Grav. 131405J.P. Bowes and P.D. Jarvis, Class. Quant. Grav. 13 (1996) 1405.","doi":""},{"text":". S Majid, Journ. Math. Phys. 342045S. Majid, Journ. Math. Phys. 34 (1993) 2045.","doi":""},{"text":". J Schwenk, J Wess, Phys. Lett. 291273J. Schwenk and J. Wess Phys. Lett. B291 (1992) 273;","doi":""},{"text":". M Fichtmüller, J Lorek, Wess, hep-th/9511106M. Fichtmüller, A Lorek, and J. Wess hep-th/9511106.","doi":"hep-th/9511106"},{"text":". P Podleś, S L Woronowicz, Comm. Math. Phys. 17861P. Podleś, S.L. Woronowicz, Comm. Math. Phys. 178 (1996) 61.","doi":""},{"text":". P Podleś, Comm. Math. Phys. 181569P. Podleś, Comm. Math. Phys. 181 (1996) 569.","doi":""},{"text":"J Lukierski, A Nowicki, Quantum Group Symposium at GROUP21. H.D. Doebner, V.K. DobrevSofiaHeron Press186q-alg/9702003J. Lukierski, A. Nowicki, in \"Quantum Group Symposium at GROUP21\", ed. H.D. Doebner, V.K. Dobrev, Heron Press, Sofia 1997, p. 186; q-alg/9702003.","doi":""},{"text":"A Nowicki, ; J Lukierski, M Mozrzymas, Warsaw, New Symmetries in the Theory of Fundamental Interactions. 43q-alg/ 9702004A. Nowicki, in \"New Symmetries in the Theory of Fundamental Interactions\", ed. J. Lukierski, M. Mozrzymas, PWN Warsaw 1997, p. 43; q-alg/ 9702004.","doi":""},{"text":"Kappa Deformed Phase Space and Uncertainty Relations. A Nowicki, math.QA/9803064A. Nowicki, \"Kappa Deformed Phase Space and Uncertainty Relations\", math.QA/9803064.","doi":"math.QA/9803064"},{"text":". J Schupp, P Watts, B , Zumino Comm, Math. Phys. 157305J. Schupp, P. Watts and B. Zumino Comm. Math. Phys. 157 (1993) 305.","doi":""},{"text":". E P Wigner, Rev. Mod. Phys. 2941E.P. Wigner, Rev. Mod. Phys. 29 (1957) 41;","doi":""},{"text":". H Salecker, E P Wigner, Phys. Rev. 109571H. Salecker, E.P. Wigner, Phys. Rev. 109 (1958) 571.","doi":""},{"text":". C Rovelli, Class. Quantum Grav. 8317ibid. 8C. Rovelli, Class. Quantum Grav. 8 (1991) 297; ibid. 8 (1991) 317.","doi":""},{"text":". G Amelino-Camelia, Mod. Phys. Lett. 131319G. Amelino-Camelia, Mod. Phys. Lett. A13 (1998) 1319.","doi":""},{"text":". S L Glashow, Nucl. Phys. (Proc. Suppl.). 70180S.L. Glashow, Nucl. Phys. (Proc. Suppl.) B70 (1999) 180.","doi":""}],"string":"[\n {\n \"text\": \". C A Mead, Phys. Rev. 135849C.A. Mead, Phys. Rev. 135B (1964) 849.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L107, and the recent review. E G See, T Padmanabhan, Class. Quantum Grav. L.J. Garay4145Int. J. Mod. Phys.See, e.g., T. Padmanabhan, Class. Quantum Grav. 4 (1987) L107, and the recent review L.J. Garay, Int. J. Mod. Phys. A10 (1995) 145.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Dopplicher, K Fredenhagen, J E Roberts, Phys. Lett. 33139S. Dopplicher, K. Fredenhagen, and J.E. Roberts, Phys. Lett. B331 (1994) 39;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Comm. Math. Phys. 172187Comm. Math. Phys. 172 (1995) 187.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Veneziano, Europhys. Lett. 2199G. Veneziano, Europhys. Lett. 2 (1986) 199;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D J Gross, P F Mende, Nucl. Phys. 303407D.J. Gross and P.F. Mende, Nucl. Phys. B303 (1988) 407;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Amati, M Ciafaloni, G Veneziano, Phys. Lett. 21641D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B216 (1989) 41;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K Konishi, G Paffuti, P Provero, Phys. Lett. 234276K. Konishi, G. Paffuti, P. Provero, Phys. Lett. B234 (1990) 276;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Yoneya, Mod. Phys. Lett. 41587T. Yoneya, Mod. Phys. Lett. A4, 1587 (1989);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. Lett. 781219Phys. Rev. Lett. 78, 1219 (1997).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Majid, Foundations of Quantum Groups. Cambridge Univ. Press5S. Majid, Class. Quant. Grav. 5 (1987) 1587; \\\"Foundations of Quantum Groups\\\" (Cambridge Univ. Press., 1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Maggiore, Phys. Lett. 30465M. Maggiore, Phys. Lett. B304 (1993) 65.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Kempf, J. Math. Phys. 354483A. Kempf, J. Math. Phys. 35 (1994) 4483;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Kempf, G Mangano, R B Mann, Phys. Rev. 521108A. Kempf, G. Mangano, and R.B. Mann, Phys. Rev. D52 (1995) 1108;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Kempf, G Mangano, Phys. Rev. 557909A. Kempf and G. Mangano, Phys. Rev. D55 (1997) 7909.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D V Ahluwalia, Phys. Lett. 339301D.V. Ahluwalia, Phys. Lett. B339 (1994) 301.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Amelino-Camelia, Mod. Phys. Lett. 91411G. Amelino-Camelia, Mod. Phys. Lett. A9 (1994) 3415; A11 (1996) 1411.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Karolyhazy, Il Nuovo Cimento A42. 390F. Karolyhazy, Il Nuovo Cimento A42 (1966) 390;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L Diosi, B Lukacs, Phys. Lett. 142331L. Diosi, B. Lukacs, Phys. Lett. A142 (1989) 331;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y J Ng, H Van Dam, Mod. Phys. Lett. 9335Y.J. Ng and H. Van Dam, Mod. Phys. Lett. A9 (1994) 335;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A10. 2801A10 (1995) 2801.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Ellis, N E Mavromatos, D V Nanopoulos, Phys. Lett. 29337J. Ellis, N.E. Mavromatos, and D.V. Nanopoulos, Phys. Lett. B293 (1992) 37;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Mod. Phys. Lett. 10425Mod. Phys. Lett. A10 (1995) 425.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Amelino-Camelia, J Ellis, N E Mavromatos, V D Nanopoulos, Int. Journ. Mod. Phys. 12607G. Amelino-Camelia, J. Ellis, N.E. Mavromatos and V.D. Nanopoulos, Int. Journ. Mod. Phys. A12 (1997) 607\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Yu, L D Alekseev, Faddeev, Comm. Math. Phys. 141413A.Yu. Alekseev, L.D. Faddeev, Comm. Math. Phys. 141 (1991) 413.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Lukierski, A Nowicki, H Ruegg, V N Tolstoy, Phys. Lett. 264331J. Lukierski, A. Nowicki, H. Ruegg and V.N. Tolstoy, Phys. Lett. B264 (1991) 331.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Lukierski, A Nowicki, H Ruegg, Phys. Lett. 293334J. Lukierski, A. Nowicki and H. Ruegg, Phys. Lett. B293 (1992) 334.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Majid, H Ruegg, Phys. Lett. 334348S. Majid and H. Ruegg, Phys. Lett. B334 (1994) 348.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Lukierski, H Ruegg, W J Zakrzewski, Ann. of Phys. 24390J. Lukierski, H. Ruegg and W.J. Zakrzewski, Ann. of Phys. 243 (1995) 90.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Amelino-Camelia, Phys. Lett. 392283G. Amelino-Camelia, Phys. Lett. B392 (1997) 283.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". See R Wijers, Nature. 393and references thereinSee R. Wijers, Nature 393 (1998) 13, and references therein.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Amelino-Camelia, J Ellis, N E Mavromatos, D V Nanopoulos, S Sarkar, Nature. 393763G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, and S. Sarkar, Nature 393 (1998) 763.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Domokos, S Kovesi-Domokos, Journ. Phys. 20G. Domokos and S. Kovesi-Domokos, Journ. Phys. G20 (1994) 1989.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J P Bowes, P D Jarvis, Class. Quant. Grav. 131405J.P. Bowes and P.D. Jarvis, Class. Quant. Grav. 13 (1996) 1405.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Majid, Journ. Math. Phys. 342045S. Majid, Journ. Math. Phys. 34 (1993) 2045.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Schwenk, J Wess, Phys. Lett. 291273J. Schwenk and J. Wess Phys. Lett. B291 (1992) 273;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Fichtmüller, J Lorek, Wess, hep-th/9511106M. Fichtmüller, A Lorek, and J. Wess hep-th/9511106.\",\n \"doi\": \"hep-th/9511106\"\n },\n {\n \"text\": \". P Podleś, S L Woronowicz, Comm. Math. Phys. 17861P. Podleś, S.L. Woronowicz, Comm. Math. Phys. 178 (1996) 61.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Podleś, Comm. Math. Phys. 181569P. Podleś, Comm. Math. Phys. 181 (1996) 569.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Lukierski, A Nowicki, Quantum Group Symposium at GROUP21. H.D. Doebner, V.K. DobrevSofiaHeron Press186q-alg/9702003J. Lukierski, A. Nowicki, in \\\"Quantum Group Symposium at GROUP21\\\", ed. H.D. Doebner, V.K. Dobrev, Heron Press, Sofia 1997, p. 186; q-alg/9702003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Nowicki, ; J Lukierski, M Mozrzymas, Warsaw, New Symmetries in the Theory of Fundamental Interactions. 43q-alg/ 9702004A. Nowicki, in \\\"New Symmetries in the Theory of Fundamental Interactions\\\", ed. J. Lukierski, M. Mozrzymas, PWN Warsaw 1997, p. 43; q-alg/ 9702004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Kappa Deformed Phase Space and Uncertainty Relations. A Nowicki, math.QA/9803064A. Nowicki, \\\"Kappa Deformed Phase Space and Uncertainty Relations\\\", math.QA/9803064.\",\n \"doi\": \"math.QA/9803064\"\n },\n {\n \"text\": \". J Schupp, P Watts, B , Zumino Comm, Math. Phys. 157305J. Schupp, P. Watts and B. Zumino Comm. Math. Phys. 157 (1993) 305.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E P Wigner, Rev. Mod. Phys. 2941E.P. Wigner, Rev. Mod. Phys. 29 (1957) 41;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Salecker, E P Wigner, Phys. Rev. 109571H. Salecker, E.P. Wigner, Phys. Rev. 109 (1958) 571.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Rovelli, Class. Quantum Grav. 8317ibid. 8C. Rovelli, Class. Quantum Grav. 8 (1991) 297; ibid. 8 (1991) 317.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Amelino-Camelia, Mod. Phys. Lett. 131319G. Amelino-Camelia, Mod. Phys. Lett. A13 (1998) 1319.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S L Glashow, Nucl. Phys. (Proc. Suppl.). 70180S.L. Glashow, Nucl. Phys. (Proc. Suppl.) B70 (1999) 180.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":70,"cells":{"text":{"kind":"string","value":"\nCosmological fluctuation growth in bimetric MOND\n24 Aug 2010\n\nMordehai Milgrom \nDepartment of Particle Physics and Astrophysics\nWeizmann Institute\n76100RehovotIsrael\n\nCosmological fluctuation growth in bimetric MOND\n24 Aug 2010\nI begin to look at the growth of weak density inhomogeneities of nonrelativistic matter, in bimetric-MOND (BIMOND) cosmology. Far from making an exhaustive study, I concentrate on one attractive cosmological scenario, which employs matter-twin-matter-symmetric versions of BIMOND, and, furthermore, assumes that, on average, the universe is symmetrically populated in the two sectors. MOND effects are totally absent in an exactly symmetric universe, apart from the significant possible appearance of a cosmological constant, Λ ∼ (a0/c) 2 . MOND effects-local and cosmological-do enter when density inhomogeneities that differ in the two sectors appear and develop. MOND later takes its standard form in systems that are islands dominated by pure matter, as are presumably the well scrutinized systems such as galaxies. I derive the nonrelativistic (weak-field-slow-motion) equations governing small-scale fluctuation growth. The equations split into two uncoupled systems, one for the sum, the other for the difference, of the fluctuations in the two sectors. The former is governed strictly by Newtonian dynamics, and describes standard growth of fluctuations. The latter is governed by MOND dynamics, which entails stronger gravity, and nonlinearity even for the smallest of perturbations. These cause the difference to grow faster than the sum, leading to anticorrelated perturbations, conducing to matter-twin-matter segregation (which continues for high overdensities). The nonlinearity also causes interaction between nested perturbations on different scales. Because matter and twin matter (TM) repel each other in the MOND regime, matter inhomogeneities grow not only by their own self gravity, but also through shepherding by flanking TM overdensities (and vice versa). The relative importance of gravity and pressure in the MOND system (analog of the Jeans criterion), depends also on the strength of the perturbation. MOND gravity, which scales as the square root of the difference perturbation, holds sway over pressure for any mass, for weak enough perturbations. The development of structure in the universe, in either sector, thus depends crucially on two initial fluctuation spectra: that of matter alone and that of the matter-TM difference. I also discuss the back reaction on cosmology of BIMOND effects that appear as \"phantom matter\" (interpreted by some as \"dark matter\"), resulting from inhomogeneity differences between the two sectors.PACS numbers: 95.35.+d\n\nI. INTRODUCTION\n\nA full fledged treatment of cosmology and of structure formation in the MOND paradigm is still wanting. Structure formation on an expanding cosmological background, in nonrelativistic (NR) versions of MOND, has been studied with N -body calculations, based on various heuristic extensions of MOND theories designed to work for isolated NR systems, e.g., in [1][2][3][4][5]. Clearly, a definite treatment of the problem has to be based on a fully relativistic theory, encompassing both cosmology and structure formation. Much work along such lines has been done in the framework of TeVeS-a relativistic MOND theory propounded by Bekenstein [6] (Sanders's stratified framework [7] is a precursor, and [8] is a recent review of TeVeS and its cosmology). For example, [9][10][11][12][13][14] considered various aspects of cosmology at large, the cosmic microwave background (CMB), and linear stages of structure formation in TeVeS-like theories. Although very encouraging, these results do not yet provide a fully satisfactory description of the phenomena at hand.\n\nA new relativistic theory for MOND (called BIMOND, for \"bimetric MOND\") was proposed recently based on a bimetric structure of space-time [15]. In this theory, gravity is described by two metrics instead of the one underlying general relativity (GR). Ordinary matter, to whose sector belongs our direct ken, couples, as in GR, only to one metric, call it the primary one. The other metric serves as an auxiliary to which the primary metric couples. Such a bimetric description of gravity has a long history, and, in particular, has been discussed recently in connection with some \"dark substance\" phenomena (see [16][17][18][19], and references therein). The novelty in BIMOND is in the special choice of interaction between the two metrics, which is particularly germane in the context of MOND, and which reproduces MOND phenomenology.\n\nIn [15], I made preliminary comments on cosmological solutions within BIMOND. Classes of cosmological solutions within certain versions of BIMOND were also considered in [20]. But, there is still much left to be done on this problem.\n\nWe are also still lacking any treatment of the important problem of the growth of small perturbations. Ideally, one should treat the problem in a fully relativistic framework, as done in [12][13][14], in the context of TeVeS-like theories. Here I take the lighter task of considering only density inhomogeneities of NR matter on scales smaller than cosmological. This limited-scope formalism, which governs the growth of perturbations under matter dominance, is still rather illuminating, and shows a variety of phenomena potentially highly pertinent to structure formation in our universe.\n\nI limit myself to a universes that on average is completely symmetric in matter and twin matter (TM). Twin matter is the putative matter that couples to the \"auxiliary\" metric as (ordinary) matter couples to the \"primary\" one. At present, the existence of TM is not required phenomenologically (in particular, it is not, in fact cannot be, a sort of \"dark matter,\" as assumed to explain galaxy dynamics); it is invoked mainly for aesthetic reasons: With TM we can obtain a simple and symmetric world picture that affords favorable cosmologies. In addition, as our present analysis shows, symmetric presence of matter and TM has very interesting consequences for structure growth. It leads to phenomena that are not only greatly different from those in GR (without dark matter), but also different from those in MOND without TM.\n\nOf course, the existence of TM would add crucial unknowns to our cosmological world view. Its properties, composition, self-interactions, amounts, etc., are not known (see [15] for more on this). To minimize these unknowns I assume here that, with one exception, the TM sector is identical to the matter sector in all respects, having the same properties (masses, etc.), the same interactions within the sector (electromagnetic, strong, etc.), the same interaction with its metric (geodesic motion, etc.), and the same average amounts of its different components present in the universe. As shown in [15] this symmetry leads to simple cosmological solutions, identical with those of GR, albeit with a naturally appearing cosmological constant (CC) of order (a 0 /c) 2 , where a 0 is the MOND constant. 1 The only difference between the two sectors I allow for, is in the amplitude and spectrum of the primordial inhomogeneities. These are putatively ascribed to quantum fluctuations in the early Universe. They could, thus, differ in the two sectors-at least in their exact distribution, if not in their statistical properties-even with all else being the same.\n\nSome preliminaries of the problem discussed here are alluded to in [21], where I considered the problem of NR, small, isolated matter-TM systems. Here, this problem is extended to the evolution of perturbations on an otherwise homogeneous and matter-TM symmetric (MTMS), expanding universe.\n\nIn section II, I briefly describe the aspects of BIMOND and of its NR limit that we need in this paper. In section III, I derive the equations controlling the evolution of small departures from exact density equality in general NR systems. This is applied to cosmological small inhomogeneities in section IV. In section V, I describe succinctly how the density differences back react on the cosmological equations, introducing modifications that can act as cosmological \"dark matter\" and \"dark energy\" (beside the CC). Section VI is a discussion.\n\n\nII. BIMOND RECAPPED\n\nBIMOND is a class of relativistic theories of gravity that involve two metrics, g µν andĝ µν . They are governed by an action of the form\nI = − 1 16πG [βg 1/2 R + αĝ 1/2R − 2(gĝ) 1/4 a 2 0 M]d 4 x + I M (g µν , ψ i ) +Î M (ĝ µν ,ψ i ),(1)\nwhere R andR are the Ricci scalars of the two metrics as appear standardly in the Einstein-Hilbert action of GR, G is Newton's constant, and I use units where c = 1. The interaction term M is a dimensionless, scalar function of the two metrics and their first derivatives. Bimetric theories have been much discussed before; the novelty and crux of BIMOND is in the choice of the interaction term. The key observation, in the MOND context, is that with two metrics and their first derivatives, we can construct tensors with the dimensions of acceleration:\nC α βγ = Γ α βγ −Γ α βγ ,(2)\nwhere Γ α βγ ,Γ α βγ are the Levi-Civita connections of the two metrics. Within the MOND paradigm, having the acceleration constant, a 0 , at our disposal, we can construct from these dimensionless scalars to serve as variables on which M depends. These scalars are obtained by contracting the dimensionless tensors a −1 0 C α βγ . 2 It befits us to concentrate on scalars that are quadratic in the C α βγ , which I do. In particular, the scalars constructed from the quadratic tensor\nΥ µν ≡ C γ µλ C λ νγ − C γ µν C λ λγ(3)\n1 So the successes of standard Big Bang cosmology, such as nucleosynthesis, are retained. The symmetry assumption entails, for example, having the same inflation, and the same mechanism for baryogenesis in the two sectors. 2 In the units where c = 1, a0 has dimensions of length −1 . In standard units, a0 is replaced everywhere by ℓ −1 ≡ a0/c 2 . Then in the NR limit only a0 appears. In standard units, C α βγ have dimensions of length −1 ; so the dimensionless tensors to be used are ℓC α βγ = (c 2 /a0)C α βγ .\n\nhave particular appeal; this tensor is symmetric under the interchange of the two metrics. The scalars I use are then\nΥ = g µν Υ µν ,Υ =ĝ µν Υ µν .(4)\nThese are obtained from each other under metric interchange, and are used symmetrically in constructing MTMS actions for BIMOND. 3 The action terms I M andÎ M are the matter actions for standard matter and for the putative TM (with their degrees of freedom denoted by ψ i andψ i ), whose existence is suggested by the double metric nature of the theory: Just as standard matter couples only to the standard metric g µν , TM couples only to the second metriĉ g µν . There is no direct interaction between the two matter sectors. The dimensionless parameters β and α permit us to use gravitational couplings in the two sectors, G ′ = G/β andĜ ′ = G/α, that differ from each other and from G itself. In the rest of the paper I shall concentrate on symmetric versions of the theory; this means taking α = β, and that the two metrics appear in a symmetric way in the interaction term. As explained in [15,21], there are good reasons to take β ≈ 1 (see below). However, for the sake of generality I shall keep a general value of β in some of the discussion. Note that without the interaction term the theory consists of two uncoupled copies of GR. The resulting field equations are of the form\nβG µν + S µν = −8πGT µν ,(5)βĜ µν +Ŝ µν = −8πGT µν ,(6)\nwhere\nG µν = R µν − 1 2 g µν R,Ĝ µν =R µν − 1 2ĝ µνR(7)\nare the Einstein tensors, and T µν ,T µν are the energy-momentum tensors (EMTs) for the two sectors. The equations of motion of matter in the two sectors, in the presence of gravity, are the standard ones since they are still controlled by the same matter actions as in standard physics.\n\nThe tensors S µν andŜ µν fall from the variation of the interaction term with respect to g µν andĝ µν , respectively. 4 The metrics appear in the interaction in three ways: (i) through their derivatives, in C α βγ , (ii) contracting the C α βγ in the quadratic scalars on which M depends, (iii) otherwise, such as in the volume density (gĝ) 1/4 , or in scalars such as g/ĝ or g µνĝ µν . As a result, S µν andŜ µν have the schematic form\nS µν = (Q{C} λ µν ) ;λ + N {CC} µν + a 2 0 P g µν , S µν = (Q{C} λ µν ) :λ +N {CC} µν + a 2 0Pĝ µν ,(8)\nwith the three term types corresponding, respectively, to the above types of appearance of the metrics. Here, {C} λ µν denotes tensors linear in C α βγ , 5 and {CC} µν denotes tensors quadratic in them; 6 they are the same in the two equations. Also, \":\" denotes covariant derivation with respect toĝ µν . Q, N, P , etc. are dimensionless, and presumably of order unity. They depend on the two metrics through the quadratic scalars, and possibly through scalars not containing the metric derivatives, such as g/ĝ or g µνĝ µν . Q and N are linear in derivatives of M with respect to the quadratic scalars, e.g., Υ andΥ defined in Eq.(4), while P gets contributions from M itself and from its derivatives with respect to scalars containing the metrics alone. This classification of the terms helps understand some interesting special cases and limits of the theory. In what follows I refer to these contributions as Q, N, P terms, respectively. 3 The special appeal in the choice of the tensor Υµν , and the two scalars constructed from it, is twofold [15]: First, it leads to a theory with a particularly simple NR limit. Second, this choice is the best equivalent to the choice of the Ricci tensor, Rµν , and the Ricci scalar, R = g µν Rµν , in the Einstein-Hilbert Lagrangian of GR. In GR we have to use the full curvature tensor, which contains the second derivatives of the metric; this is both necessary (to get a scalar), and harmless (because the second derivatives in the Einstein-Hilbert Lagrangian do not affect the field equations). In fact, it is well known that we can replace R in the GR Lagrangian by g µν (Γ γ µλ Γ λ νγ − Γ γ µν Γ λ λγ ), exactly analogous to our Υ,Υ scalars. In our case we do not add the second derivatives; this would be both harmful (as it would lead to a higher-derivative theory), and unnecessary (as the first-derivative terms are tensor in themselves). 4 They satisfy, together, one set of four Bianchi-like differential identities, stemming from the invariance of the interaction term to general coordinate transformations (details in [15]). 5 Examples of such tensors are C λ µν , δ λ µ C α αν ,ĝµνg αβ C λ αβ , etc. 6 Examples of such tensors are C γ µλ C λ νγ , C γ µν C λ λγ , Υµν , etc.\n\nFrom the symmetry of the problem we havê Q(g µν ,ĝ µν ) = −Q(ĝ µν , g µν ),N (g µν ,ĝ µν ) = N (ĝ µν , g µν ),P (g µν ,ĝ µν ) = P (ĝ µν , g µν ).\n\nBIMOND versions with β = 1 have a simple GR limit: Assume that in the limit a 0 → 0, i.e., when the quadratic scalar arguments of M go to infinity, M becomes independent of them. Then, the Q and N terms vanish. The P terms then give CC-type terms that might still couple the two metrics, but which are rather small locally. It is also possible to choose M such that the coupling disappears altogether in this limit, so the P terms in S µν collect to a 2 0 P (∞)g µν [15]. The theory then decouples into two copies of GR, possibly with a cosmological constant, if M(∞) = 0. I have not yet been able to ascertain if there is an acceptable GR limit for theories with β = 1. 7 This may be one reason to prefer theories with β = 1.\n\nWhen the two metrics are equal, we have from Eq.(9)Q = −Q ≡ −Q 0 ,N = N ≡ N 0 ,P = P ≡ P 0 , where, furthermore, all these functions of the two metrics become constants. This is because we cannot construct (nonconstant) scalars from a single metric and its first derivatives, so all the scalar arguments of these functions must become constants whenĝ µν = g µν . Clearly, Q terms vanish linearly in the metric difference, and terms of type N vanish quadratically. We are then left only with P terms, which are of a cosmological constant type with Λ ∝ a 2 0 P 0 . From this follows that, in systems with identical matter and TM distributions, and the same initial and boundary conditions, the metrics in the two sectors are the same, and are a solution of the standard Einstein equations with a cosmological constant (CC) Λ. 8,9 A. Cosmological background Preliminary considerations of cosmology in BIMOND have been described in [21] and in [20]. As mentioned in the Introduction, here I assume all along the special, but attractive, case of an MTMS cosmology. This means that matter sources in cosmology are assumed to be identical in the two sectors, apart from differences in the initial seed inhomogeneities. If we neglect these differences to lowest order, the MTMS cosmology in BIMOND is identical to the Friedmann-Robertson-Walker (FRW) cosmology in GR, with a CC, Λ ∝ a 2 0 P 0 , with G ′ = G/β instead of G, and with the standard matter content. All the standard results derived for the latter (Big Bang, Nucleosynthesis, etc.) apply here as well (with G ′ as gravitational constant).\n\nTo a higher order, the differences in the inhomogeneities in the two sectors engender departure of the background cosmology from FRW: the cosmological metric is no longer a solution of the Einstein equations. This is discussed briefly in section V. To some approximation the cosmology could still be considered an FRW one, but with nonstandard, \"phantom\" contributions to the cosmological EMT in BIMOND. This sort of feedback goes beyond the feedback due to departure from smoothness, which has been discussed extensively respecting GR cosmology. Here it is specifically due to differences between the inhomogeneities in the two sectors. As long as the statistical properties of the inhomogeneities in the two sectors are the same, the cosmological background metrics are still the same, but they satisfy the BIMOND equations, instead of the Einstein equations.\n\nThe observed 4 He abundance was used in [22] to derive a bound of |G ′ /G − 1| 0.13, where G ′ is the value that applies at the formation time. This would imply, in the present context, |β − 1| 0.13, lending further motivation for taking β = 1.\n\n\nB. The nonrelativistic limit\n\nConsider first the NR limit of BIMOND as applied to a system that is small on cosmological scales (discussed in detail in [15]). Since we assume a symmetric cosmology, with the two metrics equal, there is a coordinate frame in which locally the two metrics are nearly Minkowskian. Because of local inequality of matter and TM mass distributions, the departure from Minkowski is not the same for the two metrics, which can thus be written as\ng µν = η µν − 2φδ µν + h µν ,ĝ µν = η µν − 2φδ µν +ĥ µν ,(10)\nwith the potentials φ,φ, h µν ,ĥ µν treated to lowest order in the weak-field approximation. 10 Note, importantly, that the weak-field approximation assumes that φ,φ ≪ 1 (in our units where c = 1). This, however, does not imply that whenever φ orφ appear they raise the order in the perturbation: Unlike the case of GR, here we have an additional dimensioned constant, a 0 , and quantities such as ∇φ/a 0 or ( ∇φ − ∇φ)/a 0 are not assumed small, and do not raise the order of terms in which they appear. In this light, C α βγ as they appear in {C} λ µν and {CC} µν are first order in the small parameters; however, in their appearance in the arguments of Q, N, P , etc. a −1 0 C α βγ should not be taken as small. 11 The N terms are second order and can be neglected. The P terms contribute to lowest order as a homogeneous CC, whose value we know is observationally negligible in local NR physics. As to the Q terms, since the {C} λ µν are already first order we can equate the two metrics everywhere else with the background Minkowski metric (so, for example, covariant derivatives become normal derivatives). But note that this approximation applies only to the metrics themselves, not to their derivatives, which appear only in a −1 0 C α βγ , which are not assumed small. It can be seen 12 that to the lowest order Q = −Q, and so S µν = −Ŝ µν . Thus, the sum and difference of the two field equations are, in this approximation,\nβ(G µν +Ĝ µν ) = −8πG(T µν +T µν ), β(G µν −Ĝ µν ) + 2S µν = −8πG(T µν −T µν ).(11)\nThe sum of the Einstein tensors is linear in the sums of the potentials, and their difference, as well as S µν , depends only on the differences of the potentials.\n\nIn [15] I treated the NR limit where one further assumes that motions are slow, so that time derivatives are negligible compared with spatial ones. I then showed that with the choice of the quadratic scalar arguments given in Eq. (4), which I shall assume, 13 the field equations imply, in a certain choice of gauge: h µν =ĥ µν = 0. The two metrics thus have the form of the first-order metric in GR, but the NR potentials, φ andφ, are determined not from the Poisson equation but from field equations derived from the action\nL = − 1 8πG {β( ∇φ) 2 + β( ∇φ) 2 − a 2 0 M[( ∇φ − ∇φ) 2 /a 2 0 ]} + ρ( 1 2 v 2 − φ) +ρ( 1 2v 2 −φ).(12)\nThis follows from the fact that to lowest order we have Υ =Υ ∝ ( ∇φ − ∇φ) 2 , and so the interaction term takes the form a 2 0 M[( ∇φ − ∇φ) 2 /a 2 0 ]. (I use here the same symbol, M, for a function of a single variable: the reduction of the relativistic interaction function, also denoted M, which depends on more variables.) This action also governs the particle equations of motions in the two sectors, which take the standard form: a = − ∇φ andâ = − ∇φ. Because the metric has the GR form in terms of a single potential: g µν = η µν − 2φδ µν , light bending by slowly moving masses is expressed in terms of the potential in the same way as in GR: photons \"see\" the same potential as massive particles.\n\nThe field equations for the potentials are\n∆φ = 4πGβ −1 ρ + β −1 ∇ · [( ∇φ − ∇φ)M ′ ] ≡ 4πG(ρ + ρ p ), ∆φ = 4πGβ −1ρ − β −1 ∇ · [( ∇φ − ∇φ)M ′ ] ≡ 4πG(ρ +ρ p ),(13)\nwhere ρ p andρ p play the role of \"phantom matter\" (PM) densities for the two sectors. As detailed in [21], it is convenient to work with the two potentialsφ andφ such that\nφ = ζφ +φ,φ = ζφ −φ,(14)\nwhich satisfy\n∆φ = 4πG(ρ +ρ), ∇ · {μ(| ∇φ|/a 0 ) ∇φ} = 4πG(ρ −ρ),(15)\n10 The separation into φ and hµν is a mere convenience: One defines φ ≡ −(1 + g00)/2, and h00 = 0. 11 a0 may be viewed as a proxy for a scale length ℓ ≡ c 2 /a0. Terms such as ∇φ/a0 appear, in fact, as ℓ ∇(φ/c 2 ). The NR limit in the MOND context has to be understood as c → ∞, ℓ → ∞, with c 2 /ℓ fixed. 12 Following from Eq. (9), noting that in this approximation Q is symmetric to the interchange of the metrics, as its argument is quadratic in their difference. 13 The general form of the scalar was also discussed in [15].\n\nwith the boundary conditions at infinityφ → 0, whileφ ∝ ln(r) when M =M , andφ → 0 when M =M ; M andM are the total masses in matter and TM respectively. Here, ζ = (2β) −1 , andμ(x) ≡ 2β − 4M ′ (4x 2 ). 14 MOND and Newtonian behaviors for isolated, pure-matter systems, such as galaxies, dictate the asymptotic behaviors\nµ(x ≪ 1) ≈ x,μ(x ≫ 1) ≈ (1−ζ) −1 . Thusφ = φ N +φ N\nis simply the sum of the matter and TM Newtonian potentials, andφ is a MOND potential satisfying the nonlinear Poisson equation proposed by [23] with ρ −ρ as a source (and a modified value ofμ at large arguments).\n\n\nIII. FLUCTUATIONS IN AN ISOLATED MTMS SYSTEM\n\nConsider first the development of weak fluctuations in a cosmologically small, isolated, NR system. The two matter components are treated as fluids. The time dependent problem is governed by the NR continuity equations in the two sectors\nρ t + ∇ · (ρv) = 0,ρ t + ∇ · (ρv) = 0,(16)\nand the respective Euler\nequations v t + (v · ∇)v = − ∇φ − ρ −1 ∇p,v t + (v · ∇)v = − ∇φ −ρ −1 ∇p,(17)\nwhere p,p are the pressures in the two sectors.\n\nIn a system made predominantly of matter-as we have, purportedly, in the solar system, galaxies, etc.-these equations lead to standard MOND dynamics as they have been applied extensively in the past. In this case, the MOND acceleration is generically larger than the Newtonian one, and when the acceleration is smaller than a 0 , dynamics is controlled by the MOND limit.\n\nHowever, as discussed in [21], things are rather different in systems where the distribution of matter and TM are close to each other (i.e., |ρ −ρ| ≪ ρ). For example, when ρ =ρ the dynamics is always Newtonian, no matter how small the acceleration is. 15 Consider then weak perturbations on an arbitrary, symmetric background flow with\nρ =ρ = ρ b (r, t), v =v = v b (r, t). Then, from Eq.(15),φ = 0, and φ =φ = β −1 φ N b , where φ N b\nis the Newtonian potential of ρ b : we have a double Newtonian system with an effective G e = G/β. 16 If we seed the system with fluctuations that are the same in the two sectors, they will remain so, withφ = 0 at all times, and the development remaining quasi-Newtonian in both sectors.\n\nTaking the lowest order in the fluctuations we have\nδρ t + ∇ · (δρv b ) + ∇ · (ρ b δv) = 0, δρ t + ∇ · (δρv b ) + ∇ · (ρ b δv) = 0,(18)\nand the Euler equations\nδv t + (δv · ∇)v b + (v b · ∇)δv = −ζ ∇δφ − ∇φ − ρ −1 b ∇δp + δρρ −2 b ∇p b , δv t + (δv · ∇)v b + (v b · ∇)δv = −ζ ∇δφ + ∇φ − ρ −1 b ∇δp + δρρ −2 b ∇p b ,(19)\nwhere p b is the pressure field in the unperturbed system (assumed to be the same in the two sectors, as we assume the same equation of state for the two). The potential fluctuations are obtained from: ∆δφ = 4πG(δρ + δρ),\n∇ · (μ| ∇φ/a 0 | ∇φ) = 4πG(δρ − δρ).(20)\nIf |δρ| ∼ |δρ| ∼ |δρ − δρ|, the sources for these equations are of similar magnitudes; but this need not be so.\n\nIt is useful to work with the sums and differences:\nu = δv + δv,ū = δv − δv, q = δρ + δρ,q = δρ − δρ.(21)\nIf we can assume that in both sectors p can be taken a function of ρ; e.g., for a polytropic gas, or when the process is isentropic, we write δp = (dp/dρ)(ρ b )δρ ≡ v 2 s (ρ b )δρ and get:\nq t + ∇ · (qv b ) + ∇ · (ρ b u) = 0,(22)u t + (u · ∇)v b + (v b · ∇)u = −2ζ ∇δφ − ρ −1 b ∇(v 2 s q) + qρ −2 b v 2 s ∇ρ b ,(23)∆δφ = 4πGq,(24)q t + ∇ · (qv b ) + ∇ · (ρ bū ) = 0,(25)u t + (ū · ∇)v b + (v b · ∇)ū = −2 ∇φ − ρ −1 b ∇(v 2 sq ) +qρ −2 b v 2 s ∇ρ b ,(26)∇ · (μ| ∇φ/a 0 | ∇φ) = 4πGq.(27)\nThe barred and unbarred systems decouple, with the unbarred fluctuations developing according to standard Newtonian theory, while the barred fluctuations develope according to MOND. If we start with q andq fluctuations of a similar magnitude, and the barred fluctuations are in the MOND regime,q grows faster than q. This leads in the limit to δρ ≈ −δρ, namely segregation of the two sectors. This echoes the fact that matter and TM repel each other in the MOND regime [21].\n\n\nIV. COSMOLOGICAL FLUCTUATION GROWTH\n\nThe use in the cosmological context of the above NR equations, derived for an isolated system on a double Minkowski background, can be justified for weak perturbations in NR matter, of scale length much smaller than cosmological scales: Expand the BIMOND field equations around their cosmological solution of the FRW equations with a CC and g µν = g µν . The EMTs are expanded around their value for the homogeneous background with only the 00 components contributing. For scales much smaller than the (space-time) curvature radius the background metrics can be taken as nearly Minkowski. Then, our formalism above applies.\n\nIn this application, ρ b (t) is the ambient, homogeneous cosmological density of the specific matter component we concentrate on, and p b is its effective pressure.\n\nI shall take in what follows β = 1.\n\nAs is standard, we express the space derivatives with respect to comoving coordinates R instead of the properdistance coordinates r = a(t)R, which appear in the above equations. We also put v b = H(t)r, ∇ · v b = 3H, where H =ȧ/a is the Hubble parameter. Also, write the equations in terms of partial time derivatives at a fixed comoving position: for the field S,Ṡ ≡ S t + (v b · ∇)S. The continuity equation for the background NR matter amounts to\nρ b ∝ a −3 , so,ρ b = ∂ρ b /∂t = −3Hρ b . Define η = δρ/ρ b , ǫ = η +η = q/ρ b ,ǭ = η −η =q/ρ b .\nFor these we have:\nǫ + a −1 ∇ R · u = 0,(28)u + (ȧ/a)u = −a −1 ∇ R δφ − v 2 s a −1 ∇ R ǫ,(29)∆ R δφ = 4πGa 2 ρ b ǫ,(30)\nǫ + a −1 ∇ R ·ū = 0,\nu + (ȧ/a)ū = −2a −1 ∇ Rφ − v 2 s a −1 ∇ Rǭ ,(31)∇ R (μ| ∇ Rφ /aa 0 | ∇ Rφ ) = 4πGa 2 ρ bǭ .(32)\nThe unbarred system is identical to that in standard gravity: the perturbation sum develops fully in a Newtonian manner. Combining the time derivative of the continuity equation with the R divergence of the Euler equation we get as usualǫ\n+ 2(ȧ/a)ǫ = 4πGρ b ǫ + v 2 s a −2 ∆ R ǫ,(34)\nand for the space Fourier components defined by ǫ = d 3 kǫ k (t)e ik·r , we have the standard resulẗ\nǫ k + 2(ȧ/a)ǫ k = (4πGρ b − v 2 s k 2 /a 2 )ǫ k .(35)\nThe barred system, which is decoupled from the unbarred one, describes the evolution of the perturbation difference; it behaves as a single fluid governed by MOND, with the strength of the governing potential,φ, doubled [as expressed by the factor 2 in front of ∇φ in Eq.(32)], and with a Newtonian limit for which the effective Newton constant is G. 17 We can still eliminate the velocityū to geẗ\nǫ + 2(ȧ/a)ǫ − 2a −2 ∆ Rφ − v 2 s a −2 ∆ Rǭ = 0,(36)\nbut here ∆ Rφ is not simply expressible in terms ofǭ. The problem is nonlinear even for the smallest of perturbations, and perturbations on different scales can affect each other's growth (see below). Given some initialǭ(R, t 0 ) andū(R, t 0 ), we can calculateǫ(R, t 0 ) from the continuity equation, we then calculatē φ from the potential equation, then calculateǫ from Eq. (36), and so propagate the system in time. The velocities are propagated in time in parallel, using the Euler equation.\n\n\nA. The deep-MOND regime\n\nWith matter and TM present, there are two senses in which we can understand the term \"a deep-MOND system\": The weaker sense applies to systems where | ∇φ| ≪ a 0 so thatφ is determined by deep-MOND physics (with its peculiar symmetries, etc.). If we have an inhomogeneity of proper length scale λ that can be considered in isolation, i.e., is not subject to an external-field effect (EFE) from a larger-scale inhomogeneity of which it is a part (see below), this amounts to\n|δρ − δρ|Gλ ≪ a 0 .(37)\nIn the stronger sense of the term, one requires, also, that | ∇φ| ≪ | ∇φ| so that the full potentials φ andφ are determined by deep-MOND physics. This requires further\n|δρ + δρ|Gλ ≪ (a 0 |δρ − δρ|Gλ) 1/2 .(38)\nIn past applications of MOND, where TM is absent, the second requirement follows automatically from the first. This is also the case in the present context if |δρ − δρ| ∼ max(|δρ|, |δρ|) (for example, if the fluctuations in the two sectors are uncorrelated), but not if |δρ − δρ| ≪ max(|δρ|, |δρ|). Because here we have full decoupling of the sum and difference systems, with the latter being governed only byφ, I shall use the term in the weaker sense, which is enough to ensure that the barred system is governed by deep-MOND physics.\n\nIn the deep-MOND limit, Eq.(33) is well approximated by\n∇ R (| ∇ Rφ | ∇ Rφ ) = 4πGρ 0 a 0ǭ ,(39)\nwhere ρ 0 = a 3 ρ b . This equation is conformally invariant [21,24].\n\n\nB. Coupling of inhomogeneities on different scales\n\nThe implications of Eq.(33) for the dynamics in massive systems have been discussed extensively (e.g., in [23,25]). One clear outcome of these studies concerns the way in which the dynamics of a larger, mother system couple to those in a smaller subsystem. When the two do not differ much in size and/or mass, the interaction between them, which results from the nonlinearity of the theory, is not easy to describe. However, when the subsystem is much smaller, and much less massive, than the mother system (such as a for star in a galaxy, or a galaxy in a galaxy cluster), the interaction is simply described: The subsystem moves within the mother system as if it where a test particle, irrespective of its own intrinsic accelerations. In other words, the dynamics of the mother system is oblivious to the internal dynamics in its small subsystems, which can be treated as test particles moving in their combined mean field. On the other hand, the dynamics within the subsystem may be greatly affected by the acceleration, g ex , with which it falls within the mother system, provided this acceleration dominates the internal ones, and is in the MOND regime (g ex a 0 ). The intrinsic dynamics is then quasi-Newtonian with an effective gravitational constant G ′ ∼ Ga 0 /g ex , and some mild anisotropy induced by the external field. This is known as the MOND external-field-effect (EFE).\n\nIn the present context, the mass source in the equation for the potential, ρ bǭ , must exhibit, at any time, a continuum spectrum of fluctuations, and it is difficult to describe the mutual couplings of the different scales. We expect, however, that the evolution of a fluctuation inǭ on a given scale, will be affected mainly by larger-scale perturbations within which it is nested, in the spirit of the EFE.\n\nA difference fluctuation of magnitudeǭ and (proper) scale length λ, would have a characteristic intrinsic acceleration, when in isolation, ∼ (a 0 Gρ bǭ λ) 1/2 . Thus, if, e.g.,ǭ ∝ λ −α , for α > 1 the typical acceleration decreases with increasing scale, and we do not expect a pronounced EFE. For α < 1, the largest scales determine the dynamics within smaller ones, as described roughly by the EFE. For α = 1 the accelerations on all scales are of the same order.\n\n\nC. Pressure vs gravity\n\nFor the unbarred, fluctuation-sum system, the standard (Jeans) criterion tells us when pressure becomes unimportant in impeding gravitational growth. This applies [e.g., per Eq.(35)], when ρ b Gλ 2 ≫ v 2 s , or, equivalently, when\nM b ≫ G −3/2 v 3 s ρ −1/2 b , where M b = ρ b λ 3\nis the background mass (in either sector) in a cube of side λ. For the barred system, the Jeans criterion for gravity dominance is very different. We saw that because of the nonlinearity of this system we cannot consider separately the growth of a single perturbation with a given amplitude and scale length, as all modes are, in principle, coupled. I discuss two end cases. First, consider a perturbation of magnitudeǭ and (proper) scale λ that is not affected by larger-scale overdensities (through an analog of the EFE). From Eq.(32), or Eq.(36), the Jeans criterion is still | ∇φ| ≫ v 2 s | ∇ǭ|. From Eq.(33) we now have in the MOND limit | ∇φ| 2 /λa 0 ∼ Gρ b |ǭ|. Put together, these give the condition for gravity dominance\nM b ≫ (Ga 0 ) −1 v 4 s |ǭ|(40)\n(where gradients are substituted by division by λ; e.g.,| ∇ǭ| ∼ |ǭ|/λ). Note, in particular, the appearance of |ǭ| in the criterion, in addition to the expected characteristic, deep-MOND M ∝ v 4 relation. This occurs because the pressure increment is linear inǭ, while the potential incrementφ is proportional to |ǭ| 1/2 . This |ǭ| factor makes the criterion much easier to satisfy for weak perturbations, 18 If the particular perturbation at hand cannot be considered in isolation, but is subject to an EFE characterized by g ex , the Jeans criterion takes the form\nρ b G(a 0 /g ex )λ 2 ≫ v 2 s , or M b ≫ G −3/2 (g ex /a 0 ) 3/2 v 3 s ρ −1/2 b .(41)\nAs a concrete example, consider a baryonic, difference perturbation, of magnitudeǭ and scale λ, that may be considered unaffected by an EFE. Taking the ambient temperature appropriate for the matter-dominance era T ≈ 3 × 10 3 (a/10 −3 ) −1 K, and ambient density ρ b ≈ 4 × 10 −22 (a/10 −3 ) −3 gr cm −3 , the condition (37) for the perturbation being in the deep-MOND regime can be written\nM b /M ⊙ ≪ 2 × 10 7 a 10 −3\nThe condition (40) for gravity dominance over pressure is\nM b /M ⊙ ≫ 3 × 10 5 a 10 −3 −4 |ǭ|.(43)\nIn comparison, the Jeans criterion relevant for the sum of the perturbations is\nM b /M ⊙ ≫ 10 6 a 10 −3 −1.5 .\n(44)\n\nSo, there is a wide range of scales for which pressure stymies the growth of ǫ but not ofǭ.\n\n\nD. Later stages\n\nThe analysis above highlights the way in which small-scale, NR density inhomogeneities grow when they are still small compared with the ambient density. We have learned much about the mechanisms that are at play and how they greatly differ from those underlying perturbation growth in standard dynamics. For example, we noted the shepherding of matter overdensities by flanking TM overdensities (and vice-versa): a matter overdensity grows faster when it sits in a TM depression. A related effect is the segregation of matter and TM, as evinced by the tendency of difference fluctuations to grow faster than the sum But, beyond these qualitative observations, a numerical attack on the problem is needed to follow the growth of even weak perturbations in detail, because of the inherent MOND nonlinearity. This is also, clearly, the case when we deal with structure buildup to its presently observed stage, where the perturbations are not small. As mentioned in the introduction there have been several N -body studies of structure formation in MOND, and these could be easily adapted to include TM.\n\nWithout numerical studies, it is difficult to tell exactly how, and to what extent, these trends continue to shape structure in the later stages. Some questions that naturally arise are: \"Up to what scales are matter and TM effectively segregated today?\" In [21], I conjectured that this happens on the supercluster-void scales; namely, that matter and TM have formed interlacing cosmic webs in which TM concentrations sit in the observed matter voids, and vice-versa, and matter and TM filaments avoid each other. Another important question is: \"How rare are TM objects, such as galaxies, in matter territory, and what observable effects can they have?\" If the emptiness of matter voids is an indication, we do not expect TM galaxies within large matter concentrations. Other questions are: \"Does TM help empty matter voids more efficiently than in the CDM paradigm?\" (For possible indications for unexpectedly large and empty voids see [27][28][29][30][31][32][33].) \"Are higher very-large-scale velocities produced in the present picture than in the framework of CDM?\" (See [34][35][36] for possible indications of such unexpectedly high velocities.) \"Are objects with high peculiar velocities more common in our world picture than in the CDM paradigm?\" (See [37] for a discussion of the issue in connection with the bullet cluster, and [38] for a recent assessment, and for references to earlier work.)\n\nWe can get some guidance in pondering over these questions by considering the interaction between well separated matter and TM bodies. This question has been considered in detail in [21], which deals, in a sense, with the opposite of the situation we treat here of small departures from well mixed configurations.\n\nThe basic result found there is that matter and TM bodies repel each other in the MOND regime, and (for β = 1) they do not interact in the high-acceleration regime. For example, using the conformal invariance of the deep-MOND limit of the theory, I calculated the force between two masses M 1 and M 2 in the deep-MOND regime to be (defined as negative for attraction)\nF = − 2 3 (a 0 G) 1/2 r [(M 1 ± M 2 ) 3/2 − M 3/2 1 − M 3/2 2 ],(45)\nwhere the plus sign applies when the two masses are in the same sector (in which case F is always attractive), and the minus sign when they are different (in which case the force is always repulsive); r is the separation. We see that for equal masses the repulsive force between matter and TM masses M is (4/3)(a 0 G) 1/2 M 3/2 /r, which is ≈ 2.4 times larger than the attractive force between two masses M in the same sector. This can give us an idea of how far from each other separated bodies of matter and TM can separate over the Hubble time. 19 The characteristic separation speed for matter and TM masses M is ∼ 2.3(a 0 GM ) 1/4 ∼ 260(M/10 10 M ⊙ ) 1/4 km s −1 . In a Hubble time his produces a separation of several megaparsecs for galactic mass objects and a few tens of megaparsecs for galaxy-cluster mass objects. A test mass in one sector in the (MOND) field of a mass M of the other sector has typical separation speeds about half of that given above: a massive TM cluster can shoot out a matter bullet with a speed of several thousand km s −1 . 20 Another potentially useful result for a late-stage system made of separate matter and TM bodies is the general, 'virial' relation that holds exactly in the deep-MOND regime: For a system made of matter and TM pointlike masses m i andm j , respectively, at positions r k (k running over all masses), we have\nk r k · F k = − 2 3 (a 0 G) 1/2 [| i m i − jm j | 3/2 − i m 3/2 i − jm 3/2 j ],(46)\nwhere F k is the force on particle k. (The choice of origin is immaterial as F k = 0.) For a system with equal total masses, the first term drops, which gives an expression that is always positive; this means that on average the forces are pointing outward, bespeaking a mean repulsion. 21 For example, in a system made of N equal matter masses, m, and N equal TM masses, m, we get\n2N k=1 r k · F k = 4 3 (a 0 G) 1/2 N m 3/2 .(47)\nThus, for the accelerations we have\nr k · a k = 2 3 (a 0 Gm) 1/2 .(48)\nThe acceleration thus points outward on average, implying global self repulsion. The magnitude of the typical acceleration is ∼ (a 0 Gm) 1/2 /R, where R is the characteristic system size. This is smaller by a factor ∼ N −1/2 than the typical MOND acceleration in a pure-matter system of the same mass. Note, however, that this mean acceleration is only relevant if the matter and TM masses are very well mixed. If, in fact, they form separate matter and TM clumps within the system, as they would tend to do, these clumps would separate with higher accelerations.\n\n\nV. PHANTOM MATTER AND ITS BACK REACTION ON COSMOLOGY\n\nThere has been much discussion recently regarding back reaction of inhomogeneities on cosmological evolution in GR. The issue concerns the extent to which we err by applying FRW theory (which assumes homogeneous energy density) to the mean cosmological energy density, instead of calculating cosmological evolution correctly with inhomogeneously distributed energy, and then averaging the outcome. (For recent discussion of this issue, with references to earlier work, see, e.g., [39,40].) The exact importance of this effect is still moot.\n\nObviously, such effects exist in BIMOND as well, but they are not the subject here. In a MTMS universe governed by BIMOND there are added back-reaction effects of inhomogeneities, which, as we saw, induce the appearance of phantom matter components that should have influence on cosmology.\n\nIn a universe with different matter and TM inhomogeneities, the S µν andŜ µν terms in the BIMOND field equations (5)(6) do not vanish; so, what is their effect on the cosmological background? Strictly speaking, we then have to solve the full BIMOND cosmological equations. However, to some approximation, we may treat the resulting departure from GR perturbatively, by viewing the effects of these terms as perturbations on the MTMS, FRW cosmology: In this approximation, the S µν andŜ µν terms are viewed as the EMTs of MOND phantom matter that can effectively be added to T µν andT µν in sourcing cosmic gravity. We saw that to zeroth order in the metric difference, all that remains from these terms is a CC term, which has to be reckoned with in zeroth-order cosmology. This cosmology is then used to calculate the development of perturbations in the two sectors, starting from some seed perturbations. The next stage in such an approximation scheme would be to use the deduced local departures of the two metrics from the FRW metric, used in the lower order approximation, to calculate local contributions to S µν andŜ µν , and put them back as sources in the equations for cosmology. This program clearly deserves a thorough treatment, which, however, is beyond our scope here. In particular, the results of such an analysis is expected to depend on the details of the specific BIMOND version at hand, and there is a large variety of versions to explore. Here I only describe briefly some general properties of the different term types defined in Eq.(8), as they would be fed back into the cosmological equations.\n\nThe zeroth order of the P term (in the matter sector), a 2 0 P 0 g µν gives a genuine CC term [if P 0 = 0]. When matter and TM separate, P becomes space-time dependent, and in this regard behaves as a \"dark energy\" field. In the weakfield approximation, we can substitute in the variables on which P depends g µν ≈ĝ µν , but Υ ≈Υ ∝ ( ∇φ − ∇φ) 2 . The contribution of the P term is then\nS P µν ≈ a 0 P (z)g µν , z ≡ ( ∇φ − ∇φ) 2 /a 2 0 .(49)\nFrom Eq.(9) we see thatP = P in this approximation. The value of P in this expression varies in space-time between P 0 and P (∞). (As explained in [21], considerations of the GR limit of the theory, for a 0 → 0, dictate that P (∞) is not infinite.) Today we have in much of the volume of the universe z 10 −3 ; so in much of the volume P ≈ P 0 , but in galaxies or in galaxy clusters (within about a megaparsec of the center), z 1, and near stars P ≈ P (∞). Since the values of P around matter concentrations are the same as its values around similar TM concentrations (interchanging matter and TM amounts to interchanging φ andφ to which P is invariant), the large-scale average of P gives a time dependent effective CC, Λ(t) ∝ P space = P 0 . The next order contribution, which appears with the separation of matter from TM, comes from the lowest order in the Q terms, whose contribution to the cosmological density \"phantom\" EMT is\nT (p) µν = (8πG) −1 (Q{C} λ µν ) ;λ ,(50)\ngiving in the weak-field approximation\nT (p) µν ≈ −T (p) µν ≈ (8πG) −1 (Q{C} λ µν ) ,λ .(51)\nIn the weak-field-slow-motions approximation, where time derivatives are neglected, only the 00 component contributes and gives, simply, the totality of all the PM gotten from Eq.(13), namely a NR PM density\nρ p = (4πG) −1 ∇ · [( ∇φ − ∇φ)M ′ (z)].(52)\nIn a MTMS universe this density averages to zero over large cosmological volumes. This can be seen in two (independent) ways. First, it follows from the fact that under the interchange of matter and TM, T µν ↔T µν , we have g µν ↔ĝ µν . In the NR limit this means that ρ ↔ρ leads to φ ↔φ, which, in turns gives ρ p → −ρ p : The phantom density around a matter concentration is equal and opposite in sign to that around a similar TM concentration. In our world picture the large-scale averages of the universe are invariant to matter-TM interchange, hence the average phantom density, which changes sign, must vanish. This result holds also for a finite system with matter-TM symmetry. For example, in a system where matter-TM interchange can be achieved by a volume-preserving transformation (such as translation, reflection, rotation) the total phantom mass in a symmetric volume vanishes. We can also argue differently, noting that the phantom density is a divergence of a vector ( ∇φ − ∇φ)M ′ (z), whose absolute value is bounded (outside of point masses). As a result, its volume average vanishes in the limit of a large volume. This property of the phantom density follows from the form of the relativistic Q term contribution: We see from Eq.(50) that the trace of the phantom EMT is a covariant divergence of a vector V λ ∝ Qg µν {C} λ µν . Its integral over a space-time volume is hence a surface integral.\n\nEven if the large-scale average of the NR phantom density averages to zero it could still affect cosmology due to its inhomogeneity. In addition, there may be higher order contributions that do not average to zero.\n\nThe Q terms may also have contributions of the form (M ′ g αβ C λ αβ ) ;λ g µν , or (M ′ g αβ C λ αλ ) ;β g µν , which are of the form V λ ;λ g µν ; i.e., they contribute in cosmology as dark energy with oscillating, vanishing-average density. The PM around matter concentrations can be (and is) observed, and is interpreted by most as \"dark matter.\" As discussed in [21], the PM around TM concentrations may be observed with weak lensing of matter photons (it does not produce strong lensing), on which it acts as a repelling mass.\n\nFor the N terms, the contribution of lowest order in the departure from metric equality is the second order. In the NR limit their characteristic contribution to the phantom EMT is of the order of (8πG) −1 zN (z)(a 0 /c) 2 [z defined in Eq.(49)]. Using the well known proximity a 0 ≈ cH 0 /2π, we can write this contribution as ∼ 10 −2 zN (z)ρ c , where ρ c is the critical density today. We saw above that in most of the cosmological volume today z 10 −3 . Also we have that N (0) is of order unity (from considerations of the MOND limit, see [15,21]), and N (∞) = 0 (from considerations of the GR limit of BIMOND). N is related to the extrapolating function of MOND and is constrained to remain of order unity, or below, for all arguments. So the contribution of the N terms to the density can be at most of order ρ c (in systems where | ∇φ| ∼ a 0 ), but some orders of magnitude smaller than ρ c in most of the cosmic volume. It would be interesting to see what their tensorial form is, and to what extent they produce observable lensing effects.\n\nNote, finally, that inasmuch as the statistical properties of matter in the two sectors are the same, so are the feedback effects discussed above. 22 Namely, the effects of inhomogeneities, and the statistical properties of the PM (their cosmological mean densities, power spectra, etc.) are the same in the two sectors. The two mean background metrics thus remain the same.\n\n\nVI. DISCUSSION\n\nI have considered the dynamics of small, NR perturbations in an MTMS universe. An intriguing aspect of this world picture is that MOND dynamics, cosmological or local, are completely absent in a perfectly symmetric universe, apart from a possible appearance of a CC of order (a 0 /c) 2 . All other departure from standard physics encapsulated in MOND-as it appears in cosmology, in structure formation, and in bound systems, such as the solar system or galactic systems-is entirely an effect of \"separation\" between matter and TM, or separation between the two metrics. In a double-membrane picture of BIMOND, MOND effects are due to local separation of the membranes.\n\nTwin matter plays two roles in our narrative: In the first, it provide a counteracting medium to matter; similar to the role of a neutralizing positive ion background to an electron gas. As a result of this presence, we could assume the very appealing symmetric cosmology, identical to that of GR with a CC. In addition, the presence of matter and TM in equal average quantities leads to a well-defined MOND dynamics, in which the MOND potential is sourced only by departures from uniformity, not by the bulk. To fulfill this role, TM only has to be present in the same amount as matter; it does not necessarily have to clump.\n\nIn its second role, TM plays an equivalent part to that of matter; so it can be likened to a positron component in a globally neutral electron-positron plasma (with mutual repulsion, and no annihilation). In this role it also takes part in shaping the large-scale structure of matter.\n\nThere are several qualitative features that make the growth of fluctuations in the proposed world picture different from the standard one. Among them: (i) Two initial fluctuation spectra, not one, determine the large-scale structure at later times. For the NR problem discussed here, we can think of these as the fluctuations in matter alone, and of those in the matter-TM density difference. This is important to appreciate, for example, because we have strong constraints on matter fluctuations at the time of matter-radiation decoupling, from the CMB anisotropies; but, we have none on the difference fluctuationsǭ, at that time. So, numerical simulations have to explore all possible options. (ii) The NR fluctuations in the sum and difference of the densities form two decoupled systems controlled by different types of gravitational physics. The sum system is underlaid by standard Newtonian gravity, while the difference system is underlaid by MOND gravity, which is stronger for the same source strength. (iii) In the difference system, the competition between gravity and pressure is strongly tilted in favor of gravity, with the Jeans criterion being much more lenient than in the sum system. (iv) As a result of (ii) and (iii), the difference fluctuations tend to grow faster than the sum fluctuation, leading towards a limit where |ǭ| ≫ |ǫ|, or δρ ≈ −δρ. This implies tendency towards anticorrelation of matter and TM overdensities, or to segregation of the two sectors. This applies not only when the perturbations are weak: the fact that distinct matter and TM bodies repel each other for low accelerations implies that the process of segregation continues during the stages where the overdensities are high. (v) There is an added enhancement, due to the fact that once the fluctuations in the two sectors are anticorrelated, with maxima of each residing in minima of the other, each overdensity grows not only by the pull of its own gravity, but also by shepherding by the repulsive action of the flanking overdensities in the other sector. (vi) There can be strong coupling between the perturbations on different scales due to the nonlinearity inherent in MOND.\n\nThere is much that remains to be studied, even within the narrow paradigm of the MTMS universe. An even richer prospect opens if one considers more general cases such as matter-TM asymmetric BIMOND versions, or asymmetric universes in which the two sectors are not equally populated, or do not have exactly the same properties.\n\nFor example, we do not know why a universe that presumably started with a vanishing baryon-number density has ended up having more baryons than antibaryons, with the difference resulting in the present-day baryon density. This baryogenesis, as it is called, which requires a breakdown of symmetries (CP and baryon-number conservation), is thought to have occurred during some early phase of thermal disequilibrium. If the corresponding conditions in the two sectors where slightly different, we could have ended up with different baryon densities in the two sectors. It would be interesting to study the effects of such departure from MTMS, to lowest order in the density difference, to see what effects it has on both the cosmological evolution, and on structure growth.\n The NR limit of the theory does have a good Newtonian limit even for β = 1.8 For example, a double Schwarzschild solution is a vacuum solution of BIMOND, which corresponds to a black hole of equal matter and TM masses. Similarly, standard GR gravitational waves are also vacuum solutions of BIMOND.9 Pictorially, in a double membrane picture the Einstein-Hilbert actions for the two metrics may be viewed, as usual, as the elastic energies of the membranes-the cost of distorting them out of the flat. Then, the CC may be viewed as the contact energy density, remaining whenĝµν = gµν , and the rest of the interaction is the \"cost of separating\" the membranes.\nThe factor 4 in the argument accounts for the fact that we use the variableφ = (φ −φ)/2. 15 This is the NR expression of the reduction of BIMOND to GR for completely symmetric systems.16 This case is not to be confused with the Newtonian limit in a system made of pure matter or pure TM, in which case G itself plays the role of the Newton constant.\nSinceμ(∞) = 2,φ is half the Newtonian potential sourced by ρbǭ in this limit; the 1/2 factor canceled by above mentioned factor 2.\n|ǭ| −3 .(42)18 In[26] I gave the Jeans criterion for a MOND universe as Mb ≫ (Ga0) −1 v 4 s . This was based on a pure matter world-picture, where it was assumed, tentatively, that the source for the MOND potential includes the background density, and so the potential increment is linear in the density increment. Here, with an MTMS background, the background MOND potential vanishes, so the perturbation alone sourcesφ, and so | ∇φ| ∝ |ǭ| 1/2 .\nThis is only a rough estimates, since such bodies are hardly ever isolated on the relevant scales.\nIf the bullet test mass starts at rest a distance r0 from a TM mass M, its speed at a distance r is, assuming the system is isolated, v(r) = (4a0GM ) 1/4 ln 1/2 (r/r0) ≈ 1.6 × 10 3 (M/10 14 M⊙) 1/4 ln 1/2 (r/r0)km s −1 . 21 Such a system does not satisfy a kinematic virial relation: since Fk = mkvk, we find that d( k mkrk · vk)/dt > 0; so no steady state can be reached.\nEven if the initial fluctuations are different in the two sectors, we saw that their statistical properties are driven to equalization by the faster, MOND-like growth of the fluctuation difference.\nAcknowledgementsThis research was supported by a center of excellence grant from the Israel Science Foundation.\n. R H Sanders, Astrophys. J. 5601R.H. Sanders, Astrophys. J. 560, 1 (2001)\n\n. A Nusser, MNRAS. 331909A. Nusser, MNRAS 331, 909 (2002)\n\n. S Stachniewicz, M Kutschera, Acta Phys. Polon. B. 323629S. Stachniewicz and M. Kutschera, Acta Phys. Polon. B 32, 3629 (2001)\n\n. A Knebe, B K Gibson, MNRAS. 3471055A. Knebe and B.K. Gibson, MNRAS 347, 1055 (2004)\n\n. C Llinares, A Knebe, H S Zhao, MNRAS. 3911778C. Llinares, A. Knebe, and H.S. Zhao, MNRAS 391, 1778 (2008)\n\n. J D Bekenstein, Phys. Rev. D. 7083509J.D. Bekenstein, Phys. Rev. D 70, 083509 (2004)\n\n. R H Sanders, Astrophys. J. 480492R.H. Sanders, Astrophys. J. 480, 492 (1997)\n\n. C Skordis, Class. Quant. Grav. 26143001C. Skordis, Class. Quant. Grav. 26, 143001 (2009)\n\n. R H Sanders, MNRAS. 363459R.H. Sanders, MNRAS 363, 459 (2005)\n\n. S Dodelson, M Liguori, Phys. Rev. Lett. 97231301S. Dodelson and M. Liguori, Phys. Rev. Lett. 97, 231301 (2006)\n\n. C Skordis, D F Mota, P G Ferreira, C Boehm, Phys. Rev. Lett. 9611301C. Skordis, D.F. Mota, P.G. Ferreira, and C. Boehm, Phys. Rev. Lett. 96, 011301 (2006)\n\n. C Skordis, Phys. Rev. D. 74103513C. Skordis, Phys. Rev. D 74, 103513 (2006)\n\n. C Skordis, Phys. Rev. D. 77123502C. Skordis, Phys. Rev. D 77, 123502 (2008)\n\n. T G Zlosnik, P G Ferreira, G D Starkman, Phys. Rev. D. 7784010T.G. Zlosnik, P.G. Ferreira, and G.D. Starkman, Phys. Rev. D 77, 084010, (2008)\n\n. M Milgrom, Phys. Rev. D. 80123536M. Milgrom, Phys. Rev. D 80, 123536 (2009)\n\n. T Damour, I I Kogan, Phys. Rev. D. 66104024T. Damour and I.I. Kogan, Phys. Rev. D 66, 104024 (2002)\n\n. D Blas, C Deffayet, J Garriga, Class. Quant. Grav. 231697D. Blas, C. Deffayet, and J. Garriga, Class. Quant. Grav. 23, 1697 (2006)\n\n. D Blas, C Deffayet, J Garriga, Phys. Rev. D. 76104036D. Blas, C. Deffayet, and J. Garriga, Phys. Rev. D 76, 104036 (2007)\n\n. M Bañados, P G Ferreira, C Skordis, Phys. Rev. D. 7963511M. Bañados, P.G. Ferreira, and C. Skordis, Phys. Rev. D 79, 063511 (2009)\n\n. T Clifton, T G Zlosnik, Phys. Rev. D. 81103525T. Clifton and T.G. Zlosnik, Phys. Rev. D 81, 103525 (2010)\n\n. M Milgrom, MNRAS. 4051129M. Milgrom, MNRAS 405, 1129 (2010)\n\n. S M Carroll, E A Lim, Phys. Rev. D. 70123525S.M. Carroll and E.A. Lim, Phys. Rev. D 70, 123525 (2004)\n\n. J Bekenstein, M Milgrom, Astrophys. J. 286J. Bekenstein and M. Milgrom, Astrophys. J. 286, 7 (1984)\n\n. M Milgrom, Phys. Rev. E. 561148M. Milgrom, Phys. Rev. E 56, 1148 (1997)\n\n. M Milgrom, J. Phys. A. 351437M. Milgrom, J. Phys. A 35, 1437 (2002)\n\n. M Milgrom, Comm. Astrophys. J. 13215M. Milgrom, Comm. Astrophys. J. 13, 215 (1989)\n\n. P J E Peebles, Astrophys. J. 557495P.J.E. Peebles, Astrophys. J. 557, 495 (2001)\n\nR B Tully, arXiv:0708.0864Proceedings IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff. IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff38R.B. Tully, Proceedings IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff, 25-29 June, Bull. Am. Astron. Soc. 38, 240, (2007), arXiv:0708.0864\n\n. R B Tully, E J Shaya, I D Karachentsev, H M Courtois, D D Kocevski, L Rizzi, A Peel, Astrophys. J. 676184R.B. Tully, E.J. Shaya, I.D. Karachentsev, H. M. Courtois, D.D. Kocevski, L. Rizzi, and A. Peel, Astrophys. J. 676, 184 (2008)\n\n. A Tikhonov, A Klypin, MNRAS. 3951915A. Tikhonov and A. Klypin, MNRAS 395, 1915 (2009)\n\n. A V Tikhonov, S Gottloeber, G Yepes, Y Hoffman, MNRAS. 3991611A.V. Tikhonov, S. Gottloeber, G. Yepes, and Y. Hoffman, MNRAS 399, 1611 (2009)\n\nR Van De Weygaert, E Platen, arXiv:0912.2997proceedings COSPA2008, Pohang, Korea. accepted to Modern Physics Letters A. COSPA2008, Pohang, Korea. accepted to Modern Physics Letters AR. van de Weygaert and E. Platen, proceedings COSPA2008, Pohang, Korea. accepted to Modern Physics Letters A, arXiv:0912.2997\n\n. P J E Peebles, A Nusser, Nature. 465565P.J.E. Peebles and A. Nusser, Nature 465, 565 (2010)\n\n. R Watkins, H A Feldman, M J Hudson, MNRAS. 392743R. Watkins, H.A. Feldman, and M.J. Hudson, MNRAS 392, 743 (2009)\n\n. G Lavaux, R B Tully, R Mohayaee, S Colombi, Astrophys. J. 709483G. Lavaux, R.B. Tully, R. Mohayaee, and S. Colombi, Astrophys. J. 709, 483 (2010)\n\n. A Kashlinsky, F Atrio-Barandela, H Ebeling, A Edge, D Kocevski, Astrophys. J. 71281A. Kashlinsky, F. Atrio-Barandela, H. Ebeling, A. Edge, and D. Kocevski, Astrophys. J. 712, L81 (2010)\n\n. G W Angus, S S Mcgaugh, MNRAS. 383417G.W. Angus and S.S. McGaugh, MNRAS 383, 417 (2008)\n\n. J Lee, E Komatsu, Astrophys. J. 71860J. Lee and E. Komatsu, Astrophys. J. 718, 60 (2010)\n\n. T Clifton, P G Ferreira, Phys. Rev. D. 80103503T. Clifton and P.G. Ferreira, Phys. Rev. D 80, 103503 (2009)\n\n. T Clifton, arXiv:1005.0788T. Clifton, arXiv:1005.0788 (2010)\n"},"corpus_id":{"kind":"number","value":118806981,"string":"118,806,981"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1006.3809\",\n \"mag\": 2111678061,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevd.82.043523\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1006.3809v3.pdf\"\n ],\n \"pdf_sha\": \"ca3700ff9ebb40037d969eda92c70deaa43d8804\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1006.3809\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Cosmological fluctuation growth in bimetric MOND"},"authors":{"kind":"list like","value":["Mordehai Milgrom \nDepartment of Particle Physics and Astrophysics\nWeizmann Institute\n76100RehovotIsrael"],"string":"[\n \"Mordehai Milgrom \\nDepartment of Particle Physics and Astrophysics\\nWeizmann Institute\\n76100RehovotIsrael\"\n]","isConversational":false},"abstract":{"kind":"string","value":"I begin to look at the growth of weak density inhomogeneities of nonrelativistic matter, in bimetric-MOND (BIMOND) cosmology. Far from making an exhaustive study, I concentrate on one attractive cosmological scenario, which employs matter-twin-matter-symmetric versions of BIMOND, and, furthermore, assumes that, on average, the universe is symmetrically populated in the two sectors. MOND effects are totally absent in an exactly symmetric universe, apart from the significant possible appearance of a cosmological constant, Λ ∼ (a0/c) 2 . MOND effects-local and cosmological-do enter when density inhomogeneities that differ in the two sectors appear and develop. MOND later takes its standard form in systems that are islands dominated by pure matter, as are presumably the well scrutinized systems such as galaxies. I derive the nonrelativistic (weak-field-slow-motion) equations governing small-scale fluctuation growth. The equations split into two uncoupled systems, one for the sum, the other for the difference, of the fluctuations in the two sectors. The former is governed strictly by Newtonian dynamics, and describes standard growth of fluctuations. The latter is governed by MOND dynamics, which entails stronger gravity, and nonlinearity even for the smallest of perturbations. These cause the difference to grow faster than the sum, leading to anticorrelated perturbations, conducing to matter-twin-matter segregation (which continues for high overdensities). The nonlinearity also causes interaction between nested perturbations on different scales. Because matter and twin matter (TM) repel each other in the MOND regime, matter inhomogeneities grow not only by their own self gravity, but also through shepherding by flanking TM overdensities (and vice versa). The relative importance of gravity and pressure in the MOND system (analog of the Jeans criterion), depends also on the strength of the perturbation. MOND gravity, which scales as the square root of the difference perturbation, holds sway over pressure for any mass, for weak enough perturbations. The development of structure in the universe, in either sector, thus depends crucially on two initial fluctuation spectra: that of matter alone and that of the matter-TM difference. I also discuss the back reaction on cosmology of BIMOND effects that appear as \"phantom matter\" (interpreted by some as \"dark matter\"), resulting from inhomogeneity differences between the two sectors.PACS numbers: 95.35.+d"},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"A full fledged treatment of cosmology and of structure formation in the MOND paradigm is still wanting. Structure formation on an expanding cosmological background, in nonrelativistic (NR) versions of MOND, has been studied with N -body calculations, based on various heuristic extensions of MOND theories designed to work for isolated NR systems, e.g., in [1][2][3][4][5]. Clearly, a definite treatment of the problem has to be based on a fully relativistic theory, encompassing both cosmology and structure formation. Much work along such lines has been done in the framework of TeVeS-a relativistic MOND theory propounded by Bekenstein [6] (Sanders's stratified framework [7] is a precursor, and [8] is a recent review of TeVeS and its cosmology). For example, [9][10][11][12][13][14] considered various aspects of cosmology at large, the cosmic microwave background (CMB), and linear stages of structure formation in TeVeS-like theories. Although very encouraging, these results do not yet provide a fully satisfactory description of the phenomena at hand.\nA new relativistic theory for MOND (called BIMOND, for \"bimetric MOND\") was proposed recently based on a bimetric structure of space-time [15]. In this theory, gravity is described by two metrics instead of the one underlying general relativity (GR). Ordinary matter, to whose sector belongs our direct ken, couples, as in GR, only to one metric, call it the primary one. The other metric serves as an auxiliary to which the primary metric couples. Such a bimetric description of gravity has a long history, and, in particular, has been discussed recently in connection with some \"dark substance\" phenomena (see [16][17][18][19], and references therein). The novelty in BIMOND is in the special choice of interaction between the two metrics, which is particularly germane in the context of MOND, and which reproduces MOND phenomenology.\nIn [15], I made preliminary comments on cosmological solutions within BIMOND. Classes of cosmological solutions within certain versions of BIMOND were also considered in [20]. But, there is still much left to be done on this problem.\nWe are also still lacking any treatment of the important problem of the growth of small perturbations. Ideally, one should treat the problem in a fully relativistic framework, as done in [12][13][14], in the context of TeVeS-like theories. Here I take the lighter task of considering only density inhomogeneities of NR matter on scales smaller than cosmological. This limited-scope formalism, which governs the growth of perturbations under matter dominance, is still rather illuminating, and shows a variety of phenomena potentially highly pertinent to structure formation in our universe.\nI limit myself to a universes that on average is completely symmetric in matter and twin matter (TM). Twin matter is the putative matter that couples to the \"auxiliary\" metric as (ordinary) matter couples to the \"primary\" one. At present, the existence of TM is not required phenomenologically (in particular, it is not, in fact cannot be, a sort of \"dark matter,\" as assumed to explain galaxy dynamics); it is invoked mainly for aesthetic reasons: With TM we can obtain a simple and symmetric world picture that affords favorable cosmologies. In addition, as our present analysis shows, symmetric presence of matter and TM has very interesting consequences for structure growth. It leads to phenomena that are not only greatly different from those in GR (without dark matter), but also different from those in MOND without TM.\nOf course, the existence of TM would add crucial unknowns to our cosmological world view. Its properties, composition, self-interactions, amounts, etc., are not known (see [15] for more on this). To minimize these unknowns I assume here that, with one exception, the TM sector is identical to the matter sector in all respects, having the same properties (masses, etc.), the same interactions within the sector (electromagnetic, strong, etc.), the same interaction with its metric (geodesic motion, etc.), and the same average amounts of its different components present in the universe. As shown in [15] this symmetry leads to simple cosmological solutions, identical with those of GR, albeit with a naturally appearing cosmological constant (CC) of order (a 0 /c) 2 , where a 0 is the MOND constant. 1 The only difference between the two sectors I allow for, is in the amplitude and spectrum of the primordial inhomogeneities. These are putatively ascribed to quantum fluctuations in the early Universe. They could, thus, differ in the two sectors-at least in their exact distribution, if not in their statistical properties-even with all else being the same.\nSome preliminaries of the problem discussed here are alluded to in [21], where I considered the problem of NR, small, isolated matter-TM systems. Here, this problem is extended to the evolution of perturbations on an otherwise homogeneous and matter-TM symmetric (MTMS), expanding universe.\nIn section II, I briefly describe the aspects of BIMOND and of its NR limit that we need in this paper. In section III, I derive the equations controlling the evolution of small departures from exact density equality in general NR systems. This is applied to cosmological small inhomogeneities in section IV. In section V, I describe succinctly how the density differences back react on the cosmological equations, introducing modifications that can act as cosmological \"dark matter\" and \"dark energy\" (beside the CC). Section VI is a discussion."},{"title":"II. BIMOND RECAPPED","content":"BIMOND is a class of relativistic theories of gravity that involve two metrics, g µν andĝ µν . They are governed by an action of the form\nwhere R andR are the Ricci scalars of the two metrics as appear standardly in the Einstein-Hilbert action of GR, G is Newton's constant, and I use units where c = 1. The interaction term M is a dimensionless, scalar function of the two metrics and their first derivatives. Bimetric theories have been much discussed before; the novelty and crux of BIMOND is in the choice of the interaction term. The key observation, in the MOND context, is that with two metrics and their first derivatives, we can construct tensors with the dimensions of acceleration:\nwhere Γ α βγ ,Γ α βγ are the Levi-Civita connections of the two metrics. Within the MOND paradigm, having the acceleration constant, a 0 , at our disposal, we can construct from these dimensionless scalars to serve as variables on which M depends. These scalars are obtained by contracting the dimensionless tensors a −1 0 C α βγ . 2 It befits us to concentrate on scalars that are quadratic in the C α βγ , which I do. In particular, the scalars constructed from the quadratic tensor\n1 So the successes of standard Big Bang cosmology, such as nucleosynthesis, are retained. The symmetry assumption entails, for example, having the same inflation, and the same mechanism for baryogenesis in the two sectors. 2 In the units where c = 1, a0 has dimensions of length −1 . In standard units, a0 is replaced everywhere by ℓ −1 ≡ a0/c 2 . Then in the NR limit only a0 appears. In standard units, C α βγ have dimensions of length −1 ; so the dimensionless tensors to be used are ℓC α βγ = (c 2 /a0)C α βγ .\nhave particular appeal; this tensor is symmetric under the interchange of the two metrics. The scalars I use are then\nThese are obtained from each other under metric interchange, and are used symmetrically in constructing MTMS actions for BIMOND. 3 The action terms I M andÎ M are the matter actions for standard matter and for the putative TM (with their degrees of freedom denoted by ψ i andψ i ), whose existence is suggested by the double metric nature of the theory: Just as standard matter couples only to the standard metric g µν , TM couples only to the second metriĉ g µν . There is no direct interaction between the two matter sectors. The dimensionless parameters β and α permit us to use gravitational couplings in the two sectors, G ′ = G/β andĜ ′ = G/α, that differ from each other and from G itself. In the rest of the paper I shall concentrate on symmetric versions of the theory; this means taking α = β, and that the two metrics appear in a symmetric way in the interaction term. As explained in [15,21], there are good reasons to take β ≈ 1 (see below). However, for the sake of generality I shall keep a general value of β in some of the discussion. Note that without the interaction term the theory consists of two uncoupled copies of GR. The resulting field equations are of the form\nwhere\nare the Einstein tensors, and T µν ,T µν are the energy-momentum tensors (EMTs) for the two sectors. The equations of motion of matter in the two sectors, in the presence of gravity, are the standard ones since they are still controlled by the same matter actions as in standard physics.\nThe tensors S µν andŜ µν fall from the variation of the interaction term with respect to g µν andĝ µν , respectively. 4 The metrics appear in the interaction in three ways: (i) through their derivatives, in C α βγ , (ii) contracting the C α βγ in the quadratic scalars on which M depends, (iii) otherwise, such as in the volume density (gĝ) 1/4 , or in scalars such as g/ĝ or g µνĝ µν . As a result, S µν andŜ µν have the schematic form\nwith the three term types corresponding, respectively, to the above types of appearance of the metrics. Here, {C} λ µν denotes tensors linear in C α βγ , 5 and {CC} µν denotes tensors quadratic in them; 6 they are the same in the two equations. Also, \":\" denotes covariant derivation with respect toĝ µν . Q, N, P , etc. are dimensionless, and presumably of order unity. They depend on the two metrics through the quadratic scalars, and possibly through scalars not containing the metric derivatives, such as g/ĝ or g µνĝ µν . Q and N are linear in derivatives of M with respect to the quadratic scalars, e.g., Υ andΥ defined in Eq.(4), while P gets contributions from M itself and from its derivatives with respect to scalars containing the metrics alone. This classification of the terms helps understand some interesting special cases and limits of the theory. In what follows I refer to these contributions as Q, N, P terms, respectively. 3 The special appeal in the choice of the tensor Υµν , and the two scalars constructed from it, is twofold [15]: First, it leads to a theory with a particularly simple NR limit. Second, this choice is the best equivalent to the choice of the Ricci tensor, Rµν , and the Ricci scalar, R = g µν Rµν , in the Einstein-Hilbert Lagrangian of GR. In GR we have to use the full curvature tensor, which contains the second derivatives of the metric; this is both necessary (to get a scalar), and harmless (because the second derivatives in the Einstein-Hilbert Lagrangian do not affect the field equations). In fact, it is well known that we can replace R in the GR Lagrangian by g µν (Γ γ µλ Γ λ νγ − Γ γ µν Γ λ λγ ), exactly analogous to our Υ,Υ scalars. In our case we do not add the second derivatives; this would be both harmful (as it would lead to a higher-derivative theory), and unnecessary (as the first-derivative terms are tensor in themselves). 4 They satisfy, together, one set of four Bianchi-like differential identities, stemming from the invariance of the interaction term to general coordinate transformations (details in [15]). 5 Examples of such tensors are C λ µν , δ λ µ C α αν ,ĝµνg αβ C λ αβ , etc. 6 Examples of such tensors are C γ µλ C λ νγ , C γ µν C λ λγ , Υµν , etc.\nFrom the symmetry of the problem we havê Q(g µν ,ĝ µν ) = −Q(ĝ µν , g µν ),N (g µν ,ĝ µν ) = N (ĝ µν , g µν ),P (g µν ,ĝ µν ) = P (ĝ µν , g µν ).\nBIMOND versions with β = 1 have a simple GR limit: Assume that in the limit a 0 → 0, i.e., when the quadratic scalar arguments of M go to infinity, M becomes independent of them. Then, the Q and N terms vanish. The P terms then give CC-type terms that might still couple the two metrics, but which are rather small locally. It is also possible to choose M such that the coupling disappears altogether in this limit, so the P terms in S µν collect to a 2 0 P (∞)g µν [15]. The theory then decouples into two copies of GR, possibly with a cosmological constant, if M(∞) = 0. I have not yet been able to ascertain if there is an acceptable GR limit for theories with β = 1. 7 This may be one reason to prefer theories with β = 1.\nWhen the two metrics are equal, we have from Eq.(9)Q = −Q ≡ −Q 0 ,N = N ≡ N 0 ,P = P ≡ P 0 , where, furthermore, all these functions of the two metrics become constants. This is because we cannot construct (nonconstant) scalars from a single metric and its first derivatives, so all the scalar arguments of these functions must become constants whenĝ µν = g µν . Clearly, Q terms vanish linearly in the metric difference, and terms of type N vanish quadratically. We are then left only with P terms, which are of a cosmological constant type with Λ ∝ a 2 0 P 0 . From this follows that, in systems with identical matter and TM distributions, and the same initial and boundary conditions, the metrics in the two sectors are the same, and are a solution of the standard Einstein equations with a cosmological constant (CC) Λ. 8,9 A. Cosmological background Preliminary considerations of cosmology in BIMOND have been described in [21] and in [20]. As mentioned in the Introduction, here I assume all along the special, but attractive, case of an MTMS cosmology. This means that matter sources in cosmology are assumed to be identical in the two sectors, apart from differences in the initial seed inhomogeneities. If we neglect these differences to lowest order, the MTMS cosmology in BIMOND is identical to the Friedmann-Robertson-Walker (FRW) cosmology in GR, with a CC, Λ ∝ a 2 0 P 0 , with G ′ = G/β instead of G, and with the standard matter content. All the standard results derived for the latter (Big Bang, Nucleosynthesis, etc.) apply here as well (with G ′ as gravitational constant).\nTo a higher order, the differences in the inhomogeneities in the two sectors engender departure of the background cosmology from FRW: the cosmological metric is no longer a solution of the Einstein equations. This is discussed briefly in section V. To some approximation the cosmology could still be considered an FRW one, but with nonstandard, \"phantom\" contributions to the cosmological EMT in BIMOND. This sort of feedback goes beyond the feedback due to departure from smoothness, which has been discussed extensively respecting GR cosmology. Here it is specifically due to differences between the inhomogeneities in the two sectors. As long as the statistical properties of the inhomogeneities in the two sectors are the same, the cosmological background metrics are still the same, but they satisfy the BIMOND equations, instead of the Einstein equations.\nThe observed 4 He abundance was used in [22] to derive a bound of |G ′ /G − 1| 0.13, where G ′ is the value that applies at the formation time. This would imply, in the present context, |β − 1| 0.13, lending further motivation for taking β = 1."},{"title":"B. The nonrelativistic limit","content":"Consider first the NR limit of BIMOND as applied to a system that is small on cosmological scales (discussed in detail in [15]). Since we assume a symmetric cosmology, with the two metrics equal, there is a coordinate frame in which locally the two metrics are nearly Minkowskian. Because of local inequality of matter and TM mass distributions, the departure from Minkowski is not the same for the two metrics, which can thus be written as\nwith the potentials φ,φ, h µν ,ĥ µν treated to lowest order in the weak-field approximation. 10 Note, importantly, that the weak-field approximation assumes that φ,φ ≪ 1 (in our units where c = 1). This, however, does not imply that whenever φ orφ appear they raise the order in the perturbation: Unlike the case of GR, here we have an additional dimensioned constant, a 0 , and quantities such as ∇φ/a 0 or ( ∇φ − ∇φ)/a 0 are not assumed small, and do not raise the order of terms in which they appear. In this light, C α βγ as they appear in {C} λ µν and {CC} µν are first order in the small parameters; however, in their appearance in the arguments of Q, N, P , etc. a −1 0 C α βγ should not be taken as small. 11 The N terms are second order and can be neglected. The P terms contribute to lowest order as a homogeneous CC, whose value we know is observationally negligible in local NR physics. As to the Q terms, since the {C} λ µν are already first order we can equate the two metrics everywhere else with the background Minkowski metric (so, for example, covariant derivatives become normal derivatives). But note that this approximation applies only to the metrics themselves, not to their derivatives, which appear only in a −1 0 C α βγ , which are not assumed small. It can be seen 12 that to the lowest order Q = −Q, and so S µν = −Ŝ µν . Thus, the sum and difference of the two field equations are, in this approximation,\nThe sum of the Einstein tensors is linear in the sums of the potentials, and their difference, as well as S µν , depends only on the differences of the potentials.\nIn [15] I treated the NR limit where one further assumes that motions are slow, so that time derivatives are negligible compared with spatial ones. I then showed that with the choice of the quadratic scalar arguments given in Eq. (4), which I shall assume, 13 the field equations imply, in a certain choice of gauge: h µν =ĥ µν = 0. The two metrics thus have the form of the first-order metric in GR, but the NR potentials, φ andφ, are determined not from the Poisson equation but from field equations derived from the action\nThis follows from the fact that to lowest order we have Υ =Υ ∝ ( ∇φ − ∇φ) 2 , and so the interaction term takes the form a 2 0 M[( ∇φ − ∇φ) 2 /a 2 0 ]. (I use here the same symbol, M, for a function of a single variable: the reduction of the relativistic interaction function, also denoted M, which depends on more variables.) This action also governs the particle equations of motions in the two sectors, which take the standard form: a = − ∇φ andâ = − ∇φ. Because the metric has the GR form in terms of a single potential: g µν = η µν − 2φδ µν , light bending by slowly moving masses is expressed in terms of the potential in the same way as in GR: photons \"see\" the same potential as massive particles.\nThe field equations for the potentials are\nwhere ρ p andρ p play the role of \"phantom matter\" (PM) densities for the two sectors. As detailed in [21], it is convenient to work with the two potentialsφ andφ such that\nwhich satisfy\n10 The separation into φ and hµν is a mere convenience: One defines φ ≡ −(1 + g00)/2, and h00 = 0. 11 a0 may be viewed as a proxy for a scale length ℓ ≡ c 2 /a0. Terms such as ∇φ/a0 appear, in fact, as ℓ ∇(φ/c 2 ). The NR limit in the MOND context has to be understood as c → ∞, ℓ → ∞, with c 2 /ℓ fixed. 12 Following from Eq. (9), noting that in this approximation Q is symmetric to the interchange of the metrics, as its argument is quadratic in their difference. 13 The general form of the scalar was also discussed in [15].\nwith the boundary conditions at infinityφ → 0, whileφ ∝ ln(r) when M =M , andφ → 0 when M =M ; M andM are the total masses in matter and TM respectively. Here, ζ = (2β) −1 , andμ(x) ≡ 2β − 4M ′ (4x 2 ). 14 MOND and Newtonian behaviors for isolated, pure-matter systems, such as galaxies, dictate the asymptotic behaviors\nis simply the sum of the matter and TM Newtonian potentials, andφ is a MOND potential satisfying the nonlinear Poisson equation proposed by [23] with ρ −ρ as a source (and a modified value ofμ at large arguments)."},{"title":"III. FLUCTUATIONS IN AN ISOLATED MTMS SYSTEM","content":"Consider first the development of weak fluctuations in a cosmologically small, isolated, NR system. The two matter components are treated as fluids. The time dependent problem is governed by the NR continuity equations in the two sectors\nand the respective Euler\nwhere p,p are the pressures in the two sectors.\nIn a system made predominantly of matter-as we have, purportedly, in the solar system, galaxies, etc.-these equations lead to standard MOND dynamics as they have been applied extensively in the past. In this case, the MOND acceleration is generically larger than the Newtonian one, and when the acceleration is smaller than a 0 , dynamics is controlled by the MOND limit.\nHowever, as discussed in [21], things are rather different in systems where the distribution of matter and TM are close to each other (i.e., |ρ −ρ| ≪ ρ). For example, when ρ =ρ the dynamics is always Newtonian, no matter how small the acceleration is. 15 Consider then weak perturbations on an arbitrary, symmetric background flow with\nis the Newtonian potential of ρ b : we have a double Newtonian system with an effective G e = G/β. 16 If we seed the system with fluctuations that are the same in the two sectors, they will remain so, withφ = 0 at all times, and the development remaining quasi-Newtonian in both sectors.\nTaking the lowest order in the fluctuations we have\nand the Euler equations\nwhere p b is the pressure field in the unperturbed system (assumed to be the same in the two sectors, as we assume the same equation of state for the two). The potential fluctuations are obtained from: ∆δφ = 4πG(δρ + δρ),\nIf |δρ| ∼ |δρ| ∼ |δρ − δρ|, the sources for these equations are of similar magnitudes; but this need not be so.\nIt is useful to work with the sums and differences:\nIf we can assume that in both sectors p can be taken a function of ρ; e.g., for a polytropic gas, or when the process is isentropic, we write δp = (dp/dρ)(ρ b )δρ ≡ v 2 s (ρ b )δρ and get:\nThe barred and unbarred systems decouple, with the unbarred fluctuations developing according to standard Newtonian theory, while the barred fluctuations develope according to MOND. If we start with q andq fluctuations of a similar magnitude, and the barred fluctuations are in the MOND regime,q grows faster than q. This leads in the limit to δρ ≈ −δρ, namely segregation of the two sectors. This echoes the fact that matter and TM repel each other in the MOND regime [21]."},{"title":"IV. COSMOLOGICAL FLUCTUATION GROWTH","content":"The use in the cosmological context of the above NR equations, derived for an isolated system on a double Minkowski background, can be justified for weak perturbations in NR matter, of scale length much smaller than cosmological scales: Expand the BIMOND field equations around their cosmological solution of the FRW equations with a CC and g µν = g µν . The EMTs are expanded around their value for the homogeneous background with only the 00 components contributing. For scales much smaller than the (space-time) curvature radius the background metrics can be taken as nearly Minkowski. Then, our formalism above applies.\nIn this application, ρ b (t) is the ambient, homogeneous cosmological density of the specific matter component we concentrate on, and p b is its effective pressure.\nI shall take in what follows β = 1.\nAs is standard, we express the space derivatives with respect to comoving coordinates R instead of the properdistance coordinates r = a(t)R, which appear in the above equations. We also put v b = H(t)r, ∇ · v b = 3H, where H =ȧ/a is the Hubble parameter. Also, write the equations in terms of partial time derivatives at a fixed comoving position: for the field S,Ṡ ≡ S t + (v b · ∇)S. The continuity equation for the background NR matter amounts to\nFor these we have:\nǫ + a −1 ∇ R ·ū = 0,\nThe unbarred system is identical to that in standard gravity: the perturbation sum develops fully in a Newtonian manner. Combining the time derivative of the continuity equation with the R divergence of the Euler equation we get as usualǫ\nand for the space Fourier components defined by ǫ = d 3 kǫ k (t)e ik·r , we have the standard resulẗ\nThe barred system, which is decoupled from the unbarred one, describes the evolution of the perturbation difference; it behaves as a single fluid governed by MOND, with the strength of the governing potential,φ, doubled [as expressed by the factor 2 in front of ∇φ in Eq.(32)], and with a Newtonian limit for which the effective Newton constant is G. 17 We can still eliminate the velocityū to geẗ\nbut here ∆ Rφ is not simply expressible in terms ofǭ. The problem is nonlinear even for the smallest of perturbations, and perturbations on different scales can affect each other's growth (see below). Given some initialǭ(R, t 0 ) andū(R, t 0 ), we can calculateǫ(R, t 0 ) from the continuity equation, we then calculatē φ from the potential equation, then calculateǫ from Eq. (36), and so propagate the system in time. The velocities are propagated in time in parallel, using the Euler equation."},{"title":"A. The deep-MOND regime","content":"With matter and TM present, there are two senses in which we can understand the term \"a deep-MOND system\": The weaker sense applies to systems where | ∇φ| ≪ a 0 so thatφ is determined by deep-MOND physics (with its peculiar symmetries, etc.). If we have an inhomogeneity of proper length scale λ that can be considered in isolation, i.e., is not subject to an external-field effect (EFE) from a larger-scale inhomogeneity of which it is a part (see below), this amounts to\nIn the stronger sense of the term, one requires, also, that | ∇φ| ≪ | ∇φ| so that the full potentials φ andφ are determined by deep-MOND physics. This requires further\nIn past applications of MOND, where TM is absent, the second requirement follows automatically from the first. This is also the case in the present context if |δρ − δρ| ∼ max(|δρ|, |δρ|) (for example, if the fluctuations in the two sectors are uncorrelated), but not if |δρ − δρ| ≪ max(|δρ|, |δρ|). Because here we have full decoupling of the sum and difference systems, with the latter being governed only byφ, I shall use the term in the weaker sense, which is enough to ensure that the barred system is governed by deep-MOND physics.\nIn the deep-MOND limit, Eq.(33) is well approximated by\nwhere ρ 0 = a 3 ρ b . This equation is conformally invariant [21,24]."},{"title":"B. Coupling of inhomogeneities on different scales","content":"The implications of Eq.(33) for the dynamics in massive systems have been discussed extensively (e.g., in [23,25]). One clear outcome of these studies concerns the way in which the dynamics of a larger, mother system couple to those in a smaller subsystem. When the two do not differ much in size and/or mass, the interaction between them, which results from the nonlinearity of the theory, is not easy to describe. However, when the subsystem is much smaller, and much less massive, than the mother system (such as a for star in a galaxy, or a galaxy in a galaxy cluster), the interaction is simply described: The subsystem moves within the mother system as if it where a test particle, irrespective of its own intrinsic accelerations. In other words, the dynamics of the mother system is oblivious to the internal dynamics in its small subsystems, which can be treated as test particles moving in their combined mean field. On the other hand, the dynamics within the subsystem may be greatly affected by the acceleration, g ex , with which it falls within the mother system, provided this acceleration dominates the internal ones, and is in the MOND regime (g ex a 0 ). The intrinsic dynamics is then quasi-Newtonian with an effective gravitational constant G ′ ∼ Ga 0 /g ex , and some mild anisotropy induced by the external field. This is known as the MOND external-field-effect (EFE).\nIn the present context, the mass source in the equation for the potential, ρ bǭ , must exhibit, at any time, a continuum spectrum of fluctuations, and it is difficult to describe the mutual couplings of the different scales. We expect, however, that the evolution of a fluctuation inǭ on a given scale, will be affected mainly by larger-scale perturbations within which it is nested, in the spirit of the EFE.\nA difference fluctuation of magnitudeǭ and (proper) scale length λ, would have a characteristic intrinsic acceleration, when in isolation, ∼ (a 0 Gρ bǭ λ) 1/2 . Thus, if, e.g.,ǭ ∝ λ −α , for α > 1 the typical acceleration decreases with increasing scale, and we do not expect a pronounced EFE. For α < 1, the largest scales determine the dynamics within smaller ones, as described roughly by the EFE. For α = 1 the accelerations on all scales are of the same order."},{"title":"C. Pressure vs gravity","content":"For the unbarred, fluctuation-sum system, the standard (Jeans) criterion tells us when pressure becomes unimportant in impeding gravitational growth. This applies [e.g., per Eq.(35)], when ρ b Gλ 2 ≫ v 2 s , or, equivalently, when\nis the background mass (in either sector) in a cube of side λ. For the barred system, the Jeans criterion for gravity dominance is very different. We saw that because of the nonlinearity of this system we cannot consider separately the growth of a single perturbation with a given amplitude and scale length, as all modes are, in principle, coupled. I discuss two end cases. First, consider a perturbation of magnitudeǭ and (proper) scale λ that is not affected by larger-scale overdensities (through an analog of the EFE). From Eq.(32), or Eq.(36), the Jeans criterion is still | ∇φ| ≫ v 2 s | ∇ǭ|. From Eq.(33) we now have in the MOND limit | ∇φ| 2 /λa 0 ∼ Gρ b |ǭ|. Put together, these give the condition for gravity dominance\n(where gradients are substituted by division by λ; e.g.,| ∇ǭ| ∼ |ǭ|/λ). Note, in particular, the appearance of |ǭ| in the criterion, in addition to the expected characteristic, deep-MOND M ∝ v 4 relation. This occurs because the pressure increment is linear inǭ, while the potential incrementφ is proportional to |ǭ| 1/2 . This |ǭ| factor makes the criterion much easier to satisfy for weak perturbations, 18 If the particular perturbation at hand cannot be considered in isolation, but is subject to an EFE characterized by g ex , the Jeans criterion takes the form\nAs a concrete example, consider a baryonic, difference perturbation, of magnitudeǭ and scale λ, that may be considered unaffected by an EFE. Taking the ambient temperature appropriate for the matter-dominance era T ≈ 3 × 10 3 (a/10 −3 ) −1 K, and ambient density ρ b ≈ 4 × 10 −22 (a/10 −3 ) −3 gr cm −3 , the condition (37) for the perturbation being in the deep-MOND regime can be written\nThe condition (40) for gravity dominance over pressure is\nIn comparison, the Jeans criterion relevant for the sum of the perturbations is\n(44)\nSo, there is a wide range of scales for which pressure stymies the growth of ǫ but not ofǭ."},{"title":"D. Later stages","content":"The analysis above highlights the way in which small-scale, NR density inhomogeneities grow when they are still small compared with the ambient density. We have learned much about the mechanisms that are at play and how they greatly differ from those underlying perturbation growth in standard dynamics. For example, we noted the shepherding of matter overdensities by flanking TM overdensities (and vice-versa): a matter overdensity grows faster when it sits in a TM depression. A related effect is the segregation of matter and TM, as evinced by the tendency of difference fluctuations to grow faster than the sum But, beyond these qualitative observations, a numerical attack on the problem is needed to follow the growth of even weak perturbations in detail, because of the inherent MOND nonlinearity. This is also, clearly, the case when we deal with structure buildup to its presently observed stage, where the perturbations are not small. As mentioned in the introduction there have been several N -body studies of structure formation in MOND, and these could be easily adapted to include TM.\nWithout numerical studies, it is difficult to tell exactly how, and to what extent, these trends continue to shape structure in the later stages. Some questions that naturally arise are: \"Up to what scales are matter and TM effectively segregated today?\" In [21], I conjectured that this happens on the supercluster-void scales; namely, that matter and TM have formed interlacing cosmic webs in which TM concentrations sit in the observed matter voids, and vice-versa, and matter and TM filaments avoid each other. Another important question is: \"How rare are TM objects, such as galaxies, in matter territory, and what observable effects can they have?\" If the emptiness of matter voids is an indication, we do not expect TM galaxies within large matter concentrations. Other questions are: \"Does TM help empty matter voids more efficiently than in the CDM paradigm?\" (For possible indications for unexpectedly large and empty voids see [27][28][29][30][31][32][33].) \"Are higher very-large-scale velocities produced in the present picture than in the framework of CDM?\" (See [34][35][36] for possible indications of such unexpectedly high velocities.) \"Are objects with high peculiar velocities more common in our world picture than in the CDM paradigm?\" (See [37] for a discussion of the issue in connection with the bullet cluster, and [38] for a recent assessment, and for references to earlier work.)\nWe can get some guidance in pondering over these questions by considering the interaction between well separated matter and TM bodies. This question has been considered in detail in [21], which deals, in a sense, with the opposite of the situation we treat here of small departures from well mixed configurations.\nThe basic result found there is that matter and TM bodies repel each other in the MOND regime, and (for β = 1) they do not interact in the high-acceleration regime. For example, using the conformal invariance of the deep-MOND limit of the theory, I calculated the force between two masses M 1 and M 2 in the deep-MOND regime to be (defined as negative for attraction)\nwhere the plus sign applies when the two masses are in the same sector (in which case F is always attractive), and the minus sign when they are different (in which case the force is always repulsive); r is the separation. We see that for equal masses the repulsive force between matter and TM masses M is (4/3)(a 0 G) 1/2 M 3/2 /r, which is ≈ 2.4 times larger than the attractive force between two masses M in the same sector. This can give us an idea of how far from each other separated bodies of matter and TM can separate over the Hubble time. 19 The characteristic separation speed for matter and TM masses M is ∼ 2.3(a 0 GM ) 1/4 ∼ 260(M/10 10 M ⊙ ) 1/4 km s −1 . In a Hubble time his produces a separation of several megaparsecs for galactic mass objects and a few tens of megaparsecs for galaxy-cluster mass objects. A test mass in one sector in the (MOND) field of a mass M of the other sector has typical separation speeds about half of that given above: a massive TM cluster can shoot out a matter bullet with a speed of several thousand km s −1 . 20 Another potentially useful result for a late-stage system made of separate matter and TM bodies is the general, 'virial' relation that holds exactly in the deep-MOND regime: For a system made of matter and TM pointlike masses m i andm j , respectively, at positions r k (k running over all masses), we have\nwhere F k is the force on particle k. (The choice of origin is immaterial as F k = 0.) For a system with equal total masses, the first term drops, which gives an expression that is always positive; this means that on average the forces are pointing outward, bespeaking a mean repulsion. 21 For example, in a system made of N equal matter masses, m, and N equal TM masses, m, we get\nThus, for the accelerations we have\nThe acceleration thus points outward on average, implying global self repulsion. The magnitude of the typical acceleration is ∼ (a 0 Gm) 1/2 /R, where R is the characteristic system size. This is smaller by a factor ∼ N −1/2 than the typical MOND acceleration in a pure-matter system of the same mass. Note, however, that this mean acceleration is only relevant if the matter and TM masses are very well mixed. If, in fact, they form separate matter and TM clumps within the system, as they would tend to do, these clumps would separate with higher accelerations."},{"title":"V. PHANTOM MATTER AND ITS BACK REACTION ON COSMOLOGY","content":"There has been much discussion recently regarding back reaction of inhomogeneities on cosmological evolution in GR. The issue concerns the extent to which we err by applying FRW theory (which assumes homogeneous energy density) to the mean cosmological energy density, instead of calculating cosmological evolution correctly with inhomogeneously distributed energy, and then averaging the outcome. (For recent discussion of this issue, with references to earlier work, see, e.g., [39,40].) The exact importance of this effect is still moot.\nObviously, such effects exist in BIMOND as well, but they are not the subject here. In a MTMS universe governed by BIMOND there are added back-reaction effects of inhomogeneities, which, as we saw, induce the appearance of phantom matter components that should have influence on cosmology.\nIn a universe with different matter and TM inhomogeneities, the S µν andŜ µν terms in the BIMOND field equations (5)(6) do not vanish; so, what is their effect on the cosmological background? Strictly speaking, we then have to solve the full BIMOND cosmological equations. However, to some approximation, we may treat the resulting departure from GR perturbatively, by viewing the effects of these terms as perturbations on the MTMS, FRW cosmology: In this approximation, the S µν andŜ µν terms are viewed as the EMTs of MOND phantom matter that can effectively be added to T µν andT µν in sourcing cosmic gravity. We saw that to zeroth order in the metric difference, all that remains from these terms is a CC term, which has to be reckoned with in zeroth-order cosmology. This cosmology is then used to calculate the development of perturbations in the two sectors, starting from some seed perturbations. The next stage in such an approximation scheme would be to use the deduced local departures of the two metrics from the FRW metric, used in the lower order approximation, to calculate local contributions to S µν andŜ µν , and put them back as sources in the equations for cosmology. This program clearly deserves a thorough treatment, which, however, is beyond our scope here. In particular, the results of such an analysis is expected to depend on the details of the specific BIMOND version at hand, and there is a large variety of versions to explore. Here I only describe briefly some general properties of the different term types defined in Eq.(8), as they would be fed back into the cosmological equations.\nThe zeroth order of the P term (in the matter sector), a 2 0 P 0 g µν gives a genuine CC term [if P 0 = 0]. When matter and TM separate, P becomes space-time dependent, and in this regard behaves as a \"dark energy\" field. In the weakfield approximation, we can substitute in the variables on which P depends g µν ≈ĝ µν , but Υ ≈Υ ∝ ( ∇φ − ∇φ) 2 . The contribution of the P term is then\nFrom Eq.(9) we see thatP = P in this approximation. The value of P in this expression varies in space-time between P 0 and P (∞). (As explained in [21], considerations of the GR limit of the theory, for a 0 → 0, dictate that P (∞) is not infinite.) Today we have in much of the volume of the universe z 10 −3 ; so in much of the volume P ≈ P 0 , but in galaxies or in galaxy clusters (within about a megaparsec of the center), z 1, and near stars P ≈ P (∞). Since the values of P around matter concentrations are the same as its values around similar TM concentrations (interchanging matter and TM amounts to interchanging φ andφ to which P is invariant), the large-scale average of P gives a time dependent effective CC, Λ(t) ∝ P space = P 0 . The next order contribution, which appears with the separation of matter from TM, comes from the lowest order in the Q terms, whose contribution to the cosmological density \"phantom\" EMT is\ngiving in the weak-field approximation\nIn the weak-field-slow-motions approximation, where time derivatives are neglected, only the 00 component contributes and gives, simply, the totality of all the PM gotten from Eq.(13), namely a NR PM density\nIn a MTMS universe this density averages to zero over large cosmological volumes. This can be seen in two (independent) ways. First, it follows from the fact that under the interchange of matter and TM, T µν ↔T µν , we have g µν ↔ĝ µν . In the NR limit this means that ρ ↔ρ leads to φ ↔φ, which, in turns gives ρ p → −ρ p : The phantom density around a matter concentration is equal and opposite in sign to that around a similar TM concentration. In our world picture the large-scale averages of the universe are invariant to matter-TM interchange, hence the average phantom density, which changes sign, must vanish. This result holds also for a finite system with matter-TM symmetry. For example, in a system where matter-TM interchange can be achieved by a volume-preserving transformation (such as translation, reflection, rotation) the total phantom mass in a symmetric volume vanishes. We can also argue differently, noting that the phantom density is a divergence of a vector ( ∇φ − ∇φ)M ′ (z), whose absolute value is bounded (outside of point masses). As a result, its volume average vanishes in the limit of a large volume. This property of the phantom density follows from the form of the relativistic Q term contribution: We see from Eq.(50) that the trace of the phantom EMT is a covariant divergence of a vector V λ ∝ Qg µν {C} λ µν . Its integral over a space-time volume is hence a surface integral.\nEven if the large-scale average of the NR phantom density averages to zero it could still affect cosmology due to its inhomogeneity. In addition, there may be higher order contributions that do not average to zero.\nThe Q terms may also have contributions of the form (M ′ g αβ C λ αβ ) ;λ g µν , or (M ′ g αβ C λ αλ ) ;β g µν , which are of the form V λ ;λ g µν ; i.e., they contribute in cosmology as dark energy with oscillating, vanishing-average density. The PM around matter concentrations can be (and is) observed, and is interpreted by most as \"dark matter.\" As discussed in [21], the PM around TM concentrations may be observed with weak lensing of matter photons (it does not produce strong lensing), on which it acts as a repelling mass.\nFor the N terms, the contribution of lowest order in the departure from metric equality is the second order. In the NR limit their characteristic contribution to the phantom EMT is of the order of (8πG) −1 zN (z)(a 0 /c) 2 [z defined in Eq.(49)]. Using the well known proximity a 0 ≈ cH 0 /2π, we can write this contribution as ∼ 10 −2 zN (z)ρ c , where ρ c is the critical density today. We saw above that in most of the cosmological volume today z 10 −3 . Also we have that N (0) is of order unity (from considerations of the MOND limit, see [15,21]), and N (∞) = 0 (from considerations of the GR limit of BIMOND). N is related to the extrapolating function of MOND and is constrained to remain of order unity, or below, for all arguments. So the contribution of the N terms to the density can be at most of order ρ c (in systems where | ∇φ| ∼ a 0 ), but some orders of magnitude smaller than ρ c in most of the cosmic volume. It would be interesting to see what their tensorial form is, and to what extent they produce observable lensing effects.\nNote, finally, that inasmuch as the statistical properties of matter in the two sectors are the same, so are the feedback effects discussed above. 22 Namely, the effects of inhomogeneities, and the statistical properties of the PM (their cosmological mean densities, power spectra, etc.) are the same in the two sectors. The two mean background metrics thus remain the same."},{"title":"VI. DISCUSSION","content":"I have considered the dynamics of small, NR perturbations in an MTMS universe. An intriguing aspect of this world picture is that MOND dynamics, cosmological or local, are completely absent in a perfectly symmetric universe, apart from a possible appearance of a CC of order (a 0 /c) 2 . All other departure from standard physics encapsulated in MOND-as it appears in cosmology, in structure formation, and in bound systems, such as the solar system or galactic systems-is entirely an effect of \"separation\" between matter and TM, or separation between the two metrics. In a double-membrane picture of BIMOND, MOND effects are due to local separation of the membranes.\nTwin matter plays two roles in our narrative: In the first, it provide a counteracting medium to matter; similar to the role of a neutralizing positive ion background to an electron gas. As a result of this presence, we could assume the very appealing symmetric cosmology, identical to that of GR with a CC. In addition, the presence of matter and TM in equal average quantities leads to a well-defined MOND dynamics, in which the MOND potential is sourced only by departures from uniformity, not by the bulk. To fulfill this role, TM only has to be present in the same amount as matter; it does not necessarily have to clump.\nIn its second role, TM plays an equivalent part to that of matter; so it can be likened to a positron component in a globally neutral electron-positron plasma (with mutual repulsion, and no annihilation). In this role it also takes part in shaping the large-scale structure of matter.\nThere are several qualitative features that make the growth of fluctuations in the proposed world picture different from the standard one. Among them: (i) Two initial fluctuation spectra, not one, determine the large-scale structure at later times. For the NR problem discussed here, we can think of these as the fluctuations in matter alone, and of those in the matter-TM density difference. This is important to appreciate, for example, because we have strong constraints on matter fluctuations at the time of matter-radiation decoupling, from the CMB anisotropies; but, we have none on the difference fluctuationsǭ, at that time. So, numerical simulations have to explore all possible options. (ii) The NR fluctuations in the sum and difference of the densities form two decoupled systems controlled by different types of gravitational physics. The sum system is underlaid by standard Newtonian gravity, while the difference system is underlaid by MOND gravity, which is stronger for the same source strength. (iii) In the difference system, the competition between gravity and pressure is strongly tilted in favor of gravity, with the Jeans criterion being much more lenient than in the sum system. (iv) As a result of (ii) and (iii), the difference fluctuations tend to grow faster than the sum fluctuation, leading towards a limit where |ǭ| ≫ |ǫ|, or δρ ≈ −δρ. This implies tendency towards anticorrelation of matter and TM overdensities, or to segregation of the two sectors. This applies not only when the perturbations are weak: the fact that distinct matter and TM bodies repel each other for low accelerations implies that the process of segregation continues during the stages where the overdensities are high. (v) There is an added enhancement, due to the fact that once the fluctuations in the two sectors are anticorrelated, with maxima of each residing in minima of the other, each overdensity grows not only by the pull of its own gravity, but also by shepherding by the repulsive action of the flanking overdensities in the other sector. (vi) There can be strong coupling between the perturbations on different scales due to the nonlinearity inherent in MOND.\nThere is much that remains to be studied, even within the narrow paradigm of the MTMS universe. An even richer prospect opens if one considers more general cases such as matter-TM asymmetric BIMOND versions, or asymmetric universes in which the two sectors are not equally populated, or do not have exactly the same properties.\nFor example, we do not know why a universe that presumably started with a vanishing baryon-number density has ended up having more baryons than antibaryons, with the difference resulting in the present-day baryon density. This baryogenesis, as it is called, which requires a breakdown of symmetries (CP and baryon-number conservation), is thought to have occurred during some early phase of thermal disequilibrium. If the corresponding conditions in the two sectors where slightly different, we could have ended up with different baryon densities in the two sectors. It would be interesting to study the effects of such departure from MTMS, to lowest order in the density difference, to see what effects it has on both the cosmological evolution, and on structure growth."}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"A full fledged treatment of cosmology and of structure formation in the MOND paradigm is still wanting. Structure formation on an expanding cosmological background, in nonrelativistic (NR) versions of MOND, has been studied with N -body calculations, based on various heuristic extensions of MOND theories designed to work for isolated NR systems, e.g., in [1][2][3][4][5]. Clearly, a definite treatment of the problem has to be based on a fully relativistic theory, encompassing both cosmology and structure formation. Much work along such lines has been done in the framework of TeVeS-a relativistic MOND theory propounded by Bekenstein [6] (Sanders's stratified framework [7] is a precursor, and [8] is a recent review of TeVeS and its cosmology). For example, [9][10][11][12][13][14] considered various aspects of cosmology at large, the cosmic microwave background (CMB), and linear stages of structure formation in TeVeS-like theories. Although very encouraging, these results do not yet provide a fully satisfactory description of the phenomena at hand.\\nA new relativistic theory for MOND (called BIMOND, for \\\"bimetric MOND\\\") was proposed recently based on a bimetric structure of space-time [15]. In this theory, gravity is described by two metrics instead of the one underlying general relativity (GR). Ordinary matter, to whose sector belongs our direct ken, couples, as in GR, only to one metric, call it the primary one. The other metric serves as an auxiliary to which the primary metric couples. Such a bimetric description of gravity has a long history, and, in particular, has been discussed recently in connection with some \\\"dark substance\\\" phenomena (see [16][17][18][19], and references therein). The novelty in BIMOND is in the special choice of interaction between the two metrics, which is particularly germane in the context of MOND, and which reproduces MOND phenomenology.\\nIn [15], I made preliminary comments on cosmological solutions within BIMOND. Classes of cosmological solutions within certain versions of BIMOND were also considered in [20]. But, there is still much left to be done on this problem.\\nWe are also still lacking any treatment of the important problem of the growth of small perturbations. Ideally, one should treat the problem in a fully relativistic framework, as done in [12][13][14], in the context of TeVeS-like theories. Here I take the lighter task of considering only density inhomogeneities of NR matter on scales smaller than cosmological. This limited-scope formalism, which governs the growth of perturbations under matter dominance, is still rather illuminating, and shows a variety of phenomena potentially highly pertinent to structure formation in our universe.\\nI limit myself to a universes that on average is completely symmetric in matter and twin matter (TM). Twin matter is the putative matter that couples to the \\\"auxiliary\\\" metric as (ordinary) matter couples to the \\\"primary\\\" one. At present, the existence of TM is not required phenomenologically (in particular, it is not, in fact cannot be, a sort of \\\"dark matter,\\\" as assumed to explain galaxy dynamics); it is invoked mainly for aesthetic reasons: With TM we can obtain a simple and symmetric world picture that affords favorable cosmologies. In addition, as our present analysis shows, symmetric presence of matter and TM has very interesting consequences for structure growth. It leads to phenomena that are not only greatly different from those in GR (without dark matter), but also different from those in MOND without TM.\\nOf course, the existence of TM would add crucial unknowns to our cosmological world view. Its properties, composition, self-interactions, amounts, etc., are not known (see [15] for more on this). To minimize these unknowns I assume here that, with one exception, the TM sector is identical to the matter sector in all respects, having the same properties (masses, etc.), the same interactions within the sector (electromagnetic, strong, etc.), the same interaction with its metric (geodesic motion, etc.), and the same average amounts of its different components present in the universe. As shown in [15] this symmetry leads to simple cosmological solutions, identical with those of GR, albeit with a naturally appearing cosmological constant (CC) of order (a 0 /c) 2 , where a 0 is the MOND constant. 1 The only difference between the two sectors I allow for, is in the amplitude and spectrum of the primordial inhomogeneities. These are putatively ascribed to quantum fluctuations in the early Universe. They could, thus, differ in the two sectors-at least in their exact distribution, if not in their statistical properties-even with all else being the same.\\nSome preliminaries of the problem discussed here are alluded to in [21], where I considered the problem of NR, small, isolated matter-TM systems. Here, this problem is extended to the evolution of perturbations on an otherwise homogeneous and matter-TM symmetric (MTMS), expanding universe.\\nIn section II, I briefly describe the aspects of BIMOND and of its NR limit that we need in this paper. In section III, I derive the equations controlling the evolution of small departures from exact density equality in general NR systems. This is applied to cosmological small inhomogeneities in section IV. In section V, I describe succinctly how the density differences back react on the cosmological equations, introducing modifications that can act as cosmological \\\"dark matter\\\" and \\\"dark energy\\\" (beside the CC). Section VI is a discussion.\"\n },\n {\n \"title\": \"II. BIMOND RECAPPED\",\n \"content\": \"BIMOND is a class of relativistic theories of gravity that involve two metrics, g µν andĝ µν . They are governed by an action of the form\\nwhere R andR are the Ricci scalars of the two metrics as appear standardly in the Einstein-Hilbert action of GR, G is Newton's constant, and I use units where c = 1. The interaction term M is a dimensionless, scalar function of the two metrics and their first derivatives. Bimetric theories have been much discussed before; the novelty and crux of BIMOND is in the choice of the interaction term. The key observation, in the MOND context, is that with two metrics and their first derivatives, we can construct tensors with the dimensions of acceleration:\\nwhere Γ α βγ ,Γ α βγ are the Levi-Civita connections of the two metrics. Within the MOND paradigm, having the acceleration constant, a 0 , at our disposal, we can construct from these dimensionless scalars to serve as variables on which M depends. These scalars are obtained by contracting the dimensionless tensors a −1 0 C α βγ . 2 It befits us to concentrate on scalars that are quadratic in the C α βγ , which I do. In particular, the scalars constructed from the quadratic tensor\\n1 So the successes of standard Big Bang cosmology, such as nucleosynthesis, are retained. The symmetry assumption entails, for example, having the same inflation, and the same mechanism for baryogenesis in the two sectors. 2 In the units where c = 1, a0 has dimensions of length −1 . In standard units, a0 is replaced everywhere by ℓ −1 ≡ a0/c 2 . Then in the NR limit only a0 appears. In standard units, C α βγ have dimensions of length −1 ; so the dimensionless tensors to be used are ℓC α βγ = (c 2 /a0)C α βγ .\\nhave particular appeal; this tensor is symmetric under the interchange of the two metrics. The scalars I use are then\\nThese are obtained from each other under metric interchange, and are used symmetrically in constructing MTMS actions for BIMOND. 3 The action terms I M andÎ M are the matter actions for standard matter and for the putative TM (with their degrees of freedom denoted by ψ i andψ i ), whose existence is suggested by the double metric nature of the theory: Just as standard matter couples only to the standard metric g µν , TM couples only to the second metriĉ g µν . There is no direct interaction between the two matter sectors. The dimensionless parameters β and α permit us to use gravitational couplings in the two sectors, G ′ = G/β andĜ ′ = G/α, that differ from each other and from G itself. In the rest of the paper I shall concentrate on symmetric versions of the theory; this means taking α = β, and that the two metrics appear in a symmetric way in the interaction term. As explained in [15,21], there are good reasons to take β ≈ 1 (see below). However, for the sake of generality I shall keep a general value of β in some of the discussion. Note that without the interaction term the theory consists of two uncoupled copies of GR. The resulting field equations are of the form\\nwhere\\nare the Einstein tensors, and T µν ,T µν are the energy-momentum tensors (EMTs) for the two sectors. The equations of motion of matter in the two sectors, in the presence of gravity, are the standard ones since they are still controlled by the same matter actions as in standard physics.\\nThe tensors S µν andŜ µν fall from the variation of the interaction term with respect to g µν andĝ µν , respectively. 4 The metrics appear in the interaction in three ways: (i) through their derivatives, in C α βγ , (ii) contracting the C α βγ in the quadratic scalars on which M depends, (iii) otherwise, such as in the volume density (gĝ) 1/4 , or in scalars such as g/ĝ or g µνĝ µν . As a result, S µν andŜ µν have the schematic form\\nwith the three term types corresponding, respectively, to the above types of appearance of the metrics. Here, {C} λ µν denotes tensors linear in C α βγ , 5 and {CC} µν denotes tensors quadratic in them; 6 they are the same in the two equations. Also, \\\":\\\" denotes covariant derivation with respect toĝ µν . Q, N, P , etc. are dimensionless, and presumably of order unity. They depend on the two metrics through the quadratic scalars, and possibly through scalars not containing the metric derivatives, such as g/ĝ or g µνĝ µν . Q and N are linear in derivatives of M with respect to the quadratic scalars, e.g., Υ andΥ defined in Eq.(4), while P gets contributions from M itself and from its derivatives with respect to scalars containing the metrics alone. This classification of the terms helps understand some interesting special cases and limits of the theory. In what follows I refer to these contributions as Q, N, P terms, respectively. 3 The special appeal in the choice of the tensor Υµν , and the two scalars constructed from it, is twofold [15]: First, it leads to a theory with a particularly simple NR limit. Second, this choice is the best equivalent to the choice of the Ricci tensor, Rµν , and the Ricci scalar, R = g µν Rµν , in the Einstein-Hilbert Lagrangian of GR. In GR we have to use the full curvature tensor, which contains the second derivatives of the metric; this is both necessary (to get a scalar), and harmless (because the second derivatives in the Einstein-Hilbert Lagrangian do not affect the field equations). In fact, it is well known that we can replace R in the GR Lagrangian by g µν (Γ γ µλ Γ λ νγ − Γ γ µν Γ λ λγ ), exactly analogous to our Υ,Υ scalars. In our case we do not add the second derivatives; this would be both harmful (as it would lead to a higher-derivative theory), and unnecessary (as the first-derivative terms are tensor in themselves). 4 They satisfy, together, one set of four Bianchi-like differential identities, stemming from the invariance of the interaction term to general coordinate transformations (details in [15]). 5 Examples of such tensors are C λ µν , δ λ µ C α αν ,ĝµνg αβ C λ αβ , etc. 6 Examples of such tensors are C γ µλ C λ νγ , C γ µν C λ λγ , Υµν , etc.\\nFrom the symmetry of the problem we havê Q(g µν ,ĝ µν ) = −Q(ĝ µν , g µν ),N (g µν ,ĝ µν ) = N (ĝ µν , g µν ),P (g µν ,ĝ µν ) = P (ĝ µν , g µν ).\\nBIMOND versions with β = 1 have a simple GR limit: Assume that in the limit a 0 → 0, i.e., when the quadratic scalar arguments of M go to infinity, M becomes independent of them. Then, the Q and N terms vanish. The P terms then give CC-type terms that might still couple the two metrics, but which are rather small locally. It is also possible to choose M such that the coupling disappears altogether in this limit, so the P terms in S µν collect to a 2 0 P (∞)g µν [15]. The theory then decouples into two copies of GR, possibly with a cosmological constant, if M(∞) = 0. I have not yet been able to ascertain if there is an acceptable GR limit for theories with β = 1. 7 This may be one reason to prefer theories with β = 1.\\nWhen the two metrics are equal, we have from Eq.(9)Q = −Q ≡ −Q 0 ,N = N ≡ N 0 ,P = P ≡ P 0 , where, furthermore, all these functions of the two metrics become constants. This is because we cannot construct (nonconstant) scalars from a single metric and its first derivatives, so all the scalar arguments of these functions must become constants whenĝ µν = g µν . Clearly, Q terms vanish linearly in the metric difference, and terms of type N vanish quadratically. We are then left only with P terms, which are of a cosmological constant type with Λ ∝ a 2 0 P 0 . From this follows that, in systems with identical matter and TM distributions, and the same initial and boundary conditions, the metrics in the two sectors are the same, and are a solution of the standard Einstein equations with a cosmological constant (CC) Λ. 8,9 A. Cosmological background Preliminary considerations of cosmology in BIMOND have been described in [21] and in [20]. As mentioned in the Introduction, here I assume all along the special, but attractive, case of an MTMS cosmology. This means that matter sources in cosmology are assumed to be identical in the two sectors, apart from differences in the initial seed inhomogeneities. If we neglect these differences to lowest order, the MTMS cosmology in BIMOND is identical to the Friedmann-Robertson-Walker (FRW) cosmology in GR, with a CC, Λ ∝ a 2 0 P 0 , with G ′ = G/β instead of G, and with the standard matter content. All the standard results derived for the latter (Big Bang, Nucleosynthesis, etc.) apply here as well (with G ′ as gravitational constant).\\nTo a higher order, the differences in the inhomogeneities in the two sectors engender departure of the background cosmology from FRW: the cosmological metric is no longer a solution of the Einstein equations. This is discussed briefly in section V. To some approximation the cosmology could still be considered an FRW one, but with nonstandard, \\\"phantom\\\" contributions to the cosmological EMT in BIMOND. This sort of feedback goes beyond the feedback due to departure from smoothness, which has been discussed extensively respecting GR cosmology. Here it is specifically due to differences between the inhomogeneities in the two sectors. As long as the statistical properties of the inhomogeneities in the two sectors are the same, the cosmological background metrics are still the same, but they satisfy the BIMOND equations, instead of the Einstein equations.\\nThe observed 4 He abundance was used in [22] to derive a bound of |G ′ /G − 1| 0.13, where G ′ is the value that applies at the formation time. This would imply, in the present context, |β − 1| 0.13, lending further motivation for taking β = 1.\"\n },\n {\n \"title\": \"B. The nonrelativistic limit\",\n \"content\": \"Consider first the NR limit of BIMOND as applied to a system that is small on cosmological scales (discussed in detail in [15]). Since we assume a symmetric cosmology, with the two metrics equal, there is a coordinate frame in which locally the two metrics are nearly Minkowskian. Because of local inequality of matter and TM mass distributions, the departure from Minkowski is not the same for the two metrics, which can thus be written as\\nwith the potentials φ,φ, h µν ,ĥ µν treated to lowest order in the weak-field approximation. 10 Note, importantly, that the weak-field approximation assumes that φ,φ ≪ 1 (in our units where c = 1). This, however, does not imply that whenever φ orφ appear they raise the order in the perturbation: Unlike the case of GR, here we have an additional dimensioned constant, a 0 , and quantities such as ∇φ/a 0 or ( ∇φ − ∇φ)/a 0 are not assumed small, and do not raise the order of terms in which they appear. In this light, C α βγ as they appear in {C} λ µν and {CC} µν are first order in the small parameters; however, in their appearance in the arguments of Q, N, P , etc. a −1 0 C α βγ should not be taken as small. 11 The N terms are second order and can be neglected. The P terms contribute to lowest order as a homogeneous CC, whose value we know is observationally negligible in local NR physics. As to the Q terms, since the {C} λ µν are already first order we can equate the two metrics everywhere else with the background Minkowski metric (so, for example, covariant derivatives become normal derivatives). But note that this approximation applies only to the metrics themselves, not to their derivatives, which appear only in a −1 0 C α βγ , which are not assumed small. It can be seen 12 that to the lowest order Q = −Q, and so S µν = −Ŝ µν . Thus, the sum and difference of the two field equations are, in this approximation,\\nThe sum of the Einstein tensors is linear in the sums of the potentials, and their difference, as well as S µν , depends only on the differences of the potentials.\\nIn [15] I treated the NR limit where one further assumes that motions are slow, so that time derivatives are negligible compared with spatial ones. I then showed that with the choice of the quadratic scalar arguments given in Eq. (4), which I shall assume, 13 the field equations imply, in a certain choice of gauge: h µν =ĥ µν = 0. The two metrics thus have the form of the first-order metric in GR, but the NR potentials, φ andφ, are determined not from the Poisson equation but from field equations derived from the action\\nThis follows from the fact that to lowest order we have Υ =Υ ∝ ( ∇φ − ∇φ) 2 , and so the interaction term takes the form a 2 0 M[( ∇φ − ∇φ) 2 /a 2 0 ]. (I use here the same symbol, M, for a function of a single variable: the reduction of the relativistic interaction function, also denoted M, which depends on more variables.) This action also governs the particle equations of motions in the two sectors, which take the standard form: a = − ∇φ andâ = − ∇φ. Because the metric has the GR form in terms of a single potential: g µν = η µν − 2φδ µν , light bending by slowly moving masses is expressed in terms of the potential in the same way as in GR: photons \\\"see\\\" the same potential as massive particles.\\nThe field equations for the potentials are\\nwhere ρ p andρ p play the role of \\\"phantom matter\\\" (PM) densities for the two sectors. As detailed in [21], it is convenient to work with the two potentialsφ andφ such that\\nwhich satisfy\\n10 The separation into φ and hµν is a mere convenience: One defines φ ≡ −(1 + g00)/2, and h00 = 0. 11 a0 may be viewed as a proxy for a scale length ℓ ≡ c 2 /a0. Terms such as ∇φ/a0 appear, in fact, as ℓ ∇(φ/c 2 ). The NR limit in the MOND context has to be understood as c → ∞, ℓ → ∞, with c 2 /ℓ fixed. 12 Following from Eq. (9), noting that in this approximation Q is symmetric to the interchange of the metrics, as its argument is quadratic in their difference. 13 The general form of the scalar was also discussed in [15].\\nwith the boundary conditions at infinityφ → 0, whileφ ∝ ln(r) when M =M , andφ → 0 when M =M ; M andM are the total masses in matter and TM respectively. Here, ζ = (2β) −1 , andμ(x) ≡ 2β − 4M ′ (4x 2 ). 14 MOND and Newtonian behaviors for isolated, pure-matter systems, such as galaxies, dictate the asymptotic behaviors\\nis simply the sum of the matter and TM Newtonian potentials, andφ is a MOND potential satisfying the nonlinear Poisson equation proposed by [23] with ρ −ρ as a source (and a modified value ofμ at large arguments).\"\n },\n {\n \"title\": \"III. FLUCTUATIONS IN AN ISOLATED MTMS SYSTEM\",\n \"content\": \"Consider first the development of weak fluctuations in a cosmologically small, isolated, NR system. The two matter components are treated as fluids. The time dependent problem is governed by the NR continuity equations in the two sectors\\nand the respective Euler\\nwhere p,p are the pressures in the two sectors.\\nIn a system made predominantly of matter-as we have, purportedly, in the solar system, galaxies, etc.-these equations lead to standard MOND dynamics as they have been applied extensively in the past. In this case, the MOND acceleration is generically larger than the Newtonian one, and when the acceleration is smaller than a 0 , dynamics is controlled by the MOND limit.\\nHowever, as discussed in [21], things are rather different in systems where the distribution of matter and TM are close to each other (i.e., |ρ −ρ| ≪ ρ). For example, when ρ =ρ the dynamics is always Newtonian, no matter how small the acceleration is. 15 Consider then weak perturbations on an arbitrary, symmetric background flow with\\nis the Newtonian potential of ρ b : we have a double Newtonian system with an effective G e = G/β. 16 If we seed the system with fluctuations that are the same in the two sectors, they will remain so, withφ = 0 at all times, and the development remaining quasi-Newtonian in both sectors.\\nTaking the lowest order in the fluctuations we have\\nand the Euler equations\\nwhere p b is the pressure field in the unperturbed system (assumed to be the same in the two sectors, as we assume the same equation of state for the two). The potential fluctuations are obtained from: ∆δφ = 4πG(δρ + δρ),\\nIf |δρ| ∼ |δρ| ∼ |δρ − δρ|, the sources for these equations are of similar magnitudes; but this need not be so.\\nIt is useful to work with the sums and differences:\\nIf we can assume that in both sectors p can be taken a function of ρ; e.g., for a polytropic gas, or when the process is isentropic, we write δp = (dp/dρ)(ρ b )δρ ≡ v 2 s (ρ b )δρ and get:\\nThe barred and unbarred systems decouple, with the unbarred fluctuations developing according to standard Newtonian theory, while the barred fluctuations develope according to MOND. If we start with q andq fluctuations of a similar magnitude, and the barred fluctuations are in the MOND regime,q grows faster than q. This leads in the limit to δρ ≈ −δρ, namely segregation of the two sectors. This echoes the fact that matter and TM repel each other in the MOND regime [21].\"\n },\n {\n \"title\": \"IV. COSMOLOGICAL FLUCTUATION GROWTH\",\n \"content\": \"The use in the cosmological context of the above NR equations, derived for an isolated system on a double Minkowski background, can be justified for weak perturbations in NR matter, of scale length much smaller than cosmological scales: Expand the BIMOND field equations around their cosmological solution of the FRW equations with a CC and g µν = g µν . The EMTs are expanded around their value for the homogeneous background with only the 00 components contributing. For scales much smaller than the (space-time) curvature radius the background metrics can be taken as nearly Minkowski. Then, our formalism above applies.\\nIn this application, ρ b (t) is the ambient, homogeneous cosmological density of the specific matter component we concentrate on, and p b is its effective pressure.\\nI shall take in what follows β = 1.\\nAs is standard, we express the space derivatives with respect to comoving coordinates R instead of the properdistance coordinates r = a(t)R, which appear in the above equations. We also put v b = H(t)r, ∇ · v b = 3H, where H =ȧ/a is the Hubble parameter. Also, write the equations in terms of partial time derivatives at a fixed comoving position: for the field S,Ṡ ≡ S t + (v b · ∇)S. The continuity equation for the background NR matter amounts to\\nFor these we have:\\nǫ + a −1 ∇ R ·ū = 0,\\nThe unbarred system is identical to that in standard gravity: the perturbation sum develops fully in a Newtonian manner. Combining the time derivative of the continuity equation with the R divergence of the Euler equation we get as usualǫ\\nand for the space Fourier components defined by ǫ = d 3 kǫ k (t)e ik·r , we have the standard resulẗ\\nThe barred system, which is decoupled from the unbarred one, describes the evolution of the perturbation difference; it behaves as a single fluid governed by MOND, with the strength of the governing potential,φ, doubled [as expressed by the factor 2 in front of ∇φ in Eq.(32)], and with a Newtonian limit for which the effective Newton constant is G. 17 We can still eliminate the velocityū to geẗ\\nbut here ∆ Rφ is not simply expressible in terms ofǭ. The problem is nonlinear even for the smallest of perturbations, and perturbations on different scales can affect each other's growth (see below). Given some initialǭ(R, t 0 ) andū(R, t 0 ), we can calculateǫ(R, t 0 ) from the continuity equation, we then calculatē φ from the potential equation, then calculateǫ from Eq. (36), and so propagate the system in time. The velocities are propagated in time in parallel, using the Euler equation.\"\n },\n {\n \"title\": \"A. The deep-MOND regime\",\n \"content\": \"With matter and TM present, there are two senses in which we can understand the term \\\"a deep-MOND system\\\": The weaker sense applies to systems where | ∇φ| ≪ a 0 so thatφ is determined by deep-MOND physics (with its peculiar symmetries, etc.). If we have an inhomogeneity of proper length scale λ that can be considered in isolation, i.e., is not subject to an external-field effect (EFE) from a larger-scale inhomogeneity of which it is a part (see below), this amounts to\\nIn the stronger sense of the term, one requires, also, that | ∇φ| ≪ | ∇φ| so that the full potentials φ andφ are determined by deep-MOND physics. This requires further\\nIn past applications of MOND, where TM is absent, the second requirement follows automatically from the first. This is also the case in the present context if |δρ − δρ| ∼ max(|δρ|, |δρ|) (for example, if the fluctuations in the two sectors are uncorrelated), but not if |δρ − δρ| ≪ max(|δρ|, |δρ|). Because here we have full decoupling of the sum and difference systems, with the latter being governed only byφ, I shall use the term in the weaker sense, which is enough to ensure that the barred system is governed by deep-MOND physics.\\nIn the deep-MOND limit, Eq.(33) is well approximated by\\nwhere ρ 0 = a 3 ρ b . This equation is conformally invariant [21,24].\"\n },\n {\n \"title\": \"B. Coupling of inhomogeneities on different scales\",\n \"content\": \"The implications of Eq.(33) for the dynamics in massive systems have been discussed extensively (e.g., in [23,25]). One clear outcome of these studies concerns the way in which the dynamics of a larger, mother system couple to those in a smaller subsystem. When the two do not differ much in size and/or mass, the interaction between them, which results from the nonlinearity of the theory, is not easy to describe. However, when the subsystem is much smaller, and much less massive, than the mother system (such as a for star in a galaxy, or a galaxy in a galaxy cluster), the interaction is simply described: The subsystem moves within the mother system as if it where a test particle, irrespective of its own intrinsic accelerations. In other words, the dynamics of the mother system is oblivious to the internal dynamics in its small subsystems, which can be treated as test particles moving in their combined mean field. On the other hand, the dynamics within the subsystem may be greatly affected by the acceleration, g ex , with which it falls within the mother system, provided this acceleration dominates the internal ones, and is in the MOND regime (g ex a 0 ). The intrinsic dynamics is then quasi-Newtonian with an effective gravitational constant G ′ ∼ Ga 0 /g ex , and some mild anisotropy induced by the external field. This is known as the MOND external-field-effect (EFE).\\nIn the present context, the mass source in the equation for the potential, ρ bǭ , must exhibit, at any time, a continuum spectrum of fluctuations, and it is difficult to describe the mutual couplings of the different scales. We expect, however, that the evolution of a fluctuation inǭ on a given scale, will be affected mainly by larger-scale perturbations within which it is nested, in the spirit of the EFE.\\nA difference fluctuation of magnitudeǭ and (proper) scale length λ, would have a characteristic intrinsic acceleration, when in isolation, ∼ (a 0 Gρ bǭ λ) 1/2 . Thus, if, e.g.,ǭ ∝ λ −α , for α > 1 the typical acceleration decreases with increasing scale, and we do not expect a pronounced EFE. For α < 1, the largest scales determine the dynamics within smaller ones, as described roughly by the EFE. For α = 1 the accelerations on all scales are of the same order.\"\n },\n {\n \"title\": \"C. Pressure vs gravity\",\n \"content\": \"For the unbarred, fluctuation-sum system, the standard (Jeans) criterion tells us when pressure becomes unimportant in impeding gravitational growth. This applies [e.g., per Eq.(35)], when ρ b Gλ 2 ≫ v 2 s , or, equivalently, when\\nis the background mass (in either sector) in a cube of side λ. For the barred system, the Jeans criterion for gravity dominance is very different. We saw that because of the nonlinearity of this system we cannot consider separately the growth of a single perturbation with a given amplitude and scale length, as all modes are, in principle, coupled. I discuss two end cases. First, consider a perturbation of magnitudeǭ and (proper) scale λ that is not affected by larger-scale overdensities (through an analog of the EFE). From Eq.(32), or Eq.(36), the Jeans criterion is still | ∇φ| ≫ v 2 s | ∇ǭ|. From Eq.(33) we now have in the MOND limit | ∇φ| 2 /λa 0 ∼ Gρ b |ǭ|. Put together, these give the condition for gravity dominance\\n(where gradients are substituted by division by λ; e.g.,| ∇ǭ| ∼ |ǭ|/λ). Note, in particular, the appearance of |ǭ| in the criterion, in addition to the expected characteristic, deep-MOND M ∝ v 4 relation. This occurs because the pressure increment is linear inǭ, while the potential incrementφ is proportional to |ǭ| 1/2 . This |ǭ| factor makes the criterion much easier to satisfy for weak perturbations, 18 If the particular perturbation at hand cannot be considered in isolation, but is subject to an EFE characterized by g ex , the Jeans criterion takes the form\\nAs a concrete example, consider a baryonic, difference perturbation, of magnitudeǭ and scale λ, that may be considered unaffected by an EFE. Taking the ambient temperature appropriate for the matter-dominance era T ≈ 3 × 10 3 (a/10 −3 ) −1 K, and ambient density ρ b ≈ 4 × 10 −22 (a/10 −3 ) −3 gr cm −3 , the condition (37) for the perturbation being in the deep-MOND regime can be written\\nThe condition (40) for gravity dominance over pressure is\\nIn comparison, the Jeans criterion relevant for the sum of the perturbations is\\n(44)\\nSo, there is a wide range of scales for which pressure stymies the growth of ǫ but not ofǭ.\"\n },\n {\n \"title\": \"D. Later stages\",\n \"content\": \"The analysis above highlights the way in which small-scale, NR density inhomogeneities grow when they are still small compared with the ambient density. We have learned much about the mechanisms that are at play and how they greatly differ from those underlying perturbation growth in standard dynamics. For example, we noted the shepherding of matter overdensities by flanking TM overdensities (and vice-versa): a matter overdensity grows faster when it sits in a TM depression. A related effect is the segregation of matter and TM, as evinced by the tendency of difference fluctuations to grow faster than the sum But, beyond these qualitative observations, a numerical attack on the problem is needed to follow the growth of even weak perturbations in detail, because of the inherent MOND nonlinearity. This is also, clearly, the case when we deal with structure buildup to its presently observed stage, where the perturbations are not small. As mentioned in the introduction there have been several N -body studies of structure formation in MOND, and these could be easily adapted to include TM.\\nWithout numerical studies, it is difficult to tell exactly how, and to what extent, these trends continue to shape structure in the later stages. Some questions that naturally arise are: \\\"Up to what scales are matter and TM effectively segregated today?\\\" In [21], I conjectured that this happens on the supercluster-void scales; namely, that matter and TM have formed interlacing cosmic webs in which TM concentrations sit in the observed matter voids, and vice-versa, and matter and TM filaments avoid each other. Another important question is: \\\"How rare are TM objects, such as galaxies, in matter territory, and what observable effects can they have?\\\" If the emptiness of matter voids is an indication, we do not expect TM galaxies within large matter concentrations. Other questions are: \\\"Does TM help empty matter voids more efficiently than in the CDM paradigm?\\\" (For possible indications for unexpectedly large and empty voids see [27][28][29][30][31][32][33].) \\\"Are higher very-large-scale velocities produced in the present picture than in the framework of CDM?\\\" (See [34][35][36] for possible indications of such unexpectedly high velocities.) \\\"Are objects with high peculiar velocities more common in our world picture than in the CDM paradigm?\\\" (See [37] for a discussion of the issue in connection with the bullet cluster, and [38] for a recent assessment, and for references to earlier work.)\\nWe can get some guidance in pondering over these questions by considering the interaction between well separated matter and TM bodies. This question has been considered in detail in [21], which deals, in a sense, with the opposite of the situation we treat here of small departures from well mixed configurations.\\nThe basic result found there is that matter and TM bodies repel each other in the MOND regime, and (for β = 1) they do not interact in the high-acceleration regime. For example, using the conformal invariance of the deep-MOND limit of the theory, I calculated the force between two masses M 1 and M 2 in the deep-MOND regime to be (defined as negative for attraction)\\nwhere the plus sign applies when the two masses are in the same sector (in which case F is always attractive), and the minus sign when they are different (in which case the force is always repulsive); r is the separation. We see that for equal masses the repulsive force between matter and TM masses M is (4/3)(a 0 G) 1/2 M 3/2 /r, which is ≈ 2.4 times larger than the attractive force between two masses M in the same sector. This can give us an idea of how far from each other separated bodies of matter and TM can separate over the Hubble time. 19 The characteristic separation speed for matter and TM masses M is ∼ 2.3(a 0 GM ) 1/4 ∼ 260(M/10 10 M ⊙ ) 1/4 km s −1 . In a Hubble time his produces a separation of several megaparsecs for galactic mass objects and a few tens of megaparsecs for galaxy-cluster mass objects. A test mass in one sector in the (MOND) field of a mass M of the other sector has typical separation speeds about half of that given above: a massive TM cluster can shoot out a matter bullet with a speed of several thousand km s −1 . 20 Another potentially useful result for a late-stage system made of separate matter and TM bodies is the general, 'virial' relation that holds exactly in the deep-MOND regime: For a system made of matter and TM pointlike masses m i andm j , respectively, at positions r k (k running over all masses), we have\\nwhere F k is the force on particle k. (The choice of origin is immaterial as F k = 0.) For a system with equal total masses, the first term drops, which gives an expression that is always positive; this means that on average the forces are pointing outward, bespeaking a mean repulsion. 21 For example, in a system made of N equal matter masses, m, and N equal TM masses, m, we get\\nThus, for the accelerations we have\\nThe acceleration thus points outward on average, implying global self repulsion. The magnitude of the typical acceleration is ∼ (a 0 Gm) 1/2 /R, where R is the characteristic system size. This is smaller by a factor ∼ N −1/2 than the typical MOND acceleration in a pure-matter system of the same mass. Note, however, that this mean acceleration is only relevant if the matter and TM masses are very well mixed. If, in fact, they form separate matter and TM clumps within the system, as they would tend to do, these clumps would separate with higher accelerations.\"\n },\n {\n \"title\": \"V. PHANTOM MATTER AND ITS BACK REACTION ON COSMOLOGY\",\n \"content\": \"There has been much discussion recently regarding back reaction of inhomogeneities on cosmological evolution in GR. The issue concerns the extent to which we err by applying FRW theory (which assumes homogeneous energy density) to the mean cosmological energy density, instead of calculating cosmological evolution correctly with inhomogeneously distributed energy, and then averaging the outcome. (For recent discussion of this issue, with references to earlier work, see, e.g., [39,40].) The exact importance of this effect is still moot.\\nObviously, such effects exist in BIMOND as well, but they are not the subject here. In a MTMS universe governed by BIMOND there are added back-reaction effects of inhomogeneities, which, as we saw, induce the appearance of phantom matter components that should have influence on cosmology.\\nIn a universe with different matter and TM inhomogeneities, the S µν andŜ µν terms in the BIMOND field equations (5)(6) do not vanish; so, what is their effect on the cosmological background? Strictly speaking, we then have to solve the full BIMOND cosmological equations. However, to some approximation, we may treat the resulting departure from GR perturbatively, by viewing the effects of these terms as perturbations on the MTMS, FRW cosmology: In this approximation, the S µν andŜ µν terms are viewed as the EMTs of MOND phantom matter that can effectively be added to T µν andT µν in sourcing cosmic gravity. We saw that to zeroth order in the metric difference, all that remains from these terms is a CC term, which has to be reckoned with in zeroth-order cosmology. This cosmology is then used to calculate the development of perturbations in the two sectors, starting from some seed perturbations. The next stage in such an approximation scheme would be to use the deduced local departures of the two metrics from the FRW metric, used in the lower order approximation, to calculate local contributions to S µν andŜ µν , and put them back as sources in the equations for cosmology. This program clearly deserves a thorough treatment, which, however, is beyond our scope here. In particular, the results of such an analysis is expected to depend on the details of the specific BIMOND version at hand, and there is a large variety of versions to explore. Here I only describe briefly some general properties of the different term types defined in Eq.(8), as they would be fed back into the cosmological equations.\\nThe zeroth order of the P term (in the matter sector), a 2 0 P 0 g µν gives a genuine CC term [if P 0 = 0]. When matter and TM separate, P becomes space-time dependent, and in this regard behaves as a \\\"dark energy\\\" field. In the weakfield approximation, we can substitute in the variables on which P depends g µν ≈ĝ µν , but Υ ≈Υ ∝ ( ∇φ − ∇φ) 2 . The contribution of the P term is then\\nFrom Eq.(9) we see thatP = P in this approximation. The value of P in this expression varies in space-time between P 0 and P (∞). (As explained in [21], considerations of the GR limit of the theory, for a 0 → 0, dictate that P (∞) is not infinite.) Today we have in much of the volume of the universe z 10 −3 ; so in much of the volume P ≈ P 0 , but in galaxies or in galaxy clusters (within about a megaparsec of the center), z 1, and near stars P ≈ P (∞). Since the values of P around matter concentrations are the same as its values around similar TM concentrations (interchanging matter and TM amounts to interchanging φ andφ to which P is invariant), the large-scale average of P gives a time dependent effective CC, Λ(t) ∝ P space = P 0 . The next order contribution, which appears with the separation of matter from TM, comes from the lowest order in the Q terms, whose contribution to the cosmological density \\\"phantom\\\" EMT is\\ngiving in the weak-field approximation\\nIn the weak-field-slow-motions approximation, where time derivatives are neglected, only the 00 component contributes and gives, simply, the totality of all the PM gotten from Eq.(13), namely a NR PM density\\nIn a MTMS universe this density averages to zero over large cosmological volumes. This can be seen in two (independent) ways. First, it follows from the fact that under the interchange of matter and TM, T µν ↔T µν , we have g µν ↔ĝ µν . In the NR limit this means that ρ ↔ρ leads to φ ↔φ, which, in turns gives ρ p → −ρ p : The phantom density around a matter concentration is equal and opposite in sign to that around a similar TM concentration. In our world picture the large-scale averages of the universe are invariant to matter-TM interchange, hence the average phantom density, which changes sign, must vanish. This result holds also for a finite system with matter-TM symmetry. For example, in a system where matter-TM interchange can be achieved by a volume-preserving transformation (such as translation, reflection, rotation) the total phantom mass in a symmetric volume vanishes. We can also argue differently, noting that the phantom density is a divergence of a vector ( ∇φ − ∇φ)M ′ (z), whose absolute value is bounded (outside of point masses). As a result, its volume average vanishes in the limit of a large volume. This property of the phantom density follows from the form of the relativistic Q term contribution: We see from Eq.(50) that the trace of the phantom EMT is a covariant divergence of a vector V λ ∝ Qg µν {C} λ µν . Its integral over a space-time volume is hence a surface integral.\\nEven if the large-scale average of the NR phantom density averages to zero it could still affect cosmology due to its inhomogeneity. In addition, there may be higher order contributions that do not average to zero.\\nThe Q terms may also have contributions of the form (M ′ g αβ C λ αβ ) ;λ g µν , or (M ′ g αβ C λ αλ ) ;β g µν , which are of the form V λ ;λ g µν ; i.e., they contribute in cosmology as dark energy with oscillating, vanishing-average density. The PM around matter concentrations can be (and is) observed, and is interpreted by most as \\\"dark matter.\\\" As discussed in [21], the PM around TM concentrations may be observed with weak lensing of matter photons (it does not produce strong lensing), on which it acts as a repelling mass.\\nFor the N terms, the contribution of lowest order in the departure from metric equality is the second order. In the NR limit their characteristic contribution to the phantom EMT is of the order of (8πG) −1 zN (z)(a 0 /c) 2 [z defined in Eq.(49)]. Using the well known proximity a 0 ≈ cH 0 /2π, we can write this contribution as ∼ 10 −2 zN (z)ρ c , where ρ c is the critical density today. We saw above that in most of the cosmological volume today z 10 −3 . Also we have that N (0) is of order unity (from considerations of the MOND limit, see [15,21]), and N (∞) = 0 (from considerations of the GR limit of BIMOND). N is related to the extrapolating function of MOND and is constrained to remain of order unity, or below, for all arguments. So the contribution of the N terms to the density can be at most of order ρ c (in systems where | ∇φ| ∼ a 0 ), but some orders of magnitude smaller than ρ c in most of the cosmic volume. It would be interesting to see what their tensorial form is, and to what extent they produce observable lensing effects.\\nNote, finally, that inasmuch as the statistical properties of matter in the two sectors are the same, so are the feedback effects discussed above. 22 Namely, the effects of inhomogeneities, and the statistical properties of the PM (their cosmological mean densities, power spectra, etc.) are the same in the two sectors. The two mean background metrics thus remain the same.\"\n },\n {\n \"title\": \"VI. DISCUSSION\",\n \"content\": \"I have considered the dynamics of small, NR perturbations in an MTMS universe. An intriguing aspect of this world picture is that MOND dynamics, cosmological or local, are completely absent in a perfectly symmetric universe, apart from a possible appearance of a CC of order (a 0 /c) 2 . All other departure from standard physics encapsulated in MOND-as it appears in cosmology, in structure formation, and in bound systems, such as the solar system or galactic systems-is entirely an effect of \\\"separation\\\" between matter and TM, or separation between the two metrics. In a double-membrane picture of BIMOND, MOND effects are due to local separation of the membranes.\\nTwin matter plays two roles in our narrative: In the first, it provide a counteracting medium to matter; similar to the role of a neutralizing positive ion background to an electron gas. As a result of this presence, we could assume the very appealing symmetric cosmology, identical to that of GR with a CC. In addition, the presence of matter and TM in equal average quantities leads to a well-defined MOND dynamics, in which the MOND potential is sourced only by departures from uniformity, not by the bulk. To fulfill this role, TM only has to be present in the same amount as matter; it does not necessarily have to clump.\\nIn its second role, TM plays an equivalent part to that of matter; so it can be likened to a positron component in a globally neutral electron-positron plasma (with mutual repulsion, and no annihilation). In this role it also takes part in shaping the large-scale structure of matter.\\nThere are several qualitative features that make the growth of fluctuations in the proposed world picture different from the standard one. Among them: (i) Two initial fluctuation spectra, not one, determine the large-scale structure at later times. For the NR problem discussed here, we can think of these as the fluctuations in matter alone, and of those in the matter-TM density difference. This is important to appreciate, for example, because we have strong constraints on matter fluctuations at the time of matter-radiation decoupling, from the CMB anisotropies; but, we have none on the difference fluctuationsǭ, at that time. So, numerical simulations have to explore all possible options. (ii) The NR fluctuations in the sum and difference of the densities form two decoupled systems controlled by different types of gravitational physics. The sum system is underlaid by standard Newtonian gravity, while the difference system is underlaid by MOND gravity, which is stronger for the same source strength. (iii) In the difference system, the competition between gravity and pressure is strongly tilted in favor of gravity, with the Jeans criterion being much more lenient than in the sum system. (iv) As a result of (ii) and (iii), the difference fluctuations tend to grow faster than the sum fluctuation, leading towards a limit where |ǭ| ≫ |ǫ|, or δρ ≈ −δρ. This implies tendency towards anticorrelation of matter and TM overdensities, or to segregation of the two sectors. This applies not only when the perturbations are weak: the fact that distinct matter and TM bodies repel each other for low accelerations implies that the process of segregation continues during the stages where the overdensities are high. (v) There is an added enhancement, due to the fact that once the fluctuations in the two sectors are anticorrelated, with maxima of each residing in minima of the other, each overdensity grows not only by the pull of its own gravity, but also by shepherding by the repulsive action of the flanking overdensities in the other sector. (vi) There can be strong coupling between the perturbations on different scales due to the nonlinearity inherent in MOND.\\nThere is much that remains to be studied, even within the narrow paradigm of the MTMS universe. An even richer prospect opens if one considers more general cases such as matter-TM asymmetric BIMOND versions, or asymmetric universes in which the two sectors are not equally populated, or do not have exactly the same properties.\\nFor example, we do not know why a universe that presumably started with a vanishing baryon-number density has ended up having more baryons than antibaryons, with the difference resulting in the present-day baryon density. This baryogenesis, as it is called, which requires a breakdown of symmetries (CP and baryon-number conservation), is thought to have occurred during some early phase of thermal disequilibrium. If the corresponding conditions in the two sectors where slightly different, we could have ended up with different baryon densities in the two sectors. It would be interesting to study the effects of such departure from MTMS, to lowest order in the density difference, to see what effects it has on both the cosmological evolution, and on structure growth.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". R H Sanders, Astrophys. J. 5601R.H. Sanders, Astrophys. J. 560, 1 (2001)","doi":""},{"text":". A Nusser, MNRAS. 331909A. Nusser, MNRAS 331, 909 (2002)","doi":""},{"text":". S Stachniewicz, M Kutschera, Acta Phys. Polon. B. 323629S. Stachniewicz and M. Kutschera, Acta Phys. Polon. B 32, 3629 (2001)","doi":""},{"text":". A Knebe, B K Gibson, MNRAS. 3471055A. Knebe and B.K. Gibson, MNRAS 347, 1055 (2004)","doi":""},{"text":". C Llinares, A Knebe, H S Zhao, MNRAS. 3911778C. Llinares, A. Knebe, and H.S. Zhao, MNRAS 391, 1778 (2008)","doi":""},{"text":". J D Bekenstein, Phys. Rev. D. 7083509J.D. Bekenstein, Phys. Rev. D 70, 083509 (2004)","doi":""},{"text":". R H Sanders, Astrophys. J. 480492R.H. Sanders, Astrophys. J. 480, 492 (1997)","doi":""},{"text":". C Skordis, Class. Quant. Grav. 26143001C. Skordis, Class. Quant. Grav. 26, 143001 (2009)","doi":""},{"text":". R H Sanders, MNRAS. 363459R.H. Sanders, MNRAS 363, 459 (2005)","doi":""},{"text":". S Dodelson, M Liguori, Phys. Rev. Lett. 97231301S. Dodelson and M. Liguori, Phys. Rev. Lett. 97, 231301 (2006)","doi":""},{"text":". C Skordis, D F Mota, P G Ferreira, C Boehm, Phys. Rev. Lett. 9611301C. Skordis, D.F. Mota, P.G. Ferreira, and C. Boehm, Phys. Rev. Lett. 96, 011301 (2006)","doi":""},{"text":". C Skordis, Phys. Rev. D. 74103513C. Skordis, Phys. Rev. D 74, 103513 (2006)","doi":""},{"text":". C Skordis, Phys. Rev. D. 77123502C. Skordis, Phys. Rev. D 77, 123502 (2008)","doi":""},{"text":". T G Zlosnik, P G Ferreira, G D Starkman, Phys. Rev. D. 7784010T.G. Zlosnik, P.G. Ferreira, and G.D. Starkman, Phys. Rev. D 77, 084010, (2008)","doi":""},{"text":". M Milgrom, Phys. Rev. D. 80123536M. Milgrom, Phys. Rev. D 80, 123536 (2009)","doi":""},{"text":". T Damour, I I Kogan, Phys. Rev. D. 66104024T. Damour and I.I. Kogan, Phys. Rev. D 66, 104024 (2002)","doi":""},{"text":". D Blas, C Deffayet, J Garriga, Class. Quant. Grav. 231697D. Blas, C. Deffayet, and J. Garriga, Class. Quant. Grav. 23, 1697 (2006)","doi":""},{"text":". D Blas, C Deffayet, J Garriga, Phys. Rev. D. 76104036D. Blas, C. Deffayet, and J. Garriga, Phys. Rev. D 76, 104036 (2007)","doi":""},{"text":". M Bañados, P G Ferreira, C Skordis, Phys. Rev. D. 7963511M. Bañados, P.G. Ferreira, and C. Skordis, Phys. Rev. D 79, 063511 (2009)","doi":""},{"text":". T Clifton, T G Zlosnik, Phys. Rev. D. 81103525T. Clifton and T.G. Zlosnik, Phys. Rev. D 81, 103525 (2010)","doi":""},{"text":". M Milgrom, MNRAS. 4051129M. Milgrom, MNRAS 405, 1129 (2010)","doi":""},{"text":". S M Carroll, E A Lim, Phys. Rev. D. 70123525S.M. Carroll and E.A. Lim, Phys. Rev. D 70, 123525 (2004)","doi":""},{"text":". J Bekenstein, M Milgrom, Astrophys. J. 286J. Bekenstein and M. Milgrom, Astrophys. J. 286, 7 (1984)","doi":""},{"text":". M Milgrom, Phys. Rev. E. 561148M. Milgrom, Phys. Rev. E 56, 1148 (1997)","doi":""},{"text":". M Milgrom, J. Phys. A. 351437M. Milgrom, J. Phys. A 35, 1437 (2002)","doi":""},{"text":". M Milgrom, Comm. Astrophys. J. 13215M. Milgrom, Comm. Astrophys. J. 13, 215 (1989)","doi":""},{"text":". P J E Peebles, Astrophys. J. 557495P.J.E. Peebles, Astrophys. J. 557, 495 (2001)","doi":""},{"text":"R B Tully, arXiv:0708.0864Proceedings IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff. IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff38R.B. Tully, Proceedings IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff, 25-29 June, Bull. Am. Astron. Soc. 38, 240, (2007), arXiv:0708.0864","doi":""},{"text":". R B Tully, E J Shaya, I D Karachentsev, H M Courtois, D D Kocevski, L Rizzi, A Peel, Astrophys. J. 676184R.B. Tully, E.J. Shaya, I.D. Karachentsev, H. M. Courtois, D.D. Kocevski, L. Rizzi, and A. Peel, Astrophys. J. 676, 184 (2008)","doi":""},{"text":". A Tikhonov, A Klypin, MNRAS. 3951915A. Tikhonov and A. Klypin, MNRAS 395, 1915 (2009)","doi":""},{"text":". A V Tikhonov, S Gottloeber, G Yepes, Y Hoffman, MNRAS. 3991611A.V. Tikhonov, S. Gottloeber, G. Yepes, and Y. Hoffman, MNRAS 399, 1611 (2009)","doi":""},{"text":"R Van De Weygaert, E Platen, arXiv:0912.2997proceedings COSPA2008, Pohang, Korea. accepted to Modern Physics Letters A. COSPA2008, Pohang, Korea. accepted to Modern Physics Letters AR. van de Weygaert and E. Platen, proceedings COSPA2008, Pohang, Korea. accepted to Modern Physics Letters A, arXiv:0912.2997","doi":""},{"text":". P J E Peebles, A Nusser, Nature. 465565P.J.E. Peebles and A. Nusser, Nature 465, 565 (2010)","doi":""},{"text":". R Watkins, H A Feldman, M J Hudson, MNRAS. 392743R. Watkins, H.A. Feldman, and M.J. Hudson, MNRAS 392, 743 (2009)","doi":""},{"text":". G Lavaux, R B Tully, R Mohayaee, S Colombi, Astrophys. J. 709483G. Lavaux, R.B. Tully, R. Mohayaee, and S. Colombi, Astrophys. J. 709, 483 (2010)","doi":""},{"text":". A Kashlinsky, F Atrio-Barandela, H Ebeling, A Edge, D Kocevski, Astrophys. J. 71281A. Kashlinsky, F. Atrio-Barandela, H. Ebeling, A. Edge, and D. Kocevski, Astrophys. J. 712, L81 (2010)","doi":""},{"text":". G W Angus, S S Mcgaugh, MNRAS. 383417G.W. Angus and S.S. McGaugh, MNRAS 383, 417 (2008)","doi":""},{"text":". J Lee, E Komatsu, Astrophys. J. 71860J. Lee and E. Komatsu, Astrophys. J. 718, 60 (2010)","doi":""},{"text":". T Clifton, P G Ferreira, Phys. Rev. D. 80103503T. Clifton and P.G. Ferreira, Phys. Rev. D 80, 103503 (2009)","doi":""},{"text":". T Clifton, arXiv:1005.0788T. Clifton, arXiv:1005.0788 (2010)","doi":""}],"string":"[\n {\n \"text\": \". R H Sanders, Astrophys. J. 5601R.H. Sanders, Astrophys. J. 560, 1 (2001)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Nusser, MNRAS. 331909A. Nusser, MNRAS 331, 909 (2002)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Stachniewicz, M Kutschera, Acta Phys. Polon. B. 323629S. Stachniewicz and M. Kutschera, Acta Phys. Polon. B 32, 3629 (2001)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Knebe, B K Gibson, MNRAS. 3471055A. Knebe and B.K. Gibson, MNRAS 347, 1055 (2004)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Llinares, A Knebe, H S Zhao, MNRAS. 3911778C. Llinares, A. Knebe, and H.S. Zhao, MNRAS 391, 1778 (2008)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J D Bekenstein, Phys. Rev. D. 7083509J.D. Bekenstein, Phys. Rev. D 70, 083509 (2004)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R H Sanders, Astrophys. J. 480492R.H. Sanders, Astrophys. J. 480, 492 (1997)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Skordis, Class. Quant. Grav. 26143001C. Skordis, Class. Quant. Grav. 26, 143001 (2009)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R H Sanders, MNRAS. 363459R.H. Sanders, MNRAS 363, 459 (2005)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Dodelson, M Liguori, Phys. Rev. Lett. 97231301S. Dodelson and M. Liguori, Phys. Rev. Lett. 97, 231301 (2006)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Skordis, D F Mota, P G Ferreira, C Boehm, Phys. Rev. Lett. 9611301C. Skordis, D.F. Mota, P.G. Ferreira, and C. Boehm, Phys. Rev. Lett. 96, 011301 (2006)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Skordis, Phys. Rev. D. 74103513C. Skordis, Phys. Rev. D 74, 103513 (2006)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Skordis, Phys. Rev. D. 77123502C. Skordis, Phys. Rev. D 77, 123502 (2008)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T G Zlosnik, P G Ferreira, G D Starkman, Phys. Rev. D. 7784010T.G. Zlosnik, P.G. Ferreira, and G.D. Starkman, Phys. Rev. D 77, 084010, (2008)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Milgrom, Phys. Rev. D. 80123536M. Milgrom, Phys. Rev. D 80, 123536 (2009)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Damour, I I Kogan, Phys. Rev. D. 66104024T. Damour and I.I. Kogan, Phys. Rev. D 66, 104024 (2002)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Blas, C Deffayet, J Garriga, Class. Quant. Grav. 231697D. Blas, C. Deffayet, and J. Garriga, Class. Quant. Grav. 23, 1697 (2006)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Blas, C Deffayet, J Garriga, Phys. Rev. D. 76104036D. Blas, C. Deffayet, and J. Garriga, Phys. Rev. D 76, 104036 (2007)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Bañados, P G Ferreira, C Skordis, Phys. Rev. D. 7963511M. Bañados, P.G. Ferreira, and C. Skordis, Phys. Rev. D 79, 063511 (2009)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Clifton, T G Zlosnik, Phys. Rev. D. 81103525T. Clifton and T.G. Zlosnik, Phys. Rev. D 81, 103525 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Milgrom, MNRAS. 4051129M. Milgrom, MNRAS 405, 1129 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S M Carroll, E A Lim, Phys. Rev. D. 70123525S.M. Carroll and E.A. Lim, Phys. Rev. D 70, 123525 (2004)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Bekenstein, M Milgrom, Astrophys. J. 286J. Bekenstein and M. Milgrom, Astrophys. J. 286, 7 (1984)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Milgrom, Phys. Rev. E. 561148M. Milgrom, Phys. Rev. E 56, 1148 (1997)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Milgrom, J. Phys. A. 351437M. Milgrom, J. Phys. A 35, 1437 (2002)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Milgrom, Comm. Astrophys. J. 13215M. Milgrom, Comm. Astrophys. J. 13, 215 (1989)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P J E Peebles, Astrophys. J. 557495P.J.E. Peebles, Astrophys. J. 557, 495 (2001)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R B Tully, arXiv:0708.0864Proceedings IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff. IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff38R.B. Tully, Proceedings IAU Symp. 244: Dark Galaxies and Lost Baryons. Cardiff, 25-29 June, Bull. Am. Astron. Soc. 38, 240, (2007), arXiv:0708.0864\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R B Tully, E J Shaya, I D Karachentsev, H M Courtois, D D Kocevski, L Rizzi, A Peel, Astrophys. J. 676184R.B. Tully, E.J. Shaya, I.D. Karachentsev, H. M. Courtois, D.D. Kocevski, L. Rizzi, and A. Peel, Astrophys. J. 676, 184 (2008)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Tikhonov, A Klypin, MNRAS. 3951915A. Tikhonov and A. Klypin, MNRAS 395, 1915 (2009)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A V Tikhonov, S Gottloeber, G Yepes, Y Hoffman, MNRAS. 3991611A.V. Tikhonov, S. Gottloeber, G. Yepes, and Y. Hoffman, MNRAS 399, 1611 (2009)\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R Van De Weygaert, E Platen, arXiv:0912.2997proceedings COSPA2008, Pohang, Korea. accepted to Modern Physics Letters A. COSPA2008, Pohang, Korea. accepted to Modern Physics Letters AR. van de Weygaert and E. Platen, proceedings COSPA2008, Pohang, Korea. accepted to Modern Physics Letters A, arXiv:0912.2997\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P J E Peebles, A Nusser, Nature. 465565P.J.E. Peebles and A. Nusser, Nature 465, 565 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Watkins, H A Feldman, M J Hudson, MNRAS. 392743R. Watkins, H.A. Feldman, and M.J. Hudson, MNRAS 392, 743 (2009)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Lavaux, R B Tully, R Mohayaee, S Colombi, Astrophys. J. 709483G. Lavaux, R.B. Tully, R. Mohayaee, and S. Colombi, Astrophys. J. 709, 483 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Kashlinsky, F Atrio-Barandela, H Ebeling, A Edge, D Kocevski, Astrophys. J. 71281A. Kashlinsky, F. Atrio-Barandela, H. Ebeling, A. Edge, and D. Kocevski, Astrophys. J. 712, L81 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G W Angus, S S Mcgaugh, MNRAS. 383417G.W. Angus and S.S. McGaugh, MNRAS 383, 417 (2008)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Lee, E Komatsu, Astrophys. J. 71860J. Lee and E. Komatsu, Astrophys. J. 718, 60 (2010)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Clifton, P G Ferreira, Phys. Rev. D. 80103503T. Clifton and P.G. Ferreira, Phys. Rev. D 80, 103503 (2009)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Clifton, arXiv:1005.0788T. Clifton, arXiv:1005.0788 (2010)\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":71,"cells":{"text":{"kind":"string","value":"\nThe Decuplet Revisited in χPT\nAugust 21, 1995\n\nM K Banerjee \nDepartment of Physics\nUniversity of Maryland College Park\n20742MarylandUSA\n\nJ Milana \nDepartment of Physics\nUniversity of Maryland College Park\n20742MarylandUSA\n\nThe Decuplet Revisited in χPT\nAugust 21, 19958E78764E55066A0E2C053A298345D07BarXiv:hep-ph/9508340v1\nThe paper deals with two issues.First, we explore the quantitiative importance of higher multiplets for properties of the ∆ decuplet in chiral perturbation theory.In particular, it is found that the lowest order one-loop contributions from the Roper octet to the decuplet masses and magnetic moments are substantial.The relevance of these results to the chiral expansion in general is discussed.The exact values of the magnetic moments depend upon delicate cancellations involving ill-determined coupling constants.Second, we present new relations between the magnetic moments of the ∆ decuplet that are independent of all couplings.They are exact at the order of the chiral expansion used in this paper.* Because of the large mass difference between the ρ and the pion, as well as the quadratic nature of the chiral expansion, this issue does not arise when considering the sector of purely pseudoscalar, goldstone bosons.The effects of the ρ and other such states are incorporated as part of the needed input coupling constants of the theory.† Arguing that for a very limited region near threshold in the SU f (2) defined theory, the ∆-nucleon mass difference can be considered \"large\" * * As a self-consistent test of the assignment of hadrons to this multiplet, note that the violation to the corresponding GMO relation for this octet is only 15MeV.\n\nI. INTRODUCTION\n\nThe success of chiral perturbation theory (χPT) [1] for understanding properties of the pseudoscalar mesons is now well established [2].The approach is based on the existence of a systematic expansion in derivatives of the pion's field and the pion's mass, whereby m π divided by some large scale, generated by the theory itself and typically ∼ 4πf π , becomes the perturbative expansion parameter.For the purely mesonic sector this expansion is in fact quadratic in the pion's mass, so that even for the SU f (3) generalization, (m K /4πf π ) 2 is still a reasonablely small parameter.\n\nThe application to the baryon sector has however from the outset been confronted with a variety of difficulties [3].For example, how to handle the nucleon's mass was a problem only relatively recently solved [4,5].One unavoidable complication when including baryons is that the chiral expansion is itself more complicated [3,4] than in the purely mesonic case, and the expansion parameter is now only linear in m π (m K ).A second pertinent complication involves the issue of resonances.* Originally it was conjectured [4] that all such resonances (and most notably the ∆) need not be included as an explicit degree of freedom, i.e. that they could be \"integrated out\".While at least one notably active group [6] have maintained this viewpoint, † most researchers have subsequently found it untenable [7].The ∆ degree of freedom was first introduced into χPT by Jenkins and Manohar in Ref. [8].Recently, the importance of the Λ * (1405) for understanding threshold kaon-nucleon scattering lengths has also been realized [10,11,12].\n\nIn this paper we discuss the role of higher-multiplets for the properties of the ∆ decuplet at the one-loop level in χPT.We consider the O(p 3 ) correction to the decuplet masses and the O(p 2 ), one-loop correction to the magnetic moments of the decuplet (the electromagnetic vertex has chiral power −1 excluding whatever power maybe assigned to electric charge).Our criterion for which multiplets to consider is that the average mass-splitting, δ h , between the multiplet and the ∆ decuplet be less than the mass of the kaon,\n‡ δ h =| M H − M 10 |< m K .(1)\nThis criterion is based on the fact that when it is met, an expansion in m K /δ h is not justified so that loop effects involving these higher-mulitplet members as intermediate states cannot be absorbed into higher order terms in the chiral lagrangian.Such loop effects place a fundamental limitation on any formulation of χPT that omits the higher-multiplet as an explicit degree of freedom, even if the chiral expansion was then executed to all orders.By the convention of Ref. [14], note that the chiral power of all δ is 1.\n\nFor the case of the nucleon octet Eq. ( 1) is clearly met,\nδ N = M 10 − M 8 = 226MeV,(2)\n(M 10 = 1377MeV and M 8 = 1151MeV) and is indeed the driving phenomenological reason for expecting that the ∆ cannot be ignored for descriptions of the nucleon.\n\nWe focus here on the properties of the ∆ decuplet as opposed to those of the nucleon octet because of the simple reason that the mass splitting δ h satisfies the criterion specified by Eq. (1) for at least two resonances.\n\nWe will show that the Roper has a nontrivial effect on both decuplet mass splittings and magnetic moments.‡ mη is obtained at this order in χPT using GMOR [13].\n\nWe should also note that these results lead to a more general statement.When adding a loop one must also add resonances whose masses are within a kaon mass of the resonances already included.\n\nThe rest of this paper is organized thusly.In Section (II) we enumerate the various multiplets considered and their interactions (and experimentally obtained couplings) with the ∆ decuplet utilizing the heavy baryon formalism of Jenkins and Manohar [5].In Section (III) we present the one-loop, O(p 3 ) contributions to the decuplet masses focussing on the violation to the Decuplet Equal Spacing rule [15], the sole quantity for which χPT makes a prediction at this order in the chiral expansion [9,14].In Section(IV) we consider the one-loop, O(p 2 ) results for the magnetic moments of the decuplet, a subject first discussed in χPT in Ref. [16] where though adherence to the order of the chiral expansion was not strictly maintained.We demonstrate later that strict adherence is crucial for renormalizability.The focus in both Sections (III) and (IV) is the quantitative importance of the higher-multiplets.In addition new relations for the magnetic moments at this order in χPT are derived that are independent of the intermediate multiplets considered.Their violation would be a clear measure of the importance of higher chiral power terms in the expansion.In Section (V) we conclude with a discussion of the consequence of these results to the loop-expansion in general in χPT.\n\n\nII. HIGHER MULTIPLETS: DEFINITIONS AND COUPLINGS\n\nA number of multiplets § satisfy the criteria Eq. (1).Fortunately most of these can be eliminated due to symmetry constraints.For example, flavor singlets such as the Λ * (1405) do not couple to a decuplet via an SU f (3) octet (the goldstone bosons).Only slightly less straightfoward, a 1/2 − octet (e.g. the N(1535) multiplet) couples only through the lower components of the baryon spinors,\nL i ∼ T µ γ 5 A µ B * ,(3)\nwhich vanishes to lowest order in the heavy baryon expansion [5].(Such states would, in principle, need to be considered in higher-order calculations.)Coupling to the 5/2 − octet likewise vanishes at lowest order.With these eliminations, we obtain that only octets or decuplets of baryons with quantum numbers 1/2 + , 3/2 + , 3/2 − and 5/2 + need be considered.§ See e.g.Table 30.4 in the Particle Data Group [17].As we are at present only concerned with the average coupling of these multiplets with the ∆ decuplet, we ignore potentially interesting questions as to the exact SU f (3) composition of any particular excited state.[18] We also omit from consideration possible exotics.\n\nThe most important 1/2 + multiplet (beyond of course the nucleon's) is the octet containing the Roper, N(1440).A slight difficulty arises in determining\nδ R = M 8 (1440) − M 10(4)\nbecause one member of the Roper multiplet, the excited Cascade, has not yet been identified.To get a reasonable approximate value for its mass, we use the corresponding GMO relation [19] M\nΞ * = 3 2 M Λ * (1600) + 1 2 M Σ * (1660) − M N * (1440),(5)\nby which one estimates that M Ξ * = 1790.The average value of the Roper multiplet is thereby\nM 8 * = 1 8 (2M Ξ * + 3M Σ * + M Λ * + 2M N * ) = 1630 Mev,(6)\nfrom which one gets that\nδ R = 253MeV. (7)\nFor the ∆N * π interaction one has\nL i = C(T µ A µ B * + h.c.)(8)\nin complete analogy to the leading ∆N π interaction in the heavy fermion limit.[8] The coupling C2 can be obtained from the observed decay of the N * (1440) → ∆π which has a partial width Γ N * →∆π ≈ 90MeV.Comparing this with the decay of the ∆, Γ ∆→N π ≈ 120MeV one obtains that\nC2 ≈ 1 2 C 2 .(9)\nIn principle, one other 1/2 + multiplet meets our criteria, the octet containing the N(1710), with a δ h ≈ 470MeV.However, best estimates [17] for the partial width for N(1710)→ ∆π is ≈ 30MeV, which implies that the relevant coupling constant is significantly suppressed compared to that in Eq. ( 9).We therefore ignore this multiplet in our subsequent calculations as it amounts to only a small correction to those of the Roper octet.\n\nFor the lowest lying 3/2 − octet, we obtain directly from the experimentally measured masses [17] that * *\nδ 8 * = 296MeV. (10)\nThe interaction with the ∆ decuplet is, to leading order in the chiral lagrangian,\nL i = C * T µ γ ν A ν T * µ + h.c. .(11)\nThe coupling C * can be determined from the observed decay of the N * (1520), Γ(N * (1520) → ∆π) ≈ 25MeV, whence one finds that\nC * C 2 ≈ 1 25 .(12)\nThis relative suppression results both from the smaller branching width as well as overall kinematic factors that otherwise tend to enhance N * (1520) → ∆π w.r.t.N * (1440) → ∆π.\n\nThere are two other 3/2 − multiplets, one octet and one decuplet, listed in the Particle Data Group that could potentially satisfy our criteria, Eq. ( 1).Each are very poorly determined, containing merely one member each, the N(1700) and the ∆(1700), respectively.In the case of the N(1700), its coupling to the ∆π is experimentally negligible and hence can be safely ignored.On the other hand, the coupling with the ∆(1700) is not so readily ignored, having a decay width Γ(∆(1700) → ∆π) ≈ 120MeV.We therefore explicitly keep this decuplet, assigning for its intermultiplet spacing with the ∆ decuplet a value\nδ 10 * = 1700 − 1232 = 468MeV. (13)\nFrom the aforementioned decay width, and an interaction of the form Eq. ( 11), we obtain that for the ∆ * ∆π coupling\nH * 2 ≈ .15.(14)\nHere we have used the convention of Ref. [9] for the SU f (3) algebra factors (whereby, for the ∆∆π coupling, H 2 ≈ 4 is typical).As in the case of the N (1520) overall kinematic factors in addition to the available phase space, yields a rather suppressed value of the coupling.Indeed, as we will soon see, due to Eqs. ( 12) and ( 14) little would have been lost had we ignored the 3/2 − multiplets altogether.Nevertheless, they have been included for completeness.We come finally to the 5/2 + states.The lowest such multiplet is the N(1680) octet.It has an intermultiplet mass splitting with the ∆ decuplet of δ h = 496MeV.We conclude that there is no 5/2 + multiplet that meets our criteria Eq. ( 1).\n\nThis then concludes our examination of the relevant multiplets.By far the most important, as we will presently see, is the Roper octet.\n\n\nIII. DECUPLET EQUAL SPACING RULE\n\nThe one-loop, O(p 3 ) results for the masses of the decuplet involving intermediate ∆ decuplet and nucleon octet states have been published previously [14].The contribution, δM 10 , from the 3/2 − multiplet, for the case m > δ 8 * , is\nδM 10 = −3βδ 8 * 16π 2 f 2 π × δ 2 8 * − 1 2 m 2 1 ǫ − γ E + ln(4π) + 2 − ln m 2 µ 2 + 2δ 8 * m 2 − δ 2 8 * π 2 − tan −1 δ 8 * m 2 − δ 2 8 *(15)\nwhere β represents SU(3) algebra factors [9,14].As discussed in Ref. [14], the counterterms neccessary to renormalize these terms are either of the form δ (for the δ 8 * m 2 divergences).As the DES rule is exact for all terms through m 2 , all counterterms (divergences) cancel at this order of the chiral expansion in the violation to the DES rule.\n\nIncluding all multiplets, the one-loop, O(p 3 ), violation to the DES rule is\n(M Σ * − M ∆ ) − (M Ξ * − M Σ * ) = (M Ξ * − M Σ * ) − (M Ω − − M Ξ * ) = 1 2 {(M Σ * − M ∆ ) − (M Ω − − M Ξ * )} = 2 9 C 2 V (−δ N ) + C2 V (δ R ) + (C * ) 2 V * (δ 8 * ) − 20 81 H 2 V (0) + H * 2 V * (δ 10 * ) . (16)\n† † V and V * are given by\nV (δ) = W (m K , δ, µ) − 3 4 W (m η , δ, µ) − 1 4 W (m π , δ, µ), V * (δ) = W * (m K , δ, µ) − 3 4 W * (m η , δ, µ) − 1 4 W * (m π , δ, µ),(17)\nwherein the function\nW (m, δ, µ) is [20,21] ‡ ‡ δ = 0, W (m, δ, µ) = 1 16πf 2 π m 3 ,(18)\nm >| δ |, W (m, δ, µ) = † † Note that unlike our convention in [14], all δ h are now strictly positive and hence the explicit sign in the function V above.\n\n‡ ‡ Note the correction from [14] regarding the arctangent term in the case m >| δ |.\n1 8π 2 f 2 π (m 2 − δ 2 ) 3/2 π 2 − tan −1 δ √ m 2 − δ 2 − 3δ 32π 2 f 2 π m 2 − 2 3 δ 2 ln m 2 µ 2 | δ |> m, W (m, δ, µ) = −1 16π 2 f 2 π (δ 2 − m 2 ) 3/2 ln δ − √ δ 2 − m 2 δ + √ δ 2 − m 2 − 3δ 32π 2 f 2 π m 2 − 2 3 δ 2 ln m 2 µ 2(19)\nand\nW * (m, δ, µ) is m >| δ |, W * (m, δ, µ) = 3δ 2 8π 2 f 2 π m 2 − δ 2 π 2 − tan −1 δ √ m 2 − δ 2 − 3δ 16π 2 f 2 π (δ 2 − 1 2 m 2 )ln m 2 µ 2 | δ |> m, W * (m, δ, µ) = 3δ 2 16π 2 f 2 π δ 2 − m 2 ln δ − √ δ 2 − m 2 δ + √ δ 2 − m 2 − 3δ 16π 2 f 2 π (δ 2 − 1 2 m 2 )ln m 2 µ 2 . (20)\nAs was already mentioned in Section (II), the contribution from the 3/2 − multiplets is essentially negligible due to their suppressed coupling constants, Eqs. ( 12) and (14).Explicitly we find that,\n2 9 C * 2 V * (+δ 8 * ) ≈ −.06MeV, − 20 81 H * 2 V * (+δ 10 * ) ≈ +.3MeV(21)\nwhich are indeed negligible.Hence we omit further considerations of the 3/2 − multiplets.This is not true of the Roper octet.Evaluating, one finds that the ratio of the Roper to Nucleon multiplet contribution\nC2 V (+δ R ) C 2 V (−δ N ) = .21(22)\nwhere result (9) has been used.Taking C 2 = 2 one obtains that the Roper contribution\n2 9 C2 V (+δ R ) = −5.7MeV,(23)\nwhich is, in absolute magnitude, as large as the average, experimental value of 6.8MeV.It clearly cannot be ignored.\n\n\nIV. DECUPLET MAGNETIC MOMENTS\n\nThe topic of the magnetic moments of the decuplet in the context of chiral perturbation theory was first discussed in the work of Ref. [16].Apart from the inclusion of the Roper as an intermediate state, our work differs from Ref. [16] in the treatment of SU f (3) symmetry.In our calculation, the symmetry of the decuplet states is broken through the meson masses appearing in the one loop calculation.The meson masses are taken to be proportional to the current quark masses with the up and down quark masses being equal.The strangeness [9] and charge dependences [22] of the baryon masses are regarded as effects of chiral power 1 or more.The quantity f K − f π has chiral singularity ∼ m 2 π logm 2 π [23] and has chiral power 2. Our calculations of the decuplet mass splittings and magnetic moments are limited to chiral power O(p 3 ) and O(p 2 ), respectively.Hence, we set f K = f π and do not include in one loop calculation the sigma terms from L 1 which produce strangeness dependence of baryon masses at the tree level.We ignore charge-dependence of the masses altogether.The advantage of this strategy in the calculation of baryon masses is well-known [9,14].The counter-terms which appear at one loop level simply renormalizes the sigma terms.We find a similar result in the magnetic moment calculation at one loop level, namely, that the counter terms are strictly proportional to the baryon charge and hence renormalize the tree level decuplet magnetic moment term, Eq. ( 24) below.These advantages are lost if f K is not set equal to f π [16].\n\nThe lowest order term in the chiral lagrangian for the magnetic moment of the ith member of the ∆ decuplet is given by [24]\nL M = −ı e M N µ c q i T µ i T ν i F µν (24)\nwhere q i is the charge of the ith member.The oneloop, O(p 2 ) corrections to the magnetic moments result from vertex corrections in which the external photon attaches to the meson propagator [16] and receives contributions from intermediate states with either a 3/2 + or 1/2 + Photon attachments to the intermediate baryon are m π /M N further suppressed as are also the contributions from 3/2 − baryons.These latter are hence ignored as they form part of the higher-order contribution in the chiral expansion.Note that the η meson, being electrically neutral, also does not contribute at the order being considered.Following the notation of [24], the magnetic moment of the decuplet members, µ 10 i , at the O(p 2 ), one-loop contribution in the chiral expansion is, in nuclear magneton units (e/2M N ):\nµ 10 i = q i µ c + j=π,K M N 32π 2 f 2 π α i j 4 9 H 2 F (0, m j , µ) −β i j C 2 F (−δ N , m j , µ) + C2 F (δ R , m j , µ) . (25)\nThe function F (δ, m, µ) is ultraviolet divergent and given by\nm >| δ |, F (δ, m, µ) = δ − 1 ǫ + γ E − ln(4π) − 4 3 + ln( m 2 µ 2 ) −2 m 2 − δ 2 π 2 − tan −1 δ √ m 2 − δ 2 | δ |> m, F (δ, m, µ) = δ − 1 ǫ + γ E − ln(4π) + ln( m 2 µ 2 ) + δ 2 − m 2 ln δ + √ δ 2 − m 2 δ − √ δ 2 − m 2 . (26)\nThe expression for the nonanalytic terms in F (δ, m, µ) appeared in Ref. [16].\n\nThe coefficients α i j and β i j are simply related to the coefficients α ij and β ij of Ref. [16].We multiply the coefficients α ij by 3 so that they add up to the charge of decuplet i.Unlike Ref. [16], we use the same mass for all members of a baryon multiplet.Accordingly we add the contributions of π ± and of K ± .The sum over j in Eq. ( 25) runs over two terms -π and K.The resulting coefficients have surprizing simplicity.First, α i j = β i j .Second, they may be expressed in terms of any two of the following three -charge, q i , isospin, I i 3 , and hypercharge Y i of decuplet i.All three are traceless in any SU (3) multiplet space.We choose to use charge and isospin.\nα i π = β i π = 2 3 I i 3 ,(27)α i K = β i K = − 2 3 I i 3 + q i ,(28)α i π + α i K = β i π + β i K = q i .(29)\nEq. ( 29) is the key result for renormalizability.As a consequence of these relations, the counterterm for the ultraviolet divergences in F (δ, m, µ) (which are m independent) is simply proportional to δL M , Eq. ( 24), and therefore absorbed into a redefinition of µ c .Note that this is precisely the same procedure as for the one-loop, δ-dependent contributions to the masses.We emphasize that this procedure, and hence renormalizability, is tightly wedded to the systematics of the chiral expansion (whereby δ and f π have fixed values in Eq. ( 25)).\n\nThe simplicities of the coefficients α i j and β i j allows great simplification of Eq. (25).We introduce the combination:\nG j = M N 32π 2 f 2 π 4 9 H 2 F (0, m j , µ) −C 2 F (−δ N , m j , µ) − C2 F (δ R , m j , µ) ,(30)\nand rewrite the decuplet magnetic moments in the form:\nµ 10 i = q i (µ c + G K ) + 2 3 I i 3 (G π − G K ).(31)\nNote that the form of Eq. ( 31), in particular the modification of the coefficient of charge, reflects the choice of charge and isospin as the two traceless quantities.The form would be different if we chose charge and hypercharge or some other combination of isospin and hypercharge.The second term is present only because the SU (3) symmetry of the states is broken through the difference in π and K masses.If we had used the same masses the states would be pure decuplets and the Wigner-Eckart theorem for SU f (3) would ensure that the magnetic moments are simply proportional to the charge.At tree level the decuplet magnetic moments are proportional to the charges only.The main one-loop results is the appearance of the second term of Eq. ( 31).Now we need two magnetic moments to fix the two coefficients in Eq. ( 31), viz, µ c + G K and G π − G K .The only decuplet magnetic moment which has been measured is µ Ω − = −1.94± .17 ± .14 nbm [17].It fixes the coefficient of charge in Eq. (31).For the other magnetic moment we choose µ ∆ 0 .§ § While not measured yet, it is given entirely by the loop effect.\nµ Ω − = −µ c − G K , µ ∆ 0 = 1 3 (G K − G π ). (32)\nWe can express the magnetic moments of all other decuplets in terms of these two magnetic moments.Specifically, we derive the new relation that\nµ i = −q i µ Ω − − 2 I i 3 µ ∆ 0 .(33)\nExplicit predictions for the eight other decuplet magnetic moments at the one-loop O(p 2 ) level are listed below.\nµ ∆ ++ = −2µ Ω − − 3µ ∆ 0 µ ∆ + = −µ Ω − − µ ∆ 0 µ ∆ − = µ Ω − + 3µ ∆ 0 µ Σ * + = −µ Ω − − 2µ ∆ 0 µ Σ * − = µ Ω − + 2µ ∆ 0 µ Ξ * − = µ Ω − + µ ∆ 0 µ Ξ 0 = −µ ∆ 0 µ Σ * 0 = 0. (34)\nIndependent of the explicit mulitplets included as intermediate states, violations to these relations are strictly due to higher-order terms in the chiral expansion.Note that the magnetic moment of the Σ * 0 continues to be zero at this order in the expansion.Analogous relations follow for the quadropole moments [25].We note that these relations are not obeyed by quenched lattice QCD [26].This last result is perhaps not surprising as § § The Ω − , decaying only weakly, is sufficiently long lived to allow such measurements.Since all other members of the decuplet decay through the strong interaction, it is a challenge to extract their magnetic moments from experiment.\n\nthe quenched calculations do not contain disconnected, quark loop diagrams [27].\n\nThe explicit expression for µ ∆ 0 in terms of the functions F (δ, m, µ) is given below.\nµ ∆ 0 = M N 96π 2 f 2 π H 2 4 9 (F (0, m K , µ) − F (0, m π , µ)) −C 2 (F (−δ N , m K , µ) − F (−δ N , m π , µ)) − C2 (F (δ R , m K , µ) − F (δ R , m π , µ)) .(35)\nWith the help of Eqs.(26) it is easy to verify that µ ∆ 0 is renormalization scale independent.\n\nSince the magnetic moment of the ∆ 0 is given strictly by loop-effects, it is an appropriate measure of the relative importance of the Roper at the one-loop level.Explicitly one has from Eq. ( 35) that\nµ ∆ 0 = −.055H 2 + .229C 2 + .086 C2 ,(36)\nAs in the case of the masses [9] there is a strong cancellation between the ∆ decuplet and nucleon octet intermediate states.This implies that µ ∆ 0 is a very delicate function of H 2 and C 2 and therefore potentially very sensitive to the Roper contribution ( C2 ).Ambiguity in this regard resides in the fact that H 2 and C 2 are not sufficiently well-known that a reliable prediction of µ ∆ 0 can be made.To illustrate this point, we quote the results using two \"representative\" values for the couplings, both with and without the Roper included.For the coupling values used in Ref. [14], one obtains only a mild dependence on the Roper,\nµ ∆ 0 (H 2 = 4.4, C 2 = 2, C2 = 0) = .21, µ ∆ 0 (H 2 = 4.4, C 2 = 2, C2 = 1) = .29,(37)\nwhile for the couplings used by Ref. [16], we find a much more dramatic dependence on the Roper,\nµ ∆ 0 (H 2 = 4.84, C 2 = 1.44, C2 = 0.0) = .05, µ ∆ 0 (H 2 = 4.84, C 2 = 1.44, C2 = .72) = .12.(38)\nThe difference in µ ∆ 0 from these two parameter sets is clearly sizable, as is the relative importance of the Roper.\n\nIf instead we chose to use as input the recent, model dependent extraction [28] of the magnetic moment of the ∆ ++ , µ ∆ ++ = 4.5 ± .5 to infer µ ∆ 0 using the relations Eq. (34), one then obtains that µ ∆ 0 = −.2 ± .2.If this is indeed the data, then by Eq. (36) the contribution of the Roper is, in absolute magnitude, significant.As in the case of the mass-splittings, a formulation of χPT without the Roper as an explicit degree of freedom is intrinsically incapable of predicting such \"data\".\n\n\nV. CONCLUSIONS\n\nFrom the results of the last two sections a few points are worth discussing.\n\nFirst, that while the magnetic moment of the ∆ 0 sensitively depends on the cancellation between terms depending on relatively ill-determined coupling constants, the relations between the magnetic moments of the ∆ decuplet given in Eq. ( 34) are rigorous predictions of χPT at O(p 2 ).We urge experimental activity to confront these predictions with data.A new measurement of µ Ω − with higher precision will be most useful.At least two other decuplet magnetic moments need to be measured, hopefully with a precision of ∼ 0.1 n.m.. Second, that on the level of one-loop corrections in χPT, the contribution of the Roper octet to properties of the ∆ decuplet is as important as any other multiplet's contribution.We have seen this result quantitatively in the case of the DES rule and the magnetic moment of the ∆ 0 .Both of these quantities are good measures of the one-loop effects as they are each zero at lower-order in the chiral expansion.We expect that these results are illustrative and that they generalize to all one-loop calculations for the ∆ decuplet.Since the mass-splitting, δ R , between the Roper octet and ∆ decuplet is less than that of the kaon's mass, a Taylor expansion in m K /δ R is not permissible.Hence, these loop effects cannot be be absorbed within higher order terms of the chiral expansion of a theory not containing the Roper as an explicit degree of freedom.In such a theory it is indeed difficult to justify going to one-loop or higher without inclusion of the Roper.Phenomenologically successful results would have to be considered merely fortuitous unless shown to be a result of more general considerations (as in the relations of Eq. (34) for the magnetic moments).\n\nThird, that the above argument can be repeated in kind for the one-loop corrections to the Roper resonance.That is, even higher-multiplets, separated by δ h in mass from the Roper octet by an amount less than the kaon mass, will a priori be as important quantitatively as the ∆ decuplet for properties of the Roper at the one-loop level.Since such corrections are neccessary for a two loop calculation * * * of the baryon masses, we are led to conclude that the loop expansion in general in the baryon sector of χPT is inevitably wedded to the neccessity of including more and more multiplets in the theory as fundamental fields.While such a result may not be true in a particular limit of QCD (e.g.m u,d,s → 0 or N c → ∞), it is a consequence of the experimental fact that on the average, m π ≈ δ h .\n ACKNOWLEDGEMENTS * * * See though[14]for a discussion as to the likely feasibility of such a program.We thank Y. Umino and T. D. Cohen for useful discussions.This work was supported in part by DOE Grant DOE-FG02-93ER-40762.\n. S Weinberg, Physica. 963271979\n\n. J Gasser, H Leutwyler, Ann. of Phys. 1585391984. 1985Nucl. Phys.\n\n. J Gasser, M E Sainio, A Svarc, Nucl. Phys. 3077791988\n\n. S Weinberg, Phys. Lett. 2512881990\n\n. E Jenkins, A V Manohar, Phys. Lett. 2555581991\n\nV Bernard, N Kaiser, U Meißner, TK-95-1Chiral Dynamics in Nucleons and Nuclei. 1995Universite LouisPasteur de Strasbourg preprint CRN-95-3, and Univeritat Bonn preprint\n\n. C Ordonez, L Ray, U Van Kolck, Phys. Rev. Lett. 7219821994\n\n. E Jenkins, A V Manohar, Phys. Lett. 2593531991\n\n. E Jenkins, Nucl. Phys. 3681901992\n\n. C.-H Lee, H Jung, D.-P Min, M Rho, Phys. Lett. 326141993\n\n. M Savage, Phys. Lett. 3314111994\n\n. C.-H Lee, G E Brown, D.-P Min, M Rho, Nucl. Phys. 5854011995\n\n. M Gell-Mann, R Oakes, B Renner, Phys. Rev. 17521951968\n\nBaryon Mass Splittings in Chiral Perturbation Theory. M K Banerjee, J Milana, DOE/ER/40762-051, UMPP #95-0581994University of Maryland preprint\n\nM Gell-Mann, Proc. 11th Int. Conf. High Energy Phys. (CERN). J Prentki Editor, 11th Int. Conf. High Energy Phys. (CERN)Geneva1962CERN\n\n. M N Butler, M J Savage, R P Springer, Phys. Rev. 4934591994\n\n. Phys. Rev. 5011731994Particle Data Group\n\nFor a recent discussion, see e.g. Y. Umino and F. Myhrer, Nucl. Phys. A554. 1993593\n\nM Gell-Mann, No. CTSL-20Eightfold Way\" by ibid and Y. Ne'eman. Benjamin, NewYork1961. 1964CALTECH Synchotron LabReportreprinted in the\n\n. S Okubo, Prog. Theor. Phys. 279491962\n\nE Jenkins, A V Manohar, Effective field theories of the Standard Model. - G Ulf, Meißner, SingaporeWorld Scientific1992\n\n. V Bernard, N Kaiser, U Meißner, Z. Phys. 601111993\n\n. R F Lebed, M A Luty, Phys. Lett. 3294791994\n\n. R F Lebed, Nucl. Phys. 4302951994\n\n. E Jenkins, R F Lebed, Phys. Rev. 522821995\n\nJ F Donoghue, E Golowich, B Hosltein, Dynamics of the Standard Model. Cambridge, EnglandCambridge University Press1992\n\n. E Jenkins, M Luke, A V Manohar, M J Savage, Phys. Lett. 3024821993\n\n) and (3) of Ref. [16] makes it clear that the decuplet quadrupole moments calculated with the same treatment of SU f (3) symmetry, i.e., keeping fπ = fK and using the same mass for all members of a given baryon multiplet, would exhibit the same dependence on charge and isospin as the magnetic moments. analogy with Eq. (33), the resulting quadrupole moments would be expressible as Q 10 i = −qi Q Ω − − 2 I i 3 Q ∆ 0. A comparison of Eqs (12) and (13) with Eqs. and a set of relations analogous to Eq. (34) may be written down\n\n. D B Leinweber, T Draper, R M Woloshyn, Phys. Rev. 4630671992\n\nEssential strangeness in nucleon magnetic moments, Ohio State preprint. D B Leinweber, 1994\n\n. A Bosshard, Phys. Revd. 4419621991\n"},"corpus_id":{"kind":"number","value":5763380,"string":"5,763,380"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"hep-ph/9508340\",\n \"mag\": 1493337713,\n \"acl\": null,\n \"pubmed\": \"10021273\",\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevd.54.5804\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/hep-ph/9508340v1.pdf\"\n ],\n \"pdf_sha\": \"d2313ce999e2989bedfc95f8bb2d3e3fa05dd4e3\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://arxiv.org/pdf/hep-ph/9508340\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"The Decuplet Revisited in χPT"},"authors":{"kind":"list like","value":["M K Banerjee \nDepartment of Physics\nUniversity of Maryland College Park\n20742MarylandUSA","J Milana \nDepartment of Physics\nUniversity of Maryland College Park\n20742MarylandUSA"],"string":"[\n \"M K Banerjee \\nDepartment of Physics\\nUniversity of Maryland College Park\\n20742MarylandUSA\",\n \"J Milana \\nDepartment of Physics\\nUniversity of Maryland College Park\\n20742MarylandUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The paper deals with two issues.First, we explore the quantitiative importance of higher multiplets for properties of the ∆ decuplet in chiral perturbation theory.In particular, it is found that the lowest order one-loop contributions from the Roper octet to the decuplet masses and magnetic moments are substantial.The relevance of these results to the chiral expansion in general is discussed.The exact values of the magnetic moments depend upon delicate cancellations involving ill-determined coupling constants.Second, we present new relations between the magnetic moments of the ∆ decuplet that are independent of all couplings.They are exact at the order of the chiral expansion used in this paper.* Because of the large mass difference between the ρ and the pion, as well as the quadratic nature of the chiral expansion, this issue does not arise when considering the sector of purely pseudoscalar, goldstone bosons.The effects of the ρ and other such states are incorporated as part of the needed input coupling constants of the theory.† Arguing that for a very limited region near threshold in the SU f (2) defined theory, the ∆-nucleon mass difference can be considered \"large\" * * As a self-consistent test of the assignment of hadrons to this multiplet, note that the violation to the corresponding GMO relation for this octet is only 15MeV."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"The success of chiral perturbation theory (χPT) [1] for understanding properties of the pseudoscalar mesons is now well established [2].The approach is based on the existence of a systematic expansion in derivatives of the pion's field and the pion's mass, whereby m π divided by some large scale, generated by the theory itself and typically ∼ 4πf π , becomes the perturbative expansion parameter.For the purely mesonic sector this expansion is in fact quadratic in the pion's mass, so that even for the SU f (3) generalization, (m K /4πf π ) 2 is still a reasonablely small parameter.\nThe application to the baryon sector has however from the outset been confronted with a variety of difficulties [3].For example, how to handle the nucleon's mass was a problem only relatively recently solved [4,5].One unavoidable complication when including baryons is that the chiral expansion is itself more complicated [3,4] than in the purely mesonic case, and the expansion parameter is now only linear in m π (m K ).A second pertinent complication involves the issue of resonances.* Originally it was conjectured [4] that all such resonances (and most notably the ∆) need not be included as an explicit degree of freedom, i.e. that they could be \"integrated out\".While at least one notably active group [6] have maintained this viewpoint, † most researchers have subsequently found it untenable [7].The ∆ degree of freedom was first introduced into χPT by Jenkins and Manohar in Ref. [8].Recently, the importance of the Λ * (1405) for understanding threshold kaon-nucleon scattering lengths has also been realized [10,11,12].\nIn this paper we discuss the role of higher-multiplets for the properties of the ∆ decuplet at the one-loop level in χPT.We consider the O(p 3 ) correction to the decuplet masses and the O(p 2 ), one-loop correction to the magnetic moments of the decuplet (the electromagnetic vertex has chiral power −1 excluding whatever power maybe assigned to electric charge).Our criterion for which multiplets to consider is that the average mass-splitting, δ h , between the multiplet and the ∆ decuplet be less than the mass of the kaon,\nThis criterion is based on the fact that when it is met, an expansion in m K /δ h is not justified so that loop effects involving these higher-mulitplet members as intermediate states cannot be absorbed into higher order terms in the chiral lagrangian.Such loop effects place a fundamental limitation on any formulation of χPT that omits the higher-multiplet as an explicit degree of freedom, even if the chiral expansion was then executed to all orders.By the convention of Ref. [14], note that the chiral power of all δ is 1.\nFor the case of the nucleon octet Eq. ( 1) is clearly met,\n(M 10 = 1377MeV and M 8 = 1151MeV) and is indeed the driving phenomenological reason for expecting that the ∆ cannot be ignored for descriptions of the nucleon.\nWe focus here on the properties of the ∆ decuplet as opposed to those of the nucleon octet because of the simple reason that the mass splitting δ h satisfies the criterion specified by Eq. (1) for at least two resonances.\nWe will show that the Roper has a nontrivial effect on both decuplet mass splittings and magnetic moments.‡ mη is obtained at this order in χPT using GMOR [13].\nWe should also note that these results lead to a more general statement.When adding a loop one must also add resonances whose masses are within a kaon mass of the resonances already included.\nThe rest of this paper is organized thusly.In Section (II) we enumerate the various multiplets considered and their interactions (and experimentally obtained couplings) with the ∆ decuplet utilizing the heavy baryon formalism of Jenkins and Manohar [5].In Section (III) we present the one-loop, O(p 3 ) contributions to the decuplet masses focussing on the violation to the Decuplet Equal Spacing rule [15], the sole quantity for which χPT makes a prediction at this order in the chiral expansion [9,14].In Section(IV) we consider the one-loop, O(p 2 ) results for the magnetic moments of the decuplet, a subject first discussed in χPT in Ref. [16] where though adherence to the order of the chiral expansion was not strictly maintained.We demonstrate later that strict adherence is crucial for renormalizability.The focus in both Sections (III) and (IV) is the quantitative importance of the higher-multiplets.In addition new relations for the magnetic moments at this order in χPT are derived that are independent of the intermediate multiplets considered.Their violation would be a clear measure of the importance of higher chiral power terms in the expansion.In Section (V) we conclude with a discussion of the consequence of these results to the loop-expansion in general in χPT."},{"title":"II. HIGHER MULTIPLETS: DEFINITIONS AND COUPLINGS","content":"A number of multiplets § satisfy the criteria Eq. (1).Fortunately most of these can be eliminated due to symmetry constraints.For example, flavor singlets such as the Λ * (1405) do not couple to a decuplet via an SU f (3) octet (the goldstone bosons).Only slightly less straightfoward, a 1/2 − octet (e.g. the N(1535) multiplet) couples only through the lower components of the baryon spinors,\nwhich vanishes to lowest order in the heavy baryon expansion [5].(Such states would, in principle, need to be considered in higher-order calculations.)Coupling to the 5/2 − octet likewise vanishes at lowest order.With these eliminations, we obtain that only octets or decuplets of baryons with quantum numbers 1/2 + , 3/2 + , 3/2 − and 5/2 + need be considered.§ See e.g.Table 30.4 in the Particle Data Group [17].As we are at present only concerned with the average coupling of these multiplets with the ∆ decuplet, we ignore potentially interesting questions as to the exact SU f (3) composition of any particular excited state.[18] We also omit from consideration possible exotics.\nThe most important 1/2 + multiplet (beyond of course the nucleon's) is the octet containing the Roper, N(1440).A slight difficulty arises in determining\nbecause one member of the Roper multiplet, the excited Cascade, has not yet been identified.To get a reasonable approximate value for its mass, we use the corresponding GMO relation [19] M\nby which one estimates that M Ξ * = 1790.The average value of the Roper multiplet is thereby\nfrom which one gets that\nFor the ∆N * π interaction one has\nin complete analogy to the leading ∆N π interaction in the heavy fermion limit.[8] The coupling C2 can be obtained from the observed decay of the N * (1440) → ∆π which has a partial width Γ N * →∆π ≈ 90MeV.Comparing this with the decay of the ∆, Γ ∆→N π ≈ 120MeV one obtains that\nIn principle, one other 1/2 + multiplet meets our criteria, the octet containing the N(1710), with a δ h ≈ 470MeV.However, best estimates [17] for the partial width for N(1710)→ ∆π is ≈ 30MeV, which implies that the relevant coupling constant is significantly suppressed compared to that in Eq. ( 9).We therefore ignore this multiplet in our subsequent calculations as it amounts to only a small correction to those of the Roper octet.\nFor the lowest lying 3/2 − octet, we obtain directly from the experimentally measured masses [17] that * *\nThe interaction with the ∆ decuplet is, to leading order in the chiral lagrangian,\nThe coupling C * can be determined from the observed decay of the N * (1520), Γ(N * (1520) → ∆π) ≈ 25MeV, whence one finds that\nThis relative suppression results both from the smaller branching width as well as overall kinematic factors that otherwise tend to enhance N * (1520) → ∆π w.r.t.N * (1440) → ∆π.\nThere are two other 3/2 − multiplets, one octet and one decuplet, listed in the Particle Data Group that could potentially satisfy our criteria, Eq. ( 1).Each are very poorly determined, containing merely one member each, the N(1700) and the ∆(1700), respectively.In the case of the N(1700), its coupling to the ∆π is experimentally negligible and hence can be safely ignored.On the other hand, the coupling with the ∆(1700) is not so readily ignored, having a decay width Γ(∆(1700) → ∆π) ≈ 120MeV.We therefore explicitly keep this decuplet, assigning for its intermultiplet spacing with the ∆ decuplet a value\nFrom the aforementioned decay width, and an interaction of the form Eq. ( 11), we obtain that for the ∆ * ∆π coupling\nHere we have used the convention of Ref. [9] for the SU f (3) algebra factors (whereby, for the ∆∆π coupling, H 2 ≈ 4 is typical).As in the case of the N (1520) overall kinematic factors in addition to the available phase space, yields a rather suppressed value of the coupling.Indeed, as we will soon see, due to Eqs. ( 12) and ( 14) little would have been lost had we ignored the 3/2 − multiplets altogether.Nevertheless, they have been included for completeness.We come finally to the 5/2 + states.The lowest such multiplet is the N(1680) octet.It has an intermultiplet mass splitting with the ∆ decuplet of δ h = 496MeV.We conclude that there is no 5/2 + multiplet that meets our criteria Eq. ( 1).\nThis then concludes our examination of the relevant multiplets.By far the most important, as we will presently see, is the Roper octet."},{"title":"III. DECUPLET EQUAL SPACING RULE","content":"The one-loop, O(p 3 ) results for the masses of the decuplet involving intermediate ∆ decuplet and nucleon octet states have been published previously [14].The contribution, δM 10 , from the 3/2 − multiplet, for the case m > δ 8 * , is\nwhere β represents SU(3) algebra factors [9,14].As discussed in Ref. [14], the counterterms neccessary to renormalize these terms are either of the form δ (for the δ 8 * m 2 divergences).As the DES rule is exact for all terms through m 2 , all counterterms (divergences) cancel at this order of the chiral expansion in the violation to the DES rule.\nIncluding all multiplets, the one-loop, O(p 3 ), violation to the DES rule is\n† † V and V * are given by\nwherein the function\nm >| δ |, W (m, δ, µ) = † † Note that unlike our convention in [14], all δ h are now strictly positive and hence the explicit sign in the function V above.\n‡ ‡ Note the correction from [14] regarding the arctangent term in the case m >| δ |.\nand\nAs was already mentioned in Section (II), the contribution from the 3/2 − multiplets is essentially negligible due to their suppressed coupling constants, Eqs. ( 12) and (14).Explicitly we find that,\nwhich are indeed negligible.Hence we omit further considerations of the 3/2 − multiplets.This is not true of the Roper octet.Evaluating, one finds that the ratio of the Roper to Nucleon multiplet contribution\nwhere result (9) has been used.Taking C 2 = 2 one obtains that the Roper contribution\nwhich is, in absolute magnitude, as large as the average, experimental value of 6.8MeV.It clearly cannot be ignored."},{"title":"IV. DECUPLET MAGNETIC MOMENTS","content":"The topic of the magnetic moments of the decuplet in the context of chiral perturbation theory was first discussed in the work of Ref. [16].Apart from the inclusion of the Roper as an intermediate state, our work differs from Ref. [16] in the treatment of SU f (3) symmetry.In our calculation, the symmetry of the decuplet states is broken through the meson masses appearing in the one loop calculation.The meson masses are taken to be proportional to the current quark masses with the up and down quark masses being equal.The strangeness [9] and charge dependences [22] of the baryon masses are regarded as effects of chiral power 1 or more.The quantity f K − f π has chiral singularity ∼ m 2 π logm 2 π [23] and has chiral power 2. Our calculations of the decuplet mass splittings and magnetic moments are limited to chiral power O(p 3 ) and O(p 2 ), respectively.Hence, we set f K = f π and do not include in one loop calculation the sigma terms from L 1 which produce strangeness dependence of baryon masses at the tree level.We ignore charge-dependence of the masses altogether.The advantage of this strategy in the calculation of baryon masses is well-known [9,14].The counter-terms which appear at one loop level simply renormalizes the sigma terms.We find a similar result in the magnetic moment calculation at one loop level, namely, that the counter terms are strictly proportional to the baryon charge and hence renormalize the tree level decuplet magnetic moment term, Eq. ( 24) below.These advantages are lost if f K is not set equal to f π [16].\nThe lowest order term in the chiral lagrangian for the magnetic moment of the ith member of the ∆ decuplet is given by [24]\nwhere q i is the charge of the ith member.The oneloop, O(p 2 ) corrections to the magnetic moments result from vertex corrections in which the external photon attaches to the meson propagator [16] and receives contributions from intermediate states with either a 3/2 + or 1/2 + Photon attachments to the intermediate baryon are m π /M N further suppressed as are also the contributions from 3/2 − baryons.These latter are hence ignored as they form part of the higher-order contribution in the chiral expansion.Note that the η meson, being electrically neutral, also does not contribute at the order being considered.Following the notation of [24], the magnetic moment of the decuplet members, µ 10 i , at the O(p 2 ), one-loop contribution in the chiral expansion is, in nuclear magneton units (e/2M N ):\nThe function F (δ, m, µ) is ultraviolet divergent and given by\nThe expression for the nonanalytic terms in F (δ, m, µ) appeared in Ref. [16].\nThe coefficients α i j and β i j are simply related to the coefficients α ij and β ij of Ref. [16].We multiply the coefficients α ij by 3 so that they add up to the charge of decuplet i.Unlike Ref. [16], we use the same mass for all members of a baryon multiplet.Accordingly we add the contributions of π ± and of K ± .The sum over j in Eq. ( 25) runs over two terms -π and K.The resulting coefficients have surprizing simplicity.First, α i j = β i j .Second, they may be expressed in terms of any two of the following three -charge, q i , isospin, I i 3 , and hypercharge Y i of decuplet i.All three are traceless in any SU (3) multiplet space.We choose to use charge and isospin.\nEq. ( 29) is the key result for renormalizability.As a consequence of these relations, the counterterm for the ultraviolet divergences in F (δ, m, µ) (which are m independent) is simply proportional to δL M , Eq. ( 24), and therefore absorbed into a redefinition of µ c .Note that this is precisely the same procedure as for the one-loop, δ-dependent contributions to the masses.We emphasize that this procedure, and hence renormalizability, is tightly wedded to the systematics of the chiral expansion (whereby δ and f π have fixed values in Eq. ( 25)).\nThe simplicities of the coefficients α i j and β i j allows great simplification of Eq. (25).We introduce the combination:\nand rewrite the decuplet magnetic moments in the form:\nNote that the form of Eq. ( 31), in particular the modification of the coefficient of charge, reflects the choice of charge and isospin as the two traceless quantities.The form would be different if we chose charge and hypercharge or some other combination of isospin and hypercharge.The second term is present only because the SU (3) symmetry of the states is broken through the difference in π and K masses.If we had used the same masses the states would be pure decuplets and the Wigner-Eckart theorem for SU f (3) would ensure that the magnetic moments are simply proportional to the charge.At tree level the decuplet magnetic moments are proportional to the charges only.The main one-loop results is the appearance of the second term of Eq. ( 31).Now we need two magnetic moments to fix the two coefficients in Eq. ( 31), viz, µ c + G K and G π − G K .The only decuplet magnetic moment which has been measured is µ Ω − = −1.94± .17 ± .14 nbm [17].It fixes the coefficient of charge in Eq. (31).For the other magnetic moment we choose µ ∆ 0 .§ § While not measured yet, it is given entirely by the loop effect.\nWe can express the magnetic moments of all other decuplets in terms of these two magnetic moments.Specifically, we derive the new relation that\nExplicit predictions for the eight other decuplet magnetic moments at the one-loop O(p 2 ) level are listed below.\nIndependent of the explicit mulitplets included as intermediate states, violations to these relations are strictly due to higher-order terms in the chiral expansion.Note that the magnetic moment of the Σ * 0 continues to be zero at this order in the expansion.Analogous relations follow for the quadropole moments [25].We note that these relations are not obeyed by quenched lattice QCD [26].This last result is perhaps not surprising as § § The Ω − , decaying only weakly, is sufficiently long lived to allow such measurements.Since all other members of the decuplet decay through the strong interaction, it is a challenge to extract their magnetic moments from experiment.\nthe quenched calculations do not contain disconnected, quark loop diagrams [27].\nThe explicit expression for µ ∆ 0 in terms of the functions F (δ, m, µ) is given below.\nWith the help of Eqs.(26) it is easy to verify that µ ∆ 0 is renormalization scale independent.\nSince the magnetic moment of the ∆ 0 is given strictly by loop-effects, it is an appropriate measure of the relative importance of the Roper at the one-loop level.Explicitly one has from Eq. ( 35) that\nAs in the case of the masses [9] there is a strong cancellation between the ∆ decuplet and nucleon octet intermediate states.This implies that µ ∆ 0 is a very delicate function of H 2 and C 2 and therefore potentially very sensitive to the Roper contribution ( C2 ).Ambiguity in this regard resides in the fact that H 2 and C 2 are not sufficiently well-known that a reliable prediction of µ ∆ 0 can be made.To illustrate this point, we quote the results using two \"representative\" values for the couplings, both with and without the Roper included.For the coupling values used in Ref. [14], one obtains only a mild dependence on the Roper,\nwhile for the couplings used by Ref. [16], we find a much more dramatic dependence on the Roper,\nThe difference in µ ∆ 0 from these two parameter sets is clearly sizable, as is the relative importance of the Roper.\nIf instead we chose to use as input the recent, model dependent extraction [28] of the magnetic moment of the ∆ ++ , µ ∆ ++ = 4.5 ± .5 to infer µ ∆ 0 using the relations Eq. (34), one then obtains that µ ∆ 0 = −.2 ± .2.If this is indeed the data, then by Eq. (36) the contribution of the Roper is, in absolute magnitude, significant.As in the case of the mass-splittings, a formulation of χPT without the Roper as an explicit degree of freedom is intrinsically incapable of predicting such \"data\"."},{"title":"V. CONCLUSIONS","content":"From the results of the last two sections a few points are worth discussing.\nFirst, that while the magnetic moment of the ∆ 0 sensitively depends on the cancellation between terms depending on relatively ill-determined coupling constants, the relations between the magnetic moments of the ∆ decuplet given in Eq. ( 34) are rigorous predictions of χPT at O(p 2 ).We urge experimental activity to confront these predictions with data.A new measurement of µ Ω − with higher precision will be most useful.At least two other decuplet magnetic moments need to be measured, hopefully with a precision of ∼ 0.1 n.m.. Second, that on the level of one-loop corrections in χPT, the contribution of the Roper octet to properties of the ∆ decuplet is as important as any other multiplet's contribution.We have seen this result quantitatively in the case of the DES rule and the magnetic moment of the ∆ 0 .Both of these quantities are good measures of the one-loop effects as they are each zero at lower-order in the chiral expansion.We expect that these results are illustrative and that they generalize to all one-loop calculations for the ∆ decuplet.Since the mass-splitting, δ R , between the Roper octet and ∆ decuplet is less than that of the kaon's mass, a Taylor expansion in m K /δ R is not permissible.Hence, these loop effects cannot be be absorbed within higher order terms of the chiral expansion of a theory not containing the Roper as an explicit degree of freedom.In such a theory it is indeed difficult to justify going to one-loop or higher without inclusion of the Roper.Phenomenologically successful results would have to be considered merely fortuitous unless shown to be a result of more general considerations (as in the relations of Eq. (34) for the magnetic moments).\nThird, that the above argument can be repeated in kind for the one-loop corrections to the Roper resonance.That is, even higher-multiplets, separated by δ h in mass from the Roper octet by an amount less than the kaon mass, will a priori be as important quantitatively as the ∆ decuplet for properties of the Roper at the one-loop level.Since such corrections are neccessary for a two loop calculation * * * of the baryon masses, we are led to conclude that the loop expansion in general in the baryon sector of χPT is inevitably wedded to the neccessity of including more and more multiplets in the theory as fundamental fields.While such a result may not be true in a particular limit of QCD (e.g.m u,d,s → 0 or N c → ∞), it is a consequence of the experimental fact that on the average, m π ≈ δ h ."}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"The success of chiral perturbation theory (χPT) [1] for understanding properties of the pseudoscalar mesons is now well established [2].The approach is based on the existence of a systematic expansion in derivatives of the pion's field and the pion's mass, whereby m π divided by some large scale, generated by the theory itself and typically ∼ 4πf π , becomes the perturbative expansion parameter.For the purely mesonic sector this expansion is in fact quadratic in the pion's mass, so that even for the SU f (3) generalization, (m K /4πf π ) 2 is still a reasonablely small parameter.\\nThe application to the baryon sector has however from the outset been confronted with a variety of difficulties [3].For example, how to handle the nucleon's mass was a problem only relatively recently solved [4,5].One unavoidable complication when including baryons is that the chiral expansion is itself more complicated [3,4] than in the purely mesonic case, and the expansion parameter is now only linear in m π (m K ).A second pertinent complication involves the issue of resonances.* Originally it was conjectured [4] that all such resonances (and most notably the ∆) need not be included as an explicit degree of freedom, i.e. that they could be \\\"integrated out\\\".While at least one notably active group [6] have maintained this viewpoint, † most researchers have subsequently found it untenable [7].The ∆ degree of freedom was first introduced into χPT by Jenkins and Manohar in Ref. [8].Recently, the importance of the Λ * (1405) for understanding threshold kaon-nucleon scattering lengths has also been realized [10,11,12].\\nIn this paper we discuss the role of higher-multiplets for the properties of the ∆ decuplet at the one-loop level in χPT.We consider the O(p 3 ) correction to the decuplet masses and the O(p 2 ), one-loop correction to the magnetic moments of the decuplet (the electromagnetic vertex has chiral power −1 excluding whatever power maybe assigned to electric charge).Our criterion for which multiplets to consider is that the average mass-splitting, δ h , between the multiplet and the ∆ decuplet be less than the mass of the kaon,\\nThis criterion is based on the fact that when it is met, an expansion in m K /δ h is not justified so that loop effects involving these higher-mulitplet members as intermediate states cannot be absorbed into higher order terms in the chiral lagrangian.Such loop effects place a fundamental limitation on any formulation of χPT that omits the higher-multiplet as an explicit degree of freedom, even if the chiral expansion was then executed to all orders.By the convention of Ref. [14], note that the chiral power of all δ is 1.\\nFor the case of the nucleon octet Eq. ( 1) is clearly met,\\n(M 10 = 1377MeV and M 8 = 1151MeV) and is indeed the driving phenomenological reason for expecting that the ∆ cannot be ignored for descriptions of the nucleon.\\nWe focus here on the properties of the ∆ decuplet as opposed to those of the nucleon octet because of the simple reason that the mass splitting δ h satisfies the criterion specified by Eq. (1) for at least two resonances.\\nWe will show that the Roper has a nontrivial effect on both decuplet mass splittings and magnetic moments.‡ mη is obtained at this order in χPT using GMOR [13].\\nWe should also note that these results lead to a more general statement.When adding a loop one must also add resonances whose masses are within a kaon mass of the resonances already included.\\nThe rest of this paper is organized thusly.In Section (II) we enumerate the various multiplets considered and their interactions (and experimentally obtained couplings) with the ∆ decuplet utilizing the heavy baryon formalism of Jenkins and Manohar [5].In Section (III) we present the one-loop, O(p 3 ) contributions to the decuplet masses focussing on the violation to the Decuplet Equal Spacing rule [15], the sole quantity for which χPT makes a prediction at this order in the chiral expansion [9,14].In Section(IV) we consider the one-loop, O(p 2 ) results for the magnetic moments of the decuplet, a subject first discussed in χPT in Ref. [16] where though adherence to the order of the chiral expansion was not strictly maintained.We demonstrate later that strict adherence is crucial for renormalizability.The focus in both Sections (III) and (IV) is the quantitative importance of the higher-multiplets.In addition new relations for the magnetic moments at this order in χPT are derived that are independent of the intermediate multiplets considered.Their violation would be a clear measure of the importance of higher chiral power terms in the expansion.In Section (V) we conclude with a discussion of the consequence of these results to the loop-expansion in general in χPT.\"\n },\n {\n \"title\": \"II. HIGHER MULTIPLETS: DEFINITIONS AND COUPLINGS\",\n \"content\": \"A number of multiplets § satisfy the criteria Eq. (1).Fortunately most of these can be eliminated due to symmetry constraints.For example, flavor singlets such as the Λ * (1405) do not couple to a decuplet via an SU f (3) octet (the goldstone bosons).Only slightly less straightfoward, a 1/2 − octet (e.g. the N(1535) multiplet) couples only through the lower components of the baryon spinors,\\nwhich vanishes to lowest order in the heavy baryon expansion [5].(Such states would, in principle, need to be considered in higher-order calculations.)Coupling to the 5/2 − octet likewise vanishes at lowest order.With these eliminations, we obtain that only octets or decuplets of baryons with quantum numbers 1/2 + , 3/2 + , 3/2 − and 5/2 + need be considered.§ See e.g.Table 30.4 in the Particle Data Group [17].As we are at present only concerned with the average coupling of these multiplets with the ∆ decuplet, we ignore potentially interesting questions as to the exact SU f (3) composition of any particular excited state.[18] We also omit from consideration possible exotics.\\nThe most important 1/2 + multiplet (beyond of course the nucleon's) is the octet containing the Roper, N(1440).A slight difficulty arises in determining\\nbecause one member of the Roper multiplet, the excited Cascade, has not yet been identified.To get a reasonable approximate value for its mass, we use the corresponding GMO relation [19] M\\nby which one estimates that M Ξ * = 1790.The average value of the Roper multiplet is thereby\\nfrom which one gets that\\nFor the ∆N * π interaction one has\\nin complete analogy to the leading ∆N π interaction in the heavy fermion limit.[8] The coupling C2 can be obtained from the observed decay of the N * (1440) → ∆π which has a partial width Γ N * →∆π ≈ 90MeV.Comparing this with the decay of the ∆, Γ ∆→N π ≈ 120MeV one obtains that\\nIn principle, one other 1/2 + multiplet meets our criteria, the octet containing the N(1710), with a δ h ≈ 470MeV.However, best estimates [17] for the partial width for N(1710)→ ∆π is ≈ 30MeV, which implies that the relevant coupling constant is significantly suppressed compared to that in Eq. ( 9).We therefore ignore this multiplet in our subsequent calculations as it amounts to only a small correction to those of the Roper octet.\\nFor the lowest lying 3/2 − octet, we obtain directly from the experimentally measured masses [17] that * *\\nThe interaction with the ∆ decuplet is, to leading order in the chiral lagrangian,\\nThe coupling C * can be determined from the observed decay of the N * (1520), Γ(N * (1520) → ∆π) ≈ 25MeV, whence one finds that\\nThis relative suppression results both from the smaller branching width as well as overall kinematic factors that otherwise tend to enhance N * (1520) → ∆π w.r.t.N * (1440) → ∆π.\\nThere are two other 3/2 − multiplets, one octet and one decuplet, listed in the Particle Data Group that could potentially satisfy our criteria, Eq. ( 1).Each are very poorly determined, containing merely one member each, the N(1700) and the ∆(1700), respectively.In the case of the N(1700), its coupling to the ∆π is experimentally negligible and hence can be safely ignored.On the other hand, the coupling with the ∆(1700) is not so readily ignored, having a decay width Γ(∆(1700) → ∆π) ≈ 120MeV.We therefore explicitly keep this decuplet, assigning for its intermultiplet spacing with the ∆ decuplet a value\\nFrom the aforementioned decay width, and an interaction of the form Eq. ( 11), we obtain that for the ∆ * ∆π coupling\\nHere we have used the convention of Ref. [9] for the SU f (3) algebra factors (whereby, for the ∆∆π coupling, H 2 ≈ 4 is typical).As in the case of the N (1520) overall kinematic factors in addition to the available phase space, yields a rather suppressed value of the coupling.Indeed, as we will soon see, due to Eqs. ( 12) and ( 14) little would have been lost had we ignored the 3/2 − multiplets altogether.Nevertheless, they have been included for completeness.We come finally to the 5/2 + states.The lowest such multiplet is the N(1680) octet.It has an intermultiplet mass splitting with the ∆ decuplet of δ h = 496MeV.We conclude that there is no 5/2 + multiplet that meets our criteria Eq. ( 1).\\nThis then concludes our examination of the relevant multiplets.By far the most important, as we will presently see, is the Roper octet.\"\n },\n {\n \"title\": \"III. DECUPLET EQUAL SPACING RULE\",\n \"content\": \"The one-loop, O(p 3 ) results for the masses of the decuplet involving intermediate ∆ decuplet and nucleon octet states have been published previously [14].The contribution, δM 10 , from the 3/2 − multiplet, for the case m > δ 8 * , is\\nwhere β represents SU(3) algebra factors [9,14].As discussed in Ref. [14], the counterterms neccessary to renormalize these terms are either of the form δ (for the δ 8 * m 2 divergences).As the DES rule is exact for all terms through m 2 , all counterterms (divergences) cancel at this order of the chiral expansion in the violation to the DES rule.\\nIncluding all multiplets, the one-loop, O(p 3 ), violation to the DES rule is\\n† † V and V * are given by\\nwherein the function\\nm >| δ |, W (m, δ, µ) = † † Note that unlike our convention in [14], all δ h are now strictly positive and hence the explicit sign in the function V above.\\n‡ ‡ Note the correction from [14] regarding the arctangent term in the case m >| δ |.\\nand\\nAs was already mentioned in Section (II), the contribution from the 3/2 − multiplets is essentially negligible due to their suppressed coupling constants, Eqs. ( 12) and (14).Explicitly we find that,\\nwhich are indeed negligible.Hence we omit further considerations of the 3/2 − multiplets.This is not true of the Roper octet.Evaluating, one finds that the ratio of the Roper to Nucleon multiplet contribution\\nwhere result (9) has been used.Taking C 2 = 2 one obtains that the Roper contribution\\nwhich is, in absolute magnitude, as large as the average, experimental value of 6.8MeV.It clearly cannot be ignored.\"\n },\n {\n \"title\": \"IV. DECUPLET MAGNETIC MOMENTS\",\n \"content\": \"The topic of the magnetic moments of the decuplet in the context of chiral perturbation theory was first discussed in the work of Ref. [16].Apart from the inclusion of the Roper as an intermediate state, our work differs from Ref. [16] in the treatment of SU f (3) symmetry.In our calculation, the symmetry of the decuplet states is broken through the meson masses appearing in the one loop calculation.The meson masses are taken to be proportional to the current quark masses with the up and down quark masses being equal.The strangeness [9] and charge dependences [22] of the baryon masses are regarded as effects of chiral power 1 or more.The quantity f K − f π has chiral singularity ∼ m 2 π logm 2 π [23] and has chiral power 2. Our calculations of the decuplet mass splittings and magnetic moments are limited to chiral power O(p 3 ) and O(p 2 ), respectively.Hence, we set f K = f π and do not include in one loop calculation the sigma terms from L 1 which produce strangeness dependence of baryon masses at the tree level.We ignore charge-dependence of the masses altogether.The advantage of this strategy in the calculation of baryon masses is well-known [9,14].The counter-terms which appear at one loop level simply renormalizes the sigma terms.We find a similar result in the magnetic moment calculation at one loop level, namely, that the counter terms are strictly proportional to the baryon charge and hence renormalize the tree level decuplet magnetic moment term, Eq. ( 24) below.These advantages are lost if f K is not set equal to f π [16].\\nThe lowest order term in the chiral lagrangian for the magnetic moment of the ith member of the ∆ decuplet is given by [24]\\nwhere q i is the charge of the ith member.The oneloop, O(p 2 ) corrections to the magnetic moments result from vertex corrections in which the external photon attaches to the meson propagator [16] and receives contributions from intermediate states with either a 3/2 + or 1/2 + Photon attachments to the intermediate baryon are m π /M N further suppressed as are also the contributions from 3/2 − baryons.These latter are hence ignored as they form part of the higher-order contribution in the chiral expansion.Note that the η meson, being electrically neutral, also does not contribute at the order being considered.Following the notation of [24], the magnetic moment of the decuplet members, µ 10 i , at the O(p 2 ), one-loop contribution in the chiral expansion is, in nuclear magneton units (e/2M N ):\\nThe function F (δ, m, µ) is ultraviolet divergent and given by\\nThe expression for the nonanalytic terms in F (δ, m, µ) appeared in Ref. [16].\\nThe coefficients α i j and β i j are simply related to the coefficients α ij and β ij of Ref. [16].We multiply the coefficients α ij by 3 so that they add up to the charge of decuplet i.Unlike Ref. [16], we use the same mass for all members of a baryon multiplet.Accordingly we add the contributions of π ± and of K ± .The sum over j in Eq. ( 25) runs over two terms -π and K.The resulting coefficients have surprizing simplicity.First, α i j = β i j .Second, they may be expressed in terms of any two of the following three -charge, q i , isospin, I i 3 , and hypercharge Y i of decuplet i.All three are traceless in any SU (3) multiplet space.We choose to use charge and isospin.\\nEq. ( 29) is the key result for renormalizability.As a consequence of these relations, the counterterm for the ultraviolet divergences in F (δ, m, µ) (which are m independent) is simply proportional to δL M , Eq. ( 24), and therefore absorbed into a redefinition of µ c .Note that this is precisely the same procedure as for the one-loop, δ-dependent contributions to the masses.We emphasize that this procedure, and hence renormalizability, is tightly wedded to the systematics of the chiral expansion (whereby δ and f π have fixed values in Eq. ( 25)).\\nThe simplicities of the coefficients α i j and β i j allows great simplification of Eq. (25).We introduce the combination:\\nand rewrite the decuplet magnetic moments in the form:\\nNote that the form of Eq. ( 31), in particular the modification of the coefficient of charge, reflects the choice of charge and isospin as the two traceless quantities.The form would be different if we chose charge and hypercharge or some other combination of isospin and hypercharge.The second term is present only because the SU (3) symmetry of the states is broken through the difference in π and K masses.If we had used the same masses the states would be pure decuplets and the Wigner-Eckart theorem for SU f (3) would ensure that the magnetic moments are simply proportional to the charge.At tree level the decuplet magnetic moments are proportional to the charges only.The main one-loop results is the appearance of the second term of Eq. ( 31).Now we need two magnetic moments to fix the two coefficients in Eq. ( 31), viz, µ c + G K and G π − G K .The only decuplet magnetic moment which has been measured is µ Ω − = −1.94± .17 ± .14 nbm [17].It fixes the coefficient of charge in Eq. (31).For the other magnetic moment we choose µ ∆ 0 .§ § While not measured yet, it is given entirely by the loop effect.\\nWe can express the magnetic moments of all other decuplets in terms of these two magnetic moments.Specifically, we derive the new relation that\\nExplicit predictions for the eight other decuplet magnetic moments at the one-loop O(p 2 ) level are listed below.\\nIndependent of the explicit mulitplets included as intermediate states, violations to these relations are strictly due to higher-order terms in the chiral expansion.Note that the magnetic moment of the Σ * 0 continues to be zero at this order in the expansion.Analogous relations follow for the quadropole moments [25].We note that these relations are not obeyed by quenched lattice QCD [26].This last result is perhaps not surprising as § § The Ω − , decaying only weakly, is sufficiently long lived to allow such measurements.Since all other members of the decuplet decay through the strong interaction, it is a challenge to extract their magnetic moments from experiment.\\nthe quenched calculations do not contain disconnected, quark loop diagrams [27].\\nThe explicit expression for µ ∆ 0 in terms of the functions F (δ, m, µ) is given below.\\nWith the help of Eqs.(26) it is easy to verify that µ ∆ 0 is renormalization scale independent.\\nSince the magnetic moment of the ∆ 0 is given strictly by loop-effects, it is an appropriate measure of the relative importance of the Roper at the one-loop level.Explicitly one has from Eq. ( 35) that\\nAs in the case of the masses [9] there is a strong cancellation between the ∆ decuplet and nucleon octet intermediate states.This implies that µ ∆ 0 is a very delicate function of H 2 and C 2 and therefore potentially very sensitive to the Roper contribution ( C2 ).Ambiguity in this regard resides in the fact that H 2 and C 2 are not sufficiently well-known that a reliable prediction of µ ∆ 0 can be made.To illustrate this point, we quote the results using two \\\"representative\\\" values for the couplings, both with and without the Roper included.For the coupling values used in Ref. [14], one obtains only a mild dependence on the Roper,\\nwhile for the couplings used by Ref. [16], we find a much more dramatic dependence on the Roper,\\nThe difference in µ ∆ 0 from these two parameter sets is clearly sizable, as is the relative importance of the Roper.\\nIf instead we chose to use as input the recent, model dependent extraction [28] of the magnetic moment of the ∆ ++ , µ ∆ ++ = 4.5 ± .5 to infer µ ∆ 0 using the relations Eq. (34), one then obtains that µ ∆ 0 = −.2 ± .2.If this is indeed the data, then by Eq. (36) the contribution of the Roper is, in absolute magnitude, significant.As in the case of the mass-splittings, a formulation of χPT without the Roper as an explicit degree of freedom is intrinsically incapable of predicting such \\\"data\\\".\"\n },\n {\n \"title\": \"V. CONCLUSIONS\",\n \"content\": \"From the results of the last two sections a few points are worth discussing.\\nFirst, that while the magnetic moment of the ∆ 0 sensitively depends on the cancellation between terms depending on relatively ill-determined coupling constants, the relations between the magnetic moments of the ∆ decuplet given in Eq. ( 34) are rigorous predictions of χPT at O(p 2 ).We urge experimental activity to confront these predictions with data.A new measurement of µ Ω − with higher precision will be most useful.At least two other decuplet magnetic moments need to be measured, hopefully with a precision of ∼ 0.1 n.m.. Second, that on the level of one-loop corrections in χPT, the contribution of the Roper octet to properties of the ∆ decuplet is as important as any other multiplet's contribution.We have seen this result quantitatively in the case of the DES rule and the magnetic moment of the ∆ 0 .Both of these quantities are good measures of the one-loop effects as they are each zero at lower-order in the chiral expansion.We expect that these results are illustrative and that they generalize to all one-loop calculations for the ∆ decuplet.Since the mass-splitting, δ R , between the Roper octet and ∆ decuplet is less than that of the kaon's mass, a Taylor expansion in m K /δ R is not permissible.Hence, these loop effects cannot be be absorbed within higher order terms of the chiral expansion of a theory not containing the Roper as an explicit degree of freedom.In such a theory it is indeed difficult to justify going to one-loop or higher without inclusion of the Roper.Phenomenologically successful results would have to be considered merely fortuitous unless shown to be a result of more general considerations (as in the relations of Eq. (34) for the magnetic moments).\\nThird, that the above argument can be repeated in kind for the one-loop corrections to the Roper resonance.That is, even higher-multiplets, separated by δ h in mass from the Roper octet by an amount less than the kaon mass, will a priori be as important quantitatively as the ∆ decuplet for properties of the Roper at the one-loop level.Since such corrections are neccessary for a two loop calculation * * * of the baryon masses, we are led to conclude that the loop expansion in general in the baryon sector of χPT is inevitably wedded to the neccessity of including more and more multiplets in the theory as fundamental fields.While such a result may not be true in a particular limit of QCD (e.g.m u,d,s → 0 or N c → ∞), it is a consequence of the experimental fact that on the average, m π ≈ δ h .\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"text":". S Weinberg, Physica. 963271979","doi":""},{"text":". J Gasser, H Leutwyler, Ann. of Phys. 1585391984. 1985Nucl. Phys.","doi":""},{"text":". J Gasser, M E Sainio, A Svarc, Nucl. Phys. 3077791988","doi":""},{"text":". S Weinberg, Phys. Lett. 2512881990","doi":""},{"text":". E Jenkins, A V Manohar, Phys. Lett. 2555581991","doi":""},{"text":"V Bernard, N Kaiser, U Meißner, TK-95-1Chiral Dynamics in Nucleons and Nuclei. 1995Universite LouisPasteur de Strasbourg preprint CRN-95-3, and Univeritat Bonn preprint","doi":"TK-95-1"},{"text":". C Ordonez, L Ray, U Van Kolck, Phys. Rev. Lett. 7219821994","doi":""},{"text":". E Jenkins, A V Manohar, Phys. Lett. 2593531991","doi":""},{"text":". E Jenkins, Nucl. Phys. 3681901992","doi":""},{"text":". C.-H Lee, H Jung, D.-P Min, M Rho, Phys. Lett. 326141993","doi":""},{"text":". M Savage, Phys. Lett. 3314111994","doi":""},{"text":". C.-H Lee, G E Brown, D.-P Min, M Rho, Nucl. Phys. 5854011995","doi":""},{"text":". M Gell-Mann, R Oakes, B Renner, Phys. Rev. 17521951968","doi":""},{"text":"Baryon Mass Splittings in Chiral Perturbation Theory. M K Banerjee, J Milana, DOE/ER/40762-051, UMPP #95-0581994University of Maryland preprint","doi":"DOE/ER/40762-051, UMPP #95-058"},{"text":"M Gell-Mann, Proc. 11th Int. Conf. High Energy Phys. (CERN). J Prentki Editor, 11th Int. Conf. High Energy Phys. (CERN)Geneva1962CERN","doi":""},{"text":". M N Butler, M J Savage, R P Springer, Phys. Rev. 4934591994","doi":""},{"text":". Phys. Rev. 5011731994Particle Data Group","doi":""},{"text":"For a recent discussion, see e.g. Y. Umino and F. Myhrer, Nucl. Phys. A554. 1993593","doi":""},{"text":"M Gell-Mann, No. CTSL-20Eightfold Way\" by ibid and Y. Ne'eman. Benjamin, NewYork1961. 1964CALTECH Synchotron LabReportreprinted in the","doi":"No. CTSL-20"},{"text":". S Okubo, Prog. Theor. Phys. 279491962","doi":""},{"text":"E Jenkins, A V Manohar, Effective field theories of the Standard Model. - G Ulf, Meißner, SingaporeWorld Scientific1992","doi":""},{"text":". V Bernard, N Kaiser, U Meißner, Z. Phys. 601111993","doi":""},{"text":". R F Lebed, M A Luty, Phys. Lett. 3294791994","doi":""},{"text":". R F Lebed, Nucl. Phys. 4302951994","doi":""},{"text":". E Jenkins, R F Lebed, Phys. Rev. 522821995","doi":""},{"text":"J F Donoghue, E Golowich, B Hosltein, Dynamics of the Standard Model. Cambridge, EnglandCambridge University Press1992","doi":""},{"text":". E Jenkins, M Luke, A V Manohar, M J Savage, Phys. Lett. 3024821993","doi":""},{"text":") and (3) of Ref. [16] makes it clear that the decuplet quadrupole moments calculated with the same treatment of SU f (3) symmetry, i.e., keeping fπ = fK and using the same mass for all members of a given baryon multiplet, would exhibit the same dependence on charge and isospin as the magnetic moments. analogy with Eq. (33), the resulting quadrupole moments would be expressible as Q 10 i = −qi Q Ω − − 2 I i 3 Q ∆ 0. A comparison of Eqs (12) and (13) with Eqs. and a set of relations analogous to Eq. (34) may be written down","doi":""},{"text":". D B Leinweber, T Draper, R M Woloshyn, Phys. Rev. 4630671992","doi":""},{"text":"Essential strangeness in nucleon magnetic moments, Ohio State preprint. D B Leinweber, 1994","doi":""},{"text":". A Bosshard, Phys. Revd. 4419621991","doi":""}],"string":"[\n {\n \"text\": \". S Weinberg, Physica. 963271979\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Gasser, H Leutwyler, Ann. of Phys. 1585391984. 1985Nucl. Phys.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Gasser, M E Sainio, A Svarc, Nucl. Phys. 3077791988\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Weinberg, Phys. Lett. 2512881990\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Jenkins, A V Manohar, Phys. Lett. 2555581991\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V Bernard, N Kaiser, U Meißner, TK-95-1Chiral Dynamics in Nucleons and Nuclei. 1995Universite LouisPasteur de Strasbourg preprint CRN-95-3, and Univeritat Bonn preprint\",\n \"doi\": \"TK-95-1\"\n },\n {\n \"text\": \". C Ordonez, L Ray, U Van Kolck, Phys. Rev. Lett. 7219821994\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Jenkins, A V Manohar, Phys. Lett. 2593531991\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Jenkins, Nucl. Phys. 3681901992\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C.-H Lee, H Jung, D.-P Min, M Rho, Phys. Lett. 326141993\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Savage, Phys. Lett. 3314111994\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C.-H Lee, G E Brown, D.-P Min, M Rho, Nucl. Phys. 5854011995\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Gell-Mann, R Oakes, B Renner, Phys. Rev. 17521951968\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Baryon Mass Splittings in Chiral Perturbation Theory. M K Banerjee, J Milana, DOE/ER/40762-051, UMPP #95-0581994University of Maryland preprint\",\n \"doi\": \"DOE/ER/40762-051, UMPP #95-058\"\n },\n {\n \"text\": \"M Gell-Mann, Proc. 11th Int. Conf. High Energy Phys. (CERN). J Prentki Editor, 11th Int. Conf. High Energy Phys. (CERN)Geneva1962CERN\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M N Butler, M J Savage, R P Springer, Phys. Rev. 4934591994\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. 5011731994Particle Data Group\",\n \"doi\": \"\"\n },\n {\n \"text\": \"For a recent discussion, see e.g. Y. Umino and F. Myhrer, Nucl. Phys. A554. 1993593\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Gell-Mann, No. CTSL-20Eightfold Way\\\" by ibid and Y. Ne'eman. Benjamin, NewYork1961. 1964CALTECH Synchotron LabReportreprinted in the\",\n \"doi\": \"No. CTSL-20\"\n },\n {\n \"text\": \". S Okubo, Prog. Theor. Phys. 279491962\",\n \"doi\": \"\"\n },\n {\n \"text\": \"E Jenkins, A V Manohar, Effective field theories of the Standard Model. - G Ulf, Meißner, SingaporeWorld Scientific1992\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Bernard, N Kaiser, U Meißner, Z. Phys. 601111993\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R F Lebed, M A Luty, Phys. Lett. 3294791994\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R F Lebed, Nucl. Phys. 4302951994\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Jenkins, R F Lebed, Phys. Rev. 522821995\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J F Donoghue, E Golowich, B Hosltein, Dynamics of the Standard Model. Cambridge, EnglandCambridge University Press1992\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Jenkins, M Luke, A V Manohar, M J Savage, Phys. Lett. 3024821993\",\n \"doi\": \"\"\n },\n {\n \"text\": \") and (3) of Ref. [16] makes it clear that the decuplet quadrupole moments calculated with the same treatment of SU f (3) symmetry, i.e., keeping fπ = fK and using the same mass for all members of a given baryon multiplet, would exhibit the same dependence on charge and isospin as the magnetic moments. analogy with Eq. (33), the resulting quadrupole moments would be expressible as Q 10 i = −qi Q Ω − − 2 I i 3 Q ∆ 0. A comparison of Eqs (12) and (13) with Eqs. and a set of relations analogous to Eq. (34) may be written down\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D B Leinweber, T Draper, R M Woloshyn, Phys. Rev. 4630671992\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Essential strangeness in nucleon magnetic moments, Ohio State preprint. D B Leinweber, 1994\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Bosshard, Phys. Revd. 4419621991\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":72,"cells":{"text":{"kind":"string","value":"\nDomain Walls in AdS-Einstein-scalar Gravity\n26 Apr 2011\n\nSangheon Yun :sanhan@ewha.ac.kr \nInstitute for the Early Universe\nEwha Womans University\nDaehyun-dong, Seodaemun-gu120-750SeoulKOREA\n\nDomain Walls in AdS-Einstein-scalar Gravity\n26 Apr 2011arXiv:1012.4172v2 [gr-qc]ABJM theory; domain wall\nIn this note, we show that the supergravity theory which is dual to ABJM field theory can be consistently reduced to scalar-coupled AdS-Einstein gravity and then consider the reflection symmetric domain wall and its small fluctuation. It is also shown that this domain wall solution is none other than dimensional reduction of M2-brane configuration.\n\nIntroduction\n\nIn [1] a new progress in AdS/CFT correspondence [2] was found: the ABJM field theory, i.e. three-dimensional N = 6 superconformal U (N) ×U (N) Chern-Simons theory with Chern-Simons level (k, −k), which was conjectured to describe the low energy dynamics of a stack of N coincident M2-branes probing the singularity of a C 4 /Z k orbifold background, is holographically dual to the type IIA string theory on AdS 4 × CP 3 or to M-theory on AdS 4 × S 7 /Z k . After this discovery, there has been remarkable progress in understanding AdS 4 /CFT 3 correspondence. It is based on the large N planar limit where one is taking N, k → ∞ while holding the 't Hooft coupling λ = N k to be fixed and large. The ABJM theory has SU (4) R-symmetry, which corresponds to the symmetry of CP 3 whose isometry SU (4) preserves not only the metric, but also the Kahler form. The theory has only N = 6 manifest supersymmetry, but is superconformal, and has bifundamental matter and a specific quartic superpotential. The inverse Chern-Simons level 1/k plays a similar role to the Yang Mills coupling constant g 2 Y M . If λ is small, the superconformal Chern-Simons theory is weakly coupled and it can be studied by perturbative analysis. On the other hand, for strongly coupled regime of large λ, the supergravity is a valid description. There are still many outstanding questions, which would be valuable for research implementation in the future, and one of them is to investigate thermal aspects of the ABJM model in both the strongly and the weakly coupled regimes [3,4,5]. In studying the low energy dynamics, the compactification of the type IIA supergravity down to four dimensions is very useful. Using the resulting four-dimensional action, we may consistently study full nonlinear effects. Therefore, in this note we will first summarize the dimensional reduction of the type IIA supergravity on ABJM background and then find a solution of the low-energy effective theory other than AdS space, AdS-Schwarzschild and AdS-Reissner-Nordstrom black branes. This paper is organized as follows. In section 2 we will briefly review the type IIA supergravity and its compactification on CP 3 . In section 3 we will obtain a domain wall solution of the reduced four-dimensional Einstein gravity coupled to a scalar field and comment on other theories coupled with a few scalars. Then we will conclude this paper with discussion and future directions in section 4.\n\n\nType IIA supergravity and compactification on CP 3\n\nWe can construct the effective action of type IIA supergravity via the Kaluza-Klein mechanism by compactification of the eleven-dimensional N = 1 supergravity on S 1 and then retaining only the massless sector. Type IIA supergravity can be also obtained from the zero-slope limit of type IIA superstring theory.\n\n\nType IIA supergravity\n\nThe bosonic part of the action of ten-dimensional type IIA supergravity is written in the string frame,\nS s IIA = S s NS + S s R + S CS ,(1)\nwhere\nS s NS = 1 2κ 2 10 d 10 x √ −g e −2φ R + 4∇ M φ∇ M φ − 1 12 H MNP H MNP , S s R = 1 2κ 2 10 d 10 x √ −g − 1 4 F MN F MN − 1 48 F MNPQ F MNPQ , S CS = 1 2κ 2 10 1 2\nB [2] ∧ F [4] ∧ F [4] .\n\nHere we have used the following definitions: F [2] = dA [1] , F [4] = dA [3] . The NS-NS three form field strength is defined by H [3] = dB [2] and the gauge invariant four form field strength F [4] by F [4] = F [4] + H [3] ∧ A [1] . The ten-dimensional gravity coupling in this action is given by 2κ 2 10 = (2π) 7 ℓ 8 s g 2 s where g s denotes the string coupling. We define the dilaton field φ as its nonzero mode by subtracting its constant partφ denoting the string coupling g s = eφ.\n\nThe bosonic part of type IIA supergravity in the Einstein frame is described by the action,\nS E IIA = S E NS + S E R + S CS ,(3)\nwhere\nS E NS = 1 2κ 2 10 d 10 x √ −g R(g) − 1 2 ∇ M φ∇ M φ − 1 12 e −φ H MNP H MNP , S E R = 1 2κ 2 10 d 10 x √ −g − 1 4 e 3 2 φ F MN F MN − 1 48 e 1 2 φ F MNPQ F MNPQ . (4)\nThe string frame metric is given by the transformation g s MN = e 1 2 φ g E MN . The equations for the bosonic sector of type IIA supergravity can be given in terms of gauge invariant fields as in [3].\n\n\nAdS 4 × CP 3 background of ABJM theory\n\nThe near-horizon geometry of Nk supersymmetric M2-branes becomes AdS 4 ×S 7 with a radius R,\nds 2 = R 2 1 4 ds 2 (AdS 4 ) + ds 2 (S 7 ) ,(5)\nin which there are 32 Killing spinors. In the large Nk limit, the radius R becomes much larger than the eleven-dimensional Planck length l P and the elevendimensional supergravity description works. The metric has the SO(2, 3) conformal symmetry of AdS 4 and SO(8) symmetry of S 7 . The unit round seven-sphere S 7 can be presented as a Hopf fibration of a circle over CP 3 base,\nds 2 (S 7 ) = ds 2 (CP 3 ) + (dυ + ω) 2 ,(6)\nwhere 0 ≤ υ <